Transport Phenomena in Fires (Developments in Heat Transfer)

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Transport Phenonema in Fires

WITPRESS WIT Press publishes leading books in Science and Technology. Visit our website for the current list of titles. www.witpress.com

WITeLibrary Home of the Transactions of the Wessex Institute, the WIT electronic-library provides the international scientific community with immediate and permanent access to individual papers presented at WIT conferences. Visit the WIT eLibrary at http://library.witpress.com

International Series on Developments in Heat Transfer Objectives The Developments in Heat Transfer book Series publishes state-of-the-art books and provides valuable contributions to the literature in the field of heat transfer. The overall aim of the Series is to bring to the attention of the international community recent advances in heat transfer by authors in academic research and the engineering industry. Research and development in heat transfer is of significant importance to many branches of technology, not least in energy technology. Developments include new, efficient heat exchangers, novel heat transfer equipment as well as the introduction of systems of heat exchangers in industrial processes. Application areas include heat recovery in the chemical and process industries, and buildings and dwelling houses where heat transfer plays a major role. Heat exchange combined with heat storage is also a methodology for improving the energy efficiency in industry, while cooling in gas turbine systems and combustion engines is another important area of heat transfer research. To progress developments within the field both basic and applied research is needed. Advances in numerical solution methods of partial differential equations, high-speed, efficient and cheap computers, advanced experimental methods using LDV (laser-doppler-velocimetry), PIV (particle-image-velocimetry) and image processing of thermal pictures of liquid crystals, have all led to dramatic advances during recent years in the solution and investigation of complex problems within the field. The aims of the Series are achieved by contributions to the volumes from invited authors only. This is backed by an internationally recognised Editorial Board for the Series who represent much of the active research worldwide. Volumes planned for the series include the following topics: Compact Heat Exchangers, Engineering Heat Transfer Phenomena, Fins and Fin Systems, Condensation, Materials Processing, Gas Turbine Cooling, Electronics Cooling, CombustionRelated Heat Transfer, Heat Transfer in Gas-Solid Flows, Thermal Radiation, the Boundary Element Method in Heat Transfer, Phase Change Problems, Heat Transfer in Micro-Devices, Plate-and-Frame Heat Exchangers, Turbulent Convective Heat Transfer in Ducts, Enhancement of Heat Transfer and other selected topics.

Series Editor B. Sundén Lund Institute of Technology Box 118 22100 Lund Sweden

Associate Editors E. Blums Latvian Academy of Sciences Latvia

P.J. Heggs UMIST UK

C.A. Brebbia Wessex Institute of Technology UK

C. Herman John Hopkins University USA

G. Comini University of Udine Italy

D.B. Ingham University of Leeds UK

R.M. Cotta COPPE/UFRJ, Brazil

Y. Jaluria Rutgers University USA

L. De Biase University of Milan Italy

S. Kotake University of Tokyo Japan

G. De Mey University of Ghent Belgium

D.B. Murray Trinity College Dublin Ireland

S. del Guidice University of Udine Italy

K. Onishi Ibaraki University Japan

M. Faghri University of Rhode Island USA

P.H. Oosthuizen Queen’s University Kingston Canada

W. Roetzel Universtaet der Bundeswehr Germany

J. Szmyd University of Mining and Metallurgy Poland

B. Sarler Nova Gorica Polytechnic Slovenia

E. Van den Bulck Katholieke Universiteit Leuven Belgium

A.C.M. Sousa University of New Brunswick Canada

S. Yanniotis Agricultural University of Athens Greece

D.B. Spalding CHAM UK

Transport Phenomena in Fires

EDITORS

B Sundén Lund University, Sweden M Faghri University of Rhode Island, USA

Editors B. Sundén Lund University, Sweden M. Faghri University of Rhode Island, USA

Published by WIT Press Ashurst Lodge, Ashurst, Southampton, SO40 7AA, UK Tel: 44 (0) 238 029 3223; Fax: 44 (0) 238 029 2853 E-Mail: [email protected] http://www.witpress.com For USA, Canada and Mexico WIT Press 25 Bridge Street, Billerica, MA 01821, USA Tel: 978 667 5841; Fax: 978 667 7582 E-Mail: [email protected] http://www.witpress.com British Library Cataloguing-in-Publication Data A Catalogue record for this book is available from the British Library ISBN: 978-1-84564-160-3 ISSN: 1369-7331 Library of Congress Catalog Card Number: 2007936446 The texts of the papers in this volume were set individually by the authors or under their supervision. No responsibility is assumed by the Publisher, the Editors and Authors for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein. The Publisher does not necessarily endorse the ideas held, or views expressed by the Editors or Authors of the material contained in its publications. © WIT Press 2008 Printed in Great Britain by Cambridge Printing All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the Publisher.

Contents

Preface

xv

Chapter 1 Mathematical modelling and numerical simulation of fires .......................................... E.E.A. Nilsson, B. Sundén, Z. Yan & M. Faghri

1

1 Introduction .................................................................................................................... 2 Turbulent combustion in fires ........................................................................................ 2.1 Governing equations for turbulent reacting flows ................................................ 2.2 Chemical kinetics .................................................................................................. 2.3 Convection ............................................................................................................ 2.4 Radiation ............................................................................................................... 2.5 Burning of solids ................................................................................................... 3 Simulation and modelling .............................................................................................. 3.1 Turbulence modelling and simulation................................................................... 3.2 Combustion modelling .......................................................................................... 3.3 Pyrolysis modelling............................................................................................... 3.4 Consideration of soot formation............................................................................ 3.5 Radiation modelling .............................................................................................. 4 Numerical method .......................................................................................................... 4.1 Domain discretization ........................................................................................... 4.2 Equation discretization.......................................................................................... 4.3 Linear multi-step method ...................................................................................... 4.4 Multi-grid solver ................................................................................................... 4.5 Parallel computing ................................................................................................ 5 Boundary conditions and wall treatment........................................................................ 5.1 Boundary conditions ............................................................................................. 5.2 Wall functions ....................................................................................................... 6 Case study of upward flame spread over a PMMA board ............................................. 6.1 Problem description .............................................................................................. 6.2 Boundary and initial conditions ............................................................................ 6.3 Results and discussion of the case study...............................................................

1 2 2 4 5 6 6 6 6 10 11 11 12 12 12 13 13 14 14 14 15 15 16 18 18 19

Chapter 2 Transport phenomena that affect heat transfer in fully turbulent fires....................... S.R. Tieszen & L.A. Gritzo

25

1 Introduction .................................................................................................................... 2 Length and time scales within a fire............................................................................... 2.1 Overview ............................................................................................................... 2.2 Time and length scale range.................................................................................. 2.3 Implication for numerical simulation.................................................................... 2.4 Implications for modeling ..................................................................................... 3 Fluid dynamics within large fires................................................................................... 3.1 Quiescent conditions ............................................................................................. 3.2 Interaction with cross-winds ................................................................................. 4 Scalar transport and radiative properties ........................................................................ 4.1 Mixing ................................................................................................................... 4.2 Combustion ........................................................................................................... 4.3 Absorption properties............................................................................................ 4.4 Emission properties ............................................................................................... 5 Future of transport research in fires ...............................................................................

25 26 26 28 31 33 36 37 47 50 50 52 55 58 63

Chapter 3 Heat transfer to objects in pool fires ................................................................................ J.P. Spinti, J.N. Thornock, E.G. Eddings, P.J. Smith & A.F. Sarofim

69

1 Introduction .................................................................................................................... 1.1 Chapter outline ...................................................................................................... 2 Historical modeling approaches ..................................................................................... 2.1 Homogeneous flame.............................................................................................. 2.2 Homogeneous model and observable fire phenomena ......................................... 3 V&V as a foundation for predicting heat transfer to embedded objects in pool fires ........................................................................................................ 3.1 V&V hierarchy...................................................................................................... 3.2 Validation metric................................................................................................... 4 Surrogate fuel formulation ............................................................................................. 4.1 Validation of surrogate formulation...................................................................... 4.2 Burning rates and heat fluxes at steady state ........................................................ 4.3 Burning rates and heat fluxes for transient burning .............................................. 4.4 Effect on fuel composition changes on sooting propensity .................................. 4.5 Improved surrogate formulation ........................................................................... 5 Chemical kinetics for soot production from JP-8........................................................... 5.1 Utah Surrogate mechanism ................................................................................... 5.2 Soot formation and oxidation................................................................................ 6 Use of LES methods for pool fires................................................................................. 6.1 LES equations ....................................................................................................... 6.2 Subgrid turbulence models.................................................................................... 6.3 LES algorithm ....................................................................................................... 6.4 Large scale, parallel computing with LES ............................................................ 6.5 V&V studies of LES code/turbulence model........................................................

69 70 71 71 73 78 78 79 81 81 83 84 85 86 86 87 88 90 91 93 94 96 97

7 Combustion/reaction models .......................................................................................... 7.1 Parameterization of a reacting system................................................................... 7.2 Use of canonical reactors ...................................................................................... 7.3 Progress variable parameterization ....................................................................... 7.4 Heat loss parameterization .................................................................................... 7.5 Soot models ........................................................................................................... 8 Turbulence/chemistry interactions ................................................................................. 8.1 Validation of presumed PDF models in nonpremixed flames .............................. 8.2 Shape of presumed PDF........................................................................................ 9 Radiative heat transfer model......................................................................................... 9.1 Discrete ordinates method..................................................................................... 9.2 Radiative properties .............................................................................................. 9.3 Algorithm verification........................................................................................... 10 Heat transfer to an embedded object in a JP-8 pool fire ................................................ 10.1 Modified LES algorithm ....................................................................................... 10.2 Coupling between LES fire phase and container heat-up phase ........................... 10.3 Subsystem cases: heat transfer in a large JP-8 pool fire ....................................... 11 Prediction of heat flux to an explosive device in a JP-8 pool fire.................................. 12 Predicting the potential hazard of an explosive device immersed in a JP-8 pool fire .................................................................................................................. 12.1 Three-dimensional heat transfer, PBX combustion model ................................... 12.2 One-dimensional heat transfer, fast cook-off HMX model .................................. 12.3 Prediction of time to ignition and explosion violence .......................................... 13 Toward predictivity: error quantification and propagation ............................................ 14 Summary......................................................................................................................... Chapter 4 Heat and mass transfer effects to be considered when modelling the effect of fire on structures ........................................................................................... A. Jowsey, S. Welch & J.L. Torero 1 2 3 4

Introduction .................................................................................................................... Building fires .................................................................................................................. Methods of thermal analysis........................................................................................... The boundary condition.................................................................................................. 4.1 Gas-phase conditions ............................................................................................ 4.2 Application examples............................................................................................ 5 The compartment fire ..................................................................................................... 5.1 Compartment fire models (CFMs) ........................................................................ 6 Solid-phase phenomena.................................................................................................. 6.1 Material integrity................................................................................................... 6.2 Treatment of moisture and other chemical processes ........................................... 7 Conclusions .................................................................................................................... Chapter 5 Weakly buoyant turbulent fire plumes in uniform still and crossflowing environments....................................................................................................................... F.J. Diez, L.P. Bernal & G.M. Faeth 1 Introduction ....................................................................................................................

99 101 101 102 104 107 107 109 110 111 111 112 113 115 115 115 116 120 122 122 123 123 126 127

137 137 138 140 141 142 143 144 147 153 153 154 155

161 162

2 Structure of steady plumes in still environments ........................................................... 2.1 Introduction ........................................................................................................... 2.2 Experimental methods........................................................................................... 2.3 Theoretical methods .............................................................................................. 2.4 Results and discussion........................................................................................... 2.5 Conclusions ........................................................................................................... 3 Penetration of starting plumes in still environments ...................................................... 3.1 Introduction ........................................................................................................... 3.2 Experimental methods........................................................................................... 3.3 Theoretical methods .............................................................................................. 3.4 Results and discussion........................................................................................... 3.5 Conclusions ........................................................................................................... 4 Penetration and concentration properties of startingand steady plumes in crossflows................................................................................................................... 4.1 Introduction ........................................................................................................... 4.2 Experimental methods........................................................................................... 4.3 Theoretical methods .............................................................................................. 4.4 Results and discussion........................................................................................... 4.5 Conclusions ........................................................................................................... 5 Concluding remarks........................................................................................................ Chapter 6 Pyrolysis modeling, thermal decomposition, and transport processes in combustible solids............................................................................................................... C. Lautenberger & C. Fernandez-Pello

162 162 164 166 168 175 176 176 176 178 180 182 182 182 185 187 191 203 203

209

1 Introduction .................................................................................................................... 2 Pyrolysis modeling and fire modeling............................................................................ 2.1 Semi-empirical and fire property-based pyrolysis/gasification models................ 2.2 Comprehensive pyrolysis models: thermoplastics ................................................ 2.3 Comprehensive pyrolysis models: charring materials .......................................... 2.4 Comprehensive pyrolysis models: intumescent materials and coatings ............... 3 Decomposition kinetics and thermodynamics................................................................ 3.1 Thermal and thermooxidative stability ................................................................. 3.2 Reaction enthalpies ............................................................................................... 4 Heat, mass, and momentum transfer .............................................................................. 4.1 Solid phase heat conduction.................................................................................. 4.2 Radiation ............................................................................................................... 4.3 Convection, advection, and diffusion.................................................................... 4.4 Momentum ............................................................................................................ 4.5 Special topics: melting, bubbling, and related phenomena................................... 5 Fire growth modeling ..................................................................................................... 6 Concluding remarks........................................................................................................

209 209 211 213 217 222 224 224 229 233 233 237 243 244 244 245 247

Chapter 7 Radiative heat transfer in fire modeling .......................................................................... M.F. Modest

261

1 Introduction ....................................................................................................................

261

2 Radiative properties of combustion gases ...................................................................... 3 Radiative properties of soot............................................................................................ 4 Band models ................................................................................................................... 4.1 Traditional narrow band models ........................................................................... 4.2 Traditional wide band models............................................................................... 4.3 Narrow band k-distributions.................................................................................. 5 Global models................................................................................................................. 5.1. The WSGG method............................................................................................... 5.2 The SLW method .................................................................................................. 5.3 Full-spectrum k-distributions ................................................................................ 5.4 FSK assembly from a narrow band database ........................................................ 6 Turbulence–radiation interactions.................................................................................. 6.1 Turbulence–radiation coupling ............................................................................. 6.2 Assumed-PDF investigations ................................................................................ 6.3 Composition PDF methods ................................................................................... 6.4 Direct numerical simulations of TRIs ................................................................... 6.5 TRI effects in nonpremixed flames....................................................................... 7. Summary.........................................................................................................................

263 264 264 265 266 266 269 270 271 272 275 276 277 279 280 289 289 292

Chapter 8 Thermal radiation modeling in flames and fires............................................................. S. Sen & I.K. Puri

301

1 Introduction .................................................................................................................... 2 Basic equations............................................................................................................... 2.1 Energy conservation equation ............................................................................... 2.2 Radiative transfer equation ................................................................................... 3 Solution of the RTE........................................................................................................ 3.1 Radiative property models .................................................................................... 3.2 Radiative properties of entrained and generated particles .................................... 3.3 Solution methodologies......................................................................................... 4 Radiation from flames .................................................................................................... 5 Radiation from fires........................................................................................................ 6 Summary.........................................................................................................................

301 302 302 302 303 303 306 307 308 315 317

Chapter 9 Combustion subgrid scale modeling for large eddy simulation of fires........................ P.E. DesJardin, H. Shihn & M.D. Carrara

327

1 Introduction .................................................................................................................... 2 LES mathematical formulation ...................................................................................... 3 Combustion SGS models................................................................................................ 3.1 Filtered density function ....................................................................................... 3.2 One-dimensional turbulence ................................................................................. 4 Summary.........................................................................................................................

327 328 331 331 344 352

Chapter 10 CFD fire simulation and its recent development............................................................. Z. Yan

357

1 Introduction ....................................................................................................................

357

2 CFD simulation of conventional fire.............................................................................. 2.1 Gas phase simulation............................................................................................. 2.2 Modeling of the response of solid materials ......................................................... 2.3 Conventional fire simulation cases ....................................................................... 3 CFD simulation of spontaneous ignition in porous fuel storage .................................... 3.1 The comprehensive spontaneous ignition CFD model ......................................... 3.2 CFD simulation of spontaneous ignition experiment............................................ 4 Conclusions ....................................................................................................................

358 358 382 393 396 398 399 400

Chapter 11 The implementation and application of a fire CFD model............................................. J. Trelles & J.E. Floyd

407

1 2 3 4

Introduction .................................................................................................................... Turbulence modelling..................................................................................................... Solution speed and stability............................................................................................ Accounting for energy.................................................................................................... 4.1 Combustion modelling .......................................................................................... 4.2 Heat transfer .......................................................................................................... Liquid sprays .................................................................................................................. 5.1 Drop size distribution ............................................................................................ 5.2 Spray pattern creation ........................................................................................... 5.3 Spray momentum .................................................................................................. 5.4 Droplet heat transfer and evaporation ................................................................... 5.5 Evaporation impact on divergence........................................................................ Boundary and initial conditions ..................................................................................... The practice of modelling............................................................................................... 7.1 Preparation ............................................................................................................ Assessing the model, assessing the results..................................................................... 8.1 Verification............................................................................................................ 8.2 Validation .............................................................................................................. 8.3 Uncertainty and sensitivity analyses ..................................................................... 8.4 Certification, accreditation, quality assurance ...................................................... 8.5 Review................................................................................................................... Examples ........................................................................................................................ 9.1 Grid density ........................................................................................................... 9.2 Turbulence model.................................................................................................. 9.3 Symmetry .............................................................................................................. 9.4 Sprinklers .............................................................................................................. 9.5 Combustible material properties ........................................................................... 9.6 Radiation solver settings ....................................................................................... Conclusions ....................................................................................................................

407 409 410 411 411 415 419 420 422 422 424 425 425 426 426 428 429 429 430 432 433 433 433 433 435 435 435 437 437

Chapter 12 CFD-based modeling of combustion and suppression in compartment fires............... A. Trouvé & A. Marshall

441

1 Introduction ....................................................................................................................

441

5

6 7 8

9

10

2 Transient ignition and early fire growth......................................................................... 2.1 Modeling of PPC................................................................................................... 2.2 Simulation of the transient ignition and combustion of a fuel vapor cloud.......... 3 Smoke filling and pre-flashover fire spread ................................................................... 3.1 Modeling of fire spread ......................................................................................... 3.2 Simulation of fire spread (without flashover) ....................................................... 4 Flashover and transition to under-ventilated combustion .............................................. 4.1 Modeling of under-ventilated combustion ............................................................ 4.2 Simulation of fire spread (with flashover) ............................................................ 5 Water-based fire suppression and fire control/extinction............................................... 5.1 Models for water-based fire suppression .............................................................. 5.2 Simulation of water-based fire suppression .......................................................... 6 Conclusion......................................................................................................................

443 444 449 454 455 456 459 460 461 464 466 472 473

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Preface

Transport phenomena such as buoyant flow, momentum, convective heat and mass transfer as well as chemical reactions between combustible species and oxygen from the surrounding air play important roles in fire modeling and simulations. In addition, soot formation, soot and gas radiation, turbulent mixing are important to the mechanism of flame heat transfer that govern fire heat release rates. The mechanisms of ignition, flame spread, steady burning flame extinction and smoke transport all need to be considered in fire modeling. In addition, temperature-dependent properties are important factors for consideration. For uncontrolled fires, their evolution in time is of great concern. One aim of this book is to present the state-of-the-art modeling and numerical simulations of the important transport phenomena in fires. Another aim is to present how computational methodologies can be used in analysis and design of fire protection and fire safety. Computational fluid dynamics, turbulence modeling, combustion, soot formation, thermal radiation modeling will be demonstrated and applied to pool fires, flame spread, wildfires, and compartment fires. The first chapter presents an overview of mathematical modeling and numerical simulations in fires. It also serves as an introduction to the following chapters where specific topics are addressed in more detail. References are given to the other chapters in the book that deal with specialized topics. Specifically, it will focus on transport processes that play an important role in the fire modeling such as turbulent combustion; turbulent reacting flows, chemical kinetics, convection, radiation, pyrolysis of solid fuel and numerical simulations of turbulent reacting flows using large eddy simulation and eddy dissipation concepts. The discretization of the governing equations by control volume approach will be discussed followed by solutions of ordinary differential equations by a linear multi-step method. Multi grid iterative schemes will be introduced for solution of the algebraic equations followed by a section on parallel computing. Results will be presented for upward flame spread over vertical surfaces and turbulent combustion in pool fires using large eddy simulation and a parallel CFD fire simulation code developed by the authors. The second chapter explores transport phenomena that affect heat transfer in large (i.e. fully turbulent) fires. In this chapter the authors present the current state of knowledge as well as areas in need of additional research to enable deep understanding and quantitative prediction of hazards posed by these fires. Chapter 3 presents heat transfer to objects in pool fires. A review is presented of modeling approaches for estimating heat flux from fires and flames. This chapter describes recent research methods for addressing observed pool fire, including the multi-scale effects of soot formation and flame structure. Finally, for accurate predictions of heat flux to objects in large-scale transportation fuel fires, the importance of error quantification and propagation in the validation and verification

framework is addressed. Heat and mass transfer effects to be considered when modeling the effects of fire on structures are discussed in chapter 4. This chapter highlights the factors to be considered when doing the thermal analysis of a structure and will provide areas where future work is needed. Chapter 5 describes buoyant turbulent fire plumes in uniform still and cross-flow environments. Consideration of these flows is motivated by numerous practical applications to the unconfined flows resulting from starting and steady releases of buoyant gases and liquids from unwanted fires, from industrial exhaust stacks, from explosions and from process upsets. Chapter 6 gives an overview of pyrolysis modeling, thermal decomposition, and transport processes in combustible solids. It also discusses decomposition kinetics and thermodynamics in the solid phase due to their importance in the burning of solids. Conduction, radiation, convection, and momentum transfer within combustible solids are reviewed. Values of various material properties and pyrolysis coefficients needed for modeling are given for different materials. Radiative heat transfer in fire modeling is discussed in chapters 7 and 8. Chapter 7 presents an account of modern spectral methods for prediction of radiative heat transfer rates within combustion media consisting of strongly nongray combustion gases as well as mildly nongray soot particles. It also discusses the interactions between turbulence and radiation. Chapter 8 presents an overview of thermal radiation models for different combustion processes. The pertinent constitutive equations and associated radiative property models are discussed. Combustion subgrid scale modeling for large eddy simulation of fires is discussed in chapter 9. The objective of this chapter is to examine state-of-the-art subgrid scale combustion models for application to fire environments. The relative merits of these models for application to fire simulation are discussed with illustrative examples. The last three chapters focus on Computational Fluid Dynamics (CFD) modeling of fire simulations. Specifically, chapter 10 presents CFD fire simulation and its recent development within the framework of Reynolds Averaged Navier-Stokes (RANS), Large Eddy Simulation (LES) and Direct Numerical Simulation (DNS). Chapter 11 demonstrates the implementation and application of Fire Dynamics Simulator (FDS) developed by National Institute of Standards and Technology (NIST). Finally, chapter 12 is aimed at CFD-based modeling of combustion and suppression in compartment fires using FDS. All of the chapters follow a unified outline and presentation to aid accessibility and the book provides invaluable information for both graduate researchers and R & D engineers in industry and consultancy. We are grateful to the authors and reviewers for their contributions. We also appreciate the cooperation and patience provided by the staff of WIT Press and for their encouragement and assistance in producing this volume. We also like to thank the Wenner-Gren Center Foundation in Sweden for financial support. Mohammad Faghri and Bengt Sundén

CHAPTER 1 Mathematical modelling and numerical simulation of fires E.E.A. Nilsson1,2, B. Sundén1, Z. Yan3 & M. Faghri4 1

Department of Energy Sciences, Lund University, Lund, Sweden. Södra Cell Mörrum, Sweden. 3 Department of Building Science, Lund University, Lund, Sweden. 4 Department of Mechanical Engineering and Applied Mechanics, University of Rhode Island, USA. 2

Abstract This chapter presents an overview of mathematical modelling and numerical simulation in fires. It also serves as an introduction to the following chapters where specific topics are addressed in more detail. References are given to the other chapters in the book which deal with specialized topics. Specifically, it will outline the governing equations and briefly discuss the most important processes in fires including turbulent combustion, chemical kinetics, convection, radiation and pyrolysis of solid fuel. It also presents some basic description on modelling and simulations, the discretization of the governing equations by the control volume approach and solution by a linear multi-step method. Multi-grid iterative schemes are introduced for solution of the algebraic equations followed by a section on parallel computing. Results are presented for large eddy simulation of upward flame spread over vertical surfaces using SMAFS fire simulation code developed by one of the authors.

1â•‡ Introduction After many years of research and development in computational fluid dynamics (CFD) simulation of turbulent flow, turbulent combustion and fires, and with the advances in computer technology and further understanding of fire dynamics and fire chemistry, CFD fire simulation is becoming a routine practice, not only in the fire research community but also in practical fire safety design engineering. With CFD simulation, it is possible to gain detailed information on fires. CFD simulation of fires is a very complicated multidisciplinary subject which covers a wide range of areas including computer science, numerical methods, fluid dynamics and chemistry. A detailed description on CFD simulation of fires is a daunting task. This chapter aims to

2â•… Transport Phenomena in Fires provide an overview on this multidisciplinary subject, leaving a more detailed description on relevant topics to other chapters in this book. With this overview, readers can have some basic understanding of CFD fire simulation. This basic understanding can be particularly useful for fire engineers who would essentially be the end users of CFD fire simulation and therefore will have an interest in obtaining some basic knowledge on CFD fire simulation.

2â•‡ Turbulent combustion in fires In order to classify combustion phenomena it has been useful to introduce two types of flames, namely premixed and diffusion (non-premixed) flames. In fires, even though in some cases the premixed and non-premixed flames coexist, the non-premixed flame is usually of major importance and will be further discussed here. In most cases of non-premixed combustion, combustion is much faster than diffusion which is the rate limiting step and controls the entire process. This is the reason why those flames where the reactants are non-premixed are also called diffusion flames. In non-premixed flames, mixing takes place by convection and diffusion. Only when fuel and oxidizer are mixed at the molecular level, chemical reactions can occur. Compared to premixed flames, turbulent non-premixed flames exhibit some specific features that have to be taken into account and may lead to additional difficulties in combustion modelling. Nonpremixed flames do not propagate. They are instead located where fuel and oxidizer meet. This has consequences on the chemistry and turbulence interaction. Without propagation speed, a non-premixed flame is unable to impose its dynamics on the flow field and is more sensitive to turbulence and stretching than premixed flames. A diffusion flame is also more likely to be quenched by turbulent fluctuations. One important characteristic of turbulent combustion in fires is fire growth and flame spread over solid combustibles. For the flame to spread over solid combustibles, enough heat must be transferred from the flame to the unburned material ahead of the flame to pyrolyse the solid material. The vaporized fuel is then diffused and convected away from the surface, mixing with the oxidizer and generating a flammable mixture ahead of the flame’s leading edge, which is ignited by the flame as shown in Fig. 1. The rate of flame spread is therefore determined by the ability of the flame to transfer the necessary heat to pyrolyse the solid material and to ignite the combustible material. The heat transfer from the flame to the unburned combustible material is strongly dependent on the shape of the flame, which in turn is dependent on the characteristics of the flow. The interaction between flames and walls is another important issue in fires and other turbulent combustion applications. This issue is not normally studied in textbooks because the phenomena taking place during flame-wall interaction are not well understood. This interaction is, however, strong. The temperature decreases from burnt gas levels to wall levels occur in a near-wall layer leading to very strong temperature gradients. The flames do not usually touch walls as they quench due to the lower temperature at the wall, the radical destruction at the wall and also the blowing effects of the pyrolysis gases. The interaction influences combustion and wall heat fluxes in a significant manner and constitutes a difficult challenge in combustion studies according to Poinsot and Veynante [1]. 2.1â•‡ Governing equations for turbulent reacting flows For reacting flows, the governing equations are the continuity, momentum, energy, and species equations as well as the equation of state of the gas. These equations conserve mass, momentum,

Mathematical Modelling and Numerical Simulation of Firesâ•…

3

Oxygen mass transfer Fuel mass transfer Preheat zone Heat transfer from flame

Flame

Pyrolysis zone

Figure 1: Physical configuration of flame/wall interaction.

energy and species. The conservation equations together with the equation of state are all required to close the system of equations. For a reacting flow field, the conservation of mass is expressed by the continuity equation: ∂r ∂rui + =0 ∂t ∂xi

(1)

The equations are written in Cartesian coordinates where t represents time, r density. ui is the velocity component in the direction of the Cartesian coordinate xi. The conservation equation of momentum can be derived from Newton’s second law: ∂tij ∂ ∂ ∂p + r agi + ( rui ) + ( ru j ui ) =  ∂x j ∂ t  ∂x j ∂xi      gravity  unsteady

term

convection term

pressure gradient

(2)

term

diffusion term

where tij is the constitutive relation for a Newtonian fluid:

 ∂u ∂u j  2 ∂uk tij = m  i +  - m dij ∂x k  ∂x j ∂xi  3

(3)

This is usually called the Navier-Stokes equation of motion. In this equation, agi is the acceleration of gravity, p is the static pressure, m is the dynamic viscosity of the fluid and dij is the Kronecker delta function.

4â•… Transport Phenomena in Fires The mixture enthalpy h is conserved in the energy equation:  ∂p ∂q j ∂u ∂ ∂ ∂p  + tij i +  + u j ( rh ) + ( ru j h) =  + SQ ∂x j ∂t ∂x j ∂x j  ∂t ∂x j               heat unsteady term

convection term

diffusion term

dissipation term

compressibility term

source term

(4)

whereâ•› h = ∑ i =1 hi Yi , hi denotes the enthalpy of species i and Yi is the mass fraction of species i. SQ is a heat source term (typically from radiation in fires). qj is the heat diffusion flux defined by Fourier’s law as: n

qj = l

n  ∂Y  ∂T + r ∑ hi  Di i  ∂x j  ∂x j  i =1

(5)

The mass fraction Yi of each species i satisfies the transport equation: ∂ ∂ ∂   ∂Yi   ( rYi ) + ( ru jYi ) =  r  Di  + ∂ ∂x j ∂x j   ∂x j   t          unsteady

convection term

term

diffusion term

w i 

(6)

rate of formation source term

◊ The source term wi describes the rate of formation of each species and Di is the mass diffusivity of species i.

2.2â•‡ Chemical kinetics This section will only give a brief introduction to chemical kinetics. For more information, the reader may refer to Glassman [2], Turns [3], and Warnatz et al. [4]. The overall reaction of one mole of fuel and a moles of oxidizer to b moles of combustion products can be described by the global reaction mechanism as: Fuel + a ◊ Oxidizer → b ◊ Products

The rate at which the fuel is consumed can be expressed as:

w F =

d[ Xfuel ] = - kG (T ) [ Xfuel ]n [ Xoxidizer ]m dt

(7)

where X is used to denote the molar concentration of the species i in the mixture. The equation states that the rate of disappearance of the fuel is proportional to each of the reactants raised to a power. The global rate coefficient kG is not a constant but rather a strong function of temperature. The negative sign indicates that the fuel is consumed. The exponents n and m relate to the reaction order. The use of global reactions to express the chemistry in a specific problem is frequently called the ‘black box’ approach. In reality, many sequential processes can occur involving many intermediate species. The collection of elementary reactions necessary to describe an overall reaction is called a detailed reaction mechanism. To have a complete picture of the reaction mechanism, sometimes up to several hundred elementary reactions must be considered. One can optimize the number of necessary elementary steps to describe a particular global reaction. This method is shown in [5].

Mathematical Modelling and Numerical Simulation of Firesâ•…

5

2.3â•‡ Convection Fluid motion which is induced by body forces such as gravitational, centrifugal, or Coriolis forces is called natural convection. The flow considered in this chapter is buoyancy induced motion resulting from body forces acting on density gradients in the fluid. A typical natural convection example is shown in Fig. 2, where the vertical flat plate at temperature Tw is warmer than the surrounding at temperature T•. The heat transferred from the plate to the fluid leads to an increase of the fluid temperature close to the wall and causes a change in the density. As the density decreases with increasing temperature, buoyancy forces arise close to the wall, and warmer fluid moves up along the plate. Obviously, the effect of the plate is restricted to a thin layer close to the wall, since the additional internal energy supplied to the fluid through the wall is transported up along the wall by convection and thus cannot reach regions of fluid further away. The thickness dth of the ‘thermal layer’ (region with T > T•) is taken to be the distance from the wall at which the temperature increase has dropped to within a certain percentage (e.g. 1%) of Tw - T•. This thickness grows with the length x. This follows from a simple energy balance, according to which the total internal energy supplied through the wall up to a point x must ‘flow’ by convection of the higher temperature fluid over the cross-section where x is constant. In such a turbulent natural convection boundary layer (the natural convection often turns out to be turbulent), three regions, namely a wall layer with a linear profile of the average temperature, a buffer region or a region of intense heat transfer (where the intensity of temperature pulsations is maximum and the amplitude is a large quantity) and an external region with a small local temperature difference, considerable intermittence and the properties similar to those of flows in free streams, can be identified. Simple dimensional considerations show that the thickness of the thermal layer is smaller as the viscosity m decreases and the flow has a boundary layer character, comparable to the flows of free jets and wall jets where there is relative small or no outer flow [6]). Convection does not smooth fluctuations and gradients; therefore good algorithms for convection tend to lack numerical diffusion or the relaxation effects that are typical of local processes according to Oran and Boris [7]. u a gi

d

Tw

T − T∞

T∞

d th x y

Figure 2: Velocity and temperature distributions for non-reacting flow in natural convection over a vertical wall.

6â•… Transport Phenomena in Fires In many combustion applications such as fires, the temperature decreases from burnt gas level to wall level in a near-wall layer which is less than 1 mm thick, leading to very strong temperature gradients. Studying the interaction between the flame and the wall is extremely difficult because all interesting phenomena occur in a very thin zone near the wall. 2.4â•‡ Radiation Radiation is an important heat transfer mechanism in combustion. Inherent in combustion is the coupling between the momentum field and the scalar fields, the combustion chemistry, and the radiation. For turbulent flows, this coupling occurs simultaneously over a spectrum of length and time scales and for the most part is bidirectional. The coupling between turbulence and radiation is bidirectional through absorption. Of importance for flame spread is the absorption of radiation by solid- or liquid-fuel boundaries that subsequently pyrolyse and produce the fuel that sustains the fire [8]. For further information, the reader may refer to the book by Modest [9]. 2.5â•‡ Burning of solids The pyrolysis rate, as a function of the incident heat flux to which the solid material is subjected, is of importance in flame spread. This implies that products of fuel pyrolysis are released into the gas phase and burnt as they mix with air. A solid fuel consists of combustibles, ash and moisture. Combustibles can further be divided into fixed carbon and volatile matter. Further information on burning of solids can be found in the book by de Souza-Santos [10].

3â•‡ Simulation and modelling Numerical simulation of flows requires typically the following components. Mathematical models usually consist of the basic physical conservation laws for mass, momentum and energy. In the case of turbulent flows, averaged/filtered forms of the conservation laws are often used. These laws are usually expressed in non-linear, second-order, partial differential equations, with engineering sub-models to simplify and/or approximate the representation of certain sub-processes. To obtain adequate solutions initial and/or boundary conditions are also required. The numerical solution method is needed to discretize the model equations and approximate the equations by sets of linear algebraic equations which are solved by suitable algorithms. Pre-processing software includes methods for handling geometry and grid generation, boundary and initial conditions specification. Post-processing software helps to graphically display, analyse and understand the computed results and to derive secondary quantities (e.g. heat transfer coefficient) from the primitive variables [11]. 3.1â•‡ Turbulence modelling and simulation Most flows in nature and in engineering practice are turbulent. It is very difficult to give a precise definition of turbulence. The main characteristics of turbulent flows are described in [12]. The spectrum of length scales in a turbulent flow varies with the size of eddies, from the largest scale (determined by the geometry of the domain) down to the smallest one (determined by an energy dissipation process). Three different length scales are often referred to in non-reacting turbulent flows, namely the Kolmogorov micro-scale (h), the Taylor micro-scale (l), and the

Mathematical Modelling and Numerical Simulation of Firesâ•…

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integral-scale (ℓ), where the integral-scale is the energy containing scale that is characteristic of the flow field. The intermediate length-scale between the integral-scale and the Kolmogorov micro-scale is called the Taylor micro-scale. It can be interpreted as the distance that a large eddy convects a Kolmogorov eddy during its turnover time. The Taylor micro-scale is much smaller than the integral-scale but much bigger than the Kolmogorov micro-scale. The dissipation subrange contains the smallest scale in this group. The Kolmogorov micro-scale represents the level where turbulent kinetic energy is transferred into heat by viscous dissipation, as explained by Kolmogorov’s universal equilibrium theory of both the large-scale turbulent motion and the mean flow. Therefore, the small scales only depend upon the rate of energy supplied from the large-scale motion and the kinematic viscosity. At equilibrium, the energy transfer rate is assumed to be equal to the rate of dissipation. If the energy spectrum is measured in the entire wave number range one obtains the behaviour shown schematically in a log-log plot in Fig. 3. The spectrum attains a maximum at a wave number that corresponds to the integral-scale, since eddies of that scale contain most of the kinetic energy. For large wave numbers corresponding to the inertial sub-range, the energy spectrum decreases following the k-5/3 power law. There is a cut-off at the Kolmogorov scale (h). Beyond this cut-off, in the range called the viscous sub-range, the energy per unit wave number decreases exponentially, owing to viscous effects. The essence of the Kolmogorov theory is that turbulence generation occurs mainly at the largest scales of a flow, while viscous dissipation occurs mainly at the smallest scales. In boundary layers, turbulence also has an important effect. Turbulence in the external flow modifies the boundary layer, and the presence of a body, acting through the boundary layer, modifies turbulence. Systems may often become turbulent after a sequence of only a few instabilities at incommensurable scales. Statistical theories of turbulence were largely derived to describe fluctuations in macroscopic properties such as velocity, density, and temperature. In a statistically steady, turbulent flow, the energy density in the velocity fluctuations is denoted by E(k), where the length scale 1/k corresponds to the wave number k. For a three-dimensional flow, E(k) follows a power-law spectrum that decays at small scales according to the k-5/3 power law. This means that most of the energy in the flow is contained in the large scales.

Figure 3: Schematic representation of the turbulent kinetic energy specturm E as a function of the wave number k.

8â•… Transport Phenomena in Fires Depending on the character of the flow, Reynolds number and geometry, turbulent flows can be simulated in different ways, including Reynolds averaged Navier-Stokes (RANS) modelling, large eddy simulation (LES) and direct numerical simulation (DNS). In Figs 4 and 5 the major differences between the different models are shown. The limit of the computer resources forced many of the past attempts to model non-premixed turbulent combustion to be based on the conventional, relatively cheap averaged simulation (RANS) method, where the instantaneous equations are statistically averaged for solution and thus only the relatively smooth mean field needs to be resolved properly. Since turbulence is a property of flow, the constructed turbulence models in RANS can never be of universal character. With the advent of parallel computing technology and the rapid growth of the computing power, it is now affordable to refine the consideration of the physics in the computation, for example, to replace RANS with LES. In DNS one resolves all temporal and spatial scales, including the smallest so-called Kolmogorov micro-scales. DNS of a turbulent flow is always time dependent and one has to advance the solution long enough to obtain statistically averaged quantities. The major

Figure 4: Time evolutions of local temperature computed with DNS, RANS, or LES in a turbulent flame [1].

Figure 5: Turbulence energy spectrum plotted as a function of wave number (log-log scale).

Mathematical Modelling and Numerical Simulation of Firesâ•…

9

advantage of DNS is that no models are needed for the turbulence-combustion interaction, but DNS also has a major drawback which is its high demand for computer capacity and long computational time. A more general discussion on turbulence and simulation concepts of DNS, LES and RANS can be found in Chapter 10. 3.1.1â•‡ RANS modelling – For turbulent flows the instantaneous variable f can be decomposed into a mean component, f , and a fluctuating component, f', by a time-averaging method: f = f + f'

with

f=

1 ∆t

(8)

t + ∆t

∫

(9)

f dt

t

The average mass flow rates using Reynolds averaging may not be a conserved quantity in a steady flow. Reynolds averaging for variable density flows introduces many other unclosed correlations between the quantity f and the density fluctuations p'f' . To avoid this difficulty, mass weighted averages called Favre averaging are used as:  = rf f r

(10)

Any quantity f may be divided into mean and fluctuating components as:  + f ′′ with rf ′′ = 0 f=f

(11)

Based on this assumption, the averaged conservation equations become: ∂r ∂( r ui ) + =0 ∂t ∂xi

Continuity:

∂( r ui ) ∂( r ui u j ) ∂r ∂ tij - rui′′u j′′ + r agi + =+ ∂t ∂x j ∂ xi ∂ x j 

Momentum:

Energy:

(12)

∂u j ∂( r h ) ∂( ru j h ) ∂p ∂p ∂ q = + u j + SQ + + ( j - ruj′′ h′′ ) + tij ∂t ∂x j ∂t ∂x j ∂x j ∂xi

(13)

(14)

  ∂ ∂ ∂   ∂Yi  ′′ ′′  + w i  u Y ( rYi ) + ( r u j Yi ) = r  Di r  j i  ∂t ∂x j ∂x j   ∂x j   

Chemical species: â•…â•…â•…

(15)

3.1.2â•‡ Large eddy simulation In LES, eddies down to the inertial range are resolved properly. The contribution of smaller eddies is modelled using a sub-grid model. In LES, the large scales are explicitly calculated.

10â•… Transport Phenomena in Fires The small unresolved so-called sub-grid scales (SGSs) are modelled using sub-grid closure rules. Several SGS models have been proposed for LES. One of the first models was proposed by Smagorinsky [13] for the SGS stress tensor. The balance equations for LES are obtained by filtering the instantaneous balance equations. LES of reacting flows determines the instantaneous position of a large-scale resolved flame front, but a sub-grid model is required to take into account the effects of small turbulent scales on combustion. In LES, variables are filtered in spectral space (Fourier space) or in physical space. The filtered quantity G is defined as: G( x ) = ∫ G( x ′ )F ( x - x ′ ) dx ′

(16)

where F is the LES filter. The usual filters are described in [14], e.g. cut-off filter, box-filter and Gaussian filter. All these filters are normalized: ∞ ∞ ∞

∫ ∫ ∫ F ( x1 , x2 , x3 ) dx1dx2 dx3 = 1

-∞ -∞ -∞

(17)

For the variable density r, a mass weighted Favre filtering is introduced according to: r G ( x ) = ∫ r G( x ′ )F ( x - x ′ ) dx ′

(18)

– – The filtered quantity G is resolved in the numerical simulation, whereas Gâ•›¢ = G - G corresponds to the unresolved part (i.e. the SGS part, due to the unresolved flow motions). More information about filtering can be found in the book by Sagaut [14]. The unknown sub-grid stress tensor and sub-grid scalar flux in the filtered momentum and enthalpy equations must be modelled in terms of the properties of the resolved scales so that the equation system can be closed for solution. A SGS model should be able to properly remove the turbulence kinetic energy from resolved scales and account for backscatter of turbulence energy. Many SGS models, [13, 15-17] have been proposed for the needed modelling. Due to its simplicity and effectiveness, in spite of not allowing turbulence energy backscatter from small to large scales and being too dissipative [1], the Smagorinsky SGS model [13] remains one of the widely used models to model the unknown sub-grid stress tensor and sub-grid scalar flux in the filtered momentum and energy equations in terms of the properties of the resolved scales. More information about the Smagorinsky model can be found in [1, 13, 14].

3.2â•‡ Combustion modelling A combustion model is required to provide the source terms for the species equations. The eddy dissipation concept (EDC), devised by Magnussen and Hjerthager [18], is a simple, yet effective model which directly extends the Eddy break-up (EBU) model to non-premixed combustion. The fuel’s mean burning rate is estimated from the mean mass fractions of the fuel, oxidizer and products and depends on the turbulent mixing time, estimated from integral length scales as: r w F = CEDC r

 Y  Y Y  Y  e 1 min  YF , O , b P  ≈ CEDC r min  YF , O , b P  s s (1 + s )  (1 + s )  tt k  

(19)

Mathematical Modelling and Numerical Simulation of Firesâ•…

11

where CEDC and b are model constants. The reaction rate is limited by the deficient mean species. When b is non-zero, this deficient species may be combustion products to take into account the existence of burnt gases, providing the energy required to ignite fresh reactants. EDC is a good model in many cases but the model constants CEDC and b as well as the turbulent time tt need to be adjusted on a case by case basis. A drawback with EDC is that it cannot describe any ignition mechanism since the fuel and oxidizer are assumed to react as soon as they meet. EDC is discussed in more detail in Chapter 10. For modelling of turbulent combustion, there are other models available including the flamelet concept and the conditional moment closure method. The flamelet concept is presented in Chapters 9 and 10 which particularly focus on the SGS modelling of turbulent combustion in LES.

3.3â•‡ Pyrolysis modelling When the solid fuel is heated by exposure to heat sources, it begins to pyrolyse as the temperature reaches its pyrolysis temperature. An efficient and simple pyrolysis model described in [19-21], which is generally applicable to charring and non-charring materials, can be used to describe the burning of the solid material. It has great flexibility and can easily be used in complex cases such as those with transient incident heat flux and temperature-dependent material properties. This model is based on the numerical solution of the following equation:

∂  m ′′′( HG,T + HG,Tp ) ∂( r H ) ∂  ∂T    + m ′′′( H py + H ) + = k ∂t ∂x ∂ x  ∂ x 

(20)

. where m¢¢¢ = mass loss rate of the pyrolysing material per unit volume defined as:

m ′′′ = -

∂r ∂m ′′′ = ≥0 ∂t ∂x

(21)

The third term in equation is the energy required to heat the vaporized gas as it flows to the solid surface. This term will be zero for non-charring material. Hpy is the heat of reaction of the pyrolysis process. The details of this specific pyrolysis model including its derivation and validation are given in Chapter 10. More general information about pyrolysis and decomposition of combustible solids can be found in Chapter 6 which is devoted to this topic. 3.4â•‡ Consideration of soot formation Soot is an important contributor to thermal radiation in fires. In order to calculate the radiation accurately, soot must be considered. However, soot formation and oxidation in turbulent combustion is extremely complicated. A very brief description of soot formation and oxidation processes is given in Chapters 3 and 10. Due to the extreme complexity of the soot formation process, few very good models are currently available for soot prediction in the combustion of solid fuel, although some significant progress in soot modelling has been made in recent years [22]. Typically used models for consideration of soot in fire modelling are discussed in Chapters 3 and 10. These soot models include the empirical or semi-empirical models and the more fundamental flamelet soot model.

12â•… Transport Phenomena in Fires 3.5â•‡ Radiation modelling Radiation is an important heat transfer mechanism in a fire. Under many circumstances, the heat transfer in fires can be dominated by radiation. Radiation heat transfer is governed by the radiation transfer equation (RTE) and radiation modelling is achieved through the solution of the RTE. In the solution of the RTE, it is vitally important to properly evaluate the radiation property of the radiating species. There exist a number of models for such evaluation. The narrow band model is the most accurate method in engineering computation, but its traditional computing is unfortunately very CPU time consuming. In a study by Yan and Holmstedt [23], a fast narrow band (FASTNB) model was developed to drastically speed up the computation without really loosing any accuracy. To make the radiation property evaluation much more efficient (one order of magnitude faster), with a slight sacrifice in computation accuracy, the FASTNB can be applied in an approximate form [24]. In the case study in Section 6, the approximate version of a fast narrow-band computer model [24], which is slightly less accurate but much more efficient than FASTNB, was adopted to predict the radiation properties of the combustion products [24]. Details of FASTNB and approximated FASTNB are outlined in Chapter 10. Because of the importance of thermal radiation, besides the imbedded discussions such as those in Chapter 10, there are also two complete chapters (Chapters 7 and 8) devoted to this topic. In these two chapters, a more comprehensive discussion on the solution of the RTE and the evaluation of the radiation property can be found. In Chapter 7, the topic of turbulence-radiation interaction is also addressed.

4â•‡ Numerical method An analytical solution to the governing equations exits only for a few very special flow fields, and in general the governing equations have to be solved numerically. This section covers the discretization of the four-dimensional domain (three-dimensional space and time) and the governing equations and presents an algorithm to solve the discretized equation which is expressed in algebraic form. A good numerical method should: 1.â•… 2.â•… 3.â•… 4.â•… 5.â•… 6.â•…

be numerically stable for all cases of interest, conserve quantities that are conserved physically, be reasonably accurate, be computationally efficient, generalize to multi-dimensional cases, and be broadly applicable, that is, not specific problem dependent.

4.1â•‡ Domain discretization Domain discretization in general includes discretization in three-dimensional space and time. In space discretization, the physical space is broken into small regions, usually called computational cells. The geometry of the flow field can be represented by different computational grids such as Cartesian grids, a body-fitted structured grid or a body-fitted unstructured grid. Cartesian grids are built with rectangular cells. Therefore, the grid generation process is simple and does not require much computational storage. The drawback of Cartesian grid methods is that complex geometries are not correctly represented. The body-fitted grids represent complex geometries more accurately than the Cartesian grid. The drawbacks of the body-fitted methods

Mathematical Modelling and Numerical Simulation of Firesâ•…

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are larger computational storage requirements, the long clock times required to create the grid and the increasing complexity in creating a higher order discretization scheme. With a Cartesian grid, the computation can be very efficient according to Oran and Boris [7]. The Cartesian grid is potentially the most suitable in LES because of its economical storage requirement, less computational effort per computational cell, relatively fast numerical convergence and feasibility for constructing higher order finite difference schemes. Similar to space discretization, the time discretization discretizes the continuous time coordinate into discrete intervals. These intervals, called time steps, provide convenient increments over which to advance the numerical solution. 4.2â•‡ Equation discretization In order to solve the governing equations numerically, they have to be replaced by their discrete counterparts by means of a discretization. A successful discretization should result in zero-deviation between the discrete equations and the continuous equations when the number of grid points is increased to infinity. There are many strategies for discretization, e.g. finite differences, finite elements and finite volumes. The finite volume method divides the computational domain into a finite number of control volumes (cells). The governing equations are then integrated over each of these volumes and the resulting expressions are discretized to an algebraic set of equations which can be solved by an iterative method. The conservation law for the transport of a scalar has the general form: ∂ ( r f ) + div( r u f ) = div(Ggrad f ) + Sf ∂t

(22)

The term f is a general variable, G is the diffusion coefficient and Sf is the source term. The governing equations (eqns (1), (2), (4), and (6)) can all be rewritten in the form of eqn (22). The general integrated form of the transport equation can be written as: ∂



∫ ∂t  ∫ ( r f ) dV  dt + ∫ ∫ n ◊ ( r f u) dA dt

∆t

CV

∆t A

=

∫ ∫ n ⋅ (G

∆t A

f

grad f ) dA dt +

∫∫

∆t CV

Sf dV dt (23)

where ∆t is the discretized time interval and CV (control volume) represents an individual discretized space. Different schemes can be used for the equation discretization. In the case study in Section 6, the central difference scheme and a bounded QUICK scheme were used. For more information about difference schemes, see Versteeg and Malalasekera [25] and Lien and Leschziner [26]. 4.3â•‡ Linear multi-step method In order to solve the discretized equations, a linear multi-step method can be applied. The secondorder accurate explicit Adam-Bashford scheme was used in the case study to solve the momentum equation (eqn (13)). The transport scalar was computed using a second-order Runge-Kutta method. In practical applications, predictor-corrector methods compare favourably to RungeKutta methods. Runge-Kutta methods generally require less storage and are self-starting, because

14â•… Transport Phenomena in Fires they only require data at a single time level to begin the integration. In Runge-Kutta methods, it is also easier to vary the step size. More about the finite volume method, discretization and the linear multi-step methods can be found in Oran and Boris [7]. 4.4â•‡ Multi-grid solver The multi-grid (MG) iteration schemes are meant to guarantee a sufficiently high convergence rate, independent of the dimension of the system. The idea behind the MG solver is that the traditional iteration scheme reduces the short-wavelength error constituents and that the long-wavelength error constituents can be eliminated by a direct solver for the approximation of the problem on a much coarser grid. The MG solver methods use grids of different resolutions to damp out disturbances of different scales according to Brandt [27]. The rate of convergence depends upon the iteration procedure. Large-scale disturbances are more quickly removed on low resolution (large-scale) grids, as shown by Fletcher [28]. Some basic schemes do not smooth, instead they propagate the residuals as a moving front. For these methods, the application of MGs is intended to more rapidly convey the residual front to the outer boundaries. MG methods typically have a linear convergence. The Poisson equation for pressure was solved using a parallel MG solver in the case study in Section 6. 4.5â•‡ Parallel computing A comprehensive simulation of turbulent combustion usually includes coupled computations of turbulent flow, chemical reaction, thermal radiation, multi-phase interaction, etc., which are all highly complex. As a result, direct simulation of practical turbulent combustion is far beyond the capacity of currently available computer resources. Although the computations can be drastically simplified by introducing engineering models, the computations can still be intensive enough to challenge any modern single processor. As a developing technology, with the refinement of models and increase of problem size, computation of turbulent combustion will continue to impose tough demands on computer resources. Parallel computing has been widely adopted as a cheap and efficient methodology to explore the maximum potential of the available computer facility. By using parallel computing, a problem can be studied with increased size and/or accuracy. In the case study in Section 6, parallel computing was applied using a parallelization based on a single-program, multiple-data (SPMD) algorithm developed by Yan [29], where the computation is divided into a number of parallel tasks operating on partitioned data structures. The necessary information exchange among different tasks is carried out using a message-passing interface library, parallel virtual machine (PVM), according to Geist et al. [30]. The performance of parallel computing is affected by a number of factors such as communication overhead (time required for the necessary data exchange among different tasks), workload balance and computer memory usage. Any workload imbalance may create unnecessary idle waiting. More information about parallel computing can be found in [29].

5â•‡ Boundary conditions and wall treatment The partial differential governing equations discussed in Section 3 require boundary conditions to have a unique solution. A proper specification of boundary conditions is very important and remains a very difficult task, particularly for an open flow where artificial boundary conditions consistent with the computed unsteady flow are required. At present no adequate mathematical theory is available to ensure a correct boundary condition for the full Navier-Stokes equations.

Mathematical Modelling and Numerical Simulation of Firesâ•…

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Therefore, in order to find appropriate boundary conditions, one has to rely on physical arguments, known mathematical results and heuristic considerations according to Cebeci and Bradshaw [31]. In this section, we will briefly discuss the boundary conditions often used in fire simulation with some particular consideration delivered to wall boundary treatment. 5.1â•‡ Boundary conditions For fire simulations, the often used boundary conditions are: inlet, outlet, symmetry, free and wall boundary conditions. An inlet boundary condition specifies the fluid flow across the boundary surface into the calculation domain. For mass and momentum equations, one may specify all velocity components, or mass flow rate and direction, or total pressure. For other transport equations the values of the dependent variables must be specified. An outlet boundary condition specifies the fluid flows across the boundary surface out of the calculation domain, with a zero gradient (in streamwise direction) boundary condition often used. A symmetry boundary indicates a symmetric distribution of flow on two sides of the symmetry boundary. At free boundary, the flow is subject to a free space which usually has prescribed pressure. A solid wall boundary condition often specifies that the fluid cannot penetrate the boundary surface. That is, the component of the velocity normal to the solid boundary vanishes. The tangential components condition which is mostly accepted to be true is the no-slip boundary condition due to viscous effects. The dissipative processes are so strong that the tangential component of the velocity vanishes according to the no-slip condition. For mass and momentum equations, a no-slip condition is specified on the boundary surface. For the energy transport equation, one can impose certain prescribed heat transfer condition, such as a prescribed wall temperature. In order to directly implement such wall boundary conditions, one needs to fully resolve the wall boundary layer which is usually very thin and has very steep gradients. However, this needed resolution will not only drastically increase the computation task but also necessitate a proper consideration of combustion chemistry where the wall quenching effect on flame can be included. Due to this difficulty, the wall function was used instead. The wall function approach has been widely used in practice because of its economy of calculation. 5.2â•‡ Wall functions Within the framework of wall function, the boundary layer is not fully resolved and the flow at the first near wall node is not dominated by the viscous effects. In this case, so-called wall functions are often used to calculate the wall shear stress and the wall heat transfer. Assuming a similar flow structure between the wall and the first grid point, the wall function provides an algebraic relationship between the local wall stresses and the tangential velocities at the first grid node adjacent to the wall surface (see Fig. 6). Wall functions are derived empirically or semi-empirically. There exist a number of different wall functions for various applications. The early traditional functions are derived on forced convection, with a number of assumptions including: • small temperature difference, • negligible gradients along the wall, • fully turbulent flow, stationary in the mean,

16â•… Transport Phenomena in Fires

First grid point near the wall: velocity u1, temperature T1

tw

wall

Φ y1

Zone where the turbulence model is valid

Wall function zone

Figure 6: Principle of the wall function.

• • • • •

negligible pressure gradient, low Mach number, no chemical reactions, perfect gases and no Soret or Dufour effects, no radiative fluxes or external forces.

With the use of wall function one should be aware that it is based on several approximations, which are discussed by Schlichting [6] and Poinsot and Veynante [1]. These assumptions may become invalid in some complex flows. For example, the assumption of small temperature difference used in the traditional wall function is only valid for flows in which the temperature variations remain small. In combustion applications, this is seldom the case and the ratios between the gas temperatures to the wall temperatures can be of the order of 4-6, thus this can induce very large errors on wall friction and wall heat fluxes. In this case, non-isothermal wall functions with consideration of the effect of temperature variation in the boundary layer are expected to be better applicable. Such a non-isothermal wall function can be reformulated from the traditional isothermal wall function by taking into account the effect of high temperature gradient near the wall region, as seen in [1, 32, 33]. Recently, non-isothermal wall function was applied to a LES of natural convection along a hot surface and compared with traditional isothermal wall function [34].

6â•‡ Case study of upward flame spread over a PMMA board A LES of upward flame spread over a polymethyl methacrylate (PMMA) board has been carried out using a parallel CFD code SMAFS developed by Yan [35]. Both the turbulent combustion of the gas phase and the pyrolysis of the solid fuel were numerically simulated. In the gas phase computation, the SGS turbulence was modelled using the Smagorinsky model and the SGS turbulent combustion was modelled based on EDC described in previous sections in this chapter. The convective heat transfer was computed using wall function with the blowing effect

Mathematical Modelling and Numerical Simulation of Firesâ•…

17

of the pyrolysis gas considered. The thermal radiation was calculated using the discrete transfer method. An approximate version of fast narrow-band computer model was implemented to estimate the radiation property of the radiating medium for the solution of the radiation equation along every ray. In the solid phase, an efficient, simple and practical pyrolysis model was adopted to describe the pyrolysis of the solid fuel. In this study, as an approximation, soot was considered by assuming a constant soot conversion factor, 2%, chosen with reference to experimental measurements [36, 37]. The soot formation rate was simply assumed to be proportional to the fuel supply rate. No oxidation was considered. In order to provide proper and economical resolutions for both gas and solid phase computations, a separate grid which can be much finer than that used in the gas phase computation was used to calculate the heat conduction and pyrolysis in solid phase. The surface of a wall is subdivided into many small elements according to the gas phase grid, and along the direction which is perpendicular to the surface, each element is represented by a number of thin slices, which can be less than 1 mm thick. Furthermore, in the pyrolysis model, a local grid refinement moving with the pyrolysis front was employed to provide the needed better resolution for the pyrolysis layer, which has very steep density profile. A non-uniform grid of 96 × 96 shown in Fig. 7 was used for the gas phase computation, with clustering applied to the flame zone to provide a proper resolution. The filtered governing equations are discretized using finite volume method, with the variables at the cell faces in the finite volume discrete equations approximated by a second-order bounded QUICK scheme and the diffusion term computed based on central difference scheme.

Figure 7: The numerical grid used in the computation.

18â•… Transport Phenomena in Fires The computation was explicitly time marched, with the momentum equations solved using a second-order fractional-step Adam-Bashford scheme and the transport scalar such as enthalpy computed using a second-order Runge-Kutta method. The Poisson equation for pressure from the continuity equation was solved using a MG solver. A constant time step of 1.0 × 10-3 s satisfying these conditions was used. In order to reduce the wall clock time of the computation, the code was fully parallelized based on data decomposition. The whole computation is distributed among a group of concurrent tasks which communicate with each other through a message-passing interference library PVM. The computation was performed on a SGI Origin 2000 using four processors. 6.1â•‡ Problem description The configuration of the studied problem is shown schematically in Fig. 8. It is essentially the same as in a previous RANS study [20]. The PMMA slab is 4.5 cm thick and 114 cm high. A small propane gas burner is located at the bottom of the PMMA slab as an ignition source. The output of the burner is 10.0 kW/m. The thermal properties of PMMA were taken from literature [38-40], where ro = 1190 kg/m3, k = 2.49 × 10-7T + 1.18 × 10-7 kW/m/K, cp = 2.374 × 10-3 × T + 1.1 J/g/K, Tp = 363°C and Hc = 24.88 kJ/kg. From the heat of the gasification of thermally thick PMMA at steady state, which is Hg = 1.61 kJ/kg [38], Hpy can be calculated [20]. The size of the whole domain was 2.55 m in height and 1.97 m in width. 6.2â•‡ Boundary and initial conditions As an initial condition, the gas in the computation domain was set still with ambient temperature. At the top, applied was a convective boundary condition [41, 42] instead of the traditional Neumann boundary condition. At the free sides, static pressure boundary condition was employed. The solid fuel was assumed to be a non-slip boundary.

1.8 m Inert wall Flame

Burner

1.14 m PMMA

Figure 8: Configuration of the modelled problem.

Mathematical Modelling and Numerical Simulation of Firesâ•…

19

6.3â•‡ Results and discussion of the case study In this computation, the employed approximate FASTNB model for the important radiation calculation was first verified by comparing with original FASTNB model in a test calculation. Figure 9 compares the computed radiation heat fluxes to the virtual water-cooled heat flux meters mounted on the PMMA surface at a specific instant. As can be seen, the difference between two computations was very small. This confirms the validity of applying the approximate FASTNB in the computation. The computed location of pyrolysis front as a function of time is plotted in Fig. 10. The plot data was obtained by sampling the computation result every 500 time steps. The fluctuation on the curve reflects the transient behaviour of the turbulent combustion in the gas phase. As it can

Figure 9: Comparison of FASTNB and approximate FASTNB.

Figure 10: Pyrolysis height as a function of time.

20â•… Transport Phenomena in Fires be seen, at the early stage when the flame is of small size, the flame spread is essentially controlled by convection heat transfer and consequently the pyrolysis height increases linearly with time. When the flame becomes sufficient large, thermal radiation takes over and becomes the major heat transfer contributor. Since flame radiation can grow with flame size in a limit of black body emission, flame spread is accelerated and the flame spread velocity increases exponentially with time as shown in Fig. 11. Figure 12 compares the computed and experimentally measured flame spread velocities. In order to have a relatively smooth plot with the high frequency fluctuation removed, the computation result was sampled every 30000 time steps. As shown, the flame spread velocity is generally over-predicted. However, considering the uncertainty in experiment and the sensitivity of the flame spread to heat flux, which is very difficult to predict accurately in such a complex situation, the result is encouraging and promising. In order to identify the reason for the over-prediction of flame spread velocity, it is necessary to analyse the heat flux prediction carefully. In the experiments conducted by Orloff et al. [38], at the statistically steady-burning

Figure 11: Flame spread velocity vs. time.

Figure 12: Flame spread velocity vs. pyrolysis height.

Mathematical Modelling and Numerical Simulation of Firesâ•…

21

state, the solid surface radiant flux was measured. By introducing some assumptions, the convection flux was then calculated based on the surface energy balance. With the uncertainties kept in mind, these heat flux data provide useful reference for heat flux analysis in computation. However, due to the computer resource limit, no data of the statistically steady-burning state is available in the present computation. One important difference between LES and RANS is that LES can provide a representation of the developing of the instantaneous combustion process and thus offers a great potential of revealing detailed information. Figure 13 shows a sequence of typical instantaneous temperature profiles of the growing wall flame. A preliminary LES of upward flame spread over board has been carried out using a parallel CFD code SMAFS. Both the turbulent combustion of the gas phase and the pyrolysis of the solid fuel were numerically simulated. In the gas phase computation, the SGS turbulence was modelled using the Smagorinsky model and the SGS turbulent combustion was modelled based on EDC. The convection heat transfer was computed using wall boundary layer law. The thermal radiation was calculated using the discrete transfer method with an approximate version of fast narrow-band computer model implemented to estimate the radiation property of

Figure 13: A sequence of typical instantaneous temperature profiles of the growing wall flame.

22â•… Transport Phenomena in Fires the radiating medium. In the solid phase, an efficient, simple and practical pyrolysis model was adopted to describe the pyrolysis of the solid fuel. This preliminary computation predicts a right trend of flame spread and wall fire growth. The predicted flame spread velocity is in an encouraging agreement with measurement, although there is some considerable over-prediction.

Acknowledgements This work was financially supported by the Centre of Excellence in Combustion Science and Technology (CECOST), which is gratefully acknowledged. The Wenner-Gren Center Foundation gave financial support for the collaboration between Lund University and University of Rhode Island.

References â•‡ [1] Poinsot, T. & Veynante, D., Theoretical and Numerical Combustion, Edwards: Philadelphia, ISBN 1-930217-05, 2001. â•‡ [2] Glassman, I., Combustion, Academic Press: San Diego, California, ISBN 0122858522, 1996. â•‡ [3] Turns, S.R., An Introduction to Combustion; Concepts and Applications, 2nd edn, McGraw-Hill, ISBN 0-07-230096-5, 2000. â•‡ [4] Warnatz, J., Maas, U. & Dibble, R.W., Combustion, Springer-Verlag, ISBN 3540677518, 2001. â•‡ [5] Mauss, F., Entwicklung eines kinetischen Modells der Russbildung mit Schneller Polymerisation, PhD thesis, Rheinisch-Westfählische Technische Hochschule, 1998. â•‡ [6] Schlichting, H., Boundary Layer Theory, McGraw-Hill: New York, 2003. â•‡ [7] Oran, E.S. & Boris, J.P., Numerical Simulation of Reactive Flow, 2nd edn, Cambridge University Press, ISBN 0-521-58175-3, 2001. â•‡ [8] Babrauskas, V., Free burning fires. Fire Safety Journal, 11, pp. 33-51, 1986. â•‡ [9] Modest, M.F., Radiative Heat Transfer, Academic Press, ISBN 0125031637, 2003. [10] de Souza-Santos, M.L., Solid Fuels Combustion and Gasification (Modeling, Simulation, and Equipment Operation), Marcel Dekker, ISBN 0-8247-0971-3, 2004. [11] Scheuerer, G., An overview of the present status of future requirements for industrial CFD. Speedup Journal, 7(1), pp. 27-35, 1993. [12] Tennekes, H. & Lumley, J.L., A First Course in Turbulence, MIT Press: Cambridge, MA, 1982. [13] Smagorinsky, J., General circulation experiments with the primitive equations. Monthly Weather Review, 91, pp. 99-152, 1963. [14] Sagaut, P., Large Eddy Simulation for Incompressible Flows; An Introduction, 2nd edn, Springer-Verlag: Berlin, Heidelberg and New York, ISBN 3-540-43753-3, 2001. [15] Bardina J., Ferziger J.H. and Reynolds W.C., Improved subgrid scale models for large eddy simulation, AIAA paper, 80-0825, 1980 [16] Germano, M., Piomelli, U., Moin, P. & Cabot, W.H., A dynamic subgrid-scale eddy viscosity model. Phys. Fluids, Part A, Sec. 3, pp. 1760-1765, 1991. [17] Ghosal, S., Lund, T.S., Moin, P. & Akselvoll, K., A dynamic localization model for large eddy simulation of turbulent flows. J. Fluid Mech., 286, pp. 229-255, 1995.

Mathematical Modelling and Numerical Simulation of Firesâ•…

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[18] Magnussen, B.F. & Hjerthager, B.H., On mathematical modeling of turbulent combustion. 16th Symp. on Combustion, The Combustion Institute, Pittsburgh, pp. 719-727, 1976. [19] Yan, Z. & Holmstedt, G., CFD and experimental studies of room fire growth on wall lining materials. Fire Safety Journal, 27, pp. 201-238, 1996. [20] Yan, Z. & Holmstedt, G., CFD simulation of upward flame spread over fuel surface. Fire Safety Science; Proc. of the Fifth Int. Symp., pp. 345-356, 1997. [21] Yan, Z., Numerical Modeling of Turbulent Combustion and Flame Spread, PhD thesis, Lund University, 1999. [22] Bockhorn, H. (ed.), Soot Formation in Combustion - Mechanisms and Models, SpringerVerlag: Berlin, Heidelberg and New York, 1994. [23] Yan, Z. & Holmstedt, G., Fast, narrow-band computer model for radiation calculations. Numerical Heat Transfer, Part B, 31, pp. 61-71, 1997. [24] Yan, Z., A fast spectral approximation of narrow band model for thermal radiation calculation. Numerical Heat Transfer, Part B: Fundamentals, 46, pp. 165-178, 2004. [25] Versteeg, H.K. & Malalasekera, W., An Introduction to Computational Fluid Dynamics, the Finite Volume Method, Longman, ISBN 0-582-21884-5, 1995. [26] Lien, F.S. & Leschziner, M.A., Upstream monotonic interpolation for scalar transport with application to complex turbulent flows. Int. J. Num. Meth. Fluids., 19, p. 527, 1994. [27] Brandt, A., Multigrid Solvers on Parallel Computers, ed. M.H. Schultz, Academic Press: New York, pp. 39-83, 1981. [28] Fletcher, C.A., Computational Techniques for Fluid Dynamics, Vols 1 and 2, SpringerVerlag: Berlin, 1988. [29] Yan, Z., Parallel computation of turbulent combustion and flame spread in fires. Numerical Heat Transfer, Part B, 41, pp. 191-208, 2002. [30] Geist, A., Beguelin, A., Dongarra, J., Jiang, W., Manchek, R. & Sunderam, V., PVM: Parallel Virtual Machine, MIT Press: Cambridge, 1994. [31] Cebeci, T. & Bradshaw, P., Momentum Transfer in Boundary Layers, Hemisphere Publishing Corp.: Washington, 1977. [32] Angelberger, C., Poinsot, T. & Delhay, B., Improving near-wall combustion and wall heat transfer modeling in SI engine computations. Int. Fall Fuels & Lub. Meeting and Exposition, SAE Paper 972881, 1997. [33] Han, Z. & Reitz, R.D., A temperature wall function formulation for variable density turbulent flows with application to engine convective heat transfer modeling. Int. J. Heat Mass Transfer, 40, pp. 613-625, 1997. [34] Yan, Z. & Nilsson, A., Large eddy simulation of natural convection along a vertical isothermal surface. Heat and Mass Transfer, 41, pp. 1004-1013, 2005. [35] Yan, Z., SMAFS - Smoke Movement and Flame Spread (User Manual), 2006. [36] Mulholland, G.W., Smoke Production and Properties, The SFPE Handbook of Fire Protection Engineering, 2nd edn, Chapters 2-15, 1995. [37] Tewarson, A., Generation of Heat and Chemical Compounds in Fires, The SFPE Handbook of Fire Protection Engineering, 2nd edn, Chapters 3-4, 1995. [38] Orloff, L., De Ris, J. & Markstein, G.H., Upward turbulent fire spread and burning of fuel surface. 15thSymp. (Int.) Combustion, pp. 183-192, Combust. Inst., 1975. [39] Quintiere, J.G. & Rhodes, B., Fire Growth Models for Materials, NIST-GCR-94-647, National Institute of Standards and Technology, 1994. [40] Babrauskas, V. & Greyson, S.J., Heat Release in Fires, Elsevier Applied Science: London and New York, 1992.

24â•… Transport Phenomena in Fires [41] Akselvoll, K. & Moin, P., Large eddy simulation of turbulent confined coannular jets. J. Fluid Mech., 315, p. 387, 1996. [42] Boersma, B.J., Brethouwer, G. & Nieuwstadt, F.T.M., A numerical investigation on the effect of the inflow conditions on the self-similar region of a round jet. Physics of Fluids, 10(4), pp. 899-909, 1998.

CHAPTER 2 Transport phenomena that affect heat transfer in fully turbulent fires S.R. Tieszen1 & L.A. Gritzo2 1

Fire and Aerosol Sciences Department, Sandia National Laboratories, Albuquerque, NM, USA. 2 FM Global, Norwood, MA, USA.

Abstract Transport phenomena within large (i.e. fully turbulent) fires comprise the foundational mechanisms for several principal fire hazards including smoke production and heat transfer to engulfed and adjacent objects. These phenomena are becoming sufficiently well known that quantitative descriptions are foreseeable. In this chapter, the authors present the current state of knowledge and emphasize unknown phenomena as well as areas in need of additional research to enable deep understanding and quantitative prediction of hazards posed by these fires. The tightly coupled, nonlinear transport phenomena of large fires, as opposed to chemically reacting flows in engineered systems which have been more extensively studied by the general combustion community, are discussed. These phenomena include (1) the large length and timescale range of transport phenomena with an emphasis on the challenges of computing and experimentation; (2) fluid dynamics including turbulence and the effect of buoyancy over the length scale range including the coupling between scalar and momentum fields; and (3) radiative properties and transport including local and global characterization of the radiative emission source term. The discussion is supported by physical considerations based on analysis of data and established models. The results provide a basis to understand physical transport phenomena in large fires and lay the foundation for the understanding needed to predict fire hazards.

1 Introduction Fire is a rich multiphysics phenomenon having a significant impact on mankind from the earliest times to the present. Transport phenomena within a fire are equally rich and highly nonlinear. In order to have a coherent presentation of the transport phenomena it is useful to have both an application focus and a well defined scope. In this chapter, the focus is on heat transfer within a large fire. As such, the connection between advective and diffusive transport phenomena, and convection and radiation heat transfer, within the fire will be emphasized.

26

Transport Phenomena in Fires

By necessity, the scope of this chapter will be limited. It can be readily argued that hydrocarbon chemistry is as rich and nonlinear as the transport processes themselves. However, prioritizing here on the basis of application focus, chemistry will not be discussed except with respect to simplified characteristic time-scale arguments for comparison with transport phenomena. Similarly, the chapter will not touch on the very complex topic of fuel decomposition and/ or vaporization from liquid or solid fuels. These are very complex multiphysics processes in themselves in which both chemistry and transport are quite important. It will be assumed in this chapter that the fuel has vaporized under the incident radiative and convective loads. Further, the chapter will focus on transport within a fire. Fire induced flow, particularly in complex structures, is a rich topic in its own right, but is beyond the scope of this chapter. Finally, the chapter will largely focus on large scale fires, where the laminar to turbulent transition distance is a small fraction of the fire diameter. All three forms of heat transfer − conduction, convection, and radiation − are present in fires. In general, for fully turbulent fires, their importance is in the reverse order, with radiation being the most important and conduction the least important, subject to chemical considerations that might increase the importance of the latter, e.g. the flame phenomenology considered in Chapter 9 of this book by DesJardin, Shihn, and Carrara. In large fires, typical time-mean values of the radiative heat flux are of the order of 150 kW/m2 but can range over about an order of magnitude centered on this value. Much of the radiation is from soot with secondary radiation from the gas species in the flame [1]. Convection is secondary, but not necessarily second order. Typical time-mean temperatures in a large fire are of the order of 1300 K (compared to peak flame temperatures of 2300 K for many hydrocarbon fuels in air). Convection ranges from free to forced convection depending on local flow velocities and temperature differences. Convection coefficients in air typically range from 5 to 500 W/m2 K [2]. For a mean temperature difference between the fire and cold objects of 1000 K (1300 K fire to 300 K object), convective heat fluxes will be of the order 5 to 500 kW/m2 for a wide range of heat transfer applications. At the high end, convection can equal radiation and at the low end, it can be of second order importance. Note that the sign of the two modes can, and often will be, different. Convection can cool while radiation is heating, and vice versa. The balance depends on local environmental conditions for convection and more global conditions within the fire for radiation. In most situations convection is of secondary importance. Within the heat transfer focus and scope outlined, this chapter is structured to first discuss the large length and timescale range of transport phenomena with an emphasis on the challenges of computing and experimentation. Next the effect of buoyancy over the length scale range, including the coupling between scalar and momentum fields, will be addressed. Finally, issues that couple the flow field to radiative transport including local and global characterization of the emission source term will be discussed. The future of transport research will be touched on to conclude this chapter.

2 Length and time scales within a fire 2.1 Overview The challenges associated with understanding transport in turbulent, reacting flows are significant. Fire is an exquisitely complex chemical reaction problem, wrapped in a turbulent, buoyant plume flow problem, wrapped inside a participating media radiation heat transfer problem. The time and length scales in fires are shown in Fig. 1. For large fires, the primary coupling between (1)

Heat Transfer in Fully Turbulent Fires

27

Engineering Scale of Interest H e So at T lid ra s & nsf Fu er in els

10 3

tion vec Con

10 -3

10 -9

10 -12 10 -10

rowth

Soot

Soot G

10 -6

Fl am es

Products

Chemical Kinetics

Time Scale, seconds

10 0

of ort e nsp ectiv a r t T onv C le n rbu ive & ces u T iat r u o d S Ra

Soot Radiation

Diffusive Transport ion

diat Molecular Ra

Molecular Transport 10 -8

10 -6

10 -4 10 -2 Length Scale, meters

10 0

10 2

Figure 1: Physics coupling in fires.

the thermal radiation driving fuel vaporization and (2) the turbulent reacting flow which produces the high temperature soot (that creates the thermal radiation) can span up to 12 orders of magnitude in length scale. The smallest scales in turbulent sooty fires of direct interest are those that contribute to thermal radiation, since radiative transport couples this energy back into larger length scales and to fuel pyrolysis/vaporization. The smallest scale is determined by the electronic states of carbon atoms within soot particles O(nm) as these affect soot optical properties [3]. Soot grows from molecular length-scales O(nm) to O(100 nm) in large fires [4, 5]. Continuum approximations start at length scales of O(100s nm) depending on temperature at ambient pressure [6]. Hence, the nucleation and much of the early growth of soot is a heterogeneous, noncontinuum, process. The large end of the length scale range depends on the application. For laboratory experiments, fire sizes range from O(cm) to O(m); for building fires from O(m) to O(10s m); and for forest fires O(0.1 km) to O(kms). Another consideration in determining the largest length scale of interest is whether the primary interest is within the fire itself, or in the fire-induced flow which can exceed fire length scales by several orders of magnitude. The length scale range from nanometers to kilometers is 12 orders of magnitude. The time scales involved depend on the length scales and process rates. The shortest timescales relevant to fire applications in a theoretical sense are determined by the transit time associated with thermal radiation at the speed of light. However, as discussed in Chapter 7 of this book by Modest, the physics of the interaction between radiation transport and momentum/scalar transport is through radiation properties, not radiation transport itself. These properties vary only over transport timescales of order milliseconds, rather than nanosecond photon transport timescales. Transient timescales associated with photon transport are therefore typically ignored. Similarly, chemical kinetic (typically high-temperature radical) timescales of order nanoseconds affect heat release within flame sheets. For example, for high temperature radicals

28

Transport Phenomena in Fires

with intermolecular spacings of the order of O(100s nm), molecular velocities at high temperatures of the order of 103 m/s, with probabilities of bonding of the order of 10%, have timescales of order nanoseconds [6]. It is often assumed that these very fast timescales reach some statistical equilibrium and can be ignored with good approximation. There are a spectrum of chemical kinetic times from nanosecond to tens to hundreds of microseconds that reflect the interaction of noncontinuum molecular transport and chemical bond rearrangement. Even at ambient temperature and pressure, molecular velocities are typically of the order of 500 m/s [6]. While molecular velocities are high, continuum velocities on the other hand are quite low, even in large fires. They range from O(0.1 cm/s) to O(cm/s) at the fuel source [7] up to O(10s m/s) at the top of a large O(10 m base) fire [8]. Hence continuum transport timescales typically range from milliseconds to tens of seconds, depending on the length scales. The large end of the timescale scale range depends on the application being considered. Underground mine fires in a coal seam can burn for decades, O(108 s). Large forest fires can last for days, O(105 s). Typical large industrial fires last for hours, O(104 s). 2.2 Time and length scale range The shaded bands in Fig. 1 are obtained from partially nondimensionalizing the Navier−Stokes transport equations, which (in addition to the radiative transport equation) describe the dominant transport mechanisms in a fire. The fundamental continuum equations are expressed in terms of length and time scale gradients. The solution of the fundamental equations is the integration of these gradient based terms over the range of temporal and spatial scales determined by the physical parameters relevant for fires. By plotting the time and length scales for transport terms in the momentum equations for parameter values that occur in large fires, a visual, heuristic context is provided. The Navier−Stokes equations [9] are: ∂( ru ) + ∇ ◊ ( ruu ) = −∇P + ∇ ◊ s + r g. ∂t

(1)

For the purpose of nondimensionalization, the following reference values are defined: u r m uˆ = , rˆ = , mˆ = uref rref − r∞ mref ˆ x t xˆ = , ∇ = Lref ∇, tˆ = , Lref tref

,

P Pˆ = , Pref g gˆ = . gref

(2)

The reference values can be considered as local fire plume values where tref comparisons are being made at Lref length scales. These reference values provide the coordinate axes in Fig. 1. Substituting the reference values but leaving each term as a rate, i.e. unit of 1/time, gives: ˆ Pref m 1  ∂( rˆ uˆ )  uref ˆ ˆ ˆ )] = − [ ∇ ◊ ( ruu [ ∇Pˆ ] + 2 ref  ˆ + tref  ∂t  Lref rref uref Lref Lref rref ( r − r∞ )gref ˆ − ref [ rˆ g ]. uref rref

∇ ˆ ◊ sˆ    (3)

Heat Transfer in Fully Turbulent Fires

29

The quantities in the square brackets are assumed to be of order unity due to the nondimensionalization and the relative contribution of each term is from the reference values in front of the brackets. Comparing terms gives the following time-scale/length-scale relations:  1  Advection: tref ~  Lref  uref 

 1  2 Diffusion: tref ~  Lref .  vref 

(4)

Using the advective time scale definition gives: Buoyancy: tref

  rref ~  ( r∞ − rref )gref 

1/ 2

L1/ref2 .

(5)

Note that the same time-scale definition for buoyancy comes from a similar partial nondimensionalization of the vorticity transport equations (curl of the Navier−Stokes equations, see Najm et al. [10] for particular formulation) if the following additional reference scales are defined: ˆ wˆ = tref w, ∇ rˆ =

Lref ∇r. ( r∞ − rref )

(6)

The result is u 1  ∂wˆ  ˆ ˆ  ˆ ˆ ˆ ˆ + ref (uˆ ◊ ∇ )w + w(∇ ◊ uˆ ) − (wˆ ◊ ∇ )u  2  ∂tˆ  t L tref   ref ref ( r − r ) g  ∇ ˆ rˆ  ( r − r )  Duˆ  u m ref ref ˆ × (∇ ˆ ◊ sˆ ) + ∞ = 3ref ref ∇ × gˆ  − ∞ 2ref   .  ˆ rref Lref r Lref rref  rref tref  Dtˆ   

(7)

The last two terms are, respectively, the gravitational and baroclinic generation of vorticity. The gravitational generation corresponds to the curl of the buoyancy term in the Navier−Stokes equations. The baroclinic part results from density gradients interacting with local acceleration fields. Fires are most strongly influenced by buoyancy but near the fuel source, strong flow acceleration and steep density gradients can produce significant baroclinic generation [11]. Comparing the scaling terms for advection, diffusion and buoyancy, it can be seen that they have different length-scale dependencies. Thus, each term dominates at a different length scale as shown in Fig. 2. At small scales, diffusion is dominant because of the high molecular velocities relative to the bulk gas velocities. Representative values for viscous diffusion in air at 300 and 2300 K are shown in Fig. 2. Random direction, molecular-walk processes which define diffusion are inefficient at larger length scales and bulk advection becomes dominant. At still larger scales, buoyancy dominates. Since large fires represent turbulent-mixing-limited combustion phenomena which have a spectrum of length-scales contained within the broader length-scale spectrum of radiation transport (from noncontinuum soot emission to absorption at global application scales), both fluid transport and radiative transport contribute in overlapping length scale regimes. In general, all length scales play a role in this coupled multiphysics/multilength scale problem. Therefore, while one process may dominate at a given length scale, it cannot be said that any one of these terms dominates the entire coupled process over all length scales. The advection to diffusion ratio is the Reynolds number. In flames with fast chemistry, (Da >> 1) the balance of these processes defines the width of the diffusion flame as a function of the

30

Transport Phenomena in Fires

Figure 2: Time and length scales in fires.

imposed velocity gradient across it. A two order of magnitude increase in imposed velocity will decrease the flame thickness one order of magnitude until finite rate chemistry results in extinction. Flame widths are typically O(mm) depending on the imposed strain. Above centimeter length scales, advection and buoyancy dominate transport processes. All the transport physics normally associated with low-Mach number flows are present in a fire. For example, transport of momentum results in a turbulent cascade due to the nonlinear advection term in eqn (1) just like all other flows. Fires are also strongly affected by the buoyant source term. The characterization of the dynamic effects of the buoyant source term has received less attention in the fluid mechanics community than the turbulence generating nonlinear advection term. In eqn (1), buoyancy is a linear source term for linear momentum. In eqn (7), gravitationally produced buoyancy is a linear source term for vorticity. Equations (1) and (7) are not independent. Linear momentum generation due to buoyancy is achieved through vorticity generation as will be discussed later in this chapter. Figure 2 shows two levels of the normalized density difference, (∆r/r), of 3 and 7. The first is roughly representative of the long-time average centerline temperature (~1300 K) difference with ambient (~300 K); this gradient will exist over large length-scales in fires since this temperature difference is relatively constant over large portions of the fire [12]. The second level is related to the adiabatic flame temperature (~2300 K) and is an upper bound that exists only at small scales. The buoyant time scale is related to the reciprocal of the Brunt−Väisälä frequency [13]. It can be seen in Fig. 2, for moderate velocities typical of fires, O(1−3 m/s), and a scaled density

Heat Transfer in Fully Turbulent Fires

31

difference of 3, that advection is faster, i.e. shorter time scale at a given length scale, than buoyancy up to O(10 cm) length scales. At length scales larger than O(10 cm), buoyant time scales are shorter than advection. Experimentally, it is found that fires become transitionally turbulent for O(10 cm) base diameters and are fully turbulent at O(1−3 m) (see Drysdale [14] for a discussion and references therein), consistent with the view that buoyancy expresses itself as rotational motion whose instability induces turbulent motion. The ratio of the advective time scale to the buoyant time scale is the Richardson number. For fires, as chemically reacting flows, the ratio of fluid transport time scales to chemical and heat transfer time scales is important. Chemical time scales are dependent on temperature, composition, and specific reaction metrics (i.e. activation temperature and preexponential factors). For a given chemical time scale, comparison with the transport time scale in Fig. 2 establishes a Damkohler number, Da. Comparison can be made to diffusive time scales or advective time scales. In general, the turbulence intensities in the small-length-scale spectrum in fires are low compared to jet flames in combustors [15] and have the appearance of wrinkled flame sheets [16]. However, long chemical times may result either from low temperatures or off-stoichiometric compositions. These conditions possibly occur in two areas in large fires. (1) In the oxygen-starved vapor dome just above the fuel source, measurements [17] indicate temperatures are of the order of 1000 K. At these conditions, kinetic calculations indicate that pyrolysis reactions can occur but are fairly slow, of the order of tenths of seconds [16]. With advection velocities of O(1 m/s) to O(10 m/s) and vapor dome heights of O(m) for large fires, significant pyrolysis may occur. (2) The large rolling structures at the edges of large fires visually appear to end up filled with smoke (i.e. relatively cold soot on the air side of the flame zones), suggesting that some form of quenching has occurred that does not appear to be due to high turbulence levels. Oxygen depletion of fuel rich eddies, perhaps followed by radiative cooling, is a more reasonable hypothesis [16]. While heat transfer to fuels/objects within a fire is primarily radiative and convective, very often the internal heat transfer of the fuel/objects is limited by conduction. Figure 2 shows thermal diffusivities for a good heat conductor, aluminum, O(80 mm2/s), and a poor heat conductor, insulation O(0.3 mm2/s). From an order of magnitude perspective, conduction timescales in solids are not all that different from diffusion times in gases. Both are diffusion processes which are increasingly slow compared to advection processes as the length scale is increased. The ratio of advective to gas-phase diffusive time scales is the Reynolds number, which is O(104−106) for large fires. At large length scales, a similar disparity exists between convective heating of an object and internal conduction within the object. Due to this convective/conductive disparity, the timescale range from shortest to longest is actually longer than the length scale range in fires by several orders of magnitude. These very large spans in both length and time scales present challenges to numerical simulation of large fires. 2.3 Implication for numerical simulation The largest fire simulations run to date are of the order of millions [18] to 10s of millions of nodes [19]. Using a simple uniform-spacing scaling rule requires 10 nodes per order of magnitude of resolved length scale. For three dimensions, this means O(103) times the existing computing power for every order of magnitude of newly resolved length scale as shown in Fig. 3. Further, an additional factor of 10 increase in processor speed or number of processors is required to capture the shorter time scales associated with the incrementally resolved length scales if the computations are to be done in the advective/buoyancy controlled regime as shown in Fig. 3. In the diffusion controlled regime, where doubling the length scale quadruples the time scale requirements, capturing the time scales requires a factor of 100 increase for every

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order of magnitude resolved. Further, as the scales resolved get smaller, the number of species that participate at the short time scales increases, resulting in the need for more transport equations. Therefore, with every order of magnitude of increased resolution, the computing power of the machine needs to be 10,000 to 100,000 times as powerful. With massively parallel computing, involving O(103−105) processors, it can reasonably be expected that within the next decade an additional single order of magnitude of length scales will be resolved (i.e. billion node fire calculations). To fully simulate a problem with 12 orders of magnitude in length scale, given that we can reliably simulate 3 orders of magnitude in length scale with machines expected to be built in the near future, the computer processing power would need to be at least 10(4 per order of magnitude × 9 orders needing resolution) or 1036 times the processing power of the world’s largest machines now coming online. Assuming processing power doubles every 18 months, or roughly a factor of a hundred per decade, it will take almost 18 decades to acquire this kind of computing power. The example given is for the largest fire. For smaller laboratory scale flames, having 6 total orders of magnitude in length scale, with 3 unresolved, the machines need to be a more modest 10(4−5 orders per order of magnitude × 3 orders of magnitude) more powerful. We could see computing power reach these levels in the next 60−80 years if current trends in the rate of increase in processing power continue. The above discussion does not imply that numerical simulation is not useful. Quite the opposite, it is an extremely valuable complement to experimentation for obtaining both engineering and scientific insight. However, due to the extremely large range of length and time scales involved in fires, numerical simulation’s principal strength is not in its physics content. The strength of numerical simulation is in its diagnostics. All physics variables at all points in space and time are accessible. Correlations in both space and time are available at resolved scales, as well as direct insight into transport dynamics from transient visualization.

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In contrast, the strength of experimentation is in its physics content. For fires that have application-relevant geometries, initial conditions, and boundary conditions, the experiment contains true physics down to subatomic scales. The relative weakness of experiments is in the diagnostics. Unlike numerical simulation, experimental diagnostics are very hard to create. The amount of data recorded from a typical fire experiment is a vanishingly small fraction of the physics content present, and more often than not, fundamental transport variables cannot be measured directly but must be indirectly inferred. Furthermore, the presence of extensive intrusive diagnostics can have a significant effect on the fire physics by introducing additional heat transfer modes, and fluid mixing. Thus, numerical simulation and experimentation directly complement each other. Where one is relatively weak, the other is relatively strong. For fires, experimentation is the full truth partially exposed, while simulation is the partial truth fully exposed. Using both in combination is usually the fastest way to gain insight into either physics or engineering applications. The goal of either scientific or engineering simulation is typically to make a prediction. Engineering and science simulation use the same tools and approaches but differ on the acceptance standards for the word ‘predictive.’ It can be argued that ‘predictive’ capability already exists in the engineering sense of the term. Evidence for this argument is found in the rapid growth of the use of CFD-based numerical simulations for fire from early efforts in the 1980s to the present [20]. However, predictive in the scientific sense of the term will not be achieved until all scales are resolvable by integration of discrete approximations, or closed form solutions are found. Chapters 1, 3, 7, and, 9 in this book by Nilsson, et al., Smith et al., Modest, and DesJardin et al., respectively, deal specifically with numerical simulation in fires. 2.4 Implications for modeling The physics in Fig. 1 is continuous across length and time scales. However, it is clear from the discussion above that not all length and time scales can be resolved by solution of the discretized conservation equations. The range must be segmented into three discrete parts. Figure 4 provides a useful visualization of what processes can be captured in a given length scale range. The graph can be divided into three length-scale regimes using two length-scale cutoffs. (For example, imagine vertical lines at 10 cm and 10 m in Fig. 4, capturing two orders of magnitude). Above the larger cutoffs are length scales too large to be captured and these are represented by boundary conditions in a simulation. Below the smaller cutoffs are length scales that have to be modeled and these are represented as source or nonlinear advection terms in the transport equations. Between the boundary conditions and the source terms is the length scale range in which the transport equations are solved by discrete approximation. Implicit in the length scale cutoffs are time scale cutoffs corresponding to the time scales of the transport processes at the cutoff length scales. This splitting of the time and length scale spectrum into three regimes − boundary conditions, resolution by integration of partial differential equations, and modeling − is permitted mathematically by pre-filtering the partial differential conservation (i.e. mass, momentum, and energy) equations. Figure 4 shows graphically that the filtering process has separated the time and space regime into discrete parts with the large part of the regime (lower left) being modeled, while the upper right part of the regime is being solved by solution of discrete approximations to the transport equations. Boundary conditions are applied at the right boundary of the image (i.e. at all time scales at the largest length scale), and notionally initial conditions at the bottom boundary (i.e. all length scales at the initial, or shortest time scale).

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Filtering can be done explicitly in space or in time. As Fig. 4 shows, whichever coordinate is taken there is an implicit filter in the other. Explicit spatial filters have implicit temporal filters (e.g. over the time step which is linked by the Courant number to the spatial filter). Similarly, explicit temporal filters have implicit spatial filters. Regardless of the filter chosen, the necessity of splitting the problem in this manner creates an irreversible loss of information, perhaps best understood in the context of the three processes that occur at the filter scale at every time step and discrete element in the solution. Information is passed from the solution of the filtered partial differential equations to the model of the high spatial and temporal frequency physics that is unresolved. This model is often called a subgrid model or submodel, but should technically be referred to as a subfilter model. Based on the information passed from the partial differential equation solution, the subfilter model (by various strategies) estimates the evolution of the process within the spatial and temporal domain of the filter. After this evolution step, the model values are averaged and used to pass mean information up to the resolved solution to close either source terms or unresolved advective terms. The most serious effect of this necessary splitting procedure is the loss of information in the down-scale pass from the resolved solution to the subfilter model of the high spatial and temporal frequency physics. The subfilter model must use the information from the resolved solution as initial and boundary conditions for the spatial and temporal domain within the filter. Because the resolved solution is at the limit of its resolution, only mean values can be passed unless additional transport equations are solved. Typically a higher moment (e.g. variance equation) and perhaps a time scale for each physics are passed down. The mean, variance and a time scale are very little information to base initial and boundary condition information on for a subfilter problem that in itself may contain as many as nine orders of magnitude in length scale. Hence this downscale pass is a ‘one to many’ transfer. It is in this downscale information pass that information is irreversibly lost.

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The consequence of this loss is that no matter how accurate a subfilter model is, without fully resolved initial and boundary conditions, the mean value of the evolved process variable will contain uncertainty, the magnitude of which depends on how well the process correlates with the averaged initial and boundary conditions. For most highly nonlinear processes, the mean output is not necessarily highly correlated with the mean input. Consider the following example. Given that transport physics is well understood in a theoretical sense, it is conceivable that the subfilter domain could be solved to arbitrary accuracy using partial differential equations, assuming the initial and boundary conditions are known. For such a situation, there would be no ‘modeling error’ in that the physics in the subfilter domain can be resolved. In this case, all the uncertainty would come from ‘errors’ in applying mean (and perhaps variance) values to what would otherwise be spatially and temporally rich initial and boundary conditions. This example is theoretical due to the problems in solving the partial differential equations with high spatial and temporal frequency, or it would have been done without using filtering and incurring the errors associated with the ‘one to many’ downscale pass. Traditional modeling approaches seek to find a correlating variable that can be resolved, and tie the modeled process to that correlating variable. In this manner, the model will have minimized the output uncertainty within the context of the information it is being passed. Statistical methods are typically the tools of choice for modelers because statistical tools are well suited to find correlations. The accuracy of this approach is limited by how well the subfilter process correlates with resolved variables. Subfilter processes that are not well correlated with resolved variables often arise because important high frequency content ‘evolves’ in a weakly correlated way with the overall flow field. For situations such as these, either the grid resolution must be increased until the resolved field is better correlated with the subfilter model, or it must be modeled with a subfilter model that evolves the high frequency content. The only part of the process which does not introduce any information degradation is the upscale pass of the mean value of the subfilter process to the resolved partial differential equation, either as a source term or a nonlinear unresolved advection term. This upscale pass is of the ‘many to one’ type and is unique. Much emphasis is placed within the mathematically oriented community to ensure that filtered equations are used to clearly define the requirements of this upscale pass so that part of the process is free from errors. With the information from the subfilter processes, the resolved field is advanced a time step and the field variables updated. The uncertainties in the unresolved source or nonlinear advection terms are then propagated by the equations as the information for the next downscale pass is generated. The process repeats itself as the solution to the resolved equations evolves. In this manner, the filter and the model are linked. There are no universal models except in regimes where the process is independent of the filter scale. Arguments are often made for closing the equations in the ‘inertial’ range of turbulence, at length scales smaller than the production range but larger than the dissipation range. This argument makes the assumption that there is a broad spatial distribution between turbulence production and its dissipation. For large fires, the argument is relatively weak. As will be discussed later in this chapter, vorticity production occurs across a broad length scale spectrum because density gradients exist across broad length scale spectrums. Therefore the separation of production and dissipation present in shear flows is not present in buoyant flows. Further, even though the length scales in large fires are large, the Reynolds numbers are relatively moderate, O(104−106), because the velocities are relatively low. Note that heat transfer aspects are particularly impacted by modeling. A large fraction of the information content involved in heat transfer is tied up in correlation-based engineering subfilter

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Figure 5: Model taxonomy. models. For large fires, Fig. 4 shows that all combustion processes are of higher spatial and temporal frequency than can be grid-resolved for practical problems for the foreseeable future. All the soot generation and spatial overlap between soot and high temperature gas fields is modeled. Thus the transport terms in the radiation transfer equation have a very high basis in modeling, or ‘modeled content’. Similarly for convection, the highest frequency turbulent eddies are created in the near-wall boundary layer region of objects. Ultimately, transport is conductive in the near wall region. None of this physics can be resolved for large fires with foreseeable technology. Because of the high reliance on modeled physics, the engineering accuracy of heat transfer predictions is strongly dependent on engineering models. The basic strategy for engineering simulation of fire for the last couple of decades has been to simulate the fluid transport at large scales and model the higher spatial and temporal frequency physics. While there is no fundamental reason this approach cannot be altered, it can be expected that as machines become larger, modeling will no longer be necessary for transport physics at length scales below those just resolvable. Figure 5 provides a model taxonomy. Using this taxonomy, the first models that will be replaced will be the meso-scale mixing models, followed by models for diffusional transport processes, and finally, when machines are large enough, by molecular transport and chemical processes.

3 Fluid dynamics within large fires Like all continuum flows, momentum transport in fires is given by the Navier−Stokes equations (eqn (1)). Due to the nonlinear advection term (the second term on the left-hand side of eqn (1)), fires become turbulent for fuel sources above about one meter [14]. Large fires exhibit the full range of rich vorticity dynamics associated with turbulence including vorticity production at solid boundaries and at density gradients within the fluid. Visual evidence exists of vorticity scale change including growth in coherence length scales by vorticity rollup and pairing, and decay in coherence length scales by straining and tangling. The result of all these mechanisms is a turbulence cascade in which dynamics across the length scale spectrum discussed with respect to Fig. 2 participates in transport of momentum and scalars in a fire.

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In general, fires can occur under a very broad set of initial and boundary conditions. By common definition, they are low Mach number and have spatially separate fuel and oxidizer sources. Fires induce gas motion themselves. In enclosures, the resulting flows may be quite complex, both spatially and temporally. Thus, a complete description of the fluid mechanics of fires would require a complete description of low Mach number fluid mechanics. This breadth is beyond the scope of this treatise. Rather, what follows focuses on what makes fires fluid-mechanistically unique. To achieve this goal, two canonical flows, (1) a round plume issuing from an infinite ground plane into an otherwise quiescent fluid, and (2) a round plume issuing from an infinite ground plane into a horizontal cross flow will be discussed with respect to the turbulent dynamics. 3.1 Quiescent conditions If there is no external forcing applied, a fire is a reacting plume. As such it is a member of the family of jets and plumes as shown in Fig. 6.

(a)

(b)

(d)

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Figure 6: Fire as a reacting plume: (a) fire, (b) reacting jet [21], (c) non-reacting plume [22], and (d) nonreacting jet [23].

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3.1.1 Jet versus nonreacting plume dynamics Jets and plumes share all the momentum transport terms in common. Thus, all the nonlinearities associated with advection including turbulence are shared between jets and plumes. The difference between jets and plumes is in the source term. Isothermal jets do not have the source term in eqn (1). On the other hand, jets generally have a high value of inlet momentum. From eqn (3) and Fig. 2, it can be seen that the ratio of the buoyant source term to momentum is given by the Richardson number. Plumes have high values of Richardson numbers, while jets have low values. A popular alternative expression to the Richardson number is the Froude number, u2/gD, where g is the value of gravity, D is a characteristic length scale, and u is the vertical velocity. Sometimes the square root of this value is used. Further, if the density is used to modify the Froude number, then it is called the density modified Froude number. The Richardson number is the reciprocal of a form of the density modified Froude number. From a dynamics perspective, the difference between jets and plumes is that vorticity in isothermal jets comes entirely from the nozzle boundary layer. In an isothermal flow, when a jet enters an unconfined, uncluttered domain, all the vorticity that will ever be present in that domain comes from the source boundary layer. In plumes, the vorticity is generated from the buoyant source term in eqn (1) under the conditions expressed in eqn (7). These conditions are shown graphically in Fig. 7 with a notional fluid

Figure 7: Buoyant vorticity generation.

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element overlying a fire image. The fluid element shown would be less dense on the left-hand side than the right-hand side because the gases are hotter in the fire than outside the fire. The fluid element has a horizontal density gradient. Even in a quiescent (no velocity) condition, gravity, acting vertically through the element, will result in a force at the mass center of the element. However, the mass and geometric centers are not coincident. The mass center is biased to the right side of the geometric center in the element in Fig. 7 because it is heavier on the right. This condition will result in rotation of the element in Fig. 7 if the geometric center is fixed and allowed to rotate. For this heuristic example, the right-hand side would move downward while the left-hand side would rise, thus inducing a rotation under the force of gravity. The rotation would continue until the force through the mass and geometric centers align. In other words, the misalignment between a density gradient and an acceleration field (or equivalently, pressure gradient) will cause the generation of vorticity. More generally, all fluid elements are connected and the effect is elliptic in nature. The two parts of eqn (7) that make up the buoyant vorticity generation term can either be thought of as being due to the misalignment of density gradients with hydrostatic and hydrodynamic pressure fields, or equivalently, as explicitly stated in the form chosen for eqn (7), the misalignment of density gradients with hydrostatic and hydrodynamic acceleration fields. The two terms are called buoyant and baroclinic production of vorticity, respectively. Since vorticity is generated by the misalignment of density and acceleration fields, the length scale of the generated vorticity will be limited by the extent of either the density gradient or the accelerating field. In the case of gravity, the field is very large so that the extent of the vorticity generated is almost always limited by the extent of the density gradient. In turbulent flow fields, acceleration/deceleration is often experienced across eddy boundaries so that the baroclinic term may be limited by the coherence of the acceleration field. It may change sign rapidly in both space and time unless there is a mean acceleration of the flow. Numerical simulation suggests strong acceleration gradients at the base of the plume [11]. It should be noted that the presence of vorticity does not imply the existence of a ‘coherent vortex’ or turbulent eddy. As stated previously, plumes share all the vorticity transport dynamics inherent in jets, including the roll-up of vortex sheets, pairing of vortices by amalgamation, etc. [24], as well as the stretching and tangling of vorticity. Thus, as in Fig. 6, both jets and plumes end up with large coherent rotational structures as well as a turbulent cascade through a combination of vorticity source terms and vorticity transport terms. The vorticity transport terms in jets and plumes are shared, whereas the vorticity source terms differ between jets and plumes. The differing source terms result in quantitative differences in flow dynamics corresponding to the magnitude and length scale differences of the source terms. For example, in the far field, both jets and plumes are self-similar, and can be scaled by the same self-similarity laws as long as the overall magnitude of the source term is taken into account [25, 26]. The reason for this equivalence is that the source terms between jets and plumes differ most strongly in the near field. Isothermal jets have their highest velocity at the source and thus have all their vorticity at their injection source. Plumes accelerate due to buoyant (and baroclinic) production of vorticity. However, as a nonreacting plume moves away from its source the density gradients diminish due to turbulent mixing and diffusion. Thus the rate of production drops and the source term lessens. On the other hand, all the previously generated vorticity continues to be advected. The effect is cumulative in that the circulation grows with elevation.

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Since the ratio of production to advection of vorticity drops with elevation in buoyant plumes, at some elevation the decelerative effects of the advected vorticity exceeds accelerative affects of the buoyant vorticity source term and the plume velocity peaks. Eventually the buoyant source term becomes vanishingly small. After vorticity advection dynamics removes the ‘memory’ (i.e. details of the dynamics) of the source term, jets and plumes become similar if the magnitude of the source terms are similar [25, 26]. Between the near and far field regimes, high order turbulent statistics are more strongly affected than mean flow statistics. In particular, if the buoyant source term is not zero, vorticity will be generated across a spectrum of length scales corresponding to the spectrum of density gradients. Experimental evidence for this view can be found in nonreacting buoyant plume data [27] which shows a −3 spectral decay as opposed to a −(5/3) spectral decay in velocity over a broad spectrum of length scales above diffusive scales. It is shown that the −3 spectral decay can be obtained from scaling the ratio of buoyant and advective time scales. 3.1.2 Reacting versus non-reacting plume dynamics The discussion to this point has compared and contrasted jets and plumes. It is now appropriate to compare and contrast reacting and nonreacting plumes. Note that in eqn (1), or its vorticity equivalent eqn (7), combustion does not appear explicitly. Combustion is coupled to momentum only through density gradients under specific conditions, and the temperature dependence of viscosity. Reacting and nonreacting plumes both have in common the buoyant and baroclinic vorticity generation term in eqn (7). In theory, if a combusting flow produced the same magnitude of density gradients in the same spatial locations as a nonreacting plume, then the magnitude of this source term would be the same for each flow. In general this is not the case, resulting in quantitative differences between reacting and nonreacting plumes. Examples will be discussed in Section 3.1.3. Fuel vapor in most fires is not buoyant relative to air. When at the same temperature, only a limited number of fuels are significantly buoyant relative to air, e.g. hydrogen and methane. Carbon monoxide and the fuels with two carbon atoms are slightly to neutrally buoyant, while fuels with three or more carbon atoms are negatively buoyant. Without combustion, most fuel vapors will sink and stably stratify. Combustion products are buoyant due to their high temperature. At ambient temperatures, the products of combustion for the most part are also slightly to neutrally buoyant. Typically for alkanes, CnH2n+2, for large n, the product composition is one CO2 for every H2O. With molecular weights of 44 g/mol and 18 g/mol, respectively, an equal molar solution will give a mean molecular weight of 31 g/mol relative to air at 29 g/mol. At peak combustion temperatures that are of the order 2100 K, combustion product density is about 1/7 of air. Only hot products make a fire plume buoyant overall. Flame sheets in themselves are buoyant; however, it is the accumulation of the hot products from the flame sheets that creates an overall buoyancy in a fire plume. Figure 8 illustrates this observation. Consider two plumes, one with a plume density equal to that of air (e.g. ethene) and a second with a plume density less than that of air (e.g. products of ethene combustion), shown in Fig. 8(a) and (b), respectively. Within the flame sheet itself, hot products result in a decrease in density, creating a density gradient between the hot products and the air, and between the hot products and the plume [28]. For the case in which the plume density matches that of the air density, then the vorticity generated on each side of the plume is nominally the same (assuming the same diffusivities) since the density gradients are nominally the same. The total vorticity across such a flame zone is zero and serves only to accelerate the flame sheet at scales corresponding to the flame thickness. From a

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Figure 8: Net vorticity across a flame zone: (a) density of plume and air are equal (no net vorticity); (b) density of plume is less than air (net vorticity).

linear momentum perspective, the flame zone is a buoyant sheet and will accelerate upward under gravity. While the air and plume will be drawn into the upwardly accelerating flame, there is no net difference to create buoyancy within the plume itself. Now consider a case in which the density of the plume is less than that of the surrounding air, as in Fig. 8b. As with the first case, within the flame sheet itself, hot products result in a decrease in density, creating a density gradient between the hot products and the air, and between the hot products and the plume. Unlike the previous case, the vorticity production is not the same on each side of the flame sheet. The vorticity production on the air side will be stronger than on the plume side. This imbalance results in a net vorticity across the flame sheet, represents the buoyancy of the plume and is independent of the flame sheet itself. So again the flame sheet itself is not a net buoyancy source. A typical fire plume does not become a buoyant plume until sufficiently hot products are mixed into the core of the plume such that the overall density becomes less than that of the surrounding air. At the base of most fires, the overall plume is nonbuoyant. The vertical velocity of the fuel vapor is lower than that of the surrounding flame sheets. If fluid mechanics were solely local in nature, the vertical velocity of the fuel vapor would remain at its source value. However, the elliptic nature of the pressure field results in upward acceleration. From a vorticity dynamics perspective, the net vorticity in the buoyant part of the plume (where hot products have mixed to the core) will induce an overall velocity field which tends to accelerate the fuel upward, albeit much more slowly than the surrounding flame sheets. A quantitative example will be given in Section 3.1.3. The nature of the spectrum of turbulent production by buoyant and baroclinic vorticity generation differs somewhat between nonreacting and reacting plumes. As noted previously, the presence of vorticity does not imply the existence of a ‘coherent vortex’ or turbulent eddy. In the simplest case, a density gradient must be contiguous enough so that the vorticity formed from the gradient results in a sheet that can roll up into a coherent structure. Thus, ‘turbulence’ in the form of eddies formed by density gradients is always created at length scales larger than the density gradients creating the vorticity. Implied in Fig. 8b is that the net density gradient across the flame sheet is the density gradient that will result in the formation of a coherent eddy. This gradient is shallower, i.e. has a longer length scale, (for the same density difference) in the reacting plume than the nonreacting plume because of the presence of the flame sheet in the reacting case. In the nonreacting case, the interface can be much thinner. Thus coherent eddies resulting in dynamics that lead to turbulence will occur at smaller length scales in nonreacting flows.

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Vorticity generation in reacting flows is also suppressed by another effect − dilatation in the flame sheets. Combustion produces gradients that result in the net local divergence of the velocity field through dilatation. This effect is explicitly a sink term for vorticity. It is the second component (∇ ◊ u) of the second term in eqn (7). The term is positive on the left-hand side of the equation, and so would be a negative term if shown on the right-hand side; hence it is a sink. The dilatation term is a consequence of the local expansion of the flow due to the conversion of chemical energy to thermal energy in exothermic reactions. In these regions, the flow diverges. The term (∇ ◊ u) is actually the velocity divergence due to the dilatation. It arises due to conservation of mass (continuity equation) in a flow with density gradients [10]. A common analogy given to explain the effect is that of an ice skater in a spin. As the skater extends his arms, he slows down, as he pulls them in he speeds up. Similarly, for a point in space with vorticity present, the fluid expansion will decrease the vorticity. (Less common in fires, local fluid contraction, through condensation, for example, will increase the local vorticity.) Combustion also affects the local viscosity field, since kinematic viscosity is temperature dependent. Because the temperatures in a fire are distinctly nonuniform, kinematic viscosities will also be nonuniform. In a nonreacting plume, kinematic viscosity can be expected to monotonically vary from the plume fluid to the ambient fluid. Thus, the spatial distribution of viscosity in a nonreacting plume will be different than that of a fire, even if the nonreacting plume fluids are such that the viscosities are of similar magnitude. The first term on the right-hand side of eqn (7) is the diffusion term. It is not fully expanded so it cannot be seen that viscosity is explicit in this term. Heat release due to exothermic combustion increases the kinematic viscosity which increases the diffusion of vorticity. Locally, the term can act like either a local sink or a source term depending on whether higher or lower strength vorticity is being diffused into a region. The effect of increased viscosity is almost always discussed with respect to kinetic energy as opposed to momentum or vorticity transport. If a moment of the momentum equation is taken, i.e. taking the dot product of velocity with eqn (1), the result is the kinetic energy equation. The diffusion term in the momentum equation now becomes the dissipation term in the kinetic energy equation. Dissipation is always a sink for kinetic energy. Increasing the viscosity will decrease the turbulent kinetic energy. Gas combustion product kinematic viscosity increases with temperature. Hence, the strongest diffusion and dissipation (with all else equal) will occur within the flame zones, where the temperature is the highest. Note that these are also the locations of the highest dilatation. Physically, the effect of increasing viscosity is that random molecular motion becomes more energetic (higher velocity) with increased temperature (see Fig. 2), so that bulk or directional motion at that scale becomes less significant. In this manner, it is often said that viscosity sets the cutoff scale for turbulent motion. Dissipation is said to convert bulk motion to random motion as a means of dissipating the kinetic energy. In this manner, bulk or directed energy associated with the introduction of a plume or jet will eventually convert itself into random molecular motion at equilibrium, i.e. after the plume or jet has ‘dissipated’. There is sometimes confusion since in the momentum (and vorticity form) of the equation, diffusion is neither source nor sink, just a local means by which random walk transports momentum or vorticity. It does not ‘dissipate’ momentum; it diffuses it until it uniformly spreads to its lowest value. Higher values of viscosity will therefore result in greater transport of both momentum and vorticity. Whether viscosity change is considered as diffusion in its effect on momentum, or dissipation in its effect on kinetic energy, the scale over which it operates is defined by the velocity gradient. It is strongest at the smallest scales − in the reacting case, in the near flame zone regions.

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However, to the authors’ knowledge, detailed studies of the effect of local dilatation and enhanced viscosity in combustion zones in turbulent reacting flows have not quantified the relative effect of each term as a function of the local turbulence field. Since turbulence is often thought of in terms of the kinetic energy of turbulence, both terms are a direct sink. This observation is consistent with qualitative observations that reacting jets are ‘less turbulent’ than nonreacting jets for the same inlet conditions. Quantifying the effect of dilatation and diffusion on the turbulent spectrum is an open area of research. In the context of a turbulent flow field, many questions remain unanswered. Will an active combustion zone affect the spectral distribution of kinetic energy as more than a sink? For example, will the spectral distribution be changed? Will it be changed only near the cutoff scales, or more globally? In fires, these questions are perhaps more relevant than other forms of combustion, because the tighter coupling between the scalar and momentum fields. Further, combustion in fires is limited by the mixing. From a modeling perspective, since length scales in fires tend to be large, the scale of the grid relative to molecular scale mixing processes is also large. Therefore, in fires, a longer range of mixing scales is modeled, and details of the mixing model are more important than perhaps is the case in other reacting flow problems. It is worthwhile to note (see Chapter 9 of this book by DesJardin et al.) that virtually all combustion models to date rely on timescales, whose derivation does not take into account local temperature fluctuations. The viscosity is usually taken from a cell mean temperature. 3.1.3 Rayleigh–Taylor instabilities and buoyancy In both nonreacting and reacting plumes, momentum transport is triggered by instabilities. Fundamental observations of the transient dynamics of both nonreacting and reacting laminar and transitionally turbulent plumes [29−33] have resulted in a description of the source of plume dynamics including the puffing frequency. Recent temporal and spatially resolved experiments in turbulent nonreacting and reacting plumes have extended this description of dynamics to additional modes [22, 34]. A simple, fully turbulent helium plume taken from [22] will illustrate the near field transport dynamics of a nonreacting plume. Figure 9 shows seven planar laser-induced fluorescence (PLIF) images, each 1/6 of a ‘puff’ cycle, with the first and the seventh images being the start of a cycle. Each image only shows the left half of a 1 m diameter plume. The plume’s centerline is on the right edge of the image. Notational flow dynamics are given on each image, and detailed velocity vector plots are given in [22]. The first image shows a large coherent structure in the upper half of the image with helium entering the domain from a plume source at the base of the image. As the helium enters the domain, it is subject to a Rayleigh−Taylor instability as relatively heavy air overlies the lighter helium entering the domain. The second image in Fig. 9 shows the formation of classical bubble and spike structures, in which the helium forms ‘bubble’ structures that rise relative to the air and air forms ‘spike’ structures that sink relative to the helium. The third image in Fig. 9 shows that as these structures continue to grow, helium and air begin to mix such that by the fourth image in the sequence helium and air have formed a somewhat homogeneous mixture. The fourth image occurs at ~1/2 the puff cycle. At this point, a helium/air plume exists adjacent to the surrounding air. This is also a Rayleigh−Taylor unstable situation in which gravity is aligned with the interface instead of perpendicular. As described previously, this situation results in buoyant and baroclinic vorticity generation of the same rotational sense all along the plume air interface. In the fifth image in Fig. 9, the resulting vortex sheet begins to roll up into what will be a coherent turbulent structure. As the helium/air plume fluid rolls outward, surrounding air rolls

44

Transport Phenomena in Fires ∇P

∇r

∇P

∇P

∇r

∇r

Figure 9: Puffing and Rayleigh−Taylor instabilities. underneath the forming structure. In the sixth image in Fig. 9, the coherent structure continues to grow and begins to self-advect upward. In the process, air continues to be pulled in underneath the coherent, turbulent structure and over the top of the source helium, setting up the conditions for the next cycle. The coherent structure continues to grow until it reaches the centerline of the plume as shown in the seventh and first images in Fig. 9. As the structure continues to self-advect upward as shown in the first and second images in Fig. 9, its influence on the velocity field at the source decreases, and helium beings to re-reenter the domain and form the bubble and spike structures. Note that that the Rayleigh−Taylor instability shown in the first image in Fig.9 is not necessarily the instability in its classic form, i.e. a quiescent heavy gas layer overlying a quiescent light gas layer, and the distinction can have significant consequences for numerical simulation resolution requirements. In the first image in Fig. 9, the coherent vortex has moved away from the surface, reducing the induced radial velocity over the helium. However the effect of the moving air may still have a significant influence on the instability. In [35], it is noted that the experiments show broader air spikes than would be expected from classical Rayleigh−Taylor instability growth theory. The instabilities look to be of the order of centimeters for the 1 m helium plume shown, instead of the order of millimeters that would be expected from the fastest natural growth mode. This observation suggests that perhaps the instability is forced, not natural. An unstable interface can grow at any forced wavelength in preference to the fastest growing unforced wavelength. It can be argued that the initial condition for the instability in the first image in Fig. 9 is set up by the radial entrainment of air over the top of the helium during the last half of the previous cycle. A natural suspicion is that Kelvin−Helmholtz instabilities may result from the radial entrainment of air. However, the principal effect may still be Rayleigh−Taylor in nature. The interface between the helium source and radially entrained air forms a curved mixing layer. At the edge of the source the air has nearly horizontal velocity and near the centerline of

Heat Transfer in Fully Turbulent Fires

45

the plume, the air will have nearly vertical velocity. Chuin Wang [36] studied the effects of curvature on turbulent mixing layers, including layers with differing densities. Because the flow is not quiescent, the transition from nearly horizontal to nearly vertical creates a local acceleration field emanating from approximately the core of the large coherent turbulent structure. In this geometry, the relatively heavy air is being accelerated toward the helium source center point, thus creating a Rayleigh−Taylor unstable situation. Wang [36] studied the conditions in which the velocity between the mixing streams was the same and the density difference was of the order of 7. He notes the importance of the Rayleigh−Taylor instability in this scenario. There are differences between the flow in Fig. 9 and that studied by Wang. For example, his mixing layers were constant velocity. Here, the flow is accelerating along the helium/air mixing layer from the edge of the helium source to the centerline of the plume. In spite of the differences, Wang’s results are insightful. It may be expected that instabilities along the helium/air mixing layer will grow with time away from the source. These instabilities will grow as the air entrains with time over the surface of the helium layer in the fourth through seventh images of Fig. 9, and thus will be present in the first image in Fig. 9, and may be the cause of the larger than expected bubble and spike structures. Clearly further work is required to quantify the interactions of the various instabilities. There can be important implications. Tieszen et al. [35] report that if too coarse a numerical grid is used, the bubble and spike structure does not form, and the mean statistics can be in error by a factor of two. If the Rayleigh−Taylor instability in the first image in Fig. 9 is natural, rather than forced, the radial spacing of the bubble and spike structures does not depend on a length scale. Thus, if the spacing is centimeters for a 1 meter diameter plume, it would be centimeters for a 10 meter diameter plume. On the other hand, if the initial instability scales with the source diameter, centimeter spacing between structures in the 1 meter diameter plume would be 10 centimeter spacing in the 10 meter diameter plume. The difference in numerical resolution required to pick up the resulting bubble and spike structure is a factor of 10,000 (103 in space and 10 in time). Movies of fires in an otherwise quiescent atmosphere suggest that the dynamics of nonreacting plumes are present in fires. Figure 10 shows a visible image of a 1 m methane fire [34, 37] at two times during the puff cycle. The left image in Fig. 10 corresponds approximately with the first image in Fig. 9 while the right image in Fig. 10 corresponds to approximately the fourth image in Fig. 9. Similarly, Fig. 11 shows two still images from a video of a 20 m diameter JP-4 jet fuel fire at two times in the puff cycle. Only the left half of the fire is shown (10 m radius at base). The left

Figure 10: 1 m methane fire at different phases of the puff cycle.

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Transport Phenomena in Fires

image in Fig. 11 corresponds approximately in time with the first image in Fig. 9, while the right image in Fig. 11 corresponds approximately with the fourth image in Fig. 9. In a fire, the bubble and spike structure separates fuel and air and thus is indicated by flame sheets. Comparing the left and right images in Fig. 10 near the source, it is clear that the left image has relatively low flame surface density compared to the right image. The growth in flame surface density corresponds to air/fuel interpenetration, consistent with a Rayleigh−Taylor bubble and spike dynamics argument. The bubble and spike dynamics is perhaps not as apparent in Fig. 11 because of the smoke shielding the flame sheets. As will be discussed in Section 4.2, smoke is indicative that a fuel-rich volume has burned out the oxygen and quenched, leaving copious amounts of relatively low temperature soot behind. In this context, it indicates that a fuel bubble has mixed and burned out with an air spike. In both right-hand images in Figs 10 and 11, the trigger instability for the growth of the large turbulent coherent structure characteristic of puffing begins with the fire base having near vertical slope up to an elevation of 10−20% of the fire diameter. As with the nonreacting case in image four of Fig. 9, this geometry is unstable if the plume fluid is reasonably well mixed and buoyant. Data suggest that the two dynamic modes resulting from Rayleigh−Taylor instabilities have differing strengths for different fire sizes. For small scale (10 cm diameter and below) fires, the bubble and spike structure is either not present or relatively weak. This may be because the bubbles and spikes take time to grow into finite amplitude instabilities, and the puffing period decreases as the inverse of the square root of the diameter [30]. However small the scale, the flame sheet is nearly vertical and a fire will ‘puff’, when the products penetrate to the centerline. The amplitudes of these ‘puffs’ are barely more than bulges, but by their frequency it is clear that they are associated with this instability.

Figure 11: 20 m diameter JP-4 fire at different phases of the puff cycle.

Heat Transfer in Fully Turbulent Fires

47

At the very largest end of the fire diameter spectrum, the coherent plume structure is postulated to break down into individual fires (see Drysdale [14] for a discussion and Heskestad [38]). This regime is termed a mass fire. Scaling of the conditions for which this transition occurs is an active research topic. However, the nature of the mass fire suggests that the bubble and spike mode dominates due to the long timescales involved in large fire dynamics. For relatively low mass flow rates of fuel, it is easy to envision near complete combustion of the fuel in the bubble and spike structures and only the products of combustion are rolled into a fire plume. The greater the fuel mass flux, the longer the timescale needed to combust it and therefore the larger the scale of the fire in order for it to have the appearance of a mass fire. Up to this point in the discussion, continuous uniform fuel sources have been assumed. Studies have been conducted [39] on regular arrays of discontinuous fuel sources, i.e. heterogeneity at small scales relative to the overall fire size. The studies have been conducted in regard to what is called a ‘fire storm’, in which individual fires amalgamate into one large fire. The transition from individual fires to a fire storm and the breakdown of a fire from a continuous plume into a mass fire are likely two sides to the same transition. It is the authors’ opinion that these are very much related to the relative strengths of the bubble and spike mode versus the coherent structure mode. In this description it should be noted that nothing has been said about ‘buoyant turbulence’. This chapter is on large turbulent fires, for which it has been discussed that buoyancy plays a large role. It is the authors’ opinion that buoyancy does not cause turbulence directly. With respect to eqn (1), and its vorticity equivalent, eqn (7), buoyancy results in vorticity generation through the buoyant source term. The term is linear in nature, and it has been argued throughout this section that the instabilities are initially laminar in nature. From a linear momentum perspective, turbulence results from the nonlinear advection term (the second term on the left-hand side of eqn (1)). From a vorticity dynamics perspective, all the vorticity transport dynamics inherent in isothermal flows, including the roll-up of vortex sheets, pairing of vortices by amalgamation, etc., as well as the stretching and tangling of vorticity, are shared with plumes. The difference is in the source term, the buoyant part of which is linear. Turbulence is a result of the transport of vorticity, not its formation. Turbulence models have historically been developed to address the unresolved length scales involved in the nonlinear interaction associated with the advection term in eqn (1). For the most part, these models are dissipative in nature. They account for the fact that mechanical energy associated with advection, in the limit of equilibration, will result in the ‘energy’ being transferred to molecular motion. Modeling of unresolved but linear laminar instabilities that grow by vortex dynamics to become nonlinear and energy bearing, is a virtually nonexistent field in comparison to the vast literature on turbulence models for direct closure of the nonlinear advection terms. One aspect is clear however. Traditional dissipative closure terms will not capture the growth of high frequency (relative to the discrete solution of the partial differential equations) laminar instabilities that grow into energy bearing nonlinear structures due to vortex dynamics [35]. Note also that this process is not inherently stochastic in that the laminar instabilities are deterministic, and for them to result in energy bearing vortical structures, the growth mechanisms must also be deterministic. However, like all turbulence, the resulting turbulent velocity field may be considered stochastic in a modeling sense, just like turbulence from boundary layers. 3.2 Interaction with cross-winds This section provides a brief summary of transport dynamics. As with fires in quiescent conditions, the only difference between isothermal jets and fires is in the location and strength of the

48

Transport Phenomena in Fires

vorticity source term. Figure 12 illustrates four major types of vortical structures found in a jet in cross-flow [40]. All have counterparts in fires in cross-flow except the ‘jet shear-layer’ vortices. These shear-layer vortices are due to the boundary layer vorticity generated with the jet source. As discussed above, in a fire the azimuthal vorticity is buoyantly generated by density gradients between the fire and the surroundings. This difference in vorticity sources results in the biggest difference between a jet in cross-flow and a plume in cross-flow. Figure 13 shows a 20 m diameter JP-8 fire at three different wind speeds. As the wind speed increases, the fire plume becomes more deflected from the vertical as expected. Comparing Figs 12 and 13 highlights two significant differences. The first is that a jet is initially more vertical (less deflected by the cross-flow) than a plume, and becomes more horizontally deflected away from the jet source.

Figure 12: Vortical structures for a jet in cross-flow [40].

Figure 13: Fire in a cross-wind. Wind from the left and increases from the left image to the right image. Long-time exposure of a 20 m diameter JP-8 fire.

Heat Transfer in Fully Turbulent Fires

49

The fire plume is initially more deflected in a fire and becomes (at least within the active combustion region) less deflected away from the fuel source. The reason for this trend is that the jet has its highest vertical momentum at the source, while a fire starts with its lowest vertical momentum at its source. Only after combustion generates a buoyant plume does a fire begin to have the vertical momentum to alter the trajectory of the cross-wind as seen in Fig. 13. The second large difference between the jet and the plume is that since the plume has low vertical momentum at its source (fuel is nonbuoyant), fuel is advected downstream so that the apparent ‘source’ of the fire is elongated while the jet source maintains its original shape. This ‘flame drag’ is apparent in Fig. 13 by the elongated base of the fire in the highest wind condition compared to the lower wind cases. Fric and Roshko [40] provide a physical explanation for the large columnar vortices as the interaction of the boundary layer flow due to the cross-wind with the jet shear-layer vorticity. At the front of the jet, the rotational sense of the boundary layer and shear-layer are different and thus the boundary layer vorticity partially cancels the shear layer vorticity at the leading edge of the fire. This results in a net decrease in vertical velocity at the leading edge relative to the trailing edge. This velocity difference aligns the azimuthal vorticity in an axial direction causing the columnar vortices. Figure 14 shows columnar vortices on the downstream side of a 20 m diameter JP-4 fire. Movies indicate that the columns are not steady, but get stronger and drift downstream until their linkage back to the fuel source grows sufficiently weak such that combustion within the vortex can no longer be sustained. It appears as though the vortices alternate in strength, and the overall impression is not unlike vortex shedding from a cylinder. Figure 15 shows a wake vortex similar to that illustrated in Fig. 12. The explanation for the wake vortices given by Fric and Roshko [40] is that some of the boundary layer vorticity that is rolled around the jet into the horseshoe vortex gets caught up in the columnar vortex on one end while the other is attached to the ground. As this vorticity is stretched by the upward acceleration of the fire plume, it is strengthened. Evidence that this vorticity comes from the boundary layer as opposed to the plume is due to the flow being visible from entrainment of sand from the ground (tan color) as opposed to smoke from the plume. Hence, in Fig. 15a the wake vortex was not shed from the plume but pulled up from the ground. The fire is a 20 m diameter JP-8 fire at

Figure 14: Columnar vortices on the downwind side of a 20 m diameter JP-4 fire: (a) twin columnar vortices; (b) single columnar vortex.

50

Transport Phenomena in Fires

Figure 15: Wake vortex in a fire: (a) China lake, CA; (b) Albuquerque, NM.

China Lake, California. Wake vortices are not atypical in large fires, but for obvious reasons are most commonly seen when the surrounding terrain is dry and dusty. The major mixing structures in a fire in a cross-flow are due to boundary-layer/fire interactions. This interaction is perhaps the simplest of all interactions between fires and objects. The more general case of fire/object interactions includes bluff body dynamics leading to wakes in addition to boundary-layer dynamics. Fundamental flows along this path include flow over a backward facing step. Because of the value of these bluff body flows in inducing high mixing rates in combustors, they have been studied extensively in the combustion community and will not be reviewed here. However, it is worthwhile to note that a fire placed in a high enough crosswind in a stabilizing geometry will, in the limit, look very much like a jet engine combustor, because the transport is in effect no different.

4 Scalar transport and radiative properties 4.1 Mixing The previous sections dealt with the effect of the scalar field on the momentum field. This section examines the effect of the momentum field on the scalar field. Fire is considered a mixing limited phenomenon, that is, the rate of combustion (discussed in Section 4.2) is limited by the rate of mixing of reactants. The rate of mixing is determined by the nonlinear, elliptic growth rate of the instabilities that result in bubble and spike structures, large coherent vortical structures, and resulting turbulent cascade. The growth rate of these structures determines the rate at which air penetrates and mixes into the core of the plume. Fires are often described by the mixing and combustion characteristics as occurring in three distinct parts [41, 42]. Near the base, the combustion is persistent with continuous combustion around a vapor core. At higher elevations, the combustion is intermittent and characterized by strong turbulent mixing, resulting in complete consumption of the fuel. At still higher elevations, a turbulent, nonreacting plume exists in which surrounding air mixes with the products of combustion.

Heat Transfer in Fully Turbulent Fires

51

At the lowest elevations, where most hydrocarbon fuels are nonbuoyant, fires have a fuel vapor core just above the fuel surface termed the vapor dome [14, 41, 42]. Data from both the 1 m methane fire [34, 37] and the 20 m JP-4 fire [17] suggest that a nonbuoyant core region exists that is composed primarily of fuel. This fuel vapor region is surrounded by the high temperature products associated with the active combustion region. Qualitatively, the presence of products is indicated by the high flame surface area density in Fig. 10 and the smoke in Fig. 11. The size of the vapor dome is dependent on the scale of the mixing structures penetrating the plume. Both the bubble and spike structures and the large coherent structures result in rapid penetration of air into the plume, limiting the size of the vapor dome. As noted by Hamins [41], the time mean extent of the vapor dome is perhaps 20% of the fire height. Tieszen et al. [34] note that there is a dependency of fuel flow rate, but the time mean elevation of the vapor dome is nominally half a fire diameter. At higher elevations, the combustion is described in the literature as intermittent. Flow visualization [34] of time-resolved data sets strongly suggests that the height of the vapor dome fluctuates with time in accordance with the passage of the large coherent structures. Due to the elliptic nature of the flow field, the passage of the large eddies induces an acceleration of the vapor dome along the centerline just underneath the large coherent eddy as it self-advects upward. The large eddy structure associated with the next puff cycle grows underneath this fuel region. This intermittent lofting of fuel from the vapor dome may explain the intermittent nature of the combustion between one and two diameters above the fire. Most large fires completely deplete their fuel source along a centerline of height approximately equal to two diameters as discussed by a number of authors [43−46]. For noncircular fires, the minimum dimension is the characteristic length. This observation suggests that the majority of the fuel entrained in large turbulent vortical structures is either consumed or quenched within that structure. Above approximately two source diameters, mixing occurs between the hot products and surrounding air in the absence of combustion. From a heat transfer perspective, it is obvious that the highest rates of heat transfer will occur in the most active combustion regions, where the time-mean local temperature is the hottest. In large fires, with vapor domes that can reach meters in elevation, it is perhaps a counter-intuitive result that the lower region in the center of the fire is not the hottest place. The edge of the fire has a higher time-mean temperature until elevations are reached where the vapor dome is consumed. It is worthwhile noting that the presence of the vapor dome in hydrocarbon fires results in a quantitatively different shape of the velocity field compared to nonreacting plumes in the near source region. Figure 16 shows a comparison of the velocity fields for 1 m diameter sources with low inlet velocities [22, 34]. The fire has ‘W’ shaped vertical velocity contours near the base due to low velocities in the vapor dome relative to the high-temperature combusting surfaces at the fire edge. Not until nearly 3/4 of a fire diameter does the peak velocity occur along the centerline. The nonreacting plume on the other hand has ‘U’ shaped vertical velocity contours, and the peak velocity is along the centerline from the outset. Because the fuel core in most hydrocarbon fires is nonbuoyant, the bubble and spike structure shown in Fig. 9 for a helium plume in air does not develop in the fuel vapor dome (the exception is for hydrogen as a fuel). The unstable interface is between the hot products and the surrounding air, although clearly from Figs 10 and 11, the instability results in the creation of significant flame surface area along the edges of the fire. Time scales between fuel vaporization and combustion of that fuel above the vapor dome can reach seconds in duration in large fires. As discussed by Babrauskas [7], fuel vaporization rates for most hydrocarbon fuels fall in the 0.01−0.1 m/s range. In the vapor dome, the fuel is accelerated upward by the elliptic nature of the momentum field.

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Transport Phenomena in Fires

(a)

(b)

Figure 16: Difference in vertical velocity fields between reacting and non-reacting flows: (a) methane fire, Vo = 0.1 m/s and (b) helium plume, Vo = 0.34 m/s.

Velocities even along the centerline quickly reach the order of meter per second levels. However, in large fires, the time-mean height of the vapor dome can also reach elevations of order meters. Thermocouple measurements in large fires suggest temperatures can reach about 1000 K allowing for thermal decomposition (pyrolysis, in the absence of oxygen) of the fuel into smaller hydrocarbon fragments [16]. The pyrolyzed fuel can be more neutrally buoyant than the parent hydrocarbon. Similarly, time scales between the entrainment of fuel and air into one of the large coherent structures and its complete combustion can be of the order of seconds. The characteristic puffing period is of the order of seconds for fires greater than about 2 m diameter [29, 30]. Since large fires typically complete combustion at a couple of diameters elevation, these structures can last a couple of puff cycles, order of seconds to at most tens of seconds for the largest fires. 4.2 Combustion It has been several hundred thousand years, since humankind first learned to utilize fire to its benefit, so it is perhaps natural that much of the combustion research has focused on heating, power, and propulsion systems that directly benefit humans. Since many of the same fuels used in man-made systems are those involved in natural fires, it can be expected that the chemistry and diffusion transport issues are similar. In both fires and many man-made systems, combustion is classified as turbulent diffusion flame, in which combustion occurs along a gas-phase stoichiometric mixing surface. Descriptions can be found in a number of good, available, combustion textbooks. Perhaps three principal differences exist between man-made systems and natural fires that have an impact on combustion. Relative to man-made combusting systems, fires have longer time scales, relatively low strain rates, and tighter spatial coupling between the scalar and momentum fields. Cox [15] notes that energy release per unit volume measured in fires is lower than man-made systems’ values by factors of 10−1000. As mentioned above, fires are mixing limited and time scales for combustion are of the order of seconds to perhaps tens of seconds. For reference, for typical hydrocarbon fuels, chemical

Heat Transfer in Fully Turbulent Fires

53

timescales are of the order of tenths of milliseconds for gas phase reactions as evidenced by perfectly stirred reactor blowout timescales at stoichiometric conditions in air at ambient temperature and pressure [47]. Soot formation timescales are of the order of milliseconds to tens of milliseconds as evidenced by numerous premixed flame studies. Hence, from a chemistry perspective, soot formation is a slow process, being of the order of 10−100 times slower than principal gas phase chemistry. However, compared to the natural turbulent mixing processes in large fires that are of the order of seconds, both these timescales are short. Mixing times are perhaps 100−1000 times longer than soot formation times and 1000−100,000 times longer than primary gas phase reactions. In man-made systems, particularly propulsion systems, mixing times are much shorter, resulting in higher energy release per unit volume. One of the motivations for short mixing times in propulsion systems is to achieve combustion without significant soot formation. Related to the longer time scales in fires are lower strain rates. While large fires are fully turbulent, the time scales for the dissipation of concentration fluctuations at the small length scale end of the spectrum tend to be much longer than in jet flames [15]. In typical man-made systems, much of the turbulence comes from boundary layers (such as shed or swirled off a backward facing step into a flame holding region). Depending on the velocity gradients, the turbulent structures generated in boundary layers with relatively high kinetic energy can have small spatial frequencies relative to flame zones [48]. Hence the interaction of these high-energy, small-scale eddies with combustion zones creates local strain-induced quenching, creating a turbulent flame brush with triple flames and the like. As a result, a significant amount of combustion research has focused on these rapid transient, high-strain-rate interactions. Small scale strain rates in fires are not nearly as energetic (unless combustor-like geometries are created in cross-wind conditions). Much of the turbulence is buoyantly generated at scales larger than the flame zones. As a result, fires tend to have what look like manifold wrinkled laminar flame sheets. Figure 17 shows a typical view of the base of a medium sized (diameter of the order of meters) fire. PAH fluorescence from a laser sheet through the center of the 1 m diameter methane fire shown in Fig. 10 [37] suggests that there are perhaps of the order of 10 flame sheets between the edge and the center of the fire (exact statistics were not taken). For large fires, hundreds to perhaps thousands of flame sheets could exist between the core and edge of the fire. A third attribute of fires that significantly affects combustion is the close coupling of the combustion and mixing zones. In a typical jet flame for example, stoichiometric concentrations typical of hydrocarbons place the flame outside the core mixing layer [21]. As noted earlier, flame sheets do not result in net vorticity. It is the density gradient across the flame sheet that is the source of vorticity that drives the mixing. The flame surfaces in the right-hand images of both Figs 10 and 11 will roll up to become the large structures halfway up the images on the left-hand side, half a puff cycle later in time. These structures can be quite large as shown in Fig. 18, and of long duration. The large coherent structures can be idealized as batch reactors in the sense that fuel and air mixed into the structure will react over seconds of time to some state dependent on the ratio of fuel to air within the interfaces of the structure. For heavy hydrocarbon fuels, eddies less than about one meter in diameter burn without producing large amounts of smoke. At greater than nominally one meter, eddies visually appear to change from a tangle of reacting flame sheets to a rolling ball of smoke as they age. This effect can be seen in Fig. 19 showing the base of a large fire. Coherent structures near the base visually appear to be a tangle of flame sheets. At higher elevation (implying age of the coherent structure), the eddies appear to be smoke balls. While quantitative measurements are lacking, one explanation consistent with the observations is that structures over nominally a meter in diameter have sufficient circulation to entrain a

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Transport Phenomena in Fires

Figure 17: Typical image from the base of a medium scale JP-8 fire showing flame sheet-like combustion surfaces (see also Fig. 10).

significant fraction of plume fluid into the eddy. Because stoichiometric requirements for hydrocarbons are less than 10% by volume of the fuel, the eddies end up being fuel rich. As the oxygen in the interior of the eddy is consumed, the fuel-rich, soot-laden regions quench due to lack of oxygen. While it is reasonable to postulate that the interior of fuel-rich eddies quench due to oxygen depletion, the eddies are surrounded by air. Thus, there must be another scale-dependent quench mechanism to account for quenching the outer flame sheets in an eddy. Soot breakthrough (i.e. incomplete oxidation due to insufficient residence time) could explain smoke on the air side of the outermost flame surface. Another potential explanation is radiative quenching. T’ien [49] discusses the effect of radiation on quenching with respect to solid surfaces. He notes that as the strain rates decrease, the balance between advection of the reactants into the flame sheet and diffusion of products out of the flame sheet result in a decrease in heat release rate. On the other hand, radiative losses [42] are quite high for heavy hydrocarbon fuels. Under these conditions, it is perhaps theoretically possible for the radiative losses to exceed the low heat release rates and result in flame quenching. The resulting smoke shielding at the edge of large fires has a first order effect on radiative transport, as discussed in Chapter 7 by Modest. Clearly, soot (formed on the fuel rich side of a flame sheet) has ended up on the lean side. Therefore, some quench mechanism must be responsible. A strain-induced quench does not seem consistent with the low strain rates found in fires or the fact that the eddies appear to be populated with a tangle of flame sheets prior to becoming smoke balls. Quantifying the nature of the quenching mechanism is an open opportunity for the research community. It is both important and fundamental in nature.

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55

Figure18: Large coherent eddy in a 20 m diameter fire.

4.3 Absorption properties The temperatures and volume fraction of soot, fuel and combustion gases present in heavy hydrocarbon fires [50, 51] result in heat transfer that is dominated by soot emission and absorption [52]. Possible exceptions include heat transfer to very small items, due to the inverse dependence of convection on length scale, in the gaseous fuel-laden vapor dome region. Soot formation is a subject that has long been studied and yet is still a very active area of research due to its complexity [53−58]. The mechanisms for formation of soot, primarily for laminar flames, including models for prediction of soot nucleation (when particles first appear), growth (particles increase in size), agglomeration, and oxidation are provided in several reviews including the work by Kennedy [59]. Soot formation is strongly affected by both chemistry and transport. However, a complete description of this interaction would take a chapter in itself, and is somewhat beyond the scope of this chapter on transport phenomena in fires that affect heat transfer. Similarly, radiation transport is addressed in this book in Chapter 7 by Modest.

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Figure 19: Typical smoke shielding in large fires. However, the primary connection between fluid mechanics and heat transfer in fires is through soot formation and radiation. Thus a brief description of this interaction through soot absorption and emission is in order. Measurements from sampled soot show trends that illustrate the properties of soot in large fires as well as the physical formation mechanisms. These data [5, 60] show soot comprised of individual small primary particles typically of the order of <100 nm, agglomerated into wispy chains with fractal patterns and lengths of the order of ~1−4 microns. Analysis of the radiative properties of the aggregates [61, 62] shows some small influence of nonisotropic scattering at the wavelengths of interest, but the primary properties of the soot field are dominated by the absorption coefficient of the soot: a = g(kfvTL ).

(8)

The absorption coefficient is a function of k (the Planck mean extinction coefficient), fv (the soot volume fraction), T (the soot temperature), and L (a path length). The inverse of the absorption coefficient determines the length scale for absorption of radiative energy, as discussed in Chapter 7. The mean absorption is temperature dependent. The overlap between soot concentration and temperature will be discussed in Section 4.4. Implicit in eqn (8) is the integration over the wavelength range of interest. Spectral absorption values vary inversely with wavelength. The mean extinction coefficient of the medium in turn is dependent on the optical properties of the soot (i.e. the properties of the material irrespective of its geometric form) as well as the configuration of the soot aggregates. Optical properties of soot have been expressed in the form of the specific spectral extinction coefficient as given by Ke =

kl . fv

(9)

Heat Transfer in Fully Turbulent Fires

57

In eqn (9), l is the spectral wavelength. Although measurements of Ke are only available for limited wavelengths, where laser light can be employed in practical diagnostics, results from recent studies seem to converge to values of about 8.5 for visible and near IR, with no indication of strong wavelength dependence into the heat transfer regimes of 3−5 micron wavelengths. Fuels have a minor influence on the value, but in general trends tend to indicate that, once formed, soot is approximately the same for all hydrocarbon fires. Universally accepted models for the progression of nucleation, growth and oxidation for fires for different fuels have yet to be developed. The use of optical properties to determine the overall absorption field is still a region of active study and therefore the most common practice is to invoke the Rayleigh approximation whereby the radiative properties of the soot laden field are based solely on the optical properties and the volume fraction. More detailed theories have been proposed, but are difficult to employ in practice due to lack of widely applicable data for the fractal parameters of aggregates for fuels, fire sizes and locations within the flame zone. Additional analyses of the size distribution within aggregates [5], as well as formation time analysis from models and laminar flame data support the notion that soot is formed entirely within a single individual flame structure. Within this construct, the formation of nonabsorbing soot precursor particles, which have been observed in careful measurements in laminar flames, and their subsequent carbonization rate to absorbing particles (as studied by Dobbins [63]) clearly occurs at the flame sheet length and time scales. Oxidation of the soot however, does not universally occur as evidenced by the black tips on flamelets visible in high resolution photographs of large fire flame structures. In other words, many flame sheets exist at conditions beyond the local ‘smoke point’. Additional discussion of this issue will be presented in the section on transport phenomena effect on soot concentrations. Accordingly, the turbulent mixing region with active combustion occurring in large fires is comprised of soot within and external to reacting regions of the flame sheets. This distribution was cited as the potential explanation for comparison between high temperature soot measured by emission and the integrated effect of all soot obtained by extinction measurements in 6 m × 6 m JP-4 fires [51]. These data, as well as data from smaller fires [50], averaged over path lengths of the order of 1 cm provide volume fractions of soot of the order of 1−3 ppm. These integrated measurements are significantly lower than the 3−15 ppm shown in Table 1 as measured within a flame sheet in laminar flames by Shaddix and Smyth, [64], Santoro et al. [65], and Kent [66] when the same Ke values were employed. Use of newly measured Ke values reduces the overall volume fraction by approximately a factor of two. Table 1 also shows the volume fraction for methane flames to be notably lower, by at least an order of magnitude, in comparison with other hydrocarbon fuels. A consistent trend is observed in large scale fires burning liquefied natural gas (LNG). As noted by Luketa-Hanlin [67], levels of smoke covering the visible soot emission in the noncombusting plume region are significantly lower, resulting in higher emission from the flame zone, as will be discussed in Section 4.4. Current understanding of soot volume fraction and radiative properties for heavy hydrocarbon fires leads to absorption coefficients within the flame zone that correspond to optical thickness (i.e. path lengths for exponential absorption) of 0.1−0.3 m. Exponential attenuation of all emission occurs within three path lengths or at lengths of approximately 0.3−1.0 m. This range of paths corresponds to the same range of length scales, where large fires become fully turbulent. Accordingly, the expectation is that for significant regions within fully turbulent, large fires, the regions are optically thick. Although radiation is the dominant mode of heat transfer, and combustion and the resulting heat release are closely coupled to turbulence, there is no direct physical connection between the length scale regimes for fully turbulent and optically thick fires.

58

Transport Phenomena in Fires

Table 1: Peak soot concentrations in laminar, low-strain steady diffusion flames. Fuel

Flame

Peak fv reported

Reference

Method

Assumption

Current best estimate of fv

Methane

79 mm high coannular

3 × 10−7

[64]

HeNe extinction

Ke = 4.9 (D&S)

1.7 × 10−7 (Ke = 8.5)

Propane

85 mm high coannular

6 × 10−6

[64]

HeNe extinction

Ke = 4.9 (D&S)

3.5 × 10−6 (Ke = 8.5)

Ethylene 91 mm high coannular

13 × 10−6

[64]

HeNe extinction

Ke = 4.9 (D&S)

7.5 × 10−6 (Ke = 8.5)

Ethylene 88 mm high coannular

10 × 10−6

[65]

Ar-ion extinction (514 nm)

Ke = 4.9 (D&S)

5.8 × 10−6 (Ke = 8.5)

Ethane

88 mm high coannular

3 × 10−6

[65]

Ar-ion extinction (514 nm)

Ke = 4.9 (D&S)

1.7 × 10−6 (Ke = 8.5)

acetylene

6 mm high coannular (smoke pt)

15 × 10−6

[66]

Path-ave HeNe extinction

Ke = 3.5 (L&T)

6.2 × 10−6 (Ke = 8.5)

Ethylene 75 mm high coannular (smoke pt)

6 × 10−6

[66]

Path-ave HeNe extinction

Ke = 3.5 (L&T)

2.5 × 10−6 (Ke = 8.5)

Propane

4 × 10−6

[66]

Path-ave HeNe extinction

Ke = 3.5 (L&T)

1.6 × 10−6 (Ke = 8.5)

150 mm high coannular (smoke pt)

4.4 Emission properties The emission of radiative energy in fires is driven by the temperature and properties of the medium. Subject to the same discussion in the previous section, the relevant simplified, nonscattering, form of the radiative transport equation is given by s ◊ ∇I (s ) + aI (s ) =

asT 4 . π

(10)

In eqn (10), s is a direction, I is the irradiance, a is the absorption, s is the Stefan−Boltzmann constant, and T is the temperature. A more detailed description of radiative transport is given in Chapter 7. Given the previous discussion regarding the formation and growth of soot within the flame structures, it is evident that the position of soot within the flame structure is important. In particular, the overlap between the soot field and temperature field (note the fourth power dependence, even fifth power, when property dependence is considered) is the dominant contribution to emission. As noted previously, in large fires there are hundreds to thousands of flame sheets between the center of a fire and an edge.

Heat Transfer in Fully Turbulent Fires

59

In order to show the importance of flame structure, consider the following simplified heuristic scenario. Let all the flame sheets and intervening space between the flame sheets (filled with fuel (and products) on one side and air (and products) on the other side) be the same along a directional path of a particular irradiance direction vector. From eqn (10), it is clear that the irradiance will increase in the high temperature flame sheets and decrease as it is absorbed in the intervening relatively cool fuel, air, and product gases that may contain soot formed elsewhere and transported with bulk flow. Since in this heuristic example, the flames are repeated, it is useful to think of an average over a single unit cell (i.e. a flame sheet and the surrounding fuel and air, with whatever products they have in them). Using the notation, · Ò to represent this cell average, eqn (10) becomes 〈 s ⋅ ∇I (s )〉 + 〈 aI (s )〉 =

asT 4 . π

(11)

Since the number of sheets in large fires is quite large, let the number of flame sheets in this heuristic example to be infinite so boundary conditions are not important. Recall that property measurements indicate that optically thick conditions exist in large hydrocarbon fuel fires. Under these idealized heuristic conditions, the irradiance along the direction vector normal to the repeat flame sheets eventually reaches a ‘saturation’ condition in which the mean value of the irradiation does not change from cell to cell (even though it does change within each cell because of emission and absorption). The gradient of the mean irradiance along the direction vector is zero, i.e. (12) 〈∇I (s )〉 = 0. For the current heuristic example, assuming that the irradiance and absorption are not correlated (again see Chapter 7 by Modest for details), eqn (11) becomes 〈I 〉 =

〈 asT 4 〉 . π〈 a〉

(13)

Note that the absorption is in both the numerator and denominator. The implication is that to lead order, the irradiance is independent of the soot concentration in this heuristic example if the number of flame sheets is large enough that the irradiance saturates. This simplified case of radiative equilibrium therefore illustrates the primary importance of the overlap between the temperature and soot concentration fields. For real fires, not all flame sheets are identical as in this heuristic example. However, given that the large fires are optically thick over large regions, it is likely that radiative heat transfer is very sensitive to the overlap of the soot and temperature fields, and only moderately sensitive to the overall soot concentrations. This speculation is supported by recent measurements of emission spectra from 1 to 5 microns by Kearney [68], using fast-scanning spectroscopy from the exterior of the fires at the lower end of the fully turbulent regime. Both time averaged and time resolved spectra show Planck-like distributions at a temperature of approximately 1400 K. The results from the example (eqn (13)) also suggest that if the soot concentration overlap with the temperature field is the same between a highly sooting fuel (for example a peak soot concentration of 17 ppm in the flame sheet) and a weakly sooting fuel (for example with a peak soot concentration of 0.17 ppm in the flame sheet) in a sufficiently large fire, the radiative flux levels would be similar. Even more dramatically, what is suggested by eqn (13) is that if a low sooting propensity fuel such as methane had an overlap with the temperature field with a higher mean soot temperature (e.g. due to difficulty in forming soot), then the emission in a large fire would be even higher for this low sooting fuel.

60

Transport Phenomena in Fires

It is interesting to note that there are significant differences in the heat flux from large LNG and heavier hydrocarbon fires. Typical heavy hydrocarbon fires without smoke shielding have an effective surface emission of about 120 kW/m2. On the other hand, data from large LNG fires suggest that the surface emission from LNG fires is in the range from 150 to 350 kW/m2 (see the review by Luketa-Hanlin [67]). This result is counter-intuitive when compared to small scale fires. Hamins et al. [42] point out that the radiative fraction from weakly sooting methane fires (~20%) is smaller than that for more highly sooting fires (~33%). Smaller fires do not have enough flame sheets to be optically thick and eqn (12) does not hold. In fact in the optically thin limit where the absorption term goes to zero in eqn (10), the irradiance is directly proportional to the soot concentration and path length. So it is not surprising from Table 1, that in small fires, methane is weakly emitting compared to higher hydrocarbons, while in large fires, it is more strongly emitting. To the authors’ knowledge, it is not known if the difference in emission between large scale, heavy hydrocarbon and LNG fires is related primarily to the unique chemical stability of the methane molecule, an absence of cold soot due to an interaction between low production rates and mixing, a difference in optical properties, or a combination of these effects. Clearly, both temperature and soot concentration and, in particular, the joint soot concentration and temperature statistics, are very important for quantifying radiation transport. To lead order, temperature is strongly affected by the chemistry of the fuel but also by radiation transport. Soot concentration appears to be strongly affected by transport processes. In a series of studies, Gore, Faeth, and colleagues measured the soot concentration in turbulent diffusion flames [69] with the hope that the soot concentration would be describable by equilibrium state relations like most of the gas-phase product species. However, these studies clearly showed that soot concentration do not collapse as a function of mixture fraction alone. There is a long timescale process that prevents this feature from occurring. The search for what processes are responsible for the nonequilibrium nature of the soot concentration is currently an active area of research within the fire community. There are several possibilities. If soot is treated as any other species, each term in the continuum transport equation could be responsible for the rate limiting process. These terms include soot production/destruction, diffusional and advective transport. The relative magnitude of the terms has not been quantified for real fire conditions, so what follows is necessarily a qualitative description of possible important balances. This critical area of research is in fact in its infancy, based on what is known versus what needs to be known to quantify the relative importance of the various terms, and many possibilities exist for significant contributions, experimentally, computationally, and theoretically in this area. 4.4.1 Transport parameters affecting soot production/destruction Soot production/destruction, as it occurs within the length scales of the flame structures, is determined by soot chemistry in the context of the enthalpy and species fields. As rate processes, production and destruction compete with transport processes such as advection and diffusion that influence the enthalpy and species concentrations within the diffusion flame. In particular, the rate of soot destruction is obviously dependent on the soot concentration itself, which is discussed below. Most obviously, soot production/destruction is influenced by the advection and diffusion of gas phase species. Diffusion flames, by their very nature, are a balance between diffusion and chemistry and have very strong spatial gradients in species and temperature. For this reason, chemical kineticists have preferred to develop kinetic mechanisms in more homogeneous conditions such as in shock tubes and flat flame burners. The characteristic time scale of 1−10 ms

Heat Transfer in Fully Turbulent Fires

61

discussed above for soot production is given in this context. Obviously, away from the flame itself, where the temperature drops below 1000 K, soot production rates can be expected to completely vanish. Detailed chemical kinetic mechanisms for soot have been evaluated in steady, laminar flames, (see e.g. the work of Smooke and colleagues, Smooke et al. [70] being a recent example), but to the authors’ knowledge, not in real turbulent conditions expected in a fire. Beyond simple gas-phase diffusion, two additional diffusive transport processes may affect soot production/destruction rates. As soot is formed, it is subject to differential diffusion and thermophoretic transport. Differential diffusion occurs because soot has such a high molecular weight relative to the gas phase species surrounding it that its diffusion relative to the surrounding gases becomes effectively negligible. For example, diffusion coefficients are inversely proportional to the square root of the molecular weight. So when soot becomes larger than order 10,000 carbon atoms, the effective diffusivity of the soot drops to order 1% of the gas phase diffusivity. The lack of soot diffusion is evident from visualization studies in turbulent diffusion flames [71] that show soot as narrow streaks. Thermophoresis is another diffusional force that can affect soot production. Thermophoresis acts on particles in temperature gradients to drive them to lower temperatures. As mentioned earlier, strong temperature gradients exist in flames, of the order of 1000 K/mm. A particle in such an environment will have more energetic collisions on the high temperature side of the particle than the low temperature side of the particle. The imbalance in forces results in a particle drift velocity toward lower temperatures. For a molecule in the gas phase, the collisional imbalance is also present, but considered to be differential diffusion based on the different diffusivities at different temperatures in the gradient. The rate balance between soot production and the thermophoretic drift velocity has not been quantified for turbulent fire conditions. If significant, thermophoresis may affect the soot emission in several ways. The drift to lower temperatures can slow rates of soot growth. Both the reduced production rates and the lower temperatures directly affect emission. Dehydrogenation, a finite rate process, results in soot optical properties switching from banded properties to broadband grey-body properties. If the time scale for dehydrogenation results in significant displacement due to the thermophoretic drift-velocity before the optical properties become broadband, then the soot will effectively begin emission at a lower temperature than that at which incipient formation occurs. Finally, thermophoresis will act to slow down the passage of soot into and through the high temperature oxidation zones of the flame. To the authors’ knowledge, the effect of thermophoresis on oxidation rates is unknown. In addition to diffusive transport processes, advective processes can also strongly affect soot formation rates. Turbulence induced strain on flames acts to decrease the residence time within the flames (i.e. thinning the flame and stretching it at the same time, so that the area increases proportionally). When the residence time is less than the chemical timescale for soot production to occur, soot production will cease. The long chemical time scales for soot production relative to gas phase heat release assures that soot production ceases long before the flames themselves are extinguished. Advantage is taken of this fact in the gas turbine industry to run nearly soot free without blowout while using high soot propensity aviation fuels that would otherwise produce copious amounts of smoke, as they do in fires. To the authors’ knowledge, the degree to which soot production rates are retarded by turbulent strain rates in real fires has not been quantified. Clearly, visible evidence suggests that soot production rates are not completely suppressed to any large degree, which would be evidenced by the dominance of blue gas-phase emission over the ubiquitous red/orange/yellow soot emission. However, in regions of relatively high turbulence, such as behind a wind driven fire, there is a distinct absence of cold soot (smoke). It is not known if this absence is due to a decrease in production, or due to transport issues or both.

62

Transport Phenomena in Fires

Soot source terms are also likely affected by radiative transport. Overall, it can be expected that the enthalpy loss in the region of overlap between the highest temperatures and soot concentrations will result in slower soot formation and destruction rates. Ultimately, if the soot loading is high enough, or as discussed previously, if the strain rate is low enough, the radiative loss term can result in flame quenching (and consequently, very low production/destruction rates in the quenched region). It is also possible that soot production/destruction rates can be indirectly, but significantly, affected by the integral effect of the time-history of enthalpy loss in the gas-phase products surrounding the soot. Due to the very small size of the soot particles, it is assumed that the particle phase and gas phase equilibrate very quickly in the presence of radiative heat transfer. The combustion products lose enthalpy due to soot radiation in addition to the gas phase radiation. As these products mix into the fuel and air streams, the mixture enthalpy is less than if the gases were adiabatic. At increasing elevations, the fuel and air streams will be increasingly diluted with combustion products that have lost energy due to radiative transfer. Hot combustion products are important to maintaining a flame. On the other hand, both CO2 and H2O are in use as fire suppressants because of their high heat capacity. So with a history of radiative heat loss, their presence can be an energy sink.

4.4.2 Transport parameters affecting soot concentration From the work of Gore, Faeth and colleagues discussed above, it is clear that soot concentration is dependent on a transport process in addition to chemical processes. There has been much discussion in the community about what this slow transport process is. What follows is a possible physical explanation for turbulent fires that draws on an analogy with laminar flames. In laminar flames, it is clear that both differential diffusion between the soot and gases, and thermophoresis result in regions within the flame where the balance between soot production and soot destruction vary significantly. For example, near the base of a laminar diffusion flame, the fuel diffuses into the air much more readily than the soot being produced by the flame diffuses. However, near the top of a steady laminar diffusion flame, as the fuel is depleted, the air diffuses much more readily than the soot, forcing the soot through the flame. The turbulent flame analog of the laminar diffusion flame involves differential diffusion between the soot and gases in regions in which either air pockets, or tongues are depleted in an abundance of fuel, or fuel pockets are depleted in an abundance of air. Using the flame sheets as visualization tools to identify separation of fuel/product and air/product regions, it is clear from Figs 17 and 18, that the surface between these regions is highly convoluted. It is very unlikely that the flame sheets will be layered such that the fuel depletes at exactly the same time as the air depletes on either side of a folded flame sheet. In most cases, an eddy will be either fuel rich or fuel lean. If the eddy is fuel rich, there will be a preponderance of flame folds that hold air pockets or tongues that will be depleted before the fuel. If the eddy is lean, there will be a preponderance of flame folds that hold fuel pockets or tongues that will be depleted before the air. At the base of a large fire, where air is first entrained deep into the fuel layer by the bubble and spike structure and large eddies, it can be assumed that there is a large preponderance of air tongues that burn out. At the top of the active flaming region in a fire, where the last of the fuel is depleted in an abundance of air, the situation will be reversed. This poor mixing efficiency is perhaps the prime reason that fires have low heat release rate per unit volume compared to man-made systems as noted by Cox [15]. An analogy can be drawn between the transient depletion of air tongues, and steady laminar inverse diffusion flames, and the transient depletion of fuel tongues and normal, steady, laminar

Heat Transfer in Fully Turbulent Fires

63

diffusion flames. In the case of an air tongue, as the air is depleted, the combustion products will diffuse into the surrounding fuel at a higher rate than the soot. As a consequence, fuel counterdiffuses into the soot. In this manner soot produced by this flame is not consumed by this flame, but transported away, initially by differential diffusion, and more generally by convection, to be consumed by another flame. In analogy, in a laminar inverse diffusion flame, soot produced by that flame advects away from the producing flame. Due to differential diffusion, the combustion products diffuse away from the soot to be replaced by counter diffusing fuel. This soot will be consumed by a later flame sheet surrounded by ambient air. Since there is a preponderance of air tongues near the base of a large turbulent fire as air is initially mixed into the fuel plume, it is clear that soot concentrations can accumulate in the fuel rich regions near the base. On the other hand, near the edge or top of a fire in which the fuel is rapidly being depleted and is surrounded by an infinite air supply, the soot within the fuel will be forced to pass through a flame sheet as the fuel tongues burn up. If the soot loading is too high, or the enthalpy loss due to radiation is too high, or the combustion has too long of residence time, or a combination of the above, the flame can quench and soot will break through to the air side. Clearly from Fig. 19, soot either breaks through or quenches the flames often near the edges of large fires as can be seen by the ubiquitous smoke. The steady laminar analog of the fuel tongue burnout is the normal diffusion flame, with the quenching marked by the smoke point. The role of turbulent mixing relative to a laminar flame is to create large flame surface area that results in relatively rapid reactant depletion. However, differential diffusion followed by advection results in regions near the base of both the laminar flame and the turbulent fire in which the flames are net producers of soot, and regions near the top of both laminar flames and turbulent fires in which the flames are net destroyers of soot. Due to differential diffusion, soot can exist in all mixtures richer than stoichiometric, even if production of soot occurs in very narrow mixture fraction bands. Soot can break through to the lean side of stoichiometric, cool and become smoke. This explanation is consistent with soot morphology which suggests soot growth or destruction occurs mostly within a single flame sheet, and with the data that shows that soot concentration is not a unique function of mixture fraction alone. On the other hand, future research may show this view to be overly simplified or incorrect.

5 Future of transport research in fires From the theoretical physics perspective of deriving the fundamental mass, force and energy balances, the description of transport, particularly at continuum scales, is complete. It has been understood for about a century that the time rate of change of mass, momentum, and energy is balanced by advection, diffusion, and source terms for conserved quantities. While the usefulness of this level of understanding cannot be overestimated, it is the solution of the fundamental transport equations that will provide the basis for scientific understanding of fire phenomena. In the absence of a break-though in mathematics that would permit a closed form solution, the expression of these fundamental equations will be through numerical simulation. As noted in the beginning of this chapter, it will be decades to tens of decades of continuous computational growth before machines are large enough that direct simulation of all the length scales is possible. In the coming decades it will be necessary to formulate the problem by a combination of filtered equations with closure models, the desired nature of which is essentially the solution of the transport equations over the length scale ranges that we cannot resolve numerically. So we must have a partial solution (which we do not know in a scientific sense) to obtain the whole solution in order to be predictive.

64

Transport Phenomena in Fires

The key scientific challenge in the coming decades, then, is to develop an understanding of the nature of the chemistry and physics that is the solution of the fundamental equations in the range of length and time scales that cannot be resolved numerically. The key engineering challenge of the future is to translate that understanding into computationally efficient models that provide the best possible mean-response (given the available inputs), and to quantify the uncertainty. While the fundamental expressions are elegant in their simplicity, the expression of the physics and chemistry, while remaining elegant, is anything but simple. In all the complexity, a simple truth is sometimes forgotten: it is only through the understanding of reality that we can obtain information we need to build the partial solutions, i.e. the models that we need, in order to obtain the whole solution. While science must demonstrate its validity ultimately through experimental means, it is clear that the existence of computational tools that can provide partial solutions to the fundamental equations are a powerful step in gaining scientific understanding. Further, laser-based diagnostics developed over the last few decades have permitted unprecedented data sets and insight. In the context of the hundreds of thousands of years that humankind has interacted with fire, the next century is the blink of an eye. We are on the threshold of scientific understanding of one of humankind’s greatest threats and one of his greatest successes. Historically unique opportunities present themselves to coming generations of scientists to use the rapidly evolving power of computational science in combination with ever more advanced diagnostics to finally gain a complete understanding of fire.

Acknowledgements The authors would like to thank their many colleagues, particularly at the University of Utah and Stanford, for sharing their insights into transport phenomena in fires. The authors would also like to acknowledge contributions by Chris Shaddix and John Hewson to this chapter. Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the United States Department of Energy under Contract DE-AC04-94AL85000.

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Heat Transfer in Fully Turbulent Fires

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Zhou, X.C. & Gore, J.P., Experimental estimation of thermal expansion and vorticity distribution in a buoyant diffusion flame. 27h Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, PA, pp. 2767−2773, 1998. Soteriou, M.S., Dong, Y. & Cetegen, B.M., Lagrangian simulations of the unsteady near field dynamics of planar buoyant plumes. Phys. Fluids, 14(9), 3118−3140, 2002. Tieszen, S.R., O’Hern, T.J., Weckman, E.J. & Schefer, R.W., Experimental study of the effect of fuel mass flux on a one meter diameter methane fire and comparison with a hydrogen fire. Combustion and Flame, 139(1–2), pp. 126−141, 2004. Tieszen, S.R., Pitsch, H., Blanquart, G. & Abarzhi, S., Toward the development of a LES-SGS closure model for buoyant plumes. Proceedings of the Summer Program 2004, Center for Turbulence Research, Stanford, University, Stanford CA, pp. 341−352, 2004. Wang, C., The Effects of Curvature on Turbulent Mixing Layers, PhD Dissertation, California Institute of Technology, Pasadena, CA, 1984. Tieszen, S.R., O’Hern, T.J., Schefer, R.W., Weckman, E.J. & Blanchat, T.K., Experimental study of the flow field in and around a one meter diameter methane fire. Combustion and Flame, 129(4), pp. 378−391, 2002. Heskestad, G., A Reduced-Scale Mass Fire Experiment. Combustion and Flame, 83, pp. 293−301, 1991. Vincent, J.R. & Gollahalli, S.R., An experimental study of the interaction of multiple liquid pool fires. Journal of Energy Resources Technology, 117, pp. 37−42, 1995. Fric, T.F. & Roshko, A., Vortical structures in the wake of a transverse jet. Journal of Fluid Mechanics, 279, pp. 1−47, 1994. Hamins, A., Kashiwagi, T. & Burch, R.R., Characteristics of pool fire burning. Fire Resistance of Industrial Fluids, eds. G.E. Totten, & J. Reichel, American Society for Testing and Materials, Philadelphia, PA, ASTM STP 1284, 1996. Hamins, A., Konishi, K., Borthwick, P. & Kashiwagi, T., Global properties of gaseous pool fires. 26th Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, PA, USA, pp. 1429−1436, 1996. McCaffrey, B., Flame height (Chapter 2-1). The SFPE Handbook of Fire Protection Engineering, 2nd edn, National Fire Protection Association: Quincy, MA, pp. 2-1−2-8, 1995. Heskestad, G., Dynamics of the fire plume. Phil. Trans. R. Soc. Lond. A, 356, pp. 2815− 2833, 1998. Heskestad, G., On Q* and the dynamics of turbulent diffusion flames. Fire Safety Journal, 30, pp. 215−227, 1998. Heskestad, G., Turbulent jet diffusion flames: consolidation of flame height data. Combustion and Flame, 118, pp. 51−60, 1999. Hewson, J.C., Tieszen, S.R., Sundberg, W.D. & DesJardin, P.E., CFD modeling of fire suppression and its role in optimizing suppressant distribution. Proceedings of the Halon Options Technical Working Conference, Albuquerque, NM, 2003. White, F.M., Viscous Fluid Flow, McGraw-Hill: New York, NY, pp. 309−402, 1991. T’ien, J.S., Diffusion flame extinction at small stretch rates: the mechanism of radiative loss. Combustion and Flame, 65, pp. 31−34, 1986. Murphy, J.J. & Shaddix, C.R., Soot Properties and Species Measurements in a Two-Meter Diameter JP-8 Pool Fire: 2003 Test Series, SAND2005-0337, Sandia National Laboratories Report, 2005. Gritzo, L.A., Sivathanu, Y.R. & Gill, W., Transient measurements of radiative properties, soot volume fraction and soot temperature in a large pool fire. Combust. Sci. and Tech., 139(1–6), p. 113, 1998.

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de Ris, J. Fire radiation − a review. Seventeenth Symposium (International) on Combustion, The Combustion Institute: Pittsburgh, PA, pp. 1003−1016, 1979. [53] Street, J.C. & Thomas, A., Carbon formation in pre-mixed flames. Fuel, 34, pp. 4−36, 1955. [54] Palmer, H.B. & Cullis, H.F., The formation of carbon from gases. ed. P.L. Walker, Chemistry and Physics of Carbon, Vol. 1, Marcel Dekker: New York, pp. 265−325, 1965. [55] Wagner, H.Gg., Soot formation in combustion. Proc. Combust. Inst., 17, pp. 3−19, 1979. [56] Haynes, B.S., Wagner, H.Gg., Soot formation. Prog. Energy Combust. Sci., 7, pp. 229− 273, 1981. [57] Glassman, I., Soot formation in combustion processes. Proc. Combust. Inst., 22, pp. 295− 311, 1988. [58] Richter, H. & Howard, J.B., Formation of polycyclic aromatic hydrocarbons and their growth to soot − a review of chemical reaction pathways. Prog. Energy Combust. Sci., 26, pp. 565−608, 2000. [59] Kennedy, I.M., Models of soot formation and oxidation. Prog. Energy Combusti. Sci., 23, pp. 95−132, 1997. [60] Jensen, K.A., Suo-Anttila, J.M. & Blevins, L.G., Characterization of soot properties in two-meter JP-8 pool fires, SAND2005-0337, Sandia National Laboratories, 2005. [61] Dobbins, R.A. & Megaridis, C.M., Absorption and scattering of light by polydisperse aggregates. Applied Optics, 30(33), 1991. [62] Koylu, U.O., Faeth, G.M., Farias, T.L. & Carvalho, M.G., Fractal and projected structure properties of soot aggregates. Combustion and Flame, 100, pp. 621−633, 1995. [63] Dobbins, R.A., Soot inception temperature and the carbonization rate of precursor particles. Combustion and Flame, 130, pp. 204−214, 2002. [64] Shaddix, C.R. & Smyth, K.C., Laser-induced incandescence measurements of soot production in steady and flickering methane, propane, and ethylene diffusion flames. Combustion and Flame, 107, pp. 418−452, 1996. [65] Santoro, R.J., Yeh, T.T., Horvath, J.J. & Semerjian, H.G., The transport and growth of soot particles in laminar diffusion flames. Combustion Science and Technology, 53, pp. 89−115, 1987. [66] Kent, J.H., Quantitative relationship between soot yield and smoke point measurements. Combustion & Flame, 63, pp. 349−358, 1986. [67] Luketa-Hanlin, A., A review of large-scale LNG spills: experiments and modeling. Journal of Hazardous Materials, 132(2–3), pp. 119−140, 2006. [68] Kearney, S., Temporally resolved radiation spectra from a sooting, turbulent pool fire. Proceedings of 2001 ASME International Mechanical Engineering Congress and Exposition, New York, NY, 11−16 November 2001. [69] Gore, J.P. & Faeth, G.M., Structure and radiation properties of luminous turbulent acetylene/air diffusion flames. ASME Journal of Heat Transfer, 110, pp. 173−181, 1988. [70] Smooke, M.D., Hall, R.J., Colket, M.B., Fielding, J., Long, M.B., McEnally, C.S. & Pfefferle, L.D., Investigation of the transition from lightly sooting towards heavily sooting co-flow ethylene diffusion flames. Combustion, Theory and Modeling, 8(3), pp. 593−606, 2004. [71] Xin, Y., Gore, J.P., Nathan, G.J., Mikofski, M.A. & Geigle, K.P., Two-dimensional soot distributions in buoyant turbulent fires. Proceedings of the Combustion Institute, 30, pp. 719−726, 2005. [52]

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CHAPTER 3 Heat transfer to objects in pool fires J.P. Spinti, J.N. Thornock, E.G. Eddings, P.J. Smith & A.F. Sarofim Department of Chemical Engineering, University of Utah, USA.

Abstract In accident scenarios involving fire and the transport of explosive material, the time available for escape is dependent on the heat transfer rate from the fire to the energetic material. A review is presented of historical modeling approaches that draw on empiricism for estimating both heat flux from fires and fire hazard. While such methods can be used for conservative estimates of heat flux in determining safe separation distances, they cannot be used in situations where overestimating the heat flux may underestimate the hazard, such as the heating of high-energy explosives. Next, a large eddy simulation (LES) technique for addressing fire phenomena with embedded, heat sensitive objects is described. With the advent of high performance computing, LES is emerging as a powerful tool for resolving a large set of spatial and temporal scales in fires and for capturing observed pool fire phenomena such as visible flame structures. The development of the LES approach described here is based on verification and validation (V&V) principles, utilizing a V&V hierarchy that is focused on the intended use of the simulation. This LES approach couples surrogate fuel representations of complex hydrocarbon fuels, reaction models for incorporation of the detailed chemical kinetics associated with the surrogate fuel, soot formation models, models for unresolved turbulence/ chemistry interactions, radiative heat transfer models, and modifications to the LES algorithm for computing heat transfer to objects. The chapter concludes with an analysis of simulation and experimental data of heat transfer to embedded objects in large JP-8 pool fires and of time to ignition of an energetic device in such a fire. The analysis considers the role of validation, sensitivity analysis and uncertainty quantification in moving toward predictivity.

1â•‡ Introduction Explosives are transported via highway, rail line, and air for use in mining, space exploration, building demolition, pyrotechnics, avalanche control, and military applications. In addition, certain hydrocarbons, most notably liquefied petroleum gas (LPG, mainly composed of propane), can explode when the storage vessel is heated by an external fire, resulting in the so-called boiling liquid expanding vapor explosion (BLEVE). For these events, the time to explosion is critical as it determines the time available for first responders to intervene and for those at the scene of an accident to escape.

70â•… Transport Phenomena in Fires On August 10, 2005, a semi-trailer truck carrying 38,000 pounds of mining explosives tipped over, skidded across the pavement, caught fire, and then detonated in Spanish Fork Canyon, Utah. The driver was negotiating a sharp turn at an excessive speed when the accident occurred. Eyewitnesses estimated a time of three minutes from the start of the fire to the detonation event. The blast left a crater 30 feet deep and 70 feet wide in the road, and the truck was reduced to shards of metal, frayed pieces of tire, and an engine block. This incident and others like it provide the motivation for calculating the potential hazard of an explosive device immersed in a pool fire of transportation fuel. Heat transfer to objects in or near pool fires has been the subject of study for decades. Traditionally, the focus has been on determining a safe separation distance from the fire. Calculation with a conservatively high heat flux provided a good margin of safety. However, there are times when conservative estimations of heat flux are inadequate for determining the magnitude of the hazard, particularly when dealing with containers of energetic materials. For example, some energetic materials may detonate under slow heating (slow cook-off) conditions and deflagrate under rapid heating (fast cook-off) conditions. Overestimating the heat flux may underestimate the hazard, motivating the need for physically-based methods that accurately predict heat flux from pool fires to embedded objects. In this chapter, the hazard is characterized in terms of the time to ignition of the explosive device and the violence (measured as kinetic energy of the exploded container) of the event. Full-scale experimental investigation of heat transfer to objects in or near pool fires is limited because such experiments are expensive and difficult to instrument due to the harsh environment. Consequently, pool fire dynamics and heat transfer have been studied in small-scale, controlled laboratory settings, where detailed instrumentation yields high quality, quantitative data that is used to gain insight into the fire physics and the heat transfer process. Fire simulation tools based on computational fluid dynamics (CFD) offer a way to scale the laboratory experiments to larger, more realistic scenarios involving a variety of accidental conditions including wind speed and direction, size of the fire (1-100 m), and position of the object relative to the fire. 1.1â•‡ Chapter outline Section 2 reviews the semi-empirical modeling approaches that have been empÂ�loyedâ•‰to estimate the radiation field from hydrocarbon pool fires. The fire community has used these approaches to provide immediate and practical engineering estimates of the radiation hazard. However, these approaches are unable to predict, a priori, the effects of changing fuels, wind conditions, and fire configurations. Section 3 presents a framework for predicting heat transfer to embedded objects in pool fires based on a foundation of verification and validation (V&V). Sections 4-9 review next generation modeling tools for achieving high fidelity transportation fuel pool fire simulations within the V&V framework. Section 4 details how transportation fuels composed of complex mixtures can be represented by surrogate fuels that approximate the physical and chemical characteristics of the original fuel. Section 5 evaluates the capability of a chemical kinetic mechanism developed for such surrogate fuels to predict concentrations of soot precursors and outlines four methodologies for calculating soot from its precursors. Section 6 discusses large eddy simulation (LES), a sophisticated numerical approach that captures the dynamics of buoyant pool fires. Section 7 describes a parameterization methodology (e.g. reaction model) for reducing the degrees of freedom in detailed kinetic schemes of transporÂ�tation fuel combustion. Section 8 discusses models that account for the complex and coupled interactions between turbulence and chemistry at the unresolved scale. Together, the reaction model and the model for turbulence/chemistry interactions allow complex combustion chemistry to be coupled to the LES simulation in a realistic way.

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Section 9 provides an overview of radiation, the dominant mode of heat transfer in most large pool fires, and its complexities as a spatial and spectral phenomenon. Section 10 provides a brief overview of validation activities for heat transfer to embedded objects in transportation fuel pool fires. These activities focus on the use of a validation metric to quantify the level of agreement between experimental and simulation data. Section 11 illustrates the application of the LES fire simulation tool to the prediction of heat flux to an explosive device in a full scale hazards classification test for which data is unavailable. Section 12 demonstrates how an energetic material model can be coupled to the fire simulation tool to predict time to ignition of an explosive device. As the emphasis of this chapter is on predictive models, Section 13 concludes the chapter with a brief discussion on error quantification and predictivity.

2â•‡ Historical modeling approaches 2.1â•‡ Homogeneous flame The early models of heat transfer from flames are based on the 1959 review by Hottel [1] of Blinkov and Khudiakov’s data on burning rate and flame height as seen in Fig. 1. The data include a number of fuels in pans with diameters ranging from 0.4 cm to 30 m. The data were rationalized by equating the heat flux density, q≤, to the vaporization rate of the fuel, m· ≤, multiplied by the heat of vaporization, Δhvap. The heat flux to the fuel was decomposed into conduction, convection, and radiation contributions to give,

m ′′ Δhvap = q ′′ =

4K (TF - To ) + h(TF - To ) + s F (TF4 - To4 )(1 - e - Kad ) . d

(1)

The first term on the right-hand side of eqn (1) represents conduction from the rim of the pan at the flame temperature, TFâ•›, to the liquid at To, where K is the liquid conductivity and d is the pan diameter. The second term represents convection from the flame to the liquid, where h is

Figure 1: Correlation by Hottel [1] of burning rate and flame height from pool fires as a function of pan diameter.

72â•… Transport Phenomena in Fires the convective heat transfer coefficient. The third term represents the radiation from the flame, where F is the view factor from the flame to the pan, s is the Stefan-Boltzmann constant, K is the absorption coefficient in the flame, and a is the ratio of the mean beam length to the pan diameter. This simplified model invokes the assumption of a homogeneous flame. Hence, a turbulent flame is assumed to have homogeneous gaseous and soot concentrations at some ‘effective radiation temperature’. Hottel [1] established the framework for current semi-empirical models used to estimate radiation. For example, in Fig. 2 the fire is approximated by a cylinder at a uniform temperature and composition with a height HFâ•›, diameter DFâ•›, and temperature TFâ•›. Consider the flux per unit area, q·S, to an element at a distance RFS from the fire (eqn (2)),

q S = FFS eF sTF4 ,

(2)

where

FFS = f ( H F , DF , RFS ) ,

(3)

and

(1 - eF ) = (1 - eCO2 )(1 - eH2 O )(1 - esoot ) .

(4)

To estimate the radiation from a homogeneous flame, one needs to know the flame shape and size to compute the geometric view factor, FFS; the flame absorption coefficient/flame emissivity, computed from both gas emissivities (eCO2, eH2O) and soot emissivity (esoot); and an effective flame temperature, TFâ•›. Various semi-empirical approaches for estimating the radiation field in and around hydroÂ�carbon pool fires have been reviewed by De Ris [2] and Mudan [3]. A conical or cylindrical flame shape is usually assumed over a circular pool. A flame height can either be estimated through photographs or from the burning rate of the fuel. The nondimensional flame height (flame height to pool diameter ratio) has been found to correlate well with a nondimensional mass burning rate [3, 4]. Correlations relating the flame tilt angle from the vertical to wind velocity are also available [2].

Figure 2: Approximation of a fire by a homogeneous cylinder. Photograph taken by William Ciro, 2005.

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Once the shape and size of the fire are calculated, the radiative characteristics of the fire need to be determined. The radiative properties of the flame are often estimated in the form of gray absorption coefficients or gray emissivities by assuming a homogenous mixture of CO2, H2O, and soot. Correlations are currently available for the spectral emissivities of combustion products of hydrocarbons [5]. The relative magnitude of CO2, H2O and soot emissivities are shown in Fig. 3 for partial pressures of CO2 and H2O of 0.12, a soot volume fraction of 10-7, a mean beam length of 3 m, and a flame temperature of 1,200 K. At 1,200 K, three quarters of the blackbody spectrum is in the 2.4-4.8 μm range, a range where soot radiation dominates. Therefore, the determination of soot emission and absorption is critical in computing accurate radiant heat fluxes from flames. An alternative method of describing the fire hazard of a fuel is to estimate the total radiative output of the fire to its surroundings and report that radiative output as a fraction (cR) of the total heat of combustion. This fraction cannot be determined theoretically and is normally estimated [2]. 2.2â•‡ Homogeneous model and observable fire phenomena The major shortcoming of the homogeneous model is the evaluation of the effective flame temperature. In his review, Hottel assumed a value of 1,100 K [1]. The effective flame temperature, however, is dependent on pool size and to a lesser degree on fuel type [6, 7]. Figure 4 shows average surface emissive power as a function of pool diameter for a range of fuels. In general, the radiation is found to increase with pool diameter as a result of the increase in emissivity. However, for large pool fires, the radiation decreases as a result of the shielding of flame radiation by the outer, cooler soot layers. This phenomenon is evident in Fig. 4, where several fuels show an effective emissive

Figure 3: Spectral emissivities of CO2 (pCO2 = 0.12), H2O (pH2O = 0.12), and soot (volume fraction = 10-7) at 1,200 K for a mean beam length of 3 m.

74â•… Transport Phenomena in Fires

Figure 4: Average surface emissive power of pool fires for different fuels as a function of pool diameter [6]. power of the flame surface passing through a maximum at pool diameters of 1-10 m, with peak values near 150 kW/m2. Liquid natural gas (LNG) is the exception; being lightly sooting, its shielding effects are not yet evident. For this reason, the maximum radiation for large LNG flames exceeds that of more sooting fuels. In the review by Mudan and Croce [8], peak emissive power values of 220 kW/m2 are reported for land-based LNG fires (higher values are found on water) compared with peak values of 160 kW/m2 for LPG and 130 kW/m2 for gasoline. The shielding of the core of the flame by soot has been studied for some time, with Smith [9] first proposing models that tried to provide a mathematical framework for the observations of the periodic transport to the surface of large eddies from the hot core. The emissive powers of larger flames vary widely due to this phenomenon. Mudan [3] estimated that the luminous zones covered 20% of the surface of the flame and had emissive powers in the range of 110-130 kW/ m2, while the cooler background had an emissive power of 20 kW/m2. Thermographic cameras have been used by Schönbucher’s research group to obtain time-resolved measurements of the

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emissive power distribution in flames. They have developed probabilistic models that describe hot spots with surface emissive powers ranging from 33 to 430 kW/m2 and colder soot parcels with surface emissive powers ranging from 6 to 50 kW/m2 [7]. Radiative fluxes to the pool surface and even at locations away from the fire are likely to be influenced by the assumed flame size and shape, quantified by geometric view factors [10]. Numerical estimates of the radiative heat fluxes to the pool surface from 30 cm diameter pool fires employing the homogeneous model were found to be higher than the experimental values by 40% [2]. Most of this error was attributed to assuming a conical shape to the flame. Another shortcoming of the homogeneous model is its inability to predict the radiative feedback to the pool surface, particularly in large pool fires. Obtaining accurate estimates of the radiative fluxes to the pool surface is important for determining fuel burning rates. Hottel [1] was able to explain the trends in the burning rates of liquid fuels by relating the rate of heat transfer from the fire to the pool to the rate of fuel vaporization, but the effective emissive power of the flame that he assumed was, in effect, a fitting parameter. The cooler, unburned, sooty pyrolysis gases near the fuel surface in large fires may block part of the flame radiation from reaching the surface, similar to the effects observed for external heat transfer. Shinotake et al. [10] showed that radiation blockage significantly affects the fuel burning rates in pool fires of diameters greater than 1 m. They also observed that the experimentally measured radiative fluxes to the pool increased with increase in diameter but then quickly saturated compared to the external fluxes. They explained these observations in terms of radiation blockage by performing simple two-layer model calculations assuming conical shapes. An outer cone represented the radiative characteristics of the fire and an inner cone represented the vapor dome of pyrolysis gases. The assumption of a homogeneous flame failed to capture the observed trends in heat fluxes. However, the two-layer model calculations were found to be very sensitive to the adopted soot concentrations and soot temperatures in the flame as well as to the vapor dome. Measurements in very large pool fires also show significant gradients in the radiative heat fluxes to the pool surface, which are likely to result in significant gradients in the fuel vaporization rates within the pool [11]. In Fig. 4, the mean surface emissive power for many hydrocarbon pool fires is seen to decrease with increasing pool diameter due to smoke obscuration. Although a systematic methodology to reliably address this phenomenon is not yet available, some explanations have been proposed. The vapor dome of large fires may contain pyrolyzed fuel vapors which are at moderate temperatures relative to the reaction zone. Poor mixing and/or the slow entrainment of this stream with the air stream may result in the formation of long-lived, fuel-rich eddies that contain unoxidized fuel [12]. The smaller fluid strain rates associated with this process can reduce the diffusion rates, giving the fuel more time to pyrolyze and to form larger soot particles (smoke) that take longer to oxidize. Klassen and Gore [4] measured transient emission and absorption properties in pool fires of different fuels and sizes (maximum diameter of 1 m). They observed a relatively cold layer of soot particles near the fuel surface. Comparing their absorption and emission measurements, they showed that a large portion of the soot particles were at relatively low temperatures and did not contribute to emission. Therefore, it is important to understand both the chemical phenomena which lead to the formation of soot, and the local transport phenomena which determine the distributions of soot and soot temperature within a flame. The local soot concentration results from a time evolved history of local production and oxidation as well as convective and diffusive (thermophoretic) transport processes [13]. In fact, in laminar diffusion flames, the peak soot concentrations have been found to be slightly offset from the location of peak temperature [14]. This phenomenon is shown in Fig. 5 for a laminar C2H2 diffusion flame above a burner with a 12 mm × 96 mm fuel slot.

76â•… Transport Phenomena in Fires

Figure 5: Radial profiles of soot concentration and temperature at an axial height of 7.14 mm in a laminar C2H2 diffusion flame (Fig. 27 of ref. [14]).

The local radiant emission from a flame is linearly dependent on the soot concentration and is dependent on temperature to the fourth power. The effective emissive power at the flame surface is the integral of the local emissive power multiplied by the transmissivity to the surface and corresponds to an emission temperature that is intermediate to the maximum flame temperature (~1,960 K) and the temperature at the position of maximum soot concentration (~1,640 K). Hence, knowledge of the temperature and soot volume fraction distributions is critical in calculating the effective flame temperature across flame fronts. In pool fires, similar effects occur on a macroscopic level due to the shielding of the flame core by the cooler, external soot layers and at a microscopic level as a consequence of the soot radiation from flamelets in the combustion zone. The maximum heat flux is normally used to calculate safe separation distances from fires using metrics on damage from radiation such as those provided in Fig. 6: heat flux that causes pain to exposed humans, yields skin burns, or ignites wood for different times of exposure. Maximum tolerable heat fluxes can be established for different assumed times of exposure. Soot obscuration of radiation from fires will result in an overestimation of heat flux if flame temperature is assumed to be independent of diameter. This error will result in a conservatively safe distance of separation. For certain problems, however, overestimation of the heat flux may underestimate the fire hazard. This is particularly true when containers of high energy materials are exposed to radiation. An example of how lower heat fluxes can lead to greater hazards is shown in Fig. 7, where time to explosion of containers of the explosive PBX is plotted as a function of heat flux to the container surface. As expected, the time to explosion increases as heat flux decreases. However, at low heat flux rates, the intensity of the explosion increases as shown by the inset figures of the remnants of the container for two heat flux levels. The violence of the explosion increases as the time to explosion increases. Indeed, it is well known in the explosives community that long heating times (e.g. slow cook-off) can lead to detonations since more of the explosive material is heated to the ignition temperature. In contrast, with fast heating times (e.g. fast cook-off) only a surface layer is heated to the ignition temperature. The problem of BLEVEs with LPG is influenced by the accumulation of energy in the storage tanks that leads to the greatly enhanced strength of the explosions that result. For these reasons, conservative estimates of heat flux are no longer adequate.

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Figure 6:â•‡Skin exposure times to different heat fluxes that result in pain or burns, and flux needed to ignite wood (adapted from [8]).

Figure 7: Time to ignition for containers of the explosive PBX as a function of heat flux, using both electrical heating and external flames. Inset figures show the recovered container fragments at two heating rates. Fire modeling approaches more sophisticated than the homogeneous model are required to reliably address observed pool fire phenomena. These phenomena, including the effects of fuel type, smoke obscuration, relative locations of the flame front and of regions of high soot concentration, variation of local flame temperature, radiation blockage, and radiative feedback to the

78â•… Transport Phenomena in Fires pool surface determine the radiative heat transfer to an embedded object [15, 16]. The past two decades have seen an increasing use of CFD-based models to study fire phenomena. The progress that has been made will be discussed in the remainder of this chapter.

3â•‡V&V as a foundation for predicting heat transfer to embedded objects in pool fires The goal of this chapter is to present a physically-based method for predicting the potential hazard of an explosive device immersed in or near a pool fire of transportation fuel. To accomplish this goal, CFD-based computational tools that capture the relevant physical processes associated with the fire and the heat-up of the explosive device are employed. To move toward predictivity with these computational tools, we choose a methodology based on the V&V principles set forth by Oberkampf and Trucano [17]. Verification is the process of determining whether or not the mathematical models are implemented into computer code as the programmer intended, independent of the model’s physics. Validation determines how well the computer model matches the physical world. The process of V&V is cyclical as shown in Fig. 8, involving development of the conceptual model, verification of the model implementation, validation of the physical results, and evaluation of the conceptual model. Certification of the computer code for predictive use and quantification of error in the prediction involves the two-way coupling between the various stages of the V&V cycle. 3.1â•‡ V&V hierarchy A key tool in the V&V methodology is the construction of a V&V hierarchy. The apex of the hierarchy is the specific intended use of the simulation tool, i.e. the full system to be simulated. The remainder of the hierarchy is composed of several levels of decreasing technical complexity: subsystem cases, benchmark cases, coupled problems, unit level problems, and molecular processes. As one moves down the hierarchy, the quantity and quality of data increases and the experimental uncertainties decrease.

Figure 8: C onnectivity between verification, validation, simulation, and certification (adapted from [18]).

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At the highest level of the hierarchy, data are directly applicable to the intended use of the model but limited in scope and accuracy and often qualitative in nature. The subsystem case level is the decomposition of the overall system into simpler systems. Data with high experimental uncertainties are generally available for subsystem cases. The third level consists of benchmark cases, in which detailed experimental data from simplified but fully coupled problems are available for comparison. Coupled problems at the fourth level consist of two or more unit problems coupled together. Data available at this level include standard numerical solutions to simple problems and experimental data for coupled systems. The unit problems at the fifth level consist of isolated physical models. These models are validated as stand-alone problems with highly accurate numerical and experimental data. The lowest level in the hierarchy, molecular processes, further divides the unit problem into its fundamental components. The high fidelity data available at the unit problem level is also available at this level. A V&V hierarchy is constructed for heat transfer to explosive objects embedded in transportation fuel pool fires with V&V activities from the molecular processes level to the full system level as seen in Fig. 9. Some of these activities are highlighted in subsequent sections to demonstrate this hierarchal approach in moving toward predictivity. With one exception, the boxes at each level represent cases where experimental data sets have been identified in the context of the intended use of the simulation tool. The one exception is the ‘buoyancy driven flames’ box at the benchmark level; the desired experimental data for this box is unavailable. 3.2â•‡ Validation metric Validation of computed results in simulation science has generally consisted of comparing in graphical manner form (e.g. a two-dimensional plot) data extracted from a simulation or set of simulations with measured observables from an experiment or set of experiments. Based on such

Figure 9: V&V hierarchy for simulations of heat flux to an object embedded in a transportation fuel pool fire.

80â•… Transport Phenomena in Fires a graphical comparison, a person may declare that the computer model is ‘validated’, ‘invalidated’, or ‘requires improvement’. However, as pointed out by Oberkampf and Barone [19], these statements are qualitative in nature, leaving conclusions to the discretion of the observer. This potential for widely varying conclusions creates the need for a non-biased measurement, or metric, for determining the ‘level of agreement’ between experimental evidence and simulation results. The objective numerical values provided by the metric can then be used to formulate a value judgment of the comparison based on the level of risk one is willing to accept for the intended application. While the value judgment still requires the intervention of the biased human, the calculation of the metric does not include a notion of ‘quality’, thus making the metric itself a nonbiased participant in the validation process. Here, we briefly review a metric based on the use of statistical confidence intervals. This metric will be used in subsequent sections of this chapter to evaluate the level of agreement between LES data and experimentally measured data. Given a set of two or more experimental observations (n ≥ 2) and assuming that a population can be described by a normal distribution, a confidence interval for the true mean, μ, can be constructed using degrees of freedom ( = n -1) as

(5)

where x is the observed mean, s is the standard deviation, the level of confidence is 100(1 - a)%, and ta/2, is the student-t distribution value based on the values of a and . Now, given a set of simulation and experimental data, one may construct an estimated error,

E = ym - ye ,

(6)

__

where ym refers to the model (simulation) results and yâ•‰ â•‰ eâ•¯ refers to the sample mean of the experimental results, e.g. the average of a set of n experimental observations. Note that E is referred to as the ‘estimated’ error rather than the actual error because the true mean (μ) cannot be known given a small set of observations. Next, the true error is written as

E = ym – m.

(7)

Using the above definitions, the expression for the confidence interval for the true error is,

(8)

When computing metrics over a range of input variables such as spatial location, it is useful to reduce the collection of metrics to a single, global metric representing the system. The average relative error metric and the average relative confidence indicator, as described by Oberkampf and Barone [19], can be used to compute a global metric. The average relative error metric is defined by (9)

where xu and xl are the upper and lower bounds of the input variable. The relative average confidence indicator is given by

(10)

Heat Transfer to Objects in Pool Firesâ•…

81

The average relative error metric and the average relative confidence indicator provide a compact statement regarding the level of agreement, expressed as the global metric E / ye ± CI / ye ave . ave A few important points need to be considered relative to the metrics outlined here. First, the confidence intervals for the mean are not to be interpreted as traditional error bars on the simulation results; rather, they are the intervals in which the true error is estimated to lie with a given confidence. Second, for small numbers of experimental observations, the confidence intervals are made larger by increaÂ�sing values of ta/2, . It is thus advantageous to have multiple replications of a given experiment. Third, the confidence intervals speak to the quality of the experiÂ�mental data and may lead one to seek out more and/or improved data sets. Finally, experimental observations are given supremacy over simulation results, as acknowledged by the sole use of experimental data in the construction of the confidence intervals.

4â•‡ Surrogate fuel formulation The use of CFD to model heat flux to an explosive device in a transportation fuel pool fire raises the challenge of how to represent complex hydrocarbon fuels in computer simulations. One example of a transportation fuel is jet fuel (Jet-A or the military counterpart, JP-8), a mixture of hundreds of hydrocarbons that varies geographically and with time. It would be impractical to perform a simulation with such a complex mixture, even if the thermodynamics and detailed chemical kinetics for all the species in the mixture were available. Therefore, surrogate fuels, for which the necessary chemical and physical characteristics are known, must be developed. Different surrogates may be formulated for a given fuel depending upon the flame properties of interest. In this section, surrogate fuels are formulated for use in the simulation of a jet fuel pool fire, with particular interest in matching the burning rate and the heat transfer to objects immersed in, or in close proximity to, the fire. This work corresponds to the ‘surrogate fuel formulation’ box at the molecular processes level of the V&V hierarchy in Fig. 9. The major categories of hydrocarbons in jet fuels are normal and branched alkanes, cyclo-alkanes, and aromatics. Several investigators have developed surrogates for jet fuels [20-24] for applications other than pool fires. For the application of a transportation fuel pool fire, each surrogate component is required to have known chemical kinetics, to be representative of a main class of hydrocarbons present in jet fuels, and to be relatively inexpensive. In order to match burning rate and heat transfer to embedded objects in a pool fire, the mixture of components must match the volatility of the fuel, the sooting tendency, and the heat of combustion, and must reproduce the flame characteristics of a Jet-A/JP-8 pool fire, preferably with a small number (<10) of components. For the surrogate fuel study, the fuels tested included Jet-A, Norpar-15, and surrogates composed of various chemical reagents. Jet-A was acquired from the Salt Lake City Airport. Norpar-15, obtained from Exxon Chemicals, is a narrow-boiling range mixture of hydrocarbons. Three surrogates, Hex-11, Hex-12, and Hex-25c, each composed of six compounds, were formulated to match volatility (the boiling point distribution) and sooting tendency (smoke point) of the Jet-A/JP-8 fuel. The compositions of all five fuels are listed in Table 1. Other properties of the Jet-A and the calculated properties of the three surrogates are listed in Table 2. All three surrogates are similar to Jet-A in terms of volatility as reflected by the flash point and the average boiling point and in terms of smoke point. 4.1â•‡ Validation of surrogate formulation The ability of the surrogate formulations to match the burning rates and heat fluxes in Jet-A/JP-8 pool fires was tested by burning the jet fuel and its surrogates in a round steel pan, 0.3 m in diameter

82â•… Transport Phenomena in Fires Table 1:â•‡ Fuel compositions (in mol. % except where noted). Norpar-15

Jet-Aa

n-C8, 3.5 n-C8, 3.0 n-C8. 5.0 n-C14, 34.4 n-C12, 40.0 n-C12, 30.0 n-C12. 32.0 n-C15, 49.0 n-C16, 5.0 n-C16, 12.0 n-C16, 13.5 HMNb, 10.0 n-C17+, 3.1 Xylenes, 8.5 Xylenes, 15.0 n-HxBec, 23.0 Tetralin, 8.0 Tetralin, 13.0 PMHd, 15.0 Decalin, 35.0 Decalin, 27.0 Decalin, 15.0 Sum, 100.0 100.0 100.0 100.0

n-Paraffin, ≈28

Hex-11

Hex-12

Hex-25c

Branched paraffin, ≈29 Mono-aromatics, ≈18 Di-aromatics, ≈2 Cycloparaffin, ≈20 Nondetermined, ≈3 100.0

a

Approximate composition of Jet-A in this table is in wt. %. 2,2,4,4,6,8,8-heptamethylnonane. c n-hexylbenzene. d 2,2,4,6,6-pentamethylheptane. b

Table 2:â•‡ Properties of Jet-A and surrogate fuels. Properties

Jet-A

Hex-11

Hex-12

Hex-25c

Smoke point (mm) TSIa MW (g/mol) VABPc (°C) Flash point (°C) Latent heatd (kJ/kg) Combustion heat (MJ/kg)

24.5 26.7 173.5b 220.2 40.9 254.6 44.9

28.7 17.6 151.5 211.1 40.3 280.4 44.5

23.1 22.1 152.2 215.7 41.3 281.8 44.6

24.0 20.3 166.5 209.5 39.0 253.8 44.6

a

TSI is the threshold soot index. It is defined based on the smoke point such that its value ranges from 0 (least sooting) to 100 (most sooting). b MW (molecular weight) of Jet-A is estimated using the API empirical equation. c VABP is the volumetric average boiling point. It is the mean of the 10%, 30%, 50%, 70%, and 90% recovery temperatures determined in ASTM D86. d Latent heat is estimated at VABP.

and 0.1 m deep, placed 0.5 m above the ground. The tests were conducted in an enclosure 5 m × 5 m in cross-section and 6 m high equipped with dampers to control air infiltration and an exhaust duct. Both transient and steady-state tests were performed. Transient tests consisted of igniting and burning a batch of fuel in the pan. Steady-state tests involved continual replacement of the fuel in the pan to maintain a constant fuel level. Flame heights, shape, and puffing frequency were determined with a high-speed video camera shooting at 2,000 frames/s and with real-time video. Total heat fluxes and radiative heat fluxes were measured with gas-purged, water-cooled radiometers. Details of the test conditions can be found in [25, 26]. In classical studies of hydrocarbon pool fires [27-29], the mode of pool fire combustion (e.g. transient or steady-state) is not always stated. The tests described here indicate that the combustion mode plays a significant role in the measured fuel properties of interest.

Heat Transfer to Objects in Pool Firesâ•…

83

4.2â•‡ Burning rates and heat fluxes at steady state In the steady-state experiments, the liquid fuel density was assumed to be constant throughout the test. Gas chromatograph (GC) spectra of fuel samples taken from the burning pan show that Jet-A samples did not change in composition over time. Instantaneous volumetric burning rates were calculated and averaged over a time interval of 60 s. The results for Jet-A are plotted in Fig. 10 as a function of time. For this experiment, the steadystate burning rate of 2.07 × 10-3 m/min (0.0278 kg/m2 s) is reached 24 min after ignition. For liquid pools greater than 0.2 m in diameter, the mass burning rate (m≤, kg/m2 s) can be predicted by

m ′′ = m∞′′ [1 - exp( - k ⋅ b ⋅ d )]

(11)

where m≤∞ is the mass burning rate of an infinite diameter pool (kg/m2 s), k is the extinctionabsorption coefficient of the flame (1/m), b is the mean beam length corrector, and d is the pool diameter (m) [30]. As there are no reported constants for Jet-A pool fires, values of m≤∞ and k · b for kerosene are used (0.039 kg/m2 s and 3.5 m-1, respectively) [28]. From eqn (11), the computed mass burning rate for Jet-A is 0.0258 kg/m2 s (1.91 × 10-3 m/min), which is close to the experimental value for Jet-A reported above and to the value of 1.9 × 10-3 m/min reported for a 30 cm kerosene pool fire [1, 27]. Results from Hex-12 and Norpar-15 tests are also plotted in Fig. 10 for comparison. The steadystate regression rates for Hex-12 and Norpar-15 are 1.90 and 0.96 × 10-3 m/min, respectively. The time required for Hex-12 to reach steady state in this configuration is approximately 24 min. For the intended use of heat transfer to an explosive device, a key measure for the surrogate is its ability to yield radiation intensities that match those from a Jet-A flame. Real time heat flux measurements are shown in Fig. 11 for Jet-A and Hex-12 flames. These measurements were taken at a

Figure 10: Continuous feed, constant level 30 cm diameter pool fire surface regression rate profile.

84â•… Transport Phenomena in Fires

Figure 11: Radiative heat flux measurements from 30 cm diameter pool, continuous feed, constant fuel level experiments. height of 0.20 m above the fuel surface and 0.40 m from the center of the pan. About 20 min after ignition, the radiative heat flux reached a relatively constant value of 10.9 kW/m2 for Jet-A and 11.4 kW/m2 for Hex-12. The average radiative heat flux for Norpar-15 was 5.8 kW/m2. 4.3â•‡ Burning rates and heat fluxes for transient burning In a typical pool fire, a fixed quantity of fuel is ignited and burned to completion. Due to the complex mixture of hydrocarbons present in Jet-A/JP-8, transient behavior is observed for many of the key physical and chemical parameters. Lighter hydrocarbons are expected to vaporize and burn preferentially such that during the later stages of the fire, the fuel should be enriched in residual heavy hydrocarbons. The transient experiment was conducted by filling the pan with fresh fuel, then turning off the feed system and allowing the fuel to burn to completion. The decreasing fuel level as a function of time, used to compute the surface regression rate, was measured using an optical level sensor [25]. The surface regression rate of Jet-A in transient tests is shown in Fig. 12. The surface regression rate increases rapidly up to 10 min and reaches a peak value of 1.84 × 10-3 m/min at 11 min. This peak value is close to the burning rate obtained in the steady-state tests. The rate falls off rapidly over the range of 12-35 min and then decreases more slowly from 40-80 min until the end of burning. The Jet-A burning rate profile is successfully simulated by the surrogate fuel Hex-12 as shown in Fig. 12. The peak burning rate of Hex-12, 1.64 × 10-3 m/min, is slightly lower than that of Jet-A. The average burning rate for Hex-12 is 0.77 × 10-3 m/min, slightly lower than the average of 0.82 × 10-3 m/min for Jet-A. Two explanations have been proposed for the high burning rate soon after ignition [31]. First, in the initial burning stage, there is no heat loss to the edges of the fuel pan and the entire amount of heat transferred back from the flame to the pool surface can contribute to fuel vaporization.

Heat Transfer to Objects in Pool Firesâ•…

85

Figure 12:â•‡ Transient surface regression rate profile for 30 cm diameter pool fire.

Second, light components burn much faster initially, and the burning rate decreases after these species are depleted [28, 29]. The issue of whether the high initial burning rate is due to intense thermal feedback or to compositional variation was addressed by conducting transient burning experiments with Norpar-15, which is composed primarily of normal alkanes with 14-16 carbons (>99.8%) as shown in Table 1. After an initial transient, the surface regression rate is relatively constant, which supports the view that the initial high boiling rate for Jet-A (and its surrogates) is due to compositional change. Further confirmation was provided by the agreement of the mean fuel regression rate for Norpar-15 in the transient and steady state (Fig. 10) experiments. The radiant heat flux from the flame to the surroundings was consistent with the change in burning rate as well [25]. Finally, direct evidence of compositional change over time was demonstrated by GC analysis of Jet-A samples taken from the fuel pan [25]. 4.4â•‡ Effect on fuel composition changes on sooting propensity The composition changes described above are expected to lead to a continuous change in smoke point. Based on an ASTM D86 distillation test [32] of the Hex-12 surrogate, smoke points were calculated at different stages of volume loss and the results are plotted in Fig. 13. In addition, smoke points were measured on samples taken in a transient pool fire at different percentages of fuel volume consumption. Smoke points for Hex-12 increase with increasing volume loss for samples from both pool fire tests and distillation tests because the soot-promoting components (xylenes, tetralin, and decalin) are more volatile than n-dodecane and n-cetane. By the end of a burn, the residual surrogate fuel is nearly pure n-cetane and reaches its highest smoke point. In contrast, the smoke point of Jet-A decreases slowly with increasing burn-off/boil-off. This decrease is believed to be related to the wide spectrum of aromatics in actual jet fuel, as suggested in the detailed hydrocarbon analysis for Jet-A [33]. The existence of high molecular

86â•… Transport Phenomena in Fires

Figure 13: Smoke point variation as a function of volume of fuel burned (or distilled) for jet fuel and surrogates.

weight, high boiling point, and high sooting index naphthalenes and benzo-cycloalkanes in the actual fuel helps to maintain the smoke point relatively constant through low and intermediate percentages of fuel consumption with a slight decreasing trend at high percentages of fuel consumption. 4.5â•‡ Improved surrogate formulation Due to the challenges associated with matching sooting propensity of Jet-A over its entire boiling range (or lifetime of a transient pool fire), a more chemically complex surrogate is required. The method of structural group contributions was adapted to the formulation of surrogates [25] and was used to provide a more chemically accurate description of the fuel [26]. The method is a significant improvement over previous approaches as it does not require any experimental procedures or information on fuel properties. However, the molecular structure of the fuel molecules must be determined. The improved surrogate, Hex-25c, consists of six species. Its composition is given in Table 1, while its physical properties are given in Table 2. A comparison of the smoke point of Hex-25c with that of Jet-A is shown in Fig. 13, where it is evident that the new surrogate provides significant improvement in smoke point performance over the Hex-12 surrogate, particularly in the later stages of fuel consumption.

5â•‡ Chemical kinetics for soot production from JP-8 Two other areas of major research activity at the molecular processes level are the kinetic modeling of the gas and the solid (soot) phases. Kinetic modeling of surrogate fuels requires mechanisms and chemical kinetics for the major components of the surrogate [20-24, 34, 35]. Although the kinetic mechanisms are still under development, considerable success has been achieved by various research groups in matching experimental results. In general, these reaction mechanisms

Heat Transfer to Objects in Pool Firesâ•…

87

are tuned to meet specific objectives. For example, one focus area in engine modeling is ignition, which is influenced by chemical kinetics at low temperatures. For the purpose of calculating heat transfer to embedded objects in pool fires, soot formation, which is strongly dependent on soot precursors derived from acetylene and benzene, must be predicted. The following section describes kinetics that have been optimized to predict benzene and acetylene in premixed flames. The resulting mechanism is called the Utah Surrogate mechanism [36].

5.1â•‡ Utah Surrogate mechanism A schematic representation for soot formation, adapted from Bockhorn [37], is shown in Fig. 14. The initial steps are the ignition and consumption of the surrogate mixture, which is driven by reactions with H, O, and OH radicals (with contributions from HO2 in the ignition zone). The larger paraffinic fuel molecules decompose and cascade down to the smaller aliphatic and olefinic molecules. The key pathway to soot is the formation of benzene and then polycyclic aromatic hydrocarbons (PAH), which are the building blocks of the first particles. Acetylene is important because it is the major contributor to soot mass addition and it participates in the H-abstraction-ACetylene-Addition (HACA) mechanism [38] that leads to the growth of PAH and soot. Reaction mechanisms that address the steps shown in Fig. 14 are large, involving hundreds of chemical species and thousands of chemical reactions [22-24, 36-41]. The Utah Surrogate mechanism is formulated from detailed sub-models of n-butane, n-hexane, n-heptane, n-decane, n-dodecane, n-tetradecane, and n-hexadecane; semi-detailed sub-models of i-butane, i-pentane, n-pentane, 2,4-dimethyl pentane, i-octane, 2,2,3,3-tetramethyl butane, cyclohexane, methyl cyclohexane, tetralin, 2-methyl 1-butene, and 3-methyl 2-pentene; and aromatics

Figure 14: Schematic representation of sequential steps for soot formation and burn out (adapted from [37]).

88â•… Transport Phenomena in Fires that include benzene, toluene, and xylenes. The mechanism is available as supplemental material provided to the Combustion Institute in support of the publication by Zhang et al. [36]. The mechanism can be used to predict the fuel consumption and major combustion products for jet fuels that are comprised of mixtures of n-paraffins, i-paraffins, cyclo-paraffins, aromatics, and alkyl-substituted aromatics. Jet fuel composition can vary widely, depending upon the crude oil being refined and the refining procedures used. For example, aromatic content varies from 11% to 26%. The mechanism was built on the following foundation: • • • • • • • • • •

Marinov-Westbrook-Pitz hydrogen model [42] Hwang et al. [43] and Miller et al. [44] acetylene oxidation models Wang and Frenklach acetylene reaction set with vinylic and aromatic radicals [45] Marinov and Malte ethylene oxidation sub-model [46] Tsang propane and propene chemical kinetics [47, 48] Pitz and Westbrook n-butane sub-model [49] Miller and Melius benzene formation sub-model [50] Emdee-Brezinsky-Glassman toluene and benzene oxidation sub-model [51] Vovelle and coworkers n-heptane decomposition model [52] Pitsch i-octane decomposition model [53]

New submodels were added for a number of n-paraffins (C5, C6, C10, C12, C16), a number of iso-paraffins (i-C4, i-C5, i-C6, 2,2,3,3,-tetramethyl-C4), cyclohexane, methyl cyclohexane, butadiene, and 3-methyl 2-pentene. The mechanism is able to model a wide range of surrogates. It has been optimized for atmospheric conditions, flame studies, and soot precursors. Its ability to model the concentration of acetylene in flames of common components of surrogates, as well as a kerosene flame, is shown in Fig. 15, where results for acetylene concentration as a function of height above the burner are presented. The conditions for the flames are given in Table 3. Kinetic model predictions for the pure component fuels and for kerosene using a surrogate formulation are shown by the solid lines. The agreement with the data shows the progress that is being made in the development of kinetic models with predictive capabilities. Nevertheless, a validation metric has not been applied to this analysis, so quantitative information is not available. Similar comparisons for benzene concentration can be found in [36].

5.2â•‡ Soot formation and oxidation The Utah Surrogate mechanism provides the gas phase reactions up to the point of particle inception, as shown in Fig. 14. Several soot models are available that can provide the transition from the soot precursor molecules to soot. Four classes of models utilized in order of increasing complexity are: • Empirical models like that of Sarofim and Hottel [63] represent a combination of soot formation and destruction. Such models account for some of the major factors affecting soot contributions to radiative absorption and emission: temperature, stoichiometric ratio, and fuel type. With empirical models, a certain fraction of the fuel is converted to soot under fuel rich conditions, but no soot persists in a fuel lean environment. • The Lindstedt model [39] has four steps: nucleation, surface growth, oxidation, and coagulation. The nucleation is assumed to be an activated process that is proportional to acetylene

Heat Transfer to Objects in Pool Firesâ•…

1.5E-01 1.0E-01

3.0E-02

Bastin C2H2 Flame

2.0E-01

1.0E-01

6.0E-02

Marinov CH4 Flame Mole Fraction

Mole Fraction

2.0E-01 Mole Fraction

3.0E-01

Bockhorn C2H2 Flame

Mole Fraction

2.5E-01

2.0E-02

1.0E-02

89

Castaldi C2H4 Flame

4.0E-02

2.0E-02

5.0E-02

0

1 2 HAB (cm)

0.0E+00

3

0

(a) 2.0E-03

2.0E-02

1.5E-03 1.0E-03 5.0E-04

0.0E+00

0 0.5 1 1.5 2 HAB (cm)

0.0E+00

(e)

0

0.0E+00

0 0.2 0.4 0.6 0.8 HAB (cm)

0

0.4 0.8 1.2 HAB (cm)

(c) 2.0E-02

Vovelle nC71.0 Flame Mole Fraction

4.0E-02

0.0E+00

3

(b) 2.5E-03

Bittner C6H6 Flame Mole Fraction

Mole Fraction

6.0E-02

1 2 HAB (cm)

0.05 0.1 0.15 HAB (cm)

1.5E-02 1.0E-02 5.0E-03 0.0E+00

(f)

(d) 2.0E-02

Vovelle nC7 Flame Mole Fraction

0.0E+00

0 0.1 0.2 0.3 0.4 0.5 HAB (cm)

Vovelle Kerosene Flame

1.5E-02 1.0E-02 5.0E-03 0.0E+00

0

0.1 0.2 0.3 HAB (cm)

(h)

(g)

Figure 15: Comparison of concentrations of acetylene simulated using the Utah Surrogate mechanism with experimental results reported by different authors (see Table 3 for experimental conditions): (a) atmospheric pressure fuel rich flames of acetylene (C2H2), (b) low pressure fuel rich flames of C2H2, (c) fuel rich flames of methane (CH4), (d) fuel rich flames of ethylene (C2H4), (e) fuel rich flames of benzene (C6H6), (f) stoichiometric flames of n-heptane (nC71.0), (g) fuel rich flames of n-heptane (nC7), and (h) fuel rich flames of kerosene.

Table 3:â•‡ Experimental conditions for premixed flames in Fig. 15. Author Fuel Marinov Westmoreland Bastin Bockhorn Harris Castaldi Bittner Ciajolo Vovelle Vovelle Vovelle Vovelle a

CH4 C2H2 C2H2 C2H2 C2H4 C2H4 C6H6 C6H6 C7H16 i-C8H18 C10H22 Kerosene

N2 is used the inert rather than Ar.

Inert Ar (%) C/O P (torr) 0.453 0.05 0.45 0.55 0.656 0.578 0.3 0.752a 0.73a 0.682a 0.682a 0.684a

0.626 0.959 1.00 1.103 0.92 1.02 0.717 0.72 0.605 0.608 0.558 f = 1.7

760 20 19.5 90 760 760 20 760 760 760 760 760

Flow rate (g/(cm2 s)) -2

7.19 × 10 1.58 × 10-2 3.46 × 10-2 3.43 × 10-2 1.12 × 10-1 7.21 × 10-2 2.19 × 10-2 5.07 × 10-2 6.50 × 10-2 5.56 × 10-2 6.68 × 10-2 7.96 × 10-2

Reference [54] [55] [56] [57] [58] [59] [60] [61] [62] [62] [34] [34]

90â•… Transport Phenomena in Fires concentration and which yields nuclei of a specified size, with 100 carbon atoms being suggested. Surface growth is proportional to acetylene concentration and has a rate parameter fitted to literature rates. Soot is assumed to be oxidized by O2 with a rate constant fitted to data so as to allow for the role of OH. Coagulation is calculated using the standard equations for aerosol dynamics. • The HACA mechanism, built on the two-step process of activation of an aromatic molecule by hydrogen abstraction followed by acetylene addition, leads to both molecular weight growth and cyclization [40]. Lumping mechanisms have been developed for the PAH growth using generic rates for the different classes of reactions including acetylene addition, hydrogen abstraction, and reactions with OH and O2. Soot nucleation is assumed to occur by the dimerization of two PAHs. Although dimers of all PAH combinations can be included, it is common to use pyrene. Soot formation and growth are calculated using moment methods, with allowance for nucleation, coagulation, and surface growth, as well as oxidation reactions. The implementation of this method by Appel et al. [38] has found widespread use. • Sectional models treat the soot simultaneously with the chemical kinetics by assigning large PAH particles and soot particles to bins that have a range of carbon numbers. The transition from gas phase chemistry to particle chemistry is achieved by assigning a bin with a given carbon number as the smallest particle size. One implementation of the sectional model [41] has the smallest bin for mass numbers of 201-400, with an H/C ratio of 0.5, corresponding to a particle size of 0.85 nm. The mass limits are approximately doubled for each sequential bin; the largest of 20 bins corresponds to mass numbers of 105-210 million, an H/C ratio of 0.125, and a particle size of 68 nm. Bins are assumed to react with all other bins and with gas phase molecules. The first four bins are considered to be large PAHs and the bins from 5 to 20 to be soot particles. Soot can also be oxidized, primarily by reactions with OH, O, and O2. The rates of oxidation for most flame conditions are dominated by the reaction with OH. The rate of reaction is proportional to the collision rate of the OH with the soot surface with approximately 13% of the collisions [64] leading to the consumption of soot by the stoichiometric reaction C + OH = CO + H.

6â•‡ Use of LES methods for pool fires A major challenge in applying CFD to fires is the wide range of continuum length scales and their corresponding time scales that characterize the fire physics in large diameter (>1 m) fires. For example, important physical time and length scales range from molecular O(10-9 s, 10-10 m) to scales that are observable with the naked eye O(1 s, 1 m). This range of time and length scales prohibits the use of fully resolved, three-dimensional, direct numerical simulation (DNS) techniques. Additionally, transportation fuel fires often involve complicated interactions with the environment such as the highly unsteady processes of fluid/structure interaction, wind effects, and flame spread across fuels. Given current modeling options and the importance of unsteady effects in transportation fires, LES is the prime candidate for modeling such fires. Compared to the traditional Reynolds averaging (RANS) approach, LES captures the unsteady effects of pool fires more accurately by resolving the large length and time scales that are responsible for controlling the dynamics of the fire [65]. In fact, LES is emerging as the prevailing methodology for studying fires due to its ability to render realistic, time-resolved flows of gases, heat, and smoke throughout a domain [66]. An LES approach was employed by Schmidt et al. [67] and Kang et al. [68] to study turbulence structure in medium scale methanol pool fires. In both these efforts, reasonably good agreement

Heat Transfer to Objects in Pool Firesâ•…

91

was obtained for the mean velocity and temperature fields and their fluctuations. Xin et al. [69] conducted a study of a 7.1 cm methane pool fire that quantitatively reproduced the average scalars and velocities. Numerical simulations of pool fires employing the LES approach and accounting for participating media radiative heat transfer have also been demonstrated [70-72]. In the fire protection engineering community, a widely used fire simulation tool is Fire Dynamics Simulator (FDS), developed by McGrattan et al. at NIST [73, 74]. This LES-based tool has been used in residential and industrial fire reconstructions and in the design of fire protection systems. In the V&V hierarchy (Fig. 9), the low-Mach LES algorithm and the subgrid turbulence closure are identified as two of the unit problems. The LES algorithm is composed of the numerical differencing scheme and a solution method (algorithm) for solving the filtered set of governing equations. The subgrid turbulence model is the set of approximations that ‘close’ the set of filtered equations, effectively modeling the unresolved turbulent fluctuations. The LES algorithm and subgrid turbulence closure are closely coupled, but by separating the two, one can independently address modeling choices that affect simulation results for the intended use of the LES tool. 6.1â•‡ LES equations The essential governing equations, written in finite volume form, include the mass balance, momentum balance, mixture fraction balance, and energy balance equations. Using a boldface symbol to represent a vector quantity, the equations are: 1. The mass balance,

(12)

where r is density and u is the velocity vector.

2. The momentum balance,

(13)

∂u

where t is the deviatoric stress tensor defined as tij = 2 mSij - 23 m ∂xk dij and the symmetric k ∂uj ∂u stress tensor Sij = _â•‰â•¯12â•¯â•‰ â•‰ __ â•‰â•¯∂xiâ•¯â•‰â•¯ + __ â•‰â•¯∂x â•¯â•‰â•¯â•¯ â•‰. The second isotropic term in tij is absorbed into the pressure j i projection for the current low-Mach scheme. Also in eqn (13), g is the gravitational body force and p is the pressure.

(â•¯

)

3. The mixture fraction balance,

(14)

where f is the mixture fraction and a Fick’s law form of the diffusion term assuming equal diffusivities results in a single diffusion coefficient, D. 4. The thermal energy balance,

(15)

92â•… Transport Phenomena in Fires where h is the sum of the chemical plus sensible enthalpy and q is the radiative flux. A Fourier’s law form of the conduction term is used with a diffusion coefficient, k, and the pressure term is neglected. Now, consider a control volume, V, with surface area S. Because the equations will be solved on a computational grid, one can assume that the control volume has N faces, where unique faces are identified with their index k. The discussion is further simplified by only considering cubic volumes of length h. ___(j) Given the cubic control volume, a surface-filtered field for a variable f is defined as fâ•‰ â•‰ â•¯ (x), where the variable is filtered on a plane in the xj orthogonal direction. Then, for any surface k, the field is sampled at the face-centered location. For example, if j = 1, the surface-filtered quantity is

(16)

The volume average follows as

(17)

The bars over the variable f are labeled with superscripts ‘2d’ and ‘3d’ to distinguish between the two filters. Pope [75] identifies the proceeding definitions as using the ‘anisotropic box’ filter kernel where the resultant variables are simply averages over the interval x j - 12 h < x ′j < x j + 12 h . For convenience in isolating density in the filtered equations, a Favre-filtered quantity is defined for an arbitrary variable j as j 2d ≡

rj 2d

, r 2d

(18)

and j 3d ≡

rj 3d r 3d

.

(19)

This convention of explicitly defining the 2d and 3d filters is different than what is commonly observed in the literature, where the filtered equations from finite difference equa__2dare derived __ tions rather than finite volume equations. Thus, using r â•‰ â•¯ and â•‰r3d â•¯ in eqns (18) and (19) for surface and volume filtered densities, respectively, is appropriate for the present discussion. These definitions for filtered quantities are applied to the integral forms of the governing equations to obtain the Favre-filtered LES equations. Nevertheless, there are terms in the Favre-filtered equations that cannot be solved. These include the surface-filtered convection of momentum 2d 2d convection of mixture fraction, u u j f , and the surface-filtered conveci u j , the surface-filtered 2d tion of enthalpy, u jh . 2d 2d For the filtered momentum product, r u i u , a subgrid stress tensor is defined as, j

2d  2d  2d tijsgs = u i u j - ui u j .

Similarly, subgrid diffusion terms are defined for mixture fraction and enthalpy,

J

f

2d  2d , = u - u 2d j f jf

(20)

(21)

Heat Transfer to Objects in Pool Firesâ•… 2d h  2d  2d J = u j h . jh - u

93 (22)

Using these definitions, the final forms of the Favre-filtered equations are 1. The filtered mass balance, d 3d Sk ( r ) + nkj ( r 2d u 2d j ) = 0. dt V

(23)

2. The filtered momentum balance,

S d 3d 3d sgs 2d 2d 3d ( r ui ) = k nkj (– r 2d ui2d u 2d j + tij + tij - p dij ) + r gi . dt V

(24)

3. The filtered mixture fraction balance,

d 3d  3d S  2d + D∇f 2d + J f ). ( r f ) = k nkj (– r 2d u 2d j f dt V

(25)

4. The filtered thermal energy balance,

S 2d d 3d 3d 2d  2d ( r h ) = k nkj ( - r 2d u 2d + J h ). j h + k ∇h - q dt V

(26)

f sgs The subgrid momentum stress, â•‰tâ•‰ijâ•‰â•¯ â•‰, the subgrid mixture fraction dissipation, J , and the subh grid enthalpy dissipation, J , contain the unresolved or subgrid action of the turbulence on the transported quantities. Since these terms arise from definitions, models are introduced to include the subgrid effects that they represent. These models are discussed next.

6.2â•‡ Subgrid turbulence models Invoking an ‘eddy-viscosity’ modeling concept, the subgrid transport due to turbulent advection is treated as an enhanced diffusion term for the unclosed terms listed above. That is, the subgrid mixture fraction dissipation and subgrid enthalpy dissipation are respectively written as,

and

J f = Dt

∂f 2d , ∂x j

J h = kt

∂h 2d . ∂x j

(27)

(28)

To model Dt and kt, constant turbulent Schmidt (Sct) number, Sct = and Prandtl (Prt) number, Prt =

1 mt , r Dt

1 mt , r kt

(29)

(30)

94â•… Transport Phenomena in Fires are assumed, where mt is a turbulent viscosity. Following Pitsch and Steiner [76], the values of the turbulent Schmidt and Prandtl numbers are taken as Sct = Prt = 0.4, which is consistent with a unity Lewis number assumption. For the subgrid momentum stress tensor, tijsgs, two common LES turbulence closure models are the constant coefficient Smagorinsky model [77] and the dynamic coefficient Smagorinsky model [78]. As with the scalar subgrid modeling terms, the eddy viscosity model is again invoked for tijsgs, which is approximated by

(31)

where â‹ƒ is the filter width, t is the eddy viscosity, and | S | ≡ (2 Sij Sij )1/ 2 . For the Smagorinsky model, Cs ≈ 2 depending on the filter type, numerical method, and flow configuration [75]. For the dynamic Smagorinsky model, Cs is computed by taking a least squares approach to determine the length scale [79], (Cs Δ )2 =

ij Mij Mij Mij

,

(32)

where

 )2 | S | S , ij = 2(Cs Δ )2 | S | Sij - 2(Cs Δ ij

(33)

and

Mij ≡ 2(| S | Sij - a2 | S | Sij ).

(34)

The hat defines an explicit test filter and the angular brackets in eqn (32) conceptually represent an averaging over a homogeneous region of space that, experience has shown, is necessary for stability. Experience has also shown that averaging over the test filter width is adequate. The filter width ratio, a = Δˆ / Δ, is usually taken to be 2. 6.3â•‡ LES algorithm The set of filtered equations (eqns (23)-(26)) are discretized in space and time and solved on a staggered, finite volume mesh. The staggering scheme consists of four offset grids. One grid stores the scalar quantities and the remaining three grids store each component of the velocity vector. The velocity components are situated so that the center of their control volume is located on the face centers of the scalar grid in their respective direction. The staggered arrangement is advantageous for computing low-Mach LES reacting flows. First, since a pressure projection algorithm is used, the velocities are exactly projected without interpolation error because the location of the pressure gradient coincides directly with the location of the velocity storage location. Second, Morinishi et al. [80] showed that kinetic energy is exactly conserved on a staggered grid when using a central differencing scheme on the convection and diffusion terms without a subgrid model. Having a spatial scheme that conserves kinetic energy is advantageous because it limits artificial dissipation that arises from the differencing scheme. These conservation properties make the staggered grid a prime choice for LES reacting flow simulations. For the spatial discretization of the LES scalar equations, flux limiting and upwind schemes for the convection operator are used. These schemes are advantageous for ensuring that scalar

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95

values remain bounded. For the momentum equation, a central differencing scheme for the convection operator is used. All diffusion terms are computed with a second-order approximation of the gradient. When computing the 2d surface filtered field on the faces of the control volume, one is forced to use an interpolation from the 3d volume filtered field. This approximation is tolerated because computing the 2d surface field is not possible with the given grid scheme. An explicit time stepping scheme is chosen. A general, multistep explicit update for a variable, f, may be written as, f0 = f n , m -1

f(i ) = V ∑ (ai ,k f( k ) + Δt bi ,k L (f( k ) )),

i = 1,..., m,

(35)

k =0

f( m ) = f n +1 ,

where n is the time level, m is the substep between n and n + 1, a and b are integration coefficients, and L is a linearization operator on the convective flux and source terms. The time step is limited by Δt ≤ c Δt F.E. ,

(36)

where â‹ƒtF.E. is the forward-Euler time step limited by the Courant-Friedrichs-Levy condition and c is a constant less than or equal to 1. A higher order multistep method is derived by letting m > 1 and choosing appropriate constants for a and b. For this study, two-step and three-step, strong stability preserving (SSP) coefficients were chosen from Gottlieb et al. [81]. The coefficients for SSP-RK 2 and SSP-RK 3 are optimal in the sense that the scheme is stable when c = 1 if the forward-Euler time step is stable for hyperbolic problems. In practice, for the Navier-Stokes equations, the value of c is taken to be less than 1. Choosing an explicit time stepping scheme, rather than an implicit one, creates a challenge for solving the set of equations. The density at the n + 1 time step, which is required to__determine the __ n+1 cardinal variables, requires an estimation. Taking the estimated density for r â•‰ â•› â•¯ to be r â•‰ *â•¯ , the estima__ __ tion can be as simple as r â•‰ *â•¯ = r â•‰ â•›â•¯n. Note that the 2d and 3d filter distinction is dropped for the remainder __ of this discussion for the sake of simplicity. Another procedure includes predicting a value for r â•‰ *â•¯ from performing a forward-Euler step in time as,

r* = r n - Δt __

Sk nkj ( r u j )n . V

(37)

Ideally, one would like to know r â•‰ â†œæ¸€å±®â•¯n+1, but r is a function of the same variables that are being updated in time, namely the mixture fraction, f, and enthalpy, h. This quandary, a result of the explicit time stepping method, will not be resolved for variable density flows without using a fully implicit method. Explicit methods, however, can be advantageous, especially for large scale parallel computations. Specifically, load balancing is easier and more efficient with explicit methods because the amount of work required per processor is readily determined a priori. Explicit methods are also easier to code into a computer and to debug. For these reasons, the current algorithm discussion is limited to explicit methods. The explicit algorithm for solving the set of filtered equations is shown in Algorithm 1.

96â•… Transport Phenomena in Fires Algorithm 1 Explicit LES algorithm. for t = tmin…tmax do for RKstep = 1…N do Solve for __scalars products ( r f )n +1 and ( r h )n +1 . __n+1 Estimate â•‰ â•¯* = r â•‰__â•¯ from __ __ r __ eqn (37) if __r â•‰ â•¯* < __r â•‰ min â•¯ or â•‰râ•¯* > r â•‰ max â•¯ then â•‰ â•¯* = r r â•‰ nâ•¯ end if Compute f n +1 = ( r f )n +1 / r * and h n +1 = ( r h )n +1 / r * Compute r n+1 = f ( f n +1 , h n +1 ) Compute u *, the unprojected velocities Perform RK averaging if needed Compute correct pressure from pressure Poisson equation Project velocities with correct pressure to get u n+1 end for end for 6.4â•‡ Large scale, parallel computing with LES LES is computationally intensive because it resolves a relatively large set of spatial and temporal scales. An LES algorithm can be implemented in a serial code, but the underlying models must be simplified and/or lower resolution cases must be considered. To understand the interactions between a transportation fuel fire and embedded objects, all relevant scales require resolution. For example, the relevant scales for turbulence/chemistry interactions can be orders of magnitude smaller than the largest fire scales. Accounting for all these length and time scales requires massively parallel computations. The LES fire simulation tool described above utilizes Uintah, a component-based visual problem solving environment (PSE) that provides a framework for large-scale parallelization of different applications [82-84]. Uintah was designed and implemented to satisfy three goals: (1) to provide a general framework for massively parallel simulations of fluid and particle physics; (2) to facilitate both MPI- and thread-based parallelism; and (3) to allow scientists from outside the computer field to have an intuitive method for easily inserting their algorithms into a parallel framework without being bogged down by parallel programming details. The integration of the LES fire simulation tool in the Uintah PSE required the development of reusable, physics-based components that could be used interchangeably and interact with other components. Examples of such components include a pressure solver, a momentum solver, a scalar solver, and a subgrid scale turbulence model. Also implemented in Uintah are components developed by third parties, specifically nonlinear and linear solvers designed for complex flow problems. Realistic fire simulations must account for relevant physical processes such as turbulent reacting flow, convective and radiative heat transfer, multiphase interactions, and fundamental gas-phase chemistry. Representations of these physical processes lead to very large sets of highly nonlinear, partial differential equations (PDEs); robust nonlinear and linear solvers on massively parallel platforms are required. Hence, two suites of nonlinear and linear scalable solvers for scientific applications modeled using PDEs, Portable Extensible Toolkit for Scientific Computation (PETSc) [85] and High Performance Preconditioners (HYPRE) [86], are interfaced with Uintah.

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6.5â•‡ V&V studies of LES code/turbulence model 6.5.1â•‡ Verification using the method of manufactured solutions Both analytical and manufactured solutions are frequently used as verification tools. Analytical solutions to the Navier-Stokes equations usually involve simple systems where parts of the equations are reasonably neglected. As a result, not all parts of the equation and the corresponding discretization scheme are fully tested when compared to analytical solutions. Manufactured solutions allow for arbitrary complexity in the solutions because they have no physical meaning and can be formulated to verify all parts of the governing equations. When manufactured solutions are processed through the governing equations, the governing equation itself might not be satisfied, so an extra source term is added to account for the additional terms that arise from the manufactured solution. The method of manufactured solutions [87] is an extremely useful verification exercise for finding programming errors and ensuring expected behavior of the computer code. The convective and diffusive spatial operators as well as the pressure correction algorithm are tested in two-dimensional planes by initializing the domain with a manufactured solution for velocity and pressure (added exponential term to manufactured solution in [88]),

(38)

(39)

(40) where A is the amplitude and is the viscosity. Note that the velocity field satisfies the continuity equation, ∇ · U = 0, for constant density. To test the spatial discretization error, the advection/diffusion terms and the computed gradient of the pressure correction from the Poisson solve are evaluated at t = 0. Then, advection/diffusion terms and the correction gradient are compared to the exact solutions for each two-dimensional plane (x-y, x–z, y–z) in a three-dimensional Cartesian space. The total force vector on a fluid element is given by the sum of the individual components,

F Total = F a + F d + F ∇P ,

(41)

where Fâ•›a is the advective force, Fâ•›d is the diffusive force, and F∇P is the pressure force. Decomposing the force vector into its various components is useful for identifying programming error in individual force components, but here we consider only the total force vector. The total normalized error for the force components is measured as Normalized error =

FeTotal - F Total FeTotal

,

(42)

where the subscript e is the force computed from the manufactured solution. Figure 16 shows that the normalized error from the spatial discretization decreases at a second-order rate with increasing mesh resolution for each two-dimensional plane. 6.5.2â•‡ Verification and validation with Compte-Bellot and Corrsin data Further verification of the LES code and validation of the constant coefficient and dynamic Smagorinsky subgrid turbulence models is achieved by initializing the computational domain with the

98â•… Transport Phenomena in Fires

Normalized error

100

10−1

10−2

10−3 −1 10

x−y plane x−z plane y−z plane ∆x

100

Figure 16: Total error convergence using a manufactured solution for the spatial operators. Each two-dimensional plane in the three-dimensional Cartesian space is tested and shows second-order behavior.

Figure 17: LES code verification and turbulence subgrid model validation. Kinetic energy is reported per unit mass. experimental data of Compte-Bellot and Corrsin [89] and then marching the solution in time using the second-order SSP-RK time stepping scheme on a 323 periodic mesh. The curves generated by this technique are displayed in Fig. 17. The straight solid line represents a simulation with no subgrid turbulence model and no molecular viscosity. This line stays nearly level, with only a slight increase in kinetic energy that is added from the time stepping scheme (the energy characteristics of the SSP-RK algorithm are discussed in [90]). This result verifies that the simulation is free from

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99

numerical dissipation. The other two curves show the kinetic energy behavior obtained from the constant coefficient Smagorinsky and dynamic coefficient Smagorinsky models. Both curves generally follow the kinetic energy decay in the data, an expected result for isotropic turbulence.

6.5.3â•‡ Validation of subgrid turbulence models Additional turbulence model validation is performed using buoyant helium plume data from the ‘coupled problem’ level of the V&V hierarchy. This ‘coupled problem’ combines the effects of fluid flow and turbulence without the complications introduced by chemical reactions. The data set from the 1 m helium plume, taken in the FLAME facility at Sandia National Laboratories in Albuquerque, NM, includes time-averaged vertical velocity, horizontal velocity, and mixture fraction profiles as well as instantaneous values of these variables [91]. Simulations of the 1 m helium plume were performed on a 3 m3 computational domain using the LES code described above coupled with two types of dynamic turbulence models: the dynamic coefficient Smagorinsky model described above and a local dynamic model [92]. The purpose of the study was to determine the best turbulence model for the large buoyant plume. In the case of helium, it has been observed [93] that small Rayleigh-Taylor instabilities, on the order of 1.5 cm for a 1 m helium plume, may control the strength of the air entrainment. Failure to capture this effect leads to weak air entrainment and velocities that are too high in the centerline velocity field. Since proper mixing requires that the length scale of the Rayleigh-Taylor instability be captured on the mesh, a turbulence model that does not smear out the instability is preferred. For the simulations, the turbulent Schmidt (Sct) and Prandtl (Prt) numbers were held constant at 0.4 (eqns (29) and (30)) and the filter width (â‹ƒ) was averaged over a grid volume, (43) Δ = ( Δx ΔyΔz )1/3 . Figure 18 compares mixture fraction as a function of radial distance for the two turbulence models and three mesh resolutions at a height of 0.6 m above the inlet. The bands on the experimental data represent the 90% confidence interval constructed from the experimental data as discussed in Section 3.2. While both models overpredict the helium centerline concentration, the local dynamic model appears to perform slightly better and to converge at a lower mesh resolution than the dynamic Smagorinsky model. However, global metric values (from eqns (9) and (10)) shown in Table 4 for the mixture fraction and streamwise (u) velocity components suggest that neither turbulence model provides a distinct advantage over the other at the finest resolution (Δx = 1 cm). Further investigation of the overprediction of the centerline helium concentration is ongoing and includes understanding the effects of the density prediction in the explicit scheme and of the scalar turbulent closure.

7â•‡ Combustion/reaction models Detailed combustion modeling of turbulent flows is computationally prohibitive due to the wide range of time and length scales that are coupled through interactions between thermochemistry and fluid dynamics. The use of a detailed kinetic scheme to describe the chemistry requires the solution of a transport equation for NS - 1 species where NS is the total number of species. This requirement, coupled with the stiffness of the source terms in the transport equations, makes the computational load unmanageable for transportation fuel pool fires. Fortunately, the fluid dynamics length and time scales overlap with only a subset of the thermochemical time scales, so some degree of decoupling is possible. Indeed, a large class of combustion models relies on the

100â•… Transport Phenomena in Fires

(a)

(b)

Figure 18: Profiles of the average mixture fraction as a function of radial distance at a height of 0.6 m above the inlet for (a) the dynamic Smagorinsky model and (b) the local dynamic model.

Table 4:â•‡Global average relative errors with the average relative confidence indicator for the streamwise (u) velocity and mixture fraction. All values are percentages. u velocity

Resolution 1123 2243 3003

Mixture fraction

Dynamic Smagorinsky Local dynamic 25 ± 20 18 ± 20 11 ± 20

NA 9 ± 20 10 ± 20

Dynamic Smagorinsky

Local dynamic

91 ± 45 64 ± 45 47 ± 45

NA 41 ± 45 48 ± 45

assumption that many chemical time scales are significantly faster than the fluid dynamic scales of interest and can be decoupled. The entire thermochemical state is then represented by a small set of parameters called reaction variables. This system representation by a small set of reaction variables is only valid when the thermochemical state of the system is well-approximated by a manifold in the lower-dimensional space defined by the reaction variables [94]. The concept of a low-dimensional manifold is best explained by considering different reaction trajectories in a high-dimensional state space. Due to fast reactions, these trajectories quickly relax to a low-dimensional attracting manifold governed by the slow reactions. Once the manifold is reached, all reaction trajectories move along the manifold toward equilibrium. The ultimate goal of a manifold identification technique is to represent the chemical and molecular transport processes that control flame structure (the subgrid or microscale physics) in a macroscale simulation. This goal is achieved through parameterization of the state space (r, T, Y1, Y2,…,YS) described by the low-dimensional manifold. A transport equation is then solved on the computational mesh for each of the parameters. A model for all other thermochemical variables as a function of the resolved scale parameters provides the bridge between the resolved and the unresolved scales in the simulation. This model is called a subgrid reaction model and is located at the unit problem level in Fig. 9.

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7.1â•‡ Parameterization of a reacting system The state of a single phase reacting system with NS species requires NS + 1 variables (e.g. NS -1 mass fractions, temperature, and pressure) to uniquely specify the thermochemical state, f, of the system [95, 96]. The reaction model parameterizes f by h, a vector of parameters (reaction variables) of size n, where n < Ns + 1. The reaction model then provides a unique mapping from h to f, i.e. each fi is represented by an Nh-dimensional surface in h-space. Mathematically, the state relationship is written as

f( r, T , Y1 , Y2 ,..., YS ) ≈ f( h1 , h2 ,..., hn ) = f( h)

(44)

Given that the thermochemical state of the physical system is inherently (NS + 1)-dimensional, a unique surface may not exist in the lower-dimensional space parameterized by h. While parameterization of a low-dimensional manifold greatly simplifies the solution of a complex reacting flow by reducing the number of independent variables in the system, the choice of reaction variables is critical. The reaction variables should span both the resolved and subgrid time scales of interest and provide a reasonable representation of the subgrid scale reaction processes. In combustion applications, mixture fraction, f, is widely used as a reaction variable. Mixture fraction is defined as the local ratio of the total mass originating from the fuel stream to the total mass originating from the fuel and the oxidizer streams. For describing nonpremixed systems, mixture fraction is an obvious choice for a reaction parameter since it represents the stoichiometry of the mixture. However, it does not provide any information about the intrinsic state of the system. In the following sections, two different parameterizations are evaluated, one using DNS data [97-100] and the other using experimental data from the International Workshop on Measurement and Computation of Turbulent Nonpremixed Flames (TNF data) [101]. Both parameterizations use the concept of canonical reactors to account for the detailed chemical kinetics and subgrid transport processes. 7.2â•‡ Use of canonical reactors The two components of a reaction model as defined here are the identification of an attracting manifold in thermochemical state space and the parameterization of that manifold. Three canonical reactor models are chosen for manifold extraction: an equilibrium model, a perfectly stirred reactor (PSR) model, and a steady laminar flamelet model (SLFM). Manifolds may also be extracted from other canonical reactors such as a premixed flame reactor, a laminar diffusion flame reactor, or a reactor based on the one-dimensional turbulence model of Kerstein [102], but these reactors will not be discussed further in this chapter. The equilibrium model is based on the assumption that the chemistry is infinitely fast and hence all chemical reactions are in equilibrium. This model ignores any effects of diffusion or of transient flame behavior. The present equilibrium calculations were performed with the CANTERA solver [103], which uses Gibbs free energy minimization to find the equilibrium state. The PSR model is a mathematical approximation to a well-stirred reactor. A PSR Fortran code that predicts the steady-state temperature and species comÂ�positions [104, 105] was used to generate the results shown here. Since the PSR has a flow term the reaction trajectories account for chemical kinetics coupled to flow. The SLFM model is a one-dimensional counterflow flame configuration utilizing a coordinate transformation from physical space to mixture fraction space [106]. This reaction model accounts

102â•… Transport Phenomena in Fires for stoichiometry and diffusion simultaneously, by considering a one-dimensional coordinate in the flame-normal direction. The SLFM calculations were performed with a unity Lewis number assumption. 7.3â•‡ Progress variable parameterization The progress variable parameterization is a two-variable reaction model based on the mixture fraction and hCO2, a progress variable derived from the CO2 mass fraction. The model is generated by reparameterizing the solution to the flamelet equations by ( f, hCO2) instead of the usual parameterization by ( f, c), where c is the scalar dissipation. The advantage of the hCO2 parameterization is that the effects of extinction may be incorporated; parameterization by ( f, c) does not capture extinction because the state variables are discontinuous with respect to c at the steady extinction limit [107]. The flamelet solutions are then tabulated as functions of ( f, hCO2), with hCO2 defined as

hCO2 =

YCO2 - b a- b

,

(45)

where a = max(YCO2 | f) and b = min(YCO2 | f). 7.3.1â•‡ Generation of DNS data DNS data for reaction model validation were obtained from a DNS code that solves the compressible, reacting Navier-Stokes equations using eighth-order explicit finite-differences [108] with a fourth-order Runge-Kutta method in conjunction with a temporal error controller [109]. Mixtureaveraged transport is employed, with transport coefficients calculated from the Chemkin transport package [110]. Further details, including initial and boundary conditions, can be found in [107]. DNS calculations of a spatially evolving, two-dimensional, turbulent CO/H2/N2-air jet flame were used in this parameterization analysis [107]. The fuel stream composition in mole % was 45%CO, 5%H2, and 50% N2 at 300 K and the oxidizer stream was air at 300 K. These streams yield a stoichiometric mixture fraction of fst = 0.437. The kinetic mechanism employed for CO/ H2 oxidation included 12 species and 33 reactions [111, 112]. The mean jet velocity was 50 m/s with a co-flow velocity of 1 m/s. The Reynolds number based on the fuel stream properties (jet width and jet velocity) was 4,600. 7.3.2â•‡ Validation of progress variable parameterization Consider a set of reaction variables, h, used to parameterize the thermochemical state, f, of the system. One may project a DNS data set into h-space and determine a mean surface that the DNS data occupies by 〈f | h〉, the average value of the state variables conditioned on a given set of values of the reaction variables. This Â�concept is illustrated in Fig. 19, where the data points representing realizations of the temperature (f = T) from a DNS dataset are plotted against the mixture fraction (h = f). The thick solid line represents the conditional mean of T in mixture fraction space, 〈T | fâ•›〉, while the thick dashed line represents the temperature obtained if the system was in thermochemical equilibrium. The thin lines in Fig. 19 are explained below. Given the projected data in h-space (points in Fig. 19) and the conditional mean (thick solid line in Fig. 19), the standard deviation of f from its mean in h-space is expressed as

sfi = 〈(fiDNS - 〈 fiDNS | h 〉)2 | h 〉 ,

(46)

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Figure 19: DNS results of CO/H2/N2-air jet flame showing temperature projected into mixture fraction space. DNS data is represented by points and the conditional mean by the thick solid line. Also shown is the equilibrium solution (thick dotted line).

where fi|h represents all values of the ith state variable which correspond to the given value h, and 〈〉 indicates an average. As there may be many points in physical space that have the same h, sfi provides a quantitative measure of the best possible performance a given model parameterized by h can achieve relative to the DNS data and is henceforth referred to as the ideal model performance. The thin solid line in Fig. 19 shows sT as a function of f and provides a measure of the accuracy with which T is parameterized by f. The data deviates from an ideal model by approximately 70 K at f = 0.43, a 4% deviation. The dashed line in Fig. 19 represents the temperature predicted by the equilibrium model, which is a unique function of the mixture fraction for an adiabatic system. The deviation of the DNS data from the surface defined by the model may be defined as sf*i = 〈(fiDNS | h - fi* (h))2 〉 ,

DNS

(47)

where fâ•‰ â•‰ iâ•‰â•¯ â•‰|h is a realization of the DNS data conditioned on a specific value set of h, and f*(h) i is the ith state variable as given by the model. The thin dashed line in Fig. 19 shows the deviation,â•‰ sâ•‰T*â•‰â•‰â•¯, of the equilibrium-predicted temperature from the DNS data. The actual model performance relative to the DNS data is measured by s*fi from eqn (47). Thus, by comparing sfi and s*fi, a quantitative measure of the performance of the given model parameterized by h is obtained. Figure 20 shows the results of an ( f, c) parameterization of temperature for an ideal model generated from the DNS data as well as the SLFM reaction model. ComÂ�paring Figs. 19 and 20, it is clear that the addition of c as a second parameter allows significantly better representation of the data than the one-parameter equilibrium model, with maximum errors of 3% and 9% for the ideal and SLFM models, respectively at fst. However, the SLFM model does deviate from the ideal ( f, c) model at both low and high c. Figure 21 shows the results of an ( f, hCO2) parameterization of temperature for the same DNS and SLFM reaction model data shown in Fig. 20. The progress variable parameterization of the SLFM reaction model performs nearly ideally across the entire range of hCO2. In fact, ideal models based on

104â•… Transport Phenomena in Fires 2000

0.1 0.08

1500 DNS model σDNS σmodel

1000

Normalized σT

T,σT

DNS model

500

0.06 0.04 0.02

0 10−1

100

χo

101

0 10−1

102

100

χo

(a)

101

102

(b)

Figure 20: Parameterization of temperature by ( f, c) for the CO/H2/N2-air jet flame case. Results for (a) temperature and conditional mean and (b) normalized conditional mean from an ideal model (DNS) and the SLFM model (model) are shown.

0.025 1500 T, σT (K)

Model

1000

σDNS σmodel

500

Normalized σT

0.02 DNS

0

DNS Model

0.015 0.01 0.005

0

0.2

0.4

ηCO2

0.6

0.8

1

0

0

0.2

0.4

(a)

ηCO2

0.6

0.8

1

(b)

Figure 21: Parameterization of temperature by (â•›â†œæ¸€å±®f, hCO2) for the CO/H2/N2-air jet flame case. Results for (a) temperature and conditional mean and (b) normalized conditional mean from an ideal model (DNS) and the SLFM model (model) are shown. an ( f, hCO2)-parameterization are consistently able to represent the state variables better than ideal ( f, c)-parameterizations for this jet flame case. 7.4â•‡ Heat loss parameterization The heat loss parameterization is a two-variable reaction model based on the mixture fraction f, and g, a variable derived from enthalpy that represents fractional heat loss. It is defined as: (48)

In eqn (48), ha is the adiabatic enthalpy, h is the absolute enthalpy, Tref is the reference temperature, Tad is the adiabatic temperature, cp is the mixture-averaged specific heat from the adiabatic product composition, and ha, ref is the absolute enthalpy of adiabatic products at the reference

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temperature. The adiabatic enthalpy and temperature are the enthalpy and temperature which would exist if no energy were lost to the surroundings. The numerator is the residual enthalpy. The denominator normalizes the residual enthalpy by the sensible enthalpy of the system. When the heat loss is zero, the system is adiabatic. If heat loss is greater than zero, heat (energy) is lost from the system. If heat loss is less than zero, heat (energy) has entered the system. For unreacted fluid elements with mixture fractions near 0 or 1, the sensible enthalpy of the system is small. As a result, g can become very large near the edges of mixture fraction space. The inclusion of heat loss accounts for changes in the enthalpy of the system due to heat transfer phenomena such as radiation. By representing enthalpy changes with g, enthalpy becomes quasilinearly independent of mixture fraction. This representation also facilitates tabulation of reaction model results for implementation in a CFD code and allows the incorporation of local extinction in the constructed tables. 7.4.1â•‡ TNF data Detailed measurements were taken of a methane jet [113, 114] with a fuel composition of 22.1% CH4, 33.2% H2 and 44.7% N2 by volume. The co-flow consisted of air with 0.8% H2O entering at 292 K. The stoichiometric mixture fraction was fst = 0.167. Measurements of temperature and concentrations of N2, O2, CO, H2, CO2, H2O, OH, CH4 and NO were obtained. Axial profiles (x/d = 2.5 up to x/d = 120) and radial profiles (x/d = 5, 10, 20, 40, 60, 80) of mean and rms values, conditional statistics, and single shot data were taken. Typically, 800-1,000 samples were acquired at each location with uncertainties in the experimental measurements available in the listed references. The experimental flame data was organized into bins of ( f, g). Heat loss was calculated at each data point using eqn (48). To compute the sensible enthalpy, the adiabatic composition was obtained from an adiabatic equilibrium calculation at a reference temperature of 273.15 K. Then, each data point was placed into a bin that was characterized by an ( f, g) pair of values. The validity of the parameterization proposed in eqn (44) is assessed using this TNF data table. 7.4.2â•‡ Validation of heat loss parameterization In order to use an ( f, g) parameterization, heat loss must be present in the canonical reactor model. For the equilibrium model, heat loss was incorporated by varying the composition and enthalpy of the initial CH4/H2/N2-air mixture. For the PSR reactor, model reactor solutions were obtained for a range of mixtures (defined by the inlet equivalence ratio) at various normalized heat loss values by including heat loss from the reactor in the calculation. The volume of the reactor for the CH4/H2/N2-air case was 67.4 cm3 and the residence time was specified as 0.003 s. For the SLFM model with a unity Lewis number assumption, the adiabatic profile for enthalpy is a line connecting the enthalpy of the fuel and oxidizer streams, a direct consequence of enthalpy being a conserved scalar. To incorporate heat loss effects into the SLFM reactor model, the heat loss as defined in eqn (48) was adopted. First, the adiabatic solution was computed followed by the computation of the denominator in eqn (48). Next, the enthalpy profile was computed given a constant value of heat loss. The species flamelet equations were then solved, with temperature computed from enthalpy and composition computed using a one-equation Newton’s method. The maximum scalar dissipation rate was set at 20 s-1 since in buoyancy-driven flames, the scalar dissipation rate is low and does not vary much through the flow field. The reaction model results presented here are based on the species, thermodynamics, and detailed kinetics found in the GRI3.0 scheme, but similar results could also be obtained using the surrogate JP-8 kinetic mechanism described in Section 5.

106â•… Transport Phenomena in Fires Figure 22 shows temperature and species concentrations conditioned on various values of heat loss and plotted in mixture fraction space for the CH4/H2/N2 flame. While this flame was close to adiabatic conditions, a realizable heat loss ranging from -0.02 to 0.09 was identified [115]. These plots include the experimental data along with the results from the three canonical systems described previously. Qualitatively, the temperature manifold (Fig. 22(a)) and those of the major species, including CO2 and H2O (Fig. 22(b)), are well-represented by the PSR and SLFM reaction models [115]. Reasonable predictions for some minor species such as OH, seen in Fig. 22(c), are also achieved using the nonequilibrium models. The prediction of other minor species, including NO, could be improved with the addition of a third parameter. With this ( f,g) parameterization, qualitative analysis reveals that the nonadiabatic equilibrium calculations match the experimental data only in the lean region; significant deviations from equilibrium are noted in the near stoichiometric and rich regions of the flame. Thus, the performance of the equilibrium model relative to the TNF flame data is inferior to that of both the PSR and SLFM reaction models. Quantitative validation, although not yet completed, requires that an appropriate validation metric be applied to the results obtained from all three canonical reactors for all φi measured experimentally.

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7.5â•‡ Soot models An essential aspect of parameterization by mixture fraction in the three canonical reactor models discussed above (equilibrium, SLFM, PSR) is that all species diffuse at the same rate. However, soot is the product of a relatively slow reaction, is not in equilibrium and does not diffuse at the same rate as the molecular species. Hence, soot is not expected to correlate well with mixture fraction. Attempts to correlate the soot volume fraction with mixture fraction in calculations of turbulent diffusion flames have been carried out previously with limited success [116]. From measurements carried out in co-flow diffusion flames, Kennedy et al. [117] modeled the nucleation rate as a function of mixture fraction alone and showed the surface growth process to be the controlling mechanism in determining total soot volume fractions. A second complication presented by soot is that in strongly sooting flames, the soot can significantly alter the flame chemistry. It acts as a sink for important species such as OH and C2H2 and as a source for CO during oxidation. It also alters the heat release profile through radiative heat loss. Due to this bidirectional coupling between the soot field and the flame field, it cannot be effectively postprocessed on established flame fields as has been done with other pollutants such as NOx. Currently, there are two approaches to modeling soot formation in a multiscale fire simulation. The first approach is to solve transport equations on the computational mesh for the variables of interest in the chosen soot model. For example, if using the Lindstedt soot model [39], transport equations need to be solved for the soot volume fraction and the soot particle number density. The second approach is to include the soot formation and oxidation processes in the subgrid scale reaction model, and then parameterize these slower processes with an additional ‘time’ parameter. For example, in the SLFM approach, the slow processes such as NOx or soot formation are not accurately captured because the flamelet equations are solved to steady state [118]. To alleviate this shortcoming, the flamelet equations can be solved in unsteady form using time as an additional parameter. The transient flamelet may be thought of as moving through the computational mesh in a Lagrangian sense. Pitsch et al. [119] linked the flamelet time to axial position in a jet based on the axial jet velocity and then performed a numerical simulation of soot formation in a turbulent C2H4 jet diffusion flame. In the progress variable approach, a scalar (or combination of scalars) that correlates monotonically with the subgrid flamelet time is employed as the ‘time’ parameter and transported on the computational mesh. This approach was first employed by Desam and Smith [120] to study NOx formation in turbulent nonpremixed jet flames.

8â•‡ Turbulence/chemistry interactions Transportation fires are characterized by interactions between the length and time scales of the turbulent transport processes and the chemical reactions. These length and time scales may or may not overlap, as illustrated in Fig. 23. In this figure, the ‘mixing time scale’ refers to the time scales of the turbulent transport processes while the ‘chemistry time scale’ refers to the time scales of the reactions in the kinetic mechanism. The axis in Fig. 23 represents the time and length scales of the fire physics with the smallest scales on the left and the largest scales on the right. The scales resolved on the CFD mesh, the ‘macromixing’ region, represent only a small subset of the scales present in the fire. The ‘micromixing’ region is characterized by subgrid scale mixing phenomena and turbulence/chemistry interactions that are unresolved on the computational mesh. The LES filter scale is the boundary between these two regions. Subgrid scale models must appropriately account for these complex coupled interactions at the unresolved scale.

108â•… Transport Phenomena in Fires

Figure 23: Length and time scales of turbulent transport processes and chemical reactions. These subgrid interactions influence chemical source terms in scalar transport equations and the distribution of gas phase species and soot in the fire. A mixing model (represented by the ‘subgrid mixing model’ block at the unit problems level of the V&V hierarchy) accounts for scalar micromixing, which is the subgrid variation of the scalar field from the mean scalar value transported on the mesh, by describing the statistical distribution of the subgrid scalar field. If the joint PDF of a set of scalars j = (j1,j2,…,jn) is known, the mean value of any function of these scalars can be calculated as

(49)

where P(j1,…,jn) is the joint PDF of (j1,…,jn). Models which describe the full joint PDF of j are known as direct or transported PDF methods [121, 122]. Direct PDF methods are often used in the simulation of turbulent flows where many chemical degrees of freedom are incorporated [123, 124], although difficulties arise in modeling the diffusion terms in the PDF transport equations. Recently, Fox and coworkers have proposed the finite-mode PDF or multi-environment PDF model [125, 126]. This model is based on discretizing the joint PDF into a small number of environments or modes and then solving transport equations for the scalar concentrations in each environment along with the probability of each environment. Higher order statistics are incorporated by increasing the number of modes that are transported. In this way, joint PDFs may be discretely approximated and chemical source terms closed directly. Analogous to direct PDF methods, the primary difficulty in the multi-environment PDF approach lies in modeling the diffusion between environments.

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8.1â•‡ Validation of presumed PDF models in nonpremixed flames An alternate approach to direct PDF methods is a class of models, presumed PDF models, where the shape of the PDF is prescribed. These models represent an approximation to eqn (49). The presumed functions are typically continuous, which implies that the presumed PDF represents all statistical moments of the variable. Presumed PDF models offer significant advantages over direct PDF methods, primarily because of their relative ease of implementation into existing CFD codes. One disadvantage is that joint composition PDFs of all the reaction model parameters are not easily presumed. As a result, statistical independence is often assumed for reaction models with several parameters,

P(j1 ,..., jn ) ≈ P(j1 ) P(jn ),

(50)

where P(j1,...,jn) is the joint PDF of (j1,...,jn) and P(j1) is the PDF of (j1). With this assumption of statistical independence, the joint PDF of the reaction model parameters is represented as a product of conditional and marginal PDFs. Then, eqn (49) becomes

(51)

Despite its limitations, this class of models is widely used. Fortunately, many reaction models currently in use have only a few parameters which are often not strongly correlated. The issue of parameter independence in combustion systems was evaluated using TNF workshop data for a CO/H2/N2-air flame, a CH4/H2/N2-air flame, and a piloted CH4-air flame [127]. Two models for the joint PDF of a reaction model parameterized by ( f, g) were considered. One model assumes that the parameters are independent and that the marginal PDF of heat loss is a delta function. The other model assumes that the conditional PDF of heat loss conditioned on mixture fraction is a delta function. Both models employ a marginal mixture fraction PDF. Figure 24 shows temperature plots of the piloted CH4-air flame comparing presumed PDF model average values to experimental average values. Both PDF models use a clipped Gaussian mixture fraction PDF. For the data labeled ‘Marg. PDF’, the marginal heat loss PDF is assumed to be a delta function. For the data labeled ‘Cond. PDF’, the conditional PDF of heat loss conditioned on mixture fraction is a delta function. The plots include data from a third model, the mean value model, which assumes zero variance in heat loss and mixture fraction. Additional plots from the three flames for all measured species (N2, O2, CH4, CO, H2, CO2, H2O, OH, and NO) and temperature are found in [127]. Overall, the delta conditional heat loss PDF model predicts the mean scalar values better than the delta marginal heat loss PDF model, although application of an appropriate metric is needed to quantify the differences. The assumption that the conditional PDF of heat loss is a delta function ensures that integration occurs over all realizable space. However, the conditional PDF model does require knowledge of the conditional expectation of heat loss. A proposed shape for this function can be found in [127]. The marginal PDF model assumes that f and g are statistically independent, resulting in integration over a constant heat loss for all mixture fractions. The experimental data is not realizable for all points in f / g space, so a normalization is performed when integrating over any nonrealizable space. This normalization prevents accurate prediction of O2 and N2. The mean value model predictions are good only in regions far downstream in the flame where mixing of the fuel and air streams has occurred. The assumptions of the mean value model are poor in the near jet region where mixing is not complete.

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Figure 24: Temperature plots in a piloted CH4-air flame comparing presumed PDF model average values to experimental average values. The plots are at (a) h/D = 7.5 cm, (b) h/D = 15 cm, (c) h/D = 30 cm, (d) h/D = 45 cm, (e) h/D = 60 cm, and (f) h/D = 75 cm, where h is the height above the burner and D is the diameter of the orifice. 8.2â•‡ Shape of presumed PDF Two different presumed shapes for P(j) were considered for the pool fire simulations: the b-PDF [95, 128, 129] PDF [107, 130, 131]. These PDFs are parameterized by __ and the clipped-Gaussian 2 the mean (â•‰jâ•‰)â•¯ and variance (â•‰sâ•‰jâ•‰â•¯â•‰) of the variable__j. Given the LES formulation of the governing equations, variables transported on the mesh (â•‰jâ•‰â•¯) are implicitly filtered. Additionally, because of the variable-density nature of the flows being simulated, the Favre-filtered form of the governing equations (Section 6) is used. To compute PDF shape, the LES must supply both the Favre-filtered variable and its variance. A transport equation is typically evolved for the Favre-filtered variable, while the variance may

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be modeled in several ways [132]. The LES algorithm described in Section 6 employs a scale similarity model [128], assuming that the small-scale statistics can be inferred from the resolved scale structures in the flow. Using the standard definition of the variance, the mixture fraction variance is modeled as  s 2f = C ( f 2 - f 2 ), (52) where f is the Favre-filtered mixture fraction and the coefficient C = 0.5 [133]. Lacking a true mean as required by eqn (52), the filter is used as an approximation to the mean and then multiplied by the model parameter. By construction, the presumed PDF for j matches the mean and variance of j. For variables which range from 0 to 1, the maximum variance is given by (53) sj2,max = j (1 - j ). At maximum variance, both the b and clipped-Gaussian PDFs reduce to appropriately weighted d -functions at j = 0 and j = 1, P (j) = (1 - j )d (j) + (j )d (j - 1), sj2 = sj2,max . __ Likewise, at zero variance, they become a single -function at j = jâ•‰ â•‰ ,â•¯

(54)

P (j) = d (j - j ), sj2 = 0. (55) Both the b -PDF and clipped-Gaussian PDF become singular at zero and at maximum variances [107], but their properties (eqns (54) and (55)) insure that the PDF does not need to be constructed or integrated at these limits. Nevertheless, inteÂ�gration of the b -PDF can be very difficult (and inaccurate) when the variance is near its maximum, even when using integration schemes designed for singular functions. The clipped-Gaussian PDF creates no integration difficulties at high variances because the singularities are treated directly with a0 and a1 [107], making the clipped-Gaussian PDF easier and computationally cheaper to integrate than the b -PDF.

9â•‡ Radiative heat transfer model Radiation, the dominant mode of heat transfer in hydrocarbon fires, is incorporated in the V&V hierarchy at the unit problem level in Fig. 9. With the advent of massively parallel computers, performing realistic computations of participating media radiative transfer is increasingly tractable. In order to spatially resolve the important flow characteristics in a fire, grids containing 106-108 computational cells are used at every time step associated with the calculation. Parallelization of the radiation calculations by decomposing the radiation solution in spatial, angular, or energy domains is essential. A finite volume-based discrete ordinates radiation model that is decomposed in the spatial domain is employed. The inputs to this model are gas temperature and the concentrations of the radiatively active species (CO2, H2O, soot), which are calculated on the spatially decomposed flow grid as well as at the boundaries. The adoption of a spatial decomposition strategy for the radiation component allows easy integration with other components in the LES fire simulation tool. 9.1â•‡ Discrete ordinates method The discrete ordinates method is based on the numerical solution of the radiative transport equation (RTE) along specified directions. The total solid angle about a location is divided into a

112â•… Transport Phenomena in Fires number of ordinate directions, each assumed to have uniform intensity. Each transport equation that is solved corresponds to an ordinate direction selected from an angular quadrature set that discretizes the unit sphere and describes the variation of directional intensity throughout the domain. If zm, μm, and hm represent the direction of cosines associated with each ordinate direction, k represents the absorption coefficient and Ib represents the black body emissive power, then the differential equation governing the discrete ordinates method in the absence of scattering can be written for each direction m as [134],

zm

∂I m ∂I ∂I + mm m + hm m = - kI m + kI b . ∂x ∂y ∂z

(56)

The boundary condition associated with the eqn (56), considering the surrounding surfaces to be black, is

Im = Ib .

(57)

If the absorption coefficient and temperature within the domain and at the boundaries are specified, eqn (56) can be iteratively solved for the directional intensities (Im) throughout the domain for each direction associated with the discrete ordinates method. The variables of interest in most radiative transfer analyses are the distributions of radiative heat flux vectors (q(r)) and the radiative source terms (-∇ · q(r)). The radiative source term describes the conservation of radiative energy within a control volume and is a source term in the total energy equation, thereby coupling radiation with the other physical processes that occur in a multi-physics application. Both of these variables are direction-integrated quantities and are readily determined once the distributions of directional intensities (Im) within the domain are known [135]. When using the discrete ordinates method, integrations over solid angles to obtain q(r) and -∇ · q(r) are replaced by a quadrature of order n and an appropriate angular weight (wm) associated with each direction, m. The number of equations to be solved depends on the order of approximation, n, used. In the work described here, n = 4 (the S4 approximation). The discrete ordinates method is spatially decomposed to solve the RTE on parallel computers [136]. Mathematical libraries of robust, scalable, nonlinear and linear solvers developed by third parties are used to solve the matrices that result during the solution procedure [85]. The domain boundaries are assumed to be black walls at a temperature of 293 K. 9.2â•‡ Radiative properties In order to solve for the intensities (Im) for each direction associated with the discrete ordinates method (eqn (56)), radiative properties throughout the computational domain must be specified. It is also desirable to solve the RTE in a limited number of spectral intervals or bands in the interest of computational efficiency. Therefore, radiative property models must be selected that are appropriate for the conditions encountered in a transportation fuel pool fire, divide the spectrum of interest into a limited number of spectral intervals and provide averaged or spectrally integrated radiative properties at each interval or band. The algorithm described here requires the radiative properties in the form of an absorption coefficient. Absorption coefficients may be extracted from total or averaged transmissivity or emissivity data using Beer-Lambert’s law after specification of path length or mean beam length. However, the specification of path lengths/mean beam lengths is difficult in buoyant pool fires due to the ‘puffing’ phenomenon exhibited by such fires [137]. Also, Beer-Lambert’s

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law is not valid for an absorption coefficient that has been averaged over many spectral lines. Estimating an absorption coefficient by using a single path length and Beer-Lambert’s law for the entire spatial field results in significant error in radiative field solutions [138, 139]. Nevertheless, it is difficult to implement more rigorous procedures within the domain decomposition strategy employed here and therefore all absorption coefficients are computed using a single path length. The gray model property model that has been implemented employs total emissivity data to compute absorption coefficients. The total emissivity of CO2-H2O gas mixtures is first determined from a series of curve fit relations from Hottel charts for low temperature flames (300 K < T < 1,200 K), a weighted-sum-of-gray-gases model proposed by Coppalle and Vervisch [140] for high temperature flames (2,000 K < T < 3,000 K), and a linear interpolation between the two regimes at intermediate temperatures. Total absorption coefficients are then extracted from the total emissivity data after specification of a mean beam length. Details of this property model may be found in Adams [141]. The correlation of Sarofim and Hottel [63] for the emissivity of a sooting flame is employed to estimate the absorption coefficient of soot:

ksoot =

4 ln(1 + 350 fvTLe ), Lc

(58)

where fv is the soot volume fraction, T is the gas or soot temperature in Kelvin, and Le is the mean beam length. To determine non-gray properties, the spectral region of interest (50 to 10,000 cm-1) is divided into a number of intervals (width ≈ 25 cm-1) and spectral optical depths are determined at each interval employing a narrow band model (RADCAL) [5]. An average absorption coefficient (kh) corresponding to each interval is then obtained by dividing the spectral optical depth by a path length (L). The entire spectrum is then divided into six bands and the average absorption coefficients within each band (h) are lumped together to yield a patch mean absorption coefficient for that band according to the equation (59)

This strategy is similar to that employed by Hostikka et al. [74] for performing radiation calculations in an LES fire simulation except that a Planck mean absorption coefficient was evaluated and employed in their calculations. Krishnamoorthy et al. [142] showed the advantages of employing a Patch mean absorption coefficient over a Planck mean coefficient in comparisons against non-gray benchmark problems. The evaluation of absorption coefficients from the gray and non-gray models requires the specification of a path length or mean beam length. One-tenth of the mean beam length of the computational domain is taken as the path length by Hostikka et al. [74] in their pool fire simulations and is the mean beam length/path length used here. 9.3â•‡ Algorithm verification One case used for radiation model verification is the nonhomogeneous medium benchmark introduced by Hsu and Farmer [143]. The problem consists of an isothermal unit cube with cold black walls. The interior of the cube consists of a gray, non-scattering, absorbing/emitting

114â•… Transport Phenomena in Fires material with an optical thickness (t = absorption coefficient times the side length) distribution given by

 | x |  | y |  | z |  t( x, y, z ) = 0.9  1 11+ 0.1  0.5   0.5   0.5 

(60)

A uniform black body emissive power of unity within the domain defines the distribution of temperature. Since the radiative properties, temperature, and boundary conditions for this problem are known, the RTE can be solved to determine the distributions of the radiative fluxes and the radiative flux divergence. The root mean square error norms, also known as the L2 error norms, of both radiative flux and radiative flux divergence are shown in Fig. 25. The spherical surface symmetrical equal dividing angular quadrature scheme (SSD) [144] was employed to calculate the numerical solution accuracies plotted in Fig. 25. The results obtained by Burns and Christon [145] using the rotated LC quadrature scheme are also shown in Fig. 25

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Figure 25: Numerical accuracy of quadrature schemes as a function of spatial and angular resolution: (a) predicted radiative flux divergence along (x, 0, 0); (b) predicted radiative heat flux along (x, 0.5, 0).

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(open symbols). The number of equations that need to be solved with the SSD1a, SSD2a, and SSD3b schemes are exactly the same as those of the rotated LC4, LC6, and LC8 quadrature sets, respectively, enabling a direct comparison of the solution accuracies of the two schemes when the same number of equations is being solved. In general, the two schemes perform equally well with error norms decreasing as spatial and angular resolution increases.

10â•‡ Heat transfer to an embedded object in a JP-8 pool fire The goal of this work is to calculate the potential hazard of an explosive device immersed in a pool fire of transportation fuel. We characterize the hazard in terms of the time to ignition of the device and the violence (measured as kinetic energy of the exploded container) of the event. To accomplish this goal, a fire simulation tool for performing scalable, parallel, three-dimensional simulations of a large-scale pool fire with an embedded device has been developed. This simulation tool incorporates all the fire physics components at the unit problem level of the V&V hierarchy in Fig. 9 to accurately represent the heat transfer to the device. Coupling of this fire simulation tool with an energetics material model to predict time to ignition of an explosive device is discussed in Section 12. 10.1â•‡ Modified LES algorithm The LES equations (Section 6) are modified to account for the presence of a steel-shelled container of explosive material (PBX, HMX) in the computational domain. A law of the wall approximation [146] is used for the boundary condition for the momentum transport equation. Because radiation is the dominant mode of heat transfer in heavily sooting pool fires, radiative heat transfer between the solid and the fire is modeled in the enthalpy transport equation while convection heat transfer is neglected [147]. For the solid wall boundary conditions, the wall is considered as a black body radiating at its own temperature. The solid object heats up, so the boundary condition for the fire is time varying. The turbulent conductivity is modeled in a manner similar to the turbulent diffusivity as discussed in Section 6. The solid is modeled with the ‘material-point’ method (MPM) [148, 149], which uses material (mass) points to represent the solid and calculates stresses and heat conduction within the solid using interpolation by basis functions. The equations for the fire in the presence of an object are discretized using a finite-volume scheme, as described in Section 6. Additional details about the MPM algorithm are found in [148, 149]. 10.2â•‡ Coupling between LES fire phase and container heat-up phase Because of the wide range of time scales of the complete system (intended use) case, the simulation is decomposed into three distinct phases. For the first phase, the dynamic LES fire simulation is performed to determine a steady heat flux profile to the device. This profile is generally not symmetric and depends on such variables as the crosswind velocity, the size of the pool, and the placement of the device. This phase is characterized by simulated time scales of O(1-10 s). In the second phase of the calculation, the heat-up phase, the fire simulation is frozen. Steady heat flux values from the fire phase are applied to an MPM object representing the device embedded in or near the fire. As this phase develops, the steel shell and the explosive material heat up, with the two materials represented by a single temperature field. This phase, with time scales

116â•… Transport Phenomena in Fires of O(10 s-10 min), is continued until the explosive’s ignition criterion is reached. The third phase, the explosion phase, begins at the ignition point. The explosion phase is characterized by time scales of O(10-9-10-3 s) and represents the container breakup and the expulsion of the explosive. A second simulation decomposition strategy was also tested. In this strategy, the first phase proceeds as described above. The steady heat fluxes from the fire phase are then fed to a series of one-dimensional calculations performed in the radial direction of the cylindrical object. The onedimensional calculations compute heat transfer and pressurization along the radial direction until the ignition point of the explosive is reached, at which point the simulation terminates. This strategy does not include the details of the exploding container. 10.3â•‡ Subsystem cases: heat transfer in a large JP-8 pool fire Data sets obtained at the subsystem level of the V&V hierarchy (Fig. 9) are limited due to harsh experimental conditions and high cost, and the errors associated with such measurements are large. Nevertheless, even limited data is useful for achieving some level of validation and error quantification, particularly since the subsystem cases include the coupling of multiple physical processes and closely mimic the intended use. Here, two experimental data sets are used in a validation exercise for the LES fire phase. These data sets include heat flux measurements made at various locations in and near large JP-8 pool fires. This validation exercise is conducted using the validation metric discussed in Section 3.2. 10.3.1â•‡ Validation data sets Two experiments have been identified for subsystem validation purposes. The first experiment was conducted by Kramer et al. [150] at the Sandia National Laboratories Burn Site. The experiment was intended to measure heat fluxes from a circular JP-8 pool fire (7.16 m diameter) to a large calorimeter (4.6 m length, 1.2 m diameter, 2.54 cm wall thickness) suspended directly over the pool. After the pool was ignited, temperatures were recorded for 30 min from thermocouples fixed at various axial and azimuthal locations inside the calorimeter. From the interior thermocouple data, heat flux measurements to the outside surface of the calorimeter were deduced using the Sandia One-Dimensional Direct and Inverse Thermal (SODDIT) code [151]. In an effort to reduce wind effects, a circular wind fence (24.4 m diameter) was constructed around the fire. Wind direction and speed were measured outside the wind fence. The average wind speed was 1 m/s with a primary direction normal to the axis of the calorimeter. Despite the wind fence, the fire was observed to lean in the primary wind direction. The second experiment was conducted by Blanchat et al. [152] at the Sandia National Laboratories Burn Site to provide well-characterized environmental information relative to an open pool fire with embedded, weapon-sized calorimeters. The circular pool of JP-8 fuel measured 7.9 m. Details regarding the experimental setup can be found in [152]. Four separate tests were performed on different days with different measured wind speeds and different calorimeters. Here, the focus is on Experiment #1, wherein two small calorimeters (0.3 m diameter, 0.4 m long) were positioned over the pool at radii of 1.5 m and 2.5 m and the winds were characterized as being calm (0-2.2 m/s) in a direction normal to the axis of the calorimeters. Heat flux gauges were positioned near the ground with one gauge in the center and the remaining 48 gauges in concentric circles spaced 1 m apart along eight radial directions of the pool. No wind fence was used in this experiment; wind speeds were measured at various positions around the pool.

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10.3.2â•‡ Simulation details and results Two JP-8 pool fire simulations were performed. In the first simulation, the large calorimeter was suspended over the pool in the same configuration as [150]. The second simulation was the same configuration as the first with the exception that it did not include the calorimeter. Both simulations were run on 448 processors in a 20 m × 17 m × 20 m rectangular domain with a resolution of 200 × 170 × 200. The x-axis was taken as the vertical direction. A fuel inlet with diameter 7.16 m representing the pool surface was included on the -x face while the remaining -x face was modeled with a wall boundary condition. Fuel was introduced into the domain based on a fuel regression rate of 1.6 mm/min. On the -y vertical boundary, an inlet boundary condition was used to model the crosswind and was set to 1 m/s for both simulations. The opposing vertical side (+y) and the top of the domain (+x) were modeled with an outlet boundary condition. The remaining vertical sides (-z and +z) had pressure boundary conditions. The entire flow was initially quiescent and fuel was introduced after the simulation began. Both simulations were run until the time-averaged heat fluxes became steady. For the first simulation, heat fluxes were extracted at different axial locations on the calorimeter surface around the azimuthal direction corresponding to the thermocouple locations in the first experiment described above. For the second simulation, heat fluxes were extracted from the pool surface corresponding to the pool surface heat flux gauges of the second experiment described above. From the first simulation, azimuthal heat flux values at a location 1.96 m down the large calorimeter are presented in Fig. 26. Also shown are the experimental results with 90% confidence intervals for the mean data and the estimated error. Since only one experiment was performed, the first 10 min of the data were split into three equal parts to represent three data sets. The positions of π/2 and 3π/2 correspond to the top and bottom of the calorimeter, respectively. The windward side of the calorimeter corresponds to the π position and the leeward side to the 0 position. In Fig. 26(a), the simulation data lie within the experimental confidence intervals except for the lower half of the cylinder on the windward side where the simulation underpredicts the heat flux. The size of the confidence intervals is a strong function of the wind, even with the wind fence present. That is, the heat flux is varying wildly within the first 10 min, creating a large range in which the true mean heat flux could reside. This effect is particularly noticeable in the region of highest heat flux to the calorimeter (position π/2, bottom of the device). In Fig. 26(b), the estimated error is plotted with 90% confidence intervals. Again, the error is large for the lower half of the calorimeter on the windward side. However, the largest error range occurs in the region of highest heat flux to the calorimeter (position π/2, bottom of the device). For the second simulation, results of simulated heat fluxes to the pool surface are compared with the experimental data in Fig. 27. As with the previous data set, the temporal heat flux data were separated into four segments of equal time. The results are presented as a function of the gauge id number. Gauge #1 corresponds to the center location of the pool. Gauges #2-#9 correspond to the first ring and so on. Note that because the diameters of the simulated fire and the experimental fire were slightly different, the gauges corresponding to the 300 series from Blanchat et al. [152] are not included in this comparison. This second data set is better characterized, resulting in smaller bands for the 90% confidence interval, and most simulation data points lie within the confidence interval. Two points of higher heat flux are predicted by the simulation for Gauges #5 and #6, which correspond to the windward side of the fire. These higher simulation heat fluxes may result from holding the wind speed constant at 1m/s when the experimental wind speed varied up to 2.2 m/s. Higher wind speed results in a higher tilt to the fire, lowering heat fluxes to the windward side. From the simulation results, global metric values of the average relative error metric plus/minus the average relative confidence indicator

118â•… Transport Phenomena in Fires

(a)

(b)

Figure 26:â•‡Heat flux results for simulation 1 compared to experimental data at the 1.96 m slice of the large calorimeter. (a) Experimental data with a 90% confidence interval (Exp. Mean) and simulation mean (Comp. Data). (b) Estimated simulation error with a 90% confidence interval.

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(a)

(b)

Figure 27:â•‡Heat flux results at the pool surface for simulation 2 compared to heat flux gauge data at corresponding locations. (a) Experimental data with a 90% confidence interval (Exp. Mean) and simulation mean (Comp. Data). (b) Estimated simulation error with a 90% confidence interval.

120â•… Transport Phenomena in Fires are 11% ± 22%. In other words, the average relative error ranges from 0% to 33% with a 90% confidence.

11â•‡Prediction of heat flux to an explosive device in a JP-8 pool fire As stated in Section 3.1, the motivation for this work is to develop a simulation tool with the intended application of predicting heat transfer to an object in a large-scale transportation fire. While such scenarios are worth studying experimentally for hazard classification reasons, they remain expensive and dangerous to perform. Thus, a simulation tool built on a hierarchy of validation becomes one potential solution for negating the costs and risks associated with performing the experiment. This section focuses on the prediction of heat flux to a rocket motor in a large-scale (10-20 m) JP-8 fire for transportation hazard classification. Both the Department of Transportation (DOT) and the Department of Defense (DoD) have established testing protocols that include an external bonfire test. The DOT external fire test calls for the explosive article to be placed on a noncombustible surface (steel grate) above a fuel source of wood soaked with diesel fuel or equivalent. The fire is ignited and allowed to burn for 30 min while the material is observed for evidence of detonation, explosion, etc. [153]. The DoD testing protocol requires that the test specimen be surrounded by fuel rich flames from a large open hearth containing liquid fuel such that the heat transfer to the specimen is approximately 90% radiative. Wind speeds should not exceed 5.8 m/s [154]. Simulations of a full scale bonfire test of an explosive device under wind conditions allowable under the DoD testing protocol were performed using the LES fire simulation tool described in this chapter. One objective of the simulation was to determine if higher wind speeds allowed within the DoD protocol affected the engulfment of the rocket motor in the fire, resulting in a scenario that would not qualify under the current DoD regulations of full fire engulfment. The explosive device was represented by a 1.2 m diameter, 8 m long cylindrical steel container. The container was suspended 1 m above a 24 m × 13 m rectangular pool of JP-8 fuel. The five-component JP-8 surrogate formulation proposed by Zhang et al. [36] was used for all calculations. Simulations were run at two different wind speeds, 2.2 and 5.8 m/s, the upper limit of the testing protocol. The 5.8 m/s crosswind case was run on a 30 m × 60 m × 60 m domain with a mesh resolution of 100 × 180 × 180. The case was run on 196 processors of a massively parallel machine at Lawrence Livermore National Laboratory (LLNL). The 2.2 m/s crosswind case was run on 324 processors at LLNL on a 30 m × 30 m × 60 m domain with a mesh resolution of 150 × 150 × 220. Volume-rendered images of the temperature field at one time slice are shown in Fig. 28 for both cases. The device is not fully engulfed in the flames in either case, but in the 5.8 m/s wind condition, the fire is blown away from the container. Figure 29 shows the volume-rendered temperature field in the 2.2 m/s crosswind case from a different angle at a later time. The region of highest heat flux to the container is at a location exposed to radiation from the leeward side of the fire. As with the calorimeter experiments in Section 11, the wind speed significantly influences the azimuthal heat flux profile of the device. Table 5 lists mean heat fluxes obtained from the simulation at various locations on the surface of the device. At the lower crosswind speed (2.2 m/s), the device acts as a flame holder, leading to heat fluxes near the top of the container (position p/2) that equal or exceed those at the bottom. For the 5.8 m/s crosswind case, the heat flux at the top of the device is two orders of magnitude smaller than the heat flux at the bottom of the container. At this higher wind speed, the flame is still burning under the device but leans away from the top of the device, producing the large variation in heat flux between the top and bottom of the device.

Heat Transfer to Objects in Pool Firesâ•…

(a)

121

(b)

Figure 28:â•‡Volume-rendered images of the temperature field in the JP-8 pool fire: (a) 2.2 m/s crosswind and (b) 5.8 m/s crosswind.

Figure 29:â•‡Volume-rendered image of the temperature field in the JP-8 pool fire with a 2.2 m/s crosswind. Side view showing region of highest temperature on upper leeward side of the container surface. The low heat fluxes at the top of the device are a clear indication that the container is not even partially engulfed in the fire. Information from the LES simulation about flame location and shape, heat flux to the explosive device, and rate of device heat-up can be used to establish acceptable operating conditions

122â•… Transport Phenomena in Fires Table 5:â•‡Mean heat fluxes to the explosive device obtained from simulations at two different wind speeds.

Wind speed = 2.2 m/s

Wind speed = 5.8 m/s

Axial location

Position = p/2

Position = 3p/2

Position = p/2

Position = 3p/2

4 m 6 m

110.1 kW/m2 68.6 kW/m2

82.3 kW/m2 84.6 kW/m2

1.8 kW/m2 0.7 kW/m2

77.5 kW/m2 92.8 kW/m2

for the hazard classification bonfire test. At the time of this simulation, no data for this particular scenario existed. Thus, the LES tool was used in a predictive manner. Error bars associated with the results from these bonfire simulations must be inferred from lower hierarchical validation exercises, resulting in the qualitative statements made above regarding the effect of the wind on the flame shape and heat flux characteristics. It is recognized that for many scenarios, these types of qualitative statements are unacceptable. Indeed, in high consequence scenarios, the most valuable predictive simulation results will have quantified uncertainty. While such a simulation requirement should be considered, it is not a straightforward proposition as it involves an understanding of how errors propagate in a nonlinear fashion through the V&V hierarchy. Error quantification for multiphysics, multiscale simulations is further addressed in Section 13.

12â•‡Predicting the potential hazard of an explosive device Â�immersed in a JP-8 pool fire Ultimately, we are interested in calculating the potential hazard of an explosive device engulfed in a pool fire of transportation fuel. One metric for potential hazard is the time to explosion. This section describes two methods for computing time to explosion using heat flux data from the LES fire simulation tool (see Section 10.2). The first method represents the explosive device as a three-dimensional MPM object during the heat-up and explosion phases. With this method, the large deformations caused by the device breakup are captured on the computational mesh. The second method approximates heat transfer in the explosive device with a one-dimensional model that incorporates high fidelity reaction kinetics. Both methods simulate the response of an energetic material (HMX or PBX) in a fast cook-off environment. Here, fast cook-off is defined as ignition under confinement with the energetic material exposed to high heat fluxes. Fast cook-off is a surface phenomenon. Because the thermal conductivity of HMX is very low, large temperature gradients exist within the explosive. Only a thin layer of explosive next to the inner wall of the container experiences temperature increases high enough for chemical decomposition reactions to occur. In fact, the reaction zone is likely to occur in the region where the explosive is sandwiched next to the container wall [155]. For the purposes of this section, fast cook-off occurs when the energetic material is exposed to heat fluxes in the range of 1-100 kW/ m2, a typical range for transportation fuel pool fires. 12.1â•‡ Three-dimensional heat transfer, PBX combustion model The three-dimensional heat transfer model uses the MPM [148] infrastructure as noted in Section 10.2. Because of the potentially long time to ignition, an implicit time integration

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strategy is used to eliminate stability restrictions on the timestep [149]. A single temperature field is computed for the steel and PBX, an assumption which ignores any potential gap formation due to differential thermal expansion or pressurization due to decomposition of the explosive. Heat fluxes to the container surface obtained from the fire simulation are fit to high order polynomials, which are in turn sampled at particle locations around the surface of the container and treated as source terms in the solution of the energy equations. Once the heat-up phase reaches a preset ignition temperature, the implicit MPM code transfers the data to the explicit MPMICE code [155] for the explosion; pressurization does occur in the explosion phase. The combustion model for PBX [156] in the MPMICE code is based on a simplified two-step chemical reaction scheme introduced by Ward et al. [157] in which the solid propellant is initially converted to gas phase intermediates in a thermally activated, moderately exothermic zero-order reaction; the intermediates then react to form final products in a highly exothermic, bimolecular flame reaction having zero activation energy. As the pressure increases, the increase in rate of the second reaction moves the flame closer to the propellant surface, increasing the heat feedback and the surface temperature. The increased surface temperature increases the rate of the first reaction, which further increases the rate of gas formation. The computational model implements an iterative solver that seeks a self-consistent solution to the two closed form expressions for burn rate as a function of surface temperature and surface temperature as a function of burn rate and pressure. These models for the heat transfer and explosion phases were run using heat flux data from LES simulations of a 10 cm long, 10 cm diameter steel container of PBX immersed in 0.5-1.0 m JP-8 pool fires. 12.2â•‡ One-dimensional heat transfer, fast cook-off HMX model Heat flux data from an LES simulation of a 30 cm long, 12 cm diameter steel container immersed in a 30 cm JP-8 pool fire were extracted at 24 locations around the circumference of the steel cylinder. The 20 seconds of fluctuating heat flux data available from the simulation were assumed to be at quasi-steady state and were replicated to extend to the time required by the fast cook-off HMX model. The HMX model is spatially one-dimensional, fully transient, and consists of equations for modeling the solid (condensed) phase HMX, the gas phase, and the surrounding steel container for fast cook-off conditions [158]. The steel shell provides a thermal barrier to the external heat flux. The condensed phase HMX decomposition reactions are described by distributed kinetics (calculated throughout the condensed phase, not just at the surface). The gas phase description includes a detailed chemistry model for the combustion of HMX. Solution of the PDEs results in temperature, pressure, velocity, and species mass fractions as a function of position and time. For additional details, see [158, 159]. 12.3â•‡ Prediction of time to ignition and explosion violence By coupling both the MPM/MPMICE models and the fast cook-off HMX model with the LES pool fire simulation, time to ignition for a range of conditions (labeled ‘ignition delay’ in Fig. 30) was computed using both models as shown in Fig. 30. Also included in Fig. 30 are experimental timeto-ignition data obtained by various researchers [158]. Each point for ‘Flux at steel container’ is matched with the corresponding ‘Flux at interface’ value. The heat flux at the steel/HMX (or PBX)

124â•… Transport Phenomena in Fires 10000 1000

Ignition Delay (sec)

100

Ali et al, 99: 0.75 atm air Ali et al, 99: 1 atm air Vilynov and Zarko, 89 Strakovski (1989) Lengelle (1985) Atwood (1988) C-SAFE (1999) C-SAFE (2001)

Flux at steel container Flux at interface

10

Tang

Heat Flux to HMX Surface, 1D HMX Model LES Heat Flux to Steel Surface, 1D HMX Model LES Heat Flux to Steel Surface, MPM Model Heat Flux to PBX Surface, MPM Model

1 0.1 0.01 0.001 0.1

1

10 Heat Flux (W/cm2)

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Figure 30:â•‡Ignition delay versus heat flux showing the difference between calculated interior and calculated exterior heat flux levels. interface is always lower than the heat flux at the exterior of the steel container. In the limit as the heat flux at the exterior approaches zero, the heat flux at the interior will also approach zero and these two heat fluxes must converge. At the high heat flux end, the deviation between the two fluxes is large. As seen in Fig. 30, when the time to ignition is based on the ‘Flux at interface’ values, the model results fall in line with the experimental data. Alternatively, when the time to ignition is based on the ‘Flux at steel container’ values, the predicted values show a strong deviation from the experimental values. Hence, an important parameter for accurately predicting ignition delay is the flux that the explosive experiences, not the flux that the container experiences. In addition to time to ignition data, results from the MPMICE simulations show evidence of explosion violence; data from two cases are considered here. In case 1, the 10 cm diameter container is located 0.5 m above the edge of a 0.5 m diameter JP-8 pool fire, there is no crosswind, and the fuel regression rate is 6.4 mm/min. In case 2, the container is located 0. 25 m above the edge of a 0.5 m diameter JP-8 pool fire, the crosswind speed is 4 m/s, and the fuel regression rate is 6.4 mm/min. Polynomial fits of the azimuthal heat flux data from LES pool fire simulations of the two cases are displayed in Fig. 31 for one axial location on the container. These traces are distinctly different and produce different fragmentation patterns as observed in the three-dimensional volume renderings of the container and propellant shown in Fig. 32. A more quantitative analysis measures explosion violence by the total kinetic energy of the exploded container. Based on such an analysis, one finds that case 1 is more violent than case 2 as seen in the kinetic energy plots of Fig. 33. Experimental results have shown that lower heat fluxes produce more violent explosions, and the simulation data in Fig. 33 mirror this observation; the heat fluxes experienced by case 2 are lower than those experienced by case 1 (see Fig. 31). These time to ignition and violence of explosion predictions provide the perspective of overall trends in the simulation data and generally agree with available data. However, they do not achieve the desired predictivity as there are no associated error bars. In fact, it is unclear how the errors identified in previous sections of this chapter were propagated in a nonlinear fashion up through the hierarchy for this ‘complete system’ case. For this reason, error quantification and propagation (see Section 13) are essential areas of research in moving toward predictivity.

Heat Transfer to Objects in Pool Firesâ•… 90o

40000W/m2 60o

120o

125

Case1 Case2

30000 150o

30o

20000 10000

180o

0o

210o

330o

300o

240o 270o

Figure 31:â•‡Polynomial fits of the azimuthal heat fluxes at a single axial location on the steel container obtained from pool fire simulations of cases 1 and 2. Case 1 - no crosswind, container is located 0.5 m above the pool surface at the edge of fire. Case 2 - crosswind of 4 m/s, container is located 0.25 m above pool surface at fire’s edge.

(a)

(b)

Figure 32:â•‡Volume rendered images of container fragmentation and propellant release from simulations of a 10 cm diameter steel container of PBX embedded in a 0.5 m JP-8 pool fire simulation. (a) Case 1 - no crosswind, container is located 0.5 m above the pool surface at the edge of fire. (b) Case 2 - crosswind of 4 m/s, container is located 0.25 m above pool surface at fire’s edge.

126â•… Transport Phenomena in Fires 2.5

x 104 Case1 Case2

Kinetic energy, Joules

2

1.5

1

0.5

0

2

2.5 3 Time from ignition, sec

3.5 x 10−3

Figure 33:â•‡Total kinetic energy of all particles in the MPMICE simulation. Case 1 - no crosswind, container is located 0.5 m above the pool surface at the edge of fire. Case 2 crosswind of 4 m/s, container is located 0.25 m above the pool surface at fire’s edge.

13â•‡ Toward predictivity: error quantification and propagation The goal of the simulation at the level of the complete system is to accurately predict heat flux to a container of energetic material immersed in a transportation fuel pool fire. Despite the methodology of a V&V hierarchy, predictivity has not yet been achieved. The validation comparisons at the subsystem involve some quantification through the use of validation metrics, while the results of the ‘complete system’ simulation are qualitative in nature and do not account for uncertainties in the experimental or the simulation data. What are needed are systematic ways to represent uncertainties at lower levels of the V&V hierarchy, efficient computational algorithms to propagate those uncertainties all the way up to the complete system level, methods for identifying the parameters that control uncertainty, metrics for quantifying simulation error, and datasets for validation [160]. Ultimately, the truth comes from the experimental data; it is the window on the physical world. However, in ambitious simulations of multiphysics and multiscale simulations, it is through the tight coupling of both simulation and experimental data that predictivity with uncertainty quantification will be achieved. The field of uncertainty quantification (UQ) and error propagation in multiphysics problems is an area of active research, and it still is not clear what approach or approaches will provide the analysis tools necessary to achieve predictability. McRae [160], Marzouk and Najm [161], and Najm and coworkers [162, 163] have proposed a method for UQ based on Bayesian inference. Inferring model parameters and inputs from data is a challenging task and is known as the inverse proÂ�blem. Marzouk and Najm have focused on using Bayesian statistics as a foundation for inference [161]. Interestingly, there are strong parallels between the forward propagation of uncertainty and Bayesian approaches to inverse problems. Marzouk and Najm have formalized this connection and have successfully employed polynomial chaos expansion (PCE) techniques to propagate a wide range of uncertainty through the forward problem. In their approach, the model parameters and field variables are treated as stochastic quantities that can be modeled using

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127

PCE techniques. After sampling the resultant spectral expansion, they achieve a more efficient Bayesian solution of the inverse problem [161]. A comparison of this approach to the more conventional UQ method of sensitivity analysis and error propagation in the context of H2-O2 ignition under supercritical-water conditions was performed by Najm and coworkers [163]. The results indicate that PCE methods provide first-order information similar to that from the sensitivity analysis. In addition, the PCE methods preserve higher-order information that is needed for accurate UQ and for assigning confidence intervals on sensitivity coefficients. Analysis shows substantial uncertainties in the sensitivity coefficients, illustrating that these higher-order effects can be significant. A second approach has been proposed by Frenklach et al. [164] that relies on the concept of data collaboration. Data collaboration organizes the available experimental data and its uncertainties together with mechanistic knowledge of the physical system using the abstraction of a dataset. A dataset unit consists of ‘the measured observation, uncertainty bounds on the measurement, and a model that transforms active parameter values into a prediction for the measurement’ [164]. Note that a dataset unit includes a model prediction. In its application, the concept of data collaboration recognizes that a model is only an approximation to the truth and that the truth comes from the experimentally measured data. With this dataset abstraction, numerical analysis techniques can be used to probe the dataset. For example, consistency of the model to the measured data or of dataset units to each other can be determined with constrained optimization that utilizes solution mapping tools and robust control algorithms. Within the data collaboration framework, consistency thus becomes a quantifiable metric that can open up the model to a new level of interrogation such as what a low or moderate value of the metric means. Additionally, the uncertainties of the experimental data are transferred directly into the model. In one example of how to use the consistency metric, a consistency test was performed with the GRI-Mech 3.0 dataset [165], which is composed of 77 dataset units. The test identified two major outliers in the dataset. The researchers who collected the data re-examined their original observations and modified the reaction times they had extrapolated, removing the inconsistency in the GRI-Mech 3.0 dataset [164]. A similar consistency analysis could be applied to the model as outliers could also indicate a problem with the model. Neither the Bayesian inference nor the data collaboration approach has yet been applied to a complex, multiscale, multiphysics problem. However, in order to achieve predictivity, it is clear that these or other approaches must be implemented in more complex systems. The treatment of uncertainties must become more systematic. Additionally, to use either approach in problems involving heat transfer to an explosive device, a large number of dataset units need to be identified and compiled in a database repository including the data sets discussed in Section 10. There is clearly much work to be done both computationally and experimentally.

14â•‡ Summary The prediction of heat transfer to objects in transportation fuel pool fires using simulations requires the integration of complex methodologies. This chapter has summarized these methodologies in a manner that will assist the reader in identifying a suitable approach to this challenging problem. The high cost of large-scale experiments (both real-world and simulation), combined with the greatly reduced fidelity of experimental data at this scale, provides strong motivation for the use of a computational approach that has been validated and verified in a systematic manner and that includes the quantification and propagation of uncertainty from the unit problem level to the complete system level.

128â•… Transport Phenomena in Fires

Acknowledgments The authors wish to acknowledge the current and former members of their research groups whose work has been included in this chapter. Without their scholarship and hard work, this chapter would not have been possible. These individuals include Stanislav Borodai, William Ciro, Jim Guilkey, Todd Harman, Gautham Krishnamoorthy, Niveditha Krishnamoorthy, Seshadri Kumar, David Lignell, Randy McDermott, Rajesh Rawat, James Sutherland, Chuck Wight, Shihong Yan, Devin Yeates, and Hongzhi Zhang.

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[133]â•… Pierce, C.D. & Moin, P., A dynamic model for subgrid-scale variance and dissipation rate of a conserved scalar. Physics of Fluids, 10(12), pp. 3041-3044, 1998. [134]â•… Jamaluddin, A.S. & Smith, P.J., Predicting radiative transfer in rectangular enclosures using the discrete ordinates method. Combustion Science and Technology, 59, pp. 321340, 1988. [135]â•… Modest, M.F., Radiative Heat Transfer, 1st edn, McGraw-Hill: New York, 1993. [136]â•… Krishnamoorthy, G., Rawat, R. & Smith, P.J., Parallel computations of radiative heat transfer using the discrete-ordinates method. Numerical Heat Transfer B, 47, pp. 19-38, 2005. [137]â•… Cetegen, B.M. & Ahmed, T.A., Experiments on the periodic instability of buoyant plumes and pool fires. Combustion and Flame, 93, pp. 157-184, 1993. [138]â•… Denison, M.K., A Spectral Line-Based Weighted-Sum-of-Gray-Gases Model for Arbitrary RTE Solvers, PhD Thesis, Brigham Young University, Provo, UT, 1994. [139]â•… Liu, F., Gülder, Ö.L. & Smallwood, G.J., Non-grey gas radiative transfer analyses using the statistical narrow-band model. International Journal of Heat and Mass Transfer, 41, pp. 2227-2236, 1998. [140]â•… Coppalle, A. & Vervisch, P., The total emissivities of high-temperature flames. Combustion and Flame, 49, pp. 101-108, 1983. [141]â•… Adams, B.R., Computational Evaluation of Mechanisms Affecting Radiation in Gas- and Coal-Fired Industrial Furnaces, PhD Thesis, University of Utah, Salt Lake City, UT, 1993. [142]â•… Krishnamoorthy, G., Rawat, R. & Smith, P.J., Parallel computations of non-gray radiative heat transfer. Numerical Heat Transfer B, 48, pp. 191-211, 2005. [143]â•… Hsu, P. & Farmer, J.T., Benchmark solutions of radiative heat transfer within nonhomogeneous participating media using the Monte Carlo and YIX methods. ASME Journal of Heat Transfer, 119, pp. 185-188, 1997. [144]â•… Li, B.W., Chen, H.G., Zhou, J.H., Cao, X.Y. & Cen, K.F., The spherical surface symmetrical equal dividing angular quadrature scheme for discrete ordinates method. ASME Journal of Heat Transfer, 124, pp. 482-490, 2002. [145]â•… Burns, S.P. & Christon, M.A., Spatial Domain-Based Parallelism in Large-Scale, Participating-Media, Radiative Transport Applications, Technical report, SAND96-2485, Sandia National Laboratories, 1996. [146]â•… Sagaut, P., Large-Eddy Simulation for Incompressible Flows: An Introduction, 2nd edn, Springer-Verlag: Berlin, 2002. [147]â•… Ciro, W., Heat Transfer at Interfaces of a Container of High-Energy Materials Immersed in a Pool Fire, PhD Thesis, University of Utah, Salt Lake City, UT, 2005. [148]â•… Sulsky, D., Chen, Z. & Schreyer, H.L., A particle method for history dependent materials. Computer Methods in Applied Mechanics and Engineering, 118, pp. 179-196, 1994. [149]â•… Guilkey, J.E. & Weiss, J.A., Implicit time integration for the material-point method: quantitative and algorithmic comparisons with the finite-element method. International Journal for Numerical Methods in Engineering, 57, pp. 1323-1338, 2003. [150]â•… Kramer, M.A., Greiner, M., Koski, J.A. Lopez, C., & Suo-Anttila, A., Measurements of heat transfer to a massive cylindrical object engulfed in a circular pool fire. J. Heat Transfer, 125, pp. 110-118, 2003. [151]â•… Blackwell, B.F., Douglass, R.W., & Wolf, H., A User’s Manual for the Sandia OneDimensional Direct and Inverse Thermal (SODDIT) Code, Technical report, SAND852478, Sandia National Laboratories, 1985. [152]â•… Blanchat, T.K., Nicolette, V.F., Sundberg, W.D., & Figueroa, V.G., Well-Characterized Open Pool Experimental Data and Analysis for Model Validation and Development, Technical report, SAND2006-7508, Sandia National Laboratories, 2006

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CHAPTER 4 Heat and mass transfer effects to be considered when modelling the effect of fire on structures A. Jowsey, S. Welch & J.L. Torero BRE Centre for Fire Safety Engineering, The University of Edinburgh, UK.

Abstract The design of structural elements for fire has traditionally been done by means of furnace testing and very simple heat transfer analyses. Although many studies have been conducted in the past to understand the intricacies of the different heat and mass transport processes occurring in structural elements, knowledge is still not complete and heavily biased towards results obtained from standard furnaces. Analysis of structural behaviour by means of single element testing has been believed to provide a robust assessment of structural performance. Studies have improved on the definition of the fire by elaborating different temperature vs. time curves deemed to be more representative of realistic fires. Furthermore, equivalency methods have been developed to complement furnace tests and to translate time ratings obtained in furnaces to real fire behaviour. Recent studies following the Cardington tests have shown that the complex global behaviour of structures is strongly linked to heating regimes, thermal expansion and geometric deformations. The loading of the structural components is mostly controlled by these factors and could reach critical levels at high temperatures. In contrast, load redistributions associated with deformations can result in larger than expected structural robustness. The evolution of the material properties with temperature will mostly play a significant role close to failure. Given this information, the analysis of structural behaviour in fires requires a much deeper understanding of heat and mass transfer processes. Here, an overview of the different factors to be considered is presented. The review is not intended to provide an exhaustive compilation of the literature, but mostly to highlight the factors to be taken into account when carrying out the thermal analysis of a structure. Many aspects of this process still carry great uncertainty, thus an attempt will be made to indicate areas where future work is necessary.

1 Introduction From the perspective of fire safety, the design of a building can be approached in two different ways: the first is that the building must comply with existing ‘prescriptive’ regulations, and the second is a demonstration that certain safety goals can be achieved using validated engineering

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methods, i.e. ‘performance-based’ design. Within the scope of the former it is apparent that regulations have not been developed to fully specify the design of unique and complex buildings, nor of complex fire scenarios. The events of 11 September 2001 have highlighted the need to review many current practices. The history of structural fire safety engineering has been defined by a series of important events. The first attempts to understand structural behaviour in fire date from the early 1900s and are associated with the development of what will later become the Standard Furnace Test [1]. The Standard Furnace Test subjects single structural elements to a standard ‘temperature vs. time’ (T vs. t) curve until the structural element reaches a critical temperature. The time required to reach the critical temperature is defined as the ‘fire rating’ or ‘fire resistance’. Therefore, historically, fire resistance design of structures has been based upon single element behaviour in the standard fire resistance test. Nevertheless, the significant differences between the standard fire heating curve and a T vs. t relationship produced in a real fire have long been recognized. Details on the various standard curves and their differences with real fires are detailed by Drysdale [2] and Buchanan [3]. The early studies by Ingberg [4], Kawagoe and Sekine [5] and Magnusson and Thelandersson [6] have provided the basis for the development of compartment fire models (CFMs) as well as the time equivalence concept that have attempted to address this shortfall. The development of the later concept is associated with a series of experimental tests conducted by Pettersson et al. [7] and the analysis by Law [8]. It allows use of furnace test results to predict fire resistance in a realistic fire. A landmark example of its use is the analysis by Law and O’Brien [9], who considered the preferential heating experienced by external steel to allow the Pompidou Centre in Paris to be built with an unprotected external steel frame. A detailed summary of the knowledge associated with the thermal loading imposed by fires is given in the Society of Fire Protection Engineers (SFPE) guideline for thermal loading and in different chapters of the SFPE Handbook [10, 11]. Very little work was carried out in this area after the ‘time equivalency’ concept was developed until the Broadgate fire in London [12]. The Broadgate fire resulted in an unprotected steel frame building under construction surviving a very intense fire for several hours. The unexpected robustness of this structure was a catalyst for the Cardington frame fire tests in the 1990s [13] and the Natural Fire Safety Concept programme [14]. Nevertheless, the greatest incentive for robust design of tall buildings in fire is the World Trade Center (WTC) collapse [15, 16]. The events of 11 September 2001 highlight two different weaknesses of prescriptive design. The first is that prescriptive design is based on probable events, thus excludes extreme scenarios such as terrorist attacks, and in general the fire community will not design against such cases. The second is that it does not provide any information associated with the actual performance of a building in the event of a fire. The following sections will therefore abandon this approach, not as a criticism of prescriptive methods, but to highlight the physical processes that need to be understood to be able to predict the performance of a structure, independent of the nature of the event.

2 Building fires The schematic presented in Fig. 1 represents the possible behaviour of a building in the event of a fire. A quantitative definition of the safety objectives for a building could be expressed as a function of different characteristic times. It follows that the time to evacuate each compartment (te,i, i.e. room of origin (te,1), floor (te,2) and building (te,n)) is required to be much smaller than the time necessary to reach untenable conditions in that particular compartment, tf,i. Characteristic values

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Figure 1: Schematic of the sequence of events following the onset of a fire in a multi-storey building. The solid line corresponds to the ‘fire size’, the dotted lines to the possible outcome of the different forms of intervention (sprinkler activation, fire service). The units of ‘fire size’ could be defined as heat-release rate (kW), area of fire (m2) or any other means to quantify the magnitude of the event. The dashed lines are the percentage of people evacuated from the room, floor and building, respectively, with the ultimate goal of 100% represented by a horizontal dashed line. The dashed and dotted line corresponds to the percentage of the full structural integrity of the building.

of te,i and tf,i can be established for different levels of containment, i.e. room of origin (i = 1), floor (i = 2) and building (i = n). Furthermore, it is necessary for the evacuation time to be much smaller than the time when structural integrity starts to be compromised (tS). In summary: ∀i, i = 1 to n,

te,i << tf,i

∀i, i = 1 to n,

te,i << tS

It could be added to these goals that full structural collapse is an undesirable event however long the fire lasts, therefore: tS → ∞ Although these criteria for safety times can be considered as a simplified statement, it is clear that they describe well the main goals of fire protection. With the objective of achieving these goals a number of safety strategies are put in place. These include factors which are intended to increase tS and tf,i, such as active systems (e.g. sprinklers, or the intervention of the fire service). As shown in Fig. 1 (the dotted lines branching off below the fire curve), the success of these strategies can result in control or suppression of the

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fire. Passive protection such as thermal insulation of structural elements becomes part of the design, with the purpose of increasing tS. Finally, but most importantly, evacuation protocols and routes are designed to reduce te,i at all stages of the building evacuation. It is important to note that the safe operations of the fire service need to be included within the estimation of tS. The events following the attack on the WTC showed that these safety goals were not attained in that case. To understand why this building collapsed, it is necessary to establish the fire conditions, the interactions between the fire and the structural elements, and the sequence of the intervention and evacuation processes. Different methodologies and tools have been developed to study each of these aspects and have been applied to the forensic analysis of this event [16]. The following sections will concentrate on providing an overview of the methodologies that can be used to assess the heat and mass transfer processes associated with the boundary condition between structural elements and a fire.

3 Methods of thermal analysis Guidance for practitioners on methods of thermal analysis for structures and on representing heat transfer phenomena which define the boundary condition can be found both in industryorientated publications (e.g. SFPE Handbook of Fire Protection Engineering [11], BRE Digest series on Structural Fire Engineering Design [17]) and in the standard design guides (e.g. Structural Eurocodes [18]). Solutions can be obtained at various levels: from tabular or graphical data, through simple numerical calculations and via detailed numerical methods performed on a computer (generally referred to as ‘advanced methods’). An overview of each is given in ref. [11]. Tabular and graphical data tend to have a restricted application, i.e. they are relevant only to the exposures consistent with the standard fire resistance test itself, and relatively limited departures therefrom. For example, graphical procedures for determining protected steel member temperatures have been developed by Malhotra, Jeanes and Lie among others [11], permitting analysis of sensitivity to parameters such as the beam section size, the duration of exposure and the thermal resistances of the insulating material, in some cases via dimensionless parameters (e.g. the Fourier number), but only for certain classes of fire exposures. The equations used in simple numerical methods are derived from simplified heat transfer approaches. For example, a quasi-steady-state, lumped heat capacity analysis can provide the temperature rise of unprotected steel members, while for protected steel members, the thermal resistance provided by the insulating material can be accommodated by empirically derived correlations (cf. the Structural Eurocodes methodology [18]). These approximate methods allow the designer to use any appropriate gas-phase temperature−time curve, but their generality remains uncertain. Advanced models based on numerical heat transfer methods provide a more general approach and are becoming increasingly popular both in a research and industrial context [17, 19]. Their advantages lie in the ability to define an arbitrary gas-phase boundary condition and to perform conjugate heat transfer calculations in which the surface temperature is obtained from the balance of the gas- and solid-phase thermal conditions [20]. The analysis is typically performed in three-dimensions, thereby overcoming the limitations of simpler approaches which consider only the in-depth direction. There is a significant challenge in achieving sufficient grid resolution within the solid, where much smaller cells are normally required due to the steep thermal gradients, especially when structured computational meshes are adopted (which may demand a similar cell size in the gas and solid phases) [21]; this can be overcome using unstructured meshes, or by coupling to an independent high resolution mesh within the solid, though in the latter case

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there will still be limitations associated with the resolution of the surface cells, e.g. where these are not able to fully resolve the details of the geometry, such as an I beam. An example of the application of this approach is described in ref. [21]. In addition, temperature-dependent properties can easily be incorporated into the calculations for greater accuracy.

4 The boundary condition Fire-resistance calculations have often been conducted, both in the past and currently, on the basis of a representation of the fire by means of specified gas-phase temperature−time histories. Variations in heating arising from any feedback coupling due to the effect of the surface temperatures on the gas-phase processes are neglected (at this stage), and therefore deemed to be of second-order importance. For example, standard (e.g. ISO-834 [1]) or parametric [8] temperature−time curves can be adopted, with the actual heat transfer to the structure being defined according to the resulting convective and radiative fluxes. The energy equation of the structural element can be solved on this basis [2] and it can be defined in two different forms, depending on the thermal response capacity of the boundary material: rSc pS VS rS c pS

dT = ASqS′′ dt

∂T ∂2T = kS 2 ∂t ∂x

(Thermally thin material, e.g. steel)

(1)

(Thermally thick material, e.g. concrete)

(2)

where the boundary condition for both cases corresponds to the input from the fire and is given by: qS′′ = h(Tg − Ts ) + eg essTg4 − essTs4 qS′′ = h(Tg − Ts ) + eg es s Tg4 − es s Ts4 = − kS

(Thermally thin material) ∂T ∂x

(Thermally thick material)

(3) (4)

x=0

where Tg is the imposed temperature of the gas as defined by the temperature−time curve. The emissivity of the solid surface is given by es and that of the gas by eg. For simplicity direct heat exchanges with the environment outside of the compartment, where these components constitute an enclosure boundary, have been ignored here, but could also be included in these expressions. For the thermally thin elements AS will be the exposed area. The unexposed area can be ignored or treated as a loss to some ambient temperature. For thermally thick materials the boundary condition at the unexposed face will be fixed based on the conditions established for this side of the element. If a fire is present at the other side then a similar boundary condition will be included here; if no fire is imposed, a heat loss to an ambient temperature can be assumed. The essence of the thermal response problem is now to determine the surface temperature evolution of the structure. In the case of thermally thick materials, which are the main interest in structural fire engineering, this is a non-trivial problem due to the highly non-linear nature of the applied boundary condition (eqns (3) and (4)). In general, this can only be solved by adopting numerical heat transfer procedures which solve the coupled equation set to obtain the evolution of the thermal conditions at the surface, and hence in the solid.

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Depending on the application and the accuracy required there are also two other approaches which bypass the above complexity. First, the thermal boundary condition may be imposed by equating the surface temperature of the structural element to the local gas temperature, so that it does not need to be calculated. This approach is highly attractive due to its simplicity, and the reduction of the thermal modelling problem to a simple conduction analysis, but it will in many cases significantly overestimate the temperature of the structure, hence providing an overly conservative result. Second, in the context of validation using test data, it may be possible to use the measured surface temperatures. This procedure was adopted by many modellers, including Usmani and Lamont [22] and Moss and Clifton [23] when modelling the Cardington frame tests. Obviously, it was possible to use appropriately positioned monitoring equipment in these cases and in real fire situations this type of data is not generally available. For design methods, there is generally no alternative to undertaking a detailed thermal analysis, especially if the members are protected. 4.1 Gas-phase conditions As for the thermal analysis, there are a range of methods available for specifying the thermal exposure conditions resulting from the gas-phase phenomena. A basic approach, which has been very widely used, is simply to compute the applied heat fluxes on the basis of prescribed temperature−time curves. However, this implies a number of simplifications of which the main are: • The compartment fire temperature is homogeneous with no spatial differences worth considering. • The radiation field is in thermal equilibrium within the gas phase, i.e. there is no radiation exchange between soot particles and the gas, and thus gas temperatures can be used to establish radiative heat fluxes. • The optical depth within the gas phase is much smaller than the characteristic length-scales of the compartment, thus heat radiation can be treated as a local phenomenon. The computation of the gas-phase emissivity (eg) is also subject to various simplifications that differ according to the researcher. One common approach states that the emissivity increases exponentially with the thickness of the emitting gas and thus Pettersson et al. [7] postulate that: eg = 1 − exp( − k Lg )

(5)

where k is an emission (or absorption) coefficient and Lg is the thickness of the emitting layer. This approach carries the further assumptions that a single emitting temperature and gas-phase emissivity are sufficient to describe radiative heat exchange. In fact the radiative component needs to account for all possible sources of radiation, thus a more complete way to describe the above boundary condition is: qS′′ = h(Tg − Ts ) + q r,T ′′ − es s Ts4

(6)

where the net heat input to the structural element is q◊S″, h(Tg−Ts) is the convective contribution, −ess Ts4 is the surface re-radiation and the term q◊r,T ″ conglomerates all radiative inputs. Radiative inputs can come from the hot gases (including soot), other hot surfaces or the flame; furthermore, they are attenuated by the absorption through the gas-phase media. An example of a simple

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approximation for absorption, written as a function of the soot volume fraction and temperature through the soot absorption coefficient (k), is: k = C ⋅ fsT

(7)

where C is an empirical constant, fs is the soot volume fraction and T is the temperature. More detailed models, representing the spectral dependence of the optical properties, are reviewed by Lallemant et al. [24]. Thus if the soot volume fraction is high and the distance from the flame or other hot elements is large, eqn (5) shows that all energy from these emitting bodies will be absorbed by the smoke before reaching the target. The assumption that the hot gases adjacent to the structural element are the main contributors to its heating might then be appropriate and there would be no need to resort to eqn (6). Furthermore, if far from the flames, thermal equilibrium between soot and the gas phase in the combustion products might also be sufficiently accurate. 4.2 Application examples Some examples of methods and applications of the representation of the gas-phase conditions are described below. Lim et al. [25] simply use the ISO-834 [1] fire gas temperature as a direct thermal boundary condition for the analysis of fire-exposed floor systems. Temperature distributions within the slab are then evaluated using the finite-element program SAFIR. Usmani et al. [26] adopted a gas-phase temperature−time relationship for the design fire as described in Eurocode 1 [27], i.e. the parametric approach. Post-flashover conditions are assumed so that a single representative temperature can be used. This simple method does not take into account the different convective and radiative heat exchanges between surfaces, the fire and combustion products, nor local variations of the empirical coefficients. Similar approaches are adopted by Franssen [28] and Liew and Ma [29] using the standard, hydrocarbon and parametric fire curves in Eurocode 1, with decay periods also considered. In the context of advanced methods, the computational fluid dynamics (CFD) code FDS [30] defines a number of methods for applying thermal exposures to a structure, depending on the expected material behaviour (i.e. thermally thin sheet or a thermally thick solid). For the most relevant case of a thermally thick solid exposed to an arbitrary flux, a one-dimensional heat transfer analysis is performed across the thickness of the material. The cell resolution within the wall is non-linear, with clustering near the surface where initial thermal gradients are steeper. The surface temperature is determined according to the appropriate convective and radiative fluxes and the solid-phase conduction, i.e. it is a locally varying parameter. This treatment is best suited to planar surfaces such as enclosure boundaries where localized heating does not induce significant thermal gradients in the plane of the wall. Radiation calculations within FDS are undertaken via the solution of the radiation transport equation for a non-scattering grey gas, optionally using a wide-band model [30]. The radiation transport is based on a finite volume method using a large number of angular subdivisions. In common with DTRM methods [19], computational limits on the total numbers of discretizations can lead to a non-uniform distribution of the radiant energy, for example in the case of a target far away from a localized source of radiation like a small growing fire, a numerical error generally known as the ‘ray effect’ [19]. Prasad and Baum [31] have recognized the need for the thermal boundary condition to be defined by the appropriate incident heat flux, which in turn is determined from solutions of the radiative transport equation. This is possible within the scope of CFD calculations, which they consider as the only realistic method that one might achieve an adequate representation of the temporal and

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spatial variations associated with temperature and combustion products derived from a natural fire in a compartment. For more challenging large-scale applications, such as the WTC scenarios, with fire impingement on large truss systems, simplifications may often be needed. They present a method which considers the spatial distribution of the temperature and combustion products and also takes into account the geometry of the enclosure [31]. This involves the classification of a compartment into the hot upper layer and cool lower layer. From these layers, representative temperatures and radiative properties can be determined via CFD modelling and applied to a radiative transport model to determine heat fluxes to a target surface. Two major simplifications are embedded within this procedure; the first is use of the concept of a ‘grey gas’, the properties of which are independent of spectral frequency, hence allowing a simple approximation of the average absorption coefficient. The second is to assume a vertically stratified distribution of temperature and combustion products throughout the compartment. This two-zone classification model however ignores some important details of the spatial differences created during large-scale enclosure fires − it is these distributions that may have a marked impact on the way a structure may behave in a fire. Hence by averaging CFD output over time and space, potentially large variations in exposure conditions are inevitably lost. Other recent work has aimed at defining the thermal boundary conditions in terms of detailed spatial heat flux variation with respect to time [32]. Considerable numbers of fire safety engineers are starting to appreciate this change, although as outlined above, its determination can be complex and many times still requires significant simplifications to be made. Therefore, while current approaches can provide a reasonable representation of worst-case scenarios, real fire situations demand greater attention to detail to accurately define the thermal-structural boundary conditions. Only through increased understanding of the definition of the boundary condition can one determine with confidence the detailed thermal conditions evolving within a specific element in order to undertake a full structural analysis. The relevance of each modelling assumption can be evaluated for every specific scenario, but to better understand the validity of the various simplifications it is worth briefly reviewing some basic concepts of compartment fires; this is done in the following section.

5 The compartment fire A fire can have a significant effect on a structure but the characteristics of the compartment that encloses the flames can also have an impact on the fire development itself. In particular, temperatures within the compartment are affected by heat transfer with the compartment boundaries, but they in turn influence fuel generation via the energy feedback from the flames and hot surfaces to the combustible surfaces, thereby affecting the growth of the fire. A fire is often represented as undergoing a series of processes from its inception, through spread and growth to its fully developed stage, and finally decay. A singularity in the growth process is the event of ‘flashover’. Here, ‘flashover’ is defined as a transition, usually rapid, in which the fire grows to a ‘fully developed’ state in the compartment. This state is where all of the fuel available is involved to its maximum extent, according to oxygen or fuel limitations. During the growth phase a fire may be represented by a two-zone model, where the fire entrains air through a cold lower zone, and generates products of combustion which migrate to an upper layer. Pressure in a compartment fire is considered to be approximately atmospheric and flows occur at vents due to small hydrostatic pressure differences [33, 34]. Following ‘flashover’ the fire flows can often be modelled via a single zone representation and the use of the ideal gas law, in conjunction with the conservation equation for energy and mass.

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In fully developed fires, the fuel limit is determined by the free burning rate and the ventilation limit is generally related to an opening factor parameter that is associated with the geometry of the vents to the compartment. Although significant research has been done to establish the characteristics of such fires [35, 36] there are many uncertainties associated with real fire incidents. Considering again the uniformity of the thermal exposures, there may be simplifying factors in some cases. For example, the thermal inertia of structural elements is always significantly larger than that of the gas phase; thus characteristic times for temperature changes within the solids are very much longer than those of the gas phase. Furthermore, the presence of thermal insulation can result in very slow temperature rises in the protected material throughout the entire fire growth period. Due to the great difference in the solid and gas-phase timescales, this may in some cases allow the use of time-averaged values for the gas-phase temperatures and the assumption that the fire can be considered as fully developed for all thermal calculations related to the structure. If it is further assumed that the effects of spatial variations in heating average out over the duration of the burning then a first-order approximation to representing the fire exposures would be to simply assume a homogeneous temperature throughout the compartment. This translates to defining the characteristic length scale of eqn (5) as the characteristic size of the compartment (Lg). Considering the definitions of temperature−time curves, a number of fundamental studies have been undertaken. Pettersson et al. [7] made a significant effort to study the different stages of the fire. Their tests and computations resulted in a method of defining of temperature−time curves, considering the effects of different fuel loads, ventilation conditions and thermal properties of the compartment boundaries. However, there are some inconsistencies in these methods when adopted to represent natural fires since only one characteristic temperature is defined, which is clearly inappropriate for most of the development phase of a fire. The Conseil International du Bâtiment (CIB) took a different approach in their research on compartment fires, which was one of the most comprehensive studies on the subject [37, 38]. Wood cribs were used as fuel and although this arrangement has its own particular burning characteristics the observations illustrate the main factors controlling a fully developed fire. This study used reduced-scale room geometries (H = 0.5−1.5 m), and the cribs covered nearly the entire floor area. For wooden cribs in a compartment, the area of the vertical shafts of the crib, ___ quantified by the factor (HAo/A)crib, and the ventilation factor of the compartment, A/Ao √Ho , control the oxygen flow through the crib, H being the height of the vertical shafts of the crib, Ho the height of the compartment opening, Ao the area of the vertical shaft or the compartment opening and A the surface area of the crib or the room floor. For limited oxygen conditions the ventilation factor controls the burning rate and this is independent of vertical shaft areas. With sufficient oxygen, the exposed surface area of the sticks controls the burning rate and therefore the burning rate increases with (HAo/A)crib. If the burning rate can be established, and knowing the heat of combustion, the energy release rate can be calculated and thus the temperature of the compartment estimated. Then a correction can be made to establish the fraction of the energy that remains within the compartment. Figure 2 represents the curve fit presented by Thomas and Heselden [37] that gives estimates of the temperatures that could be expected for wood cribs in small-scale (1 m high) compartments. The actual data has some scatter, which Law and O’Brien [9] suggest to be a result of some particularly extreme experimental conditions. The results are expressed in terms of the ventilationfactor and surface area and are hoped to be scale independent. As can be seen in Fig. 2, this study again only provides a single average temperature for each condition, instead of a spatial evolution of the temperature. Despite being less information this is consistent with the assumptions of the thermal model adopted. The extent of the period characterized by the peak temperature can be defined as a function of the empirical burning rates and the duration of the decay can be estimated using a simple energy balance for the compartment.

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1200 1000

T (oC)

800 600 400 200 0 0

10

20 AT/AH

30 1/2

(m

-1/2

40

50

)

Figure 2: Time mean temperature near the ceiling, where AT is the total area excluding floor and opening, A the window area and H the height of the window. The fuel loads for these tests are in the range 20−40 kg/m2, which is smaller but nevertheless comparable to what would be expected in a modern office [37].

Overall, the CIB work consisted of a parametric study that included more than 100 experiments, thus allowing a reasonable level of confidence to be associated with the data. Nevertheless, the data presented is limited to average values and does not address the spatial temperature distributions within the compartment nor the proximity of the flames to the structural elements. Drastic temperature variations within the compartment have been suspected for many years [38]. Further evidence of temperature gradients is reported by Welch et al. [39]. A series of fire tests were conducted in 1999 as part of a European collaboration supported by the European Coal and Steel Community (ECSC) to develop a new fire safety concept based on the behaviour of natural fires. A total of eight tests were performed and the reader is referred to [39] for a detailed description of the experiments. Parameters investigated in the tests were the location of the ventilation opening, type of fire load and thermal properties of the compartment linings. The quantity of fuel load, area of ventilation openings and size of compartment were fixed for all tests. Tests were conducted in a compartment of dimensions 12 × 12 × 3 m. In each test, the value of fire load was taken to be 40 kg/m2 of wood for the full floor area. The tests used a fire load of 100% timber and also 80% timber/20% plastic by calorific value. A variation in ventilation was provided through openings at the front and back. In all tests there is strong evidence of stratification and temperature gradients close to the openings. Despite the evidence of temperature gradients, very few experiments exist to demonstrate their significance. However, numerical modelling can provide a useful insight into this, as described below. Figure 3 shows the results of illustrative CFD simulations, using FDS, in which we examined an identical fire embedded in compartments with three different aspect ratios, for ‘postflashover’ conditions. It can be observed that temperature variations greater than 600oC exist throughout the compartment in each case. Furthermore, analysis of the predicted soot volume fractions also shows well-defined distributions. These observations seem to further establish that

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Figure 3: Example of three FDS calculations of a compartment fire where the contour lines represent 100°C intervals. For all three cases the compartment cross section is 4 m × 4 m and the lengths are (a) 4 m, (b) 8 m and (c) 16 m. For all cases the heat-release rate per unit area is a 1,000 kW/m2 propane fire distributed across the whole floor surface. All solid surfaces are made out of concrete and the grid size is approximately 0.3−0.5 m in all directions. The ventilation opening is 4 m wide by 2.5 m in height.

the basic premise of a single compartment temperature might be over-simplified, even beyond the growth phase. The obvious consequence of this is the need to compute the local temperatures and to solve the radiative transport equation to establish true boundary fluxes. This can only be done using appropriate CFMs or through experimental characterization of the radiative fluxes to the different surfaces [31]. Section 5.1 will discuss the state-of-the-art in compartment fire modelling. 5.1 Compartment fire models (CFMs) The current practice for input to thermal and structural calculations is normally to ignore the fire growth period. As a first order of magnitude approach, this might be appropriate but if CFMs are going to be used to define the heating, this simplification is unnecessary. The objective of a CFM will be to provide a much more detailed evolution of the conditions within the compartment where the fire originated and in adjacent areas. For a particular fire scenario it is possible that flashover might be attained within the compartment of origin before any structural element has

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undergone significant heating; nevertheless none of the adjacent compartments will be expected to have reached fully developed conditions. Furthermore, growth beyond the compartment of origin will generally be within the same timescales as the heating of structural elements. Under the principle that fire resistance is given directly by the temperature of the structural elements, neglect of the growth period might not matter and would result simply in conservative requirements for insulation. But if the behaviour of the structure is to be studied dynamically and in an integral manner [40] then the results are uncertain. If the objective is to integrate CFMs with structural analysis, significant effort is necessary to establish realistic timescales and characteristic conditions of fire growth beyond the compartment of origin. Experimental validation should follow because little or no useful data exists. It is of critical importance to note that whilst extensive experimental data has been gathered on the evolution of the temperatures within a compartment very little information exists on the evolution of the heat fluxes imposed on a compartment surface. Among the few studies where fluxes were measured under post-flashover conditions are the large compartment fire tests reported by Welch et al. [39]. Since the early 1970s a wide variety of CFMs have been developed. Initially the term ‘Model’ referred to either analytical or empirical formulations that allowed simple calculations associated with the growth of a fire within a compartment. Computer-based models rapidly followed and were developed on the framework established by these analytical expressions and experimental data. Computer tools available for fire modelling in the 1970s favoured the development of zonemodels. Zone models require simple computations; therefore, they were an appropriate solution given the computational constraints of the time. A number of variants emerged and their use became generalized towards the end of this decade. Only since the 1980s has advancement in computer technology made CFD or field models a viable alternative for fire-related calculations [41]. Currently, a wide range of CFD computer models, including fire-dedicated programs, exists and they compete well with traditional zone models. Analytical and experimental formulations are still used to gain insight into the behaviour of fires within compartments, but due to the multiple variables and complexities of the problem, quantitative predictions are now mostly obtained from numerical computations. CFMs are discussed in more detail below, and for models it will be understood ‘computer models’. Special emphasis is given to the evaluation of the models pertaining to the proper quantification of eqn (6) and to the depth of understanding of these models which is required to guarantee proper use. By dissecting the particular application of CFMs to structural analysis the advantages of this approach will be introduced and a number of limitations of the different methodologies will be highlighted. Furthermore, this section attempts to provide a brief overview of the applicability of current models, with consideration of gaps in pertinent predictive capabilities, input data requirements, required assumptions and their effect on predictive results. Given that this is not a review, the objective is not a comprehensive analysis but to simply highlight some relevant considerations. 5.1.1 General remarks on computer-based CFMs Numerous reviews on computer-based CFMs have been made in the past [42–46] and it is not the objective of this section to provide a new one. These models have been traditionally divided in two groups: zone models and CFD or field models. This division is still relevant and will be used here. Reviews available in the literature are of two types: surveys and summaries of features. Surveys collect data on all existing models and provide a list of them with some brief description of the code, its developers and application. Friedman [42] has published the most comprehensive

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survey of this type, which has more recently been updated and extended by Olenick and Carpenter [43]. Both surveys provide a complete list of all known existing models, their sources and applications. No critical review of the models is provided. Summaries of features have been published in much greater number, the last of them being the reviews by Walton [44] and Cox and Kumar [41], covering zone and CFD models, respectively. These summaries of features expose the basic principles of the CFMs, numerical techniques and applications. In most cases limitations are presented only within a general context. The rapid increase on the usage of CFD codes for fire has prompted more detailed reviews recently, of which the more comprehensive are those of Novozhilov [45] and Luo [46]. As mentioned above two different fundamental methodologies can be used for prediction of compartment fire behaviour. The first is zone models and the second is CFD; these are described below. 5.1.2 Zone models Zone models generally treat compartments as a control volume subdivided into two smaller control volumes. It is assumed that within each control volume all properties and conditions are homogeneous. One control volume considers the combustion products mixture and the other the fresh air. Flow, temperature and species fields within these control volumes are not spatially resolved. Any adjacent compartments are linked by mass and energy transfer between them. The solution of the flow, which is the most computationally intense aspect of these calculations, is thus avoided by this simple two-zone approach. All heat transfer related quantities within these codes are established in an empirical manner; therefore, no general comments on the limitations of these codes will be provided at this stage. Significant experimental validation of the principles of this methodology has been generated in the last three decades and its limitations have been many times described. The reader is referred to Walton [44] for detailed information. Two-zone models are by definition limited when analysing heat transport from the gas phase to the solid phase. They avoid the solution of the fluid mechanics equations, thus allow for faster computations and more complex scenarios. Nevertheless, they rely on empirical correlations at all levels of heat and mass transfer. These empirical correlations have in general no direct link with the burning conditions, thus the convective heat transfer coefficients and radiation heat transfer representations used for a small fire will be the same as for a large fire. Calculation of the convective coefficient (assumed to be natural convection) is via correlations for walls, ceilings and floors (hot surface up or cold surface down) and ceilings and floors (cold surface up or hot surface down). A series of examples extracted from the technical description of the CFAST zone model are presented below for illustration [47, 48]; similar approaches can be found in other zone models. The convective transfer coefficient is generally defined in terms of the Nusselt number [47]: h=

NuL k = C ⋅ RaLn L

(8)

where the Rayleigh number is defined as: RaL = GrL ⋅ Pr =

g b (Ts − Tg ) L3 na

(9)

This number is based on a characteristic length, L, of the geometry. The power n is typically 1/4 and 1/3 for laminar and turbulent flow, respectively. All properties are evaluated at the film

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temperature: Tf = (Ts + Tg)/2. The thermal diffusivity and thermal conductivity of air are also defined as a function of the film temperature, from data in Atreya [49]: a = 1.0 × 10 −9 Tf(7 / 4) [m 2 /s]

(10)

 0.0209 + 2.33 × 10 −5 Tf  k=  [W/mK] 1 − 0.000267Tf  

(11)

Table 1 presents the different correlations employed in CFAST. In two-zone models the Reynolds number cannot be calculated properly since velocity fields are not determined within the two zones. For this reason, the convective heat transfer and the boundary layer have to be calculated in a different way. The thickness of the boundary layer is determined by the temperature difference between the gas zone and the wall or object being heated [47]; thus, convective heat transfer is calculated based only on the temperature difference between the zone and the object. From the principles of the model it is impossible to improve this approach; nevertheless it is important to establish if these correlations could be accurately used to establish the thermal evolution of structural elements in fire. To the knowledge of the authors, there is no systematic evaluation of the performance of zone models when predicting heating of structural elements in compartment fires. Thermal radiation tends to be treated in a more complex manner. Methods such as the fourwall algorithm, derived by Siegel and Howell [50], that solve the net radiation equation, are present in zone models. The objects that participate in the radiation exchange are walls, gases and fires. Heat exchange between layers is also possible. The zones and surfaces are assumed to radiate and absorb like a grey body. Gas layer absorption can be calculated. This method can show adequate results when appropriate absorbance coefficients are applied (e.g. 0.5 and 0.01 for the upper and lower layers, respectively). These coefficients represent reasonable approximations for fires with sooty upper layers and clean lower layers; they are nevertheless fully empirical. For fully developed conditions these coefficients have never been validated [51]. Zone models generally do not include pyrolysis models, thus the user must set pyrolysis rates. Approximate pyrolysis rates for pre-flashover fires are defined by empirical heat-release rates and abundant data is available in the literature. For fully developed fires heat-release rates are ultimately defined by ventilation and a very restricted set of data is available. This is very important because the burning time, and thus the total heat transfer, is strongly influenced by the pyrolysis rates. The emissivity and optical properties of eqns (3) and (4), will also be strongly

Table 1: Different heat transfer correlations employed in CFAST [47, 48]. Geometry

Correlation

(

Restrictions

0.387Ra 1/6

)

2

Walls

L NuL = 0.825 + ______________ ≈ 0.12Ra1/3 9/16 8/27

None

Ceilings and (floors hot surface up or cold surface down) Ceilings and floors (cold surface up or hot surface down)

NuL = 0.13Ra1/3

2 × 108 ≤ RaL ≤ 1011

NuL = 0.16Ra1/3

108 ≤ RaL ≤ 1010

1+[(0.492/Pr)

]

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influenced by the soot absorption coefficient and thus by the soot yield of the fire under each specific condition. This information is currently very limited for fully developed fires. The validity of these models depends on the general applicability of the two-zone representation and of the relevant empirical correlations. The limits of the two-zone approximation have not been studied extensively. Among the empirical correlations, such as those corresponding to the fire plume, the expressions for entrainment rates are critical and their validation under conditions other than free axi-symmetric or line fires is limited [52]. The use of any of these models for complicated geometries has not been validated and thus is questionable or inadmissible for extreme cases. Because of the enforced condition of constant properties in each zone, complicated geometries have to be treated in the same way as less complicated ones. The absence of velocity fields and lack of a turbulence model implies that the convective heat transfer will in any case not be affected by any details of a complicated geometry. Zone model assumptions have also been found to break down in flashover fire scenarios [45], leading to predicted heat-release rates that are lower than the actual ones. This problem has been treated by others using models which allow transition to a single zone representation. A good example is the model OZone [53] that has been successfully used to predict the boundary condition for finite element computations [28]. Finite element computations are used both to do detailed heat transfer calculations and structural modelling. It is important to note that the heat transfer calculations are simple conductive analyses thus will not be addressed in detail here. These calculations are subject to strong simplifications such as neglecting moisture transport in concrete and insulating materials. This will be discussed briefly later. In summary, zone models are inherently limited by their basic assumptions. Nevertheless, they are simple to use, robust in nature and can provide good insight on fire development for simple scenarios. Extensive validation is available in the literature and clear estimates of error can be generated. Even so, the intrinsic limitations of zone models are clearly of great importance when addressing the application to modelling of structures in fires. 5.1.3 CFD codes The main aspect that differentiates amongst different categories of CFD code is the way in which turbulence is modelled. Thus, CFD codes can simplistically be divided into three groups, with models based on Reynolds-averaged Navier−Stokes (RANS), large eddy simulation (LES) and direct numerical simulation (DNS). For the modelling of an environment such as a compartment fire, and given the computational resources currently available, DNS simulations are not feasible for a number of reasons. DNS requires the grid resolution to be as fine as the Kolmogorov microscale. All eddies, down to the dissipation length scale, must be simulated explicitly with sufficient accuracy. The number of DNS grid points required for the resolution of all scales increases approximately with the cube of the Reynolds number (Re3). The Reynolds number for typical fire and smoke movement in a compartment is approximately 105, hence the total number of cells necessary for solving fire and smoke movement in a room is of the order of 1013. Current supercomputers have the capability to provide a grid resolution not greater than 108 cells. Therefore, current computing technology is still completely insufficient to solve such detailed fire flows. DNS, therefore, cannot be used to simulate complicated fire spread and smoke movement in a full-scale compartment. Since full resolution of the Navier−Stokes equations is not practically possible, it is necessary to model some aspects of the flow. The choices of which aspects of the flow will be modelled, and thus the approach to be followed, is difficult and implies inevitable subjectivity. RANS takes the option of solving the ensemble-averaged Navier−Stokes equations by using turbulence modelling, thereby approximating the fluid flow fluctuations via a modelled state. In a RANS solution,

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all dynamical degrees of freedom smaller than the size of the largest (energy-containing) eddies are averaged, so there is no dynamic information about the smaller scales. RANS can be further divided into turbulent viscosity models (such as the k−e model) and Reynolds-stress models. The most widely used turbulent viscosity model in fire is the standard buoyancy-modified k−e model. LES, pioneered in the 1960s−70s by Smagorinsky [54] and Deardorff [55], assumes that turbulent motion can be separated into large-eddies and small-eddies. The motion of the large eddies (super-grid scale) is directly simulated while that of the small eddies (sub-grid scale) is approximated. Since LES solves time-dependent flow, it can provide detailed information on turbulence, such as 3D instantaneous velocities. The key step in both LES and RANS is the derivation of the underlying dynamical equations averaged over small scales. The fundamental difference between LES and (transient) RANS is the definition of small scales. In LES, the small length-scales are smaller than the grid size and in RANS small length-scales are smaller than the largest eddies. If the grid size of an LES simulation is taken to be progressively larger, self-consistency requires that the LES results approach the RANS results [56]. LES techniques always need to be 3D and must have a time-step short enough to capture most of the important turbulent motion. Because of this, LES is computationally more expensive than RANS, especially for the case of essentially time-invariant problems where RANS codes can be run in a quasi steady-state mode. Nevertheless, recent advances in computer performance and numerical methods have meant that LES is becoming more feasible for such fire and smoke flow problems. Some general limitations for both RANS and LES approaches to the modelling of turbulent flows relevant to fires can be established as described below. RANS codes average locally over time/space, thus all dynamic information for scales smaller than the large turbulence scales is lost. For the calculation of the thermal response of structural elements this might not be significant, since the time-scales of solid heating are much larger than those of gases. Nevertheless, the loss of dynamic information could significantly affect the predictions of fire growth and therefore needs to be handled with great proficiency. LES does not average over time so it allows modelling the time evolution of the sub-grid scales. This can be translated into a better resolution of the time evolution of the fire. To achieve computations within reasonable time constraints, this requires an increase in the cell size; thus large grid cells typically characterize LES solutions. The grid cells can be much larger than the flame thickness; therefore the temperatures of each cell typically represent an average of reactive and non-reactive regions. Thus, the capability of these codes to properly predict the temperatures of combusting mixture, and thus radiative heat transfer, is questionable. Furthermore, LES modelling implies a proper definition of the grid size that is consistent with the model parameters and with the computational constraints. A reduction of the grid does not always produce an improvement in precision. Determination of the grid requires pre-acquired empirical knowledge, or independent computations [45]. RANS relies on numerous empirical model coefficients (between 7 and 12 different parameters) that will describe turbulent viscosity and fluid wall interactions. These functions are well defined for high Reynolds numbers with homogeneous turbulence but difficult to establish for transitional flows with constraint boundaries such as those to be expected close to the surfaces of structural elements. Wall functions have been established to address these areas but their accuracy and generality is still questionable [57]. Diffusion flames representative of fires are generally considered (spatially) thick, thus the validity of the direct application of RANS flamelet models and simple LES combustion models could also be questioned [57]. Despite this statement, proficient use of these models can provide adequate results. LES also relies on empirical model coefficients (e.g. Smagorinsky constant) but their calibration is easier and is expected to be independent of the Reynolds number. In fact, the Smagorinsky

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constant can be avoided entirely with a dynamic sub-grid model [56]. Calibration of the model coefficients has been done for a multiplicity of scenarios but these rarely include conditions typical of fires [56]. Dynamic sub-grid models are beginning to appear in the fire literature but are still a topic of ongoing research and are not fully validated. Proper combustion models are necessary to generate correct heat-release rates (thus temperatures) and species distributions. To achieve proper temperature predictions it is also necessary to adequately represent radiative heat transfer. For radiation to be properly modelled the most important aspects are temperature and soot concentrations/morphology. Significant work on the development of combustion models, radiation models, and soot models is currently underway [41, 45, 46]. These models are being incorporated into numerical tools on a constant basis. Currently, existing models have been validated only in simple scenarios, for simple fuels and with very limited diagnostics. Common validations rely on simple comparison with temperature measurements [58, 59] that in many cases are decades old [60]. These validation exercises are clearly not sufficient to determine the adequacy of the complex models proposed. The relatively long timescales which are relevant to structural behaviour imply in most cases fully developed fires. Although validation exercises under these conditions exist [61, 62], they still remain insufficient. The data available for post-flashover, fully developed fires, are generally in the form of average point measurements of temperature [37, 38, 63, 64] which are more suited for the validation of zone models than CFD codes. Combustion and soot models are greatly sensitive to the burning conditions; therefore, the capability of existing models to provide reasonable predictions under fully developed fire conditions remains largely untested. Independent of the model used, all numerical tools are severely limited by any improper definition of the fundamental properties of materials controlling fire growth. An analysis of the input variables for all flammable materials shows a fundamental dependence on values in databases which are typically very simple and approximate [2, 3, 65, 66]. The errors that can be induced by an improper or incomplete selection of material properties can be more important than those generated by an incorrect specification of the parameters of the numerical flow and combustion models. In addition, the conclusion of a recent ‘Round Robin’ assessment of fire modelling was that the effect of user choices on uncertain inputs, including reaction-to-fire properties, can have a dominant effect on the outputs obtained from numerical codes in general [67]. These general limitations to such codes are by no means insurmountable, but improvement and confidence can only be achieved with systematic and careful validation and by improved guidance and training for users. In their current state, all CFD codes require great proficiency in their use and by far the biggest challenge is to guarantee that the users are making a proper use of the tools.

6 Solid-phase phenomena Numerous other effects have to be considered when modelling the temperature evolution of structural elements. Many of these are stochastic in nature, thus are very difficult to describe in detail. In a similar way, some other processes have well defined physical variables but their complexity makes their modelling intractable. The mechanical breakdown of insulating coatings corresponds to the former type and the behaviour of intumescent coatings and moisture migration correspond to the latter. 6.1 Material integrity Good examples of the impact of mechanical failure of fire proofing are illustrated in the WTC analysis [16], where the aircraft impact seemed to have resulted in significant dislodging of the

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fire proofing, and on the experimental study of cellular beams reported by Bailey [68]. In the latter study the fireproofing broke off around the perforations of cellular beams resulting in more severe heating of the web. The uncertainty associated with these processes makes unjustifiable a more detailed analysis and suggests that reliance on empirical data is necessary.

6.2 Treatment of moisture and other chemical processes This section covers briefly the heating of concrete and thermal insulation such as plasterboard and focuses mostly on the treatment of moisture. Concrete is not a homogenous material like steel, for example, but it is composed of cement gel, aggregate and, frequently, steel or polymer fibre reinforcements. Each of these components has a different reaction to heat in itself, and the behaviour of the composite system in fire is not easy to define or model. The heating process of concrete occurs within a changing porous matrix where phase changes and complex chemical processes coexist [69]. Thus, it is common for design codes to focus only on basic heat transfer and average properties and bypass the complexity. The result is the simple specification of a certain depth of concrete cover to the reinforcement bars in a concrete structure to provide an insulating effect upon the steel. The depths are generally defined on the basis of empirical data. There are a number of physical and chemical changes which occur in concrete subjected to heat. Some of these are reversible upon cooling, but others are non-reversible and may significantly weaken the concrete structure after a fire [70, 71]. Most porous concretes contain a certain amount of liquid water. This will obviously vaporize if the temperature significantly exceeds the moisture plateau range of 100−140°C or so, possibly causing a build-up of pressure within the concrete. If the temperature reaches about 400°C, the calcium hydroxide in the cement will dehydrate, generating further water vapour and also bringing about a significant reduction in the physical strength of the material. Other changes may occur in the aggregate at higher temperatures, for example quartzbased aggregates increase in volume, due to a mineral transformation, at about 575°C and limestone aggregates will decompose at about 800°C. When modelling high temperature behaviour of concrete, in general, these processes will be treated as volumetric heat sources or sinks, nevertheless very little work is available on this subject. One of the most poorly understood processes in the reaction of concrete to high temperatures or fire is that of ‘explosive spalling’. This is the phenomenon whereby chunks of concrete are ejected from the surface of the concrete slab, often at fairly high velocities. This process is generally assumed to occur at high temperatures, yet it has also been observed to occur in the early stages of a fire and at temperatures as low as 200°C. If significant amounts of spalling occur in a fire, the layer of concrete covering the reinforcement bars may be greatly reduced and the reinforcement may be exposed to high temperatures, thus the importance of this process on heat transfer calculations relevant to structural response is great. The mechanism leading to spalling is generally thought to involve large build-ups of pressure within the porous concrete which the structure of the concrete is not able to dissipate, so fractures occur and chunks of the material are forced outward [72]. Thermal insulation is commonly used to protect structural elements. Currently, many different materials are used for this purpose, therefore it is difficult to address here specific heat and mass transfer processes involved in the calculation of the protection to fire exposures that these materials can provide. Most methods are based on empirical results obtained from furnace tests [1] and are summarized by Milke in ref. [11]. Dry wallboard or ‘Gypsum’ wallboard is a commonly used insulation material that exhibits many of the common issues associated with fire exposure of

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insulation, thus will be used as an example. ‘Gypsum’ wallboard is a multi-layered material formed by external layers of paper and a calcium sulphate dihydrate (CaSO4·2H2O) porous core. Calculation of heat transfer through this material requires quantification of the dehydration endothermic reactions that result in the release of chemically bonded water. Heat transfer from the fire to the structure is in many cases limited by the presence of this heat sink. Kukuck and Prasad [73] indicate that water content in US produced ‘Gypsum’ wallboard consists of approximately 24% by mass. Currently, there are several models that treat the effects of moisture on the thermal response of wallboard to fire, but in most cases they need to resort to either empirical formulations or strong simplifying assumptions [74].

7 Conclusions Heat and mass transport effects have been explored to provide the reader with the necessary material and references to understand the complexity of the transfer processes involved in the modelling of structural behaviour in fire. Emphasis has been given to the definition of the boundary condition as opposed to the more conventional conductive heat transfer calculations in the solid. A review of the different approaches used to establish the thermal boundary condition required to make a detailed analysis of a structure in the event of a fire has been presented. A series of general comments on the validity and limits of the different current approaches has been provided. These general comments give a guideline to areas that need further attention. From this evaluation it emerges that the only general way to precisely model the thermal boundary condition is via numerical models. Other techniques need to resort to strong simplifications. The impact of those simplifications on accuracy could be minor, but has not been fully established and will be highly dependent on application. Numerical models can be of significant use in assessing the relevance of many of the assumptions embedded in current calculation methodologies. A review of the most commonly used modelling approaches then reveals that currently these techniques also have limitations. Many of these can be circumvented by proficient and experienced use, but the lack of detailed validation still remains a serious problem. The afore-mentioned limitations, together with the complexity of the models, imply that at this stage computer-based models for compartment fires are at a level of development that enables their effective exploitation only by qualified users. In this respect, special mention has to be made of CFD-based tools where improper definition of the input parameters and user variables can result in output of unknown accuracy. Finally, it is also important to note that these are generally complex tools and therefore specific improvements will have to be studied within the context of specific tools.

Acknowledgements The numerous comments and contributions of Prof. Asif Usmani from the University of Edinburgh and Dr Barbara Lane and Dr Susan Lamont from Arup are gratefully acknowledged.

Nomenclature cp fs g

specific heat capacity (J/kg K) soot volume fraction (ppm) gravitational acceleration (m/s2)

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Transport Phenomena in Fires

h k . q'' t x A Gr H L Nu Pr Ra T V

heat transfer coefficient (W/m2 K) thermal conductivity (W/m K) heat flux (W/m2) time (s) distance into depth of material (m) area (m2) Grashof number (−) height of openings (m) characteristic length (m) Nusselt number (−) Prandtl number (−) Rayleigh number (−) temperature (K) volume (m3)

Greek letters a b e k r n

thermal diffusivity (m2/s) volumetric thermal expansion coefficient (K−1) emissivity (−) absorption coefficient (m−1) density (kg/m3) kinematic viscosity (m2/s)

Subscripts and superscripts b e f g n o r s L S T

building evacuation floor, film gas, gas layer power index opening radiative surface characteristic length structure temperature, total

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Heat and Mass Transfer Effects

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CHAPTER 5 Weakly buoyant turbulent fire plumes in uniform still and crossflowing environments F.J. Diez1, L.P. Bernal2 & G.M. Faeth2,† 1

Department of Mechanical and Aerospace Engineering, Rutgers, The State University of New Jersey, USA. 2 Department of Aerospace Engineering, University of Michigan, USA.

Abstract Recent theoretical and experimental studies of the temporal and steady properties of weakly buoyant round turbulent plumes (i.e. turbulent plumes where effects of thermal radiation are small and physical property variations can be properly described by the Boussinesq approximations) in uniform still and crossflowing environments are reviewed. Consideration of these flows is motivated by numerous practical applications to the unconfined flows resulting from starting and steady releases of buoyant gases and liquids (fluids) from unwanted fires, from industrial exhaust stacks, from explosions, and from process upsets, among others. The present discussion of turbulent plumes properties emphasizes results far from the source, where effects of source disturbances are lost and the plumes approximate self-preserving buoyant turbulent flows. Then, the plumes are dominated by their conserved properties and appropriately scaled plume properties are independent of streamwise distances from the source. Findings for plumes in uniform still environments show that the penetration properties of both starting and steady plumes in uniform still environments exhibit self-preserving scaling at streamwise distances from the source greater than 20−30 source diameters. Mean and fluctuating flow structure properties, however, do not achieve self-preserving behavior until streamwise distances exceed 80−90 source diameters from the source. Findings for plumes in uniform crossflowing environments also show that they exhibit self-preserving behavior at somewhat greater distances from the source. Initial streamwise source velocities decay relatively rapidly for plumes in uniform crossflows, however, so that they eventually deflect into the nearly crosstream direction for all values of the crosstream velocity. As a result, their self-preserving structure consists of two counter-rotating vortices whose axes are nearly aligned with the crosstream direction, and move away from the source in the streamwise direction due to the action of buoyancy. The onset of self-preserving behavior requires that the

†

Professor G.M. Faeth, our colleague, mentor, and friend, passed away unexpectedly in January 2005. This chapter is dedicated to his memory.

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axes of the counter-rotating vortex system be nearly aligned with the crosstream direction. In turn, achieving this alignment is a strong function of the source-crossflow velocity ratio.

1 Introduction Recent progress toward understanding and modeling the structure and penetration properties of round buoyant turbulent starting plumes and steady plumes (denoted starting plumes and steady plumes in the following) in still and crossflowing unstratified uniform environments is reviewed. These flows are of interest for several reasons: they have practical applications to interrupted, developing, and steady nonreactive gas and liquid releases caused by process upsets, explosions, and fires; they are simple classical flows that are relatively easy to interpret in order to better understand the properties of steady and unsteady buoyant turbulent flows; and they involve uncomplicated geometries having well-defined initial and boundary conditions that provide measurements useful for evaluating models of practical buoyant turbulent flows. Study of these flows generally has been limited to conditions far from the source where analysis based on the self-preserving turbulent flow approximation has proven to be surprisingly successful. This approach has provided convenient ways to interpret and correlate the properties of relatively complex flows such as starting plumes and steady plumes in still and crossflowing environments. Motivated by these observations, the present review will consider the properties of these flows emphasizing self-preserving conditions. The review begins with consideration of the structure properties of round buoyant turbulent flows, where past work has largely been limited to steady plumes in still environments so that measurements are reasonably tractable. Subsequent discussion of flow penetration properties will involve unsteady flows in both still and crossflowing unstratified uniform environments, considering the penetration properties of the following flow configurations: starting plumes in still environments, and starting and steady plumes in crossflows. Discussion of the various flows can be read independently and each is organized as follows: introduction, experimental methods, theoretical methods, results and discussion, and conclusions. Detailed methods of modeling these flows have been addressed by others (see Baum et al. [1−3] and references cited therein); therefore, present considerations emphasize measurements and approximate analysis needed to find self-preserving scaling laws. Finally, several reviews of early work concerning buoyant turbulent flows have appeared (see Turner [4, 5], Tennekes and Lumley [6], Hinze [7], Chen and Rodi [8], List [9], and references cited therein); therefore, the present discussion emphasizes more recent studies.

2 Structure of steady plumes in still environments 2.1 Introduction The structure of steady plumes in still environments is an important fundamental problem that has attracted significant attention since the classical study of Rouse et al. [10]. The earliest work of Rouse et al. [10], Morton et al. [11], and Morton [12] concentrated on the scaling of mean flow properties within fully developed plumes, where flow properties become independent of source disturbances and source momentum and satisfy self-preserving scaling laws that provide simple correlations of appropriately normalized flow properties. Measurements of mean properties generally have satisfied the resulting scaling relationships, however, there are considerable differences among the various determinations of centerline property values, radial distributions

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of flow properties, and flow widths (see refs. [8−27] and references cited therein). Aside from problems of experimental methods in a few instances, List [9] and Papanicolaou and List [22, 23] attribute these differences to problems of reaching fully developed, self-preserving steady plume conditions. One parameter that is helpful for estimating when steady plumes become self-preserving is the distance from the virtual origin normalized by the source diameter, (x – xo)/d. This parameter is a measure of conditions where distributions of flow properties appropriate for the often confined conditions of a source, have adjusted to reach distributions appropriate for an unconfined plume. The value of (x – xo)/d for self-preserving behavior depends on the nature of the flow, the properties of the source, and the property of the flow for which self-preserving behavior is sought. For example, based on results for nonbuoyant round turbulent jets, values of (x – xo)/d greater than roughly 40 and 100 should be required to obtain self-preserving flow structure properties, based on distributions of mean and fluctuating (turbulent) properties, respectively [6]. By these measures, the past measurements of plume structure [10−25], which generally involved buoyant jets for the plume source, probably involve transitional plumes, e.g. they generally are limited to (x – xo)/d ≤ 62. The main reason for not reaching large values of (x – xo)/d for plumes, similar to jets, is that scalar properties decay much faster for plumes than for jets, e.g. proportional to (x – xo)−5/3 for plumes compared to (x – xo)−1 for jets [9−12, 26, 27]. Thus, an unusually large dynamic range for measuring scalar properties is required to maintain experimental uncertainties at reasonable values far from the source within plumes. Contributing factors are that plume velocities are relatively small compared to jet velocities so that using velocity measurements to define self-preserving plume properties is also problematical and controlling room disturbances far from the source is more difficult for plumes than for jets due to the relatively small streamwise momentum levels of plumes. A second parameter that is helpful for estimating when steady plumes become self-preserving is the distance from the virtual origin normalized by the Morton length scale, (x – xo)/ℓM, as a measure of conditions where the momentum of the plume resulting from effects of buoyancy is much larger than the momentum of the flow at its source. For general buoyant jet sources the Morton length scale is defined as follows [9, 12, 21]: M = M o3 / 4 / B o1/ 2

(1)

For steady round plumes with uniform properties defined at the . source (similar to the flows of interest in the following), the source specifi c momentum fl ux, M o, and the source specific buoy. ancy flux, Bo, can be computed as follows [9, 28]: M o = (π / 4)d 2uo2

(2)

B o = (π / 4)d 2uo g ro − r∞ / r∞

(3)

where an absolute value has been used for the density difference in eqn (3) in order to account for both rising and falling plumes. Substituting eqns (2) and (3) into eqn (1) yields the following expression for the Morton length scale of steady round plumes having uniform source properties: M / d = (π / 4)1/ 4 ( r∞ uo2 /( gd | ro − r∞ |))1/ 2

(4)

The ratio, ℓM/d, is proportional to the source Froude number, which is defined as follows for conditions appropriate for the use of eqn (4) [9]: Fro = (4 / π)1/ 4 M / d = ( r∞uo2 /( gd | ro − r∞ |))1/ 2

(5)

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The source Froude number is a convenient measure of the dominance of buoyancy at the source, e.g. Fro = 0 and • for purely buoyant and purely nonbuoyant sources, respectively. Thus, (x – xo)/ℓM, is a reasonable measure of conditions where effects of source momentum have been lost and the momentum of the flow is dominated by effects of buoyancy. For example, purely buoyant sources have Fro = ℓM = 0 and are immediately dominated by effects of buoyancy whereas purely nonbuoyant sources have Fro = ℓM = • and are never dominated by effects of buoyancy. Papanicolaou and List [21, 22] suggest that the buoyancy-dominated conditions for mean and fluctuating quantities in plumes are reached when (x – xo)/ℓM is greater than roughly 6 and 14, respectively. A greater proportion of the measurements of mean properties in refs. [10−27] exceed this criterion, however, the effects of transitional plume behavior (in terms of (x – xo)/d) on these observations raises questions about the accuracy of these values of (x – xo)/ℓM for the onset of self-preserving behavior for plumes. A third criterion also should be satisfied when self-preserving conditions are reached; namely, that the density of the plume fluid should be linearly related to the degree of mixing of ambient and source fluid [29]. At self-preserving conditions in plumes, which generally are far removed from the complexities of practical sources such as fires, all scalar properties are conveniently represented by functions of the mixture fraction (which corresponds to the mass fraction of source material in a sample), called state relationships [28−30]. Typically, the mixture fraction, f << 1, at self-preserving conditions so that the state relationship giving the density as a function of mixture fraction can be linearized as follows [28−30]: r = r∞ + f r∞ (1 − r∞ / ro ),

f << 1

(6)

This provides a useful relationship for the driving potential for effects of buoyancy, as follows: d(ln r) / df

f →0

= ro − r∞ / ro

(7)

Virtually all the studies in refs. [10−27], seeking to observe self-preserving steady plume properties, satisfy the f << 1 criterion required for a linear relationship between mixture fraction and density. The preceding discussion suggests that the measurements of mean and fluctuating properties within plumes [10−27] mainly involved transitional plumes combined with experimental difficulties in some instances [28]. Prompted by this observation, a series of studies were undertaken by Dai et al. [28−31], seeking to better identify the conditions required to observe self-preserving round plumes in still and unstratified environments, and to determine the mean and turbulent structure of the flow at these conditions, including mixture-fraction, velocity, and velocity/mixturefraction statistics. These studies will be described in the following sections, considering experimental methods, theoretical methods, results and discussion, and conclusions, in turn. The following discussion is brief; see Dai et al. [28−31] for more details. 2.2 Experimental methods A sketch of the experimental apparatus for observations of steady round plumes in still and unstratified environments appears in Fig. 1. The test plumes were within a screened enclosure that was located in turn within an outer plastic enclosure. The screened enclosure could be traversed in order to accommodate rigidly mounted optical instrumentation. The plume sources were long round tubes having various diameters, inlet flow straighteners, and length-to-diameter ratios of 50:1. The sources could be traversed in the vertical direction to allow measurements at different streamwise positions. The source flows were either carbon dioxide or sulfur hexafluoride

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165

Figure 1: Sketch of the test apparatus for steady plumes in still environments (from Dai et al. [28]).

to provide negatively buoyant gaseous plumes that were exhausted from the test area using a dispersed exhaust system located at the floor. Source gas flows were controlled and measured using pressure regulators in conjunction with critical flow orifices. Mean and fluctuating mixture fraction properties were measured using laser-induced fluorescence (LIF) of iodine vapor seeded into the source flow. Seeding was done by passing a portion of the source flow through a bed of iodine crystals and then mixing this flow back into the main source flow. The LIF signal was produced by the unfocused beam of an argon-ion laser (at the 514.5 nm green line) which is absorbed by iodine and causes it to fluoresce at longer wavelengths in the visible yellow portion of the spectrum. The LIF signal was separated from scattered laser light using long-pass filters having 520 nm cut-off wavelengths. The detector outputs were sampled at constant time intervals and processed using a digital laboratory computer. Effects of preferential diffusion of iodine relative to carbon dioxide and sulfur hexafluoride were small; see Dai et al. [28] for additional details about monitoring and processing LIF signals and estimates of experimental uncertainties. Mean and fluctuating velocity properties were measured using dual-beam, frequency-shifted laser velocimetry (LV), based on the 514.5 mm line of an argon-ion laser. The optical axis of the LV passed horizontally through the flow with signal collection at right angles to the optical axis.

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The beam plane was rotated in order to measure the streamwise and crosstream components of velocity. The detector output was processed using a burst counter and the low pass-filtered analog output of the processor was sampled at equal time intervals in order to avoid velocity bias, whereas directional bias and ambiguity were controlled by frequency shifting. The source flow was seeded with oil drops with seeding levels controlled so that step noise contributed less than 3% to determinations of velocity fluctuations; see Dai et al. [30] for additional details about processing LV signals and experimental uncertainties. Finally, combined measurements of mixture fractions and velocities were undertaken using combined LIF/LV as described by Dai et al. [29]. Source conditions for the carbon dioxide and sulfur hexafluoride plumes were as follows: source diameters of 9.7 and 6.4 mm, mean velocities of 1.74 and 1.89 m/s, Reynolds numbers of 2,000 and 4,600, Froude numbers of 7.80 and 3.75, and values of ℓM/d of 7.34 and 3.53. Ambient air properties for these flows involved pressures and temperatures of 99 ± 0.5 kPa and 297 ± 0.5 K which implies an ambient density and kinematic viscosity of 1.16 kg/m3 and 14.8 mm2/s. 2.3 Theoretical methods Analysis has been undertaken to provide the self-preserving scaling laws for mean and fluctuating mixture fractions, velocities, and combined mixture-fraction/velocity properties. A sketch of the steady plume arrangement considered appears in Fig. 2. The following assumptions were made about the flow: the flow is steady and axisymmetric in the mean, the source flow is known and is aligned with the gravitational vector, the ambient gas is known and is motionless and unstratified, the state relationship for density is linear according to eqns (6) and (7) and satisfies the Boussinesq approximation (e.g. |r – r•|/r• << 1), the boundary layer approximations apply, Reynolds numbers are sufficiently large so that molecular transport can be neglected, and the

Figure 2: Sketch of a steady plume in a still environment.

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167

flow is self-preserving. Under these assumptions, conservation principles and the state relationship for density imply that the buoyancy flux in the streamwise (vertical) direction is conserved for buoyant turbulent plumes. Under the previous assumptions, mean mixture fractions and streamwise velocities can be scaled as follows in the self-preserving region, based on analyses by Chen and Rodi [8], List [9], Rouse et al. [10], Morton et al. [11], and Morton [12]: fg d(ln r ) / df

f →0

( x − xo )5 / 3 / B o2/3 = F (r /( x − xo ))

u (( x − xo ) / B o )1/ 3 = U (r /( x − xo ))

(8) (9)

where the virtual origin location of the source, xo, has been introduced in order to extend the region where self-preserving behavior is observed as much as possible toward the source. Notably, except for the virtual origin location, the only source properties appearing in eqns (8) and . (9) are the plume buoyancy flux, Bo, and the source density, ro; this behavior follows because the details of the source are not important in the self-preserving region, only the buoyancy flux, which is a conserved plume property, and ro which affects the driving potential for buoyancy effects. Finally, recalling eqn (7), eqn (8) can be rewritten to more directly indicate the roles of source and ambient density on flow mixing properties, as follows: fg 1 − r∞ / ro ( x − xo )5/3 / B o2/3 = F (r /( x − xo )) (10) . Viewed in this form, eqns (9) and (10) clearly indicate that Bo and ro are the only relevant properties of the source in the self-preserving region of the plumes. All other variables in the self-preserving region of plumes can be normalized in terms of the scaled − variables of eqns (8)−(10), or equivalently in terms of the centerline mixture fraction f c and center– line velocity uc, which is typical of methods for reducing turbulence properties for self-preserving flows (see Tennekes and Lumley [6], Hinze [7], Chen and Rodi [8], List [9], and references cited __ __ __ – – therein). Examples along these considered in the following include f ¢/fc, rv–/((x–xo)uc), u ¢/uc, ______ _____ lines – __ _____ – __ – __ f ¢w¢/(fc uc), f ¢v¢ (fc uc), and f ¢v¢/ (fc uc), where primes denote fluctuating quantities. The functions F(r/(x – xo)) and U(r/(x – xo)) in eqns (8)−(10) are appropriately scaled radial distribution functions of mean mixture fractions and streamwise velocities, that are universal functions in the self-preserving region far from the source where eqns (8)−(10) apply. The radial distribution functions typically are approximated by Gaussian fits, as follows [8−10]: F (r /( x – xo )) = F (0)e

– k 2f ( r /( x – xo ))2

U (r /( x − xo )) = U (0)e − ku (r /( x − xo )) 2

2

(11) (12)

where k f = ( x − xo ) / f ,

k u = ( x − xo ) / u

(13)

are constant plume width coefficients in the self-preserving region and ℓf and ℓu are characteristic – – __ __ plume radii where f /fc = u/uc = exp(−1), respectively. Finally, defining a characteristic plume Reynolds number as follows: Rec = lu uc / v∞

(14)

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__

Then substituting for uc and ℓu in eqn (14) from eqns (9) and (13), yields: (15) Rec = U (0)( B o ( x − xo )2 )1/ 3 /(ku v∞ ) . Noting that U(0) and ku are universal constants, whereas B o and v• are constants for any given plume, eqn (15) shows that the Reynolds numbers for self-preserving plumes progressively increase with increasing distance from the source, proportional to (x – xo)2/3, implying progressively larger ranges of length scales, and ratios of macro/micro scales of the turbulence, as the distance from the source increases [6]. 2.4 Results and discussion Measurements of steady plumes undertaken by Dai and coworkers [28−31], in order to define conditions required to achieve self-preserving behavior and to find the properties of self-preserving plumes, will be considered in the following. A picture of the development of transitional plumes toward self-preserving conditions that can be obtained from the radial distributions of mean mixture fractions for both carbon monoxide and sulfur hexafluoride source flows is illustrated in Fig. 3. In this case, the scaling parameters of eqn (10) are used so that the ordinate is equal to F(r/(x – xo)). The measurements are plotted for various streamwise distances with (x – xo)/d ≥ 7. The radial distributions of mean mixture fractions exhibit a progressive narrowing,

Figure 3: Development of radial distributions of mean mixture fractions as a function of distance from the source for steady plumes in still environments (from Dai et al. [28]).

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with the value of F(0) progressively increasing, with increasing distance from the source. Selfpreserving conditions, where the scaled mean mixture fraction distributions no longer vary with increasing streamwise distance, are observed when (x – xo)/d ≥ 87, which also corresponds to (x – xo)/ℓM ≥ 12. This range of conditions also corresponds to characteristic plume Reynolds numbers of 2,300−5,900, which are reasonably large for unconfined turbulent flows [6−8], e.g. this range is comparable to the largest wake Reynolds numbers where measurements of round steady turbulent wakes have been reported, whereas turbulent wakes exhibit self-preserving turbulence properties at characteristic wake Reynolds numbers as small as 70 [32]. Within the self-preserving region, the radial distributions of mean mixture fractions were reasonably approximated by a Gaussian fit, as mentioned in connection with the discussion of eqn (11); the parameters of this fit will be discussed later so that mean streamwise velocity properties can be considered at the same time. Finally, a surprising feature about the results of Fig. 3 is that reaching self-preserving conditions for mean quantities is significantly delayed beyond the region generally considered during past studies of the self-preserving properties of steady plumes [10−25], and are also significantly delayed from conditions required for self-preserving mean properties within round nonbuoyant turbulent jets, (x – xo)/d = 40, discussed by Tennekes and Lumley [6]. The approach of steady plumes to self-preserving turbulence properties is illustrated in Fig. 4, where radial distributions of rms mixture fraction fluctuations (denoted mixture fraction fluctuations

Figure 4: Development of radial distributions of rms mixture fraction fluctuations as a function of distance from the source for steady plumes in still environments (from Dai et al. [28]).

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in the following) are plotted in terms of self-preserving variables at various streamwise distances from the source. Near the source, the distributions are rather broad and exhibit a dip near the axis, much like the behavior of nonbuoyant jets [21, 22, 26, 33]. The mixture fraction fluctuation distributions evolve, however, with both the width and the magnitude of the dip near the axis gradually decreasing with increasing streamwise distance. Eventually, self-preserving conditions are reached for mixture fraction fluctuations at conditions similar to the self-preserving conditions for mean mixture fractions, e.g. (x – xo)/d ≥ 87 and (x – xo)/ℓM ≥ 12. This behavior is comparable to conditions required for self-preserving turbulence properties in nonbuoyant jets, (x – xo)/d = 100 [6, 26]. This result is also not surprising because self-preserving conditions for mean properties are generally a necessary condition for self-preserving conditions for fluctuating properties for turbulent flows [6]. The gradual disappearance of the dip in mixture fraction fluctuations is an interesting feature of the results illustrated in Fig. 4. The development of the flow from source conditions where mixture fraction fluctuations are smaller than 1% is certainly a factor in this behavior. In addition, the presence of effects of turbulence production due to buoyancy, as (x – xo)/ℓM increases, is also a factor. In particular, nonbuoyant jets have reduced mixture fraction fluctuations near the axis because turbulence production is small in this region due to symmetry requirements; in contrast, effects of buoyancy provide turbulence production near the axis of plumes in spite of symmetry due to buoyant instability in the streamwise direction [21, 22, 33], e.g. the density always approaches the ambient density in the streamwise direction. The radial distributions of mean streamwise velocities in plumes within the self-preserving region, (x – xo)/d ≥ 87 and (x – xo)/ℓM ≥ 12, are illustrated in Fig. 5. These results are plotted according to the self-preserving variables of eqn (9) so that the ordinate is equal to U(r/(x – xo)). The variation of U(r/(x – xo)) was reasonably approximated by a Gaussian fit, as mentioned in connection with the discussion of eqn (12); the parameters of this fit will be discussed later. Consideration of conservation of mass for round turbulent plumes satisfying the Boussinesq approximation shows that rv /(( x − xo )uc ) = V (r /( x − xo ))

(16)

where V(r/(x – xo)) is a universal function for self-preserving plumes [30]. Mean radial velocities in plumes based on the measurements of Dai et al. [30] are plotted in Fig. 6 for self-preserving conditions, (x – xo)/d ≥ 87 and (x – xo)/ℓM ≥ 12. These measurements are compared with results obtained indirectly from measurements of mean streamwise velocities through the conservation of mass expression. The results indicate good internal consistency of the measurements of mean streamwise and radial velocities because they agree with estimates obtained from measurements of mean streamwise velocities through the conservation of mass equation. In addition, the fact that V(r/(x – xo)) is clearly universal for the range of the measurements illustrated in Fig. 6 supports the existence of self-preserving behavior for these conditions. These results also yield direct information about flow entrainment properties that play an important role in simplified integral theories of turbulent plumes [11, 12]. In particular [29]: dQ / dx = (r v )∞ = Eo u uc

(17)

where Eo is the entrainment coefficient of integral theories. Introducing the appropriate self-preserving plume properties into eqn (17) and solving for the entrainment coefficient then yields: Eo = 5 /(6 ku )

(18)

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Figure 5: Radial distributions of mean streamwise velocities for self-preserving steady plumes in still environments (from Dai et al. [30]).

The values of Eo found from the velocity measurements of Dai et al. [30] will also be discussed subsequently. Radial distributions of streamwise velocity fluctuations measured in the self-preserving region of plumes by Dai et al. [30] are illustrated in Fig. 7. These results are plotted in terms of the selfpreserving normalization of velocity fluctuations discussed earlier. Similar to the other plume variables, the streamwise velocity fluctuations are seen to properly satisfy the requirements for self-preserving behavior. In this case, the presence of the dip in streamwise velocity fluctuations near the axis seen in Fig. 7 is similar to behavior seen in nonbuoyant jets (see ref. [22] and references cited therein). Consideration of combined correlations involving mixture fraction and velocity fluctuations also support self-preserving behavior for the same range of conditions found for pure mixture fraction and velocity statistics. Evidence of this behavior is provided in Fig. 8, where radial distributions of turbulent mass fluxes are plotted according to the expectations of self-preserving behavior for (x – xo)/d ≥ 87 and (x – xo)/ℓM ≥ 12. These results show that tangential turbulent mass fluxes are negligible as they should be for an axisymmetric flow, and that direct determinations of radial tangential mass fluxes are internally consistent with measurements of streamwise

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Figure 6: Radial distributions of mean radial velocities for self-preserving steady plumes in still environments (from Dai et al. [30]).

tangential mass fluxes through the governing equation for conservation of mean mixture fractions. In addition, the measurements illustrated in Fig. 8 clearly agree with the expectations of self-preserving behavior. Finally, the contribution of the streamwise turbulent mass flux to the total streamwise mixture fraction transport is significant (roughly 15%) and must be considered for accurate determination of the buoyancy flux of the plume and for conservation checks, e.g. integral theory considerations of plumes based only on transport by mean quantities will yield errors on the order of 15% in the self-preserving portion of the flow due to effects of streamwise turbulent mass fluxes that are generally ignored for these theories. Measurements of temporal power spectral densities of mixture fraction fluctuations of plumes were also reported by Dai et al. [28], considering (x – xo)/d ≥ 25 and r/(x – xo) of 0.0−0.2. The spectra exhibited an initial decay according to the conventional −5/3 power of frequency [6, 7], followed by a region where the decay was more rapid, according to the −3 power of frequency (see Dai et al. [28] for figure and further details). The latter fast-decay region has been observed during several investigations of buoyant turbulent flows but has not been observed in nonbuoyant flows [21, 22]. Notably, the spectra through the −3 decay region could all be plotted in a universal manner, based on self-preserving variables. An exception was that effects of progressively

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Figure 7: Radial distributions of streamwise rms velocity fluctuations for self-preserving steady plumes in still environments (from Dai et al. [30]).

increasing Reynolds numbers with increasing distance from the source (see eqn (15)) naturally modified microscales and thus the high-frequency portions of the spectra, which can never be self-preserving in plumes. The behavior of the low frequency portions of the spectra is notable as another example of the fact that various flow properties reach self-preserving behavior at various distances from the source of the flow, and that some variables (e.g. temporal power spectra) exhibit self-preserving behavior quite close to the source. Other examples of behavior of this type will be found subsequently when the penetration properties of various types of round buoyant turbulent flows are considered. The measurements of Dai et al. [28−31] generally were carried out farther from the source than earlier measurements of self-preserving plume properties and yield narrower distributions and larger values near the axis (when appropriately scaled) of flow properties than earlier results in the literature. This behavior is quantified in Table 1, where the medium of the flow (gas or liquid), the range of streamwise distances considered for measurements of the radial distributions – – , 2 2 of self-preserving __plume __ properties, and the corresponding values of kf , ℓf /(x – xo), F(0), (f ¢/ f )c ku ,ℓu /(x–xo), U(0), (u¢/u)c, and Eo are summarized by representative recent studies. Past measurements generally satisfy the criterion for buoyancy-dominated flow, i.e. (x – xo)/ℓM > 6 [21, 22].

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Figure 8: Radial distributions of turbulent mass fluxes for self-preserving steady plumes in still environments (from Dai et al. [29]).

Except for the measurements of Dai et al. [28−31], however, the measurements summarized in Table 1 were obtained at values of (x – xo)/d that are not normally associated with self-preserving conditions for round turbulent buoyant jet sources. Similar to the tendency for transitional plumes – – to have broader radial distributions and smaller scaled values at the axis for f and f ¢ than self– – 2 preserving plumes in Figs 3 and 4, values of kf , F(0) and (f ¢/ f )c all tend to decrease, whereas values of ℓf /(x – xo) tend to increase, as the maximum streamwise values of (x – xo)/d of the measurements is decreased. This behavior yields a reduction of the characteristic plume radius, and an increase of the scaled mean mixture fraction at the axis of 30%, when approaching self-preserving conditions over the range of maximum streamwise distances considered in Table 1. The behavior of streamwise velocities is similar, yielding a reduction of the characteristic plume radius of 40%, and an increase of the scaled mean streamwise velocity at the axis of 30%. Discrepancies between transitional and self-preserving plumes of this magnitude have a considerable impact on the empirical parameters obtained by fitting turbulence models to measurements. For example, Pivovarov et al. [34] suggest that the standard set of constants used in empirical turbulence models is inadequate based on the assumption of self-preserving plumes in conjunction with past measurements of transitional plumes; however, their predictions using standard constants are in reasonably good agreement with the measurements of self-preserving plume

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Table 1: Summary of self-preserving properties of round turbulent steady plumes in unstratified still fluids.

Parameter

Dai et al. [28, 31]

Medium (x − xo)/d (x − xo)/ℓM kf2 ℓf /(x − xo) F(0) – – (f ¢/f )c ku2 ℓu/(x − xo) U(0) __ __ (u ¢/u)c Eo

Gas 87−151 12−43 125 0.09 12.6 0.45 93 0.10 4.3 0.22 0.086

Papanicolaou and List [22] Liquid 22−62 9−62 80 0.11 14.3 0.40 90 0.11 3.9 0.25 0.080

Papanicolaou and List [21] Liquid 12−20 >5 80 0.11 11.1 0.40 58 − − − −

Shabbir and George George Ogino [24] et al. [15] et al. [19] Gas 10−25 6−15 68 0.12 9.4 0.40 55 0.13 3.4 0.32 0.109

Gas 8−16 6−12 65 0.19 9.1 0.40 51 0.14 3.4 0.28 0.112

Liquid 6−36 5−15 − − − − 0.14 3.4 − 0.117

Round buoyant turbulent plumes in still and unstratified environments. The range of streamwise distances are for conditions where quoted self-preserving properties were found from measurements over the cross-section of the plumes. Entries are ordered in terms of decreasing ku.

properties of Dai et al. [28−31] that were just discussed (see Dai [35] for the details of this evaluation). 2.5 Conclusions The properties of steady round turbulent buoyant plumes in still and unstratified environments were reviewed, yielding the following major conclusions: 1. Dai et al. [28−31] observed self-preserving behavior for all plume properties for (x – xo)/d ≥ 87 and (x – xo)/ℓM ≥ 12, which is significantly farther from the source than earlier measurements of self-preserving plume properties that were limited to (x – xo)/d ≤ 62 [10−25]. Within the self-preserving region, observations of mean properties of plumes by Dai et al. [28−31] yielded plume widths that were 30−40% narrower, with scaled mean values at the axis 30% larger, than the earlier observations in refs. [10−25]. 2. Differences in the properties of plumes of the magnitude just mentioned, due to failure to achieve self-preserving conditions, can have a considerable effect on the properties of approximate models of buoyant turbulent flows. For example, evaluation of turbulence models of steady selfpreserving plumes by Pivovarov et al. [34] found considerable deficiencies of conventional model constants using the measurements from refs. [10−25], whereas similar evaluations by Dai [35] found good performance of conventional model constants using the measurements from Dai et al. [28−31]. In view of these observations, past evaluations of models of buoyant turbulent flows, based on the assumption of self-preserving flows within buoyant turbulent plumes using the measurements from refs. [10−25], should be reconsidered. 3. Achieving self-preserving behavior for buoyant turbulent flows is affected by the nature of the flow, the properties of the source, and the flow property considered. For example, Dai et al.

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[28−31] found that the low-frequency portions of temporal power spectra were self-preserving for (x – xo)/d as small as 25 (the smallest value that they considered) whereas self-preserving behavior for all properties of these flows was not observed until (x – xo)/d ≥ 87, as noted earlier. 4. Streamwise turbulent mass fluxes are quite large near the axis of steady plumes where corresponding mixture-fraction/velocity fluctuation correlation coefficients reach values of roughly 0.7. This behavior is responsible for the large turbulent mass flux contribution to the total streamwise buoyancy flux in plumes (roughly 15%). This contribution is often ignored and leads to errors of flow properties estimated using integral models; it also leads to errors of conservation of buoyancy flux checks for other models of plumes.

3 Penetration of starting plumes in still environments 3.1 Introduction The penetration properties of starting plumes in still environments is an important fundamental problem relevant to the unconfined and unsteady turbulent flows resulting from fluid releases caused by process upsets, explosions, and unwanted fires. Due to its simplicity, this flow is also of interest as a classical turbulent buoyant flow that helps to illustrate the development of unsteady turbulent buoyant flows. Finally, due to well-defined initial and boundary conditions, the starting plume is useful for providing data needed to evaluate methods for predicting the properties of turbulent buoyant flows, particularly at self-preserving conditions where detailed definition of source properties is no longer needed. Similar to thermals, however, starting plumes have received less attention than steady plumes due to the measurement problems of unsteady flows. The few past investigations of round turbulent starting plumes in still and unstratified environments (denoted simply as starting plumes in the following) include studies reported by Turner [36], Middleton [37], Delichatsios [38], Pantzlaff and Lueptow [39], and references cited therein. Similar to past investigations of thermals, these studies have provided self-preserving scaling rules that describe the main features of turbulent buoyant starting plumes; however, corresponding measurements of the self-preserving properties of these flows are surprisingly limited and involve concerns about whether selfpreserving conditions were actually achieved. Prompted by these observations, Diez et al. [40] undertook a study of starting plumes in still and unstratified environments using methods similar to Sangras et al. [41]. The objectives were to determine conditions required for self-preserving behavior, to find the vertical and radial penetration properties of the flows at self-preserving conditions, and to use the measured penetration properties of these flows to evaluate self-preserving scaling and to find empirical factors needed to correlate the penetration properties of these flows. This study will be described in the following sections, considering experimental methods, theoretical methods, results and discussion, and conclusions, in turn. The following discussion is brief; see Diez et al. [40] for more details. 3.2 Experimental methods The experiments involved salt-water modeling of buoyant turbulent flows as suggested by Steckler et al. [42]. A sketch of the test apparatus appears in Fig. 9. The apparatus consisted of a plexiglass water bath with source fluid injection at the top. The dense salt-containing source liquid settled naturally to the bottom of the tank and was removed from time-to-time using the water

Weakly Buoyant Turbulent Fire Plumes

177

Figure 9: Sketch of the test apparatus for starting plumes in still environments (from Sangras et al. [41]).

bath drain. The source fluid for the plumes was injected into the tank using smooth round glass tubes having length/diameter ratios greater than 50 to help insure fully developed turbulent pipe flow for sufficiently large injector Reynolds numbers [43]. The source liquid was supplied to the injectors using syringe pumps that were computer controlled to deliver liquid at preselected rates. Calibration of pump performance indicated nearly constant delivery rates with relatively short start [41]. Salt-water source liquids were prepared by adding highly purified salt to given weights of water to reach desired source liquid densities. The densities of the present test fluids agreed with the tabulation of Lange [44] based on density measurements using precision hygrometers. A Cannon/Fenske viscometer was used to measure liquid viscosity. Finally, red vegetable dye was added to the source liquid in order to facilitate flow visualization of self-preserving behavior. Starting plumes penetration properties were measured as a function of time using video records of the flow. The video records were analyzed to provide the maximum mean streamwise penetration distances (taken as the average of the largest streamwise distance of injected source liquid from the jet exit) and the mean maximum radial penetration distance as a function of time from the start of injection, and the mean radial penetration distances of the plume-like portion of these flows as a function of distance from the injector exit at various time after the start of injection. These mean values were obtained by averaging the results of three separate tests at a particular

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jet exit condition. (Note that each test condition provided numerous data points, so that only three tests at each condition provided rather dense plots of the data within the experimental uncertainties noted in the following.) Experimental uncertainties (95% confidence) of the measurements from the video records are as follows: less than 7% for times from the start of injection, less than 8% for mean maximum streamwise penetration distances and less than 15% for mean maximum radial penetration distances. The test conditions for the starting plume studies can be summarized as follows: source diameters of 3.2 and 6.4 mm, source densities of 1,071 and 1,198 kg/m3, ambient densities of 998 kg/m3, source Reynolds numbers of 6,000−12,000, and source Froude numbers of 10−82. 3.3 Theoretical methods Assuming that the flow is in the self-preserving regime, expressions for the streamwise flow penetration distance for starting plumes are available from Turner [4, 5, 36], List [9], Middleton [37], Delichatsios [38], Pantzlaff and Lueptow [39], and references cited therein. The configuration they considered for starting plumes is illustrated in Fig. 10. The source flow enters from a passage having a diameter d, with a density ro, and flows into an environment having a density r•; properties of interest include the maximum vertical and radial penetration distances, xp and rp, illustrated in the figure. Major assumptions for analysis to find self-preserving flow scaling [9] are as follows: physical property variations are small (i.e. the flows are weakly buoyant so that density variations are linear functions of the degree of mixing as discussed earlier), sources are assumed to start

Figure 10: Sketch of a starting plume in a still environment (from Diez et al. [40]).

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179

instantly and to maintain constant flow rates when the source fluid is flowing (extrapolated temporal origins were used to handle actual start processes for measurements considered in the following), virtual origins were used to maximize conditions where self-preserving behavior is observed, and source flow properties are assumed to be uniform so that eqns (4) and (5) can be used to find ℓM/d and Fro. Under these assumptions, the temporal variation of maximum vertical penetration distance can be expressed as follows within the self-preserving region of starting plumes: ( xp − xo ) / d = C x ((t − td ) / t *)n

(19)

where xo and td are the extrapolated virtual origin to treat initiation of pump flow to the source, and the time of flow required to create the source is assumed to be small compared to values of (t – td) of interest. The corresponding temporal variation of the maximum radial penetration distance can be expressed most conveniently in terms of the vertical penetration distance, as follows: rp /( xp − xo ) = Cr

(20)

The same general form can be applied to a variety of round transient flows in still fluids − puffs, thermals, starting jets, starting plumes, etc. − with values of Cx, Cr, t*, and n varying depending upon the particular flow that is being considered. The values of Cx and Cr are best-fit empirical parameters of the self-preserving analysis and will be considered later when the measurements are discussed. The values of t* and n, however, follow from the requirements for self-preserving flow and can be expressed as follows for round starting plumes [9]: t* = (d 4 / B o )1/ 3 , n = 3 / 4, starting plume,

(21)

. where Bo is the source specific buoyant flux in the plume. Under the present assumption of uniform source properties, the source specific buoyant flux for a plume can be found from source properties, as follows [9]: B o = Q o g | ro − r∞ | / r∞ , starting plume,

(22)

where the absolute value has been used for the density difference, as before, to account for both rising and falling plumes. The analogous formulation for nonbuoyant starting jets can be found in Sangras et al. [41]. The expressions for vertical penetration distance of starting plumes in still environments, eqns (19), (21), and (22), are convenient for illustrating the development of plumes toward self-preserving behavior, and their subsequent penetration properties in the self-preserving portion of the flow. This formulation is misleading, however, because it involves the source diameter which is not a relevant variable of self-preserving plumes. This is apparent because d cancels out of eqns (19) and (21), to yield the following expression for the vertical penetration distance: ( xp − xo ) / x* = C x ,

(23)

where x* is a characteristic streamwise distance that involves a conserved property of the flow. The value of x* can be expressed as follows for round starting plumes: x* = ( B o1/ 3 (t − td ))3 / 4 , starting plumes,

(24)

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where the theoretical value, n = 3/4, for self-preserving starting plumes has been used in eqn (24). Values of n, t*, and x* for plumes in still fluid are summarized in Table 2 for convenience. 3.4 Results and discussion The normalized vertical penetration distances of starting plumes are plotted according to the selfpreserving scaling of eqns (19) and (21) in Fig. 11. Near-source behaviors vary depending upon Table 2: Summary of self-preserving properties of round turbulent starting plumes in unstratified still fluids. Conserved property ◊ ◊ Bo = Qo g|ro − r•|/(r•) Reo Diez et al. [45] 4,000−11,000 Turner [36] − a

n 3/4

t* ◊ (d /Bo)1/3

x* ◊ 1/3 (Bo (t − td))3/4

n

(xρ − xo)/d a

Cx

Cr

3/4

110

2.7 (0.05)

0.16 (0.006)

7.0 (1.7)

3/4

−

−

0.18 (0.03)

−

4

xo/d

Maximum vertical penetration distance observed. Experimental uncertainties and 95% confidence in parentheses.

Figure 11: Maximum vertical penetration distances as a function of time for starting plumes in still environments (from Diez et al. [40]).

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181

source properties, however, present flows generally correspond to over-accelerated flows (Fro > 5) so that the flows generally decelerate at first before self-preserving conditions are approached. All the measurements are seen to follow the self-preserving scaling at large dimensionless times, e.g. at (t – td)/t* > 20 which corresponds to (xp – xo)/d > 40 and (xp – xo)/ℓM > 5. Both these distances are much nearer to the source than the values of (x – xo)/d > 80 and (x – xo)/ℓM > 10 needed to reach self-preserving behavior for round turbulent buoyant plumes based on measured mean and fluctuating mixture fraction and velocity distributions discussed in connection with Figs 3−8. This observation highlights the fact that conditions for self-preserving behavior depend on the flow and on the property observed whereas radial flow widths tend to reach self-preserving behavior slower than other properties of most flows. Normalized maximum radial penetration distances (found near the jet tip) of starting plumes are plotted according to the self-preserving scaling of eqn (20) in Fig. 12. More near-source points are plotted in Fig. 12 than in Fig. 11 because some test conditions were omitted in Fig. 11 in order to reduce overlap and improve clarity of this figure. The normalized maximum radial penetration distance has relatively large values in the region nearest the source where measurements were made; this is expected, however, because this property becomes unbounded at the virtual origin. The normalized maximum radial penetration distance decreases with increasing streamwise distance and becomes relatively constant in the self-preserving region where (xp – xo)/d ≥ 40. As just

Figure 12: Maximum radial and normalized vertical penetration distances as a function of maximum vertical penetration distances for starting plumes in still environments (from Diez et al. [40]).

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noted, this radial flow parameter approaches self-preserving behavior at comparable conditions to streamwise penetration distance considered in Fig. 11. For self-preserving conditions, rp/(xp – xo) = Cr = 0.15. In addition, measurements plotted according to the normalized expression for the vertical penetration distance from eqns (23) and (24) are illustrated in Fig. 12; as already noted, the vertical penetration for starting plumes reaches self-preserving behavior at distances comparable to the radial penetration, e.g. at (xp – xo)/d > 40. A summary of the values of n, Cx, and Cr for starting plumes is provided in Table 2 considering the measurements of Diez et al. [40] and Turner [36]. The results for n for both studies agree with the expectations of self-preserving theory. The values of Cr are somewhat smaller for the measurements of Diez et al. [40] than for the measurements of Turner [36]. Based on the results in Fig. 12 it seems likely that this discrepancy occurs because the measurements of Diez et al. [40] were carried out farther from the source than those of Turner [36]. 3.5 Conclusions The properties of round turbulent buoyant starting plumes in still and unstratified environments were reviewed, yielding the following major conclusions, mainly based on the observations of Diez et al. [40]: 1. The flows became turbulent within five diameters of the source exit; although near-source behavior varied significantly with source properties, self-preserving behavior generally was observed for (xp – xo)/d ≥ 40 and (xp − xo)/ℓM ≥ 5. 2. Within the self-preserving region, the vertical dimensionless penetration distance, (xp – xo)/d, generally varied as a function of time in agreement with anticipated behavior for self-preserving starting plumes with maximum vertical penetration distances varying according to dimensionless time to the 3/4 power, yielding Cx = (xp – xo)/x* = 2.7. 3. Within the self-preserving region, the normalized maximum radius of the flow grew as a function of time in the same manner as the normalized streamwise penetration distance, yielding Cr = rp/(xp – xo) = 0.15, which is smaller than the earlier measurements of Turner [36] probably because the latter results were not obtained sufficiently far from the source to reach selfpreserving conditions.

4 Penetration and concentration properties of starting and steady plumes in crossflows 4.1 Introduction Recent studies of the temporal and steady penetration properties of round turbulent puffs, thermals, starting and steady jets, and starting and steady plumes in both still fluids and uniform unstratified crossflows [40, 45−47], and the mixing properties of steady round nonbuoyant turbulent jets in uniform crossflows [48], were extended by Diez et al. [49] to consider the mixing properties of starting and steady round buoyant turbulent plumes in uniform unstratified crossflows (denoted as ‘starting and steady turbulent plumes in crossflows’ in the following sections) and will be reviewed next. The penetration properties of starting plumes in crossflowing environments is an important fundamental problem relevant to the dispersion of heat and harmful substances due to accidental releases because releases are generally of extended duration so that the thermal approaches self-preserving starting plume behavior whereas releases occur in

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the presence of significant crossflow more often than not. Similar to the other unsteady flows already considered, however, starting plumes in crossflow have not been measured very often due to the complexities of providing well-defined crossflows and the difficulties of measuring the properties of three-dimensional unsteady flows. Interest in steady turbulent plumes in crossflows is motivated by practical applications to the dispersion of harmful releases of heat and substances into atmospheric crosswinds. Similar to the study of the mixing properties of steady turbulent jets in crossflows due to Diez et al. [48], the present study emphasized flow properties far from the source, where effects of source disturbances are lost, where the flows are largely controlled by their conserved properties and where flow properties approximate self-preserving turbulent flow behavior, and where appropriately scaled flow properties became independent of the distance from the source. This region is of particular interest because the properties of selfpreserving turbulent flows provide a compact presentation of measurements that substantially simplifies the interpretation of flow behavior. In order to fix ideas, a visualization of a typical steady turbulent plume in crossflow appears in Fig. 13. This flow actually involves the injection of dye-containing salt water (the more dense salt water flowing vertically downward) into a fresh water crossflow (flowing from left to right) in a water channel facility. Following past practice [45, 47], however, the vertical direction has been

Figure 13: Visualization of the penetration properties of a steady turbulent plume in a uniform crossflow (d = 6.4 mm, Reo = 5,000, ro/r• = 1.150, Fro = 223, uo/v• = 7). The upper figure is a side view; the lower figure is a top view (from Diez et al. [49]).

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inverted to show the flow as an upwardly injected turbulent plume with the source density smaller than the ambient density because most individuals are more familiar with positively buoyant upwardly flowing plumes than with negatively buoyant downwardly flowing plumes. This practice will be adopted throughout the present review. The images appearing in Fig. 13 consist of side and top views of a typical turbulent plume in crossflow obtained when steady flow conditions have been reached. The length scales that appear on the images are appropriate for streamwise (vertical, or in the same direction as the plume source flow) and crosstream (horizontal, or in the same direction as the ambient crossflow) directions. It will be shown later that the streamwise velocities of steady turbulent plumes in crossflows progressively decrease with increasing streamwise distance from the source so that the streamwise velocity eventually becomes small and the trajectory of steady turbulent plumes in crossflows become nearly horizontal far from the source where self-preserving behavior is approached. When this condition is reached, the streamwise penetration of the flow approximates a two-dimensional horizontal line thermal in a still fluid. Then the streamwise motion of the line thermal, retarded along its sides in the streamwise direction by the uniform ambient crossflow, naturally leads to the flow becoming two nearly horizontal counter-rotating vortices whose axes are aligned along the axis of the plume as a whole (and thus are nearly horizontal as well). Evidence for this behavior is provided by the top view of the flow which is the lower image in Fig. 13, where the darker regions associated with the two vortices are separated by a significantly lighter region dominated by the presence of dyefree ambient fluid that is entrained by the vortex system along its plane of symmetry. Another interesting feature of the visualization appearing in Fig. 13 is that turbulent distortions of the lower surface of the flow (the side facing the source) are smoothed out because this region is stable to buoyant disturbances whereas the turbulent distortions of the upper surface of the flow (the side facing away from the source) are enhanced because this region is unstable to buoyant disturbances. Notably, corresponding visualizations of steady turbulent jets in crossflow do not exhibit this behavior but instead exhibit similar degrees of distortion on the lower and upper surfaces (the sides toward and away from the source, respectively) of the flow because mechanisms of buoyant stability and instability are absent in this case, see Diez et al. [48]. In order to develop the objectives of the present investigation, earlier studies of turbulent plumes will be discussed in the following sections. Most practical releases of turbulent plumes are exposed to crossflow; therefore, there have been a number of attempts to extend the results just discussed for turbulent plumes in still fluids to corresponding turbulent plumes in crossflows (see Diez et al. [45, 47], Fischer et al. [50], Lutti and Brzustowski [51], Andreopoulos [52], Alton et al. [53], Baum et al. [1], Hasselbrink and Mungal [54], and references cited therein). These studies generally have shown that motion in the crossflow direction satisfies the no-slip convection approximation and that the deflection of the plume toward the crossflow direction results in the development of a counter-rotating vortex system over the cross-section of the flow, as discussed in connection with Fig. 13. Measurements of the mixing structure of these flows, however, generally have been limited to the region near the source where the flow undergoes most of its deflection toward the crosstream direction. Studies of the potential self-preserving behavior of this flow show that the decay of streamwise velocities with increasing distance from the source is relatively rapid so that the flow eventually becomes nearly aligned with the horizontal direction for all source/crossflow velocity ratios, uo/v•, i.e. the general appearance of the flow illustrated in Fig. 13 is typical of flows of this type. Thus, self-preserving behavior for these flows eventually involves no-slip convection in the crossflow direction combined with the motion of a line thermal in the streamwise direction [45, 47, 50]. Relationships for the self-preserving transient and steady penetration properties of these flows have been confirmed by measurements, obtaining results similar

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to corresponding flows in still environments. In particular, the flows become turbulent within five source diameters from the source in the streamwise direction, and become self-preserving at streamwise distances greater than 40−50 source diameters from the source for uo/v• < 35 [45, 47]. On the other hand, the rates of mixing and the structure properties of these flows at selfpreserving conditions have not yet received any attention. Prompted by these observations, a study was undertaken by Diez et al. [49] seeking to extended past work concerning the self-preserving penetration properties of steady turbulent plumes in crossflows [45, 47], in order to develop an improved understanding of their self-preserving mixing structure properties, as follows: Measure the self-preserving mixing structure of these flows, including the trajectories of the axes of the counter-rotating vortices, and the distributions of the mean and rms fluctuations of source fluid concentrations within the counter-rotating vortex system, for steady flow and for source and crossflow conditions typical of practical applications. Exploit the new measurements of these flows in order to evaluate the effectiveness of selfpreserving scaling relationships developed by Diez et al. [45, 47] for penetration properties (specifically for the penetration properties of the vortex axes that have not been considered before) and by Fischer et al. [50] for flow mixing structure properties, and to determine the empirical parameters associated with the various scaling relationships. The present description of the research begins with a discussion of experimental methods and the self-preserving scaling properties of the flows; measured scaling results are then described, considering flow penetration properties and flow mixing structure properties in turn. The following discussion is brief; see Diez et al. [45, 49] for more details. 4.2 Experimental methods 4.2.1 Test apparatus The experiments of Diez et al. [45, 49] adopted methods analogous to the salt/fresh-water modeling experiments for buoyant turbulent flows suggested by Steckler et al. [42]. Somewhat different source and ambient fluids were required, however, for measurements of source flow penetration properties by visualization of a dye-containing source fluid and measurements of source flow mixing structure properties by planar-laser-induced-fluorescence (PLIF) records of a dye-containing source fluid because the latter measurements required matching the indices of refraction of the source and ambient fluids in order to avoid scattering the laser beam away from the buoyant flow. Thus, the source and ambient fluids for the two types of experiments differed as follows: a salt (NaCl) water source containing a red vegetable dye was injected into an unstratified uniform fresh water crossflow for flow visualization measurements of source flow penetration properties, and a salt (sodium phosphate, KH2PO4) water source containing Rhodamine 6G dye was injected into an unstratified uniform ethyl-alcohol/water crossflow for matched refractive index PLIF measurements of source flow mixing (structure) properties. The unstratified and uniform crossflow was produced by a water channel facility. The test section of the water channel had cross-section dimensions of 610 × 760 mm and a length of 2,440 mm. The sides and bottom of the test section were constructed of 20 mm thick acrylic and float glass panels, respectively, to provide optical access. The crossflow in the channel was driven by a propeller pump to yield crossflow velocities of 40−300 mm/s. The properties of the crossflow were characterized using a pure water flow as described by Diez et al. [48]. A contraction of ten-toone (involving a fifth-order polynomial having a zero slope and curvature at the entrance and the exit), flow straighteners, screens, etc., of the water channel combined to yield a flow nonuniformity of less than 1.5% and turbulence intensity levels less than 1%, in the test section.

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The source flows had a density larger than the crossflows for present conditions and were injected vertically downward into the channel flow to obtain steady turbulent plumes in crossflows. The source flows passed through smooth round injector tubes having inside diameters of 2.1, 3.2, and 6.4 mm. The source injector tubes had length/diameter ratios of 200, 100, and 50, respectively, to help insure fully developed turbulent pipe flow at the source exit for sufficiently large source Reynolds numbers, as discussed by Wu et al. [43]. The source injector tubes were mounted vertically and discharged roughly 25 mm below the liquid surface. The source injector tubes passed through a plane horizontal Plexiglas plate (508 × 914 mm in plan dimension × 12 mm thick) with a tight fit. The source injector tube exits were mounted flush with the lower surface of the Plexiglas plate in order to provide well-defined entrainment conditions at the source exit. The source liquid was supplied to the tubes using either a syringe pump (Harvard Apparatus, PHD2000, Model 70−2000, with four 150 cc syringes having volumetric accuracies of ±1% mounted in parallel) for small flow rates, or a peristaltic pump (Masterflux L/S Digi-Staltic Dispersion, Model 72310-0) with two flow dampers for large flow rates. The pumps were calibrated by collecting liquid at timed intervals. 4.2.2 Penetration measurements Observations of dye-containing source liquids were obtained using CCD video cameras similar to earlier studies of the penetration properties of starting and steady turbulent plumes in still and crossflowing fluids [45, 47]. The video records were analyzed to provide flow penetration properties as follows: maximum mean streamwise (xp) and crosstream (yp) penetration distances (taken as averages of the largest streamwise and crosstream distances of the injected source liquid from the jet exit), and the maximum mean radial distance (rp) and maximum mean half-width (wp). All these mean parameters were obtained by averaging the results of three separate tests at a particular test condition; this yielded acceptable experimental uncertainties because each test condition provided numerous data points and rather dense plots of the data. A portion of these results [45, 47] for steady turbulent plumes in crossflows will be considered for completeness. Experimental methods for these measurements are described by Diez et al. [45, 47] which should be considered for these details. Experimental uncertainties (95% confidence) of the measurements from the video records were as follows: less than 8% for mean maximum streamwise and crosstream penetration distances and less than 15% for mean maximum radial and half-width penetration distances. Experimental uncertainties of all penetration distances were largely governed by sampling errors due to the irregular turbulent boundaries of the present turbulent flows but also include fundamental accuracies of distance calibrations and measurements. Finally, the PLIF measurements of flow structure were used to find the streamwise distance to the center of the counter-rotating vortex system and the horizontal distance between the axes of the counter-rotating vortices; the experimental uncertainties (95% confidence) of these measurements were less than 8 and 15%, respectively. 4.2.3 Structure measurements The PLIF arrangement was similar to the arrangement used by Diez et al. [48] for studies of the structure of steady turbulent jets in crossflows, except for features required to match the refractive indices of the source and crossflowing fluids discussed by Ferrier et al. [55], and Alahyari and Longmire [56]. The arrangement consisted of a laser, optics for scanning the laser beam across the image area, and a digital camera for recording the image. Rhodamine 6G dye at a concentration of 5.0 × 10−6 mol/l was used for the PLIF signals in the source liquid, see Ferrier et al. [55] for a discussion of the properties of this dye. An argon-ion laser (Coherent Innova 90-4) operated in the single-line mode at 514.5 nm with an optical power of 3,200 mW and a beam diameter of 1.5 mm (at the e−2 intensity locations) was used to excite the fluorescence.

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A mirror located roughly 1,000 mm downstream from the imaged cross-section of the flow allowed the camera to view the PLIF image. The camera was a Redlakes Inc., Model Mega Plus ES 1020. This monochrome 10-bit CCD camera has a 1,004 × 1,004 pixel array with a 7.4 × 7.4 mm active sensor area. A PC-Cam Link frame grabber from Coreco Imaging transferred the camera images to a computer for processing and storage. The images were corrected for vignetting, sweep geometry, attenuation (by water, Rhodamine 6G, sodium phosphate, and ethanol) and background effects as discussed by Ferrier et al. [55]. Finally, the use of potassium phosphate (KH2PO4) to increase the density of the source fluid along with an appropriate concentration of ethyl alcohol in the crossflowing water which decreased its density, matched the refractive indices of the source and ambient fluid as discussed by Ferrier et al. [55] and Alahyari and Longmire [56]. Close control of the temperature differences between the source and ambient fluid, however, was also required for proper matching of refractive indices; this was done using a heater for the source fluid that limited temperature differences between the source and ambient fluids to less than 0.10 K. The mean and fluctuating concentrations of source fluid were obtained over cross-sections of the flow by averaging 4,000 images. The experimental uncertainties (95% confidence) were less than 7% and 15% for mean and rms fluctuating concentrations, respectively, at each point in the flow. These uncertainties were largely governed by sampling errors due to the finite number of measurements of concentration properties that were averaged at each point in the flow. 4.2.4 Test conditions Overall test conditions were as follows: source diameters of 2.1, 3.2, and 6.4 mm; corresponding source passage length/diameter ratios of 200, 100, and 50; source/crossflow velocity ratios of 4−96; source Reynolds numbers of 2,500−15,000; source Froude numbers of 6−211; streamwise (vertical) penetration distances of 0−202 source diameters; and crosstream (horizontal) penetration distances of 0−620 source diameters. Due to the modification of source and crossflow properties in order to match refractive indices for the PLIF measurements, source/ambient density ratios for the flow penetration test were 1.073 and 1.150 and for the flow structure measurements was 1.024. 4.3 Theoretical methods 4.3.1 Conditions for self-preservation The present discussion of scaling for steady turbulent plumes in crossflows addresses three aspects of scaling, in turn: (1) conditions required for self-preservation, (2) self-preserving penetration properties, and (3) self-preserving structure properties. The discussion is limited to the behavior of steady turbulent plumes in crossflows; see Diez et al. [45, 47] for consideration of the penetration properties of starting plumes in still and crossflowing fluids. A parameter that is frequently used to estimate when steady turbulent unconfined flows reach self-preserving behavior is the distance from the effective (virtual) origin of the flow normalized by the source diameter, (x – xos)/d, taken to be in streamwise (vertical) direction (parallel to the source flow) for present conditions, where the subscript ‘os’ is used to denote conditions at the virtual origin for a steady plume. This normalized distance is a measure of conditions where distributions of flow properties appropriate for the often confined conditions of a source have adjusted to reach distributions appropriate for an unconfined flow. The value of (x – xos)/d needed for self-preserving behavior depends on the nature of the flow, the properties of the source, and the property for which self-preserving behavior is sought. For example, results for steady turbulent plumes in still fluids suggest that values of (x – xos)/d greater than 40−50 source diameters

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are needed in order for flow penetration properties to reach self-preserving behavior, whereas values of (x – xos)/d greater than 80−100 source diameters are needed for the structure of flow concentration properties to reach self-preserving behavior based on the observations of Dai [28]. Measurements of steady turbulent jets in still fluids also indicate that the L /d of the source passage, as well as the Reynolds number of the flow through the passage, can have a profound effect on transition to turbulence and, accordingly, on the conditions required for self-preserving behavior to be observed [43]; it is probable that similar behavior would be observed for steady round turbulent buoyant plumes in still fluids. Finally, the crossflow was found to have a strong effect on the conditions required for onset of self-preserving behavior during the present investigation; this effect and the corresponding flow regime map that identifies conditions required for selfpreserving flow will be discussed subsequently. 4.3.2 Penetration properties Scaling relationships for the penetration properties of starting and steady turbulent plumes in crossflows were developed and evaluated successfully based on measurements of dye-containing source liquids as discussed by Diez et al. [45]. Diez et al. [49] extend those results to consider the penetration properties of the axes of the vortices of the counter-rotating vortex system observed for steady turbulent plumes in crossflows based on measurements of flow structure properties using PLIF. The development of scaling relationships for the penetration (geometrical) properties of starting and steady turbulent plumes in crossflows is discussed by Diez et al. [45]. The configuration of steady turbulent plumes in crossflows considered in the following is sketched in Fig. 14. The source flow enters from a round passage normal to the crossflow and flows into an environment having a uniform crossflow velocity. As discussed by Diez et al. [45], the streamwise velocity decays rapidly with increasing streamwise distance for this flow; for example, the streamwise velocity is proportional to (t – tos)−1/3 when the crossflow velocity is large compared to the streamwise velocity and the steady plume is nearly horizontal, as the self-preserving region is approached far from the source [45]. In addition, the flow approximates no-slip convection in the crosstream direction [45, 47]. This behavior implies that the plume eventually is deflected so that its axis is nearly aligned with the crosstream direction. At this condition, the initial streamwise source specific buoyancy flux continues to be conserved so that the flow approximates the behavior of a line thermal. Then the streamwise source specific buoyancy flux per unit length of the line thermal causes a counter-rotating pair of vortices to form, leading to a somewhat flattened shape of the flow cross-section. The properties of interest for starting plumes in crossflow include the maximum vertical, crossflow (horizontal), and radial penetration distances, xp, wp, and rp, respectively,

Figure 14: From Diez et al. of a steady turbulent plume in a uniform crossflow (from Diez et al. [49]).

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illustrated in Fig. 14. In addition, the steady penetration properties of the flow are of interest, including the farthest streamwise penetration distance, xps – xos, and the transverse penetration width of the flow, wps, both as functions of the crosstream distance, y – yos. The geometrical properties of interest for this flow include the trajectory of the axis of the vortex system, (xc – xos)/d, as a function of (y – yos)/d, after allowing for virtual origins to extend the self-preserving region as near to the source as possible, as well as the separation between the axes of the two counter-rotating vortices, wc. Major assumptions used to find scaling relationships for the geometrical features of the flow, e.g. the trajectories of the vortex system axis and the counter-rotating vortex axes, as well as the various penetration properties of the flow, are as follows: physical property variations are small as normally considered within the Boussinesq approximation of buoyant flows, i.e. the physical properties of the source and ambient fluids are the same except for modest density differences that are responsible for the effects of buoyancy in the flow; the flows are self-preserving so that effects of source disturbances are lost; the streamwise source specific buoyancy flux per unit length of the line thermal is conserved; sources are assumed to start instantly and subsequently to maintain a constant flow rate (using an extrapolated temporal origin to handle the actual start process similar to starting plumes in still environments), virtual origins are used in both the vertical and crosstream directions to maximize conditions where self-preserving behavior is observed. Thus, the maximum crosstream (horizontal) penetration distance of starting plumes in crossflow was found from the no-slip convection approximation, as shown by Diez et al. [45], which implies that the flow moves in the crosstream (horizontal) direction at the crosstream (horizontal) velocity as follows: ( yp − yo ) / d = C y (v∞ (t − td ) / d )

(25)

The diameter d factors out of eqn (25) immediately to provide an equation in the self-preserving region that is independent of d. Penetration in the streamwise, xp, and radial, rp, directions was still given by eqns (19), (20), and (23). The conserved property in the streamwise direction is given by the source specific buoyancy flux per unit length of the line thermal, B′, o associated with motion in this direction. Values of n, t*, and x* were found in the same manner as for starting plumes in still fluid; these properties of starting plumes in crossflow are summarized in Table 3. Finally, radial penetration in the horizontal plane at self-preserving conditions represented by the half-width parameter wp is given by an expression analogous to eqn (20) for rp, as follows: wp /( xp − xo ) = Cw

(26)

Table 3: Summary of equations and empirical parameters for the self-preserving penetration properties of starting plumes in unstratified and uniformly crossflowing fluids. Conserved property ◊ B′o = Qo g|ro − r∞|/(r∞n•) Source Diez et al. [45]

n

t* 3

2/3

x* ( B′o (y − yos)/n•)2/3

1/3

1/2

(d / B′o )

Cx

Cr

Cw

xo/d

1.5 (0.03)

0.24 (0.005)

0.41 (0.009)

17.9 (3.1)

Self-preserving behavior summarized here was observed for starting plumes (xro − xos)/d > 40−50. Experimental uncertainties and 95% confidence in parentheses. Also Cy = Cys = 1 and yo = yos = 0.0.

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In addition, eqns (4) and (5) can be used to find ℓM/d and Fro, similar to a starting plume in a still environment under present approximations. Finally, the steady-state trajectory of a starting plume in crossflow can be observed by letting the flow run for a time after the tip of the plume has passed a given location. Thus, the flow in the streamwise direction, after the axis of the counter-rotating vortex system is nearly aligned with the crossflow direction, approximates a line thermal having a conserved streamwise source specific buoyancy ◊ flux per unit length B′o = Qogrο–r•/(r•v•). Then the trajectory for steady plumes in crossflow can be obtained by noting that (xps – xos), rps, and wps are now functions of the displacement in the crosstream direction, (y – yo), rather than just a particular penetration distance. Then eliminating time from the expression for x* for starting plumes in crossflow in Table 3, noting that the no-slip convection approximation in the crosstream direction implies that (t – td) = (y – yo)/v•, and recalling that Cy = 1, a new expression for the streamwise penetration distance for a steady plume in crossflow is found as follows: ( xps − xos ) / xs* = C xs

(27)

where the associated conserved property in the streamwise direction, B′o, and n and x*s are summarized in Table 4. Similarly, the other steady penetration properties of steady plumes in crossflow are found from equations analogous to eqns (20) and (26), as follows: rps /( xps − xos ) = Crs

(28)

wps /( xps − xos ) = Cws

(29)

and

In addition, the properties of the axis of the vortex system as a whole and the spacing between the axes of the two counter-rotating vortices, as defined in Fig. 14, are given by expressions similar to eqns (27), (28), and (29), as follows: ( xc − xos ) / xs* = C xcs

(30)

wc /( xc − xos ) = Cwcs

(31)

and

Table 4: Summary of equations and empirical parameters for the self-preserving penetration properties of steady plumes in unstratified and uniformly crossflowing fluids. Conserved property ◊ B′o = Qo g|ro − r∞|/(r∞n•) Cxs Diez et al. [45]

n

xs*

2/3

( B′o (y − yos)/n•)2/3 1/2

Crs

Cws

xos/d

1.9 (0.08)

0.36 (0.008)

0.49 (0.015) 25.6 (4.5)

Self-preserving behavior summarized here was observed for starting plumes beyond (xro − xos)/d > 40−50. Experimental uncertainties and 95% confidence in parentheses. Also Cy = Cys = 1 and yo = yos = 0.0.

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4.3.3 Structure properties Similar to penetration properties, present considerations of structure properties are limited to steady flows. Analyses to find the self-preserving behavior of the mean and rms fluctuating concentration distributions in self-preserving steady turbulent plumes in crossflows are described by Fischer et al. [50]. The results of these analyses in terms of the present notation are as follows: cg( xc − xos )2 /(co Bo′ ) = F[( x − xc ) /( xc − xos ), z /( xc − xos )]

(32)

c ′g( xc − xos )2 /(co Bo′ ) = F ′[( x − xc ) /( xc − xos ), z /( xc − xos )]

(33)

and

__

The form of eqn (33) for c ¢ is used because it provides a compact notation that avoids defining an additional empirical parameter for the maximum mean concentration of source fluid over a __ cross-section of the flow, cm. 4.4 Results and discussion 4.4.1 Overview As noted earlier, even though present flows involved downwardly injected negatively buoyant turbulent plumes in crossflows, plots of the results of these flows have all been inverted to show them as upwardly injected positively buoyant steady turbulent plumes in crossflows, similar to the approach taken for the visualization illustrated in Fig. 13. This was done due to the greater familiarity of most individuals with upwardly flowing plumes. Measurements of starting and steady plumes in crossflow undertaken by Diez et al. [45, 48, 49], in order to define conditions required to achieve self-preserving behavior and to find the properties of self-preserving plumes in crossflow will be considered in the following sections. 4.4.2 Flow regime map for self-preservation The conditions required to observe self-preserving behavior are influenced by the type of flow being considered, the properties of the source, and the property for which self-preserving behavior is being sought. For the present steady turbulent plumes in crossflows, having relatively long source passages (L/d ≥ 50) and relatively large source Reynolds numbers (Reo ≥ 2,500), the most conservative property for the onset of self-preserving behavior was the concentration structure of the flow. Thus, consideration of the range of conditions required for the flow to exhibit selfpreserving behavior will be based on the concentration structure of the flow. In addition to the properties just mentioned, observations during the present investigation indicated that the source/crossflow velocity ratio, uo/v•, substantially influenced conditions at the onset of self-preserving behavior for the concentration structure of steady turbulent plumes in crossflows. This behavior follows because self-preservation was only observed when the axes of the counter-rotating vortex system were nearly aligned with the crossflow direction. At this condition, the plume was nearly horizontal so that the full effect of the streamwise source specific buoyancy flux of a line thermal could act upon the deflected plume in order to develop the full strength of the counter-rotating vortex system. In turn, conditions when the axes of the counterrotating vortex system were nearly horizontal were a strong function of uo/v•. In particular, nearly horizontal alignment of the axes of the counter-rotating vortex system was delayed when uo/v• was large because a large streamwise distance was required before uo decayed to a value that was small compared to v• so that the plume could be deflected into the nearly horizontal direction.

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Present experimental observations of the onset of self-preserving behavior for the concentration structure of steady turbulent plumes in crossflows were based on mean concentration distributions plotted according to the self-preserving scaling of the flows due to Fischer et al. [50], as shown in eqn (32). The actual approach involved plotting the mean values of the concentrations according to eqn (32) for several paths over the cross-section of the plumes (along a horizontal line passing through the axes of the two counter-rotating vortices, along the two vertical lines passing through the axis of each vortex, and along a third vertical line in the plane of symmetry of the counter-rotating vortex system). Given these results for a particular value of uo/v•, the condition was assumed to be self-preserving when the distributions along all these paths did not depart from the average distributions for all the paths over the self-preserving region by more than 5%. The resulting self-preserving flow regime map for steady turbulent plumes in crossflows is illustrated in Fig. 15. This map was constructed based on tests specifically conducted to determine the onset of self-preservation, with conditions prior to and within the self-preserving region denoted by appropriate symbols. The flow regime map of Fig. 15 shows the developing flow and self-preserving regions of steady turbulent plumes in crossflows in terms of the streamwise distance from the source, (xc – xos)/d, for uo/v• in the range 4−100. Note that the values of uo/v• smaller than four were not considered because the plume deflects immediately upon leaving the source tube so that its properties are affected by the walls of the water channel. The tendency for increased values of uo/v• to delay the onset of self-preserving flow is evident with onset reached

Figure 15: Flow regime map of the developing flow and self-preserving regions for steady turbulent plumes in crossflows (from Diez et al. [49]).

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at (xc – xos)/d of 10−20 for uo/v• of four (the smallest value of uo/v• considered) but increasing to 160−170 for uo/v• of 100 (the largest value of uo/v• considered.) 4.4.3 Penetration properties As described by Diez et al. [45], measurements of the crossflow motion in plumes showed that the no-slip convection approximation was quite reasonable, with eqn (25) satisfied by Cy = 1.0 and yo/d = 0. Since a similar behavior was observed for jets, puffs, plumes, and thermals, Diez et al. [45] showed, as a typical case, only the results for thermals in crossflow. This is reproduced in Fig. 16. In addition, the virtual origin in the crosstream direction was essentially zero, similar to earlier findings for thermals in uniform crossflow. Normalized radial, half-width, and streamwise penetration distances of starting plumes in crossflow are plotted according to the self-preserving relationships of eqns (20), (23), and (26), with n = 2/3 and x* for starting plumes in crossflow from Table 3 in Fig. 17. Near-source behavior varies depending on the source properties for each test condition but the starting plumes become self-preserving for all properties when (xp – xo)/d > 40−50, similar to the other transient flows in crossflow that were considered. Best-fit values of Cx, Cr, and xo/d are readily obtained from the measurements and are summarized in Table 3 for comparison with the corresponding results for other flows. The order of magnitude of these parameters is similar to the other flows

Figure 16: Crosstream (horizontal) penetration distance of thermals in crossflow as a function of time. Similar behavior was observed for plumes, jets, and puffs (from Diez et al. [45]).

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Figure 17: Radial, half-width, and streamwise (vertical) penetration distances of starting plumes in crossflow as functions of streamwise (vertical) penetration distance (from Diez et al. [45]). studied during the present investigation. The details differ, however, because the counter-rotating vortex structure caused Cr to be larger for starting plumes in crossflow than in still fluids. A visualization of a typical plume of this nature has already been discussed in connection with Fig. 13. The locations of the axes of the vortices of the counter-rotating vortex system were found by averaging 4,000 PLIF images at each streamwise location. The fine details of the dynamics of the mixing pattern of the source and ambient fluids can be seen from the sequence of PLIF images illustrated in Fig. 18. In order to achieve adequate spatial resolution for these PLIF images, the diameter of the laser beam sweeping the cross-section of the flow was reduced to 0.5 mm for these images. The time between images was 50 ms, which implies a crosstream distance between images that is relatively large compared to the integral length scales of the flow. The instantaneous images appearing in Fig. 18 show the largely distorted presence of the two counter-rotating vortices separated at the plane of symmetry by deeply penetrating ambient

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Figure 18: Instantaneous PLIF images of the cross-section of a steady turbulent plume in a uniform crossflow (d = 3.2 mm, Reo = 10,000, ro/r• = 1.038, Fro = 223, uo/v• = 47, (xc – xos)/d = 120, y/d = 98 and ∆t = 50 ms between frames) (from Diez et al. [49]).

fluid. In addition, the presence of ambient fluid being transported deep into the vortex system along its plane of symmetry clearly has an important effect on the flow structure, as mentioned earlier in connection with the discussion of the flow visualization of Fig. 13. Finally, by averaging 4,000 images, each similar to those illustrated in Fig. 18, at each crosstream test condition, it was possible to locate the axes of the vortices (axes of maximum concentration/intensity) with an experimental uncertainty (95% confidence) of less than 4% of the transverse distance between the vortex axes, wcs. Present measurements of the penetration properties of the concentration structure of steady plumes in crossfl ow involved the trajectories of the counter-rotating vortex system, (xc – xos) / x*s = 1/2 (xc – xos)/Bo¢ (y – yo)/v•)2/3, and the normalized spacing between the axes of the two counter-rotating vortices, wcs/(xc – xos), as given by eqns (30) and (31). These properties are plotted as a function of the normalized streamwise vortex core penetration distance, (xc – xos)/d, in the bottom two plots of Fig. 19. These parameters are rather scattered in the developing flow region at small streamwise distances from the source but eventually approach constant values, Cxcs = 1.5 and Cwcs = 0.46, for self-preserving flow at large streamwise distances from the source. Notably, these measurements in the developing flow region of Fig. 19 generally involve uo/v• < 50 and were obtained for (xc – xos)/d < 100, which generally agrees with the developing flow region of the flow regime map of Fig. 15. Measurements of the penetration properties of the boundaries of the source fluid of steady plumes in crossflow from Diez et al. [45] involved the farthest normalized streamwise penetration distance, (xps – xos)/xs* = (xps – xos)/(Bo¢1/2(y – yo)/v•)2/3, the normalized radial penetration distance, rps/(xps – xos), and the normalized lateral penetration width, wps/(xps – xos), as given by eqns (27), (28), and (29). These properties are plotted as a function of the normalized streamwise penetration

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Figure 19: Radial, half-width, and streamwise (vertical) penetration distances of steady plumes in crossflow as functions of streamwise (vertical) penetration distance (from Diez et al. [45]).

distance, (xps – xos)/d, in the top three plots of Fig. 19. These parameters are rather scattered in the developing flow region at small streamwise distances from the source but eventually approach constant values Cxs = 1.9, Crs = 0.36, and Cws = 0.49, for self-preserving flow at large streamwise distances. Notably, the measurements in the developing flow region of Fig. 19 generally involve uo/v• < 25 and were obtained for (xps – xos)/d < 75, which generally agrees with the developing flow region of the flow regime map of Fig. 15. Thus, the fact that penetration properties of the boundaries of the source fluid reach self-preserving behavior sooner than the penetration properties of aspects of the concentration structure of the plumes in crossflow comes about largely due to the smaller values of uo / v• that were used in the source fluid penetration tests.

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Table 5: Summary of empirical parameters for the self-preserving flow and vortex axes penetration properties of steady round turbulent jets and plumes in uniform crossflowing fluids. Flow

Cxs

Crs

Results for flow penetration properties from Diez et al. [45] Plume 1.9 (0.08) 0.36 (0.008) Jet 2.3 (0.08) 0.23 (0.005) Flow

Cxcs

Cws

xos/d

0.49 (0.015) 0.31 (0.011)

25.6 (4.5) 6.1 (2.7)

Cwcs

xos/d

Results for vortex axes penetration properties from Diez et al. [48] and from the present study Plume 1.5(0.06) 0.46(0.013) 0.0 Jet 1.7(0.08) 0.36(0.017) 0.0 Experimental uncertainties and 95% confidence in parentheses; also Cys = 1 and yos = 0.0.

Finally, for convenient reference, the measured parameters associated with the penetration properties of steady turbulent plumes in crossflows are summarized in Table 5. Experimental uncertainties for flow penetration properties (95% confidence) are as follows: less than 5% for Cxs, less than 3% for Crs, less than 4% for Cws, less than 18% for xos/d, and less than 5% for both Cxcs and Cwcs. For comparison, the corresponding parameter values for steady turbulent jets in crossflows from Diez et al. [48] are also provided in the table; the experimental uncertainties (95% confidence) of these parameters for jets are similar to the present values for plumes. It is of particular interest to compare the values of Crs, Cws, and Cwcs for jets and plumes in crossflow because larger values of these parameters imply larger penetration of the source flow into the crossflow, which implies faster rates of mixing. The results in Table 5 show that these parameters are 30−60% larger for plumes than for jets, suggesting increased rates of mixing for the plumes due to their enhanced motion as a result of buoyancy. On the other hand, the streamwise penetration coefficients are larger for jets than plumes, however, these parameters are not directly comparable because the functional forms of xs* for these two flows are fundamentally different, see Diez et al. [45]. Finally, the counter-rotating vortex system that develops for steady turbulent plumes in crossflows promotes the mixing properties of the flow as a function of streamwise penetration distance compared to steady turbulent plumes in still fluids where the axis of the flow is aligned with the direction of penetration of the flow. In particular, the penetration of the flow normal to the axis is much greater for steady turbulent plumes in crossflows than for corresponding plumes in still fluids, e.g. rps/(xps – xos) and wps/(xps – xos) are 0.36 and 0.49 for present plumes in crossflow whereas rp/(xp – xo) is 0.16 for the measurements of Diez et al. [47] for plumes in still fluids. Thus, penetration of the flow normal to its axis is 2−3 times faster for steady turbulent plumes in crossflows than for similar plumes in still fluids for similar streamwise distances and initial source conditions. Notably, this behavior is qualitatively similar to results observed by Diez et al. [48] for steady turbulent jets in crossflows compared to similar jets in still fluids. This behavior suggests that there is much more effective mixing between the source and the ambient flows when the axes of the source flow are perpendicular to the streamwise direction of penetration, which is the case for steady turbulent plumes and jets in crossflows than when the axis of the source flow is aligned with the direction of penetration which is the case for similar plumes and jets in still fluids.

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4.4.4 Structure properties The development of distributions of mean concentrations of source fluid over cross-sectional planes of steady turbulent plumes in crossflows is illustrated in Fig. 20. This plot involves mean and fluctuating concentrations of source fluid plotted according to the self-preserving structure variables of eqns (32) and (33) along a horizontal line crossing the flow that intersects the axes of the two counter-rotating vortices for various test conditions including different values of (xc – xos)/d. On this plot, results from the developing flow region are designated by open and partially open symbols whereas results for self-preserving flow are shown as dark symbols that represent the average of all the results measured within the self-preserving flow region. Notably, results in the developing flow region of Fig. 20 for both mean and rms fluctuating concentrations were obtained for uo/v• = 24−50 and (xc – xos)/d ≤ 62, which is within the developing flow region of the flow regime map of Fig. 15. Similarly, all results for the self-preserving flow region of Fig. 20 for both mean and rms fluctuating concentrations involve values of uo/v• and (xc – xos)/d that are within the self-preserving flow region of the flow regime map of Fig. 15. Mean concentrations of source fluid reach a maximum at the vortex axes, somewhat analogous to reaching

Figure 20: Plots of the development of mean and fluctuating concentrations of source fluid in terms of self-preserving variables for transverse paths through the vortex axes over the cross-section of the flow for steady turbulent plumes in crossflows (from Diez et al. [49]).

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maximum values of mean concentrations at the axis of steady turbulent plumes in still fluids [28]. In contrast, the maximum values of rms concentration fluctuations along these paths reach maximum values at somewhat larger radial distances than the axes of the counter-rotating vortices, which also is analogous to the position where rms concentration fluctuations reach a maximum in steady turbulent plumes in still fluids [28]. Finally, the region between the counter-rotating vortices involves a broad minimum of rms concentration fluctuations, caused by the entrainment of ambient fluid between the axes of the vortices seen in the dye visualization image of Fig. 13 and the PLIF images of Fig. 18. Figures 21 and 22 are illustrations of mean and rms fluctuating concentrations of source fluid plotted according to the self-preserving structure variables of eqns (32) and (33). These results are for various paths over the flow cross-section (horizontally through the axes of the two counter-rotating vortices, vertically through the axes of the counter-rotating vortices,

Figure 21: Plots of mean concentrations of source fluid in terms of self-preserving variables for various vertical and horizontal paths over the cross-section of the flow for steady turbulent plumes in crossflows within the self-preserving region (from Diez et al. [49]).

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Figure 22: Plots of rms concentration fluctuations of source fluid in terms of self-preserving variables for various vertical and horizontal paths over the cross-section of the flow for steady turbulent plumes in crossflows within the self-preserving region (from Diez et al. [49]).

and vertically along the plane of symmetry of the counter-rotating vortex system). All these results were obtained at test conditions within the self-preserving flow region of the flow regime map of Fig. 15; the averages of these results have been plotted in Figs 21 and 22. The variations of appropriately scaled mean and rms fluctuations of the concentrations of source fluid for these conditions were well within experimental uncertainties, e.g. less than 11% (95% confidence) for |z|/(xc – xos) < 0.45 and |x – xc|/(xc – xos) < 0.40. The distributions of mean and rms fluctuating concentrations over the cross-section of steady turbulent plumes in crossflows in the self-preserving region are quite complex and cannot be reduced to a simple empirical formula similar to that obtained by Dai et al. [28] for steady turbulent plumes in still fluids. Instead, present measurements of mean and rms fluctuating concentrations over cross-sections in the self-preserving region were reduced in terms of self-preserving

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201

variables and plotted as a function of location in the cross-section according to self-preserving streamwise and transverse variables indicated by eqns (32) and (33), e.g. (x – xc)/(xc – xos) and z/(xc – xos), respectively. Results of this nature are illustrated in Figs 23 and 24. The mean locations of the two axes of the counter-rotating vortex system are shown as white crosses on these plots, for reference purposes. On both plots, the self-preserving scaled values of mean and rms fluctuating concentrations are divided into 12 gray scales, for ranges of mean and rms fluctuating source fluid concentration values in terms of self-preserving variables of 0−70 and 0−35, respectively. The counter-rotating vortex system is seen to contribute to the two-lobed structure of the flow with the entrainment of ambient fluid along the plane of symmetry from the side of the flow opposite to the plume source tending to displace maximum mean concentrations along this plane in the streamwise (upward) direction (i.e. in the direction of penetration of the counter-rotating vortex system). A particularly surprising feature of this flow is its unusually large streamwise and transverse penetration distances of (x – xc)/(xc – xos) of approximately +0.6 to −0.3 and z/(xc – xos) of approximately ±0.5. These values are 2−3 times larger than the corresponding radial dimensions of steady round buoyant turbulent plumes in still fluids [28], where r/(x – xo) is approximately 0.16. In addition, the concentration field for steady turbulent plumes in crossflows decays according

Figure 23: Contour plots of mean concentrations of source fluid in terms of self-preserving variables over the cross-section of the flow for steady turbulent plumes in crossflows within the self-preserving region (from Diez et al. [49]).

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Figure 24: Contour plots of rms concentration fluctuations of source fluid in terms of selfpreserving variables over the cross-section of the flow for steady turbulent plumes in crossflows within the self-preserving region (from Diez et al. [49]).

__

to cm ∼ (x – xos)−2, which is slightly faster than for steady turbulent plumes in still fluids which __ decays according to cm ∼ (x – xos)−5/3 [28]. This highlights the capability of vortex structures in crossflow to promote effective mixing between source and ambient fluids. These results can be compared with the data obtained by Smith and Mungal [57] in air. Their detail concentration measurements in the counter-rotating vortex cross-section included nonbuoyant jets with source/crossflow velocity ratio uo/v• = 5, 10, 15, 20, 25 (sets of 400 images) and vertical penetrations as far as 75 source diameters for their uo/v• = 25 case. The presented review complements those measurements by providing data for the first time in the self-preserving region of the flow. The mean concentrations obtained by Smith and Mungal [57] are consistent with the present results even though their measurements were done in the developing region. They do not report rms values of the concentration fluctuation. Estimates based on their probability density function, p.d.f. data also suggest reasonable agreement. It should be noted that crossflow jets and plumes show much higher rms values of the concentration fluctuation compared to free jets. Thus, present results show that mixing in crossflow jets [48] and plumes is much faster than in free jets and therefore Schmidt number effects may not be as important.

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4.5 Conclusions Scaling relationships for the penetration (geometrical) and structure properties of steady turbulent plumes in crossflows have been evaluated based on measurements of the mixing properties of salt water sources injected into fresh water and ethanol/water crossflows. Major conclusions are as follows: 1. The flows become turbulent at streamwise distances of 0−5 source diameters from the source. The onset of self-preserving behavior required that axes of the counter-rotating vortex system be nearly aligned with the crossflow direction whereas this condition was strongly affected by the source/crossflow velocity ratio, uo/v•. As a result, self-preserving behavior is observed at streamwise distances greater than 10−20 source diameters from the source for uo/v• = 4 (the smallest value of uo/v• considered), increasing to streamwise distances greater than 160−170 source diameters from the source for uo/v• = 100 (the largest value of uo/v• considered). The detailed flow regime map of the developing flow and self-preserving regions for steady turbulent plumes in uniform crossflows of Fig. 15, however, should be consulted for the details of the variation of (xos − xcs)/d at the onset of self-preserving behavior as a function of uo/v•. 2. Combining the no-slip convection approximation in the crosstream direction with self-preserving scaling for a horizontal line thermal in a still fluid in the streamwise direction yielded good predictions of both steady penetration properties and the steady structure properties (consisting of the mean and rms fluctuations of the concentrations of source fluid) of the flow within the self-preserving region of steady turbulent plumes in crossflows. 3. Diez et al. [49] observed that the self-preserving structure of steady turbulent plumes in crossflows involves a counter-rotating vortex system whose axes are nearly aligned with the crossflow and thus are nearly horizontal. The nearly crossflow orientation of the axes of the counter-rotating vortex system promotes unusually rapid mixing due to the approximately crossflow motion in the streamwise direction of steady turbulent plumes in crossflows compared to steady turbulent plumes in still fluids where the flow axis is aligned with the streamwise direction, e.g. (rps and wps)/(xps – xos) = 0.36 and 0.49 for steady turbulent plumes in crossflow compared to rp/(xp – xs) = 0.16 for steady turbulent plumes in still fluids. The rapid onset of self-preserving behavior for plumes in crossflow at small values of uo/v•, where the axes of the counter-rotating vortex system becomes aligned with the crossflow almost immediately upon leaving the source and the onset of self-preserving behavior is observed at streamwise distances of 10−20 source diameters from the source, provides further evidence of unusually rapid mixing due to the crossflow motion of the plume, e.g. steady turbulent plumes in still fluids where mixing is limited to the longitudinal direction only exhibit self-preserving behavior at streamwise distances greater than 80−100 source diameters from the source [28]. The observations of conclusion (3) for steady turbulent plumes in crossflows, combined with similar observations from Diez et al. [48] for steady turbulent jets in crossflows, highlight the enhanced mixing and transport that occur when the geometry involves crossflow rather than parallel flow; this behavior is observed frequently in many areas of fluid mechanics, e.g. the response of hot wires mainly to crossflow rather than coflow, and the highly effective primary breakup properties of nonturbulent liquid jets subjected to crossflow rather than coflow, among others.

5 Concluding remarks The findings of the present review suggest that significant progress has been made toward gaining a better understanding of round buoyant turbulent plumes in unstratified still and crossflowing

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environments. The main observations are that achieving self-preserving behavior depends upon the flow and the property under consideration; that complete self-preserving behavior is achieved farther from the source and generally involves narrower scaled flow widths than previously thought; that thermal and plume motion in crossflow satisfies the no-slip convection approximation; and that salt-water simulations appear to provide results at laboratory scale that are relevant to large-scale practical flows. However, many issues about these flows must still be resolved: the structure and mixing properties of unsteady thermals and plumes are not well understood due to problems of making measurements of transient flows; the structure and mixing properties of steady plumes in crossflow are not well understood due to problems of making measurements in three-dimensional flows; the behavior of buoyant turbulent flows in stratified environments has received very little attention so that even simple flow penetration measurements for this flow would be helpful; and baseline information about the penetration, structure, and mixing properties of turbulent nonbuoyant puffs and jets should be developed in order to better understand effects of buoyancy in corresponding turbulent buoyant thermals and plumes.

Acknowledgments The authors’ research concerning buoyant turbulent flows was supported by the United States Department of Commerce, National Institute of Standards and Technology, Grant Nos. 60NANB8D0081 and 60NANB1D006, with H.R. Baum of the Building and Fire Research Laboratory serving as Scientific Officer; Z. Dai and L.-K. Tseng also contributed to this research.

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[50] [51] [52] [53]

[54]

Weakly Buoyant Turbulent Fire Plumes

[55] [56]

[57]

207

Ferrier, A.J., Funk, D.R. & Roberts, P.J.W., Application of optical techniques to the study of plumes in stratified fluids. Dyn. Atmos. Oceans, 20, pp. 155−183, 1993. Alahyari, A. & Longmire, E.K., Particle image velocimetry in a variable density flow: application to dynamically evolving microburst. Experiments in Fluids, 17, pp. 434−440, 1994. Smith, S.H. & Mungal, M.G., Mixing, structure and scaling of the jet in crossflow. J. Fluid Mech., 357, pp. 83−122, 1997.

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CHAPTER 6 Pyrolysis modeling, thermal decomposition, and transport processes in combustible solids C. Lautenberger & C. Fernandez-Pello University of California, Berkeley, USA.

Abstract In a fire, combustion occurs when gaseous fuel liberated from solid materials mixes with the surrounding oxidizer and reacts with oxygen, releasing heat and combustion products. This heat in turn supports further gasification of the fuel. Therefore, condensed-phase processes are one of the primary factors controlling ignition, burning, and flame spread in fires. This chapter reviews several aspects of condensed-phase processes that affect a material’s overall reaction to fire, with an emphasis on modeling. The various pyrolysis modeling strategies that have been used to simulate the burning of solids are summarized. An overview of decomposition kinetics and thermodynamics in the solid phase is given due to their importance in the burning of solids. Conduction, radiation, convection, and momentum transfer within combustible solids are reviewed. Wherever possible, values of material properties and pyrolysis coefficients needed for modeling are given for different materials.

1 Introduction Property-based first principles fire modeling of the end-use configuration is currently considered an appropriate long-term goal of fire research [1, 2]. This requires an understanding of the processes and transport phenomena occurring in both the gaseous and the condensed phases. The latter is the focus of this chapter. The primary solid phase processes that control a fire’s development are the rates at which combustible solids heat up and generate gaseous fuel that becomes available for combustion. These phenomena are interrelated and driven largely by external heat transfer from flames, a hot smoke layer, and nearby heated surfaces. The goal of this chapter is to provide a brief overview of the condensed-phase heat and mass transfer processes that are relevant to fires, with an emphasis on modeling. Wherever possible, literature values of the properties needed to characterize common materials are given.

2 Pyrolysis modeling and fire modeling Most ‘fire modeling’ performed to date should probably be called ‘fire consequence modeling’ because rarely is the fire itself modeled in detail. Instead, the fire is specified a priori as a

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time-history of heat and chemical species generation rates (usually, a heat release rate (HRR) and associated product yields). A fire model then predicts the effects, or consequences, of this particular fire on the space under consideration. This approach is suitable for designing a building’s egress and life safety systems to mitigate the threat from a specific fire, but it cannot be used to estimate how a fire would develop from a given initiating fire. Due in part to recent increases in computer power, fire modeling has reached the point that fire development can actually be predicted. An essential component of these predictions is ‘pyrolysis modeling’, the process through which the condensed-phase processes that control ignition, flame spread, and ultimately fire growth are simulated. In the chemical engineering literature, pyrolysis generally refers to the anaerobic thermal decomposition of solids; as it is often used by the fire community, pyrolysis refers generically to the liberation of gaseous volatiles from a solid fuel, regardless of the presence or absence of oxygen. Although they all have slightly different connotations, we will use the terms pyrolysis, gasification, degradation, and decomposition throughout this chapter to refer to the generation of gaseous components from a condensed-phase fuel. Generically, a pyrolysis model is an algorithm that quantifies the rate at which solid combustible surfaces heat up and generate gaseous pyrolysate when thermally stimulated. It may be a standalone entity or coupled to a computer fire model that calculates gas-phase combustion and transport phenomena. A crucial aspect of pyrolysis modeling involves quantitatively describing a material or assembly in terms of the parameters (or ‘material properties’) that are needed to calculate its temperature and fuel production rate as a fire develops. The pyrolysis models proposed to date range from simple empirical formulations that rely heavily on fire test data to highly complex models that attempt to simulate microscale physical and chemical processes in exhaustive detail. Due to the difficulties associated with obtaining reliable property data, highly complex pyrolysis models are not practical for fire problems where a variety of fuels are encountered. In fact, the biggest challenge of pyrolysis modeling for fire applications is not formulating a comprehensive set of governing equations and then coding a computer program that solves those equations. Instead, the challenge is making enough simplifications and approximations that the number of empirical or adjustable parameters is kept manageable without compromising the generality of the model or neglecting relevant physical phenomena. If a model is intended for use outside of a research environment, then it is equally important that the required properties can be estimated either by consulting the literature, by direct measurement or inference from laboratory test data, or through an optimization exercise where model predictions are matched to experimental data [3, 4]. Most pyrolysis models intended for fire applications fit into one of two main categories. The first category comprises semi-empirical or material fire property-based formulations that relate burning and flame spread rates directly to bench-scale fire test data. This class provides a macroscale description of the burning process without considering the individual micro-scale physical and chemical processes that collectively contribute to a material’s overall reaction to fire. The next category includes comprehensive models that consider (with a widely varying level of detail) the actual small-scale processes and transport phenomena occurring within the solid. All pyrolysis models begin (explicitly or implicitly) with universally applicable statements of conservation of mass, energy, species, and sometimes momentum. However, the approach taken by most authors when postulating a model is to make approximations and simplifications that reduce these general conservation laws to a simplified set of governing equations that are applicable only to one class of materials. Therefore, most comprehensive pyrolysis models can be further divided into thermoplastic polymer, charring, or intumescent formulations. In Section 2.1, semi-empirical material fire property-based pyrolysis models and comprehensive pyrolysis models for thermoplastic, charring, and intumescent materials are reviewed.

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The emphasis here is on the modeling strategy; later the controlling kinetics and transport phenomena are examined in greater detail. In this short chapter, it is not possible to do justice to the broad literature amassed on this topic, and the reader is referred to several reviews [5−10] for additional details.

2.1 Semi-empirical and fire property-based pyrolysis/gasification models The fire scientist’s primary tools for quantifying the fire behavior of combustible solids are benchscale laboratory tests that measure the mass loss rate (MLR) and HRR of small-scale (~0.01 m2) fuel samples exposed to a well-characterized thermal stimulus, usually radiant heating. These experiments provide an overall measure of a material’s fire behavior or ‘reaction to fire’ − and can provide insight into the transport phenomena that affect the decomposition of combustible solids. Although effective material fire properties such as the effective thermal inertia and apparent heat of gasification can be estimated from these tests, fundamental material properties (e.g. temperature-dependent thermal conductivity and specific heat capacity) cannot. Some of the earliest fire growth modeling studies were motivated by the possibility that the outcome of standardized full-scale fire tests could be predicted with a model formulated in terms of property data that can be obtained directly from existing bench-scale fire tests. Since the cost of this bench-scale fire testing is a fraction of that associated with full-scale fire testing, this approach has been advocated as cost-effective alternative to full-scale fire testing. Several simple models have been postulated that require input data that can be directly measured or inferred from widely used bench-scale fire tests such as the cone calorimeter [11] or LIFT apparatus [12]. One modeling approach [13−15] assumes that a material’s burning rate is zero until its surface is heated to its pyrolysis temperature Tp, sometimes taken as equal to the ignition temperature, Tig. The reason for this equivalence is that a solid’s pyrolysis rate is very sensitive to temperature so small increases in temperature can cause large increases in the pyrolysis rate. The time at which a material element reaches Tp is determined by solution of the transient heat conduction equation. This can be accomplished many ways, for example by assuming that the material is a one-dimensional constant-property semi-infinite inert solid and applying Duhamel’s theorem [13]: net,mod ′′ ( t) t q  1 dt for t < t p T0 + ∫ πk rc t = 0 t −t Ts (t ) =  T for t ≥ tp  p

(1)

In eqn (1), Ts is the calculated surface temperature, T0 is the initial solid temperature, krc is the apparent thermal inertia, and tp is the time at which the surface first reaches Tp. q◊ net,mod ″ is the net heat flux to the material’s surface calculated by the model (accounting for convective and radiative losses) and may include contributions from an ignition burner and hot smoke layer. Throughout this section, the subscript ‘mod’ is used to differentiate a modeled quantity from an experimentally measured quantity (denoted with the subscript ‘exp’ as in eqn (3) below). Ignition and subsequent burning is assumed to occur after the material’s surface is heated to Tp. After ignition, it is assumed that the surface temperature remains constant and equal to Tp [13]. However, it should be emphasized that this is an approximation because experimental measurements show that the surface temperature of burning solids is generally higher than the pyrolysis (or ignition) temperature [16, 17]. For thermally thick noncharring solids burning under

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steady-state conditions [18], the MLR per unit area is proportional to the net heat transfer to the solid surface divided by the effective heat of gasification ∆Hg. For now, we will consider ∆Hg as a material fire property. As will be discussed in further detail in Section 3.2, ∆Hg is the quantity of heat required to generate unit mass of volatiles at temperature Tp from unit mass of solid initially at T0. Thus, the HRR of a solid element after it ignites can be calculated from the net heat flux to the surface multiplied by ∆Hc/∆Hg [13, 15]. for t < t p 0  Qmod ′′ (t ) =  ∆H c (t ) for t ≥ t p ′′  ∆H q net,mod  g

(2)

◊ Here, Qmod ″ is the HRR per unit area calculated by the model. ∆Hc is the heat of combustion, and the ratio ∆Hc/∆Hg is another material fire property sometimes called the combustibility ratio [19] or heat release parameter [20]. Equation (2) is strictly valid only for noncharring thick solids burning under steady-state (thermally thick) conditions, but it has been applied to other burning regimes [13, 15] with good results. As an alternative to eqn (2), a solid’s HRR is sometimes related directly to transient HRR or MLR measurements obtained from small-scale fire tests [21, 22] or an approximate curve fit to this data [23]: Q mod ′′ (t ) = Q exp ′′ (t )

(3)

Q mod exp( − lt ) ′′ (t ) = Q peak,exp ′′

(4)

In eqn (3), used by Brehob et al. [21] and Tsai and Drysdale [22], the modeled HRR history ◊ ◊ (Qmod ″ (t)) of a burning element is assumed identical to the measured HRR history (Qexp ″ (t)) from ◊ the cone calorimeter. In eqn (4), proposed by Karlsson [23], Qpeak,exp ″ is the peak HRR measured experimentally, e.g. in the cone calorimeter, and l is a fitting parameter that controls the assumed exponential decay in HRR. The primary shortcoming of eqns (3) and (4) is that they implicitly assume that the net heat flux history which the material in the model ‘feels’ is identical to its exposure in the laboratory test. Therefore, this type of model technically cannot accommodate any difference in heat flux history between the experiment and the model, such as an increase in the burning rate due to external heating (e.g. from hot layer radiation). Additionally, it has been shown experimentally that the MLR is affected by the total (cumulative) heat absorbed by the solid [24]. It may be possible to obtain reasonable results by performing fire tests at multiple irradiance levels and developing some sort of an interpolation scheme to extend the data to an arbitrary heating history, but this type has not yet been widely demonstrated. As an attempt to remedy this deficiency, an ‘acceleration’ function has been introduced [25, 26] where the modeled HRR is related essentially to the total heat flux absorbed by the solid, summarized in eqn (5): q net,mod (t ) ′′

x (t ) =

q net,exp (t ) ′′

q (t ) = ∫

t

t′=0

x(t ′ ) dt ′

Q mod ′′ (t ) = x(t )Q exp ′′ (q (t ))

(5a)

(5b) (5c)

Condensed-Phase Processes in Combustible Solids

213

Although these modeling approaches are crude in comparison to some of the more refined treatments of the solid phase, it will be shown in Section 5 that this simple description of solid phase processes has been successful at predicting full-scale fire behavior. The primary advantage of ◊ these modeling approaches is that all required input parameters (krc, Tig, ∆Hg, ∆Hc, and Qexp ″ (t)) can be obtained from existing bench-scale fire tests using well-established data reduction techniques. As greater levels of complexity are added, establishing the required input parameters (or ‘material properties’) for different materials becomes an onerous task. 2.2 Comprehensive pyrolysis models: thermoplastics Thermoplastic materials include many of the widely used commodity polymers such as polythylene (PE), polypropylene (PP), polystyrene (PS), and polymethylmethacrylate (PMMA). Unless fire retardants are added, thermoplastics usually do not char. Instead, they burn completely and leave minimal residue. Thermoplastics melt to various degrees, and the models considered in this section do not explicitly consider melting; however, melting and related phenomena will be discussed further in Section 4.5. Laboratory scale combustion experiments show that after an initial transient period, noncharring thermoplastics exhibit a quasi-steady-state burning rate that depends primarily on the applied irradiance level. However, this steady-state burning period is observed only for thick materials that are not affected by heat losses from the back (unexposed) face of the sample. Figure 1, adapted from Babrauskas [27], shows the HRR of PMMA samples having different thicknesses measured in cone calorimeter [11] combustion experiments. It can be seen that steady-state burning is not achieved, except for the thicker samples. This indicates that a material’s burning behavior is affected by heat transfer at the unexposed side of the sample, i.e. the insulating effect of the substrate is a factor. There is no mechanism included in eqns (1) and (2) above to account for this behavior, but it can be captured with comprehensive pyrolysis models that treat the heat transfer aspect of the problem in greater detail. 1200

1.5 mm 3 mm

Heat release rate (kW/m2)

6 mm

10 mm

12.5 mm

900

20 mm 25 mm

600

300

0

0

300

600 Time (s)

900

1200

Figure 1: PMMA burning in the cone calorimeter. Effect of thickness (adapted from Babrauskas [27]).

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Transport Phenomena in Fires

The simplest class of comprehensive pyrolysis models for polymers is ablation models [28−32] that are basically refinements to eqns (1) and (2). Rather than using Duhamel’s theorem to calculate the surface temperature of a semi-infinite inert solid (eqn (1)), the temperature profile of the finite-thickness solid is determined either by a numerical finite difference solution [31, 32] or integral methods [28−30] wherein a functional form of the temperature profile is assumed a priori. This makes it possible to handle materials that do not exhibit thermally thick behavior. With ablation models, the pyrolysis rate is zero until Tp is reached, at which point the surface temperature is maintained at Tp (as in eqn (2)). Thus, it is assumed that the pyrolysis kinetics are much faster than heat diffusion, and that the latter is the limiting process. The rate of mass loss is calculated from a heat balance at the sample surface where it is assumed that all mass loss occurs. The temperature distribution in the solid T(z,t) is calculated by solving the one-dimensional heat conduction equation for an opaque constant density inert solid: rc

∂T ∂  ∂T  = k  ∂t ∂z  ∂z 

(6)

Solution of eqn (6) requires specification of one initial condition and two boundary conditions. The initial condition describes the temperature profile in the solid at time t = 0, and the ‘back-face’ boundary condition describes the rate of heat transfer from the back-face as a function of temperature. This back-face boundary condition makes it possible to capture the upturn in the HRR after most of the material has been consumed, e.g. Fig. 1. The remaining boundary condition is applied at the front face, and it takes a slightly different form depending on whether or not the material has ignited (started to gasify). Denoting tp as the time at which the surface temperature reaches Tp, the front-face boundary condition at z = 0 is: −k

−k

∂T = q net ′′ for t < t p ∂z

(7a)

T = Tp for t ≥ t p

(7b)

∂T = qnet ′′ − m ′′ ∆Hvol for t ≥ t p ∂z

(7c)

The primary quantity of interest is usually the fuel generation rate (equivalent to the MLR), m· ″, which is determined from eqn (7c). ∆Hvol is the heat of volatilization, often called the heat of vaporization by the fire community (see Section 3.2 for further explanation), and is not the same as the heat of gasification ∆Hg. ∆Hvol is the quantity of heat required to generate unit mass of volatiles at Tp from unit mass of solid at Tp and is positive for an endothermic process. As will be discussed in Section 3.2, ∆Hg is equal to ∆Hvol plus the sensible enthalpy required to raise unit mass of solid from its initial temperature T0 to its pyrolysis temperature Tp. The primary advantage of this approach compared with more detailed models discussed later is its simplicity, being not much more complicated than eqns (1) and (2) above. The decomposition process is characterized by a single parameter (Tp), making approximate analytical solutions possible [28]. Finite thickness materials that are influenced by the back-face boundary condition can be readily handled. With a finite-difference solution method, temperature-dependent material properties (k, r, c) can be incorporated. Despite its simplicity, this approach is capable of accurately reproducing burning rates in bench-scale combustion experiments [30, 32], as shown in Fig. 2. One disadvantage of the ablation approach compared with the semi-empirical models discussed earlier is that the properties required to characterize a particular material cannot be directly determined from bench-scale fire tests. Individual values of k, r, and c are needed rather than the

Condensed-Phase Processes in Combustible Solids

215

40

Mass loss rate (g/m2-s)

30

20

Experiment Model

10

0

0

200

400

600

800

1000

Time (s)

Figure 2: PE mass loss rate burning in a cone calorimeter at 70 kW/m2 irradiance. Comparison of experimental results and predictions of integral ablation model [30].

product krc (which can be estimated from fire tests [12]). Furthermore, the heat of volatilization ∆Hvol is also needed; it cannot be easily estimated from bench-scale fire test data (only ∆Hg can). Related to the physics, disadvantages include the assumption that ignition occurs when the solid surface temperature reaches Tp and then remains constant. In actuality, the surface temperature at ignition depends on the environmental conditions and pilot strength/location. As mentioned above, the temperature of a burning solid does not remain constant at its ignition temperature [16, 17]. Unless an advective term is added to eqn (6) to accommodate surface regression, there is no mechanism to account for a change in thickness as the sample burns away. Since the calculations in Fig. 2 are based on a model that does not account for surface regression, the model calculations deviate from the experimental data after ~800 s due to the insulating effect of the substrate and the decreasing thickness of the solid. Finite rate pyrolysis models, which usually involve a single-step nth-order Arrhenius reaction, represent the next level of complexity. Pyrolysate generation has been treated as occurring only at the surface [33, 34] or more frequently, as a distributed in-depth reaction [35−44] to account for sub-surface fuel generation. With ablation models [28−32] or finite-rate kinetics models that relate the fuel generation rate to the surface temperature [33, 34], all fuel generation occurs at the surface. However, once a finite-rate distributed reaction is introduced (see eqns (8)−(10) below), fuel generation also occurs in-depth. The decomposition or pyrolysis process is characterized by three parameters: pre-exponential factor (A), activation energy (E), and reaction order (n), although n is frequently assumed to be 1. The volumetric decomposition rate (kg of volatiles generated per unit volume of condensed phase per second) is a function of temperature and the ‘conversion’ a which can be thought of as the reaction progress (discussed in greater detail in Section 3.1, see eqn (19)). This type of model can be summarized briefly as: rc

∂T ∂T ∂  ∂T  + rwc =  k  − w g′′′∆H vol − Q s′′′−g ∂t ∂z ∂z  ∂z 

(8)

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Transport Phenomena in Fires

 E  w g′′′= r A(1 − a)n −1 exp  −  RT  z

m g′′( z) = ∫ w g′′′ dz d

(9)

(10)

It has been assumed in writing eqns (8) and (10) that all fuel generated in-depth escapes instantaneously and no pyrolysate vapor is stored as bubbles in the molten polymer. Quantities without a subscript refer to the condensed phase, and the subscript ‘g’ denotes the gaseous volatiles. The second term on the LHS of eqn (8) includes the advective velocity w because as the condensed phase is volatilized, it is assumed that the molten polymer instantaneously fills the voids, thereby giving rise to surface regression. Here, m◊ g″(z) is the local mass flux of gaseous fuel, taken as negative when flowing toward the surface since the +z direction points into the solid. It is assumed that the condensed-phase density is constant (i.e. density invariant with temperature) so from mass conservation the advective mass flux of the molten polymer is related to the mass flux of gaseous fuel as rw(z) = −m◊ g″(z). The divergence of the volatile mass flux is the local volumetric fuel gen∂m ′′

eration rate, i.e. g = w g′′′. Due to temperature gradients, the temperature at which pyrolysate is ∂z generated in-depth is generally different from the temperature of the condensed-phase material through which it must pass to reach the surface. Thus, there will be heat transfer between the gaseous and condensed phases. This is accounted for through the term Q s′′′− g, the volumetric rate of heat transfer from the solid phase to the gas phase. This term is sometimes modeled by assuming thermal equilibrium between the gaseous and condensed phases, giving rise to a convective ∂Tg term of the form m◊ g″cg ___ . However, this term is sometimes omitted [34] on this basis that it is ∂z small except at high heat flux levels with steep temperature gradients. One challenge is simulating the mechanism through which pyrolysate vapors generated in-depth escape from the solid. Although it is usually assumed that the vapors instantaneously escape with no flow resistance, a few studies have included the effect of bubbling, ranging from simplified [42] to detailed [45−47] treatments. One shortcoming of this modeling approach is that the effect of oxygen concentration on the decomposition rate is not explicitly included. In general, the decomposition kinetics and thermodynamics (as well as the composition of the volatiles generated) are sensitive to oxygen concentration. In a fire, combustible solids can be exposed to oxygen concentrations ranging from those of the ambient oxidizer to close to zero. The sensitivity of solid decomposition to oxygen has been demonstrated by Kashiwagi and Ohlemiller [48]. They measured the MLR of PMMA irradiated (under nonflaming conditions) at 17 and 40 kW/m2 in atmospheres ranging from pure nitrogen to 40% oxygen by volume. Their results are reproduced in Fig. 3. It can be seen that the MLR increases with the oxygen content of the atmosphere, but the oxygen sensitivity is more noticeable at 17 kW/m2 irradiance (Fig. 3a) than 40 kW/m2 (Fig. 3b). This observation can be explained as follows: the oxygen concentration in the vicinity of the sample surface is reduced as the MLR increases because gas-phase oxygen from the oxidizer stream is displaced by the gaseous pyrolysate ‘blowing’ from the sample surface. Since the MLRs are higher at 40 kW/m2, the surface is better protected by blowing and it is more difficult for oxygen to penetrate into the polymer. This reveals some of the transport phenomena affecting solid decomposition: oxygen must be making its way into the molten polymer by either molecular diffusion or penetration into burst pores created by bubbling. As will be discussed in Section 3.1, we still have a limited quantitative understanding of these phenomena.

Condensed-Phase Processes in Combustible Solids

217

3.5 40% Oxygen

Mass loss rate (g/m2-s)

3.0 2.5 20% Oxygen

2.0 1.5

10% Oxygen

1.0 5% Oxygen 0.5 Nitrogen 0.0

0

300

600

900

Time (s)

(a) 14

Mass loss rate (g/m2-s)

12 10

40% Oxygen

8 6 20% Oxygen 4

10% Oxygen

2 Nitrogen 0 (b)

0

30

60

90 Time (s)

120

150

180

Figure 3: Effect of ambient oxygen concentration on gasification rates of PMMA: (a) 17 kW/m2; (b) 40 kW/m2 (adapted from Kashiwagi and Ohlemiller [48]).

2.3 Comprehensive pyrolysis models: charring materials A large number of materials encountered in practice exhibit charring, either naturally (wood, thermoset polymers, phenolic composites) or due to addition of fire retardants. In contrast to thermoplastic materials where most of the fuel generation occurs near the surface (even for a distributed reaction), the primary fuel generation zone in charring materials can be located well below the surface at a reaction front that separates the char layer from the virgin layer. For a fixed thermal exposure, thermoplastics show an increasing or steady-state MLR/HRR until the material

218

Transport Phenomena in Fires

is consumed. However, charring materials generally show a decaying MLR/HRR after an initial peak is reached. This is because a low-density porous char helps protect the virgin material from further heat transfer (char can be quite effective at limiting carbon transfer from the condensed phase). Some materials show a second peak if the reaction zone ‘feels’ the insulating effects of the underlying substrate. This dual-peak shape is characteristic of charring materials, but it is not always observed − whether or not two peaks occur depends on the thickness, heat flux, and substrate material. Figure 4 shows the MLR of particleboard measured in nitrogen when irradiated at six heat flux levels between 14 and 39 kW/m2 [49]. At higher heat flux levels the distinctive twopeak curve becomes apparent, but this does not occur at the lower heat flux levels. The change in shape of the MLR curve with heat flux is the combined effect of several phenomena: heat transfer in the virgin and char layers and to the underlying substrate as well as chemical kinetics, which control the rate at which the virgin material is converted to char. In addition to general reviews that cover some aspects of modeling charring degradation [5−7, 9], modeling the fire behavior of charring materials has been specifically covered in review papers published in 2000 [8] (wood and polymers) and 2005 [10] (lignocellulosic fuels). These papers provide a good assessment of the current status of modeling the decomposition of charring solids. Models of charring pyrolysis usually use numerical solution of the governing equations. An exception is the asymptotic analysis of Wichman and Atreya [50] wherein approximate formulas are developed for the MLR of a charring solid in the limit of large activation energy. In the simplest class of numerical models for charring pyrolysis, it is assumed that an infinitely thin reaction zone (or pyrolysis front) separates the char layer from the virgin material [51−61], analogous to the Stefan problem where phase change occurs at a thin interface. This is a reasonable approximation at high heat flux levels, but can become questionable at lower heat fluxes. A single reaction is considered, and infinitely fast or finite rate kinetics can be used. In some models, the conversion of virgin material to char is assumed to occur at a fixed pyrolysis temperature [51−56, 59, 60] and the velocity at which the front propagates into the solid is determined by a heat

15 39 kW/m²

Mass loss rate (g/m2-s)

12 31 kW/m² 9

25 kW/m²

6 18 kW/m² 16 kW/m²

3 14 kW/m² 0 0

300

600 Time (s)

900

1200

Figure 4: Mass loss rate of particleboard in nitrogen (adapted from [49]).

Condensed-Phase Processes in Combustible Solids

219

balance at the pyrolysis front. Thus, the kinetics are infinitely fast, analogous to the thermoplastic ablation models discussed earlier. In other formulations, finite rate kinetics are used, and the propagation velocity follows the usual Arrhenius form [34, 57, 58]. A recent paper [61] compared the infinite-kinetics (fixed pyrolysis temperature) approach to the finite kinetics approach. Integral models [51, 52, 54, 55, 59, 60] have the advantage that the governing partial differential equations are transformed to ordinary differential equations, but numerical solution is generally still required. Several models have been postulated that do not rely on the assumption of an infinitely thin pyrolysis front separating the char layer from the virgin material [62−69]. When a single global reaction is considered, they are all essentially variations of Kung’s 1972 model [62], originally developed for wood. Although this model is quite simple compared to some of the more recent modeling efforts, it illustrates the main characteristics of the newer models that generally include more sophisticated submodels for transport phenomena or reaction chemistry. Kung’s model [62] describes the decomposition process as a single-step endothermic Arrhenius reaction where virgin wood is converted to char and volatiles, with the volatiles escaping instantaneously. The density, thermal conductivity, and volumetric heat capacity are assumed to vary linearly with the extent of conversion between virgin wood and char. Heat transfer due to movement of volatiles through the char layer is accounted for by assuming that thermal equilibrium exists between the solid and gas phases. Kung originally posed the model in terms of an ‘active material’ generated from a virgin material. However, a slightly different formulation is presented here wherein the mass fraction of each solidphase ‘species’ (i.e. virgin material or char) is tracked. This makes it straightforward to extend the model to more detailed cases where multiple species are tracked. For simplicity, the moisture content is assumed to be negligible. Let Yc and Yv designate the local mass fractions of char and virgin material in the solid (Xc and Xv are the analogous volume fractions). The bulk density of the virgin material is denoted rv, and the bulk density of the fully reacted char is rc. It is assumed here that rv and rc are constant for a particular material, i.e. they do not depend on temperature. The sensible specific enthalpies of the virgin and char are denoted hv and hc. Then, the weighted bulk density and specific enthalpy are defined as: r = Xv rv + Xc rc

(11a)

h = Yv hv + Yc hc

(11b)

Consider a single reaction that converts virgin fuel to char and volatiles. It is assumed to be first order in the remaining virgin material, with the reaction rate following the usual Arrhenius dependency on temperature:  E  w ′′′ = rYv A exp  −  RT 

(12)

◊ ) are determined The volumetric formation rate of gaseous pyrolysate (w◊″¢ g ) and solid char (w″¢ c from eqn (12) and the ratio of the char and virgin bulk densities as:  r  w g′′′=  1 − c  w ′′′ rv   w c′′′=

rc w ′′′ rv

(13a)

(13b)

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Transport Phenomena in Fires

Assuming there is no net shrinkage or swelling during the charring process and that the storage of gaseous pyrolysate in the char pores is negligible in comparison to its formation rate, conservation of mass, species, and energy can be summarized as:

r

∂m g′′ ∂r =− = − w g′′′ ∂t ∂z

(14)

∂( rYc ) r = w c′′′= c w ′′′ ∂t rv

(15)

∂h ∂  ∂T  = k  − w g′′′∆H p − Qs′′′−g ∂t ∂z  ∂z 

(16)

In eqn (14), m◊ g″ is the mass flux of volatiles, taken as negative when flowing toward the surface (because the +z direction points into the solid). It has been assumed that the volatiles escape instantaneously with no flow resistance and that no net shrinkage or swelling occurs. For clarity of presentation, eqn (16) has been written assuming that both solid species have equal specific heat capacities; the more general case gives rise to terms having the form of a volumetric reaction rate multiplied by enthalpy differences between species but is not instructive. The Q s′′′− g term in eqn (16) accounts for heat transfer from the solid phase to the gas phase, analogous to eqn (8). ∆Hp is the ‘heat of pyrolysis’, the analogous quantity to the heat of volatilization ∆Hvol discussed earlier with reference to thermoplastic materials. As with ∆Hvol, ∆Hp is positive if the reaction is endothermic. Note that the conservation equation for __the virgin mass fraction is obtained from solution of eqn (15), since Yc + Yv = 1. In eqn (16), k is the effective thermal conductivity. It depends on the local state of the material (i.e. Yc and Yv) and perhaps temperature, particularly due to radiative transfer across pores (these issues are discussed further in Section 4). Although it is difficult to accurately estimate the thermal properties of partially degraded materials, the simplest approach is to assume that k varies linearly with the local mass fractions and is independent of temperature: k = X v kv + Xc kc

(17)

where kv and kc are constants corresponding to the thermal conductivity of the virgin material and the completely charred material respectively. Similarly, the mass-weighted specific heat capacity is: c = Yv cv + Yc cc

(18)

Even for this relatively simple model, ten model constants are required to characterize a particular material: kv, rv, cv, kc, rc, cc, A, E, ∆Hp, and cg. Of these, only rv is readily attainable by direct measurement. Recognizing this difficulty, de Ris and Yan [70] developed an optimization method that determines a set of ‘equivalent properties’ which maximize the agreement between the predictions of a linearized version of Kung’s model and experimental data. This methodology for determining the model constants has been applied infrequently [71], and only a few sets of parameter values were found in the literature. Some of those are listed in Table 1. More complex analyses of charring pyrolysis usually use a basic modeling approach similar to that embodied in eqns (11)−(18). What differs from model to model is the reaction mechanism (i.e. multi-step, reaction order other than unity), the number of condensed-phase species, and treatment of the transport phenomena, i.e. calculation of the effective thermal properties, inclusion of an internal flow-resistance, swelling/shrinkage, or description of bubbling and related phenomena.

Condensed-Phase Processes in Combustible Solids

221

Table 1: Literature values of Kung [62] char model parameters. Pacific Maple [66] kv (W/m K) rv (kg/m3) cv (J/kg K) kc (W/m K) rc (kg/m3) cc (J/kg K) cg (J/kg K) A (s−1) E (kJ/mol) ∆Hp (J/kg) e (−)

Particle Board [66]

0.16 0.126 530 663 2420 2520 0.16 + 8.2 × 10−5(T − T0) 0.126 106 133 1.0 2520 − − 5.25 × 108 5.25 × 107 110 125.6 0 0 − 0.9

White Pine [67]

Red Oak† [71]

0.157 + 0.0003(T − T0) 0.15−0.21 − 660−700 2140 + 4.19(T − T0) 1400−1800 0.084 + 0.002(T − T0) 0.18−0.27 − 170−200 1928 + 1.98(T − T0) 2500−3000 2000 − 7.49 × 109 − 145‡ − 3.0 × 105 1.0 − 6.8 × 105 − 0.88 − 1.0

†

A and E given for linearized pyrolysis reaction [70, 71]. Listed as ‘1.45E8 J/(mol K)’.

‡

For example, the influence of the porous char structure on flow of volatiles has been investigated by adding solution of the momentum equation using Darcy’s law [63, 64]. Fredlund [64] formulated a detailed two-dimensional model (most models are one-dimensional) that included flow of both liquid and gaseous water, with Darcy’s law for flow through the porous components. The model also included a surface reaction. The model predictions were compared to extensive experimental data, including temperature, density, and pressure measured at several in-depth locations with generally good results. This set of experimental data [64] remains one of the most comprehensive available in the literature. These more complex models have rarely been applied to fire situations, although they are used extensively in chemical engineering for purposes of optimizing energy conversion techniques, particularly from biomass. The MLR of charring materials generally increases with oxygen concentration. Oxidative reactions occurring at the surface of the char layer may substantially increase the surface temperature above that in inert environments, and under some circumstances char oxidation may account for ~10% of the HRR under flaming conditions [72]. As with the thermoplastic model above, when using eqns (11)−(18), there is no way to account for the influence of oxygen on the pyrolysis reaction, unless A, E, and ∆Hp are made explicit functions of the ambient oxygen concentration or an additional oxidative reaction is added. To illustrate the importance of oxygen concentration on the decomposition of charring materials, the MLR and surface temperature of white pine irradiated at 40 kW/m2 is shown in Fig. 5 at three different oxygen concentrations [73]. It can be seen from Fig. 5a that the MLR in air is approximately double that in nitrogen. The effect of char oxidation is evident in Fig. 5b, where it can be seen that the surface temperature of the sample tested in air is approximately 150ºC greater than that of the sample tested in nitrogen. Related to fundamental transport phenomena, these data indicate that the decomposition kinetics and/or thermodynamics (endothermic vs. exothermic reactions) are affected by oxygen concentration. Surface heating due to char oxidation is a critical factor affecting autoignition because gasphase combustion reactions are initiated by the hot surface. The surface temperature of decomposing solids is very difficult to measure accurately, and measurements are scarce, but recent work using optical methods is promising [17, 74].

222

Transport Phenomena in Fires 12 Air

Mass loss rate (g/m2-s)

9 10.5% Oxygen 6

3 Nitrogen

0

0

120

240

(a)

360 Time (s)

480

600

720

600

720

700 Air 600

Temperature (°C)

500 10.5% Oxygen

Nitrogen

400 300 200 100 0

(b)

0

120

240

360 Time (s)

480

Figure 5: (a) Mass loss rate and (b) surface temperature of white pine irradiated at 40 kW/m2 at three different oxygen concentrations (adapted from [73]). 2.4 Comprehensive pyrolysis models: intumescent materials and coatings An intumescent material or coating swells when heated to form a porous low-density char, thereby reducing heat transfer to the underlying virgin material. Intumescence is an effective mechanism for providing a high degree of thermal resistance while minimizing thickness of the protective skin. Intumescent coatings are sometimes applied to steel structural members to achieve the fire resistance ratings required by building codes, although the US Navy has concluded that intumescent coatings do not provide a level of fire resistance equivalent to traditional insulation, largely due to poor

Condensed-Phase Processes in Combustible Solids

223

adhesion characteristics [75]. In addition to being applied as a coating, intumescents are sometimes grafted into organic composite materials to improve fire performance. Given the environmental problems associated with traditional halogenated fire retardants, ecological concerns may lead to an increased usage of intumescence or char promoters, which are thought of as ecologically friendly processes [76]. Modeling intumescence is considerably more complicated than modeling thermoplastic or charring pyrolysis due to large changes in density and volume. The earliest model for the thermal response of an intumescent material in the open literature was presented by Cagliostro and Riccitiello [77] to help develop design guidelines for a NASA coating. The semi-empirical model used measured temperature-dependent property data where available. When quantitative data were not available, ‘reasonable values’ were chosen. Decomposition was modeled by a three-step Arrhenius reaction. Since the emphasis was on the ability of the intumescent coating to insulate a substrate from thermal insult, the only quantitative comparison of prediction and experiment was for the measured substrate temperature. Anderson and Wauters [78] used an approach similar to Cagliostro and Riccitiello [77] to model a different intumescent coating used by the US Navy. Thermogravimetric (TG) and differential scanning calorimetry (DSC) data (see Sections 3.1 and 3.2) were used to establish the kinetics and thermodynamics of the decomposition process. An interesting feature of their approach is that the TGA experiments were modeled with a 31 term Fourier series rather than the traditional Arrhenius approach. A Lagrangian formulation was adopted, and the change in volume of an element was related directly to its change in mass. They compared model predictions to experimental data for the substrate temperature and expansion factor. They found that the model was able to capture the main features of the experiments, but there were quantitative discrepancies between the measured and calculated substrate temperature. Buckmaster et al. [79] later argued based on experimental observation that intumescence occurs at a thin front. Adopting an Eulerian description, they modeled the reaction as occurring at fixed temperature at an infinitely thin interface between the intumescent char layer and the unreacted material, reducing their model to a Stefan problem. The results were compared only qualitatively with experimental data because their emphasis was on the mathematical description of the problem rather than making quantitative predictions. Henderson and Wicek [80] developed a detailed model of an expanding phenolic composite that included gas flow by Darcy’s law. Temperature dependent thermal properties were used, along with an Arrhenius decomposition reaction in which the activation energy, pre-exponential factor, and reaction order varied with the extent of conversion. The only quantitative comparison between the model predictions and experimental data was for temperatures measured at four different locations within the solid, and very good agreement was obtained. An interesting feature of their model is that it predicted internal overpressures greater than 40 atm. Shih et al. [81] extended the model developed earlier by Buckmaster et al. [79] and treated intumescence as a phase change occurring over a finite temperature range using the concept of a ‘pseudo latent heat’ to account for the endothermicity of the intumescent reaction. Their model was capable of reproducing the ‘bending’ behavior seen in the experimentally measured substrate temperature profiles. Similar to Shih et al. [81], Bourbigot et al. [82] treated the intumescent process in a polypropylene intumescent material as a phase change process, but with an Arrhenius reaction rate. The most detailed models to date are probably due to Di Blasi and Branca [83] and Di Blasi [84]. They simulated the experiments of Cagliostro and Riccitiello [77] with a three-step reaction mechanism. The model predictions were very sensitive to the submodel used to calculate the effective thermal conductivity of the char. Quantitative agreement between the model predictions and the substrate temperature measurements [77] was possible only when using a conductivity

224

Transport Phenomena in Fires

model developed specifically for intumescent coatings with a modification to account for radiation heat transfer across pores. The model of Wang et al. [85] is notable because the three-dimensional problem was considered (all other models are one-dimensional). However, their model was not ‘fully’ three-dimensional because gas flow and swelling were permitted only in one direction. Most authors simulating the decomposition of intumescent materials have used substrate temperature measurements as the only metric against which the predictive capabilities are judged. This is partly due to the dearth of available experimental data. However, it can be misleading to conclude that a particular model ‘works’ on the basis of a comparison of a single temperature measurement. Also important are the MLR or HRR and the degree of swelling predicted by the model. The paper by Griffin et al. [86] gives TG and differential thermal analysis (DTA) data for three different intumescent coatings as well as measurements of substrate temperature, expansion factor (degree of swelling), and HRR in the cone calorimeter. This may prove to be a useful source of validation data for modelers.

3 Decomposition kinetics and thermodynamics Many of the pyrolysis models discussed above either treat the decomposition as being infinitely fast or use a single lumped reaction that approximates the ‘global’ decomposition behavior. In actuality, the production of gaseous volatiles from a heated combustible solid is the macroscopic net result of hundreds of microscale reactions occurring simultaneously. Oxygen is involved in some reactions, as in the case of char oxidation, whereas other reactions can occur in the absence of oxygen, as in pyrolysis under nitrogen. As a result, both exothermic and endothermic reactions occur, sometimes simultaneously. Although techniques exist for measuring the rate constants of elementary gas-phase reactions, analogous techniques have not yet been developed for kinetics in solids. Nonetheless, some fairly advanced diagnostic tools are used in the field of thermal analysis to investigate the kinetics and thermodynamics of decomposing solids. Experimental techniques that aim to study the decomposition kinetics and thermodynamics of solids use very small samples (on the order of a few mg) to reduce heat and mass transfer effects. The assumption is that all gradients become negligible and the degrading sample can be treated as homogeneous (isothermal). 3.1 Thermal and thermooxidative stability One of the most important factors contributing to a combustible solid’s overall fire hazard is its thermal decomposition kinetics. Since ignition of solid materials is usually kinetically controlled (i.e. very sensitive to temperature near the heated surface), ignitability is strongly influenced by thermal stability. By viewing the flame spread process as a sequence of piloted ignitions, it can be seen that a material’s propensity to propagate a flame is also affected by its thermal stability. Thermogravimetric analysis (TGA) is the most widely used experimental technique for quantifying the thermal stability of solids. A high-precision scale is used to measure the mass of a small sample (usually no more than a few mg) as it is exposed to an atmosphere with specified temperature and composition. TGA experiments may be isothermal, or more frequently, expose the sample to an atmosphere having a temperature that increases linearly with time. Due to the small sample size, the sample temperature is taken as equal to the temperature of the atmosphere (low Biot number). For a sample with initial mass m0, the conversion a is defined as: a=

m0 − m m0 − m∞

(19)

Condensed-Phase Processes in Combustible Solids

225

where m∞ is the sample mass at the end of the experiment. For materials that leave no residue (such as noncharring polymers), m∞ = 0 and therefore a = 1 − m/m0. TGA data are usually analyzed within the framework of a kinetic model of the form: da = k (T ) f (a) dt

(20)

In eqn (20), k(T) is a function carrying the temperature-dependency of the reaction rate and f(a) is the ‘reaction model’, often assumed to be: f (a) = (1 − a)n

(21)

where n is called the ‘reaction order’ (in the remaining solid mass). With few exceptions [78, 87], the function k(T) is assumed to take an Arrhenius form:  E  k (T ) = A exp  −  RT 

(22)

where A is the frequency factor, or pre-exponential factor (sometimes denoted Z), and E is the activation energy. After combining eqns (20−22), the time rate of change of a becomes: da  E  (1 − a)n = A exp  −  RT  dt

(23)

In nonisothermal experiments, the atmosphere temperature usually increases linearly with time at a constant heating rate (e.g. 20ºC/min) denoted b. By assuming that the sample temperature is equal to the atmosphere temperature, the transformation dt = dT/b can be made, and eqn (23) can be written as: da A  E  (1 − a)n = exp  −  RT  dT b

(24)

The three model parameters (A, E, and n, sometimes called the ‘kinetic triplet’) are determined from a plot of a or da/dT as a function of T. A thermogravimetric (TG) curve is a plot of a vs. T, whereas a differential thermogravimetric (DTG) curve is a plot of da/dT vs. T. Several techniques have been proposed for extracting the three model parameters (A, E, and n) from TG and DTG curves. They are all essentially nonlinear curvefitting exercise that seek to minimize the residual error between eqn (24) (or its integral) and experimental data. As an example, Fig. 6 shows the experimentally measured da/dT for high density polyethylene compared with the calculation of eqn (24) using A = 3.85 × 1015 s−1, E = 252.8 kJ/mol, and n = 0.582 [88]. Ideally, kinetic parameters should be determined for variable thermal conditions (heating rates). There is some debate regarding the interpretation of the parameters A and E as well as the physical correctness of modeling the rate constant using an Arrhenius form. Vyazovkin and Wight [89] suggest that the physical interpretation of the Arrhenius function as applied to solid decomposition is supported by a sound theoretical foundation. Taking the opposite view, Agrawal [90] states, ‘Although the Arrhenius equation has little physical significance in solid-state reactions, it may be assumed as a two-parameter model to correlate the data thereby minimizing the number of adjustable parameters.’ Simon [87, 91] has also suggested that A and E have no physical meaning and should be interpreted merely as adjustable model parameters. Similarly, Parker [92] wrote that for complex materials ‘not much physical significance should be attached to such parameters’.

226

Transport Phenomena in Fires 0.030

0.025

20 °C/min, model 20 °C/min, exp. 10 °C/min, model 10 °C/min, exp.

da /dT (K-1)

0.020

0.015

0.010

0.005

0.000 650

675

700

725 Temperature (K)

750

775

800

Figure 6: Single-step decomposition of high density polyethylene [88]. Individual points are experimental data and the solid line is calculated using eqn (24). Regardless of the physical significance of the parameters in the single-step Arrhenius equation, it is capable of adequately reproducing experimental thermogravimetric data for a variety of materials that exhibit single-step decomposition. Values of A, E, and n that have been reported in the literature for a few representative materials are listed in Table 2. The atmosphere under which the experiments were conducted is listed in Table 2 because the decomposition kinetics of many materials are sensitive to oxygen concentration. One must be careful when applying literature values of thermokinetic parameters. For ignition studies, it is more appropriate to use values obtained in an oxidative environment (e.g. air) than nitrogen. For estimating burning rates, values obtained under nitrogen are probably more appropriate because during flaming combustion the oxygen concentration near the solid surface is low since most oxygen is consumed at the diffusion flame front. Modeling flame spread and fire growth is more complicated because mass burning occurs in a largely inert environment (pyrolysis zone) but preheating and flame spread occur under oxidative conditions. Due to the sensitivity of decomposition kinetics to oxygen concentration and the wide range of oxygen concentrations encountered in fires, a kinetic equation that explicitly accounts for the presence of oxygen on the decomposition rate could be useful. However, this has not yet been applied to fire scenarios and has only been used in research environments. Esfahani [97] modeled the thermo-oxidative degradation of a PMMA slab using a kinetic equation of the form: da  E  (0.8 + X O∞2 )8 = A exp  −  RT  dt

(25)

where X ∞O2 is the freestream oxygen concentration. Other workers [94, 98] have used a similar equation: da  E  (1 − a)n X O∞ m = A exp  − 2  RT  dt

(26)

Condensed-Phase Processes in Combustible Solids

227

Table 2: Literature values of reaction order, pre-exponential factor, and activation energy for single-step decomposition of solid materials. Material

Ref.

Atmosphere

b (ºC/min)

n

ln A (ln (s−1))

E (kJ/mol)

Cellulose Cellulose PE PE PE PE (HD) PE (LD) PE (LD) PET POM PP PP PP PP PS PS PS

[93] [93] [93] [93] [94] [95] [95] [95] [94] [96] [93] [93] [95] [95] [93] [93] [95]

N2 Air N2 Air 5% O2/95% N2 N2 N2 N2 Air N2 N2 Air N2 N2 N2 Air N2

5 5 5 5 5−20 − − − 5−20 4 5 5 − − 5 5 −

1 1 1 1 1.3 1 1 1 0.9 1 1 1 1 1 1 1 1

54.0 47.6 25.2 29.7 18.4 30.6 34.5 27.6 32.1 26.0 26.4 18.8 35.7 26.1 35.3 11.7 31.1

317 277 203 181 138 220 241 201 222 118 205 127 244 188 240 110 204

where m is an exponent characterizing the material’s decomposition sensitivity to oxygen. Senneca et al. [94] determined the constants m and n for PE and PET. Jun et al. [98] used eqn (26) to study the thermooxidative decomposition of polypropylene. However, n varied with heating rate for a fixed value of m and E and A varied with oxygen concentration and heating rate. One shortcoming of using eqns (25) or (26) to model the decomposition of thick solids is that the reaction rate within the solid should depend on the local oxygen concentration within the solid, not the freestream value. However, modeling the penetration of oxygen into a solid to determine the local oxygen concentration is a difficult task. Additionally, a material’s decomposition kinetics may not change monotonically with oxygen concentration, as implied by eqns (25) and (26). For example, below ∼270ºC the decomposition of PMMA is actually faster in nitrogen than in air, but it becomes faster in air at higher temperatures [99]. The decomposition process of many solids is too complex to be characterized by a single-step reaction because multiple reactions become active over different temperature ranges. As an example, an experimental DTG curve for flexible polyurethane foam is shown in Fig. 7, along with the calculation of a five step reaction mechanism developed by Rein et al. [4]. Table 3 lists a few references where multiple-step reactions have been developed for various materials. Interestingly, many examples exist in the literature where different workers have found that the same generic material (i.e. PMMA or PE) shows a different number of reaction steps. As an example, the decomposition of PMMA under nitrogen has been observed to proceed as a singlestep [108] and four steps [100]. These differences are probably attributed to differences in polymer synthesis or sample preparation. Lyon and Walters [109] point out that the ‘heat release capacity’ of polymers (as determined by pyrolysis combustion flow calorimetry) can vary by ±20%, depending on the source of the sample. As with the number of decomposition steps, it is common for different values of reaction order to be reported for the same generic material, and there is much discussion in the thermal analysis

228

Transport Phenomena in Fires 0.006 Experiment, 5 °C/min Model, 5 °C/min

0.005

Experiment, 20 °C/min Model, 20 °C/min

d(m/m0 )dt (1/s)

0.004

0.003

0.002

0.001

0.000 0

100

200

300

400

500

600

Temperature (°C)

Figure 7: Decomposition of polyurethane foam in air. Individual points are thermogravimetric data and solid lines are reaction mechanism of Rein et al. [4].

Table 3: Literature sources for multi-step reaction decomposition mechanisms. Material

Ref.

Douglas fir PMMA PVC Intumescent coatings Polyurethane Polyurethane Epoxy resin Chestnut wood Lodgepole pine Pine (wet) FR white pine

[92] [100] [101] [86] [102] [4] [103] [104] [105] [106] [107]

Atmosphere Argon He:O2 (variable) Air, N2 Air Air, N2 Air

Nitrogen

No. of steps 4+ 4 3 or 7 3+ 3 5 2 7 5 6 3

literature regarding how to determine reaction order. Gao et al. [108] show that reaction order can be estimated by the value of a at the maximum reaction rate. So-called ‘model-free’ or ‘isoconversional’ methods have been developed [91, 110] which permit the activation energy to be estimated independent of the reaction order. However, the activation energy depends on the extent of conversion, and the pre-exponential factor cannot be estimated without assuming a reaction order. While TGA is a useful tool for quantifying a solid’s thermal stability, it has several limitations. TGA’s relevance to fires has been questioned [103] because typical TGA heating rates (between

Condensed-Phase Processes in Combustible Solids

229

0.1 K/min and 30 K/min) are much lower than can be encountered in fires (sometimes 500 K/min or greater). One difficulty associated with using higher heating rates in TGA is that the thermal lag between the sample temperature and the atmosphere temperature increases with heating rate, especially if the decomposition process is endothermic. The magnitude of this thermal lag is difficult to accurately quantify. Undetected thermal lag may be responsible for the ‘compensation effect’, which refers to linear dependence of ln A on E frequently found in thermogravimetric studies [108, 111]. Consequently, the thermokinetic parameters found by TGA are a function of the heating rate. Carrasco and Pagès [112] found that the pre-exponential factor depends only on the heating rate (compensation effect) but also on the mass of the sample, indicating that the effects of heat and mass transfer are not completely absent from the TGA experiments.

3.2 Reaction enthalpies Whereas the rate at which a material burns once ignited is usually not strongly sensitive to its thermal decomposition kinetics, it is quite sensitive to the enthalpies of reaction. At solid temperatures typical of flaming combustion, the decomposition kinetics of most solids become so fast that they are no longer the limiting factor in the gasification process. Instead, the mass burning rate is determined by a balance between the applied heat flux, surface heat losses, heat conduction to the interior of the solid, and the heat absorbed or released in chemical reactions within the solid. However, since decomposition kinetics affect this heat balance, they do have a secondary effect on mass burning rates.This heat balance is the basis for the semi-empirical pyrolysis models discussed in Section 2.1. More specifically, the steady-state MLR can be related to the heat (or enthalpy) of gasification (∆Hg) as: m ′′ =

q net ′′ ∆H g

(27)

where q◊ net ″ is the net rate of heat transfer to the material’s surface. It can be seen from eqn (27) that (under steady-state conditions) m◊ ″ plotted against q◊ net ″ should have slope 1/∆Hg, provided ∆Hg does not depend on q◊ net ″ . This permits ∆Hg to be measured directly from bench-scale combustion experiments [18], and ∆Hg is widely viewed as a material fire property. Experimental values of ∆H g for different fuels have been tabulated elsewhere [9, 20] and are not reproduced here. Equation (27) was originally applied to the steady burning of polymers [18]. Following Lyon and Janssens [9], ∆Hg can be defined more precisely as: Tp

∆H g = ∫ c(T )dT + ∆H m + ∆H d + ∆H v T0

(28)

where ∆Hm is the latent heat of melting (if melting occurs between T0 and Tp), ∆Hd is the bond dissociation energy, and ∆Hv is the heat of vaporization of the decomposition products. ∆Hd can be thought of as the heat required to break a polymer molecule into fragments by thermal decomposition, and ∆Hv is the heat required to subsequently vaporize those decomposition products. Thus, the heat of gasification is the difference between the enthalpy of the solid fuel at T0 and the enthalpy of its volatiles at Tp. In other words, ∆Hg is the quantity of heat required to generate unit mass of volatiles at temperature Tp from unit mass of solid initially at T0.

230

Transport Phenomena in Fires

Although ∆Hg is usually treated as an effective value that is constant for a particular material, there are several reasons that ∆Hg is technically not constant. Due to the upper bound of the integral on the RHS of eqn (28), ∆Hg depends on the temperature at which volatilization occurs (Tp). As discussed earlier, real materials do not instantaneously volatilize at a fixed temperature, but rather over a finite temperature range. Additionally, a spectrum of decomposition products is formed during thermal degradation. The composition of these volatiles depends on temperature, atmospheric composition, and perhaps heating rate. As an example, the monomer yield decreases with temperature for PMMA and PS, but increases with temperature for PP [113]. Partially oxidized fuel fragments may appear in the decomposition products for decomposition under air, but not inert environments unless the solid contains oxygen. Since the heat required to break large molecules into fragments and then vaporize these fragments depends on their chemical composition, ∆Hd and ∆Hv are expected to vary with the volatile composition. In the context of eqn (28), ∆Hd and ∆Hv will vary with the atmospheric composition, and they should be interpreted as average values over the range of temperatures at which decomposition occurs in that environment. For these reasons, strictly speaking a material cannot be assigned a single value of ∆Hg [114]. However, this is more of a technicality than a practical consideration because the effective heat of gasification measured from combustion experiments has been shown to be a good predictor of steady-state burning rates [30]. Consistent terminology and nomenclature have not yet been adopted for discussing the various terms in eqn (28). In the fire safety literature, the term ‘heat of vaporization’ is frequently used [18, 30, 34, 114]. In the above context, this is approximately equal to ∆Hd + ∆Hv. Less frequently, the terms ‘heat of volatilization’ [113] and ‘heat of decomposition’ [115] have been used. For wood, the terms ‘heat of pyrolysis’ [116] or the more general ‘heat of reaction’ [62, 106, 117, 118] have been used, but rarely precisely defined. References [93, 119] report experimental values for the ‘heat of pyrolysis’ of several materials. Since the solid phase energy conservation equation includes a source term that generally appears as a reaction rate multiplied by a heat of reaction/pyrolysis/ vaporization/volatilization/decomposition, it is important to be sure that values from the literature are not misinterpreted. The field could benefit from some consensus in this area because it is difficult to interpret literature values from different workers that may or may not be referring to the same basic quantity. Following Frederick and Mentzer [113], we use the term ‘heat of volatilization’. This quanity is defined for a polymer that decomposes according to a single-step reaction (see eqns (8) and (9)) as: ∆H vol = ∆H d + ∆H v

(29)

The heat of pyrolysis ∆Hp is the analogous quantity for a charring material that decomposes by a single-step reaction (see eqns (12) and (16)). Both ∆Hvol and ∆Hp are global values that apply to the decomposition process as a whole. They imply a single-step reaction and do not include a sensible enthalpy contribution. A positive value designates an endothermic reaction and a negative value designates an exothermic reaction. Where multiple reactions are considered, the term ‘heat of reaction’ (∆Hr) is recommended. For example, Alves and Figueiredo [106] developed a six step reaction mechanism for wood, with each reaction carrying its own heat of reaction. In the discussion above, the units of ∆Hvol and ∆Hp are joules per kilogram of gases liberated from the condensed phase. However, there exists some confusion regarding the units of ∆Hr. The reason for this is that some authors use a heat of reaction that implies units of joules per kilogram of reactants consumed, while others use a ∆Hr having units of joules per kilogram of gases liberated from the condensed phase. Viewing ∆Hr as an empirical quantity rather than one defined strictly from a thermodynamic basis, both definitions are acceptable. In fact, one can be converted

Condensed-Phase Processes in Combustible Solids

231

to the other given knowledge of the reaction stoichiometry. However, care must be taken when interpreting literature values of the heat of reaction. For practical applications of comprehensive solid pyrolysis models, it is the values of ∆Hvol, ∆Hp, or ∆Hr that are needed, as opposed to ∆Hg. While the heat of gasification ∆Hg can be estimated from bench-scale combustion experiments conducted at multiple heat flux levels, ∆Hvol, ∆Hp, or ∆Hr cannot. For this reason, these quantities are usually treated as adjustable parameters [120] or estimated experimentally from techniques similar to TGA such as DTA or DSC [121]. DSC is a thermal analysis technique that can be used to measure the enthalpy of reaction in a constant mass (nonvolatilizing) solid. Similar to TGA, DSC exposes milligram size samples to a programmed atmospheric composition and temperature, usually increasing linearly with time. DSC devices measure the difference in the rate of heat flow to the sample of interest and a reference sample with well-known thermal properties. Then, the apparent specific heat capacity of the sample can be calculated as [121]: cs =

1 ms

 Q s − Q r  cr mr +  b  

(30)

Here, the subscript ‘s’ denotes the sample of interest and subscript ‘r’ denotes the reference sam◊ ple. Mass is denoted by m, rate of heat flow by Q, and the linear heating rate by b. Glass transitions, phase change, and chemical reactions all affect the apparent specific heat capacity. Thus, the apparent specific heat measured by DSC is not the ‘real’ specific heat capacity, which is attributed to heat storage by molecular vibrations. The heat associated with a physical change or chemical reaction is determined from a DSC plot of cs vs. T as the area under a ‘peak’ minus the ‘baseline’ specific heat capacity. The latter is the specific heat that would have been recorded in the absence of the glass transition, phase change, or chemical reaction of interest [121]. It cannot be exactly determined, and must be estimated. As an example, Fig. 8 shows a simulated DSC curve (converted to apparent specific heat capacity) for a hypothetical material with a reaction centered at 650 K. The hatched area represents the heat of reaction. 2.0

Specific heat capacity (kJ/kg-K)

Baseline Apparent 1.8

1.6

1.4

1.2

1.0 500

550

600

650

700

750

800

Temperature (K)

Figure 8: A sample DSC curve showing an endothermic reaction centered at 650K.

232

Transport Phenomena in Fires

With DSC, the sample mass is not monitored unless it is used as part of a simultaneous thermal analysis (STA) device. Therefore, non-STA DSC is well-suited for quantitatively measuring heats of reaction or heats of transition that do not involve volatilization, but its accuracy is reduced when volatilization occurs [121]. Few values of ∆Hvol or ∆Hp for different materials were located in the literature, and this search was confounded by the inconsistent terminology mentioned above. In fire applications, the heat of pyrolysis of wood is frequently assumed to be zero [92, 122]. Table 4 lists some heats of volatilization obtained for different materials from DSC and STA. As mentioned earlier, it is expected that ∆Hvol will change with atmospheric composition. Peterson et al. [99] found that for PMMA, ∆Hvol decreases from approximately 1080 J/g under nitrogen to 550 J/g under air (both endothermic). Dakka [123] presented DTA data suggesting the decomposition reaction for PMMA is endothermic under nitrogen, but exothermic in the

Table 4: Literature values of heat of volatilization (see eqn (29)). Material

Ref.

Atmosphere

b (ºC/min)

PE PE PP PP PMMA PMMA PMMA PMMA PMMA PS PS PS POM POM Nylon 66 Nylon 66 Nylon 6 Polychloral P(a-M-S) PVC PAN PBT BPC II-polyarylate PET PPO HDPE PTFE PC PI Kevlar PBZT

[113] [18] [113] [115] [113] [18] [115] [99] [99] [113] [18] [115] [18] [115] [18] [115] [113] [115] [115] [115] [115] [115] [115] [115] [115] [115] [115] [115] [115] [115] [115]

Nitrogen Nitrogen Nitrogen Nitrogen

10 − 10 10 10

Nitrogen Nitrogen Air Nitrogen

10 20 20 10

Nitrogen

10

Nitrogen

10

Nitrogen Nitrogen Nitrogen Nitrogen Nitrogen Nitrogen Nitrogen Nitrogen Nitrogen Nitrogen Nitrogen Nitrogen Nitrogen Nitrogen Nitrogen Nitrogen

10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10

∆Hvol (kJ/kg) 665 961 631 370 803 1007 687 1080 550 819 1012 683 1720 937 564 140 786 380 443 140 −444 281 −302 174 150 256 447 111 62 228 338

Condensed-Phase Processes in Combustible Solids

233

presence of oxygen. However, in a later paper [124] the same author presents additional DTA data using smaller particles indicating the PMMA decomposition reaction is endothermic.

4 Heat, mass, and momentum transfer The preceding sections emphasized the pyrolysis modeling strategies used by different workers as well as solid phase decomposition kinetics and thermodynamics. This section takes a closer look at the relevant transport phenomena occurring within the solid. 4.1 Solid phase heat conduction The one-dimensional heat conduction equation for a constant property inert solid that is opaque to thermal radiation is: ∂ T ( z, t ) k ∂ 2 T ( z, t ) = ∂t r c ∂z 2

(31)

Note the similarity to eqn (6) in which no assumptions regarding the temperature dependency of the thermal conductivity or specific heat are made. Most analytical approaches to modeling solid fuel ignition and flame spread are based on eqn (31). Due to the nonlinear surface radiant emission term that appears in the surface boundary condition, numerical solution is normally required. However, analytical solutions exist if this surface radiant emission is linearized. Four exact solutions to eqn (31) for different sets of boundary conditions are listed in Table 5. The first two solutions (A and B) are useful because they can be used to make quick engineering estimates (e.g. how long will it take for a surface to reach Tp, given a particular heat flux?) without resorting to numerical methods. Unfortunately, a computer program is generally required to evaluate the second two solutions (C and D) due to the infinite summation and the eigenvalues ln that must be determined from a transcendental equation. Solutions C and D are useful for verifying whether a finite difference solution to eqn (31) has been correctly implemented. The solutions in Table 5 are valid only if k, r, and c do not vary with temperature. Although this approximation is usually made, the themophysical properties of real materials generally vary with temperature. Numerical methods can easily accommodate temperature-dependent thermal properties. Room temperature values of k, r, and c have been tabulated for most polymers [9, 125], with simple scaling relations suggested for the variation of these properties with temperature. The density of polymers generally decreases with temperature, by as much as 20% between room temperature and 350ºC (a typical ignition temperature). Temperature-dependent densities of several polymers have been compiled by Orwoll [126]. The temperature-dependent specific heat of most polymers can be found in the series of papers by Gaur et al. [127−134]. However, the data generally do not extend much above 300ºC. Temperature-dependent thermal conductivities of polymers are scattered throughout the literature. The papers by Zhang et al. [135, 136] are a good source of data (k and rc) for commodity polymers. Since PMMA is widely used in flammability studies, Steinhaus [137] conducted a detailed investigation of its thermophysical properties. A few experimental measurements for the temperature dependency of density, specific heat capacity, and thermal conductivity of common polymers are listed in Tables 6−8 (temperature in units of °C). Whereas moisture content has only a minor effect on the thermal properties of polymers, it can affect the thermal properties of wood significantly. A further complication is that the thermal

234

Transport Phenomena in Fires

Table 5: Solutions to heat conduction equation for initial condition T = T0, ambient temperature T∞ = T0, and radiation exposure at z = 0. Boundary conditions ∂T z = 0: −k __ = q. ″

A B C

∂z

e

z → ∞: T → T0 ∂T z = 0: −k __ = q.e″− h T(T−T0) ∂z z → ∞: T → T0 ∂T z = 0: −k __ = q. ″ z = L:

D

Solution to eqn (31) T–T0 ___ q· ≤e d/k

∂z ∂T __ = ∂z

1__ = __ exp( − ( _dz )2 ) − _dz erfc( _dz ) √π

((

))

__

( √)

T–T0 z _t _t ____ _z ___ _z q· ≤e/hT = erfc( d )−exp k/hT + tc erfc d + tc T–T0 ___ = q· ≤ d/k

e

e

0

∂T z = 0: −k __ = q.e″− hT(T−T0) ∂z

T–T0 ___ = 1− q· ≤e/hT

∂T z = L: __ =0 ∂z

∞

2n + z / L + ierfc ( ________ )] ∑ [ ierfc ( ______ d/L ) d/L 2(n +1) –z / L

n=0

∞

∑

n=1

[ __________ cos (l (L − z))exp( − __ l t ) ] 4sin (lnL) 2lnL + sin (2lnL)

n

k rc

2 n

k ln is eigenvalue found from: cot (lnL) = __ h T ln

A: Semi-infinite solid exposed to constant net radiative heat flux (no surface heat losses). B: Semi-infinite solid exposed to constant incident radiative heat flux with surface heat losses by Newtonian cooling to ambient at temperature T0. C: Solid of thickness L exposed to constant net radiative heat flux (no surface losses) and perfectly insulated at back face. D: Solid of thickness L exposed to constant incident radiative heat flux with surface heat losses by Newtonian cooling to ambient at temperature T0 and perfectly insulated at back face. ______ d = √4kt/rc 2

tc = krc / hT

exp (−x 2)

__ ierfc(x) = ______ − x erfc(x) √π

properties of wood also vary with temperature and grain orientation; a good deal of information is contained in refs [138−144]. Most experimental measurements indicate that the specific heat of wood is fairly independent of species. The following relation has been suggested for the specific heat capacity of generic oven dry virgin wood [139]: cv0 (T ) = 1160 + 3.87T

(32)

where T has units of °C and cv0 has units of joules per kilogram kelvin. The variation of wood’s specific heat with moisture content and temperature can be estimated as [139]: cv ( M , T ) =

cv0 (T ) + 4186 M + ∆cv ( M , T ) 1+ M

∆cv ( M , T ) = M (23.6T − 1326 M + 238)

(33a) (33b)

In eqns (32) and (33) the subscript ‘v’ denotes virgin wood, not constant volume, and M is the moisture content on an oven dry mass basis, defined as: M=

m −1 m0

(34)

where m is the mass of a wood sample at a given moisture content and m0 is the oven dry mass of that same wood sample.

Condensed-Phase Processes in Combustible Solids

235

Table 6: Temperature-dependent density of several polymers. Material

r(T) (kg/m3)

Ref.

Rubber (natural)

[126]

Nylon 6

[126]

Nylon 6,6

[126]

PC

[126]

PE (branched)

[126]

PE (linear)

[126]

PET

[126]

PMMA

[126]

POM

[126]

PP (atactic)

[126]

PP (isotactic)

[126]

Polystyrene

[126]

PTFE

[126]

PVC

[126]

T r T r T r T r T r T r T r T r T r T r T r T r T r T r

0 921 240 1176 260 1100 40 1192 120 801 140 785 140 1172 40 1181 100 1063 80 827 180 764 40 1040 360 1548 100 1352

20 909 260 1165 280 1086 80 1180 140 790 160 774 160 1156 80 1171 120 1048 100 816 200 754 80 1026 380 1504 120 1338

280 1154 300 1071 120 1167 160 780 180 762 180 1140 120 1153 140 1033 120 802 220 744 120 1005

300 1143

160 1150 180 769 200 751 200 1125 160 1126 160 1018

200 1123 200 759

240 1095 220 749

280 340 1067 1025

200 1097 180 1004

220 1082 200 990

240 260 1067 1052 220 976

240 734 160 984

260 724 200 961

280 714 240 939

300 705 280 916

320 893

140 1322

Equations (32)−(34) apply below approximately 200°C. At higher temperatures, thermal decomposition begins and the wood is transformed to a carbonaceous char generally having properties different from the virgin wood. The rate at which the virgin wood is converted to char depends on the kinetics of the decomposition reactions (see Section 3.1). The temperature dependency of wood char specific heat (cc) is similar to that of amorphous graphite, increasing with temperature [144]: cc ≈ 714 + 2.32T − 0.0008T 2 − 3.69 × 10 −7 T 3

(35)

In eqn (35), T has units of °C and cc has units of joules per kilogram kelvin. Little information is available regarding the specific heat capacity of partially degraded wood, but the temperature-dependent

236

Transport Phenomena in Fires

Table 7: Temperature-dependent specific heat capacity of several polymers. Material

Ref.

PE (c)

[127]

PE (a)

[127]

PP (c)

[129]

PP (a)

[129]

PMMA

[137]

PS

[130]

PTFE

[132]

PVC

[132]

Nylon 6,6

[133]

Nylon 6

[133]

PET

[133]

POM

[128]

PC

[134]

c(T) (J/kg K) T c T c T c T c T c T c T c T c T c T c T c

17 1515 17 2176 17 1563 17 2067 17 1434 17 1179 7 976.4 17 922.2 17 1416 17 1451 17 1136

47 1639 27 2206 47 1756 27 2103 47 1564 47 1317 37 1023 37 978.2 47 1566 37 1559 67 1322

77 1804 77 2361 77 1970 77 2284 77 1694 77 1460 107 1109 57 1038 50 2223 40 2404 69 1736

107 2151 127 2516 107 2197 127 2464 97 1781 127 1935 187 1236 77 1102 127 2383 87 2468 107 1792

137 2585 177 2670 127 2354 177 2643 107 2180 177 2063 247 1357 81 1424 177 2486 147 2549 147 1851

157 2889 227 2824 147 2514 227 2824 167 2333 227 2190 327 1328 87 1457 227 2590 207 2630 197 1924

177 3213 277 2979 167 2679 277 3005 227 2486 277 2317 367 1379 97 1513 277 2693 267 2711 257 2013

187 3382 327 3134 187 2850 327 3183 277 2613 327 2445 447 1475 107 1569 327 2797 327 2792 317 2101

T c T c

−3 1152 17 1168

17 1237 57 1328

27 1284 107 1534

47 1387 145 1695

67 1500 147 1891

87 1619 187 1982

107 1740 237 2096

117 1800 287 2210

(a) amorphous; (c) crystalline.

Table 8: Temperature-dependent thermal conductivity of several polymers. Material

Ref.

PC

[135]

PE

[135]

PP

[135]

PS

[135]

PMMA

[137]

k(T) [W/m-K] T k T k T k T k T k

28 0.24 17 0.34 71 0.25 35 0.16 0 0.2

83 0.25 45 0.31 105 0.24 44 0.16 105 0.2

119 0.26 107 0.23 116 0.23 89 0.17 275 0.16

146 0.26 118 0.22 138 0.2 108 0.17

169 0.26 129 0.22 146 0.19 115 0.16

204 0.25 139 0.22 156 0.13 163 0.16

225 0.23 160 0.22 222 0.13 216 0.16

248 0.23 214 0.21 234 0.13 238 0.16

Condensed-Phase Processes in Combustible Solids

237

analogue of eqn (18) (i.e. linear interpolation between the cv and cc) can be used as a first approximation. The thermal conductivity of wood depends primarily on oven dry density and moisture content. The following relation was found to provide a good correlation to experimental thermal conductivity measurements [139]: kv ≈ 0.019 +

r0 (0.194 + 0.406 M ) 1000

(36)

In eqn (36), r0 is the oven dry density (kg/m3). Temperature has a relatively minor effect on the thermal conductivity of wood [138]. It has been suggested that the thermal conductivity increases by approximately 10% for each 50°C increase in temperature [139]. Little information is available regarding the thermal conductivity of charred wood, particularly partially degraded wood. Alves and Figueiredo [106] experimentally measured the thermal conductivity of pine char between 30°C and 220°C, recommending the following correlation: kc ≈ 0.113 + 8.2 × 10 −5 T

(37)

where T has units of °C. It is unlikely that eqn (37) applies at all temperatures because, as discussed in Section 4.2, radiation transport across char pores increases the ‘effective’ or ‘apparent’ thermal conductivity at high temperatures. Several models for the effective thermal conductivity in porous materials such as chars have been proposed [82, 142, 145−148]. However, the predictive capabilities of these models have not been rigorously evaluated, particularly due to the difficulties associated with accurately measuring the temperature-dependent thermal conductivity of chars. For example, Cagliostro and Riccitiello [77] generated four different samples of an intumescent char under identical furnace conditions and found inter-sample variations of ± 100% in the measured thermal conductivity. 4.2 Radiation Radiation is the dominant mode of heat transfer through the gas phase in large fires [149]. Since radiation drives burning rates, it is important to understand the radiative characteristics of solid materials. A fraction of the radiation incident upon a material may be reflected from the surface. The remainder can be absorbed by or transmitted through the material. In a fire, one of the most important radiative characteristics of a material is its absorptivity, defined as the fraction of the incident radiation that is absorbed by the material. The absorptivity is strongly wavelength-dependent. For example, at wavelengths below 1 µm the absorptivity of clear PMMA is close to zero, but above 3 µm it approaches unity. The absorptivity at a single wavelength l is denoted al. However, in a fire we are usually interested in the ‘integrated’ absorptivity a: ∞

∫ al qe,′′l dl a= 0 ∞ ∫0 qe,′′l dl

(38)

It can be seen from eqn (38) that the integrated absorptivity depends on the spectral energy distribution of the radiation source. Therefore, a material cannot be assigned a single value of integrated absorptivity because the spectral distribution of the incoming radiation depends on the temperature of the emitter even if it behaves as a blackbody (Wien’s displacement law). In fires, the temperature of radiation sources ranges from approximately ~600 K (smoke layer, hot

238

Transport Phenomena in Fires

surfaces) to ~2000 K (flames). Additionally, certain bench-scale fire tests use tungsten-filament heaters that operate at temperatures near 3000 K. Thus, the effect of source temperature on the integrated (or effective) absorptivity has relevance for both real fires and bench-scale fire testing. Hallman’s 1971 PhD dissertation [150] and subsequent publications [151, 152] remain some of the best sources of information on the change of polymers’ integrated surface absorptivity with the temperature of the emitter for polymers. Hallman measured the spectral absorptivity of several solids and then determined the integrated surface absorptivity for hexane flames, blackbodies between 1000 K and 3500 K, and solar energy. His absorptivity data are reproduced in Table 9. Note that the integrated absorptivity of some materials is relatively insensitive to the temperature of the radiation source (black PMMA) but others are quite sensitive. For example, the absorptivity of clear PMMA decreases from 0.85 for a 1000 K blackbody to 0.25 for a 3500 K blackbody. Similar measurements were made by Wesson et al. [153] for undegraded wood. Their results are reproduced in Table 10. During a fire, a material’s radiative characteristics may change. Although the integrated absorptivities from Wesson et al. [153] (reproduced in Table 10) are relatively low, the absorptivity of charred wood is generally not the same as that of virgin wood. Janssens [154] suggested that blackening causes the absorptivity of wood to increase from ∼0.76 (based on ref. [153]) to approximately unity as the surface temperature approaches the ignition temperature. He therefore used an average value of 0.88 in his ignition analyses, and recommends using an integrated absorptivity of 1.0 during flaming combustion [144]. Wood is not the only class of materials that exhibits a change in radiative characteristics during a fire. Under nonflaming conditions, low density polyethylene has been observed to change from visually opaque to transparent, eventually followed by a darkening of the surface [48]. This indicates that a change in the material’s radiative characteristics occurred (at least in the visible range). Modak and Croce [155] reported that for clear PMMA, 39% of flame radiation is transmitted through the surface, but for ‘charred’ PMMA (previously exposed to a fire environment and then cooled) no radiation penetrates in depth. Bubbling occurring near the surface of polymers can change their radiative characteristics, but this effect has not yet been reliably quantified. In a real fire, materials may become coated in soot from flames or a smoke layer, causing their absorptivities to approach unity. For a material that is opaque to thermal radiation, almost all absorption occurs within a micron of the surface, and absorption can be treated as a surface phenomenon. This is the usual assumption in fire problems, but at certain wavelengths some solids are semi-transparent and absorb radiation in-depth. This diathermancy is usually modeled using an absorption coefficient kl that describes the rate of attenuation of radiation at a particular wavelength: ∂q r,′′l ∂z

= − kl q r,′′l

(39)

where for simplicity, incident radiation applied only normal to the surface and ‘one-way’ radiation transport have been assumed. More complicated treatments of in-depth radiation absorption than eqn (39) have also been used [156–159]. As with surface absorptivity, the absorption coefficient is strongly wavelength-dependent. Similar to the integrated surface absorptivity, an integrated absorption coefficient can be defined as: ∞

∫ kl qe,′′l dl a= 0 ∞ ∫0 qe,′′l dl

(40)

Table 9: Integrated surface absorptivities for polymers and rubber from Hallman [151]. Blackbody emitter temperature (K) Generic name

1000

1500

2000

2500

3000

3500

Flame

Cycolac® Uvex®

0.91 0.84 0.64 0.91 0.93 0.90 0.87 0.92 0.94 0.85 0.91 0.92 0.86 0.87 0.75 0.86 0.92 0.81 0.90 0.88 0.91 0.92 0.92 0.88 0.91 0.79

0.86 0.71 0.56 0.88 0.90 0.86 0.83 0.88 0.94 0.69 0.86 0.86 0.78 0.83 0.60 0.75 0.89 0.65 0.90 0.87 0.90 0.93 0.93 0.82 0.92 0.66

0.77 0.56 0.49 0.85 0.86 0.81 0.78 0.82 0.95 0.54 0.78 0.78 0.70 0.78 0.46 0.63 0.83 0.49 0.89 0.86 0.89 0.93 0.94 0.76 0.93 0.58

0.71 0.43 0.46 0.82 0.82 0.77 0.75 0.77 0.95 0.41 0.70 0.71 0.63 0.74 0.35 0.53 0.77 0.38 0.89 0.85 0.88 0.93 0.94 0.72 0.93 0.54

0.65 0.34 0.44 0.80 0.75 0.75 0.72 0.72 0.95 0.31 0.62 0.64 0.57 0.70 0.28 0.45 0.72 0.30 0.89 0.84 0.87 0.93 0.95 0.69 0.93 0.52

0.61 0.27 0.44 0.79 0.71 0.75 0.71 0.68 0.95 0.25 0.56 0.59 0.53 0.68 0.22 0.40 0.68 0.24 0.89 0.83 0.86 0.93 0.95 0.68 0.93 0.53

0.92 0.88 0.60 0.91 0.93 0.91 0.88 0.93 0.94 0.89 0.92 0.93 0.88 0.86 0.78 0.88 0.93 0.85 0.91 0.88 0.92 0.92 0.92 0.89 0.91 0.79

Formica® Bakelite Lexan® Plexiglas® Plexiglas® Plexiglas® Delrin®

Styrolux® Texin®

Kydex® Kydex®

Condensed-Phase Processes in Combustible Solids

Acrylonitrile butadiene styrene Cellulose acetate butyrate Cork Melamine/formaldehyde Nylon 6/6 Phenolic Polycarbonate (rough surface) Polyethylene (low density) Polymethylmethacrylate (black) Polymethylmethacrylate (clear) Polymethylmethacrylate (white) Polyoxymethylene Polyphenylene oxide Polypropylene Polystyrene (clear) Polystyrene (white) Polyurethane thermoplastic Polyvinyl chloride (clear) Polyvinyl chloride (gray) PVC/acrylic (gray, rolled) PVC/acrylic (red cast) Rubber (Buna-N) Rubber (Butyl IIR) Rubber (natural, gum) Rubber (neoprene) Rubber (silicone)

Trade name

239

240

Transport Phenomena in Fires

Table 10: Integrated surface absorptivity for wood from different emitters (from Wesson et al. [153]). Wood Alaskan cedar Ash Balsa Birch Cottonwood Mahogany Mansonia Maple Oak Redgum Redwood Spruce White pine Masonite

Flame radiation

Tungsten lamp radiation

Solar radiation

0.76 0.76 0.75 0.77 0.76 0.76 0.76 0.76 0.77 0.77 0.77 0.76 0.76 0.75

0.44 0.46 0.41 0.47 0.48 0.49 0.47 0.49 0.56 0.52 0.51 0.45 0.49 0.52

0.36 0.36 0.35 0.39 0.40 0.52 0.51 0.44 0.49 0.56 0.55 0.35 0.43 0.61

Compared to measurements of surface absorptivity, there is less information available regarding the in-depth absorption of thermal radiation in combustible solids. At wavelengths greater than 2.5 µm, the radiative absorption depth is less than 2 mm in PE and less than 1 mm in PMMA [48]. Several workers have included the effects of in-depth radiation absorption in their models [43, 44, 97, 156−159]. Obtaining accurate property data that characterizes the in-depth absorption (normally, the ‘gray’ absorption coefficient) can be difficult. Modak and Croce [155] reported that the gray absorption coefficient of clear PMMA for its flame radiation is 124 m−1. Progelhof et al. [160] give band-mean absorption coefficients for PMMA and poly(4-methylpentene-1) as a function of wavelength (and developed exact solutions for the temperature profiles resulting in semi-transparent solids). Table 11 gives the absorption coefficient and absorptivity for PMMA as determined experimentally by Manohar et al. [159] over 14 different wavelength ‘bands’. Reduction in the effective surface absorptivity or in-depth absorption coefficient both increase the time to ignition. Therefore, ignition times at the same applied heat flux level from different thermal radiation sources are not necessarily the same. Figure 9 shows Hallman’s data [150] for the ignition time of PE and PS from benzene flames and a tungsten lamp. At a given heat flux, the ignition times are generally longer using the tungsten lamp because the integrated surface absorptivity is lower for the tungsten lamp than for the benzene flame (see Table 9). Thomson and Drysdale [161] also found differences in the ignition times of PMMA and PP at the same heat flux level (as measured with a Gardon heat flux gauge) depending on whether the heat flux was varied by holding the heater temperature constant and changing its position, or by holding the heater position constant and changing its temperature. These spectral effects have not been extensively studied by fire researchers, although it appears to be important, particularly when interpreting experimental data from bench-scale flammability tests. In some standardized tests, the irradiated surface is coated with a thin layer of carbon black or paint to ensure that the applied radiant heat flux is absorbed at the surface. For example, the specimen preparation protocol in ASTM E2058-03 [162] requires that ignition/combustion samples are sprayed with a

Condensed-Phase Processes in Combustible Solids

241

Table 11: Absorption coefficient and surface absorptivity for clear PMMA over 14 wavelength bands (from Manohar et al. [159]). l1 (µm)

l2 (µm)

kl (m−1)

al (−)

1.67 1.77 1.87 1.99 2.13 2.29 2.47 2.68 2.93 3.24 3.62 4.09 4.71 5.56

268 555 274 170 226 1277 2407 870 2165 2453 2474 2864 3585 3895

0.994 0.991 0.990 0.990 0.987 0.770 0.927 0.981 0.385 0.957 0.436 0.976 0.934 0.670

1.59 1.67 1.77 1.87 1.99 2.13 2.29 2.47 2.68 2.93 3.24 3.62 4.09 4.71

400 PE (benzene flame) PE (tungsten lamp) PS (benzene flame) PS (tungsten lamp)

Ignition time (s)

300

200

100

0 0

25

50 75 100 Applied radiant heat flux (kW/m2)

125

150

Figure 9: Effect of heater type on time to ignition for PE and PS [150].

single coat of high temperature flat black paint. Babrauskas (p. 306 of ref. [163]) has cautioned that a surface coating of graphite powder may affect ignition times. It may be possible to improve a material’s fire performance with additives that reduce its surface absorptivity or in-depth absorption coefficient. It has been found that addition of a small amount of carbon nanotubes to polypropylene reduces its ignition time because the in-depth radiation absorption coefficient was increased [164].

242

Transport Phenomena in Fires

Radiation may be an important or even dominant mode of heat transfer in a porous medium even if the solid material itself is effectively opaque (k → ∞). Consider a single pore embedded in a porous material in which a temperature gradient exists. Energy is transferred by thermal radiation through the gas filling the pore from the hotter side of the pore to the colder side. The magnitude of this heat transfer depends on the temperature gradient, the size and shape of the pore, its absorptivity, and radiation attenuation by any participating gases contained in the pore. See refs [165, 166] for comprehensive reviews of radiative transfer in porous media. The simplest engineering treatment of radiative transfer in porous media involves the concept of a ‘radiant conductivity’. The heat transfer due to radiation is calculated as: q r′′ = − kr

∂T ∂z

(41)

where the radiant conductivity kr varies with the third power of temperature: kr = 4 Fd sT 3

(42)

In eqn (42), d is the pore diameter and F is the radiative exchange factor [166] which, for the purposes of this chapter, can be considered an empirical parameter related primarily to the pore structure. Both are difficult to determine theoretically or experimentally, although the pore diameter may be estimated by high-resolution microscopy. Most workers have used some variation of eqn (42) to calculate radiant conductivities, and a few of the expressions that have been used are listed in Table 12 (where e is emissivity and y is porosity): The radiant thermal conductivity calculated with eqn (42) is plotted in Fig. 10 for F = 4/3 and pore diameters of 100 µm and 10 µm. For comparison, a typical solid phase thermal conductivity for many polymers and cellulosic materials is ∼0.1−0.3 W/m-K. Figure 10 suggests that the radiant thermal conductivity may become of comparable magnitude to the solid thermal conductivity at relatively low temperatures, particularly for the case of 100 µm pore diameter. Di Blasi and Branca [83] found that radiation transfer through a porous char was the dominant mode of heat transfer in simulation of an intumescent coating, but Kantorovich and Bar-Ziv [146] have suggested that the radiant contribution to the thermal conductivity can be neglected for temperatures lower than 1000 K. Based on the limited information available in the literature, it is difficult to draw any conclusions regarding when the radiant conductivity should be included in calculations and what the ‘correct’ values of F and d to use for a particular material or class of materials. An alternative to rigorously attempting to establish F and d independently is to lump them together into a fitting parameter g, i.e. kr = gsT 3 where g is approximately in the range 10−5 m < g < 10−2 m. Table 12: Radiant conductivity expressions. Ref.

Material

kr

d

[102]

PU foam

16 __ sdT 3

[105] [167] [83]

Lodgepole pine Intumescent coating Intumescent coating

13.5sdT _______

Virgin: 50 µm Charred: 1300 µm 40 µm ~ 100 µm Unreacted: 5 µm Reacted: 325 µm

3

3

e

e2 sdT 3 3

13.5sdT _______ ey

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Radiant thermal conductivity (W/m-K)

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Figure 10: Radiative conductivity calculated using eqn (42) for F = 4/3. 4.3 Convection, advection, and diffusion For the purposes of this chapter, the distinction will be made between convection and advection in the interior of a decomposing solid as follows: advection is bulk motion of the condensed phase, and convection is heat transfer between the gaseous and condensed phases within the solid. The difference between the two can be illustrated by considering a pyrolyzing slab of a noncharring solid material. As volatiles escape from the interior of the condensed phase to the ambient atmosphere, the condensed phase (e.g. molten polymer) instantaneously fills the voids left by escaping volatiles. This causes surface regression, i.e. the thickness of the solid decreases with time due to the resultant advection. The same effect occurs due to a change in bulk density with temperature (swelling or shrinkage) or due to an intumescent reaction. This bulk motion of the condensed phase is advection. In comparison, convection occurs when volatiles generated in-depth move toward the surface and pass through condensed-phase material that is not necessarily at the same temperature. This gives rise to convective heat transfer between the volatiles and the condensed phase. For the case of a solid irradiated at its surface, the temperature in the region where volatiles are produced in-depth is lower than the temperature closer to the surface (unless there is in-depth radiation absorption, heating at the back-face, or exothermic reactions occurring in-depth). Thus, as the volatiles flow toward the surface, heat is transferred from the solid to the volatiles, in effect cooling the solid. The temperature difference between the condensed and solid phases (‘thermal nonequilibrium’) and its effect on the heat transfer rates have been investigated extensively in the field of heat transfer in porous media [168, 169]. However, in most fire-related studies it is assumed that the solid and gaseous phases are in thermal equilibrium due to the much smaller volumetric heat capacity of the volatiles [62−64]. One exception is Florio et al. [170], who investigated the effect of thermal nonequilibrium during the degradation of an ablative composite. They assumed that the rate of heat transfer between the condensed and gaseous phases was proportional to a volumetric heat transfer coefficient multiplied by the temperature difference. The authors [170] found differences of as much as 200 K between the gaseous and condensed phases.

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However, the applied heat flux was 280 kW/m2, considerably greater than typical fire-level heat fluxes. Florio et al. [170] also considered the case where the solid and condensed phases were in thermal equilibrium. Although there were not drastic differences between the temperature profiles calculated with the thermal equilibrium or nonequilibrium models, Florio et al. [170] advocate using thermal nonequilibrium for similar materials and boundary conditions. However, at boundary conditions imposed by typical fire-level heat fluxes, the assumption of thermal equilibrium between the gaseous and condensed phases is probably acceptable, and this eliminates one adjustable model parameter (the internal volumetric heat transfer coefficient) from the problem. The convection term is sometimes omitted from models altogether [65] on the basis that it is small. Diffusion of ambient oxygen into a decomposing solid may be an important effect to consider if the reactions in the solid are sensitive to the local oxygen concentration. To date, there has been little work investigating this effect. However, gaseous diffusion inside decomposing solids probably becomes less important at high heating rates or high heat flux levels where transport of volatiles takes place primarily by convection and blowing from the surface limits penetration of ambient oxygen into the decomposing solid. Diffusion of gases due to concentration gradients inside a decomposing solid is sometimes modeled using the dusty gas flux equation developed for multicomponent gas transport in porous media (such as a char layer). This approach has been used to simulate energy recovery processes such as flash pyrolysis [171], but it has not yet been applied to simulate practical fire problems. It is difficult to justify the inclusion of such complexity in fire modeling at the present time given the lack of knowledge regarding the composition of gaseous pyrolysate and secondary gas−solid reactions inside the char layer. 4.4 Momentum By invoking the assumption that the decomposition products move from the condensed phase to the adjacent gas phase with negligible internal resistance, the momentum equation is usually not solved. However, the empirical Darcy’s law is sometimes used to model the internal pressure distribution and the resultant velocity of the escaping decomposition products [63, 64, 80, 170]. With this approach, the velocity of the volatiles is proportional to the internal pressure gradient and the material’s permeability divided by the dynamic viscosity of the gas. This implies a Stokes flow where inertial terms are negligible. Any accumulation of gases inside the solid results in an increase in the internal pressure, and the pressure evolves according to a transient diffusion equation similar to the heat conduction equation. The model of Henderson and Wicek [80] predicted overpressures of as much as 50 atm in an expanding polymer composite. Overpressures of this magnitude could have a significant effect on the structural integrity of a material. Lee et al. [172] measured overpressures of 0.3 atm in wood when heated perpendicular to the grain, but only 0.003 atm when heated parallel to the grain. Under most fire scenarios, unless one is interested in estimating the structural response of a material, the assumption of instantaneous escape of volatiles is advantageous because it can reduce the solution complexity. 4.5 Special topics: melting, bubbling, and related phenomena Crystalline solids have a well-defined melting temperature, a common example is ice. Amorphous materials, such as glass, generally soften when heated and melt over a range of temperatures rather than at a single temperature. Most thermoplastic polymers are semi-crystalline,

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containing a mixture of amorphous and crystalline components. Many polymers begin softening or melting at temperatures lower than typical piloted ignition temperatures. Therefore, melting usually occurs prior to ignition and becomes even more pronounced during flaming combustion. Melting influences the combustion behavior of polymers through latent heat absorption (i.e. ∆Hm in eqn (28)) and changes in thermal properties (thermal conductivity and specific heat capacity) in the vicinity of the melting temperature. From a practical standpoint, polymers that are rigid at room temperature may melt, flow, and drip. Pool fires formed by melt flow at the base of vertical walls have been observed to dominate upward flame spread rates [173], and efforts are underway to model this phenomenon [174]. Melting allows for increased penetration of oxygen into the polymer by molecular diffusion and also makes bubbling possible. One of the best descriptions of the bubbling process in polymers is given by Kashiwagi and Ohlemiller [48], who studied the behavior of PE and PMMA irradiated at two heat flux levels and several ambient oxygen concentrations. PMMA irradiated in a nitrogen atmosphere at 17 kW/m2 formed ∼1 mm diameter bubbles at depths as much as 3 mm below the surface. Bubbles that formed within 1 mm of the surface were seen to vent through small holes; bubbles that formed deeper below the surface occasionally burst through necklike holes to the gas phase, violently ejecting vapor and molten polymer. This violent bursting process has also been observed in microgravity [175]. The presence of gas-phase oxygen reduces the viscosity of the molten polymer, leading to higher bubbling frequency and a less-violent bursting process. The ∼1 mm holes formed by the bubbles allow oxygen to penetrate as much as 1 mm into the polymer, thereby increasing the depth of the oxygen affected region beyond that which is possible by diffusion. At higher heat fluxes, the bubbles are smaller and closer to the surface. Despite the importance of melting and bubbling to polymer flammability, there has been only a handful of modeling studies aimed at better understanding these phenomena. A model that includes melting has been developed and used to successfully predict the time to ignition of a polypropylene/glass composite [176]. More recently, polymer melt flow behavior in laboratory-scale experiments has been modeled with encouraging results [174]. However, modeling of bubbling is still a research topic. Wichman [45] developed a model that describes the effect of bubbling on the transport of volatiles under steady-state conditions. No direct comparison with experimental data was possible, but the model is in qualitative agreement with experimental observations. In Butler’s model [46], the bubble layer is assumed to be perfectly mixed. The model predicted a MLR that was approximately constant with time, whereas experimental data showed an increasing MLR. A more recent model [47] includes a more detailed description of bubbling, but the author concludes that a better representation of the bursting process is still needed.

5 Fire growth modeling The preceding sections have provided the reader with a feel for the various approaches taken for pyrolysis modeling, identified some of the strengths and weaknesses of each, and examined the fundamental transport phenomena occurring within combustible solids. Although microscale transport phenomena and small-scale laboratory experiments were emphasized, the ultimate goal of pyrolysis modeling (at least in the fire field) is the prediction of large-scale fire behavior. For this reason, we highlight here a few examples of fire growth modeling wherein pyrolysis models have been coupled to gas-phase models that handle combustion, heat transfer, and fluid mechanics to predict large-scale fire behavior. To date, there have been few rigorous attempts at validating fire growth models. Most fire model validation work has involved ‘gas burner’ type problems where the movement of heat and

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smoke from a fire having a predetermined HRR is predicted and compared to experimental measurements, e.g. the US Nuclear Regulatory Commission’s reports [177]. Typically, a fire growth model is evaluated by comparing its predictions of large-scale behavior to experimental measurements of HRR, thermocouple temperatures, or pyrolysis front position. However, it is difficult to assess how well a pyrolysis submodel actually ‘works’ based solely on this type of comparison because there may be compensating factors at play. For example, an overprediction of flame heat fluxes combined with a pyrolysis model that underpredicts mass burning rates could give results that match the experimental HRR history. Clearly, the overall predictive capabilities of a fire growth model depend not only the pyrolysis model itself, but also on several additional aspects such as its treatment of gas-phase fluid mechanics, turbulence, combustion chemistry, and convective/radiative heat transfer. It is important to bear this in mind for the discussion below. The simplest class of fire growth models is self-contained standalone models that do not interface with a zone or field model for prediction of heat fluxes or compartment effects. As an example, Weng and Hasemi [178] combined a simple treatment of the gas phase with a one-dimensional integral charring pyrolysis model to simulate flame spread beneath a combustible medium density fiberboard ceiling, with good results. Flame heat fluxes to the ceiling were estimated from experimental data obtained from propane line burner experiments. Properties for the integral pyrolysis model were taken from the literature for white pine, where available, with the remaining properties estimated by calibration. The model predicted well the experimental flame length and pyrolysis front position, with the HRR slightly underpredicted, most likely due to underestimated flame heat fluxes. The primary advantage of this type of model is simplicity, but they lack flexibility and can usually only be applied only to a single class of problems (in this case, flame spread under ceilings). The semi-empirical ‘fire property’ based approach represented by eqn (2) has been used [13−15] for simulation of standardized room/corner fire scenarios such as ISO 9705. This type of fire growth modeling has practical importance for materials development or fabrication because it can be used as a screening tool, i.e. the outcome of expensive large-scale fire tests (required by some building codes and other regulations) can be estimated on the basis of smallscale fire test data that can be obtained relatively inexpensively. To account for compartment effects, i.e. the accumulation of a hot layer, pyrolysis models have been combined with an empirical correlation for the upper layer temperature [13] or zone fire models [15]. Although more general than standalone models, this approach can usually be directly applied only to the prediction of flame spread from a particular ignition burner because empirical measurements or approximations to the heat transfer from the ignition burner and wall/ceiling flames are required. In addition to sensitivity to the assumed flame heat fluxes, it has been shown [14] that predictions of this type of model are sensitive to the data reduction technique used to determine the required input parameters from fire test data. Despite the simplified nature of this type of fire growth modeling and its inherent uncertainties, good agreement between calculated and measured HRRs has been demonstrated [15, 23] even for nonsimple materials such as fire retarded composites [15]. Figure 11 gives a comparison of the calculated and measured HRR for a fire retarded vinylester composite tested in the ISO 9705 room/corner test [15]. There are certain circumstances under which the HRR during post-flashover or fully developed burning is of interest, e.g. the design of smoke management systems in an underground tunnel. The compartment heat transfer problem is simpler during post-flashover burning than pre-flashover burning due to a higher degree of homogeneity. Thus, a one-layer zone model can be combined with a simple pyrolysis model (e.g. eqn (2)) to estimate post-flashover burning rates from the exposed surface area of combustibles. This has been applied to railcars [179, 180]. A more complex integral

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Figure 11: Comparison of predicted and measured heat release rate (HRR) for fire retarded vinylester composite material burning in ISO 9705 room corner test (from [15]). pyrolysis model has been coupled to a single-layer zone fire model for calculating MLRs in fully developed fires [181]. The most promising long-term prospect for modeling flame spread and fire growth at building scales is the coupling of first principles based condensed-phase fuel generation models to computational fluid dynamics models that simulate the gas-phase fluid mechanics, combustion, and heat transfer aspects of a fire. The primary advantage of this approach is its flexibility, and it has been suggested [182] that this type of fire growth modeling will become an ‘invaluable tool for researchers and engineers’ due to this flexibility. With a coupled pyrolysis/CFD fire model, it should be possible to consider complex geometries and ignition scenarios, evaluate the impact of design changes on expected fire behavior, and perhaps assist in forensic fire reconstruction. To date, the level of complexity included in condensed-phase models has generally lagged that of the gas phase where a full solution of the Navier−Stokes equations is normally used, sometimes with detailed chemistry [183]. Solid phase pyrolysis models have been coupled to CFD for simulating bench-scale fire tests [66, 184, 185], primarily two-dimensional upward flame spread [186, 187], reduced-scale compartment fires [32, 187, 188], and building-scale compartment fires [189−199]. Noted difficulties include strong sensitivity of model predictions to solid phase properties [192] and grid size [191, 194, 196, 198]. The latter appears to be particularly problematic. Figure 12 shows the experimentally measured HRR from a room/corner test on spruce panels compared to the predictions of a CFD-based fire growth prediction at three different grid resolutions [191]. It can be seen that the predicted HRR is sensitive to the underlying grid spacing.

6 Concluding remarks This chapter has reviewed several aspects of the condensed-phase processes that affect a material’s overall reaction to fire, with an emphasis on modeling. An overview of solid phase decomposition

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0.067 m grid

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Figure 12: CFD-based prediction of spruce panel fire growth in room corner fire test at three different grid spacings compared to experimental data [191].

kinetics and thermodynamics is given due to their importance for the burning of solids. Conduction, radiation, convection, and momentum transfer within combustible solids are briefly discussed, and the various pyrolysis modeling strategies that have been used to simulate the burning of solids are summarized. Since prediction of large-scale fire development is an appropriate long-term goal, a few examples of combined condensed/gaseous-phase fire growth modeling are presented to give the reader a sense of the capabilities of current models. The pyrolysis models examined here are generally formulated in a way that makes them applicable only to one class of materials, i.e. noncharring, charring, intumescent, etc. However, since the conservation equations on which these models are based apply universally, it is possible to formulate a generalized pyrolysis model that can be applied to most solid combustibles. Then, a particular material could be simulated by specifying a set of input parameters (thermophysical properties, reaction mechanisms, etc.) rather than reformulating the entire model. The flexibility to invoke submodels for various transport phenomena is an important feature because there may be little consequence to omitting a particular phenomenon from a simulation other than reducing the computational expense and the number of parameters that must be specified to characterize a material. A major obstacle impeding real world application of fire growth modeling is the difficulty associated with determining the input parameters or material properties required to characterize different materials. Due to the lack of readily available material property data suitable for pyrolysis modeling, in this chapter we have presented literature values for thermal properties, kinetics coefficients, etc. of several common materials. However, the reader is cautioned that when property values obtained from microscale tests (i.e. most of those presented in this chapter) are used as input to a pyrolysis model, there is no guarantee that the predicted macroscale behavior will be ‘accurate’. Another confounding issue is that different samples of the same generic material may not necessarily have the same properties.

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There are several key areas in which progress is still needed. Formulation of a computationally inexpensive generalized pyrolysis model that can simulate most materials (when provided with proper input parameters) would be a major contribution. The authors have taken initial steps in this area [200], but additional model validation is needed. It is also important to understand which transport phenomena are important to include in a simulation under which circumstances, thereby keeping the number of required input parameters to a minimum. Also needed is the continued development of self-consistent and cost-effective methodologies that can be used to establish the input parameters required to characterize different materials, i.e. from fire test data [3], specialized small-scale tests [4, 201], or some combination thereof. The field would also benefit greatly from a databank of ‘validation’ experiments to help evaluate the predictive capabilities of pyrolysis and fire growth models. Finally, the grid-dependency of CFD-based predictions of large-scale fire growth has been noted by several authors, but it has not yet been systematically investigated to the point that any practical guidelines can be made; such recommendations would be useful.

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CHAPTER 7 Radiative heat transfer in fire modeling M.F. Modest Department of Mechanical Engineering, Pennsylvania State University, USA.

Abstract In recent years it has been increasingly recognized that thermal radiation plays an important if not dominant role in fires and that reliable predictions of combustion behavior is not possible without a sophisticated radiation model. And only today is it also becoming apparent that interactions between turbulence and radiation, to date always neglected, tend to be of great importance in large turbulent flames. In this chapter an account of modern spectral methods is presented for the prediction of radiative heat transfer rates within combustion media consisting of strongly nongray combustion gases as well as mildly nongray soot particles, and perhaps accompanied by larger scattering particles. Modern narrow band methods are discussed, such as the statistical narrow band model (SNB) and k-distribution method. Emphasis is given to state-of-the art global models, including the weighted-sum-of-gray-gases (WSGG) model, the spectral-line-based WSGG model (SLW), and the full-spectrum k-distribution method (FSK). Probability density function methods have been found to be effective tools for the study of turbulence−radiation interactions (TRIs). A brief account of such methods is given as applied to the fledgling state of the art of TRI modeling in diffusion flames, where TRIs are important, and what turbulence moments need to be considered to capture them.

1 Introduction It is well-recognized today that thermal radiation is an important and often the dominant heat transfer mechanism in fires. This is caused by the high temperatures encountered during burning combined with the fact that combustion gases (such as carbon dioxide, water vapor, hydrocarbons) soot and other particles (fuel, ash) strongly absorb and emit in the infrared part of the spectrum. Because of the difficulties associated with the prediction of radiative heat transfer rates, thermal radiation has, in the past, been commonly neglected in combustion models, or has been accounted for through very simplistic models. Today we know that neglecting radiation in medium-to-large-sized flames may lead to overprediction of temperature levels by several hundred degrees Celsius, while the use of a simplistic gray model may lead to underprediction of temperatures by 100°C or more [1]. Very recently it has been found that the interaction

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Transport Phenomena in Fires

between radiation and the turbulent flow field can also have profound effects on radiative heat transfer values in large turbulent flames, with their neglect leading to overpredicted temperatures of 100°C or more. Therefore, inclusion of an adequate radiation model is essential to the success of a mathematical model of a fire, particularly in large systems (with medium-to-large optical thickness). There are several severe challenges, which need to be overcome before a successful model for large scale fires can be generated. Firstly, the governing radiative transfer equation (RTE) is a five-dimensional integro-differential equation (three space and two direction coordinates), which is very difficult to solve and extremely computer time intensive. Many researchers have devoted much of their lives to the development of RTE solution methods, the most popular ones today being the discrete ordinates method (DOM) and its modern cousin, the finite volume method, the spherical harmonics method (particularly the lowest-level P-1 approximation), and the statistical Monte Carlo method. The reader is referred to standard text books for a detailed discussion of RTE solvers [1, 2]. The second challenge is the strong spectral variations of the radiative properties of combustion gases, soot and other particles present in flames, making their experimental measurements, theoretical determination, and their efficient integration with RTE solution methods an extremely difficult task [1]. Much progress has been made in recent years in modeling molecular gas and soot radiation, and this topic will be one focus of the present review chapter. Finally, a third new challenge has appeared with the modeling of turbulence−radiation interactions (TRI). Limited experimental data plus simple numerical analysis has shown that such interactions can significantly enhance radiative heat loss from turbulent flames. TRI cannot be modeled with the standard moment methods usually applied in turbulence models (such as the common k−e and Reynolds stress models) [3, 4], making such predictions a truly daunting task. Modeling of these important effects is still at an early stage of development and a review of its present state of the art will be a second focal point of this chapter. Models dealing with spectral variations in radiative heat transfer can be loosely grouped into the following three categories (in order of decreasing complexity or, rather, computational intensiveness): (i) line-by-line calculations, (ii) band models, and (iii) global models. LBL calculations (i.e. solving the spectral RTE for an extremely large number of spectral locations) are the most accurate, but require vast computer resources. This has lead to a number of band models, in which pertinent radiative properties are averaged over small (narrow band models) or large (wide band models) parts of the spectrum, or even over the entire spectrum (global models). First a short description is given of the nature of the absorption coefficients of important combustion gases and of soot, followed by a brief review of traditional band models. Modern k-distribution methods and state-of-the-art global methods are then discussed in somewhat greater detail. In turbulent fires the velocity fluctuations cause fluctuations in species concentrations and temperature. Consequently, the radiation field, which is directly related to species concentration and temperature, will fluctuate as well. In a numerical simulation fluctuations in the radiation field interact with the fluctuations of the flow field, causing the so-called turbulence−radiation interactions. Models dealing with TRI can also be loosely grouped into three categories, in order of increasing complexity: (i) assumed probability density function (PDF) methods, (ii) calculated PDF models, and (iii) direct numerical simulation (DNS). The fact that accurate TRI calculations requires the evaluation of many turbulence moments (and, thus, coupled differential equations) has prompted early investigators to use assumed PDFs (i.e. primitive models to estimate turbulence), and their work will be reviewed. Most serious TRI models today solve some form of the transport equation for the PDF, usually by stochastic means, and this work and their results will be discussed in some more detail. Finally, very recently some preliminary work on TRI using DNS has been carried out, and will also be briefly reviewed.

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263

2 Radiative properties of combustion gases When a gas molecule absorbs or emits radiative energy, this raises or lowers the vibrational and/or rotational energies of the molecule. Since these energy levels are quantized this leads to many thousands of discrete spectral lines, forming so-called vibration−rotation bands in the infrared. The precise photon energies required for these transitions are altered a little by a number of effects, primarily due to molecular collisions and molecular movement (Doppler effect), leading to slight broadening of the spectral lines. A single spectral line at a certain spectral position is fully characterized by its strength and its line width (plus knowledge of the broadening mechanism, i.e. collision and/or Doppler broadening). Locations, strengths and widths of spectral lines have been collected in modern databases, notably the HITRAN and HITEMP databases [5, 6], which also contain directions on how to calculate the resulting absorption coefficient. An example is given in Fig. 1, showing the pressurebased absorption coefficient kph for the most important wavenumber range of the strong 4.3 µm band of carbon dioxide. The strong spectral variations of the absorption coefficient are clearly visible in the form of about 50 dominant broadened lines, although the given range contains more than 5,000 lines in the HITRAN database (most of them fairly weak and overlapping). At lower total pressure the spectral variations become amplified, since lines are broadened less (higher peaks and narrower widths). At the high temperatures common in fires and combustion applications many more spectral lines appear, the so-called ‘hot lines’, generated by energy transitions from molecules populating higher vibrational energy levels. For example, the HITEMP database [6] (assumed accurate to 1000 K) contains 27,000 lines for the spectral interval given in Fig. 1, while the CO2 database CDSD1000 [7, 8] (claiming accuracy to 3000 K) has 36,000. The resulting absorption coefficient then resembles electronic noise [1].

400 T = 300 K, p = 1 bar, pCO2 = 0 bar

κpη, cm–1bar –1 (=κη/pCO2)

300

200

100

0 2300

2325

2350 η, cm–1

2375

Figure 1: Pressure-based spectral absorption coefficient for small amounts of CO2 in nitrogen; 4.3 µm band at p = 1.0 bar, T = 300 K.

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Transport Phenomena in Fires

3 Radiative properties of soot Soot particles are produced in fuel-rich parts of flames, as a result of incomplete combustion of hydrocarbon fuels, and make up a large part of the smoke observed in fires. As shown by electron microscopy, soot particles are generally small and spherical, ranging in size between approximately 5 and 80 nm [9, 10]. While mostly spherical in shape, soot particles may also appear in agglomerated chunks and even as long agglomerated filaments. It has been determined experimentally in typical diffusion flames of hydrocarbon fuels that the volume fraction of soot, fv, generally lies in the range between 10−6 and 10−8 [11−13]. Since soot particles are very small, they are generally at the same temperature as the flame and, therefore, strongly emit thermal radiation in a continuous spectrum over the infrared region. Experiments have shown that soot emission often is considerably stronger than the emission from the combustion gases. In order to predict the radiative properties of a soot cloud, it is necessary to determine the amount, shape and distribution of soot particles, as well as their optical properties, which depend on chemical composition and particle porosity. Early work on soot radiation properties concentrated on predicting the absorption coefficient kl for a given flame as a function of wavelength l. For all but the largest soot particles the size parameter x = πd/l (based on soot diameter d) is very small for all but the shortest wavelengths in the infrared, so one may expect that Rayleigh’s theory for small particles will, at least approximately, hold. This condition would lead to negligible scattering and an absorption coefficient of  m 2 − 1  6πfv fv 36 πnk kl = bl = −ℑ  2 = 2 ,  2 2 2 2 l l (n − k + 2) + 4 n k  m + 2 

(1)

where bl is the extinction coefficient and m = n − ik is the complex index of refraction. Experiments have confirmed that scattering may indeed be neglected [14]. It is customary to approximate the soot absorption coefficient by kl =

Cfv la

,

(2)

where C and a are empirical constants; values of the dispersion exponent a incorporate the spectral dependence of the complex index of refraction, ranging from 0.7 to as high as 2.2. However, the optical properties of soot material have also been measured directly by a number of experimenters. The most reliable ones today are perhaps those obtained by Chang and Charalampopoulos [15] for propane soot, which have been corroborated by several other studies, and which have been curve-fit in polynomial form. Agglomeration of soot into chunks or long chains renders the assumption of nonscattering soot questionable. The prediction of agglomeration requires complicated models, mostly due to Frenklach and coworkers [16−22]. The radiative properties of agglomerated soot have also been measured and modeled by a significant number of researchers [23−27]. A brief review of such models and of how to estimate radiative properties of agglomerated soot has been given in the book by Modest [1].

4 Band models Because of the strong spectral variation of radiative properties, as shown for CO2 in Fig. 1, evaluation of radiative flux requires, in principle, many spectral solutions to the RTE, up to one million

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265

and more if LBL calculations are made. Until recently detailed spectral knowledge of radiative properties was not known, nor were computers powerful enough to carry out such calculations. This has led to a number of approximate band models, with which ‘line-of-sight’ gas transmissivities and emissivities can be estimated. These models use statistical averaging procedures and are limited to nonscattering media and black walls. Modern band models, known as k-distributions, employ the new high-resolution databases and reorder, rather than average, the absorption coefficient. These methods can be applied directly to the RTE and are, thus, also valid for scattering media and for nonblack walls. 4.1 Traditional narrow band models The spectral intensity leaving a homogeneous gas layer of thickness s, bounded by a black wall at s = 0, is [1] I h (s ) = I b (Tw )th (s ) + I bh (T )eh (s ),

(3)

where Tw and T are the temperatures of the boundary wall and the gas, respectively, and th ( s) = 1 − eh ( s) = e

− kh s

,

(4)

are the transmissivity th and emissivity eh, respectively, for a gas column of length s. Forming a narrow band average (with a ∆h = 4/25 cm−1), and noting that the blackbody intensity essentially remains constant over such small spectral range, we obtain Ih =

1 1 I h dh = ( I bwh th + I bh eh ) dh I bwh th + I bh eh , ∫ h ∆ ∆h ∆h ∫∆h

(5)

where th =

1 −k s e h dh ; ∆h ∫∆h

eh =

1 −k s (1 − e h ) dh = 1 − th , ∆h ∫∆h

(6)

are narrow band-averaged transmissivities and emissivities, respectively. We note from Fig. 1 that the absorption coefficient undergoes many oscillations across any narrow band, but that − if __ __ simple approximations for th and eh can be found − the total intensity (or radiative flux) can be obtained in a (relatively) straightforward fashion from ∞

∞

I (s ) = ∫ I h (s )dh = ∫ I h(s )dh. 0 0 Several different narrow band models have been proposed, viz., the Elsasser model (assuming spectral lines to be of equal strength as well as equally spaced, as often the case for diatomic gases) and a number of statistical models (assuming different forms of randomness for line strength and spacing). It is known today that the Malkmus statistical model [1, 28] best represents multi-atomic combustion gases. In this model the placement of lines is random, while line strengths are picked from a probability distribution that accounts for the many weak lines that are always present. Using this model, the narrow band emissivity is evaluated from  b 4t   − 1  , eh ( L ) = 1 − exp  −  1 + b    2 

(7)

266

Transport Phenomena in Fires

where L is the length of the gas column, b is a line overlap parameter, and t (not to be confused with transmissivity th) is the average optical depth, the latter two are defined by b=

πb ; d

t=

S L, d

(8)

where b is the spectral lines’ half-width at half-maximum (measured in cm−1), d is the average line spacing (in cm−1), and S is the average line strength (in cm−2). For sufficiently small narrow bands the Malkmus model can predict transmissivities very accurately over wide ranges of parameters. Nonhomogeneous gas layers are somewhat problematical, but reasonable accuracy can be achieved with the so-called Curtis-Godson approximation [1], by defining path-averaged values for b and t as L S 1 LS (9) t = ∫ dx , b = ∫ b dx. 0 d t 0 d Two narrow band databases are available for engineers; the first being based on experimental data [29], and the second formed from a high-resolution database [30]. The major limitation of traditional narrow band models is the fact that they can only predict line-of-sight transmissivities and emissivities, i.e. they cannot be incorporated into the RTE, precluding their use in scattering media and/or systems with reflecting surfaces. 4.2 Traditional wide band models Traditional narrow band models predict a simple transmissivity for a small wavenumber range of about ∆h 10 / 25cm –1 , thus requiring several hundred RTE evaluations. Given the substantial computer requirements and the fact that earlier, experimentally-based values had limited accuracy, a number of models were developed in the 1960s to reduce the transmissivity of an entire vibration−rotation band to a single value. By far the most successful of these so-called wide band models was the exponential wide band model by Edwards and coworkers [31−33]. Despite its limited accuracy (perhaps 30%) the model enjoyed wide popularity until the mid 1990s, because better models were simply not available. Since the advent of high-resolution databases and the development of k-distribution methods detailed in the following sections, their use has declined considerably. Therefore, details of the models will not be presented here, and the reader is referred to standard textbooks [1, 2]. 4.3 Narrow band k-distributions It was recognized some time ago by meteorologists that, for a homogeneous medium, the spectrally oscillating absorption coefficient can be reordered into a monotonically increasing function, greatly simplifying spectral integration of radiative fluxes. In a homogeneous medium the absorption coefficient, while varying spectrally, is spatially constant. The RTE for such an emitting, absorbing and scattering medium is [1] dI h ds

= kh I bh − (kh + ss )I h +

ss I h ( sˆ ′ ) Φ( sˆ , sˆ ′ ) dΩ ′, 4π ∫4π

(10)

where ss is the medium’s scattering coefficient, Φ is its scattering phase function, Ω denotes solid angle, and sˆ is a unit direction vector. Let eqn (10) be subject to the boundary condition at a wall I h = I wh = e w I bwh + (1 − e w )

1 I h | nˆ ⋅ sˆ | dΩ, π ∫nˆ ⋅sˆ < 0

(11)

Radiative Heat Transfer in Fire Modeling

267

where Iwh is the spectral intensity leaving the wall, ew is the wall’s emittance and nˆ is a unit surface normal pointing into the medium. Equation (11) assumes all surfaces to be diffuse, but generalization to more complicated surface properties is straightforward. In the k-distribution scheme it is assumed that the spectral variation of scattering properties (ss, Φ) and surface emittances (ew) is much more benign than that of the gas, i.e. we will assume that these properties are constant (gray) across small parts of the spectrum (narrow band). Use is also made of the fact that across a narrow band the Planck functions Ibh (medium) and Ibwh (wall) remain essentially constant. Therefore, it is apparent that each location across the narrow band where the absorption coefficient has one and the same value kh = k will result in identical intensities Ih. The absorption coefficient can be reordered into a monotonically increasing function, making sure that a correct fraction of the narrow band contains an absorption coefficient k ≤ kh ≤ k + dk, for all k, as indicated in Fig. 2a. Mathematically, this is achieved by multiplying equations (10) and (11) with the Dirac delta function d(k−kh)/∆h, followed by integration over the narrow band. Thus dI k s = kf (k )I b − (k + ss )I k + s ∫ I k (sˆ ′ ) Φ(sˆ , sˆ ′ ) dΩ ′, ds 4π 4π

(12)

with boundary condition I k = I wk = e w f (k )I bw + (1 − e w )

1 I k | nˆ ⋅ sˆ | dΩ, π ∫nˆ ⋅sˆ < 0

(13)

where ∞

I k = ∫ I h d (k − kh ) dh

(14)

0

is the intensity Ih collected over all spectral locations where kh = k (per dk), and f (k ) =

1 d (k − kh ) dh ∆h ∫∆h

(15)

10

1

2320 (a)

102

k, cm–1 bar –1

κpη, cm–1bar –1 δkj

102

1

10

2322 δηi (kj ) η, cm–1

2324

0 (b)

0.25

0.5

g

0.75

1

Figure 2: Extraction of k-distributions from spectral absorption coefficient data (here for small amounts of CO2 in nitrogen, across a small part of its 4.3 µm band at p = 1.0 bar, T = 300 K: (a) actual absorption coefficient; (b) reordered, equivalent k-distribution.

268

Transport Phenomena in Fires

is known as the k-distribution, which is a PDF for the absorption coefficient. The problem is further simplified by using the cumulative k-distribution, g, k

g(k ) = ∫ f (k ) dk = 0

1 H (k − kh ) dh, ∆h ∫∆h

(16)

where H() denotes Heaviside’s unit step function. Physically, g(k) is the fraction of the narrow band over which kh ≤ k. In practice, of course, the cumulative k-distribution is evaluated numerically, using k-bins of finite width dk, as shown in Fig. 2a for the jth bin; A small step in g is then evaluated as dg j = f ( k j ) d k j =

1 dh [ H ( k j + d k j − kh ) − H ( k j − kh )]. ∑ ∆h i dkh

(17)

i

The redistributed absorption coefficient for the spectrum of Fig. 2a is shown in Fig. 2b. Note that both figures have identical maximum and minimum values for the absorption coefficient, and it is observed that absorption coefficients between 19 cm−1 and 20 cm−1 (dkj in Fig. 2a) occupy about 1.5% of the narrow band spectrum (equal to the sum of dhj in Fig. 2a). Once the cumulative k-distribution has been found and inverted to yield k(g), the ‘spectral’ intensity Ig is found from the RTE as dI g ds



= k ( I bh (T ) − I g ) − ss  I g − 

 1 I g ( sˆ ′ ) Φ( sˆ , sˆ ′ ) dΩ ′  , ∫ π 4 4π 

(18)

with the boundary conditions I g = I wg = e w I bwh + (1 − e w )

1 I g | nˆ ⋅ sˆ | dΩ, π ∫nˆ ⋅sˆ < 0

(19)

where I g = I k /f (k ) =

1 I h d (k − kh )dh f (k ), ∆h ∫∆h

(20)

and the narrow band-averaged intensity is evaluated from Ih =

∞ 1 1 I h dh = ∫ I k dk = ∫ I g dg. ∫ ∆ 0 0 h ∆h

(21)

In the original development of k-distributions the aim was to obtain simplified expressions for narrow band transmissivities and emissivities, similar to the traditional models. Applying the k-distribution technique to eqn (6) leads to th =

∞ 1 1 −k s e h dh = ∫ e − ks f (k ) dk = ∫ e − ks dg. ∫ 0 0 ∆ h ∆h

__

Inspection of this equation shows that th is the Laplace transform of f(k), as was first recognized by Domoto [34]. This implies that k-distributions can not only be generated from high-resolution databases, but from also low- and medium-resolution experimental data. As for the statistical models, application of the reordering concept to spatially nonhomogeneous absorption coefficients is somewhat problematical. It turns out the k-distribution approach is exact for a correlated absorption coefficient: at every wavenumber where kh(r1) at one location has one and the same value, k, the

Radiative Heat Transfer in Fire Modeling

269

1 0.9

transmissivity, τη; emissivity, εη

0.8 0.7 0.6 0.5

∆η = 25cm LBL scaled k correlated-k -1

0.4 0.3 0.2 0.1 0 1000

1500

2000

2500

wavenumber, η, cm-1

Figure 3: Narrow band transmissivities for two-temperature slab, as calculated by the LBL, scaled-k, and correlated-k (CK) methods; 6.3 µm band of H2O with pH2O = 0.2 bar.

absorption coefficient kh(r2) at a different location always also has one unique value k* (which may be a function of k but not h). If the ratio k*/k is constant for all h across the narrow band (not a function of k) the absorption coefficient is scaled, i.e. spatial and spectral dependence are separable. Details on these restrictions are found in [35]. As an illustration, a simple (but severely nonhomogeneous) example is given in Fig. 3, showing transmissivities through a hot layer (50 cm width at 1000 K) adjacent to a cold slab (50 cm width at 300 K) of a 20% H2O−80% N2 mixture at a total pressure of 1 bar. The 6.3 µm vibration−rotation band of water vapor with a narrow band resolution of 25 cm−1 is shown. Correspondence between exact LBL and k-distribution results is seen to be excellent (except for slight differences at a few wavenumbers) despite the severe nonisothermality. A very compact database of narrow band k-distributions for CO2 and H2O has been collected by Soufiani and Taine [30]; a larger, high-accuracy database has recently been given by Wang and Modest [36].

5 Global models Global models deal with the entire spectrum at once, trying to reduce the RTE solutions to a small number. The earliest global model was the weighted sum of gray gases (WSGG) introduced by Hottel for his zonal method [37], and was based on experimental data for total gas emissivities. After Modest [38] showed that the WSGG method could be used with arbitrary RTE solvers, the model quickly became the method of choice for nongray media. The method gained further popularity when Denison and Webb [39−43] showed how high resolution data (such as HITRAN and HITEMP) could be used rather than the dated emissivity data, calling

270

Transport Phenomena in Fires

it spectral-line-based WSGG (SLW). A similar method was also developed in France, called the absorption distribution function (ADF) method [44−46]. Finally, Modest and coworkers [35, 47−50] have been able to extend the narrow band k-distribution concept to the entire spectrum, calling it the full-spectrum k-distribution (FSK) method. It was found that the SLW/ADF approaches are low-level implementations of the FSK approach. 5.1. The WSGG method In the WSGG method the spectrally integrated form of eqn (3) is considered, i.e. ∞

I (s ) = ∫ I h dh = I bw (Tw ) [1 − e (Tw , s )] + I b (T )e (T , s),

(22)

0

where e (T , s ) =

∞ 1 −k s (1 − e h )I bh (T ) dh, I b (T ) ∫0

(23)

is the total emissivity of a gas column of length s. Hottel approximated this total emissivity by a weighted sum of gray gases, i.e. N

e (T , s ) =

∑ an (T )(1 − e − k s ),

(24)

n

n=0

where the gray-gas absorption coefficients kn are constants, while the weight factors an may be functions of source temperature; neither kn nor an are allowed to depend on path length s. Depending on the medium, the quality of the fit, and the accuracy desired, an N value of 2 or 3 usually gives results of satisfactory accuracy [51]. For an infinitely thick medium, the emissivity approaches unity and, thus, N

∑ an (T ) = 1.

(25)

n=0

Because pure molecular gases have ‘spectral windows’ (i.e. kh ⯝ 0 between vibration−rotation bands) k0 = 0 by convention; in the presence of absorbing particles the n = 0 term is dropped. Sticking eqn (24) into eqn (22) leads to N

I (s ) =

N

∑ I n (s) = ∑ [an I b ](Tw )e − k s + [an I b ](T ) 1 − e − k s   ,

n= 0



n

n



(26)

n= 0

i.e. each In is the solution to the RTE for a gray medium with absorption coefficient kn, but using a weighted Planck function [an Ib]. WSGG parameters for various media can be found in the literature, mostly based on experimental data [38, 52−54]. As an example Fig. 4 shows the heat loss from an isothermal slab of varying thickness L. The slab is at a temperature of T = 1000 K, a total pressure of p = 1 atm, it consists of a mixture of 70% N2, 20% H2O and 10% CO2 (by volume), and is bordered by cold black walls. Cases with (volume fraction fv = 5 × 10−6) and without soot are considered using the WSGG method together with Truelove’s [54] parameters, and are compared with LBL results using the HITEMP database [6]. Here four gray gases without soot and eight with soot (approximated by a constant index of refraction m = 1.89 − 0.92i in the LBL

Radiative Heat Transfer in Fire Modeling

271

1.1 1 0.9 WSGG LBL and FSK

0.8 q/σT 4

0.7

with soot

0.6 0.5 0.4 without soot

0.3 0.2 0.1 0 –2 10

–1

10

0

L (m)

10

1

10

Figure 4: Nondimensional heat loss from an isothermal N2−H2O−CO2 mixture with and without soot.

calculations) have been used, leading to very good agreement, at least for small-to-medium slab thickness. The small discrepancy for small L with soot is due to the different (but unknown) soot index of refraction used by Truelove. Through the developments accompanying the SLW and FSK methods, it is known today that the WSGG may be applied also to scattering media and/or reflecting walls, i.e. eqns (10) and (11) may be used to find the In, simply by replacing kh and Ibh(T) by kn and [anIb](T), respectively. Note that, as for any global model, it is assumed that scattering and surface properties (ss, Φ, ew) are gray. The greatest limitation of the WSGG method is its restriction to spatially constant absorption coefficients. No successful WSGG parameters for nonhomogeneous media appear to exist. 5.2 The SLW method Physically, the WSGG approach may be interpreted as a medium with spectrally varying absorption coefficient, which − while allowed to vary wildly across the spectrum − can attain only N different values kn. The weight factors an are then the fraction of the Planck function Ib for which kh = kn, i.e. Ibh integrated only over those wavelengths, divided by Ib. Denison and Webb [39−43] and Soufiani et al. [44−46] recognized that these parameters could, therefore, also be obtained from modern spectroscopic databases. If one denotes the fraction of the spectrum with kh < k, weighted by Ibh, by g(k), one obtains a reordered absorption coefficient much like the narrow band k-distributions. One such distribution is shown in Fig. 5 for a mixture containing 10% CO2 and 90% N2 (by volume) at 1000 K and 1 bar, evaluated from the HITEMP database [6]. Setting nominal values for kn (k0 = kmin, k1,…, kN = kmax; with kmin and kmax being the minimum and maximum values of kh across the spectrum) and corresponding values for gn(g = 0, g1,…, gN = 1), they solve the modified RTE dI n s = kn [ an I b ] − (kn + ss ) I n + s ∫ I n (sˆ ′ ) Φ(sˆ , sˆ ′ ) dΩ ′, ds 4π 4 π

(27)

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Transport Phenomena in Fires

10

0

10% CO2 + 90% N2 mixture

k, cm

–1

x

10

–1

10

–2

10

–3

10

–4

FSK/HITEMP FSK/Correlation (D&W) FSK/Correlation (Z&M) Gaussian quadrature points SLW/HITEMP SLW/Correlation (D&W)

x

x x

10–5 10

x

x x

–6

x x

10–7 0

0.25

0.5 g

0.75

1

Figure 5: Planck-function-weighted cumulative k-distributions for 10% CO2 in nitrogen for gas and Planck function temperatures of 1000 K, as evaluated from the HITEMP database and the correlations by Denison and Webb [42] and Zhang and Modest [55].

where kn = kn kn −1 ,

an = gn – gn –1.

(28)

One possible choice for the set of kn is also included in Fig. 5 as a thin dash-dotted line. The corresponding values for heat loss from such a mixture are shown in Fig. 6. It is seen that the SLW does reasonably well compared to LBL calculations, at least for small to moderate slab thickness L. For large values of L the slab is opaque for large k, making small values more important, which are approximated crudely in this example (see Fig. 5). Also shown in Figs 5 and 6 are results from simple engineering correlations for k(g) given by Denison and Webb [42] and Zhang and Modest [55]. Such correlations bypass the need for lengthy calculations from a high-resolution database. 5.3 Full-spectrum k-distributions The new mathematical definition of narrow band k-distributions [35], using Dirac-delta functions, as given by eqns (12) and (13) makes it possible to reorder the absorption coefficient across the entire spectrum. Multiplying eqns (10) and (11) by the Dirac-delta function d(k−kh), followed by integration over the entire spectrum, and assuming scattering and surface properties (ss, Φ, ew) to be gray, leads to dI k s = kf (T , k ) I b − (k + ss ) I k + s ∫ I k ( sˆ ′) Φ( sˆ , sˆ ′) dΩ ′, ds 4π 4π

(29)

Radiative Heat Transfer in Fire Modeling

273

0.25

10% CO2 + 90% N2 mixture

LBL/HITEMP, FSK/HITEMP FSK/Correlation (D&W) FSK/Correlation (Z&M) SLW/HITEMP SLW/Correlation

0.2

q/σT

4

0.15

0.1

0.05

0 –2 10

10

–1

10

0

10

1

L (m)

Figure 6: Heat loss from an isothermal slab of 10% CO2 in nitrogen at T = 1000 K, as evaluated from the LBL, FSK, and SLW models.

with the boundary condition I k = I wk = ew f (Tw , k )I bw + (1 − ew )

1 I k | nˆ ⋅ sˆ | dΩ, π ∫nˆ ⋅sˆ < 0

(30)

where ∞

I k = ∫ I h d (k − kh ) dh 0

(31)

and f (T , k ) =

1 Ib

∞

∫0

I bh (T ) d (k − kh ) dh

(32)

is now a Planck-function-weighted k-distribution, which is a function of the gas state at which the absorption coefficient is evaluated and of temperature T through the Planck function. Again, it is more convenient to cast the RTE in terms of the cumulative k-distribution, now defined by k

g(T , k ) = ∫ f (T , k )dk = 0

1 Ib

∞

∫0

I bh (T )H (k − kh )dh.

(33)

As already indicated in the discussion of the SLW method, g is the fraction of the spectrumintegrated Planck function with an absorption coefficient kh < k. Since the full-spectrum k-distribution is a function of temperature, one cannot simply divide eqns (29) and (30) by f(T, k) as was done for narrow bands. Instead, one must define a reference temperature T0 , and the equations are divided by f (T0, k), leading to dI g ds



= k  a(T , T0 , g )I b (T ) − I g  − ss  I g − 

 1 I g (sˆ ′ ) Φ(sˆ , sˆ ′ ) dΩ ′  , ∫ π 4 4π 

(34)

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Transport Phenomena in Fires

subject to the boundary condition I g = I wg = ew a(Tw , T0 , g )I bw + (1 − ew )

1 I g | nˆ ⋅ sˆ | dΩ. π ∫nˆ ⋅sˆ < 0

(35)

Here I g = I k /f (T0 , k ) = ∫

∞

0

I h d (k − kh ) dh f (T0 , k ), k

g(T0 , k ) = ∫ f (T0 , k )dk , 0

a(T , T0 , g ) =

f (T , k ) dg(T , k ) = , f (T0 , k ) dg(T0 , k )

(36) (37) (38)

and the total intensity is evaluated from ∞

∞

1

0

0

0

I = ∫ I h dh = ∫ I k dk = ∫ I g dg.

(39)

It is seen that the a-function acts as a nongray stretching function. The FSK method is exact for homogeneous media; its accuracy depending only on the quality of the integration of eqn (39), e.g. by using a Gaussian quadrature with the quadrature points indicated in Fig. 5. Thus, the lines labeled LBL/FSK in Figs 4 and 6 coincide. In the simplest case of approximating Ig by a step function (i.e. constant values of Ign across a ∆gn with constant kn), the FSK method reduces to the SLW method with In = (Ig∆g)n. For nonhomogeneous media the FSK remains valid if the absorption coefficient is correlated or scaled as outlined in Section 4.3. A reference state is chosen (based on total emission from the medium) where f0 = (T0, p0 , x0) is a vector containing all state variables influencing the absorption coefficient, such as temperature T, total pressure p, and species mole fractions x. At this state the absorption coefficient is evaluated exactly and is used for the calculation of the reference k-distribution f(T0 , k) and the stretching function a(T, T0 , g). In the full-spectrum correlated-k (FSCK) method the absorption coefficient at other (nonreference) states is assumed to be correlated, and the k in eqn (34) for a spatially invariable absorption coefficient is replaced by k(T0, f, g), i.e. the k vs. g distribution found using the absorption coefficient evaluated at the local state f, and the Planck function at the reference temperature T0. In the full-spectrum scaled-k (FSSK) approach the absorption coefficient is assumed to be scaled, and k is replaced by k(T0, f0, g)u(f, f0), i.e. k vs. g evaluated at reference state and reference Planck function temperature, multiplied by a scaling function u. The latter approach tends to be somewhat more accurate, since the scaling function can be optimized. Details of both methods can be found in Modest [35]. An example for a mixture with extreme inhomogeneities is given in Fig. 7, considering a mixture of 10% CO2−20% H2O−70% N2 (by volume) confined between two cold, black plates. The mixture is at a total pressure of 1 bar and consists of a hot, isothermal layer of fixed width Lh = 50 cm and a temperature of Th = 1000 K, and a cold, isothermal layer (Tc = 300 K) of variable width Lc. Shown is the radiative heat flux leaving from the cold layer using the FSCK and FSSK methods, and LBL values are included for comparison, all three methods using the HITEMP database [6]. It is observed that both FSCK and FSSK results coincide with LBL data for Lc = 0, since the methods become exact. For Lc > 0 the FSCK method consistently underpredicts the heat loss, with a maximum error of about 25% at intermediate Lc, due to the assumption of a correlated absorption coefficient while, in fact, it is not, particularly due to ‘hot lines’ at elevated temperatures.

Radiative Heat Transfer in Fire Modeling 0.4

275

0.16 LBL FSSK FSCK 0.12

0.3

q/σ Th4

Th = 2000K 0.2

0.08

0.04

0.1 Th = 1000K

0

0 0

50

100

150

200

Lc, cm

Figure 7: Heat loss from the cold column of a two-column 10% CO2−20% H2O−70% N2 mixture at different temperatures (Th = 1000 K and 2000 K, Lh = 50 cm; Tc = 300 K, Lc variable; uniform p = 1 bar, cold and black walls), from LBL, FSSK and FSCK models, all using the HITEMP database. The FSSK method can partially compensate for this lack of correlation, with a maximum error of only about 10% at intermediate Lc. The case of Th = 2000 K is also included in Fig. 7 (Tc remains at 300 K), making hot lines much more important and, thus, further decorrelating the absorption coefficients. Maximum relative errors are seen to increase slightly to about 30% (FSCK) and 15% (FSSK). It should be realized that this example is extreme; in realistic combustion applications the errors rarely exceed a few percent. 5.4 FSK assembly from a narrow band database Full-spectrum k-distributions can provide answers rivaling the accuracy of LBL calculations, but at a minuscule fraction of the computational cost (about 1:100,000). However, assembling these k-distributions as functions of Planck function temperature and state of the gas for every point in a three-dimensional enclosure is a tedious task at best. Thus, it would be highly desirable to have a permanent database of such k-distributions available. Since creating a database of the infinite number of possibilities for gas−soot mixtures is not possible, accurate mixing schemes need to be found to assemble mixture k-distributions from those for individual species. Such mixing is best carried out on a narrow band level, followed by collecting full-spectrum distributions from their narrow band counterparts. This concept was first explored by Modest and Riazzi [50] who also proposed a new, accurate mixing scheme. Assuming the absorption coefficients of gas species in a mixture to be uncorrelated they obtained, for a mixture of I species, gmix (kmix ) = ∫

1

g1=0

∫

1

g1=0

H[ kmix − (k1 + + kI )] dgI dg1.

(40)

While this mixing scheme can also be applied at the full-spectrum level, it was found to be more accurate at the narrow band level. Soot is easily included at the narrow band level, since the soot

Transport Phenomena in Fires 900000

1 0.9 0.8

5 % CO2–5 % H2 O–90 % N2 T=1200 K, p=1 bar, fv=10–7

g

0.6 0.5

Direct FSK FSK, NB

0.4

dq /dx (W/m3)

0.7

0.3

0.5

L=20cm

Error (%)

276

850000

0

800000

–0.5

750000 LBL FSK, NB FSK, Direct

700000 650000

0.2 600000

0.1 0

550000 10–3

(a)

10–2

k (cm–1)

10–1

100

0

0.25

(b)

0.5 x/L

0.75

1

Figure 8: Gas mixtures with nongray soot: (a) full-spectrum k vs. g distributions; (b) gradient of heat flux inside a homogeneous slab.

absorption coefficient ksoot, h can be assumed constant across the jth narrow band [but not across the entire spectrum, see eqn (1)]. Then, for a soot-gas mixture kmix ( gj ) = ksoot , j + kgas ( gj).

(41)

The FSK is then collected as g(T , k ) =

∑

j e all narrow bands

I bh j (T ) I b (T )

g j (k ),

(42)

where Ibhj is the Planck function Ibh integrated across the jth narrow band ∆hj. An example is given in Fig. 8a, showing the FSKs for a mixture consisting of 5% H2O−5% CO2−90% N2 at T = 1200 K and p = 1 bar and soot with fv = 10−7 (using the correlation of Chang and Charalampopoulos [15] for the soot’s complex index of refraction). Figure 8b shows the radiative source, dq/dx, for such a mixture in a one-dimensional slab of L = 20 cm width, bounded by cold, black walls. The FSK method is exact for such a medium, the only errors stemming from (i) inaccuracy of the individual narrow band k-distributions; (ii) errors due to interpolation; (iii) inaccuracy of the mixing model, eqn (40); and (iv) the neglected spectral variation of Ibh and ksoot, h across individual narrow bands. The accuracy is seen to be better than 0.5%, i.e. is comparable to the accuracy of the LBL calculations themselves. Modest and Riazzi [50] also showed how a narrow band database can be employed to accurately and efficiently deal with the problem of nongray scattering and/or surface reflection by assembling part-spectrum k-distributions. A compact, high-accuracy, narrow band k-distribution database for these models has recently been provided by Wang and Modest [36] (presently limited to H2O and CO from HITEMP [6], and CO2 from CDSD [7]).

6 Turbulence–radiation interactions In a similar way as molecular diffusion is aided by turbulent motion, giving rise to the so-called eddy-diffusivities, radiative flux is also enhanced by the nonlinear interaction between fluctuating intensity and fluctuations of local radiative properties. Determination of such TRIs requires simultaneous consideration of turbulence, chemical reactions, and thermal radiation, with each of them posing formidable challenges by themselves. In the following we will first outline how TRIs arise

Radiative Heat Transfer in Fire Modeling

277

in turbulent fires. Then we will review older, more approximate work, followed by a discussion of state-of-the-art models, using stochastic methods to predict turbulent PDFs, and with them TRIs. Finally, a brief review will be given on some early direct numerical simulations of TRIs. 6.1 Turbulence–radiation coupling Turbulent motion is random and irregular, and has a broad range of length scales and time scales. With the exception of direct numerical simulations for very simple problems, at present time calculations are restricted to the determination of averaged quantities (e.g. mean velocities, mean species concentrations) in turbulent flows. In turbulent reacting flows, conventional time averaging (also called Reynolds averaging) and mass-weighted averaging (also called Favre averaging) are generally used to formulate the problem. In this section, conventional means will be represented by angle braces, and Favre means will be denoted by tildes. Fluctuations about them are designated by a single prime and double primes, respectively. Thus, for any quantity Q we have Q = Q + Q ′ = Q + Q ′′.

(43)

The general relationship between the conventional and Favre means is rQ Q = , r

Q = r Q /r ,

(44)

where r represents density. In problems of an engineering nature, when the radiant energy density is much smaller than the total energy density of the fluid, radiation pressure is much smaller than the pressure of the fluid. Also, when the fluid velocity is much smaller than the velocity of light, radiation does not contribute to the mass and momentum conservation equations [56]. These equations retain their classical form, and only the energy equation is modified in a radiatively participating fluid. The thermal energy equation for a fire can be written as [57] r

∂h ∂h ∂ p ∂ + r u i − = ∂t ∂xi ∂t ∂xi

 ∂h  Γ T + S  ∂xi 

radiation

,

(45)

where h is the total enthalpy of the mixture, ΓT is the turbulent diffusivity, and 〈S〉radiation is the source term due to thermal radiation. In the total enthalpy formulation temperature T must be deduced implicitly from the total mixture enthalpy, i.e. h = ∑ Yi hi = ∑ Yi  ho, i + ∫ c p, i (T )dT  ,

(46)

where Yi and hi are the mass fraction and the total enthalpy of species i in the mixture, respectively, and ho,i and cp,i are the enthalpy of formation and the specific heat of species i. Also, in this formulation the heat release due to chemical reactions is included within the enthalpies of formation. The radiative source term can be interpreted as a local source/sink of thermal energy, due to the local volumetric rate of radiant energy gain/loss owing to the difference between emission and absorption. This source term can be expressed as [1] ∞



Sradiation = −∇ ⋅ q = ∫ kh  ∫ R

0

 I dΩ − 4 πI bh  dh , 4π h

(47)

278

Transport Phenomena in Fires

where the spectral absorption coefficient kh may be a function of temperature T and species concentrations of the radiating medium Y _ , and the radiative intensity Ih requires the solution of the RTE, eqn (10), which for an absorbing-emitting but nonscattering gas reduces to, ( sˆ ⋅ ∇)I h = kh ( I bh − I h ),

(48)

where the first term on the right-hand side represents augmentation due to emission and the second term is attenuation due to absorption. The time constant for radiative transport in an enclosure of characteristic dimension of a meter is of the order of a few tens of nanoseconds. This is considerably faster than the fastest chemical reaction, whose time-constants are typically of the order of microseconds. Therefore, the transient term in the radiative equation of transfer has been neglected [1]. To include radiation effects in conventional turbulence calculations, eqns (47) and (48) need to be time averaged, resulting in S

radiation

∞ = ∫  ∫ kh I h dΩ − 4π kh I bh  dh , 0  4π 

(sˆ ⋅ ∇) I h = kh I bh − kh I h .

(49)

(50)

Solving the averaged governing equations, one can obtain the Favre-averaged enthalpy and species concentration, h and Y , and the time-averaged spectral radiation intensity 〈Ih〉. However, due to their strongly nonlinear dependence on the temperature and species concentrations, 〈khIh〉 does not equal kh(h , Y )〈Ih〉 and 〈khIbh〉 does not equal kh(h , Y )Ibh(h , Y). In other words, these two terms are unclosed. 〈khIh〉 represents a correlation between the spectral absorption coefficient and the spectral incident intensity, and 〈khIbh〉 represents a correlation between the spectral absorption coefficient and the spectral blackbody intensity. Complete information of the statistics among the composition variables is needed in order to determine these correlations. For the convenience of later discussion, these two correlations are loosely called ‘absorption coefficient− incident intensity correlation’ and ‘absorption coefficient−blackbody intensity correlation’. The time-averaging procedure can be applied to any solution technique for the RTE, as described in earlier sections, and different unclosed terms may arise for different chosen spectral models and solution methods in the averaging process. However, all of them can be categorized into two groups: (a) correlations that can be calculated from scalars f directly or indirectly only, and (b) correlations that cannot. The set of scalars f is defined as f = (Y , h) = (f1, f 2, ...., fs),

(51)

where s is the total number of scalar variables (number of species plus the enthalpy) and the last scalar, fs, is reserved for enthalpy. The unclosed term 〈khIbh〉 belongs to group (a), since both kh and Ibh are functions of variables in set f only. The unclosed terms 〈khIh〉 belong to group (b), because Ih is involved in this correlation, since it is not a function of only the local values of f. One of the most common approximations made in the open literature for TRIs is the optically thin Fluctuation approximation (OTFA) (sometimes also called optically thin eddy approximation) as described by Kabashnikov and Myasnikova [58], who suggested that if the mean free path for radiation is much larger than the turbulence length scale, then the local radiative intensity is only weakly correlated with local absorption coefficient, i.e. 〈 kh I h 〉 〈 kh 〉〈 I h 〉,

(52)

Radiative Heat Transfer in Fire Modeling

279

in which case 〈kh〉 is loosely defined as ‘absorption coefficient self-correlation’. The rationale behind these assumptions is that the instantaneous local intensity at a point is formed over a path traversing several turbulent eddies. Therefore, the local intensity is weakly correlated to the local radiative properties. Kabashnikov and Myasnikova provided several conditions for the validity for the thin-eddy approximation. In general however, the thin-eddy approximation depends on the assumption that the optical thickness of the turbulent eddies is small, kl << 1,

(53)

where l is the turbulent eddy length scale. The validity of this assumption depends on the eddy size distribution and the radiation properties of the absorbing gases. In a numerical simulation of combustion chambers, Hartick et al. [59] showed that, although the thin-eddy assumption may not be valid over some highly absorbing parts of the spectrum, these spectral zones affect the total radiation exchange only slightly, thus allowing straightforward application of the thin-eddy assumption in their simulation. On the other hand, this assumption may well be violated in strongly sooting fires. Since to date there are no reliable soot models available (particle size, number, shape, makeup, radiative properties, etc.) only nonsooting flames will be considered in this section and, thus, the thin-eddy assumption will be employed here. As a result, all correlations needed to capture TRIs belong to group (a), requiring only correlations between composition variables. Note that this slightly approximate treatment of TRIs makes the radiative source term evaluation equivalent to the evaluation of the chemical reaction source term. Consequently, any method devised for the averaging of the chemical reaction source is immediately applicable also to TRI determination. 6.2 Assumed-PDF investigations As mentioned earlier, accurate predictions of TRIs require the modeling of and solution for many turbulence moments. This can be avoided by specifying, rather than calculating, the PDF for turbulence fluctuations, and this was done in all earlier (as well as some present day) work. Through a Taylor series expansion of the Planck function, Cox [60] estimated that the contribution from temperature fluctuations to radiative emission may dominate the contribution from the mean temperature field when the temperature fluctuation intensity exceeds approximately 40%. Faeth et al. [61] showed through experiments that actual radiative fluxes may be two times or more larger than would be expected based on the mean values alone. The earliest numerical calculations were undertaken by Germano [62]. His calculations were based on the assumptions that the scalar fluctuations are random and the medium is gray. Although far from the truth, calculations based on these assumptions did prove the importance of TRIs. Pearce and Varma [63] performed similar calculations for the 4.3 µm CO2 band. In their approach, the optical paths were broken up into a number of statistically independent homogeneous segments. The two-point statistical correlations were then described by assuming exponential correlation functions. This approach was later adopted by Chan and Chern [64] and by Faeth et al. [61]. The method is usually used to calculate radiative intensity along a given line of sight. Although this method is effective in showing the importance of TRIs, it cannot be used in a coupled flow−radiation calculation. A similar line-of-sight radiation calculation has been performed by McDonough and Mengüç [65, 66], in which chaotic-map theory was used to model temperature and species concentration fluctuations. Song and Viskanta [67] have investigated a turbulent premixed flame inside a two-dimensional furnace. While TRIs were considered, in order to obtain closure for their governing equations, correlation functions for gaseous properties had to be assumed. Gore et al. [68] and Hartick et al. [59]

280

Transport Phenomena in Fires

applied this approach to the study of diffusion flames, in which extended k−e−g models were used. Ripoll [69] outlined how TRIs can be evaluated using the simple M1-radiation solver (a somewhat more sophisticated derivative of the popular spherical harmonic P1-method) together with Gaussian assumed PDFs. A more advanced approach was taken by Coelho et al. [70] to model Flame D from the International Workshop on Measurement and Computation of Turbulent Nonpremixed Flames [71], for which experimental data are posted on a website. They used a Reynolds stress model for turbulence modeling, the laminar flamelet model for combustion, the discrete ordinates method as radiation solver, together with the SLW model for nongray combustion gas properties, and the assumed PDFs of Song and Viskanta [67] for the TRIs. Coelho [72] also calculated TRIs for a nonpremixed methane−air turbulent jet flame based on experimental data for temperature and concentrations and using a stochastic semi-causal model, reporting a nearly 50% increase in radiative heat loss. 6.3 Composition PDF methods PDF methods have the unique feature that many nonlinear interactions can be treated exactly [73], and have been widely used in the modeling of reacting flows in the absence of radiation, to evaluate chemical source terms, no matter how complicated they are [74, 75]. Since the radiative source term in the presence of turbulence is mathematically equivalent to the chemical source term and, thus, can be evaluated exactly by these methods, PDF methods have been introduced to the study of TRIs by Mazumder and Modest [76] and by Li and Modest [77, 78]. Mazumder and Modest [76] employed the velocity−composition joint PDF method in their simulation of a bluff body combustor and found inclusion of the absorption coefficient−temperature correlation alone may increase radiative heat flux by 40−45%. For the purpose of capturing TRIs, the composition PDF method is as rigorous as the velocity−composition joint-PDF method, but computationally more robust and more efficient. Its use for the study of TRIs was demonstrated by Li and Modest [77, 78]. PDF methods have initially been offered as alternatives by Dopazo and O’Brien [79], and later were further developed by Pope [73], Kollmann [74] and Dopazo [75]. Pope [73] has given a comprehensive description of the methodology; in particular, the stochastic modeling and the particle simulation technique, two key elements for the successful application of the PDF approach, are introduced and demonstrated as a viable alternative to conventional deterministic closure models and numerical techniques. The composition joint-PDF equation is not a self-contained model. The mean momentum equations must be solved for velocity, u , and a turbulence model (say, the k−e method) is needed to determine both turbulent diffusivity and the mixing rate used in the stochastic mixing model. The philosophy of the composition PDF approach is to consider the thermo-fluid quantities as random variables and consider the transport of their PDFs rather than their finite moments. The advantage of this method is the fact that for any quantity, Q, as long as it is a function of the scalar field, f, e.g. the mixture density r, specific heat cp, gas absorption coefficient k, chemical reaction source term Sreaction, radiative emission Ib, etc., its mean can be evaluated directly from the PDF as follows: ∞

〈Q 〉 = ∫

0

〈 r 〉Q (f ) = ∫

∞

0

f (y)Q(y)dy,

(54)

f (y) r(y)Q(y)dy.

(55)

Radiative Heat Transfer in Fire Modeling

281

In these equations, y represents the composition space variable, y ≡ (y1, y2,…, ys), and f(y) is defined as the probability density of the compound event f = y (i.e. f1 = y1, f2 = y2,…, fs = ys), so that, f (y)dy = Probability(y ≤ f ≤ y + dy).

(56)

The composition PDF, f(y), defined informally by eqn (56), is the simplest form of the PDF methods, since it carries information only about the scalar variables, f. Also, it governs the probability distribution only at a single point. However, since it contains all of the statistical information about the scalars, its determination is in many ways more useful than that of the mean values. In a general turbulent reactive flow, the composition PDF is also a function of space, x, and time, t. The transport equation for the composition PDF has been derived by Pope for nonradiating reactive flows [73]. For radiating reactive flows, the transport equation for the mass density composition PDF, F(y, x, t) = r(y, x, t)f(y, x, t), can be similarly derived, leading to ∂F ∂ ∂ ∂ + [u iF ] + [ Sreaction (y)F ] = − [〈u ′′| y 〉F ] ∂t ∂xi ∂ya ∂xi i +

 ∂  1 ∂Jia ∂    y F−  〈 Sradiation /r y 〉F  , y ∂ya  r ∂xi ∂  s  

(57)

where i and a are summation indices in physical space and composition space, respectively; and variables with tildes and double primes are the Favre means of the variables and the fluctuations about them, respectively. The notation of 〈A|B〉 is the expectation of the conditional probability of event A, given that event B occurs. On the left-hand side of eqn (57), the first two terms represent the rate of change of the PDF when following the Favre-averaged mean flow. The third term is the transport of the PDF in composition space by chemical reactions. The processes represented by these terms are accounted for exactly. In contrast, the terms on the right-hand side need to be modeled. The first term on the right represents transport in physical space due to turbulent convection. Since the joint composition PDF contains no information on velocity, the conditional expectation of 〈u≤i |y〉 needs to be modeled. Generally, a gradient-diffusion model with information supplied for the turbulent flow field by a flow solver is employed [73], −〈ui′′| y 〉 F Γ T

∂F , ∂xi

(58)

where ΓT is the turbulent diffusivity and is usually estimated by ad-hoc turbulent closures such as [73]: ΓT =

c m 〈 r 〉k 2 sf e

,

(59)

where k, e, cµ, and sf are, respectively, the Favre-averaged turbulent kinetic energy, Favre-averaged dissipation rate of turbulent kinetic energy, a modeling coefficient in a standard two-equation k−e turbulence model, and the turbulent Schmidt or Prandtl number. Such gradient transport models are, of course, approximate and subject to many objections, especially when applied to

282

Transport Phenomena in Fires

variable-density reactive flows [80]. For example, it detracts from its usefulness for premixed combustion where counter-gradient diffusion is known to occur [81], but it should perform well in fire modeling. The second term on the right-hand side of eqn (57) represents transport in scalar space due to molecular mixing. This term has been found to be crucial, and many mixing models have been proposed, such as the interaction-by-exchange-with-the-mean (IEM) model [82], also known as the linear mean square estimation model, Pope’s particle-pairing model [73], and the Euclidean minimum spanning trees (EMST) mixing model proposed by Subramaniam and Pope [83]. The IEM model has the effect of moving the value of the instantaneous composition toward the mean composition at a controlled rate and thus reduces the distribution in the composition values. In homogeneous isotropic turbulence, the IEM model predicts that the shape of the composition PDF does not always relax to a Gaussian distribution, but instead preserves the shape of the initial PDF [73]. The particle-pairing model can also be used in PDF calculations but its computational requirements are higher and their superiority has not been shown for reacting flow simulations. The EMST mixing model is perhaps the most advanced scheme to date, in which Euclidean Minimum Spanning Trees are employed to construct the definition of neighboring particles in composition space for particle interaction. For the results presented later the IEM model was used: 1 J ia 1 y Cf w(ya − f a), r ∂xi 2

(60)

where Cf is a model constant and w is the mixing frequency, which is calculated using the k−e model, i.e. w = e/k. The third term on the right-hand side of eqn (57) represents the contribution from thermal radiation. A photon Monte Carlo method can potentially account for all TRIs including the absorption coefficient−incident intensity correlation. A first attempt in that direction was made by Tessé et al. [84], who calculated radiative transfer for a sooty, turbulent ethylene−air diffusion jet flame using a photon Monte Carlo method together with a CK model for the gas properties. However, in their TRI model the radiative properties of the assumed homogeneous turbulent structures are randomly obtained from an ad-hoc multidimensional PDF of the reaction progress variable, mixture ratio and the soot volume fraction, and smaller scales of turbulence were neglected. They concluded that TRIs yielded an increase of about 30% in radiative heat loss. All other RTE solvers require the use of the optically thin eddy approximation. If the FSSK method of the previous section is employed, reduces the radiative source to −

M ∂   = ∑ w j ∂  k j u(4 πa j I b − G j )/r y F  S r y F / radiation   ∂ys  ∂ys  j =1 M

= ∑ 4 πw j k j j =1

M ∂ ∂  (ua j I b /r )F  − ∑ w j k j uG j /r y F  ,   ∂ys ∂ y s j =1

(61)

where G = ∫4π IdΩ is the incident radiation, and the spectral integration (over g) was replaced by Gaussian quadrature with M quadrature points and weights wj. The first term on the right-hand side of the above equation represents radiative emission and can be considered exactly. The second term can be closed by adopting the optically thin eddy approximation, as discussed earlier. This leads to M

M ∂  ∂ (u〈G j 〉/r )F  . uG j /r y F  ≈ ∑ w j k j   ∂ y s s j =1

∑ w j k j ∂y j =1

(62)

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As a result, the modeled transport equation for the composition mass density PDF can be written as M ∂F ∂ ∂ ∂ (ua I b /r)F  [u iF ] + [ Sa, reaction (y)F ] − ∑ 4 πw j k j +  ∂t ∂xi ∂ya ∂ys  j j =1

=

∂ ∂x i

M  ∂F  1 ∂ ∂ (u〈G j 〉/r )F  , [(ya − f a)F ] − ∑ w j k j Γ T  + Cf w   x 2 y y ∂ ∂ ∂ a i  s  j =1

(63)

which is closed and contains all necessary information for all composition variables. When compared to the exact transport equation, it is seen that the terms on the right-hand side of the equation are contributions due to conditional expectations of velocity fluctuations, molecular mixing, and absorption of radiation, respectively. 6.3.1 Hybrid finite volume/PDF Monte Carlo method The composition PDF can be solved, in principle, by traditional finite volume methods. This approach has been taken by Janicka et al. [85] for a jet diffusion flame. But, generally, the PDF is a function of a large number of independent variables (three spatial variables, one temporal variable and s composition variables), and finite-difference or finite-element algorithms become prohibitively expensive because numerical effort for solution grows rapidly with the number of independent variables. Monte Carlo approaches for the solution of the PDF equation, so-called PDF/Monte Carlo methods, have been developed by Pope and coworkers [73]. The basic idea is to represent the PDF by a sufficiently large number of stochastic particles. Each particle can be interpreted as an independent realization of the flow or as a delta function discretization of the PDF and it evolves in time according to a set of stochastic differential equations. Pope [86] has shown that there is a one-to-one correspondence between the modeled PDF equation, particleevolution equations and the modeled Eulerian governing equations for the field means. For the modeled mass density PDF, eqn (63), the corresponding particle equations for location, x, and scalar quantities, f are governed by the following stochastic differential equations: d x (t ) = [u + ∇Γ T /〈 r 〉]x ( t ) dt + [2Γ T /〈 r 〉]1x/2(t ) dW ,

(64)

dfa (t ) = Sa, reaction dt − 0.5Cf (fa − fa )e/kdt M

− das ∑ w j k j u (4 πa j I b − 〈G 〉) /r dt ,

a = 1, … , s.

(65)

j =1

Here the variables with an asterisk refer to the values of a Lagrangian particle and W is an isotropic vector Wiener process, which is used to mimic a turbulent diffusion process. The composition PDF contains all information about scalar variables, and the transport equation for any scalar mean or its higher moments can be derived directly from it. For example, the governing equation for any scalar mean, fa can be obtained by multiplying the PDF transport equation, i.e. eqn (63), by the composition space variable ya, followed by integration over the entire y space. That is, ∂[ 〈 r 〉fa ] ∂[〈 r 〉u i fa ] fa  ∂  + = Γ T  + 〈 r 〉 S a, reaction ∂t ∂xi ∂xi  ∂xi  M

+ das ∑ w j k j 〈u〉〈G j 〉 − 4π〈ua j I b 〉  . j =1

(66)

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Transport Phenomena in Fires

In PDF/Monte Carlo simulations, instead of solving the partial differential equation (eqn (66)) directly, as is done in conventional moment methods, a large number of particles are traced according to eqns (64) and (65). The PDF is then obtained approximately as a histogram of the particles’ properties in sufficiently small neighborhoods in physical space. The Monte Carlo scheme is employed for integration of the set of particle equations and computation of field means from particle properties. Conventional particle-tracing schemes use a constant time step for the integration of particle evolution equations, which tends to waste computer time, since, to capture sharp gradients, very thin cells need to be used in such regions. The time step would then be dictated by the small cells, leading to an extremely small value, which would have particles in large cells move only a short distance in a single time step compared to their cell size. Li and Modest developed an adaptive time-step-splitting scheme for the integration of particle equations [87]. The main idea is to ensure that each particle in the flow field does not change its physical position too much or too little in one move in comparison with the cell size in which it resides. The global time step for each individual particle (i.e. the time step between updates of the global velocity and scalar fields) is divided into several sub-steps. Thus, from time level k to level k + 1, every particle experiences a series of moves, interacting with all cells it passes through rather than only the one in which it ends up at the end of the time step. In regions with strong gradients particles update their properties more frequently, and particles always reflect the local flow-field information. Another complication of using cell systems with large cell-to-cell volume disparities is the need to control the number of particles over cells since small cells tend to contain too few particles and large cells tend to hold too many. This imbalance makes the particle tracing inefficient because tracing more particles than necessary in a cell wastes computer time, while tracing too few leads to large statistical errors (since the error scales as the inverse of square root of the number of particles in a cell). This imbalance problem is particularly severe when solving a problem in cylindrical coordinates, in which cells near the axis tend to suffer from low particle counts due to their small volume while outside cells tend to hold too many particles. In many such flows, sharp gradients of the flow field occur near the axis, requiring cells to be smaller still, making things even worse. The need to control particle number density has been realized for some time [73], but has been given little attention in the literature. Subramaniam and Haworth [88] have proposed a particle splitting and combination scheme to control the particle number density over cells, and a similar scheme has been developed by Li and Modest [87]. The number of particles in each cell is checked after every global time step. If the number in one cell is less than a prescribed minimum, a particle splitting process is initiated, and if the number of particles exceeds the prescribed maximum, a particle combination process is carried out. During this process, the particles’ distribution and their number may change before and after the particle splitting and combination procedure, but the two sets of particles should be equivalent in a statistical sense, and field means (including mass, scalars and their higher-order moments) extracted from particle field should be preserved. The PDF/Monte Carlo (PDF/MC) method must be coupled with a flow solver for the calculation of the velocity field. In the resulting hybrid finite volume/PDF Monte Carlo method the velocities, pressure, turbulent kinetic energy and its dissipation rate, are solved in the flow solver and composition variables, including species concentrations and temperature, are solved within the PDF/MC solver, while the two codes communicate with each other during every iteration. For the results presented here a commercial code, FLUENT [89], was used as the flow solver. The entire numerical simulation can be divided into two parts: the initialization part and the main iteration loop. In the initialization stage, the flow solver is run first, in which the mesh system, which is used in both the flow solver and the PDF/MC scalar solver, is generated, the problem is set up and a flow field is prepared for the PDF/MC scalar solver. The energy equation and the set

Radiative Heat Transfer in Fire Modeling

285

of species concentration equations are not solved in the flow solver. The PDF/MC solver obtains the mesh layout from the flow solver and then prepares connectivity information of the mesh for later use in the particle-tracing process. The PDF/MC solver and the flow solver run consecutively during every iteration. Using the velocity field information, the PDF/MC solver obtains temperature, species concentration and density fields. It passes the density field to the flow solver and the flow solver obtains an updated flow field. The updated flow fields are fed back to the PDF/MC solver and a new iteration is started. Due to the nature of the PDF/MC method, even for a statistically stationary problem, many global iterations are required until a statistically stationary solution is obtained. 6.3.2 Chemical reaction submodel Although PDF methods allow the use of detailed chemical reaction mechanisms in principle, computational intractability has limited their application. In practice, reduced mechanisms are often used in the PDF calculations. A wide range of reduced mechanisms of chemical reactions for hydrocarbon fuels is available in the literature [90], and a simple methane flame with a singlestep skeletal mechanism was used to obtain the results presented here. This takes the form: CH 4 +2O2 → CO2 + 2H 2 O.

(67)

Westbrook and Dryer [91] provided an Arrhenius relationship for the reaction rate of methane as: d[CH 4 ] = − A exp( − Ea /Ru T )[CH 4 ]a [O2 ]b , dt

(68)

where the quantities within square brackets represent molar concentrations; Ru is the universal gas constant, and Ea is the activation energy of the methane. A, a, and b are constants in the general Arrhenius equation, which may be obtained from Westbrook and Dryer. The main reasons for choosing a methane flame are (i) the availability of experimental data for comparison, and (ii) the nonsooting nature of such flames, because accurate soot-radiation predictions do not yet appear possible. However, all submodels described in this section, including the spectral model for radiation, can immediately be applied to sooting flames, as soon as accurate predictions of soot-radiative property fields become available. 6.3.3 TRIs in jet flames Turbulence−radiation interactions exists in both reactive and nonreactive flows. A nonreactive hot mixture of radiatively participating species, typically carbon dioxide and water vapor, may be found in the exhaust sections of almost all combustors, including the regions above fires. Because scalar fluctuations in such nonreactive flows are usually much smaller than in flames, it is commonly believed that the effects of TRIs are negligible in such media. Mazumder and Modest substantiated this belief by conducting a series of numerical simulations [92]. An important conclusion of their study was that the role of TRIs depends largely on how the temperature fluctuations correlate with the concentration fluctuations of carbon dioxide and water vapor. In most nonreactive flows the fluctuations are found to be uncorrelated and, thus, TRIs tend to be negligible. In reactive flows, when a blob of a fuel−air mixture is burnt, it produces high temperatures and high concentrations of carbon dioxide and water vapor locally, resulting in a positive temperature−concentration correlation. Such a positive correlation has a profound impact on radiation calculations, because flame emission is always enhanced. The current study is only focused on reactive flows.

286

Transport Phenomena in Fires

As mentioned earlier, the present section will consider only nonsooting flames, even though many real flames have significant amounts of soot, often dominating radiative heat transfer. Very preliminary calculations by Mazumder [93] appear to indicate that TRIs decrease in the presence of soot and may even become negligible. This issue needs to be re-examined as soon as accurate soot models become available: the methods described here can be used without change once reliable predictions of soot radiative property fields become possible. Flame optical thickness has a strong influence on radiative heat transfer. Three jet flames with different optical thickness will be compared, where flame optical thickness is defined as (69) t = kP L, where kP is an average Planck mean absorption coefficient of the participating medium, i.e. the combustion products of H2O and CO2, and L is the flame length. For turbulent jet flames, flame length is approximately a linear function of jet diameter [94] and, in this study, is estimated to be L = 40 dj. The baseline flame is Sandia’s Flame D, which has been well documented [95, 96]. Flame optical thickness for this flame is 0.237 by eqn (69). The other two considered (artificial) flames were derived from Flame D by doubling and quadrupling the jet diameter, bringing their flame optical thickness to 0.474 and 0.948, respectively. For future reference these three flames will be denoted as kL.1, kL.2 and kL.3, respectively. In order to study TRIs, three different scenarios were considered for each flame. In the first scenario, radiation is completely ignored in order to study the importance of radiation in flame simulations in general. In the second and third scenarios, radiation is considered but TRIs are ignored and considered, respectively. The importance of TRIs can be assessed by comparing numerical results from these two scenarios. By ignoring TRIs, it is implied that the two unclosed terms 〈u〉 and 〈uaIb〉 are evaluated based on the cell means; while by considering it, these two terms are treated exactly. When comparing numerical results of these three scenarios the most obvious difference is that the flame gets colder and colder as radiation without TRIs and radiation with TRIs are considered. This is universally true for every flame although the trend is more obvious for flames with large optical thickness. Flame peak temperatures for different flames are tabulated in Table 1. To facilitate the discussion, drops in temperature as a result of considering radiation with/without TRIs are also listed in the table. While the peak temperature drops only 64 and 18 K for a small optical thickness flame, it drops 145 and 64 K, respectively, for a medium flame, and by 327 and 117 K for a large optical thickness flame. While peak temperature applies only to a single point, it usually characterizes the entire temperature field. Figure 9 shows the computed temperature contours for Flame kL.3. To examine the differences in more detail, temperature profiles at the axis are shown in Fig. 10. From these figures, it is seen that temperature levels have fallen globally as a result of consideration of radiation and TRIs (with the exception of some radiative preheating just ahead of the flame front). From these comparisons, it is clear that radiation cannot just be conveniently ignored in fire simulations, since this would lead to overpredicted flame temperature, which is especially true for large flames, such as Flame kL.3. Moreover, TRIs account for about one third of the total temperature drop due to radiation and, thus, TRIs generally cannot be neglected if radiation is going to be considered in a turbulent flame simulation. The important quantities that describe the overall radiation field of a flame are the total radia⋅ ⋅ tive emission (Qem), the net radiative heat loss (Qnet) from the flame, and its normalized variable, the ‘radiant fraction’ (frad), which is defined as the ratio of the net radiative heat loss to the total heat released during combustion, i.e. frad ≡

Q net , m fuel∆H comb

(70)

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287

Table 1: Computed flame peak temperature. Flame

Tno rad (K)

Tno TRI (K)

TTRI (K)

∆Trad (K)

∆TTRI (K)

2165 2161 2169

2101 2016 1842

2083 1952 1725

−64 −145 −327

−18 −64 −117

κL.1 κL.2 κL.3

without considering radiation

(K) 2000 1800 1600 1400 1200 1000 800 600 400

without considering TRI

considering TRI 10

r/dj

8 6 4 2 0

0

10

20

30

40

50

60

70

x/dj

Figure 9: Temperature structure for Flame kL.3. where m⋅ fuel is the mass flow rate of fuel, and ∆Hcomb is the heat of combustion. In every simulation, these quantities were calculated and the results are shown in Table 2. As the flame’s optical thickness is increased, the flame’s radiant fraction increases quickly and the flame gets colder as discussed earlier. In the current study optical thickness was varied by changing the size of the flame. The total potential chemical energy that a fluid particle can release is fixed. Thus, as the flame gets larger, the flow residence time becomes longer, which implies that an average fluid particle will lose more energy through radiation. As a result, the radiant fraction increases as flame size increases. The radiant fraction is only about 5% for Flame kL.1, but as high as 18% for Flame kL.3. This also explains why temperature levels drop more significantly in optically thick flames. The table also shows how the TRIs enhance radiative heat transfer. For Flame kL.1, the net radiative heat loss from that flame is increased from 0.534 kW to 0.798 kW, indicating a 49% increase as a result of TRIs. In contrast, total radiative heat loss increases by 32% for Flame kL.2 and by only 4.6% for Flame kL.3 as a result of considering TRIs. As the flame gets optically thicker, the actual values of radiative heat loss, ignoring TRIs and considering TRIs, become

288

Transport Phenomena in Fires

2400 L. 1, no rad κL. 1, no TRI κL. 1, with TRI κL. 2, no rad κL. 2, no TRI κL. 2, with TRI κL. 3, no rad κL. 3, no TRI κL. 3, with TRI

Temperature (K)

2000 1600 1200 800 400 0

10

20

30 40 50 Axial location (x/d j )

60

70

Figure 10: Centerline temperature profiles for kL -series flames.

Table 2: Summary of radiation calculation results.

Flame kL.1 kL.2 kL.3

Without TRIs ⋅ ⋅ Qem Qnet (kW) (kW) 0.624 4.12 21.68

0.534 2.98 12.12

With TRIs 1 f rad

2

2 f rad

(%)

Qem (kW)

Qnet (kW)

(%)

3.05 8.51 17.3

0.928 5.33 20.94

0.798 3.92 12.68

4.56 11.2 18.1

1

frad – frad ______ 1 frad (%)

49 32 4.6

closer and closer. This does not mean that considering TRIs is less important for optically thick flames. Radiative heat loss is strongly dependent on flame temperature and the temperature level is greatly decreased as a result of TRIs. Because of the different temperature levels, comparison of only the radiative loss quantities would be misleading. Li and Modest investigated several other aspects of TRIs not shown here. In one study [97] they took a frozen stochastic particle field to study the impact of different contributions to the overall TRIs. They found that, contrary to early belief, the temperature self-correlation 〈Ib〉/Ib(〈T〉) is not the dominating part of TRIs, especially for nongray media. Instead, they found that the absorption coefficient − Planck function correlation, 〈uaIb〉/uaIb(〈T〉,〈Y〉), by far dominates the TRIs. Thus, the complicated interaction of temperature as well as concentration fluctuations must be accounted for. Effects of various flame parameters, such as Reynolds number Re, Damköhler number Da (the ratio of flow and chemical reaction time scales), Froude number Fr (buoyancy effects) and flame

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optical thickness kL were also investigated [98]. It was found that TRIs are rather insensitive to changes in Da and Fr. As expected TRIs are strongly influenced by flame optical thickness: radiant heat loss and TRIs rapidly increase with increasing optical thickness. The opposite is true for increasing Re: if optical thickness kL is held constant, this implies increased velocity and, thus, decreased flame residence time, resulting in less radiation losses as well as TRIs. 6.4 Direct numerical simulations of TRIs Since PDF simulations include a number of modeled (i.e. approximate) terms, their accuracy must be verified, and this is most conveniently done through DNS. In addition, PDF methods in general use simplified RTE solvers, which cannot resolve the absorption coefficient−intensity correlation 〈kI〉 (an exception being the recent work by Tessé et al. [84], who used a complicated Monte Carlo scheme to capture TRIs, including 〈kI〉). Thus, DNS can also be used to quantify 〈kI〉 and test the accuracy of the optically thin eddy approximation. Wu et al. [99, 100] considered a one-dimensional premixed flame with single-step finite-rate Arrhenius chemistry (in terms of temperature and a simple progress variable). The underlying DNS code uses third order Runge−Kutta time integration, and a sixth-order compact scheme for spatial discretization. To obtain the radiative source term for the energy equation a high-order photon Monte Carlo scheme was developed, commensurate with the order of the underlying DNS code. Emission and absorption of photon bundles, using the energy partitioning method [1], was done via an adaptive scheme, which uses up to sixth-order polynomials in regions of strong gradients. Using the same DNS and photon Monte Carlo code Deshmukh et al. [101] simulated the transient development of a nonpremixed flame. In these early studies only a gray, nonscattering medium was considered with a temperature and progress variable dependent Planck-mean absorption coefficient, fashioned after the water vapor model suggested in [71]. For the cases studied radiation required about ten times the CPU time of the underlying simple chemistry DNS code. One of the advantages of a Monte Carlo code is that, when nongray effects are considered, the radiation effort can be expected to grow very little. On the other hand, if detailed chemistry is taken into account one may expect the DNS calculations to dominate the CPU requirements (and demanding a computer of great power). 6.5 TRI effects in nonpremixed flames To give a qualitative picture of how TRI effects can be extracted from DNSs, some of Deshmukh et al. [101] results will be presented here. In their work a statistically one-dimensional, nonstationary (decaying), turbulent nonpremixed system is considered (fuel occupying one half of the computational domain, oxidizer the other half). The simulations were carried out using a 65 × 64 × 64 computational grid, and tracing 107 photon bundles per time step. A parallel implementation using MPI was used to speed up the photon Monte Carlo calculations. Mean quantities were estimated by averaging over all grid points in the y−z plane for each x-location for this statistically one-dimensional configuration. The simulations proceed from the initial condition; fuel and oxidizer react to form products and, due to the imposed turbulence, the flame wrinkles. Results are shown for a nondimensional time t/t = 1.36 (where t is the initial eddy turnover time), at which time the flame covers roughly one third of the computational domain as seen in Fig. 11, which shows the temperature (and thus also the flame thickness) at that time. Figure 12 shows that the effects of emission TRIs, on a percentage basis, are relatively unaffected by optical thickness, similar to the prediction of Li and Modest using PDF methods [97, 102]. Fluctuations of

290

Transport Phenomena in Fires κP L = 10.0 κP L = 1.0 κP L = 0.1 No radiation

8

6

4

–3

–2

–1

0

1

2

3

x

Figure 11: Mean temperature along the x-direction for three values of optical thickness at t/t = 1.36 and for the case without radiation.

κP L = 10.0 κP L = 1.0 κP L = 0.1

<κP′Ib′>/<κP>

2

1.5

1

0.5

0

–3

–2

–1

0

1

2

3

x

Figure 12: Absorption coefficient−Planck function correlation factor along the x-direction for three values of optical thickness at t/t = 1.36. temperature and species are maximum in the diffusion region due to the transport of hot products from the flame, the preheating of reactants due to the chemical heat release and the radiative emission. Since the plot is an instantaneous snapshot in time, the two peaks are not equal, indicating the flame wrinkling more on one side than the other. Also, the peak values decrease with

Radiative Heat Transfer in Fire Modeling

291

increasing optical thickness. This indicates that in addition to decrease in temperature, the fluctuations are smoothened out with increasing optical thickness. Absorption TRIs, in contrast to emission TRIs, is strongly influenced by optical thickness, as seen in Fig. 13: being negligible for small optical thickness, it strongly increases as optical thickness rises. For very large optical thickness, absorption TRIs tend to counter emission TRIs. Thus, net TRIs are expected to be maximum at some intermediate optical thickness, and this is demonstrated in Fig. 14. κP L = 10.0 κP L = 1.0 κP L = 0.1

2.5

<κP′G′>/<κP>

2

1.5

1

0.5

0

–3

–2

–1

0

1

2

3

x

[<∇⋅qrad>TRI – <∇⋅qrad>NO TRI]/<κPIb>NO TRI

Figure 13: Absorption coefficient−intensity correlation factor along the x-direction for three values of optical thickness at t/t = 1.36.

κP L = 10.0 κP L = 1.0 κP L = 0.1

2.5

2

1.5

1

0.5

0 –1

–0.5

0

0.5

1

x

Figure 14: Net TRIs along the x-direction for three values of optical thickness at t/t = 1.36.

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Transport Phenomena in Fires

The DNS results presented here are very preliminary and use a highly idealized model. Qualitatively, however, they nicely complement and corroborate the PDF model conclusions. For a meaningful comparison it must be remembered that Figs 12 and 13 show local TRI correlation values, while those given in Tables 1 and 2 are flame averages.

7. Summary In this chapter a review of modern spectral methods for the prediction of radiative heat transfer rates in a fire with strongly nongray combustion media has been given and a brief account of the important emerging field of TRIs has also been provided. While the WSGG method yields respectable accuracy, it depends on inaccurate and incomplete experimental data. That, together with the emergence of more modern approaches, has caused its popularity to wane in favor of statistical narrow band analyses and, most recently, FSKs and its simplest implementation, the SLW method. In the FSK method the absorption coefficient of the combustion medium is reordered across the entire spectrum into a smoothly varying, monotonically increasing function. FSKs for arbitrary mixtures can efficiently and accurately be assembled for a large array of combustion applications, but at a minuscule fraction of the computational cost. The computational PDF method was shown to be a powerful tool to study radiating reactive turbulent flows. The method is able to treat TRIs in a rigorous way: many TRI terms that remain unclosed in conventional moment methods can be calculated exactly. Simulations show that, by ignoring TRIs, radiation heat losses are always severely underpredicted and, consequently, temperature levels are generally overpredicted substantially. In addition, numerical results show that, in order to determine TRIs, consideration of the temperature self-correlation alone is not sufficient (although nonlinearity of the Planck function with temperature is the severest among other functions); the absorption coefficient−Planck function correlation must also be considered. Finally, preliminary DNS calculations show that while the common optically thin fluctuation assumption is indeed valid for most turbulent flames, absorption TRI must be accounted for in the presence of optically thick eddies.

Nomenclature a b d d f fv g h I k k l L m n nˆ

weight function for WSGG, SLW and FSK methods spectral line half-width (cm−1) spectral line spacing (cm−1) soot diameter (cm) k-distribution function (cm); probability density function volume fraction cumulative k-distribution enthalpy (J/kg) radiative intensity (W/m2 sr) absorption coefficient variable (cm−1); turbulent kinetic energy absorptive index turbulent length scale (m) geometric length (m) complex index of refraction refractive index unit normal vector

Radiative Heat Transfer in Fire Modeling

N p q s sˆ, sˆ ¢ S T u x x, x Yi

number of gray gases pressure, bar radiative heat flux (W/m2) distance along path (m) unit direction vectors spectral line strength; source function temperature (K) scaling function for absorption coefficient, velocity scattering size parameter mole fraction (vector) mass fraction (of species i)

Greek symbols b b e e h ∆h l k f y Φ r ss t t w Ω

extinction coefficient (cm−1) spectral line overlap parameter gas emissivity, or wall emittance dissipation rate of turbulent energy wavenumber (cm−1) narrow band wavenumber range (cm−1) wavelength (µm) absorption coefficient (cm−1) composition variable vector composition space variable vector scattering phase function density (kg/m3) scattering coefficient (cm−1) gas transmissivity spectral line optical depth mixing frequency solid angle (sr)

Subscripts 0 b g j k max min w h, l

reference state blackbody emission spectral, with cumulative k-distribution as spectral variable bin or narrow band spectral, with absorption coefficient as spectral variable maximum minimum wall spectral

Overscores − ∼

narrow band average spatial average narrow band value, Favre average

293

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Transport Phenomena in Fires

References [1] Modest, M.F., Radiative Heat Transfer, 2nd edn, Academic Press: New York, 2003. [2] Siegel, R. & Howell, J.R., Thermal Radiation Heat Transfer, 4th edn, Taylor and FrancisHemisphere: Washington, 2002. [3] Mazumder, S., Numerical Study of Chemically Reactive Turbulent Flows with Radiative Heat Transfer, PhD Thesis, The Pennsylvania State University, University Park, PA, 1997. [4] Mazumder, S. & Modest, M.F., A PDF approach to modeling turbulence−radiation interactions in nonluminous flames. International Journal of Heat and Mass Transfer, 42, pp. 971−991, 1999. [5] Rothman, L.S., Rinsland, C.P., Goldman, A., Massie, S.T., Edwards, D.P., Flaud, J.M., Perrin, A., Camy-Peyret, C., Dana, V., Mandin, J.Y., Schroeder, J., McCann, A., Gamache, R.R., Wattson, R.B., Yoshino, K., Chance, K.V., Jucks, K.W., Brown, L.R., Nemtchinov, V. & Varanasi, P., The HITRAN molecular spectroscopic database and HAWKS (HITRAN atmospheric workstation): 1996 edition. Journal of Quantitative Spectroscopy and Radiative Transfer, 60, pp. 665−710, 1998. [6] Rothman, L.S., Camy-Peyret, C., Flaud, J.M., Gamache, R.R., Goldman, A., Goorvitch, D., Hawkins, R.L., Schroeder, J., Selby, J.E.A. & Wattson, R.B., HITEMP, the high-temperature molecular spectroscopic database. Available through http://www.hitran.com, 2000. [7] Tashkun, S.A., Perevalov, V.I., Bykov, A.D., Lavrentieva, N.N. & Teffo, J.L., Carbon dioxide spectroscopic databank (CDSD). Available from ftp://ftp.iao.ru/pub/CDSD-1000, 2002. [8] Tashkun, S.A., Perevalov, V.I., Teffo, J.L., Bykov, A.D. & Lavrentieva, N.N., Cdsd-1000, the high-temperature carbon dioxide spectroscopic databank. Journal of Quantitative Spectroscopy and Radiative Transfer, 82(1–4), pp. 165−196, 2003. [9] Singer, J.M. & Grumer, J., Carbon formation in very rich hydrocarbon−air flames − I: Studies of chemical content, temperature, ionization and particulate matter. Seventh Symposium (International) on Combustion, The Combustion Institute, pp. 559−572, 1959. [10] Wersborg, B.L., Howard, J.B. & Williams, G.C., Physical mechanisms in carbon formation in flames. Fourteenth Symposium (International) on Combustion, The Combustion Institute, pp. 929−940, 1972. [11] Sarofim, A.F. & Hottel, H.C., Radiative transfer in combustion chambers: Influence of alternative fuels. Proceedings of the Sixth International Heat Transfer Conference, Vol. 6, Hemisphere: Washington, DC, pp. 199−217, 1978. [12] Kunugi, M. & Jinno, H., Determination of size and concentration of soot particles in diffusion flames by a light-scattering technique. Eleventh Symposium (International) on Combustion, The Combustion Institute, pp. 257−266, 1966. [13] Sato, T., Kunitomo, T., Yoshi, S. & Hashimoto, T., On the monochromatic distribution of the radiation from the luminous flame. Bulletin of Japan Society of Mechanical Engineers, 12, pp. 1135−1143, 1969. [14] Becker, A. Annalen der Physik, 28, p. 1017, 1909. [15] Chang, H. & Charalampopoulos, T.T., Determination of the wavelength dependence of refractive indices of flame soot. Proceedings of the Royal Society (London) A, 430(1880), pp. 577−591, 1990. [16] Frenklach, M., Wang, H. & Rabinowitz, M.J., Optimization and analysis of large chemical kinetic mechanisms using the solution mapping method − combustion of methane. Progress in Energy and Combustion Science, 18, pp. 47−73, 1992.

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CHAPTER 8 Thermal radiation modeling in flames and fires S. Sen1 & I.K. Puri2 1

Department of Mechanical Engineering, Jadavpur University, India. Department of Engineering Science and Mechanics, Virginia Polytechnic Institute and State University, USA. 2

Abstract An overview of thermal radiation models for different combustion processes is presented. The pertinent constitutive equations and associated radiative property models are discussed. Various solution techniques for the radiative transfer equation are described. Finally, the implications for flames and fires are discussed in the context of the role of different thermal radiation models.

1 Introduction Radiation is the dominant energy transport mechanism to surrounding surfaces during many combustion processes, particularly when entrained particles are present. It plays an important role in practical systems such as furnaces [1], during fundamental flame phenomena, such as the radiation-induced extinction of flames at low stretch [2], radiation-turbulence interactions [1], and in fires [3]. Typically, the radiation intensity on a wall becomes very significant when the combustion length scale approaches a meter [4]. Thermal radiation is strongly coupled with soot kinetics and the flame structure through the local temperatures. Characterizing radiative energy transport is therefore a crucial element in modeling combustion systems. However, this is also a very complex problem. For example, in a typical coal fired furnace, radiation includes contributions from particulates (like coal/char, soot, ash) as well as from gases (such as CO2, H2O). The accuracy of a radiation simulation depends on a combination of the accuracy of the calculation method and the accuracy with which the radiative properties of the media and surfaces are known [5]. The current focus for radiative heat transfer models is on describing radiative interactions with a participating medium. The process is characterized by absorption, emission, and scattering of radiant energy, and mathematical models and computer codes are necessary to solve various radiation problems. Products of combustion gases, such as carbon dioxide, water vapor, and their mixtures with soot are considered within these radiation problems. An important issue of gas radiation is the description of the radiative properties of these gases (or the so-called nongray gases) [6].

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2 Basic equations 2.1 Energy conservation equation The mass, momentum, energy, and species balance equations must be solved to address combustion problems. These equations retain their usual form, while the energy equation addresses thermal radiation in a participating medium through an appropriate term, i.e. [6] ∂h ˆ ) = −∇ ⋅ qˆ + S, + ∇ ⋅ ( rvh (1) ∂t . where h denotes the enthalpy of the fluid and S the sources of heat generation, viscous heat dissipation, pressure, and body force work. The heat flux vector r

N

qˆ = − K ∇T + r ∑ hi wi vˆi + Fˆ . i =1

(2)

The terms on the right-hand side represent heat conduction, inter-species heat diffusion, and radiative heat flux that can be written as: Fˆ (rˆ, t ) =

∞

∫ ∫

I l (rˆ, sˆ, t )sˆdΩdl.

0 w=4π

(3)

2.2 Radiative transfer equation The spectral intensity of radiation, Il (rˆ, sˆ , t), at wavelength l is found from the radiative transfer equation (RTE). The interaction of electromagnetic radiation with matter that absorbs, emits, and scatters thermal radiation is described by the RTE. For the sake of brevity the derivation of RTE has not been repeated here, since it is well explained in the literature [7]. If we consider the local thermodynamic equilibrium of a medium capable of absorbing, emitting, and scattering radiation, the spectral radiation intensity Il (rˆ, sˆ, t) of the radiation field is [7] sˆ ⋅ ∇I l = kl I bl − bl I l +

ss l 4π

∫ I l (sˆi )Φl (sˆi , sˆ)dΩi ,

4π

(4)

where k denotes the absorption coefficient, b the extinction coefficient, s the scattering coefficient, and φ the scattering phase function that can be interpreted as the ratio of scattered radiative intensity in a given direction to the scattered radiative intensity in the same direction by isotropic scattering. The time dependent term for the radiation intensity can be usually neglected (as we have done). Equation (4) is the radiation balance for an infinitesimal pencil of rays in which absorption, extinction and scattering terms are considered. Thus, in order to obtain a volume balance, the equation must be integrated over all solid angles, i.e. ss l I (sˆ )Φ (sˆ , sˆ )dΩi dΩ. 4 π 4∫π l i l i 4π

∫ sˆ ⋅ ∇I l dΩ = ∫ kl I bl dΩ − ∫ bl I l dΩ + ∫

4π

4π

4π

(5)

Thermal Radiation Modeling in Flames and Fires

303

After rearranging, this relation can be written in the form [7] ∇ ⋅ ql = 4 πkl I bl − bl

∫ I l (sˆ)dΩ + ssl ∫ I l (sˆi )dΩi .

4π

4π

(6)

Using the relation kl = bl – ssl, the spectral relationship   ∇ ⋅ ql = kl  4 πI bl − ∫ I l dΩ = kl (4 πI bl − Gl ),   4π

(7)

where Gl denotes the incident radiation. Integration over the entire spectrum can be carried out to obtain the divergence of radiation heat flux, Fˆ.

3 Solution of the RTE The solution of the RTE can be substituted in the thermal energy equation to determine the temperature distribution in a reacting flow. This requires knowledge of the spectral radiative properties of the gases and an efficient method to solve the RTE. The evaluation of the radiative flux vector Fˆ or its divergence represents a fundamental problem that is related to the treatment of the spectral radiative properties of gases, the solution of the RTE, and integration over the spectrum. 3.1 Radiative property models The accuracy of any solution of the RTE depends upon an accurate knowledge of the radiation properties of the combustion product gases and the entrained/generated particles. Models used to define the radiative properties of combustion gases in radiation calculations can be roughly sorted in three groups [8], i.e. 1. Spectral line-by-line (LBL) models; 2. Spectral band models; and 3. Global models. Each has its merits and drawbacks that prescribe its area of application. Historically, the oldest and the simplest concept for the prediction of radiation by a gas is the gray gas model [9]. It assumes that the gas absorption coefficient is constant and therefore, when compared with other models, provides predictions with poorer accuracy [10]. The LBL model provides the best accuracy. With this method the RTE is integrated over the detailed molecular spectrum for the gases [11]. Because of the enormous computational requirements, this model is used mainly for benchmark solutions. It has two main disadvantages. First, the required spectral resolution should be smaller than the line width (i.e. 10−3−10−2 cm−1), which corresponds to roughly 106 spectral discretization intervals to cover the entire infrared spectrum. Second, at higher temperatures, accurate descriptions of a large number of lines associated with high energy levels and their quantum states are required. Recent developments in gas molecular spectroscopy have improved the database of radiative properties. Rothman et al. [12] have successfully implemented the LBL method to predict radiative transfer in the low temperature atmosphere. The 1992 version of the HITRAN database [12] contains information about spectroscopic transition line parameters for 32 molecules at the reference

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temperature of 296 K. The molecules included in the database are H2O, CO2, O3, N2O, CO, CH4, O2, NO, SO2, NO2, NH3, HNO3, OH, HF, HCl, HBr, ClO, OCS, H2CO, HOCl, N2, HCN, CH3C, HCl, H2O2, C2H2, C2H6, PH3, COFr, SF6, H2S, and HCO2H. It covers a spectral range from 10−6 to 22,656 cm−1 (i.e. 0.4414−1,010 µm). LBL calculations have been reported for H2O−CO2−CO semitransparent gas mixtures up to 2,000 K [13, 14]. In this model the spectral absorption coefficient for a gas mixture kl = ∑∑ klij , i

(8)

j

where the subscript i refers to the absorbing species and subscript j to all the lines of all the bands in these species. The line mixing and collision effects can be considered using phenomenological functions [12, 15]. The key problem is to obtain all center locations, lower level energy, and intensities of all lines contributing to absorption. The HITRAN database [12] is available for atmospheric and higher temperatures [16, 17]. Approximate models that have been developed to characterize the spectral structure of radiation can be classified into two groups, namely, (1) band models and (2) global or total models. In band models, the entire spectrum is divided into a number of bands. Thereafter, radiative properties, averaged over each band, are calculated from the absorption spectra or statistical properties of a line. These bands are assumed to be sufficiently narrow such that variations of radiative properties can be neglected. When a narrow band model is used, the RTE must be averaged over a band to yield the bandaveraged radiation intensity. To obtain spectrally-averaged or narrow band values of the absorption coefficient, some information must be available on the spacing of individual lines within the group and their strengths. A number of models have been proposed for this purpose and the statistical model is most widely used. It assumes that the spectral lines are not equally spaced within a narrow band. This can be a true representation for complex molecules for which lines from different rotational modes overlap in an irregular fashion [18]. An extensive description of narrow band models can be found in refs [15, 18]. These models require a database containing the measurements of both the reciprocal mean line spacing parameter and the mean absorption coefficient over the entire infrared spectrum at different temperatures. The most complete set, of measured narrow band parameters for H2O and CO2 is that of Ludwig and coworkers. Improved narrow band parameters have been published by Taine and coworkers [13, 14, 19] and Phillips [20, 21]. To date, the general dynamic database enables radiative property calculations for H2O, CO2, CO, OH, NO, HF, and CN. For CH4, C2H2, and soot, a narrow band model calculation can be performed using the approaches outlined by Brosmer and Tien [22, 23] and Ludwig et al. [18]. The statistical narrow band (SNB) model is more accurate for predictions of radiative transfer in high temperature gases. It provides the spectral transmissivity averaged over a narrow band. However, because of this it is difficult to couple the model with a solution methodology such as the finite volume method (FVM), which requires values of spectral (monochromatic) absorption coefficients or their averages over wavelength intervals. Another disadvantage is that the model requires a large number of bands and is therefore computationally very expensive. The wide band model (WBM) is a simplification of the SNB model. Instead of spectral lines, it considers bands and so is more economical but less accurate. It yields wideband absorptance while the solution of RTE by FVM operates either with spectral or averaged absorption coefficients. Therefore the WBM cannot easily and simply be incorporated in a FVM. In addition, WBM requires knowledge of the path length in the model as well as the associated spectral parameters.

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The concept underlying the exponential wide band model (EWBM) [24, 25] stems from the experimental observation that absorption by gases is mainly due to four or five strong absorption bands located in the near infrared and infrared regions (l−20 µm). The EWBM does not look at a narrow band but rather at the whole absorption band. This model provides fairly simple mathematical expressions to predict the temperature and pressure dependence of the most important absorption bands of H2O, CO2, CO, CH4, NO, SO2, N2O, NH3, and C2H2. The EWBM can also be utilized to predict the homogeneous total emissivity of gas-soot mixtures provided the soot concentration is known. This model could be utilized to calculate radiation in nonhomogeneous gas mixtures. From a practical perspective, the requirement of an almost insignificant database is the main advantage of EWBMs as compared to SNB models. The model does not require excessive computational power and possesses all the generality required for the radiation simulations in combustion at atmospheric and elevated pressures. Its major shortcoming is in treating radiation problems in enclosures, since it does not readily account for wall interactions. This problem originates from the formulation, which requires the calculation of the total band absorptance prior to those of the band transmissivity and bandwidth. Since the width of each absorption band varies with temperature, pressure, and pathlength, a different division of the infrared spectrum is required for each path along which the RTE is solved. However, to properly account for gas-wall interactions, the same spectral division for the wall and the gas phase should be used. The division of the spectral emissivity at the wall can only be setup once the RTE is solved along each path within the gas volume. This procedure requires a larger computational memory than narrow band models and is more computationally expensive. Some of the main impediments to the coupling of the EWBM with classical solution methods for the RTE have been discussed by Edwards [26]. In correlated-k (CK) methods, for any radiative quantity that is solely dependent on the gas absorption coefficient (which is true for a narrow band over which the blackbody function can be treated as a constant), the integration over wavenumber can be replaced by integration over the absorption coefficient. When integration over wavenumber is replaced with one over the gas absorption coefficient, the spectral gas absorption coefficient kl is denoted by k, since it now plays the role of an independent variable that is no longer a function of wavenumber. This is equivalent to the concept of the reordered absorption coefficient introduced by Lee et al. [27]. The distribution function f(k) can be determined in two ways. One is by analyzing the HITRAN database [28] and another by performing the inverse Laplace transformation of the gas transmissivity of a narrow band model, as in the SBN-based correlated-k (SNBCK) method described by Lacis and Oinas [29]. Denison and Webb [30−32] have developed polynomial correlations to calculate the cumulative distribution functions for water vapor and carbon dioxide. A set of 30 correlations was developed for H2O. The k-distributions were selected to cover the spectral range 400−12,000 cm−1 (0.83−25 µm) with intervals of 400 cm−1. The correlations were obtained by fitting spectral line predictions of the cumulative distribution in the temperature and volumetric ranges 400− 2,500 K and 0−1, respectively. The k-distribution model suffers from the disadvantage that new correlations must be computed whenever the total pressure or the temperature is changed (unlike narrow band models and EWBMs). The most widely used global model method is the weighted sum of gray gases (WSGG). The concept was first presented by Hottel and Sarofim [9]. In this method, the nongray gas is replaced by a number of gray gases. For the gray gases, the heat transfer rates can be calculated independently. The total heat flux is then found by summing these heat fluxes with appropriate weights.

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This approach can be used in a directional equation of transfer, as shown by Modest [33], and can therefore be used with any solution technique for the RTE. 3.2 Radiative properties of entrained and generated particles The evaluation of radiative properties of entrained and generated particles is important in combustion processes, since in almost all the cases, the flow is laden with particles like soot, coal, or ash. The radiative properties required for an entrained particle phase are the absorption coefficients and the scattering phase function which depend on the particle concentration, size distribution, and effective complex refractive indices. However, the optical properties of particulates, such as soot and coal, are not well characterized [34, 35] and considerable uncertainties exist, e.g. regarding the size and concentration of soot and the refractive index of ash. The absorption and scattering efficiencies are strongly dependent on the concentration and size distributions. Generally, to simplify the calculation of radiative properties, particles are assumed to be spherical and homogeneous, which is an oversimplification, although the radiative characteristics of a cloud of irregularly shaped particles are not very sensitive to the geometric shapes of the particles (as for pulverized coal) [4]. Given these assumptions, the absorption cross-sections can be calculated using Mie theory [36, 37] based on a specified particle size distribution, wavelength of radiation, and the complex refractive index. Once the absorption and scattering efficiencies for individual particle sizes are known from Mie theory, the absorption and scattering coefficients of the particulates can be evaluated [5]. Unfortunately, Mie theory is strictly applicable only to isolated particles interacting with plane waves. Because of this limitation and due to the highly forward-scattering properties of entrained or combustion-generated particulates, scattering intensities are often approximated by phase functions. The scattering phase function can be modeled using the Dirac-delta approximation [38]. The overall absorption coefficient for the volume can then be obtained as k = kp + kg ,

(9)

where kp and kg are the absorption coefficients for particle and gas, and the total radiative source for the gas enthalpy equation can be written as   Fˆ = k  ∫ Idw − 4 E b  ,  4 π 

(10)

where Eb denotes the blackbody emission of the gas. For a gray analysis, the blackbody emissive power for the gas is provided by Stefan−Boltzmann law of radiation [39]. Particle radiative properties and radiative emission can be determined using the source-in-cell technique [40] for Eulerian radiation field calculations. Almost all the soot models assume that the soot number density decreases as a result of particle agglomeration into spherical aggregates. The only exception is perhaps the study of Ezekoye and Zhang [41] who investigated the effect of particle agglomeration. It has been established that soot aggregates consist of more or less identical primary soot particles but the primary soot particle number density remains almost constant in the growth region. For either a Mie or a Rayleigh−Debye−Gans calculation, the overall properties must be integrated over either particle or aggregate size distribution. However, the fractal aggregate nature of soot has not found its way into most computational studies.

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3.3 Solution methodologies The RTE is an integro-differential equation for which exact solutions are not available for practical engineering applications. Multi-dimensionality, nonhomogeneous media, and the spectral variation of radiative properties make solutions of the RTE quite difficult. However, reasonably accurate numerical solutions can be obtained by introducing certain approximations. Because it is not possible to develop a single solution method that is equally applicable to a wide variety of different systems, several methods with varying degrees of approximation have been developed according to the nature of the physical systems, characteristics of the medium, degree of accuracy required, and the availability of computer facilities [4]. The major approaches are: (1) statistical or Monte Carlo methods; (2) zonal methods; (3) flux methods including the discrete-ordinates approximation; (4) moment methods; (5) spherical harmonics approximation; and (6) hybrid methods. Description of the statistical methods for radiative heat transfer calculations have been provided by Modest [7] and Haji-Sheikh [42]. These can be used for complex geometries and spectral effects can be accounted for without much difficulty. In its simplest form, a statistical approach simulates the histories of a finite but very large number of photons which originate from specified volume/surface elements, propagate in all directions, and are absorbed and scattered based on local values for absorption and scattering coefficients. Different researchers have applied these methods to combustion problems [43−45]. The zonal method (usually referred to as Hottel’s zonal method) is a widely employed model for calculating radiative transfer in enclosures such as combustion chambers. Hottel and Sarofim [9] and Hottel and Cohen [46] first described this approach. In this method, the surface and volume of the combustion chamber are divided into a number of zones, each of which has a uniform temperature and radiative properties. An energy balance is written for each zone, which leads to a set of simultaneous equations for the unknown heat fluxes or temperatures. The radiative heat flux generated by the exchange between the zones is determined using a radiosity method based on appropriate shape factors. The radiosities of the zones are determined by solving simultaneous equations. When an absorbing−emitting medium is involved, the calculation of direct exchange areas becomes complicated by the attenuation of radiation along a path connecting two zones. Because the approach is practical and powerful, it is attractive for many engineering calculations. This method is not however computationally efficient when coupled with finite-volume reactive fluid flow approaches usually used in comprehensive combustion models [47, 48]. Flux methods are based on separating the angular dependence of the radiation intensities, which arise from the spatial dependence of the in-scattering source term. By employing the assumption that intensities are uniform over defined intervals of the solid angle, the integro-differential equation can be simplified into a series of coupled, linear, differential equations expressed in terms of average radiative intensities or fluxes. Different flux models arise based on the number of solid angles used to approximate the directional dependence of the radiant intensity, such as two-flux approach (i.e. forward and backward scattering) for a one-dimensional approximation, four-flux for two dimensions, or six-flux models for three dimensions. Within each of the different flux models, various approximations and simplifications are employed to relate the angular dependence of the fluxes and characterize scattering phase functions to arrive at a closed set of solvable equations. The flux methods have been particularly effective for simultaneous use with reacting flow field solutions. The discrete-ordinates approach is a particular case of a flux method that was originally developed by Chandrasekhar [49] for astrophysical applications. It has also been used extensively in analyzing neutron transport [50, 51]. In the discrete-ordinates model, a quadrature scheme is used to integrate the in-scattering term. The quadrature set consists of ordinates

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(radiant energy directions) and weights which are chosen by applying appropriate constraints, such as symmetry and moments matching, as well as conserving radiant energy within a control angle and the total control volume. One quadrature scheme commonly used is the SN approach, in which the entire solid angle is divided into N(N + 2) angular divisions, where N can be evaluated on the basis of the order of the quadrature scheme. Applications of the SN approach have been shown by Fiveland [48, 52, 53] to produce accurate and computationally efficient results as compared with other approaches. Raithby and Chui [54, 55] and Chai et al. [56] have presented quadrature approaches based on a control volume scheme for fluid mechanics and convective heat transfer calculations. Control volumes can be based on either structured or unstructured grids. Because of its capabilities to consider common control volumes and grid structures to couple the radiation and reactive fluid flow solutions, the discrete ordinates method is the method of choice in comprehensive combustion models where thermal radiation has an important role. In other approximations of the RTE, the radiative intensity is expressed as a series of products of angular and spatial functions. With it, the integral part of the equation can be eliminated and a series of equations in terms of different orders of moments can be generated. A moment is defined as the integral of intensity multiplied by a power of a directional cosine over a predetermined solid angle division. If the angular dependence is expressed using a Taylor power series expansion, then the method is called the moment method, and if spherical harmonics are used to express the intensity, the spherical harmonics (PN) approximation results. The first moment, the first order spherical harmonics (P1), and the S2 discrete ordinates approximation are essentially identical. Applications of the moment method have been discussed by several researchers [57−60]. The general equations for the solution of the P1 and P3 approximations for absorbing, emitting, and anisotropically scattering in cylindrical and three-dimensional rectangular enclosures were developed by Menguc and Viskanta [61, 62]. The P1 approximation is particularly simple, since it can be cast as a single, second-order differential equation, but this simplicity is at the expense of accuracy. The prediction error can be as large as 50% for media with small optical thicknesses. Finally, combinations of various methods for solving the RTE (described above) have been used to formulate hybrid methods, which attempt to compensate for the flaws of one approach with the strengths of another. Several of these hybrid approaches have been summarized by Viskanta and Menguc [4]. As an example of a hybrid approach, the ‘discrete transfer’ model proposed by Lockwood and Shah [58] combines the advantages of the zonal, Monte Carlo, and discrete ordinates methods. Although designed for computing radiation in absorbing, emitting, and scattering media, no results have been reported for scattering in media in multi-dimensional enclosures. This method has been applied in furnace simulations [63, 64] and has been adapted for coupling with complex fluid mechanics and heat transfer grid topologies [65].

4 Radiation from flames Although the importance of radiation heat transfer as a flame heat-loss mechanism is recognized, the radiation absorption and nongray radiation models have not been adequately implemented in combustion simulations. It has become almost common practice, mainly due to computational constraints, to assume (often without justification) that the optically thin approximation (OTA) is valid to simulate various laboratory (sooting or nonsooting) flames due to their relatively small optical dimensions [66]. In fact, several numerical calculations of such flames using detailed chemistry and the OTA for radiation heat transfer have been performed [67−72].

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Kennedy et al. [67] developed a model for a laminar soot-laden ethylene diffusion coflow flame and compared it with measurements in sooting and nonsooting flames, studied by Santoro et al. [73]. They incorporated the divergence of the radiative heat flux term in the energy equation and calculated it using an optically thin model for emission from soot alone [74, 75]. They ignored the gas radiation in view of the strong luminosity of the soot particles and compared calculated-temperature values for the nonsooting flame with experimentally obtained data [76]. This comparison shows good agreement at a lower height above the burner. At a large distance above the burner, their model predicts a higher temperature. Since the model considers radiation from only soot, it predicts a smaller radiation loss than if gas radiation had also been included. Smooke et al. [68] included an optically thin radiation model in their calculations of laminar coflow flames. They assumed that for methane−air mixtures the significant radiating species are H2O, CO, CO2, and soot. The OTA method was also employed by Hall [77, 78] whose results predict the general shape and structure of flames (e.g. locations of peak flame temperatures) reasonably well. Temperature is a critical flame parameter since it has a direct influence on virtually all the other flame properties. The Hall model underpredicts experimental measurements along the centerline by about 100 K at all heights within the flame. While the location and value of the peak temperature are reasonably well-described by the model up to a certain height for a flame, temperatures are overpredicted at the outside flame edge (i.e. predicted temperatures do not fall off as rapidly as the data with increasing radius). In addition, the laboratory flame closes to the centerline more rapidly than their model predicts. Figure 1 shows the temperature profiles at different heights in the flame from ref. [68].

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One possible explanation for the low centerline temperatures is the lack of inclusion in the model of absorption by methane of the energy radiated from the flame front. This lack of participation by different components induces a smaller radiation loss from the outer surface of the flame. Without this additional radiation component, the predicted peak temperatures are higher by an increment that is large enough to affect predicted soot levels and species concentrations. The significance of thermal radiation in a lightly sooting coflow flame arises fromresidence times that are relatively long compared to those for counterflow diffusion flames, in which radiation effects are significant only for extremely low strain rates [79]. In nonbuoyant flames, radiation plays an important role since it is the major heat transfer mechanism (in absence of convection). Walsh et al. [71] performed a numerical and experimental investigation of an axisymmetric laminar diffusion flame to assess the role of buoyancy and dilution on various flame properties, such as temperature, fuel, and oxygen concentration, and soot volume fraction. Their predicted temperature profiles are in excellent agreement with measurement in both normal gravity and microgravity flames for low dilution levels. Kaplan and Kailashnath [70] investigated flowfield effects on soot formation in normal and inverse methane− air diffusion flames. Because methane is a relatively low sooting fuel, the radiation transport submodel is based on the simplifying assumption that the medium is optically thin. The absorption coefficient for soot was assumed based on Kent and Honnery [80], while that for CO2 and H2O was derived from Magnussen and Hjertager [81]. These were combined to provide an overall Planck mean absorption coefficient. Microgravity experiments on flame spread over thermally thick fuels have been important for assessments of fire hazards in orbiting spacecraft [82]. Such experiments on flame spread have been conducted over thermally thick foam fuels (that have relatively low densities and thermal conductivities, and thus higher spread rates as compared to dense fuels such as PMMA [83]). These experiments suggest that steady spread can occur over thick fuels in quiescent microgravity environments, especially when a radiatively active diluent such as CO2 is employed. This is due to the dominance of radiation transfer from the flame to the fuel surface over conduction heat transfer from the flame to the fuel bed. Radiative effects become more significant at microgravity due to a larger flame thickness volume of radiating combustion products. Liu et al. [84] investigated an atmospheric coflow laminar moderately sooting ethylene-air diffusion flame. They calculated radiation heat transfer using both the optically thin model and the discrete-ordinates method coupled with a SNB model. The radiation source term in the energy equation was obtained using the discrete-ordinates method in an axisymmetric cylindrical geometry as described by Truelove [85]. The T3 quadrature [86] was used for the angular discretization. Spatial discretization of the transfer equation was achieved using the FVM along with the central difference scheme. The SNBCK-based WBM developed by Liu et al. [87, 88] was employed to obtain the abso rption coefficients of the combustion products containing CO, CO2, and H2O at each wide band. They assumed the spectral absorption coefficient of soot to be 5.5fv /l (fv being the soot volume fraction and l the wavelength). The wide bands considered in the calculations were formed by lumping 10 successive uniform narrow bands of 25 cm−1 to obtain a bandwidth of 250 cm−1 for each wide band. The blackbody intensity at each wide band was evaluated at the band center. The SNB parameters for CO, CO2, and H2O were those compiled by Soufiani and Taine [89] based on LBL calculations. At overlapping bands, the approximate treatment based on the optically thin limit developed by Liu et al. [90] was employed. To further speed up the calculations without losing accuracy, the 4-point Gaussian−Legendre quadrature was used to invert the cumulative distribution function to obtain the absorption coefficients based on Liu et al. [91]. The radiation source term was calculated by summing up contributions of all the 36 wide bands (from 150 to 9,150 cm−1) considered in the calculations.

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To evaluate the results of the optically thin model using those of the WBM discussed above, numerical calculations were also conducted [84] using the optically thin radiation model. Under OTA, the radiation source term Fˆ = −Cfv T 5 − kp 4sT 4 ,

(11)

where C is a constant that is calculated based on the spectral absorption coefficient of soot, s the Stefan−Boltzmann constant, and kp the Planck mean absorption coefficient of the gas mixture including contributions from CO, CO2, and H2O. The Planck mean absorption coefficients of these three species were calculated based on the SNB model given by Ju et al. [92]. The predicted temperature field with both gas and soot radiation accounted for (using SNBCK) was in qualitative agreement with measurements, as shown in Fig. 2. However, the predicted peak flame and centerline temperatures were more than 100 K lower than the corresponding measured values. In addition, the predicted maximum temperature annulus was found to be thinner than the measurement. The causes of these discrepancies were attributed to the use of a simplified soot model. The peak temperature predicted using the optically thin model was only about 5 K lower than that of the band model. The centerline temperatures were also underpredicted due to neglect of radiation absorption by CO2 (and to a lesser degree by CO), especially in CO burnout regions just above the flame tip where the concentration of CO2 is high. The optically thin model underpredicted the temperatures in these regions by more than 50 K compared to the band model. When radiation by gases was neglected, the predicted temperature levels were in reasonably good agreement with experimental values. However, the peak flame temperature was now higher, indicating that radiation by gases is also important and should be accounted for. The results [84] suggest that soot radiation is more important than gas radiation in such a flame, since its inclusion has a greater impact on the predicted temperature distributions. The comparison of measured and predicted soot volume fraction is shown in Fig. 3. (a) experiment

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Figure 3: Comparison of measured and predicted soot volume fraction distributions with the peak values indicated. (Reprinted with permission from Elsevier [84]. Copyright 2002 Elsevier.) Sivathanu and Gore [93] investigated the effect of gas band radiation on soot kinetics in a laminar coflow methane-air diffusion flame. The local radiative heat loss fraction was obtained using the solution of the energy equation including soot and gas-band radiation [94]. They concluded that the contribution of the participating gases (CO2 and H2O) dominates the soot radiation by an order of magnitude in their methane/air flames and the local radiative heat loss/gain strongly influences the soot nucleation, formation, and oxidation rates. The measurements and predictions of area-integrated soot volume fractions as a function of axial distance using three different methods of estimating the local temperatures are shown in Fig. 4 from ref. [93]. The calculations are offset by 20 mm so that soot growth starts at approximately the same location for both the measurements as well as the predictions. These three sets of calculations correspond to: (1) a fully coupled calculation where the local temperatures are obtained by considering the local energy loss or gain by radiation from gas molecules and soot particles, (2) an uncoupled calculation where adiabatic flame temperatures are prescribed at all locations as a function of the local mixture fraction, and (3) an uncoupled calculation where the radiative heat loss fraction was assumed to be constant at 19% (close to the experimentally observed value) and temperatures are prescribed at all locations as a function of the local mixture fraction. Qin et al. [95] characterized gravity and radiation effects on the structure of laminar methaneair partially premixed flames through detailed simulations. They modeled radiation using an optically thin assumption that provides a limiting value for the radiation heat transfer. The overall effect of radiation on the structure of 1-g flame is less significant than for the corresponding 0-g flame. Due to radiation effects, the flame heights of 0-g flames increase, and the heat release rate intensity near the premixed reaction zone tip decreases. When radiation effects are not included in the simulations, the peak temperatures are nearly the same for the 1-g and 0-g flames. With radiation the difference in these temperatures is significant. The decrease in the peak temperature due to radiation for their 0-g flame is nearly five times larger than for the 1-g flame. The value of the radiation loss fraction for 0-g flames without coflow can be as large as 50%, although

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Figure 4: Axial variation of the measured and predicted soot volume fraction in a methane/air diffusion flame using different radiation models. (Reprinted with permission from Elsevier Science [93]. Copyright 1997 by The Combustion Institute.) it drops significantly in the presence of a coflow. While the flowfields upstream of the inner premixed reaction zone are nearly identical for 1-g and 0-g double flames, they are markedly different in the regions between the two reaction zones as well as downstream of the outer nonpremixed reaction zone. Lock et al. [96] also discussed important aspects of thermal radiation effects in microgravity flames. Gas radiative properties vary strongly with wavelength so that use of the gray-gas model or the OTA often introduces large errors. Although LBL models provide very accurate results of spectral radiation heat transfer, they are not practical for coupled multi-dimensional calculations of radiation, fluid flow, and chemical reactions due to large computational time required. In the absence of LBL results, results of the SNB model are often sufficiently accurate, even for benchmark solutions for the evaluation of other approximate nongray-gas radiation models. Direct implementation of the SNB model in multiple dimensions is computationally very demanding since it is formulated in terms of the transmissivity instead of the absorption coefficient [97]. Goutière et al. [98] have shown that the SNBCK method is an efficient alternative to the SNB model with essentially the same accuracy. The computational time for the standard SNBCK method (using a bandwidth of 25 cm−1 and seven-point Gauss−Lobatto quadrature), though substantially smaller than for the SNB model (6 times), is still quite large, especially in multi-dimensional problems involving two or more radiating gases. In several subsequent studies, Liu et al. [88, 90, 91] considerably improved the computational efficiency of the SNBCK method with only minimal loss of accuracy by developing approximate treatments for overlapping bands, introducing the bandlumping strategy, and using an optimized quadrature set. Band models of different band-widths can be formulated by easily lumping different numbers of narrow bands. In addition, the order of Gauss quadrature used in SNBCK calculations can also be readily changed. The improved SNBCK

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method also offers greater flexibility to accommodate different levels of compromise between efficiency and accuracy without reformulation of the model. Recently, Goutière et al. [99] proposed the so-called optimized band-lumping (regrouping) strategy based on the four large absorbing bands of CO2. This band-lumping method offers better computational efficiency than the uniform lumping strategy originally proposed by Liu et al. [87, 100]. The spectral-line-based weightedsum-of-gray-gases model developed by Denison and Webb [31, 32], though computationally more efficient, can be less accurate than the SNBCK methods in two-dimensional gas radiation calculations [98, 99]. The effects of radiative transfer on the structure and extinction limits of counterflow H2/O2/N2 diffusion flames have been investigated [101]. The radiative properties of the main emitting species, H2O and OH in these flames were computed using an SNB model. As expected radiative losses decrease the flame temperature and width, and significantly reduce the minor species (such as NO) that are sensitive to temperature. The amplitude of the radiative effects increases with the ratio of the global radiative loss to chemical heat release. Quick estimation of this parameter is useful for determining whether radiative transfer can be neglected in the prediction of small-scale flames. No effect of radiation on the extinction of high-strain rate extinction limit was found in the study, but suggested a value for the radiation-limited extinction for low strain rate flames. An early review of radiation from turbulent flames [102] suggests that the calculation of thermal radiation in turbulent combustion involves three key factors: the solution of the RTE, the spectral variation of radiative properties, and the evaluation of turbulence−radiation interactions. Because of the inherent difficulties in radiation calculations, the common practice for turbulent flames has been to neglect turbulence−radiation interactions and to use the OTA or a gray-medium assumption, even for luminous sooting flames [103−105]. Although the OTA is useful, its application produces errors, since it neglects self-absorption effects [106, 107]. Radiation, along with soot, is such a complex and nonlinear phenomena that gray or OTA models generally are inadequate to describe it, particularly in oxygen-enriched flames. Wang et al. [108] implemented two radiation models with self-absorption effects (one that accounts for nongray-gas properties and the other that does not) for an oxygen-enriched, propane-fueled, turbulent, nonpremixed jet flame. One important aspect for combustion studies is the prediction of radiative transport due to particles and gases within flames. The RTE is generally used to solve for the radiative heat transfer in a semi-transparent media. In order to make proper calculations, information regarding the radiative properties of the medium must be calculated. Soot, or the polycyclic aromatic hydrocarbon particulates formed during coal combustion, can be a major contributor to radiative transport. Some calculations indicate that soot can contribute as much as 15% to the total radiative flux [109]. Other research indicates that temperature predictions in certain combustor locations can vary by as much as several hundred degrees kelvin based on the inclusion of soot predictions [110, 111]. While this may not be important for the overall energy balance in a combustor, it could be significant for pollutant calculations. Minor changes in temperatures can make a substantial impact on pollutant formation predictions, which tend to be highly temperature-dependent. Since the near burner region is of great interest for pollutant prediction calculations (and also tends to be the region of maximum soot concentration), predicting the radiative contribution of soot becomes even more important. Adams and Smith [110] developed a simple empirical model for turbulent soot formation that related the soot volume fraction to the local equivalence ratio. Soot was assumed to exist where the local equivalence ratio was unity and above, increasing linearly to a maximum value at an equivalence ratio of 2.0 and above. The maximum soot concentration value was calculated as a

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direct function of the amount of volatile carbon calculated to exist at that point. The soot volume fraction was then related to the radiative properties. They concluded that the inclusion of a soot radiation model increases predicted radiative transfer; however, the maximum local temperature difference between predictions with and without the soot model was lower than expected (about 50 K). Ahluwalia and Im [109] developed a similar model, and reported that soot significantly enhanced the radiative heat transfer. Both models use an empirical survivability constant of 0.1 which serves to reduce the predicted soot concentrations. A detailed discussion on coal combustion and gasification can be found in ref. [112]. Another important aspect is study of the flames immersed in porous media. There are many interesting radiation problems in such system. Interested readers can consult the excellent review paper by Howell et al. [113].

5 Radiation from fires Two types of fire analyses are mainly found in the fire safety literature: for compartment fires (also called enclosure fires) and for outdoor fires (such as forest or wildland fires). Although thermal radiation plays a significant role in these phenomena, radiative mechanisms are often only grossly accounted for in the associated models [3]. Fire simulations use fluid dynamic models for chemically reacting and radiating flows. Due to the complexity of the phenomena, fire modeling has been carried out with different degrees of simplification ranging from strictly empirical models to correlations that are variously based on dimensional analysis, semi-empirical models, and theoretical models. The trend in fire simulations is to use software packages based on CFD codes for chemically reacting flows. A number of fire simulation codes were developed in the 1980s of which some are based on commercially-available codes [114, 115]. In fire modeling, the spread of heat and smoke in a complex geometry is required. The complex structure of the combustion products, composed of molecular gases and soot particles, makes the treatment of radiative transport very difficult. Hence, researchers initially used grossly simplified models for combustion and/or radiation [116]. The fire dynamics simulator (FDS) is most popular among these codes. It uses the large eddy simulation (LES) FVM with improved radiation and combustion models [117−119]. An FVM submodel solves the RTE by using a narrow band gas radiation model (RADCAL). There have been many enhancements to RADCAL after its earlier development, e.g. Fuss et al. [120, 121] incorporated high temperature flammable gases into RADCAL. A more sophisticated radiative submodel is available in the JASMINE code [114], which is based on the discrete transfer method (DTM) combined with a simplified model of gas radiative properties. The radiation model works in conjunction with a SIMPLE pressure correction algorithm and the standard k−ε method to compute turbulence. The simulation of fires in enclosures (SOFIE) code was developed to predict fire propagation in enclosures [122]. It uses either a gray six-flux algorithm or a DTM combined with a WSGG to solve the RTE [123]. A Monte Carlo solution method for the gray-formulated RTE has been implemented in a 3D CFD model to investigate compartment fires. Consalvi et al. [124] developed a CFD FVM code to simulate fires in enclosures with internal obstacles or partitions. They used a modified Chai et al. blocked-off discrete ordinates model [56] to solve a gray formulation of the RTE. The FDS code has also been used recently with some modifications in the combustion and soot models [125] and by incorporating the computation of the radiative flux emitted towards the fuel surface and upstream of it [126]. The structure of soot and its radiative properties are important parameters for analyses of fires. These radiative properties have been extensively investigated by Faeth and coworkers in the

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context of fire modeling [127−131]. The work by Sivathanu et al. [132] on the in situ determination of the absorption coefficient of soot particles represents another contribution. The estimation of optical and radiative properties of soot can be carried out using the Rayleigh−Debye−Gans theory for fractal aggregates with acceptable accuracy [133]. Gas radiative properties are also important parameters for fire modeling. The use of LBL calculations is formidable for fire applications since the absorption spectra of most combustion gases contain a large number of spectral lines for evaluating gas radiative properties. Modak [134, 135] proposed a method for computing gray gas radiative properties for isothermal, homogeneous mixtures of carbon dioxide, water vapor, and soot. This method is popular due to its simplicity and reasonable accuracy and is still being used in a number of fire simulation codes. Since the isothermal assumption is not valid in large-scale fires, Modak’s model might not provide highly accurate results for such cases [136]. Grosshandler [137] proposed remedial corrections to Modak’s simplified method and suggested an SNB model based upon the Goody approach [15] and the Curtis−Godson approximation to deal with nonhomogeneous effects. His code uses the data of Ludwig et al. [18] and the parameters for the primary infrared bands of water vapor and carbon dioxide are computed according to Malkmus [138, 139]. Grosshandler’s SNB approach used in conjunction with the DTM were recently evaluated together with three other gas radiative property models, Modak’s approach [135], Truelove’s mixed gray gas model [85], and Edward’s WBM [24]. The test cases were an idealized 1D case with parabolic profiles of temperature, CO2 and H2O concentrations, and two experimental cases, i.e. a laboratory-scale flame and a field-scale natural gas jet fire [140]. The comparison in terms of radiative fluxes shows a satisfactory level of agreement. However, the performances of all models are diminished when compared to the narrow band model while dealing with complex non-Cartesian meshes. Encouraged by this, researchers have developed ‘fast narrow band models (FASTNB)’ [141, 142]. Used in conjunction with the DTM, FASTNB [143] is 20 times faster than RADCAL for comparable results (and deviations smaller than 1%). Other useful alternatives to SNB approaches that are based on the absorption coefficient concept are the WSGG model, CK model, and their recently improved versions (SLW, ADF, ADFFG, FSCK). Detailed presentations of these models can be found in Modest [7]. Most of these predict small/medium scale fire behavior (like that of compartment fires) satisfactorily. The large number of grid points required for large/outdoor fires requires a large amount of computational time and this, in turn, limits the complexity of the radiation model that is employed [144]. Discrete Transfer Models are therefore less attractive to solve the RTE for large fires and thus simplifications are required in the associated radiation models. Usually, the radiation models employed in case of large fires are gray and scattering is generally neglected [3]. An overview of the recent research on wildland and forest fires has been provided by Albini [145] and regarding which Grishin and coworkers have made noteworthy contributions [146− 148]. Radiation has been unsurprisingly identified as the controlling heat transfer mechanism that fixes the rate of spread of wildland fires [149−153]. Fire growth models are now being used in campaign fire strategic planning. Research is also being performed to develop a gray nonscattering DOM for solving the RTE and couple it to a CFD model for wildland fire propagation [154, 155]. Research is also focused on developing radiation models to simulate fire spread in urban areas. Mainly due to the complex geometries, most such models are zonal. Radiative transfer is approximated by means of gray gas, gray surface emissivities, and radiative exchange areas or view factors between the cells [156]. The urban−wildland interface fire is another subject of concern. Again, radiative transfer is mostly treated through rough simplified models. Predictions of the impact of fire on structures in the urban−wildland interface have been obtained using sophisticated simulation

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tools, such as DOM with SLW or gray nonscattering gas models, that have been coupled with CFD codes [157, 158]. The gray gas model (as compared to SLW) has been found to be reasonably accurate. On the other hand, savings in computational time (six times faster than SLW) allow the use of this model as a fire operational tool [157, 158]. Simeoni et al. [159] reached a compromise between rapidity and accuracy, by developing a semi-physical model of fire spread across a fuel bed, which is unsteady and two-dimensional along the ground shape. In this model, they assumed that radiation is the prevailing heat transfer mechanism involved in fire spread [160]. Nevertheless, it was unable to correctly predict the high wind effects on the rate of spread.

6 Summary Thermal radiation is an important and often dominant heat transfer mode during many combustion processes. However, the complexities of the RTE and the associated radiative properties have made the modeling effort quite challenging. Among the different gas property models, the LBL method makes computations time intensive and it is therefore mainly used to obtain benchmark results. Global models are very popular for their simplicity and acceptable accuracy. However, a compromise between the two can be found in spectral band models and their different variations which have found extensive use by the researchers. The radiative and optical properties of entrained particles (like soot or ash) play an important role during combustion. These properties are usually found using the Mie theory which, however, is not valid for particle aggregates. Hence other theories such as the Rayleigh−Debye−Gans theory are employed to describe the radiative properties of particulate aggregates. Various types of solution methods are used to solve the integro-differential RTE. It is not possible to develop a single solution method that is equally applicable for different systems. Therefore, several solution methods (with varying degrees of approximation) have been developed and can be applied according to the nature of the physical system, characteristic of the medium, the degree of accuracy required, and the availability of computer facilities.

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[129] Koylu, U.O. & Faeth, G.M., Optical properties of overfire soot in buoyantturbulent diffusion soot flames at long residence times. J. Heat Transf., 116, pp. 152−159, 1994. [130] Koylu, U.O. & Faeth, G.M., Spectral extinction coefficient of soot aggregates from turbulent diffusion flames. J. Heat Transf., 118, pp. 415−421, 1996. [131] Wu, J.S., Krishnan, S.S. & Faeth, G.M., Refractives indices at visible wavelengths of soot emitted from buoyant turbulent diffusion flames. J. Heat Transf., 119, pp. 230−237, 1997. [132] Sivathanu, Y.R., Gore, J.P., Jansen, J.M. & Senser, D.W., A study of in situ specific absorption coefficients of soot particles in laminar flames. J. Heat Transf., 115, pp. 653− 669, 1993. [133] Baillis, D. & Sacadura, J.F., Thermal radiation properties of dispersed media: theoretical prediction and experimental characterization. J. Quant. Spec. Radiat. Transf., 67, pp. 327−363, 2000. [134] Modak, A.T., Thermal radiation from pool fires. Combust. Flame, 29, pp. 177−192, 1977. [135] Modak, A.T., Radiation from products of combustion. Fire Res., 1, pp. 339−361, 1978/79. [136] De Ris J., Fire radiation−a review. Proc. Combust. Inst., 17, pp. 1003−1016, 1979. [137] Grosshandler, W.L., Transfer in non-homogeneous gases: a simplified approach. Intl. J. Heat Mass Transf., 23, pp. 1447−1459, 1980. [138] Malkmus, W., Infrared emissivity of carbon dioxide (2.7 µm band). General DynamicsAstronautics, AE63−0047, 1963. [139] Malkmus, W., Infrared emissivity of carbon dioxide (4.3 µm band). J. Opt. Soc. Am., 53, pp. 951−961, 1963. [140] Cumber, P.S., Fairweather, M. & Ledin, H.S., Application of wide band radiation models to non-homogeneous combustion systems. Intl. J. Heat Mass Transf., 4, pp. 1573−1584, 1998. [141] Yan, Z. & Holmstedt, G., CFD simulation of upward flame spread over fuel surface. Proceedings of the 5th International Symposium on Fire Safety Science, Melbourne, Australia, pp. 345−356, 1997. [142] Yan, Z. & Holmstedt, G., Three-dimensional computation of heat transfer from flames between vertical and parallel walls. Combust. Flame, 117, pp. 574−578, 1999. [143] Yan, Z. & Holmstedt, G., Fast narrow-band computer model for radiation calculations. Num. Heat Transf., Part B, 31, pp. 61−71, 1997. [144] Baum, H.R. & McGrattan, K.B., Simulation of large industrial outdoor fires. Proceedings of the 6th International Symposium on Fire Safety Science, Poitiers, France, pp. 611−622, 1999. [145] Albini, F.A., An overview of research on wildland fire. Proceedings of the 5th International Symposium on Fire Safety Science, Melbourne, Australia, pp. 59−74, 1997. [146] Grishin, A.M., Mathematical Modeling of Forest Fires, Publishing House of the University of Tomsk, Russia, 1981. [147] Grishin, A.M., Zverev, V.G. & Shevelev, S.V., Steady state propagation of top crown forest fire. Fizika Goreniya i Vzryva, 22, pp.101−108, 1986. [148] Grishin, A.M. Mathematical modeling of forest fires and new methods of fighting them. Ed. Publishing House of the University of Tomsk: Russia, 1997. [149] Emmons, E.W., Fire in the forest. Fire Res. Abstr. Rev., 5, pp.163−178, 1964. [150] Hottel, H.C., Williams, G.C. & Steward, F.R., Modeling of fire spread through a fuel bed. Proc. Combust. Inst., 10, pp. 997−1009, 1964.

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[151] Telitsin, H.P., Flame radiation as a mechanism of fire spread in forests. Heat Transfer in Flames, eds. N.H. Afgan & J.M. Beer, Wiley: New York, pp. 441−449, 1974. [152] Albini, F.A., Wildland fire spread by radiation − a model including fuel cooling by natural convection. Combust. Sci. Technol., 45, pp. 101−113, 1986. [153] Carrier, G.F., Fendell, F.E. & Wolff, M.F., Wind-aided fire spread across arrays of discrete fuel elements. Combust. Sci. Technol., 75, pp. 31−51, 1991. [154] Porterie, B., Morvan, D., Loraud, J.C. & Larini, M., Firespread through fuel beds: modeling of wind-aided fires and induced hydrodynamics. Phys. Fluids, 12, pp. 1762−1782, 2000. [155] Morvan, D. & Dupuy, J.L., Numerical simulation of the propagation of a surface fire through a Mediterranean schrub. Proceedings of the 7th International Symposium on Fire Safety Science, Worcester, MA, USA, pp. 557−567, 2002. [156] Himoto, K. & Tanaka, T., A physically-based model for urban fire spread. Proceedings of the 7th International Symposium on Fire Safety Science, Worcester, MA, USA, pp. 129−140, 2002. [157] Porterie, B., Nicolas, S., Consalvi, J.L., Loraud, J.C., Giroud, F. & Picard, C., Modeling thermal impact of wildland fires on structures in the urban interface—Part 1: Radiative and convective impact of flames representative of vegetation fires. Num. Heat Transf., Part A, 47, pp. 471−489, 2005. [158] Consalvi, J.L., Porterie, B., Nicolas, S., Loraud, J.C. & Kaiss, A., Modeling thermal impact of wildland fires on structures in the urban interface—Part 2: Radiative impact of a flre front. Num. Heat Transf., Part A, 47, pp. 491−503, 2005. [159] Simeoni, A., Santoni, P.A., Larini, M. & Balbi, J.H., Reduction of a multiphase formulation to include a simplified flow in a semi-physical model of fire spread across a fuel bed Intl. J. Thermal Sciences, 42, pp. 95−105, 2003. [160] Morandini, F., Santoni, P.A. & Balbi, J.H., Analogy between wind and slope effects on fire spread across a fuel bed - modelling and validation at laboratory scale. Proceedings of the 3rd International Seminar on Fire and Explosion Hazards, Edinburgh, Scotland, UK, 2000.

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CHAPTER 9 Combustion subgrid scale modeling for large eddy simulation of fires P.E. DesJardin, H. Shihn & M.D. Carrara Department of Mechanical and Aerospace Engineering, University at Buffalo, The State University of New York, USA.

Abstract While large eddy simulation of engineered combustion systems is starting to become a reality, its extension to fire environments presents a new set of challenges. The scale of fire systems are often very large, placing additional requirements on mesh resolution. In addition, the finite-rate chemistry processes of soot and toxin formation and flame quenching that are extremely difficult to model in a turbulent flow are also critical aspects of characterizing a fire environment. The objective of this chapter is to examine state-of-the-art subgrid scale combustion models for application to fire environments with a focus on presumed (via flamelet) and transported filtered density function approaches and one-dimensional turbulence modeling. The relative merits of these models for application to fire simulation are discussed with illustrative examples.

1 Introduction The financial loss to developing nations due to fires is enormous – accounting for upwards of 1% of the gross domestic product, which can easily translate into billions of dollars. It is surprising then that while fires have been in existence since the beginning of mankind, the underlying science of fires has only now started to be fully understood. The fundamental difficulty is that the physics of fires depends on the intimate coupling of turbulent flow, all modes of heat transfer (i.e. conduction, convection, and radiation), and chemical reactions that potentially occur in several material phases. Certainly any one of these ingredients of a fire can be an entire topic all by itself. In recent times, with the advent of faster computers and advanced diagnostic tools, researchers are starting to unravel the mysteries of this very complex dynamical system that will hopefully lead to better fire safety design practices. The challenge in understanding the dynamics of fires lies in the intimate non-linear coupling of chemical reactions, radiative heat transfer, and molecular and turbulent transport processes over a very wide range of time and length scales. Since it is only feasible to computationally resolve a limited range of length and time scales, subgrid scale (SGS) engineering models must

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be introduced. These models account for many of the most important thermo-chemical processes that define the development of a fire, consisting of the four major stages: (1) ignition, (2) growth, (3) flashover, and finally (4) decay [1]. The initiation of a fire begins with ignition resulting in the pyrolyzing of organic material which forms char and releases volatile gases that will mix and burn with the surrounding air. Following ignition, the fire may grow depending on the local ventilation environment. For a fire in a room, the ventilation is defined by the local heating, ventilation, and air conditioning system in addition to time-dependent damage state of the building. For instance, broken windows or large voids in walls due to local structural failure will provide unanticipated ventilation pathways that will significantly enhance the growth rate of a fire. As a fire grows, a thick layer of hot smoke is formed that serves to further preheat the surrounding surfaces through radiative heat transfer. Finally, if there is enough oxygen in the room, flashover will occur resulting in a rapid increase in the growth of a fire to a fully developed state. At this stage of the fire, most of the surfaces in the room are burning and an explosive increase in pressure may result causing further damage to the structure. While large disparities in length and time scales occur in almost any turbulent reacting flow, it is especially exacerbated in fire environments since the soot formed at sub-micron scales is directly responsible for the thermal radiative loading to the surroundings at the macroscales. Finite rate chemistry processes leading to soot formation are, in turn, sensitive to the local turbulent mixing environment which in turn is determined by the room geometry and ventilation. As an example, for a large room of O(10 m) the computational grid requirements to resolve all relevant convective processes are of the order O((10 m/0.1 µm)3) = O(1 × 1024) grid points. Adaptive meshing may help alleviate some of this burden by reducing the overall number of grid points by an order of magnitude in each direction, but the resulting number of grid points will still far exceed the ability to compute and store data from first principles-based simulations (i.e. direct numerical simulation (DNS)) for years to come. The need for engineering SGS models to account for the effects of combustion processes is therefore a necessity. Since a fire is inherently an unsteady process, the large eddy simulation (LES) based on either temporal filtering (unsteady RANS), or spatial filtering, is ideally suited for this problem. The objective of this chapter is to summarize some of the current state-of-the-art SGS modeling techniques used in LES. The mathematical formulation of LES based on spatial filtering is first summarized, setting the framework for the needs of SGS modeling and specifically the filtered reaction rates. Three types of combustion SGS modeling approaches are discussed starting from the simplest description to perhaps the most complex. The first is the flamelet modeling description using a presumed filtered density function (FDF) for mixture fraction. The second is transported FDF approaches that readily allow for finite rate chemistry effects to be included. The third approach is to include multi-point SGS stochastic process descriptions using one-dimensional turbulence (ODT) modeling concepts.

2 LES mathematical formulation A complete description of fire physics would, in general, require a multiphase formulation to account for processes of soot formation and fire suppression using liquid agents (e.g., water sprays/mists). While existing formulations for multiphase LES are available [2–4], this level of complexity is not considered here. Rather, the following is restricted to single-phase flows since the focus here is on the modeling of homogeneous reactions for turbulent flows. The starting point for this description is the Navier–Stokes equations, supplemented with equations for multicomponent species transport and an energy equation for a reacting system. These equations

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329

are Favre (density) filtered using a positive definite filtering function, G (i.e. G(∆f , x –x¢) > 0 ∀ x¢), with the usual normalization and symmetry properties,

∫ G(∆ f , x − x ′)dx ′ = 1

(1a)

G( ∆ f , −( x − x ′)) = G( ∆ f , x − x ′)

(1b)

D

where ∆f is the filter width which is often directly related to the CFD mesh size. The process of pre-filtering the governing equations explicitly separates out information that is resolved on the CFD grid, and information falling below the grid must be modeled. The information on the grid is usually expressed in terms of density weighted (Favre) spatial averages over the SGS volume, f˜ , and is defined by the following convolution integral. rf 1 f ( x) = = r r

∫ r( x ′)f( x ′)G(∆ f , x − x ′)dx ′

(2)

D

Application of the filtering operation to the governing equations results in the following set of transported equations for mass, momentum, energy, and species mass fraction. ∂r + ∇ ⋅ ( r u ) = 0 ∂t

(3)

∂( r u ) + ∇ ⋅ ( r u ⊗ u ) = ∇ ⋅ ( − pI + t + Tuu ) + r g ∂t

(4)

N ∂[ r (et + Q)] + ∇ ⋅ ( r u h t ) = ∇ ⋅ (T uh + T uu⋅u + u ⋅ t − q ) − ∑ h ofm w m′′′ + r u ⋅ g ∂t m =1

(5)

∂( r Y m ) + ∇ ⋅ ( r Y m  ) = ∇ ⋅ (T uYm + q m) + w m′′′ ∂t

(6)

Often implicit in the filtering process is the assumption that ∆f is constant. This assumption allows for the commutation of differentiation with the filtering operation. In practice, non-uniform meshes are often employed which in principle introduces commutation error [5]. This error could potentially be avoided by using commutative preserving filters [6], but in practice this error is often simply ignored. __ __ ) is In eqns (3)–(6) r is the mixture density, u is the velocity, p is the pressure, e t (= ht − RT the total resolved energy, Ym is the mass fraction of the mth species, and ht (= h + u ⋅ u /2) is the ⋅ u)/2) total enthalpy including the resolved sensible enthalpy (h ) and kinetic energy. Q (= (u ⋅ u − u is the subgrid kinetic energy and may be assumed negligible since the flow for fire applications __ is often at very low speed. The filtered viscous stress tensor (t ), species diffusion (qm), and heat __ fluxes (q) may be modeled using Newton’s, Fick’s, and Fourier’s laws, respectively, in terms of resolved quantities thereby neglecting subgrid fluctuations. Assuming equal diffusivities of all species, simple expressions may be determined for these relations. 2 t = − m(T ) I ∇ ⋅ u + m(T )(∇ ⊗ u + (∇ ⊗ u )T ) 3

(7a)

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Transport Phenomena in Fires

m(T ) ∇Y m Sc

(7b)

1  1    1 −  ∇T + ∇h  + qrad Le Sc 

(7c)

q m = −

C q = − m(T )  P  Pr

The molecular viscosity (µ), Schmidt (Sc), Prandtl (Pr), and Lewis (Le) numbers in eqn (7) are functions of temperature and often are assumed to be close to that of air. Consistent with the assumption of negligible molecular SGS effects, the term u . t on the right-hand side of eqn (5) is approximated as u ⋅ t , where t is given in eqn (7a). The rest of the mixture weighted thermo dynamic properties such as the specific heat, CP, can be readily determined using the polynomial curve fits from existing data bases such as the CHEMKIN library [7]. For lack of established models, all SGS contributions to mixture weighted thermodynamic properties associated with filtering are often neglected, e.g. h = ∑ Ym hm = ∑ Ym hm (T). __ Several terms in eqns (4) through (6) require explicit SGS modeling. The term qrad in eqn (7c) is the contribution of the radiative heat flux to the total heat flux. The divergence of _________________ this term can __ be in turn expressed in terms of the local emission and irradiation as ∇ · qrad = –kP(4sT 4–G), where s is the Stefan–Boltzmann constant. The main contributor to the absorptivity, kP , is from the presence of soot for which Rayleigh scattering may____be assumed and therefore kP ∝T. The product of kP with the emission term then results in a T 5 correlation term. The closure of this term along with the irradiation, G, is the subject of other chapters in this book and will also be discussed later in this chapter with regard to FDF. The remaining second-order correlation quantities, Tab , in eqns (4)–(6) represent unknown T ≡ − r (a b − a b ). An abundance of SGS correlation for variables a and b, and are defined as: ab models are currently available for closing these terms. The most common is the use of dynamic Smagorinsky and gradient diffusion models [8]. The details of the implementation of these SGS models have shown to work reasonably well for a wide range of flow conditions and have been successfully applied in recent times to large-scale, non-reacting helium–air [8], and reacting methane–air plumes [9] that are representative of fire flow conditions. Modeling challenges still remain, nevertheless, especially with regard to the modeling of Tuu when insufficient grid resolution is available to capture near-field plume instability modes. As an example, Fig. 1 shows representative LES results from a 1-m diameter (D) helium plume for two stages of a puff cycle. Two instability modes may be observed. The first is the classical puffing mode instability with frequency 1.5/ D . The second instability is the appearance of ‘finger’ type of instabilities that form near the base of the plume that are readily apparent in Fig. 1(a). The formation and growth of these structures is still the subject of ongoing research; however, what is known is that if these small-scale flow features are not resolved, the near-field mixing processes are not captured and, consequently, the flow field predications are in error. For very large fires, it is conceivable that the near-field dynamics of the fires cannot be adequately resolved and therefore the burden of capturing these processes will fall onto the modeling of T uu. final term, The m , in eqns (5) and (6) accounts for the average production/destruction of w′′′ species. It is the modeling of this term which poses the greatest challenge to combustion simulation because of its exponential dependence on temperature. It is the closure of this term that is the focus of this chapter.

Combustion Subgrid Scale Modeling

(a)

331

(b)

Figure 1: Instantaneous snapshots of LES results for a 1-m diameter helium–air plume from DesJardin et al. [8]. The isosurfaces correspond to vorticity magnitude at 5% of the maxi mum with the superimposed vorticity transport equation source term, r∞ | ∇ r × g | / r 2, during the (a) early and (b) late stages of a puff cycle.

3 Combustion SGS models Many SGS modeling approaches for turbulent combustion have been pursued for use with LES that includes the eddy breakup [10], flamelet or conserved scalar approaches [9, 11–14], conditional moment closure (CMC) [15], FDF methods [16–20], ODT and/or linear eddy modeling (LEM) [21–26] approaches. Extensive up-to-date developments in turbulent combustion modeling may be found in the excellent works by Poinsot and Veynante [27], Fox [28], and Peters [11], and in the recent review articles by Givi [29], Veynante and Vervisch [30], Novozhilov [31], and Bilger et al. [32, 33]. The goal of the following discussion is not to provide an exhaustive review of available SGS modeling approaches, but rather to focus on a select few that show promise with regard to fire modeling for which the authors have experience using. The discussion of the SGS modeling is tailored towards non-premixed systems that are typical of fire environments during their growth stage. Later time flashover events when premixed or partially premixed combustion processes are important are not explicitly considered, although many of the combustion models that will be examined either have been, or can be, extended for use in these combustion regimes. 3.1 Filtered density function The first two SGS combustion models considered fall into the class of FDF approaches. FDF represents the probability distribution of the subgrid composition at a particular point in space at a given time and is constructed by a density weighted sampling over the filtering volume. Mathematically, this sampling is constructed by a collection of Dirac delta functions which map the physical variable, f, to the corresponding value in composition space defined by the variable, y, as illustrated in Fig. 2 for which only five samples are considered.

332

Transport Phenomena in Fires

Figure 2: Illustration of mapping of physical variables to composition space to define the FDF as defined in eqn (8).

The FDF is defined as the superposition of the Dirac delta functions. In the limit where every point in the SGS volume is considered, the superposition results in a weighted volume integration over the SGS volume, defined as, 1 F (f) = r

∫ r( x′, t )d (f ( x′, t ) − y)G( x′ − x)dx′

(8)

D

where G is the same as that defined in eqn (1). By construction, the FDF has the normalization property, 1

∫ F dy = 1.

(9)

0

It is clear from this definition that subgrid gradient information is lost with the sampling procedure and therefore must be supplemented via a SGS model. More specifically, later in the discussion it is shown that filtered averages of gradient products (i.e. condition dissipation) appear in the evolution equation for F for which mixing models are introduced. Once the FDF is known, all statistical moments of the flow-field can be determined via convolution integrals with the FDF. The first moments of the FDF are simply the LES filtered quantities. Higher-order moments of the FDF provide information on the higher-order correlations. Perhaps the greatest advantage of FDF approaches is that all non-linear source terms involving chemical reactions and radiation emission appear in closed form! These advantages are off-set by the requirements that much of the SGS mixing processes must be modeled. The expectation is that modeling of SGS mixing processes are easier than defining closures for the SGS quantities in eqns (3)–(6) directly. Before discussing the details of various FDF approaches, it is worth contrasting FDF approaches with probability density function (PDF) methods that have been used in RANS formulations [34]. The main conceptual difference is how the SGS probability distribution is constructed and interpreted. In PDF formulations, the PDF is rigorously defined through an ensemble of flow realizations for a statistically stationary flow. While the FDF has many of the mathematical properties of a PDF, strictly speaking, is not a PDF since the ensemble of realizations cannot be rigorously defined. Rather, the FDF is a representative set of SGS realizations at a particular time and location in space [35].

Combustion Subgrid Scale Modeling

333

3.1.1 Presumed FDF with flamelet The first SGS combustion model considered is a flamelet model with an assumed functional distribution for the FDF. This type of approach is attractive due to the simplicity and relatively low cost of implementation [11, 30]. In a flamelet modeling approach, the turbulent combustion SGS is assumed to be composed of an ensemble of small laminar diffusion flames referred to as ‘flamelets’ [11]. These flames can be either unsteady or steady. In the simplest description, a flamelet is assumed to be quasi-steady (i.e. for a single strain rate) for which the species mass fraction Ym can be expressed solely in terms of the mixture fraction Z, i.e. Ym(Z). The mixture fraction being defined locally as the amount of mass which originated from the ‘fuel stream’. For fire environments, this may be the fuel from a hydrocarbon pool or a burning solid surface. The relation, Ym(Z), can be shown to be exact if the diffusivities of all reactive scalars are assumed to be the same in the limits of pure mixing, infinitely fast chemistry (thin-flame sheet) and for chemical equilibrium. However, it also has been experimentally observed that the validity of this assumption may encompass a wider range of flow conditions [36]. Relating the species mass fractions to mixture fraction greatly simplifies the combustion modeling task by eliminating the need to solve eqn (6) for each species. Rather, in its place, the distribution of mixture fraction ( FZ ) within an SGS volume is tracked, and the filtered species composition is determined from it by the convolution integral of the product of Ym(Z) and FZ over the composition space, 0

Y m = ∫ Ym (x )F Z (x )dx.

(10)

1

In addition, in the presumed FDF approach, the functional form of the FZ is presumed, hence its name. Previous DNS studies indicate that a beta function is a reasonable approximation for FZ over the entire range of mixing states [37], defined as, F Z (x) =

Γ ( b1 + b2 ) b1 −1 x (1 − x) b2 −1 Γ ( b1 )Γ ( b2 )

(11)

)( Z (1 − Z ) /s 2 − 1)) are (Z (1 − Z ) /s 2 − 1)) and b2 (= (1 − Z where Γ is the gamma function, b1 (= Z Z Z parameters for the beta function that depend on Z and its variance sZ2 . The value of Z is often determined from a modeled transport equation, ∂( r Z ) + ∇ ⋅ ( r Z u) = ∇ ⋅ (TuZ − qZ ) ∂t

(12)

where standard closures can be used to model TuZ, similar to that used in eqns (5) and (6). The variance required in eqn (12) can be modeled using several approaches based on either a scalesimilarity approximation [14], gradient diffusion approximation [38] or solved using an modeled evolution equation [30]. With FZ now defined, the filtered species composition can be determined from eqn (10) if the state-relations, Ym(Z), are known. In the simplest case for fire modeling, we can assume infinitely fast chemistry for which the Ym(Z) is assumed to be a piecewise linear function,

Ym =

 Z Y | + (Ym |Z = Zst −Ym |Z = 0 )  m Z =0 Zst    Z − Zst + (Ym |Z =1 −Ym |Z = Z ) Y | st  m Z = Z st 1 − Z st 

for Z ≤ Zst (13) for Z > Z st

334

Transport Phenomena in Fires

where Zst is the value of the mixture fraction at stoichiometric conditions. Substituting eqn (13) in eqn (10), the following closed-form expression for the filtered composition can be obtained [13], Y m =

Ym |Z = Zst − Z st Ym |Z =1 1 − Z st +

+ (Ym |Z = 0 −Ym |Z =1 )I Zst ( b1 , b2 )

Ym |Z = Zst −Ym |Z = 0 + Z st (Ym |Z = 0 −Ym |Z =1 ) Z st (1 − Z st )  Z st (Ym |Z =1 −Ym |Z = Zst )  I Zst ( b1 , b2 ) + Ym |Z = Zst −Ym |Z = 0 + Z st (Ym |Z = 0 −Ym |Z =1 ) 

×

b1 b1 + b2

−

Γ ( b1 + b2 ) Z stb1 (1 − Zst ) b2   Γ ( b1 )Γ ( b2 ) b1 

(14)

where I Zst is the incomplete beta function evaluated at Z = Zst. As an example, Fig. 3 shows the ~ filtered CO2 mass fraction, YCO2, versus Z for several values of sZ2 for the single-step methane reaction, CH4 + 2(O2 + 3.76N2)→CO2 + 2H2O + 2(3.76 N2). As shown, increasing sZ2 results in lower values of composition mimicking the effects of SGS mixing. Along with the filtered species composition, the filtered reaction rate also requires evaluation for use in eqns (5) and (6). Under the flamelet modeling assumptions, the instantaneous reaction rate can

~ Figure 3: Filtered CO2 mass fraction versus Z with s2Z using single-step methane–air infinitely fast chemistry state relationships.

Combustion Subgrid Scale Modeling

335

be derived by re-expressing the spatial and time derivatives of eqn (6) in terms of changes in Z [36]: 1 w m′′′ = − rc  d 2Ym /dZ 2  2

(15)

where c (= 2Dm∇Z · ∇Z) is defined as the scalar dissipation rate. A more thorough mathematical justification for eqn (15) is detailed by Peters, where a variable transformation is used to re-express the conservation equations in terms of a local coordinate system attached to the flame surface [11, 39]. Using order of magnitude estimates of various terms and assuming the flamelet to be steady, Peters obtained the same expression given in eqn (15) as well as unsteady versions. The major assumptions for the use of the flamelet concepts for modeling turbulent flows are that (1) transport process along the flame surface are negligible, (2) the flamelets do not interact, and (3) the turbulent flame is uniquely represented as an ensemble of these smaller laminar flamelets. While some of these assumptions have recently come into serious question [32], flamelet modeling continues to find widespread use. Under these assumptions, the turbulent flame is described by the distribution of the flamelets which are in turn characterized by their mixture fraction and dissipation rate within the SGS volume. The filtered reaction rate required in eqns (5) and (6) may then be determined by a convolution integration of eqn (15) with the joint FDF of the mixture fraction and its dissipation rate, c [9], w m′′′ = −

r 2

cmax 1

∫∫ 0 0

F Z c c

d 2Ym dx 2

dx dc = −

r 2

1

∫ F Z 〈 c | x 〉 0

d 2Ym dx 2

dx

(16)

where the 〈… | x〉 notation represents a mass weighted conditionally filtered quantity on mixture fraction [18]. With ( FZ ) and Ym(Z) already defined, the remaining challenge is the determination of 〈c | Z〉. Physically, the conditional dissipation rate represents the average flame strain rate at the SGSs and therefore is a very important parameter controlling the overall reaction rate [11]. The challenge in modeling 〈 c | Z 〉 is evident by the large number of currently available approaches for closing this term that may be categorized into two general approaches. In the first approach, a statistical independence assumption between the mixture fraction (Z) and dissipation rate (c) is invoked allowing for 〈 c | Z 〉 to be approximated in terms of its mean (filtered) value, 〈 c | Z 〉 c , and a model for c is then proposed. Models for c include the use of a turbulent diffusivity model to account for SGS fluctuations [40, 41], scale-similarity approximations [14], and gradient diffusion models [42]. Using these approximations, the following closed-form expression for the filtered reaction rate can be obtained using the state-relations defined in eqn (13) [13], w m′′′ =

rc Ym |Z = Zst −Ym |Z = 0 + Z st (Ym |Z = 0 −Ym |Z =1 ) Γ ( b1 + b2 ) × Z st( b1 −1) (1 − Z st )( b2 −1) (17) 2 Z st (1 − Z st ) Γ ( b1 )Γ ( b2 )

Figure 4 shows an example using this approach where the production rate of CO2 is plotted as a function of Z for a single-step methane-air reaction. As shown, the filtered reaction rate decreases below the laminar limit (as sZ2 → 0) with increasing SGS mixing, i.e. increasing sZ2 . As discussed by DesJardin, results using eqn (17) significantly underpredict the extent of reaction resulting in low values of temperature and stream-wise velocities when compared to

336

Transport Phenomena in Fires

~ Figure 4: Normalized filtered CO2 reaction rate versus Z with increasing SGS variance (s2Z) for single-step methane–air infinitely fast chemistry state relationships.

experimental data for large fire plumes [9]. Various alternatives [41, 42] to modeling c were also explored in this study but resulted in similar disappointing results. It appears that there is a more fundamental issue at the root of the problem. The basic problem is that the assumption of statistical independence between Z and c is not valid for transitionally turbulent flows. Even for very large, highly turbulent pool fires, the flow undergoes a laminar to turbulent transition near the ground where most of the reactions take place. This process is different from high Reynolds number combusting jets. For jet flows, transition to turbulence from shear forces occurs far upstream of the location where the bulk of combustion processes occur. Models for c that rely on statistical independence are therefore expected to perform better in application to jets than for pool fires. An improvement to modeling 〈 c | Z 〉 is using an assumed functional form 〈 c | Z 〉 c st f ( Z ), where f (Z) defines the functional dependence of c on mixture fraction and is usually modeled using analytical solutions from opposed jets [43] or mixing layers [39]. Alternatively, models for 〈 c | ξ〉 can be directly derived from the assumed FDF. For homogeneous turbulent flows the time evolu~ tion for FZ is given as, ∂ F Z 1 ∂2 =− (F Z 〈 c | x 〉) 2 ∂x 2 ∂t

(18)

where, as evident in eqn (18), 〈 c | ξ〉 explicitly appears in the transport of FZ , therefore if FZ is known then 〈 c | ξ〉 can be determined by the integration of eqn (18) twice over the mixture fraction space. In other words, the choice of 〈c | Z 〉 and FZ are not mutually exclusive and are intimately related through the transport equation for FZ . This relationship has been exploited extensively in

Combustion Subgrid Scale Modeling

337

the context of homogeneous turbulence for which analytical relations can be derived for a variety of assumed FDFs [28].    c (t )  1/ 2  1 + sin(2πsZ2 (t ))   −1 2 〈 c | x 〉 =  c (t )  exp[ −2(erf (2x − 1)) ] 2 − π s 1 sin(2 ( t ))  Z     c (t ) 1  1 + 2 | x − 〈 Z 〉 |     sZ (t ) 2  

for a Gaussian FDF for an AMC FDF

(19)

for a Laplace FDF

In addition, the conditional scalar dissipation for the assumed beta function used here also can be determined but requires numerical integration [28]. For non-homogeneous turbulent flows, a transport equation analogous to eqn (18) may be derived for FZ [9], r

 ∂F  D F Z = r  Z + u ⋅ ∇ F Z  Dt  ∂t  = − r 〈 u ′′ | x 〉 ⋅ ∇F Z + ∇ ⋅ ( r Dm ∇F Z ) −

r ∂2 ( F Z 〈 c | x 〉) 2 ∂x 2

(20)

where 〈u″ | ξ〉 is the conditionally filtered velocity fluctuation. This term can be modeled using a gradient diffusion model, i.e. − r 〈 u ′′ | x 〉∇ F Z ∇ ⋅ ( r DT ∇ F Z ), where DT is the turbulent diffusion coefficient. Again, 〈c | ξ〉 explicitly appears in the transport of FZ . Integration of eqn (20) twice over the mixture fraction space becomes more complicated and an analytical expression is not attainable; however, a transport equation for 〈c | ξ〉 can be obtained [9], 〈 c | x〉 = −

 2  D ΦZ  r ( D + D )∇Φ   − ∇ ⋅ r m T Z   r F Z  Dt 

(21)

The function ΦZ in eqn (21) is defined as the integration of the cumulative distribution function (CDF) and is related to FZ by the following relations. x

Φ Z (x ) = ∫ CDFZ (z )dz 0

(22a)

z

CDFZ (z ) = ∫ F Z (ς )dς 0

(22b)

Substitution of eqn (21) in eqn (16), along with the state relationships from eqn (13) results in the following closed-form evolution equation for the filtered reaction rate [9]. w m′′′ =

Z st (Ym |Z = 0 −Ym |Z =1 ) − (Ym |Z = Zst −Ym |Z = 0 ) Z st (1 − Z st )  D ΦZst    × r − ∇ ⋅  r ( Dm + DT )∇ΦZst   Dt  

(23)

338

Transport Phenomena in Fires

Representative LES results using the SGS flamelet models using eqns (17) (SLFM #1) and (23) (SLFM #2) from the study of DesJardin [9] are shown in Figs 5 and 6 for a large 1 m diameter methane–air plume for the experimental conditions of Tieszen et al. [44]. Figure 5 is an instantaneous snapshot of the flow-field for an isocontour of the vorticity magnitude with the buoyancy production of vorticity, r∞ | ∇r × g | /r 2 , superimposed. As shown, much of the vorticity generation occurs near the base of the fire plume where the mismatch of the density and pressure gradients are the largest. Figure 6 shows comparisons of time averaged and RMS velocity profiles from the LES using SLFM models #1 and #2 with comparison to experimental data at a downstream location of x = 0.6 m. The vertical bars on the experimental data (shown with symbols) define the uncertainty bounds of the experimental measurements of ±20%. The use of SLFM #1 results in a gross under prediction of both the average and RMS stream-wise velocity with differences as large as 40%. Comparisons using SLFM #2 with the data show a dramatic improvement in agreement, with a maximum error of less than 10%. The reason for the dramatic failure of SLFM #1 is attributed to the laminar-to-turbulent flow transition that occurs at the base of a fire for which the assumption of statistical independence is no longer valid. Accounting for this transition using SLFM #2 results in very good agreement in LES predictions of mean and RMS velocity to experimental data. The modeling of the conditional dissipation is therefore extremely important when using flamelet models with application to fire environments. In summary, assumed FDF approaches work well in predicting the local heat release rate which defines the overall temperature and velocity distributions for fire flow environments. Recently, Raman et al. have also explored relaxing the prescribed FDF assumption by solving for FZ explicitly via transported FDF approaches (to be discussed) in the context of bluff body

Figure 5: Instantaneous LES snapshot from DesJardin [9] showing an isosurface of vorticity magnitude at 5% of the maximum with the vorticity transport equation source term r∞ | ∇ r × g | / r 2 superimposed.

Combustion Subgrid Scale Modeling

(a)

339

(b)

Figure 6: Comparison of (a) time-averaged mean and (b) RMS stream-wise velocity profiles at a downstream location x = 0.6 m from DesJardin [9]. Solid lines are LES predictions and symbols are experimental data. stabilized combustion [45]. Their results comparing transported FDF with assumed beta FDF show very little difference in predictions of temperature and reactive species. Therefore, assuming a beta function for FDF may not be overly restrictive; however, this assumption has not yet been explored for fire environments. As will be discussed, the real advantage of transported FDF approaches is that chemical reactions appear in closed form and therefore no additional modeling is required. 3.1.2 Transported FDF Transported FDF approaches have largely been pioneered by Givi et al. for application to jet and free-shear flows [16–20]. In this approach, closure is shifted from unresolved correlation quantities, as in traditional LES, to unclosed conditional averages in the FDF transport equation whose closure is probabilistic in nature. The use of standard models from PDF formulation for these terms has been applied with some success [16–20]. Solution of the FDF equation requires Monte-Carlo numerical approaches involving the numerical integration of a system of stochastic differential equations (SDEs) based on a Wiener–Levy process that reproduce the same statistical moments as the original FDF [29]. The advantage of using this approach is that, in principle, there are no limitations to the modeling approach to either the type of combustion regime or the number of reactions that may be included. In addition, as stated previously, all non-linear source or sink terms appear in closed form. We begin our discussion of transported FDFs by first considering the relaxation of the presumed FDF assumption discussed previously. While the presumed FDF may be reasonable for some class of flows, it is expected to be limited in regions of the flow that are highly intermittent. It may therefore be desirable to solve for the evolution of FZ that is given in eqn (20) directly. In this case, a model for 〈c | Z〉 must be specified. Available models include the interaction exchange with the mean (IEM) [46], the coalesce and dispersion model (CD) [47], the Euclidean minimum spanning tree (EMST) [48] and the Fokker–Planck (FP) model [49]. An excellent review of these and additional models along with general guidance on the construction of new mixing models is given by Fox [28]. The simplest of these is IEM where the term containing 〈c | ξ〉 is modeled as: −

∂2  r  ∂  ΩM (x − Z ) r F Z  〈 c | x 〉 F Z  2 2 ∂x   ∂x

(24)

340

Transport Phenomena in Fires

where ΩM is defined as the stirring frequency that can be related to the filtered dissipation rate, 〈c〉, and variance as ΩM = 〈c〉/s2Z [46]. Under conditions of local turbulence equilibrium, 〈c〉 is proportional to the time rate in change of the variance e = CΩs2Z/t, where CΩ is a constant of proportionality and t is time scale for small-scale mixing processes. For LES, a time scale that can be used for t–1 is (Dm + DT)/∆2f which when used in the definition of e and substituted into ΩM results in ΩM = CΩ(Dm + DT)/∆2f , where CΩ 1 − 3 [16]. Substituting eqn (24) in eqn (20) provides a modeled equation for FZ . To simplify this equation for developing Monte-Carlo solution methods, a mass-weighted FDF, FZ, is introduced, defined as FZ = 〈r〉 FZ . ∂FZ ∂2 Z ) + ∇ ⋅ [ r ( DT + Dm )∇( FZ /r )] + 2 (〈 c | x 〉 FZ ) = −∇ ⋅ (uF ∂t ∂x     1 = −∇ ⋅   u + ∇[ r ( DT + Dm )] FZ  + ∇2 [( DT + Dm )FZ ] r    

+

∂ [ΩM (x − Z )FZ ]. ∂x

(25)

Equation (25) contains derivatives in time, space, and composition space. While the dimensionality of this equation is amenable to standard discretization methods for PDEs, there are several practical limitations. More commonly it is advantageous to exploit the probabilistic nature of the FDF transport equation and use the ‘principle of equivalent systems’ to develop an equivalent system of SDEs that are solved using Lagrangian Monte-Carlo methods. This system of SDEs is established by first comparing eqn (25) to a generalized FP equation for the PDF, p. (See discussion on pp. 95–98 of ref. [50] on Ito’s formula and connection between the FP equation and the associated SDE.) n ∂p ∂ 1 n n ∂  = −∑ Ai ( x, t ) p]+ ∑ ∑ B( x, t ) BT ( x, t ) [ ∂t 2 i =1 j =1 ∂ri ∂rj  i =1 ∂ri

(

)

i, j

p  

(26)

where A and B are defined as the drift vector and diffusion matrix, respectively. The variable ri (26) is a vector of random variables on which p depends. This may include both appearing in eqn spatial variables (x) as well as properties of the flow such as temperature, mixture fraction, etc. By comparing eqn (25) with eqn (26), a direct connection can be established for the drift vector with r = {xi, x}T and A = {u j + (∂( r ( DT + Dm ))/∂x j )/r ,Ω M (x − Z )}T . As for the diffusion matrix, if B is 2 2 2 = B22 = B33 = 2( DT + Dm ). Making these substituassumed to be diagonal then B = BT and B11 the same equation as eqn (25), with F replaced by p. The advantions in eqn (26) results in exactly Z tage of establishing a connection between the FDF and FP equations is the wealth of established mathematical theory and numerical methods that have been developed for the solution of the latter [50]. More precisely, for the FP equation it can be shown that an equivalent system of SDEs may be constructed (within the context of Ito calculus) which will reproduce exactly all the statistical moments for the evolution of p given in eqn (26) that has the following functional form, (27) dr + = A(r + , t )dt + B( r + , t ) ⋅ dW (t ). In eqn (27) Wi is a Wiener–Levy process with the property: dWi = ζi(dt)1/2, where ζi is a Gaussian random variable with zero mean and unity variance. For the simple case considered, substitution of A and B results in the following system of SDEs, dx + = {u + ∇ [r ( Dm + DT )]/r }dt + 2( Dm + DT )dW (28a)

Combustion Subgrid Scale Modeling

dx + = −ΩM (x + − Z )dt .

341 (28b)

When particle-based methods are used to solve the system of SDEs, interpolation of the properties on the RHS of eqn (28) from an Eulerian grid is required. The collection of Monte-Carlo particles can then be used to construct the local composition of the reacting flow field via flamelet relations. There are numerous details involved in obtaining a stable numerical solution using this hybrid approach. The most important of which is the statistical noise that is introduced constructing the local composition field, by virtue of the finite number of Monte-Carlo particles, and the propagation of this noise as the system of equations are integrated in time [45]. Hybrid finite-difference FDF-flamelet approaches have recently been used by Sheikhi et al. [51] for application to turbulent diffusion flames and by Raman et al. [45] for application to bluff body stabilized flames. Representative results from Sheikhi et al. are shown in Fig. 7 with comparison to Sandia experimental data for Flame D [52]. As shown, the use of the hybrid method avoids much of the numerical dispersion errors obtained using center-based finite difference methods. The agreement to the experimental data for mixture fraction PDF in Fig. 7(b) is exceptional. To the authors’ knowledge the hybrid LES-FDF approach has not yet been applied for use in fire applications; however, there is no reason to believe that a comparable level of success cannot be achieved. The use of the FDF approach can also be readily extended to include finite rate chemistry effects that are important in the prediction of fire flashover and ignition events. The use of FDF to include finite-rate chemistry is a straightforward extension of the previous example. In this case, the FDF comprises the entire scalar composition space consisting of species mass fractions for each of the reactive scalars and temperature, i.e. F = F(f) where f = {Ym,T} is the array of primitive variables and is governed by the following generalized transport equation [18], ∂( rfa ) + ∇ ⋅ ( rufa ) = ∇ ⋅ (g∇fa ) + rSa ∂t

(a)

(29)

(b)

Figure 7: Representative results from Sheikhi et al. [51] using a hybrid finite-difference FDFflamelet modeling of the Sandia D flame showing (a) an instantaneous snapshot of filtered mixture fraction and (b) comparisons of predicted (solid line) and experimentally measured (symbols) PDF of mixture fraction.

342

Transport Phenomena in Fires

for which a low Mach number approximation has been employed that allows for a decoupling of the pressure and density so the energy equation can be written in the above form. The corresponding mass-weighted FDF modeled transport equation is [18],     (g + gm )  1 ∂F F = −∇ ⋅   u + ∇ gT + gm   F  + ∇2  T r r ∂t     

+

∂〈 Sa | fa = ya 〉 ∂ . [ Ω M (ya − fa )F ] − ∂ya ∂ya

(30)

Comparing eqns (25) and (30) shows that the modeled mixing term now contains all the reactive scalar components (repeated a implies summation). In addition, the last term on the righthand side of eqn (30) represents the production or destruction of chemical species from chemical reactions. For homogeneous gas-phase reactions, the chemical reactions are solely a function of temperature and composition. For this case, the conditional average of the source is simply the source term expressed in terms of the composition variables, i.e. 〈Sa | fa = ya〉 ≡ Sˆa(ya), hence this term is in closed form and no additional modeling assumptions are required [34]. This feature of composition based FDF approaches is extremely appealing since it is these non-linear source/sink terms which have historically plagued moment-based modeling approaches. This property also can be taken advantage of when it comes to closing the radiative emission. Since the emission term is a function of composition and temperature only, it too appears in closedform [53]. Considering these advantages, composition-based FDF approaches appear to be well suited for fire applications for which the flow is both reacting and strongly radiating. The equivalent system of SDEs for eqn (30) is a direct extension of the single passive scalar case, dx + = {u + ∇(gm + gT ) /r } dt + 2(gm + gT )/r dW

(31a)

dfa+ =  −ΩM (fa+ − fa ) + Sˆa (f + ) dt

(31b)

Up to this point, only the species mass fractions and temperature (or enthalpy) have been considered in the composition space of the joint FDF. A further generalization of the FDF approach is to include velocity into the composition space resulting in a full velocity-scalar formulation [20, 51]. The advantage of including the velocity in the composition space is to eliminate the need for modeling the conditional velocity. The increase in modeling fidelity is off-set by the increase in computational cost, therefore this approach has only been applied to fairly simple flows. The full joint mass-density FDF transport equation is quite lengthy but functionally it is similar to eqn (30). However, there is one distinct difference. In the full FDF approach, mixed second-order derivatives appear in the modeled form of the equation, e.g. ∂2( )/∂xj∂ya and ∂2( )/∂uj∂ya. The appearance of these terms results in off-diagonal terms in the diffusion matrix of the equivalent system of SDEs given in eqn (27). In principle, there are many variations in B that will result in the same B BT used in eqn (26). It is, therefore, quite difficult to start with a the equivalent system of SDEs. Rather, in practice it is often modeled FDF equation and deduce useful to start with a parameterized system of SDEs and the equivalent FP equation is then determined. The coefficients of the parameterized FP equation are then matched to the transported FDF equation through trial and error. Details of this approach are given in ref. [20]. Current research in FDF methods includes the extension to multiphase flows [2, 54]. For fire applications, this will be necessary for modeling soot formation and fire suppression processes using liquids. A two-fluid extension of the full velocity-scalar FDF approach for multiphase

Combustion Subgrid Scale Modeling

343

flows has been pursued by Carrara and DesJardin [2]. In this approach, a two-phase fine-grain density function is defined using a phase indicator that is related to a level set function. In addition to the transport terms given in eqn (30) and additional terms resulting from the mixed second-order derivatives (noted above), phase interaction terms also arise that require closure. However, these terms, can be shown to be directly related to the source terms that arise in standard phase-averaging approaches for multiphase flows and are therefore known for a given twophase flow problem. Figure 8 shows the results obtained using this approach for a 2D temporally developing mixing layer containing evaporating droplets. Figure 8(a) shows droplet temperature with superimposed droplet number density isolines. Droplets can be seen to congregate in regions of large shear, consistent with previous DNS studies. Figure 8(b) shows comparisons of momentum thickness with DNS data – the agreement is very good. In addition to the extension to multiphase flow, mixing models are continually being improved for use with the FDF method. Fox suggests that next generation mixing models should also be sensitive to the type of chemical kinetics – reinforcing a key property that transport processes and chemical kinetics are really not separate processes but rather are intimately linked together. (See discussion on p. 270 of ref. [28].) The sensitivity of the mixing processes to the species that are present can be seen by a more careful examination of the modeled FDF transport equation given in eqn (30) where the following IEM-based mixing model was employed.  ∂fa ∂fb F ∂2 ∂ Ω M (ya − f )F  |y  gm =− a  ∂ya ∂yb  ∂xi ∂xi r  ∂ya 

(32)

It is clear from eqn (32) that the unknown conditional dissipation mixing term contains information regarding all the participating species. However, the IEM mixing model neglects these interdependencies and expresses mixing of species only in terms of its local mean. Fox summarizes several models that are improvements to IEM but suggests that developing a comprehensive mixing model to account for all inter-species mixing sensitivities may be as challenging as solving the original reacting Navier–Stokes equations from first principles [28]. It may then be desirable to consider modeling approaches for which the details of the molecular mixing and chemical reactions are treated together without approximation. Such an approach needs to include mixing models that

(a)

(b)

Figure 8: Temporally developing shear layer using two-phase LES-FDF approach showing (a) instantaneous snapshot of droplet temperature with superimposed droplet number density isolines and (b) momentum thickness with comparison to DNS results.

344

Transport Phenomena in Fires

include multi-point information. One example of such a model is the ODT methodology developed by Kerstein [55], which is discussed next. 3.2 One-dimensional turbulence In the previously discussed presumed and transported FDF approaches, the statistical nature of the flow is represented as a single-point statistical quantity. As a consequence, any process defined in terms of a spatial gradient, or more specifically its statistics, requires an explicit model. Molecular diffusion processes therefore have to be modeled at some level. Recent studies have highlighted the importance of molecular transport properties in addition to detailed kinetics [56]. It is desirable to have a SGS modeling methodology that can incorporate both detailed chemistry as well as molecular transport processes without approximation. One approach is the ODT modeling of Kerstein [55]. In this approach, a one-dimensional domain is used to resolve all relevant turbulent and molecular processes. The trade-off of this simplicity means that the threedimensional turbulent mixing characteristics of the flow must be modeled. In ODT, the effect of turbulent mixing is treated by performing a collection of re-arrangement eddy mapping events that serve to transport physical quantities as well as to increase the local scalar gradients. Mapping events occur at different locations, length scales, and temporal frequencies as determined by a stochastic model and the local instantaneous state of the one-dimensional domain. Although several different mappings have been explored, the ‘triplet map’ is currently preferred. An illustration of a triplet mapping event is shown in Fig. 9. The mapping consists of the replacement of a 1D profile on the sampled segment (eddy length, l) by three identical copies compressed to one-third of their original length, with the middle copy inverted. The result of an eddy event then maps the scalar f(x)→f(f(x)) where f is the mapping function with the following definition [55], 3( x − xo ) 2l − 3( x − x )  o f ( x ) ≡ xo + 3( x − x ) − 2l o   x − xo

if xo ≤ x ≤ xo + l /3 if xo + l /3 ≤ x ≤ xo + 2l /3 if xo + 2l /3 ≤ x ≤ xo + l

(33)

otherwise

where xo is the starting point for the triplet map. The rate and location of the triplet mapping events are assumed to follow the Poisson processes. The probability that an eddy within the size range [l, l + dl] and location range [xo, xo + dxo] will take place during [t, t + dt] is equal to l(l; xo, t)dldxo dt where the eddy rate distribution, l is defined as: l = 1/l 2t [55]. The time scale, t(l; xo, t), in turn depends on the flow-field and is determined based on a phenomenologically

Figure 9: Turbulent convective stirring from a single triplet-mapping event.

Combustion Subgrid Scale Modeling

345

based balance equation for the turbulent kinetic energy generated from eddy events, (l/t)2, which includes production/dissipation terms that are based on scaling arguments. The exact functional form of the driving mechanisms for eddy production is, in general, problem dependent. For fire applications, the principal mechanisms for generating turbulent kinetic energy are from buoyancy forces and velocity shear. Following previous simple implementations of ODT, a balance equation for eddy turbulent kinetic energy may be specified as [57, 58], 2

l   2 Ad u ) +  t(l; x, t )  = (

shear production ∆KE

 lg dr  B  r 

buoyancy production

2

−

 l  Z   td 

(34)

viscous dissipation

__

where r is the average density of the eddy. The terms on the right-hand side of eqn (34) represent the production of turbulent kinetic energy from shear and buoyancy forces and a sink term from viscous dissipation. The motivation for the functional form of the shear production and viscous dissipation terms is discussed by Kerstein [55] where td (= l 2/16n) is a viscous time scale, and du and dr represent bulk differences in velocity and density across the eddy and are defined in terms of averages across the right and left sides of the eddy as: 2 du (or dr ) =  l 

xo + (l / 2)

∫

 u( x, t ) (or r( x, t ))dx   xo + (l / 2) xo + l

u( x, t ) (or r( x, t ))dx −

xo

∫

(35)

The buoyancy production term in eqn (34) is unique to buoyancy driven flows simulated on a horizontal ODT domain and is included to account for the physical processes of air engulfment from a long wave length instability modes, such as plume puffing seen in Fig. 1 [58] (a different buoyancy representation is used on a vertical ODT domain [59–61]). The motivation for the functional form of this term first comes from considering the production term in the vorticity transport equation for buoyancy driven flows, Dw ∇r × g ∼ r∞ Dt r2

(36)

for which a time scale for eddy production can be estimated for a given eddy. 1 t2

1 dr g r l

(37)

Multiplying eqn (37) by l2 then provides a measure of the energy production. Using eqn (34), the eddy rate distribution can then be determined and is given as, l=

1 l t 2

=

n l 1 = 4 3 tl l

A2 Rel2 + B

r∞ Grl − 162 Z r

(38)

where Rel (≡ dul/n) and Grl (≡ gbdTl 3/n 2) are the eddy Reynolds and Grashof numbers, respectively. The remaining constants A, B, and Z in eqn (38) are, in general, problem dependent but presumably do not vary too much from one problem to another. As discussed by Kerstein, the probability density given in eqn (38) can, in principle, be sampled by first constructing the distribution for all possible values of l and xo and then sampling from that distribution [55]. However, the cost of implementing such an approach is prohibitive and therefore a

346

Transport Phenomena in Fires

generalized rejection method is pursued as discussed in ref. [55]. In this case a trial joint PDF for eddy size and location is first assumed and used to obtain, lassumed = p(l)g(xo)/∆tstir, where p(l) and g(xo) define the probability density for the size and location, respectively and ∆tstir is the stirring time step [55]. The rejection method is implemented by first sampling from p(l) and g(xo) to determine the size and location of a trial eddy. Once a trial eddy is selected then l is computed for that eddy using eqn (38), and an acceptance probability Pa = l/lassumed = l∆tstir/(p(l)g(xo)), is determined. Another random number, RN, is then sampled. If RN is less than Pa then the eddy is implemented, otherwise the eddy is rejected. This approach for implementing eddy events may be viewed as a generalized rejection approach to sample a given PDF. (See discussion on pp. 229–230 of ref. [62].) Since the use of the rejection method is probably the most challenging aspect of the ODT model to understand, a further discussion of the rejection method is warranted. In a rejection method, an invertible surrogate PDF (Pref) is used to sample from a known, but non-invertible PDF (Pdesired). As a simple illustration of this approach, Fig. 10 shows examples of sampling from a presumed PDF using a different ‘trial’ PDF using the rejection method. Figure 10(a) shows the construction of a top-hat distribution by sampling from a normal distribution with zero mean and unit variance. The PDFs for these distributions are as follows,  − x2  exp   2π  2 

1

Pref = Pnormal =

  1 1   Pdesired = Ptop-hat =  H  x +  − H  x −    2 2   

(39a)

(39b)

where H is the Heaviside function. In this case, both PDFs are actually invertible so that the following relations may be derived for sampling. + xnormal = 2erf −1 (2 RN − 1)

(40a)

+ xtop-hat = RN − 1/2

(40b)

where 0 ≤ RN ≤ 1 is a random number. However, sampling of the top-hat PDF distribution given in eqn (40b) may also be achieved by first sampling the normal distribution given in eqn (40a)

Desired pdf Assumed pdf Sampled pdf

1

pdf

0.8 0.6 0.4 0.2

(a)

0 −3

−2

−1

0 x

1

2

3

(b)

Figure 10: Illustration of rejection method for sampling used in ODT showing (a) construction of a top-hat PDF distribution using a normal distribution and (b) construction of the LEM PDF distribution using a normal distribution. Note y-axis is log-scale.

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347

which will provide a trial value for x+. A second random number, RN2, is then sampled and compared to the acceptance probability defined as: Pa =

Pdesired /Pref ( Pdesired /Pref ) |max

(41)

where (PdesiredPref)|max is the maximum of the ratio of the unknown to surrogate PDF and provides a normalization factor such that 0 ≤ Pa ≤ 1 [62]. If RN2 ≤ Pa then the sample is accepted, otherwise it is rejected. For this ___example, (Pdesired/Pref)|max occurs at |x| = 1/2 and has a value of (Ptop-hat/ Pnormal)|max = (1/2)/[(1/√2π )exp(–1/8)] = 1.42 (crossing location of Ptop-hat and Pnormal is shown in Fig. 10(a)). The symbols shown in Fig. 10(a) comprise a re-constructed Ptop-hat using this approach. The number of accepted trial samples divided by the total number of samples is directly proportional to [(Pdesired/Pref)|max]–1 = 0.704; therefore, the closer the shape of the surrogate PDF shape is to the actual PDF of interest, the greater the number of samples that will be accepted. For this case, approximately three-quarters of the total samples are accepted. The advantage of this approach is that not only does Pdesired not have to be invertible, but in addition, it only has to be evaluated at the composition location that is used to sample Pref; therefore Pdesired does not need to be constructed if it is not known. The trade-off for this level of generality is that many more samples are required than simply using eqn (40b) directly to sample Pdesired. This can become problematic if the shapes of the Pref and Pdesired are very different. Consider the results shown in Fig. 10(b) showing the rejection method using a Gaussian PDF to sample the LEM PDF given as [63], Pdesired = PLEM =

3− p

l   3 /p −1   L  ReT − 1  L 

p −4

.

(42)

In eqn (42) ReT | = (L/hK)p) is a turbulence Reynolds number assumed to be equal to 90, p is a high Reynolds number scaling factor taken usually as 4/3, and L is the integral turbulence length scale. In this case, the ratio of [(Pdesired/Pref)|max]–1 = 0.0484, which is much lower than the previous example, and therefore requires many more samples to achieve the same number of accepted samples for constructing statistics. The challenge in applying the rejection method for use with ODT is that Pdesired/ Pref |max is, in general, not known a priori. However, as long as Pdesired/Pref |max is chosen to be sufficiently large, the statistics constructed from accepted samples are invariant. Or in other words, as long as Pa is sufficiently small, the statistics are independent of Pdesired/Pref |max. This generalization of the rejection method is precisely the method employed in the ODT. To ensure that Pa(=l∆t/(p(l) g(xo))) is sufficiently small, ∆tstir is reduced until the statistics of interest no longer change. Kerstein suggests that Pa < 0.1 is sufficient for this to be the case. Stand-alone implementations of ODT usually involve the construction of ensemble or timeaveraged statistics, depending on if the implementation is a temporal or spatially dependent formulation. A sequence of realizations of the flow-field is then simulated in the context of a Monte-Carlo simulation. Previous studies using ODT and its predecessor (LEM) [21, 22, 64–66] have shown this modeling approach to be successful at reproducing single-point statistical moments of flow variables when compared to DNS and experimental data for simple flows [55]. Recent application of ODT to buoyancy driven vertical slot convection shows that the simulation results agree very well with DNS for mean temperature and velocity, and has also reproduced established Nusselt number scaling effects on Rayleigh number (Ra) The exploration of ODT for use in buoyancy driven flows, characteristic of fire environments, has only recently been initiated. An example of such an approach is the work of Shihn and DesJardin for simulating turbulent flows in the near-wall region of a vertical wall [58]. The goal of this research is to eventually predict fire

348

Transport Phenomena in Fires

spread along vertical surfaces. Fire spread is highly sensitive to the conjugate heat transfer involving the heating and off-gases of the decomposing material and the combustion of these gases in the near-wall region, which in turn serve to enhance the heat flux to the wall. This process is further complicated by local turbulent mixing processes whose exact influence on flame spread is still not well understood. As a first step for using ODT as an SGS model in an LES, both temporal and spatial stand-alone ODT models are explored. In both versions the one-dimensional domain is oriented normal to the wall as illustrated in Fig. 11. For the temporal implementation, the ODT domain is regarded as a moving Lagrangian __ domain along the wall with a mean velocity, v, as shown in Fig. 11(a). On this domain conservation equations for temperature and velocity are solved. ∂v ∂2 v g b =n 2 + (T − T∞ ) ∂t r ∂x

(43a)

∂T ∂ 2T =a 2 ∂t ∂x

(43b)

In eqns (43a) and (43b) all transport properties are assumed constant and a Boussinesq approximation is used to relate changes in density to temperature. These equations are the same as those used in the heated cavity study of Dreeben and Kerstein [67] except in this case, the mean pressure gradient is set equal to zero. Dirichlet boundary conditions are imposed for the temperature and velocity at the ends of the domain. Equations (43a) and (43b) are integrated in time using standard Euler time advancement with second-order centered differences. The spatial location of the mov_ ing ODT domain is determined by assuming the entire domain is advected at a bulk velocity, n b, which can then be used to determine the downstream location, y, using the following relation, t

y(t ) = ∫ vb (t )dt

(44)

0

___

For this study, vb(t) is defined as the ratio of momentum flux integral to the mass flux integral across the 1-D domain consistent with previous temporal implementations of ODT [23]. L

vb (t ) =

∫ rv

2

0 L

( x, t )dx (45)

∫ rv( x, t )dx 0

(a)

(b)

Figure 11: Stand-alone ODT implementation for a heated vertical wall using (a) a temporal formulation and (b) a spatial formulation.

Combustion Subgrid Scale Modeling

349

In the second implementation, a collection of fixed ODT domains is considered forming a two-dimensional Cartesian grid, as shown in Fig. 11(b). For this configuration, a parabolic formulation is pursued similar to the approach of Wei [68]. In this approach, a statistically stationary state is attained at each level, yielding time-averaged properties and convection terms that serve as inflow conditions for the next downstream ODT domain. The spacing between successive downstream ODT domains, ∆y, is much larger than the viscous and thermal sub-layer thicknesses (smallest relevant length scales), therefore time-averaged quantities are the only meaningful information to be exchanged between ODT domains. Decreasing ∆y to much smaller distances would effectively require DNS resolution in the near-wall region, negating the computational savings of using the one-dimensional approximation in the first place. The instantaneous equations for an ODT domain are expressed in a parabolized formulation, resulting in the following continuity and velocity and temperature transport equations. ∂u ∂v =− ∂x ∂y

(46a)

∂v ∂2 v g b  ∂v   ∂v  = − u  − v  + v 2 + (T − T∞ )  ∂x   ∂y  r ∂t ∂x

(46b)

∂T ∂2T  ∂T   ∂T  = −u −v +a 2    ∂x   ∂y  ∂t ∂x

(46c)

The above equations are the same as the basic equations in the Boussinesq approximation for the natural convection in the laminar regime. The only exception is the temporal term in the transport __ __ equations. _ In eqns (46b) and (46c), u and v are the time averaged velocity at each ODT node, where v is constructed using an iterative time relaxation procedure at each downstream location. __ __ The continuity equation is then used to get u given the converged value for v. In this approach, the transport equations are numerically discretized as follows, ∗  f i , j − f i , j −1  Γ∆t n+1 + fi ,nj+1 = fi ,nj − ∆t  v i , j f − 2fi ,nj+1 + fi −n1+,1j + Sfn   ∆x 2 i +1, j ∆y  

(

)

(47)

where Γ = v,a is the generalized transport coefficient ___ and f = v, T is a scalar that has a general advection, diffusion, and source terms. In eqn (47) f∗i,j–1 is time-averaged f at the nearest upstream ___ ∗ location. The value of f represents a tentative time-averaged value of f at the current i, j node. i,j ___ To determine f∗i,j, eqn (47) is first integrated in time using an implicit solver until a statistically stationary state for the scalar fields are obtained. Time-averages are then constructed to determine ___ f∗i,j which is then substituted ___ ___ back into eqn (47) and the processes repeated until convergence is obtained for which f∗i,j → fi,j. It should be noted that other parabolic ODT formulations are also possible. Recently, Ashurst and __ Kerstein reformulated the temporal-based sampling procedure by replacing ∆tstir with ∆y/ v and reformulating the governing to include only spatially dependent quantities and constructing a far-field boundary to account for entrainment. (See the extensive discussion in Appendix B of ref. [69].) In both the temporal and spatial implementations of ODT for this study, the trial PDF for p(l) used for the rejection-based sampling is the LEM PDF given in eqn (42) which may be readily inverted to allow for sampling using the following expression [70], 1



 p −3





l + = L RN  1 − ReT3 /p −1  + ReT3/p −1 

while the probability for the location g(xo) is assumed to be a top-hat uniform distribution.

(48)

350

Transport Phenomena in Fires

As discussed by Kerstein, the eddy selection procedure may occasionally result in the occurrence of non-physically large eddies that will dominate the overall scalar evolution. To remedy this issue, a large-scale eddy suppression mechanism is introduced. In this study, the median model is implemented [55]. In this approach, a linear profile across the eddy range is first constructed with a slope corresponding to the median of |dv/dx| across the eddy range. This velocity gradient is used to determine a minimum reference eddy rate probability, lmin. If lassumed < lmin, then the selected eddy is rejected. Results using the stand-alone ODT models with comparison to the experiments of Tsuji and Nagano [71] are summarized in Figs 12 and 13. A total of 1,000 realizations are used to con__ struct ensemble mean and RMS statistics. Comparisons __ of normalized time-averaged velocity, v TN TN (x, y)/vO , and normalized time averaged temperature, (T(x, y) – T∞/TO , against dimensionless TV TV TN __ TN downstream distances, x/LOV and x/LOT , respectively. The quantities vO = vmax(y), TO = Twall – T∞, TN TN LOV = dv(y) and LOT = dT(y) are the outer scalings proposed by Tsuji and Nagano where dT(y) = L

__

L

__

∫0 (Twall – T(x,y))/(Twall – T∞)dx and dv(y) = ∫0 (v (x ,y)/vmax(y))dx are the integral thermal and momentum boundary layer thicknesses, respectively [71].

(a)

(b)

Figure 12: Comparisons of (a) instantaneous and (b) average stream-wise velocity using ODT near-wall modeling for a heated wall with data from Tsuji and Nagano [71].

(a)

(b)

Figure 13: Comparisons of (a) ensemble-averaged temperature and (b) Nu along the wall using ODT near-wall modeling for a heated wall with data from Tsuji and Nagano [71].

Combustion Subgrid Scale Modeling

351

Figure 12(a) and (b) shows an instantaneous snapshot of velocity and average velocity profiles, respectively. Several downstream profiles are plotted, showing that the velocity becomes self-similar for y > 6 m downstream. The temporal and spatial formulations predict nearly the same result, with the spatial implementation agreeing slightly better with the experimental data near the entrainment boundary. Figure 12(a) shows that the size of the implemented eddy events gradually grows in size with increasing downstream distance – mimicking a laminar-to-turbulent transition; however, the extent to which ODT can actually capture this phenomenon is not clear. As shown in Fig. 12(b), the agreement between the ODT predictions and measured velocity are excellent. One of the primary reasons for exploring the use of ODT as a near-wall SGS model is for accurate predictions of heat transfer. Figure 13(a) and (b) shows comparisons of the average temperature profiles and the Nusselt number, respectively. Consistent with the velocity comparisons, excellent agreement between the normalized temperature profiles are also observed. Figure 13(b) shows that while the predictions of Nu exhibit the proper Gr1/3 scaling behavior, they are lower by a factor of approximately 2. These results are consistent with the Nu results of Dreeben and Kerstein [67] for flow in a heated channel. (See Fig. 14 of ref. [67].) While the source of these discrepancies is still a subject of research, further improvements are expected with the current stand-alone ODT model by introducing multidimensional effects in the context of a vector formulation [72] as well as including the effect of variable density [69]. In addition, LEM and ODT have also been recently explored as a possible LES SGS model for application in turbulent boundary layers by Schmidt et al. [73]. Figure 14 shows representative results from their study. The influence of the recent small eddy events in high-shear flow regions is clearly visible in

80 LES/ ODT DNS: Moser et al. (1999) Inner law, Log la w

Re τ = 10,000

70

Re τ = 4800 60

Re τ = 2400 50

Re τ = 1200 40 25 LES/ODT overlap region

ODT inner region recent small eddy events

20

u+ Re τ = 590

30

DNS Calculations (Moser et al., 1999) 15 u+

two example instantaneous profiles

Re τ = 395

20

time avg. profile sharp

10

+

+

u = 2.44 ln(y ) + 5.2 smoothed by molecular diffusion

u + = y+

10

5 Reτ = 1200 (Re2h = 49,336)

(a)

0 0.7

0 1

0.75

0.8 0.85 0.9 y'/h (distance from centerline)

0.95

1

(b)

10

1 00

10 00

10 4

y+

Figure 14: LES-ODT results from Schmidt et al. showing (a) the instantaneous streamwise velocity and (b) the time-averaged profiles in comparison with DNS and experimental data [73].

352

Transport Phenomena in Fires

Fig. 14(a). With time, these events are smoothed out by molecular diffusion processes. Comparisons of time-averaged normalized velocity compared to DNS and experiments are exceptional. In summary, ODT and LEM approaches are extremely advantageous for modeling SGS chemically reacting processes because all of the turbulent length and time scales are explicitly resolved. The extension of these approaches to fire applications is still very much an evolving topic but appears to be very promising. Current research issues in this area concern exchanging information between the ODT domain and the filtered flow field that is solved on the CFD mesh for which several coupling strategies are being explored [73, 74].

4 Summary In this chapter three SGS modeling approaches for LES of fires are summarized ranging from the simplest (flamelet with a presumed FDF) to perhaps one of the most complex (ODT). It is fair to say that no universal SGS modeling approach has been accepted and the degree of modeling sophistication largely depends on the aspect of the fire that is of interest. However, a few rules of thumb can be established on the level of SGS modeling complexity required for specific fire applications. For predicting the overall flow dynamics, temperature distribution, and major species of combustion from an isolated pool fire or fire plume, then a presumed FDF-flamelet approach is most likely sufficient. Using this type of model would be helpful for understanding the overall plume flow dynamics in complex geometries and predicting smoke transport. If the primary interest in simulating the fire is to examine the formation of toxins and initial level of soot present, then the presumed FDF with the flamelet modeling approach presented may be too limiting. In this case, unsteady flamelet modeling techniques (not discussed) could be pursued in conjunction with a presumed FDF. As discussed, recent studies on the use of transported FDF approaches with flamelet models show little advantage in transporting the mixture fraction FDF when compared to simply using a presumed beta function. A transported FDF approach therefore doesn’t appear necessary in this context. However, if the problem requires the knowledge of very slow kinetics and/or radiation is of primary concern, then transported FDF scalar approaches appear to be well suited for these purposes since both the finite-rate chemistry and radiative emission terms appear in closed-form. These types of models will be helpful for predicting the initial toxin and soot levels in the near field of a fire plume; however, they may be too computationally expensive to simulate multiple room environments. Since the SGS scalar gradient information is lost in FDF approaches, the near-wall heat transfer processes must be entirely modeled. Developing mixing models for complicated near-wall treatments, such as a pyrolyzing wall, may be too difficult. For this case, a near-wall ODT modeling approach may be better suited. The 1D ODT formulation naturally fits with near-wall modeling treatments for which complicated molecular diffusion processes, such as flame spread, are important. Near-wall ODT SGS models have proven to be useful for predicting wall heat transfer and could be used in an LES context to determine the sensitivity of structural failure for a given fire environment.

Acknowledgments The authors gratefully acknowledge the financial support from Sandia National Laboratories, the National Science Foundation (CTS-0348110) and the Office of Naval Research (Grant No.

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N00014-03-1-0369 and N00014-06-0623) for supporting their research. The first author is grateful for the extensive discussions with Drs Sheldon Tieszen and Alan Kerstein of Sandia National Laboratories and Dr Lou Gritzo of FM Global who have provided guidance and inspiration for the research in fire simulations. The authors are also thankful for the contributions from Professor Peyman Givi, Dr Rod Schmidt and Dr Alan Kerstein on state-of-the-art results using FDF and ODT SGS modeling approaches. Computer resources for the results presented by the authors are provided by the Center for Computational Research (CCR) at the University at Buffalo.

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CHAPTER 10 CFD fire simulation and its recent development Z. Yan Department of Building Science, Lund University, Sweden.

Abstract This chapter presents an outline on computational fluid dynamics (CFD) fire simulations within the framework of Reynolds averaged Navier−Stokes (RANS), large eddy simulation (LES), and direct numerical simulation (DNS). First the general simulation concepts regarding RANS, LES, and DNS are briefly discussed with emphasis on the major theoretical difference between these three simulation concepts. This is followed by discussions on modeling of different essential sub-processes in fires. These essential sub-processes include turbulence, turbulent combustion, thermal radiation, soot formation, heat transfer inside solid, and pyrolysis of combustible solid fuel, etc. Within the discussion of sub-process modeling, some recent developments in sub-process modeling based on the author’s research are presented. Besides the discussion on the simulation of conventional fires, this chapter also presents techniques and some interesting recent developments on CFD simulation of spontaneous ignition in porous fuel storage.

1 Introduction Computational fluid dynamics (CFD) simulation plays an important role in fire research. It provides a new efficient, reliable, and economic path for fire research and has become an essential fire research tool. Before the advent of CFD simulation, fire study was limited to experimental investigation and empirical correlation. Due to its high expense and practical difficulty, experimental data is usually very limited, if not unavailable. With the limited experimental data as validation base, a properly validated CFD simulation tool can provide much more information and thus extensively extrapolate the limited experimental data. In many cases, CFD simulation can also be used as pre-investigation of a to-be-performed experimental test and provide guidance for experiment. With the wide adoption of performance-based fire safety design, CFD simulation is becoming a routine practice for obtaining necessary fire design information. With new development in modeling techniques, fast increase of computing power, and quick drop of hardware price, it is expected for CFD simulation to continuously gain popularity in the fire community. This chapter attempts to provide an outline on CFD simulation techniques for both conventional fires and spontaneous ignition in porous fuel storage.

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2 CFD simulation of conventional fire Figure 1 gives a simple illustration of a conventional fire. For CFD simulation of conventional fires, in most cases, turbulent combustion in gas phase, processes in bulk solid phases, and phase interaction need to be simultaneously taken into account. The turbulent combustion in gas phase is extremely complex. The processes in bulk solid may in general include the highly complicated internal heat, mass, and momentum transfer and chemical reaction. The interaction between gas and bulk solid concerns heat, mass, and momentum exchange. 2.1 Gas phase simulation The gas phase process in the conventional fire considered in this chapter is in essence an unwanted turbulent combustion. The simulation of turbulent combustion in a conventional fire can in principle be carried out using the Reynolds averaged Navier−Stokes (RANS), large eddy simulation (LES), and direct numerical simulation (DNS) concepts. 2.1.1 Some basics of turbulence and turbulent combustion Before we discuss RANS, LES, and DNS of gas phase turbulent combustion, it is necessary to present some very important basic physics of turbulence and turbulent combustion. Turbulence remains one of the most challenging topics nowadays. It can be considered as a deterministic random hydrodynamic system. It is deterministic in a sense that for a unique set of initial and boundary conditions, the system is believed to hold a unique solution. It is random in a sense that when the Reynolds number of the flow is sufficiently high, the system becomes unstable. The flow solution in this case turns out to be very sensitive to external disturbances including the initial and boundary conditions. Although it is very difficult to give an exact definition of turbulence, its general characteristic features can be identified [1]. The irregularity is one of the most important features of turbulence,

Figure 1: Simple illustration of a conventional fire.

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which is in contrast to the regularity of laminar flow. The random motion of turbulence enhances the mixing process drastically and turbulence is thus diffusive. Turbulence comes from the instability of the viscous flow at high Reynolds number, which results from the interaction between the viscous and non-linear inertial terms of the Navier−Stokes equations. The interaction is very complex and makes it a prohibitive task to fully analyze the flow instability. Turbulence holds a continuous spectrum of scales ranging from integral scales to Kolmogorov scales. It obtains its kinetic energy from the mean flow through shear and buoyancy production. The large eddies which have dimensions comparable to those of the mean flow contain most of the energy and thus are the energy containing eddies. The kinetic energy will largely be transferred to smaller eddies when the smaller eddies are stretched by the larger eddies. Most of the turbulence kinetic energy will be dissipated at the smallest eddies. Therefore, turbulence is always dissipative. In the inertial range of eddies if we ignore the small dissipation the energy received from the larger eddies is equally transferred to the smaller eddies. This gives the picture of the turbulence kinetic energy cascade. The rotational nature of turbulence and the turbulence maintaining mechanism of vortex stretching bring turbulence to a three-dimensional space, because vortex stretching does not exist in a two-dimensional space. The dissipative character of turbulence is closely related to rotational motions. The rotation of the flow favors the creation of many regions of large gradients and thus enhances the dissipation. Turbulence has a wide range of scales. The turbulence eddy size can range from the integral length scale to the Kolmogorov length scale. The integral length scale l is comparable to a physical dimension of the problem under consideration. The Kolmogorov length scale is the length scale of the smallest eddies in a turbulent flow. According to Kolmogorov’s theory [1−3], the size of smallest eddies can be estimated as h = (n3/e)1/4, where n is the kinematic viscosity and e is the turbulence kinetic energy dissipation rate which can be estimated as e = (u¢ )2/(l/u¢ ) = (u¢ )3/l. The Kolmogorov length scale can be as small as 0.1 mm. Turbulence time scales also vary from the integral time scale of the large eddies to the Kolmogorov time scale of the smallest eddies. The integral time scale is of the order l/u¢ , where u¢ is the rms value of velocity fluctuations. The Kolmogorov time scale can be estimated as t = (n/e)1/2. The ratio of the integral time scale to the Kolmogorov time scale is the square root of the Reynolds number which is usually quite large. In turbulent reacting flow, chemical reaction is another very complicated factor. Even for simple fuels, the chemistry may comprise thousands of reactions. In the reactions, the time scale can also vary by several orders of magnitude, going from the fast heat release reactions to the slow pollution formation reactions such as the NOx and soot formation. Meanwhile there is a strong interaction between combustion and turbulence. Both of them have dual effects on each other. The heat released by combustion will cause the thermal expansion of the fluid mixture, thus driving a flow. The viscous flow may loose its stability when the Reynolds number of the flow is sufficiently high and the transition from laminar flow to turbulent flow may happen. On the other hand the reduction of the fluid density has a damping effect on vorticity, diminishing turbulence. Meanwhile, turbulence also has dual effects on combustion. In a diffusion flame, the fuel and oxidant need to be mixed at the molecular level first before the reaction can happen. In a premixed flame, the flame propagation depends on the heat conduction and the diffusion of radicals from the burned region to the unburned region. Many chemical reactions have much smaller time scales than the physical mixing process. Thus the mixing is often crucial for combustion and the turbulent combustion often turns out to be mixing controlled in a general sense. As a result, the turbulence, which can enhance mixing, may intensify combustion, in both diffusion and premixed flames. However, strong turbulence may also affect the flame structure and increase the flame heat loss rate to such an extent that the flame may be extinguished by the high strain rate [4−6].

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Depending on the relation between flow and reaction scales, turbulent combustion can fall into several distinguished regimes. The ratios of the corresponding flow scales and reaction scales define several non-dimensional numbers, including the Reynolds number, Damköhler number, and Karlovitz number. These numbers have their clear physical interpretation and thus give some indication on the status of the combustion. Borghi’s diagram gives a good conceptual description on this point for a premixed flame (see [6−8] for details). Accurate modeling of turbulent combustion requires a proper consideration of all the important physics and chemistry. The physical aspects briefly discussed above give some important clues for the modeling strategy and possible simplifications. In turbulence modeling, the spectrum of the turbulence scales explains why conventional turbulence modeling cannot be universally applicable. The length and time scales of the large eddies are comparable to those of the mean flow. As a result, turbulence is a property of the flow. Any conventional turbulence model, which is tuned for certain types of flow, may fail in other situations. In LES, since the modeled small eddies do not have a strong direct interaction with the mean flow, the LES modeling can be expected to be more universal. Concerning the combustion modeling, when the time scale of the chemistry is much smaller than that of mixing, the combustion is mixing controlled and one may reasonably assume that combustion happens once the fuel and oxidant are mixed. The mixing control assumption allows the combustion analysis to be significantly simplified. This is the starting point of Magnussen’s eddy dissipation concept (EDC) combustion model [9, 10] which will be discussed later. The fast chemistry assumption is also one of the key points in the development of the flamelet combustion model [11−17], where the turbulent flame is considered to be an ensemble of wrinkled laminar flamelets which have a well-defined structure. 2.1.2 CFD simulation of turbulent combustion in a conventional fire CFD simulation of fires corresponds to numerically solving a set of governing equations which describe the physics of interest. For the gas phase turbulent combustion in a conventional fire, the governing equations, comprising continuity, momentum, energy, and species equations, are: ∂r ∂( rui ) + =0 ∂t ∂ xi   ∂u ∂u j  2 ∂uk  m i +  + r agi  − mdij ∂ xk    ∂ x j ∂ xi  3 ∂uk  ∂p* ∂   ∂ui ∂u j  2 =− + + m  + ( r − r∞ )agi  − mdij ∂ xi ∂ x j   ∂ x j ∂ xi  3 ∂ xk 

(1)

∂( rui ) ∂( rui u j ) ∂p ∂ + =− + ∂t ∂x j ∂ xi ∂ x j

(2)

∂( rh) ∂( ru j h) ∂  m ∂h  + = + Sh ∂t ∂x j ∂ x j  Pr ∂ x j 

(3)

∂( rYi ) ∂( ru j Yi ) ∂  m ∂Yi  + =   + Ri ∂t ∂x j ∂ x j  Sc ∂ x j 

(4)

where p* = p − p∞ + r∞agi xi is the pressure minus its hydrostatic value, xi is the space coordinate vector, t is the time, r is the density, ui is the velocity vector, h is the enthalpy, µ is the dynamic viscosity, Sc is the Schmidt number, Pr is the Prandtl number, Yi is the mass fraction for chemical

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species i, dij is the Kronecker delta tensor, agi is the gravity acceleration vector, Ri is the reaction rate, Sh is the energy source term resulting from the radiation, and T   h = ∑ Yi hi = ∑ Yi  h0,i + ∫ c p,i dT    T0

(5)

in which h0,i is the heat of formation of species i at temperature T0. The radiation equation and the thermal state relations provide the necessary auxiliary equations. With this set of established mathematical equations, the task left is to build up proper initial and boundary conditions for a concerned particular case and using numerical methods to find out the solution. While pursuing a numerical solution of the above governing equations, one has to deal with space and time discretization. This raises the resolution issue. This resolution issue is critically important for CFD computations. Numerical resolution must be very carefully examined with regard to the physical scales described by the equations to be solved. The principle which has to be followed for a numerical computation of a turbulent reacting flow is that the adopted numerical resolution must be fine enough to resolve all the concerned physical scales described by the governing equations to be solved. This principle will be explained in detail in the following sections. Any numerical computation performed without following this principle looses its fundamental base and thus does not really have much physical meaning, even though the computation may still be able to produce colorful pictures. In CFD, due to different reasons and variable concerns, the physical scales that need to be resolved may vary from case to case. Basically three different simulation concepts can be constructed depending on what physical scales are to be resolved. These three simulation concepts are the well-known RANS, LES, and DNS concepts which will be discussed below. 2.1.2.1 Direct numerical simulation In DNS, the Navier−Stokes equations are solved in original form without any pre-treatment. With regard to turbulence, the physical scales described in the Navier−Stokes equations in original form cover the whole turbulence spectrum, ranging from the integral scales to Komogrov scales. Each individual eddy in a turbulent flow can be characteristically seen as a ‘kingdom’. Therefore, in order to implement DNS, one has to resolve the smallest scales properly, in both space and time coordinates. As discussed before, Kolmogorov length scale can be estimated as h = (n3/e)1/4 and is related to the integral length scale as h/l = Re−3/4, where Re is the Reynolds number Re = u¢ l/n which is usually quite large. Similarly, one can easily show that the ratio of the integral velocity scale to the Kolmogorov velocity scale v is the one-fourth power of the Reynolds number, and the ratio of the integral time scale to the Kolmogorov time scale is the square root of the Reynolds number. Therefore, in a three-dimensional turbulent flow computation, with an increase in Reynolds number, the grid size decreases according to Re−3/4 and consequently the grid number increases according to (Re3/4)3 = Re9/4. In a DNS computation, an explicit scheme is normally adopted and the time step size is numerically limited by the Courant−Friedrichs−Levy (CFL) stability condition u∆t/∆l ≤ 1, where u is flow velocity, ∆t is the allowed time step size and, ∆l is the grid size. This indicates ∆t ≤ ∆l/u. Meanwhile, approximately, u ≈ u¢ . For the smallest eddies, we also have t = h/v = h/u¢ (Re−1/4) = h(Re1/4)/u¢. Since the computational grid needs to resolve the smallest eddies one can write ∆l ≈ h. Therefore, ∆t < t. This indicates that with explicit time marching the time step size limited by the CFL condition must be small enough to resolve the Kolmogorov

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time scale. Consequently, assuming the CPU time for solving algebraic equations is linearly proportional to the number of unknowns, the computational effort can be indicated by the product of grid number and the number of time steps which is given by t/∆t = (l/u)/(∆l/u). Based on the above discussion, the computation effort can be shown to be proportional to Re9/4(l/u)/(∆l/u) = Re9/4l/∆l ≈ Re9/4l/h = Re9/4Re3/4 = Re3. As a result, the resolution required in DNS turns out to be very computer resource demanding and thus creates resolution problem for DNS. At present, even with massive parallel computers, DNS is only possible for some simple flows. While doing DNS, the resolution must be fine enough to capture the smallest eddies. However, this is not yet enough. There is also some requirement on the employed numerical scheme. A high order numerical scheme is needed to minimize numerical diffusion which otherwise may overpower the physical diffusion associated with the fluid. It should be noted that the viscosity for normal gas such as air has a very small value and thus can easily be distorted if the numerical scheme is not adequate. Furthermore, a turbulent flow system is unstable. The flow solution is very sensitive to external disturbances including the initial and boundary conditions. Therefore, the treatment of initial and boundary conditions must be very carefully examined. For DNS of fires, consideration must be given to the additional sub-processes such as chemical reaction. In fires, both premixed and non-premixed flame may exist. In chemical reactions, the time scale can also vary by several orders of magnitude going from the fast heat release reactions to the slow pollution formation reactions such as soot formation. The time scales of all the relevant chemical reactions must be properly resolved. Meanwhile, spatial resolution consideration also needs to be given to flame thickness which can be very small. Strictly speaking, to really implement DNS for fires, the detailed chemistry must also be known and included. This brings extra difficulty. Currently, it is prohibitive to do any DNS on practical fires. 2.1.2.2 Reynolds averaged Navier−Stokes To overcome the resolution problem associated with DNS, one has to relax the to-be-resolved physical scales. The to-be-resolved physical scales are associated with the governing equations. To have the needed relaxation, one way is to do some pre-treatment on the governing equations. Perhaps we can call a simulation concept within this framework a relaxed simulation. In this sense, both RANS and LES can be seen as relaxed simulation methods, although the pre-treatments on the governing equations in RANS and LES are different. With the pre-treated governing equations, the associated physical scales get relaxed accordingly. Meanwhile, this also provides flexibility with the numerical schemes and initial and boundary conditions. However, one should note that there is one requirement for the pretreatment. That is the relaxed solution should be able to deliver the intended information. RANS is one commonly used cheap relaxation method where the pre-treatment applied on the original instantaneous equations corresponds to statistical averaging. This pre-treatment is widely adopted in engineering computations because the most concerned is the mean flow property in many practical engineering applications. Since the instantaneous governing equations are statistically averaged for solution, only the relatively smooth mean field needs to be resolved properly. Therefore, the to-be-resolved physical scales are much larger than those found in DNS. In this way, the resolution issue can be very much relaxed so that simulation of practical complex turbulent flows can be handled using currently available computers. However, it should be noticed that the statistical averaging pre-treatment does not really help wipe out the problem associated with DNS. Instead, this pre-treatment is a kind of problem shift mechanism. The resolution problem is removed at the price of a closure problem. By doing statistical averaging, extra terms representing the contributions of fluctuations to the averaged field show up in the pre-treated governing equations. Models must be constructed to represent these contributions.

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In conventional fires where flow velocity is of the order of several meters per second (Mach number << 1.0), with bulk viscosity and Dufour and Soret effects ignored, the set of Favre-averaged governing equations for fire simulation can be written as: ∂r ∂( r ui ) + =0 ∂t ∂ xi

(6)

∂( r ui ) ∂( r ui u j ) + ∂t ∂x j   ∂ui ∂u j  2  ∂uk + − r ui′′ u′′j  + r agi m  − mdij ∂ xk   ∂ x j ∂ xi  3   ∂uk ∂ p* ∂   ∂ui ∂u j  2 =− + + − r ui′′ u′′j  + ( r − r∞ )agi m  − mdij ∂ xi ∂ x j   ∂ x j ∂ xi  3 ∂ xk  

=−

∂p ∂ + ∂ xi ∂ x j

∂( r h) ∂( ru j h) ∂ + = ∂t ∂x j ∂x j

(7)

 m ∂h  ′′ ′′  Pr ∂ x − r u j h  + Sh j  

(8)

∂( rYi ) ∂( ru jYi )  ∂  m ∂Yi + = − r u′′j Yi′′  + Ri  ∂t ∂x j ∂ x j  Sc ∂ x j 

(9)

where the over bar represents Reynolds averaging, the tilde represents Favre averaging, and the double prime represents the fluctuation from Favre averaging. __ ___ __ The mean enthalpy source term S , mean reaction rate Ri, and the second moments such as −r _______ h _________ ui′′uj′′ and −ruj′′h′′ arise from the averaging operation of the governing equations. They need to be modeled and will be discussed in modeling section. Figure 2 gives a typical temperature profile from a RANS simulation of a buoyant flame. Obviously, the presented plot represents an averaged profile. The main advantage of RANS is its simplicity and computational efficiency. However, because the energy containing large eddy scales within the turbulence spectrum are comparable to the scales of the mean flow, turbulence is a property of the flow. As a result, the constructed turbulence models in RANS can never be universal. Any conventional turbulence model, which is tuned for certain types of flow, may fail for others.

2.1.2.3 Large eddy simulation LES is another relaxed simulation method, which lies between DNS and RANS. In LES, the pre-treatment involves spatial filtering. This filtering provides a scale separation where the turbulence scales are separated into resolved scales and unresolved scales. In LES, eddies down to the inertial sub-range are properly resolved. The contribution of smaller eddies is modeled using sub-grid models. In a turbulent flow, large energy containing eddies are directly subject to geometry and boundary conditions. These large eddies are highly flow-dependent and play a major role in transport. They are resolved in LES. The remaining small unresolved eddies are not tuned in the same range of frequency of the mean flow. Therefore, sub-grid modeling of contribution from small eddies can be expected to be relatively more universal than the turbulence modeling in RANS. Since the resolution goes down to the inertial

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Figure 2: A typical temperature profile from a RANS simulation of a buoyant flame. sub-range, the needed resolution can be rather fine. Due to the adopted fine spatial and temporal resolutions, LES also reveals more detailed flow dynamics and structure. However, these advantages come at a price. Because of the adopted fine resolution, LES is much more computer resource demanding. Meanwhile, compared to the RANS equations, the filtered transport equations in LES are much less dissipative and consequently the solution of the filtered equations in LES is usually much more sensitive to the specified boundary and initial conditions. Both the initial and boundary conditions may have a significant effect on the simulation. For example, a numerical study in [18] reported a strong effect of initial conditions on LESs of a thermal plume. Unfortunately, a proper specification of initial and boundary conditions for LES remains a difficult task. During past years, much research has been done to find out proper specifications of boundary conditions for different boundaries in flow computations [19−22]. It should be emphasized that the resolution must be enough to resolve down to the inertial sub-range in an LES computation. As a very rough guidance,__________ this required resolution can be __ estimated by the Taylor microscale which is defined as l = √ u 2 /(∂u/∂x)2 . In isotropic turbulence, the Taylor microscale is related to dissipation rate as e = 15nu2/l2, where n is the kinematic viscosity and u is the rms value of velocity fluctuations. Without proper resolution, a computation cannot be accepted as a LES computation and does not have a scientific basis from a LES point of view. Before presenting a LES computation, the numerical resolution needs to be examined to show it is adequate. This examination can be made by analyzing the spectrum of the turbulence kinetic energy to make sure that eddies down to the inertial sub-range are properly resolved. Figure 3 shows a typical turbulence energy spectrum at a location on the center line in a three-dimensional computation of a buoyant plume [23]. As can be seen in this computation, with some substantial inertial sub-range predicted, the theoretical Kolmogorov −5/3 decay was fairly well captured in the computation. This indicates that the spatial resolution used in the computation is adequate. Large eddy simulation of conventional fires is based on the solution of a set of density weighted spatially filtered conservation equations, comprising continuity, momentum, energy, and species equations. Turbulence and turbulent combustion sub-grid scale (SGS) models overcome the

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prediction 0.0 -5/3 decay -2.0 ln (E(f))

-4.0 -6.0 -8.0 -10.0

height = 1.5 m, at central line

-12.0 -3

-1

1 ln (f)

3

5

Figure 3: Turbulence energy spectrum at a location of the center line in a simulated thermal plume.

closure difficulty resulting from filtering the partial differential equations. The radiation equation and the thermal state relations provide the necessary auxiliary equations. Assuming that the error arising from the commutation of filtering operation and differentiation for a used non-uniform grid is absorbed in the SGS models, the filtered equations have a form that is similar to those in RANS and can be written as: ∂r ∂( r ui ) + =0 ∂t ∂ xi

(10)

∂( r ui ) ∂( r ui u j ) + ∂t ∂x j   ∂u ∂u j  2  ∂uk + tij  + r agi m i +  − mdij ∂ xk   ∂ x j ∂ xi  3   ∂uk ∂ p* ∂   ∂ui ∂u j  2 =− + + + tij  + ( r − r∞ )agi m  − mdij ∂ xi ∂ x j   ∂ x j ∂ xi  3 ∂ xk  =−

∂p ∂ + ∂ xi ∂ x j

 ∂( r h ) ∂( r u j h ) ∂  m ∂h + = + q j ,h  + Sh  ∂t ∂x j ∂ x j  Pr ∂ x j  ∂( rYi ) ∂( r u j Yi ) ∂ + = ∂t ∂x j ∂x j

 m ∂Yi  + q j ,Yi  + Ri   Sc ∂ x j 

(11)

(12)

(13)

where the over bar and the tilde represent spatial filtering __and Favre spatial filtering, respectively. ___ In the above equations, the filtered enthalpy source term Sh, the filtered reaction rate Ri, the SGS stress tij, and the SGS scalar fluxes including qj,h and q j ,Yi need to be modeled and will be discussed in the modeling section.

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The spatial filtering operation on a variable j is given by j (r , t ) =

∫ G(| r − r ′ |)j(r ′)dr ′

(14)

V

and the Favre spatial filtering operation is defined as j (r , t ) =

∫ G( r − r ′ )j(r ′) r(r ′)dr ′ / ∫ G(|r − r ′|) r(r ′)dr ′ = rj / r

V

V

(15)

where r and r' are space vectors, V is the volume of filtering, and G( | r − r' | ) is a filter function which must satisfy the normalization condition ∫v G( | r − r' | )dr' = 1. The SGS stress tensor tij includes the Leonard, cross and Reynolds SGS stresses which can i u j + r ui u j , − r u i u ′′j − r u j and − r u be written as − r u i′′u i′′uj′′. Similarly, the SGS flux qj,j for a j ′′ and − r u general scalar j has three components − r u j + r u j , − r u ′′j − r u ′′j ′′. j

j

j

j

j

In LES, instantaneous distributions are computed like the one shown in Fig. 4, which may look more realistic than the profile obtained in RANS according to one’s real life impression. However, for practical engineering application, in many cases, people need to know the mean distributions (statistics). In LES, the statistics are obtained by averaging a large number of instantaneous profiles over a sufficient time period. The higher the order of the desired statistics, the longer the time period required for averaging. Some discussions on averaging and the averaging time can be found in [23]. 2.1.3 Modeling of gas phase sub-processes in a conventional fire When pursuing the simulation of a conventional fire using computers, a set of mathematical equations are numerically solved. In the governing and the necessary auxiliary equations, for certain terms

Figure 4: A typical instantaneous temperature profile from a LES of a buoyant diffusion flame.

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and/or variables, due to limits in scientific knowledge, there may be a lack of proper fundamental description. In this case, models are needed to provide approximations for those unknowns which may either already exist in the original equations constructed from physical laws or come from the pre-treatment of the original equations. The models can be constructed in the form of mathematical formula(s) and/or through a set of additional model differential equations. For CFD fire simulations, depending on one which simulation concept is used, the gas phase sub-process modeling mainly covers turbulence, turbulent combustion, radiation, and soot formation. 2.1.3.1 Turbulence modeling As was already discussed, modeling is explicitly needed for consideration of turbulence when the RANS or LES simulation concept is adopted. The main purpose of turbulence models is to provide modeling for the unknown terms arising from the pre-treatment of the Navier−Stokes equations. In DNS, the original instantaneous equations are solved, turbulence is fully accounted for in those equations and thus no explicit model should be used. 2.1.3.1.1 Turbulence model in RANS The turbulence modeling closure problem in RANS can be addressed by constructing turbulence models, which are used to somehow feed back the contribution of fluctuations to the mean fields. Many different turbulence models, of varying complexities and applicabilities, have been proposed [24], such as the mixing length model, the k−e model, the k−w model, the algebraic stress model (ASM), and the Reynolds stress model (RSM). Consideration should be given to both applicability and simplicity when selecting a specific turbulence model for a given application. Due to the complexity, the closure is often constructed heavily based on dimensional arguments with the closure ‘constants’ (if there is any) derived from relatively welldefined flows. The lack of full physics prevents the closure to be universal. As a result, a model or closure, which is applicable to certain types of flow, may fail when applied to other types of flow. The confidence of applying a turbulence model to a specific type of flow can only be obtained through extensive validation and testing. In particular, a turbulence model should be used with care when it is applied to a flow with a condition under which the closure assumption introduced in the model development can seriously be violated. When applying a turbulence model, for numerical reasons, the simplicity and numerical stability of the turbulence model is also highly concerned. Therefore, the choice of a turbulence model mainly depends on the simulation objectives and the compromise between accuracy and simplicity. The standard k−e model: The standard k−e model is perhaps one of the most widely used models. Like all the other turbulence models it has its own serious defects such as limited applicability to flow of strong streamline curvature or strong rotation [24]. The defects of the standard k−e model originate from the closure assumptions including the adopted Boussinesq assumption which assumes a linear relation between the Reynolds stress and the mean strain rate. The Boussinesq assumption may fail when the flow is subject to a sudden change of strain rate or a strong rotation. When scalar transport is involved, the simple gradient modeling procedure is normally adopted for the scalar fluxes within the framework of the standard k−e model. This introduces additional defects and leaves space for improvement. However, the standard k−e model has its important advantages that it is simple, numerically stable, and has been proved successful in many practical applications. Strategies are suggested to abandon the Boussinesq assumption, for example, by constructing a non-linear relation between Reynolds stress and the mean strain rate, solving algebraic Reynolds stress equations or differential Reynolds stress equations. However, all of these alternatives have their own deficiencies with respect to their complexities and modeling uncertainties. For example, in the RSM up to seven additional differential equations need to

368

Transport Phenomena in Fires

be solved and the modeling of the pressure strain redistribution term is still a challenge. Therefore, the standard k−e model remains attractive. When applied to buoyant flows, the standard k−e model needs to be modified to include the important buoyancy effect on turbulence and turbulent transport. In this buoyancy modified k−e model, two turbulence quantity equations are solved ∂( r k ) ∂( r u j k ) ∂  mt ∂k  + =   + r ( P + G − e) ∂t ∂x j ∂ x j  sk ∂ x j 

(16)

∂( re ) ∂( r u j e ) ∂  mt ∂e  e e2 + =   + c1e r ( P + G )(1 + c3 e R ′f ) − c2 e r ∂t ∂x j ∂ x j  se ∂ x j  k k

(17)

_______

∂u˜i ∂u˜i ∂u˜j __ ∂u˜i and is modeled as P = υt __ + __ , Rf′ is the modified flux Richardson where P = −ui′′uj′′ __ ∂x ∂x ∂x ∂x j

_______

(

j

i

)

j

ut __ ∂T˜ number −G/(G + P), G = −bui′′T ′′ agi and is modeled as G = b __ st ∂x agi, b is the thermal expansion __

i

∂r

coefficient − _1__r __˜ . ∂T The solution of the partial differential equations for turbulence kinetic energy and its dissipation rate provides an estimate of the rms value of the velocity fluctuation, u, and a characteristic turbulence length scale, l, as: u ~ k0.5 and l ~ k1.5/e. With the dimensional arguments and an analogy to molecular mixing, they can be used to compute the eddy viscosity µt ~ rul, which is used in the _______

(

)

__ ∂u˜k ∂u˜i __ ∂u˜ Boussinesq approximation to model the Reynolds stress tensor −rui′′uj′′ = mt __ + ∂xj − _23 __ d − _23 ∂x ∂x ij _________

__

j

i

k

__ ∂u˜i __ ∂u˜ rkdij. The Reynolds stress modeling reduces to −rui′′uj′′ = mt __ + j − _23 rkdij when the divergence ∂xj ∂xi

(

)

of averaged velocity field is ignored._______ This provides the closure for the averaged momentum equa__ _________ tion. The turbulent scalar fluxes −ruj′′Yi′′ and −ruj′′h′′ are modeled similarly by using the simple gradient modeling method with turbulent Prandtl/Schmidt numbers introduced. Values of the various constants used in the model and the corresponding references are listed in Table 1.

Table 1: k−e model constants. Constant

Value

Reference

Cm C1e C2e C3e

0.09 1.44 1.92 (a) 0.0 (b) 0.8 1.0 1.3 0.7 0.7 2.0

13 13 13 13

σk σe σT σj Cg

Note: (a) is for vertical flow and (b) is for horizontal flow.

13 13 14 14 14

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An improved k−e model: The modeling procedure in the above standard buoyancy-modified k−e model is simple, but has been found inadequate for the modeling of buoyant turbulent flows. It cannot capture counter gradient diffusion and tends to significantly under-predict the spreading rate of vertical thermal plumes [25, 26] and over-predict the spreading rate of horizontal, stablystratified flows [27]. Counter gradient diffusion is an important point which should be considered in the modeling of buoyancy driven turbulent flows. The heated gas has a low mass density, thus can be preferentially accelerated by pressure difference, and consequently may feature counter gradient diffusion [28]. The incorrect prediction of the spreading rates can bring serious deficiencies to the prediction of the important velocity and scalar profiles. A modified k−e two-equation turbulence model was developed in [29], to improve the consideration of buoyancy effects. This modified model was found to be stable, computationally economic and applicable to complex situations. It is capable of capturing counter gradient diffusion and well predicts the plume spreading rates and velocity and temperature profiles. In this improved model, the generalized gradient diffusion hypothesis is adopted to replace the _______ simple gradient diffusion for modeling of turbulent scalar fluxes. In particular, the u ′′ _______ i T ′′ term in the buoyancy production of turbulence kinetic energy G = −bui′′T ′′ agi is modeled using generalized gradient diffusion as k ∂T ui′′ T ′′ = – ct ui′′ u′′j e ∂x j

(18)

where ct = _32 cm / sT. _______ It was found in [29] that the generalized gradient diffusion modeling of u″ T″ provides a better representation of G. _______ To consider a direct effect of buoyancy on −ui′′uj′′ as indicated by the ASM formula, the turbulence kinetic energy shear production term P, which appears in both k and e equations, is computed as P = ( P )std (1 − R f )

(19)

∂u˜j __ ∂u˜i ∂u˜i where (P)std = ut __ + __ is the turbulence kinetic energy shear production modeled in the ∂x ∂x ∂x

(

j

i

)

j

standard buoyancy-modified k−e model. _______ _______ _______ In consistency with modeling of the heat flux ui′′T ′′, all the scalar fluxes such as uj′′h ′′ and uj′′Yi′′ are modeled in a generalized gradient diffusion form. All the empirical model constants keep their values from the standard k−e model except for c3e, which is directly related to buoyancy effect. In the standard buoyancy-modified k−e model, different values of c3e were recommended [30], varying with flow conditions from about 0.0 to 1.0. In this improved model, a value of 0.6 is recommended for c3e, largely based on numerical experiments. When compared with the standard buoyancy-modified k−e turbulence model, this model gives significantly improved numerical results as illustrated in Fig. 5. 2.1.3.1.2 Sub-grid scale turbulence model in LES In LES, the unknown sub-grid stress tensor and sub-grid scalar fluxes in the filtered momentum and scalar transport equations must be modeled in terms of the properties of the resolved scales so that the equation system can be closed. A good SGS model should be able to properly remove the turbulence kinetic energy from the resolved scales and account for backscatter of turbulence energy. Smagorinsky proposed the

Transport Phenomena in Fires 450 400 350 300 250 200 150 100 50 0 -0.4

4.0

0.9 m above burner Exp.

0.9 m above burner Exp. standard model This model

3.5

standard model This model

3.0 Velocity, m/s

Temperature, C

370

2.5 2.0 1.5 1.0 0.5

-0.2

0

0.2

0.0 -0.4

0.4

Distance from burner center, m

-0.2

0

0.2

0.4

Distance from burner center, m

Figure 5: Comparison of predicted and measured temperature and velocity profiles at 90 cm above a buoyant C3H6 diffusion flame [29] (©Elsevier Science Ltd, with permission).

first SGS model in 1963 [31]. Since then, many SGS models such as the similarity model [32] and the dynamic model [33, 34], have emerged. Although the Smagorinsky SGS model does not allow turbulence energy backscatter from small to large scales, it remains popular due to its simplicity and effectiveness. It has been used in many LESs. Considering its popularity, the Smagorinsky model is briefly presented below. In the Smagorinsky SGS model, the SGS stress tensor is modeled by tij −

dij 3

tkk = 2 msgs Sij

(20)

which is based on the eddy viscosity concept similar to that in RANS. The strain rate tensor is given by 1  ∂u ∂u j  Sij =  i +  2  ∂x j ∂xi 

(21)

An approximation of the SGS eddy viscosity µsgs can be obtained by assuming local equilibrium of small eddies. The dissipation rate can be estimated by e ≈ ksgs/t, where ksgs is the sub-grid kinetic energy and t is the sub-grid time scale. The sub-grid time scale can be represented by t ≈ 0.5 ∆/ksgs , where ∆ is the filter width. Meanwhile, the SGS kinematic viscosity can be approximated 0.5 4/3 as nsgs ≈ ksgs ∆. Therefore, we have nsgs ≈ e1/3∆ . The production of SGS turbulence kinetic __ ∂ r ∂T˜ 1 1 __ energy can be given by PK = nsgs 2S˜ijS˜ij − ___r __˜ __ a Finally, by assuming the equilibrium ∂T sT ∂xi gi condition e = PK, after some simple mathematics, we have the buoyancy-modified Smagorinsky model where the SGS viscosity can be approximated as

)

(

nsgs = Cs2 ∆ 2 (2 Sij Sij + PB )0.5 __

(22)

∂ r 1 __ ∂T˜ where PB is the buoyancy production term − ___1r __˜ __ a and Cs is the case dependent Sma∂T sT ∂xi gi gorinsky ‘constant’. The standard Smagorinsky model can be recovered from eqn (22) by simply excluding the buoyancy production term PB. The case dependent Smagorinsky ‘constant’ Cs usually has a value between 0.1 and 0.2.

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The SGS turbulent scalar flux qj,j for scalar j arising from the filtering of a scalar transport equation is modeled using a simple gradient transport hypothesis, as q j ,j =

msgs ∂j sj ∂ x j

(23)

where sj is the SGS turbulent Prandtl/Schmidt number for scalar j. 2.1.3.2 Turbulent combustion models Turbulent combustion models are adopted to cope with the closure problem created by pre-treating the non-linear reaction rates. There are different combustion models available. In this chapter, the discussion is limited to two basic strategies popularly employed for fire simulation. One is to solve the pre-treated transport equations of the non-conserved mass fractions with the pre-treated chemical reaction rates directly modeled. Another one is to solve the transport equation of a conserved scalar and then relate the concerned pre-treated non-conserved mass fractions to the conserved scalar using the laminar flamelet concept and a probability density function. They are briefly discussed below. 2.1.3.2.1 Eddy dissipation concept in RANS The eddy dissipation concept proposed by Magnussen and Hjertager [9] is a popular representative method in RANS to model the mean reaction rate directly. It has gained its popularity since it is simple and widely applicable. In a diffusion flame, the fuel and oxidant need to be mixed at the molecular level before a reaction can happen. In a premixed flame, the flame propagation is dependent on the mixing of hot products with the unburnt mixture and the diffusion of radicals. The chemical kinetics determines how the reaction will proceed in the mixture. Therefore, combustion depends on both mixing and chemical kinetics in general, but is essentially controlled by the slower of these two sequential processes, particularly when the slower process is much slower than the faster one. When the chemistry is much faster than the mixing, combustion turns out to be mixing-controlled. In this case, the mixing rate can well represent the combustion rate. The mixing process in a turbulent flow is largely dependent on the property of the turbulence. In a turbulent flow, there exists a kinetic energy cascade, which was already discussed earlier. The turbulence kinetic energy is extracted by the large eddies from the mean flow and dissipated mostly at the smallest eddies through molecular viscous dissipation. Therefore, the turbulence kinetic energy dissipation rate is closely related to the molecular mixing. The turbulence kinetic energy divided by its dissipation rate, k/e, indicates a dissipation time scale. By dimensional j˜ * e 2 argument we may estimate the dissipation rate of a general variable j as cj ___ k where j* = j″ and cj is an empirical proportion coefficient. __ (j˜ )0.5e

* Based on these arguments, the mean combustion rate can be estimated as cjr____ k , with j representing the mass fraction. For a general reaction represented by

aA + bB + cC + → dD + eE + fF +

(24)

where a, b, c, d, e, f are the reaction coefficients, A, B and C represent the reactants (normally A represents fuel), and D, E and F represent the products, assuming the fluctuation is simply related to the mean value, the mean reaction rate is modeled in EDC [9] by the turbulence dissipation rates of reactant and/or product eddies as RA = −cEDC r

e a a   min  YA , YB , YC , …   k b c

(25)

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Transport Phenomena in Fires

or RA = −cEDC r

  e a a a a a min  YA , YB , YC , … , YD , YE , YF , … k  b c d e f 

(26)

where cEDC is an empirical model constant. Usually, a value of about 4.0 can be used for cEDC and in many cases the simulation is not very sensitive to the value used. Obviously, the applicability of the above modeling procedure is limited by the introduced assumptions especially by the fast chemistry assumption. EDC was later extended [10] for possible inclusion of chemical kinetics, after introducing a series of assumptions. 2.1.3.2.2 Eddy dissipation concept in LES The eddy dissipation concept was originally proposed for RANS. However, this concept can be easily extended for use in LES. In the modeled reaction rate formula shown in eqns (25) and (26), _ke is the inverse of an estimated time scale. In LES, since neither k nor e is computed, _ke is not available. However, we can alternatively estimate the relevant time scale from the velocity field as 2/[((∂u˜i/∂xj) + (∂u˜j/∂xi))((∂u˜i/∂xj) + (∂u˜j/∂xi))]0.5. With the averaged variables replaced by corresponding filtered variables and the time scale replaced by the new estimation, eqns (25) and (26) can be used in a straight way to estimate the needed filtered reaction rates in LES. 2.1.3.2.3 Flamelet combustion model in RANS Flamelet models provide another approach to deal with the closure problem created by pre-treating the non-linear reaction rates. In this section, flamelet combustion models will be presented for non-premixed combustion. The structure of laminar diffusion flames has been found from measurements to be quite universal when it is plotted against the local equivalence ratio [35]. Assuming an infinitely fast chemistry, it can be shown that there exists a unique relation between mass fractions and mixture fraction for a two-stream system with equal mass diffusivities. When the chemistry is fast, but not infinitely fast, as it is in many practical considerations, to leading order the flame structure can still be regarded as well defined in a conserved scalar space, which is usually, but not necessarily, selected as the mixture fraction. Typically, the steady flamelet structure is given by Yi = Yi(f,cst), T = T(f,cst), where f is the mixture fraction, c is the scalar dissipation rate, and the subscript ‘st’ denotes stoichiometry. This flamelet structure can be obtained from detailed experimental measurements [35−37] or theoretical laminar flame calculations [38, 39]. The above discussion implies that in the modeling of turbulent non-premixed combustion, if we assume that the turbulent flame is an ensemble of laminar flamelets of well-defined structure, the detailed chemistry computation could be separated from the flow field calculation, and the modeling can proceed by finding out the flamelet statistics, which can be represented by a probability density function (PDF) p˜(f,cst). The mean mass fraction in a turbulent flame is evaluated from the laminar flamelet structure and the statistics of the related conserved scalar and the scalar dissipation rate as Yi =

∞1

∫ ∫ Yi ( f , cst ) p ( f , cst ) dcst

(27)

00

If f and cst are assumed statistically independent, then the above equation becomes Yi =

∞

∫ 0

1   ∫ Yi ( f , cst ) p ( f )df  p ( cst )dcst  0 

(28)

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373

In a turbulent reacting flow the local PDF is very difficult to obtain but can be approximately prescribed or estimated from the solution of a carefully constructed PDF transport equation [40−43]. The prescribed PDF, p˜(f ), can be constructed from the mean mixture fraction and its variance, for example, using the normalized beta [12, 14, 17], clipped Gaussian function [44] or top hat functions [45]. Very often, a prescribed normalized beta function PDF is used. The normalized beta function has only two parameters and can approximate reasonably well the usual PDF shape found in a jet when the parameters are properly given. In some situations, the simulation of a turbulent reacting flow may not be sensitive to the specific PDF shape [45]. The PDF constructed using the normalized beta function can be written as p( f ) =

f a−1 (1 − f ) b −1 1

∫f

a −1

(1 − f ) b −1 df

(29)

0

where the function parameters a and b can be related to the mean mixture fraction and its variance as a = ˜f [ ˜f (1 − ˜f )/g˜ − 1], b = a(1 − ˜f )/ ˜f , where g˜ is the variance of the mixture fraction f˜ ″2. Both mean mixture fraction and its variance are given by the solution of their transport equations  ∂( r f ) ∂( ru j f ) ∂  m ∂ f + = − r u′′j f ′′  ∂t ∂x j ∂ x j  Sc ∂ x j 

(30)

 ∂( r g ) ∂( r u j g ) ∂  m ∂g ∂ f e + = − r u ′′j g − 2 r u ′′j f ′′ − cg r g  ∂t ∂x j ∂ x j  Sc ∂ x j ∂x j k 

(31)

The third hypothesis of Kolmogorov states that the logarithm of the dissipation rate, averaged in a space much smaller than the integral scale, has a normal distribution [2]. Assuming a lognormal distribution of c, the prescribed p˜(c) can be constructed as [12, 14] p ( c ) =

1 cs 2 π

 1  exp  − 2 (ln c − m)2   2s 

(32)

where the parameters µ and s are related to c˜ as c = exp( m + 0.5s 2 )

(33)

in which s can be estimated from s2 = 0.5 ln(0.1 Re0.5), and c˜ can be modeled by c = cg 2

g e k

(34)

In this approach, the transport equation for the mean mass fraction is not solved and thus the mean reaction rate does not need to be directly modeled. 2.1.3.2.4 Flamelet combustion model in LES In LES, the flame is typically not spatially resolved and a SGS combustion model is needed to consider the interaction between turbulence and chemistry. The SGS models of non-premixed combustion based on the laminar flamelet concept are similar to the flamelet combustion models in RANS. At the SGS level, it is assumed that the turbulent flame is an ensemble of wrinkled laminar flamelets which have a well-defined structure. The interactions between turbulence and chemistry can be included

374

Transport Phenomena in Fires

by treating the turbulent reacting flow as a random process and introducing a SGS probability density function. Typically, as in RANS, a prescribed normalized beta PDF can be used. In order to parameterize the SGS normalized beta PDF, a modeled transport equation is solved to obtain the filtered mixture fraction ˜f and a scale-similarity model can be employed to determine the SGS mixture fraction variance g˜ [46]. The transport equation for the mixture fraction can be written as ∂( r f ) ∂( r u j f ) ∂ + = ∂t ∂x j ∂x j

 m ∂ f  + q j, f    Sc ∂ x j 

(35)

where qj,f is the SGS flux for mixture fraction which can be modeled as discussed in the LES turbulence modeling section. The mixture fraction variance is given by g = K ( f − f )

(36)

where K is the model constant chosen as 3.0 [46]. In fires, the gas velocity is usually low (of the order of 1.0 m/s) and the local strain rate at the flame is low. If we assume that the effect of local extinction due to the flame stretch is of minor importance, the Favre spatially filtered mass fractions of the chemical species are then obtained from 1

Yi = ∫ Yi ( f ) p( f ) df

(37)

0

2.1.3.3 Soot modeling Soot is of great interest for various reasons. The soot emitted from combustion poses a pollution threat to the environment and plays a role in smoke inhalation which is usually the main cause of death in fires. It can enhance the desired radiation heat transfer in industrial furnaces and the undesired radiation heat transfer in fires. Soot is also used as indication of fire in many fire detection systems. Soot is a product of incomplete combustion of hydrocarbon fuels. In premixed flames, the hydrocarbon fuel will break into small hydrocarbon radicals when passing through the high temperature zone. From these radicals, small hydrocarbon molecules like acetylene can be formed. Under fuel rich conditions, the small molecules and radicals can react to form aromatic rings. Through ‘planar growth’ and coagulation, small soot particle nuclei can be formed from polycyclic aromatic hydrocarbons (PAHs). This process is the particle inception. The small particles that are formed will quickly coagulate and pick up gas phase component for surface growth. The coagulation determines the final soot particle size, while the surface growth is critical for the final soot volume fraction. The surface growth is mainly through the hydrogen abstraction carbon addition (HACA) mechanism and its rate is largely dependent on the number of active sites on the soot particle surface. The activity of soot particle surface can decrease quickly with residence time. The formed soot might be oxidized by O2 and OH when transported to the lean region. The oxidation may be frozen when the temperature drops below about 1,300 K [47], and then the unoxidized soot will be emitted. In diffusion flames, the soot formation process is similar after the formation of the first ring. It is largely the initial stage that makes difference. In diffusion flames, since soot is formed in the fuel rich and relatively low temperature zone, the molecular structure of the parent fuel has more importance for soot tendency than in premixed flames [47, 48].

CFD Fire Simulation and Its Recent Development

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Detailed modeling of soot formation and oxidation is a prohibitive task. In general, despite significant progress in recent years [49], soot modeling is still in its infancy. For simple fuels, different flamelet-based soot models have been proposed. These models can work reasonably well when used with care within their limitations. In fires, normally involved is complex fuels such as woods and plastics. For these complicated fuels, one may still have to rely on empirical formulations to provide an approximate consideration of soot formation. In the following sections, both the flamelet soot model and empirical soot model are presented. It should be noted that by taking the pre-treated variables respectively as averaged variables in RANS and filtered variables in LES, the presented soot models apply in the same form for both RANS and LES. 2.1.3.3.1 Flamelet-based soot model Since surface growth and oxidation are heterogeneous slow processes, there is no strict state relation between the soot volume fraction and the mixture fraction and the classical flamelet concept is thus not applicable. However, it may be argued that the soot formation and oxidation rates can be approximated as a function of mixture fraction. Therefore, with the reaction source term provided, soot can be modeled within the framework of the flamelet concept by solving additional transport equations. In recent years, a number of flamelet based soot models have been proposed. Moss has presented a two-variable soot model [50−53] which includes the essential physics. With the model parameters carefully adjusted through calibration, his model has been shown to be quite successful. A similar two-variable soot model was proposed by Lindstedt [54, 55] and was quite successfully applied to turbulent soot modeling [56, 57]. In this soot model, soot formation is related to the chemical intermediate acetylene rather than to the parent fuel which was used in the soot model suggested by Moss et al. [50−53]. Recently, Balthasar et al. [58] have presented a soot model in which the transport equation for soot volume fraction was solved with the source terms calculated in the mixture fraction/scalar dissipation rate space for laminar flamelets and tabulated in a library. In these models, based on the flamelet concept, the reaction rates of soot formation and oxidation are eventually considered as a function of mixture fraction alone. For brevity, the details of these models are not discussed here. Figure 6 shows preliminary result from a simulation of a C3H6 buoyant diffusion flame using Balthasar et al.’s soot model. The application of the above-mentioned types of soot model is still limited to simple fuels. In [59], the flamelet soot model proposed by Lindstedt [54, 55] was employed for soot modeling in a CFD calculation of propane flame heat transfer. For complex fuels, as an estimation, simple empirical models based on measurements can be used [60, 61]. 2.1.3.3.2 Empirical soot model As stated before, for complicated fuels, one may still have to rely on empirical formulations to provide an approximate consideration of soot formation. This kind of empirical soot models can be constructed based on measurement data. For example, as an approximation, soot can be considered by assuming that a certain amount of fuel is simply converted to soot with an empirical soot conversion factor [60, 61] which can be chosen with reference to some experimental measurements. Different conversion factors should be used for different fuels. The soot conversion rate can be simply assumed to be locally proportional to either fuel supply rate or fuel combustion rate which can be modeled by the combustion model. The formed soot is subject to transport. A scalar transport equation is solved to calculate the soot mass concentration distribution. This transport equation can be written as ∂( rYsoot ) ∂( r u j Ysoot ) ∂ + = ∂t ∂x j ∂x j

 m ∂Ysoot  − r u ′′j Ysoot ′′  + Rsoot   Sc ∂ x j 

(for RANS)

(38)

376

Transport Phenomena in Fires

Figure 6: Predicted soot volume fraction distribution in a C3H6 flame. ∂( rYsoot ) ∂( r u j Ysoot ) ∂ + = ∂t ∂x j ∂x j

 m ∂Ysoot  + q j,Y  + Rsoot  soot  Sc ∂ x j  ___

(for LES)

(39)

___

In the above two rate Rsoot can be modeled as Rsoot = cconversion × (fuel ___ equations, the conversion ___ ___ supply rate) or Rsoot = cconversion × Rfuel, where cconversion is the empirical conversion factor and Rfuel is the local fuel combustion rate which can be provided by the combustion model such as EDC. 2.1.3.4 Radiation modeling Due to its strong dependence on temperature and the strong radiating behavior of combustion products, thermal radiation is an important heat transfer mechanism in many combustion systems, particularly in fires. 2.1.3.4.1 Radiation transfer equation (RTE) and its solution Thermal radiation can be defined as electromagnetic waves emitted by a medium solely due to its temperature. The wavelength important to heat transfer is normally limited between 0.1 µm (ultraviolet) and 100 µm (mid-infrared). For a medium in local thermodynamic equilibrium, the differential equation of the spectral radiation can be written as [62, 63] ∂I w, s ∂s

= kw,s I w0 ,s − bw,s I w,s +

sw , s 4π

∫ I w,s (si )Φ(si , s ) dΩi

4π

(40)

where the superscript 0 stands for black body radiation, kw,s is the effective spectral absorption coefficient which represents induced transition including both induced absorption and induced emission, sw,s is the spectral scattering coefficient, bw,s is the extinction coefficient bw,s = kw,s + sw,s, si , and s are the direction vectors, and Φ( si , s ) is the phase function, which is equal to 1.0

CFD Fire Simulation and Its Recent Development

377

for isotropic scattering. The three terms on the right-hand side represent the augmentation due to emission, the attenuation due to absorption and scattering, and the augmentation due to scattering, respectively. Since the light speed in a medium is usually large, the radiation time scale is much smaller than almost all the other practical time scales concerned in fires, the transient term has been ignored in the above equation. The spectral radiation transfer equation is usually rewritten using non-dimensional optical coordinate as ∂I w, s ∂ xw

+ I w,s = (1.0 − ww )I w0 ,s +

ww 4π

∫ I w,s (si )Φ(si , s ) dΩi = Sw ( xw , s )

(41)

4π

s

s

w,s where the single scattering albedo ww is defined as ww = ___ , x = ∫0 bw,s ds and the source funcbw,s w w 0 w __ tion Sw (xw, s ) = (1.0 − ww) Iw,s + 4π ∫4π I w,s ( si )Φ( si , s )dΩi . The rewritten RTE is an integral−differential equation. Multiplying this RTE by exw results in

∂( I w , s e x w ) ∂ xw

= Sw ( xw , s )e xw

(42)

which can be integrated along the radiation path from an arbitrary point s1 to another arbitrary point s2 to give I w,s2 e

xw ,s2

xw ,s2

=

∫

x Sw ( xw , s )e xw dxw + I w,s1 e w ,s

(43)

1

xw ,s1

This integration can be rewritten as xw ,s2

∫

I w, s2 =

−( x Sw ( xw , s )e w ,s

− xw )

2

dxw + I w, s e

( xw ,s1 − xw ,s2 )

1

xw ,s1 xw ,s2

∫

=

−( x Sw ( xw , s )d(e w ,s

2

− xw )

) + I w, s e

( xw ,s1 − xw ,s2 )

(44)

1

xw ,s1 –(x

–x )

where e w ,s w is the transmissivity from point s to point s2. The above solution is a third order integral equation, where the source function itself is an integration of unknown radiation intensity. From the radiation intensity, the spectral heat flux vector and radiation energy source term (Sh in eqn (3)) can be readily calculated as qw = ∫ I w (s )sdΩ (45) 2

4π

Sh = ∇ ⋅ qw = ∇ ⋅ ∫ I w (s )sdΩ = 4π

=



∫  kw,s I w,s − bw,s I w,s +

4π

0

∫

4π

sw , s 4π

  = kw,s  4πI w0 − ∫ I w (s )dΩ   4π

s ⋅ ∇I w (s )dΩ =

∂I w ( s ) ∫ ∂s dΩ 4π 

∫ I w,s (si )Φ(si , s )dΩi  dΩ

4π

(46)

378

Transport Phenomena in Fires

Due to its complexity, an exact solution for radiation can only be obtained for a few simple cases. For most practical applications where the radiation path is usually non-gray and non-homogeneous, solutions are obtained using engineering approximation methods. There are a number of approximate methods to solve the RTE and obtain Sh and the radiation heat flux to surfaces, including the spherical harmonics method (PN approximation), the discrete ordinates method, the discrete transfer method, the finite volume method, the zonal method and the Monte Carlo method. The discussion of these methods can be found in [63, 64]. Since radiation has a highly non-linear dependence on the radiating medium’s thermal properties including temperature and mass compositions, the pre-treatment of the radiation within the RANS and LES concepts is extremely complicated. As an approximation, in normal practice, the mean/filtered radiation flux and radiation energy source term are modeled simply based on radiating medium’s mean/filtered thermal properties. 2.1.3.4.2 Radiation property evaluation The radiation properties of the radiating medium must be evaluated when solving the radiation equation. Commonly adopted radiation property evaluation methods are the total absorptivity method [65], the weighted sum of gray gas (WSGG) method [65], the spectral line weighted sum of gray gas (SLWSGG) [66] method, the wide band model [67], and the narrow band model [68]. In fires, radiation comes from both gases and particles. The radiating gases normally include water vapor, carbon dioxide, hydrocarbon fuels, and carbon monoxide, etc. The particles mainly include soot. Radiating gas and soot have quite different radiation behavior. At the moderate temperature, in the gas, bound-free and free−free transitions are rare and the energy level transitions mainly happen as bound−bound transition. According to quantum theory, the molecular energy levels are quantitized. As a result, the radiating gases are highly non-gray showing strong band radiation. Water vapor and carbon dioxide are usually the main radiating gases in a combustion system, having their most important infrared radiation bands centered at 1.38, 1.87, 2.7, and 6.3 microns for water vapor, 2.7 and 4.3 microns for carbon dioxide. The diatomic molecules including oxygen and nitrogen do not radiate at normal combustion temperature. In some situations, the gas fuels and their intermediates may play an important role in radiation. According to Mie’s scattering theory, the scattering efficiency factor is proportional to the fourth power of the particle’s size parameter. Since the gas molecules are much smaller than the thermal radiation wavelength, their contribution to the scattering is negligible. Soot particle radiation includes both scattering and absorption/emission, but the scattering can also be ignored when the particle is sufficiently small. Soot shows a spectrally continuous radiation and is quite gray since its spectral absorption coefficient is a weak function of the wavelength (approximately proportional to the inverse of wavelength when the particles are small). Spectral radiation of gases: The spectrum of a radiating gas consists of a large number of individual radiation lines. Each individual line corresponds to a transition between two molecular quantitized energy levels. However, the individual line is not exactly monochromatic. It is broadened by nature broadening, collision broadening, and Doppler broadening. Thus, each individual line has some finite width. The shape of collision broadened line is the same as that of nature broadened line, but differs from that of Doppler broadened line. The width of the broadened line is dependent on the broadening mechanism and the mixture’s thermal state. Since collision broadening results from molecule’s collisions which are proportional to molecule’s number __ p density (proportional to _T ) and average speed (proportional to √ T ), it is not surprising that the p __ . The Doppler line’s half-width is, on the half-width of the collision__ line is proportional to __ √T contrary, proportional to √T and the wave number. When calculating radiation, all the broadening

CFD Fire Simulation and Its Recent Development

379

mechanisms should be considered in principle. But in many cases the collision broadening is much more important and the problem can be simplified. When the collision line’s half-width is 2 or 3 times more than that of the Doppler line, the Doppler broadening can be ignored. At high temperature and/or low pressure, due to the different variation behaviors of the half-widths of collision and Doppler lines, and that the shift of the important part of Planck function to large wavenumber spectrum, the Doppler line can become important. Spectral radiation of soot particle: The radiation behavior of soot particles varies with the chemical composition of the particle. This variation can be very complex, and is thus often ignored. Assuming the soot particles are sufficiently small so that the scattering can be ignored, we have [63]  m 2 − 1  2 πr Qabs ≈ −4 ℑ  2 ⋅  m + 2 l

(47)

∞  m2 − 1  π ∞ 4 3 kl = ∫ πr 2 n p (r ) Qabs dr = −6 ℑ  2  ∫ πr n p (r ) dr  m + 2 l 0 3 0

(48)

where Qabs is the absorption efficiency factor, r is the radius of the idealized spherical particle, np(r) ∞ is the particle number density, ℑ denotes the imaginary part, and ∫ 0 _43 πr3np(r)dr gives the soot volume fraction fv. The above equation can further be written as  m 2 − 1  6π fv   fv 36πnk kl = −ℑ  2  l = 2 2 2 2 2 l  ( n − k + 2) + 4 n k   m + 2

(49)

which indicates that the spectral radiation absorption coefficient of a soot cloud of small particles is proportional to the soot volume fraction and inversely proportional to the wave length. The complex refractive index is dependent on the particle’s chemical composition, which is expected to vary with fuel type. Hottel and Sarofim [65] suggested a value of about 7.0 for 36πnk ____________ _fv 2 2 2 2 2 . Therefore, we approximately have kl ≈ 7.0 l . (n − k + 2) + 4n k

Narrow band model for a constant parameter path: The spectrum of a radiating gas is usually highly complex. Although the exact solution of gas radiation can be obtained in principle by lineby-line calculations, it requires a huge computation effort and all the detailed information of the individual lines, and is thus prohibitive. This problem necessitates model approximations. The narrow band model is the most accurate engineering method for the evaluation of radiation properties. In the narrow band model, the evaluation is carried out not on every individual line but on small spectrum intervals which are normally at 5−50 cm−1. Within each narrow band, there are many individual lines. Therefore, the purpose of narrow band models is to approximate the mean radiation properties within the small spectrum intervals. In order to extract their mean radiation properties, the information about the strength distribution, spacing, width, and shape of the individual lines are needed. This information usually turns out to be very complicated. As a result, approximations are introduced when constructing the narrow band models. The line broadening mechanism describes the line shape and its width. Most important assumptions are related to line strength distribution and line spacing. Different approximations on line distribution and line spacing result in different narrow band models. The choice of a specific narrow band model can be based on accuracy and mathematical simplicity. The accuracy of a particular narrow band model relies on how well its assumptions comply with the spectrum structure of the radiating

380

Transport Phenomena in Fires

molecule being considered. The regular model (Elasser model) is more appropriate for simple molecules which have approximately equal line spacing and line strength, particularly when the Q branch is not important or even presented. For complex molecules which have irregular line strength distribution and line spacing, the random (statistical) model can be more applicable. The spectral absorption coefficient at any wavenumber can be calculated as a sum of the contributions from all the individual lines kw = ∑ kwi .

(50)

i

The mean spectral (narrow band) absorption coefficient and emissivity can be calculated from w+

1 kw = ∆w w+

1 ew = ∆w

∆w 2

∫ w−

∆w 2

∆w 2

∫

∆w w− 2

kw ′ dw ′

(51)

  s  1 − exp  − ∫ kw′ ds  dw ′.   0  

(52) __

__

Since e is a non-linear function of k, ew cannot be simply written as ew(kw), even for a homogeneous and isothermal path. This is similar to the problem found in turbulent combustion modeling where the mean reaction rates cannot be given solely by the mean temperature and mean concentrations. __ __ Narrow band models give approximations for kw and ew. Depending on the approximation of line distribution, different narrow band models can be proposed. In particular, the following discussion of FASTNB and approximate FASTNB is based on a narrow band model where the so-called Goody model is adopted for the collision line and a random model of equal line strength is adopted for Doppler line [68]. Fast narrow band (FASTNB) model: The narrow band model is the most accurate engineering approximation method, but it is usually expensive to implement. The computation speed of narrow band computations is of critical importance. In [69], a FASTNB computer model which predicts the radiation intensity in a generally non-isothermal and non-homogeneous combustion environment was developed. Due to the complexity of the radiation, there is no analytical expression for the narrow band calculation of the radiation intensity. Therefore, the calculation must rely on a numerical solution. The line path is broken into a number of elements and the thermal radiation spectrum is divided into many (several hundred) small intervals. The spectral and spatial integrations are thus replaced by summations over all the spectrum intervals and elements, respectively. The radiation transfer equation then becomes M

I=

M

N

∑ I w ,s ∆wm = ∑ ∑ I w0 , j −0.5 (tw , j→ N − tw , j −1→ N ) ∆wm

m =1

m

m =1 j =1

m

m

m

(53)

The main task is to calculate the transmissivity twm , j → N . In the narrow_____ band equations, in the evaluation of transmissivity, there are two important band parameters, S/d and 1/d. These two parameters are functions of both temperature and wavenumber, and can be evaluated by molecular models or by interpolation from tabulated data. This evaluation consumes a large amount

CFD Fire Simulation and Its Recent Development

381

of CPU time and it is repeated in the case of non-homogeneous paths with multiple elements. The CPU time would significantly be reduced if this evaluation could be avoided. One possible method of doing this is to pre-evaluate these two parameters and create a once-and-for-all database for them [69]. As mentioned above, the thermal_____ radiation spectrum is divided into many intervals. We can now compute the band parameters S/d and 1/d in advance at all these corresponding discrete wavenumbers and temperatures, by the molecular model or the_____ interpolation of tabulated data. Therefore, a once-and-for-all database can be created for both S/d and 1/d. In the computer program, the data for each parameter can be stored in a two-dimensional array and can be used directly. By pre-creating a database, the evaluation of band parameters can be avoided during narrow band calculations and the computation speed can be significantly improved. FASTNB provides an accurate calculation at reasonably fast speed. When compared with Grosshandler’s narrow band model RADCAL [70], which has been verified quite extensively against experimental measurements, FASTNB is substantially faster and gives almost exactly the same result as RADCAL, as shown in Fig. 7. The details of FASTNB can be found in [69]. Approximated fast narrow band (approximated FASTNB) model: The approximated FASTNB was developed in [71] to further speedup the narrow band model computations at a price of small loss in accuracy. Considering a calculation of spectral radiation intensity along a single radiating path which is spatially discretized into N elements in a narrow band model including FASTNB since the non-gray effects in a narrow spectral interval are modeled, in order to calculate the spectral radiation intensity Iwm,n for the nth element of the path according to eqn (53), there will be n evaluations of spectral transmissivity. Therefore, to compute a complete radiation intensity distribution along the whole path, the total number of the spectral transmissivity evaluation along this radiating path will be 1 + 2 + 3 + + N = (1 + N )* N / 2

(54)

Radiation intensity (kW/m2sr)

which is proportional to N2.

90

Speed ratio=25

75

RADCAL

60

FASTNB

45 30 15 0 0

0.2

0.4

0.6

0.8

1

1.2

Distance (m)

Figure 7: Comparison of total intensities, for a path of parabolic minimum temperature and concentrations, with a uniform soot volume fraction of 2.0 × 10−6 [69] (©Taylor & Francis).

382

Transport Phenomena in Fires

Assuming locally gray behavior in a small spectral interval in each spatial element which has a local mean spectral absorption coefficient kwm ,n, , the computational effort can significantly be reduced to be proportional to N. In this case, the spectral transmissivity twm , j → n can be approximated as follows  n   n −1  twm , j → n = exp  − ∑ kwm ,i ui  = exp  − ∑ kwm ,i ui  × exp( − kwm ,n un )  i= j   i= j  = twm , j → n −1 exp( − kwm ,n un ) = twm , j → n −1twm ,n −1→ n

(55)

Therefore, we have n

I wm , n = ∑ I w0 m , j − 0.5 (twm , j → n − twm , j −1→ n ) j =1

n −1

n

j =1

j=n

= ∑ I w0 m , j − 0.5 (twm , j → n − twm , j −1→ n ) + ∑ I w0 m , j − 0.5 (twm , j → n − twm , j −1→ n ) n −1

= ∑ I w0 m , j − 0.5 (twm , j → n −1 − twm , j −1→ n −1 ) × exp( − kwm , n un ) + I w0 m , n − 0.5 (1.0 − twm , n −1→ n ) j =1

= I wm ,n −1twm ,n −1→ n + I w0 m ,n − 0.5 (1.0 − twm ,n −1→ n )

(56)

which shows that the spectral intensity for a spatial element n is directly related to the spectral intensity for the spatial element n − 1. If we march the computation along the radiating path from spatial element 1 to element N, for each element, we will only need to evaluate one new local spectral transmissivity. As a result, the total computational effort on spectral transmissivity evaluation is linearly proportional to N. The local spectral transmissivity can still be evaluated using the same formulas as those in the original narrow band model [71]. The deviation caused by the approximation is analyzed in [71]. In general, we can expect the loss in accuracy in the calculated radiation intensity to be small, particularly when a radiating path is contaminated by soot. Figures 8 and 9 show comparisons between the approximate FASTNB and FASTNB models for typical paths. As analyzed before, the computational effort is proportional to N2 in the original FASTNB and to N in the FASTNB approximation. This implies that the speedup will approximately increase linearly with N. This can clearly be seen in Fig. 10. Therefore, a high speedup ratio can be expected in a heavy computation where N is large. More details about the FASTNB approximation can be found in [71]. 2.2 Modeling of the response of solid materials In conventional fires, the bulk solid material responds and feeds back to the state variation of its surrounding gases. For a combustible solid material, the most important processes inside the solid include both heat conduction and pyrolysis, which will be discussed in the following sections. In most cases, simulation of a conventional fire is of interest only during the fire period before flashover. During this short period, for each surface of a solid exposed to fire, the heat

CFD Fire Simulation and Its Recent Development Speed ratio = 14.3

155

Rad. Intensity (kW/m2sr)

383

150

145

140 FASTNB approximation

135

Original FASTNB 130 0.0

1.0

2.0

3.0

Distance (m)

Rad. Intensity (kW/m2sr)

Figure 8: Comparison of total intensities, 50 spatial cells along the path, without soot [71] (©Taylor & Francis).

210

FASTNB approximation

180

Original FASTNB

150 120 90 60 Speed ratio = 14.3

30 0 0.0

0.2

0.4

0.6

0.8

1.0

Distance (m)

Figure 9: Comparison of total intensities, 50 spatial cells along the path, with uniform soot volume fraction of 20 × 10−6 [71] (©Taylor & Francis).

wave penetration is limited to a thin layer (usually less than a few centimeters). In the following sections, for convenience of discussion, we generally denote such a surface layer as a wall. Please note that such a general wall can but does not have to be a real building wall. It can represent for example the surface layer of a wood table.

384

Transport Phenomena in Fires 100.0

Speed up ratio

80.0

60.0

40.0

20.0

0.0 0

50

100

150

200

250

300

350

Number of spatial elements

Figure 10: Variation of speedup ratio with number of spatial elements [71] (©Taylor & Francis).

2.2.1 Heat conduction inside a solid wall Heat conduction inside a wall determines the internal temperature distribution and provides the necessary boundary condition for the gas phase calculation. For an inert wall, the heat balance is described by the following transient heat conduction equation ∂( r H ) = ∇ ⋅ ( k ∇T ) ∂t

(57)

T

where H is the enthalpy given by H = H0 + ∫T0 cpdT. The specific heat cp and conductivity k are usually temperature dependent and can be represented by polynomial functions. In a typical conventional fire such as a building fire, the walls of interest are usually connected and each wall is exposed to gas on two surface sides. In principle, the heat conduction should be considered in three-dimensional space with coupling between walls included. However, simplifications can usually be made. In many cases, the heat conduction from one wall to another through their inter-connection is of minor importance in comparison with the heat transfer between the wall and its surrounding gas. Furthermore, if the heat conduction along the direction perpendicular to wall surface dominates the problem can further be simplified to be one-dimensional. Consequently, the multidimensional heat conduction equation reduces to its one-dimensional form, which is ∂( r H ) ∂  ∂T  = k  ∂t ∂x  ∂x 

(58)

The initial and boundary conditions can be given by T (t = 0, x ) = T0 ( x ) k

∂T ∂x

surface =

hc (Tgas − Tsurface ) + Rflux

(59) (60)

CFD Fire Simulation and Its Recent Development

385

where Rflux represents the net radiation flux, Tgas is the temperature of the gas close to the wall surface, and hc is the convection heat transfer coefficient. These parameters are provided by the gas phase modeling of turbulent combustion. Due to space limitations, the important convective heat transfer is not included in the above discussion of turbulent combustion modeling. Normally because the boundary layer is very thin and it is very expensive to resolve the boundary layer in simulations, the convective heat transfer is calculated using wall functions. Information on wall functions and their application in calculating convective heat transfer in fire simulation can be found in [61, 72, 73] and the references cited therein. When a solid wall is exposed to fire, it may be subjected to intensive heating. Meanwhile, most practical solid wall materials are not good heat conductors. Along the direction perpendicular to the wall surface, the temperature gradient in a solid can be very high particularly near the wall surface. The accuracy requirement of the numerical solution of this heat conduction equation necessitates adoption of a grid which can be much finer (typically around 1.0 mm) than that used in the gas phase computation. Therefore, the presently affordable gas phase numerical grid cannot be directly adopted for solid phase computations and a separate grid system is needed for the solid phase calculation. In the simplified one-dimensional case, the surface of a wall is subdivided into many small elements according to the gas phase grid and along the direction which is perpendicular to the surface each element is represented by a number of thin slices which can be less than 1 mm thick. Therefore, a special grid arrangement needs to be made where separate grids are used for the gas phase computation and the solid phase temperature calculation. Figure 11 illustrates a grid system which can be adopted. In this grid system, the surface of a solid boundary is subdivided into many small elements according to the gas phase grid, and along the direction perpendicular to the surface, each element is represented by a number of very fine slices used for solid calculation. 2.2.2 Pyrolysis modeling In conventional fires, many involved solid walls are combustible. When sufficiently heated, these combustible walls start to degrade and pyrolyze. Since the pyrolysis is usually highly complex, the detailed pyrolysis modeling of solid fuels is a very difficult task. Fortunately, in fire simulations

T1

T2

T3

T4

T5

…...

Depth direction

Figure 11: Illustration of a separate grid system [74] (©Taylor & Francis).

386

Transport Phenomena in Fires

particularly when the energy output is of major concern the pyrolysis modeling can be simplified without loosing the main physics. The pyrolysis reaction can be described in a simplified form as Virgin material → Volatile products + Char For a non-charring material, char will not be produced during the pyrolysis. For a pyrolyzing material, in a local very small finite volume represented by a control volume used in numerical computation, virgin material, char, and volatile products may coexist. If we denote their volume fractions by gvir, gchar, and gvol, respectively and assume one-dimensional mass flow along the direction perpendicular to the wall surface, considering that chemical reactions do not affect the total enthalpy in a reacting system, the energy balance equation in a pyrolyzing material can be written as ∂( rvir gvir hvir + rchar gchar hchar + rvol gvol hvol ) ∂(m ′′hvol ) + = ∇ ⋅ ( k ∇T ) ∂t ∂x

(61)

where the subscripts vir, char, and vol denote the virgin material, char, and volatile products, respectively, x is the coordinate along the direction perpendicular to the wall surface, h is the total enthalpy, and m◊ ″ is the mass flux of the volatile products. The three terms in the above equation represent the energy storage, energy convection due to the flow of the volatile products, and heat conduction, respectively. The effects of pressure variation and flow kinetic energy are ignored. The above equation can be reorganized as ∂( rvir,bulk hvir + rchar,bulk hchar + rvol,bulk hvol ) ∂t

+

∂(m ′′hvol ) = ∇ ⋅ ( k ∇T ) ∂x

(62)

where rvir,bulk = rvirgvir is the local bulk density of virgin material, rchar,bulk = rchargchar is the local bulk density of char, and rvol,bulk = rvolgvol is the local bulk density of volatile products. As already mentioned, in many cases, the heat conduction along the direction perpendicular to the wall surface dominates and can thus be simplified to one-dimensional as well. The above energy balance equation then becomes ∂( rvir,bulk hvir + rchar,bulk hchar + rvol,bulk hvol ) ∂t

+

∂(m ′′hvol ) ∂  ∂T  = k  . ∂x ∂x  ∂x 

(63)

For convenience, the discussion that follows will be based on the above equation with onedimensional heat conduction. However, it is straightforward to extend the discussion to a case with three-dimensional heat conduction. The mass conservation gives ∂( rvir,bulk + rchar,bulk + rvol,bulk ) ∂t

+

∂m ′′ =0 ∂x

(64)

For simplicity, we assume an instant escape of the volatile products from the solid. Thus, we have ∂rvol,bulk _____ = 0 and rvol,bulk = 0. If we further write rvir,bulk + rchar,bulk as rs and assume hvir = hchar = hs, ∂t where the subscript s denotes the solid, the above two equations can be rewritten as ∂( rs hs ) ∂(m ′′hvol ) ∂  ∂T  + = k  ∂t ∂x ∂x  ∂x 

(65)

∂rs ∂m ′′ + =0 ∂t ∂x

(66)

CFD Fire Simulation and Its Recent Development

387

Combining eqns (65) and (66) gives ∂[ rs (hs − hs,0 )] ∂t

+

∂[ m ′′(hvol − hs,0 )] ∂x

=

∂  ∂T  k  ∂x  ∂x 

(67)

where hs,0 is the total enthalpy of the solid at a reference temperature T0 and hs − hs,0 is the sensible T T enthalpy given by hs − hs,0 = ∫T0 cp,s dT . For convenience, we define Hs = hs − hs,0 =∫T0 cp,s dT and T similarly Hvol = hvol − hvol,0 = ∫T0 cp,vol dT. The total enthalpy of volatile products can be written as T

hvol = hs,Tp + H py + ∫ cp,vol dT = hs,Tp + H py + H vol,T − H vol,Tp Tp

(68)

where Hpy is the chemical conversion heat associated with a unit mass of volatile products, and can be calculated by the difference in total enthalpy of the virgin material and the volatile products at temperature Tp, i.e. H py = hvol,Tp − hs,Tp = hvol,Tp − hvir,Tp = hvol,Tp

Tp   −  hvir,T0 + ∫ cp,vir dT    T0

(69)

It is worth pointing out that Hpy is approximately constant for a specific material and is different from the heat of gasification, Hg, where H g = qnet / m total =

hc (Tg − Tx = 0 ) + Rflux m total

(70)

which is a local and transient value and changes considerably during the pyrolysis process [75]. For the thermally thick vaporizing material, at steady state Tp

H g = hvol,Tp − hvir,T0 = H py +

∫ cp,vir dT .

T0

(71)

Using eqn (68) and hs,Tp = hs,0 + Hs,Tp , the left-hand side of eqn (67) can be reorganized as ∂[ rs (hs − hs,0 )] ∂t

+

∂[ m ′′(hvol − hs,0 )] ∂x

  ∂( rs Hs ) ∂  m ′′( Hs,Tp + H py + H vol − H vol,Tp ) + ∂t ∂x     ∂( rs Hs ) ∂  m ′′( Hs,Tp + H py ) ∂  m ′′( H vol − H vol,Tp ) = + + ∂t ∂x ∂x ∂  m ′′( H vol − H vol,Tp ) ∂( rs Hs )  = + m ′′′( Hs,Tp + H py ) +  ∂t ∂x =

(72)

∂m◊ ″ where m◊ ¢″ = ___ , representing the mass loss rate of the pyrolyzing material per unit volume. ∂x

388

Transport Phenomena in Fires

Since the pyrolysis happens in a narrow temperature range around Tp, Hs,Tp can be approximated by Hs when m◊ ¢″ is of importance, we can therefore rewrite the above equation approximately as (for convenience, the subscript s will be dropped from r and H in the following equation) ∂  m ′′( Hvol − Hvol,Tp )  ∂( r H )  = ∂  k ∂T  + m ′′′( H py + H ) +    ∂t ∂x ∂x  ∂x 

(73)

where the third term is the energy required to heat the vaporized gas as it flows to the solid surface. This term has no important effect here, thus it will be ignored. But it can be very easily included. ∂m◊ ″ From the mass conservation equation and the definition m◊ ¢″ = ___ , one can have ∂x m ′′′ = −

∂r . ∂t

(74)

∂r ∂(rH) ∂r ∂r ∂H ∂T Noting that ____ = H __ + r __ = H __ + rcp __ and m◊ ″¢ = __ , the energy equation can be rewritten as ∂t ∂t ∂t ∂t ∂t ∂t

r cp

∂T ∂  ∂T  + m ′′′H py = k  ∂t ∂x  ∂x 

(75)

Assumptions have been introduced during the derivation of the simplified energy conservation equation. These assumptions can be better justified for non-charring materials. To be more general, assumptions can be lifted at the expense of increasing the complexity. The eqns (74) and (75) are the final mass conservation and energy conservation equations to be solved for modeling of pyrolysis. To implement the numerical solution, eqn (75) needs to be discretized first. The discretization can be based on either an implicit or explicit scheme. When the pyrolysis model is coupled with CFD for flame spread simulations, the time marching scheme in the pyrolysis model should fit with the time marching scheme in CFD. Typically, for the pyrolysis model, an implicit scheme is used when the RANS CFD concept is employed and an explicit scheme is adopted when the LES CFD concept is used. For convenience, we will present the discussion with one typical discretization which is based on the fully implicit backward time stepping and the central space difference scheme. With this typical discretization, the discretized energy equation can be written as Tn − Tn′ T −T T − Tn −1 d x + m d′′x H py = k( n +1)( n ) n +1 n − k( n )( n −1) n (76) ∆t dx dx where the prime indicates the previous time step, m◊ ″ is the mass loss rate per unit area of the d rcp

dx

x

thick strip, and the subscript of the conductivity k denotes the two temperature nodes between which the surface of the control volume is located. For convenience, we reorganize the above discrete equation as ApTn = Ae Tn +1 + Aw Tn −1 + Su

(77)

where Ae =

k( n +1)( n ) dx

;

Aw =

k( n )( n −1) dx

Ap = Ae + Aw + rcp d x / ∆t;

(78) (79)

CFD Fire Simulation and Its Recent Development

Su = rcp d xTn′ / ∆t − m ∂′′x H py

389 (80)

Since the conductivity, k, is generally a function of temperature, it may vary with x and it is not necessary for Ae to be equal to Aw. In order to get a reasonable result for the mass loss rate, a very fine grid was required to resolve the density profile in the pyrolyzing layer and locate the pyrolysis front. However, this very fine grid is unnecessary and very expensive for the temperature solution. This inconsistency is overcome by introducing a dual mesh concept where a relatively coarser grid is defined for the temperature solution and then refined into a second grid for the mass loss rate calculation, as shown in Fig. 12. The temperature of the refined grid m of the coarser grid n (we will denote this grid as grid (n,m) later), Tn,m, is obtained by interpolation, as shown in Fig. 12, assuming a linear distribution between Tn and Tn+1. In the above discussion, the mass and energy balance equations and their discretizations are presented. However, from the mass and energy balance equations, one can easily find out that in these two equations, there are three dependent variables, namely r, m◊ ¢″, and T. The equation system is not closed. This is because the pyrolysis chemistry is not yet considered to relate the mass loss rate m◊ ¢″ with the other two dependent variables. As presented below, depending on how the mass loss rate m◊ ¢″ is evaluated with r and T, one can basically have two different pyrolysis modeling procedures within the above overall framework. 2.2.2.1 Thermo-pyrolysis modeling For most combustible solid materials, the activation energy of the pyrolysis is large [76]. Due to the large activation energy and the endothermic feature of the pyrolysis process, the pyrolysis mainly happens in a narrow temperature range. The approximate constancy of the pyrolysis temperature suggests that below the pyrolysis temperature Tp, the pyrolysis reaction proceeds at a negligible rate, but above the pyrolysis temperature, the chemistry becomes so fast that the pyrolysis rate is then essentially determined by the physical heat transfer. Therefore, as far as the mass release rate is concerned, the pyrolysis can well be described by the heat balance without the need of going to the details of the chemical reaction rates which may involve many unknown chemical mechanism parameters.

Figure 12: Temperature solution grid and its refinement (e.g. with N = 5, M = 10, where N is the coarser grid number and M is the refined grid number in a coarser grid) [60] (©Elsevier Science Ltd., with permission).

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As mentioned before, the pyrolysis rate is slow when the material temperature is below its pyrolysis temperature, Tp, and becomes fast when the temperature is above Tp. As an approximation, we can reasonably assume that the material will start to pyrolyze only when its temperature reaches the pyrolysis temperature. Once started, the endothermic pyrolysis process will adjust its rate to keep the material temperature at the pyrolysis temperature until the material is completely pyrolyzed. Thus we have rcp

∂T ∂  ∂T  = k  ∂t ∂ x  ∂ x 

m ′′′H py =

∂  ∂T  k  ∂x  ∂x 

when T ≤ Tp or r ≡ rchar

when T ≥ Tp

and r > rchar

(81)

(82)

During the iteration, Tn,m may exceed the pyrolysis temperature. In and only in this case, the endothermic pyrolysis will be induced and then it will adjust itself to limit the local temperature to the pyrolysis temperature. Thus, from eqn (77), for an arbitrary refined grid (n,m), the energy available for pyrolysis can be approximated as H n,m = max 0.0, Ap (Tn,m − Tp ) / M 

(83)

where M has been defined in Fig. 12. The above energy excess allows the pyrolyzing layer to spread over a number of refined grids and provides one necessary, but not sufficient, factor to determine the pyrolysis rate. In order to finally calculate the pyrolysis rate, one also needs to know the mass of the volatile material remaining in the refined grid (n,m). The volatile mass remaining in a refined grid can be easily calculated by monitoring its density history. However, this would require much memory storage, particularly when dealing with a large number of pyrolyzing solid elements in flame spread simulations. In order to minimize the memory usage and make the data structure of the computer program more tidy, only the variation of the average density of the coarser grids is followed. The density of a refined grid is calculated by assuming a specific density distribution in the coarser grid. For a coarse temperature grid of an average density which is equal to rchar or rvir, the density of its refined grid will simply be the same as the average density. For a partly pyrolyzed coarser grid, if we assume that the char layer and the virgin material is separated by a single partly pyrolyzed refined grid, the density of a refined grid will be either rchar, rvir or rmix = Mr − (m0 − 1)rchar − (M − m0)rvir. In the rmix formula, r is the average density of the coarse temperature grid, the integer m0 locates the assumed partly pyrolyzed refined grid and can be determined by requiring r < rmix < rvir. For an arbitrary refined grid (n,m), the density can be generalized by a single formula rn,m = min[ rvir ,max( rchar , rmix, m )]

(84)

where rmix,m = Mr − (m − 1)rchar − (M − m)rvir. The mass of the volatizable material remaining in the grid (n,m) is given by massvol =

dx dx ( r − rchar ) = min{rvir − rchar ,max[0.0,( rmix − rchar )]} M n, m M

(85)

The mass loss rate from grid (n,m) is thus finally determined by m n,m = min{H n,m / H py ,massvol / ∆t}.

(86)

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The overall pyrolysis rate can be obtained by the summation over all the grids and expressed as m = ∑∑ m n,m = ∑∑ min( H n,m / H py ,massvol / ∆t ). n

m

n

(87)

m

The corresponding heat release rate is represented by c Q = mH

(88)

where Hc is the heat of combustion. As a basic test, this thermo-pyrolysis model was used successfully to simulate the Cone Calorimeter tests of both charring material (particle board) and non-charring material (PMMA). An example simulation result is shown in Fig. 13. This pyrolysis model is very fast and describes the essential physics, in so far as is needed to predict the correct mass loss and heat release rates. It can easily be used in complex cases such as those with a transient incident heat flux and temperature dependent material properties. It is applicable to both charring and non-charring materials and can automatically consider the regression of the surface of the non-charring solid material during its pyrolysis. Using this pyrolysis model, an ‘equivalent properties’ optimization program can be developed to analyze and fit the Cone Calorimeter test results. A database of the ‘equivalent properties’ of the materials tested in the Cone Calorimeter can thus be created. By using the optimized equivalent properties, this pyrolysis model can be expected to be applicable to realistic composite materials and be used as an alternative to the more complex and expensive models [77]. 2.2.2.2 Kinetic-pyrolysis modeling In a situation where the assumptions in the above thermo-pyrolysis modeling are not applicable, one may need to consider the chemistry directly and relate the mass loss rate with the local solid density and temperature based on the chemical kinetics of pyrolysis. Normally the pyrolysis chemistry can be very complicated and the reaction

Figure 13: Example result from simulation of Cone Calorimeter tests for PMMA [60] (©Elsevier Science Ltd., with permission).

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Transport Phenomena in Fires

path may vary under different external conditions such as the imposed heating and the available oxygen supply. For different fuels, the pyrolysis chemistry can be totally different. For simple solid combustibles such as PMMA, one may assume a first order of pyrolysis reaction and correspondingly express the mass loss rate as a function of density and temperature in an Arrhenius formula. Typically, for such a reaction, one can have  E  m ′′′ = A( r − rchar ) exp  − A   RT 

(89)

where A is the pre-exponential factor, EA is the activation energy, and R is the universal gas constant. If more than one reaction is involved in the pyrolysis, one needs to sum up over all the relevant reactions. For a numerical implementation of the kinetic-pyrolysis modeling, essentially all the numerical methods used in the above thermo-pyrolysis modeling can be used in the same way, including the temperature and density interpolation for the refined grid. With the interpolated local density and temperature for a refined grid (n,m), for a reaction described by eqn (89), the mass loss rate for that refined grid can be simply given by  EA  m (′′′n,m ) = A( r( n,m ) − rchar ) exp  − .  RT( n,m ) 

(90)

Using kinetic data measured in [78], this pyrolysis model was applied to simulate a Cone test for particle board and the simulation result was compared in Fig. 14 with results obtained using the thermo-pyrolysis model. The comparison shows that these two models give rather similar results. This indicates that for this case, the very efficient thermo-pyrolysis model is quite valid.

Thermo-Pyrolysis Model 2.5E+05 Kinetic-Pyrolysis Model

RHR (W/m2)

2.0E+05

1.5E+05

1.0E+05

5.0E+04

0.0E+00 0

100

200

300

400

500

600

Time (s)

Figure 14: Comparison of results from simulations of a Cone Calorimeter test for a particle board using kinetic and thermo models.

CFD Fire Simulation and Its Recent Development

393

2.2.2.3 Consideration of moisture For many combustible materials encountered in a conventional fire, such as wood, the internal moisture content can play an important role in the material’s pyrolysis process. Water absorbs a considerable amount of energy during evaporation, and will thus cause some delay in ignition. For a moisturized solid, when heated, the moisture at a local point may first evaporate and then be convected away to the surrounding. Some water vapor may be transported to the cold part of the solid and then recondense there. The transport of water vapor is heavily dependent on the pressure distribution inside the solid and the permeability of the solid. Typically, the permeability of a solid is highly dependent on the internal structure of the solid and can be inhomogeneous and non-isotropic. To consider all these, a detailed flow dynamic modeling is needed. This can be very complicated. Instead, as a first approximation, one may turn to a simple method. In a crude approach, the moisture can be described in a way that is similar to the way pyrolysis is described, where one just needs to replace the pyrolysis temperature with water’s boiling temperature, which is 100°C at a pressure of 1.0 atm, and the pyrolysis heat with water’s latent heat. By assuming 10% of initial moisture, the Cone Calorimeter tests of particle board were resimulated. Figure 15 gives a typical result which clearly demonstrates the moisture effect. 2.3 Conventional fire simulation cases Although still under development, CFD fire simulations have been widely used now for different purposes in both research and practical applications. Several fire simulation CFD codes have been developed including KAMELEON [79], SOFIE [80], FDS [81], and SMAFS [82], etc. To briefly illustrate the usage of the conventional fire simulation techniques discussed above, three typical case studies based on these simulation methods will be presented in the following sections. All the simulations discussed in this chapter were performed using the software package SMAFS [82] developed by the author.

1.5E+05

10% moisture

No moisture

RHR (W/m2)

1.0E+05

5.0E+04 Radiating flux: 25 kW/m2 0.0E+00 0

100

200

300

400

500

600

Time (s)

Figure 15: Moisture effect in a simulation of a Cone Calorimeter test for a particle board (a kinetic pyrolysis model was used).

394

Transport Phenomena in Fires

2.3.1 CFD simulation of flame spread in room corner fire This case study comes from an early attempt made to simulate the flame spread in room corner fires, in both 1/3 and full scale scenarios. This simulation was a comprehensive study based on the RANS concept, in which essentially all the main physical sub-processes mentioned above were considered. In this case study, the gas temperature, solid temperature, heat release history, and char layer development are all analyzed and compared with experimental measurement. The simulation results are rather promising. The predicted flame spread pattern and heat release history in the 1/3 scale scenario are shown and compared with experimental measurements in Figs 16 and 17. (In Fig. 17, the total heat release includes the energy released from combustion inside the room and the energy associated with the fuel which leaves the room unburnt. This unburnt fuel may burn outside the room.) The details of this study are presented in [60].

RHR (kW)

Figure 16: Flame spread pattern in the 1/3 scale scenario, indicated by surface temperature (in kelvin) (t = 300 s, threshold = 600 K).

180

exp (by B. Andersson)

160

exp (this work)

140

pre - c(total)

120

pre - c(inside)

100

pre - p(total) pre - p(inside)

80 60 40 20 0 0

50

100

150

200

250

300

350

Time (s)

Figure 17: Calculated and measured heat release rates in the 1/3 scale scenario. The symbol (c) denotes using Cone data input method and (p) the pyrolysis model [60] (©Elsevier Science Ltd., with permission).

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395

2.3.2 CFD simulation of complex practical fire This is another RANS simulation of fire and flame spread. Unlike the previous case which is mainly for research development and validation, this case study deals with a practical real life fire which happened in Gothenburg, Sweden, in 1998. This fire caused big losses in both of property and people’s life. Figure 18 shows an overview of the building. A fire was started in a half-flight of stairs in the emergency exit stairwell while the door on the ground floor was left partially open. After a short while, through the emergency exit door, the fire quickly spread into the dance hall from the stairwell. Due to the strong heating, some combustible in the dance hall was ignited and the fire became more severe. After the fire, an extensive investigation including this CFD simulation was organized by the Swedish Board of Accident Investigation and the police, to gather necessary information for the perception and interpretation of events, actions taken as well as the evacuation process. Due to its complexity and high demand in CPU time, this computation was performed in parallel on a SGI 2000 machine using a parallel algorithm presented in [74]. The results of this CFD simulation correspond with the real fire pattern very well and help explain observed fire phenomena. Figure 19 shows typical plots of heat flux and window flame temperature. The details of this simulation can be found in [83].

Figure 18: Different views of the building [74] (©Taylor & Francis).

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Transport Phenomena in Fires

Figure 19: Stairwell heat flux and window flame temperature [74] (©Taylor & Francis).

2.3.3 LES of a turbulent buoyant diffusion flame This case study presents a LES of non-premixed turbulent combustion in a pool fire. The subgrid stresses were modeled using the Smagorinsky model. The SGS scalar fluxes in the filtered scalar transport equations were modeled based on simple gradient transport hypothesis with assumed constant SGS Prandtl/Schmidt numbers. The SGS randomness of the turbulent combustion was taken into account based on a flamelet concept with a prescribed SGS probability density function. With the nucleation, surface growth, coagulation, and oxidation considered, sooting was modeled using a flamelet-based soot model where the balance equations for soot mass fraction and soot number density are solved. The instantaneous thermal radiation was calculated using the discrete transfer method with the radiation property of the participating medium evaluated by an approximated FASTNB model. The details of the models have already been discussed in previous sections. The configuration of the problem is simple. Fuel (propane) was injected into an open environment from a square burner located at the bottom center of the domain. The size of the whole computation domain was 0.475 m (x) × 1.25 m (y, vertical direction) × 0.475 m (z). A non-uniform grid of 96 × 128 × 96 was used in the computation, with clustering applied to the flame zone to provide an optimum resolution. Figure 20 presents a short sequence of the instantaneous temperature profile from LES showing a buoyant flame’s puffing.

3 CFD simulation of spontaneous ignition in porous fuel storage This section presents the recently developed CFD simulation techniques for prediction of spontaneous ignition in porous fuel storage. Unlike the already discussed conventional fires, spontaneous combustion in porous fuel storage (including bio-fuel storage and other fuel storage such as coal storage) is mainly due to heat generation by low temperature oxidation and bio-activity. When the generated heat cannot be adequately dissipated into the surrounding environment, the temperature rises and in turn further speeds up the exothermic oxidation process to eventually result in a self-ignition. Spontaneous ignition of fuel storage is a complicated problem which is dependent on many physical and chemical processes. The processes involved mainly include fluid flow, heat transfer, mass transfer, water condensation and evaporation, bio-activity, and chemical reactions, etc. Because of the large

CFD Fire Simulation and Its Recent Development

397

Figure 20: A short sequence of the instantaneous temperature profile from LES showing a buoyant flame’s puffing.

number of involved processes, spontaneous ignition relies on many different parameters which may affect the involved processes. Some well-known typical parameters which have an important influence on spontaneous ignition are storage size, moisture content, permeability, porosity, fuel particle size, and fuel’s low temperature reactivity. Environmental conditions such as the ambient temperature and wind flow also have determining effects on spontaneous ignition. Spontaneous ignition of a porous fuel storage is of great importance in the fuel industry and poses a serious safety and economic problem which can harm the environment and cause big economic losses. Due to economic and safety concerns, it is of great interest to investigate how these parameters and conditions can affect spontaneous ignition and thus to find out under which conditions the spontaneous ignition can occur or be avoided. Because of its practical importance, much experimental and theoretical effort [84−91] has been devoted to this area. Among the theoretical analysis, the most notable pioneer work is the wellknown classical Frank-Kamenetskii theory [84]. In the classical Frank-Kamenetskii analysis, in order to obtain an analytical solution on such a complicated problem, heavy simplifications are made in many aspects regarding the heat and mass transfer, chemical reaction, geometry, and boundary conditions. The advantage of the classical Frank-Kamenetskii analysis is that it explicitly gives the representative relations between some important parameters controlling the spontaneous ignition. Unfortunately, these assumptions are highly invalid in many practical situations.

398

Transport Phenomena in Fires

This seriously limits the applicability of the Frank-Kamenetskii analysis. CFD modeling is a good strategy to abandon assumptions which are purely imposed by the difficulty of obtaining a mathematical solution of the differential equations and thus improve the applicability of the theoretical analysis. CFD numerical modeling has the possibility to solve a set of coupled partial differential equations and thus offers great advantage of high flexibility to simultaneously take into account different coupled processes. In the numerical modeling, the assumption and simplification can be limited to the aspects of the physical understanding. 3.1 The comprehensive spontaneous ignition CFD model Self-ignition in a porous fuel storage such as biomass fuel or coal storage is fully dependent on the competition between heat generation inside the fuel storage and its heat loss to the environment. Normally, in a biomass fuel storage, heat can be generated due to bioactivity which can play a major role when temperature is less than about 350 K and chemical reaction which becomes dominant as temperature increases. Usually, the bio-activity is also heavily dependent on local moisture content and oxygen concentration. The exothermic chemical reaction is largely proportional to the local oxygen concentration. The local moisture content and oxygen concentration are governed by mass transport. For moisture, the condensation and evaporation can also be critically important. On the other hand, for heat loss, the heat transfer inside the fuel storage and the heat exchange between the fuel storage and its environment are governed by radiation, convection, and diffusion. In order to perform a theoretical study of self-ignition in a porous fuel storage, consideration must be given to all the important governing processes. In a CFD modeling, a set of coupled governing partial differential equations are solved to obtain the storage’s state evolution. The governing equations, which include the continuity equation, momentum equation, mass transport equation, and energy equations for both gas and solid phases, are listed below. Continuity equation ∂r ∂( r ui ) + = Sgas ∂t ∂ xi

(91)

Momentum equation j

∂( rg ui ) ∂t

+

∂( rg ui u j ) ∂x j

 ∂ p m ∂ = j2 − − ui + ∂ xi k j∂ x j 

   ∂u   cF rg  m  i   − 0.5 (ui ui )0.5 ui + rg agi    ∂ x j   k 

(92)

Mass conservation equation for chemical species j

∂rgYi ∂t

+

∂ ∂ ( rgYi u j ) = j ∂x j ∂x j

 m ∂Yi    + SYi  Sc ∂x j 

(93)

 m ∂H g    + SH g  Pr ∂x j 

(94)

Energy conservation equation for gas phase j

∂ rg H g ∂t

+

∂ ∂ ( rg H g u j ) = j ∂x j ∂x j

CFD Fire Simulation and Its Recent Development

399

Energy conservation equation for solid phase ∂ rs H s ∂ = ∂t ∂x j

 ls ∂H s    + SHs  C p ∂x j 

(95)

∂W = rw ∂t

(96)

The moisture balance in solid phase − rs

In the above equations, j is the porosity, k is the permeability, cF is the form-drag constant. In order to close the above equations, the source terms must be given based on the solved variables. The details of these source terms are omitted here and can be found in [92]. 3.2 CFD simulation of spontaneous ignition experiment The spontaneous ignition experiments simulated are basket heating tests. The biomass fuels used include wood sawdust, and wood pellets of 6 mm in diameter. The main equipment used was a temperature-controlled oven of 0.34 m (depth) × 0.40 m (height) × 0.40 m (width) with recirculating air. A stainless-steel 0.6 mm mesh basket filled with solid fuel was suspended in the oven. The size of the basket is 0.1 m × 0.1 m × 0.1 m. In order to trigger and speedup the spontaneous ignition process, the oven was heated up and maintained to have a temperature of 180°C for the sawdust case and 200°C for the wood pellets case. The temperature evolution inside the basket was monitored using five 0.25 mm type K thermocouples which were placed between the volume center of the basket and center of the basket at one side surface. The basket tests were simulated using SMAFS [82]. Figure 21 shows the comparison between predicted and measured temperature histories for the five locations of the sawdust case. The level-off phenomenon in the temperature history is due to moisture effects. At about 250 min, the temperature curves cross each other. This indicates a high potential of spontaneous ignition. As can be seen in Fig. 21, the prediction reproduces the experimental measurements very well. The temperature rise pattern, the ‘level-off’ temperature and temperature crossing time and values are all well-predicted. This indicates that all the important processes were well-captured by

Simulation

160

Point 1 Point 2

120

Experiment

200

Point 3 80

Point 4

40

Point 5

Temperature (C)

Temperature (C)

200

160 Point 1 120

Point 2

80

Point 3 Point 3

40

Point 5

0

0 0

60

120

180

Time (minutes)

240

300

0

60

120

180

240

300

Time (minutes)

Figure 21: Predicted and measured temperature histories at five locations [93] (©IAFSS).

400

Transport Phenomena in Fires Point 3

Temperature (C)

200 160 120 80

Measurement

40

Simulation 0 0

60

120

180

240

300

Time (minutes)

Figure 22: Point comparison of predicted and measured temperatures [93] (©IAFSS).

Figure 23: Spatial distribution of moisture content at t = 120 min [93] (©IAFSS). the numerical simulation. To provide a better data comparison, the predicted temperature evolution is compared with measurement for each individual point. Figure 22 gives an example comparison for point 3. To have an overview of the spatial distribution of the important moisture content, a typical visualization of moisture content at a time of 120 min is presented in Fig. 23 showing a heart pattern distribution. As indicated earlier, the solid fuel of 6 mm wood pellets was also studied. The results are similar with those for sawdust and good agreement between prediction and measurement is also obtained. More details of this simulation can be found in [93].

4 Conclusions This chapter presents a review of CFD simulations of both conventional fires and spontaneous ignition in a porous fuel storage. For conventional fire simulations, the discussion is focused on

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401

modeling of different essential sub-processes in fires including turbulence, turbulent combustion, thermal radiation, soot formation, heat transfer inside solids, and pyrolysis of combustible solid fuel, etc. For simulations of spontaneous ignition in a porous fuel storage, some interesting recent developments are presented.

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[3] [4]

[5] [6] [7] [8]

[9]

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[16] [17] [18]

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CHAPTER 11 The implementation and application of a fire CFD model J. Trelles & J.E. Floyd Hughes Associates, Inc., Baltimore, MD, USA.

Abstract The previous chapters have presented the details of various fire transport phenomena. The present chapter demonstrates how to put these theoretical and experimental building blocks together into a viable computational fluid dynamics (CFD) model and then how to effectively use the model. The emphasis is on the Fire Dynamics Simulator (FDS), a leading model in the field of fire CFD which also has the highest usership. First, the aerodynamic fundamentals such as the low Mach number equations for expandable flow and the manipulation of the equations for the purposes of discretization are presented. Two different approaches to turbulence are discussed: direct and large eddy simulations (LESs). The emphasis then shifts to heat sources and heat sinks for the CFD models. These are what really make this a fire CFD model. Two methods are presented for distributed heat input from unconfined combustion sources: solution of transport equations with Arrhenius terms for direct simulations and a mixture fraction model for LESs. The radiation model in FDS allows for heat transfer from flames and hot layers without doubling the computational overhead. Heat extraction methods are implemented for a variety of fire protection systems. These include sprinklers, water mist systems, and smoke exhaust. The last topic encompasses the effective application of what is really a complicated model. The first step is appropriate modelling of the scenario in question. The collection of input data is always a challenge in fire CFD because many items, such as the composition of a fuel, are unknown. Good results are often obtained with a sufficiently simple model that captures all the important physical contributions. Checking the work against known results from the literature is important to ensure reasonable predictions. Examples are given of how a good model can be changed into a bad model by the injudicious choice of one input variable.

1 Introduction The preceding chapters have detailed the physical and chemical fundamentals of fire phenomena. In this chapter, it is shown how these concepts are integrated into a comprehensive fire

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dynamics simulation. Recall that a computational fluid dynamics (CFD) model is based on the Navier−Stokes equations, ∂r = ∇ ⋅ ( rv ), ∂t r

Dv = −∇p + m∇2 v + r g, Dt

(1)

(2)

Du = −∇ ⋅ q ′′ + q ′′′ − ∇ ⋅ rv + ∇ ⋅ (t ⋅ v ) + v ⋅ r g. (3) Dt Here t is the time, ∇ is the gradient with respect to the space vector x , r is the density, v is the velocity vector, p is the pressure, µ is the viscosity coefficient, g is the acceleration of gravity, u is the volumetric heat generation term, and __is the internal energy, q′′ is the heat flux vector, q′′′ t is the stress tensor. When using these equations to model a fire, a striking observation is made after considering the , in these equations is phenomena to be addressed. In a fire, the volumetric energy source term, q′′′ combustion. A fire is a complex series of hundreds of chemical reactions occurring on 10−6 m or less length scales and 10−6 s or less timescales. Density changes and velocity can be driven by either local temperature changes or by pressure waves. Therefore, the velocity term can range in magnitude from buoyancy driven flows, 10 m/s, to sonic flows, 103 m/s. If the problem of interest is, for example, a wildland fire, then the overall problem length can be 104 m, and the time can be 106 s, 1 week. To create a computer model of fire, therefore, means efficiently solving the Navier− Stokes equations over potentially 12 or more orders of magnitude in time and 10 or more orders of magnitude in length. It is clear that simplifications must be made to make a solution computationally feasible. Even when simplified, a number of complex issues remain. Sources and sinks, such as volumetric heat addition, momentum impact terms induced by the transport of condensed phases, and various boundary conditions (BCs), serve to convert a basic aerodynamic solver into fire physics platform. A number of additional physical models are required to populate the variables in the Navier−Stokes equations. The heat release rate (HRR) term must be determined. Heat transfer to surfaces must be determined. Additionally, a fire protection engineer will wish to include the effects of suppression or the response of installed detection systems. Each of these additional models can be made very complex or very simple. To create a fire model, therefore, is an exercise in determining the appropriate level of complexity in the various physical models to reach the desired solution accuracy while consuming a reasonable quantity of computational resources (CPUs and/or time). For concreteness, the emphasis will be on the Fire Dynamics Simulator (FDS), a large eddy simulation (LES) developed and maintained by the National Institute of Standards and Technology (NIST) [1]. FDS is a multidimensional, multiphysics CFD simulation. It can handle isothermal or thermally variable flows. It has options for the direct simulation of turbulence or for LESs. It can accommodate axisymmetric cylindrical, two-dimensional and three-dimensional Cartesian coordinates. The approach can be summarized by three maxims: r

1. The required computing power should be affordable to a typical fire protection firm ($1,000s rather than $10,000s). 2. The required computing time should lie within the length of time available to a typical fire protection project (days or weeks rather than months).

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3. Computational time for a submodel should be proportional to the importance of that submodel in the final solution (this has been referred to as Baum’s rule). This chapter will also address the practice of CFD modelling. Obtaining a good series of simulations is a challenging objective in general that, for fire problems, is exacerbated by the uncertainty associated with characterizing such important items as fuels and HRRs. The technique of good modelling is detailed along with the methods of analysis that can be employed to certify the fidelity of a solution set. Since the readily available FDS theory manual [1] and the user’s manual [2] form an excellent introduction to the FDS, the authors of this chapter have made a conscious effort to avoid topics covered therein, instead concentrating, as much as possible, on relevant material not developed elsewhere.

2 Turbulence modelling Fires require air to burn. Air and fuel are mixed via turbulent eddies and diffusion in the flow field. Too much mixing will result in short flame heights and cooler plumes. Too little mixing and the opposite will happen. Turbulence is formed by shear within the flow resulting from either wall friction or density gradients. Turbulent structures within a flame can be less than 1 mm, whereas those within a large plume can be greater than 1 m. Thus, if the primary concern is the flame then very small turbulent structures must somehow be resolved. If the fire plume or other larger-scale fire phenomena are the main interest then only the larger eddies need be resolved. Within FDS, both options are available to the user. Direct numerical simulation (DNS) can be used when the grid resolution is capable of resolving the smallest eddies. LES can be used for larger grids. When performing a DNS computation, no submodels are used for the purpose of creating turbulence in the computed flow field. Provided sufficient grid resolution is used and appropriate BCs are specified, the proper flow field will be computed. While using DNS can avoid the uncertainties of a turbulence submodel containing empirical constants, it imposes a significant cost on the user. When performing a CFD computation, the solution for the next time step uses the information from the prior time step. If that information traverses multiple grid cells in a time step, velocity × time step > grid size, then the solution can become unstable. For a DNS fire simulation with the grid resolution of the order of a millimetre and flow speeds of the order of 10 m/s, the time step will be < 0.1 ms. Therefore, if one could update a grid cell per 0.1 µs, then using DNS on a single CPU to compute a fire in a 1 m3 cube for one minute would take almost 2 years. DNS, therefore, is not practical for fire protection applications. Nonetheless, a few exemplars exist such as the laboratory bench-scale studies published in [3] and [4] and the slot-burner examples presented below. The other option available to a user of FDS is LES. In an LES computation the grid is selected to resolve the dominant eddy structures in the flow, which for a fire are driven by the plume, and all smaller eddy structures are handled by a subgrid scale (SGS) model. Multiple SGS models exist, and FDS uses a fairly simple model: the Smagorisnky SGS model. In the Smagorisnky model, the effect of SGS turbulence is accounted for by computing an effective local viscosity. This effective viscosity is given by 2   mLES = r(Cs ∆ )2  2(def u ) ⋅ (def u ) − (∇ ⋅ u )2  .   3

(4)

The parameter ∆ is the mean length of the local grid cell, def is the symmetric gradient operator, and the parameter Cs is an empirical constant. FDS uses a constant Cs. Experimentally Cs will

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vary depending upon where in the flow one is evaluating the turbulent viscosity. However, for the large-scale fires typically modelled with FDS the assumption of a constant Cs has yielded good results in the validation studies [37] that have been performed to date. One question that often arises is how small or large can the grid, and hence ∆, be and still have an accurate solution. For large-scale fire simulations it is recommended [1] to resolve the characteristic size of the fire, D*, with as many cells as possible, 2

 5 Q D* =   .  ra Ta cp g 

(5)

This, however, is a rule of thumb and is not a substitute for a grid study to verify a converged solution.

3 Solution speed and stability CFD fire models, while capable of producing highly resolved output and supporting realistic renderings of mass and energy flows, are also very slow when compared to other methods of computation. This speed can often eliminate the use of CFD as a tool for a particular commercial project or greatly limit the range of scenarios to be examined. Therefore, assumptions and approximations that can reduce the computational time are in order. In many CFD codes, the computational time is driven by the pressure solver. For example, in FDS, the perturbation pressure is determined by solving a Poisson equation with an involved right-hand side. Since FDS uses a two stage Runge−Kutta method for each time step update, the Poisson solver is called twice at each time step. The pressure field is tightly coupled with the velocity field. The influence of the pressure field is felt in the form of internal waves and acoustic waves. Internal wave speeds are of the same order as typical non-premixed fire-related flow speeds. These internal waves can be associated with the pressure potential that is driving the lowMach number flow. However, the speed of sound of about 350 m/s is substantially greater than typical compartment fire flows. In spite of this velocity dominance, for low-Mach number flows the speed of sound is related to acoustic waves which have little impact on the flow and the surroundings (unlike shock waves which would dominate the flow). In other words, for the compartment fire flow regime, the acoustic waves are essentially naturally decoupled from the ensuing velocity field. On the other hand, FDS’s explicit time-stepping implies that, because of the Courant−Friedrichs−Levy (CFL) condition, the time step is inversely proportional to the fastest speed. FDS addresses this dilemma by first using a subset of the Navier−Stokes equations which eliminate acoustic waves. The expandable gas equations [1, 6] can formally be derived using a Mach number expansion. Equations (1)−(3) are non-dimensionalized according to the parameters outlined in [7]. The dependent variables are expanded in powers of Mach number, M [7, 8]. By making the appropriate substitutions and taking the limit M→0, it is established that the lowest order terms for the pressure are only functions of time while the next order term is a function of x and t. Hence the following decomposition of the pressure results, p( x, t ) = P0 (t ) + p ( x, t ). (6) Note that p˜ ( x ,t) is often decomposed into –gzêz + pˆ ( x ,t) in order to facilitate outdoor simulations where the background weather dominates the flow. Equation (6) is responsible for filtering out sound waves in the resulting system of equations. It is also a generalization of the commonly

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411

used gauge pressure decomposition. In fact, for open domains, P0(t) is constant and the standard gauge pressure decomposition is obtained. Rehm and Baum [6] worked with inviscid equations and used different non-dimensionalization and expansion parameters. Reference [6] contains a dispersion equation obtained from the expandable equations’ limit. It is shown to be consistent with the dispersion relation for a fluid with internal waves but no acoustic waves. Hence the two manifestations of the pressure field were not only decoupled but the undesirable acoustic waves were eliminated. The low Mach number family of equations have also been investigated for mathematical well-posedness and short-time existence of solutions [9−11]. It has already been mentioned that eliminating acoustic waves increases the time steps allowed by the CFL condition. In order to get an idea of just how pervasive acoustic waves are, consider the example of a fire within a single compartment with a slightly open window and an initially closed door. The act of simply opening the door sends out acoustic waves. The fire crackle is also a source of constant sound waves. Typically, fire investigators are not interested in tracking the dynamics and fronts of these waves. The disadvantage to the expandable gas equations is that strong deflagrations, detonations, and high speed jets cannot be modelled. The rule of thumb for applicability [1] is to stay within the incompressible regime for which M < 0.3 [12]. FDS’s procedure for updating the pressure is closer to that used in projection methods rather than that used in SIMPLE algorithms. The resulting Poisson equation does not have constant coefficients so an approximation is made that allows an efficient, fast Fourier transform (FFT)-based solver to be used. The choice of Poisson solver has had a large impact on the design of FDS. It has resulted in an orthogonal coordinate system that can only have two axes with variable gridding. In order to accommodate mixed BCs, the pressure solver is implemented with deferred corrections based on the previous time step’s velocity field. When more flexibility was required in the types of domains that could be modelled, the desire to maintain the same Poisson solver resulted in multiblocked domains and further corrections in order to match the pressures across subdomain interfaces.

4 Accounting for energy Modelling a fire entails the incorporation of various energy sources and sinks and heat transfer to and from the gas phase of the computational domain. The energy sources are the fire itself as well as other potential sources that may be present in the simulation such as appliances, radiant/ convective heat sources (stoves, radiators). Energy sinks within fire modelling can be evaporative cooling (sprinklers, water mist) and heat transfer to surfaces. Energy is transferred from the gas phase by convection to surfaces, by advection out of the domain, and by radiant transfer. 4.1 Combustion modelling As mentioned in Section 1, combustion, even for the simplest fuels, will involve a multitude of chemical reactions and intermediate species and free radicals. For example, hydrogen combustion in air (a nitrogen−hydrogen−oxygen mixture) has 82 reactions [13]. While there are times when one desires to model this level of complexity, for a fire simulation, where the detailed chemistry is not critical, a much lower level of detail is advised. Within FDS, two combustion models are used: a single-step Arrhenius model and a single-parameter mixture fraction model. In the Arrhenius reaction, separate species are defined for fuel, nitrogen, oxygen, and the desired major and minor combustion products. Note that each species will require its own conservation of mass equation. The equations to model the combustion are the chemical equation

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Transport Phenomena in Fires

for the combustion of the fuel and a reaction rate equation. The equations are shown below where F is fuel, O is oxidizer, P are products, a and b are stoichiometric coefficients, C is concentration, B and E are Arrehnius coefficients, R is the ideal gas constant, and T is the temperature. Since only one chemical equation is used, the model is referred to as a single-step model. aF + bO → P,

(7) E

− dCF = − BCFa COb e RT . dt

(8)

The mixture fraction model in FDS is a single-step model which assumes that the chemistry is infinitely fast and that it always occurs regardless of the local temperature. This phenomenon is succinctly summarized in the phrase ‘mixed is burnt’. That is fuel and oxidizer cannot coexist. Reformulating the single-step reaction allows for either excess fuel or excess oxidizer, B  A   A B   AF + BO → min  ,  P + max  0, A − a F + max  0, B − b O.  a b   b  a 

(9)

Since the chemistry is infinitely fast, only the right-hand side can appear in the computational domain rather than the original unreacted mixture of fuel and oxidizer. In the original unreacted mixture, if one considers any given volume of gas for which the temperature is known, then knowing the quantity of fuel determines the quantity of oxidizer and vice versa (i.e. any mass in a location that was not originally fuel must have been originally oxidizer). Therefore, these two assumptions lead to the conclusion that all possible combinations of products can be determined by a single parameter: the original fuel or oxidizer mass fraction. Thus, by defining a term called the mixture fraction, Z, that represents the amount of mass in location that was originally fuel, a series of state relations (such as the one given in Fig. 1 for hydrogen and air) can be generated. 1.0 0.9

Mass Fraction (kg/kg)

0.8 0.7 0.6 N2 O2 H2 H2O

0.5 0.4 0.3 0.2 0.1 0.0 0.0

0.1 Zf

0.2

0.3

0.4 0.5 0.6 Mixture Fraction (Z)

0.7

0.8

0.9

1.0

Figure 1: State relationships for hydrogen−air combustion. Zf denotes the location of the flame surface.

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This concept works surprising well for well-ventilated flames as can be seen in Fig. 2, which shows state relationships for methane plotted along with measured data from a Wolfard−Parker slot burner [14−16]. It also works fairly well for data collected during compartment fires [17] (Fig. 3). However, in the compartment fire case, the infinitely fast, mixed is burnt assumption cannot capture the full variability of CO and CO2 in underventilated cases. In the mixture fraction approach, combustion occurs wherever fuel and oxygen meet. Since the two cannot coexist, the only location at which this occurs is the stoichiometric surface where 0.25 CH4 O2 N2 H2O CO2

Mass Fraction (kg/kg)

0.20

0.15

0.10

0.05

0.00 0.00

0.02

0.04

0.06

0.08 0.10 0.12 Mixture Fraction (Z)

0.14

0.16

0.18

0.20

Figure 2: Methane−air state relationships and slot burner data. 0.30 C3H8 O2 N2 CO2 H2O CO

Mass Fraction (kg/kg)

0.25

0.20

0.15

0.10

0.05

0.00 0.00

0.05

0.10

0.15 0.20 Mixture Fraction (Z)

0.25

Figure 3: Propane−air state relationships and compartment fire data [17].

0.30

414

Transport Phenomena in Fires

the post-reaction mass fractions of both fuel and oxygen are zero. The mixture fraction value at this surface is Zf, as was shown in Fig. 1. At this surface oxygen will exist on the lean side and fuel will exist on the fuel rich side. The HRR is then a function of how quickly the two can meet. The transport of fuel and oxygen is shown below in the transport equations for mixture fraction and oxygen. DZ = ∇ ⋅ r D∇Z , Dt

(10)

DO2 = ∇ ⋅ r D∇YO + m O′′′ . 2 2 Dt

(11)

r r

By applying the chain rule to the oxygen derivative and summing the equations, a single equation is obtained that related the change in oxygen to the mixture fraction. This equation is shown below. − m O′′′2 = ∇ ⋅ r D

dYO2 dZ

∇Z −

dYO2 dZ

∇ ⋅ r D ∇Z .

(12)

This particular formulation, while analytically correct, is not well suited for a numerical solution. However, consider that the entire computational domain can be divided into two domains, a fuel

(

)

dYO2

rich domain where there is no oxygen ___ dZ = 0 and a fuel lean domain. The divergence theorem can be applied to eqn (12) to convert it from a volume representation to a surface representation. This results in: m O′′ = − 2

dYO2 dZ

r D ∇Z ⋅ n Z = Z .

(13)

f

In the above equation, the mass loss rate of oxygen is determined by the rate at which mixture fraction diffuses across the stoichiometric surface. This is equivalent to saying that the oxygen is consumed as fast as it can enter the flame. Oxygen consumption is then related to the HRR by the heat of combustion per unit mass of oxygen. The above equation is advantageous numerically as it only has a single first-order, space derivative. Furthermore, this equation only needs to be evaluated at grid cell faces that separate a fuel rich cell from a fuel lean cell (indicating the flame sheet cuts through one of the two cells). This approach has some limitations to it, though. Two limitations are numerical. A third limitation is physical. The HRR is a function of the gradient. Therefore, if the gradient is poorly resolved, then the local heat release will also be poorly resolved. Although conservation of fuel mass guarantees the is grid dependent. In other words, ∫ q ′′′ d V can be right global HRR, the local value of q′′′ expected to be accurate even though q ′′′( x, t ) is a sensitive variable. This fact can result in two outcomes. Overly large gradients, as might occur near a burner that is spanned by a small number of grid cells, result in local HRRs that exceed those seen in real fires. Excessive local heat release leads to locally elevated temperature levels which subsequently impact both the buoyancy and the radiation source terms. On coarse grids, the mixture fraction is smeared out by excess artifi . A related limitation is that cial diffusion, leading to lower flame heights and increased local q′′′ a coarse grid will result in the flame sheet occupying too small of a volume. This will result in lower flame heights which will again lead to buoyancy and radiation source term errors. The flame height error can be corrected by selecting a different surface to integrate over. Since dYO mixture fraction is a conserved quantity, any value of Z can be selected, provided that ___ dZ is 2

The Implementation and Application of a Fire CFD Model

415

appropriately adjusted, and the same heat release will result, provided of course that the new surface lies within the computational domain. Excessive local HRRs can also be accounted for. There is an upper bound to the amount of energy per unit area of a diffusion flame. This upper bound is in part a function of the type of fuel being burned. FDS assumes, with the option for user modification, that combustible materials typical to the built environment are to be used. The assumption is then made that Heskestad’s flame height correlation [18] is an adequate characterization of the minimum volume occupied by fire. If it is further assumed that the shape of the fire is a cone then the surface area of the fire can be derived from Heskestad’s correlation. The end result of combining the cone area with the fire size is a HRR per unit area. By default FDS will use both of the aforementioned corrections. Using the grid size, the heat release per unit area is converted to a volumetric HRR during the initialization routines. The Z value of the flame sheet is updated at each time step based on the resolution of the fire size (i.e. as the fire grows larger, the effective grid resolution increases). The volumetric heat release in each grid cell is computed using the effective Z value. Heat release in any grid cell exceeding the volumetric limit is clipped with a running tally made of the clipped heat releases. After all cells are computed, the total clipped heat release is then added to those cells with a non-zero HRR proportional to the heat release of that cell. 4.2 Heat transfer Heat transfer within a fire model is an important phenomenon that can have a profound impact on all the solution variables. Surface heating by convective or radiative heating can result in the ignition of other materials. Conduction into an object can result in ignition on another face of the object or it can result in a failure of the object. The removal of heat via water evaporation can act to mitigate the effects of/suppress a fire. 4.2.1 Convection heat transfer Convection heat transfer to a surface is determined by how fast energy can diffuse from the free stream of a flow field, through the boundary layer, and into the surface (see Fig. 4). In a DNS computation, where the boundary layer is resolved, the heat transfer is determined by applying the conduction equation to the temperature gradient normal to the wall: q ′′ = (kgas ∇Tgas ⋅ n ) |wall .

(14)

In an LES computation, the boundary layer is not resolved. Therefore, the temperature gradient at the wall does not reflect the actual heat transfer that is occurring. A correlation must be used, therefore, to compute the heat transfer. Correlations for convective heat transfer are typically divided into two categories: forced convection and free convection. While fire driven flows result from free convection, sizeable velocities can be achieved and a forced convection correlation may be more appropriate. To account for this, FDS computes the heat transfer assuming both forced and free convection and then picks the larger of the two. The correlations for the heat transfer coefficient [19] are given below with the free convection on the left: 1 4 1  k 3 5 h = max C | ∆T | , 0.0037 Re Pr 3  . L  

(15)

416

Transport Phenomena in Fires

TW

TB

→

v

Figure 4: Turbulent convective heat transfer to a surface is characterized by a parallel average velocity component that increases with distance away from the wall. In this example, the temperature profile decreases as the perpendicular coordinate moves away from the wall. The opposite trend is possible, the occurrence being problem specific.

Since the Reynolds number is proportional to the characteristic length, L, the forced convection is only weakly dependant on length. To simplify the computation, L in FDS is assumed to be 1 m. 4.2.2 Conduction heat transfer The general equation for time-dependent heat transfer in a solid material is given by: ∇ ⋅ k ∇T + q ′′′ =

1 ∂T . r c ∂t

(16)

Here, r is the wall density, T is the wall temperature, c is the wall specific heat, k is the wall ther represents sources terms that can be obtained from phenomena such as mal conductivity, and q′′′ pyrolysis. For many of the materials common to building construction, either k, the conductivity, tends to be fairly small (wood, masonry) or the thickness of the material is very thin in comparison to its exposed surface area (sheet metal). Thus, in general, the primary concern is heat transfer into a material as opposed to across a material’s face. The above equation can therefore be simplified to transfer across the dimension normal to the surface: ∂ ∂T 1 ∂T k + q ′′′ = . ∂x ∂x r c ∂t

(17)

The above equation is used by FDS to model conduction heat transfer. The general approach used is to discretize the surface into a number of nodes and solve the resulting set of linear equations

The Implementation and Application of a Fire CFD Model

417

using a Crank−Nicolson scheme in time. To reduce computational expense, if the material thickness and anticipated heat flux are small enough, then one can consider the material to be isothermal in space. This is the lumped mass or thermally thin approximation. This approximation is appropriate to use when the Biot number, Bi =

h∆x , k

(18)

is less than 0.1. For a fire simulation, the heat transfer coefficient, h, represents the effective coefficient resulting from the maximum combined radiative and convective heat transfer that might occur to a surface. For example, if a 1500 K gas temperature and 300 K surface temperature are assumed, an 2 ∆x effective h of 250 W/(m2·K) is computed. The ratio __ k < 0.0004 (m ·K)/W can be used to determine if a material is thermally thin or thermally thick. Hence a 1 cm slab of steel, for which k = 50 W/(m·K), is thermally thin but a 0.1 mm sheet of hardwood, with k = 0.1 W/(m·K), is thermally thick. Solution of the heat conduction equation for a surface requires specification of the BCs on each side of the surface. These are given by the net convective and radiative fluxes to the surface as discussed in Sections 4.2.1 and 4.2.3. To keep the solution of the equation stable, especially for thin insulating materials, it is desirable that the heat fluxes used as BCs at the beginning of a time step still be valid at the end of a time step after the surface temperature has changed. The convective heat flux is driven by the temperature difference between the surface and the gas adjacent to it. It is therefore linear with surface temperature as is the conduction equation. It is trivial to incorporate this into the formulation of the BC. The radiative flux, however, depends on the difference of the fourth power of the gas and surface temperatures. The net radiative flux is given by the following equation: qr,net ′′ = (1 − sw )qr,in ′′ − swTw4 .

(19)

The fourth power dependence can be removed by a Taylor series expansion Tw over a time step. So that Tw can be given as: (Twn + 1 )4 = 4(Twn )3 (Twn + 1 − Twn ) ,

(20)

where n + 1 and n are time steps. 4.2.3 Radiation heat transfer In Chapter 7 it was established that the radiation transport equation (RTE) is s ( x , l) s ⋅ ∇I l ( x , s ) = − [ k( x , l) + ss ( x , l)]I l ( x , s ) + B( x, l) + s Φ(s , s ′ )I l ( x, s ′ )dΩ . 4 π 4∫π

(21)

Here s is the direction vector for the radiation pencil, Ω is the solid angle, Il is the intensity in units of power per wavelength per steradian, k is the mass absorption coefficient, ss is the scattering coefficient, B is the spontaneous emission term, and Φ is the phase function. This equation states that the local gradient of radiant intensity for a specific wavelength in a specific direction (the left-hand side) is the sum of the loss by absorption and scatter, the gain by local emission, and the gain by in-scatter from other directions. A number of approaches exist to solve this equation.

418

Transport Phenomena in Fires

One approach is to randomly pick at each time step a subset of surface and volume cells, compute a number of rays of emission in random directions, and follow the ray until it impacts a surface or leaves the domain. If enough cells are selected and rays generated, summing the results for each ray will result in a reasonable approximation of the solution. This method is referred to as a Monte-Carlo method. If enough random samples are made, it will yield very accurate results including properly capturing the phenomenon of shadowing. For a steady-state computation or computing surface-to-surface view factors, Monte-Carlo methods are computationally affordable. For a complex, multidimensional, time-dependent simulation with a volumetrically distributed radiative source (fire, surfaces, and hot gas layer), Monte-Carlo methods become computationally expensive. A second approach is to utilize, Legendre polynomials to simplify the angular dependence. The simplest such use is the P1 approximation, which uses the first two Legendre polynomials, P0(x) = 1 and P1(x) = x. Applying the P1 approximation to the angular dependence of the RTE, converts eqn (21) to a diffusion equation where radiation is diffused throughout the computational domain. This formulation can be advantageous over the Monte-Carlo method as one simultaneously solves for the entire domain. Therefore, the computational expense of solving the diffusion equation can be less than the expense of the Monte-Carlo method, when the radiant source geometry is complex (e.g. would require a large number of rays to resolve). The main disadvantage of this method is that it diffuses radiation; therefore, shadowing is not handled well as radiation streaming past an obstacle will diffuse around the backside of it. A third approach which attempts to capture the ray-like behaviour of radiation using a computationally affordable algorithm is the finite volume method (FVM). First, the RTE is simplified by assuming a non-scattering gas and by limiting the wavelengths to a small number of bands: s ⋅ ∇I n ( x, s ) = kn ( x )[ I b, n ( x ) − I n ( x, s )], n = 1, … , N , (22) where n indicates a band and Ib is the source term given by: sT 4 I b,n ( x ) = Fn π

(23)

where Fn is the fraction of the source term emitted into band n. These simplifications have eliminated the scattering integral and have also limited the wavelength spectrum from a large number of discrete wavelengths to a small number of bands. However, there is still the direction vector which spans the 4π spherical angle. If the direction vector is grouped into a finite number of control angles (like wavelengths were grouped into bands), dΩl, and the resulting equation is integrated over each grid cell, then the following is obtained: (24) ∫ ∫ s ⋅ ∇I n ( x, s ) = ∫ ∫ kn ( x )[ I b,n ( x ) − I n ( x, s )], n = 1, …, N . Ωl

Ωl

Vijk

Vijk

Using the divergence theorem, the volume integral on the left-hand side can be replaced by a surface integral over each of the six faces of the grid cell. This resulting equation is: 6

∑ Am I l ∫ (s ⋅ nˆ m ) = kb,ijk [ I b,ijk − Iijkl ] Vijk dΩl ,

m =1

m,n

n = 1, … , N .

(25)

Ωl

This is the equation implemented in FDS. A number, l, of angles are selected which is typically about 100. To update the radiative intensity, the radiative solution for each angular direction for each band is solved and the results summed over each band to obtain the net radiative intensity.

The Implementation and Application of a Fire CFD Model

419

Since scattering has been ignored, all that matters along any given direction is the net radiation that is transmitted through a grid cell. The solution method, therefore, is as follows. For each angle the upwind corner of the computational domain is determined. The grid is then swept towards the downwind direction. For each grid cell the radiant intensity in that cell is the sum of the intensities from upwind cells, plus any radiative source term in that cell, minus any loss due to absorption. This process is repeated for all angles and all bands. It is clear that this process involves a computation that loops over the entire domain many times. This can become computationally expensive. Two things can be done to reduce the computational expense of this method. The first is to reduce the number of bands. This can typically be done without a significant impact on the radiation solution. For combustion products the species of interest are CO, CO2, H2O, fuel, and soot. The frequency dependence for these species can be collapsed to 6 or 10 bands depending on the importance of radiant absorption by the fuel. For many common combustible materials, soot dominates the radiative emission and absorption which allows a single band to be used. The second is to reduce the number of angles. However, since radiation is only transmitted along the angles, too few angles and the solution will show significant ‘hot’ spots along the angles and significant ‘cold’ spots between the angles. There is, however, another method to effectively reduce the number of angles, and that is to only update a subset of the angles at each time step. Consider a typical CFD simulation of a compartment fire. In general, the geometric distribution of temperature and heat release changes slowly; at least in comparison to the time step size. Under these conditions the radiation source term does not vary greatly and the solution to the RTE also does not vary greatly in time. Thus, only updating a portion of the angles in each time step taking a small number of time steps to fully update the RTE (in effect reducing the number of angles) will not have a significant impact on the solution accuracy. For example, for dt ≈ 0.01 s, updating the radiation calculation every ~0.1 s is adequate. Solution of the RTE requires determining the absorption coefficient, k, and the radiative source term, Ib. For a grey gas the source term is typically given as ksT 4. When a coarse grid is used, one that does not resolve the flame, the gas temperature inside of a grid cell with combustion is not likely to be representative of a flame temperature. With the fourth power dependence, using a lower temperature can result in greatly underpredicting the radiative source in grid cells with combustion. To avoid this FDS uses ksT 4 in all grid cells without combustion. In grid cells with combustion it computes both ksT 4 and a radiative source term given by the local HRR multiplied by a user specified radiative fraction, and then FDS uses the larger of the two as the source term. k is function of the local species mass densities. Since species are uniquely determined by the mixture fraction, k is therefore a function of the local Z and T and the frequency band. k is precomputed using RADCAL [20] and stored in a table.

5 Liquid sprays FDS includes the ability to model the effect of water or fuel sprays. This is done using a Lagrangian superdrop model. A Lagrangian model follows the path of each droplet individually as opposed to an Eulerian model which would represent all drops using a scalar quantity much like gaseous species are handled. The superdrop indicates that each droplet being track represents a much larger number of droplets. Since a sprinkler nozzle may discharge tens of thousands of droplets per second, to track every droplet would be computationally prohibitive. A liquid spray of either fuel or water will impact each of the major CFD conservation equations. Evaporation of liquid from the drops and attenuation of radiant heat transfer by the spray

420

Transport Phenomena in Fires

will impact the energy equation. Lastly, the evaporated mass must be accounted for in the mass equation. 5.1 Drop size distribution Based on the findings of Chan [21], FDS uses a Rosin−Rammler/log-normal distribution to determine the size of the individual droplets coming out of a nozzle. The formula for this cumulative size distribution is 1   ln[ d / dm ]    1 + erf    if d ≤ dm (Log-normal),  2s   2  F (d ) =  g  d  – ln 2  d   m 1 – e if d > dm (Rosin-Rammler). 

(26)

Note that the Rosin−Rammler term is identical to FRR (d ) = 1 − 2

 d −   dm 

g

(27)

.

The operator erf() designates the error function [22]. Figure 5 shows experimental data for F(d). Whenever data like this is available, a non-linear least squares solver can be used to determine g, s, and, if need be, dm. Profiles for F(d), however, are rarely provided by the manufacturers.

100

200

300

400

500

600

700

800

900 1.1

1.0

1.0

0.9

0.9

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4 F(d) RR/LN Distribution F(d) Weighted Data U(d) from RR/LN Distribution

0.3 0.2

0.3

Cumulative number fraction U(d)

Cumulative volume fraction F(d)

0 1.1

0.2

0.1

0.1

0.0

0.0 0

100

200

300

400 500 d (µm)

600

700

800

900

Figure 5: Cumulative volume and number fractions for the Securiplex Velomist water mist nozzle derived from the Rosin−Rammler/log-normal distribution.

The Implementation and Application of a Fire CFD Model

421

Increasingly, though, the drop diameter when F = 0.1, designated Dv[10], the drop diameter when F = 0.5, (Dv[50] or dm), and the drop diameter when F = 0.9, (Dv[90]) are being published. These values, along with eqn (26), suffice to determine g, s, and, dm. For the Securiplex Velomist water mist nozzle shown in Fig. 5, the mean droplet diameter, dm, was 175 µm, d = 50 µm when F = 0.1, and d = 400 µm. From these data, along with inverses of eqn (26), it was determined that g = 1.45 and s = 0.978. The results are plotted in Fig. 5. The theoretical Rosin−Rammler/log-normal distribution closely resembles the experimental one that accompanies it. FDS actually determines the diameter of each introduced droplet from the cumulative number fraction, U(d). It is defined as d

U (d ) = ∫ f ( d )dd ,

(28)

0

where f(d) is the probability density function, F ′( d )

f (d ) =

∞

d

3

∫ 0

F ′( d ) dd d3

. (29)

The expression for U(d) can now derived. The derivative of the cumulative size distributions is  ln[ d / dm ]  −  1 e  2s   2 πs d F ′( d ) =  g  d g −1 − ln 2    d  dm  g ln 2 e g  dm 2

if d ≤ dm

(Log-normal), (30)

if d > dm

(Rosin–Rammler).

The integral of this function that forms the basis for U(d) is  9s 2 / 2  3s 2 + ln[ d / dm ]  erf  e  2s     3 2 dm F ′( d )  g = d d  ∫ d3  d  g 3  − ln 2        d 3 d  dm  3/g  3[ln 2]   Γ  − , ln 2    − g e   g d d     m m     3 gd 

if d ≤ dm ,

(31)

if d > dm .

The operator Γ(,) denotes the incomplete gamma function [22]. Evaluated over the whole range of possibilities, the result is ∞

∫ 0

F ′( d ) d3

dd =

e9s

2

/2

2dm3

  3s    erf   + 1 +   2 

 3  g e − ln 2 − 3[ln 2]3 / g Γ  − , ln 2  g  g dm3

.

(32)

422

Transport Phenomena in Fires

Making the substitutions, it is evident that the cumulative number fraction is,   3s 2 + ln[ d / dm ]   erf   + 1 2s     for d ≤ dm ,  3s   2  − ln 2 3/g  3 − 3[ln 2] Γ  − , ln 2   + 1 + g e  g  2  g  e9s

U (d ) = e

9s2 / 2

   erf  

2

/2

(33)

and, 



d g − ln 2    d   3  d    dm  3[ln 2]3 / g   Γ  − , ln 2    − g e  dm   g  dm   3

g

g d3

U (d ) = e9 s / 2 2 dm3 2

 3  g e − ln 2 − 3[ln 2]3 / g Γ  − , ln 2   3s    g  erf   + 1 + 3 g dm 2  

+1

for d > dm .

(34)

Equations (26), (33), and (34) are plotted, along with the experimental data for F(d), in Fig. 5. Clearly the Rosin-Rammler/log-normal distribution provides a good fit to the experimental data. Note how quickly U(d) levels off. In a large enough sample, FDS would provide the correct mix of large and small droplets. The large droplets tend to dominate the trajectories while the smaller droplets have more rapid evaporation. Choosing droplets solely based on cumulative number fraction is insufficient for water mist in a numerical method that transports far fewer droplets than is actually the case. In reaction to the findings of Hunt et al. [23], NIST implemented a bin selection algorithm in order to introduce more large droplets at each injection cycle. 5.2 Spray pattern creation The spray pattern from a nozzle is fully defined by a spherical drop size distribution, a spherical mass flux distribution, and a spherical velocity distribution. That is if one envisions a sphere drawing around the nozzle at some distance from its orifice (to allow the liquid stream to break into drops), at any given point on that sphere there will be a mass flux, a drop size distribution, and a velocity vector. At the dark patches in Fig. 6, droplets are injected at the flow rate fraction of the user-specified frequency. The user specified radius for the sphere is employed to convert the flux specified at a face to the corresponding flow rate through that face. This approach allows data obtained from the detailed diagnostic methods of Putorti et al. [24] to be directly translated into a numerical characterization of the spray. It has also been effective in recreating volumetric flux maps obtained in experiments where the water streaming from the nozzle was collected in a uniform array below the nozzle [25]. This flexibility is most welcome when spray fires are to be modelled, as some models have non-circular orifices. 5.3 Spray momentum The equation of motion for a drop is obtained by applying the conservation of momentum to the drop. That is the change in velocity for a drop is given by applying the force of gravity and drag

The Implementation and Application of a Fire CFD Model

423

Figure 6: A nozzle or sprinkler is characterized by the faces of the user-specified number of solid angles. In this example, the darkened solid angles were proposed to numerically characterize a high pressure multiport water mist nozzle.

to the drop. Since the gas around the drop may also be in motion, this must also be accounted for. This results in the following equation where CD is the drag coefficient: d 1 (md ud ) = md g − rCD πrd2 (ud − u ) ud − u . dt 2

(35)

Evaluating CD is key to properly accounting for changes in a drop’s velocity. This is not necessarily a simple task. For a single hard ball sphere falling through the air, well accepted correlations for CD exist. A water spray, does not, however consist of isolated single hard ball spheres. Water drops can have shapes ranging from near spherical to pancake like shapes depending on the drops size and velocity, that is whether or not surface tension pulling the drop into a sphere is stronger than the drag force attempting to pull the drop apart. Furthermore, a droplet from a spray nozzle is surrounded by many other droplets. Since, the flow field resulting from flow over a single sphere is different than the flow field that results from flow over a group of spheres of varying size, the drag forces will be different as well. FDS uses a hard ball drag force correlation which does not presently account for either multiple drop effects or drop shape effects.

424

Transport Phenomena in Fires

5.4 Droplet heat transfer and evaporation A liquid drop will exchange mass and energy with its surroundings. It can absorb and emit radiant energy, it can convect energy to or from the gas around it, and it a can add or remove energy through mass exchanged by evaporation or condensation. Radiative absorption by the droplets is determined in the FVM radiation solver. During initialization, the MieV code is used to compute absorption and scattering cross sections as a function of droplet radius and wavelength. In the radiation solver the droplet absorptivity is determined by doing a table lookup using the average drop radius and average drop density for each grid cell. The resulting total absorptivity is added to the gas absorptivity used in the FVM solver. After solving for the new radiative intensity, the droplet absorptivity is then used to determine the amount of radiant energy absorbed by droplets in each grid cell. This quantity is then transferred to the individual drops in a gas cell weighted by surface area. Re-radiation from the drops is not accounted for. Convective heat transfer to the drops is computed within the droplet update routine. A heat transfer correlation, which is a function of the Nusselt number defined on the droplet radius, is applied. The actual heat transfer to the drop is then the most limiting of: the heat transfer required to completely evaporate the drop, the heat transfer required to have the drop and the gas in its grid cell be at thermal equilibrium, or the heat transfer given by the heat transfer coefficient and time step size. The droplet evaporation model used in FDS is a quasi-equilibrium model. Droplets will evaporate in an effort to reach vapour equilibrium based on the temperature of the gas cell the droplet is located in. Since evaporation will remove energy from the drop that may not be replaced by heat transfer from the gas, in any given time step a droplet may not reach equilibrium with its gas cell. The rate of evaporation is determined using a Sherwood number correlation that accounts for the difference between the droplet equilibrium vapour concentration and the current vapour concentration of the gas cell. Condensation is not accounted for. Equilibrium vapour concentration is given by the Clausius−Clapeyron relationship: peq = p0 e

−

hv M  1 1 − R  Td Tb 

,

(36)

where the peq is the equilibrium vapour pressure, hv is the heat of vaporization, M is the molar mass, Td is the droplet temperature and Tb is the boiling temperature. The temperature change of a droplet as it evaporates is a function of the energy required to evaporate the drop, the heat being transferred to the drop, and the mass of the drop remaining. If insufficient heat is available to maintain the drops temperature, it will cool as it evaporates. This means that the end of time step drop temperature will be lower and it is possible that the evaporation rate given by the Sherwood correlation will result in evaporating too much mass (the actual end of time step vapour pressure being higher than the end of time step equilibrium pressure). Since the equilibrium pressure involves the exponential of the inverse of the drop temperature, a simple analytic solution does not exist. To avoid the potential need to iterate for each droplet in the simulation, a linearized solution to the energy balance is used to limit the droplet evaporation. Following the computation of heat and mass transfer for a droplet, the gas cell temperature, density, and vapour mass fraction are updated. If the droplet is a fuel droplet, the mass added becomes available for combustion in the combustion routine. Note however, that if the mixture fraction combustion model is being used, and the amount of vapour added does not raise the gas concentration above Zf , then no additional combustion will occur. It is also noted that within the droplet routine, the heat transfer, position update, and evaporation computation are performed sequentially for each droplet. Therefore, in the event that multiple drops

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exist in a grid cell, the convective heat transfer and evaporation of the droplets will not be correctly computed. The first drop will see the initial gas temperature and vapour concentration and evaporate accordingly, the second drop will see the updated temperature and concentrations, and so on. However, this phenomenon typically occurs in cells near the spray head, where most of the other assumptions break down as well. 5.5 Evaporation impact on divergence As a drop evaporates it has combined set of effects on the gas surrounding it. The addition of mass to the gas phase is an expansive effect, e.g. it acts to raise the local pressure. Depending upon the drop temperature, local vapour concentration, and local gas temperature, the mass added by the evaporation may be at either a higher or lower temperature which will cause respectively an expansive or contractive effect. This combined effect is added to the divergence equation in FDS as follows:   Yi  dT R  T ∇⋅u = + + m H′′′2 O  ,  r∑   g p0  i  Mi  dt MH2 O 

(37)

where the first term in the parenthesis accounts for the net energy exchange between the drop and the gas, and the second term accounts for the mass exchange.

6 Boundary and initial conditions One of the limits of a CFD simulation is that its extent is bounded in physical space. Even a laboratory experiment inside a building does not have this stringent a limitation. For example, smoke that escapes the laboratory through a window encounters outdoor flows which could lead to the recirculation of smoke by the same window. Smoke that leaves the computational domain cannot return unless the developer has added a method by which to accomplish this. The flow BC in FDS accommodates flows across the outer boundary with either a specified flow rate or an open BC based on the pressure difference. Specified flow BCs are effective means for modelling exhaust points for forced smoke control systems. Keep in mind that these specified flow rate BCs, unlike fans, are insensitive to surrounding flow field. The velocity will accelerate or slow down to match the BC. Each open boundary has a user specified set of conditions for temperature, species concentration, etc. The gas coming in would have these properties. For radiation, the open boundaries are treated as black walls, where the incoming intensity is the black body intensity of the ambient temperature. Open BCs can be used for heat and smoke vents. They do not account for phenomena such as vena contracta. In order to capture these effects, the computational domain must be extended beyond the compartment limits, the open BC vent replaced by a wall with an opening, and the now extended boundary will all be open. FDS has mirror BCs that reflect the solution. For radiation, the intensities leaving the wall are calculated from the incoming intensities using a predefined connection matrix. The modeller needs to use mirror BCs with care since the reflection process constrains the turbulence of a fire up against the mirrored BC to a symmetry that it would otherwise not have. The user should also be aware of the implication of a domain with symmetry boundaries. For example, a domain with a fire in the centre and two orthogonal mirror boundaries actually implies four fires and four times the calculated HRR. Using a reflected BC for the ceiling or for the floor can be strange indeed. Mirror BCs on opposing walls are to be avoided.

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Thermal BCs were covered in Section 4.2. To give the user an idea of the impact of thermal BCs, consider once again the example of a single fire in a suitably ventilated room. Adiabatic BCs would cause the heat within the compartment to rise too quickly, potentially leading to problems with the radiation transport calculation. Constant temperature BCs (the FDS default) can lead to low upper layer temperatures. Specified material BCs work best. The difference in the temperature field obtained by changing from gypsum to wood wall properties is much less dramatic than would be the case if the change were to adiabatic or isothermal BCs. For this reason, the current FDS algorithms for thermal BCs are being improved to provide even more realistic response. The need for dependable thermally decomposing BCs is equally great but, as of this writing, this goal is still in the realm of active research. When specifying a fire of known HRR emanating from a surface, make sure that the implied mass loss rate per unit area is consistent with published sources [26]. A large mass flux per area can have jet-like dynamics. It will also result in run-time performance degradation because of the CFL condition. An excessively small mass flux per unit area will lead to ‘flame’ dynamics reminiscent of weakly buoyant flows. FDS requires consistent initial conditions just as differential algebraic equation (DAE) solvers do [27]. The flow field and the pressure field are tightly coupled in FDS. Hence every flow boundary or internal condition must be consistent with the pressure at that surface. This requirement is also relevant to the initial conditions. Specifying an initial velocity field without a consistent initial pressure will lead to numerical difficulties. Often, the best initial state for a series of simulations is obtained by running the simulation for a while to establish prevailing flow patterns. Examples include (a) flow past a building with openings, (b) stack effects in both buildings and stairwells, (c) contributions from heating, ventilation, and air conditioning (HVAC) equipment, and (d) significant buoyant flows coming off devices such as hot plates and running machinery. The goal is to get beyond the initial transients so that the streamlines resulting from the various possible sources fill the space(s) in question with a reasonably steady flow. From that point on, the items that distinguish one run from another would be introduced into each member of the simulation suite.

7 The practice of modelling The development of a viable numerical simulation is a recognized difficult undertaking. Performing a successful regimen of simulations presents a host of challenges that are often not as well known. It is quite simple to make just one mistake in a simulation that will run for three weeks and obtain results that are useless. Any of the common errors listed in Table 1 can ruin an FDS simulation. One of the key steps in the realization of a successful modelling project is preparation. Good preparation entails review and research, categorization, focus, and elimination. At the end of this process a set of scenarios should be arrived at which would be the analogue of an experimental test series. The choices must be made in full knowledge of the limitations of the model. Assessment techniques are presented in Section 7.1. Once results are obtained, they should be carefully reviewed for correctness and applicability. 7.1 Preparation 7.1.1 Review and research Any modelling effort begins with the identification of the problem and an assessment that CFD is an appropriate form of engineering analysis. Typically the modeller will be skilled with one simulation.

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Table 1: Common mistakes in FDS modelling. Problem No floor Internal open vent Vent sign error Placed solid surface where vent should be Modelled outer walls as obstacles on outer domain boundary Obstacle (partially) covering the fire bed Placed sprinkler/detector within obstacle Used the default thermal BCs Used the default emissivity (e = 1) in enclosure with reflective walls Set smoke properties without considering that visibility ∝ ∆Hc/Ys Extremely coarse computational grid Fine resolution in light of the CFL condition Did not make the number of cells in any given coordinate a multiple of 2, 3, and 5 Did not define enough outputs

Manifestation Flow across floor Not recognized Smoke not exhausted Sealed room Numerous Crash or low HRR No participation in simulation Low temperature field Incorrect radiative transport Insufficient conservatism Many Slow run time Slow run time Wasted simulation

The assessment would specify if that model is appropriate for the problem at hand. The next step is the review process. Here, the relevant available information sources are concentrated and examined for pertinent inputs. Typical sources are engineering drawings, fire and police department reports, eye witness accounts, client requests, applicable government regulations and requirements, military specifications, and the various fire, mechanical, and municipal codes. These usually suffice to obtain an initial computational domain. Research is then appropriate in order to supply the remaining missing information and to familiarize the modeller with antecedents. For example, obtaining material properties or fuel combustion data frequently require research of some type. It is not unheard of that experiments were performed in order to obtain the necessary inputs for modelling. The modeller also needs to familiarize herself with the history of the problem. Searches through the technical literature should result in a set of papers where similar experiments and/or simulations were performed and where the relevant basic science is expounded. The rule of thumb is that you, the modeller, should know what to expect before you start the simulation process. 7.1.2 Categorize inputs The review process will result in a list of concerns. These need to be categorized in order of importance. The technique is analogous with the selection process of parameter based asymptotic expansions [28, 29]. In this approach, variables are expanded in terms of a parameter that is large or small in some limit. These expansions are introduced into the governing equations and the limit is taken at the required order. So the process typically begins (ignoring logarithmic terms) with zeroth order terms, then first order terms, etc. In a properly designed expansion, each successive order provides a smaller, corrective addition to the terms preceding it. The important point for modelling is that the variables where introduced together based on their order. So CFD inputs should also be introduced into the computational domain based on their order of impact. For example, flow rates from a ventilation system would be included because they are of the same order of magnitude as

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the characteristic velocity of the fire but the flow induced the computer fans would be neglected because they are of lower order. If, subsequently, it is determined that they are necessary, then all flows of this order should be introduced into the simulation. Just as asymptotic expansions have a limit beyond which adding extra terms no longer improves the accuracy, the level of input detail for a CFD model is limited by the resolution of the simulation. 7.1.3 Focus and eliminate CFD simulations can become burdened with unnecessary information. Subgrid minutiae are a common example. If subgrid information is important to the simulation, a way must be found to model their impact (see Section 3). This implies a development exercise if the model does not currently accommodate the effect. Frequently, though, a great deal of information will be added about the features of a commercial space that either overlap in a fixed grid code or lead to a large number of fine cells in a code with automatic mesh generation and unstructured grids. Preparation is the key to focusing the scope and determining what can be eliminated. A hazard assessment of the impact of wind on a burning building provides a good example. The choices for a simplified wind model include direction (three variables), magnitude, thermal stratification, momentum stratification, and wind turbulence parameters (at least four). Clearly a parameter study that varies one variable at a time is not feasible. A better approach would be to identify important zones that, given an unfavourable wind direction, would be affected by the fire. Then study the wind patterns for the vicinity based on the nearest available soundings. Next, consider the topography of the region. The outcome of this assessment would be a set of weather conditions of interest and another set of secondary importance. Focus also applies to the goal of the study, including its budget and deadline restrictions. The wind impact study provides another example. The building in question is a paper mill located on the banks of a river in a V-shaped valley that runs from west to east. For years, the mill was isolated, using the river as the chief conduit for raw material (logs) and a service road to transport the finished products out of the area. Now, as is the case in many areas, retreat homes are springing up in the valley and the plant’s insurers are requiring the hazard assessment. The consultant’s initial assessment found that westerly and easterly winds provided the greatest hazard. Their impact was quantified via modelling. It was also found that the wind perpendicular to the valley tended to trap the smoke in a recirculating pattern that mostly affected the plant. However, a pollution problem was also discovered when the plants’ emissions were coupled with the emissions from a sufficiently large valley population. The last discovery is outside the scope of the initial investigation. The client should be informed of the situation but the matter should not be pursued any further for the present study.

8 Assessing the model, assessing the results In many ways, a simulation can be like a black box, even to the developer. Several approaches have been developed to assess the limits and capabilities of the model. Verification, validation, and error estimation are the most frequently used techniques. Although verification and validation are synonymous as far as the standard thesaurus is concerned, they have taken on distinct meanings in the field of numerical modelling. Verification is the process by which it is established that the model is implemented correctly and that it is correctly solving the desired equations [30]. Consider the example of a set of quadratic equations to be solved. The goal of the verification exercise would be to determine problems such as the algorithm unintentionally accessing an address outside of the array bounds or ferreting out an improper implementation of the equations, e.g. using b3 under the radical instead of b2. CFD verification exercises are clearly more involved. The procedure

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encompasses source code review, collection of a series of know solutions, determination of a series of runs, and quantitative comparison with the exact solutions. Validation is the process by which it is shown that the simulation is correctly reproducing the desired physico-chemical phenomena [30]. The approach typically involves comparing simulation results with experimental data. Ideally, error estimation would produce a priori quantitative assessments of the model’s uncertainties. In practice, this is hard to achieve. What are typically done are sensitivity analyses. These involve the variation of a select group of simulation parameters in order to assess the impact on the final solution. The techniques selected so far are of local scope. In other words, the simulation can reproduce the results of an exact solution to a given tolerance, is shown to lie within the error bars of the experimental data, or shown to converge as the grid is refined. These outcomes instil a sense of confidence in the code but no more. As Roache [30] pointed out, these exercises confirm the appropriateness of the simulation for the problems that were addressed but they do not prove that the code has been globally verified and validated. Certification is a process by which a simulation is judged to be appropriate for a given set of problems. The synonyms ‘accreditation’ and ‘quality assurance’ have been used to describe the same process as well. The approaches leading to certification are the same as those introduced above. The certification process is intended to be comprehensive and can take a great deal of effort to achieve. The US Department of Defense provides comprehensive guidelines the verification, validation, and certification of computer software [31]. This guide provides an excellent introduction into all aspects of the practice. It does, however, cover areas that are not yet relevant to fire CFD models such as software for automatic control and interactive programs. ASTM E 1355 [32] provides specific guidelines for fire models. For modelling projects, once a set of results are obtained, they should be subjected to a comprehensive review process. For example, if the model was used in a flow regime where it was not accredited, a careful review of the results is warranted. This could go as far as implementing the techniques of verification and validation. For problems within the scope of the recognized capabilities of the model, checking that the solution is within bounds and that the initial modelling assumptions were accurate may suffice. 8.1 Verification Many journals these days require some verification and/or validation as a condition for publication. As was mentioned before, CFD simulations are often applied to situations not envisioned during the development process. Besides meeting requirements, verification can endow the modeller a sense of confidence that the CFD code is appropriate for the task at hand. FDS has been shown to reproduce the centreline results for a fire plume [33]. The approach used is typical of many CFD verifications: the results were time-averaged so that a comparison could be made with the analytical equations. The choice of verification exercise should be made as close to the problem at hand as possible. For example, if the problem involves flow past a storage tank, a simulation using flow past a cylinder is a relevant verification. The wake in Fig. 7 would have to be appropriately time averaged so that it could be compared with the exact solutions published in [34, 35]. 8.2 Validation The goals of validation are the same as those of verification. Typically, though, validation provides the modeller with more options to choose a suitably close example to the problem at hand.

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Figure 7: Verification exercise showing Karman vortex street past a circular cylinder.

The preparation process should have yielded various relevant references. Often the data published therein makes an excellent choice for a validation exercise. When the budget allows, combining the modelling effort with a series of experiments provides another avenue for obtaining validation data. As part of a program to certify fire models for nuclear reactor safety, FDS has undergone a comprehensive verification and validation process for fire concerns relevant to that industry. The details can be found in [5, 37]. The FDS theory manual [1] contains a long list of verification and validation publications for FDS. 8.3 Uncertainty and sensitivity analyses Convergence studies and the more broadly defined sensitivity analyses are desirable exercises with often thought-provoking consequences. They are typically not pursued because of the time required. An example of a problem signalling an excessively coarse grid is low maximum temperature in a domain with a fire. Roache [30] gives post-processing techniques that can be used to check the order of a simulation. Although it is not yet computationally practical to perform a formal uncertainty analysis of a comprehensive fire simulation such as FDS, the methodology will be outlined. First, consider the algebraic function F which depends on N parameters, Pj, F = F ( P1 , … , Pj , … , PN ),

j = 1,… , N .

(38)

The uncertainty, U , is the standard deviation composed of the terms from the Taylor expansion of eqn (38), 2

N N  ∂F  2 ∂F ∂F 2 U = ∑ SPj Pk +  SPj + ∑∑ j =1  ∂Pj  j =1 k =1 ∂Pj ∂Pk 2

N

(39)

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where SPj is the standard deviation of Pj and SPjPk is the covariance of Pj and Pk. The partial derivatives, ∂F/∂Pj, are a vector known as the sensitivity coefficients. The relative uncertainty, u = U / F,

(40)

is used as well. In practice, eqn (39) is truncated to the first term and the measurement or instrument uncertainty, ∆Pj, is substituted for SPj. Hence 2

 ∂F  2 U ≈ ∆F = ∑   ∆Pj . ∂ P   j j =1 N

2

2

(41)

For a set of M functions Fi = Fi ( P1 ,… , PN ),

i = 1, … , M ,

(42)

the uncertainty becomes a vector 2

 ∂F  ∆Fi ≈ ∑  i  ∆Pj2 , j =1  ∂Pj  N

2

(43)

and the sensitivity coefficients form a tensor. Now consider a system of ordinary differential equations (ODEs), dFi = Gi ( F1 ,… , FM , P1 ,… , PN ), dt

i = 1,… , M .

(44)

Partial differentiation of this system with respect to Pj results in the dynamic sensitivity equations d  ∂Fi  M  ∂Gi   ∂Fk   ∂Gi    =∑   + , dt  ∂Pj  k =1  ∂Fk   ∂Pj   ∂Pj 

i = 1,… , M , j = 1,… , N .

(45)

In order to determine how the uncertainty varies with time, eqns (44) and (45) are solved simultaneously. At each time step, eqn (43) is used to determine the uncertainty in Pj. The spatially discretized equations in FDS can be considered to form a system of DAEs, i.e. a combination of eqns (38) and (42). One possible way of estimating the uncertainty is to proceed as was indicated above, solving the DAE + sensitivity system for each cell and then calculating the uncertainty of at each cell using eqn (43). Although this approach is not comprehensive (for example, the uncertainty in the BCs have not been considered and the uncertainty in the perturbation pressure cannot be addressed in this fashion), it does indicate the immense computational scope required for a comprehensive uncertainty analysis of a CFD program. Techniques can be employed to reduce the scope. For example, the uncertainty in the HRR can be quite high. Since the HRR has a profound impact on the results, a one parameter uncertainty analysis can still yield useful results. Typically, one is interested in results (such as temperature) at a discrete set of points. By deriving the sensitivity equations for the corresponding cells, a tractable methodology is obtained by which FDS plus a limited set of sensitivity equations can be used to determine the uncertainty in the temperature at a given number of points. Sensitivity analyses demonstrate how the solution changes, when one feature of the simulation is varied. Convergence studies have already been presented as an example of grid and/or time

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Figure 8: Heat release rates resulting from running a flame spread problem on four different platforms.

step sensitivity analyses. Another fruitful technique for identifying potential problems is platform sensitivity. Figure 8 shows the outcome of running a poorly conceived FDS flame spread simulation on four different platforms. Concepts such as order of accuracy give the modeller an idea of the numerical error that would be expected from a simulation. It is much more difficult to estimate how uncertainties in the inputs would manifest themselves in the results. For example, if the HRR is 5 MW ± 0.5 MW, how does this affect the temperature reading at any given place? The formal method briefly introduced above indicates that considerable development work would be required for a model that does not have native sensitivity analysis capabilities. The question can be answered with no further development by performing a set of simulations with the code at 4.5, 5, and 5.5 MW. Again, as with model development itself, the parameters chosen for sensitivity analyses must be chosen carefully for budget and scheduling reasons. Ordering the parameters in relevance aids in the selection process. Hamins and McGrattan [38] demonstrate how scaling and empirical relations can be used to obtain good uncertainty bounds for certain variables such as the temperature in the upper layer. 8.4 Certification, accreditation, quality assurance As was mentioned earlier, guidelines for certification can be found in [31]. Some commercial CFD software are accredited by independent agencies such as the United Kingdom Accreditation Service and Lloyd’s Register of Quality Assurance according to quality assurance standards such as ISO 9001. FDS has undergone a variety of certification programmes. The first example occurred when the Alaska Department of Environmental Conservation adopted nomographs resulting from FDS calculations into their in situ oil spill burning decision tree [36]. FDS has been used to determine entrainment rates in balcony spill plumes associated with atrium smoke management systems. FDS results were compared with existing correlations and experimental data and used to develop recommendations for calculating entrainment rates for balcony spill plumes that may be included in future ASHRAE handbooks. FDS is currently being used to investigate the impact of make-up air on fire plumes in atria with mechanical exhaust systems. The goal of this project is to determine

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the impact of the make-up air velocity on the fire plume and the hot layer and to determine how large the maximum make-up velocity can be before it can reduce the effectiveness of the exhaust system. FDS passed the SPEC benchmark suite. FDS has been accepted as a hazard analysis tool for the nuclear industry [37]. 8.5 Review The review process for the results of a series of simulations should embody the same rigor as the evaluation of an experimental regimen. All the statistical analysis techniques [39−41] are applicable. The results should be checked for evidence of common mistakes such as those outlined in Table 1. An important assessment is the adequacy of the assumptions and the approach. Using FDS as an example, the default thermal BC (constant wall temperature) is often used in simulations. If the review process shows lower upper layer temperatures than expected, this constant wall temperature should be suspected and new runs performed using a more representative thermal BC are in order. As has been emphasized repeatedly, the modeller should be particularly wary if the simulation was used outside the limits of its intended development goals or outside of the certified flow regime. Checking that the solution is within the bounds of relevant analytical solutions or that the data is sufficiently similar to that published in the literature is prudent. If none are found then sufficiently relevant models should be developed that can be tested against published results.

9 Examples 9.1 Grid density The issue of grid density is often raised in any CFD analysis. One wishes to obtain accurate and converged results which leads one to use more grid cells. However, if one doubles the grid cells in each direction the computational time will increase by a factor of 8, a factor of two for each axis and a fourth factor of two from halving the time step to preserve the CFL condition. This leads one to use fewer grid cells. As an illustration a simulation of a 200 kW pool fire in the open is performed using a range of grid sizes. From eqn (5), the rule of thumb suggests grid sizes in the range of 5−10 cm. Figure 9 below plots the centreline temperature and velocity profile for each of the four grids. The plots indicate that at a resolution of 10 cm, that grid is not converged for this fire. Even at 5 cm, the grid is not completely converged; however, decreasing the grid size 20 % to 4 cm only results in a small change in the centreline quantities. The rule of thumb is a good approximation, but it does not replace the need to perform a proper grid study. 9.2 Turbulence model In this set of examples, choosing the appropriate turbulence model is examined. Two sets of two cases are simulated with each set being the same except for the choice of turbulence model. Two of the cases model a 2 MW fire in a 4 m × 4 m domain with 4 cm grid cells. The other two cases are a 20 kW fire in a 0.4 m × 0.8 m domain with 4 mm grid cells. For all cases the default fuel and surface properties in FDS are used with all boundaries except for the floor being open. Contours of HRR per unit volume for each case are shown in Fig. 10. At coarse grid resolutions, running the simulation in DNS mode results in non-physical heat release distribution while the LES mode

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600

5

C e n te rlin e Ve lo c ity (m /s )

C e n te rlin e Te m p e ra tu re (°C )

800

400

200

4

3 20 cm grid 10 cm grid 5 cm grid 4 cm grid

2

1

0

0 0

0.5

1

1.5

Elevation (m)

2

2.5

3

0

0.5

1

1.5

2

2.5

3

Elevation (m)

Figure 9: Plume centreline temperatures and velocities for a 200 kW propane fire using four grid resolutions. LES

DNS

4 mm grid

4 cm grid

Figure 10: Contours of heat release rate per unit volume (kW/m3) for LES and DNS simulations at fine (4 mm) and coarse (4 cm) grid resolutions. results in heat release distribution that like is a snapshot of a diffusion flame. At fine grid resolutions, both modes result in HRR distributions that look like snapshot of a diffusion flame. However, the LES result is binary in nature. Combustion appears at essentially one volumetric intensity while the DNS result shows a much wider range of values.

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9.3 Symmetry There are problems where symmetry will appear to exist. An example of this would be determining detection times or sprinkler activation times for a fire beneath a flat ceiling where the distance from the fire to the walls is large. One would be tempted to treat this problem as a twodimensional axisymmetric problem or as a three-dimensional axisymmetric problem by using a quarter of the domain with symmetric boundaries. However, symmetry in this case only exists in the time averaged sense. The reality is that the mass flow in the plume will rotate about the plume’s centre. This rotation is a significant factor in the entrainment of the plume. To illustrate this, three FDS simulations were performed. Each simulation is of a plume beneath an unbounded ceiling. The first simulation treats the plume as a 2D axisymmetric problem. The second treats the plume as a 3D problem with symmetry conditions on two sides (one quarter of the domain). The third models the entire domain without any symmetry conditions. The computational domain consists of a 4 m × 4 m × 3 m domain with open sides and 200 kW fire at the centre of the floor. A 7.5 cm grid resolution was used. Along the ceiling at radial distances of 0, 1, and 2 m were a triplet of heat detectors with a 74°C setpoint and response time indexes of 50, 100, and 200 (m/s)1/2. Activation times for the three simulations are shown in the Table 2 below where DNA indicates the detector did not activate. 9.4 Sprinklers As discussed in Section 5, FDS uses a superdrop, Lagrangian model for sprinklers. There are a number of issues associated with this model that the user must be aware of. These are parameters determining the injection of drops into the flow field, the grid size around the sprinkler, and the definition of the sprinkler head’s spray pattern. The first issue involves the injection of drops into the flow field. Three parameters in FDS determine how drops are inserted. These are the time interval between insertions, the number of drops inserted per head in a time interval, and the total number of drops allowed. If the number of drops inserted per time interval is too small, then the mass of the superdrop can become too large and overly perturb the gas flow. If the time interval between insertions is too large, the gas cooling and mass flow impacts from the water spray will not have a continuous impact. Both will result in a poor resolution of the desired spray pattern. 9.5 Combustible material properties Surfaces within FDS can be defined as combustible. This can be done by a couple of methods. In the first method, the surface is assigned an ignition temperature and a predetermined HRR curve. In Table 2: Detector activation times (in seconds). DNA indicates the detector did not activate.

Case 2D Quarter Full

0 m distance

1 m distance

2 m distance

50 100 200 (m/s)1/2 (m/s)1/2 (m/s)1/2

50 100 200 (m/s)1/2 (m/s)1/2 (m/s)1/2

50 100 200 (m/s)1/2 (m/s)1/2 (m/s)1/2

12 17 24

7 9 13

5 6 6

86 143 103

43 71 55

22 37 29

DNA DNA DNA

136 DNA 168

67 DNA 91

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the second method, the surface is assigned an ignition temperature and a heat of pyrolysis. In this case the HRR is determined based on the radiative and convective feedback to the burning surface. Using the first method, errors in computing the correct feedback are not a concern; however, applying a single heat release curve globally across the cells of an object may not properly capture the true behaviour of the burning object. Also, the test data used to generate the heat release curve may not be indicative of the thermal environment seen in the simulation. The second method, while having the appearance of greater physical accuracy, is sensitive to the input values and the grid used. An example of the sensitivity of the FDS burning rate as a function of the inputted material properties and computational grid is discussed below. These simulations make use of the room fire data file that is distributed with FDS. In all simulations the room fire geometry (a single room with an open door that is filled with a variety of furniture shown in Fig. 11) is modified to change the material for all the furniture to the polyurethane fuel definition distributed with FDSv4. Four simulations are performed. The first three have the same grid definition but vary the ignition temperature using 280°C, 285°C, and 290°C. The fourth simulation used an ignition temperature of 280°C, but increased the grid density by 50%. Figure 12 shows the pyrolysis rates for the four simulations. As can be seen the burning rate behaviour for the 280°C and 285°C ignition temperature cases are very similar; however, the 290°C ignition temperature case shows a markedly different behaviour. The difference of 10°C is only a small percentage of the change in ignition temperature and well within the experimental errors associated with devices such as the cone calorimeter used to measure the ignition temperature. The denser grid case, while using a 280°C ignition temperature, has burning rate behaviour similar to that of the 290°C case. In general it is currently very difficult to predict flame spread in a complex geometry with any certainty.

Figure 11: Roomfire4 geometry.

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0.12

Burning Rate (Kg/m2s)

0.10

0.08 280 °C 285 °C 290 °C 280 °C,densegrid

0.06

0.04

0.02

0.00

0

100

200

300 Time(s)

400

500

600

Figure 12: Roomfire burning rates.

9.6 Radiation solver settings Since the radiation solver uses a finite number of angles, there are preferential directions for radiation heat transfer, which are given by how the angles line up with the computational grid. When the radiant source is geometrically large, any preferential heat transfer from a single source cell is washed out when all source cells are accounted for. This is not the case when the radiant source is geometrically small. In Fig. 13 below, are the results of four simulations. In all four simulations the domain is a 4 m × 4 m × 4 m cube with the ceiling open and the remaining surfaces fixed at 20°C. Two pairs of simulations were run, one with 50 angles in the radiation solver and one with 500 angles. In each pair a 1 m2 and a 4 m2 hot surface was placed at the centre of the ceiling. One would expect to see five hot spots on the surfaces: four slightly below the top centre of each wall and one centred on the floor. As shown in the figure, the 4 m2 source does show five hot spots for both 50 and 500 angles although some directional bias is shown with only 50 angles. The 1 m2 source with 50 angles shows four hot spots on each wall rather than one while the 500 angle result is as expected.

10 Conclusions The implementation of an effective computational fire dynamics simulation entails a balance of computer resources, efficient deterministic models, fast numerical methods, and effective human interaction. Currently, CFD models such as FDS perform well on smoke transport problems and inconsistently on flame spread problems. One of the steps that made the successes possible was the incorporation of submodels to correct for resolution inadequacies. Verification and validation help to identify for developers the areas where further effort needs to be expended. For problems that pass the verification and validation process, the code can move on to certification of the simulation for that application.

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500 Angles

4 m2 Source

1 m2 Source

Figure 13: Relationship between number of radiation angles and size of radiant source. For the modeller who is not primarily a developer, the simulation should be used for the problems it was created for and for problems that it was accredited for. Good modelling practice is just as challenging as developing the simulation. Careful preparation and thorough review of results are instrumental in achieving a successful modelling endeavour. The modeller needs to keep in mind that just one error, such as any of those listed in Table 1, can suffice to ruin the results.

Acknowledgments The authors would like to express their gratitude to Prof. M. Faghri of the University of Rhode Island for his excellent editorship and to the staff of the WIT Press for all their support.

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McGrattan, K.B. (ed.), Fire Dynamics Simulator (Version 4), Technical Reference Guide, NIST Special Publication 1018, National Institute of Standards and Technology: Gaithersburg, MD, September 2005. McGrattan, K.B. & Forney, G.P., Fire Dynamics Simulator (Version 4), User’s Guide, NIST Special Publication 1019, National Institute of Standards and Technology: Gaithersburg, MD, July 2004. Mell, W.E., Olson, S.L. & Kashiwagi, T., Flame spread along free edges of thermally thin samples in microgravity. Twenty-Eighth Symposium (International) on Combustion, eds P.A. Candel et al., Combustion Institute: Pittsburgh, PA, pp. 2785−2792, 2000. Lautenberger, C.W., de Ris, J.L., Dembsey, N.A., Barnett, J.R. & Baum, H.R., A simplified model for soot formation and oxidation in CFD simulation of non-premixed hydrocarbon flames. Fire Safety Journal, 40(2), pp. 141−176, 2005.

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Najafi, B., Salley, M.H., Joglar, F. & Dreisbach, J., Verification and Validation of Selected Fire Models for Nuclear Power Plant Applications, Volume 1: Main Report, U.S. Nuclear Regulatory Commission, Office of Nuclear Regulatory Research (RES), Rockville, MD, and Electric Power Research Institute (EPRI), Palo Alto, CA, NUREG-1824 and EPRI 1011999, May 2007. Rehm, R.G. & Baum, H.R., The equations of motion for thermally driven, buoyant flows. Journal of Research of the NBS, 83, pp. 297−308, 1978. Majda, A. & Lamb, K.G., Simplified equations for low mach number combustion with strong heat release. Dynamical Issues in Combustion Theory, eds P.C. Fife & F.W. Williams, Springer-Verlag: New York, pp. 167−211, 1991. Majda, A. & Sethian, J., The derivation and numerical solution of the equations of zero mach number combustion. Combustion Science and Technology, 42, pp. 185−205, 1985. Embid, P., Well-posedness of the nonlinear equations for zero mach number combustion. Reacting Flows: Combustion and Chemical Reactions, ed. G.S.S. Ludford, American Mathematical Society: Providence, RI, pp. 253−271. Embid, P., Well-posedness of the nonlinear equations for zero Mach number combustion. Communications in Partial Differential Equations, 12, pp. 1227−1285, 1987. Embid, P., On the reactive and non-diffusive equations for zero mach number flow. Communications in Partial Differential Equations, 19, pp. 1249−1281, 1989. Anderson, J.D., Jr., Introduction to Flight, 2nd edn, McGraw Hill: New York, 1985. Linteris, G.T. & Williams, F.A., Asymptotic and numerical analysis of a premixed laminar nitrogen dioxide−hydrogen flame. Combustion Science and Technology, 105, pp. 165− 182, 1995. Norton, T.S., Smyth, K.C., Miller, J.H. & Smooke, M.D., Comparison of experimental and computed species concentration and temperature profiles in laminar, two-dimensional methane/air diffusion flames. Combustion Science and Technology, 90(1–4), pp. 1−34, 1993. Smyth, K. C., NO production and destruction in a methane/air diffusion flame. Combustion Science and Technology, 115, pp. 151−176, 1996. Data collected by K.C. Smyth, available at http://www.bfrl.nist.gov. Wieczorek, C., Vandsburger, U. & Floyd, J., An evaluation of the global equivalence ratio concept for compartment fires: Part I − Effect of experimental measurements. Journal of Fire Protection Engineering, 14(1), pp. 9−32, February 2004. Heskestad, G., Fire plumes, flame height, and air entrainment. SFPE Handbook of Fire Protection Engineering, 3rd edn, eds P.J. DiNenno et al., Society of Fire Protection Engineers: Quincy, MA, pp. 2-1−2-17, 2002. Holman, J.P., Heat Transfer, 7th edn, McGraw-Hill, Inc.: New York, NY, 1990. Grosshandler, W., RadCal: A Narrow Band Model for Radiation Calculations in a Combustion Environment, NIST Technical Note TN 1402, National Institute of Standards and Technology: Gaithersburg, MD, 1993. Chan, T.-S., Measurements of water density and drop size distributions of selected ESFR sprinklers. Journal of Fire Protection Engineering, 6(2), pp. 79−87, 1994. Zhang, S. & Jin, J., Computation of Special Functions, Wiley: New York, 1996. Hunt, S.P., Floyd, J.E. & Cutonilli, J.A., Fire suppression modeling applied to shipboard design. Talk given at the Workshop on Fire Suppression Technologies, Mobile, AL, February 2003. Putorti, A.D., Everest, D. & Atreya, A., Simultaneous Measurements of Drop Size and Velocity in Large-Scale Sprinkler Flows Using Particle Tracking and Laser-Induced

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Fluorescence, Technical Report NIST GCR 03-852, National Institute of Standards and Technology: Gaithersburg, MD, 2003. Trelles, J., Mawhinney, J.R. & DiNenno, P.J., Characterization of a high-pressure multijet water mist nozzle for the purposes of computational fluid dynamics modeling. Computational Simulation Models in Fire Engineering and Research, ed. J.A. Capote Abreu, GIDAI: Santander, Spain, pp. 261−270, 20 October 2004. Babrauskas, V., Heat release rates. SFPE Handbook of Fire Protection Engineering, 3rd edn, eds P.J. DiNenno et al., Society of Fire Protection Engineers: Quincy, MA, pp. 3-82− 3-161, 2002. Brown, P.N., Hindmarsh, A.C. & Petzold, L.R., Consistent initial condition calculation for differential-algebraic systems. SIAM Journal on Scientific Computing, 19(5), pp. 1495− 1512, 1998. Kevorkian, J., Partial Differential Equations, Analytical Solution Techniques, 2nd edn, Springer-Verlag: New York, 2000. Van Dyke, M., Perturbation Methods in Fluid Mechanics, Parabolic Press: Stanford, CA, 1975. Roache, P.J., Quantification of uncertainty in computational fluid dynamics. Annual Review of Fluid Mechanics, 29, pp. 123−160, 1997. Department of Defense, Verification, Validation and Accreditation (VV&A) Recommended Practice Guide, Office of the Director of Defense Research and Engineering Defense Modeling and Simulation Office, November 1996. ASTM, Standard Guide for Evaluating the Predictive Capability of Deterministic Fire Models, E 1355, American National Standards Institute: New York, 1998. McGrattan, K.B., Baum, H.R. & Rehm, R.G., Large eddy simulation of smoke movement. Fire Safety Journal, 30, pp. 161−178, 1998. Hinze, J.O., Turbulence, 2nd edn, McGraw-Hill: New York, 1975. Libby, P.A., An Introduction to Turbulence, Taylor & Francis: Washington, DC, 1996. McGrattan, K.B., Baum, H.R., Walton, W.D. & Trelles, J., Smoke Plume Trajectory from in Situ Burning of Crude Oil in Alaska – Field Experiments & Modeling of Complex Terrain, NISTIR 5958, National Institute of Standards and Technology: Gaithersburg, MD, January 1997. McGrattan, K., Verification and Validation of Selected Fire Models for Nuclear Power Plant Applications, Volume 7: FDS, U.S. Nuclear Regulatory Commission, Office of Nuclear Regulatory Research (RES), Rockville, MD, and Electric Power Research Institute (EPRI), Palo Alto, CA, NUREG-1824 and EPRI 1011999, May 2007. Hamins, A. & McGrattan, K., Verification and Validation of Selected Fire Models for Nuclear Power Plant Applications, Volume 2: Experimental Uncertainty, U.S. Nuclear Regulatory Commission, Office of Nuclear Regulatory Research (RES), Rockville, MD, and Electric Power Research Institute (EPRI), Palo Alto, CA, NUREG-1824 and EPRI 1011999, May 2007. Wilson, E.B., An Introduction to Scientific Research, Dover: Mineola, NY, 1952. Holman, J.P., Experimental Methods for Engineers, 2nd edn, McGraw-Hill: New York, 1971. Taylor, B.N. & Kuyatt, C.E., Guidelines for Evaluating and Expressing the Uncertainty of NIST Measurement Results, Technical Note NISTIR 1297, Gaithersburg, MD, 1994.

CHAPTER 12 CFD-based modeling of combustion and suppression in compartment fires A. Trouvé & A. Marshall Department of Fire Protection Engineering, University of Maryland, College Park, USA.

Abstract This chapter is aimed at illustrating contemporary computational fluid dynamics (CFD) capabilities for compartment fire applications. We choose to use the fire dynamics simulator (FDS) for illustration purposes. FDS is developed by the National Institute of Standards and Technology, USA, and is one of the leading fire simulation software available to fire protection engineers and scientists; it is both representative of current capabilities as well as indicative of future trends. The material in this chapter is organized around a presentation of the different stages of a typical enclosure fire, from localized ignition to its fully developed state. We present simulations of model problems that serve to illustrate the performance and limitations of CFD-based descriptions for: (1) transient ignition phenomena and early fire growth; (2) smoke filling and pre-flashover fire spread; (3) possible flashover and transition to under-ventilated combustion; (4) activation of water-based fire suppression systems and subsequent fire control and/or extinction.

1 Introduction Fire safety is one of the engineering design problems considered during the construction and/or refurbishing of houses or buildings. Fires occurring in confined spaces exhibit unique features associated with smoke accumulation, restricted air ventilation, interactions with solid walls, and in many cases interactions with automatic fire suppression systems. In the scientific literature, these wall-confined fires belong to a special class of problems usually referred to as compartment or enclosure fires. A typical compartment fire scenario involves the following successive stages [1−3]: (1) ignition and early growth; (2) pre-flashover growth featuring a well-ventilated (i.e. fuel-limited) fire and a hot smoke layer that develops near the compartment ceiling; (3) flashover that corresponds to a dramatic increase in the amount of burning liquid/solid materials; (4) post-flashover, fully developed fire dynamics featuring a ventilation-controlled (i.e. oxygen-limited) fire. We briefly review below the main physical features observed during the different stages of compartment fires.

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Stage 1 is difficult to describe in general terms because fire problems feature a wide variety of possible ignition scenarios. The list includes scenarios dominated by heterogeneous processes (for instance, smoldering combustion as may occur in porous solid flammable materials) in which the incipient gas-phase combustion phenomena are strongly coupled with the thermally driven physical and chemical changes taking place within the liquid/solid fuel sources. It also includes scenarios dominated by gas-phase processes (for instance, processes occurring during ignition of fuel vapor clouds) in which transient combustion events correspond to a wide range of flame regimes, including premixed, non-premixed, and partially premixed combustion (PPC) modes. After ignition has occurred and a fire is established (stage 2), flow confinement and buoyancy forces lead to a natural vertical stratification of the fire room environment into two layers: a ceiling layer resulting from the accumulation of high temperature combustion products mixed with entrained air; and a floor layer corresponding to fresh air at ambient (or slightly preheated) temperature. The high temperatures found in the ceiling layer contribute to the intensification of the heat feedback experienced by flammable objects/materials present in the fire room, and thereby to a faster fire growth. Fire growth is gradual at first, until critical conditions are reached, at which point it becomes very rapid. This rapid increase in fire size is called flashover (stage 3); flashover may be interpreted as a series of spontaneous ignition events driven by super-critical levels of irradiation from super-hot ceiling layer gases (i.e. gases with temperatures in excess of 800−900 K). After flashover, virtually all flammable objects and materials present in the room are involved in the fire. The amount of gaseous fuel mass generated by the thermal degradation processes occurring in liquid or solid flammable sources (i.e. as a result of evaporation or pyrolysis processes) may then be so large that the combustion dynamics become fuel rich, i.e. oxygen-limited (stage 4). Under oxygen-limited fire conditions, the flame may experience a dramatic change and migrate from the fuel sources to the compartment vents location; this transition is similar to the flame opening process observed in Burke−Schumann type laminar diffusion flames when going from fuel-lean to fuel-rich conditions. In addition, in typical under-ventilated fire situations, large sections of the flame are supplied with vitiated air, i.e. with a mixture of pure air and re-circulating combustion products. Sufficient levels of air vitiation will result in sub-critical oxygen concentrations and consequent partial or total flame extinction. Air vitiation effects stress the importance of oxygen depletion for the flame dynamics, and explain why a flame that develops in a sealed or poorly ventilated space will ultimately experience quenching. Finally, during stage 4, since only a fraction of the fuel mass generated in the fire room actually burns there, the excess (unburnt) fuel mass may be transported into adjacent rooms through vents and openings. This leakage of fuel mass may in turn lead to burning outside the room of fire origin, and thereby contribute to fire spread to adjacent spaces. In this chapter we examine the ability of current computational fluid dynamics (CFD) approaches to simulate the different physical features observed in compartment fires. We choose to use the fire dynamics simulator (FDS) for illustration purposes. FDS is developed by the Building and Fire Research Laboratory (BFRL) of the National Institute of Standards and Technology (NIST), USA; it is available as freeware [4] and is oriented towards fire applications [5−7]. It is worth emphasizing that our selection of FDS is not meant as an endorsement of FDS over other CFD tools also available for simulations of building fires (see refs [8, 9] for examples of results obtained using other CFD tools). FDS is adopted in this study because of its popularity and availability, and because it is representative of current fire modeling capabilities. The main features of FDS include a large eddy simulation (LES) approach to treat turbulent flow motions (based on the turbulent eddy diffusivity concept and the classical Smagorinsky

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closure model); a low Mach number formulation to handle flow compressibility (and to filter out fast moving and computationally demanding acoustic waves); a second-order finite difference scheme for spatial discretization; a second-order predictor−corrector explicit scheme for time integration; a multi-block, rectangular Cartesian grid capability; a parallel computing capability based on the message passing interface (MPI) protocols. The official release of FDS includes a mixture-fraction-based model proposed to describe non-premixed turbulent combustion; this model includes a flame extinction capability due to air vitiation [5, 6]. In this study, we also used a modified in-house version of FDS that has enhanced modeling capabilities and can treat multimode (i.e. partially premixed) combustion. All computational results presented herein were generated by running FDS in parallel mode using five processors on a Linux PC cluster; the cluster is equipped with 2.4−3.8 GHz Intel Xeon processors and was made available to us by BFRL/ NIST. The material in this chapter is organized around a presentation of the different stages of a typical enclosure fire, from localized ignition to its fully developed state. We present simulations of model problems that serve to illustrate the current performance and limitations of CFD-based descriptions for: transient ignition phenomena and early fire growth (Section 2); smoke filling and pre-flashover fire spread (Section 3); possible flashover and transition to under-ventilated combustion (Section 4); activation of water-based fire suppression systems and subsequent fire control and/or extinction (Section 5).

2 Transient ignition and early fire growth In this section, we examine the feasibility of a CFD approach to simulate transient ignition events, as might occur in fire and explosion safety scenarios. Note that out of the long list of possible ignition scenarios, we limit the scope of our discussion hereafter to gas-phase processes. We consider a case corresponding to the hazardous accumulation of fuel vapor in a room (Fig. 1); this accumulation may be the result of an accidental liquid fuel spill, assuming that the liquid fuel has a low flashpoint temperature and undergoes spontaneous evaporation under room ambient conditions. We choose to assume that the room is closed, although many of the following developments would also apply to scenarios occurring in vented rooms or unconfined spaces. We also assume in the following that: there is a significant delay between the start of the fuel build-up and the ignition event, thereby allowing the formation of a sizeable fuel vapor cloud prior to combustion; the fuelair mixing process results in some large portion of the fuel vapor cloud being within the fuel-air flammability limits; ignition occurs at some location in the flammable portion of the fuel vapor cloud. Following ignition, the combustion will proceed in part as a thin deflagration or detonation wave that propagates across the flammable portions of the fuel vapor cloud. We focus in the following on the deflagration scenario, in which the premixed flame propagates at subsonic speeds and pressure remains quasi-uniform across the combustion zone (pressure will vary with time, however, as a result of the room confinement). We also focus on a scenario in which the volume of the fuel source is large, the duration of the fuel release process is long, and the premixed flame successfully flashes back to the fuel source location. This flash back triggers in turn the formation of a turbulent diffusion flame attached to the fuel source. From a combustion theory perspective, the scenario presented in Fig. 1 corresponds to a transition from initially premixed to subsequently non-premixed turbulent combustion. The subject of accidental combustion in fuel vapor clouds has received significant interest in the scientific literature. Previous CFD modeling studies typically belong to one of the following two categories: studies in which flammable conditions are assumed across the bulk of the fuel vapor

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Figure 1: Case C1 corresponding to the ignition and combustion of a vertically stratified fuel vapor cloud located at floor level in a closed room.

cloud, and combustion is described as premixed [10−12]; and studies in which ultra-rich conditions are assumed across the bulk of the fuel vapor cloud, and combustion is described as nonpremixed [13−17]. Clearly the ignition/deflagration/light-back/diffusion-flame sequence occurring in case C1 (Fig. 1) requires a more general formulation in which combustion can be described as both, simultaneously or sequentially, premixed, and non-premixed. We present below such a formulation and focus on specific issues resulting from the coupling of premixed and non-premixed turbulent flame models. This coupling has received growing interest in recent years, primarily driven by the need to adapt combustion formulations for a CFD treatment of lifted turbulent diffusion flames [18−20]. The burning regime in the stabilization region of lifted diffusion flames is usually referred to as partially premixed combustion. We adopt below the PPC formulation proposed in refs [19, 20]; the LES PPC model is described in Section 2.1. The performance of the model is then evaluated in a numerical simulation of case C1; results are presented and discussed in Section 2.2. 2.1 Modeling of PPC 2.1.1 Deflagration modeling We start from the classical description of premixed combustion based on the concept of a reaction progress variable c: c = 0 in the fresh reactants, c = 1 in the burnt products, and the flame is

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the (thin) region where c goes from 0 to 1 [21−23]. The c-framework is general and flexible, and it has been previously adapted to a LES treatment of propagating turbulent flames [24−34]. The treatment is based on a transport equation for the LES-filtered reaction progress variable c . We adopt in the following the closure models of refs [26, 27] and write: ∂ ∂ ∂  ∂c  ∂ ( r c ) + ( r ui c ) = + rD  ∂t ∂xi ∂xi  ∂xi  ∂xi

 nt ∂c   r Sc ∂x  + w c + w ign t i

(1)

where r is the mass density, ui is the xi-component of the flow velocity vector, D is the molecular diffusion _ _ coefficient, nt is the turbulent eddy diffusivity, Sct is the turbulent Schmidt number, and w◊c and w◊ign are combustion source terms; and where the overbar symbol denotes straight LESfiltered quantities, and the tilde symbol Favre-weighted (i.e. mass-weighted) LES-filtered quantities. The first term on the right-hand side of eqn (1) represents transport of c due to molecular diffusion; the second term represents transport of c due to subgrid-scale convective motions; the third and fourth terms represent production of c due to self-sustained flame propagation and flame ignition, respectively. The subgrid-scale convective transport term has been expressed assuming gradient transport and using the classical turbulent eddy viscosity concept [22, 23]. _ Following Veynante and co-workers [26, 27], the chemical reaction term w◊c is expressed using a classical flamelet closure expression:  6  c(1 − c ) w c = ru sL × Ξ  4  ∆c  π

(2)

where Ξ is a non-dimensional number (Ξ ≥ 1; Ξ = 1 for a laminar flame) known as the subgridscale flame wrinkling factor and ∆c is the length scale defined as the LES filter size for the c-equation. In refs [26, 27], the molecular transport term is also re-formulated according to the realizability requirement that under laminar flow conditions, the flame propagates at the laminar flame speed (considered as a known quantity): rD =

ru sL ∆ c 16 6 / π

(3)

where ru is the unburnt gas mass density and sL is the laminar flame speed. Given these choices, the expression for the LES-filtered heat release rate (HRR) per unit volume is: q p = (w c + w ign ) × (YFm − YFeq )∆H F    6  c(1 − c ) =  ru sL × Ξ  4 + w ign  × (YFm − YFeq )∆H F   π  ∆c  

(4)

where YFm is the value of the fuel mass fraction in the unburnt gas, YFeq is its value in the burnt gas, and ∆HF is the heat of combustion (per unit mass of fuel). YFm is an input quantity to the combustion problem that characterizes the pre-combustion state of the reactive mixture; YFeq is a quantity that characterizes the post-premixed-flame state; both quantities are discussed below. Equation (1) is a classical convection-diffusion-reaction partial differential equation and may be easily handled by CFD solvers. It is also possible to gain further insight into the model formulation presented in eqns (1)−(3) by using the analytical methods of the Kolmogorov−Petrovskii− Piskunov (KPP) theory [35−38]. The theory applies to a simplified, yet revealing, case corresponding to

446

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a statistically one-dimensional planar flame propagating into frozen homogeneous turbulent flow. We find that in this case, eqns (1)−(3) imply the following expression for the propagation velocity of the LES-filtered flame:  nt  6 st = sL Ξ ×  1 + 16 π Sct sL ∆ c  

(5)

This expression conveniently illustrates some of the main features of the present filtered-c approach. First, as mentioned above, in the absence of subgrid-scale turbulence (nt = 0 and Ξ = 1), the flame propagates at the laminar speed sL. Second, in the presence of subgrid-scale turbulence (nt > 0 and Ξ > 1), st is increased by a variable factor and this factor depends on local turbulence properties, as measured by the flame wrinkling factor and the turbulent eddy diffusivity. Elaborate closure model expressions have been recently proposed in the scientific literature to describe the flame wrinkling factor [39, 40]. In the following, however, we adopt a temporary expedient, and neglect subgrid-scale flame wrinkling entirely, and assume Ξ = 1. The local flame propagation velocity is in that case a unique function of the pseudo-Reynolds number (sL∆c/vt). References [33, 34] present a detailed discussion of the grid resolution requirement of the model formulation in eqns (1)−(3), and of the relationship between the LES filter size ∆c and the computational_grid cell size ∆. Using the methodology proposed in ref. [41], it can be shown that the thickness d f of the LES-filtered flame is: nt  6 π  1 + 16  6  π Sct sL ∆ c 

(6)

Yk = (1 − c ) × Ykm ( Z ) + c × Ykeq ( Z )

(7)

df = ∆ c

_ This expression shows that d f is of order ∆c (and is an increasing function of the subgrid-scale turbulence intensity) and therefore suggests that the flame is correctly resolved on the computational grid for values of (∆c/∆) significantly larger than one [24−29, 33, 34]. The numerical tests performed in refs [33, 34] quantify this statement and show that the filtered-c model in FDS requires a filter-to-grid length scale ratio equal to or greater than 4, (∆c/∆) ≥ 4. We now turn to an extension of the filtered-c model to the case of fuel-air mixtures with variable composition. Such an extension is required to treat the problem presented in Fig. 1, in which the fuel vapor cloud features variable fuel-air ratios ranging from rich-flammable to ultra-lean conditions. The extended formulation is based on: a two-variable description of the combustion ~ process, using the reaction progress variable c and the mixture fraction Z as principal ~ variables; and a description of the laminar flame quantities sL, YFm , and YFeq as functions of Z . We start by a brief discussion of the gaseous mixture composition. Upstream of the deflagration front, c =0 and the mixture composition corresponds to the pure mixing solution, Yk= Ykm (Z) [21−23] (Yk is the mass fraction of species k); downstream of the deflagration front, c=1 and the mixture composition may be approximated by the classical Burke−Schumann equilibrium solution, Yk = Ykeq (Z) [21−23]. Figure 2 presents typical variations of YFm and YFeq with mixture fraction; these variations are obtained assuming global single-step chemistry (i.e. fuel and oxygen in air being transformed into carbon dioxide and water vapor) and propane-air combustion. In the flame region, the mixture composition undergoes a transition from the inert mixing solution to the equilibrium chemistry solution. This transition may be described via modified state relationships that give Yk as a weighted average between YFm and YFeq , using c as a weight coefficient:

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447

Figure 2: Variations of the mixture composition with mixture fraction Z. The left plot corresponds to the pure mixing solution, Yk = Ykm (Z); the right plot corresponds to the Burke−Schumann equilibrium solution, Yk = Ykeq (Z) (the plot assumes that the fuel is propane).

Next, we discuss the variations of laminar flame speed with fuel-air ratio, i.e. the variations of ~ ~ sL with Z . The description sL(Z ) plays a central role in the deflagration model as it contains information on the flammability limits of the fuel-air mixture. The variations of sL with mixture strength may be obtained from experimental measurements, or from numerical calculations of the inner structure of laminar premixed flames, using detailed descriptions of chemical kinetics ~ and molecular transport [21−23]. We propose here a convenient alternative in which sL(Z ) is described via an ad-hoc analytical expression parametrized in terms of four input variables, called ZLFL, ZUFL, Zst and sL,st. ZLFL and ZUFL are the values of Z at the lower and upper flammability limits; Zst and sL,st are the stoichiometric values of Z and sL. We present in Fig. 3 a piece~ wise second-order polynomial function that approximates the variations of sL with Z ; the proposed approximation vanishes at ZLFL, ZUFL, is maximum at Zst, and features a peak value equal to sL,st. 2.1.2 Diffusion flame modeling We start from the classical Burke−Schumann theory of diffusion flames in which infinitely fast chemistry is assumed and the flame structure is described in terms of mixture fraction. It is worth emphasizing that in many fire problems, the turbulent motions are buoyancy-driven and the turbulence intensities remain low-to-moderate. Under such conditions (and assuming well-ventilated conditions) flame extinction remains unlikely and the assumption of infinitely fast chemistry may be considered as an acceptable simplification. As shown in ref. [42], the Burke−Schumann theory also produces explicit expressions for the chemical reaction rates; for instance, the LES-filtered (non-premixed) fuel mass reaction rate may be written as:  Y∞   1  w F = −  F   rcst  p ( Z st )   1 − Zst   2

(8)

c st is the LES-filtered value of scawhere YF∞ is the fuel mass fraction in the fuel supply stream, ~ lar dissipation rate (averaged along the subgrid-scale flame surface contour Z = Zst), and p~(Zst) is

448

Transport Phenomena in Fires

Figure 3: Variations of laminar flame speed sL with mixture fraction Z. The flammable domain corresponds to ZLFL ≤ Z ≤ ZUFL.

the stoichiometric value of the (Favre-weighted) probability density function (pdf) that describes subgrid-scale variations in Z. We now introduce additional simplifications for the description of the conditional mixing rate ~ c st and the pdf value p~(Zst). First, we assume that c~st may be approximated by the unconditional scalar ~ dissipation rate c~; we write [43]: c~ ª 2(vt/Sct)∇Z 2.~Second, we assume that p~(Zst) may be approximated using a d-pdf closure expression: p~(Zst) = d(Z −Zst). The d-pdf approximation is a crude presumed pdf model in which subgrid-scale variations in Z are simply neglected. While clearly questionable, this closure model is adopted here because it complies with the simple realizability requirement that under well-ventilated conditions, all the fuel mass coming from the fuel source is actually consumed by the turbulent flame [44]. As shown in ref. [44], more elaborate presumed pdf expressions do not necessarily satisfy this realizability requirement and therefore fail to predict the correct global HRR. The corresponding expression for the LES-filtered HRR per unit volume is:  Y∞   n 2 qd =  F   r t ∇Z  d ( Z − Z st ) × ∆H F   1 − Z st   Sct

(9)

Equation (9) is the non-premixed combustion model currently used in FDS version 4.0 [4−7]. 2.1.3 Modeling of partially premixed combustion As mentioned above, the description of the combustion dynamics occurring in case C1 (Fig. 1) requires a formulation in which combustion can be described as both, simultaneously or sequentially, premixed and non-premixed. We now turn to a description of the coupling interface between the premixed and non-premixed flame models discussed in Sections 2.1.1 and 2.1.2. The formulation of the coupling interface includes the modified state relationships presented in eqn (7), as

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well as a modified HRR model. The modified HRR model is based on an identification of the locally dominant combustion mode (premixed versus non-premixed) using the concept of a LESresolved flame index FI [19, 20]. Following ref. [19], we define the flame index as: FI =

  1 ∇YF ⋅ ∇YO2  + 1 2  ∇YF × ∇YO  2

(10)

~ where Y F and YO2 are the grid-resolved fuel and oxygen mass fractions (as given by eqn (7)). Note that this expression differs slightly from that in ref. [19]: the FI-expression in ref. [19] includes a subgrid-scale contribution; this contribution is neglected in eqn (10). As seen in eqn (10), FI is a non-dimensional field quantity that varies between 0 and 1: inert mixing between cross-diffusing fuel and oxygen corresponds to FI = 0; a diffusion flame configuration in which fuel and oxygen penetrate the diffusive/reactive layer from opposite directions also corresponds to FI = 0; in contrast, a premixed flame configuration corresponds to FI = 1. In regions ~ where the fuel or oxygen mass is homogeneously distributed (i.e. in regions where ∇Y F = 0 or ∇ YO 2 = 0), FI is set to 0. In eqn (11), we adopt the PPC closure model of ref. [19] and describe HRR as a weighted average between the premixed and non-premixed contributions, using FI as a weight coefficient: q = FI × q p + (1 − FI ) × fign × qd

(11)

where fign is an ad hoc ignition factor. fign is introduced in eqn (11) so that the diffusion flame model remains inactive when inert mixing is taking place (fign = 0 when c = 0), and is only activated as a post-premixed-flame event (fign = 1 when c = 1). We use the expression: fign = 0.5 + 0.5 tanh((c−0.6)/0.05). The model summarized in eqns (1)−(4), (7), (9), (10), and (11) has been implemented into an inhouse version of FDS. The next section presents a numerical simulation of case C1 aimed at illustrating the current performance of the PPC model. 2.2 Simulation of the transient ignition and combustion of a fuel vapor cloud The numerical configuration presented in Fig. 1 corresponds to a square-shaped fuel leak of size D × D = (0.5 × 0.5) m2, located at floor level, and releasing heptane fuel vapors into a sealed compartment. The compartment is 4 m long, 4 m wide, and 3 m high; the walls are made of concrete. Heptane is a liquid fuel at normal temperature and pressure conditions, and is characterized by a low flash point temperature (equal to −4°C). Thus, when exposed to ambient air at 20°C, heptane will spontaneously evaporate and lead to a flammable gaseous fuel-air mixture. The initial fuelair mixture used in case C1 corresponds to a one-dimensional, vertically stratified spatial distribution; this distribution features flammable fuel-rich conditions at vertical elevations below z = 0.5 m (where the mixture fraction is equal to 0.12, or equivalently the equivalence ratio is equal to 2.1), and pure air conditions at elevations above 0.5 m (where the mixture fraction or equivalently the equivalence ratio is equal to 0). The corresponding fuel mass and combustion energy stored in the initial fuel vapor cloud are 1.26 kg and 56 MJ, respectively (the heat of combustion per unit mass of heptane fuel is taken as ∆HF = 44.745 MJ/kg). The leak mass flow rate m◊ F (also called the fuel mass loss rate or MLR hereafter) is prescribed in case C1. MLR is initially negligibly small, so that quasi-quiescent conditions are _maintained prior to ignition. Ignition is triggered at time t = 2 s by activating a numerical ignitor w◊ign located

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at an off-center position at (x; y; z) = (1; 0; 0.5) (Fig. 1); the ignition source duration is 0.1 s. The input parameters for laminar heptane-air deflagrations are approximated as: ZLFL = 0.03, ZUFL = 0.15; Zst = 0.062; and sL,st = 0.5 (Fig. 3). Also, after ignition has occurred, MLR is increased to m◊ F ~ ~ 11.2 g/s. This rate is chosen in anticipation of the previously discussed ignition, deflagration, and light-back sequence, and subsequent transition to a diffusion flame attached to the fuel source; the corresponding size of the diffusion flame is (m◊ F ∆HF) = 500 kW. The computational grid is a uniform cubic mesh; the mesh size is (160 × 160 × 120), which corresponds to 3,072,000 grid cells; the grid cell size is ∆ = 2.5 cm. In order to both accelerate the calculations and avoid memory limitations associated with single-processor computing, the simulation is performed in parallel mode. The computational domain is decomposed into five nonoverlapping blocks, and the computational load is distributed over a network of five processors. While the selected grid is deemed acceptable for simulations of a fully developed turbulent diffusion flame, an ongoing study of the grid requirement associated with the PPC formulation presented in Section 2.1 indicates that the computational grid is still too coarse (by a factor of 2−3 in each coordinate direction) for an accurate simulation of the transient ignition and combustion process [45]. The PPC grid requirement is significantly higher than typical requirements established in previous LES studies of premixed or non-premixed combustion, a finding that may be understood as a requirement that the LES premixed flame remains thin in mixture fraction space. Consistent with the analysis presented in ref. [45], it is found that a direct application of the PPC model fails to provide a correct description of the partially premixed combustion sequence. ◊ In particular, the premixed flame component of the global (i.e. spatially averaged) HRR, Qp_= ∫∫∫V _ ◊ ◊ (FI × q p)dV, does not vanish, and the diffusion flame component, Qd = ∫∫∫V ((1−FI) × fign × q◊ d)dV, remains below its theoretical steady state value of 500 kW. To overcome this difficulty, an ad-hoc modification is proposed with the intent of avoiding the prohibitive grid requirement of the original formulation. The modified PPC model uses a fuel source light-back criterion and a second mixture fraction variable Z2. The light-back criterion is based on monitoring the temperature Ts at the fuel source location and is used to define a lightback time t. t is defined as the time required for Ts to become larger than a certain value (we use 900 K). The variable Z2 is then used to mark the fuel mass that originates from the leak, after lightback has occurred. Z2 is calculated like a regular mixture fraction variable, except that the boundary condition applied in the fuel stream is now time-dependent: Z2 = 0 for t ≤ t; Z2 = 1 for t >_ t. Z_2 is conveniently used to enforce a strict transition to diffusion burning, we locally impose q◊ = q◊ d whenever and wherever Z2 is greater than a small value selected below the lower flammability limit (we use Z2 ≥ 0.01). It is worth emphasizing that this forced transition is only applied to the spatio-temporal domain that corresponds to regions of the compartment reached by the source fluid after the light-back event. ◊ ◊ ◊ Figure 4 presents the time variations of the global HRR, Q = (Qp + Qd), as well as those of its premixed and diffusion flame components, as obtained using the modified PPC formulation. These variations are consistent with the expected ignition/deflagration/light-back/diffusion-flame sequence. For instance, Fig. 4 shows that ignition occurs at t = 2 s, followed by a PPC phase during which both premixed and non-premixed flame modes co-exist (2 s ≤ t ≤ 15 s), and followed in turn by a transition to pure diffusion combustion (t ≥ 15 s). Note that the flame extinction model has been de-activated in the present simulation, HRR is consequently insensitive to oxygen mass depletion and diffusion combustion is (incorrectly) predicted to be sustained indefinitely. The total HRR is maximum shortly after ignition, at t ~ ~ 4 s, and reaches a peak value of approximately 15 MW; at that time, 70% (30%) of the burning is premixed (non-premixed) and the HRR is dominated by its deflagration component. Interestingly, the transient PPC phase comes out as

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Figure 4: Time variations of the global heat release rate. The plot shows the total heat release rate ◊ (thick solid line) as well as its premixed and diffusion flame components, QF (circles) ◊ and Qd (squares), respectively (case C1). relatively short and intense: premixed burning becomes negligible after t = 7 s and completely ◊ ceases at t = 15 s. In addition, light-back occurs at t = t ~ ~ 5 s and shortly after (t ≥ 7 s), Qd achieves the expected steady state value of 500 kW. Thus we see that the modified LES-PPC formulation can provide an overall description of case C1 that is physically sound. Additional insight into the combustion process may be obtained by studying the flame structure at different_ times. We recall the decomposition of HRR per unit volume proposed_ in eqn (11), in which q◊ is written as the sum of two premixed and non-premixed components, q◊ = FI × _ _ ◊q + (1 − FI) × f × q◊ . Figures 5−8 present instantaneous snapshots of surfaces corresponding p ign d to iso-levels of the premixed and non-premixed components of HRR. The iso-levels are selected somewhat arbitrarily in order to facilitate the graphical display. The snapshots reveal the shape and location of the premixed and diffusion flame zones, and a comparison between successive snapshots provide some understanding of the overall combustion dynamics. For instance, Fig. 5 shows the initial flame structure shortly after ignition, at time t = 2.5 s. The flame is seen to expand from the ignition point location in both horizontal and (upward) vertical directions. The horizontal spread is associated with the (sombrero-shaped) premixed flame (i.e. the deflagration or flash fire), whereas the vertical spread is associated with the (mushroom-shaped) diffusion flame (i.e. the fireball). The early development of a diffusion flame may be explained as follows. Since the premixed flame propagates into a fuel-rich mixture, products of premixed combustion correspond to carbon dioxide and water vapor, mixed with nitrogen and (unburnt) excess fuel. The excess fuel mass found in post-premixed-flame gases subsequently mixes with upperlayer air and burns in a diffusion flame mode. Note that the intensity of the diffusion flame is initially much smaller than that of the deflagration; this will change in the course of the simulation: the diffusion flame contribution to global HRR is 30% at t ≈ 4 s, and 100% at t ≥ 15 s (see Fig. 4).

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Figure 5: Instantaneous iso-level surfaces showing the spatial distribution of the premixed (left) and non-premixed_ (right) combustion components of_the heat release rate (see eqn (11)). Left: (FI × q◊ p) = 5 MW/m3; right: ((1−FI) fign × q◊ d)=100 kW/m3. Case C1, time t = 2.5 s.

Figure 6: See Fig. 5 for details. The rear vertical planes at x = −2 m and y = −2 m show iso-contours of gas temperature. Time t = 3 s.

Also, Fig. 5 suggests that the diffusion flame is strongly affected by buoyant motions as it assumes the classical shape of a buoyant puff (i.e. the classical shape of a fireball). Figure 6 shows the flame structure at time t = 3 s. The flame is seen to continue spreading in both horizontal and vertical directions, and this spread results in significant flame-wall interactions: the deflagration interacts with the vertical wall located at x = 2 m, while the fireball impinges on the ceiling located at z = 3 m. A comparison between Figs 5 and 6 suggests that the deflagration propagates at a speed of approximately 1−1.5 m s−1, while the fireball rises at a speed larger than 3 m s−1. The buoyant flow acceleration and the impingement of the fireball onto the ceiling come out as energetic dynamical events that are powerful enough in the simulation to force a transition to turbulence.

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Figure 7: See Fig. 6 for details. Time t = 3.5 s.

Figure 8: See Fig. 6 for details. This plot shows the non-premixed combustion component of HRR. Time t = 8 s.

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Figure 7 shows the flame structure at time t = 3.5 s, close to the moment of maximum HRR. Compared to Fig. 6, the fireball has grown further in size and has now spread across the entire ceiling; at this time, the flame zone (including both the deflagration and fireball regions) is seen to occupy most of the compartment volume. Figure 7 also displays iso-contours of gas temperature plotted in the rear vertical planes of the compartment. These iso-contours give a convenient graphical representation of the location of the hot ceiling layer. The elevation of the ceiling layer is seen to range from z = 1.5 to 2 m. For 4 s ≤ t ≤ 7 s, the topology of the combustion zone becomes complex and chaotic-like, and the individual premixed and non-premixed components do not show any clear spatial structure. Figure 8 shows the diffusion flame structure at time t = 8 s. The deflagration is not shown because its intensity has now weakened considerably, and there is only residual premixed burning. At t = 8 s, light-back has already occurred and the diffusion flame assumes the classical cone-like shape of a flame attached above a fuel source. Furthermore, the iso-contour plots of gas temperature suggest that the ceiling layer has now descended to the floor level. The diffusion flame may be sustained for some time in this environment but will ultimately be affected by air vitiation and oxygen starvation effects. While not part of the present simulation (since as mentioned previously, the flame extinction model is not activated), combustion in case C1 would ultimately lead to flame extinction. In summary, we have presented in this section a CFD-based treatment of a model problem corresponding to ignition and transient combustion of a fuel vapor cloud. The simulation is shown to provide physically sound results as well as original insights into the complex flame dynamics. While much work remains to be done, we feel that a description of ignition using PPC concepts offers a number of new exciting possibilities for CFD applied to fire and explosion safety problems. Possible areas of application include confined or unconfined, flash fires, and fireball events, as well as backdraft events. Finally, while we intend to leave this section on a positive note, it is worth emphasizing again that ignition and early fire growth still remain challenging topics for CFD solvers, because of both the wide range of possible scenarios, and the small length scales that are typically involved.

3 Smoke filling and pre-flashover fire spread We now consider the fire dynamics occurring during stage 2 (see Section 1). In this section, we assume that ignition has already occurred and start the discussion from an initial fire of moderate size. We examine the feasibility of a CFD approach to simulate fire growth, i.e. the smoke build-up and the fire spread from a localized initial source to surrounding flammable objects and materials. We consider the case of a fire growing in a room featuring a single doorway opening (Fig. 9). The initial flame is fuelled by a liquid heptane fuel pan located at floor level in the center of the room; the combustion is then observed to spread to the surrounding floor made of wood. It is worth emphasizing that the growth of a fire corresponds to a complex closed-loop heat feedback mechanism in which the MLR that results from the evaporation and/or pyrolysis processes is driven by the gas-phase thermal environment. The closed-loop mechanism may be explained as follows. Let us consider a certain amount of combustible mass released by a fuel source; this combustible mass mixes with ambient air, then burns and contributes to raise the gas temperatures; the hot gases in turn generate a certain level of thermal loading (due to radiative and/or convective heat transfer) on surrounding liquid/solid surfaces (including back to the fuel source), and thereby contribute to sustain or intensify the burning process. Thus, in contrast to classical turbulent combustion configurations, in which the fuel mass flow rate into the combustion zone is controlled and considered as an input quantity (these configurations are

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Figure 9: Case C2 corresponding to an initial fire located in the center of the room and spreading to the surrounding wooden floor. The room features a single doorway opening.

representative of engine applications), the MLR in fire safety applications is unknown and must be considered as a problem variable. A CFD treatment of case C2 (Fig. 2) requires: a fuel source model to describe MLR; a combustion model; a soot formation model; a radiative heat transfer model; and a convective heat transfer model. Soot formation is an important ingredient as soot concentrations often dominate the thermal radiation properties of the compartment gases. Also note that convective heat transfer is an ingredient that is often overlooked: while the contribution of convective heat transfer is small when the flame is located far from fuel sources, it becomes large in the presence of significant flame-wall interactions (for instance when the flame is located in the boundary layer of flammable walls). Section 3.1 below presents a brief review of the different sub-models used in FDS for MLR, combustion, soot, thermal radiation, and convective heat transfer. The overall performance of the models is then evaluated in a numerical simulation of case C2; results are presented and discussed in Section 3.2. 3.1 Modeling of fire spread We start with the fuel source model and consider the case of a non-charring flammable solid material. In FDS (version 4.0), the fuel source model uses a classical gas−solid heat transfer formulation in which the MLR becomes a function of the gas-to-solid wall heat flux [4, 5].

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The formulation treats the solid wall as a thermally thick volume, and evaluates the in-wall temperatures from the solution of a (one-dimensional) heat equation (in the wall normal direction) and a modified boundary condition at the gas−solid interface: q w′′ = − ks

∂TS + m F′′∆H v ∂n

(12)

w is the wall heat flux, ks is the solid wall thermal conductivity, Ts is the solid wall temwhere q′′ F is the fuel mass loss rate per unit fuel surface perature, n is the normal distance to the wall, m′′ area, and ∆Hv is the heat of gasification (per unit mass of fuel). Equation (12) represents a balance between the gas-to-solid heat flux due to convection and radiation, the conduction heat flux responsible for heat transfer to the wall interior, and the rate of energy consumption associated with the endothermic fuel gasification process. In this description, fuel gasification is assumed to take place at the gas−solid interface. To complete the formulation, the MLR in eqn (12) is treated F (Ts) (this function is empirically determined and typias a known function of temperature, m′′ cally takes an Arrhenius form), so that eqn (12) may be interpreted as a boundary condition for Ts [4, 5]. The outputs of the model are the wall surface distributions of temperature, MLR, and normal transpiration velocity. Let us now consider the combustion and soot formation models. The diffusion flame model adopted in FDS (version 4.0) has been previously described in Section 2.1.2 [4−6]. The soot formation model uses a crude description based on the concept of a soot yield (defined as the mass of soot produced per unit mass of fuel consumed by combustion). This model is cost-effective but also unrealistic in many ways. For instance, it assumes a strong correlation between parent fuel mass and soot mass; it also assumes a strong correlation between soot mass and mixture fraction; both assumptions are not supported by experimental studies. The formulation of more accurate, yet tractable, soot models remains an open problem in CFD descriptions of combustion systems (see ref. [46] for a general review, and ref. [47] for an example of current modeling efforts oriented towards fire safety applications). Next, we briefly discuss the thermal radiation model. In FDS, thermal radiation transport is treated via a solution of the radiative transfer equation (RTE) [4−6]. The RTE is formulated assuming a nonscattering gas, and using either a gray model (a low-resolution approach in which radiation properties are integrated over the electromagnetic spectrum) or a wide band model (a higher-resolution approach that considers wavelength dependencies). The RTE also incorporates a simple model for subgrid-scale fluctuations in radiation intensities (and in particular a simple model for subgrid-scale flame−radiation interactions). In FDS, the RTE is solved using a finite volume method [4−6]. Finally, we turn to the convective heat transfer model. In FDS, convective heat transfer is simply described using standard correlations developed for flat plate boundary layers. This approach is questionable and does not apply to the case of flame−wall interactions. The formulation of more accurate wall boundary layer models remains an open problem in CFD descriptions of fire configurations (see refs [48, 49] for examples of current modeling efforts oriented towards fire safety applications). 3.2 Simulation of fire spread (without flashover) The numerical configuration presented in Fig. 2 corresponds to a square-shaped liquid fuel pan of size D × D = (0.5 × 0.5) m2, located at floor-level, and releasing heptane fuel vapors into a ◊ vented compartment. The MLR in the heptane pool is prescribed: m◊ F ~ ~ 11.2 g/s and (mF∆HF) = ◊ 500 kW, for t ≤ 1,200 s; and mF = 0, for t >1,200 s. Heptane depletion is simply simulated by deactivating fuel evaporation after an arbitrarily selected dry-up time, t = 1,200 s.

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The compartment is 4 m long, 4 m wide, 3 m high, and is vented by a single doorway. The doorway is 1 m wide, 2 m high, and located in the middle of the x = 2 m vertical wall. The ceiling and vertical walls are made of concrete; the floor is made of pine wood. Pine wood is treated as a flammable non-charring material; the parameters of the fuel release model are: an ignition temperature equal to 390°C; a heat of gasification ∆Hv,wood = 2.5 MJ kg−1; and a heat of combustion ∆HF,wood = 12.044 MJ kg−1. The thickness of the floor is assumed large and effects associated with wood burn-out are neglected. The computational grid is a uniform cubic mesh; the mesh size is (40 × 40 × 30), which corresponds to 48,000 grid cells; the grid cell size is ∆ = 10 cm. The simulation is performed using a decomposition of the computational domain into five non-overlapping blocks, and using a parallel network of five processors. Note that the grid size used in case C2 is 64 times smaller than that used in the simulation of case C1. The decision to resort to a lower grid resolution is motivated by the need to simulate longer physical time scales in case C2. For instance, fire growth occurs over time scales of the order of several minutes (or even several tens of minutes). In addition, heptane depletion occurs at 20 min. It is interesting to simulate the response of the compartment fire to the drying up of the heptane pool as this serves to distinguish between two regimes: a first regime in which the fire has intensified slightly but remains sustained by the ignition source; a second regime in which the fire has intensified considerably and has reached self-sustaining conditions (i.e. conditions that do not depend on the continued presence of the ignition source). In order to observe the post-dry-up regime, the simulation duration is chosen to be 1,800 s. The decision to resort to a lower grid resolution in case C2 will result in a loss of accuracy. The magnitude of this loss is unknown and was not investigated in the present series of simulations. While considered beyond the scope of the present discussion, a methodical evaluation of computational grid requirements remains an open problem in CFD descriptions of compartment fires, particularly for scenarios featuring small scale physics (for instance, flame ignition or flame extinction). Figure 10 presents the time variations of the global HRR and MLR inside the compartment. It is seen that in the course of the simulation, the fire approximately doubles in size (up to 1 MW), while the MLR is multiplied by a factor 5 (up to 50 g s−1) (note that the difference between the variations in HRR and MLR is simply due to the relatively low value of the heat of combustion of

Figure 10: Time variations of the global heat release rate (left) and the fuel mass loss rate (right) inside the compartment (case C2).

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wood; for instance, 1 kg of gas produced by wood pyrolysis has approximately the same energetic value as 0.25 kg of heptane vapors). While significant, this increase in fire size remains sub-critical in several ways. First, the combustion remains well-ventilated; we estimate that at time t = 1,200 s, the value of the compartment global equivalence ratio is 0.25. Second, when the heptane pool dries up, the fire experiences complete extinction; the fire regime in case C2 may therefore be described as assisted burning. Figure 11 presents a view of the flame shape and location at time t = 1,000 s. In this figure, the flame is identified as the iso-level surface corresponding to stoichiometric mixture fraction. The flame is tilted towards the back wall, both because of the blowing effect of the incoming air, and because of fire spread to the floor region located between the liquid fuel pan and the back wall. Figure 11 also displays iso-contours of gas temperature plotted in the rear vertical planes of the compartment. These iso-contours give an instructive visual representation of the two-layer stratification of the compartment gases. The elevation of the ceiling layer is seen to range from z values 0.5 to 0.8 m. Additional insight into the fire spread mechanism may be obtained by examining the variations w and the wall surface temperature Tw. Figure 12 presents the time variaof the wall heat flux q′′ w and Tw, as measured by four different numerical probes located at floor level at (x; y) tions of q′′ = (−0.8; 0), (−1.3; 0), (−1.3; −1.3) and (−1.3; 1.3) (in the figures, these locations are identified as

~ Figure 11: Instantaneous iso-contour plot of stoichiometric mixture fraction, Z = Zst. The rear vertical planes at x = −2 m and y = −2 m show iso-contours of gas temperature. Case C2, time t = 1,000 s.

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Figure 12: Time variations of the wall heat flux (left) and wall surface temperature (right) measured at four different floor-level probe locations (case C2). targets 1, 2, 3, and 4, respectively). Figure 12 reveals large inhomogeneities in the intensity of the thermal feedback from the growing fire to the wooden floor: floor locations in the immediate w ≥ 50 vicinity of the fire (targets 1 and 2 in Fig. 12) experience high levels of thermal loading ( q′′ kW/m2), whereas floor locations that are adjacent (targets 3 and 4) experience considerably lower w ≤ 5 kW/m2). Furthermore, the temperature variations plotted in Fig. 12 indicate an levels ( q′′ early ignition (i.e. Tw ≥ 390°C) at target 1, a delayed ignition at target 2, and no ignition at targets 3 and 4. Note, however, that while ignition is not achieved at targets 3 and 4, this result is not general as it depends on the heptane pool dry-up time. The floor temperatures at targets 3 and 4 are increasing functions of time, and a scenario that would allow for longer-duration fires is likely to lead to a wider floor burning region. In summary, we have presented in this section a CFD-based treatment of fire growth based on a fuel release model and a variable MLR formulation. The simulation corresponds to pre-flashover fire dynamics, i.e. to a fire regime characterized by well-ventilated combustion, a two-layer stratification, and assisted burning conditions. We present in the next section a different case that corresponds to flashover dynamics and a fire regime characterized by under-ventilated combustion and self-sustaining burning conditions.

4 Flashover and transition to under-ventilated combustion We now consider the fire dynamics occurring during stages 3 and 4 (see Section 1) and examine the feasibility of a CFD approach to simulate flashover and post-flashover fire conditions. We consider again the case of a fire growing in a room featuring a single doorway opening (Fig. 9). Like in case C2, the initial flame is fuelled by a liquid heptane fuel pan located at floor level in the center of the room; in contrast to case C2, the combustion is observed to spread not only to the surrounding floor, but also to the vertical walls and ceiling, all made of wood. In the following, we refer to this new case as case C3. The description of flashover and post-flashover fire conditions is a challenging task, since it combines the difficulties found in the description of fire growth (as discussed in the previous section) with those found in the description of under-ventilated combustion. The modeling difficulties found in the description of under-ventilated combustion include flame extinction due to air vitiation (Section 1) and the emission of products of incomplete combustion (i.e. carbon monoxide, unburnt hydrocarbons,

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soot, hydrogen) [1−3, 50−58]. We limit the scope of our discussion here to flame extinction effects (see refs [59−61] for examples of recent or current modeling efforts oriented towards modeling of carbon monoxide emissions). The FDS model for flame extinction is presented in Section 4.1. The overall performance of the model is then evaluated in a numerical simulation of case C3; results are presented and discussed in Section 4.2. 4.1 Modeling of under-ventilated combustion We start from the diffusion flame model presented in Section 2.1.2 and summarized in eqns (8) and (9). The model assumes infinitely fast chemistry and needs to be modified in order to treat air vitiation effects and flame extinction due to oxygen starvation. Following refs [5] and [61], the reduction in flame strength resulting from smoke-air mixing is incorporated into the model via the introduction of a flame extinction factor (FEF):  Y∞   n 2 qd =  F   r t ∇Z  d ( Z − Z st ) × ∆H F × (1 − FEF) 1 Z Sc −    st  t

(13)

where FEF is the locally defined probability of finding inactive flame elements in a given LES computational grid cell: FEF = 0 for a fully burning flame; FEF = 1 for a fully extinguished flame. The model formulation for FEF uses the following ingredients: a critical flame temperature Tc, below which extinction is predicted to occur; a lower oxygen index, that characterizes limiting oxygen levels for flames supplied with diluted air at ambient temperature, T∞ = 300 K; and a model for the flame temperature Tst. The critical flame temperature model may be viewed as a simplified version of a classical description based on critical values of the scalar dissipation rate [21−23]; we use Tc ≈ 1,700 K [62]. The lower oxygen index is also described as an empirical input quantity and is specified as YO2 ,c ≈ 0.17 (mass fraction) [54, 55]. The flame temperature model is based on a classical Burke−Schumann expression: Tst = T1

YO2 ,2 rsYF,1 + YO2 ,2

+ T2

rsYF,1 rsYF,1 + YO2 ,2

+

∆H F YF,1YO2 ,2 c p rsYF,1 + YO2 ,2

(14)

where T1 and T2 are the temperatures in the fuel and oxidizer streams feeding the flame, YF,1 and YO2 ,2 are the mass fractions of fuel and oxygen in those feeding streams, rs is the stoichiometric oxygen-to-fuel mass ratio, and cp is the specific heat of the reactive mixture at constant pressure (assumed constant). Equation (14) provides a useful expression of Tst as a function of the oxidizer stream properties YO2 ,2 and T2. Next, we combine the flame temperature model in eqn (14) with the concepts of a critical temperature Tc and an oxygen limit YO2 ,2 . We get after some algebraic manipulations: Tst = Tc + (Tc − T∞ )

rsYF,1 rsYF,1 + YO2 ,2

 YO2 ,2 (Tc − T2 )  −    YO2 ,c (Tc − T∞ ) 

(15)

where T1 = T∞ has been assumed. This expression may now be conveniently used to construct a flammability diagram in terms of the vitiated air variables YO2 ,2 and T2 (see Fig. 13). In Fig. 13, flammable conditions correspond to super-critical flame temperatures, i.e. flame temperatures such that Tst ≥ Tc, or ( YO2 ,2 / YO ,c ) − (Tc − T2)/(Tc − T∞) ≥ 0, whereas non-flammable conditions correspond 2

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Figure 13: Flammability diagram for a pure-fuel/vitiated-air diffusion flame as a function of the vitiated air properties YO2 ,2 and T2, as predicted by the criterion in eqn (16). The white region corresponds to FEF = 0; the gray region to FEF = 1. to sub-critical flame temperatures, i.e. flame temperatures such that Tst ≤ Tc, or ( YO2 ,2 / YO ,c ) − 2 (Tc − T2)/(Tc − T∞) ≤ 0. And the following binary expression for the flame extinction factor is obtained:  (T − T2 ) YO2 ,2  FEF = H  c −   (Tc − T∞ ) YO2 ,c 

(16)

where H is the Heaviside function, H(x) = 1 if x ≥ 0, H(x) = 0 if x < 0. Equation (16) is a closure model for FEF, provided that the variables YO2 ,2 and T2 are known. Note that the oxidizer stream properties correspond to unresolved conditional information, and should ~ not be confused with the LES grid-resolved oxygen mass fraction and temperature, YO2 and T . The estimation of YO2 ,2 and T2 in eqn (16) is based on a simple search algorithm applied to all computational grid cells in which heat release is taking place. The search algorithm interrogates neighboring cells and identifies among them the cells that are both non-reacting (q◊d = 0) ~ ~ and located on the lean side of the flame (Z ≤ Zst); the values of YO2 and T in those oxidizer cells are then used to estimate the vitiated air conditions at the LES flame location. With this scheme, eqns (13) and (16) provide an extended HRR model. 4.2 Simulation of fire spread (with flashover) The numerical configuration used in case C3 is identical to that used in case C2 (see Section 3.2) with the difference that the vertical walls and ceiling are now made of wood (instead of concrete).

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This change clearly increases the risk of flashover. For instance, the flame in case C2 shows signs of interactions with the back wall (Fig. 11); consequently, as we modify the configuration and change the material of the back wall from inert to flammable, we can anticipate a larger fire size and a faster fire growth. In order to highlight the impact of air vitiation on the simulated fire dynamics, we perform the simulation of Case C3 twice, with and without the (optional) flame extinction model. We start our discussion with the results obtained without flame extinction (i.e. using a formulation in which FEF remains equal to 0). Figure 14 presents the time variations of the global HRR and MLR inside the compartment. These variations are dramatically different from those plotted in Fig. 10. It is seen that the fire rapidly increases in size at time t ≈ 500 s and reaches a steady state value of 4 MW, while the MLR is multiplied by a factor 80 (up to 800 g s−1). This increase in fire size corresponds to super-critical conditions and transition to flashover. First, the combustion clearly becomes under-ventilated: complete burning of the fuel produced by wood pyrolysis would correspond to a fire size of approximately 10 MW (including the heptane pool fire); the observed level of 4 MW indicates that the compartment global equivalence ratio is approximately 2.5. Second, when the heptane pool dries up, the fire experiences no noticeable change; the fire regime in case C3 may therefore be described as self-sustaining burning. Figure 15 presents a view of the flame shape and location at time t = 1,000 s. In contrast to the flame topology observed in case C2 (Fig. 11), the flame is now detached from the fuel pan and stabilized close to the doorway, i.e. close to the air stream coming into the compartment. This provides further evidence of the under-ventilated conditions that are prevalent in this post-flashover fire regime. Figure 15 also displays iso-contours of gas temperature. These iso-contours indicate that the ceiling layer has now descended to the floor and fills the entire volume of the compartment. Additional insight into the fire spread mechanism may be obtained by examining the varia w and wall surface temperature Tw, at selected probe locations tions of the wall heat flux q′′ (Fig. 16). The probes are identical to those used previously in case C2 (Fig. 12). We see that the pre-flashover dynamics in case C3 is similar to that observed in case C2. In contrast to case C2, however, ignition is observed at targets 3 and 4, and takes place at the time of flashover. After w and Tw change significantly and become quite homogeneous; flashover, the variations of q′′

Figure 14: Time variations of the global heat release rate (left) and the fuel mass loss rate (right) inside the compartment. Case C3, without flame extinction.

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~ Figure 15: Instantaneous iso-contour plot of stoichiometric mixture fraction, Z = Zst. The rear vertical planes at x = −2 m and y = −2 m show iso-contours of gas temperature. Case C3, without flame extinction. Time t = 1,000 s.

Figure 16: Time variations of the wall heat flux (left) and wall surface temperature (right) measured at four different floor-level probe locations. Case C3, without flame extinction.

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Figure 17: Time variations of the global heat release rate (left) and the fuel mass loss rate (right) inside the compartment. Case C3, comparison between results obtained with (solid line) and without (symbols) flame extinction.

this change is driven both by the smoke filling process and by the migration of the flame to the doorway. We now turn to a brief discussion of the results obtained in the simulation that includes flame extinction (i.e. using a formulation with a variable FEF, see Section 4.1). Figure 17 presents a comparison of the time variations of HRR and MLR obtained with and without flame extinction. We see that air vitiation has a significant impact on the fire dynamics. As a result of flame extinction, both HRR and MLR exhibit large fluctuations and reduced intensities during a long transient period that lasts for approximately 15 min (500 s ≤ t ≤ 1,400 s). During that period, the fire size is reduced by a factor 2, and the MLR is reduced by a factor 2−2.5 (so that interestingly enough, during this transient regime, the compartment global equivalence ratio is quasi-stoichiometric). After time t = 1,400 s, the flame successfully migrates to the doorway; once located at the doorway, the flame is no longer exposed to vitiated air, and consequently flame extinction is no longer observed. In summary, we have presented in this section a CFD-based treatment of a model problem corresponding to fire growth, flashover, and the subsequent transition to under-ventilated combustion. In both Sections 3 and 4, the MLR is treated as a problem variable and is determined by the heat feedback from the gas-phase thermal environment. While the results are encouraging and demonstrate the feasibility of a variable-MLR approach, much work remains to be done. For instance, the following technical areas are areas in which physical models remain under-developed: fuel pyrolysis, soot formation, flame−radiation interactions, and flame−wall interactions.

5 Water-based fire suppression and fire control/extinction Finally, we discuss in this section the dynamics of a suppressed fire occurring during stage 4 (see Section 4). Just as in Section 4, we assume that ignition has already occurred creating a fire of moderate size. This fire is allowed to spread and grow; however, two fire suppression devices have been placed in the compartment as shown in Fig. 18. We refer to this suppressed fire simulation as case C4. The fire suppression devices are fitted with thermal fire detectors and are placed on either side of the door at (x; y) = (0.5; −0.5), (0.5; 0.5) m, and 0.2 m below the ceiling. The fire suppression devices are medium pressure water mist nozzles having a fire detector response time index

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Figure 18: Case C4 corresponding to an initial fire located in the center of the room and spreading to the surrounding wooden floor and walls. The room is protected with two overhead fire suppression nozzles.

RTI = 150 (ms)1/2. A K-factor of 3.5 l min−1 bar−1/2 and operating pressure of 12.8 bar are also specified for both nozzles. In the simulation, we allow the fire to grow and to activate the fire suppression nozzles.We then evaluate the fire dynamics during suppression giving special attention to the important mechanisms contributing to fire suppression and to the ability of the spray to control the fire. Water-based fire suppression systems are of particular interest because of their extensive use in a variety of fire protection applications. These systems are required to perform effectively over a wide range of extremely harsh and complex operating conditions. The fire suppression performance of water sprays depends on the initiation, formation, dispersion, and surface cooling characteristics of the sprays created by these devices. The elementary suppression mechanisms for these sprays are clearly understood; however, detailed physical models to describe and predict their behavior are only now emerging due to the complex transport mechanisms associated with the fire−spray interaction. It is possible to simulate the gas (or continuous phase) behavior of fires with a high degree of fidelity as demonstrated in the previous three sections. Yet in fire suppression problems, the strong coupling between the continuous phase and the dispersed phase, evidenced by the very existence of suppression, makes accurate dispersed phase models essential. The discussion included in Section 5.1 introduces the important physical models for fire suppression including models for spray activation, atomization, dispersion, and surface wetting. Results from a suppressed fire simulation using these models are then evaluated in Section 5.2.

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5.1 Models for water-based fire suppression 5.1.1 Spray activation The activation time for linked suppression nozzles is well-characterized in the literature and in the fire protection engineering practice [63, 64]. In many fire suppression nozzles, water flow is initiated when a temperature sensitive deformable plug (i.e. the link) is displaced upon reaching its activation temperature, Ta. The predictive model for activation of these devices is based on a simple lumped-capacitance transient heat transfer analysis, considering the heat stored in the nozzle activation link, the convective heat transfer from the hot gases to the link, conduction losses from the link to the sprinkler mount, and evaporative cooling from droplets originating from adjacent nozzles: |u| dTl C C = (Tg − Tl ) − (Tl − Tm ) − 2 b | u | dt RTI RTI RTI

(17)

where T1 is the nozzle link temperature, Tg is the local gas temperature, and Tm is the nozzle mount temperature typically taken as ambient. The RTI groups the physical characteristics of the nozzle link and quantifies its thermal inertia. Making use of the RTI, and only considering convection, the activation time ta is given by: ta =

RTI  Tg − Tm  ln   u  Tg − Ta 

(18)

The ratio of RTI divided by the square root of the gas velocity represents the time constant of the system. The activation time is delayed by higher activation temperatures, large RTI (thermal inertia), or small convection velocities. The activation time in eqn (18) is of course increased according to the conduction losses and evaporative cooling effects quantified by the empirically derived conduction and evaporation C-factors. Although the RTI can vary with nozzle orientation, RTI, C, and C2 are considered nozzle properties and are specified as such in the CFD simulation. 5.1.2 Modeling atomization for fire suppression sprays Upon activation, the suppression nozzle injects water into the room. This water is introduced as a continuous jet; however to improve the dispersion of this water, the nozzle acts to break this continuous volume of fluid into small discrete drops forming a spray. This process is referred to as atomization. Atomization models use inlet conditions (including injector geometry) for the injected volume of fluid to predict the initial drop size, velocity, and location of the spray. The initial specification of the spray is important for accurately predicting the dispersion of the drops and ultimately the suppression of the fire. An atomization model has been formulated specifically for fire suppression devices [65, 66] based on the atomization stages illustrated in Fig. 19 and listed below: (1) surface interaction with the deflector resulting in a radially expanding sheet; (2) wave instabilities and fragmentation of the radially expanding sheet resulting in ligaments; (3) wave instabilities and fragmentation of the ligaments resulting in drops. The modeling approach introduced in this section addresses each stage of the atomization process with physics-based sub-models. The velocity, U, and thickness of the liquid sheet, hdef, are critical parameters that govern the atomization process. If the viscous interaction between the deflector and the impinging jet is

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Jet

Growth of Waves

Deflector Sheet Formation

Sheet → Ligament

Ligament → Drop

Figure 19: Illustration of atomization physics for impinging jet suppression nozzles (left) and the predicted initial drop size distribution for the suppression nozzles used in this study (right).

assumed to be small in an ‘ideal’ sprinkler without tines, the fluid velocity along the deflector is easily determined from the injection pressure, U = (2∆p/r1)1/2, and the thickness of the film at the edge of the deflector is given by: hdef =

K ∆p1/ 2 πDdef U

(19)

Alternatively, the effects of viscous interactions with the deflector can be estimated using free surface impinging jet theory proposed by Watson [67]. This viscous formulation will result in a somewhat smaller velocity and a correspondingly larger sheet thickness at the deflector edge [65, 66]. The central mechanism for atomization in water-based suppression injectors is the breakup of the liquid sheet formed by the injector into ligaments. To describe the liquid sheet breakup process, the wave instability concept is used which assumes that the disintegration of a liquid sheet occurs when the waves imposed by the surrounding atmosphere reach a critical amplitude. This concept was used by Dombrowski [68] to describe the disintegration of viscous liquid sheets emanating from fan nozzles. In the present atomization model for fire suppression devices, the same concept is used assuming that waves persist and grow on the free surface of the unconfined expanding liquid generated by the deflector. The disintegration of the sheet occurs when the wave amplitude reaches a critical value. At this point, the sheet breaks forming ring shaped ligaments and later drops are produced as the ligaments disintegrate. In the atomization model, sinusoidal waves are assumed to travel on the surface of the liquid sheet. A force balance is performed on the undulating sheet considering inertial, pressure, viscous, and surface tension forces. After considerable reformulation and simplification, the force balance can be expressed in terms of the growth rate of the waves present on the liquid sheet [68]: 2

ml 2  ∂f  2( ra nU 2 − s n2 )  ∂f  =0   + n   − ∂t ∂t rl rl h

(20)

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where f is the dimensionless total growth of the wave, s is the surface tension, n is the wavenumber of the disturbance imposed on the liquid stream (n = 2π/l), l is the wavelength, ra is the air density, r1 is the liquid density, U is the velocity of the sheet (determined previously), h is the thickness of the liquid sheet, t is the time, and µ1 is the liquid viscosity. Because the wave with the maximum growth leads to the breakup of the sheet, the corresponding critical wavenumber is of interest. For inviscid analysis it can be shown that (ninv)crit = roU2/2s. Since the wavelength is inversely proportional to the wavenumber, the critical wavelength which leads to the breakup of the sheet increases as the liquid surface tension increases, but decreases as the air density or sheet velocity increases. With the knowledge of the most unstable wave, (ninv)crit, the sheet breakup time, tbu,sh can be determined by integrating eqn (20) with respect to time to find the time taken to reach a critical dimensionless amplitude, assuming that the sheet thickness and velocity are known. This critical dimensionless amplitude can be determined experimentally and has been found in other studies not to depend on operating conditions; however, it may depend on the general injector configuration [68]. A constant value (f = 12) is applied in this model [68]. The sheet is assumed to breakup into a ring-shaped ligament having a characteristic width equal to exactly 0.5lcrit. The ligament diameter and mass can be determined from this assumption. A more detailed viscous analysis can be found in Wu [65, 66]. The radially expanding sheet thickness and velocity are not only critical in determining the instability and breakup of the sheet, but also in determining the trajectory of the sheet and ultimately the initial ligament location. A trajectory for radially expanding sheets has been proposed by Ibrahim and McKinney [69] and is currently being incorporated into the atomization model [70]. In this model the continuity and simplified 2D momentum equations are solved in a curvilinear coordinate system along the sheet. The trajectory along with tbu,sh are used to determine the initial ligament location. Details of the trajectory model are not included in the present discussion, but can be found in ref. [70]. For the analysis of case C4, the sheet and ligament trajectories were specified. The ligaments formed from the sheet breakup are also unstable and subject to the growth of waves that lead to ligament fragmentation into drops. Weber [71] has analyzed the properties of these waves where surface tension forces predominate, the critical ligament breakup wave number can be calculated by: 1/ 2

ndlig

1  3 ml  = +  2 2( rl s dlig )1/ 2 

(21)

It is assumed that each fragment will have a length equal to the critical ligament break up wavelength and that these fragments will contract into a single droplet. Conserving fragment mass, the characteristic droplet diameter, ddrop, is: 3lcrit,lig  2/3  ddrop = dlig  2   

1/ 3

(22)

The number of drops that are formed after ligament breakup can be expressed as: N=

6mlig 3 rl πddrop

(23)

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determined by conserving mass between the ligament and the drops. Weber [71] also provides an expression for the breakup time as:  2r  t bu,lig = 24  l   s 

1/ 2

 dlig   2   

3/2

(24)

The distance that it takes for the ligaments to disintegrate into drops is easily calculated from the ligament velocity, U, and tbu,lig. The initial drop location, corresponding to the total distance the liquid travels until drops are formed, is given by starting at the initial ligament location and integrating from tbu,sh to tbu,lig assuming that the ligament travels along the same trajectory and at the same speed as the sheet at breakup. The initial spray drop size, ddrop, and initial spray location, rdrop, are defined by eqns (22) and (24), while the velocity (neglecting viscous interaction with the deflector) is given by U = (2∆p/ r1)1/2. These quantities are determined from the nozzle geometry (K, rd), injection pressure (∆p), surrounding flow gas phase conditions (ra, µa), and liquid properties (s, r1, µ1). It should be noted that for the current formulation, the velocity of the gas in the vicinity of the sheet was assumed to be zero; however, the velocity of the fire and even the spray induced flow would change the relative velocity of the sheet. This relative velocity could replace the sheet velocity in eqn (20). These atomization relationships provide characteristic initial spray conditions for a given nozzle geometry and injection pressure, fire condition, and liquid suppressant. Of course in real applications, a multitude of drops with different sizes are created. In order to model this behavior a stochastic analysis should be introduced as proposed by Rizk and Mongia [72]. Only the deterministic equations have been provided in this paper to illustrate the physics of the atomization process. Details of the stochastic model can be found in refs [65, 66]. A predicted example distribution has been provided in Fig. 19 showing a mass fraction histogram along with cumulative volume fraction curves. It should also be noted that preliminary validation of the atomization model has been performed through comparisons with sprinkler measurements [65]. Predicted and measured characteristic drop sizes for the initial spray demonstrated favorable agreement. By combining the atomization model and the particle tracking capabilities of CFD, the dispersion of distributed sprays can be simulated for suppression analysis. Note, however, that the atomization model presented in this Section is not yet fully coupled with FDS. Currently, the initial gas phase temperature and velocities are assumed and input into the atomization model for predictions of initial drop properties. But in a fully coupled model, the calculated gas phase properties in the atomization region and the relative velocity of the sheet would be continuously updated for recalculation of the transient spray development. Nevertheless, the atomization model developed in this chapter still reveals some interesting insight into the spray behavior; it can provide overall statistical quantities of the spray and it is able to provide spray inputs for CFD models. 5.1.3 Modeling spray dispersion The spray dynamics are strongly coupled with the continuous phase dynamics; therefore, equations for both phases should be solved simultaneously to obtain accurate solutions of the spray dispersion. The conservation equations of mass, momentum, and energy are typically solved using an Eulerian formulation for the continuous phase, while these equations are normally solved using a Lagrangian formulation for the dispersed phase [73]. The governing equations for the continuous phase are the well-known equations of continuum mechanics. The conservation equations for the drops can be determined from mass, momentum, and energy balances on the droplet assuming uniform properties [74]. These equations are integrated using a time marching approach starting from specified

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initial conditions. The Lagrangian formulation of the dispersed phase is very sensitive to these initial conditions and reliable estimates for the initial droplet characteristics are required for accurate dispersion predictions. The initial conditions for the drop dispersion equations are provided by the atomization model previously discussed. It should also be noted that drops are introduced into the flow at a specified interval representing drops of a certain size class, as it is impractical to track the entire myriad of droplets present in an actual spray. The acceleration of the droplet is described by the momentum conservation equation: d( md (ui )drop ) dt

=

md fdrag tv

(ui − (ui )drop ) + md gi

(25)

where md is the mass of the drop, fdrag is the friction factor describing the ratio of the drag coefficient to Stokes drag given by fdrag = (1 + 0.15 Rer0.687 ) [75], tv is the velocity response time given 2 by tv =r1 ddrop /18 µA, ui is the gas velocity, (ui)drop is the drop velocity, gi is the gravitational acceleration vector, Rer is the Reynolds number based on the relative velocity, rL is the liquid density, ddrop is the drop diameter, and µA is the gas viscosity. In turn, the acceleration of the air due to the drops must also be considered. This is accounted for by determining the reaction from the viscous force of all of the drops present in each computational cell and including this as a source term in the gas phase momentum equation. Figure 20 shows predictions from the atomization

Figure 20: Dispersion of the liquid water spray in a room without fire. Mass flux of water (kg s−1m−2) at the floor (z = 0). Case C4, without fire.

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and dispersion models in the absence of fire. The floor mass flux distribution from the two suppression nozzles is relatively concentrated toward the center of the room. This concentrated distribution is consistent with the full cone angle of 90° specified for the nozzles. In actual fire suppression applications, the evaporation of droplets must be considered. Droplet evaporation is described by the mass conservation equation: dmdrop dt

(

= Shπddrop rl Dv YH2 O,∞ − YH2 O,s

)

(26)

where the Sherwood number is Sh = hmddrop/Dv, hm is the convective mass transfer coefficient, Dv is the mass diffusivity, YH O,∞ is the mass fraction of water vapor in the gas environment, and 2 YH 2O,s is the mass fraction of water vapor at the droplet surface. YH2 O,s is given by the Clausius− Clapeyron equation: h M X H2 O,s = exp  L W R 

 1 X H2 O,s 1   −   ; YH2 O,s = X H2 O,s + (1 − X H2 O,s ) M A / M W  Tb Tdrop  

(27)

where R is the universal gas constant, Tb is the boiling temperature of the water, MA is the molecular weight of air, MW is the molecular weight of water, and Tdrop is the temperature of the drop. The heating of the droplet is described by the energy conservation equation: dTdrop dt

=

ddrop qR′′ Nu 1 Sh 1 Pr hL + (T∞ − Tdrop ) + (YH 2 O,∞ − YH 2 O,s ) 2 kA tT 2 tT 2 tT Sc cA

(28)

where Td is the droplet temperature; q′′ R is the net radiative heat flux to the drop; kA is the ther2 mal conductivity of the gas; tT = c1r1 ddrop /12kA is a characteristic heating time for the droplet; cl is the specific heat of the liquid; the Nusselt number is given by Nu = hTddrop/kA, where hT is the convective heat transfer coefficient; the Prandtl number is given by Pr = υA/aA; the Schmidt number is given by Sc = υA/Dv; T∞ is the gas temperature; hL is the latent heat of vaporization; and cA is the specific heat of the gas. In sprays, the radiative term is typically determined from gas phase control volume analysis employing Mie Theory and associated local drop number, drop size, and absorption coefficient distributions [76]. This analysis is not presented here, but the radiation model is used in the suppression simulation presented in Section 5.2. It should also be noted that the exchange of mass and energy due to evaporation must also be accounted for in the gas phase conservation equations and the details of treatment of these effects depends on the formulation of the gas phase equations. 5.1.4 Simulating surface wetting Once the water droplets reach solid surfaces exposed to the thermal radiation from the fire and to convective heat transfer from the hot gases, they provide evaporative cooling thus reducing the average surface temperature. By keeping the surface temperature low, pyrolysis of the solid materials is curtailed and the solid is protected. Since no more fuel becomes available for burning in this situation, the fire is contained and suppression is achieved. Heat exchange between hot surfaces and droplets is extremely complex and the physics is not explicitly handled in FDS. Currently, once droplets reach the surface they continue to be tracked and the drops are allowed to travel parallel to the surface in a random direction at a specified low velocity of 0.5 m s−1 [5]. The surface is indirectly cooled through modeled gas-droplet exchange

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near the wall. However for reacting surfaces, an empirical model has been included in FDS to account for the reduced burning rate due to wetting [5, 77]. This model applies a correction factor to the local ‘dry’ mass loss rate determined for the surface. The burning rate correction factor is a function of the surface materials sensitivity to water application and the local water mass flux at the surface. The correction factor is continuously modified based on the cumulative water mass flux. 5.2 Simulation of water-based fire suppression We now discuss a simulation including suppression effects (case C4) using the dispersed phase and suppression models previously described. The numerical configuration used in case C4 is identical to that used in case C3 (see Section 4.2), except for the presence of two water-based fire suppression nozzles (Fig. 18). The computational grid is also significantly finer; the mesh size is (80 × 80 × 60), which corresponds to 384,000 grid cells; the grid cell size is ∆ = 5 cm. The atomization model predicts a spray having a volume median diameter of dv50 = 221 mm as shown in Fig. 19. The previous dispersion analysis performed in a room without fire also revealed that the spray was relatively narrow as demonstrated in Fig. 20. In the fire configuration, a water mass balance indicates that the spray remains confined to a relatively narrow region with only 5% of the spray hitting the sidewalls and 95% of the spray reaching the floor. The small drop size of this spray allows it to be classified as a water mist. Yet, despite this relatively small drop size, the water mass balance revealed that droplet evaporation is negligible as the amount of water evaporated was too small to quantify. The simulated behavior is consistent with the medium pressure class of water mist nozzles modeled in this simulation. These nozzles are designed to have good fire penetration and floor wetting performance, and in doing so sacrifice some of the evaporative cooling performance. Nevertheless, the water is very effective at controlling the fire as shown in Fig. 21. The model predicts activation of the suppression nozzles at 90 s leaving plenty of margin to prevent flashover, as the flashover time in the uncontrolled fire (case C3) was approximately 500 s. In the suppressed fire, the heat release rate remains relatively constant at about 500 kW until the pilot flame is turned off at 1,200 s at which time the fire extinguishes. The extinction of the fire upon removal of the pilot indicates that the flame spread is not self-sustaining. The suppression nozzles easily overwhelm the fire generated by ignited regions of the floor. This behavior sharply contrasts the approximately 4 MW heat release rate of the uncontrolled fire after flashover. Based on the water mass balance, suggesting negligible evaporation, the primary suppression mechanism is wetting of the floor. The water delivered to the floor keeps the floor cool confining burning to a very small region as shown in Fig. 22. Although the fire spreads onto the floor from the pilot, the burning region is still quite small with a correspondingly small increase in the energy release rate even after 1,100 s. It should be noted that the previously discussed models are capable of capturing the gas phase evaporative cooling and oxygen depletion suppression mechanisms; however, they were not observed in this simulation. Perhaps, if the doorway ventilation was reduced or if the nozzles were activated very late in the fire (post-flashover), higher compartment temperatures and a larger smoke layer would have caused these suppression mechanisms to become important. In summary, we have presented in this section a CFD-based treatment of a model problem corresponding to a successful application of water sprays to fire spread control and flashover prevention. Fire suppression is achieved in this problem by fuel cooling (i.e. by wetting burning surfaces) and by flame spread inhibition (i.e. by pre-wetting adjacent combustible surfaces). This model problem illustrates the potential of a CFD approach for the design of fire suppression systems.

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Figure 21: Time variations of the global heat release rate inside the compartment. Comparison between results obtained for an uncontrolled fire (case C3) and for a fire with suppression (case C4). In case C4, nozzle activation occurs at time t = 90 s.

6 Conclusion This chapter presents a series of numerical simulations that correspond to several generic compartment fire configurations and serve to illustrate the performance and limitations of current CFD tools. The series includes: a problem corresponding to ignition of a fuel vapor cloud; a problem corresponding to fire growth from an initial liquid pool fire to surrounding wooden walls, with and without flashover; and a problem corresponding to control of fire growth by a water−mist system. The discussion is focused on a presentation of combustion modeling concepts and simulation results from what may be considered as feasibility tests. It does not include comparisons with experimental data (i.e. validation tests). For some of the models that are presented, some preliminary or partial validation tests have been performed; for others, validation tests are planned or are in progress. In our view, the computational models for ignition, flame spread, under-ventilated combustion, and fire suppression are still in early stages of development, and while clearly desirable, a more complete discussion including validation tests is still premature. The discussion suggests that many of the challenges found in CFD-based modeling of compartment fire configurations are similar to those found in modeling of thermal engine configurations. Technical areas in which there is significant overlap between fire modeling and engine modeling include the following topics: turbulence, turbulent fuel-air mixing, turbulent combustion (including multi-mode combustion), pollutant formation (including soot particles), and convective heat transfer (including flame−wall interactions).

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Figure 22: Floor burning area in suppressed fire case. Case C4, time t = 1,100 s.

Yet, it is important to recognize that the challenges found in modeling of compartment fires are also significantly different from those found in modeling of engine combustion. Technical areas that are specific to fire modeling include the following topics: fuel pyrolysis (including descriptions of both the MLR and fuel composition), buoyancy-driven turbulence (including effects of low-to-moderate flow Reynolds numbers), fuel-rich (i.e. under-ventilated) combustion, and radiative heat transfer (including flame−radiation interactions). Additional challenges come from the wide range of possible scenarios (this is particularly true for ignition), from the long time scales that are typically associated with fire events (from minutes to hours), and from the thermal coupling with the building structure. While much remains to be done to add fidelity and realism to CFD simulations of compartment fires, current capabilities available to fire protection engineers, are already routinely used for design and analysis tasks, and are responsible for profound changes in the FPE professional practice. More changes may be expected as the domain of application of CFD-based fire modeling continues to expand, as the FPE work force becomes better trained in CFD tools and concepts, and as scientific computing technology continues to enjoy sustained growth. Areas of high potential for CFD-based fire modeling include: performance-based design, forensic applications, fire-fighter training, sensor-driven real-time emergency management, and risk analysis.

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Acknowledgments This work was supported in part by the US National Institute of Standards and Technology, Building and Fire Research Laboratory. Fruitful interactions with Drs K. McGrattan and A. Hamins are gratefully acknowledged.

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