Modeling of Combustion Systems A Practical Approach
Joseph Colannino
Boca Raton London New York
A CRC title, part of the Taylor & Francis imprint, a member of the Taylor & Francis Group, the academic division of T&F Informa plc.
© 2006 by Taylor & Francis Group, LLC
Published in 2006 by CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2006 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group No claim to original U.S. Government works Printed in the United States of America on acid-free paper 10 9 8 7 6 5 4 3 2 1 International Standard Book Number-10: 0-8493-3365-2 (Hardcover) International Standard Book Number-13: 978-0-8493-3365-1 (Hardcover) Library of Congress Card Number 2005053146 This book contains information obtained from authentic and highly regarded sources. Reprinted material is quoted with permission, and sources are indicated. A wide variety of references are listed. Reasonable efforts have been made to publish reliable data and information, but the author and the publisher cannot assume responsibility for the validity of all materials or for the consequences of their use. No part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www.copyright.com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC) 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe.
Library of Congress Cataloging-in-Publication Data Colannino, Joseph, 1957Modeling of combustion systems : a practical approach / by Joseph Colannino. p. cm. Includes bibliographical references and index. ISBN 0-8493-3365-2 (alk. paper) 1. Combustion chambers--Mathematical models. I. Title. TJ254.7.C65 2006 621.402’3--dc22
2005053146
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Dedication
First, to Judy
My children and family
My father,
Frank Colannino, 12-May, 1923 to 4-August, 2004†
†
Ed udii una voce dal cielo che diceva: “Scrivi questo: Da adesso saranno beati quelli che moriranno nel Signore! ‘Sì’, dice lo Spirito, ‘perché possono riposare dalle loro fatiche, e le loro opere li seguiranno in cielo!’”(Apocalisse 14:13, La Parola è Vita, © 1997, International Bible Society)
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Contents List of Tables List of Figures Concordance of Nomenclature About the Author Prologue
1 1.1
1.2
1.3
1.4
1.5
Introduction to Modeling .............................................................. 1 Model Categories...........................................................................................1 1.1.1 Model Validation...............................................................................2 1.1.2 Fundamental Theoretical Models...................................................2 1.1.3 Simulations.........................................................................................2 1.1.4 Semiempirical Models ......................................................................3 1.1.5 Dimensionless Models......................................................................3 1.1.6 Empirical Models ..............................................................................3 1.1.7 Problems with Post Hoc Models .....................................................4 Kinds of Testing .............................................................................................4 1.2.1 No Physical Testing ..........................................................................4 1.2.2 Scale Testing .......................................................................................5 1.2.3 Full-Scale Testing...............................................................................5 Analytical Methods .......................................................................................5 1.3.1 Qualitative Analysis..........................................................................6 1.3.2 Dimensional Analysis.......................................................................8 1.3.3 Raleigh’s Method ..............................................................................9 1.3.3.1 Cautions Regarding Dimensional Analysis ................ 11 1.3.4 Function Shape Analysis................................................................15 1.3.5 The Method of Partial Fractions...................................................18 1.3.5.1 Limitations of Function Shape Analysis......................22 Perceiving Higher Dimensionality ...........................................................23 1.4.1 A View from Flatland .....................................................................23 1.4.2 Contour Surfaces .............................................................................24 1.4.3 Orthogonal Directions ....................................................................26 1.4.4 Visualization with Cubic Regions ................................................26 1.4.5 The Use of Color .............................................................................28 Basic Data Classifications ...........................................................................29 1.5.1 Level of Scale ...................................................................................29 1.5.2 Data Quality.....................................................................................32 1.5.3 Planned Experiments......................................................................32 1.5.4 Unplanned Experiments ................................................................33 1.5.5 Source Classifications .....................................................................33 1.5.6 Functional Classifications ..............................................................33
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A Linear Algebra Primer ............................................................................35 1.6.1 Matrix Addition...............................................................................35 1.6.2 The Transpose Operator.................................................................36 1.6.3 Multiplication by a Constant ........................................................37 1.6.4 Matrix Multiplication .....................................................................37 1.6.5 Distributive Property of Multiplication over Addition ............38 1.6.6 Symmetric Matrices ........................................................................39 1.6.7 The Identity Matrix.........................................................................40 1.6.8 The Unity, Zero, and Constant Vectors .......................................41 1.6.9 The Inverse .......................................................................................42 1.6.10 Elementary Row Operations .........................................................42 1.6.11 Solving for the Inverse ...................................................................45 1.6.12 The Determinant .............................................................................46 1.6.13 Orthogonality...................................................................................47 1.6.14 Eigenvalues and Eigenvectors ......................................................52 1.7 Important Concepts and Notation............................................................55 1.7.1 Summation and Matrix Notation .................................................55 1.7.2 Converting between Summation and Matrix Notation............56 1.7.3 Averages: Mean, Mode, and Median...........................................57 1.7.4 Various Means and the Generalized Mean.................................58 1.8 Least Squares ................................................................................................62 1.8.1 The Method of Least Squares........................................................62 1.8.2 The Method of Least Squares: The Calculus..............................65 1.8.3 Least Squares for Continuous Intervals ......................................69 1.8.4 Least Squares as a Filter.................................................................72 1.8.5 A Misconception about Least Squares.........................................75 1.8.6 Transforming Equations for Least Squares Fitting of the Parameters ............................................................................75 1.8.7 Constrained Polynomials...............................................................77 1.8.8 Orthogonal Polynomials ................................................................80 1.8.9 General Definition of Orthogonal Polynomials .........................83 1.8.9.1 Discrete MOPs and Real Data .......................................90 1.9 Addendum....................................................................................................93 1.9.1 Proof That M0 Reduces to the Geometric Mean ........................93 1.9.2 Proof of the Monotonicity of Mp...................................................95 1.9.3 Proof That Mp Approaches xmax as p → ∞................................... 98 1.9.4 Proof That Mp Approaches xmin as p → – ∞................................ 99 1.9.5 Proof xmin ≤ Mp ≤ xmax for x > 0......................................................99 1.9.6 Proof That Mp Increases with Increasing p and the Converse ...........................................................................................99 References.............................................................................................................100 1.6
2 2.1
Introduction to Combustion ...................................................... 101 General Overview......................................................................................102 2.1.1 The Burner......................................................................................102 2.1.1.1 The Fuel System.............................................................103
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2.2
2.3
2.4
2.1.1.2 About Fuels ....................................................................104 2.1.1.3 Fuel Metering .................................................................105 2.1.1.4 Turndown........................................................................105 2.1.1.5 The Air System...............................................................106 2.1.1.6 The Flame Holder..........................................................108 2.1.1.7 Stabilizing and Shaping the Flame.............................108 2.1.1.8 Controlling Emissions...................................................109 2.1.2 Archetypical Burners ....................................................................109 2.1.2.1 Round-Flame Gas Diffusion Burners ......................... 111 2.1.2.2 Round-Flame Gas Premix Burners ............................. 111 2.1.2.3 Flat-Flame Gas Diffusion Burners .............................. 113 2.1.2.4 Flat-Flame Premix Burners .......................................... 114 2.1.2.5 Flashback......................................................................... 115 2.1.2.6 Use of Secondary Fuel and Air ................................... 115 2.1.2.7 Round Combination Burners....................................... 116 2.1.2.8 Burner Orientations....................................................... 119 2.1.2.9 Upfired............................................................................. 119 2.1.2.10 Downfired .......................................................................120 2.1.2.11 Side-Fired ........................................................................121 2.1.2.12 Balcony Fired..................................................................121 2.1.2.13 Combination Side and Floor Firing............................121 Archetypical Process Units ......................................................................123 2.2.1 Boilers..............................................................................................123 2.2.1.1 Firetube Boilers ..............................................................123 2.2.1.2 Watertube Boilers...........................................................123 2.2.1.3 Fired Heaters and Reactors..........................................123 2.2.1.4 Vertical Cylindrical........................................................124 2.2.1.5 Cabin Style......................................................................124 2.2.1.6 Fired Reactors.................................................................126 2.2.1.7 Hydrogen Reformers ....................................................126 2.2.1.8 Ammonia Reformers.....................................................126 2.2.1.9 Ethylene Cracking Units (ECUs) ................................127 Important Factors and Responses...........................................................127 2.3.1 The Traditional Test Protocol ......................................................127 2.3.2 Instability, Thermoacoustic and Otherwise ..............................128 2.3.3 Quarter-Wave Behavior................................................................129 2.3.4 Half-Wave Behavior......................................................................131 2.3.5 Helmholtz Resonator Behavior...................................................131 2.3.6 Mechanism for Thermoacoustic Coupling................................132 2.3.7 Comments Regarding Thermoacoustic Resonance .................133 2.3.7.1 Resonance in the Field..................................................134 Mass Balance for Combustion in Air .....................................................135 2.4.1 Wet vs. Dry Measurements .........................................................137 2.4.2 Flue Gas Relations for Hydrocarbons .......................................137 2.4.3 Accounting for Moisture..............................................................140 2.4.4 Addition of Molecular Hydrogen to the Fuel..........................143
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Addition of Flue Gas Components to Fuel ..............................145 Substoichiometric Combustion ...................................................148 2.4.6.1 Lead-Lag Control...........................................................148 2.4.6.2 Substoichiometric Equations .......................................148 2.4.7 Conservation of Mass for Flow in a Furnace ...........................154 2.4.8 Simplifying Assumptions (SAs)..................................................155 2.4.9 Ideal Gas Law ................................................................................158 2.4.10 Dilution Correction .......................................................................159 2.5 Conservation of Energy ............................................................................164 2.5.1 Heat and Related Quantities .......................................................164 2.5.2 Work ................................................................................................165 2.5.3 Heating Value ................................................................................166 2.5.4 Adiabatic Flame Temperature.....................................................167 2.5.5 Heat Capacity as a Function of Temperature...........................169 2.5.6 Adiabatic Flame Temperature with Preheated Air..................171 2.6 Mechanical Energy Balance .....................................................................173 2.6.1 Work Terms ....................................................................................173 2.6.2 Theoretical Mechanical Models ..................................................174 2.6.2.1 Units of Pressure............................................................174 2.6.2.2 Natural Draft Model .....................................................175 2.6.2.3 Draft Pressure in a Furnace .........................................175 2.6.2.4 Air Velocity Due to Natural Draft ..............................177 2.6.2.5 Airflow through a Diffusion Burner ..........................177 2.6.2.6 Airflow through Adjustable Dampers .......................182 2.6.2.7 Unknown Damper Characteristics .............................183 2.6.2.8 Fuel Flow as a Function of Pressure ..........................184 2.6.2.9 Compressible Flow ........................................................185 2.6.2.10 The Fuel Capacity Curve Revisited............................186 2.6.2.11 Airflow in Premix Burners...........................................188 2.6.2.12 Gas Jets Entraining Flue Gas .......................................189 References.............................................................................................................189 2.4.5 2.4.6
3 3.1
3.2 3.3
Experimental Design and Analysis .......................................... 191 Some Statistics............................................................................................192 3.1.1 Statistics and Distributions..........................................................193 3.1.2 The Normal, Chi-Squared (χ2), F, and t Distributions ............194 3.1.2.1 The Normal Distribution..............................................195 3.1.2.2 Probability Distribution for Galton’s Board..............196 3.1.2.3 Pascal’s Triangle.............................................................197 3.1.2.4 The Chi-Squared Distribution .....................................200 3.1.2.5 The F Distribution .........................................................201 3.1.2.6 The t Distribution ..........................................................202 The Analysis of Variance (ANOVA).......................................................203 3.2.1 Use of the F Distribution .............................................................206 Two-Level Factorial Designs....................................................................209 3.3.1 ANOVA for Several Model Effects............................................. 211
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General Features of Factorial Designs .......................................212 Construction Details of the Two-Level Factorial .....................213 Contrast of Factorial and Classical Experimentation..............216 3.3.4.1 Statistical Properties of Classical Experimentation.....219 3.3.4.2 How Factorial Designs Estimate Coefficients...........221 3.3.4.3 The Sneaky Farmer .......................................................222 3.3.5 Interpretation of the Coefficients................................................229 3.3.6 Using Higher-Order Effects to Estimate Experimental Error.................................................................................................232 3.3.6.1 Normal Probability Plots for Estimating Residual Effects..............................................................232 3.4 Correspondence of Factor Space and Equation Form.........................234 3.5 Fractional Factorials ..................................................................................240 3.5.1 The Half Fraction ..........................................................................241 3.5.2 Quarter and Higher Fractions.....................................................242 3.6 ANOVA with Genuine Replicates ..........................................................245 3.6.1 Bias Error ........................................................................................248 3.6.2 Center-Point Replicates ................................................................250 3.6.2.1 Degrees of Freedom Entries.........................................251 3.6.2.2 Sum-of-Squares Entries.................................................254 3.6.3 Standard Errors and the t Test....................................................257 3.6.4 The Value of Orthogonal Designs with ANOVA..................... 258 3.6.5 Rotatability .....................................................................................259 3.7 Randomization ...........................................................................................260 3.7.1 Hysteresis .......................................................................................260 3.7.2 Lurking Factors..............................................................................261 3.8 About Residuals.........................................................................................263 3.8.1 Residuals vs. Run Order ..............................................................263 3.8.2 Other Residual Plots .....................................................................263 3.8.3 Full and Block Randomization ...................................................264 3.8.4 Blocking ..........................................................................................265 3.8.5 Random vs. Fixed Effects.............................................................265 3.9 Screening Designs......................................................................................269 3.9.1 Simplex Designs ............................................................................269 3.9.2 Highly Fractionated Factorials....................................................272 3.9.3 Foldover ..........................................................................................274 3.10 Second-Order Designs ..............................................................................275 3.10.1 Central Composites.......................................................................275 3.10.1.1 Quadratic Bias Only......................................................277 3.10.1.2 Orthogonal Components ..............................................278 3.10.1.3 Adjusting the Axial Component .................................280 3.10.2 Box–Behnken Designs ..................................................................283 3.10.3 Multilevel Factorials .....................................................................283 3.11 Sequential Experimental Design .............................................................286 3.11.1 Augmenting to Less Fractionated Factorials............................287 3.11.2 Method of Steepest Ascent ..........................................................287 3.3.2 3.3.3 3.3.4
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3.11.3 Augmenting to Second-Order Designs .....................................289 References.............................................................................................................291
4 4.1
4.2
4.3
4.4
4.5 4.6
Analysis of Nonideal Data ........................................................ 293 Plant Data....................................................................................................294 4.1.1 Problem 1: Events Too Close in Time ........................................294 4.1.2 Problem 2: Lurking Factors .........................................................295 4.1.3 Problem 3: Moving Average Processes......................................295 4.1.4 Some Diagnostics and Remedies................................................297 4.1.5 Historical Data and Serial Correlation ......................................297 Empirical Models.......................................................................................298 4.2.1 Model Bias from an Incorrect Model Specification .................301 4.2.2 Design Bias .....................................................................................303 Ways to Make Designs Orthogonal........................................................305 4.3.1 Source and Target Matrices: Morphing Factor Space .............306 4.3.2 Eigenvalues and Eigenvectors ....................................................308 4.3.3 Using Eigenvectors to Make Matrices Orthogonal .................316 4.3.4 Canonical Forms............................................................................318 4.3.4.1 Derivation of A Canonical Form.................................318 4.3.4.2 Derivation of B Canonical Form .................................319 4.3.4.3 Canonical Form and Function Shape.........................320 Regression Statistics and Data Integrity ................................................324 4.4.1 The Coefficient of Determination, r2 ..........................................324 4.4.2 Overfit .............................................................................................325 4.4.3 Parsing Data into Model and Validation Sets ..........................326 4.4.4 The Adjusted Coefficient of Determination, rA2 .......................327 4.4.5 The PRESS Statistic .......................................................................328 4.4.6 The Hat Matrix ..............................................................................329 4.4.7 The Coefficient of Determination, Predicted, rp2......................330 4.4.8 Extrapolation..................................................................................331 4.4.8.1 Failure to Detect Hidden Extrapolation ....................336 4.4.9 Collinearity.....................................................................................337 4.4.9.1 Reparameterization in Noncorrelated Factors..........339 4.4.9.2 Variance Inflation Factor ..............................................342 4.4.10 Beta Coefficients ............................................................................343 4.4.11 Confidence and Prediction Intervals .........................................346 Residual Analyses......................................................................................348 Categorical Factors ....................................................................................349 4.6.1 Multilevel Categorical Factors ....................................................349 4.6.2 Accounting for Multiple Blocks..................................................352 4.6.3 Accounting for Hard-to-Change Factors...................................356 4.6.3.1 The Longest Duration Experimental Series ..............357 4.6.3.2 The Shortest Duration Experimental Series ..............358 4.6.3.3 Experimental Units........................................................361 4.6.3.4 The Split-Plot Design ....................................................362 4.6.4 Expected Mean Squares (EMS) ...................................................367
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Methodology for Deriving EMS for Balanced Data ..........................................................367 4.6.4.2 EMS for the Factorial Design.......................................373 4.6.4.3 EMS for a Split-Plot Design .........................................374 4.6.4.4 Split-Plot Structure with Multiple Whole-Plot Factors..............................................................................378 4.6.4.5 Nested Factors................................................................378 4.7 Categorical Response Values ...................................................................383 4.7.1 Conversion from Qualitative to Quantitative Measures ........384 4.7.2 Using the Logit and Probit Functions to Categorize Flame Quality ................................................................................385 4.8 Mixture Designs.........................................................................................386 4.8.1 Simplex-Centroid...........................................................................388 4.8.2 Simplex-Lattice ..............................................................................390 4.8.3 Simplex-Axial.................................................................................391 4.8.4 Generalizing to Higher Dimensions ..........................................392 4.8.5 Fuels of Many Components ........................................................395 4.8.6 Fuel Chemistry ..............................................................................395 4.8.6.1 Hydrogen ........................................................................396 4.8.6.2 Hydrocarbon Chemistry...............................................396 4.8.6.3 Bonding ...........................................................................397 4.8.6.4 Saturates ..........................................................................397 4.8.6.5 Olefins..............................................................................399 4.8.6.6 Coke Formation .............................................................399 4.8.6.7 Mono-Olefins..................................................................400 4.8.6.8 Di-Olefins ........................................................................400 4.8.6.9 Acetylenes .......................................................................401 4.8.6.10 Aromatic Hydrocarbons ...............................................401 4.8.6.11 Cyclo Hydrocarbons .....................................................402 4.8.7 Representing Gaseous Fuel Blends ............................................402 4.8.7.1 Chemical Bond Method................................................403 4.8.7.2 Equivalent Oxygen Method.........................................407 4.8.7.3 Component Ranges .......................................................408 4.8.7.4 Pseudo-Components .....................................................410 4.8.8 Orthogonal Mixture Designs.......................................................410 4.8.8.1 Ratios of Mixture Fractions.......................................... 411 4.8.9 Combining Mixture and Factorial Designs...............................413 4.8.9.1 Mixtures within Factorial .............................................414 4.8.9.2 Mixture within Fractional Factorial............................414 4.8.9.3 Fractionated Mixture within Fractional Factorial ....415 References.............................................................................................................420 4.6.4.1
5 5.1
Semiempirical Models................................................................ 421 NOx and Kinetics ......................................................................................422 5.1.1 NOx: Some General Comments..................................................422 5.1.2 The Thermal NOx Mechanism ...................................................422
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The Fuel-Bound Nitrogen Mechanism ......................................424 The Prompt NOx Mechanism .....................................................426 Chemical Kinetic Effects for NOx in Diffusion Flames..........427 5.1.5.1 NOx Response to Air in Diffusion Flames................427 5.1.5.2 Dimensional Units for NOx.........................................432 5.1.5.3 The Relation of Referent and Objective Forms ........434 5.1.5.4 NOx Response to Temperature in Diffusion Flames..............................................................................435 5.1.5.5 NOx Response to Fuel Composition..........................438 5.1.5.6 Chemical NOx When Prompt NOx Is Important ....439 5.1.6 Chemical Kinetic Effects for NOx in Premixed Flames..........440 5.1.6.1 NOx Response to Temperature in Premixed Flames..............................................................................440 5.1.6.2 NOx Response to Air in Premixed Burners..............440 5.1.6.3 Solving for ζ as a Function of αw................................441 5.1.6.4 Solving for T as a Function of αw ...............................441 5.1.6.5 Log NOx as a Function of αw ......................................442 Overview of NOx Reduction Strategies ................................................443 5.2.1 Low Excess Air (LEA) Operation ...............................................443 5.2.2 Air Staging .....................................................................................445 5.2.3 Overfire Air ....................................................................................445 5.2.4 Burners out of Service (BOOS) ...................................................446 5.2.5 Fuel Staging....................................................................................446 5.2.6 Fuel Blending .................................................................................447 5.2.7 Flue Gas Recirculation..................................................................447 5.2.7.1 Mass-Based Relations....................................................447 5.2.7.2 Molar and Volumetric Definitions ..............................450 5.2.8 Fuel Dilution, Flue Gas Inducted Recirculation (FIR) ............453 5.2.9 Steam or Water Injection..............................................................456 5.2.10 Selective Noncatalytic Reduction (SNCR) ................................456 5.2.11 Selective Catalytic Reduction (SCR) ..........................................457 NOx Models ...............................................................................................458 5.3.1 Categorization of Emissions Reduction Strategies ..................460 5.3.2 Temperature Reduction Strategies .............................................460 5.3.2.1 Fuel Blending or Fuel Dilution ...................................460 5.3.2.2 Flue Gas Inducted Recirculation.................................460 5.3.2.3 Flue Gas Recirculation ..................................................461 5.3.2.4 Steam or Water Injection ..............................................462 5.3.2.5 Air Staging......................................................................462 5.3.2.6 Fuel Staging ....................................................................467 5.3.2.7 Overfire Air.....................................................................467 5.3.2.8 Burners out of Service ..................................................469 5.3.3 Concentration Reduction Strategies...........................................469 5.3.3.1 Low Excess Air (LEA) Operation ...............................469 5.3.3.2 Air Staging with Fuel-Bound Nitrogen .....................470 5.3.3.3 Fuel Staging with Fuel-Bound Nitrogen ...................471 5.1.3 5.1.4 5.1.5
5.2
5.3
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Reagent Injection Strategies.........................................................471 5.3.4.1 Selective Noncatalytic Reduction (SNCR).................471 5.3.4.2 Selective Catalytic Reduction (SCR)...........................476 5.3.4.3 Limestone Injection .......................................................476 5.4 CO Models ..................................................................................................477 5.4.1 Cold CO ..........................................................................................478 5.4.2 Hot CO............................................................................................479 5.4.3 General Behavior of Hot CO.......................................................479 5.4.4 Equilibrium Considerations ........................................................481 5.4.5 Arrested Oxidation of CO (via Ammoniacal Poisoning of OH Catalysis) ............................................................................483 5.5 Response Transformations .......................................................................483 5.5.1 Empirical Considerations for Transformation of the CO Response .........................................................................................483 5.5.2 Empirical Considerations for Transformation of NOx Response .........................................................................................486 5.6 Heat Flux.....................................................................................................486 5.6.1 Heat Flux Profile ...........................................................................487 5.6.1.1 The Normalized Heat Flux Equation.........................489 5.6.1.2 Data Normalization.......................................................491 5.6.1.3 Data Smoothing .............................................................492 5.6.1.4 Renormalization.............................................................497 5.6.1.5 The Heat Flux Model ....................................................500 5.6.2 Heat Flux as a Function of Furnace Temperatures .................500 5.6.3 Qualitative Behavior of zmax ........................................................503 5.6.3.1 The Effect of Air Preheat ..............................................506 5.6.3.2 The Effect of Air/Fuel Ratio........................................507 5.6.3.3 The Effect of Fuel Pressure ..........................................507 5.6.4 Heat Flux Profile in Terms of Fractional Heat Release...........509 5.6.4.1 The Effect of the Heat Sink (Process)......................... 511 5.6.4.2 Final Heat Flux and Process Efficiency......................512 5.6.4.3 Run Length and Flux Profile Curvature....................512 5.6.4.4 Factors Affecting the Initial Heat Flux.......................513 5.6.4.5 Similarity and Scaling of Heat Flux Curves .............515 5.7 Flame Shape................................................................................................515 5.7.1 Flame Measurements....................................................................516 5.7.2 Flame Length .................................................................................517 5.8 Visible Plumes ............................................................................................521 5.8.1 Bisulfite Plumes .............................................................................521 5.8.2 Ammonium Chloride Plumes.....................................................522 5.8.3 Sulfur Oxides .................................................................................524 5.8.3.1 Equations for Dew Point Elevation ............................525 References.............................................................................................................528 5.3.4
Epilogue................................................................................................ 531 References.............................................................................................................532
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Appendices A
Fuel and Combustion Properties .............................................. 533
B
Mechanical Properties ................................................................ 555
C
Units Conversions....................................................................... 573
D
Properties of the Elements......................................................... 577
E
Statistical Tables ......................................................................... 601
F
Numbers in Binary, Octal, and Hexadecimal Representations ........................................................................... 609
G
Kinetics Primer............................................................................ 613
H
Equilibrium Primer..................................................................... 617
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List of Tables
Table 1.1 Table 1.2 Table 1.3 Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table
1.4 1.5 1.6 2.1 2.2 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10 3.11 3.12 3.13 3.14 3.15 3.16 3.17 3.18 3.19 3.20 3.21
Table Table Table Table Table Table Table Table
3.22 3.23 3.24 3.25 3.26 3.27 3.28 3.29
Qualitative Analysis of NOx Formation .......................................7 Some Dimensionless Groups.........................................................15 Some Potential Factors Affecting NOx Response from a Burner...................................................................................34 Hypothetical Data ...........................................................................50 Classical Orthogonal Polynomials ...............................................85 Theoretical vs. Fitted Coefficients ................................................93 Burner Sampling for One Manufacturer ................................... 111 Reactants and Products for Example 2.3...................................136 The Naked ANOVA......................................................................204 A Single Factor Example..............................................................208 ANOVA for Example 3.1..............................................................209 A Factorial Design in Three Factors...........................................209 Basic ANOVA for Table 3.4..........................................................212 Partitioned ANOVA for Table 3.4...............................................212 Contrast of Classical and Factorial Designs .............................216 A Factorial Design in Three Factors...........................................228 Partitioned ANOVA......................................................................234 A Curious Experimental Design.................................................235 A Half Fraction in Five Factors...................................................241 Replicate Data ................................................................................247 ANOVA for Replicate Data .........................................................248 ANOVA with Pooled Effects .......................................................248 The Naked ANOVA with Replicates .........................................250 ANOVA with Replicates ..............................................................250 A 22 Factorial Design with Center Points .................................251 ANOVA for Orthogonal Pooled or Unpooled Entries............252 t Test ................................................................................................258 A Nonrandomized Experiment ..................................................261 Analysis of Variance for Data for a Nonrandomized Experiment .....................................................................................262 A Randomized Experiment .........................................................262 Analysis of Variance for a Randomized Experiment..............263 A Randomized 23 Factorial Design in Two Blocks ..................266 ANOVA for a 23 Factorial Design in Two Blocks ....................267 Data for 23 Factorial in Two Blocks............................................267 ANOVA for the 23 Factorial Design in Two Blocks.................268 ANOVA for the 23 Factorial Design without Blocks ...............269 A 2 × 3 × 4 Factorial Design ........................................................284
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Table 3.30 Half Fraction of a 4 × 4 Factorial Derived from the 1/2 24 Design........................................................................... 286 Table 3.31 A 12-Factor Screening Design in 16 Runs and 2 Blocks.........290 Table 3.32 Additional Points for a Central Composite Design ................291 Table 4.1 A Classical Design in Three Factors ..........................................303 Table 4.2 Random Data .................................................................................325 Table 4.3 r2 for All Possible 80/20 Validations..........................................327 Table 4.4 PRESS and Other Statistics for the Data of Table 3.17 ...........331 Table 4.5 Is This Extrapolation?...................................................................332 Table 4.6 Outlier Statistics ............................................................................335 Table 4.7 A Poor Experimental Design in Three Factors.........................337 Table 4.8 A Weird Experimental Design in Two Factors.........................339 Table 4.9 A Factorial Design in Original Metrics .....................................345 Table 4.10 Regression Statistics, Analysis for Coded Factors ...................346 Table 4.11 Regression Statistics, Analysis for Original Metrics................346 Table 4.12 A 24 Factorial Design in Four Blocks .........................................353 Table 4.13 Annotated XTX Matrix for the 24 Design in Four Blocks .......354 Table 4.14 ANOVA for 24 Design in Four Blocks .......................................356 Table 4.15 Responses and Factors of Interest ..............................................357 Table 4.16 Longest Runtime Order ...............................................................359 Table 4.17 Preparing to Construct Shortest Setup Time............................359 Table 4.18 Run Sequence for Shortest Setup Time .....................................360 Table 4.19 Longest and Shortest Setup Times for Furnace Experiment .....................................................................................361 Table 4.20 ANOVA for Two-Factor Factorial Design .................................363 Table 4.21 ANOVA for Two-Factor Split-Plot Design ................................364 Table 4.22 Split-Plot Design............................................................................365 Table 4.23 Split-Plot Design............................................................................366 Table 4.24 Expected Mean Squares Table, Step 1 .......................................368 Table 4.25 Expected Mean Squares Table, Step 2 .......................................368 Table 4.26 Expected Mean Squares Table, Step 6 .......................................371 Table 4.27 Expected Mean Squares Table, Step 7 .......................................372 Table 4.28 ANOVA with Expected Mean Squares, Step 8 ........................373 Table 4.29 A Factorial Design for Multilevel Factors.................................374 Table 4.30 Expected Mean Squares for Split Plot, Step 6 ..........................375 Table 4.31 Expected Mean Squares Table, Step 7 .......................................376 Table 4.32 ANOVA for Two-Factor Split-Plot Design ................................376 Table 4.33 ANOVA for Two-Factor Fully Randomized Factorial Design .............................................................................................377 Table 4.34 ANOVA for Split-Plot Design with Two Factors in the Whole Plot ..........................................................................380 Table 4.35 Expected Mean Squares Table for Example 4.8 .......................382 Table 4.36 ANOVA for Example 4.8..............................................................383 Table 4.37 Formulas for Bond Types.............................................................401 Table 4.38 Sample Refinery Gas ....................................................................406
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Table 4.39 Source Refinery Gas: Augmented with Numbers of Bond Types ................................................................................406 Table 4.40 Fuel Composition Ranges............................................................408 Table 4.41 Fuel Compositions within Constraints......................................410 Table 4.42 A 22 Factorial Mixture Design in Two Orthogonal Blocks ..............................................................................................416 Table 4.43 A Four-Factor Central Composite Design in Three Orthogonal Blocks.........................................................................418 Table 5.1 SO2 vs. Limestone Injection Rate................................................476 Table 5.2 ANOVA for SO2 Capture .............................................................477 Table 5.3 ANOVA for SO2 Capture, Pooled Residual ..............................477 Table 5.4 Heat Flux vs. Normalized Elevation .........................................493 Table 5.5 ANOVA for Fourth-Order Smoothing.......................................493 Table 5.6 ANOVA for Example 5.14............................................................495 Table 5.7 A Normalized Heat Flux Curve .................................................498 Table 5.8 Heat Flux Comparison .................................................................503 Table 5.9 Heat Flux Equation Summary for y = aln(1 + z) – bz + c ......514 Table 5.10 Dew Point Elevation Data ...........................................................527 Table A.1 Physical Constants of Typical Gaseous Fuel Mixture Components ...................................................................................534 Table A.2 Combustion Data for Hydrocarbons .........................................535 Table A.3 Chemical, Physical, and Thermal Properties of Gases: Gases and Vapors, Including Fuels and Refrigerants, English and Metric Units .............................................................537 Table A.4 Physical and Combustion Properties of Fuels .........................543 Table A.5 Thermodynamic Properties of Selected Compounds .............547 Table A.6 Heat Capacity vs. Temperature for Selected Compounds, J/mol K ...........................................................................................548 Table A.7 Adiabatic Flame Temperatures ...................................................552 Table A.8 Volumetric Analysis of Typical Gaseous Fuel Mixtures.........553 Table A.9 Physical Constants of Typical Gaseous Fuel Mixtures ...........554 Table B.1 Areas and Circumferences of Circles and Drill Sizes .............556 Table B.2 Physical Properties of Pipe..........................................................565 Table B.3 K Factors .........................................................................................570 Table C.1 Common Conversions..................................................................574 Table C.2 Unit Dimensions for Some CombustionRelated Quantities.........................................................................575 Table D.1 Periodic Table of the Elements....................................................578 Table D.2 Standard Atomic Weights 1981...................................................579 Table D.3 Properties of Saturated Steam and Saturated Water...............584 Table D.4 Properties of Superheated Steam ...............................................591 Table E.1 Normal Probability Function ......................................................602 Table E.2 Students t Distribution .................................................................603 Table E.3 χ2 Distribution................................................................................604 Table E.4 F-Distribution, 99%, 95%, and 90% Confidence.......................606
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Table Table Table Table
F.1 F.2 F.3 F.4
Positional Number Representation in Base 10 .........................609 Positional Number Representation in Base 8 ...........................609 Positional Number Representation in Base 2 ...........................610 Base Equivalents............................................................................ 611
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List of Figures
Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure
1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.10 1.11 1.12 1.13 1.14 1.15 1.16 1.17 1.18 1.19 1.20 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 2.10 2.11 2.12 2.13 2.14 2.15 2.16 2.17 2.18 2.19
A graph with asymptotes ..............................................................16 Basic function shapes .....................................................................16 Excess air vs. oxygen for CH4 .......................................................21 Two hmh curves ...............................................................................23 An observer in Flatland .................................................................24 Contour surfaces for a hyperellipsoid in four dimensions......25 A four-dimensional hypercube .....................................................27 Coordinate space in three dimensions ........................................27 Visualizing multiple dimensions with cubic regions................28 Data types.........................................................................................30 An ordinal scale for flame quality ...............................................31 The least squares line .....................................................................63 Linear algebra with spreadsheets.................................................66 A least squares approximation of a continuous function ........71 A filter analogy for least squares..................................................73 A bimodal distribution...................................................................88 The first four MOPs ........................................................................90 The first 16 spectral coefficients....................................................91 Discrete data case............................................................................92 The generalized mean and its derivatives ................................100 A typical industrial burner ..........................................................103 A typical capacity curve...............................................................105 The tile ledge as flame holder.....................................................108 A gas burner in operation............................................................109 A gas premix floor burner ........................................................... 112 A flat-flame gas diffusion burner ............................................... 114 Floor-fired flat-flame burners...................................................... 115 A typical heat flux profile ............................................................ 116 A flat-flame diffusion burner ...................................................... 117 Wall-fired diffusion burners in operation ................................. 117 A flat-flame premix burner.......................................................... 118 Venturi section of a premix burner ............................................ 118 An oil gun ...................................................................................... 119 A combination burner ..................................................................120 A downfired burner for hydrogen reforming ..........................121 Sidewall burners in operation.....................................................122 Balcony burner...............................................................................122 Some process heater types...........................................................124 Resonant modes for furnaces......................................................129
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Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure
2.20 2.21 2.22 2.23 2.24 2.25 2.26 2.27 2.28 2.29 2.30 2.31 2.32 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10 3.11 3.12 3.13 3.14 3.15 3.16 3.17 3.18 4.1 4.2 4.3
Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure
4.4 4.5 4.6 4.7 4.8 4.9 4.10 4.11 4.12 4.13
Flue gas relations...........................................................................141 Wet–dry flue gas relations ...........................................................142 A process heater ............................................................................144 Substoichiometric combustion relations....................................152 A furnace control volume ............................................................156 Concentration vs. time for well-mixed behavior.....................158 Dilution correction ........................................................................159 Simplified airflow analysis of burner ........................................177 A typical airside capacity (cap) curve .......................................181 Single-blade damper.....................................................................182 Airflow vs. damper function.......................................................183 Probit and logit functions ............................................................184 Thermodynamic relations ............................................................186 The Galton board ..........................................................................195 The normal distribution ...............................................................196 Pascal’s triangle .............................................................................197 Classical and factorial experiments............................................217 Information fraction for a factorial and classical design........221 The sneaky farmer ........................................................................223 A 23 factorial design......................................................................227 Representations of NOx response ..............................................231 Normal probability plot ...............................................................233 Some two-factor designs..............................................................235 A curious experimental design ...................................................236 Illustration of degrees of freedom..............................................254 Information fraction for factorial with center points..............259 Residual vs. run order..................................................................264 A central composite in two factors ............................................276 Comparison of central composite designs................................282 Box–Behnken design for p = 3 ....................................................283 Method of steepest ascent............................................................288 A municipal solid waste boiler ...................................................295 A moving average with random data .......................................296 Graphical representation of various experimental designs ............................................................................................305 Various response surfaces............................................................321 On overfitting data........................................................................326 Hidden extrapolation ...................................................................333 Failure to detect extrapolation ....................................................336 A poor experimental design........................................................338 Augmented design........................................................................341 A better experimental design ......................................................341 Actual vs. predicted with confidence interval .........................348 Factor space for a ternary mixture.............................................386 Ternary coordinate systems.........................................................387
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Figure 4.14 Figure 4.15 Figure 4.16 Figure 4.17 Figure 4.18 Figure 4.19 Figure 4.20 Figure 4.21 Figure 4.22 Figure 4.23 Figure 4.24 Figure 4.25 Figure 4.26 Figure 5.1 Figure 5.2 Figure 5.3 Figure 5.4 Figure 5.5 Figure 5.6 Figure 5.7 Figure 5.8 Figure 5.9 Figure 5.10 Figure 5.11 Figure 5.12 Figure 5.13 Figure 5.14 Figure A.1 Figure A.2
The simplex-centroid ....................................................................389 Some designs and their information fractions .........................392 Some simplexes in c – 1 dimensions..........................................393 Structural features of some alkanes ...........................................397 Constrained mixtures ...................................................................409 Transformation to orthogonal factors ........................................ 411 Orthogonal ratios ..........................................................................412 Evenly spaced ratios .....................................................................413 Mixed coordinates.........................................................................413 Factor space: mixtures within a factorial ..................................414 Factor space: mixtures within a fractional factorial ................415 Independent mixture factors .......................................................416 Central composite in three orthogonal blocks .........................419 An ECU simulator.........................................................................436 A CO trim strategy .......................................................................444 A burner using fuel dilution and staging .................................447 Mass balance for a typical FGR system ....................................448 A mass balance for a typical FIR system ..................................453 Comparison of FGR and FIR strategies ....................................454 CO behavior ...................................................................................479 Actual vs. predicted CO...............................................................482 Box–Cox response transformation .............................................485 Box–Cox response transformation .............................................485 Schematic of a heat flux probe....................................................486 The y*-z* plot for heat flux ..........................................................490 Graph of data for Example 5.7 ...................................................496 Time-averaged flames...................................................................518 Heat capacity vs. temperature for polyatomic gases. .............549 Heat capacity vs. temperature for triatomic and other gases ................................................................................................550 Figure A.3 Heat capacity vs. temperature for diatomic and other gases ................................................................................................551
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Concordance of Nomenclature
lower case Roman e eb er [s] []
error (collective or pure) bias error residual error concentration of species s dimensionless
lower case Roman italic a, b, c, … a a0 a1, a2, … ak b b b b0, b1, … b0, b1, … b0, b1, … c c c c c0, c1, … cj d d d dL2 dnn do do,k e e1 e2
arbitrary exponents coefficient in normalized heat flux equation arbitrary constant arbitrary constants kth coefficient Arrhenius activation constant arbitrary offset coefficient in normalized heat flux equation arbitrary constants blocking factor coefficients mixture coefficients coefficient in normalized heat flux equation number of columns in matrix or vector speed of sound number of mixture components arbitrary constants jth coefficient of characteristic equation arbitrary column or number of columns in matrix or vector number of dimensions number of points along an edge of a simplex leverage distance nth diagonal element orifice diameter kth orifice diameter 2.71828… number of pure components (simplex vertices) in mixture design number of binary blend combinations (simplex edges) in mixture design
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e3 e4 ek f fa fb fk fo fs g gc g(x) h h hj,k hk,k i i(x) j k k kf kr l m m m m m m a m b m f m mk mjk r m n n n n n n n
number of ternary blend combinations (simplex faces) in mixture design number of quaternary blend combinations (simplex volumes) in mixture design number of k blend combinations in mixture design (simplex hypervolumes when k ≥ 5) number of factors fraction of air staging fraction of burners out of service kth factor fraction of overfire air stoichiometric mixture fraction gravitational acceleration constant unit dimensional constant for U.S. customary units spectral function height horizontal asymptote (function shape analysis) arbitrary element of the H matrix kth diagonal element of the H matrix imaginary unit, −1 information fraction of x an index an index constant of proportionality forward rate constant reverse rate constant linear asymptote (function shape analysis) limit (in summation operator) mass max or min (function shape analysis) mean square of quadratic factors slope mass flowrate mass flowrate of air to burner mass flowrate out of windbox or plenum mass flowrate of fuel kth mean value arbitrary matrix element flue gas mass flowrate recirculated to burner limit (in summation operator) number of burners number of independent factors number of factors in simplex number of moles number of observations mass flowrate
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n a nb n b nc nC nf n f n g nH nk no nr n r p p q q q q0 q0 q1 qg qmax qp r r r r r2 rA2 rc rI2 rk rk 2 rO 2 rP2 s s s s2 sC–C sC–H sH–H sk t tCH4
molar flowrate of air to burner number of burners out of service molar flowrate out of windbox or plenum number of centerpoint replicates number of carbon atoms in fuel molecule number of factorial points molar flowrate of fuel molar flowrate of flue gas out of the stack number of hydrogen atoms in fuel molecule number of levels of the kth factor number of orifices number of replicates flue gas molar flowrate recirculated to burner generic stoichiometric product coefficient number of parameters in model generic stoichiometric product coefficient orthogonalization factor instantaneous heat release at z fractional heat release at the floor instantaneous heat release at z = 0 fractional heat release at the roof fractional thermal power in flue gas maximum fractional thermal power in flue gas fractional process heat-release arbitrary row or number of rows in matrix or vector coefficient of correlation generic reactant stoichiometric coefficient reaction rate coefficient of determination adjusted coefficient of determination critical pressure ratio coefficient of inlying kth replicate kth coefficient of collinearity coefficient of outlying coefficient of prediction arbitrary row or number of rows in matrix or vector estimated standard deviation generic reactant stoichiometric coefficient estimated variance number of C–C bonds in source fuel number of C–H bonds in source fuel number of H–H bonds in source fuel kth standard deviation time fraction of CH4 in target fuel
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tC3H8 tH2 t(n, z) tr( ) u u v v v vo w wk w(x) x x x x xCO xNO xS x1, x2, … xC,k xF x xk xR y y y y yˆ y0 y1 ya ya,s yb y* yˆ *k yCO yNO yNO,0 yNO,r yO2,b yO2,bt yO2,v
fraction of C3H8 in target fuel fraction of H2 in target fuel t-distribution of z trace operator unbounded curve (function shape analysis) number of unique points in factor space velocity (fluid flow) vertical asymptote (function shape analysis) mean value of a vector velocity at the jet orifice mass fraction of subscripted species kth transformed source factor particular weight function for an orthogonal polynomial arbitrary factor (independent variable) mass fraction mole fraction dissolved substance transformed factor conversion of CO conversion of NOx conversion of species S arbitrary factors kth Cartesian coordinate fixed factor (effect) arithmetic mean kth arbitrary factor random factor (effect) arbitrary response mole fraction normalized heat flux mean response predicted, true, or fitted response normalized heat flux at floor normalized heat flux at roof mole fraction of species a mole fraction of species a at the stack exit mole fraction of species b reduced form of normalized heat flux predicted response with the kth residual deleted mole fraction CO mole fraction NOx initial mole fraction NOx mole fraction NOx reduced or converted by postcombustion strategy mole fraction of oxygen in the windbox CO breakthrough point oxygen at the venturi outlet
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yx,0 yx,ref ymax yˆ max z z z z* z z1 z−1 zC,k zk zmax
mole fraction of species x at 0% O2 mole fraction of species x at reference oxygen conditions maximum value of normalized heat flux before renormalization unadulterated maximum value of normalized heat flux before renormalization mixture factor normal ordinate normalized elevation reduced form of normalized elevation average burner distance height of first combustion zone, normalized harmonic average burner distance kth equilateral triangular coordinate kth distance or elevation normalized elevation of maximum heat flux
UPPER CASE ROMAN [S] []
concentration of species S dimensionless
UPPER CASE ROMAN ITALIC A A A A Ak A(x) B B C C Cc Co Cp Cˆ
area Arrhenius pre-exponential coefficient in heat flux equation generic numerator in partial fractions kth spectral coefficient fitted least-squares response, continuous data coefficient in heat flux equation generic numerator in partial fractions arbitrary constant coefficient in heat flux equation coefficient for critical flow initial centerline concentration isobaric heat capacity
Cp Cv Cˆ
specific isobaric heat capacity isometric heat capacity
p
v
Cv Cx D
molar isobaric heat capacity
molar isometric heat capacity specific isometric heat capacity centerline concentration of species x diameter
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DF DFB DFE DFK DFM DFR DFT DS DS EMS EMSCH(x) E(y) F F F(m, n, z) G H Hk(x) HHV I(x) J J K K Kk Kdry Keq Kwet Kwg L L* L1 L2 LCL LHV Lk Lk LS L(x) M M M-1 M0 M2 Mp
diameter of a cylindrical furnace degrees of freedom, bias (error) degrees of freedom, (pure) error degrees of freedom, blocks degrees of freedom, model degrees of freedom, residual (error) degrees of freedom, total diameter of a cylindrical stack degree of saturation for hydrocarbons expected mean square expected mean square column heading for x expected value of y degrees of freedom force F-distribution of z Gibbs free energy enthalpy kth Hermite polynomial higher heating value information function of x number of items indexed by j arbitrary integer for design fractionation K-factor for fluid flow constant of proportionality kth K-factor for fluid flow fuel factor, dry arbitrary equilibrium constant fuel factor, wet equilibrium constant for water-gas shift reaction length dimensionless flame length lower limit of integrand upper limit of integrand lower confidence limit lower heating value level of kth factor in a factorial design length of the fuel jet from the kth orifice length of the stack logit function ammonia/NOx ratio, initial arithmetic mean harmonic mean geometric mean root mean square generalized mean
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MSB MSE MSK MSM MSR N(y) NEu NF NFr,f NMa NRe P P P(y) Pc Pr Pref Q QF Q Q 1 Q g Q p Q max R RFG ,i RFG ,e Rˆ FG ,i Rˆ FG ,e S SSB SSE SSK SSM SSR SST S(x) S(x) T T* T0 T Tb Tc Tmax
mean squared bias (error) mean squared (pure) error mean squared, block mean squared, model mean squared residual (error) normal distribution of y Euler number friction factor, Darcy–Weisbach formulation Flame Froude number Mach number Reynolds number pressure P-factor (probability that the null hypothesis is true) probability distribution of y critical pressure reduced pressure reference pressure generalized mean flame quality according to Figure 1.11 heat release heat loss from first combustion zone resident thermal power of the flue gas at a given elevation thermal power transferred to the process resident thermal power of the flue gas at a given elevation universal gas constant internal mass ratio of flue gas recirculation external mass ratio of flue gas recirculation internal molar ratio of flue gas recirculation external molar ratio of flue gas recirculation entropy sum of squares, bias (error) sum of squares, (pure) error sum of squares, blocks sum of squares, model sum of squares, residual (error) sum of squares, total arbitrary continuous function squash (probit) function temperature dimensionless temperature (fourth-order) flue gas temperature at furnace floor ambient temperature normal boiling point of solvent critical temperature maximum flue gas temperature
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U UCL V(y) V ( y) TDP TDPw Tr TWP TWPw V Vˆ V VIFk W X Y Y Ymax Z Z Zmax Z Z1 ZH
internal energy upper confidence limit variance of y mean variance of y total dry products, volume total dry products, weight reduced temperature total wet products, volume total wet products, weight volume molar volume volumetric flowrate kth variance inflation factor molecular weight dimensionless independent factor arbitrary response (dependent variable) heat flux, mV (Ch 5) maximum heat flux, mV collision frequency elevation elevation of maximum heat flux Z-factor for gases length of first combustion zone length of radiant zone
bold lower case Roman a a asa b c e k ryx sk2 s2y u uT v vT x xT yˆ
column vector of least squares constants norm of vector a unit vector for path of steepest ascent Eigenvector column vector of identical constants error column vector an Eigenvector associated vector for y estimated variance-covariance matrix estimated response variance matrix generic column vector generic row vector generic column vector generic row vector arbitrary column vector arbitrary row vector column vector of predicted, fitted, or true responses
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BOLD UPPER CASE ROMAN 0 1 A A B B C D F G H I J K M N P Rxx S T U X XC ZE
zero column vector unit column vector A canonical form matrix generic matrix B canonical form matrix generic matrix generic matrix diagonal matrix source-target transformation matrix transformation-filter matrix, filters and transforms a from y hat matrix, filters yˆ from y identity matrix error-pass filter matrix, filters e from y eigenvector matrix generic matrix generic matrix generic matrix correlation matrix for X source matrix target matrix orthogonalized matrix arbitrary matrix Cartesian coordinate matrix equilateral triangular coordinate matrix
Arial Font [L] [M] [N] [T]
generic generic generic generic
length unit mass unit mole unit temperature unit
lower case Greek α α α α αs αsteam αwater αw β β
axial orthogonalization factor constant in quadratic formula molar H2/H2O ratio molar air/fuel ratio stoichiometric air/fuel molar ratio any factor proportional to the steam injection rate any factor proportional to the water injection rate air/fuel mass ratio constant in quadratic formula molar CO/CO2 ratio
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β0 β1 βk γ γ δ δk δ 0 , δ 1 , δ2 , … ε ζ ζ η θ [θ] θk κ λ λ λn λkk μ μ ν νk ξ ξ1 , ξ 2 , … ξ ξ– ξ+ xk π π π( ) ρ ρ ρ1 ρ2 ρ01 ρ12 ρ13 ρ31 ρo σ σ σ2 σ 2K
special coefficient, Eqn. 5-37 special coefficient, Eqn. 5-37 kth standardized (beta) coefficient constant in quadratic formula molar flue-gas/air ratio relative roughness of pipe kth dummy factor dummy factors excess air natural log, collision frequency wet oxygen ratio, flue gas/windbox process efficiency arbitrary angle generic time unit kth A canonical form coefficient ratio of specific heats, Cp/Cv arbitrary power transformation wavelength nth eigenvalue (latent root) kth quadratic coefficient in A or B canonical form viscosity true mean value frequency kth normalization constant arbitrary untransformed factor arbitrary untransformed factors arithmetic mean of untransformed factor minimum untransformed factor maximum untranformed factor kth arbitrary factor 3.14159… probability of success probability operator density ambient density mixture ratio, H2/CH4 mixture ratio, H2/C3H8 source bond ratio, H–H/C–H mixture ratio, H2/(H2+CH4) mixture ratio, H2/(H2+C3H8) source bond ratio, C–C/C–H density at the orifice true deviation Stefan–Boltzmann constant true variance block variance
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σ 2r 2 σ SP 2 σ WP φ φ( ) φk( ) φj,k χ χ χ 2 ( n, z ) χS ψ ψk ω
replicate variance sub-plot variance whole-plot variance transformed y-axis implicit function operator kth implicit function kth term of the jth orthogonal polynomial H/C molar ratio for H2-hydrocarbon fuel mixtures reaction coordinate chi-squared distribution conversion of species S hydrogen/carbon molar ratio kth pseudo-component Wobbe index
bold lower case Greek β σ 2k
beta coefficient matrix true variance–covariance matrix
UPPER CASE GREEK
()
Γ n Δ ΔG ΔH ΔH ΔH c ΔH c , f ΔH vap ΔL ΔP ΔTb ΔTF Θ Σ Π Φ Ψy Ω
gamma function difference operator Gibbs free-energy difference enthalpy difference volumetric higher heating value specific heat of combustion heat of combustion of undiluted fuel latent heat of vaporization acoustic correction for stack length pressure drop boiling point elevation temperature rise due to combustion linearized radiation coefficient summation operator product operator stoichiometric coefficient standardized response success-failure ratio
BOLD UPPER CASE GREEK Λ Φ
eigenvalue matrix matrix of orthogonal functions
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Sub
scripts
0, 1, 2, … 0 1 -0 -1 a A AFT b b b BWT c c d e f f F K g i I k O P r r r r r R ref min max o o v v vap
Super
′ ~
distinguishing ordinals or states initial condition final condition first column or row deleted corresponding to a0 harmonic average air property adjusted relating to the adiabatic flame temperature burner or windbox boiling point statistical bias bridgewall temperature combustion property critical property damper effluent property fuel property forward reaction fixed effect relating to a block effect flue-gas property influent property inlying kth value of A outlying prediction reduced property relating to the statistical residual reverse reaction replicate observation relating to recirculation random effect reference state minimum value of A maximum value of A orifice overfire air belonging to control volume relating to the venturi outlet relating to vaporization
scripts prime, indicates similar but different function tilde, volumetric property
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ˆ ¯ ¯ · – +
* T
hat, molar property overbar, mass property overbar, mean property dot, rate (time derivative) minus, minimum value plus, maximum value asterisk, special transform transpose operator
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About the Author
Joseph Colannino earned his degree in chemical engineering from the California Polytechnic University at Pomona. He has more than 20 years experience in the combustion and fuels industries. Much of his experience was garnered as a combustion consultant working on a wide variety of combustion equipment including boilers of all stripes (black liquor recovery, fluidized-bed, utility, waterand fire-tube) firing on a wide variety of fuels (coal, municipal solid waste, heavy and light oils, waste gases, landfill gas, refinery gas, and PSA gas). As principal of Colannino Consultants, he pioneered or applied many novel NOx remediation strategies and modeling techniques to operating equipment and has worked for many industries including paper and pulp, petroleum and petrochemical, municipal solid waste-to-energy, electrical generation, and various industrial and commercial concerns. He also has many years of experience working with petroleum and petrochemicals plants for a wide variety of reactors and fired heaters including ethylene-cracking units, hydrogen-reforming units, and various process heaters. Joseph Colannino has written and reviewed problems for the National Council of Examiners for Engineering and Surveyors (NCEES); its professional engineering exams are used in all 50 states for engineering licensure. He is a registered professional engineer in the State of California. More recently, he served as director of engineering for the burner group at John Zink, LLC, and is currently the manager of knowledge systems there. He is a current or past member of the following institutions and societies: Air and Waste Management Association (AWMA), American Chemical Society (ACS), American Institute of Chemical Engineers (AICHE), American Statistical Association (ASA), Combustion Institute, Instrument Society of America (ISA), and the National Society of Professional Engineers (NSPE). He is widely published, having contributed to many articles and several texts including Air Pollution Control Equipment Selection Guide (Schifftner/ Lewis Publishers), Industrial Burners Handbook (Baukal/CRC Press), and The John Zink Combustion Handbook (Baukal/CRC Press) and is listed in several Who’s Who compilations.
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Prologue A process engineer at a refinery may ask, “How can I model flame quality, or NOx and CO emissions, or unit performance as a function of various fuel and air preheat scenarios?” A controls engineer may wonder, “How can I construct a model for a feedforward algorithm from available plant data?” A control room operator at an electrical utility may ask, “How does the H/C ratio in my fuel affect the unit’s feed rate and flue gas emissions?” Others may ponder, “How will an increase in air preheat temperature affect the radiant heat flux of my process unit?” or “How can I design an experimental series to give me the answers I need?” Most of these practitioners will have two things in common: they will have only days or weeks to conclude their analysis, and they will not have a Ph.D. in combustion modeling. This book is written for them — the practicing engineer or operator. This text introduces semiempirical combustion modeling to practicing engineers and operators of fired equipment. This equipment includes boilers, process heaters, and reactors such as ethylene cracking units and ammonia and hydrogen reformers, among others. This text illustrates its examples from the field of combustion and provides the following how-to information: • How to formulate basic theoretical combustion-related models from first principles where possible • How to augment these models with some adjustable parameters as necessary, or develop wholly empirical models when theory is lacking • How to fit adjustable parameters from data • How to validate such models • How to design experiments to collect the most data in the shortest possible time and at the lowest possible cost • How to model real unit behavior with engineering accuracy There are many books on modeling in general, and some on combustion modeling in particular. However, there seems to be no furnace and combustion modeling texts that show how to generate usable models from collected data. The approach we advocate is a simple one. First, derive a mathematical model of the system from first principles to the extent possible. We derive many such models for fuel flow, airflow, NOx, CO, flame length, radiant heat flux, etc., as a function of excess air rate, fuel flow and composition, air preheat temperature, and the like. These are applicable to a wide variety of fired equipment, including boilers, process heaters, and fired reactors such as hydrogen reformers or ethylene cracking units. A number of fully worked examples support the text, along with step-by-step derivations.
© 2006 by Taylor & Francis Group, LLC
Second, declare gaps in knowledge with adjustable parameters and error terms, and fit these with actual data from the unit itself. The data may be historical or require on-purpose experiments. In either case, we show how to analyze the data, perform statistical tests, and fit the model parameters. If the practitioner must decide which experiments to perform, we provide guidance and an overview of statistical experimental design principles. We show how to generate the maximum data from the fewest experiments, how to analyze them, and how to determine the parameter values. Sometimes, one cannot collect all the data or fully randomize the experiments. We show how to derive accurate models nonetheless that reduce the response of interest to a semiempirical mathematical model. Others may be just starting their career at a municipal solid waste facility, utility, refinery, or petrochemical plant. Their undergraduate education has not emphasized burners and fired heaters. What is a wicket tube process heater or a radiant wall burner, anyway? Accordingly, this text provides brief overviews to the many kinds of burners, fired heaters, and reactors that one may encounter. The work is organized into five chapters. At its conclusion, an engineer or scientist will be able to understand and apply the concepts in this text to real combustion problems of interest. The book provides most of the recipes for constructing the modeling equations; application to related fields using the method will be straightforward. Throughout the text, we reinforce the material with fully worked examples. We also work nearly all of the examples using spreadsheet software — not because this is the ideal platform (often it is not), but because access to such a computer tool is widespread. Dedicated statistical or mathematical software has major advantages and occasionally we use some. The overwhelming emphasis is on practical analysis with widely available personal computing tools. Chapter 1 introduces modeling and includes basic model categories and analytical methods, such as dimensional analysis, the method of least squares, and a primer on linear algebra. These techniques are seminal for the type of modeling we use. Those with no prior experience will learn all they need to know about them in this chapter. Readers with a thorough understanding of these concepts may skip it. Chapter 2 introduces combustion. It describes most of the combustion equipment one is likely to encounter, including all manner of burners, process heaters, boilers, and some reactors. We have tried to focus on adding descriptions for practical equipment that is not described very well elsewhere. Our focus is on the combustion side rather than the process side. References at the end of the chapter will point the way to more thorough information. The second part of the chapter provides mathematical models for most of the important mechanical and combustion relations. Chapter 3 introduces experimental design. It covers the standard fare for design of experiments, including important statistics, the analysis of variance, two-level factorial and fractional factorial designs, second-order designs, and sequential design strategies. References at the chapter’s end indicate some
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of the many excellent design-of-experiment texts. The contribution here is to apply the methods to combustion experiments in particular. Chapter 4 shows how to analyze nonideal data, such as plant and historical data, nonorthogonal data, and some diagnostics for hidden extrapolation, and some postanalytical procedures. We then treat analysis of designs with restricted randomization, such as nested and split-plot designs, and rules for constructing expected mean squares for performing a valid analysis of variance on balanced data. We treat mixture designs here as well, with particular application to fuel chemistry and refinery gases, including ways to simplify multicomponent mixtures to ternary or quaternary blends, and construct simulation blends for complicated mixtures. We also show some methods for embedding mixture designs within factorial ones. Chapter 5, “Semiempirical Models,” draws together the individual chapters to combine theoretical models with experimental and analytical strategies for constructing accurate semiempirical models. These include models for NOx and CO; many NOx reduction strategies, such as flue gas recirculation; premixing, staging, blending, and postcombustion strategies, such as selective catalytic and noncatalytic reduction; etc. The text also treats radiant flame profiles and flame lengths and other combustion-related models. Such models represent a concise expression of real-world behavior with sufficient engineering accuracy to be useful for control, optimization, and descriptive and predictive purposes. A series of appendices give important physical and chemical properties for fuels and combustion-related quantities, mechanical properties for pipe and drill bits, K factors for fittings and geometries, some common unit conversions and dimensions, and statistical tables. An unabridged table of nomenclature for the text precedes this prologue. As this is an introductory text in combustion modeling, it is written for those without any special expertise in this field. However, the author has not avoided detailed mathematics, nor would it be possible for such a text. Some of the derivations require calculus, knowledge of thermodynamics, mechanical energy balances, and the like. However, readers with at least some college engineering courses should see their way through these. If not, one can always take the final equation on faith. Though that is less satisfying, the final outcome will be the same — accurate semiempirical models developed in short order for the process unit of choice. In my experience as a combustion consultant for nearly 20 years, and now as a burner designer these last 7 years, my respect for practicing engineers and equipment operators has only increased. I hope that this text will not disappoint them. In that vein, I would appreciate any suggestions, data, or problems that readers may think beneficial for future editions. I will also be very grateful to those who care enough to contact me with any errata at http://www.combustion-modeling.com. Joseph Colannino Luxembourg
© 2006 by Taylor & Francis Group, LLC
1 Introduction to Modeling
Chapter Overview We begin this chapter by looking at various model categories and associated kinds of experiments. We then survey various analytical methods such as qualitative analysis and dimensional analysis and introduce function shape analysis. Because many times we consider multidimensional data, we devote a section to perceiving greater than three dimensions. We then define and discuss basic data classifications, i.e., nominal, ordinal, interval, and ratio data. Distinguishing among these data types is important — some mathematical operations have no meaning for certain data types. In other cases, we must change our analytical methodology. In closing, we provide a primer on linear algebra and least squares, and some proofs regarding a generalized mean.
1.1
Model Categories
We consider three possible kinds of mathematical models, each having two subdivisions: • Theoretical models – Fundamental – Simulations • Semiempirical models – General models with adjustable parameters – Dimensionless models with adjustable parameters • Empirical models – Quantitatively empirical –
Qualitatively empirical 1
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2 1.1.1
Modeling of Combustion Systems: A Practical Approach Model Validation
Validation is the testing of the model with data from the situation of interest. All models must be validated, but for different reasons. Theoretical models require validation to define their applicable range. For example, the ideal gas law is wrong at high pressures and low temperatures. Newton’s second law of motion is wrong at high speeds approaching the velocity of light. But within their spheres of applicability they are highly accurate. Simulations require validation because there may be errors in the computer code, illconditioned problems, improper convergence, etc. Semiempirical models require some data to determine values of the adjustable parameters. They cannot even begin without some valid data. But a well-formulated semiempirical model may or may not be valid for new situations, so one must validate these. Empirical models build from particular data. As a rule, extrapolation to new conditions will yield erroneous results. However, if one is within the bounds of the original data set, empirical models are excellent interpolators and give valid estimations. Extrapolation is not always obvious for a complex multidimensional data set (termed hidden extrapolation; Chapter 3 gives some tools for finding it). Another way of looking at things is to consider that extrapolations of the various kinds of models reference different domains. For fundamental physical models, anything within the system of physics is an interpolation. For semiempirical models, data representing similar systems are an interpolation. For empirical models, interpolation is constrained to values bounded by the original data set. Therefore, regardless of the model type, validation is always a good practice and one should carefully assess whether or not the results represent extrapolation. 1.1.2
Fundamental Theoretical Models
If we have a very good understanding of the system, it may be possible to formulate a theoretical model. By theoretical model, we mean a model with no adjustable parameters, i.e., an a priori model with known form, factors, and coefficients. A valid theoretical model represents the highest level of understanding. A theoretical model, insofar as it represents the actual physics involved, can make accurate generalizations about new situations. 1.1.3
Simulations
A simulation is a computer-generated result based on the solutions of fundamental physical equations. An example is computational fluid dynamics (CFDs), which solves simultaneous equations for mass, momentum, and heat transfer. It is not possible to do this without some simplifications, so one must validate the results. However, once validated, the simulations are quite accurate and good extrapolators for new problems of the same kind. For example, CFD has shown itself to be valid for fluid flow and heat transfer
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in combustion systems. However, it is not generally valid for quantitative combustion kinetics (e.g., as NOx and CO formation), though it may indicate trends. At present, simulation science is a dedicated profession requiring detailed domain knowledge. 1.1.4
Semiempirical Models
The next level of modeling is semiempirical modeling — the main subject of this text. A semiempirical model is a model whose factors and form are known a priori, but where some or all of the coefficients are determined from the data. The form of the model may arise from theoretical considerations or dimensional considerations, or both. Semiempirical models can also correlate simulation data in order to make a smaller, faster, and more portable model. This kind of empirical fitting requires some special considerations because simulation data have no random error, only bias error. 1.1.5
Dimensionless Models
Dimensionless models are semiempirical models that arise from a consideration of the units involved in the system physics. Dimensions in this sense are synonymous with fundamental units, e.g., mass [M], moles [N], length [L], temperature [T], and time [θ]. (This text uses Arial or Greek typeface enclosed in square brackets to denote a fundamental unit dimension. If we wish to specifically note a dimensionless quantity, we will use empty square brackets [ ].) Dimensionless models assess the system physics only in the sense of understanding the units associated with the phenomena. Moreover, we must presume the model form — generally it is a power law relation. That is, the model has the form n
Y=C
∏X
ak k
= CX1a1 X2a2 X3a3 ...Xnan
(1.1)
k =1
where Y is the dependent variable, a dimensionless group, whose behavior we wish to correlate; C is a constant; k indexes the n dimensionless coefficients; X is a dimensionless independent factor or factor group; and a1 to an are the exponents for the dimensionless groups. If the dimensionless model is valid, it provides a scaling law for testing, enabling a study of system behavior at other than full scale. 1.1.6
Empirical Models
Finally, we may consider empirical models. We distinguish two types. In a quantitatively empirical model, we know most or all of the factors a priori, but we do not know the form of the model or the coefficient values. However,
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Modeling of Combustion Systems: A Practical Approach
we do know qualitatively which factors belong to the model. One could dare to define an even lesser type of an empirical model: in a qualitatively empirical model, we know nothing but the response with certainty; we do not even know qualitatively most of the factors that are important, though we may have a menu of possibilities (candidate factors). The data themselves determine the model form a posteriori in some post hoc procedure.
1.1.7
Problems with Post Hoc Models
All post hoc procedures have pitfalls; there is a good chance that one may develop a senseless model. Consider the analogy of the drunken shooter who takes the bet that even in his inebriated state, he can hit a target with good accuracy during the pitch-black evening. Amid the company’s sporadic laughter, all agree that the drunken shooter will fire five rounds into his vacant barn and all will retire for the evening to examine the results at daylight. About 10:00 A.M. they awaken. Upon inspection, and to all but the shooter’s amazement, three of five rounds are in the center of the target, one is not too far off the mark, and only one has strayed a considerable distance. He wins the bet handily. Now only the shooter is laughing. What happened? At first light the shooter walked outside and drew a target around his best three-shot group before stumbling back to bed to reawaken with the others. This is the pitfall of any post hoc procedure. If we decide what the model is after the fact, we are prone to commit this kind of error. The problem grows worse with smaller data sets and larger numbers of candidate factors, as these elevate the probability of finding a senseless model that fits the data. Senseless factors or coefficients with the wrong signs often betray this fallacy.
1.2
Kinds of Testing
There are three kinds of testing: no testing, scale testing, and full-scale testing.
1.2.1
No Physical Testing
A thorough and complete theoretical knowledge requires no additional testing. This is the case for certain fluid flow problems, freefall of bodies, etc. These mathematical models signify the highest level of modeling and represent great cost, design, and time advantages. Simulations based on rigorous solution of fundamental physics also fall into this category in the sense that they model behavior without physical testing. However, no simulation perfectly captures all the physics, and therefore, we must validate such models with some confirmatory data. Depending on the speed of coding the problem into the computer, the time for convergence of the simulation, and
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the cost of computer time, it may actually be less expensive to perform physical experiments.
1.2.2
Scale Testing
Scale models are physical models at other than actual size. If the investigator is successful in specifying the necessary and sufficient factors that determine the system response, then one may determine appropriate dimensionless groups or other similarity parameters. If one understands which similarity parameters are actually necessary and sufficient to characterize the system, then one may construct a physically similar system at other than actual size (scale model). Scale testing often inures significant cost and time advantages. While an accurate theoretical model represents the ultimate in terms of reduced cost and time saving, scale testing often represents a significant cost and time advantage over full-scale testing. Moreover, a scale unit has greater flexibility than the actual unit.
1.2.3
Full-Scale Testing
Full-scale testing represents the lowest level of understanding in the sense that we are so unsure of the results that we will only believe a full-scale test. However, full-scale testing does not always represent the greatest test burden. For example, many full-scale units are sufficiently instrumented and historical data so well preserved that one requires only limited additional data. This may (or may not) be the case for so-called plant data. By plant data, we mean the historical data records of an operating process unit. Plant data often suffer from a number of statistical maladies that we will address later. Notwithstanding, very often plant data play a very useful role in model development, and the actual process unit is the ultimate target for the model development in the first place. Full-scale testing is not always the most expensive alterative. For many of the processes we consider, full-scale testing is required because a theoretical model is simply intractable and scale testing is not credible. Although process units are dumb and mute, they can physically solve what we can barely formulate: highly coupled nonlinear systems of differential equations comprising simultaneous chemistry and heat, mass, and momentum transfer.
1.3
Analytical Methods
Associated with and derivative of our degree of knowledge are several analytical methods. A theoretical analysis is an a priori analysis from first principles. It will consider first principles, physics, and domain knowledge
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Modeling of Combustion Systems: A Practical Approach
(detailed knowledge of a particular physical system) in order to come up with a theoretical model, or at least a model form. We make a determined effort to provide such models for most features of combustion, including fuel flow and airflow, emissions such as NOx and CO, flame length, and heat flux. We introduce these as the need arises beginning in Chapter 2. A dimensional analysis is an assessment of a plausible system model based on dimensional consistency. Usually we will need some understanding of the system in order to arrive at appropriate candidate factors. A first step for dimensional analysis is often a qualitative analysis. A qualitative analysis is a derivation of the model factors based on domain knowledge. If we know at least the general shape of the factor response relation, we may perform a function shape analysis. A function shape analysis is a derivation of a plausible model form based on response behavior. In this section we first treat qualitative analysis, then dimensional analysis, and finally introduce the reader to function shape analysis.
1.3.1
Qualitative Analysis
A qualitative analysis will seek to identify the important factors and assign some sign to the candidates (+, 0, –) based on what we know about the system; that is, will an increase in the factor increase the value of the response (+), leave it unchanged (0), or decrease it (–)? If the candidate does not change the response (0), then we remove the factor from the model. This is a valuable analysis to perform prior to any modeling effort because it forces one to think about the system and advance a hypothesis. Now one of several things will happen: 1. The data support the model and the investigator has confirmed his intuition and understanding of the system. 2. The model coefficients are not what the investigator expected (the wrong sign or unexpected magnitudes), in which case: a. The model is wrong and the investigator must revise it; thereby learning occurs b. The model is right and the design of the experiment or the collection of the data has errors, which the investigator must find. All of these outcomes are beneficial.
Example 1.1
Qualitative Analysis
Problem statement: Consider NOx as a function of the following factors: • x1, the furnace temperature • x2, the furnace excess oxygen concentration
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Introduction to Modeling • x3, the • x4, the • x5, the • x6, the • x7, the
7
hydrogen content in the fuel fuel pressure burner spacing, centerline to centerline air preheat temperature absolute humidity
Qualitatively, what will be the effect of these factors? Solution: Solving such a problem requires expert knowledge. For the practitioner that is new to the field, this will require brief interviews with experts. This is usually an extremely valuable exercise. In the case where interviewees agree, the interviewer can say that he has established consensus. Although consensus is not always right, it is usually right, so we start here. Mixed opinions force a more critical evaluation of the system. In such cases, open disagreement should not only be tolerated but embraced, because only what we do not know constitutes new learning and knowledge. Even the seasoned engineer should consider outside opinions. The opinions of juniors or new employees can be a good source of new and creative thinking. Such employees have had less exposure to the status quo. In such encounters, at least one party will learn something. Table 1.1 comprises the author’s opinion. TABLE 1.1 Qualitative Analysis of NOx Formation Factor 1 2 3 4 5 6 7
Factor Description Furnace temperature Excess oxygen concentration Hydrogen content in the fuel Fuel pressure Burner spacing Air preheat temperature Absolute humidity
Sign
Strength
+ + + – – + –
Strong Moderate Moderate Weak Weak Strong Weak
Consider another example of qualitative analysis regarding a common plant operation: fluid flow in a pipe.
Example 1.2
Qualitative Analysis for Fluid Flow in a Pipe
Problem statement: Postulate candidate factors that might be important to correlate pressure drop associated with friction of a flowing fluid.
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Modeling of Combustion Systems: A Practical Approach Solution: Most engineers have adequate exposure to fluid flow and conservation of mechanical energy. We delay these topics until Chapter 2. However, intuitively, we may understand that the following factors are important and that they affect the value of the response (pressure drop): • v, the velocity of the fluid [L/θ] • ρ, the density of the fluid [M/L3] • • • •
D, the diameter of the pipe [L] μ, the viscosity of the fluid [M/Lθ] δ, the roughness of the pipe [L] L, the length of the pipe [L]
One might imagine other fluid factors to be important, such as surface tension, diffusion coefficient, etc. However, it turns out that these are not important considerations for macroscopic pipe flow.
1.3.2
Dimensional Analysis
Dimensional analysis takes an appropriate qualitative analysis a step farther by refining the equation form. A dimensional analysis is a method of establishing a model from consideration of the dimensions (units) of the important factors. See Appendix C, especially Table C.2, for dimensions of common combustion-related factors. Dimensional analysis finds dimensionless factor ratios. The method can drastically reduce the number of factors required for fitting and correlation. Buckingham Pi theory gives the degrees of freedom of the system as a function of the number of factors (f ) and dimensions (d) according to F= f −d
(1.2)
If the dimensionless parameters are the proper ones, then we have discovered similarity and we gain an ability to perform scale testing. This combination of reducing the scale of our testing and the scope (by reducing the required number of factors needing investigation) has great economic benefits. Generally, our experimentation (cost and time, C) will be proportional to some base (b) to the exponent of the number of factors (nf ), C ∝b
nf
(1.3)
with b = 1.4 as a typical value. To see the kind of reductions that are possible, consider the following example.
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Introduction to Modeling
Example 1.3
9
Reduction in the Degrees of Freedom from a Dimensional Analysis
Problem statement: Use a Buckingham Pi theory to decide if dimensional analysis can reduce the system of Example 1.2 to fewer factors. If so, how many dimensionless factors will there be? Estimate the possible savings in cost and time. Solution: From Equation 1.2 we need to find f and d. We have six factors (ΔP is not a factor but a response). We find d by listing the unit dimensions for all factors and the response. A matrix of exponents organizes these most conveniently:
μ δ v D ρ L ΔP
M 1 0 0 0 1 0 1
L −1 1 1 1 −3 1 −1
θ −1 0 −1 0 0 0 −2
For example, the first row of the above matrix specifies μ = M Lθ . Collectively, the above factors comprise three dimensions (d = 3: M, L, and θ). Therefore, we can combine these six factors into F = f – d = 6 – 3 = 3 groups. So yes, dimensional analysis can dramatically reduce the system from six factors to three. The approximate reduction in cost and time is (1.46 − 1.43 ) 1.46 = 64%. In other words, if the foregoing is true, we can cut our experiments by nearly two thirds.
1.3.3
Raleigh’s Method
The Raleigh method is a technique to determine dimensionless groups. The general idea is as follows: 1. Use a qualitative analysis to develop a system of candidate factors. 2. Write a power law model according to Equation 1.1.
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Modeling of Combustion Systems: A Practical Approach 3. Construct a dimensional matrix with undetermined exponents for all n factors and the response. 4. For each dimension, write an equation in the exponents. 5. Choose f – d independent exponents to comprise as many dimensionless groups. Select exponents corresponding to factors you believe will constitute separate dimensionless groups. 6. Solve for the remaining exponents in terms of the f – d exponents. 7. Group terms under each exponent. 8. Determine the independent exponent values from experiment.
We illustrate with an example.
Example 1.4
The Raleigh Method for Dimensional Analysis
Problem statement: For the pipe flow problem of Example 1.2, use the Raleigh method to determine the proper form for the exponents. Solution: Step 1: From Example 1.2 we obtain the candidate factors. Step 2: We construct the following power law model: ΔP = Cμ a1 δ a2 v a3 D a4 ρa5 La6 Step 3: We construct the following matrix, augmented with coefficients:
μ δ v D ρ L ΔP
M 1 0 0 0 1 0 1
L −1 1 1 1 −3 1 −1
θ −1 0 −1 0 0 0 −2
a1 a2 a3 a4 a5 a6
Step 4: For each dimension, we obtain the following equations: For M:
1 = a1 + a5
For L:
–1 = – a1 + a2 + a3 + a4 – 3a5 + a6
For θ:
–2 = – a1 – a3 or 2 = a1 + a3
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Step 5: There are six factors and three dimensions, leaving 6 – 3 = 3 degrees of freedom. We select μ, δ, and L as factors that we wish to have in separate dimensionless groups. These correspond to the exponents a, a2, and a6, respectively. The selection is arbitrary. We could have selected any three factors. In fact, we can multiply any two dimensionless groups we choose to create a new dimensionless group after the fact if we desire. Step 6: From the first equation, we obtain a5 = 1 – a1, and a3 = 2 – a1 from the third. Solving the second equation for a4 gives a4 = –1 + a1 – a2 – a3 + 3a5 – a6. Substituting the above equations into this gives a4 = –1 + a1 – a2 – (2 – a1) + 3(1 – a1) – a6 or a4 = –(a + a2 + a6). Step 7: Collecting factors under common exponents we obtain a
a
⎛ Δp ⎞ ⎛ μ ⎞ 1⎛ δ ⎞ 2⎛ L⎞ C = ⎜⎝ ρv 2 ⎟⎠ ⎜⎝ Dvρ ⎟⎠ ⎜⎝ D ⎟⎠ ⎜⎝ D ⎟⎠
a6
The response is the Euler number NEu, the second dimensionless group is the reciprocal of the Reynolds number, 1/NRe, the third quantity is the relative pipe roughness, and the fourth is the length-to-diameter ratio of the pipe. Since the exponents are undetermined anyway, for convenience we will substitute a1 for –a1 and write the following: a2
a1 ⎛ δ ⎞ ⎛ L ⎞ N Eu = CN Re ⎜⎝ D ⎟⎠ ⎜⎝ D ⎟⎠
a6
(1.4)
If a power law is applicable and we have selected the appropriate factors, we should be able to correlate pressure drop due to friction with only three factors: NRe, δ/D, and L/D. Moreover, systems having identical values of these three factors should behave identically. That is, NRe, δ/D, and L/D are the similarity parameters and define similarity for any system scale. 1.3.3.1 Cautions Regarding Dimensional Analysis This is quite a substantial reduction in experimental effort. It is all the more amazing considering we have derived a model knowing nothing but the fundamental units of the important parameters. However, let us list our heretofore tacit modeling assumptions: • We have specified all of the important factors. • A power law model is appropriate. • The model has physical significance.
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Modeling of Combustion Systems: A Practical Approach
By no means should we take these assumptions for granted. It turns out from experiment and theoretical considerations that N Eu =
1 ⎛ L⎞ NF ⎜ ⎟ ⎝ D⎠ 2
where δ⎞ ⎛ N F = φ ⎜ N Re , ⎟ ⎝ D⎠ Here, NF is the Darcy–Weisbach form of the friction factor — a function of the Reynolds number and the relative roughness of the pipe — and φ( ) denotes an implicit function of the enclosed factors with respect to the response, NF . That is, Equation 1.4 conforms to the present case when C = 1/2 and h = 1 if a ⎛ δ ⎞ N F = N Re ⎜⎝ D ⎟⎠
b
(1.5)
We can correlate data with the above equation, so in the present case the method will work. However, the equation ⎛ 1 δ 1 2.51 ⎞ = −2 log ⎜ + ⎟ NF ⎝ 3.7 D N Re N F ⎠ is the most widely used equation to correlate the friction factor data.* Inverting gives NF =
−1 ⎛ 1 δ 2.51 ⎞ 2 log ⎜ + ⎟ ⎝ 3.7 D N Re N F ⎠
(1.6)
Clearly, Equation 1.6 requires an iterative solution for NF; it is not conformable to Equation 1.5. In the present case, we have only correlated the pressure drop due to friction. It turns out that Equation 1.7 correlates the total pressure drop, friction + fluid motion: N Eu =
( ∑ K ) + 12 N ⎛⎜⎝ N
1 1+ 2
* This is known as the Colebrook equation.
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F
Re
,
δ⎞ D ⎟⎠
(1.7)
Introduction to Modeling
13
where ΣK is the sum of K-factors (friction factors) for all flow disturbances, such as elbows, tees, sudden contractions and enlargements, etc. (See Table B.3 in the Appendices for such K-factors.) Since Equation 1.7 has additive terms, it is not strictly a power law model and a dimensional analysis may not arrive at this form. Furthermore, unless we know a lot about the system we are studying, there is a very real possibility that we will miss some important factors. For example, we know that the molar volume of a gas is a function of the pressure, and temperature only over quite a wide range. Can we perform a dimensionless analysis and reduce the system?
Example 1.5
The Raleigh Method Applied to an Ideal Gas
Problem statement: For ideal gases, the molar volume of a gas is a function of the pressure and temperature only. Use a dimensional analysis to propose useful dimensionless groups for correlating the data. Solution: We organize our data in a matrix:
P V T
M 1 −1 0
L −1 3 0
T 0 0 1
θ −2 0 0
However, the matrix has more dimensions than factors. This means we have F = 3 – 4 = –1 degrees of freedom. But unless F ≥ 0, the matrix is overspecified and insoluble. How can this be? It can be because we have neglected a very important parameter — the universal gas constant, R. Without it, we cannot develop a dimensionally consistent equation. Augmenting the matrix with R, we obtain a soluble matrix, F = 4 – 4 = 0:
P V T R
M 1 −1 0 0
L −1 3 0 2
T 0 0 1 −1
θ −2 0 0 −2
We have no degrees of freedom; therefore, only one dimensionless group is possible, and it is PV RT . The power law model would be α
⎛ PV ⎞ ⎜ RT ⎟ = k ⎝ ⎠
© 2006 by Taylor & Francis Group, LLC
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Modeling of Combustion Systems: A Practical Approach and we would evaluate the constants, k and α, from the data. If the data were from a combustion system, the ideal gas law would be adequate and our data would generate α = 1 and k = 1.
In the above example, we could not form the proper dimensionless relationship because we did not have all of the starting parameters. However, even if the starting factors can form a consistent dimensional equation, that does not guarantee that the dimensionless equation will be appropriate: for real gases at low temperatures or high pressures, the dimensionless formulation in the above example would not be adequate. For example, if we define Z = PV RT, we can rewrite the ideal gas law as PV = ZRT. From statistical mechanics, one may write a virial equation: a a Z = a0 + ˆ1 + ˆ 22 + ... = a0 + V V
∞
∑ Va
k k
k =1
where a0 – ak are constant coefficients. If the virial equation is the governing equation, then we have improperly specified our slate of starting factors because we have excluded terms having powers of reciprocal volumes. Below is another model. The principle of corresponding states says that real gases behave very much like one another if they comprise the same ratio with their respective critical pressures (Pc) and temperatures (Tc). Such critical properties are unique for each pure gas, and Z = φ(Tr , Pr), where Tr =
T P , Pr = Tc Pc
and φ( ) designates an implicit (unspecified) functionality. We refer to Tr and Pr as the reduced temperature and pressure, respectively. Since it did not occur to us to list the critical pressure and temperature in our factor list for the dimensional analysis, we did not derive these dimensionless parameters. We conclude our treatise on dimensional analysis with a final caution: cramming four factors together in a dimensionless group does not hold the universe hostage. Unless one has done experiments validating all the factors over several levels, one has not established general similarity. While it is true that a valid equation must be dimensionally consistent, the converse is not true; dimensionally consistent equations may be nonsense. For example, P = μ 2 L is dimensionally consistent, but does not have physical meaning for macroscopic flow in a pipe. The author recommends dimensional analysis — it can be a big time and cost saver. Use the technique, but be aware of its pitfalls and let the data validate your conclusions. Dimensional analysis is a very powerful tool for establishing similar systems and correlating data. Many engineers use it in
© 2006 by Taylor & Francis Group, LLC
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a wide variety of disciplines. But it is only accurate if our starting model is appropriate (e.g., a power law model) and provided that we have specified and varied all the physically significant factors. Table 1.2 summarizes a few dimensionless groups that are often important in combustion. TABLE 1.2 Some Dimensionless Groups Dimensionless Number
Ratio
Symbol
Euler Reynolds Froude
Pressure/inertial Inertial/viscous Inertial/gravitational
NEu = ΔP/ρv2 NRe = Lvρ/μ NFr = v2/gL
Prandlt
Viscous Momentum diffusivity = Thermal Thermal diffusivity
NPr = Cpμ/k
Force
Group
Buoyant Gravitational Inertial Pressure
ΔρgL ρgL ρv2 ΔP k ρv ⋅ ρCp L μ ρv ⋅ ρ L
Thermal Viscous
Note: P is pressure [M/Lθ2]; ρ is density [M/L3]; v is velocity [L/θ]; L is characteristic length, height, or diameter [L]; μ is viscosity [ML/θ]; g is gravitational acceleration [L/θ2]; Cp is isobaric heat capacity [L2/θ2T]; and k is thermal conductivity [ML/θ3T].
1.3.4
Function Shape Analysis
Here we introduce the reader to a new kind of analysis — function shape analysis. By function shape analysis, we mean a technique for developing equations by direct inspection of the curvilinear form. This also is a weak form of analysis because we do not consider any underlying physics, except as expressed by the shape of the function. Notwithstanding, function shape analysis can provide some insight or at least obviate an incorrect model form. Consider the graph of Figure 1.1. Per the usual convention, we graph the dependent variable (y) on the vertical axis (positive direction up) and the independent variable (x) on the horizontal axis (positive direction right). It has two asymptotes: a vertical one at x = 0 and a horizontal one at y = 0 given by the dashed lines. Obviously, the data are not linear or even parabolic; e.g., models of the form y = a0 + a1x or y = a0 + a1x + a2 x 2 are not appropriate. Traditionally, practitioners characterize functions by the monikers linear, second order, third order, etc., referring to the highest power of the independent variables. These are quite familiar shapes (lines, parabolas, cubics, etc.), and they are used in all technical disciplines. However, this
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Modeling of Combustion Systems: A Practical Approach
y = f (x) 1 f (x) = x 0 < x < ∞ y
x
FIGURE 1.1 A graph with asymptotes. The graph has asymptotes indicated by the dashed lines — the curve approaches but never reaches these lines for any value of x or y. In the present case, the vertical line corresponds to x = 0 and the horizontal line corresponds to y = 0.
kind of characterization is not what we are after. Function shape analysis uses a different nomenclature based only on the shape of the function, not the underlying equation. The presumption is that we do not know the underlying equation; we know the function shape. We shall consider 12 basic shapes per Figure 1.2. vh 1 (x) = x 0 < x <∞ (a)
(b) l
(c) umu
(x) = –x
(x) = x2 –∞ < x < ∞
–1≤ x≤ 1
–∞ < x < ∞ (e) vmh 1 1 (x) = 2– x x 0 < x <∞
hmh 1 1+x2 –∞ < x < ∞ (i)
(x) =
(f ) vl 1 (x) = –x x 0 < x <∞
(j) vml x2 (x) = 1+x –1 < x < ∞
(d) m2 (x) =±√1–x 2
(g) vu 1 (x) = + x2 x –∞ < x < 0
(k) vmu 1 (x) = + x 2 x 0 < x <∞
(h) v 2 x (x) = –1+x2 –1 ≤ x ≤ 1
(l) vmv x2 (x) = 1–x2 –1 ≤ x ≤ 1
FIGURE 1.2 Basic function shapes. The chart shows shapes for functions that do not exceed second order and have no more than one relative extremum. In some cases the graphs have been scaled for convenience of viewing. For graphs with poles, the pertinent part is shown.
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We use the following five descriptors: • v indicates a vertical asymptote. • u indicates an unbounded curve. • m indicates a maximum or minimum point in the curve (i.e., maxima), either local or absolute. • l indicates a linear asymptote or straight line. • h indicates a horizontal asymptote. Therefore, we characterize the curve of Figure 1.1 as vh because it has a vertical asymptote followed by a horizontal asymptote. The 12 figures shown in Figure 1.2 represent all the simple two-dimensional function shapes. By simple, we mean continuous curves with no more than one descriptor each for near-, middle-, and far-field behavior. The above nomenclature is sufficient to characterize more complicated forms, but the simple shapes are more than enough for our purposes. In addition, the function y = φ( x) may undergo various transformations. If a function shape retains its nomenclature after a transformation, we refer to the function as being invariant under that transformation. The above function shapes are invariant under the following transformations: • Translation: – Vertical translation by +a units: y = φ( x) + a – Horizontal translation by +a units: y = φ( x − a) • Reflection: – Across the y axis: y = φ(− x) – Across the x axis: y = −φ( x) – Across the line y = x: x = φ( y) • Rotation: – By any counterclockwise angle θ : ξ( x) = − x cos(θ) + φ( x)sin(θ) Φ( x) = − x sin(θ) − φ( x)cos(θ) – – –
where φ and ξ are the new vertical and horizontal axes, respectively. One may substitute –θ for clockwise angles. For the special case of θ = 90° rotation (π/2 radians): x = φ(− y). For the special case of a 45° (π/4 radians) rotation: ξ( x ) = ⎡⎣φ( x ) − x ⎤⎦ Φ( x ) = − ⎡⎣φ( x ) + x ⎤⎦
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2 2
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Modeling of Combustion Systems: A Practical Approach
• Scaling: – Vertical scaling [stretch (a > 0) or shrink (a < 0) vertically by a times]: y = a ⋅φ( x) – Horizontal scaling [stretch (a > 0) or shrink (a < 0) vertically by a times]: y = φ x a
( )
However, functions with arguments greater than zero remain invariant under positive power transformations. The function shapes are not necessarily invariant under the following transformations: • Power scaling by a: y = φ a ( x) • Exponential scaling: y = e φ ( x) In general, a modified Laurent series having terms in order ranging from –2 to 2 can produce all the basic shapes given by Figure 1.2. By modified Laurent series, we mean specifically Equation 1.8: y=
b2
+
(x − c ) ( 2
2
b1 + a0 + a1x + a2 x 2 x − c1
)
(1.8)
We can fit the constants b2, b1, a0, a1, and a2 with least squares if we know c1 and c2. If either b1 or b2 is nonzero, then y will have a vertical asymptote at x = c1 or x = c2, respectively; thus, we can estimate c1 and c2 by inspection of the curve. That is not to say that other functional forms cannot give the same shapes — they can. However, the Laurent series provides the most convenient method for generating them. 1.3.5
The Method of Partial Fractions
In order to arrive at the modified Laurent series, it may be necessary to use the method of partial fractions. For those readers unfamiliar with the method, we show it here. It may come about that we wish to transform an equation having a product in the denominator to one having separate denominators, e.g., transform y=
(
1 x+1 x−2
)(
)
to y=
1 1 − 3 x−2 3 x+1
(
)
(
)
The reader may verify that these two equations are identical for all x. We may find the latter expression via the method of undetermined coefficients. That is, we presume the identity
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Introduction to Modeling
19
1
( x + 1)( x − 2 )
≡
A B + x+1 x−2
and then solve for A and B. According to a theorem in linear algebra, this identity always exists. Starting with
1
( x + 1)( x − 2 )
≡
A B + x+1 x−2
(
) (
)
we multiply by the denominator at left, 1 = A x − 2 + B x + 1 . Now if the foregoing relationship is true for all x, then we may select convenient values at our choosing. Thus, if x is 2, then the first term vanishes, and when x is –1, the second term vanishes. For x = 2, we obtain 1 = B 2 + 1 , or B = 1 3 , and when x = –1, we obtain 1 = A −1 − 2 , or A = − 1 3 . In general, we can write
(
)
(
)
A B 1 ≡ + φ1 ( x)φ2 ( x) φ1 ( x) φ2 ( x)
(1.9)
φ 3 ( x) φ ( x) φ ( x) ≡A 3 +B 3 φ1 ( x) g( x) φ1 ( x) φ 2 ( x)
(1.10)
or
where A=
1
()
φ2 ⎡⎣φ1−1 0 ⎤⎦
and B=
1 −1 ⎡ φ1 ⎣φ2 0 ⎤⎦
()
In the above equations φ1(x), φ2(x), and φ3(x) are polynomial expressions having no repeated roots, and φ1−1 ( x), φ−21 ( x), and φ−31 ( x) are the inverse functions. In all cases, the degree of the denominator must exceed that of the numerator. If it does not, we simply divide by the denominator to obtain a number plus a remainder. Then we expand the remainder as a partial fraction. For repeated roots we may write φ 3 ( x) φ ( x) φ ( x) φ ( x) ≡ A1 3 + B1 3 + B2 23 + φ1 ( x) φ 2 ( x) φ1 ( x)φn2 ( x) φ 2 ( x)
+ Bn
φ 3 ( x) φn2 ( x)
(1.11)
and use the method of undetermined coefficients to solve for A1, and B1 to Bn.
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Modeling of Combustion Systems: A Practical Approach
Example 1.6
Partial Fraction Expansion
Problem statement: Use Equation 1.9 to determine the partial fraction expansion of y=
3 x 2x − 1
(
)
Solution: We rewrite the above equation as ⎡ ⎤ 1 y = 3⎢ ⎥ ⎣ φ1 ( x)φ2 ( x) ⎦ to conform to Equation 1.9. Therefore, φ1 ( x) = x and φ2 ( x) = 2 x − 1. We find the inverse functions by solving φ1(x) and φ2(x) for x: φ1−1 = x = φ1 ( x) a n d φ−21 = x = [φ 2 ( x ) + 1] 2 . T h e n φ1−1 (0) = 0 a n d φ−21 (0) = 1 2 . Then according to Equation 1.9, A=
1 1 1 = = −1 = −1 φ 2 ⎡⎣φ1 ( 0 ) ⎤⎦ φ 2 ( 0 ) −1
and B=
1 φ1 ⎡⎣φ
−1 2
( 0)⎤⎦
=
1 =2 ⎛ 1⎞ φ1 ⎜ ⎟ ⎝ 2⎠
leading to ⎡ 1 3 2 ⎤ 6 3 = 3 ⎢− + − ⎥= x 2 x 1 x − x 2x − 1 2 x − 1 ⎥⎦ ⎣⎢
(
)
(
)
or more succinctly, y=
3 6 3 = − 2x − 1 x x 2x − 1
(
)
Now, it is obvious that vmh is equivalent to hmv, being only a reflection across the y axis. So vmh and hmv are synonymous, i.e., two different terms with the same meaning. This differs from equivocal — calling two different things by the same name. Synonyms present fewer problems than equivocations, but we can eliminate even these by adopting a reverse alphabetic order. Notwithstanding, m will always need to come between two letters when it occurs.
© 2006 by Taylor & Francis Group, LLC
Introduction to Modeling
21 90%
20% 18%
85% 16% 14%
80%
12% 75%
10% 8%
70% 6% 4%
65%
2% 60%
0% 0%
10%
20%
30%
40%
50%
60%
70%
80%
90%
100%
Excess Air ε FIGURE 1.3 Excess air vs. oxygen for CH4.
Example 1.7
Function Shape Analysis
Problem statement: For a process heater combusting fuel in air, the excess air (ε) and oxygen (yO2) have the following general relations: ε = 0 when yO2 → 0 and ε → ∞ as yO2 → 21%. So the function shape is vh . Figure 1.3 gives an example for CH4 combustion in air. Use function shape analysis to select an appropriate explicit function when 0 < yO 2 < 0.21 . Solution: From Figure 1.2, one possible solution for a vh function shape is y = φ( x) = 1 x . We will use the generic function shape variables, letting y = ε and x = yO2 . However, we must transform it to the desired form. First, we reflect it across the y axis: y = φ(− x) = 1 − x = − 1 x . Then we translate it so that it has a vertical asymptote at 21%: y = φ(− x + a) = φ(− x + 0.21) = 1 (0.21 − x). As x → – ∞, y → 0; this is a horizontal asymptote. However, at x = 0, y = 100 21 , which is not the proper value. To obtain the proper value we can adjust it with a vertical translation. For the vertical translation we have y = φ( x ) − a = 1 (0.21 − x ) − 1 0.21 . This has the limiting value of y = − 1 0.21 as x → ∞. However, this is not a significant limitation, as we have already decided that 0 < x < 0.21.
© 2006 by Taylor & Francis Group, LLC
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Modeling of Combustion Systems: A Practical Approach Another way to force the function to zero at x = 0 is to multiply by x. That is, y = x (0.21 − x) . From either of these forms, we may add adjustable parameters to force the function to intermediate behaviors. For example, ⎛ x ⎞ y = a1 ⎜ ⎝ 0.21 − x ⎟⎠ or ⎛ 1 1 ⎞ y = a1 ⎜ − ⎝ 0.21 − x 0.21 ⎟⎠ In fact, from theory (Chapter 2) the actual relation for combustion of fuel in air resembles the first of these equation forms. Specifically, ⎛ ⎞ yO 2 ε = K⎜ ⎝ 0.21 − yO 2 ⎟⎠ where K is a function of the fuel composition.
1.3.5.1 Limitations of Function Shape Analysis One should always view the concept of something for nothing with great suspicion. Earlier we found that dimensional analysis was not a something-fornothing technique. Both the power and poverty of dimensional analysis lie in its presumptions; likewise, for function shape analysis. We know nothing about the system that the shape of the curve does not reveal, and we are examining 2 it only qualitatively. Consider the normal probability curve given by y = e −( x /2 ). In our terminology, this would be an hmh curve. But going backwards, we may not arrive at this equation. For example, y = 1 ( x 2 + 1) also gives the same qualitative curve (hmh). Other equations of this type are ⎛ 1 ⎞ y = a0 ⎜ 2 ⎝ x + 1 ⎟⎠
a1
We can fit a1 to the data using the method of least squares described later in this chapter, if the log transform is appropriate, i.e.,
(
)
ln y = a0′ − a1 ln x 2 + 1
where a′0 = ln a0. 2 Figure 1.4 compares and contrasts the equations y = 1 ( x 2 + 1) and y = e −( x /2 ). The agreement is not spectacular, but the curves are qualitatively the same hmh form. The point is that qualitative analysis will arrive at a qualitatively
© 2006 by Taylor & Francis Group, LLC
Introduction to Modeling
23 1.2 1.0 0.8 0.6 0.4 0.2
-2.5
-2.0
-1.5
-1.0
-0.5
0.0 0.0
0.5
1.0
1.5
2.0
2.5
FIGURE 1.4 Two hmh curves. The solid line is the normal ordinate curve y = exp(–x2/2), while the dashed line is an imposter, y = a0 + a1/(1 + x2). The agreement is not spectacular, but both curves share the same qualitative features.
correct form, but there may be important quantitative differences between the source and target curves. This is an important limitation of the method. Therefore, it is always preferable to develop a theoretical model or to use function shape analysis in concert with theoretical modeling. Notwithstanding, a straight-line fit for this curve would have been futile. A polynomial of second or higher order would have approximated the near-field behavior well, but not the far-field behavior. In contrast, the proper qualitative curve is likely to have better results over all.
1.4
Perceiving Higher Dimensionality
One may extend the method of function shape analysis to higher dimensions. In three dimensions, the curves become surfaces. Greater than three dimensions generates hypersurfaces. The number and variety of possible surfaces grow exponentially as the dimensionality increases. The method loses convenience in greater than two dimensions. Notwithstanding, it is a general benefit to have some perception of higher dimensions, so we collect some methods here. Dedicated texts have addressed this problem in a variety of ways.1–4 With some standard perceptual tricks one may envision data in greater than three dimensions. 1.4.1 A View from Flatland Probably the most famous treatise about multiple dimensions is Edwin A. Abbott’s Flatland: A Romance of Many Dimensions.5 We will follow Abbott’s
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Modeling of Combustion Systems: A Practical Approach
initial approach in expanding our dimensional thinking by first imagining a two-dimensional creature who has no notion of a three-dimensional world until a sphere enlightens him. Figure 1.5 shows such a creature as a sphere penetrates his world.
t4 t3 t2 t1 t0 X
FIGURE 1.5 An observer in Flatland. (1) The sphere is traveling at a slow but steady speed on its way toward a two-dimensional world. (2) The sphere first contacts the plane at t0. The observer (happy face) will see a point in his plane. (3) The point will grow into a circle as the sphere continues its journey. At t1 the observer will see a growing circle. (4) At t2 the circle will reach its maximum diameter and then begin to shrink. At t3 the observer will see the diameter shown on the upper circle. (5) Finally, the circle will shrink back to a point at t4 just before disappearing. The sphere will have now passed completely through the two-dimensional world.
The sphere enters the plane at a point, expanding as a circle. Once the sphere’s equator passes through the plane, the circle appears to shrink to a point and then vanishes. Thus, the two-dimensional creature does not apprehend the sphere as a whole, but as a process in time; time functions as a third dimension. In the same way, a four-dimensional hypersphere entering our three-dimensional world would appear as a point growing as a sphere, then shrinking to a point and vanishing. In our mind’s eye, we can replay this “movie” at will using time as a fourth dimension. 1.4.2 Contour Surfaces A contour map has one less dimension than the object it represents. Thus, two-dimensional contour lines represent a surface having extension in three dimensions. The most familiar is the typical hiking map with contour lines for elevation. The contour lines represent vertical slices of terrain at equal intervals of elevation. Close lines mean steep hills; far apart lines mean shallow ones. So, the three-dimensional contour surfaces for a hypersphere would resemble a series of spheres of various dimensions. Figure 1.6 shows the contour surfaces for a four-dimensional hyperellipsoid.
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Introduction to Modeling
25 fourth dimension
FIGURE 1.6 Contour surfaces for a hyperellipsoid in four dimensions. A hyperellipsoid is represented as a series of three-dimensional contour surfaces. The surface is spherical in three dimensions and elliptical in a fourth dimension.
We may write analytical expressions up to second-order in multiple dimensions without too much trouble. For example, for y = φ(x1, x2, x3, x4), one solves for the variable of interest, say x1, as x1 = φ(x2, x3, x4, y). Each constant contour surface is found by setting y to a desired value (constant) and then tracing x1 = φ(x2, x3, x4) to produce a three-dimensional contour of the hypersurface; x1 = φ(x2, x3) creates contour lines for a surface extending in three dimensions.
Example 1.8
Contour Lines for y = f(x 1, x 2)
Problem statement: Find the general expression to generate contour lines for y = a0 + a1x1 + a2 x2 + a11x12 + a12 x1x2 + a22 x22
(1.12)
Solution: We may choose either x1 = φ( x2 , y) or x2 = φ2 ( x1 , y) as our contour equation. If we choose the latter, the expression becomes a0 + a1x1 + a2 x2 + a11x12 + a12 x1x2 + a22 x22 − y = 0 . We may rearrange this to ( a22 )x22 + ( a2 + a12 x1 )x2 + ( a11x12 + a1x1 + a0 − y) = 0 . Letting A = a22, B = a2 + a12 x1, and C = a11x12 + a1x1 + a0 − y , we may write the above equation as Ax22 + Bx2 + C = 0, having the general solution
x2 =
− B ± B2 − 4 AC = β ± β2 − γ 2A
where β = −B 2 A and γ = C A , or in terms of the original factors,
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Modeling of Combustion Systems: A Practical Approach
2
⎛a +a x ⎞ ⎛a +a x ⎞ a x 2 + a1 x1 + a0 − y x2 ( x1 , y ) = ⎜ 2 12 1 ⎟ ± ⎜ 2 12 1 ⎟ − 11 1 ⎝ 2 a22 ⎠ ⎝ 2 a22 ⎠ 2 a22
(1.13)
Now we have x2 = φ( x1 , y) and we may plot this with axes for x2 and x1. We obtain a contour line by setting y to any desired value. We may repeat the procedure for several values of y to generate a series of contour lines. The plot thus represents three dimensions in a two-dimensional plot space. Contour surfaces can allow for a four-dimensional representation in two-dimensional space. Spreadsheet software allows convenient plots of contour lines though not contour surfaces. Dedicated plotting software provides for better options. One may solve analytically for general polynomials up to the fourth order.6 General polynomials of fifth order and higher are analytically insoluble. However, if one knows the coefficients, one can find the roots numerically. Moreover, one may solve higher-order polynomials, providing they are less than fourth order in form. For example, ax6 + bx3 + c is quadratic in form (let u = x3) even though it is a sixth-order equation.
1.4.3
Orthogonal Directions
A point (zero dimensions), moved in a single direction, traces a line. The line, moved in an orthogonal direction, sweeps out a square. The square, moved in another orthogonal direction, sweeps out a cube. In three dimensions, we run out of orthogonal directions. However, if a fourth dimension existed, we could move the cube in a fourth direction mutually orthogonal to the previous three and sweep out the hypervolume of a cube. Figure 1.7 depicts the process.
1.4.4
Visualization with Cubic Regions
Cubic regions constitute the most intuitive method for visualizing spatial coordinates in multiple dimensions. The method requires no special printing techniques or the use of color, which is an advantage. A high and low value for each direction form a series of coordinates. For convenience, we define a unit cube as having coordinates at ±1. We may omit the 1 as redundant if we are speaking merely of vertices and indicate only the sign. For example, a coordinate in three dimensions would look something like (+ + –), telling us that x1 and x2 are at their maximum and x3 is at its minimum position. This coordinate would represent one vertex of a cube. Allowing for noninteger values, one may represent any point on the surface or within the volume of the cube (Figure 1.8).
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Introduction to Modeling
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1
4
2
3
FIGURE 1.7 A four-dimensional hypercube. Take a zero-dimensional point and move it from station 1 to station 2, tracing the one-dimensional line (1–2). Move line 1–2 in an orthogonal direction (toward station 3), generating the two-dimensional square in the foreground. Move the square in the direction 3 → 4, sweeping out the three-dimensional volume shown by the cube in the foreground, some of whose vertices are 1, 2, 3, and 4. At this point we run out of dimensions. However, if a fourth dimension existed, we could follow the path moving the cube from 4 in the indicated direction and sweep out the four-dimensional hypervolume shown. The hypercube is distorted for clarity but has 16 vertices, 24 faces, and 32 edges.
(+ + –) vertex +x2
0)
0 (+ face (+ 0 +) edge
+x3
+x1 (0 0 0) interior
FIGURE 1.8 Coordinate space in three dimensions. A series of three coordinates represent each vertex of a cubic space. Vertices have the form (±1 ±1 ±1). The center of the cube is at (0 0 0). Coordinates may represent any point within the unit cube. For example, edge points will have two ±1 coordinates and one coordinate between –1 and +1. Points on the surface of the unit cube will have only one ±1 coordinate and two coordinates between –1 and +1. Points within the volume of the cube will have no coordinates at ±1.
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Modeling of Combustion Systems: A Practical Approach
One may easily extend the method to multiple dimensions. Consider two cubes; four coordinates (+ + – +) represent a point in four dimensions. The first three coordinates tell us what vertex on a cube we will choose; the fourth coordinate tells us on which cube the point resides (–1 = first cube, +1 = second cube). In this fashion, we may select space in any number of dimensions. Figure 1.9 shows the procedure for up to six dimensions, though we need not stop here.
3D
4D
5D 6D
FIGURE 1.9 Visualizing multiple dimensions with cubic regions. The top left corner of the figure shows a three-dimensional surface, each vertex of the cube represents some combination of high and low levels. If the cube is moved in an orthogonal direction, one obtains a four-dimensional representation. The first coordinate specifies on which cube the point is located, and the remaining three coordinates specify where on the cube the point is located. One may repeat the procedure indefinitely to visualize higher dimensions.
The vertices indicate extreme levels. Edges represent travel along a single dimension, faces represent two dimensions, volumes represent three dimensions, and two or more cubes represent hypervolumes. Extension to more dimensions is straightforward. Factor space is the region bounded by orthogonal factor ranges. Generally, we represent only the factor space in this way and indicate the response value either by drawing contours or, most commonly, by showing it at each vertex, leaving the mind’s eye to interpolate the results.
1.4.5
The Use of Color
Sometimes three-dimensional surface graphs use color as a redundant indication of elevation. However, if the color on the three-dimensional graph corresponds to a value in a fourth dimension, then color constitutes a means
© 2006 by Taylor & Francis Group, LLC
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29
of distinguishing a hyperdimension. It is possible to graduate the colors in a rainbow scheme, remembered by the mnemonic ROY G. BIV (red, orange, yellow, green, blue, indigo, and violet, respectively). However, this color scheme is not intuitive and takes some practice to interpret. A better way is to pick a single color and vary its saturation (brightness). One may then use additional colors for additional dimensions. The normal human eye has three color receptors (RGB). This allows for a maximum of three colors with color blending, indicating a general surface in up to six dimensions. The reader should note that RG color blindness is a fairly common perceptual deficiency. Approximately 10% of males have it, leading to the failure to distinguish between red and green. So, one should not use these two colors if their contrast is meant to convey information. Notwithstanding, with a careful choice of color, the method is a viable one for representing high dimensionality and for hypersurfaces or hypervolumes that do not bend in more than six dimensions simultaneously.
1.5
Basic Data Classifications
We may classify data in several ways: • • • •
Level of scale Quality of data collection and care in the experimental design Classification by source Classification by function
We discuss each in turn. With respect to scale and level, data may be either discrete or continuous and characterized by four basic levels. The scales are, in order of increasing level: 1. 2. 3. 4.
Nominal Ordinal Interval Ratio
Figure 1.10 illustrates this fundamental scheme. 1.5.1
Level of Scale
The first basic distinction is to identify discrete vs. continuous data. Discrete data are those data that can take on only particular and definite values. For example, a pipe is available only in discrete sizes. In contrast, continuous data
© 2006 by Taylor & Francis Group, LLC
30
Modeling of Combustion Systems: A Practical Approach
Data Types Discrete Nominal ➢Batches or lots e.g., • raw materials • fuels,… Examples: ➢Equipment types • burner types • process unit types ➢Equipment operators • Judy, Chris, Nathan, Jamie…
Characteristics:
Lack of rank or order
Worst
Ordinal ➢Pipe sizes • 2”, 3”, 4”, …
➢Fuel oil grades
• #2, #4, #5, #6 …
➢Comparisons
• good, bad • pass, fail ➢Ordering or ranking • poor, fair, good, • No.1, No. 2, …
Possess rank with non-uniform intervals
Continuous Interval ➢Some temperature
Ratio ➢Absolute temperature
scales • Fahrenheit • Celsius ➢Some state functions • enthalpy • free energy ➢Some other engineering scales • degrees API
scales • Rankine • Kelvin ➢Most engineering units • heat capacity • density • pressure • energy • power • etc.
Equal intervals between levels. Possess a zero relative to a reference state (relative zero)
Possess a true (absolute) zero. Ratios have meaning
Best
FIGURE 1.10 Data types. Factors may be classified per the above categories. The highest order of data is ratio data, followed by interval data, ordinal data, and nominal data. Failure to recognize the factor type can lead to serious errors in analysis.
are data that can take on any intermediate value over an applicable range. We may also measure data on a variety of scales. The nominal scale is that which identifies category or quality only. Data reported on a nominal scale have no order or rank. All nominal data are necessarily discrete. For example, we may classify burner types by firing orientation: sidewall (horizontal) (S), floor fired (vertically up) (F), or roof fired (vertically down) (R). But there is no intrinsic order or rank among the types. The ordinal scale is that which allows order or rank among the data, but nothing more. The ordinal scale is the next higher level of data. For example, engineers classify fuel oil (by boiling point ranges) into various categories, e.g., fuel oil 2, 4, 5, 6. We can assign order or intrinsic rank to the data. We know that 2 is lighter than 6, but it is not meaningful to say that fuel oil 6 is triple fuel oil 2. Likewise, one may classify flame quality as excellent (0), very good (1), good (2), fair (3), poor (4), or very poor (5). Figure 1.11 gives such a classification. But flame quality = 5 is not necessarily 2.5 times worse than flame quality = 2, and the difference between levels (the interval) is not necessarily the same between all adjacent levels and, for some ordinal scales, may even lack quantitative meaning. Ordinal data are ordinarily discrete. The interval scale is the next higher level of data. The interval scale is that scale which quantifies deviation from a relative zero. A relative zero is a zero
© 2006 by Taylor & Francis Group, LLC
Introduction to Modeling
31
TYPE
EXTENT
Spatial
Temporal
CHARACTER Softness (0)
Rollover (1)
Contact (2)
High TMT (3)
None (0)
Never (0)
0
1
2
3
Some (1)
Sometimes (1)
1
2
3
4
Nearly all (2)
Nearly always (2)
2
3
4
5
FIGURE 1.11 An ordinal scale for flame quality. Flame quality degrades first by becoming soft at the flame tail, then rolling over toward the process tubes, then contacting them, and finally, elevating the tube metal temperature (TMT). We characterize this with an ordinal factor named character, having the following levels (horizontal in figure): softness, rollover, contact, and high TMT. Next we decide the extent of the problem in either time or space (vertical categories). A flame quality of 3 means that we never have high TMT, we sometimes have flame contact with the tubes, but we nearly always have rollover. We consider 3 to be the minimal acceptable flame quality.
point set by convention but not necessity. A relative zero is not unique. For example, we may define 0°C as the freezing point of water, but one could just as easily choose another convenient reference, for example, the freezing point of a saltwater ice bath (0°F). An interval scale has more than mere rank or order. In principle, interval scales are continuous scales, so addition and subtraction have quantitative meaning, e.g., 30°C is 10°C warmer than 20°C. However, because convention rather than necessity defines the zero, ratios have no real meaning. For example, in the Celsius scale, the ratio between 5 and 95°C is 95/5 = 19. However, we cannot say that 95°C is 19 times higher than 5°C. If we were to perform the ratio in the Fahrenheit scale, we would obtain 203/41°F = 4.95. Since the temperature ratio has changed based on scale, we conclude that one cannot meaningfully ratio quantities on an interval scale. This is important because convention defines interval scales for many thermodynamic properties (e.g., ΔH, ΔG; see Chapter 2). The ratio scale represents the highest level of data. The ratio scale is that scale which quantifies deviation from an absolute zero. For example, Rankine and Kelvin are ratio scales K = °C + 273.14°C, R = °F + 459.59°F. Performing the previous calculation in these units gives 366.14 K/276.14 K = 662.6 R/ 500.6 R = 1.323. So, we see that the temperature ratio is meaningful and identical on an absolute scale. The zero (known as absolute zero in this case) is unique. In principle, the ratio scale is continuous over some range.
© 2006 by Taylor & Francis Group, LLC
32 1.5.2
Modeling of Combustion Systems: A Practical Approach Data Quality
We should use the highest level and quality of data possible for our analysis. We have already implicitly defined that level of scale refers to whether the data are measured on nominal, ordinal, interval, or ratio scales. And we have ranked such data from best to worst according to level: ratio > ordinal > interval > nominal (thus level of scale is itself an ordinal scale). Quality is another ordinal scale for classifying data. Quality is the level of care taken in the design of the experiment and the collection of the data. The old adage “garbage in, garbage out” (GIGO) applies: if our data are of poor quality, they will set a ceiling on the amount of information we can glean from them. No amount of post-data analysis can rectify deficiencies in data quality. Statistical analysis after the fact cannot improve data quality (but it can allow us to glean the maximum information from the data and it can make data deficiencies apparent). This is of extreme importance because the only thing worse than poor data quality is unrecognized poor data quality. Poor data quality results in wasted time and effort, but if unrecognized, it results in unsubstantiated conclusions upon which we base decisions involving even larger quantities of time and money. Our presumption is that the investigator has used due care and diligence in collecting the data (by all means verify this). But even if it is true, care in the data collection is not sufficient to generate quality data as defined. Care and diligence must characterize not only the collection of the data, but the design of the experimental program as well.
1.5.3
Planned Experiments
A planned experiment is a deliberate (on-purpose) experimental series. Statistical experimental design (SED) is a method for planning experiments that maximizes the probability of obtaining accurate information. Statistically, no experimentation is more efficient (information per datum) than SED. This is not to say that planned experiments that do not use SED methodology are worthless — far from it. Scientists have discovered great and fundamental truths without SED. Indeed, as a methodology, SED occurs only in the latter half of the 20th century. At least one statistician has gone so far as to say that SED is not responsible for a single fundamental scientific truth.* Perhaps. However, SED is a power tool allowing one to squeeze the most information from the data and to plan the most efficient experimental data sets. SED is no substitute for intuition or brilliance, but it can lead to insights that one might otherwise overlook. On the data quality scale, appropriately planned experiments represent the highest level of data quality. * David, F.N., “No discovery of some importance would have been missed by the lack of statistical knowledge,” as quoted by Rao, C.R., in “Statistics: Reflections on the Past and Visions for the Future,” Amstat News, American Statistical Association, Alexandria, VA, September 2004, p. 2 (http:www.amstat.org).
© 2006 by Taylor & Francis Group, LLC
Introduction to Modeling 1.5.4
33
Unplanned Experiments
Not all experiments are planned, and great discoveries occur from unplanned events. Others have said that “life is the stuff that happens in between the plans that we make” and that “chance favors the prepared mind.”* But chance is not a generally useful experimental strategy. The job of an investigator is to find the reason for experimental outcomes, even unexpected ones, and here statistical analysis is a valuable ally. 1.5.5
Source Classifications
Another useful classification scheme for factors is a relational one. Generally, factors come from four sources: inherited, designed, uncontrollable, and unknown. Because this is a source classification scheme, it is a relative classification (rather than one that is absolute or intrinsic to the factors themselves). Inherited factors are those factors that come from a functional specification by others. They are typically nonnegotiable or negotiable only within narrow limits. For example, a heater manufacturer may dictate to a burner manufacturer the desired flame shape, the heat release, the excess oxygen, the desired heat flux, the maximum limits for combustion-related emissions such as NOx and CO, etc. Vendors inherit such factors from their customers, such as refineries and end users. Designed factors are those factors whose levels we can set. Our control may not be absolute, there may be physical limitations to the factor ranges, but we have greater freedom to manipulate them compared to inherited factors. Uncontrollable factors are those factors that due to their nature cannot be controlled. We wish to understand their influence but we cannot control them. In the case of combustion, these include humidity, ambient temperature, and barometric pressure. Finally, unknown factors (also called lurking factors) are factors of which we are unaware. We are powerless to control them because we are not even aware that they exist. However, because they are influential, we must make provision for them in the experimental design. This may seem like quite a trick — to minimize the effect of something we know nothing about, but it can be done. Two techniques are to randomize the order of experimental runs and orthogonally block experiments. We discuss these in detail in Chapter 3. 1.5.6
Functional Classifications
We can also classify factors by function: operation-related factors, burner-related factors, furnace-related factors, and ambient factors. This classification scheme is more objective than the former because it does not depend on the vendor–customer relationship, which changes based on the investigator’s position in the supply chain. Table 1.3 classifies the most common combustionrelated factors in this way. * Attributed to Louis Pasteur (1822–1895).
© 2006 by Taylor & Francis Group, LLC
34
TABLE 1.3
Operating Factors (Constrained by Process)
Burner Factors (Controlled by Burner Manufacturer)
Degree of air preheat Firing rate Fuel composition Furnace temperature Fuel pressure Oxygen concentration Process fluid flow rate
Burner throat diameter Degree of air staging Degree of fuel staging Fuel port arrangement Fuel port diameter Multiple combustion zones Number of fuel ports
© 2006 by Taylor & Francis Group, LLC
Furnace Factors (Controlled by Heater Design)
Ambient Factors (Uncontrollable)
Available air-side pressure drop Burner-to-burner spacing Burner-to-furnace wall spacing Heat release/furnace vol. ratio
Ambient humidity Ambient air temperature Barometric pressure
Modeling of Combustion Systems: A Practical Approach
Some Potential Factors Affecting NOx Response from a Burner
Introduction to Modeling
1.6
35
A Linear Algebra Primer
The method of least squares is our last subject for discussion in this chapter. Before going on, we digress to recount some properties of matrices since they play an important role in obtaining least squares solutions. Readers familiar with matrices may skip this section. Ordinary numbers, constants, or variables are single valued, and we refer to them as scalars. Linear algebra is the study of matrices and vectors. A matrix in this sense is a two-dimensional array of numbers. A vector is a onedimensional array. For example, for ⎛1 M=⎜ ⎝1
2⎞ 3⎟⎠
⎛ 1⎞ v = ⎜⎜ 3⎟⎟ ⎜⎝ 2 ⎟⎠
M is a matrix and v is a vector. Here the lowercase bold type (v) identifies a vector, while the uppercase bold type (M) identifies a matrix (two-dimensional array). We use this convention throughout the text. Rows extend in the horizontal direction, columns in the vertical. A column vector comprises a single column with multiple rows (e.g., v). This text uses lowercase bold type to refer only to a column vector. A row vector comprises a single row with multiple columns. If we wish to refer to a row vector, we shall use the designation vT where the Arial superscript T indicates the transpose operation: a swapping of rows and columns. Thus, for the above example, vT = (1 3 2) is a row vector. For vectors, we can indicate individual elements with a single subscript, e.g., vT = (v1 v2 v3). For matrices we will require two subscripts to identify any particular element. We shall use the notation mrc to specify a particular element, where M is the matrix and mrc is the element of M in row r and column c. Thus, in the above example ⎛1 M=⎜ ⎝1
2 ⎞ ⎛ m11 = 3⎟⎠ ⎜⎝ m21
m12 ⎞ m22 ⎟⎠
For example, m22 = 3. If the number of rows equals the number of columns, then the matrix is said to be square. 1.6.1
Matrix Addition
If and only if two matrices have the same number of rows and columns can we add or subtract them element by element. For example, ⎛1 ⎜⎝ 2
3⎞ ⎛ 1 + 1⎟⎠ ⎜⎝ 2
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−1⎞ ⎛ 1 + 1 = 2 ⎟⎠ ⎜⎝ 2 + 2
3 − 1⎞ ⎛ 2 = 1 + 2 ⎟⎠ ⎜⎝ 4
2⎞ 3⎟⎠
36
Modeling of Combustion Systems: A Practical Approach
Matrix addition is both commutative and associative.
1.6.2
M+N=N+M
(1.14)
M + (N + P) = (M + N) + P
(1.15)
The Transpose Operator
We signify the transpose operator by T. It swaps rows and columns. As examples, consider the following: T
⎛ 1⎞ ⎜ 2⎟ = 1 ⎜ ⎟ ⎜⎝ 3⎟⎠
(
⎛1 ⎜⎝ 4
2 5
2
3
⎛1 T 3⎞ = ⎜⎜ 2 7 ⎟⎠ ⎜⎝ 3
) 4⎞ 5⎟⎟ 7 ⎟⎠
Or in general, ⎛ a11 ⎜a ⎜ 21 ⎜ ⎜ ⎝ am1
a12 a22 am 2
T
⎛ a11 a1n ⎞ ⎟ ⎜a a2 n ⎟ = ⎜ 12 ⎟ ⎜ ⎟ ⎜ amn ⎠ ⎝ a1n
a21 a22 a2 n
am1 ⎞ am 2 ⎟⎟ ⎟ ⎟ amn ⎠
(1.16)
The transpose of the sum is the sum of the transposes*:
(M + N)
T
= MT + NT
(1.17)
The transpose of a product is the product of the transposes in reverse order:
(MN)
T
= N TM T
(1.18)
* Technically, it is also correct to say that the transpose of a sum is equal to the sum of the transposes in reverse order. However, since matrix addition is commutative, we may reorder the sums per Equation 1.17.
© 2006 by Taylor & Francis Group, LLC
Introduction to Modeling 1.6.3
37
Multiplication by a Constant
To multiply or divide by a constant, one performs the operation on each matrix element. For example, ⎛1 3⎜ ⎝3
1.6.4
2⎞ ⎛ 3 ⋅ 1 = 4⎟⎠ ⎜⎝ 3 ⋅ 3
3 ⋅ 2⎞ ⎛ 3 = 3 ⋅ 4⎟⎠ ⎜⎝ 9
6⎞ 12 ⎟⎠
Matrix Multiplication
One may multiply two matrices if and only if the number of columns in the first is identical to the number of rows in the second. Each cell of the result matrix will equal the sequential sum of the row–column products. The resulting matrix will have the same number of rows as the first matrix and the same number of columns as the second; that is, Mrc Nsd = Prd
(1.19)
Here one may say that M premultiplies N, or that N postmultiplies M; in this case, either statement refers to the equivalent operation. For example,
(
2
⎛ 1⎞ 7 ⎜⎜ 2 ⎟⎟ = 2 ⋅ 1 + 4 ⋅ 2 + 7 ⋅ 3 = 31 ⎜⎝ 3⎟⎠
)
4
(
)
(1.20)
Since the multiplicand (first matrix) has only one row and the multiplier (second matrix) has only one column, the product matrix comprises a single row and column, i.e., a single value, 31. Here is another example: ⎛1 ⎜2 ⎜ ⎜⎝ 3
4⎞ ⎛6 5⎟⎟ ⎜ ⎝9 7 ⎟⎠
⎛ 1⋅6 + 4 ⋅ 9 8⎞ ⎜ = 2 ⋅6 + 5⋅9 11⎟⎠ ⎜⎜ ⎝ 3⋅6 + 7 ⋅9
1 ⋅ 8 + 4 ⋅ 11⎞ ⎛ 42 2 ⋅ 8 + 5 ⋅ 11⎟⎟ = ⎜⎜ 57 3 ⋅ 8 + 7 ⋅ 11⎟⎠ ⎜⎝ 81
52 ⎞ 71 ⎟⎟ 101⎟⎠
Note that in general, matrix multiplication is not commutative. That is, A ⋅ B ≠ B ⋅ A , in general
(1.21)
For example, ⎛ 1⎞ ⎜ 2⎟ 2 ⎜ ⎟ ⎜⎝ 3⎟⎠
(
4
© 2006 by Taylor & Francis Group, LLC
⎛ 1⋅ 2 7 = ⎜⎜ 2 ⋅ 2 ⎜⎝ 3 ⋅ 2
)
1⋅ 4 2⋅4 3⋅4
1⋅7 ⎞ ⎛ 2 2 ⋅ 7 ⎟⎟ = ⎜⎜ 4 3 ⋅ 7 ⎟⎠ ⎜⎝ 6
4 8 12
7⎞ 14⎟⎟ 21⎟⎠
(1.22)
38
Modeling of Combustion Systems: A Practical Approach
Equation 1.20 clearly does not equal Equation 1.22. However, matrix multiplication is associative. That is,
(
) (
)
A ⋅ B⋅C = A ⋅B ⋅C
(1.23)
For example, ⎛1 ⎜⎝ 1
2 ⎞ ⎡⎛ 2 ⎢ 3⎟⎠ ⎢⎣⎜⎝ 2
1⎞ ⎛ 1 2 ⎟⎠ ⎜⎝ 2
3⎞ ⎤ ⎛ 1 ⎥= 3⎟⎠ ⎥⎦ ⎜⎝ 1
2⎞ ⎡ 4 ⎢ 3⎟⎠ ⎣ 6
9 ⎤ ⎛ 16 ⎥= 12 ⎦ ⎜⎝ 22
⎡⎛ 1 ⎢⎜ ⎢⎣⎝ 1
2⎞ ⎛ 2 3⎟⎠ ⎜⎝ 2
1⎞ ⎤ ⎛ 1 ⎥ 2 ⎟⎠ ⎥⎦ ⎜⎝ 2
3⎞ ⎡ 6 =⎢ 3⎟⎠ ⎣8
5⎤ ⎛ 1 ⎥ 7 ⎦ ⎜⎝ 2
3⎞ ⎛ 16 = 3⎟⎠ ⎜⎝ 22
33⎞ 45⎟⎠ 33⎞ 45⎟⎠
Perhaps the best way to think of matrix multiplication is visually, with a horizontal box drawn around the matrix row of interest in the multiplicand and a vertical box drawn around the column of interest in the multiplier. The boxes, if superimposed in the product matrix, will select the particular matrix element to which we assign the result; that is, the bottom left corner of the resulting matrix is derived from multiplication of the bottom row of the multiplicand (horizontal box) and the first column of the multiplier (vertical box): (3)(1) + (5)(2) = 13. ⎛ 2 1 ⎞ ⎛ 1 1⎞ ⎛ 4 3 ⎞ ⎟ ⎜ 3 5 ⎟ ⎜ ⎟ =⎜ ⎝ ⎠ ⎝ 2 1⎠ ⎝ 13 8 ⎠
1.6.5
Distributive Property of Multiplication over Addition
Matrix multiplication over addition is distributive. That is, M(N + P) = MN + MP
(1.24)
For example, ⎛1 ⎜⎝ 1
2⎞ ⎡⎛ 2 ⎢ 3⎟⎠ ⎢⎣⎜⎝ 1
1⎞ ⎛ 2 + 2 ⎟⎠ ⎜⎝ 1
2⎞ ⎤ ⎛ 1 ⎥= 3⎟⎠ ⎥⎦ ⎜⎝ 1
2⎞ ⎡ 2 ⎢ 3⎟⎠ ⎣ 1
1⎤ ⎛ 1 ⎥+ 2 ⎦ ⎜⎝ 1
2⎞ ⎡ 2 ⎢ 3⎟⎠ ⎣ 1
⎛1 ⎜⎝ 1
2⎞ ⎡⎛ 2 ⎢ 3⎟⎠ ⎣⎢⎜⎝ 1
1⎞ ⎛ 2 + 2 ⎟⎠ ⎜⎝ 1
2⎞ ⎤ ⎛ 1 ⎥= 3⎟⎠ ⎥⎦ ⎜⎝ 1
2⎞ ⎡ 4 ⎢ 3⎟⎠ ⎣ 2
3⎤ ⎛ 8 ⎥= 5 ⎦ ⎜⎝ 10
13⎞ 18⎟⎠
© 2006 by Taylor & Francis Group, LLC
2⎤ ⎛ 8 ⎥= 3 ⎦ ⎜⎝ 10
13⎞ 18⎟⎠
Introduction to Modeling 1.6.6
39
Symmetric Matrices
A matrix is symmetric if akj = ajk for all elements. For example, ⎛ 1 ⎜ 2 ⎜ ⎜⎝ −4
2 3 1
−4⎞ 1 ⎟⎟ 2 ⎟⎠
is a symmetric matrix. Often a symmetric matrix is written as ⎛ 1 ⎜ ⎜ ⎜⎝ sym
2 3
−4⎞ 1 ⎟⎟ 2 ⎟⎠
where sym reflects the elements across the principle diagonal. The principal diagonal is the largest diagonal running from top left to bottom right of a matrix, e.g., 1 3 2 in the above matrix. The matrix ⎛ 1 ⎜ 2 ⎜ ⎜⎝ −4
2 3 1
5⎞ 1⎟⎟ 2 ⎟⎠
is not a symmetric matrix because a13 ≠ a31. The addition of a matrix and its transpose always results in a symmetric matrix: ⎛ 1 ⎜ 2 ⎜ ⎜⎝ −4
2 3 5
3⎞ ⎛ 1 1⎟⎟ + ⎜⎜ 2 2 ⎟⎠ ⎜⎝ 3
2 3 1
−4⎞ ⎛ 2 5 ⎟⎟ = ⎜⎜ 4 2 ⎟⎠ ⎜⎝ −1
4 6 6
−1⎞ ⎛ 2 6 ⎟⎟ = ⎜⎜ 4 ⎟⎠ ⎜⎝ sym
4 6
−1⎞ 6 ⎟⎟ 4 ⎟⎠
The multiplication of a matrix and its transpose always results in a symmetric matrix: ⎛ 1 ⎜ 2 ⎜ ⎜⎝ −4
2 3 5
3⎞ ⎛ 1 1⎟⎟ ⎜⎜ 2 2 ⎟⎠ ⎜⎝ 3
2 3 1
−4⎞ ⎛ 14 5 ⎟⎟ = ⎜⎜ 2 ⎟⎠ ⎜⎝ sym
11 14
However, MMT ≠ MTM in general, per Equation 1.21:
© 2006 by Taylor & Francis Group, LLC
12 ⎞ 9 ⎟⎟ 45⎟⎠
40
Modeling of Combustion Systems: A Practical Approach ⎛ 1 ⎜ 2 ⎜ ⎜⎝ −4 ⎛1 ⎜2 ⎜ ⎜⎝ 3
1.6.7
2 3 5
2 3 1
3⎞ ⎛ 1 1⎟⎟ ⎜⎜ 2 2 ⎟⎠ ⎜⎝ 3 −4⎞ ⎛ 1 5 ⎟⎟ ⎜⎜ 2 2 ⎟⎠ ⎜⎝ −4
−4⎞ ⎛ 14 5 ⎟⎟ = ⎜⎜ 11 2 ⎟⎠ ⎜⎝ 12
2 3 1
3⎞ ⎛ 21 1⎟⎟ = ⎜⎜ −12 2 ⎟⎠ ⎜⎝ −3
2 3 5
12 ⎞ ⎛ 14 9 ⎟⎟ = ⎜⎜ 45⎟⎠ ⎜⎝ sym
11 14 9
−3⎞ ⎛ 21 19 ⎟⎟ = ⎜⎜ 14 ⎟⎠ ⎜⎝ sym m
−12 38 19
11 14
−12 38
12 ⎞ 9 ⎟⎟ 45⎟⎠ −3⎞ 19 ⎟⎟ 14 ⎟⎠
The Identity Matrix
A matrix whose principal diagonal elements are 1 and all others are zero is known as the identity matrix: ⎛1 ⎜ I=⎜ ⎜ ⎜ ⎝
⎞ ⎟ ⎟ ⎟ ⎟ 1⎠
1
(1.25)
Here, the blank elements are understood to be zero. A matrix whose only nonzero elements are in the principal diagonal is known as a diagonal matrix. Clearly, the identity matrix is a diagonal matrix. The identity matrix has the following property: A ⋅I = I ⋅A = A
(1.26)
For example, ⎛1 ⎜ ⎜ ⎜⎝
⎞ ⎛3 ⎟ ⎜8 ⎟⎜ 1⎟⎠ ⎜⎝ 1
1
⎛1
3 ⎜⎜ ⎜⎝
1
1 4 7
⎞ ⎛3 ⎟ =⎜ ⎟ ⎜ 1⎟⎠ ⎜⎝
⎛3 For example, if we let a = 3 and A = ⎜⎜ ⎜⎝
© 2006 by Taylor & Francis Group, LLC
2⎞ ⎛ 3 4⎟⎟ = ⎜⎜ 8 6 ⎟⎠ ⎜⎝ 1 ⎞ ⎟ ⎟ 3⎟⎠
3
3
⎞ ⎟ ⎟ 3⎟⎠
1 4 7
2⎞ 4⎟⎟ 6 ⎟⎠
Introduction to Modeling
41
then A = aI, and either is identical for the purposes of multiplication. For example, ⎛3 2 ⎜⎜ 8 ⎜⎝ 1
1.6.8
1 4 7
2⎞ ⎛ 2 4⎟⎟ = ⎜⎜ 6 ⎟⎠ ⎜⎝
2
⎞ ⎛3 ⎟ ⎜8 ⎟⎜ 2 ⎟⎠ ⎜⎝ 1
1 4 7
2⎞ ⎛ 6 4⎟⎟ = ⎜⎜ 16 6 ⎟⎠ ⎜⎝ 2
2 8 14
4⎞ 8 ⎟⎟ 12 ⎟⎠
The Unity, Zero, and Constant Vectors
We designate a vector whose elements are all unity as the unity vector, 1: ⎛ 1⎞ ⎜ 1⎟ 1=⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ 1⎠
(1.27)
We designate a vector whose elements are all zero as the zero vector, 0: ⎛ 0⎞ ⎜ 0⎟ 0=⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ 0⎠
(1.28)
We designate a constant as c. A vector of constants has the same value in each row: ⎛ c⎞ ⎜ c⎟ c=⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ c⎠
(1.29)
The above is fundamentally different than a vector of variables (e.g., v), where each element may be different. ⎛ v1 ⎞ ⎜v ⎟ v = ⎜ 2⎟ ⎜ ⎟ ⎜ ⎟ ⎝ vn ⎠ Be very careful about these distinctions.
© 2006 by Taylor & Francis Group, LLC
(1.30)
42
Modeling of Combustion Systems: A Practical Approach
1.6.9
The Inverse
We use the superscript –1 to indicate the inverse operator for matrix multiplication. The multiplicative inverse has the following property: A –1A = AA –1 = I
(1.31)
Multiplication with an inverse is commutative. For example, the matrix ⎛1 M=⎜ ⎝1
2⎞ 3⎟⎠
has the inverse ⎛ 3 M−1 = ⎜ ⎝ −1
−2 ⎞ 1 ⎟⎠
We verify this by noting ⎛1 MM−1 = ⎜ ⎝1
2⎞ ⎛ 3 3⎟⎠ ⎜⎝ −1
⎛ 3 −2 ⎞ = M−1M = ⎜ ⎟ 1⎠ ⎝ −1
−2 ⎞ ⎛ 1 1 ⎟⎠ ⎜⎝ 1
2⎞ ⎛ 1 = 3⎟⎠ ⎜⎝ 0
0⎞ 1⎟⎠
If a square matrix has a multiplicative inverse, that inverse is unique. However, not all square matrices have inverses. For example, ⎛1 M=⎜ ⎝2
2⎞ 4⎟⎠
has no inverse. A square matrix without an inverse is termed singular. In this case M is singular because the second row is merely a multiple of the first, 2(1 2) = (2 4), and therefore adds no independent information. The number of independent rows in a matrix is referred to as its rank. If the rank of a square matrix is equal to its rows, then it is termed full rank and it will have a unique inverse. We may use elementary row operations to determine the rank of a matrix.
1.6.10
Elementary Row Operations
Consider the matrix equation y = Xa. We will encounter many matrix equations of this form. The simplest hand calculation procedure is to use elementary row operations to determine the rank. Elementary row operations do not interchange columns. Therefore, these operations affect the elements of y and X, but not the elements of a (which are multiplied by respective
© 2006 by Taylor & Francis Group, LLC
Introduction to Modeling
43
columns whose order does not change). Matrix algebra allows for the following elementary row operations: 1. Row swapping: One may swap any row with any other row. For example, if ⎛ y1 ⎞ ⎛ x11 ⎜y ⎟ = ⎜x ⎜ 2 ⎟ ⎜ 21 ⎜⎝ y ⎟⎠ ⎜⎝ x 3 31
x12 x22 x32
x13 ⎞ ⎛ a1 ⎞ x23 ⎟⎟ ⎜⎜ a2 ⎟⎟ x ⎟⎠ ⎜⎝ a ⎟⎠
⎛ y1 ⎞ ⎛ x11 ⎜y ⎟ = ⎜x ⎜ 3 ⎟ ⎜ 31 ⎜⎝ y ⎟⎠ ⎜⎝ x 2 21
x12
x13 ⎞ ⎛ a1 ⎞ x33 ⎟⎟ ⎜⎜ a2 ⎟⎟ x23 ⎟⎠ ⎜⎝ a3 ⎟⎠
33
3
then
x32 x22
2. Row addition: One may add or subtract any row to any other row in the matrix. To pick one example, if ⎛ y1 ⎞ ⎛ x11 ⎜y ⎟ = ⎜x ⎜ 2 ⎟ ⎜ 21 ⎜⎝ y ⎟⎠ ⎜⎝ x 3 31
x12 x22 x32
x13 ⎞ ⎛ a1 ⎞ x23 ⎟⎟ ⎜⎜ a2 ⎟⎟ x33 ⎟⎠ ⎜⎝ a3 ⎟⎠
then it is also true that ⎛ y1 ⎞ ⎛ x11 ⎜ y ⎟ =⎜ x 2 21 ⎟ ⎜ ⎜ ⎜⎝ y2 + y3 ⎟⎠ ⎜⎝ x21 + x 31
x13 ⎞ ⎛ a1 ⎞ x23 ⎟⎟ ⎜⎜ a2 ⎟⎟ x23 + x33 ⎟⎠ ⎜⎝ a3 ⎟⎠
x12 x22 x22 + x32
3. Row multiplication: One may multiply or divide any row by any constant except zero. For example, if ⎛ y1 ⎞ ⎛ x11 ⎜y ⎟ = ⎜x ⎜ 2 ⎟ ⎜ 21 ⎜⎝ y ⎟⎠ ⎜⎝ x 3 31
x12 x22 x32
x13 ⎞ ⎛ a1 ⎞ x23 ⎟⎟ ⎜⎜ a2 ⎟⎟ x ⎟⎠ ⎜⎝ a ⎟⎠ 33
3
then ⎛ y1 ⎞ ⎛ x11 ⎜ 3 y ⎟ = ⎜ 3x ⎜ 2 ⎟ ⎜ 21 ⎜⎝ y3 ⎟⎠ ⎜⎝ x31 is also true.
© 2006 by Taylor & Francis Group, LLC
x12 3 x22 x32
x13 ⎞ ⎛ a1 ⎞ 3 x23 ⎟⎟ ⎜⎜ a2 ⎟⎟ x33 ⎟⎠ ⎜⎝ a3 ⎟⎠
44
Modeling of Combustion Systems: A Practical Approach
The goal of elementary operations will be to put the matrix into echelon form, where the xkk element is equal to 1 or 0, and all elements below and to the left of xkk are zero. For example, the matrix ⎛1 ⎜ ⎜ ⎜ ⎜ ⎝
x1n ⎞ x2 n ⎟⎟ ⎟ ⎟ 1 ⎠
x12 1
is in echelon form. In echelon form, the sum of the diagonal elements determines the rank of the matrix.
Example 1.9
Determining Rank via Elementary Row Operations
Problem statement: Determine the rank of the following matrix using elementary row operations: ⎛ y1 ⎞ ⎛ 1 ⎜y ⎟ =⎜ 3 ⎜ 2⎟ ⎜ ⎜⎝ y3 ⎟⎠ ⎜⎝ −1
3⎞ ⎛ a1 ⎞ 2 ⎟⎟ ⎜⎜ a2 ⎟⎟ 4⎟⎠ ⎜⎝ a3 ⎟⎠
2 1 3
Solution: The above matrix has three rows and columns. Therefore, full rank is equivalent to rank = 3. We shall leave the elements of the y vector symbolic, to evidence the row operations. We generate the following sequence, driving the matrix to echelon form. For clarity, we bracket the targeted matrix elements that we shall drive to 0 or 1. ⎛ y1 ⎞ ⎛ 1 ⎜ y ⎟ = ⎜ ⎡3⎤ ⎜ 2⎟ ⎜ ⎣ ⎦ ⎜⎝ y ⎟⎠ ⎜ 3 ⎝ −1 ⎛ y1 ⎞ ⎛ 1 ⎜ y + 3y ⎟ = ⎜ 0 3⎟ ⎜ ⎜ 2 ⎜⎝ y3 ⎟⎠ ⎜ ⎝ −1 ⎛ y1 ⎜ 1 ⎜ y2 + 3 y3 ⎜ 10 ⎜ y3 ⎝
(
© 2006 by Taylor & Francis Group, LLC
3
3 ⎞ ⎛ a1 ⎞ ⎟ 14⎟ ⎜⎜ a2 ⎟⎟ ⎜ ⎟ 4 ⎟⎠ ⎝ a3 ⎠
2 ⎡⎣10 ⎤⎦
⎞ ⎛ 1 ⎟ ⎟ =⎜ 0 ⎟ ⎜ ⎟ ⎜⎝ ⎡⎣ −1⎤⎦ ⎠
)
3⎞ ⎛ a1 ⎞ ⎟ 2 ⎟ ⎜⎜ a2 ⎟⎟ ⎜ ⎟ 4⎟⎠ ⎝ a3 ⎠
2 1
3 2 1 3
3 ⎞ ⎛ a1 ⎞ ⎟ 1.4⎟ ⎜⎜ a2 ⎟⎟ 4 ⎟⎠ ⎜⎝ a3 ⎟⎠
Introduction to Modeling
45
⎛ y1 ⎜ 1 ⎜ y2 + 3 y3 ⎜ 10 ⎜ y +y ⎝ 3 1
(
⎛ y1 ⎜ 1 ⎜ y2 + 3 y3 ⎜ 10 ⎜ 1 y + y1 ⎜ ⎝ 5 3
(
(
)
⎞ ⎛1 ⎟ ⎟ = ⎜0 ⎟ ⎜ ⎟ ⎜⎝ 0 ⎠
2 1 ⎡⎣ 5 ⎤⎦
3 ⎞ ⎛ a1 ⎞ ⎟ 1.4⎟ ⎜⎜ a2 ⎟⎟ 7 ⎟⎠ ⎜⎝ a3 ⎟⎠
⎞ ⎟ ⎛1 ⎟ ⎜ ⎟ = ⎜0 ⎟ ⎜⎝ 0 ⎟ ⎠
2 1 ⎡⎣1⎤⎦
3 ⎞ ⎛ a1 ⎞ ⎟ 1.4⎟ ⎜⎜ a2 ⎟⎟ 1.4⎟⎠ ⎜⎝ a3 ⎟⎠
)
)
At this point the matrix obviously has rank = 2, as there are only two independent equations, but continuing, we obtain the echelon form: ⎛ y1 ⎜ 0.1 y2 + 3 y3 ⎜ ⎜1 1 y2 + 3 y3 ⎜ y3 + y1 − ⎝5 10
( )
(
⎞ ⎟ ⎛1 ⎟ = ⎜0 ⎟ ⎜⎜ ⎟ ⎝0 ⎠
)
(
)
3 ⎞ ⎛ a1 ⎞ 1.4⎟⎟ ⎜⎜ a2 ⎟⎟ 0 ⎟⎠ ⎜⎝ a ⎟⎠
2 1 0
3
The sum of the principal diagonal is 2. Since rank = 2 but rows = 3, the matrix is singular; i.e., it does not have a multiplicative inverse.
1.6.11
Solving for the Inverse
By noting that M−1M = I , we obtain ⎛ a11 ⎜⎝ a
21
a12 ⎞ ⎛ 3 a22 ⎟⎠ ⎜⎝ −1
−2 ⎞ ⎛ 1 = 1 ⎟⎠ ⎜⎝ 0
0⎞ 1⎟⎠
where ⎛a M−1 = ⎜ 11 ⎝ a21
a12 ⎞ a22 ⎟⎠
Multiplying the matrices gives the following equations; sequential substitution readily solves them: 3 a11 − a12 = 1 3 a21 − a22 = 0 then
© 2006 by Taylor & Francis Group, LLC
−2 a11 + a12 = 0 −2 a21 + a22 = 1
46
Modeling of Combustion Systems: A Practical Approach 3 a11 − 2 a11 = 1 3 a21 = a22
a12 = 2 a11 −2 a21 + 3 a21 = 1
leading to a11 = 1 a22 = 3
a12 = 2 a21 = 1
and ⎛ a11 ⎜⎝ a
21
a12 ⎞ ⎛ 1 = a22 ⎟⎠ ⎜⎝ 1
2⎞ 3⎟⎠
Spreadsheets are powerful but underutilized matrix solvers. Section 1.8 gives Excel™ spreadsheet functions for all the operations below except where noted. The inverse of a product is equal to the product of the inverses taken in reverse order. That is, (MN)–1 = N–1M–1: ⎡⎛ 1 ⎢⎜ ⎢⎣⎝ 1 ⎛5 ⎜⎝ 7
2⎞ ⎛ 1 3⎟⎠ ⎜⎝ 2 8⎞ 11⎟⎠
1.6.12
−1
−1
2⎞ ⎤ ⎛1 ⎥ =⎜ ⎟ 3⎠ ⎥⎦ ⎝2
⎛ −11 =⎜ ⎝ 7
−1
2⎞ ⎛ 1 3⎟⎠ ⎜⎝ 1
2⎞ 3⎟⎠
−1
⎛ −3 =⎜ ⎝ 2
2⎞⎛ 3 –1⎟⎠ ⎜⎝ –1
–2⎞ ⎛ −11 = 1 ⎟⎠ ⎜⎝ 7
8⎞ , or –5⎟⎠
8⎞ –5⎟⎠
The Determinant
One may also determine the rank of a matrix using the determinant. This is a generally tedious process by hand for matrices having more than three rows and columns. By hand, the row–echelon form is the preferred format. Spreadsheets cannot typically solve for determinants without the use of third-party programs. Mathematical solvers such as MathCAD™ can solve for determinants. A vertical bar on each side of a matrix indicates the determinant operation. The determinant of any singular matrix (noninvertible) is zero. Thus, 1 2
2 =0 4
For a two-by-two matrix, the determinant is a11 a21
a12 = a11a22 − a21a12 a22
For a three-by-three matrix, the determinant is
© 2006 by Taylor & Francis Group, LLC
Introduction to Modeling
a11 a21 a31
a12 a22 a32
47
a13 a23 = a11 a22 a33 − a32 a23 − a21 a12 a33 − a32 a13 + a31 a12 a23 − a22 a13 a33
(
)
(
)
(
)
We can consider this by breaking the determinant down into smaller steps: a11 a21 a31
a12 a22 a32
a13 a a23 = a11 22 a32 a33
a23 a − a21 12 a33 a32
a13 a + a31 12 a33 a22
a13 a23
The cofactor is the determinant of each submatrix. The cofactor is the determinant formed by the elements that do not share a row or column with the multiplicand. If the row and column subscripts of the multiplicand differ by an even number, then the cofactor is positive; otherwise, it is negative. Continuing in this way, one may derive determinants for ever-larger matrices. For example, a11 a21 a31 a41
a12 a22 a32 a42 a12
a13 a23 a33 a43
a14 a22 a24 = a11 a32 a34 a42 a44
a13
a14
+ a31 a22
a23
a42
a43
a24 − a41 a44
a12 a22 a32
a23 a33 a43 a13 a23 a33
a24 a12 a34 − a21 a32 a44 a42
a13 a33 a43
a14 a34 a44
a14 a24 a34
One then reduces each of the cofactors to multiplicands and further cofactors, continuing in series until deriving the final expression.
1.6.13
Orthogonality
All orthogonal matrices have nonzero elements exclusively along the principal diagonal. In analytical geometry, such matrices represent vectors at mutual right angles, hence the term orthogonal. Orthogonal comes from the Greek and literally means “right angled.” Thus, we spoke of mutually perpendicular (orthogonal) directions earlier, in our visit to Flatland. In this text, we will distinguish three senses of the word orthogonal in linear algebra: the lenient sense, the stricter sense, and the strictest sense. Definition 1 (the lenient sense): If X is a matrix and XTX is a diagonal matrix (D), then X is an orthogonal matrix in the lenient sense. Mathematically,
© 2006 by Taylor & Francis Group, LLC
48
Modeling of Combustion Systems: A Practical Approach ⎛ d11 ⎜ D=⎜ ⎜ ⎜ ⎝
d22
⎞ ⎟ ⎟ ⎟ ⎟ dnn ⎠
(1.32)
and XT = DX–1
(1.33a)
XTX = D
(1.33b)
For example, the following matrix is orthogonal in the lenient sense because it produces a diagonal matrix (D): ⎛1 ⎜1 X=⎜ ⎜1 ⎜ ⎝1
−1 −1 0 0
−1⎞ ⎛4 −1⎟⎟ T −1 , X X = DX X = D = ⎜⎜ 0 −2 ⎟ ⎜⎝ 0 ⎟ 0⎠
0 2 0
0⎞ 0⎟⎟ 6⎟⎠
Definition 2 (the stricter sense): If X is a matrix and XTX is a diagonal matrix comprising identical elements, then X is an orthogonal matrix in the stricter sense. Mathematically, XT = cX–1
(1.34a)
XTX = cI
(1.34b)
where c is a constant and I is the identity matrix. For example, the following matrix is orthogonal in the stricter sense because the principal diagonal of the matrix comprises identical values: ⎛1 ⎜1 X=⎜ ⎜1 ⎜ ⎝1
−1 1 −1 1
−1⎞ ⎛1 −1⎟⎟ T −1 , X X = cX X = cI = 4 ⎜⎜ 1⎟ ⎜⎝ ⎟ 1⎠
1
⎞ ⎛4 ⎟ =⎜ ⎟ ⎜ 1⎟⎠ ⎜⎝
4
⎞ ⎟ ⎟ 4⎟⎠
Definition 3 (the strictest sense): If X is a matrix and XTX is the identity matrix, then X is an orthogonal matrix in the strictest sense. Mathematically, XT = X–1 XT X = I
© 2006 by Taylor & Francis Group, LLC
(1.35a) (1.35b)
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49
For example, the following matrix is orthogonal in the strictest sense because the transpose and the inverse are identical. Therefore, the product of the matrix’s transpose and itself is the identity matrix. ⎛1 ⎜2 ⎜ ⎜1 ⎜2 X=⎜ ⎜1 ⎜ ⎜2 ⎜1 ⎜ ⎝2
−
1
−
2 1
−
2
−
6 1 6 2
0
6
0
⎛ 1 ⎜ 2 ⎜ ⎜− 1 ⎜ 2 X −1 = ⎜ ⎜ − 1 ⎜ 6 ⎜ 1 ⎜ ⎜⎝ − 12
1 ⎞ 12 ⎟⎟ 1 ⎟ − ⎟ 12 ⎟ 1 ⎟ − ⎟ 12 ⎟ 3 ⎟ ⎟ 12 ⎠
1
0
1 2 1 − −
1 ⎞ 2 ⎟ ⎟ 0 ⎟ ⎟ ⎟ 0 ⎟ ⎟ ⎟ 3 ⎟ ⎟ 12 ⎠
1 2 0
2 1
2
6 1
6 1
−
12
⎛1 ⎜ X TX = X −1X = I = ⎜ ⎜ ⎜ ⎝
12
1 1
⎞ ⎟ ⎟ ⎟ ⎟ 1⎠
(1.36a)
(1.36b)
(1.36c)
If desired, one may put the radicals in proper form (i.e., rational denominators where possible): 1 2
=
2 2
or 2 6
=
2 2 2 3
=
2 3
One may also choose to reduce improper fractions (i.e., those whose numerator is greater than their denominator) to proper form, e.g., 15/4 = 3-3/4.
© 2006 by Taylor & Francis Group, LLC
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Modeling of Combustion Systems: A Practical Approach
However, the author prefers leaving them alone. They are far more informative and comparatively useful in their improper forms. Whenever we refer to orthogonal matrices without qualification, we shall always mean orthogonal in the lenient sense. Orthogonal matrices have advantages because if y = Xa and X is orthogonal, then the least squares coefficients will be uncorrelated. In other words, in general, XTXa represents a system of simultaneous equations. However, if XTX is orthogonal, then XTXa represents a system of independent equations, each one of which separately determines an independent coefficient. Besides the computational ease, this also ensures that the coefficients do not bias one another. Since the orthogonality of XTX depends exclusively on the X matrix alone (and not y), with proper planning, we can create an experimental design that generates unbiased coefficients.
Example 1.10 Orthogonal Experimental Designs Problem statement: Table 1.4 shows hypothetical data sets from two different investigators. Suppose that y = a0 + a1x1 + a2x2 + a12x1x2 for both sets of data; that is, y = Xa where ⎛1 ⎛ y1 ⎞ ⎜ ⎜y ⎟ ⎜1 2 y=⎜ ⎟, X=⎜ ⎜ y3 ⎟ ⎜1 ⎜ ⎟ ⎜ ⎝ y4 ⎠ ⎜⎝ 1
(x ) (x ) (x ) (x ) 1
1
1
2
1
3
1
4
( x2 )1 ( x1x2 )1 ⎞ ⎛ a0 ⎞ ⎟ ⎜a ⎟ ( x2 )2 ( x1x2 )2 ⎟ 1 ⎟, a=⎜ ⎟ a ⎜ x x x 2⎟ ( 2 )3 ( 1 2 ) 3 ⎟ ⎜ ⎟ ⎟ ⎝ a12 ⎠ ( x2 )4 ( x1x2 )4 ⎟⎠
and the subscripts outside the parentheses correspond to the subscript for y. Compare the orthogonality of each investigator’s design. TABLE 1.4 Hypothetical Data y 3.5 5.0 2.0 2.5 y 2.3 3.1 4.4 3.4
© 2006 by Taylor & Francis Group, LLC
Investigator 1 x1 x2 1.0 1.0 –1.0 –1.0
–1.0 1.0 –1.0 1.0
Investigator 2 x1 x2 –0.6 –0.5 0.5 1.0
–1.0 0.8 1.0 –1.1
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51
What do you note about the designs? Use the inverse to calculate a. Suppose x1x2 is found to be an insignificant term in the model. Recalculate the coefficients of the truncated model for both data sets. Compare and contrast any differences. Solution: Multiplying both sides of the equation by XT, we obtain X T y = X TXa . These reduce to the following models for each investigator’s data: Investigator 1 ⎛ 13⎞ ⎛ 4 ⎜ 4⎟ ⎜ ⎜ ⎟ =⎜ ⎜ 2⎟ ⎜ ⎜ ⎟ ⎜ ⎝ 1⎠ ⎝
⎞ ⎛ a0 ⎞ ⎟⎜a ⎟ ⎟ ⎜ 1⎟ ⎟ ⎜ a2 ⎟ ⎟⎜ ⎟ 4⎠ ⎝ a3 ⎠
4 4
Investigator 2 ⎛ 13.20⎞ ⎛ 4.00 ⎜ 2.670⎟ ⎜ 0.40 ⎜ ⎟ =⎜ ⎜ 0.840⎟ ⎜ −0.30 ⎜ ⎟ ⎜ ⎝ −1.400⎠ ⎝ −0.40
0.40 1.86 −0.40 −1.01
−00.30 −0.40 3.85 0.79
−0.40⎞ ⎛ a0 ⎞ −1.01⎟⎟ ⎜⎜ a1 ⎟⎟ 0.79⎟ ⎜ a2 ⎟ ⎟⎜ ⎟ 1.98⎠ ⎝ a3 ⎠
Clearly, the design for investigator 1 is orthogonal, while that of investigator 2 is not. From Section 1.6.9 we learned that M−1M = I, and by definition MI = M. Therefore, if we let M = X TX, then M−1X T y = M−1Ma = a . Substituting for M we have
(X X) T
−1
XTy = a
From this, we may solve for a. In the present case, this leads to Investigator 1
Investigator 2
⎛ a0 ⎞ ⎛ 3.25⎞ ⎜ a ⎟ ⎜ 1.00⎟ ⎜ 1⎟ = ⎜ ⎟ ⎜ a2 ⎟ ⎜ 0.50⎟ ⎜ ⎟ ⎜ ⎟ ⎝ a3 ⎠ ⎝ 0.25⎠
⎛ a0 ⎞ ⎛ 3.27 ⎞ ⎜ a ⎟ ⎜ 0.98⎟ ⎜ 1⎟ = ⎜ ⎟ ⎜ a2 ⎟ ⎜ 0.52 ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ a3 ⎠ ⎝ 0.24⎠
In fact, the coefficients differ only by round-off error. Since the data are hypothetical, the author is in a position to declare that the model of investigator 1 is the true model. However, the data of investigator 2 leads to substantially the same model. So far, so good; however, the solution for the XTX matrix of investigator 2 requires inverting the matrix with the procedure given in Section 1.6.9. (Later in Section 1.8 we show a convenient method for using
© 2006 by Taylor & Francis Group, LLC
52
Modeling of Combustion Systems: A Practical Approach a spreadsheet to invert the matrix numerically.) By contrast, the inversion of the data of investigator 1 is nearly trivial. Since the matrix is orthogonal, the inverse is merely the reciprocal of the XTX matrix. That is, a0 = 13/4, a1 = 4/4, a2 = 2/4, and a1 = 1/4, or in other words,
(X X) T
−1
⎛ 0.25 ⎜ =⎜ ⎜ ⎜ ⎝
0.25 0.25
⎞ ⎟ ⎟ ⎟ ⎟ 0.25⎠
Now if we truncate the model for investigator 1, none of the remaining coefficients will change value. However, this is not the case for the data of investigator 2. We present the respective equations below. Investigator 1 ⎛ 13⎞ ⎛ 4 ⎜ 4⎟ =⎜ ⎜ ⎟ ⎜ ⎜⎝ 2 ⎟⎠ ⎜⎝
4
Investigator 2
⎞ ⎛ a0 ⎞ ⎟⎜a ⎟ ⎟ ⎜ 1⎟ 4⎟⎠ ⎜⎝ a2 ⎟⎠
⎛ 13.20⎞ ⎛ 4.00 ⎜ 2.67 ⎟ = ⎜ 0.40 ⎟ ⎜ ⎜ ⎜⎝ 0.84⎟⎠ ⎜⎝ −0.30
⎛ a0 ⎞ ⎛ 3.25⎞ ⎜ a ⎟ = ⎜ 1.00 ⎟ ⎟ ⎜ 1⎟ ⎜ ⎜⎝ a2 ⎟⎠ ⎜⎝ 0.50⎟⎠
0.40 1.86 −0.40
−0.30⎞ ⎛ a0 ⎞ −0.40⎟⎟ ⎜⎜ a1 ⎟⎟ 3.85⎟⎠ ⎜⎝ a2 ⎟⎠
⎛ a0 ⎞ ⎛ 3.26⎞ ⎜ a ⎟ = ⎜ 0.86⎟ ⎟ ⎜ 1⎟ ⎜ ⎜⎝ a2 ⎟⎠ ⎜⎝ 0.56⎟⎠
For investigator 2, the estimators are less efficient. This is especially so for a1 because the experimental design had a relatively small range in the x1 direction. This would not be a good experimental design, especially for determining a1. We will always prefer orthogonal designs for this purpose. In turn, orthogonal designs are balanced and symmetrical designs, as we shall show in Chapter 3.
1.6.14
Eigenvalues and Eigenvectors
Suppose ⎛ y1 ⎞ ⎛ x11 ⎜y ⎟ ⎜x ⎜ 2 ⎟ = ⎜ 21 ⎜ ⎟ ⎜ ⎜ ⎟ ⎜ ⎝ y n ⎠ ⎝ x n1
x12 x22 xn 2
© 2006 by Taylor & Francis Group, LLC
…
x1n ⎞ ⎛ a1 ⎞ ⎛ u11 x2 n ⎟⎟ ⎜⎜ a2 ⎟⎟ ⎜⎜ u21 = ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ xnn ⎠ ⎝ an ⎠ ⎝ un1
u12 u22 un2
…
u1n ⎞ ⎛ b1 ⎞ u2 n ⎟⎟ ⎜⎜ b2 ⎟⎟ ⎟⎜ ⎟ ⎟⎜ ⎟ unn ⎠ ⎝ bn ⎠
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53
then y = Xa = Ub. Further suppose that Λ = UT U. Then U is an orthogonal matrix, Λ is the eigenvalue matrix comprising the elements λ1, λ2, … λn. When y = Xa = Ub, then Ub has the advantage that its coefficients are independent. In other words, while Xa represents a system of simultaneous equations, Ub represents an orthogonal system of independent equations. To illustrate the concept, consider y = Xa for the following values: ⎛ 2.19⎞ ⎛ 1 ⎜ 2.70⎟ ⎜ 1 ⎜ ⎟ =⎜ ⎜ 2.95⎟ ⎜ 1 ⎜ ⎟ ⎜ ⎝ 2.83⎠ ⎝ 1
−0.5 1.5 −0.5 −0.5
−0.5 −0.5 1.5 −0.5
−0.5⎞ ⎛ 2.668⎞ −0.5⎟ ⎜ 0.255 ⎟ ⎟⎜ ⎟ −0.5⎟ ⎜ 0.380 ⎟ ⎟⎜ ⎟ 1.5 ⎠ ⎝ 0.320 ⎠
We may also express the same vector as y = Ub where U = XK, b = KTa, and K = eigenvectors (XTX): ⎛ 2.19⎞ ⎛ 1 ⎜ 2.70⎟ ⎜ 1 ⎜ ⎟ =⎜ ⎜ 2.95⎟ ⎜ 1 ⎜ ⎟ ⎜ ⎝ 2.83⎠ ⎝ 1
0 1.619 −0.991 −0.628
−0.866⎞ ⎛ 2.668⎞ −0.289⎟⎟ ⎜⎜ −0.082 ⎟⎟ −0.289⎟ ⎜ −0.032 ⎟ ⎟⎜ ⎟ −0.289⎠ ⎝ 0.5551⎠
0 −0.210 −1.298 −1.5 507
For the present case, the reader may verify that ⎛4 ⎜ Λ = UTU = ⎜ ⎜ ⎜ ⎝
4 4
⎞ ⎟ ⎟ . ⎟ ⎟ 1⎠
Thus, eigenvalues can transform a square matrix of full rank into an orthogonal one. Eigenvalues multiplied by any constant (c) are still eigenvalues:
( cΛ ) ⎛⎜⎝ 1c b⎞⎟⎠ = Λ′ b′ = Λb Thus, there is no unique set of eigenvalues and eigenvectors for a given matrix. To establish one, we use normalized eigenvalues and their corresponding eigenvectors. To normalize the eigenvectors, we choose c such that ∑ λ 2k = 1. If we order normalized eigenvalues by magnitude, then we have a unique set. Regrettably, most spreadsheets do not generate eigenvalues and eigenvectors with a stand-alone function, and the solution of eigenvector problems becomes tedious by hand. However, modestly priced software such as Mathcad does solve for eigenvectors.
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Modeling of Combustion Systems: A Practical Approach
Eigenvalues may be found as the solution to an nth-order polynomial (characteristic) equation, where n is the number of rows, presuming the starting matrix is a nonsingular, square one. However, the procedure becomes unwieldy for matrices greater than 3 × 3, and dedicated software is really a must. Statistical software is the best option. One may find the characteristic equation from the determinant of the subject matrix augmented with an undetermined variable (λ):
c0 + c1λ + c2 λ 2 +
cnλ n =
a11 − λ a21
a12 a22 − λ
a1n a2 n
a21
a3 n ann − λ
an1
a21
=0
This generates an nth-order characteristic equation. For real symmetric matrices (the only kind we need to consider in this text), the solution to the characteristic equation gives n real roots, though some may be repeated. Another method, which is generally easier and more amenable to spreadsheets, is to make use of the trace of the matrix to find the eigenvalues. The trace of a matrix is the sum of the diagonal elements. We may also define traces for higher-order square matrices: n
( ) ∑m
t n = tr Mn =
n kk
(1.37)
k =1
In the above equation, we are relying on context to obviate equivocation for n: for tn and mnkk, the superscript n is mere nomenclature, referencing the trace or elements of Mn, respectively; for Mn, the superscript is an authentic exponent. n
Mn =
∏ M = (M)(M) (M)
(1.38)
k =1
Thus, M2 = MM (not MTM). The characteristic equation and its solutions follow directly from tn: n
∑c λ k
k
=0
(characteristic equation)
(1.39)
k=0
⎧ 1 if j = n ⎪ cj = ⎪ ⎨ n− j − 1 oth herwise ⎪ −1 cn− k t n− j− k ⎪n − j k=0 ⎩
∑
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(coefficient solutions)
(1.40)
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We use the terms latent roots, eigenvalues, or characteristic roots for the values of λ that generate zeros for the characteristic equation. Each eigenvalue is associated with an eigenvector that we find by manipulating matrices. Section 4.3.2 gives an example of using eigenvalues and finding their associated eigenvectors to orthogonalize a matrix.
1.7 1.7.1
Important Concepts and Notation Summation and Matrix Notation
Generally, we may express operations with arrays and columns of numbers using either summation or matrix notation. Summation notation uses a capital sigma that may be subscripted, superscripted, both, or none. It represents a series of addition operations per the following equation: n
∑y
k
= y1 + y2 +
+ yn
(1.41)
k =1
• Here, k is the index and n is the limit. The letter i (for index) is the most common designation, with letters immediately following to indicate distinctions among indices, if necessary. We shall use k because it is easier to see in print and cannot be confused with the number 1, and because i often represents the imaginary unit, i = √ –1. • The index increments by 1, until reaching the limit, n. The limit may be finite (e.g., n = 7) or infinite (∞, that is, the series does not terminate). If the limit is infinite, the sum is referred to as an infinite series. • We may omit the limit, index, or both from the notation when context obviates ambiguity. The analogous operation with multiplication is known as product notation. It looks like this: n
∏ y = ( y )( y ) ( y ) k
1
2
(1.42)
n
k =1
The log and exponential functions convert between the forms for logs of any positive real number: n
∑ k =1
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⎛ yk = ln ⎜ ⎜⎝
n
∏ k =1
⎞ e yk ⎟ = y1 + y2 + ⎟⎠
+ yn
(1.43)
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Modeling of Combustion Systems: A Practical Approach
n
∑ ln( yk )
n
∏y
k
= e k =1
( )( ) ( y )
= y1 y2
k =1
n
(1.44)
Also note that for positive real numbers the sum of the logs is equal to the log of the products: n
⎛ ln yk = ln ⎜ ⎜⎝
∑ ( ) k =1
⎞
n
∏ y ⎟⎟⎠ k
(1.45)
k =1
Example 1.11 Summation and Product Notation Problem statement: For y1 = 1, y2 = 3, and y3 = 5 calculate ∑ y and ∏ y. Solution: From the problem statement it is obvious that k = 1 and n = 3:
∑ y = 1 + 3 + 5 = 9 , ∏ y = (1)(3)(5) = 15 1.7.2
Converting between Summation and Matrix Notation
Often, algebra is less cumbersome with matrix notation than with summation notation. The following identities convert between the forms: n=
1 = n
v=
∑k ≡ 1 1 T
1
∑
k
( )
≡ 1 T1
(1.46)
−1
∑ v ≡ (1 1) T
n
−1
(1.47)
1T v
(1.48)
Note that the overbar indicates a sum (Σv) normalized by the number of elements (n), also known as the arithmetic average, arithmetic mean, or simply the mean:
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57
∑u ≡ c u = u c
(1.49)
∑u ≡ 1 u = u 1
(1.50)
∑ uv ≡ u v = v u
(1.51)
∑v
(1.52)
c
T
T
T
2
T
T
T
≡ v Tv
Please pay attention to the difference between vector and scalar notation: c is a scalar constant comprising a single number, whereas c is a vector of constant values, each equal to c. Likewise, u and v are vectors comprising a column of different numbers, but u and v are single variables. Finally, n and v are special constants; the former is the total number of elements in a vector and the latter is the arithmetic average (mean) value of all the values of v.
1.7.3
Averages: Mean, Mode, and Median
We may sometimes characterize a vector of values with three kinds of averages. The mean, unqualified by any other term, is the sum of vector elements normalized by the number of elements; it is also called the arithmetic average. The mode is the data value that occurs most frequently within the vector. The mode may not be unique. The median is the central value of ordered vector elements; if the number of elements is even, then the median is the mean of the two central values.
Example 1.12 Working with Averages Problem statement: Calculate the mean, mode, and median for the following: xT = (–1 2 3 3 5 18). Suppose –1 and 18 are errant values and should be 1 and 8. Recalculate the averages and comment on what you notice. Solution: Mean = (–1 + 2 + 3 + 3 + 5 + 18)/6 = 5 Mode = 3 Median = (3+3)/2 = 3 For the vector xT = (1 2 3 3 5 8) we have:
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Modeling of Combustion Systems: A Practical Approach Mean = (1 + 2 + 3 + 3 + 5 + 8)/6 = 3.67 Mode = 3 Median = (3 + 3)/2 = 3 We see that the median is much less sensitive to errant values than the mean. The mode has limited utility, especially for small data sets, though in this case it is single-valued.
1.7.4
Various Means and the Generalized Mean
There are various operations for determining the average. The best known and used is the arithmetic average, which we have already defined. Mathematically, Equation 1.48 defines the arithmetic mean; whenever we use the term mean without qualification, we refer to the arithmetic mean. However, there are more means than the simple arithmetic mean. The next several equations list the most common kinds. We use the generalized nomenclature, M p, to refer to each type of mean. 1
M −1 =
M0 =
M=
M2 =
1 n
∑
1 v
n 1 1 + + v1 v2
=
n
∏v
1 n
∑ v = v + v +n
k
=
n
1
1 n
∑v
2
=
1 + vn
( v )( v ) ( v ) 1
2
2
n
+ vn
v12 + v22 + n
+ vn2
Harmonic mean
(1.53)
Geometric mean
(1.54)
Arithmetic mean
(1.55)
Root mean square (RMS) (1.56)
Example 1.13 Calculation of Various Means Problem statement: For the values vT = (1 10 100), calculate the harmonic, geometric, and arithmetic means, as well as the RMS value. Give the results to two decimal places. What do you notice? Repeat the calculation for vT = cT = (10 10 10). What further observations can you make? Repeat the calculation for vT = (–3 –2 1). What do you notice now? Finally, repeat with vT = (3 –1). Can you make any general hypotheses from these data sets?
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59
Solution: From Equations 1.53 through 1.56 we have the following, to two decimal places: M −1 =
3 1 1 1 + + 1 10 100
=
2.70
M0 =
(1)(10)(100)
=
31.62
=
37.00
=
58.03
M=
1 + 10 + 100 3 12 + 10 2 + 100 2 3
M2 = Observations:
• We see that in all cases, the mean is somewhere between the extreme values (e.g., between 1 and 100); that is, the mean tends toward the center of the values, termed central tendency. • Second, for the positive data set given, as p increases, M p tends toward larger values; conversely, as p decreases, M p tends toward smaller values. • These behaviors seem to indicate some relation among the above means. This raises the possibility of some grander generalized definition for the mean. For vT = cT = (10 10 10) the means become 3 1 1 1 + + 10 10 10
M −1 =
M0 =
M=
(10)(10)(10)
10 + 10 + 10 3
M2 =
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3
102 + 102 + 102 3
=
10
=
10
=
10
=
10
60
Modeling of Combustion Systems: A Practical Approach The fact that all the means are the same (i.e., M −1 (c) = M 0 (c) = M(c) = M 2 (c) = c ) is not so surprising. If all the means lie somewhere between their minimum and maximum for positive values, and these are identical, then the mean must also take that value, i.e., M p (c) = c . Now, we consider vT = (–3 –2 1): 3 1 1 1 + + −3 −2 1
M −1 =
M0 =
M=
3
( −3) ( −2 )(1)
( −3) + ( −2 ) + (1)
M2 =
3 −32 + −2 2 + 12 3
=
18
=
1.82
=
−1.33
=
2.16
This is strange. For the data set vT = (–3 –2 1), only M0 and M give values between –3 < v < 1. In fact, M2 cannot possibly give a negative number no matter what the data set. Continuing, for vT = (–3 1) we have M −1 =
2 1 1 + −3 1
=
−3
M0 =
( −3)(1)
=
3i
M=
−3 + 1 2
M2 =
−3 2 + 12 2
=
−1
≈
2.24
The result for M0 is not even a real number.
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61
The above considerations lead us to a definition of a generalized mean as follows:
Mp ≡
p
1 n
n
∑x
p k
Generalized mean
(1.57)
k =1
To avoid confusion, note that p on the left side of the equation is a mere nomenclature, not a true exponent. (By way of analogy, consider that in trigonometric functions, e.g., sin–1(x), the –1 merely signifies the inverse, i.e., arcsine(x), not the reciprocal, i.e., sin–1(x) ≠ 1/sin(x).) However, on the right side of the equation, p functions as a true exponent or root. The definition of Equation 1.57 is perfectly general, but if we want to keep M p in the domain of real numbers, we will sometimes need to restrict the range to x > 0. The generalized mean has the following six properties: 1. The generalized mean reduces to the geometric mean when p approaches zero:
M0 ≡
lim p→ 0
n
p
∑ k =1
xkp n
n
=
n
∏x
k
(1.58)
k =1
2. The generalized mean increases monotonically with p for x > 0: dM p ≥ 0 for x > 0 and −∞ < p < ∞ dp
(1.59)
3. The minimum and maximum values bound the generalized mean if x > 0 for all x: xmin ≤ M p ≤ xmax if x > 0 for all x
(1.60)
4. The generalized mean approaches the maximum x value as p increases without bound for all x, if x > 0: lim M p = xmax if x > 0 for all x p→ ∞
(1.61)
5. The generalized mean approaches the minimum x value as p decreases without bound if x > 0 for all x:
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Modeling of Combustion Systems: A Practical Approach
lim M p = xmin if x > 0 for all x p → −∞
(1.62)
6. The generalized mean increases with increasing p (and conversely decreases with decreasing p) if x > 0 for all x:
{
}
∀ ( p2 > p1 ) ∩ ( x > 0 )
∃
{M
p2
, M p1
}
∋
{M
p2
> M p1
}
(1.63)
(The above is read for every p2 greater than p1 and x greater than zero, there exist M p2 and M p1 such that M p2 is greater than M p1 .) The restriction x > 0 for all x is sufficient but not necessary. For example, for positive, odd-integer powers (including M), the range of x may extend over all real values, −∞ < x < ∞.
1.8
Least Squares
The technique of least squares (LS) is a powerful method for determining the best curve to a given set of data. LS is the appropriate fitting method if the errors are normally distributed and one knows the factor values.
1.8.1
The Method of Least Squares
The traditional introduction to this subject usually involves the fit of a straight line to data. We will begin here as well. However, we will show shortly that we can fit much more than straight lines with the least squares technique. We will make use of linear (matrix) algebra because it is the simplest way to obtain solutions to least squares problems. To begin, let us consider a scatter diagram with data values given in Figure 1.12. We have decided to fit a straight line: y = a0 + a1 x
(1.64)
Now we desire to find the constants a0 and a1. For the five data points we can write y1 = a0 + a1x1 y2 = a0 + a1x2 y3 = a0 + a1x3
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Introduction to Modeling 12
x 1 2 3 4 5
10
63
y 1.81 4.27 6.01 7.30 9.15
8 area equals square of vertical distance to regression line
6
regression, y = a0 + a1x
4
data points
2
0 0
1
2
3
4
5
6
FIGURE 1.12 The least squares line. The least squares line is the line that minimizes the sum of the squared distances from the data points to the regression line. One may generalize the method to any number of dimensions.
y4 = a0 + a1x4 y5 = a0 + a1x5 where the subscripts index each point for the x-y data. We may represent these data in matrix form as ⎛ y1 ⎞ ⎛ 1 ⎜ y ⎟ ⎜1 ⎜ 2⎟ ⎜ ⎜ y3 ⎟ = ⎜ 1 ⎜ ⎟ ⎜ ⎜ y4 ⎟ ⎜ 1 ⎜⎝ y ⎟⎠ ⎜⎝ 1 5
x1 ⎞ x2 ⎟⎟ ⎛ a0 ⎞ x3 ⎟ ⎜ ⎟ ⎟ ⎝ a1 ⎠ x4 ⎟ x5 ⎟⎠
or
⎛ 1.81 ⎞ ⎛ 1 ⎜ 4.27 ⎟ ⎜ 1 ⎜ ⎟ ⎜ ⎜ 6.01 ⎟ = ⎜ 1 ⎜ ⎟ ⎜ ⎜ 7.30 ⎟ ⎜ 1 ⎜⎝ 9.15 ⎟⎠ ⎜⎝ 1
1⎞ 2 ⎟⎟ ⎛ a0 ⎞ 3⎟ ⎜ ⎟ ⎟ ⎝ a1 ⎠ 4⎟ 5⎟⎠
(1.65)
From Equation 1.65, we have five equations but only two unknowns (a0 and a1). So, the system is overspecified. To account for this overspecification, we will add an error vector, e: ⎛ y1 ⎞ ⎛ 1 ⎜ y ⎟ ⎜1 ⎜ 2⎟ ⎜ ⎜ y3 ⎟ = ⎜ 1 ⎜ ⎟ ⎜ ⎜ y4 ⎟ ⎜ 1 ⎜⎝ y ⎟⎠ ⎜⎝ 1 5
⎛ e1 ⎞ x1 ⎞ ⎜e ⎟ ⎟ x2 ⎟ ⎛ a0 ⎞ ⎜ 2 ⎟ x3 ⎟ ⎜ ⎟ + ⎜ e 3 ⎟ ⎟ ⎝ a1 ⎠ ⎜ ⎟ x4 ⎟ ⎜ e4 ⎟ ⎟ x5 ⎠ ⎝⎜ e5 ⎟⎠
© 2006 by Taylor & Francis Group, LLC
or
⎛ 1.81 ⎞ ⎛ 1 ⎜ 4.27 ⎟ ⎜ 1 ⎜ ⎟ ⎜ ⎜ 6.01 ⎟ = ⎜ 1 ⎜ ⎟ ⎜ ⎜ 7.30 ⎟ ⎜ 1 ⎜⎝ 9.15 ⎟⎠ ⎜⎝ 1
⎛ e1 ⎞ 1⎞ ⎜e ⎟ ⎟ 2⎟ ⎛ a0 ⎞ ⎜ 2 ⎟ 3⎟ ⎜ ⎟ + ⎜ e3 ⎟ ⎟ ⎝ a1 ⎠ ⎜ ⎟ 4⎟ ⎜ e4 ⎟ ⎟ ⎜⎝ e ⎟⎠ 5⎠ 5
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Modeling of Combustion Systems: A Practical Approach
Then in matrix format we have y = Xa + e
(1.66)
From this point forward, the easiest way to follow along will be to put the matrices in a spreadsheet and name them as y and X, respectively (e and a are unknown so far). Premultiplying y and X by XT gives the following: X Ty = X TXa + X Te
(1.67)
One can do this easily in Excel™ using the transpose command: XT = transpose(X), which we label XT.* Then using the command for matrix multiplication we have XTy = mmult(XT,y) and XTy = mmult(XT,X); we label these as XTY and XTX, respectively, in the spreadsheet. We do not know the values for e. However, we can see that XTX is a square symmetric matrix with the number of rows and columns exactly equal to the number of parameters for which we are solving. Therefore, letting XTe = 0 leaves X Ty = X TXa
⎛ ⎜ ⎜ ⎝
(1.68a)
∑ y ⎞⎟ = ⎛⎜ N ∑ x ⎞⎟ ⎛ a ⎞ ⎜ ⎟ ∑ xy⎟⎠ ⎜⎜⎝ ∑ x ∑ x ⎟⎟⎠ ⎝ a ⎠ 0
2
(1.68b)
1
15 ⎞ ⎛ a0 ⎞ 55⎟⎠ ⎜⎝ a1 ⎟⎠
⎛ 28.54 ⎞ ⎛ 5 ⎜⎝ 103.33⎟⎠ = ⎜⎝ 15
(1.68c)
Inverting XTX and then premultiplying both sides of the Equation 1.68a by this quantity gives a:
(X X) T
−1
(
X Ty = X TX
)
−1
X TXa = a
(1.69)
* To label a range in Excel, place the mouse on the top left corner of the matrix, left click the mouse and drag to highlight the entire range. Then type alt + i n d and define the range name. To put results in a field of cells, first highlight the original calculation cell, then drag the mouse right and down over the output range, press F2, and then ctrl + shift + enter to apply the calculation to the highlighted range.
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65
One may code for (XTX)–1 in the spreadsheet as minverse(XTX); we label this inv. Finally, a = mmult(inv,XTY). Now, the predicted values for y (which we designate as yˆ ) are given by Equation 1.70, which is Equation 1.66 without the error term: yˆ = Xa
(1.70)
The spreadsheet command becomes yˆ = mmult(X,a) and the error vector ˆ Figure 1.13 shows the completed spreadsheet, showing both is e = y – y. values and formulas. The reader may wish to verify that X Te = 0 by naming the error vector e and finding the result of X Te = mmult(XT,e). For our particular tworegressor case, we have ⎛ 0⎞ X Te = ⎜ ⎟ ⎝ 0⎠ Equations 1.68a to c are termed the normal equations. The matrix form of Equation 1.68a is completely general for any least squares solution. For the case of two regressors, it reduces to Equation 1.68b, and for our particular case, it reduces to Equation 1.68c. From Figure 1.13 we see that, rounded to two decimal places, ⎛ a ⎞ ⎛ 0.40⎞ a = ⎜ 0⎟ ≈ ⎜ ⎝ a1 ⎠ ⎝ 1.77 ⎟⎠
(1.71)
Substituting the results of Equation 1.71 back into our starting equation (Equation 1.64) gives us the desired result: y = 0.40 + 1.77 x
1.8.2
The Method of Least Squares: The Calculus
Figure 1.12 shows the data and the regression line as well as several square regions. The area of each region is equal to the square of the vertical distances. Suppose we wish to adjust our regression line to minimize the sum of these squares. (By definition, this would be the least squares solution.) We shall begin with the assumption that we can specify any independent variable (x) without significant error, but that most of the error (e) occurs when we perform an experiment to measure the dependent variable, y. We shall formulate the error associated with y as y − y* = e , where y is what we measure, y * is the true value, and e is the difference between what we got (y) and what we should have gotten ( y *).
© 2006 by Taylor & Francis Group, LLC
66
y_hat 2.17 3.94 5.71 7.48 9.25
XTY 28.54 103.33
a 0.40 1.77
y 1.81 4.27 6.01 7.3 9.15
X 1 1 1 1 1
ε -0.36 0.33 0.30 -0.18 -0.10
y_hat =MMULT(X,a) =MMULT(X,a) =MMULT(X,a) =MMULT(X,a) =MMULT(X,a)
ε =y-y_hat =y-y_hat =y-y_hat =y-y_hat =y-y_hat
X 1 1 1 1 1
1 1 1 1 1
1 2 3 4 5
XT 1.00 1.00
1.00 2.00 XTX 5.00 15.00
1.00 3.00
15.00 55.00
1.00 4.00
1.00 5.00
(XTX)-1 1.10 -0.30
-0.30 .10
XT =TRANSPOSE(X) =TRANSPOSE(X) =TRANSPOSE(X) =TRANSPOSE(X) =TRANSPOSE(X) =TRANSPOSE(X) =TRANSPOSE(X) =TRANSPOSE(X) =TRANSPOSE(X) =TRANSPOSE(X) XTX (XTX)-1 =MMULT(XT,X) =MMULT(XT,X) =MINVERSE(XTX) =MINVERSE(XTX) =MMULT(XT,X) =MMULT(XT,X) =MINVERSE(XTX) =MINVERSE(XTX)
XTX a =MMULT(XT,y) =MMULT(Inverse,XTY) =MMULT(XT,y) =MMULT(Inverse,XTY) FIGURE 1.13 Linear algebra with spreadsheets. The upper matrix shows the generated matrices for the data shown in Figure 1.12. The lower matrix gives the actual spreadsheet formulas used using named ranges.
© 2006 by Taylor & Francis Group, LLC
Modeling of Combustion Systems: A Practical Approach
y 1.81 4.27 6.01 7.30 9.15
Introduction to Modeling
67
In the method of least squares, we seek to adjust the coefficients so that they minimize the sum of the squared error to its lowest possible value: ∑ e2 → min . Now, going backwards is a bit tricky because the true model is usually unknown and perhaps unknowable. If we do not know y*, how can we minimize e2? If we cannot know the true model, we must conjecture at least the model form based on other considerations. If we do not know the form of the model, our final hope will be a polynomial approximation of the true function. Whatever our conjecture, we shall use yˆ to indicate that this is our best guess for the proper value. That is, we will presume that y − yˆ = e. If yˆ ≈ y * (and more importantly, we have data that are not unduly biased or careless), then we have a good chance to arrive at an acceptable estimate for the true value. For the case of a straight line, we have yˆ = a0 + a1x . We can minimize the squared error by setting the first derivatives to zero (the maximum is unbounded, so the first derivative gives the minimum):
∑ε
∂
2
∂a0
∑ε
∂
∂a1
=
2
=
∑ ( y − yˆ )
∂
2
∂a0
=
∑ ( y − a − a x ) = ∑ 2 ( y − a − a x)( −1) ∂a 2
∂
0
1
0
1
0
∑ ( y − yˆ )
∂
∂a1
2
=
∑( y − a − a x) = ∑ 2 ( y − a − a x)(− x) ∂a 2
∂
0
1
0
1
1
These derivatives simplify to ∑ y = a0 N + a1 ∑ x and ∑ y = a0 ∑ x + a1 ∑ x 2 , or in matrix form (Equation 1.68b), ⎛ ⎜ ⎜ ⎝
∑ y ⎞⎟ = ⎛⎜ N ∑ x ⎞⎟ ⎛ a ⎞ ⎜ ⎟ ∑ xy⎟⎠ ⎜⎜⎝ ∑ x ∑ x ⎟⎟⎠ ⎝ a ⎠ 0
2
1
Remarkably, linear algebraic techniques earlier in the chapter gave us this very same equation. Thus, the linear algebraic solution is the least squares solution. This is true in general.
Example 1.14 Calculation of Least Squares Coefficients Problem statement: Given the following data, find the least squares solution for a0, a1, and a2 in the model y = a0 + a1x + a2x2.
© 2006 by Taylor & Francis Group, LLC
x
y
1.00 2.00 3.00 4.00 5.00
1.81 4.27 6.01 7.30 9.15
68
Modeling of Combustion Systems: A Practical Approach Solution: For the model y = a0 + a1x + a2x2, the matrix equation (y = Xa) becomes ⎛ 1.81 ⎞ ⎛ 1 ⎜ 4.27 ⎟ ⎜ 1 ⎜ ⎟ ⎜ ⎜ 6.01 ⎟ = ⎜ 1 ⎜ ⎟ ⎜ ⎜ 7.30 ⎟ ⎜ 1 ⎜⎝ 9.15 ⎟⎠ ⎜⎝ 1
1⎞ 4 ⎟⎟ ⎛ a0 ⎞ ⎜ ⎟ 9 ⎟ ⎜ a1 ⎟ ⎟⎜ ⎟ 16 ⎟ ⎝ a2 ⎠ 25⎟⎠
1 2 3 4 5
The solution procedure is identical to that used to derive Equation 1.71; it yields ⎛ a0 ⎞ ⎛ −0.44 ⎞ a = ⎜⎜ a1 ⎟⎟ = ⎜⎜ 2.49⎟⎟ ⎜⎝ a ⎟⎠ ⎜⎝ −0.12 ⎠⎟ 2 Note that we could have just as easily fit y = ( a−1 x) + a0 + a1x , or any other model with five or less linear coefficients. For the record, the values in this latter case would have been ⎛ 1.81 ⎞ ⎛ 1 ⎜ 4.27 ⎟ ⎜ 1 / 2 ⎜ ⎟ ⎜ ⎜ 6.01 ⎟ = ⎜ 1 / 3 ⎜ ⎟ ⎜ ⎜ 7.30 ⎟ ⎜ 1 / 4 ⎜⎝ 9.15 ⎟⎠ ⎜⎝ 1 / 5
1 1 1 1 1
1⎞ 2 ⎟⎟ ⎛ a−1 ⎞ ⎜ ⎟ 3⎟ ⎜ a0 ⎟ ⎟ 4⎟ ⎝⎜ a1 ⎟⎠ 5⎟⎠
yielding ⎛ a−1 ⎞ ⎛ 2.37 ⎞ a = ⎜⎜ a0 ⎟⎟ = ⎜⎜ 1.41 ⎟⎟ ⎜⎝ a1 ⎟⎠ ⎜⎝ −1.95⎟⎠ Note that the order of the columns is immaterial so long as they match the order of the rows in a. Which model is best? We will adopt three rules.
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Rule 1: The rule of rationality: A model must make sense. Rule 2: The rule of parsimony (Occam’s Razor): All things being equal, the simpler model is the better one. Rule 3: The rule of accuracy: All things being equal, the model that better explains the data is the better one. First of all, there is no compelling reason to use quadratic or reciprocal terms in the model (rule 1). Inspection of the model from a straight-line fit does not show any obvious problems with the linear model, and there are no theoretical reasons to augment the model. Second, the linear model is the simpler model and preferred from that standpoint (rule 2). Third, we expect that a model with additional parameters will do a better job of fitting the data. Certainly, with a least squares fit, the more parameters we add, the closer the approximation. In fact, if we use a five-parameter model for the five data points given, our regression would exactly fit the data. However, this violates the “all things being equal” clause of rule 3. In short, we conclude that the best model is the linear model we gave earlier. In Chapter 3, we develop some quantitative measures later to guide our preference.
1.8.3
Least Squares for Continuous Intervals
Thus far, we have examined the calculus of least squares for discrete data. However, it may be that we desire to approximate a more complicated but continuous function with a simpler function. For example, say we want to approximate sin(x) with a quadratic function (a0 + a1x + a2x2) over the interval 0 to π: S( x) = sin( x)
(1.72)
A( x) = a0 + a1x + a2 x 2
(1.73)
The least squares equation for continuous intervals is ∂ ∂a
x2
⌠ ⎮ ⌡
2
⎡⎣S( x) − A( x) ⎤⎦ dx = 0
(1.74)
x1
Here, a stands for the adjustable parameters in the model, with each differentiated, in turn. The lower and upper limits of the interval are x1 and x2, respectively. Note that this is analogous to the least squares criteria for discrete data, with the integral replacing the summation operator.
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70
Modeling of Combustion Systems: A Practical Approach
The derivatives of the integral are equal to the integral of the derivatives for functions that are everywhere differentiable within the continuous interval. That is, x2
⌠ ⎮ ⎮ ⌡
x1
2 ∂ ⎡⎣S( x) − A( x) ⎤⎦ dx = 0 ∂a
Expanding the derivative via the chain rule and noting that (∂ ∂a)S( x) = 0 (because it has no adjustable parameters) gives x2
⌠
2 ⎮⎮ ⎡⎣S( x) − A( x) ⎤⎦ ⌡
x1
∂ A( x) dx = 0 ∂a
This reduces to x2
⌠ ⎮ ⎮ ⎮ ⌡
⎞ ⎛ ∂ ⎜⎝ ∂a A( x)⎟⎠ S( x) dx =
x1
x2
⌠ ⎮ ⎮ ⎮ ⌡
⎞ ⎛ ∂ ⎜⎝ ∂a A( x)⎟⎠ A( x) dx
(1.75)
x1
This procedure minimizes the integrated squared error (ISE). For the case at hand, Equation 1.75 reduces to the following matrix form: ⎛ sin x dx ⎞ ⎛ dx ⎜ ⎟ ⎜ ⎜ ⎟ ⎜ ⎜ x sin x dx ⎟ = ⎜ xdx ⎜ ⎟ ⎜ ⎜ x 2 sin x dx ⎟ ⎜ x 2 dx ⎜⎝ ⎟⎠ ⎜⎝
∫ ∫ ∫
∫ ∫ ∫
⎞
∫ xdx ∫ x dx⎟ ⎛ a ⎞ ⎟ x dx x dx ⎟ ⎜⎜ a ⎟⎟ ∫ ∫ ⎟⎜a ⎟ ⎟⎝ ⎠ ∫ x dx ∫ x dx⎟⎠ 2
0
2
3
1
3
4
(1.76)
2
Equation 1.75 is the continuous form of the least squares normal equations (Equations 1.68a to c); the integrals have replaced the summation operator. The method is general for any number of factors and dimensions so long as they are linear in the coefficients.
Example 1.15 A Least Squares Approximation of a Continuous Function Problem statement: Use Equation 1.75 to approximate the sine function with a second-order polynomial over the interval 0 to π.
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Introduction to Modeling
71
Solution: We have already reduced Equation 1.75 for the sine function in Equation 1.76. Substituting the limits from 0 to π and solving numerically gives us the following normal matrix equation with values rounded to three decimals: ⎛ 2.000 ⎞ ⎛ 3.142 ⎜ 3.142 ⎟ = ⎜ 4.935 ⎟ ⎜ ⎜ ⎜⎝ 5.870 ⎟⎠ ⎜⎝ 10.335
4.935 10.335 24.352
10.335 ⎞ ⎛ a0 ⎞ 24.352 ⎟⎟ ⎜⎜ a1 ⎟⎟ 62.204 ⎟⎠ ⎜⎝ a2 ⎟⎠
Solving for a, we obtain the following results rounded to three decimal places: ⎛ a0 ⎞ ⎛ −0.050 ⎞ ⎜ a ⎟ = ⎜ 1.312 ⎟ ⎟ ⎜ 1⎟ ⎜ ⎜⎝ a2 ⎟⎠ ⎜⎝ −0.418 ⎟⎠ Figure 1.14 shows the relations graphically. The dotted line is the least squares solution. We may also write it as the matrix equation 1.0
0.8
0.6
0.4
0.2
0
-0.2 0
0.2π
0.4π
0.6π
0.8π
π
FIGURE 1.14 A least squares approximation of a continuous function. The solid line represents S(x) = sin(x), while the dashed line represents an approximating function (least squares parabola) representing the minimum integrated squared error, A(x) = a0 + a1x + a2x2. See the text for the numerical values of the coefficients.
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72
Modeling of Combustion Systems: A Practical Approach
(
A( x ) = x a = 1 T
1.8.4
x
⎛ a0 ⎞ x ⎜ a1 ⎟ ⎜ ⎟ ⎜⎝ a ⎟⎠ 2 2
)
Least Squares as a Filter
We can look at least squares through the lens of linear algebra in yet another way. We can suppose that y, the vector comprising the source data, contains two components — the unadulterated signal, yˆ (pure information), and some contamination or adulteration, e. Our objective is to filter out the noise, e, and leave behind the pure information contained in the source data, yˆ . For this purpose, we define a function G = (XTX)–1XT
(1.77)
Then, according to Equation 1.69, a = Gy
(1.78)
The vector a actually codes for the pure information, yˆ , according to Equation 1.70. Substituting Equation 1.78 into Equation 1.70 gives us yˆ = XGy. We may define H = X(XTX)–1XT
(1.79)
We note by inspection that H = XG and yˆ = XGy. Then we may also write yˆ = Hy
(1.80)
H is known as the hat matrix because it converts y into yˆ (y hat). It has some very interesting properties. The reader may verify that H = HTH = H2 = Hn where n is any positive integer. This property is known as idempotency; i.e., H is an idempotent matrix. The sum of the principal diagonal (also called the trace) is equal to the number of columns in X. Finally, we note that H, like G, comprises only manipulations on the X matrix. The model we choose to fit will completely determine X, and thus H. Presuming our least squares model is correct, H acts as a filter per Equation 1.80; it strains out the error from the data, leaving only information. Finally, we may define J = I – H, where I is the identity matrix comprising the same number of rows and columns as XTX. By substitution with the earlier relations we have
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Introduction to Modeling
73 J = I – H = I – X(XTX)–1XT
(1.81)
The J matrix filters out all the useful information from the data and leaves behind only the error: e = Jy
(1.82)
H is standard nomenclature, G and J are not standard, but we use them because we believe they are helpful tools. Figure 1.15 shows these relations graphically and some corresponding and useful matrix equations.
1
a
X
2
G
y ˆ y=y+e
5
1T 1Tyˆ 0 9
7
yˆ
3
e
4
H J
XT
1. a=Gy 2. a=G yˆ 3. yˆ =Hy 4. e=Jy 5. yˆ =Xa 6. XTyˆ =XTy 7. 1Tyˆ =1Ty 8. 0=XTe 9. 0=1Te
6
8
XTyˆ
0
G H I J
= (XTX)-1XT = JX = (XTX)-1XTX = identity matrix = I – H = I – (XTX)-1XTX
FIGURE 1.15 A filter analogy for least squares. Moving from left to right represents filtering of the source data (y), where y comprises both the unadulterated information ( yˆ ) and the noise (e, random error). G and H filters reject noise but allow information to pass. However, the noise is lost; subsequent filtering cannot restore it. At the opposite extreme, the J filter rejects the signal and allows only noise to pass. All filters are functions of the original data matrix, X. Zero represents the loss of all information. The numbers above each path correspond to the equation legend at right. Important vectors are shaded.
G, H, and J are not really transforms in the true sense of the word, because the result is an alteration, not merely a transformation of the source matrix;
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74
Modeling of Combustion Systems: A Practical Approach
so, the filter analogy is conceptually more correct. For example, if we filter white light through a blue filter, we end up with only blue light. If the information we are looking for is blue and the rest is noise, then well and good. One can even say that blue is better than white because the signal is unadulterated. Nonetheless, we cannot get the other colors back from the blue light; the filter has excluded them. In a similar analogy, if we filter the hiss from the music, we have better music, but we cannot get the hiss back. If we reject the music and leave only the hiss, we cannot get the music back. For the case at hand, it is not possible to reconstruct useful information from the error vector. That is, one might suppose that if e = Jy, then y = J–1e. However, one cannot invert G, H, or J; J is not necessarily square (its columns are usually greater than its rows) and G and H are not necessarily full rank. Nor should this be possible, because it is not possible to get information from randomness. Likewise, if we filter out yˆ from y, thereby rejecting e, we cannot get the original error back and reconstitute y. The vector a behaves as a true transform between X and yˆ ; that is, yˆ = Xa. For y = yˆ + e, we can reconstitute y (from yˆ and e) or, conversely, decompose y into its components ( yˆ and e). But we cannot transform yˆ to e or vice versa using some function of X alone. Finally, we mention some other identities for least squares models:
∑ y = ∑ yˆ , or equivalently, 1 y = 1 yˆ
(1.83)
X T y = X T yˆ
(1.84)
X Te = 0
(1.85)
T
T
∑ e = 0, or equivalently, 1 e = 0 T
∑ yˆ
2
= yˆ T yˆ = y T yˆ
(1.86)
(1.87)
The last relation may be proved by noting y = yˆ + e and substituting
(
)
T
(
)
y T yˆ = yˆ + e Xa = yˆ T + eT Xa = yˆ TXa + eTXa But as eTXa = 0 and Xa = yˆ , we have y T yˆ = yˆ T yˆ . Equations 1.83 through 1.87 are true for least squares with any real data set whatever. Figure 1.15 also depicts these relations.
© 2006 by Taylor & Francis Group, LLC
Introduction to Modeling 1.8.5
75
A Misconception about Least Squares
Now it is time to clear up an important misconception. The method of least squares uses matrix algebra, also called linear algebra. This seems to imply for some that least squares only fits straight lines. However, the method of least squares refers to matrix formulations that are linear in the coefficients, not necessarily the variables; in a regression we are not solving for the variables, we are solving for the coefficients. Thus, an equation such as y = a0 + a1 sin( x1 ) + a2 x2 is certainly not linear in x1 or x2; however, it is linear in a0, a1, and a2. Therefore, we can find the coefficients using linear least squares. This gives the method great practical power. For the record, in such a case, Equation 1.68a reduces to ⎛ ⎜ ⎜ ⎜ ⎜ ⎜⎝
∑ sin(x ) ∑ x ∑ y ⎞⎟ ⎛⎜ N ∑ y sin(x )⎟⎟ = ⎜⎜ ∑ sin(x ) ∑ sin (x ) ∑ sin(x ) ⎟ ⎜ ∑ y x ⎟⎠ ⎜⎝ ∑ x ∑ sin(x ) x ∑x 1
2
2
1
1
2
1
2
1
1
2
2
⎞ ⎟ ⎛ a0 ⎞ x2 ⎟ ⎜⎜ a1 ⎟⎟ ⎟ ⎟ ⎜⎝ a2 ⎟⎠ ⎟⎠
and one computes the coefficients in the usual way via Equation 1.69. We leave the derivation of the above expression as an exercise for the reader. However, spreadsheet or statistical software will automatically populate the matrix with the proper values when we solve for a = ( X TX )−1 X Ty .
1.8.6
Transforming Equations for Least Squares Fitting of the Parameters
If our model form is not linear in the parameters, it may be possible to transform the equation so that it is. As examples, linear least squares fits the following equations:
y = f ( x) = a0 +
y = f ( x) =
a1 + a2 sin( x) + a3 x + e x
1 a0 + a1x + a2 x 2 + e
x x y = f ( x1 , x2 ) = a0 a1 1 a2 2 e e Here are the respective matrices:
© 2006 by Taylor & Francis Group, LLC
(1.88a)
(1.89a)
(1.90a)
76
Modeling of Combustion Systems: A Practical Approach ⎛ y1 ⎞ ⎛ 1 ⎜y ⎟ ⎜ ⎜ 2⎟ = ⎜1 ⎜ ⎟ ⎜ ⎜ ⎟ ⎜ ⎝ yn ⎠ ⎜⎝ 1
1 / x1
sin( x1 )
1 / x2
sin(xx2 )
1 / xn
sin( x3 )
⎛ 1 / y1 ⎞ ⎛ 1 ⎜ 1/ y ⎟ ⎜ 1 2⎟ ⎜ =⎜ ⎜ ⎟ ⎜ ⎜ ⎟ ⎜ ⎝ 1 / yn ⎠ ⎝ 1
x1 x2 xn
( ) ( )
x2 ,1
( )
xn,1
⎛ ln y1 ⎞ ⎛ 1 ⎜ ⎟ ⎜ ⎜ ln y2 ⎟ ⎜ 1 ⎜ ⎟ =⎜ ⎜ ⎟ ⎜ ⎜⎝ ln yn ⎟⎠ ⎝ 1
x1,1
x1 ⎞ ⎛ a0 ⎞ ⎛ e1 ⎞ ⎟⎜ ⎟ ⎜ ⎟ x2 ⎟ ⎜ a1 ⎟ ⎜ e2 ⎟ ⎟⎜a ⎟ +⎜ ⎟ ⎟ ⎜ 2⎟ ⎜ ⎟ xn ⎟⎠ ⎝ a3 ⎠ ⎝ en ⎠
⎛e ⎞ x12 ⎞ ⎛ a0 ⎞ ⎜ 1 ⎟ 2⎟ x2 ⎟ ⎜ ⎟ ⎜ e 2 ⎟ a + ⎟ ⎜ 1⎟ ⎜ ⎟ ⎜ ⎟ ⎝ a2 ⎟⎠ ⎜ ⎟ ⎝ en ⎠ xn 2 ⎠ ⎛ e1 ⎞ x1,2 ⎞ ⎛ ln a0 ⎞ ⎟ ⎟ ⎜ e2 ⎟ x2 , 2 ⎟ ⎜ ⎜ ln a1 ⎟ + ⎜ ⎟ ⎟⎜ ⎟ ⎜ ⎟ ⎟ ⎝ ln a2 ⎠ ⎜ ⎟ xn , 2 ⎠ ⎝ en ⎠
( ) ( ) ( )
(1.88b)
(1.89b)
(1.90b)
Technically, in Equations 1.89b and 1.90b we are not solving the original equation, but once we derive the coefficients, we can transform the results back to the original form. However, least squares cannot properly fit the following equations: y = f ( x) = a0 +
y = f ( x) =
a1 + a2 sin( x) + a3 x e x
1 +e a0 + a1x + a2 x 2
x x y = f ( x1 , x2 ) = a0 a1 1 a2 2 e
(1.88c)
(1.89c)
(1.90c)
Why not? Take a closer look at the error term. Equations 1.88a, 1.89a, and 1.90a can be linearized to result in an additive error term, and XTXe = 0. However, Equations 1.88c, 1.89c, and 1.90c do not have an additive error term and one cannot linearize them to provide one. But how are we to know how the error relates to the model? In the absence of information to the contrary, one should hope for the best and plan for the worst. So, once we fit the equation, we shall perform a postanalysis to validate our assumptions. If our postanalysis invalidates them, we must abandon them and use different techniques.
© 2006 by Taylor & Francis Group, LLC
Introduction to Modeling 1.8.7
77
Constrained Polynomials
It may happen that we wish to constrain the approximating polynomial in some important way. In general, minimizing ISE will minimize n – t coefficients, where n is the number of coefficients and t is the number of constraints. For the case at hand, as we have used an approximating function having three coefficients, we are free to specify up to three constraints. If we specify three, then we will have no degrees of freedom left to minimize ISE; the constraints will completely determine the coefficients. If we specify two constraints, we may use ISE to determine one coefficient. If we specify one constraint, then ISE will determine the remaining two. For example, suppose we have the following general function: y( x) = a0 + a1φ1 ( x) + a2φ2 ( x) + In a regression, we are solving for the constants a1, a2, etc. We always know y(x) and the function values φ1 ( x), φ2 ( x) because these comprise the data. Then as soon as we specify a constraint such as a1 = 3, we can immediately reduce the equation to something like y x − a1φ1 ( x) = a0 + a2φ2 ( x) + because we fully know the left-hand side of the equation. The matrix equation then reduces to
()
⎛ ⎜ ⎜ ⎜ ⎜ ⎝⎜
∑ ⎡⎣ y(x) − a φ (x)⎤⎦ ⎞⎟ ⎛⎜ N ∑ φ (x) ∑ φ (x) ⎡⎣ y(x) − a φ (x)⎤⎦⎟⎟ = ⎜⎜ ∑ φ ( x) 1 1
2
2 2 2
1 1
⎟ ⎟⎠
⎜ sym ⎜⎝
⎞ ⎟ ⎛ a0 ⎞ ⎟⎜a ⎟ ⎟ ⎜ 2⎟ ⎟ ⎜⎝ ⎟⎠ ⎠⎟
and we eliminate a1 from the right-hand side of the equation. We illustrate with some examples.
Example 1.16 Constraining Least Squares Equations Problem statement: Consider the linear equation y = a0 + a1x . Give the normal equations for regressing the parameters. Then rewrite the normal matrix equation for the following constraints: a0 = 0, a1 = 1, and both constraints above, simultaneously. Solution: As we have seen, the normal equations are ⎛ ⎜ ⎜ ⎝
© 2006 by Taylor & Francis Group, LLC
∑ y ⎞⎟ = ⎛⎜ N ∑ x ⎞⎟ ⎛ a ⎞ ⎜ ⎟ ∑ xy⎟⎠ ⎜⎝ ∑ x ∑ x ⎟⎠ ⎝ a ⎠ 0
2
1
78
Modeling of Combustion Systems: A Practical Approach If we know that a0 = 0, then the equation becomes y = a1x . This reduces the normal matrix equation to (∑ xy) = (∑ x 2 )a1 or, more simply, a1 = ∑ x 2 ∑ xy . Similarly, if we know that a1 = 1, then y = a0 + x or, equivalently, y − x = a0 . Then the normal matrix equation becomes ∑ ( y − x) = ( N )a0 or, more simply, a0 = ∑( y − x) N . Finally, if we know that both a0 = 0 and a1 = 1, the equation becomes y = x . As there are no adjustable parameters to solve, there is no normal matrix equation.
It is also possible to specify constraints on the response values. For these cases, we can often solve for the coefficients in the usual way and adjust them after the fact. One common example is specifying that the final equation attains some maximum value. We illustrate with an example.
Example 1.17 Minimizing ISE Subject to a Constraint Problem statement: Minimize the following equation subject to the constraint that the curve attains unity at its maximum:
(
)
y = a0 − a1x + a2 ln x + 1
(1.91)
Solution: First, we will solve for the unconstrained function, Y, using the normal equations: ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣
∑
∑Y ∑ xY
⎤ ⎛ N ⎥ ⎜ ⎥ ⎜ ⎥=⎜ ⎥ ⎜ Y ln ( x + 1) ⎥ ⎜⎝ sym ⎦
∑x ∑x
∑ ln ( x + 1) ⎞⎟ ⎛ A ⎞ −∑ x ln ( x + 1)⎟ ⎜ A ⎟ ⎟⎜ ⎟ ⎟⎜A ⎟ ∑ ln ( x + 1) ⎟⎠ ⎝ ⎠
−
0
2
1
2
2
This determines A0, A1, and A2. The function is not yet normalized to have a maximum value of 1, but the ratio Y( x) Y ′ ( xmax ) must equal 1 when x = xmax. Now one may find Y( xmax ) by noting dY that the xmax will occur when = 0 ; that is dx
(
)
d ⎡ A0 − A1x + A2 ln x + 1 ⎤⎦ = 0 dx ⎣
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Introduction to Modeling
79
This reduces to dY A = 0 = − A1 + 2 x +1 dx yielding xmax = y=
=
=
=
or
A2 − 1 as the location of the extremum. This gives A1
( (
)
A0 − A1x + A2 ln x + 1 Y( x) = Y xmax A0 − A1xmax + A2 ln xmax + 1
(
)
(
)
)
A0 − A1x + A2 ln x + 1
⎛A ⎞ ⎛A ⎞ A0 − A1 ⎜ 2 − 1⎟ + A2 ln ⎜ 2 − 1 + 1⎟ ⎝ A1 ⎠ ⎝ A1 ⎠ A0 − A1 x + A2 ln ( x + 1) ⎛A ⎞ A0 + A1 − A2 + A2 ln ⎜ 2 ⎟ ⎝ A1 ⎠
()
Y x
(
Y xmax
)
=
(
)
A0 − A1x + A2 ln x + 1
(
)
A0 + A1 + A2 ln A2 − ln A1 − 1
(
)
y = a0 − a1x + a2 ln x + 1
,
(1.92)
where a0 =
A0 A0 + A1 + A2 ln A2 − ln A1 − 1
(
)
a1 =
A1 A0 + A1 + A2 ln A2 − ln A1 − 1
a2 =
A2 A0 + A1 + A2 ln A2 − ln A1 − 1
(
)
(1.92a)
(1.92b)
and
(
From this, we note that b0 a0 b0 a0 , = = b1 a1 b2 a2
© 2006 by Taylor & Francis Group, LLC
)
(1.92c)
80
Modeling of Combustion Systems: A Practical Approach and b1 a1 = b2 a2 as we would expect if normalizing the curve by a constant value. Section 5.6 revisits this equation form in detail to model normalized heat flux.
1.8.8
Orthogonal Polynomials
Many times, we desire a spectral representation of a continuous curve. For this purpose, one may use orthogonal polynomials. Consider a Taylor series for a single continuous variable, x: y = a0 + a1x + a2 x 2 + . For the continuous interval L1 ≤ x ≤ L2 , the normal equations will give the least squares solutions for a0, a1, etc.: ⎛ L2 ⎞ ⎛ L2 ⎜ ydx ⎟ ⎜ dx ⎜ ⎟ ⎜ ⎜ L1 ⎟ ⎜ L1 ⎜ L2 ⎟ ⎜ ⎜ xydx ⎟ ⎜ ⎜ ⎟ =⎜ ⎜ L1 ⎟ ⎜ ⎜ L2 ⎟ ⎜ ⎜ x 2 ydx ⎟ ⎜ sym ⎜ ⎟ ⎜ ⎜ L1 ⎟ ⎜ ⎜⎝ ⎟⎠ ⎝⎜
∫
L2
∫
∫
L2
xdx
L1
L1
L2
∫
∫
∫
x 2 dx
L2
x 2 dx
L1
∫ x dx 3
L1 L2
∫ x dx
∫
4
L1
⎞ ⎟ ⎟ ⎟ ⎟ ⎛ a0 ⎞ ⎟⎜a ⎟ ⎟ ⎜ 1⎟ ⎟ ⎜ a2 ⎟ ⎟⎜ ⎟ ⎟⎝ ⎠ ⎟ ⎟ ⎟⎠
These are not necessarily orthogonal. However, suppose an orthogonal function exists, y = a0φ0 ( x) + a1φ1 ( x) + a2φ2 ( x) + , where φ0 ( x) = b00 , φ1 ( x) = b10 + b11x , φ2 ( x) = b20 + b21x + b22 x 2 , etc. Then, our task will be to find b00, b10, b11, etc., such that the matrix is orthogonal in the lenient sense. That is, we know that the normal equations will yield ⎛ L2 ⎞ ⎛ L2 ⎜ φ0 ydx ⎟ ⎜ φ20 dx ⎜ ⎟ ⎜ ⎜ L1 ⎟ ⎜ L1 ⎜ L2 ⎟ ⎜ ⎜ φ ydx ⎟ ⎜ ⎜ 1 ⎟ =⎜ ⎜ L1 ⎟ ⎜ ⎜ L2 ⎟ ⎜ ⎜ φ ydx ⎟ ⎜ sym ⎜ 2 ⎟ ⎜ ⎜ L1 ⎟ ⎜ ⎜⎝ ⎟⎠ ⎜⎝
∫
∫
∫
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∫
L2
∫
L2
φ0φ1dx
L1
L1
L2
∫ φ dx 2 1
L1
∫
φ0φ2 ( x)dx L2
∫ φ φ dx 1 2
L1
L2
∫ φ dx 2 2
L1
⎞ ⎟ ⎟ ⎟ ⎟ ⎛ a0 ⎞ ⎟⎜a ⎟ ⎟ ⎜ 1⎟ ⎟ ⎜ a2 ⎟ ⎟⎜ ⎟ ⎟⎝ ⎠ ⎟ ⎟ ⎟⎠
(1.93)
Introduction to Modeling
81
And we seek coefficients to drive the nondiagonal integrals to vanish; so, L2
L2
∫ φ φ dx = 0, ∫ φ φ dx = 0, etc., 0 1
0 2
L1
L1
or, more generally, L2
∫
L2
φjφ k dx = 0 and
L1
∫
L2
φ kφ k dx =
L1
∫ φ dx ≠ 0 2 k
L1
for all j and k from 0 to ∞. We have the liberty to choose the limits of interest for the integrals and the starting function f0(x). Since we will truncate the infinite series from left to right, we can also solve for the coefficients in this order. This will ensure that the series is orthogonal for any truncated length. The procedure is known as Gram–Schmidt normalization. For convenience, let us choose −1 ≤ x ≤ 1 and φ0 = b00 = 1. If we truncate the series to just two terms, Equation 1.93 becomes ⎞ + b11x dx ⎟ ⎟ ⎛ a0 ⎞ −1 ⎟⎜ ⎟ 1 ⎟ ⎝ a1 ⎠ 2 b10 + b11x dx ⎟ ⎟⎠ −1
1 ⎛ ⎞ ⎛ ⎜ ⎟ ⎜ 2 ydx ⎜ ⎟ ⎜ −1 ⎜1 ⎟ =⎜ ⎜ ⎟ ⎜ ⎜ b10 + b11x ydx ⎟ ⎜ sym ⎜⎝ ⎟⎠ ⎝⎜ −1
1
∫
∫(
∫ (b
)
∫(
)
10
)
where 1
∫ (b
10
)
+ b11x dx = 0
−1
and 1
∫ (b
10
)
2
+ b11x dx ≠ 0
−1
Now it is an easy matter to determine the coefficients: 1
∫ (b
10
)
+ b11x dx = 0 =
−1
© 2006 by Taylor & Francis Group, LLC
1
∫
−1
1
1
−1
−1
x2 b10dx + b11xdx = b10 x + b11 2
∫
= 2b10
82
Modeling of Combustion Systems: A Practical Approach
giving b10 = 0 for any value of b11. For convenience, we choose b11 = 1. Expanding to the next term we have 1 ⎛ ⎞ ⎛ ⎜ ⎟ ⎜ 2 ydx ⎜ ⎟ ⎜ −1 ⎜ ⎟ ⎜ 1 ⎜ ⎟ ⎜ xydx ⎜ ⎟ =⎜ ⎜ ⎟ ⎜ −1 ⎜1 ⎟ ⎜ ⎜ ⎟ ⎜ 2 ⎜ b20 + b21 x + b22 x ydx ⎟ ⎜ sym ⎜⎝ ⎟⎠ ⎜⎝ −1
⎞ b20 + b21 x + b22 x 2 dx ⎟ ⎟ −1 ⎟⎛a ⎞ 1 ⎟ 0 2 x b20 + b21 x + b22 x dx ⎟ ⎜ a1 ⎟ ⎟ ⎜⎜ ⎟⎟ −1 ⎟ ⎝ a2 ⎠ 1 ⎟ 2 b20 + b21 x + b22 x 2 dx ⎟ ⎟⎠ −1 1
∫(
)
∫ x dx ∫ (
)
∫
1
∫
∫(
2
−1
)
∫(
)
Solving for 1
∫ (b
)
+ b21 x + b22 x 2 dx = 0
20
−1
gives 1
0=
∫b
20
1
∫
1
1
∫
dx + b21xdx + b22 x dx =
−1
−1
2
∫
−1
−1
1
x3 b22 x dx = b20 + b22 3 1
∫ x (b
from which we obtain b22 = −3b20 and
= 2b20 +
2
20
−1
2 b22 3
)
+ b21x + b22 x 2 dx = 0. For conve-
−1
3 nience, we will let 2b20 = 1, then b22 = − . Thus, our orthogonal polynomial 2 has become ⎛ 3⎞ y = a0 + a1x + a2 ⎜ x 2 − ⎟ + 2⎠ ⎝
(1.94)
It has a normal matrix equation that is clearly orthogonal: 1 ⎛ ⎞ ⎛1 ⎜ ⎟ ⎜ dx ydx ⎜ ⎟ ⎜ −1 −1 ⎜ ⎟ ⎜ 1 ⎜ ⎟ ⎜ ⎜ ⎟ ⎜ xydx ⎜ ⎟ =⎜ ⎜ 1 −1 ⎟ ⎜ ⎜ ⎛ ⎟ ⎜ ⎞ 3 ⎜ ⎜ x 2 − ⎟ ydx ⎟ ⎜ sym 2⎠ ⎜ −1 ⎝ ⎟ ⎜ ⎜ ⎟ ⎜ ⎝ ⎠ ⎝
∫
∫
∫
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∫
1
∫ x dx 2
−1
1
∫
−1
2
⎛ 2 3⎞ ⎜⎝ x − 2 ⎟⎠ dx
⎞ ⎟ ⎟ ⎟⎛a ⎞ ⎟ 0 ⎟ ⎜ a1 ⎟ ⎟⎜ ⎟ ⎟ ⎜ a2 ⎟ ⎟ ⎜⎝ ⎟⎠ ⎟ ⎟ ⎟ ⎠
Introduction to Modeling
83
Since we have more coefficients than constraints, we can choose them to be convenient, or such that all the diagonal elements have the same value, or such that all the functions have identical values at the limits, say, ⎧−1 φ k L1 = ⎨ ⎩⎪ 1
( )
( )
if k is odd , and φ k L2 = 1 if k is even
This last option is typical for this orthogonal series (the Legendre polynomial series).
1.8.9
General Definition of Orthogonal Polynomials
Suppose y is a function of x: y( x). It is often possible to represent y( x) in a spectral series, g( x) , such that g( x) = y( x) and g( x) are the sum of orthogonal components. ∞
g( x) =
∑ A w(x)φ (x) = w(x) ⎡⎣ a φ (x) + a φ (x) + k
k
0 0
1 1
k=0
⎤⎦
(1.95)
where Ak is the kth coefficient, w( x) is a weight function, described shortly, and φ k ( x) is the kth orthogonal polynomial. In order to be orthogonal over a continuous interval, φ k ( x) must have the following properties: L2
∫ w(x)φ (x)φ (x)dx = 0 j
j≠k
k
(1.96)
L1
L2
∫ w(x) ⎡⎣φ (x)⎤⎦ dx = ν 2
k
k
≠0
(1.97)
L1
We shall call νk the unit normalization factor, because if we divide Equation 1.97 by this number, we shall obtain unity; i.e., L2
∫
L1
2 w( x) ⎡⎣φ k ( x) ⎤⎦ dx = 1 νk
We can take the weight and normalization functions into our definition by defining
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84
Modeling of Combustion Systems: A Practical Approach
φ′k ( x) =
w( x) φ k ( x) νk
Then n=∞
y( x) =
∑ A′φ′ (x) = a′ φ′ (x) + a′φ′ (x) + k k
0 0
1 1
(1.95′)
k=0
L2
∫ φ′(x)φ′ (x)dx = 0 j
j≠k
k
(1.96′)
L1
L2
∫ ⎡⎣φ′ (x)⎤⎦ dx = 1 2
k
(1.97′)
L1
We derive Ak by evaluating L2
Ak =
∫ y(x)φ (x) dx k
(1.98)
L1
or equivalently, L2
Ak′ =
∫ y(x)φ′ (x) dx k
(1.98′)
L1
Sometimes we may prefer to leave νk out of the definition or use something different. Regardless, these schemes transform the original functions in x to a function in φ(x) or φ′k ( x) having orthogonal coefficients Ak or Ak′ , respectively. This then is a spectral representation for y(x). We usually truncate the infinite series of Equation 1.95 at some convenient point. Sometimes, we can represent an infinite series with a finite series if we select the proper orthogonal polynomial. Table 1.5 lists orthogonal transforms for various polynomial series. These are taken from Abramowitz and Stegun7 and modified to suit. Specific to this text we will refer to the classical orthogonal polynomials of Table 1.5 as COP functions and the modified orthogonal polynomials of Equations 1.95', 1.96', and 1.97' as MOP functions. These phrases are peculiar to this text. We see that COPs and MOPs are essentially spectral representations of continuous curves. Often, if we choose the proper orthogonal polynomial, the coefficients die out rapidly.
© 2006 by Taylor & Francis Group, LLC
Classical Orthogonal Polynomials (COPs) Name
L1
L2
(1 x ) 2
Gegenbauer
Sk
w(x) a
U2(
1 2
(
(1 x ) (1 + x ) a
b
a, b > 1
Legendre
Tchebychev, I
Tchebychev, II
–1
1
( 1 2 x ( ) ¨ ( )( )
(
) ( ) ¬ ¼ 2 2 k a , k a + + ( ) ® ( )¾ 2
(
2
2
1 2k
U 2
j
k2j
j= 0
( (
) , a+ kj
)
)
j ! k 2j !
( k + a)!( k + b )!( x 1) ( x + 1) kj
k
j
¨ j ! ( k j )! ( b + j )! ( k + a j )! j= 0
)
if a = 0 if a | 0
U U 2
© k¹ floo or ª º « 2»
1 , a
2
2 2k + 1
1 x
1 x
) ()
)
k !, k + b , k + a b + 1
1
+1
(
) , k + 2a
k ! k + a ¬, a ¼ ® ¾
a>1 2
Jacobi, I
1 2 a
Explicit Expressions
Introduction to Modeling
TABLE 1.5
1 2k
k 2
© k¹ floor ª º « 2»
( 1) ¬2 ( k j)¼! ¨ j!( k ®j)!(2j k¾)! j
j= 0
© k¹ floor ª º « 2»
( 1) ( k j 1)! ¨ j!( k 2j)! (2x)( j
k2j
)
j= 0
© k¹ floor ª º « 2»
( 1) ( k j)! j
( k 2j)
¨ j ! ( k 2 j )! ( 2 x ) j= 0
85
© 2006 by Taylor & Francis Group, LLC
86
(1 x ) x a
Jacobi, II
(
a, b > 1 Shifted Legendre
( k + a + b + 2) ( k + b + 2) ) k ! a + 1 + 2 ( k + b + 1)
a+ 2 b +1 2 ( ) a + 1+ 2 k + b + 1
b
(
1 1
(
x 1 x
(
Shifted Tchebychev, II
j
)
a + b + 2k + 1
j= 0
x 1 x
)
if a = 0 if a 0
2
)
Not listed separately
8
( 1) k ! x j ! ( k j )! j
1
x
e
j= 0
j
0 Generalized Laguerre
a
xe
x
( k + a + 1)
k
k!
j= 0
floor
Hermite
e
x2
k2
k
k! j= 0
© 2006 by Taylor & Francis Group, LLC
)
)
k
Laguerre
k j
( 1) ( k + a)! x ( k j )! ( j + a )! j
k 2
( 1) 2( j !( k j
k 2j
j
) x( k 2 j )
)
2j !
Modeling of Combustion Systems: A Practical Approach
Shifted Tchebychev, I
( 1) k ! ( a + b + 2k j + 1) x( j !( k j )! ( b + k j + 1)
k
1 2k + 1
1 0
(
(b + k + 1)
Introduction to Modeling
87
By examining the weight function (or its square root), we can usually select an appropriate series for a spectral representation. For example, Table 1.5 shows that w(x) = 1 for the Legendre polynomials. Therefore, they are an appropriate transform for simple (Taylor/Maclaurin series) polynomials. Laguerre polynomials orthogonalize Boltzman, Poisson distribution, and Arrhenius functions. Hermite polynomials orthogonalize normal distribution functions. We have not listed the Fourier series or Bessel functions due to widespread coverage in other texts.8 However, they are orthogonal polynomials as well. Although the polynomials of Table 1.5 are the classical orthogonal ones, nothing stops us from defining custom orthogonal polynomials (e.g., in sin–1(x), tanh(x), or whatever) to suit our purposes using the Gram–Schmidt orthogonalization procedure. Notwithstanding, we can usually find a COP series to suit our purposes. COPs are well-studied and have well-understood properties. For example, suppose we theorize that the molecular weight of a liquid fuel is a function of various chemical reactions in the refinery. Some of them increase the molecular weight and others decrease it. Suppose the reactions are adding and subtracting various compounds along a continuous molecular weight distribution curve and that each addition or deletion occurs as a normal distribution centered at a particular molecular weight. From Table 1.5, 2we 2 see that the Hermite polynomials have w( x) = e − x . Therefore, w( x) = e − ( x /2 ) , which is the unit normal probability distribution. Then, we might define a 2 MOP with w( x) = e − ( x /2 ) to give a spectral representation for a series of normal distributions. We shall illustrate with a sample bimodal distribution.
Example 1.18 Spectral Representations: A Hermite Polynomial to Represent a Bimodal Distribution Problem statement: Derive a spectral representation in modified Hermite polynomials for the bimodal distribution of Equation 1.99, shown in Figure 1.16:
y = a1e
⎛ x− m ⎞ 1 ⎜ ⎟ ⎝ 2 s1 ⎠
2
+ a2 e
⎛ x − m2 ⎞ ⎜ ⎟ ⎝ 2 s2 ⎠
2
(1.99)
where a1 = 1, m1 = 0, s1 = 1/2, a2 = 1, m2 = 2, and s2 = 2. Solution: The most parsimonious representation of Equation 1.99 is the equation itself in six parameters: a1, m1, s1, a2, m2, and s2. However, no convenient transformation linearizes the parameters of Equation 1.99, so we must use dedicated software and a nonlinear (iterative) routine to fit them. The computational intensity increases rapidly as the number of terms increases. As an alternative, we shall perform a spectral decomposition of Equation 1.99.
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88
Modeling of Combustion Systems: A Practical Approach 1
2
a
X
y
G
ˆ y=y+e 5
1T 1Tyˆ 0 9
7
yˆ
3
e
4
1. a=Gy 2. a=G yˆ 3. yˆ =Hy 4. e=Jy 5. yˆ =Xa 6. XTyˆ =XTy 7. 1Tyˆ =1Ty 8. 0=XTe 9. 0=1Te
H J
XT 6
8
G H I J
Ty ˆ
X
0
= (XTX)-1XT = JX = (XTX)-1XTX = identity matrix = I – H = I – (XTX)-1XTX
FIGURE 1.16 A bimodal distribution. The addition of two normal probability curves creates a bimodal distribution. In the general case, all constants differ. The case shown has a1 = 1, m1 = 0, s1 = 1/ 2, a2 = 1, m2 = 2, and s2 = 2.
Therefore, we define Equation 1.100 as our MOP (see the Hermite polynomials in Table 1.5 for the source of this derivation):
H k′ ( x ) =
Hk ( x ) 2k k ! π
e
−
x2 2
(1.100)
From the table we see that ⎛ k⎞ floor⎜ ⎟ ⎝ 2⎠
k!
∑ j= 0
( −1) 2( ) x( j ! ( k − 2j )! j
k−2 j
k−2 j
)
where floor (k/2) defines the largest integer less than k/2; e.g., floor (5/2) = 2. From this we may derive
Hk ( x) =
© 2006 by Taylor & Francis Group, LLC
−
⎛ k⎞ x 2 floor⎜ ⎟ ⎝ 2⎠ 2
4
π
e
∑ j =0
( −1)
j
⎛k ⎞ ⎜ − 2 j⎟
k ! 2⎝ 2 ⎠ x ( j !( k − 2 j ) !
k− 2 j )
(1.101)
3365_book copy.fm Page 89 Thursday, February 16, 2006 10:12 AM
Introduction to Modeling
89
The first several MOPs are H 0 ( x) =
1 4
π
e
−
x 2
⎛ 1 −x ⎞ H1′ ( x) = ⎜ e 2⎟ x 2 ⎝4π ⎠ ⎛ 1 −x ⎞ ⎛ 2 1⎞ H 2′ ( x) = ⎜ e 2⎟⎜x − ⎟ 2 2⎠ ⎝4π ⎠⎝ ⎛ 1 −x ⎞ ⎛ 3 3 ⎞ 2 H 3′ ( x) = ⎜ e 2 ⎟ ⎜ x − x⎟ 2 ⎠ 3 ⎝4π ⎠⎝ ⎛ 1 −x ⎞ ⎛ 4 3⎞ H 4′ ( x) = ⎜ e 2 ⎟ ⎜ x − 3x 2 + ⎟ 4 4⎠ ⎝ π ⎠⎝
2 3
⎛ 1 −x ⎞ ⎛ 5 15 ⎞ 2 H 5′ ( x) = ⎜ e 2 ⎟ ⎜ x − 5x 3 + x⎟ 4 4 ⎠ 15 ⎝ ⎝ π ⎠ We have done nothing more than incorporate the square root of the weight and unit normalization parameters into the Hermite polynomial. Therefore, the relations of Equations 1.95', 1.96', and 1.97' hold. Figure 1.17 graphs the first several of these functions. Now we may derive a spectral representation, ∞
∑ A′ H ′ ( x) k
k
k=0
for the original function. That is,
y = a1e
⎛ x− m ⎞ 1 ⎜ ⎟ ⎝ 2 s1 ⎠
2
+ a2 e
⎛ x − m2 ⎞ ⎜ ⎟ ⎝ 2 s2 ⎠
2
∞
=
∑ A′ H ′ ( x) k
k=0
k
90
Modeling of Combustion Systems: A Practical Approach 1.0 0.8
H'0
0.6
H'2
H'1
0.4 0.2 0.0
H'3
– 0.2 – 0.4 – 0.6 – 0.8
–3
–2
–1
0
1
2
3
x FIGURE 1.17 The first four MOPs. These particular MOPs are derived from Hermite polynomials with the weight factor incorporated and normalized to unit area. The are useful for spectral representations of sums of normal distributions.
where Equation 1.98' determines Ak′ . The first 16 spectral coefficients follow, truncated to three decimal places: A0′ = 1.876
A1′ = 0.744
A2′ = 0.426
A3′ = 0.574
A4′ = 0.541
A5′ = 0.368
A6′ = 0.261
A7′ = 0.215
A8′ = 0.167
A9′ = 0.118
A10 ′ = 0.084
A11 ′ = 0.062
A12 ′ = 0.045
A13 ′ = 0.031
A14 ′ = 0.022
A15 ′ = 0.015
…
We show these coefficients graphically in Figure 1.18. Although the series is infinite, the first eight coefficients comprise virtually all of the variation. 1.8.9.1 Discrete MOPs and Real Data If we derive the coefficients from real data, we often find that coefficients beyond some threshold characterize only the noise. Then, we may truncate the model and retain the few remaining coefficients as significant. If so, one
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Introduction to Modeling
91
2.0 1.8 1.6 1.4 1.2
A'k 1.0 0.8 0.6 0.4 0.2 0.0 0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
k FIGURE 1.18 The first 16 spectral coefficients. These coefficients are the spectral representation of Figure 1.16. Although the series is infinite, these coefficients — when multiplied by their respective MOPs — will reconstitute Figure 1.16 so that the two curves are indistinguishable by eye.
is able to represent a complex continuous curve with only a few coefficients. These are usually much handier than the original continuous and nonlinear functions. Since all the terms are orthogonal, they cannot bias one another. Therefore, the coefficients are unique, and they do not change if we truncate the model. In fact, the coefficients will be the least squares solutions, and one can make MOPs orthogonal in the strictest sense. In practice, we will not know the underlying distribution and must estimate the function from the data, which will be discrete rather than continuous. Accordingly, x is now discretely valued over n intervals at xj = L1 +
L2 − L1 j n
(1.102)
where j is an index from 0 to n. That is, L1 ≤ xj ≤ L2. Accordingly, Equation 1.98′ becomes n
Ak′ =
∑ y φ′ ( x ) j k
j
(1.98″)
j =1
Equation 1.98″ is perfectly general. However, if we prefer, we could recast the equation in one of two matrix notations. The first would be to define Ak′ = y Tφ′k for each of the k coefficients. We could also define φj,k in a row–column matrix and write a′ = y TΦ for the vector of results, and this would be a second formulation.
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92
Modeling of Combustion Systems: A Practical Approach
As an illustration of the procedure, reconsider the function of Equation 1.99. We shall make this a discrete function by taking n = 100 data points at even x intervals between L1 = –10 and L2 = 10 and adding an error, ej, that is uniformly distributed. Figure 1.19 shows the data for –0.2 ≤ ej ≤ 0.2. This is significant error and represents an error band that is nearly 15% of the maximum value of the function.
FIGURE 1.19 Discrete data case. The solid line is the original curve. Contaminating the curve with uniform error gives the crosses. The dashed line is the curve reconstituted from the discrete and adulterated data. Despite significant error and a discrete sample, the reconstituted curve is a reasonable approximation to the underlying continuous distribution.
Remarkably, despite the serious error, the coefficients for the MOP are identical to three places for the first eight coefficients. These first eight spectral coefficients give rise to the reconstituted curve shown (dashed line). For reference, we also show the unadulterated curve (solid line). The agreement is quite good, even in the wake of the large errors we have imposed. Moreover, the spectral representation (eight coefficients) is nearly as parsimonious as the original equation (six coefficients). Thus, despite serious adulteration, we are able to recover accurate values for the spectral representation and a reasonable estimate of the unadulterated function. Table 1.6 shows the coefficients and the error by which they differ. It is possible to generalize orthogonal polynomials to two or more factors by solving simultaneous equations for the undetermined coefficients using a Gram–Schmidt orthogonalization procedure.9
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Introduction to Modeling
93
TABLE 1.6 Theoretical vs. Fitted Coefficients
1.9
Coefficient
Theoretical
Fitted
Error
A0′ A1′ A2′ A3′ A4′ A5′ A6′ A7′ A8′ A9′ A10 ′ A11 ′ A12 ′ A13 ′ A14 ′ A15 ′
1.876 0.744 0.426 0.574 0.541 0.368 0.261 0.215 0.167 0.118 0.084 0.062 0.045 0.031 0.022 0.015
1.876 0.744 0.426 0.574 0.541 0.368 0.261 0.215 0.156 0.132 0.087 0.063 0.033 0.074 0.036 0.035
<0.0005 <0.0005 <0.0005 <0.0005 <0.0005 <0.0005 <0.0005 <0.0005 +0.011 –0.014 –0.003 –0.001 +0.012 –0.043 –0.014 –0.020
Addendum
This section compiles various proofs regarding the geometric mean for positive x. 1.9.1 Proof That M0 Reduces to the Geometric Mean We first note that a Taylor series expansion of eu at u = 1 becomes u2 e = 1+ u+ + 2! u
∞
=
∑ uk ! k
k =1
Since yλ = e
( ) = e λ ln y
ln y λ
we may expand it as yλ = e
( ) = e λ ln y = 1 + λ ln y + λ 2 ln 2 y +
ln y λ
2!
∞
=
∑ k =1
Rearranging λ ln 2 y yλ − 1 = ln y + + λ 2!
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∞
=
∑ k =1
λ k −1 ln k y k!
λ k ln k y k!
94
Modeling of Combustion Systems: A Practical Approach
and applying the limit gives the natural log:10 lim ⎛ y λ − 1 ⎞ = ln y λ → 0 ⎜⎝ λ ⎟⎠
(1.103)
To avoid confusion between p as a name (as in Mp ) and p as a true exponent, we will rename Mp as Q for the duration of this and the subsequent proofs. Subsequently, p functions in the usual sense of an exponent. That is,
Q≡
p
n
∑x
1 n
p k
k =1
Then, raising both sides of this defining equation and multiplying by n, we obtain n
nQ p =
∑x
p k
k =1
where p is a true exponent on both sides of the equation. Subtracting n and dividing by p gives
nQ p − n = p
⎛ ⎜ ⎜⎝
n
⎞
∑ x ⎟⎟⎠ − n p k
k =1
p
This may be factored as ⎛ Qp − 1 ⎞ n⎜ ⎟= ⎝ p ⎠
⎛ xkp − 1 ⎞ ⎜ p ⎟ ⎝ ⎠ k =1 n
∑
Upon rearrangement we obtain ⎛ Qp − 1 ⎞ 1 ⎜ p ⎟=N ⎝ ⎠
⎛ xkp − 1 ⎞ ⎜ p ⎟ ⎝ ⎠ k =1 N
∑
(1.104)
But the first term in Equation 1.104 is identical in form to Equation 1.103. Therefore, taking the limit as p → 0 gives
© 2006 by Taylor & Francis Group, LLC
Introduction to Modeling
95
lim ⎛ Q p − 1 ⎞ lim 1 = ⎜ ⎟ p → 0⎝ p ⎠ p → 0 N
⎛ xkp − 1 ⎞ ⎜ p ⎟ ⎝ ⎠ k =1 N
∑
Reinstituting our nomenclature, we define
ln M 0 ≡
lim ⎛ Q p − 1 ⎞ p → 0 ⎜⎝ p ⎟⎠
which reduces to 1 ln M = n 0
n
∑ ln x
k
k =1
But the sum of the logs is equal to the log of the products (Equation 1.45), giving us 1 ⎛ ln ⎜ n ⎜⎝
ln M 0 =
n
∏ k =1
⎞ xk ⎟ ⎟⎠
Taking the antilog, we obtain the final equation: n
M0 =
n
∏x
k
k =1
Thus, we have thus proved our assertion: M0 is the geometric mean.
1.9.2
Proof of the Monotonicity of M p
Starting once again from n
Q≡
p
∑ xn
p k
k =1
we take logs to obtain ln Q =
© 2006 by Taylor & Francis Group, LLC
1 ⎛ ln ⎜ p ⎜⎝
n
∑ k =1
xkp ⎞ ⎟ n ⎟⎠
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Modeling of Combustion Systems: A Practical Approach
and then differentiate implicitly: ⎛ 1⎞ d⎜ ⎟ ⎛ ⎝ p⎠ d ln Q = ln ⎜ ⎜⎝ dp dp
n
∑ k =1
⎛ n xp ⎞ k d ln ⎜ ⎟ ⎜ ⎟⎠ p⎞ n ⎝ k =1 xk 1 ⎟+ dp N ⎟⎠ p
∑
Rearranging terms we obtain
⎛ ⎜ ⎜⎝
x ⎞ ⎟ n ⎟⎠
n
∑
p k
k =1
−
1 p
dQ 1 ⎛ = − 2 ln ⎜ dp p ⎝⎜
n
∑ k =1
x ⎞ 1 ⎟+ n ⎟⎠ p p k
n
1 N
∑ k =1
xkp n
∑ k =1
⎛ xp ⎞ d⎜ k ⎟ ⎝ n⎠ dp
If we let y = xp, then ln y = p ln x and d ln y 1 dy dy = = ln x or = x p ln x dp y dp dp Substituting this result into the above equation gives ⎛ ⎜ ⎜⎝
N
∑ k =1
x ⎞ ⎟ n ⎟⎠ p k
−
1 p
dQ 1 ⎛ = − 2 ln ⎜ dp p ⎝⎜
N
∑ k =1
xkp ⎞ 1 ⎟+ n ⎟⎠ p
N
1 N
∑ xn
p k
∑ x lnn x p k
k
k =1
k =1
which reduces to
dQ = dp
⎛ ⎜ ⎜⎝
1
p x ⎞ ⎟ n ⎟⎠ k =1 p2 n
∑
p k
−1
⎡ ⎢ ⎢ ⎢ ⎣⎢
n
∑(
xkp ln xkp
k =1
n
⎛ ⎜ ⎜ −⎜ ⎝
n
) ∑x k =1
n
p k
⎞ ⎟ ⎛ ⎟ ⎟⎠ ln ⎜ ⎜⎝
n
∑ k =1
⎤ ⎥ xkp ⎞ ⎥ ⎟⎥ n ⎟⎠ ⎥ ⎦
or if we prefer, d ⎡⎣ M p ⎤⎦ dp
1 ⎛1 = 2⎜ p ⎜⎝ n
n
∑ k =1
xkp ⎞ ⎟ N ⎟⎠
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1 −1 p
⎡1 ⎢ ⎢n ⎣
n
∑(x k =1
n
p k
⎛
n
⎞⎤
) n1 ∑ ( x ) ln ⎜⎜⎝ n1 ∑ x ⎟⎟⎠ ⎥⎥
ln xkp −
p k
k =1
p k
k =1
⎦
(1.105)
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97
Now that we have found the first derivative, we may examine its behavior to see if d ⎡⎣ M p ⎤⎦ dp
≥0
for all p. For x > 0 we note that the multiplier cannot be negative, 1 ⎛1 ⎜ p2 ⎜⎝ n
n
∑ k =1
xkp ⎞ ⎟ n ⎟⎠
1 −1 p
≥0
Then, the proof reduces to the hypothesis that if x > 0, then 1 n
n
∑(
)
xkp ln xkp ≥
k =1
1⎛ ⎜ n ⎜⎝
n
∑ k =1
⎞ ⎛1 xkp ⎟ ln ⎜ ⎟⎠ ⎜⎝ n
n
⎞
∑ x ⎟⎟⎠ p k
k =1
which may be further simplified to 1 n
n
∑(x
p k
ln x
p k
k =1
)
1⎛ ≥ ⎜ n ⎜⎝
n
∑ k =1
( )
⎞ xkp ⎟ ln xkp ⎟⎠
(1.106)
or
( )
xkp ln xkp ≥ xkp ln xkp
(1.107)
That is, the mean of xln(x) is greater than or equal to the product of the mean and the log of the mean. An equivalent formulation of Equation 1.106 is 1 n
n
∑ x ⎡⎢⎣ln x p k
k =1
p k
( )
− ln xkp ⎤ ≥ 0 ⎥⎦
which we may further simplify to n
∑ k =1
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⎛ p x xkp ln ⎜ k ⎜⎝ x p k
⎞ ⎟ ≥0 ⎟⎠
(1.108)
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Modeling of Combustion Systems: A Practical Approach
Now whenever xip > xip , then
n
∑ k =1
⎛ p x xkp ln ⎜ k ⎜⎝ x p k
⎞ ⎟ >0 ⎟⎠
and the quantity is positive. Since xkp > xip multiplies this quantity, greater values weight the positive quantities. Hence, Equation 1.108 must be true. In fact, the only way one can cause Equation 1.108 to equal zero is if and only if all xkp are identically valued, that is, xkp = c . In that case, ⎛ p x ln ⎜ k ⎜⎝ x p k
⎞ ⎟ =0 ⎟⎠
and the equality holds.
1.9.3
Proof That M p Approaches xmax as p → ∞
By definition for nonnegative x, p
xp = x
(1.109)
Now, we may rank a series of xk values from lowest to highest as xmin ≥ x 1 ≥ x2 ≥ ≥ xmax and for x > 0. Now p → ∞ does not disturb the p p ≥ x1p ≥ x2p ≥ ≥ xmax ranking. That is, xmin . However, xmax is multiplied by a larger value (itself, p times) than all others. Therefore, as p → ∞, xmax is weighted more and more heavily, becoming a larger part of the mean. That is, lim ⎛ 1 p → ∞ ⎜⎝ n
⎞
∑ x ⎟⎠ = x p k
max
Then, substituting this result into Equation 1.109, we obtain
p
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lim ⎛ 1 p → ∞ ⎜⎝ n
⎞
∑ x ⎟⎠ = p k
p
xmax p = xmax
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99
Therefore, as p increases without bound, M p approaches xmax. Proof That M p Approaches xmin as p → – ∞
1.9.4
p p For x > 0 and xmin ≥ x 1 ≥ x2 ≥ ≥ xmax we know that xmin ≤ x1p ≤ x2p ≤ ≤ xmax as p → –∞. That is, negative exponents reverse the rank order; xmin becomes the maximum value. Then
lim ⎛ 1 p → −∞ ⎜⎝ n
⎞
∑ x ⎟⎠ = x p k
min
Substituting this result into Equation 1.109 we obtain lim ⎛ 1 p → −∞ ⎜⎝ n
p
⎞
∑ x ⎟⎠ = p k
p
xmin p = xmin
Therefore, as p decreases without bound, M p approaches xmin. Proof xmin ≤ M p ≤ xmax for x > 0
1.9.5
For x > 0 the following is true: •
p
lim ⎛ 1 p → ∞ ⎜⎝ n
•
p
lim ⎛ 1 p → −∞ ⎜⎝ n
⎞
∑ x ⎟⎠ = p k
p
⎞
∑ x ⎟⎠ = p k
xmax p = xmax
p
xmin p = xmin
• M p is monotonic. Therefore, x
1.9.6
min
≤ M p ≤ xmax.
Proof That M p Increases with Increasing p and the Converse
This follows by definition of a monotonically increasing function. For d [ M p ] dp ≥ 0 must be everywhere increasing with increasing p. Therefore, for any p2 > p1, then M p2 > M p 1 if it exists. Since M p exists for all x > 0, then M p increases with increasing p. The converse is necessarily true. That is, for any p2 > p1, then M p2 > M p 1 for x > 0 (see Figure 1.20).
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Modeling of Combustion Systems: A Practical Approach 1.2 Generalized Mean
1 Data Set: x1, x2, x3 = {0.1,1.0,10.0}
0.8 0.6 0.4
First Derivative
0.2
Power, p -3.0
-2.0
-1.0
0 0.0
1.0
2.0
3.0
4.0
5.0
-0.2 Second Derivative
-0.4 -0.6 FIGURE 1.20 The generalized mean and its derivatives. For positive values, the generalized mean remains positive; as the power p increases, so does the generalized mean; the extrema lie at ± ∞, with a point of inflection in between.
References 1. Cleveland, W.S., Visualizing Data, Hobart Press, Summit, NJ, 1993. 2. Tufte, E.R., The Visual Display of Quantitative Information, Graphics Press, Chesire, CT, 1983. 3. Tufte, E.R., Envisioning Information, Graphics Press, Chesire, CT, 1990. 4. Harris, R.L., Information Graphics: A Comprehensive Illustrated Reference, Oxford University Press, New York, 1999. 5. Abbott, E.A., Flatland: A Romance of Many Dimensions, available online at http:// www.geom.uicuc.edu/~banchoff/Flatland/ 6. Gellert, W. et al., Eds., The VNR Concise Encyclopedia of Mathematics, American Edition, Van Nostrand Reinhold Company, New York, 1977, p. 588. 7. Abramowitz, M. and Stegun, I.A., Eds., Handbook of Mathematical Functions, with Formulas, Graphs, and Mathematical Tables, Dover Publications, New York, 1965. 8. Spiegel, M.R., Advanced Mathematics for Engineers and Scientists, Fourier Series, Schaum’s Outline Series, McGraw-Hill, New York, 1971, chap. 7, p. 182. 9. Davis, H.F., Fourier Series and Orthogonal Functions, Dover Publications, New York, 1963, pp. 50f. 10. Box, G.E.P and Draper, N.R., Empirical Model-Building and Response Surfaces, John Wiley & Sons, New York, 1987, pp. 288–293. (This limit was shown to be useful for indicating the log transformation of a response.)
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2 Introduction to Combustion
Chapter Overview This chapter overviews some practical and theoretical aspects of combustion. We first discuss burners generically, including fuel and air metering, flame stabilization and shaping, and some fundamental techniques for control of emissions such as NOx. So prepared, we move on to consider archetypical burners — those representative of the traditional classes of burners one might find in a refinery or petrochemicals plant, or that provide heat for steam. We build upon this foundation by next considering archetypical process units such as boilers, process heaters of various types, and reactors such as hydrogen reformers and cracking units. In order to lay the groundwork for more detailed combustion modeling, the chapter considers important combustion-related responses such as NOx emissions, flame length, noise, etc., and the factors that influence them. Historically, practitioners have defined a traditional test protocol for quantifying these effects, which we present. We also consider some aspects of thermoacoustic instability; this has become a more important topic with the advent of ultralow NOx burners employing very fuel lean flames. In the latter third of the chapter, we develop stoichiometric and mass balance relations in considerable mathematical detail. We also consider energy-related quantities such as heat and work, adiabatic flame temperature, and heat capacity. As well, we relate the practical consequences of a mechanical energy balance as applied to combustion equipment. Such things include draft pressure, incompressible airflow, compressible fuel flow, and practical representations thereof — capacity curves for air and fuel.
101
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102
2.1
Modeling of Combustion Systems: A Practical Approach
General Overview
Combustion is the self-sustaining reaction between a fuel and oxidizer characterized by a flame and the liberation of heat. Usually, but not always, the flame is visible. A flame is the reaction zone between fuel and oxidizer; it typically comprises steep thermal and chemical gradients — the flame is often only a millimeter or so thick. On one side of the flame, there is fuel and oxidizer at low temperature; on the other side are combustion products at high temperature. Hydrogen and hydrocarbons in some combination are the typical fuels in the petrochemical and refining industries. Occasionally, due to some special refining operations, we find carbon monoxide in the fuel stream. Oxygen (in air) is the usual oxidizer. In practice, combustion reactions proceed to completion with the fuel as the limiting reagent — that is, with air in excess. A burner is a device for safely controlling the combustion reaction. It is typically part of a larger enclosure known as a furnace. A process heater is any device that makes use of a flame and hot combustion products to produce some product or prepare a feed stream for later reaction. Examples of such processes are the heating of crude oil in a crude unit, the production of hydrogen in a steam–methane catalytic reformer, and the production of ethylene in an ethylene cracking unit. A boiler is a device that makes use of a flame or hot combustion products to produce steam. The furnace is the portion of the process unit or boiler encompassing the flame. The radiant section comprises the furnace and process tubes with a view of the flame. In contrast, the convection section is the portion of the furnace that extracts heat to the process without a line of sight to the flame. Every industrial combustion process has some thermal source or sink.
2.1.1
The Burner
A burner is a device for safely controlling the combustion reaction. A diffusion burner is one where fuel and air do not mix before entering the furnace. If fuel and air do mix before entering the furnace, then the device is a premix burner. Premix burners may mix all or some of the combustion air with the fuel. If one desires to distinguish between them, a partial premix burner is one that mixes only part of the combustion air, with the remainder provided later. Most pilots are of the partial premix type to ensure that they will light under high excess air conditions typical of furnace start-up. Diffusion burners supply most of the heating duty in refinery and boiler applications; therefore, we discuss them first. Figure 2.1 shows the main features for accomplishing this. The particular version of burner shown in Figure 2.1 is a natural draft burner. That is, a slight vacuum in the furnace (termed draft — 0.5 in. water column below atmospheric pressure is a typical figure) and a relatively large opening (burner throat) allow enough air to enter the combustion zone to
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Introduction to Combustion
Burner Throat Refractory Tile
Furnace Floor
103
Primary Fuel Tip
Flame
Secondary Fuel Tips Burner Front Plate Cone (Throat Restriction)
Fuel Risers Inlet Damper
Noise Muffler Air Plenum FIGURE 2.1 A typical industrial burner. The typical industrial burner has many features, which can be classified in the following groups: air metering, fuel metering, flame stabilization, and emissions control. Refer to the text for a discussion of each.
support the full firing capacity. The diffusion burner comprises a fuel manifold, risers, tips, orifices, tile, plenum, throat restriction, and damper. Each diffusion burner type may differ in detailed construction, but all will possess these main functional parts. We discuss each in turn. 2.1.1.1 The Fuel System A fuel manifold is a device for distributing fuel. In the figure, one fuel inlet admits fuel to the manifold while several risers allow the fuel to exit. A riser is a fuel conduit. (In the boiler industry, risers are sometimes termed pokers, but the function is the same.) Each riser terminates in a tip; for some boiler burners the tip is called a poker shoe or just a shoe. A tip is a device designed to direct the fuel in a particular orientation and direction. It has one or more orifices, holes, or slots drilled at precise angles and size. An orifice is a small hole or slot that meters fuel — for a particular design fuel pressure, composition, and temperature, the orifice restricts the flow to the specified rate. Together, these parts comprise the fuel system of a gas burner.
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2.1.1.2 About Fuels There are two main gaseous fuels for combustion processes: natural gas and refinery gas. Commercially available natural gas has a stable composition comprising mostly methane (see Appendix A, Table A.9). Refinery gas, on the other hand, is capable of considerably more variation. In a sense, refinery gas is the “garbage dump” for the gas products in the refinery. Generally, whatever the refinery cannot use for some higher value-added process it consumes as fuel. It is quite typical for refineries to specify several different refinery fuels for combustion equipment — one representing normal conditions, another representing a normal auxiliary case, and perhaps two or three upset scenarios. The scenarios will vary in hydrogen concentration, typically from 10 to 60%, giving the fuels quite different combustion properties. It is very important that the refinery weigh the likelihood of scenarios that represent widely varying heating values on a volumetric basis. For example, suppose a particular process unit can receive fuel according to four different scenarios: • • • •
Fuel A: 620 Btu/scf, normal fuel representing 10% of run time Fuel B: 760 Btu/scf, stand-by fuel representing 84% of run time Fuel C: 840 Btu/scf, start-up fuel representing 5.98% of run time Fuel D: 309 Btu/scf, high hydrogen upset case representing 0.02% of run time
Since fuel C is a start-up case, it does not matter how infrequently it occurs, the burners must operate on fuel C. However, the difference in volumetric flow rate among the fuels A, B, and C is small. On the other hand, fuel D represents a significant difference in hydrogen concentration. This will markedly affect major fuel properties such as flame speed, specific gravity, and flow rate through an orifice. Later, we shall develop the flow equations that show that fuel D represents the maximum flow (and maximum fuel pressure) condition. One can obtain a burner to meet all these fuel conditions. However, for fuels A, B, and C, the pressure will necessarily be lower. Some possible consequences are lower fuel momentum and “lazier” longer flames when the facility is not running fuel D. In severe cases, the flue gas momentum will control the flame path. Thus, flames may waft into process tubes and will be generally poorer in shape — all for the sake of preserving good operation for an operating case representing only 0.02% of the run time. A far better scenario would be to design the burner to handle fuels A, B, and C. Then fuel B is the maximum pressure case, but the facility will not have enough pressure to make maximum capacity with fuel D. Therefore, 99.98% of the time the flames will be fine and the unit will operate properly; 0.02% of the time the unit will not be able to fire the full firing rate. In the opposing scenario, the operators will struggle with the unit 99.98% of the time, and the burner will run optimally only 0.02% of the time.
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105
2.1.1.3 Fuel Metering A fuel control valve upstream of the burner generally does the fuel metering. A riser or poker delivers fuel to the burner tip. The facility specifies the pressure the burner will receive at maximum (design) firing rate. The burner manufacturer then sizes the fuel orifices in the tips to ensure that burner will meet the maximum fuel capacity at the specified conditions. The burner manufacturer provides a series of capacity curves (one for each fuel) that show the firing rate vs. the pressure. Figure 2.2 gives an example. 3.5 l A uel B uel C F F
Fue
Heat Release, MW
3.0
2.5
sonic flow 2.0 transition to sonic flow
1.5
subsonic flow
1.0
0.5
0.0
0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
1
1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2
Fuel Pressure, bar(g) FIGURE 2.2 A typical capacity curve. Fuel capacity curves give heat release as a function of fuel pressure. They are quite accurate for a given fuel composition. However, if there are a range of fuels, each needs its own capacity curve. This example shows three. The typical range is two to five fuel scenarios, all represented on the same graph. As the flow transitions to sonic, the capacity curve becomes linear.
In natural draft burners, air is generally the limiting factor, and its controlling resistance is the burner throat. Thus, the only way to increase the overall firing capacity of the burner is to increase the throat (i.e., burner) size. This may or may not be possible depending on the available space in the heater and if the heater can handle the extra flue gas and heat that result. If not, the entire unit may require modifications, not just the burner. 2.1.1.4 Turndown Burners operate best at their maximum capacity. One measure of the flame stability of a burner design is the turndown ratio. The turndown ratio is the
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Modeling of Combustion Systems: A Practical Approach
ratio of the full firing capacity to the actual firing capacity. The maximum turndown ratio is the max/min firing ratio. The turndown ratio is higher if one modulates the air in proportion to the fuel. If the air dampers are manually controlled, then one is interested in the maximum unmodulated turndown ratio, because this is the more conservative case. A typical maximum turndown ratio for premix burners is 3:1. Diffusion burners often have turndown ratios of 5:1 without damper modulation, i.e., leaving the damper fully open despite lower fuel flow. One typically achieves 10:1 turndown ratios with automatic damper and fuel modulation. Multiple burner furnaces usually require a turndown ratio of somewhere between 3:1 and 5:1 with the air dampers fully open. To achieve greater turndown for the unit as a whole, one isolates some of the burners (fuel off, dampers closed). Turndown is easiest for a single fuel composition. Multiple fuel compositions always reduce the available pressure for some fuels and reduce the maximum turndown ratio. 2.1.1.5 The Air System For natural draft burners, air is the limiting reactant. That is, sufficient fuel pressure is available to allow the burner to run at virtually any capacity, but only so much air will flow through the burner throat at the maximum draft. The burner throat refers to the minimum airflow area; it represents the controlling resistance to airflow. Therefore, the airside capacity of the burner determines the burner’s overall size. Air may enter from the side (as shown in Figure 2.1) or in line with the burner. Analogous to the fuel orifice, there are two metering devices for the airside. The first is the damper, upstream of the plenum. A damper assembly is a variable-area device used to meter the air to the burner. This is necessary because the maximum airside pressure drop is limited and the firing rate modulates. Therefore, one must modulate the air to maintain the air/fuel ratio. The damper may require manual adjustment, or one may automate it by means of an actuator. The inlet damper unavoidably creates turbulence and pressure fluctuations behind it. The plenum is the chamber that redistributes the combustion airflow before allowing it to enter the burner throat. This redistribution does not need to be perfect, and there is a trade-off between uniform flow (larger plenum) and burner cost. In some cases, one may shorten the plenum by means of a turning vane — a curved device designed to redirect the airflow (not present in the figure). Burners are available in discrete standard sizes. To accommodate the infinite variety of potential capacities, manufacturers adjust the limiting airflow by means of a restriction in the burner throat. A choke ring is an annular blockage from the outside diameter inward. A baffle plate is a flat restriction originating from the center outward. A cone is an angled restriction originating from the center of the burner throat. Figure 2.1 shows a cone, but baffle plates or choke rings are also very common. The purpose of the throat
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107
restriction is to provide a point of known minimum area and pressure drop characteristic. Some codes actually specify what portion of the draft (airside pressure drop across the burner) that the damper and the burner throat must take. For example, one guideline says that the burner must use 90% of the available pressure and take 75% of the pressure drop across the throat.* The 90/75 rule, as it has come to be known, aims to create turbulence in the burner throat rather than at the damper to aid fuel–air mixing. In the author’s opinion, this is a misguided approach and the industry should abandon it for several reasons. First, if the burner performs well, the degree of turbulent mixing in the throat is immaterial. Second, if the burner performs well, it does not matter where the pressure drop occurs so long as the total pressure drop is correct. Third, this rule increases the time and cost of burner testing. More importantly, changes to the burner for the sake of meeting the 90/75 rule may actually make the burner perform worse. Therefore, one is consuming resources to meet an essentially useless rule. Very often, the end user will have to clean or inspect tips while the furnace is running. In multiple-burner furnaces comprising many burners, there is usually little danger in shutting off a single burner. In petroleum refineries, the usual practice is to specify a burner having replaceable tips that do not require burner removal from the heater (of course, one must still shut off fuel flow to the individual burner before removing the tips). One must also take care to close the air damper during this procedure; otherwise, the furnace will admit tramp air. Tramp air is air admitted out of place. Air entering the furnace should participate fully in the combustion process, and tramp air enters the combustion process too late to oxidize the fuel properly. Tramp air may come not only through unfired burners, but also through leaks in the furnace. One possible sign of tramp air is a high CO reading even with supposedly sufficient excess oxygen. CO is a product of incomplete combustion. Depending on the furnace temperature, 1 to 3% oxygen should represent enough excess air, and CO should be quite low under such conditions. But with tramp air, significant CO (>200 ppm) may still occur — even with 3% oxygen (or more) in the flue gas. The oxygen has entered the furnace somewhere, but it is not participating fully in the combustion reaction. As long as there is tramp air, the furnace will require higher oxygen levels — enough to provide both effectual air through the burner and ineffectual tramp air. In severe cases of tramp air leakage, even a wide-open damper cannot provide enough air to the burner and CO persists, though stack oxygen levels are 5% or more. The exit of the furnace radiant section is the relevant place to measure emissions for combustion purposes. The exit of the stack is the relevant place to measure emissions for compliance purposes. The stack exit is not generally useful for understanding what is happening in the combustion zone. * This requirement appears as a footnote to API Standard 560, Fired Heaters for General Refinery Service, 3rd ed., Washington, DC, American Petroleum Institute, May 2001, p. 65.
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2.1.1.6 The Flame Holder Anchoring the flame to the burner is essential for the sake of performance and safety. A flame holder is a device designed to keep the leading edge (root) of the flame stationary in space. There are many different devices for accomplishing this. The most common device is the burner tile. A burner tile is a refractory flame holder designed to withstand the temperature of direct flame impingement. The tile ledge is the portion of the tile that anchors the flame (Figure 2.3).
FIGURE 2.3 The tile ledge as flame holder. (From Baukal, C.E., Jr., Ed., The John Zink Combustion Handbook, CRC Press, Boca Raton, FL, 2001.)
One type of flame holder is the bluff body — a nonstreamlined shape in the flow path — to present an obstruction to some of the flowing fuel–air mixture; the tile ledge qualifies. This obstruction generates a low-pressure, low-velocity zone at its trailing edge. The flame holder affects only a small portion of the flow, reducing its velocity to well below the flame speed. The velocity upstream of the flame holder is very low. Thus, the hot combustion products recirculate there, continually mixing fresh combustion products with an ignition source — the hot product gases. In this way, the flame holder anchors and stabilizes the flame over a wide turndown ratio. Figure 2.4 shows a 2-MW round-flame burner employing a tile-stabilized flame. The shape of the burner and position of the fuel ports mold the flame into the desired shape. 2.1.1.7 Stabilizing and Shaping the Flame A flame has a very fast but finite reaction rate. One measure of the reaction rate is the flame speed. The laminar flame speed is the flame propagation rate [L/θ] in a combustible mixture of quiescent fuel and air. If the air and fuel mixture exceed the flame speed, then the flame will travel in the direction of the stream — a phenomenon known as liftoff. If the liftoff continues, the
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109
FIGURE 2.4 A gas burner in operation.
flame will be transported to a region of high flue gas concentration that cannot support combustion and the flame will extinguish. Air and fuel velocities in a typical industrial burner far exceed laminar flame speeds. But in the vicinity of the bluff body, the fluid speed is low; therefore, the flame anchors over wide turndown range and the burner is quite stable throughout its entire operation. 2.1.1.8 Controlling Emissions In the past, the only emission of concern was CO because it indicated incomplete combustion, combustion inefficiency, or a safety hazard. Nowadays, life is more complex and other emissions such as nitric oxides are important due to their role in the formation of ground-level ozone and photochemical smog. Technically, noise is also a regulated emission (e.g., <85 dBA). Emissions control is an active subject of interest. Staging the combustion into distinct zones is one strategy (termed staging) to lower certain emissions such as NOx. If NOx emissions are not a concern, then the secondary fuel tips of Figure 2.1 may be unnecessary and the burner carries out combustion using only primary fuel tips. 2.1.2 Archetypical Burners When taxonomists classify things, they speak of slots and filler. Slots are the classes or categories, and filler is the stuff that populates the class. With respect to major considerations that affect burner design, we shall list five:
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• Fuel state: – Gas – Liquid – Solid • Flame shape: – Round – Flat • Fuel–air mixing strategy: – Fuel and air premixed (premix burner) – Separately metered fuel and air (diffusion burner) • Firing orientation – Upfired (the burner fires from the floor upward) – Downfired (the burner fires from the roof downward) – Side-fired (the burner fires from the wall sideway) • Emissions – The burner design reduces combustion-related emissions. – The burner design has no special features for reducing combustion-related emissions. These five characteristics generally fix the burner design, and they will define an archetypical burner. Neglecting solid-fired burners for now (e.g., wood, municipal solid waste, pulverized coal, etc.), each of the above categories has two possibilities, except for firing orientation, which has three. This leads to 24·3 = 48 different slots. However, as is typical, not every slot has a filler, and some burner models fill more than one slot. For our purposes, about a dozen burner types are of importance. We should add that there are many kinds of esoteric designs for special reactions, but as regards traditional fuel–air combustion, these categories will do. Table 2.1 shows the slots and how one burner manufacturer has chosen to fill them. It is, of course, possible to entertain other considerations. For example, • Type of draft: Is the motive force for air due to natural convection in the heater (natural draft), from a fan outlet upstream of the burner (forced draft), from a fan inlet downstream of the burner (induced draft), or both inlet and outlet fans (balanced draft)? • More fuel-state variations. Will the burner fire liquid and gaseous fuels at the same time or separately? • Service: Will the heater serve in a boiler to generate steam, or will it serve in a process heater to refine petroleum or make petrochemicals? However, for the most part, design variants of the enumerated burner types will accommodate all of the above categories. So, we will describe the
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TABLE 2.1 Burner Sampling for One Manufacturera Mounting Orientation Up
Fuel Type Gas Oil
Down
Gas Oil
Horizontal
Gas Oil
Flame Shape Round Flat Round Flat Round Flat Round Flat Round Flat Round
Diffusion Conventional Low NOx
Premix Conventional Low NOx
SFG SFFG LNC + EA
SMR HEVD XMR, PSFFR DeepStar + HERO
LMX
MDBP/PFLD
DSMR
HEVD
DSMR MKIII
HEVD PMS
LMX LPMW
SFG SMR PSFFG HMVF FPMR LNC + EA DeepStar + HERO
Flat a
All referenced burner models are trademarks of John Zink LLC, Tulsa, OK.
burner types first and then make general remarks about other features where appropriate. 2.1.2.1 Round-Flame Gas Diffusion Burners Burners in these categories are gas fired and designed to produce round flames. These comprise the lion’s share of fired duty. These also comprise the largest single burner capacities — a typical refinery size fires about 2.5 MW (~8 MMBtuh). However, they can be as large as 8 MW (~35 MMBtuh), though this is uncommon — the traditional approach in the refinery has been to use more but smaller burners rather than a few large burners. In the power generation industry, opposite sensibilities prevail. Package boilers represent the middle ground. A 3-MW floor-fired burner is about as big around as a man can circle his arms, and roughly his height. Each weighs about 500 kg. All kinds of process heaters and boilers use them. Electrical utilities and cement kilns use the larger sizes (>10-MW heat release per burner). 2.1.2.2 Round-Flame Gas Premix Burners Figure 2.5 shows an example of a round-flame floor-fired premixed burner. Some furnaces use premix burners in larger upfired applications requiring round flames. However, this has fallen into disfavor because the burners are usually loud (due to fuel jet noise) and sensitive to hydrogen concentration variations. In premix burners, air and fuel mixing occur prior to entering the furnace. Premixing has several advantages. First, premixed burner flames tend to be short, crisp, and well defined. The fuel jet provides momentum for fuel mixing and air entrainment prior to the burner exit. An advantage of this
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Secondary air
Primary air
Pilot Gas FIGURE 2.5 A gas premix floor burner. (Courtesy of the American Petroleum Institute, Washington, DC.)
arrangement is that increased fuel flow results in increased airflow. However, premixed burners also have major disadvantages in the wrong application. First, the momentum of the fuel stream depends on the molecular weight of the gas and the fuel pressure. As the H/C ratio of the fuel changes, so does the molecular weight, the fuel pressure for a given heat release, and the amount of educted air. Higher pressures tend to educt less than the proportional air, while lower fuel pressures educt higher than ideal airflows. Thus, the air/fuel ratio is not as constant as one might hope. Another serious drawback of premixed combustion is the potential for flashback. Flashback is the upstream propagation of flame into the premix chamber of the burner. In a diffusion burner, it is impossible for a flame to propagate back into the fuel riser because there is no oxygen there to support combustion. However, in a premix burner, there is a combustible mixture inside the burner tip. Premix burners also have a much larger tip and orifices because these have to accommodate a high-volume, low-pressure fuel–air mixture. If the flame speed significantly exceeds the fuel–air exit velocity, then the flame may flash back into the burner tip.
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Burner internals cannot withstand the high temperatures of combustion for long. Flashback is very sensitive to hydrogen concentration because the laminar flame speed of hydrogen is about three times that of hydrocarbons. Various techniques moderate flashback. One important consideration is the quench distance — a characteristic length for a given orifice geometry through which a flame cannot propagate. For small-diameter orifices, the edges will abstract sufficient heat from a propagating flame to extinguish it. The quench distance varies with orifice diameter — smaller orifices are more effective than larger ones. Appendix A, Table A.4 gives critical diameters and minimum slot widths for various fuels. Generally, for a given area, a circular orifice is more efficient for quenching flames than a rectangular one. However, smaller rectangular slots may have manufacturing advantages. In practice, both geometries are commercially available. The removal of heat from the flame via thermal conduction through the tip is a mechanism for quenching the flame and eliminating flashback. Since the tip temperature depends in some measure on the temperatures of the furnace, fuel, and combustion air, the tip’s ability to conduct heat away from a propagating flame also changes with temperature. Moreover, at higher temperatures, the orifices in the tip expand and one must take account of this effect in burner design as well. 2.1.2.3 Flat-Flame Gas Diffusion Burners Figure 2.6 gives a typical example. These burners are usually floor fired against a flat wall that radiates heat to the process tubes (Figure 2.7). In other ways, these burners are similar to round gas diffusion burners. One uses these burners to maximize radiant heat transfer from the wall to the process. For example, high-temperature processes — such as production of hydrogen or ethylene — use the hot wall to radiate to process tubes containing feedstock. Since the reaction occurs along the length of the tube, the so-called heat flux profile can be important. Figure 2.8 gives an example. The burner manufacturer adjusts the heat flux profile according to furnace vendor specifications by changing the angle, size, and distribution of the fuel jets. NOx reduction with these burners is a challenge because the process operates at very high furnace temperatures (1000 to 1250°C). Another type of flat-flame diffusion burner is side-fired. The architecture of the burner resembles that of the side-fired premix burner, but the burner meters the air and fuel separately. Figure 2.9 gives one example, and Figure 2.10 shows some in operation. Ethylene reactors and wall-fired hydrogen reformers (described later) use these burners. The overall chemistry for hydrogen production is CHψ + 1/2 O2 → CO + ψ/2 H2
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FIGURE 2.6 A flat-flame gas diffusion burner. The burner creates a flat flame used to heat a wall that radiates heat to the process.
Hydrogen plants often use pressure-swing adsorption (PSA) to purify hydrogen and separate it from the CO2 by-product. The off-gas from this stream is mostly CO2, with some hydrogen, and to a lesser extent hydrocarbons. The high concentration of CO2 dramatically lowers the flame temperature. Flat-flame side-fired diffusion burners have excellent stability, cannot flash back, and NOx emissions below 20 ppm are possible even with 400°C air preheat and 1200°C furnace temperatures. As Figure 2.9 and Figure 2.10 show, the fuels travel down a central riser; the slotted tip projects the fuel in a radial plane parallel to the wall. Preheated air comes through the large annular gap and enters the furnace in the same orientation. At the high furnace temperatures, the separate fuel and air streams react to generate a flame with very low NOx and uniform radiation. 2.1.2.4 Flat-Flame Premix Burners Flat-flame premix burners comprise side-fired ethylene or steam–methane reforming service (hydrogen production) almost exclusively. Figure 2.11 shows a common design. The burner tip is roughly 4 in. in diameter and 8 in. long. Slots or holes cover the end and admit premixed fuel and air to the furnace. Premix burners are prone to flashback, though proper design will ameliorate this.
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FIGURE 2.7 Floor-fired flat-flame burners. The burners, John Zink Model PXMR, are shown firing against a wall in an ethylene cracking furnace. The wall radiates heat to the process tubes (not shown).
2.1.2.5 Flashback Flashback can only occur in premixed burners because they are the only type that has a combustible mixture inside the tip. Once the flame flashes back into the burner tip it can destroy it in minutes. Combustion inside the tip increases the mixture temperature downstream of the eductor. This backpressures the eductor and further reduces the air/fuel ratio. The richening of the fuel mixture and higher temperatures increase the flame speed. Therefore, once flashback occurs, there is no mechanism for moving the flame back to the furnace side. The burner immediately experiences lower mass flow due to the reduced density of the gas during flashback; one may hear a gurgling sound from the combustion rumble in the tip. 2.1.2.6 Use of Secondary Fuel and Air The tip receives its premixed fuel and air from a venturi (Figure 2.12). A fuel jet ahead of the venturi induces surrounding air via the fuel’s forward momentum. Under some conditions the fuel educts only a portion of the combustion air. In that case, secondary air slots allow additional air to bypass the venturi (shown in Figure 2.11). In some cases, the premix burner may have some nonpremixed fuel in addition to the fuel–air mixture, thus staging the fuel (Figure 2.11). Fuel not added to the immediate combustion zone — the primary zone — is termed secondary fuel or staged fuel.
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process tubes
116
% of max heat flux 90 80 70 60
50
40 30 20 10
< floor burners > FIGURE 2.8 A typical heat flux profile. Shown is a side view of one cell of a dual-cell floor-fired ethylene cracking unit (ECU) with its associated heat flux profile. The proper heat flux profile is a tradeoff between reduced fouling (flat heat flux profile) and maximized efficiency (heat release skewed toward furnace bottom). Burner vendors design floor-fired burners to provide the required flame shape for a given flux profile.
Secondary fuel injection is one technique for reducing NOx. Side-fired premixed burners are typically much smaller than floor-fired burners and weigh a mere 50 kg or so, including the tile. Firing rates for these burners are about 1 MMBtuh or 1/3 MW. 2.1.2.7 Round Combination Burners In some facilities, fuel oil can be a significant fuel stream. Normally, a refinery will want to burn as heavy a fuel as possible because other liquid fuels have greater value (e.g., transportation fuels for automobiles, trucks, and aviation). When sold commercially, fuel oils are widely available and graded as either number 2 or 6, with intermediates formed by blending. Fuel oil 2 is similar to automotive diesel. Fuel oil 6 is much heavier (also called residual fuel oil or, archaically, Bunker C oil). Marine and stationary boilers and some process heaters burn this fuel. Sometimes, the liquid fuel comprises rejected oil from other processes (waste oil) in whole or part. One can also burn pitch — a nondescript fuel from a variety of sources that is solid at room temperature. One must heat these fuels to reduce their viscosity in order for them to burn efficiently. Heavy liquid fuels do not atomize well even under pressure
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FIGURE 2.9 A flat-flame diffusion burner. Radial orifices admit fuel through the center pipe, while combustion air flows through the outer pipe. Unlike premix designs, this radiant wall burner cannot flash back. The design accommodates high forced draft air preheat applications.
FIGURE 2.10 Wall-fired diffusion burners in operation. The photo shows an ethylene cracking furnace equipped with John Zink Model FPMR burners. The process tubes (right) are receiving heat radiated from the burner firing along the wall. (Photo courtesy of John Zink LLC, Tulsa, OK.)
(so-called mechanical atomization), so fuel guns make use of pressurized steam to produce the requisite atomization. Mechanical atomization is sufficient for light oils such as fuel oil 2. Sometimes, light liquid fuels use compressed air for atomization. This is the case if steam atomization could be detrimental or there is insufficient fuel oil pressure for mechanical atomization. For
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FIGURE 2.11 A flat-flame premix burner. The flame heats the refractory, which in turn radiates heat to the process tubes inside the furnace. Secondary air allows for a higher capacity, as the eductor need not inspirate all of the combustion air. Also shown is a secondary fuel nozzle at the burner tip. Some burners do not have all these features.
INLET BELL
THROAT
EXPANSION SECTION
EDUCTED AIR FUEL NOZZLE
FUEL/AIR MIXTURE TO TIP
EDUCTED AIR
VENTURI FIGURE 2.12 Venturi section of a premix burner. The Venturi (more generically, an eductor) comprises an inlet bell, throat, and expansion section. The fuel jet induces a low-pressure zone along the jet surface. The surrounding atmospheric pressure pushes air into the low-pressure zone. The fuel and air mix and exit the venturi toward the tip outlet.
example, light naphtha fractions can prevaporize in the fuel oil gun. Prevaporization is unwanted because it leads to slug flow in the fuel gun, that is, alternate slugs of liquid and gaseous fuel going to the burner. This causes erratic flow and performance.
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Practitioners use the term oil gun to refer to the liquid fuel delivery and atomization assembly. Figure 2.13 shows a typical design.
FIGURE 2.13 An oil gun. Steam vaporizes oil droplets allowing for uniform combustion.
Steam shears heated oil into fine droplets — the fuel oil vapor is the phase that actually burns. The burning vapor provides heat, vaporizing even more droplets and recharging the combustion zone. Very often, one burns fuel oil and gas together (in so-called combination burners). This is sometimes to add fuel flexibility — perhaps the gas and the liquid fuel are available in different seasons. However, the more typical practice is to use the less expensive heavy oil with the gas fuel serving to make up the required process heat. Thus, both fuels fire simultaneously. A combination burner is a gas-fired burner augmented with a fuel oil gun. Figure 2.14 shows one common arrangement. The typical combustion scenario is a single fuel oil gun in the center of the burner with gas firing at the periphery. When both fuels fire at once, flame lengths tend to be longer than when either fires alone. This is due to the peripheral combusting gas reducing the available oxygen for the fuel oil stream. To minimize (but not eliminate) this effect, separate air registers provide individualized airflow to each zone. 2.1.2.8 Burner Orientations One may fire burners in four basic orientations: up, down, sideways, and balcony fired (horizontally mounted to the wall but having a vertical flame at right angles to the burner). 2.1.2.9 Upfired This is the most typical firing arrangement (see Figures 2.1, 2.4, and 2.7). The burner mounts to the floor. Pillars support the heater and allow sufficient clearance for personnel to walk under the burners and inspect and maintain them. A 1.8-m gap from the lowest part of the burner to the ground is adequate for personnel to walk under the heater without stooping. However, some units do not have this kind of clearance. Clearance is important for initial installation and because one requires sufficient room to extract, clean, and reinsert fuel oil guns, risers, and tips.
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REGEN TILE GAS RISERS (FOR COMBINATION FIRING)
OIL GUN TERTIARY AIR CONTROL
PRIMARY TILE
GAS PILOT
AIR INLET PLENUM PRIMARY AIR CONTROL
SECONDARY AIR CONTROL
GAS RISER MANIFOLD (FOR COMBINATION FIRING) FIGURE 2.14 A combination burner. John Zink Model PLNC combination gas–oil burner. One may fire the burner on gas only, oil only, or both. (Rendering courtesy of John Zink LLC, Tulsa, OK.)
2.1.2.10 Downfired Most burner models can be adapted to fire downward with some kind of tile case or support to hold the tile in place at the roof. Figure 2.15 shows a generic schematic. Hydrogen and ammonia reformers of this type are large furnaces often with more than 200 burners, each firing at ~2 MW. With so many burners in a furnace, first cost is important. Downfired operation affects the flame shape because the firing direction is opposite the buoyant force. Hence, forced draft is the preferred option for these burners. Forced draft operation helps to minimize the flame bending toward the tubes. The greater momentum of the forced air helps to overcome the buoyant effects and furnace currents. Furnace currents can be quite complicated. Downfiring increases NOx emissions by 15% or so as the burner receives hotter convective air at the roof than at the floor. Space is limited and roof burners are more difficult to access than floor burners. There are also limitations on the total burner weight because the furnace roof can support only so much. Therefore, a simple, reliable design is the order of the day.
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FIGURE 2.15 A downfired burner for hydrogen reforming. This burner is equipped with a center gun for waste gas from a pressure-swing adsorption (PSA) unit.
2.1.2.11 Side-Fired Some steam–methane reformers and ethylene cracking units (ECUs) use side-firing. The general idea is to present a uniform heat flux to the reactor tubes by using many small burners (<1 MMBtuh) in the furnace wall. A large ECU may have hundreds of sidewall burners. Figure 2.16 shows some in operation. The flame travels parallel to the plane of the wall. This path is necessary because the reactor tubes are less than 2 m away; flame impingement would overheat them. Reactor temperatures for these kinds of units are some of the highest temperatures found in a petrochemical plant, 1200 to 1250°C. 2.1.2.12 Balcony Fired Figure 2.17 shows a typical balcony burner, also known by a lengthier and more descriptive moniker — horizontally mounted, vertically fired (HMVF). These burners penetrate the side of a furnace; however, the firing direction is up. A 90° bend in the air passage accomplishes this. 2.1.2.13 Combination Side and Floor Firing Some heater vendors fire ECU and related units with both floor and wall firing. The advantage of wall firing is a superior ability to tailor the heat flux profile. The heat flux profile is the radiant heat distribution along the vertical reactor dimension. A heat flux that decreases with elevation can improve unit efficiency. However, a perfectly even heat flux distribution maximizes the tube life and the conversion of the feed can be easier to model. Sidewall burners
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FIGURE 2.16 Sidewall burners in operation. The radial combustion pattern of the burners at left radiates along the wall, which in turn heats the process tubes (far right). The two rectangular spots on the end wall are sight ports.
FIGURE 2.17 Balcony burner. This burner is horizontally mounted but vertically fired.
ensure an even heat flux profile. However, they represent a higher first cost (due to the many burners), and operators must adjust the air doors row by row, making process changes and start-up more labor-intensive. One hundred percent floor firing does not have these drawbacks. The burner vendor can adjust the heat flux profile in the bottom two thirds of
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the furnace by changing tip drillings, in essence, changing the heat release distribution. However, heat flux in the top third of the furnace is difficult to influence with tip geometry at the floor. A heater comprising both floor and wall firing is one compromise to reduce first cost and operating labor while retaining good control of the reactor heat distribution. In practice, one sees all three firing scenarios.
2.2
Archetypical Process Units
Boilers, process heaters, and reactors comprise three main categories of process units. One may further differentiate among them as follows.
2.2.1
Boilers
A boiler is a device for generating steam. There are two main configurations for fired units: firetube and watertube. 2.2.1.1 Firetube Boilers In firetube boilers, the flue gas flows through the tubes and out the stack, transferring heat to surrounding water. Nowadays, firetube boilers are generally smaller units generating saturated steam. They are usually fully automatic and unattended. These provide facility steam and heat for schools, hospitals, and other commercial needs. The household water heater is a firetube configuration. However, because it does not generate steam, it is not a boiler. 2.2.1.2 Watertube Boilers Watertube boilers are considerably larger and provide process steam for refineries, pulp mills, electrical generation, etc. As the name implies, watertube boilers generate steam inside the tube. In this respect, they are similar to process heaters but with water in the tube rather than process fluid. Largecapacity, high-pressure, and superheated steam units are invariably of the watertube design because small tubes can withstand higher internal pressure than large shells for a given thickness. Watertube boilers can grow to be quite large. Coal-fired utility boilers are the largest watertube boilers. Those in the petroleum refinery and industrial plants are many times smaller. 2.2.1.3 Fired Heaters and Reactors A fired process heater is a combustion unit for heating any process fluid other than water. In a refinery, they comprise hot-oil heaters, crude heaters, vacuum heaters, and the like. A fired reactor is a process combustion unit
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designed to effect some thermochemical transformation. One major distinction among fired units is their shape. Figure 2.18a to 2.18i gives a sampling of the many process heater and fired reactor configurations. We further discuss some of them below. 2.2.1.4 Vertical Cylindrical Vertical-cylindrical (VC) units have tall right-circular shells (see Figure 2.18a and b). The helical coil is the rarer of the two. The heater designer may arrange the tubes helically or vertically, and may accommodate several separate passes through the heater, depending on the process needs. 2.2.1.5 Cabin Style A cabin (or box) unit has a rectangular profile and is usually shorter than a VC, though it can be many times wider (Figure 2.18c to i). Tubes in a cabin heater typically run horizontally at the walls (e.g., Figure 2.18d and e); they usually fire with one row of burners down the center of the heater floor (not shown). However, there may be two rows of burners, especially if there are center tubes (Figure 2.18d). Some process heaters are also end fired (Figure 2.18e).
Convection Section
Side View
Radiant Section
Process Tubes
Burners
Burners
Radiant Section
Radiant Section
Burners
Burners
Radiant Section
Burners
Process Tubes
Plan View (a)
(b)
(c)
(d)
(e)
FIGURE 2.18 Some process heater types. Process and convection tubes are shaded. Heaters may or may not have convection sections. Round heaters are known as vertical cylindrical (VC), and rectangular heaters are known as cabin heaters. (a) VC, (b) VC with helical coil. Cabin type: (c) wicket tube or arbor coil, (d) floor fired, (e) end wall fired.
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Radiant Section
125
Radiant Section
Radiant Section
Radiant Section
Burners
Burners
Burners
Plan View (f)
(g)
Convection Section
Burners
Burners
Side View
Radiant Section
Radiant Section
Burners
Burners
Radiant Section
Radiant Section
Burners
Plan View (h)
(i)
(j)
FIGURE 2.18 (continued) (f) Vertical tube, sidewall fired; (g) vertical tube, floor fired; (h) floor + wall fired — usually two or three rows of sidewall burners + floor burners; (i) downfired (one of many cells), (j) terrace wall (horizontally mounted, vertically fired).
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2.2.1.6 Fired Reactors In the petroleum and petrochemicals industry, most chemical reactors use fired duty to effect their transformations. We shall discuss ethylene cracking units and hydrogen and ammonia reformers as illustrative. 2.2.1.7 Hydrogen Reformers Hydrogen reformers exist in side-fired (Figure 2.18f), downfired (Figure 2.18i), or terrace wall-fired (Figure 2.18j) configurations. Hydrogen is an important commodity in fuels upgrading. The general chemistry is CH4 + H2O → 3H2 + CO
(2.2)
That is the reforming reaction. The reaction temperature inside the process tubes is ~815°C for this step.1 Therefore, the bridgewall temperature of the reactor must be higher (~1050°C). The following water–gas shift reaction (or simply, shift reaction) increases the H2 yield: CO + H2O → CO2 + H2 The shift reaction can occur at low temperatures (200 to 230°C) or high temperatures (300 to 450°C), depending on the catalyst in the tube and the desired conversion efficiency. Thermodynamically, the low-temperature shift reaction is more efficient. However, nowadays, pressure-swing adsorption (PSA) is the separation process of choice, and its natural companion is the high-temperature shift reaction because one can burn any unconverted CO as fuel. As the name implies, PSA is a pressure cycle. Solid adsorbents adsorb impurities at high pressure and release them at low pressure. Using two adsorbent vessels, the process may produce product on a continuous basis. The depressurization cycle gives CO, CO2, N2, and some H2 as a fuel stream at low pressure. For this purpose, hydrogen reformer burners are equipped with a PSA tip. For example, Figure 2.15 shows a burner equipped with a large center tip to use the PSA tail gas for process heat. PSA tail gas tends to form very little NOx (owing to the inert content of the fuel) and good flames (owing to fast flame speeds of H2 and CO). 2.2.1.8 Ammonia Reformers Most ammonia reformers resemble downfired hydrogen reformers (Figure 2.18i) in shape and size. The chemical reaction is reversible: N2 + 3H2 ↔ 2NH3
(2.3)
Since the reactants comprise four moles and the products two, pressure (which always favors the denser phase) increases the yield of NH3. Fertilizer comprises the largest market for ammonia.2
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2.2.1.9 Ethylene Cracking Units (ECUs) Figure 2.18g gives an example of a floor-fired ethylene cracking unit (ECU); Figure 2.18h shows an ECU that is fired by both floor and wall burners. In such a configuration, the floor burners have 70 to 80% of the heat release, with the wall burners comprising the balance. Figure 2.18f gives an example of a wall-fired steam–methane reformer, and Figure 2.18i shows one cell of a downfired hydrogen reformer. Such reformers comprise several cells and hundreds of burners. Modern ECUs contain tubes 10 to 12 m in length. Each tube hangs from the roof with a single U-bend at the bottom. Burners on both sides of the process tube heat it. The heat converts the feed hydrocarbon into ethylene, C2H4. Ethylene is the largest volume organic chemical, and almost all ethylene production is by the thermal process.2 High heat causes abstraction of hydrogen and formation of a double bond. For example, CH3—CH3 → CH2=CH2 + H2
(2.4)
Ethane, propane, and naphtha are the most common hydrocarbon feedstocks. These reactors have the highest bridgewall temperatures of any common petrochemical process, 1200 to 1250°C.
2.3
Important Factors and Responses
The combustion-related responses that we wish to model are things like NOx and CO emissions, CO breakthrough point, flame length, heat flux profile, and the like. These will be a function of one or more of the following factors: fuel composition, oxygen concentration, burner type, degree of staging, furnace temperature at some convenient point such as the bridgewall or floor, etc. Tests for these kinds of things are now fairly standard. 2.3.1
The Traditional Test Protocol
The American Petroleum Institute (API) formally defines various burner tests.3 We numerically index the important ones below. To this, we add point 0, as it does not specifically appear in the API publication, but is normally tested. 0. Cold light-off. This refers to ignition of the burner in a cold furnace. One should use the same method for ignition that is available in the field. This may comprise manual ignition by a torch, ignition by a burner pilot, or manual spark ignition. The idea is to simulate the field start-up condition, where the burners start up for the first time. One looks for attached and stable flames, and a smooth transition as the burner ramps up. The damper position at this condition is termed the light-off position. It is usually of interest and therefore recorded.
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1. Normal. This point refers to burner operation at the specified normal firing rate and design excess air. The damper position at the normal firing rate is termed the normal position and is of interest. 2. Minimum. This point refers to the maximum turndown of the burner with its damper in the normal position. 3. Maximum. This point refers to the increased firing rate that causes the excess oxygen to reach its minimum acceptable limit (often 1%) with the damper in the normal position. 4. CO breakthrough. This point refers to the firing rate greater than maximum, where CO just becomes greater than 200 ppm — the CO breakthrough point or oxygen at CO breakthrough refers to the percent oxygen at this point. 5. Absolute minimum. This point is the maximum achievable turndown with adjustment of the air damper to give the design excess air. The API guidelines confirm suitability for purpose and are the usual and sufficient tests to predict acceptable burner operation in the field. However, these points will not provide enough information for detailed analysis, testing, and characterization of burners. Later chapters provide other experimental designs for these purposes. 2.3.2
Instability, Thermoacoustic and Otherwise
API 535 characterizes unacceptable burner operation as burner instability at any of the test points; it further characterizes instability as3 a. Pulsation or vibration of burner flame, burner, or furnace. b. Uncontrollable fluctuations in the flame shape. c. Significant combustibles in the flue gas, in other words, over 2000 ppmvd [parts per million on a volume dry basis] CO or over 0.5 percent unburned combustibles. d. Flashback into the venturi section of premix burners. e. Loss of flame. While the API document has burner instabilities in mind, not all instabilities are due to burner design proper. Specifically, thermoacoustic coupling between the burner and furnace can sometimes be the problem. The latter case may not indicate any problem with the burner per se, but may be due to a coupling between the test furnace and the burner — a thermoacoustic problem. Thermoacoustic coupling is any resonant phenomenon propagated by a thermal source. The issue is a complex one and a specialty discipline in its own right. The predominant modes of burner–furnace interaction are quarterwave and Helmholtz resonation. Generally, if heat adds to an acoustic wave
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in phase with a pressure rise, it will reinforce the resonance. One can also reinforce resonance by removing heat in phase coincident with a falling pressure wave.
2.3.3
Quarter-Wave Behavior
pressure oscillation
If the stack is large and the damper is wide open, the furnace may behave acoustically as a tube closed at one end (Figure 2.19).
LS
mass-spring analogy
L LF
pressure
pressure
oscillation
oscillation
(a)
(b)
(c)
FIGURE 2.19 Resonant modes for furnaces. (a) A furnace and stack in quarter-wave mode. The pressure oscillates between minimum and maximum at the heater floor. In half-wave mode (b) the pressure has maximum oscillations at the top and bottom end of the heater. In Helmholtz resonant mode (c) the furnace acts as a spring and the stack as a piston. In principle, the energy from the combustion reaction can excite any of these modes. However, the heat release pattern in a furnace tends to excite modes with (a) or (c), most commonly.
Since the damper is open, the pressure at the stack exit must be atmospheric. Acoustically, the stack behaves as if it were a bit longer than it actually is, so the node of the pressure wave actually extends a bit beyond the boundary of the stack by ΔL. However, such a correction for end effects is usually small, generally, π ΔL 4 < < 8 DS 3π where DS is the stack diameter.4
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The bottom end of the heater (where the burners reside — not shown) corresponds to the closed end of a tube. This would be the case when the burner throat represents a very small fraction of the total cross-sectional area of the heater, as is typical in furnace design. At the closed end of the tube, the velocity must be zero. Since velocity and pressure are 90° out of phase, the absolute value of the pressure wave will be at a maximum (the pressure antinode). That is, the pressure wave will alternate between maximum and minimum at the heater floor. If such an acoustic phenomenon were to occur, the length of the resulting wave would be one fourth of the heater + stack height. We can convert this to a frequency with the formula c = λν
(2.5)
where c is the speed of sound [L/θ], λ is the wavelength [L], and ν is the frequency [1/θ]. We may also calculate the speed of sound from
c=
γRT W
(2.6)
where γ is the heat capacity ratio, R is the gas constant [L2/θ2T], T is the temperature [T], and W is the molecular weight (~28 for flue gas). For flue gas, γ ≈ 7/5; for fuels, γ ≈ 4/3.* Substituting Equation 2.6 into Equation 2.5 and solving for ν (Greek letter nu) gives
ν=
1 λ
γRT c = W λ
(2.7)
If the process unit length represents one quarter of the resonant wavelength, then λ = 4L. Substituting this into the above gives the expected frequency of the unit resonating in a quarter-wave mode:
ν=
Example 2.1
1 4L
γRT c Quarter-wave behavior = W 4L
(2.8)
Calculation of Resonant Frequency for a Quarter-Wave Resonator
Problem statement: Suppose an enclosed flare behaves as a quarterwave resonator. Calculate the resonant frequency if the average * The heat capacity ratio is approximately 5/3 for monatomic gases, 7/5 for diatomic gases like air, and 4/3 for simple polyatomic gases such as CH4, H2O, and CO2.
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flue gas temperature is 1450°F and the height of the unit is 60 ft tall. Solution: From Equation 2.6 we have ⎡ psia ⋅ ft 3 ⎤ 1.38 ⋅ 10.73 ⎢ ⎥ 1450 + 460 ⎡⎣°R ⎤⎦ ⎡ in 2 ⎤ ⎡ ft ⎤ ⎡ lbm ⋅ ft ⎤ ⎣ lbmol ⋅°R ⎦ c= ⋅ 144 ⎢ 2 ⎥ = 2164 ⎢ ⎥ ⋅ 32.2 ⎢ 2 ⎥ ⎡ lbm ⎤ ⋅ ft lbf s ⎦ ⎣s⎦ ⎣ ⎣ ⎦ 28 ⎢ ⎥ lbmol ⎣ ⎦
(
)
This is about double the speed of sound at room temperature in air. From Equation 2.7 we have ⎡ ft ⎤ 2164 ⎢ ⎥ ⎣ s ⎦ = 9.0 ⎡ 1 = Hz ⎤ ν= ⎢s ⎥ 4 ⋅ 60[ft ] ⎣ ⎦ Thus, the resonant frequency will be 9 Hz. This is below the threshold of human hearing (20 to 20,000 Hz). Therefore, observers may feel such resonance but cannot hear it. It may correspond to the resonant frequency of large structures such as homes. In such a case, residents have been known to mistake the phenomena for an earthquake.*
2.3.4
Half-Wave Behavior
If the tube is closed (or open) at both ends, then the resonant frequency will correspond to a half wave (Figure 2.19b). This could be the case if the damper is nearly closed or if the convection section behaves as an acoustically opaque body. In such a case, the heater will resonate at double the previously calculated frequency. ν=
2.3.5
1 2L
γRT c Half-wave behavior = W 2L
(2.9)
Helmholtz Resonator Behavior
If the stack area is small compared with the cross-sectional area of the heater, then the heater may behave as a Helmholtz resonator. In this mode, the stack * Private conversation with Tim Hogue, supervising engineer at the Hyperion Wastewater Treatment Facility in Los Angeles, 1993.
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acts as a piston and the furnace volume as a spring. Equation 2.10 represents such a case:
ν=
1 ⎛ AS ⎞ ⎛ γRT ⎞ c AS = Helmholtz resonator ⎟ ⎜ ⎜ ⎟ 2 π ⎝ VF LS ⎠ ⎝ W ⎠ 2 π VF LS
(2.10)
where AS is the cross-sectional area of the stack, LS is the length of the stack, and VS is the volume of the furnace. For a cylindrical furnace and stack we may write
ν=
⎛ DS ⎞ ⎜ ⎟ 2 π LF Ls ⎝ DF ⎠ 1
⎛ DS ⎞ = ⎜ ⎟ 2 π LF Ls ⎝ DF ⎠
γRT W Helmholtz resonator, VC heater
(2.11)
c
The frequency of a heater in Helmholtz resonance will usually be lower than that for a half- or quarter-wave resonance.
Example 2.2
Calculation of Resonant Frequency for a Helmholtz Resonator
Problem statement: Repeat Example 2.1 for a VC heater behaving as a Helmholtz resonator with a furnace height of 40 ft and a stack height of 20 ft. Presume that the stack diameter is 20% of the furnace diameter. Solution: The speed of sound will be identical to our first example: 2165 ft/sec. Then Equation 2.11 gives ⎡ ft ⎤ 2164 ⎢ ⎥ ⎣s⎦ ν= 0.2 = 2.4 ⎡⎣ Hz ⎤⎦ 2 π 40 ⎡⎣ ft ⎤⎦ 20 ⎡⎣ ft ⎤⎦
( )
2.3.6
Mechanism for Thermoacoustic Coupling
Earlier, we stated that the heat must add in phase with the pressure wave in order for a burner to couple thermoacoustically with a furnace. Upon reflection, this seems counterintuitive. It seems that an increase in pressure near the burner would attenuate the flow of air, and thus the heat release
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would be out of phase with an acoustic pressure wave. (We can usually neglect any variation in fuel pressure because the fuel flow across an orifice is usually in critical flow — also termed sonic or choked flow. It is impossible for a pressure wave to propagate upstream of a critical flow nozzle.) To understand the mechanism for thermoacoustic coupling, we must examine the transfer of heat. The velocity wave must be out of phase with the pressure wave because when a high velocity encounters a wall, the velocity must drop to zero (velocity node) and the pressure must rise to maximum (pressure antinode). Therefore, the waves are 90° out of phase. Now, imagine we are at the position of the flame and pressure and velocity are fluctuating there. Let us focus on the velocity for a moment. The average velocity is the mean flow rate through the heater divided by the cross-sectional area. An acoustic wave superimposed on the mean flow will alternately enhance or attenuate the flow as the velocity alternates. Now in the case of enhanced flow, the flue gas is colder than the mean temperature because it comprises a greater portion of cool influent air. In the case of attenuated flow, the flue gas temperature is warmer than average because less cold air flows into the furnace during that period. Clearly, we transfer more heat when the thermal gradient is larger, that is, when the flowing fluid is cooler than average. In other words, more heat transfer results during the enhanced flow cycle than during the attenuated flow cycle. Now since pressure and velocity are not in phase, we can find a position in the heater for the flame where the increase in heat transfer will coincide with an increase in the pressure wave. For a furnace resonating in quarterwave mode, that position is L/4. That is, if the position of our heat source (flame) is at L/4, then we will enhance thermoacoustic coupling. Not only can we add heat at L/4, but also, if we remove heat at 3L/4, we will subtract heat during falling pressure, which amounts to the same thing, as Rijke showed.4 So in fact, having a flame at L/4 and a convection section at 3L/4 reinforces thermoacoustic coupling. The minor miracle seems not to be why some heaters experience instability, but why not all heaters experience instability. The answer seems to lie in the following facts: 1. Flames are not concentrated at L/4. 2. Surfaces and volumes within the heater can cause destructive interference of the fundamental resonant mode. 3. Resistances in the heater, such as convection tubes, etc., reduce the efficiency of thermoacoustic coupling.
2.3.7
Comments Regarding Thermoacoustic Resonance
Burner–furnace interactions are a cause for concern if the test furnace is acoustically similar to the field unit — even if the burner itself is not the
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problem. However, the test unit almost never bears acoustic similarity to the field unit. Therefore, thermoacoustic resonance in the test furnace does not usually signal a concern in the field, but burner instability does. How can we know the difference? To be sure, one must construct an acoustically similar system, either physically or mathematically. This is not trivial and requires expert assistance. However, if the following remedies are effective at eliminating or attenuating the instability, then thermoacoustic coupling is the likely culprit. 1. Reduce the stack damper opening and compensate with a steam sparger or induced draft (ID) fan, if available. By closing the damper, one may possibly convert the furnace from quarter-wave to halfwave mode and decouple the burner and the furnace. 2. Switch test furnaces. If the same burner is stable in a furnace with different dimensions, the culprit is likely thermoacoustic resonance. 3. Measure the frequency of the instability. Calculate the expected frequencies of the resonant phenomenon for quarter-wave and Helmholtz behavior. If the frequencies match, this is a strong indication that thermoacoustic behavior is an issue. 4. Is there an induction period? Does the problem build? Resonance is the constructive addition of multiple pressure waves. Theoretically, the pressure waves add forever, becoming infinitely strong. However, nonlinear behavior and attenuating mechanisms put a ceiling on the maximum amplitude. Notwithstanding, during the induction phase, pressure waves grow over time. Usually some seconds are required to reach the maximum. Although one cannot hear such frequencies by ear, the “huffing” of air in the 2- to 20-Hz range can be heard or felt. Theoretically, the velocity lags the pressure wave by 90°, but the frequency is the same and thus an appropriate indicator. 2.3.7.1 Resonance in the Field Sometimes, heaters resonate. Even if due to thermoacoustic resonance, it is easier to redesign the burner rather than the furnace, although the addition of quarter-wave tubes on the furnace roof can help. These reflect the pressure wave 180° out of phase with the resonant frequency. The destructive interference eliminates the resonance. The author has used this method successfully for a process heater,* and the literature reports successful application to boilers.5 Redesign of the burners could include changing the flame dimensions. Sometimes, shortening of the flame below the L/4 criterion can be effective; however, shorter flames may elevate NOx.
* Together with Mahmoud Fliefil, acoustic engineer, John Zink LLC, Tulsa, OK.
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Introduction to Combustion
2.4
135
Mass Balance for Combustion in Air
With this brief discussion of combustion equipment concluded, we turn our attention to the combustion reaction itself. Reaction 2.12 shows the combustion reaction for methane: CH4 + 2O2 → CO2 + 2H2O
(2.12)
The single arrow (→) indicates a reaction proceeding unilaterally from reactants to products at right. Reaction 2.12 encodes many important aspects of the combustion reaction. First, the stoichiometry (law of proportions) is apparent: one volume of methane (CH4) reacts with two volumes of oxygen (O2) to form one volume of carbon dioxide (CO2) and two volumes of water (H2O). Second, the combustion reaction alters molecular entities, but not atomic ones. That is, reactants and products each have the same number of C, H, and O atoms. However, the identity of the molecular entities has changed from CH4 and O2 to CO2 and H2O. Therefore, combustion reactions conserve mass: 1 kg of reactants will yield 1 kg of products without fail. We may augment Reaction 2.12 to account specifically for the nitrogen in the air. Nitrogen does not take part in combustion chemistry to any appreciable extent. Combustion in air does form some nitrogen oxides in part per million quantities, but they are a trivial part of the mass balance, although they form an important class of regulated compounds, which we discuss later. Reaction 2.13 augments the combustion reaction to account for nitrogen. The nitrogen dilutes both reactants and products. The ratio of nitrogen to oxygen in air is approximately 79/21 by volume. CH4 + 2O2 + 2(79/21) N2 → CO2 + 2H2O + 2(79/21) N2
(2.13)
Reaction 2.13 gives the stoichiometric amount of air required for combustion of one volume of CH4. However, the typical industrial practice is to use some quantity of extra air to ensure complete combustion. In principle, if one were to mix the air and fuel thoroughly during the combustion process, then no excess air would be required. However, adding a little excess air is the most cost-effective way to ensure complete combustion. Reaction 2.14 accounts for this excess air (ε): CH4 + 2(1 + ε)O2 + 2(79/21)(1 + ε) N2 → CO2 + 2H2O + 2(79/21)(1 + ε)N2 + 2ε O2
(2.14)
For the first time, oxygen appears in the products. The reader may verify that atomic entities are still equal for both products and reactants.
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One final step will transform Reaction 2.14 to a general equation for hydrocarbon combustion: we rewrite CH4 as CHψ, where ψ [ ] is a generalized subscript used to account for any desired H/C ratio (ψ): CHψ + (1 + ψ/4)(1 + ε) O2 + (79/21)(1 + ψ/4)(1 + ε) N2 → CO2 + (ψ/2) H2O + (79/21)(1 + ψ/4)(1 + ε) N2 + ε(1 + ψ/4) O2
(2.15)
Example 2.3 shows the equation’s utility.
Example 2.3
Refinery Gas Combustion: Wet and Dry Oxygen Calculations
Problem statement: Use Reaction 2.15 to account for the combustion of a refinery gas with ψ = 2.5 and 15% excess air. What are the dry and wet oxygen concentrations that one would measure in a furnace operating under these conditions? What is the moisture fraction in the flue gas? Solution: With ψ = 2.5 and ε = 0.15, Reaction 2.15 becomes CHψ=2.5 + (1 + 2.5/4)(1 + 0.15)O2 + (79/21)(1 + 2.5/4)(1 + 0.15) N2 → CO2 + (2.5/2) H2O + (79/21)(1 + 2.5/4)(1 + 0.15)N2 + 0.15(1 + 2.5/4)O2 We may conveniently organize the results in Table 2.2. We shall presume a basis of 1 kgmol CHψ, though any basis would do. From the table we may calculate that the flue gas comprises 0.24/ 9.52 = 2.56% oxygen on a wet volume basis, and 0.24/(9.52 – 1.25) = 2.95% on a dry volume basis. The moisture fraction in the flue gas is 1.25/9.52 = 13.1% on a wet volume basis. TABLE 2.2 Reactants and Products for Example 2.3 Species CHψ O2 N2 CO2 H2O
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Mol. Wt.
Input kgmol kg
Output kgmol kg
14.54 32.00 28.02 44.01 18.02
1.00 1.87 7.03 0.00 0.00
14.54 59.80 196.98 0.00 0.00
0.00 0.24 7.03 1.00 1.25
0.00 7.80 196.98 44.01 22.53
Sum
9.90
271.32
9.52
271.32
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In addition, the reader may verify the following atomic quantities for both product and reactants: C = 1.0, H = 2.5, O = 3.74, N = 14.06. Atomic species (and therefore mass) are conserved. However, moles are not conserved. We begin with 9.9 kg/mol and end with 9.52 kg/ mol. However, mass is conserved at 271.32 kg/kg/mol_CHψ. If not otherwise qualified, this text presumes excess oxygen refers to oxygen on a dry volume basis. One should not confuse the terms excess air and excess oxygen; they differ by a factor of five, roughly; that is, 3% excess oxygen and 15% excess air specify approximately the same air/fuel ratio.
2.4.1
Wet vs. Dry Measurements
Why do the wet and dry readings differ? Since the wet products include water, the wet volume is larger than the dry volume. On the same volume basis, water vapor reduces the other species’ concentrations. Although there will be no liquid water in the furnace, nonetheless, the oxygen concentration is still referred to as wet. So-called in situ oxygen analyzers measure oxygen on a wet basis with a probe in contact with the actual flue gas. Extractive analyzers first extract a sample of flue gas through a heated sample line. Usually, the sample splits to several analyzers, e.g., O2, CO, and NOx (NO + NO2). A heated sample line keeps water from condensing. This is important because NO2 is very water soluble; liquid water in the sample line will remove it before it reaches the analyzer, artificially lowering the NOx reading. For boilers and process units, NO2 is less than a few percent of the total NOx. However, for units that operate with high excess air, such as combustion turbines, about half the NOx may be NO2. Just before the analyzers, a sample conditioning section quickly extracts the water from the heated sample. It does so in a way that minimizes both the contact surface and contact time between the water and extracted gas. Therefore, an extractive oxygen analyzer will report higher oxygen concentration than an in situ analyzer, as it is measuring species on a dry volume basis. Extractive and in situ analyzer readings will always differ so long as there is hydrogen in the fuel; the difference will be greater for larger H/C ratios.
2.4.2
Flue Gas Relations for Hydrocarbons
One advantage of having a single entity (ψ) represent the H/C ratio for the fuel is that it allows the construction of flue gas species charts using ψ = H/C as a single parameter. Reaction 2.15 provides all the information we need to derive flue gas species as a function of excess air. From this, it is easy to develop the equations for the concentrations of each species for wet and dry conditions. The total wet products (TWP) and total dry products (TDP) are
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TWP =
100 ⎛ ψ⎞ ψ 1+ ⎟ 1+ ε + ⎜ 21 ⎝ 4⎠ 4
(2.16a)
TDP =
100 ⎛ ψ⎞ ψ 1+ ⎟ 1+ ε − ⎜ 21 ⎝ 4⎠ 4
(2.16b)
(
(
)
)
TWP is the volumetric flue-gas-to-fuel ratio. The volume or mole fractions for each species become ⎛ ψ⎞ 1 yO 2 ,dry = ε ⎜ 1 + ⎟ 4 ⎠ TDP ⎝
(2.17a)
⎛ ψ⎞ 1 yO 2 ,wet = ε ⎜ 1 + ⎟ 4 ⎠ TWP ⎝
(2.17b)
yN 2 ,dry =
1 79 ⎛ ψ⎞ 1+ ⎟ 1+ ε ⎜ 4⎠ TDP 21 ⎝
yN 2 ,wet =
1 79 ⎛ ψ⎞ 1+ ⎟ 1+ ε ⎜ 4⎠ TWP 21 ⎝
(
(
) )
(2.18a)
(2.18b)
yCO 2 ,dry =
1 TDP
(2.19a)
yCO 2 ,wet =
1 TWP
(2.19b)
1 ψ TWP 2
(2.20)
yH 2O =
Here, y refers to the mole fraction of the subscripted species. Usually, the excess air is unknown and we must calculate it from the oxygen concentration and the fuel stoichiometry. We can also calculate flue gas concentrations on a mass basis by multiplying by the molecular weight for each species. We will use w to indicate mass fractions (as opposed to mole or volume fractions) and W to indicate the molecular weight of various species. This leads to the following: TDPw =
⎛ ψ⎞ W WCO2 79 ⎛ ψ⎞ WN2 + ⎜1+ ⎟ 1+ ε + ε ⎜ 1 + ⎟ O2 4 ⎠ WCHψ WCHψ 21 ⎝ 4⎠ WCHψ ⎝
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(
)
(2.21a)
Introduction to Combustion
TWPw =
139
⎛ WN2 WCO2 ψ WH2O 79 ⎛ ψ⎞ W ψ⎞ + + ⎜1+ ⎟ 1+ ε + ε ⎜ 1 + ⎟ O2 WCHψ WCHψ 2 WCHψ 21 ⎝ 4 ⎠ WCHψ 4⎠ ⎝
(
)
(2.21b)
where TDPw and TWPw [ ] are the mass ratios of total dry and wet products to fuel, respectively. Then the mass fractions become wO2 ,dry = ε
WO2 WCHψ
⎛ ψ⎞ 1 ⎜⎝ 1 + 4 ⎟⎠ TDP w
(2.22a)
wO2 ,wet = ε
WO2 WCHψ
⎛ ψ⎞ 1 ⎜⎝ 1 + 4 ⎟⎠ TWP w
(2.22b)
wN2 ,dry =
1 WN2 79 ⎛ ψ⎞ 1+ ⎟ 1+ ε ⎜ 4⎠ TDPw WCHψ 21 ⎝
wN2 ,wet =
1 ψ⎞ WN2 79 ⎛ 1+ ⎟ 1+ ε 4⎠ TWPw WCHψ 21 ⎜⎝
(
(
) )
(2.23a)
(2.23b)
wCO 2 ,dry =
WCO2 1 WCHψ TDPw
(2.24a)
wCO 2 ,wet =
WCO 2 1 WCHψ TWPw
(2.24b)
WH2O 1 ψ WCHψ TWPw 2
(2.25)
wH 2O =
The following equations give the molecular weight of the flue gas, Wg, on dry and wet bases: Wg ,dry = yCO2, dryWCO2 + yN2,dryWN2 + yO2,dryWO2
(2.26a)
Wg ,wet = yCO2 ,wetWCO2 + yH2O ,wetWH2O + yN2 ,wetWN2 + yO2 ,wetWO2
(2.26b)
We may rearrange Equation 2.17 to solve for excess air. Conveniently, for hydrocarbon combustion the excess air is a function of two separable quantities: fuel composition and excess oxygen. We may collect the fuel composition effects into a single constant, K.6 These provide convenient relations for oxygen and excess air.
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140
Modeling of Combustion Systems: A Practical Approach ⎛ 0.79ψ + 4 ⎞ K dry = ⎜ , ⎝ ψ + 4 ⎟⎠
⎛ 1.21ψ + 4 ⎞ K wet = ⎜ ⎝ ψ + 4 ⎟⎠
(2.27)
⎛ ⎞ ⎛ ⎞ yO 2 ,dry yO 2 ,wet ε = K dry ⎜ = K wet ⎜ ⎟ ⎟ ⎝ 0.21 − yO 2 ,wet ⎠ ⎝ 0.21 − yO 2 ,dry ⎠
(2.28)
yO 2 ,dry =
yN 2 ,dry =
yCO 2 ,dry =
(
0.21ε K dry + ε
(
0.79 1 + ε K dry + ε
0.84 ψ + 4 K dry + ε
)(
yO 2 ,wet =
)
yN 2 ,wet =
0.21ε K wet + ε
yCO 2 ,wet =
)
(
0.79 1 + ε K wet + ε
(
(2.29)
)
0.84 ψ + 4 K wet + ε
)(
(2.30)
)
(2.31)
Figure 2.20 shows the equations graphically. Reaction 2.15 also gives the molar air/fuel ratio, α: α=
100 ⎛ ψ⎞ 1+ ⎟ 1+ ε 21 ⎜⎝ 4⎠
(
)
(2.32a)
Multiplying by the molecular air/fuel ratio (Wa /Wf ) gives the ratio on a mass basis. αW =
100 Wa ⎛ Ψ⎞ 1 + ⎟ (1 + ε ) ⎜ 21 W f ⎝ 4⎠
(2.32b)
Dividing the volumetric flue gas/fuel ratio (TWP) by α gives the molar flue gas/air ratio, γ. ⎛ 21 ⎞ ψ γ = 1+ ⎜ ⎟ ⎝ 100 ⎠ 4 + ψ 1 + ε
(
)(
)
(2.33a)
To find γW , we note that γ W = (1 + αW )/αW , giving γW = 1+
21 W f 100 Wa
4 ( 4 + Ψ )(1 + ε )
(2.33b)
If one adjusts the flue gas volume to some standard temperature and pressure, then γ is the volumetric flue gas/air ratio at standard temperature and pressure.
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141
20%
90% Dry N2
85%
16% H2O
14%
80%
12% CO2
10%
Wet N2
75%
8%
Vol % N2
Vol % species (except N2)
18%
70%
6% O2
4%
65%
2% 0%
Vol % dry species > Vol % wet species in all cases 0%
10%
20%
30%
40%
50%
ε
60%
70%
80%
90%
60% 100%
FIGURE 2.20 Flue gas relations. The figure gives relations for air in excess with combustion of a typical refinery gas comprising 25% H2, 50% CH4, and 25% C3H8.
2.4.3
Accounting for Moisture
One may also derive an equation for the moisture fraction of Table 2.2: yH 2O =
42 ψ ψ = 100 ψ + 4 1 + ε + 21ψ 2 ψ + 4 K wet + ε
(
)(
)
(
)(
)
(2.34)
The following formula relates the wet and dry measurements: ywet 100(ψ + 4)(1 + ε) − 21ψ K dry + ε = = = 1 − yH 2O ydry 100(ψ + 4)(1 + ε) + 21ψ K wet + ε
(2.35)
Here, y is the mole fraction of any species of interest and the subscripts distinguish the wet or dry bases. From Equation 2.35, one can convert from dry to wet concentrations or vice versa if one knows the hydrocarbon ratio and the excess air. If the H/C ratio in the fuel varies (e.g., municipal solid waste (MSW), waste gases, refinery fuel gases, landfill and digester gases, etc.), one may use ratios of the above equations to solve for ψ. For example, if we know moisture and CO2 in the flue gas, we can find ψ directly, or if we know any two of the three, the following relations will determine the third: ψ=2
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yH2O yCO2 ,wet
⎛ 2 ⎞ ⎛ yH2O ⎞ =⎜ ⎟⎜ ⎟ ⎝ yCO2 ,dry ⎠ ⎝ 1 − yH2O ⎠
(2.36)
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Modeling of Combustion Systems: A Practical Approach
yH2O = 2 ψyCO2 ,wet =
ψyCO 2 ,dry 2 + ψyCO2 ,dry
(2.37)
⎛ 2⎞ yCO2 ,wet = ⎜ ⎟ yH2O ⎝ ψ⎠
(2.38a)
⎞ ⎛ 2⎞⎛ y yCO2 ,dry = ⎜ ⎟ ⎜ H2O ⎟ ⎝ ψ ⎠ ⎝ 1 − yH2O ⎠
(2.38b)
Figure 2.21 shows these graphically.
20% 18%
Ψ=1
Ψ=3
Ψ=2
Ψ=4
CO2, dry basis
16% 14%
Ψ=5 12%
Ψ=6 10%
Ψ=7 Ψ=8 Ψ=9 Ψ=10
8% 6% 6%
8%
10%
12%
14% 16% 18% % moisture
20%
22%
24% 26%
FIGURE 2.21 Wet–dry flue gas relations. Knowing ψH 2O and ψCO 2 dry, one may determine ψ, the H/C ratio of the fuel, using this chart. CO2 is given on a dry basis because CO2 analyzers usually report results on this basis. The chart requires a moisture determination. One may do this with a physical moisture analysis or knowing oxygen on both a wet and dry basis. See text for cautions with this approach.
To determine the moisture in flue gas, one draws a known volume of gas and condenses the water. However, in principle from Equation 2.35, one can find the moisture concentration if one measures both wet and dry oxygen: yH2O = 1 −
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yO 2 ,wet yO 2 ,dry
(2.39)
Introduction to Combustion
Example 2.4
143
Determination of Moisture Content from Wet and Dry O 2 Measurements
Problem statement: The dry and wet oxygen concentrations of a flue gas are 2.95% and 2.56%, respectively. Determine the moisture content by Equation 2.39 above and compare it with the moisture content given in the table. Solution: From Equation 2.39, we know that yH2O = 1 −
2.56 − 1 = 13.1% 2.95
This agrees with the listed value in Example 2.3. However, despite this agreement, measurement errors can affect the calculation severely, as described presently. One should exercise caution in the application of Equation 2.39. The equation is perfectly accurate. However, oxygen measurements are subject to some error. The largest error comes when the analyzers sample wet and dry oxygen from different locations (typical). Very often, one measures wet oxygen in the furnace and dry oxygen in the stack (Figure 2.22). For negative and balanced draft units, air in-leakage can bias the results wildly. For example, even though we listed the dry oxygen as 2.95%, air inleakage could cause the measurement in the stack to be 5.0%. If we were to repeat Example 2.4 with this dry O2, the result would be 1 – 2.56/5.00 – 1 = 48.8% moisture, an absurd result. Even if we measure the wet and dry oxygen at the same location, small errors can still make a significant difference. For example, suppose the dry O2 were biased upward by a 0.1% reading, and the wet O2 were biased downward by the same amount; then 1 – 2.46/3.05 – 1 = 19.3% moisture. In short, the procedure is subject to considerable error. Notwithstanding, it gives a good indication of moisture in the furnace gas if we draw the wet and dry samples from the same location and calibrate the analyzers before measurement.
2.4.4
Addition of Molecular Hydrogen to the Fuel
Refinery fuels may comprise more than just hydrocarbons; molecular hydrogen (H2) is a common addition. We distinguish molecular hydrogen from fuel-bound hydrogen, i.e., hydrogen that is part of a hydrocarbon fuel component (CHψ). A typical simulation of refinery gas comprises 25% H2, 50% natural gas, and 25% C3H8. These fuels are readily available and easily blended. In the case of molecular hydrogen, one could adjust ψ to account for both contributions (i.e., r H2 + CHψ = CHψ*, where ψ* = ψ + 2r). Then ψ* represents
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stack
convection section
stack probe
convection tubes
furnace probe
shock tubes “arch” or “bridgewall” radiant section
radiant tubes
measure emissions here for regulatory purposes
measure emissions here for control purposes
burners
FIGURE 2.22 A process heater. The typical sections of a process heater comprise a (a) radiant section — the portion of the furnace that can “see” the flame; (b) arch or bridgewall — the transition section between the radiant and convection sections; (c) convection section — where the primary heat transfer mechanism is via convection; and (d) the stack — the outlet conduit for the furnace. For control purposes, one must measure O2 and CO at the bridgewall. For regulatory purposes, one must measure emissions in the stack.
the total molar H/C ratio where H is counted from both CHψ and H2. This is a straightforward concept with intuitive physical meaning; however, if we combust pure hydrogen, then ψ → ∞. If we desire a single equation to account for a blend of hydrogen and hydrocarbons — including possibly pure hydrogen — then we should transform x from its unbounded range, 0 < ψ < ∞, to another variable, say χ, that has finite bounds (e.g., 0 < χ < 1). We accomplish this by defining χ as the total mole fraction of hydrogen in the fuel, having the following relation to ψ*: χ=
ψ* H = 1 + ψ * H+ C
(2.40a)
χ H = 1− χ C
(2.40b)
ψ* =
Then we may recast our equations in terms of χ for mixtures of hydrocarbons and hydrogen via the substitution given in Equation 2.40b. This leads to the following:
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⎛ 4 − 3.21χ ⎞ K dry = ⎜ , ⎝ 4 − 3χ ⎟⎠
⎛ 4 − 2.79χ ⎞ K wet = ⎜ ⎝ 4 − 3χ ⎟⎠
(2.41)
The K factor form of the equations remains valid with either Equation 2.40a or Equation 2.40b defining the K factor. Recasting the remaining equations in terms of χ gives the following: χ=
yH2O yH2O = 2 yCO 2 ,wet + yH2O 2 1 − yH2O yCO 2 ,dry + yH2O
(
yH2O =
yCO 2 ,dry =
2.4.5
)
2 χyCO 2 ,dry 2 χyCO 2 ,wet = 1− z 1 + χ(2 yCO 2 ,dry − 1)
⎛ 1 − χ ⎞ ⎛ yH2O ⎞ (1 − χ)yCO 2 ,wet =⎜ ⎝ 2 χ ⎟⎠ ⎜⎝ 1 − yH2O ⎟⎠ 1 − χ 1 + 2 yCO 2 ,wet
(
)
(2.42)
(2.43)
(2.44)
Addition of Flue Gas Components to Fuel
Besides hydrogen and hydrocarbons, other significant (nontrace) quantities in refinery fuel gas sometimes include carbon monoxide and diluents such as nitrogen and carbon dioxide. Sometimes oxygen and water vapor are also present, especially for certain waste streams or for digester or landfill gas — a 50/50 mixture of CO2 and H2O from anaerobic decomposition diluted with some air. In the case of landfill gas, a series of subterranean collectors exert a vacuum and extract gas from the landfill. If the vacuum is too high, air will migrate through the soil and dilute the landfill gas. If the vacuum is too low, landfill gas will escape, releasing methane to the atmosphere. The customary practice results in some small addition of air to the landfill gas, and this is the safer alternative to fugitive methane emissions. H2 and CO significantly increase certain fuel properties, such as flame speed and temperature (see Appendix A). Hydrogen reduces the heating value on a volumetric basis owing to its very low molecular weight. The heating value on a volumetric basis (rather than a mass basis) is the usual value of interest because burners meter fuel on a volumetric basis using a fuel pressure differential across orifices. For this purpose, the Wobbe index (ω) is a useful ratio: Wa ω = ΔH Wf
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(2.45)
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where ΔH is the volumetric higher heating value (HHV; e.g., Btu/scf), Wa is the molecular weight of air (29 g/mol), and Wf is the molecular weight of the fuel (see Appendix A for tabulations of these quantities for some common fuels). The ratio Wf Wa is known as the specific gravity of the fuel (abbreviated sp. g.). Thus, an alternate form for Equation 2.45 is ω =
ΔH . Thesp. g.
oretically, the heat release from a fuel orifice scales as the Wobbe index. If H2 and CO concentrations in the fuel vary during the course of normal operation, it may be better to make a separate accounting for them in the mass balance. As we have already seen, the addition of molecular hydrogen adds no real complexity to the mass balance. However, the addition of O2, N2, CO2, CO, and H2O to the fuel complicates matters because the species now appear on both sides of the combustion equation. In that case, the stoichiometric equation becomes more involved: c CO2 + d N2 + f O2 + g H2O + r H2 + s CO + CHψ + (1+ r/2 + s/2 + x/4 – f)(1 + ε) O2 + (79/21)(1+ r/2 + s/2 + x/4 – f)(1 + ε) N2 → (1 + c + s) CO2 + (g + r + x/2) H2O + ε(1 + r/2 + s/2 + x/4 – f)O2 + [(79/21)(1 + r/2 + s/2 + x/4 – f)(1 + ε) + d] N2 (2.46) Here, c is the CO2/CHψ mole ratio, d is the N2/CHψ mole ratio, f is the O2/CHψ mole ratio, g is the H2O/CHψ mole ratio, r is the H2/CHψ mole ratio, s is the CO/CHψ mole ratio, and ψ is the H/C ratio in the hydrocarbon stream only. We emphasize that these refer to the component ratios in the fuel, not the combustion products. Equation 2.46 is the most general form of the stoichiometric equation we will consider. Other combustibles such as H2S and NH3 may be present in trace amounts in the fuel stream, and we could modify Equation 2.46 to account for them. However, as regards the stoichiometry, trace quantities are insignificant. H2S and NH3 do affect SO2 and NOx emissions, respectively, so we will consider these species in this context later in this chapter. Since we are accounting for molecular hydrogen separately from the hydrocarbon portion of the fuel, there is no need to use a variable transform such as χ. Equations cast in terms of ψ or K have the following general form: y=
a + bε K+ε
(2.47)
The reader may verify that if c, d, f, g, r, and s = 0, the following expressions will reduce to the original (simpler) relations:
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(
) ) (
)
) ) (
)
⎡ 4 c + d + f + g + 2(s − r ) − ψ ⎤ ⎦ K dry = 1 + 0.21 ⎣ ⎡4 1 − f + 2 r + s + ψ ⎤ ⎣ ⎦
(
(
⎡ 4 c + d + f + g + 2(r + s) + ψ ⎤ ⎦ K wet = 1 + 0.21 ⎣ ⎡4 1 − f + 2 r + s + ψ ⎤ ⎣ ⎦ f 4 1− f + 2 r + s + ψ
aO2 =
ε=
(
) (
)
(2.48b)
(2.49a)
yO 2 ,dry =
0.21ε + aO 2 ε + K dry
(2.49b)
yO 2 ,wet =
0.21ε + aO 2 ε + K wet
(2.49c)
yO 2 ,dry K dry − aO 2 yO 2 ,wet K wet − aO 2 = 0.21 − yO 2 ,dry 0.21 − yO 2 ,wet 84d + 0.79 4 1− f + 2 r + s + ψ
aN2 =
(
) (
)
(2.50a, b)
(2.51a)
yN 2 ,dry =
0.79ε + aN 2 ε + K dry
(2.51b)
yN 2 ,wet =
0.79ε + aN 2 ε + K wet
(2.51c)
( ) ) ( )
(2.52a)
yCO 2,dry =
aCO2 ε + K dry
(2.52b)
yCO 2,wet =
aCO2 ε + K wet
(2.52c)
aCO2 =
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(
(2.48a)
(
84 1 + c + s
4 1− f + 2 r + s + ψ
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Modeling of Combustion Systems: A Practical Approach Substoichiometric Combustion
So far, we have considered air in excess and fuel as the limiting reagent. This is the usual (and safe) combustion practice. Substoichiometric combustion is the fuel–oxygen reaction having oxygen as the limiting reagent. Substoichiometric combustion is extremely dangerous in conventional furnaces because large volumes of hot combustibles may accumulate, find some air, and explode — a heater explosion can have remarkable destructive potential. When the combustion first becomes substoichiometric, the flame does not go out. Burners will stay lit even without the full complement of air because what air is available will combust enough fuel to keep the combustion reaction going and establish a flame. However, under these circumstances burners can become unstable: the burners may go out, relight, or the flame may detach and reattach haphazardly. If one discovers substoichiometric combustion, the conventional wisdom is not to add more air, and especially not to add it quickly. Adding air to a hot furnace full of combustibles increases the likelihood of explosion. Rather, one should slowly reduce the fuel until oxygen reappears and then reaches its target level in the flue gas; only then should one add additional air and increase the firing rate. This is a good general rule of thumb, but it is not foolproof. So be warned, this situation is extremely dangerous and potentially deadly. The only safe option is to avoid substoichiometric combustion in the first place. Related to avoiding substoichiometric combustion is a lead-lag control strategy. 2.4.6.1 Lead-Lag Control Whenever the unit is to increase in firing rate, the air increase should lead the fuel increase; that is, the air increase should occur first in time. Whenever the unit must decrease in firing rate, the air decrease should lag the fuel decrease; that is, the air decrease should occur last in time. This is the leadlag control scheme for combustion. This practice keeps the unit from encountering substoichiometric combustion. 2.4.6.2 Substoichiometric Equations It is possible to develop equations for substoichiometric combustion, recognizing the following limitations: 1. Practical combustion systems are not supposed to be operated substoichiometricly. 2. Equilibrium calculations neglect the formation of soot. 3. Some soot formation is likely. Even so, such equations can at least be instructive, and for these reasons we develop the relations that follow. (Note: Some reactors — partial oxidation reactors — deliberately employ substoichiometric combustion. However,
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partial oxidation reactors have specific design features to carry out the substoichiometric combustion safely. We do not consider them in this text per se. But if the partial oxidation reaction is with air, the following equations will be applicable nonetheless.) Earlier in the chapter, we defined the quantity 1 + ε as being the ratio of theoretical air required for combustion. When ε > 0, some excess air exists in the furnace and the flue gas contains some O2. When ε < 0, there is no relation between excess air and O2 because yO2 = 0 for all ε < 0. Accordingly, we define Φ as the fuel/air ratio (also known as the stoichiometric ratio), having the following relationship to ε: 1 1+ ε
(2.53a)
1− Φ Φ
(2.53b)
Φ=
ε=
When Φ < 1 (ε > 0), then the system has more than the required air for combustion and we term the combustion superstoichiometric. In superstoichiometric combustion, we know that all H will oxidize to H2O and all C to CO2. However, in substoichiometric combustion, H will go to H2O — until it consumes much of the oxygen — and to H2 thereafter. Likewise, C will first go to CO and thereafter to CO2. So, let us define the following molar ratios: α=
H2 H2O
(2.54a)
β=
CO CO 2
(2.54b)
Then we may recast Equation 2.15 in terms of α and β:
CH ψ +
1 ⎡β + 2 79 ⎞ β ψ ⎤⎛ 1 CO 2 + CO + N2 → ⎢ ⎥ O2 + 2 ⎢ β + 1 2 1 + α ⎥ ⎜⎝ 21 ⎟⎠ β+1 β+1 ⎣ ⎦
(
)
(2.55a)
⎛ 1 ⎞ ⎛ 79 ⎞ ⎡ β + 2 ψ ⎤ 1⎛ 1 ⎞ 1 ⎛ αψ ⎞ + H 2O + ⎜ + ⎜ H2 + ⎜ ⎟ ⎜ ⎟ ⎢ ⎥ N2 ⎟ ⎟ 2 ⎝ ψ +α⎠ 2 ⎝ 1+ α⎠ ⎝ 2 ⎠ ⎝ 21 ⎠ ⎢⎣ β + 1 2 1 + α ⎥⎦
(
)
Turns7 notes that α and β are related by the equilibrium constant Kwg for the water–gas shift reaction: CO + H2O = CO2 + H2, and Kwg = 0.19 provides
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good agreement with more rigorous equilibrium calculations for propane. That is, K wg =
α ≈ 0.19 β
(2.56)
Then we may recast Equation 2.55a as a function of β:
CH ψ +
ψ 1 ⎡β + 2 ⎢ + 2 ⎢ β + 1 2 1 + K wgβ ⎣
(
)
⎤⎛ 1 β 79 ⎞ ⎥ ⎜ O2 + N2 → CO 2 + CO β+1 β+1 21 ⎟⎠ ⎥⎦ ⎝
(2.55b)
⎞ ⎛ 1 ⎞ ⎛ 79 ⎞ ⎡ β + 2 1⎛ 1 1 ⎛ K βψ ⎞ ψ H2 + ⎜ ⎟ ⎜ ⎟ ⎢ + ⎜ + H 2O + ⎜ wgg ⎟ ⎟ 2 ⎝ ψ + K wgβ ⎠ 2 ⎝ 1 + K wgβ ⎠ ⎝ 2 ⎠ ⎝ 21 ⎠ ⎢ β + 1 2 1 + K wgβ ⎣
(
)
⎤ ⎥ N2 ⎥⎦
Now we know that the total amount of oxygen consumed must by definition be ⎛ ψ + 4⎞ 1 ⎛ ψ + 4⎞ = 1+ ε ⎜ ⎟ ⎜ Φ⎝ 4 ⎠ ⎝ 4 ⎟⎠
(
)
Equating this with the coefficient for O2 in Equation 2.55 yields 1 ⎛ ψ + 4⎞ 1 ⎡β + 2 ψ = ⎢ + Φ ⎜⎝ 4 ⎟⎠ 2 ⎢ β + 1 2 K wgβ + 1 ⎣
(
or
(
)
Φ β, ψ =
(K
)
⎤ ⎥ ⎥⎦
(2.57)
)( )( ) 2 ( K β + 1) (β + 2 ) + (β + 1) ψ β+1 β+1 ψ + 4
wg
(2.58)
wg
Then Φ or ε is a function of only one parameter, β. From Equation 2.55 we may calculate all the required species as a function of β. Therefore, we may cast the total dry and wet products (TDP and TWP, respectively) as functions of β and ψ alone:
(
)
TDP β, ψ = 1 +
(
1 ⎛ K wgβψ ⎞ 79 1 ⎡ β + 2 ψ ⎢ + + 2 ⎜⎝ 1 + K wgβψ ⎟⎠ 21 2 ⎢ β + 1 2 K wgβ + 1 ⎣
(
)
TWP β, ψ = 1 +
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)
ψ 79 1 ⎡ β + 2 ψ ⎢ + + 2 21 2 ⎢ β + 1 2 K wgβ + 1 ⎣
(
)
⎤ ⎥ ⎥⎦
⎤ ⎥ ⎥⎦
(2.59)
(2.60a)
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Making use of Equation 2.57, we recast TWP as a function of Φ and ψ. ⎛ 79 1 ⎞ ⎛ ψ⎞ ψ 1+ ⎟ + TWP Φ, ψ = ⎜ 1 + ⎟ ⎜ 21 Φ ⎠ ⎝ 4⎠ 4 ⎝
(
)
(2.60b)
The species expressions become yO2 ,dry = 0
(2.61a)
yO2 ,wet = 0
(2.61b)
yN2 ,dry =
⎞ 79 ⎛ 1 ⎞ ⎛ β + 2 ψ + ⎜ ⎟ ⎟ ⎜ 21 ⎝ 2 TDP ⎠ ⎝ β + 1 2 K wgβ + 1 ⎠
(2.62a)
yN2 ,wet =
⎞ 79 ⎛ 1 ⎞ ⎛ β + 2 ψ + ⎜ ⎟ ⎟ ⎜ 21 ⎝ 2 TWP ⎠ ⎝ β + 1 2 K wgβ + 1 ⎠
(2.62b)
yCO2 ,dry =
1 ⎛ 1 ⎞ TDP ⎜⎝ β + 1 ⎟⎠
(2.63a)
yCO2 ,wet =
1 ⎛ 1 ⎞ TWP ⎜⎝ β + 1 ⎟⎠
(2.63b)
(
(
)
1 ⎛ ψ ⎞ 2 TWP ⎜⎝ K wgβ + 1 ⎟⎠
(2.64)
yH2,dry =
1 ⎛ K wgβψ ⎞ 2 TDP ⎜⎝ K wgβψ + 1 ⎟⎠
(2.65a)
yH2,wet =
1 ⎛ K wgβψ ⎞ 2 TWP ⎜⎝ K wgβψ + 1 ⎟⎠
(2.65b)
1 ⎛ β ⎞ TDP ⎜⎝ β + 1 ⎟⎠
(2.66a)
yH2O =
yCO,dry =
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)
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yCO,wet =
1 ⎛ β ⎞ TWP ⎜⎝ β + 1 ⎟⎠
(2.66b)
Combining Equations 2.57 and 2.66b gives
(
)
yCO,wet β, ψ =
⎛ β ⎞ ⎟ ⎜ ⎛ 79 1 ⎞⎛ ψ ⎞ ψ ⎝ β + 1⎠ ⎜1+ ⎟ ⎜1+ ⎟ + 21 Φ β, ψ ⎠ ⎝ 4⎠ 4 ⎝ 1
(
(2.66c)
)
20% Vol % species (except N2)
18% 16%
DryN2
H2O
14% 12% 10%
CO2 CO
8% Wet N2
6% 4% 2% Vol % dry > Vol % wet species in all cases 0% 0.50 0.60 0.70 0.80
O2 H2 0.90
1.00
1.10
1.20
1.30
1.40
90% 88% 86% 84% 82% 80% 78% 76% 74% 72% 70% 68% 66% 64% 62% 60% 1.50
Vol % N2)
Equation 2.66c gives yCO,wet (β, ψ ) and Equation 2.58 gives Φ(β, ψ ) . In principle, we could combine these to give Φ( yCO,wet , ψ ) and yCO,wet (Φ, ψ ) , but this results in a messy analytical expression. The easier solution is to use β as a parameter and plot Φ against ψ as β varies, or solve for β numerically for a given Φ or yCO. As noted, we may use β as a parameter in Equation 2.58 and the species concentration equations above to yield the equilibrium relations. Combining the substoichiometric and superstoichiometric relations, we may estimate wet or dry species for any hydrogen–hydrocarbon blend. If desired, we may use the substitution of Equation 2.40b for very high hydrogen cases. Figure 2.23 shows an example for refinery gas. From the above equations or the figure, we make the following and immediate observations:
Φ FIGURE 2.23 Substoichiometric combustion relations. The figure gives relations for a typical refinery gas comprising 25% H2, 50% CH4, and 25% C3H8.
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• No matter how substoichiometric the combustion, oxygen is always zero. Therefore, the oxygen concentration cannot estimate ε or Φ, except merely to say that ε < 0 and Φ > 1. • For hydrocarbon fuels, CO is present whenever the combustion is substoichiometric. In principle, a CO analyzer could estimate Φ in the furnace. However, CO analyzers for combustion monitoring read no more than 5000 ppm. This represents only a very small amount of substoichiometric combustion. Therefore, this kind of CO measurement is not a useful quantitative measure of the degree of air starvation. However, CO measurement does provide a valuable early warning signal — CO skyrockets before entering and while in a dangerous substoichiometric operating region.
Example 2.5
Calculation of Φ and ε for Substoichiometric Combustion
Problem statement: For a refinery gas having 25% H2, 50% CH4, and 25% C3H8, calculate Φ and ε if the furnace gas shows 5000 ppm CO. Estimate the CO level if Φ = 1.1. Comment on the suitability of a CO or combustibles analyzer having an upper limit of 5000 ppm for this kind of service. Solution: Since this reading comes from the furnace, we may presume it is an in situ reading and that the CO is on a wet basis. Solving for ψ we obtain ψ=
0.25(2) + 0.50( 4) + 0.25(8) = 3.6 0.25(0) + 0.50(1) + 0.225(3)
Using Equation 2.66c to give yCO,wet (β, ψ ) and Equation 2.58 to give Φ(β, ψ ) , we solve for β such that yCO,wet (β, ψ ) = 5000 ppm and ψ = 3.6 using an iterative routine. Spreadsheets usually contain one or two such routines (e.g., Goal Seek™ in Excel™). This gives β = 0.0318 and Φ = 1.011. Thus, 5000 ppm CO represents only a small level of substoichiometric combustion (about Φ – 1 = 1.1% substoichiometric). If we use Equation 2.53b to calculate ε, we obtain the same result ε=
1 − Φ 1 − 1.011 = = −0.0109 = −1.1% Φ 1.011
Using the same process, if Φ = 1.1 (ε = –0.10 = –10%), then β = 0.3197 and yCO,wet ≈ 41,000 ppm. Thus, 10% substoichiometric operation
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Modeling of Combustion Systems: A Practical Approach results in more than 40,000 ppm CO. Therefore, a typical combustion analyzer having an upper limit of 5000 ppm would not be useful to indicate this level of substoichiometric combustion; however, it would allow us to know when we exceed ~1% substoichiometric combustion, giving us an early warning that something has gone awry.
Although we desire to have excess oxygen present, too much excess air robs efficiency from the unit. CO evidences how little O2 is too little and quantifies a healthy balance between good efficiency and good combustion. For these reasons, we recommend that furnaces be equipped with wellmaintained and calibrated CO and O2 analyzers. 2.4.7
Conservation of Mass for Flow in a Furnace
As shown previously, combustion conserves total mass and individual atomic species but not molecular entities. However, we may write a general conservation equation for all molecular species, including those that the reaction consumes or produces. In a strict sense, we do not conserve such species, but rather account for them. However, the standard nomenclature is conservation of mass, not accounting of mass. Heuristically, the general conservation equation is in + gen = out + acc
(2.67)
where in is the flow rate into an arbitrary volume [M/θ], gen is the amount of substance produced per unit time within the volume [M/θ], out is the flow rate out of the volume [M/θ], and acc is the amount of substance accumulating per unit time [M/θ]. In order to know the difference between in and out we need a system boundary. We divide the universe into two pieces: system and surroundings. The system is any portion of the universe we are interested in; the surroundings comprise the rest. We shall always pick a convenient reference — one where we know the boundary conditions. Our sole purpose is to define a system so that we may solve the conservation equation and produce results that are meaningful to us. Mathematically, Equation 2.67 becomes
(
d ρv xv ρi xiVi + rV = ρe xeVe + V dt In Equation 2.68 we have the following entities:
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)
(2.68)
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ρ is the density [M/L3]. x is the mass fraction [ ]. V is the volumetric flow rate [L3/θ]. i is a subscript indicating influent flow. e is a subscript indicating effluent flow. v is a subscript indicating properties belonging to the control volume. r is the reaction rate [M/L3θ]. V is the system volume [L3]. t is the time [θ]. Therefore, the units of Equation 2.68 are [M/θ]. We use the dot superscript to refer to flow rates in general. For example, V is the volumetric flow rate, is the mass flow rate, N is the molar flow rate, etc. Time is the unit in the m denominator for all rate equations.
2.4.8
Simplifying Assumptions (SAs)
Here we list some conditionals that can simplify Equation 2.68: 1. If there is no reaction, gen = 0 (rV = 0). 2. If the process is time invariant with position (steady state), then
(
)
⎛ d ρv xv ⎞ acc = 0 ⎜ V = 0⎟ . dt ⎝ ⎠ 3. If the process is well mixed, then ρv = ρe and xv = xe . 4. If the density is time invariant, then V
Example 2.6
(
d ρv xv dt
) = ρ V dx v
v
dt
.
An Example of a Non-Steady-State Mass Balance
Problem statement: Consider a furnace with a control volume as shown in Figure 2.24. Take the system to be that as shown by the dotted boundary in the figure. Beneath the system boundary is a combustion process producing some emission of concern. Above the system boundary is a sensor that measures the emission. Presume that the system volume is well mixed. What will be the time-varying behavior of the analyzer to a sudden step change in the emission rate from a process upset? In other words, develop an equation for xθ, where xθ is the analyzer concentration at time t = θ.
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effluent flue gas sample probe ρexeQe
V
d(ρv xv) dt
influent flue gas
ρi xi Qi
control volume boundary
FIGURE 2.24 A furnace control volume. The furnace diagram shows a control volume and elements of the associated mass balance. For ease of computation, the control volume is drawn such that there is no reaction within its boundary.
Solution: The process is well mixed, so simplifying assumption (SA) 3 applies. We have drawn the system volume above the reaction zone in the flame, so SA 1 applies. We shall presume that automated controls will operate to keep the temperature and composition constant. Therefore, SA 4 applies. Furnaces do not accumulate mass; otherwise, the furnace would become a pressure vessel. Since the acc and gen terms are zero, in = out, so Vi = Ve . Dropping the subscripts for the variables that do not need to be distinguished, Equation 2.68 becomes dx ρxiV = ρxV + ρV dt Dividing by ρV and multiplying by dt gives xi
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V V dt = x dt + dx V V
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We can easily separate the variables and integrate the equation at once: V − V
xθ
θ
∫
dt =
0
⌠ ⎮ ⎮ ⌡
x0
dx x − xi
This yields the equation −
⎛ x − xi ⎞ V θ = ln ⎜ θ V ⎝ x0 − xi ⎟⎠
(2.69)
or equivalently, V
− θ xθ − xi =e V x0 − xi
(2.70)
Equations 2.69 and 2.70 are dimensionless forms. We may rearrange the latter to give xθ in terms of the other variables.
(
xθ = xi 1 − e
V − θ V
)+ x
0
(2.71)
In reactor engineering, V V is known as the space velocity, having units of reciprocal time [θ–1]. Conversely, V V is known as the space-time [θ]. In process control engineering, V V is known as the first-order gain, while V V is known as the system time constant. (V V )θ gives the number of volume changes for the furnace in a given time, where V is a volume of interest [L3], usually the furnace volume. The residence-time is the time that a quantity of fluid of interest stays within a specified volume. For well-mixed volumes of the type considered in the above example, the space-time and the residencetime are synonymous. Figure 2.25 shows the relation graphically. Equation 2.71 and Figure 2.25 point out some interesting facts. First, furnaces behave as integrators. The effluent is a time average of input conditions. The time constant, V V , is the time it takes for the outlet value to achieve 1 − 1 e ≈ 63.2% of the final value. Second, after making a change, one must allow sufficient time to observe it. The time constants for furnace volumes are typically only a few minutes. However, they tie to processes that can have much longer time constants and that communicate to the furnace volume via a control system. Suppose the time constant for a furnace and process is 15 minutes; i.e., V V is 15 minutes. After 30 minutes, the system has reached about 85% of its final value. After 45 minutes, the system reaches
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Modeling of Combustion Systems: A Practical Approach 1.0 1-e-3= 95.0%
0.9 xθ – x0 xi
1-e-2= 86.5%
Normalized Concentration
0.8 0.7 1-e-1= 63.2%
0.6 0.5 0.4 0.3 0.2 0.1 0.0 0
1 Time Constant,
Q
2
3
V
FIGURE 2.25 Concentration vs. time for well-mixed behavior. Concentration shows a law of diminishing returns over time. The effluent concentration, xθ , approaches but never quite reaches the new inlet concentration, xi.
95% of its final value. So it would be futile to make sense of an observation 20 minutes after effecting a process change. Many furnaces require 45 minutes to return to steady state after a significant step change, as a rule of thumb. However, depending on the magnitude and nature of the disturbance, it may take even longer.
2.4.9
Ideal Gas Law
Except for very special applications, industrial combustion is a high-temperature, low-pressure affair. Therefore, the ideal gas law is applicable and quite accurate: PV = nRT (2.72) where P is the pressure [ML/θ2], V is the volume [L3], n is the number of moles [N], R is the gas constant [L2/θ2T], and T is the temperature [T]. In some cases, one may prefer to use the molar volume Vˆ = V/n [L3/N], in which case Equation 2.72 takes the form of Equation 2.73: PVˆ = RT
(2.73)
Some cases will call for mass rather than molar quantities: PW = ρRT
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(2.74)
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where W is the molecular weight [M/N] and ρ is the density [M/L3].
2.4.10 Dilution Correction Suppose a local agency regulates NOx to 40 ppm. What does this mean? Technically, it means that for every million volumes of flue gas, 40 will be NOx. If the NOx limit were not otherwise qualified, there would be two ways to meet it. One way would be to reduce the amount of NOx the combustion system generates; another way would be to increase the flue gas volume by adding dilution air. But this latter option does not reduce the amount of NOx that enters the atmosphere. Indeed, this dilution may occur without our even knowing it. For example, if the stack pressure is below atmospheric pressure, then any leakage will allow air to infiltrate the stack gas and reduce the NOx concentration. For this reason, air quality districts reference the NOx concentration to some oxygen percentage. For boilers and process heaters the typical reference is 3% oxygen. For gas turbines, the typical reference is 15%. Some districts use 0% O2 as the reference. A mass balance allows us to develop a simple equation to correct any actual concentration to a reference concentration. Figure 2.26 shows the general logic. effluent
. Ve yNO,e yo2,e
total mass balance: . . . Ve = Vi + Va
. Va yo2,a
mass balance on O2: . . . yo2,eVe = yo2,iVi + yo2,aVa
air in-leakage
mass balance on. NOx: . yNO,eVe = yNO,iVi . Vi yNO,i yo2,i
influent
FIGURE 2.26 Dilution correction. A mass balance provides a method for calculating the amount of air inleakage and the influent NOx concentration.
Now we can do a total mass balance as follows: in + gen = out + acc If the air in-leakage does not react with the emissions of concern (that is, it functions only as a diluent), then gen = 0. For a steady-state process, acc = 0. Therefore, in = out and we obtain i +m a = m e m
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(2.75)
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i is the influent mass flow rate [M/θ], m a is the mass flow of inwhere m e is the effluent mass flow rate [M/θ]. We leaking (diluting) air [M/θ], and m can use volumetric flow rates if we correct them to some standard temperature and pressure (STP), say 60°F and 1 atm. In that case, we have Vi + Va = Ve
(2.76)
where Vi is the influent volumetric standard flow rate [L3/θ], Va is the volumetric standard flow rate of in-leaking air [L3/θ], and Ve is the effluent volumetric standard flow rate [L3/θ], all at STP. Now a mass balance for NOx will be yNO ,iVi + yNO , aVa = yNO ,eVe , but yNO , a = 0 ; therefore, the equation becomes yNO ,iVi = yNO ,eVe
(2.77)
We may solve Equation 2.77 for Vi Ve directly, leading to Vi yNO ,e = Ve yNO ,i
(2.78)
With respect to oxygen, the mass balance yields yO 2 ,iVi + yO 2 , aVa = yO 2 ,eVe Rearranging Equation 2.79, we have V V yO 2 ,i i + yO 2 , a a = yO 2 ,e Ve Ve But we know from Equation 2.76 that Vi + Va =1 Ve Therefore, Va V = 1 − i Ve Ve Substituting this into the foregoing equation gives
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(2.79)
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⎛ V V ⎞ yO 2 ,i i + yO 2 , a ⎜ 1 − i ⎟ = yO 2 ,e Ve Ve ⎠ ⎝ Collecting Vi Ve yields Vi yO 2 ,i − yO 2 , a + yO 2 , a = yO 2 ,e Ve
(
)
Multiplying both sides by –1 and solving for Vi Ve , we obtain Vi yO 2 , a − yO 2 ,e = Ve yO 2 , a − yO 2 ,i
(2.80)
Equating Equations 2.78 and 2.80 to eliminate Vi Ve and inverting the fractions gives yNO ,i yO 2 , a − yO 2 ,i = yNO ,e yO 2 , a − yO 2 ,e
(2.81)
The above equation gives us the inlet concentration ( yNO ,i ) if we know the effluent concentration ( yNO ,e ). In fact, we can correct even for hypothetical influent concentrations, and this is the real power of Equation 2.81. Let us replace yO 2,i and yNO ,i by yO 2,ref and yNO ,ref , respectively. In this way, we expressly declare that we are correcting our results to a hypothetical reference concentration. Then Equation 2.81 becomes ⎛y − yO 2 ,ref yNO ,ref = ⎜ O 2 , a ⎝ yO 2 , a − yO 2 ,e
⎞ ⎟ yNO ,e ⎠
(2.82)
This is the traditional form for correction to a hypothetical reference condition. Indeed, this dilution correction philosophy is perfectly generic for any emission of interest. ⎛y − yO 2 ,ref yx ,ref = ⎜ O 2 , a ⎝ yO 2 , a − yO 2 ,e
⎞ ⎟ yx ,e ⎠
(2.83)
where the subscript x refers to any emission of interest that is not found in the dilution air.
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Example 2.7
Emissions Corrections to Specified Reference Conditions
Problem statement: Emissions measurement at the stack reveals NOx and CO fractions of 32 and 18 ppm, respectively, at an outlet concentration of 5% oxygen. The legal limit is 40 ppm NOx and 200 ppm CO, corrected to 3% oxygen. Calculate the corrected values. Is the unit in compliance? Solution: The form of Equation 2.82 is perfectly generic for correcting the concentration of any species not found in the dilution air. Therefore, we have ⎛y − yO 2 ,ref yNO ,3% = ⎜ O 2 , a ⎝ yO 2 , a − yO 2 ,e
⎞ ⎛ 21 − 3 ⎞ ⎟ yNO ,e = ⎜⎝ 21 − 5 ⎟⎠ 32 ⎡⎣ ppm ⎤⎦ = 36 ⎡⎣ ppm ⎤⎦ ⎠
⎛y − yO 2 ,ref yCO ,3% = ⎜ O 2 , a y ⎝ O 2 , a − yO 2 ,e
⎞ ⎛ 21 − 3 ⎞ ⎟ yCO ,e = ⎜⎝ 21 − 5 ⎟⎠ 18 ⎡⎣ ppm ⎤⎦ = 20.3 ⎡⎣ ppm ⎤⎦ ⎠
So the emissions are well within the legal limits. Dilution correction normalizes all emissions to the same reference basis and eliminates the dilution effects of air. Note: Many regulatory districts have their own opinion about how much oxygen air comprises. Some districts use 20.9% for the oxygen concentration, some use 20.8%. Be sure to replace 21 by whatever the district specifies. The following example highlights how different references affect the reported values.
Example 2.8
Corrections to Various Bases
Problem statement: One district corrects its NOx readings to 3% oxygen, while another corrects to 15% oxygen. Yet another corrects its readings to 0% oxygen. Show the equivalence of 40 ppm corrected to 3% with these other bases. Solution: We begin with NOx at 40 ppm corrected to 3%. If we correct this to 15% oxygen we have ⎛ 21 − 15 ⎞ yNO,15% = ⎜ 40 ⎡ ppm ⎤⎦ = 13.3 ⎡⎣ ppm ⎤⎦ ⎝ 21 − 3 ⎟⎠ ⎣
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If we correct to 0% we have ⎛ 21 − 0 ⎞ yNO,15% = ⎜ 40 ⎡ ppm ⎤⎦ = 46.7 ⎡⎣ ppm ⎤⎦ ⎝ 21 − 3 ⎟⎠ ⎣ Therefore, to convert emissions referenced at 3 to 15%, we divide by 3 (or multiply by 1/3), that is, ⎛ 21 − 15 ⎞ 6 1 ⎜⎝ 21 − 3 ⎟⎠ = 18 = 3 To convert emissions referenced at 3% to a base of 15%, we multiply by 7/3, that is, ⎛ 21 − 0 ⎞ 7 ⎜⎝ 21 − 3 ⎟⎠ = 3 The point of dilution correction is that dilution-corrected emissions are invariant to dilution by air. An example illustrates the point.
Example 2.9
Invariance of Dilution-Corrected Emissions
Problem statement: Dilution-corrected emissions are invariant to dilution by air. Prove this by considering 100 volumes of flue gas containing 100 ppm NOx at 0% O2. Then add 100 volumes of air to this and determine the oxygen and NOx at this condition. Correct both cases to 3%. What do you note? Solution: In the first case, we have 100 ppm of NOx. Doubling the volume by adding air will reduce the NOx emissions to 100/2 = 50 ppm at 21/2 = 10.5% oxygen. Correcting the first case to 3% oxygen, we obtain ⎛ 21 − 3 ⎞ yNO,3% = ⎜ 100 ⎡⎣ ppm ⎤⎦ = 85.7 ⎡⎣ ppm ⎤⎦ ⎝ 21 − 0 ⎟⎠ corrected to 3% oxygen. Correcting the second case to 3% oxygen, we have ⎛ 21 − 3 ⎞ yNO,3% = ⎜ 50 ⎡ ppm ⎤⎦ = 85.7 ⎡⎣ ppm⎤⎤⎦ ⎝ 21 − 10.5 ⎟⎠ ⎣ corrected to 3% oxygen.
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Modeling of Combustion Systems: A Practical Approach Therefore, in any case, the corrected emissions are identical, as the cases differ only by air dilution. This is the point of dilution correction.
2.5
Conservation of Energy
Mechanical energy (E) is force times distance (d), E = Fd. In turn, force is mass times acceleration, F = ma. Thus, E [=] [ML2/θ2], where [=] is read has units of. However, all energy terms have the same units. The fundamental SI unit for energy is the Joule (J) = kg m2/sec2. The U.S. customary unit is the British thermal unit (Btu). The general conservation equation applies to energy as well as mass. There are two basic divisions for energy: work and heat. First, we consider heat and related quantities.
2.5.1
Heat and Related Quantities
Heat is thermal energy in transit. Internal energy is thermal energy stored in a body. If a body acquires thermal energy, we no longer call it heat; it is internal energy. We may not know the absolute internal energy of a body. However, one may calculate the difference in internal energy from one state to another if we know the initial and final states of the body. Temperature is a measure of the average random kinetic energy of atoms and molecules in a body. At absolute zero, the average random kinetic energy is zero; this does not mean that all atomic and subatomic motion stops, only random kinetic motion. It is not true that absolute zero is unattainable or that calamity awaits us when we get there. However, we cannot get there by purely thermal means because temperatures above absolute zero can never average to zero. One must not confuse the concepts of temperature, internal energy, and heat. They are all different but related things. To relate them, it is useful to first distinguish between intensive and extensive variables. An intensive variable describes some property of a substance or system that does not depend on mass. For example, pressure and temperature are inherently intensive variables. The temperature of a substance will tell us nothing of its extent, e.g., its mass, number of moles, or volume. Volume, energy, and power are examples of extensive variables. An extensive variable describes some property of a substance or system that depends on mass (extent). Twice the amount of hot water contains double the volume and double the internal energy. Therefore, both internal energy and volume are extensive variables. We can convert any extensive variable to an intensive one by dividing by some measure of extent. We will use the following nomenclature: U, internal
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energy (Btu or kJ, [ML2/θ2]), an extensive property; U = U m , specific internal energy (e.g., Btu/lbm or kJ/kg [L2/θ2]), an intensive property; Uˆ = U n , molar internal energy (e.g., Btu/lbmol or J/mol [ML2/Nθ2]), an intensive ~ property; and, U = U V , volumetric internal energy (e.g., Btu/ft3 or kJ/m3, [M/Lθ2]) an intensive property. We shall use the same superscript symbols whenever we want to distinguish the molar or volumetric forms of any property, variable, or function; however, when unit dimensions are clear from context we may omit the superscripts. Equation 2.84 relates the internal energy to the temperature through a constant of proportionality known as the isometric (constant volume) heat capacity: Δ 21Uˆ = Cˆ v Δ 21T
(2.84)
where the prefix Δ21 is a difference operator signifying the difference between state 2 and state 1 of the succeeding variable (e.g., Δ21U = U2 – U1), Uˆ is the molar internal energy, Cˆ v is the isometric heat capacity (e.g., Btu/lbmol °F or J/mol K [ML2/NTθ2]), and Δ21T is the temperature difference from state 2 to state 1 (T2 – T1). 2.5.2
Work
If the substance expands or contracts upon heating, then it performs work against the surrounding pressure. Work is nonrandom energy in transit. Energy in transit that is not heat is work, and vice versa. The work may or may not do something useful. Regardless, it is a form of energy, and we must account for it in the energy conservation equation. A flame certainly produces heat. However, less obvious in a constant-pressure system is that the flame expands against the atmosphere and therefore performs pressure–volume work. Often, we harness some of this work to pump air in natural draft systems, i.e., natural draft. To account for both heat and pressure–volume work we define enthalpy. Enthalpy is internal energy plus pressure–volume work: H = U + PV
(2.85)
where H is the enthalpy [ML2/θ2], P is the pressure [M/Lθ2], and V is the system volume [L3]. For a constant-pressure system (e.g., a furnace), we may write Δ 21Hˆ = Cˆ p Δ 21T
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(2.86)
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where Δ 21Hˆ is the molar enthalpy difference from state 2 to state 1 (e.g., Btu/ lbmol or J/mol [ML2/Nθ2]) and Cˆ p is the isobaric (constant-pressure) heat capacity (e.g., Btu/lbmol °F or J/mol K [ML2/Nθ2T]), which accounts for differences in both the internal energy and volume. For an ideal gas Cˆ p = Cˆ v + R The reader should note that the ideal gas constant has the same units as the molar heat capacity, e.g., (lit·atm/mol·K) or (psia·ft3/lbmol·°R), i.e., [ML2/ Nθ2T].
2.5.3
Heating Value
Combustion enthalpy (more commonly called heating value) is the state function used to the account for energy in constant-pressure combustion systems such as furnaces and boilers. State functions are path-independent functions; they depend only on the initial and final states. State functions offer great convenience because we are free to construct the path in any theoretical way we choose, so long as we start and end at the specified initial and final states. The heating value is the total enthalpy difference of the fuel and flue gas products referenced to a standard state. An initial state comprises a quantity of fuel and stoichiometric oxygen at standard conditions (usually 25°C and atmospheric pressure) and ends with products at the same temperature and pressure. Thus, heating values comprise interval data and ratios of heating values have no intrinsic meaning, depending as they do on an arbitrary reference state. The lower heating value presumes that water vapor generated in the combustion reaction does not condense, even at the reference condition. The higher heating value presumes that the water of combustion does condense, thereby adding the enthalpy of vaporization to the total heating value. Hence, the higher and lower heating values differ by the heat of vaporization of water. Fuels with no hydrogen (e.g., CO) have identical higher and lower heating values, as their combustion produces no water. Because thermal energy can pass through solid walls (something mass cannot do), we need to modify the generic conservation equation to account for this feature. Then the generic equation for conservation of energy becomes in + gen + trans = out + acc
(2.87)
where trans denotes the energy transferred without mass flow across the system boundary into the system [ML2/θ3]. The following two processes deserve special attention:
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1. An isothermal processes is one whose temperature does not change. In such a case, the inlet and outlet temperatures must be equal (Ti = Te). 2. An adiabatic process is one that does not gain or lose heat. In such a case, trans = 0. It is possible for a system to be both adiabatic and isothermal, for example, nonreacting flow in a perfectly insulated pipe. It is also possible for a system to be isothermal but not adiabatic, for example, water and steam in a boiler tube. Additionally, it is possible for a system to be adiabatic but not isothermal, for example, reacting flow in a perfectly insulated pipe. Finally, a system may be neither adiabatic nor isothermal, for example, reacting flow in an imperfectly insulated pipe. Simplifying assumptions: 1. If there is no reaction, gen = 0. 2. If the process is time invariant with position (steady state), then acc = 0. 3. If the process is adiabatic, then trans = 0. 4. If the process is isothermal, then the inlet and outlet temperatures are equal (Ti = Te).
2.5.4
Adiabatic Flame Temperature
Suppose we wish to estimate the adiabatic flame temperature. Then we presume an adiabatic steady-state process, and the energy balance reduces to in + gen = out or
(
)
(
)
(
)
(
f Cp , f Tf − Tref + m aCp , a Ta − Tref + m f ΔH c = m f +m a Cp , g TAFT − Tref m
)
(2.88)
f is the flow of fuel [M/θ]; Cp , f is the specific heat capacity of the where m fuel, Btu/lbm °F or kJ/kg K [L2/θ2T]; Tf is the temperature of the fuel; Tref is a is the flow of air; Cp , a is the heat an arbitrary reference temperature; m capacity of the air; Ta is the temperature of the air; ΔH c is the specific heat of combustion of the fuel; Cp , g is the heat capacity of the flue gas; and TAFT is the adiabatic flame temperature for which we wish to solve. We may put this equation into dimensionless form also: Cp , f Tf − Tref C T −T ΔH c + α w p , a a ref + = 1 + αw Cp , g TAFT − Tref Cp , g TAFT − Tref Cp , g TAFT − Tref
(
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)
(2.89)
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where a m mf
αw =
At the high temperatures found in combustion, species such as CO2 and H2O dissociate in the flame — enough to reduce the adiabatic flame temperature. However, for our purposes, we only need an approximate calculation, so we will neglect this effect. We can introduce an additional simplification by presuming that the heat capacity on a mass basis [L2/θ2T] is approximately the same for hydrocarbon fuels, air, and flue gas, Cp = Cp , f = Cp , a = Cp , g (0.25 Btu/lbm °F is one typical f Cp , f (1 + α w ) and solving for TAFT − Tref we have value). Dividing m Tf − Tref ⎛ α w ⎞ ΔH c +⎜ Ta − Tref + = TAFT − Tref ⎟ 1 + αw ⎝ 1 + αw ⎠ Cp 1 + α w
(
)
) (
)
(2.90)
⎞ ΔH c = 1 + αw ⎟+ ⎠ Cp , g TAFT − Tref
(2.91)
(
Or in dimensionless form, ⎛ Tf − Tref ⎜ T −T ⎝ AFT ref
⎞ ⎛ Ta − Tref ⎟ + αw ⎜ T − T ⎠ ⎝ AFT ref
(
)
Now if the air and fuel are at ambient temperature, and we use the ambient temperature for the reference temperature, then Tf = Tref and Ta = Tref , and the first two terms in Equation 2.90 vanish and we have TAFT − Tref =
ΔH c Cp 1 + α w
(
)
(2.92)
We may rearrange this to solve for TAFT: TAFT =
ΔH c + Tref Cp 1 + α w
(
)
(2.93)
Now αw is purely a function of the fuel composition and excess air. We may derive it as a function of both by noting the following: 1. The mass ratio of wet flue gas to fuel must equal the mass of fuel plus the mass of air divided by the mass of fuel, i.e., g m +m a m = f = 1 + αw mf mf
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(2.94)
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2. We may derive this quantity directly from Reaction 2.15:
1 + αw =
WCO2 ψ WCO 79 WN2 + + WCHψ 2 WCHψ 21 WCHψ
⎛ ψ⎞ WO2 ε ⎜⎝ 1 + 4 ⎟⎠ 1 + ε + W CHψ
(
)
(2.95)
So, substituting Equation 2.93 into Equation 2.95 gives the adiabatic flame temperature as a function of fuel composition (ψ) and excess air (ε). If we want a more precise value for the adiabatic flame temperature, we can use individual heat capacities. TAFT =
ΔH c + Tref (2.96) WN2 ⎛ ψ⎞ WO2 WCO2 ψ WCO 79 1 + ⎟ 1 + ε + Cp ,O2 ε Cp ,CO2 + Cp ,CO + Cp ,N2 WCHψ ⎜⎝ 4⎠ WCHψ WCHψ 2 WCHψ 21
(
)
We can also obtain αw directly from the volumetric air/fuel ratio, α, and the molecular weight of the fuel, Wf [M/N], and air, Wa [M/N]. α
2.5.5
Wa = αw Wf
(2.97)
Heat Capacity as a Function of Temperature
Heat capacity is a weak function of temperature. However, for combustion calculations, the temperature difference between the influent air (and fuel) and the flue gas emerging from the flame is quite large. Therefore, we consider heat capacity as a function of temperature. In general, we may estimate the heat capacity of an ideal gas at any temperature by the formula Cp = a0 + a1T + a2T 2 + a3T 3
(2.98)
Appendix A, Table A.5 gives the constants for calculation of some important flue gas species. Then the heat capacity of the flue gas species will be n
Cp , g =
∑w (a
© 2006 by Taylor & Francis Group, LLC
(2.99b)
+ b1 ,kT + b2 ,kT 2 + b3 ,kT 3
∑ y (b k =1
)
0 ,k
n
k
(2.99a)
+ a1 ,kT + a2 ,kT 2 + a3 ,kT 3
k =1
Cˆ p , g =
)
0 ,k
k
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where wk is the mass fraction and yk is the mole or volume fraction of the kth flue gas component. The overall heat capacity of the flue gas will be Cp , g = α 0 + α 1T + α 2T 2 + α 3T 3 or Cˆ p , g = β 0 + β1T + β 2T 2 + β 3T 3 , where n
αj =
∑w a
k j,k
k =1
and n
βj =
∑y b
k j,k
k =1
Since all the mass and mole fractions are functions of ε and ψ only, α and β are also functions of ε and ψ only. Therefore, for a given temperature Cp , g = φ(ε , ψ ), and we may calculate Cp,g for any stoichiometry. We may do the same for molecular weight calculations. This will greatly simplify thermodynamic calculations with flue gas.
Example 2.10 Calculation of Adiabatic Flame Temperature Problem statement: Find the approximate adiabatic flame temperature for a fuel having W = 16 lbm/lbmol, ψ = 4, and ΔH c = 22,000 Btu/lbm in 15% and 10% excess air for Cp = 0.25 Btu/lbm °F. Solution: From Equation 2.32, α=
100 ⎛ 100 ⎛ 4⎞ ψ⎞ 1+ ⎟ 1+ ε = 1 + ⎟ 1 + 0.15 = 10.95 ⎜ ⎜ 21 ⎝ 4⎠ 21 ⎝ 4⎠
(
)
(
)
We convert this to αw by multiplying by the molecular weight ratio of fuel and air according to Equation 2.97. That is, α w = 10.95 29 16 = 19.85 . From Equation 2.93
(
)(
)
TAFT =
ΔH c + Tref Cp 1 + α w
(
)
Substituting all this into Equation 2.93 gives
TAFT =
ΔH c Cp 1 + α w
(
)
⎡ Btu ⎤ 22 , 000 ⎢ ⎥ ⎣ lbm ⎦ + Tref = + 60 °F = 4, 280 °F ⎡ Btu ⎤ 0.255 ⎢ ⎥ 1 + 19.85 ⎣ lbm °F ⎦
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(
)
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At 10% excess air α=
100 ⎛ 100 ⎛ 4⎞ ψ⎞ 1+ ⎟ 1+ ε = 1 + ⎟ 1 + 0.10 = 10.48 ⎜ ⎜ 21 ⎝ 4⎠ 21 ⎝ 4⎠
(
)
(
)
and ⎛ 29 ⎞ α w = 10.48 ⎜ ⎟ = 18.99 ⎝ 16 ⎠
(
)
Then the calculation becomes
TAFT =
ΔH c Cp 1 + α w
(
)
⎡ Btu ⎤ 22 , 000 ⎢ ⎥ ⎣ lbm ⎦ + Tref = + 60 °F = 4460 °F ⎡ Btu ⎤ 0.255 ⎢ ⎥ 1 + 18.99 ⎣ lbm °F ⎦
(
)
Thus, the adiabatic flame temperature has risen about 180°F due to the lower excess air.
2.5.6
Adiabatic Flame Temperature with Preheated Air
If the air is preheated, then we may no longer consider the combustion air temperature equal to the reference temperature, though Tf = Tref still applies. Then Equation 2.90 becomes ⎛ αw ⎞ ΔH c TAFT − Tref = ⎜ Ta − Tref + ⎝ 1 + α w ⎟⎠ Cp 1 + α w
(
)
(
)
(2.100)
)
(2.101)
or in dimensionless form, TAFT − Tref ⎛ α w ⎞ ΔH c =⎜ + ⎟ Ta − Tref ⎝ 1 + α w ⎠ Cp 1 + α w Ta − Tref
)(
(
and ⎛ αw ⎞ ΔH c TAFT = ⎜ Ta − Tref + + Tref ⎝ 1 + α w ⎟⎠ Cp 1 + α w
(
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)
(
)
(2.102)
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Note that if Ta = Tref, then the first term on the right vanishes and the equation reduces to Equation 2.93.
Example 2.11 Adiabatic Flame Temperature with Preheated Air Problem statement: Repeat the calculation of Example 2.10 with 400°C air preheat. Solution: This is very high air preheat — about as high as it gets in industry. The calculation of interest likely references a hightemperature reactor such as a hydrogen reformer. In the interest of efficiency, one must recover this energy by heat exchange with some process stream. One such convenient stream is the inlet combustion air. To begin, we will convert the air preheat temperature from the problem statement to degrees Fahrenheit so that we have directly comparable results with the previous example: 400[°C](1.8)[°F/°C] + 32[°F] = 752°F. Then from Equation 2.102 we have ⎛ αw ⎞ ΔH c TAFT = ⎜ Ta − Tref + + Tref ⎝ 1 + α w ⎟⎠ Cp 1 + α w
(
)
(
)
The latter term is the adiabatic flame temperature without air preheat. So the results will differ only by the first term: ⎛ αw ⎞ ⎛ 19.85 ⎞ ⎜⎝ 1 + α ⎟⎠ Ta − Tref = ⎜⎝ 1 + 19.85 ⎟⎠ 752 − 60 °F = 659°F w
(
)
(
)
Adding this product to the previous example’s results gives 4280°F + 660°F = 4940°F at 15% excess air and 4460°F + 660°F = 5120°F at 10% excess air These results ignore dissociation, which would lower the temperatures. Notwithstanding, as the problem shows, air preheat can significantly increase the flame temperature.
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173
Mechanical Energy Balance
Besides thermal energy, combustion equipment makes use of mechanical energy. Mechanical energy for burners primarily includes air and fuel flow under pressure. Conservation of energy allows us to relate pressure and flow.
2.6.1
Work Terms
Since area times distance is volume (Ad = V), pressure times volume (PV) has units of [ML2/θ2]. Therefore, PV work is an energy term. As we have stated, any energy or work term will have dimensions of [ML2/θ2]. For example, the kinetic energy of a moving body or fluid is 1/2 mv2 [ML2/θ2], and the potential energy of a fluid under gravitation is mgh [ML2/θ2], where g is the gravitational constant [L/θ2] and h is the height of the fluid column [L]. We shall consider three kinds of mechanical energy: pressure–volume work ( PV ), gravitational work (mgh), and kinetic work (1 2 mv 2 ), each having units of [ML2/θ2]. If there are no other sources or sinks of energy, then the mechanical energy balance comprises all the energy the system has to perform work. Combustion reactions conserve energy. Therefore, in going from state 1 to state 2, we have P1V +
1 1 mv12 + mgh1 = P2V + mv22 + mgh2 2 2
(2.103)
Since mass [M] appears in all energy terms, each is an extensive variable. To cast them as intensive variables, we normalize by the mass. Specific energy is the term for energy normalized by mass. Dividing by m, the specific energy terms become P1 v12 P v2 + + gh1 = 2 + 2 + gh2 ρ 2 ρ 2 or more succinctly, ΔP Δv 2 + + gΔh = 0 2 ρ
(2.104)
This famous equation, the Bernoulli equation,8 applies to steady flow and neglects nonidealities such as compressible flow or friction. Nonetheless, it is a very useful starting point and it applies to real situations with some modifications, which we will discuss shortly.
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The reader should note carefully the distinction between Δv 2 = v22 − v12 and Δ v = (v2 − v1 )2. The above equation uses the former designation and meaning. The units for Equation 2.104 are not energy units; they are specific energy units [L2/θ2], often referred to as head. For example, 30 ft of head actually denotes 30 ft-lbf/lbm. It is the equivalent potential energy of a 1-lb weight at a height (head) of 30 ft above a reference plane. Notwithstanding, industry convention calls Equation 2.104 the mechanical energy balance, not a head balance, or specific energy balance. 2
2.6.2
Theoretical Mechanical Models
In this section, we derive theoretical models relevant to combustion equipment. We begin from first principles of momentum, mass, and energy conservation. 2.6.2.1 Units of Pressure From Newton’s second law, gravitational force (F) is equal to mass (m) times gravitational acceleration (g): F = mg
(2.105)
The SI unit for force is the Newton (N), N = 1 kg m/sec2. The customary unit is the pound-force (lbf). All force units have dimensions of [ML/θ2]. By definition, pressure (P) is force divided by area (A): P = F/A
(2.106)
The SI unit for pressure is the kilopascal (kPa), kPa = 1000 kg/m sec2; the customary unit is the pound-force per square inch (psi). All pressure units have dimensions of [M/Lθ2]. There are three different references for pressure, e.g., psi absolute (psia), psi gauge (psig), and psi differential (psid). • The absolute pressure is the only pressure with a true zero — it is a ratio variable. The gauge pressure has a relative zero at 1 atm at sea level (e.g., 101 kPa or 14.7 psig). • The gauge pressure is equal to absolute pressure minus atmospheric pressure. Gauge pressures are interval variables; therefore, ratios of psig or kPa(g) do not necessarily have meaning. • The final pressure category is the differential pressure. It is also an interval variable. In the case of differential pressure, the zero is the downstream pressure. Thus, if a fuel jet at 40 psia passes across an orifice such that its downstream pressure is 22 psia, then one may measure 18 psid (40 – 22 = 18).
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Note that so long as the upstream and downstream pressures have the same relative zero, it makes no difference which unit one uses to derive the differential measure. For example, to repeat the previous calculation in gauge units, 25.3 psig – 7.3 psig = 18 psid. 2.6.2.2 Natural Draft Model Equating PV work to kinetic energy determines the draft pressure and maximum draft velocity that a flame or furnace can create. Natural draft is the pressure gradient created by a temperature (i.e., density) difference. In our case, the temperature–density difference is between the system (furnace) and its surroundings (ambient air). Ultimately, draft units are pressure units, no matter how well disguised. The two most common are inches water column (“w.c., in. w.c., i.w.c., or "H2O) and millimeters water column (mm w.c. or mm H2O), referring to the differential pressure required to elevate a column of water so many inches or millimeters. Since we wish to know what velocity or draft pressure is possible due to natural draft, we equate the terms PV = 1 2 mv 2 = mgh . In order to cast the equation in terms of intensive variables, we divide by m: P v2 = = gh ρ 2
(2.107)
Equation 2.107 has units of [L2/θ2]. 2.6.2.3 Draft Pressure in a Furnace We shall solve first for the draft pressure, by multiplying Equation 2.107 by ρ: P=
ρv 2 = ρgh 2
(2.108)
To specify the gradient, we apply the difference operator, Δ 21P = Δ 21ρgh
(2.109)
and obtain P2 − P1 = (ρ2 − ρ1 ) gh . Note that g is invariant over the height of the furnace, and h is implicitly Δh because one always measures height relative to a floor or zero datum. (We shall use state 2 to signify the furnace and state 1 to signify the surroundings. However, when the difference in states is obvious from context, we shall omit the subscripts.) Thus, a negative draft pressure corresponds to a furnace pressure that is lower than the surroundings. By the ideal gas law
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Modeling of Combustion Systems: A Practical Approach ⎛ PM ⎞ ⎛ 1 1 ⎞ − Δρ = ⎜ ⎝ R ⎟⎠ ⎜⎝ T2 T1 ⎟⎠
the final equation becomes ⎛ PM ⎞ ⎛ 1 1 ⎞ − ΔP = ⎜ gh ⎝ R ⎟⎠ ⎜⎝ T2 T1 ⎟⎠
(2.110)
Therefore, if we know the furnace temperature (T1), the ambient temperature (T2), and the height of the furnace (h), we may determine the draft at the floor or any other furnace elevation. Often, heater vendors express draft units in terms of an equivalent column of water at standard temperature and pressure under normal gravitation. There are 406.8 in. w.c. in a standard atmosphere. This unit looks more like a distance rather than a unit of pressure, but Equation 2.109 shows what is unstated, with ρ being the density of water and g the gravitational acceleration. A typical draft pressure is –0.5 in. w.c. It is amazing what this seemingly feeble force per unit area can do.
Example 2.12 Calculation of Draft Pressure Problem statement: What is the maximum draft pressure at the floor of a 10-m furnace operating at 800°C? Presume that the control system adjusts the pressure at the top of the furnace to a draft of 3 mm w.c. via a stack damper. Solution: From Equation 2.110, at 1 atm of pressure (101 kPa) we have ⎛ ⎡ g ⎤⎞ ⎜ 101 ⎡⎣ kPa ⎤⎦ 29 ⎢ ⎥⎟ ⎛ ⎞ ⎡m⎤ 1 1 ⎣ mol ⎦ ⎟ ΔP = ⎜ − ⎜ ⎟ 9.8 ⎢ 2 ⎥ 10 ⎡⎣ m ⎤⎦ 3 ⎜ ⎡ m ⋅ Pa ⎤ ⎟ ⎝ 800 + 273 ⎡⎣ K ⎤⎦ 25 + 273 ⎡⎣ K ⎤⎦ ⎠ ⎣s ⎦ ⎜ 8.314 ⎢ ⎟ ⎥ ⎜⎝ ⎟ ⎣ mol ⋅ K ⎦ ⎠
(
)
(
)
= −54.6 ⎡⎣ Pa ⎤⎦ = −5.6 ⎡⎣ mm w.c. ⎤⎦ The negative sign reminds us that the pressure is below atmospheric. Adding the –3 mm w.c. pressure (draft) we have at the bridgewall gives a total of –8.6 mm w.c. pressure at the floor (or 8.6 mm w.c. draft).
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2.6.2.4 Air Velocity Due to Natural Draft We may solve for the maximum air velocity in terms of the available draft pressure. We are usually interested in the velocity of the air into the furnace; there, the density of the air is approximately that of the surroundings (ρ). With these assumptions, we obtain ΔP = − ρv 2 2 yielding v = 2ΔP ρ. The maximum velocity occurs when there is no resistance to flow (dampers wide open, frictionless flow). In practice, one can realize the theoretical draft pressure because it occurs at the no-flow condition, i.e., without frictional losses. However, one cannot attain the maximum theoretical velocity because flowing air is subject to friction. In order to approximate inlet losses, one usually uses a flow coefficient, Co [ ], with values typically ranging from 0.6 to 0.9 for simple orifices.
v = Co
2ΔP ρ
(2.111)
2.6.2.5 Airflow through a Diffusion Burner Consider Figure 2.27.
FIGURE 2.27 Simplified airflow analysis of burner. A simplified flow analysis shows the many twists and turns that the air must take. Flow around the center riser has been neglected, and only the restriction of the cone is taken into account. Also, we presume that normal operation of the burner is with the damper wide open, so we neglect the flow disturbance across the damper. The velocity at Pt 9 is presumed to be expanded by the furnace temperature to a higher velocity. The momentum of the fuel from the center riser (which will help inspire airflow) is also neglected for this simplified analysis. Also, as the duct is short, we presume that the skin friction (NF ) is negligible.
The continuity equation gives the mass flow in terms of flow area and density:
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Modeling of Combustion Systems: A Practical Approach = ρAv m
(2.112)
is the mass flow [M/θ], A is the area [L2] of the controlling resistance where m (e.g., burner throat or damper), and v is the air velocity [L/θ]. Combining Equations 2.108 and 2.112, we obtain the mass flow relation for a given draft = A 2ρ1Δ 21P. However, this is the maximum possible mass flow. pressure: m The actual mass flow will be less because when gases flow across an orifice, the flow is not perfectly streamlined. Some of the energy goes to sending wakes of gas around in circles (known as eddies). So we augment the equation with a loss coefficient, Cb: = Cb A 2ρ1Δ 21P m
(2.113)
Theoretically, we can determine Cb from the burner geometry alone using loss factors for various geometries, but as the figure shows, the flow path in burners can be quite complicated. Appendix B, Table B.3 lists some examples of flow obstructions. The K factor is a dimensionless parameter that modifies the kinetic energy term to account for friction. We begin with Equation 2.109 and modify it to include two important kinds of resistance: skin friction and form drag. Skin friction is the resistance due to surfaces parallel to the flow path, while form drag is the resistance due to bodies and fittings normal to the flow path. Equation 2.109 becomes ΔP v 2 ⎛ L = ⎜ 1 + NF + ρ 2 ⎝ D
⎞
n
∑ K ⎟⎠ k
(2.114)
k =1
where NF is the frictional factor [ ] and Kk [ ] represents the loss for the kth fitting or obstruction to flow. Rewriting the equation in terms of velocity gives v=
2 ΔP ρ
1 L 1 + NF + D
n
∑K
(2.115)
k
k =1
In Equation 2.115, the first quantity to the right of the equal sign is the burner coefficient for natural draft, Cb. That is, Cb =
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1 L 1 + NF + D
(2.116)
n
∑K k =1
k
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Thus, we may also write Equation 2.115 as
ΔP =
1 ρv 2 Cb2 2
(2.117)
Example 2.13 Calculation of Pressure Loss in a Burner Problem statement: Consider the burner shown in Figure 2.27. Calculate the pressure loss resulting from an airflow through the burner of 3 m/sec. Presume that ambient temperature is 25°C and that the furnace temperature is 800°C. What is the burner airflow coefficient? Solution: Figure 2.27 shows a simplified analysis of the effects. Equation 2.116 becomes
Cb =
1 ⎛ 800 + 273 ⎞ 1 + 0 + 3.5 + 1.0 ⎜ ⎝ 25 + 273 ⎟⎠
= 0.35
Note that we have adjusted the last term for the greater velocity of the expanded gas because the Cb references the ambient temperature and pressure. Note also that Cb is much lower than Co for simple orifices because of the tortured flow path and sudden expansion caused by heating of the air from the furnace. We have neglected skin friction losses through the duct because losses from expansions and flow direction changes dwarf them. Then, from Equation 2.117
ΔP =
1 ρv 2 Cb2 2
The density of air is approximately 1.18 kg/m3 at 25°C. Therefore, 2
ΔP =
1
( 0.35)
2
⎡ kg ⎤ ⎛ ⎡ m ⎤⎞ 1.18 ⎢ 3 ⎥ ⎜ 3 ⎢ ⎥⎟ ⎣ m ⎦ ⎝ ⎣ s ⎦⎠ = 43.3 ⎡⎣ Pa ⎤⎦ = 4.4 ⎡⎣ mm w.c. ⎤⎦ 2
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Modeling of Combustion Systems: A Practical Approach The foregoing calculations provide a good a priori estimate of burner capacity, but accurate airside capacity relations require experimental determination due to a number of factors. First, the maximum resistance to flow occurs in the burner throat, so one should evaluate the density and pressure drop there. A better procedure would be to integrate along the entire flow path. However, the factors vary along the flow path and depend in a complex way on the burner design. Reradiation from the burner tile and the burner internals and momentum effects from the fuel jets affect the mass flow of air into the burner. Therefore, the practical procedure is to evaluate the factors at convenient locations and adjust Cb to match the actual measured flow. Generally, we take A to be the burner throat area (without restriction or baffle), ρ is the air density at the inlet, and ΔP is the natural draft pressure at the furnace floor. A semiempirical correlation is an input–output relation whose model form is theoretically derived, but which has one or more experimentally determined coefficients (adjustable parameters). For example, if we determine Cb experimentally, then Equation 2.117 is a semiempirical equation. Figure 2.28 gives an example of capacity curves based on a semiempirical model. Often, the manufacturer presents the curves on a log–log scale to linearize the relation. This permits easier extrapolation and interpolation. To correct burners to actual conditions, one may make use of the following dimensionless relation:
⎛ Δ P⎞⎛ P ⎞⎛ ρ ref ref ref ⎜ ⎟⎜ ⎟⎜ ⎜⎝ ΔP ⎟⎠ ⎜⎝ P ⎟⎠ ⎜⎝ ρ
⎞ ⎛ T ⎞ ⎛ C ⎞ ⎛ A ⎞ ⎛ 1 + ε ⎞ ⎛ Q ⎞ b , ref ref =1 ⎟⎜ ⎟⎠ ⎝ Tref ⎟⎠ ⎜⎝ Cb ⎟⎠ ⎜⎝ A ⎟⎠ ⎜⎝ 1 + ε ref ⎟⎠ ⎜⎝ Q ref ⎟⎠
(2.118)
where subscript ref refers to the reference conditions required by the capacity curve. For example, suppose the burner capacity is 8 MMBtuh (millions of British thermal units per hour) and one wants to know what will happen to the burner capacity if the ambient air temperature decreases from 100 to 10°F, ceteris paribus. This may reference daytime summer temperatures to nighttime winter temperatures for a particular locality. Then all ratios in Equation 2.118 become unity except for the Q and T ratios. Therefore, Equation 2.118 reduces to ⎛ T ⎞ ⎛ Q ⎞ ⎜ ⎟⎜ ⎟ =1 ⎝ Tref ⎠ ⎝ Qref ⎠
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For model XYZ burner requiring 15% excess air at sea level and 60˚F 10 9 8 7
20 19 18
Burner Throat Size, inches
Heat release, MMBtuh
6 5 4 3
2
17 16 15 14 13 12
1
.1
.2
.3
.4
.5 .6 .7 .8 .9 1
2
3
4
5 6 7 8 9 10
Air side pressure drop FIGURE 2.28 A typical airside capacity (cap) curve. The log-log scale linearizes the square root relation between burner capacity (heat release) and pressure drop. Thus, the capacity curve allows one to easily select the appropriate burner size as a function of the available pressure drop. One must correct the cap curve for altitude, air temperature, or excess air conditions that are different than the stated reference conditions. Due to differences among burner coefficients for various models, the cap curve is specific for sizes within a particular burner family only.
Solving for Q one obtains T Q = Q ref ref T Therefore, the burner capacity rises to 460 + 100 Q = 8 ⎡⎣ MMBtuh ⎤⎦ = 8.7 ⎡⎣ MMBtuh ⎤⎦ 460 + 10 To correct for barometric pressure at elevation, one may use the relation of Equation 2.107 in differential form and integrate it between some reference height (zref) and some elevation z: P
−
∫
Pref
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dP =g ρ
z
∫ dh
zref
182
Modeling of Combustion Systems: A Practical Approach The negative sign is necessary because barometric pressure decreases with elevation. Substituting Equation 2.74 into this leads to RT − M
P
∫
Pref
dP =g P
z
⎛ P ⎞ Wa g ⎟ = − RT z − zref ref ⎠
(
∫ dh → ln ⎜⎝ P
zref
⎛P ⎞ Wg ln ⎜ ref ⎟ = a z − zref ⎝ P ⎠ RT
(
)
)
→
Wa g Pref ( z− zref ) → = e RT P
→
Wa g ⎛ z − zref ⎞ ⎟ 2 ⎠
⎜ Pref RT =e ⎝ P
(2.119)
One can then substitute Equation 2.119 into Equation 2.118 to correct for barometric pressure at another elevation. To use Equation 2.119, one must presume some average temperature for the air column, say 60°F. 2.6.2.6 Airflow through Adjustable Dampers We may model a damper as a variable orifice. Consider the damper shown in Figure 2.29. Here the free damper area (A = Ao – AN) is related to the damper angle normal to the flow direction by the following formula: = m
Cd A 2ρΔP 1 − cosθ
(2.120)
. m = ρAv1 = ρANv2 = ρA(1– cos θ)v2 . Co A m=
1– cos θ
√ 2ρΔP
θ AN =A(cos θ)
Ad
Ao
FIGURE 2.29 Single-blade damper. The free damper area (A) is the total duct area (Ao) minus the normal area projection of the damper (AN).
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where Cd is the damper coefficient, which we presume constant, and A is the free area of the duct with the damper fully open. We reference the damper angle normal to the flow so that 0° corresponds to a closed damper and 90° (π/2) to a fully open one. We may express Equation 2.120 in the following dimensionless form: ⎧⎪ 1 ρv 2 1 m = = ⎨ 2 ΔP 2 NEu ⎩⎪ A 2ρΔP
(
⎫⎪ ⎞ ⎛ 1 ⎬ = Cd ⎜ ⎝ 1 − cos θ ⎟⎠ ⎭⎪
(2.121)
)
Therefore, a plot of 1 2 NEu vs. 1 (1 − cos θ) will yield a line of slope Cd . We refer to 1 (1 − cos θ) as the damper function. It generally does a good job of linearizing airflow through a damper (Figure 2.30). 3500
80°° 70°° 60°° ▲
3000
Air flow, Nm3/h
■
▲
■
2500
50°°
▲
2000
◆
θ
airflow
ΔP=20 =20 mm ▲ water column ■
▲
■
◆
30°°
40°°
ΔP=15 mm w.c. ◆
■
◆
ΔP=10 mm w.c.
◆
▲
1500
■
Damper function 1/(1-cos θ), θ=deg. from vertical
◆
1000 1
2
3
4
5
6
7
8
FIGURE 2.30 Airflow vs. damper function. The damper function does a good job of linearizing airflow across a damper.
2.6.2.7 Unknown Damper Characteristics For a complex damper characteristic having a sigmoid shape, one may use an empirical function to linearize it. The squash or probit function is an empirical equation that does a good job of linearizing the flow characteristic: y = S( x) =
ex 1 + ex
−∞ < x < ∞
0 < S( x) < 1
The inverse of the squash function is the logit function:
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(2.122a)
184
Modeling of Combustion Systems: A Practical Approach ⎛ y ⎞ x = L( y) = ln ⎜ ⎝ 1 − y ⎟⎠
0< y<1
−∞ < L( y) < ∞
(2.122b)
Logit Function
We graph both in Figure 2.31.
1.5
1.0 Probit Function 0.5
0 -0.5
0
0.5
1
1.5
-0.5
FIGURE 2.31 Probit and logit functions. The probit and logit functions are inverses. The probit function squashes the real number line between 0 and 1. The logit function blows up the real number interval between 0 and 1 to ± ∞. These functions are used for a variety of purposes, including neural networks, categorical data analysis, and linearization of sigmoid functions.
2.6.2.8 Fuel Flow as a Function of Pressure Incompressibility refers to the resistance of a fluid to density changes from pressure. Water is one of the most incompressible fluids known. So far, we have treated airflow in natural draft and forced draft burners as incompressible flow. This may strike the reader as odd, especially so in light of the fact that earlier we gave a universal relation between pressure and density for any gas (ideal gas law) as PW = ρRT If pressure affects density, how can we ignore this effect for burner airflow? In fact, for such devices, the pressure is simply too low to cause any significant compression of the air. Even in a forced draft situation, one has no more than 8 in. w.c. air pressure. A standard atmosphere is 406.8 in. w.c. and 414.8/406.8 ≈ 1. On the other hand, fuel gas is usually at much higher pressures.
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When it is at low pressure, such as some waste streams or PSA tail gas, one can use the equations developed earlier for incompressible flow. However, as the pressure approaches a critical pressure corresponding to sonic velocity, we may no longer treat a fluid as incompressible. 2.6.2.9 Compressible Flow Heat capacities have the following definitions and relations according to thermodynamics: ⎛ ∂S ⎞ ⎛ ∂H ⎞ = T⎜ Cp = ⎜ ⎝ ∂T ⎟⎠ P ⎝ ∂T ⎟⎠ P
(2.123)
⎛ ∂S ⎞ ⎛ ∂U ⎞ = T⎜ Cv = ⎜ ⎝ ∂T ⎟⎠ V ⎝ ∂T ⎟⎠ V
(2.124)
In the above cases, the first equality is the thermodynamic definition while the second derives from thermodynamic laws. These derivations are sometimes complex; the heuristic of Figure 2.32 gives the interested reader some memory devices for deriving the common ones. We shall consider some aspects of sonic flow as isentropic (ΔS = 0). If we want to express (∂H ∂T )P in terms of entropy (S), we force the issue by writing ⎛ ∂H ⎞ ⎛ ∂S ⎞ ⎛ ∂H ⎞ ⎜⎝ ∂T ⎟⎠ = ⎜⎝ ∂S ⎟⎠ ⎜⎝ ∂T ⎟⎠ P P P Noting from Figure 2.32 that T = (∂H ∂S)P , we derive Equation 2.123 in one step. By suitable manipulations for isentropic flow of an ideal gas (constant entropy, i.e., adiabatic and frictionless flow), we derive the following equations:8,9 κ
κ
⎛ T ⎞ κ −1 P2 ⎛ ρ1 ⎞ =⎜ ⎟ =⎜ 2⎟ P1 ⎝ ρ2 ⎠ ⎝ T1 ⎠
⎛ κ + 1⎞ rc = ⎜ ⎝ 2 ⎟⎠
⎛ κ ⎞ ⎜⎝ κ −1 ⎟⎠
(2.125)
⎛ ⎜
Cp
⎞ ⎟
⎛ C + Cv ⎞ ⎝ Cp −Cv ⎠ =⎜ p ⎟ ⎝ 2Cv ⎠
(2.126)
where κ = Cp Cv . Equation 2.126 gives the critical pressure ratio, rc, that results in sonic flow.
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Modeling of Combustion Systems: A Practical Approach
⎩
H=U+PV=G+TS G=H–TS=A+PV
U S or reflected across the diagonal
⎩
U=H–PV=A+TS
T
⎩
A
PVT Relations
⎩
T
⎩∂U ⎩∂H ⎩ ∂S ⎩V =⎩ ∂S ⎩P ⎩∂H ⎩∂G V= ⎩∂P ⎩S =⎩∂P⎩T ⎩∂A ⎩∂G –S= ⎩∂T ⎩V =⎩ ∂T⎩P ⎩∂A ⎩∂U –P= ⎩∂V ⎩T =⎩∂V⎩S T=
⎩
A=U–TS=G–PV
P S or rotated
⎩
Definitions of State Functions
S P or rotated
⎩
S H
V
⎩
V
T
⎩
P
A
Maxwell Relations
⎩
T
⎩
dG = VdP–SdT
⎩
U
dH=–VdP–TdS
⎩
G
dU = TdS–PdV
⎩
dA=–SdT–PdV
V
T
⎩∂T ⎩∂P ⎩∂V ⎩S= ⎩ ∂S ⎩V ⎩∂V ⎩ ∂T – ⎩ ∂S ⎩P= –⎩∂P⎩V ⎩ ∂S ⎩∂V ⎩∂P ⎩T= –⎩ ∂T⎩P ⎩∂P ⎩ ∂S ⎩∂T ⎩V= ⎩∂V⎩T –
⎩
Derivatives of State Functions
A
⎩
186
V U S
or reflected across the diagonal
FIGURE 2.32 Thermodynamic relations. All the thermodynamic relations shown may be derived from the single figure at top left. “All vegetables understand sunshine helps plants grow thermodynamically” provides an acronym for the clockwise letter sequence. Use the indicated patterns to derive the relations. If the bolded letter is below the GU line, then the quantity is negative; the definitions of state functions are the exceptions: if the second term is below the first, then the sign that follows will be negative.
Then for air, κ = 7/5 and rc = [(7 + 5) (2·5)](7/2 ) = 1.89. For CH4, a polyatomic gas, κ = 8/6 = 4/3 and rc = [(8 + 6 2 · 6)]8 2 = 1.85. So whenever the fuel pressure nears double atmospheric pressure or greater (>1.85 to 1.89), we will have compressible flow and sonic velocity at the fuel orifice. 2.6.2.10 The Fuel Capacity Curve Revisited Mass is a conserved quantity in combustion reactions as presented earlier: = ρAv m
(2.112)
If one repeatedly integrates with respect to velocity, one obtains a heuristic for deriving the conserved quantities. For example, mass integrated with → mv . respect to velocity gives momentum, also a conserved quantity: ∫ mdv
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In a similar manner, if momentum is integrated with respect to velocity, then → 1 2 mv 2 . Repeated application generates one obtains kinetic energy: ∫ mv 3 4 n 6 , mv the series mv n ! . However, practitioners have not found 24, … mv any physical relevance for these higher homologs. Notwithstanding, the heuristic is convenient for remembering the conserved quantities m, mv, and 1/2 mv2, and therefore for deriving the respective dimensionality for mass, momentum, and energy: [M], [ML/θ], and [ML2/θ2]. When the pressure ratio exceeds the critical pressure ratio, then the exit velocity becomes sonic. No matter what the upstream pressure in excess of the critical pressure, the velocity from a simple orifice cannot exceed sonic velocity. (It is possible to obtain supersonic flow from a converging–diverging nozzle; however, manufacturers do not include them in burner designs because they are expensive to machine and because they create shock waves and loud noise. However, some steam-assisted flares use supersonic nozzles to increase the air entrainment rate.) Despite the limitation of sonic velocity, the mass flow always increases with increasing upstream pressure, even for simple orifices in choked (sonic) flow. From Equation 2.112 with v = c and constant area, the resulting increase in mass flow can only result from compression of the fluid to higher density. The Mach number relates the actual velocity to the sonic velocity: NM =
v c
(2.127)
If NM << 1, then we may treat the fluid as incompressible; otherwise, it is compressible. In incompressible flow, the mass has a square root relation to the pressure via Equation 2.113. If NM << 1 for fuel flow (P/Pc << rc), then the fuel behaves incompressibly. Since the fuel mass flow relates to the heat release by a ∝ v ∝ ΔP or constant (the heating value), then we may write Q ∝ m Q = C ΔP Subsonic, incompressible flow
(2.128)
where C is a constant for subsonic flow. This is the square root portion of the fuel capacity curve of Figure 2.2. The square root dependence on pressure comes from the Bernoulli relation of Equation 2.107 that gave v 2 ∝ ΔP . However, at the critical pressure, the velocity no longer changes, and any mass flow increase comes from a directly proportional change of density with pressure. Therefore, the capacity curve becomes linear beyond the critical pressure: Q = Cc ΔP + C0 Choked flow
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(2.129)
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where Cc is the flow coefficient for the choked flow condition and C0 is some offset in accordance with the equation of a straight line. As there is no discontinuity at the transition, we may write dQ = Cd ΔP = Cc dΔP or dQ C = =C dΔP c 2 Δ c P
(2.130)
Substituting this equation into Equation 2.129 at the critical point and solving for C0 gives C0 =
C ΔcP 2
(2.131)
Substituting Equations 2.130 and 2.131 into Equation 2.129 gives the final relation for choked sonic flow ΔP + Δ c P Q = C 2 ΔcP Combining this relation with the incompressible one gives the fuel capacity curve: ⎧ C ΔP ⎪ Q = ⎪ ⎨ ⎪C ΔP + Δ c P ⎪ 2 ΔP c ⎩
if rc < 1 (2.132) if rc ≥ 1
2.6.2.11 Airflow in Premix Burners A premixed burner will often have two dampers: an air door and a secondary damper. The primary air door is the device that regulates the airflow to the venturi. The secondary air door regulates additional airflow that bypasses the venturi (Figure 2.11). The function of the primary air door is to provide some or all of the combustion air. When the draft is insufficient, one opens secondary air doors to allow additional air in to complete the combustion. This may happen at higher elevations in the heater, where the draft is low, or to increase the capacity of the burner. We may treat the secondary air door as a gate valve modeled as a variable-area orifice: ⎛A ⎞ m = Cd ⎜ o ⎟ 2ρΔP ⎝ Ad ⎠
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The primary damper is more complex because the motive force for the air comes from the draft across the burner and the momentum of the fuel jet. For mass entrainment by a fuel jet, the following equation applies:9 a ⎛ x ⎞⎛ ρ ⎞ m = C⎜ ⎟ ⎜ a ⎟ f m ⎝ d0 ⎠ ⎝ ρ f ⎠
(2.133)
where subscripts a and f refer to air and fuel quantities, respectively, C is a constant (~0.32), x is the distance from the nozzle exit, and d0 is the nozzle diameter. 2.6.2.12 Gas Jets Entraining Flue Gas The form of Equation 2.133 remains the same for a fuel jet entraining flue gas. We merely change the subscript a to subscript g to reference flue gas rather than air. g m ⎛ x ⎞⎛ρ ⎞ = C⎜ ⎟ ⎜ g ⎟ f m ⎝ d0 ⎠ ⎝ ρ f ⎠
(2.134)
References 1. Meyers, R.A., Ed., Handbook of Petroleum Refining Processes, 2nd ed., McGrawHill, New York, 1996, pp. 6.21f. 2. Grayson, M., Ed., Kirk-Othmer Concise Encyclopedia of Chemical Technology, John Wiley & Sons, New York, 1985. 3. API Publication 535, Burners for Fired Heaters, 1st ed., American Petroleum Institute, Washington, DC, July 1995. 4. Rijke, as cited by Rayleigh, J.W.S., The Theory of Sound, Vol. 2, Dover Publications, London, 1945, p. 198. 5. Putnam, A.A., Combustion-Driven Oscillations in Industry, American Elsevier Company, New York, 1971. 6. Dukelow, S.G., The Control of Boilers, 2nd ed., Instrument, Systems, and Automation Society (ISA), Research Triangle Park, NC, 1992, p. 68. 7. Turns, S.R., An Introduction to Combustion, Concepts and Applications, McGrawHill, New York, 1996, pp. 47–51. 8. Hughes, W.F. and Brighton, J.A., Fluid Dynamics, 3rd ed., Schaums Outline Series, McGraw-Hill, New York, 1999, p. 49. 9. Berg, L., Bussman, W., and Henneke, M. Fundamentals of fluid dynamics, in The John Zink Combustion Handbook, Baukal, C.E., Jr., Ed., CRC Press, Boca Raton, FL, 2001, chap. 4.
© 2006 by Taylor & Francis Group, LLC
3 Experimental Design and Analysis
Chapter Overview How do fuel composition, excess oxygen, and furnace temperature fluctuations affect some important response such as process conversion rate or emissions? What factors affect NOx and CO? Are burners in similar process units behaving differently or alike? What is the statistical uncertainty that the proposed model is correct? To score models for these kinds of investigations, we need some understanding of statistics. This chapter begins with some elementary statistics and distributions, and progresses to its seminal tool for separating information from noise — the analysis of variance. Next, the chapter covers factorial designs — the foundation of statistically cognizant experiments. We find that simple rules produce fractional factorials and reduce the number of required experiments. We show that by modifying the alias structure, we can clear factors of certain biases. We then discuss the importance of replication in obtaining an independent estimate of statistical error, and we show how blocking can reduce it further. Further discussion shows how orthogonality eliminates mutual factor biases. The chapter moves on to consider how to mute certain adulterating effects, including hysteresis and lurking factors, and how to validate analytical integrity with residuals plots. For looking at many factors with few experiments, we introduce screening designs such as simplex and highly fractionated designs. The reader then learns how random and fixed effects differ, and how they affect the analysis. To show how one may assess curvature in factor space, a discussion of second-order designs follows. The chapter concludes by considering the sequential assembly of the various experimental designs.
191
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Some Statistics
A statistic is a descriptive measure that summarizes an important property of a collection of data. For example, consider the group of numbers in braces: {1, 10, 100}. Though there are only three data values, we could define an unlimited number of statistics related to them. Here are a few: • The maximum, 100, is a statistic because it summarizes a property of the data, namely, that all data are equal to or below a certain value, 100. • The minimum, 1, is also a statistic, defining the magnitude that all data meet or exceed. • The half range, 49.5, that is, (100 – 1)/2, is a statistic. It is a measure of the dispersion of the data. • The count, 3, tells us the number of data points. If the data were repeated measures of the same quantity differing only by measurement error, the count would relate to a measure of certainty. Intuitively, we would expect that the more replicates we measure, the more certain we become of the true value. • The median, 10, is the middle value of an ordered data set. It measures central tendency. Presuming the data comprise replicate observations, one intuitively expects the true value to be closer to the middle than the extremes of the observations. There are ever more statistics, but let us pause here to answer some interesting questions: • Can we really describe three observations with five or more statistics? Yes. • How can we have five statistics for only three observations? Not all of the statistics are independent. In fact, no more than three can be independent if we are deriving our statistics from these particular data — the number of independent statistics cannot exceed the number of data points. The reason for so many statistics is that we have so many questions that we want to ask of our data; for example: – What limit represents a safe upper bound for a NOx prediction? – How likely are we to exceed this upper limit? – What confidence do we have that our predicted value represents the true value? – – –
How precisely does our model fit the data? Is a particular point within or beyond the pale of the data? How large a margin should we state if we want 99.9% of all future data to be below it?
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For every important question, it seems someone or several have invented one or more statistics. In this chapter, we shall describe important statistics that relate to modeling in general and combustion modeling in particular.
3.1.1
Statistics and Distributions
Suppose we wish to measure a response (y) contaminated by a randomly distributed error term (e). We would like to separate the information (μ) from the noise (e). One option would be to repeatedly measure the response at the same condition and average the results. In summation notation we have ∑ y = ∑ μ + ∑ e, or ∑ y = nμ + ne , where n is the number of replicate measurements. We may divide by the total number of measurements to give
∑ y = nμ + ne n
n
n
But in Chapter 1 we defined this as the arithmetic mean (Equation 1.55). Here we will designate it with an overbar. That is, y=
∑y
(3.1)
n
Now if e (the error vector) were truly random, we would expect the longrun average to be zero. This occurs when n → ∞. We refer to long-run results as expected values and designate them with the expectation operator, E( ). We will refer to the true value for y as μ. Therefore, E(y) = μ and y is an unbiased estimator for μ. Intuitively, we would expect all the y values to distribute about the true value, differing only by e. Since our best estimate of μ from the data is y , then y is a measure of central tendency — the inclination of the average of repeated measures to converge to the true value. The mean is an important statistic — one might dare say the most important statistic — but it is insufficient to characterize certain aspects of some populations. For example, suppose the average height of 100 adult male humans is 1.80 m (5.9 ft). How many will be 0.01 m (<1 in.) tall? How many will be 3.59 m (11.8 ft) tall? We know from experience that there are no adult male humans at either of these extremes. Yet the average of these two numbers is 1.80 m. Therefore, as important as central tendency statistics are, we are also interested in other measures. That is, we would also like some measure of dispersion. Dispersion indicates how values differ from the mean. One statistic that quantifies dispersion is the variance. Let us define the variance (V) of a sample (y) as follows: V ( y) =
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∑( y − y )
2
(3.2)
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Then the following equation gives the mean variance:
∑( y − y ) V ( y) =
2
(3.3)
n
The long run average of the mean variance (V ) is
σ = 2
lim n→ ∞
V ( y) =
∑( y − y )
lim n→ ∞
2
n
(3.4)
However, if we are using the sample mean derived from a finite data set to estimate the variance, V( y) tends to overestimate σ2 unless n is large. The reason for the overestimation is that we have already used the data to determine y . Therefore, y plus n – 1 data points exactly determine the nth data value; i.e., V( y) is not a completely independent measure of dispersion. So the proper denominator to estimate σ2 in Equation 3.3 is n – 1, not n. In other words, we use up (lose) one degree of freedom when we use a finite data set to estimate y . Thus, n – 1 are the degrees of freedom for the estimated variance. We shall use the symbol s2 to denote this quantity:
s
2
∑( y − y ) =
2
(3.5a)
n−1
This is also called the sample-adjusted variance. Obviously, Equation 3.5a and Equation 3.3 become identical as n → ∞. One problem with Equation 3.5a is that the units differ from y because the variance uses the squared value of the response. For this reason, we define the sample standard deviation as
s=
∑( y − y )
2
n−1
(3.6)
It has the same units as the response and it is an unbiased estimator for the true standard deviation, σ. Now s will tell us something about the dispersion of possible values about the true mean. To find out what, we need to know something about how values distribute themselves in the long run.
3.1.2
χ2), F, and t Distributions The Normal, Chi-Squared (χ
To develop the idea of distributions further, let us consider Figure 3.1.1
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1
1 1 3
1 5
1 1
Ball
6 7
1
1
8
15
4
20
1 5
10
35
56
1
6 10
21
28
3
4
1
Peg
1
2
15 35
70
1 6
21
56
Bin 1
7
28
1
8
1
FIGURE 3.1 The Galton board. The Galton board comprises a vertical arrangement of pegs such that a ball may take one of two possible paths at each peg, finally arriving at a bin below. The numbers between the pegs show the number of paths leading through each space. The numbers follow Pascal’s triangle (superimposed numbers). The total number of paths for this Galton board sums to 256. Thus, for the ball shown arriving at the bin, the probability is 56/256 = 21.9%. One of 56 possible paths leading to that bin is shown (dotted line). The distribution approaches the normal probability distribution as the number of rows in the Galton board increases.
Galton’s board looks a bit like a pinball machine. It comprises a vertical slate with pegs arranged in a triangle pattern that widens from top to bottom. A ball dropped onto the topmost peg may fall either to the left or to the right, whereupon it strikes the next peg and again may fall to either the left or the right. The ball continues in this fashion until it ultimately drops into one of the bins below. What is the probability that a dropped ball will fill any particular bin? To answer the question, we begin by calculating the distribution of the possibilities. 3.1.2.1 The Normal Distribution Most of us are familiar with the normal distribution — the so-called bellshaped curve — perhaps it is more nearly cymbal shaped. At any rate, Equation 3.7 gives the mathematical representation:
N ( y) =
1 2πσ 2
e
−
1⎛ y − μ⎞ 2 ⎝⎜ σ ⎠⎟
2
(3.7)
Here, N(y) is the frequency of y and e is Euler’s constant, e = 2.71828.… The probability of finding y between two limits, – ∞ < a and b < + ∞, is given by b
P( y ) =
⌠ ⎮ ⎮ ⎮ ⌡ a
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1⎛ y − μ⎞ σ ⎟⎠
1 − 2 ⎜⎝ e 2π
2
⎛ y−μ⎞ d⎜ ; 0 < P( y ) < 1 ⎝ σ ⎟⎠
(3.8)
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The statistics μ and σ2 completely characterize the normal distribution. One may standardize the normal distribution using the coding z = ( y − μ) σ.
N ( z) =
1 2π
e
−
z2 2
(3.9)
Figure 3.2 depicts the above equation.
FIGURE 3.2 The normal distribution. The so-called bell-shaped curve has a maximum at z = 0, y = 1 and points of inflection at z = ±1, y = 1/ 2 πe . The area under the curve sums to unity.
The normal distribution has the following properties: • It is symmetrical about its center at z = 0. • It has an inflection point where the curve changes from concave down to concave up (at z = ±1). • The area under the curve sums to unity. 3.1.2.2
Probability Distribution for Galton’s Board
Galton’s board is a very good simulator of random error even though Newtonian physics dictate the ball’s motion. Yet, we have no way of predicting what bin the ball will fall into on any given trial because very small variations affect the path of the ball. Such variations include: • The elasticity and roundness of the ball’s surface • The elasticity, angle, and roundness of each peg • The mutual interactions among balls At each peg, the distribution is a binary one: the ball will fall either to the left or to the right. In no way can we consider this a normal distribution. It is an equiprobable binary distribution. Notwithstanding, statistical considerations allow us to do the following: • Calculate the ultimate distribution of the balls into the slots • Calculate the probability of any given ball falling into a particular slot • Show that the ultimate distribution is a normal probability distribution
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To derive the probability distribution for Galton’s board, we proceed as follows. First, we count the total number of paths through each space. At the first peg, we have one path to the left and one path to the right. So the possible paths from left to right are distributed as {1, 1}. At the second row of pegs, we may take one path to the left and fall outside the far left peg. But if the ball jumps left and then right, it will fall between the two pegs on the second row. Likewise, if the ball falls to the right of the first peg and to the left of the second peg, it will also fall between the two pegs of the second row; therefore, there are two paths leading between the two pegs of the second row. Finally, if the ball takes a right jump at the first peg and then a right jump at the second peg, it will fall to the right of the right peg. Therefore, the number of paths from left to right at this level is {1, 2, 1}. Now the total number of paths between any two pegs will be the sum of the paths communicating with it. For Galton’s board there are two such paths overhead and to the left and right. Thus, the distribution of paths for the next row of pegs is {1, 3, 3, 1}. We may continue in this fashion all the way down the board. 3.1.2.3 Pascal’s Triangle We know the pattern {1}, {1 1}, {1 2 1}, {1 3 3 1} … as Pascal’s triangle (Figure 3.3). Pascal’s triangle is a numerical triangle having outside edges of 1. The 1 1 1 1 1 1 1 1
6 7
2
4
21
3
10
10
16
1 5
15 35
8
1 4
20 35
4
1
6
15
2
1
3
5
1
6 21
32
1
64
1 7
1
128
256 1 8 28 56 70 56 28 8 1 ⠄ ⠄ ⠄ ⠄ ⠄ ⠄ ⠄ ⠄ ⠄ ⠄ ⠄ ⠄ ⠄ ⠄ ⠄ ⠄ ⠄⠄ ⠄⠄ ⠄ ⠄⠄ ⠄ ⠄⠄ ⠄ ⠄⠄ ⠄ ⠄⠄ ⠄ ⠄⠄ ⠄ ⠄⠄ ⠄ ⠄⠄ ⠄ ⠄⠄ ⠄ ⠄ FIGURE 3.3 Pascal’s triangle. Each term is calculated by adding the two terms above it. In some versions, the second row of ones (1 1) is omitted, but we include it here for consistency. Horizontal rows (f) are numbered starting with zero at top and incrementing by 1. Entries (k) in each row are numbered starting from 0 at left and incrementing by 1. Thus, the coordinates (f, k) = (4,2) correspond to the value 6. f also indicates the number of factors in a factorial design discussed presently. The sum of any horizontal row equals the number of terms in the saturated factorial (2f ). k indicates the overall order of the term. One may calculate the value of entry k in row f directly using the formula f !/[k! (f – k)!], e.g., 4!/[2!(4 – 2)!] = 6.
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sum of the two numbers immediately above forms each lower entry. The sum of the numbers in a horizontal row is always n = 2f
(3.10)
where f is the number of the row starting from top down; all counting for rows and entries begins with zero, i.e., 0, 1, 2, … . Equation 3.11 gives the kth entry in row f directly: m=
f! k! f − k !
(
)
(3.11)
where m is the number contained in the kth entry of the f th row. For reference, we have superimposed Pascal’s triangle onto Galton’s board in Figure 3.1. Each number represents the number of possible paths traversing through the interstice. As shown, the board has eight rows of pegs. At the bottom, we have nine slots and the distribution of paths is {1, 8, 28, 56, 70, 56, 28, 8, 1}. The total number of paths is 1 + 8 + 28 + 56 + 70 + 56 + 28 + 8 + 1 = 256 = 28. So, the probabilities for a ball falling into any given slot from left to right are 1/256, 8/256, 28/256, 56/256, 70/256, 28/256, 8/256, and 1/256, whose fractions sum to 1. This is a binomial frequency distribution. We may find this directly by the ratio of Equations 3.10 and 3.11: B( f , k ) =
1 f! f 2 k !( f − k ) !
(3.12)
where B(f, k) is the probability of the ball finding its way to the kth interstice (counting from zero) under the f th peg. For reference, we have superimposed a bar graph in Figure 3.1 for each bin. The bar is proportional to the probability of a ball finding its way to that particular bin. The distribution approaches the normal probability distribution as f → ∞. But even after several rows, the resemblance to Equation 3.7 is unmistakable. In fact, ⎤ lim ⎡ 1 f! ⎢ ⎥ = N (μ , σ ) = f → ∞ ⎢⎣ 2 f k !( f − k ) ! ⎥⎦
1 ⎛ y−μ ⎞ σ ⎠⎟
− ⎜ 1 e 2⎝ 2πσ
2
(3.13)
where μ = f 2 and σ = f 4 and y = k. This is part of an even broader concept known as the central limit theorem: as the number of independent and identically distributed random variables increases, the aggregate distribution approaches the normal probability distribution. With the following substitution, ⎛ y−μ⎞ z=⎜ ⎝ σ ⎟⎠
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(3.14)
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Letting μ = f 2 and σ = 1, Equation 3.13 reduces to Equation 3.9:
N ( z) =
1 2π
e
−
z2 2
(3.9)
We call Equation 3.9 the probability density function. Then the cumulative probability function for –a < z < a is
P ⎡⎣ N ( z) ⎤⎦ =
1 2π
a
⌠ ⎮ ⌡
e
−
z2 2
(3.15)
dz
−a
We call the variable, z, the standard unit variate, and when the limits of the integration are from – ∞ to + ∞, the integral attains unity. Equation 3.15 implies a two-tail test because we are asking for the probability for z being between –a and a. If we were only interested in P[N(z)] being greater than –a or less than a, we would only be interested in one tail of the distribution. Since the distribution is symmetrical, the probability of the one-tailed test is exactly half that of the two-tailed test. Most computer spreadsheets have functions to calculate this. Excel™ has several related functions. The function normdist(x,m,s,TRUE) evaluates Equation 3.8, where x is a particular value, m the mean, and s the standard deviation. The function normdist(z,m,s,FALSE) evaluates 1 – normdist(z,m,s,TRUE). The function normsdist(z)— note the s in the middle of this function name — evaluates Equation 3.15. For example,
normsdist(1.96) =
1 2π
1.96
⌠ ⎮ ⌡
e
−
z2 2
dz ≈ 0.975
−1.96
Many statistical tests are strictly valid only for normally distributed errors. However, according to the central limit theorem, even if the parent distribution is not normally distributed, the accumulation of several levels of randomly distributed deviations will tend toward a normal distribution. This was the case with Galton’s board. The parent distribution was binary (highly nonnormal), yet the final distribution approached the normal distribution quite closely. So we expect estimates of the mean to distribute around the arithmetic mean, approaching the normal distribution. In fact, according to Equation 3.13, μ and σ completely determine the shape of the normal probability curve. Thus, from these two statistics alone, we can derive any other property or statistic for normally distributed data. If data do not distribute normally, often one can transform them to such. For example, emissions data such as CO and NOx are always nonnegative, and therefore they do not
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distribute normally. However, the logarithms of these quantities are normally distributed (e.g., ln(NOx)). As we shall see, this will permit us to estimate: • How likely we are to exceed some upper limit • What limit represents a safe upper bound for a prediction • What confidence we have that our predicted value represents the true value 3.1.2.4 The Chi-Squared Distribution Another important distribution is the distribution of variance. The variation will never be negative because variance is a squared quantity. Thus, variance cannot distribute normally. In fact, it distributes as a chi-squared distribution. Knowing something about this distribution will allow us to develop a list of additional statistics related to the model, such as: • The goodness of fit • The confidence that a particular factor belongs in the model • The probability that we can accurately predict future values The chi-squared distribution has the following form:
χ ( n, z ) = 2
z
n− 2 z − 2 2
e ⎛ n⎞ 2 Γ⎜ ⎟ ⎝ 2⎠ n 2
(3.16)
where n is the degrees of freedom (an integer); z is the standard variate, defined in Equation 3.14; and Γ(n) is the gamma function, defined as
()
Γ n =
∞
⌠ ⎮ t n−1e − t dt ⎮ ⌡ 0
Excel has several related functions. The function gammaln(z) will return the natural log of the gamma function for positive arguments of z. To obtain the gamma function itself, one uses exp(gammaln(z)). Equation 3.17 gives the cumulative probability function for the chi-squared distribution: a
P ⎡⎣ χ ( z, n) ⎤⎦ = 2
⌠ ⎮ ⎮ ⎮ ⎮ ⎮ ⎮ ⌡ 0
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z
n− 2 z − 2 2
e dz ⎛ n⎞ 2 Γ⎜ ⎟ ⎝ 2⎠ n 2
(3.17)
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One may use the Excel function CHIDIST(z,n) to calculate this. The gamma function has some interesting properties. One way to think of it is as a generalization of the discrete factorial function. That is,
() (
)
Γ n = n−1 !
(3.18)
where n is an integer. For example, Γ(4) = (4 – 1)! = 3! = (3)(2)(1) = 6. Other properties include
(
⎛ 2n + 1 ⎞ = π Γ⎜ ⎝ 2 ⎟⎠
)
()
Γ n + 1 = nΓ n
(3.19)
⎛ 1⎞ Γ⎜ ⎟ = π ⎝ 2⎠
(3.20)
n
∏ k =1
(
) ⎤⎥
⎡ 1 ⋅ 3 ⋅ 5$ 2 n − 1 2k + 1 = π⎢ n 2 2n ⎢⎣
(3.21)
⎥⎦
For example, ⎛ 7 ⎞ 1⋅ 3 ⋅ 5 15 π= π Γ⎜ ⎟ = 3 8 ⎝ 2⎠ 2 ⎛ 1 − 2n ⎞ = −1 Γ⎜ ⎝ 2 ⎟⎠
( )
n
n
π
∏ k =1
( )
2n = −1 2k − 1
n
⎡ 2n π⎢ ⎢⎣ 1 ⋅ 3 ⋅ 5$ 2 n − 1
(
)
⎤ ⎥ ⎥⎦
(3.22)
For example, 4 ⎛ 7⎞ 24 16 Γ ⎜ − ⎟ = −1 π= π 1⋅ 3 ⋅ 5 ⋅7 105 ⎝ 2⎠
( )
3.1.2.5 The F Distribution Often, in deciding the significance of certain terms in our mathematical models, we will examine the ratio of two variances. The ratio of variances cannot distribute normally because it comprises a ratio of two χ2 distributions and the result must always be nonnegative. Equation 3.23 gives the F distribution:
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m
(
)
⎛m
⎞
⎛ n + m ⎞ ⎛ m ⎞ 2 ⎜⎝ 2 −1⎟⎠ z Γ⎜ ⎝ 2 ⎟⎠ ⎜⎝ n ⎟⎠
F m , n, z =
⎛ m⎞ ⎛ n⎞ ⎛ mz ⎞ Γ⎜ ⎟ Γ⎜ ⎟ ⎜1+ n ⎟⎠ ⎝ 2 ⎠ ⎝ 2⎠ ⎝
(3.23)
m+ n 2
The related cumulative probability function is a
(
)
P ⎡⎣ F m, n, z ⎤⎦ =
⌠ ⎮ ⎮ ⎮ ⎮ ⎮ ⎮ ⎮ ⎮ ⎮ ⌡
m
⎛m
⎞
⎛ n + m ⎞ ⎛ m ⎞ 2 ⎜⎝ 2 −1⎟⎠ z Γ⎜ ⎝ 2 ⎟⎠ ⎜⎝ n ⎟⎠ ⎛ m⎞ ⎛ n⎞ ⎛ mz ⎞ Γ⎜ ⎟ Γ⎜ ⎟ ⎜1+ n ⎟⎠ ⎝ 2 ⎠ ⎝ 2⎠ ⎝
m+ n 2
dz
(3.24)
0
and is given by the Excel function FDIST(m,n,z). 3.1.2.6 The t Distribution The F(1, n) distribution is a special distribution called the t distribution.* It accounts for deviation from the normal distribution owing to less than an infinite number of independent trials.
( )
t n, z =
⎛ n + 1⎞ Γ⎜ ⎝ 2 ⎟⎠ ⎛ n⎞ ⎛ z2 ⎞ nπ Γ ⎜ ⎟ ⎜ 1 + ⎟ 2⎠ ⎝ 2⎠ ⎝
n+ 1 2
(3.25)
The t distribution approaches the normal distribution as n→∞. In general, the t distribution has a flatter peak and broader tails than the normal distribution. The t distribution adjusts the normal probability function for the uncertainty of having less than an infinite number of samples. For n > 20, the t and normal distributions are practically identical. The associated cumulative probability function is
* W.S. Gosset writing under the pen name “Student” while working for the Guiness Brewing Company derived the t distribution that bears his pen name — Student’s t distribution. Details are available in any statistics text. See, for example, Mendenhall, W., Scheaffer, R.L., and Wackerly, D.D., Mathematical Statistics with Applications, 3rd ed., PWS Publishers, Boston, 1986, p. 273.
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a
( )
P ⎡⎣ a, t n, z ⎤⎦ =
⌠ ⎮ ⎮ ⎮ ⎮ ⎮ ⎮ ⎮ ⎮ ⌡
−a
⎛ n + 1⎞ Γ⎜ ⎝ 2 ⎟⎠ ⎛ n⎞ ⎛ z ⎞ nπ Γ ⎜ ⎟ ⎜ 1 + ⎟ 2⎠ ⎝ 2⎠ ⎝ 2
n+ 1 2
dz
(3.26)
The Excel function TDIST(z,n,1) gives the single-tailed function. TDIST(z,n,2) gives the two-tailed test of Equation 3.26.
3.2
The Analysis of Variance (ANOVA)
The F distribution allows us to estimate probabilities for ratios of variances. We use it in an important technique known as the analysis of variance (ANOVA). ANOVA is one of the most important concepts in statistical experimental design (SED). It is based on an amazing identity:
∑( y − y )
2
≡ ≡
SST
∑ ( yˆ − y )
2
SSM
+ +
∑ ( y − yˆ )
2
(3.27)
SSR
where SST stands for sum of squares, total; SSM is the sum of squares, model; and, SSR is the sum of squares, residual term. (The reader should note that although the statistical literature uses these abbreviations often, some texts use slightly different acronyms.)* The residual error includes whatever variation the model does not explain. SSM and SSR are the estimators for the model and the residual variance, respectively. The above identity seems to break the rules of algebra, by ignoring the cross product. But in fact, the cross product vanishes for least squares solutions. This is easy to show:
∑ ( y − y ) = ∑ ⎡⎣( y − yˆ ) + ( yˆ − y )⎤⎦ 2
=
2
∑ ( y − yˆ ) + 2∑ ( y − yˆ )( yˆ − y ) + ∑ ( y − y ) 2
2
* For example, Montgomery uses SST (mean sum of squares, treatments) in lieu of our SSM and he uses SSTO (sum of squares, total) in lieu of our SSM: Montgomery, D.C., Design and Analysis of Experiments, 5th ed., John Wiley & Sons, New York, 2001, pp. 531–535. However, other texts are consistent with our nomenclature here. See Box and Draper2 for example.
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Now, noting that
∑ ( y − yˆ )( yˆ − y ) = e ( Xa − y ) T
we have
∑ ( y − yˆ )( yˆ − y ) = e Xa − e y T
T
But both terms are identically 0 for least squares (see Equations 1.86 and 1.87). Therefore, the cross product is also identically zero. We may construct a table making use of the identity of Equation 3.27. It will have the slots shown in Table 3.1. TABLE 3.1 The Naked ANOVA Term
SS
DF
MS
F1–P(DFM, DFR)
P(F,DFM,DFR)
M R
SSM SSR
DFM DFR
MSM MSR
MSM/MSR
P
T
SST
DFT
The column headers are M = model, R = residual, and T = total. The row headers are SS (sum of squares), DF (degrees of freedom), F (F ratio). The entries, defined by appending the row header to the column header, are SSM, SSR, SST, DFM (degrees of freedom, model), DFR (degrees of freedom, residual), and DFT (degrees of freedom, total). Those are the slots, now for the filler. Consider the following model: yˆ = a0 + a1x
(3.28)
We desire to know if a1 is significant. In other words, does x really influence the response or not? If not, then the experiments amount to repeated measures of y, which differ only by experimental error yˆ = y ; this would be equivalent to a0 = y and a1 = 0 for Equation 3.28. We shall call yˆ = y the null hypothesis. For the null hypothesis, a0 is the only model parameter. Then the total degrees of freedom becomes n – 1, where n is the total number of observations. Therefore, Equation 3.28 reduces to Equation 3.5b and is equivalent to the total sum of squares divided by the total degrees of freedom:
SST = s2 = DFT
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∑( y − y ) n−1
2
(3.5b)
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The alternative hypothesis is that |a1| > 0. If |a1| > 0, then Equation 3.5b is not the correct measure of random variance because the total variance includes a nonrandom contribution from the model. If the alternative hypothesis is true, the appropriate estimation of experimental error will be the variance left over after subtracting the model from the data. In other words, SSR s = MSR = = DFR 2
∑ ( y − yˆ )
2
(3.5c)
n− p
Here, p is the total number of model parameters. For the current case, p = 2, i.e., a0 and a1. Note that SSR = SST – SSM and DFR = DFT – DFM. SSR is the sum of squared residual error. The mean squared residual (MSR) gives an estimate of the error variance. The residual is what remains after subtracting the portion of the total variance that belongs to the model, SSM. Equation 3.29 gives the mean squared model variance: SSM MSM = = DFM
∑ ( yˆ − y )
2
(3.29)
p−1
This accounting is easy to remember. The model variance, SSM, is the variance over and above the mean; the residual variance (SSR) is the total variance (SST) minus the model variance (SSR = SST – SSM). If we add these two contributions (SSM + SSR), we obtain the total variance (SST = SSM + SSR) — the variance of the actual data over and above the mean. Perhaps vector notation is more straightforward and easier to remember: SSM = yˆ T yˆ − y T y (model – mean)
(3.30)
SSR = y T y − yˆ T yˆ (actual – model)
(3.31)
SST = y T y − y T y (actual – mean)
(3.32) n
These relations are easy to prove. Note that ( y − yˆ )T ( y − yˆ ) = ∑ ( yk − yˆ )2 . Expanding, this yields ( y − yˆ )T ( y − yˆ ) = y T y − y T yˆ − yˆ T y + yˆ T yˆ , but k for least squares, y T yˆ = yˆ T yˆ (see Equation 1.96), so ( y − yˆ )T ( y − yˆ ) = y T y − yˆ T y . One may also substitute yˆ = Xa giving yˆ T yˆ = a TX TXa if desired. Expanding SST gives: n
SST =
∑( y − y ) = ∑ y 2
k
k
=
∑y
2
2
− 2y
∑y+∑y = ∑y
− ny 2 = y T y − y T y
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2
2
− 2 ny 2 + ny 2
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Therefore, n
SST =
∑( y − y ) = ∑ y 2
k
2
− ny 2 = y T y − y T y
k
Finally, SSM =
∑ ( yˆ − y )
2
which we may expand as
∑ ( yˆ − y ) = ∑ yˆ − 2 y∑ yˆ + ∑ y 2
2
But note that
∑ yˆ = ∑ y = ∑ y because
∑ y = ∑ yˆ + e = ∑ yˆ + ∑ e = ∑ yˆ + 0 Thus,
∑ ( yˆ − y ) = ∑ yˆ 2
2
− ny 2 = yˆ T yˆ − y T y = a TX TXa − y T y
We may use either the summation or matrix equations to determine SSM, SSR, and SST.
3.2.1
Use of the F Distribution
The proper measure of random error depends on knowing whether or not the null hypothesis is true. But how can we know? Recall that a ratio of variances distributes as an F distribution. Now if MSM/MSR ~ 1, then the null hypothesis is true, and if MSM/MSR >> 1, the null hypothesis is false, leaving the alternative hypothesis. But what if the ratio is 1.5? Should we accept the null hypothesis then? A ratio greater than 1 could occur merely by chance. Now we are 100% confident that if our ratio were infinitely large, then MSM would be significant. If we were willing to be less confident, say
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95% confident, then our theoretical ratio would not need to be as large as infinity. A normal distribution needed μ and σ to determine its shape. An F distribution needs three things: the degrees of freedom in the numerator of the variance ratio, the degrees of freedom in the denominator of the variance ratio, and the confidence with which we desire to say that the ratio differs from chance. Here is the general procedure. Once we have the sum-of-squares relations, we calculate the mean sum of squares by the degrees of freedom. Thus, MSM = SSM/DFM and MSR = SSR/DFR. The mean squares are variances, and they distribute as chisquared variables. Therefore, the ratio of mean squares distributes as an F distribution. Again, to determine an F distribution we need three things. 1. The degrees of freedom used to determine MSM (i.e., DFM) 2. The degrees of freedom used to determine MSR (i.e., DFR) 3. Decide the probability (P) we will use in judging that MSM/MSR ≠ 1; that is, the ratio differs from 1 by more than chance alone. We shall call C = 1 – P the confidence. Thus, if P ≤ 0.05, then we have C ≥ 95% that MSM/MSR > F. (Most texts use a lowercase p to indicate probability. However, this text uses a lowercase p to indicate the number of model parameters. Therefore, we use an uppercase P so as not to confuse the two.) Step 1: Specify P. We must do this before beginning any analysis. We will certainly want P to be less than 0.10, indicating that we have 90% confidence or more (1 – 0.10 = 0.90) that our correlation is not due to chance. The typical test is P ≤ 0.05 (95% confidence), denoted F95 (m, n), where the subscript indicates the percent confidence, and m and n are the DFM and DFR, respectively. Step 2: Determine MSM/MSR, m, and n. Step 3: Compare MSM/MSR with the minimum F ratio. If MSM/ MSR > FC(m, n), then reject the null hypothesis and accept the alternative hypothesis — that the model is significant at the C × 100% confidence level. Appendix E, Table E.4 gives the minimum ratio that this is so, depending on C (or alternatively P), m, and n. For example, suppose we want to be sure that a term belongs in the model with 95% confidence. Let us say that m = DFM = 3 and n = DFR = 2 and C = 95% (P = 0.05). Then according to the table, FC (m, n) = 19.2. (We may also find this from the Excel function FINV(0.05,3,2)=19.2.) If MSM/MSR F95(3, 2) = 19.2, then we shall reject the null hypothesis and conclude that MSM is a significant effect. Statistical programs will give the P value directly and obviate the need for the table. Spreadsheets will also do the same. In Excel, the command FDIST(F,m,n) gives the P value for F1–P. Let us consider an example.
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Example 3.1
ANOVA for a Single-Factor Investigation
Problem statement: Derive the ANOVA for the following hypothetical data of Table 3.2. TABLE 3.2 A Single Factor Example x
y
–3 –2 –1 0 1 2 3
–19.5 –17.9 –12.7 –6.7 0.5 2.4 10.2
Compare the MSM/MSR ratio with that of an F95 distribution having the same degrees of freedom and determine whether the null hypothesis is valid at the given confidence level. Use a spreadsheet to assess the value of P. Solution: We solve for the model y = a0 + a1x using least squares and generate ⎛ −21.6 ⎞ ⎜ −17.9 ⎟ ⎜ ⎟ ⎜ −12.7 ⎟ ⎛ a0 ⎞ ⎛ −6.24⎞ ⎜ ⎟ ⎜⎝ a ⎟⎠ = ⎜⎝ 5.10⎟⎠ and yˆ = ⎜ −6.7 ⎟ 1 ⎜ 0.5 ⎟ ⎜ ⎟ ⎜ 2.4 ⎟ ⎜⎝ 1.2⎠⎟⎠ From these and Equations 3.30 to 3.32, we determine the following entries to the ANOVA table: SSM = 729.30, SSR = 14.78, and SST = 744.08. Now DFM = 1 (a1), DFT = 6 (7 – 1 DF for the mean), leaving DFR = 5 (6 – 1). This gives the following mean squares: MSM = 729.30/1 = 729.30, MSR = 14.78/5 = 2.96. Finally, we have F = MSM/MSR = 729.30/2.96 = 246.77. This gives Table 3.3. Now the ratio for F95(1, 5) = 246.77 according to Appendix E, Table E.4. Therefore, we conclude that the effect is significant. Using the Excel function we have FDIST(246.77,1,5)= 1.9E-05, or P < 0.0001. At any rate, the model y = a0 + a1x is statistically significant and differs from the alternative (null) hypothesis that yˆ = y .
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TABLE 3.3 ANOVA for Example 3.1 Term
3.3
SS
DF
MS
F(1, 5)
P
Model, M Residual, R
729.30 14.78
1 5
729.30 2.96
246.77
<0.0001
Total, T
744.08
6
Two-Level Factorial Designs
A two-level factorial is an experimental design that investigates each factor at two levels: high and low. Factorial designs generate orthogonal matrices and allow us to assess separately each effect in a partitioned ANOVA. To begin, let us consider the effect of three factors on NOx: excess oxygen, O2 (x1); air preheat temperature, APH (x2); and furnace bridgewall temperature, BWT (x3). To investigate every possible combination of high and low factors requires a minimum of 2f points. For example, consider Table 3.4. TABLE 3.4 A Factorial Design in Three Factors Point 1 2 3 4 5 6 7 8
x1
x2
x3
y
– – – – + + + +
– – + + – – + +
– + – + – + – +
1.81 2.67 2.47 3.34 3.92 5.58 4.59 6.25
Table 3.4 gives all possible combinations of high and low factor levels along with the response ln(NOx). We use + and – to signify high and low levels in the table. Actually, we can let + and – refer to +1 and –1, respectively, by coding all values as follows: xk =
ξ k − ξk ξk
(3.33)
where k is an index from 1 to p, xk is the kth-coded factor, ξ k is the kth factor in the original metric, ξ k is the average value defined by Equation 3.34, and ξ k is the half range, defined by Equation 3.35:
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(3.34)
ξ + − ξ −k ξˆ k = k 2
(3.35)
ξk =
where ξ +k is the high level of the kth factor in the original metric and ξ −k is the low value. Equivalently, we may combine Equations 3.33 through 3.35 in a single equation: xk =
(ξ
k
) (
− ξ +k + ξ k − ξ −k
(ξ
+ k
−ξ
− k
)
)
(3.36)
They are linear transforms that make xk dimensionless, with zero mean and unit half range in either direction. They give convenient numerical properties, help minimize round-off error in later matrix calculations, and establish a uniform dimensionless metric. One may easily invert them: ξ k = ξ k + ξˆ k xk
(3.37)
These are the typical coding relations for experimental design. However, two other single-factor linear transforms may be useful in some situations: 0/1 coding and deviation-normalized coding: xk ,0/1 =
ξk − ξ− (0/1 coding) 2 ξˆ
(3.38)
k
ξ k = 2 ξˆ k xk ,0/1 + ξ − (inverse for 0/1 coding)
xk ,s
(ξ =
k
− ξk sk
) , where s
k
=
∑(ξ
k
− ξk
n−1
)
(3.39)
2
(deviation-normalized coding) (3.40)
ξ k = sk xk ,s + ξ k (inverse for deviation-normalized coding)
(3.41)
It follows that the linear transforms have the following relations: ξ k = sk xk ,s + ξ k = 2 ξˆ k xk ,0/1 + ξ − = ξ k + ξ k xk
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(3.42)
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One may use Equation 3.42 to convert among transforms. For factorial designs, we will generally use ±1 coding. For the regression of historical data sets from unplanned experiments, deviation-normalized coding may be preferable. 0/1 coding will not generate orthogonal matrices, but sometimes the investigator desires a model where the low level corresponds to zero. For example, in classical experimentation, 0/1 coding better highlights that we are exploring along a single-factor axis at a time. Also, 0/1 coding has some advantage for representing a categorical factor with multiple levels (see Chapter 4).
Example 3.2
Factor Coding
Problem statement: If the factors have the following ranges, give the equations to code their values to ±1: oxygen, 1% to 5%; air preheat temperature, 25 to 325°C; and furnace temperature, 800 to 1100°C. Give the inverse relations also. Solution: From Equations 3.33 through 3.35 we have x1 =
ξ1 − 3 ⎡⎣% ⎤⎦ 2 ⎡⎣% ⎤⎦
, x2 =
ξ 2 − 175 ⎡⎣ o C ⎤⎦ 150 ⎡⎣ o C ⎤⎦
, x3 =
ξ 3 − 950 ⎡⎣ o C ⎤⎦ 150 ⎡⎣ o C ⎤⎦
and the inverse relations are ξ1 = 3 ⎡⎣% ⎤⎦ + 2 ⎡⎣% ⎤⎦ x1 , ξ 2 = 175 ⎡⎣ o C ⎤⎦ + 150 ⎡⎣ o C ⎤⎦ x2 , ξ 3 = 950 ⎡⎣ o C ⎤⎦ + 150 ⎡⎣ o C ⎤⎦ x3
3.3.1
ANOVA for Several Model Effects
If X is orthogonal, we can assess each effect separately in the ANOVA table. In Table 3.4, the model we are attempting to fit is y = a0 + a1x1 + a2 x2 + a3 x3 and it has the following normal matrix equation: ⎛ 30.63⎞ ⎛ 8 ⎜ 10.05 ⎟ ⎜ ⎜ ⎟ =⎜ ⎜ 2.67 ⎟ ⎜ ⎜ ⎟ ⎜ ⎝ 5.05 ⎠ ⎝
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8 8
⎞ ⎛ a0 ⎞ ⎟⎜a ⎟ ⎟ ⎜ 1⎟ ; ⎟ ⎜ a2 ⎟ ⎟⎜ ⎟ 8⎠ ⎝ a3 ⎠
⎛ a0 ⎞ ⎛ 3.83⎞ ⎜ a ⎟ ⎜ 1.26 ⎟ ⎜ 1⎟ = ⎜ ⎟ ⎜ a2 ⎟ ⎜ 0.33⎟ ⎜ ⎟ ⎜ ⎟ ⎝ a3 ⎠ ⎝ 0.63⎠
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From the ANOVA, we may also derive a statistic to measure overall goodness of fit. We shall call it the coefficient of determination and represent it with the symbol r2. It has the following definition: r2 =
SSM SSR = 1− SST SST
(3.43)
If we desire, we may augment the ANOVA table with r2, though strictly speaking it is not part of the accounting. However, the table has some empty space, so why not? For that matter, we can also show s = √MSR. Table 3.5 gives the consolidated form of ANOVA that we will use. TABLE 3.5 Basic ANOVA for Table 3.4 Term
SS
DF
MS
F
P
M
16.70
3
5.57
70.5
0.0006
R
0.32
4
0.08
r2
98.1%
T
17.02
7
s
0.28
However, because each effect in a factorial design is orthogonal to every other, we may partition SSM and DFM further, as shown in Table 3.6. TABLE 3.6 Partitioned ANOVA for Table 3.4 Term
SS
DF
MS
F
P
Model a1 a2 a3
12.63 0.89 3.19
1 1 1
12.63 0.89 3.19
159.8 11.3 40.3
0.0002 0.0283 0.0031
R
0.32
4
0.08
r2
98.1%
T
17.02
7
s
0.28
We see now that the SSM in Table 3.5 partitions as 12.63 + 0.89 + 3.19 = 16.70, where the sums are for x1, x2, and x3, respectively. We can also see that though Table 3.5 showed that the model as a whole was significant at greater than 99.9% confidence (P = 0.0006), not all effects are significant at exactly the same confidence level. In this particular case, it happens that all effects are significant with greater than 95% confidence (P < 0.05). However, this need not be the case. Even if the model is very significant, some individual effects may not be. 3.3.2
General Features of Factorial Designs
A factorial design uses all possible high- and low-factor combinations of f factors and comprises 2f experimental points. Therefore, the factorial design can fit at most 2f terms according to
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Experimental Design and Analysis
y = a0 +
213
∑ a x + ∑∑ a x x + ∑∑∑ a k k
jk j k
j< k
x x x +$
hjk h j k
h< j
k
j< k
(3.44)
k
The last term in Equation 3.44 will be the f-factor interaction. For example, if f = 3, the last interaction (the eighth term, 23 =8) will be a123x1x2x3. Also, in Equation 3.43, the number of summands indicates the overall order of the term. For example,
∑∑∑ a
x xx
hjk h j k
h< j
j< k
k
comprises all the third-order interaction terms for a given factorial design, presuming f = 3 or greater. To construct the X matrix for all terms, we may make use of Equation 3.44. We may also use Figure 3.3 to determine the number of terms that are first, second, and third order, etc. Using Equation 3.11, the value of the mth parameter indicates the number of terms (m) of overall order f. For example, for f = 2 we have entries (1, 2, 1) indicating that there is 1 zero-order term (k = 0), 2 first-order terms (k = 1), and 1 second-order (overall) term (k = 2). Since the factorial has only high and low values of each factor, no term may contain factors having an individual order above 1. Therefore, factorial terms that overall are second order are of the form xjxk, terms that overall are third order have the form xhxjxk, and so forth. If we want to know the number of third-order terms for the 25 factorial design, we can use Equation 3.11 directly to find m=
5! = 10 3! 5 − 3 !
(
)
or we can view Figure 3.3 and note that for f = 5 and p = 3, m = 10.
3.3.3
Construction Details of the Two-Level Factorial
To construct the X matrix for a two-level factorial design, we use the following rules. 1. There will be n = 2f design points, where f is the number of factors. 2. Construct factor columns by alternating between low (–) and high (+) factor values in blocks of 2f–k, where k is the factor subscript. 3. Continue for f factors. The resulting matrix comprises the design matrix of factor coordinates.
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Example 3.3
Construction of a 2 3 Factorial Design
Problem statement: Construct a 23 factorial design using the foregoing procedure. Solution: As this is a 23 factorial design, f = 3 and n = 23 = 8. For the first factor, k = 1; therefore, we alternate the first row in blocks of 2(3–1) = 4: Pt 1 2 3 4 5 6 7 8
x1 − − − − + + + +
For the second factor, k = 2, and we alternate signs in blocks of 2(3–2) = 2: Pt 1 2 3 4 5 6 7 8
x1 − − − − + + + +
x2 − − + + − − + +
Finally, we add the third factor in blocks of 2(3–3) = 1: Pt 1 2 3 4 5 6 7 8
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x1 − − − − + + + +
x2 − − + + − − + +
x2 − + − + − + − +
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This is the 23 factorial design having eight data points comprising all possible high/low combinations of three factors. One may also derive the factorial matrix from binary counting. For readers unfamiliar with binary and related bases, see Appendix F. For full factorial designs, conversion of the whole numbers from 0 to 2f –1 to binary gives the sign pattern directly. Binary order is the sign pattern of the whole numbers expressed in binary. The method also allows one to jump immediately to the nth-factor pattern by expressing n – 1 as a binary whole number. Some authors refer to this as standard order; others use the phrase to refer to the reverse sign pattern. On occasion, we will renumber our points starting from 1 rather than zero.
Example 3.4
Construction of a 2 3 Factorial Design, Binary Counting Method
Problem statement: Construct a 23 factorial design using binary counting. For a 26 design in binary order, what is the sign pattern of the 40th entry? Solution: This is a 23 factorial design. Therefore, the whole numbers from 0 to 23 – 1 are 0, 1, 2, 3, 4, 5, 6, and 7. Converting these to binary we have 0 = 000 = – – – 1 = 001 = – – + 2 = 010 = – + – 3 = 011 = – + + 4 = 100 = + – – 5 = 101 = + – + 6 = 110 = + + – 7 = 111 = + + + If we prefer to number our points starting from 1 rather than 0, we add one to the above entries. In matrix form (annotated), we would have
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Modeling of Combustion Systems: A Practical Approach
Pt
x1
x2
x3
1 2 3 4 5 6 7 8
⎛− ⎜− ⎜ ⎜− ⎜ ⎜− ⎜+ ⎜ ⎜+ ⎜+ ⎜ ⎝+
− − + + − − + +
−⎞ + ⎟⎟ −⎟ ⎟ +⎟ −⎟ ⎟ +⎟ −⎟ ⎟ +⎠
To calculate the sign pattern of the 40th factor in a 26 factorial design we have 40 – 1 = 39 = 1001112 = + – – + + +; that is, Pt 40
x1
(+
x2
x3
x4
x5
x6
−
−
+
+
+
)
Note that the numerical subscript following the main number refers to its base (see Appendix F). 3.3.4
Contrast of Factorial and Classical Experimentation
Traditionally, experimentalists held constant all factors but one in order to assess the individual affect of that factor on the desired response. Proponents have even misstated this strategy as a requirement of the scientific method; it is not. In fact, factorial designs vary all factors at once. So then, how do we know which factors correlate with the response if several vary simultaneously? Actually, if one is clever about the experimental design, not only can one vary all factors at once, but also the strategy is more efficient than the classical one-factor-at-a-time approach, with better statistical properties (such as better estimates for the factor’s effect on the response). For example, let us contrast classical and factorial designs for two factors with some hypothetical data. Table 3.7 gives the factor patterns. TABLE 3.7 Contrast of Classical and Factorial Designs Classical Design ξ1 ξ2 y
Factorial Design x1 x2 y
0 0 0 1 2
– – + +
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0 1 2 1 1
2.5 0.0 –2.5 0.5 0.5
– + – +
2.5 –2.5 0.5 3.5
Experimental Design and Analysis
217
In the classical design, we suppose that an investigator, seeking only to vary one factor at a time, starts at some point in ξ1·ξ2 factor space (0, 0), which is the current optimum and maximal response, 2.5 units (see bottom left corner of Figure 3.4). We shall further suppose that the graph represents the entire operating space; that is, the operating constraints do not allow for excursions beyond the boundaries shown. The investigator begins by increasing ξ2 one unit, keeping ξ1 constant, and recording a value of 0. Obviously, the response is going in the wrong direction, but just to be sure, he increases ξ2 by another unit. The response drops to –2.5. At this point, he has reached an operating constraint and has not been successful in increasing the response value.
1 -2
-2
3
-1 2
0.5 0
x2 x1 0
-1
-0.5
1
0
0.5
1
-0.5
ξ2 1
ξ1
2 -1
FIGURE 3.4 Classical and factorial experiments. In the classical design (circles) the investigator starts at some origin (bottom left corner) and changes one factor while holding the other constant. This results in the vertical path along the left axis. Finding no increase in response (note contours), he returns to the origin and increases x1 with x2 constant, proceeding along the bottom right of the figure. He concludes (falsely) that the origin is the place of maximum response. In fact, a factorial design (squares) would have generated the proper model and found that the higher response within constraints is in the upper right-hand corner.
Returning to the historic maximum, he holds ξ2 constant and increases ξ1 by one unit. Regrettably, the response drops from 2.5 to 0.5 units. He decides to increase ξ1 by another unit, but the response stays the same at 0.5 units and he has reached the maximum value for ξ1. Based on his data, he concludes the following:
© 2006 by Taylor & Francis Group, LLC
218
Modeling of Combustion Systems: A Practical Approach
• A least squares fit of his data in the given metric is
(
)
y = 2.5 + ξ1 ξ1 − 3 − 2.5ξ 2
(3.45)
• Increases in ξ2 are a disaster; they reduce the response precipitously. • Increases in ξ1 are also a bad idea as they decrease the response. However, the decline flattens out as ξ1 increases. • The response is a maximum. One cannot increase things any further within the operating constraints. As we show presently, the last conclusion is incorrect. However, it is not due to an error in analysis, but to the limitations of the experimental design itself. A factorial design would have allowed for different conclusions. A factorial design uses all possible high and low combinations of the factors. For two factors, there are 22 = 4 possible combinations. The investigator runs them in random order. We describe randomization in detail in Section 3.7. For now, we merely mention that randomizing the run order is essential to negate the effects of bias from unknown factors correlated in time (e.g., humidity, air temperature, batch cycles, etc.). If we know that a factor is influential, we can account for it in the experiment. But to mute the potential effect of those factors we do not know about, we randomize. In accordance with standard practice, we transform the factors. This bounds the operating region by –1 < x1 < 1 and –1 < x2 < 1. With this design, the investigator comes to the following conclusions: • The regression equation in transformed coordinates is y = 1 + x1 −
x2 + 2 x1x2 2
(3.46)
• Because the matrix is orthogonal, one may directly compare the coefficients. There is a strong interaction term, meaning that if x1 and x2 move together, the response increases. • Ultimately, the surface slopes downward along the diagonal running from bottom right to top left (see Figure 3.4), and upward along the diagonal running from bottom left to top right. • There is a region of improved response within the design constraints (top right corner of the figure, response = 3.5). • Figure 3.4 maps both investigations in factor space. In the figure, the circles represent the classical design points and the squares represent the factorial approach. The investigator could not come to the proper conclusions because his experimental strategy was flawed. His analysis of the data is correct as far as it goes, but the distorted factor space compromised the results.
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Experimental Design and Analysis
219
3.3.4.1 Statistical Properties of Classical Experimentation Generally, classical experimental strategies have poor statistical properties for the following reasons: • The information is concentrated along a few axes rather than spread over the entire factor space. We will always have more certainty of information near the design points or for interpolations among them, compared to distant and extrapolated regions. Box and Draper present an information function for gauging the certainty of estimating a response function:2 ⎛ σ2 ⎞ 1 I (x) = ⎜ ⎟ ⎝ n ⎠ x T X TX
(
)
(3.47)
−1
x
where σ 2 is the variance, n is the total number of points in the design, x is the position vector that we describe shortly, and X is the matrix of the presumably true model. For our purposes, we shall be interested only in the fraction of maximal information as we traverse the factor space,
() i (x) = I (x) I x
=
max
( ) x (X X)
x Tmax X TX T
−1
T
x max
−1
(3.48)
x
which we shall call i(x), the information fraction; x Tmax is the vector in factor space having the greatest certainty of response estimation. We can find x Tmax for
(
1
x T X TX
(
when x T X TX
)
−1
)
−1
x
x is a minimum, i.e., for
(
∂ ⎡ T T x X X ∂x ⎢⎣
)
−1
⎤ x⎥ = 0 ⎦
For the factorial design, the presumed model is x T = (1 x1 x2 x1x2 ), and the design gives
(X X) T
© 2006 by Taylor & Francis Group, LLC
−1
⎛4 ⎜ =⎜ ⎜ ⎜ ⎝
4 4
⎞ ⎟ ⎟ ⎟ ⎟ 4⎠
220
Modeling of Combustion Systems: A Practical Approach Then
(
x T X TX
)
−1
(
x = 1
x1
(
⎛4 ⎜ x1x2 ⎜ ⎜ ⎜ ⎝
)
x2
)(
= 4 1 + x12 1 + x22
4 4
⎞⎛ 1 ⎞ ⎟⎜ x ⎟ ⎟⎜ 1 ⎟ ⎟ ⎜ x2 ⎟ ⎟ ⎟⎜ 4⎠ ⎝ x1x2 ⎠
)
Also,
(
)(
)
x Tmax =
∂ ⎡ 4 1 + x12 1 + x22 ⎤ = 0 ⎦ ∂x ⎣
⎛ x1 ⎞ ⎜⎝ x ⎟⎠
⎡ 8 x1 1 + x22 ⎤ ⎛ 0⎞ ⎥= =⎢ ⎢ 8 1 + x12 x2 ⎥ ⎜⎝ 0⎟⎠ ⎣ ⎦
leading to
2
max
(
(
)
)
Therefore, ( x1 , x2 ) = (0, 0), which is obvious even by inspection. max Thus, x Tmax = (1 0 0 0) and x Tmax (X TX)−1 x max = 4 . Then, for the factorial design, the information fraction becomes
(
)
iF x1 , x2 =
1
(1 + x ) (1 + x ) 2 1
(3.49)
2 2
where the subscript F denotes the factorial design. Using the same method, we arrive at the information fraction for the classical design:
(
)
iC ξ1 , ξ 2 =
1 2 2 ⎧ ⎫ ξ1 ξ 2 + 2 + ⎪ ⎪ ⎪ ⎡ ξ14 + ξ 24 − ξ1ξ 2 3 − ξ1 − ξ 2 − 2 ξ1 + ξ 2 + ⎤ ⎪ ⎨ ⎢ ⎥⎬ ⎪3 ⎢ ⎥⎪ 3 3 2 2 3 x1x2 − 4 ξ1 + ξ 2 + 5 ξ1 + ξ 2 ⎪ ⎢ ⎥⎪ ⎦⎭ ⎩ ⎣
(
) ( )(
) (
(
) (
(3.50)
)
)
where the subscript C denotes the classical design. Figure 3.5 presents the results graphically. The factorial design has better and more even-handed certainty in estimating the response. The classical design has highly asymmetric information contours and poor certainty for about two thirds of the operating region — basically, anywhere but near the coordinate axes. In other words, even if Equation 3.45 were appropriate, the classical design would not be the best way to estimate it from real data.
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Experimental Design and Analysis
221
Factorial Design
Classical Design 2.0
1.0 40%
40%
100% x
0.0
1.0
85%
-0.5 40% -1.0 -1.0
77% 62%
1.5
0.5
70%
40%
55% -0.5
0.0
31% 62% 77% 46% 92% 100% x
0.5
62% 77%
0.0 0.5
1.0
0.0
0.5
1.0
1.5
2.0
FIGURE 3.5 Information fraction for a factorial and classical design. The factorial design has its maximum certainty at the center of the operating region (100%, x marks the spot). It also has symmetrical information contours. The classical design has a very asymmetrical information distribution. The design provides adequate certainty for estimating the response function only in close proximity to the coordinate axes. It provides poor certainty for most (about two thirds) of the operating region.
• The classical one-factor-at-a-time strategy fails in the case of strong interaction. Strong interaction is quite typical for combustion problems due to inherent nonlinearities and complex response surfaces (e.g., NOx and CO emissions, excess air relations, etc.). Equation 3.45 has no interaction term, nor can the design support one. Unless one varies both factors at once, it is impossible to estimate interaction. 3.3.4.2 How Factorial Designs Estimate Coefficients At this point, it may seem like a minor miracle that one can vary several factors at once and come to any conclusions, let alone sound ones. If all the factors vary at the same time, how can one know which factor or factors have changed the response? To see, let us compare the classical and factorial designs and note their similarities rather than their differences. In a classical design, one varies a single factor. For example, suppose we measure a response in two places, y0 and y1, corresponding to two positions along the factor x1 axis, namely, x1,0 and x1,1, respectively. Now, we would like to fit the model y = a0 + a1x1. Then y1 = a0 + a1x1,1 and y0 = a0 + a1x1,0. Subtracting y0 from y1 gives y1 − y0 = a1 ( x1,1 − x1,0 ) . Thus, a1 =
© 2006 by Taylor & Francis Group, LLC
y 1 − y0 x1,1 − x1,0
222
Modeling of Combustion Systems: A Practical Approach
Beginning our experiments at the design origin (x1,0 = 0) and substituting this into our original equation gives a0 = y0. This is the classical design strategy. Now suppose we want better certainty and we perform our experiments a number of times at these two points. Then clearly
n
a0 =
∑y
n
y 1, k
k =1
a1 =
(3.51a)
n
n
∑
0,k
k =1
n
−
∑y
0,k
k =1
x1,1 − x1,0
n
(3.51b)
Suppose n = 4. Then we have made eight measurements, four for y0 and four for y1. Although we may calculate a0 with much better certainty, it seems superfluous to perform eight experiments to fit only two coefficients. In three factors, the number of experiments mushrooms to many more; in the classical strategy, we would start from some origin and move in one of three factor directions. Since all the factor axes coincide at the origin, we may write x1,0 = x2,0 = x3,0 = 0 and call this point x0,0. Then we have a total of four design points: x0,0, x1,1, x1,2, and x1,3. If we replicate each point four times, then we perform 4·8 = 32 experiments. Thirty-two experiments for four coefficients is not an efficient strategy. However, if we are clever, we can perform only eight experiments yet replicate each point four times. To know how, consider the sneaky farmer. 3.3.4.3 The Sneaky Farmer Suppose a farmer’s land must show at least five rows of mature palm trees in order to qualify for a tax break, with the law requiring at least four palm trees per row. The farmer could purchase 20 mature trees and plant them in a five-by-four grid, but mature palms are expensive, especially since the new law has taken effect. So the farmer decides to plant the trees in a star pattern (Figure 3.6). This only takes 10 trees because the farmer uses every tree in two rows. Factorial designs are similar in the sense that every point serves multiple duty. For the case of a three-factor factorial, the design uses each point in eight averages. To see this, we shall examine the XTX matrix for a 23 factorial design in some detail. Here it is:
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Experimental Design and Analysis
223
(a) five rows of four trees each
(b) five rows of four trees each
FIGURE 3.6 The sneaky farmer. Option (a) gives the conventional solution to planting 5 rows of 5 trees each and requires 20 trees. The sneaky farmer chooses option (b), planting only 10 trees. Yet, undeniably, there are five rows of four trees each. Each tree has been made to count twice, being in two rows at once. Factorial designs are like that. In a 23 factorial design, each point is used in eight different averages to give eight different coefficients. Classical one-factor-at-a-time experiments are more like option (a) and require more points for the same statistical certainty.
Pt
x1
x2
x3
y
1 2 3 4 5 6 7 8
− − − − + + + +
− − + + − − + +
− + − + − + − +
y1 y2 y3 y4 y5 y6 y7 y8
Consider that we shall fit the following model from the above eight experiments.
y = a0 + a1x1 + a2 x2 + a3 x3 + a12 x1x2 + a13 x1x3 + a23 x2 x3 + a123 x1x2 x3
In matrix form, this becomes
© 2006 by Taylor & Francis Group, LLC
(3.52)
224
Modeling of Combustion Systems: A Practical Approach ⎛ y1 ⎞ ⎛ + ⎜y ⎟ ⎜+ ⎜ 2⎟ ⎜ ⎜ y3 ⎟ ⎜ + ⎜ ⎟ ⎜ ⎜ y4 ⎟ = ⎜ + ⎜ y5 ⎟ ⎜ + ⎜ ⎟ ⎜ ⎜ y6 ⎟ ⎜ + ⎜y ⎟ ⎜+ ⎜ 7⎟ ⎜ ⎜⎝ y ⎟⎠ ⎝ + 8
− − − − + + + +
− − + + − − + +
− + − + − + − +
+ + − − − − + +
+ − + − − + − +
− ⎞ ⎛ a0 ⎞ + ⎟⎟ ⎜⎜ a1 ⎟⎟ + ⎟ ⎜ a2 ⎟ ⎟ ⎟⎜ − ⎟ ⎜ a3 ⎟ + ⎟ ⎜ a12 ⎟ ⎟ ⎟⎜ − ⎟ ⎜ a13 ⎟ − ⎟ ⎜ a23 ⎟ ⎟ ⎟⎜ + ⎠ ⎜⎝ a123 ⎟⎠
+ − − + + − − +
(3.53)
Then the normal equations are ⎛N ⎛ ∑y ⎞ ⎜ ⎜ ⎟ ⎜ ⎜ ∑ x1 y ⎟ ⎜ ⎜ ∑ x2 y ⎟ ⎜ ⎜ ⎟ ⎜ ⎜ ∑ x3 y ⎟ = ⎜ ⎜ ∑ x1x2 y ⎟ ⎜ ⎜ ⎟ ⎜ ⎜ ∑ x1x3 y ⎟ ⎜ ⎜ ∑x x y ⎟ ⎜ 2 3 ⎜ ⎟ ⎜ ⎜⎝ ∑ x x x y ⎟⎠ ⎜ 1 2 3 ⎜ ⎝
∑ x12 ∑ x22 ∑ x32
(
∑ x1x2
)
2
(
∑ x1x3
)
2
(
∑ x2 x3
)
2
⎞ ⎟ ⎛ a0 ⎞ ⎟⎜ ⎟ ⎟ ⎜ a1 ⎟ ⎟⎜ a ⎟ 2 ⎟⎜ ⎟ ⎟ ⎜ a3 ⎟ ⎟⎜ a ⎟ ⎟ ⎜ 12 ⎟ ⎟ ⎜ a13 ⎟ ⎟⎜ ⎟ ⎟ ⎜ a23 ⎟ ⎟ ⎜⎝ a123 ⎟⎠ 2 ∑ x1x2 x3 ⎟⎠
(
(3.54)
)
Now since X is orthogonal (XTX diagonal), all the equations are independent and have unbiased least squares coefficients. We may solve them immediately. ∑y N
(3.55a)
a1 =
∑ x1 y ∑ x12
(3.55b)
a2 =
∑ x2 y ∑ x22
(3.55c)
a3 =
∑ x3 y ∑ x32
(3.55d)
a0 =
a12 =
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∑ x1x2 y
(
∑ x1x2
)
2
(3.55e)
Experimental Design and Analysis
a13 =
a23 =
a123 =
225 ∑ x1x3 y
(
∑ x1x3
)
(3.55f)
2
∑ x2 x3 y
(
∑ x2 x3
)
(3.55g)
2
∑ x1x2 x3 y
(
∑ x1x2 x3
)
2
(3.55h)
In fact, all the denominators sum to 8. We may also write the coefficients in terms of the response values by considering the actual signs of the summands in the above equations, or as detailed in Chapter 1 (recall Equation 1.78: a = Gy, where G = (XTX)–1XT). a0 = y =
y1 + y2 + y3 + y4 + y5 + y6 + y7 + y8 8
⎛ y5 + y6 + y7 + y8 ⎞ ⎛ y1 + y2 + y3 + y4 ⎞ ⎟⎠ ⎟⎠ − ⎜⎝ ⎜⎝ 4 4 a1 = 2 ⎛ y3 + y4 + y7 + y8 ⎞ ⎛ y1 + y2 + y5 + y6 ⎞ ⎟⎠ ⎟⎠ − ⎜⎝ ⎜⎝ 4 4 a2 = 2 ⎛ y2 + y4 + y6 + y8 ⎞ ⎛ y1 + y3 + y5 + y7 ⎞ ⎟⎠ ⎟⎠ − ⎜⎝ ⎜⎝ 4 4 a3 = 2 ⎛ y1 + y2 + y7 + y8 ⎞ ⎛ y3 + y4 + y5 + y6 ⎞ ⎟⎠ ⎟⎠ − ⎜⎝ ⎜⎝ 4 4 a12 = 2 ⎛ y1 + y3 + y5 + y8 ⎞ ⎛ y2 + y4 + y6 + y7 ⎞ ⎟⎠ ⎟⎠ − ⎜⎝ ⎜⎝ 4 4 a13 = 2
© 2006 by Taylor & Francis Group, LLC
(3.56a)
(3.56b)
(3.56c)
(3.56d)
(3.56e)
(3.56f)
226
Modeling of Combustion Systems: A Practical Approach ⎛ y1 + y4 + y5 + y8 ⎞ ⎛ y2 + y3 + y6 + y7 ⎞ ⎟⎠ ⎟⎠ − ⎜⎝ ⎜⎝ 4 4 a23 = 2
a123
⎛ y2 + y3 + y5 + y8 ⎞ ⎛ y1 + y4 + y6 + y7 ⎞ ⎟⎠ ⎟⎠ − ⎜⎝ ⎜⎝ 4 4 = 2
(3.56g)
(3.56h)
Comparing Equations 3.55a to h with Equations 3.51a and b shows that the form is identical. It is as if we replicated each point eight times. For a0, we use an average comprising all eight points, as is the case for the classical experimentation. Yet, we have only run eight experiments. This is the power of statistical experimental design. Figure 3.7 graphically displays the logic. Because the design is orthogonal, truncating the model to fewer terms will not change the value of the remaining ones. This is not true for least squares, in general, but it is true with orthogonal designs. This latter fact will allow us to independently test effects for statistical significance. For nonorthogonal matrices, one may need to consider all possible models before passing judgment to retain or reject a model term. In the present case there are eight possible terms in the model (a0 – a123) generating 28 = 256 possible models. Although hierarchical strategies may allow us to reduce the search,* this will require dedicated software.
Example 3.5
Results for 2 3 Factorial Data Set
Problem statement: Calculate the value of the coefficients for the factorial data set given in Table 3.8. The data show the dependence of NOx on excess oxygen (O2), air preheat temperature (APH), and bridgewall temperature (BWT) for a particular burner and furnace. 1. Calculate the coefficients for the full factorial using the relations of Equation 3.54. 2. Repeat the calculation using matrix algebra. 3. Then calculate the coefficients for the reduced model: y = a0 + a1x1 + a2 x2 + a12 x1x2. 4. Comparing the coefficients, what do you notice?
* A mixed strategy, where one alternately adds terms that most improve or subtracts terms that least harm model significance, is likely to result in optimum models. For a discussion, see Draper, N.R. and Smith, H., Selecting the ‘best’ regression equation, in Applied Regression Analysis, 3rd ed., John Wiley & Sons, New York, 1998, chap. 15, pp. 327f.
© 2006 by Taylor & Francis Group, LLC
y 1+y 2+y 3+y 4+y 5+y 6+y 7+y 8 8
2
a1 =
4
(y5+y6+y7+y8)–(y1+y2+y3+y4) 8
2
x3 6
4
6
x1 1
a12 =
8
7
5 a13 =
4
x1 1
5
(c)
2
a23 =
6
4
7
8 x1 1
5
(f)
x2 x1 1
3 7
5
2 6
7
a123 =
(y2+y3+y5+y8)–(y1+y4+y6+y7) 8
2
4 x3
8
6
8 x2
x2 1 x1 1
3 7
3
(d)
4
x2 3
8
x3
x2 x1 1
4
6
(y1+y4+y5+y8)–(y2+y3+y6+y7) 8
x3 8
(e)
8
3
(y1+y3+y5+y8)–(y2+y4+y6+y7) 8
x3
2
x2
7
(b)
(y2+y4+y6+y8)–(y1+y3+y5+y7) 8
x3
6
x1 1
3
2
5
4
x2
(y1+y2+y7+y8)–(y3+y4+y5+y6) 8
6
2
a3 =
x3
x2
(a)
(y3+y4+y7+y8)–(y1+y2+y5+y6) 8
x3 8
5
a2 =
5
(g)
Experimental Design and Analysis
a0 = y =
1x 1
3 7
5
(h)
3 7
FIGURE 3.7 A 23 factorial design. The shaded areas depict contrasts among the response values that are used to determine each respective coefficient. All eight points are used to calculate each coefficient.
227
© 2006 by Taylor & Francis Group, LLC
228
Modeling of Combustion Systems: A Practical Approach TABLE 3.8 A Factorial Design in Three Factors Factor Point
1 O2, %
2 APH, F
3 BWT, F
y ppm NOx
1 2 3 4 5 6 7 8
1 1 1 1 5 5 5 5
25 25 325 325 25 25 325 325
800 1100 800 1100 800 1100 800 1100
7.8 9.5 13.4 15.0 12.8 16.0 20.0 23.9
Solution: First we apply the following coding transforms to center and scale all factors to a uniform dimensionless metric of ±1:
x1 =
O 2 − 3% 2%
x2 =
APH − 175°F 150°F
x3 =
BWT − 950°F 150°F
1. Using the relations of Equation 3.54 we obtain the following estimates rounded to the decimal accuracy given: a0 = 14.8, a1 = 3.38, a2 = 3.27, a3 = 1.28, a12 = 0.51, a13 = 0.46, a23 = 0.08, a123 = 0.10 2. Using matrix algebra we obtain the following normal matrix equation: ⎛ 118.36 ⎞ ⎛ 8 ⎜ 27.08 ⎟ ⎜ ⎜ ⎟ ⎜ ⎜ 26.12 ⎟ ⎜ ⎜ ⎟ ⎜ ⎜ 10.21 ⎟ = ⎜ ⎜ 4.08 ⎟ ⎜ ⎜ ⎟ ⎜ ⎜ 3.71 ⎟ ⎜ ⎜ 0.67 ⎟ ⎜ ⎜ ⎟ ⎜ ⎝ 0.80 ⎠ ⎝
© 2006 by Taylor & Francis Group, LLC
8 8 8 8 8 8
⎞ ⎛ 14.80 ⎞ ⎟⎜ ⎟ ⎟ ⎜ 3.38 ⎟ ⎟ ⎜ 3.27 ⎟ ⎟⎜ ⎟ ⎟ ⎜ 1.28 ⎟ , ⎟ ⎜ 0.51 ⎟ ⎟⎜ ⎟ ⎟ ⎜ 0.46 ⎟ ⎟ ⎜ 0.08 ⎟ ⎟⎜ ⎟ 8⎠ ⎝ 0.10 ⎠
Experimental Design and Analysis
229 ⎛ a0 ⎞ ⎛ 14.80 ⎞ ⎜ a ⎟ ⎜ 3.38 ⎟ ⎜ 1 ⎟ ⎜ ⎟ ⎜ a2 ⎟ ⎜ 3.27 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ a3 ⎟ = ⎜ 1.28 ⎟ ⎜ a12 ⎟ ⎜ 0.51 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ a13 ⎟ ⎜ 0.46 ⎟ ⎜ a ⎟ ⎜ 0.08 ⎟ ⎜ 23 ⎟ ⎜ ⎟ ⎜⎝ a ⎟⎠ ⎝ 0.10 ⎠ 123
thus
3. Truncating the model we obtain ⎛ 118.36 ⎞ ⎛ 8 ⎜ 27.08 ⎟ ⎜ ⎜ ⎟ =⎜ ⎜ 26.12 ⎟ ⎜ ⎜ ⎟ ⎜ ⎝ 4.08 ⎠ ⎝
8 8
⎛ a0 ⎞ ⎛ 14.80 ⎞ ⎞ ⎛ 14.80 ⎞ ⎜a ⎟ ⎜ ⎟ ⎟ ⎜ 3.38 ⎟ ⎟⎜ ⎟ , thus ⎜ 1 ⎟ = ⎜ 3.38 ⎟ ⎜ a2 ⎟ ⎜ 3.27 ⎟ ⎟ ⎜ 3.27 ⎟ ⎜ ⎟ ⎜ ⎟ ⎟⎜ ⎟ 8⎠ ⎝ 0.51 ⎠ ⎝ a12 ⎠ ⎝ 0.51 ⎠
4. In the second case, the coefficients are identical to the first. In the third case, the coefficients for the remaining effects are identical to the respective coefficients in the first two cases. For a factorial design, one may omit any number of terms from the model and the remaining coefficients retain their original values. This property is not true for least squares solutions in general, but it is true with factorial designs. This is how one may vary many factors at once, yet obtain increased certainty for individual coefficients, as if we had taken repeated measures along a single-factor axis.
3.3.5
Interpretation of the Coefficients
Factorial designs also generate coefficients with sensible interpretations. For example, a0 is the average response. This is also the expected response value at the design center (0, 0, 0 for the case at hand). Note that the least squares method derives each coefficient from all the response values, resulting in the highest certainty for each estimate. When ak > 0, the coefficients represent the increase in the response value as xk moves one unit in the positive direction. Conversely, if ak < 0, then the response decreases. Since we have coded xk at ±1 for the boundaries of the
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factorial design, ak represents are the first-order effects at the design extremes. Thus, examining the signs of these coefficients will tell us if an increase in the factor will increase or decrease the response. Since we have standardized all the factors to unit range and zero mean, they are unbiased; we may compare the coefficients directly. The ajk coefficients represent the interaction terms. Interaction terms account for synergy or moderation of the factors. The statistical literature uses the term effect as a synonym for a model term. If effects interact, such interactions will show up as an interaction coefficient. An ajk coefficient represents a binary interaction between xj and xk. An ahjk coefficient represents a ternary interaction among xh, xj, and xk. The overall order of an effect is the sum of the factor exponents expressed as an ordinal number (e.g., first, second, third, etc.). The individual order of a factor within an effect is the ordinal number of its exponent. Therefore, the product xh xj xk is third order overall and first order in xh, xj, and xk. Third- or higher-order terms are rarely significant in factorial designs. If they are insignificant, then they represent an estimate of experimental error, e.g., ahjk xh xj xk ≈ e. If so, Equation 3.52 becomes y = a0 + a1x1 + a2 x2 + a3 x3 + a12 x1x2 + a13 x1x3 + a23 x2 x3 + e
Example 3.6
(3.57)
Interpretation of the Example 3.5 Coefficients
Problem statement: Examine and interpret the coefficients of Example 3.5. Solution: Example 3.5 gave the following coefficients. a0 = 14.8, a1 = 3.38, a2 = 3.27, a3 = 1.28, a12 = 0.51, a13 = 0.46, a23 = 0.08, a123 = 0.10 Recall that the subscripts have the following references: 1 = oxygen, 2 = air preheat temperature, and 3 = furnace temperature. For discussion purposes, we start with a graphical depiction of the NOx relation. Figure 3.8 gives the pictures. Since NOx is a function of three factors, the actual response surface is four-dimensional. We may represent this as a series of contour slices or as a series of superimposed three-dimensional surfaces. For an overall qualitative understanding of the NOx surface, the stacked surfaces are better. For more quantitative results, the contours are better. For exact numbers and computer programs, Equation 3.52 is best. From the coefficients, we understand the following:
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Hi Furnace Temp
NOx Contour Lines
16 17 18 19 20 2122 23 15
Med Furnace Temp 15 16 17 18 19 20 21 14
14 Air Preheat
13 12 11 10
Oxygen
Lo Furnace Temp 14 15 16 17 18 19 13
13 12
Air Preheat
Air Preheat
Experimental Design and Analysis
11 10 9
Oxygen
NOx Contour Surfaces
11 10 9 Oxygen
Furnace Temp
NOx
Hi Med Lo
12
t ea
eh
A
r ir P
Oxygen
FIGURE 3.8 Representations of NOx response. The figures show NOx as a function of air preheat temperature, oxygen concentration, and furnace temperature. The contour surfaces (left) give a good overall picture, but the contour lines (above) work better for more quantitative estimates. This figure was adapted from output via the statistical program JMP.
1. The average response is 14.8 ppm NOx. 2. An increase in oxygen from 3 to 5% increases the NOx by just over 3.38 ppm. Conversely, a decrease in oxygen from 3 to 1% decreases NOx by this same amount. Increasing furnace temperature adds up to 1.28 ppm to the NOx. (The reader should keep in mind that this is a 2.6 ppm NOx increase from lowest to highest levels because the difference between –1 and +1 is 2 units.) However, when interactions exist, we must consider both the interactions and the main effects to get a true picture of the response surface. 3. For the time being, we will presume that we have no thirdorder interactions. Therefore, a123 = 0.10 probably represents experimental error, e.g., e ≈ 0.10. If so, we should reject a23 as well. Replicating some points would allow for an independent estimate for experimental error and a basis for rejecting model terms. We will discuss this shortly. For now, we note that even if the effects were statistically significant,
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Modeling of Combustion Systems: A Practical Approach they are not practically significant because they influence the response by not more than 0.1 ppm. Statistical significance refers to something that is greater than the experimental error by some a priori margin. Practical significance denotes something that is greater than some a priori response threshold. 4. With respect to oxygen, we have two interactions to consider: x12 and x13. The a12 coefficient tells us that the NOx increases by an additional 0.51 ppm with increasing oxygen and air temperature. This effect is in addition to the a1 and a2 coefficients, so some synergy exists here. However, if both oxygen and air preheat are low, 0.51 ppm is still added, thus moderating the response (if x1 = –1 and x2 = –1, then x1x2 = +1 and the contribution is still positive). If oxygen and air preheat differ in sign (high O2, low air preheat, or vice versa) then the interaction lowers NOx by 0.51 ppm. 5. These interaction effects account for the curved contour lines and the uneven spacing in Figure 3.8. Visually these interactions appear to be slight; mathematically, the largest interaction is only about 40% of the smallest first-order effect. 6. From this examination, we have learned that NOx increases with an increase in air temperature, furnace temperature, and oxygen concentration. Positive interactions between the temperatures and oxygen concentration exacerbate these effects.
3.3.6
Using Higher-Order Effects to Estimate Experimental Error
The model of Example 3.5 does not have any term for experimental error, and our general assumption is that experimental error is important. Without any estimate for experimental error, we will not be able to decide quantitatively if a coefficient represents random noise or is truly significant. Often, higher-order interactions are negligible, or we may notice a group of terms tending toward the same low coefficient values; these may represent no real effect at all. In such a case, we could pool them into the residual and use them to estimate σ2 with more degrees of freedom (and thus improved certainty). Plotting coefficients in a normal probability plot is one formal way to decide if coefficients belong in the noise or information category. 3.3.6.1 Normal Probability Plots for Estimating Residual Effects In Example 3.6, we suggested that perhaps the x1x2x3 and x2x3 terms were not statistically significant. If not, we could put these two terms in the residual error part of the ANOVA and use them to obtain an improved estimate of σ2. Besides inspection, we would like some formal way of deciding.
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One tool is the normal probability plot, best done with statistical software. Figure 3.9 shows such an output. 4 +APH+02
3
Estimate
2 +BWT
1 +
+
0
+
+
-1 -2 -3 -4 -3
-2
-1
0 Normal Quantile
1
2
3
FIGURE 3.9 Normal probability plot. The labeled points deviate from a straight line on the normal probability plot. Only the main effects are significant according to this procedure. This particular plot comes from the statistical program JMP.
If the terms are merely taking up random error but no more, they will tend to fall in a straight line on normal probability paper. Those factors that deviate from a straight line we gauge as significant. According to this plot, only the main effects are significant — none of the interactions plays a role. If one does not have statistical software, or normal probability paper, one may create a similar plot by doing the following: 1. Order the coefficients from lowest to highest. These shall form the ordinate values for the plot. Exclude a0. 2. Number each of the coefficients, from first to last, as k = 1 … p. 3. Transform the numbers according to zk = (k – 0.5)/p; this approximates the expected value for the order statistics.* 4. Calculate the inverse of the cumulative probability distribution for zk. The Excel function normsinv(zk) will do this. 5. Plot the ordered coefficients against these values. When the effects are orthogonal, one may calculate the sum of squares for the kth model term, SSMk, according to SSMk = (XTy)k·(a)k * This procedure was adapted from Nelson, P.R., Design and analysis of experiments, in Handbook of Statistical Methods for Scientists and Engineers, Wadsworth, H.M., Ed., McGraw-Hill, New York, 1989, chap. 14.
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where (XTy)k is the kth row of the XTy vector and (a)k is the kth row of the coefficient vector. On the assumption that only the main effects are significant, we generate the partitioned ANOVA of Table 3.9. TABLE 3.9 Partitioned ANOVA Term
SS
DF
MS
F(1, 4)
P
Model a1 a2 a3
91.63 85.31 13.02
1 1 1
91.6 85.3 13.0
93.3 86.8 13.3
0.0006 0.0007 0.0220
R
3.94
4
1.0
r2
98.0%
T
193.91
7
s
0.99
We use the remaining effects (residual) to estimate the error for use in the F tests. We see that all three main effects have P < 0.05, meaning all are significant at the 95% confidence level.
3.4
Correspondence of Factor Space and Equation Form
If each factor represents a direction in space, then we refer to the line, area, volume or hypervolume collectively defined by the factor or factors as factor space. As usual, the XTX matrix provides the link between the factor space and the mathematical model. To keep things simple, let us consider twofactor space. Figure 3.10 shows various two-factor designs and the mathematical model having the lowest possible order comprising the maximum number of terms. We do not endorse all of these designs; the purpose here is to fix firmly in the reader’s mind the correlation between the factor space and the associated mathematical models. By inspection, we may formulate the following general rules: 1. The number of model terms cannot exceed the number of design points. In general, we prefer mathematical models comprising fewer terms than design points because we wish to reserve some degrees of freedom to estimate experimental error. 2. The maximum order in any factor direction cannot exceed the number of levels minus 1 in that factor direction. One counts levels by projecting the design onto the factor axis. If there are Lk levels in the k direction, the order of the kth factor cannot exceed Lk–1; i.e., xkLk −1 will be the highest order for factor k in any term. For this reason, Figure 3.10ee has an x12 term (L1 = 3) but cannot have an x22 term (because L2 = 2 and x2L2 −1 = x2 ; that is, the largest term for x2 is first order). However,
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x1 a y = a0 + a1x1 + a2 x2 + a12x1x2
x2
e
冦
a0 + a1x1 + a2 x2 + a11x12 + y= a12 x1 x2 + a112 x12 x2
b a0 + a1x1 + a2 x2 + a12 x1x2 + y= a2211(x22 + x12)
f a0 + a1x1 + a2 x2 + a11x12 + y= a12x1 x2 + a112 x12 x2 + a2211(x22 + x12)
冦
冦
c y = a0 + a1x1 + a2 x2 + a2211(x22 + x12)
g a0 + a1x1 + a2 x2 + a12x1 x2 + a11 x12 + y= a22x22 + a112 x12 x2 + a122 x1 x22
冦
d y=
冦
h a0 + a1x1 + a2 x2 + a12x1 x2 + a11 x12 + y= a22x22 + a112 x12 x2 + a122 x1 x22 + a1122 x12 x22
a0 + a1x1 + a2x2 + a11x12 + a22 x22
冦
FIGURE 3.10 Some two-factor designs. Here are various designs along with some particular associated mathematical models: the models represent the lowest-order models comprising the maximum number of terms. Not all designs are ideal or orthogonal.
the model can have an x12x2 term, as this does not violate the rules for either x1 or x2. 3. One may determine coefficients for any model with a nonsingular inverse matrix. That is, if (XTX)–1 is full rank, the data and the model are compatible.
Example 3.7
Factor Space and Model Form
Problem statement: Given the data of Table 3.10, find the lowestorder model comprising the maximum number of terms corresponding to the experimental series. Do you find anything surprising?
TABLE 3.10 A Curious Experimental Design x1
x2
–1.0 0.5 –0.5 1.0
–0.5 –1.0 1.0 0.5
Solution: Figure 3.11 shows the factor space. Since the design has four experimental points, the model cannot exceed four terms. However, orthogonal projections of the points
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x´1
x1 1 x´2
0.5 1
-0.5 0 -1 -0.5
y = a0 + a1x1 + a2x2 + a12x1x2 2
3
2
3
x2
0.5
0
y = a0 + a1x1 + a11x1 + a111x1
-1
y = a0 + a2x2 + a22 x2 + a222 x2
y = a0 + a1(x1 + x2) + a2(x1 + x2)2 + a3(x1 + x2)3 y = a0 + a1(3x1 + x2) + a2(x2 – 3x1) + a12(3x1 + x2)(x2 – 3x1) FIGURE 3.11 A curious experimental design. Orthogonal projections from the data points intersect each axis in four places. Therefore, one may fit a model up to third order in either x1 or x2 provided the number of terms does not exceed the number of data points. The listed equations show some of the possibilities. The first and the last are orthogonal with unbiased coefficients. The last equation has a form identical to the first when the coordinates are rotated to x1′ – x2′.
show that each axis has four projections. According to rule 2, we are free to fit up to a third-order equation. This seems bizarre since the design is nothing more than a rotated factorial design, and it was impossible for the 22 design to fit anything greater than first order in either factor (compare also Figure 3.10a and Figure 3.10c), but let us continue. Now any equation that does not violate rules 1 and 2 we can subject to rule 3. If the XTX matrix is invertible, then we can regress the model coefficients from the data. Some possibilities among many are y = a0 + a1x1 + a2 x2 + a12 x1x2
(3.58)
y = a0 + a1x1 + a2 x12 + a3 x13
(3.59)
y = a0 + a1x2 + a2 x22 + a3 x23
(3.60)
(
)
(
y = a0 + a1 x1 + x2 + a2 x1 + x2
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)
2
(
+ a3 x1 + x2
)
3
(3.61)
Experimental Design and Analysis
(
237
)
(
y = a0 + a1 x1 − x2 + a2 x1 − x2
)
2
(
+ a3 x1 − x2
)
3
(3.62)
Of all these equations, only the first has unbiased coefficients (i.e., an orthogonal X matrix). However, an equation that corresponds to the factor space a bit better is
(
)
(
)
(
)(
y = a0 + a1 3 x1 + x2 + a2 x2 − 3 x1 + a12 3 x1 + x2 x2 − 3x1
)
(3.63)
It too has unbiased coefficients and is equivalent to y = a0 + a1x1′ + a2 x1′ + a12 x1′ x2′
(3.64)
where x1′ = 3 x1 + x2 and x2′ = x2 − 3 x1; in fact, if one multiplies x1′ and x2′ by 2/5, one obtains the traditional 22 factorial design. The factors x1′ and x2′ correspond to the rotated coordinate axes shown in the figure. Of course, this is merely the 22 factorial design in x1′ ⋅ x2′ factor space. For that matter, a normal 22 factorial design can have any of the equation forms given in this example if one is willing to rotate axes and define new factors. However, this is a bit like turning diamond into graphite — possible but unprofitable.* In Chapter 4 we show how to generate simpler equations from complicated ones at will. From the foregoing, we understand that the mathematical models given in Figure 3.10 are not unique unless one imposes some rule such as filling hierarchical models from lowest to highest order. In other words, y = a0 + a1x1 + a2 x2 + a12 x1x2 is one possible model for the Figure 3.10a design. But so is y = a0 + a12 x1x2 + a111x13 + a222 x23 . Of course, the higher-order terms are less likely to be important within the factor space. Remembering our qualitative rules for evaluating competing models in Chapter 1, rule 1 states that the model must make sense. Rule 2 states that we prefer the simplest model. We want to include all significant terms, but no more. Without some strong theoretical reasons to the contrary, we would reject the higher-order model in favor of the lower-order one. To better see the link between experimental design and the mathematical model, we make use once again of the XTX matrix.
* The conversion of diamond to graphite occurs rapidly above 1500°C. See Masterson, W.L. and Slowinsky, E.J., Chemical Principles, 3rd ed., W.B. Saunders Co., Philadelphia, 1973, p. 227.
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Example 3.8
Possible Mathematical Models from a Given Experimental Design
Problem statement: For the Figure 3.10a experimental design, derive the XTX matrix. Show which higher-order terms bias which lower-order ones. Solution: We begin with the model corresponding to the infinite Maclaurin series.
) (
(
)
⎧ a0 + a1x1 + a2 x2 + a11x12 + a12 x1x2 + a222 x22 + ⎪ y = φ( x1 , x2 ) = ⎨ 3 2 2 3 ⎪⎩ a111x1 + a112 x1 x2 + a122 x1x2 + a222 x2 + $
(
)
Then XTXa becomes ⎛ N ⎜ ⎜ X TXa = ⎜ ⎜ ⎜ ⎜ ⎝ sym
∑x ∑x ∑x ∑x x ∑x 1
2
2 1
1 2 2 2
$⎞ ⎟ ⎛ a0 ⎞ $⎟ ⎜ a1 ⎟ ⎟⎜ ⎟ ⎟⎜a ⎟ $⎟ ⎜ 2 ⎟ ⎟⎝ % ⎠ '⎠
or for the experimental design of Figure 3.10a, ⎛1 ⎜ ⎜ ⎜ ⎜ ⎜1 ⎜ ⎜ ⎜1 ⎜ ⎜ ⎜ T X Xa = 4 ⎜ ⎜ ⎜ ⎜ ⎜1 ⎜ ⎜ ⎜1 ⎜ ⎜ ⎜1 ⎜ ⎝%
1
1
1
1 1
1
1 1
1
1 1
1 1
1 1
1 1
1
1
1
1
%
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%
1 1
1
1 %
1 1 1
1 %
1
1
1
1 %
1
1
1
1
1 1
1 1
1
1 1
1
1
1
1
1
1
1
%
%
%
%
1 %
%
1 1
1 %
%
1 %
$⎞ ⎛ a0 ⎞ $⎟⎟ ⎜⎜ a1 ⎟⎟ $⎟ ⎜ a2 ⎟ ⎟ ⎟⎜ $⎟ ⎜ a11 ⎟ $⎟ ⎜ a12 ⎟ ⎟ ⎟⎜ $⎟ ⎜ a22 ⎟ $⎟ ⎜ a111 ⎟ ⎟ ⎟⎜ $⎟ ⎜ a112 ⎟ ⎟ ⎟⎜ $⎟ ⎜ a122 ⎟ $⎟ ⎜ a222 ⎟ ⎟ ⎟⎜ $⎟ ⎜ a1111 ⎟ $⎟ ⎜ a1112 ⎟ ⎟ ⎟⎜ $⎟ ⎜ a1122 ⎟ $⎟⎟ ⎜⎜ a1222 ⎟⎟ $⎟ ⎜ a2222 ⎟ ⎟ ⎟⎜ '⎠ ⎝ % ⎠
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239
For the case at hand, one may see the biases by examining either rows or columns. Reporting only the subscripts (e.g., 1222 refers to x12x22), we have the following: 0 ↔ 12 ↔ 22 ↔ 14 ↔ 1222 ↔ 24 ↔ …
(3.65a)
1 ↔ 13 ↔ 122 ↔ …
(3.65b)
2 ↔ 23 ↔ 122 ↔ …
(3.65c)
12 ↔ 132 ↔ 123 ↔ …
(3.65d)
The double arrows identify terms having identical numerical entries in the XTX matrix. There are four such factor series corresponding to the four points in the Figure 3.10a design. Perhaps this will become clearer if we partition XTX/4 by the overall order of its terms and augment the matrix with subscripts for rows and columns. If there is a nonzero entry in the matrix, then the factor denoted by that row and column has collinearity. 0 0 1 2 12 12 22 13 12 2 12 2 23 14 132 12 2 2 12 3 24 %
1
2
1
12
12
1
22
13
12 2
12 2
23
14
1
1
1 1
1
1
1 1
1
1
1
1
1 1
1
1
1
1 1
1
1
1 1
$ $ $ $ $ $ $ $ $ $ $ $ $ $ $ 1
1
1 1
1
1
1 1
1
$
1
1
1
24
1
1 1
1 1
1 1
1
1
1 1
12 3
1
1
1
12 2 2
1 1
1
132
1 1
1
1
1 1 1 1
1
1
As we have only four points, the maximum number of terms we may fit from the data is four — one term from each factor series. Using the lowest-order term in each set, we obtain the following equation (and this is what we show in the figure beside each data set): y = a0 + a1x1 + a2 x2 + a12 x1x2
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(3.66)
240
Modeling of Combustion Systems: A Practical Approach However, we could have just as easily fit the following (absurd and unjustified) model: y = a2222 x24 + a111x13 + a222 x23 + a1222 x1x23 In fact, the coefficients in this latter model will be identical to those of the former for any associated set of response values. We must use good judgment to specify the appropriate mathematical model; the general rule is to include the lowest-order terms possible (Occam’s razor, Chapter 1). The original data matrix for the Figure 3.10a design provides another way of seeing that the Example 3.8 models are identical. ⎛1 ⎜1 y = Xa = ⎜ ⎜1 ⎜ ⎝1
−1 1 −1 1
−1 −1 1 1
1⎞ ⎛ a0 ⎞ −1⎟⎟ ⎜⎜ a1 ⎟⎟ −1⎟ ⎜ a2 ⎟ ⎟⎜ ⎟ 1⎠ ⎝ a12 ⎠
Multiplying the second (x1) and third (x2) columns to produce x1x2 gives the last column in X. One will obtain exactly this XTX matrix with any substitution of the relations in Equation 3.65. For example, the reader may verify that ⎛1 ⎜1 y = Xa = ⎜ ⎜1 ⎜ ⎝1
−1 1 −1 1
−1 −1 1 1
1⎞ ⎛ a0 ⎞ ⎛ 1 −1⎟⎟ ⎜⎜ a1 ⎟⎟ ⎜⎜ 1 = −1⎟ ⎜ a2 ⎟ ⎜ 1 ⎟⎜ ⎟ ⎜ 1⎠ ⎝ a12 ⎠ ⎝ 1
−1 1 −1 1
−1 −1 1 1
1⎞ ⎛ a2222 ⎞ −1⎟⎟ ⎜⎜ a111 ⎟⎟ −1⎟ ⎜ a222 ⎟ ⎟ ⎟⎜ 1⎠ ⎝ a1222 ⎠
When different terms have identical X matrices (and necessarily the same XTX matrices and coefficients), we refer to such terms as aliases. We refer to the set of all such relations as the alias structure. Thus, the relations of Equation 3.65 give the alias structure for the design of Figure 3.10a, and the double-headed arrow (↔) in Equation 3.65 may be read “is aliased with.”
3.5
Fractional Factorials
Factorial designs have excellent statistical properties and we recommend them for designed experiments. However, they suffer from a few faults:
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1. First, the number of required experiments grows exponentially with the number of factors. Full factorials may require too many experiments, e.g., f ≥ 5. 2. Second, we are generally not interested in three-factor interactions or higher. So we should be able to reduce the total number of experimental points if we are willing to sacrifice estimating higher-order interactions. 3. Finally, since each factor is only at two levels, we cannot use factorial designs to evaluate pure quadratic coefficients, e.g., terms like a11x12, etc. Fractional factorials greatly reduce the number of required runs, having 2f–k experimental design points (or alternatively expressed as 1 2 k 2 f ), where f is the number of factors and 1 2 k is the fraction. Thus, k = 0 specifies the full factorial, k = 1 specifies the half factorial, k = 2 specifies the quarter factorial, etc. 3.5.1 The Half Fraction Suppose we wish to study five factors (f =5). Then a full factorial design would require 25 = 32 runs. If experiments are expensive in time or money, we may not be able to afford 32 experiments. But if we were able to afford 16 experiments, we could use the half fraction. We construct it by first making the full factorial in four factors to generate 16 experimental design points (Table 3.11). TABLE 3.11 A Half Fraction in Five Factors Pt
x1
x2
x3
x4
x5
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
– – – – – – – – + + + + + + + +
– – – – + + + + – – – – + + + +
– – + + – – + + – – + + – – + +
– + – + – + – + – + – + – + – +
+ – – + – + + – – + + – + – – +
The columns headed x1 through x4 look like the standard factorial design. To generate the last column we use x5 = x1x2x3x4. Since all the factors have been coded to ±1, the value of x5 will be negative if there are an odd number
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of negative signs in x1x2x3x4 and +1 otherwise. There is another half fraction corresponding to x5 = –x1x2x3x4, and we could have just as easily used it. We call the relation x5 = x1x2x3x4 a blocking relation because making the result to be +1 or –1 divides the design into two blocks. Normally, we will refer to such relations by their ordered subscripts only, e.g., 5 = 1234. However, we have not gotten something for nothing. We have given up some resolution of higher-order interactions. For example, we will not be able to separately assess the difference between the x1x2x3x4 interaction term and x5. If that interaction actually has a strong influence on the result, we will falsely ascribe this effect to x5. We say that x5 is aliased or confounded with the 1234 interaction. That is, for our particular design, factor x5 is just another name for the x1x2x3x4 interaction. We cannot separate which effect is really influential. If both effects are influential, this design will not be able to discriminate between them. And that is not the only alias. In fact, we can write the full alias structure as 1 ↔ 2345, 2 ↔ 1345, 3 ↔ 1245, 4 ↔ 1235, 5 ↔ 1234, 12 ↔ 345, 13 ↔ 245, 14 ↔ 235, 15 ↔ 234, 23 ↔ 145, 24 ↔ 135, 25 ↔ 134, 34 ↔ 125, 35 ↔ 124, 45 ↔ 123 There is an easy way to generate this alias structure without having to look at each factor pattern, or the XTX matrix. We simply take the blocking relation we started with, 1 = 2345, and rewrite it as the blocking generator 12345. To find the factor’s alias, we multiply the blocking generator by whatever factor or factor interaction that interests us and cancel even powers. For example, if we think that the 25 interaction may be important and we want to know what interaction might confound our understanding of that effect, we write (25)(12345) = 1223452 = 134. Then 25 ↔ 134. If the reader wishes to reassure himself that this is so, he may list the signs of the 25 and 134 interactions and compare them. Both patterns are identical. Now we could have selected any blocking generator, not merely 12345. For example, we could have selected 15 as the blocking generator. However, that would have been a horrible choice because the effect of x1 would be indistinguishable from that of x5. In general, when choosing a single blocking generator, we want it to be as large as possible so that main effects are aliased with interactions that have as high an order as possible. We prefer that single factors are clear of aliases among one another and among two-factor interactions if possible.
3.5.2
Quarter and Higher Fractions
Now suppose that we need to assess the effects of six factors, but we cannot afford more than 16 experiments. In such a case, we can use the quarter fraction, i.e., 26–2 = 24 = (1/4)(26) = 16. We construct this design using a full factorial for factors 1234. For factors 5 and 6 we will use the blocking generators 1235 and
© 2006 by Taylor & Francis Group, LLC
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2346. Why choose these two blocking generators? Why not choose 12345 and 12346, for example? Did we not say that we wanted the blocking generators to be as large as possible? Yes, but for more than one generator, we must consider all possible interactions — even those between other blocking generators. If we were to use 12345 and 12346 as blocking generators we would also have the tacit blocking generator (12345)(12346) = 1222324256 = 56. In other words, the blocking generators 12345 and 12346 would result also in a 56 blocking generator, meaning that the x5 and x6 would have exactly the same factor pattern. Try it. It would be impossible to distinguish the effect of factor 5 from factor 6. In contrast, the blocking generators we have recommended for the quarter fraction give (1235)(2346) = 1456. This is not so bad. At least all first-order effects will be aliased with only third- or higherorder effects, and two-factor interactions will be aliased only with other twofactor or higher-order interactions. In factorial designs, we are usually only concerned with interactions up to second order because third- and higher-order interactions are very rarely significant within the design region. So in the case of multiple blocking generators, select them such that they are as large and with as little overlap as possible. This will create blocking generators that are reasonable. Also, since we are holding the blocking generators to be +1 (or –1) the blocking generators will always be aliased with a0, which also has this pattern; e.g., 0 ↔ 1235 ↔ 2346 ↔ 1246. Usually there is no preference over whether we should use 1235, 2346, 1456, or any other similar structure, say 1234, 3456, and 1256. In this latter case, we would write a 24 design using factors 1, 2, 3, and 5 and use the blocking generator 1234 to determine the sign pattern for x4 and 1256 to determine the factor pattern for x6. The numerical factor descriptions for blocking generators are termed words and the number of factors in each word is known as the word length. The resolution of the design is the word length of the smallest blocking generator. Thus, the blocking generators comprise the words 1234, 3456, and 1245 and the design is a resolution 4 design (traditionally expressed by uppercase roman numerals, e.g., IV). Now it may happen that it is impossible to create a fractional factorial with blocking generators that all have equal word lengths. In that case, some factors will have better resolution than others. If one has some a priori knowledge (or even a hunch) regarding an important interaction, one could construct the design so that the selected effects are as free and clear of one another and the remaining factors as possible. In such case, the author finds it easiest to construct the design first and afterward assign the factor identities: 1. Construct the original design using the first f – k factors. 2. Define the blocking generators to determine remaining k factor patterns. 3. If the resulting design does not split the associations the way the investigator desires, reassign the factors after constructing the design.
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This requires less mental effort than reblocking the design to achieve predetermined associations.
Example 3.9
The Quarter Fraction in Five Factors
Problem statement: 1. Design an eight-run experiment for five factors so that the alias structure is as uniform as possible. 2. Is there a factor or factors that have a preferred alias structure? If so, specify it or them. 3. Suppose the five factors are oxygen concentration (O), furnace temperature (T), hydrogen content in the fuel (H), air preheat temperature (A), and fuel pressure (P) and the investigator is particularly interested in the OT and OH interactions? To which factors should we assign O, T, and H? 4. What is the overall resolution of the design? 5. Comment on the suitability of the design. Solution: For this design, f = 5 and the total number of runs is 23. Therefore, k = 2. So we construct the design in eight runs using the first three factors. 1. For the alias structure, we select the largest blocking generators having the smallest possible overlap. No matter how we try, the best we can do is to have one four-factor block and two three-factor blocks. Any other choice will result in a design that has lower resolution. No matter what, the alias structure is unequally distributed. Suppose we choose 1234 and 125, resulting in the tacit blocking generator 345. Then the alias structure becomes 0 2 4 13
↔ 1234 ↔ 125 ↔ 345 ↔ 134 ↔ 15 ↔ 2345 ↔ 123 ↔ 1245 ↔ 35 ↔ 24 ↔ 235 ↔ 145
1 ↔ 234 ↔ 25 ↔ 1345 3 ↔ 124 ↔ 1235 ↔ 45 5 ↔ 12345 ↔ 12 ↔ 34 14 ↔ 23 ↔ 245 ↔ 135
2. All main factors are aliased with a two-factor, three-factor, and four-factor interaction except for x5. It is aliased with two two-factor interactions and one six-factor interaction. Since two-factor interactions are often important, x5 has a greater chance of being confounded with an important twofactor interaction.
© 2006 by Taylor & Francis Group, LLC
Experimental Design and Analysis
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3. Here are all the one- and two-factor interactions: 1 ↔ 25, 2 ↔ 15, 3 ↔ 45, 4 ↔ 35, 5 ↔ 12 ↔ 34, 13 ↔ 24, 14 ↔ 23. Since single-factor effects (also called main effects) are the most likely to be important, we prefer to assign OT to either 13 ↔ 24 or 14 ↔ 23 and let OH take whichever remains. Any other assignment will end up with OT and OH confounded with main effects. Having no particular preference, the author chooses to assign O to x1 and T to x3. This leaves H as x4. Then neither OT nor OH is aliased with a main effect. 4. The design is resolution 3 overall because this is the word length of the shortest blocking generators (125 and 345). More precisely, one could report the arithmetic average word length of the blocking generators, 3-1/3. 5. Resolution 3 designs have main effects confounded with two-factor interactions. We were able to choose our factor assignments in a way that gave us sufficient resolution for a couple of interactions we thought might be important. But this will not always be the case. Even if we can do it, we may not be able to do it for all or many two-factor interactions. For example, it would not have been possible to separate out OT from AH in the present case. If one knows a priori that two-factor interactions are unlikely, then a resolution 3 design is acceptable. If one knows a priori that many two-factor interactions are likely to be important, a resolution 3 design is unacceptable. In general, we prefer designs that are resolution 4 or higher so that main effects are not aliased with two-factor interactions.
3.6
ANOVA with Genuine Replicates
A genuine replicate is a repeated point that is subject to the same sources of error as the other points in the design. Genuine replicates differ only by pure error (also called experimental error or random error). The usual strategy is to salt the experimental design with genuine replicates. One then randomizes the run order of all the experimental points. Now if there were no error, the replicates would necessarily give identical responses. Therefore, we can use error estimates derived from the replicates to independently test if higherorder effects are significant. If they are not, we can pool them to obtain even more degrees of freedom to estimate experimental error.
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Example 3.10 Genuine Replicates
Problem statement: 1. Given the data in Table 3.12, decide which effects are significant. Use an ANOVA table and separately assess pure error. 2. According to an F test with P ≤ 0.10, which factors are significant? What is the standard error and correlation coefficient? 3. On the assumption that the nonsignificant effects merely represent experimental error, pool these effects into the error term and reestimate r2 and σ. Are there any differences between these and the former estimates? Why or why not?
Solution: 1. Using the methods given in the text, we obtain Table 3.13. Note that in the table we have relabeled the residual in the ANOVA, calling it the pure error. 2. From the table we see that only the main effects are significant (bolded type), with greater than 90% confidence (P ≤ 0.10). From s = √MSE, we have s = 2.07 and r2 = 94.1%, approximately. 3. On the presumption that the nonsignificant effects represent random error, we pool them into the error term. This gives the ANOVA with pooled effects of Table 3.14. From this, we calculate s = 2.08 and r2 = 90.0%. Now we notice that s is virtually identical to its former value, while r2 has dropped from 94.1 to 90.0%. This is quite sensible. Regarding s, if the pooled effects are really just noise, then the MSE should be about the same before and after pooling. However, we will know it with better certainty because we are dividing by more degrees of freedom. In fact, the F tests are now more sensitive and the confidence factors have changed slightly for the better. Regarding r2, it has declined and this is also sensible. A least squares model with only four effects must do a poorer job of fitting the data than one with 16 effects. The fact that r2 has not declined much even after eliminating 12 terms from the model is another indication that the additional effects were statistically insignificant.
© 2006 by Taylor & Francis Group, LLC
Replicate Data Data Point
x1 Fuel Split
x2 Excess Oxygen
x3 Fuel Pressure
x4 Burner Size
1a 1b 2a 2b 3a 3b 4a 4b 5a 5b 6a 6b 7a 7b 8a 8b
– – – – – – – – – – – – – – – –
– – – – – – – – + + + + + + + +
– – – – + + + + – – – – + + + +
– – + + – – + + – – + + – – + +
y NOx
Data Point
x1 Fuel Split
x2 Excess Oxygen
x3 Fuel Pressure
x4 Burner Size
y NOx
10.8 6.6 12.9 14.5 9.9 11.7 19.3 17.9 17.7 19.9 27.1 24.0 21.7 23.3 23.9 25.8
9a 9b 10a 10b 11a 11b 12a 12b 13a 13b 14a 14b 15a 15b 16a 16b
+ + + + + + + + + + + + + + + +
– – – – – – – – + + + + + + + +
– – – – + + + + – – – – + + + +
– – + + – – + + – – + + – – + +
12.5 15.5 19.2 18.0 14.1 15.4 21.0 22.0 28.6 21.4 28.5 28.2 26.9 22.1 26.5 28.4
Experimental Design and Analysis
TABLE 3.12
247
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Modeling of Combustion Systems: A Practical Approach TABLE 3.13 ANOVA for Replicate Data Effect
DF
SS
MS
F
P
Model x1 x2 x1x2 x3 x1x3 x2x3 x1x2x3 x4 x1x4 x2x4 x1x2x4 x3x4 x1x3x4 x2x3x4 x1x2x3x4
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
117.43 728.67 1.49 18.76 7.51 10.24 0.14 195.53 2.26 9.57 0.23 0.00 1.40 11.88 2.70
117.43 728.67 1.49 18.76 7.51 10.24 0.14 195.53 2.26 9.57 0.23 0.00 1.40 11.88 2.70
27.28 169.30 0.35 4.36 1.74 2.38 0.03 45.43 0.52 2.22 0.05 0.00 0.33 2.76 0.63
0.0001 <0.0001 0.5648 0.0532 0.2052 0.1425 0.8602 <0.0001 0.4794 0.1554 0.8210 0.9799 0.5760 0.1161 0.4397
Pure error
16
68.87
4.30
Total
31
1176.66
r2
94.1%
s
2.07
TABLE 3.14 ANOVA with Pooled Effects Effect
3.6.1
DF
SS
MS
F
P
Model x1 x2 x3 x4
1 1 1 1
117.43 728.67 18.76 195.53
118.06 728.78 18.51 194.87
27.28 169.30 4.36 45.43
<0.0001 <0.0001 0.0464 <0.0001
Error
27
116.28
4.31
Total
31
1176.66
r2
90.1%
s
2.08
Bias Error
Error, broadly speaking, is the difference between our model and reality. This difference may be due to bias error, pure error, or some combination of both. If bias exists, it will accumulate in the residual error term along with any random error. Suppose that we fit a response with a full factorial: n
y = a0 +
n− 1
n
j< k
k =1
n− 2 n− 1
n
∑ a x + ∑∑ a x x + ∑∑∑ a k k
k =1
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jk j k
x x x $+ e
ijk i j k
i<j
j< k
k =1
(3.67)
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For the moment, let us presume that this is the true model. Then its error term, e, must comprise pure error only. Now, if we elect to fit the simple first-order model instead, n
y = a0 +
∑a x + e k k
r
k =1
then er (the residual error) will have contributions from not only experimental error, but also from bias (eb) caused by an incorrect model specification. Thus, er = eb + e. Let us partition the residual error into two components, the pure error and the bias error: n
y = a0 +
∑a x + e + e k k
(3.68)
b
k =1
Then subtracting Equation 3.66 from Equation 3.67 gives the bias error: n− 1
eb =
∑∑ j< k
n− 2 n − 1
n
k =1
ajk xj xk +
n
∑∑∑ a
xxx $
ijk i j k
i<j
j< k
(3.69)
k =1
In other words, eb comprises all the nonrandom stuff we should have accounted for in the model specification. Since e is a random variable, its variance should distribute as a chi-squared distribution and estimate σ2. Likewise, if it turns out that there is no significant bias, then the associated variance, σb2, should also distribute as a chi-squared distribution. Thus, we should be able to refer σb2/σ2 to an appropriate F distribution and find that σb2/σ2 is not significantly greater than unity. On the other hand, if Equation 3.69 is meaningful, then σb2/σ2 should be significantly greater than 1. We can obtain estimates for both σb2 and σ2 from the ANOVA. Since eb is the sum of orthogonal factorial effects, it is also orthogonal to the model terms. Then we may also break this out separately in the ANOVA. Finally, we note that genuine replicates can differ only by pure error (σ). We shall call this variance the sum-of-squares error (SSE). Then subtracting SSE from SSR will give us the sum-of-squares bias, SSB = SSR – SSE. Likewise, we may do the same for the degrees of freedom: DFB = DFR – DFE. Then, of course, MSB = SSB/DFB and MSE = SSE/DFE. But MSB ~ σb2 and MSE ~ σ2 (where ~ means “is asymptotically equal to,” i.e., equal in the long run). Table 3.15 gives the slots and shows what fill them. We may now divide SSR into two orthogonal components, SSB and SSE, and make two F tests, one for F1–P (DFM, DFE) = MSM/MSE and another for F1–P (DFB, DFE) = MSB/MSE. If SSB does not differ significantly from SSE, then we have no significant bias and SSR estimates the pure error. If so, we may pool our estimates into a single error term having more degrees of freedom.
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Modeling of Combustion Systems: A Practical Approach TABLE 3.15 The Naked ANOVA with Replicates Term
DF
M
SS T
T
T
F1–p = MSM/MSE
a1(X y)1 a2(XTy)2
a1(X y)1 a2(XTy)2
a1(X y)1/MSE a2(XTy)2/MSE
1 DFB
an(XTy)n SSB
an(XTy)n MSB
an(XTy)n/MSE MSB/MSE
DFE
SSE
MSE
DFT
SST
P
T
1 1
R
MS
r2 = SSM/SST s = √SSE
TABLE 3.16 ANOVA with Replicates Term M
R
T
DF
SS
MS
F1–p = MSM/MSE
P
x1 x2 x3 x4 B
1 1 1 1 11
117.43 728.67 18.76 195.53 47.42
117.43 728.67 18.76 195.53 4.31
27.28 169.30 4.36 45.43 1.00
0.0001 <0.0001 0.0532 <0.0001 0.4851
E
16
68.87
4.30
r2
90.1%
31
1176.66
s
2.07
Since factorial models have orthogonal terms, one may also divide SSM into orthogonal components — one for each model term in the case of the factorial designs. Table 3.16 shows the figures with the data of Example 3.10 represented this way. From the table we note that MSB = 4.31, while MSE = 4.30. Clearly, the P value of 0.49 is nowhere near a threshold value of 0.10 or 0.05, so MSB is not significantly different than MSE. Thus, we may dismiss the notion that the model has significant bias and leave the residual pooled. We discuss bias error in additional detail in Chapter 4. 3.6.2 Center-Point Replicates For a 23 factorial design, we could replicate all eight points. But this may be too expensive in terms of time or money. We could replicate some rather than all points. However, if we do not replicate points symmetrically about the design center, then we will destroy the balance of the design and its orthogonality. A clever way of measuring pure error with as few replicates as possible is to augment the factorial design with center points. Since center points are located at the origin of the design, all powers and interactions are identically zero. Thus, they do not harm a factorial design’s orthogonality. They also give us information about a new point in factor space. Since they are replicates, they also lead to estimates for pure error.
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Let us once again consider the 22 factorial in order to fit the first-order model: y = a0 + a1x1 + a2 x2 + e
(3.70)
Since we have four points and three coefficients, we have an additional degree of freedom left to estimate experimental error, e. However, if the x1x2 interaction is significant, then the actual model is y = a0 + a1x1 + a2 x2 + a12 x1x2 and there are no degrees of freedom left to estimate experimental error. Moreover, if we attempt to fit Equation 3.70 and falsely ascribe a12x1x2 to the error term, we will inflate the error term. This may cause us to reject significant coefficients falsely when we perform our F test. Replicate center points avoid this dilemma. Consider the factorial design plus center-point replicates given in Table 3.17. TABLE 3.17 A 22 Factorial Design with Center Points Point
x1
x2
1 2 3 4 5 6
– – + + 0 0
– + – + 0 0
Examining the XTX matrix, we see that X remains orthogonal: ⎛6 X X = ⎜⎜ ⎜⎝ T
4
⎞ ⎟ ⎟ 4⎟⎠
Only the first element of the principal diagonal increases. Center-point replicates also allow us to test for model bias (also called lack of fit). The ANOVA is a straightforward extension of earlier versions. Table 3.18 shows them. If the model terms are orthogonal, then we can make separate entries for them in the ANOVA as shown. Otherwise, we could pool the effects into things like SSM and SSR, because at the very least these will be orthogonal for any set of data whatsoever. 3.6.2.1 Degrees of Freedom Entries 3.6.2.1.1 Model Terms By definition, there are p model terms, one of which estimates the mean; therefore, DFM = p – 1 (general)
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(3.71)
252
TABLE 3.18 ANOVA for Orthogonal Pooled or Unpooled Entries Effect
SS
MS
F
P
x1, j yj
1
SSx1/1
MSx1/MSE or MSx1/MSR
P(1, DFE) or P(1, DFR)
x2 , j yj
1
SSx2/1
MSx2/MSE or MSx2/MSR
P(1, DFE) or P(1, DFR)
x p , j yj
1
SSxp/1
MSxp/MSE or MSxp/MSR
P(1, DFE) or P(1, DFR)
DFM = p–1
MSM = SSM/(p – 1)
F(DFM, DFE)
P(DFM, DFE) or P(DFM, DFR)
DFB = u–p
MSB = SSB/(u – p)
F(DFB, DFE)
P(DFB, DFE)
DFE = n–u
MSE = SSE/(n – u)
DFR = n–p
MSR = SSR/(n – p)
n
x1
a1·(XTy)1 = a1
if orthogonal
Model, M
j =1 n
x2
a2·(XTy)2 = a2 j =1
n
xp
ap·(XTy)p = ap j =1
p
or
M
n
a–1TX–1Ty =
ak
x k , j yj
k =1
j =1
Residual, R
u
Model bias, B
(
rk yˆ k
SSB =
yk
)
2
k =1 u
Pure error, E
rk
(y
SSE =
yj
j,k
)
2
j =1 k =1
or
R
SSB + SSE = SSR n
Total, T
(y
SST = k =1
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k
y
)
2
DFT = n–1
r2 = SSM/SST s = MSE
Modeling of Combustion Systems: A Practical Approach
DF
Experimental Design and Analysis
253
If the model effects are orthogonal, the ANOVA can include separate entries for each parameter and DFxk = 1
(3.72)
where k is the factor index going from 1 to p – 1. 3.6.2.1.2 Residual Terms The residual comprises whatever degrees of freedom the model does not: DFR = n – p (general)
(3.73)
where n is the total number of observations, including replicate observations. We may further subdivide the residual into other components such as model bias and experimental error. 3.6.2.1.3 Pure Error In the case of genuine replicates, we can separately assess the pure error. If there are u unique points in factor space and n points in total, then the number of replicates available to assess pure error must be n – u + 1. Since we are making all of our measurements relative to a mean, then the degrees of freedom for pure error are 1 less than n – u + 1: DFE = n – u (for replicated data, generally)
(3.74)
If the total number of replicates are center-point replicates, nc, then clearly nc = n – u + 1
(3.75)
Substituting Equation 3.74 into Equation 3.73 for the case of center-point replicates gives DFE = nc – 1 (for replicated center points)
(3.76)
3.6.2.1.4 Model Bias Model bias must include whatever degrees of freedom the residual has left over after one accounts for pure error. Since DFR = DFE + DFB, DFB = DFR – DFE
(3.77)
Substituting Equations 3.73 and 3.74 into Equation 3.77 for DFE and DFR, respectively, gives DFB = u – p (for replicated data, generally)
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(3.78)
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Modeling of Combustion Systems: A Practical Approach
p=6 u=9 n = 12
DFM = p –1 = 5 DFB = u – p = 3 DFE = n – u = 3 DFT = n –1 = 11
y = a0 + a1x1 + a2x2 + a11x12 + a12x1x2 + a22x22 p=6 n = 12 nc = 4
DFM = p –1 = 5 DFB = (n – nc ) – (p –1) = 3 DFE = nc –1 = 3 DFT = n –1 = 11
FIGURE 3.12 Illustration of degrees of freedom. The design is a central composite. Using the equations given in the chapter, one arrives at the appropriate degrees of freedom for the model, the bias, and the pure error. The top set of equations are the general ones. The bottom set are for designs whose replicates are exclusively center points, as is the present case. The center points, though coincident, are offset for clarity.
If our only replicates are center points, then we may solve for u in Equation 3.75 and substitute this into Equation 3.78: DFB = (n – p) – (nc – 1) (for center-point replicates)
(3.79)
From Equation 3.78, we see that the number of unique points in a design increases the degrees of freedom for assessing bias. Thus, center-point replicates are valuable in that they provide yet another coordinate in factor space. To underscore the arithmetic, consider Figure 3.12. 3.6.2.2 Sum-of-Squares Entries 3.6.2.2.1 Model Terms Equation 3.80 gives the sum of squares for the entire model: p
SSM = a–0TX–0Ty =
⎛ ak ⎜ ⎜⎝
n
∑ ∑ k =1
j =1
⎞ xk ,j yj ⎟ (general case) ⎟⎠
(3.80)
where the subscript –0 refers to a model missing the effect a0. (Equivalently, one could compute a TX T y − ny 2 , since a −0 TX −0 T y = a TX T y − ny 2 .) This will be the sum of squares for all model terms combined. Equation 3.80 is equivalent to the sum of the products of the individual entries in the XTy matrix multiplied by their respective coefficients, excluding the entries for a0. If the model has orthogonal effects, one may partition SSM into its individual orthogonal components. Each orthogonal term is
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Experimental Design and Analysis
255
n
SSxk = ak
∑x
y
k ,j j
j =1
We do this for each of p – 1 model parameters; we do not count a0, i.e., the mean, because the total variance is the sum of squared differences from the mean. 3.6.2.2.2 Residual Terms The residual comprises whatever variance the proposed model does not. If the model is accurate, the experiments carefully performed, etc., then the residual comprises only pure error. However, if the model is incorrect (e.g., a linear model when the true function is quadratic), then the residual will also contain bias due to an incorrect model. The only way to estimate pure error separately is to perform replicate measurements. 3.6.2.2.3 Pure Error In the general case for replicates, the replicate points occur at various locations in factor space, and we seek the variance between each point of the replicate set and the replicate mean. Consider the jth replicate set having a replicate mean yj . Then the variance for the kth point in the replicate set with its mean is ( yjk − yj )2 . If there are rj replicate points in the jth replicate set, then the sum of squares for that replicate set is rj
∑( y
jk
− yj
k =1
)
2
For example, for the four center points shown in Figure 3.12, rj = 4 and the mean response there is yj . Then the sum of squares (variance) for that replicate set is rj = 4
∑( y k =1
j,k
− yj
) = (y 2
j ,1
− yj
) + (y 2
j ,2
− yj
) + (y 2
j ,3
− yj
) + (y 2
j,4
− yj
)
2
Now in Figure 3.12, this is the only replicate set, and thus this calculation would comprise all of the SSE. However, in the general case there may be more replicate sets. If so, then we must repeat this summation for each replicate set. Then if there are nr replicate sets, we have nr
SSE =
rk
∑ ∑( y j =1 k =1
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j,k
− yj
)
2
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In fact, we could repeat this procedure for every unique point in factor space — not just the points with replicates — because unreplicated points are their own mean: yjk ≡ yj whenever yjk is not replicated. Thus for unreplicated points, rk
∑( y
) = ∑( y − y )
− yj
j,k
1
2
j
k =1
j
2
≡0
k =1
Therefore, if we do not wish to introduce a new parameter, nr , we may write u
SSE =
rk
∑ ∑( y
j,k
− yj
j =1 k =1
)
2
(general case)
(3.81)
If we have a design with only center-point replicates, then nc
SSE =
∑( y k =1
c,k
− yc
)
2
(center points replicated only)
(3.82)
where nc is the number of center points, yc,k is the value of the kth center point, and yc is the center-point mean. 3.6.2.2.4 Model Bias If experimental error were nonexistent, then bias would be any difference between what our model predicts and what we actually obtain, yˆ − y, and the sum of squares, bias would be n
∑ ( yˆ − y ) k
2
k
k =1
In the face of experimental error, we can use replicate measurements as an estimate of the true value. Then instead of taking the difference yˆ k − yk, we will instead take yˆ k − yk , where yˆ k is the predicted value by the model of the kth point and yk is the mean at that point. Now the mean and predicted values are invariant for a given position in factor space, so if one replicates them rk times, the sum of squares will become u
SSB =
∑ r ( yˆ − y ) k
k =1
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k
k
2
(3.83)
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where u is the total number of unique points in factor space, k indexes them, and rk is the number of replicates at the kth point in factor space. If we come to a point that is not replicated, then rk = 1 and yk ≡ yk. However, ( yˆ k − yk )2 is not necessarily zero even for unreplicated points. 3.6.2.2.5 Total Sum of Squares The total sum of squares is the sum of the variance of every point from the grand mean, y : n
SST =
∑( y − y )
2
(3.84)
k
k =1
The residual is whatever we have left over after we have fit our model. Thus, SSR = SST – SSM
(3.85)
Or in terms of the quantities we have derived, n
SSR =
∑( y − y ) − a 2
k
T −0
X T−0 y
(3.86)
k =1
3.6.3
Standard Errors and the t Test
Equation 3.87 gives the variance–covariance matrix for least squares:
(
σ 2k = σ 2 X TX
)
−1
(3.87)
where σ 2k is a matrix of values according to ⎛ σ 20 ⎜ σ 2k = ⎜⎜ ⎜ ⎝⎜ sym
σ 0σ 1
$
σ
$
2 1
'
σ 0 σ p −1 ⎞ ⎟ σ 1σ p − 1 ⎟ % ⎟⎟ σ 2p−1 ⎟⎠
(3.88)
The diagonal elements are the variances; the nondiagonal elements are the covariances; σjσk measures the mutual bias of aj and ak coefficients upon one another, where the subscripts refer to the respective factors. For orthogonal matrices, the nondiagonal elements vanish, meaning that the regression coefficients are unbiased. We estimate σk2 from sk2 from the actual data using Equation 3.89.
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(
s2k = s2 X TX
)
−1
(3.89)
where s2 = MSE, which comes from the ANOVA. Then the diagonal elements of Equation 3.89 are the standard errors for each coefficient, often written as ak [±sk]. The ratio of the coefficient value to the standard error is the t ratio: t=
ak sk
(3.90)
where sk is defined by the square root of the diagonal elements of Equation 3.89. If we want to know if a particular ak is significant, we can use the F test as already described, or we can use the equivalent t test. Statistical programs generate this regression statistic. One may also use the Excel function tdist(x,DF,2), where x is the estimate for the coefficient, DF is the degrees of freedom (n – p), and 2 performs the two-tailed test, that is, that the interval –sk·tn–p < ak < sk·tn–p does not include zero. As we have already noted, the t statistic is identical to √F(1, n). Tests with this statistic necessarily give exactly the same confidence intervals, etc. Table 3.19 reports results for Table 3.4, but using t tests on the effects rather than F tests on the mean squares. TABLE 3.19 t Test Coefficient
Estimate
Std Err
t Ratio
P
a0 a1 a2 a3
3.8288 1.2563 0.3338 0.6313
0.0994 0.0994 0.0994 0.0994
38.53 12.64 3.36 6.35
<0.0001 0.0002 0.0283 0.0031
However, s, as reported in Table 3.6, is not identical to sk in Table 3.19 because the former is the estimate of the response (s = √MSE) while the latter is the estimate on the kth effect (sk). Equation 3.89 relates them. If we want to estimate the response variance, sy2 , we can write a relation in terms of the hat matrix, H = X(XTX)–1XT: s2y = s2H
3.6.4
(3.91)
The Value of Orthogonal Designs with ANOVA
If X is not orthogonal, we cannot assess each effect separately. For a model having p adjustable coefficients we will have 2p possible models, and strictly speaking, we must use separate ANOVA tables for each. For example, the
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model y = a0 + a1x1 + a2 x2 + a3 x3 will generate 23 submodels (presuming each model retains a0; otherwise, there are 24 submodels including y = 0). The 23 models are 1. y = a0
2. y = a0 + a1x1
3. y = a0 + a2 x2
4. y = a0 + a3 x3
5. y = a0 + a1x1 + a2 x2
6. y = a0 + a1x1 + a3 x3
7. y = a0 + a2 x2 + a3 x3
8. y = a0 + a1x1 + a2 x2 + a3 x3
This is not a problem for a few effects. However, if we are screening 12 possible factors for significance, 212 = 4096; these are a lot of models to fit. There are generally reliable ways to find the best models without a complete regression of all possible models, but they require dedicated statistical software. However, orthogonal designs eliminate the requirement to fit multiple models because we can examine all orthogonal effects at once with a partitioned ANOVA table.
3.6.5
Rotatability
Adding additional center points not only preserves orthogonality, but also preserves rotatability. We refer to designs with radially symmetrical information fractions as rotatable designs, and the property itself as rotatability. Figure 3.13 gives the information fraction for this first-order design. 1.0 37.5% 50.0% 62.5% 75.0% 87.5% 100% x
0.5
0.0
-0.5
-1.0 -1.0
-0.5
0.0
0.5
1.0
FIGURE 3.13 Information fraction for factorial with center points. This factorial design (open circles) has its maximum certainty at the center of the operating region (100%, x marks the spot). It also has radially symmetrical information contours. It is orthogonal and rotatable. The center points are coincident but have been jittered for clarity.
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The equation is i( x1 , x2 ) = 2 (2 + 3 x12 + 3 x22 ). Perhaps the most striking feature of the design is that the information fraction is radially symmetrical. We may see this by expressing i(x1, x2) as a function of the number of center-
(
)(
)
−1
point replicates, nc, i( x1 , x2 ) = ⎡1 + nc n f + 1 x12 + x22 ⎤ , where nc is the num⎣ ⎦ ber of center points and nf is the number of factorial points. Since x1 and x2 scale identically with nc, the function remains radially symmetrical in x1·x2 factor space regardless of the number of center points. That is, the design’s rotatability is invariant with nc . The advantage of a rotatable design is that the information fraction does not depend on the orientation of the coordinate axes. This is important because the design factors may not be the true factors. For example, the disappearance for a reactant in a gas-phase chemical reaction is a function of important factors such as the collision frequency, the reaction mechanism, the concentration of reactants and products, etc. We shall call these the penultimate factors. However, one cannot measure such factors directly. Instead, one measures pressure, temperature, and the like, because these related measures are more convenient. We shall call these the conventional factors. Guided by theory, we may be able to express penultimate factors in terms of conventional ones. But even if not, a rotatable design will preserve statistically beneficial properties if the unknown penultimate factors are linear combinations of the conventional factors. In essence, the design remains invariant under a rotation of axes. In Chapter 4, we show how to express such penultimate factors in terms of linear combinations of conventional factors. However, rotatability is not an all-or-nothing proposition. Practically speaking, if a design is approximately rotatable, that will be good enough for our purposes.
3.7
Randomization
In Section 3.3.3, we mentioned that randomizing the run order is essential to negate serial correlation. Serial correlation is order-dependent influence. For example, certain equipment or process cycles, atmospheric conditions, etc., may influence our response unbeknownst to us. If we do not plan for such an eventuality, we may falsely ascribe it to a particular factor or factors. That could cause us to falsely include an uninfluential factor or falsely exclude an influential one from our model. Then our model will lack predictive power. Serial correlation may also result from hysteresis. 3.7.1 Hysteresis Hysteresis is the memory effect of a response or factors whereby prior levels affect subsequent responses. Examples of systems that exhibit hysteresis are dead (inactive) time in controller response (process control applications) or
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dead (empty) volume in a vessel, or processes that change over time (e.g., human or animal learning). Play or slop in a gear (loose fit) will also cause hysteresis. Randomization will tend to distribute the effects of hysteresis over all the factors, inflating the error but introducing no bias. 3.7.2 Lurking Factors Serial correlation may also result from a lurking factor — an unknown variable that is influential but not part of the experimental study. If a lurking factor influences the response, then randomization will tend to distribute this unknown influence among all the factors. This will inflate the overall variance, but it will not bias our coefficients. On the other hand, lurking factors, by definition, are not controlled factors. Therefore, they will tend to vary with a plant or diurnal cycle, or whatever is producing them. It is likely that these will trend up or down over time. If we run our experiments in other than random order, we are likely to alias a lurking factor with a design factor. Therefore, we will not distribute the unknown factor influence, but project it onto a design factor. Then we will wrongly ascribe the effect of the lurking factor to a design factor even when the design factor has no influence on the response. For example, suppose that the ambient temperature is an important influence on a particular response, but that we do not know that ambient temperature is an important factor. Table 3.20 gives some hypothetical results. TABLE 3.20 A Nonrandomized Experiment Temp, °C 11 13 15 17 19 20 22
x 3 2 1 0 –1 –2 –3
y 21.7 25.6 30.0 33.8 37.7 39.9 43.1
Please note that x is an independent factor that may be set arbitrarily at the whim of the investigator. However, he has foolishly chosen not to randomize the run order; now x and ambient temperature (the real factor of influence) are hopelessly aliased. Suppose that out of convenience, the investigator has run the design starting with a high level of factor x and ending a low level. He wishes to fit the following model: y = a0 + a1x + e Now if he starts his experiments in the morning and finishes them in the afternoon, the ambient temperature will have steadily increased. The investigator supposes he is measuring the effect of x on y and derives the following expression:
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He dutifully summarizes the data in Table 3.21. TABLE 3.21 Analysis of Variance for Data for a Nonrandomized Experiment Effect Model, M Residual, R Total, T Effect a0 a1
SS
% SS
DF
MS
360.72
99.2
1
360.72
2.99
0.8
5
0.60
363.71
100.0
6
Estimate 33.11 –3.59
604.15
P <0.0001
r2 = 99.2% s = 0.77
Standard Errors Std Err Std Beta 0.29 0.15
F(1, 5)
t Ratio
0.00 –1.00
113.3839 –24.5795
P <0.0001 <0.0001
From the data we see that the model has r2 = 99.2%, and the investigator has greater than 99.99% confidence (p < 0.0001) that the model is statistically significant. Be that as it may, the analysis is nonsense. The reader has the benefit of the author’s third-person omniscient narrative for this story. But in real life, we will never know when serial correlation may come knocking. Therefore, the prudent course of action is to randomize the run order. The wide scatter in a randomized experiment should alert us to look for other sources of variation, including lurking factors. 3.7.2.1 Ameliorating Lurking Factors Suppose we randomize the run order. We cannot randomize unknown effects, so the ambient temperature behaves as before, increasing with run order. But this time, the run order is randomized. Table 3.22 shows the new run order. TABLE 3.22 A Randomized Experiment Temp, °C
x
y
11 13 15 17 19 20 22
2 –1 1 –2 0 3 –3
21.7 25.6 30.0 33.8 37.7 39.9 43.1
Table 3.23 is markedly different from its predecessor. Now the lurking factor is no longer falsely aliased to the design factor and our r2 drops from 99.2 to 8.5%. This indicates that the model has no significant correlating ability, which is the proper conclusion.
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TABLE 3.23 Analysis of Variance for a Randomized Experiment Effect Model, M
% SS
DF
MS
F(1, 5)
P
0.46
0.5261
30.87
8.5
1
30.87
Residual, R
332.84
91.5
5
66.57
Total, T
363.71
100.0
6
Effect a0 a1
3.8
SS
Estimate
Standard Errors Std Err Std Beta
33.11 –1.05
3.08 1.54
0.00 –0.29
r2 = 8.5% s = 8.16 t Ratio
P
10.7382 –0.6810
0.00012 0.52613
About Residuals
Residuals provide all that we can know statistically about errors between our model and reality. Therefore, any analysis is incomplete unless one also analyzes the residuals.
3.8.1
Residuals vs. Run Order
One diagnostic that one should always perform is to plot residuals vs. run order. If the residuals comprise only random error, they should show no discernable pattern. Figure 3.14 shows the plot for the randomized data. Even though the data were randomized, we see a linear trend of residuals with run order, indicating serial correlation. Now, no diagnostic is foolproof. However, these two simple things — randomizing the experimental run order and examining the residuals— will often keep the investigator from being led astray.
3.8.2
Other Residual Plots
If our factors account for all the significant sources of variation, then the residuals will result from a plethora of insignificant influences. We know from the central limit theorem that these will distribute as a normal distribution having a particular mean and a standard deviation. Then a plot of residuals against each factor should show no discernable pattern. In particular, they should not show a linear trend, a curvilinear or sinusoidal trend, or any tendency to converge or diverge (funnel shape).
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Residual (Actual – Predicted)
15.0 10.0 5.0 0.0 -5.0 -10.0 -15.0 Run Order FIGURE 3.14 Residual vs. run order. The validity of our model depends on the error being randomly distributed. One should always check this assumption. Standard checks comprise plotting the residual as a function of time, run order, each factor, and the response. We expect to observe only random behavior. However, this plot clearly shows a linear trend, indicating nonrandom response with run order. So alerted, the investigator should proceed to find the reason for this behavior. Some possibilities are lurking factors, collinear data, serial correlation, and model bias, to name a few.
Another residual plot we should always examine is a residuals vs. fitted response plot. The residuals should show random behavior. Plots such as e vs. xk and e vs. yˆ should show a constant error variance (i.e., they should not diverge, converge, cycle, or trend in any direction. (Please note that one should not plot residuals against the actual response values. Since the actual response contains the residual, a plot of residuals against actual response will show a trend. This is normal and does not indicate problem data.)
3.8.3
Full and Block Randomization
Were we to randomize the order of the experimental series (including the replicates), the replicates would be just as likely to experience the same variation in error as any other design point. If we do not randomize our experimental order, we will not account for all sources of variation. Were we to run the replicates in sequence, we would not have genuine replicates, we would not be exposing the replicates to the full design variation. However, there are times when it is not possible or desirable to run the design in a single block and randomize all the experiments including replicates. There may be so many experiments that it will take days or weeks to complete them all, and something influential may change over this length of time. In such cases, it will be advantageous to break the experimental series into several
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orthogonal pieces, randomize each piece, and sprinkle each piece with replicate points.
3.8.4
Blocking
As we have learned, randomization may inflate the variance but keeps the coefficients unbiased. However, an inflated variance can also be a problem because it makes the F test less sensitive by increasing the denominator in the F ratio. This may lead us to falsely accept the null hypothesis and remove a significant effect from consideration. Blocking can reduce the error variance and make our F tests more sensitive. Blocking is the segregation of factors into groups such that most of the expected variance occurs between groups rather than within them. As such, it is a restriction on randomization. For example, suppose we want to study three factors in a 23 factorial design — let us say NOx from an oil burner as a function of oxygen concentration, steam/fuel ratio, and furnace temperature. However, constraints on the furnace time mean that we can only conduct four experiments now and four experiments 3 weeks from now when the furnace is again unencumbered. How should we conduct our experiments? We should first note that many potentially influential factors are out of our control. For example, the fuel oil may change over time — it may form sediment, lose volatiles, oxidize, or who knows what. Of course, it would be useful to remove as many sources of variation as possible. But in all likelihood, there will be differences in the fuel oil today vs. fuel oil 3 weeks from now. We also cannot know what the weather will be; ambient humidity and temperature can affect important combustion-related responses. Suppose someone will need this particular burner in the meantime. They may change out parts or modify the burner with the intent to restore it to its original condition later. However, the restored condition may differ from the original in an unknown and unintended way. At any rate, it would seem best not to have all these possible sources confound our assessment of the subject factors. Blocking can remove these sources of variation from the residual. The best kind of blocking will be blocking that is orthogonal to the model effects. Orthogonal blocking allows independent block and factor entries on the ANOVA. Actually, we already know how to block factorial designs orthogonally. We merely select a blocking generator as we did in creating fractional factorials. Only this time, we will run both fractions — half now, half later. We can decide which fraction we will run first with a coin toss.
3.8.5
Random vs. Fixed Effects
We refer to the block effect as a random effect, while factor effects (e.g., x1, x2, …) are known as fixed effects. Fixed effects represent actual levels of influential factors. However, random effects represent a randomly distributed range of
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less influential factors. The block effect is uncontrolled and uncontrollable. We cannot set it as we can fixed effects. We are not even interested in the actual level of the blocking factor, per se. We are mostly trying to control a nuisance. Therefore, the block effect is qualitatively different from the model effects.
Example 3.11 Orthogonal Blocking of a 2 3 Factorial Design Problem statement: Block the 23 factorial design into two orthogonal blocks. Fit the model y = a0 + a1x1 + a2 x2 + a3 x3 + ab xb + e , where abxb is the block effect. Give a skeleton of the ANOVA for this design. Solution: We shall use the defining contrast 123, referring to the factor numbers. For half the design points, 123 = +1, for the other half, 123 = –1. We write the design in binary order, segregate by blocks, and then randomize within each block. Table 3.24 gives one such randomization. TABLE 3.24 A Randomized 23 Factorial Design in Two Blocks Run Order
Pt (binary order)
1
1 2 3 4
1 7 2 4
– + – +
– + + –
+ + – –
+ + + +
5 6 7 8
6 0 5 3
+ – + –
+ – – +
– – + +
– – – –
Factor Pattern 2 3
Block
We used the following approach: 1. Randomize the design. For this purpose, one can use tables of random numbers, statistical software, or a random number generator. For example, the Excel function 7*RAND( ) gives numbers between 0 and 7, randomly. One may then run the function repetitively until assigning an order to every number. 2. Express each number in the sequence in binary format, e.g., 0 = – – –, 1 = – – +, etc. 3. Segregate the runs by block and randomly choose which to run first by a coin toss, etc.
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Table 3.25 gives a skeleton of the ANOVA. TABLE 3.25 ANOVA for a 23 Factorial Design in Two Blocks Effect
SS
DF
MS
F(1, 3)
x1 x2 x3 xb R
SSx1 SSx2 SSx3 SSb SSR
1 1 1 1 3
MSx1 = SSx3 MSx2 = SSx3 MSx3 = SSx3 MSb = SSb/1 MSR = SSR/3
MSx1/MSR MSx2/MSR MSx3/MSR MSb/MSR
T
SST
7
Example 3.12 A 2 3 Factorial in Two Blocks Problem statement: Table 3.26 shows flame lengths for a burner using a blend of hydrogen and natural gas at various fuel flow and air/fuel ratios.
TABLE 3.26 Data for 23 Factorial in Two Blocks Run Order
Pt (binary order)
Heat Release, MMBtuh
Factor O2 in Flue Gas, %
H2 in Fuel, %
Block = 123
Flame Length, ft
1 2 3 4
1 7 2 4
3 12 3 12
2 10 10 2
100 100 0 0
+ + + +
17.5 12.6 5.4 10.3
5 6 7 8
6 0 5 3
12 3 12 3
10 2 2 10
0 0 100 100
– – – –
5.0 9.9 17.1 12.2
The table gives the fuel flow as MMBtuh (millions of British thermal units per hour), and one may relate the final oxygen concentration in the flue gas to the air/fuel ratio. Perform the regression, build the ANOVA, and estimate σ. Repeat the procedure but include the SS blocks in the residual. What does this do to the error and the F tests? Solution: The regression equation is y = 11.59 + 3.84 x1 − 2.71x2 − 2.47 x3
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Modeling of Combustion Systems: A Practical Approach where x1, x2, and x3 are the heat release, oxygen, and hydrogen concentrations, respectively, transformed to ±1. It says that the flame length grows with the heat release and shrinks with a higher air/fuel ratio and the addition of hydrogen to the fuel. These are well-known effects. The investigator performed the original experiments in two separate trials separated by some weeks. However, since the blocks are orthogonal to the remaining effects, we can separately assess any bias correlated with time. Such will end up in the sum-of-squares blocks entry and not in the residual. With such tests, there are many opportunities for bias. Others may have done experiments with the same burner, but not restored it properly. Or the test may be with a similar but not identical unit. In fact, this ANOVA shows that there is significant bias. Actually, the data come from two different projects using the same burner model, at different times. So the block error also includes the effect of manufacturing differences from burner to burner. Notwithstanding, we are still able to get good estimates for the main effects. Table 3.27 shows that the firing rate, oxygen concentration, and hydrogen content are all very significant (P < 0.0002). TABLE 3.27 ANOVA for the 23 Factorial Design in Two Blocks Effect
SS
DF
x1 x2 x3 Blocks
118.07 58.88 48.73 33.64
1 1 1 1
0.29
3
259.60
7
Residual Total
MS
F(1, 3)
P
118.1 58.9 48.7 33.6
1239.6 618.1 511.6 353.2
0.0001 0.0001 0.0002 0.0003
r2 =
86.9%
s=
0.31
0.10
We also have an estimate for σ of ±0.31 ft. This gives our F tests the precision they need. Repeating the analysis, but this time without the sum-of-squares blocks entry, we see a marked difference. Table 3.28 shows that only the heat release (x1) is significant at the 95% confidence level. Our estimate for σ is badly inflated to ±2.9 ft because the residual contains bias from differences over time. Because this is a large difference we should attempt to find out what has changed between the two test series. We should point out that since the sum-of-squares blocks entry is orthogonal to the effects, none of the coefficients were changed. That is, our model is still y = 11.59 + 3.84 x1 − 2.71x2 − 2.47 x3 , but now only a1 = 3.84 appears
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TABLE 3.28 ANOVA for the 23 Factorial Design without Blocks Effect x1 x2 x3 Residual Total
SS
DF
MS
F(1, 3)
P
118.07 58.88 48.73
1 1 1
118.1 58.9 48.7
13.9 6.9 5.7
0.0203 0.0579 0.0746
33.93
4
8.48
r2 =
86.9%
259.60
7
s=
2.91
significant. The residual has been inflated and the F tests have lost their sensitivity. A famous rule of thumb is “Block what you can, randomize what you cannot.”3
3.9
Screening Designs
Suppose that one has a large slate of candidate factors. Then even 1/2 or 1/4 fractions may entail too many experimental points. If we may properly presume that first-order effects predominate, we may use screening designs. These comprise highly fractionated factorials or fully saturated designs (such as saturated factorials and simplex designs). Saturated designs are those that use all their degrees of freedom for the initial estimating of effects. Screening designs can whittle the candidate factors down to a few important ones for further scrutiny. 3.9.1 Simplex Designs Simplex designs are screening designs and use the absolute minimum number of points. They are easy to construct, but they generate designs having nonuniform coding limits and number of levels. They also are more difficult to augment than fractional factorials. Consider a linear equation in two factors: y = a0 + a1x1 + a2x2. Theoretically, we ought to be able to determine the three coefficients with only three experiments. It is possible even to have an orthogonal design for this — the simplex design. To construct it, do the following.4 Step 1: Construct an inceptive matrix in n – 1 rows, where n is the total number of coefficients. The principal diagonal comprises the counting numbers, 1, 2, …, n – 1, in order. Above the principal diagonal are –1 values. Below are 0 values. Then, for the case at hand, n = 3 and the matrix looks like this: ⎛1 ⎜⎝ 0
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Step 2: To this matrix add an additional row above the first, comprising –1 values:
⎛ −1 ⎜ 1 ⎜ ⎜⎝ 0
−1⎞ −1⎟⎟ 2 ⎟⎠
Step 3: Normalize each column vector to unit extension using the square root of the sum of squares:
⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝
−1 2 −1 2 0
−1 ⎞ ⎟ 6⎟ ⎛ −0.707 −1 ⎟ ⎜ ≈ 0.707 ⎟ 6 ⎟ ⎜⎜ ⎝ 0.000 2 ⎟ ⎟ 6⎠
−0.4408⎞ −0.408⎟⎟ −0.816⎟⎠
From left to right, the columns comprise the factor patterns for x1 and x2. Step 4: Augment the matrix with an initial column of 1 values. Here is the X matrix for the simplex, followed by a diagonal XTX, indicating that the matrix is orthogonal:
⎛1 Xa = ⎜⎜ 1 ⎝⎜ 1
−0.707 0.707 0.000
⎛3 −0.408⎞ ⎛ a0 ⎞ ⎟ ⎜ ⎟ T −0.408⎟ ⎜ a1 ⎟ , X X = ⎜⎜ ⎜⎝ −0.816⎟⎠ ⎜⎝ a2 ⎟⎠
1
⎞ ⎟ ⎟ 1⎟⎠
Example 3.13 A Simplex in Six Factors Problem statement: Construct a simplex to determine the following equation using only seven experiments: y = a0 + a1x1 + a2 x2 + a3 x3 + a4 x4 + a5 x5 + a6 x6 . Calculate XTX for the simplex matrix to show that it is orthogonal.
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Solution: Step 1: ⎛1 ⎜0 ⎜ ⎜0 ⎜ ⎜0 ⎜0 ⎜ ⎝0
−1 2 0 0 0
−1 −1 3 0 0
−1 −1 −1 4 0
−1 −1 −1 −1 5
0
0
0
0
−1⎞ −1⎟⎟ −1⎟ ⎟ −1⎟ −1⎟ ⎟ 6⎠
⎛ −1 ⎜ 1 ⎜ ⎜ 0 ⎜ ⎜ 0 ⎜ 0 ⎜ ⎜ 0 ⎜⎝ 0
−1 −1 2 0 0 0 0
−1 −1 −1 3 0 0 0
−1 −1 −1 −1 4 0 0
−1 −1 −1 −1 −1 5 0
−1⎞ −1⎟⎟ −1⎟ ⎟ −1⎟ −1⎟ ⎟ −1⎟ 6⎟⎠
Step 2:
Step 3: ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝⎜
−1
−1
−1
−1
−1
2 1
6 −1
12 −1
20 −1
30 −1
2
6 2
12 −1
20 −1
30 −1
6
12 3
20 −1
30 −1
12
20 4
30 −1
20
30 5
0 0
0
0
0
0
0
0
0
0
0
0
0
0
© 2006 by Taylor & Francis Group, LLC
30 0
−1 ⎞ 42 ⎟⎟ −1 ⎟ ⎟ 42 ⎟ −1 ⎟ ⎟ 42 ⎟ −1 ⎟ ⎟ 42 ⎟ −1 ⎟ ⎟ 42 ⎟ −1 ⎟ ⎟ 42 ⎟ ⎟ 6 ⎟ ⎟ 42 ⎠
272
Modeling of Combustion Systems: A Practical Approach Step 4: ⎛ ⎜1 ⎜ ⎜ ⎜1 ⎜ ⎜ ⎜1 ⎜ ⎜ ⎜1 ⎜ ⎜ ⎜1 ⎜ ⎜ ⎜1 ⎜ ⎜ ⎜1 ⎜⎝
−1
−1
−1
−1
−1
2 1
6 −1
12 −1
20 −1
30 −1
2
6 2
12 −1
20 −1
30 −1
6
12 3
20 −1
30 −1
12
20 4
30 −1
20
30 5
0 0
0
0
0
0
0
0
0
0
0
0
0
0
⎛7 ⎜ ⎜ ⎜ ⎜ T X X=⎜ ⎜ ⎜ ⎜ ⎜⎝
−1 ⎞ 42 ⎟⎟ −1 ⎟ ⎟ 42 ⎟ −1 ⎟ ⎟ 42 ⎟ −1 ⎟ ⎟ 42 ⎟ −1 ⎟ ⎟ 42 ⎟ −1 ⎟ ⎟ 42 ⎟ ⎟ 6 ⎟ ⎟ 42 ⎠
30 0
1 1 1 1 1
⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ 1⎟⎠
Thus, the matrix is orthogonal in the lenient sense. (Multiplying all columns except the first by √n where n is the number of points in the simplex would yield an orthogonal design in the stricter sense. Multiplying the resultant matrix by 1/√n would yield an orthogonal matrix in the strictest sense. However, these operations are unnecessary for our purposes.) 3.9.2 Highly Fractionated Factorials In general, one cannot always construct an n-run fractional factorial to determine n coefficients. However, one can construct designs having 2f–k ≥ n points. When the equality holds we have a saturated design. We can construct highly fractionated factorials according to 2f–k where k = ceil[log2(n)]. Here, ceil is the ceiling operator whose output is the smallest integer greater than its argument. For example, ceil(2.8) = 3. To evaluate log2(n) note that log(n) ln(n) = = log 2 (n) log(2) ln(2)
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Experimental Design and Analysis
273
For n = 7 we have k = ceil[log2(7)] = ceil[2.81]= 3. Then, using 26–3 (or 1/8 fraction of the 26 factorial), one can construct an eight-run design for six factors. When log2(n) is not an integer, our screening design will not be a saturated one but a highly fractionated factorial design. For our example, the blocking generators 124, 135, and 236 (the k largest generators having minimal overlap) give the following design: ⎛+ ⎜+ ⎜ ⎜+ ⎜ + Xa = ⎜ ⎜+ ⎜ ⎜+ ⎜+ ⎜ ⎝+
+ + + + − − − −
+ + − − + + − −
+ − + − + − + −
+ + − − − − + +
+ − + − − + − +
+⎞ ⎛ a0 ⎞ − ⎟⎟ ⎜ ⎟ a1 −⎟ ⎜ ⎟ ⎟ ⎜ a2 ⎟ +⎟ ⎜ ⎟ a3 +⎟ ⎜ ⎟ ⎟ ⎜ a4 ⎟ −⎟ ⎜ ⎟ ⎜ a5 ⎟ −⎟ ⎜ ⎟ ⎟ ⎝ a6 ⎠ +⎠
(3.92)
Clearly, the matrix is orthogonal. ⎛8 ⎜ ⎜ ⎜ ⎜ X TXa = ⎜ ⎜ ⎜ ⎜ ⎜⎝
8 8 8 8 8
⎞ ⎛ a0 ⎞ ⎟⎜a ⎟ ⎟ ⎜ 1⎟ ⎟ ⎜ a2 ⎟ ⎟⎜ ⎟ ⎟ ⎜ a3 ⎟ ⎟ ⎜ a4 ⎟ ⎟⎜ ⎟ ⎟ ⎜ a5 ⎟ 8⎟⎠ ⎜⎝ a6 ⎟⎠
Consideration of the blocking generators gives the following alias structure for two-factor interactions: 1 ↔ 24 ↔ 35
4 ↔ 12 ↔ 56
2 ↔ 14 ↔ 36
5 ↔ 13 ↔ 46
3 ↔ 15 ↔ 26
6 ↔ 25 ↔ 34
16 ↔ 25 ↔ 34
We have only listed interactions up to second order, as third- and higherorder interactions are usually negligible. This is a resolution 3 design, and one may properly use it only if second-order interactions pale compared to the first-order effects. We call designs of this type screening designs because the idea is that only a few of the effects will be significant.
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Modeling of Combustion Systems: A Practical Approach
Highly fractionated designs have several advantages over simplex designs: 1. 2. 3. 4.
3.9.3
All factors have values of ±1 rather than various levels. One may easily augment the designs to increase the resolution. The alias structure is easy to determine. There are some degrees of freedom left over to estimate experimental error.
Foldover
It may be that we find, after an initial investigation, that we need to increase the design resolution. We can augment fractional factorials easily for this purpose. For example, suppose we need to increase the resolution of the previous design from a 1/8 26 to a 1/4 26 design. We may do this with a procedure known as foldover: constructing another design with the signs of the blocked factors reversed. For example, the following matrix is created by folding over (or reversing) the signs of the blocked factors of Matrix 3.91: ⎛+ ⎜+ ⎜ ⎜+ ⎜ + Xa = ⎜ ⎜+ ⎜ ⎜+ ⎜+ ⎜ ⎝+
+ + + + − − − −
+ + − − + + − −
− + − + − + − +
− − + + + + − −
− + − + + − + −
−⎞ ⎛ a0 ⎞ + ⎟⎟ ⎜ ⎟ a1 +⎟ ⎜ ⎟ ⎟ ⎜ a2 ⎟ −⎟ ⎜ ⎟ a3 −⎟ ⎜ ⎟ ⎜ ⎟ a4 ⎟ +⎟ ⎜ ⎟ ⎜ a5 ⎟ +⎟ ⎜ ⎟ ⎟ ⎝ a6 ⎠ −⎠
(3.93)
Recall that the blocking generators for Matrix 3.91 were 124, 135, and 236. These in turn generated the tacit blocking factors 2345, 1346, and 1256. Now if we leave the signs of the unblocked factors unchanged (1, 2, and 3) and then reverse the signs of x4 we get a full factorial in x1, x2, x3, and x4. So the blocking factors 2345, 1346, and 1256 determine the sign pattern for x5 and x6. The reader may construct this design from scratch using a 24 factorial and then constructing the sign patterns for x5 and x6 using the above three blocking generators. He will note that the design is identical to that generated by foldover. 124 is no longer a blocking generator, as 4 is now an independent factor. Therefore, the design is now resolution 4; the first-order effects are not confounded with any two-factor interactions, i.e., 1 ↔ 256 ↔ 346
2 ↔ 156 ↔ 345
3 ↔ 146 ↔ 245
4 ↔ 136 ↔ 235
5 ↔ 126 ↔ 234
6 ↔ 125 ↔ 134
© 2006 by Taylor & Francis Group, LLC
Experimental Design and Analysis
275
In resolution 4 designs, two-factor interactions alias with other two-factor interactions but not with any main effects: 12 ↔ 56
13 ↔ 46
14 ↔ 36
15 ↔ 26
16 ↔ 25 ↔ 34
23 ↔ 45
Thus, foldover is an easy way to increase the resolution of fractional factorial designs. Rather than keep the factor pattern of the unblocked factors constant, we could have chosen not to reverse the signs of some of the blocked factors to create a specific alias structure. This is useful if particular interactions specifically interest the investigator. However, in the absence of new knowledge, the foldover technique usually creates the most desirable properties.
3.10 Second-Order Designs Factorial designs cannot estimate pure quadratics, i.e., effects like a11x12, a22x22, etc. However, the central composite and Box–Behnken designs can estimate these. We first discuss central composites.
3.10.1
Central Composites
One can augment factorial and fractional factorial designs with axial (also called star) points to obtain estimates for pure quadratics. Axial points have coordinates comprising all zeros but one, e.g., (0, 0, …, a). For example, one may augment the 22 factorial design with four points (0, ±√2) and (±√2, 0) to create the design shown in Figure 3.15. We have chosen axial coordinates at ±√2 units from the design center to give a rotatable design. That is, the resulting error structure will be circular (or spherical or hyperspherical, as the case may be). We can know this from the information fraction (see Equation 3.48). If we later find that our axes should be rotated, then the existing design accommodates this with no need for further experiments. This design also has another advantage: it measures the response at five different levels for each factor (–√2, –1, 0, 1, √2). Rotatable designs have symmetric statistical properties, but in fact, if the design is approximately rotatable, that is good enough. It never hurts to have a rotatable design, but for some practical cases where linear combinations of the factors have no real meaning, rotatable designs are less beneficial. Generally, we augment the design with several center-point replicates. We have partitioned the X matrix to highlight the factorial, axial, and center points:
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x1 y = a0 + a1x1 + a2x2 + a11x12 + a12x1x2 + a22x22 + eb + e
x2
FIGURE 3.15 A central composite in two factors. The design is shown augmented with three center-point replicates (bolded circles). The replicates are coincident but jittered for clarity. The solid circles comprise the factorial points. The hollow circles at the periphery are the axial or star points. The model, as shown, has 2 DF to measure pure error, 6 DF for model parameters, and 3 DF to measure bias. If there is no significant bias, the two error terms may be pooled to a residual error term having 5 DF to better estimate s.
⎛1 ⎜1 ⎜ ⎜1 ⎜ ⎜1 ⎜1 ⎜ Xa = ⎜ 1 ⎜ ⎜1 ⎜ ⎜1 ⎜1 ⎜ ⎜1 ⎜ ⎝1
1 1 −1 −1
1 −1 1 −1
1 1 1 1
1 −1 −1 1
0
2
0
0
0
− 2
0
0
2
0
2
0
− 2 0 0 0
0 0 0 0
2 0 0 0
0 0 0 0
⎛ 11 ⎜ ⎜ ⎜ T X Xa = ⎜ ⎜ ⎜ ⎜ ⎝
© 2006 by Taylor & Francis Group, LLC
8 8 8 12 sym
4
1⎞ 1⎟⎟ 1⎟ ⎟ 1⎟ ⎛ a0 ⎞ ⎜a ⎟ 2⎟ ⎜ 1 ⎟ ⎟⎜ a ⎟ 2 2⎟ ⎜ ⎟ ⎟ ⎜ a11 ⎟ 0⎟ ⎜ ⎟ ⎟ a12 0⎟ ⎜⎝ a ⎟⎠ 22 2 0⎟⎟ 0⎟ ⎟ 0⎠ 8 ⎞ ⎛ a0 ⎞ ⎟⎜ a ⎟ ⎟⎜ 1 ⎟ ⎟ ⎜ a2 ⎟ ⎟⎜ ⎟ 4 ⎟ ⎜ a11 ⎟ ⎟ ⎜ a12 ⎟ ⎟⎜ ⎟ 12 ⎠ ⎝ a22 ⎠
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277
One should also note that the central composite design is not orthogonal with or without center-point replicates even in the lenient sense. The design is orthogonal in the main effects and the interaction term, but the pure quadratics bias one another and the pure constant (a0). We have four options. 1. 2. 3. 4.
Tolerate it. Restrict the bias to be among quadratics only. Construct orthogonal components. Change the design value for the axial points to generate an orthogonal design.
3.10.1.1 Quadratic Bias Only We may subtract the mean square value, m = and fit the following model:
∑x n
2 k
n
from the quadratics
y = a0 + a1x1 + a2 x2 + a11 ( x12 − m) + a12 x1x2 + a22 ( x22 − m) For our example, m = 8/11, giving ⎛1 ⎜1 ⎜ ⎜1 ⎜ ⎜1 ⎜1 ⎜ X = ⎜1 ⎜ ⎜1 ⎜ ⎜1 ⎜1 ⎜ ⎜1 ⎜ ⎝1
1 1 −1
1 −1 1
3 / 11 3 / 11 3 / 11
1 −1 −1
−1
−1
3 / 11
1
2
0
14 / 11
0
− 2
0
14 / 11
0
0
2
−8 / 11
0
0 0 0 0
− 2 0 0 0
−8 / 11 −8 / 11 −8 / 11 −8 / 11
0 0 0 0
⎛ 11 ⎜ ⎜ ⎜ X TX = ⎜ ⎜ ⎜ ⎜ ⎝
© 2006 by Taylor & Francis Group, LLC
8 8 68 / 11 sym
4
3 / 11⎞ 3 / 11⎟⎟ 3 / 11⎟ ⎟ 3 / 11⎟ −8 / 11⎟ ⎟ −8 / 11⎟ ⎟ 14 / 11⎟ ⎟ 14 / 11⎟ −8 / 11⎟⎟ −8 / 11⎟ ⎟ −8 / 11⎠ ⎞ ⎟ ⎟ ⎟ ⎟ 13 / 11⎟ ⎟ ⎟ 68 / 11⎠
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Modeling of Combustion Systems: A Practical Approach
This gets rid of the biases for the pure quadratics with the constant, but it does not remove the bias among the quadratics. Then in our ANOVA, the sum of the quadratics will be orthogonal to the rest of the design, but we will need separate ANOVAs to assess the model with various combinations of quadratic terms. 3.10.1.2 Orthogonal Components Following option 3, one can construct a component that is orthogonal to all others using the terms x1 + bx2 − c and x2 + bx1 − c . Then the model becomes
(
)
(
y = a0 + a1x1 + a2 x2 + a11 x12 + bx22 − c + a12 x1x2 + a22 x22 + bx12 − c
)
This will generate the following XTX matrix: ⎛N ⎜ ⎜ ⎜ ⎜ ⎜ T X X=⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝
∑(x
+ bx22 − c
∑(x
+ bx22 − c
2 1
∑x
∑(x
)
2 1
∑x
2 2 2 1
)
∑(
2
sym
∑( x x )
2
1 2
)
⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ x12 + bx22 − c x22 + bx12 − c ⎟ ⎟ ⎟ ⎟ ⎟ 2 2 2 ⎟⎠ x2 + bx1 − c 2 2
+ bx12 − c
)(
∑(
)
)
In order to make the nondiagonal elements vanish, we must solve for b and c such that
∑(x ∑(x
2 j
2 j
)
+ bxk2 − c = 0
)(
(3.94)
)
+ bxk2 − c xk2 + bxj2 − c = 0
where j ≠ k. The first equation gives
∑ ( x ) + b∑ ( x ) − nc = 0 2 j
2 k
But for a balanced design,
∑x = ∑x 2 j
© 2006 by Taylor & Francis Group, LLC
2 k
= nm
(3.95)
Experimental Design and Analysis
279
where m is the mean square:
∑x
m=
2 k
n
Therefore, we have c = (b − 1)m . Expanding Equation 3.94 gives
∑ ⎣⎡(1 + b ) x x + b ( x 2
2 2 j k
) ( )(
)
+ xk4 − b + 1 c xj2 + xk2 + c 2 ⎤ ⎦
4 j
Performing the summation gives
( 1 + b ) ∑ x x + b∑ ( x 2
2 2 j k
4 j
) ( ) ∑(x
+ xk4 − b + 1 c
2 j
)
+ xk2 + nc 2
Let q=
∑(x ) = ∑( x ) 4 j
4 k
n
n
Then the equation simplifies to
(1 + b ) ∑ x x 2
2 2 j k
(
)
+ 2 nbq − 2 b + 1 nmc + nc 2
Substituting Equation 3.96 into this gives
(1 + b ) ∑ x x 2
(
)(
)
(
)
+ 2 nbq − 2 b + 1 b − 1 nm2 + nm2 b − 1
2 2 j k
2
This gives a quadratic equation in b: b 2 ⎡⎢ ⎣
∑ ( x x ) − nm ⎤⎦⎥ + b ⎡⎣2n( q − m )⎤⎦ + ⎡⎣⎢3nm + ∑ ( x x )⎤⎦⎥ = 0 . 2 2 j k
2
2
2
2 2 j k
Solving for b, one obtains
b=−
(
∑(
)
⎛ ± ⎜ ⎜ xj2 xk2 − nm2 ⎝
n q − m2
)
© 2006 by Taylor & Francis Group, LLC
(
∑(
)
2
⎞ ⎟ − xj2 xk2 − nm2 ⎟⎠
n q − m2
)
∑ ( x x ) + 3nm ∑ ( x x ) − nm 2 2 j k
2 2 j k
2
2
(3.96)
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Modeling of Combustion Systems: A Practical Approach
where
∑(x ) = ∑( x ) q= 4 j
4 k
n
n
and m=
∑x = ∑x 2 j
n
2 k
n
and one takes the solution such that –1 ≤ b ≤ 1. We recall that
(
)
c = b−1 m
(3.97)
One may also solve for Equations 3.96 and 3.97 numerically. This procedure gives an orthogonal design. 3.10.1.3 Adjusting the Axial Component The third alternative (and in the author’s opinion the best) is to adjust the value of the axial coordinate so that the quadratics do not bias one another or the other coefficients. For central composite designs, this is simple to do numerically because all the star points are the same distance from the design center. Therefore, one iterates a single variable only. Statistical programs will also calculate this third option. One can also calculate it algebraically. In this case, the XTX matrix is ⎛N ⎜ ⎜ ⎜ ⎜ ⎜ T X X=⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝
∑x
2 1
∑x
2 2
∑( x
2 1
−m
)
2
∑x x
2 2 1 2
sym
Then our only criterion is that
∑(x © 2006 by Taylor & Francis Group, LLC
2 j
)(
)
− m xk2 − m = 0
∑(
⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ 2 2 x1 − m x2 − m ⎟ ⎟ ⎟ ⎟ 2 ⎟ x22 − m ⎠
∑(
)(
)
)
Experimental Design and Analysis
281
This expands to
∑x x
− nm2 = 0
2 2 j k
Back-substituting for m, we have
∑x
2 k
n
∑(x x )
2
j k
=
n
(3.98)
When this is so, X will become orthogonal. Now, we need only find out how
∑x
2 k
n and
∑(x x )
2
j k
n relate to α, the distance of the star points from the center of the design. We know that xk2 = 1 for every factorial point. We also know that xk2 = α 2 for every axial point along the k factor axis. If there are f factors, then there are precisely 2f axial points, but only two along any one axis. Therefore,
∑x
2 k
=
n
n f + 2α 2 n
Now ( xj xk )2 = 0 for all axial and center points and ( xj xk )2 = 1 for factorial points. Then
∑(x x )
2
j k
n
=
nf n
Substituting these expressions into Equation 3.98 gives n f + 2α 2 = n Solving for α, we obtain
© 2006 by Taylor & Francis Group, LLC
nf n
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Modeling of Combustion Systems: A Practical Approach
nn f − n f
α=±
(3.99)
2
For a standard design, we may also write nf = 2f and n = 2f + 2f +nc, where nc is the number of centerpoints. For example, if we have a central composite in two factors with three center points, then we obtain
(2
α=±
2
)
+ 2 ⋅ 2 + 3 22 − 22 2
=
11 − 2 ≈ 1.147
This design is not rotatable, but it is a greater advantage in the author’s opinion to have meaningful unbiased coefficients than strict rotatability. Figure 3.16 compares information fractions for this design, α = 1.147, and a standard central composite, α = 1.414. 18%
18%
100% X
x1 100%
x1
X
80% 75% 95% 70% 90% 100% 95% 90% 65% 85%
x2
84% 92% 100% X
18%
84% 67% 50% 34%
x2
100% X 92% 75% 59% 42% 26%
18%
Fully Orthogonal, Quasi-Rotatable
Quasi-Orthogonal, Fully Rotatable
FIGURE 3.16 Comparison of central composite designs. The design at the left is with axial points to give a fully orthogonal matrix. However, the design is no longer rotatable. The design at the right is the standard central composite. It is not fully orthogonal — the quadratics bias one another. However, it is rotatable. If linear combinations of the factors have no meaning, it is better to use the fully orthogonal design.
Both designs have three center-point replicates. The standard central composite has better statistical properties, including rotatability and a symmetric information fraction. However, if linear combinations of the original factors have no clear meaning, it is better to use the fully orthogonal design and sacrifice some rotatability.
© 2006 by Taylor & Francis Group, LLC
Experimental Design and Analysis 3.10.2
283
Box–Behnken Designs
It may happen that physical constraints do not allow axial points at different or greater levels than the main factors. In such a case, a Box–Behnken design may be appropriate. Rather than choose the vertices of a p-dimensional hypercube, one may choose the middle edge points. Since there are more edges than vertices, this results in more experiments. However, no one experiment is ever at the extreme range of all factors simultaneously. Figure 3.17 gives an example.
3 Point 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
x1
x2
x3
-1 -1 -1 -1 0 0 0 0 1 1 1 1 0 0 0
-1 0 0 1 -1 -1 1 1 -1 0 0 1 0 0 0
0 -1 1 0 -1 1 -1 1 0 -1 1 0 0 0 0
8
x3
6
11 1
4 15
13
x2
14
9
x1 2
12
7
5 10
FIGURE 3.17 Box–Behnken design for p = 3. The design comprises edge points in three dimensions. The center points are coincident but jittered for clarity. This second-order design avoids extreme points, which may be difficult to achieve in practice.
3.10.3
Multilevel Factorials
We are not limited to two-level factorials. For example, we could construct a factorial at various levels in three factors. Suppose one factor was at two levels, one at four, and one at three levels. How would we construct such a factorial and how many points would it have? By definition, a full factorial has every possible combination of each factor level. Therefore, our factorial must have 2 × 4 × 3 = 24 points. If we want to construct the design in binary order, we would do the following: 1. Reorder the factor columns according to increasing levels from left to right. Let Lk be equal to the number of levels of the kth factor and xk be the name of the kth factor. Then L1 = 2, L2 = 3, and L3 = 4. Thus, the factors have been reordered and renamed x1 (two levels), x2 (three levels), and x3 (four levels). The left-most factor, x1, has two levels; the factor to the right, x2, has three levels; and the right-most factor, x3, has four levels.
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2. Cycle each factor through its levels in blocks of r
∏L
r − k +1
k =1
where r is the number of factors to the right and L0 = 1 by definition. Then for x1 there are two factors to the right (r = 2) and it will cycle through its levels in blocks of L2·L1 = 3·4 = 12. x2 has one factor to the right (r = 1) and it will cycle through its levels in blocks of L1 = 4. Finally, x3 has no factors to the right (r = 0) and it will cycle through its levels in blocks of L0 = 1 (by definition). This generates the design of Table 3.29. TABLE 3.29 A 2 × 3 × 4 Factorial Design Pt
x1
x2
x3
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2
1 1 1 1 2 2 2 2 3 3 3 3 1 1 1 1 2 2 2 2 3 3 3 3
1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4
3. Center and scale the factors and develop the desired X matrix. We know that the factorial design has 2 × 3 × 4 levels. From Section 3.4 this can fit a model with up to 24 coefficients providing no effect has any factor at greater than Lk–1 order. The lowest-order model comprising all 24 coefficients would have the following factor patterns:
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Experimental Design and Analysis 0
285
1
12
122
1223
12233
2
13
123
1233
12333
3
22
133
1333
22333
23
223
2233
33
233
2333
122333
333 If our factors are nominal at several levels, then we may have to run something like a 2 × 3 × 4 factorial. For example, if we evaluate the effect of temperature on three different burner types (e.g., A, B, and C) burning four different kinds of fuels (start-up, design, upset, and waste gas), we may run a 2 × 3 × 4 factorial. But if the factors are continuous (temperature, tip angle, and hydrogen concentration), then we can make use of two-level fractional or full factorials, central composites, etc. Two-level and related designs have advantages. A design like 2 × 3 × 4 is difficult to fractionate. However, we can fractionate any continuous factor in 2k levels by representing the k levels of a single factor as k two-level factors. For example, if we wish to code for two factors, each at four levels, we would need a 4 × 4 design comprising 16 points. We could do this using a 24 factorial design. The first two factors will code for the levels of the first factor, and the second two for the second factor. We can use binary arithmetic to calculate each level. For example, – – = 002 = 0
– + = 012 = 1
+ – = 102 = 2
+ + = 112 = 3
Then two factors code for four levels, – 0, 1, 2, and 3 (we could also use 1, 2, 3, and 4 by simply adding 1 to the number); likewise for the second two factors. The advantage of using several two-level factors to code for multiple levels of a single factor is that we can fractionate the design easily.
Example 3.14 Fractionating a 4 × 4 Factorial Problem statement: Fractionate the 4 × 4 design into a half fraction comprising eight runs. Solution: This design is equivalent to the 24 factorial design. Fractionating this design into the 1/2 24 fractional factorial according to Section 3.5 gives the design of Table 3.30. In the table, w1 through w4 denote the two-level factors used to derive the four-level factors x1 and x2.
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w1
w2
w3
w4
x1
x2
1 2 4 7 8 11 13 14
–1 –1 –1 –1 1 1 1 1
–1 –1 1 1 –1 –1 1 1
–1 1 –1 1 –1 1 –1 1
1 –1 –1 1 –1 1 1 –1
0 0 1 1 2 2 3 3
1 2 0 3 0 3 1 2
For an orthogonal matrix in x1 and x2 , we would code the levels as –1, –1/3, 1/3, and 1, using the transform of Equation 3.33.
3.11 Sequential Experimental Design Paradoxically, the best time to design an experiment is after one has analyzed the data. Only then does one know what one should have studied. Since investigators are not clairvoyant, this creates a dilemma. Sequential experimental design attempts to reduce unproductive experimental time. The idea is to commit no more than 1/4 to 1/3 of the experimental resources to the first set of experiments because subsequent knowledge will likely make future experiments more productive. For large experiments, a general order that is often fruitful is to use a screening design as a first experimental series, followed by fractional or full factorials, and ultimately second-order designs such as central composite or Box–Behnken designs. To begin, we recommend screening designs comprising highly saturated fractional factorials. If one cannot run the entire design under homogeneous conditions, then block the design orthogonally. For f factors, the total number of runs must be greater than or equal to f + 1: n = 2f –k ≥ f + 1
(3.100)
Thus, we select k such that 2 f −k ≥1 f +1 As we noted in Section 3.9.2, if we want the most highly saturated design, we choose k such that k = f – ceil[log2 (f + 1)]
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(3.101)
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where ceil is the ceiling operator indicating the smallest integer greater than or equal to log2 (f + 1). We can derive this by taking the logarithm of Equation 3.100 and solving for k. We can use any convenient base of logarithms, because they differ only by a factor. For example,
(
)
log 2 f + 1 =
(
) = log ( f + 1)
ln f + 1 ln(2)
10
log10 (2)
Basically, we select designs having n > f, where n = 2J, J being an integer.
3.11.1
Augmenting to Less Fractionated Factorials
The idea behind a screening design is to see which main effects are important. After we determine that, we can increase the resolution for the important effects via foldover. This would allow us to de-alias two-factor interactions. We examined foldover in Section 3.9.3. Now at this juncture it may be that we have to move the design center to a new location. One way of deciding which direction will prove profitable is to use the method of steepest ascent.
3.11.2
Method of Steepest Ascent
With screening designs, we will only be able to assess first-order coefficients. However, when these provide an adequate approximation to the surface, they will also suggest an appropriate direction to optimize the response. For example, consider a nonlinear surface such as the maximum given by Figure 3.18. A factorial or screening design far from the design center gives a firstorder screening model: y = a0 + a1x1 + a2 x2 . For the direction of steepest ascent, we set the first derivative of the equation to zero, dy = a1dx1 + a2 dx2 = 0 , and obtain the following vector of steepest ascent: ⎛ ∂y ∂x1 ⎞ ⎛ a1 ⎞ ⎜⎝ ∂y ∂x ⎟⎠ = ⎜⎝ a ⎟⎠ 2 2 If we desire a unit vector, we may normalize it by the vector length a12 + a22 For a screening model, we have
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true paths of steepest ascent
3
contour lines 2 of e at nt tim ce es s n st a sig pe de tee s
1
FIGURE 3.18 Method of steepest ascent. The actual surface is nonlinear. In the figure, contour lines are solid and orthogonal projections (paths of steepest ascent) are dashed. Although the surface is nonlinear, a factorial design (1) estimates the path of steepest ascent and leads to the design marked (2). One continues the procedure until the linear approximation is no longer valid (3). At that point, one may use a suitable second-order design.
p−1
y = a0 +
∑a x
k k
(3.102)
k =1
The unit vector for the path of steepest ascent asa becomes ⎛ a1 ⎞ ⎜ ⎟ 1 ⎜ a2 ⎟ a sa = a ⎜ % ⎟ ⎜ ⎟ ⎝ ap−1 ⎠
(3.103)
where p−1
a =
∑a
2 k
k =1
the norm of the vector — a scalar. (If we desire to minimize the response rather than maximize it, the path of steepest descent is merely –asa.) We continue in this direction until a linear approximation is no longer valid. At that point, we would need to augment to a higher-order design to account for interaction, or a quadratic behavior.
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Experimental Design and Analysis 3.11.3
289
Augmenting to Second-Order Designs
Screening designs easily augment to higher-order factorials, and these in turn easily augment to central composite designs. One can run the axial points in a separate block if necessary. Other second-order designs, such as Box–Behnken designs and higher-order simplexes, are not as easy to add. Of course, one can always run such designs regardless of the past experimental history, but it is more efficient to build on already existing design points. For example, for f = 3 one could run the 1/2 23 with two center-point replicates. If one needs interaction terms, the complementary half fraction will provide the full factorial. One may also add a couple of center points. The blocks are orthogonal to one another. If second-order behavior is important, one could add the axial points and end up with a full central composite design.
Example 3.15 Sequential Design Problem statement: 1. An investigator wants to study 12 possible candidate factors on three responses, but can run no more than eight experiments in a single block. Suggest a screening design. 2. Suppose x1, x5, and x7 appear to be important. Augment the design to investigate two-factor interactions and examine possible curvature. 3. If curvature is important, what other design points will convert the experimental plan into a central composite? Solution: 1. One might begin with a 212–8 factorial as a screening design. This would require 16 runs. Statistical software is really a must for these kinds of investigations. Developing the blocking generators by hand for so many factors is tedious. JMP™ suggests the following generators in two blocks:* I = 12345 = 2346 = 1347 = 348 = 1249 = 24A = 14B = 123C = 23K. We have written the factor effects in base 12 numbering, where A = 10, B = 11, and C = 12. K is the block. Table 3.31 shows the design. 2. Now if only 1, 5, and 7 are important factors, then we may consider the design to be two replicates of a 23 full factorial for x1, x5, and x7. This design has the following alias structure * JMP is a registered trademark of SAS, SAS Campus Drive, Cary, NC 27513; phone, 877-59GOJMP (U.S.) or +1 919 677 444 (international);
[email protected] or www.jmp.com.
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Modeling of Combustion Systems: A Practical Approach TABLE 3.31 A 12-Factor Screening Design in 16 Runs and 2 Blocks Pt
Block
1
2
3
4
5
a b c d e f g h i j k l m n o p
1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2
– – – – + + + + – – – – + + + +
– – + + – – + + – – + + – – + +
+ + – – + + – – – – + + – – + +
– + – + – + – + – + – + – + – +
– + – + + – + – + – + – – + – +
Factor 6 7 + – + – + – + – – + – + – + – +
+ – – + – + + – – + + – + – – +
8
9
10
11
12
– + + – – + + – + – – + + – – +
– + + – + – – + – + + – + – – +
+ – – + + – – + + – – + + – – +
+ – + – – + – + + – + – – + – +
+ + + + – – – – – – – – + + + +
in pertinent parts: 15 ↔ 6, 17 ↔ 8, and 57 ↔ 2. Since we did not detect 6, 8, or 2 as important effects, it is likely that the 15, 17, and 57 interactions are also not important. Of course, there is always a small possibility that the interactions act in the opposite direction to their aliases, and thus their effects diminish to the point of insignificance. This is a possible but small likelihood. If we were actually concerned about this, we could rerun a 23 factorial in factors 1, 5, and 7, with the remaining factors held at their center-point levels, but odds are that this is not necessary. 3. To examine curvature, we could run some center-point replicates in 1-5-7 factor space. Adding six points + two centerpoint replicates to the design would give a central composite in 1-5-7 factor space. For rotatable factor space we would use α = 1.682. For orthogonal factor space we would use α = 1.287. Table 3.32 gives the additional points.
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TABLE 3.32 Additional Points for a Central Composite Design Pt
Block
q r s t u v w x
3 3 3 3 3 3 3 3
Factor 1 0 0 0 0 –α +α 0 0
5
7
0 0 –α +α 0 0 0 0
–α +α 0 0 0 0 0 0
References 1. Gellert, W. et al., Eds., The VNR Concise Encyclopedia of Mathematics, American Edition, Van Nostrand Reinhold Company, New York, 1977, p. 588. 2. Box, G.E.P. and Draper, N.R., Empirical Model-Building and Response Surfaces, John Wiley & Sons, New York, 1987, p. 484. 3. Box, G.E.P., Hunter, W.G., and Hunter, J.S., Statistics for Experimenters: An Introduction to Design, Data Analysis, and Model Building, John Wiley & Sons, New York, 1978, p. 103. 4. Box, G.E.P. and Draper, N.R., Empirical Model-Building and Response Surfaces, John Wiley & Sons, New York, 1987, p. 506.
© 2006 by Taylor & Francis Group, LLC
4 Analysis of Nonideal Data
Chapter Overview In this chapter, we show how to obtain information from less than ideal data. Thus far, we have studied statistically cognizant experimental designs yielding balanced, symmetrical data with ideal statistical properties. Statistical experimental design (SED) has great advantages, and whenever we have an opportunity to use SED, we should. However, there will be many occasions when the data we receive are historical, or from plant operating history, or other nonideal sources with much less desirable statistical properties. But even poorly designed (or nondesigned) experiments usually contain recoverable information. On rarer occasions, we may not be able to draw firm conclusions, but even this is preferable to concluding falsehoods unawares. We begin our analysis with plant data. With the advent of the distributed control systems (DCSs), plant data are ubiquitous. However, they almost certainly suffer from maladies that lead to correlated rather than independent errors. Also, bias due to an improper experimental design or model can lead to nonrandom errors. In such cases, a mechanical application of ANOVA and statistical tests will mislead; F ratios will be incorrect; coefficients will be biased. Since furnaces behave as integrators, we look briefly at some features of moving average processes and lag plots for serial correlation, as well as other residuals plots. The chapter shows how to orthogonalize certain kinds of data sets using source and target matrices and, more importantly, eigenvalues and eigenvectors. Additionally, we discuss canonical forms for interpreting multidimensional data and overview a variety of helpful statistics to flag troubles. Such statistics include the coefficient of determination (r2), the adjusted coefficient of determination (rA2), the prediction sum of squares (PRESS) statistic and a derivative, rP2, and variance inflation factors (VIFs) for multicollinear data. We also introduce the hat matrix for detecting hidden extrapolation. In other cases, the phenomena are so complex or theory so lacking that we simply cannot formulate a credible theoretical or 293
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Modeling of Combustion Systems: A Practical Approach even semiempirical model. In such a case, it is preferable to produce some kind of model. For this purpose, we shall use purely empirical models, and we show how to derive them beginning with a Taylor series approximation to the true but unknown function. This chapter also examines categorical factors and shows how to analyze designs with restricted randomization such as nested and split-plot designs. This requires rules for deriving expected mean squares, and we provide them. On occasion, the reader may need to fit parameters for categorical responses, and we touch on this subject as well. The last part of the chapter concerns mixture designs for fuel blends and how to simulate complex fuels with many fewer components. This requires a brief overview of fuel chemistry, which we present. We conclude by showing how to combine mixture and factorial designs and fractionate them.
4.1
Plant Data
Plant data typically exhibit serial correlation, often strongly. Serial correlation indicates errors that correlate with run order rather than the random errors we subsume in our statistical tests. Consider a NOx analyzer attached to a municipal solid waste (MSW) boiler, for example. Suppose it takes 45 minutes for the MSW to go from trash to ash, after which the ash leaves the boiler (Figure 4.1). Then the natural burning cycle of the unit is roughly 45 minutes or so. If we pull an independent NOx sample every 4 hours, it is unlikely that there will be any correlation among the data. Except in the case of an obvious malfunction, the history of the boiler 4 hours earlier will have no measurable effect on the latest sample. However, let us investigate what will happen by merely increasing the sampling frequency.
4.1.1
Problem 1: Events Too Close in Time
DCS units provide a steady stream of continual (and correlated) information. Suppose we analyze NOx with a snapshot every hour. Will one reading be correlated with the next? How about every minute? What about every second? Surely, if the previous second’s analysis shows high NOx, we would expect the subsequent second to be high as well. In other words, data that are very close in time exhibit positive serial correlation. Negative serial correlation is possible, but rarer in plant environments. However, it can occur in the plant when one effect inhibits another. Nor is this the only cause of serial correlation.
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Analysis of Nonideal Data
295
FLUE GAS
EXHAUST STACK
MUNICIPAL SOLID WASTE
NOx REDUCTION ZONE AMMONIA INJECTION FEED CHUTE COMBUSTION ZONE STOKER GRATE ASH
UNDERGRATE COMBUSTION AIR
FIGURE 4.1 A municipal solid waste boiler. It takes roughly 45 minutes for the trash-to-ash cycle. This particular unit is equipped with ammonia injection to reduce NOx. (From Baukal, C.E., Jr., Ed., The John Zink Combustion Handbook, CRC Press, Boca Raton, FL, 2001.)
4.1.2
Problem 2: Lurking Factors
Lurking factors are an important cause of serial correlation. For example, O2 concentration affects both NOx and CO emissions. If we were so naïve as to neglect to measure the O2 level, we could easily induce a serial correlation. For example, air temperature correlates inversely to airflow, and the former relates to a diurnal cycle. Therefore, we can also expect airflow with fixed damper positions, e.g., most refinery burners, to also show a diurnal cycle. Every effect must have a cause. If we account for all the major sources of fixed variation, then the multiple minor and unknown sources should distribute normally according to the central limit theorem and collect in our error term. Therefore, it behooves us to find every major cause for our response because major fixed effects in the errors can result in correlated rather than normally distributed errors.
4.1.3
Problem 3: Moving Average Processes
If we consider the boiler furnace as an integrator, then flue gas emissions and components comprise a moving average process — and moving averages are
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highly and positively correlated. To see this, consider a random distribution — a plot of xk against the next data point in time, xk+1 (Figure 4.2a). 0.6 2
R =0.9% s=±0.28
0.2
R 2=84.0% s=0.09
xk+1 0.4
ξ k+1
10
0.1
ξ k = Σ xk k=1
0.2
-0.6
-0.4
0.0 -0.2 0.0
ξk xk -0.3 0.2
0.4
0.6
-0.2
-0.1
0.0 0.0
0.1
0.2
-0.1
-0.2 -0.2
-0.4 -0.6
-0.3
(a) Nearest Neighbor Plot, Uniform Random Distribution (b) Nearest Neighbor Plot, Moving Average FIGURE 4.2 A moving average with random data. Figure 3.7a shows data from 100 points generated by a uniform random number generator, –0.5 < x < 0.5. The graph plots each data point against its neighbor (xk+1 vs. xk). The correlation is, as expected, nearly zero (r2 = 0.009). Figure 3.7b shows the same data as 10-point moving averages. Plotting the moving average data in the same fashion gives noticeably less dispersion (s = 0.09 vs. 0.28) and high correlation, despite the fact that the moving averages comprise uniform random data. In the same way, integrating processes such as combustion furnaces can have emissions with serially correlated errors.
The first plot shows 100 nearest neighbors from a uniform random distribution plotted one against the other. The data were generated with the Excel™ function RAND( )-0.5, representing a uniform distribution with zero mean between –0.5 and 0.5. The nearest-neighbor plot shows no correlation to speak of (r2 = 0.009), the mean is essentially zero ( y = 0.04), and the standard deviation is s = 0.28. These are very close to the expected values for these statistics, and it is not so surprising that random data show no trend when plotted against nearest neighbors. But Figure 4.2b tells a different story. To create the second plot, we performed a moving average using the 10 nearest neighbors: n=10 ⎛ ⎞ 1 xk ⎟ ⎜ ξk = ⎜⎝ 10 k =1 ⎟⎠
∑
where k indexes each point sequentially. Note that the correlation of ξk with ξk+1 in Figure 4.2b has an r2 of 84.0% despite being drawn from an originally
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uniform random population with zero mean. Also note that the standard deviation of the process has fallen by a factor of three (from 0.28 to 0.094). The deflation of the standard deviation by a factor of three is not a coincidence, for the denominator in the calculation of standard deviation is n − 1 , or 10 − 1 = 3 . However, the mean values for both data sets are virtually identical at ~0.0. Since the mean values are unaffected, we may perform regressions and generate accurate values for the coefficients. However, as the moving average process deflates s, our F test will errantly lead us to score insignificant effects as significant ones. That is, failure to account for serial correlation in the data set before analysis will result in inflated F tests. An analysis showing many factors to be statistically significant is a red flag for the deflation of variance from whatever cause.
4.1.4
Some Diagnostics and Remedies
Here are a few things we can do to warn of serial correlation and remedy it: 1. Always check for serial correlation as revealed by an xk vs. xk+1 plot and time-ordered residuals. 2. Make sure that the data are sufficiently separate in time and each run condition sufficiently long to ensure that the samples are independent. 3. Carefully consider the process, not just the data. Since the serially correlated data have both fixed and random components, the problem becomes assessing which are which. One could make an a priori estimate for a moving average process using a well-stirred model of the furnace per the transient mass balance for the boiler in Chapter 2. Using such results, we could adjust the sampling period to be sufficiently large.
4.1.5
Historical Data and Serial Correlation
For historical data, we do not have the privilege of changing how the data were collected. Therefore, we must do our best to note serial correlation and deal with it after the fact. Once we recognize serial correlation, the problem becomes recovering independent errors from correlated ones and using only the former in our F tests. As we have noted, most serial correlation will evaporate if we can identify lurking factors or the actual cause for the correlation. We then put that cause into a fixed effect in the model. If there are cyclical trends, an analysis of batch cycles within the plant may lead to the discovery of a lurking factor. Failing this, one may be able to use time series analysis to extract the actual random error term from the correlated
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one.1,2 This is not so easy. Such models fall into some subset of an autoregressive-integrated-moving average (ARIMA) model, with the moving average (MA) model being the most likely. Time series analysis is a dedicated discipline in its own right. Often one will have to do supplemental experiments to arrive at reasonable estimates and models.
4.2
Empirical Models
The main subject of this text is semiempirical models, i.e., theoretically derived models with some adjustable parameters. These are always preferable to purely empirical models for a variety of reasons, including a greater range of prediction, a closer relation to the underlying physics, and a requirement for the modeler to think about the system being modeled. But in some cases, we know so little about the system that we are at a loss to know how to begin. In such cases, we shall use a purely empirical model. For the time being, let us presume that we have no preferred form for the model. That is, we have sufficient theoretical knowledge to suspect certain factors, but not their exact relationships to the response. For example, suppose we know that oxygen (x1), air preheat temperature (x2), and furnace temperature (x3) affect NOx. We may write the following implicit relation: y = φ(ξ1 , ξ 2 , ξ 3 )
(4.1)
where ξ represents the factors in their original metric and φ is the functional notation. Although we do not know the explicit form of the model, we can use a Taylor series to approximate the true but unknown model. Equation 4.2 represents a general Taylor series: p
y = φ(ξ1 , ξ 2 ,..., ξ f ) = φ( a1 , a2 ,..., an ) +
∑ ∂∂ξφ ( ξ k =1
p−1
p
∑ ∑ ∂ξ∂ ∂φξ j< k
k =1
2
j
(ξ − a )(ξ j
k a ,a j k
j
k
)
− ak +
k
)
− ak +
k a k p
∑ ∂∂ξφ k =1
2
2 k a k
(
ξ k − ak 2!
)
(4.2)
2
+ ...
Here ξ refers to the factors, subscripted to distinguish among them. We reference the Taylor series to some coordinate center in factor space (a1, a2, … , ap), where each coordinate is subscripted per its associated factor. The farther we move from the coordinate center, the more Taylor series terms we require to maintain accuracy. For Equation 4.1, the Taylor series of Equation 4.2, truncated to second order, gives the following equation:
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Analysis of Nonideal Data
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⎧ ∂φ ∂φ ⎪φ( a , a , a ) + ∂φ ξ1 − a1 + ξ 2 − a2 + ξ 3 − a3 + ⎪ 1 2 3 ∂ξ1 ∂ ξ ∂ ξ3 a 2 a a1 2 3 ⎪ ⎪ 2 ∂ 2φ ∂ 2φ ⎪ ∂φ ξ1 − a1 ξ 2 − a2 + ξ1 − a1 ξ 3 − a3 + ⎨ ∂ ∂ ξ ξ ∂ ξ ∂ ξ ∂ ξ 1 3 a ,a 2 ∂ξ 3 ⎪ 1 2 a1 , a2 1 3 ⎪ 2 2 2 ⎪ 2 ξ1 − a1 ξ 2 − a2 ξ 3 − a3 ∂ 2φ ∂ 2φ ⎪∂ φ + 2 + 2 2 ⎪ ∂ξ1 a1 2! 2! 2! ∂ξξ 2 a2 ∂ξ 3 a3 ⎩
(
y≈
(
(
)
)(
(
)
)
)
(
(
(
)
)
)(
(
)
(ξ
2
)(
)
− a2 ξ 3 − a3 +
a2 , a3
)
Now if we code the factors to ±1 with the transforms given earlier, the Taylor series becomes the simpler Maclaurin series, which by definition is centered at the origin (0, 0, 0): 3
y ≈ φ(0, 0, 0) +
∑ k =1
∂φ xk + ∂xk 0
2
3
∑∑ j< k
k =1
∂ 2φ ∂xj ∂xk
3
xj x k +
∑ k =1
0 ,0
∂ 2φ xk 2 ∂xk2 0 2 !
Equations 4.3 and 4.4 give heuristics for the infinite Maclaurin and Taylor series terms, respectively. For our purposes, we will usually truncate them at n ≤ 2:
(
)
n=∞
y = φ x1 , x2 ,$ xφ = φ(0 ) +
(
)
T
y = φ ξ1 , ξ 2 ,$ , ξ f = φ(a ) +
∑ ∑ p =1
n=∞
T
1 ⎡ ∂φ ⎢ p ! ⎢ k =1 ∂ξ k ⎣ f
∑ ∑ p=1
p
f ⎤ 1 ⎡ ∂φ ⎢ xk ⎥ Maclaurin series (4.3) p ! ⎢ k =1 ∂xk 0 ⎥ ⎣ ⎦
( ak
p
⎤ ξ k − ak ⎥ Taylor series ⎥⎦
)
(4.4)
In the above equations, φ( ) is the functional notation; x and ξ are the independent variables (factors), the former being scaled and centered and the latter not — i.e., in their original or customary metrics; k indexes the factors from 1 to f; f is the number of factors in the model; p indexes the order of the series from 1 to n; n is the overall order of the series (for an infinite series n = ∞), and 0 and a are vectors — the former is a vector of f zeros and the latter a vector of f constant terms (a1, a2, … , af)T. For nonlinear models, when n < ∞, the series is no longer exact but approximate. In such a case we replace the equality (=) by an approximate equality (≈). We illustrate the use of Equations 4.3 and 4.4 with an example.
Example 4.1
Derivation of the Maclaurin Series for Two Factors
Problem statement: Use Equations 4.3 and 4.4 to derive the Maclaurin and Taylor series for y = φ( x1 , x2 ), truncated to third order. What would the corresponding fitted equation look like?
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Modeling of Combustion Systems: A Practical Approach Solution: For f = 2 and n = 3, Equation 4.3 becomes n= 3
y ≈ φ(0, 0) +
f =2 ⎤ 1 ⎡ ∂φ ⎢ xk ⎥ p ! ⎢ k =1 ∂xk 0 ⎥ ⎣ ⎦
∑ ∑ p =1
p
Proceeding step by step, we have the following: 1
⎞ ⎞ ∂φ ∂φ 1 ⎛ ∂φ 1 ⎛ ∂φ y ≈ φ(0, 0) + ⎜ x1 + x2 ⎟ + ⎜ x1 + x2 ⎟ ∂x1 0 ⎠ ∂x1 0 ⎠ 1 ! ⎝ ∂x1 0 2 ! ⎝ ∂x1 0 +
⎞ 1 ⎛ ∂φ ∂φ x1 + x2 ⎟ ⎜ 3 ! ⎝ ∂x1 0 ∂x1 0 ⎠
2
3
⎧ ⎞ ⎛ ∂φ ⎞ 1 ⎛ ∂ 2φ ∂φ ∂ 2φ ∂ 2φ ⎪φ(0, 0) + ⎜ x1 + x2 ⎟ + ⎜ 2 x12 + 2 x1x2 + 2 x22 ⎟ + ∂x1 0 ⎠ 2 ! ⎝ ∂x1 0 ∂x1∂x2 0 ∂x2 0 ⎠ ⎪⎪ ⎝ ∂x1 0 y≈⎨ ⎪ ∂ 3φ ∂ 3φ ∂ 3φ 3 ⎞ 1 ⎛ ∂ 3φ 3 2 2 x + 3 x x + 3 x x + x2 ⎟ ⎜ 1 2 1 1 2 ⎪ 3 ! ⎝ ∂x13 0 ∂x12 ∂x2 0 ∂x1∂x22 0 ∂x23 0 ⎠ ⎪⎩ ⎧ ⎛ ∂φ ⎞ ⎛ ∂ 2φ x 2 ∂ 2φ ∂ 2φ x 2 ⎞ ∂φ ⎪φ(0, 0) + ⎜ x1 + x2 ⎟ + ⎜ 2 1 + x1x2 + 2 1 ⎟ + ∂x1 0 ⎠ ⎝ ∂x1 0 2 ! ∂x1∂x2 0 ∂x2 0 2 ! ⎠ ⎪⎪ ⎝ ∂x1 0 y≈⎨ ⎛ ∂ 3φ x 3 ⎪ ∂ 3φ x12 x2 ∂ 3φ x1x22 ∂3φ x23 ⎞ 1 + 3 + + ⎜ ⎟ ⎪ 3 2 ∂x2 0 3 ! ⎠ ∂x1∂x22 0 2 ! ⎝ ∂x1 0 3 ! ∂x1 ∂x2 0 2 ! ⎪⎩ If we were to evaluate the above equation numerically from a data set, we could fit the third-order model
(
) (
)
⎧ a0 + a1x1 + a2 x2 + a11x12 + a12 x1x2 + a22 x22 + ⎪ y≈⎨ a1111x13 + a112 x12 x2 + a122 x1x22 + a222 x23 ⎪⎩
(
)
Here, we have grouped the terms in parentheses by overall order. We may derive the Taylor series in the same manner, replacing xk by ξk – ak and 0 by ak .
(
) (
)
⎧b0 + b1ξ1 + a2 ξ 2 + b11ξ12 + b12 ξ1ξ 2 + b22 ξ 22 + ⎪ y≈⎨ b111ξ13 + b112 ξ12 ξ 2 + b122 ξ1ξ 22 + b222 ξ 32 ⎪⎩
(
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)
Analysis of Nonideal Data 4.2.1
301
Model Bias from an Incorrect Model Specification
In the previous section, we constructed a model comprising a finite number of terms by truncating an infinite Taylor series; therefore, if higher-order derivatives exist, then they will bias the coefficients. We introduced the reader to this concept in Chapter 3 beginning with Section 3.4. Here we explore additional considerations. For example, let us suppose that Equation 4.5 gives the true model for NOx: ln y = A −
b T
(4.5)
where y is the NOx, A and b are constants, and T is the furnace temperature. Further, suppose that due to our ignorance or out of convenience or whatever, we fit the following (wrong) model: y = a0 + a1T
(4.6)
where x is centered and scaled per our usual convention, i.e., x=
T −T Tˆ
ˆ + T. Then T = Tx The Maclaurin series becomes y=φ + 0
dφ d 2 φ x 2 d 3φ x 3 x+ 2 + +$ dx 0 dx 0 2 ! dT 3 0 3 !
where φ( x) = e
b A− ˆ Tx +T
We may also write this as y = a0 + a1x + a2 x 2 + a3 x 3 + a4 x 4 + $
(4.7)
where a0 = φ , a1 = 0
dφ 1 d 2φ 1 d 3φ , a2 = , a3 = $ 2 dx 0 2 ! dx 0 3 ! dT 3 0
So long as the series remains infinite, there is a one-to-one correspondence between the coefficients and the evaluated derivatives. However, once we truncate the model, this is no longer strictly true: higher-order derivatives
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will bias the lower-order coefficients. Yet, near zero, higher-order terms will vanish more quickly than lower-order ones. So, if x is close to zero then the model has little bias. We refer to the error caused by using an incorrect mathematical expression as model bias. At x = 1 each term is weighted by its Maclaurin series coefficient. As x grows beyond 1, then the higher-order terms exert larger and larger influence; so mild extrapolation leads quickly to erroneous results. This would not be the case if the model were correct. Notwithstanding, even for the incorrect empirical model, this bias may be nil so long as we are within the bounds of our original data set (coded to ±1). For x >> 1, we need to add many additional terms for the empirical model to adequately approximate the true model. As x grows larger and larger, we need more and more empirical terms. This is so, despite the fact that the true model comprises only two terms. This is why it is much more preferable to generate a theoretical or semiempirical form rather than a wholly empirical one. Nonetheless, an empirical model of second order at most (and usually less) is sufficient for interpolation. In other words, empirical models are very good interpolators and very poor extrapolators. This is true for all models in the sense that we may never have exactly the right model form, but it is especially so for empirical models. Suppose that we could expand our model to comprise an infinite number of terms (which would require an infinite data set to fit). Then we could evaluate the coefficients for Equation 4.7, generating the following normal equations: ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝
∑ y ⎞⎟ ⎛⎜ N ∑x ∑x ∑ xy ⎟⎟ ⎜⎜ ∑x ∑x ⎟ ⎜ ∑ x y⎟⎟ = ⎜⎜ ∑ x ∑x ∑x ∑ x y⎟⎟ ⎜⎜ ∑x ∑x ∑ x y⎟⎟ ⎜⎜ ∑ x ∑x ∑x 2
2
4
4
3
⎠⎟
⎜⎝
%
6
%
6
6
4
4
%
4
2
2
$⎞ ⎟ ⎛ a0 ⎞ $⎟ ⎜ ⎟ ⎟ a1 ⎟⎜ ⎟ $⎟ ⎜ a2 ⎟ ⎟ ⎜ a ⎟ (4.8) $⎟ ⎜ 3 ⎟ ⎟ ⎜ a4 ⎟ ⎜ ⎟ $⎟ ⎝ % ⎠ ⎟ '⎟⎠
4
%
%
8
%
Because we centered x, the sum of the odd powers is zero, but the sum of the even powers is not. Since our approximate model comprises only two terms — a0 and a1 of Equation 4.6 — the higher-order terms will bias them. A careful examination of Equation 4.8 shows that the even terms bias a0 and the odd terms bias a1. We are actually fitting an equation something like
(
) (
)
y = b0 c0 + c2 x 2 + c4 x 4 + $ + b1 c1 + c3 x 3 + c5 x 5 + $ x
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(4.9)
Analysis of Nonideal Data
303
where bk and ck are constants accounting for the contributions of the higherorder derivatives. Again, for 0 < x < 1 the sum of the higher powers will likely be negligible, and for this reason, empirical models are excellent interpolators. Nonetheless, a good theoretical model would eliminate this bias and would require fewer terms for an adequate fit to the data.
4.2.2
Design Bias
We have seen from the previous section that an improper model specification is a problem if we extrapolate beyond the bounds of the experimental design. The proper model derived from theoretical considerations ameliorates this problem. We have also seen that a purely empirical model will do a very good job within the design boundaries even if it is wrong. However, even with the proper model, an improper experimental design may still bias the coefficients. We refer to errors introduced by a less than ideal X matrix as design bias. Conversely, proper experimental design can eliminate this bias. TABLE 4.1 A Classical Design in Three Factors y = ln(NOx)
x1 = O2, %
x2 = APH, °F
x3 = BWT, °F
2.19 2.70 2.95 2.83
1 5 1 1
25 25 500 25
1000 1000 1000 2000
Consider a classical one-factor-at-a-time design given in Table 4.1. Here, x1 is the excess oxygen concentration in the furnace, x2 is the air preheat temperature (APH) of the combustion air, and x3 is the furnace temperature, measured at the bridgewall of the furnace (BWT). Let us represent this factor space by S: ⎛1 ⎜1 S=⎜ ⎜1 ⎜ ⎝1
1 5 1 1
25 25 500 25
1000 ⎞ 1000 ⎟⎟ 1000 ⎟ ⎟ 2000⎠
(4.10a)
Then ⎛ 4 ⎜ 8 S TS = ⎜ ⎜ 575 ⎜ ⎝ 5000
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8 28 675 251, 875
575 675 251, 875 600, 000
5000 ⎞ 9000 ⎟⎟ 600, 000 ⎟ ⎟ 7 , 000, 000⎠
(4.10b)
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This is not a promising start, as STS contains not a single zero value; everything mutually biases everything else. Coding will zero some of the off-diagonal values. Using the coding transforms, we have ⎛1 ⎜1 X=⎜ ⎜1 ⎜ ⎝1 ⎛4 ⎜0 X TXa = ⎜ ⎜0 ⎜ ⎝0
−0.5 1.5 −0.5 −0.55 0 3 −1 −1
−0.5 −0.5 1.5 −0.5
−0.5⎞ −0.5⎟⎟ −0.5⎟ ⎟ 1.5 ⎠
(4.11a)
0 −1 3 −1
0⎞ ⎛ a0 ⎞ −1⎟⎟ ⎜⎜ a1 ⎟⎟ −1⎟ ⎜ a2 ⎟ ⎟⎜ ⎟ 3⎠ ⎝ a3 ⎠
(4.11b)
(We show a merely to give the coefficient references.) These coded data are better. At least a0 is unbiased, but a1 to a3 still bias one another. Figure 4.3a depicts the classical design. It forms a right-angled tetrahedron in factor space. Since it is neither scaled nor centered, the edges are not equal lengths, nor does the design center (centroid of the tetrahedron) coincide with the center of the factor space (centroid of the cubic region). Figure 4.3b is the same design scaled to 0/1 coordinates, but not centered. Since it is not centered, the design center is not coincident with the center of the factor space. Figure 4.3c shows the design in ±1 coordinates. The design and coordinate centers are now coincident. However, the design is still not orthogonal because it is not balanced about the coordinate center. Figure 4.3d is an example of a fractional factorial design. It is centered and scaled, and since it is balanced about the origin, it also gives an orthogonal matrix. Let us represent this factor space by T. Then, ⎛1 ⎜1 T=⎜ ⎜1 ⎜ ⎝1 ⎛4 ⎜ TTT = ⎜ ⎜ ⎜ ⎝
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−1 −1 1 1
−1 1 −1 1
4 4
1⎞ −1⎟⎟ −1⎟ ⎟ 1⎠
(4.12a)
⎞ ⎟ ⎟ ⎟ ⎟ 4⎠
(4.12b)
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305
Coordinate APH: 25 – 500 F Center Design Center
x
O2:1– 5%
x BWT: 1000 – 2000 F
A Design Point (1 of 4) (a) Design in Original Coordinates (distorted right-angle tetrahedron)
(b) Design in 0/1 Coordinates (right-angle tetrahedron)
st
s3
t2
x t3
(c) Design in ±1 Coordinates (centered right-angle tetrahedron)
t1 s2
(d) Orthogonal Design in ±1 Coordinates (regular tetrahedron)
FIGURE 4.3 Graphical representation of various experimental designs. (a) The classical design in the original coordinates. The coordinate center does not coincide with the center of the design. (b) The design coded to 0/1 coordinates. This conformally shrinks the factor directions to uniform dimension. (c) The design in ±1 coordinates. The design and coordinate center are now coincident (X). (d) A design that is orthogonal and centered in the new coordinates.
If we could transform our design to these coordinates, we would have an orthogonal design. In fact, we can.
4.3
Ways to Make Designs Orthogonal
We have two basic remedies to make designs orthogonal: we can either change the design or morph the factor space. Changing the design means that before we begin the experiment, we think about what factors are important and how we can arrange the test matrix to be orthogonal. This generates a balanced design having an equal number of high and low values for each factor equidistant from zero in each factor direction, e.g., factorial designs. The advantage of using orthogonal designs is that one can examine independent factors with clear meaning and perform a number of statistical tests, etc. The only “disadvantage” is that it requires up-front thinking. Remember Westheimer’s discovery: “a couple of months in the laboratory will save you a couple of hours at the library.”
© 2006 by Taylor & Francis Group, LLC
306 4.3.1
Modeling of Combustion Systems: A Practical Approach Source and Target Matrices: Morphing Factor Space
Suppose we want to convert a source matrix (S) that is nonorthogonal but full rank and square, such as Matrix 4.10a, into an orthogonal target matrix (T), such as Matrix 4.12a. We could postmultiply by some transformation (F) matrix: SF = T
(4.13)
F = S–1SF = S–1T
(4.14)
Then, solving for F we have
So long as the source matrix is full rank, it will have an inverse. It does not matter whether we use the original matrix or first transform the matrix using a linear transform. Accordingly, let us first scale the numbers by coding the high and low values to 0 and 1, respectively, using the following transform:
wk =
ξ k − ξ −k ξ +k − ξ −k
(4.15)
Again, we could just have easily used the original matrix, the above 0/1 coding, or the traditional ±1 coding. But as this is a classical design, onefactor-at-a-time investigations usually proceed from some origin, which is more conveniently coded as the coordinate center. ⎛1 ⎜1 S=⎜ ⎜1 ⎜ ⎝1
⎛4 ⎜1 S TS = ⎜ ⎜1 ⎜ ⎝1
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0 1 0 0
0⎞ 0⎟⎟ 0⎟ ⎟ 1⎠
0 0 1 0
1 1 0 0
1 0 1 0
1⎞ 0⎟⎟ 0⎟ ⎟ 1⎠
(4.16a)
(4.16b)
Analysis of Nonideal Data
307
We would like to transform S in Matrix 4.16a into T of Matrix 4.12a. We will do this with a transformation matrix: ⎛1 ⎜1 T = SF = ⎜ ⎜1 ⎜ ⎝1
0 1 0 0
0 0 1 0
0⎞ 0⎟⎟ F 0⎟ ⎟ 1⎠
(4.17)
To find F, we apply Equation 4.14 and obtain ⎛1 ⎜0 F=⎜ ⎜0 ⎜ ⎝0
−1 2 0 2
−1 0 2 2
−1⎞ 2 ⎟⎟ 2⎟ ⎟ 0⎠
(4.18)
We now observe that indeed ⎛1 ⎜1 T = SF = ⎜ ⎜1 ⎜ ⎝1
0 1 0 0
0 0 1 0
0⎞ ⎛ 1 0⎟⎟ ⎜⎜ 0 0⎟ ⎜ 0 ⎟⎜ 1⎠ ⎝ 0
−1 2 0 2
−1 0 2 2
−1⎞ ⎛ 1 2 ⎟⎟ ⎜⎜ 1 = 2⎟ ⎜ 1 ⎟ ⎜ 0⎠ ⎝ 1
−1 1 −1 1
−1 −1 1 1
−1⎞ 1⎟⎟ 1⎟ ⎟ −1⎠
(4.19)
Before the transformation, we have something like y = b0 + b1s1 + b2 s2 + b3s3 in s1·s2·s3 factor space. After the transformation, we have y = a0 + a1t1 + a2t2 + a3t3 in t1·t2·t3 factor space. This latter function is orthogonal in t1, t2, and t3. In other words, if y = Ta = Sb and SF = T, (where F maps s1, s2, and s3 onto t1, t2, and t3) then SFa = Sb. So, on the one hand, we have gained independent coefficients. On the other hand, we are not sure what they mean. In other words, we are trading a non-orthogonal design in orthogonal s1·s2·s3 factor space for an orthogonal design in distorted t1·t2·t3 space. If the distorted space has no physical meaning, we have gained little. We see that after the fact, it may be possible to find combinations of the original factors that represent an orthogonal design. However, this is a much weaker approach than conducting a proper design in the first place, because the factor combinations often have no real meaning. On the other hand, sometimes a linear combination of factors does have meaning and the linear combination may actually be the penultimate factor.
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Modeling of Combustion Systems: A Practical Approach
For example, kinetic expressions (those determining the rate of appearance or disappearance of a species like NOx or CO) are really a function of collision frequency (Z). But it is not possible to directly observe molecular collisions and hence Z. However, Z is related to the temperature (T), pressure (P), and concentration (C) — all increase the collision frequency. Suppose, for the sake of argument, that the actual production rate of an important species, y = f(ζ), were actually a function of the log of the collision frequency, ζ = ln(Z), and that Z is given by Equation 4.20: Z = b0 P a1 e
− a2 T
C a3
(4.20)
Then ζ = a0 + a1x1 + a2 x2 + a3 x3 where a0 = ln(b0), x1 = ln(P), x2 = − 1 T , and x3 = ln(C). So for y = φ(ζ), the most parsimonious model would actually be a linear combination of x1, x2, and x3. In such a case, orthogonal components may be useful to spot such relations in the data. However, we do not want to distort the original factors. We seek only to rotate the axes to expose these relations. Eigenvectors and eigenvalues can do this for us.
4.3.2
Eigenvalues and Eigenvectors
One may use eigenvalues and eigenvectors to decompose a matrix into orthogonal components, and they are the best alternative for that purpose because they do not distort the factor space as the source–target method may Λ) and eigenvectors (K) are defined for a square matrix do. Eigenvalues (Λ (M) of full rank as follows: Λ = MK KΛ
(4.21)
where K is the eigenvector matrix of M, and Λ is the diagonal eigenvalue matrix for M. Λ is a diagonal matrix of the eigenvalues (λ) of M: ⎛ λ1 ⎜ Λ=⎜ ⎜ ⎜ ⎝
λ2 '
⎞ ⎟ ⎟ ⎟ ⎟ λn ⎠
For real symmetric matrices, we shall derive eigenvectors that are orthogonal in the strictest sense. That is,
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Analysis of Nonideal Data
309 K TK = K −1K = I
(4.22)
Theoretically, eigenvalues are solutions of an nth-order polynomial (characteristic) equation, where n is the number of rows in the starting matrix, presuming it is nonsingular and square. Matrix algebra texts give the procedure.3 However, the mechanics can become unwieldy and dedicated software is really a must for this procedure. Regrettably, Excel does not have a standard function for this, but software such as MathCAD™ does. Dedicated statistical software is the best option. The procedure can be done in a spreadsheet, but it is tedious, as we show now. We may make use of the trace of the matrix to find the eigenvalues. The trace of a matrix is the sum of the diagonal elements. We may also define traces for higher-order square matrices. n
( ) ∑m
t n = tr Mn =
n kk
k =1
In the above equation, we are relying on context to obviate any equivocation for n, (for Mn the superscript is an authentic exponent). n
M = n
∏ M = (M)(M)$(M) k =1
Thus, M2 = MM. However, for tn and mnkk, the superscript n is mere nomenclature. Once we have tn, the characteristic equation and its solutions follow: n
∑c λ k
k
= 0 (characteristic equation)
(4.23)
k=0
⎧ 1 if j = n ⎪ ⎪ (coefficient solutions) cj = ⎨ n− j − 1 n− j − k ⎪ −1 oth herwise cn− kt ⎪n− j k=0 ⎩
∑
(4.24)
where λ are the latent roots (also called eigenvalues, proper values, or characteristic values). To clarify these concepts, we illustrate with an example.
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Modeling of Combustion Systems: A Practical Approach
Example 4.2
The Characteristic Equation Using the Trace Operator
Problem statement: Given Matrix 4.16b, find the characteristic equation and the eigenvalues. Solution: Matrix 4.16b is a full-rank (nonsingular) matrix having four rows (n = 4). We solve for tn in the following manner: Let ⎛4 ⎜1 M=⎜ ⎜1 ⎜ ⎝1
1 1 0 0
1⎞ 0⎟⎟ 0⎟ ⎟ 1⎠
1 0 1 0
Then ⎛4 ⎜1 M2 = ⎜ ⎜1 ⎜ ⎝1
1 1 0 0
1 0 1 0
1⎞ ⎛ 4 0⎟⎟ ⎜⎜ 1 0⎟ ⎜ 1 ⎟⎜ 1⎠ ⎝ 1
⎛ 91 ⎜ 24 M3 = ⎜ ⎜ 24 ⎜ ⎝ 24 ⎛ 436 ⎜ 115 M4 = ⎜ ⎜ 115 ⎜ ⎝ 115
1 1 0 0
1⎞ ⎛ 19 0⎟⎟ ⎜⎜ 5 = 0⎟ ⎜ 5 ⎟ ⎜ 1⎠ ⎝ 5
1 0 1 0
24 7 6 6
24 6 7 6
115 31 30 30
115 30 31 30
24⎞ 6 ⎟⎟ 6⎟ ⎟ 7⎠ 115⎞ 30 ⎟⎟ 30 ⎟ ⎟ 31 ⎠
Now the traces of each matrix become
( )
t = tr M = 4 + 1 + 1 + 1 = 7
( )
t 2 = tr M2 = 25
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5 2 1 1
5 1 2 1
5⎞ 1⎟⎟ 1⎟ ⎟ 2⎠
Analysis of Nonideal Data
311 t 3 = 112 t 4 = 529
Solving for the coefficients of the characteristic matrix according to Equation 4.24, we have c4 = 1 (the nth coefficient of the characteristic equation is always 1) c3 = − c4t = −7
c2 = −
c1 = −
c0 = −
(
)
( )( ) ( )( )
1 1 c3t + c4 2 t = − ⎡⎣ −7 7 + 1 25 ⎤⎦ = 12 2 2
(
)
( )( ) ( )( ) ( )( )
1 1 c2t + c3 2 t + c4 3 t = − ⎡⎣ 12 7 + −7 25 + 1 112 ⎤⎦ = −7 3 3
(
)
( )( ) ( )( ) ( )( ) ( )( )
1 1 c1t + c2 2 t + c3 3 t + c4 4 t = − ⎡⎣ −7 7 + 12 25 + −7 112 + 1 529 ⎤⎦ = 1 4 4
and the characteristic equation is 1 – 7λ + 12λ2 – 7λ3 + λ4. Fortunately, this equation factors as (λ2 – 5λ + 1)(λ- 1)2 = 0 with the solutions ⎧⎪ λ = ⎨1, ⎩⎪
1,
5 − 21 , 2
5 + 21 ⎫⎪ ⎬ 2 ⎭⎪
Since these are solutions for a single variable, one may also use numerical procedures such as the goal seek algorithm in Excel to solve for them. Also, the rational roots (if they exist) will always be factors of the constant. In our case, the constant is 1, so we would try ±1, finding 1 to be a double root, as shown above. This rational roots procedure can often help to factor the equation and reduce the order of the remainder, simplifying the final solution.
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312
Modeling of Combustion Systems: A Practical Approach Analytically, one can always find the solutions for polynomials up to fourth order using various procedures.*
Each eigenvalue has an associated eigenvector such that (M – λI)k = 0
(4.25)
where k is an eigenvector. The eigenvectors are not unique in the sense that any scalar multiple of an eigenvector will itself be an eigenvector. To resolve this problem, we shall reduce the eigenvectors to unit magnitude, i.e., n− 1
∑k
2 j
=1
j= 0
For real, symmetric matrices (the only kind we need to consider in this text), the eigenvectors are always orthogonal. That is, jTk = 0
(4.26)
where j and k are any two different vectors in the K matrix. For the case at hand, Equation 4.25 reduces to
(
⎛4−λ ⎜ 1 M − λI b = ⎜ ⎜ 1 ⎜ ⎝ 1
1 1− λ
)
0 0
1 0 1− λ 0
1 ⎞ ⎛ k0 ⎞ ⎛ 0 ⎞ 0 ⎟⎟ ⎜⎜ k1 ⎟⎟ ⎜⎜ 0⎟⎟ = 0 ⎟ ⎜ k2 ⎟ ⎜ 0 ⎟ ⎟⎜ ⎟ ⎜ ⎟ 1 − λ ⎠ ⎝ k3 ⎠ ⎝ 0 ⎠
(4.27)
We illustrate the procedure for one of the eigenvalues in the next example.
Example 4.3
Finding an Eigenvector from an Eigenvalue
Problem statement: For the Matrix 4.27, we have shown that the characteristic equation is (λ2 – 5λ + 1)(λ- 1)2 = 0, having solutions ⎧⎪ λ = ⎨1, ⎪⎩
1,
5 − 21 , 2
5 + 21 ⎫⎪ ⎬ 2 ⎪⎭
Find the eigenvector associated with the eigenvalue =
5 − 21 ≈ 0.2087 . 2
* Any standard mathematical text will have solutions for up to fourth-order polynomials. See, for example, Gellert, W. et al., Eds., The VNR Concise Encyclopedia of Mathematics, American Edition, Van Nostrand Reinhold Company, New York, 1977, pp. 80-101. General equations of fifth order and higher have been proven impossible to solve, though many special equations of arbitrary order are solvable; e.g., the triquadratic equation ax6 + bx3 + c = 0 may be reduced to a quadratic equation with the substitution u = x3.
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Analysis of Nonideal Data
313
Solution: We can find the eigenvector numerically using a spreadsheet. Step 1: First, we substitute a selected eigenvalue, e.g., λ =0.2097:
(
⎛ 3.7913 ⎜ 1 M − λI k = ⎜ ⎜ 1 ⎜ ⎝ 1
)
1 0.7913 0 0
1 ⎞ ⎛ k0 ⎞ ⎛ 0 ⎞ 0 ⎟⎟ ⎜⎜ k1 ⎟⎟ ⎜⎜ 0⎟⎟ = 0 ⎟ ⎜ k2 ⎟ ⎜ 0 ⎟ ⎟⎜ ⎟ ⎜ ⎟ 0.7913⎠ ⎝ k3 ⎠ ⎝ 0⎠
1 0 0.7913 0
Step 2: Now we arbitrarily set k3 = 1, ⎛ 3.7913 ⎜ 1 ⎜ ⎜⎝ 1
1
1
0.7913 0
0 0.7913
⎛ k ⎞ ⎛ 0⎞ 1⎞ ⎜ 0 ⎟ ⎜ ⎟ k 0 0⎟⎟ ⎜ 1 ⎟ = ⎜ ⎟ ⎜ k ⎟ ⎜ 0⎟ 0⎟⎠ ⎜ 2 ⎟ ⎜ ⎟ ⎝ 1 ⎠ ⎝ 0⎠
and reduce the matrix by one column and the eigenvector and solution vector by one row, so that the system becomes soluble. ⎛ 3.7913 ⎜ 1 ⎜ ⎜⎝ 1
1 0.7913 0
⎛ 1⎞ ⎞ ⎛ k0 ⎞ ⎟ ⎜ ⎟ 0 ⎟ ⎜ k1 ⎟ = − ⎜⎜ 0⎟⎟ ⎜⎝ 0⎟⎠ 0.7913⎟⎠ ⎜⎝ k2 ⎟⎠ 1
Step 3: Premultiplying by the inverse of the matrix we have ⎛ 3.7913 ⎛ b0 ⎞ ⎜b ⎟ = −⎜ 1 ⎜ ⎜ 1⎟ ⎜⎝ 1 ⎜⎝ b2 ⎟⎠
1 0.7913 0
1 ⎞ 0 ⎟⎟ 0.7913⎟⎠
−1
⎛ 1⎞ ⎛ −0.791⎞ ⎜ 0⎟ = ⎜ 1 ⎟ ⎟ ⎜ ⎟ ⎜ ⎜⎝ 0⎟⎠ ⎜⎝ 1 ⎠⎟
But we had arbitrarily set k3 = 1, so the full vector is ⎛ k0 ⎞ ⎛ −0.791⎞ ⎜k ⎟ ⎜ 1 ⎟ ⎜ 1⎟ = ⎜ ⎟ ⎜ k2 ⎟ ⎜ 1 ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ k3 ⎠ ⎝ 1 ⎠ and this is an eigenvector associated with λ = 0.2097.
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314
Modeling of Combustion Systems: A Practical Approach Step 4: Normalizing this by root of the sum of squares, 12 + 12 + 12 + 0.79132 = 1.9042 , we obtain the unit eigenvector associated with λ = 0.2097: ⎛ −0.416⎞ ⎜ 0.525⎟ ⎟ k3 = ⎜ ⎜ 0.525⎟ ⎜ ⎟ ⎝ 0.525⎠ where the subscript denotes the column of the column vector in the eigenvectors’ matrix K.
So long as the eigenvectors are distinct, this method will lead to the associated eigenvectors. The major advantage of this method is that spreadsheets can do all the calculations. However, if the eigenvectors are not distinct (e.g., multiple roots), we will end up with a problem — two different eigenvectors associated with two identically valued eigenvalues. We can continue without problem to obtain an eigenvector associated with (5 + 21 ) 2 ≈ 4.7913. ⎛ 0.910⎞ ⎜ 0.240⎟ ⎟ k4 = ⎜ ⎜ 0.240⎟ ⎜ ⎟ ⎝ 0.240⎠ But we run into trouble almost immediately, solving for the eigenvectors associated with the double root, λ = {1, 1}, generating the matrix ⎛3 ⎜1 ⎜ ⎜1 ⎜ ⎝1
1 0 0 0
1 ⎞ ⎛ k0 ⎞ ⎛ 0 ⎞ 0⎟⎟ ⎜⎜ k1 ⎟⎟ ⎜⎜ 0⎟⎟ = 0 ⎟ ⎜ k2 ⎟ ⎜ 0 ⎟ ⎟⎜ ⎟ ⎜ ⎟ 0⎠ ⎝ 1 ⎠ ⎝ 0⎠
1 0 0 0
It reduces to the following equations: 3k0 + k1 + k2 = –1 and k0 = 0. Substituting one into the other, we obtain k1 + k2 = –1, from which we may evaluate the remaining two eigenvectors. ⎛ 0 ⎜ K′ = ⎜ a ⎜ −(1 + a) ⎜ ⎝ 1
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0 b −(1 + b) 1
−0.416 0.525 0.525 0.525
0.910⎞ ⎟ 0.240⎟ 0.240⎟ ⎟ 0.240⎠
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Here a and b are undetermined coefficients. We use a and b because the remaining two eigenvectors cannot be the same; eigenvectors for real symmetric matrices are always mutually orthogonal. Note that we have not yet normalized the first two vectors in K to unit magnitude, so for now we label the eigenvector matrix as K′′ rather than K. Now if the first two column vectors in K′′ (let us call them k0 and k1) are mutually orthogonal, then k0Tk1 = 0, giving ab + (a + 1)(b + 1) + 1 = 0, which reduces to a = − (b + 2) (2b + 1) . Arbitrarily choosing b = 1 gives a = –1. Substituting into the matrix gives
⎛ 0 ⎜ K′ = ⎜ 1 ⎜ −2 ⎜ ⎝ 1
0 −1 0 1
−0.416 0.525 0.525 0.525
0.910⎞ ⎟ 0.240⎟ 0.240⎟ ⎟ 0.240⎠
Normalizing the first two vectors to unit magnitude gives
⎛ 0 ⎜ K = ⎜ 0.408 ⎜ −0.816 ⎜ ⎝ 0.408
0 −0.707 0 0.707
−0.416 0.525 0.525 0.525
0.910⎞ ⎟ 0.240⎟ 0.240⎟ ⎟ 0.240⎠
MathCAD gives the following solution, which the reader may verify is equally correct, yielding the relations given in Equations 4.21, 4.22, and 4.25. (Multiple roots do not have unique associated eigenvectors.)
⎛ 0 ⎜ K = ⎜ −0.272 ⎜ 0.803 ⎜ ⎝ −0.531
0 0.770 −0.149 −0.620
−0.416 0.525 0.525 0.525
0.910⎞ ⎟ 0.240⎟ 0.240⎟ ⎟ 0.240⎠
(4.28)
At any rate, once we obtain the eigenvalues and eigenvectors, we can move on to making real symmetric matrices orthogonal. Least squares solutions always generate real symmetric matrices; thus, they are amenable to this treatment. Recall that for real symmetric matrices, eigenvectors are orthogonal in the strictest sense. And so it follows for our example that
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⎛ 0.000 ⎜ 0.000 ⎜ ⎜ −00.416 ⎜ ⎝ 0.910 ⎛1 ⎜ ⎜ ⎜ ⎜ ⎝
0.408 −0.707 0.525 0.240
1 1
−0.816 0.000 0.525 0.240
0.408 ⎞ ⎛ 0.000 0.707 ⎟⎟ ⎜⎜ 0.408 0.525 ⎟ ⎜ −0.816 ⎟⎜ 0.240 ⎠ ⎝ 0.408
0.803
−0.531⎞ ⎛ 0.000 −0.620⎟⎟ ⎜⎜ −0.272 0.525⎟ ⎜ 0.803 ⎟⎜ 0.240⎠ ⎝ −0.531
0.000 −0.707 0.000 0.707
−0.416 0.525 0.525 0.525
0.910⎞ 0.240⎟⎟ = 0.240⎟ ⎟ 0.240⎠
⎞ ⎟ ⎟ ⎟ ⎟ 1⎠
or ⎛ 0.000 ⎜ 0.000 ⎜ ⎜ −0.416 ⎜ ⎝ 0.910 ⎛1 ⎜ ⎜ ⎜ ⎜ ⎝
4.3.3
−0.272 0.770 0.525 0.240
1 1
−0.149 0.525 0.240
0.000 0.770 −0.149 −0.620
−0.416 0.525 0.525 0.525
0.910⎞ 0.240⎟⎟ = 0.240⎟ ⎟ 0.240⎠
⎞ ⎟ ⎟ ⎟ ⎟ 1⎠
Using Eigenvectors to Make Matrices Orthogonal
Premultiplying Equation 4.21 by KT gives Λ ≡ Λ ≡ KTMK KTKΛ
(4.29)
Given y = Xa, we seek another system of factors giving linear combinations of X such that y = Ub, and also where UTU = D, a diagonal matrix. Here is the procedure: Step 1: Express the first specification mathematically: y = Xa = Ub
(4.30)
M = X TX U = XK
(4.31) (4.32)
b = KTa
(4.33)
Step 2: Define the following:
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Step 3: The above equations complete the transformation: substitution of these relations into Equation 4.30 gives Ub = XKKTa = Xa because KKT = I; therefore, y = Ub represents an alternate system of factors and coefficients for y = Xa. To see this, consider the necessary properties of UTU. Premultiplying y = Ub by UT gives UTy = UTUb. Substituting in terms of X gives UTU = (xK)TXK = KTXTXK. But M = XTX, and in light of Equation 4.29, this substitution gives UTU = KTMK = Λ. Therefore, U is a diagonal matrix — the eigenvector matrix of XTX to be exact. Collecting these equations: y = Xa = Ub K = eigenvectors (XTX)
(4.34)
U = XK b = KTa UT U = Λ
(4.35)
ΛKT XTX = KΛ
(4.36)
KTXTXK = Λ
(4.37)
Equations 4.28 and 4.30 amount to the following in light of Equation 4.32: u1 = –0.272 x1 + 0.803 x2 – 0.531 x3 u2 = 0.770 x1 – 0.149 x2 – 0.620 x3 u3 = –0.416 + 0.525 (x1 + x2 + x3) u4 = 0.910 + 0.240 (x1 + x2 + x3) Thus, the uk represent linear combinations of xk, and either system will give identical values for y: y = a0 + a1x1 + a2x2 + a3x3 = b1u1 + b2u2 + b3u3 + b4u4. The advantage of the eigenvalue procedure over the source–target matrix procedure is that: 1. We have merely rotated axes, not distorted factors. 2. Both the original and new coordinate axes are orthogonal. 3. We may apply the procedure to any nonsingular matrix, even when X is nonsquare, because M = XTX will always be square.
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4. The new system, y = Ub, is orthogonal. Therefore, we estimate b without design bias. 5. If the linear combinations of factors have meaning, they may represent a more parsimonious model and help to identify an important underlying relationship.
4.3.4
Canonical Forms
Consider a second-order surface given by the general second-order equation in summation form for f factors. f
y = a0 +
∑
ak xk +
k =1
f
f
j≤ k
k =1
∑∑ a x x
jk j k
(4.38)
In this formulation, the inequality in the index of the second-order terms (j ≤ k ) includes the pure quadratics akk xk2 . Now this surface may take several possible forms, depending on the coefficients. However, by rotating and/or translating axes, we can always simplify the equation to either of two forms: f
y = a0 +
∑
f
θ k uk +
k =1
∑λ
2 kk k
A canonical form
(4.39a)
u B canonical form
(4.40a)
u
k
f
y=
∑λ
2 kk k
k =1
Box and Draper4 call the first the A canonical form and the second the B canonical form. The A canonical form represents a rotation of axes. The B canonical form represents both a rotation and a translation to a new design center. If we are far from the design center, the A canonical form will be more useful. If we are close to the design center, we shall prefer the B canonical form. 4.3.4.1 Derivation of A Canonical Form We may rewrite Equation 4.38 in matrix form as y = a0 + xTa + xTAx where xT = (x1 x2 … xf), aT = (a1 a2 … af), and
© 2006 by Taylor & Francis Group, LLC
(4.41)
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319
⎛ 2 a11 ⎜ 1 A= ⎜ 2⎜ ⎜ ⎜⎝ sym
a12
$
2 a22
$
a1 f ⎞ ⎟ a2 f ⎟ % ⎟ ⎟ 2 a ff ⎠⎟
'
a matrix containing all the terms that are second-order overall. That is,
(
y = a0 + x1 x2
⎛ a12 a ⎞ $ 1n ⎟ ⎛ ⎞ ⎜ a11 ⎛ a1 ⎞ x1 2 2 ⎜ ⎟⎜ ⎟ ⎜a ⎟ a x 2 ⎜ a22 $ 2 n ⎟ ⎜ 2 ⎟ $ xn ⎜ ⎟ + x1 x2 $ x f ⎜ ⎟ 2 ⎜ %⎟ ⎜ %⎟ ⎜ ⎟ ⎜ ⎟ ' % ⎟ ⎜⎝ x ⎟⎠ ⎜ ⎝ af ⎠ f ⎜⎝ sym a ff ⎟⎠
(
)
)
Letting uT = xTK and θ = KTa
(4.42a,b)
we can recast Equation 4.41 as y = a0 + (xTK)(KTa) + (xTK)(KTAK)(KTx) where K is the eigenvector matrix of A, KTx = u, KTa = θ, and Λ = KTAK. y = a0 + u Tθ + u T Λu
A Canonical Form
(4.39b)
or
(
y = a0 + u1 u2
⎛ λ 11 ⎞ ⎛ u1 ⎞ ⎛ θ1 ⎞ ⎜ ⎟⎜u ⎟ ⎜θ ⎟ λ 22 ⎟ ⎜ 2⎟ $ u f ⎜ 2 ⎟ + u1 u2 $ u f ⎜ ⎜ ⎟⎜ % ⎟ ⎜ %⎟ ' ⎜ ⎟⎜ ⎟ ⎜ ⎟ λ ff ⎠ ⎝ u f ⎠ ⎝ θn ⎠ ⎝ sym
)
(
)
Now all second-order coefficients vanish except the pure quadratics. 4.3.4.2 Derivation of B Canonical Form We may also write Equation 4.41 as y = xTBx
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(4.43)
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where xT = (1 x1 x2 … xf) and ⎛ 2 a0 a1 a2 ⎜ 2 a11 a12 ⎜ 1⎜ B= 2 a22 2⎜ ⎜ sym ⎜ ⎝
af ⎞ ⎟ $ a1 f ⎟ $ a2 f ⎟ ⎟ ⎟ ' ⎟ 2 a ff ⎠ $
Then, letting uT = xTK, where K is the eigenvector matrix of B, and Λ = KTAK, we have y = a0 + uTΛu
B canonical form
(4.40b)
That is,
(
y = 1 u1 u2
⎛ 1⎞ ⎛ λ 11 ⎞⎜ ⎟ ⎜ ⎟ ⎜ u1 ⎟ λ 22 ⎟ ⎜ u2 ⎟ $ uf ⎜ ⎜ ⎟⎜ ⎟ ' ⎜ ⎟ % λ ff ⎠ ⎜⎜ ⎟⎟ ⎝ sym ⎝ uf ⎠
)
Equation 4.40b corresponds to a rotation of axes and a translation to a new coordinate center. By setting first derivatives to zero, we see that Λ represents a set of coordinates in u1·u2···un space that correspond to the extremum (maximum, minimum, or saddle point, collectively referred to as the stationary point). If the stationary point is close to our design center, we will prefer the A canonical form. If not, we shall prefer the B canonical form. If we can assign a clear meaning to the linear combinations of the factors derived from the canonical analysis, we may well prefer to keep our regression in the canonical space rather than the original space. 4.3.4.3 Canonical Form and Function Shape One may use these canonical forms to simplify second-order equations of the type given in Equation 4.41 by either rotating (A form) or rotating and translating (B form) the axes. By examining the second-order coefficients of either form, one may determine what kind of surface one is dealing with. If all λkk are positive or negative, then one is dealing with a minimum or maximum surface, respectively. If they are of differing signs, then the surface has a min-max or saddle shape. If some of the λkk are close to zero and the associated θk are positive, then the surface is a rising ridge. If the associated θk are near zero, then the surface is a stationary ridge. Figure 4.4 shows some of these forms for f = 2.
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321
Stationary Ridge λj < 0, λk = 0
Rising Ridge λj < 0, θk > 0
Maximum λj < 0, λk < 0
Saddle λj < 0, λk > 0
First-order + Interaction θj · θk ≠ 0; λj, λk = 0
Tilted Plane θj, θk ≠ 0; λj, λk = 0
FIGURE 4.4 Various response surfaces. Values of λ and θ allow the investigator to quickly determine the shape of the response surface in any number of dimensions. When all λs are of the same sign, then the response surface is a maximum or minimum; when the signs differ, the response surface is a saddle (min-max or col) shape. When λ is close to zero, then the design is first order in that response factor.
Example 4.4
Canonical Forms
Problem statement: Consider the second-order equation y = 1.029 + 0.326 x1 – 0.085 x2 + 0.282 x12 – 0.031 x1x2 – 0.127 x22 (a) Explicitly declare the equation according to the forms of Equation 4.39a. Use an eigenvector analysis to reduce it to the A canonical form and give the explicit equations. What can you tell about the surfaces by inspection of the coefficients? (b) Repeat the analysis using the B canonical form beginning with Equation 4.40a. Solution: a) From the problem statement we find
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322
Modeling of Combustion Systems: A Practical Approach ⎛ a0 ⎞ ⎛ 1.029 ⎞ ⎜ a ⎟ ⎜ 0.326 ⎟ ⎜ 1⎟ ⎜ ⎟ ⎜ a2 ⎟ ⎜ −0.085 ⎟ ⎜ ⎟ =⎜ ⎟ ⎜ a11 ⎟ ⎜ 0.282 ⎟ ⎜ a12 ⎟ ⎜ −0.031 ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ a22 ⎠ ⎝ −0.127 ⎠ According to Equation 4.39a, we have
⎛a ⎞ x2 ⎜ 1 ⎟ + x1 ⎝a ⎠
(
)
y = a0 + x1
2
(
⎛ ⎜ a11 x2 ⎜ ⎜ a12 ⎜⎝ 2
)
a12 ⎞ 2 ⎟ ⎛ x1 ⎞ ⎟⎜ ⎟ ⎝x ⎠ a22 ⎟⎟ 2 ⎠
or, explicitly
(
y = 1.029 + x1
⎛ 0.326⎞ x2 ⎜ + x1 ⎝ −0.085⎟⎠
)
(
⎛ 0.282 x2 ⎜ ⎝ −0.016
)
−0.016 ⎞ ⎛ x1 ⎞ −0.127 ⎟⎠ ⎜⎝ x2 ⎟⎠
Making use of y = a0 + (xTK)(KTa) + (xTK)(KTAK)(KTx), or equivalently, y = a0 + uTθ + uTΛu, we find K = eigenvectors (A). This turns out to be ⎛ 0.999 K=⎜ ⎝ −0.038
0.038⎞ 0.999⎟⎠
Therefore, ⎛ 0.999 x1 − 0.038 x2 ⎞ ⎛ u1 ⎞ ⎛ 0.329⎞ , u = KTx = ⎜ = ⎜ ⎟ , θ = K Ta = ⎜ ⎟ ⎝ −0.073⎟⎠ ⎝ 0.038 x1 + 0.999 x2 ⎠ ⎝ u2 ⎠ and ⎛ 0.282 Λ = K TAK = ⎜ ⎝
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⎞ −0.128⎟⎠
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323
So, the A canonical form becomes ⎛ 0.329⎞ u2 ⎜ + u1 ⎝ −0.073⎟⎠
(
)
y = 1.029 + u1
(
⎛ 0.282 u2 ⎜ ⎝
)
⎞ ⎛ u1 ⎞ −0.128⎟⎠ ⎜⎝ u2 ⎟⎠
The reader may verify that this equation gives exactly the same values as our starting equation. The u vector represents a rotation of axes that zeroes out the non-diagonal elements for Λ. By examining the coefficients of Λ, we see that they are of opposite sign, indicating a saddle shape oriented along the axes of u1 and u2. The saddle is steeper in the u1 direction as indicated by a coefficient about double that of u2. b) Likewise, from Equation 4.40a, we may write our equation as
(
y= 1
⎛ ⎜ a0 ⎜ a x2 ⎜ 1 ⎜ 2 ⎜ ⎜ a2 ⎜⎝ 2
)
x1
a1 2 a11 a12 2
a2 ⎞ 2 ⎟⎛ 1⎞ ⎟ a12 ⎟ ⎜ ⎟ x1 2 ⎟ ⎜⎜ ⎟⎟ ⎟ ⎝ x2 ⎠ a22 ⎟⎟ ⎠
or, explicitly,
(
y= 1
x1
⎛ 1.029 x2 ⎜⎜ 0.163 ⎜⎝ −0.043
)
0.163 0.282 −0.016
−0.043 ⎞ ⎛ 1 ⎞ −0.016 ⎟⎟ ⎜⎜ x1 ⎟⎟ −0.127 ⎟⎠ ⎜⎝ x ⎟⎠ 2
The eigenvectors and eigenvalues for this system are ⎛ −0.205 K = ⎜ 0.979 ⎜ ⎜⎝ −0.018
0.978 0.205 −0.038
⎛ 0.248 0.033⎞ T ⎟ 0.025 , Λ = K BK = ⎜⎜ ⎟ ⎜⎝ 0.999⎟⎠
1.605
Therefore, ⎛ −0.205 x1 + 0.979 x2 − 0.018 x3 ⎞ ⎛ u1 ⎞ u = K x = ⎜ 0.978 x1 + 0.205 x2 − 0.038 x3 ⎟ = ⎜ u2 ⎟ ⎟ ⎜ ⎟ ⎜ ⎜⎝ 0.033 x + 0.025 x + 0.999 x ⎟⎠ ⎜⎝ u ⎟⎠ 3 1 2 3 T
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⎞ ⎟ ⎟ −0.129⎟⎠
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(
y = a0 + u1
4.4
u2
⎛ 0.248 u3 ⎜⎜ ⎜⎝
)
1.605
⎞ ⎛ u1 ⎞ ⎟ ⎜u ⎟ ⎟ ⎜ 2⎟ −0.129⎟⎠ ⎜⎝ u3 ⎟⎠
Regression Statistics and Data Integrity
We may derive other statistics related to the ANOVA. Some of the more important ones are the coefficient of determination (r2), the adjusted coefficient of determination (rA2), the prediction sum of squares (PRESS) statistic, and a coefficient we shall call the coefficient of prediction (rP2). We can also define the variance inflation factor (VIF) and leverage statistics. We shall speak to each in turn.
4.4.1
The Coefficient of Determination, r2
Chapter 3 (Equation 3.43) defined an important ratio — the best known (and most misused) statistic for goodness of fit: the coefficient of determination, r2: r2 =
SSM SSR = 1− SST SST
The statistic r2 gives the fraction of the total variance accounted for by the model. An r2 of 0.9 (or 90%) means that the model accounts for 90% of the total variation of the data. For the purposes of combustion modeling, we desire r2 > 0.8, according to the following scale: • • • • •
0.9 < r2 < 1.0 — very strong correlation 0.8 < r2 < 0.9 — strong correlation 0.7 < r2 < 0.8 — good correlation 0.6 < r2 < 0.7 — fair correlation 0.5 < r2 < 0.6 — weak correlation
If r2 = 1, then the model accounts for all of the variation and fits the data perfectly. A related measure is r, the coefficient of correlation, r = r 2 . Since 0 < r2 < 1, then r ≥ r 2.
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Analysis of Nonideal Data 4.4.2
325
Overfit
Now r or r2 will always increase as the number of adjustable parameters in the model increases. Continuing to add adjustable model parameters eventually results in a condition known as overfit. Overfit is the unjustified addition of model parameters resulting in the fitting of random error to false factor effects. This is a statistical no-no, because random effects cannot be related to nonrandom variables. An example will help make this clear.
TABLE 4.2 Random Data x
y
1 2 3 4 5
0.60 0.93 0.68 0.07 0.94
Suppose we have the hypothetical data given by Table 4.2. Most spreadsheets and calculators have random number (uniform distribution) generators. The Excel command RAND( )simulates a uniform random distribution between 0 and 1. Therefore, the underlying model to these data is y = 0.5 + e . Suppose we fit the following models to the first four data points: Model 0: y = a0 Model 1: y = a0 + a1x Model 2: y = a0 + a1x + a2 x 2 Model 3: y = a0 + a1x + a2 x 2 + a3 x 3 All of these models except model 0 are nonsense. Notwithstanding, here are the least squares solutions and the associated r2 values: Model 0: y = 0.5699 ; r2 = 0.0000 Model 1: y = −0.1827 + 1.0266 x ; r2 = 0.4237 Model 2: y = −0.1572 + 1.0011x − 0.2368 x 2 ; r2 = 0.9931 Model 3: y = −0.5666 + 1.6522 x − 0.5292 x 2 + 0.0390 x 3; r2 = 1.0000 Figure 4.5 graphs the results.
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1.0
0.6 0.4 0.2
Random Value
0.8
yˆ = a0
Model 0
Model 1
ˆy = a0 + a1x
Sequence #
0.0 1
2
3
4
5
-0.2
yˆ = a0 + a1x + a2x2 + a3x3
-0.4 -0.6 -0.8 -1.0
Model 3
Random Data, curve fit Random Data, not curve fit Predicted from curve fit Deviation from actual
ˆy = a0 + a1x + a2x2
Model 2
-1.2 FIGURE 4.5 On overfitting data. The data were generated by the Excel function RAND( ), which gives a uniform distribution between 0 and 1. Therefore, the true model is y = 0.5 + e; i.e., model 0 is the only sensible model. Model 1 is a first-order fit to the first four data points (diamonds); likewise, model 2 is a second-order fit and model 3 a third-order fit. Data point 5 (square) is next in the random sequence. All models but model 0 are biased and give absurd results for point 5; they represent overfit — the ascription of random data to nonrandom effects. Despite this, all models have higher r2 than model 0. Therefore, r2 is not a useful statistic for revealing overfit. In this case, the model with the lowest r2 is actually the best.
If we were to judge based on r2 only, we would prefer model 3. Clearly, we have gotten too carried away and fit random behavior as if it were determined by the factor x. Our job, using least squares, is to find the underlying model. Here the underlying model is yˆ = 0.5 . But by continuing in a least squares frenzy, we have gone far beyond finding the true model and have force-fit random behavior as a function of x. Random behavior is not a function of any known factor (else it would not be random). Now the expected value of the fifth observation is 0.5. Model 0 comes the closest to predicting the true value; model 0:0.57. The rest of the models are way off — model 1: 0.11, model 2: –1.07, and model 3: –0.66. With each adjustable parameter to the model the r2 has gone up, but the predictive power is lower than for model 0. 4.4.3 Parsing Data into Model and Validation Sets If we fit random data to a factor, we overfit the model and generate nonsense. This is a great danger in judging models with only r2. One way to catch this behavior (presuming we have enough data points) is to do the following:
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1. Parse the data set into two sets, a model set (comprising about two thirds of the points) and a validation set (comprising one third of the points). 2. Randomly assign the points to the data sets. 3. Fit coefficients to the model set. 4. Gauge r2 using the validation set. Because a different set validates the data, the lower limit for r2 is no longer zero and the difference between r2 coefficients for the whole and parsed data sets is an indicator for overfit. We also hope for stable model coefficients. However, other data maladies (e.g., influential and errant response values) can cause large differences in r2. In other words, this is a severe test for regressed data. Nonetheless, if the data pass this test, one can be reasonably sure that the regression is trustworthy. For illustration purposes, and as we have already begun with an 80/20 split of the data, let us extend this analysis for all five possible model/ validation sets. We shall use four points in the model set and one in the validation set. Table 4.3 shows the results; the statistic rk2 is the r2 statistic based on the kth point for validation. TABLE 4.3 r2 for All Possible 80/20 Validations r2
0
r12 r22 r32 r42 r52
0.992 0.677 0.996 –4.467 0.644
Model Order 1 2 0.916 0.625 0.996 –5.737 –0.754
–4.177 0.343 0.858 –5.894 –9.309
3 –4.126 0.599 0.858 –0.711 –5.545
The largest entries are bolded. From examining the table, one sees that in four out of five instances, no model is superior to model 0. Therefore, the majority report from this analysis is that model 0 is the best model. As well, we could have parsed the data into a 60/40 model/validation split. This would have generated 10 rjk2 statistics: r122, r132, r142, r152, r232, r242, r252, r342, r352, and r452 and resulted in the same conclusion. (In general, there are 2n possible r2 statistics for all possible parsings into two groups. This means that a data set comprising only 20 values could prepare and analyze over a million such statistics. Obviously, this is would be overkill.)
4.4.4
The Adjusted Coefficient of Determination, rA2
The foregoing discussion prods us to search for better goodness-of-fit statistics. Practitioners have developed several statistics to gauge more fairly the
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trade-off between increased fit and reduced degrees of freedom and to alert the investigator to overfit. One such statistic is the adjusted coefficient of determination, rA2: rA2 = 1 −
MSR SSR ⎛ p − 1 ⎞ = 1− MST SST ⎜⎝ n − p ⎟⎠
(4.44)
Thus, r2 makes use of the sum of squares while rA2 makes use of the mean squares. This compensates partially for reducing the degrees of freedom as we add more terms to the model. However, rA2 still overstates the case, though not as badly as r2.
4.4.5
The PRESS Statistic
A better statistic for gauging overfit is the PRESS statistic. When we overfit data, we increase goodness of fit by correlating random error. What we really have is noise, but we add a coefficient to the model and treat it as if it were information. In other words, overfit adds bias by equivocating noise with information. This bias may have great influence, especially at the outskirts of the model, or beyond. What we would really like to know is what the residual would be if we were to encounter points that were not in our original data set. We have already described one method for detecting this — splitting the data set into model and validation portions. Though a milder test, the PRESS statistic requires less effort and gives good results. Many statistical programs perform it. Effectively, we do the following: 1. Delete the first point from the model and calculate the variance between the deleted point and the model prediction. 2. Do this for all n points. 3. Cumulate the variance. We shall call this the sum of squares residual, predicted, or SSRp, but the common term in the statistical literature is the PRESS statistic. PRESS is a composite acronym and stands for prediction sum of squares. n
SSR p =
∑ ( y − yˆ * ) k
k
2
(4.45)
k =1
In Equation 4.45, yˆ k * is the predicted value for yk but regressed from the data with the kth response deleted. Likewise, we may define the sum of squares model, predicted, SSMp as SSMp = SST – SSRp
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(4.46)
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For a well-behaved model, we would expect that SSRp ≈ SSR and SSMp ≈ SSM. Calculating n – 1 regressions does not appear to be a less tedious procedure than parsing the data set. However, we can make use of an identity to calculate the PRESS statistic in a single regression run using the hat matrix.
4.4.6
The Hat Matrix
Recall from Chapter 1 that Equation 1.79 defines the hat matrix: H = X(XTX)–1XT The matrix has some remarkable properties: • It is square symmetric (and singular, except for saturated designs where it becomes an n-row identity matrix). • It has as many rows and columns as there are y values in the data set — that is, n rows and n columns. • It codes for yˆ (y hat) in terms of linear combinations of y (Equation 1.80): yˆ = Hy • It is idempotent. H2 = HH = H
(4.47)
• Since it is symmetrical, H = HTH = HHT = H2 = Hn. This last point is so because if H2 = H, then we may also write H3 = H(H2) = HH = H. • The diagonal elements of H are always between 0 and 1: 0 ≤ hk , k ≤ 1
(4.48)
where hk,k are the diagonal elements of H and k indexes them. • The sum of the diagonal elements of H is equal to the number of parameters, p, in the model: n
∑h
k ,k
=p
k =1
• The elements in any row or column sum to unity.
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(4.49)
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n
n
∑h = ∑h j,k
k ,j
k =1
=1
(4.50)
k =1
One may use the diagonal hat matrix elements to transform the normal residual to the deleted residual used in Equation 4.45: yk − yˆ k * =
yk − yˆ k 1 − hk , k
(4.51)
This last property is the one that allows for direct calculation of SSRp in a single regression, thus avoiding n regressions: ⎛ yk − yˆ k ⎞ ⎜ 1− h ⎟ ⎝ k ,k ⎠ k =1 n
SSR p =
∑
2
(4.52)
We can use the PRESS statistic to derive a modified coefficient of determination that we shall call the coefficient of determination, predicted.
4.4.7
The Coefficient of Determination, Predicted, rp2
We may use SSRp to estimate a goodness of fit for the predicted values. We define the coefficient of determination, predicted (rp2) as rp2 = 1 −
SSR p SSM p = SST SST
(4.53)
When the data are not badly overfit, deleted observations will have little effect on SSRp and rp2 will be close to r2. Of necessity, rp2 ≤ r 2 ; however (and in contrast to r2), rp2 will not necessarily have a lower bound of zero because deleted points have inflated SSR to SSRp. The more strongly the data are overfit, the lower rp2. For example, let us compare r2, rA2, and rp2 for the hypothetical data of Table 4.2. This is best done with dedicated statistical software. Table 4.4 shows them. We notice several things: • First, we note that r2 continually increases as we add more parameters to the model. This will always be the case for any data set. This is why we cannot look at r2 alone to decide on an appropriate model. • The cubic model (model 3) is particularly good at capturing the variation with an r2 of 92.5%. However, the PRESS statistic alerts us to a problem.
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TABLE 4.4 PRESS and Other Statistics for the Data of Table 3.17 r2 r2 rA2 rp2 SSEp SSE
0 0.000 0.000 –0.562 0.785 0.502
1
Model Order 2
0.006 –0.325 –2.004 1.508 0.499
0.081 –0.840 –14.047 7.556 0.462
3 0.925 0.701 –10.252 5.650 0.037
• Comparing PRESS (SSEp) to SSE shows that in all cases, the PRESS statistic exceeds the SSE. This is typical. What is more important is that the magnitude of this difference is quite large in some cases and increases with model order. This should alert us to a potential problem regarding the predictive ability of our model. • Examining rA2 shows that models 1 and 2 are not preferred over model 0. However, rA2 for model 3 is quite respectable at 70.1%. It is less than r2 of 92.5%, but not sufficiently. Were we to choose a model based on rA2, it would be model 3. • In contrast, the rp2 statistic clearly shows the worsening condition with increasing model order. Were we to gauge the predictive ability of the model using this statistic, we would conclude that no model is better than model 0. Now, suppose the data were not random and that model 3 was actually a valid model. Our statistics would be no different, rp2 would still be very negative, and this would lead us to question the validity of model 3. But such skepticism would be appropriate. If we were actually trying to fit a model with four parameters (model 3), it would be best to have more than five data points. Five data points leave only one degree of freedom to assess random error. The rp2 statistic would prod us to gather more data and let us know that deletion of points makes a big change in the model form. As such, we could say that rp2 is a measure of robustness of fit. Although no single statistic is a panacea, by looking at a variety of statistics we get a good idea of a model’s fitness for purpose. So then, let us continue to build our arsenal of appropriate statistics to help us understand how well behaved our model and data are.
4.4.8
Extrapolation
As we have shown, empirical models are very poor extrapolators but excellent interpolators. Semiempirical models fare a bit better in this regard; they at least have a model form based on the physical system of study. However,
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even these have empirical parameters. In all cases, we would prefer to avoid extrapolation, or at least know when it is occurring. Extrapolation beyond individual factor ranges is easy to detect. For example, if –1 < x < 1, then x = 2 is clearly an extrapolation; if the new data point lies outside any of the individual factor boundaries, we can be sure that it is an extrapolation no matter what the shape of the data cloud in factor space. Now refer to Table 4.5. TABLE 4.5 Is This Extrapolation? x1
x2
1.00 0.50 0.50 –0.25 –0.25 –1.00
1.00 0.10 –1.00 0.00 –0.50 –1.00
Is the data point below an extrapolation or interpolation regarding the above data set? –0.70
0.80
Before reading on, hazard a guess. Does the last data point comprise an extrapolation compared to the previous ones? The data point is clearly within the range of each individual factor. However, Figure 4.6 shows that the data point actually lies outside the joint region defined by both factors considered simultaneously. Hidden extrapolation refers to data that are outside the joint region of factors but not outside the range of any individual factor. For two factors, one may easily detect hidden extrapolation by plotting data as we have done in Figure 4.6. For three or more factors, one may plot every possible two-factor projection. However, even this may not detect all possible extrapolations. Ideally, we prefer a statistic that is easy to calculate, generates a single score, and reliably flags all extrapolation, hidden or otherwise. A variant of the hat matrix will do the trick. It turns out that H defines an ellipsoid or hyperellipsoid in p-dimensional factor space that bounds the cloud of data. If our new data point lies outside this ellipsoid, then we have an extrapolation. The diagonal elements of H measure the distance between the kth value of X and the mean value of all the X values. We can think of them as the distances in p-factor space from the point to the center of the data cloud. We know from Equation 4.49 that the diagonal elements of H sum to p, so the mean value of the diagonal elements of H must be p/n. Thus, if we have a new value and want to test for it being an outlier, it seems sensible to calculate xT(XTX)–1x and compare it to p/n. If xT(XTX)–1x ≈ p/n, then the point represents an interpolation — it
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2
1
X2 0
–1
–2 –2
–1
0 X1
1
2
FIGURE 4.6 Hidden extrapolation. The solid circle represents an extrapolated value. Although it lies within the range of both x1 and x2 for the original data set, when one views the joint range as in the above plot, the point is clearly seen as outlying. This kind of extrapolation is even harder to detect as the number of factors increases.
is within the data cloud. However, if xT(XTX)–1x >> p/n, then we have an outlier. We shall consider a point an outlying point if x T (X TX)−1 x >
2p n
(4.54)
Therefore, Equation 4.54 shall be our test for outliers. We shall call this value the leverage value. The name comes from the influence that the particular point exerts on the residual error based on its position in factor space. In the hat matrix, the higher the leverage (diagonal element), the smaller the residual variance. We can also recast the leverage value as a statistic nominally between 0 and 1 for inlying data by multiplying Equation 4.54 by n/2p. This gives a kind of normalized distance: ⎛ n⎞ dL2 = ⎡⎣ x T (X TX)−1 x ⎤⎦ ⎜ ⎟ ⎝ 2p ⎠
(4.55)
When 0 ≤ dL2 ≤ 1 we shall consider the point an inlier, and an outlier otherwise. If we wish to compress the whole universe of potential values between 0 and 1, we may also define a coefficient of outlying, 0 ≤ rO2 ≤ 1:
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rO2 =
⎡ x T (X TX)−1 x ⎤ dL2 ⎣ ⎦ = 1 + dL2 ⎡ T T −1 ⎤ 2 p ⎣ x (X X) x ⎦ + n
(4.56)
If rO2 > 0.5, then the point is outlying. In the same fashion, we may also identify a coefficient of inlying having antithetical behavior: 2p 1 n r = 1− r = = 1 + dL2 ⎡ T T −1 ⎤ 2 p ⎣ x (X X) x ⎦ + n 2 I
2 O
(4.57)
If rI2 > 0.5, then the point is inlying, and outlying otherwise. This may be more sensible in that low values of rI2 are bad (less inlying) and high values of rI2 are good, as is the case for r2, rA2, and rp2. These last two equations — rO2 and rI2 — are not standard statistics; they are peculiar to this text, but engineers often find it helpful to use measures between 0 and 1 to quantify behavior, and these are offered in that vein.
Example 4.5
Testing for Outliers
Problem statement: Apply Equation 4.54 to the data of Table 4.5 and test if the final value is an outlying one. Perform the tests on the following models: linear, linear with an interaction term, and full quadratic. Explain any differences. Solution: For two factors, x1 and x2, we have the following models, corresponding x vectors, and hat matrices. • Linear model:
y = a0 + a1x1 + a2 x2
x Tk = 1
⎛ 0.7031 ⎜ ⎜
