International Journal of Numerical Methods for Heat & Fluid Flow (Vol. 17, nº 3, 2007) Selected papers from the International Conference on Computational Heat and Mass Transfer, Paris, May 2005

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ISSN 0961-5539

Volume 17 Number 3 2007

International Journal of

Numerical Methods for Heat & Fluid Flow Selected papers from the International Conference on Computational Heat and Mass Transfer, Paris, May 2005 Guest Editor: Professor Rachid Bennacer

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International Journal of

ISSN 0961-5539

Numerical Methods for Heat & Fluid Flow

Volume 17 Number 3 2007

Selected papers from the International Conference on Computational Heat and Mass Transfer, Paris, May 2005 Guest Editor Professor Rachid Bennacer

CONTENTS

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243

Editorial advisory board __________________________

244

Guest editorial ___________________________________

245

A numerical model for the thermocapillary flow and heat transfer in a thin liquid film on a microstructured wall A. Alexeev, T. Gambaryan-Roisman and P. Stephan __________________

247

An evaluation of synthetic jets for heat transfer enhancement in air cooled micro-channels Victoria Timchenko, John Reizes and Eddie Leonardi _________________

263

Computational investigation of turbulent jet impinging onto rotating disk A.C. Benim, K. Ozkan, M. Cagan and D. Gunes _____________________

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284

CONTENTS continued

A computational fluid dynamics analysis of a PEM fuel cell system for power generation Elena Carcadea, H. Ene, D.B. Ingham, R. Lazar, L. Ma, M. Pourkashanian and I. Stefanescu _______________________________

302

Modelling of the thermosolutal convection and macrosegregation in the solidification of an Fe-C binary alloy Z.Q. Han, R.W. Lewis and B.C. Liu ________________________________

313

Modeling natural convection with the work of pressure-forces: a thermodynamic necessity M. Pons and P. Le Que´re´ ________________________________________

322

Buoyancy effects on upward and downward laminar mixed convection heat and mass transfer in a vertical channel Youssef Azizi, Brahim Benhamou, Nicolas Galanis and Mohammed El-Ganaoui _________________________________________

333

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EDITORIAL ADVISORY BOARD M. Bellet CEMEF, Ecole Nationale Supe´rieure des Mines de Paris, Sophia Antipolis, Valbonne 06560, France

M. Napolitano Istituto di Macchine ed Energetica, Politecnico di Bari, Via Re David 200, 1-70125 Bari, Italy P. Nithiarasu School of Engineering, University of Wales Swansea, Singleton Park, Swansea SA2 8PP

G.F. Carey College of Engineering, University of Texas at Austin, Austin, Texas 78712-1085, USA

C. Nonino Dipartimento di Energetica e Macchine, Universita` degli Studi di Udine, Via delle Scienze 208, 33100 Udine, Italy

R. Codina Resistencia de los Materiales y Estructuras en Ingenierı´a, Universitat Politecnica de Catalunya, Jordi Girona 1-3, Edifici C1, 08034 Barcelona, Spain

A.J. Nowak Institute of Thermal Technology, Silesian University of Technology, Konarskiego 22, Gliwice 44-101, Poland

Gianni Comini Dipt di Energetica e Macchine, Universita` degle Studi di Udine, Via delle Scienze 208, 33100 Udine, Italy R.M. Cotta Department of Mechanical Engineering, EE/COPPE/UFRJ, CX Postal 68503, Cicade Universitaria, Rio de Janeiro, RJ, Brazil Marcela Cruchaga Departamento de Ingeneria Mecanica, Universidad de Santiago de Chile, Santiago de Chile G. De Vahl Davis University of New South Wales, Sydney, NSW, Australia 2052 E. Dick Department of Machinery, State University of Ghent, Sint Pietersnieuwstraat 41, B-9000 Ghent, Belgium Amir Faghri Mechanical Engineering Department, University of Connecticut, 191 Auditorium Road, U-139, Storrs, Connecticut 06269-3139, USA

J. Peiro Dept of Aeronautics, Imperial College of Science & Tech, Prince Consort Road, London SW7 2BY D. Peric Civil and Computational Engineering Research Centre, School of Engineering, University of Wales Swansea, Swansea I. Pop Department of Mathematics, University of Cluj, R-3400 Cluj, Romania Dr D.A.S. Rees Department of Mechanical Engineering, University of Bath, Claverton Down, Bath BA2 7AY, UK B. Sarler Faculty of Mechanical Engineering, University of Ljubljana, Askerceva 6, 1000 Ljubljana, Slovenia K.N. Seetharamu MSR School of Advanced Studies, Bangalore, India Wei Shyy Department of Aerospace Engineering, University of Michigan, 1320 Beal Avenue, Ann Arbor, MI 48109, USA

D. Gethin Department of Mechanical Engineering, University of Wales Swansea, Singleton Park, Swansea SA2 8PP, UK

D.B. Spalding CHAM, Bakery House, 40 High Street, Wimbledon Village, London SW19 5AU, UK

Dan Givoli Faculty of Aerospace Engineering, Technion – Israel Institute of Technology, 32000, Haifa, Israel

B. Sunden Lund Institute of Technology, Heat Transfer Division, Box 118, S-221 00 Lund, Sweden

D.B. Ingham Department of Applied Mathematical Studies, University of Leeds, Leeds LS2 9JT, UK

K.K. Tamma Department of Mechanical Engineering, 125 Mech. Engng, University of Minnesota, 111 Church Street SE, Minneapolis, Minnesota 55455, USA

Mike Keavey Faculty of Computing, Engineering and Mathematical Sciences, University of the West of England, Frenchay Campus, Coldharbour Lane, Bristol 8S16 1QY, UK T.G. Keith Jr Department of Mechnical Engineering, The University of Toledo, Toledo, Ohio 43606, USA R.E. Khayat Dept of Mechanical & Materials Engineering, University of Western Ontario, London, Ontario, Canada N6A 5B9 R. Lohner GMU/CSI, MS 5C3 Dept of Civil Engineering, George International Journal of Numerical Mason University, Fairfax, VA 22030-4444, USA Methods for Heat & Fluid Flow Vol. 17 No. 3, 2007 p. 244 q Emerald Group Publishing Limited 0961-5539

Professor K. Vafai Department of Mechanical Engineering, University of California Riverside, A363 Bourne Hall, Riverside, CA 92521 0425, USA J.A. Visser Department of Mechanical Engineering, University of Pretoria, Pretoria 0002, South Africa V.R. Voller Civil Engineering, University of Minnesota, 500 Pillsbury Drive, Minneapolis, Minnesota 55455-0220, USA L.C. Wrobel Department of Mechanical Engineering, Brunel University, Uxbridge 4BS 3PH, UK

Guest editorial The special issue of the International Journal of Numerical Methods for Heat & Fluid Flow (IJNMHFF) brings together selected papers from the International Conference on Computational Heat and Mass Transfer held in Paris 17-20 May 2005 (ICCHMT 05). The conference covers a relative broad spectrum of topics, ranging from applied mathematics and computational analysis to various applications and design optimization. It is a rapidly evolving research field, as well as an irreplaceable design tool in modern engineering practice. We would like to express our gratitude to the IJNMHFF Editor, Professor Roland W. Lewis, for dedicating a special issue of this journal to high quality papers presented in the ICCHMT 05 conference. The contributions were selected from the 300 presented papers. The objective of the meeting was to address the state-of-the-art on the CFD-CHMT interaction in several domains involving developers of model, tools, and to focus on some specific applications. Much work is devoted to complex systems and coupled heat and mass transfer, the latter also including air quality and pollutant transport. We note also a new point of interest centred on microfluidic and on molecular and lattice Boltzmann methods with applications related to various scales. The computational fluid dynamic and the computational heat transfer dealing with the transport phenomena appeared to be in exponential growth. The complex local treatment is supported by huge computer power ability. In the near future, the continuous modelling from local to global size will be possible by using heterogeneous and hybrid approach (coupling: finite volumes, finite elements, boundary elements, spectral, Lattice-Boltzmann, molecular dynamic). Such processes involving fluid flow, heat and mass transport are encountered in very diverse fields, such as chemistry, biomedical, biochemistry, engineering, botany, geology, and medicine. Specific practical examples of engineering interest abound, including metallurgical process, powder and drug delivery systems, chemical reactors, filters, leaf transpiration and cropping, nuclear waste disposal, underground pollutant transport, oil/gas recovery, building insulation, alveolar respiration and capillary circulation among many others. The several process are involved as the major phenomena by those studying transport phenomena in materials, the vastness of the applicative field is also a weakness for hindering the development of uniform community, theories and standards. The state-of-the-art, trends and perspectives in the few fields are overviewed by the selected papers, presented by internationally recognized research groups. The present issue is dedicated to complex coupled no-linear problem dealing with the microscales (papers by Alexeev et al. and Timchenko et al.) where the natural convection, and the mixed convection and the surface tension effects control and impose the resulting heat and mass transfer and the flow structure. The dynamic behaviour remains one of the challenges in the flow stability treatment and moving interface (as presented in the papers by Benim et al., Carcadea et al. and Han et al.). It is followed by particular studies dealing with the humidity or compressibility

Guest editorial

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International Journal of Numerical Methods for Heat & Fluid Flow Vol. 17 No. 3, 2007 pp. 245-246 q Emerald Group Publishing Limited 0961-5539

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(papers by Pons and Le Que´re´ and by Azizi et al.) which are generally considered as minor effects and neglected in heat and/or mass transfer. It is pointed out that under these specific conditions, such effects modify the obtained results. We hope that the volume provides an overview of the state-of-the-art of chosen topics in heat and mass transfer and will thus serve to many readers from academic and industrial communities as an up-to-date source of information, as well as a useful resource for further research. Rachid Bennacer Guest Editor

The current issue and full text archive of this journal is available at www.emeraldinsight.com/0961-5539.htm

A numerical model for the thermocapillary flow and heat transfer in a thin liquid film on a microstructured wall A. Alexeev Department of Chemical Engineering, University of Pittsburgh, Pittsburgh, Pennsylvania, USA, and

Thermocapillary flow and heat transfer 247 Received 1 January 2006 Accepted 9 July 2006

T. Gambaryan-Roisman and P. Stephan Darmstadt University of Technology, Darmstadt, Germany Abstract Purpose – This paper aims to study thermocapillarity-induced flow of thin liquid films covering heated horizontal walls with 2D topography. Design/methodology/approach – A numerical model based on the 2D solution of heat and fluid flow within the liquid film, the gas above the film and the structured wall is developed. The full Navier-Stokes equations are solved and coupled with the energy equation by a finite difference algorithm. The movable gas-liquid interface is tracked by means of the volume-of-fluid method. The model is validated by comparison with theoretical and experimental data showing a good agreement. Findings – It is demonstrated that convective motion within a film on a structured wall exists at any nonzero Marangoni number. The motion is caused by surface tension gradients induced by temperature differences at the gas-liquid interface due to the spatial structure of the heated wall. These simulations predict that the maximal flow velocity is practically independent from the film thickness, and increases with increasing temperature difference between the wall and the surrounding gas. It is found that an abrupt change in wall temperature causes rupture of the liquid film. The thermocapillary convection notably enhances heat transfer in liquid films on heated structured walls. Research limitations/implications – Our solutions are restricted to the case of periodic wall structure, and the flow is enforced to be periodic with a period equal to that of the wall. Practical implications – The reported results are useful for design of the heat transfer equipment. Originality/value – New effects in thermocapillary convection are presented and studied using a developed numerical model. Keywords Volume measurement, Fluid dynamics, Numerical control, Convection, Heat transfer Paper type Research paper

Nomenclature A Bi C C~

¼ wall structure amplitude ¼ Biot number ¼ color function ¼ averaged color function

cp d F Fs

¼ specific heat capacity ¼ wall structure period ¼ body forces ¼ body force due to surface tension

The authors would like to acknowledge the generous support of the German Science Foundation, DFG, through the Emmy Noether Program. A. Alexeev is grateful to Virgil Stoica for his assistance in the conduction of the experiments.

International Journal of Numerical Methods for Heat & Fluid Flow Vol. 17 No. 3, 2007 pp. 247-262 q Emerald Group Publishing Limited 0961-5539 DOI 10.1108/09615530710730139

HFF 17,3

248

hg hw Dh * Dh k M M cr n Nu p S t T DT u umax

¼ gas layer thickness ¼ wall structure height ¼ average thickness of liquid film ¼ minimal thickness of liquid film ¼ interface curvature ¼ Marangoni number ¼ critical Marangoni number ¼ unit normal to interface ¼ Nusselt number ¼ pressure ¼ interface “height” function ¼ time ¼ temperature ¼ temperature drop ¼ velocity ¼ maximal liquid velocity at interface

Greek symbols a ¼ thermal diffusivity

ds l m r s sT t w

¼ Dirac distribution function at interface ¼ thermal conductivity ¼ dynamic viscosity ¼ density ¼ surface tension ¼ temperature coefficient of surface tension ¼ shear stress tensor ¼ wall structure angle

Superscripts n ¼ iteration level Subscripts 0, g ¼ gas 1 ¼ liquid i, j ¼ computational cell index w ¼ wall

Introduction Surface tension is crucial in the dynamics of thin liquid films on substrates of different topography, which are frequently encountered in many engineering applications, including the thermal management of electronic devices, food processing, chemical engineering, MEMS. Normally, the surface tension of a liquid is a decreasing function of temperature. If the temperature varies at the gas-liquid interface, surface tension gradients cause thermocapillary (Marangoni) flow (Colinet et al., 2001). If a thin liquid film is heated on a planar substrate of a uniform temperature, a conducting solution exists, which implies that the film is motionless and the free surface of the liquid is isothermal. This solution is stable for sufficiently small temperature gradients across the liquid layer. If the temperature gradient exceeds a critical value, the conducting solution loses stability and convective patterns are developed (Pearson, 1958). If the substrate has a structure on its surface, the convection prevails for any temperature difference (Alexeev et al., 2005). It occurs due to the temperature inhomogeneity, which is imposed on the interface by the spatial structure of the substrate. We develop a numerical model to describe the motion of a thin liquid film on a heated structured wall. We deploy a finite difference algorithm to integrate the Navier-Stokes and energy equations. To cope with the movable gas-liquid interface, we apply the volume-of-fluid (VOF) method (Scardovelli and Zaleski, 1999). The calculations are performed simultaneously through the whole computational domain containing the gas and liquid regions, while the surface force at the interface is included into the momentum balance equations via a volumetric force (Kothe and Mjolsness, 1992). The energy equation is solved within the solid wall as well. There are only few previous studies where the Marangoni flows in a cavity are considered in the framework of the VOF. Sasmal and Hochstein (1994) calculated the Marangoni convection induced by a temperature difference between the sidewalls of a rectangular cavity. They studied heat transfer within the cavity and the effect of the contact angle on the flow patterns. More recently, Wang (2002) applied a VOF model to

investigate the Marangoni convection in trapezoidal cavities. In these works, the temperature gradient was caused by a temperature difference between the sidewalls. Thus, the heat flux was mostly directed along the gas-liquid interface, in that way it was justified to neglect its component normal to the interface by considering the adiabatic condition at the free surface. In contrast, in a flow on a heated structured wall, the heat flux is practically perpendicular to the gas-liquid interface resulting in a strong temperature gradient in that direction. In this case, the thermocapillary force is induced by a relatively small variation of the liquid temperature along the interface. Hence, a very accurate calculation of the temperature is required to avoid unphysical flow currents due to inaccuracy in the temperature gradient evaluation. In present work, we study thermocapillary motion within a thin film of a low volatility liquid on a heated highly thermal conductive wall with 2D microscale topography. We consider a situation where the liquid layer covers a horizontal wall, and its thickness is comparable with the amplitude of the wall microstructure. We neglect the effect of gravity. Our solutions are restricted to the case of periodic wall structure, and the flow is enforced to be periodic with a period equal to that of the wall. Numerical model Governing equations The incompressible flow is governed by the continuity equation: 7 · u ¼ 0; and the Navier-Stokes equations:   ›u þ u · 7u ¼ 27p þ 7 · t þ F; r ›t

ð1Þ

ð2Þ

where u is the velocity, r the density, p the pressure, F the body forces, and t time. Moreover, t is the shear stress tensor given by:   m ›uj ›ui t ij ¼ þ ; ð3Þ 2 ›xi ›xj where m is the dynamic viscosity. The equations are coupled with the energy equation given by:   ›T þ u · 7T ¼ 7 · l7T; r cp ›t

ð4Þ

where T is the temperature, cp the specific heat capacity, and l the thermal conductivity. To track the moving gas-liquid interface, we utilize the VOF technique (Scardovelli and Zaleski, 1999). A color function C is introduced, which equals to 1 within the liquid and to 0 within the gas. The color function is governed by a transport equation:

›C þ u · 7C ¼ 0: ›t

ð5Þ

We are looking for the solutions, which are characterized by a period equal to that of the wall structure, d (Figure 1). Thus, we impose a symmetry boundary condition at

Thermocapillary flow and heat transfer 249

HFF 17,3

y hg

Gas

Liquid

250

ϕ Wall

Tw

hw

∆h*

Tw−∆T

x

Figure 1. Outline of the structured wall from the experiments

d

x ¼ 0 and x ¼ d=2. We also impose T ¼ T w at y ¼ 0 and T ¼ T w 2 DT at y ¼ hw þ Dh * þ hg . Moreover, a free flow condition for the velocity ð›u=›y ¼ 0Þ is applied at y ¼ hw þ Dh * þ hg . Numerical method The hydrodynamic equations (1)-(3) are solved with a finite difference algorithm on a rectangular staggered grid using the projection method (Ferziger and Peric, 2002). The projection method consists of three steps. First, the prediction velocity field u* due to the advective and diffusive terms in equation (2) is to calculate semi-implicitly: Du 1 1 1 þ u n · 7Du 2 n 7 · Dt ¼ 2u n · 7u n þ n 7 · t n þ n F n ; Dt r r r

ð6Þ

where Du ¼ u* 2 u n and Dt ¼ t* 2 t n . Then, the Poisson equitation, which is obtained using equation (1), is solved to calculate the pressure field:  7·

1 7p nþ1 rn

 ¼

7 · u* : Dt

ð7Þ

Finally, the velocity field is corrected to the time level n þ 1: u nþ1 ¼ u* þ

Dt 7 · u* : rn

ð8Þ

Using the velocity field u nþ1 , the color function is advected by solving: DC þ u nþ1 · 7DC ¼ 2u nþ1 · 7C n ; Dt

ð9Þ

where DC ¼ C nþ1 2 C n . The final stage of the numerical solution involves calculation of the temperature field:

Thermocapillary flow and heat transfer

DT 1 þ u nþ1 · 7DT 2 7 · l nþ1 7DT Dt ðcp rÞnþ1 ¼ 2u nþ1 · 7T n þ

1 7 · l nþ1 7T n : ðcp rÞnþ1

ð10Þ

Here, DT ¼ T nþ1 2 T n . The advection terms in the r.h.s. of equations (6), (9) and (10) are solved with the third order essentially non-oscillatory (ENO) scheme (Shu and Osher, 1988), while the terms in the advection terms in the l.h.s. of these equations as well as the viscous and conductivity terms are approximated with the second order finite differences. The ENO scheme is used since it provides good tracking of the discontinuity-like interfaces. To treat implicit parts of equations (6), (9) and (10), we utilize the approximate factorization approach and solve the equations separately along the x- and y-directions. The overall accuracy of our method is of the second order in space. The multigrid technique (Wesseling, 1991) is applied to solve the Poisson equitation for pressure (equation (7)). We use v-cycle and the number of multigrid levels K is given by 3 £ 2K ¼ minðN x ; N y Þ, where N x and N y are the grid size in the x- and y-directions, respectively. The use of the multigrid technique reduces the overall computational time by an order of magnitude as compared to the standard iterative methods. To impose the non-slip velocity condition at the liquid-solid boundary, we utilize the immersed boundary approach. We set the x and y velocity components within the solid domain at the nodes right next to the liquid-solid interface such that the linearly interpolated velocity at the interface equals to zero. We also set at these nodes a zero gradient for pressure, while solving equation (7). Following the VOF approach, we calculate the values of density and viscosity used in equations (6)-(10) as: ~ r0 þ C~ r1 ; r ¼ ð1 2 CÞ

~ m0 þ C~ m1 m ¼ ð1 2 CÞ

ð11Þ

Hereafter, the index 1 stands for the liquid, while the index 0 denotes the gas properties. Moreover, C~ is the averaged color function (Alexeev et al., 2005). Accurate calculation of the temperature distribution along the gas-liquid interface is critical for a correct modeling of thermocapillary driven flows. When the heat flux is directed across the interface, the difficulty arises due to the discontinuity of properties of the fluids across the interface. In this case, the cell average values cannot provide a satisfactory description for the fluid properties at the interface. In particular, our simulations show that the use of an averaging, either algebraic or geometric, for the calculation of l causes spurious currents in the fluids. Mehdi-Nejad et al. (2005) have recently developed an approach for a more accurate calculation of the convection terms in the energy equation. They successfully applied this approach to study heat transfer in molten tin drops during their fall. In the case of thermocapillary driven flows on heated walls, however, the heat flux across the interface is typically dominated by the diffusive rather than convective terms. We, therefore, propose a simple approach to calculate the temperature flux across the interface. Consider an interface that is located at the cell ði; jÞ such that the upper and bottom boundaries of the cell are along the interface (Figure 2). To resolve the diffusive

251

HFF 17,3

y Ci,j+1=0 Ti,j+1

Dy

Ti,j+1 i,j+1

252

T 0i,j

Dy0

T ii,j Ti,j

Dy1

Figure 2. Schematic diagram of computational cells near the interface and an approximation of the temperature distribution across the interface

T 1i,j

i,j Ci,j-1=1 Dy

interface Ti,j-1

Ti,j-1

T

i,j−1

terms in equation (10), we approximate the heat fluxes across the bottom and upper boundaries of ði; j Þ as: f2 y ¼ 2l1

T 1i;j 2 T i;j21 T i;jþ1 2 T 0i;j þ and f ¼ 2 l ; 0 y Dy þ Dy 1 Dy þ Dy 0

ð12Þ

respectively. Here, Dy 1 ¼ C i; j Dy, Dy 0 ¼ ð1 2 C i; j ÞDy and Dy is the computational grid step. The temperatures T 0i; j and T 1i; j are calculated using two conditions: continuity of the temperature at the interface:   Dy þ 2Dy 0 T ii;j ¼ T i;jþ1 þ T 0i;j 2 T i;jþ1 ; Dy þ Dy 0

ð13aÞ

  Dy þ 2Dy 1 : T ii;j ¼ T i;j21 þ T 1i;j 2 T i;j21 Dy þ Dy 1

ð13bÞ

and energy conservation within the cell ði; jÞ:

rcp T i;j ¼ ðrcp Þ0 T 0i;j ð1 2 C i;j Þ þ ðrcp Þ1 T 1i;j C i;j ;

ð14Þ

rcp ¼ ð1 2 C i;j Þðrcp Þ0 þ C i;j ðrcp Þ1 :

ð15Þ

where:

T 0i;j

T 1i;j

and can be readily calculated. These Combining equations (13)-(15), temperatures are also used to evaluate the temperature gradients within the liquid near the gas-liquid interface needed to calculate the thermocapillary force acting at the interface. To estimate the heat flux component, which is directed along the interface at the cell ði; jÞ, an algebraic averaging for l is used (equation (10)). At the liquid-solid interface,

where we do not need an accurate value of the temperature gradients, the geometric averaging is used to assess the vertical heat flux, while the algebraic averaging is applied in the horizontal direction. To illustrate the method for the temperature calculation, we solve equation (10) for a test problem in which the interface between two motionless fluids ðu ¼ 0Þ is slightly inclined and the heat flux is due to a temperature difference between the upper and lower walls (Figure 3(a)). In Figure 3(b) and (c), we present the x and y components of the temperature gradient along the interface, respectively. We are interested in the temperature gradient since the thermocapillary force is directly proportional to its magnitude. To compare with our approach, we also include the results for the temperature gradients calculated with the algebraic and geometric averaging of the thermal conductivity within the numerical cells at the interface. One can expect for the considered problem (Figure 3(a)) that the interface temperature changes monotonically, meaning that the magnitude of the temperature gradient may not oscillate along the interface. Nevertheless, both the algebraic and geometric averaging results in strong oscillations of the temperature gradient with a period which is correlated with the numerical grid spacing. These oscillations eventually cause spurious currents along the interface induced by the unphysical variations in the thermocapillary force. In contrast, our approach gives a much better approximation of the gradients along the interface. Although there is still some noise in ›T=›x due to the discretization, it can be reduced by applying an appropriate smoothing. Our simulations, however, show that this noise practically does not affect the results. To include the effect of surface tension into the momentum equations, we adopt the continuum surface force approach (Kothe and Mjolsness, 1992). The body force due to surface tension is given by:

Fs ¼

2rds ½skn þ ð1 2 n^nÞ7s
; ð r0 þ r1 Þ

ð16Þ

~ is the Dirac distribution function at the where s is the surface tension, ds ¼ j7Cj interface, k is the curvature of the interface, and n is the unit normal to the interface. Moreover, 7s ¼ sT 7T, where sT is the temperature coefficient of surface tension. To obtain 7T at the gas-liquid interface, we calculate the temperature gradients at the cells near the interface, which are filled with the liquid, and then extrapolate the gradients to the interface. To assess the unit normal n and the curvature of the interface k, we utilize a reconstruction algorithm (Sussman, 2003), which is based on reconstructing the “height” function S directly from the color function C. Although the ENO scheme, which is used to calculate C provides good tracking of the interface, it causes some numerical “foam” around the interface, which can be accumulated during long time calculations. We, therefore, restore C near the interface in such a way that the “height” functions S and the normal n remain unchanged, while the “foam” is eliminated. In fact, this procedure breaks the global mass conservation. Our simulations show, however, that the change of the mass is rather small and usually does not exceed 102 3 of its initial value even for long calculations.

Thermocapillary flow and heat transfer 253

dT/dx=0

Gas

254

Tw−∆T hmax

Interface hmin

dT/dx=0

HFF 17,3

Liquid d Tw (a) 0.001

dT/dx d/∆T

0.0005 0 −0.0005 −0.001

0

0.2

0.4

0.6

0.8

1

x/d (b) 0.06

dT/dy d/∆T

0.05 0.04 0.03 0.02 0.01 0

0.2

0.4

0.6

0.8

1

x/d (c)

Figure 3. Test problem for the solution of equation (4) for an inclined interface between two domains of an equal average thickness

Notes: The domains have thermal properties of a gas (air) and a liquid (water). Grid size is 96 × 96; domain size is h = 0.5 (hmin + hmax) =1 mm, ∆h=hmax – hmin = 0.1 mm (~ 5∆y), d = 5mm; ∆T = –10K: (a) schematic of the test problem; (b) and (c), respectively, represent the horizontal and vertical components of the temperature gradient along the interface calculated for different approximations for l . The solid lines show the approximation of equations (12)-(15), the dotted and dashed lines are for the algebraic and geometric averaging, respectively

Computational parameters We carry out the simulations for two liquids, which are water and silicon oil, while the gas is air. Their properties are chosen at Tw ¼ 238C. The calculations are performed for two types of wall structures. The first wall (Figure 1) corresponds to the experiments reported in Alexeev et al. (2005) with d ¼ 1 mm, hw ¼ 0.5 mm and f ¼ 308. The second wall is given by:   2px yw ¼ A 1 2 cos ; d

255



ð17Þ

where A ¼ hw =2 is the wall structure amplitude. Properties of the wall material are those of copper. We also set in all our calculations hg ¼ 0.4 mm. We perform the calculations for a half of the groove. The computational domain is 0 # x # d=2 and 0 # y # ðhw þ Dh * þ hg Þ (Figure 1). Our rectangular computational grid usually consists of 96 £ 96 cells. To test the grid quality its density was increased, indicating that an increase in grid density practically does not affect the solution. The calculations are started with zero initial velocities and continued up to the moment when a steady state solution is obtained. Results and discussion Model validation We first consider thermocapillary convection in a rectangular cavity due to a temperature difference between the sidewalls. In the limit of thin film within a wide cavity, this problem can be solved analytically (Levich, 1962) and, therefore, can serve as a test case for our numerical model. Figure 4(a) and (b) shows the velocity and temperature distributions within the cavity for silicon oil and water, respectively. As expected, the thermocapillary force induces vortexes within the fluids in which the flow near the interface is directed toward the wall having lower temperature. Note that the isotherms within the silicon oil (Figure 4(a)) are practically vertical that corresponds to the conducting solution, while in Figure 4(b) for water, they are considerably distorted by the flow. This difference arises because water has a lower Prandtl number as compared to the silicon oil. Figure 5 shows the pressure distribution along the gas-liquid interface the solutions shown in Figure 4 as well as the theoretical prediction (Levich, 1962). As seen, there is good agreement between the numerical solutions and the theory. Some discrepancy near the sidewalls can be attributed to the fact that the free surfaces are deformed by the flow, while the theory neglects this effect. To verify our numerical model for the case when the thermocapillary convection is driven by a vertical temperature gradient, we first consider a rectangular cavity and estimate the critical Marangoni number M cr corresponding to the onset of thermocapillary convection (Colinet et al., 2001). The Marangoni number is given by: M¼

sT DT 1 Dh ; r1 n1 a 1

Thermocapillary flow and heat transfer

ð18Þ

0.8

0.2

0.4

0.6

256

where DT 1 ¼ DTBið1 þ Bi Þ21 is the temperature drop over the liquid, a the thermal diffusivity, Dh the average thickness of the liquid film and Bi ¼ l0 Dh=l1 hg the Biot number. In our simulations, we found good agreement with the linear theory (Colinet et al., 2001). Namely, for M less than the theoretical value of M cr (Colinet et al., 2001), an initial disturbance introduced into the conducting temperature distribution decays and the fluid flow stops after a transient, while for M . M cr , steady vortexes develop. For more thorough model validation in the case of heated grooved walls, we conducted experiments with a thin film of silicon oil (5cSt) on a wall with a structure shown in Figure 1. In the experiments, we measured temperature of the film as well as

0.4

HFF 17,3

Gas

0.8

0.6

0.4

0.2

0.2

y/d

0.3

Liquid 0.1 0.8

0.6

0.8

0.8

0.6

0.8

0.6

0.8

0.4

0.6

0.2

0.6

0.4

0

0.2

0

1

0.2

0.4

0.4

x/d (a)

Gas

0.2

0.4

0.2

y/d

0.3

Liquid

0

0

0.2

0.4

0.2

Figure 4. Velocity and temperature distributions within a liquid film in a rectangular cavity due to a thermocapillary flow induced by a temperature difference between the sidewalls

0.1

0.4

0.6 0.8 1 x/d (b) Notes: ∆T = 1K, h = 0.1 mm, d = 0.5mm. The liquids are: (a) silicon oil; and (b) water. The arrows show the velocity field. The thin lines represent isotherms. The temperature is normalized as (T – Tw) /∆T

Thermocapillary flow and heat transfer

5 CFD Silicon oil ∆pd/(∆TσΤ)

2.5

CFD Water Theory

0

257 −2.5 −5

0

0.2

0.4

0.6

0.8

1

x/d Notes: ∆T = 1K, h = 0.1 mm, d = 0.5 mm

Figure 5. Pressure distribution along the gas-liquid interface in a rectangular cavity due to a temperature difference between the sidewalls

its maximal velocity. A detailed description of the experimental setup and the procedure can be found elsewhere (Alexeev et al., 2005). Figure 6 shows a numerically calculated flow pattern for M < 1 ! M cr . The simulation predicts the formation of a vortex, in which the liquid near the free surface moves toward the groove trough at x ¼ 0. This convection is induced by the thermocapillary force due to a temperature gradient along the gas-liquid interface, which is originated from the topography of the heated wall. Moreover, in agreement with the experiments (Alexeev et al., 2005), our simulations predict that the convection in films on structured walls arises for any temperature difference across the film. In Figure 7(a), we compare the experimentally measured maximal values of the liquid velocity at the gas-liquid interface, umax , with the predictions of the numerical model. 1 0.2

0.2

Gas

0.8

0.4 0.5 0.6

0.6

0.7 0.8

0.7 0.8 .9 0

y/d

0.6

0.4 0.5

0.4 0.9

Liquid

0.2

Wall 0 0

0.2

0.4 x/d

Notes: ∆h* = 0.1mm, ∆T = 1.5 K, M ≈ 1. The arrows show the velocity field. The thin lines represent isotherms. The temperature is normalized as (T – Tw + ∆T) /∆T

Figure 6. Velocity and temperature distributions in a silicon oil film on a structured wall (Figure 1)

HFF 17,3

2

umax, mm/s

258

M=Mcr

Experiment Numerical results

1.5 1 0.5 0 0

10

20 Tw–T∝, K (a)

30

40

2 Experiment Numerical results: with convection conduction only

Tw–Tint, K

1.5

Figure 7. (a) Maximal velocity of the liquid as a function of wall temperature; (b) temperature drop over the liquid layer as a function of wall temperature

M=Mcr

1 0.5 0 0

10

20

30 Tw–T∝, K

40

50

60

(b) Notes: Silicon oil, ∆ h* = 0.5 mm

One can see that there is reasonable agreement between the model and the experiments, although almost everywhere the model prediction is slightly below the experimental data. When the temperature exceeds the value corresponding to M cr , the velocity increases much faster than that for the lower temperatures. It is an effect of the convective motion within the liquid layer on heat transfer. Indeed, the circulations in the layer bring hotter liquid from the hot wall to the interface, thus increasing the temperature gradient and the thermocapillary force that in turn increases the velocity. Figure 7(b) shows the measured and calculated temperature drop across the liquid film as a function of the wall temperature. To demonstrate the effect of the convection on heat transfer, the temperature drop is also calculated when the convective terms are omitted from equation (4). In this case, the temperature drop increases linearly with T w , while in the calculations with the convection terms and in the experiments, the temperature drop declines from the straight line for M . M cr . This result suggests that there is an increase in heat transfer due to the Marangoni convection within the liquid. We conclude that the overall agreement between our calculations and the theory and experiments is rather convincing, and our numerical model may be applied to study thermocapillary flows within thin films on structured walls. In what follows, we present some numerical solutions, which are characteristic of such flows.

Flow velocities Figure 8 shows the dependence of umax on the Marangoni number. In Figure 8(a), the temperature drop is fixed at DT ¼ 10 K and M changes due to Dh * . In Figure 8(b), M increases due to the increasing temperature difference, while the other parameters are constant. As seen in Figure 8(a), all the data collapse into a single curve. For M . 100, umax , a1 Dh 21 M 1=2 . Taking into account that DT ¼ const, we obtain that M changes as Dh 2 , and, therefore, umax , const. It means that for larger M , umax practically does not depend on the thickness of the liquid layer. The temperature difference DT, however, does modify the velocity as shown in Figure 8(b). Note that umax grows exponentially with increasing M . It is interesting that for different Dh * , umax follows a common curve. Our calculations show that for Dh * ¼ const, u , DT 2=3 .

Thermocapillary flow and heat transfer 259

umax∆h / α1

100

10

A=0.25mm A=0.125mm A=0.25mm hw=0.5mm

1 10

100

1000

d=2mm d=2mm d=1mm d=1mm

10000

M (a)

umax∆h / α1

1000

100

∆h*/d=0.025

∆h*/d=0.1

∆h*/d=0.05 ∆h*/d=0.2 ∆h*/d=0.5

∆h*/d=0.2 ∆h*/d=0.4 ∆h*/d=0.6

10

1 10

100

1000 M (b)

10000

100000

Notes: The liquid is water. The empty markers denote calculations for a sinusoidal wall profile, while the full markers stand for calculations with a wall shown in Figure 1: (a) ∆T = 10K; (b) A = 0.25 mm, d = 2 mm

Figure 8. Maximal velocity of liquid vs the Marangoni number

HFF 17,3

260

Temperature transient The simulations show that when the wall temperature increases suddenly, the liquid film may be ruptured as shown in Figure 9. In this simulation, uniform initial temperature equal to T w 2 DT is imposed. At t ¼ 0, T w is applied at y ¼ 0. The rupture is caused by a large temperature gradient along the free surface, which is induced when the thermal boundary layer initially formed along the liquid-wall interface reaches the gas-liquid interface near the groove crest. We note that for the parameters in Figure 9, a steady state solution may be obtained either by a steady increase in the wall temperature or when the temperature change takes place at gas above the film. In the latter case, a thermal boundary layer is formed first at the gas-liquid interface. The temperature propagates perpendicular to the free surface, preventing the appearance of strong temperature gradients causing the rapture. Heat transfer Figure 10 shows the Nusselt number as a function of Dh * for water and silicon oil. In these calculations, we consider a wall having a structure shown in Figure 1, which is either heated or cooled. When DT ¼ 210 K, i.e. the wall is cooled, Nu exceeds unity by few percents for small Dh * only, while for larger Dh * , Nu is practically equal to unity. Thus, the effect of convection on heat transport from a cooled structured wall is relatively weak and prevails for small Dh * only. For a heated wall, however, our simulations predict that heat transfer can be significantly enhanced by the Marangoni convection within the film. In this case, Nu 0.5

Gas 0.4

0.1

0.1

y/d

0.2

Liquid

3 0.40. 6 . 6 . 0 0

0.2 0.1 0 Figure 9. Velocity and temperature distribution in a liquid film just before rupture due to an abrupt increase in wall temperature

Wall

0.1 00.02.4.3 0.6 0.8

0

0.1

0.4 0.6

0.1

0.3

0.2 0.3

0.2

0.3

0.4

0.5

x/d Notes: The temperature drop ∆T = 87K (M = 1,435) corresponds to the minimal temperature at which a water film with ∆h* = 0.1 mm is ruptured on a sinusoidal wall with A = 0.25 mm, d = 2 mm. The arrows show the velocity field.The thin lines represent isotherms. The temperature is normalized as (T – Tw + ∆T )/∆T

1.7

Thermocapillary flow and heat transfer

∆T= 10K ∆T= 10K

1.6

Nu

1.5 1.4

261

1.3 1.2 1.1 1 0

0.2

0.4

0.6

0.8

1

1.2

Film thickness, ∆h*/d Notes: The liquid is water. The empty markers denote calculations for a sinusoidal wall with A = 0.25 mm, d = 1 mm. The full markers stand for calculations with a wall shown in Figure 1

has maximum, which occurs for about the same optimal film thickness Dh*opt for both liquids. Thus, Dh*opt corresponds to a layer thickness, for which the effect of convection on heat transfer is most pronounced. Our simulations show that if Dh * . Dh*opt , a stagnant layer of liquid is formed under the vortexes attached to the gas-liquid interface. This stagnant layer reduces the convective heat transfer from the wall, resulting in a decrease in Nu. We also note in Figure 10 that the sinusoidal wall provides better convective heat transport compared to the experimental wall. Thus, an appropriate choice of the structure can enhance heat transfer in thin liquid films on structured walls. Summary We study thermocapillarity-induced flow of thin liquid films covering heated horizontal walls with 2D topography. To this end, we develop a numerical model based on the integration of the Navier-Stokes and energy equations by a finite difference algorithm. The mobile gas-liquid interface is tracked with the VOF technique. The numerical model is verified by comparison with a theory and experiments showing good agreement. We demonstrate that convective motion within a film on a structured wall exists at any nonzero Marangoni number. The motion is caused by surface tension gradients induced by temperature differences at the gas-liquid interface due to the spatial structure of the heated wall. Our simulations predict that the maximal flow velocity, which occurs at the gas-liquid interface, is practically independent from the thickness of the liquid layer, and increases according to a power-law with increasing DT. It is found that an abrupt change in wall temperature causes rupture of the liquid film near the structure crest. The rupture may occur at the same value of DT, for which a steady state solution exists and can be obtained either by a gradual increase in wall temperature or cooling gas above the liquid. We show that the thermocapillary convection notably enhances heat transfer in liquid films on heated structured walls. An optimal film thickness exists for which Nu attains the maximal value for a specific temperature drop.

Figure 10. Numerically calculated Nusselt number as a function of film thickness

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262

References Alexeev, A., Gambaryan-Roisman, T. and Stephan, P. (2005), “Marangoni convection and heat transfer in thin liquid films on heated walls with topography: experiments and numerical study”, Physics of Fluids, Vol. 17, pp. 062106-13. Colinet, P., Legros, J.C. and Velarde, M.G. (2001), Nonlinear Dynamics of Surface-Tension-Driven Instabilities, Wiley, New York, NY. Ferziger, J.H. and Peric, M. (2002), Computational Methods for Fluid Dynamics, Springer, New York, NY. Kothe, D.B. and Mjolsness, R.C. (1992), “RIPPLE: a new model for incompressible flows with free surfaces”, AIAA Journal, Vol. 11, pp. 2694-700. Levich, V.G. (1962), Physicochemical Hydrodynamics, Prentice-Hall Inc., Englewood Cliffs, NJ. Mehdi-Nejad, V., Mostaghimi, J. and Chandra, S. (2005), “Modeling interfacial heat transfer from single or multiple deforming droplets”, International Journal of Computational Fluid Dynamics, Vol. 19, pp. 105-13. Pearson, J.R.A. (1958), “On convection cells induced by surface tension”, Journal of Fluid Mechanics, Vol. 4, pp. 489-500. Sasmal, G.P. and Hochstein, J.I. (1994), “Marangoni convection with a curved and deforming free surface in a cavity”, ASME Journal of Fluids Engineering, Vol. 116, pp. 577-82. Scardovelli, R. and Zaleski, S. (1999), “Direct numerical simulation of free-surface and interfacial flow”, Annual Review of Fluid Mechanics, Vol. 31, pp. 567-603. Shu, C.-W. and Osher, S. (1988), “Efficient implementation of essentially non-oscillatory shock-capturing schemes”, Journal of Computational Physics, Vol. 77, pp. 439-71. Sussman, M. (2003), “A second order coupled level set and volume-of-fluid method for computing growth and collapse of vapor bubbles”, Journal of Computational Physics, Vol. 187, pp. 110-36. Wang, G. (2002), “Finite element simulations of free surface flows with surface tension in complex geometries”, ASME Journal of Fluids Engineering, Vol. 124, pp. 584-94. Wesseling, P. (1991), An Introduction to Multigrid Methods, Wiley, New York, NY. Corresponding author P. Stephan can be contacted at: [email protected]

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An evaluation of synthetic jets for heat transfer enhancement in air cooled micro-channels Victoria Timchenko, John Reizes and Eddie Leonardi School of Mechanical & Manufacturing Engineering, The University of NSW, Sydney, Australia

Synthetic jets for heat transfer enhancement 263 Received 1 January 2006 Accepted 9 July 2006

Abstract Purpose – The development of novel cooling techniques is needed in order to be able to substantially increase the performance of integrated electronic circuits whose operations are limited by the maximum allowable temperature. Air cooled micro-channels etched in the silicon substrate have the potential to remove heat directly from the chip. For reasonable pressure drops, the flow in micro-channels is inherently laminar, so that the heat transfer is not very large. A synthetic jet may be used to improve mixing, thereby considerably increasing heat transfer. This paper seeks to address this issue. Design/methodology/approach – CFD has been used to study the flow and thermal fields in forced convection in a two-dimensional micro-channel with an inbuilt synthetic jet actuator. The unsteady Navier-Stokes and energy equations are solved. The effects of variation of the frequency of the jet at a fixed pressure difference between the ends of the channel and with a fixed jet Reynolds number, have been studied with air as the working fluid. Although the velocities are very low, the compressibility of air has to be taken into account. Findings – The use of a synthetic jet appreciably increases the rate of heat transfer. However, in the frequency range studied, whilst there are significant changes in the details of the flow, due primarily to large phase changes with frequency, there is little effect of the frequency on the overall rate heat transfer. The rates of heat transfer are not sufficiently large for air to be a useful cooling medium for the anticipated very large heat transfer rates in future generations of microchips. Research limitations/implications – The study is limited to two-dimensional flows so that the effect of other walls is not considered. Practical implications – It does not seem likely that air flowing in channels etched in the substrate of integrated circuits can be successfully used to cool future, much more powerful microchips, despite a significant increase in the heat transfer caused by synthetic jet actuators. Originality/value – CFD is used to determine the thermal performance of air flowing in micro-channels with and without synthetic jet actuators as a means of cooling microchips. It has been demonstrated that synthetic jets significantly increase the rate of heat transfer in the micro-channel, but that changing the frequency with the same resulting jet Reynolds number does not have an effect on the overall rate of heat transfer. The significant effect of compressibility on the phase shifts and more importantly on the apparently anomalous heat transfer from the “cold” air to the “hot” wall is also demonstrated. Keywords Heat transfer, Fluid dynamics, Jets, Simulation Paper type Technical paper

The authors wish to thank the Australian Research Council for providing financial support for this work.

International Journal of Numerical Methods for Heat & Fluid Flow Vol. 17 No. 3, 2007 pp. 263-283 q Emerald Group Publishing Limited 0961-5539 DOI 10.1108/09615530710730148

HFF 17,3

264

Introduction Since, one of the performance limiting factors of microchips is the maximum allowable temperature, and since it appears that the presently available temperature control methods will not be adequate in future microchip technology, the development of novel cooling techniques is necessary in order to sustain the rate of development of integrated electronic circuits. Forced air convection in micro-channels etched in the silicon substrate has the potential to remove heat directly from the chip and has been used as one of the strategies for cooling integrated circuits (Tuckerman and Pease, 1981). Since, for reasonable pressure drops across the channel, the flow in micro-channels is inherently laminar and the heat transfer is not, in fact, very large. In addition, if the upper surfaces of the micro-channels are hot, the heat transfer on that surface may be further reduced by separations near the inlet. Some method of providing better mixing is therefore, necessary in order to increase the rate of heat transfer. It is proposed to use synthetic jets to “disrupt” the laminar flow in the channels thereby creating better mixing with the mean flow, whilst at the same time interfering with the stagnant zones. A synthetic jet (Glezer and Amitay, 2002) can be developed in the cooling fluid by a micro-pump actuator (Mallinson et al., 2003). This consists of an oscillating membrane in a cavity with a small orifice in the face opposite the diaphragm. When “steady conditions” on the average over a cycle have been reached, the actuator has a zero net mass flow through the orifice, but a non-zero net momentum transfer over an entire period of the diaphragm oscillation. Under appropriate operating conditions this actuator provides a fluctuating flow away from the orifice into the cooling channel, thereby creating a “quasi-turbulent” flow. As a result, it may be possible to significantly increase the rate of heat transfer when this jet impinges on a hot surface. This process is quite different to that which is obtained when the synthetic jet is discharged into a quiescent atmosphere (Timchenko et al., 2004). An experimental study of periodic disturbances on the penetration and mixing of synthetic jets in cross-flow was performed by Eroglu and Breidenthal (2001). They observed that periodic forcing of the jet stream affects mixing by creating vortex loops whose strength and spacing are determined by the frequency of the forcing jet and the jet-to-cross-flow velocity ratio. The main parameter characterizing the jet in cross-flow was found to be the jet-to-cross-flow momentum ratio. The effect of periodic disturbances on the coherent flow structures and heat transfer in the case of an impinging air jet was studied experimentally by Mladin and Zumbrunnen (2000). Their results suggest that the large coherent flow structures associated with a high amplitude, high frequency flow pulses increase the Nusselt number at the stagnation line by surface renewal effects. Similar effects had been previously demonstrated by Kataoka et al. (1987) for steady impinging flows. The most significant effect on the enhancement of heat transfer corresponded to high values of the product St AN in which St is the Strouhal number and AN is the ratio of the amplitude of the velocity pulsation to the time-averaged velocity at the jet nozzle exit. Their results for a free (non-impinging) jet agreed with those of Ho and Huang (1982) who had demonstrated that in the presence of an external periodic disturbance, large flow structures have their frequency fixed by the forcing disturbance. They also showed that the large structures, formed within the mixing layers at 1/f time intervals travel with a constant velocity. Since, the frequency of the large structures matched the forcing frequency, f, it follows that their wavelength and size decreased with forcing frequency.

All previous studies of periodic disturbances and their effect on heat transfer have been performed on macro-scale turbulent flows. In this work we intend to introduce the same sort of disturbances using synthetic jets, in laminar flows, in micro-sized devices. We study numerically the behaviour of a synthetic jet in cross-flow constrained in a micro-channel with a hot top wall and determine its effect on the distribution of heat flux. Computations of unsteady compressible laminar flow are performed for the two-dimensional channel including the fluid motion and temperature variations within the synthetic jet actuator cavity. The amplitude of membrane deflection was varied at different frequencies so as to maintain an approximately constant the average velocity at the exit the orifice. The effectiveness of the proposed technique for cooling microchips is evaluated by comparing the heat transfer rates when the synthetic jet is operating, with those obtained when the synthetic jet is not used and with each other at different frequencies.

Synthetic jets for heat transfer enhancement 265

Mathematical and numerical model A two-dimensional micro-channel 200 mm high and 4.2 mm long has been used. It is open at either end with the top surface hot and all the other walls adiabatic. Parts of the regions outside the channel have been included in the calculations as may be seen in Figure 1. For micro-sized devices the flow is dependent on the Knudsen number, Kn, which is the ratio of the mean free path of the gas molecules to a characteristic length, for example, Kn ¼ l/d in the orifice. Since, in this study flows with Kn , 0.01 have been modelled, the continuum approach using conventional conservation equations is valid (Karniadakis and Beskok, 2002). The transient compressible flows generated by the constant 250 Pa pressure difference between the ends of the channel and by the synthetic jet actuator have been simulated using a commercial package, CFX-5.7. The set of the unsteady conservation equations for laminar flow comprises the continuity equation:

›r ~ ¼ 0; þ 7ðrVÞ ›t

ð1Þ

›rV~ ~ ¼ 27p þ 7ðmð7V~ þ ð7VÞ ~ T ÞÞ þ rg~ þ 7ðrV~ VÞ ›t

ð2Þ

the Navier-Stokes equation:

and the energy equation:

opening

opening

L hot wall

opening

opening

inlet

W

D w

outlet

opening

adiabatic wall

H oscillating membrane

opening

Figure 1. Schematic diagram of channel and actuator

HFF 17,3

› rha ~ a Þ 2 7ðl7TÞ ¼ 0; þ 7ðrVh ›t

266

~ p, m, ha, T, l, and t denote the density, the velocity vector, the gauge in which, r, V, pressure, the dynamic viscosity, the specific enthalpy, the absolute temperature, the thermal conductivity and time, respectively. The compressibility of the air is taken into account by treating it as an ideal gas with:

r¼

ð p þ pref Þ ; RT

ð3Þ

ð4Þ

in which R is the specific gas constant and pref is the absolute ambient pressure. The displacement of the membrane Ym was assumed to be a parabola, which varies sinusoidally in time, viz.:
2 ! 2x Ym ¼ A 1 2 ð5Þ sinð2pftÞ W in which A is the centreline amplitude, f is the frequency of oscillation, respectively, and W is the width of the diaphragm (Figure 1). To simulate the displacement of the membrane an actual mesh deformation in the vicinity of the membrane has been applied accordingly to equation (5), so that computational domain has been reconstructed at each time step. A second order backward Euler differencing scheme was used for the transient term, whereas a central differencing scheme was employed for the advection terms in the Navier Stokes equation. At each time step of a cycle, equal to one hundredth of the period of a cycle, the internal iterations were continued until the mass and momentum residuals had been reduced to 102 6. Validation of mathematical and numerical model has been performed in our earlier paper (Timchenko et al., 2004) for the case of an incompressible synthetic jet in an axisymmetric configuration. Computed instantaneous stream-wise and transverse velocity profiles across the orifice exit were in very good agreement with numerical results (Mallinson et al., 2003) validated against experimental data. In general, when one refers to compressible flows it is understood that flows of Mach number greater than about 0.3 are to be discussed. However, in our case, even for the flow with Mach number much lower then 0.3 (Ma ¼ 4.5 £ 102 2), pressure and temperature differences lead to sufficiently large density changes which result in resonance effects thereby affecting the heat transfer results. It follows that an estimate of the natural frequency and the phase shifts to be expected is required, before evaluating the operation of synthetic jet actuators in micro channels in which air is used as the cooling fluid. Natural frequency and phase shift in the cavity and orifice of the actuator The determination of the natural frequency is derived from a linearised lumped parameter model of the cavity and the orifice. Typical flow within the cavity and the orifice obtained in this study at all frequencies are shown in Figure 2. It appears that a column of air stretching from the lower edge of the main channel to the diaphragm with a width approximately equal to the width of the orifice, moves as

an oscillating body. Suppose therefore, that the column of air marked as ABCDEFGH in Figure 3, is moved down by a small distance Y from its equilibrium position, the mass m moved by the process is: m ¼ rðwbÞðH þ hÞ;

ð6Þ

in which the geometric variables w, b, h and H are shown in Figures 1 and 3. It follows that the force, Fi, required to accelerate this mass is given by: F i ¼ rðwbÞðH þ hÞ

Synthetic jets for heat transfer enhancement

d2 Y : dt 2

267 ð7Þ

If the pressure outside and inside the cavity are initially the same, the restoring force is provided by the pressure difference, Dp, between the cavity and the outside air in the main channel. Let the instantaneous pressure in the cavity be pc and the pressure in the main channel at the orifice be p so that: ð8Þ

Dp ¼ pc 2 p:

If p is assumed to remain constant, the driving pressure is dependent only on pc which can be determined if state path law is known. Assume that the expansion and compression processes in the cavity are isentropic, the state path law may be written: pc V gol ¼ C;

ð9Þ

in which, Vol is the volume of the cavity excluding the oscillating column, g is the ratio of the specific heat capacities and C is a constant. It follows from equation (9) that:

Figure 2. Example of flow in the cavity and orifice passage at 10 kHz at the lowest position of the membrane

B

C

b A

D

h w

F

G

H E

H

Figure 3. Schematic diagram of air column in the cavity

HFF 17,3

dp c dV ol þg ¼ 0; pc V ol

ð10Þ

in which d indicates a small change. Now, since dV ol ¼ Ywb and V ol ¼ ðW 2 wÞHb, dpc can be evaluated from equation (10) as:

268 dpc ¼ 2

gpc w Y: ðW 2 wÞH

ð11Þ

The restoring force, Fr, is therefore given by: F r ¼ dpc wb ¼

gpc w 2 b Y: ðW 2 wÞH

ð12Þ

All that remains is to determine the damping force, Fd, which consists of forces required to overcome friction in the orifice conduit and to generate and maintain the fluid motion in the cavity. The lowest estimate for Fd would therefore be the force required to overcome friction in the orifice passage. Because of the complex flows which occur in the orifice, it is not easy to obtain an estimate of the instantaneous losses in the orifice, so that the minimum possible approximation of Fd would be the steady state frictional force at the same instantaneous velocity, viz.: Fd ¼

12mbh dY ; w dt

ð13Þ

in which dY/dt represents the average velocity through the orifice. The equation describing the motion of the air column can then be written as: Fi þ Fd þ Fr ¼ Fe

ð14Þ

in which Fe is the external exciting force, in this case the force generated by the diaphragm and which can be written as: F e ¼ Kð f Þbð2pf Þ2 sinð2pftÞ

ð15Þ

in which K( f ) is a constant related to the amplitude of the diaphragm motion in a complex way and is a function of the frequency f. The substitution of equations (7), (12), (13) and (15) in equation (14) for Fi, Fr, Fd and Fe, respectively, leads to:

rðwbÞðH þ hÞ

d2 Y 12mbh dY gpw 2 b þ Y ¼ Kð f Þbð2pf Þ2 sinð2pftÞ: þ dt 2 w dt ðW 2 wÞH

ð16Þ

Since, the rate of change of the flow rate in the main channel is proportional to the frequency (Eroglu and Breidenthal, 2001), as the frequency becomes larger the accelerations and hence the pressure gradients in the main channel need to increase in the same way. This means that the amplitude of the pressure variations in the main channel will rise with the frequency. It follows that whilst K( f ) can be assumed to be a constant at a particular frequency of the diaphragm, it will vary as that frequency is changed and that is the main reason for writing K( f ) as a function.

Equation (16) can be simplified to: d2 Y dY þ ð2pf n Þ2 Y ¼ K 1 ð2pf Þ2 sinð2pftÞ þ 2k dt 2 dt

ð17Þ

Synthetic jets for heat transfer enhancement

in which: ð6mhÞ ; þ hÞÞ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ðgpwÞ fn ¼ 2p ðrðW 2 wÞH ðH þ hÞÞ k¼

ðrw 2 ðH

ð18Þ

ð19Þ

is the natural frequency, which when employing the equation of state for the perfect gas becomes: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ðgRT c wÞ fn ¼ ð20Þ 2p ððW 2 wÞH ðH þ hÞÞ and: K1ð f Þ ¼

Kð f Þ ðrwðH þ hÞÞ

ð21Þ

The general solution to equation (21) (Kreyszig, 1972) is: Y ¼ e2kt ðAsinð2pftÞ þ Bcosð2pftÞÞ K 1 ð f Þð2pf Þ2 þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sinð2pft 2 bÞ; ðð2pf Þ2 2 ð2pf n Þ2 Þ2 þ 4k 2 ð2pf Þ2 in which A and B are constants and the phase angle b is given by:
 4pkf 21 b ¼ tan : ð2pf n Þ2 2 ð2pf Þ2

ð22Þ

ð23Þ

Once the initial transient has passed, equation (23) reduces to: K 1 ð f Þð2pf Þ2 Y ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sinð2pft 2 bÞ ðð2pf Þ2 2 ð2pf n Þ2 Þ2 þ 4k 2 ð2pf Þ2

ð24Þ

so that the analytically average velocity at the exit of the orifice, v
O can be obtained by differentiation, viz.: K 1 ð f Þð2pf Þ3 v
O ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cosð2pft 2 bÞ: ðð2pf Þ2 2 ð2pf n Þ2 Þ2 þ 4k 2 ð2pf Þ2

ð25Þ

Now the velocity of the diaphragm, vm, is obtained by differentiating equation (5) yielding:

269

HFF 17,3

270




2x vm ¼ A 1 2 W

2 ! ð2pf Þcosð2pftÞ:

ð26Þ

It may be seen by comparing equations (25) and (26) that b is the phase angle between the velocity of the diaphragm and the velocity through the orifice and that the orifice velocity lags the velocity at the diaphragm by an angle b. Thus, it is now possible to evaluate the natural frequency and phase angle variations and discuss numerical results. Results and discussion To study the abilities of the synthetic jet to disrupt laminar flows in micro channels, an orifice 50 mm wide and 100 mm long was placed 1.2 mm downstream from the inlet. The width of the diaphragm was 2 mm and the cavity depth was 400 mm. Since, the inflow temperature and velocity distributions affect the heat transfer in the channel, external domains 1 mm long and 1.5 mm high were included at either end of the channel (Figure 1). A 250 Pa pressure was imposed at the inlet of the left external domain with the temperature set to 208C. At the outlet, the right external domain, the pressure was programmed at zero, pref fixed at 100 kPa and the temperature set to 308C. The upper wall of the channel was isothermal at 508C. Grid points 50 £ 20 in the stream-wise and transverse directions, respectively, were used in the orifice of the synthetic jet generator, so mesh size was kept to 2.5 mm in the orifice. Outside the orifice the grid was gradually expanded with a maximum mesh size equal to 5 mm. The total number of mesh points was equal to 164,316. To describe the behaviour of pulsed jets in cross-flow, Eroglu and Breidenthal (2001) used a mean jet-to-cross-flow velocity ratio. Gordon and Soria (2001) characterized the jet in the cross-flow by the jet-to-cross-flow momentum ratio. Following Smith and Glezer (1998) we define the computed average velocity, V
j through the orifice only over the discharge period of the cycle, that is: Z t0 þt=2 Z w=2 2 dt V j dx ð27Þ V
j ¼ tw t 0 2w=2 in which, t0 is the time of beginning of discharge from the orifice, t ¼ 1=f , is the period of diaphragm oscillation and Vj is the velocity at a point in the exit of orifice passage. Thus, the jet Reynolds number, Rej, and the jet-to-cross-flow momentum ratio Cj, can be defined as Rej ¼ V
j w=n and: Z w=2 Z t0 þt=2 rV 2j ðx; tÞdx dt 2w=2 t C j ¼ Z D Z0 t ð28Þ 2 rU ch ð y; tÞdy dt 0

0

In this work the jet-to-cross-flow momentum ratio, Cj < 8 has been chosen as preliminary tests with the synthetic jet operating at Cj , 8 indicated that there was little effect on the heat transfer rate. Indeed, under these conditions the synthetic jet hardly reached the hot wall at all. As a consequence, an average jet velocity approximately 42 ms2 1 was employed.

Steady flow results The effectiveness of the proposed cooling strategy was evaluated by comparing the heat transfer effectiveness when the synthetic jet was operating with that obtained with the jet switched off. Temperature and vorticity contours with the jet not active may be seen in Figure 4. The imposed pressures yielded an average velocity in the channel of 7.75 ms2 1, that is a Reynolds number defined as Rech ¼ U
ch D=n was equal to 103. The horizontal contours of vorticity indicate a laminar, parabolic flow with no mixing. This is confirmed by the velocity distributions shown in Figure 5(a). The flow enters main channel with almost top hat distribution which gradually develops to a fully developed laminar flow. The lack of mixing is also illustrated in the steady flow temperature distribution shown in Figure 5(b). The temperature in the lower half of the channel has increased by less then 15 K whereas the potential temperature rise is 30 K which points to the fact that it should be possible to significantly increase them average heat transfer flux of 7.3 kWm2 2 from the hot upper wall. In order to evaluate the frequencies to be used in determining the effectiveness of introducing the synthetic jet to enhance the heat transfer rate, the natural frequency needs to be determined.

Synthetic jets for heat transfer enhancement 271

(a)

Figure 4. (a) Temperature, 208C # T # 508C; (b) vorticity, maximum positive 105 s 2 1, minimum negative 2 105 s 2 1. Steady flow

(b)

(b) 50

(a)

45 Temperature,°C

U velocity, m/s

10

inlet outlet 5

40 35 30

inlet outlet

25 20

0

0

0.05

0.1 Y, mm

0.15

0.2

15

0

0.05

0.1 Y, mm

0.15

Figure 5. (a) Horizontal velocity and (b) temperature profiles at the inlet and outlet of 0.2 channel. Steady flow

HFF 17,3

272

Natural frequency and phase angle calculations With T c < 300 K and geometric parameters given above fn was calculated from equation (19) as 19.88 kHz. Given the accuracy of the estimate, it may be assumed that fn ¼ 20 kHz with velocity at the orifice (equation (25)) lagging of velocity of the diaphragm (equation 25) by 908. This is exactly the situation shown in Figure 6(a), in which at a diaphragm frequency of 20 kHz, the numerical results for the vertical velocity at the centre point of the exit of the orifice, lag the diaphragm velocity at the centre point by 908. In Figure 6 time is normalised by the period of oscillation t. Thus, the above simple theory is surprisingly accurate at predicting the natural frequency. Since, the true transients have to be calculated that is small time steps had to be used, the run times were large, in the order of weeks, despite the significant computer resources deployed for the calculations. It follows that only two other frequencies were used, namely 10 and 30 kHz, one above and one below the natural frequency. Because of the phase shifts, valid comparisons between heat transfer distributions at various frequencies can only be performed if they are at the same point in cycle of the air motion at the orifice and not at the same point in the cycle as indicated by the motion of the diaphragm. It follows that it was necessary to determine the phase shift at each of the frequencies. Note, that amplitudes of oscillation were adjusted to keep the average jet velocity approximately the same and equal to 42 ms2 1. The damping factor k did not have to be determined to evaluate the phase angle at a frequency of 20 kHz, because the denominator in equation (23) becomes zero at resonance, however in order to estimate b at 10 and 30 kHz, k needs to be evaluated. From equation (18), k ¼ 7.34 £ 103 s2 1. When this value is substituted into equation (18), b < 4:58 results at a frequency of 10 kHz, whereas b < 378 was obtained from the numerical experiments as may be seen in Figure 6(b). This difference was expected because of the underestimate of k discussed above in the development of the theory. Now, a more appropriate estimate of the value of k may be obtained by using the actual value of b at 10 kHz, namelyb < 378 which leads to a value of k ¼ 7.1 £ 104 s2 1, nearly 10 times the value obtained from equation (18). The fluid motion in the cavity is generated and sustained by the movement the column of air (Figure 2). This process extracts energy from the air column. Further, the very complex flow in the orifice passage (Figure 2) leads to very significant “losses” not taken into account in equation (18). It is therefore not surprising that the actual value of k is very much larger than the initial estimate based on steady flow pressure losses. When the larger value of k is used in equation (23) to estimate the phase angle at 30 kHz, b < 1268 results. This value is remarkably close to b < 1248, the angle obtained in the CFD calculations as may be seen in Figure 6(c). Now that the phase changes are known, it is possible to discuss the details of the flow and the thermal fields and their effects on the heat transfer in main channel. Fluid flow details The computed instantaneous velocity vectors are shown in Figure 7 at different instants in the fourth cycle from the beginning of motion of the membrane when the diaphragm is operated at a frequency of 10 kHz. The structure of events remains approximately the same at all frequencies; the most significant change being the phase lag discussed above caused by compressibility effects in the cavity.

b = 90

Vertical velocity, m/s

(a) 100

membrane velocity orifice exit velocity

Synthetic jets for heat transfer enhancement

50

273 0

−50 0

1

2

3

4 5 time f=20kHz and A=20 m m b = 37

6

7

membrane velocity orifice exit velocity

Vertical velocity, m/s

(b) 100

50

0

−50 0

1

2

4 5 time f=10kHz and A=40 m m b = 124

(c) 100

Vertical velocity, m/s

3

6

7

membrane velocity orifice exit velocity

50

0

−50 0

1

2

3

4 5 time f=30kHz and A=40 m m

6

7

Figure 6. Vertical velocity at centre point of membrane and centre point of exit of orifice

HFF 17,3

274

(a)

(b)

(c)

Figure 7. Velocity vectors (a) t ¼ 3.0; (b) t ¼ 3.25; (c) t ¼ 3.5; (d) t ¼ 3.75 for f ¼ 10 kHz and A ¼ 40 mm

(d)

The velocity vectors in Figure 7(a), occur at a time t ¼ 3.0, when the diaphragm is moving at its maximum velocity towards the orifice. By this stage the expelled air from the orifice has reached the upper wall of the channel thereby interfering with the boundary layer in the region in which heat transfer occurs. The emerging jet from the orifice generated the clockwise vortex immediately down stream of the jet and at the same time it generated a small anticlockwise near the upper surface on the downstream side of the jet. Since, the flow is two-dimensional, as may be seen in Figure 7(a), when the jet impinges on the upper surface the flow in the main channel is separated into two parts with no flow from the region upstream of the dividing jet to that down stream. This leads to a complete flow reversal in the area upstream of the jet so that a complex flow pattern begins to develop, caused by the interaction of the flow reversal and the developing vortex. Downstream of the impinging jet the motion is sustained by the jet and the remnant of the vortices from the previous cycle. As may be seen in Figure 7(b), the impinging jet is reflected from the upper surface thereby strengthening the clockwise vortex generated when the jet emerging from the orifice on the downstream side of the orifice and forming a counter-clockwise vortex on

the downstream side of the first reflection from the upper wall. This new, counter clockwise vortex is strengthened by the reflection of the jet from the lower side. This is repeated a number of times so that a series of geared vortices are created which are separated by the main flow from the jet. During the suction stage the geared vortices persist but weaken as time progresses while the axial velocity is reduced, as may be seen in Figure 7(c) and (d). What is not clear in Figure 7(c) that the vortex which was upstream of the orifice in Figure 7(b) has moved to be just downstream of the orifice. Similarly, the other vortices, shown in Figure 7(b) have moved downstream with significant loss of strength. The vortices continue to move downstream during the remainder of the suction phase as may be seen in Figure 7(d). Further, during the suction phase, as may be seen in Figure 7(c) and (d) most of the fluid enters the cavity rather than continuing along the main channel, so that there is a significantly reduction in velocity in the main channel during the suction phase of the actuator cycle. It follows, that the velocity in the main channel, averaged over a cycle, is significantly reduced, in this case to 6.3 ms2 1, from its steady flow value of 7.75 ms2 1. It is interesting to note that the structure of the synthetic jet in a confined cross-flow is completely different to that found in synthetic jets in still atmosphere. Synthetic jets in still atmosphere may appear to be continuous, vortices are generated at the orifice and travel in a direction parallel to the jet weakening as they move away from the orifice (Timchenko et al., 2004). Under the same circumstances it is shown here that in a constrained cross flow the synthetic jet is not continuous. Moreover, whilst the vortices are convected in the direction of the main flow, they are generated both upstream and downstream of the orifice as well as by the reflections of the jet from the solid boundaries. When the frequency of the diaphragm is increased to 20 and 30 kHz with the exit velocity from the orifice maintained at approximately the same value, the structure of the flow and the vortices at the same phase of the cycle relative to the jet velocity at the exit of the orifice passage, remains the same and need not be discussed any further. When a quasi equilibrium condition is reached, the mass flow rate into the channel must be exactly equal that leaving the channel averaged over a cycle, so that the synthetic jet actuator becomes a zero net mass flow device, but in all the cases here this condition was not reached. The mass flow rate averaged over a cycle was always higher than the mass flow rate into the channel, indicating that the air in the cavity was warming up. Even after six cycles, the difference between the outflow and inflow mass flow rates was still around 4 per cent. However, we thought this to be sufficiently close to a quasi equilibrium condition to allow us to evaluate the heat transfer enhancement capabilities of synthetic jets. Heat transfer enhancement The instantaneous local heat flux variation on the hot surface immediately above the centre of the orifice passage at a frequency of 30 kHz is shown in Figure 8 as a function of non-dimensional time from the beginning of the calculations which were started from the steady flow results. There is a is a long transient period, lasting approximately six actuator cycles, before the instantaneous heat transfer rate as a function of time can be said to repeat during an actuator cycle. At all frequencies, the results presented in this section have therefore been selected from the seventh cycle from the beginning of the calculations.

Synthetic jets for heat transfer enhancement 275

HFF 17,3

orifice exit velocity heat flux

f = 78

1.20E+05

100

1.00E+05

276

Heat flux,W/m2

6.00E+04 0

Velocity, m/s

50

8.00E+04

4.00E+04

2.00E+04 −50

Figure 8. Vertical velocity at (x ¼ 0, y ¼ 0) and surface heat flux at (x ¼ 0, y ¼ 200 mm), f ¼ 30 kHz

0.00E+00 1

2

3

4

5

6

7

time

The heat flux fluctuations as a function of time are rather large; varying from a maximum of 103 kWm2 2, which is about 13 times larger the steady flow heat transfer flux at the same location, to a minimum of 6 Wm2 2. In fact, the heat flux at the point on the upper wall immediately above the centre of the orifice passage, averaged over the 7th cycle, is 43 kWm2 2, which is approximately 5.6 times the steady flow heat flux. Clearly, there has been a very large enhancement in the heat transfer rate at this location. In fact, as may be seen from Figure 9, the maximum heat flux does not occur at the point immediately above the centre of the orifice, but, depending on the frequency can occur on either side of it. Further, it is interesting to note by comparing Figure 9(a), (b) and (c) that the non-dimensional time at which the maximum heat flux occurs is also dependent on the frequency. Heat flux distributions as a function of time As may be shown in Figure 8, there is a significant time-lag between the centreline velocity of the jet at the exit of the orifice and the heat flux on the hot wall. This is obviously due to the time taken by the jet, which starts with a zero velocity at the beginning of the ejection phase of the actuator cycle and gradually increases to a maximum. The time-lag at 30 kHz when expressed as a phase angle is approximately 788. Since, the height of the channel was the same in all the cases considered here, the actual time required for the jet to reach the upper wall would inversely proportional to

(a) 1.0E+05 steady case t = 5.85, end of suction t = 6.1, max orifice velocity t = 6.15, max local flux

Heat flux,W/m2

8.0E+04 6.0E+04

Synthetic jets for heat transfer enhancement 277

4.0E+04 2.0E+04 0.0E+00 –1

0

1 X, mm

2

(b) 1.0E+05 steady case t = 6.0, end of suction t = 6.25, max orifice velocity t = 6.4, max local flux

Heat flux,W/m2

8.0E+04 6.0E+04 4.0E+04 2.0E+04 0.0E+00 –1

0

1 X, mm

2

(c) 1.0E+05 steady case t = 6.1, end of suction t = 6.35, max orifice velocity t = 6.6, max local flux

Heat flux,W/m2

8.0E+04 6.0E+04 4.0E+04 2.0E+04 0.0E+00 –1

0

1 X, mm

2

Figure 9. Local heat flux at different time instants of the cycle, (a) f ¼ 10 kHz A ¼ 40mm; (b) f ¼ 20 kHz A ¼ 20mm; (c) f ¼ 30 kHz A ¼ 40 mm

HFF 17,3

278

the maximum jet velocity at the exit of the orifice. Since, the phase angle is directly proportional to the ratio of the actual time taken to reach the upper wall to the period of the actuator cycle, it follows that the phase angle ff 2 at a frequency f2 would be given by:

ff 2 ¼

f 2 vj1 ff f 1 vj2 1

ð29Þ

in which the phase angle ff 1 is known at a frequency f1 and vj1 and vj2 are the maximum velocities at the jet exit at the two frequencies. Although a reasonable effort was expended to obtain a constant value of the maximum jet exit velocity at all frequencies, because of the long transient and the unknown form of K1( f ) in equation (25), it was very difficult to adjust the amplitude of the diaphragm to ensure that this condition was met in all cases. As a result, the value of the maximum velocity at the exit from the orifice varied from a high of 104 ms2 1 at 10 kHz to a low of 89 ms2 1 at 20 kHz. Since, the aim of the research was to determine whether synthetic jet actuators are appropriate for enhancing heat transfer in micro-channels carrying air, a 17 per cent changes in the maximum velocity of the jet would not affect the evaluation of the performance of the device. When the values at 10 and 20 kHz are substituted in equation (29), phase lags of 23 and 508, respectively, are obtained. These values are in remarkably close agreement with 22 and 508, respectively, extracted from the numerical calculations. The phase shift due to the time taken for the jet to reach the upper surface has to be added to the phase shift due to compressibility effects discussed earlier, in order to obtain the phase shift relative to the diaphragm motion. Since, the non-dimensional time is based on the movement of the membrane of the actuator, the maximum heat flux must occur at different non-dimensional times at the various frequencies shown in Figure 9. Similarly, different phase shifts occur at different frequencies for other events, such at the time of the end of suction, and need to be taken into account if a consistent set of results for the spatial distribution of the heat flux is to be presented. Spatial heat transfer distribution At 10 kHz at the time of the maximum heat transfer rate, there appears to be two maxima of the heat flux occur, one on either side of the orifice, whereas at the other two frequencies there was only one maximum (Figure 9). A stagnation region was detected immediately opposite the orifice at 10 kHz. Interestingly, as the frequency is increased the location of the point at which the maximum local heat flux occurs moves towards the inlet of the main channel, as is clearly shown in the three cases in Figure 9. Further, the horizontal extent of the large maximum heat flux spike depends on the frequency, changing from a maximum of 0.43 mm at 10 kHz to a minimum of 0.29 mm at 30 kHz. Whilst at the time of maximum heat flux the distributions of the local heat flux appear to be similar, there are significant differences which are due to the frequency at which the actuator is operating. Since, the duration of the ejection phase of the actuator cycle is inversely proportional to the frequency and the average velocity in the channel reduces as the frequency increases, the spacing between adjacent vortices downstream of the orifice will, per force, be reduced as may be seen in Figure 9 from the distances between successive maxima of the heat flux distribution. However, it is not apparent

as to why the increased heat flux spreads further downstream as the frequency becomes larger, as is readily noticeable from a comparison of Figure 9(a) with Figure 9(c). Because of the lack of mixing at the low frequency of 10 kHz (Figure 9(a)) the heat flux towards the exit falls below the heat flux in the same area when steady flow prevails. Whereas, as can be seen at t ¼ 6.15, in the region downstream of the large spike, the heat flux is higher than that obtained in steady flow. At 20 kHz, however, because of the increased mixing, the heat flux, in the region near the exit, is higher than that obtained in steady flow at the time of its maximum. In fact, the heat flux at t ¼ 6.4 is significantly higher at all points downstream of the beginning of the large heat transfer spike, than at 10 kHz and that in steady flow (Figure 9(b)). The heat flux at t ¼ 6.6 at a frequency of 30 kHz (Figure 9(c)) is very much larger in the region downstream of the large spike than at 20 kHz and very much larger than the steady flow heat flux. Upstream of the large spike the heat flux at 10 kHz at t ¼ 6.15 is slightly lower than the heat flux which prevails at steady flow (Figure 9(a)). At 20 kHz at the time of the maximum heat flux, t ¼ 6.4, Figure 9(b), the heat flux upstream of the large spike is approximately the same as that in steady flow, that is slightly higher that that at 10 kHz at the same point in the cycle. As was the case downstream of the large spike at 30 kHz, with t ¼ 6.6, the heat flux upstream of the spike is larger than at 10 and 20 kHz and larger than that which obtains in steady flow. Thus, it may be concluded, that in the range of frequencies investigated, at the maximum heat transfer stage of the cycle, the heat transfer rate increases with the frequency. As may be seen in Figure 9, the time when the maximum jet velocity occurs at the orifice the form of spatial distribution of the instantaneous local heat flux for all frequencies, is similar to that which occurs at the time of the maximum heat flux, but, with the heat flux significantly lower than that which pertains at the later time. The lowest heat transfer occurs at the end of the suction phase. At 10 kHz, despite there being some regions in which the heat flux is larger that that obtained in steady flow, most of the upper surface yields a heat flux lower than the steady flow heat flux (Figure 9(a)). Similar remarks apply to the heat flux at 20 kHz (Figure 9(b)), which is even lower than at 10 kHz. The results at a frequency of 30 kHz are totally unexpected as may be seen in Figure 9(c). Here the heat flux is very low, with only two small regions in which the heat flux exceeds the heat transfer rate in steady flow. What is remarkable, and therefore totally unexpected, is that downstream of x < 0:4mm the heat flux is negative, that is, the heat transfer is from the air to the isothermal wall. In order to be able to transfer heat from the fluid to the wall, the temperature of the fluid in the vicinity of the wall would have to be higher than the temperature of the wall. Since, the hottest boundary is the upper wall, it seems impossible that the fluid could reach a higher temperature. This is indeed true for an incompressible fluid, but is not the case for a compressible fluid. The accelerations in the main channel are a function of the frequency squared, since the velocities are a function of the frequency. As a result the pressure gradients and hence the pressure changes at a particular point in the main channel are also proportional to the frequency squared. That is, during the deceleration phase at 30 kHz, the maximum difference in pressure between the pressure at a point in the main channel, downstream of the orifice, and the external domain, would need to be about 2(1/4) times that which exists at 20 kHz. This is indeed

Synthetic jets for heat transfer enhancement 279

HFF 17,3

280

the case with the maximum pressure difference being 7.2 kPa at 30 kHz, whereas it is 2.9 kPa at 20 kHz; a ratio of 2.5. An isentropic compression from 100 to 107 kPa would lead to a 2 per cent increase in temperature. Now, during the suction phase there is little flow in the main channel downstream of the orifice, as may be seen Figure 4(d). In fact flow is directed upstream, particularly near the hot, upper wall. That is the hottest air near the hot wall is forced to move upstream getting further heated. A 2 per cent increase in temperature, mentioned above, with the air initially at 323 K, the hot wall temperature, would lead to a temperature 6.3 K higher than the wall temperature. This would result in a significant heat transfer from the air to the hot wall. Now, since the compression is not isentropic because there is heat transfer, the temperature would not rise as much. In fact, the highest temperature near the wall is 4 K higher than the wall temperature; a temperature rise equal to approximately 2/3 of the isentropic temperature rise predicted above. It follows that the pressure rise during deceleration phase in the main channel, coupled with the fact that air is compressible, is responsible for the apparently anomalous heat transfer from the fluid to the hot wall. The heating process occurs to a lesser extent at 20 kHz, so that there are no regions of negative heat transfer as may be shown in Figure 9(b) at t ¼ 6.0. The effect is even smaller at 10 kHz so that the heat transfer at the end of suction, is not much less than the steady state heat transfer. As is shown in Figure 9, the much larger changes in pressure at 30 kHz are therefore responsible for the very bigger fluctuations in the heat flux in time than those at 10 or 20 kHz. Work is required to increase the pressure and temperature in the main channel and this is supplied by work, additional to that which would be provided by the diaphragm, if the actuator were operating in a still ambient atmosphere. The magnitude of the extra work, and its consequences in evaluating the viability of using synthetic jet actuators, has not been determined because, at this stage the effectiveness of the synthetic jet as a means of heat transfer enhancement is to be established. Overall heat transfer rates There is a significant increase in the overall heat transfer averaged over the whole length of the channel and over an actuator cycle as is shown in Figure 10. The heat flux shown by the lower curve in Figure 10 increases as the frequency becomes larger. This is despite the fact that at 30 kHz there is a heat transfer from the air to the hot wall for part of the cycle, which does not happen at the lower frequencies. The heat flux at 30 kHz is 37 per cent larger than the heat flux obtained when the flow is steady. However, the increase in heat flux when the frequency is changed from 10 to 30 kHz is only 18 per cent. It is interesting to note that the heat flux increase has occurred despite a 19 per cent reduction in the inlet air velocity as the frequency is increased from zero to 30 kHz. It may be seen in Figure 9 that the effect of the synthetic jet diminishes beyond x ¼ 2 mm so that it is more appropriate to evaluate the effect of the jet over the region from the entrance x ¼ 2 1.1 mm, to x ¼ 2 mm. The upper curve in Figure 10 shows the heat flux averaged over this region and over one cycle of the membrane motion. Here, the enhancement in the average flux is somewhat larger then when the heat flux is averaged over the whole length of the channel, being 42 per cent. Interestingly, the gradient of the upper line in Figure 10 is smaller than the lower curve. This means that

Synthetic jets for heat transfer enhancement

12,000

10,000

281

Heat flux,W/m2

8,000

3 mm 4.2 mm 4.2 mm - steady 3 mm - steady

6,000

4,000

2,000

0

0

5

10

15

20

25

30

Frequency, kHz

the enhancement in the heat flux smaller when the frequency is increased for 10-30 kHz, when the heat flux is averaged over the shorter region. This is in agreement with the fact, mentioned above, that as the frequency is increased, the effect of the mixing seems to affect the heat transfer further downstream. In any case, the heat transfer rates which need to be handled in microchips have to be orders of magnitude higher that the maximum heat flux averaged over time and space of about 11 kWm2 2 obtained in this study, so that it seems unlikely that synthetic jets can provide sufficient heat enhancement to make air cooled micro-channels a viable method of heat removal in future generations of microchips. Conclusions A model has been developed of the flow and heat transfer processes in forced convection an air cooled micro-channel with an isothermal hot wall and in which a synthetic jet was used to enhance the heat transfer. Here, the compressibility effects of the air were included and the motion of the diaphragm of the actuator was directly simulated. It was demonstrated that compressibility of air is very significant and a simple method was found for determining the natural frequency of the cavity as well as allowing a reasonable estimates of the phase angle between the velocity of the air at the exit from the orifice of the actuator and the diaphragm motion, once a reliable value of the damping could be established. Once the actuator was switched on, a flow pattern consisting of geared vortices occurred downstream of the jet during the ejection phase of the actuator cycle. Surprisingly, the vortices persisted during the ingestion part of the cycle. The distance

Figure 10. Time and surface averaged heat flux for different frequencies

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between adjacent vortices decreased as the frequency increased. Upstream of the orifice, large fluctuations in velocity occurred, including flow reversals. As expected, the heat flux results as a function of time, at a point on the hot wall immediately above the orifice, lag in the air velocity at the exit of the. At that point on the hot wall, and adjacent points, there are very large fluctuations in the heat transfer flux which increase with the frequency, however, on the average, there is a very significant enhancement of the heat transfer from its value in steady flow, for example at 30 kHz the average heat transfer flux in this area is 5.6 times the steady flow value. Although superficially the same, at the time of maximum heat transfer, the better mixing which results from the larger density of vortices as the frequency increases results in larger heat fluxes downstream of the orifice. An apparent anomaly occurs at the beginning of the ejection cycle at 30 kHz down stream of the orifice. At that time, in the majority of the channel, there is heat transfer from the “cooling” air to the “hot” isothermal surface. The large accelerations required to adjust fluid velocities at 30 kHz, result in pressure changes, which, because of the compressibility of air, are sufficiently large to increase the temperature of the air near the isothermal boundary to above that of the wall, leading to heat transfer from the fluid to the wall. It is seen here that if air is used for cooling in situations in which large pressure changes may occur, this may lead to a significant decrease, or even a reversal, as occurs at 30 kHz, of heat transfer. Despite the reversal of heat transfer at some points during part of the actuator cycle, the heat flux averaged over time and space, increases as the frequency increases. The enhancement is, however, only 37 per cent over that obtained in steady flow. Since, the heat fluxes required in the next generation of microchips are orders of magnitude larger than those found in this study, it seems unlikely that air forced convection in micro-channels will lead to viable devices, even if the cooling is enhanced by synthetic jets. References Eroglu, A. and Breidenthal, R.E. (2001), “Structure, penetration, and mixing of pulsed jets in crossflow”, AIAA Journal, Vol. 39 No. 3, pp. 417-23. Glezer, A. and Amitay, M. (2002), “Synthetic jets”, Ann. Rev. Fluid Mech., Vol. 34, pp. 503-29. Gordon, M. and Soria, J. (2001), “Scalar mixing of zero-net-mass-flux jets in crossflow”, Proc. 14th Australasian Fluid Mech. Conf., pp. 729-32. Ho, C.M. and Huang, L.S. (1982), “Subharmonics and vortex merging in mixing layers”, J. Fluid Mech., Vol. 119, pp. 443-73. Karniadakis, G. and Beskok, A. (2002), Micro Flows: Fundamentals and Simulation, Springer-Verlag Inc., New York, NY. Kataoka, K., Suguro, M., Degawa, H., Maruo, K. and Mihata, I. (1987), “The effect of surface renewal due to large scale eddies on jet impingement heat transfer”, Int. J. Heat Mass Transfer, Vol. 30, pp. 559-67. Kreyszig, E. (1972), Advanced Engineering Mathematics, Wiley, New York, NY. Mallinson, S.G., Kwok, C.Y. and Reizes, J.A. (2003), “Numerical simulation of micro-fabricated zero mass-flux jet actuators”, Sensors and Actuators A, Vol. 105, pp. 229-36. Mladin, E. and Zumbrunnen, D. (2000), “Alterations to coherent flow structures and heat transfer due to pulsations in an impinging air-jet”, Int. J. Term.Sci., Vol. 39, pp. 236-48.

Smith, B.L. and Glezer, A. (1998), “The formation and evolution of synthetic jets”, Physics Fluids, Vol. 10 No. 9, pp. 2281-97. Timchenko, V., Reizes, J. and Leonardi, E. (2004), “Compressibility effects in micro synthetic jets”, paper presented at 2nd Int. Conference on Microchannels and Minichannels, Rochester, New York, 17-19 June, Paper 32-1D. Tuckerman, D.B. and Pease, R.F.W. (1981), “High performance heat sinking for VLSI”, IEEE Electron Device Letters, Vol. EDL-2 No. 5, pp. 126-9. Corresponding author Victoria Timchenko can be contacted at: [email protected]

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A.C. Benim, K. Ozkan and M. Cagan Department of Mechanical and Process Engineering, Duesseldorf University of Applied Sciences, Duesseldorf, Germany, and

Received 1 January 2006 Accepted 9 July 2006

D. Gunes Mechanical Engineering Faculty, Istanbul Technical University, Istanbul, Turkey Abstract Purpose – The main purpose of the paper is the validation of a broad range of RANS turbulence models, for the prediction of flow and heat transfer, for a broad range of boundary conditions and geometrical configurations, for this class of problems. Design/methodology/approach – Two- and three-dimensional computations are performed using a general-purpose CFD code based on a finite volume method and a pressure-correction formulation. Special attention is paid to achieve a high numerical accuracy by applying second order discretization schemes and stringent convergence criteria, as well as performing sensitivity studies with respect to the grid resolution, computational domain size and boundary conditions. Results are assessed by comparing the predictions with the measurements available in the literature. Findings – A rather unsatisfactory performance of the Reynolds stress model is observed, in general, although the contrary has been expected in this rotating flow, exhibiting a predominantly non-isotropic turbulence structure. The best overall agreement with the experiments is obtained by the k-v model, where the SST model is also observed to provide a quite good performance, which is close to that of the k-v model, for most of the investigated cases. Originality/value – To date, computational investigation of turbulent jet impinging on to “rotating” disk has not received much attention. To the best of the authors’ knowledge, a thorough numerical analysis of the generic problem comparable with present study has not yet been attempted. Keywords Turbulence, Jets, Rotational motion, Modelling Paper type Research paper

Nomenclature Cf d e

International Journal of Numerical Methods for Heat & Fluid Flow Vol. 17 No. 3, 2007 pp. 284-301 q Emerald Group Publishing Limited 0961-5539 DOI 10.1108/09615530710730157

h k Nud NuR QJ

¼ skin friction coefficient (Cf ¼ 2tw/rU 2) ¼ jet diameter (m) ¼ distance between jet and disk axes (m) ¼ distance between jet exit to disk (m) ¼ turbulence kinetic energy (m2/s2) ¼ Nusselt number based on jet diameter (Nud ¼ ad/l) ¼ Nusselt number based on disk radius (NuR ¼ aR/l) ¼ time averaged jet flow rate (m3/s)

¼ radial coordinate (m) ¼ disk radius (m) ¼ disk Reynolds number (ReD ¼ VR 2/n) Re ¼ rotational Reynolds number defined by impingement radius (Ree ¼ Ve2/n) ReJ ¼ jet Reynolds number (ReJ ¼ Ud/n) u, v, w ¼ axial radial and azimuthal velocities (m/s) U ¼ time averaged bulk jet velocity y ¼ wall distance (m) r R ReD

yþ z

¼ non-dimensional wall distance pffiffiffiffiffiffiffiffiffiffi ð y þ ¼ ð y=nÞ tw =rÞ ¼ distance from disk surface (axial coordinate) (m)

Greek symbols a ¼ time averaged heat transfer coefficient (W/m2K) b ¼ ratio of Re numbers (b ¼ ReJ/ReD) 1 ¼ dissipation rate of k (m2/s3) l ¼ thermal conductivity (W/mK) n ¼ kinematic viscosity (m2/s) r ¼ density (kg/m3) tw ¼ time averaged wall shear stress (Pa)

v V

¼ turbulence frequency (1/s) ¼ disk rotational speed (rad/s)

Superscripts 0 ¼ instantaneous value ðÞ ðÞ ¼ time-averaged value Abbreviations RANS ¼ Reynolds averaged Navier-Stokes equations RSM ¼ Reynolds stress model SST ¼ shear stress transport WF ¼ wall functions

Introduction The turbulent impinging jet is of great importance in many engineering problems, concerning processes such as cooling, heating and drying. We are particularly interested in gas turbine cooling, as discussed in Benim et al. (2004). During the manufacture of gas turbines, a detailed insight of the heat transfer mechanisms is very important especially in designing the highly thermally stressed components such as the turbine disk. There is a variety of schemes used for this purpose, an overview of which is provided in Owen and Rogers (1989). An efficient disk cooling arrangement consists of squirting a cooling jet at the rotating disk. In pre-swirl systems, a similar flow situation can occur for some designs and operation conditions (Benim et al., 2004; Owen and Rogers, 1989). Thus, the present work focuses on the generic problem of turbulent jet impinging onto a rotating disk, for assessing the predictive capability of computational procedures in such applications. Owing to vast applications, many experimental and computational studies have been reported on turbulent impinging jets. Most of them consider the problem of impinging jet onto a stationary surface with or without cross flow (Jambunathan et al., 1992; Cooper et al., 1993; Craft et al., 1993; Behnia et al., 1999). However, the case of jet impingement onto a rotating surface exhibits different characteristics. The boundary layer established by disk rotation and pumping attaining its maximum velocity at the wall is skewed, and subject to a centrifugal force field. A comparably smaller amount of experimental work has been published on the problem of turbulent jet impinging on a rotating surface. Earlier studies, such as those of Metzger and Grochowsky (1977), Popiel and Boguslawski (1986) and Metzger et al. (1989) were mainly concentrated on heat transfer measurements and some flow visualization. Rather recently, more detailed LDA measurements of the flow field, including turbulence statistics have been published in Brodersen et al. (1996) and Minagawa and Obi (2003). Computational investigation of the phenomenon has received an even smaller attention. To the best of the authors’ knowledge, a thorough numerical analysis of the generic problem has not yet been attempted. Thus, the purpose of the present study is the computational investigation of the generic problem of turbulent jet impinging onto

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a rotating disk, and a validation of the computational procedures by comparisons with the available experiments. An overview of the applied mathematical and numerical modeling For numerical investigations, the general-purpose CFD code Fluent (2004) has been used as basis, which utilizes a finite volume method to discretize the governing equations, and a pressure-correction formulation to handle the velocity-pressure coupling. The Reynolds (Re) averaged continuity, Navier-Stokes and convective-diffusive energy transport (modeled according to Re’s analogy) equations (Durbin and Reif, 2003) have been solved for the incompressible, quasi-steady, turbulent flow, together with the additional transport equations for the turbulence quantities. For the material properties, constant values have been used, which correspond to a mean temperature of the considered problem. Within the framework of a RANS formulation, a broad range of two-equation turbulence models, namely, the standard (Launder and Spalding, 1974), the RNG (Yakhot and Orszag, 1986) and realizable (Shih et al., 1995) k-1 models, as well as the k-v model (Wilcox, 1993), and the SST model (Menter, 1994) have been used. Additionally, the Re Stress Model (RSM), using a quadratic modeling of the pressure-strain term (Speziale et al., 1991), has been investigated. Mainly, the low Re number (low-Re) versions (Shima and Launder, 1989) of the models have been applied, adopting amendments such as the two-layer-zonal approach (Wolfstein, 1969), which are able to resolve the near-wall region. For comparison, high-Re number (high-Re) versions (Launder and Spalding, 1974) of some models have also been applied adopting a wall-function approach (WF) for the near-wall flow. For this purpose, the non-equilibrium wall functions (Kim and Choudhury, 1995) have been employed, which are principally superior to the standard ones (Launder and Spalding, 1974). For all applications, always the original sets of model constants have been used. In generating the computational grids, special care has been taken for a convenient near-wall resolution, resulting in adequate near-wall y þ values, which are compatible with the assumptions of the respective turbulence model (Benim and Arnal, 1994). A second order upwind scheme (Barth and Jespersen, 1989) has been used for the discretization of the convective terms of the governing equations of all convective diffusively transported variables, for attaining a high numerical accuracy (for Re stress components, a first-order scheme has been used for some cases). Test cases The investigated test cases have been based on the experiments of Minagawa and Obi (2003) (MO), Popiel and Boguslawski (1986) (PB) Metzger et al. (1989) (MBB) and Metzger and Grochowsky (1977) (MG). The generic configuration of the disk and jet arrangements for the test cases with the indication of boundary types conditions are shown in Figure 1. Table I summarizes the geometric parameters and operation conditions for the investigated test cases. Please note that the cases MG-1 and MG-2, each, represent 5-6 cases with varying jet Re number for a given disk Re number.

Computational investigation of turbulent jet

d inlet U boundary A

h

computational domain H

boundary D h

e

287

rotating boundary B wall boundary C

z r Ω

R L

Boundary conditions The applied boundary types are shown in Figure 1 and Table I. At the inlet of the jet flow (Figure 1), all variables (velocity components, turbulence quantities, temperature) are prescribed. In all experiments, quite long lances were used, so that practically fully developed conditions towards the end of the pipe should be expected. Thus, the prescription of an empirical turbulent fully developed pipe flow velocity profile (e.g. 1/7 law) at the domain inlet (Figure 1) could be thought to provide sufficient accuracy. Nevertheless, instead of this approach, the boundary conditions at the domain inlet (Figure 1) have been obtained by preliminary computations of the developing pipe flow problem, separately for each case, and for each turbulence model used, aiming at an even higher accuracy of the inlet boundary conditions, especially for the various turbulence quantities required by different turbulence models. In these preliminary computations of the developing pipe flow, uniform distributions have been applied at the inlet of the pipe, using the experimentally known velocities and temperatures, and assuming a turbulence intensity of 4 percent and a mixing length of 30 percent pipe diameter. An inspection of Table I shows that the jet Re number indicates a laminar flow within the pipe, for a few cases of MG-1 and MG-2 (where, but, the flow becomes turbulent in the outer region, near the rotating disk). In computing those cases, using low-Re turbulence models, a very low turbulence intensity has been prescribed at the pipe inlet (0.1 percent) in order to cope with this situation. For better accounting for a possible interaction of the jet flow with the flow in the outer domain, in the vicinity of the jet exit, the inlet boundary of the solution domain has not been placed exactly at the jet outlet, but at a certain distance upstream. The latter has normally been taken to be equal to the jet-to-disk-distance h (Figure 1), which was taking values about 2-5d (Table I). At the disk wall, the rotational velocity and the wall temperature have been prescribed, as these values were provided by the experiments. The jet-disk arrangements were placed in quite large plenums for most of the experiments, namely for those of Minagawa and Obi (2003), Popiel and Boguslawski (1986) and Metzer and Grochowsky (1977) (characteristic dimension of the confining box much larger than the disk diameter (ten times, or more)). Thus, for these cases, the effect of the confinement can be neglected and the configuration can be assumed to be free in space.

Figure 1. Geometry, solution domain, boundaries

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Table I. Definitions of the investigated cases

Case

h/d

R/d

e/R Modeling h/H R/L bd.A bd.B bd.C bd.D ReJ

ReD

MO-1 MO-2 PB-1 PB-2 MBB-1 MBB-2 MG-1 MG-2

5 5 5 5 2.13 2.13 2 2

6 6 7.56 7.56 21.3 21.3 10 10

0 0 0 0 0 0 0.8 0.8

235,650 251,400 1,209,600 1,209,600 271,000 464,000 40,000 120,000

2D 2D 2D 2D 2D 2D 3D 3D

0.5 0.5 0.5 0.5 1 1 0.5 0.5

,1 ,1 ,1 ,1 1 1 ,1 ,1

p p p p w w p p

p p p p o o p p

s s s s w w s s

s s s s s s i i

14,500 14,500 178,000 46,000 23,170 23,170 200-20,000 200-20,000

Notes: Abbreviations: MO, Minagawa and Obi (2003); PB, Popiel and Boguslawski (1986); MBB, Metzger et al. (1989); MG, Metzger and Grochowsky (1977); o, outlet boundary; p, pressure boundary; s, symmetry axis or plane; w, solid wall; i, domain interior

Consequently, in computations based on these experiments, instead of modeling the whole plenum geometry, which would require an unnecessarily too large grid, a cylindrical section (radius ¼ L, height ¼ H, Figure 1) around the apparatus has been defined as the solution domain. The enclosure of this domain is, then, has been declared to be pressure boundaries (Table I), with a prescribed constant ambient pressure. This is expected to be an accurate enough formulation, provided that the pressure boundaries are placed sufficiently far away from the impingement region. The latter has been ensured by preliminary computations. At the pressure boundaries, the ambient values of the temperature and turbulence quantities need additionally to be prescribed, since, there, an entrainment of the ambient fluid across these boundaries is expected. As the temperature is given by the experiments, uncertainties exist for the turbulence quantities. For the latter, a turbulence intensity of 1 percent and a viscosity ratio (turbulent viscosity divided by laminar viscosity) of 1 have been assumed. Additional computations have been performed by reducing and increasing these values within reasonable ranges, where no remarkable changes of the results have been observed. This confirms the relative insensitivity of the results on these quantities around the prescribed values, and, thus, the convenience of the prescribed values. In the experiments of Metzger et al. (1989), the jet-disk arrangement was placed in a rather small confinement, where the walls of the confining box was aligned with the disk edge and the jet outlet. For this case, the boundaries of the enclosure (Figure 1) have been defined to be solid walls (Table I), of course, where an outlet boundary has also been modeled as a part of the boundary (Figure 1) aligning with the disk edge, according to the geometry given in Metzger et al. (1989). In all considered test cases, except those based on the experiments of Metzger and Grochowsky (1977), co-axial jet and disk arrangements have been considered. These test cases give rise to a 2D-axisymmetric problem formulation, which has been utilized in the computations, of course (Figure 1, Table I). In the test cases based on the experiments of Metzger and Grochowsky (1977), the jet and the disk have not been co-axial, but eccentric. Thus, for these test cases, a 3D analysis have been applied (Figure 1, Table I).

Grid generation Structured grids have been employed. Special care has been taken for obtaining adequate y þ values for near-wall cells. For low-Re turbulence model applications, grids in near-wall regions have been constructed so fine that sufficiently small local y þ values have been obtained. For all such applications, the local y þ values have been varying between the minimum of , 0.1 and maximum of , 0.5, having the value of , 0.3 in the mean. In addition, a rather mild grid expansion has been applied, which resulted in a quite fine resolution of the near-wall layer (about ten cells in the region 0 , y þ , 5). For high-Re turbulence model applications with WF, the near-wall grid has been made coarse enough to obtain sufficiently large y þ values. For these applications, the local y þ values have been varying between the minimum of , 20 (rather small regions) and a maximum of , 100, exhibiting a mean value of , 70. A grid independency study has been performed for the 2D-axissymmetric case of MO-1 (Table I), using 100 £ 100, 200 £ 200, 300 £ 300 and 400 £ 400 grid resolutions (using the SST model). It has been observed that even 100 £ 100 grid provides a sufficient grid independency, provided that the wall layers are resolved adequately and a second order discretization scheme is used. However, for excluding any uncertainty, computations have been performed using the finer, 300 £ 300 grid for the 2D cases, where the total number of grid points were not that critical with respect to the computational overhead. Here, the grid lines have been concentrated in the impingement area, with a mild expansion ratio in the radial direction beyond this region. In the axial direction, 200 cells have been used for the region between the jet exit position and the disk and 100 cells for the region between the pressure boundary and the jet exit location. A formal grid independency study has not been performed for the remaining cases, where the construction of the grids have been led by the experience gained for the first case and the by the requirements of obtaining an adequate near-wall resolution. For the 2D cases (MG.1, MG-2, Table I) a grid consisting about 800,000 cells has been used, some views of which are shown in Figure 2. Convergence criteria Special attention has been paid to obtain results with a high convergence level. The default convergence criterion of fluent is that scaled residuals of all equations fall below 102 3, whereas a tolerance of 102 6 is required for the energy equation. In the present computations 100 times smaller tolerances have been required. For some RSM computations, it was not possible to obtain that low residual values, although a reasonable convergence, well below the default tolerances, was, in any case, obtained. For such cases, additional transient runs were performed to ensure this behavior were not due to an eventual suppression of any important transient phenomena. In any case, upon the fulfillment of the respective convergence criteria, before judging for the convergence, several hundreds of additional iterations have been performed for ensuring that all important variables do not show any significant changes more. The final results have been obtained using rather high under-relaxation factors (velocities: 0.7, k, 1, v: 0.8, Re stresses: 0.5, other variables: 1.0). Results Applying the above-mentioned turbulence models to the test cases summarized in Table I, a rather huge amount of data has been produced. A limited selection will be

Computational investigation of turbulent jet 289

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(a)

(b)

290

(c)

Figure 2. The 3D grid (a) disk surface grid; (b) detail of impingement region; (c) grid distribution in axial direction

presented here. The results of the standard, realizable and the RNG versions of the k-1 model were mostly quite similar, whereas the RNG version has been observed to provide slightly better results for some cases. Thus, here, only the RNG version results are presented for the k-1 model, which are designated as “k-1” in figure legends. For all models, the presented results are the ones, obtained using the low-Re versions of the models, resolving the near-wall regions, with one exception: results using the wall-functions are additionally presented only for the high-Re, k-1 model, for comparison, which are designated as “k-1-WF” in figure legends. Cases MO-1, MO-2 For the test cases MO-1 and MO-2 (Table I) based on the measurements of Minagawa and Obi (2003), experimental data were available for velocities and turbulence

quantities. Heat transfer measurements were not performed. Radial profiles of the radial velocity along an axial section at z/d ¼ 0.032 are shown in Figure 3, for the case MO-1. All models underestimate the peak velocity around r/d ¼ 1, where the k-v model shows the best agreement with the experiments. The largest discrepancy for the peak velocity is observed for the k-1 model, which predicts a rather flat variation throughout. The k-1 -WF (high-Re) results are poorer compared to the k-1 (low-Re) ones. In this comparison (k-1 vs k-1-WF) the discrepancy is rather small. But, please note that the displayed results for k-1-WF are the cell-center values of near-wall cells, but not their (linear) interpolation to the wall (this is thought to be rather in the sense of the wall-functions philosophy, which, strictly speaking, would require a logarithmic interpolation). The SST model shows the second best agreement with the experiments, exhibiting a performance between k-1 and k-v models, as one would expect. The peak velocity captured by RSM is similar to that of the SST model, and, thus, shows a better agreement with experiments than the k-1 model. However, at larger radii, RSM shows a poorer agreement with experiments (Figure 3). Variations of radial velocity with z/d, at the radial position r/d ¼ 5.8, for a slightly increased disk speed (MO-2, Table I) are shown in Figure 4. An inferior agreement with the experiments is shown by k-1-WF (the curve is drawn up to the center of the near-wall cell, but not between the cell-center and the wall). Among the low-Re models, the k-1 model performs inferiorly compared to the others. In predicting the near-wall peak value, the k-v model shows the best performance, which, however, exhibits a deviation from the measured values for larger z/d. The SST model under-predicts the near-wall peak value compared to the k-v model, but shows a better performance at higher z/d. The near-wall peak value is well predicted by RSM, which, however, over-predicts the measurements away from the wall (Figure 4).

Computational investigation of turbulent jet 291

1.2 Exp. k−ω

1 SST k−ε

0.8 v/U

k−ε−WF

RSM

0.6

0.4

0.2

0

0

1

2

3 r/d

4

5

Figure 3. Radial variation of near-wall radial velocity at z/d ¼ 0.032 (MO-1)

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0.5 Exp. k-ω 0.4

SST k-ε

292

k-ε-WF

0.3 z/d

RSM 0.2

0.1

Figure 4. Radial velocity as function of z/d at r/d ¼ 5.8 (MO-2)

0 0

0.1

0.2

0.3

0.4

v/U

Axial profiles of the normal components of the Re stresses, as predicted by RSM, at the radial position r/d ¼ 5.8, for the case MO-2 are shown in Figure 5. Although a qualitative agreement of the predicted Re stresses with the measurements can still be claimed in so far that the tangential component is predicted to be larger than the other components, which is qualitatively in agreement with the measured vales, it is obvious that the measured strong anisotropy of the Re stresses is strongly underpredicted by RSM, quantitatively (Figure 5). 0.5

u 2 / u2 Exp. v 2 / v2 Exp. w 2 / w2 Exp.

0.4

u 2 / u2 Pred. v 2 / v2 Pred. w 2 / w2 Pred.

z/d

0.3

0.2

0.1

Figure 5. Reynolds stresses as function of z/d at r/d ¼ 5.8 (MO-2)

0

0

0.05 0.1 0.15 normalized Reynold stresses

0.2

Cases PO-1, PO-2 For the test cases PB-1 and PB-2 (Table I) based on the experiments of Popiel and Boguslawski (1986), heat transfer measurements were available, whereas velocity field measurements were not presented. Radial variations of the Nu number along the disk surface are shown in Figure 6, for the case PB-1. The k-v and SST models show the best overall agreement with the experiments. The SST model performs better in predicting the peak values in the impingement region, whereas the k-v model show a better agreement with the measurements for larger radii. Greater discrepancies between the models are observed rather in the central, i.e. in the core of the impingement region. For higher values of r/d, all models predict rather quite close values and slightly overpredict the experiments, except the k-v model, which shows a better performance (Figure 6) throughout. Figure 7 shows the radial variation of the Nu number along disk surface for the case PB-2. Also for this case, the predictions obtained by different models show the largest discrepancies between each other in the central, stagnation region, in the vicinity of r/d ¼ 0. For all values of r/d, the SST model shows the best agreement with the experiments. The k-v model delivers quite close results to the SST ones, and shows, quite closely, the second best agreement. For intermediate values of the radius, i.e. for approx. 1.5 , r/d , 4.5, the experiments show a local minimum, where the Nu number first decreases up to r/d < 3, and then increases again, quite abruptly. None of the models could sufficiently predict this behavior. RSM may be considered to show a weak qualitative similarity to this variation, which, however, shows a strong underprediction at low radii, and an overprediction at large radii, quantitatively (Figure 7). A much better performance of the RSM was expected at the beginning of the present study, since it can cope much better with the predominantly non-isotropic turbulence structure of swirling flows compared to the turbulence viscosity-based models.

Computational investigation of turbulent jet 293

Exp. 1000

K-ω SST K-ε

800

Nud

K-ε- WF

RSM

600

400

200

0 0

1

2

3

4 r/d

5

6

7

Figure 6. Radial variation of Nu number along disk surface (PB-1)

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500 Exp. K-ω SST

400

K-ε

294

K-ε- WF

300 Nud

RSM

200

100

Figure 7. Radial variation of Nu number along disk surface (PB-2)

0 0

2

4

6

r/d

Thus, the present comparably rather poor performance of RSM (Figures 3-7) is an unexpected behavior, an explanation of which would require further analysis and more detailed comparisons with turbulence data. Within the present study, additional RSM computations have been performed using the linear (Launder et al., 1975) modeling of the pressure-strain term, with and without considering the so-called wall reflection terms, which have led to no improvement of the RSM performance. A inspection of the boundary conditions of the cases PB-1 and PB-2 (Table I) implies that rotational effects in comparison to impingement effects should be greater for the case PB-2 compared to the case PB-1, as the jet Re number of the latter is smaller than that of the former for the same disk Re number. The discrepancies in the central stagnation region put aside, the discrepancies between the measurements and the predictions for larger radii, which are observed to be larger for PB-2, may be attributed to the stronger rotational effects for this case. For this co-axial arrangement, it is obvious that the effects of rotation start to dominate at a certain distance away from the axis, whereas the near-field of the jet impingement region is expected to be impingement dominated. In Popiel and Boguslawski (1986), the following empirically-based expression:    2=3 i r ReJ R 2 h ¼ 1 2 2 £ 1024 ReJ ð1Þ d ReD d was suggested for defining the transition point between the impingement and rotation dominated regions, depending in the Re numbers and the geometries of the jet and the disk. According to this criterion, the transition form the impingement dominated region to the rotation dominated region occurs at r/d < 3.1, for the case PB-1 and at r/d < 1.6, for PB-2. These numbers may be considered to show some correspondence to the regions where rather larger discrepancies between the predictions and

measurements occur or start to occur are observed at larger radii, which is especially apparent for the more strongly rotation dominated case PB-2 (Figure 7). The radial distribution of the predicted skin friction coefficient for the case PB-2 is shown in Figure 8. One can observe that the skin friction curve shows a local minimum at r/d < 1.6, which corresponds quite well with the criteria given by equation (2) for the transition position from the impingement dominated regime into the rotation dominated one. This finding may be seen to imply that the transition location between the impingement and rotation dominated regimes is indicated by the local minimum of the wall shear stress. In Figure 8, the results for the impinging jet with a stationary disk (A) and the rotating disk without a jet (B), as well as the superposition of A and B are also plotted, for comparison. It is interesting to note that the superposition of the cases A (jet with stationary disk) and B (rotating disk without jet) produces a curve very close to the curve obtained for the “combined” problem of impinging jet onto the rotating disk, i.e. for the case PB-2 (Figure 8).

Computational investigation of turbulent jet 295

Cases MBB-1, MBB-2 For the test cases MBB-1 and MBB-2 based on the experiments of Metzger et al. (1989), the jet-disk arrangement was placed in a low confinement (Table I), resulting in a different flow situation (Table I) than that of a free-disk arrangement of all other test cases. Predicted and measured radial variations of the Nu number along disk surface are shown in Figures 9 and 10. For the case MBB-1 (Figure 9), with comparably lower disk Re number (lower rotational effects), the predictions obtained by all turbulent viscosity-based models lie quite close to each other, which show a rather good agreement with the measurements.

0.045 PB-2 0.04

Jet on stationary disk (A) rotating disk without jet (B)

0.035

A+B superposition 0.03

Cf

0.025 0.02 0.015 0.01 0.005 0 0

2

4 r/d

6

8

Figure 8. Radial variation of skin friction coefficient along disk surface (PB-2)

HFF 17,3

500

Exp. k-ω SST

400

k-ε k-ε-WF

296

RSM

NuR

300

200

100

Figure 9. Radial variation of Nu number along disk surface (MBB-1)

0 0

0.2

0.4

0.6

0.8

1

r/R

500

Exp. k-ω SST

400

k-ε k-ε-WF

300 NuR

RSM

200

100

Figure 10. Radial variation of Nu number along disk surface (MBB-2)

0

0

0.2

0.4

0.6

0.8

1

r/R

The results of the RSM show deviations from the experiments and the rest of the predictions, qualitatively and quantitatively (Figure 9). For the case MBB-2 (Figure 10), with higher disk Re number (higher rotational effects), all low-Re turbulence viscosity-based models show a similar agreement to each other, which, however, show a greater discrepancy from the measurements compared

to the previous case (MBB-1, Figure 9). The high-Re, k-1 model with wall functions show a deviation form the results of the low-Re models and a greater discrepancy to the experiments. The discrepancy to the experimental results become even larger using the RSM (Figure 10). Cases MG-1, MG-2 The test cases MG-1 and MG-2, which are based on the experiments of Metzger and Grochowsky (1977) majorly differ from the previous ones through the fact that the jet and disk are arranged eccentrically (Figure 1, Table I). Thus, a 3D problem arises. The modes of interaction between the jet and the disk become also different, where the rotational effects may be dominating even at the impingement position, depending on the Re number ratio. For the present test cases, only the models have been applied, which consistently have shown the best overall agreement with the predictions so far, namely the k-v and SST models. Metzger and Grochowsky (1977) did not provide measurements of the velocity field. However, some velocity field data on a different, but quite similar test rig was provided by Brodersen et al. (1996), which are used, here, for a qualitative comparison. Figure 11 shows the measured (Brodersen et al., 1996) and predicted velocity vector fields in a plane close to disk surface (z/d ¼ 0.03) for different values of the Re number ratio b. Although the experimental and computational Re number ratios are not identical, similar qualitative trends are indicated by the predictions and the experiments. For small values of b the jet does not seem to fully penetrate into the wall boundary layer. With increasing b the jet becomes more capable of disturbing the wall flow and is able to expand even in against the direction of disk rotation. Metzger and Grochowsky (1977) had also performed flow visualization experiments (which, however, were not (a)

Computational investigation of turbulent jet 297

(b)

2 m/s−

(c) 2 m/s−

(d)

Figure 11. Velocity vector fields at z/d ¼ 0.03: (a) experiments (Brodersen et al., 1976), b ¼ 0.0724; (b) present prediction, b ¼ 0.056; (c) experiments (Brodersen et al., 1976), b ¼ 0.1306; (d) present prediction, b ¼ 0.128

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298

documented in their paper) and defined “impingement dominated” and “rotation dominated” regimes, depending on the resulting velocity patterns, where a strong jet penetration (similar to Figure 11(c) and (d)) is used to define the “impingement dominated” regime, whereas a weak penetration and a strong deflection of the jet (similar to Figure 11(a) and (b)) is used to mark the “rotation dominated” regime. They have found that the transition between the two regimes correlates to the flow rate ratio QJ/QP, where QP is defined to be the empirical “pumping flow rate” at the impingement radius for a free rotating disk, which is given by Schlichting (1968) as: QP ¼ 0:886pneRe0:5 e

ð2Þ

The transitional values of QJ/QP were found to additionally depend on the rotational Re number (Metzger and Grochovsky, 1977). For the present test cases, the experimental Nu numbers (Metzger and Grochowsky, 1977) were provided as average values for the whole disk surface. Figures 12 and 13 display the predicted and measured Nu numbers (average values for the whole disk surface) as a function of the flow rate ratio QJ/QP, for the cases MG-1 and MG-2, respectively. In the figures, the impingement and rotation dominated regimes after Metzger and Grochowsky (1977) are also indicated. For the case MG-1, with lower disk Re number one can observe (Figure 12) that the k-v and SST model predictions are quite close to each other and show a similar agreement with the experiments. For the case MG-2, with higher disk Re number, the k-v model show throughout a much better agreement with the experimental values than the SST model. In both cases, the predictions agree better with the experiments in the impingement dominated regime, compared to the rotation dominated regime and the transition zone. 300 Exp. 250

K−ω SST

NuR

200

150

100

50

Figure 12. Variation of average disk Nu number with QJ/QP (MG-1)

rotation dominated 0 0.01

impingement dominated

transition

0.1

1 QJ / QP

10

Computational investigation of turbulent jet

300 Exp. 250

K−ω SST

NuR

200

299

150

100

50 rotation dominated 0 0.01

transition

0.1

impingement dominated 1

10

QJ / QP

Conclusions Flow and heat transfer for turbulent jet impinging onto rotating disk have computationally been investigated, using different turbulence models. Co-axial and eccentric arrangements of the jet and the disk have been analyzed, which resulted in 2D axisymmetric and 3D formulations. Results have been compared with the experiments. It has also been demonstrated/confirmed that the low-Re turbulence models resolving the wall-layer are potentially superior to the high-Re models based on wall functions, as this discrepancy has been quantified within the framework of the test cases. The best overall agreement with the experiments has been observed to be provided by the k-v model, where the SST model has been observed to show a similar overall performance for most of the cases. It has been observed that the agreement between the predictions and measurements generally deteriorates for increasing rotational effects, i.e. in rotation dominated regimes and cases, as well as in transitional regimes between impingement and rotation dominated zones. It was initially expected that the RSM would provide a superior performance especially for these cases, since it can much better accommodate for the non-isotropic turbulence structure dominating the rotational flows. The presents results have shown, however, that RSM performed, on the contrary, generally poorer compared to the turbulence viscosity-based models. The investigation of the reasons for this unexpected behavior, which would require a more detailed comparisons with turbulence data, is intended to be performed as part of the future work.

References Barth, T.J. and Jespersen, D. (1989), “The design and application of upwind schemes on unstructured meshes”, AIAA Paper: AIAA-89-0366.

Figure 13. Variation of average disk Nu number with QJ/QP (MG-2)

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Behnia, M., Parneix, S., Shabany, Y. and Durbin, P.A. (1999), “Numerical study of turbulent heat transfer in confined and unconfined impinging jets”, International Journal of Heat and Fluid Flow, Vol. 20, pp. 1-9. Benim, A.C. and Arnal, M. (1994), “A numerical analysis of the labyrinth seal flow”, in Wagner, S., Hirschel, E.H., Pe´riaux, J. and Piva, R. (Eds), Computational Fluid Dynamics’94, Wiley, New York, NY, pp. 839-46.

300

Benim, A.C., Brillert, D. and Cagan, M. (2004), “Investigation into the computational analysis of direct-transfer pre-swirl systems for gas turbine cooling”, ASME Paper: ASME-GT2004-54151. Brodersen, S., Metzger, D.E. and Fernando, H.J.S. (1996), “Flows generated by the impingement of a jet on a rotating surface: Part I – basic flow patterns”, ASME Journal of Fluids Engineering, Vol. 118, pp. 61-7. Cooper, D., Jackson, D.C., Launder, B.E. and Liao, G.X. (1993), “Impinging jet studies for turbulence model assessment – I. Flow-field experiments”, International Journal of Heat and Mass Transfer, Vol. 36, pp. 2675-84. Craft, T.J., Graham, R.H. and Launder, B.E. (1993), “Impinging jet studies for turbulence model assessment – II. An examination of the performance of four turbulence models”, International Journal of Heat and Mass Transfer, Vol. 36, pp. 2685-97. Durbin, P.A. and Reif, B.A.P. (2003), Statistical Theory and Modelling for Turbulent Flows, Wiley, New York, NY. Fluent (2004), “Fluent 6.1”, User Manual, Fluent Inc., Lebanon, NH. Jambunathan, K., Lai, E., Moss, M.A. and Button, B.L. (1992), “A review of heat transfer data for single circular jet impingement”, International Journal of Heat Fluid Flow, Vol. 13, pp. 106-15. Kim, S.E. and Choudhury, D. (1995), “A near wall treatment using wall functions sensitised to pressure gradient”, Separated and Complex Flows ASME FED, Vol. 217. Launder, B.E. and Spalding, D.B. (1974), “The numerical computation of turbulent flows”, Computer Methods in Applied Mechanics and Engineering, Vol. 3, pp. 269-89. Launder, B.E., Reece, G.J. and Rodi, W. (1975), “Progress in the development of a Reynolds-stress turbulence closure”, Journal of Fluid Mechanics, Vol. 68, pp. 537-66. Menter, F.R. (1994), “Two equation eddy-viscosity turbulence models for engineering applications”, AIAA Journal, Vol. 32, pp. 1598-695. Metzger, D.E. and Grochowsky, L.D. (1977), “Heat transfer between an impinging jet and a rotating disk”, ASME Journal of Heat Transfer, Vol. 99, pp. 663-7. Metzger, D.E., Bunker, R.S. and Bosch, G. (1989), “Transient liquid crystal measurement of local heat transfer on a rotating disk with jet impingement”, ASME Paper: ASME-89-GT-287. Minagawa, Y. and Obi, S. (2003), “Turbulence impinging jet onto a co-axial rotating disk”, in Hanjalic, K., Nagano, Y. and Tummers, M.J. (Eds), Turbulence, Heat and Mass Transfer 4, Begell House, New York, NY, pp. 747-54. Owen, J.M. and Rogers, R.H. (1989), Flow and Heat Transfer in Rotating Disk Systems, Vol. I: Rotor-Stator Systems, Wiley, New York, NY. Popiel, C.O. and Boguslawski, L. (1986), “Local heat transfer from a rotating disk in an impinging round jet”, ASME Journal of Heat Transfer, Vol. 108, pp. 357-64. Schlichting, H. (1968), Boundary Layer Theory, 6th ed., McGraw-Hill, New York, NY.

Shih, T.H., Liou, W.W., Shabbir, A., Yang, Z. and Zhu, J. (1995), “A new k-1 eddy-viscosity model for high Reynolds number turbulent flows – model development and validation”, Computers & Fluids, Vol. 24, pp. 227-38. Shima, N. and Launder, B.E. (1989), “Second moment closure for the near-wall sublayer: development and application”, AIAA Journal, Vol. 27, pp. 1319-25. Speziale, C.G., Sarkar, S. and Gatski, T.B. (1991), “Modelling the pressure-strain correlation of turbulence: an invariant dynamical systems approach”, Journal of Fluid Mechanics, Vol. 227, pp. 245-72. Wilcox, D.C. (1993), Turbulence Modeling for CFD, Technical Report, DCW Industries Inc., La Canada, CA. Wolfstein, M. (1969), “The velocity and temperature distribution of one-dimensional flow with turbulence augmentation and pressure gradient”, International Journal of Heat and Mass Transfer, Vol. 12, pp. 301-18. Yakhot, V. and Orszag, S.A. (1986), “Renormalization group analysis of turbulence: I. Basic theory”, Journal of Scientific Computing, Vol. 1, pp. 1-51. Corresponding author A.C. Benim can be contacted at: [email protected]

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Computational investigation of turbulent jet 301

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A computational fluid dynamics analysis of a PEM fuel cell system for power generation

302

Elena Carcadea

Received 1 January 2006 Accepted 9 July 2006

National Research Institute for Isotopic & Cryogenic Technologies, Rm.Valcea, Romania

H. Ene Mathematical Institute, Romanian Academy of Sciences, Bucharest, Romania

D.B. Ingham Department of Applied Mathematics, University of Leeds, Leeds, UK

R. Lazar National Research Institute for Isotopic & Cryogenic Technologies, Rm.Valcea, Romania

L. Ma and M. Pourkashanian Centre for Computational Fluid Dynamics, University of Leeds, Leeds, UK, and

I. Stefanescu National Research Institute for Isotopic & Cryogenic Technologies, Rm.Valcea, Romania Abstract

International Journal of Numerical Methods for Heat & Fluid Flow Vol. 17 No. 3, 2007 pp. 302-312 q Emerald Group Publishing Limited 0961-5539 DOI 10.1108/09615530710730166

Purpose – This paper aims to present a three-dimensional computational fluid dynamics (CFD) model that simulates the fluid flow, species transport and electric current flow in PEM fuel cells. Design/methodology/approach – The model makes use of a general-purpose CFD software as a basic tool incorporating fuel cell specific submodels for multi-component species transport, electrochemical kinetics, water management and electric phase potential analysis in order to simulate various processes that occur in a PEM fuel cell. Findings – Three dimensional results for the flow field, species transport, including waster formations, and electric current distributions are presented for two test flow configurations in the PEM fuel cell. For the two cases presented, reasonable predictions have been obtained, and this provides an insight into the effect of the flow designs to the operation of the fuel cell. Research limitations/implications – It is appreciated that the CFD modeling of fuel cells is, in general, still facing significant challenges due to the limited understanding of the complex physical and chemical processes existing within the fuel cell. The model is now under further development to improve its capabilities and undergoing further validations. Practical implications – The model simulations can provide detailed information on some of the key fluid dynamics, physical and chemical/electro-chemical processes that exist in fuel cells which are crucial for fuel cell design and optimization.

One of the authors, Elena Carcadea would like to express her thanks to the University of Leeds and to the EU Marie Curie Fellowship scheme.

Originality/value – The model can be used to understand the operation of the fuel cell and provide and alternative to experimental investigations in order to improve the performance of the fuel cell. Keywords Fluid dynamics, Fuels, Flow measurement

Computational fluid dynamics

Paper type Research paper

303

Nomenclature a c D I j j0 p R S T u~ V Y

¼ water vapor activity ¼ molar concentration ¼ species diffusivity ¼ current density, A/m2 ¼ current density, A/m3 ¼ exchange current density ¼ pressure, atm ¼ gas constant, 8.314 J/mol K ¼ source terms ¼ temperature ¼ velocity vector, m/s ¼ cell potential, V ¼ species mass fraction

Greek symbols 1 ¼ porosity h ¼ overpotential, V

f r m s l

¼ phase potential, V ¼ density, kg/m3 ¼ viscosity, kgm/s ¼ electric conductivity, S/m ¼ water content of the membrane

Superscript eff ¼ effective value sat ¼ saturation value Subscripts a ¼ anode c ¼ cathode e ¼ electrolyte i ¼ species ref ¼ reference value

1. Introduction The demand for a friendly environment and a reliable power source continues to increase at a rapid rate. One of the emerging technologies in power generation is the fuel cell technology. A fuel cell is an electrochemical device that converts the chemical energy of a fuel into electrical energy through chemical reaction instead of combustion. The high efficiency, high reliability and flexibility and ultra-low emission of the fuel cell makes it a strong candidate as an alternative power generation for both stationery and transportation applications. However, modeling of fuel cells still face substantial challenges and this is primarily due the our limited knowledge on various processes that occur in the fuel cell (Ma et al., 2005). This paper presents a computational fluid dynamics (CFD) model that simulates the fluid flow, species transport and electric current flow in PEM fuel cells. The model simulations can provide detailed information on some of the key fluid dynamics, physical and chemical/electro-chemical processes that exist in fuel cells which are crucial for the optimization of fuel cells. A basic PEM fuel can be made of three components, namely an anode that accommodates the fuel, a cathode that supplies oxidant, and an electrolyte which separates the anode and the cathode and provides a passage for the transport of ions. Both the anode and the cathode have three distinct components: the catalyst layer, the gas diffusion layer and the gas channel (Figure 1). The fuel and the oxidant are distributed through the gas channel to the diffusion layer across the fuel cell. The gas diffusion layer and the catalyst layer are made of porous materials so that the fuel and the oxidant can be further transported from the diffusion layer to the catalyst layer where electro-chemical reactions take place in order to generate electricity.

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Current Collector

304 z

y GCa GDLa CLa M

x

CLc GDLc

Current Collector

Figure 1. The geometry of a fuel cell with straight channels (up), and interdigitated channels (down)

z

y GCa x

GDLa CLa M

CLc GDLc

The purpose of the development of the CFD model is to numerically investigate various major physical and chemical processes that occur in the fuel cell, in particular, the effectiveness of the multi-component transport of the fuel and the oxidant across the fuel cell and its effects on the electrochemical kinetics and the overall performance of the fuel cell. 2. Model description When developing the model, it is assumed that the fuel cell operates under steady and isothermal conditions, the fuel and the air flows can be treated as incompressible and laminar, and the porous media, such as the gas diffusion layer and the catalyst layers, are isotropic and homogeneous. Under these assumptions, the fluid flow, reactant species transport and the electrical potential in the fuel cell can be expressed by following conservative equations (Bernardi and Verbrugge, 1991): 7 · ð1ru~ Þ ¼ 0

ð1Þ

7 · ð1ru~ u~ Þ ¼ 217p þ 7 · ð1m7u~ Þ þ Su
 7Y 7 · ð1u~ Y i Þ ¼ 7 Deff þ Si i i

ð2Þ

  7 · seeff 7fe þ Sf ¼ 0

ð3Þ ð4Þ

where u~ , Yi and fs denote the velocity vector of the fluid flow, the mass fraction of the ith species and the electric potential, respectively, within the fuel cell. For the fuel cell operated with hydrogen and air, typically four species may be considered in the system including hydrogen, H2, oxygen, O2, water, H2O, and nitrogen, N2. The source terms in equations (2 –4) are due to the presence of the porous matrix, the chemical reactions and the ionization process. The matrix of the porous elements of the fuel cell produces a significant resistance to the fluid flow. Typically this effect is modeled by the Darcy Law and this result in a momentum source in the momentum equation (2). Since, the resistance of the matrix is usually the dominant force in the flow system, the convective acceleration and the diffusion terms, appearing in equation (2), are relatively small and thus can often be ignored. However, it should be noted that the use of the Darcy law only has its limitations when modeling fluid flows in multi-component fuel cells when the porosities in different layers vary significantly and the Darcy law does not take into consideration the effect of the boundaries between the layers. The consumption of the fuel and oxidant in the catalyst layers result in a concentration gradient across the fuel cell and the delivery of the reactants to the reaction site significantly rely on the process of species diffusion, which is primarily driven by the concentration gradients. The species diffusion transport is modeled in equation (3) with a species diffusivity coefficient. The presence of the porous matrix has a significant impact on the species diffusion and in this model this is modeled by employing the Bruggemann correction to the mass diffusivity coefficient as follows (Um and Wang, 2000): 1:5 Deff i ¼ 1 Di

ð5Þ

In this model we assume isotropic and homogenous porous media. It should be noted that in practice, due to the nature of the material of which the gas diffusion layer of the PEMFC is made, as well as the manufacturing process taken to make the fuel cell, the matrix is usually not isotropic and the pores are far from being homogenous. Further, since at least three species are present in the system then the species transport in the fuel cell is a multi-component diffusive process in which the flux of one component is influenced by the concentration gradient of other components. The Maxwell-Stefan equations have been used to account for the cross-coupling between multi-species components and the binary mass diffusion coefficient are calculated using the following equation (Fuller and Newman, 1993): N  ›X k X XlXm  i ¼ vl 2 vim D ›x i lm l¼1

ð6Þ

where X is the species mole fraction, v denotes the component of the diffusive velocity of the species and Dlm is the binary diffusivity of any two species l and m. The source term for the phase potential equation (4) only exits for the catalyst layer. This is because the ionization only takes place within the catalyst layer which produces the electric current. The transfer current densities are given by the Butler-Volmer equation (Um and Wang, 2000), as follows:

Computational fluid dynamics

305

HFF 17,3

janode ¼

jref 0;a

jcathode ¼

306

jref 0;c

cH 2 cref H2

!1=2 

cO 2 cref O2

   aa F ac F exp hact;a 2 exp 2 hact;a RT RT

! exp



    aa F ac F hact;c 2 exp 2 hact;c RT RT

ð7Þ

ð8Þ

where aa and ac are the transfer coefficients, hact,a and hact,c are the activation ref over-potential and jref 0;a and j0;c are the reference exchange current density at the anode and cathode side, respectively. The over-potentials should be a function of the transfer current density and in this model they are supplied as inputs into the model which, at the moment have to be obtained experimentally. Water management is a critical issue for the performance of the PEM fuel cell since the polymer membrane must be in a highly hydrated state to facilitate proton transport. Water transport within the polymer membrane is controlled by two, usually opposite processes, namely the electro-osmotic drag and the back diffusion. When the PEMFC is in operation, hydrogen ions moving through the electrolyte from the anode to the cathode pull the water molecules with them. The larger the number of ions transferred then the more water will be dragged along with them. Thus, the water content in the electrode is determined by both the water generation due to the oxygen reduction reaction and the water flux to/from the polymer membrane. In the model presented, the molar flux of water, a, is calculated based on the electro-osmotic drag coefficient nd, water content, l, and water activity, a, and these values are given by the following expressions (Bernardi and Verbrugge, 1991; Fuller and Newman, 1993; Nguyen and White, 1993):

a¼

nd · j F

ð9Þ

nd ¼ 0:0028l þ 0:05l 2 3:5 £ 10219 (

l¼

0:043 þ 17:18a 2 39:85a 2 þ 36:0a 3

for 0 , a # 1

14 þ 1:4ða 2 1Þ

for 1 # a # 3 a¼

X H2 O p p sat

log10 p sat ¼ 22:1794 þ 0:02953ðT 2 273:15Þ 2 9:1837 £ 1025 ðT 2 273:15Þ2 þ 1:4454 £ 1027 ðT 2 273:15Þ3

ð10Þ ð11Þ

ð12Þ

ð13Þ

A summary of the source terms in equations (2)-(4) is given in Table I. In order to solve the governing set of equations (1)-(4) incorporating the supplementary equations (5)-(13), we employed a single computational domain that covers every component of the fuel cell. As a result, no condition at the interface between the different components of the fuel cell is required, except that a special treatment is implemented for equation (4) where a no-flux of electrons has been assumed across the membrane. Boundary conditions are only required at the external surfaces of the fuel cell and typically they

include the fluid flow conditions at the inlets and outlets of the gas channels and the operational potential of the fuel cell. As a preliminary test of the model, we have used a straight channel fuel cell unit (Um and Wang, 2000) (Figure 1). It consists of an anode and a cathode sandwiched with a polymer electrolyte. Within each electrode there are two parallel straight gas channels, a gas diffusion layer and a very thin catalyst layer. Porous graphite has been used to support the electrodes and act as a current collector. In the simulation, two types of flow configurations have been investigated, namely, the straight flow configuration and the interdigitated flow configuration. In both configurations co-flow arrangements have been used, i.e. the fuel and air are fed in from the same end and flow in the same direction in the fuel cell. The interdigitated flow configuration is formed by injecting the air into the bottom channel from one end and forcing it out from the top channel at the other end. The other end of the channels are blocked (Figure 1(b)). Figure 2 shows a typical computational domain that has been employed with a coarse grid consisting of about 106,000 cells. A finer grid has been tested which consists of about 530,000 computational cells and the results obtained are graphically indistinguishable between the two meshes. Since, the Reynolds number of the fluid flow in the fuel cell is very small, a laminar flow is assumed throughout the cell. The phase potential is set to be zero on the anode side and a constant value on the cathode side. The rates of fuel and air flows have been specified at the inlets of the gas channels and at the outlets a pressure condition has been specified. The details of the physical and operational parameters employed are listed in Table II. Governing equations

Volumetric source terms

Momentum transport Hydrogen transport (anode) Oxygen transport (cathode) Water transport (anode) Water transport (cathode) Phase potential

Su ¼ 2ðm=kÞ1u SH2 ¼ 2ðM H2 =2FÞ · janode SO2 ¼ 2ðM O2 =4FÞ · jcathode SH2 O ¼ 2M H2 O · a SH2 O ¼ M H2 O · a þ ðM H2 O =2FÞ · jcathode Sf ¼ 2j

H2

H2

Computational fluid dynamics

307

Table I. Source terms of conservation equations

Air

Air

Figure 2. Computational domain with a coarse grid

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308

Table II. Physical and operating parameters

Property Cell length (cm) Gas channel height (cm) Gas channel width (cm) Current collector width (cm) Anode/cathode GDL thickness (cm) Membrane thickness (cm) Anode catalyst layer thickness Cathode catalyst layer thickness Porosity of GDL Porosity of CL Cell temperature (K) Inlet N2/O2 mole fraction ratio Pressure at the anode/cathode gas channel Velocity(m/s) Mass fraction of H2 Mass fraction of H2O Mass fraction of O2 Inlet temperature (K)

Value 7.112 0.0762 0.0762 0.0762 0.0254 0.0178 0.001 cm 0.001 cm 0.8 0.4 353 0.79/0.21 3/5 (atm) Anode 0.5 0.406 0.594 0 353

Cathode 0.35 0 0.08534 0.23 353

3. Results and discussions Because the length of the cell (in the y-direction) is very large compared with the other dimensions of the cell, we have scaled the y-direction with a factor 0.1 for all the figures presented, in order to achieve a good visualization of the results. Figure 3 shows the velocity contours in two representative planes within the gas channels and across the diffusion layers for both straight and interdigitated configurations. It can be seen that for the straight channel configuration, Figure 3(a), in the anode side the velocity is maximum at the inlet and gradually decreases along the flow channel due to the effect of flowing into the diffusion layer and the consumption of the fuel. A slight increase in the flow speed may be observed near to the exit. In the cathode side, an increase in the gas velocity may be observed due to the formation of water that increases the mass flow rates of the flow. Fluid velocities within the porous regions are generally very small compared to that in the gas channel. Further, significant secondary flows are observed both in the gas channel and across the porous regions. For the interdigitated configuration, Figure 3(b), in the cathode side the fluid velocity are very small at the dead ends of both the upper and the bottom channels while high speed flows appear at the inlet and out let of the channel. The air is forced to penetrate the porous current collector between the lower and the upper channels and this is beneficial to an efficient delivery of the oxygen across the region. Figure 4 shows the distribution of oxygen in the fuel cell in terms of mass fractions. The contours of oxygen mass fractions are shown in a reaction surface within the catalyst layer and two sections across the gas channels in the cathode side for the straight flow configuration, Figure 4(a), and for the interdigitated flow configuration, Figure 4(b). In both configurations, air flow is coming in form the left hand side and discharged from the right hand side of the channel. The concentration of the oxygen is decreasing downstream of the channel for both cases due to

0.83 0.79 0.75 0.71 0.66 0.62 0.58 0.54 0.50 0.46 0.41 0.37 0.33 0.29 0.25 0.21 0.17 0.12 0.08 0.04 0.00

Computational fluid dynamics

309 Y Z

Y X

Z

Y

Y

Z X

Z

(a)

0.30 0.29 0.27 0.25 0.24 0.23 0.21 0.20 0.18 0.17 0.15 0.14 0.12 0.11 0.09 0.08 0.06 0.05 0.03 0.02 0.00

X

X

(b)

Figure 3. Velocity contours above/under current collector for a cell with: (a) straight and (b) interdigitated fuel channels

z X

Y

z Y

X

Figure 4. The O2 mass fraction for a cell with: straight fuel channels (up), interdigitated fuel channels (down)

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310

Figure 5. The phase potential profiles for the cell with: straight fuel channel (up), interdigitated fuel channel (down)

chemical/electrochemical reactions that consume oxygen. In the case of the straight channel, the oxygen decreases monotonically along the gas channels as the electrochemical reaction proceeds and the profile of oxygen concentration shows a symmetrical pattern about the z-axis due to the symmetrical flow arrangement. It is noted that due to the effect of the porous current collector, a low oxygen concentration regions exists underneath the current collector around the axis of symmetry. Under a straight flow configuration, the convective flow is in parallel to the gas channel and the oxygen delivery to the region covered by the current collector is mainly through diffusion processes and this substantially limits the efficient delivery of the reactants to the reaction site. However, this situation may be changed by using the interdigitated flow configuration, Figure 4(b), where the air is forced to flow through the porous current collector before it discharged at the top channel. As we can observe in Figure 4 that the overall distribution of the oxygen in the interdigitated flow configuration is very different from the case in the straight flow configuration. Oxygen concentration in the region underneath the current collector is improved significantly and lowest oxygen concentration is near to the dead end of the bottom air channel. The dead zone formed at the end of the channel limits the transfer of the oxygen to the region. The change in the air delivery between the two flow configurations has a significant impact on the performance of the fuel cell. Figure 5 shows the predicted cathode side electrical potentials in a plane within the catalytic reaction layer. For the straight flow configuration, Figure 5(a), the profile of the phase potential is symmetric and the maximum phase potential occurs at the inlet of the channel, where the highest oxygen concentration exists. Downstream of the channel the potential decreases with the 0.80028 0.80027 0.80025 0.80024 0.80022 0.80021 0.80020 0.80018 0.80017 0.80015 0.80014 0.80013 0.80011 0.80010 0.80008 0.80007 0.80006 0.80004 0.80003 0.80001 0.80000

Z Y

X

Z Y

X

decrease in the oxygen supply in the gas stream. The lowest potential occurs in the region covered by the current collector. In the case of interdigitated flow (Figure 5(b)), oxygen supply in the region improves and thus the phase potential. In general, a relatively higher phase potential is predicted for the interdigitated flow configuration compared to the straight flow configuration. In order to keep the polymer electrolyte active, moisture is usually introduced in the fuel gas flow to compensate the water losses due to the electro-osmotic action in the electrolyte. The predicted water transportation in the anode and the cathode of the fuel cell are shown in Figure 6. In the anode side, the water concentration steadily decreases along the gas flow while in the cathode it increases due to the water production in the reaction and the influx of water from the electrolyte. At the exit of the gas channel on the cathode side of the interdigitated flow configuration, Figure 6(b), the gas flow contains more water than it does in the straight flow configuration, Figure 6(a), showing an improve performance of the fuel cell. It should be noted that in some situations water back diffusion from the cathode to the anode may occur if the water content at the cathode is much higher than it is at the anode side, particularly when a low ionization-efficiency exists downstream of the gas flow channel. It should also be noted that liquid water may form if the water partial pressure is higher than the saturated vapor pressure. The existence of the liquid water in the catalyst and the gas diffusion layers can block the effective reacting surface and the pores of the diffusion layer. In this situation, employing a single-phase flow model will be inaccurate.

Computational fluid dynamics

311

0.594 0.561 0.528 0.495 0.462 0.429 0.396

Z

0.363 Y

X

0.330 0.297 0.264 0.231 0.198 0.165 0.132 0.099 0.066 0.033 0.000

Z Y

X

Figure 6. Water mole fraction profile for a cell with: straight fuel channels (up), interdigitated fuel channels (down)

HFF 17,3

312

4. Conclusions This paper presents results from a three-dimensional, steady state, single-phase model of a PEM fuel cell that has been developed using the fluent CFD software as a basic tool. The fuel cell specific sub-models have been developed which incorporate the electrochemical kinetics and multi-dimensional fluid flow and multi-component species transport. Water management and electric fields under typical PEMFC operation conditions have been simulated. For the two test cases presented, reasonable predictions have been obtained for both the reactant distributions, including water formations across the cell as well as the cell potentials predictions. The model is now under further development to improve its capabilities and under going further validations. It is appreciated that the CFD modeling of fuel cells is, in general, still facing significant challenges due to the limited understanding of the complex physical and chemical processes existing within the fuel cell. However, with the further development of the modeling capabilities, the modeling of fuel cells using CFD techniques can be an important alternative to the experimental measurements in providing information that is critical to the fuel cell design and optimization. References Bernardi, D. and Verbrugge, M.W. (1991), “A mathematical model of the solid-polymer-electrolyte fuel cell”, J. Electrochem. Society, Vol. 139 No. 9, p. 2477. Fuller, T.F. and Newman, J. (1993), “Water and thermal management in solid-polymer-electrolyte fuel cells”, J. Electrochem. Society, Vol. 140, pp. 1218-25. Ma, L., Ingham, D.B., Pourkashanian, M.C. and Carcadea, E. (2005), “Review of the computational fluid dynamics modeling of fuel cells”, Journal of Fuel Cell Science and Technology, Vol. 2 No. 4, pp. 246-57. Nguyen, T.V. and White, R.E. (1993), “A water and thermal management model for proton exchange membrane fuel cells”, J. Electrochem. Society, Vol. 140, pp. 2178-86. Um, S. and Wang, C.Y. (2000), “Computational fluid dynamics modeling of proton exchange-membrane fuel cells”, J. Electrochem. Society, Vol. 147, pp. 4485-93. Corresponding author D.B. Ingham can be contacted at: [email protected]; [email protected]

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Modelling of the thermosolutal convection and macrosegregation in the solidification of an Fe-C binary alloy Z.Q. Han Department of Mechanical Engineering, Tsinghua University, Beijing, China

Thermosolutal convection and macrosegregation 313 Received 1 January 2006 Accepted 9 July 2006

R.W. Lewis School of Engineering, University of Wales Swansea, Singleton Park, Swansea, UK, and

B.C. Liu Department of Mechanical Engineering, Tsinghua University, Beijing, China Abstract Purpose – The motivation for this work is to establish a model that not only includes the main factors resulting in macrosegregation but also retains simplicity and consistency for the sake of potential application in casting practice. Design/methodology/approach – A mathematical model for the numerical simulation of thermosolutal convection and macrosegregation in the solidification of multicomponent alloys is developed, in which the coupled macroscopic mass, momentum, energy and species conservation equations are solved. The conservation equations are discretized by using the control volume-based finite difference method, in which an up-wind scheme is adopted to deal with the convection term. The alternative direction implicit procedure and a line-by-line solver, based on the tri-diagonal matrix algorithm, are employed to iteratively solve the algebraic equations. The velocity-pressure coupling is handled by using the SIMPLE algorithm. Findings – Based on the present study, the liquid flow near the dendritic front is believed to play an important role in large-scale transport of the solute species. The numerical or experimental results in the literatures on the formation of channel segregation, especially those about the location of the initial flow as well as the morphology of the liquidus front, are well supported by the present investigation. Research limitations/implications – The modelling is limited to dealing with the thermosolutal convection of two-dimensional cases. More complicated phenomena (e.g. crystal movement) and 3D geometry should be considered in future research. Practical implications – The present model can be used to analyze the effects of process parameters on macrosegregation and, with further development, could be applied as a useful tool in casting practice. Originality/value – The numerical simulation demonstrates the capability of the model to simulate the thermosolutal convection and macrosegregation in alloy solidification. It also shows that the present model has good application potential in the prediction and control of channel segregation. Keywords Thermodynamics, Convection, Numerical control, Simulation, Solidification, Alloys Paper type Research paper

International Journal of Numerical Methods for Heat & Fluid Flow Vol. 17 No. 3, 2007 pp. 313-321 q Emerald Group Publishing Limited 0961-5539 DOI 10.1108/09615530710730175

HFF 17,3

Nomenclature cp Cl Dl

314

fl fs g K K0 k kp L m p T Tf t

¼ specific heat, J/(kg 8C) ¼ species mass fraction in liquid, percentage ¼ species diffusion coefficient in liquid, m2/s ¼ mass fraction of liquid ¼ mass fraction of solid ¼ gravity acceleration, m/s2 ¼ permeability, m2 ¼ permeability coefficient, m2 ¼ thermal conductivity, W/(m 8C) ¼ equilibrium partition ratio ¼ latent heat, J/kg ¼ liquidus slope, 8C/percentage ¼ pressure, Pa ¼ temperature, 8C ¼ fusion temperature of pure metal, 8C ¼ time, s

u, v ¼ x- and y-component of superficial velocity of liquid, m/s u ¼ superficial velocity of liquid, m/s x, y ¼ coordinates, m r ¼ density, kg/m3 bT ¼ thermal expansion coefficient, 1/8C bC ¼ solutal expansion coefficient, 1/percentage ml ¼ liquid viscosity, Pa s w ¼ general dependent variable G ¼ diffusion coefficient S ¼ source term Superscript i ¼ species Subscript ref ¼ reference

Introduction Macrosegregation is a type of chemical inhomogeneity in solidified castings, which is considered in some particular situations as a serious defect that deteriorates mechanical properties. Thermosolutal convection, induced and driven by uneven distribution of temperature and species in alloy solidification, is closely associated with the formation of macrosegregation. Therefore, the fluid flow, heat and mass transfer in casting solidification have significant impact on the casting quality. Modelling of macrosegregation in castings has more than 40 years of history. At the end of the 1960s, Flemings and Nereo (1967, 1968) and Flemings et al. (1968) proposed the famous local solute redistribution equation (LSRE), which has been considered the beginning of modelling on macrosegregation. In the succeeding 20 years, a combination of the LSRE and Darcy’s law that describes fluid flow in a porous medium was a basic approach for macrosegregation modelling. In the late 1980s, a continuum model (Bennon and Incropera, 1987a, b) and a volume-averaged model (Beckermann and Viskanta, 1988) were proposed, which have been recognized as an important milestone in the history of this field. In 1990s, a volume-averaged two-phase model (Ni and Beckermann, 1991; Schneider and Beckermann, 1995) and a multiscale/multiphase model (Wang and Beckermann, 1996; Beckermann and Wang, 1996) were developed, in which the effect of nucleation, crystal movement, and microscopic interfacial undercooling on macrosegregation were taken into account. Channel segregation (also termed as freckles) is a typical macrosegregation occurring in unidirectional solidification of, for example, nickel-base superalloys (Giamei and Kear, 1970). The formation of this segregation is attributed principally to the thermosolutal double-diffusive convection. Copley et al. (1970) were the first investigators who found the presence and the role of double-diffusive convection in a bottom-chilled NH4Cl-H2O system. Based on direct observation, they concluded that freckles are caused by upward flowing liquid jets that result from a density inversion in the mushy zone. However, Sample and Hellawell (1984) stated that channels

originate, not within the dendritic array at any depth, but immediately ahead of the growth front as a result of perturbation from the less dense boundary layer into the bulk liquid. This conclusion was well supported by the work of Neilson and Incropera (1991) and Felicelli et al. (1991) who carried out numerical simulation on NH4Cl-H2O and Pb-Sn alloys, respectively. In recent years, mathematical models for freckle formation in multi-component alloys have been developed (Schneider et al., 1997; Felicelli et al., 1998). In this paper, a mathematical model for describing the transport phenomena in multi-component alloys is presented. The motivation for this work, as a potential application in casting practice, is to establish a model that not only includes the main factors resulting in macrosegregation but also retains simplicity and consistency. By using the developed model, the solidification of a Fe-C alloy in rectangular domains is simulated, in which two cases are considered: (1) side-cooling boundary condition case, where the distribution of flow vector, temperature and solute concentration is presented; and (2) bottom-cooling condition case, in which the origin and development of channel segregation are discussed. Mathematical model The mathematical model is developed based on mainly the continuum conservation equations proposed by Bennon and Incropera (1987a), and the following assumptions are invoked: . laminar, constant viscosity, Newtonian flow in the liquid phase; . equal and constant phase densities except for variations in the buoyancy terms, and validity of the Boussinesq approximation; . stationary solid without deformation and internal stress; . isotropic permeability in the mushy zone; . negligible flow induced by phase transformation shrinkage; . equal and constant phase specific heat and thermal conductivity; . local thermal and phase equilibrium in the mushy zone; and . negligible species diffusion in the solid phase. Based on these assumptions, the mass, momentum, energy, and species conservation can be described by the following equations: Continuity :

7u ¼ 0

ð1Þ

› ml ›p ðruÞ þ 7ðruuÞ ¼ 7ðml 7uÞ 2 u 2 ð2Þ ›t K ›x 2 3  X
i › ml ›p i ðrvÞþ7ðruvÞ ¼ 7ðml 7vÞ2 v2 þ rg4bT ðT 2T ref Þþ biC C l 2C l;ref 5 ð3Þ ›t K ›y i   › k L ›f s Energy : ðrTÞ þ 7ðruTÞ ¼ 7 ð4Þ 7T þ r ›t cp cp › t Momentum :

Thermosolutal convection and macrosegregation 315

HFF 17,3

Species :

     ›  ›  i rC l þ 7 ruC il ¼ 7 rf l Dil 7C il þ 1 2 kip rf s C il ›t ›t

ð5Þ

In the mushy zone, the energy and species conservation equations are fully coupled via the following equation:

316

ð6Þ T ¼ T f þ Si m i C il In the conservation equations, the superficial velocity of liquid, u, is defined as: ð7Þ u ¼ f l ul where ul is the intrinsic velocity of the liquid. The permeability in the mushy zone is calculated by using the Kozeny-Carman equation: f 3l ð8Þ ð1 2 f l Þ2 where K0 is a parameter depending on the morphology and size of the dendrites. K ¼ K0

Solution methodology The above conservation equations can be written in the following unified form: › ðrfÞ þ 7ðrufÞ ¼ 7ðG7fÞ þ S ð9Þ ›t where w is a general dependent variable, G is the diffusion coefficient, and S is a source term. The unified conservation equation is discretized by using the control volume-based finite difference method, in which an up-wind scheme is adopted to deal with the convection term. A detailed description of the discretization procedure can be found in the literature (Patankar, 1980). The solid fraction field is updated using an approach based on the coupling of temperature and liquid concentrations in the mushy zone, by equation (6). The alternate direction implicit procedure and a line-by-line solver, based on a tri-diagonal matrix algorithm, are employed to iteratively solve the algebraic equations. The velocity-pressure coupling is handled by using the SIMPLE algorithm (Patankar, 1980). Results and discussion The solidification of a Fe-C alloy in rectangular domains is simulated. Two cases, i.e. side- and bottom-cooling boundary conditions are studied. The domain geometry and boundary conditions are shown in Figure 1. In both cases, the initial temperature and carbon concentration of the alloy are 1,5108C and 0.8 per cent (mass fraction), respectively, and the initial state of the alloy is assumed to be quiescent. Non-slip boundary conditions are imposed on all of the domain boundaries. The main parameters used in the simulation are listed in Table I. Case 1. Side-cooling boundary condition In the early cooling stage, no solid phase appears and no solute redistribution happens, thus the solute distribution is uniform in the whole domain. However, the heat dissipation at the side boundary results in unevenness in temperature, and consequently an uneven distribution of liquid density in the considered domain. Figure 2 shows the calculated flow vector and temperature distribution at different times in the early cooling stage. It can be seen that at the beginning, a downward flow appears near the

Thermosolutal convection and macrosegregation 317

h=52W/m2C T=25 C

adiabatic

symmetric line

adiabatic

h=52W/m2C T=25 C

Figure 1. The domain geometry and boundary conditions (a) for Case 1; (b) for Case 2

50×100 mm (b)

250×250 mm (a)

Parameter

Value 3

7.3 £ 103 7.23 £ 102 28.4 2.7 £ 105 6.0 £ 102 4 2.0 £ 102 9 0.3 255.0 1.1 £ 102 2 2.0 £ 102 4 1533.0 5.56 £ 102 11

Density, kg/m Specific heat, J/(kg 8C) Thermal conductivity, W/(m 8C) Latent heat, J/kg Liquid viscosity, Pa s Mass diffusivity in liquid, m2/s Partition ratio Liquidus slope, 8C/percentage Solutal expansion coefficient, 1/percentage Thermal expansion coefficient, 1/8C Fusion temperature of pure iron, 8C Permeability coefficient, m2

20 mm/s

20 mm/s

T 1508.77 1507.55 1506.32 1505.10 1503.87 1502.65 1501.42 1500.20 1498.97 1497.74 1496.52 1495.29 1494.07 1492.84 1491.62

(a) 5s

20 mm/s T 1508.77 1507.55 1506.32 1505.10 1503.87 1502.65 1501.42 1500.20 1498.97 1497.74 1496.52 1495.29 1494.07 1492.84 1491.62

T 1508.77 1507.55 1506.32 1505.10 1503.87 1502.65 1501.42 1500.20 1498.97 1497.74 1496.52 1495.29 1494.07 1492.84 1491.62

(b) 10s

Table I. The parameters used in the numerical simulation

(c) 20s

cooling boundary where the liquid density increases due to heat dissipation, and eddying flows result at the bottom corner of the domain. Then, the liquid flows along the base of the domain. When this flow encounters that coming from the opposite direction, the flow direction is changed and an upward flow is formed at the symmetric line. In the further cooling process, the fluid flow exhibits complex multi-eddy features. The variation of the temperature distribution basically reflects the development of the

Figure 2. The flow vector and temperature distribution in the early cooling stage

HFF 17,3

318

fluid flow. Meanwhile, a low temperature region appears at the bottom corner of the domain. A solid phase appears firstly at the bottom corner of the domain, and a mushy zone forms near the lower part of the side boundary. Owing to the large resistance of the solid dendrites to the fluid flow, the bulk liquid changes its flow direction and starts to move along the newly formed solidifying front (Figure 3). It has also been found that an upward flow occurs in the mushy zone due to the local accumulation of carbon, however, the flow is quite slow as compared with that in the bulk liquid. On the other hand, after commencement of the solidification, the temperature distribution in the bulk liquid tends to become even whereas in the mushy zone a large temperature gradient develops in the normal direction of the isothermals. Figure 4 shows the variation of local averaged solute concentration (solid plus liquid) during the solidification stages. The simulation results show that in the region where the solid phase appears earlier, the local averaged solute concentration is lower than the initial concentration of the alloy. This implies that the rejected solute during solidification is carried away by the inter-dendritic liquid convection and diffusion, thus resulting in a local solute depletion. Additionally, it is seen that the rejected solute is carried into the bulk liquid mainly by the flow near the solidifying front. Therefore, the liquid flow near the dendritic front is believed to play an important role in large-scale transport of the solute species. Case 2. Bottom-cooling boundary condition In this case, the simulation and discussion focus mainly on the origin and development of channel segregation in the unidirectional solidification process. Convection onset and channel origin. Figure 5 shows the calculated velocity vectors and solute concentration in the liquid phase during the early stages of solidification. The solid fraction contours during this stage are shown in Figure 6. It can be seen that 20 mm/s

20 mm/s

T

1507.08 1504.30 1501.52 1498.74 1495.97 1493.19 1490.41 1487.63 1484.85 1482.07 1479.29 1476.52 1473.74 1470.96 1468.18

Figure 3. The flow vector and temperature distribution during solidification process (a) 120s

(a)120s

T

1507.08 1504.30 1501.52 1498.74 1495.97 1493.19 1490.41 1487.63 1484.85 1482.07 1479.29 1476.52 1473.74 1470.96 1468.18

(b) 140s

1507.08 1504.30 1501.52 1498.74 1495.97 1493.19 1490.41 1487.63 1484.85 1482.07 1479.29 1476.52 1473.74 1470.96 1468.18

(c) 160s

CM 0.8061 0.8018 0.7975 0.7932 0.7890 0.7847 0.7804 0.7761 0.7718 0.7675 0.7632 0.7589 0.7546 0.7503 0.7460

CM 0.8061 0.8018 0.7975 0.7932 0.7890 0.7847 0.7804 0.7761 0.7718 0.7675 0.7632 0.7589 0.7546 0.7503 0.7460

CM 0.8061 0.8018 0.7975 0.7932 0.7890 0.7847 0.7804 0.7761 0.7718 0.7675 0.7632 0.7589 0.7546 0.7503 0.7460

Figure 4. The distribution of local averaged solute concentration during solidification process

20 mm/s

T

(b)140s

(c)160s

5mm/s

5mm/s

Thermosolutal convection and macrosegregation

5mm/s

319 CL 0.8071 0.8035 0.8000

(a) 65s

(b) 80s

5 mm/s

(c) 90s

5 mm/s

FRS 0.6582 0.6143 0.5704 0.5265 0.4826 0.4388 0.3949 0.3510 0.3071 0.2633 0.2194 0.1755 0.1316 0.0878 0.0439

(a) 65s

CL 0.8071 0.8035 0.8000

CL 0.8071 0.8035 0.8000

5 mm/s

FRS 0.6582 0.6143 0.5704 0.5265 0.4826 0.4388 0.3949 0.3510 0.3071 0.2633 0.2194 0.1755 0.1316 0.0878 0.0439

(b) 80s

Figure 5. The velocity vector and solute concentration in the liquid phase during the channel origin stage

FRS 0.6582 0.6143 0.5704 0.5265 0.4826 0.4388 0.3949 0.3510 0.3071 0.2633 0.2194 0.1755 0.1316 0.0878 0.0439

(c) 90s

in the early stage of solidification, an uneven distribution of liquid composition develops at the dendritic front, namely, the liquid at a lower position has a higher carbon concentration, which results in an inverse distribution of the liquid density, i.e. the liquid density near the dendritic front is less than that of the bulk liquid. This is really an instable configuration in the system and once the instability increases to a certain level, an onset of upward fluid motion occurs, as shown in Figure 5(b). Further development of the initial fluid motion, after a short time, is shown in Figure 5(c). This simulation result perfectly supports the explanation given by Sample and Hellawell (1984) for the mechanism of channel origin. The initial flow occurs just ahead of the liquidus front, instead of at a deep location in the mushy zone. The simulation results (not shown here due to length limitation) also show that the temperature decreases monotonously from the top to the bottom of the domain, which could counteract in some extent the inverse distribution of liquid density. However, it may finally be seen that the distribution of liquid concentration dominates the process. Thermosolutal convection development and channel segregation. Figure 7 shows the velocity and local averaged concentration during the full development of the channel

Figure 6. The solid fraction contours during the channel origin stage

HFF 17,3

Figure 7. The velocity vector and the local averaged concentration during the channel development stage

(a) 100s

CM 0.8086 0.8069 0.8051 0.8034 0.8016 0.7999 0.7981 0.7963 0.7946 0.7928 0.7911 0.7893 0.7876 0.7858 0.7841

CM 0.8086 0.8069 0.8051 0.8034 0.8016 0.7999 0.7981 0.7963 0.7946 0.7928 0.7911 0.7893 0.7876 0.7858 0.7841

CM 0.8086 0.8069 0.8051 0.8034 0.8016 0.7999 0.7981 0.7963 0.7946 0.7928 0.7911 0.7893 0.7876 0.7858 0.7841

320

5 mm/s

5 mm/s

5 mm/s

(b) 250s

(c) 400s

flows. The developed convection is characterized by fountain-like flow patterns, namely, upward liquid jets at the centre of the channels are surrounded by downward flowing liquid. This feature was observed by Sample and Hellawell (1984) in NH4Cl-H2O system. Furthermore, in the present study, it was found that the liquidus front bends downwards at the locations where the channel flow is fully developed. This feature was also captured by Schneider et al. (1997) and Felicelli et al. (1998) who called these concaves “volcanoes”. In addition, intensive convection only occurs in the region above the liquidus front. It was reported that the liquid velocities in the mushy zone are about 2-3 orders of magnitude lower than those in the bulk liquid (Neilson and Incropera, 1991) due to the large resistance of the solid skeleton to inter-dendritic liquid flow. This is probably the main reason why the channels originate near the liquidus front. Conclusions A mathematical model for describing the fluid flow, heat and mass transfer in the solidification of multicomponent alloys is developed. The solidification of a Fe-C alloy in rectangular domains with side- or bottom-cooling conditions was simulated and discussed, which shows the capability of the model to simulate the thermosolutal convection and macrosegregation in alloy solidification. The modelling of the origin and development of channel segregation shows that the present model has good application potential in predicting and controlling channel segregation. References Beckermann, C. and Viskanta, R. (1988), “Double diffusive convection during dendritic solidification of a binary mixture”, PhysicoChemical Hydrodynamics, Vol. 10, pp. 195-213. Beckermann, C. and Wang, C.Y. (1996), “Equiaxed dendritic solidification with convection: Part III. Comparisons with NH4Cl-H2O experiments”, Metallurgical and Materials Transaction, Vol. 27A, pp. 2784-95. Bennon, W.D. and Incropera, F.P. (1987a), “A continuum model for momentum, heat and species transport in binary solid-liquid phase change systems, Part I. Model formulation”, International Journal of Heat and Mass Transfer, Vol. 30, pp. 2161-70.

Bennon, W.D. and Incropera, F.P. (1987b), “A continuum model for momentum, heat and species transport in binary solid-liquid phase change systems, Part II. Application to solidification in a rectangular cavity”, International Journal of Heat and Mass Transfer, Vol. 30, pp. 2171-87. Copley, S.M., Giamei, A.F., Johnson, S.M. and Hornbecker, M.F. (1970), “The origin of freckles in unidirectionally solidified castings”, Metallurgical Transactions, Vol. 1, pp. 2193-204. Felicelli, S.D., Heinrich, J.C. and Poirier, D.R. (1991), “Simulation of freckles during vertical solidification of binary alloys”, Metallurgical Transactions, Vol. 22B, pp. 847-59. Felicelli, S.D., Poirier, D.R. and Heinrich, J.C. (1998), “Modelling freckle formation in three dimensions during solidification of multicomponent alloys”, Metallurgical and Materials Transactions, Vol. 29B, pp. 847-55. Flemings, M.C. and Nereo, G.E. (1967), “Macrosegregation: Part I”, Transactions of AIME, Vol. 239, pp. 1449-61. Flemings, M.C. and Nereo, G.E. (1968), “Macrosegregation: Part III”, Transactions of AIME, Vol. 242, pp. 50-5. Flemings, M.C., Mehrabian, R. and Nereo, G.E. (1968), “Macrosegregation: Part II”, Transactions of AIME, Vol. 242, pp. 41-9. Giamei, A.F. and Kear, B.H. (1970), “On the nature of freckles in nickel base superalloys”, Metallurgical Transactions, Vol. 1, pp. 2185-92. Neilson, D.G. and Incropera, F.P. (1991), “Unidirectional solidification of a binary alloy and the effects of induced fluid motion”, International Journal of Heat and Mass Transfer, Vol. 34, pp. 1717-32. Ni, J. and Beckermann, C. (1991), “A volume-averaged two-phase model for transport phenomena during solidification”, Metallurgical Transactions, Vol. 22B, pp. 349-61. Patankar, S.V. (1980), Numerical Heat Transfer and Fluid Flow, Hemisphere, New York, NY. Sample, A.K. and Hellawell, A. (1984), “The mechanism of formation and prevention of channel segregation during alloy solidification”, Metallurgical Transactions, Vol. 15A, pp. 2163-73. Schneider, M.C. and Beckermann, C. (1995), “Formation of macrosegregation by multicomponent thermosolutal convection during the solidification of steel”, Metallurgical and Materials Transactions, Vol. 26A, pp. 2373-88. Schneider, M.C., Gu, J.P., Beckermann, C., Boettinger, W.J. and Kattner, U.R. (1997), “Modelling of micro- and macrosegregation and freckle formation in single-crystal nickel-based superalloy directional solidification”, Metallurgical and Materials Transactions, Vol. 28A, pp. 1517-31. Wang, C.Y. and Beckermann, C. (1996a), “Equiaxed dendritic solidification with convection: Part I. Multiscale/multiphase modelling”, Metallurgical and Materials Transactions, Vol. 27A, pp. 2754-64. Wang, C.Y. and Beckermann, C. (1996b), “Equiaxed dendritic solidification with convection: Part II. Numerical simulation for an Al-4wt pct alloy”, Metallurgical and Materials Transaction, Vol. 27A, pp. 2765-83. Corresponding author Z.Q. Han can be contacted at: [email protected]

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Thermosolutal convection and macrosegregation 321

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HFF 17,3

Modeling natural convection with the work of pressure-forces: a thermodynamic necessity

322

M. Pons and P. Le Que´re´ CNRS, Orsay Cedex, France

Received 1 January 2006 Accepted 9 July 2006

Abstract Purpose – This paper aims to present and then resolve the thermodynamic inconsistencies inherent in the usual Boussinesq model, especially with respect to the second law, and to highlight the effects of the correction. Design/methodology/approach – The Boussinesq model (i.e. still assuming fv ¼ 0) is made thermodynamically consistent by maintaining in the heat equation, primarily the work of pressure forces, secondarily the heat generated by viscous friction. Numerically speaking, the modifications are very easy and hardly affect the computing time. However, new non-dimensional parameters arise, especially the non-dimensional adiabatic temperature gradient, f. Findings – There are presented and interpreted results of systematic numerical simulations done for a two-dimensional square differentially-heated cavity filled with air at 300K, with Rayleigh number ranging from 3,000 to 108 and f ranging from 102 3 to 2. All configurations are stationary and the fluid is far from its critical state. Nevertheless, the pressure-work effect (similar to the piston effect) enhances the heat transfer while diminishing the convection intensity. The magnitude of this effect is non-negligible as soon as f reaches 0.02. Practical implications – The domain where the thermodynamic Boussinesq model must be used encompasses configurations relevant to building engineering. Originality/value – Exact second-law analyses can be developed with the so-corrected model. Keywords Thermodynamics, Convection, Heat transfer, Numerical control, Modelling Paper type Research paper

Nomenclature Ar cp g H L NI Nu NW P International Journal of Numerical Methods for Heat & Fluid Flow Vol. 17 No. 3, 2007 pp. 322-332 q Emerald Group Publishing Limited 0961-5539 DOI 10.1108/09615530710730184

¼ aspect ratio of the cavity, H/L ¼ heat capacity (Jkg2 1K2 1) ¼ gravity (ms2 2) ¼ height of the cavity (m) ¼ width of the cavity (m) ¼ number of irreversibility ¼ Nusselt number ¼ non-dimensional work rate ¼ pressure (Pa)

¼ Prandtl number ¼ heat-rate (W) ¼ heat-flux-density vector, (Wm2 2) ¼ heat-rate locally generated by viscous friction (Wm2 3) Ra ¼ Rayleigh number s ¼ specific entropy ( Jkg2 1) t ¼ time (s) T ¼ temperature (K)

Pr Q q qv

This work has been done in the double framework of (1) the PRI CARNOT (Communaute´ d’Analyse et Recherche sur les Nouvelles Orientations de la Thermodynamique) (2) the GAT Baˆtiment, both funded by the Programme Interdisciplinaire E´nergie of CNRS. The numerical calculations were done on the NEC-SX5 of the Institut du De´veloppement et des Ressources en Informatique Scientifique (CNRS-IDRIS) Orsay, France.

u * ¼ specific internal energy (Jkg2 1) u ¼ non-dimensional horizontal component of velocity v ¼ specific volume, 1/r (m3kg2 1) v ¼ velocity vector, (ms2 1) Vz ¼ vertical component of fluid velocity (ms2 1) w ¼ non-dimensional vertical component of velocity x ¼ non-dimensional horizontal position 1 xc ¼ position of the cold wall, ¼ A2 r z ¼ non-dimensional vertical position Greek symbols a ¼ thermal diffusivity (m2s2 1) b ¼ isobaric expansion coefficient (K2 1) DT ¼ temperature difference Th 2 Tc, (K) f ¼ non-dimensional adiabatic temperature gradient

F

u r s S t

¼ heat-rate locally dissipated by friction ¼ non-dimensional temperature ¼ density (m3kg2 1) ¼ local rate of entropy production (WK2 1m2 3) ¼ total rate of entropy production (WK2 1) ¼ non-dimensional time

Subscripts 0 ¼ reference state c ¼ cold side C ¼ Carnot h ¼ hot side m ¼ mechanical energy q ¼ conductive (heat diffusion) v ¼ viscous l ¼ purely conductive system

Introduction More than a century ago, Oberbeck (1879) and Boussinesq (1903) established the famous Oberbeck-Boussinesq equations, which have been very helpful since then for modeling buoyancy-driven natural convection, see for instance Bejan (1984) or Gebhart et al. (1988) among numerous authors. As long as those equations are used for simulations dedicated to comparison with experimental data or comprehensive studies like the benchmark of De Vahl Davis (1983), they surely are pertinent. However, the current studies about natural convection are more and more refined and theoretical, involving second law analyses, stability analyses, or multiple solutions, so that one may wonder whether those equations are still adapted to the intended purposes. Several authors (Tritton, 1988; Gray and Giorgini, 1976; Velarde and Perez-Cordon, 1976) showed that the Boussinesq approximation is valid as long as the temperature difference is sufficiently limited for the fluid density (and the other thermophysical properties) to be assumed as uniform and constant. When this condition is not fulfilled, the problem is said non-Boussinesq; such problems are investigated since the with low-Mach-number models, e.g. by Paolucci (1982) or more recently Vierendeels et al. (2001). The present concern is completely different. It originates in the difference between two entropy balances, that of real natural convection in steady-state, and that of the system simulated with the usual Boussinesq (UB) equations. Indeed, the two systems (real and UB ones) do not have same entropy balances. The consequences of this fundamental thermodynamic inconsistency become visible when the temperature difference is very small, i.e. rather close to thermodynamic equilibrium (Pons and Le Que´re´ 2004, 2005a, b). The present study describes that thermodynamic inconsistency and investigates its consequences. Natural convection and entropy balance Non-dimensional quantities are well-known for energies, e.g. the Nu number is the ratio of the effective heat flux in steady-state or in average and that transferred by the

Modeling natural convection

323

HFF 17,3

purely conductive system (fluid at rest): Nu ¼ Q/Ql. The same purely conductive system, more exactly its entropy production Sl, can also be the reference for entropy balances, thus yielding non-dimensional entropy-productions or changes. This non-dimensionalization straightforwardly leads to an equality that must exist in steady-state (or in average) between the number of total irreversibility (NI ¼ S/Sl) and the Nu number, as demonstrated here under:

324

21 S Q ðT 21 Q c 2 Th Þ ¼ ¼ Nu ¼ 21 Sl Ql ðT c 2 T 21 Q Þ l h

NI ¼

ð1Þ

In thermally-induced natural convection without diffusion (i.e. in pure substances), the sources of irreversibility are heat diffusion by conduction and viscous friction. Non-dimensionalizing the corresponding entropy productions yields the respective numbers of irreversibility NIq and NIv, that must satisfy the following condition: N I ¼ N Iq þ N Iv ¼ Nu

ð2Þ

Thermodynamic inconsistency in the usual Boussinesq equations Combining the Gibbs equation (Tds ¼ du * þ pdv), the entropy balance in its local form [r Ds/Dt ¼ 2 f(T 2 1q) þ s ], and the UB assumptions, i.e. dv ¼ ¼ 0 and (r Du */Dt ¼ 2 fq), results in s ¼ qf(T 2 1). In other words, the thermodynamic system described by the UB equations (and called “the UB system” in the following) recognizes only heat conduction (and not viscous friction) as a source of irreversibility. Indeed, any UB calculation leads to equality between NIq and Nu, where: N Iq

1 ¼ Ar

Z

1

0

1 Nu ¼ 2 Ar

Z

xc

0

Z 0

1




2
2 ›u=›x þ ›u=›z
2 dx dz; 1 þ uDT=T 0



›u ›x



and ð3Þ

dz x¼0

The UB system is thus in contradiction with equation (2). This contradiction with thermodynamics exists for two reasons. The first one, more visible but less significant, is the neglect in the heat equation of the heat generated by viscous friction, while the corresponding loss of kinetic energy is accounted for in the momentum equation. Thermodynamically speaking, in the UB system some kinetic energy (work) is lost in viscous friction but not transformed into heat. Remembering the words of Lavoisier: “Rien ne se perd” (nothing disappears), one deduces that that work lost in friction is necessarily released as work outside the system. Any exact transformation of work into work is a reversible process. This analysis shows that viscous friction does not create entropy in the UB system (when it does in the real world). Notice that losing kinetic energy is not sufficient for creating entropy in a convective system, irreversibility actually consists in the transformation of the lost work into heat, regardless of the magnitude of this heat rate compared to the other ones, conductive or advective. Another feature shows that the UB systems cannot include viscous friction in its entropy balance: the number of viscous irreversibility is given by:

N Iv

1 bgH T 0 ¼ Ar cp DT

Z 1Z 0

0

xc


2
2
2 2 ›u=›x þ ›u=›z þ ›w=›x þ2 ›w=›z
 dx dz; ð4Þ 1 þ uDT=T 0

where appears a parameter, independent from Ar, Pr, and Ra, and which is absent from the UB problem: bgHT 0 =ðcp DT Þ: How could a model lying on those three parameters only describe a phenomenon that involves a fourth one? Tritton (1988) noticed that that fourth parameter is the adiabatic temperature gradient (bgT0/cp) non-dimensionalized in the problem framework (by DT/H). As this fourth parameter is very often mentioned in the following it will be denoted by the symbol f. Bejan (1984), and Gebhart et al. (1988) as well, mention that f might easily be comparable to one, so that the viscous irreversibility “is not necessarily negligible” as written in both textbooks. The second reason for inconsistency between the UB system and thermodynamics is that the usual heat equation does not contain any term involving transformation between heat (internal energy) and work within the fluid. This absence raises the question: where does the UB system find kinetic energy (work) for compensating the continuous viscous loss? The answer is: if not from the fluid itself, then from outside. In other words, the UB system somehow continuously receives mechanical energy from outside in order to compensate a continuous transfer of work (equating the viscous loss) to the outside. After this simple analysis, the energy transfers of the UB system in steady-state can schematically be represented as shown in Figure 1: in addition to the two equal heat fluxes exchanged with the heat sources, this system also exchanges two equal and opposite fluxes of mechanical energy with the surrounding, and there is absolutely no exchange between those two kinds of energy. Is it really natural convection? As a consequence of this whole development the heat equation of the usual Boussinesq model (i.e. DT=Dt ¼ af 2 T ) should be questioned.

Modeling natural convection

325

The thermodynamic Boussinesq model Oberbeck’s and Boussinesq’s approach consisted in deriving approximations of the three basic transport equations (mass, momentum and energy) by neglecting terms of lower orders of magnitude. The same approach is widely used nowadays for building numerical models. About 46 years ago, Spiegel and Veronis (1960) developed +NWm

Nuh

Work

–NWv

Heat

Nuc

Notes: Energy diagram describing thermodynamic system defined by the usual Boussinesq equations in steady-state. In this UB system, which exchanges work with its surrounding, the fluxes of thermal energy (represented at the lower level) and of kinetic energy (upper level) are completely disconnected. Nuh = Nuc, and NWm = NWv, in steady-state

Figure 1.

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326

a different approach: they considered the thermodynamic meaning rather than the orders of magnitude. They obtained what they called “the thermodynamic Boussinesq model” (denoted as TB in the following). This model can be called Boussinesq, because the temperature difference is still assumed small enough for neglecting the changes in fluid density and other thermophysical properties, except for the buoyancy term in the momentum equation. It is a thermodynamic model, because none of the processes, which are intrinsic to natural convection are discarded in the heat equation, especially the work of pressure stress and the heat generated by viscous friction. As a result, the heat equation in its enthalpic form is:   DT qv T › ðr 21 Þ DP 2 ¼ a7 T þ ð5Þ þ Dt cp r cp ›T P Dt The equation of state considered herein for the fluid is the usual one: r ¼ r0[1 2 b(T 2 T0)]. As the present study focuses on steady-states, only the hydrostatic pressure field is considered for the last term on the RHS, which thus finally transforms pffiffiffiffiffiffi into: 2 (bg/cp)TVz. Lastly, taking the cavity height H, the speed V * ¼ ð Raa=H Þ, and DT as respective references for distances, velocity, and temperature difference (T 2 T0), the non-dimensional form of equation (5) is:   Du 1 bgH F 2 ¼ 7 uþ 2 uw 2 fw ð6Þ cp Dt Ra 1=2 Ra 1=2 Note that this equation does involve the parameter f. The other involved parameter, bgH/cp, is very small (of the order of 102 5), while f is “not necessarily negligible”. In addition, when the development from the Gibbs equations to the entropy productions is now derived with equation (5) as heat equation, one correctly finds two causes of irreversibility: heat diffusion and viscous friction. The boundary conditions considered herein are extremely common; fixed temperatures (^0.5) on the vertical walls, adiabatic horizontal walls, no slip on the four walls. Numerical implementation The modification of the heat equation is implemented into an initially usual Boussinesq model already described by Gadoin et al. (2001). The two complementary terms are treated like the other non-linear terms, i.e. implicitly through linear extrapolation. The extra CPU-cost of this modification is absolutely negligible. We investigate herein the square two-dimensional differentially heated cavity filled with air at 300K (Ar ¼ 1, Pr ¼ 0.71). The grid is regular and staggered with a 256 £ 256 mesh (the cases with Ra . 108 are calculated with 512 £ 512). The configurations are stationary (Ra ranges from 3,000 to 108) and f ranges from very small values (f ¼ 102 3, i.e. small cavities where the UB approximation is valid) to large ones (f ¼ 2, for, which the largest value of H is 3.8 m). Thermodynamic balances All the calculations done with the TB model yield Nuh ¼ Nuc, and NWm ¼ NWv (the rates of work or heat are non-dimensionalized by the heat flux of the purely conductive system Ql). In addition, the second law balance now agrees with equation (2): full thermodynamic consistency is obtained.

The work of pressure forces and its effect At given Ra number, the heat flux now depends on f, i.e. on the cavity height. The dependence of Nu on f for 3,000 # Ra # 108 is shown in Figure 2 where the Nu number calculated with the UB model, NuUB, is taken as reference (Table I). It can be seen that the actual Nu number (calculated with the TB model) is always larger than NuUB, the difference can be quite significant when f is of the order of unity. When f is small compared to one, the relative difference is 0.3 £ f (in other words, if f ¼ 0.1, the UB model underestimates the Nu number by 3 percent). The term in DP/Dt in equation (5) describes the work exerted on the fluid by the hydrostatic pressure field. When non-dimensionalized into equation (6), the (by far) main part of that work is 2 fw, where w is the vertical component of velocity. When the fluid flows from regions of high pressure (bottom of the cavity) to regions of low pressure (top of the cavity), this term acts as a heat sink. In other words, when it flows upward close to the hot wall, the fluid generates work (just like in a turbine) and its internal energy is reduced by as much. On the other side of the cavity, close to the cold wall, the fluid flows downward, it receives work (just like in a compressor) and its internal energy increases: the term 2 fw acts as a heat source. Globally, the combination of these two opposite exchanges between heat and work produces a heat transfer by a process, which operates

Modeling natural convection

327

1.6

Nu / NuUB

1.4

1.2

1 0

0.5

1 φ

1.5

2

Figure 2. Dependence of the Nu number on the parameter f for different Ra numbers

Notes: +: 3000; : 104; : 105; O: 106; : 107; ×: 108. The Nu numbers NuUB calculated with the usual Boussinesq model, i.e. with f = 0, are taken as reference (values are given in Table I ). The dashed line corresponds to: (Nu–NuUB) /NuUB = 0.3 × f

Ra

3,000

104

105

106

107

108

Nu

1.504

2.245

4.524

8.841

16.62

30.75

Note: Values of the Nu number calculated with f ¼ 0 (usual Boussinesq model)

Table I.

HFF 17,3

328

in parallel to conduction plus advection. We will simply call this process the pressure-work (PW) effect. This process is similar to the piston effect identified by Onuki et al. (1990) and Zappoli et al. (1990), but somehow different. Indeed, the piston effect is due to fluid expansion and contraction, i.e. volume changes; when the fluid specific volume remains constant, the fluid internal energy changes like cvT. The pressure-work effect is, in steady-state, due to motion in the hydrostatic pressure-field; when the fluid remains at constant pressure, the fluid specific enthalpy changes like cpT. Actually, temperature-, volume- and pressure-changes all three co-exist in natural convection, and so do the piston and pressure-work (PW) effects. The distinction between them is not fundamental but it must be done when they are quantified. The proportion of the total heat transfer, which is due to the PW effect is obtained by subtracting from the total heat flux that due to conduction plus advection (with cp as fluid heat-capacity) through the vertical mid-plane (x ¼ 0.5). Non-dimensionalization yields a Nu number due to the PW effect, NuPW, and the ratio (NuPW/Nu) is the proportion of the Nu number, which is due to PW effect. This proportion is presented as a function of f in Figure 3 for the same Ra numbers as in Figure 2. It can be noticed that as soon as convection is developed (Ra $ 104), all the curves practically merge into a single correlation: f is a pertinent parameter for describing this phenomenon. As expected, the PW effect is quite significant when f is of the order of unity. Velarde and Perez-Cordon (1976) and Gray and Giorgini (1976) had already mention that the UB approximation requires f to be small for being valid. Those authors recommend f to be less than 0.1. However, the contribution of the PW effect to the total heat transfer in cavities has never been calculated before, even for small values of f. Figure 3 shows that this contribution can be very significant and as high as 1.2 £ f when f is small. In other words, when f ¼ 0.1, the PW effect is responsible for 12 percent of the total heat transfer. Compared to the UB system, the work of pressure forces is also responsible for a strong reduction in convection intensity, so that the global changes in Nu number shown in Figure 2 actually result from the combination of a reduced convection with the PW effect. 1

NuPW/Nu

0.8

Figure 3. Contribution of the pressure-work effect to the total heat transfer, as a function of f and for different Ra numbers (same convention as in Figure 2)

0.6

0.4

0.2

0 0

0.5

1 φ

1.5

2

Notes: The dashed line corresponds to: NuPW / Nu = 1.2 × f

From this analysis, the energy transfers occurring in natural convection (and accounted for in the TB equations) can be represented as shown in Figure 4. The exchanges between heat and work occurring within the fluid have a double result: first, a net production of kinetic energy (work) that compensates the loss in viscous friction; second, a heat transfer by the PW effect, which is larger by several orders of magnitude than the net produced work. It must be here emphasized that all those results are obtained for steady-states and a fluid, which is far away from its critical point. The PW effect does not exist only in transient evolutions and in near-critical fluids. This statement also applies to the piston effect.

Modeling natural convection

329

Validity domains for the different models Thermodynamically speaking, there is no good reason for neglecting the work exerted by the hydrostatic pressure field on the flow and the viscous heat generation, because these are intrinsic components of buoyancy-induced natural convection. Nevertheless, validity of the UB approximation is a frequent issue. The validity limit stated by Gray and Giorgini (1976) or Velarde and Perez-Cordon (1976) was f , 0.1. The results presented in the previous section show that when f ¼ 0.1, a process as large as 12 percent of the total heat flux is completely discarded by the UB model. It seems much more reasonable (especially when considering the huge progresses done in computing technologies during the last 30 years) to position the validity limit around f ¼ 0.01 or 0.02. Dimensionless quantities surely are fundamental. It is however interesting to consider the physical meaning of the different limits mentioned above, either between the Boussinesq and non-Boussinesq cases, or between the usual and thermodynamic Boussinesq cases. Figure 5 shows a diagram (DT, H), in log-log axes, established for air at 300K. Each value of Ra corresponds to a line with negative slope (solid lines); each value of f corresponds to a line with positive slope (dashed lines). We have separated the whole domain in three regions. First, on the right, the region of large DT’s, i.e. of non-Boussinesq configurations where Low-Mach-number models must be used. +NWm

−NWv Compression

Pressure reduction

Nuh Th

Nuc Tc

Notes: Energy diagram describing natural convection and the thermodynamic Boussinesq system in steady-state (same conventions as in Figure 1). The large triangles symbolize the internal energy transformations between heat and work induced by the pressure forces

Figure 4.

Non-Boussinesq

rime nts

Bo

10 4

0.1

φ= 10 −6

10 − 5

c mi

ine

uss

yna

Expe

H [m]

Th

10 5

erm

od

1

sq

10 6

330

10 − 4

φ= 2

Ra =1 08 10 7

10 − 3

10 − 2

. 1.

10

0.1

HFF 17,3

Usual Boussinesq

0.01 0.01

Figure 5. Diagram (H, DT) showing the different domains of natural convection

0.1

1 ∆T [K]

10

100

1000

Notes: The lines for prescribed values of Ra and f are established for air at 300K. The region with large ∆T ’s (and – at fixed Ra – relatively small cavities and small f ’s) is the one for Non-Boussinesq cases. The region with f larger than 0.02 (and – at fixed Ra – relatively large cavities and small ∆T ’s) must be studied with the thermodynamic Boussinesq model. In between is the region of validity of the usual Boussinesq model

Second, in the upper left corner, the region of large f’s, i.e. of thermodynamic Boussinesq configurations where the TB model must be used. In between, lies the validity region of the usual Boussinesq model. The open ellipse drawn in the latter region shows where are located typical experimental configurations: cavity size between 10 cm and 2 m; DT between 5 and 20K, i.e. large enough for being controlled (Salat et al., 2004). It can be seen that the corresponding values of f are very weak (102 4-102 3), which means that the PW effect is so small in experiments that it cannot be observed. It also results from this smallness that the UB model has always been well-adapted for comparing numerical calculations to experimental data; indeed, the agreement between the UB model and experiments is generally good. However, this does not prove that the UB model is universal. The non-Boussinesq problems received some attention during the last years in literature. For instance, the open circle drawn in this region (H < 7 cm, DT < 700K, Ra < 107) approximately shows a configuration simulated by Vierendeels et al. (2001). Compared to UB calculations, the non-Boussinesq effects (non-solenoidal flow, non-uniform viscosity) change the Nu number by 2-3 percent. What would be configurations where the pressure-work effect would modify the Nu number as much. The above results (Figure 2) show that such a change in Nu is obtained when f is lies

around 0.06-0.09. Keeping the same Ra number as above (< 107), these values of f correspond to a DT of 0.1-0.15K applied to a 0.9-1 m high cavity. Yes, although the causes are very different, the configuration with a DT of 0.1K applied to a 1 m high enclosure departs from the usual Boussinesq case as much as a DT of 700K applied to a box of 7 cm. This simple example shows that some configurations looking very common however present a cavity height large enough for making the PW effect non negligible. Relatively high cavities can be found in architecture and housing engineering, so that configurations with a DT of 1K applied to a room of some meters (see the thick double-arrow in Figure 5) belong to the TB region. This fact, which is even more true for smaller DT’s, received no attention until now. Conclusions The usual Boussinesq equations do not exactly represent buoyancy-induced natural convection. Indeed, the thermodynamic system actually simulated by those equations exchanges mechanical energy with its surrounding, so that it recognizes only heat diffusion as irreversibility. It results that the entropy balances obtained from UB models are not correct. Thermodynamic consistency is retrieved when both work of pressure forces and heat generated by viscous friction are accounted for in the heat equation: this is the thermodynamic Boussinesq model. Numerically, this correction is very simple and its cost in computing time is negligible. However, it re-introduces into the system a phenomenon, which is intrinsic to buoyancy-induced natural convection but unfortunately discarded by the usual Boussinesq equations: the exchanges between heat and work occurring within the fluid due to flow inside the hydrostatic pressure field. This pressure-work effect, which induces a heat-transfer in parallel to advection þ conduction, is controlled by the adiabatic temperature gradient non-dimensionalized in the problem framework, f ¼ bgHT0/(cpDT). This parameter f is one of the control parameters of natural convection. Systematic numerical calculations done with the thermodynamic Boussinesq model show that the magnitude of the pressure-work effect can be as large as 1.2 £ f, i.e. non negligible as soon as f . 0.01 or 0.02. Moreover, any theoretical study about buoyancy-induced natural convection (second law analyses, very probably stability analyses as well) should be done with the thermodynamic Boussinesq model. References Bejan, A. (1984), Convection Heat Transfer, Wiley, New York, NY. Boussinesq, J. (1903), The´orie Analytique de la Chaleur, Gauthier-Villars, Paris. De Vahl Davis, G. (1983), “Natural convection of air in a square cavity: a bench mark numerical solution”, Int. J. Numer. Meth. Fluids, Vol. 3, pp. 249-64. Gadoin, E., Le Que´re´, P. and Daube, O. (2001), “A general methodology for investigating flow instabilities in complex geometries: application to natural convection in enclosures”, Int. J. Numer. Meth. Fluids, Vol. 37, pp. 175-208. Gebhart, B., Jaluria, Y., Mahajan, R.L. and Sammakia, B. (1988), Buoyancy-induced Flows and Transport, Reference ed., Hemisphere Pub. Corp., New York, NY. Gray, D.D. and Giorgini, A. (1976), “The validity of the Boussinesq approximation for liquids and gases”, Int. J. Heat Mass Transfer, Vol. 19, pp. 545-51.

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Oberbeck, A. (1879), “U¨ber die Wa¨rmeleitung der Flu¨ssigkeiten bei Beru¨cksichtigung der Stro¨mungen infloge von Temperatur Differenzen”, Ann. Phys. Chem., Vol. 7, pp. 271-92. Onuki, A., Hong, H. and Ferrell, R.A. (1990), “Fast adiabatic equilibration in a single-component fluid near the liquid-vapor critical point”, Phys. Rev. A, Vol. 41, pp. 2256-9. Paolucci, S. (1982), On the Filtering of Sound from the Navier-Stokes Equations, SAND82-8257, Sandia National Laboratories, Livermore, CA. Pons, M. and Le Que´re´, P. (2004) in Gobin, D. et al. (Eds), Les e´quations de Boussinesq et le second principe, Socie´te´ Franc¸aise de Thermique, Paris, pp. 229-34. Pons, M. and Le Que´re´, P. (2005a), “An example of entropy balance in natural convection, Part 1: the usual Boussinesq equations”, Comptes Rendus Mecanique, Vol. 333, pp. 127-32. Pons, M. and Le Que´re´, P. (2005b), “An example of entropy balance in natural convection, Part 2: the thermodynamic Boussinesq equations”, Comptes Rendus Mecanique, Vol. 333, pp. 133-8. Salat, J., Xin, S., Joubert, P., Sergent, A., Penot, F. and Le Que´re´, P. (2004), “Experimental and numerical investigation of natural convection in a large air-filled turbulent cavity”, Int. J. Heat Fluid Flow, Vol. 25, pp. 824-32. Spiegel, E.A. and Veronis, G. (1960), “On the Boussinesq approximation for a compressible fluid”, Astrophys. J., Vol. 131, pp. 442-7. Tritton, D.J. (1988), Physical Fluid Dynamics, 2nd ed., Oxford University Press, Oxford. Velarde, M.G. and Perez-Cordon, R. (1976), “On the (non-linear) foundations of Boussinesq approximation applicable to a thin layer of fluid. II. Viscous dissipation and large cell gap effects”, J. de Physique, Vol. 37, pp. 177-82. Vierendeels, J., Merci, B. and Dick, E. (2001), “Numerical study of natural convective heat transfer with large temperature differences”, Int. J. Numer. Meth. Heat Fluid Flow, Vol. 11, pp. 329-41. Zappoli, B., Bailly, D., Garrabos, Y., Le Neindre, B., Guenoun, P. and Beysens, D. (1990), “Anomalous heat transport by the piston effect in supercritical fluids under zero gravity”, Phys. Rev. A, Vol. 41, pp. 2264-7. Corresponding author M. Pons can be contacted at: [email protected]

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Buoyancy effects on upward and downward laminar mixed convection heat and mass transfer in a vertical channel

Heat and mass transfer in a vertical channel

Youssef Azizi and Brahim Benhamou

Received 1 January 2006 Accepted 9 July 2006

LMFE, Physics Department, Faculty of Sciences Semlalia, Marrakech, Morocco

333

Nicolas Galanis THERMAUS, De´partement de ge´nie me´canique, Universite´ de Sherbrooke, Que´bec, Canada, and

Mohammed El-Ganaoui SPCTS, De´partement de Physique, Faculte´ des Sciences et Techniques, Limoges, France Abstract Purpose – The objective of the present study is to investigate numerically the effects of thermal and buoyancy forces on both upward flow (UF) and downward flow (DF) of air in a vertical parallel-plates channel. The plates are wetted by a thin liquid water film and maintained at a constant temperature lower than that of the air entering the channel. Design/methodology/approach – The solution of the elliptical PDE modeling the flow field is based on the finite volume method. Findings – Results show that buoyancy forces have an important effect on heat and mass transfers. Cases with evaporation and condensation have been investigated for both UF and DF. It has been established that the heat transfer associated with these phase changes (i.e. latent heat transfer) may be more or less important compared with sensible heat transfer. The importance of these transfers depends on the temperature and humidity conditions. On the other hand, flow reversal has been predicted for an UF with a relatively high temperature difference between the incoming air and the walls. Originality/value – Contrary to most studies in channel heat and mass transfer with phase change, the mathematical model considers the full elliptical Navier-Stokes equations. This allows one to compute situations of flow reversal. Keywords Heat transfer, Convection, Flow measurement Paper type Research paper

Nomenclature b C D Dh f g

¼ channel width (m) ¼ dimensionless mass fraction, ¼ (w 2 w0)/(ww 2 w0) ¼ mass diffusion coefficient (m s2 2) ¼ hydraulic diameter, ¼ 2b (m) ¼ friction factor ¼ gravitational acceleration (m s2 2)

Grashof number, GrM ¼ solutal ¼ gb *D3h(ww 2 w0)/n 2 GrT ¼ thermal Grashof number, ¼ gbD3h(Tw 2 T0)/n 2 þ Gr ¼ effective Grashof number, ¼ GrT þ GrM hfg ¼ latent heat of vaporization/

International Journal of Numerical Methods for Heat & Fluid Flow Vol. 17 No. 3, 2007 pp. 333-353 q Emerald Group Publishing Limited 0961-5539 DOI 10.1108/09615530710730193

HFF 17,3

334

condensation (J kg2 1) ¼ thermal conductivity (W m2 1 K2 1) ¼ channel height (m) ¼ molecular mass of air (kg kmol2 1) ¼ molecular mass of water vapor (kg kmol2 1) NuS ¼ local Nusselt number for sensible heat transfer NuL ¼ local Nusselt number for latent heat transfer Pm ¼ dimensionless pressure, ¼ ð p 2 r0 gxÞ =r0 U 20 Pr ¼ Prandtl number q ¼ heat flux (W m2 2) Re ¼ Reynolds number, ¼ U0Dh/n Sc ¼ Schmidt number, ¼ n/D Sh ¼ Sherwood number T ¼ temperature (K) U, V ¼ dimensionless velocity components Ve ¼ dimensionless transverse vapor velocity W ¼ mass fraction ((kg of vapor) (kg of

k L Ma Mv

mixture)2 1) x, y ¼ axial and transverse co-ordinates (m) X, Y ¼ dimensionless co-ordinates Greek symbols b ¼ coefficient of thermal expansion, ¼ 1/T0 (K2 1) b * ¼ coefficient of mass fraction expansion, ¼ Ma/Mv 2 1 g ¼ aspect ratio of the channel, ¼ b/L u ¼ dimensionless temperature, ¼ (T 2 T0)/(Tw 2 T0) n ¼ kinematic viscosity (m2 s2 1) r ¼ density (kg m2 3) f ¼ relative humidity (%) Subscripts ¼ at the inlet ¼ mean value m ¼ at the wall w ¼ relative to latent heat transfer L ¼ relative to sensible heat transfer S 0

1. Introduction Engineering applications of confined flows with simultaneous thermal and mass diffusion are many. Cooling of electronic equipment, desalination, cooling towers and air conditioning (i.e. evaporative cooling) are some of these applications. Many analyses of combined heat and mass transfer convection in channels are available in the literature. Nelson and Wood (1989) studied a developing laminar natural convection flow in a vertical parallel-plates channel. They used a boundary-layer approximation model to derive a correlation for heat and mass transfer coefficients in the case of uniform temperature and concentration plates. Yan and Lin (1990) analyzed the effect of latent heat transfer of finite liquid film wetting the channel walls on laminar natural convection heat and mass transfer. They considered uniform temperature walls wetted with ethanol or water films. Their numerical study concluded that the assumption of extremely thin film is valid when the liquid mass flow rate is small. The same result was obtained for downward laminar mixed convection in the case of uniform wall heat flux by Yan (1992) and in the case of uniform wall temperature by Yan (1993). An interesting conclusion of the former study is that, under certain conditions, the upward buoyancy forces may result in flow reversal of the gas stream. Owing to his mathematical model of boundary-layer type, the author did not investigate this situation of flow reversal. Desrayaud and Lauriat (2001) have examined the condensation of laminar natural convection flow of humid air in a vertical parallel-plates channel. The plates are maintained at a temperature of 58C and the airflow inlet temperature was 25 , 358C. Their numerical results show the conditions for condensation occurrence. Condensation has also been studied by Yan and Lin (2001) in the case of laminar natural convection in vertical annuli with asymmetric isothermal and wetted walls.

The authors concluded that large amount of water evaporation was obtained with higher wall temperature, but this also results in condensation closer to the inlet at the wall whose temperature is low. Salah El Din (2003) has performed a numerical investigation of the thermal and buoyancy effects on mixed convection in a vertical channel with uniform wall heat and mass fluxes. It is interesting to mention the experimental work of Yan et al. (1995) on the evaporative cooling of the falling liquid film through interfacial heat and mass transfer in a vertical channel. Boulama and Galanis (2004) have obtained an analytical solution for upward, fully developed, laminar, mixed convection heat and mass transfer in a vertical channel. Recently, Jang et al. (2005) have conducted a numerical study on mixed convection heat and mass transfer with evaporation in an inclined square duct. They have established that heat and mass transfer related with film evaporation enhance transfer rates for systems with a lower humidity or a higher wetted wall temperature. Most of the cited studies considered situations where both heat and mass were added to the air stream, thus the main objective was liquid film cooling. Furthermore, numerical studies neglected axial diffusion of heat, momentum and chemical species. This approach results in a set of parabolic equations resolved by the marching technique (Patankar, 1980). On the other hand, the axially parabolic model is less accurate when the buoyancy forces are important and it cannot predict flow reversal. In the present study, the adopted model includes the axial diffusion terms and, therefore, the equations are elliptical in all directions. Moreover, the conditions under investigation correspond to those encountered in air humidifier, humidification-dehumidification desalination systems or air evaporative coolers, where hot air is in direct contact with cold water. In a previous paper (Ait Hammou et al., 2004), the effects of simultaneous cooling and mass transfer on the downward laminar flow of humid air in an isothermal vertical channel with wet walls have been studied. It was established that, depending on the inlet conditions of the flowing humid air, mass transfer may result in film evaporation or condensation of water vapor. On the other hand, it has been shown that thermal and solutal buoyancy forces have significant effects on flow characteristics. The main objective of the present study is to investigate the effect of thermal and buoyancy forces on both upward flow (UF) and downward flow (DF). The former is of interest in counter-flow air humidifiers. This paper is an extension of a recent conference communication by the authors (2005). 2. Problem formulation Consider a vertical channel formed by two parallel plates (Figure 1). Air enters the channel with uniform velocity U0, temperature T0, and relative humidity f0. The plates are maintained at the same constant temperature and wetted by thin liquid water films. Here, we consider that these films are extremely thin so that they can be treated as a boundary condition (Yan and Lin, 1990; Yan, 1992, 1993). Steady state conditions are considered and the flow is supposed to be laminar. Viscous dissipation, radiation heat transfer and other secondary effects (such as pressure work, energy transport by the inter-diffusion of species, Dufour and Soret effects) are negligible (Gebhart and Pera, 1971). Finally, the physical properties are taken to be constant except for the density in the body forces which is considered to be a linear function of temperature and mass fraction (Boussinesq approximation):

Heat and mass transfer in a vertical channel 335

HFF 17,3

(U0, T0, w0) Air

y Liquid water film

336

Tw

Tw

ww

ww

L

b

Figure 1. Schematic representation of the physical system (g ¼ þg x in the case of a DF and g ¼ 2g x in the case of an UF) x

r ¼ r0 ½1 2 bðT 2 T 0 Þ 2 b * ðw 2 w0 Þ


ð1Þ

The following reference quantities are used for non-dimensionalisation: L for linear dimensions, U0 for the velocity components, Tw 2 T0 for the temperature difference T 2 T0 and ww 2 w0 for the mass fraction difference w 2 w0. With this formulation and the above assumptions, the governing equations of the problem can be written. Continuity equation:

›U ›V þ ¼0 ›X ›Y Streamwise momentum equation:     ›U ›U ›P m 2 g ›2 U › 2 U 1 U þV þ þ ðGr T u þ Gr M CÞ ^ ¼2 2 2 2gRe 2 ›X ›Y ›X Re ›X ›Y Spanwise momentum equation:     ›V ›V ›P m 2g ›2 V ›2 V þV þ þ U ¼2 ›X ›Y ›Y Re ›X 2 ›Y 2

ð2Þ

ð3Þ

ð4Þ

Energy equation:  U

›u ›u þV ›X ›Y

 ¼

2g Pr Re



›2 u ›2 u þ ›X 2 ›Y 2

 ð5Þ

Heat and mass transfer in a vertical channel

Species equation:  2    ›C ›C 2g › C ›2 C þV þ U ¼ ›X ›Y Sc Re ›X 2 ›Y 2

337 ð6Þ

Note that in equation (3), the plus sign in front of the buoyancy term corresponds to the case of the UF and the minus sign to the case of a DF. These equations show that six dimensionless groups define the problem: g, Sc, Pr, Re, GrT and GrM. Instead of the Grashoff number, it is possible to consider the Richardson number, which indicates the relative intensity of the buoyancy force compared with the inertia force. Two Richardson numbers are to be considered: the thermal Richardson number RiT ¼ GrT/Re 2 and the solutal Richardson number RiM ¼ GrM/Re 2. The boundary conditions for the problem under consideration are: At the inlet ðX ¼ 0Þ : U ¼ 1 and V ¼ C ¼ u ¼ 0

ð7Þ

›U ›V ›u ›w ¼ ¼ ¼ ¼0 ›X ›X ›X ›X

ð8Þ

At the outlet ðX ¼ 1Þ :

At the walls ðY ¼ 0 and Y ¼ gÞ : U ¼ 0; V ¼ ^V e ; C ¼ u ¼ 1

ð9Þ

where the non-dimensional transverse velocity at the interface is (Bumeister, 1993):     22g ›C Ve ¼ ð10Þ ðww 2 w0 Þð1 2 ww Þ21 Re Sc ›Y Y ¼0 The mass fraction at the wall ww corresponding to the saturation conditions at Tw, is calculated by assuming that air-vapor mixture is an ideal gas mixture. Heat transfer between the wet wall and the humid air is the sum of a sensible and a latent component (Lee et al., 1997):     ›T ›w 2rDhfg ð1 2 ww Þ21 ð11Þ qT ¼ qS þ qL ¼ 2k ›y y¼0 ›y y¼0 Therefore, the Nusselt number: NuT ¼

hDh qt D h ¼ ¼ NuS þ NuL k kðT w 2 T m Þ

ð12Þ

where: NuS ¼ 22gð1 2 um Þ

21



›u ›Y

 ð13aÞ Y ¼0

HFF 17,3

NuL ¼ 22gSð1 2 um Þ21



›C ›Y

 ð13bÞ Y ¼0

and:

338

S¼

rDhfg ðww 2 w0 Þð1 2 ww Þ21 ðT w 2 T 0 Þ21 k

ð13cÞ

S represents the relative importance of energy transport through species diffusion to that through thermal diffusion. The Sherwood number characterizes mass transfer at the interface:   hM D h 21 ›C Sh ¼ ¼ 22gð1 2 C m Þ ð14Þ D ›Y Y ¼0 while the friction factor is:   ›U fRe ¼ 4g ›Y Y ¼0

ð15Þ

3. Numerical solution The solution of the PDEs modeling the flow field is based on the finite volume method. The velocity-pressure coupling is treated with the SIMPLER algorithm (Patankar, 1980). Convergence of this iterative procedure is declared when the relative variation of any dependent variable is less than 102 4 and if the mass source residual falls below 102 6 at all the grid points. The grid is non-uniform in both the streamwise and transverse directions with greater node density near the inlet and the walls where the gradients are expected to be more significant. To check the adequacy of the numerical scheme and the developed code the results for the case of forced heat convection were first obtained. Good agreement was found in comparison with results from the literature (Shah and London, 1978). In addition, excellent agreement was found between the present calculations and those of Yan and Lin (1989) for mixed convection heat and mass transfer. Furthermore, different grid sizes were considered to ensure that the solution was grid-independent. The details of these validations and grid sensibility study are presented in a previous work (Ait Hammou et al., 2004). 4. Results and discussion The thermophysical properties are taken to be constant and evaluated at a temperature and a concentration given by the one-third rule. This special way of evaluating the properties has been found to be appropriate for the analysis of heat and mass transfer problems (Chow and Chung, 1983; Lin et al., 1988). Indeed, Chow and Chung (1983) have shown that the one-third rule works well, even at high temperature, when the stream is mostly air. The properties of air, water and their mixture are evaluated by formulas given by Fuji et al. (1977). All the results presented here have been calculated with an aspect ratio L/2b ¼ 65.

The two Grashof numbers in equation (3) are not independent since ww is related to Tw. So we choice T0 and f0 as independent variables instead of GrT and GrM. Since, we are interested in studying the effect of the inlet conditions on the flow field, we have fixed the following conditions: Pr ¼ 0:7; Sc ¼ 0:58; Re ¼ 300; T w ¼ 208C ðhence ww ¼ 14:5 g kg21 Þ

ð16Þ

Heat and mass transfer in a vertical channel 339

In light of practical situations, the selected combinations of T0 and f0 are specified in Table I. The corresponding values of the two Grashof numbers and other relevant parameters are also indicated. Negative Grashof numbers indicate that the corresponding buoyancy force acts in the direction of gravity and so aids the entering DF or opposes the entering UF. Note that the direction of the thermal buoyancy force is the same for all the studied cases (GrT , 0). On the other hand, the direction of the solutal buoyancy force changes: it acts in the direction of gravity in cases 3 and 5 (GrM , 0) and in the opposite direction otherwise. Furthermore, the values of the thermal Richardson number are more important than those of the solutal Richardson numbers, indicating that the dominant buoyancy force here is the thermal one. For each of the combinations of T0 and f0 in Table I, the system of PDEs has been solved twice: once in the case of an UF and a second time in the case of a DF. The case of pure forced convection (GrM ¼ GrT ¼ 0) has also been considered for comparison purposes. By comparing the results of these numerical experiments it is possible to identify the effects of the buoyancy forces on the flow field and heat-mass transfers. Figure 2(a) and (b) show the effect of T0 and f0 on the axial evolution of the average air temperature, Tm, for DF and UF. We notice that the air is being cooled in all cases. In Figure 2(a), it should be noted that, at a given axial position and given T0, Tm is lower in the case of a DF. Thus, in this kind of flow, air is being cooled more rapidly and this tendency is increased for higher values of T0. Indeed, buoyancy forces are essentially aiding for a DF. These forces accelerate the flow and this results in an increasing of heat transfer between air and channel walls. In the case of an UF, buoyancy forces are essentially opposing and so the flow is decelerated and heat transfer decreases. A comparison of Figure 2(a) and (b) shows that the effect of T0 is more important than that of f0. This is due to the fact that the buoyancy force induced by mass diffusion is less important than that induced by thermal diffusion (Table I). The axial evolution of the average vapor mass fraction wm is shown in Figure 3(a) and 3(b). In cases 1 and 4 wm increases with x, while it decreases in cases 3 and 5.

Case No. 1 2 3 4 5

T0 (8C)

f0 (percent)

w0 (g kg2 1)

GrT

GrM

RiT

RiM

40 40 40 45 70

10 30 50 10 10

4.6 13.9 23.3 6.0 19.6

274,576 274,717 274,860 289,975 2147,610

7,142 559 26,123 6,042 23,045

20.8 20.8 20.8 21.0 21.6

0.08 0.01 2 0.07 0.07 2 0.03

Table I. Values of parameters for the studied cases

HFF 17,3

12 DF, T0= 40˚C #1 DF, T0= 45˚C #4 DF, T0= 70˚C #5 UF, T0= 40˚C #1 UF, T0= 45˚C #4 UF, T0= 40˚C #5 Gr = 0

11

340

Sh

10

9

8

7

6

0

0.25

0.5 X (a)

0.75

1

9 DF, φ0=10% #1 DF, φ0=30% #2 DF, φ0= 50% #3 UF, φ0= 10% #1 UF, φ0= 30% #2 UF, φ0= 50% #3 GR=0

8.5

Sh

8

7.5

Figure 2. Axial evolution of the average air temperature: (a) f0 ¼ 10 percent; (b) T0 ¼ 408C; DF, downward flow; UF, upward flow

7

0

0.25

0.5 X (b)

0.75

1

Heat and mass transfer in a vertical channel

40 DF, T0= 40˚C #1 DF, T0= 45˚C #4 DF, T0= 70˚C #5 UF, T0= 40˚C #1 UF, T0= 45˚C #4 UF, T0= 70˚C #5

341

NuT

20

0

–20 0

0.25

0.5 X

0.75

1

(a) 40 DF, φ0=10% #1 DF, φ0=30% #2 DF, φ0=50% #3 UF, φ0=10% #1 UF, φ0=30% #2 UF, φ0=50% #3

NuT

20

0

–20

0

0.25

0.5 X (b)

0.75

1

Figure 3. Axial evolution of the average mass fraction: (a) f0 ¼ 10 percent; (b) T0 ¼ 408C; DF, downward flow; UF, upward flow

HFF 17,3

342

In case 2, wm stays essentially constant. In all cases, its value at the channel outlet tends towards the imposed wall condition (ww ¼ 14.5 g kg2 1). The increase of wm indicates that water vapor is transferred from the water film to the airflow. So evaporation of water film occurs in cases 1 and 4. While the decrease of wm indicates that water vapor is transferred from the airflow to the water film. So condensation of water vapor takes place in cases 3 and 5. These results are consistent with the relative values of w0 and ww: when ww . w0 (as in cases 1 and 4) vapor flows from the liquid film to the air stream, otherwise (ww , w0 as in cases 3 and 5) it flows in the opposite direction. Finally, when ww < w0 (case 2) mass transfer between the wall and the fluid is negligible and wm remains essentially constant. In cases where evaporation occurs (1 and 4, Figure 3(a)), it is noticed that at a given axial position in the entrance region, ww is slightly more important for a DF in comparison with the upward one. On the other hand, in the cases with condensation (3 and 5, Figure 3), at a given axial position ww is less important for a DF in comparison with the upward one. So, in both cases, the values of ww for DF approach the final equilibrium condition earlier than for UF. This is due to the fact that the combined buoyancy force, which acts downwards in all five cases (since GrT þ GrM is always negative), accelerates the fluid near the wall for DF and decelerates it for UF. Therefore, near the wall more water vapor is convected downstream in the case of DF and, as a result, the corresponding dimensionless mass fraction is always lower than for UF. Hence, in light of the results in Figures 2 and 3 it is clear that air is being cooled, whether with evaporation (cases 1 and 4), with condensation (cases 3 and 5) or without mass transfer (case 2). It is believed that this situation is due to the specific conditions of the present problem, i.e. to the fact that sensible heat transfer between the warm air and the cool walls is more important than the latent heat associated with the phase change. This point will be more discussed hereafter. A recent study conducted by Marmouch et al. (2005) reveals a similar conclusion for analogous conditions. This study concerns heat and mass transfer in a humidifier consisting of a channel whose walls are maintained humid by a water film. It is based on one-dimensional model and takes into account the water film thickness. The axial evolution of the friction factor is shown in Figure 4. For comparison purposes, the case of pure forced convection (GrT ¼ GrM ¼ 0) is also reported. We notice an important effect of buoyancy forces on this parameter. Also the effect of air hygrometry at the entrance is noticeable, especially the effect of T0 (Figure 4(a)). The values of fRe for a DF are higher than those for pure forced convection flow. The opposite is true for an UF. These effects can be explained by examining the axial velocity profiles shown in Figure 5. This figure reveals that the axial velocity profiles are significantly modified by natural convection (i.e. buoyancy forces). The aiding buoyancy forces near the channel walls accelerate the downward airflow. This results in an increase of the velocity gradient at the wall and of the corresponding wall shear stress and friction factor (Figure 4(a)). This increase is more important as T0 augments. Indeed, the magnitude of the buoyancy forces increases with T0. This magnitude, given by the effective Grashof number (Gr þ ¼ GrT þ GrM), is about 7 £ 104 in case 1 and 15 £ 104 in case 5 (Table I). It should be noted that the acceleration of the airflow near the channel walls is counterbalanced by a deceleration at the channel axis, so that the maximum velocity does not occur at the channel mid-point for a DF (Figure 5(a)). On the other hand, the opposing buoyancy

Heat and mass transfer in a vertical channel

12 DF, T0= 40˚C #1 DF, T0= 45˚C #4 DF, T0= 70˚C #5 UF, T0= 40˚C #1 UF, T0= 45˚C #4 UF, T0= 70˚C #5 Gr=0

11

343

Nus

10

9

8

7

6

0

0.25

0.5 X (a)

0.75

1

9 DF, φ0= 10% #1 DF, φ0= 30% #2 DF, φ0= 50% #3 UF, φ0= 10% #1 UF, φ0= 30% #2 UF, φ0= 50% #3 Gr=0

Nus

8.5

8

7.5

7

0

0.25

0.5 X (b)

0.75

1

Figure 4. Axial evolution of the friction factor: (a) f0 ¼ 10 percent; (b) T0 ¼ 408C; DF, downward flow; UF, upward flow

HFF 17,3

30 DF, T0= 40˚C #1 DF, T0= 45˚C #4 DF, T0= 70˚C #5 UF, T0= 40˚C #1 UF, T0= 45˚C #4 UF, T0= 70˚C #5

20

344 Nul

10

0

–10

–20

–30

0

0.25

0.5 X (a)

0.75

1

30 DF, φ0= 10% #1 DF, φ0= 30% #2 DF, φ0= 50% #3 UF, φ0= 10% #1 UF, φ0= 30% #2 UF, φ0= 50% #3

20

Nul

1

0

–10

–20

Figure 5. Velocity profiles at X ¼ 0.082: (a) f0 ¼ 10 percent; (b) T0 ¼ 408C; DF, downward flow; UF, upward flow

–30

0

0.25

0.5 X (b)

0.75

1

forces near the channel walls decelerate the UF. This results in a significant decrease of fRe (Figure 4(a)). A careful check of this figure reveals that in case 5 fRe becomes negative at some axial locations. The represented velocity profiles in Figure 5(a) correspond to an axial position with fRe , 0. These negative values of the friction factor indicate that flow reversal occurs near the channel walls. The opposing buoyancy forces induce this flow reversal. In case 5, these forces have an important magnitude (Gr þ ¼ 15 £ 104) so that they overcome inertia forces; this results in an axial velocity in the opposite direction of the main upward airflow (negative U in Figure 5(a)). The zone where negative values of U prevail constitutes a recirculation cell. The dimensions of this cell can be appreciated in Figure 4(a). The minimum of fRe in this figure corresponds to the maximum magnitude of buoyancy forces. As the airflow moves beyond this axial position, the buoyancy force becomes weaker. Thus, the inertia force pushes the air upwards and its axial velocity becomes positive, so the recirculation cell switches off. It is important to note that Yan and Lin (1990) have mentioned this phenomenon of flow reversal in a problem similar to the present one. However, he was not able to compute it because of his boundary layer type model. Also, we mention the study of Salah El-Din (1992) and Boulama and Galanis (2004). These authors derived a criterion for flow reversal by means of an analytical study of fully developed heat and mass transfer in a vertical channel. As far as we can ascertain, the present study is the first to compute flow reversal in a developing flow with heat and mass transfer. The axial evolution of the sensible Nusselt number is shown in Figure 6. In the case of a DF NuS decreases monotonically to the asymptotic value of 7.54, which corresponds to a fully developed flow (Shah and London, 1978). On the other hand, for an UF NuS decreases towards a minimum and then increases to the same asymptotic value. The minimum of NuS occurs at the same axial position as the minimum of fRe (Figure 4). An important effect of T0 on NuS is revealed by Figure 6(a). In the case of a DF, NuS is higher than that for a forced convection flow and it increases with T0. In this kind of flow, as it is shown above, buoyancy forces are aiding and their intensity increases with T0. Thus, these forces accelerate the airflow near the channel walls (Figure 5(a)). This results in an important air temperature gradient near the walls (as it is revealed by u vs y profiles, not shown here) and an important decrease of the mean air temperature Tm (Figure 2(a)). Thus, according to equation (13a), sensible heat transfer is increased by buoyancy forces for a DF. The opposite is true in the case of an UF where buoyancy forces are opposing for all the cases considered here. It is important to notice that differences of NuS between a DF and an UF are up to 38 percent in case 5. The effect of air humidity at the entrance on NuS is less important than that of T0 as we can see in Figure 6(b). However, this effect is clearer in the case of DF. Here, we can observe that NuS increases with f0. Indeed, increasing f0 involves a decrease in GrM and a small increase of GrT, so that the magnitude of the effective Grashof number Gr þ increases (jGr þ j ¼ 67,434 for fo ¼ 10 percent and jGr þ j ¼ 80,983 for fo ¼ 50 percent, see Table I). Thus, buoyancy forces, which are aiding here, increase sensible heat transfer. Figure 7 shows the axial evolution of the latent Nusselt number. Contrary to NuS, the sign of NuL changes from negative to positive depending on the direction of the

Heat and mass transfer in a vertical channel 345

HFF 17,3

55 DF, T0= 40˚C, #1 DF, T0= 45˚C, #4 DF, T0= 70˚C, #5 UF, T0= 40˚C UF, T0= 45˚C UF, T0= 70˚C Gr = 0

45

346 fRe

35

25

1

15

0 5 –1

0.05

–5

0

0.25

0.5 X (a)

0.09

0.75

0.13

1

2.1

1.7

U

1.3

DF, T0= 40˚C, #1 DF, T0= 45˚C, #4 DF, T0= 70˚C, #5 UF, T0= 40˚C UF, T0= 45˚C UF, T0= 70˚C

0.9

0.5

0.001

0.1 0

Figure 6. Axial evolution of the sensible Nusselt number: (a) f0 ¼ 10 percent; (b) T0 ¼ 408C; DF, downward flow; UF, upward flow

-0.001

–0.3

0

0.00015

0

0.005

0.01 Y (b)

0.015

Heat and mass transfer in a vertical channel

60 DF, φ0= 10% #1 DF, φ0= 30% #2 DF, φ0= 50% #3 UF, φ0= 10% #1 UF, φ0= 30% #2 UF, φ0= 50% #3 Gr = 0

50

347

fRe

40

30

20

10

0

0.25

0.5 X (a)

0.75

1

1.8

1.5

U

1.2

0.9 DF, φ0= 10% #1 DF, φ0= 30% #2 DF, φ0= 50% #3 UF, φ0= 10% #1 UF, φ0= 30% #2 UF, φ0= 50% #3

0.6

0.3

0 0

0.005

0.01 Y (b)

0.015

Figure 7. Axial evolution of the latent Nusselt number: (a) f0 ¼ 10 percent; (b) T0 ¼ 408C; DF, downward flow; UF, upward flow

HFF 17,3

348

latent heat flux. According to equations (13b) and (13c), negative NuL indicates that the latent heat flux is directed from the liquid film to the air (ww . w0 so there is evaporation) while positive NuL indicates that the latent heat flux is directed from the air to the liquid film (ww , w0 so there is condensation). It follows that in cases 1 and 4 water is evaporated and in cases 3 and 5 vapor is removed form the airflow and condensed on the film. In case 2, NuL is essentially zero indicating that phase change and interfacial mass transfer are negligible. Indeed, this case corresponds to w0 < ww (Table I). Figure 7 also shows that, near the channel inlet, for a given T0 and f0, the absolute value of NuL is higher for a DF compared to an upward one. This effect of buoyancy forces can be explained by examining the expression of NuL (equation (13b)), which reveals that this effect is exerted on NuL by means of two parameters: the dimensionless mean temperature and the mass fraction gradient at the wall. The first parameter is higher in the case of DF because, in this case, airflow is being cooled more rapidly as it is clearly seen in Figure 2. The non-dimensional mass fraction profiles (not presented here) show that the second parameter is steeper in a DF. It is important to mention that differences of NuL between a DF and an UF are as high as 37 percent (case 5). These differences are of the same order as those for NuS. Careful observation of Figure 7 further indicates that the difference in NuL between the two flow directions is negligible beyond a certain axial location which is about X < 0.5 for the studied cases. At this axial location NuL has the same value for the two flow directions and beyond it the absolute value of NuL becomes slightly more important for an UF. This inversion in NuL tendency is attributed to the dimensionless mass fraction gradient. It is important to mention that variations of NuL between a DF and an UF are up to 6 percent (case 5) for X . 0.5. Another interesting effect shown in Figure 7 is that of the air hygrometry at the entrance (T0 and f0) on the transport of latent heat associated with evaporation. As T0 or f0 are increased, NuL decreases in absolute value (compare cases 1 and 4 in Figure 7(a) and cases 1 and 2 in Figure 7(b)). This trend is attributed to the S factor (see equations 13(b) and 13(c)), which diminishes in absolute value as T0 or f0 are increased. Lee et al. (1997) have also reported this effect and explained it by the fact that the species diffusion mechanism is more effective at lower concentration levels. This is in agreement with our analysis considering that S factor represents the relative importance of energy transport through species diffusion to that through thermal diffusion. The total Nusselt number NuT, i.e. the sum of NuS and NuL, is shown in Figure 8. An overall inspection of these figures shows that, as the airflow moves downstream, NuT decreases in cases 3 and 5 and increases in cases 1 and 4. These axial trends show that the minimum of NuS which occurs in the case of an UF (Figure 6) is not reflected in NuT. On the other hand, NuT, is positive in all cases except for case 1. At this stage, it is useful to point out the meaning of the Nusselt number signs in connection with heat flux directions. The channel walls are maintained at a temperature, which is lower than that of the airflow, so sensible heat flux is always directed towards the channel walls and NuS is positive. The direction of latent heat flux is the same if condensation occurs and then NuL is positive (cases 3 and 5 in Figure 7). If evaporation takes place, the direction of latent heat flux changes to the opposite and NuL is negative (cases 1 and 4, Figure 7(a)).

Heat and mass transfer in a vertical channel

0.02 DF, T0= 40˚C #1 DF, T0= 45˚C #4 DF, T0= 70˚C #5 UF, T0= 40˚C #1 UF, T0= 45˚C #4 UF, T0= 70˚C #5

Wm

0.016

349

0.012

0.008

0.004

0

0.25

0.5 X (a)

0.75

1

0.024 DF, φ0= 10% #1 DF, φ0= 30% #2 DF, φ0= 50% #3 UF, φ0= 10% #1 UF, φ0= 30% #2 UF, φ0= 50% #3

Wm

0.018

0.012

0.006

0

0.25

0.5 X (b)

0.75

1

Figure 8. Axial evolution of the total Nusselt number: (a) f0 ¼ 10 percent; (b) T0 ¼ 408C; DF, downward flow; UF, upward flow

HFF 17,3

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A careful inspection of NuT in case 1 (Figure 8(a)) leads to the following remarks about the importance of each heat flux. It is clear that, in this particular case, NuT is negative near the channel entrance and positive at the exit. Thus, it takes a zero value near the channel mid-point (X < 0.6). At this axial position, NuS and NuL have the same absolute value so that their sum NuT is zero. This feature corresponds to a situation of a zero heat balance: sensible heat flux supplied by the airflow to the channel walls is counterbalanced by latent heat flux gained by airflow by means of water film evaporation. Thus, based on the sign of NuT in case 1, it can be deduced that near the channel entrance NuL and the corresponding latent heat flux predominate (NuT and NuL have the same sign). On the other hand, beyond the channel mid-point, NuS and the sensible heat flux get the upper hand. In cases 2 and 4 NuT is positive, indicating that the sensible heat flux is predominant (NuL is negative in case 4 and essentially zero in case 2). Regarding the effects of combined buoyancy forces on NuT in the two kinds of flow (upward and downward), and by examining the relative variations between these flows, two extreme cases are to be considered. The first is case 5 where these variations are of the same order (about 33 percent) in NuS, NuL and NuT. The second is case 1 where the relative variations in NuS and NuL are of the same order (about 14 percent), otherwise the variations in NuT are much weak (about 3 percent). In the first case (case 5), both NuS and NuL are positive, thus there is an accumulation of buoyancy effect in their sum, NuT. In the second case (case 1) NuS is positive while NuL is negative, thus buoyancy effects in NuS and NuL are antagonistic and so these effects are rather weak in their sum, NuT (NuT for DF and UF collapse in case 1, Figure 8). The axial evolution of the Sherwood number, shown in Figure 9, is similar to that of NuS (Figure 6) as a result of the close values of the Prandtl and Schmidt numbers in this study. Overall, the buoyancy forces increase the Sherwood number (i.e. the mass transfer at the wall) in the case of a DF. For such flows, larger values of Sh are obtained for higher T0 or f0 due to the larger combined buoyancy effects (i.e. larger values of Gr þ ). On the other hand, the buoyancy forces decrease the Sherwood number in the case of an UF. For such flows, Sh decreases as T0 or f0 increase. 5. Conclusion The effects of natural convection on laminar heat and mass transfer between a stream of warm air and the cooler wetted walls of a vertical channel have been investigated numerically by solving the coupled elliptical partial differential conservation equations. Cases with evaporation and condensation have been investigated for both UF and DF. For the conditions under investigation, sensible heat is always directed from the air towards the walls while latent heat transfer depends on the difference between the vapor mass fraction at the channel inlet and at the walls. In four of the five investigated cases the former is more important than the latter. The results show that natural convection can increase or decrease heat and mass transfer fluxes depending on the direction of the flow. Flow reversal has been predicted for an UF with a relatively high temperature difference between the incoming air and the walls. This phenomenon is of considerable importance because it can influence flow stability and induce transition to turbulence (Tam et al., 2004).

Heat and mass transfer in a vertical channel

70 DF, T0= 40˚C #1 DF, T0= 45˚C #4 DF, T0= 70˚C #5 UF, T0= 40˚C #1 UF, T0= 45˚C #4 UF, T0= 70˚C #5

60

351 Tm

50

40

30

20

0

0.25

0.5 X (a)

0.75

1

40 DF, φ0= 10% #1 DF, φ0= 50% #3 UF, φ0= 10% #1 UF, φ0= 50% #3

Tm

35

30

25

20

0

0.25

0.5 X (b)

0.75

1

Figure 9. Axial evolution of the Sherwood number: (a) f0 ¼ 10 percent; (b) T0 ¼ 408C; DF, downward flow; UF, upward flow

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References Ait Hammou, Z., Benhamou, B., Galanis, N. and Orfi, J. (2004), “Laminar mixed convection of humid air in a vertical channel with evaporation or condensation at the wall”, Int. J. Thermal Sciences, Vol. 43, pp. 531-9. Boulama, K. and Galanis, N. (2004), “Analytical solution for fully developed mixed convection between parallel vertical plates with heat and mass transfer”, J. Heat Transfer, Vol. 126, pp. 381-8. Bumeister, L.C. (1993), Convective Heat Transfer, 2nd ed., McGraw-Hill, New York, NY. Chow, L.C. and Chung, J.N. (1983), “Evaporation of water into laminar stream of air and superheated steam”, Int. J. Heat Mass Transfer, Vol. 26, pp. 373-80. Desrayaud, G. and Lauriat, G. (2001), “Heat and mass transfer analogy for condensation of humid air in a vertical channel”, Heat and Mass Transfer, Vol. 37, pp. 67-76. Fuji, T., Kato, Y. and Bihara, K. (1977), “Expressions of transport and thermodynamic properties of air, steam and water”, Sei San Ka Gaku Ken Kuu Jo, Report No. 66, Kyu Shu University, Kyu Shu. Gebhart, B. and Pera, L. (1971), “The nature of vertical natural convection flows resulting from the combined buoyancy effects of thermal and mass diffusion”, Int. J. Heat Mass Transfer, Vol. 14, pp. 2025-50. Jang, J.-H., Yan, W.M. and Huang, C.-C. (2005), “Mixed convection heat transfer enhancement through film evaporation in inclined square ducts”, Int. J. Heat Mass Transfer, Vol. 48, pp. 2117-25. Lee, K.T., Tsay, H.L. and Yan, W.M. (1997), “Mixed convection heat and mass transfer in vertical rectangular ducts”, Int. J. Heat Mass Transfer, Vol. 40, pp. 1621-31. Lin, T.F., Chang, C.J. and Yan, W.M. (1988), “Analysis of combined buoyancy effects of thermal and mass diffusion on laminar forced convection heat transfer in a vertical tube”, ASME J. Heat Transfer, Vol. 110, pp. 337-44. Marmouch, H., Benhamou, B., Orfi, J. and Ben Nasrallah, S. (2005), “E´tude nume´rique d’un syste`me de climatisation e´vaporative”, 7e`me Congre`s de Me´canique, Casablanca, Avril 19-22. Nelson, D.J. and Wood, B.D. (1989), “Combined heat and mass transfer natural convection between vertical parallel plates”, Int. J. Heat Mass Transfer, Vol. 32, pp. 1779-87. Patankar, S.V. (1980), Numerical Heat Transfer and Fluid Flow, Hemisphere/McGraw-Hill, New York, NY. Salah El-Din, M.M. (1992), “Fully developed forced convection in vertical channel with combined buoyancy forces”, Int. Comm. Heat Mass Transfer, Vol. 19, pp. 239-48. Salah El Din, M.M. (2003), “Effect of thermal and mass buoyancy forces on the development of laminar mixed convection between parallel plates with uniform wall heat and mass fluxes”, Int. J. Thermal Sciences, Vol. 42, pp. 447-53. Shah, R.K. and London, A.L. (1978), Laminar Flow Forced Convection in Ducts, Academic Press, New York, NY. Tam, C.T., Maiga, S.E., Landry, M., Galanis, N. and Roy, G. (2004), “Numerical investigation of flow reversal and instability in mixed laminar vertical tube flow”, Int. J. Thermal Sciences, Vol. 43, pp. 797-808. Yan, M.W. (1992), “Effects of film evaporation on laminar mixed convection heat and mass transfers in a vertical channel”, Int. J. Heat Mass Transfer, Vol. 35, pp. 3419-29.

Yan, W.M. (1993), “Mixed convection heat transfer in a vertical channel with film evaporation”, Canadian J. Chemical Engineering, Vol. 71, pp. 54-62. Yan, W.M. and Lin, T.F. (1989), “Effects of wetted wall on laminar mixed convection in a vertical channel”, J. Thermophysics, Vol. 3, pp. 94-6. Yan, W.M. and Lin, T.F. (1990), “Combined heat and mass transfer in natural convection between vertical parallel plates with film evaporation”, Int. J. Heat Mass Transfer, Vol. 33, pp. 529-41. Yan, W.M. and Lin, D. (2001), “Natural convection heat and mass transfer in vertical annuli with film evaporation and condensation”, Int. J. Heat Mass Transfer, Vol. 44, pp. 1143-51. Yan, W.M., Lin, T.F. and Tsay, Y.L. (1995), “Evaporative cooling of liquid film through interfacial heat and mass transfer in a vertical channel-I. Experimental study”, Int. J. Heat Mass Transfer, Vol. 38, pp. 2905-14. Further reading Azizi, Y., Benhamou, B., Galanis, N. and El-Ganaoui, M. (2005), “Heat and mass transfer in a vertical channel with phase change”, Proceedings of 4th Int. Conf. Computational Heat Mass Transfer ICCHMT, Paris, France, May 17-20, pp. 750-5. Corresponding author Brahim Benhamou can be contacted at: [email protected]

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Heat and mass transfer in a vertical channel 353