Heat transfer in multiscale materials is ubiquitous in natural and engineered systems. These materials

are often modeled at a macroscopic scale, where microscopic details are filtered out to reduce numerical and

physical complexity. Here, we use the method of volume averaging to upscale heat transfer equations for a

saturated porous medium with non-linear bulk and surface sources. This approach leads to the development of a

variety of macroscopic models, including a two-temperature model with a second order closure that extends

previous results from Quintard and Whitaker [2000]. Effective properties are calculated for model unit-cells (1D,

2D and 3D) and also for a realistic pore-scale geometry obtained using X-ray tomography. The model further

features a distribution coefficient that indicates the distribution of the surface heat between the two phases at the

macroscale. By comparing computational results for the two-temperature model against direct numerical

simulations, we show that this effective distribution coefficient captures well the partitioning of heat, even in the

transient regime

Davit, Yohan

Quintard, Michel

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   Open Archive TOULOUSE Archive Ouverte (OATAO) OATAO is an open access repository that collects the work of Toulouse researchers and makes it freely available over the web where possible. This  is  an  author-deposited  version  published  in  :  http://oatao.univ-toulouse.fr/ Eprints ID : 14427To cite this version : Davit, Yohan and Quintard, Michel Heat Transfer in Porous Media: Second-Order Closure and Nonlinear Source Terms. In: 17 èmes Journées Internationales de Thermique (JITH 2015), 28 October 2015 - 30 October 2015 (Marseille, France). Any correspondance concerning this service should be sent to the repository administrator: staff-oatao@listes-diff.inp-toulouse.fr17èmes Journées Internationales de Thermique (JITH 2015)Marseille (France),  28 - 30 Octobre 2015_________________________________________________________Heat Transfer in Porous Media: Second-Order Closure and NonlinearSource TermsYohan DAVIT1,2 et Michel QUINTARD1,21Université de Toulouse ; INPT, UPS ; IMFT (Institut de Mécanique des Fluides deToulouse) ; Allée Camille Soula, F-31400 Toulouse, France2CNRS ; IMFT ; F-31400 Toulouse, Franceydavit@imft.fr ; Michel.Quintard@imft.frAbstract: Heat transfer in multiscale materials is ubiquitous in natural and engineered systems. These materialsare often modeled at a macroscopic scale, where microscopic details are filtered out to reduce numerical andphysical  complexity. Here,  we use the method of volume averaging to upscale heat transfer equations for asaturated porous medium with non-linear bulk and surface sources. This approach leads to the development of avariety of macroscopic models, including a two-temperature model with a second order closure that extendsprevious results from Quintard and Whitaker [2000]. Effective properties are calculated for model unit-cells (1D,2D and 3D) and also for a realistic pore-scale geometry obtained using X-ray tomography. The model furtherfeatures a distribution coefficient that indicates the distribution of the surface heat between the two phases at themacroscale.  By  comparing  computational  results  for  the  two-temperature  model  against  direct  numericalsimulations, we show that this effective distribution coefficient captures well the partitioning of heat, even in thetransient regime.Mots clés : porous media, averaging, heat sources1. IntroductionWe consider heat transfer in the porous medium schematically represented Figure 1 where a fluid phase,,  flows  in  a  -phase  solid  skeleton.  Homogeneous  heat  sources  may  be  found  in  both  phases,  while  aheterogeneous heat source may also be considered at the  interface. Such heat sources may be due to variousphysical and chemical phenomena: chemical reactions, radioactivity, etc… The pore-scale heat transfer problemwhich is considered in this paper is written below as(1)(2)(3)(4)This problem has to be solved together with the following total mass and momentum equations(5)(6)(7)In this paper, the assumption is made that density and viscosity do not vary with temperature. Therefore,equations 5 through 7 may be solved independently from the heat transfer problem.VLFigure 1 : Titre de la figureThe upscaling of the heat transfer problem has received a lot of attention in the literature. It is emphasizedin [1] that several models may be developed: direct numerical simulation and heuristic models [2, ...] one-temperature, local equilibrium models [3, 4, ...], two-temperature, local non-equilibrium models, with linear driving force [5, 6, ...] or more sophisticatedmathematical expressions involving, for instance, convolution products [7, 8, ...], various  one-temperature  representations  of  non-equilibrium  situations  [9,  10,  11,  ...],  such  as  theasymptotic behavior of two-temperature models, fractional derivatives, wave equations [12], etc... hybrid or mixed models coupling macro-scale equations with micro-scale pore-scale submodels.In this paper, we particularly discuss the way the heat sources appear in the macro-scale models. In thecase of a local equilibrium model, a total source term is introduced in the equation for the mixture temperature,Tβσ,  under the form(8)where av=Aβσ/V  is the specific area of the porous medium. The macro-scale source term is simply the average ofthe pore-scale source terms. In the case of local non-equilibrium models, the homogeneous source terms, as faras non-linearities are not concerned, do not pose a problem and averages appear in the right-hand sides of therespective phase equations. This is another matter for the heterogeneous source term. In the engineering practice,for  instance  catalytic  burners  [13],  the  total  produced  heat  flux is  assigned to  the solid  phase.  However,  athorough pore-scale analysis or upscaling results do not necessarily support this choice as it is emphasized in thispaper.The objective of this work is two-fold.  First, following [1,  14],  the theory that leads to a local  non-equilibrium model incorporating the effect of the heat sources terms is briefly developed. Then some quantitativeapplications of the theory that emphasize the applicability of the proposed model are presented.2. Theory Averages are defined classically as [15](9)for any variable ψβ defined in the β-phase. The phase intrinsic average is defined as(10)with the β-phase volume fraction, εβ, defined as(11)Deviations to the averaged values are also classically defined as(12)The mixture temperature, Tβσ, may now be defined precisely as(13)with(14)2.1. averaged equations and deviation equationsThe upscaling theory is not described with all details in this paper, the reader is referred to the citedliterature for an introduction to averaging, thermal dispersion, etc... Since the emphasis here is on heat sources,the appropriate references are taken from a first development given in [1] with a subsequent improvement in[14], especially in terms of a second order closure. Only major steps are outlined below.The first step starts with the averaging of the pore-scale equations. Making use of the averaging theoremssuch as(15)(16)the averaging of Eq. 1 leads to(17)A similar equation may be written for the σ-phase, i.e.,(18)The temperature deviations are solutions of the pore-scale problem transformed using the decompositionEq.  12 and subtracting the averaged equations. After some algebra, the governing equations for the deviationsmay be written as(19)(20)(21)(22)There are several source terms in these equations that will produce deviations of the temperature fieldsfrom the averaged values. A closure of the coupled macro-scale and micro-scale equations will require to find anapproximate expression linking the deviations to the averaged values through the source terms. The source termin Eq.  20 and the first two ones in Eq.  21 will lead to classical thermal dispersion, tortuosity and interphaseexchange effects, which have been dealt with at length in many papers (see for instance discussion in [1,  14,15]). In this work, we focus on the impact of the heat source terms.If the heat source terms are constant, as is the case for instance in nuclear safety problems where they areproduced by radioactivity, the homogeneous source terms in the bulk equations for the deviations disappear anddo not play longer a role in the closure problem while the heterogeneous term in BC2 remains. Developing thesource term by a Taylor's expansion around the averaged temperature, one obtains(23)In this work, we will assume that the deviation part in Eq.  23 is small compared to  . Thevalidity of such an assumption will depend on the process under consideration. An example can be found in [16]in the case of source terms produced by an Arrhenius reaction rate. In this reference, it is shown that the aboveassumption works relatively well as soon as some Damköhler number remains small. Significant discrepanciesarise otherwise. Neglecting higher order terms in Eq. 23 leads to the following approximation(24)which is used in the further developments.As a consequence, the only remaining source term in the closure problem due to the heat sources is theone associated to the heterogeneous source term.2.2. closure and macro-scale equationsIt is beyond the scope of this paper to discuss the various models (transient closure, hybrid models, quasi-steady closure, asymptotic models, etc...) that can be developed from the averaged and deviation equations asdiscussed in [14]. Following [1, 14], a quasi-steady closure (closure is intended here as an approximate solutionof the coupled macro and micro-scale equations), can be developed under the following form(25)(26)where second order terms have been kept. The mapping variables, i.e., , , etc..., obey governing equationswhich realize an approximate solution of the coupled macro micro-scale system of equation. The first three termsin Eqs. 25 and 26 corresponds to the classical theory for the first order two-temperature model and will not bediscussed  here.  The reader  can  refer  to  [6]  for  resolution of  the three  corresponding  closure  problems andcalculations of the related effective properties, i.e., thermal dispersion tensors, heat exchange coefficient, etc...The other terms are not “traditional” and represent the contribution of the heterogeneous source, which will bediscussed later, and of second order terms. The resulting macro-scale equations are obtained by substituting Eqs. 25 and 26 into Eqs. 17 and 18. Oneobtains(27)(28)In these macro-scale equations, the effective parameters are calculated from the mapping variables. Forinstance, the exchange coefficient is expressed as(29)The  macro-scale  terms  associated  to  the  homogeneous  heat  source  terms  are  simply  the  pore-scalefunction calculated at the average temperature weighted by the phase volume fraction. This is a consequence ofneglecting high non-linearities as shown in Eq. 23. On the contrary, the heterogeneous heat source term is notentirely affected to a particular phase equation but, instead, has to be distributed between the two phase macro-scale equations. This is achieved through the introduction of a distribution coefficient which can be calculatedfrom the pore-scale properties by solving the following closure problem:(30)(31)(32)(33)(34)(35)where the distribution coefficient ξ is given by(36)In the next section, some properties of this distribution coefficient are discussed.3. Applications: the distribution coefficientIn this section, the closure problem for the heterogeneous heat source term is solved for various pore-scale geometry and physical parameters. Then, a 0D macro-scale problem is solved to emphasize some peculiarfeatures of this heat distribution problem.3.1. Results for the distribution coefficientThe closure problem expressed by Eqs. 30 through 36 has been solved analytically for stratified unit cellsand 2D ou 3D simple unit cells[14]. In this paper, results are presented for a more realistic porous mediumobtained from x-ray tomography [1].In the case of a stratified unit cell such as the one represented Figure 2a, the distribution coefficient canbe obtained analytically. Its value does not depend on the velocity and is given by(37)An example  of  calculation  of  the  distribution  coefficient  in  the  case  of  more  complex  unit  cells  isprovided below. The pore-scale geometry is taken from an X-ray microtomograph and the solid phase indicatoris represented Figure 2b. σ x yβ σlβl /2βl /2β(a)                                                               (b)Figure 2. Unit Cells (UC)The evolution of the distribution coefficient as a function of the thermal conductivity ratio is providedFigure 3 for three different unit cells (UC). They all have the same general trend:(38)In  other  words,  the  heterogeneous  heat  source  has  the  tendency  to  be  distributed  towards  the  mostconductive phase. If the practical problem under consideration involves a highly conductive solid phase, which isthe case for instance in metallic catalytic porous burners, the heat source is affected to the σ-phase equation,which is the  common engineering practice. In between these two limits, the detailed variation depends on thepore-scale  geometry. The results  emphasize  the  fact  that  the  resulting  distribution  coefficient  value  is  verysensitive to the pore-scale geometry and cannot be predicted accurately from results for simple unit cells.The impact of the velocity field has been studied in [14]. Results for simple 2D unit cells are providedFigure 4. The Péclet number is defined as(39)where  lc is the array unit vector length (very close here to the grain diameter). The results show a transitionregime for Pe between 1 and 100 in between two constants for Pe → 0 and Pe → ∞. In the range of exploredvalues, the distribution coefficient varies between 10 to 20%, which is less significant than the influence of thethermal conductivity ratio.ξ00.20.40.60.81kσ / kβ0.01 0.1 1 10 100Strat. FVM (CC: 813)Image (1503)Figure 3. Distribution coefficient as a function of  (Strat.=stratified UC; FVM=Cubic Centered; Image=X-ray tomography)1.00.80.60.4ξ0.20.00.1 1 10 100PeinlinestaggeredFigure 4. Distribution coefficient as a function of the Péclet number for two types of arrays of cylinders andvarious values of the thermal conductivity ratio.3.2. Influence of the distribution coefficient on the temperature differenceAnother common sense view of the problem is that the hottest phase, macroscopically speaking, is the one withthe larger diffusivity. Let us explore that notion by looking at the following macro-scale 0D problem.(40)(41)(42)Solving  analytically  this  problem,  it  can  be  shown  that,  after  a  transient  behavior,  the  two  macro-scaletemperatures increase with time but the temperature difference is constant and given by(43)and the condition for obtaining a zero difference is(44)In order to better understand the implications, let us consider the case of the stratified material representedFigure 2. Solving analytically the closure problem in that case gives(45)and(46)The condition Eq. 44 gives(47)We see that, if the volume fractions are the same, the change in the sign of the temperature differenceoccurs indeed when the diffusivities are equal. However, this is no longer true for different volume fractions. Infact, the material with the lowest diffusivity can have a higher averaged temperature if it is thin enough to catchup with the thicker phase! This is another feature, apparently counter intuitive, of this heterogeneous heat sourceproblem.ConclusionThe introduction of  heat  source  terms in macro-scale  equations in  the case  of  local  non-equilibriummodels requires the resolution of more complex closures. An example is provided in this paper in the case of aquasi-steady closure which generally leads to the classical two-temperature model. This model is modified herewith the introduction of a distribution coefficient which insert a portion of the heterogeneous heat source into theeach macro-scale phase equation. This coefficient varies rapidly for thermal conductivity ratios between 0.01 and100, and the value is relatively sensitive to the pore-scale geometry as emphasized by the calculations on simpleunit cells and also on a tomographic image.This  concept  has  already  been  used  in  various  contexts:  for  instance  for  heat  sources  coming fromArrhenius reactions [16], for a local non-equilibrium heat transfer model including radiative heat transfer [17]. Itproved very useful, which suggests that improvements should be sought in various different directions: higherorder closures for non-linear systems, multi-physic coupling, etc...Nomenclature Symbol Name, unitcp heat capacity, J/kg.Kg gravitational acceleration, m/s2k thermal conductivity, W/m.KK effective thermal dispersion, W/m.KT temperature, Kav specific area, m-1v velocity, m/sV averaging volume, m3R homogeneous heat source, J/m3.sGreek Symbols refers to -phaseε volume fractionμ dynamic viscosity, N.s/m2ξ distribution coefficientρ density, kg/m3Ω heterogeneous heat source, , J/m2.sIndiceseq equilibriumβσ mixture variableReferences[1] Y. Davit and M. Quintard, Theoretical analysis of transport in porous media: Multi-Equation and HybridModels for a Generic Transport Problem with Non-Linear Source Terms, ch. 7 in Handbook of Porous Media ed.by K. Vafai, Taylor & Francis, 2015.[2] E.U. Schlünder , Equivalence of one- and two-phase models for heat transfer processes in packed beds: one-dimensional theory. Chem Eng Sci, Volume 30, Pages 449-452, 1975.[3] H.I. Ene and E. Sanchez-Palencia , Sur la propagation de la chaleur dans les milieux poreux. C. R. Acad. Sci.Paris Sér. II Méc. Phys. Chim. Sci. Univers Sci. Terre, Volume 292, Pages 1181-1184, 1981.[4] I. Nozad, R.G. Carbonell and S. Whitaker , Heat conduction in multiphase systems I: Theory and experimentfor two-phase systems. Chem Eng Sci, Volume 40, Pages 843-855, 1985.[5]  F. Zanotti  and R.G. Carbonell,  Development  of  transport  equations for  multiphase systems -  I:  Generaldevelopment for two-phase systems. Chem Eng Sci, Volume 39, Pages 263-278, 1984.[6] M. Quintard, M. Kaviany and S. Whitaker , Two-Medium Treatment of Heat Transfer in Porous Media:Numerical Results for Effective Properties. Advances in Water Resources, Volume 20, Pages 77-94, 1997.[7] C. Moyne , Two-Equation Model for a Diffusive Process in Porous Media Using the Volume AveragingMethod with an Unsteady State Closure. Advances in Water Resources, Volume 20, Pages 63-76, 1997.[8] Y. Davit, B. Wood, G. Debenest and M. Quintard , Correspondence Between One- and Two-Equation Modelsfor  Solute  Transport  in  Two-Region  Heterogeneous  Porous  Media.  Transport  in  Porous  Media,  SpringerNetherlands, Volume 95, Pages 213-238, 2012.[9] C. Moyne, S. Didierjean, H.P.A. Souto and O.T.D. Silveira , Thermal Dispersion in Porous Media: One-Equation Model. International Journal of Heat and Mass Transfer, Volume 43, Pages 3853-3867, 2000. [10] M. Quintard, F. Cherblanc and S. Whitaker , Dispersion in heterogeneous porous media : one-equation non-equilibrium model. Transport in Porous Media, Volume 44, Pages 181-203, 2001.[11] Y. Davit, M. Quintard and G. Debenest , Equivalence between volume averaging and moments matchingtechniques for mass transport models in porous media. International Journal of Heat and Mass Transfer, Volume53, Pages 4985 - 4993, 2010.[12] L. Wang and X. Wei  ,  Equivalence  between dual-phase-lagging and two-phase-system heat  conductionprocesses. International Journal of Heat and Mass Transfer, Volume 51, Pages 1751-1756, 2008.[13]  M.  Tancrez,  Modélisation  du  rayonnement  et  transferts  couplés  dans  des  milieux  poreux  réactifs.Application aux brûleurs radiants à gaz. Châtenay-Malabry, Ecole Centrale de Paris, 2002.[14] M.  Quintard and S. Whitaker, Theoretical  analysis of transport in porous media. In: Vafai,  K. (Ed.),  inHandbook of Porous Media ed. by K. Vafai, Marcel Decker Inc., 2000.[15] S. Whitaker, The Method of Volume Averaging. Kluwer Academic Publishers, 1999.[16] C. Yang, M. Quintard and G. Debenest , Upscaling for Adiabatic Solid–Fluid Reactions in Porous MediumUsing a Volume Averaging Theory. Transport in Porous Media, in press, Pages 1-33, 2015.[17] V. Leroy, B. Goyeau and J. 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Heat Transfer in Porous Media: Second-Order Closure and Nonlinear Source Terms

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Heat Transfer in Porous Media: Second-Order Closure and Nonlinear Source Terms

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