Theoretical predictions of the effective thermodiffusion coefficients in porous media

International Journal of Heat and Mass Transfer 53 (2010) 1514–1528

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Theoretical predictions of the effective thermodiffusion coefficients in porous media H. Davarzani, M. Marcoux *, M. Quintard Université de Toulouse, INPT, UPS, IMFT (Institut de Mécanique des Fluides de Toulouse), GEMP (Groupe d’Etude des Milieux Poreux) Allée Camille Soula, F-31400 Toulouse, France CNRS, IMFT, F-31400 Toulouse, France

a r t i c l e

i n f o

Article history: Received 13 March 2009 Received in revised form 11 October 2009 Accepted 15 October 2009 Available online 8 January 2010 Keywords: Thermal diffusion Diffusion Soret effect Effective coefficient Porous media Volume averaging technique

a b s t r a c t This study presents the determination of the effective Darcy-scale coefficients for heat and mass transfer in porous media including the thermodiffusion effect using a volume averaging technique. The closure problems related to the pore-scale physics and providing effective coefficients are solved over periodic unit cells representative of the porous structure. The results show that, for low Péclet numbers, the effective Soret number in porous media is the same as the one in the free fluid and that it does not depend on the solid to fluid conductivity ratio. On the opposite, in convective regimes, the effective Soret number decreases. In this case, a change of conductivity ratio will change the effective thermodiffusion coefficient as well as the effective thermal conductivity coefficient. The macroscopic model obtained by this method is validated by comparison with direct numerical simulations at the pore-scale. Then, coupling between forced convection and Soret effect for different cases is investigated. Ó 2009 Elsevier Ltd. All rights reserved.

1. Introduction It is well established, see for instance [12], that a multicomponent system under non-isothermal condition is subject to mass transfer related to coupled-transport phenomena. This has strong practical importance in many situations since the flow dynamics and convective patterns in mixtures are more complex than those of one-component fluids due to the interplay between advection and mixing, solute diffusion, and the Soret effect (or thermal diffusion) [45]. The Soret coefficient may be positive or negative depending on the direction of migration of the reference component (to the cold or to the hot region). There are many important processes in nature and technology where thermal diffusion plays a crucial role. Thermal diffusion has various technical applications, such as isotope separation in liquid and gaseous mixtures [35,36], polymer solutions and colloidal dispersions [45], study of compositional variation in hydrocarbon reservoirs [11], coating of metallic items, etc. It also affects component separation in oil wells, solidifying metallic alloys, volcanic lava, and in the Earth Mantle [14]. Platten and Costeseque [24] searched the response to the basic question:” is the Soret coefficient the same in a free fluid and in a porous medium?” They measured separately four coefficients: iso-

* Corresponding author. Address: Institut de Mécanique des Fluides de Toulouse, UMR n°5502 CNRS/INPT/UPS, Groupe d’Etude sur les Milieux Poreux, Allée Prof. Camille Soula, F-31400 Toulouse, France. Tel.: +33(0)5 61 28 58 76; fax: +33(0)5 61 28 58 99. E-mail address: [email protected] (M. Marcoux). 0017-9310/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijheatmasstransfer.2009.11.044

thermal diffusion and thermodiffusion coefficients, both in free fluid and porous media. They measured the diffusion coefficient in free fluid by the open-ended-capillary (OEC) technique, then they generalized the same OEC technique to porous media. The thermodiffusion coefficient in the free system has also been measured by the thermogravitational column technique [23]. The thermodiffusion coefficient of the same mixture was determined in a porous medium by the same technique, except that they filled the gap between two concentric cylinders with zirconia spheres. In spite of the small errors that they had on the Soret coefficient due to measuring independently diffusion and thermodiffusion coefficient they announced that the Soret coefficient is the same in a free fluid and in porous media [24]. The experimental study of Costeseque et al. for a horizontal Soret-type thermodiffusion cell, filled first with the free liquid and next with a porous medium showed also that the results are not significantly different [8]. Saghir et al. have reviewed some aspects of thermodiffusion in porous media; including the theory and the numerical procedure which have been developed to simulate these phenomena [37]. In many other works on thermal diffusion in a square porous cavity, the thermodiffusion coefficient in free fluid almost has been used instead of an effective coefficient containing tortuosity and dispersion effects. Therefore, there are many discrepancies between the predictions and measurements separation. The effect of dispersion on molecular diffusion coefficients is now well established (see for example Saffman [53], Bear [48],. . .) but this effect on thermal diffusion has received limited attention. Fargue et al. searched the dependence of the thermal diffusion coefficient on flow velocity in a packed thermogravitational

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Nomenclature Abr Abe Abr AS bCb bSb bTb cp cb hcb ib e cb c0 Da Db DTb DTb Db g I kb kr Kb   kb ; k l lUC lb L Le nbr pb hpb ib Pe Pr

area of the b–r interface contained within the macroscopic region, m2 area of the entrances and exits of the b–r phase associated with the macroscopic system, m2 area of the b–r interface within the averaging volume, m2 segregation area, m2 e b, m mapping vector field for C e b, m mapping vector field for C mapping vector field for Te b , m constant pressure heat capacity, J.kg/K total mass fraction in the b-phase intrinsic average mass fraction in the b-phase spatial deviation mass fraction in the b-phase initial concentration Darcy number binary diffusion coefficient, m2/s thermal diffusion coefficient, m2/s K total thermodiffusion tensor, m2/s K total dispersion tensor, m2/s gravitational acceleration, m2/s unit tensor thermal conductivity of the fluid phase, W/m K thermal conductivity of the solid phase, W/m K permeability tensor, m2 total thermal conductivity tensors for no-conductive and conductive solid phase, W/m K characteristic length associated with the microscopic scale, m characteristic length scale associated with a unit cell, m characteristic length for the b-phase, m characteristic length for macroscopic quantities, m characteristic length for re, m unit normal vector directed from the b-phase toward the r-phase pressure in the b-phase, Pa intrinsic average pressure in the b-phase, Pa cell Péclet number Prandtl number

column. Their results showed that the behaviour of the effective thermal diffusion coefficient looks very similar to the effective diffusion coefficient in porous media [10]. The numerical model of Nasrabadi et al. [21] in a packed thermogravitational column was not able to reveal the dispersion effect on the thermodiffusion process, mainly due to low velocities [21]. In the macroscopic description of mass and heat transfer in porous media, the convection–diffusion phenomena (or dispersion) in a porous medium are generally analyzed using an up-scaling method, in which the complicated local situation (transport by convection and diffusion at the pore scale) is finally described at the macroscopic scale. At this level, dispersion can be characterized by effective tensors [20]. There are several different ways of upscaling macroscopic properties in a porous medium: among others, the method of moments [5], the volume averaging method [6] and the homogenization method [18] are the most used techniques. In this work, we shall use the volume averaging method to obtain the macro-scale equations that describe thermodiffusion in a homogeneous porous medium [9]. It has been extensively used to predict the effective transport properties for many processes including transport in heterogeneous porous media [32], two-phase flow [29], reactive media [44], solute transport with adsorption [1] multi-component mixtures [30]. The considered media can also be

r0 r Sc ST ST t t* Tb hT b ib Te b TH, TC vb hv b ib

v~ b Vb V x, y z

radius of the averaging volume, m position vector, m Schmidt number Soret number effective Soret number time, s characteristic process time, s temperature of the b-phase, K intrinsic average temperature in the b-phase, K spatial deviation temperature, K hot and cold temperature mass average velocity in the b-phase, m/s intrinsic average mass average velocity in the b-phase, m/s spatial deviation mass average velocity, m/s volume of the b-phase contained within the averaging volume, m3 local averaging volume, m3 Cartesian coordinates, m elevation in the gravitational field, m

Greek symbols eb volume fraction of the b-phase or porosity j kr/kb, conductivity ratio lb dynamic viscosity for the b-phase, Pa s tb kinematic viscosity for the b-phase, m2/s qb total mass density in the b-phase, kg/m3 s scalar tortuosity factor u arbitrary function w separation factor or dimensionless Soret number Subscripts, superscripts and other symbols b fluid-phase r solid-phase b–r interphase br be fluid-phase entrances and exits * effective quantity hi spatial average intrinsic b-phase average hib

subjected to thermal gradients coming from natural origin (geothermal gradients, intrusions,. . .) or from anthropic anomalies (waste storages,. . .). Thermodiffusion has rarely been taken completely under consideration, coupled effects being generally forgotten or neglected in most descriptions. However, the presence of temperature gradient in the media can generate a mass flux. This can modify species concentrations of fluids moving through the porous medium and lead to local accumulations [8]. For modelling mass transfer by thermodiffusion, the effective thermal conductivity must be first determined. Different model have been investigated for two-phase heat transfer systems depending on the validity of the local thermal equilibrium assumption. When one accepts this assumption, macroscopic heat transfer can be described correctly by a classical one-equation model [15,31,29,33]. The reader can look at [2] for the possible impact of non-equilibrium on various flow conditions. For many initial boundary-value problems, the two-equation model shows an asymptotic behaviour that can be modelled with a ‘‘non- equilibrium” oneequation model [46,47]. The resulting thermal dispersion tensor is greater than the one-equation local-equilibrium dispersion tensor. It can also be obtained through a special closure problem as shown in [19]. These models can also be extended to more complex situations like two-phase flow [20], reactive transport [27,34,39].

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However, for all these models many coupling phenomena have been discarded in the upscaling analysis. This is particularly the case for the possible coupling with the transport of constituents in the case of mixtures. A model for Soret effect in porous media has been proposed by Lacabanne et al. [16]. They used a homogenization technique for determining the macroscopic Soret number in porous media. They assumed a periodic porous medium with the periodical repetition of an elementary cell. In this model, the effective thermodiffusion and isothermal diffusion coefficient is calculated by only one closure problem while, in this paper, two closure problems have to be solved separately to obtain effective isothermal and thermal diffusion coefficients. They have also studied the local coupling between velocity and Soret effect in a tube with a thermal gradient. The results of this model showed that when convection is coupled with Soret effect, diffusion removes the negative part of the separation profile [16]. However, they calculated the effective coefficients for a purely diffusive regime for which one cannot observe the effect of force convection and conductivity ratio as explained later in this paper. In addition, these results have not been validated with experimental results or a direct pore-scale numerical approach. The aim of this study is to characterise the modifications induced by thermodiffusion on the description of mass transfer in porous media. It especially consists in the determination of the effective thermodiffusion coefficient using a volume averaging technique. Effective properties will be calculated for a simple unit-cell but for various physical parameter, in particular the Péclet number and the thermal conductivity ratio. Finally the obtained macroscopic model is validated by comparison with direct numerical simulations at the pore-scale.

2. Governing microscopic equation We consider in this study a binary mixture fluid flowing through a porous medium subjected to a thermal gradient. This system is illustrated in Fig. 1, the fluid phase is identified as the b-phase while the rigid and impermeable solid is represented by the r-phase. From the thermodynamics of irreversible processes as originally formulated by Onsager [50,51] the diagonal effects that describe heat and mass transfer are Fourier’s law which relates heat flow to the temperature gradient and Fick’s law which relates mass flow to the concentration gradient. There are also cross effects or

coupled-processes: the Dufour effect quantifies the heat flux caused by the concentration gradient and the Soret effect, the mass flux caused by the temperature gradient. In this study, we neglect the Dufour effect, which is justified in liquids [25] but in gaseous mixtures the Dufour coupling becomes more and more important and can change the stability behaviour significantly in comparison to liquid mixtures [13]. Therefore, the transport of energy at the pore level is described by the following equations and boundary conditions for the fluid (b-phase) and solid (r-phase)

@T b þ ðqcp Þb r  ðT b v b Þ ¼ r  ðkb rT b Þ; in the b-phase @t BC1 : T b ¼ T r ; at Abr ðqcp Þb

BC2 : nbr  ðkb rT b Þ ¼ nbr  ðkr rT r Þ; at Abr @T r ¼ r  ðkr rT r Þ; in the r-phase ðqcp Þr @t

ð1Þ ð2Þ ð3Þ ð4Þ

where Abr is the area of the b–r interface contained within, the macroscopic region. The component pore-scale mass conservation is described by the following equation and boundary conditions for the fluid phase [4]

@cb þ r  ðcb v b Þ ¼ r  ðDb rcb þ DTb rT b Þ; in the b-phase @t BC1 : nbr  ðDb rcb þ DTb rT b Þ ¼ 0; at Abr

ð5Þ ð6Þ

where cb is the mass fraction of one component in the b-phase, Db and DTb are the molecular isothermal diffusion coefficient and thermodiffusion coefficient. We assume in this work that the physical properties of the fluid and solid are constant. To describe completely the problem, the equations of continuity and motion have to be introduced for the fluid phase. We use Stokes equations for the flow motion at the pore-scale, assuming classically negligible inertia effects in porous media. The Stokes equation, the continuity equation, and the noslip boundary condition are then written as

r  v b ¼ 0; in the b-phase

ð7Þ

0 ¼ rp þ lb r  ðrv b Þ þ qb g; in the b-phase

ð8Þ

BC1 : nbr  v b ¼ 0; at Abr

ð9Þ

In this problem, it is assumed that the solid phase is rigid and impervious to solute diffusion and the thermal and solutal expansions have been neglected. 3. Volume averaging method Because the direct solution of the convection–diffusion equation is in general impossible due to the complex geometry of the porous medium, equations describing average concentrations and velocities must be developed [42]. The associated averaging volume, V is shown in Fig. 1. The development of local volume averaged equations requires that we define two types of averages in terms of the averaging volume [44]. For any quantity ub associated with the b-phase, the superficial average is defined according to

hub i ¼

1 V

Z Vb

ub dV

ð10Þ

while the second average is the intrinsic average defined by

hub ib ¼ Fig. 1. Problem configuration.

1 Vb

Z Vb

ub dV

ð11Þ

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Here we have used Vb to represent the volume of the b-phase contained within the averaging volume. These two averages are related by

hub i ¼ eb hub ib

ð12Þ

in which eb is the volume fraction of the b-phase or porosity. The spatial averaging theorem in the divergence form for any arbitrary ub-field associated with the b-phase and for the b–r system is given by

1 hrub i ¼ rhub i þ V

Z Abr

nbr ub dA

ð13Þ

Following classical ideas [44] we will try to solve approximately the problems in terms of averaged values and deviations. The pore-scale fields deviation in the b-phase and r-phase are respectively defined by

~ b and ur ¼ hur ir þ u ~r ub ¼ hub ib þ u

ð14Þ

The classical length-scale constraints (Fig. 1) have been imposed by assuming

lb << r 0 << L

ð15Þ

After performing the volume averaging on the original boundary value problem and solving the associated closure problems, the final form of the transport equations contains local averages, rather than micro-scale point values. Thus, the microscopic equations that hold for a point in space are developed into the appropriate macroscopic equations, which hold at a given point for some volume in space of the porous medium. 3.1. Darcy’s law If we assume that density and viscosity are constants, the flow problem can be solved independently from the heat and constituent transport equations, and the change of scale for Stokes flow equation and continuity has already been investigated and this leads to Darcy’s law and the volume averaged continuity equation [40,43] written as

hv b i ¼ 

 Kb   rhpb ib  qb g ; in the porous medium

lb

r  hv b i ¼ 0; in the porous medium

ð16Þ ð17Þ

where Kb is the permeability tensor. For now on, we will assume that the pore-scale and, subsequently, the macro-scale velocity fields are known vector fields. 3.2. Local thermal equilibrium Since we have neglected Dufour effect, the heat transfer problem may be solved independently from Eqs. (5) and (6). This question has received a lot of attention in the literature. The conditions for the validity of a one-equation conduction model have been investigated by Quintard et al. [52]. They have examined the process of transient heat conduction for a two-phase system in terms of the method of volume averaging. Using two equation models, they have explored the principle of local thermal equilibrium as a function of various parameters, in particular the conductivity ratio, micro-scale and macro-scale dimensionless times and topology [52]. In this work we will use a local equilibrium model, considering fast thermal exchange between the different regions, characterized by

hT b ib  hT r ir

ð18Þ

and this allows to introduce a macro-scale temperature such as

ðeb ðqcp Þb þ er ðqcp Þr ÞhTi ¼ eb ðqcp Þb hT b ib þ er ðqcp Þr hT r ir i:e: hTi ¼ hT b ib ¼ hT r ir

ð19Þ

Under these conditions it is possible to introduce a quasi-steady representation of the temperature deviations in terms of the gradient of the average temperature [6]

Te b ¼ bTb  rhTi Te r ¼ bT r  rhTi

ð20Þ ð21Þ

in which bTb and bTr are referred as the closure variables for solid and liquid respectively. Therefore, they are specified by the following boundary value problem [29,44] Problem I:

eb þ ðqcp Þb v b  rbTb þ ðqcp Þb v

Z e1 b V

nbr  kb rbTb dA

Ab r

¼ kb r2 bTb

ð22Þ

ð23Þ BC1 : bTb ¼ bT r ; at Abr BC2 : nbr  kb rbTb ¼ nbr  kr rbT r þ nbr  ðkb  kr Þ; at Abr ð24Þ Z e1 r nrb  kr rbT r dA ¼ kr r2 bT r ð25Þ V Abr Periodicity : bTb ðr þ li Þ ¼ bTb ðrÞ & bT r ðr þ li Þ ¼ bT r ðrÞ; i ¼ 1; 2; 3

ð26Þ

Averages : hbTb ib ¼ 0;

hbT r ir ¼ 0

ð27Þ

This way of writing the problem, i.e., under an integro-differential form, is reminiscent of the fact that this must be compatible with the full two-equation model as described, for instance, in [31]. However, following the mathematical treatment described also in this paper (using the decomposition described by Eqs. (20)) in Ref [31], this problem reduces to (the proof involves the use of periodicity conditions) the following problem. Problem I:

e b ¼ kb r2 bTb ðqcp Þb v b  rbTb þ ðqcp Þb v

ð28Þ

BC1 : bTb ¼ bT r ; at Abr

ð29Þ

BC2 : nbr  kb rbTb ¼ nbr  kr rbT r þ nbr  ðkb  kr Þ; at Abr ð30Þ 0 ¼ kr r2 bT r

ð31Þ

Periodicity : bTb ðr þ li Þ ¼ bTb ðrÞ & bT r ðr þ li Þ ¼ bT r ðrÞ;

i ¼ 1; 2; 3 ð32Þ

r

b

Averages : hbT i ¼ eb hbTb i þ er hbT r i ¼ 0

ð33Þ

In fact, the resulting field is also compatible with Eq. (27), which is consistent with the local-equilibrium closure being compatible with the one from the two-equation model, as a limit case. The closed form of the convective–dispersion governing equation for hTi is therefore

ðeb ðqcp Þb þ er ðqcp Þr Þ

  @ hTi þ ðqcp Þb r  eb hv b ib hTi @t



¼ r  ðk  rhTiÞ

ð34Þ

*

where k is the thermal dispersion tensor given by 

k ¼ ðeb kb þ er kr ÞI þ e b bTb i  ðqcp Þb h v

ðkb  kr Þ V

Z

nbr bTb dA

Abr

ð35Þ

As an illustration of such a local-equilibrium situation, we will compare a direct simulation of the pore-scale equations with a macroscale prediction. The geometry is an array of NUC of the periodic Unit Cell (UC) shown Fig. 2. The initial temperature in the domain is a

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Fig. 2. Representative unit cell (eb = 0.8).

constant, TC. The fluid is injected at x = 0 at temperature TH. The temperature is imposed at the exit boundary and is equal to TC. This latter boundary condition has been taken for a practical reason: we have ongoing experiments using the two-bulb method, which is closely described by this kind of boundary-value problem. In addition, this particular problem will help us to illustrate some theoretical considerations given below. The parameters describing the case were:

N UC ¼ 120; lUC ¼ 2  103 m; kb ¼ 1W=m:K; kr ¼ 4 W=m:K;

eb ¼ 0:615; ðqcp Þb ¼ 4  106 J=m3 K; ðqcp Þr ¼ 4  106 J=m3 K;  hmb ib ¼ 1:4  104 m=s; k ¼ 1:607 W=m:K ð36Þ An example of comparison between the averaged temperatures obtained from the direct simulation and theoretical predictions is given Fig. 3. We considered three stages:  a short time after injection, which is often the source of a discrepancy between actual fields and macro-scale predicted ones, because of the vicinity of the boundary,  an intermediate time, i.e., a field less impacted by boundary conditions,  a long time typical of the steady-state condition associated to the initial boundary-value problem under consideration. We see on these figures a very good agreement between the direct simulations and the predictions with the local equilibrium model. This illustrates the fact that the local-equilibrium model does allow to represent correctly the system behaviour for moderate contrasts of the pore-scale physical properties. What happens when this contrast becomes dramatic, i.e., when the pore-scale characteristic times are very different? To illustrate the problem, we designed such a case by taking the following parameters:

N UC ¼ 480; ‘UC ¼ 2  103 m; kb ¼ 1 W=m:K; kr ¼ 0:01 W=m:K;

eb ¼ 0:615; ðqcp Þb ¼ 4  106 J=m3 K; ðqcp Þr ¼ 4  106 J=m3 K; 

hv b ib ¼ 6:95  105 m=s:k ¼ 0:455 W=m:K ð37Þ The comparison between the averaged temperatures obtained from direct numerical simulations and the theoretical predictions of the local-equilibrium model are presented Fig. 4 for three different times. At early stages, we see a clear difference between the averaged temperatures of the two phases, and also a clear difference

Fig. 3. Normalized temperature versus position, for three different times (triangle, direct numerical simulation=ðhT b ib  T C Þ=ðT H  T C Þ; circles, direct numerical simulation =ðhT r ir  T C Þ=ðT H  T C Þ; solid line, local-equilibrium model=ðhTi  T C Þ= ðT H  T C Þ.

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with the local-equilibrium predictions. This difference is also visible for intermediate times, and one sees that the local-equilibrium model has an effective conductivity too small. However, at steady-state, it is remarkable to see that the temperature fields revert to the local-equilibrium conditions and that, despite the steep gradient near the boundary, the local-equilibrium models offers a very good prediction. It must be pointed out that this possibility has not been documented in the literature, and this may explain certain confusion in the discussion about the various macro-scale models. Without going into many details, we may summarize the discussion as follows:  for moderate thermal properties contrasts, the local-equilibrium predictions are very good, and not very sensitive to boundary conditions or initial conditions,  the situation is much more complex for higher contrasts, which lead to non-equilibrium conditions. If the local equilibrium assumption does not hold, the different stages for the typical problem considered here are the following:

Fig. 4. Normalized temperature versus position, for three different times (triangle, direct numerical simulation=ðhT b ib  T C Þ=ðT H  T C Þ; circles, direct numerical simulation =ðhT r ir  T C Þ=ðT H  T C Þ; solid line, local-equilibrium model=ðhTi  T C Þ= ðT H  T C Þ.

 early stages: initial conditions with sharp gradients and the vicinity of boundaries create non-equilibrium situations that are difficult to homogenize. They may be modelled through modified boundary conditions [22,7], mixed models (i.e., a small domain keeping pore-scale description such as in [3]),  two-equation behaviour: in general, the initial sharp gradients are smoothed after some time and more homogenizable conditions are found. Different models may be used: mixed models, different types of two-equation models (see a review and discussion in [32]), or more sophisticated equations in [41]. Two-equation models may be more or less sophisticated, for instance, two-equation models with first order exchange terms [5,31,29,46,47] or two-equation models with more elaborate exchange terms like convolution terms that would model nonlocal and memory effects [19]. This is beyond the scope of this paper to develop such a theory for our double-diffusion problem,  asymptotic behaviour: if the medium has an infinite extent (this can also be mimicked by convective conditions at the exit for a sufficiently large domain), cross diffusion may lead to a so-called asymptotic behaviour which may be described by a one-equation model with a different effective thermal conductivity, larger than the local-equilibrium value. This asymptotic behaviour for dispersion problems has been investigated by several authors and the link between the one-equation model obtained and the properties of the two-equation model well documented [47,1,17]. The one-equation non-equilibrium model may be derived directly by a proper choice of the averaged concentration/temperature and deviations as in [17,20],  complex history: it must be emphasized that non-equilibrium models corresponding to the asymptotic behaviour require special situations to be valid. If events along the flow path change due to forcing terms like source terms, heterogeneities, boundaries, the conditions leading to the asymptotic behaviour are disturbed and a different history develops. This is what happened in our test case. The boundary effects dampened the asymptotic behaviour that has probably taken place in our system (in the absence of an interpretation with two-equation models or one-equation asymptotic models, we cannot distinguish between the two possibilities, while the large extent of the domain has probably favoured an asymptotic behaviour) and this led to a steady-state situation well described by the local-equilibrium model. This possibility has not been seen by

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many investigators. However, it must be taken into account for practical applications. Hence, for our test case, it would be better to use a two-equation model, which truly embeds the one-equation local-equilibrium model, than the asymptotic model that would fail to catch the whole history. Given our objectives, it is beyond the scope of this paper to develop all the models discussed above, and we will only work with the local-equilibrium assumption. Hence, we now have at our disposal mapping vectors that give the local temperature fields in terms of the averaged value. It is important to remark that the upscaling of the heat equation problem has been solved independently from the solute transport problem. This feature is a key approximation that will simplify the treatment of the solute transport equation as explained in the next section. 3.3. Transient convection and diffusion mass transport In this section we have applied the volume averaging method to solute transport with Soret effect in the case of a homogeneous medium in the b-phase. We begin our analysis with the definition of two spatial decompositions for local concentration and velocity

cb ¼ hcb ib þ ~cb ;

v b ¼ hv b ib þ v~ b

ð38Þ

Therefore, the volume average theorem [44] can be applied to mass transport equation (5) to yield

  @hcb i þ hv b ib  rhcb ib ¼ r  Db rhcb ib þ e1 b r @T ! Z Db  nbr ~cb dA þ e1 b reb V Abr b

 ðDb rhcb ib Þ þ r  ðDTb rhT b ib Þ ! Z DTb 1 e nbr T b dA þ eb r  V Abr b

1 b

ð39Þ

Here, we have assumed that, as a first approximation, ~ b i ¼ 0 and h~cb i ¼ 0. Therefore, the volume averaged convective hv transport has been simplified to

ð40Þ

Subtracting Eq. (39) from Eq. (5) yields the governing equation for ~cb

@ ~cb ~ b  rhcb ib ¼ r  ðDb r~cb Þ þ r  ðDTb r Te b Þ þ v b  r~cb þ v @t ! Z Db ~ r  n dA c þ e1 br b b V Abr

DTb V

<< r  ðDb r~cb Þ

nbr ~cb dA

Abr

Z

ð42Þ

! nbr Te b dA

<< r  ðDTb r Te b Þ

ð43Þ

Ab r

The fourth and the sixth terms in the right hand side of Eq. (41) are diffusion and thermodiffusion sources. On the basis of the lengthscale constraint given by lb  Le we can discard these two terms. This constraint is automatically satisfied in homogeneous porous media for which Le is infinite [44]. The last term in the right hand side of this equation is the nonlocal convective transport and can be neglected whenever lb  L. The closure problem for ~cb will be quasi-steady whenever the constraint

l2b 1 Db t

ð44Þ

is satisfied. Therefore, the quasi-steady closure problem for the spatial deviation concentration takes the form

vb  r~cb þ v~ b  rhcb ib ¼ r  ðDb r~cb Þ þ r  ðDTb r Te b Þ

ð45Þ

and the associated boundary conditions as

BC1 : nbr  ðDb r~cb þ DTb r Te b Þ ¼ nbr  ðDb rhcb ib þ DTb rhT b ib Þ; at Abr BC2 : ~cb ¼ f ðr; tÞ; at Abr

ð46Þ ð47Þ

We wish to solve the closure problem in some representative region. So, we must discard the boundary condition given by (47) and replace it with some local condition associated with the representative region. This naturally leads us to treat the representative region as a unit cell in a spatially periodic model of a porous medium [44]. Therefore, boundary condition given by Eq. (47) takes the form

i ¼ 1; 2; 3

ð48Þ

~cb ¼ bCb  rhcb ib þ bSb  rhTi

ð49Þ

in which bCb and bSb are referred to as the closure variables which are specified by the following boundary value problems Problem IIa

vb  rbCb þ v~ b ¼ Db r2 bCb

ð50Þ

BC : nbr  Db rbcb ¼ nbr Db ; at Abr

ð51Þ

Periodicity : bcb ðr þ li Þ ¼ bcb ðrÞ; i ¼ 1; 2; 3

ð52Þ

b

Averages : hbcb i ¼ 0

ð53Þ

Problem IIb

vb  rbSb ¼ Db r2 bSb þ DTb r2 bTb

b 1 þ e1 b reb  ðDb rhc b i Þ þ eb r ! Z DTb  nbr Te b dA V Abr

ð54Þ

BC : nbr  ðDb rbSb þ DTb rbTb Þ ¼ nbr  DTb ; at Abr

ð55Þ

Periodicity : bSb ðr þ li Þ ¼ bSb ðrÞ;

ð56Þ

Averages : hbSb ib ¼ 0

i ¼ 1; 2; 3

ð57Þ

e b from decomposition equations into Eq. By substituting ~cb and T (39) and imposing the local equilibrium condition the closed form of the convection–double diffusion equation can be expressed by

b þ e1 b reb  ðDTb rhT b i Þ

~ ~ þ e1 b r  hc b v b i

e1 b r

!

Z

Here the single non-homogeneous term in the local closure problem is proportional to rhcijx and rhTijx so we can express ~cb as

þ e reb  ðDTb rhT b i Þ þ e r

~ bi hcb v b i ¼ eb hv b ib hcb ib þ h~cb v

e

Db r V

Periodicity : ~cb ðr þ li Þ ¼ ~cb ðrÞ;

1 b

~bi  h~cb v

1 b

ð41Þ

The non-local diffusion and thermodiffusion terms can be discarded on the basis of

    @heb cb ib þ r  eb hv b ib hcb ib ¼ r  eb Db  rhcb ib þ eb DTb  rhTi @t ð58Þ

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Db ¼ Db I þ 1 Vb

DTb ¼ Db

1 Vb Z

!

Z

~ b bCb ib nbr bCb dA  hv

Abr

!

nbr bSb dA þ DTb I þ

Abr

1 Vb

ð59Þ Z

! nbr bTb dA

Ab r

b

~ b bSb i  hv

ð60Þ

4. Results In order to illustrate the main features of the proposed multiple scale analysis, we have solved the dimensionless form of the closure problems (I, IIa, IIb) on the simple unit cell, Fig. 2, to determine the effective properties. If we treat the representative region as a unit cell in a spatially periodic porous medium, we can replace the boundary condition imposed at Abe with a spatially periodic condition on ~cb [44]. The unit cell used to compute the effective coefficients is the symmetrical cell shown in Fig. 2, for an ordered porous medium (in line arrangement of circular cylinders). This type of geometry has already been used for many similar problems [30,44]. Then, the large scale effective properties are determined by Eqs. (35), (59), and (60). For this illustration we have fixed the fluid mixture properties at ðqcp Þr =ðqcp Þb ¼ 1. The numerical simulations have been done using COMSOL Multiphysics finite elements code. In this study, we have just calculated the longitudinal coefficients which will be needed to simulate a test case for the macroscopic, onedimensional equation.

Here, the parameters stared are the effective coefficients and the others are the coefficient in the free fluid. This relationship is similar to the one obtained for effective diffusion and thermal conductivity in the literature [30,44]. Therefore, we can say that the tortuosity factor acts in the same way on Fick diffusion coefficient and on thermodiffusion coefficient. In this case, the tortuosity is defined as

I

s

¼Iþ

1 Vb

Z Abr

a

DTb

¼

eb kb

¼

ð63Þ

1.E+00

1.E-01

1.E+00

1.E+01

1.E+02

Pe

b K *β ε β kβ

1.E+02 1.E+01 1.E+00 1.E-01 1.E-02

1.E-01

1.E+00

1.E+01

1.E+02

Pe

c

Diffusive regime 5

-5

1.E-02

DTβ

I

ð61Þ

s

ð62Þ

1.E+01

D

¼

 kb

nbr bCb dA

Abr

1.E+02

* Tβ

Db

DTb

Z



1.E-01 1.E-02

In this section, the solid thermal conductivity is assumed to be very small and will be neglected in the equations. The equivalent of the closure problem in this case (kr  0) is presented in Appendix A. One can find in the literature several expressions for the effective diffusion coefficient base on the porosity, such as Wakao and Smith [54] Db ¼ eb Db , Weissberg [55] Db =Db ¼ 1=ð1  0:5 ln eb Þ, Maxwell (1881) Db =Db ¼ 2=ð3  eb Þ (see Quintard [29]). For isotropic systems one may write Db =Db as I/s, where s is the scalar tortuosity of the porous matrix. Fig. 5 shows our results of the closure problem resolution (A.I, A.IIa and A.IIb) in the case of pure diffusion (Pe = 0). As shown in Eq. (61), we have found that the effective thermodiffusion coefficient also can be estimated with this single tortuosity coefficient.

1 Vb

Db kb 1 þ 0:0234Pe1:70 ¼ ¼ Db eb kb 1:20

4.1. No-conductive solid-phase (kr  0)

Db

nbr bTb dA ¼ I þ

The results with convection (Pe – 0), are illustrated in Fig. 6. One can see that for low Péclet number (diffusive regime) the ratio of effective diffusion coefficient to molecular diffusion coefficient in the porous medium is almost constant and equal to the inverse of the tortuosity of the porous matrix, which is consistent with previously published results. On the opposite, for high Péclet numbers, the above mentioned ratio changes following a powerlaw trend after a transitional regime. The curves of dispersion (Fig. 6a) and thermal diffusivity (Fig. 6b) have the classical form of dispersion curves [44]. In our case, they can be represented by

D*β Dβ

where the total dispersion and total thermal–dispersion tensors are defined by

1.E-01

1.E+00

1.E+01

1.E+02

-15 -25

Convective regime

-35 -45 -55

0.6

0.6

S T* ST

ε β kβ DTβ

I D*Tβ

Dβ

= D*β

1.0

d

0.8

=

k *β

=

τ

1.0

0.4 0.2

Diffusive regime

0.2 -0.2 1.E-02 -0.6

1.E-01

1.E+00

-1.0

0.0 0.0

0.2

0.4

0.6

0.8

1.0

1.E+01

1.E+02

Convective regime

Pe

(Porosity) Fig. 5. Effective diffusion, thermodiffusion and thermal conductivity coefficients at Pe = 0.

Fig. 6. Effective, longitudinal coefficients as a function of Péclet number (kr  0): (a) molecular diffusion coefficient, (b) thermal diffusivity coefficient, (c) thermodiffusion coefficient and (d) Soret number.

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where the dimensionless Péclet number is defined as

Pe ¼

hv b ib lUC Db

ð64Þ

The dispersive part of the effective thermal diffusion coefficient decreases with the Péclet number (Fig. 6c), and for high Péclet number it becomes negative. As we can see in Fig. 6c there is a change of sign of the effective thermodiffusion coefficient. This phenomenon may be explained by the fact that, by increasing the fluid velocity, the gradient of bTb (or temperature field) changes gradually its direction to the perpendicular flow path which could lead to a reversal bsb distribution, and, as a result, a change of the DTb sign. This curve can be fitted with a correlation as

DTb 1 ¼  0:0052Pe2:00 DTb 1:20

ð65Þ

The results in terms of Soret number, which is the ratio of isothermal diffusion coefficient on thermodiffusion coefficient, are original. Fig. 6d shows the ratio of effective Soret number to the Soret number in free fluid as a function of the Péclet number. The results show that, at diffusive regime, one can use the same Soret number in porous media as the one in the free fluid ðST =ST ¼ 1Þ. This result agrees with the experimental results of Platten and Costesèque [24] and Costeseque et al. [8] but, for convective regimes, the effective Soret number is not equal with the one in the free fluid. For this regime, the Soret ratio decreases with increasing the Péclet number, and for high Péclet number it becomes negative. To test the accuracy of the numerical solution, we have solved the steady-state vectorial closures A.I, A.IIa and A.IIb analytically for a plane Poiseuille flow between two horizontal walls separated by a gap H. For this case, we found the following relation between the effective thermal diffusion and the thermal diffusion in the free fluid (eb = 1)

DTb Pr Pe2 ¼1  Sc 210 DTb

ð66Þ

Fig. 7 shows the dependence of the effective tensors with the conductivity ratio, for different Péclet number. As shown in Fig. 7a, the effective conductivity initially increases with an increase in j and then reaches an asymptote. As the Péclet number increases, convection dominates and the effect of j on  k =eb kb is noticeably different. The transition between the high and low Péclet number regimes occurs around Pe = 10 (see also  [15]). For higher Péclet numbers (Pe > 10), k =eb kb is enhanced by lowering j, as shown in Fig. 7a for Pe = 14.  Our results for DTb =DTb have a similar behaviour as k =eb kb . Fig. 7b shows the influence of the conductivity ratio on the effective thermodiffusion coefficients for different Péclet numbers. One can see that increasing the solid thermal conductivity increases the value of the effective thermodiffusion coefficient for low Péclet number. Contrary, for high Péclet number (Pe > 10) increasing thermal conductivity ratio decreases the absolute value of the effective thermodiffusion coefficient. As shown in Fig. 7b thermal conductivity ratio has no influence on thermodiffusion coefficients for pure diffusion case (Pe = 0). To summarize our findings, the results show that for low Péclet numbers the effective thermodiffusion coefficient is the same as the effective diffusion coefficient and that it does not depend on the conductivity ratio. However, in this regime, the effective thermal conductivity changes with the conductivity ratio. On the opposite, for high Péclet numbers, both the effective diffusion and thermal conductivity increase following a power-law trend, while the effective thermodiffusion coefficient decreases. In this regime, a change of the conductivity ratio will change the effective thermodiffusion coefficient as well as the effective thermal conductivity coefficient. Seeing that, the ‘‘solid–solid contact effect” has a great consequence on the effective thermal conductivity [29,38], it can also change the effective thermodiffusion coefficient. In particular, the results may be completely different, when the fluid phase is completely trapped in the solid phase. Study of this phenomenon or in more general case thermodiffusion, in fractured porous media can be important enough to deserve a specific study.

and the longitudinal dispersion is given [56] by

Db Pe2 ¼1þ Db 210

ð67Þ

a

3.1

Pe=0 b

2.7

b

Here, Pe is defined as Pe ¼ hv zb i H=Db where hv zb i is the z-component of the intrinsic average velocity of the fluid. The predicted values agree with the analytical results.

Pe=5

k * kβ

2.3

4.2. Conductive solid-phase (kr – 0)

Pe=8

1.9

Pe=10

1.5

Pe=14

1.1 0.7 1.E-03

1.E-02

1.E-01

1.E+00

1.E+01

1.E+02

kσ k β

b DT* β DTβ

In the previous section, we made the assumption kr – 0 only for simplification whereas for example, the soil thermal conductivity is about 0.52 W/m K, and it varies with the soil texture. The thermal conductivity of most common non-metallic solid materials is about 0.05–20 W/m K, and this value is very large for metallic solids [15]. Values of kb for most common organic liquids range between 0.10 and 0.17 W/m K at temperatures below the normal boiling point, but water, ammonia, and other highly polar molecules have values several times as large [26]. The increase of the effective thermal conductivity when increasing the phase conductivity ratio, j, is well established from experimental measurements and theoretical approaches ([15,44]) but the influence of this ratio on thermodiffusion is yet unknown. In this section, we study the influence of the conductivity ratio on the effective thermodiffusion coefficient. To achieve that, we solved numerically the closure problems with different conductivity ratios.

0.9 0.7 0.5 0.3 0.1 -0.2 -0.4 -0.6 -0.8 -1.0 1.E-03

Pe=0 Pe=5 Pe=8 Pe=10 Pe=14

1.E-02

1.E-01

1.E+00

1.E+01 1.E+02

kσ k β Fig. 7. The influence of conductivity ratio (j) on (a) effective, longitudinal thermal conductivity and (b) effective thermodiffusion coefficients.

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y

TH = 1

Initial condition ( T0 = 0 & c0 = 0 )

TC = 0

x

Danckwerts B.C. for concentration field

Danckwerts B.C. for concentration field

Fig. 8. Schematic of a spatially periodic porous medium (TH: hot temperature and TC: cold temperature).

ð68Þ

BC2 : x ¼ 15; nbr  ðrcb þ wrT b Þ ¼ 0 and T ¼ T C ¼ 0 IC : t ¼ 0; c ¼ c0 ¼ 0 and T ¼ T 0 ¼ 0

ð69Þ ð70Þ

Mass fluxes are taken equal to zero on other outside boundaries and on all fluid–solid boundary surfaces. Zero heat flux was used on the outside boundary except at the entrance and exit boundaries where we have imposed a thermal gradient. In the case of a conductive solid-phase, classical continuity conditions were imposed on the fluid–solid boundary surface while these surfaces will be adiabatic for a non-conductive solid-phase. Macroscopic fields are also obtained using the dimensionless form of Eqs. (16), (17), (34), and (58). We obtained, from a method for predicting the permeability tensor [28], a Darcy number equal to Da ¼ K b =l2UC ¼ 0:25, for the symmetric cell shown in Fig. 2. The boundary condition at the exit and entrance of the macroscale domain were taken similar to the pore scale expressions but in terms of the averaged variables. Depending on the pressure boundary condition and therefore Péclet numbers, we can have different flow regimes. First, we assume that the solid phase is not conductive (Section 5.1) and we compare the results of the theory with the direct simulation. Then, the comparison will be done for a conductive solid-phase (Section 5.2) and different Péclet numbers. In all cases the micro-scale values are cell averages obtained from the micro-scale fields. 5.1. Non-conductive solid-phase (kr  0) In this section, the solid thermal conductivity is assumed to be very small and will be neglected in the equations and the solidphase energy equation is not solved. The various contributions of the fluid flow including dispersion can be expressed.

Averaged micro

a

Mass fraction at the exit

BC1 : x ¼ 0; nbr  ðrcb þ wrT b Þ ¼ 0 and T ¼ T H ¼ 1

Prediction macro

ψ =1 0.5 0.4 0.3 0.2 0.1

ψ=0

0.0 -0.1

0 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300

Time

b

Volume averaged temperature

In order to validate the theory developed by the up-scaling technique in the previous sections, we have compared the results obtained by the macro-scale equations with direct simulations. The porous medium is made of an array of unit cell described in Fig. 2. The array is chosen with 15 unit cells, as illustrated in Fig. 8. In the macro-scale problem, the effective coefficients are obtained from the previous solution of the closure problem. The macroscopic, effective coefficients are the axial diagonal terms of the tensor. Given the boundary and initial conditions, the resulting macro-scale problem is one-dimensional. Calculations have been carried out in the case of a binary fluid mixture with simple properties such that, w ¼ DT  DTb =Db ¼ 1 and ðqcp Þr ¼ ðqcp Þb . Microscopic scale simulations, as well as the resolution for the macroscopic problem, have been performed using COMSOL Multiphysics. The 2D pore-scale dimensionless equations and boundary conditions to be solved are Eqs. (1)–(9). Velocity was taken to be equal to zero (no-slip) on every surface except at the entrance and exit boundaries. Danckwerts condition [49] was imposed for the concentration at the entrance and exit (Fig. 8). In this dimensionless system, we have imposed a thermal gradient equal to one.

5.1.1. Pure diffusion (Pe = 0) We have first investigated the Soret effect on mass transfer in the case of a static homogeneous mixture. In this case, we have imposed a temperature gradient equal to one, in the dimensionless system, for the microscopic and macroscopic models, and we have imposed a Danckwerts boundary condition for concentration at the medium entrance and exit. The porosity of the unit cell is equal to 0.8 and, therefore, in the case of pure diffusion, the effective coefficients (diffusion, thermal conductivity and thermodiffusion) have been calculated with a single tortuosity coefficient equal to 1.20 as obtained from the solution of the closure problem shown in Fig. 5. Fig. 9a shows the temporal evolution of the concentration at the exit for the two models, microscopic and macroscopic, with (w = 1) and without (w = 0) thermodiffusion. One can see that thermodiffusion modifies the local concentration and we cannot ignore this effect. The maximum modification at steady-state is equal to w. We also see that the theoretical

Lines: prediction macro Points: averaged micro

0.9 0.7 t=1 0.5 0.3

t=10

t=30 t=300

0.1 -0.1

t=0

0

1

2

3

4

5

6

7

8

9 10 11 12 13 14 15

x

c

Volume averaged concentration

5. Macroscopic simulation and validation (comparison with numerical experiments)

0.5

Lines: prediction macro Points: averaged micro

0.4 0.3 0.2

t=10

t=1

0.1 0

t=300

t=30

t=0

-0.1 -0.2 -0.3 -0.4 -0.5

0

1

2

3

4

5

6

7

8

9 10 11 12 13 14 15

x Fig. 9. Comparison between theoretical and numerical results at diffusive regime and j = 0, (a) time evolution of the concentration at x = 15 and (b and c) instantaneous temperature and concentration field.

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H. Davarzani et al. / International Journal of Heat and Mass Transfer 53 (2010) 1514–1528

results are here in excellent agreement with the direct simulation numerical results. Fig. 9b and c shows the distribution of temperature and concentration in the medium at given times. Here also, one can observe the modification of the concentration profile generated by the Soret effect compared with the isothermal case (c = 0). The macroscopic model also perfectly fits the microscopic results. 5.1.2. Diffusion and convection (Pe – 0) Next, we have imposed different pressure gradients on the system shown in Fig. 8. The temperature and concentration profiles at Pe = 1 for different times are shown in Fig. 10a and b, respectively. The results show a significant change in the concentration profile because of species separation when imposing a thermal gradient. Here, also, the theoretical predictions are in very good agreement with the direct simulation of the micro-scale problem. Comparison of the concentration evolution curves at x = 0.5, 7.5 and 13.5 in Fig. 10c between the two regimes (with and without thermodiffusion) also shows that the evolution curve for no-thermodiffusion is different from the one with thermodiffusion. The shape of these curves are very different from the pure diffusion case (Fig. 9a) because, in this case, the thermodiffusion process is affected by forced convection. One also observes a very good agree-

Lines: prediction macro Points: averaged micro t=50

1

t=16

0.8

t=13

0.6

t=10 t=4 t=7

0.2 t=0

0

-0.2

0

2

4

6

8

10

12

14

b

Volume averaged concentration

x 0.8

Lines: prediction macro Points: averaged micro

0.6

t=4

we know that for pure diffusion, increasing the conductivity ratio increases the effective thermal conductivity. According to Fig. 7a  for Pe = 0 and j = 10, we obtain k ¼ 1:72eb kb . Figs. 11a and 9b show the temporal change in temperature and concentration profile for both models. The symbols represent the direct numerical results (averages over each cell) and the lines are the results of one-dimensional macro-scale model. One sees that, for a conductive solid-phase, our macro-scale predictions for concentration and temperature profiles are in excellent agreement with the micro-scale simulations.

a

t=13

t=1 t=0

0

5.2.1. Pure diffusion In this example, the pure diffusion (Pe = 0) problem has been  solved for a ratio of conductivity equal to 10 j ¼ kr =kb ¼ 10 . In this condition the local thermal equilibrium is valid as shown in Quintard et al. [52]. Then, we can compare the results of the micro-scale model and the macro-scale model using only one effective thermal conductivity (local thermal equilibrium). We have shown in Section 5.2 that the thermal conductivity ratio has no influence on the effective thermodiffusion coefficient for diffusive regimes. Therefore, we can use the same tortuosity factor for the effective diffusion and the thermodiffusion coefficient that the one used in   the previous section Db ¼ 0:83Db and DTb ¼ 0:83DTb . Whereas,

t=10 t=7

0.4 0.2

Thermal properties of the solid matrix have also to be taken into account in the thermodiffusion process. In this section, the heat diffusion through the solid-phase is considered. Therefore, the comparison has been done for the same micro-scale model but with a conductive solid-phase. First, we compare the results for a pure diffusion system and then we will describe the local dispersion, coupling with Soret effect.

t=16 t=50

-0.2 -0.4 -0.6 -0.8 0

2

4

6

8

10

12

14

c

Volume averaged concentration

x 0.8

Macro (x=0.5)

0.6

Macro (x=7.5)

Lines: prediction macro Points: averaged micro

0.9 0.7 t=1

0.5 t=30

t=10

0.3

t=300

0.1 -0.1 0

t=0

1

23

45

67

8

9 10 11 12 13 14 15

x

Macro (x=13.5)

0.4

Micro (x=0.5)

0.2

Micro (x=7.5)

0.0

Micro (x=13.5) Isothermal

-0.2 -0.4 -0.6 -0.8

Volume averaged temperature

t=1

0.4

5.2. Conductive solid-phase (kr – 0)

0

5

10 15

20 25 30

35 40

45 50

Time

b

Volume averaged concentration

Volume averaged temperature

a

ment between the micro-scale simulations and the macro-scale predictions.

0.5 0.4 0.3 0.2 0.1

t=10

t=1

t=30

t=0

0

-0.1 -0.2 -0.3 -0.4 -0.5 0

t=300

Lines: prediction macro Points: averaged micro

1

2

3

4

5

6

7

8

9 10 11 12 13 14 15

x Fig. 10. Comparison between theoretical and numerical results, j = 0 and Pe = 1, (a) time evolution of the concentration at x = 0.5, 7.5 and 13.5 (b and c) instantaneous temperature and concentration field.

Fig. 11. Comparison between theoretical and numerical results at diffusive regime and j = 10, temporal evolution of (a) temperature and (b) concentration profiles.

H. Davarzani et al. / International Journal of Heat and Mass Transfer 53 (2010) 1514–1528

In order to better understand the effect of the thermal conductivity ratio on the thermal diffusion process, we have plotted in Fig. 12 the temperature and concentration profiles for different thermal conductivity ratios, at a given dimensionless time (t = 10). As shown in this figure, a change in thermal conductivity modifies the temperature profiles (Fig. 12a) and, consequently, the concentration profiles (Fig. 12b). Since we showed that the thermal diffusion coefficient is constant in the pure diffusion case, we conclude that modifications in concentration for different thermal conductivity ratio come from changes in the temperature profiles. These modifications are especially distinguishable in Fig. 12c which shows the time evolution of the concentration at x = 15 for different thermal conductivity ratios. 5.2.2. Coupling between forced convection and Soret effect A comparison has been made for Pe = 1 and a thermal conductivity equal to 10. The temperature and concentration profiles for different times are shown in Fig. 13a and b, respectively. Here, also, the theoretical predictions are in very good agreement with the direct simulation of the micro-scale problem. Fig. 14a shows the effect of the Péclet number on the axial temperature distribution in the medium.

Volume averaged temperature

a

At small Pe, the temperature distribution is linear but as the pressure gradient (or Pe) becomes large, convection dominates the axial heat flow. In Fig. 14b the steady-state distribution of the concentration is plotted for different Péclet numbers. One can see clearly that the concentration profile changes with the Péclet number. For example, for Pe = 2, because the medium has been homogenized thermally by advection in most of the porous domain, the concentration profile is almost the same as in the isothermal case (without thermodiffusion). Near the exit boundary, there is a temperature gradient which generates a considerable change in the concentration profile with an optimum point. This peak is a dynamic one resulting from coupling between convection and Soret effect. If we define a new parameter, AS, named segregation area defined as the surface between isothermal and thermodiffusion case on concentration profiles we can see that increasing the Péclet number decreases the segregation area. As shown in Fig. 15 the concentration profile, and consequently the peak point, not only depends on the Péclet number, but also it is been changed by the conductivity ratio (j) and separation factor, w. The concentration profile in the case of Pe = 0.75 and j = 1 for different separation factors, w, has been plotted in Fig. 15a. One

Prediction macro,

t=10

0.9

1525

Prediction macro, 0.7

Prediction macro, Averaged micro,

0.5

Averaged micro,

0.3

Averaged micro, 0.1 -0.1

0

1

2

3

4

5

6

7

8

9 10 11 12 13 14 15

x

b Volume averaged concentration

0.5

t=10

0.4 0.3

Prediction macro,

0.2

Prediction macro,

0.1

Prediction macro,

0 -0.1

Averaged micro,

-0.2

Averaged micro,

-0.3

Averaged micro,

-0.4 -0.5 0

24

6

8

10

12

14

x

Mass fraction at the exit

c

Prediction macro,

0.5 0.4

Prediction macro,

0.3

Prediction macro,

0.2

Averaged micro,

0.1

Averaged micro,

0.0 -0.1

Averaged micro, 0

40

80

120

160

200

240

280

Time Fig. 12. Effect of thermal conductivity ratio at diffusive regime on (a and b) instantaneous temperature and concentration field at t = 10 and (b) time evolution of the concentration at x = 15.

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H. Davarzani et al. / International Journal of Heat and Mass Transfer 53 (2010) 1514–1528

Lines: prediction macro Points: averaged micro

Volume averaged temperature

a

t=70

1 0.8

t=16

0.6 0.4

t=4 t=7

0.2

t=0

0 -0.2

t=13

t=10 t=1

0

2

4

6

8

10

12

14

x

b

Lines: prediction macro Points: averaged micro

Volume averaged concentration

0.8 0.6

t=10 t=7

0.4

t=13 t=16

t=4 0.2

t=1 t=0

0

t=70

-0.2 -0.4 -0.6 -0.8

0

2

46

8

10

12

14

x

Volume averaged concentration

c

0.8

Macro (x=0.5)

0.6

Macro (x=7.5) Macro (x=13.5)

0.4

Micro (x=0.5)

0.2

Micro (x=7.5)

0.0

Micro (x=13.5) Isothermal

-0.2 -0.4 -0.6 -0.8

0

5

10

15

20

25

30

35

40

45

50

Time Fig. 13. Comparison between theoretical and numerical results, j = 10 and Pe = 1, (a) time evolution of the concentration at x = 0.5, 7.5 and 13.5 (b and c) instantaneous temperature and concentration field.

can see that increasing the separation factor increases the local segregation area of species. Fig. 15b shows the influence of conductivity ratio for a fixed Péclet number and separation factor (Pe = 2 and w = 1) on the concentration profile near the exit boundary (x between 10 and 15). The results show that a high conductivity ratio leads to smaller optimum point but higher segregation area than the ideal non-conductive solid-phase case. This means that the segregation area will be a function of wPej. This specific result should be of importance in the analysis of species separation and especially in thermogravitational column, filled with a porous medium.

6. Conclusion In this study, we have determined the effective Darcy-scale coefficients for heat and mass transfer in porous media using a volume averaging technique. We showed that the effective Soret number may depart from the micro-scale value because of advec-

tion effects. For these convective regimes, it is shown that the effective thermodiffusion coefficient depends in a complex manner of the pore-scale properties (geometry, conductivity ratio). The results have shown that, for low Péclet numbers, the effective Soret number in porous media is the same as the one in the free fluid and that it does not depend on the conductivity ratio. On the opposite, in convective regimes, the effective Soret number decreases. In this case, a change of conductivity ratio will change the effective thermodiffusion coefficient as well as the effective thermal conductivity coefficient. A good agreement has been found between macro-scale resolutions and micro-scale, direct simulations, which validates the proposed theoretical model. We have presented a situation illustrating how variations of Péclet number, conductivity ratio and separation factor coupled with Soret effect can change locally the segregation of species in a binary mixture. This may be of a great importance when evaluating the concentration in applications like reservoir engineering, waste storage, and soil contamination.

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a Volume averaged temperature

1

Macro (pe=0.001) Macro (pe=0.1)

0.8

Macro (pe=0.25) Macro (pe=0.75)

0.6

Macro (pe=2) Micro (pe=0.001)

0.4

Micro (pe=0.1) Micro (pe=0.25)

0.2 0

Micro (pe=0.75) Micro (pe=2)

0

2

4

6

8

10

12

14

16

x

b Volume averaged concentration

0.5

Macro (pe=0.001) Macro (pe=0.1)

=1

0.3

Macro (pe=0.25) Macro (pe=0.75)

0.1

Macro (pe=2) Micro (pe=0.001)

-0.1

Micro (pe=0.1) Micro (pe=0.25)

-0.3

Micro (pe=0.75) Micro (pe=2)

-0.5

0

2

4

6

8

10

12

14

16

x Fig. 14. Influence of Péclet number on (a) temperature and (b) concentration profiles (j = 10).

Volume averaged concentartion

a 0.2

0 -0.2

Pe=0.75 =1

-0.4 -0.6 -0.8 0

3

6

9

12

15

13

14

15

x

Volume averaged concentartion

b

0.2 0.1 0 -0.1

Pe=2 =1

-0.2 -0.3 -0.4 10

11

12

x Fig. 15. Influence of (a) separation factor and (b) conductivity ratio on pick point of the concentration profile.

Appendix A. The closure problem in the case of kr  0 In this appendix, we have listed the closure problems and effective coefficients in the case of kr  0. Problem A.I: the closure problem for effective thermal conductivity coefficient

e b ¼ kb r2 bTb ðqcp Þb v b  rbTb þ ðqcp Þb v

ðA:1Þ

BC1 : nbr  rbTb ¼ nbr ; at Abr

ðA:2Þ

Periodicity : bTb ðr þ li Þ ¼ bTb ðrÞ; b

Averages : hbTb i ¼ 0

i ¼ 1; 2; 3

ðA:3Þ ðA:4Þ

1528

H. Davarzani et al. / International Journal of Heat and Mass Transfer 53 (2010) 1514–1528

Problem A. IIa: the closure problem for effective diffusion coefficient

v b  rbCb þ ve b ¼ Db r2 bCb BC : nbr  Db rbCb ¼ nbr  Db ; at Abr Periodicity : bCb ðr þ li Þ ¼ bCb ðrÞ; i ¼ 1; 2; 3

ðA:5Þ ðA:6Þ ðA:7Þ

Averages : hbCb ib ¼ 0

ðA:8Þ

Problem A.IIb: the closure problem for effective thermodiffusion coefficient

v b  rbSb ¼ Db r2 bSb þ DTb r2 bTb BC : nbr  ðDb rbSb þ DTb rbTb Þ ¼ nbr  Db ; at Abr Periodicity : bSb ðr þ li Þ ¼ bSb ðrÞ; i ¼ 1; 2; 3

ðA:9Þ ðA:10Þ ðA:11Þ

Averages : hbSb ib ¼ 0

ðA:12Þ

and the effective coefficients are calculated as

! Z 1 e b bTb i ¼ kb eb I þ nbr bTb dA  ðqcp Þb h v ðA:13Þ V Ab r ! Z 1 e b bCb ib nbr bCb dA  h v ðA:14Þ Db ¼ Db I þ V b Abr ! ! Z Z 1 1 nbr bSb dA þ DTb I þ nbr bTb dA DTb ¼ Db V b Ab r V b Ab r e b bSb i ðA:15Þ  hv

 kb

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