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