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International Journal of Thermal Sciences 80 (2014) 108e117

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International Journal of Thermal Sciences

journal homepage: www.elsevier .com/locate/ i j ts

Experimental, theoretical and numerical interpretationof thermodiffusion separation for a non-associating binary mixturein liquid/porous layers

Amirhossein Ahadi a, C. Giraudet b, H. Jawad a, F. Croccolo b, H. Bataller b, M.Z. Saghir a,*aMechanical and Industrial Engineering Department, Ryerson University, 350 Victoria St, Toronto, ON M5B 2K3, Canadab Laboratoire des Fluides Complexes et leurs Réservoirs e UMR 5150, Université de Pau et des Pays de l’Adour, BP 1155, F-64013 Pau Cedex, France

a r t i c l e i n f o

Article history:Received 10 August 2013Received in revised form5 February 2014Accepted 9 February 2014Available online 13 March 2014

Keywords:ThermodiffussionBinary liquidsPorous mediaMulti-layerOptical interferometry experimentCFD

* Corresponding author.E-mail addresses: zsaghir@ryerson.ca, zsaghir@gm

http://dx.doi.org/10.1016/j.ijthermalsci.2014.02.0031290-0729/� 2014 Elsevier Masson SAS. All rights re

a b s t r a c t

Thermodiffusion in a hydrocarbon binary mixture has been investigated experimentally and numericallyin a liquid-porous cavity. The solutal separation of the 50% toluene and 50% n-hexane binary mixtureinduced by a temperature difference at atmospheric pressure has been performed in a new thermo-diffusion cell. A new optimized cell design is used in this study. The inner part of the cell is a cylindricalporous medium sandwiched between two liquid layers of the same binary hydrocarbon mixture.Experimental measurement and theoretical estimation of the molecular diffusion and thermodiffusioncoefficients showed a good agreement. In order to understand the different regimes occurring in thedifferent parts of the cell, a full transient numerical simulation of the solutal separation of the binarymixture has been performed. Numerical results showed that the lighter species, which are of n-hexanemigrated toward the hot surface, while the denser species, which is toluene migrated towards the coldsurface. Also, it was found that a good agreement has been reached between experimental measure-ments and numerical calculations for the solutal separation between the hot and cold surface fordifferent medium porosity. In addition, we used the numerical results to analyse convection anddiffusion regions in the cell precisely.

� 2014 Elsevier Masson SAS. All rights reserved.

1. Introduction

The physical process of concentration gradient in a mixture dueto a temperature gradient is called thermodiffusion or the Soreteffect [1,2]. Soret effect in binary mixture is a subject of greatattention amongst scientists, researchers and engineers due to itswide range of applications in many engineering [3,4], natural [5,6]and geophysical applications [7]. These include oil reservoirs,storage of nuclear wastes, mineral migration and mass transfer inliving matters and many other applications. The ratio of the ther-modiffusion coefficient (DT), to the molecular diffusion coefficient(D), represents the Soret coefficient (ST), which can be stated as thefollowing [8,9]:

ST ¼ DT

D(1)

Buoyancy convection in a binary mixture is noticeably morecomplicated than in pure fluid due to variation of the density in the

ail.com (M.Z. Saghir).

served.

system. Consequently, an interaction between convection, ther-modiffusion and mass diffusion occurs when thermal gradientwould be applied in the system. The orientation of temperature andconcentration gradients plays a major role on the dynamics ofconvection in binary mixture and can be totally different fromthose induced by the thermal buoyancy [3,10,11]. Huke et al.concluded that the numerical investigation of convection in binarymixture is more complicated compared to pure fluid when theLewis number (Le) is much less than 0.01, which causes a noticeablenarrow boundary layer behaviour of the concentration field [12].The transport of solute due to the Soret effect for a mixture can beexpressed by the separation ratio and can be calculated from thefollowing equation:

q ¼ CB=ð1� CBÞCT=ð1� CT Þ

(2)

where CB and CT are the concentration of the lighter component atthe bottom and top horizontal surfaces, respectively. Saghir et al.reported that when the flow characteristic time equal to thethermal characteristic time, maximum separation occurs and

A. Ahadi et al. / International Journal of Thermal Sciences 80 (2014) 108e117 109

permeability and porosity plays a major role [13]. Due to theimportance of Soret effect in binary mixtures, the measurement ofthe Soret coefficient has gained an extensive attention assigned inboth experimental and theoretical studies [14e16]. Eslamian, Sri-nivasan and Saghir have listed some useful experimental worksperformed on binary hydrocarbon mixtures [3,15].

Van Vaerenbergh et al. measured the Soret coefficients formulti-components species and concluded that increasing thenumber of mixture components causes the theoretical model per-formance of the Soret effect to be less accurate [17e20]. Charrier-Mojtabi et al. concluded that for a cavity heated from above andwhen the separation ratio is greater than zero, the equilibriumsolution is linearly stable and when separation ratio is less thanzero a stationary or an oscillatory bifurcation occurs [21,22].Al-Salem et al. presented the effects of the addition of porous layeron heat transfer in a cross flow over a heated cylinder and foundthat with the increase of porosity in the porous layer increases theaverage Nusselt number [23]. The effect of permeability on con-centration distribution was investigated by Abbasi et al. [24].Results showed that at low permeability (0.0001e0.1 md) the Soreteffect phenomenon is dominant and as the permeability increases,the concentration distribution is greatly affected by mixing due toconvection.

The objective of the present study is to investigate the Soreteffect in a porous layer superimposed between two liquid layersboth experimentally and numerically and to study the effect oftemperature on separation process in the presence of thermo-diffusion in a hydrocarbon binary mixture. In addition a newoptimised design of Soret cell is introduced and examined usingexperimental and numerical result. The “I” shape cylindrical cellwith the specific aspect ratio is used in this study to minimizethe convection pattern inside the porous layer. Moreover, this cellconfiguration provides an improved a linear variation of thetemperature inside the cell. The solutal separation of the 50%toluene and 50% n-hexane binary mixture induced by a tem-perature difference at atmospheric pressure has been performed.The experimental procedure allows following the kinetic of theseparation, but only between two points for different porosity inthe system. However, it is possible to extract the diffusion coef-ficient and the Soret coefficient of the binary mixture. In order tounderstand the different regimes occurring in the different partsof the cell and their interaction and also to analyse thethermal performance of the cell design, a full transient numericalsimulation of the solutal separation in the experimental set-uphas been performed. In Section 2 the experimental set-up isdescribed, in Section 3 the experimental procedure isexposed, Section 4 explains the numerical approach, Section 5examines the theoretical estimation of the Soret coefficient andSection 6 discuss the results and the conclusion is presented inSection 7.

2. Experimental set-up

The experimental apparatus has been described in more detailsin previous works of Croccolo et al. [25e27]. The cell consists of astainless steel cylinder with two water circuits at its top andbottom. Pure water coming from two distinct thermostatic bathscirculates through these loops to keep the top and bottom of thecell at two distinct temperatures with a stability of about �0.1 Kover many days. Most of the interior volume of the cell is filled by aporousmedium consisting of a monolithic silica cylinder having thesame diameter as the cell interior. At the two extremities of theporous medium, two free volumes, which we refer to as deadvolumes (DVs) allow refractive index measurements by means ofinterferometry. The optical access to the two DVs is provided by

two opposing sapphire windows for each DV letting a laser beampass through the fluid perpendicular to the cell axis. As reportedelsewhere by Croccolo et al. the value of the optical path has beenobtained from the analysis of experimental data taken with puretoluene, hexane andwater, and it is d*¼ 21.5� 0.3 mm [25,26]. Thecell is designed to maintain a liquid mixture in a pressure rangebetween 0.1 and 100 MPa and in a temperature range between 5and 40 �C. The cell is externally covered by a ceramic insulator, inorder to limit heat transfer between the cell walls and theenvironment.

The cell is filled by means of a filling system consisting of arotary vacuum pump able to evacuate most of the air from the cellbefore filling operations; a fluid vessel at atmospheric pressure; amanual volumetric pump and a number of valves used to facilitatethe procedure. After a low vacuum is made inside the cell, themixture to be studied is transferred to the cell by letting it enterfrom its bottom side. In this phase, visual inspection through thesapphire windows is needed to check bubble presence. The cell isthen abundantly fluxed with about 100 ml of the fluid mixture.At the end of the procedure a valve at the end of the circuit isclosed and the volumetric pump is operated to modify the liquidpressure within the cell and perform the experimental runs. Amanometer (Keller, PAA-33X/80794, pressure range: 0.1e100 MPa,precision �0.03 MPa) is connected between the volumetric pumpand the cell in order to constantly check the pressure of the fluidmixture [26]. At the top and bottom of the cell, two K-type ther-mocouples are positioned within the two DVs in contact with theliquid.

The cell is inserted in a Mach-Zehnder interferometer, eachhalf of the laser beam crossing one DV as explained by Croccoloet al. [25]. The light source consists in an HeeNe laser (MellesGriot, 25 LHP 151e230) operating at a wavelengthofl0 ¼ 632.8 nm. The beam is deflected by a metallic mirror and ismade divergent by means of a positive lens (f ¼ 2 cm). A 50/50beam splitter further divides the beam into two beams of equalintensity, each beam crossing one DV in order to measure therefractive index difference between the two DVs. After the beamof the bottom is bent by a mirror then both beams overlap at asecond 50/50 beam splitter and eventually propagate to the CCDcamera (Cohu, 7712-3000) after being captured by a microscopeobjective [28].

All tests have been made with the cell in the vertical positionwhile heating from above; this allows having a stable configurationfrom the thermal point of view and on composition point of view, atleast for all samples with positive Soret coefficients. The entireoptical set-up is mounted on an optical table.

3. Experimental procedure

Initially, the two thermoregulated baths are set to the sametemperature (mean temperature of the experimental run). Even-tually for time t¼ 0, the top side of the cell is heated to temperatureTh while the bottom side is cooled to temperature Tc, resulting in atemperature difference of DT ¼ Th � Tc. Interference patterns arerecorded to evaluate the phase difference variation as a function oftime. This can be written as [25]:

DwðtÞ ¼ �2pd*

l0DnðtÞ (3)

where Dn represents the total variation of the refractive index be-tween the hot and cold side of the cell. For binary systems, thevariation of refractive index due to temperature and concentrationvariation are expressed as follows [8]:

A. Ahadi et al. / International Journal of Thermal Sciences 80 (2014) 108e117110

Dn ¼ vnvT

DT þ vnvc

Dc (4)

where Dc is the concentration difference of the densest compo-nent between the hot and the cold side of the cell. Coefficients vn/vT and vn/vc are the so-called contrast factors. Further detailscould be found in literature [26,29,30]. At the initial stage, thethermal kinetics provides a significant change in the magnitude ofthe phase difference. On the contrary, in a second stage the con-centration kinetics becomes dominant in generating phase dif-ference change. Hence, the two effects are decoupled enough toclearly distinguish the solutal contribution from the temporalpoint of view as explained by Zhang et al. [31,32] and Ahadi et al.[9,33,34].

The concentration difference that is measured after the estab-lishment of the temperature gradient can be written using Eqs. (3)and (4) as:

Dc ¼ � l0Dw

2pd*vnvc(5)

More details about this experimental method are presented byZhang et al. [31,32] and by Croccolo et al. [25].

Fig. 1. Configuration model and boundary conditions of the problem.

Table 1Physical properties of 50% toluene and 50% n-hexane mixture at 25 �C.

Property Symbol Theoreticalvalue

Experimental value

Kinematic viscosity (m2/s) g 5.15 � 10�7 4.96 � 10�7

Dynamic viscosity (Pa s) n 0.386 � 10�3 0.371 � 0.004 � 10�3

Thermal diffusivity (m2/s) a 8.963 � 10�8 e

Diffusion coefficient (m2/s) D 2.78 � 10�9 2.77 � 10�9

Density (kg/m3) r 751.19 748.09 � 0.01Soret coefficient (1/K) ST 4.915 � 10�3 (4.92 � 0.13) � 10�3

Thermal expansion (1/K) bT 1.26 � 10�3 (1.22 � 0.03) � 10�3

Solutal expansion bC 2.88 � 10�1 0.38 � 0.1Prandtl number Pr 5.74 e

Schmidt number Sc 333.45 e

Thermal characteristictime (s)

sth 1115.7 e

4. Problem formulation for numerical study

4.1. Model configuration and boundary conditions

A numerical model as sketched in Fig. 1 is considered in thisstudy, with a vertical porous layer superimposed between twoliquid layers in accordance to the experimental cell’s dimensions.The larger width of the closure is 21.5 mm in diameter (D) and thesmaller width is 10.0 mm in diameter (d) with a total height of45 mm (L). The porous layer has a height of 32.2 mm (d3), while,each of the fluid layers have a height of 6.40 mm (d1 and d2). Theexternal walls of the closure are assumed to be adiabatic. Thetemperature of the top surface, being the hot surface is set equal toTh and the temperature of the bottom surface, being the cold sur-face is set equal to Tc and the temperature difference (DT ¼ Th � Tc)will be applied in order to study the effect of the temperaturedifference on the solute. It is assumed that the liquid and the porouslayer are in thermal equilibrium. The porous layer has a porosity f

equal to 0.64, and the permeability has been set equal to4.4� 10�12 m2 based on the experimental measurement. It must bementioned that the n in Fig. 1 represents the direction perpendic-ular to each walls.

The porous medium is assumed to be homogeneous andisotropic with neglected effects of inertia. The density of the bi-nary mixture that saturates the porous matrix is modelled usingBoussinesq approximation. According to other numerical studiesin the literature, this approximation provides accurate densityvariation for this problem [9,35,36]. Thus, the fluid is assumed asa Newtonian and viscous mixture and the physical properties ofthe fluid are assumed constant except the density in the buoy-ancy term which varies linearly with the local temperature andthe concentration of the fluid of the denser component (seeTable 1).

Diffusive characteristictime (s)

sD 3.6 � 104 e

Thermal conductivity(W/m K)

k 0.128851 e

Thermal conductivity ofporous material(W/m K)

ks 0.833666 e

Lewis number Le 58.03 e

Specific heat (J/kg K) Cp 1923 e

4.2. Governing equations

The complete continuity, momentum balance, energy balanceand mass balance equations are solved numerically using the finiteelement technique. The following terms are used to establishgoverning equations in the non-dimensional form:

A. Ahadi et al. / International Journal of Thermal Sciences 80 (2014) 108e117 111

U ¼ uu

V ¼ v

uW ¼ w

uq ¼ T � Tc

T � T

o o o h c

R ¼ rL

Z ¼ zL

C ¼ c� coDc

s ¼ tuoL

P ¼ pLmuo

N ¼ bcDcbTDT

Da ¼ k

L2uo ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffigbTDTL

p

Gr ¼ g$bT$DT$r2$L3

m2Pr ¼ n

aSc ¼ n

DRe ¼ uoL

n

where U, V and W are the non-dimensional radial, circumferentialand axial component of velocity, R, Z and 4 are the non-dimensionalradial, axial and angular coordinates, respectively. P is the non-dimensional pressure term and q is the non-dimensional temper-ature term. The characteristic length and velocity are denoted by Land u0. In the non-dimensional analysis, several other parametersappear, such as the Reynolds number Re, the Prandtl number Pr, theDarcy number Da and the Schmidt number Sc. The non-dimensional forms of the governing equations in the fluid layersbased on Navier-Stocks formulation are:

Continuity balance:

1R

v

vRðRUÞ þ 1

RvVv4

þ vWvZ

¼ 0 (6)

Momentum balance e R direction:

Re�vUvs

þ UvUvR

þ VRvUv4

þWvUvZ

� V2

R

�

¼ �vPvR

þ�V2U � U

R2� 2R2

vVv4

�(7)

Momentum balance e 4 direction:

Re�vVvs

þUvVvR

þVRvVv4

þWvVvZ

þUVR

�¼ �1

RvPv4

þ�V2Vþ 2

R2vVv4

� VR2

�(8)

Momentum balance e Z direction:

Re�vWvs

þUvWvR

þ VRvWv4

þWvWvZ

�¼ �vP

vZþhV2W

iþGrRe

½q�N:C�

(9)

Heat transfer equation:

Re Pr�vq

vsþ U

vq

vRþ V

Rvq

v4þW

vq

vZ

�¼

"v2q

vR2þ 1R2

v2q

v42 þv2q

vZ2

#(10)

Mass transfer equation:

�vCvs

þ UvCvR

þ VRvCv4

þWvCvZ

�¼ 1

Re1Sc

"v2CvR2

þ 1R2

v2Cv42 þ

v2CvZ2

#

þ a

ScRe

"v2q

vR2þ 1R2

v2Cv42 þ

v2q

vZ2

#

(11)

Here aT is the thermal diffusion factor. The non-dimensionalforms of the continuity and mass balance governing equations inthe porous layers are the same as in the fluid layers. The mo-mentum and energy balance equations in the non-dimensionalform for the porous layer using the Brinkman model are:

Momentum balance e R direction:

Re�vUvs

�þ 1Da

U ¼ �vPvR

þ v

vR

�1R

v

vRðRUÞ

�þ 1R2

v2U

vq2� 2R2

vVvq

þ v2UvZ2

(12)

Momentum balance e 4 direction:

Re�vVvs

�þ 1Da

V ¼ �1RvPvq

þ v

vR

�1R

v

vRðRVÞ

�þ 1R2

v2V

vq2þ 2R2

vUvq

þ v2VvZ2

(13)

Momentum balance e Z direction:

Re�vWvs

�þ 1Da

W ¼ �vPvZ

þ v

vR

�1R

v

vRðRWÞ

�þ 1R2

v2W

vq2þ v2W

vZ2

þ GrRe

½q� N:C�(14)

Heat transfer equation:

Re Pr�vq

vsþ U

vq

vRþ V

Rvq

v4þW

vq

vZ

�¼ G

"v2q

vR2þ 1R2

v2q

v42 þv2q

vZ2

#

(15)

where G is the non-dimensional thermal conductivity and can beexpressed as:

G ¼ kekf

¼ Bkf þ ð1�BÞ$kskf

(16)

where ks is the thermal conductivity of the porous medium and kfis the thermal conductivity of the binary fluid. It has been shownthat the thermodiffusion coefficient (DT) and the mass diffusioncoefficient (D) in porous layer are related to the one in liquid layervia the tortuosity. However, the Soret coefficient is identicalwhether it is a liquid system or a porous system. The tortuositywas set equal to 1.8 as reported experimentally [25]. The finiteelement code used for this particular study was validated andshowed a good agreement with the study performed by Jawadet al. [37]. The computation was conducted over a period of 72 h. Itwas observed that at such time, the system have reached a steadystate condition.

Jawad et al. conducted extensive mesh sensitivity and wasadopted in the current study [37]. In addition, convergence criteriahave been set as the maximum difference between two iterationsfor all variables decided by initial value of each variablemust be lessthan 10�4. Thus, the optimummesh resolution of total 31 elementsin the horizontal direction and 135 elements in the vertical direc-tion was used in the current simulation.

5. Theoretical estimation of Soret coefficient

According to the non-equilibrium thermodynamics studied byEslamian et al. [3,38] and by Ahadi et al. [34,39], in the linearresponse regime, in an n-component mixtures in mechanicalequilibrium subjected to a thermal gradient, the molar diffusionflux of a component i, Ji, with respect to the molar average velocity,can be expressed as:

Ji ¼ �1T

Xn�1

k¼1

Lik

240@�Q*k � Q*

n

�T

1AVT þ

Xn�1

j¼1

vmkvcj

Vcj

35 (17)

Table 2Porosity, permeability and tortuosity measured by mercury porosimetry.

Filling medium f (%) K (Da) Tortuosity s

Case 1 Two porous layers 62.5 7.045 1.74Case 2 Two porous layers 69.6 1.118 2.53Case 3 Free medium 100% e e

Fig. 2. a) Measured difference of temperature difference and b) phase difference be-tween DVs versus time.

A. Ahadi et al. / International Journal of Thermal Sciences 80 (2014) 108e117112

where Lik and Liq are phenomenological coefficients respectivelyconnected to mass and thermodiffusion, T the temperature, cj andmj respectively the molar fraction and the chemical potential ofcomponent j and Q*

i is the net heat transport of the ith component.In addition, the mass flux for n-component can be written as:

Ji ¼ �r

0@Xn�1

j¼1

DijVcj þ DT ;icicjVT

1A (18)

where r is the total mole density, Dij are the Fick’s mass diffusioncoefficients. By comparing Eqs. (17) and (18) based on Srinivasanet al. [40,41] and Eslamian et al. [42] it allows us to express thethermodiffusion coefficient in terms of their phenomenologicalcounterpart as follows;

DTi ¼ Liq

cT2and Dij ¼

1cT

Xn�1

k¼1

LikXn�1

l¼1

�dkl þ

xjxn

�vmlvxj

(19)

where Liq is equal to

Liq ¼Xn�1

k¼1

�Q*k � Q*

n

�Lik (20)

As mentioned in the previous section, the computation of mo-lecular and thermodiffusion coefficient requires the estimation ofthe net heat of transport, Q*

i , for each of the i components in themixture. The physical meaning of Q*

i has been interpreted by Sri-nivasan et al. as the net amount of energy which must be absorbedby the region per mole of a component while diffusing out in orderto maintain the constancy in the temperature and pressure of themixture [3,41,43,44]. Using the above approach, the Soret coeffi-cient was calculated and compared to the experimental data. It wasfound that at 25 �C the experimental and theoretical results are ingood agreement. Table 1 shows the physical properties of themixture using the experimental and theoretical approach. In thepresent computation it was decided to adopt the thermodiffusiondata obtained from the theoretical calculation. Exception is madefor the permeability, porosity and tortuosity where the experi-mental data was used.

6. Results and discussion

6.1. Experimental results

As a first step we have performed an experiment at atmosphericpressure by imposing a fixed temperature difference between thetwo thermostatic baths, with the concentration C ¼ 0.5 of the firstcomponent. The mean temperature in the cell is set atTmean ¼ 25 �C. In Table 1 the values of the Soret coefficient and thetortuosity of the binary mixture are reported.

Here, three different case studies have been used to analyse theeffect of different porous layers and to survey the performance ofthe new cell for various conditions. While the mixture is keptsimilar for all cases, the characteristics of the porous medium at themiddle part of the cell have been changed (see Table 2 for moredetail).

In Fig. 2a) the temperature difference between the two DVs as afunction of time is illustrated. As shown, themeasured temperaturedifference between the two DVs is much smaller than the oneapplied to the two baths and it reaches the steady state value ofabout 4.4 K due to a consistent loss of energy between the thermalbaths and the cell. The contrast factor used in the image analysis at25 �C were vn/vT ¼ �5.5 � 10�4 (1/K) and vn/vc ¼ 0.1228 [31].

In Fig. 2b) the phase difference between the beam that crossedthe hot DV and the beam that crossed the cold one is plotted. It canbe seen that while a fast variation is obtained due to the thermaleffect (first kinetic) eventually a smaller and slower variation isobserved due to the Soret-induced concentration gradient (secondkinetic). As expected, these two kinetics are temporarily wellseparated. The sign of the phase variation of the two kinetics is thesame as one can expect if sample properties [31] and Eqs. (3) and(4) are considered. In fact, in this case vn/vT, and Dc are negative(for a positive Soret coefficient, the denser component concentratesin the cold zone of the cell) while vn/vc is positive, then both termsin the sum in Eq. (4) have the opposite sign respect to the tem-perature gradient.

Table 3Porosity, permeability and tortuosity measured by mercury porosimetry as func-tions of the cycles of intrusion and extrusion.

Cycle 1 2 3

f (%) 64.9 63.1 62.5K (Da) 268 262 260Tortuosity s 1.72 1.74 1.75

A. Ahadi et al. / International Journal of Thermal Sciences 80 (2014) 108e117 113

Little oscillations can be detected over the phase signal inFig. 2b). While it seems that the interferogram does not obtain thesteady condition at the end of the experiment in Fig. 2b); carefulanalysis of these oscillations indicates that they are mainly due tothe ambient temperature oscillations (not shown). This effect canbe filtered out from the raw data, by performing a de-convolutionof the phase data with the ambient temperature ones. Eventuallythe phase change of the second kinetics is considered. The effects ofthese thermal disturbances can be observed in the concentrationprofile of the cell. In Fig. 3 the absolute value of difference in themass concentration of the denser component as a function of thetime is reported. However, the concentration fluctuations due tothe fair performance of the thermal unit are evident between thehot and cold sides; after 10 days of application of thermal gradient,the average value of the maximum separation across the cellremained constant.

At the end of separation, the value of the difference in the massconcentration of the toluene can to be estimated at(5.4 � 0.2) � 10�3. The molar weight of the toluene is 92.14 g/moland for the hexane 86.18 g/mol. As a first approximation, the dif-ference in the molar concentration can to be considered the sameas the difference in the mass concentration.

Eventually, the porous medium was removed from the cell andanalysed by gas porosimetry (nitrogen) with a Tristar instrumentfrom Micromeritics. The specific surface was evaluated to be266.93 m2/g. The mercury porosimetry was performed with anAutopore also from Micromeritics, based on the capillary law gov-erning mercury penetration into the pores as shown by Leon [45].In one cycle of this measurement, the pressure is increased fromatmospheric pressure until 200 MPa and then decreased. The re-sults of three intrusion/extrusion cycles are reported in Table 3. Thevalue of the tortuosity is in good agreement with the value of thetortuosity measured with the thermodiffusion experiment.

From the analysis of the second kinetic of the phase differencebetween the two laser beams crossing the cell, we can recover thevalues of the molecular diffusion and Soret coefficients, as reportedin previous works by Croccolo et al. [26]. However, the experi-mental procedure only permits to follow the kinetic of the sepa-ration between two points in the totally of the concentration field.In order to understand the different regimes occurring in thedifferent parts of the cell and their interaction, a full transient

Fig. 3. Absolute mass concentration difference of the toluene versus time for case 1.

numerical simulation of the solutal separation in the experimentalset-up has been performed. As indicated, the thermal control of theinterferometer around the thermodiffusion cell has been signifi-cantly improved for the two last cases. This improvement was anessential requirement for these two cases because of the smallerseparation value across the cell. In Figs. 4 and 5, we report theevolution of the difference in concentration of toluene as a functionof time for cases 2 and 3 respectively.

As seen in Fig. 4, the experiment for case 2 has been repeatedtwice in order to be sure of the accuracy of the improved thermalunit. Less amount of fluctuations is evident is this figure for bothruns. It must be noted that although these two runs do not providean identical result; the steady state value of the separation for bothruns are similar. Moreover, if one considers the best fitted curve toeach of these separation trends, the same Soret behaviour andcoefficient would be obtained.

Eventually, we performed the experiment without porous me-dium in the cell (case 3). The temporal separation result of this caseis demonstrated in Fig. 5. Two important observations can be madeby comparing the cases with and without porous medium. The firstobservation is the faster separation for case 3 in the free medium incomparison with the other cases. It can be seen that after about 3days the maximum separation has been reached across the cell forfreemedium case. The second observation is smaller absolute valueof the maximum separation for this case. It can be concluded thatwhile porous layer delays the separation, it increases the separationsimultaneously. Finally, if one compares the curve in Figs. 4 and 5with the new one included in Fig. 3, it would be found that dataare less disturbed confirming the improvements made in thethermal regulation of the interferometer.

6.2. Numerical results

Three-dimensional simulation has been performed to study theeffect of the temperature and porosity on the thermodiffusion

Fig. 4. Absolute mass concentration difference of the toluene versus time for case 2.

Fig. 5. Absolute mass concentration difference of the toluene versus time for case 3(free medium).

A. Ahadi et al. / International Journal of Thermal Sciences 80 (2014) 108e117114

phenomenon in a hydrocarbon binary mixture which consists of50% toluene and 50% n-hexane. Prandtl number was found to beequal to 5.74 and the Schmidt number was found to be equal to33.45. Table 1 highlights the physical properties of the liquid ob-tained experimentally and theoretically using the equation of State.In a system configuration with heating from the top, the densitywill decrease in the opposite direction of the gravity force causing adiffusive regime; therefore, a solute transport due to thermodiffu-sion in the porous layer may occur. The thermal characteristic timerepresents the time required for the thermal heat to diffuse throughthe system in order to create the applied temperature field in thatsystem.

There is a big difference between the thermal characteristic timeand the diffusive characteristic time in the system. This time dif-ference is owing to the slower rate of the mass flux of n-hexanethan the heat diffusion rate in the system as a result of the tem-perature difference between both top and bottom surfaces. Becausethe mass flow characteristic time (sD) is greater than the thermalcharacteristic time (sth) (as shown in Table 1), thermodiffusionbecomes the major contributor for the separation of the mixturecomponents. The experimental time used in our analysis is 14 days(1.21 �106 s) which is greater than both the thermal characteristictime (1115.7 s) and the diffusive characteristic time (3.6 � 104 s) toensure that the system reaches the steady state. In binary mixtures,the buoyancy force produced is due to both the temperature andspecies concentration gradients. Thermal and solutal porous Ray-leigh numbers have been calculated for the porous layer for a DT of4.4 K, using the following equations:

RaT ¼ gbTDTkd3ga

(21)

RaS ¼ gbcDcd3kga

(22)

where g, bT, bC g, and a are the gravitational force, the solutalexpansion coefficient, the thermal expansion coefficient, the kine-matic viscosity and the thermal diffusivity of the fluid respectively.While k is the permeability, DT is the temperature difference inporous layer, Dc is the concentration difference in the porous layer

and d3 is height of the porous layer. In our current analysis it wasfound that RaS and RaT in the porous layer were found to be equal to253 and 10 respectively.

When heated from above as in this case, the flow is purelydiffusive and convection is minimized. The thermal Rayleighnumber RaT is found to be less than RaS which denotes that massdiffusion is the dominant force. The Soret Coefficient ST is greaterthan zero, which causes the lighter component of the mixture tomigrate in the direction of the hot surface (top), then a convectivemotion occurs in the bottom fluid layer until the concentrationreaches a steady state condition. For a DT ¼ 4.46 K as obtainedexperimentally and numerically, it was noticed that the species ofthe n-hexane which is the lighter component moved in the direc-tion of the hot surface (the top horizontal surface), and the toluenewhich is the heavier component moved towards the cold surfacedue to the thermodiffusion. These opposite movements of themixture species are due to the positive Soret coefficient of themixture.

As shown in Fig. 6, eight steady convection cells in the liquidlayer of the cylindrical cavity are observed. Four of these convectivecells show low intensity. However, no convection was observed inthe porous layer. This is evident by examining the temperaturedistribution in the porous layer. A linear temperature variation isobvious in the porous layer in Fig. 6a. It must be noted that theacceptable linear variation of the temperature profile insidethe porous layer is obtained as a result the specific geometry of theSoret cell.

Fig. 7(a) and (b) shows the concentration difference of n-hexaneat the steady state condition in the porous and free mediums be-tween the top and bottom horizontal surfaces for cases 1 and 3respectively. According to these figures a pure and almost lineardiffusive regime can be observed in the porous layer for both cases;however, the separation for free medium case was noticeableweaker than case 1.

The difference in concentration obtained numerically is in thesame range to the one obtained experimentally. In fact the totalseparation in the porous layer from the experiment was reported tobe equal to 5.5 � 10�3 whereas numerically as shown in Fig. 7 thetotal separation DC is equal to 5.27 � 10�3 for the case 1.Amaximum 4% deviation between the experimental and numericalresults is observed. This deviation may be caused due to some errormeasurement in the experiment. However, this shows a goodagreement between the experimental and the numerical tech-nique. The summary of the experimental and numerical result arereported in Table 4.

7. Conclusion

Numerical and experimental methods have been conducted tostudy the Soret effect for system of porous media sandwiched be-tween two layers of the same binary hydrocarbon mixture.A mixture of 50% toluene and 50% n-hexane which has a positiveSoret effect has been used. It was found that for a binary liquid witha positive Soret coefficient (ST > 0), the lighter components of thefluid moved into the direction of the hot surface. Numericalsimulation demonstrates eight steady convection cells in the liquidlayer of the cylindrical cavity; however, in the porous layer at themiddle of the cell no convection cell was observed. A linear tem-perature variation is found in the porous layer due to the specificgeometry of the designed Soret cell. This design resulted in a pureand almost linear diffusive regime at the middle zone of the cell forboth cases with and without porous layer; however, the separationfor free medium case was weaker. It can be concluded that inporous layer separation require longer time to reach steady state.Finally, the smooth trend of the separation was obtained

Fig. 6. Temperature and streamline contours of the n-hexane concentration.

Fig. 7. Concentration distribution in the Soret cell for cases 1 and 3, Cmin ¼ 4.964 � 10�1, Cmax ¼ 5.036 � 10�1, DC ¼ 3.5 � 10�4 (between each two lines).

A. Ahadi et al. / International Journal of Thermal Sciences 80 (2014) 108e117 115

Table 4Comparison between the numerical and experimental result.

Filling medium f (%) K (Da) Tortuosity s Dcexp � 10�3 DcCFD � 10�3 DTexp [K] DTCFD [K]

Case 1 Porous layer 62.5 7.045 1.74 5.35 5.27 4.5 4.46Case 2 Porous layer 69.6 1.118 2.53 4.08 4.12 4.5 4.41Case 3 Free medium 100 e e 3.34 3.38 4.5 4.51

A. Ahadi et al. / International Journal of Thermal Sciences 80 (2014) 108e117116

experimentally when the improvements made to the thermalregulation of the interferometer. A comparison between experi-mental and numerical calculation showed that both approachesobtained the same separation with a difference of 4%. In addition acomparison between experimental measurement and theoreticalestimation of the Soret coefficient proved the accuracy of thetheoretical approach.

Nomenclaturec concentration of the fluidC non-dimensional concentration of the fluidCB concentration of the lighter component at the bottomCT concentration of the lighter component at the(Cp)f specific heat of liquid at constant pressure (J/Kg K)d diameter of glass beads (m)d1 height of the bottom fluid layer (m)d2 height of the top fluid layer (m)d3 height of the porous layer (m)D1 width of the cavity (m)Da Darcy number ¼ k/L2

D mass diffusion coefficient in liquid (m2/s)DT thermal diffusion coefficient in liquid (m2/s k)g gravitational acceleration (m/s2)k Permeability of porous media (m2)kf conductivity of the fluid (W/m k)ks conductivity of the solid (porous media) (W/m k)ke effective thermal conductivity (W/m k)L characteristic length of the cavity in Y-

direction ¼ d1 þ d2 þ d3 (m)Le Lewis number ¼ D/aN buoyancy ratiop pressure (Pa)P non-dimensional pressurePr Prandtl number ¼ n/aq separation ratioR non-dimensional radial directionRaS solutal Rayleigh number in porous layerRe Reynolds numberSc Schmidt numberST Soret coefficient (1/k)t time (s)T temperature (k)u radial velocity (m/s)u0 characteristic velocityU non-dimensional velocity component in the R

direction ¼ u/uov circumferential velocity (m/s)V non-dimensional circumferential velocityw axial velocity (m/s)W non-dimensional axial velocityz dimensional axial directionZ non-dimensional axial direction

Greek symbolsa thermal diffusivity ¼ kf/r0.(Cp)f (m2/s)aT thermal diffusion factor ¼ TST

bC solutal expansion coefficientbT thermal expansion coefficient (1/k)q non-dimensional temperatures non-dimensional timesD diffusive characteristic time (s)sth thermal characteristic time (s)m dynamic viscosity (kg/m s)g kinematic viscosity (m2/s)r density (kg/m3)r0 density of the fluid at reference temperature To(kg/m3)f porosity4 circumferential direction

Subscripts0 referencec colde effectivef fluidh hotL liquidP porous

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