International Journal of Heat and Mass Transfer 73 (2014) 239–249

Contents lists available at ScienceDirect

International Journal of Heat and Mass Transfer

journal homepage: www.elsevier .com/locate / i jhmt

Heat and mass transfer modeling for multicomponent multiphase flowwith CFD

http://dx.doi.org/10.1016/j.ijheatmasstransfer.2014.01.0750017-9310/� 2014 Elsevier Ltd. All rights reserved.

⇑ Corresponding author. Tel.: +55 4837216409.E-mail address: cintia.soares@ufsc.br (C. Soares).

Natan Padoin a, Adrieli T.O. Dal’Toé a, Leonardo P. Rangel b, Karolline Ropelato b, Cíntia Soares a,⇑a Laboratory of Processes Control, Department of Chemical and Food Engineering, Federal University of Santa Catarina (UFSC), Florianópolis 88040-900, Santa Catarina, Brazilb Engineering Simulation and Scientific Software Ltda. (ESSS), Florianópolis 88032-700, Santa Catarina, Brazil

a r t i c l e i n f o

Article history:Received 23 July 2013Received in revised form 29 January 2014Accepted 30 January 2014

Keywords:CFDEulerian two-fluid modelMultiphase flowMulticomponent mass transferMaxwell–Stefan’s equationsHeat transfer

a b s t r a c t

Heat and mass transfer take place in a large number of processes. These phenomena are encountered insystems comprised of two or more phases, in which at least one of them is a mixture of many chemicalspecies. The predictability of such multiphase and multicomponent systems plays a major role in the effi-cient design and operation of equipment and processes, where CFD has been frequently applied success-fully over the past decade. Modeling multicomponent flow remains a challenge in relation to both microor macro systems. In this study, simulations were carried out with the commercial code ANSYS� CFD(FLUENT�), version 14.0, and customized functions developed to predict the equilibrium compositionsand temperature of a vapor–liquid system. A preliminary study on a binary mixture (water/air) was con-ducted in order to validate the results obtained with the commercial code using the data obtained from astandard psychrometric chart. In addition, simulations were carried out for a mixture of four pure hydro-carbons (methane, n-pentane, n-hexane and n-octane). Thus, a complete multicomponent mass transfertheory, based on Maxwell–Stefan’s equations, was applied as a customized function code, which can beused to calculate high flux corrections and the convective mass flux. The results were verified with pre-dicted values obtained using the steady-state process simulator PRO/II�, version 8.2.

� 2014 Elsevier Ltd. All rights reserved.

1. Introduction

Non-reactive coupled heat and mass transfer in gas–liquid sys-tems is widely encountered in the context of several operations,including distillation, extraction, absorption and drying. However,although efforts have been focused on understanding and predict-ing the underlying physics in such systems, this problem remainsto be solved. In particular, the numerical simulation of thephenomena associated with the transport between fluid phasescontaining mixtures of n chemical species is challenging andrequires further investigation.

According to Lakehal et al. [1], Hassanvand and Hashemabadi[2] and Ishii and Hibiki [3], two approaches are commonly usedfor the modeling of the transport phenomena in multiphase sys-tems using the Eulerian approach, i.e., with a fixed grid: modelsbased on (i) the direct numerical simulation of the interface, alsoknown as one-fluid models, and (ii) the interpenetrating continuahypothesis, known as two-fluid models. The direct numericalsimulation (DNS) of the position of the interface provides local

information on the flow properties. However, this approachrequires very fine grids and small time steps, which leads to highcomputational times. On the other hand, the local conservationequations can be averaged by means of the interpenetrating coin-tinua hypothesis. This approach can be applied when the shape ofthe interface is not known or not relevant [1]. Relatively coarsegrids and time steps are acceptable, which reduces significantlythe computational time. However, empirical correlations areneeded for the calculation of the interphase transport coefficients.Thus, the accuracy of the solution is dependent on the appropriateset up of these closure models.

Several authors have investigated the direct approach, in whichthe description of the interface motion is mainly carried out in con-juction with the volume of fluid (VOF) method, along with the levelset [4–6] and the front tracking [7] methods. However, it should benoted that in these studies the modeling of the interphase masstransfer was performed using Fick’s law, which was derived forbinary systems. Thus, the applicability of such models in thecontext of multicomponent mixtures relies on the assumption ofdiluted mass transfer, i.e., only one species in the mixture has a fi-nite concentration. Moreover, it is interesting to note that the VOFapproach has several limitations: the fluid phases share the same

240 N. Padoin et al. / International Journal of Heat and Mass Transfer 73 (2014) 239–249

temperature and velocity field, only one phase can enter or exit atany inlet or outlet, a higher quality mesh is needed in the regionwhere the interface is expected (the accuracy of the modeldecreases as the interface length scale becomes closer to the meshscale), and so on [8]. In this context, the Eulerian–Eulerian two-fluid model arises as a generic description of the flow propertiesof the system, since the governing equations are solved indepen-dently for each phase and the interphase interaction (exchangeof momentum, mass and energy) is modeled through source termsin the respective equations. This makes the two-fluid modelsuitable for cases such as those in which there is relative velocityand temperature between the fluid phases.

Haelssig et al. [9] applied the VOF method to the direct simula-tion of vapor-liquid flow with heat and mass transfer. A 2D domainwas used for the analysis of fluid–fluid contact in countercurrenton a wetted-wall in a short horizontal channel. Although theydeveloped the general form of the governing equations for thesimulation of multicomponent mass transfer in tridimensionalgeometries, only a binary ethanol/water mixture was effectivelystudied. In the same context, Banerjee [10,11] applied the VOFmethod to the prediction of coupled heat and mass transfer invapor-liquid stratified flow in a 2D inclined channel. An ethanol/iso-octane binary mixture was used and the classical Fick’s lawwas employed for the modeling of the diffusive flux at the inter-face. Banerjee [12] also studied the evaporation of the ethanol/iso-octane mixture in a single droplet using the VOF method.

On the other hand, few studies have addressed the use of theEulerian–Eulerian two-fluid model for the prediction of interphaseheat and, especially, mass transfer, even in binary mixtures. Chahedet al. [13] applied the Eulerian–Eulerian two-fluid model to the pre-diction of interphase momentum interaction in gas–liquid bubblyflows. They implemented a transport equation of the Reynoldsstress tensor used in the closure of average equations, whichaccounted for the turbulence contribution to the force exerted bythe liquid on the bubbles. Kalteh et al. [14] studied the effect ofrelative velocity and temperature between the phases on the heattransfer in two-phase nanofluid (copper/water) flow inside anisothermally heated microchannel.

Given the possibility of modeling each phase independently andthe reduced computational resources required when compared tothe direct approach, it is worth considering the application of theEulerian–Eulerian two-fluid model for studies on the coupled heatand mass transfer in multiphase flow. Moreover, there is a consen-sus that the multicomponent mass transfer models based onMaxwell–Stefan’s equations should be used for the adequatedescription of mass transport in mixtures, when the simplifiedassumptions of Fick’s law are not strictly valid. Such an approachallows for the modeling of the notable aspects of the multicompo-nent transport phenomena, e.g. the high flux correction issue [15]and the strong influence of all species on the mixture on the flux ofeach species i.

In this regard, the work of Cui et al. [16] should be highlighted,in which a kinetic model based on Fick’s generalized law was usedfor the simulation of direct contact heat and mass transfer in avapor-liquid flow in an inclined channel at sub-atmospheric pres-sure. A mixture of five petroleum pseudocomponents, NBP-(154,188, 225, 258 and 296), was studied. Additionally, studies wereperformed on the binary mixture NBP-(154/188) and the resultswere compared with the predictions obtained through empiricalcorrelations. However, it should be noted that the calculation ofthe matrix of the binary diffusion coefficients was based on theChapman-Enskog equation for the vapor phase and theMatthews-Akgerman equation for the liquid phase. In other words,the phenomenological theory of the Maxwell–Stefan’s equationswas not fully contemplated. Furthermore, Song et al. [17] providedan important contribution to the modeling of multicomponent

mass transfer through the implementation of closure relationsfor mass and energy balances based on the Maxwell–Stefanequations in the commercial process simulator AspenOne�. Theextractive distillation process involving a nonideal azeotropicmixture (n-butane, 1-butene, cis-2-butene, trans-2-butene andwater) was studied. To date, no studies in which the full phenom-enological theory of multicomponent mass transfer [18], based onMaxwell–Stefan’s equations and Fick’s generalized law, was usedin the context of computational fluid dynamics have beenpublished.

The aim of this research was to study the coupled heat and mul-ticomponent mass transfer in two-phase vapor-liquid flows basedon the Eulerian–Eulerian two-fluid model. To this aim, a code builtin-house (User Defined Function), written in C programminglanguage [8] and based on the script developed by Ropelato et al.[19], was implemented in the commercial code ANSYS� CFD(FLUENT�), version 14.0. A preliminary study was conducted, inwhich the model was evaluated for the prediction of the equilib-rium composition and temperature of a binary water/air mixture,where evaporation of water droplets into unsaturated air was car-ried out. The results predicted by the CFD approach were verifiedconsidering data presented in the standard psychrometric chart.In addition, the model was evaluated in a full multicomponentmixture composed of four pure hydrocarbons (methane, n-pentane,n-hexane and n-octane) and the equilibrium compositions andtemperatures were verified with results obtained in simulationsconducted in the steady-state process simulator named PRO/II�,version 8.2. Moreover, the influence of the diffusive and convectiveterms, as well as high flux corrections, on the total flux of the chem-ical species was evaluated. Also, different thermodynamic modelswere evaluated for the calculation of the vapor-liquid equilibrium(VLE) at the interface.

1.1. Theoretical background

1.1.1. Multicomponent mass transferThis section presents a summary of the fundamental theory

associated with modeling multicomponent mass transfer in multi-phase systems, upon which this study was based. It is important tohighlight that, although the fluxes calculated in the ANSYS� CFDcode should be expressed on a mass basis, the derivations herepresented will be expressed in molar terms, according to the for-malism frequently encountered in the development of such models.The conversion to the appropriate mass form was performed in thecalculations employing the code built in-house used in thesimulations.

In general, the molar flux of a chemical species i can bedescribed by a diffusive and a convective term. In simple binarysystems, the diffusive term is generally represented by Ficks’slaw, which states that the flux of the ith species is dependent onlyon its own composition gradient, scaled by a proportionality con-stant, the diffusion coefficient D, according to Eq. (1) [18]:

Ji ¼ �cDrxi; i ¼ 1;2 ð1Þ

Furthermore, the molar flux of the species i, Ni, is calculatedwith the addition of the contribution of the convective flow, ex-pressed as the sum of all individual fluxes, to the diffusive term[18]:

Ni ¼ �cDrxi þ xi

X2

j¼1

Ni; i ¼ 1;2 ð2Þ

However, in multicomponent systems there may be unnotice-able interaction effects in simple binary cases, such as diffusion bar-riers, reverse diffusion and osmotic diffusion. The diffusion barriercorresponds to the case where the diffusive flux of a species i is null

N. Padoin et al. / International Journal of Heat and Mass Transfer 73 (2014) 239–249 241

even if there is a composition gradient for that species. The reversediffusion corresponds to the movement of the molecules of species iagainst its positive composition gradient. Finally, osmotic diffusionrelates to the diffusion of a chemical species when its compositiongradient is zero. Thus, in multicomponent systems, each species iinteracts strongly with all other species in the mixture.

The most fundamental model for the description of materialtransport in multicomponent systems is the Maxwell–StefanEquations, based on the momentum conservation in collisionsbetween different kinds of molecules [20,18,21,17,22,23]. More-over, the driving force for the molecular transport of a species i isthe chemical potential gradient [18,24]:

ðJÞ ¼ �c½B��1ðdÞ ð3Þ

where the elements of the B matrix are given by Eqs. (4) and (5):

Bii ¼xi

D0inþXn

k ¼ 1i–k

xk

D0ikð4Þ

Bij ¼ �xi1

D0ij� 1

D0in

!ð5Þ

and the driving force, (d), is defined by Eq. (6):

di �xi

RTrlijT;P ð6Þ

In practical terms, handling chemical potential requires physi-cal detailing. Thus, this quantity is conveniently converted to ameasurable form, such as the composition of species i. For systemscomprising non-ideal fluids, Eq. (6) can be manipulated to give thefollowing relation:

di �Xn�1

j¼1

Cijrxj ð7Þ

where the quantity Cij, the matrix of thermodynamic factors, is ex-pressed by Eqs. (8) and (9), when evaluated in the liquid and vaporphases, respectively:

Ccij ¼ dij þ xi

@lnci

@xjð8Þ

Cuij ¼ dij þ xi

@lnui

@xjð9Þ

Thus, the diffusive flux of a chemical species i applying the Max-well–Stefan approach can be represented by Eq. (10):

ðJÞ ¼ �c½B��1½C�ðrxÞ ð10Þ

On comparing Eqs. (1) and (10), it is possible to recognize a rela-tion between the diffusive flux calculated by the Maxwell–Stefanapproach and that calculated by Fick’s law. This relation allowsfor Fick’s law to be rewritten for multicomponent systems, origi-nating the generalized Fick’s law, expressed by Eq. (11):

ðJÞ ¼ �c½D�ðrxÞ ð11Þ

Thus, the generalized Fick’s law can be explicitly related toMaxwell–Stefan’s Equation through Fick’s matrix of diffusion coef-ficients, D, according to Eq. (12):

½D� ¼ ½B��1½C� ð12Þ

Eq. (11) can be conveniently expanded in order to represent then-1 independent equations of the generalized Fick’s law:

ðJ1Þ ¼ �cD11rx1 � � � � � cD1;n�1rxn�1

ðJ2Þ ¼ �cD21rx1 � � � � � cD2;n�1rxn�1

..

.

ðJiÞ ¼ �cDi1rx1 � � � � � cDi;n�1rxn�1

ðJn�1Þ ¼ �cDn�1;1rx1 � � � � � cDn�1;n�1rxn�1

ð13Þ

Furthermore, in multiphase systems analyzed in steady-state, itcan be assumed that all resistance to the interfacial mass transfercan be concentrated in a thin layer adjacent to the interface I,according to the film model. For a two-phase system, composedof phases p and q, the flux of each species i and, consequently,the total flux, must be equal in the film adjacent to phase p andto phase q, in order to avoid accumulation. Thus, it is convenientto consider that all resistance to mass transfer is concentrated injust one phase (one of the two sides of the film, taking the interfaceas a reference). According to Taylor and Krishna [18] and Alopaeuset al. [20], transport in the interior of the film occurs only throughmolecular diffusion, i.e., turbulent diffusion is negligible, althoughin the outer region molecular diffusion is negligible and thehypothesis of an ideal mixture can be assumed. In this scenario,the mass transfer coefficient can be conveniently defined, with re-spect to a reference located in the bulk of the phase, by Eq. (14):

kb ¼ limNiarrow0

Ni;b � xi;bNt

cðxi;b � xi;IÞð14Þ

In practical terms, the introduction of this quantity allows forconsiderable simplification in the calculations, since the gradientsnon-aligned to the normal vector pointing toward the interface Ican be neglected and only the difference in the composition ofspecies i at the edges of the film are taken into account, scaledby the film thickness. Based on this scenario, for a multicomponentsystem, the material flux of species i can be described by Eq. (15):

ðJÞ ¼ �c½D�‘ðDxÞ ¼ �c½k�ðDxÞ ð15Þ

However, since the film thickness, ‘, is often unknown, the masstransfer coefficients can be calculated through empirical correla-tions, based on dimensionless numbers [25]. Such correlationsassume the general form of Eq. (16):

Sh ¼ b � Scp ð16Þ

Furthermore, for the case of multicomponent systems, Eq. (16)can be rewritten in a matricial form, according to Eq. (17):

½Sh� ¼ b � ½Sc�p ð17Þ

Alopaeus and Nordén [25] proposed a simplified form for thecalculation of the matrix exponentials of Eq. (17). In this method,the diagonal and off-diagonal elements of a generic matrix B,which is equivalent to matrix A raised to a fractional power,according to Eq. (18), are expressed by Eqs. (19) and (20),respectively:

½A�n=m ¼ ½B� ð18Þ

Bii ¼ Apii ð19Þ

Bij ¼ Aij

Apii � Ap

jj

Aii � Ajjð20Þ

Through the mathematical manipulation of the parameters inEq. (17), it is possible to obtain an explicit relation for the matrixof mass transfer coefficients, defined by [20], in the following form:

½k� ¼ c½D� þ d½D�1�p ð21Þ

Table 1Models for the estimation of the multicomponent effects in the generalized Vignesrelation [32].

Model Expression Ref.

Wesselingh and Krishna D1ij ¼ ðD1ij D1ij Þ

1=2 [33]

Kooijman and Taylor D1ij ¼ ðD1ik D1jk Þ

1=2 [34]

Krishna and van BatenaD1ij ¼ ðD

1ik Þ

xi=xiþxj ðD1jk Þxj=xiþxj [35]

Krishna and van BatenbD1ij ¼

xj

xiþxj ðD1ik Þ þ

xixiþxj ðD

1jk Þ [35]

Rehfeldt and Stichlmair D1ij ¼ ðD1ik D1jk D1ij D1ij Þ

1=4 [36]

a Vignes-like.b Darken-like.

242 N. Padoin et al. / International Journal of Heat and Mass Transfer 73 (2014) 239–249

However, it should be noted that the mass transfer coefficientobtained in Eq. (21) is valid for negligible fluxes of species i. Thissituation typically occurs in binary systems, when, due to equimo-lar counterdiffusion, the total flux of the species is zero. Neverthe-less, when the net flux of the species in the mixture assumes finitevalues, a common scenario in multicomponent cases, it is neces-sary to use correction factors. Accordingly, Eqs. (22) and (23) ex-press the matrix of high flux-corrected mass transfer coefficients,considering a composition reference in the bulk and at the inter-face, respectively:

½k�b� ¼ ½kb�½Nb� ð22Þ

½k�I � ¼ ½kI�½NI� ð23Þ

The matrix of high flux corrections introduces the effect of thecurvature on the composition profiles in the mass transfer region,due to the convective flow and the diffusive interactions. Differentmethods can be used to obtain the elements of the matrix of highflux corrections ½N�. In this regard, the most rigorous approach con-sists of the calculation of the high flux correction factors from theexact solution of the Maxwell–Stefan equations. In this case, ½N� isdefined according to Eqs. (24) and (25), considering the referencecomposition in the bulk of phase x and at the interface of thephases x and y, respectively:

½Nb� ¼ ½W�½exp½W� � ½I���1 ð24Þ

½NI� ¼ ½W�½exp½W��½exp½W� � ½I���1 ð25Þ

where the elements of the matrix of mass transfer factors, [W], aredefined according to Eqs. (26) and (27), respectively:

Wii ¼Ni

cD0in=‘þXn

k¼1i – k

Nk

cD0ik=‘ð26Þ

Wij ¼ �Ni1

D0ij=‘þ 1

cD0in=‘

!ð27Þ

Moreover, the linearized theory can be applied [26–29]. In thisapproach, the elements of the Fick matrix of diffusion coefficients,[D], are assumed to be constant (independent of composition)through the diffusive path and average diffusion coefficients canbe used in the calculations. Thus, the matrices of high flux correc-tions, defined by Eqs. (24) and (25), can be rewritten according toEqs. (28) and (29):

½Nb� ¼ ½W�½exp½W� � ½I���1 ð28Þ

½NI� ¼ ½W�½exp½W��½exp½W� � ½I���1 ð29Þ

where the matrix of linearized mass transfer factors is expressed byEq. (30):

½W� ¼ Nt‘

c½D��1 ¼ Nt

c½k��1

ð30Þ

In this approach, the matrix of high flux corrections, ½N�, can beexpanded in a power series, according to Eq. (31):

½N� ¼ ½W�X1k¼0

½W�k

k!� ½I�

" #�1

¼X1k¼1

½W�k�1

k!

" #�1

ð31Þ

Eq. (31) can be approximated by a linearization near the origin,resulting in a generalized and relatively simple expression for thematrix ½N�, presented in Equations (32) and (33):

½N� ¼ ½I� � a½W� ð32Þ

½k�½N� ¼ ½k� � Ntac½I� ð33Þ

The parameter a can be defined by statistical methods. Alopaeusand Nordén [30] applied a normally distributed function to thecharacteristic values (eigenvalues) of the matrix of linearized masstransfer coefficients in the interval [-1,1], with different levels ofstandard deviations. Following this approach, the authors obtainedvalues of 0.46 < a < 0.50 for the film model and the value ofa = 0.48 was then suggested. It should be emphasized that theparameter a is null when the high flux correction is neglected.

However, it is worth noting that the reliable estimation of thediffusion coefficients is a key aspect in the successful modelingof multicomponent mass transfer phenomena. In this regard, theestimation Maxwell–Stefan’s diffusion coefficients, D0, is oftenbased on the Vignes relation, originally written for binary systemsin the form of Eq. (34) [31]:

D0ij ¼ D1ij� �xi

D1ij� �xj

ð34Þ

In multicomponent systems this relation can be generalized totake the interaction of all species in the mixture into account, thusassuming the form of Eq. (35):

D0ij ¼ D1ij� �xi

D1ij� �xj

D1ij� �xk

ð35Þ

The last term of Eq. (35) is difficult to estimate, since it is non-accessible experimentally [31]. Thus, a wide range of models havebeen proposed for the calculation of this term [32], some of whichare summarized in Table 1:

In the context of this study, it is worth highlighting the contri-bution of Leahy-Dios and Firoozabadi [31] in relation to the calcu-lation of the binary Maxwell–Stefan’s diffusion coefficients, D0, inliquid and vapor phases as well as in supercritic mixtures of non-polar species. In this approach, the diffusion coefficients are calcu-lated based on the generalized Vignes relation and the Kooijmanand Taylor model, according to Eq. (36):

D0ij ¼ D1ij� �xi

D1ij� �xjYn

k¼1k – i;j

D1ij� �xk=2

ð36Þ

A least-squares fit of the general model expressed by Eq. (37)with a large set of experimental data (889 points), obtained fromthe literature, resulted in the correlation presented in Eq. (38) [31]:

cD1

ðcD0Þ¼ f ð l

l0 ; Tr; Pr;xÞ ð37Þ

cD121

ðcD0Þ¼ A0

Tr;1Pr;2

Tr;2Pr;1

� �A1 ll0

� �½A2ðx1 ;x2ÞþA3ðPr ;TrÞ�

ð38Þ

where the constants A0 to A3 are given by Eqs. (39)–(42),respectively:

N. Padoin et al. / International Journal of Heat and Mass Transfer 73 (2014) 239–249 243

A0 ¼ ea1 ð39Þ

A1 ¼ 10a2 ð40Þ

2 ¼ a3ð1þ 10x1 �x2 þ 10x1x2Þ ð41Þ

3 ¼ a4ðP3a5r;1 � 6Pa5

r;2 þ 6T10a6r;1 Þ þ a7T�a6

r;2 þ a2ðTr;1Pr;2

Tr;2Pr;1Þ ð42Þ

where:

a1 ¼ �0:0472

a2 ¼ 0:0103

a3 ¼ �0:0147

a4 ¼ �0:0053

a5 ¼ �0:3370

a6 ¼ �0:1852

a7 ¼ �0:1914

Furthermore, the diffusion coefficient for finite concentrations,D, can be calculated from a wide range of correlations available inthe literature. However, these must be selected according the char-acteristics of the system, as well as the availability of parameters. Ingaseous systems, the Gilliland’s correlation, expressed by Eq. (43),can be applied successfully. It should be noted that, even thoughthere are more accurate correlations available, such as Füller’s cor-relation [37,38], there may be difficulties in the estimation of theirparameters, which can prevent their application in complex sys-tems, for instance, those containing petroleum pseudocomponents.

D0ij ¼ 4:3� 10�6 T3=2

PðV1=3i þ V1=3

j Þ2 ð

1Miþ 1

MjÞ

1=2

ð43Þ

In addition, the low-pressure viscosity in Eq. (38) can be calcu-lated employing the Stiel and Thodos correlation [39], according toEq. (44), which produces good results especially for non-polarmixtures.

l0i ni ¼

34� 10�8ðTr;iÞ0:94; if Tr;i < 1:5

17:78� 10�8ð4:58Tr;i � 1:67Þ5=8; if Tr;i > 1:5

(: ð44Þ

where:

ni ¼T1=6

c;i

M1=2i ð0:987� 10�5Pc;iÞ

2=3 ð45Þ

Finally, the diluted vapor viscosity can be calculated through aweighted average according to Eq. (46) [31]:

l0 ¼ l01M1=2

1 þ l02M1=2

2

M1=21 þM1=2

2

ð46Þ

1.1.2. Vapor–liquid equilibrium (VLE)According to classical thermodynamics, in a two-phase system

the phases p and q are in equilibrium if there is equality of temper-ature ðTp ¼ TqÞ, pressure ðPp ¼ PqÞ and chemical potential ðlp

i ¼ lqi Þ

at the interface. In other words, the thermal, mechanical andchemical gradients in each phase tend to disappear and an equilib-rium value is reached, at which the system is stable [40].

As stated in Section 1.1.2, the chemical potential can be conve-niently defined in terms of the composition of the species, cor-rected for any non-ideal behavior of the mixture. At the interface,

the composition of species i in phase p can be related to thecomposition of the same species i in phase q according to Eq. (47):

Ki ¼Yp

i

Yqi

ð47Þ

Different approaches (e.g., c� c; c�u and u�u) and modelscan be used for further correction of Eq. (47) in order to includenon-ideal effects of the mixture [40,41].

2. Method

The CFD simulations were performed in the commercial codeANSYS� CFD (FLUENT�), version 14.0, in a pseudo-1D domain with10 to 30 m length, depending on the case study, and 0.05 m height.This geometry was adopted to approximate the CFD studies to theconditions of the simulations carried out in the process simulatorPRO/II�. Two spatial discretization (meshes) were carried out:

(I) elements with 50.0 mm, uniformly distributed along thelongitudinal direction, and

(II) elements with 12.5 mm, uniformly distributed in the longi-tudinal and the axial directions.

Fig. 1 shows details of the domain and the meshes. Mass flowrate and specified velocity were imposed at the inlet, while null(gauge) pressure was defined at the outlet. Furthermore, all otherboundaries were assigned with symmetry conditions to avoid walleffects on the flow.

The inlet of the system consisted of vapor (primary/continuousphase) and liquid (secondary/dispersed phase) streams enteringwith different flow rates, temperatures and compositions, butsharing the same pressure.

The Eulerian–Eulerian two-fluid model [8] was used for thetwo-phase flow calculations. Thus, the local governing equationswere averaged based on the interpenetrating continua hypothesis[42,3] and the equations for conservation of mass, species, momen-tum and energy assumed the form of Eqs. (48)–(51) [8]:

� Conservation of Mass:

1qrq

½ @@tðaqqqÞ þ r � ðaqqqvqÞ ¼

XNf

p¼1

ð _mlk � _mklÞ� ð48Þ

� Conservation of Chemical Species:

@

@tðaqqqYi;qÞþr�ðaqqqvqYi;qÞ

¼�r� ðaqJi;qÞþaqRi;qþaqSi;qþXNf

p¼1

ð _mpiqj � _mqjpi ÞþR ð49Þ

� Conservation of Momentum:

@

@tðaqqqvqÞ þ r � ðaqqqvqvqÞ

¼ �aqrpþr � tauq þ aqqqgþXNf

p¼1

½Kpqðvp � vqÞ

þ _mpqvpq � _mqpvqp� þ ðFq þ Flift;q þ Fvm;qÞ ð50Þ

� Conservation of Energy:

@

@tðaqqqhqÞ þ r � ðaqqquqhqÞ ¼ aq

@pq

@tþ sq

: ruq �r � qq þ Sq þXNf

p¼1

ðQ pq þ _mpqhpq � _mqphqpÞ ð51Þ

Fig. 1. Scheme of (a) the computational domain used in the CFD studies and (b, c) the two mesh refinements.

244 N. Padoin et al. / International Journal of Heat and Mass Transfer 73 (2014) 239–249

The term ð _mpiqj � _mqjpi Þ in Eq. (49) represents the source termfor the mass transfer of individual species between the vapor andliquid phases (due to evaporation and condensation) and is mod-eled according to Eq. (52):

_mliv j ¼ �qkAðYvi � KeqYl

iÞ ð52Þ

In particular, the term k in Eq. (52) was calculated according tothe theory presented in Section 1.1.1 based on the Maxwell–Ste-fan’s model. Therefore, k was calculated, in matricial form, accord-ing to Eq. (53):

½k� ¼ ½Sh�½D�‘

ð53Þ

where the matrix of diffusion coefficients [D] was calculatedaccording to Eq. (12). Moreover, the diagonal and off-diagonal ele-ments of matrix [B�1 appearing in Eq. (12) were defined accordingto Eqs. (4) and (5). The matrix of thermodynamic correction factors,[C], can be defined according to Eqs. (8) and (9). However, since themixture evaluated can be assumed as ideal, the term [C] was equalto the identity matrix. In addition, the high flux-corrected masstransfer coefficients were evaluated according to Eq. (33).

It should be noted that the Maxwell–Stefan’s diffusion coeffi-cients were calculated according to Eq. (36), making use of thecorrelation developed by Leahy-Dios and Firoozabadi [31] whichis expressed by Eqs. (37)–(42). Furthermore the parameters repre-sented by Eqs. (43)–(46) were used in these calculations.

When convection was taken into account, the term ðYilP _mliv j Þwas added to Eq. (52). In addition, the mass source terms pre-sented in Eqs. (48), (50) and (51) correspond to the sum of the indi-vidual species sources (i.e., _mlv ¼

P_mliv j ).

The discretization of the governing equations was performedwith the finite volume method through a pressure-based solver[8]. The pressure–velocity coupling was solved in a segregatedfashion with the phase coupled SIMPLE algorithm [43]. The gradi-ents were evaluated with the least squares cell-based method. Inaddition, the first-order upwind scheme was used for the spatialdiscretization. According to [44], in cases where the flow is alignedwith the grid, e.g., in a rectangular duct modeled with quadrilateralor hexahedral cells, which is the case of this study, the first-orderupwind discretization scheme may be acceptable. Furthermore,the domain used is an idealized one, since the wall effects were ne-glected due to the assignment of the symmetry condition for allboundaries apart from the inlet and outlet boundaries.

The heat transfer between the phases, the term Qpq in Eq. (51),was modeled by Eq. (54), with the heat transfer coefficient, h, givenby Ranz-Marshall’s correlation [45] (Eq. (55)).

Q pq ¼ h � DT ð54Þ

h ¼ k1dp� ½2:0þ 0:6 � ðqdpjvp � vj

lÞ

1=2

� ðCplk1Þ

1=3

� ð55Þ

The momentum exchange between the phases was consideredin the calculations and the drag coefficient, the term Kpq in Eq.(50), was modeled using the Schiller-Naumann’s [46] correlation.However, the external, lift and virtual mass forces, correspondingto the terms Fq; Flift;q and Fvm;q in Eq. (50), were neglected.

Since the system was studied in steady state, it was assumedthat all resistance to the interfacial mass transfer was concentratedin the vapor phase and an in-house User Defined Function [8,47]was built for the calculation of the thermodynamic equilibriumat the interface and the mass fluxes. The complete description ofthe multicomponent mass transfer phenomena, as described inSection 1.1.2, was adopted.

In particular, the thermodynamic equilibrium at the interfacewas calculated by the c�u and Raoult’s approaches, accordingto Eqs. (56) and (57) [40,41]:

Kc�ui ¼ ciP

sati usat

i

Puið56Þ

KRaoulti ¼ Psat

i

Pð57Þ

where the liquid-phase activity coefficient, ci, was calculated usingScatchard–Hildebrand’s theory and the vapor phase fugacity coeffi-cient, ui, was calculated through Soave–Redlich–Kwong’s equationof state. The saturation pressure of each species i in the liquid phasewas calculated employing Antoine’s equation [41].

The Sh number was assumed to be constant and equal to 6.0.Parallel tests were carried out and the results indicated that theconstant Sh number adopted was suitable to represent the systemunder study. Moreover, it should be noted that the derivatives inthe thermodynamic correction factors expressed by Eqs. (8) and(9) were neglected, i.e., the factors were considered equal to Kro-necker’s delta ðCij ¼ dijÞ and the mixtures were assumed to beideal.

2.1. A preliminary study – heat and mass transfer in a binary mixture

A preliminary study was carried out with a binary mixture com-posed of water and air. Table 2 presents the values set at the inletof the domain for both phases.

The interfacial length, ‘, was set as 5�10�4 a value definedbased on the equivalent diameter of the duct. For the criteriumadopted, the interfacial length was set as 1.0% of the equivalentdiameter of the computational domain. It should be noted thattests were carried out in which this parameter was evaluatedand the value adopted was the optimum condition for the opera-tional conditions of this study. The surface tension was assignedthe value of 7.2�10�2 N�m�1. The simulation was carried out atstandard atmospheric pressure (101 325 Pa), the vapor phasewas modeled as an ideal gas and a volume fraction of 10% of liquidphase was set at the inlet of the domain.

N. Padoin et al. / International Journal of Heat and Mass Transfer 73 (2014) 239–249 245

The predicted results were verified with data obtained from astandard psychrometric chart and the deviations between the ref-erence values and those obtained by the CFD simulations were cal-culated according to Eq. (58).

Errorð%Þ ¼ ð/CFD � /reference

/referenceÞ � 100 ð58Þ

The Reynolds number calculated for this case yielded the max-imum value of approximately 1.26�103. Thus, the flow regime waslaminar and there was no need to take a turbulence model intoaccount.

Moreover, the convergence criteria adopted for this study wereabsolute residuals of less than 1:0� 10�8 for the mass and momen-tum conservation equations in both phases, as well as for the vol-ume fraction of the liquid phase, 1:0� 10�4 for the energyconservation in both phases and, finally, 1:0� 10�3 for the speciesequation in both phases.

Furthermore, it should be noted that in this case the mass frac-tion of air in the liquid phase was set as zero, according to Table 2,and, thus, only the saturation pressure of water was calculated inthis phase by means of Antoine’s equation.

Table 3

2.2. Heat and mass transfer in a multicomponent mixture

After the model had been evaluated in the simplified systemand the predicted values were validated with the reliable data ob-tained from the psychrometric chart, a complete multicomponentstudy was carried out. In particular, a mixture of four pure hydro-carbons (n-pentane, n-hexane, n-octane and methane) was evalu-ated. This system was selected since the primary goal was to studythe heat and mass transfer in the context of the petrochemicalindustry processes. Four pure hydrocarbons of low molecularweight were adopted to serve as a model for direct comparisonwith the data provided by the PRO/II� method.

In order to assess the accuracy of the model for the prediction ofthe behavior of multicomponent mixtures, comparative tests werecarried out with results obtained in PRO/II�, version 8.2, a commer-cial steady-state process simulator widely used in the engineeringactivities of many companies around the world. In particular, anadiabatic flash simulation was conducted. Table 3 summarizesthe mass flow rate of the feed streams as well as the equilibriumstreams of both phases obtained in this simulation. The liquidand vapor phases entered the flash tank at 313.15 and 423.15 K,respectively, while the temperature inside the equipment was keptat 337.50 K. Moreover, the pressure inside the vessel was assumedto be equal to 98 066.49 Pa.

A conversion of the mass flow rate was perfomed in order tomake the values used in the simulation of the real case suitablefor the dimensions of the pseudo-1D domain used in the CFD stud-ies. Based on the relation of the total mass flow rates of the liquidand vapor phases given in Table 3, the corresponding feed streamsin the CFD studies were assigned to 1:0 kg � s�1, for both the liquidand vapor phases. However, a ratio between velocities of thephases was defined at the inlet, based on Eq. (59):

Table 2Conditions set at the inlet of the computational domain for the simulation of heat andmass transfer in a binary mixture composed of water and air.

Phase Species v ðm � s�1Þ T (K) Y i

Liquid Water 0.05 295.15 1.00000Air 0.00000

Vapor Water 0.50 303.15 0.01325Air 0.98675

v liq=vap ¼_mliq � qliq

_mvap � qvapð59Þ

which yielded the value v liq=vap ¼ 1:744698� 10�3.Table 4 summarizes the compositions of the liquid and vapor

streams assigned at the inlet of the pseudo-1D domain, calculatedfrom the real case simulated in PRO/II�. The temperatures of bothphases are also shown.

The interfacial length, ‘, was set to the same value used in thebinary studies described in Section 2.1. However, for the multi-component mixture, the value adopted for the surface tensionwas 1:7� 10�2 N m�1.

It is important to note that backflow was avoided in all calcula-tions and thus the conditions specified at the outlet did not influ-ence the results.

The results for the equilibrium compositions and temperaturein both phases were verified through comparison with those ob-tained with the PRO/II�, according to Eq. (58). In these studies,three different scenarios were evaluated in the CFD simulations:

(i) only diffusion at the interface was computed;(ii) the high flux correction and the convective contribution was

considered, together with the assumption made in (i).

Furthermore, the Reynolds number calculated for this caseyielded the approximate value of 2:34� 105. Thus, the flow regimewas turbulent and the SST k-x model [8] was applied.

Finally, it is worth highlighting that the convergence criteriaadopted for this study were absolute residuals of less than1:0� 10�5 for the mass and energy conservation equations, in bothphases, 1:0� 10�4 for the momentum and species conservationequations, in both phases, and 1:0� 10�3 for the turbulence vari-ables k and x.

2.3. Physical properties

The physical properties of the mixture in the liquid and vaporphases were calculated from the properties of the pure compo-nents. As previously mentioned, the vapor phase was consideredto be an ideal gas. Thus, the density of this phase was calculatedaccording to Eq. (60) [41,44]:

q ¼ Pop þ PRTPn

k¼1Yi=Mw;ið60Þ

On the other hand, a composition-dependent density was as-signed to the liquid phase according to the volume-weighted mix-ing law presented in Eq. (61) [41,44]:

q ¼ 1Pnk¼1Yi=qi

ð61Þ

Furthermore, the viscosity and thermal conductivity of bothphases were determined by the mass-weighted mixing law ex-pressed by Eqs. (62) and (63), respectively [41,44]:

Results for the mass flow rate ðkg � h�1Þ obtained in the adiabatic flash simulationcarried out in PRO/II�.

Stream Species Liquid Vapor

Feed n-Pentane 0.0000 900.0028n-Hexane 1800.0004 900.0002n-Octane 1800.0048 900.0024Methane 0.0000 900.0027

Equilibrium n-Pentane 74.9137 825.0892n-Hexane 516.6323 2183.3683n-Octane 1681.3646 1018.6312Methane 0.7989 899.2038

Table 4Composition and temperature of the fluid phases at the inlet of the computationaldomain.

Phases Species Y i T (K)

Vapor n-Pentane 0.25 423.15n-Hexane 0.25n-Octane 0.25Methane 0.25

Liquid n-Pentane 0.00 313.15n-Hexane 0.50n-Octane 0.50Methane 0.00

246 N. Padoin et al. / International Journal of Heat and Mass Transfer 73 (2014) 239–249

l ¼Xn

k¼1

Yili ð62Þ

k ¼Xn

k¼1

Yiki ð63Þ

Finally, the heat capacities of the liquid and vapor phases werecalculated using the mixing-law expressed in Eq. (64) [41,44]:

Cp ¼Xn

k¼1

YiCp;i ð64Þ

3. Results and discussion

3.1. Heat and mass transfer in a binary mixture

Table 5 summarizes the results obtained in the preliminarystudies for the prediction of heat and mass transfer during waterevaporation from pure water droplets into unsaturated air, for boththe thermodynamic approaches, c-u and Raoult, and meshschemes.

It can be observed that the results obtained from the CFD sim-ulations were very similar to those presented in a standard psy-chrometric chart analyzed under the same conditions in whichthe numerical study was conducted. In particular, excellent agree-ment was observed for the equilibrium temperature. Moreover, inthis simple binary case there was little dependence of the resultsobtained from the CFD simulations on the thermodynamicapproach used for the modeling of the equilibrium at the interfaceas well as the mesh refinement. Therefore, both mesh schemeswere suitable for the calculations.

The complete multicomponent description of mass transferimplemented in the code built in-house was suitably reduced toa binary system, in which most of the terms of the equations de-scribed in Section 1.1.2 were cancelled out. In other words, in thiscase the flux of each species was only dependent on its own com-position gradient and the high flux corrections were neglected.

Table 5Comparison between the predicted values for the equilibrium temperature andcomposition for a binary vapor-liquid system and the data in the standardpsychrometric chart for different thermodynamic equilibrium models and meshschemes.

Mesh Model Variable CFD Reference Error (%)

I c�u T (K) 295.187 295.150 +0.0125Y H2O 0.0159 0.0162 �1.8519

Raoult T (K) 295.193 295.150 +0.0146Y H2O 0.0158 0.0162 �2.4691

II c�u T (K) 295.187 295.150 +0.0125Y H2O 0.0159 0.0162 �1.8519

Raoult T (K) 295.193 295.150 +0.0146Y H2O 0.0159 0.0162 �2.4691

Thus, the diffusive and total flux of the species i were reduced tothe standard form of Fick’s law, according to Eqs. (1) and (2),respectively. Therefore, under such conditions, the UDF evaluatedin this study assumed the same hypothesis adopted by Ropelatoet al. [19].

Figs. 2 and 3 present the profiles for the water mass fraction inthe vapor phase and temperature in both phases. It is worth high-lighting that, although the calculation occurred across the whole ofthe domain (10.0 m), only 2.0 m are shown for better visualizationof the profiles, since the composition equilibrium was reached ataround 1.0 m.

For the system studied, the temperature equilibrium wasreached at a value close to that of the inlet thermal condition forthe liquid phase. It is possible to observe in Fig. 3 that there wasa decrease of around 8.0 K for the vapor phase, while the temper-ature of the liquid phase varied by only 0.04 K. Moreover, the massfraction of water in the vapor phase increased from 0.013 2 to0.015 9, reaching a value close to that of the reference data(0.016 2). Since the air was initially non-saturated (relative humid-ity of 50.0%) and with a temperature slightly higher than that ofthe water droplets, there was a heat and mass transfer gradientthat allowed the movement of water molecules from the dropletsto the vapor phase, until the composition equilibrium was estab-lished for the temperature reached in the entire system. Neverthe-less, it should be noted that there was heat transfer from the air tothe droplets, due to the thermal gradient, which was used to heatup the droplets (sensible heat) and promote the evaporation ofwater (latent heat). The main conclusion here was that the temper-ature of the vapor phase was decreased by convective heat transferand evaporative cooling.

3.2. Heat and mass transfer in a multicomponent mixture

Figs. 4 and 5 present a direct comparison between the values forthe equilibrium compositions obtained in the simulations carriedout in PRO/II� and in ANSYS� CFD (FLUENT�) for the vapor and li-quid phases, respectively.

Excellent agreement was observed between the values obtainedfrom the steady state simulations in PRO/II� and in ANSYS� CFD(FLUENT�). For the two cases tested, described in Section 2.2, aswell as for the two thermodynamic models evaluated (c�u andRaoult), the results for the equilibrium composition in both phaseswere the same, indicating that the high flux correction and theconvective term did not have a great influence on the mass transferat the interface.

Fig. 2. Composition profiles for the water in the vapor phase as a function of thedomain length for the two mesh schemes adopted.

Fig. 3. Temperature profiles for the vapor and liquid phases as a function of thedomain length.

Fig. 5. Compositions of the four species in the liquid phase in the feed stream andthe equilibrium values obtained with the simulations carried out in ANSYS� CFD(FLUENT�) and PRO/II�.

N. Padoin et al. / International Journal of Heat and Mass Transfer 73 (2014) 239–249 247

In the vapor phase, the mass fraction of n-pentane, n-octaneand methane decreased from 0.25 to 0.166 9, 0.209 2 and 0.1828, respectively, while the mass fraction of n-hexane increased from0.25 to 0.441 1. On the other hand, in the liquid phase the massfraction of n-hexane decreased from 0.50 to 0.231 9, while thecomposition of n-octane increased from 0.50 to 0.733 6. Further-more, the mass fraction of n-pentane and methane increased from0.00 to 0.034 4 and 0.000 1, respectively. Moreover, the composi-tion equilibrium in both phases was reached near the inlet of thedomain, as can be observed in Figs. 6 and 7.

It can be observed that the results obtained for the equilibriumtemperature were not dependent on the thermodynamic modeland the assumptions made in Section 2.2. Table 6 summarizesthe results obtained.

For all cases the equilibrium temperature predicted by the CFDsimulations was very similar to the results obtained with the pro-cess simulator PRO/II�, yielding a relative error of approximately2.0%. In this regard, neither the mesh refinement scheme nor thethermodynamic equilibrium model, used for the computation ofthe compositions at the interface, had a strong influence on theresults. However it should be noted that the turbulence modeldirectly affected the results. Studies were conducted without thecomputation of a turbulence model, i.e., considering the flow re-gime as laminar (despite the high Reynolds number). Under theseconditions, the influence of the convective contribution on theinterface heat transfer was clearly observed. In the tests whereonly diffusive mass transfer at the interface was computed the

Fig. 4. Compositions of the four species in the vapor phase in the feed stream andthe equilibrium values obtained with the simulations carried out in ANSYS� CFD(FLUENT�) and PRO/II�.

relative error between the CFD predictions and the data obtainedfrom the process simulator was approximately 10%, and whenthe complete model was considered, i.e., taking into account thehigh flux correction and the convective contribution to the interfa-cial mass transfer, the relative errors were approximately 3.5% and1.8% for the c-/ and Raoult thermodynamic equilibrium ap-proaches, respectively.

It is worth highlighting that in the case where the turbulencemodel was not taken into account, the composition profiles werethe same as those presented in Figs. 4–7. Thus, the influence ofthe turbulence was observed only on the equilibrium temperature.This effect may be related to the greater mixing between thephases provided by the turbulence model.

It should be noted that even though this observation illustratesthe influence of the high flux correction and the convectivecontribution on the overall model, the results presented in Table 6,i.e., considering a turbulence model in the calculations, should beconsidered as the best representation of the system studied, sincethe flow pattern in the computational domain was, indeed,turbulent.

Moreover, the mass flow rate for both phases at the inlet of thedomain was reduced by a factor of 10:1 ð0:1 kg � s�1Þ, keeping thevelocity ratio, defined by Eq. (59), constant. The results obtainedwere very similar to those presented in Figs. 4 and 5 and Table 6.

It is worth highlighting that only one temperature value is givenin Table 6 for the software PRO/II� since this process simulatoronly calculates values of equilibrium (e.g., for composition and

Fig. 6. Composition profiles for the four species in the vapor phase obtained fromthe simulations carried out in ANSYS� CFD (FLUENT�).

Fig. 7. Composition profiles for the four species in the liquid phase obtained fromthe simulations carried out in ANSYS� CFD (FLUENT�).

Table 6Results for equilibrium temperature obtained in ANSYS� CFD (FLUENT�) and PRO/II�

for different assumptions regarding multicomponent mass transfer and thermody-namic models.

Case Mesh Eq. Model CFD PRO/II� Error (%)

i I c�u 330.094 337.50 �2.19Raoult 330.593 �2.04

II c�u 330.203 �2.16Raoult 330.766 �2.00

ii I c�u 330.195 �2.16Raoult 330.646 �2.03

II c�u 330.000 �2.22Raoult 330.500 �2.07

248 N. Padoin et al. / International Journal of Heat and Mass Transfer 73 (2014) 239–249

temperature). In particular, an adiabatic flash simulation was car-ried out in PRO/II� with the four pure hydrocarbons and the ther-modynamic equilibrium temperature is given in Table 6.

4. Conclusions

The complete multicomponent mass transfer theory based onMaxwell–Stefan’s equations was implemented in a code built in-house implemented in the commercial code ANSYS� CFD (FLU-ENT�) for the prediction of heat and mass transfer phenomena intwo-phase systems. Initial studies were based on the simulationof the binary air–water mixture and the results were validatedthrough comparison with data obtained from a standard psychro-metric chart. Additionally, a multicomponent study was carriedout with four hydrocarbons and the results predicted by the CFDsimulations were verified based on the predictions of a standardsteady-state process simulator, widely used in industrial projectdesigns. For both studies, excellent agreement was observedbetween the predictions of the equilibrium compositions and tem-perature obtained with the CFD code and the reference values.

The studies carried out with the binary and the multicompo-nent mixtures showed that the predicted values for the equilib-rium composition and temperature were not dependent on themesh and the thermodynamic model used for the calculation ofthe compositions at the interface. Moreover, in the multicompo-nent case, the results obtained when only diffusive mass transferwas considered at the interface and when the complete modelwas considered, i.e., assuming the contributions of the high fluxcorrection and the convective term, were very similar (maximumdeviation of approximately 11%). The turbulence model playedan important role. When turbulence was not taken into account,

the values for the equilibrium temperature in the case where thecomplete model was considered were in better agreement withthe reference values compared to the case where only diffusionwas accounted for, especially for the Raoult thermodynamicapproach.

Although several studies have dealt with numerical simulationsof heat and mass transfer in multiphase systems, to date fewauthors have addressed the issue of multicomponent mixtures intheir calculations, based on simplifying assumptions that allowthe description of the problem using the classical Fick’s law whichis valid for binary mixtures. Moreover, the studies published in thisfield based on the Eulerian description of the fluid flow have fo-cused mainly on the application of interface tracking methodsand on the exact solution of the conservation equations. Althoughsuch approaches are of interest to the scientific community, it iswell known that the computational resources required are toodemanding for practical applications. Furthermore, to the best ofour knowledge, no studies in which the fundamental multicompo-nent mass transfer theory based on Maxwell–Stefan’s equationswas implemented and evaluated in the commercial CFD code havebeen previously published. Thus, this study contributes to thedevelopment of strategies for the prediction of heat and masstransport based on the continuum description of the multiphaseflow and taking into account all of the inherent characteristics ofmulticomponent mixtures, which introduce considerable complex-ity into the system when compared to the more commonly appliedbinary solution, such as the high flux correction issue, the contribu-tion of all composition gradients to the flux of each species i, thediffusion barrier, reverse diffusion and osmotic diffusion.

It is worth highlighting that efforts towards the enhancement ofsuch full multicomponent computational models are very signifi-cant with regard to the development of best practices in the designof relevant processes and equipment, allowing for a reduction inthe time and costs of testing different designs, as well as of opti-mizing the designs. In this regard, further studies will be carriedout by this group, in which different numerical approaches willbe applied. Moreover, it should be noted that different scenariosneed to be evaluated based on the code developed, such as tridi-mensional studies in which wall effects and the turbulence contri-bution are relevant, transient analysis based both on the filmmodel and on the penetration theory, variation in the discretephase diameter according to a statistical-phenomenological model(e.g., population balance approaches), evaluation of the Sh numberbased on correlations specific for ranges of operational conditionsand processes and a mixture of species with strong differences inthe physical properties (polarity, molecular weight, criticalconstants, vaporization temperature, etc.).

Acknowledgments

The authors acknowledge Engineering Simulation and ScientificSoftware Ltda. (ESSS) and CNPq/CAPES for the scholarships.

References

[1] D. Lakehal, M. Meier, M. Fulgosi, Interface tracking towards the directsimulation of heat and mass transfer in multiphase flows, Int. J. Heat FluidFlow 23 (3) (2002) 242–257.

[2] A. Hassanvand, S. Hashemabadi, Direct numerical simulation of interphasemass transfer in gas–liquid multiphase systems, Int. Commun. Heat MassTransfer 38 (7) (2011) 943–950.

[3] M. Ishii, T. Hibiki, Thermo-Fluid Dynamics of Two-Phase Flow, SpringerScience+ Business Media, 2011.

[4] F. Beux, B. Knowlton, S. Banerjee, A three-dimensional levelset method fordirect numerical simulation of two-phase flows in variable gravityenvironments, in: Proceedings of the Fourth Microgravity Fluid Physics andTransport Phenomena Conference, Clevland, 1998.

[5] C. Yang, Z.-S. Mao, Numerical simulation of interphase mass transfer with thelevel set approach, Chem. Eng. Sci. 60 (10) (2005) 2643–2660.

N. Padoin et al. / International Journal of Heat and Mass Transfer 73 (2014) 239–249 249

[6] J. Wang, P. Lu, Z. Wang, C. Yang, Z.-S. Mao, Numerical simulation of unsteadymass transfer by the level set method, Chem. Eng. Sci. 63 (12) (2008) 3141–3151.

[7] D. Juric, G. Tryggvason, Computations of boiling flows, Int. J. Multiphase Flow24 (3) (1998) 387–410.

[8] Fluent, version 14.0 Theory Guide, Ansys Inc, 2011.[9] J.B. Haelssig, A.Y. Tremblay, J. Thibault, S.G. Etemad, Direct numerical

simulation of interphase heat and mass transfer in multicomponent vapour–liquid flows, Int. J. Heat Mass Transfer 53 (19) (2010) 3947–3960.

[10] R. Banerjee, A numerical study of combined heat and mass transfer in aninclined channel using the vof multiphase model, Numer. Heat Transfer, PartA: Appl. 52 (2) (2007) 163–183.

[11] R. Banerjee, Turbulent conjugate heat and mass transfer from the surface of abinary mixture of ethanol/iso-octane in a countercurrent stratified two-phaseflow system, Int. J. Heat Mass Transfer 51 (25) (2008) 5958–5974.

[12] R. Banerjee, Numerical investigation of evaporation of a single ethanol/iso-octane droplet, Fuel 107 (2013) 724–739.

[13] J. Chahed, V. Roig, L. Masbernat, Eulerian–Eulerian two-fluid model forturbulent gas–liquid bubbly flows, Int. J. Multiphase Flow 29 (1) (2003) 23–49.

[14] M. Kalteh, A. Abbassi, M. Saffar-Avval, J. Harting, Eulerian–Eulerian two-phasenumerical simulation of nanofluid laminar forced convection in amicrochannel, Int. J. Heat Fluid Flow 32 (1) (2011) 107–116.

[15] V.J. Pohjola, J. Tanskanen, V. Alopaeus, The issue of high flux correction inmulticomponent reactive and ordinary distillation, Comput. Chem. Eng. 26 (9)(2002) 1295–1302.

[16] X. Cui, X. Li, H. Sui, H. Li, Computational fluid dynamics simulations of directcontact heat and mass transfer of a multicomponent two-phase film flow in aninclined channel at sub-atmospheric pressure, Int. J. Heat Mass Transfer 55(21) (2012) 5808–5818.

[17] Y. Song, J. Song, M. Gong, B. Cao, Y. Yang, X. Ma, Modeling of mass transfer innonideal multicomponent mixture with Maxwell–Stefan approach, Chin. J.Chem. Eng. 18 (3) (2010) 362–371.

[18] R. Taylor, R. Krishna, Multicomponent Mass Transfer, John Wiley and Sons Inc.,New York, NY (United States), 1993.

[19] K. Ropelato, A. Castro, M. Mori, W. Geraldelli, Empty spray sections of vacuumtowers: heat and mass transfer with a CFD approach, Chem. Prod. ProcessModel 5 (1) (2010).

[20] V. Alopaeus et al., Calculation of multicomponent mass transfer betweendispersed and continuous phases, Helsinki University of Technology, 2001.

[21] V. Dilman, V. Lotkhov, O. Kashirskaya, Experimental verification of the Stefan–Maxwell equations, Theor. Found. Chem. Eng. 43 (3) (2009) 288–292.

[22] V. Alopaeus, J. Aittamaa, Appropriate simplifications in calculation of masstransfer in a multicomponent rate-based distillation tray model, Ind. Eng.Chem. Res. 39 (11) (2000) 4336–4345.

[23] K.S. Gandhi, Use of Fick’s law and Maxwell–Stefan equations in computation ofmulticomponent diffusion, AIChE J. 58 (11) (2012) 3601–3605.

[24] J. Wesselingh, R. Krishna, Mass Transfer in Multicomponent Mixtures, DelftUniversity Press, Delft, 2000.

[25] V. Alopaeus, H.V. Nordén, A calculation method for multicomponent masstransfer coefficient correlations, Comput. Chem. Eng. 23 (9) (1999)1177–1182.

[26] H. Toor, Solution of the linearized equations of multicomponent mass transfer:I, AIChE J. 10 (4) (1964) 448–455.

[27] H. Toor, Solution of the linearized equations of multicomponent mass transfer:II. Matrix methods, AIChE J. 10 (4) (1964) 460–465.

[28] L.W. Smith, R. Taylor, Film models for multicomponent mass transfer: astatistical comparison, Ind. Eng. Chem. Fundam. 22 (1) (1983) 97–104.

[29] T.C. Young, W.E. Stewart, Comparison of matrix approximations formulticomponent transfer calculations, Ind. Eng. Chem. Fundam. 25 (4)(1986) 476–482.

[30] V. Alopaeus, J. Aittamaa, H.V. Nordén, Approximate high flux corrections formulticomponent mass transfer models and some explicit methods, Chem. Eng.Sci. 54 (19) (1999) 4267–4271.

[31] A. Leahy-Dios, A. Firoozabadi, Unified model for nonideal multicomponentmolecular diffusion coefficients, AIChE J. 53 (11) (2007) 2932–2939.

[32] X. Liu, T.J. Vlugt, A. Bardow, Maxwell–Stefan diffusivities in liquid mixtures:using molecular dynamics for testing model predictions, Fluid Phase Equilib.301 (1) (2011) 110–117.

[33] J. Wesselingh, R. Krishna, Elements of mass transfer, 1990.[34] H.A. Kooijman, R. Taylor, Estimation of diffusion coefficients in

multicomponent liquid systems, Ind. Eng. Chem. Res. 30 (6) (1991) 1217–1222.

[35] R. Krishna, J. Van Baten, The darken relation for multicomponent diffusion inliquid mixtures of linear alkanes: an investigation using molecular dynamics(md) simulations, Ind. Eng. Chem. Res. 44 (17) (2005) 6939–6947.

[36] S. Rehfeldt, J. Stichlmair, Measurement and calculation of multicomponentdiffusion coefficients in liquids, Fluid Phase Equilib. 256 (1) (2007) 99–104.

[37] E.N. Fuller, P.D. Schettler, J.C. Giddings, New method for prediction of binarygas-phase diffusion coefficients, Ind. Eng. Chem. 58 (5) (1966) 18–27.

[38] E.N. Fuller, K. Ensley, J.C. Giddings, Diffusion of halogenated hydrocarbons inhelium. The effect of structure on collision cross sections, J. Phys. Chem. 73(11) (1969) 3679–3685.

[39] L.I. Stiel, G. Thodos, The viscosity of nonpolar gases at normal pressures, AIChEJ. 7 (4) (1961) 611–615.

[40] J.M. Prausnitz, R.N. Lichtenthaler, E.G. de Azevedo, Molecular Thermodynamicsof Fluid-phase Equilibria, Prentice Hall, 1998.

[41] B.E. Poling, J.M. Prausnitz, O. John Paul, R.C. Reid, The Properties of Gases andLiquids, vol. 5, McGraw-Hill, New York, 2001.

[42] M. Wörner, A compact introduction to the numerical modeling of multiphaseflows, Forschungszentrum Karlsruhe, 2003.

[43] S. Vasquez, V. Ivanov, A phase coupled method for solving multiphaseproblems on unstructured meshes, in: Proceedings of ASME FEDSM00: ASME2000 Fluids Engineering Division Summer Meeting, Boston, 2000.

[44] Fluent, version 14.0 User’s Guide, Ansys Inc, 2011.[45] W. Ranz, W. Marshall, Evaporation from drops, Chem. Eng. Prog. 48 (3) (1952)

141–146.[46] L. Schiller, A. Naumann, A drag coefficient correlation, Vdi Zeitung 77 (1935)

318–320.[47] Fluent, version 14.0 UDF Manual, Ansys Inc.