Lu, J. H., Lei, H. Y., & Dai, C. S. (2019). Lattice Boltzmann equation for mass transfer in multi solvent systems. International Journal of Heat and Mass Transfer, 132, 519–528.

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Lattice Boltzmann equation for mass transfer in multi solvent systems

J.H. Lu, H.Y. Lei, C.S. Dai

^⇑

Key Laboratory of Efficient Utilization of Low and Medium Grade Energy, MOE, Tianjin University, School of Mechanical Engineering, Tianjin University, 135 Yaguan Road, Jinnan District, Tianjin 300050, China

a r t i c l e i n f o

Article history:

Received 9 June 2018

Received in revised form 22 November 2018 Accepted 2 December 2018

Keywords:

Mass transfer Multi solvent system Lattice Boltzmann equation Henry’s law

Fick’s law

a b s t r a c t

The key problem for simulating mass transfer in multi solvent systems is the concentration jump at the phase interface described by Henry’s law. In the present paper, after analyzing the shortages of Fick’s law in describing mass transfer in multi solvent systems, the revised Fick’s law that can be applied to mass transfer in multi solvent systems is proposed. The corresponding governing equation can describe the concentration jump directly without any modification and doesn’t need to transform the scalar or parameters. To solve the governing differential equation based on the revised Fick’s law, a multi-relaxation-time (MRT) lattice Boltzmann equation (LBE) is constructed. Five test cases containing both the transient and steady mass transfer problems in multi solvent systems with straight or curved interfaces are conducted to validate the proposed MRT LBE. The results show that proposed MRT LBE is easy in implementation and capable of simulating both steady and transient mass transfer in multi solvent systems. In addition, the comparison indicates that the MRT LBE could have better accuracy than single-relaxation-time (SRT) LBE by keeping an optimal relation between two relaxation times.

1. Introduction

Mass transfer between different solvents is a very common physical phenomenon that relates to numerous industrial applications, such as the gas-liquid slurry reactors[1], bioreactors[2–4], proton exchange membrane (PEM) fuel cells[5], etc. The key fea- ture of this phenomenon is the concentration jump described by Henry’s law at the phase interface. Moreover, the different properties between different solvents also increase the difficulty of numerical simulation. In the present paper, attentions are paid to the general conjugate mass transfer between different solvents, the simplified situations [6–8] where the mass transfer in one phase is neglected after some simplifications and the mass transfer with chemical reaction[9–11]are not discussed.

The mass transfer phenomenon in general is described by Fick’s law!q

m¼ DrC, and the corresponding governing equation can be written as

@C

@t ¼rðDrCÞ r!uC

ð1Þ For a gas-liquid conjugate mass transfer problem, the interface conditions are determined by the continuity of mass flux and Henry’s law that describes the concentration jump at the interface [12], which can be respectively expressed as

DGrCG¼ DLrCL ð2Þ

He¼CG=CL ð3Þ

whereDGandDLare the diffusivity coefficients in the gas and liquid phases, respectively,CGandCLare the solute concentrations in the gas and liquid sides, respectively,Heis the distribution coefficient determined by the properties of the solute and the solvents. Note that the description for gas-liquid system above can be extended to liquid-liquid system as well.

To simulate mass transfer in multi solvent systems, many numerical methods have been proposed. Those numerical methods can be classified into two groups, which are the unified scheme that solves the whole concentration field directly and the discrete scheme that solves the governing equations at different solvents separately.

The unified scheme has been widely used to simulate the mass transfer in multi solvent systems owing to its simplicity and effi- ciency. For instance, Davidson and Rudman[12]developed a volume of fluid (VOF) method for this problem. Coupling with the deforming mesh for flow field, they simulated the mass transfer between a rising drop and the surrounding liquid. However, only a continuous equilibrium concentration distribution between different solvents was considered. Onea et al. [13] proposed an approach to simulate the mass transfer in multi solvent systems by adding an additional source term to the governing equation, and then made a qualitative computational study on the mass

⇑ Corresponding author.

E-mail address:[email protected](C.S. Dai).

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j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / i j h m t

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transfer in upward bubble chain flow through square and rectangular mini-channels. Based on VOF method as well, some other researchers dealt with the concentration jump at the interface by adding virtual mass flux to the governing equation at the interface area. Haroun et al.[14,15]derived a unified equation to simulate the mass transfer in different solvents, in which the virtual mass flux induced by the gradient of fluid fraction is added to approximately ensure the concentration jump at the interface. More recently, Marschall et al. [16] proposed a similar algorithm to model the conjugate mass transfer in free-surface flows by using the software OpenFOAM. What distinguishes this algorithm from that of Haroun et al.[14,15]is the average law of the parameters at the diffusive interface area. Both of the two algorithms have been validated by comparing the calculated results with the exact solutions, respectively, and no obvious deviations have been observed. Following the algorithm proposed by Haroun et al., Nieves-Remacha [17] made a research on the conjugate mass transfer in an advanced-flow reactor with VOF method.

In addition, some researchers simulated mass transfer in multi solvent systems by introducing continuous scalars. Yang and Mao [18] simulated the conjugate mass transfer of a droplet moving in a continuous immiscible liquid with level set method. The concentration field is solved through the one-fluid approach by introducing a new scalar that corresponds toC_LHe¹⁼²in the liquid phase andCG=He¹⁼²in the gas phase. The scalar transformation makes the transformed scalar continuous at the interface, however, several parameters, such as local velocity and diffusivity, have to be transformed as well to satisfy the mass flux continuity at the interface.

Bothe et al.[19]simulated the mass transfer of oxygen from single bubbles and bubble chains rising in aqueous solutions by using the VOF method. The concentration jump described by Henry’s law was considered by introducing a new continuous scalar related to the concentration as well.

Another approach to simulate mass transfer in multi solvent systems is the discrete scheme. Through the level set method, Deshpande and Zimmerman [20] simulated the mass transfer across the moving droplet, where the mass transport equations in different phases are solved separately. Molaeimanesh and

Akbari [5] used the lattice Boltzmann method (LBM) to model the oxygen transfer across the gas-electrolyte interface in a cathode catalyst layer of a PEM fuel cell. In their method, double distribution functions are adopted to simulate the oxygen transfer in the gas phase and the electrolyte phase, respectively. Henry’s law at the interface is satisfied by adjusting the two distribution functions and taking the bounce-back approximation. The shortage of their method is that the interface must locate at the computational nodes, which in general cannot be ensured for a moving interface, even for a curved stationary interface. In addition, Li et al.[21]proposed a difference algorithm for conjugate heat and mass transfer in LBM, however, the concentration jump condition was not taken into account. Following the difference algorithm mentioned above, they developed their work[22]to deal with temperature (concentration) jump and heat (mass) flux jump at the interface. However, the concentration jump value in their difference algorithm is spec- ified while the actual value is variable with the concentration distribution at the two sides of the interface.

As a brief summary, major researches introduced above are based on the governing equation written as Eq.(1), therefore, have to satisfy the concentration jump at the interface through different approaches, such as adding a source term, virtual mass flux and so on. Obviously, it brings additional computation cost to the numerical simulations. Compared with these methods, introducing a new continuous scalar[18,19]to describe the concentration jump at the interface is more reasonable. However, several parameters, such as local velocity and diffusivity, have to be transformed as well, and the physical meaning is not clear.

In the present paper, the shortages of Fick’s law in describing mass transfer in multi solvent systems are discussed at first. Then, a new governing equation based on the revised Fick’s law is proposed to simulate mass transfer in multi solvent systems. As a consequence, the concentration jump can be directly ensured without additional treatments for the phase interface. In addition, the revised governing equation does not need to transform the scalar or parameters. To solve the revised governing equation, a novel LBE, which can recover the revised governing equation through the Chapman-Enskog expansion analysis, is constructed.

Nomenclature

cs lattice sound speed (m=s) C concentration (mol=m³) Cav^e average concentration

CG concentration in the gas (mol=m³)

C^eq_G equilibrium concentration in the gas (mol=m³) Ch high concentration (mol=m³)

Cl low concentration (mol=m³) CL concentration in the liquid (mol=m³)

C^eq_L equilibrium concentration in the liquid (mol=m³) d diameter (m)

D diffusivity coefficient (m²=s)

Dequiv equivalent diffusion coefficient (m²=s) DG diffusivity coefficient in the liquid (m²=s) DL diffusivity coefficient in the gas (m²=s)

!e

i discrete velocity in directioni(m=s) g_i distribution function (mol=m³) g^{ð Þ}_i^eq distribution function (mol=m³)

H length (m)

He distribution coefficient L side length (m)

NL the amount of solute across the interface from the liquid phase within a unit time (mol=s)

NG the amount of solute across the interface from the gas phase within a unit time (mol=s)

q_e mass flux of the east face (mol=m²s ) q_m mass flux (mol=m²s

)

q_n mass flux of the north face (mol=m²s ) q_s mass flux of the south face (mol=m²s

) q_w mass flux of the west east face (mol=m²s

) r distance to the central point (m)

R0;R1,R2 radius (m)

t time (s)

t dimensionless time

!u

velocity (m=s) w weight coefficient x abscissa (m) y ordinate (m)

Greek symbols

D^t lattice time step (s) D^x lattice space (m)

s

g dimensionless relaxation time

/ a dimensionless parameter that proportional to the average diffusion velocity

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2. Analyses on Fick’s law

Fick’s law has been widely used to simulate mass transfer problems. However, we believe that it is not reasonable enough in describing mass transfer in multiphase systems, because Henry’s law has to be artificially introduced to describe the concentration jump at the interface. If we use the governing equation based on Fick’s law, Eq.(1), to describe a conjugate mass transfer problem between the gas and liquid, it is easy to get a continuous concentration distribution. In fact, there is a concentration jump described by Henry’ law at the interface, which brings extra difficulties to the numerical simulation. It is an obvious shortage of Fick’s law in describing conjugate mass transfer. However, many researchers mentioned above neglected it and considered it as a special fea- ture. By comparing Fick’s law with Fourier’s law of heat conduc- tion, which has a similar expression with Fick’s law, it can be seen that Fourier’s law can be extended to multi solvent systems directly, while Fick’s law cannot.

Beside the shortage mentioned above, another shortage of Fick’s law is that the equivalent diffusion coefficient defined by Fick’s law is variable with the concentration distribution. Here, a one- dimensional conjugate mass transfer problem shown inFig. 1is considered. It is an area with a length ofH filled with liquid in the left side and gas in the right side. The left and right walls are imposed with constant low concentration (Cl) and constant high concentration (Ch), respectively. After taking the summation of Eqs.(1)(3), the analytical solution at steady state can be expressed as

C¼

2ðD_GC_hDGHeC_lÞ DLþDGHe

ð ÞH xþCl in liquid

C_hDLClDLHe D_LþDGHe

2x

Hþ^HeC^h^D^G_D^þ2HeD^L^C^l^C^h^D^L

LþDGHe in gas 8<

: ð4Þ

For the entire area, the mass flux across the interfaceq_mand the equivalent diffusion coefficientDequivare given as

qm¼ Dj rCj ¼2DGDLðChHeClÞ DLþDGHe

ð ÞH ð5Þ

Dequiv¼ Hq_m ChCl

¼ChHeCl

ChCl

2DGDL

DLþDGHe ð6Þ

It can be seen that the equivalent diffusion coefficientDequivvar- ies with the concentration distribution. The dependence property ofDequiv on concentration distribution may cause inconveniency and difficulties in analyzing the equivalent diffusion coefficient for a multi solvent system.

The two shortages cannot be seen as the special features of conjugate mass transfer. On the contrary, a more reasonable equation which can be directly extended to multi solvent systems is required. According to the discussions above, it can be concluded that the key problem of Fick’s law is taking concentration as the driving potential in conjugate mass transfer. For an equilibrium concentration distribution between liquid and gas, there is no mass flux across the interface while has a concentration difference determined by Henry’s law. It certainly proves that it is unreason-

able to take concentration as the driving potential in conjugate mass transfer.

To analyze the driving potential in conjugate mass transfer, the physical mechanism of Henry’s law is previously discussed. The one-dimensional equilibrium conjugate mass transfer between liquid and gas is shown inFig. 2. It is an area with an infinite length filled with liquid in the left side and gas in the right side. We define NLandNGas the amount of solute across the interface from liquid phase and gas phase within a unit time, respectively. Then, the net mass flux across the interface can be expressed as

q_m¼NLNG¼0 ð7Þ

Assuming that the solute molecules are independent and satisfy the same diffusion velocity distribution,fð Þ, in each solvent, then

v

we can discuss the expressions ofNLandNG. Note that the diffusion velocity is the average velocity in the time scale of diffusion, and the solute molecules may collide with the solvent molecules many times in a time step of this time scale. For the solute molecules moving in the liquid, the probability of a solute molecule in the velocity range of^ð

v

^;

v

^þ^d

v

^Þcan be given asf_L^{ð Þd}

v v

^{. For the}

solute molecules in the velocity range of^ð

v

^;

v

^þ^d

v

Þ, they are able to cross the interface within a time intervalD^tif the solute molecules are in the position range ofð

v

^D^t; 0Þ. So thatNLcan be evaluated as

NL¼ R_þ1

0 C^eq_L

v

D^tfL^{ð Þd}

v v

D^t ^¼

Z _þ1

0

C^eq_L

v

^f^L^{ð Þd}

v v

^ð8Þ

Define the average diffusion velocity^{h i}

v

Lin the liquid as

v

h i_L¼ Z _þ1

0

v

^f^L^{ð Þd}

v v

^ð9Þ

Thus,NLcan be expressed asNL¼C^eq_L^{h i}

v

L. Similarly, we have

NG¼C^eq_G^{h i}

v

G ð10Þ

where^{h i}

v

Gis the average diffusion velocity in the gas. Substituting Eqs.(8)(10)into Eq.(7), we can get

He¼C^eq_G C^eq_L ¼^{h i}

v

L

v

h i_G ð11Þ

It can be seen that the mass flux in the conjugate mass transfer is not only related to concentration, but also related to the average diffusion velocity in that phase. Therefore, it is more reasonable to take the product of the concentration and the average diffusion velocity, Ch i, as the driving potential. Correspondingly, Fick’s

v

law should be modified as qm¼ D

v

h ir^ðC^{h i}

v

^Þ ^ð12Þ

To keep writing in a similar format with the original Fick’s law, we introduce a dimensionless parameter/that proportional to^{h i.}

v

The value of/is given as /¼ 1 in gas

He in liquid

ð13Þ

Fig. 1.Conjugate mass transfer between liquid and gas. Fig. 2.Conjugate mass transfer between liquid and gas in equilibrium.

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Finally, Fick’s law can be revised to

!q

m¼ D

/rðC/Þ ð14Þ

The corresponding governing equation can be expressed as

@C

@t ¼r D

/rðC/Þ

r!uC

ð15Þ

The interface conditions of the governing equation are C/

ð Þ_L¼ðC/Þ_G ð16Þ

D /rðC/Þ

G

¼ D /rðC/Þ

L

ð17Þ

They are equivalent with the interface conditions given by Eqs.

(2) and (3), respectively, however, can be satisfied naturally in the new governing equation, while Henry’s law is artificially added to Eq.(1). As a consequence, the concentration jump can be described directly by using the new governing equation.

By takingC/as the driving potential of mass transfer in a multi solvent system, the equivalent diffusion coefficient of the physical problem described inFig. 1can be given as

D

/ equiv¼ Hq_m ChHeCl

¼ 2HDGDL

DLþDGHe ð18Þ

which is independent of concentration distribution. It indicates that the other shortage, which is the variable equivalent diffusion coefficient in the multi solvent system, can be removed as well in the new governing equation.

In conclusion, the proposed equation, Eq.(14), is equivalent with Fick’s law in describing mass transfer in a single phase. How- ever, it can be extended to multi solvent systems directly. Coupling with Henry’s law, the derived governing equation, Eq. (15), can ensure the concentration jump at the phase interface so that it is more reasonable than the original Fick’s law in describing mass transfer in multi solvent systems.

3. Numerical method

To solve the proposed governing equation, a lattice Boltzmann equation (LBE) is proposed. As an alternative numerical method for solving N-S equation and convection-diffusion equation, the lattice Boltzmann method based on Boltzmann equation in statis- tical mechanics has been developed rapidly in recent years. It has been proven to be an effective and very simple method in simulating multiphase fluid flows[23–28]. It has also been applied to simulate heat transfer in multiphase systems [29] and solid-liquid phase change problems[30].

3.1. SRT collision scheme

The single-relaxation-time (SRT) lattice Boltzmann equation (also known as BGK model) for mass transfer in multi solvent systems is written as follows:

g_i ! þx !e

iDt;tþD^t

¼g_i!x;t 1

s

^g ^gⁱ ^!^x^;^t

g^{ð Þ}_i^eq!x;t

h i

ð19Þ whereg_iis the discrete concentration distribution function,g^{ð Þ}_i^eq is the equilibrium concentration distribution function and

s

^g ^{is the}

single relaxation time.Here, two-dimensional model with nine discrete velocities (D2Q9 model) is adopted for simulation. Note that the two-dimensional model can be extended to three-dimensional

easily. The discrete velocities!e

iand equilibrium concentration distribution functions are, respectively, given as follows:

ei¼ 0 1 0 1 0 1 1 1 1

0 0 1 0 1 1 1 1 1

c ð20Þ

g^{ð Þ}_i^eq ¼

1/

ð ÞCþw0C/ i¼0 wiC /þ!e

i!u

c²_s

i–0 8<

: ð21Þ

where!u is the velocity,wiis the weight coefficient which is given as w₀¼4=9, w_1;2;3;4¼1=9 and w_5;6;7;8¼1=36, c_s is the lattice sound speed calculated by cs¼1= ffiffiffi

p3

c, / is the dimensionless parameter that proportional to the average diffusion velocity. Note that the values of/in the liquid/L and gas /G are arbitrary in theory when Eq.(11) is satisfied. However, in practical tests, it is found that they have to be confined in the range of maxf/G;/Lg61 to ensure the numerical stability of proposed LBE. Therefore, the values of /L and /G in LBE are given as /G¼1=max 1ð ;HeÞ and /L¼He=max 1ð ;HeÞ, respectively. The relaxation time

s

^g is related to the diffusivity coefficient through the following equation:

D

/¼

s

^g0:5

c²_sDt ð22Þ

And the bulk concentration is obtained by C¼X

i

gi ð23Þ

Through the Chapman-Enskog expansion analysis, the LBE can recover the revised governing equation, Eq.(15), up to second- order accuracy. Firstly, via a Taylor expansion in time and space, the lattice Boltzmann equation, Eq.(19), can be written in a continuous form as

@tþ!e

ir

giþD^t 2 @tþ!e

ir

2

giþODt²

¼ 1

s

^gD^t ^gⁱ^g

eq i

ð24Þ

Then, we expandg_ias

g_i¼g^{ð Þ}_i⁰ þ

e

^g^{ð Þ}i¹ þ

e

²^g^{ð Þ}i² þO

e

³ ^ð25Þ

where

e

is a small parameter proportional to the Knudsen number.

At the same time, two macroscopic time scalest1¼

e

^t,^t2¼

e

²^t^and

a macroscopic length scalex1¼

e

^x are introduced to recover the control equation, thus we have

@t¼

e

@t1þ

e

²@t2þO

e

³ ; r¼

e

^r¹ ^ð26Þ

Substituting the expansions in Eqs.(25) and (26)into Eq.(24), we can rewrite Eq.(24)in the different orders of the small expansion parameter

e

as follows:

e

⁰^: ^g^{ð Þ}i⁰ ¼g^{ð Þ}_i^eq ð27aÞ

e

¹: @t1þ!e

ir1

g^{ð Þ}_i⁰ ¼ 1

s

gD^t^g

ð Þ1

i ð27bÞ

e

²: @t2g^{ð Þ}_i⁰ þ @t1þ!e

ir1

1 1 2

s

^g

g^{ð Þ}_i¹

¼ 1

s

^gDtg^{ð Þ}_i² ð27cÞ Through some algebraic computation, it can be seen that the equilibrium distribution function g^{ð Þ}_i^eq satisfies the following formulations:

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X⁸

i¼0

g^{ð Þ}_i^eq ¼C; X⁸

i¼0

g^{ð Þ}_i^eq!e

i¼C u!; X⁸

i¼0

g^{ð Þ}_i^eq!e

i!e

i¼C/c²sI ð28Þ

whereI is a unit matrix. Thus, by combining Eqs.(23), (25) and (27a), we can get

e

ⁿ: X⁸

i¼0

g^{ð Þ}_iⁿ ¼0; nP1 ð29Þ

Taking the summation of Eqs.(27b) and (27c)about indexi, respectively, we can obtain the corresponding macroscopic equations:

e

¹: @t1Cþr1!uC

¼0 ð30aÞ

e

²: @t2Cþr1 1 1 2

s

^g

X⁸

i¼0

!e

ig^{ð Þ}_i¹

" #

¼0 ð30bÞ

By multiplying both sides of Eq.(27b)by!e

i, then substituting it into Eq.(30b)and neglecting the higher order terms, Eq.(30b)can be further simplified as

e

²: @t2Cr1 c²_s

s

^g0:5

D^tr1ð/CÞ

¼0 ð31Þ

Combining the equations at the orders of

e

¹^and

e

², i.e., Eqs.(30a) and (31), we can obtain the macroscopic equation.

@tCþr!uC

¼r c²_s

s

^g0:5 D^trð/CÞ

ð32Þ By defining^D_/¼c²_s

s

g0:5

Dt;the macroscopic governing equation, Eq.(15), can be recovered

3.2. MRT collision scheme

In SRT collision scheme, the evolutions of distribution functions of different discrete velocities are independent, and the numerical stability is confined. To improve the SRT collision scheme, d’Humieres proposed the Generalized LBE with multi-relaxation- time collision scheme, which is also called as the MRT LBE. The later research [31] proved that the MRT collision scheme has advantages in choosing adjustable parameters and numerical stability. The MRT LBE for conjugate mass transfer is expressed as

g_i ! þx !e

iDt;tþD^t

E

g_i!x;tE

¼ hM¹SMi

g_i!x;t

E

g^{ð Þ}_i^eq!x;tE

ð33Þ where the subscriptjidenotes column vector and½denotes matrix.

Here,½ M is the matrix that projects a vector onto the moment space j i, which can be expressed asm [30]

M¼

1 1 1 1 1 1 1 1 1

4 1 1 1 1 2 2 2 2

4 2 2 2 2 1 1 1 1

0 1 0 1 0 1 1 1 1

0 2 0 2 0 1 1 1 1

0 0 1 0 1 1 1 1 1

0 0 2 0 2 1 1 1 1

0 1 1 1 1 0 0 0 0

0 0 0 0 0 1 1 1 1

2 6666 6666 6666 6666 664

3 7777 7777 7777 7777 775

ð34Þ

The corresponding equilibrium momentjm^eqiis given as m^eq

j i ¼C;4Cþ2C/; 4C2C/; uxC;uxC;uyC; uyC; 0; 0 ð35Þ

The relaxation matrix ½ , which is a diagonal matrix, can beS expressed as

S¼diag s0; se; s_e; sj; sq; sj; sq; se; se

ð36Þ The same with the SRT collision scheme, the macroscopic parameters are also obtained by Eqs.(22) and (23).

Among the relaxation parameters in the diagonal relaxation matrix, the values ofs0andsjare set ass0¼1 andsj¼1=

s

^g^{, respec-}

tively, and the rest relaxation parameters, which can be adjusted to achieve better performance in real application, are variable in a range of 0<s_e;_e_;q<2. In the present paper, the relation given by Eq.(37)is kept in the MRT collision scheme.

1 se

1 2

1 sj

1 2

¼1

4 ð37Þ

By keeping the relation, the better numerical accuracy and stability can be achieved[32]. It has been proved by Huang et al.[30]

that the selected parameters can significantly eliminate the numerical diffusion across the interface in a solid-liquid phase change problem with discontinuous heat flux across the interface.The detailed procedures to solve the SRT LBE and MRT LBE are given as follows:

(a) Collision step. For SRT collision scheme, the equation is g_i!x;tþD^t^¼^gi!x;t

1

s

^g ^gⁱ ^!^x^;^t

g^{ð Þ}_i^eq!x;t

h i

ð38Þ

while for MRT collision scheme, the equation is gi!x;tþDt

E

¼gi!x;tE

hM¹SMi

gi!x;t

E

g^{ð Þ}_i^eq!x;tE

ð39Þ (b) Streaming step.

gi ! þx !e

iDt;tþDt

¼gi!x;tþDt

ð40Þ (c) Update the bulk concentration and the equilibrium distribu-

tion function through Eqs.(23)and(21), respectively.

3.3. Control volume method (CVM)

For validation of the proposed method, the control volume method (CVM) is applied to make numerical simulation as well.

The proposed governing equation, Eq. (15), is discretized at the uniform mesh. As depicted inFig. 3, the blue¹color area is the control volume of nodeð Þ,i;j q_n,q_s,q_wandq_eare the mass fluxes of the north, south, west and east faces, respectively. For time discretization, predictor-corrector scheme is adopted, thus the discrete equation can be written as

C^p¼Cⁿþqn Cⁿ þqs Cⁿ þqwCⁿ þqe Cⁿ Dt=Dx C^nþ1¼Cⁿþ¹₂q_n Cⁿ þq_s Cⁿ þq_wCⁿ þq_e Cⁿ

Dt=D^x þ¹₂q_n C^p þq_s C^p þq_w C^p þq_e C^p

Dt=D^x 8>

<

>: ð41Þ

whereC^pis the predicted concentration. The mass flux of north face q_ncan be expressed as

qnð Þ ¼ C ½u i;ð jþ1ÞC i;ð jþ1Þ u i;ð ÞC i;j ð Þj=2 þ 2D i;ð ÞD i;j ð jþ1Þ

D i;ð jþ1Þ/ð Þ þi;j D i;ð Þ/j ði;jþ1Þ ð/ði;jþ1ÞC i;ð jþ1Þ /ð ÞC i;i;j ð ÞjÞ

Dx ð42Þ

1For interpretation of color in Fig. 3, the reader is referred to the web version of this article.

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Here, central difference scheme is adopted for discretizing the convection term. The expressions of the mass fluxes of the rest three faces are similar with Eq.(42).

4. Results and discussion

To validate the proposed unified LBE, five test cases for conjugate mass transfer are calculated, which include: (1) steady mass transfer between liquid and gas; (2) transient conjugate mass transfer with only diffusion; (3) transient conjugate mass transfer with both diffusion and convection; (4) steady conjugate mass transfer in a two-layer annulus; (5) transient conjugate mass transfer with a curved interface. The results are compared with the analytical solutions or the results calculated by the control volume method (CVM) to verify the correctness of the MRT LBE.

4.1. The steady mass transfer between two solvents

In this part, a one-dimensional conjugate mass transfer problem between liquid and gas is considered. As depicted in Fig. 4, the computational domain for simulation is filled with half liquid and half gas in the left and right sides, respectively. The left and right boundaries are imposed with constant low concentrationC_l and high concentrationCh, respectively.

In the simulation, the diffusivity coefficient ratio is set as D_G=D_L¼2, while the distribution coefficient He is variable in a range from 0.01 to 100. For practical applications like the absorp- tion of CO2, O2and H2S in water, the distribution coefficients are about 1.1, 30 and 0.4 [15], respectively. Therefore, the present selected range ofHeis enough for most of practical applications.

For simplicity, the values of Cl and Ch are set as Cl¼0 and Ch¼1, respectively.

To make comparison with the analytical solutions, the steady concentration distributions at differentHeare calculated and compared with the corresponding analytical solutions. The mesh size and minimum relaxation time in the two phases used in numerical simulations areD^x^¼^H=100 and 0.53, respectively. The analytical solution of this problem is

C¼

2ðDGC_hDGHeC_lÞ D_LþDGHe

ð ÞH xþCl x<H=2

ChDLClDLHe DLþDGHe

2x

Hþ^HeC^h^D^G_D^þ2HeD^L^C^l^C^h^D^L

LþDGHe x>H=2

8<

: ð43Þ

The results calculated by the MRT LBE are summarized inFig. 5, it is obvious that all the results obtained by the MRT LBE show very good agreement with the analytical solutions at steady state. The concentration jumps at the gas-fluid interface for different He can be simulated correctly without any additional modification.

It proves that the proposed LBE is capable of simulating the steady conjugate solute diffusion. In addition, it shows good numerical stability against a large range ofHe.

4.2. Transient conjugate mass transfer with only diffusion

To test the MRT LBE for transient conjugate mass transfer, a one-dimensional conjugate mass transfer problem in an infinite system is simulated in this part. As shown inFig. 6, it is an infinite system filled with liquid in the left side and gas in the right side.

The interface locates atx¼0, and the liquid phase has a uniform low concentrationCl¼0 while the gas phase has a uniform high concentration Ch¼1 at the initial timet¼0. The length of the computational domain is H, the diffusion coefficient ratio is DL=DG¼0:2 and the distribution coefficient isHe¼0:5. For gener- ality, the time is made dimensionless as

t¼t= H²=DL

ð44Þ To simulate an infinite system problem with finite meshes, the computation time should be short enough before the mass flux has obvious influence on the two sides of the interface. Therefore, the dimensionless time is limited to be less than 210³in the simulation. The relaxation times in liquid and gas are set as

s

g L¼0:5108 and

s

g G¼0:527, respectively, the mesh size is Dx¼H=100, the values ofCh andClare set asCh¼1 andCl¼0, respectively. For validation, the concentration distributions at four different transient periods including t¼0, t¼410⁴, t¼110³,t¼210³are captured and then compared with

Fig. 4.Mass transfer between liquid and gas.

0.0 0.2 0.4 0.6 0.8 1.0

-0.1 0.6 1.3 2.0

C

x/H

He=0.01, analytical He=0.1, analytical He=1, analytical He=10, analytical He=100, analytical He=0.01, MRT LBE He=0.1, MRT LBE He=1, MRT LBE He=10, MRT LBE He=100, MRT LBE

Fig. 5.Comparison of concentration distributions between the MRT LBE results and analytical solutions at differentHe.

Fig. 3.Discretization scheme of CVM at uniform mesh.

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the corresponding analytical solutions. The analytical solution of the problem has been given in the Ref.[15], as shown in the following equation:

C¼

ClþðChHeClÞ^{1þerf x=} ffiffiffiffiffiffiffi

4DLt

p

Heþ ffiffiffiffiffiffiffiffiffiffi

DL=DG

p in liquid

ChþðClCh=HeÞ^{1erf x}⁼ ffiffiffiffiffiffiffiffi

4D_Gt

p

1=Heþ ffiffiffiffiffiffiffiffiffiffi

DG=DL

p in gas

8>

><

>>

: ð45Þ

The comparisons between the MRT LBE and the analytical solutions are displayed inFig. 7. It can be seen that the results obtained by the MRT LBE agree well with the analytical solutions at different transient states. Especially, the concentration jumps at the interface can be described precisely when sharp interface is adopted.

To compare the accuracy of the MRT LBE and SRT LBE, their convergence orders are analyzed. The physical model introduced above is calculated by the two LBE schemes under different mesh sizes. The relaxation times in liquid and gas are set as

s

g L¼0:5108 and

s

g G¼0:527, respectively. The corresponding errors of the concentration distribution are displayed in Fig. 8.

The error is defined as follows:

Error1¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi PN

j¼1C^c_j C^a_j2

r

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi PN

j¼1 C^a_j ²

r ð46Þ

whereC^c_j is the calculated concentration whileC^a_j is the analytical solution at pointj, andN is the total number of computational nodes. The errors are estimated at dimensionless time t¼210⁴ when the boundaries are far from being fluctuated by mass flux.

Based on the error analysis of the concentration field, the convergence orders of the two LBE schemes can be obtained. As shown in Fig. 8, the SRT LBE scheme processes a convergence order of 1.953 while the convergence order of the MRT scheme is 2.117.

At the same time, the error of the MRT LBE is obviously less than that of the SRT LBE. It indicates that the LBE of MRT scheme could have better accuracy than SRT scheme by keeping an optimal relation between two relaxation times.

In addition, the local error distributions of the two LBEs are given inFig. 9to compare the local error. The local error is defined asError2¼CC^a=jC_hC_lj. As shown inFig. 9, for both LBEs, the maximum errors appear in the area near the interface. Compared with the SRT LBE, the maximum error of the MRT LBE is smaller, but the local error of MRT LBE exceeds SRT LBE at some area.

4.3. Transient conjugate mass transfer with both diffusion and convection

To validate the proposed LBE further, a transient conjugate mass transfer problem with both diffusion and convection is considered in this part. The physical model is the same with the one in Section4.2at the initial time, however, the whole system will move toward right with a constant velocity u¼20DG=H when the time t>0. For the moving interface, the volume of fluid (VOF) method is adopted and the position is captured implicitly by the fraction of fluidF. For such a simple 1D case, the distributions ofFat different time can be previously given as follows:

Fig. 6.1D conjugate mass transfer in an infinite system with a stationary interface.

-0.50 -0.25 0.00 0.25 0.50

0.0 0.3 0.6 0.9 1.2

C

x/H

t^*=0, analytical t^*=4×10^-4, analytical t^*=1×10^-3, analytical t^*=2×10^-3, analytical t^*=0, MRT LBE t^*=4×10^-4, MRT LBE t^*=1×10^-3, MRT LBE t^*=2×10^-3, MRT LBE

Fig. 7.Comparison ofCbetween the MRT LBE results and the analytical solutions at different periods for one-dimensional transient conjugate mass transfer with only diffusion.

-2.6 -2.4 -2.2 -2.0 -1.8 -1.6

-4.0 -3.5 -3.0 -2.5 -2.0 -1.5

slope=2

slope=2.117

lg ( Er ro r 1)

lg( x/H )

SRT MRT slope=1.953

Fig. 8.Error versus the lattice interval of two LBE schemes.

0.3 0.4 0.5 0.6 0.7

-0.002 0.003 0.008 0.013 0.018

Error 2

x/H

SRT MRT

Fig. 9.Local error distributions of the two LBE schemes.

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F¼

1 x<utRm

1^xutþR_2R ^m

m utRm6x6utþRm

0 x>utþRm

8>

<

>: ð47Þ

where R_m is the half width of the interface with a value of Rm¼0:5Dx. Thus, the parametersDand/at different time can be determined by Eqs.(48a) and (48b), respectively.

/¼½F=/Lþð1FÞ=/G¹ ð48aÞ

D¼/ F DL

/L 1

þð1FÞ DG

/G

" 1#1

ð48bÞ

The analytical solution of this problem can be obtained from Eq.

(45)through transforming the coordinatextoxut. Thus, the corresponding analytical solution can be expressed as follows:

C¼

ClþðChHeClÞ^1þerf ^ð^xut^Þ=

ffiffiffiffiffiffiffi

4DLt

p

Heþ ffiffiffiffiffiffiffiffiffiffi

DL=D_G

p x<ut ChþðClCh=HeÞ^1erf ^ð^xut^Þ= ffiffiffiffiffiffiffiffi

4DGt

p

1=Heþ ffiffiffiffiffiffiffiffiffiffi

DG=DL

p x>ut 8>

><

>>

: ð49Þ

Similarly, the concentration distribution calculated by the MRT LBE at four different transient periods including t¼0, t¼410⁴,t¼110³,t¼210³are output and compared with the corresponding analytical solutions. The relaxation times in liquid and gas are set as

s

g L¼0:512 and

s

g G¼0:53, respectively, and the mesh size isD^x^¼^H=100. The comparisons between the MRT LBE results and the analytical solutions are shown in Fig. 10. It can be seen that the concentration distributions at different periods calculated by the MRT LBE match well with the corresponding analytical solutions, except the concentration jump at the moving interface cannot be described precisely due to the use of the diffusive interface.

4.4. Steady conjugate mass transfer in a two-layer annulus

To test the MRT LBE against the conjugate mass transfer with a curved interface, a two-dimensional conjugate mass transfer problem in a two-layer annulus is simulated. The model shown in Fig. 11is a two-layer annulus filled with liquid and gas. The defini- tion ofris the distance from a point to the center point and the area R06r6R1 belongs to liquid while the area R16r6R2

belongs to gas. The geometry parameters are set asR1¼2R0and R2¼3R0, the diffusivity coefficient ratio is set as DL:DG¼1:1

while the distribution coefficientHeis a variable with values of 0.01, 0.1, 1, 10, 100. The outer and inner circles are kept at constant high concentrationCh¼1 and low concentration Cl¼0, respectively. A difference method for boundary treatment[33]is adopted to satisfy the boundary conditions. For the interface between the two annuluses, the VOF method is adopted and the liquid fraction is approximately evaluated as

F¼

1 r<R1Rm

1^rR_2R¹^þR_m ^m R1Rm6r6R1þRm

0 r>R1þRm

8>

<

>: ð50Þ

where Rm is the half width of the interface with a value of Rm¼0:5Dx.

The concentration distributions at steady state are compared with the corresponding analytical solutions. The minimum relaxation time in the entire area is set as 0.53, and the mesh size is D^x^¼^R2=120. The analytical solution at steady state can be expressed as

C¼

D_GIn rð=R₀Þ

DLIn Rð2=R1ÞþDGHeIn Rð1=R0Þ R06r<R1 DLIn rð=R1ÞþDGHeIn Rð1=R0Þ

DLIn Rð2=R1ÞþDGHeIn Rð1=R0Þ R1<r6R2

8<

: ð51Þ

As shown in Fig. 12, obviously, good agreement can be observed. In addition, for conjugate mass transfer with a curved interface, the MRT LBE shows good numerical stability against a large range ofHeas well.

-0.50 -0.25 0.00 0.25 0.50

0.0 0.3 0.6 0.9 1.2

C

x/H

t^*=0, analytical t^*=4×10^-4, analytical t^*=1×10^-3, analytical t^*=2×10^-3, analytical t^*=0, MRT LBE t^*=4×10^-4, MRT LBE t^*=1×10^-3, MRT LBE t^*=2×10^-3, MRT LBE

Fig. 10.Comparisons ofCbetween the MRT LBE results and the analytical solutions at different periods for one-dimensional transient conjugate mass transfer with both convection and diffusion.

Fig. 11.Conjugate mass transfer between liquid and gas with a curved interface.

1 2 3

-0.1 0.4 0.9 1.4 1.9

C

He=0.01, present He=0.01, analytical He=0.1, present He=0.1, analytical He=1, present He=1, analytical He=10, present He=10, analytical He=100, present He=100, analytical

r/R

₀

Fig. 12.Comparison of concentration between the MRT LBE results and analytical solutions at differentHe.

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4.5. Transient conjugate mass transfer with a curved interface

Aiming to validate the proposed LBE further, another transient conjugate mass transfer problem with a curved interface is simulated. As described inFig. 13, it is a square cavity containing a liquid drop at the center. The side length of cavity is L and the diameter of the liquid drop isd¼0:8L. All the boundaries of the cavity is set at constant concentrationCh¼1, and the entire area has a uniform initial concentrationCl¼0. The distribution coefficient is set asHe¼2 while the diffusion coefficient ratio is given

asDG=DL¼0:2. For the circular interface between the liquid drop and gas, the VOF method is adopted as well.

This model is calculated by both the control volume method (CVM) and MRT LBE. The mesh number and time step size used in the two methods are 200200 and 210⁶L²=DL, respectively.

The calculated contours of concentration at the transient time Fig. 13.Conjugate mass transfer between a liquid drop and the surrounding gas.

(a1) t

^*

= 0 . 2 , MRT LBE (a2) steady, MRT LBE

(b1) t

^*

= 0 . 2 , CVM (b2) steady, CVM

Fig. 14.Comparisons between the MRT LBE (a1, a2) and CVM (b1, b2) at transient periodt¼0:2 (left) and steady state (right).

0.0 0.2 0.4 0.6 0.8 1.0

0.0 0.2 0.4 0.6 0.8

C

ave

t

^*

MRT CVM

Fig. 15.Comparison of transient average concentration of the entire area between the MRT LBE and CVM.

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t¼t= L²=DL

¼0:2 and the steady state are compared inFig. 14.

As expected, the contours calculated by MRT LBE match well with those calculated by CVM at both the transient and steady states.

To compare the results of the two methods in more detail, the curves of the average concentration of the entire areaCaveversus the dimensionless time are given. As shown inFig. 15, the transient average concentration obtained by the MRT LBE shows good agreement with that calculated by CVM as well. It indicates that the MRT LBE can be easily implemented to deal with curved interface the by adopting the diffusive interface.

5. Conclusion

In the present paper, the shortages of Fick’s law in describing mass transfer in multi solvent systems are analyzed. Then, the revised Fick’s law that can be extended to multi solvent systems is proposed and the corresponding governing equation can be constructed. To solve the proposed governing equation, a unified LBE is constructed. It doesn’t need to artificially add modification for the concentration jump at the interface. Through the Chapman-Enskog expansion, the LBE can recover the proposed governing equation.

Five test cases containing both transient and steady conjugate heat transfer problems with straight or curved interfaces are calculated to validate the proposed MRT LBE. The results show that the proposed MRT LBE has good performances in simulating conjugate mass transfer at both the transient and steady periods, with both flat and curved interfaces.

In conclusion, the proposed MRT LBE is easy in implementation.

It can simulate the conjugate mass transfer with concentration jump at the interface directly without needs to add modification for the interface and transform the scalar or parameters. The curved interface can be easily treated by coupling the LBE with VOF method. In addition, the MRT LBE shows good numerical stability in a large range of concentration distribution coefficientsHe from 0.01 to 100, and it could have better accuracy than the SRT scheme by keeping an optimal relation between two relaxation times.

Conflict of interest statement

We declare that we have no financial and personal relationships with other people or organizations that can inappropriately influence our work, there is no professional or other personal interest of any nature or kind in any product, service and/or company that could be construed as influencing the position presented in, or the review of, the manuscript entitled, ‘‘Lattice Boltzmann equation for mass transfer in multi solvent systems”.

Acknowledgements

This work is financially supported by the National Natural Science Foundation of China (Grant Nos. 41574176 and 41672234).

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