distillation column internals

Chemical Engineering Science 62 (2007) 1839–1850www.elsevier.com/locate/ces

A modified model of computational mass transfer for distillation column

Z.M. Sun, K.T. Yu, X.G. Yuan∗, C.J. LiuState Key Laboratory for Chemical Engineering (Tianjin University) and School of Chemical Engineering and Technology, Tianjin University, Tianjin 300072,

People’s Republic of China

Received 9 October 2005; received in revised form 4 December 2006; accepted 11 December 2006Available online 27 December 2006

Abstract

The computational mass transfer (CMT) model is composed of the basic differential mass transfer equation, closing with auxiliary equations,

and the appropriate accompanying CFD formulation. In the present modified CMT model, the closing auxiliary equations c2.�c [Liu, B.T.,2003. Study of a new mass transfer model of CFD and its application on distillation tray. Ph.D. Dissertation, Tianjin University, Tianjin, China;Sun, Z.M., Liu, B.T., Yuan, X.G., Liu, C.J., Yu, K.T., 2005. New turbulent model for computational mass transfer and its application to acommercial-scale distillation column. Industrial and Engineering Chemistry Research 44, 4427–4434] are further simplified for reducing thecomplication of computation. At the same time, the CFD formulation is also improved for better velocity field prediction. By this complexmodel, the turbulent mass transfer diffusivity, the three-dimensional velocity/concentration profiles and the efficiency of mass transfer equipmentcan be predicted simultaneously. To demonstrate the feasibility of the proposed simplified CMT model, simulation was made for distillationcolumn, and the simulated results are compared with the experimental data taken from literatures. The predicted distribution of liquid velocityon a tray and the average mass transfer diffusivity are in reasonable agreement with the reported experimental measurement [Solari, R.B.,Bell, R.L., 1986. Fluid flow patterns and velocity distribution on commercial-scale sieve trays. AI.Ch.E. Journal 32, 640–649; Cai, T.J., Chen,G.X., 2004. Liquid back-mixing on distillation trays. Industrial and Engineering Chemistry Research 43, 2590–2597]. In applying the modifiedmodel to a commercial scale distillation tray column, the predictions of the concentration at the outlet of each tray and the tray efficiencyare satisfactorily confirmed by the published experimental data [Sakata, M., Yanagi, T., 1979. Performance of a commercial scale sieve tray.Institution of Chemical Engineers Symposium Series, vol. 56, pp. 3.2/21–3.2/34]. Furthermore, the validity of the present model is also shownby checking the computed results with a reported pilot-scale tray column [Garcia, J.A., Fair, J.R., 2000. A fundamental model for the predictionof distillation sieve tray efficiency. 1. Database development. Industrial and Engineering Chemistry Research 39, 1809–1817] in the bottomconcentration and the overall tray efficiency under different operating conditions. The modified CMT model is expected to be useful in thedesign and analysis of distillation column.� 2007 Published by Elsevier Ltd.

Keywords: Computational mass transfer (CMT); c2.�c model; Turbulent mass transfer diffusivity; Simulation; Sieve tray

1. Introduction

Distillation, the most commonly used separation process forthe liquid mixture, has been widely used in the chemical andallied industries due to its reliability in large-size column ap-plication and its maturity in engineering practice. Among thedistillation equipments, the tray column is popularly employedin the industrial production for its simple structure and lowcost of investment. However, the estimation of distillation tray

∗ Corresponding author. Tel.: +86 22 27404732; fax: +86 22 27404496.E-mail address: [email protected] (X.G. Yuan).

0009-2509/$ - see front matter � 2007 Published by Elsevier Ltd.doi:10.1016/j.ces.2006.12.021

efficiency, which might vary significantly from one to anotherand is extremely influential to the technical–economical behav-iors of a column, has long been relying on experience, and thedesign of distillation columns is essentially empirical in nature(Zuiderweg, 1982; Lockett, 1986). The lack of in-depth under-standing of the processes occurring inside a distillation columnis known to be the major barrier to the proper estimation andimprovement of the column performance (AIChE, 1998).

With the development of computer technology and theadvancement in numerical methods, it becomes possible toinvestigate the transfer process numerically with the chemi-cal engineering and cross-disciplinary theories. The numerical

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1840 Z.M. Sun et al. / Chemical Engineering Science 62 (2007) 1839–1850

approaches have many advantages, such as offering more in-depth information than that upon experiments, shortening thecycles for process and equipment development by visualizingand comparing the results of virtual trials and modifications.The computational fluid dynamics (CFD) has been used suc-cessfully in the field of chemical engineering as a tool.

In the simulation of distillation process and equipment, Yu(1992) and Zhang and Yu (1994) presented a two-dimensionalCFD model for simulating the liquid phase flow on a sievetray, in which the k.� equations were employed to achievethe closure of the equation system and a body force of vaporwas included in the source term of the momentum equation toconsider the interacting effect of vapor and liquid phases. Onthis basis, Liu et al. (2000) developed a model describing theliquid-phase flow on a sieve tray with consideration of boththe resistance and the bubbling effect generated by the uprisingvapor. Later on, Wang et al. (2004) further developed a three-dimensional model considering the effect of vapor by addingdrag force, lift force, virtual mass force and body force in themodel, and simulated a 1.2-m-diameter column with 10 sievetrays under total reflux. The CFD application to distillationwas also made by Krishna et al. (1999) and van Baten andKrishna (2000). They used fully three-dimensional transientsimulations to describe the hydrodynamics of trays, and gaveliquid volume fraction, velocity distribution and clear liquidheight for a rectangular tray and a circular tray, respectively,via a two-phase flow transient model. Also, Fischer and Quarini(1998) proposed a three-dimensional heterogeneous model forsimulating tray hydraulics. Mehta et al. (1998) and Gesit et al.(2003) predicted liquid velocity distribution, clear liquid height,froth height, and liquid volume fraction on trays using CFDtechniques.

The idea of using CFD to incorporating the prediction ontray efficiency relies on the fact that the hydrodynamics is anessential influential factor for mass transfer in both interfacialand bulk diffusions, which could be understood by the effectof velocity distribution on concentration profile. This in factopens an issue on the computation for mass transfer predictionbased on the fluid dynamics computation.

The key problem of this approach is the closure of the dif-ferential mass transfer equation, as two unknown variables, theconcentration and the turbulent mass transfer diffusivity, beinginvolved in one equation. The turbulent mass transfer diffu-sivity depends not only on the fluid dynamic properties (e.g.turbulence viscosity of the fluid) but also on the fluctuationof concentration in turbulent flow. Liu (2003) proposed a two-equation model with a concentration variance c2 equation andits dissipation rate �c equation as a measure to the closure of thedifferential mass transfer equation. Liu’s computational masstransfer (CMT) model has been applied successfully to predictthe turbulent mass transfer diffusivity and efficiency of a com-mercial scaled distillation column by Sun et al. (2005).

However, Liu’s model is of prototype as its initial form iscomplicated and the computation is tedious. In the present pa-per, the c2.�c model is simplified and the model constants areascertained. At the same time, the CFD equation is modified indescribing the interaction between the vapor and liquid phases

to improve the velocity modeling, which is influential to thecomputed tray efficiency. To testify the validity of the sim-plification and improvement, the computed results are com-pared with the experimental data taken from literatures. Theagreement between them demonstrates that the modified CMTmethod can be used effectively in analyzing the performanceof existing distillation column as well as assessing the tray ef-ficiency before construction.

2. Proposed model for CMT

2.1. Simplification of c2.�c model

The instantaneous equation of turbulent mass transfer can bewritten as follows in the tensor form for avoiding complicatedmathematical expression:

�C

�t+ Uj

�C

�xj

= D�2C

�x2j

+ SC , (1)

where U and C are the instantaneous velocity and concentration,respectively. If both U and C are expressed by the time averagevalues U and C, the foregoing equation is transformed to thefollowing Reynolds average form for the transport of averageconcentration:

�C

�t+ Uj

�C

�xj

= �

�xj

(D

�C

�xj

− uj c

)+ SC . (2)

Similar to Boussinesq’s assumption, the turbulent mass fluxuj c in Eq. (2) can be expressed in terms of turbulent masstransfer diffusivity Dt and concentration gradient

−uj c = Dt

�C

�xj

. (3)

Since the turbulent mass transfer diffusivity Dt is regarded asdirect proportional to the product of the characteristic velocityand the characteristic length, we have,

Dt = Ctk1/2Lm. (4)

With the relationship Lm =k1/2�m, �m =√��c, and the defini-

tion of two timescales (Colin and Benkenida, 2003) �� = k/�,

�c = c2/�c, the turbulent mass transfer diffusivity Dt can bewritten as

Dt = Ctk

(k

�

c2

�c

)1/2

. (5)

In Eq. (5), c2 is the concentration variance and �c is the dissipa-tion rate of concentration variance, which can be expressed as

�c = D�c

�xj

�c

�xj

. (6)

For the closure of the turbulent mass transfer, or the elimi-nation of diffusivity Dt , two auxiliary equations are developedas follows. Substituting C = C + c and U = U + u into Eq. (1)

Z.M. Sun et al. / Chemical Engineering Science 62 (2007) 1839–1850 1841

and subtracting Eq. (2), the transport equation for the concen-tration fluctuation can be obtained. After mathematical treat-ments, the precise expression for c2 equation is as follows:

�c2

�t+ Uj

�c2

�xj

= �

�xj

(D

�c2

�xj

− uj c2

)− 2uj c

�C

�xj

− 2�c.

(7)

Taking the derivative of Eq. (1) with respect to xk and mul-tiplying by 2D�c/�xk and averaging, the �c equation is givenbelow:

��c�t

+ Uj

��c�xj

= �

�xj

(D

��c�xj

− uj �c

)

− 2D�c

�xj

�uk

�xj

�C

�xk

− 2Duj

�c

�xk

�2C

�xj�xk

− 2D�c

�xk

�c

�xj

�Uj

�xk

− 2D�c

�xj

�uj

�xk

�c

�xk

− 2D2 �2c

�xj�xk

�2c

�xj�xk

. (8)

Because of the presence of unknown covariance terms, theforegoing two equations cannot be used for direct computa-tion unless they are further simplified. Applying the treatmentsimilar to the Reynolds stress, the turbulent diffusion termsuj c2 and uj �c can be expressed by the following gradient typeequations:

−uj c2 = (Dt/�c)�c2/�xj , (9)

−uj �c = (Dt/��c )��c/�xj . (10)

The complicated Equation (8) should be simplified to theform suitable for computation. In this paper, the method ofmodeling is employed, giving a simplified new expression asfollows. Let the second, third, and fourth terms on the right-hand side of Equation (8) to be the production part as shownbelow:

P�c = − 2D�c

�xj

�uk

�xj

�C

�xk

− 2Duj

�c

�xk

�2C

�xj�xk

− 2D�c

�xk

�c

�xj

�Uj

�xk

. (11)

And the following two terms on the right-hand side of Eq. (8)be the dissipation part:

��c = −2D�c

�xj

�uj

�xk

�c

�xk

− 2D2 �2c

�xj�xk

�2c

�xj�xk

. (12)

The simplification of the �c equation might resemble thetreatment of � equation in the conventional CFD. The modelingof the production part of � equation in CFD by Zhang (2002)is given by

production part of � equation

= C�11

�× production part of k equation, (13)

where � is the timescale and can be expressed as k/�in CFD.

Similarly, the production part of �c equation can be modeledin the following manner:

production part of �c equation

= Cc11

�× production part of c2 equation, (14)

where the concentration timescale c2/�c is used to express �.The production part of c2 equation is uj c�C/�xj , then the finalform of the production part of �c equation can be written as

P�c = −Cc1�c

c2uj c

�C

�xj

. (15)

Since the dissipation part of � equation in CFD can be modeledas

dissipation part of � equation

= C�21

�× dissipation part of k equation. (16)

In the same way, the dissipation part of �c equation can bewritten as

dissipation part of �c equation

= Cc21

�× dissipation part of c2 equation. (17)

According to the postulation by Launder (1976), the combi-nation of two timescales of velocity (k/�) and concentration(c2/�c) are used for expressing � for the case involving masstransfer. Since the dissipation part of c2 equation is �c, thenEq. (12) can be modeled by the following form:

��c = −Cc2�2c

c2− Cc3

��ck

. (18)

The auxiliary equations for closing the differential masstransfer equation are finally to be

�c2

�t+ Uj

�c2

�xj

= �

�xj

[(D + Dt

�c

)�c2

�xj

]− 2uj c

�C

�xj

− 2�c,

(19)

��c�t

+ Uj

��c�xj

= �

�xj

[(D + Dt

��c

)��c�xj

]

− Cc1�c

c2uj c

�C

�xj

− Cc2�2c

c2− Cc3

��ck

. (20)

Normally, the model constants are determined by experi-ments. At the present, in view of lacking experimental data un-der the condition of mass transfer, as a substitute, the follow-ing manner of determination is employed. The value of Ct inEq. (5) was defined as follows:

Ct = C�

Sct

√R

, (21)


where C� is the coefficient in the CFD modeling equation �t =C�k2/�, which is adopted generally to be 0.09. In consideringthe approximate turbulent Schmidt number is generally takenas Sct = �t /Dt = 0.7 and the timescale ratio R = �c/�� =(c2/�c)/(k/�)=0.9. (Lemoine et al., 2000), we obtain Ct =0.14.By the analogy between mass transfer and heat transfer, theconstants �c and ��c in Eqs. (9) and (10) are both assigned to beunity, which are consistent with the assumption by Elghobashiand Launder (1983). According to the analogy and the researchwork of Colin and Benkenida (2003) on the concentration fieldof a combustion device, we choose Cc1 to be 2.0.

Similar to the treatment of Nagano and Kim (1988), theconstants Cc2 and Cc3 are related as follows:

Cc2 = R(C�2 − 1), (22)

Cc3 = 2/R, (23)

where C�2=1.92, which is taken from standard k.� model (Eqs.(33) and (34)), and R=0.9, which is the timescale ratio betweenconcentration and velocity as given above. Consequently, weobtain Cc2 and Cc3 to be 0.83 and 2.22, respectively.

In summary, the model constants in the present model aregiven below:

Ct = 0.14, Cc1 = 2.0, Cc2 = 0.83,

Cc3 = 2.22, �c = 1.0, ��c = 1.0.

Furthermore, by comparing the Cc2 and Cc3 and consideringthe value of timescale ratio R, it is found that the numericalvalue of Cc3��c/k is about 3 times greater than that of Cc2�2

c/c2,

and therefore the latter term may be neglected without affect-ing the numerical result of simulation as demonstrated in thesubsequent section. The final form of simplified �c equationbecomes

��c�t

+ Uj

��c�xj

= �

�xj

[(D + Dt

��c

)��c�xj

]

− Cc1�c

c2uj c

�C

�xj

− Cc3��ck

. (24)

In order to testify the validity of simplification, comparisonbetween the simplified and original models were made. Thesimplified model by using Eqs. (19) and (20) is referred toas Model I hereinafter, and that with Eq. (24) is referred toas present model or Model II. The use of Eq. (19) and thefollowing equation for �c is referred to as original model:

��c�t

+ Uj

��c�xj

= �

�xj

[(D + Dt

��c

)��c�xj

]

− Cc1�c

c2uj c

�C

�xj

− Cc2�2c

c2

− Cc3uiuj

�Ui

�xj

�ck

− Cc4��ck

+ DDt

�2C

�xj�xk

, (25)

where the model constants are: Ct =0.11, Cc1 =1.8, Cc2 =2.2,Cc3 = 0.72, Cc4 = 0.8, �c = 1.0, ��c = 1.0.

2.2. Application of the proposed CMT model to distillationcolumn

The proposed CMT model as applied to distillation iscomposed of two parts. The first part is the respective differ-ential CFD equations describing the velocity distribution onthe distillation tray. The second part is the mass transfer equa-tions including the basic differential equation together with thesimplified c2.�c model for its closure as derived above. In thispaper, the quasi-single liquid phase flow model is adopted forthe first part, the CFD computation.

2.2.1. The CFD equationsFor simulating the velocity profile on distillation tray, the

equations of the steady-state continuity and momentum forthe liquid phase in two-phase flow are adopted. In the presentmodel, the liquid volume fraction is considered and the inter-action between the vapor and liquid phases is attributed to thesource term and is also implicitly involved in the velocity of theliquid phase, which is an improvement to the former pseudo-single-phase model (Wang et al., 2004). The CFD model canbe written as

��LUj

�xj

= 0, (26)

Ui

��LUj

�xi

= − �L

1

L

�p

�xj

+ �Lg

+ �

�xi

[�L�

�Uj

�xi

− �Luiuj

]+ SMj , (27)

where SMj is the source term representing the momentumexchange between vapor and liquid phases; and uiuj is theReynolds stress, for which the Boussinisque’s relation isapplied:

−uiuj = �t

(�Ui

�xj

+ �Uj

�xi

)− 2

3ij k, (28)

where �t = C�k2/�. We assume that the liquid volume fraction�L is not varying with the position, and is given by the corre-lation of Bennett et al. (1983):

�L = exp

[−12.55

(Us

√G

L − G

)0.91]

. (29)

For the source term in Eq. (27), Wang et al. (2004) consideredin their model the drag force, lift force, virtual mass force andbody force. However, except for the drag force, which has beenemployed to interpret the interaction between individual bubbleand liquid by a number of researchers (Krishna et al., 1999;Gesit et al., 2003; Wang et al., 2004), there are no commonconsensus in the literatures on the uses of other forces i.e., liftforce, virtual mass force and body force. Liu et al. (2000) gave


a fairly good prediction for a two-dimensional and two-phaseflow on distillation tray with only considering the body forcegiven previously by Zhang and Yu (1994). Such considerationis adopted in the present work for the source term in the x andy coordinates:

SMj = −GUs

Lhf

Uj (j = x, y), (30)

where the froth height is estimated by the correlation hf =hL/�L, in which the clear liquid height hL is calculated byAIChE (1958) correlation:

hL = 0.0419 + 0.189hw − 0.0135Fs + 2.45qL/lw (31)

and the liquid volume fraction �L is estimated by Eq. (29). Forthe source term in the z coordinate, the drag force expressedby Krishna et al. (1999) is chosen:

SMz = (1 − �L)3

U2s

g(L − G)| �UG − �UL|(Us − ULz). (32)

In closing Eq. (27), the following standard k.� method is used:

�k

�t+ Uj

�k

�xj

= �

�xj

[(� + �t

�k

)�k

�xj

]− uiuj

�Ui

�xj

− �, (33)

��

�t+ Uj

��

�xj

= �

�xj

[(� + �t

��

)��

�xj

]

− C�1�

kuiuj

�Ui

�xj

− C�2�2

k. (34)

The model parameters are customary chosen to be C� = 0.09,C1 = 1.44, C2 = 1.92, �k = 1.0, �� = 1.3.

2.2.2. The mass transfer equationThe equation governing concentration profile of distillation

tray is

Uj

��LC

�xj

= �

�xj

(�LD

�C

�xj

− �Luj c

)+ SC , (35)

where SC is the source term for mass transfer between vaporand liquid phases. The steady form of Eqs. (5), (19) and (20)(or Eq. (24)) are used to close Eq. (35), that is to eliminatethe unknown mass transfer diffusivity Dt in order to obtain theconcentration profile.

The source term SC in Eq. (35) is commonly known to be

SC = KOLa(C∗ − C), (36)

where KOL is the overall liquid phase mass transfer coefficient,a is the effective vapor–liquid interfacial area and C

∗is the liq-

uid composition in equilibrium with the vapor passing throughthe liquid layer on the tray.

The overall liquid phase mass transfer coefficient KOL canbe expressed by the conventional relationship:

KOL = 1

1/kL + 1/mkG

, (37)

where kL and kG are the film coefficients of mass transfer onliquid side and gas (vapor) side, respectively, m is the coefficientof distribution between two phases which can be obtained fromthe vapor–liquid equilibrium data.

The simulated result by the proposed model depends on thechoice of mass transfer coefficients and effective vapor–liquidinterfacial area. A number of correlations developed for kL andkG can be found from the literatures. Several correlations havebeen used and checked the simulated results with the experi-mental data of a commercial scale distillation column reportedby Sakata andYanagi (1979). It was found that applying the cor-relations presented by Zuiderweg to calculate the mass trans-fer coefficients for simulating the commercial scale distillationcolumn concerned gave the least deviation with the experimen-tal data. It can be understood that the Zeiderweg’s correlationsare based on the data mostly from the commercial columns.The corresponding equations for kL and kG are given below:

kG = 0.13

G

− 0.065

2G

(1.0 < G < 80 kg m−3), (38)

kL =(

1

LPR− 1

)mkG, (39)

where LPR is the liquid phase resistance which is 0.37(Zuiderweg, 1982), G is the vapor density. The average valueof m covering the range of concentration under considerationwas found to be 0.0055.

The effective vapor–liquid interfacial area was calculated bythe correlation presented by Zuiderweg (1982).

The numerical computation is begun from the top of thecolumn. As only the compositions of reflux and the vapor leav-ing the top are known and the composition of entering vapor tothe top tray is unknown, the following trial-and-error method isused to start the computation. An entering vapor composition isassumed and then the trial value of C∗ can be obtained, whichis in equilibrium with the average vapor composition betweenentering and leaving. The amount of mass transfer in the toptray is calculated by Eq. (34). By material balance, the liquidcomposition leaving the top tray can be found, which shouldbe equal to the assumed composition of entering vapor underthe condition of total reflux. If not, make the trial again untilthe error is not more than 2%. For all the trays below, similarmethod are used to obtain the compositions of vapor enteringthe tray and the liquid leaving the tray.

2.2.3. The boundary conditionsThe inlet conditions of the present CMT model are: U =

U in, C = Cin and that for the k.� equations is followed the

conventional formulas (Nallasamy, 1987) to be kin =0.003U2xin

and �in = 0.09k3/2in /(0.03 × W/2).

The inlet conditions of c2.�c equations, deducted by Liu(2003) and Sun et al. (2005), are given below:

c2in = [0.082 · (Cin − C∗)]2, (40)

�cin = R

(�in

kin

)c2

in, (41)


where R represents the timescale ratio of concentration to ve-locity and equals to 0.9 as shown in previous section.

At the outlet, we have p = 0, and �C/�x = 0.The boundary conditions at the tray floor, the outlet weir and

the column wall are considered as non-slip, and the conven-tional logarithm law expression is employed.

At the interface of the vapor and liquid, all the stresses areequal to zero, so we have �Ux/�z=0, �Uy/�z=0, and Uz =0.Similarly, both at the wall and the interface, the concentrationflux is equal to zero.

3. Computational result of CMT model for distillationcolumn

3.1. Velocity distribution

To assess the validity of the CFD part of the proposed CMTmodel, the velocity distribution on a 1.2-m-dia. sieve tray issimulated for the comparison with experimental data reportedby Solari and Bell (1986). The model geometry and boundariesare shown in Fig. 1. Solari and Bell (1986) measured the linearliquid-velocity along two lines perpendicular to the liquid flowdirection on a plane 0.038 m above the tray floor. In the simu-lated computation, air–water system is used. Figs. 2 and 3 showthe predicted liquid horizontal velocity and the experimentaldata of Solari and Bell (1986). From the figures, we can see thatthe predictions agree reasonably with the experimental data inspite of having some deviations. The discrepancy between themmay be due to the following reasons. Firstly, the experimen-tal work was under the condition of two-phase flow, while thequasi-single-phase model is used for the simulation. Secondly,the experiment is one-dimensional, namely the measured ve-locity is the linear velocity of the tracer dye from one probeto the next, while the present simulation is three-dimensional,and the computed liquid phase velocity shown in Figs. 2 and 3are the velocity component in x direction. Obviously, the com-parison between experimental data and prediction is not ex-actly on the same basis. Thirdly, the inlet velocity distribu-tion in present simulation is assumed to be uniform, while theexperimental condition might deviate from such assumption.Fig. 4(a) and (b) show the liquid-velocity vector plot. It can beseen that the velocity is uniform in the main flow area. The cir-culating flow is found near the corner of the inlet weir, whichhas been observed in many experimental works (Yu and Huang,1981; Porter et al., 1992; Biddulph, 1994; Yu et al., 1999,

Fig. 1. Flow geometry and boundary conditions.

Fig. 2. Liquid-velocity profile, QL = 6.94 × 10−3 m3 s−1, FS = 1.015 m s−1

(kg m−3)0.5: (a) upstream profile; (b) downstream profile.

Liu and Yuan, 2002). The existence of circulating flow can beexplained as follows. When the liquid passes through the inlet,the flow area suddenly expands, leading to the separation of theboundary layer and forming the eddy current. The circulationflow increases the extent of fluid mixing, which is reflected onthe increase of turbulent mass transfer diffusivity Dt as shownin the later section.

3.2. Turbulent mass transfer diffusivity distribution

As a result of the present CMT simulation, Figs. 5–7 showthe turbulent mass transfer diffusivity profiles, which weregiven separately by using the Original Model, Models I and II(present model) for simulating a commercial scaled distillationtray operated with cyclohexane-n-heptane system at 165 kPaand outlet weir liquid load at 0.013 m−3 s−1 m−1. Since theturbulent mass transfer diffusivity Dt , which is highly affected


Fig. 3. Liquid-velocity profile, QL = 6.94 × 10−3 m3 s−1, FS = 1.464 m s−1

(kg m−3)0.5: (a) upstream profile; (b) downstream profile.

by the velocity and concentration fields, represents the inten-sity of back-mixing, the larger local value of Dt correspondsthe lower local mass transfer efficiency. It can be seen fromthe figures that the distribution of Dt is quite diverse. If wetake the volume average value of Dt , the order of magnitudeis about 10−2.10−3, which is close to those reported in the lit-eratures (Barker and Self, 1962; Yu et al., 1990, Cai and Chen,2004). Comparing the three figures, we can see that the shapeof Dt profile obtained by different models are similar, as seen inFigs. 6 and 7. Fig. 8 shows that the volume average values ofDt computed by Models I and II are in good agreement withthe average experimental data for commercial scaled columnreported by Cai and Chen (2004), while the computed resultsby using Original Model are much lower. It demonstrates thatthe simplified model can give better results than the original oneas far as in predicting the turbulent mass transfer diffusivity isconcerned.

Fig. 4. Liquid-velocity vector plot on the x–y plane, z = 0.038 m. (b) Localview of the circulation area (rectangle area in (a)).

Fig. 5. Turbulent mass transfer diffusivity profile at 20 mm above the floor(Original Model).

3.3. Concentration distribution

The following computation aims at the simulation of a com-mercial scale distillation column reported by Sakata and Yanagi(1979). The separating system is cyclohexane-n-heptane at the


Fig. 6. Turbulent mass transfer diffusivity profile at 20 mm above the floor(Model I).

Fig. 7. Turbulent mass transfer diffusivity profile at 20 mm above the floor(Model II).

operating pressure of 165 kPa. The liquid rate is 30.66 m3 h−1

and vapor rate is 5.75 kg s−1. More detailed data about the col-umn and the average physical properties of the systems areavailable in the literature (Sakata and Yanagi, 1979). The liq-uid in the downcomer is assumed to be completely mixed andthe computation followed a tray-by-tray scheme to simulate thetray cascade. The grids and the coordinates for computation areshown in Fig. 1. The trays should be numbered 2–9 from thetop of the column, while the reflux is designated as tray 1.

As a sample of the computed results, Figs. 9–11 show thecomputed concentration distribution on tray 2. It can be seenthat the concentration profiles computed by the three differentmodels are similar. Unfortunately, no experimental data on theconcentration field of a tray is available at the present in the

Fig. 8. Experimental vs. computational of turbulent mass transfer diffusivity.

Fig. 9. Concentration profile of x–y plane on tray 2 at 20 mm above the floor(Original Model).

literature for the comparison. However, we may compare indi-rectly by means of the outlet concentration of each tray. FromFig. 12, it can be seen that the computed outlet concentrationof each tray is in good agreement with the experimental mea-surement except for the tray 6. As we understand for the totalreflux operation, the outlet concentration should form a smoothcurve on the plot. The deviation on tray 6 is likely to be due toexperimental error or some other unknown reasons. The aver-age deviation of the outlet composition is 3.77%.

The Murphree efficiency for each tray is also computed andcompared with experimental data as shown in Fig. 13. Exceptfor trays 6 and 7, the predicted results are in agreement withthe measurement. The deviation at trays 6 and 7 is probablycoming from using different outlet concentration at tray 6 forcalculating EMV . The overall tray efficiency can be evaluatedby the Fenske–Underwood equation. The predicted overall tray


Fig. 10. Concentration profile of x–y plane on tray 2 at 20 mm above thefloor (Model I).

Fig. 11. Concentration profile of x–y plane on tray 2 at 20 mm above thefloor (Model II).

efficiency is 83.34% by Original Model, 81.46% by Model Iand 80.68% by Model II, while the experimental measurementis 89.4%.

To further demonstrate the feasibility of applying the sim-plified Model II, simulation is also made for the bottom con-centration and overall tray efficiency of another distillation col-umn, a pilot-scale distillation column as described by Garciaand Fair (2000), which is 0.429 m in diameter with eight sievetrays of 0.457 m tray spacing operated under total reflux at dif-ferent F -factors (Fs = us

√G). The separating system is the

cyclohexane-n-heptane mixture at 165 kPa. As we know, theKOL is related with the structure and size of the sieve tray. Itwas found that the correlations of kL and kG by Hoogendoornet al. (1988) is applicable to the pilot-scale column, and was

Fig. 12. Predicted concentration vs. experimental measurement.

adopted for the simulation concerned:

kL = 8D0.5L , (42)

kG = 0.625kL

(DG

DL

)0.5

. (43)

From Figs. 14 and 15, the computed bottom concentrations arefound to be somewhat less than the experimental measurementsand the overall tray efficiency is slightly higher. The averagedeviation of the bottom concentration is 6.5%. The cause ofdiscrepancy may be attributed to the ideal operational condi-tions concerned in the simulation, such as no weeping, no en-trainment and perfect construction, which an existing columnmay not achieve.

4. Conclusion

The original c2−�c model (Liu, 2003), which is used to closethe differential mass transfer equation, is further simplified andthe model constants are ascertained. An improved CFD equa-tion is employed to predict the velocity field. To test the valid-ity of the improvement, the proposed simplified CMT modelis applied to two distillation columns. The computed resultsare compared with the respective experimental data taken fromthe literatures. The comparison with the experimental data foran industrial scale distillation column reported by Sakata andYanagi (1979) reveal that the simplified models can give betterpredictions on the turbulent mass transfer diffusivity than theoriginal one, while the computed concentrations at the outlet ofeach tray and the tray efficiency by these two models are in sat-isfactory agreement. In addition, the comparison is also madeto a pilot-scale distillation column described by Garcia andFair (2000), the predicted bottom concentration and the overalltray efficiency under different F -factors of a pilot-scale sievetray column are confirmed reasonably with the experimentaldata. The proposed simplified CMT model has demonstrated


Fig. 13. Predicted EMV vs. experimental measurement.

Fig. 14. Predicted concentration vs. experimental measurement (0.429 mcolumn).

to be a prospective tool to predict the turbulent mass transferdiffusivity, concentration profile on a tray as well as the trayefficiency of a distillation column.

Notation

a effective vapor–liquid interfacial area, m2 m−3

c fluctuating concentration (mass fraction)c2 concentration varianceCt , Cc1,

Cc2, Cc3

turbulence model constants for the concentrationfield

Fig. 15. Overall tray efficiencies under different F -factors (0.429 m column).

C�, C�1,

C�2

turbulence model constants for the velocity field

C instantaneous concentration (mass fraction)C time average concentration in liquid phase (mass

fraction)C

∗time average concentration in liquid phase inequilibrium with concentration in gas phase(mass fraction)

D molecular mass transfer diffusivity, m2 s−1

DG vapor-phase molecular mass transfer diffusivity,m2 s−1


DL liquid-phase molecular mass transfer diffusivity,m2 s−1

Dt turbulent mass transfer diffusivity, m2 s−1

EMV Murphree efficiency of vapor phaseFs F -factor (Fs = us

√G)

g acceleration due to gravitation, m s−2

hf froth height, mhL clear liquid height, mhw weir height, mk turbulent kinetic energy, m2 s−2

kG vapor-phase mass transfer coefficient, m s−1

kL liquid-phase mass transfer coefficient, m s−1

KOL overall liquid phase mass transfer coefficient,m s−1

lw weir width, mLm Prandtl mixing length, mm distribution coefficientp time average pressure, PaP�c production term in the �c equationqL volumetric flow of liquid flow, m3 s−1

R timescale ratioRe Reynolds numberSC source of interphase mass transferSC time average source of interphase mass transferSMj source of interphase momentum transfert time, su fluctuating velocity, m s−1

U instantaneous velocity, m s−1

U time average velocity, m s−1

Us superficial vapor velocity, m s−1

Greek letters

�L liquid volume fractionij Kronecker delta� turbulent dissipation, m2 s−3

�c dissipation rate of c2, s−1

�t turbulent viscosity, m2 s−1

density, kg m−3

�c, ��c ,

�k, ��

turbulence model constants for diffusion of c2,�c, k, �

��c dissipation term in the �c equation� time scale, s��, �c timescales of velocity and concentration fields, s�m mean time scale, s

Subscripts

G gasin inleti, j, k tensor symbolsL liquidx, y, z x, y, and z coordinates

Acknowledgments

The authors wish to acknowledge the financial support by theNational Natural Science Foundation of China (No. 20136010),

and the assistance by the staffs in the State Key Laboratoriesof Chemical Engineering (Tianjin University).

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