Chemical Engineering Science 62 (2007) 1839 – 1850 www.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. Liu State 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 2006 Available 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 a commercial-scale distillation column. Industrial and Engineering Chemistry Research 44, 4427–4434] are further simplified for reducing the complication of computation. At the same time, the CFD formulation is also improved for better velocity field prediction. By this complex model, the turbulent mass transfer diffusivity, the three-dimensional velocity/concentration profiles and the efficiency of mass transfer equipment can be predicted simultaneously. To demonstrate the feasibility of the proposed simplified CMT model, simulation was made for distillation column, and the simulated results are compared with the experimental data taken from literatures. The predicted distribution of liquid velocity on 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 modified model to a commercial scale distillation tray column, the predictions of the concentration at the outlet of each tray and the tray efficiency are 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 shown by checking the computed results with a reported pilot-scale tray column [Garcia, J.A., Fair, J.R., 2000. A fundamental model for the prediction of distillation sieve tray efficiency. 1. Database development. Industrial and Engineering Chemistry Research 39, 1809–1817] in the bottom concentration and the overall tray efficiency under different operating conditions. The modified CMT model is expected to be useful in the design and analysis of distillation column. 䉷 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 for the liquid mixture, has been widely used in the chemical and allied industries due to its reliability in large-size column application and its maturity in engineering practice. Among the distillation equipments, the tray column is popularly employed in the industrial production for its simple structure and low cost 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 䉷 2007 Published by Elsevier Ltd. doi:10.1016/j.ces.2006.12.021
efficiency, which might vary significantly from one to another and is extremely influential to the technical–economical behaviors of a column, has long been relying on experience, and the design of distillation columns is essentially empirical in nature (Zuiderweg, 1982; Lockett, 1986). The lack of in-depth understanding of the processes occurring inside a distillation column is known to be the major barrier to the proper estimation and improvement of the column performance (AIChE, 1998). With the development of computer technology and the advancement in numerical methods, it becomes possible to investigate the transfer process numerically with the chemical engineering and cross-disciplinary theories. The numerical
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approaches have many advantages, such as offering more indepth information than that upon experiments, shortening the cycles for process and equipment development by visualizing and comparing the results of virtual trials and modifications. The computational fluid dynamics (CFD) has been used successfully 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-dimensional CFD model for simulating the liquid phase flow on a sieve tray, in which the k. equations were employed to achieve the closure of the equation system and a body force of vapor was included in the source term of the momentum equation to consider the interacting effect of vapor and liquid phases. On this basis, Liu et al. (2000) developed a model describing the liquid-phase flow on a sieve tray with consideration of both the resistance and the bubbling effect generated by the uprising vapor. Later on, Wang et al. (2004) further developed a threedimensional model considering the effect of vapor by adding drag force, lift force, virtual mass force and body force in the model, and simulated a 1.2-m-diameter column with 10 sieve trays under total reflux. The CFD application to distillation was also made by Krishna et al. (1999) and van Baten and Krishna (2000). They used fully three-dimensional transient simulations to describe the hydrodynamics of trays, and gave liquid volume fraction, velocity distribution and clear liquid height 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 for simulating 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 CFD techniques. The idea of using CFD to incorporating the prediction on tray efficiency relies on the fact that the hydrodynamics is an essential influential factor for mass transfer in both interfacial and bulk diffusions, which could be understood by the effect of velocity distribution on concentration profile. This in fact opens an issue on the computation for mass transfer prediction based on the fluid dynamics computation. The key problem of this approach is the closure of the differential mass transfer equation, as two unknown variables, the concentration and the turbulent mass transfer diffusivity, being involved in one equation. The turbulent mass transfer diffusivity depends not only on the fluid dynamic properties (e.g. turbulence viscosity of the fluid) but also on the fluctuation of concentration in turbulent flow. Liu (2003) proposed a twoequation model with a concentration variance c2 equation and its dissipation rate c equation as a measure to the closure of the differential mass transfer equation. Liu’s computational mass transfer (CMT) model has been applied successfully to predict the turbulent mass transfer diffusivity and efficiency of a commercial scaled distillation column by Sun et al. (2005). However, Liu’s model is of prototype as its initial form is complicated and the computation is tedious. In the present paper, the c2 .c model is simplified and the model constants are ascertained. At the same time, the CFD equation is modified in describing the interaction between the vapor and liquid phases
to improve the velocity modeling, which is influential to the computed tray efficiency. To testify the validity of the simplification and improvement, the computed results are compared with the experimental data taken from literatures. The agreement between them demonstrates that the modified CMT method can be used effectively in analyzing the performance of existing distillation column as well as assessing the tray efficiency before construction. 2. Proposed model for CMT 2.1. Simplification of c2 .c model The instantaneous equation of turbulent mass transfer can be written as follows in the tensor form for avoiding complicated mathematical expression: jC jC j2 C + Uj = D 2 + SC , jt jxj jxj
(1)
where U and C are the instantaneous velocity and concentration, respectively. If both U and C are expressed by the time average values U and C, the foregoing equation is transformed to the following Reynolds average form for the transport of average concentration: jC jC jC j D = − uj c + S C . (2) + Uj jt jxj jxj jxj Similar to Boussinesq’s assumption, the turbulent mass flux uj c in Eq. (2) can be expressed in terms of turbulent mass transfer diffusivity Dt and concentration gradient −uj c = Dt
jC . jxj
(3)
Since the turbulent mass transfer diffusivity Dt is regarded as direct proportional to the product of the characteristic velocity and the characteristic length, we have, Dt = Ct k 1/2 Lm .
(4)
√ With the relationship Lm = k 1/2 m , m = c , and the definition of two timescales (Colin and Benkenida, 2003) = k/, c = c2 /c , the turbulent mass transfer diffusivity Dt can be written as 1/2 k c2 . (5) Dt = Ct k c In Eq. (5), c2 is the concentration variance and c is the dissipation rate of concentration variance, which can be expressed as c = D
jc jc . jxj jxj
(6)
For the closure of the turbulent mass transfer, or the elimination of diffusivity Dt , two auxiliary equations are developed as follows. Substituting C = C + c and U = U + u into Eq. (1)
Z.M. Sun et al. / Chemical Engineering Science 62 (2007) 1839 – 1850
and subtracting Eq. (2), the transport equation for the concentration fluctuation can be obtained. After mathematical treatments, the precise expression for c2 equation is as follows: jc2 jc2 jc2 j jC 2 D + Uj = − uj c − 2uj c − 2c . jt jxj jxj jxj jxj (7) Taking the derivative of Eq. (1) with respect to xk and multiplying by 2Djc/jxk and averaging, the c equation is given below: jc jc j jc D = − uj c + Uj jt jxj jxj jxj j2 C
jc juk jC jc − 2D − 2Duj jxj jxj jxk jxk jxj jxk − 2D
j2 c j2 c . jxj jxk jxj jxk
−uj
= (Dt /c
)jc2 /jx
(8)
j,
(9)
−uj c = (Dt /c )jc /jxj .
(10)
The complicated Equation (8) should be simplified to the form suitable for computation. In this paper, the method of modeling is employed, giving a simplified new expression as follows. Let the second, third, and fourth terms on the righthand side of Equation (8) to be the production part as shown below: Pc = − 2D − 2D
jc juk jC jc j2 C − 2Duj jxj jxj jxk jxk jxj jxk jc jc jU j . jxk jxj jxk
(12)
The simplification of the c equation might resemble the treatment of equation in the conventional CFD. The modeling of the production part of equation in CFD by Zhang (2002) is given by production part of equation = C1
1 × production part of k equation,
= Cc1
1 × production part of c2 equation,
(14)
where the concentration timescale c2 /c is used to express . The production part of c2 equation is uj cjC/jxj , then the final form of the production part of c equation can be written as Pc = −Cc1
c c2
uj c
jC . jxj
(15)
= C 2
1 × dissipation part of k equation.
(13)
(16)
In the same way, the dissipation part of c equation can be written as dissipation part of c equation = Cc2
1 × dissipation part of c2 equation.
(17)
According to the postulation by Launder (1976), the combination of two timescales of velocity (k/) and concentration (c2 /c ) are used for expressing for the case involving mass transfer. Since the dissipation part of c2 equation is c , then Eq. (12) can be modeled by the following form: c = −Cc2
2c c2
− Cc3
c . k
(18)
The auxiliary equations for closing the differential mass transfer equation are finally to be jc2 Dt jc2 jc2 j jC + Uj D+ − 2uj c = − 2c , jt jxj jxj c jxj jxj
(11)
And the following two terms on the right-hand side of Eq. (8) be the dissipation part: jc juj jc j2 c j2 c c = −2D − 2D 2 . jxj jxk jxk jxj jxk jxj jxk
production part of c equation
dissipation part of equation
Because of the presence of unknown covariance terms, the foregoing two equations cannot be used for direct computation unless they are further simplified. Applying the treatment similar to the Reynolds stress, the turbulent diffusion terms uj c2 and uj c can be expressed by the following gradient type equations: c2
where is the timescale and can be expressed as k/ in CFD. Similarly, the production part of c equation can be modeled in the following manner:
Since the dissipation part of equation in CFD can be modeled as
jc jc jU j jc juj jc − 2D jxk jxj jxk jxj jxk jxk
− 2D 2
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(19) jc jc j + Uj = jt jxj jxj − Cc1
c c2
uj c
D+
Dt c
jc jxj
2 c jC − Cc2 c − Cc3 . 2 jxj k c
(20)
Normally, the model constants are determined by experiments. At the present, in view of lacking experimental data under the condition of mass transfer, as a substitute, the following manner of determination is employed. The value of Ct in Eq. (5) was defined as follows: Ct =
C √ , Sct R
(21)
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where C is the coefficient in the CFD modeling equation t = C k 2 /, which is adopted generally to be 0.09. In considering the approximate turbulent Schmidt number is generally taken as 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, the constants c and c in Eqs. (9) and (10) are both assigned to be unity, which are consistent with the assumption by Elghobashi and Launder (1983). According to the analogy and the research work of Colin and Benkenida (2003) on the concentration field of a combustion device, we choose Cc1 to be 2.0. Similar to the treatment of Nagano and Kim (1988), the constants Cc2 and Cc3 are related as follows: Cc2 = R(C2 − 1),
(22)
Cc3 = 2/R,
(23)
where C2 =1.92, which is taken from standard k. model (Eqs. (33) and (34)), and R=0.9, which is the timescale ratio between concentration and velocity as given above. Consequently, we obtain Cc2 and Cc3 to be 0.83 and 2.22, respectively. In summary, the model constants in the present model are given below: Ct = 0.14,
Cc1 = 2.0,
Cc3 = 2.22,
c = 1.0,
Cc2 = 0.83, c = 1.0.
Furthermore, by comparing the Cc2 and Cc3 and considering the value of timescale ratio R, it is found that the numerical value of Cc3 c /k is about 3 times greater than that of Cc2 2c /c2 , and therefore the latter term may be neglected without affecting the numerical result of simulation as demonstrated in the subsequent section. The final form of simplified c equation becomes jc jc Dt jc j D+ = + Uj jt jxj jxj c jxj − Cc1
c c2
uj c
c jC − Cc3 . jxj k
(24)
In order to testify the validity of simplification, comparison between the simplified and original models were made. The simplified model by using Eqs. (19) and (20) is referred to as Model I hereinafter, and that with Eq. (24) is referred to as present model or Model II. The use of Eq. (19) and the following equation for c is referred to as original model: jc Dt jc jc j D+ = + Uj jxj jxj c jxj jt − Cc1
c c2
uj c
− Cc3 ui uj + DD t
2 jC − Cc2 c jxj c2
jU i c c − Cc4 jxj k k
j2 C , jxj jxk
(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 distillation column The proposed CMT model as applied to distillation is composed of two parts. The first part is the respective differential CFD equations describing the velocity distribution on the distillation tray. The second part is the mass transfer equations including the basic differential equation together with the simplified c2 .c model for its closure as derived above. In this paper, the quasi-single liquid phase flow model is adopted for the first part, the CFD computation. 2.2.1. The CFD equations For simulating the velocity profile on distillation tray, the equations of the steady-state continuity and momentum for the liquid phase in two-phase flow are adopted. In the present model, the liquid volume fraction is considered and the interaction between the vapor and liquid phases is attributed to the source term and is also implicitly involved in the velocity of the liquid phase, which is an improvement to the former pseudosingle-phase model (Wang et al., 2004). The CFD model can be written as jL U j = 0, jxj Ui
jL U j 1 jp = − L + L g jxi L jxj jU j j + L − L ui uj + SMj , jxi jxi
(26)
(27)
where SMj is the source term representing the momentum exchange between vapor and liquid phases; and ui uj is the Reynolds stress, for which the Boussinisque’s relation is applied: jU j jU i 2 −ui uj = t − ij k, + (28) jxj jxi 3 where t = C k 2 /. We assume that the liquid volume fraction L is not varying with the position, and is given by the correlation of Bennett et al. (1983): 0.91
G L = exp −12.55 Us . (29) L − G For the source term in Eq. (27), Wang et al. (2004) considered in their model the drag force, lift force, virtual mass force and body force. However, except for the drag force, which has been employed to interpret the interaction between individual bubble and liquid by a number of researchers (Krishna et al., 1999; Gesit et al., 2003; Wang et al., 2004), there are no common consensus in the literatures on the uses of other forces i.e., lift force, virtual mass force and body force. Liu et al. (2000) gave
Z.M. Sun et al. / Chemical Engineering Science 62 (2007) 1839 – 1850
a fairly good prediction for a two-dimensional and two-phase flow on distillation tray with only considering the body force given previously by Zhang and Yu (1994). Such consideration is adopted in the present work for the source term in the x and y coordinates: SMj = −
G Us Uj L h f
(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 by AIChE (1958) correlation: hL = 0.0419 + 0.189hw − 0.0135Fs + 2.45qL / lw
(31)
and the liquid volume fraction L is estimated by Eq. (29). For the source term in the z coordinate, the drag force expressed by Krishna et al. (1999) is chosen: SMz =
(1 − L )3 g( L − G )|UG − UL |(Us − ULz ). Us2
(32)
In closing Eq. (27), the following standard k. method is used: jk t jk jk jU i j + − ui uj + Uj = − , (33) jt jxj jxj k jxj jxj j t j j j + = + Uj jt jxj jxj jxj jU i 2 − C1 ui uj − C 2 . k jxj 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 equation The equation governing concentration profile of distillation tray is jL C j jC L D Uj = − L uj c + S C , (35) jxj jxj jxj where SC is the source term for mass transfer between vapor and liquid phases. The steady form of Eqs. (5), (19) and (20) (or Eq. (24)) are used to close Eq. (35), that is to eliminate the unknown mass transfer diffusivity Dt in order to obtain the concentration profile. The source term SC in Eq. (35) is commonly known to be SC = KOL a(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 liquid composition in equilibrium with the vapor passing through the liquid layer on the tray. The overall liquid phase mass transfer coefficient KOL can be expressed by the conventional relationship: KOL =
1 , 1/kL + 1/mk G
(37)
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where kL and kG are the film coefficients of mass transfer on liquid side and gas (vapor) side, respectively, m is the coefficient of distribution between two phases which can be obtained from the vapor–liquid equilibrium data. The simulated result by the proposed model depends on the choice of mass transfer coefficients and effective vapor–liquid interfacial area. A number of correlations developed for kL and kG can be found from the literatures. Several correlations have been used and checked the simulated results with the experimental data of a commercial scale distillation column reported by Sakata and Yanagi (1979). It was found that applying the correlations presented by Zuiderweg to calculate the mass transfer coefficients for simulating the commercial scale distillation column concerned gave the least deviation with the experimental data. It can be understood that the Zeiderweg’s correlations are based on the data mostly from the commercial columns. The corresponding equations for kL and kG are given below: 0.13 0.065 − 2 (1.0 < G < 80 kg m−3 ), G G 1 kL = − 1 mk G , LPR kG =
(38)
(39)
where LPR is the liquid phase resistance which is 0.37 (Zuiderweg, 1982), G is the vapor density. The average value of m covering the range of concentration under consideration was found to be 0.0055. The effective vapor–liquid interfacial area was calculated by the correlation presented by Zuiderweg (1982). The numerical computation is begun from the top of the column. As only the compositions of reflux and the vapor leaving the top are known and the composition of entering vapor to the top tray is unknown, the following trial-and-error method is used to start the computation. An entering vapor composition is assumed and then the trial value of C ∗ can be obtained, which is in equilibrium with the average vapor composition between entering and leaving. The amount of mass transfer in the top tray is calculated by Eq. (34). By material balance, the liquid composition leaving the top tray can be found, which should be equal to the assumed composition of entering vapor under the condition of total reflux. If not, make the trial again until the error is not more than 2%. For all the trays below, similar method are used to obtain the compositions of vapor entering the tray and the liquid leaving the tray. 2.2.3. The boundary conditions The inlet conditions of the present CMT model are: U = U in , C = C in and that for the k. equations is followed the 2
conventional formulas (Nallasamy, 1987) to be kin = 0.003U xin 3/2 and in = 0.09kin /(0.03 × W/2). The inlet conditions of c2 .c equations, deducted by Liu (2003) and Sun et al. (2005), are given below: c2 in = [0.082 · (C in − C ∗ )]2 , in cin = R c2 in , kin
(40) (41)
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where R represents the timescale ratio of concentration to velocity and equals to 0.9 as shown in previous section. At the outlet, we have p = 0, and jC/jx = 0. The boundary conditions at the tray floor, the outlet weir and the column wall are considered as non-slip, and the conventional logarithm law expression is employed. At the interface of the vapor and liquid, all the stresses are equal to zero, so we have jU x /jz =0, jU y /jz =0, and U z =0. Similarly, both at the wall and the interface, the concentration flux is equal to zero. 3. Computational result of CMT model for distillation column 3.1. Velocity distribution To assess the validity of the CFD part of the proposed CMT model, the velocity distribution on a 1.2-m-dia. sieve tray is simulated for the comparison with experimental data reported by Solari and Bell (1986). The model geometry and boundaries are shown in Fig. 1. Solari and Bell (1986) measured the linear liquid-velocity along two lines perpendicular to the liquid flow direction on a plane 0.038 m above the tray floor. In the simulated computation, air–water system is used. Figs. 2 and 3 show the predicted liquid horizontal velocity and the experimental data of Solari and Bell (1986). From the figures, we can see that the predictions agree reasonably with the experimental data in spite of having some deviations. The discrepancy between them may be due to the following reasons. Firstly, the experimental work was under the condition of two-phase flow, while the quasi-single-phase model is used for the simulation. Secondly, the experiment is one-dimensional, namely the measured velocity is the linear velocity of the tracer dye from one probe to the next, while the present simulation is three-dimensional, and the computed liquid phase velocity shown in Figs. 2 and 3 are the velocity component in x direction. Obviously, the comparison between experimental data and prediction is not exactly on the same basis. Thirdly, the inlet velocity distribution in present simulation is assumed to be uniform, while the experimental condition might deviate from such assumption. Fig. 4(a) and (b) show the liquid-velocity vector plot. It can be seen that the velocity is uniform in the main flow area. The circulating flow is found near the corner of the inlet weir, which has been observed in many experimental works (Yu and Huang, 1981; Porter et al., 1992; Biddulph, 1994; Yu et al., 1999,
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 be explained as follows. When the liquid passes through the inlet, the flow area suddenly expands, leading to the separation of the boundary layer and forming the eddy current. The circulation flow increases the extent of fluid mixing, which is reflected on the increase of turbulent mass transfer diffusivity Dt as shown in the later section. 3.2. Turbulent mass transfer diffusivity distribution
Fig. 1. Flow geometry and boundary conditions.
As a result of the present CMT simulation, Figs. 5–7 show the turbulent mass transfer diffusivity profiles, which were given separately by using the Original Model, Models I and II (present model) for simulating a commercial scaled distillation tray operated with cyclohexane-n-heptane system at 165 kPa and outlet weir liquid load at 0.013 m−3 s−1 m−1 . Since the turbulent mass transfer diffusivity Dt , which is highly affected
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Fig. 4. Liquid-velocity vector plot on the x–y plane, z = 0.038 m. (b) Local view of the circulation area (rectangle area in (a)).
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 intensity of back-mixing, the larger local value of Dt corresponds the lower local mass transfer efficiency. It can be seen from the figures that the distribution of Dt is quite diverse. If we take the volume average value of Dt , the order of magnitude is about 10−2 .10−3 , which is close to those reported in the literatures (Barker and Self, 1962; Yu et al., 1990, Cai and Chen, 2004). Comparing the three figures, we can see that the shape of Dt profile obtained by different models are similar, as seen in Figs. 6 and 7. Fig. 8 shows that the volume average values of Dt computed by Models I and II are in good agreement with the average experimental data for commercial scaled column reported by Cai and Chen (2004), while the computed results by using Original Model are much lower. It demonstrates that the simplified model can give better results than the original one as far as in predicting the turbulent mass transfer diffusivity is concerned.
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 commercial scale distillation column reported by Sakata and Yanagi (1979). The separating system is cyclohexane-n-heptane at the
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Fig. 8. Experimental vs. computational of turbulent mass transfer diffusivity. Fig. 6. Turbulent mass transfer diffusivity profile at 20 mm above the floor (Model I).
Fig. 9. Concentration profile of x–y plane on tray 2 at 20 mm above the floor (Original Model). 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 column and the average physical properties of the systems are available in the literature (Sakata and Yanagi, 1979). The liquid in the downcomer is assumed to be completely mixed and the computation followed a tray-by-tray scheme to simulate the tray cascade. The grids and the coordinates for computation are shown in Fig. 1. The trays should be numbered 2–9 from the top of the column, while the reflux is designated as tray 1. As a sample of the computed results, Figs. 9–11 show the computed concentration distribution on tray 2. It can be seen that the concentration profiles computed by the three different models are similar. Unfortunately, no experimental data on the concentration field of a tray is available at the present in the
literature for the comparison. However, we may compare indirectly by means of the outlet concentration of each tray. From Fig. 12, it can be seen that the computed outlet concentration of each tray is in good agreement with the experimental measurement except for the tray 6. As we understand for the total reflux operation, the outlet concentration should form a smooth curve on the plot. The deviation on tray 6 is likely to be due to experimental error or some other unknown reasons. The average deviation of the outlet composition is 3.77%. The Murphree efficiency for each tray is also computed and compared with experimental data as shown in Fig. 13. Except for trays 6 and 7, the predicted results are in agreement with the measurement. The deviation at trays 6 and 7 is probably coming from using different outlet concentration at tray 6 for calculating EMV . The overall tray efficiency can be evaluated by the Fenske–Underwood equation. The predicted overall tray
Z.M. Sun et al. / Chemical Engineering Science 62 (2007) 1839 – 1850
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Fig. 12. Predicted concentration vs. experimental measurement. Fig. 10. Concentration profile of x–y plane on tray 2 at 20 mm above the floor (Model I).
adopted for the simulation concerned: kL = 8DL0.5 ,
(42)
kG = 0.625kL
DG DL
0.5 .
(43)
From Figs. 14 and 15, the computed bottom concentrations are found to be somewhat less than the experimental measurements and the overall tray efficiency is slightly higher. The average deviation of the bottom concentration is 6.5%. The cause of discrepancy may be attributed to the ideal operational conditions concerned in the simulation, such as no weeping, no entrainment and perfect construction, which an existing column may not achieve. 4. Conclusion
Fig. 11. Concentration profile of x–y plane on tray 2 at 20 mm above the floor (Model II).
efficiency is 83.34% by Original Model, 81.46% by Model I and 80.68% by Model II, while the experimental measurement is 89.4%. To further demonstrate the feasibility of applying the simplified Model II, simulation is also made for the bottom concentration and overall tray efficiency of another distillation column, a pilot-scale distillation column as described by Garcia and Fair (2000), which is 0.429 m in diameter with eight sieve trays 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, the KOL is related with the structure and size of the sieve tray. It was found that the correlations of kL and kG by Hoogendoorn et al. (1988) is applicable to the pilot-scale column, and was
The original c2 −c model (Liu, 2003), which is used to close the differential mass transfer equation, is further simplified and the model constants are ascertained. An improved CFD equation is employed to predict the velocity field. To test the validity of the improvement, the proposed simplified CMT model is applied to two distillation columns. The computed results are compared with the respective experimental data taken from the literatures. The comparison with the experimental data for an industrial scale distillation column reported by Sakata and Yanagi (1979) reveal that the simplified models can give better predictions on the turbulent mass transfer diffusivity than the original one, while the computed concentrations at the outlet of each tray and the tray efficiency by these two models are in satisfactory agreement. In addition, the comparison is also made to a pilot-scale distillation column described by Garcia and Fair (2000), the predicted bottom concentration and the overall tray efficiency under different F -factors of a pilot-scale sieve tray column are confirmed reasonably with the experimental data. The proposed simplified CMT model has demonstrated
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Fig. 13. Predicted EMV vs. experimental measurement.
Fig. 14. Predicted concentration vs. experimental measurement (0.429 m column).
Fig. 15. Overall tray efficiencies under different F -factors (0.429 m column).
to be a prospective tool to predict the turbulent mass transfer diffusivity, concentration profile on a tray as well as the tray efficiency of a distillation column.
C , C1 , C 2 C C
Notation
C
a c c2 Ct , Cc1 , Cc2 , Cc3
effective vapor–liquid interfacial area, m2 m−3 fluctuating concentration (mass fraction) concentration variance turbulence model constants for the concentration field
∗
D DG
turbulence model constants for the velocity field instantaneous concentration (mass fraction) time average concentration in liquid phase (mass fraction) time average concentration in liquid phase in equilibrium with concentration in gas phase (mass fraction) molecular mass transfer diffusivity, m2 s−1 vapor-phase molecular mass transfer diffusivity, m2 s−1
Z.M. Sun et al. / Chemical Engineering Science 62 (2007) 1839 – 1850
DL Dt EMV Fs g hf hL hw k kG kL KOL lw Lm m p P c qL R Re SC SC SMj t u U U Us
liquid-phase molecular mass transfer diffusivity, m2 s−1 turbulent mass transfer diffusivity, m2 s−1 Murphree efficiency of vapor phase √ F -factor (Fs = us G ) acceleration due to gravitation, m s−2 froth height, m clear liquid height, m weir height, m turbulent kinetic energy, m2 s−2 vapor-phase mass transfer coefficient, m s−1 liquid-phase mass transfer coefficient, m s−1 overall liquid phase mass transfer coefficient, m s−1 weir width, m Prandtl mixing length, m distribution coefficient time average pressure, Pa production term in the c equation volumetric flow of liquid flow, m3 s−1 timescale ratio Reynolds number source of interphase mass transfer time average source of interphase mass transfer source of interphase momentum transfer time, s fluctuating velocity, m s−1 instantaneous velocity, m s−1 time average velocity, m s−1 superficial vapor velocity, m s−1
Greek letters L
ij c t c , c , k , c , c m
liquid volume fraction Kronecker delta turbulent dissipation, m2 s−3 dissipation rate of c2 , s−1 turbulent viscosity, m2 s−1 density, kg m−3 turbulence model constants for diffusion of c2 , c , k, dissipation term in the c equation time scale, s timescales of velocity and concentration fields, s mean time scale, s
Subscripts G in i, j, k L x, y, z
gas inlet tensor symbols liquid x, y, and z coordinates
Acknowledgments The authors wish to acknowledge the financial support by the National Natural Science Foundation of China (No. 20136010),
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