Ind. Eng. Chem. Res. 2005, 44, 4427-4434
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New Turbulent Model for Computational Mass Transfer and Its Application to a Commercial-Scale Distillation Column Z. M. Sun, B. T. Liu, X. G. Yuan,* C. J. Liu, and K. T. Yu State Key Laboratory for Chemical Engineering (Tianjin University) and School of Chemical Engineering and Technology, Tianjin University, Tianjin 300072, China
A new c2-c model is proposed in this paper for solving the turbulent mass-transfer equation in order to obtain the concentration profile of chemical equipment, like the distillation column, without relying on the experimental measurement of the diffusivity of mass transfer. The computation is performed simultaneously with the computational fluid dynamics model, so that the velocity and concentration profiles as well as the efficiency of the equipment can be predicted at once for the equipment with nonideal flow and distributed mass transfer. The advantage of the present computational method is to discover the imperfections of flow and concentration distributions at the early stage of process design or to assess technically the effectiveness of the existing equipment, so that the necessary adjustment could be made for achieving higher efficiency. The feasibility of the present method is demonstrated by predicting the performance of a commercial-scale distillation column with eight sieve trays as reported by Fractionation Research, Inc. The computed results on the outlet concentration of each tray and the overall tray efficiency under various F factors as well as the turbulent diffusivity of mass transfer are respectively in satisfactory agreement with the experimental data published by Sakata and Yanagi (Inst. Chem. Eng. Symp. Ser. 1979, 56, 3.2/21-34) and Cai and Chen (Ind. Eng. Chem. Res. 2004, 43, 2590) for the FRI distillation column. Introduction Computational fluid dynamics (CFD) is becoming a powerful tool in chemical engineering. Over the last 10 years, the CFD method was rapidly applied to the areas of chemical engineering research, development, design, and production. Among those, we can find examples from the numerical simulation of a distillation tray,2-5 a fluidized bed,6 and many others. The great success of applying the CFD in chemical engineering fields is undoubtedly evident. However, the CFD method is applied only for the computation of the velocity distribution at present but is not yet extended to the prediction of the concentration field. Because the concentration distribution in chemical equipment, like the distillation column, is the basis for determining its operating efficiency, the investigation of the computational method for predicting the in-depth behavior of a mass-transfer process is unquestionably interesting to chemical engineers. As the starting point, the generalized equation of momentum and heat and mass transfer is shown:
(
)
∂Θi ∂Θi ∂Θi ∂ ) Γ - ujθi + Sθi +U hj ∂t ∂xj ∂xj ∂xj -ujθi ) Γt
∂Θ ∂xj
(1)
The source term Sθi is related to the specific chemical engineering problem and can be evaluated by theoretical formulation or modeling, and Γt is the turbulent diffu* To whom correspondence should be addressed. Tel.: +86 22 27404732. Fax: +86 22 27404496. E-mail: yuanxg@ tju.edu.cn.
sivity of the process concerned, which is usually an unknown parameter. For the simulations of hydrodynamics and heattransfer processes, nearly all of the works in the past were focused on the treatments of the source term Sθi. In the 1960-1970s, the k- turbulent two-equation model was established to eliminate the turbulent diffusivity Γt of momentum transfer for solving the hydrodynamic equation. Later, Nagano et al.7-9 developed several two-equation models to simulate the heat transfer for turbulent flow near the wall. For solution of the generalized equation (1) for the mass-transfer process, the source term in the equation is obviously the rate of mass transfer per unit volume. For the unknown turbulent diffusivity of mass transfer, it is difficult to determine with chemical engineering theory but depends on an experimental or empirical correlation. Nevertheless, in practically all of the experimental determinations of the process diffusivity, the tracer technique with an inert tracer was employed. Obviously, such a measurement is different from those involving mass transfer. For the purpose of overcoming the difficulty of determining the turbulent diffusivity in the mass-transfer process, a new method of solving the turbulent masstransfer equation by the addition of auxiliary modeling equations, to be termed the c2-c model, is proposed in this paper. By this method, the combination of the proposed mass-transfer model and the CFD equations enables us to calculate the turbulent diffusivity and solve the mass-transfer equation. Consequently, applying this technique could broaden the application of the computational method to the field of chemical engineering. For instance, it may follow that the concentration field as well as the tray efficiency of a distillation tray
10.1021/ie049382y CCC: $30.25 © 2005 American Chemical Society Published on Web 05/10/2005
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can be predicted. As demonstrated in this paper, by using this model together with quasi-single-phase CFD equations, the flow and mass-transfer behaviors of a commercial-scale distillation column with eight trays under total reflux, which is identical with that described by Sakata and Yanagi,1 are simulated. The predicted liquid concentrations of each tray outlet, tray efficiency, and overall column efficiency under various F factors as well as the turbulent diffusivity of mass transfer are reasonably checked with the reported experimental data.1,24 c2-Ec Model for Computational Mass Transfer Assuming the concentration to be a passive scalar, the governing equation of turbulent mass transfer is as follows:
∂ujc ∂C h ∂C h h ∂C +U hj )γ + Sc ∂t ∂xj ∂xj ∂xj ∂xj
(
2γuj
∂C h -ujc ) Dt ∂xj
terms ujc2 and ujc can be expressed by the following equations of gradient type:
Dt ) Ctk1/2Lm
c2/c, the turbulent diffusivity of mass transfer can be written as follows:
( )
1/2
(5)
The fluctuating concentration variance and the dissipation rate of 1/2 concentration variance are defined by eqs 6 and 7.
c2 ) cc
(6)
(
c ) γ
)
∂c ∂c ∂xj ∂xj
(7)
The precise expressions for c2 and c are given as follows:
(
)
∂C h ∂c2 ∂2c2 ∂c2 )γ - ujc2 - 2ujc - 2c (8) +U hj ∂t ∂xj ∂xj ∂xj ∂xj and
(10)
-ujc ) (Dt/σc) ∂c/∂xj
(11)
jj ∂uj ∂c ∂c ∂2c ∂2c ∂c ∂c ∂u - 2γ - 2γ ) ∂xj ∂xk ∂xj ∂xk ∂xj ∂xk ∂xk ∂xk ∂xj ∂xk c c ∂U hj c2 c ∂C h - C2 uiuj - C3 - C4 -C1 ujc (12) ∂xj k ∂xj k c2 c2
The other generation term associated with the mean concentration field can be approximated as follows:
(4)
With the relationship Lm ) k1/2τm, where τm ) xτµτc, and the definition of the time scales7 τµ ) k/ and τc )
k c2 Dt ) Ctk c
-ujc2 ) (Dt/σc) ∂c2/∂xj
For the generation term, the treatment by Newman et al.10 is employed:
-2γ2
Because Dt is regarded as directly proportional to the product of the characteristic velocity and the characteristic length, we have
∂c ∂uj ∂c ∂2c ∂2c - 2γ2 (9) ∂xj ∂xk ∂xk ∂xj∂xk ∂xj∂xk
Because of the presence of unknown covariance terms, the foregoing two equations cannot be used in practice unless they are further modified. Applying a treatment similar to the Reynolds stress, the turbulent diffusion
(2)
(3)
hj ∂c ∂2C h ∂c ∂c ∂U - 2γ ∂xk ∂xj∂xk ∂xk ∂xj ∂xk 2γ
2
Similar to the conventional treatment, the foregoing Reynolds concentration flux ujc is expressed in terms of turbulent mass-transfer diffusivity Dt as follows:
)
∂c ∂c ∂c ∂ ∂c ∂uj ∂C h ) γ - cuj - 2γ +U hj ∂t ∂xj ∂xj ∂xj ∂xj ∂xj ∂xk
-2γuj
( )
∂c ∂2C h ∂2 C h ) γDt ∂xk ∂xj ∂xk ∂xj ∂xk
2
(13)
The useful two equations for the c2-c model are as follows:
[( ) ] [( ) ]
∂c2 ∂ ∂c2 +U hj ) ∂t ∂xj ∂xj
γ+
Dt ∂c2 ∂C h - 2ujc - 2c (14) σc ∂xj ∂xj
c Dt ∂c ∂C h - Cc1 2ujc σ c ∂x 2 c j ∂c c2 ∂U h i c c ∂2 C h - Cc4 + γDt (15) Cc2 2 - Cc3uiuj ∂x k k ∂x ∂x c j j k
∂c ∂c ∂ +U hj ) ∂t ∂xj ∂xj
γ+
The parameters of the present model are as follows:11 Ct ) 0.11, Cc1 ) 1.8, Cc2 ) 2.2, Cc3 ) 0.72, Cc4 ) 0.8, σc ) 1.0, and σc ) 1.0. The details of the derivation of this model can be found in ref 12. Simulation of a Multiple-Tray Distillation Column (1) Simulated Distillation Column, Separating System, and Operating Conditions. The commercialscale distillation, as described by Sakata and Yanagi,1 is taken for the simulation. Figure 1a is a schematic diagram of the sieve column with eight trays. The
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Figure 1. (a) Schematic diagram of the column. (b) Mesh diagram of the column. Table 1. Tray Dimension Details column diameter (m) tray spacing (mm) hole diameter and spacing (mm × mm) outlet weir height and length (mm × mm)
1.2 610 12.7 × 38.1 51 × 940
clearance under the downcomer (mm) downcomer area (m2) effective bubbling area (m2) hole area (m2)
38 0.14 0.859 0.118
Table 2. Average Physical Properties of the System under Operating Conditions
To solve eq 17, the modified k- method13 is used, where the k equation is
pressure (kPa) 165 liquid viscosity (mPa s) 0.23 vapor density (kg m-3) 4.31 relative volatility 1.57 -3 -1 liquid density (kg m ) 673 surface tension (mN m ) 13.5
U hj
detailed dimensions of the simulated distillation column are given in Table 1. The separating system is cyclohexane-n-heptane at an operating pressure of 165 kPa. The average physical properties for the system at the operating pressure are given in Table 2. The column was operated at total reflux with a liquid reflux rate at 30.66 m3 h-1. The cyclohexane concentration of reflux was 93.1%. Considering that the flow is symmetrical and to save computer time and machine memory, only half of the tray was simulated. The liquid in the downcomer is assumed to be completely mixed; in other words, the average concentration of the outlet liquid of each tray is also the concentration of the inlet liquid to the tray below. The simulated distillation column, the grids, and the coordinates for computation are shown in Figure 1b. (2) CFD and Mass-Transfer Equations. In the formulation of the equation set for computation, the conventional CFD equations are adopted for simulating the velocity profile as follows:
∂U h i/∂xi ) 0 U hj
(16)
(
)
∂U hi ∂U hi j 1 ∂p ∂ )+ ν - uiuj + SMi ∂xj F ∂xi ∂xj ∂xj
(17)
where SMi is the source term representing the momentum change between the vapor and liquid phases and uiuj is the Reynolds stress, for which the Boussinisque’s relation is applied:
(
)
hj ∂U h i ∂U 2 -uiuj ) νt + - δijk ∂xj ∂xi 3 in which νt ) Cµk2/.
∂k ∂ ) ∂xj ∂xj
[( )( )] ( ν+
νt ∂k σk ∂xj
+ νt
and the equation is given by
U hj
∂ ∂ ) ∂xj ∂xj
[( ) ( )] [ ( ) ν+
C1νt
νt σ
∂ ∂xj
+
]
h j ∂ui ∂U h i ∂U (20) + + C1G - C2 ∂xj ∂xi ∂xj k
where G ) 1.28 × 10-4UsL-0.75, and the model parameters are customarily chosen to be Cµ ) 0.09, C1 ) 1.44, C2 ) 1.92, σk ) 1.0, and σ ) 1.3. The equation-governing concentration profile of a tray is represented by
U hj
(
)
∂ ∂C h ∂C h ) γ - ujc + SC ∂xj ∂xj ∂xj
(21)
where SC is the source term for mass transfer between the vapor and liquid phases; the Reynolds concentration flux is expressed by eq 3. Equations 5, 14, and 15 in the steady state are used to eliminate the unknown mass-transfer diffusivity Dt in order that eq 21 can be solved to obtain the concentration distribution. The source term SMi in eq 17 can be represented by the following equation:2
SMi ) (18)
)
h j ∂ui ∂U h i ∂U + + ∂xj ∂xi ∂xj G - (19)
FGUsi U h FLhf i
(22)
where the liquid height is equal to hf ) hw + how and how can be estimated by the following correlation:14
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how )
()
2.84 Lh 1000 lw
2/3
(23)
The mass-transfer process on a tray is governed by eqs 5, 14, 15, and 21. The source term SC in eq 21 can be expressed by15
h) SC ) kLa(C* - C
(24)
where the liquid mass-transfer coefficient kL is taken from the literature16 as follows:
kL )
2.6 × 10-5 ηL0.25
(25)
and the contacting area of the bubble and liquid per unit volume is given by
a ) a′/hf
(26)
[
]
k2 1 R ) Ct Dt
2
The foregoing equation is valid for the concentration ranging from C h ) 0 to C h ) C*. As an approximation, we may assume that R is a constant at the inlet, and its value can be determined conveniently by the experimental data obtained under the “end condition” of C h ) h ) 0 for a sieve tray has been 0. The diffusivity Dt at C measured by many investigators with the tracer technique and is generally regarded as the “coefficient of dispersion” or “coefficient of backmixing”. Our calculation shows that the predicted results by various published formulas are quite diverse, although the order of magnitude is all around 10-3-10-2. By taking their average values and substitution into eq 33a, it was found that the value of R was around 0.9. The boundary condition from eq 33 at the inlet for cin therefore can be given by
cin ) 0.9(in/kin)c2in
where16,17
a′ )
(
)
2 43 Fbba hLFP σ F0.3
0.53
(27)
FP ) (FG/FL)0.5
(28)
hL ) 0.6hw0.5p0.25(FP/b)0.25
(29)
(3) Boundary Conditions. The inlet conditions are h )C h in, and for the k- equations, the U h )U h in and C following conventional formulas18 are used:
kin ) 0.003U h xin2 in ) 0.09kin3/2/0.03
(30) W 2
(31)
The inlet condition of the c2 equation should be further discussed and investigated. Strictly speaking, the extent of concentration fluctuation at the inlet should be experimentally determined. Because the experimental data are lacking at present, we may take advantage of utilizing the similarity between heat and mass transfer. By the work of Tavoularis and Corrsin19,20 and Ferchichi and Tavoularis21 in heat transfer, the extent of temperature fluctuation induced by heating is roughly 0.082 times the heat flow. Similarly, we might postulate that
c2in ) [0.082(C* - C h in)]2
(32)
The inlet condition of the c equation is a complicated problem and could be determined by the following steps. Let R represent the ratio of “the time needed for the dissipation of turbulent energy fluctuation to that for the dissipation of concentration fluctuation”, namely
R)
k/
(33)
2
c /c The value of R is affected by many factors and varies over the whole field under consideration. Combining eqs 5 and 33, we have
(33a)
(34)
At the outlet, we have p ) 0 and ∂C h /∂x ) 0. The boundary conditions at the tray floor, the outlet weir, and the column wall are considered as nonslip, and the conventional logarithm law expression is employed. At the interface of the vapor and liquid, all of the stresses are equal to zero, so we have ∂ux/∂z ) 0, ∂uy/∂z ) 0, and uz ) 0. Similarly, at both the wall and the interface, the concentration flux is equal to zero. Result and Discussion The computation was performed by using commercial software FLUENT for three-dimensional simulation. The computed results are analyzed by taking the profiles on trays 6 and 8 as an example. (1) Outlet Concentration of Each Tray. In Figure 2, the concentration of cyclohexane at the outlet of each tray obtained by the present modeling is compared with the experimental data by Sakata and Yanagi.1 It can be seen that the simulation results are in good agreement with the experimental data. The obvious deviation at tray 4 may be attributed to experimental error. For a distillation column separating a binary mixture under total reflux, the curve connecting the outlet concentration of all trays should be nearly smooth. The experimental point at tray 4 is clearly outside of the curve representing all other points. The Murphree efficiency22 EMV ) (yn - yn+1)/(y/n yn+1) for each tray is also computed. Figure 3 shows the measured Murphree efficiency and the predicted ones. Except for trays 3 and 4, the predicted results are in agreement with the measurement. The reason for the obvious deviation at trays 3 and 4 is probably due to experimental error, as can be seen from the distribution of experimental points. When the simulation was undertaken, two methods were used for comparison. Method I is a tray-by-tray computation, and method II is used to solve the equation set of the whole column and obtain the results for all trays. Figures 2 and 3 show that the predicted results by using methods I and II are practically no different. Method I has the advantage of requiring fewer grids and saving computer memory, while the advantage of method II is to be able to obtain all of the results at once at the
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Figure 2. Predicted concentration vs experimental measurement.
Figure 5. Velocity profile of the x-y plane on tray 8 at 70 mm above the floor (method II).
Figure 6. Velocity profile of the x-y plane on tray 6 at 20 mm above the floor (method I). Figure 3. Predicted EMV vs experimental measurement.
Figure 4. Velocity profile of the x-y plane on tray 8 at 20 mm above the floor (method II).
expense of requiring larger computer memory. Therefore, when the number of trays is large, the choice of using method I is preferable. (2) Velocity Profile. Figures 4-7 are the velocity profiles on one of the x-y planes on trays 6 and 8. It was found that the flow pattern on tray 8 is nearly the same as that on the other seven trays, except having a slight difference around the gap of the liquid inlet. The reason is that the liquid inlet at the top tray of the column is assumed to be uniformly distributed, while the other tray is not. In these figures, different extents of the liquid backflow can be found near the wall. The
Figure 7. Velocity profile of the x-y plane on tray 6 at 70 mm above the floor (method I).
velocity profiles predicted by methods I and II are practically no different for all trays. (3) Concentration Profile. Figures 8-11 give the concentration contours on one of the x-y planes on trays 6 and 8. The concentration patterns of the other six trays are similar to that of tray 6. The tendency for concentration change along the flowing direction is similar for all trays, except the top tray where the inlet liquid is assumed to be uniformly distributed, which is different from the other trays. (4) Turbulent Mass-Transfer Diffusivity Profile. The computed profiles for the distribution of turbulent mass-transfer diffusivity Dt are shown in Figures 12 and
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Figure 8. Concentration profile of the x-y plane on tray 8 at 20 mm above the floor (method II).
Figure 9. Concentration profile of the x-y plane on tray 8 at 70 mm above the floor (method II).
Figure 10. Concentration profile of the x-y plane on tray 6 at 20 mm above the floor (method I).
13. Because the turbulent mass-transfer diffusivity Dt represents the extent of backmixing, it is seen that the backmixing at the regions around the tray inlet, outlet, and wall is serious. These unfavorable phenomena are related to the velocity distribution because there exists circulation of flow. The figures also show that the distribution of turbulent mass-transfer diffusivity is diverse. However, we may take their volume average values for the purpose of comparison with the published data on the backmixing of the sieve tray. Nevertheless, most of the reported investigation on turbulent masstransfer diffusivity is based on one-dimensional model-
Figure 11. Concentration profile of the x-y plane on tray 6 at 70 mm above the floor (method I).
Figure 12. Turbulent mass-transfer diffusivity profile on tray 6 at 20 mm above the floor (method II).
Figure 13. Turbulent mass-transfer diffusivity profile on tray 6 at 70 mm above the floor (method II).
ing, and the experimental diffusivities obtained are quite different, with their order of magnitude ranging from 10-2 to 10-3. For instance, Zuiderweg16,23 gave a correlation for estimating the turbulent mass-transfer diffusivity as follows:
Dt ) 8.3FGus2hL2/FL(qL/l)
(35)
where hL is calculated from eq 29. The predicted Dt from the above correlation for the simulated distillation column is 0.0137 m2 s-1, while the volume average Dt calculated from the present model is 0.011 m2 s-1. The recent data on Dt measured for the FRI column as
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Figure 14. Overall tray efficiencies under different F factors.
described in this paper, reported by Cai and Chen,24 is around 0.015-0.020 m2 s-1 at an F factor (Fs ) usxFG) ranging from 1.0 to 2.25. Our predicted value of Dt is resonably close to the experimental measurement. It follows that a feature of the present new model is able to predict the mass-transfer diffusivity instead of the experimental finding. Furthermore, from the threedimensional analysis of the mass-transfer diffusivity on a tray, the discovery of the region’s existing serious backmixing may enable us to make necessary changes to the tray design to reduce the unfavorable effect in order to achieve higher tray efficiency. (5) Overall Tray Efficiency. The overall tray efficiency can be calculated by the Fenske-Underwood equation25 as follows:
Eo ) NT/NA log NT )
[
(36)
]
xT(1 - xB)
xB(1 - xT) log R
(37)
To further demonstrate the feasibility of applying the present model, the overall tray efficiencies under different F factors (Fs ) usxFG) are also computed. It can be seen from Figure 14 that the predicted efficiencies are in satisfactory agreement with the experimental results, although the predicted data are not all close to the experimental ones. It can be explained that the mass-transfer coefficient kL and the effective contacting area a used in computing the rate of mass transfer in the present simulation are evaluated by using the correlations taken from the literature, the error of which is the main source of discrepancy between the prediction and the experimental measurement. Conclusions A new model accompanied by CFD equations is developed for predicting the three-dimensional velocity/ concentration profiles, the turbulent mass-transfer diffusivity Dt, and the efficiency of chemical equipment. The use of the proposed model is demonstrated by applying to a commercial-scale distillation column with eight trays as reported by FRI, and the computed results are briefly presented, including the velocity and concentration profiles at various levels above the tray deck as well as the overall tray efficiency at different F factors. The computed results of the concentration at the outlet of each tray and the tray efficiency are in
satisfactory agreement with the published experimental data by Sakata and Yanagi.1 The computational result also reveals that the mass-transfer diffusivity varies considerably throughout the whole tray, and its value is on the order of magnitude between 10-3 and 10-2. The volume average of the computed mass-transfer diffusivity is reasonably checked with the experimental data by Cai and Chen.24 The present computational method may help the designer and operator to find out the imperfections of the equipment, such as the existence of uneven velocity distribution, low concentration zone, and regions of serious backmixing on a distillation tray, so that the necessary changes could be made for achieving higher efficiency. Concerning the strategy of simulating a distillation column, the computed results using the tray-by-tray method and the whole-column method show practically no difference. Acknowledgment The authors acknowledge financial support by the National Natural Science Foundation of China (Contract No. 20136010) and assistance by the staff in the State Key Laboratories of Chemical Engineering (Tianjin University). Nomenclature a ) specific vapor-liquid contacting area (m-1) a′ ) interfacial area per unit bubbling area b ) weir length per unit bubbling area (m-1) C h ) mean concentration (mass fraction) C* ) equilibrium concentration (mass fraction) Ct, Cc1, Cc2, Cc3, Cc4 ) turbulence model constants for the concentration field Cµ, C1, C2 ) turbulence model constants for the velocity field c ) fluctuating concentration (mass fraction) c2 ) concentration variance Dt ) eddy diffusivity (m2 s-1) EMV ) Murphree efficiency of the vapor phase Eo ) overall tray efficiency F ) fraction hole area per unit bubbling area Fbba ) vapor F factor on the bubbling area [m s-1 (kg m-3)0.5] Fs ) F factor (Fs ) usxFG) FP ) flow parameter hf ) height of the liquid layer (m) hL ) liquid holdup (m) hw ) weir height (m) k ) turbulent kinetic energy (m2 s-2) kL ) liquid mass-transfer coefficient (m s-1) Lh ) liquid rate in the column (m3 h-1) Lm ) Prandtl mixing length (m) l ) average liquid flow width across the bubbling area (m) lw ) weir width (m) NA ) actual number of trays NT ) theoretical number of trays p ) pitch of holes in the sieve plate (m) p j ) mean pressure (Pa) qL ) volumetric flow of liquid across the plate (m3 s-1) R ) time-scale ratio Re ) Reynolds number Sθ ) source term SC ) source of interphase mass transfer SM ) source of interphase momentum transfer t ) time (s) U h ) mean velocity (m s-1) Us, us ) superficial vapor velocity (m s-1)
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u ) fluctuating velocity (m s-1) W ) weir length (m) xB ) mole fraction of the more volatile component in the liquid phase at the bottom of the section of trays xT ) mole fraction of the more volatile component in the liquid phase at the top of the section of trays yn ) mole fraction of the more volatile component in the vapor phase leaving tray n yn+1 ) mole fraction of a component in the vapor phase leaving tray n + 1 y/n ) mole fraction of the more volatile component in the vapor phase in equilibrium with the liquid leaving tray n Greek Symbols R ) relative volatility Γ, γ ) molecular diffusivity (m2 s-1) Γt ) turbulent diffusivity (m2 s-1) δij ) Kronecker delta ) turbulent dissipation (m2 s-3) c ) turbulent dissipation of the concentration (m2 s-3) ηL ) liquid viscosity (Pa s) Θ, θ ) any vector or scalar ν ) molecular viscosity (m2 s-1) νt ) turbulent viscosity (m2 s-1) F ) density (kg m-3) σ ) surface tense (N m-1) σc, σc, σk, σ ) turbulence model constants for diffusion of c2, c, k, and τm ) characteristic time scale for turbulent mass transfer τµ, τc ) time scales of the velocity and concentration fields Subscripts G ) gas L ) liquid in ) inlet i, j, k ) x, y, and z coordinates x, y, z ) coordinates
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Received for review July 14, 2004 Revised manuscript received December 9, 2004 Accepted April 8, 2005 IE049382Y
