Ind. Eng. Chem. Res. 1998, 37, 697-703
697
RESEARCH NOTES Simulation and Optimization of the Waste Nitric Acid Recovery Process Sea Cheon Oh,† Young Se Oh,‡ and Yeong-Koo Yeo*,† Department of Chemical Engineering, Hanyang University, 17 Haengdang-dong, Sungdong-ku, Seoul 133-791, Korea, and Technology, Research and Development Center, Hanil Synthetic Fiber Company Ltd., 222 Yangduk-dong, Hoiwon-ku, Masan 630-791, Korea
This paper deals with the simulation and optimization of composite distillation columns for the waste nitric acid recovery process. The composite distillation columns which consist of a multistage vacuum column and an atmospheric pressure column, half of which, consists of a packed bed, were modeled by using an equilibrium stage method and a nonequilibrium stage method. The required physical properties of a nitric acid solution for simulation were obtained from correlations based on experimental data. Simulation results using the nonequilibrium model showed better agreement with actual plant data than those of the equilibrium model. Based on the simulation results, the optimal operation conditions were studied. In the optimization reflux ratio was employed as the key variable to maximize the operating profit. Introduction In the production of acryl fiber the waste effluent contains a lot of nitric acid solution and the recovery of the waste nitric acid solution is imperative to protecting environmental contamination. The distillation column studied in this work is in fact a concentration process of dilute nitric acid flowing out from the acryl fiber plant and characterized by composite distillation columns which consist of a multistage vacuum column and an atmospheric pressure column, half of which consists of a packed bed. Modeling of multistage distillation columns is well-known technology (Henley and Seader, 1981). But the simulation of the distillation column which consists of a packed bed requires the nonequilibrium analysis of packed-bed operational data and the physical properties of components in the distillation column. The use of the equilibrium stage model in the analysis of actual packed columns results in a wide discrepancy with actual operation data (Schweitzer, 1979). The lower half of an atmospheric pressure column studied in this work consists of a packed bed, and the required physical properties of a nitric acid solution were not available. For these reasons the nitric acid distillation column has not been studied in depth and the authors could find only a few published results on the analysis of nitric acid recovery processes. Usually two methods are employed in the analysis of the nonequilibrium model for a packed column. In one method the continuous contacting area of the packed * Author to whom correspondence is addressed. Telephone: 82-2-290-0488. Fax: 82-2-291-6216. E-mail: ykyeo@email. hanyang.ac.kr. † Hanyang University. ‡ Hanil Synthetic Fiber Co., Ltd. Telephone: 82-551-903201. Fax: 82-551-90-3209.
column is divided into many subdivisions and each of them is considered as a tray (Holland, 1975). In the other method the packed column is assumed to be composed of tiny intervals and mass and energy balances are obtained by solving differential equations set up for each tiny interval (Kelly et al., 1984; Cho and Joseph, 1983a,b). Krishnamurthy and Taylor (1985) presented a nonequilibrium model based on the former method which employed mass and energy transfers at liquid and vapor phases as well as interfacial equilibrium relations. In this work the nonequilibrium model based on Krishnamurthy and Taylor’s method with simplified heat-transfer relations was used. The simplifying assumptions and new relation equations for the simulation using the nonequilibrium model were provided and employed in the analysis the of nonequilibrium model. To obtain the required physical properties of the nitric acid solution for simulation, experimental studies were performed. The optimization was studied for maximization of the operating profit, and the reflux ratio was employed as a key variable. Process Description Figure 1 shows the process diagram of the nitric acid recovery process which consists of composite distillation columns. The dilute nitric acid (35 wt %) flowing out from the acryl fiber production process is concentrated into strong acid (66.5 wt %) in the recovery process. From the first distillation at the vacuum column (K-1) strong acid (56 wt %) is produced as a bottom product and is fed into the atmospheric pressure column. From the second atmospheric pressure column (K-2) strong acid (66.5 wt %) is obtained as a bottom product and top product is utilized as an energy source for the first vacuum column with bottom product.
S0888-5885(97)00153-X CCC: $15.00 © 1998 American Chemical Society Published on Web 01/09/1998
698 Ind. Eng. Chem. Res., Vol. 37, No. 2, 1998 Table 2. Values of Coefficients of Eq 5 concentration (wt %)
a
b
c
35 45 55 65
3.0084 4.0534 3.5486 3.5923
-0.013 756 -0.020 086 -0.016 586 -0.016 919
0.204 62 × 10-4 0.297 40 × 10-4 0.233 37 × 10-4 0.231 69 × 10-4
the nitric acid heat capacity was studied to develop the heat capacity equation which can be used in the simulation of the nitric acid distillation column. The heat capacity was correlated from the current and voltage recorder at the interval of 2 °C in the temperature range of 30-100 °C. As an energy source we used a 12 V dc electrical source. The heat loss rate to environment was predicted experimentally from an energy balance equation based on water. Basically the heat loss rate is a function of time and temperature. As starting relations we can set up the following:
Figure 1. Process diagram of waste nitric acid recovery. Table 1. Operating Conditions Used in the Simulations vacuum column column pressure, mmHg abs reflux ratio temperature of feed, °C top flow rate, kmol/h bottom flow rate, kmol/h feed flow rate, kmol/h sidestream flow rate, kmol/h feed composition, wt % HNO3 condenser heat duty, kcal/h reboiler heat duty, kcal/h plates number of rectifying section plates number of stripping section diameter of rectifying section, m diameter of stripping section, m height of rectifying section, m height of stripping section, m
102 0.3058 69 576.8 577.1 1167.3 13.4 35 7 539 359 8 147 757 11 6 3.38 3.38 4.95 2.20
atmospheric column 760 2.099 115 148.2 428.9 577.1 56 4 172 400 8 278 200 10 packed bed 1.56 1.35 4.00 6.40
The upper half of the atmospheric pressure column from the feed stage is multistage column and the lower half of the column is a packed bed. The type of multistage part is a sieve tray, and the packed bed is packed irregularly with special ceramic material. In the analysis of the packed bed the packed material was assumed to be ceramic Raschig rings with the average size of 2 in. Table 1 shows operation conditions used in the simulation. Physical Properties of the Nitric Acid Solution The required physical properties of the nitric acid solution were not available and so the relationships for the physical properties of the nitric acid solution were developed based on the experimental results and the literature (Perry and Chilton, 1973) data. Heat Capacity. In order to simulate the nitric acid distillation column, heat capacity data applicable to the various areas of the operating conditions of the distillation column is necessarily required. However, the heat capacity data of a nitric acid solution which can be obtained from the literature were restricted within narrow limits. Thus, in this work the measurement of
dqloss dqinput dT ) - mcp dt dt dt
(1)
dqinput ) 0.239VI dt
(2)
dqloss ) f(T,t) dt
(3)
In the development of a rate equation we have to consider the differences in the heat-transfer areas of experimental systems. An iterative procedure was employed for the interpretation of experimental results using pure water. The model of the heat loss rate was obtained as follows:
dqloss ) 1.6342∆T0.51648(t/T)-0.61886m0.22781 dt
(4)
The relations between time and temperature of nitric acid solutions were obtained from the experiments. Using these results and heat loss rate model (4) we can calculate the heat capacity of the nitric acid solution based on eq 1. From the results of calculations we get the heat capacity eq 5. Table 2 shows values of
cp ) a + bT + cT2
(5)
coefficients of eq 5 as function of the concentration of nitric acid solutions. The maximum error between the values predicted by eq 5 and the experimental data given in the literature (Perry and Chilton, 1973) is 3.8%, which means that we can use the relation (5) in the simulation study. Latent Heat. The vapor enthalpy or latent heat of the nitric acid solution was not available, and so the relationships for the latent heat of the nitric acid solution were developed based on the Perry and Chilton (1973) data. The vapor pressure of pure materials can be represented by the Antoine equation of the form
ln P ) A -
B T+C
(6)
From the Perry and Chilton (1973) data for the vapor pressure of a nitric acid solution, we get Table 3 which shows values of coefficients of eq 6. In order to calculate the latent heat of a nitric acid solution from the vapor
Ind. Eng. Chem. Res., Vol. 37, No. 2, 1998 699 Table 3. Values of Coefficients of Eq 6 for T g 50 °C concentration (wt %)
A
B
C
20 25 30 35 40 45 50 55 60 65 70
20.5875 20.5114 20.1581 20.8602 21.0175 22.0070 21.0741 20.4037 20.5094 19.5939 19.7423
5343.3388 5358.0566 5156.8301 5697.2337 5822.2589 6557.4041 5814.7472 5357.8012 5448.6583 4898.2728 5087.8959
279.6055 280.8317 273.9755 290.9412 292.9544 312.6932 286.5751 270.3515 272.1186 255.8691 265.9306
Table 4. Values of Coefficients of Eq 8 concentration (wt %)
A
B
C
35 45 55 60 65
4.5988 5.8515 6.1154 7.5384 6.1219
2618.7857 3908.2196 3879.3295 4289.7445 3618.3435
-588.5373 -704.1058 -682.8750 -641.8720 -645.6455
pressure, we used eq 7 obtained from eq 6 and the Clausius-Claperyron equation.
λ)
BRGT2 (T + C)2
(7)
Viscosity. In this experiment the viscosity of the nitric acid solution was measured by using an Ostwald viscometer in the temperature range of 70-120 °C. From the results of experiments we get the viscosity equation (8). Table 4 shows values of coefficients of eq
B ln η ) A + T+C
(8)
8 as a function of the concentration of nitric acid solutions. K-values of a nitric acid solution were directly obtained from the vapor pressure data on temperature and composition change in the literature (Perry and Chilton, 1973). Other physical properties of the mixture, such as density and thermal conductivity, were evaluated using the mixing rules suggested by Reid et al. (1977). Simulation of Distillation Columns Nonequilibrium Stage Model. Krishnamurthy and Taylor (1985) presented a nonequilibrium model based on heat and mass transfer between liquid and vapor phases and the equilibrium relationship in the interface. But in their model all the equations are nonlinear and an error in one variable cause errors in other variables. For the calculation of the vapor and liquid phase energy transfer to simulate the distillation column by using a nonequilibrium model, Krishnamurthy and Taylor (1985) used the assumed function to correct the heat-transfer coefficient of the vapor phase for the influence of mass transfer rate and ignored the heat-transfer correction factor for the liquid phase. However, in this work the heat-transfer correction factor for the liquid phase cannot be safely ignored. Further, for the vapor phase even if the various heat-transfer correction factors were employed to correct the heattransfer coefficient, a reasonable heat-transfer coefficient cannot be obtained. It is assumed that this is
Figure 2. Comparison of field and predicted temperature for the vacuum column using the equilibrium model.
why the nitric acid solution is very nonideal. Therefore, in this work it is very difficult to calculate the interfacial temperature from the energy balance at the interface. In the present study a simple assumption for the temperature relations at the interface was introduced as follows:
TIj ) TVj ) TLj (for multistage column)
(9)
TIj ) 0.5(TVj + TLj ) (for packed column)
(10)
In the calculation of mass-transfer rates, various methods such as film theory are used. In the present study a modified effective diffusivity method based on Wilke’s formula (Sherwood et al., 1975) was used. The rate equations can be written by V a(yVi - yI) + yVi Nt NVi ) kieff
V kieff a)
1 - y* i
(11) (12)
c
V (y* ∑ i /k*ik a) k)1 V In the above equations yi is the average value of yij+1
and y*i, and y*i is the average value of yVi and yI. In this work, to calculate the effective diffusion coefficient, we employed the following method.
k*ijVa ) kikaΦVi
(
ΦVi ) exp
Vi Vi + Li
(13)
)
(14)
In the calculation of the above equations the binary mass-transfer coefficients (kika) were obtained using Gilliland-Sherwood’s correlation and Onda’s correlation (Sherwood et al., 1975; Krishnamurthy and Taylor, 1985) and for the multistage and packed columns, respectively. The mass transfer in the liquid mixture should be analyzed using chemical potential gradients rather than concentration gradients as in the vapor phases (Krish-
700 Ind. Eng. Chem. Res., Vol. 37, No. 2, 1998
Figure 3. Comparison of field and predicted composition for the vacuum column using the equilibrium model.
Figure 5. Comparison of field and predicted composition for the atmospheric pressure column using the nonequilibrium and equilibrium models.
hL ) kLavCLp (LLe )0.5
Figure 4. Comparison of field and predicted temperature for the atmospheric pressure column using the nonequilibrium and equilibrium models.
namurthy and Taylor, 1985; Krishna and Standart, 1979). But the use of chemical potential gradients requires a lot of experimental data, and so Wilke’s effective diffusivity method was used as equal to the vapor phase mass-transfer rate. The heat-transfer rate in the liquid mixture was obtained from the modified version of Krishna and Standart’s (1979) heat-transfer rate equation. The heattransfer rate is given by
eLj
)
hLj a
ψhj
c
exp(ψhj ) - 1
(TIj - TLj ) +
NiHLi ∑ i)1
(15)
c
ψhj )
L (NiCpi /hLj a) ∑ i)1
(16)
The heat-transfer coefficient was obtained from the following equation as in the case of Krishnamurthy and Taylor (1985).
(17)
As previously stated, it is very difficult to obtain a reasonable heat-transfer coefficient for the vapor phase. An incorrect heat-transfer coefficient generates significant numerical errors in the simulation of the nitric acid distillation column. Therefore, in this work the heattransfer rate in the vapor mixture was calculated by using the energy balance equation at the interface and the heat-transfer rate in the liquid mixture. Simulation Results. The simulation results of the vacuum column and atmospheric pressure column were analyzed and compared with actual plant operation data. The two columns were separately simulated by fixing the feed to each column using data given in Table 1. To solve the nonlinear problem such as the nonequilibrium model presented in this work, Newton’s method or fixed point algorithm is widely used (Krishnamurthy and Taylor, 1985). But Newton’s method requires reasonable initial conditions and the existence of inverse Jacobian matrices. Finite-difference approximations of the derivatives for the computation of Jacobian matrices can greatly increase the computing load because more evaluations of physical properties are required. Moreover, it is very difficult to find appropriate initial guesses of mass- and heat-transfer coefficients. To avoid these problems, we employed a fixed-point algorithm. Figures 2 and 3 show the results of calculations of temperature and composition distribution for the vacuum column based on the equilibrium model as well as actual operation data. The vaporization efficiency was used in the simulation of the equilibrium model. There is little discrepancy between the two cases. Temperature distribution for the atmospheric pressure column based on the equilibrium and nonequilibrium models is shown in Figure 4. Considering the fact that the upper half of the column is the multistage part and the lower half is the packed column, we can see that the equilibrium model can be applied only to the multistage part while nonequilibrium model shows good agreement with operation data in the whole column section. The McCabe-Thiele method was used in the calculation of the number of theoretical plates for the packed column. The usefulness of the nonequilibrium model is also demon-
Ind. Eng. Chem. Res., Vol. 37, No. 2, 1998 701
Figure 6. Comparison of flow rate for atmospheric pressure column using the nonequilibrium and equilibrium models.
Figure 8. Condenser cooling load of the atmospheric pressure column on change of reflux ratio.
Figure 7. Predicted temperature of liquid and vapor for the atmospheric pressure column using the nonequilibrium model.
Figure 9. Reboiler heat requirements of the vacuum column on change of reflux ratio.
strated in Figure 5 which shows the predicted composition distribution according to column depth. In actual operation the measurement of compositions is possible only at the top and at the bottom of the column. Figure 6 shows the dependency of flow rates on the column depth. The vapor flow rate at the right corner in Figure 6 was obtained from the actual reboiler heat requirement. In order to confirm the assumption introduced in the interfacial temperature for the nonequilibrium model, we predicted the vapor and liquid phase temperature distributions according to column depth based on the nonequilibrium model, results of which are shown in Figure 7. As can be seen, the difference of temperature between the vapor and liquid phases is not large, which shows the effectiveness of the assumption.
little effect on the objective function of the optimization problem because the distillation column studied in this work is the recovery process of the waste nitric acid solution flowing out from the acryl fiber plant. Reboiler heat input and cooling steam input are effected mainly by reflux ratios for the constant feed rate. Thus, reflux ratio was employed as the key variable to maximize the operating profit. Relation of Reflux Ratios. The nitric acid recovery process studied in this work consists of a vacuum column and an atmospheric pressure column. The top and bottom product from the atmospheric pressure column is utilized as an energy source for the vacuum column and so the operating reflux ratio of the vacuum column is affected by the reflux ratio of the atmospheric pressure column. However, the change of the reflux ratio of the atmospheric pressure column has little effect on the flow rate of the bottom product. Thus, the relation of reflux ratios of two columns was obtained by the heat balances of two columns related to the top product of the atmospheric pressure column. Figures 8 and 9 show the simulation results for the change of the cooling load of an atmospheric pressure column and
Optimization of the Waste Nitric Acid Recovery Process In this work we considered only the determination of the optimal operating conditions. Usually, the manipulated variables are heat input, cooling stream inputs, and product flow rate. But the product flow rate has
702 Ind. Eng. Chem. Res., Vol. 37, No. 2, 1998 Table 5. Values for the Optimization of Waste Nitric Acid Recovery Process symbol
description
value
CS CC CB CF F
reboiler heat cost of the atmospheric column condenser cooling cost of the vacuum column value of 66.5 wt % HNO3 value of 35 wt % HNO3 feed flow rate
1.27 $/105 kcal 0.00 $/105 kcal 1.28 $/kmol 0.00 $/kmol 1167.3 kmol/h
Table 6. Comparison of the Optimization Results and the Actual Operating Data product flow rate (66.5 wt % HNO3), kmol/h reboiler heat duty of the atmospheric column, kcal/h condenser heat duty of the vacuum column, kcal/h reflux ratio of the atmospheric column reflux ratio of the vacuum column operating profit, $/day
the reboiler heat requirement of a vacuum column on various reflux ratios. For the calculation of Figures 8 and 9 the program starts with a given reflux ratio and distillate. Using the result of simulation, new distillate is calculated using eq 18.
D xF - xB ) F xD - xB
optimization results
actual operating data
429.4 8 983 136 7 692 933 2.082 0.3016 10453.11
428.9 9 069 360 7 713 933 2.099 0.3058 10411.47
rR ) nitric acid sales (66.5 wt %) - utility costs raw materials ) CBB - (CSQB + CCQC) - CFF ) CBB - CSQB subject to
0.09 e Rv e 0.628
(18)
1.31 e Ra e 3.396
The initial distillate is incremented proportional to its difference by using eq 19, and the procedure is repeated.
D ) D + Kc(Dc - D)
(19)
Gain Kc determines the size of the increment and hence the rate of convergence. When successive values of distillate agree within 0.1%, iteration is terminated. In eq 19 Dc is the new distillate. Using Figures 8 and 9 the relation between the reflux ratio and the change of the cooling load for the atmospheric pressure column can be written as
∆QC ) QS - QC ) 3.0832 - 1.4689Ra(106 kcal/h) (20) where QS is the cooling load calculated using the operating reflux ratio (Ra ) 2.099) in Table 1. As in the case of the atmospheric pressure column, we get the relation for the change of the reboiler heat requirement as
∆QB ) QS - QC ) 1.8083 - 5.9134Rv(106 kcal/h) (21) where QS is the reboiler heat requirement calculated using the operating reflux ratio (Rv ) 0.3058) in Table 1. By the heat balance of two columns, from eqs 20 and 21 we have the relation of reflux ratios as follows:
Rv ) 0.2484Ra - 0.2156
Rv ) 0.2484Ra - 0.2156
(22)
Optimization Result. The raw material cost CF has values of 0 because raw material is the waste nitric acid solution recovered from the production unit of acryl fiber. Moreover, the condensing cost Cc is very small compared with other costs. Based on these facts, the optimization problem can be formulated as follows:
xD e 0.2 wt % HNO3
(23)
The constraint on the top product in eq 23 is imperative to protect environmental contamination. The inequality constraints of reflux ratios in eq 23 were obtained from minimum and maximum reflux ratios. Minimum reflux ratios of two columns were calculated by the McCabe-Thiele method, and maximum reflux ratios were obtained from the design heating capacities of reboilers considering the safety factor of 80%. On the basis of the data in Table 5, we maximized fR with respect to Ra using the region elimination method (Edgar and Himmelblau, 1988). For the optimization the atmospheric pressure column was simulated with the data obtained from the simulation results of the vacuum column. The optimization results as well as the actual operating data are shown in Table 6. From the optimization results we can see that there is little discrepancy between the optimum reflux ratio and the operating reflux ratio. This is due to the fact that the quality constraint of the top product is xDX e 0.2 wt % HNO3. Conclusion A nitric acid recovery process which consists of a vacuum column and an atmospheric pressure column was simulated by using a nonequilibrium model and an equilibrium model. From the results of simulations we can see that the upper half of the atmospheric pressure column and the vacuum column, consisting of a plate column, can be modeled using the equilibrium model. But the nonequilibrium model showed much better results for the lower half of the atmospheric pressure column, which are packed parts. The nonequilibrium model developed in this study showed good agreement with actual operation data. For optimization the relation of reflux ratios of a vacuum column and an atmospheric pressure column was obtained based on the heat balances of two columns. From the results of optimization we obtained an optimal reflux ratio.
Ind. Eng. Chem. Res., Vol. 37, No. 2, 1998 703
Acknowledgment The authors are grateful to Graduate School of Advanced Materials and Chemical Engineering at Hanyang University for a fellowship.
eff: effective value of property i, k: component number j: stage or section number t: total
Literature Cited Nomenclature a: interfacial area on a stage, m2 c: number of components C: cost, $ cp: specific heat, kJ/(kg °C) Cp: specific heat, kJ/(kmol K) D: distillate flow rate, kmol/h ∆T: sample temperature - surrounding temperature, °C t/T: time/temperature, s/°C e: energy transfer rate, kJ/s F: feed flow rate, kmol/s fR: objective function h: heat-transfer coefficients, kJ/(m2 s K) H: enthalpy, kJ/kmol I: electric current, A k: mass-transfer coefficient, kmol/(m2 s) L: liquid flow rate, kmol/s Le: Lewis number, kd/dcpDAB m: mass, kg N: mass-transfer rate, kmol/s P: pressure, mmHg q: heat flow rate, kcal/s Qc: condensing load of the condenser, kcal/h QB: heating requirement of the reboiler, kcal/h RG: gas constant, kcal/(kmol K) Ra: reflux ratio of an atmospheric pressure column Rv: reflux ratio of a vacuum column T: temperature, K t: time, s V: vapor flow rate, kmol/s V: voltage, V x: liquid phase composition y: vapor phase composition Greek Letters ψ: heat-transfer correction factor Φ: mass-transfer correction factor λ: latent heat, kcal/kmol η: viscosity, cP Subscripts av: average property
Cho, Y. S.; Joseph, B. Reduced-Order Steady-State and Dynamic Models for Separation Processes : Part I. Development of the Model Reduction Procedure. AIChE J. 1983a, 29, 261. Cho, Y. S.; Joseph, B. Reduced-Order Steady-State and Dynamic Models for Separation Processes : Part II. Application to Nonlinear Multicomponent Systems. AIChE J. 1983b, 29, 270. Edgar, T. F.; Himmelblau, D. M. Optimization of Chemical Processes; McGraw-Hill: New York, 1988. Henley, E. J.; Seader, J. D. Equilibrium-Stage Separation Operations in Chemical Engineering; John Wiley & Sons Inc.: New York, 1981. Holland, C. D. Fundamentals and Modeling of Separation Processes; Prentice-Hall: Englewood Cliffs, NJ, 1975. Kelly, R. M.; Rousseau, R. W.; Ferrell, J. K. Design of Packed, Adiabatic Absorber: Physical Absorption of Acid Gases in Methanol. Ind. Eng. Chem. Process Des. Dev. 1984, 23, 102. Krishna, R.; Standart, G. L. Mass and Energy Transfer in Multicomponent Systems. Chem. Eng. Commun. 1979, 3, 201. Krishnamurthy, R.; Taylor, R. A Nonequilibrium Stage Model of Multicomponent Separation Processes: Part I. Model Description and Method of Solution. AIChE J. 1984a, 31, 449. Krishnamurthy, R.; Taylor, R. A Nonequilibrium Stage Model of Multicomponent Separation Processes: Part II. Comparison with Experiment. AIChE J. 1984b, 31, 456. Krishnamurthy, R.; Taylor, R. Simulation of Packed Distillation and Absorption Columns. Ind. Eng. Chem. Process Des. Dev. 1985, 24, 513. Perry, R. H.; Chilton, C. H. Chemical Engineers Handbooks, 5th ed.; McGraw-Hill: New York, 1973. Reid, R. C.; Prausnitz, J. M.; Sherwood, T. K. The Properties of Gases and Liquids; McGraw-Hill: New York, 1977. Schweitzer, P. A. Handbook of Separation Techniques for Chemical Engineers; McGraw-Hill: New York, 1979. Seader, J. D. The Rate-Based Approach for Modeling Staged Separation. Chem. Eng. Prog. 1989, Oct, 41. Sherwood, T. K.; Pigford, R. L.; Wilke, C. R. Mass Transfer; McGraw-Hill: New York, 1975.
Received for review February 17, 1997 Revised manuscript received November 3, 1997 Accepted November 4, 1997X IE9701534
X Abstract published in Advance ACS Abstracts, December 15, 1997.
