I n d . E n g . Chem. Res. 1988,27, 1198-1203
1198
in contact with the calcining gases per unit length is given by 6Q,t A = (Yavgsin LY + (1/2)KV2t2)dppp (d) Calciner Specifications. The dimensions are slope = 16 mm/m and rpm = 2.38 min-l; other dimensions are specified in Figure 1. (e) Data Used for Simulation. The data are as follows: C, = 0.276(900 - 1000k),0.21(45012) kcal/(kg K); C, = 0.271 kcal/(kg K); d p = m; hf = 20 kcal/(m2 K h); K , = 9 X lo7 h-l; U,, = 45 kcal/(m2 K h), near the inlet end = 30 kcal/(m2K h), for the rest of the calciner (7); ef = 0.19; eg = 0.20 (for the first 6 m), 0.70 (for the rest of the calciner); e,, = 0.50; AH = 75 kcal/kg of trona; X = 540 kcal/kg; and pp = 10989 kg/m3.
Literature Cited Coulson, J. M.; Richardson, J. F. Unit Operations, 3rd ed.; Pergamon: Oxford, 1978; Vol. 2, pp 715-717. Dumont, G.; Belanger, P. Rf. Znd. Eng. Chem. R o c . Des. Dev. 1978, 17, 107-117. Green, E. W. “Conversion Factors and Miscellaneous Tables”. In Chemical Engineers’ Handbook; Perry, T., Ed.; McGraw-Hill: New York, 1984; pp 1-26. Kim, N. K.; Lyon,J. E.; Suryanarayana, N. V. Znd. Eng. Chem. R o c . Des. Deu. 1986, 25, 843-849. Manitius, A.; Kurcyusz, E.; Kawecki, W. Ind. Eng. Chem. R o c . Des. Deu. 1974, 13, 132-142. Schofield, F. R.; Glikin, P. G. Trans. Znst. Chem. Eng. 1962, 40, 183-190. Received for review September 14, 1987 Revised manuscript received February 16, 1988 Accepted March 5, 1988
SEPARATIONS Mass Transfer in a Kuhni Extraction Column Ani1 Kumar and S t a n l e y H a r t l a n d * Department of Chemical Engineering and Industrial Chemistry, Swiss Federal Institute of Technology, Zurich, Switzerland
Concentration profiles in the Kuhni column were simulated by using the stagewise backflow model to obtain values of true mass-transfer coefficients. The correlated values are presented in terms of the drop Reynolds number. For given interfacial conditions, these depend only on the agitation intensity within the column, so scale-up is facilitated if the variation in backmixing with column diameter is known. T h e effects of the shape of the equilibrium curve, number of optimized parameters, and number of stages employed on the mass-transfer coefficient are also described.
A quantitative relationship between system properties, process variables, and equipment characteristics is necessary for prediction of extractor performance. Precise mathematical models, employing parameters which can be calculated from physical properties and operating variables, can be written to describe a system; the parameters can then be independently examined experimentally. In this paper, mass transfer in a Kuhni extraction column is studied. Analysis of the experimental data involves the evaluation of mass-transfer coefficients using the measured parameters, especially concentration profiles, for a given system and column dimensions. Once the mass-transfer coefficient is known, then using operating variables for that particular column geometry concentration profiles can be simulated and the column may be designed. The true number of transfer units is obtained from models which take into account the axial mixing effects. Axial mixing flattens the concentration profiles and reduces the driving force available for mass transfer. Jumps in the inlet concentrations are also obtained when axial mixing is present. Five parameters are involved: the extraction factor, separation factor, number of transfer units, and Peclet numbers in both phases. Simplified models consider the dispersed liquid phase as a continuum in which the concentration is constant over any cross section, the conditions for which are discussed 0888-5885/88/2627-1198$01.50/0
by Miyauchi and Vermeulen (1963). They are fulfilled for a strongly coalescing system and also in noncoalescing systems if drop sizes are uniform and the distribution coefficient, holdup, and mass-transfer coefficient are constant. Dispersion and stagewise models which take the effects of axial mixing into account are the most *idely used. Normally for preliminary design estimates, analytical solutions or simple graphical methods can be exployed. A more precise approach, however, requires numerical solutions, which are also necessary when more than one solute is present, the phases are mutually soluble, the equilibrium relationship is nonlinear, and axial variation in hydrodynamic properties is to be taken into account. This study makes use of the stagewise backflow model in which the column is divided into N hypothetical stages (which is not necessarily equal to the actual number of stages). The stages are perfectly mixed, but equilibrium is not assumed (Steiner and Hartland, 1980). In addition to the forward flowing streams, two backflow streams are defined due to entrainment and eddies. The backflow model has been studied in detail by Young (19571, Sleicher (1960), Miyauchi and Vermeulen (1963), and Hartland and Mecklenburgh (1966). The two backflow streams are defined in terms off and g, the amounts of heavy phase and light phase flowing 0 1988 American Chemical Society
Ind. Eng. Chem. Res., Vol. 27, No. 7, 1988 1199 backwards, expressed as fractions of the inlet flows. If the column consists of a series of perfectly mixed stages, the number of stages (N) for the model may be conveniently taken as equal to the actual number in the column. However, a greater number of stages is usually required when axial mixing is high. Mathematical treatment becomes more complex as the number of chosen stages increases due to an increase in the number of equations to be solved. In practice, a number between 20 and 200 is chosen. Assuming f and g to be constant over the contactor length, a material balance around the nth stage of height h, for the continuous x phase gives fVcXn-1 + Vc(1 + f)xn+lfvcxn- (1+ f)vcxn+ KocaAh,(Xn* - X n ) / p c= Ah, dX,/dt (1- t) (1) which under steady-state conditions become
and similarly for the dispersed phase
where
The boundary conditions under steady-state conditions obtained from material balances around the end stages are for the first stage: (4) ud
-[Yin hm for the Nth stage:
+ gY2 - (1 + g ) Y J = - -rlt
(5)
(7)
Numerical Solution of Backflow Model Boundary iteration may be used to solve this set of finite difference equations (Mecklenburgh and Hartland, 1969). However nonconvergence occurs when backmixing is present in both phases or has a low value (Mecklenburg and Hartland, 1967). The direct matrix solution is therefore preferred. The set of 2N material balance equations in the linear equilibrium relationship case and 3N (material balance and equilibrium) equations in the nonlinear equilibrium relationship case can be solved simultaneously in matrix form. The normal iterative procedure of Newton-Raphson is commonly used to solve the block matrices. Ricker et al. (1981) extended the solution of McSwain and Durbin (1966) to multistage operation and solved the equations in three diagonal matrix form. An extension of the same method for partially miscible solvents in three diagonal matrix form was given by Spencer et al. (1981), who used the method of Marquardt (1963) to find the value of the chosen parameter by iteration, using the least-squares fit of the calculated and experimental concentrations. Numerical solution of the backflow model may be obtained more efficiently by writing the material balances and in terms of a new variable, 2 (defined by X n =
Table I. Dimensions of Kuhni Extraction Column Type E 150/18 column diameter, Dk,m active column height, Hk,m no. of compartments compartment height, H,,m stator plate (fractional free area), 6 shrouded six-bladed turbine: diameter, D,, m height H,, m end sections: length HE, m
diameter, DE,m packing (in bottom end section)
0.150 1.260 18 0.07 0.235, 0.40 0.085 0.010
0.600 (upper) 0.550 (lower) 0.200 Melapack (Sulzer AG, Switzerland)
Y , and ZZn(Steiner, 1984)). Here, the resulting system of equations, for both phases, can be’expressed in the form of a pentadiagonal matrix. Conte (1965) gives an algorithm for the solution of the pentadiagonal matrix without consideration of the unoccupied elements. The calculated number of theoretical stages using plug flow can be used as the initial guess to obtain the concentration profiles for the linear equilibrium case. These concentrations can then be used to get the concentration-dependent values of the distribution coefficient m, in each stage. The matrix is now solved repeatedly for different numbers of transfer units until the difference in the value of the distribution coefficients from two consecutive iterations becomes less than some acceptable value, e.g., C A m l N I In the model version used (Fischer, 1983), the real process is closely approached. The model allows for the variation of the hydrodynamic properties along the length of the column, the use of a curved equilibrium line, and the presence of end sections. Inlet Effects. The top and bottom end sections of the column are designed differently from the main part of the column to facilitate coalescence and reduce entrainment. As shown in Table I, the diameter of the end sections is 33% more than the actual column diameter. The end sections may also contain special packing on which entrained drops coalesce and return to the column, thus increasing ita capacity. The different mass-transfer rates in the end sections were obtained by adding stages to the active part of the column following the procedure of Fischer (1983). The mass-transfer coefficients could be calculated from the measured concentrations at the end of the active column section together with the known inlet and outlet concentrations. It is also assumed that the backflow of dispersed phase in the bottom section is equal to that in the first stage and the backflow of the continuous phase in the top section is equal to that in the Nth stage. Material balance equations analogous to that for the active column can now be written. The output matrix from the active part of the column consists of 2(N- 2) equations; the first and the Nth stages are left out as the end concentrations X I , Y l , XN, and YN are known. Additional matrices can be written for the end sections at the top and bottom of the column. Solution Procedure The values of the hydrodynamic parameters (drop size, drop size distributions, holdup, axial mixing) are obtained experimentally. The experimental concentration profiles are then used together with the experimental values of E, and Ed (expressed in terms off and g ) or those obtained from a correlation to solve the matrix and obtain values of the product Koca. If the variation of holdup and drop
1200 Ind. Eng. Chem. Res., Vol. 27, No. 7, 1988
size along the column length is known, the true value of the mass-transfer coefficient may then be obtained using actual values of the specific area a. The method described above requires only one parameter optimization. In the absence of experimental axial mixing results, the parameter optimization feature of the program may be used to simulate values of K,, f , and g. The reliability of the results obtained from multiparameter optimization is limited by the interdependence of the parameters. The equations constituting the matrix are nonlinear due to the dependence of distribution coefficient on concentration. They are first linearized and then solved as follows: (i) Assume a number of hypothetical perfectly mixed stages. (ii) Assign numbers of hypothetical stages to the top and bottom sections. (iii) Enter the following values: column dimensions, throughput, phase ratio, holdup in different stages, drop size in different stages, values of concentration in different stages, estimated number of theoretical stages (NTS), average (linearized) value of the distribution coefficient, physical properties of the system, concentration dependence of the distribution coefficient, and values of the axial mixing coefficients E, and Ed. (iv) Calculate backflow factors f and g, drop size, and holdup in each stage by linearization. (v) Define iteration limits and required precision. (vi) Solve matrix to obtain concentration profiles starting with linearized equilibrium curve and estimated number of theoretical stages (NTS) as initial guess for actual number of transfer units, T. Solve pentadiagonal matrix for chosen initial values of parameters assuming parameters are concentration independent. (vii) Obtain chosen parameter t (or t , f , and g) by using a nonlinear optimizingsubroutine (e.g., Marquardt (1963)). (viii) Use concentrations so obtained to determine concentration-dependent parameters. If deviation in the new distribution coefficient mnmwfrom the previous value lies below the defined limit,
C abs (mnew- m o l d ) / N
I
calculate deviation predicted from experimental concentration values. Goodness of fit y is then given by comparing simulated and experimental concentration values at all k measurement points: 1 k =
5 2
IXi,aim
- Xi,expl
XL,exp
- YL,erpl + IYwimYi,exp
(8)
If deviation is unacceptably large, resolve matrix system using mn,new* (ix)Use optimized value o f t to evaluate the value of the mass-transfer coefficient, KO,= t V,p,/ah,.
Experimental Section A pilobscale experimental arrangement, shown in Figure 1, was used for study of the extraction process in a 150mm-diameter Kiihni extraction column. The inlets and outlets of the column are connected to four glass tanks each of 500-L capacity. Tanks TIand T2were used for the aqueous phase and T, and T, for the organic phase. The feed tanks (TIand T4)provided with heaters, coolers, and thermostats maintain the temperature of the inlet streams at 20 i 0.5 "C. The flow rates through the rotameters are automatically controlled by flow controllers using photocells to locate the position of the rotameter floats. The liquid streams are filtered before entry to the column.
Figure 1. Flow diagram for Kuhni extraction column.
Figure 2. Typical Kuhni extractor stages. 1, Bearing; 2, turbine; 3, shaft; 4,perforation; 5, shaft support.
Detailed specifications of the extraction column are given in Table I. The column consists of 18 compartments, each 70 mm high. Stator plates with different free cross-sectional areas (23.5% and 40%) were used to influence the operation of the column. Agitation in each compartment was achieved with 85-mm-diameter shrouded six-bladed turbines. Figure 2 shows a typical stage. Two glass sections each of 200" diameter constitute the end sections of the column. The column operation was studied both with solute transfer taking place and in the absence of solute transfer. The system used for study was o-xylene-water during no solute transfer and o-xylene-acetone-water for studies during mass transfer, o-xylene being always the dispersed phase. Agitation intensity was varied between 100 and 300 rpm in steps of 40 rpm. For this column diameter and design and high interfacial tension system, agitator speeds lower than 100 rpm were not found to be of practical significance, and for speeds higher than 300 rpm, problems due to emulsion formation were encountered. The phase ratio V,/V, for experiments without mass transfer was varied in a wide range (typically between 1:5 and 5:l). However, under solute-transfer conditions, the phase ratio v d / v, was kept constant at 1 5 1 , to avoid a pinch point occurring. However, some experiments were also performed at extreme phase ratios of 1to 5 and 5 to 1,with mass transfer taking place.
Results Theoretical stages were stepped off between the straight operating line and the curved equilibrium line for a volumetric phase ratio of Vd/V, = 1.5. These were used as initial estimates in the computer program for optimizing
Ind. Eng. Chem. Res., Vol. 27, No. 7, 1988 1201 Table 11. Comparison of Mass-Transfer Coefficients from Simulation of Concentration Profiles for Curved and Linear Equilibrium Lines eauilibrium line K,. ka/(m2.s) Tlm 2.22 x 10-2 8.99 (i) curved (ii) linearized in operating range 2.11 x 10-2 8.58 (iii) linearized through origin 1.13 X 4.62
the mass-transfer coefficients. During mass transfer from the disperse phase to the continuous phase, large drops with low interfacial area per unit volume lead to a smaller number of stages than when solute is transferred in the opposite direction. During solute transfer from the continuous phase to the dispersed phase, solute equilibrium is quickly established between the drops and the continuous phase film between them, whereas solute transfer continues over the rest of the drop surface. This creates an interfacial tension gradient which opposes the forces causing drainage in the film (Marangoni, 1871, 1878), thus reducing coalescence. During solute transfer in the opposite direction, the increased concentration of the solute in the film between the two drops creates an interfacial tension gradient which complements the drainage forces, thus enhancing coalescence of the drops. For both types of plates, throughput has negligible effect on NTS at low agitator speeds, whereas at higher rpm’s NTS increases with increasing throughput. For all agitator speeds, however, NTS decreases near the flooding point. A maximum of four stages per meter column length was observed for the case of mass transfer from the dispersed phase to the continuous phase. When the mass-transfer direction is from the continuous phase to the disperse phase, the number of theoretical stages initially increases up to 7.5 per meter with agitation but lower values occur at higher speeds due to increased axial mixing. A larger number of theoretical stages can be obtained if’plates with smaller free area (which lower the axial mixing) are used. This, however, reduces the capacity of the column and increases the residence times.
Parameter Estimation (Simulation of Concentration Profiles) A Hewlett-Packard (HP9825 Model) desk-top computer with 64-kbyte memory was used for simulation of the concentration profiles taking the end effects into account. The relative error between the simulated and experimental concentrations was determined at all It measurement points by using eq 8. Choice of Equilibrium Curve. The solution to the model equations is greatly simplified if the equilibrium curve can be approximated by a straight line. Experimental concentration profiles were simulated to see the effect of varying the shape of the equilibrium line, three forms being considered: (i) slightly curved; (ii) straight line approximation, Y * = m X + C, in operating region; and (iii) straight line approximation, Y * = m X ,passing through the origin. Optimized values of the mass-transfer coefficient obtained in this way are compared in Table 11, which shows good agreement between the first two cases. Deviations for the third case become more critical as the operating line approaches the equilibrium line at higher agitation intensities. Effect of Number of Hypothetical Stages. To see the effect of the chosen number of stages on model fit, the column was divided into different numbers of hypothetical stages. The values of the mass-transfer coefficient K, alone were optimized with the experimental values, the
E 4
t
0
A B
0.20.40.60.8 T Height C-I Exp. 430, 4 - X , D - Y , c: d=l: 1. 5 B = 6 m3/m2h, N = 220rpm d32= 0.79 m m , holdup = 16. 49% CI = 1.27E 03 m2/m3 Ec = 3. 08E-04 m2/s (experimental) Ed = 6. 00E-04 m2/s (experimental) K x = 5. 56E-03 k g / m Z s , T= 12. 92
241 u 6
’*
3
0
t
I
I
B
0.20.40.60.8 T H e i g h t [-I Exp. 430, < - X , D - Y , c: d=l: 1. 5 B = 6 m3/m2h, N = 220rpm d32= 0. 79 mm, h o l d u p = 16. 49% o = 1. 27E 03 m2/m3 Ec = 5. 70E-04 m2/s ioptlmized) E d = 1. 26E-03 n2/s (optimizedl K X = 5. 93E-03 k g / d s , T= 13.78
Figure 3. Simulated concentration profiles taking axial variation of holdup and drop size into account. Reduced height of active column section measured upward is shown. (a, top) One-parameter optimization using experimental values of axial mixing coefficients. (b, bottom) Three-parameter optimization.
parameters E, and Ed (Kumar et al., 1984) held constant. For this comparison, average values of holdup and volume/surface mean diameter d32of the drop size distribution were used. This relates both the ratio of the dispersed-phase solute capacity to the mass transfer and the gravitational driving force to the drag force acting on the drops. The results are presented in Table 111. The computing time for the desk-top computer used is proportional to the chosen number of stages. It is seen from Table 111 that the values of K, agree within f 5 % for N 1 24. The number of hypothetical stages ( N )must be greater than P,/2 and Pd/2to ensure that the backmixing factors f and g used in eq 1-6 are positive; otherwise, the results have no physical meaning. The number should be large enough to accurately reproduce the measured profile, but very large values of N should be avoided as much more computer time is then required. Concentration profiles for the present work were simulated using N = 36. Number of Optimized Parameters. The program also allows the optimization of axial mixing coefficients. Three-parameter optimization gives a better fit of the experimental and simulated profiles than one-parameter
1202 Ind. Eng. Chem. Res., Vol. 27, No. 7, 1988 Table 111. Effect of Chosen Hypothetical Number of Stages on Mass-Transfer Coefficient for E, = 3.08 X a n d Ed = 6.00 X m2/s re1 IO~K,, computno. of stages kg/(mZ.s) re1 error, % (eq 8) ing time 12 24 36 12 144
6.11 5.83 5.80 5.96 5.90
2.69 1.55 1.63 1.39 1.50
1 2 3 6 12
optimization as demonstrated in Figure 3 in which the experimentally measured concentrations are indicated by open triangles. The relative error defined by eq 8 is 0.98% and 1.6070, respectively. As discussed by Wolschner (1980), Spencer et al. (19811, and Fischer (1983), the concentration profiles are not sensitive to the eddy diffusion coefficients, but values of the eddy diffusion coefficients determined from the concentration profiles are very sensitive to small errors in the experimental measurements, particularly in the end sections, and thus will tend to differ from the experimentally measured values of E, and Ed. Parts a and b of Figure 3 are thus similar despite the fact that the values of E, and Ed differ considerably for the two cases. Eddy diffusion is, however, probably the controlling factor in the important question of scale-up provided the flow regime in the larger column is similar to that in the smaller column and is not affected by other factors such as dead zones and macroscopic circulation of the phases.
True Mass-Transfer Coefficients The backflow model was used to simulate the concentration profiles and calculate the mass-transfer coefficients. The average value of the overall mass-transfer coefficient, K,,,, for the complete column length was then obtained by averaging the values obtained in each stage. Masstransfer coefficients obtained in the top and bottom sections are not included in K,,,,: N
-
E Ko,n
n=l
KO,= N
where
in which
a,, =
6~/d32
local values of the holdup and mean drop size are interpolated from the experimental measurements along the column. Higher values of the mass-transfer coefficients were observed during mass transfer from the organic (dispersed) phase to the aqueous phase. These higher values are a result of increased mass-transfer rates in drops of bigger sizes due to the presence of oscillations created by coalescence between the droplets which is enhanced by the Marangoni (1871, 1878) effect as explained above. In Figure 5, the mass-transfer results are presented in terms of dimensionless numbers Sh, and Re, where
in which Dc is the diffusion coefficient of acetone in water and V , is the relative velocity between the phases.
C
=
cont. phase
-
(water)
D = disp. p h a s e (0-xylene)
100 100
101R e ,
4
r
102
4
.
103
4
Figure 4. Dependence of overall Sherwood number on Reynolds number in the Kiihni column. (1) Models of Rozen and Bezzubova (1968) and Garner and Tayeban (1960). (2) Models of Wellek and Skelland (1965) and Higbie (1935). (3) Equation 9. I
'
'
" '
'
'
" '
'
'
" I
Re, r. 5c, c
Figure 5. Relationship between Sherwood number Sh, and Re,&,: comparison of mass transfer in different extractors.
Results when the solute transfer direction was from the continuous phase to disperse phase could be correlated by using the single drop model of Skelland and Wellek (1964) and Higbie (1935) as shown by curve 2 in Figure 4. Equation 9 was obtained by using a least squares fit and correlates the experimental values well when Re, > 6, (curve 3-Figure 4): Sh,, = -40 + 23Re,'I3 (9) I t must be noted that since Re, is based on the drop velocity in a swarm, it cannot be compared to the Reynolds number based on the terminal velocity of a single drop. The Sherwood number obtained by using the models of Rozen and Bezzubova (1968) and Garner and Tayeban (1960), i.e., for oblate-prolate drops undergoing oscillation at higher Reynolds number, is shown by curve 1 in Figure 4 and only poorly correlates the experimental results during mass transfer from the dispersed phase to the continuous phase. The relative high scatter in the results only allows a rough estimate of the Sherwood number, for which eq 10 may be used. Sh,, = 200 + 0.047Re,'.44 for 50 < Re, < 1000 (10) Mass-transfer results obtained in different extraction columns have been presented by Fischer (1983), where the Kiihni column is shown to cover a small range only. These results are for the acetic acid-ethyl acetate-water system. The Kuhni column can, however, cover a far wider range as can be seen in Figure 5 where the results of the present work on the o-xylene-acetone-water system for both
Ind. Eng. Chem. Res., Vol. 27, No. 7, 1988 1203 = real dispersed-phase velocity in column = Vd/E, m/s V, = continuous-phase mean superficial velocity based on empty column cross section, m/s v d = dispersed-phase mean superficial velocity based on empty column cross section, m/s V, = relative velocity, V, = u, + ud, m/s X,,= solute concentration in heavy phase leaving stage n, mass fraction Y, = solute concentrationin light phase leaving stage n, mass fraction 2 = dimensionless column height
mass-transfer directions are also included.
ud
Conclusions Mass-transfer rates achieved in the Kuhni column are usually comparable and may be greater than those obtained in other types of extraction columns. A maximum of 7.5 theoretical stages per meter column length was obtained in a Kuhni extractor with 23.5% free area. The stagewise backflow model predicts the Kuhni column behavior well. Analysis of the concentration profiles shows that the shape of the equilibrium line and axial variation of holdup and drop size should be taken into account for precise modeling. Little error is introduced if a linear approximation of the curved equilibrium line in the operating region is used. Sufficient hypothetical stages should be used to simulate the concentration profiles to ensure the backmixing coefficients are positive. Multiparameter optimization may be used to determine accurate values of the true mass-transfer coefficient from the experimentally measured concentration profiles in the active column section, but the backmixing coefficients may only be so obtained if the concentration profiles in the end sections of the column are also available. The masstransfer coefficient depends on the mass-transfer direction which is affected by the interdrop coalescence but for a given system is only dependent on the interfacial conditions and agitation intensity. Scale-up is thus facilitated if the variation in the backmixing Coefficients with column diameter is known.
Greek Symbols 4 = fractional plate free area pc = continuous-phase density, kg/m3 Pd = dispersed-phase density, kg/m3 k, = continuous-phase viscosity, kg/ (m-s) k d = dispersed-phase viscosity, kg/(m.s) E = dispersed phase holdup y = fractional error between correlated and experimental concentrations
Nomenclature a = specific interfacial area, m2/m3 A = active column area of cross section, m2 C = intercept of equilibrium line dS2= volume/surface mean drop diameter, m D, = diffusion constant, m2/s Dk = column diameter, m D, = agitator diameter, m E, = axial mixing coefficient of continuous phase, m2/s Ed = axial mixing coefficient of dispersed phase, m2/s f = continuous-phase backflow coefficient g = dispersed-phase backflow coefficient = hypothetical stage height, m H = heavy-phase flow rate, kg/s H , = stage height, m H = active column height, m KO,= overall mass-transfer coefficient based on continuous phase, kg/ (m2.s) k = number of experimental points L = light-phase flow rate, kg/s m = distribution coefficient between light phase and heavy phase n = agitator speed, rev/s N = number of stages NTS = number of theoretical stages P, = V&/E,(l - E),Peclet number in continuous phase P d = VdH,/EdE, Peclet number in dispersed phase r, = rate of mass transfer in stage n, l / s Re, = Reynold number based on the relative velocity =
em
Vrd32Pc/Pc
S c , = Schmidt number based on continuous phase = p c / D s c Sh,, = Sherwood number based on the overall phase mass-
transfer coefficient = K,d3,/p$, t = K,,ah,/ V,p,, number of transfer units per stage T = total number of transfer units u, = real continuous-phase velocity in column = V,/(lmls
E),
Subscripts
n = typical stage in the column
in = inlet sim = simulated exp = experimental opt = optimized Superscript
* = equilibrium concentration Literature Cited Conte, S. D. Elementary Numerical Analysis; McGraw-Hill: New York, 1965. Fischer, A. Ph.D. Dissertation 5016, Swiss Federal Institute of Technology, Zurich, 1973. Fischer, v. E. Ph.D. Dissertation 7220, Swiss Federal Institute of Technology, Zurich, 1983. Garner, F. H.; Tayeban, M. Anal. Real. SOC.Espan. Fis. Quim. (Madrid) 1960 B56,479. Higbie, R. Trans. AIChE J . 1935, 31, 365. Hartland, S.; Mecklenburgh, J. C. Chem. Eng. Sci. 1966, 21, 1209. Horvath, M. Ph.D. Dissertation 5774, Swiss Federal Institute of Technology, Ziirich, 1976. Kumar, A.; Steiner, L.; Hartland, S. Proceedings 4th Chemical Engineering Conference, Grado, Italy, 1984. Marangoni, C. Nuovo Cimento 1871, 2, 239. Marangoni, C. Nuovo Cimento 1878, 3, 97. Marquardt, D. W. J. SOC.Ind. Appl. Math. 1963,11, 431. McSwain, C. V.; Durbin, L. D. Sep. Sci. 1966, 1 , 677. Mecklenburgh, J . C.; Hartland, S. IChE Symp. Ser. 1967,26, 127. Mecklenburgh, J. C.; Hartland, S. Can. J. Chem. Eng. 1969,47,453. Miyauchi, T.; Vermeulen, T. Ind. Eng. Chem. Fundam. 1963,2,113, 304. Reissinger, K. H. Proceedings 3rd Chemical Engineering Congress, Graz, Italy, 1982; Vol. I, pp 412-423. Ricker, N. L.; Nakashio, F.; King, C. J. AIChE J. 1981, 27, 277. Rozen, A. M.; Bezzubova, A. I. Theor. Found. Chem. Eng. (Engl. Transl.) 1968, 2, 715. Skelland, A. H.; Wellek, R. M. AIChE J. 1964, 10, 491. Sleicher, C. A. AIChE J . 1960, 6 , 529. Steiner, L. Lectures ETH Zurich, April 1984. Steiner, L.; Hartland, S. Chem.-Zng.-Technol. 1980, 52, 819180. Spencer, J . L.; Steiner, L.; Hartland, S. AIChE J . 1981, 27, 1008. Wellek, R. M.; Skelland, A. H. P. AZChE J. 1965, 11, 557. Wolschner, B. H. Ph.D. Dissertation, Technische Universitat, Graz, Italy, 1980. Young, E. F. Chem. Eng. 1957,2, 241. Received for review March 11, 1986 Revised manuscript received March 20, 1987 Accepted January 26, 1988
