Nomenclature
A E
= = k = m = S =
frequency factor activation energy firsborder rate constant exponent in severity correlation severity factor
GREEKLETTERS = residence time Y = mol of product pler mol of pentene cracked
T
literature Cited
Barabanov, N. L., Muk.hina, T. N., Petrol. Chem. USSR,1,227 (1962).
Eshevska a M. P., Refiner, 19, 264 (1940). Kunugi, Sakai, T., Soma, K., Sasak, Y., Ind. Eng. Chem., Fundam., 8, 374 (1969). Kunzru, D., Ph.D. Thesis, University of Pittsburgh, 1972. Kunzru, D., Shah, Y. T., Stuart, E. B., Ind. Eng. Chem., Process Design Develop., 11,605 (1972). Kunzru, D., Shah, Y. T., Stuart, E. B., Ind. Eng. Chem., Process Design Develop., 12, 339 (1973). Rice, F. O., Polly, 0. L., J . Chem. Phys., 6, 273 (1938). Rice, F. O., Rice, K. K., “The Aliphatic Free Radicals,” Johns Hopkins Press, Baltimore, Md., 1935, p 91-107. Robinson, K. K., Weger, E., Ind. Eng. C i a . , Fundam., 10, 198 (1971). RECEIVED for review September 28, 1972 ACCEPTEDFebruary 26, 1973
5,:
The financial help by Gulf Educational Fund is appreciated.
Mass Transfer in a n Agitated Vessel Brahmi D. Prasher and George B. Wills* Chemictzl Engineering Department, Virginia Polytechnic Znstitute and State University, Blacksburg, Virginia 24061
The applicability of Lamont’s eddy cell model to gas-liquid transfer in turbulent flow in an agitated vessel is examined. According to this model the very small scales of turbulent motion in the equilibrium range are considered to be! controlling the mass transfer process. Gas-liquid absorption data in an agitated vessel indicate that the rate of surface renewal in the expression for the mass transport coefficient is predictable on the basis of the easily measured hydrodynamic quantities, viz., the rate of energy dissipation and the liquid kinematic viscosity, as predicted by the Lamont model.
v a r i o u s models have been offered to predict mass transfer around gas-liquid interfaces. Most of the models proposed so far are based on the concept of a rigid interface or an interface where some sort of surface renewal occurs through the displacement of liquid a t the interface or a combination of these concepts. Surface renewal theories presented so far have received greater acceptance for free surface mass transfer. However, in the case of most of these models, whether those based on the concept of a rigid interface (e.g., Lewis and Whitman, 1924), or those based on the concept of surface renewal (e.g., Higbie, 1935; Danckwerts, 1951; Kishinevsky, 1955, 1956) or those based on the modified film-penetration theories (e.g., Dobbins 1956; Marchello and Toor, 1963; Toor and Marchello, 1958, Harriott, 1962), the hydrodynamic parameters in them cannot yet be determined without recourse to observations of mass transfer rates. Certainly, if a theoretical basis for determining any of these hydrodynamic parameters in terms of easily measured hydrodynamic quantities could be found, then the value of the model would be infinitely increased. I n recent years there have been a number of attempts to relate mass transfer coefficients with the conditions of agitation in stirred vessels in both gas-liquid and solid-liquid systems. I n the case of solid-liquid mass transfer, a number of workers (Calderbank, 1961; Harriott, 1962; Keey, 1966; Sykes and Gomezplata, 1967; Nienow, 1969; Miller, 1971; Levins, 1972; Levich, 1962; Middleman, 1965) have tried to explain the mechanics of particle-liquid mass transfer by in-
voking Kolmogorov’s theory of local isotropic turbulence, though some of them found that some additional parameters are necessary to predict k ~ All . approaches based on this theory of local isotropy make the implicit assumption that equal power input per unit mass is the right method to be used in mixer scale-up. As regards to gas-liquid transfer in agitated vessels, the results of the investigation of the dependency of the mass transfer coefficient on the conditions of agitation are so unclear that further investigations in this direction are necessary. Calderbank’s data (1961) indicate that the liquid-phase mass transfer coefficient in gas-liquid transfer is independent of agitation intensity. Hyman (1962) also reported that the mass transfer coefficient is relatively constant, even a t high rates of shear; Davies, et al. (1964), studied gas absorption rates in exceedingly clean water in a stirred cell and found that the mass transfer rates are dependent on the agitation intensities. Yoshida (1963) working with different types of impellers also found that the mass transfer coefficients in agitated vessels are dependent on the agitation intensity. It is the purpose of this paper to examine whether the energy input in a gas-liquid system in agitated vessels affect the liquid side mass transfer coefficient and also to examine if any model can be used to predict the mass transfer coefficients on the basis of easily measurable hydrodynamic quantities. Eddy Cell Model
Calderbank (1961) working with particle-liquid transfer in Ind. Eng. Chem. Process Des. Develop,, Vol. 12, No. 3, 1973
351
agitated vessels empirically derived a relationship for the mass transfer coefficient k~ in terms of energy of dissipation having the form
kL = 0 . 2 ~ ~ ~ - 2 / 3 ( ~ ~ ) 1 / 4
(1)
Lamont (1966, 1970) developed an eddy cell model, postulating that mass transfer a t both fixed (gas-solid) and free (gas-liquid) interfaces is mainly dependent on the motions of small-scale eddies in the equilibrium range of turbulence. The small eddies, in spite of their low energy, cause mixing within the very surface of a large eddy. Lamont solved the Navier-Stokes equation for small Reynolds numbers using the appropriate boundary conditions a t the free interface and by considering a spatially periodic solution for the velocity field. The resulting velocity distributions were used to determine the concentration distribution of a solute in the velocity field. Lamont's calculations give the following relationships for the mass transfer coefficients a t fixed and free interfaces, respectively.
kL a
Nso-1/2(~~)1/4
(3)
The gas was sparged from the bottom of the tank through a '/*-in. diameter orifice. For the case of the absorption of carbon dioxide in the nonreacting system, the concentration of carbon dioxide in the bulk of the liquid is maintained near zero by introducing caustic into the tank a t the same rate as the physical absorption is taking place. For the case of absorption of carbon dioxide in 0.085 M caustic the concentration of caustic is maintained by introducing concentrated caustic a t the same rate that it is depleted in the tank. The whole procedure involves trial and error and the absorption time for both situations is kept as short as 1 min after a steady state is achieved as regards to the dynamics of bubbles in the tank, so as not to alter significantly the physiocheinical properties of the solutions. The mass transfer coefficients are evaluated by making use of the experimental ratio of the rate of absorption of COz in caustic to the rate of absorption in water. A computer program, using a numerical technique similar to the one used by Pearson (1962-1963), was used to obtain the theoreticalvalues of this ratio of the rates of absorption based on the penetration theory. This ratio R should be given by a function of the following form
The value of the constant of proportionality in eq 2 and 3 is approximately 0.4, so that eq 2 and 3 can be written as
k~
N
0.4Ns,-2/a(e~)1/4
k~
N
0.4Nsc"'2(ev)1'4
(4)
respectively. Ranerjee, et al. (1968), considering an idealized situation in the eddy cell, obtained a similar result to Lamont, except that the value of the constant is equal to 1.0 in their case. If either the Danckwerts surface renewal model or the Higbie penetration model, both of which give an exponent 0.5 for the diffusivity dependency, is assumed, then comparison of these models and eq 5 result in the following expression for the relationship of the rate of surface renewal s (or contact time t ) with E and Y . s = (4/?rt)1'2= 0 . 1 6 ( e / ~ ) ~ ' ~
(6)
Thus if eq 6 is a reasonable depiction of the rate of surface renewal, then the surface renewal for many hydrodynamic situations can easily be predicted since e and Y are easily determined hydrodynamic quantities. This, in our opinion, is the chief attractiveness of the Lamont eddy cell model, in addition to the fact that it is a more realistic model which takes into account the fluctuating turbulent velocity fields near the interface. Experimental Apparatus and Procedure
The absorption experiments involving the absorption of carbon dioxide in distilled water and 0.085 M strength caustic solutions was conducted in an 11.5 in. diameter plexiglass vessel. The relative dimensions of the vessel and the turbine impeller were similar to that used by Rushton, et al. (1950), and Calderbank (1958). A flat six-bladed radial-flow impeller was chosen. The overall diameter of the impeller used was one-third the diameter of the tank. The dimensions of each of the blades are D/4 and D/5, respectively. The average height of the liquid in the tank was kept the same as the diameter of the tank. The impeller was placed centrally in the tank a t one impeller diameter above the tank bottom. The vessel was equipped with four symmetrically mounted baffles, whose width corresponded to a tenth of the diameter of the tank. 352 Ind. Eng. Chem. Process Des. Develop., Vol 12, No. 3, 1973
R
=
k ~ ' / k=~f(D.4, DB, t )
(7)
With the experimental values of R obtained for similar dynamic conditions in the case of absorption in the reacting and nonreacting cases, the contact time t was obtained from the plot of the curve of theoretical value R us. t. Using this contact time t the liquid mass transfer coefficient was calculated for absorption in water according to the equation k~
=
(8)
2 ( D ~ / d 'I2 )
The absorption experiments were conducted using five different agitation speeds ranging from 150 to 350 rpm and also five different gas superficial velocities ranging from 0.29 to 1.2 cni/sec. Further details of this work are given elsewhere (Prasher, 1970). Estimation of Energy of Dissipation
For a fully baffled tank, the energy of dissipation per unit mass of liquid in an agitated system in turbulent flow is given by CNaDK eo =
-
T2H
The constant C has been shown to be virtually independent of of the ratio D I T for turbine impellers under the conditions of baffling generally employed. Since the agitator-tank system used in this case is nearly similar to the one used by Rushton et al., a value of C of 8.0 is quite appropriately used in this case. The effect of gas flow resulting in the decrease of energy input in the system can easily be estimated using the correlations given by Calderbank. Applying this correction, e can be written as
z = 8.0 N3D6@ T2H
where $J =
1.0 - 1.26 N -!?-D a
Q < 3.5
for aN - D
X 10-2
or
(11)
9 = 0.62
-
Q 1.85-3 ND
Q > 3.5 X for 3ND -
I
9 8 -
5
-$0 546 7 ;
s *I I
1-1
40
60
I00
200
400
1
1
I
I
1000
2000
4000
I
-
3 -
0
-
N
0
J
I
I
I
I
I
.
EO. (13)
X
Y
x/i I
I
2 -
-
-
z cm?/sec? Figure 1 . Plot of liquid mass transfer coefficient vs. average energy dissipation for the absorption C02 in water in an agitated tank Results and Discussiori
Figure 1 shows the mass transfer coefficients plotted against ;estimated from eq 10. Examination of the graph in Figure 1 indicates that k~ increases as Z increases, which is also indicated by the equation given by the Lamont model. If the exponent of 0.25 is assumed for ;to indicate the relationship of l c and ~ ?, then eq 5 from Lamont's eddy cell model may be written as
k~
==
CYDA~/~(C/V)~/~
(12)
where CY is a constant to be determined from experimental data. The constant CY in eq 12 was determined by a least-squares linear regression analysis procedure. The result of this analysis gives k~ = 10.592D~''*( e/v)".25
(13)
and the value of CY is 0.592 0.033 within the 95% confidence interval. The analysis of variance for the data gave an F ratio of 891 for 1 and 23 degrees of freedom, which indicates that the results obtained :&revery highly significant. Figure 2 represents a plot of k~ us. D ~ ~ ~ ~ ( e / vwhere ) " * D112(e/v)0.26 ~~, has been rearranged slightly. The data for this work correlate well with eq 13. The mean deviation for the data from this correlation is of the order of 14%, the maximum deviation being less than 3196. The data of this work indicate that the value of a for the eddy cell model is close to the value given by Lamont as compared to the value given Banerjee, et al., obtained from an idealized case. Hence it can be assumed without too much risk that the value of 0.4 is likely to be more correct than the value of 1.0. I n fact, 11s has been shown elsewhere from gas absorption data in another equipment by one of the authors of this paper (Prasher, 1972), CY is almost definitely equal to or less than 0.4. The value of a obtained in the present work is higher than 0.4 possibly due to the gross inequality in the distribution of the energy of dissipation in the agitated vessel. Cutter (1960) using a turbine impeller similar to the relative dimensions of the one used in the present case, found that for flow in the bulk of the vessel (e/;) = 0.26. In the impeller stream, which occupies 10% of the tank volume, (e/?) = 7.7. If the mass transfer is assumed to occur preponderantly in and around the impeller stream (and this is reasonable to assume since a number of workers optically found that the interfacial area is higher in and around the impeller stream than elsewhere in the bulk), then assumption of the energy of dissipation by a factor of 5 or 6 of the average dissipation should almost account for the difference in the value of CY obtained from 0.4. ii
2
I
I
I
I
3
4
5
6 7 8910
l
l
l
l
I 15
(D,/V)"~ (Cv)0'25x IO2 (cm./sec.)
Figure 2. Plot of
kL
vs. ( D A / V ) ~ . ~ ( Z V )for O . *the ~ absorption of
C02 in water in an agitated tank number of workers (Levins and Glastonbury, 1972) have noticed that k~ increases with the D I T ratio and, therefore, for scale-up purposes k~ should have the form
k~
= f(?, DIT)
(14)
limitations
(1) It has already been pointed out in the preceeding discussion that energy dissipation in an agitated vessel is grossly nonhomogeneous. Though the Lamont theory predicts that k~ is dependent on c, for design purposes k~ would have to depend on some other parameters like the D I T ratio as has been indicated by Levins and Glastonbury (1972). (2) I n the present study, the gas has been sparged from a single orifice in the bottom of the vessel and hence the gas has accumulated in large globs under the impeller, where it is sheared by the impeller into smaller bubbles. One would guess that the mass transfer rate has not been significantly affected by the sparging process, since the transfer area associated for the sparging is small and the gas velocities involved are rather large. But sparging through a sieve plate having multiple orifices could make the analysis more complicated. (3) The gas rate into the vessel influences the rate of energy uptake by the impeller. The greater the gas rate, the lower the energy dissipation in the vessel and hence the lower would be the value for k L as is shown by the Lamont model. (4) The gas rate also affects the bubble sizes and dynamics. Rut if the Lamont model is assumed to be reasonably realistic, and hence mass transfer in the liquid is affected preponderantly by the smaller eddies which are much smaller in scale than the gas bubbles, then the size of the gas bubble should not be a very critical parameter for the estimation of kL.
Conclusions
The following conclusions can be offered on the basis of this study. (1) Mass transfer coefficients in gas-liquid systems in agitated vessels are dependent on the energy input into the system. (2) The eddy cell model of Lamont is a good model for predicting mass transfer rates on the basis of easily estimable hydrodynamic parameters e and Y . (3) While the eddy cell model is extremely realistic in describing mass transfer phenomena in terms of the eddies in the equilibrium range, it must be recognized that energy dissiInd. Eng. Chem. Process Des. Develop., Vol. 12, No. 3, 1973
353
pation is not homogeneous in most contacting equipment. Also the energy dissipation around the gas-liquid interface is probably a fraction of that in the bulk. Therefore, some other factor, as D I T ratio in the case of agitated systems, must be considered for purposes of predicting k~ and equipment scaleUP * Nomenclature
D DA
diameter of tank, cm molecular diffusivity of solute gas, cm2/sec Dg = molecular diffusivity of reactant, cmz/sec H = height of liquid in tank, cm kL = liquid side mass transfer coefficient in nonreacting case, cm/sec kL‘ = liquid side mass transfer coefficient in reacting case, cm/sec N = agitator revolutions per sec N s c = Schmidt number = V ~ D A Q = gas rate into tank, cm3/sec R = ratio defined by eq 7 s = random surface renewal rate, sec-’ T = tank diameter, cm = =
(196.1\
GREEKLETTERS
\ - _ _ _
constant in eq 12 e = energy of dissipation by turbulence per unit mass, cmk/sec3 go = average energy of dissipation by turbulence per unit mass without gas flow in tank, cm2/sec ; = average energy of dissipation, cm2/seca v = kinematic viscosity, cm2/sec @ = factor defined by eq 11 ?r = 3.1459 CY
Cutter, L. A., Thesis, Columbia University, New York N. Y., (1960). Danckwerts, P. V.,Znd. Eng. Chem., 43, 1400 (1951) Davies, J. T., Kilmer, A. A., Ratcliff, G. A., Chem. Eng. Sci., 19, 583 (1964). Dobbins, W. E., “Biological Treatment of Sewage and Industrial Wastes, W. E. McCabe and W. W. Eckenfelder, Ed., Reinhold, New York, N. Y . , 1956. Harriott, P., Chem. Eng. Sca., 17, 149 (1962). Harriott, P., AIChE J., 8,93 (1962). Higbie, R., Tram. Inst. Chem. Eng., 35,365 (1935). Hyman, D., “Advances in Chemical Engineering,” T. B. Drew, Ed., Academic Press, New York, N. Y., 1962. Keey, R. B., Glen, J. B., AIChE J., 12,401 (1966). Kishinevsky, M. K., J . Appl. Chem. USSR., 28,881 (1955). Kishinevsky, M. K., Serebryansky, V. T., J . A p p l . Chem. USSR, 29,29 (1956). Lamont, J. C., Ph.D. Thesis, University of British Columbia, 1966. Lamont, J. C., Scott, D. S., AZChE J., 16,513 (1970). Levich, V. G., “Physiochemical Hydrodynamics,” Prentice-Hall, Englewood Cliff, N. J., 1962. Levins, D. M., Glastonbury, J. R., Trans. Inst. Chem. Eng., 50, 132 (1972). Levins, D. M., Glastonbury, J. R., Chem. Eng. Sci., 27,537 (1972). Lewis, W. K., Whitman, W. G., Znd. Eng. Chem., 16,1215 (1924). Marchello, J. M., Toor, H. L., Znd. Eng. Chem., Fundam., 2, 8
=
literature Cited
Banerjee, S., Scott, D. S., Rhodes, E., Znd. Eng. Chem., Fundam., 7,22 (1968). Calderbank, P. H., Trans. Znst. Chem. Eng., 36,443 (1958). Calderbank, P. H., Moo-Young, M. B., Chem. Eng. Sci., 16, 39 (1961).
Middleman, S., AZChE J., 11,750 (1965). Miller, D. H., Ind. Eng. Chem., Process Des. Develop., 10, 365 i\ Ai .av ‘7A i, .)
Nienow, A. W., Can. J . Chem. Eng., 47,248 (1969). Pearson, J. R. A., Appl. Sci. Res., Sec. A , 11, 321 (1962-1963). Prasher, B. D., Ph.D. Thesis, Virginia Polytechnic Institute and State University, 1970. Prasher, B. D., Chem. Eng. Sci., in press. Rushton, J. G., Costich, E. W., Everett, H. J., Chem. Eng. Progr., 46.395 (1950). Sykis, P., Gomeeplata, A,, Can. J . Chem. Eng., 45, 189 (1967). Toor H. L., Marchello, J. A., AZChE J., 4,97 (1958). YoshTida, F., Miura, Y., Znd. Eng. Chem., Process Des. Develop., 2,263 (1963). RECEIVED for review October 4, 1972 ACCEPTEDJanuary 31, 1973
Separation of Aromatics and Naphthenes by Permeation through Modified Vinylidene Fluoride Films F. P. McCandlerr Department of Chemical Engineering, Montana State University, Bozeman, Monl. 59715
T h e penetrant flux of a compound through a membrane is considered to be due to a combination of two effects: (1) its diffusivity arising from its size or molecular configuration; and (2) the change in chemical potential between the two sides of the membrane. I n the case of permeation of a n organic material through a film, the change in chemical potential is proportional to the solubility of the material in the film if a vacuum is maintained on the downstream side. Thus, the selectivity of a membrane is usually greatly influenced by the relative solubility of the components in the film. I n view of this, it seemed appropriate to investigate the inclusion of an aromatic extraction agent as a plasticizer in a relatively 354 Ind. Eng. Chem. Process Des. Develop., Vol. 12, No. 3, 1973
impermeable polymer in a study of the separation of aromatics and naphthenes by vapor permeation. This would be attractive because a relatively small amount of extraction agent could be used to separate a much larger volume of feed. Experimental Section
Apparatus. A diagram of the test cell is shown in Figure 1. It was fabricated from two stainless steel blank flanges, 6 / * in. thick and 4.5 in. in diameter. Two cavities were machined in the flanges for the feed circulation and membrane support. The membrane was supported by a porous stainless
