Ind. Eng. Chem. Process Des. Dev. 16, Interim Rept. No. 3, Book 2, July 1964-Mar 1968a. Curran, 0. P., Fink, C. E., Gorin, E., "Phase 11: Bench Scale Research on CSG Process--Operatlon of the Bench Scale Continuous Gasification Unl", OCR R and D Rept. No. 16, Interim Rept. No. 3, Book 3, Dec 1967-July 1968b. Curran, G. P.. Clancey, T. J., Fink, C. E., "Development of the COP Acceptor Pr0cess Directed toward Low Sulfur Boiler Fuels", Consol. Coal Co., Library, Pa., Res. Div. APTD 1157 EPA EHSD7115 (Nov 1971). Curran, G. P., Clancey, J. T., f'asek, C. E., "Production of Clean Fuel Gas from Bituminous Coal", Consol. Coal Co., Library, Pa., Res. Dlv. EPA Report No. 65012-73-049, EPA EHSD71 15 (Dec 1973). Dedman, A. J., Owen, A. J., Trans. Faraoky Soc., 58, 2027 (1962). Durai-Swamy, K.. Che. S., Knell, E., Green, N. W., Zahradnlk, R., Abstr., Am. Chern. SOC. Div. Fuel Chem. Prepr., 24(2), 177 (1979). Hoke, R. C., Berbarid, R. R., Nutkls, M. S.,Kinzler, D. D., Ruth, L. A,, Gregory, M. W., "Studies of the Pressurized Fluidlzed-Bed Coal Combustion Process", Exxon Research arid Eng. Co., Prepared for EPA, Contract No. 66-02-1312 and 66-02-1451 (1976). Jones, J. F., Schmid, M. R., Eddinger, R. T., Chem. Eng. Prog., 60(6), 69 (1964). Jones, J. F., Eddlnger, R. T., Seglin, L., Chem. Eng. Prog., 62(2), 73 (1966). Jones, J. F., Schoemann, F. H., Harnshar, J. A., Eddinger, R. T., "COED", FMC Corp., Chem. Res. and Dev. Center, Princeton, N.J., 1971. Neavel. R., Exxon Research arid Engineering, Baytown, Texas, personal communication, 1978. Pell, M., Ph.D. Thesis, C i University of New York, New York, N.Y., 1971. Prlestley, J. J., Cobb, J. W., Gas J., 182, 95 (1928).
1980, 19, 653-665
653
Squires, A. M., Graff, R. A., Pell, M., Chern. Eng. Pmg. Symp. Ser., 67(115), 23 (1971). Strom, A. H., Eddinger, R. T., Chem. Eng. Prog.,67(3), 75 (1974). Suuberg, E. M., Sc.D. Thesis, Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, Mass., 1977. Suuberg, E. M., Peters, W. A,, Howard, J. B., Seventeenth Symposium (International) on Combustion", p 117, The Combustion Institute, Ptttsburgh, Pa., 1979. Turkdogan, E. R., Olsson, R. G., Wriedt, H. A,, Darken, L. S., Trans. Soc. Mining Eng., AIM€, 354, 10 (1973). Vestal, M. L., Essenhigh, R. H., Johnston, W. H., Am. Chem. SOC. Div. Pet. Chem. Prepr., 15(4), A153 (1970). Yeboah, Y. D., Sc.D. Thesis, Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, Mass., 1979. Yergey, A. L., Lampe, F. W., Vestal, M. L., Day, A. G., Fergusson, G. L., Johnston, W. H., Snyderman, J. S., Essenhigh, R. H., Hudson, J. E., Ind. Eng. Chem. Process Des. Dev., 13, 233 (1974).
Received for review February 4, 1980 Accepted May 23, 1980 Financial support for this research was provided by the United States Department of Energy under Contract No. EX-76-A-012295, Task Order No. 27, and is gratefully acknowledged.
Analysis of Batch Crystallizers Narayan S. Tavare and John Garside' Department of Chemical and Biochemical Engineering, University College London, London WC 1E 7JE, United Kingdom
Madhav R. Chivate Deparlment of Chemical Technology, University of Bombay, Matunga, Bombay 4000 19, India
The analysis of crystal size distribution (CSD) in batch operated crystalliiers is considered for various types of operation (cooling, evaporation, and dilution) using moment transformation of the population balance. Semiempirical power law nucleation kinetics independent of magma properties and size-independent or linear sizedependent power law growth kinetic models are assumed. For different modes of operation solution of the population balance coupled with mass balance and moment equations enables the CSD to be defined as a function of time and size. The analytical procedure is illustrated by numerical examples.
Introduction With the development of population balance theory the analytical description of continuous crystallizers has become well established. Many industrial operations, however, are carried out in batch crystallizers. Such systems are useful in small-scale operations, especially when working with chemical systems which are difficult to handle due perhaps to their toxic or highly viscous properties. They are simple, flexible, require less investment, and generally involve less process development. Supersaturation in batch crystallizers is usually generated by one of three methods and analysis of the three corresponding cryst(a1lizer types is similar. In cooling crystallizers supersaturation is generated because of the reduction in solubility with temperature; the volume of the system remains approximately constant. In evaporative crystalkers supersaturation is produced by loss of solvent with the subsequent reduction of volume with time; the solubility of the salt remains almost constant as the operation may be assumed isothermal. In dilution crystalh e r s supersaturation is generated by the added diluent reducing the solute solubility and the volume of the system 0196-4305/80/1119-0653$01 .OO/O
consequently increases with time. An understanding of crystal size distribution (CSD) in batch crystallizers is of great importance since this interacts strongly with the method of operation and determines the end uses of the product. However, analytical information regarding CSD from batch crystallizers is very limited. Previous work (e.g., Mullin and Nyvlt, 1971; Jones and Mullin, 1974; Jones 1974; Tavare, 1978) has generally been restricted to the investigation of a particular system in a given type of operation. The object of the present paper is to develop a generalized analytical technique by which the CSD from different types of batch crystallizer operated under different modes can be evaluated. Normally batch operations are flexible but it is rather difficult to decide on optimum operating conditions to achieve the required product specifications. Nevertheless, recent studies on batch dilution and evaporative crystallizers (Tavare, 1978; Tavare and Chivate, 1979a; Tavare et al., 1979) operating in a given mode with specified constraints and objectives have shown that it is possible to assign, a priori, values to the operating variables in order to achieve better crystallizer performance. The scope of the present
0 1980 American
Chemical Society
654
Ind. Eng. Chem. Process Des. Dev., Vol. 19, No. 4, 1980
paper does not extend to these optimization procedures but is restricted to the presentation and generalization of analytical methods for evaluating the batch population density function with respect to time and size. Several numerical examples are included to illustrate application of the equations to specific crystallizer designs. Kinetics of Crystallization Nucleation. Formation of new crystals may result from any one, or a combination, of different nucleation mechanisms (primary homogeneous or heterogeneous and secondary nucleation) or by attrition. The nucleation rate per unit mass of solvent will be expressed by the semiempirical equation
The nucleation rate constant, kb, depends on a number of variables. Temperature, hydrodynamics and perhaps trace amounts of impurity may affect the nucleation characteristics of the system. Equation 1 is conventionally used to represent nucleation rates in industrial situations. For systems where secondary nucleation is important, the nucleation rate constant may be modified and should normally show the effect of terms such as energy input per unit volume of suspension and some moment of the CSD. Power law terms of impeller tip speed and suspension density may be incorporated in the rate constant of the present model. For simplicity the effects of such terms could be lumped together with the nucleation rate constant using average values of the variables over the batch time. Growth. The overall growth rate will be expressed by the empirical equation dL G = - = k,A@ dt In general, the overall growth rate constant, k,, depends on variables such as temperature, crystal size, hydrodynamics, and impurities in the system. In some inorganic systems the growth rate depends on crystal size and this may be represented by many empirical equations. In the present work, both the size-independent growth rate model (eq 2) and a simple linear size-dependent model (Canning and Randolph, 1967) represented by G = G(Ac)G(L)= k,'Acg(l + a L ) (3) are used. Other empirical size-dependent models could be linearized over the size range of interest. Population Balance Moment Equations. As the working volume of a batch crystallizer may be time varying, it is convenient to define the population density function, A , based on the total solvent capacity a t any time as ii = nS (4) For the perfectly mixed batch crystallizer with negligible attrition and agglomeration the population balance equation reduces to aii - + - =anG () (5)' at aL To simplify solution of this equation a new variable, y, will be defined such that y = J t G ( t ) dt and so
(6)
t =
XY&
(7)
The variable y can be thought of as the size-of a crystal at any time t, which was originally nucleated at time t = 0. With y as a variable, the population balance for a crystal system in which growth rate is independent of size becomes
an - + - = an o
aL The moment equations obtained by moment transformation of eq 8 are ay
For the linear size-dependent growth kinetic model (eq 3) a new variable, x , may be defined as
so that
For this case the population balance equation with x as a new variable is aii aii - (1 f f L ) ffii = 0 ax: aL and the moment equations are dN _ - ii(x,O) dx
+ +
+
_ dA - 2kaL + 2aA dx
E = dx
+ 3aw
Batch Population Density Function. To determine the batch population density as a function of time and size, the population balance equation must be solved under some known initial conditions. Becker and Larson (1969) solved this equation for zero initial moments by defining the generalized moment equations, inverting them by Dirichlet's formula for multiple integrals and comparing their results with a generalized definition of moments. They also suggested that the solution may be obtained by the method of function characteristics. Tavare and Chivate (1973) used the Laplace transform technique to define the solution of the partial differential equation under different known functional forms of the initial conditions. Their results are here extended to a more generalized functional form of initial conditions and are reported in Table I. Two cases for size-independent growth kinetics are presented in Table I. The first case (la) corresponds
Ind. Eng. Chem. Process Des. Dev., Vol. 19,No. 4, 1980 655
Table I. Batch Population Density Functionn initial conditions for y or x
for L
population density, n ( y , L )
1. Size-Independent Growth Rate: G a.nO(oJ0) 0 no(O,Lo)u(w + Lo) b. n(0,L) n(y,O)S(y) n(O,-w) [l - u ( w ) ] t
With y as a new variable this may be modified to dAc - + -dc* + - - = o1 d W (23) dY dY dY where
s
n(w,o)s(w)u(w) 2. Size-Dependent Growth- Rate: G ( A c ) G(L) a. E o ( o , L o ) 0 no(O,Lo) [G(Lo)IGW)Iu(z) b. no(O,Lo) n(x,OM3:) no(O,~o)[G(~o)IG(L)l +
W , O )S(z)u(z 1 G(L) n w = y - L ; z = x - ( l / a r ) l n [ G ( L ) ] ; G ( L ) = ( l t aL).
to seeding with crystals of uniform size, Lo,with negligible nucleation. The initial population density function is assumed to have a point value of iio (0, Lo)at size Lo and will move with respect toy, i.e., t, as shown in Table I. For the second case (lb) an initial size distribution, ii (0, L ) , and nuclei population density, nCy, 0) S Cy), are assumed. Derivation of the batch population density as a function of the variable y and size L for this generalized set of initial conditions is presented in the Appendix and the results are given in Table I. Two similar cases for linear size dependent growth kinetics (2a and 2b) are also included in Table I. This evaluation of the population density function in terms of x or y reduces the time dimension into the size domain as defined by eq 6 and 13. An analytical solution of the population balance equation is then sometimes possible in the time domain. In subsequent sections the relationship between time and these new variables will be defined which in turn enables the batch population density to be determined with respect to time and crystal size. Cooling Crystallizers Batch operated perfectly mixed:ooling crystallizers are widely used in the chemical industry. Systems having a large positive temperature-solubility coefficient are normally crystallized in this way. The temperature-solubility relationship may be expressed by a variety of empirical equations. Examples are those of the Arrhenius type c* = c'exp(-4H/R8) N c'(1 - AH/R8) (20) and use of a constant temperature coefficient over the operating range dc* _ (21) d8 - k, The rate constants appearing in eq 1-3 will generally be functions of temperature. In the present work, changes in these rate constants are assumed to be insignificant. If the operating temperature range is sufficiently large to produce substantial changes in the rate constants a different procedure might have to be used to take account of the effect. Batch cooling crystallizers are flexible and can be operated with several different cooling modes. Some of these will be considered here. It will be assumed throughout that the working volume of the system is constant. Natural Cooling. Although in practice most cooling crystallizers are operated in this way, an analytical solution becomes intractable because of its complexity. The supersaturation balance for this mode is dAc dc* 1dW = 0 - -t - + -dt dt S dt
and the temperature coefficient (dc*/d8) may be obtained from either eq 20 or 21; dW/dy may be evaluated by numerical solution of the moment equations. With the assumption of constant population density of nuclei, no, an analytical solution may be possible. However, this assumption is somewhat questionable and will only be valid if changes in supersaturation are small or the relative kinetic order is close to unity. With the assumption of constant no, dW/dy may be represented by
Using the above equation the relationship between y and t can be estimated by solving the supersaturation balance (eq 23) numerically. This solution of y with respect to t defines the population density as a function of time. If the system growth kinetics are governed by eq 3 the analy_sismay be carried out in an analogous way, the values of d W/dx being evaluated by solving the moment equations (eq 16 to 19) numerically, or analytically if constant iio is assumed. If numerical values are used in the moment equations the analytical solution may be greatly simplified. Constant Cooling Rate. In this case the cooling rate is kept at a constant value determined by the constant level of heat removal. This type of operation was studied by Ayerst and Phillips (1969) for the ammonium perchlorate-water system. They reported the supersaturation variation with time and the final CSD at different cooling rates. Analysis of this mode is analogous to natural cooling operation. The only change necessary is in evaluating the time rate of change of temperature (deldt), which now becomes constant and equal to ke. When the solubilitytemperature curve has a constant temperature coefficient (k,) over the range of interest, the supersaturation balance modifies to d Ac dt 1dW kok,-=0 dY dY s dY dW/dy may be determined by solution of the moment equations and the supersaturation balance equation can then be solved to define the relationship between y and t. This in turn may be used to calculate the population density as a function of time. Example. Derive the relation between y and t, and hence calculate the population density as a function of size at the end of the batch time, for an unseeded batch cooling crystallizer operated at constant cooling rate given the following specification. (a) System kinetics: G = 104Ac m/s; B = 106Ac2no./s kg of solvent; (b) physical parameters: p = 2 X lo3 kg/m3; k, = 2.5 X kg of solute/kg of solvent K; ko = -1/60 K/s; k, = 0.5; k, = 3.5; 7 = lo4 s; (c) initial conditions: So = 1.0 kg of solvent; Aco = 0.01 kg of solute/kg of solvent; moments = 0.0. To simplify the analysis it will be assumed that changes in supersaturation are negligible with respect to y; i.e., dAc/dy = 0. This will probably be a realistic assumption at higher values of time due to the inherent self-regulating character of batch crystallizers. From the system kinetics the initial nuclei population density
+
+
Ind. Eng. Chem. Process Des. Dev., Vol. 19, No. 4, 1980
650
Assuming the nuclei population density to be constant at this value, eq 24 gives
-d W- - 6kVpfio0y3= 0.5 x 2 x dY
6
103 x 1010~3 = 1013y3kg / m
Assuming dAc/dy = 0, eq 25 gives dt _ dY
-
Substituting this value into eq 27 gives d4c -+ml=O dY4 where m l= 6kvpKRyGi-1.The solution of eq 29 for c will be Y4 24
c - co = -ml-
-1013~3 dW 1 dY Sk&% (1.0)(-1/60)(2.5 X
2.4
X
1017y3s/m
Solving for t withy = 0 at t = 0, t = 6 X 1016y4s or y = 6.38 X 10-5t0.25m. Thus at t = 104s,y = 6.38 X m. Under the above assumptions the population density at t = lo4 s can be calculated from Table I as A = 10'Ou (6.38 X lo4 - L ) no./m. Note that as defined by eq 4, ri is based on the total solvent capacity. Constant Level of Supersaturation, It is well known that batch crystallizers operated under uncontrolled conditions generally yield a nonuniform product of poor quality. The crystallizer should, therefore, be operated under controlled conditions. As the supersaturation level is of paramount importance in determining the performance characteristics it is desirable to operate the crystallizer at a controlled level of supersaturation. This level of supersaturation has a strong influence on the product CSD and may alter the quality, habit, and purity of the product crystals. Mullin and Nyvlt (1971) demonstrated the potential utility of this mode of operation by crystallizing potassium and ammonium sulfate from aqueous solutions and showed that the mean crystal size was always higher than that obtained with uncontrolled conditions. In their mathematical development the nucleation process was represented in the supersaturation balance by a series of uniform monodisperse pulses, each comprising the number of new crystals formed during a burst of nucleation. Using this approach, Jones and Mullin (1974) analyzed batch cooling crystallizers operated under different programming modes and compared their analytical results with those obtained from experiments for the potassium sulfatewater system. A similar configuration was also analyzed by Tavare and Chivate (1973) assuming a continuous nucleation rate while Jones (1974) presented the theory of controlled cooling crystallizers based on moment transformation of the population balance coupled with material and energy balances. In this section the temperature-time relationship necessary to achieve a constant level of supersaturation over the period of the batch is calculated and the crystal size distribution determined. The mass balance with y as a variable may be represented as dc dW s-+-=o dY dY Using moment equations and combining them with the mass balance it can be shown (Tavare and Chivate, 1973) that d4c S- + 6kvpSn(y,0)= 0 (27) dY4 If the nucleation and growth kinetics are represented by eq 1and 2, respectively, the nuclei population density may be written as
+ "(0)-Y36 + E(0)-Y22 + C(0)y
(30)
and therefore dc _ - -ml-Y3 + F(0)-Y2 + c(0)y + c(0) dY
6
2
(31)
For a constant level of supersaturation -d(Ac) - - d(c - c*) = O (32) dt dt and therefore dc* - dc _ (33) dY dY Using the temperature-olubility relationship as given by eq 20 c'AH d8 -dc= - - dc* = - -d6 (34) R82 dy dy do dy which on substituting eq 31 and solving for 8 under 8 = 80 at y = 0 (i.e., t = 0) gives
y is defined by eq 6 and for constant supersaturation becomes y = Gt
(36)
Equation 35 is the generalized cooling curve for a crystallizer operated at constant supersaturation for a system with a solubility curve described by eq 20. If the initial CSD is known, the initial moments will be determined from which the initial values c(O), c(0) and C(0) can be calculated with the aid of the mass balance and moment equations. However, if the initial moments are arbitrarily assumed to be zero, the values of c(O), c(0) and C(0) will be zero and after using the final conditions (Of, tf) to eliminate the constant, the cooling curve simplifies to (37)
If the system has a constant temperature-solubility coefficient (eq 21) the generalized cooling curve may be derived as
+
E ( @ - Y2 + C(0)y (38) 6 2 24 which with the arbitrary assumption of zero initial moments and utilizing the final operating conditions takes the form m1y4 + ~ ( 0Y3 )-
k,(8 - 80) = --
(39)
Ind. Eng. Chem. Process Des. Dev., Vol. 19, No. 4, 1980 657
The cooling curve represented by eq 39 was suggested by Mullin and Nyvlt (197:l) for this simplified programmed cooling operation. All these solutions suggest that the initial cooling rate should be slow but should then increase with time. In the case of natural cooling obeying Newton's cooling curve, however, the initial cooling rate is normally high and decreases with time. If the growth kinetics are governed by eq 3 the growth rate arising from the supersaturation, G(Ac) will remain constant throughout the operation. The mass balance equation with x as a variable is dc + dW=0 s-dx
clx
where x is defined by eq 13 and which for this case is x = G(Ac)t
(41)
No = Lo =
A,
Lmiio dL = iiook
smri& 0
dL= iio0k2
= k , i m q , L 2dL = 2k,iiO0k3
W o= k v p i m i i & 3 dL
= 6k.,pii2k4
For k = m and using the given values of ago,k,, k,, and p , No = lo2;Lo = lo4 m; A. = 7 X m2; and Wo =6X kg. Initial derivatives of c are now determined from the mass balance (eq 26) and moment equations (eq 9-12), again for k = lo-@m
If the solubility curve for the system is given by eq 21 over the operating range the mass balance relation modifies to d0 Sks,dx
dW +=0 dx
lo-" kg/kg m
Integrating eq 42 with respect to x under 0 = 0, and W = Woat x = 0 (i.e., t = 0) gives the generalized cooling curve for the system under this mode
-Sk,(19 - 0,) = W - Wo
kg/kg m2
(43)
where the values of W !would be defined by solving the set of moment equations (eq 16-19) numerically with suitable initial conditions. However, if the population den_sity of nuclei, Eo, is constant the analytical solution of W is
-6
X
lo5 kg/kg m3
The cooling curve can now be evaluated from eq 38 ml = 6kvpiioo/S= 6
G = 10-6Ac =
X
10l3kg/m4kg
m/s for Ac =
kg/kg of solvent
and so
No + t-o+- A,) -+ no0
9a3
CY'
CY
ka
exp(3ax)
]+
-400(2.5 X 1012y4+ 105y3+ 3
Wo exp(3ax) (44)
with the value of the nuclei population density, iioo,being defined as
K h [ G (Ac) ] '-'S
: I
X
10-3y2+ 6 X 10-l'~)K
From eq 36, y = m and (0 - 0,) = ( t 4 + 4t3 + 12t2 24t) K. At 7 = 2 X lo4 s, y = 2 X m and the population density function for k = lo-* m is
+
(45)
Use of numerical values may simplify these analytical derivations and it would in general be better to start the derivations with given numerical values. After substituting eq 44,45, and 41 in eq 43 and finding initial moments from the initial CSD, eq 43 will represent the temperature variation with time within the system necessary to maintain the desired constant supersaturation. Example 2. Derive the cooling curves which will maintain a constant level of supersaturation and calculate the final CSD obtained in a seeded batch cooling crystallizer. Use the specification given for Example 1 except for the following: 7 = 2 X lo4 s; CSD of seed crystals given by i i o = ii: exp(-Llk)., where a,," = lolo no./m. Consider the cases for k = 0, and m. The initial moments are first calculated from the initial CSD, iio = iiooexp(-L/k). Thus
1010
4 2 x 10-4 - L)
The cooling curves for this and other initial CSDs are plotted in Figure 1 and the corresponding population density function in Figures 2a-b. Also included for comparison are the curves calculated for a supersaturation level of 0.0075 kg/kg of solvent, ii,," = 0.75 X 1O1O no./m and k = 10" m. The dominating effects of both the initial CSD and level of supersaturation on the required cooling curves and the product CSD are clearly seen. Example 3. Derive the cooling curve and product CSD for an unseeded batch cooling crystallizer operated at a constant level of supersaturation. Assume the specifications given in Example 2 except for the growth kinetics which are now G = 104(1 + 104L)Ac m/s with L in m. From the system kinetics, G(Ac) = 1O*Ac = lo4 m/s; G(L) = 1 + CYLwhere CY = lo4 m-l; iioo= SB/G(Ac) = 10l2Ac=
658
Ind. Eng. Chem. Process Des. Dev., Vol. 19, No. 4, 1980
which is plotted in Figure 2c. Cumulative undersize weight percent distributions representing the final CSDs for the conditions discussed in Examples 2 and 3 are shown in Figure 2d. These were calculated by integrating the corresponding population density functions over the size range 0-2000 pm using a step length of 1pm. The effect of different conditions on the mass median size and the shape of the distribution can be clearly seen.
-
-10
-Y
Evaporative Crystallizers
:I
a, a,
-20
-- ----
G
I
\
lo-' m/s (Example 2 )
G = 0 75 x
mls (Example 2 1
G = 10-8 ( 1 + i o L L ) m/s (Example 31
-30 I 0
I
I
I
I
5
IO
15
20
Time, t , xI0-l
I
(SI
Figure 1. Time variation of solution temperature required to give a constant level of supersaturation-the programmed cooling curve (Examples 2 and 3).
1O1O no./m. As the supersaturation and system volume remain constant, the nuclei population density, noo,will also remain constant. Under this condition with zero initial moments W is obtained from eq 44 as
3 x 0.5 x 2 x 103 x 1010 (104)4 104x 1 -+ exp(104x)- 2 exp(2 x 1 0 4 ~ ) 3
91 exp(3ax)
)
=
+
Supersaturation may be generated in a crystallizer by evaporation of solvent, and systems having a flat solubility curve are normally crystallized in this way. Larson and Garside (1973) have analyzed such a crystallizer while Tavare and Chivate (1977) showed that results for a batch evaporative crystallizer handling potassium sulfate were analogous to those of cooling crystallizers under otherwise similar conditions in different modes of operation (Mullin and Nyvlt, 1971;Jones and Mullin, 1974;Jones 1974). The study was extended to cover a wide range of control variables, and suitable values of operating variables were suggested in order to achieve better performance when a larger size of the seed with minimum CV of the product was desired (Tavare and Chivate, 1979a). Two modes of operation will be considered here: first that where the evaporation rate is constant, and second for conditions where a constant level of supersaturation is required. As supersaturation is generated by evaporation of solvent there may be significant changes in the system volume which consequently becomes time varying. The crystallizer contents are assumed to be well mixed, operation is isothermal, and since the systems handled have flat solubility curves, the solute solubility remains essentially constant during the run. Constant Evaporation Rate. In practice this mode of operation is most frequently encountered in industry. The energy input to the system is kept at a constant value so that constant evaporation rate might be achieved. The supersaturation balance equation with y as a variable is now dAc dY
-d S = - - d S dt = k,-dt
1 - exp(3 x 10-4t) 9
dy
Using eq 43 the required cooling curve is
i.e. exp(W4t)
+ 1 exp(2 x
This cooling curve is plotted in Figure 1. The CSD at = 2 X lo4 s is given from Table I as ii
= [l0lo/(1 + 1o4~)lU[2.Ox
dW dY
where
+ e ~ p ( l o - ~-t )-21 exp(2 x 10-4t) +
dS dY
-+ c * - + - = o
From eq 41, x = 1 0 3 and so
-
+
7
In (1 104L)] no./m
dt dy
dy
The supersaturation balance equation can be solved numerically to determine the relationship between y and t. This relationship is then used to define the population density as a function of time and size. Example 4. Derive the relation between y and t for an unseeded batch evaporative crystallizer operated at constant evaporation rate. Hence determine the final batch population density function. The following conditions are given: (a) system kinetics: G = 10-5Ac m/s; B = 109Ac3 no./s kg of solvent; (b) physical parameters: p = 2.5 X lo3 kg/m3; c* = 0.2 kg of solute/kg of solvent; k , = -5 X kg of solvent/s kg of solvent; k , = 0.5; 7 = lo4 s; (c) initial conditions: So = 1.0 kg of solvent; Aco = 0.0075 kg of solute/kg of solvent; moments = 0. As in Example 1, we assume dAc/dy = 0. The reservations noted earlier therefore also apply to this solution. The nuclei population density based on the initial supersaturation level is ii: = lOI4Ac2 = 5.6 X lo9 no./m.
Ind. Eng. Chem. Process Des. Dev.,Vol. 19, No. 4, 1980 30
30
_--_--seed
-
I
I
I
I
------
CSD
-
product CSD
I
859
I
seed CSD i k
=
mi
product CSD
20
20
IC
IC
-
C
C
IO
10
I
b
I 0
I
0
200
I
I
I
I
400
600
800
1000
0 1:
!
I
0
200
Crystal size, L (pml
30
L
I
I
I
I
I
I
LOO
600
800
1000
1200
1000
1200
Crystal size, L ( prn )
I
Unseeded ,
k
:0
Size independent growth
/ i Example 2 I 20
IC
C
IO
C 0 Crystal size,
L lkm)
200
LOO
600
800
Crystal size, L ( p m l
Figure 2. Crystal size distributions at t = 2 X lo4 s: a, effect of seed size (Example 2); b, effect of supersaturation (Example 2); c, effect of size dependent growth (Example 3); d, product cumulative weight distributions (Examples 2 and 3).
Assuming this value to be constant throughout the run, eq 24 gives -d W- - 6kvpfio0Y3= 0.5 x 2.5 x 103 x 5.6 x 109y3 = dy 6 7 X 1012y3kg/m Substituting into eq 46
therefore 7 x 10'2 dt _ y 3 = 6.75 X 1017y3s/m dy (0.0075 + 0.2)(!5 x which can be solved with y = 0 at t = 0 to give t = 1.68 X 10"y4 s or y = 4.9 X 10-5t0.25 m. The population density a t T = lo4 s therefore becomes ii = 5.6 X 109u(4.9 x - L)no./m, which is of identical form to that calculated in Example 1. Note that in this case the volume of solvent varies with time and n = i i / S where the value of S at any time is given
by S = So (1 + k,t). A t T = lo4 5, S = S ( T )= 0.5 kg of solvent and so n = 2 n no./m kg of solvent. Constant Level of Supersaturation. In this mode, the level of supersaturation is maintained constant by adjusting the evaporation rate. To evaluate the required variation of solvent volume with time the mass balance, coupled with the moment equations, needs to be solved. The mass balance equation for this case is dS d W c-+-=o (47) dY dy As the system is operated at constant supersaturation both the growth rate and the population density of nuclei will remain constant throughout the batch time. If the mass balance is coupled with the moment equations it can be shown that
d4S + 4a4S = 0 dY4
(48)
where 4a4 = 6kvpKRyGi-'/c.Solution of this equation using S = So at y = 0, i.e., t = 0 gives
000
Ind. Eng. Chem. Process Des. Dev., Vol. 19, No. 4, 1980
S(y) = So COS (ay) cash (UY) + 2a2S(O) - S(0) cos ( a y ) sinh ( a y ) +
[ [
4a3
2a2S(O) + 3(0) 4a3
] ]
I
1 .o
I
I
I
I
I
I .oo
-
I
-cn sin ( a y ) cosh (ay) +
S(0) 2a2 while the relationship between y and t is given by eq 36. If the initial CSD is known, initial moments may be evaluated and from the moment equations, and mass balance values of the initial solvent mass derivatives can be determined. Thus eq 49 along with eq 36 defines the time variation of volume needed to maintain a constant supersaturation over the duration of the batch, while the energy input necessary to maintain the constant supersaturation is given by
0.8
0.98
v)
-Y" v)
- sin ( a y ) sinh ( a y ) (49)
where the time rate of change of solvent mass can be obtained from eq 49 and 36. If the growth rate kinetics are governed by eq 3, the time variation of solvent required to maintain the system at a constant level of supersaturation will be given by c(S - So) + (W- Wo) =0 (51) where W is defined by eq 44, x by eq 41, and the nuclei population density by eq 45. The energy input requirement is again given by eq 50 in which the time rate of change of volume may be derived from eq 51. Example 5. Calculate the volume variation with time for a seeded batch evaporative crystallizer such that it will operate at constant supersaturation. Also determine the final population density function. The specifications are: (a) system kinetics: G = 10-4Ac2m/s; B = 10"Ac4 no./s kg solvent; (b) physical parameters: p = 2 X lo3 kg/m3; c* = 0.14 kg of solute/kg of solvent; k, = 0.5; k, = 3.5; 7 = 2 X lo4 s, (c) initial conditions: So = 1.0 kg of solvent; Aco = 0.01 kg of solute/kg of solvent; CSD of seed crystals, iio= ii: exp(-L/k), where E," = 10" no./m and k = 0, and m. As in Example 2, the initial moments are first calculated. The required expressions are shown in Example 2 and for m give ITo = lo3,Lo = m, A, = 7 X m2, k= and W o= 6 X 10-l8 kg. Initial derivatives of S can be calculated from the mass balance (eq 47) and moment equations (eq 9-12), which m give for k =
S(0) =
-(
%)Ao
= -4 X
kg/m
c 0.0
\
1
I
I
0
5
IO
k
:0
\
I \
15
and lo-',
I
10.90
20
Time, t , ~ 1 0 (' 5~)
Figure 3. Time variation of solvent capacity required to give a constant level of supersaturation (Example 5).
Le., a = 5.6 X lo3. Further, since G = 10-4Ac2= m/s and y = Gt = 10-8t m (from eq 36) the variation of solvent mass with time can be calculated from eq 49
S ( t ) = cos (5.6 X 10-5t) cosh (5.6 X 10-5t) + 5.7 X cos (5.6 X 10-5t) sinh (5.6 X 10-5t) - 5.7 X sin (5.6 X 10-5t) cosh (5.6 X 10-5t) - 6.4 X sin (5.6 X 10-5t) sinh (5.6 X 10-5t) kg This is plotted in Figure 3 together with the evaporation rate program for the other initial CSDs. For 7 = 2 X 104s, y = 2 X 10-4mand so the final population density based on the total working volume is
1 0 9 ( 2 x 10-4 - ~ ) 4 x2 10-4 - L ) Dilution Crystallizers Supersaturation can be generated by the addition of a diluent to the system. Tavare (1978) analyzed the crystallization of potassium sulfate from aqueous solution using ethanol as a diluent, assuming known size dependent growth rate and power law nucleation rate kinetics. Different operating modes were studied. Two important modes of operation will be analyzed here: constant rate of change of diluent concentration and constant level of supersaturation operation. Since addition of a third component (the diluent) takes place, the crystallizer volume is time varying although in some cases addition of the diluent may be relatively small and volume changes may be neglected. Changes in volume due to mixing will be neglected. Operation of the well-mixed crystallizer is assumed to be isothermal and the kinetics of the system are governed by eq 1 and 2 or 3. The solubility of solute in the original solvent and added diluent will be represented as a function of the diluent concentration by (Mullin, 1972) c* = co* exp(-kdM)
(52)
Other empirical solubility relationships could also be used and treated in an analogous way. Since the diluent is assumed miscible with the solvent, S now represents the capacity of solvent plus diluent. Constant Rate of Change of Diluent Concentration. In this mode of operation, a constant value for the rate of change of diluent concentration (i.e., dM/dt = kM) is maintained. The supersaturation balance equation for the
Ind. Eng. Chem. Process Des. Dev., Vol. 19, No. 4, 1980
system with variable volume and y as a variable is dAC -I-dSc* + -d= W (53) dY dY dY which after integrating with respect to y gives (SAC - SoAco) + (SC* - Soco*) + (W- Wo)= 0 (54) Using eq 54 and 2 in eq 7 it is possible to deduce the relationship between ,y and t as shown in the following example. This relation will enable the population density to be defined as a function of time and size. For size-dependent growth the analysis can be treated in an analogous way. For a constant rate of diluent addition a similar treatment is possible and for the case when volume changes are negligible this is identical with the example described above. Example 6. Derive the relation between y and t for an unseeded batch dilution crystallizer operated with a constant rate of change of diluent concentration. Assume the same kinetics and initial conditions as in Example 1with, in addition: (b) physical parameters: co* = 0.25 kg of solute/kg of solvent; k , = 10.0 kg (solvent and diluent)/kg of diluent; k M = lo4 \rg of diluent/kg of solvent s Assume that volume and solubility changes are negiigible and that the nuclei population density based on the initial supersaturation level is constant throughout the batch. Thus ii? = 10l2Ac= 10'O no./m. As in Example 1, dW/dy = 1013y3kg/m. Now iif S N So, c* N co* and Ac N Aco the supersaturation balance may be modified to
and so dt _ dy
1013,y3 [(0.25 x 9) -t0.01]10-6
= 4.4 x 1oi8y3s/m
Solving with y = 0 a t t = 0 gives y = 3.08 X 10-5t0.25 m. Constant Level of Supersaturation. In order to achieve a constant supersaturation level the diluent addition rate must vary with respect to time. Crystallizers operated in such a way would be expected to yield a better product than those where the supersaturation is uncontrolled. The mass balance equation withy as a variable is now
881
constant supersaturation. In this equation W may be defined from the moment equations, y by eq 36, and the population density of nuclei by eq 28. The volume variation may be described as
S=S0+M
(59)
If the initial moments are assumed to be zero, the volume changes to be negligible during the operation and the initial diluent concentration zero, then the time variation of diluent concentration to maintain constant supersaturation may be derived as exp(-kdM) +
kvpKRGi+3t4 4co* = I
(60)
The situation may be still further simplified by assuming that the solubility curve is linear over the operating range of concentration C* =
~ o * ( l- kdM)
(61)
when the concentration curve takes the form
M-Mo
( t ) =iGx& This is analogous to the simplified cooling curve (eq 39) and represents the concentration variation of diluent required to maintain the system at a constant level of supersaturation. If the growth rate kinetics of the system are governed by eq 3 the mass variation of diluent species may be defined by eq 58 in which W is defined by eq 44 and n by eq 41. Example 7. (a) Derive the relation between diluent concentration and time assuming a constant volume seeded, dilution batch crystallizer to be operated at constant supersaturation. (b) For the same configuration devise a programming policy for the variation of diluent with time assuming a linear solubility relation. Use the data of Example 5 with co* = 0.25 kg of solute/kg of solvent, kp = 10.0 kg (solvent and diluent)/kg of diluent, and Mo = 0. (a) For a constant volume system with finite initial CSD moments the diluent concentration can be calculated by substituting the integrated form of eq 24 into eq 58. co*[l - exp(-kdl/i] =
(55)
As operation is isothermal, the solubility of solute in the original solvent, co*, will remain constant. The working volume is assumed to 'be time varying and the solubility may be assumed as E* = co* exp(-k,'IH)
(56)
Since the supersaturation will be constant it can be shown that (57) Using eq 56 and 57 in the mass balance and integrating W = Wo the resulting differential equation under M = Mo, a t y = 0, i.e., at t = 0 gives Eo* [exp(-kdIH) - exp(--k,'Mo)]
+ ( W - Wo)= 0
Using the initial moments calculated in Example 5 for k = m and noting that y = 1 0 3 gives M_ = .
(58)
Equation 58 represente the generalized variation of mass of diluent species in the system in order to maintain a
1 - 2.4
X
10-17(
&+
+ + t)]kg/kg
Figure 4 depicts the variation represented by this equation together with the corresponding curve for the other initial CSDs. (b) From eq 61 the linearized solubility relation may be written C* = ~ 0 * ( 1- kdM) = ~ o * [l k,(S - So)/S] thus dc* -- co*kpSo dS _ dY
s 2
dY
662
Ind. Eng. Chem. Process Des. Dev., Vol. 19, No. 4, 1980 Time, t
0
, xlO-'
5
IO
I
I
(
0.15
s1 15
I
I
I
I
0.03
20
-
10-2
0 .IO
0.02
m Y
-m
cn"
up
-
IO-)
Y
I
I
v,
v,
10 - I
0.05
0.01
5
0 0
5
IO
Time, t
20
15
, x IO-) ( s )
Figure 4. Time variation of diluent concentration required to give a constant level of supersaturation (Example 7a).
For a constant level of supersaturation the supersaturation balance becomes dS dW -[Ac + ~0*(1- k,)] + - = 0 dY dY Using the moment equations and assuming constant throughout the batch time -d4S _ a4S = 0 dY4 where 6kvpiioo
a4 = -
AC
+ co*(l
-
kp)
1
[
[ "k'
+
a2
a3
1
s;o]
0.5 -- - sin (ay) + 0.5 From the supersaturation balance, the moment equations and the initial moments calculated in Example 5, the initial solvent capacity derivatives may be calculated for k = m S(0) = -
S(0)= S(0) = -
3kvP
6kVp AC + co*(l - k,) 6kvP
AC + co*(l - kp)
- = 2.67
1 1
to = 2.67
X
10-lo kg/m
X
kg/m2
No = 2.67 X lo6 kg/m3
20
0
Time, t, x ~ O - ~ ( S )
As in Example 5, G = lo* m/s; noo= 10" no./m; y = 10-8t m, and 0.5 X 2 X lo3 X 10" 0.01 + 0.25 (1 - 10)
X
Le., a = 4.04 X lo3. Thus the variation of diluent volume with time is given by S ( t ) = So = 0.25 exp(-4.04 X 10-5t) 0.25 exp(4.04 X W 5 t )- 0.5 sin (4.04 X 10-5t) + 0.5 cos(4.04 X 10-5t) kg This relationship is plotted in Figure 5 together with the curves for other values of k. Crystallization from Previously Supersaturated Solutions Finally, crystallization from previously supersaturated solutions will be analyzed. In this mode, the achievement of supersaturation is first carried out by any one or a combination of means such as cooling, evaporation, or dilution, without any growth and nucleation taking place. Growth and nucleation then occur without any subsequent generation of supersaturation. Normally such an operation proceeds via seeding. As the supersaturation is generated before growth and nucleation take place, the system will have constant volume and, under isothermal conditions, the equilibrium concetration of the solute in solution will remain essentially constant. Such an operation is typically used in laboratory studies to establish growth kinetics (e.g., Mullin and Jones, 1973; Misra and White, 1971; Tavare and Chivate, 1979b). Becker and Larson (1969) analyzed such a system to a limited extent by neglecting the effects of initial conditions and the results were evaluated for fiistand second-order relative kinetics. For no generation of supersaturation and a constant volume system the mass balance equation is given by eq 26 which, on integration with respect toy, is modified to S(AC- A c ~ )+ ( W -Wo)= 0 (63) where W may be defined from the moment equations. These equations may be solved numerically or an analytical solution may be possible if a constant nuclei pop-
+
The time variation of diluent volume is thus described by S=S,+M= sh"' S(0) S(0) 0.25 So --- - exp(-ay) +
15
Figure 5. Time variation of diluent volume required to give a constant level of supersaturation (Example 7b).
a4 = - 6
no = noo =
10
+
Ind. Eng. Chem. Process Des. Dev., Vol. 19, No. 4, 1980 663
ulation density is assumed. Equation 63 represents the variation of supersaturation as a function of y. Using eq 2 and 7 the relationship between y and t may be evaluated numerically although with certain numerical values for the parameters an analytical solution may be possible. Now if the initial moments are assumed to be zero arbitrarily and a constant value of iiooassumed the supersaturation variation may be defined as
-
AC :001 k g / k g Ac I 0 0075 k g l k g
20
X
On using eq 7 for first-order growth kinetics, the relationship between t and y is then given by 1 b-ay 1 aY t=111 - -tan-' (65) 4ab3 b 2ab3 b
+
where a4 = k,pkgfio0/4S and b4 = k,Aco. Size-dependent growth kinetics may be treated in a similar manner. Example 8. Derive the relation between y and t and define the final CSD for a seeded batch crystallizer charged with previously supersaturated solution. Use the data in Example 2. The mass balance equation is d W + -dAc -= ldY dY With the moment equations and system kinetics this can be modified to give
-d4G
IO
5
0 (s1
Time, t , x
Figure 6. Variation of y with time for seeded batch crystallizer (Example 8). I
I
I
k
I
I
0 and lo-' rn
10-8
+ 4a4G = 0
dly4 where 4a4 = 6kvpk&~. The solution of this equation with G = Go at y = 0 is G(y) = Go cos (ay) cosh (ay) + Cr(0) cos (ay) sinh (ay) +
[ 202G(041; -1
al
4-
L
IO-^
----0
The initial growth rate derivatives with respect to y can be calculated using the mass balance and moment equations (eq 9-12). They are evaluated here for k = using the moments of the initial CSD given in Example 2
5
\
Ac = 00075 kg/ kg
IO
15
Time, t
,x
20
25
s) Figure 7. Variation of growth rate with time for seeded batch crystallizer (Example 8). (
The relation between y and t is defined by eq 7. Integration of this was performed using the Euler formula with m and the results are shown in Figure 6 . The dy = corresponding variation of G with t is shown in Figure 7. For k = m and 7 = 2 X lo4 s, y = 1.87 X lo-* m. The population density function at this point is then
1018G(y- L)u(y - L ) no./m With Go = 4a4 = 6 X y is thus G(y) =
m/s and KR = 1Ols no. s/m2 kg of solvent, i.e., a =: 6.2 X lo3. The variation of G with
+
cos (6.2 >I: lo3 y) cosh (6.2 x 103y) 6.3 x cos (6.2 X l103y) sinh (6.2 X 103y)- 6.3 X sin (6.2 X 103y)cosh (6.2 X 103y)- 7.8 X sin '(6.2X 103y)sinh (6.2 X 103y)m/s
where y = 1.87 X m. This is plotted in Figure 8. Also shown in Figures 6 to 8 are the results of calculations for Aco = 0.0075 kg of solute/kg of solvent. Conclusions Batch crystallizers are frequently used in industry because of their simplicity, flexibility, and economic viability. A CSD analysis for batch cooling, evaporative, and dilution crystallizers operated under different operating modes has
664
Ind. Eng. Chem. Process Des. Dev., Vol. 19, No. 4, 1980 30
~ ( p =d LT b(y,O) S (Y)J 1 ----Solving eq A3 for ii(p,L) seed
-
644)
CSD
product CSD
L
S, i
~ ( P , Lexp(pL) ) - iib,~) =
i
~exp(pL) ) d
~4
or L
i i ( p , ~=) exp(-pL)J
0
i i ( 0 , ~ exp(pL) ) d
~ 4+
ii(p,O) exp(-pL) (A5)
Taking the inverse Laplace transform for eq A5 ii(y,L) = ii(0,L - y ) [ l - u(y - L ) ] + n(Y - L,O)SCy - L ) u ( ~- L) (A6)
I
0
’ 0
200
LOO
I
600
1
800
1
1000
1200
Crystal size, L (pml
Figure 8. Crystal size distributions at t = 2
X
lo4 s (Example 8).
been presented for power law size independent and linear size dependent growth and power law but magma property independent nucleation rate models. This analysis should help as a starting point in the rational design, understanding and better utilization of batch crystallizers. It is possible, in principle, to determine the population density function under specified constraints, initial conditions, mode of operation, and fundamental system kinetics. In real systems it may often be difficult to assign realistic values for initial conditions since in any batch crystallizer initial nucleation may occur by several mechanisms and often takes place as an initial shower. Under these circumstances it would be advisable to determine the initial CSD by experimentation. Crystallizers operated under a constant level of supersaturation normally yield a better product. Such operations may find utility in understanding the effects of other parameters since the strong effect of supersaturation on crystallizer performance is eliminated. The analysis presented here may also be useful in identifying deviations from ideal behavior in continuous crystallizers since the description of plug flow crystallizers will be similar to that given here if the time variable is replaced by length. Finally, analysis of real crystallizers operated under a combination of cooling, evaporation, and dilution may be an interesting and useful problem for investigation.
Appendix Solution of Eq 8 by Laplace Transform Technique for Case l b in Table I
whereu(y-L)=lify>Landu(y-L)=Oify L; = 0 if y < L UR = ratio of heat transfer coefficient to heat capacity, m-2 S-1
u(y
-
L) = delta dirac input function; = 1 if y = L ; = 0 if y
# L
w=y-L W = mass of crystals in suspension, kg/kg of solvent x = variable defined by eq 13, m y = variable defined by eq 6, m z = x - (l/a)In [ G ( L ) ] Greek Letters a = constant, m-l p = crystal density, kg/m3 7
with
= batch time, s
X = latent heat of vaporization, kJ/kg of solvent 8 = temperature, K 8, = cooling water temperature, K
Superscripts
Taking the Laplace transform of eq 8 and A2
- = quantities based on total solvent (and diluent, if
capacity * = equilibrium = derivatives with respect to t , x , y
. and
Subscripts f = final
present)
Ind. Eng. Chem. Process Des. Dev. 1980, 19, 665-671
0 = initial
Literature Cited Ayerst, R. P., Phillips, M. I., iin "Industrial Crystallization", (Symposium Proceedings) p 56, Institution of Chemical Engineers, London, 1969. Becker, G. W., Larson, M. A., Chem. Eng. Prog. Symp. Ser. No. 95, 65, 14 (1969). Canning, T. F., Randolph, A. D., AIChE J., 13, 5 (1967). Jones, A. G., Chem. Eng. Sor., 29, 1075 (1974). Jones, A. G., Mullin, J. W., Chem. Eng. Sci., 29, 105 (1974). Larson, M. A., Garside, J., Ci'rem. Eng. (London), 318 (June 1973). Misra, C., White, E. T., Chem. Eng. frcg. Symp. Ser. No. 7 70,67, 53 (1971). Mullin, J. W., Nyvk, J., Chem. Eng. Sci., 26, 369 (1971).
865
Mullin, J. W., "Crystallization", 2nd ed, p 48, Butterworth, London, 1972. Mullin, J. W., Jones, A. G., Trans. Inst. Chem. Eng., 51, 302 (1973). Tavare, N. S., Chivate, M. R., Chem. Age(India), 24, 751 (1973). Tavare, N. S., Chivate, M. R., Chem. Eng. J., 14, 175 (1977). Tavare, N. S., Ph.D. Thesis, University of Bombay, 1978. Tavare, N. S., Chivate, M. R., Indian J . Techno/., 17, 404 (1979a). Tavare, N. S., Chivate, M. R., Trans. Inst. Chem. Eng., 57, 35 (1979b). Tavare, N. S., Palwe, B. G.,Chivate, M. R., Chem. Eng. Commun., 3, 127 (1979).
Received f o r review March 6 , 1980 Accepted July 7 , 1980
Liquid-Liquid Extraction with Interphase Chemical Reaction in Agitated Columns. 1 Mathematical Models Sukharnoy Sarkar, Clive J. Mumford,' and Colin R. Phllllps+ Department of Chemical Engineering and Applied Chemistry. University of Toronto, Toronto, Ontario M5S lA4 and Department of Chemic,al Engineering, University of Aston in Birmingham, Birmingham, United Kingdom
For solvent extraction with interphase chemical reaction in batch and continuous agitated columns, mathematical models are developed in terms of the film, penetration, and Danckwerts models for first- or second-order reactions categosrized as slow, fast, or instantaneous. For large Peclet numbers, a simple and numerically solvable form of the model equation is developed in terms of dimensionless concentrations.
Introduction Although liquid-liquid extraction with interphase chemical reaction is commonly used in the chemical industry in the recovery of metals from leach liquors and in aromatic nitrations, few fundamental studies have been made of it, and the phenomena involved are not well understood, largely because of the greater complexity relative to conventional mass transfer operations, and uncertainty concerning the chemical kinetics. Individual studies have dealt with either a specific application, correlation of data for a specific duty, or the prediction of specific cliaracteristics for a particular system. Limited data are available from single drop studies, but no precise mechanism has been established for extraction with chemical reaction in the practical case of a swarm of drops in a turbulent continuum. The complexities involved include multiple hydrodynamic regimes of the dispersed phase, complex residence time distribution, and unpredictable mass transfer characteristics of the dispersion due to coalescence-redispersion phenomena. Axial and radial mixing effects also arise in operation in columns. Direct application of single-drop data is therefore of limited value. In extraction with chemical reaction, the two reactive species are present in two different, distinct phases. One phase is continuous and the other dispersed. The reactive species must therefore diffuse to a reaction zone, or interface, and the reaction product must diffuse away to the selective phase, or phases, to allow fresh reactive elements to continue the process. The reaction zone may be in either phase, or extendl to both phases. In the limit it may reduce to a reaction plane, dependent upon the reaction 'University of Aston.
type and the mode of contracting. The rate of extraction is controlled both by the kinetics of the reaction and by the diffusional characteristics of the system. However, under certain circumstances, a process may be either entirely diffusion-controlled or entirely kinetically controlled. For a very slow reaction accompanied by high mass transfer rates, the overall extraction rate is determined by the kinetics of reaction, whereas for a very fast reaction, the rate of diffusion controls the overall rate. Most extraction processes were originally classified as entirely mass-transfer controlled and only recently have the implications of kinetics received attention. Kinetic effects tended to be ignored for the following reasons (Hanson et al., 1974): (a) Extractors were designed by calculation of the number of equilibrium stages for a given separation. Design from equilibrium isotherms automatically neglects kinetic effects. (b) Many processes involving chemical reactions were performed in mixer-settlers. The mixer comprised an agitated tank; since this was generally over-designed, the resultant long residence times ensured equilibrium between the phases. (c) Extractions carried out in differential contactors have been of the type involving a rapid chemical reaction, that is, a mass transfer controlled process. Conversely, many liquid-liquid reactions in organic synthesis were classed as kinetically controlled. For example, the rates of aromatic nitration reactions, involving interaction between an aromatic hydrocarbon and an aqueous solution of nitric and sulfuric acids, were considered to depend under all conditions on the kinetics of reaction. This is rather surprising since such reactions can be so rapid that control is difficult. Recent work has shown that most extraction processes involving interphase reaction in agitated systems will be diffusion-controlled under the influence of coalescence-
0196-4305/80/1119-0665$01.00/0@ 1 9 8 0 American Chemical Society
