ai = initial activity af = final activity d = mean activity a ( t ) = optimal activity progression with time C = concentration; CA, of reactant; CR, of product CR = regeneration cost C R ~= fixed regeneration cost CR= ~ varying regeneration cost d = order of deactivation E = activation energy for main reaction; E d , for deactivation FAO = reactant feed rate Go, = netincome rate I_ = operation index I = optimum operation index J = regeneration index K = lumped parameter defined in the text k = rate constant for main reaction; k d , for deactivation; h,, for regeneration k o = preexponential factor for k; k d O for hd m = order of concentration dependency n = order of reaction p = species causing deactivation, eq 1 = profit = optimumprofit P,,, = maximum profit R = product T = temperature of reactor T* = maximum allowable temperature T = optimal temperature progression t = time t o , = operation time
e
t R = regeneration time t R f = fixed regeneration time t~~ = varying regeneration time W = weight of catalyst in reactor X A = fractional conversion of reactant XAO = optimum conversion X A =~ transition conversion defined in Figure 1 XA* = conversion a t T* XA'= local conversion z = fractional weight of catalyst Greek Letters = valueofproduct /3 = ratio of the total operational cost rate to a F ~ 0see ; eq 2 y = ratio of activation energies, Ed/E w = constants defined in eq 6 (Y
Literature Cited Chou, A., Ray, W. H., Aris, R.. Trans. Inst. Chern. Eng., 45, T153 (1967). Crowe, C. M., Can. J. Chern. Eng., 48, 576 (1970). Khang, S. J., Levenspiel, O., lnd. Eng. Chern., Fundarn., 12, 185 (1973). Levenspiel, O., "Chemical Reaction Engineering", Wiley, New York, N.Y., 1972. Miertschin, G. N., Jackson, R., Can. J. Chern. Eng., 48, 702 (1970). Ogunye, A. F., Ray, W. H., Trans. Inst. Chern. Eng., 46, T225 (1968). Park, J. Y., Ph.D. Thesis, Oregon State University, 1976. Szepe, S., Ph.D. Thesis, Illinois Institute of Technology, 1966. Szepe, S., Levenspiel, O., "Proceedings of the Fourth European Symposium on Chemical Reaction Engineering", Brussels, 1968, p 265, Pergamon, London, 1971. Weekman, V. W., Jr., Ind. Eng. Chern., Process Des. Dev., 7, 252 (1968).
Receiued for review November 11, 1975 Accepted May 7,1976
A Simplified Analytical Design Method for Stagewise Extractors with Backmixing H. R. C. Prall' Universify of Newcastle upon Tyne, England
Exact analytical solutions for the concentration profiles in stagewise extractors with backmixing, assuming a linear equilibrium relationship, are derived in forms comparable with those of Miyauchi (1957) for differential contactors. These are then used to obtain simplified approximate forms which are solved to obtain the number of stages required to give a specified performance; the resulting expressions closely resemble those previously obtained for differential contactors (Pratt, 1975). Values of the number of stages given by the approximate expressions have been compared with the exact computed solutions over a wide range of variables. The error was found to be acceptably small for six or more stages, or even less in some circumstances, even with very high backmixing ratios.
Introduction The recognition of backmixing as an important factor in determining the performance of liquid-liquid extraction equipment is now firmly established, and two models describing this are available. Of these, one relates to differentially continuous columns in which longitudinal back-diffusion is superimposed upon plug flow of the phases. This leads to a fourth-order differential equation which can be solved analytically for the case of a linear equilibrium relationship to give the concentration profile in the form of a sum of expo-
* Present address: Department of Chemical Engineering, University of Melbourne, Parkville, Vic. 3052, Australia. 544
Ind. Eng. Chern., Process Des. Dev., Vol. 15, No. 4, 1976
nentials. The application of this result to the design of such columns, i.e. the calculation of the length, involves tedious iteration, but a simple approximate solution, which in fact is of adequate accuracy for all practical purposes, has recently been obtained (Pratt, 1975). The other model relates to stagewise contactors and describes the backmixing by a discrete backward flow of the phases. This leads to a fourth-order difference equation, the general solution to which gives the concentration profile for the linear equilibrium case as a sum of power functions involving the number of stages. This solution, together with the four boundary conditions which determine the constants, was first obtained by Sleicher (19601, who gave the results of computer calculations for a large number of parameter com-
binations in terms of a contactor "efficiency"; the case was also considered in which part of the mass transfer occurred in the settler. The results were expressed by an empirical relation, and analytical solutions were also given for less complex cases (see also Vermeulen et al., 1966). I t was concluded that fairly high values of the backmixing, i.e., up to 40% entrainment of the phases, has relatively little effect on the efficiency of mixer-settlers. However, it is to be noted that backmixing ratios as high as 2-4 are possible with some types of stagewise column, e.g., rotary disk, pulsed plate, and Scheibel columns, in which case this conclusion does not apply. Hartland and Mecklenburgh (1966) have also given the full analytical solutions to the stagewise case, presenting these side by side with those for the differential case. These are complex, however, and difficult to use for design purposes where the number of stages is required. I t was therefore decided to seek a simplified approximate solution comparable with one recently described for differential contactors (Pratt, 1975). Such a solution has now been obtained, and the derivation is given below, starting from a more convenient form of the general solution similar to that given by Miyauchi (1957) and Miyauchi and Vermeulen (1963a) for the case of differential contactors.
General Solution The basic difference equations, obtained by material balances around stage n of a cascade of N stages, may be written as follows (Sleicher, 1960, Hartland and Mecklenburgh, 1966) (1 + a . r ) C x , i n - l )
where the ai are given by
(10)
Equation 10 is obtained by elimination of Nox between eq 9 and 11-14. The p, in eq 7 and 8 are the roots of the characteristic equation, which may conveniently be expressed as follows (p
- 113 - C Y ( p- 1)' - P ( -~ 1) -
=
o
(11)
where a=
(1 + NOXI) - (1 - Nox'F) (1+
QX
(12)
CY))
The roots (pi - 1)of this equation are given by the standard formula, as follows (pi - 1) = CY 2 v 5 cos 3
+
where h = 0, 120°, and 240' for p 2 , p3, and p4 respectively
- (1+ 2 a x ) c x , n + a x C x , i n + l ) u = cos-'(q/p"*)
taking the angle between 0 and 1.This solution applies only if q 2 - p3 < 0 Assuming a linear equilibrium line represented by
c, = mcy
+q
eq 1 and 2 may be put in dimensionless form by introducing the dimensionless concentrations X and Y, viz.
X=
+
c, - (mc, 1 4) . - (mc,l+ 4) '
cxo
y= c,o
m(c, - c q l ) - (mc, 1 4)
+
(3)
which is normally the case in the present application. The constants A, in eq 7 and 8 are determined by the four boundary conditions, obtained by writing material balances for each component a t each end of the cascade. For this purpose it is convenient to add fictitious end stages 0 and N 1, which are in effect after-settlers in which no mass transfer occurs (Sleicher, 1960; Hartland and Mecklenburgh, 1966). On this basis the boundary conditions become, in dimensionless form; n = 1:
+
Hence eq 1 and 2 become (1
+ CYx)X,-1
- (1 + 2a,)X,
+ axXn+l= Noxl(X, - Y,) (4)
CY>Y,-l - (1 + 2CY,)Y,
Equations 4 and 5 are put into A operator form and Y is eliminated to give a 4th-order difference equation in X . Expressed in E operator form, where A = ( E - l),this becomes
- [(I + @,)(I + ~ ~ ~ 1 ) - u x (-i N ~ , ~ F( )E]- 113 - [I + N ~ , ~ ( ~+ , F - F + 2 ) ] ( E- 1)' - Nox'(l - F ) ( E - l)]X, = 0 (6)
+
114
The solution of eq 6 gives the X-profile, and combination of this result with eq 4 gives the Y-profile, as follows
+ A'pz" + A 3 p s n + A4fiq" Y, = A1 + Aza2pf + A3a3pp + A4a4pqn X, = A1
(16)
YO = Yo = Y1
(17)
X' = X.y+1 = XN
(18)
Y.V+l - a, (Y.v - YN+d = 0
(19)
n = N:
+ (1+ ay)Yn+l= -Nox'F(Xn - Yn) ( 5 )
jcvx(iQ , ) ( ~-
xo - CYX(X1- XO) = 1
(7) (8)
Substitution of eq 7 and 8 in eq 16-19 gives four simultaneous linear algebraic equations in the A , , the solution to which may conveniently be expressed in determinantal form as follows (20)
The solutions for X, and Y , become
Y, = A 1
+ A 2 a 2 k z n + A 3 a 3 ~ +3 ~A 4 ( n+ Nox %)
(32)
On substituting eq 31 and 32 into the boundary condition eq 16-19 to obtain the A i and simplifying as before, the final result is
(b) F # 1.0, ax # 0, ay = 0. For this case ~3 = 0 and the characteristic equation reduces to a quadratic as follows Equations 7 and 8 are not convenient for rapid design purposes, i.e., for the calculation of the total number of stages required to achieve specified exit concentrations, since extensive iteration is required. However a simplification can be effected from a consideration of the properties of the roots p r of the characteristic equation, as follows: (i) ~2 is relatively large and positive, so that terms containing p z - N can be disregarded if N is not too small; (ii) 0 5 p 3 5 1,so that terms containing ygN can be disregarded if N is not too small; (iii) p 4 = 1.0 when F = 1.0; also ( p 4 - 1)is small and positive when F > 1.0, and small and negative when F < 1.0. Equations 7 and 8 can therefore be simplified, provided N is reasonably large (> about 4-6) by writing them out in full using values of the A, given by eq 20 to 25, and dividing numerator and denominator by p z N . Terms containing up-^ and ~ 3 ' " ' are then neglected, when the expression for Y , becomes
a x h
-
- [1+ Noxl(l + CYXF)l(P - 1) - Nox'(l - F ) = 0
(34)
in which ~2 is the larger root and 114 the remaining root. Consequently only three boundary conditions are required, and the final simplified solution for N becomes
( c ) F # 1.0, a x = 0, ay # 0. The characteristic equation for this case also reduces to a quadratic; thus (1
+ a y ) ( l+ N o x ' ) ( p - 1)' + [l
+ Noz1(2 - F + a y ) ] ( / ~- 1)+ No,'(l
- F ) = 0 (36)
The two roots correspond to fig and w4, of which w3 is the smaller. The number of stages is then
YF2P4"(P3
+ F b Z P 4 ' " ' ( P 3 - 1)(P4 - 1)P2n--N - 1 ) [ ( 1 4 - 1 ) P f - ( P 3 - 1 ) ~ 4 ~ I ) (26) - ~ ) ( I LZ ~ 4 +) Q ( P Z - 1 ) ( P 4 - ~ 3 )
- 1)(Pz - P4) f a 4 b z
F2p4"(fi3
At the end of the cascade where n = 0 the exit composition is given by Y = Yo;the term containing p 2 n - N can then be neglected, while 13" = ~ 4 , = 1. Equation 26 therefore becomes
Y-
- 1)(P2 - P 4 ) + F a 4 ( 1 1 2 - 1 ) ( P 4 - P 3 ) (27) F 2 P 4 " ( P 3 - 1)(P2 - 114) + a 4 b 2 - 1 ) ( P 4 - P 3 )
F2P4"(P3
Solving for pcqNand taking logarithms then gives the desired expression for the number of stages
If the outlet X-phase composition X1 is specified, the corresponding value of Yo is given by the overall material balance
YO = F(1 - XI)
(29)
Special Cases (a) F = 1.0, ax and ay # 0. On substituting F = 1in eq 6 it is found that 1.~4= 1.0 and the characteristic equation reduces to a quadratic, as follows
546
(d) F = 1.0, a x or ay = 0. Putting F = 1 in eq 34 and 36 gives w4 = 1 in both cases together with the remaining root p2 or j ~ 3 Substitution . of the latter in the simplified equations for N then gives the following expression, applicable to either case
Ind. Eng. Chem., Process Des. Dev., Vol. 15, No. 4, 1976
where a, relates to a, or aiy, as appropriate.
Discussion Accuracy of Solutions. The accuracy of the proposed equations was tested by computing the exact concentration profiles for a large range of parameter values and substituting the resulting values of YO into the appropriate equation to give the approximate number of stages. For the general 5-parameter case, Le., with N , a,, CY>, No, and F as variables, eq 7,8, 15, and 20-25 were used and calculations were carried out for 4,6,8, and 16 stages with No, values ranging from 0.05 or 0.10 up to 1.0 and F values of 0.50, 1.50,and, in certain cases, 0.25, 3.00, and 4.00. Similar calculations were conducted for the 4-parameter case in which F = 1.00. In selecting an appropriate range of values for the backmixing coefficients, it was noted that Sleicher (1960) has studied the effect of entrainment ratios up to 40% (Le., CY, values of 0.40) on the performance of mixer-settlers and found that the effect on overall efficiency was relatively small. Much higher values of the a J , from 1.50 to 4.00 (150 to 400% entrainment) were therefore assumed in the present study since it was intended that the results should be representative of
Table I. Accuracy of Solutions a % Error
N 4.0
QX
aY
Nox
1.5 2.0
1.5 2.0 3.0 4.0 1.5
0.10-1.00 0.10-1.00 0.10-1.00 0.10-1.00 0.10-1.00 0.10 0.25 0.50
3.0 4.0
6.0
8.0
1.5 2.0 2.0 2.0 2.0 3.0 4.0 4.0 4.0
4.0
4.0
4.0
1.5 2.0 3.0
1.5 2.0 3.0 4.0
1.5
4.0 1.5
1.5-4.0 1.5 4.0
a
4.0 4.0
4.0 4.0
16.0
2.0
2.0 2.0 2.0 3.0
1.5-4.0b
4.0 1.5
F = 0.25
F
=
0.50
