Ind. Eng. Chem. Res. 1993,32, 1169-1173
1169
Generalized Solution of the Transient Backflow Model Equations for Tracer Concentration in Stagewise Liquid Extraction Columns Kishor R. Dongaonkar; H. R. Clive Pratt, and Geoffrey W. Stevens' Department of Chemical Engineering, University of Melbourne, Parkville, Victoria 3052, Australia
A generalized solution is presented of the transient backflow model equations, for use in determining the backmixing ratio for stagewise extraction columns from pulse tracer injection data obtained over a wide range of operating conditions. Results obtained with this model agreed with those given by earlier solutions for restricted boundary conditions. The generalized solution was also used successfully to predict the tracer response for a Kiihni column in which a tracer pulse was injected into the end stage and samples were withdrawn from two downstream stages. It was observed that numerical instability can occur in all solutions a t short response times for low values of the backflow ratio, a,and a relatively large number of stages. Introduction The backflow model (Sleicher, 1960; Pratt and Baird, 1983) has been widely used to describe the effect of backmixing in stagewise extraction columns, e.g., in the pulsed plate and rotary agitated types. A commonly used method for the determination of the backmixing coefficient, a,in such columns involves the injection of a salt or color tracer, usually as a pulse (Dirac 6 function), and measurement of the response at one or two downstream locations (Figure 1). However, most workers using this technique have interpreted their results in terms of the diffusion model, with the backmixing expressed in terms of the axial Peclet number, ULIE, where E is an eddy axial diffusion coefficient. This model was developed originally for differential-type extractors such as the packed column (Sleicher, 19591,but has also been much used for stagewise extractors, presumably because the solution to the transient diffusion model is well-known and easy to apply. Although there is a simple relationship between a and the Peclet number, this applies strictly QJ.It is apparent, only as the number of stages, N therefore, that such tracer response data should preferably be interpreted in terms of the backflow rather than the diffusion model. Solutions to the dynamic backflow equations have been reported for the following cases (cases 2 and 4 will be described later): Case 1. Pulse of tracer injected at the column inlet and measurement of the response at the outlet (Roemer and Durbin, 1967;Haddad and Wolf, 1967). Case 3. Pulse of tracer injected at an intermediate stage, m,with measurement at a further stage, n (Sawinsky and Hunek, 1981). However, no solutions are available for the injection of a pulse of tracer at the inlet with the measurement at any stage n (case 2), or for the general case of an arbitrary tracer input at an intermediate stage m in the column (e.g., resulting from a pulse of tracer at the column inlet) with measurement at a stage n further along the column (case 4). For the continuous phase, the extent of backmixing varies in the end stages, especially close to the dispersed-phase inlet where asignificant number of stages is required for the drop size distribution to become stabilized (Garg and Pratt, 1984). Thus the solutions of Roemer and Durbin and Haddad and Wolf are of limited value in this case. Also it is experimentally difficult to
J. -n
d
L
A-
+
-
+ Preeent address: C/- Computer System and Services,Bharat Petroleum Corp. (R),Mahal,Bombay 400 074, India.
0-
Figure 1. Pulse tracer injection with two downstream responses.
introduce a pulse of tracer at a point in a column without distributing the column hydrodynamics, thus making the solution of Sawinsky and Hunek (case 3)difficult to apply. A generalized solution is developed below, which is shown to reduce to the solution given by Roemer and Durbin (1967) on incorporation of the appropriate boundary conditions, and is also applicable to the solution of other cases on making appropriate substitutions.
Development of Model Equation Basic Assumptions. The system is assumed to consist of a cascade of N perfectly mixed stages of equal volume
0888-5885/93/2632-1169$04.00/00 1993 American Chemical Society
1170 Ind. Eng. Chem. Res., Vol. 32, No. 6, 1993
0 = Ft/NV,
stage N + 1: V i . . .V,. . .VwV/N vo =vN+, =o
c,
F - Net forward flow f - Backflow a=f/F - Backmixing coefficient ci-Inlet concentration of tracer , . .c,+,-Concentration of tracer in stages 0 t o N+l
Equation 2 represents the transient response of tracer at stage n for any input C" at the inlet and can be solved in conjunction with boundary conditions given by eqs 1and 3. Transforming eqs 1-3 into the Laplace domain gives
C" = (1+ a)C0- aC1
+ CY)^,, - (1+ 2a)C, +
(1
CN f
=d,/N
=
(5) (6)
where
$$
C, = C,(S) + Kexp(-sB) C,(@ de
f
and s is the Laplace operator. Equation 5 can be now rewritten as
C,+l
N+l
(4)
I".+,
V-N+l
+ (1+ ")e,,
- (1- 2a + s/N)c,
- - - J
Figure 2. Mass balance for given phase in stagewise extraction column.
(V, = V/N) with a net volumetric flow rate, F (see Figure 2). There is a backflow, f, of the phase under consideration from each cell to the preceding cell, each of which is wellmixed; the backmixing coefficient is defined as a = f/F. Two extra stages (0 and N + 1)of negligible volume, in which backflow but no mass transfer occurs, are added to the cascade, one at each end, to simplify the overall boundary conditions. A quantity of tracer, m, of negligible volume is introduced instantaneously into a given stage at or near the inlet of the phase under consideration. The impulse response is measured in one or more stages downstream of the injection point and gives, in effect, a measure of the residence time distribution. Dynamic Mass Balance Relations. The material balance equations for stages 0 through N + 1can be written as follows, with the concentrations expressed in dimensionless form, Le., as C, = c,V/m:
=0
(7)
The characteristic equation for this set of linear secondorder difference equations is as follows: ax2- (1- 2a
+ s/N)X + (1+ a) = 0
(8)
Solving for X gives A=
-
(1 + 2a + s/N)f [(l+ 2a + s/N)~4a(l + 2a
+
a
+
1 2a + s/N [4a(1 + a)]'/2
4a(l + a) On making the following substitutions cos($) =
1+2a+s/N 2141 + a)31/2
and
stage 0:
+
a = [(l a)/a]'/2
eq 9 becomes Substituting
a = f / F and
noting that VO = 0 gives
C"(t) = (1+ a)C0- aC1
(1)
stages 1In IN:
= a[cos(+) f i sin ($11 = a exp(fi$)
A standard solution to the set of difference equations represented by eq 12 is
This becomes (1+ a)C,,
where
c, = klX,n + k2X2, - (1+ 2a)C,
+ ac,+l= dCn N dB
(2)
(13)
where A1 and X2 are the roots of the characteristic equation, i.e., eq 8, as given by eq 12. Substituting these values of A1 and A2 into eq 13 then gives
Ind. Eng. Chem. Res., Vol. 32, No. 6,1993 1171
= a,(k,[cos(n$) + i sin(n$)l + k,[cos(n$) - i sin(n$)l)
a"[Kl cos(n$) + K, sin(n$)]
(14)
CN = [aNsin($)~/[aa(asin[(^ + I)$]- 2 sin(N$) + (l/dsin[(N- l)$l)l (20) This transfer function can be inverted by the method of residues to give the tracer response at the outlet. Thus, for 0 < a < m, N real, distinct poles of the transfer function are obtained as follows:
+
+
si = N(Z[(Y(I a)]"' cos($i) - (1 2a)j for 1 Ii IN
where
(21) The N values of $i are the zeros of the denominator of eq 20 in the interval 0 < $ < P, i.e., of The constants K I and K Zare obtained by solving eq 14 using the boundary conditions given by eqs 4 and 6. The boundary condition at the inlet, given by eq 4, is
eo= -eo l+ff ff
cy
DEN($) = (a sin[(N + 1)$1- 2 sin(N$) + (l/a) sin[(N- 1W1) (22) The transient response at the Nth stage (outlet) is then given by N
- e,
cN(@
=
exp(sie)
(23)
r=l
= a2Co- C,
where
= K l [ a 2- a cos($)] - K,a sin($)
Therefore
CO K, = a[a2 - a cos($)]
K,a sin($) + a2- a cos($)
(15)
Substitution of the values of CN and &+I from eq 14 into eq 6, the boundary condition at the outlet, gives
aN+'(K,cos[(N + l)$l
+ K , sin [ ( N + l)$l) = aN[K,cos(N$) + K , sin (N$)I
Case 2. Transient Response at Any Stage n for an Impulse Tracer Input at the Inlet. At the inlet, we have Co = 1. Therefore n ) $ ~- a sin[(N+ 1- ~ ) $ I ) I / [{a sin[(N + 1)$1- 2 sin(N$) + (l/a) sin[(N- 1)$1)1 (24)
C, = [(-a"'){sin[(N-
The transfer function can be inverted in the same manner as in case 1,giving N
K2 is obtained by equating eqs 15 and 16; further simplification then gives K2 = [C& COS[(N + l)$]- COS(N$))]/[C~-U~)(U X sin[(N + 1)$]- 2 sin(N$) + (l/a) sin[(N- 1)$1)1 (17) K1 is then obtained by substitution of K2 into eq 16; thus K , = [Co{sin(N$)- a sin[(N + 1)$l}1/[cr(-a2){a sint(N + l)$l- 2 sin(N$) + (l/a) sin[(N- 1)$1)1 (18) Finally, C, is obtained by substituting K1 and KZinto eq 14, as follows: a sin[(N + 1- n)$l)]/ [a(-a2)(asin[(N + I)$] - 2 sin(N+) + ( ~ a ) sin[(N- W1)l (19)
C, = [Coa"{sin[(N- n)$1-
By making appropriate substitutions, followed by inversion, eq 19 yields the solutions for a number of cases, as described below. Case 1. Transient Response at the Outlet for an Impulse Tracer Input at the Inlet. For an impulse tracer input at the inlet, eo= 1and n = N, so that eq 19 becomes
where S i is given by eq 21 and Ai by
-
Ai = [2aW1sin($i)(sin[(N- n)$J a sin[(N + 1- n)$il~l/[d[DENl+,+/d$l (25a)
Case 4. General Case: Transient Response at Any Stage,n,with Respect to That at Any UpstreamStage, myfor an Arbitrary Tracer Input at the Inlet. The ratio of the transfer functionsfor stages n and m is obtained from eq 19, as follows: a sin[(N + 1- n)+]) {sin[(N- m)$l- a sin[(N + 1- m)$l)
C, - aWm(sin[(N-n)$l-
C,-
(26)
This transfer function yields N - m values of $i as the zeros of the denominator in the interval 0 < $i < T , which are to be used in the inversion of the Laplace transform. The transient response at any stage n to an arbitrary tracer input at the inlet is expressed in terms of the response at upstream stage m, the backmixing coefficient, and other relevant parameters, as follows:
where si is given by eq 21, and
1172 Ind. Eng. Cham. Res., Vol. 32, No. 6, 1993
Ai = [(-2Na"-"'+' sin($i)(sin[(N - n)qi3 - a sin[(N + 1 - n)$il}l/[d(sin[(N - m)$I - a sin[(N+ 1 - m)$I)+~+/d$l(27a)
Backflow Model .--
;(E
6 1 5 h
c1 I
Comparison of Solutions
As stated earlier, a number of solutions to the dynamic backflow model have been described in the literature, and these are compared below with the present solution. Roemer and Durbin's Solution (1967). These workers presented a solution for the transient response at the extractor outlet for an impulse (Dirac6 function) input at the inlet. This can be identified with case 1 described earlier, and is therefore a subset of the present general solution as given by eq 27. Haddad and Wolf's Solution (1967). An approximate analyticalsolution for the residence time distribution of the tracer at stage N for a S input at the inlet has been obtained by Haddad and Wolf; this is similar to Roemer and Durbin's solution, i.e., case 1,without the assumption of fictitious stages at the two ends. Their final expression in terms of dimensionless time B was written as follows:
E M
v
e
3 t 2 t 1
a c = 1.0
0
0
5
10
15
20
25
35
30
40
45
50
55
Time (min)
Figure 3. Transient tracer responses predicted by bacMow model for a = 0.2,0.6, and 1.0 (H/W = Haddad and Wolf (1967);R/D = Roemer and Durbin (1967);S/H = Sawinsky and Hunek (1981)). 5 , 0
Experimental
Predicted Response
wetel'
where Si is given by eq 21 and the N values of $i are the roots of the following transcendental equation:
)
sin($i) = ia a - cos($;) for i = 1,2, ...,N (29) Sawinsky and Huneks Solution (1981). A solution of somewhat different form was obtained by Sawinsky and Hunek (1981) for the dynamic response at stage n to an impulse input of tracer at an upstream stage m (i.e., case 3); this was as follows: $@+ 1)
+ 2 tan-(
N
CJB) =
~ N ~ u "exp(-siB) - ~ Z D ~(30) am1
where
EiGi Di = (-l)i+l1
+ si
Ei= a sin[(N + 1- n)$il - sin[(N - n)qi3 (30b) Gi = a sin(m$i) - sin[(m - l)$il
(30~)
The values of si in eq 30 are given by eq 21, in which $i is the ith root of the following equation: (a - 1) tan($
-
y)
= (a
+ 1) tan($)
(31)
Numerical Results Cases 1 and 2. The solutions for cases 1-3 were compared numerically to obtain the responses for the continuous phase backmixing in a 7.25-cm-diameter K-i extraction column of 27 stages with a continuous-phase flow rate of 0.091 cm/s, over a range of arc values of 0.2-1.0. The responses were calculated at the continuous-phase exit for a pulse injection of the tracer at the inlet. The results indicated that for cyc = 0.2 (or less) the numerical solutions were unstable, whereas for ac= 0.5 or greater the solutions converged to give similar results for
3 78 crn3/.cc
MlBK 5 66 crn3/aec Rolor speed 120 rpm
0
5
IO
15
20
25
Optimum a 0
15
30
35
40
Time ( m i n )
Figure 4. Experimental and predicted tracer responses in a 27stage Kiihni extraction column.
all cases (Figure 3). The response curves were calculated using double precision on a 32-bit computer, and although there is no apparent reason for the instability, it is to be noted that all of the solutions contain the term aN = ((1 + c y , ) / ~ w , ) ~ / ~ as a multiplication factor, and that at low values of acand high values of N this term attains very high values. Thus for aC= 0.2 and N = 27, this term has a value of 3.2 X lolo. Also, an improvementin the accuracy of the determination of the roots of the characteristic equation, Le., from seven to nine significant figures, reduced the extent of oscillation of the solution. However, no instability was observed for values of N up to 12 as used by Roemer and Durbin (1967). Case 4. Figure 4 shows experimental data obtained for a Kiihni column in which tracer was injected at the continuous-phase inlet and the concentration responses were measured at the fourth and eleventh stages. Here, case 4 was used by a trial-and-error procedure, involving the use of the convolution integral, to obtain an optimum value of ac based on the differences between the two responses. The value of 1.55 obtained by this method compared well with that obtained using steady-state tracer injection of 1.51 (Dongaonkar et al., 1991),and is therefore capable of predicting the tracer response in the eleventh stage accurately. Discussion It is clear from the foregoing that all four solutions give satisfactory results provided that the condition of a small
Ind. Eng. Chem. Res., Vol. 32, No. 6, 1993 1173 backmixing coefficient combined with a large number of stages does not apply. This is a particular disadvantage with cases 1-3, where the use of a relatively large number of stages is desirable to minimise end effects in the column. Case 4 is more satisfactory in this regard, especially for the continuous phase as the response can be determined beyond the region adjacent to the dispersed-phase inlet in which the droplet size distribution is changing. Cases 1-3 all involve a pulse (Dirac 6 function) input of tracer, which must be mixed virtually instantaneouslywith the stream into which it is introduced. This condition is very difficult, if not impossible, to achieve in practice. It is therefore recommended that case 4 should be used, despite the greater numerical complexity involved.
Nomenclature Ai = coefficients defined by eqs 23a, 25a, and 27a a = parameter defined as a = [(l+ ( U ) / ( Y I ~ / ~ C, = c,V/m, dimensionless tracer concentration in stage n C o ( t ) = dimensionless tracer concentration at inlet as a function of time C = Laplace transform of C c, = tracer concentration in stage n in phase under consideration, kg m-3 Di = parameter defined by eq 30a DEN(+) = defined by eq 22 E = axial dispersion coefficient of phase, m2 s-1 Ei = parameter defined by eq 30b F = net forward volumetric flow rate of phase, m3 s-1 f = backflow rate of phase, m3 s-l Gi = parameter defined by eq 30c It, = compartment height, H/N, m H = height of column, m K1, K2 = parameters defined by eqs 16 and 17 L = length of test section of column, m3 m = mass of tracer injected, kg N = total number of stages (Le., compartments) in column P e = Peclet number, ULIE s = Laplace operator si = parameter defined by eq 21 U = superficial velocity of phase, m s-l
V = total volume of test section of column, m3 2Vn = volume of single stage, m3 Greek Symbols CY = dimensionless backmixing coefficient, f l F 0 = dimensionless time, FtIV X = roots of characteristic equation (eq 8) rp = fractionalvolumetric holdup of phase under consideration = parameter defined by eq 10
+
Subscripts 0 = phase inlet; fictitious inlet stage i = number of pole of eq 21 m = intermediate stage n = intermediate stage (downstream of m if appropriate)
N = phase exit stage N + 1 = fictitious exit stage
Literature Cited Dongaonkar, K. R.; Pratt, H. R. C.; Stevens, G. W. Mass Transfer and Axial Dispersion in a KOhni Extraction Column. AIChE J. 1991, 37, 694. Garg, M. 0.; Pratt, H. R. C. Measurement and Modelling of Droplet Coalescenceand Breakage in a Pulsed Plate Column. AIChE. J. 1984, 30, 432. Haddad, A. J.; Wolf, D. Residence Time Distribution Function for Multi-Stage Systems with Backmixing. Can.J. Chem.Eng. 1967, 45,loo. Pratt, H. R. C.; Baird, M. H. I. Axial Dispersion. In Handbook of Solvent Ertraction; Lo, T. C., Baird, M. H. I., Hanson, C., Eds.; Wiley: New York, 1983;Chapter 5. Roemer, M. H.; Durbin, L. D. Transient Response and Momenta Analysis of Backflow Cell Model for Flow System with Longitudinal Mixing. Ind. Eng. Chem. Fundam. 1967,6, 120. Sawinsky, J.; Hunek, J. Methods for Investigating Backmixing in the Continuous Phase of Multiple-Mixer Extraction Columns. Tram. Znst. Chem. Eng. 1981,59,64. Sleicher, C. A. Axial Mixing and Extraction Efficiency. AIChE J. 1959,5, 145. Sleicher, C. A. Entrainment and Extraction Efficiency of MixerSettlers. AIChE J. 1960, 6, 529.
Receiued for reoiew October 30, 1992 Accepted February 25, 1993
