2712
Ind. Eng. Chem. Res. 1994,33, 2712-2716
Diffusion from Spherical Particles in a Continuous Flow Stirred Tank Train Charles Cozewith Exxon Chemical Co., 5200 Bayway Drive, Baytown, Texas 77520
Stripping of residual solvent from polymer particles is often carried out in a train of continuous flow stirred tanks. In this paper, design equations are derived to calculate fractional solvent removal from spherical particles as a function of operating conditions for a n arbitrary number of tanks in series. The results can be used to optimize the size and number of vessels needed to obtain a given level of residual solvent in the polymer.
Introduction In a previous study, Matthews et al. (1986a,b) developed equations to describe the steam stripping process used to remove solvent from synthetic rubbers made by solution polymerization. The porous elastomer particle produced by steam flocculation was modeled as an assemblage of small, solid polymer spheres with an effective diameter, R, and it was shown that the Fick’s law diffusion equation could be used to calculate the solvent removal rate for batch stripping. These results were then extended to solvent stripping in the commercial process which involves passing a slurry of rubber particles suspended in hot water through a continuous flow stirred tank (CFST). Due to environmental and cost pressures, much higher levels of solvent removal from the polymer are needed than in the past, and tanks in series are often used to obtain low residual solvent concentrations. In this paper we derive general equations for calculating the rate of solvent removal from spherical particles in a train of continuous flow stirred tank strippers containing an arbitrary number of vessels. The equations were obtained by averaging the batch diffusion equation over the residence time distribution stage by stage, starting at the last stage and working backwards toward the first. This technique produces simple, versatile design equations which can account for different diffusion coefficients in each stage.
Table 1. Residence Time Distribution Function, E W , for Tanks in Series value of E(t) no. of equal tanks residence time“ unequal residence timeb ~~~~~
a
~~
0 = total residence time for all tanks
=J0j.
mri
solvent concentration in the particle and D is invariant, eq 2 describes the amount of solvent removed from any one particle with a residence time t in the system. To get the mean solvent concentration in the particle slurry leaving the system, eq 2 can be averaged over the residence time distribution via the equation
Equation Development The diffusion rate from a spherical particle is given by dC/dt = (l/r2)d[Dr2dC/drYdr
(1)
For a constant diffusion coefficient and an initially uniform solvent concentration throughout the particle, integration of this equation yields the result (Crank, 1956):
(X,- X)/(X, - X,,)
= 1m
(exp(-Dn2&R2))/n2
(6/7&
(2)
n=l
The diffusion coefficient in polymerlsolvent systems is usually a function of the solvent concentration and not a constant. In this situation we still apply eq 2 to calculate the stripping efficiency but use an integral average value for D, as described in the previous work (Matthews et al., 1986b). For continuous flow strippers in which the particles enter the system with a uniform
where E ( t ) is the residence time distribution function. We will limit our discussion to solvent removal from slurries of polymer particles of uniform size in a CFST train in which the particles are perfectly mixed in each vessel and no particle agglomeration or fragmentation occurs. For the case of a single CFST, E(t) is equal to (l/O)e-t’o, and integration of eq 3 gives the design equation obtained by Matthews et al. (1986a,b). The same methodology can be extended to a series of CFSTs when the equilibrium solvent concentration and D are unchanged along the stripper train. The E ( t )function for equal volume tanks in series is given by Himmelbau and Bischoff (19681, and by using the Laplace transform method they describe, E(t)for unequal tanks can readily be obtained. Inserting these E(t)functions for tanks in series with either equal or unequal residence times (see Table 1)into eq 3 produces the expressions for stripping efficiency in a 2, 3, a n d j vessel train shown in Table 2. Direct integration of eq 3 is only possible with the restriction that the diffusion coefficient, D, is invariant in the system. Because solvent concentration, and perhaps also temperature, will vary from vessel to vessel, D will most likely also differ in each tank. When
0888-588519412633-2712~Q4.5010 0 1994 American Chemical Society
Ind. Eng. Chem. Res., Vol. 33, No. 11, 1994 2713 Table 2. Series Stripper Efficiency Equation from Integration of Eq 2
D I R= ~ same in all stages Xeq= same in all stages
no. of tanks
a
value of (XO- X,)/(Xo - XeqP unequal residence timeC
equal residence timeb
Xj = exit concentration from tank j . Xo = inlet concentration to first tank.
l/n(l@,/ei) j
cP=
-
= 1m1 - eI/0J(l - 0JOJ
B = Dn2n2/R2. 0 = total residence time =jOj.
... (1- @pi)].
m=l mti
this occurs, integration of eq 3 over the residence time distribution for the system is no longer possible. To obtain stripper design equations for the case of variable D,a stage by stage integration is required. However, eq 2 for X is only applicable in the first vessel of the train since a uniform initial solvent concentration across the particle radius does not exist for the feed entering later tanks. To get around this problem, the following approach is used to derive an expression for Xj that can be integrated along the stripper train. For two vessels in series, consider a single particle that is in the first stripper for tl s and the second stripper for t 2 s before leaving the system. We will initially assume both strippers are a t identical condiThe amount of tions (Dl= DZ= D,Xeql= Xeq2= Xeq). solvent removed in the second stripper, XI- X2,is equal to the amount of solvent removed in a time (tl t 2 ) minus the amount of solvent removed in time tl. From eq 2
thus
A similar result was obtained by Markelov et al. (1992) for the case of diffusion in a semi-infinite flat sheet. For the entire population of particles of all ages in the second stripper, the residence time distribution function E(t2)is (1/02)
[email protected], the average amount of solvent in the particles leaving the second vessel that were in the first stripper for exactly tl s is
+
X,- X,= AX (1- (6/2)ce-Bt1/n2)
( X 2 4= hw(1/02)e-t2/"2 X,d t ,
where XZis given by eq 5. By averaging ( X 2 ) t l over the residence time distribution of the particles leaving the first stripper, we obtain the average solvent concentration leaving the second vessel:
X, - X,= AX (1- (6/2)Ce-P't1ft2)/n2) where we have introduced the definitions
p =D ~ ~ ~ / R ~
X, = ~ ~ ( 1 / O l ) e - t " " 1 ( ~ 2dt,) t l
Since X I is equal to
n=l
Subtracting these two equations yields
(7)
Equations 6 and 7 are the key expressions for calculating the overall amount of stripping in two strippers in series. Substituting (5) into (6) and integrating gives
AX=x, -xeq
c=c
(6)
then
2714 Ind. Eng. Chem. Res., Vol. 33, No. 11, 1994
(XJtl = (X,- AX) - (6Ax/71%5O,e-~~~/n~ (1t
Inserting (10) into (7) and integrating the second time gives the final result for solvent removal in two series strippers at identical conditions:
(X,- X2)/Ax= 1 - ( 6 / 2 ) C l/[(l+ p02)(l+ PO,)n2I (11)
c.
U
0)
X
0
E 3. X I
0
Equation 11 appears to differ considerably from the equivalent expression in Table 2 for two unequal tanks that was derived by integration of eq 3. In fact, both equations give identical numerical results. However, eq 11 is preferable in view of its simpler form. The same methodology is used to obtain the stripping equation when the conditions are different in each tank. In this case, the particle is in the first stripper for a time tl at conditions D1 and Xeql and then in the second for t 2 at conditions D2 and Xeqp. Assume the concentration ratio, (XO- XJ(X0 - Xeql),reaches the value shown by point a in the schematic plot of eq 1 in Figure 1 at tl. When the particle enters the second stripper X,, changes from Xeql to Xeq2 and the value of the concentration ratio shifts to (XO- Xl)/(Xo- Xeq2) given by point b. This then represents the initial state of the particle in the second vessel following time period tl. Also shown in Figure 1is the stripping curve at diffusion coefficient D2 in the second stripper. We wish t o determine how much additional stripping occurs in the second time period t 2 following the period tl. The solvent concentration ratio in the particle at point b would have been obtained in a time tl’ (see Figure 1) if the diffusion coefficient were D2 instead of D1. Thus, the amount of stripping in period t 2 can be calculated from the amount of stripping in time tl’ t 2 minus the amount of stripping in time tl’. The appropriate stripping equations are
+
5
TIME
Figure 1. Schematic of sequential batch stripping with differing
D values. From the form of eq 2, it can be seen that to get the concentration ratio at point a with the diffusion coefficient D2 in time tl’, the following relationship must hold
Dltl = D2t1’ Using this result to eliminate tl’ in eq 12 gives
Equation 13 is next ins_ertedinto eq 6, and evaluation of the integral gives (X2Itl as a function of X I and tl. The expression for X I (eq 9) is then inserted into this equation, and then the integration in eq 7 is performed to give the final result:
(X, - X2)/Ax= 1 - ( 6 / 2 )
X,- XI = Ax2(1- ( 6 / 2 ) xe-b2t’/n2) and subtracting
t 2 ”+ t 3
t 1 #’+t 2
t i
x
+
l/[n2 (plOl 1)( 8 2 0 2
+ 1)1 (14)
For three strippers in series the same procedure is used to obtain an expression for X3. We again assume in the derivation that AX1 = AX2 = AX3 = AX. At the end of batch stripping for a period tl at D1 and t 2 at D2 ,X2 is given by
(X,, - X2)/Ax= 1 - ( 6 / 2 ) ~e-(b1t1+bztz)/n2 (15) For commercial stripping vessels, the initial solvent concentration in the rubber particles is usually on the order of several weight percent and conditions in the stripping vessel are set t o give low values for the equilibrium concentration, Xeq(Matthews et al., 1986b). Consequently,
XO>> Xeq To simplify the expressions that follow, we will assume that
x,- Xeq1= x,- xeq2... = xo- xew = Ax ’
although it is possible to derive the desired operating equations without this restriction. With this approximation points a and b in Figure 1 coincide.
The amount of batch stripping in an additional period t3 with diffusion coefficient D3 will equal the stripping that occurs in a time t2’ t3 minus the stripping in time t2’, where t2’ is the time required to reach the X2 value in eq 15 with a diffusion coefficient of D3 (See Figure 1). As before, to obtain the concentration ratio represented by point c in Figure 1 with the diffusion coefficient equal to 0 3 requires that
+
D3t; = Dltl
+ D2t2
so the batch stripping expressions for X3 is
X 3 = X 2 - Ax(6/2)[x(exp-(P,tl z(exp-(Pltl
+ P2t2))/n2-
+ P2t2+ ,@3>)/n21(16)
Ind. Eng. Chem. Res., Vol. 33, No. 11, 1994 2716 Table 3. Recommended Diffusion Equations for Tanks in Series"
Table 4. Effect of Vessel Configuration on Stripping Performancea no. of vessels el, s e2, s %x0b 1 batch 600 0.077 1 CFSR 600 0.220 2 CFSR 480 120 0.170 2 CFSR 390 210 0.157 2 CFSR 300 300 0.153 a
D/R2 = 0.000 35/s. Xes = 0.0.
obtained by use of eq 12 to subtract result is
from &-I.
m
- XJ/AX=
( 6 / 2 ) [ x l / ( n 2 ( 1+p101) ... (1
The
+
n=l
" X O= same in all stages,pi = Din2n2/R2,j = tank number.
Equation 17 is then inserted in the averaging equation
pj-lOj-l)) - C l / ( n 2 ( 1+ plOJ ... (1 + pj0,))l Collecting terms in the series and carrying out algebraic simplification gives the final result
(xJ-l - T)/M= 6(D/R2) Oj
x m
1/(1
+ plOl)(l +
n=l
P 2 0 , ) ... (1
which is integrated 3 times to give the result
( X , - X3)/AX = 1 - (6/2)Cl/[n2(Plo1+ 1)(p202+ 1)(p3@3 1>1 (17) By comparison of eqs 14 and 17, we can deduce the result for j vessels in series
+
+ + 111 (18)
( X , - Q A X = 1 - ( 6 / 2 ) x 1/[n2(pIOl 1)(p202 1) ... (pjOj
These equations for series stripping in 2,3, and j vessels with unequal diffusivities are collected in Table 3.
Discussion Since eq 18 is a single equation applicable to tanks in series with equal or unequal residence times and equal or unequal diffusivities in each vessel and, in addition, is simpler in form than the equations in Table 2, this is the recommended design equation for calculating the stripping efficiency in a train of CFSTs. For tanks in series with constant diffusivity eq 18 simplifies to m
(X, -
q)/AX= 1 - ( 6 / 2 ) z l / [ n 2 ( p 0 1+ 1)(/3O, + n=l
1)... (POj
+ 1)1 (19)
If in addition the residence time is equal in each tank then the equation becomes m
(X,- X,)/AX = 1 - ( 6 / 2 ) C l/[n2(/30j+ 1Y1
(20)
n=l
Although eqs 19 and 20 have a markedly different form from the equations presented in Table 2 for stripping in a j tank train with unequal and equal residence times, the two sets of equations are mathematically equivalent and produce identical numerical results. The stripping efficiency for any one stage in a train can be
+ pjoj, (21)
Equation 21 should be useful for estimating the efficiency expected from adding additional stages to a train. Matthews et al. (198613) indicate that typical values for R and D in steam stripping of ethylene-propylene rubber particles are
R = 0.07 cm D = 0.17 x
cm2/s
D/R2 = 0.00035/s With these parameter values, eqs 2 and 18 were used to calculate the residual solvent level in rubber particles for batch stripping and stripping in one and two stirred tanks. As shown by the results in Table 4, batch stripping gives considerably more solvent removal than stripping for an equivalent time (600 s) in a single CFST. Using two CFSTs in series with the same total residence time provides an improvement over the single CFST case, with the best results obtained when the two vessels have equal residence time. It can be seen from the form of eq 18 that stripping efficiency is independent of the order in which series tanks are arranged. The equations derived in this work are generally applicable to calculating diffusion rates from spherical particles passing through a series of stirred tanks and so may be useful for mass transport calculations besides solvent removal from polymer particles. Also, the techniques we used to derive the equations may have applications in other situations where the boundary conditions for the batch rate expressions prevents straightforward stage by stage averaging starting with the first stage in a stirred tank train.
Nomenclature C = concentration, wt/vol D = diffusion coefficient, cm2h E = residence time distribution function n = summation index j = index for the number of tanks in series r = radial position in particle
2716 Ind. Eng. Chem. Res., Vol. 33, No. 11, 1994
R = particle radius, cm t = time, s = weight fraction of solvent in a particle X = average weight fraction of solvent in the particle slurry
X
Greek
p = Dn2n21R2 0 = vessel residence time Subscripts
1,2,3,...,j = number of vessel in a train 0 = inlet to first vessel eq = equilibrium value
Literature Cited Crank, J. The Mathematics of Diffusion; Oxford Press: London, 1956.
Himmelbau, D. M.; Bischoff, K. B. Process Analysis and Simulation; John Wiley and Sons: New York, 1968. Markelov, M.; et al. Migration Studies of Acrylonitrile from Commercial Copolymers. Ind. Eng. Chern. Res. 1992,31,21402146. Matthews, F. J.; et al. Solvent Removal from Ethylene-Propylene Elastomers. 1.Determination of Diffusion Mechanism. Ind. Eng. Chem. Prod. Res Dev. 1986a,25, 58-64. Matthews, F. J.; et al. Solvent Removal from Ethylene-Propylene Elastomers. 2. Modeling of Continuous Flow Stripping Vessels. Ind. Eng. Chem. Prod. Res Dev. 1986b,25, 65-68. Received for review March 7, 1994 Revised manuscript received June 28, 1994 Accepted July 8, 1994@
* Abstract published in Advance ACS Abstracts, September 15, 1994.
