Ind. Eng. Chem. Fundam., Vol. 18, No. 3, 1979 260
VI = dimensionless phase velocity, VI = Vp/ao x = position vector, cm Ax = growth distance defined by eq 43 Greek Letters a = term defined by eq 38 (no units) @ = particle-to-fluid drag coefficient, dyn s/cm4 = value of /3 in the uniform bed, dyn s/cm4 tc? = rate of change of @ with respect to voidage e, evaluated for e = eo, @( = (d@/dt),,dyn s/cm4 y = angle between direction of unperturbed flow u oand the wave vector k,rad e = volume fraction of voids, i.e., the voidage td = volume fraction of voids in the dumped bed eo = volume fraction of voids in the uniform bed el = perturbation of the voidage, tl = t - to iil = amplitude of the voidage distribution function eI = term defined by eq 73 (no units) 11 = magnitude of the imaginary part of the complex frequency s, s-1
6 = angle between the applied field vector H o and the wave
number vector k
h = wavelength of the disturbance pattern, cm p
= fluid viscosity, g/cm-s, P
t = growth factor representing real part of the complex frequency s, s-l
normal oriented in the i direction, dyn/cm2 = element of magnetic stress tensor, dyn/cm2 tiks= element of solids’ stress tensor, dyn/cm2 ps = mass density of the solids, g/cm3 $ m = magnetic anisotropy factor (no units); see eq 76 xo = chord susceptibility based on properties of the magnetic solids, G/Oe 2 = differential susceptibility based on properties of the magnetic solids, G/Oe tikm
Literature Cited Anderson, T. B.; Jackson, R. Ind. Eng. Chem. Fundam. 1967a, 6, 478. Anderson, T. B.; Jackson, R. Ind. Eng. Chem. Fundam. 1967b, 6, 527. Anderson, T. B.; Jackson, R. Ind. Eng. Chem. Fundam. 1968, 7, 12. Cowley, M. D.; Rosenswelg, R. E. J . FluMMech. 1987, 30, 671. Jackson, R. Trans. Inst. Chem. Eng. 1963, 41, 13. Jackson, R. Chem. Eng. Rog. Symp. Ser. No. 1051970, 66, 3. Kunii, D.; Levenspiel, 0. “FluldiratlonEngineering”,Wlley: New York, 1969; PP 3,9. Murray, J. D. J . F/uM Mech. 1965, 27, 465. Mutsers, S. M. P.; Rietema, K. Powder Techno/. 1977, 78, 239. Penfield, P., Jr.; Haus, H. A. ”Electrodynamics of Moving Medla”, Research Monograph No. 40; The M.I.T.Press: Cambridge, Mass., 1967; Table C.2, p 255. Pigford, R. L.; Baron, T. Ind. Eng. Chem. Fundam. 1965, 4 , 81. Rice, W. J.; Wllhelm, R. H. AIChEJ. 1956, 4 , 423. Rosenswelg, R. E. “Fwohy&&ymmics”, in “EncycbpaedcDictbnery of physics," Suppl. 4, Pergammon: 1971; pp 111-117. Rosenswelg, R. E. Science 1979, 24, 57.
tr = dimensionless growth factor defined by eq 70 Eik
Received f o r review September 13, 1978 Accepted March 7, 1979
= element of fluid stress tensor representing component of stress in k direction experienced by a surface having its
Steady-State Decoupling of Distillation Columns T. J. McAvoy Department of Chemlcal Engineering, University of Massachusetts, Amherst, Massachusetts 0 1003
Steady-state sensitivity of one-way decoupling to errors in decoupler gain is defined using Bristol’s relative gain array, and applications to distillation are given. Columns controlled via manipulation of reflux, and boilup, conventional control, are prone to sensitivity problems whereas material balance controlled columns are not. In some columns sensitivities can be so high that one-way decoupling appears to be impossible to achieve. Past studies on distillation decoupling are examined in the l i t of the sensitivity results presented. One-way decoupling of conventional h m u r i t y columns is shown to be approximately equivalent to material balance control.
Introduction A number of authors have discussed the subject of decoupling distillation composition control loops (Luyben, 1970; Luyben and Vinante, 1972; Toijala and Fagervik, 1972; Wood and Berry, 1973; Schwanke et al., 1977). In all of these studies complete, two-way decoupling was attempted. In a recent article (Jafarey et al., 1978) it is shown that steady-state two-way decoupling probably cannot be achieved in distillation systems and that only one-way decoupling seems feasible. In this paper the subject of steady-state decoupler sensitivity is discussed. In any real world decoupling application, errors in decoupler gains are inevitable. Decoupler sensitivity is concerned with the practical question of how much interaction resulb from errors in decoupler gains. It is shown that even one-way steady-state decoupling may not be achievable in certain distillation columns. Initial, incisive results on decoupler sensitivity were published by Shinskey (1977a). His approach was based on the relative gain array, RGA (Bristol, 1966), and made use of a model published by Toijala and Fagervik (1972). Shinskey showed that columns controlled via manipulation 0019-7874/79/1018-0269$01 .OO/O
of reflux and boilup, conventional control, are more likely to have sensitivity problems than material balance controlled columns. After outlining the results and limitations of Shinskey’s sensitivity approach, an alternate approach is discussed. This alternate approach also shows that conventionally controlled columns will be the most sensitive to decoupler errors. By using new analytical results for the RGA (Jafarey et al., 1979), it is shown that the most sensitive conventionally controlled columns are those with high reflux ratios and/or high product purities. The new approach to decoupler sensitivity also indicates, in agreement with Shinskey (1977a), that material balance controlled columns may not have sensitivity problems. It is shown that past studies on decoupling conventionally controlled columns, both experimental and simulational, have dealt primarily with low sensitivity columns. Lastly, it is shown that one-way decoupling of high-purity, conventionally controlled columns is approximately equivalent to material balance control. Decoupler Sensitivity To discuss the concept of decoupler sensitivity the system shown in Figure 1 will be used. In the analysis 0 1979 American Chemical Society
270
Ind. Eng. Chern. Fundarn., Vol. 18, No. 3, 1979
0-as 1.11, which agrees almost exactly with their dynamic
XI
1 -
x2
Figure 1. Model illustrating decoupler sensitivity.
which follows the various ai, and k,’s will be taken as pure gains. The RGA which is considered here is m2
m,’ xi(
hmi’,rn?’
xz (1 - Ami
’,rn20
With no decoupling (kij = 0 and mi = mi) it can be shown (Shinskey, 1977a) that A,, is given by
1--
a11a22 The primes have been removed from m land m2to indicate that no decoupling is used. With decoupling (kij # 0 and mi’ # mi)it has been shown (Shinskey, 1977a) that 1 (3) Am~,,rn~, = (a21 + k21azz)(a12 + klzall) 1(a22 + klzazl)(all + kzlalz) For perfect decoupling (k12 = -a12/a11and/or kzl = -a21/az?) equals 1. For any real world application there will be some error in the k,’s. If small errors in the k,’s produce large errors in A,l,,,p then the decoupled system is highly sensitive and perfect decoupling cannot be achieved in practice. It should be noted that if Amtl,m,2 becomes large and positive, or approaches 0.5, the resulting system is highly interacting (Witcher and McAvoy, 1977). I. Shinskey’s Approach to Decoupler Sensitivity. Shinskey’s (1977a) approach to decoupler sensitivity is based upon results published by Toijala and Fagervik (1972). For a simulated column with the decoupling configuration shown in Figure 1, these authors ran a series of tests where both decoupler gains were changed by a common multiplying factor p klz = -palz/all
(4)
= -paz,/azz
(5)
kP1
For /3 = 1.1,Toijala and Fagervik’s control system broke into a large amplitude oscillation with a period which was 11times the natural period. Their system was much more tolerant of values of p less than 1.0. By using eq 3 Shinskey was able to explain why oscillations can be expected for increasing p. By substituting eq 4 and 5 into eq 3, it is possible to solve for the condition which makes Aml,,m12-00 as (Shinskey, 1977a) k12kZl = 1 (6) In terms of /3 eq 6 becomes
simulation results. As pointed out by Shinskey (1977a), the numerical results of McAvoy (1977) show that A,, is always positive and greater than 1.0 when reflux and boilup are manipulated. Large values of A, cause to be close to 1.0. Since /? = 1.0 is necessary for perfect decoupling, columns with large values of A, ,m2 will be highly sensitive to decoupling errors. Shinskey (1977a) also noted that McAvoy’s (1977) numerical results show that A,,? for material balance control is always a positive fraction. Thus, for material balance control no real value of Pmexists. Shinskey (1979) also presents a graph of Am!lFl vs. p for several values of A,,. This graph can be used to assess decoupler sensitivity. While the above approach yields some very useful results and insights, there are questions which can be raised about it. The major difficulty is that it involves two decouplers, each of which is adjusted by the same common multiplier. Jafarey and McAvoy (1978) have recently pointed out that steady-state two-way decoupling may not be achievable in practice for distillation columns. Shinskey (1977a) has noted that most of the benefits of decoupling can be obtained by introducing only one decoupler. If either klz or kzl is 0, eq 6 shows that there is no finite value of the -00. Thus, for other decoupler gain for which A,,, one-way decoupling Pm-*. Another point to be noted is that for two-way decoupling using eq 4 and 5, values of p exist which cause Xmimh to approach 0.5 and the resulting control system to be highly interacting (Witcher and McAvoy, 1977). These values of P can be calculated using eq 3 to 5 or read from Shinskey’s chart (1979). 11. An Alternate Approach to Decoupler Sensitivity. An alternate approach to decoupler sensitivity which applies to one-way decoupling is possible. This approach can also be extended and used to categorize and compare past studies on decoupling control. If kzl is assumed to be 0 in Figure 1, then the sensitivity of the control system to klz can be defined as dAm;,m,
I
Amt,,m~z (Xmll,mlZ = 1
SI2 3
(8)
The significance of the definition of decoupler sensitivity given by eq 8 is as follows. If s12is large, then small percentage errors in klz produce large percentage changes in A,~,,,l,. The result is that the actual system is not decoupled but interacts. If kZ1is set to 0 and eq 3 is differentiated with respect to k12, s12can be evaluated as 812 = 1 (9) Equation 9 like eq 7 and Shinskey’s chart (1979) shows that the sensitivity of a decoupled system depends upon the magnitude of Ami,,, for the nondecoupled system. It is straightforward to show that if k12 were 0 and kzl were active in Figure 1,then dArn~~,rn~,l
(7)
om
Equation 7 gives the important result that depends upon the magnitude of for the nondecoupled system. For Toijala and Fagervik’s column, Shinskey calculated
Thus k12and kzl have the same sensitivity. Equations 9 and 10 are general expressions valid for any 2 X 2 system. Their application to dual composition control of distillation columns is discussed below.
Ind. Eng. Chem. Fundam., Vol. 18, No. 3, 1979
$?A-
Table I. Decoupling Studies with L and V as Manipulative Variables
L=333
I
REFLUX
=,75
'L,Y
k,2:
HIN REFLUX Id qF = =E Il 55
F :I
Sl2
v =3 83
xw=
271
~~
2
467 OBI
-45r
02
Figure 2. Highly interacting distillation system.
Sensitivity of Columns Where Reflux a n d Boilup Are Manipulated Several authors have discussed the properties of the when L and V are RGA and presented values of,,,,X, manipulated (XL,v) to achieve dual composition control in distillation columns. As discussed by Shinskey (1977a,b), XL,v for such control is always greater than 1.0 and usually large. McAvoy (1977) has presented exact values of XL,V which were calculated numerically for 24 binary columns. Recently, Jafarey et al. (1979) presented approximate analytical expressions for XL," which explain all of the trends in McAvoy's numerical data. McAvoy's numerical data will initially be used to illustrate the use of eq 9 and then the analytical expressions will be used to analyze trends in si,. Of the 24 columns considered by McAvoy, 14 had XL,$S greater than 10. The largest value reported by McAvoy is 58.48 and the smallest is 3.54. Large values of XL,v are also found in commercial columns. For an azeotropic distillation column Nisenfeld and Stravinski (1968) have reported a value of 27.7. As eq 9 and 10 show, the larger hL,", the more one can expect difficulty in implementing a decoupler when L and V are manipulated. To illustrate the significance of large XL,J's consider the column shown in Figure 2. The gains for this system are those of one of the most interacting columns studied by McAvoy (1977). The sensitivity of this column to k12 is ~ 1 = 2 1 - 46.7 = -45.7 (11) If klz were to be in error by only f l %, then Amll,m,2 would change as Xmtl,m,2 3: 1 f ~ 1 (0.01) 2 1 7 0.457 (12) If Ak12/k12were positive then the apparently decoupled system would show significant interaction (Xmtl,mt, 3: 0.543). (The actual value of can be calculated from eq 3 as 0.554.) It has been pointed out recently (Jafarey and McAvoy, 1978) that it appears to be impossible to achieve perfect two-way decoupling in distillation control, i.e., simultaneous implementation of both k12 and kZl. Since errors in decoupler gains are inevitable in practice, what decoupler sensitivity indicates for the column shown in Figure 2 is that one would not be able to achieve even perfect one-way decoupling. The important conclusion which emerges from this analysis is that a certain amount of interaction must be tolerated in high sensitivity, conventionally controlled columns. The analytical expressions for XL,v published by Jafarey et al. (1979) can be used to analyze when large siJ's can be expected. One of the approximations published by these authors for liquid feeds and a binary system is ~ ( X D- x w ) R ( R x ~ 1)
author
year
Luyben and Vinante
1972 1973 1970 1977 1972 1970
Wood and Berry Luyben (5195) Schwanke e t al. Toijala and Fagervik Luyben (2198)
type of study A,, * , , both 1.78 2.01 experimental 3.30 simulation simulation 3.78 5.22 simulation simulation 10.67
A similar expression holds for vapor feeds. The denominator of eq 13 is always positive and greater than 1.0. Although the magnitude of the denominator affects XLJ, the numerator has a much more significant effect, assuming that xw and 1 - xD are on the same order of magnitude. The most significant variable in terms of giving large XL,v)s is the reflux ratio, R. As can be seen, XL,V is a quadratic function of R. Large values of R occur for columns separating close boiling components where a 1.0. For such columns XL,v and siJwill be very large. As discussed by Jafarey et al. (1979), NT in eq 13 is a loga1.0 and xw rithmic function of xw and 1 - xD. As XD 0, the 1- XD term in the numerator of eq 13 goes to zero faster than NT goes to infinity, with the result that XL,v become very large. Thus, columns with high product purities are also highly sensitive to decoupler errors. Although eq 13 applies to a binary system, the conclusions drawn should be valid for multicomponent systems. Shinskey's (1977b) expression for X L , ~for multicomponent columns shows that XL,v increases as the heavy key concentration in the distillate decreases. Also, XL,v increases with R. It can be noted that eq 13 is more accurate than Shinskey's XL,v expression for binary columns, especially for predicting the behavior with increasing R (Jafarey et al., 1979). Previous Decoupling Studies on Columes Where L a n d V Are Manipulated Decoupler sensitivity is also useful as a means of classifying previous studies. Table I summarizes the results of five previous studies on distillation decoupling. In all of these studies L and V were ultimately manipulated and two decouplers were used. In general, with the exception of Luyben and Vinante's (1972) study, the results reported in these studies support the conclusion that decoupling improves system performance. For his 2/98 case Luyben reported difficulty in implementing exact decoupling. He did, however, report success with the simplified decoupling scheme, shown in Figure 1. Since two decouplers were used, eq 9 does not apply to these past distillation studies. Both the presence of kZ1 and any error in kZl have an effect on the sensitivity of a decoupled system to errors in k12. If kZl is specified as
-
-
a21
k2l = -I) a22
-
(14)
then the definition of s12given by eq 8 can still be used. For J/ = 1, kzl has no error while I) # 1 corresponds to errors in kzl. If eq 14 is substituted into eq 3 and s12is evaluated from eq 8, one obtains
+
Equation 15 is plotted in Figure 3. As can be seen, the value of I) has a pronounced effect on s12.For J/ = 1perfect decoupling is achieved in spite of errors in klz. For other values of I), s12can be either positive or negative and it
272
Ind. Eng. Chem. Fundam., Vol. 18, No. 3, 1979 50
LZ3.33
m
40
q =I
30
F *I
x
t
I
1 3 l . " W = 50
Figure 4. Material balance control.
20
10
$
I
,'
U
0
"W
Figure 5. Partial decoupling scheme for column shown in Figure
-10
2.
- 20
-30
-40
-50 10
20
3b
\ 40
50
Aml,mz
Figure 3. Sensitivity of k12vs. A,, kz1.
as a function of the error in
increases with increasing .,,,,X, Even for small deviations is of IC, from 1, large sensitivities can result when, , , ,X, large. An examination of Table I shows that previous decoupling studies have primarily treated columns with small values of A, m2. Using Figure 3 one can categorize these columns as low sensitivity columns. Thus, one of the reasons for the reported success of decoupling may be that low sensitivity columns were studied in the past. There are several questions which the results presented here as well as those presented by Jafarey and McAvoy (1978) can help bring into focus. For columns with high values of Xm,1, e.g., 50, can any decoupling be achieved? Secondly, the results of Jafarey and McAvoy indicate that perfect two-way decoupling may not be feasible in practice and yet the studies outlined in Table I have reported success with two-way decoupling. It can be asked whether or not perfect two-way decoupling was achieved in past studies or the decouplers simply added favorable compensation to the system. Thirdly, the results shown in Figure 3 indicate that for columns with small X m l , m 2 ' ~ two-way decoupling is less sensitive to errors than one-way decoupling and therefore it may be more advantageous than one-way decoupling. This point needs to be investigated. The results presented here as well as by Jafarey and McAvoy are based on steady-state considerations only. What is needed to answer the questions raised above is a dynamic simulation of decoupling in distillation control. Work is currently underway in this direction. Sensitivity of Columns under Material Balance Control The sensitivity analysis can also be applied to material
balance control which is achieved by manipulating either distillate and boilup, or bottoms and reflux. As originally discussed by Shinskey (1977a,b), for material balance control is always a positive fraction for binary columns. The recent analytical results of Jafarey et al. (1979) confirm this fact and Shinskey (1977b) demonstrates the same result for multicomponent columns. Since eq 9 and 10 are valid for material balance control si, will always be a fraction for such control. Consequently a 1% change in decoupler gain will produce less than a 1%change in and perfect one-way decoupling may be possible for material balance control. If the gains for material balance control are calculated for the same column treated in Figure 2, then the results shown in Figure 4 are obtained. Also shown are k12 and kZl. As can be seen, k12is large and kzl is small. In general, the decoupler gain from the material balance variable (D or W) to the nonmaterial balance variable (V or L ) will be large and the other gain will be small. For the column shown in Figure 4 one would not implement k12in practice because it is so large. If it were implemented, small changes in bottoms flow would produce intolerable changes in reflux. Of the 24 columns considered by McAvoy the largest kI2 is -78.7 (case illustrated in Figure 4) while the smallest is -4.17. The average value of k,, is -23.7. As relative volatility decreases, reflux ratio increases and product purity increases, the magnitude of k12increases. The fact that in most columns klz is large agrees with the conclusion that two-way decoupling in distillation columns does not appear to be feasible in practice (Jafarey and McAvoy, 1978). Since kzl is a fraction, it is a candidate for implementation. However, since kzl is so small, it may be prone to large percentage errors which in turn would cause large changes in Xm~l~mi2.Although decoupling sensitivity is low for k21, it may or may not be feasible to implement it in cases such as the one shown in Figure 4. Whether kzl can be implemented successfully and one-way decoupling shown to be feasible for all material balance controlled columns are points under investigation. Again what is needed is a dynamic study of the points raised here.
Equivalence of One-way Decoupling in Conventional Columns and Material Balance Control A one-way decoupling scheme for the conventionally controlled column treated in Figure 2 is shown in Figure 5. As can be seen, klz = 0.987. Shinskey (197713) has
Ind. Eng. SUBCOOLED REFLUX
REfLUX,L
STEAM
Figure 6. Column controlled with steam and subcooled reflux.
pointed out, based on Luyben's (1970) results, that when L and V are manipulated and product purities are high k21 1.0 (16) If k I z in Figure 5 were exactly 1.0, then the output of GC1, mil, would be L - V or -D, the distillate flow. However, using -D and V to control compositions in a column is a form of material balance control (Shinskey, 1969, 1977b). Thus, one-way decoupling a high purity column controlled via manipulation of L and V appears to be almost equivalent to material balance control. This suggests that one should have started with material balance control in the first place. It should be noted that two-way decoupling in conventional columns does not result in an approximate form of material balance control. If the scheme shown in Figure 1 were used on the same column treated in Figure 2 and 5 , then changes in mtl are proportional to changes L 0.987V or approximately proportional to changes in -D. Similarly, changes in mf2are proportional to changes in V - 0.99.L or approximately proportional to changes in -B. Control achieved by changing -D and -B is not material balance control. A point about eq 16 should be noted. In actual columns steam flow is often manipulated rather than V itself. Also in some columns subcooled reflux is returned (Vinante and Luyben, 1972). Both of these effects add additional gains to the system as shown in Figure 6. If one were to calculate decoupler gains for the two actual manipulated variables in Figure 6, namely steam flow and subcooled reflux flow, the relationship given by eq 16 would not hold. However, since L and V are ultimately manipulated, the preceding conclusion on approximate material balance control resulting from one-way decoupling is still valid. One could include the two additional gains in Figure 6 with Gcl and Gc2and replace the system shown there with an equivalent one where L and V are manipulated. The additional gains in Figure 6 do not change the RGA since they cancel in the calculation. Conclusion This paper has treated the question of how sensitive a decoupled system is to decoupler errors. A steady-state approach has been taken and applications to dual composition control of distillation columns given. It has been shown that the sensitivity of a decoupled system depends upon the size of the relative gain elements of the nondecoupled system. Conventionally controlled columns have been shown to be much more sensitive to decoupler errors than material balance controlled columns. Particularly high sensitivities arise in conventional columns with large reflux ratios and high product purities. Sensitivities are so high in some cases that perfect one-way decoupling may not be achievable in practice. Previous studies on distillation decoupling have dealt primarily with
Chem. Fundam., Vol. 18, No. 3, 1979
273
low sensitivity columns and this may be the reason for the success which they report for decoupling. It has also been shown that one-way decoupling a conventionally controlled column is approximately equivalent to material balance control. Two qualifying points should be noted about the results presented here. First, and most importantly, these results are based upon steady-state considerations only. Secondly, they are based upon using the RGA to measure interaction. Thus, the conclusions presented should be taken as being tentative until they can be verified by dynamic simulations and/or experimentally. Work on dynamic simulations is currently underway. Acknowledgment This work was supported by the National Science Foundation under Grant ENG-76-17382. Nomenclature ai, = process gains D = distillate flow Gci = feedback controller k , = decoupler gain L = reflux flow mi = manipulated variable m'i = controller output in decoupled system NT = column trays (theroretical) R = reflux ratio si. = sensitivity of decoupler kij defined by eq 8 and 10 V' = vapor flow W = bottoms flow xi = controlled variable or composition Greek Letters a =
relative volatility
0 = multiplying factor Dm = defined by eq 7 ic/ = defined by eq 14
h,,,,, = relative gain element for nondecoupled system Am~,,m~2= relative gain element for decoupled system AL," = relative gain for conventional column control
Subscripts D = distillate F = feed
i=lor2 j=lor2 W = bottoms Literature Cited Bristol, E. H., I€€€ Trans. Autom. Confrol, AC-11. 133 (1966). Jafarey, A., McAvoy, T. J., Douglas, J. M., Znd. Eng. Chem. Fundom., 18, 181 (1979). Jafarey, A., McAvoy, T. J., Znd. Eng. Chem. Process Des. Div., 17, 485-490 (1978). Luyben, W. L., AZChEJ.. 18, 198 (1970). Luyben, W. L., Vinante, C. D., Kem. Teollisuus, 28, 499-514 (1972). McAvoy. T. J., ZSA Trans., 18(4), 83-90 (1977). Nisenfeid, A. E., Stravinski, G., Chem. Eng., 227, 227-236 (Sept 23, 1698). Shinskey, F. G., Oil Gas J . , 87, 76-83 (July 14, 1969). Shinskey, F. G., "The Stability of Interacting Control Loop With and Without Decoupiing", Proceedings, IFAC Multivariable Technological Systems Conference, 4th International Symposium, University of New Brunswick, pp 21-30 July 4-8, 1977a. Shinskey, F. G., "Distillation Control for Productivity and Energy Conservation", McGraw-Hill, New York, Chapter 10, 1977b. Shinskey, F. G., "Process Control Systems", 2nd ed, McGraw-Hill, New Ywk, pp 216-220, 1979. Schwanke, C. D., Edgar, T. F., Hougen, J. O., ZSA Trans., 16(4), 69-81 (1977). Toijala, K., Fagervik, K., Kem. Teolhsuus, 28, 1-12 (1972). Witcher, M. F., McAvoy, T. J., ZSA Trans., 16(3), 35-41 (1977). Wood, R . K., Berry, M. W., Chem. Eng. Sci., 28, 1707-1727 (1973).
Received for review September 18, 1978 Accepted March 1, 1979
