An Analytical Solution of Cyclic Mass Transfer Operations Orlando R. Rivas Chemical Engineering Department, University of California, Berkeley, California 94 720
Simple analytical equations which can be used to calculate the ideal number of trays for cyclic countercurrent processes such as cyclic distillation, absorption, and stripping are derived in this work. These equations represent an accurate solution of the set of ordinary differential equations which result from making material balances on each plate of a cyclic column. They simplify greatly the preliminary design of these columns since they are similar and as easy to use as the well known Kremser-Souder-Brown equations used in conventional processes.
Cyclic countercurrent mass transfer processes were first introduced by Cannon (1961) about 25 years ago. A cyclic column has an operating cycle consisting of two parts: a vapor flow period when vapor flows upward through the column and liquid remains stationary on each plate, and a liquid flow period when vapor is stopped, reflux and feed liquid are supplied, and liquid is dropped from each tray to the tray below. Several workers, especially Horn (1967) and Robinson and Engel (1967) have analyzed this type of process mathematically showing that cyclic operation does increase the performance of a plate column considerably. These authors derived equations for the overall stage efficiency of a periodically operated column when the number of trays goes to infinity (asymptotic efficiency). Experimental work, especially that done by Gerster and Schull (1970) and McWhirter and Lloyd (1963), also shows that a considerable improvement in performance is obtained when a plate column is operated in the cyclic mode. In all of the previous theoretical work, the emphasis has been on comparing cyclic columns with respect to conventional ones rather than in providing methods for the design of cyclic processes. Therefore, in this work we develop several equations which can be used for this purpose. They are useful for calculating the ideal number of trays to give a desired separation once a set of operating conditions is given.
Scope and Simplifying Assumptions The purpose of this work is to develop a short cut method for the preliminary design of ideal cyclic distillation columns. The method will be strictly valid for an idealized column which meets the following conditions: (1) column vapor holdup negligible in comparison to liquid holdup; (2) constant vapor flow rates during vapor flow period; (3) constant and equal liquid holdup for rectifying section and stripping section; (4) binary mixture; (5) equilibrium stages; (6) linear equilibrium relationship; (7) plug flow of liquid from stage to stage during the liquid flow period. Conditions 1 to 5 are the usual simplifying assumptions made when designing ideal conventional columns. This means that all the assumptions and shortcomings usually associated with the McCabe-Thiele method are also inherent in our analysis. Condition 6 is usually met in processes such as absorption and stripping, but it is in general not valid for distillation. However, we can treat this problem by representing the whole equilibrium curve encountered in a distillation column by several straight lines. This will be shown later in an example. Condition 7 states that there is no mixing of liquid from two adjacent plates during the liquid flow period. McWhirter was 400
Ind. Eng. Chem., Process Des. Dev., Vol. 16, No. 3, 1977
able to show that this condition was attained in columns as large as 6 in. in diameter. His work was done using packed plates, and it has been shown that columns with plates of large free area perform similarly to packed plate columns. Some mixing is inevitable, however, and this will tend to destroy the concentration gradients across the column which in turn will decrease the separating ability of the column.
Operating Line for Cyclic Column Consider the cyclic column shown in Figure 1. In this column the vapor is transported uniformly during the vapor flow period with instantaneous flow rate Vi (full line). During the liquid flow period fraction 4 of the liquid holdup of each stage ( H ) is moved to the stage below (dashed line) and mixed there with the remaining fraction (1- 4)of the liquid holdup of this plate. The overall vapor and liquid flow rates over an entire cycle are given by the equation
L =H4t
(1)
C
v = vi (?) where t , is the total cycle time and t , is the vapor cycle time. During the vapor flow period the liquid is at rest, and a mass balance around the envelope shown in Figure 1yields dxl dx dx V i ( > l o - ~ , ) = dt H - + f L dt + . . . + H - dt l (3) We now define the dimensionless time L
(4)
T = -
t” T varies from 0 to 1 during the vapor flow period. In terms of the new variable eq 3 can be written as Vit,(yo - y,)dT = H(dx1
+ dXp + . . + dx,) ,
(5)
Integrating both sides of the above equation between T = 0 and T = 1, and noting that s,I
YndT = 7, = (average vapor composition over the vapor cycle)
gives Vit,(yo - y n ) = H(x1” - X I ’
+ x*” - xp‘ + . . . +
X,”
- xn’) (6)
where x,‘ is the liquid composition at the begining of the vapor flow period, and x,” is the composition a t the end of the vapor flow period.
t
I
1
Figure 2. Composition profile for the A’th stage of a cyclic column.
(1)During the vapor flow period each plate is continuously depleted of the more volatile component. Moreover, the mass transfer gradients are greater a t the begining of the period than a t the end. Therefore, the composition profile in any stage will appear as in Figure 2. In other words, the composition of the more volatile component decays with time without maximum, minimum, or inflection points. (2) The differential equation for plate “1” can be solved because the composition of the vapor flowing into this plate ( y o ) is constant. We assume that over a small range of composition the equilibrium curve is given by a straight line
i L
I Vi
Figure 1. Schematic representation of a cyclic column.
The periodicity condition for cycling columns is given by x,‘
= @,+I”
+ (1 - f$)xn”
(7)
Substituting eq 7 into eq 6 and rearranging, we arrive at
V J ,(40-- y, ) = Hq5 (x 1” - x,+ I ” ) (8) From eq 1 and 2 we obtain in terms of the overall flow rates
V (7”- 4,) = L
(XI’’
- X,+l”)
=
L
( x n + 1 ” - xout) + Y O V
i’)
(xn+1-
Thus for plate “1” we have the following differential equation
V
dxl
+@(Kxl) = constant dT L whose general solution is of the form
where
x1)
+ YO
(17)
(10)
Equation 10 is the operating line equation for a cyclic countercurrent stage column. If in this equation we substitute y n by y n and x,+l” and XI” (xouf)by x,+1 and X Irespectively, , we obtain the familiar operating line equation for a conventional countercurrent stage column >‘n =
(14)
(9)
and noting that XI’’= x < , ~ ~ Sn
y=Kx+b
(11)
Since in the conventional case y , and x, are in equilibrium, eq 11 together with the equilibrium equation y = f ( x ) can be used to predict the performance of a conventional column. On the other hand, and x,,” are not in equilibrium in the cyclic case. Therefore, the operating line eq 10 together with the equilibrium equation are not enough to predict the performance in this case. Knowledge of the transient behavior of the column during the cycle is necessary.
-
(3) For a very large column, the profile of the last plates ( n a) mus; approach a straight line since the composition a t the begining and a t the end of the period are nearly equal. Thus
(18)
Based on arguments (2) and (3) we now choose to represent the profile of any stage by a general function of the form xn = A,
+ B,T + C,e-XT
(19)
Equation 19 will represent the limiting profiles as stated in (2) and (3) if B1 = C, = 0. The coefficients A,, B,, and C, for each plate are determined by making the function satisfy the following conditions: (a) the concentration at the begining of the period x,(O)
Composition Profile in a Cyclic Column A mass balance around stage n of a cyclic column during the vapor flow period yields
+ B,r
= A,
x,
= x,)
(20a)
(b) the concentration a t the end of the vapor flow period x , ( l ) = X,”
(20b)
(c) the area under the curve or average composition
J 1x,dT
In terms of the overall flow rate and the dimensionless time
= JE,
From the above conditions we obtain
x,”
-
Xn
= A,
= A n+
+ C, + B , + C,e-X
= A,
x,’
Since y n and Y , , - ~ are both unknown functions of time, the solution of the above differential equation is far from trivial. However, we can still treat the problem analytically if we notice the following properties of the composition profiles.
1 -B, 2
1
+ -x (1- e-X)C,
Ind. Eng. Chern., Process Des. Dev., Vol. 16, No. 3, 1977
(214 (21b) (21c) 401
Solving for A,, B,, and C,, we arrive a t
A , = x,’ (1 +
): + 2x,” d
d
f - Ad (->
x,+1”
+ (A
- dx,
d
At this point we introduce the periodicity condition (7) into eq 31
C, = dX, - - x,” - - x n f 2 2
x,’ = f$Xn+1”
where
Xn’’ = xn-1’
f = d ( 1 - e-X)
Upon substituting eq 32a and 32b into eq 31 and rearranging, we arrive a t x,‘ = P
Equation 25 must hold at all times between Thus, at T = 0 we have
40
[(f-hd)(y-
-b
1
1
X
= 0 and
T
+ -LV 4yn‘ = -LV c$yn-lI
= 1.
(344
1)
1)
T
)]
[ (t-1) d + (x-2) f + ( A - 111 [ (1 - %, + (f - ] Q= [ (4- d + (i - f) f + - l ) ]
(25)
C$~,-~(T)
(33)
where
ds Substituting eq 24 into the mass balance differential eq 13 gives
B, - XC,
+ Qxn-l‘
P=
+ -LV f$y,(r) = -LV
(32b)
= x,’
Xn+1”
d x n = B, -
-
(32a)
or
By comparing with computer solutions we found that eq 19, with coefficients given by eq 22, does not exhibit maximum, minimum, or inflection points in the interval 0 5 T 5 1. Therefore, this general function also satisfies property (1)of the profile. This added to the fact that we forced eq 19 to go through points x,’ and x,“ and to have an area under the curve equal to X,, ensures that it will represent the composition profile of any stage accurately. From eq 19 we obtain that
B,
(7)
For simplicity we first divide the problem into two separate cases: when I$ = 1,and when f$ # 1. Case 1: 4 = 1. For this case eq 7 reduces to
1
d=
+ (1 - f $ ) x n ”
(34b)
(A
We now apply eq 33 to stages 1 , 2 , 3 , . . . , N to obtain
+ QXO’ ~ 2 ‘= P + QXl’ = P + PQ + Q2xo’
XI’
(26)
=P
but yn’ = Kx,’
+b
~ ~ - = 1 Kxn-1’ ’
XN’
+b
Therefore, eq 26 becomes
B, - XC,
+ Xf$x,‘
= P(1+ Q
XN’
= X4~,-1‘
+ Q2 + . . + QN-1) + QNxo’ ,
or
(27)
Substituting eq 22b and 22c into the above expression and rearranging, yields
=P
N- 1
2 8’ + QNXo’
(35)
i=O
For I Q 1
< 1the sum of power series is given by (36)
Equation 35 then becomes
Recalling that XN’ = X N + ~ ” = xin, and rearranging, we finally arrive at: for I Q 1 < 1
From the equilibrium equation we know that
N = In [(l- Q)xi, - PI - In [ ( l- Qlxo’ - PI ( 3 8 4
if we eliminate yn in eq 29 by using the operating line expression for a cyclic column (eq IO),we obtain (30) Equation 30 is then substituted into eq 28 to give 402
Ind. Eng. Chem., Process Des. Dev., Vol. 16, No. 3, 1977
In Q For I Q I > 1 we divide eq 35 by QN-1 and follow the same procedure as in the preceding case to obtain: for I Q 1 > 1
N = In [ ( Q- l)xi,
+ P ] - In [(Q - 11x0’+ PI
In Q For Q = 1 it is very simple to show that
(38b)
In eq 38, xo’ represents a fictitious concentration because stage “0” does not exist, and the difference equation (33) is not applicable to stage “1”. We can derive an expression for xo’ by applying eq 28 to plate “1”with 4 = 1
(f - Ad),,
(46a)
+ ( A + A -d2 - -2f - 1 ) x i 1
(f1;d
=
1
XI”+
A,,
(39)
, where Yo is the liquid concentration in equilibrium with the composition of the vapor entering the column. Equation 39 is the true mass balance equation for plate “1”. The difference equation (33) was derived by substituting the periodicity equation ( 7 ) into eq 31. For plate “1” this will imply that x,” = xo’ = T o
(7)
which is not true since plate “0” does not exist. “TO”will be equal to x 1” (xout) only for the especial case when the number of plates goes to infinity. Substituting (7) into (39) gives (f
where
- Xd)Tl+ ( A
The constants C1 and C2 can be found by applying eq 45 to plates “0” and “1”.In this way we obtain
c,=----B2x O’I (82
(62
Xout
- 01)
(P2
- P1)
(P2
-P
- 1)Q
M 2
- p1 - Po) (474
c2 =
( P i - 1)Q xout -~P I X O f ‘ + ( 0 2 - Pl)(P2- P1 - Po) ( 6 2 - 61) ( P 2 - P1)
(47b) In the above equations the apparent concentration xo” can be calculated by using the periodicity condition
We now apply eq 45 to plate N to arrive at
+ A d-2 - -2f - 1 ) x1’ = ( A + - -f
A -d- 1 ) xo’ (40) 2 2 We now redefine xo’ in such a way as to make eq 40 valid for plate “1”.This is done by equating the right-hand sides of eq 39 and 40. Thus
To calculate N from eq 49 we need trial and error methods. However, P 2 is usually negligible with respect to 61, and a very good approximation to eq 49 is
- 1 (50)
N=
xl”+A~o=
It is obvious that the above equations are not valid when (P2 - P 1 - PO)= 0. For this case, which occurs when A = 1,we have
Solving for X O ’ , and recalling that xi’’ = xOut,yields
Since P 2 can be neglected, eq 51 can be rewritten as Equations 38 with X O ’ given by eq 42 are more useful in a design problem where the separation is specified (xOut)but N is unknown. Case 2: 4 # 1. For this case the periodicity condition is given by x,’ = l#lXn+1” xn-l’ = 4 x n “
+ (1 - $ ) x n ”
+ (1 - 4)xn-1’’
I
(7)
Substituting the above two equations into eq 31 and rearranging yields
P2xn+i” - Pix,” - Poxn-l” = Q
(43)
where
p2 =
[ (x-1 24) f + ( A f
P I = [ ( i d - A) d
+ (1 -
5)
+ ( 6 4 - 1141
(44a)
+ (2Ad - A - I)@]
(44b)
- 1) d f
Equation 43 is a difference equation of second order with constant coefficients. It is shown in Perry (1963) that the general solution of this equation is of the form
N = ( X i n - C1)(P2 + PO)-
Q
(52)
For this case
cl=------P2xo” (P2
e2
=
- 1)
Xout
(P2
- 1)
- xo”) (02 - 1 )
(Tout
+ ( P 2 - 1 )QP n + Po)
-
8 (62
- O(P2 + Po)
(534 (53b)
Comparison with Other Works The equations developed in this work were used to calculate the overall stage efficiency of cyclic columns for several values of A and N . The calculated results were compared with those obtained by May and Horn (1968) and Gerster and Scull (1970) for ideal columns (Murphree efficiency = 1).The values obtained by the different methods were in all cases practically the same. This indicates that the equations derived here are accurate and mathematically sound. Distillation Columns The preceding equations were developed for cyclic countercurrent stage processes such as absorption and stripping. For distillation these equations have to be slightly modified in order to include the special conditions at the top, feed, and Ind. Eng. Chem., Process Des. Dev., Vol. 16, No. 3, 1977
403
I. I
-4
.4t J
Q v5
L--
Figure 4. Equilibrium curve for the benzene-toluene binary system.
, Vw
-
w,xw,
Figure 3. Schematic diagram of a cyclic distillation column.
bottom trays. Consider the cyclic distillation column shown in Figure 3. The operation of this column is as follows: vapor from plate N in the rectifying section is condensed and collected in the accumulator during the vapor flow period. During the liquid flow period part of the condensate is removed as top product, and part is returned to plate N as reflux. Vapor from the reboiler is allowed to enter the column only during the vapor flow period. Feed is introduced into the column during the liquid flow period if the feed is saturated liquid, and during the vapor flow period if it is saturated vapor. We will now discuss how to use the preceding equations for the design of cyclic distillation columns. Case 1: @ = 1. Rectifying Section. For this section the previous equations apply without change as long as we use the appropriate values for inlet and outlet concentrations and separation factor A. The value for 70 to be used in eq 34 can be calculated by making a mass balance for a complete cycle around the rec= ~ N S the , tifying section of the column. Noting that mass balance gives
where the subscript r refers to the rectifying section. For a total condenser, the condition a t the top tray is:
During the liquid flow period we mix the feed (composition Z F ) with the liquid from the tray immediately above the feed entrance (composition XNR”) to give the initial composition of the tray below the feed (xNs’). The optimum feed-stage location corresponds to the tray where XKR”
(57)
= X N S ’ = ZF
That is, the tray where there is no mixing of flows with different compositions. Saturated Vapor Feed. For saturated vapor feed, the feed flow is introduced during the vapor flow period, and the optimum feed-tray location corresponds to YNS
= ZF
(58)
Since in this case both the feed and the vapor from stage “NS” flow into stage “NR”, eq 33 does not apply to this plate. Therefore, the concentration XNR” to be used in the design equations is an apparent concentration which can be determined by using the expression
The derivation of eq 59 is similar to the derivation of eq 42. Noting that XNS’ = (xouJr we can calculate XNS‘ from eq 54. Thus (60)
(XinIr = X D
We can also note that (XouJr
=
(XO’)r
= ZNR”
Stripping Section. For the stripping section, eq 34,38, and 39 apply. We only have to replace 70by y, and xin by X N S ’ , and calculate xoUtby using the boundary condition a t the bottom tray. For a partial reboiler this condition is (55) where y, is the vapor concentration in equilibrium with x,. In order to use the above equations to determine the equilibrium stages of a cyclic distillation column, one more condition must be specified. This condition is the composition of the feed stage, and usually it is desirable to set the composition to correspond to optimum feed-stage location. Feed T r a y Compositions. S a t u r a t e d Liquid Feed. For saturated liquid feed, the feed is introduced during the liquid flow period and XNS’ is given by the following expression 404
Ind. Eng. Chern., Process Des. Dev., Vol. 16, No. 3, 1977
Case 2: 4 # 1. The equations for this case need not be presented since they easily follow from eq 49 and the discussion for the previous case. Example. Determine the number of equilibrium stages for the cyclic distillation of benzene-toluene subject to the following conditions: 4 = 1;X D = 0.95 (concentrations in terms of benzene mole fraction); x , = 0.05; ZF = 0.50; ( L / D ) ,= 1.57; ( V / D ) ,= 2.57; ( L / D ) ,= 3.57; ( V / D ) ,= 2.57. The equilibrium relationship is given in Figure 4. Solution. As can be seen in Figure 4 the equilibrium curve can be closely approximated by four straight lines. These lines are: Line 1: y = 2 . 2 ~0; I x I 0.18; Line 2: y = (3/2)x 0.08; x I 0.35; Line 3: y = 0.85~ 0.30; 0.35 I x I0.60;Line 0.18 I 4: y = 0 . 4 7 5 ~ 0.525; 0.60 Ix I 1.0. Because our development is rectricted to a linear equilibrium relationship, we have to calculate the number of trays in the stripping section, N,, in three parts and Nrin two. Stripping Section. Since = 1,eq 38 apply to this case. Part 1. For this part we have K = 2.2; b = 0.0; ( V / L ) ,=
+
+
+
( V l D ) , / ( L l D ) ,= 2.57/3.57 = 0.72. From eq 17 we know that X = K ( V I L ) = 2.2 X 0.72 = 1.58; = yw = K x , = 2.2 X 0.05 = 0.11; ( x l n ) = ~ 0.18 (upper valid limit for line 1).Subscript “1” refers to part 1. (xOut)lis calculated using eq 55. L , = W V,; :. ( W I L ) , = 1 - ( V / L ) ,= 1 - 0.72 = 0.28. Therefore, (xoUt)l= 0.28 X 0.05 0.72 X 0.11 = 0.093. From eq 23 we find that d = -9.96 and f = -7.91. Using eq 34a and 34b we calculate P and Q: P = 0.043; Q = 2.77. From eq 42 we obtain that xo’ = 0.0779. Since IQI > 1we use eq 38b to calculate From this equation we obtain: (N,)1 = 0.68. Part 2. In this case we have K = %; b = 0.08; X = 0.72 X (%) = 1.08. Here we are in a midsection of the column. Therefore (x& = b o ’ ) * = (xln)l= 0.18, and ( x , , ) ~= 0.35. We can also note that (yo)2 = (J,,)l. (Y,)l can be obtained by using eq 10 and noting that ( X , + ~ ” ) I = (xln)l = 0.18. From this equation = 0.231. With the above values we it follows that (?,)I = use the same equations as in the preceding case to obtain (Ns)2 = 0.82. Part 3. In this part we arrive at the feed tray. Since we have saturated liquid feed eq 57 applies.
+
+
(xln)3 = XNS’ = ZF
= 0.50
Also, K = 0.85; b = 0.30; X = 0.612; ( x o ’ ) ~= (xoUJ3 = ( x I n ) 2= 0.35; ( j j o ) ~= (7,,)2 = 0.467 (from eq 10). Substituting these values into the appropriate equations we arrive a t ( N s ) 3= 1.1.The total number of trays in the stripping section is then
N,= ( N A i + (Nsh + ( N J 3 = 2.6 Rectifying Section. Part 1. K = 0.85; b = 0.30; ( V / L ) , = ( \’/D),/(L/D),= 2.571157 = 1.64. Hence X = 1.39. Also (xOuJl
= ( x o ’ ) ~= XNR” = 0.50, and tain (Y0)l
=
( x l n ) l=
(0.59,,“.95)
0.60. From eq 54 we ob-
+ 0.95 = 0.675
Using these values we get ( N , ) 1 = 0.54. Part 2. K = 0.475; b = 0.525; X = 0.78; (xOut)2= ( ~ 0 ’ 1 2 = 0.60; (.Zin)l = X D = 0.95. From eq 54, (jj0)2 = 0.736. Substituting into the design equations we obtain (N,)2= 2.13. Hence N,= 0.54 2.13 = 2.67. Total Number of Trays.
+
N
=
N,+ N,
= 5.27
The total number of trays needed for this separation in a conventional column is 11.5 without including the reboiler King (1971). This indicates that in this case cyclic operation increases the theoretical performance by almost two and a half times.
Acknowledgment The author is grateful to the Venezuelan Fund for Research in Hydrocarbons (FONINVES) for financial support during his studies a t Berkeley, and to Edgar Rasquin, graduate student at the University of California, who helped to improve the manuscript. Nomenclature A, = constant in general composition profile eq 19; defined by eq 22a b = intercept of linear equilibrium curve, eq 14 B , = constant in general composition profile eq 19; defined by eq 22b C1,Cp = constants in design eq 49 and 51; defined by eq 47 or 53
C, = constant in general composition profile eq 19; defined by eq 22c d = constant delined by eq 23a f = constant defined by eq 23b H = liquid holdup of any stage K = slope of equilibrium curve, eq 14 L = overall liquid flow rate N = number of trays of a column N, = number of trays in the rectifying section of a distillation column N, = number of trays in the stripping section of a distillation column P = constant in difference eq 33; defined by eq 34a Po,P1,P2 = const,.nts in difference eq 43; defined by eq 44 Q = constant in difference eq 33; defined by eq 34b Q = constant in difference eq 43; defined by eq 44d t = time t , = total cycle time t , = length of vapor flow period V = overall vapor flow rate V , = instantaneous vapor flow rate X D = condensate composition x,, = composition of the liquid entering a column or section of a column x, - = instantaneous liquid composition of stage n x, = average liquid composition of stage n X N R ” = liquid composition of stage NR a t the end of the vapor flow period; defined by eq 57 or 59 XNS’ = liquid composition of stage NS a t the begining of the flow period, eq 57 or 60 -x o vapor = liquid composition in equilibrium with jjo xo’ = fictitious liquid composition of stage “0” at the begining of the vapor flow period; defined by eq 42 x0” = fictitious liquid composition of stage “0” a t the end of the vapor flow period; defined by eq 48 x w = bottom composition. 2, = instantaneous composition of vapor leaving stage n 2” = average vapor composition of stage n yo = average composition of vapor entering a column or section of a column y , = vapor composition in equilibrium with x w ZF = feed compcsition Greek Letters = roots of difference eq 43 characteristic equation; defined by eq 46 = fraction of liquid holdup that flows during the cycle X = separation factor, eq 17 T = dimensionless time
PI,&
Subscripts n = refers to stage n of a column r = refers to the rectifying section s = refers to the stripping section Superscripts ’ = refers to value a t the begining of vapor flow period ” = refers to value at the end of vapor flow period
Literature Cited Cannon, M. R.,Ind. Eng. Chem., 53, 629 (1961). Gerster, J. A., Scull, H. M., AIChE J., 16, 108 (1970). Horn, F. J. M., Ind. Eng. Chem., Process Des. Dev., 8, 30 (1967). King, C.J., “Separation Processes,” pp 339-340, McGraw-Hill, New York, N.Y., 1971. May, R. A . , Horn, F. J. M., Ind. Eng. Chem., Process Des. Dev., 7,61 (1968). McWhirter, J. R.,Lloid, W. A., Chem. Eng. Prog., 59, 58 (1963). Perry, J. H., Ed., “Chemical Engineers’ Handbook,” 4th ed. p. 2-45, McGraw-Hill, New York, N.Y.. 1963. Robinson, R. G., Engel, A. J., Ind. Eng. Chem., 59, 22 (1967).
Received for review August 26,1976 Accepted March 24,1977
Ind. Eng. Chem., Process Des. Dev., Vol. 16, No. 3, 1977
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