For rich feed compositions ( 20.70) the approximate optimal solutions showed no advantage over constant distillate-rate operation. These cases were based on the assumption that all initial tray compositions were equal to the feed composition ; thus a t low feed purity, the column was best operated a t high reflux ratio during the initial period of operation. Here as one might expect, the approximate solutions show some gain. The operating policies and resulting overhead compositions for run 7 are presented in Figure 1. A comparison with the no-holdup policy is presented in Figure 2. I n addition to the initial low yield period they also differ in that the case with holdup yields a significantly lower total product, the distillate rate always being less. Since the approxima1.e-optimal solution in no way resembles on-off control, we are forced to conclude that for the cases studied, the maximum yield is relatively insensitive to distillate policy. The approximate method, in addition to providing a solution, has a t the same time provided some information concerning the sensitivity of the objective function to the control.
S T
= = = =
t
=
V = Xi =
X, = y = y* =
X
=
refers to top tray ith adjoint variable still holdup duration of operation time vapor rate mole fraction of more volatile component in liquid on tray i mole fraction of more volatile component in still liquid mole fraction of more volatile component in distillate mole fraction of more volatile component in final product Lagrange multiplier
literature Cited (1) Bellman, R. E., “Adoptive Control Processes,” Princeton University Press, Princeton, N. J., 1961. (2) Converse, A. O., Preprint 116, AIChE Meeting, December 1962. to be uublished. ( 3 ) Converse,’A. O., Gross, G. D., IND.ENG.CHEM.FUNDAMENTALS 2, 217-21 (1963). (4) Huber, C., M. S . thesis, Carnegie Institute of Technology,
1964.
A. 0. CONVERSE’ C. I. HUBER
Nomenclature
D
n
pi
Carnegie Institute of Technology Pittsburgh 73, Pa.
= distillate rate
F = G = HI = H2 = h j K
defined by Equation 6 defined by Equation 7 Hamiltonian function when holdup is neglected Hamiltonian function when holdup is considered = tray holdup = refers to tray number = equilibrium constant
RECEIVED for review March 16, 1964 ACCEPTED May 26, 1965 A.1.Ch.E. Meeting, Houston, Tex., December 1963. Present address, Dartmouth College, Hanover, N. H.
COM MUNI CATION
BINARY COPOLYMER AZEOTROPES Azeotropic copolymer compositions present a unique route to uniform polymers. This simple methodology should not be bypassed when seeking products with sharply characteristic properties. The method for calculating binary azeotropes is reviewed and graphs are presented for simplifying computations. A table of useful examples is included.
language of polymer chemistry, an azeotrope exists when monomer and polymer mole fractions are identical. I n general this is an unusual occurrence. Walling and Briggs (7) have shown that in multicomponent systems only one possible “azeotrope copolymer’’ is possible. This occurrence is frequent enough to merit further attention. I n particular the gener,al two-component system consisting of monomers m- 1 and m- 2, whose coreactivities are related by reactivity ratios, r l and 1 2 , where r l = k l l / k l 2 and r 2 = k 2 2 / k 2 1 , and where ml and m2 are mole fractions in the polymer of monomers whose mole fractions are M1 and M2,can be represented by the differential equation :
I
N THE
d ( M d - ( M I ) (rlM1 d(Md (M2) (r2Mz
+ Mz) +MI)
same monomers in the resulting polymer, this leads to the well known copolymer composition equation ( 2 ) .
As it is known that
Since the ratios of the two monomers’ addition rates into a well mixed reactor a t steady state are also the ratios of these
+
m2
= 1,
Equation 2 can be rearranged to the more common form:
ml =
(1 1
ml
+
rlM12 MiMz rlM? 2MM2 rzM2
+
+
(3)
Using the knowledge that M B= 1 - M I , substituting ml = M1 for azeotropic conditions, and rearranging into quadratic form VOL. 4
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477
~~
Table 1.
Binary Polymer Azeotropes Mob
% m1
m-
m-
1
Acrylamide Acrylonitrile Butyl acrylate Crotonic acid Diethyl fumarate Diethyl fumarate Diethyl maleate Ethyl acrylate Methacrylic acid Methacrylonitrile
Methyl acrylate Methyl methacrylate Styrene
Vinyl acetate Acrylic acid
9, Figure 1.
(rl
MI vs. 4
+ r2
-
2)m?
2
Styrene Styrene Vinylidene chloride Styrene Vinylidene chloride Vinyl acetate Styrene Vinyl acetate Vinyl chloride Vinyl acetate Acrvlonitrile Styiene Styrene n-Butyl methacrylate Methyl methacrylate 2-methyl styrene Styrene Styrene Butyl methacrylate Ethyl acid maleate Methyl acid maleate Styrene
rl 0.35 0.04 0.91
in Areorz trope 0.22 54.5 0.40 38.5 0.37 87.4
0.17 0.52
0.63 0.85
0.01 0.33 0.07 0.30 0.444 0.011 0.47 0.12 0.043 0 . 1 7 0.95 0.44 0.70 0.15 0.25 0.25 0.51 0.69
40.4 43.0 64.0 62.4 46.5
0.70
0.70
50.0
0.25 0.20 0.46 0.66
0.17 0.75 0.52 0.59
52.5 23.8 47.7 54.7
0.13
0.035
52.7
0.522
0.035 66.9
0.03
0.42
+ (3 - 2r2 - r1)ml + (rz-1)
Solutions to this quadratic equation are 6 ) where
1/(1
29.8 23.8
+
ml
=
91.9
73.8 50.0 38.8
37.5
0
(4)
= 1 and m l =
T h e root rnl = 1 is trivial. However, when the latter root is used, useful results are obtained. By using Figures 1 and 2 one can quickly obtain the equilibrium azeotrope between two monomers, if such a n azeotrope exists. Example I
m-l is methyl methacrylate. r l = 1.35
m-2
r2
is acrylonitrile
= 0.18
0.35
4 = -= -0.427 -0.82
but 4 must be 7 0 . Therefore, no azeotrope exists. Example II m-1 is methyl methacrylate. rl
= 0.415
+=-
m-2
= p-chlorostyrene
r2 = 0.89
-0.585 = 5.81 =0.110
1
m l = - = 0.147 6.81
9, Figure 2. 478
I&EC FUNDAMENTALS
r l vs.
4
Therefore an azeotrope exists a t 14.7 mole methacrylate.
yo
methyl
Example 111 m-1
is acrylonitrile. 71
m-2
is styrene
=: 0.04
rz = 0.40
-0.96 +=-0.60
= 1.60
=
= 0.385
m 1
2.60
Therefore a n azeotrope exists a t 38.5 mole yo acrylonitrile. By making use of the Alfrey-Price Q and e parameters, generalized azeotropic composition calculations can be made covering a wide variety of polymer compositions. Evaluations of these approximate correlations and other proposed methods have been well covered in the literature (7-6). Table I contains a list of potentially useful binary polymer azeotropes. Most of the reactivity data used were taken from Young ( 8 ) . Acrylate-styrene azeotropes contain 20 to 30% acrylate, whereas methacrylate-styrene azeotropes contain approximately 45% methacrylate. Acrylonitrile forms azeotropes with styrene a t about the 40% level and with ethyl acrylate a t about the 10% level.
Acknowledgment
T h e author thanks H. L. Gerhart, S. W. Gloyer, and J. J. Reis for their support of work in this area.
literature Cited
(1) Alfrey, T., Bohrer, J. J., Mark, H., “Copolymerization,” Interscience, New York, 1952. (2) Alfrey, T., Price, C. C., J . Polymer Sci.2, 101 (1947). (3) Bamford, C. H., Jenkins, A. D., Zbid.,53, 149 (1961). (4) Bamford, C. H., Jenkins, A. D., Johnson, R., Trans. Faraday Soc. 5 5 , 418 (1959). (5) Charton, M., Capato, A. J., J . Polymer Sci. 2 (3), 1321 (1964). (6) Kawabata, N., Tsuruto, T., Furukawa, J., Makromol. Chem. 51, 70 (1962). (7) Walling, C., Briggs, E. R., J . Am. Chem. Soc. 67, 1774 (1945). (8) Young, L. J., J . Polymer Sci. 54, 411 (1961). J. A. SEINER Pittsburgh Plate Glass Co. Springdale, Pa.
Nomenclature = mole fraction monomer 1 in monomer rnl = mole fraction monomer 1 in polymer
MI
RECEIVED for review July 20, 1964 ACCEPTED June 9, 1965
CO M MUN ICAT I ON
ACTIVITY COEFFICIENTS FROM LIQUID SOLUBILITY DATA FOR T H E n-BUTANE-I-BUTENE-WATER SYSTEM A ternary, three-suffix Margules equation, adapted for liquid-liquid systems, has been used to calculate binary and ternary coefficients from solubility data for the n-butane-1 -butene-water system. The temperature-dependence constants of these coefficients have been used to calculate partial molal enthalpies and entropies.
CTIVITY
coefficients for partially miscible liquid systems
A can be predicted from solubility data by use of the Margules
equation, provided the system is adequately described by this expression. I t was confidently felt by the author that the three-suffix Margules equation (6) should adequately describe all water-hydrocarbon cL,ystems. T h e procedure for predicting Margules constants (limiting activity coefficients) from liquid-liquid solubility data is described by Severance (4) for binary systems, and was first proposed by Carlson and Colburn (7) for van Laar equations. This report deals with a ternary system, so the ternary Margules equation was utilized. Wehe and McKetta (5) have collected and published solubility data for the n-butane-1 -butene-water system. Solubilities were reported for both the aqueous and hydrocarbon phases. Using these data we have computed Margules binary and ternary constants for the above-mentioned system. In these calculations component 1 is n-butane, component 2
is 1-butene, and component 3 is water. I t is felt confidently that the value of A12 should equal that of A21 and that both should be very close to zero. I n our calculations, we assumed that A12 = A21 but did not equate this value to zero. We also tested the four-suffix equation to see if the addition of the third binary constants ( 0 1 2 , 0 1 3 , 0 2 3 ) increased the goodness of fit of the experimental data. I t was found that the addition of these constants did not significantly affect the goodness of fit. Table I shows the Margules constants which were obtained by the method just described. T h e data on the last line of Table I represent the work of Gerster and coworkers ( 3 ) ,in which component 1 is isobutane instead of n-butane. Also, the value of 0.00 for A12 was assumed rather than determined. However, the results of our computations are in good agreement with those of Gerster, and we feel that this agreement justifies our method of treatment of McKetta’s data. Temperature-dependence expressions were calculated for VOL. 4
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