Example 111 m-1
is acrylonitrile. 71
m-2
is styrene
=: 0.04
rz = 0.40
-0.96 +=-0.60
=
m 1
=
2.60
=
1.60 0.385
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
MI
= mole fraction monomer 1 in monomer rnl = mole fraction monomer 1 in polymer
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 c stems. all water-hydrocarbon L,y 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
NO. 4
NOVEMBER 1 9 6 5
479
Table 1.
Temp., K. 31 1 344 377 41 1 31 1
Margules Constants for the n-Butane-1-Butene-Water System
A23
A32
Ala
A31
3.69
2.79
4.28
3.18
each of the coefficients listed in Table I. T h e temperature effect of each constant is adequately described by the equation :
A12 =
AH
=
-RT2
(F)
d In y
(y)
is known, where y represents a finite activity coefficient. T h e A’s in Table I1 are logarithms of activity coefficients a t infinite dilution ; therefore partial heats of solution are calculated from these data. The parenthetical factor of Equation 2 can be obtained by differentiating Equation 1 with respect to temperature. This manipulation gives the expression dln y
dT
-
dA dT
-
-Ez T2
(3)
Table II. Temperature-Dependence Constants Margules Coe.cients E1 1.622 658.8 -3.098 1833.6 1.979 735.7 -3.716 2146.3 -0.286 107.4 4.902 -1177.4 4.290 -1015.0
Table 111. Partial Heats and Entropies of Solution at 25’ C. for All Possible Binary Combinations in the n-Butane-1Butene-Water System
Binary Composition 1-Butene inf. dil. in water Water inf. dil. in butene n-Butane inf. dil. in water Water inf. dil. in n-butane n-Butane inf. dil. in 1-butene
Correspon din,g Margulgs Coeficient ’423
A32 A13 A31
A12
~
480
l&EC FUNDAMENTALS
1.13
C* 1 07
Combining Equations 2 and 3 we obtain for partial heat of solution the expression :
AH This equation presumes the heats of solution to be independent of temperature, which is not true. However, fairly good linearity is observed over the indicated range if one plots the Margules coefficients of Table I as a function of temperature. This means that heats of solution for the indicated binaries do not vary appreciably over the designated temperature range. In Table I1 the temperature-dependence constants of Equation 1 are listed for all the binary and ternary Margules coefficients obtained. From these data, heats of solution can be obtained. From basic thermodynamics the expression :
C
Ail
0.043
= 2.303 RE2
The factor 2.303 is included in Equation 4 because the A’s of Table I are common logarithms of the limiting activity coefficients. Since AF = 2.303 R T log y, the Margules coefficient, Aij, is a n evaluation of the partial free energy of solution of component i infinitely dilute in component j . Using this fact with the results of Equation 4 the entropy of solution can be calculated from the expression :
AF
=
AH - TAS
AS, Cal./Mole -6.7 13.6 -8.7 14.8 1.3
(5)
Using Equations 4 and 5 partial heats and entropies of solution have been calculated for all possible binary combinations of n-butane, 1-butene, and water. T h e calculations were also made for each component of the binary a t infinite dilution in the other component. Table I11 summarizes these calculations. The values of AH corresponding to A32 and A31 in Table I11 are very close to those calculated for the same system using the Hildebrand method (2). Also the absolute values of AS, except for the n-butane-1-butene binary, are close to AS values arrived a t by Raoult’s law calculations assuming XZ= 1. The change from positive to negative entropy of mixing when going from a system of water infinitely dilute in hydrocarbon to one where hydrocarbon is infinitely dilute in water may be attributed to a difference in solvation effects between the two systems. This would mean that a system of water infinitely dilute in hydrocarbon would tend to solvate, representing an increase in disorder, while a system of hydrocarbon infinitely dilute in water would tend to repulse solvation, thus tending toward a more ordered system. T h e values of AH and AS in Table I11 corresponding to A12 probably have no meaning, since A12 was so close to zero a t all temperatures represented. There should be very little nonideality between two hydrocarbons, so their activity coefficients should approximate unity. Acknowledgmenl
The author acknowledges the assistance of G. W. Harris and J. Pignato in programming and calculating the Margules constants.
AH, KcallMole 3.03 8.44 3.38 9.38 0.50
(4)
Nomenclature
A
= Margules first and second binary constants = Margules ternary constant D = Margules third binary constant temperature-dependence constants for A El, Ez s = partial molal entropy of mixing
c
literature Cited
(1) Carlson, H. C., Colburn, A. P., Znd. Eng. Chem. 34, 581 (1942). (2) Hildebrmd, J. H., Scott, R. L., “Solubility of Nonelectroly’es,” 3rd ed., pp. 119-33, Reinhold, New York, 1950. (3) Jordan, D., Gerster, .1. A., Colburn, A. P., Wohl, K., Chem. Eng. Progr. 46, 601 (1950). (4) Severance, \V. .4.N., Akell, R. B., Fitzjohn, J. L., IND.ENG. 2, 246 (1963). CHEM.FUNDAMENTALS
(5) Wehe, A. H., McKetta, J. J., J . Chem. Eng. Data 6 , 167 (1961). (6) Wohl, K., Trans. Am. Znst. Chem. Engrs. 42, 215 (1946); Chem. Eng. Progr. 49, 218 (1953). H. M . SMILEY Union Carbide Cork. South Charleston, W . Va.
RECEIVED for review December 21, 1964 ACCEPTEDMay 27, 1965
CO M M UN I CAT1ON
U S E OF E X P E R I M E N T A L D A T A I N D E S I G N OF M U L T I V A R I A B L E CONTROL Multivariable (control systems designed on the basis of experimental input-output data can be extremely sensitive to approximation errors. The practice of fitting an approximate mathematical model to the data for analytical solution of the control problem i s critically examined. Numerical examples illustrate that control elements designed in this fashion may be unlike those needed. An alternative method is suggested which yields more accurate results by delaying all curve-fitting approximations until the final step in the design. HE design of control elements for multivariable systems Thas been studied analytically by a number of workers (2-5, 7). A detailed linearized treatment of a chemical reactor was discussed by Bollinger and Lamb (7). These studies show that within the limitations of stability and realizability, it is feasible to design a noninteracting multivariable controller matrix, K(s), that will produce an arbitrary closedloop transmission matrix, T(s), if a good linear model of the system is available. I n dealing with processes of chemical engineering interest it is common, however, to encounter nonlinearities, distributed parameters, or time-varying coefficients in complex combinations. As a result it is usually necessary to use linear, lumped parameter approximations, or to rely heavily on experimental data. Frequency response testing is a well known method of obtaining the requisite information for each inputoutput pair, but a number of alternative linear approaches are possible. By whatever technique it is obtained, the result is usually expressed as the plant matrix, P(s), or its equivalent, a set of Bode plots. Implicit in this treatment is the supposition that the approximations made in data smoothing and curve fitting are acceptable, or at any rate are as good as can be achieved. Indeed, bvith regard to one system studied by Bollinger and Lamb ( 2 ) , it was concluded that a controller designed from reasonable initial plant approximations would provide a satisfactory control quality. This communication shows that such approximations can be misleading, producing in some cases controller designs considerably in error.
Analytic Approach
Consider a dynamic system described by the matrix equation
where c(s) is an n vector of outputs, x(s) is a n n vector of controllable inputs, and y ( ~ )is a n m vector of uncontrollable disturbance inputs. P(s) and D(s) are n X n and n X m
matrices, respectively, representing the dynamic relations between inputs and outputs. A likely control scheme for this system is shown in Figure 1, where E(s) and K(s) are matrices of control elements of the indicated order. Other control configurations are possible (6) but do not alter the nature of the results. An equation relating the output vector, c , to the input vectors, r and y, may easily be derived from the block diagram. T h e result is c =
(I
-
PK)-‘Pr
+ (I - PK)-’(PE
- D)y
(2)
If it is desired that the control be noninteracting-Le., each of the n outputs be independently controlled by one of the manipulative inputs-and that the disturbance inputs be compensated by feedforward control, the requirement is : c =
Tr
(3)
where T(s) is a diagonal n X n matrix containing the desired
n
Figure 1.
I
Block diagram of control configuration VOL. 4
NO. 4
NOVEMBER 1 9 6 5
481
