significant change within the temperature range examined. By means of these experiments, Sodasorb was found to be 60% more efficient than Baralyme above 60°F. At 40"F, the efficiency of Sodasorb was 80% greater than Baralyme. Lithium hydroxide was shown to be three times as efficient as Baralyme and two times more efficient than Sodasorb a t room temperatures. At about 40"F, lithium hydroxide was found to be four times more efficient than Sodasorb and eight times more efficient than Baralyme. Acknowledgments The author thanks Grace Treadway for her assistance in performing some of the experimental procedures. The author also thanks the Harbor Branch Foundation Laborato-
ry for providing the financial support needed to complete this program. Literature Cited Cook, R. B.. "Temperature and Pressure Effects on Sodasorb and Baralyme.' USNA-TSPR No. 25, 1972. Miller, R. R.. "Lithium Hydroxide and Soda Lime." The Present Status Of Chemical Research in Atmosphere Purification and Control on Nuclear-Powered Submarines, NRL Report 5465, Apr 21, 1960. Rousseau, J., "Carbon Dioxide Management Subsystems." Section 11, Atmosphere Control System for Space Vehicles, March AD 405-766. 1963.
Tsen C. Wang
Harbor Branch Foundation Laboratory Fort Pierce, Florida 33450
Received for reuiew August 6 , 1974 Accepted J a n u a r y 8 , 1976
Noninterac t ing Control
The analysis of an experimental heat exchanger study provides additional insight into the nature of nonlinear noninteraction control.
Hutchinson and McAvoy (1973) examined the application of on-line multivariable control methods to a n experimental heat transfer facility. While the emphasis of the study was on practical implementation of several multivariable schemes, the work was so effectively presented that it can be used directly to provide a better look a t noninteracting control. The process consisted of a closed, stirred-tank heat exchanger in which water was heated by an immersed steam coil. The controlled variables were the water temperature and level in the tank, while the manipulative variables were the feed flow rate and the steam pressure. A nonlinear, interactive system model was developed as the pair of differential equations
_-
dh __ 1 dt -
dT dt - 60Ah
(1)
- G,)
60pc G, = K C P
(3)
and
$)
+
Since the partial derivatives are fairly constant ove; the range of interest, as shown in Figure 1, and the T responses of the two models are similar, the level responses should be similar. The differences are probably due to a decimal error in the h coefficient, which was given in the cited reference as 0.0681. Noninteracting Control
One of the control schemes studied was the nonlinear derivative diagonalization approach of Liu (1967). In this approach, it is assumed that the controlled process behaves as the noninteracting system
- G,T + G I T ) (2)
where
A P = exp ( a -
level responses of the two models were quite different. The approximation involved in the linearization of eq 1 is
Po(+-)(:
I
:Q)
+
dT dt where a1 and a2 are to be as large as possible without violating constraints. By equating these equations with eq l and 2 , the "nonlinear noninteraction controllers" are obtained directly as - = a,(T, - T)
GI
This can be linearized about the steady state, assuming that Pa 14.7 psia, to give a linear process model in the form dJz = 0.08316,
dt
_ dT _ -- -13.826,
dt
- 0.000030?
- 0.00681~
(5)
- 0.0319T
(6)
In the cited study, the open-loop linear and nonlinear
+
6OA~i(h,- h)
(10)
Some interesting features of this control can be noted if the control functions themselves are linearized about the steady state to give
6,= (3.7 x - 0.0194T'
G,
1
?
lo-')?
+
(0.082 - 12a,)i
= (1.639 - 51.4a2)?
+
712G1
(12) (13)
These "linearized noninteraction controllers" can be comInd. Eng. Chem.. P r o c e s s Des. Dev., Vol. 14, No. 2, 1975
193
Figure 3. Uncoupling controller design logic.
.?
.e
.9
to
h
Figure 1 . Temperature and level dependence of the outlet flow.
Figure 4. Structural-analysis control logic.
controllers with the linearized noninteraction controllers of eq 12 and 13. While the actual nonlinear noninteraction controllers (eq 10 and 11) provide diagonalization and suitable response dynamics in a combined interactive effort, the linearization separates these roles much as the structural-analysis approach does. However, eq 12 and 13 Figure 2. Linearized heat exchanger model. provide only proportional-mode loop feedback and would cause offset. This will also be true of the nonlinear noninpared with the controllers obtained by the structural-anateraction controllers if model error or disturbances exist. lysis approach. The structural-analysis primary feedback controllers, on the other hand, can be specified as PI controllers and can Structural-Analysis Control be made to give zero offset. In fact, if these conventional A structural-analysis control design will now be develfeedback controllers were properly tuned, the structuroped for this example. Structural-analysis control (Greenal-analysis control of this moderately nonlinear heat exfield and Ward, 1967) is a linear uncoupling approach that changer should be more effective than the nonlinear nonuncouples the state derivatives of the linearized process interaction control. model. The noninteraction approach might be useful if an acThe linearized process model, given by eq 5 and 6 curate nonlinear process model is available. However, in above, can be represented as the structure illustrated in its present form, it still presents the following difficulFigure 2. The two intercoupling terms a21 and blz can be ties. compensated by two controller elements f 2 l M M and F1zMC, 1. The arbitrary diagonalization form used (eq 8 and 9) as shown in Figure 3. The application of the invariance is probably not a generally effective choice. condition to the derivative nodes gives the elements di2 . There is no straightforward way to select suitable rectly as values of a1 and a2. The original approach of Liu involved either an off-line iterative solution of the system equations fzlMM = % I (14) for a complete trajectory or an on-line iterative solution a22 for each time increment so as to follow the constraint flzMC = boundary. This last, which is equivalent to a saturation (15) ai 1 treatment, may not provide suitable control without modification. The “uncoupled” subsystems are then controlled by pri3. The calculation of the controller functions corremary feedback controllers f l l ” C and f2z‘C. The form of sponding to eq 10 and 11 above may become prohibitively these is not specified by the method. Rather, the form and difficult in more complex process situations. parameters can be determined by conventional single-loop 4. The diagonalization choice used will always give offmethods. In this sense, the method can be viewed as a diset if there is model error in the nonlinear model. While rect extension of conventional multiloop feedback control. other choices are possible (Rich, et al., 1974), these will The complete structural-analysis control for this example, generally involve additional controller parameters. AS in as illustrated in Figure 4, is defined by the case of a1 and u2 above, there does not appear to be a k1 = (3.7 x 10-4)T + f i l M C h (161 straightforward design available for these additional parameters. Incidentally, it is possible that the linear optimal con?‘ = f22Mc? + 7126, (17) trol method used by Hutchinson and McAvoy could lead Discussion to better results than indicated in the cited study. Since It is interesting to compare these structural-analysis the linearized process model is used in this optimal con-
d
3
194
Ind. Eng. Chern., Process Des. Dev., Vol. 14, No. 2, 1975
trol design, the use of the correct a slightly better design.
h coefficient might
give
Nomenclature The symbols used correspond to those of Hutchinson and McAvoy, with the following additions. a pperscript indicating a perturbation variable ( i . e . , A T = T - T,) fllMC = primary feedback controller relating and f z Z M C = primary feedback controller relating T and T' f#C = uncoupling controller relating T and GL fzlMM = uncoupling controller relating G1 and T' a, b = coefficients in the linearized process model D = disturbance
h
41
Literature Cited Greenfield, G.G.. Ward, T. J , /nd. Eng. Chem., Fundam., 6, 571 (1967). Hutchinson, J. F.. McAvoy, T. J . , Ind. Eng. Chem., Process Des Develo p . , 12, 226 (1973). Liu, s., h d . h?. Cham., Process Des. Deveiop., 6,460 (1967). Rich, S . E.. Law, V . J.. Weaver, R. E. C., "Model Characteristics and the Control Of Nonlinear Multivariate Processes," Proceedings, 1974 Joint Automatic Control Conference. AIChE, New York, N.Y., 1974.
Yang-chu Cheng Thomas J . Ward*
Chemical Engineering Department Clarkson College of Technology Potsdam, New York 13676
Received f o r review August 19, 1974 Accepted November 25, 1974
Prediction of Binary Azeotropes
A method is presented for prediction of binary azeotropic points on P-T-x space as a good approximation for conditions well removed from the critical state. The total pressure to vapor pressure ratio is obtained using the Clausius-Clapeyron equation and is related to the compositions by a correlation for the activity coefficients. The azeotropic temperature and composition are thus calculated from the boiling points and latent heats of vaporization of the pure substances, and the correlation constants for the binary system. The predicted results compare well with the experimental data for binary mixtures comprising polar and nonpolar components.
Introduction Formation of azeotropes is often encountered in industrial separation processes which necessitates study of their pressure-temperature-composition relationship. The azeotropic data available in the literature (Horsley, 1952; Hala, et al., 1968) are mostly a t atmospheric pressure. The approximate method suggested by Carlson and Colburn (1942) for estimating the effect of temperature on azeotropic composition is based on the assumption that the ratio of the activity coefficients is independent of temperature, and the vapor phase can be treated as an ideal gas. Skolnic (1951) developed a graphical method for predicting azeotropic pressure-temperature relationship using the nomographs of Lippincott and Lyman (1946). This method is empirical in nature and is useful only at subatmospheric pressures. The method has a few limitations, namely, the normal boiling temperature of the azeotrope must be available for prediction, and accordingly the method fails for systems which form azeotropes at pressures above atmospheric, Also, the method cannot be applied to systems comprising components belonging to the same group. In this work a method is developed for quick prediction of azeotropic temperature and composition from the boiling points and heat of vaporization of the pure components, and a correlation for the activity coefficient such as the Redlich-Kister (1948) and Van Laar (1913) equations.
In this expression it is assumed that the temperature variation of the molal heat of vaporization is insignificant over the temperature range in question. If the excess free energy is calculated using any one of the commonly used correlations such as the Redlich-Kister equation (31
gE = RTXjX2[AI + Bj(xl - x,)] then the activity coefficients are given as
In y t l = ( A ,
+
In y'z = ( A , -
3Bj)Xz2 - ~ 3B1)Xj2
+
B
~
4B1Xl3
x ~ (4 (5 1
Combining eq 1, 2,4, and 5 one gets T / T 1 = 1 - T(Rxz2/AH1)(Aj+ 3Bi - 4B1X2) ( 6 ) T / T z = 1 - T(RX,*/AHz)(Aj - 3Bj f 4BiXi) (7)
The azeotropic temperature and composition are then calculated from simultaneous solution of eq 5 and 6. Similarly, if the activity coefficients are repted to composition by the Van Laar equation as (8)
(9) Met hod Assuming vapor phase to be an ideal gas, the activity coefficient of an azeotropic system is given as
Then the azeotropic temperature and composition are obtained from simultaneous solution of eq 10 and 11.
From the Clausius-Clapeyron equation, the total pressure to vapor pressure ratio (Malesinski, 1965) is expressed as Ind. Eng. C h e m . , P r o c e s s D e s . D e v . , Vol. 1 4 , No. 2, 1975
195
~
