High-Pressure Va pot-liq uid Equilibria. Calculation of Partial Pressures from Total Pressure Data. Thermodynamic Consistency Kwang W. Won and John M. Prausnitz* Chemical Engineering Department, University of California, Berkeley, Calif. 94720
Barker's method for reduction of isothermal total-pressure data i s extended to binary systems containing one supercritical component. The isothermal, nonisobaric Gibbs-Duhem equation i s used to calculate y-x results from experimental P-x data at constant temperature. This calculation has two applications: if P-x data are available, vapor compositions can be calculated and need not be measured; or, if P-x-y data are available, they can be checked for thermodynamic consistency b y comparing y (experimental) with y (calculated from P-x data alone). Illustrative calculations are given for several binary systems.
T o reduce experimental effort, it is frequently common practice to calculate vapor-liquid equilibria a t low or moderate pressures from isothermal total-pressure data (see, for example, Orye and Prausnitz, 1965). This procedure, based on the Gibbs-Duhem equation, can be carried out in several ways (Van ?Jess, 1964) but the most convenient method is that of Barker (1953). The total-pressure method reduces experimental work because the composition of only one phase (either vapor or liquid) needs to be measured; the composition of the second phase is calculated from the total-pressure data. However, if experimental data a t constant temperature are available for the compositions of both phases in addition to the total pressure, it is possible to test the data for thermodynamic consistency, as discussed in several books (see, for example, Hougen, et al., 1959; Prausnitz, 1969; Van Ness, 1964); many of these tests are based on the equal-area criterion (Chueh, et al., 1965; Herington, 1947; Redlich and Kister, 1948; Thompson and Edmister, 1965). It has recently been shown by Van Ness (1973) that this type of test is not satisfactory and that the most reliable procedure for testing data is to calculate y's from isothermal P-x data and then to compare the calculated y's with those measured. For thermodynamically consistent data this comparison yields negligible difference between calculated and observed y's. The purpose of this work is to extend to high-pressure vapor-liquid equilibria the ideas briefly summarized above. This extension is not trivial for three reasons. (1) I n most highpressure equilibria one of the components is supercritical. It is therefore preferable to use the unsymmetric convention of normalization for activity coefficients which then requires calculation of Henry's constant. ( 2 ) Vapor-phase nonidealities are highly significant. They cannot be estimated by the simple virial equation used in Barker's method. (3) The isothermal differential Gibbs-Duhem equation includes a term with the differential of the total pressure. This term is negligibly small a t low or moderate pressure, but is significant a t high pressures, especially near the critical region. It is desirable to make the extension to high-pressure systems for two applications. First, since it is difficult to withdraw truly representative samples for chemical analysis from a high-pressure equilibrium apparatus, it is advantageous to
sample only one phase, rather than two. Second, if both phases are sampled for analysis, it is often of interest to determine the reliability of the data. Testing for thermodynamic consistency provides a valuable tool for this purpose. Extension of Barker's Method
We consider a binary system a t constant temperature; the heavy component is designated by subscript 1 and the light component by subscript 2. Assuming that P-x data are available, our task is t o calculate y. To do so we utilize the GibbsDuhem equation for isothermal but not isobaric conditions and therefore we must also know the molar volume of the liquid mixture as a function of 5 along the saturation line. We assume that this volume is available either from experimental measurements or else that it can be satisfactorily calculated from a well-established correlation for mixtures of normal fluids (Prausnitz and Chueh, 1968). The total pressure is the sum of the two partial pressures
P
=
YlP
+ y2P
(1)
Each partial pressure is related to an activity coefficient
r by
(3) where 4 is the vapor-phase fugacity coefficient and fo is the standard-state fugacity in the liquid phase. For component 1, the standard-state fugacity chosen is the fugacity of pure, saturated liquid 1 a t system temperature fi0
=
(4)
Pl"1S
where PlS is the saturation (vapor) pressure and $ l a is the fugacity coefficient a t saturation of pure liquid 1. For component 2, the standard-state fugacity chosen i s Henry's constant for solute 2 in solvent 1 a t system temperature and a t PI".Thus f2Q
=
42Y2P H2,13 lim __ 0'?2
x?
Ind. Eng. Chern. Fundam., Vol. 12, No. 4, 1973
459
I600
I
1400
I
I
I
'4
I
I
1
1
1
Z E X PFITTED ERIMENTAL
EXPERIMENTAL FITTED
0
PREDICTED
0 I
I I
P I
P
0
I
m200 0
0
0.1
0
0.2 0.3 0.4 0.5 0.6 0.7 MOLE FRACTION METHANE
Figure 1. P-x-y diagram for propane ( 1 )-methane (2) at 100°F (data of Reamer, et a/., 1950)
The asterisk on I?** indicates that the activity coefficients used here are normalized by the unsymmetric convention
rl -+ 1 rz*-,1
( 2 , + 1) (x2+ 0)
(6)
The activity coefficients used here are designated by the capital letter r (rather than by the conventional lower-case letter y) to emphasize that a t constant temperature they depend not only on x but also on P . W e now choose an arbitrary function F2(xz)to represent the variation of the logarithm of rz*with x along the saturation line In
rz*= F z ( x z )
Since uL is assumed to be known along the saturation line, the function G can be determined readily and contains no unknown coefficients. We now obtain an expression for In rl from the GibbsDuhem equation
- X?)
d In
___
dxz
+
d In 2 2
~
rZ* _- - -uL
dxz
--qA' 0
0.2
0.4
dP RT dxz
(9)
rl = F1(x2)
(10)
where
0.8
1.0
Figure 2. P-x-y diagram for propane (1)-methane (2) at - 7 5 O F (data of Wichterle and Kobayashi, 1972)
The unknown coefficients appearing in FZ (and, through eq 11 in 8'1)are determined by fitting experimental total pressures to those calculated from eq 12. Iteration Procedure
TO obtain a fit of experimental and calculated pressures it is necessary to iterate because the fugacity coefficients are functions of y in addition to pressure and temperature. I n the first iteration, let $1 = $2 = 1 and FZ = 0 for all X Z . Then, with F I given by eq 11 y1 =
X1jlO - exp F 1 P
y2 = 1 -
(13) (14)
y1
Using the y's from eq 13 and 14 we proceed to estimate fz0 from experimental P-x data by
-
where x z is chosen to be small (typically x2 0.01). Assuming some F2(x2)we again find F 1 from eq 11 and now we proceed to fit the P-x data (eq 12) by adjusting the constants in F2 to give the best fit. The next set of y's is calculated from X I j10
- exp F I
yl =
91p
Rearranging and integrating, we obtain In
0.6
MOLE FRACTION METHANE
(7)
The function Fz contains a number of unknown coefficients which will be determined from the total-pressure data as shown later. Further, we represent the experimental (or calculated, see below) molar liquid volume of the mixture by an arbitrarily chosen function of pressure G ( P )
(1
I
d
100-
y2 =
XZjiO exp F z 9ZP ~
Equations 16a and 16b are not explicit in y because I$ depends on y. The y's found from eq 16a and 16b may not add to unity; for the calculation of I$ they are normalized by
The function F 1 contains only those unknown coefficients which are in the function F z of eq 7 . When eq 2, 3, 7 , and 10 are substituted into eq 1 we obtain the working equation
P
XlfIO
= - exp $1
F 1 + X?fiO - exp F Z $2
460 Ind. Eng. Cham. Fundam., Vol. 12, No. 4, 1973
(12)
I n each subsequent iteration, a new fzo is found from fzO
=
4ZYZP lim ___ x2
22-0
(18)
Table I. Representation of Isothermal P-x Data Av pressure dev, psia"
Tezp,
F
System
C
B
A
2 35 - 3 04 45 75 -- 00 937 0 498 3 3 -lis 545 0 648 100 1 449 4 179 -10 0 10 2 160 1 76 -2 37 7 90 0 78 1s defined by a75 = (l/n)z, a Average pressure deviation IP,,,,I - P c a l e d l , where n = number of experimental points. CHd-CaH8 CHd--C3Hs CH,-H,S C02-C4Hio
100
Table II. Henry's Constants, Critical Pressures, and Deviation in Vapor-Phase Compositions Critical System
O F
CHI-C3Hs CHd-CyHs CH4-H2S C02-C,Hio
100 -75 100 160
Av dev in
pressure, psia -
Temp,
x
H , psia
Calcd
Exptl
I
2333 1016 7009 1633
1351 950 1891 1185
1353 944 1907 1184
10.9 7 10 8.3
103"
II
4.3 3.4 5.6 5.5b
The average deviation is defined by & = (l/n)& 1 yerDti , where n is the number of experimental points. Column I is calculated with all experimental points. Column I1 is calculated with all experimental points except the critical point and the one closest to it. b Calculated with linear rule for v,,~.
0
600 800 1003 1x0 PRESSURE, psia
200 400
Figure 3. G(P) for CaHs (1)-CH4 (2) at -75°F
(predicted)
a
yonled
where the 62 and yi are takeii from the previous iteration. The limit is obtained by linear extrapolation to x2 = 0. I n each iteration the constants in F2 are adjusted to fit the P-x data, as represented by eq 12. Iterations are continued until the adjustable constants no longer change. Illustrative Examples
To illustrate the procedure 11e have made calculations for foul systems: methane-propane a t 100"F, methane-propane a t - 75°F; methane-hydrogen sulfide a t 100°F; and carbon dioxide-butane a t 160°F For hi ri*a e assume a function similar to one used earlier by Prausnitz and Chueh (1968) 111
r2*=
()'
+ ~ ( a -~ 4
[ i(apL2 - 2a2)
11.
-+
4/3a23)
c ( w - 6/aai5)1 (19) ahere A , B , a i d C are empirical constants to be found from fitting P-x data. The \ olume fraction a2is defined by a2
X2LC,
= XlUC,
+ xzvc,
where u, is the critical volume. We can now obtain a n expression for ln Duhem equation (eq 11). It is
(20)
rl from the Gibbs-
Liquid-phase volumetric data along the saturation line are given by a n empirical expression of the form vL
-
RT(1 - 2 , )
where vo, a, b, and c are constants determined by fitting to the experimental data. For each isotherm two sets of constants are used, one for the region remote from the critical and one for the region near the critical. (For the region remote from critical, uo = v18.) When no experimental volumetric data are available, they are estimated by the method described by Prausnitz and Chueh (1968). I n our calculations, vapor-phase fugacity coefficients were obtained from a modified Redlich-Kwong equation as discussed by Prausnitz and Chueh (1968). Equations 19, 21, and 22 appear to be suitable for fitting isothermal P-x data of nonpolar systems. Table I gives the empirical coefficients A , B , and C for all four systems and also indicates the small deviation between calculated and observed total pressures. As expected, the fit is least good in the immediate critical region. Calculated critical pressures are, however, in good agreement with those observed as indicated in Table 11. Henry's constants are also given in Table Figure 1 shows results for met'hane-propane a t 100°F and Figure 2 shows results for the same system a t -75°F. Phaseequilibrium and volumetric data a t the higher temperature are from Reamer, et al. (1950), and phase-equilibrium data a t the lower temperature are from Wichterle and Kobayashi (1972). Volumetric data a t - 75"F, calculated from Chueh's method, are shown in Figure 3. The solid lines in Figures 1 and 2 show the fit of the liquidphase P-x data. The dashed lines shorn the predicted vaporphase P-y curve. Both figures indicate very good agreement between calculated and observed vapor-phase behavior. Figure 4 gives results for the system methane-hydrogen sulfide a t 100°F; phase equilibrium and volumetric data are from Reamer, et al. (1951). Again predicted and observed vapor-phase compositions are in very good agreement. Finally, Figure 5 gives results for the system carbon dioxide n-butane a t 160°F based on P-x data reported by Olds, et al. (1949). Volumetric data for the liquid phase, shown in Figure 6, are also from Olds, et al. (1949). For this binary system, the calculated vapor compositions are sensitive to details in the calculation of fugacity coefficients. Since the components in this system differ markedly in size [vc(C02) = 94 while v,(C4H10) = 255 cc/mole] the fugacity coefficients depend on Ind. Eng. Chem. Fundam., Vol. 12,
No. 4, 1973
461
----
I
I6’
I800 -
1
I
I
I
I
I
EXPERIMENTAL POINTS -FITTED
II
1
0.9
.-
I
0.7
0.6
0.3 0.2
--0
2001
i
EXPERIMENTAL FITTED PREDICTED
0
0
0.1
0.2 0.3 0.4 0.5 0.6 0.7 MOLE FRACTION METHANE
Figure 4. P-x-y diagram for hydrogen sulfide ( 1 )-methane (2) at 100°F (data of Reamer, et a/., 1951)
ol
0
I 0.1
1 1 I I I I 0.2 0.3 0.4 0.5 0.6 0.7 MOLE FRACTION CARBON DIOXIDE
L
0.1
Figure 5 . P-x-y diagram for n-butane (1)-carbon dioxide (2) at 160°F (data of Olds, et a/.,1949)
the mixing rule used to obtain the mixture parameter vC,?. We notice that better results are obtained when an arithmetic mean, rather than a Lorentz mean, is used to calculate this parameter. This difference in results points out once again the difficulty of calculating thermodynamic properties of dense fluids containing molecules of appreciably different size. Prediction of Vapor Compositions and Thermodynamic Consistency
With only isothermal P-x-vLdata, vapor-phase compositions were calculated as shown in Figures 1-4. With the exception of the region very close to the critical, predicted and observed vapor-phase compositions are in good agreement as indicated in Table 11. It appears, therefore, that extension of 462
Ind. Eng. Chem. Fundam., Vol. 1 2 , No. 4, 1973
XK)
4co
600 800 IC00 1200
PRESSURE,
Psi0
Figure 6. G(P) for n-C4Hlo(l)-COt (2) at 160°F (data of Olds, et a/.,1949)
Barker’s method to high pressures may be useful for reducing the experimental effort required to obtain high-pressure vaporliquid equilibria. Since the calculation is based on the Gibbs-Duhem equation, a comparison of calculated arid observed vapor-phase composition provides a test of thermodynamic consistency. Provided that the equations chosen give a good representation of the experimental P-x data, and provided that the vapor-phase equation of state gives reliable fugacity coefficients, serious disagreement between calculated and exp erimental vapor compositions indicates lack of thermodynamic consistency due to experimental error. The favorable results given in Table I1 suggest that the data considered here are satisfactorily consistent. There is no fundamental difficulty in finding a set of thermodynamically consistent equations to fit the P--2: data. Equations 19 and 22 were arbitrarily chosen in this work and justified empirically. Other empirical equations may be chosen to obtain as close a fit of the P-x data as desired. However, a more fundamental difficulty arises in the choice of a vaporphase equation of state. The large importance of this choice is shown in Figure 5 ; in some cases, especially in those where the components differ appreciably in chemical nature or in molecular size, small changes in the equation-of-state mixing rules can have large effects on the fugacity coefficients and hence on calculated vapor-phase compositions. Therefore i t is not possible to reject a set of P-2-y data as thermodynamically inconsistent unless it can be shown that the vaporphase fugacity coefficients used give a correct representation of vapor-phase properties. Nevertheless, for many binary high-pressure systems of industrial interest, it is possible to test P-x-y data for thermodynamic consistency because vapor-phase volumetric data are often available. I n that event it is possible to make a n independent check of the fugacity coefficients and hence to make a n unambiguous comparison between calculated and observed vapor compositions. literature Cited
Barker, J. A., Aust. J . Chem. 6, 207 (1953). Chueh, P. L., Muirbrook, X. K., Prausnitz, J. AI., A.I.Ch.E. J . 11, 1097 (1965). Herington, E. F. G., Nature (London)160, 610 (1947).
Hougen, 0. A., Watson, K. M., Ragatz, R. A., “Chemical Process Principles,” Part 11, p 874, Wiley, Yew York, Y. Y.,
19.59. Olds, R. H., Reamer, H. H., Sage, B. H., Lacey, W. X., Ind. Eng. Chem. 41, 47.5 (1949). Orve, 1%.V., Prausnitz, J. M.,Trans. Faraday SOC.61, 1338 (1965). Prausnitz, J. M,, “Molecular Thermodynamics of Fluid-Phase Equilibria,” p 214, Prentice-Hall, Englewood Cliffs, N. J., 1969. Prausnitz, J. >I., Chueh, P. L., “Computer Calculations for High-Pressure Vapor-Liquid Equilibria,” p 83, Prentice-Hall, Englewood Cliff?, N. J., 1968. Reamer, H. H., Sage, B. H., Lacey, W. N., Ind. End. Chem. 42, 343 (1950).
Reamer, H. H., Sage, B. H., Lacey, W. N., Ind. Eng. Chem. 43, 976 (1951).
Redlich, O., Kister, A. T., Ind. Eng. Chem. 40, 345 (1948). Thompson, R. E., Edmister, W. C., A.I.Ch.E. J . 1 1 , 457 (1965). Van Ness. H. C.. “Classical Thermodvnamics for Son-Electrolyte Solutions,’; p 107, Pergamon, Eimsford, N. Y., 1964. Van Ness, H. C., A.I.Ch.E. J . 19, 238 (1973). Wichterle, I., Kobayashi, R., J . Chem. Eng. Data 17, 9 (1972). RECEIVED for review March 12, 1973 ACCEPTED June 26, 1973 For financial support the authors are grateful to the Sational Science Foundation, the American Petroletim Institute and the Xatural Gas Producers Association.
Parallel Cascade Control William 1. Luyben Department of Chemical Engineering, Lehigh University,Bethlehem, Pa. 18015
The differences between conventional “series” cascade control systems and configurations that are of a “parallel” type are explored. Quantitative analysis and controller design for each type of system shows the improvements in load responses. Second-order processes, with and without deadtime, are studied over a range of time constants.
C a s c a d e control systems are widely used in industry for improving the dynamic response of systems. Cascade systems are particularly useful in reducing the effects of load disturbances that are introduced into the secondary or slave loop. The majority of these systems are of the series type; Le., the output of the secondary loop process transfer function is the input to t,he primary loop process transfer function. d typical example (Figure 1) is tray temperature control in a distillation column cascaded to reboiler steam flow control. Steam flow rate is the output of the secondary loop process transfer function, and steam flow is the input to the primary loop process transfer function (temperature/steam flow). These series cascade systems were quantitatively studied by Franks and Worley (1956). Cascade control is sometimes used in process systems where the primary and secondary process transfer functions are not in series but are in parallel. Jauffret (1973) cited one example, the temperature control of subcooled reflux by cascade control of exit cooling water temperature in a condenser. The manipulative variable, cooling water flow, affects both exit cooling water temperature and reflux temperature through parallel transfer functions. Ainother example (Figure 1) is the overhead composition control of a distillation column by cascade control of a tray temperature. The manipulative variable, reflux flow, affects overhead compositioii and tray temperature t,lirough two parallel process transfer functions. The purpose of this paper is to explore quant,itatively the differences between the series and parallel configurations. Stability Analysis
Figure 1 shows two cascade control loops on a distillation column. The lower temperature control loop maintains a
temperature on some tray in the stripping section of the column by changing the setpoint of a steam flow controller. The secondary or slave process transfer fuiictiou Gs in this system is the valve transfer function, the relationship between the flow controller output and the steam flow rate. The primary or master process transfer function G ~ is I the relationship between steam flow and t,ray temperature. These two process transfer functions are in series, as illustrated in Figure n
L.
The stability of the slave loop depends on the roots of the closed loop characteristic equation 1
+ BsGs = 0
(1)
The stability of the master loop in this conventional series cascade system, with the slave loop on automatic, depends on the roots of the closed loop characteristic equation
Without cascade control, the closed loop characteristic equation for the system is
1
+ BMGMGS 0 =
(3)
The upper temperature control loop in Figure 1 illustrates a parallel cascade process. Overhead vapor composition is controlled by changing the setpoint of a tray t’emperature controller. The manipulative variable in this system is reflux flow rate. It affects both overhead composition and tray temperature through two distinct transfer functions. Thus the process transfer functions are not connected in series. Reflux has parallel effects on temperature and overhead composition. These effects are, of course, interdependent and interacting. Reflux does not affect temperature first, which then affects Ind. Eng. Chem. Fundam., Vol. 12, No. 4, 1973
463
