Conclusions
and the Hamiltonian is given by
R(y,x,u) = (bu’ [b(u1 + u2)
-
+ (-
x1’)yl’
- x121y12 +
(-
I u1 I
u1 11’2
- x21)yz’ -] ’ .I 1’2 - ~ 2
+ ~ ) y 2(18”) ~
The auxiliary variables are defined as before
with solutions
ytf
There may be an advantage to considering discrete steadystate optimization problems as transient problems. By visualizing a perturbation in the neighborhood of the steady-state solution of the system, the problem may be cast into one which uses the continuous maximum principle wit!iout the requirement of adding convexity specifications. Two explicit systems have been analyzed here with positive results. The approach should be of general applicability unless it is impossible to make the transition from the discrete to the continuous formulation.
= C,.‘ec
literature Cited
From the fact that we maximize x?, it follows that the only nonzeroyj(t1) isyZ2(tl) = - (-1) = 1 and thus, that y22 =
with all other 18” becomes
3.2
= 0.
fl = - ( 1
u111/2
el--21
>0
(34)
Thus, the Hamiltonian in Equation
+ ’ u2
I 112
+ xz2)el--11
(35)
Now we wish to find that u* = [u’ u z ] * which minimizes fl. Since e‘-‘1 > 0, we need only consider the term in the parentheses of Equation 35. By inspection we see that R possesses a minimum a t u1 = 0 and u2 = 0. I n this case, we cannot differentiate R to find u1 and u2, since a t the origin fl is not differentiable (consider u as a scalar, form d ’ u 1 l / z / d u and note the consequence of positive and negative u’s approaching the origin).
Arimoto, S., J . Math. Anal. Appl. 17, 161 (1967). Athans, M., ZEEE Trans. Auto. Control 11, 580 (1966). Chane, S. S. L., ZEEE Trans. Auto. Control 12, 121 (1967). Franc A,, Lapidus, L., Chem. Eng. Progr. 62,’No. 6. 66 (1966). Halkin, H., J . SZAMControl4, 90 (1966). Holtzman. J. M.. ZEEE Trans. Auto. Control 11. 30 (1966a). Holtzman; J. M.; ZEEE Trans. Auto. Control 11; 273(1966b). Holtzman, J. M., Halkin, H., J . SZAM Control 4, 263 (1966). Horn, F., Jackson, R., IND. END. CHEM.FUNDAMENTALS 4 , 110 ~~
(1965).
Jackson, R., Horn, F., Intern. J. Control 1, 389 (1965). Jordan, B. FV., Polak, E., J . Elect. Control 17, 697 (1964). Pearson, J. D., Intern. J . COnt70l 2, 117 (1965). Pontryagin, L. S., Boltyanskii, V. G., Gamkrelidze, R. V., Mishchenko, E. F., “The Mathematical Theory of Optimal Processes,” Interscience, New York, 1962. Rosenbrock, H. H., Storey, C., “Computational Techniques for Chemical Engineers,” Pergamon Press, New York, 1966.
RECEIVED for review November 20, 1967 ACCEPTED June 14, 1968
ESTIMATION OF MORSE POTENTIAL PARAMETERS FROM THE CRITICAL CONSTANTS AND THE ACENTRIC FACTOR D A N I E L D. KONOWALOW AND STEVEN L. GUBERMAN’ Dejartment of Chemistry, State Uniuersity of New York at Binghamton, Binghamton, N . Y .
73901
Reliable estimates of the most likely Morse potential parameters are obtained from the same data (T,, P,, and a single vapor pressure datum) needed to calculate Pitzer’s acentric factor, w, by invoking corresponding states arguments. The errors in reproducing experimental viscosities and second virial coefficients are typically 1.5 times larger for the estimated potentials than for the most likely ones. For normal fluids these properties are reproduced with comparable efficacy by Morse and Kihara potentials which are estimated from corresponding states relations. The estimated parameters E and (r for the two potentials agree within 5y0for substances whose acentric factor is in the range 0 5 w 5 0.3.
Konowalow (1967) showed that the dimensionless R Morse potential parameter, c, could be estimated simply ECENTLY,
from Pitzer’s (1955) acentric factor, w . This enables us to estimate the remaining Morse parameters, e / k and 6,from the critical constants T, and P, by invoking corresponding states arguments. Thus, we are able to estimate the three Morse potential parameters from the same set of data (T,, P,,and a single vapor pressure datum) needed to calculate w . Present address, Department of Chemistry, California Institute of Technology, Pasadena, Calif. 91 109. 622
l&EC FUNDAMENTALS
The Morse function $ ( r ) = 44y2
-y)
y = exp[c(l
- r/u)]
(1)
(2)
was first used as a n intermolecular potential by Mayer and Careri (1952). Later, Konowalow et al. (1961) developed algorithms for calculating B ( T ) and its first two temperature derivatives. Love11 and Hirschfelder (1962) and Smith and M u m (1964) provided the means to calculate various transport properties. Since then, a number of applications of the Morse
potential to the equilibrium and transport properties of dilute gases have appeared. This paper is restricted to those Morse potentials obtained by simultaneously fitting experimental values of the viscosity and the second virial coefficient according to the E1 procedure (of Konowalow and C a d (1965). At present, such most likely Morse potentials are available for 27 substances; 24 of them are considered here. Estimating and Testing the Morse Potentials
Table I contains two groups of 12 substances. The first, termed the basic group, comprises essentially the same collection of normal fluids which was used by Konowalow (1967) to establish the relation G
= 5
+ llw
(3)
between the most like1,y values of the Morse parameter, c, and Pitzer’s (1955) acentric factor. The second group of 1 2 substances, termed the test group, comprises a wide variety of structural types of molecules. Attention was restricted to corresponding states correlations of the type e l k = f ( c , T,) and u = g(c, c / k , P,). Many functional forms for these relations were considered. The best such expressions were found by a least-squares fitting of the critical constants to the known, most likely values of the Morse parameters for the basic group of substances. The results are:
e l k == 2.5 T,(1
- 3/c)
= [2i/2.2 d k ~ , (~ 4) 1113
(4) (5)
Here, d = 1 K./atm.-cu. A. is a dummy constant introduced for dimensional consistency. T h e constants (2.5, 3, 2.2, and 4) are dimensionless when. the remaining quantities appearing in Equations 4 and 5 are expressed in their usual units, as indicated in the Nomenclature section. The least-squares analysis actually yielded irrational constants which differed by less than 0.0570from the rational ones cited above. T h e substitution of c from Equation 3 into Equations 4 and 5 yields
e l k = 2.5 T,[1
- 3/(5
+ llw)]
better than another. For Xe, isobutane, n-hexane, benzene, COZ, N20, and SO?,Table I shows that one of the properties is described better by the estimated potential than by the most likely potential. This is possible, since the most likely potentials are defined to provide a particular compromise fitting cjf B ( T ) and v( T ) data. The compromise offered here is ease in estimating Morse potentials a t the cost of accuracy with which they describe the two properties. The first two lines of Table I1 summarize the fitting errors for the individual substances listed in Table I. For both the basic group and the basic-plus-test group of substances, the average r.m.s. per cent deviation for q ( T ) is about 1.5 times larger for the estimated potentials than for the most likely potentials; the corresponding factor is about 2 for the r.m.s. deviations for B( T ) . T h a t these factors are similar for both the basic and the test group of substances indicates that Equations 3, 6, and 7 are applicable to a wide variety of substances. The r.m.s. per cent deviation is a more meaningful error measure than the r.m.s. deviation for substances whose B ( T ) data neither span nor closely approach the Boyle temperature. For the 16 substances listed in Table I for which the per cent error is appropriate, the average of the r.m.s. per cent error in fitting B ( T ) is 2.7% for the most likely potentials and 4.2% for the estimated potentials. Thus, for both B ( T ) and o ( T ) , the average r.m.s. per cent error is about 1.5 times larger for the estimated potentials than for the most likely potentials.
(6)
+ 1 i w ) ( 2 + 1iw)/2.2 ~ P , ( I + 1iw)11/3 (7)
[2.5 ~ , ( 5
Equations 3, 6, and 7 reproduce the most likely values of c, elk, and u with average deviations of 3.1, 3.4, and 1.770, respectively, for the basic group, and 6.4, 7.4, and 3.670, respectively, for all the substances listed in Table I. There it is seen that the agreement between the estimated and the most likely parameters is poorest for dipolar molecules such as CO, N 2 0 , and ”3. This is no surprise, since neither the correlations nor the potential rnodel account specifically for the multipolar nature of such molecules. By employing the well known relations Z, = P,V,/RTc, and 2, = 0.291 - O.O8Ow, Equations 3, 6, and 7 may be modified trivially to assume other forms. Such variants are not considered here, since they are expected to be less useful than Equations 3, 6, and 7. The usefulness of Equations 3, 6, and 7 can be tested by comparing how well the estimated and the most likely potentials reproduce certain bulk properties. The comparison is restricted to the experimental second virial coefficient, B ( T ) ,and viscosity, ?( T ) , d a t a which were used initially to determine the most likely Morse potentials (Konowalow, 1966, 1968; Konowalow and C a r r i , 1965). T h e errors in reproducing the B- data for trans-2-butene are unaccountably large. Generally, however, the estimated potentials perform nicely. No bias was apparent in fitting one property or one temperature range
Comparison with Other Correlations
Here, comparison is made with the results of others which are based on the Kihara spherical core model and the LennardJones (L-J) 12-6 model potentials. The Kihara potential is obtained by inserting the expressionp = [(a - 2a)/(r - 2a)I6 into Equation 1. The L-J model is recovered by setting a = 0. Tee et al. (1966a, b) determined Lennard-Jones 12-6 potentials (1966a) and Kihara spherical core potentials (1966b) by a weighted least squares fitting of experimental q ( T ) and B ( T ) data for 14 normal fluids. Since the method used to determine them differs only in details from that used to fix the most likely Morse potentials, these potentials are also termed “most likely.” Tee et al. (1966a, b) also determined additional sets of potential parameters by imposing corresponding states restrictions in their fitting procedure. Here, these are called “estimated potentials.’’ Most likely and estimated potentials based on the Morse, Kihara, and L - J models are available for 13 normal fluids: Ar, K r , Xe, Nz,COS, CH4, C2H4, C2H6, CsHs, n-CIHlo, nC5H1?,n-C7H16, and C6H6. The results for these individual species are not strictly comparable, since the Kihara and L-J potentials are based on somewhat different, and differently weighted, collections of B ( T ) and q ( T ) data from those used for the Morse potentials. I t is more reasonable to compare the averages of the fitting errors for these substances that are given in lines 3 to 5 of Table 11. These averages are expected to be relatively insensitive to differences in the data selection for the bireciprocal and the Morse potentials. The results for the L-J and the Kihara models have already been compared by Tee et al. (1966b). Lines 3 and 4 of Table I1 show that the Kihara and Morse potentials estimated from corresponding states correlations reproduce experiment lvith roughly comparable accuracy. T h e Morse results appear somewhat better for B, and slightly poorer for 7 than the Kihara results. T h a t the Morse correlations fare so well is a bit surprising. The correlations of Tee et al. (1966a, b) would ordinarily be expected to be considerably more accurate on two counts. First, their correlations were obtained by fitting the B ( T ) and VOL. 7
NO. 4
NOVEMBER 1 9 6 8
623
Substance Basic group Kr
Source a
b
Xe
a
Methane
a
Ethane
C
b
b
b
Propane
C
n-Butane
d
n-Pentane
d b
n-Heptane
E
n-Octane
c
1-Butene
d b
trans-2-Butene
d b
Isobutane
d b
Test group Ar
b b
b
b
Table I. Comparison of Most Likely and Estimated Morse Potentials R.M.S. Deviations For B , For B , For 7, .,/k c, K. C cc. /mole A. % %
199.59 208.0 274.00 291.7 221 .03 198.2 391.09 391.4 478.25 509.1 617.80 621.1 714.80 722.1 904.60 893.8 966.03 968.5 608.38 594.2 683.65 644.4 569.84 582.5
4.905 4.978 4.989 5.022 5,561 5.143 6.096 6.155 6.313 6.672 7.283 7.211 7.820 7.772 9.070 8.872 9.612 9,400 7.211 7.15 8.430 7.53 7.253 6.98
3.5609 3.540 3.9477 3.831 3.5097 3.571 4.0035 4.018 4.5620 4.51 5.0212 4.958 5.4434 5.404 6.3000 6.257 6.8092 6.641 4.8578 4,797 4.9915 4.840 5.1485 4.936
132.64 5.064 3.5540 149.7 4.978 3.299 e 780 8.30 6.0 n-Hexane b 803.7 8.19 5.805 d 6.162 379.74 3.7992 Ethene b 6.21 3.869 365.1 d 6,424 4,3573 490.32 Propene b 4.354 486.8 6.43 c 6.519 4.8376 804.32 Benzene b 7.365 5.008 833.1 d 8.128 3.3434 238.90 co b 5.46 3.428 149.8 d 8.371 509.13 3.5188 con b 7.475 455.3 3.574 a 5.238 120.27 3.7153 Nz b 5.440 141.3 3.402 d 155.21 5.370 3.3877 0 2 b 5,266 165.9 3.215 d 5.311 3.1804 416.37 NzO b 6.76 3.563 430.5 d 7.831 3.7171 745.41 so2 b 8.18 681.5 3.994 d 9,344 3.0112 874.80 NH 3 b 7.750 3,429 622.1 Calculated from Equations 3, 6, and 7 a Konowalow and Carrd (7965). 0
b
1 .so 9.3 0.88 3.1 1.9 4.7 1.4 1.7 3.3 4.2 12.6 13.3 13.2 16.4 18.8 29 . O 25.1 49.3 5.6 37.6 2.8 73 20 21
... ...
...
2.1 2.3 2.4 1.9 1.5 2.4
1.1
1.1
1.2 2.8 4.0 2.4 2.4 2.1 2.4 1.5 1.8 1.8 2.6 1.2 7.4 0.3 7.1 5.4 4.9
1.2 0.5 2.1 2.7 3.5 0.32
, . .
... ...
... 3.7 ... 6.7 28.4 2.0 2.1 30.7 4.7 7.7 11.6 4.8 3.5 1.6 6.4 2.2 15 3.7 12 1.5 0.37 ... 3.4 ... 11.3 ... ... 13.3 ... 1.4 2.7 ... 1.5 ... 2.1 ... 5.6 6.4 4.7 3.1 2.8 6.4 6.8 29 3.4 2.8 22 13 Konowalow (7966).
Temperature Range, " K . For B For 1)
108-873
144-366
273-973
278-555
108-623
122-366
272-51 0
189-478
274-611
300-466
283-511
27 8-3 66
298-573
311-366
3.4 3.8 2.3 4.4 2.3 4.0 2.8 4.3
349-623
373-523
373-573
373-523
273-420
293-393
243-333
298-393
1 .o
283-511
300-389
6.3 9.8 10.4 6.3 1.9 2.4 1.5 1.6 3.5 4.6 4.0 5.9 3.5 3.4 6.3
85-873
1 .o
6.7
89-1978
318-410
343-578
181-473
200-578
223-523
292-393
295-629
355-578
273-423
89-1 400
204-873
200-1344
90-673
111-1711
11.9
90-373 6.7 100-1300 7.2 243-423 13.4 189-1 244 13.6 10.0 265-473 266-1089 9.6 8.4 273-573 244-678 17.0 Konowalow (7968). e Equations 2, Konowalow
(1966).
Item 1 2 3 4 5 c
Table II. Comparison with Experiment for Most likely and Corresponding States Estimates for Morse, Kihara, and 1-J Potentials Average of R.M.S., Deviations For B, Cc./Mole For 7, 70 Most Corresp. Most Corresp. Potential Substances likely states likely states 8.9 21.9 1.9 3.1 Morse Basic group 8.2 17.0 4.1 5.5 Morse Basic-plus-test group 7.2 9.8 2.7 3.9 Morse Normal fluids" Kiharab Normal fluids" 7.2 13.6 1.8 3.5 Normal fluidsa 28.1 59.5 1.6 2.6 L-Jc
a Ar, Kr, Xe, N 2 , COP,CH4, c2H.1,C P H ~ CsH,, , nTC4H1o,n-CbHlP, n-C7H16,and CeH6. Results from Tee et al. (1966a), Table I V , Correlation zx.
624
I&EC FUNDAMENTALS
Results from T e e et al. (1966b), Table I V , Correlation iii.
,;(T) data directly, in contrast to the indirect method used in this paper to establish the Morse correlations. Second, the 13 substances considered in this section comprise all but C(CH3)4 of the fluids treated by Tee et al. (1966a, b). Only eight of the 13 substances were in the basic group of this paper; the remaining five-Ar, Ns, COS, c2H4 and CGH6-are from the test group. Thus, a comparison of the average errors listed in lines 3 to 5 of Table I1 constitutes a more severe test of the correlations for the Morse model than for the bireciprocal models. I n part, the relative success of the corresponding states correlations for the Morse potential may be attributed to their nonlinear dependence on u. Expressions linear in w , and valid for small w , can be obtained from Equations 6 and 7 for comparison with the linear relation:; of Tee et al. (1966a, b) : Morse.
e/kTc
Kihara.
t/kT,
L- J.
E/kT,
Morse.
+ 3.3 w - O(w’), w
