I n d . E n g . Chem. Res. 1987,26, 601-606
position during a test. Stencel et al. (1986) indicated that the carbonization of the iron/manganese catalyst was related to the potassium present in the catalysts. Summary The effects of various activations of an iron-manganese catalyst on synthesis gas conversion and product selectivity were investigated in a slurry reactor. Although the activation temperature and pressure did affect catalyst activity, olefin hydrogenation and isomerization reactions still appeared to be significant in all cases. The activating gas composition had the most dramatic effect, with the catalyst exhibiting no activity after a hydrogen activation. Both XPS and chemical analyses results corroborated the catalyst inactivity after the hydrogen activation. A high level of potassium promotion of the iron-manganese catalyst was needed to increase and stabilize the olefin selectivity of the product. The initial activity of the catalyst also increased, but potassium promotion caused an increase in carbon formation on the catalyst, as verified by XPS and chemical analysis of the catalyst. Although the potassium promotion stabilized the olefin selectivity and increased the water-gas-shift activity of the catalyst, the hydrocarbon distribution of the product spectrum shifted to a higher average molecular weight. Acknowledgment We thank R. R. Schehl, R. E. Tischer, R. D. H. Chi, S. R. Miller, R. A. Anderson, M. J. Mima, and D. H. Finseth for contributions to this project. Reference in this report
.
60 1
to any specific commercial product, process, or service is to facilitate understanding and does not necessarily imply its endorsement or favoring by the US Department of Energy. This paper was presented in Part at the Division of Fuel Chemistry, 189th National ACS Meeting, Miami Beach, Florida, May 1985. Registry No. K, 7440-09-7; Fe, 7439-89-6; Mn, 7439-96-5; CO, 630-08-0; H2CeHCH2CH3,106-98-9; H,CCH=CHCH,, 107-01-7.
Literature Cited Bussemeier, B.; Frohning, C. D.; Horn, G.; Kluy, W. German Patent 2 518964, 1976. Gibbon, G. A,; Hackett, J. P.; Feldman, J. J . Chromatogr. Sci. 1985, 23(7), 285. Kolbel, H.; Ralek, M. Catal. Rev. Eng. 1980, 21, 225. Kolbel, H.; Tillmetz, K. D. US Patent 4 177 203, 1979. Maiti, G. C.; Malessa, R.; Baerns, M. Appl. Catal. 1983, 5 , 151. Mobil Final Report, US DOE Contract No. DE-AC22-80PC30022, June 1983. Pennline, H. W.; Zarochak, M. F.; Tischer, R. F.; Schehl, R. R. Appl. Catal. 1986, 21, 313. Richard, M. A.; Soled, S. L.; Fiato, R. A.; DeRites, B. A. Mater. Res. Bull. 1983, 18(7), 829. Schulz, H.; Gokcebay, H. Proc.-Org. React. Coni., Charleston, S.C. 1982, 1. Stencel, J. M.; Diehl, J. R.; Anderson, R. A.; Zarochak, M. F.; Pennline, H. W. unpublished results, 1986. van Dijk, W. L.; Niemantaverdriet, J. W.; Van Der Kraan, A. M.; Van Der Baan, H. S. Appl. Catal. 1982, 2, 273. Zarochak, M. F.; Pennline, H. W.; Schehl, R. R. DOE/PETC/TR84/5, 1984; US Department of Energy.
Received for review February 24, 1986 Accepted September 12, 1986
Temperature-Dependent Parameters and the Peng-Robinson Equation of State Zhong Xu* and Stanley I. Sandler Department of Chemical Engineering, University of Delaware, Newark, Delaware 19716
It is suggested that two temperature-dependent parameters be used in the Peng-Robinson equation of state. Parameters obtained from supercritical volume data and subcritical saturated liquid volumes and vapor pressures are reported for nine hydrocarbons, argon, oxygen, nitrogen, carbon monoxide, carbon dioxide, water, and methanol. These new parameters are correlated as a function of reduced temperature in the subcritical and supercritical temperature range and as a cubic-spline function in the near-critical region. Remarkably good predictions are obtained for the volumes and saturated vapor pressures of the pure components studied by using the Peng-Robinson equation with the proposed parameters. T h e predictions are much better than those obtained with the standard, generalized Peng-Robinson parameters. Since van der Waals (1873) proposed his cubic equation of state
many modifications have been made to improve agreement with experimental data. Most of the modifications have been to the second or attractive term and are of the form to (i) change its volume dependence and (ii) introduce a temperature dependence into the a parameter, as suggested by Zudkevitch and Joffe (1970),Joffe et al. (1970),
* Permanent address: Jiao Tong University, Sian, People’s Republic of China. 0888-5885/87/2626-0601$01.50/0
Chang and Lu (1970), and others to give accurate purecomponent vapor pressure predictions. An example of such a modified cubic equation of state, and the one we will consider here, is the Peng-Robinson equation (1976) a(T) p = - -RT (2) u - b U(U + b ) + b(u - b ) To obtain the parameters in this equation, Peng and Robinson (1976) used the conditions that at the critical point
which leads to 0 1987 American Chemical Society
602 Ind. Eng. Chem. Res., Vol. 26, No. 3, 1987
a(TJ = $,(Tc)R2Tc2/P
(4)
b ( Tc)= $b(Tc)RTc/ p c
(5)
and 0 6
0.005
with
$,(TC)= 0.45724
(6) 0.5
and
$b(Tc)= 0.077 80
*b
(7)
At temperatures other than the critical temperature, they assumed, following Soave (1972), that $,(TI = $,(TC)[l+ 41 - (T/Tc)''2)12
0.080
$0
0.4
(8) 0.075
and 0.3
Further, by analyzing vapor pressure data for the alkanes from methane to n-decane, nitrogen, hydrogen sulfide, and carbon dioxide as a function of the acentric factor, they found that the K parameter could be correlated as a function of the acentric factor w , K
= 0.37464
+ 1.54226~- 0 . 2 6 9 9 2 ~ ~
(10)
Note that while eq 4-7 are a general result of having chosen the equation of state, eq 2, and invoking the critical point conditions (eq 3), eq 8-10 are, to some extent, arbitrary. In particular, there is no reason to believe that the $b parameter should, for all fluids, be independent of temperature or that the temperature dependence of $,( r ) in eq 8 and the acentric factor dependence of K in eq 10 should be applicable for any other fluids than those included in the data base for which the correlation was developed. This, of course, has been recognized by others. For example, Haman et al. (1977) and Yarborough (1979) correlated $, and $b in terms of T, = T / T, and w for cubic equations of state but limited their correlation to the subcritical region. It should be noted that Yarborough (1979) and others have suggested the use of various temperature dependencies below the fluid critical point and the constant critical point value for T, > 1, which results in a discontinuity in first derivatives of the a and b parameters, adversely affecting the enthalpy and fugacity coefficient. Turek et al. (1980) also developed a correlation for equation-of-state parameters which is limited to the subcritical region and to mixtures containing carbon dioxide. The most recent correlation is that of Morris and Turek (1986). I t is complicated and retains the first derivative discontinuity a t the critical point. The purpose of the present paper is to present information on the temperature dependence of the two parameters in the Peng-Robinson equation of state which lead to greater accuracy in vapor pressure and density predictions than when eq 8-10 are used. To do this, we have obtained $, and $b from saturated liquid volumes and vapor pressures a t subcritical conditions, the critical point conditions, and from volumetric data a t supercritical conditions. In addition, we have also shown that good accuracy is obtained for the density and vapor pressure of water, methyl alcohol, and other components by using the Peng-Robinson equation of state, provided the specialized correlations developed here are used rater than eq 8-10.
Correlation of the Parameters, $, a n d $b As mentioned above, the parameters were obtained from saturated liquid volumes, vapor pressures, and supercritical
0 2
05
1.5
1.0
2.0
2 5
3.0
Tr
Figure 1. Temperature-dependent characteristics of
$a
and $* of
methane.
volume data. The references for the original data are in Table I. The parameters were obtained by minimizing the objective function
for subcritical temperatures and the function
in the supercritical temperature region. In these functions, A denotes the difference between the calculated and measured properties. The use of these objective functions was found to result in unique values of the $a and $b parameters. The parameters, $a and $b, so obtained were correlated with equations of the form 2
$,I
= Cai'T,' i=O
(12)
and 2
$2 = i=O Cb,'T,'
(13)
where i refers to the species and I denotes either subcritical or supercritical conditions. The four sets of parameters, aisub,aisuP, b y b , and b?P, are given in Table I. The subcritical parameters are applicable at reduced temperatures below 0.985, while the supercritical parameters are to be used at reduced temperatures above 1.015. Unfortunately, the temperature dependence of the parameters is sufficiently complicated, as shown in Figure 1, that two different correlations were needed. In the temperature transition region, 0.985 < T, < 1.015, the two parameters, $, and $b, are represented with the cubic-spline functions which are continuous and smooth a t the critical temperature. The function used is of the form
Ind. Eng. Chem. Res., Vol. 26, No. 3, 1987 603
for i = a and b and 0.985 < T , < 1 and r
1
-5
1
c
-10 0.4
il \
het;
p,ropose,d pora,meter; the originol p o r o m e t e r s
0.5
0.6 0.7
0.8
0.9
1.0
Tr Figure 2. Comparison of predicted molar liquid volumes for saturated methane.
for i = a and b and 1 < Tr < 1.015 In these equations, the subscripts, 0, 1, and 2, refer to values a t TI = 0.985,1, and 1.015, respectively. The terms, mio,.mil,and mi2, are the first derivatives of the function, +i( I = a and b ) , a t T I = 0.985, 1, and 1.015. The values of mi, and mi2are obtained from eq 12 and 13, while mil is obtained from 1 1 3 -mio + 2mi1+ -mi2 = -(+iz 2 2 2AT,
- +io)
(16)
Finally, ATr = TI, - T,, = T I , - T f i = 0.015. The temperature dependences of and #b for methane using eq 12-16 are shown in Figure 1. Similar behavior is found for the other components in Table I. For five components in Table I, there are insufficient supercritical volume data to fit the parameters, a?P and b p . In this, eq 8 and 9 are used to obtain the +i2 and mi2 values and, then, to calculate mil from eq 16. Finally, we note that the relatively complicated form of the spline-fit function could be avoided by ignoring the critical conditions and smoothly connecting the T I < 0.985 and T, > 1.015 functions. We have not explored this possibility here.
Results In Tables I1 and 111,we show the average absolute deviations (A.A.D.) of the predicted values from experimental volumes (both liquid and vapor) and vapor pressures using the temperature-dependent parameters we have proposed. For the purpose of comparison, we also include the errors in predictions which result from the use of the ai and bi parameters proposed by Peng and Robinson. (Table 11 does not include the results of the five components stated above, because few data exist in the supercritical regions for these components.) From these tables we see that the average absolute deviation for the vapor pressure is approximately one-sixth as great when the proposed param-
5
1
a
--
the proposed poromeiers the o r i g i n a l
0 IO0
I60
I30
I90
T,'K
Figure 3. Comparison of predicted molar vapor volumes for saturated methane.
eters are used as when the original Peng-Robinson parameters (eq 4-9) are used. Similarly,the average absolute deviation in the volume is only approximately 40% of that which results from using the original Peng-Robinson parameters. What is also impressive is that with the proposed fluid-specificparameters, the errors in volume predictions for water and methanol are reduced by almost an order of magnitude, so that the errors for these distinctly nonhydrocarbon fluids, using the Peng-Robinson equation of state, are of the same order as those for the hydrocarbons. Also, with the revised parameters, the errors in vapor pressure predictions for the polar fluids water and methanol are only slightly higher than for the hydrocarbons. Note that the overall improvement is so great with the parameters proposed here, that the errors in the volume and vapor pressure predictions for water and methanol are
604 Ind. Eng. Chem. Res., Vol. 26, No. 3, 1987 Table I. Values of the Coefficients. a; and b;.of Eauation 7 alBUb aZBUb component aOSUb argon methane oxygen nitrogen carbon monoxide ethane propane propylene carbon dioxide water methanol n-butane n-pentane n-hexane n-heptane benzene component
0.77967501 -0.25453925 0.78201729 -0.27407181 0.81113070 -0.30379444 0.81578445 -0.30059186 0.86680776 -0.42204800 0.90577853 -0.50785202 0.90789413 -0.46744743 0.88366586 -0.40379918 0.96666628 -0.56434441 0.93420982 -0.76686120 1.10695362 -0.94499099 0.91873884 -0.46234739 0.94832230 -0.51094002 0.95035815 -0.48550135 0.98651439 -0.55592757 0.90427643 -0.46275631 bosub
argon methane oxygen nitrogen carbon monoxide ethane propane propylene carbon dioxide water methanol n-butane n-pentane n-hexane n-heptane benzene
0.07055458 0.07193621 0.07041253 0.06643935 0.06948082 0.07288811 0.05900056 0.05674857 0.05211300 0.07250322 0.06785157 0.05295983 0.05351885 0.04500325 0.04419140 0.04933474
-0.07904863 -0.06522629 -0.060501 18 -0.07145700 0.00264424 0.03821815 -0.00456938 -0.03836720 0.03308220 0.23555398 0.23239106 -0.02272222 -0.00656694 -0.03735375 -0.00314254 -0.00965121 blsub
0.06781454 0.05896984 0.06752864 0.07644470 0.06692074 0.04967674 0.07981175 0.08818834 0.09057645 -0.01848893 -0.00032905 0.09096103 0.08346366 0.10090445 0.09713437 0.09270781
alsuPer
aOsuPe'
0.55789668 0.57954848 0.55698574 0.57483929 0.65531725 0.58773327 0.65156901 0.17525676 0.73021990 0.85802239 1.02034271
a2wer
-0.12271010 0.00830337 -0.13215594 0.00207479 -0.12928048 0.01059158 -0.13690966 0.00939819 -0.23016444 0.02314564 -0.09623815 -0.03622254 -0.19203129 -0.00266536 0.61039585 -0.33620256 -0.31995407 0.04391478 -0.60228193 0.13341056 -0.82705432 0.21474762
bZBUb -0.06336407 -0.05671475 -0.06285726 -0.06844313 -0.06121870 -0.05013840 -0.06612156 -0.07103167 -0.07025131 0.01061862 -0.00423735 -0.07182065 -0.06549629 -0.07521132 -0.07089569 -0.07051162
ref Angus and Armstrong, 1971 Angus et al., 1978 Vargaftik, 1983 Angus et al., 1979 Vargaftik, 1983 Vargaftik, 1983 Vargaftik, 1983 Angus et al., 1980 Angus et al., 1976 Keenan et al., 1978 Smith, 1948 Das et al., 1973 Das et ai., 1977 Vargaftik, 1983 Vargaftik, 1983 Vargaftik, 1983
bOSuPer
b,wer
b2wer
0.07076953 0.04627120 0.06907655 0.06841091 0.07081967 0.05394017 0.05966436 -0.16915888 0.07012508 0.11334278 0.11838132
0.00646783 0.03862211 0.00802093 0.00975966 0.00613727 0.03253532 0.02331238 0.40465274 0.00764327 -0.07292674 -0.07858859
-0.00037742 -0.00898702 -0.00037580 -0.00073262 -0.00007378 -0.00945264 -0.00565655 -0.16072746 0.00041225 0.02218090 0.02329863
Table 11. Summary of Deviations between Calculated an Literature Va-Jme Values" A.A.D. '70 component argon methane oxygen nitrogen carbon monoxide ethaneb propane propylene carbon dioxide water methanol'
T, K 84-1300 100-620 75-1300 64-1100 70-670 160-500 230-600 150-575 220-1100 273-1589 308-1026
P, bar 10-1000 10-1000 10-1000 10-1000 10-1000 10-500 10-600 10-1000 10-1000 10-1034 12-1193
no. of pts 545 644 655 616 258 243 232 462 624 587 303
origC 4.03 5.23 5.00 7.01 5.45 5.27 3.69 6.16 1.59 12.35 22.72
propd 1.83 2.16 1.93 2.96 2.16 2.30 1.77 2.79 0.99 1.69 3.95
