75
Ind. Eng. Chem. Fundam. 1986, 25, 75-84 Boer, J. H. de A&. Catal. Re/. Subj. 1957, 9 , 472. Boudart, M. CHEMTECH 1978. 8 , 231. Boudart, M. Catal. Rev.-Sci. Eng. 1981, 2 3 , 1. Boudart, M. J. Phys. Chem. 1983, 8 7 , 2786. Boudart, M.; Dj6ga-Mariadassou, G. “Kinetics of Heterogeneous Catalytic Reactions”; Princeton University Press: Princeton, NJ, 1984. Boudart, M.; Egawa, S.; Oyama, S. T.; Tamaru, K. J. Phys. Chem. 1982, 7 8 , 987. Boudart, M.; Loffier, D. G.; Gottifredi, J. C. I n t . J. Chem. Kinet. 1985, 77, 1119. Christlansen, J. A&. Catal. Re/. Subj. 1953, 5 , 311. Cimino. A.; Boudart, M.; Taylor, H. S. J. Phys. Chem. 1954, 58, 796. De Donder, Th. “L’AftlnitB”; Gauthier-Viiiars: Paris, 1927; p 43. Emmett, P. H. “New Approaches to the Study of Catalysis”; Phi Lambda Upsilon: University Park, PA, 1962; Chapter 5. Eyring, H. J. Chem. Phys. W35, 3, 107. Garfinkle, M. J. Chem. Phys. 1983, 7 9 , 2779. Glassman, I . “Combustion”; Academic Press: New York, 1977. Gcddard, W. A., 111 Science 1985, 227, 917. Hampton, R. F., Ed. J. M y s . Chem. Ref. Data 1973, 2 , 267. Happel, J. Catal. Rev. 1972, 6 . 221. Herbo, CI. Bull. SOC. Chim. Be@. 1942, 5 1 , 44. Hjortkjaer, Jes ”Rhodium Complex Catalyzed Reactions”; Polyteknisk Vorlag: Lyngby, Denmark, 1982; Voi. 2. Horiuti, J. J. Res. Inst. Catal., Hokkaido Unlv. 1953, 2 , 87. Horiutl, J. Ann. N.Y. Acad. Sci. 1973, 273, 5. Horiuti. J.; Nakamura, T. Adv. Catal. Re/. Sub]. 17, 1. Johnston, H. S. “Gas Phase Reaction Kinetics of Neutral Oxygen Species”; U.S. Government Printing Office: Washington, D.C., 1968; NSRDS-NBS 20. p 12.
Kembali, C. Discuss. F a r a h y SOC. 1066, 41, 190. Kondrat’ev, V. N. “Rate Constants of Gas Phase Reactions”; Nauka: Moscow, 1970. Madix, R. J. “Chemistry and Physics of Solid Surfaces”; Vanseiow, R., Ed.; CRC Press: Boca Raton, FL, 1979; Vol. 2, p 63. Mars, P.; Schoten, J. J. F.; Zwietering, P. “The Mechanism of Heterogeneous Catalysis”; de Boer, J. H., et ai., Eds.; Elsevier: Amsterdam, 1960; p 66. North, A. M. “The Kinetics of Free Radical Polymerization”; Pergamon Press: Oxford, 1966. Ozaki, A.; Aika, K. “Catalysis, Science and Technology”; Anderson, J. R., Boudart, M., Eds.; Springer-Veriag: Heidelberg, 1981; Voi. 1, p 87. Prigogine, I.; Outer, P.; Herbo, CI. J. Phys. Colloid Chem. 1948, 5 2 , 321. Satterfieid, C. N.; Roberts, G. W. AIChE J. 1988, 14, 159. Sinfelt, J. H.; Hurwitz, H.; Shuiman, R. A. J. Phys. Chem. 1960, 64, 892. Temkin, M. I . I n t . Chem. Eng. 1971, I f , 709. Temkin, M. I.; Pyzhev, V. Acts Physicochim. URSS 1940, 12, 217. Ti”, B. Proc. I n t . Congr. Catal., 8th, 1984 1985 1, 7. Vannice, A. “Catalysis, Science and Technology”; Anderson, J. R., Boudart, M., Eds.; Springer-Veriag: Heidelberg, 1982; Voi. 3, p 139. Van Tiggekn, A. “Oxidations et Combustions”; Technip: Paris, 1968; Voi. 11, Chapter 12. Wagner, C. Adv. Catal. Re/. Subj. 1970, 2 1 , 323.
Received f o r review June 21, 1985 Accepted October 17, 1985
This paper was presented at the National Meeting of the American Chemical Society, Miami, April 1985.
GENERAL ARTICLES Cubic Chain-of-Rotators Equation of State Hwayong Kim,+ Ho-Mu Lln, and Kwang-Chu Chao’ School of Chemical Engineering, Purdue lJnivers&
West Lafayette, Indiana 47907
A cubic equation of the perturbation type is developed to express pressure as being made up of contributions due to repulsive, rotational, and attractive forces. Use of the equation requires p,, T,, and w of a substance to be known. Calculated pvT, vapor pressure, and enthalpy are compared with data and with the Soave equation and the Peng-Robinson equation for a variety of substances over wide ranges of temperature and pressure. The equation is extended to mixtures by using van der Waals one-fluid mixing rules for the equation parameters. Gas-liquid equillbria of fluid mixtures are calculated for low-pressure symmetric mixtures as well as for highpressure asymmetric mixtures of a heavy solvent with a light gas such as hydrogen, methane, carbon dioxide, and nitrogen. Calculated pvT of mixtures is illustrated with two binary systems for gas and liquid states%p to the critical point.
Introduction Equations of state are useful for the calculation of fluid thermodynamic properties. This usefulness has prompted a continual development of new equations. The perturbation type of approach is noteworthy for being productive of some very useful equations of which the cubic equations have received much attention. Equations such as Redlich-Kwong (1949), Soave (1972), and Peng-Robinson (1976) are in wide use due to their simplicity and generality combined with reasonable accuracy. AU of these equations contain van der Waals excluded-volume expression for the repulsive pressure RT/(u - b). The excluded-volume concept is valid for a dilute gas but breaks down at high densities such as those of liquids (Vera and Prausnitz, 1972; Gubbins, 1973; Abbott, 1979; Henderson, 1979). In place of van der Waals form, Carnahan and Starling (1969, 1972) obtained an expression for the repulsive pressure based on molecular dynamics calculations. Donohue and Prausnitz (1978) developed a perturbation ‘Department of Chemical Engineering, University of Delaware, Newark. DE. 0196-4313/86/1025-0075$01.50/0
equation of state to include Carnahan and Starling’s repulsive pressure and, additionally, to account for the rotational motion of polyatomic molecules in terms of equivalent translational degrees of freedom. Chien et al. (1983) obtained an expression for the rotational pressure from Boublik’s (1975) equation for hard dumbbell molecules. The chain-of-rotators (COR) equation of state that Chien et al. developed using the Carnahan and Starling repulsive pressure expression and their new rotational pressure expression appears to be accurate, but complex. In this work we develop a cubic equation as a simplified form of the COR equation of Chien et al. in order to provide the computational ease of cubic equations while retaining the structure of the COR equation. The repulsive and rotational pressure contributions are simulated with simpler functions. The attractive pressure expression is also simplified. The equation is then fitted to the vapor pressure and saturated liquid density at subcritical temperatures and the pu isotherms at supercritical temperatures. The CCOR equation requires the three constants T,,p,, and o for the substance to be described. These constants are known for a large number of substances, in contrast 0 1986 American Chemical Society
78
Ind. Eng. Chem. Fundam., Vol. 25, No. 1, 1986 14
12
,',r
r
I
l
t
N
-
New equation
i/
1'5
/
/
0.5 "O1
0
0
0.1
0.2
0.3
0.4
0.5
9 ---
-0
01
Figure 2. Compressibility factor of a pure rotator of a = 1.078.
Carnahan-Starling (1969, van d e r Woals (18733 New equation
02
0 3
n o Doto
'05d
0 5
04
I
7
from Gibbons
o n d Keutjler(1968)
In
Figure 1. Repulsive compressibility factor.
to the COR equation constants P , V,, and cR which must be determined for each substance of interest. The Equation of State The cubic chain-of-rotators (CCOR) equation of state is =
RT(1 + 0.77b/v) u - 0.42b
+ CR 0.055RTb/u u - 0.42b
a U(V
+ C)
bd U(U
+ C)(U
- 0.42b)
(1)
Equation 1 expresses the pressure in terms of repulsive, rotational, and attractive force contributions. The repulsive pressure function R T ( l + 0.77b/v)/(v - 0.42b) approximates the molecular dynamic calculations of hard-sphere fluids which are accurately represented by the Carnahan-Starling equation zhs
= (1 + 7
+ 7' - q3)/(i -
(2)
where 7 = b/(4v) and.b/4 is the hard-core volume. Figure 1 shows the close agreement of eq 2 with the repulsive compressibility factor of eq 1, i.e. Zhs = (U + 0 , 7 7 b ) / ( ~ - 0.42b) = (1 + 3.087)/(1 - 1.687) (3) Figure 1also shows the repulsive pressure of van der Waals RT/(v - b) term expressed as a repulsive compressibility factor Zhs = V / ( U - b) = 1/(1- 47) (4) Equation 4 departs widely from eq 2 and 3 at high densities. van der Waals repulsive pressure form is in serious error at liquid densities. The second term on the right-hand side of eq 1expresses the rotational contribution to pressure of polyatomic molecules. It is a simplification of the rotational pressure expression of Chien et al. (1983) ZI,t
= cR-
( a - 1) [37+ 3 4 - (a + 2
i)731/(1
- 713 (5)
cR is the number of rotational degrees of freedom each with
a rotational strength a. Chien et al. obtained eq 5 from Boublik's (1975) equation of state of hard dumbbell molecules. In reducing eq 5 to the second term on the right-hand side of eq 1, we have assigned a = 1.078 which
is the rotational strength of a C-C segment. In eq 1, cR is then the effective number of such rotators contained in the molecule of interest. Figure 2 shows the comparison of eq 5 with the simplified form in eq 1. The last two terms of eq 1 represent the attractive pressure. The attractive pressure was calculated from experimental pvT and vapor pressure data by using eq 1 in which the repulsive and rotational pressures had been determined a priori as described above. Values of a and c were then calculated and correlated as functions of TI and w . Equation 1is cubic in v and contains the five parameters a, b, c , d, and cR. All five parameters have been expressed in terms of T,, p,, w, and T, (Lin et al., 1983b). The CCOR equation is thus a three-parameter equation requiring T,, p,, and w to be known for the substances to be described. p v T and Vapor Pressure Calculations of puT and vapor pressure with the CCOR equation reported here are based on the use of three parameters, T,, p,, and w , for each substance. Figure 3 shows the calculated pv isotherms and gasliquid saturation envelope for argon in comparison with the table values of Gibbons and Keubler (1968). Figure 4 shows a similar comparison for propane with data taken from Goodwin and Haynes (1982). Generally good agreement is obtained with the exception of temperatures and pressures immediately above the critical. The puT behavior in the immediate vicinity of the critical state is nonanalytic (Levelt-Sengers, 1976) while the CCOR equation is analytic. An analytic equation is fundamentally deficient in the near-critical region. Vapor pressure is calculated at a given temperature by searching for the pressure at which the computed fugacities
Ind. Eng. Chem. Fundam., Vol. 25, No. 1, 1986 77 Table I. ComDarison of Calculated VaDor Pressure for Some Heavy Non-Paraffinic Substances
T,,K pe, bar
compd furan cyclohexane methylcyclohexane thiophene pyridine m-cresol tetralin bicyclohexyl naphthalene thianaphthene diphen ylmethane 1-methylnaphthalenle quinoline fluorene phenanthrene
490.2 553.4 572.1 579.4 620.0 705.8 716.5 731.4 748.4 752.1 767.0 772.0 782.0 870.0 878.0
55.00 40.73 34.71 56.90 56.30 45.60 33.49 26.14 40.53 38.81 29.79 35.67 43.57 47.00 32.30
no. of data points 22 21 24 28 30 19 22 28 11 30 26 19 20 12 11
w
0.209 0.213 0.233 0.196 0.243 0.450 0.315 0.388 0.302 0.283 0.471 0.334 0.311 0.344 0.437
AAD,% T range, K Soave 366-483 1.1 333-533 323-533 422-577 450-610 388-662 423-693 424-703 441-728 424-630 490-700 425-700 431-724 490-640 510-628
P-R
CCOR
2.0
1.5 1.5 0.8 1.5 1.0 2.2 1.4 2.5 1.1 1.6 1.6 1.2 2.6 2.6 3.2
0.8 0.8 1.5 0.7 3.2 1.7 3.2 1.3 2.2 2.4 2.0 1.6 3.2 3.1
data source Kobe and Mathews (1970) Washburn (1928); Zwolinski (1953) Francis (1957); Zwolinski (1953) Kobe and Mathews (1970) Kobe and Mathews (1970) Simnick (1979); Zwolinski (1977) Simnick (1979) Wieczorek and Kobayashi (1980) Wilson et al. (1981) Wieczorek and Kobayashi (1980) Simnick (1979); Timmermans (1950) Simnick (1979); Zwolinski (1953) Simnick (1979); Jones (1977) Sivaraman and Kobayashi (1982) Wilson et al. (1981)
4-
2-
lOOK
h
0-
Y
n 0
-0
-2-
-
-4
o L i t e r a t u r e Data
In)'''
I
d 1.19 + 0 . 0 5 5 ~ ~ 0.42 0.42(c 0.42b)RT u - 022b U(C 0.42b) - bd u In In -+ z u c(c 0.42b)RT u c 1.0 - In z (6) =
[
+
+
+ +
+
Figures 5 and 6 show vapor pressures and saturated liquid volumes for n-paraffins from methane to n-hexadecane calculated from the CCOR equation. The data source, number of data points, and the temperature range of data used in the calculation were reported by Lin et al. (1983b) for each substance. For the entire temperature range of the liquid state from the triple point to the critical point the absolute average deviation (AAD) generally amounts to 1-270 for the vapor pressure and slightly less for the saturated liquid volume. Comparison with the Soave equation and the Peng-Robinson (P-R) equation has been made by Lin et al. (1983b) for the substances of Figures 5 and 6 and for argon, nitrogen, carbon dioxide, benzene, toluene, and m-xylene. Vapor pressures calculated by the three equations are generally comparable with the CCOR equation being the best by a slight margin. Much better results are obtained from the CCOR equation for n-hexadecane, indicating a significant improvement of this equation for large molecules. The CCOR equation is superior in representing saturated liquid volumes. Table I shows a comparison of the calculated vapor pressures with data for some heavy non-paraffin substances. The liquid volume data are generally unavailable for these substances. The comparison is, therefore, limited to vapor pressure. Critical properties and acentric factors
c4
Eq
--COR
a
c2 \
C-4
-61
I
I
I
I
0
3
6
9
12
1 0 0 0 / T , K-'
Figure 5. Vapor pressure of n-paraffins.
" [ - 1000
k
500
2 2001
o
Literature Data Eq
--COR
~~
0
400
200
600
T, K
Figure 6. Saturated liquid molal volume of n-paraffins.
used in the calculations are reported in the table inasmuch as these values are subject to some uncertainty and diverse
78
Ind. Eng. Chem. Fundam., Vol. 25, No. 1, 1986 o
Data from Goodwin(l977) Eq
- CCOR
0
Data from Cediel et 01.(1982)
--COR
, b
-
2
-
:
L
-
3
7
Ea
0
. 200
:
100
-8
1 2, lo
I 1 I
5
I
,
102
2
5
,03
, 2
I
5
I
,
,04
2
5
P , kPa
Figure 7. Enthalpy of propane.
values are found in the literature. The CCOR equation again appears to be the best by a slight margin.
Enthalpy From eq 1 we obtain the following enthalpy deviation function
u - 0.42b U
c(u
+ c)
-
c(c
-10 10’
1 o3
104
P , kPa
Figure 8. Enthalpy of toluene.
where 8 stands for either a, b, c, d, or cR. Fori = j , eii= g iwhich is just the parameter of pure component i. For i # j , the cross parameters eijare given by
+
+ 0.42b)(u + c) + PU - R T (7)
where h” is the ideal gas molal enthalpy. Figure 7 shows the calculated enthalpies of propane in comparison with Goodwin’s (1977) table of values. A very wide range of temperatures is covered from close to the triple point to well above the critical. Figure 8 shows calculated enthalpies of toluene in comparison with the data of Cediel et al. (1982). All the temperatures are subcritical, but the pressure extends to well above the critical. Generally good agreement is obtained between the calculated values and data except at temperatures and pressures immediately above the critical. As a result of the representation of CY’, the scaled temperature function of a, by two different functions joined at the critical temperature (Lin et al., 1983b), eq 7 is discontinuous at T,. The derivative d a l d T is the source of this discontinuity. The discontinuity does not cause any significant errors but can lead to nonconvergence of computer search calculations involving enthalpy, such as in isenthalpic calculations. We suggest that the discontinuity be removed in computer programs for enthalpy calculations by interpolating between T,- e and T,+ e at temperatures very close to T,,Le., when IT - Tcl < e where e is a suitably small number. Mixing Rules The CCOR equation is extended to mixtures upon introducing mixing rules for the equation parameters. In this work we adopt the van der Waals one-fluid mixing rules as follows
4, = (dIId,,)l’z CIIR
(12)
= (CUR+ CjjR)/2
(13)
Two binary interaction parameters k, and k,,are introduced in eq 9 and 11, respectively, to describe mixture properties and are determined by referring to experimental data on mixtures. We have also performed extensive calculations of mixture properties by using an alternate set of interaction parameters, kagJand kb, (Kim, 1984). The results obtained from both sets of parameters are about the same. We report here only the result of the set It,,, and kCv.
+
Density of Saturated Mixtures of Methane Propane and Methane n -Pentane Figure 9 shows the calculated density of saturated gas and liquid mixtures of methane + propane from the CCOR equation at four temperatures. The calculated results agree well with experimental data of Reamer et al. (1950) up to and including the critical states. The interaction parameters k, and k,,used in the calculations are determined from vapor-liquid equilibrium data described in the next section. Figure 10 shows a similar comparison for mixtures of methane + n-pentane. Again the interaction parameters employed in the CCOR equation calculations are obtained from vapor-liquid equilibrium data.
+
Vapor-Liquid Equilibrium A major interest in an equation of state lies in the usefulness of the equation for the representation of mixture fluid-phase equilibria. To apply the CCOR equation to
Ind. Eng. Chem. Fundam., Vol. 25, No. 1, 1986
Table 11. CCOR Equation Calculated VLE of Hydrogen Mixtures no. of Trange, p range, data solvent K MPa points koI2 220-290 103-173 278-361 328-394 310-395 277-478 463-544 462-584 462-664 433-534 462-575 462-582 462-662 462-663 462-702 462-730 462-702 462-702 462-702
CO2 methane propane n-butane isobutane n-hexane n-octane n-decane n-hexadecane benzene toluene m-xylene m-cresol tetralin quinoline 1-methylnaphthalene bicyclohexyl thianaphthene diphenylmethane
1-20 1-11 3-55 3-17 3-21 3-28 2-15 2-25 2-25 2-18 2-25 2-25 2-25 2-25 2-25 2-25 2-25 2-25 2-25 0
1.3726 1.2662 1.3181 1.4289 1.5017 1.4144 1.6046 1.5151 1.5420 1.5194 1.4344 1.4993 1.5469 ).5155 1.5420 1.5234 1.5363 1.5652 ).5426
30 28 34 60 21 56 45 26 29 49 24 27 41 24 27 36 28 27 27
AAD, % K2 7.1 2.3 5.2 3.7 4.4 6.4 3.4 2.8 4.5 8.5 9.3 9.7 2.9 2.9 3.8 9.8 3.9 15.1 1.8 2.1 4.8 4.8 4.5 5.6 4.6 5.2 3.7 4.4 6.3 4.0 5.8 3.8 5.9 6.7 5.6 6.4 6.3 6.5
K1
ke12
0.0122 -0.0331 0.0208 0.1045 0.1649 0.0247 0.1003 0.0615 -0.0125 0.0924 0.0418 0.0472 0.0506 0.0175 0.0331 0.0357 0.0479 0.0061 0.0235
Data trom Reamer et al. (1950)
e
-
79
source Suano et al. (1968) Sigara et al. (1972) Buriss et al. (1953) Klink et al. (1975) Dean and Tooke (1946) Nichols et al. (1957) Connolly and Kandalic (1963) Sebastian et al. (1980d) Lin et al. (1980) Connolly (1962) Simnick et al. (1978b) Simnick et al. (1979a) Simnick et al. (1979a) Simnick et al. (1977) Sebastian et al. (1978a) Yao et al. (1977) Sebastian et al. (1978~) Sebastian et al. (197813) Simnick et al. (1978a)
Data from Sage et a1.(1942) CCOREq
Critical
K
uI 2
0.4 0.6 0.8 Mole Fraction of Methane
1.0
Figure 9. Density of saturated mixtures of methane
+ propane.
0
0.2
+
+
a(2CNkCik - C ) cRT(v c )
+
2CNkCik - C bd
(c
In
2dCNkbik
+[
c
+ 2bCNkdik - bd
+ 0.42b
+ 0.42(2xNkbik - b) + 0.42b)2
u - 0.42b U
-1
75
$
u - 0.42b u
+
Data from Scatchard et a1.(1939)
- CCOR Eq w i t h ka.. = 0.1693 kc;: i0.1652
-
1[
0.4216RT
00
-
+ 0
1
d(2CNkbik - b ) [ 1 (c + 0.42b)RT 0.42b
70t
00
15
1 u In cRT v+c -
--
1.0
80
+
[
0.8
Figure 10. Density of saturated mixtures of methane + n-pentane.
vapor-liquid equilibrium (VLE), we derive the following fugacity formula from the mixing rules of eq 8-13. O.ll(xNkCikR- CR) 1.19 0 . 0 5 5 ~ ~ In ai = X 0.42 u - 0.42b (1.19 0.0%CR)(2CNkbik - b ) In u u - 0.42b 2xNkaik a(2CNkcik - C) cRT c2RT ]In&-
+
0.4 0.6 Mole Fraction of Methane
0.2
0
Data t r o m Mentzer e t a l . (1982) CCOR Eq
0.2 0.4 0.6 0.8 1.0 Mole Fraction ot Cyclohexone
Figure 11. Vapor-liquid equilibrium in mixtures of cyclohexane +
u - 0.42b
]-1nz
(14) The calculation of low-pressure vapor-liquid equilibrium of symmetric mixtures from the CCOR equation is illustrated with mixtures of cyclohexane + benzene and n-
benzene.
hexane + benzene. Figures 11 and 12 show the calculated results in comparison with experimental data. Data of exceptional quality have been reported on the former mixture by Scatchard et al. (1939) and Mentzer et al. (1982) each for a number of temperatures. With the in-
80
Ind. Eng. Chem. Fundam., Vol. 25, No. 1, 1986
'
1 1
0
Data from LI 8 Lu(1973) w l t h ka,,= 0.0838. k c , = 0.0666
- CCOR Eq
/ /
t
I
1 3-
-
0
t 0
-
CCOR Eq
---
SOAVE Eq ( k 1 , = 0 . 7 2 9 )
0
Data f r o m Simnick e t a 1 . ( 1 9 7 7 )
I 0.2 0.4 0.6 0.8 1.0 Mole Fraction of n-Hexane
Figure 12. Vapor-liquid equilibrium in mixtures of n-hexane benzene.
11 1
+
teraction constants shown in Figure 11,all of the data from both sources are well represented by the CCOR equation. The highest temperature and the lowest temperature are presented in the figure. For the n-hexane + benzene mixtures shown in Figure 12, Li and Lu (1973) reported only experimental bubble point conditions. Their results are well represented by the CCOR equation a t all temperatures for which data are reported. Mixtures of this type are conventionally represented with vapor pressure and activity coefficient. The CCOR equation is shown here to be capable of describing such mixtures. The use of an equation of state in place of vapor pressure and activity coefficient for the description of vapor-liquid equilibrium offers the advantage of a uniform and consistent method which works at low pressure and high pressure, for mixtures of condensables only and for mixtures containing noncondensables. Hydrogen-containingmixtures are of much technological interest in connection with hydrofining and other processes. Table I1 shows the representation of vapor-liquid equilibrium for a number of hydrogen-containing mixtures with the CCOR equation. The average absolute deviation of the calculated K values from the experimental is reported for each component of a mixture. Subscript 1refers to the light gas, and 2, the solvent. An average absolute deviation of about 5% is observed for both components for most of the mixture systems. Interaction parameters are reported in the table for all the mixtures studied. Table I1 gives a global picture of the comparison of the CCOR with data on many mixtures. The comparison is, however, deficient as it lacks details. We, therefore, show detailed comparison with figures for a few selected mixtures. Figure 13 shows the CCOR equation calculated K values of hydrogen in mixture with tetralin for comparison with experimental data by Simnick et al. (1977). The interaction parameters used in the calculation are included in Table 11. We include the Soave equation calculated results in the figure because of the wide use of the equation. The Soave equation appears to diverge substantially from the data. Figure 14 shows similar results for the K value of tetralin in the same mixture of Figure 13. The comparison of the
1
3
5
7
P,
1 0 MPa
50
30
Figure 13. K values of hydrogen in mixtures with tetralin.
G623K
01
Ooo5
1
-CCOREq ---SOAVE Eq 0
'\,4628K
Data from
Simnlck et al.(1977)
0002-"-'
t l ' l '
1
3
5 7 1 0
I
1
30
I
/ I I l / ,
50
100
P,MPa
Figure 14. K values of tetralin in mixtures with hydrogen.
CCOR equation and the Soave equation with experimental data also appears to be similar to that shown in Figure 13. Table I11 presents CCOR equation calculated VLE of methane mixtures. The average absolute deviation of the calculated K values from the experimental amounts to about 3% for both methane and the solvent for most of the mixtures. Interaction parameters are reported in the table for all the mixtures studied. Figure 15 shows the K values of methane + propane mixtures calculated with the CCOR equation in comparison with experimental data of Reamer et al. (1950). The mixture of methane and a much heavier solvent, n-decane, is shown in Figure 16. The CCOR equation appears to agree equally well with data for both mixtures. Table IV presents CCOR equation calculated VLE of C 0 2 mixtures. The average absolute deviation of the calculated from the experimental K values amounts to about 4 % for both C02 and the solvent for most of the mixtures. Interaction parameters are reported in the table for the mixtures studied.
Ind. Eng. Chem. Fundam., Vol. 25, No. 1, 1986
Table 111. CCOR Equation Calculated VLE of Methane Mixtures
AAD, %
Trange, K
p range, MPa
no. of data points
ka12
ke12
K,
Kz
source
COP ethane
230-270 158-283
1.5-8.5 0.2-7
33 38
0.2381 ._ _ 0.1211
0.0227 0.1615
2.8 10.3
2.8 8.2
propane n-butane isobutane n-pentane isopentane neopentane n-hexane n-heptane n-octane n-decane n-hexadecane cyclohexane benzene toluene m-xylene m-cresol tetralin quinoline 1-methylnaphthalene diphenylmethane
277-361 294-394 311-378 311-444 344-411 344-411 311-412 411-511 348-423 511-583 462-703 378-444 421-501 422-543 461-582 462-663 462-665 462-703 464-704 462-703
0.7-10 0.5-12 0.5-11 0.1-15 3-15 2-12 0.5-2 1-18 2-7 3-19 2-25 1-20 2-24 2-25 2-20 2-25 2-25 2-25 2-25 2-25
40 43 41 33 21 18 16 19 19 23 24 24 18 25 22 25 24 28 18 25
0.2308 0.2500 0.2424 0.2256 0.2603 0.2755 0.2979 0.2824 0.4119 0.3181 0.2908 0.2569 0.2442 0.2614 0.2836 0.3017 0.3120 0.2961 0.2993 0.3038
0.1204 0.1019 0.1081 0.0796 0.1192 0.1452 0.1142 0.0821 0.1407 0.0716 -0.0056 0.0587 0.0081 0.0131 0.0447 0.0008 0.0190 0.0125 0.0157 0.0280
Davalos et al. (1976) Price and Kobayashi (1959) Wichterle and Kobayaahi (1972) Reamer et al. (1950) Sage et al. (1940) Olds et al. (1942) Sage et al. (1942) Prodany and Williams (1971) Prodany and Williams (1971) Gunn et al. (1974) Reamer et al. (1956) Kohn and Bradish (1964) Lin et al. (1979) Lin et al. (1980) Reamer et al. (1958) Lin et al. (1979) Lin et al. (1979) Simnick et al. (197913) Simnick et al. (1979b) Sebastian et al. (1979) Simnick et al. (1979~) Sebastian et al. (1979) Sebastian et al. (1979)
solvent
Table IV. CCOR Equation Calculated VLE of CO, Mixtures no. of T range, p range, data solvent K MPa points kat* H2S 273-353 2-8 32 0.1827 ethane 223-293 0.6-6 55 0.1371 propane 277-344 7-68 38 0.2048 310-411 n-butane 0.7-8 34 0.2486 311-394 isobutane 0.7-7 29 0.2328 n-pentane 310-394 1-10 23 0.1000 n-hexane 313-393 0.8-11 39 0.2813 463-584 n-decane 2-5 16 0.3619 n-hexadecane 16 463-664 2-5 0.3505 cyclohexane 473-533 2-10 31 0.2577 393-543 to1uen e 2-5 20 0.2845 m-xylene 462-583 2-5 16 0.3290 m-creso1 463-665 14 2-5 0.3334 tetralin 462-665 2-5 16 0.3943 quinoline 462-703 2-5 15 0.3320 1-methylnaphthalene 463-704 15 2-5 0.4664 463-704 diphenylmethane 16 2-5 0.3129
kc12
0.0798 0.0294 0.1802 0.1973 0.2064 0.2239 0.2305 0.1739 0.0664 0.1989 0.1470 0.1605 0.1007 0.1507 0.1032 0.1774 0.0999
3.9 4.0 2.2 2.6 2.4 3.2 1.4 3.1 6.5 1.8 4.8 1.1 1.6 3.0 2.2 1.7 2.6 2.0 1.9 2.2
1.7 2.7 2.1 2.9 4.2 3.5 3.7 3.1 3.9 2.9 16.0 3.8 3.6 4.8 3.3 3.4 8.5 3.9 7.3 9.9
AAD,% K1 KZ 7.2 3.3 7.0 6.2 5.1 4.0 9.5 3.0 5.3 2.2 3.2 3.0 3.9 3.2 7.2 3.7 2.5
4.5 2.8 4.3 4.2 4.5 12.4 10.2 3.9 9.6 0.8 6.0 3.0 2.1 3.2 2.2 4.9 6.8
source Bierlein and Kay (1953) Fredenslund and Mollerup (1974) Reamer et al. (1953) Olds et al. (1949) Besserer and Robinson (1973) Poettmann and Katz (1945) Li et al. (1981) Sebastian et al. (1980e) Sebastian et al. (1980e) Krichevskii and Sorina (1960) Sebastian et al. (19800 Sebastian et al. (19800 Sebastian et al. (1980a) Sebastian et al. (1980~) Sebastian et al. (1980a) Sebastian et al. (1980b) Sebastian et al. (1980b)
511 K&.
OO€
0
Data from Reamer et a1.(1950)
ODAO
et o l . (f 1r o Data 97 m9 )Lin
CCOR Eq
5Y
Methane
1-
-
t
0.11 0.2
'
'
"
'
1
0.5
8
I
'
5
1
10
I
0.11 1
3
P , MPa
Figure 15. K values of methane
+ propane mixtures.
5
P , MPa
Figure 16. K values of methane
1
10
+ n-decane mixtures.
1
30
81
82
Ind. Eng. Chem. Fundam., Vol. 25, No. 1, 1986
Table V. CCOR Eauation Calculated VLE of Nitrogen Mixtures no. of T range, data P range, solvent K MPa points kd2 methane 125-183 0.3-5 104 0.1726 n-hexane 311-444 2-35 52 0.2800 n-heptane 453-497 5-28 0.3776 21 2-25 n-hexadecane 462-703 0.3856 27 methylcyclohexane 311-478 0.4-17 0.3278 28 m-cresol 2-25 462-664 0.3583 26 tetralin 2-25 464-663 0.3911 25 quinoline 2-25 27 462-704 0.3850 2-25 462-703 27 1-methylnaphthalene 0.3946
AAD, 9i kc12
K1
0.1516 0.0008 0.0342 -0.0357 0.0183 -0.0373 -0.0073 -0.0220 -0.0162
6.0 3.7 2.1 3.1 2.6 3.4 2.6 3.8 3.3
Kz 6.9 9.1 6.3 13.5 6.2 2.5 6.4 4.8 6.0
source Stryjek et al. (1974) Poston and McKetta (1966) Brunner et al. (1974) Lin et al. (1981) Robinson et ai. (1981) Kim et al. (1983) Kim et al. (1983) Kim et al. (1982) Lin et al. (1983a)
100
40 o m 0
-
501 I
D a t a f r o m Sebastian (1980e) CCOR Eq
k
20/
0
Data from Kim e t a l . (1983)
10
t
Nitrogen
\
I L l l J
1 .o
0.3
IO
5.0
P , MPa
Figure 17. K values of COz + n-decane mixtures.
'1
'1
oom
Data f r o m B i e r l e i n e t 0 1 (1953)
303Kk
- CCOR
v;
Eq
Figure 19. K values of nitrogen
323K
2Y
1
I
+ tetralin mixtures.
experimental data. There appears a greater tendency for the K value of H,S to increase with decreasing pressure than predicted by the CCOR equation, indicating a strongly varying activity of H2S in dilute liquid solution in C02. Table V presents CCOR equation calculated VLE of nitrogen mixtures. The average absolute deviation of the calculated K values from the experimental amounts to about 4% for nitrogen and 5% for the solvent. Interaction parameters are reported in the table for the mixtures studied. Figure 19 shows the K values of nitrogen tetralin mixtures calculated with the CCOR equation in comparison with experimental data of Kim et al. (1983). Both components seem to be well described by the CCOR equation. In all the VLE calculations with the CCOR equation, two interaction parameters were used. The introduction of the second parameter ItCY(or Itb,), in addition to It,$,, appears to improve significantly the accuracy of the equation, particularly for asymmetric mixtures. The calculations with the Soave equation used only one interaction parameter, k,,,. El-Twaty and Prausnitz (1980) suggested a new temperature-dependent a'(T) function for the Soave equation for temperatures above the critical and modified the composition dependence of the covolume
+
0.5 2
3
4
5
7
10
P , MPa
Figure 18. K values of COz + H2S mixtures.
Figure 17 shows the K values of C02 + n-decane mixtures calculated with the CCOR equation in comparison with experimental data of Sebastian et al. (1980e). Both components are well described by the CCOR equation. Figure 18 shows the K values for COz + H2Smixtures calculated with the CCOR equation in comparison with
Ind. Eng. Chem. Fundam., Vol. 25, No. 1, 1986 83
parameter b by introducing an interaction parameter E,, to replace k, for hydrogen + heavy hydrocarbon mixtures. However, the calculated results with their modification did not improve appreciably over those with the original Soave equation.
Discussion and Conclusion A cubic equation of state, called the CCOR equation, has been obtained by simplifying the COR equation to facilitate repetitive and iterative computations. The cubic equation can be solved algebraically for u at a given T and p . Such a solution, always required in the calculation of fugacity and other derived thermodynamic properties, is repeated numerous times in engineering calculations. The algebraic solution greatly simplifies the procedures. The new equation requires only T,, p,, and w of a substance to be known which is a convenience in contrast to the COR equation whose parameters P,Vo, and cR must be determined for each substance of interest. The CCOR equation quantitatively represents the vapor pressure and liquid density from the triple point to the critical point. Dependable calculation of enthalpy is also demonstrated. We report here the results of mixture calculations using one-fluid mixing rules for the equation parameters. Any mixture calculation with an equation of state must necessarily be dependent on the mixing rules of which a number are available and more are being developed. We use the simplest of them in this work as a start. The CCOR equation of state appears capable of describing puT and phase equilibrium of mixtures of nonpolar and slightly polar substances in good agreement with experimental data. An extensive comparison with VLE data has been made for mixtures of diverse types including symmetric mixtures of condensables and asymmetric mixtures containing a light gas and a heavy solvent. The CCOR equation gives almost uniformly good representation for low-pressure systems as well as high-pressure systems. The alternative to equation-of-state calculation of VLE is to represent condensables with activity coefficient and vapor pressure and to represent the noncondensable component with activity coefficient and Henry's constant. The equation-of-state approach offers a uniform method for all components. The CCOR equation appears to implement such an approach with accuracy for nonpolar and slightly polar mixtures. The representation of polar substances and their mixtures with an equation of state remains an active and fascinating area of investigation. Acknowledgment Financial support for this work was provided by Electric Power Research Institute through research project RP-367 and by the National Science Foundation through Grant CPE-8209624. T. M. Guo assisted in the VLE calculation of mixtures.
Nomenclature a = parameter in eq 1 b/4 = hard-core volume per mole c = parameter in eq 1
cR = number of equivalent rotational degrees of freedom d = parameter in eq 1 f = fugacity
h = enthalpy per mole ho = ideal gas enthalpy per mole
k = interaction parameter = equilibrium vaporization ratio N = mole fraction p = pressure R = universal gas constant
K
T = temperature P = characteristic temperature in COR equation u = volume per mole Vo = closest packed volume z = compressibility factor Greek Letters a = rotational strength of a rotator in eq 5 (Boublik's spherocylinder constant) a' = scaled temperature function of a t = a small number 11 = reduced density b / ( 4 u ) 8 = equation constant = fugacity coefficient o = acentric factor Subscripts a = for constant a b = for constant b c = for constant c c = critical-state property H = hydrogen hs = hard-sphere property i = component j = component k = component m = mixture r = reduced property rot = rotational contribution T = tetralin 1 = solute 2 = solvent
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--.
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Received for review September 8, 1983 Revised manuscript received December 17, 1984 Accepted March 19,1985
Part of this work was presented in the Third International Conference on Fluid Properties and Phase Equilibria for Chemical Process Design, Callaway Gardens, GA, April 10-15,1983 ( R o c . Conf., Fluid Phase Equilib. 1983,13, 143).
Modeling of the Impregnation Step To Prepare Supported Pt/A1,03 Catalysts Osvaldo A. Scelza, Albert0 A. Castro, D a h R. Ardlles, and Josd M. Parera' Institufo de Investig8ciones en Cat6lisis y Petroqdmica, INCAPE, Santiago del fstero 2654, 3000 a n t a Fe, Argentina
This paper reports a mathematical model that describes the process of impregnating porous particles with a solution that contains two or more adsorbable components. The parameters of the model (adsorption rate constants and diffusivities of the different solution species) can be estimated from the variation of the concentration of different adsorbable specles wlth the impregnation time. With these parameters, the profiles of deposited species into the particles are predicted and compared with those experimentally obtained. The model was applied to alumina impregnation with H,RCI, and HCI.
Introduction One of the most commonly used methods to prepare supported catalysts is the impregnation of particles of a porous support with solutions that contain species which are adsorbed on the support. Sometimes all the adsorbed species are catalytically active, but in other cases some of them only modify the distribution of the other species on the support surface. 0196-4313/86/1025-0084$01.50/0
The behavior of supported catalysts usually depends on the spatial distribution profiles of the species deposited on the internal surface of the support. Hence, the factors which define the distribution profiles must be known in order to design supported catalysts. The Maatman and Prater (1957) paper raised interest in the study of the impregnation step. Since then, much research has been done on this subject. Several mathe0 1986 American Chemical Society
