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Ind. Eng. Chem. Fundam. 1982, 21, 56-58
Relations among the Virfal Coefficients of the Series‘ for the Compressibitky Factor Zand for 2-‘ and Log 2 George S. Kell DMsion of Chemistry. National Research Council of Canada, Ottawa, Canada K1A OR9
The relations between the virial coefficients of both the pressure and density series for the compressibility factor Z and for Z-‘ and log Z are given for up to the fifth virial coefficient in a way that displays the patterns of the transformations between them.
+ B2p + B3p2 + B4p3 +... Z-1(4) = 1 + c24 + c342 + c443 + ... Z-’(p) = 1 + D2p + D3p2 + D4p3 + ... In Z(4) = Ez4 + + E443+ ... In Z(p) = F2p + F3p2 + F4p3 + ...
If, to simplify the relations to be written later in this paper, 4 is written in place of PIRT, the equation of state of a gas may be written 4 = PZ (1) where p is the density and Z is the compressibility factor, which is defined by this equation. At moderate densities Z for an isotherm io often expressed as a power series in pressure or density, and the coefficients of these series are called virial coefficients. If it is desired to express p as a function of 4, eq 1 becomes p = $2-1
Z(p) = 1
(6) (7) (8)
(9) There are 30 sets of relations among the coefficients of these six equations, but only 13 different forms. All transformations can be obtained simply-where “simply” means by the relations given in the appendix-from three that are suitably chosen, say eq 10,12, and 17; but all 13 relations are needed to show the symmetries of the transformations. Relations Involving 2 and 2-’ The relations between coefficients A and B have been given by many authors (Scatchard, 1930; Epstein, 1952; Putnam and Kilpatrick, 1953; Altunin and Spiridonov, 1967; Kilpatrick and Ford, 1969; Mavrides, 1976). The relations are A2 = B2
(2)
The expansion of Z-’ as a pressure series has been used by McLaurin and Kell (1980). Alternatively, eq 1 may be written in logarithmic form to give log 4 = log p + log z (3) a transformation which is always possible at low densities where 4, p, and 2 are positive. Log Z may be expanded as a power series in pressure or density, and both expansions were treated in a systematic way by Silberberg et d. (1959). Briano and Glandt (1981) report that log Z as a series in density shows good convergence. Kell(l981) has shown that only the use of the pressure expansion of log Z leads to an analysis of the Burnett experiment that is consistent with the assumptions of linear least squares. The choice of one form of the equation of state over another may be made on the basis of economy in representing the data for a particular substance, or to have linear coefficients in the preferred equation, either for integration in thermodynamic relations or to obtain a model suitable for the linear theory of the treatment of data. The need for the transformations is accepted as shown elsewhere. The purpose of this note is simply to give the interrelations of the coefficients of the series. Several authors have discussed the degree to which the expansions for Z are not equivalent (Eubank and Angus, 1973; Cox and Lawrenson, 1973). There appears to have been no discussion of the propagation of error when the coefficients of one series are expressed in terms of another, but as the transformations are nonlinear, the linear theory of error must be used with caution. For this reason only terms up to the fifth virial coefficients are written here. The series under consideration may be written as follows. Z(4) = 1 + A24 + A3@ + + ... (4)
A3 = B3 - B: A4 = B4 - 3B&
+ 2B,3
A5 = B5 - 4B& - 2B32+ lOB3BZ2- 5B24
(10)
The inverse transformation is Bz = A2
B, = A3 + AZ2 B4 = A4 B5 = A5
+ 4A&
+ 3A3A2 + A23 + 2A32+ 6A3A2’ + A;
(11) We denote the relation between sets of coefficients by (A) (B),where the arrow points toward the coefficient on the left side of the equations. The relations between C and D can be determined by successive substitutions by the Scatchard (1930) method, except that here the substitutions are based on p = 4Z-l rather than on 4 = pZ. The result is that (A) (B) = (D) (C). The relations for coefficients A and C are easily derived by noting that 22-’= 1 (Van Orstrand, 1910; Abramovitz, 1964). C2 = -A2
-
-
C3 = -A3
-
+ AZ2
+ 2A3A2 - A2, C5 = -A5 + 2A4A2 + A32- 3A3Az2 + AZ4 C4 = -A4
‘NRCC No. 19808. 0196-4313/82/1021-0056$01.25/0
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0
1982 American Chemical Society
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57
Ind. Eng. Chem. Fundam., Vol. 21, No. 1, 1982
Z (9
t.' Z% Figure 1. Transformations relating to the Coefficients of 2 and 2'. The numbers refer to the equations giving the transformation.
-
-
-
-
And showing the same pattern are {AI (C)= (Cl {A) = {D) {B)= (B] (D). Combining the preceding results gives the following for B and C C2 = -B2 C3
-
E4 = B4 - 4B3B2 + 10/3B23
-
D2
= -A2
0 3
= -A3
B2 = E2
B5 = E5
-
(14)
And (D) (A)= (A) {D). Figure 1 shows schematically how the five transformations, eq 10 to 14, relate the coefficients of eq 4 to 7. Relations Involving In 2 The relations between E and F can be obtained by the Scatchard method basing the substitution on 9 = p exp(ln Z(P)I. E2 = F2
+ 5E&
-
F3 = A3 + f/2A22
+ 2A3A2 + Y3Az3 F5 = A5 + 3A4A2 + 3/2A32+ 3A3A2' + 5hA; F4
= E6
(15)
= E2
+ 3E& + 3 / 8 2 '
+ 4E4E2 + 2E32+ 8E3EZ2+ 8/3E2'
(16)
The relation between the coefficients of Z and ln Z can be found either by applying the expression for In (1+ x ) to Z or by expanding exp(1n 2). E2 A2
E3 = A3 - f/2A22 E4 = A4 - A3A2
A3 = F3 - 1/FZ2
A3 = E3 + A6 = E5
+
Y3 = sX3+ aXZ2
+ fX3XZ2 + gX24(23)
Write a second transformation as 2 2 = UY2
Z3 = uY3 + aYZ2
+ E3E2 + 1/6E23
+ E4Ez + 7 8 3 , + f/2E3EZ2+ 1/243z4
(22)
And (A) {F)= {D] (E). The interrelations involving eq 15 to 22 are shown in Figure 2. It may be noticed that the structure of the terms of these expressions would be slightly more transparent if the subscript on each virial coefficient were lowered by one. Appendix The form of transformation treated in the main text may be displayed more abstractly. Let s = f l be the sign of the leading term in the transformation, which can be written as Y2 = sx,
Ys = sX5 + dX4Xp + (17)
A2 = E2 A4 = E4
-
+ 9/2F3F22- 9/$24
Y4 = sX4+ bX3X2+ c X , ~
+ '/3A23
E5 = A5 - A4A2 - f/2A32+ A3AZ2 - 1/4A24
(21)
A2 = F2
AS = F5 - 3F4Fz - 3/2F3'
F3 = E3 + E22
F5
= A4
A4 = F4 - 2F3F2 + Y3F2'
E4 = F4 - 3F3F2 + 3/2F23 E5 = F5 - 4F4F2- 2F32+ 8F3F2, - 8/3F24
-
(20)
And (B) {E)= {C) {F). The remaining relations are F2 = A2
E3 = F3 - F:
= E4
+ 4E&, + 8/3323 + 5/2E32 + 25/2E3E,2+ '25/,&z4
B4 = E,
= -A5 - 2A4A2 - Aa2 - A3AZ2
F4
(19)
B3 = E3 + 3/2EZ2
= -A4 - A3A2
F2
-
E5 = B5 - 5B4BZ - 5/2B32+ 15B3Bz3- 35/4B24
Similarly, the relations between
D4
-
+ 5 B a 2- 5B23
-p).
0 5
-
E3 = B3 - 3 / 2 B Z 2
C5 = -B5 + 6B4B2 + 3B32- 21B3B2 + 14B24 (13) And (B) {C]= (C) A and D are
-
And (E] {A)= {F) (B)= (F) (Dl = {El IC). I t is necessary to be able to express all coefficients in terms of B. For the transformations between E and B, by combining eq 10 and 17 we obtain E2 = B2
+ 2B22
= -B3
C4 = -B4
Figure 2. Transformations involving the coefficients of In 2.
+ BY3Y2 + -yY23 Z5 = uY5 + bY4Y2 + tYS2 + {Y3Y22+ qY2* Z4 = uY4
(18)
(24)
Ind. Eng. Chem. Fundam. 1082,27, 58-63
58
Y , = -x,
Noting that s2 = 1, substitution of eq 23 in eq 24 gives 2 2 = usx2
+ (aa + C K ) X ~ ~ Z4 = a ~ X 4+ (ab + p)X3X2 + (UC + SY + s@)XZ3 Z5 = asX5 + (ad + 6)X4X2 + (ae + t ) X 3 2+ (uf + s l + sb6 + 2 s a ~ ) X ~+ X(ag ~+ ~ + sc6 + a2t + a l ) X Z 4 Z3 = asX,
TJ
Z X for a transformation and its inverse, of which eq 10 and 11,15 and 16,17 and 18, and 19 and 20 are examples. Noting that u = s, s-l = s, and that a = -sa, etc., the inverse of eq 23 is xz = SY, x3
= sY3 - say22
+ (sub - c ) Y ~ ~ X 5 = sY, - sdY4Y2- seY3, + (sbd + 2sae - f)Y3YZ2 + X 4 = sY4 - sbY3Yz
{ ( c - sab)d - su2e + a f - sglYz4
Except for the trivial case where a = b = ... = 0, in transformations where the forward and reverse transformations have the same coefficients, such as eq 12,13, and 14, s must have the value -1. The coefficients are not all independent, and c and f can be expressed in terms of the other coefficients.
Y3 = -x3
+ ax22
+ bX3Xz - f/,abXz3 Y5 = -X5 + d X 4 X z+ eXS2- (ea + Y2bd)X3X: + gX24 Y4 = -X4
Literature Cited Abramowitz, M. “Handbook of Mathematical Functions”; Abramowitz. M.; Stegun, I. A.. Eds.; National Bureau of Standards, Applied Mathematics Serbs, No. 55, 1964; pp 14-16. Altunln, V. V.; Spkidonov, G. A. H@h Temp. 1967, 5 , 1011-1013. Brleno, J. G.; Glendt, E. D. Fluld phase EquWb. 1981, 5 , 207-223. Cox, J. D.; Lawrenson, I. J. “Speclalist Periodical Reports: Chemical Thermodynamics”; McGlashan, M. L., Senior Reporter; The Chemical Society; London, 1973 pp 162-203 (see esp. p 168). Epsteln. L. F. J . Chem. phvs. 1952, 20. 1981-1982. Eubank, P. T.; Angus, S. J . Chem. Eng. Dsfa 1973, IS, 428-430. Kell, G. S. Physica A , 1981, 105A, 536-551. Kllpatrlck, J. E.; Ford, D. I. Am. J . phvs. 1969, 37, 881-887. Mavrkles, A. Chimka Chronka, New serfes, 1976, 5, 333-335. McLaurln, G. E.; Kell, G. S. “Water and Steam (Proceedings of the 9th International Conference on the Ropertles of Water and Steam, Munich, September 1979)”, Straub, J.; Scheffler, K., Eds.; Pergamon Press: Oxford, 1980 pp 185-190. Putnam, W. E.; Kllpatrlck. J. E. J . Chem. phvs. 1953, 21, 951. Scatchard, G. Roc. NaN. Aced. Sei. 1930, 16, 811-813. Silberberg, I.H.; Kobe, K. A.; McKetta, J. J. J . Chem. Eng. Dsfa 1859, 4 , 314-323. Van Orstrand, C. E. Phli. Mag. 1910, 19, 366-376.
Received for reuiew November 10,1980 Accepted September 22, 1981
Poisoning in Catalytic Tubular Reactors with Significant Axial Dispersion. An Analytical Approach Duong D. Do Department of Chemical Englneering, California Institote of Technology, Pasadena, California 9 1125
Ralph H. Weiiand’ Department of Chemical €n@neerhg, Ciarkson College of Technology, Potsdam, New York 13676
The evaluation of performance of a tubular reactor operating at moderate Peclet number and undergoing catalyst poisonlng by parallel and series deactivation mechanisms is analyzed. The techniques used are the application of finite integral transforms to the nonlinear kinetics In a novel way, followed by a singular perturbation analysis in the transform domain using generalized multiscaling. Analytical descriptions of the influence of Peclet number, reaction number, intraparticle dmwion resistance and poisontng mechanism on the time dependence of conversion are obtained for the dispersion model. The methods can be extended readily to any nonlinear kinetic rate form.
Introduction The dynamic response of fiied bed reactors to catalyst poisoning by reactant or product has been investigated by Do and Weiland (1981a,b) for first- and nth-order kinetics, respectively, in the limit of large Peclet number based on reactor length. A quasicontinuum model (cf. Hlavacek and Votruba, 1977) was used and although axial dispersion was included in the analysis of the startup period, it had to be neglected during the time of catalyst deactivation. The assumption Pe >> 1 allowed solutions to be developed asymptotically and the first term in the solution was obtained. Peclet numbers based on reactor length may not
always be large, however, so such an assumption is unnecessarily restrictive. We have recently developed a way of using finite Sturm-Liouville integral transforms in solving a class of nonlinear differential equations (Do and Weiland, 1981~) and the method has been used to analyze single catalyst pellet problems (Do and Weiland, 1981d,e). The requirement is that the differential operators themselves be linear, but highly nonlinear behavior of the reaction rate term can be effectively handled. Here we apply the technique to poisoning in fiied-bed catalytic reactors with modest Pe. Pore diffusional resistance is included by
0196-4313/82/102 l-OO58$O1.25/O 0 1982 American Chemical Society
