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 Y 5 = -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
Ind. Eng. Chem. Fundam., Vol. 21, No. 1, 1982 59
allowing catalyst activity to appear nonlinearly in the rate expression for the decline of activity, in the manner of Khang and Levenspiel (1973). First-order main kinetics is considered, although the approach taken readily extends to other kinetic forms. The method of multiple scales (cf. Cole, 1968; Nayfeh, 1973) is applied to the transformed equations and this resulta in analytical solutions uniformly valid in time from startup to final loss of activity. Multiscaling appears to have been rarely if ever used by chemical engineers; because of its extreme power it is discussed in some detail. Formulation For a first-order reaction in an isothermal tubular reactor, the nondimensional material balance equation on reactant is
where a is the reaction number kL(1- ef)(l - ep)(uefl and we have assumed the dimensionless concentrations A and activity a to have uniform radial distributions. Catalyst activity declines according to &/at = -eaAad, (parallel) &/at = - e a ( l - A)a, (series)
(2a) (2b)
where e = kdCo[ef + (1- ef)p]/[k(l- ef)(l - e,)]
