Stationary-State Solutions for Coupled Reaction-Diffusion and Temperature-Conduction Equations I. Infinite Slab and Cylinder with General Boundary Conditions

J. Brindley, N. A. Jivraj, J. H. Merkin, S. K. Scott
1990 Proceedings of the Royal Society A  
A simple model for the exotherm ic oxidation of a solid re a c ta n t or reaction in a cataly st bed is considered. A gaseous com ponent, e.g. 0 2, diffuses through a porous m edium where it reacts releasing heat. The concentration of th e gaseous species in the surrounding reservoir is held co n stan t as is the surrounding (am bient) tem perature. H e a t tran sfer w ithin th e reaction zone occurs by conduction, leading to an internal te m p eratu re-p o sitio n profile. The reaction rate a
more » ... any po in t depends on the local tem p eratu re thro u g h an A rrhenius rate-law and is first order in th e gaseous species concentration. Consum ption of th e solid is ignored in this paper. S tatio n arystate solutions are governed by the dimensionless coupled reaction-diffusion equations V2v -aA( 1 +^)e x p 1 +ew)] = 0, V2/m + A(1 + v) exp = 0, where u and v are tem p eratu re and concentration respectively, A, a and e are param eters and V2 is the laplacian operator. The b o undary conditions considered here are of Robin fo rm : du/dn + pu = 0, d v /d u + vv = 0 a t where p and v are the N usselt and Sherwood (or Biot) num bers. The reactio geom etry is restricted to the infinite slab and infinite cylinder for which V2 = 02/0 x 2 + (j/x)0/0# w ith j = 0 and 1 respectively. Of particu lar interest are the dependences of th e sta tio n a ry -sta te tem p eratu re and concentration profiles on the p aram eter A and th e w ay in which these dependences are unfolded as a, e, p and v are varied. U p to five branches o solutions m ay be encountered, although this requires 1. The developm ent of the tem p eratu re-p o sitio n and reactio n -rate-p o sitio n profiles along th e different branches is determ ined both num erically and analytically.
doi:10.1098/rspa.1990.0101 fatcat:35mvgb6b5jci3kxnc3ypx3smny