Int. J. Heat Mass Transfer. Vol. 33, No. 6, pp. 1081-1085, 1990 0017-9310/90 $3.00+0.00 Printed in GreatBritain ~ 1990 PergamonPress plc An extremum variational principle for some non-linear diffusion problems TEODOR M. ATANACKOVI(~ and DJORDJE S. DJUKI(~ Faculty of Technical Sciences, University of Novi Sad, 21000 Novi Sad, Yugoslavia (Received 19 December 1988 and infinalform 8 May 1989) Alntract--An extremum variational principle for two non-linear boundary value problems is formulated, The first boundary value problem corresponds to the coupled diffusion reaction with high-order kinetics, The second boundary value problem describes zero-order chemical kinetics in a single catalyst pellet with Robin boundary conditions at the pellet's outer surface. For both problems, approximate solutions and their error estimates for several values of the parameters are obtained. INTRODUCTION Cm~MICAL reaction and diffusion problems often lead to non-linear boundary value problems for ordinary and partial differential equations. For example, steady-state problems with a single reaction are de- scribed by a boundary value problem of the form [1] and dy dy forx=0 ~=0; forx=l -~=Bi(l--y). (2) In equations (1) and (2) y denotes the non-dimen- sional concentration, x the (single) space coordinate, s depends on the problem geometry and has the values 0, l, 2 for a slab, cylinder and sphere, respectively, and Bi is the Blot number. The functionf0(y) may have a variety of forms and in what follows we shall assume that it is given by ~(l+k)f (i) f0(y) = k+y ~ (3) and 0i) fo(y) = 02.v ". (4) In equations (3) and (4) ~b 2 is the Thiele modulus, n the reaction order and k the non-dimensional par- ameter that measures the influence of the catalyst on the process. Thus equations (1) and (2) with fo given by (i) describe the diffusion reaction with nth order kinetics inside a single catalyst pellet, while equations (1) and (2) with f0 given by (ii) describes the diffusion reaction with nth order kinetics. Different aspects of the boundary value problem, equations (1) and (2), withfo given by equation (3) were studied, for example, in refs. [1-5]. 1081 In what follows we shall treat the boundary value problems (1), (2), (3) and (1), (2), (4) by a variational procedure developed in ref. [6]. Thus we shall first construct an extremum variational principle for both problems. Then this principle will be used to obtain an approximate solution to the problem. Finally, the error in the approximate solution will be estimated. The error estimating procedure presented here is somewhat different from the procedure presented in ref. [6]. VARIATIONAL PRINCIPLE (i) First we consider diffusion inside a single catalyst pellet. In this case fo(Y) is given by (i), that is the diffusion process is described by d (x~dy) ~2(l+k)f =0 (5) x-~x\" ~x)- k+f dy dy d'-~ = 0 for x = 0, ~xx = Bi(l-y) for x = 1. (6) The variational method developed in ref. [6], when applied to equations (5) and (6), shows certain diffi- culties in the error estimating procedure. Therefore, we transform equations (5) and (6) by introducing a new independent variable by the relation t = x t+'. (7) Then equations (5) and (6) become d f ~/(I +,> dy~ Ay" dr LDt d-tJ- k +f = 0 (8) lim r '/° + "> ----~ = 0 ,~o dt ~/JD (1 -y(l)) dy(l) --87-= (9)