Abstract
A simple model for the exothermic oxidation of a solid reactant or reaction in a catalyst bed is considered. A gaseous component, e. g. O
2
, diffuses through a porous medium where it reacts releasing heat. The concentration of the gaseous species in the surrounding reservoir is held constant as is the surrounding (ambient) temperature. Heat transfer within the reaction zone occurs by conduction, leading to an internal temperature-position profile. The reaction rate at any point depends on the local temperature through an Arrhenius rate-law and is first order in the gaseous species concentration. Consumption of the solid is ignored in this paper. Stationary-state solutions are governed by the dimensionless coupled reaction-diffusion equations ∇
2
v
-
αλ
(1 +
v
) exp [
u
/(1+
εu
)] = 0, ∇
2
u
+
λ
(1 +
v
) exp [
u
/(1 +
εu
)] = 0, where
u
and
v
are temperature and concentration respectively,
λ, α
and
ε
are parameters and ∇
2
is the laplacian operator. The boundary conditions considered here are of Robin form: ∂
u
/∂
n
+
μu
= 0, ∂
v
/∂
u
+
νv
= 0 at
x
= 1, where
μ
and
ν
are the Nusselt and Sherwood (or Biot) numbers. The reaction geometry is restricted to the infinite slab and infinite cylinder for which ∇
2
= ∂
2
/∂
x
2
+ (
j
/
x
) ∂/∂
x
with
j
= 0 and 1 respectively. Of particular interest are the dependences of the stationary-state temperature and concentration profiles on the parameter
λ
and the way in which these dependences are unfolded as
α, ε
,
μ
and
v
are varied. Up to five branches of Stationary-state solutions may be encountered, although this requires σ =
μ/v
< 1. The development of the temperature-position and reaction-rate-position profiles along the different branches is determined both numerically and analytically.
Reference10 articles.
1. Aris R. 1975 The mathematical theory of diffusion and reaction in 'permeable catalysts vol. 1: The theory of the steady state. O xford: Clarendon Press.
2. Steady states of the reaction-diffusion equations, P art 1: Questions of existence and continuity of solution branches. J. Aust. math;Burnell J. G.;Soc. B,1983
3. 5 Steady states of the reaction-diffusion equations, P art 2: Uniqueness of solutions and some spherical cases. J. Aust. math;Burnell J. G.;Soc. B,1983
4. Steady states of the reaction-diffusion equations, P art 3: Questions of multiplicity and uniqueness of solutions. J. Aust. math;Burnell J. G.;Soc. B,1985
5. Communications on the theory of diffusion and reaction — IV Combined effects of internal and external diffusion in the non-isothermal case
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