52
R. S. Lebelo and O. D. Makinde
Equations (5.3) and (5.4) now have the following forms (Eq. (5.6)):
∂θ
∂t
=
∂
2
θ
∂r 2 +
1
r
∂θ
∂r
+ α(1 + εθ)
m e
[θ/(1+εθ)]
+ αω(1 + εθ)
m e
[θ/(1+εθ)]
− ϕθ.
(5.6)
The initial and boundary conditions are shown in Eq. (5.7):
θ (r, 0) = θ 0 ;
∂θ
∂r
(0, t) = 0; θ (1, t) = 0.
(5.7)
Here θ and θ 0 are the respective dimensionless temperature and initial temperature.
The dimensionless parameters α, ε, r and are Frank-Kamenetskii (reaction rate),
activation energy, radial distance, and activation energy ratio, respectively. Lastly,
the dimensionless parameters for two-step reaction and heat loss are represented by
ω and ϕ.
5.2.1 Numerical Approach
The numerical solution for the governing equation using the semi-implicit FDM
followed the following route.
θ
N +1
− θ
N
t
=
∂
2
∂r 2 θ
N +τ
+
1
r
∂
∂r
θ
N
+ α
(1 + εθ)
m e
[θ/(1+εθ)]
N
+ αω
e
[εθ/(1+εθ)]
N − ϕθ
N
(5.8)
The τ is an arbitrary number such that 0 ≤ τ ≤ 1 and τ = 1 for the convenience
of using larger time steps. Rearranging Eq. (5.8), multiplying through by t and
taking γ =
t
r 2 yielded the expression for θ
N +τ as follows:
−τ γ θ
N +1
j+1 + (1 + 2τ γ )θ
N +1
j
− τ γ θ
N +1
j−1 = −γ (1 − τ )θ
N
j+1 +
1 − 2γ (1 − τ ) − ϕ
θ
N
j
− γ (1 − τ )θ
N
j−1 +
1
2r
γ
θ
N
j+1 − θ
N
j−1
+ ααt
(1 + εθ)
m e
[θ/(1+εθ)] + ωe
[θ/(1+εθ)]
N
(5.9)
A tri-diagonal matrix system was derived from Eq. (5.9) and Maple software was
used to solve the system.