Page 278 - Numerical Methods for Chemical Engineering

A spherical catalyst pellet 267

negligible. When ≥ 1, the opposite is true and mass transfer resistance becomes rate-
dominant. β is a measure of the relative importance of the heat of reaction, so that for β
> 1 there is significant internal heating, T (r) > T S , and when β< −1, significant internal
cooling. γ is the dimensionless activation energy, and a large γ means that the reaction rate
is very sensitive to the local temperature.

Finite differences on a nonCartesian, nonuniform grid

To solve (6.44), we use finite differences on a grid 0 <ξ 1 <ξ 2 <···<ξ N < 1 and require
that at each ξ j , (6.44) be satisfied locally:

d 2 dϕ A 2 2 γβ(1 − ϕ j )

ξ − ξ exp ϕ j = 0 (6.47)
j
dξ dξ 1 + β(1 − ϕ j )
ξ j
where ϕ j = ϕ A (ξ j ). As we expect strong gradients near ξ = 1 when ≥ 1, we use a
nonuniform grid with closer grid points near the surface. Defining the mid-points in the
intervals between grid points,
1
1
ξ j+1/2 = (ξ j + ξ j+1 ) ξ j−1/2 = (ξ j + ξ j−1 ) (6.48)
2 2
we use a central difference approximation for the second derivative,
) * ) *

ξ 2 j+1/2 dϕ A − ξ 2 j−1/2 dϕ A
d 2 dϕ A dξ ξ j+1/2 dξ ξ j−1/2
ξ ≈ (6.49)
dξ dξ (ξ j+1/2 − ξ j−1/2 )
ξ j
For the first derivatives, we use similar approximations,

dϕ A ϕ j+1 − ϕ j dϕ A ϕ j − ϕ j−1
≈ ≈ (6.50)
dξ ξ j+1 − ξ j dξ ξ j − ξ j−1
ξ j+1/2 ξ j−1/2
to obtain from (6.49) and (6.50) the finite difference approximation

d 2 dϕ A

ξ ≈ A j, j−1 ϕ j−1 + A jj ϕ j + A j, j+1 ϕ j+1 (6.51)
dξ dξ
ξ j
where
(lo) (lo) (hi) (hi)
A j, j−1 = α A j, j =− α + α A j, j+1 = α
j j j j
(6.52)
ξ 2 ξ 2
(lo) j−1/2 (hi) j+1/2
α = α =
j j
(ξ j − ξ j−1 )(ξ j+1/2 − ξ j−1/2 ) (ξ j+1 − ξ j )(ξ j+1/2 − ξ j−1/2 )
For each interior point j = 2, 3,..., N − 1 that does not neighbor a grid point at the
boundary, the nonlinear algebraic equation obtained from (6.44) by finite differences is
γβ(1 − ϕ j )

2
2
f j = A j, j−1 ϕ j−1 + A jj ϕ j + A j, j+1 ϕ j+1 − ξ exp ϕ j = 0 (6.53)
j
1 + β(1 − ϕ j )
Treatment of Dirichlet and von Neumann boundary conditions
The boundary conditions (6.45) are of Dirichlet-type (specified ϕ) at the surface and of von
Neumann-type (specified dϕ/dξ) at the center. At the last grid point ξ N < 1, we enforce