Page 280 - Numerical Methods for Chemical Engineering

A spherical catalyst pellet 269

elements of the Jacobian in each interior row j = 2, 3,..., N − 1 are

J j, j−1 = A j, j−1 J j, j+1 = A j, j+1
(6.63)
d γβ(1 − ϕ)
2
2
J jj = A jj − ξ g(ϕ j ) g(ϕ) = exp
j
dϕ 1 + β(1 − ϕ)
In the first and last rows, the nonzero elements are
2
2
J 11 = A 11 + a 1 A 10 − ξ g(ϕ 1 )
1 J 12 = A 12 + a 2 A 10 (6.64)
2
2
J NN = A NN − ξ g(ϕ N )
J N,N−1 = A N,N−1
N
The linear system comprising (6.63) and (6.64) is thus tridiagonal, and elimination is per-
formed rapidly, even for large N.
Definition of the effectiveness factor
From the solution of the BVP (6.44) and (6.45), we use the resulting concentration and
temperature fields to compute the total rate of reaction within the pellet,
' R
2
R tot = k(T )c A (r)(4πr )dr (6.65)
0
Rewriting this integral in terms of the dimensionless quantities yields
' 1
4 3 γβ(1 − ϕ A (ξ)) 2
R tot = π R (k s c A,s ) 3 ϕ A (ξ)exp ξ dξ (6.66)
3 0 1 + β(1 − ϕ A (ξ))
The product in the first set of square brackets is the total reaction rate if there were no con-
centration (or temperature) gradients within the catalyst particle. We define the effectiveness
factor η from the relation
4 3

R tot = π R (k s c A,s )η (6.67)
3
such that
' 1 γβ(1 − ϕ A (ξ))
2
η = 3 ϕ A (ξ)exp ξ dξ (6.68)
0 1 + β(1 − ϕ A (ξ))

Numerical solution in MATLAB
catalyst nonisothermal scan.m plots η vs. for fixed γ and various values of β.For γ = 1,
and ξ = 0.01 in the interior and ξ = 0.001 near the surface, the results are shown in
Figure 6.5.
For zero or moderate reaction heating, β ≈ 0, increasing reduces η. This is easily
understood, as slow diffusion causes a depletion zone of low A concentration to form in the
center of the pellet, where the reaction rate is consequently low. By contrast, if there were
significant heating relative to conduction, β ≥ 1, for ≈ 1, η would be greater than 1;
i.e., the rates of reaction are higher than in the absence of internal transport resistance. This