Page 281 - Numerical Methods for Chemical Engineering

270 6 Boundary value problems

1
β 1

η
1
eectiveness actr β 1

β

β 1

1 2 1 1 1 1 1 1 2
iee ds Φ

Figure 6.5 Effectiveness factor vs. Thiele modulus for nonisothermal first-order chemical reaction
within a spherical catalyst pellet.

effect arises due to an increase in k(T ) by (6.36). In practice, this behavior is not observed,
as solids are better transmitters of heat than mass. Therefore, η< 1 when internal mass
transfer resistance is strong.

Finite differences for a convection/diffusion equation

Above we have considered problems in which transport is solely diffusive. We now consider
the treatment of convection terms that introduce new numerical difficulties. We consider
the simple PDE
2
∂ϕ ∂ϕ ∂ ϕ
=−v + + s(z, t,ϕ) (6.69)
∂t ∂z ∂z 2
At steady state and with no source term, this equation becomes
2
dϕ d ϕ
− v + = 0 (6.70)
dz dz 2
With the Dirichlet boundary conditions
ϕ(0) = ϕ 0 ϕ(L) = ϕ L (6.71)
the solution for 0 ≤ z ≤ L is
e z(Pe)/L − 1
ϕ(z) = ϕ 0 + (Pe) (ϕ L − ϕ 0 ) (6.72)
e − 1
where the dimensionless Peclet number is defined as
vL strength of convection
Pe = = (6.73)
strength of diffusion
This solution is plotted in Figure 6.6 for various Pe values. When Pe 1, diffusion is dom-
inant and we observe a linear increase from ϕ 0 = 0to ϕ L = 1. When Pe 1, convection