Page 277 - Numerical Methods for Chemical Engineering

266 6 Boundary value problems

grid that is nonuniform, with closer spacing between the points near the surface than deep
within the interior of the pellet.
Actually, for this problem if we assume constant D A , λ, and H, we can reduce the BVP
to a single equation; however, we return to the question of multiple fields later. Let us divide
(6.33) by − H,
d 2 λ dT 2
r + r (kc A ) = 0 (6.37)
dr (− H) dr
and add (6.37) to (6.32),
d d λ dT
2 dc A 2
r D A + r = 0 (6.38)
dr dr dr (− H) dr
Integrating this equation once yields
dc A λ dT
r 2 D A + = C 1 (6.39)
dr (− H) dr
2
Dividing by r gives
d λ C 1
D A c A + T = (6.40)
dr (− H) r 2
From the symmetry boundary condition (6.35), C 1 = 0, and a second integration yields
λ λ
D A c A + T = C 2 = D A c AS + T S (6.41)
(− H) (− H)
Thus, there exists a linear relation between c A (r) and T (r),
D A ( H)
T (r) − T S = [c A (r) − c AS ] (6.42)
λ
that allows us to solve only a single equation, (6.32).

Dimensionless formulation

We next reduce the number of independent parameters by rephrasing the problem in terms
of the dimensionless quantities
r c A (r = ξ R) T (r = ξ R)
ξ = ϕ A (ξ) = θ(ξ) = (6.43)
R c AS T S
After some manipulation, the BVP takes the dimensionless form
d 2 dϕ A 2 2 γβ(1 − ϕ A )
ξ − ξ exp ϕ A = 0 (6.44)
dξ dξ 1 + β(1 − ϕ A )

dϕ A
ϕ A (1) = 1 = 0 (6.45)
dξ
ξ=0
There are now only three dimensionless parameters
+
k(T S ) D A (− H)c AS E a
= R β = γ = (6.46)
D A λT S RT S
is the Thiele modulus, and represents the strength of internal mass transfer resistance.
When 1, diffusion is so fast compared to reaction that mass transfer resistance is