Page 282 - Numerical Methods for Chemical Engineering

Finite differences for a convection/diffusion equation 271

1
e 1
e 1
e 1
e 1

ϕ

increasin e

2
1

2 1

Figure 6.6 Analytical solution of a 1-D convection/diffusion equation at various Peclet numbers.

is dominant, and the flow (which is from left to right) “pushes” the incoming fluid at z = 0
downwind to the right and causes a sharp increase in ϕ near the exit outflow at z = 1.

Central difference scheme (CDS)

To solve this problem numerically, we apply the finite difference method for a grid of points,
0 < z 1 < z 2 <···< z N < L, and require the differential equation to be satisfied locally at
each point,
2

dϕ d ϕ
− v + = 0 (6.74)
dz dz 2
z j z j
For simplicity, we place the grid points uniformly with z = L/(N + 1), z j = j( z). For
the second derivative, we have the approximation
2
d ϕ ϕ j−1 − 2ϕ j + ϕ j+1
≈ (6.75)
dz 2 ( z) 2
z j
For the first derivative, we consider two alternatives. In the CDS, we choose the more
accurate approximation
(CDS)
dϕ ϕ j+1 − ϕ j−1 2
= + O[( z) ] (6.76)
dz 2( z)
z j
which yields the linear equation for grid point j,

ϕ j+1 − ϕ j−1 ϕ j−1 − 2ϕ j + ϕ j+1
− v + = 0 (6.77)
2( z) ( z) 2
2
We multiply by −2( z) / to obtain
v( z)
[ϕ j+1 − ϕ j−1 ] − 2ϕ j−1 + 4ϕ j − 2ϕ j+1 = 0 (6.78)