Page 285 - Numerical Methods for Chemical Engineering

274 6 Boundary value problems

1 1
e c 1 e 1
ϕ e c 1 e 1 ϕ
1 1

e c 1 e e c 21 e 11
ϕ ϕ
2 2
1 1

e c e 2 e c 1 e 1
ϕ ϕ
2 2
1 1

Figure 6.8 UDS solution of 1-D convection/diffusion equation at various Peclet numbers (N = 50).

2 2
Recognizing the factor within the square brackets as an approximation of −d ϕ/d z| z j ,we
have the relation
(UDS) (CDS) 2
dϕ dϕ z d ϕ
= − (6.87)
dz dz 2 dz 2
z j z j z j
Thus, the UDS discretization of the equation
(UDS) 2
dϕ d ϕ
v − = 0 (6.88)
dz dz 2
z j z j
is equivalent to a CDS discretization with “extra” diffusion,
(
(CDS) 2 2
dϕ z d ϕ d ϕ
v − − = 0
dz 2 dz 2 dz 2
z j z j z j
(6.89)
(CDS) 2
dϕ v( z) d ϕ
v − + = 0
dz 2 dz 2
z j z j
Upwind differencing is equivalent to adding additional numerical diffusion,or artificial
diffusion, to obtain an effective diffusion constant
v( z)
eff = + (6.90)
2
such that the effective local Peclet number is always less than 2:
v( z) v( z) 2
Pe loc,eff = = = (6.91)
eff + v( z)/2 1 + 2/Pe loc
We could achieve the same effect if we used the CDS equations and increased the effec-
tive diffusion constant by a sufficient amount to avoid oscillations. In multiple spatial