Page 283 - Numerical Methods for Chemical Engineering
P. 283
272 6 Boundary value problems
1
e 1 e 1
ϕ ϕ c
e c 1 e 1 2
1 1
1
e c 1 e e c 21 e 11
ϕ ϕ
2
1 1
1 1
e c e 2 e c 1 e 1
ϕ ϕ
−
1 1
Figure 6.7 CDS solution of a 1-D convection/diffusion equation for various values of the local Peclet
number below and above the critical value of 2 (N = 50).
We next define the local Peclet number
v( z) ( z) Pe
Pe loc ≡ = (Pe) = (6.79)
L N + 1
to write the CDS equation for grid point j as
− (Pe loc + 2)ϕ j−1 + 4ϕ j + (Pe loc − 2)ϕ j+1 = 0 (6.80)
With α ≡ Pe loc − 2,β ≡ Pe loc + 2, the CDS linear system is
4 α
ϕ 1 βϕ 0
−β 4 α
ϕ 2 0
.
. . .
−β 4 . . . = . . (6.81)
. .
. . 0
. . α ϕ N−1
−β 4 ϕ N −αϕ 1
While for all Pe loc ≥ 0wehave β> 0, α changes sign at Pe loc = 2. We expect some
qualitative change in the solution when this occurs. The CDS solution is plotted in Figure 6.7
for various values of Pe loc for a grid of 50 points. When Pe loc >2, the numerical solution
exhibits oscillations that are not present in the true solution. Also, as this equation models the
convection and diffusion of a density field, it is physically unrealistic for ϕ to take negative
values, but it does so in the numerical solution when Pe loc > 2. To remove such spurious
oscillations, we can make the grid finer, thus decreasing Pe loc for fixed Pe. However, often
we do not know a priori how small to make the grid to avoid the oscillations, whose effect
on the convergence of numerical algorithms may be severe.