Page 288 - Numerical Methods for Chemical Engineering

Finite differences for a convection/diffusion equation 277

caracteristic ine 2
se v
tis rein neiter
inences nr rein
is inenced inenced

rein tat
inences t caracteristic ine 1

se
tis rein neiter
t inences nr
is inenced
t
Figure 6.9 Space-time diagram of the 1-D convection equation showing characteristic lines.

infinitesimally close together,

z q = z p + dz t q = t p + dt (6.103)

we use a truncated Taylor expansion to relate the field values at P and Q,

∂ϕ ∂ϕ
ϕ q − ϕ p = dt + dz (6.104)
∂t ∂z
(P) (P)
We apply similar expansions to the first derivatives,
2 2

∂ϕ ∂ϕ ∂ ϕ ∂ ϕ
− = dt + dz
∂t (Q) ∂t (P) ∂t 2 (P) ∂z∂t (P)
(6.105)
2 2

∂ϕ ∂ϕ ∂ ϕ ∂ ϕ
− = dt + dz
∂z ∂z ∂t∂z ∂z 2
(Q) (P) (P) (P)
We now combine these two expansions of the first derivatives with the condition that the
differential equation be satisfied at P ,
2 2 2
∂ ϕ ∂ ϕ ∂ ϕ
a + b + c + f (P) = 0 (6.106)
∂t 2 ∂t∂z ∂z 2
(P) (P) (P)
to obtain the linear system
 2 
∂ ϕ
 

∂ϕ ∂ϕ
∂t −
 2 
dt dz 2 ∂t 
   (P)   ∂t
  (Q)
∂ ϕ  (P) 
dt dz    ∂ϕ ∂ϕ  (6.107)
 =
  
 ∂t∂z (P)   − 
a b c  2   ∂z (Q) ∂z (P) 
 ∂ ϕ 
− f (P)

∂z 2
(P)