Oxidation of boiler steels - RWTH Aachen University

7 th GTT-Workshop, May 2005 Computer Modelling
∂c
∂t
=
∂
∂x
Mathematical Treatment of the Diffusion Process
Diffusion Equation
∂c
∂t
⎛
⎜⎝
=
∂c
D
∂x
D
analytical solution:
only for some special cases;
complex thermodynamic
treatment implementation
“impossible”
UNIVERSITÄT
SIEGEN
⎞
⎟⎠
∂
2
∂x
D=f(T)
c
2
∂c
∂t
=
2
∂ c
D 2
∂x
numerical solutions:
explicit method:
c
− 2c
D∆t
∆x
D∆t
∆x
⎠
j j j
j+
1 j
i+
1 i i−1
j ⎛ ⎞ j
j
ci = ci
+ D∆t
⋅
= ci
1 1 ci
c
2
2 + + ⎜ − + 2 ⎟
2 i−1
∆x
+ c
implicit method: Crank - Nicholson technique:
c
j+
1
i
∆t
− c
j
i
=
2-dim.
∂c
∂t
j j j j+
1 j
D ⎛ ci−1
− 2ci
+ ci+
1 ci−1
− 2ci
⋅⎜
⎜
+
2
2 ⎝ ∆x
∆x
Institut für Werkstofftechnik
Lehrstuhl für Materialkunde und Werkstoffprüfung
=
D
x
∂
⎝
2
∂x
c
2
+
D
+ 1
2
y
∂
D∆t
∆x
+ c
2
∂y
c
2
j+
1
i+
1
⎞
⎟
⎠