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