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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 />

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