Some simple solutions to the diffusion equation at steady state

3 Some simple solutions to the diffusion equation at steady state 

Steady state may be reached in a diffusion problem proceeding for a long time in a finite medium. Steady state means that the concentration at any point does not change with time any more, i.e., 

For steady-state solution, the initial condition does not matter because the steady state does not depend on the initial condition. Only the boundary condition is necessary for solving the steady-state diffusion equation. The three-dimensional diffusion equation at steady state is 

In Cartesian coordinates, the equation takes the following form  

C can be solved from f C)=Ax + B. One application of this solution is for an insulated metal rod with one end in ice water (0°C) and the other end in boiling water (100°C). 

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