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    Steady state diffusion through three layers system

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    Consider a system that consists of a flat layer of material of thicness L that creates radioactive particles surrounded on each side by a very thic layer of material that absorbs the partices that are produced by the radioactive layer.... see attached

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    https://brainmass.com/math/functional-analysis/steady-state-diffusion-through-three-layers-system-577351

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    A condensed version of the solution is provided below. Please see attached for full workings, with equations.

    (a) The equations are:
    (1.1)
    Since the system is invariant under translation in the xy plane, the concentration profile is a function of time (t) and altitude (z) only.
    In a steady state, the concentration profile, by definition, does not change with time:
    (1.2)
    The steady state concentration profile is a function of z alone So the equations for the steady state become standard ordinary differential equations:
    (1.3)

    For equation (1.3.1) we get:
    (1.4)


    Since are constants, integrating twice with respect to z yields the general solution:

    ...

    Solution Summary

    The solution shows how to formulate the steady state equation, solve them and apply physical and mathematical conditions to obtain the concentration profile.

    $2.19

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