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© BrainMass Inc. brainmass.com March 5, 2021, 1:17 am ad1c9bdddf
A condensed version of the solution is provided below. Please see attached for full workings, with equations.
(a) The equations are:
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:
The steady state concentration profile is a function of z alone So the equations for the steady state become standard ordinary differential equations:
For equation (1.3.1) we get:
Since are constants, integrating twice with respect to z yields the general solution:
The solution shows how to formulate the steady state equation, solve them and apply physical and mathematical conditions to obtain the concentration profile.