Substituting U = r R into the spatial part of the equation transforms the spherical Laplacian operator into a simple second-order differential equation:
That is identical with the equation for one-dimensional diffusion through a slab, provided that the initial and boundary conditions reflected in U are
If a solution for U can be found, then the corresponding solution for the sphere is:
The first example is that for a homogeneous sphere and shell. The shell has no radon at time t = 0. The purpose of this calculation is to show how fast radon would escape from an undisturbed sphere and through a shell by diffusion only. The dimensions and initial and boundary conditions are as follows:
A solution that satisfies those conditions is:
Figure B.1 shows a graph of C(r)/Co as a function of radius from the center of the sphere with a diffusion coefficient for radon in water of D = 1 × 10-5 cm2s-1 (Tanner 1964). The radon concentration decreases near the boundary of the sphere but is almost unchanged at a radius less than 3 cm. Even after 1 h, there is considerable radon remaining in the sphere. This illustrates that diffusion alone is not sufficient to transfer all the available radon through the stomach wall in the amount of time corresponding to normal residency in the lumen.
The model was then revised to take into account removal of radon from the lumen by means other than diffusion. The contents of the stomach are considered