A solution is given for unsteady state concentration distribution of diffusing substance in an infinitely long circular cylinder around an impermeable rod. In the case in which the circular cylinder has zero initial concentration and the surface r=b is maintained at constant C0 for t>0 while the other surface r=a is impermeable, the equations for the concentration are where D is diffusion coefficient.
The final result is where Jm and Ym are Bessel functions of the first and second kind respectively, of order m (m=0, 1) and βn are the positive roots of
The expression for the amount of diffusing substance entering the circular cylinder in time t is given by
The curves of C/C0 and Mt/M∞ are plotted for different values of a/b. It seems that the diffusion in the circular cylinder is approximated by that in a plane sheet with thickness 2(b-a) when the value of a/b is larger than 0.95.
Dye concentration distributions in nylon 6 were obtained by a film roll on a glass rod. The concentration distributions agreed with the theoretical curves calculated from eq. (5).