Mean transit time and mean residence time for linear diffusion ...

46J. Waniewskiwith the coefficients j ¼ JL=D, a ¼ AL 2 =D, s ¼ SL 2 =D. Equation (25) is now› 2 t›z 2 þ j ›t 2 at ¼ 21 ð49Þ›zwith (›t/›z)(0) ¼ 0 and t(1) ¼ 0. The solution of equation (49) for j, a – 0istðzÞ ¼ 1 u coshðuz=2Þþj sinhðuz=2Þ1 2 expðjð1 2 zÞ=2Þau coshðu=2Þþj sinhðu=2Þpwhere u ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffij 2 þ 4a.Important special cases of solution (50) areð50Þ(1) for j – 0, a ¼ 0(2) for j ¼ 0, a . 0(3) for j ¼ 0, a ¼ 0tðzÞ ¼ 1 j1 2 z 2 expð2jÞðexpðjð1 2 zÞÞ 2 1ÞjtðzÞ ¼ 1 a1 2 coshð pffiffia zÞpffifficoshð a ÞtðzÞ ¼ 1 2 ð1 2 z 2 Þð51Þð52Þð53Þand, fort 2 ðzÞ ¼ 1 52 6 2 z 2 1 2 z 2 6ð54ÞvarðzÞ ¼2 1 ð 1ðt 2 tÞ 2 dmðt; zÞ ¼t 2 2 t 2 ;M 0 0varðzÞ ¼ 1 6 ð1 2 z 4 Þ:ð55ÞFor the Dirichlet boundary conditions at both boundaries for the main equation (49) onehas the boundary conditions for t:t(0) ¼ 0 and t(1) ¼ 0. Then, the solution of equation (49)for j, a – 0ispwhere u ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffij 2 þ 4a.tðzÞ ¼ 1 a1 2 eðu2jÞz=2 þðe u=2 2 e j=2 2jz=2 sinhðuz=2ÞÞesinhðu=2Þð56Þ