Explicit solutions for boundary value problems in diffusion of heat and mass

The Fourier's infinite series solutions of transient diffusion in finite solids, with convection boundary conditions, contain eigenvalues, which are defined by implicit transcendental equations. The eigenvalue equations have to be evaluated numerically, making the use of the Fourier's solution difficult. Using the successive substitution method, explicit expressions were derived for the first eigenvalue λ1, in plates, cylinders and spheres, as a function of the Biot number. The derived expressions for λ1(Bi), together with the one-term Fourier series solution, yield simple explicit solutions for temperature or concentration profiles in plates, cylinders and spheres. The explicit solutions are valid for Fo>0.2.