Numerical evaluations of transcendental equations for transient experiments

Transcendental equations have to be solved in order to extract surface exchange and diffusion coefficients from transient techniques. In this work, numerical evaluations of the equation βntanβn = A, frequently used for a thin plane sheet geometry, have been studied in detail. Based on the Newton–Raphson method and the characteristics of the tangent function, a fast and stable algorithm was suggested to solve this equation for all values of A, which has so far not been the case in the literature. Moreover, a numerical approach to the transcendental equation used in a spherical geometry was also brought up.

► The transcendental equation βntanβn = A has been evaluated numerically.
► The only algorithm in the literature can not handle small values of A.
► Based on the Newton–Raphson method, a fast numerical algorithm has been developed.
► A numerical approach for a spherical geometry was also briefly brought up.