Extension to nonlinear adsorption isotherms of exact analytical solutions to mass diffusion problems

•A new methodology gives new exact analytical solutions to mass diffusion problems.•We couple arbitrary initial profile, finite volume effect, film boundary resistance.•We present the first exact analytical expansion for piecewise linear isotherms.•Variant includes first exact analytical expansion for short contact times.•Piecewise isotherms successfully approximates Langmuir and Oswin nonlinear isotherms.

Analytical solutions with algebraic expressions are proposed for one-dimensional nonlinear mass diffusion problems. The solutions proposed to solve sorption/desorption problems are rigorously exact for piecewise linear isotherms. They provide an efficient methodology to devise analytical solutions to nonlinear isotherms, and instantly time-varying boundary conditions, with arbitrary accuracy. An exact solution is calculated for each linear piece of the isotherm and is then propagated iteratively to the next linear piece. For end-user convenience and efficiency, three exact analytical expansions are proposed: a new short and intermediate contact time expansion obtained using the Laplace transform (erfc solution), a modified version of Eq. (33) in Sagiv (2002) with improved stability (Sagiv solution), and a new expansion with a decreasing energy norm (energy solution). All expansions are compared in terms of their accuracy and number of terms for typical nonlinear isotherms, mass Biot numbers and volume ratios covering a broad range of applications. When the thermodynamic conditions are changed at the interface, only the “erfc” and the “energy” solutions retain overall accuracy (machine precision) with few terms (<30). New strategies to enable the simultaneous estimation of diffusion coefficients, isotherms and mass Biot numbers are finally derived.