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.