Computational Mathematics and Modeling, Vol. 29, No. 3, July, 2018
NUMERICAL DETERMINATION OF TWO SORBENT CHARACTERISTICS FROM
S. R. Tuikina
For a mathematical model with external-diffusion kinetics, we consider an inverse problem of determin-
ing the inverse isotherm and a kinetic coefficient from two dynamic output curves observed at two
points in a single experiment. A gradient-type iterative method utilizing the adjoint problem technique
is proposed for this inverse problem, and numerical results are reported.
Keywords: inverse problems, mathematical model of sorption, numerical methods
Sorption and ion-exchange processes are widely used in various chemical-engineering technologies and in
chemical analysis. Ion-exchange and sorption processes in columns are described using mathematical models
that allow for various types of kinetics and longitudinal diffusion [1–3]. The analysis of these processes requires
the determination of some process characteristics (the sorption isotherm, the kinetic coefficient, the diffusion
coefficient). These are the coefficients in a system of quasilinear partial-differential equations that specify the
sorption-dynamics model; they are usually described by functions that depend on concentration, i.e., on a func-
tion which is one of the components of the solution of the initial–boundary-value problem for a quasilinear par-
tial-differential system. Experimental methods for direct determination of the sorption isotherm and the kinetic
coefficient are quite difficult and time-consuming and in some cases cannot be applied altogether. The most
accessible information about the sorption process is provided by the dynamic curve, which plots the concentra-
tion at fixed points of the sorption column for all time instants. This leads to inverse problems that involve de-
termination of the solution-dependent coefficients of the quasilinear partial-differential system given additional
information about the solutions of the initial–boundary-value problem specified at a fixed point in space for all
Solution methods for inverse problems that determine one of these characteristics are considered in [4–7].
Unique solvability of such inverse problems is investigated, for instance, in [8–10]. Inverse problems and solu-
tion methods for simultaneous determination of two coefficients from dynamic curves obtained in two experi-
ments on the boundary are studied in [10–13].
In the present article, we assume a mathematical model with external-diffusion kinetics and consider the in-
verse problem of determining the inverse isotherm and the kinetic coefficient from two output dynamic curves
observed at two points in a single experiment. A gradient-type iterative method using the adjoint problem meth-
odology is proposed for this inverse problem. The reported numerical results make it possible to assess the ac-
curacy of the proposed method.
The Inverse Problem
Consider a mathematical model of sorption with external-diffusion kinetics in the absence of longitudinal
diffusion when the kinetic coefficient depends on concentration:
Faculty of Computational Mathematics and Cybernetics, Moscow State University, Moscow, Russia; e-mail: email@example.com.
Translated from Prikladnaya Matematika i Informatika, No. 56, 2017, pp. 54–60.
1046–283X/18/2903–0299 © 2018 Springer Science+Business Media, LLC 299