A Fast and Accurate Semi-Analytical Solution for Lumped Models of Co-Current Moving Bed Reactors

Moving bed reactors (MBRs) are widely used in various industrial processes, making the development of mathematical models crucial for their design, optimization, and control. This study presents a semi-analytical solution (SAS) for a lumped parameter kinetic and heat transfer model of a tubular MBR, where a first-order chemical reaction occurs uniformly within the particles. SAS is developed using the concepts of the finite analytic method: decomposition of the problem domain into small intervals, keeping the terms as linear and evaluated under the conditions at the beginning of each interval, and obtaining local analytical solutions in these intervals. SAS is a fast, consistent, and unconditionally stable numerical scheme that can handle very stiff systems (SR=10^305). A comparison of the SAS results with those of traditional ODE solvers – explicit Euler, Heun, Ralston, and Runge-Kutta of 3rd, 4th, and 5th order – shows excellent agreement. Moreover, for a case study with Biot number less than 0.11 and fourth Damköhler number less than 1.5, the comparison of SAS with a numerical solution (using the finite difference method) of a distributed parameter model, shows a maximum relative error of 0.17%, 0.04% and 6.3% for particle temperature, fluid temperature and conversion, respectively. SAS is (partially) validated by comparison with experimental data from thermogravimetric analysis of kaolinite calcination. In addition, a specific methodology for error analysis is presented, which allows rounding and linearization errors to be estimated.