Limit Solutions for a Lumped Parameter Model of Heat Transfer in a Diluted and Indirectly Heated Co-Current/Counter-Current Moving Bed Heat Exchanger with Heat Loss to Surroundings

Limit solutions and solutions by analogy play an important role in the interrelation of engineering problems. However, research with this approach is extremely rare in heat transfer in particulate systems. Given this scarcity, a series of limit solutions are developed for a lumped capacity (LP) model of a co/counter-current tubular moving bed heat exchanger (MBHE), vertically arranged, indirectly heated, diluted and with heat losses to the surroundings. In the solution methodology, the problem is formulated by analyzing the model parameters and the eigenvalues of the system operator, and by identifying indeterminate forms (INDs). Then, INDs are usually calculated by algebraic rearrangement and/or L'Hopital's rule. Among the results, there are almost a dozen new solutions, the systematization of INDs in the counter-current flow and the verification of agreement between experimental data and the LP model for infinitely diluted bed provided that the simplifying hypotheses are observed. Finally, the interconnection of LP models of MBHE in a decreasing order of complexity and in a historical perspective provides a broad understanding of the evolution of the LP modeling of this important industrial equipment.