Solution by the self-adjoint operator method for a model of a co/counter-current moving bed heat exchanger with heat losses

Abstract

A solution by the self-adjoint operator method (SAO) (still not widely used in engineering) is presented for a lumped parameter model (LM) of a co/counter-current, indirect heated, diluted, tubular, vertical moving bed heat exchanger (MBHE), with heat loss to surroundings. The self-adjoint formulation of the problem and several mathematical techniques enabled the development of an ingenious methodology capable of solving the difficult problem of establishing the character of the eigenvalues of the MBHE operator - only partially known to date - for co/counter flow.For co-current flow, it is demonstrated that the self-adjoint solution corresponds to a solution from the literature - obtained by the Laplace transform - but with the character of the roots not yet established. Therefore, the work aims to, through analysis, study the character of the eigenvalues of the MBHE operator to generalize results from the literature, allowing the use of the solution in a comprehensive parametric range, as well as for different flow arrangements, discussed within the scope of case studies. From the analysis of the eigenvalue equation, five thermal behaviors of the MBHE could be identified. The model was also solved numerically, and the results show excellent agreement in both calculation methods. The obtained analytical solution reveals a possible inconsistency in a literature solution of the analogous heat transfer problem in a triple tube heat exchanger (the) with thermal losses at the outer surface. The study generalizes previous results and allows its extension, by analogy, to the solution for the TTHE with thermal losses at the outer surface.