Co-current and counter-current vertical pipe moving bed heat exchangers: Analytical solutions

• Co- and counter-current moving bed heat exchanger energy equations are presented.
• Develop novel analytical solutions via Laplace transforms and expansion theorem.
• Solutions in counter-flow depend strongly on the capacity ratio.
• Consistency in the solution is demonstrated at the limit of dimensionless groups.
• Graphical analysis demonstrates agreement with physical expectations.

Recently, analytical solutions for parallel-plate moving bed heat exchangers have been obtained by means of integral transform methods. This study extends the analysis to vertical pipe geometries due to their important industrial applicability. Steady-state energy equations, for systems operating under co- and counter-current conditions, are formulated and nondimensionalized. Laplace transforms and various forms of the expansion theorem are then used to solve the problems, resulting in temperature functions for the solids and fluid domains. Limiting cases are then analyzed and the solutions are shown to simplify to various expressions in the literature. A graphical analysis is also presented, depicting representative behaviors of the solutions and addressing their physical consistency.