Symbolic mathematics-based simulation of cylinder spaces for regenerative gas cycles

Torben Möller

doi:10.1007/s40095-015-0163-3

Abstract

Regenerative gas cycles, including the Stirling engine, are sealed machines using pistons within cylinders to transfer energy from a heat source to a colder reservoir using a gas as working substance. For the optimal design of these cycles, we need a detailed description of gas dynamic behavior. This contribution deals with the simulation of cylinder spaces without internal combustion (as we find for regenerative gas cycles). For the simulation, we suggested a symbolic mathematics-based strategy to describe the dynamic system behavior based on partial non-linear differential equations for the conserved quantity. The renunciation of numerical approximation gives the advantage that the underlying physical mechanisms are characterized by exact expressions and parameters. Using some assumptions, the dynamic behavior of the gas within the cylinder is already described by ordinary non-linear differential equations. Depending on the selected boundary conditions analytical solutions can be obtained for some cases. Finally, the developed solution is based on it and will be received as a series expansion. Additionally, for the simulation-based optimization of the processes it is possible to calculate directly the periodical-steady state of the system with the help of the symbolic solution. The simulation is suitable for fundamental theoretical investigations, as well as for the implementation in simulation software for different regenerative gas cycles.

Highlights

The process configuration of a regenerative gas cycle always consists of a heat exchanger circuit and a combination of cylinder piston assemblies
The working gas inside a machine of a regenerative gas cycle is pushed through the heat exchangers and the regenerator by cylinder piston assemblies
For the optimal design of these components one requires the information of the gas mass flow emitting and receiving by the cylinder spaces

Summary

Introduction

The process configuration of a regenerative gas cycle always consists of a heat exchanger circuit and a combination of cylinder piston assemblies. For the optimal design of these components different methods for modeling and simulation are used (cf [4]). The differential equations for the conserved quantities: mass, energy, and momentum are usually not directly solvable, i.e., do not have closed form solutions. Solutions must be approximated using numerical methods. For purposes of simplified calculations, one may assume that the process takes place from initiation to completion without a change in a special state quantity of the system. In this case analytic solutions exist, such as the Schmidt model for Stirling engines [11]. One more model that has occasionally been used is the model for the regenerator due to Hausen [2]

Objectives

Conclusion