A new isothermal theory for Stirling machine analysis and a volume optimization using the concept of ‘ancillary’ and ‘tidal’ domains

Abstract

Theoretical studies of Stirling cycle machines have always utilized a topological system view that goes back to Schmidt's isothermal analysis, where the process is analysed by reference to the expansion space volume variations. Due to this idiosyncrasy in the formulation, it has been difficult to deduce meaningful design criteria from the results. In this paper an alternative visualization is presented, using the newly introduced concepts of a ‘tidal phase angle’ and overlapping ‘tidal’ and ‘ancillary’ domains. With vectorial parameters and a centralized reference basis, a non-dimensional parameter Rcaronfr;tcr, the ‘tidal compression ratio’, equal to the ratio of the average masses in the tidal and ancillary domains, is derived. This number uniquely characterizes the operation of equivalent machines and is therefore akin to the compression ratio in internal combustion engines. On the basis of this, a second new parametric grouping emerged to enhance the usefulness of the resultant integrated equations for use with dimensional analysis. It was defined as the ‘specific performance’ Rcaronfr;sp and is proportional to the output per unit mass, the gas constant and the operating temperature range. It is applicable to engines, heat pumps and refrigerators. Prior attempts at optimizing the proportions of a Stirling engine have not yielded usable results and consequently nearly all Stirling cycle machines built up to the present time have expansion and compression spaces of equal size. The new analysis shows that this is not the most appropriate configuration and it readily yields an optimization of the component volumes. One single analytical conditional equation for the optimum relative sizes of the constituent spaces was obtained from the new formulation for performance that quantifies the condition for an optimized proportioning of any Stirling cycle machine. It has three distinct usable solutions, one of which is an analytical confirmation of a postulate that has previously been published by the author without proof, equating VE/ VC and also Vh/ Vk to the temperature ratio TE/ TC. A numerical verification of this rule based on the proportions of the United Stirling V-160 engine compares it with 12 equivalent re-proportioned derivative engines, all with equal charge masses and operating at precisely the same conditions. This shows a substantial increase in the ideal performance through the use of the derived criteria. The main conclusion is that this theory may lead to a re-examination of the overall layout of Stirling cycle machines and the emergence of a new class of machines with superior performance.