The Pressure-Volume Relationship and Hysteresis Loss in Stirling Refrigerators

Abstract

Stirling refrigerators consist of several spaces in which a volume of gas undergoes expansion and compression. This is also known as a ‘gas spring’. In such a space, heat transfer occurs due to the cyclic temperature difference between the working gas and the adjacent walls. This causes cyclic heat dissipation, known as hysteresis loss. Hysteresis loss is one of the many losses within Stirling machines that are not completely understood.

This project investigated the underlying mechanisms of hysteresis loss, by examining the relationships between temperature, heat transfer, pressure in single and multiple space Stirling refrigeration systems. This was carried out with a thorough investigation and analysis with mathematical models, a Sage single cylinder model, and single cylinder experimental tests. An experimental validation of a Twinbird 40 W cooler, a beta-type Stirling refrigerator, was also presented. Finally, the Sage model of an alpha Stirling refrigerator and a simple cylinder with regenerator material was used to explore hysteresis loss within the multiple spaces of a Stirling refrigerator.

A simple, closed form equation was developed to show the relationship of net P-V work for given pressure and volume amplitudes with a specified pressure phase shift for sinusoidal motion, which worked for both single cylinder and Stirling refrigerator models. The sinusoidal Schmidt equations were used to show that there will always be a pressure phase shift even in ideal situations for any temperature ratio other than 𝜏 = 1.

The pressure phase shift is shown to be in both sinusoidal and discrete execution of the Stirling cycle. An effective pressure phase shift in the discrete Stirling cycle is presented, and an applied cycle is discussed to show how the isochoric processes impose this effective phase shift. It was found that there will always be a pressure phase shift if there is a net heat or P-V work transfer, in both multiple and single space systems.

The Peclet number is found to be an insufficient quantity to predict hysteresis loss in Stirling refrigerators. Hysteresis loss, or the net heat transfer, is not always from the gas to the wall; in the alpha Stirling model it was found to be a net heat gain from 10 to 20 Hz. It is proposed that hysteresis is not always a loss in multiple space systems which transfer heat or do work.

The design implications from the study are to increase the hydraulic diameter where heat transfer is beneficial. Hysteresis loss was found to vary with the working gas at different frequencies. At 10 Hz, the hysteresis loss in a single space experiment with helium as a working gas was five times that of air. However, it is suggested not to base the working gas selection on hysteresis loss minimisation as the regenerator already minimises the effect of hysteresis loss, and the fact that net heat transfer is still required in the heat exchangers.

The regenerator and how it reduces heat transfer and therefore hysteresis within Stirling refrigerators was also explored. It was found that the presence of the regenerator ‘isothermalises’ the system by increasing the overall heat transfer amplitude and decreasing the net heat transfer, therefore reducing the pressure phase shift.