Realistic Second-Order Analysis of Two-Stage Stirling Cryocooler with Real Gas Modeling and Enhanced Regenerator Method

Abstract

Stirling cryocoolers (SCs) are widely used in space infrared detection and superconducting electronics due to their compactness, high reliability, and capability to reach cryogenic temperatures. Although the single-stage SCs are effective for moderate temperature ranges, achieving lower temperatures with sufficient cooling power requires a two-stage configuration. The second-order analysis of SC has traditionally relied on idealized assumptions for evaluating cooling power and coefficient of performance (COP). While useful for preliminary design, these methods often introduce significant inaccuracies, particularly in modeling regenerator performance. For single-stage SCs, simplified temperature variations of the regenerative heat exchanger by employing logarithmic mean temperatures have led to systematic over-predictions of cycle efficiency. When extended to a two-stage configuration, such approximations further amplify errors because of inter-stage thermal coupling and the compounded effect of regenerator losses. To overcome these limitations, the present study develops a corrected second-order model for two-stage Stirling cryocoolers that explicitly incorporates the one-dimensional temperature variation of both regenerators. The governing heat balance equation is solved while accounting for variations in mass flow rate and thermophysical properties of the working fluid across each stage. A key enhancement is the integration of the CoolProp thermophysical property database within Python, which provides accurate, real-gas evaluations of hydrogen properties over the wide cryogenic temperature range encountered between compressor inlet, inter-stage, and final cold-end conditions. This eliminates the ideal-gas simplifications and ensures realistic variations in the thermophysical properties, such as specific heat, thermal conductivity, and viscosity. The corrected approach predicts lower cooling effects and COP values than traditional analyses, with reductions of approximately 3–4% and 0.4–0.6%, respectively. Further, the model reveals that the cooling power at the second stage is highly sensitive to the choice of inter-stage expansion temperature and to the thermal effectiveness of the first-stage regenerator. Moreover, Parametric studies also highlight that the optimal operating phase angle lies in the range of 85°–95°, and a balanced inter-stage temperature yields the most efficient distribution of cooling capacity between the two stages. Overall, the inclusion of real-fluid property data via CoolProp, combined with the corrected regenerator modeling, significantly enhances the predictive capability of second-order analysis.