A one-dimensional mathematical model is derived for a three-stage pulse-tube refrigerator (PTR) that is based on the conservation laws and the ideal gas law. The three-stage PTR is regarded as three separate single-stage PTRs that are coupled via proper junction conditions. At the junctions there are six fluid flow possibilities each defining its own boundary conditions for the adjacent domains. Each single stage cools down the gas in the regenerator to a lower temperature such that the system reaches its lowest temperature at the cold end of the third stage. The velocity and pressure amplitudes are decreasing towards the higher stages and there is an essential phase difference between them at different positions. The system of coupled PTRs is solved simultaneously first for the temperatures and then for the velocities and the regenerator pressures. The final result is a robust and accurate simulation tool for the analysis of multi-stage PTR performance.

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