Gifford-McMahon cycle — a theoretical analysis

doi:10.1016/0011-2275(86)90217-1

Cryogenics

Volume 26, Issue 3, March 1986, Pages 177-188

https://doi.org/10.1016/0011-2275(86)90217-1 Get rights and content

Abstract

A theoretical analysis of the Gifford-McMahon cycle is presented. Expressions for the ideal refrigeration produced and the figure of merit are developed. Various losses occurring in a real machine are considered and equations to account for these losses are derived. Results are presented in graphical form.

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On the cryocooling by ideal gases: A review (by the unified model of cryocoolers)
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Cryocooling by ideal gases is the counter version of cryocooling based on real gas properties. In view of the unified model, coolers of ideal gas refrigerants (Stirling, Solvay, reverse Brayton, Gifford-McMahon, Pulse Tube, Simon) become completely analogous; their similarity is expressed by a common formulation of performance and that is the ground of the present review. Ideal gas cools down only by work extraction, whether externally (by loading) or internally. The extracted work, may or may not be recovered. The so obtained elementary temperature reduction, is proportional with the instantaneous temperature. The proportionality factor is the sole parameter that distinguishes between all these coolers. The unified model of cryocoolers provides closed form expression for the entire group. So are interrelated the constructional parameters: compression ratio, size of recuperator/regenerator, rate of compression and the species of gas, with the operational parameters: the lowest reachable temperature, cooling capacity, and coefficients of performance.
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The development of a two-stage Pulse Tube Cryocooler (PTC) which produces a no-load temperature of ∼3 K and delivers a refrigeration power of ∼250 mW at 5 K is reported in this work. The system uses stainless steel meshes along with lead (Pb) granules and combinations of Pb, Er₃Ni and HoCu₂ in layered structures as the first and second stage regenerator materials respectively. With Helium as a working fluid, the pressure oscillations are generated using a 6 kW water-cooled Helium compressor along with an indigenous rotary valve. Different configurations of pulse tube systems have been experimentally studied, by both varying the dimensions of pulse tubes and regenerators as well as the second stage regenerator material composition. The pulse tube Cryocooler has been numerically analyzed by using both the isothermal model and the model based on solving the energy equations. The predicted refrigeration powers as well as the temperature profiles have been compared with the experimental results for specific pulse tube configurations.
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The dynamic characteristics of a pneumatically driven Gifford-McMahon cryorefrigerator are modelled when the cryorefrigerator is operating in the steady state regime. The mathematical model, which includes dynamic, fluid dynamic and thermodynamic effects, consists of a set of six simultaneous differential equations. The model variables were determined to be the gas masses in the various chambers of the cryocooler's two stages, and the displacer's position and velocity. The non-linear time-varying system of differential equations is solved by numerical integration routines that use the Adams - Moulton method with an adaptive integration step size and a user specified tolerance. A direct comparison of the algorithm predicted results and actual experimental data indicates that the algorithm models the trends of the physical phenomena in an accurate fashion.

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PaperGifford-McMahon cycle — a theoretical analysis

Abstract

Cryogenics

Cryogenics

Progress in refrigeration science and technology

Progr Cryog

Adv Cryog Eng

Adv Cryog Eng

Paper
Gifford-McMahon cycle — a theoretical analysis