This paper studies systems of partial differential equations modelling pulse-tube refrigerators. Through a regular asymptotic expansion, the pressure is found to be a known function of time. Therefore, models for heat and mass transfer are derived neglecting the equation for conservation of momentum; the cooling mechanism being highlighted in terms of this model. New approximate formulas are derived for the velocity in the tube and the non-linear thermal wave speed in the porous medium (known as the regenerator) on the short (piston oscillation) time-scale. The temperature dependence of the thermal wave speed is proposed as one measure of the efficiency of the transport of heat away from the cold heat exchanger. Progress is limited on the long time-scale, however, the importance of the difference between the gas temperature averaged over an oscillation and heat exchanger temperature is identified. (C) 2001 Elsevier Science Ltd. All rights reserved.