Transient temperature profile inside thermoacoustic refrigerators

The linear theory used to calculate the thermal quantities inside the stack in the classical thermoacoustic refrigerators always overestimates those measured. The causes of these discrepancies have to be found in the complex processes of thermal exchanges. The analytical study of the transient response should provide an interpretation of these complex processes. This present paper provides such analytical modelling. This modelling remains within the framework of the classical linear theory. It includes the effects of the thermoacoustic heat flux carried along the stack, the conductive heat flux returning in the solid walls of the stack and through the fluid inside the stack, the transverse heat conduction in the stack and the heat leakages through the duct walls, the heat generated by viscous losses in the stack, the heat generated by vorticity at the ends of the stack, and the heat transfer through both ends of the stack. A modal analytical solution for the temperature profile is proposed, assuming the usual approximations in such thermal problems to avoid intricate calculations and expressions. The theoretical transient response of a thermoacoustic refrigerator is compared with experimental data. A good qualitative agreement is obtained between analytical and experimental results after fitting empirical coefficients.