The study of thermoacoustic effects is a relatively new area, particularly in application to thermoacoustic engines. For thermoacoustic engines to be commercially viable, there are still many aspects to be investigated, not only practical aspects but also at the fundamental level of physics. Particularly lacking is research on heat exchangers in thermoacoustic engines, despite the fact that this is one of the most important components, for which a design methodology does not yet exist. The primary aim of this work was to investigate the design methodology for heat exchangers in thermoacoustic devices to improve their efficiency. In this work, second law analysis was chosen as the design methodology and was applied to a simplified model of heat exchangers in thermoacoustic engines and its validity was examined. However, for the analysis to be useful to design practical devices, further knowledge of the heat transfer mechanism in oscillatory, compressible flow, and on the development of boundary layers under such conditions are required. This is not currently available for thermoacoustic devices. The commercial software PHOENICS was used to investigate this oscillatory heat transfer problem numerically. To test the capability of the software for simulating thermoacoustic phoenomena, two dimensional standing waves and thermoacoustic couples were simulated at various operating conditions and geometries, including conditions very close to those at heat exchangers in thermoacoustic engines. The results were compared with existing analytical solutions and the results of numerical simulations from others and showed that PHOENICS is capable of simulating thermoacoustic effects. However, the accuracy of second order effects, such as heat flux induced by thermoacoustic effects, was limited by the capability of PHOENICS and the results should be interpreted with this in mind. Energy and flow fields from thermoacoustic couple simulations were investigated from plots of energy vectors, energy lines, instantaneous velocity fields, particle traces and energy dissipation.The dependence of such quantities on plate spacing, plate length and Mach numbers are presented. One important result from these test which is relevant to the design of regenerators or heat exchangers in thermoacoustic engines was that a net heat pumping effect appears only near the edges of thermoacoustic couple plates, within about a particle displacement distance from the edges. Also it was observed that the energy dissipation near the plate is proportional to the plate surface area but increases quadratically as the plate spacing is reduced. The results also indicated the presence of larger scale vortical motion outside the plates which disappeared as the plate spacing was reduced. The presence of such vortical motion did not seem to influence the heat transfer to the plates. In order to simulate heat exchangers in thermoacoustic engines without simulating the whole device, boundary conditions representative of those near the ends of the regenerator plate were considered and tested. Although in some test cases, the simulation converged to a solution with minimal energy imbalances, there was a major discontinuity in the energy flux vectors near the boundary. Further investigations (both numerical and experimental) are required to provide further insight into the boundary conditions which need to be specified for future simulations of heat exchangers in thermoacoustic engines.