Time-average temperature distribution in a thermoacoustic stack

The three-dimensional time-average temperature distribution in a pore of a thermally isolated thermoacoustic stack is calculated. A boundary-value problem is formulated in the acoustic and short-stack approximations from the equation of conservation of energy using literature results for the time-average energy flux. In the central region of the pore, the solution for the time-average temperatures of the wall, Tw, and of the gas along its center line, Tg, share a common profile, linear in the axial coordinate, z. Near the pore ends, where the energy flux approaches zero, the axial gradient of Tg approaches the critical temperature gradient over a distance of order the acoustic displacement amplitude. The axial gradient of Tw approaches zero over a much smaller distance, provided the wall has small thermal conductivity. The transverse heat-flux density, q, is nonzero only near pore ends. Under certain conditions, q=h1(Tg−Tw), where h1 is proportional to the thermal conductivity of the gas divided by the thermal penetration depth. The constant of proportionality, of order unity, depends on pore width and Prandtl number. Results agree favorably with recently published numerical calculations.