Friction factor and Nusselt number for thermoacoustic transport phenomena in a tube

The transport phenomena for a viscous compressible oscillating flow (with a zero mean velocity) in a tube subjected to a prescribed cycle-steady axial temperature gradient are analyzed. The governing equations are linearized under the conditions of high oscillating frequencies, small amplitudes, and in a tube with a high length-to-radius ratio. Based on a linearized theory, an analytical expression is obtained for the local friction factor, which depends on the prescribed cycle-steady axial temperature and independent of time. The local friction factor is shown to be a complex number indicating a phase shift between the cross-sectional averaged velocity and the local pressure gradient. Closed-form analytical expressions are also obtained for the temperature distribution and for the Nusselt number of solid/fluid interfacial heat transfer by solving the energy equations of the fluid and solid phases. The Nusselt number is also a complex number indicating a phase shift between the heat flux and the temperature difference between the wall and the oscillating fluid. The magnitude of the Nusselt number is also dependent on the prescribed axial cycle-steady temperature and independent of time, and is related to dimensionless thermal property parameters, dimensionless geometrical parameter, and dimensionless operation conditions parameters. To understand the momentum transport and energy transport characteristics, the radial distributions of axial velocity and temperature of the fluid are presented for different ratios of the inner radius with respect to fluid's viscous penetration depth. Particular attention is given to the transport phenomena in the following two limiting cases: 1) a viscous compressible oscillating flow in a porous medium based on a capillary-tube model, and 2) a viscous compressible oscillating flow in a resonant tube of a thermoacoustic refrigerator or in a pulse tube of a Stirling-type pulse-tube refrigerator.