Thermoacoustic Theory of Viscous Fluid, Part 1

Energy Conversion and Energy Flux of Small Cycles

Abstract

Discussed are energy conversion and energy flux accompanied to oscillations of viscous fluid. The energy conversion per unit volume is shown to be ρ_m(∂T/∂p)_S«p⋅dS/dt»+«u⋅∇p» for a small cycle, for which variations of pressure p and entropy S are small. ρ_m is time average of the density, u is the velocity and ∇p is the pressure gradient. « » indicates double operations of spatial average over crosssectional area of flow-channel and time average. «u⋅∇p» is a negative quantity corresponding to dissipation due to viscosity of fluid. The average of energy flux is shown to be enthalpy flux ρ_m«H⋅u» approximately, in the case that the velocity u and the mass flux «ρ⋅u» are small and the wave length is long. ρ and H are variations of the density and the enthalpy respectively. The enthalpy flux for small cycle is shown to be summation of heat flux and work flux: ρ_m«H⋅u»=ρ_mT_m«S⋅u»+«p⋅u» where «p⋅u» is the work flux and ρ_mT_m«S⋅u» is the heat flux. Final discussions are on the energy conversion, the work flux and the heat flux of adiabatic reversible small-cycle, isothermal small-cycle, polytropic small-cycle, isobaric small-cycle and isochoric small-cycle.