# Friction Factor Correlation for Regenerator Working
## 1 Introduction  
The underlying mechanism of the thermoacoustic energy conversion processes is a heat transfer interaction taking place between the gas parcels undergoing oscillatory motion and a solid material along which the gas oscillations occur. In thermoacoustic prime movers (engines), a spontaneous acoustic wave is excited which causes the gas parcel oscillations coupled with their compression and expansion, thus providing a means of transporting heat from the hotter to the cooler place of the solid. In thermoacoustic coolers, the acoustically induced displacement/compression/expansion of gas parcels leads to heat pumping effects in the solid. The solid material used in such applications often takes the form of a porous structure--for example, stacked layers of woven mesh screens--which by analogy to classic Stirling engines is referred to as \"regenerator\". Figure 1a explains the interaction of gas parcels with a solid material (in this example, a plate with an imposed temperature gradient). The gas experiences thermal expansion during the displacement towards the higher temperature and thermal contraction during the displacement towards the lower temperature. In this way, the correct time phasing is obtained to meet the [PERSON]'s criterion [1] to produce power. Figure 1b illustrates the thermodynamic cycle in comparison to the classic idealStirling cycle, while Figure 1c shows one of the known configurations of thermoacoustic engines referred to as Thermo-Acoustic Stirling Heat Engine, TASHE [2]. Figure 1: (**a**) Schematic of a gas parcel oscillating in the vicinity of a solid plate with an imposed temperature gradient, undergoing stages of compression and expansion; (**b**) pressure-volume (P-V) diagram illustrating the nature of the thermodynamic cycle; and (**c**) schematic of a possible configuration of a thermoacoustic travelling-wave engine. Strictly speaking, the thermoacoustic core (a regenerator placed between the hot and cold heat exchangers) works simply as a power amplifier. To complete this Stirling-like thermodynamic process, acoustic power has to be fed to the ambient end of the regenerator with a near travelling-wave phasing. This leads to practical designs having feedback resonator types (closed loops) with various acoustic network elements such as inertance and compliance to control the correct/preferred phasing between pressure and velocity oscillation; further discussions are available in reference [3]. Thermoacoustic systems have gained considerable attention due to their environmental friendliness arising from the use of inert/noble gases as the working media and the lack of moving parts when executing the thermodynamic cycle which makes them of simple design and potentially of high reliability and low cost. As part of the requirement to achieve high performance, the regenerator is used in travelling-wave thermoacoustic device with an aim to provide sufficient thermal contact with the working medium but also cause as little viscous loss as possible. Thermal contact is improved due to the large surface area of the porous and tortuous structure of the regenerator. However, care is needed to avoid high viscous losses. From the practical point of view, and to illustrate these problems, the regenerators are typically constructed from metal wire mesh screens by compacting a large number of wire mesh disks (tens or hundreds) along the direction of wave propagation. The conflict between efficient thermal contact and unwanted pressure losses of the porous and tortuous structure of the regenerator requires careful investigation. In the context of thermoacoustics, the matter is difficult to address not only because the oscillatory nature of the flow but also because the correctly defined phasing between pressure and velocity needs to be considered. However, most of the past experiments have not considered specific phase angles between pressure and velocity oscillations. Therefore, their effect on the regenerator hydrodynamic condition is still unknown.