# Friction Factor Correlation for Regenerator Working
## 6 Conclusions  
Modelling the full state of a regenerator may be very computationally demanding due to the very small size of pore and the tortuosity of the structure. Hence, in this study, an economical way of modelling a regenerator is developed. A two-dimensional model has been developed for the regenerator working in a travelling-wave setting. The structure is modelled as a porous medium. Modelling a porous medium requires a proper closure model to represent the momentum losses and thermal inertia of the flow due to the presence of the structure. In this study, the momentum losses are modelled with pressure drop data gained from the experiment conducted in a well-controlled travelling-wave condition. The pressure drop of the porous tortuous structure of the regenerator has been modelled and validated by the results obtained from experiment carried out in an apparatus where a control could be achieved over the phase angle between velocity and pressure thus enabling a travelling-wave condition to be imposed. The values of computationally determined permeability coefficient, \(K\), that represents the hydrodynamic loss of the regenerator investigated, are shown in Table 3. Friction factor correlation has been developed based on a porous medium model. A simplification offered by [PERSON]'s law has been used for the low speed condition studied. The results from this correlation appear to match the results calculated through the correlation proposed by [PERSON] and [PERSON] [7]. This indicates that steady flow assumptions, used by them when developing their correlation, apply to the travelling-wave conditions investigated here. This study suggests that the difference in friction correlation between oscillatory flow and steady flow, as reported in many previous studies, may be related to the \"phase shifting\" between pressure and velocity which are used for predicting the friction factor. For a travelling-wave system, when  
Figure 9: Pressure drop (dP) and velocity at regenerator (Vr) for: (**a**) #200; (**b**) #180; (**c**) #94; and (**d**) #30.  
the pressure and velocity are in-phase, the pressure drop and velocity relate to each other in a similar way as in a steady flow. Results of this investigation show that the pore size of the mesh screen used in a travelling-wave system should not be too large. Otherwise, difficulties in sustaining the travelling-wave phase angle within the regenerator may arise. Should this happen the friction correlation may need to be revised to account for the effect of the \"phase shifting\" phenomenon. Unfortunately, the results obtained from the experiment are somewhat limited because of the design of the apparatus. Flow within samples made of fine mesh screen was limited to low Reynolds number, while the flow within samples made of coarse mesh screen was at a relatively high Reynolds number (otherwise a sufficient pressure drop could not be recorded). Phase shift was observed between pressure and velocity within the regenerator made of coarse mesh screen (#94 and #30). The phase shifting phenomenon may be related to the inertial effect described by the Forchheimer model. This may be the reason for the discrepancy between numerical and experimental results at Reynolds number higher than 60. The central focus of this study is pressure drop. However, thermal condition of the regenerator is also an important criterion to consider when selecting a regenerator. This can be done for example on the basis of Equation (24) in reference [3], while similar insights are also available in references [7; 24]. A regenerator with a coarse mesh screen may be favourable in that it provides a smaller pressure drop. Nevertheless, the thermal contact may not be as good as for the fine mesh. Further investigation on this subject is strongly suggested to cover the whole range of the thermoacoustic phenomena, including the heat transfer within the regenerator.