## 3 Results and Discussion
### Simulation Results
Therefore, a net refrigeration power is given by:  
\[\dot{Q}_{\mathrm{net}}=\left\langle P\dot{V}\right\rangle_{\mathrm{c}}-\left \langle\dot{H}\right\rangle_{p}-\dot{Q}_{\mathrm{reg}}-\dot{Q}_{\mathrm{cond}} -\dot{Q}_{\mathrm{pt}}-\dot{Q}_{\mathrm{rad}} \tag{11}\]  
where \(\left\langle P\dot{V}\right\rangle_{\mathrm{c}}\) is the acoustic power at the cold end; \(\left\langle\dot{H}\right\rangle_{p}\) is the enthalpy flow associated with the enthalpy pressure dependence (real gas effect); \(\dot{Q}_{\mathrm{reg}}\) is the thermal loss associated with enthalpy flow caused by imperfect heat transfer and limited heat capacity in the regenerator (regenerator ineffectiveness); \(\dot{Q}_{\mathrm{cond}}\) is the conduction heat leak; \(\dot{Q}_{\mathrm{pt}}\) is the loss associated with an imperfect pulse tube or any irreversible expansion process at the cold end; and \(\dot{Q}_{\mathrm{rad}}\) is the radiation heat leak to the cold end, which is ignored in this work. Considering the lack of accurate theoretical calculation for pulse tube losses, an estimated value of pulse tube losses is adopted, as the pulse tube expansion efficiency is assumed to be 80%. The loss characteristics under our operating conditions are calculated and the proportions of various losses in \(\left\langle P\dot{V}\right\rangle_{\mathrm{c}}\) are compared. Figure 10 gives the distribution of losses in a PTR, which indicates that regenerator ineffectiveness is one of the main reasons contributing towards refrigeration losses. By using the above correlation, the ineffectiveness of regenerators with different diameters and lengths are estimated to analyze their effects on the regenerator performance. With respect to the effect of diameter on the regenerator performance, five regenerators are studied, with diameters of 12, 14, 16, 17.5 and 20 mm, respectively, and the results are shown in Figure 11. With the increase of diameter, the ineffectiveness of the regenerator shows a tendency towards increase at a smaller diameter, reaching the maximum near the diameter of 14 mm, and then decreases quickly before being stabilized at the diameter of 20 mm. The occurrence of the ineffectiveness peak can be explained as follows: the heat exchange between fluid and solid matrix is associated with heat transfer coefficient and heat transfer area. On the basis of this principle, under the condition of constant mass flow, a large fluid velocity accompanied by a small diameter, leads to a large heat transfer coefficient and thus minimal ineffectiveness; a large heat transfer area accompanied by a large diamet  
Figure 10: Contribution to the system losses from real gas effect, regenerator ineffectiveness, conduction heat leak and irreversible expansion. ineffectiveness; when the diameter of regenerator is intermediate, the combination of fluid velocity and heat transfer area would lead to the worst heat exchange, and as a result, the ineffectiveness reaches the maximum value. In order to obtain the effect of the length on the regenerator performance, simulations are performed on five regenerators with different lengths: 20, 40, 65, 80 and 100 mm, and the results are shown in Figure 12. It can be observed that the ineffectiveness decreases monotonically with increasing length of the regenerator. The variations become smaller with further increase of length beyond 65 mm, and finally tend to stabilize at the length of 100 mm. It can be explained as follows: the larger heat transfer area of the longer-length regenerator enhances the heat transfer, and thus, reduces the ineffectiveness. When the regenerator length is large enough, the heat transfer between fluid and solid matrix has reached the limit, and any further increase of length contributes little to the heat transfer enhancement, as well as the ineffectiveness reduction.