## 4 Results and Discussion
(2017) showed that the net heat transfer rate between solid and fluid states mainly occurred on the inlet and outlet faces of the heat exchanger with parallel plates. When the fin length is increased, the heat flux on inlet and outlet faces can be positive or negative, leading to a decrease in the net heat flux. As a result, it is also possible that the heat transfer performance of the oscillating flow in the heat exchanger with a length of 15 mm, is much better than the others due to the increased heat transfer rate per unit area. Figures 8-10 present the Nusselt number variations of \(A_{r}\) when Valensi numbers are 150, 200, and 250, respectively. These figures also show that for heat exchangers with the same hydraulic diameter, but different length, the Nusselt number decreases as the heat exchanger length is increased. It can be seen that when \(Re\) and \(Va\) are controlled, heat exchangers with a length of 20 mm (A2 and B2) have about a 30% lower \(Nu\) number than those with a length of 15 mm (A1 and B1), and heat exchangers with a length of 30 mm (A3 and B3) have about a 50% lower \(Nu\) number than those with a length of 15 mm. When considering those with a length of 15 mm (A1 and B1), it is evident that better heat transfer performance mainly occurs when \(A_{r}\) is larger than 1, supporting the conclusion offered by [PERSON] (1988) and [PERSON] (1998). In fact, when \(Re\) and \(Va\) are controlled constants, the relative fluid displacement amplitude \(A_{r}\) is equivalent to the reciprocal of \(1/d_{h}\). Compared to Figures 5-7, the \(Nu\) variations of \(A_{r}\) reflect the same physical pattern. However, the impact of relative fluid displacement amplitude is actually a result of comprehensive effects; an aspect which has been neglected in the previous study. [PERSON] (2018) numerically studied the relationship between the mean convective heat transfer coefficient \(h\), and the relative displacement amplitude  
Figure 8: Nusselt number variation with \(A_{r}\), \(Va=150\). Figure 9: Nusselt number variation with \(A_{r}\), \(Va=200\). \(A_{r}\), under the same pressure ratio based on energy conservation. Results showed that remarkably, \(h\) increased with increasing \(A_{r}\) when \(A_{r}>1\), while \(h\) remained constant when \(A_{r}\leq 1\), which does not reflect the results concluded in this paper. As \(A_{r}\) includes the geometrical parameters, as well as the impacts of \(Re\) and \(Va\), it is difficult to explain the direct relationship between the oscillating-flow heat transfer and each parameter. In comparison, \(l/d_{h}\), instead of \(A_{r}\), can be chosen as an important geometrical parameter, which should be taken into account when considering the empirical correlation for the f  
Figure 11: Heat transfer rate variation with \(A_{r}\), \(Va=150\). Figure 4.0 Nusselt number variation with \(A_{r}\), \(Va=250\). In order that the experimental data can be conveniently applied to the design of finned heat exchangers working under oscillating conditions in regenerative thermal engines, further data fitting was implemented to obtain a relatively comprehensive empirical correlation. The Reynolds number, Valensi number, Mach number, Prandtl number, specific heat ratio, ratio of length and hydraulic diameter, and pressure ratio \(PR\), are considered as the main similarity criterions relating to the oscillating-flow heat transfer in heat exchanging channels [5]. This experiment adopted helium gas as the working fluid, so the specific heat ratio and Prandtl number can be considered as nearly constant. In addition, the Mach number implies the compressibility of fluid, which was kept at a value much lower than one in the experiment.