## 2 Experimental Methods  
In order to simulate the actual working conditions of heat exchangers in regenerative thermal engines, an experimental setup is established to measure the heat transfer characteristics of the finned heat exchangers with oscillating flow conditions. The working fluid is helium. Figure 1 presents the schematic diagram of the experimental setup, and more details can be found in (15). A linear compressor (Manufactured by Lihan Thermoacoustic Technologies Co., Ltd., Shenzhen, China) is used to generate pressure wave in oscillating flow. The compressor is driven by variable-frequency power supplies so as to indirectly control the oscillating frequency of the flow by adjusting the frequency of the compressor. The input work of the compressor is controlled within 2.5 kW and the operating current is controlled below 11 A. A variable-frequency power supply allows the voltage and current limits to be set according to load characteristics required by programmable logic controllers. A frequency range of 0 Hz to 650 Hz, and a resolution of 0.01 Hz, are accessible. The combination of an adjusting valve and a gas reservoir structure is used to control and measure the velocity amplitude of the flow. More specifically, for oscillating flow, the adjusting valve and the gas reservoir can be analogous to resistance and capacity impedance, in series with the required values achieved by changing the valve opening, which controls the velocity amplitude into and out of the reservoir. By measuring the pressure wave in the gas reservoir, combined with the volume of the gas reservoir and the physical properties of the working fluid, the magnitude of the oscillating flow velocity amplitude can be determined. The heat exchanger, connected channels, and water pipes, are all thermally insulated by polystyrene foam in order to minimize the heat leakage to the surroundings. Systematic experimental design is carried out and is based on the Box-Behnken Design (BBD), a standard method of response surface methodology (RSM) (20). RSM is a method used to optimize the  
Figure 1: Schematic diagram of the experimental apparatus (1—Linear compressor; 2—Cooling water loop; 3—Testing section; 4—Adjusting valve; 5—Reservoir; 6—Thermostat). experimental conditions, suitable for solving nonlinear data processing-related problems, including techniques such as experimental design, modeling, testing the suitability of the model, searching for the best combination of conditions, and so on. Through the process of regression fitting and contour drawing, the corresponding response value of each factor's level can be easily identified. Based on the response values, the optimal response values and the corresponding experimental conditions can be obtained. At the same time, RSM can also be used to fit complex unknown function relations in a small area, with a simple first or second order polynomial model. The relatively simple calculation reduces the development cost and improves the product quality forsolving the practical problems in data processing. The software Design-Expert (Version 8.0.6, Stat-Ease Inc., Minneapolis, MN, USA) is applied in order to choose the maximum Reynolds number, the Valensi number, and \(l/d_{h}\) as the first-level influencing factors. After setting the corresponding names, units, and ranges, experimental operating points are suggested as a guidance for obtaining the correct experimental results, and are listed in a table with 18 typical testing points. It is proposed that these are carried out first for a quicker and better identification of each parameter's influence and trend of optimal results. In the experiments designed by RSM, the mean pressure was fixed at 3 MPa and the pressure ratio was fixed at 1.2. The maximum Reynolds number ranged from 400 to 1200 (velocity amplitude ranged from 1.0 m/s to 3.8 m/s, correspondingly), and the Valensi number ranged from 150 to 250 (working frequency ranged from 40 Hz to 90 Hz, correspondingly).