## 2 Materials and Methods
### Cost-Effective Data Acquisition System
#### 2.3.1 Pressure Sensors  
As previously mentioned, the engine is designed to operate at a 23 Hz frequency, which is a critical factor in selecting the appropriate measurement instruments. Once the system achieves stability and oscillates consistently around the resonance frequency, data acquisition and, consequently, recording of the engine's operational cycles begin. Before specifying the placement of each sensor, it is essential to clarify the concept of acoustic power, crucial for understanding the monitoring points within the system. Figure 7 illustrates the acoustic power generated during the Stirling cycle at the engine's core (\(\dot{E}_{core}\)), which is distributed in two main pathways: the transit through the feedback branch and the instantaneous work exerted on the power extraction branch. The feedback branch absorbs power (\(\dot{E}_{feedback}\)), which is employed for feedback purposes, enabling the traveling wave to return to its initial starting point at the beginning of the duty cycle. The power diverted to the power extraction branch has two contributions (\(\dot{E}_{junction}+\dot{E}_{RC}\)). The power that can be measured at the RC load is considered the effective power (\(\dot{E}_{RC}\)), which can be extracted and used. This paper specifically analyzes the effective power. The amount of acoustic power dissipated in the RC load varies depending on the position of the valve regulating the flow resistance to the acoustic cavity, while the remainder is used to drive the resonating piston (\(\dot{E}_{junction}\)). This action must overcome friction within the cylinder to achieve displacement. To achieve maximum power dissipation in the load, the system operates with the valve open just before the point at which the piston ceases to move. It is important to point out that the RC load was installed in the TA-SLICE system specifically for the experimental evaluation of the power output. In normal conditions, i.e., if the TA-SLICE were functioning as an energy converter (from heat to mechanical power), the RC load would be fully closed, or simply taken apart. Figure 8 illustrates the three low-cost Farnell (P1.1, P1.2, and P2) piezoresistive absolute pressure sensors (MPX5100 AP) used to measure the two pressure oscillations in the power extraction branch of the mechanical resonator. For double validation of the results, sensors P1.1 and P1.2 are positioned near the inlet of the needle valve, while sensor P2 is located in the tank. The RC load pressure is monitored with a single low-cost sensor, as double validation in this region would be redundant. The data from the other sensors provide sufficient conditions to extrapolate behavior to this region. At this stage, it is relevant to remember that the engine oscillates around 23 Hz. Consequently, the sensor must capture values at a minimum sampling frequency of \(F_{s}\geq 46\) Hz to prevent aliasing, thereby ensuring that the acquisition frequency is at least double the original sampled frequency. In this work, the sampling frequency was set at \(F_{s}=600\) Hz, significantly exceeding the Nyquist-Shannon sampling requirement, thereby adequately fulfilling this criterion. Figure 8: Location of the absolute pressure sensors P1 and P2 used to measure the difference in pressure oscillations in the resonator and in the RC load, respectively. The distinction between P1.1 and P1.2 is made since they are two identical sensors used to monitor very close points in the vicinity of the resonator and each of them is linked to a high- and low-cost acquisition system, respectively. Figure 7: Power flux generated in the core branch of the TA-SLICE \(\dot{E}_{core}\) and its distribution into two main pathways: the part absorbed by the feedback branch \(\dot{E}_{feedback}\) and the instantaneous work delivered into the power extraction branch \(\dot{E}_{junction}+\dot{E}_{RC}\). This understanding guides the strategic positioning of sensors to effectively capture the system’s operational dynamics.