# Study of Standing-Wave Thermoacoustic Electricity Generators for Low-Power Applications
## 4 Results and Discussion
### One Stage Engine
For the working fluid modeled in the present study (air at atmospheric pressure) the above condition is matched at frequencies \(\geq\)200 Hz. So, when the frequency is lowered below 180 Hz, the reduction in the performance of the stack dominates over the increase in the ability of the alternator in absorbing acoustic power and a fall in \(\eta_{ha}\) is observed. As for the acoustic-to-electric conversion efficiency of the engine it can be seen that it exhibits a maximum at slightly lower values of \(f\) (and \(R_{L}\)) (\(f\approx 165\) Hz, \(R_{L}\) = 33.4 \(\Omega\)). Simulations reveal (results not shown) that that the optimal load resistances are related to the working frequency by the same functional dependence shown in Figure 1. This guarantees that in each configuration the LA is working at its highest acoustic-to-electric conversion efficiency that explains the behavior of \(\eta_{ae}\) vs. \(f\). Note that the phase angle \(\phi_{pU}\) between the driving pressure and volume velocity is quite high (82\({}^{\circ}\)). In this case, the regulation operated by the back volume is evidently ineffective to fulfill at the same time the requirement of a high impedance (dictated by the LA position) and of a low \(X_{a}\) (dictated by acoustic power enhancement purposes). On the other hand, the last requirement (low \(X_{a}\)) is not compatible with the dynamics of the gas oscillations (standing-wave oscillations). The operation point (\(f\approx 174\) Hz, \(R_{L}\) = 39.2 \(\Omega\)) where the global \(\eta_{he}\) maximizes (\(\eta_{he}\approx 4.6\%\)) is clearly a compromise between the operative conditions maximizing \(\eta_{ha}\) and the ones maximizing \(\eta_{ae}\) \begin{table}
\begin{tabular}{c c c c c} \hline
**TAEGs** & **Working Frequency (Hz)** & **\(R_{L}\) (\(\Omega\))** \\ \hline One-stage TAEG & & 174 & 39.2 \\ Two-stage TAEG with LA in push-pull mode (A) & & 200 & 81.6 \\ Two-stage TAEG with LA in side branch (B) & & 194.4 & 19 \\ \hline One stage engine & Length (m) & Diameter (m) & Porosity & \(R_{h}\) (m) & Taper angle (deg) & Volume (m\({}^{3}\)) \\ \hline Hot duct & 0.044 & 0.13 & - & \\ stack & 0.0523 & & 0.881 & 0.00024 & \\ HHX & 0.0197 & & 0.592 & 0.0005 & \\ AHX & 0.0287 & & 0.592 & 0.0005 & \\ Ambient duct & 0.795 & 0.13 & & \\ Connecting cone & 0.03 & 0.13-0.147 & & 31.5 & \\ Connecting duct & 0.01 & 0.147 & & 0.0005823 \\ Back volume & & & & \\ \hline Two-stage engine (A) & Length (m) & Diameter (m) & Porosity & \(R_{h}\) (m) & Taper angle (deg) & Volume (m\({}^{3}\)) \\ \hline Hot duct & 0.0862 & 0.13 & & \\ stack & 0.0168 & & 0.881 & 0.00024 & \\ HHX & 0.0117 & & 0.592 & 0.0005 & \\ AHX & 0.01 & & 0.592 & 0.0005 & \\ Ambient duct & 0.04 & 0.13 & & \\ Connecting cone & 0.03 & 0.16-0.147 & & 31.5 & \\ Connecting duct & 0.01 & 0.147 & & & \\ \hline Two-stage engine (B) & Length (m) & Diameter (m) & Porosity & \(R_{h}\) (m) & Taper angle (deg) & Volume (m\({}^{3}\)) \\ \hline Hot duct & 0.0446 & 0.13 & & - & \\ stack & 0.047 & 0.881 & 0.00024 & & \\ HHX & 0.0179 & & 0.592 & 0.0005 & \\ AHX & 0.0251 & & 0.592 & 0.0005 & \\ Ambient duct & 0.018 & 0.13 & & & \\ Connecting cone & 0.189 & 0.13-0.11 & & 7 & \\ Connecting duct & 0.294 & 0.11 & & & 0.00151 \\ Back volume & & & & & \\ \hline \end{tabular}
\end{table} 
Table 2: Geometrical parameters of the TAEG models and operation conditions. The high \(W_{a}\) values found at high and low frequencies simply reflect the drop in performance of the engine in these frequency ranges.