## 3 Evaluated Performance
### Efficiency and Coefficient of Performance  
In Figure 1b, the thermoacoustic cooler is schematically re-illustrated from a thermodynamical point of view. The subscripts \(e\), \(c\)\(A\), \(H\), and \(C\) denote engine stack, cooler stack, ambient end, hot end, and cooler end, respectively. For example, \(\dot{W}_{e,A}\) denotes acoustic power at the ambient end of the engine stack. [PERSON] et al. demonstrated [1] that, when the temperature \(T_{H}\) exceeds a critical value, the gas inside the looped tube including inside the stacks spontaneously oscillates. As a result, the spontaneously generated acoustic wave travels along the looped tube and transports energy. The acoustic power \(\dot{W}_{e,A}\) of the generated acoustic wave is input from the ambient end of the engine stack and is amplified in the engine stack. The amplified acoustic power \(\dot{W}_{e,H}\) is emitted from the hot end of the engine stack. Therefore, the gain in the acoustic power of the engine stack, \(\Delta\dot{W}_{e}\), is expressed as Equation (3) as follows:  
\[\Delta\dot{W}_{e}=\dot{W}_{e,H}-\dot{W}_{e,A}. \tag{3}\]  
In order to amplify the acoustic power, it is necessary for the hot heat exchanger to supply acoustical thermal power \(\dot{Q}_{H}\). Hence, the efficiency of the engine stack is expressed as Equation (4) as follows:  
\[\eta_{e}=\frac{\Delta\dot{W}_{e}}{\dot{Q}_{H}}. \tag{4}\]  
A part of acoustic power \(\dot{W}_{e,H}\) is dissipated along the tube between the hot heat exchanger attached to the engine stack and the ambient heat exchanger attached to the cooler stack. The remainder of \(\dot{W}_{e,H}\), denoted as \(\dot{W}_{e,A}\), enters the cooler stack from its ambient side. This is used to pump heat from the cold to ambient heat exchangers along the cooler stack. Subsequently, \(\dot{W}_{e,C}\) is output from the cold heat exchanger. The acoustic power is also dissipated in the other part of the looped tube between the cold heat exchanger of the cooler stack and the ambient heat exchanger of the engine stack. The thermal power pumped from the cold heat exchanger is defined as \(\dot{Q}_{C}\). Because the acoustic power used in the cooler stack is \(\Delta\dot{W}_{c}=\dot{W}_{c,A}-\dot{W}_{c,C}\), the coefficient of performance of the cooler stack is expressed as  
\[\text{COP}_{c}=\frac{\dot{Q}_{C}}{\Delta\dot{W}_{c}}. \tag{5}\]  
Acoustic power \(\dot{W}_{e,C}\) is delivered to the ambient end of the engine stack and is re-amplified. The efficiency of the looped tube as a transmission line of acoustic power is defined as follows:  
\[\eta_{tube}=\frac{\dot{W}_{e,A}-\dot{W}_{e,C}}{\Delta\dot{W}_{e}}. \tag{6}\]  
When \(\eta_{tube}=1\), dissipation does not occur along the tube with the exception of the engine and cooler stacks, and all of the generated acoustic power in the engine stack is utilized in the cooler stack.  
The total coefficient of the performance of the total cooler system is expressed as follows:  
\[\text{COP}_{total}=\frac{\dot{Q}_{C}}{\dot{Q}_{H}}. \tag{7}\]  
It is also expressed by using \(\eta_{e}\), \(\text{COP}_{c}\), and \(\eta_{tube}\) as follows:  
\[\text{COP}_{total}=\eta_{e}\cdot\text{COP}_{c}\cdot\eta_{tube}. \tag{8}\]