## 1 Introduction  
In 2001, [PERSON] et al. constructed and tested a thermoacoustic cooling system that has no moving parts and can be driven by various types of heat sources such as sunlight and waste heat [1]. As shown schematically in Figure 1a, their system is composed of a looped tube, an engine stack, a cooler stack, and four heat exchangers. The stacks include several narrow flow channels and are sandwiched by two heat exchangers. Experimental results indicated that an increase in the temperature of the hot heat exchanger (see Figure 1) spontaneously produces an acoustic wave due to the thermoacoustic effect [2,3] and the wave travels in the looped tube. The acoustic wave causes a thermoacoustic heat-pumping effect [2] in the cooler stack (see Figure 1), such that the temperature of the cold heat exchanger decreases. Additionally, [PERSON] et al. measured the acoustic pressure and velocity along the looped tube and experimentally demonstrated that both the excitation of the acoustic wave and the thermoacoustic heat pumping are performed through thermodynamic cycles in a manner similar to Stirling and reversed Stirling cycles [4], respectively. These cycles are known to be inherently reversible, and, thus, it is expected that the cooler becomes an efficient system.  
The thermoacoustic cooling system constructed by [PERSON] et al. includes the engine and cooler stacks in the looped tube as previously mentioned. It is widely known that the design parameters of a stack, such as the flow channel radius and the installation position, affect the efficiency of thermoacoustic energy conversions [5, 6, 7, 8]. Furthermore, there is a possibility that the parameters interact with each other in the thermoacoustic cooling system. Thus, it is necessary to simultaneously optimize the parameters of the engine and cooler stacks to improve the performance of the system. However, there is a paucity of extant studies in which the optimization of the previously mentioned parameters is applied.  
In this study, we have numerically investigated the performance of the thermoacoustic cooling system shown in Figure 1 by using the thermoacoustic theory initially proposed by [PERSON] [9; 10] and advanced by [PERSON] [2] and [PERSON] [3; 11]. The efficiency of the engine stack, coefficient of performance (COP) of the cooler stack, and acoustical transmission loss along the looped tube were calculated by changing three parameters, namely the flow channel radius in the engine stack, the flow channel radius in the cooler stack, and the relative position between the stacks. The results indicated that the total efficiency that is defined as the ratio of the cooling power and input thermal power became 0.40 when the three parameters were optimized. This value corresponds to 13% of the thermodynamically ideal value and implies that the efficiency of the engine stack became 75% of [PERSON]'s efficiency, the COP of the cooler stack became 53% of [PERSON]'s COP, and the efficiency of the looped tube as an acoustical power transmission line became 33%.  
The next section presents the model of the thermoacoustic cooling system. The following section describes the definitions of the evaluated values showing the performance of the engine and cooler stacks and the loss of the looped tube. Then, the numerical method to calculate the values is shown. Finally, the calculation results and discussion are presented.  
Figure 1: A thermoacoustic system with engine and cooler stacks [1] in a looped tube.