## 1 Introduction
They found the following results: that the refrigeration temperature could be lowered as porosity increased, but that the power input of the precooling stage increased significantly; that lower average pressure yielded a lower refrigeration temperature, as well as a lower power input of the second precooling stage; that the performance of the final stage regenerator had a strong dependence on the operating frequency, especially at a low average pressure; and that the regenerator void volume in a multi-stage Stirling type PTR had a significant impact on the distribution of cooling capacity over different stages of regenerator. [PERSON] and [PERSON] [12] proposed a friction factor correlation applicable to a larger mesh range for a regenerator in a travelling-wave thermoacoustic system, based on experimental and CFD studies. They also studied the effects of temperature field, gravity and device orientation on the oscillatory flow [13]. [PERSON] and [PERSON] [14] carried out an experimental study on the heat flow in different mesh screen regenerators and compared the experimental result with four empirical equations, and concluded that all four empirical equations reproduce the measured heat flow rates to a similar degree. In the previous studies, Eulerian representation has been widely employed to describe the oscillating flow and heat transfer in the regenerators. Recently, [PERSON] and [PERSON] [15; 16] applied the lattice Boltzmann method (LBM) to the study of porous media flow and heat transfer. They verified the LBM model by simulating the flow across a cylinder, and then used the model to investigate the oscillating flow and heat transfer in porous media. The results showed that LBM was an effective method for the optimization of the PTR. In previous studies, the Lagrange description-based model has been adopted. In 1994, [PERSON] [17] employed the Lagrangian representation for the governing equations of energy conservation. They solved the regenerator problem using the \(\varepsilon\)-NTU (number of transfer units) method, and analyzed the effect of NTU and solid-gas heat capacity ratio on the performance of the regenerator. However, the momentum equations were ignored in their study, and details of the oscillating flow in regenerators are still not well understood. [PERSON] et al. [18] used the Lagrange description to explain the working mechanism of PTR and proposed the thermodynamic asymmetry effect. However, they focused on the pulse tube, and the regenerator was assumed to be perfect. [PERSON] [19] concluded that the Lagrangian method was more suitable for analyzing the heat transfer in the regenerative duct of free-piston type Stirling engines than other theories. In 2014, [PERSON] et al. [20] developed a one-dimensional model of a regenerator based on the Lagrange description, and investigated the enthalpy flow, entropy flow on the Euler description and the gas parcels motion and loss on the Lagrange description, respectively. This work is helpful in understanding the regenerative process, but the flow resistance and real gas properties are neglected, and the heat transfer coefficient is also set constant. In this study, Lagrangian representation is used to describe the thermodynamic cycle process of gas parcels inside the regenerator, and track the flow and heat transfer of a single parcel, leading to a better understanding of the operation principles and characteristics of the regenerator. A one-dimensional transient model of the regenerator in the PTR operating at 50 Hz is developed based on Lagrangian representation. The SIMPLER algorithm is used to solve the governing equations. The porous Darcy-Brinkman-Forchheimer model is applied in the momentum equation, and the thermal non-equilibrium model is implemented in the energy equation. The simulation results of mass flow at the cold end agree well with the experimental data, which verifies the reliability of this model.