Performance analysis of free piston Stirling engine based on the phasor notation method

The free piston Stirling engine (FPSE) is a couple system of dynamics and thermodynamics. Due to the complicated and interactive relationships between the dynamic parameters and thermodynamic parameters, the performance of the FPSE is always difficult to predict and evaluate. The phasor notation method is proposed based on a thermodynamic-dynamic coupled model of a beta-type FPSE in this paper. The output power and efficiency under the different heating temperature and charging pressure are analysed and compared. In addition, based on the Sage numerical model, the influences of heating temperature and charging pressure on the pistons’ displacement amplitudes, power work and efficiency are revealed. This study can provide the assistance for the performance analysis, prediction and optimization of the FPSE.


Introduction
The free piston Stirling engine (FPSE) is invented by Beale in 1964 [1,2] , which eliminates the motor as well as the crankshaft. For the features of compactness, long-term life and light mass, it has been applied in various areas such as aerospace technologies, military industry, solar power generations and so on [3] . However, such mechanical simplification comes with a price and high costs. Without the connecting rods and crankshafts, the movements of the driving piston and the displacer become less predictable because the pneumatic force they sustain is affected by various factors such as the geometry of the components, charge pressure and spring stiffness. Some unexpected factors could result in substantial deviation of their movement amplitudes and phases, which could lead to a significant decrease in output power and thermal efficiency.
Nevertheless, the temptation of such mechanical simplification is so huge that great efforts has been made to model and predict the movement of the power piston and displacer with various mathematical models. In most basic analysis of FPSE dynamics, the Schmidt isothermal model is adopted to couple the thermodynamics and dynamics of the FPSE [4] . Redlich and Berchowitsz applied linear dynamics model to plot a root locus under various parameters and operating conditions [5] . Benvenuto and de Monte studied a linear methodology to optimize and evaluate the performance of FPSE dynamics for space applications. The effects of temperature variations of the compression and expansion spaces on the spring stiffness were analyzed and the gas spring space were included as linear damping terms [6] . Rogdakis presented the matrix model eigenvalues to analysis a stability based on a linear dynamic model [7] . Ulusoy applied the nonlinear effects on FPSE using isothermal and non-isothermal modeling, and studied the nonlinear damping term, nonlinear pressure loss [8] . In this study, an idealized mathematical model is established and some important parameters are discussed based on this model, which would analyse the coupling dynamic and thermodynamic characteristics of FPSE.

Table1. Explanation of physics notations in figure1
Physics Notation Explanation

A subsection Object and assumptions
The mathematical model is established upon a drawing of a β type free-piston Stirling engine which is shown in Figure 1. This Stirling engine is mainly comprised of two chambers, two pistons and three heat exchangers. The two chambers are expansion space and compression space, the two pistons are displacer and power piston, and the three heat exchangers are the heater, cooler and regenerator. The displacer is connected to the blade springs with a rod extruded into the bounce space, which is separated with power piston by an ideal clearance seal. The spring forces are exerting on the displacer as well as the power piston to adjust their motion. A1 is greater than A2 due to the existence of the rod. Detailed explanations of the physics notations in Fig.1 have been listed in Table1. To establish the model, some assumptions are made as follows. ·The instantaneous pressure of working fluid inside each components are always equal and uniform; ·The temperature of working fluid inside the expansion space, the compression space and regenerator are equal to TH, TL and (TH+TL)/ln(TH/TL), respectively; ·Neither leakage nor friction occurs in the clearance seals; · During the operation, the instantaneous pressure in expansion space and compression space are equal and the pressure inside the bounce space remains constant and equals to pm; · The displacer and power piston execute simple harmonic motion, while the pressure oscillates in a simple harmonic manner, the phase angle of pressure is set to be zero; · The working fluid is ideal gas;

The dynamic analysis of displacer and power piston
The Newton's Second law is applied in analyzing the dynamic characteristics of displacer and power piston, the forces that subjected to the displacer includes the gas force, spring force and damping force, which can be presented as eq(1), where p, pm, and xd represent the instantaneous pressure, the average pressure and the displacement of the displacer, respectively, which can be represented as follows based on harmonic assumptions, so the phasor form of eq(1) can be reduced as eq (4), where the boldface physics notations represents the phasors of p, xd and xp. The phasor diagram is represented in Fig.2.

Fig. 2. Phasor diagram of displacer.
Similarly, the identical derivation process can be applied to the power piston's dynamic analysis. However, the power piston is also subjected to the reversed magnetic force produced by the linear alternator that can be calculated by eq (6), where FB means the reversed magnetic force, B, L and R means the magnetic intensity, the loop length and its resistance, repectively.
According to eq(6), the magnetic force is proportional to the displacer's velocity, which can be merged with damping force, so the dynamic equation of power piston can be given as follow, the cp' means the equivalent damping coefficient, and phasor diagram of power piston is represented in figure 3. Something should be noted is that the magnetic force subjects to the power piston in this study, it can also subject to displacer, which is determined by the structure of FPSG system.

The mass conservation analysis of working fluid
FPSE is a closed thermal system with the high frequency oscillating operating conditions, the total mass of working fluid is conservative. It means the sum of the gas mass in the expansion space, compression space and regenerator when the system is working remains a constant, which is equal to the initial charged gas mass, and is also equal to the total gas mass when the system is terminated. VH, VL represent the instantaneous volume of expansion space and compression space. VH0, VL0 represent the volume of expansion space and compression space at equilibrium state, respectively.
After excluding all the second-order terms and applying phasor notation, eq(9) can be given as follow, and three parameters B1,B2,B3 are defined, Because the mathematical model is under the isothermal assumptions, the heat absorbed by expansion space transforms into the heat rejected by heat sink, damping dissipation work and output power, The output power produced by FPSG during a cycle can be derived based on law of sines and presented as eq(18), therefore, the thermal efficiency of FPSE system is given as follows, due to A2 is less than A1, so the thermal efficiency is less the Carnot efficiency. Eq(19) reveals that because of the necessary damping forces of displacer and power piston, the thermal efficiency of FPSE is actually the upper limit of the FPSE's efficiency. Therefore, FPSE is inherently of lower efficiency. Additionally, the cross-sectional area difference between displacer and power piston is necessary to the production of power.

Discussion
Since the idealized mathematical model is established upon various assumptions, it is necessary to compare this model with a more practical numerical model which is established with a commercial software Sage. Fig.5 to Fig.8

Conclusion
This paper has established an idealized mathematical model based on a series of assumptions. The expression of output power, damping dissipation work and thermal efficiency as well as basis dynamic equations of FPSE have been derived. In summary, following conclusions can be achieved. · The damping forces of displacer and power piston are necessary for the production of output work, but they also consume a part of absorbed heat, which cannot be avoided and make FPSE inherently less efficient; · The the cross-sectional area difference between displacer and power piston is also necessary to the production of power. ·Due to the viscous flow resistance and the incomplete heat transfer in the regenerator, the discrepancy are caused between the results of thermodynamic-dynamic coupled model and Sage model. In general, Sage numerical model can provide rather precise results applicable to practical designs and manufactures. For the mathematical model, although based upon more simplifying assumptions and less accurate, it provide formulae which can reveal the nature of FPSE and help us understand the mechanism behind the numerical results. So, the future work should be devoted to establishing an optimal design method based on the idealized model, which would work as a preliminary optimization before further numerical simulation.