A simplified dynamic model for the prediction of displacer's free stroke in Hofbauer cycle machine

Highlights

• A simplified dynamic model is developed for prediction of displacer's free stroke.

• Experiments on Hofbauer cycle machine is carried out.

• A simplified method of determining the masses in chambers with leakage is proposed.

• Relative error of predicted pressure is in the range of -0.26% ∼ 0.85%

• Relative error of predicted free stroke is in the range of -2.45% ∼ +3.02%.

Abstract

Hofbauer cycle machine is one promising technology for heating. It has many advantages compared to conventional Vuilleumier machines due to its non-continuous motions. However, its non-continuous motions also bring challenges for control. In this work, a simplified dynamic model is proposed to predict the free stroke for control. Based on assumption of a mass-spring system for the displacers, a simplified solution is deduced. The simplified solution is based on constant position of pressure balance points and a linear change of gas mass in the working chamber. Experiments on a prototype machine were carried out to validate the model. Motions of displacers are analyzed and causes for abnormal motions are revealed. Then, the predicted free stroke is obtained based on the simplified model and compared with the measured results. The relative error is in the range of -2.45% ∼ +3.02%. Thus, the simplified model could provide a reference for quick control in Hofbauer cycle machines.

Introduction

Global warming and climate change are critical issues facing human beings. European Union and China have made schedules for carbon neutrality (Pietzcker et al., 2021; Li et al., 2021). Thus, greenhouse gas emission will face more and more limitations in future. As conventional heating systems provides heat by direct combustion of fuel in boiler, it contributes much of greenhouse gas emission. In China, CO₂ emission from residential heating is 715.4 million tons (Xia et al., 2014). Moreover, combustion of fuel such as coal, which is the major fuel for residential heating in China, will lead to air pollutions. Therefore, it is essential to reform the form of heating and increase the use of energy-saving and eco-friendly technologies, such as heat pumps (Kühn, 2013; Johra et al., 2019).

Thermally driven heat pump attracts increasing attention due to its advantages in low grade thermal energy utilization (Xu et al., 2020; Chen et al., 2021) and its adoption of environmentally benign refrigerants with zero Global Warming (Wu et al., 2014; Dogkas and Rogdakis, 2018; Luo et al., 2020; Sun et al., 2020). At present, there are several thermally driven heat pumps such as absorption heat pump (Wu et al., 2014), Vuilleumier heat pump (Dogkas and Rogdakis, 2018), duplex Stirling heat pump (Luo et al., 2020) and thermoacoustic heat pump (Sun et al., 2020). The last three heat pumps are Stirling engine driven Stirling heat pumps. Compared to absorption heat pump, this kind of heat pumps have some advantages such as wider operating temperature and lifting temperature range (Bakker et al., 2010). The differences among them are the couplings that is working fluid in Vuilleumeir machines, piston in duplex Stirling machines and sound waves in thermoacoustic machines. At present, Vuilleumier heat pumps received more attentions among the three Stirling engine driven Stirling heat pumps (Dogkas and Rogdakis, 2018). Theoretical results show that Vuilleumier heat pump could save 34% ∼ 44% source heating energy for space heating compared to an 80% efficient gas furnace (Woods and Bonnema, 2019). Thus, it is a promising technology for applications in heating.

In the 1980s and early 1990s, Vuilleumier heat pump received a lot of attentions from universities and companies in Europe and Japan (Dogkas and Rogdakis, 2018). Related prototypes such as crank shaft Vuilleumier machine (Carlsen, 1989) and free piston Vuilleumier machine (Thomas and Schulz, 1991) were built and studied. The experimental results of 20 kW crank shaft Vuilleumier machine built by Carlson achieved 1.52 heating COP at 600 °C hot temperature, 55 °C warm temperature and 5 °C cold temperature (Carlsen, 1994). The results showed the potential for applications in residential heating. Compared to crank shaft Vuilleumier machines, free piston Vuilleumier machines have high reliability and long life due to elimination of any wear mechanisms (Schreiber, 2006). However, it could have lower COP at partial load due to resonate frequency (Kawajiri et al., 1997).

A few studies were carried out in the late 1990s and early 2000s. However, it receives attention again in recent years. Dogkas studied the thermodynamic and flow characteristics such as heat flow and fluid flow (Dogkas et al., 2019) and the effect of rotational speed on performance (Dogkas et al., 2019) in crank shaft Vuilleumier machines. Hofbauer made a huge innovation to free piston Vuilleumier machines (Hofbauer, 2016; Hofbauer, 2017). In Hofbauer’ Vuilleumier machines, the displacers will be controlled to be stationary at its dead point alternately. Thus, sequential motions for displacers were achieved, which is significantly different to sinusoidal and continuous motions employed in conventional Vuilleumier machines. Chen's work (Chen and Longtin, 2018) showed that Hofbauer’ Vuilleumier machines have 28% higher heat output at given condition. Furthermore, the partial load can be easily achieved by prolonging the dwell time of displacers (Chen and Longtin, 2018). In order to distinguish Hofbauer’ Vuilleumier machines from conventional Vuilleumier machines, Hofbauer cycle machines is used to name the machines in this work. At present, a few studies on Hofbauer cycle machines have been carried out. Luo developed a simplified adiabatic model (Liao et al., 2019), an analytical model (Guo et al., 2020), and a hybrid model (Zou et al., 2020) for thermodynamic performances of Hofbauer cycle machines. Chen (Chen et al., 2019; Chen et al., 2020) investigated the effects of parameters on Hofbauer cycle machines’ thermodynamic and dynamic performance. Overall, present studies on Hofbauer cycle machines are focused on thermodynamic performance.

As the sequential motions in Hofbauer cycle machines require the displacer to be stationary at its dead point, the achieved stroke is required to be nearly the design stroke. However, the achieved stroke could be slightly larger (overshooting) or smaller (undershooting) than the design stroke under some operating cases. As overshooting will produce work loss and noise while undershooting will consume electricity for pulling the displacer to run the rest stork, dynamic performance should be studied for prediction of achieved stroke during the operation.

There have been related publications (Chen et al., 2019; Matsue et al., 2008; Kawajiri et al., 1997; Matsue et al., 2002) for prediction of achieved stroke. These studies are generally obtained by solving the following equations with iterations:

$$m_h·{\ddot{x}}_h+c_h·{\dot{x}}_h+k_h·x_h=(P_{wg}-P_{gs,c})A_{r,h}+F_{g,h}+F_{e,h}$$

$$m_c·{\ddot{x}}_c+c_c·{\dot{x}}_c+k_c·x_c=(P_{wg}-P_{gs,c})A_{r,c}+F_{g,c}+F_{e,c}$$

where m is the mass of displacer, c is the damping coefficient, k is spring coefficient, P is the pressure, A is the area, F is the force, and subscripts “wg”,“gs”,“h”, “c”, “g” and “e” represent working chamber gas, gas spring chamber, hot, cold, gravity and electromagnetic. Furthermore, Eqs. (1-2) are usually coupled with thermodynamic equations, which determines the pressure and damping coefficient by iterations. Matsue (Matsue et al., 2008) investigated dynamic performance of free piston Vuilleumier cycle heat pumps. The largest error for hot and cold displacers’ amplitudes between measured and predicted values is 0.1 mm. Thus, such model has very high accuracy for prediction of stroke. However, it generally requires tens of minutes for calculations. As the operating condition could be significantly changed during the calculations, it is not suitable for being directly used to predicted stroke for control during operation and is essential to develop a simplified model, which could provide predicted stroke during operation in seconds or milliseconds (Schulz and Thomas, 1992).

However, undershooting in conventional free-piston Vuilleumier machine has much less adverse effects since the displacer will not be pulled by electricity as that in Hofbauer cycle machine. Thus, there is only need of preventing overshooting. It can be achieved by temperature control based on a map, which is built by solving complicated thermodynamic and dynamic equations (Matsue et al., 2002). Therefore, few works have been carried out to develop a simplified model to predict the stroke for control during operation in conventional free-piston Vuilleumier machine. Only Schulz (Schulz and Thomas, 1992) developed a simplified free-piston Vuilleumier model for purpose of quick calculation in an early design stage. The model was based on simplified calculation of pressure and presupposing accurate gas temperature and flow friction factor. The largest error for hot and cold displacers’ amplitudes between measured and predicted values in one operating condition was 2.6 mm. However, it needs accurate gas temperature and mean pressure. Furthermore, the model lacks of further validation in various operating condition. Overall, simplified model could provide quick estimation of achieved stroke with satisfied accuracy.

In this work, a simplified dynamic model and a simplified solution is developed for quick prediction. Furthermore, a method to determine the masses in working chamber and gas spring chamber with leakage between the two chambers is proposed. Then the predicted free stroke is investigated based on the developed model. Besides, experiments on a prototype machine is carried out to validate the model.