Stirling Engine Research Articles

Exploring entanglement effects in a quantum stirling heat engine

Abstract This study presents the development of an entangled quantum Stirling heat engine based on a two-qubit Heisenberg model. We investigate the impact of entangled states of the working substance on the energy exchange and the performance of the Stirling cycle. The results show an improvement in the cycle's performance, which is accomplished by changing the entangled states. Furthermore, the results suggest that the operational modes of the quantum cycle can be controlled by considering changes in the entangled states of the working material and the temperatures of the reservoirs.

CFD Simulation of Stirling Engines: A Review

Stirling engines (SEs) have long attracted the attention of renewable energy researchers due to their external combustion design and flexibility in operating with various heat sources. The mathematical analysis of these devices is conducted by using a broad range of models ranging from basic zero-order to highly detailed fourth-order models, which are implemented through Computational Fluid Dynamics (CFD) simulations. The unique features of this last approach, combined with the increase in computing power, have promoted the use of CFD as a tool for analyzing SEs in recent years, significantly reducing the costs associated with prototype construction. However, Stirling CFD simulations are sophisticated due to the variety of physical phenomena involved, such as volume change, conjugated heat transfer, turbulent compressible fluid dynamics, and flow through porous media in the regenerator. Furthermore, there is currently no comprehensive review of CFD simulations of SEs in the literature; therefore, this contribution aims to fill that gap. Emphasis has been placed on identifying the type of engine, the physical phenomena modeled, the simplifying assumptions, and specific numerical aspects, such as mesh type, spatial and temporal discretization, and the order of the numerical schemes used. As a result, it has been found that in many cases, CFD numerical reports lack sufficient detail to ensure the reproducibility of the simulations. This work proposes guidelines for reporting CFD studies on Stirling engines to address this issue. Additionally, the need for a sufficiently detailed experimental benchmark database to validate future CFD studies is stressed. Finally, the use of Large Eddy Simulations on coupled key engine components—such as compression and expansion spaces, pistons, displacer, and regenerator—is suggested to provide further insights into the specific flow and heat transfer characteristics in Stirling engines.

Experimental demonstration of a liquid piston Stirling engine with a wet regenerator

In this study, a liquid piston Stirling engine with a water-absorbing wick to keep the regenerator wet (LPSE(wet)) was designed and built. The feasibility of operating the LPSE (wet) at lower temperatures to increase power output compared with that of a liquid piston Stirling engine with a dry regenerator (LPSE (dry)) was experimentally verified. First, the critical temperatures of LPSE (wet) and LPSE (dry) were measured. The experimental results revealed that LPSE (dry) operated at 100 °C, whereas LPSE (wet) operated at 65 °C. Thus, in this study, LPSE (wet) can be operated at a critical temperature 35 °C lower than that of LPSE (dry). The LPSE (wet) could be operated continuously for 900 min, demonstrating the effectiveness of the water-absorbing wick mechanism for long-duration operation. Next, the specific acoustic impedance distribution, the phase difference between the pressure and the cross-sectional mean velocity distribution, and the acoustic power distribution inside the device were measured for LPSE (dry) and LPSE (wet) during each operation. The results confirmed that both LPSE (dry) and LPSE (wet) realized a traveling wave field in which the phase difference between the pressure and cross-sectional mean velocity was close to zero and the specific acoustic impedance more than 30 times the characteristic impedance of a sound wave propagating in free space, which are required to obtain high thermal efficiency, near the regenerator position. The ratio of acoustic power before and after the regenerator was 1.18 (at a hot end temperature of 130 °C) for LPSE (dry) and 1.54 (at a hot end temperature of 90 °C) for LPSE (wet). Under the same conditions, the acoustic power of LPSE (wet) was 10 times or above that of LPSE (dry). These results confirmed that LPSE (wet) designed in this study could operate at lower temperatures and exhibit higher-power output than LPSE(dry).

Evaluation of heat transfer enhancement effect at the hot/cold end of a Stirling engine using performance improvement factor

The driving force of a Stirling engine comes from the heating expansion and cooling compression of working medium. The enhancement of heat transfer capacity of the hot/cold end is the key to improve the Stirling engine performance, while how to objectively evaluate the heat transfer enhancement effect is a precondition. Herein, a polytropic model of a Stirling engine thermal cycle process was firstly established. A method for evaluating the heat transfer enhancement performance under oscillating flow was proposed based on the efficiency and power improvement of a Stirling engine. This method was further used to evaluate the heat transfer enhancement effect of a novel four-leaf helical heating tube under oscillating flow, the efficiency and power improvement factor of which was reached 13.5 % and 6.1 %, respectively. The results show that the efficiency and power improvement evaluation criterion was more comprehensive and intuitive in evaluating the heat transfer enhancement performance under oscillating flow compared with the conventional evaluation indictor like performance evaluation criterion and equivalent tube length and tube number criterion.

Simulation of Ground Power Unit – 3 (GPU3) Stirling Engine with Air as Working Fluid

Abstract The immediate need to mitigate climate change presents a chance to move civilization in the direction of a more sustainable future. A Stirling Engine has multi-fuel capability such as Biomass, Solar Thermal and Waste Heat and hence can contribute significantly to the energy mix of fuel sources. The most common working fluids for Stirling engines are hydrogen, helium, and air, with air being the least expensive and safest. Studies analysing Stirling Engine performance with 3D CFD are limited, and even fewer use air as the working fluid. This research presents a novel 3D CFD analysis of the Ground Power Unit – 3 (GPU3) Stirling Engine with air as the working fluid using ANSYS Fluent. The fluid domain was modelled in SolidWorks and 1/8th of the geometry was used for simulation with realizable EWT k-ε as eddy viscosity model. It was found that on average there was a reduction in power output by 50% when air was used as working fluid against helium as working fluid. The engine's power output decreases as the engine's speed increases. The impinging effect contributes to vortex formation and temperature variation within the engine components was non-sinusoidal, this is inline with similar studies performed on GPU-3 Stirling engine.

Two‐Dimensional Solidification Simulation of PCM Molten Salt as a Thermal Energy Storage for Stirling Engine

ABSTRACTThermal energy storage technologies have been widely used to mitigate intermittency from renewable energy sources such as solar energy. Phase change material (PCM) is a material that can be used as a heat storage medium and is available in a wide range of operating temperatures. Molten salt is one of the PCMs that has the advantage of a very high operating temperature. The PCM solidification simulation based on HitecXL molten salt using COMSOL Multiphysics software was carried out with variations in heat absorption of 1–5 kW/m2, assuming constant heat absorption. The results showed that the PCM solidification process started from the surface of the Stirling engine heat exchanger pipe. The part of the PCM that is solidified falls due to gravity, causing a phenomenon similar to a droplet. The flow that occurred was natural, driven by the buoyancy force resulting from density changes due to temperature gradients in the solidification process. The time required for the PCM to completely solidify was closely related to the amount of heat absorption; the greater the heat absorption from the pipe, the faster the PCM is fully solidified.

The experimental investigation of performance behaviors of a beta-type Stirling engine with bell-crank motion mechanism

The current study aims to develop a novel power generation system that is capable of working with a Stirling engine. Within this context, a β-type Stirling engine design has been created with a bell-crank motion mechanism and a 365 cm3 swept volume. The engine has been manufactured, and then a detailed assessment has been conducted to determine the impact of the motion mechanism on engine performance characteristics. The designed and manufactured engine has been tested using a range of working fluids, such as air, argon, carbon dioxide, helium, and nitrogen gases. The performance tests of this engine have been carried out at 1000 K (±10) heater and 300 K (±5) coolant temperatures. Based on the outcomes of the experimental studies, the highest engine power and torque values have been obtained at a charge pressure of 4 bar using helium gas, with 143.5 W at 267 rpm and 7.75 Nm at 100 rpm, respectively. Moreover, the maximum engine power values obtained from other tests with nitrogen, air, carbon dioxide and argon gases have been compared with helium gas. Helium gas has been found to outperform nitrogen, air, carbon dioxide, and argon gases in tests by 67.3%, 73.9%, 197.1%, and 200.2%, respectively. Finally, the highest thermal efficiency value has been obtained with helium gas as 48.7% at a charge pressure of 4 bar.

Development of dynamic simulation model for 20 kWe micro heat pipe fission battery with dual power conversion system

Recent advancements in politics, technology, and economics have sparked interest in small-scale nuclear power generation. Heat-pipe micro-reactors, a type of small modular reactor (SMR), are ideal for remote and specialized applications like deep-sea and deep-space exploration. Traditionally, these reactors have used thermoelectric generators (TEGs) or Stirling engines for power conversion. However, TEGs suffer from low efficiency, and while Stirling engines have improved, they still have lower long-term stability compared to TEGs. Furthermore, comprehensive studies on system-level transients, dynamic behavior, and thermal responses are lacking. This study designs and models a dual power conversion system integrating TEGs and Free-Piston Stirling Generators (FPSGs) for micro heat pipe reactors to enhance efficiency, stability, and reliability. By integrating the two systems in parallel, the limitations of each technology are addressed, offering a robust solution for small, portable nuclear power generation. A dynamic simulation model incorporating the reactor core, heat pipes, and power conversion system enabled a comprehensive study of system transients and interactions. The core was modeled using thermal energy balance equations, and a thermal resistance approach was applied to the heat pipe. The TEG model included the Thomson and Seebeck effects, while neutronics feedback used a point kinetics model. The nonlinear analysis model was applied to simulate the FPSG. The system reached a stable operation after about 1000 s, generating 20kWe with 29.39% efficiency. It maintained stability without external control during transients within a specific range, demonstrating its feasibility. The TEG and FPSG exhibited independent yet complementary behaviors, ensuring reliable performance across scenarios.

Thermodynamic quantum Fokker–Planck equations and their application to thermostatic Stirling engine

We developed a computer code for the thermodynamic quantum Fokker–Planck equations (T-QFPE), derived from a thermodynamic system–bath model. This model consists of an anharmonic subsystem coupled to multiple Ohmic baths at different temperatures, which are connected to or disconnected from the subsystem as a function of time. The code numerically integrates the T-QFPE and their classical expression to simulate isothermal, isentropic, thermostatic, and entropic processes in both quantum and classical cases. The accuracy of the results was verified by comparing the analytical solutions of the Brownian oscillator. In addition, we illustrated a breakdown of the Markovian Lindblad-master equation in the pure quantum regime. As a demonstration, we simulated a thermostatic Stirling engine employed to develop non-equilibrium thermodynamics [S. Koyanagi and Y. Tanimura, J. Chem. Phys. 161, 114113 (2024)] under quasi-static conditions. The quasi-static thermodynamic potentials, described as intensive and extensive variables, were depicted as work diagrams. In the classical case, the work done by the external field is independent of the system–bath coupling strength. In contrast, in the quantum case, the work decreases as the coupling strength increases due to quantum entanglement between the subsystem and bath. The codes were developed for multicore processors using Open Multi-Processing (OpenMP) and for graphics processing units using the Compute Unified Device Architecture. These codes are provided in the supplementary material.

Classical and quantum thermodynamics in a non-equilibrium regime: Application to thermostatic Stirling engine

We have developed a thermodynamic theory in the non-equilibrium regime, which we describe as a thermodynamic system–bath model [Koyanagi and Tanimura, J. Chem. Phys. 160, 234112 (2024)]. Based on the dimensionless (DL) minimum work principle, non-equilibrium thermodynamic potentials are expressed in terms of non-equilibrium extensive and intensive variables in time derivative form. This is made possible by incorporating the entropy production rate into the definition of non-equilibrium thermodynamic potentials. These potentials can be evaluated from the DL non-equilibrium-to-equilibrium minimum work principle, which is derived from the principle of DL minimum work and is equivalent to the second law of thermodynamics. We thus obtain the non-equilibrium Massieu–Planck potentials as entropic potentials and the non-equilibrium Helmholtz–Gibbs potentials as free energies. Unlike the fluctuation theorem and stochastic thermodynamics theory, this theory does not require the assumption of a factorized initial condition and is valid in the full quantum regime, where the system and bath are quantum mechanically entangled. Our results are numerically verified by simulating a thermostatic Stirling engine consisting of two isothermal processes and two thermostatic processes using the quantum hierarchical Fokker–Planck equations and the classical Kramers equation derived from the thermodynamic system–bath model. We then show that, from weak to strong system–bath interactions, the thermodynamic process can be analyzed using a non-equilibrium work diagram analogous to the equilibrium one for given time-dependent intensive variables. The results can be used to develop efficient heat machines in non-equilibrium regimes.

Optimized multi-criteria performance of a poly-generation layout including a Stirling engine and a supercritical Brayton cycle using biogas and methane as two potential fuels of a topping gas turbine cycle

The combination of a supercritical Brayton cycle, an absorption chiller, a Stirling engine, a reverse osmosis desalination system, and a proton exchange membrane electrolyzer is investigated for waste heat recovery of a topping gas turbine cycle. The required electricity for hydrogen and freshwater production is supplied by the Stirling engine and supercritical Brayton cycle, respectively. In addition, the output power is provided by the gas turbine cycle, and the exiting cold stream of the Stirling engine is considered hot water for domestic utilization. A comparison is made between biogas and pure methane as two possible fuels for the system. Energy and exergy analyses, economic analysis using the specific exergy costing approach, and environmental analysis considering the carbon dioxide emissions are employed in the study. The three-objective optimization of the configuration discloses an exergy efficiency (ηex) of 39.49 % and 39.85 % in the cases of biogas and methane, respectively. The proposed system can improve ηex by 6.18 % points compared to the stand-alone GTC. The specific cost of poly-generation (cpoly) and the total cost rate (C˙tot) are obtained as 25.92 $GJ−1 and 249.5 $h−1 in the biogas mode, while these values are calculated as 36.75 $GJ−1 and 336.7 $h−1 in the methane case. The environmental cost rate (C˙env) of the system is 24.78 $h−1 in the case of biogas and 17.47 $h−1 in the methane mode. The results confirm the superiority of the methane case from the environmental viewpoint over biogas. However, the low values of cpoly and C˙tot in the biogas case indicate that biogas is superior to methane from a general standpoint. The utilization of the present setup for waste heat recovery of biogas-driven GTCs is recommended due to the higher ηex than the previous layouts.

A design method based on neural network to predict thermoacoustic Stirling engine parameters: Experimental and theoretical assessment

A neural network-based design method for thermoacoustic Stirling engines (TASE) is presented in this work. The main goal of the article is to predict the design parameters (Length of resonator (stub) (LR), Length of inertance tube (Lin), Mean pressure (Pm)) of the thermoacoustic Stirling engine in such a way that the dynamic instability is guaranteed. In this regard, the governing equations of the engine are presented first. Next, with the help of the data extracted from the simulation of the governing equations of the engine, a neural network with the structure of 9–35-3 is trained. The effectiveness of the artificial neural network (ANN) structure is explored via regression and MSE (mean square error) analyses. Next, in order to evaluate and validate the neural network, the experimental data of the constructed thermoacoustic Stirling engine (SUTech-SR-4) is considered. It is important to note that the SUTech-SR-4 is introduced for the first time in this work. Investigations have shown that the neural network could estimate the engine design parameters well and with an acceptable approximation in such a way that the dynamic instability of the engine is also established. Besides, the value of the estimated design parameters for the engine made by the neural network was close to the experimental values. Following, the developed engine is introduced and its performance is discussed. The developed engine has been able to produce a power equivalent to 0.346 W at a pressure of 1 bar (with air working fluid) and a working frequency of 15.9 Hz. It should be noted that the method presented in this article can be used for other types of thermoacoustic Stirling engines. Moreover, the selection of design parameters for the presented method will depend on the physics of the engine and the available facilities.

Quantifying damping coefficients in a rhombic-drive β-type Stirling engine based on a novel CFD-mechanism dynamic model and experimental data

Effectively exploiting renewable thermal energy sources and reducing the operating costs of renewable thermal systems remain challenging. Stirling engines, which serve as the heart of some renewable thermal systems, significantly impact both the maintenance and overall efficiency of these systems. However, the lack of information on friction behaviors in Stirling engines hinders the achievement of these goals. Therefore, this study clarifies the behaviors of damping coefficients in a rhombic-drive β-type Stirling engine. A novel CFD-mechanism dynamic model is developed to compute numerical values of cyclic-averaged engine speed corresponding to various loading torques. By employing the steepest descent method, the unknown values of the damping coefficients are adjusted to ensure the best match between numerical and experimental variations of loading torque with cyclic-averaged engine speed. Consequently, the study sheds light on the variation of damping coefficients with the instantaneous engine speed. The damping coefficient between the cylinder and piston is consistently lower than that between the displacer and cylinder. Additionally, the damping coefficient between the cylinder and piston ranges from 15.8 to 63.4 N s/m, while the coefficient between the cylinder and displacer increases from 50.0 to 195.5 N s/m as the instantaneous engine speed decreases from 1650 to 450 rpm.

Low temperature rotary Stirling engine: Conceptual design and theoretical analysis

The use of low-temperature energy sources for electricity generation demands a dual focus: a substantial enhancement in the efficiency of energy conversion devices and a reduction in system production costs. Particularly in scenarios where low-temperature energy sources are scarce, this factor can be pivotal in facilitating widespread adoption of such technologies. The Stirling engine emerges as a promising solution capable of meeting these articulated expectations, owing to its straightforward design and utilization of non-toxic, non-flammable, and cost-effective working mediums. This paper introduces a novel concept of a rotary Stirling engine, exhibiting significant potential for operation with low-temperature energy sources. Additionally, an analytical model of the engine is presented, enabling simulations of its operation under varying supply temperatures and geometric configurations. To analyse the impact of internal leaks on the net efficiency and net power of the engine, a modified adiabatic model was introduced. It was observed that utilizing identical heat exchangers for heat supply at 250 °C and 100 °C could lead to a decline in net efficiency from 8 % to 3 % for the worst case. Furthermore, an analysis was performed to assess the impact of the heater’s overall heat transfer coefficient and engine rotational speed on both net efficiency and net mechanical power for a heat supply temperature of 200 °C.

A calculation algorithm for the heater of a Stirling engine

Abstract The paper presents a calculation algorithm for the heater of a Stirling engine. In the current context of the energy crisis, Stirling engines can be used to produce mechanical work from renewable energy sources. The heat exchangers are one of the key elements regarding the efficiency of a Stirling engine. So, a calculation algorithm for the heater of a Stirling engine can be used for the dimensioning of this component, according to the requirements for the functioning of the system. To validate this algorithm, a dedicated program for the calculation of the heat exchangers based on the finite element analyse was used.

Numerical study of electric power generation from a liquid piston thermoacoustic Stirling engine

Numerical study of electric power generation from a liquid piston thermoacoustic Stirling engine

Electricity and hydrogen fuel generation based on wind, solar energies and alkaline fuel cell: A hybrid IWGAN–AVOA approach

Addressing the pressing need for sustainable energy solutions, this paper proposes a novel hybrid energy system integrating wind, alkaline fuel cells and solar energies. This paper introduces a pioneering hybrid energy system designed to produce electricity and hydrogen fuel through the integration of wind, solar energies, and an alkaline fuel cell. The proposed approach, coined as IWGAN–AVOA technique, synergizes the improved Wasserstein generative adversarial network (IWGAN) and the African vultures optimization algorithm (AVOA). The primary aim is to delineate a distinctive energy cycle reliant on renewable sources, featuring a Stirling engine, electrolyzer, alkaline fuel cell, wind turbine, and solar photovoltaic system, with the objective of generating hydrogen fuel and energy. The AVOA enhances fuel cell capabilities, simulates optimal system component sizes, and rectifies erroneous configurations based on geographical specifics. By validating the efficacy of IWGAN, the study achieves optimal configurations for device capacities and enhances power flow efficiency. Comparative analyses reveals that the IWGAN–AVOA approach surpasses existing techniques, like particle swarm optimization (PSO), wild horse optimizer (WHO), and heap based optimizer (HBO), with a remarkable efficiency rate of 98%, outperforming current methods. The last figure illustrates the effectiveness comparison, showcasing efficiencies of 62%, 79%, and 85% for HBO, PSO, and WHO, respectively, against the superior 98% efficiency achieved by the IWGAN–AVOA approach, affirming its superiority in energy system optimization.

Effects of internally charged hydrostatic gas bearing on performance of gas springs in free piston stirling engine

The pneumatic supporting system consisting of gas springs and self-charged hydrostatic bearings can suppress vibrations and improve durability of free piston Stirling engines. In this paper, a free piston Stirling generator with pneumatic supporting system on the displacer was developed based on a prototype of plated spring scheme. A series of experiments were carried out to evaluate the impacts of self-charging gas bearing on the performance of gas springs. During the experiments, the linear alternator was applied an external AC power supply and the pressure fluctuations in working chamber were ensured to be consistent so that the pneumatic supporting system was separated from the entire engine. Through blocking the orifices and ducts of gas bearings with plugs, the gas springs were decoupled with the gas bearing. Comparative results show that the internally charged hydrostatic bearing can effectively reduce the mechanical damping of the displacer with a reduction from 117 N/m to 83 N/m after the bearing is switched on. While the hydrostatic bearing has side effects on the gas springs that increase the hysteresis loss from 78.48W to 337.11W and decrease the axial stiffness significantly.

Experimental study on liquid piston Stirling engine combined with self-rectifying turbine.

A liquid piston Stirling engine is an external combustion engine that uses air and water under atmospheric pressure as its working fluids. Resulting from its uncomplicated design and the capacity to operate under relatively low temperature differentials of less than 100 °C, it has attracted considerable attention in recent years. This paper presents the fundamental characteristics of the liquid piston engine combined with a self-rectifying turbine for the advancement of thermal generators. When the turbine is installed in the water region rather than in the air region, it exhibits unidirectional rotation with a rotational speed directly proportional to the velocity amplitude of the reciprocating axial flow. Additionally, the acoustic impedance within the duct section containing the turbine is determined, demonstrating that the real part of impedance rises with increasing axial velocity, indicating a loss mechanism similar to the minor loss. Furthermore, the installation of the turbine results in a breakdown of symmetry in the engine oscillation mode. To maintain symmetry and improve system design, future developments must consider the installation of a turbine in each unit. These findings can pave the way to the design of liquid piston Stirling engines and their applications in thermal energy conversion.

Experimental Demonstration of a Solar Powered High Temperature Latent Heat Storage Prototype

The challenge of imbalances between renewable energy supply and grid demand underscores the significance of energy storage in microgrids. This research presents an empirical assessment of the operational capabilities of a full-scale Electrical Thermal Energy Storage (ETES) prototype system named Thermal Energy Storage Power On Demand (TES.POD®), in solar-abundant and harsh desert conditions. The system incorporates a high-temperature commercial-scale latent heat thermal energy storage, integrated with a Stirling engine. Over a continuous span of 10 days, from September 26th to October 6th, 2022, the input and output power, as well as the heat transfer fluid temperatures during charging and discharging were monitored to assess the power block and system efficiencies. Results from the experiments reveal that this prototype effectively maintains a near-constant electricity production rate of 10.5 ± 1 kW for a discharge duration of 13 hours. Average cycle efficiency stands at 23%, while power block efficiency reaches 25%. These findings collectively suggest the system's potential for applications involving long duration thermal energy storage.