Stirling Engine Research Articles

The advantages and applications of the stirling engine

Stirling engines are a kind of heat engine that achieve their function via compressing and expanding the working fluid at temperatures that are entirely different from one another. The Stirling cycle consists of four processes, two of which are isothermal and the other two are isochoric. Stirling engines may be assembled in one of three distinct configurations: alpha, beta, or gamma. They are superior to traditional heat engines, with high efficiency, low noise, and low pollution. Stirling engines may be found in various systems that produce power and those used for mechanical propulsion, heating and cooling, and other similar applications. In today's world, programs such as this are really helpful. On the other hand, they have some drawbacks, such as a low power density, a high price tag, a sluggish beginning-of-operation and reaction time, and restricted availability. This page covers a variety of topics pertaining to Stirling engines, including their history, composition, characteristics, applications, and use, as well as some of the likely next advancements in the field.

Research on the application of heat engine efficiency in reducing energy consumption

In recent years, the problem of global warming and environmental pollution has become increasingly serious, so improving the efficiency of heat engines is of great significance to reduce energy consumption. This paper uses the method of literature review to study the current situation and prospects of heat engine efficiency in reducing energy consumption in three aspects. Firstly, the heat engine and its efficiency are summarized from the theoretical derivation. Secondly, the working principles of the Carnot heat engine and the Stirling heat engine are explained in detail through formula derivation and comparative analysis. Finally, taking the air-source heat pump as an example, we explain the basic principle, operation mechanism, and technical ways to improve efficiency in detail.

Numerical and experimental investigations on coiled resonance tube of a duplex Stirling cryocooler

Duplex Stirling cryocooler is promising in mitigating methane emission due to its environmentally-friendly working gas, high thermal efficiency and high adaptivity to varying methane concentration. In this work, a coiled resonance tube is introduced to supplant the conventional straight tube in a resonance tube coupled duplex free-piston Stirling cryocooler (RT-DFPSC) to reduce footprint and elevate power density. At first, an experimental setup was built employing resistance-compliance acoustic load to produce the analogical acoustic field. It was observed from experiments that on a coiled resonance tube, an acoustic power transfer efficiency of 35% was obtained. Based on the experiments, a three-dimensional computational fluid dynamics (CFD) model was subsequently developed to pinpoint the loss mechanism. Due to the interaction of centrifugal, viscous and inertial force, there are two secondary flow vortices in the cross-section of the coiled resonance tube during the reversing flow, accounting for the primary loss. Furthermore, the acoustic loss of the coiled resonance tube increases with the decrease of the coiled diameter, while the spiral pitch has minimal effect on the acoustic and flow field. Typically, the acoustic loss of a coiled resonance tube with a diameter of 0.8 m increases by approximately 9% compared to that of a straight tube. Finally, a one-dimensional equivalent model of the coiled resonance tube was established and integrated into a one-dimensional system-level model. Compared to the straight RT-DFPSC, a system with a coiled tube resonance tube demonstrates a cooling capacity and overall exergy efficiency reduction by 465.2 W and 2.4%, respectively.

Thermodynamic and economic evaluation of a CCHP system with biomass gasifier, Stirling engine, internal combustion engine and absorption chiller

This work proposes a combined cooling, heating and power (CCHP) system based on biomass gasification with an internal combustion engine (ICE), a Stirling engine and an absorption chiller. The energy and exergy efficiencies and economic performance of the CCHP are evaluated under different working conditions by thermodynamic model and net present value method. The Stirling engine recovers the high temperature potential of syngas during cooling process and improves the CCHP output power by 14 % along with ICE, while the thermal efficiency of ICE reaches 39 % at 5000 rpm. In this work, about half of the total exergy destruction comes from ICE rather than gasifier due to the utilization of Stirling engine as an additional mover to produce electricity. The recycling of weak solution in absorber is adopted in order to increase the COP (Coefficient of performance) of single-effect absorption chiller. The COP could be increased by 7 % and reach 76 % as half of the weak solution recycling to absorber. The electric efficiency and total CCHP efficiency of this system reach 37 % and more than 60 %, respectively. The payback period is estimated to be less than 6 years at an ICE rotational speed of over 4000 rpm and an operation time of more than 4000 h/year.

Investigating dynamic characteristics and thermal-lag phenomenon in a thermal-lag engine using a CFD-mechanism dynamics model

Thermal-lag engines (TLE) are external combustion engines with a single piston that offer the potential for reduced maintenance and production costs compared to traditional Stirling engines. Despite these promising prospects, studies on the physical behavior of this engine type remain limited. In this study, three novelties are presented: (1) a novel computational fluid dynamics-mechanism dynamics (CFDMD) model with high complexity for the TLE; (2) an analysis of the work generation in relation to the phase difference between pressure and volume; and (3) an analysis of the thermal-lag phenomenon based on the full spectrum of the phase difference between temperatures and piston position. In the CFDMD model, there are two key components: a computational fluid dynamics (CFD) model that captures the temporal evolution of the three-dimensional thermo-fluid fields and a mechanism dynamics model that solves the transient dynamics of the crank-drive mechanism. The CFDMD model effectively simulates the complex interplay between the thermodynamic behavior of the working gas and the dynamic behavior of the engine mechanism. The study additionally investigates the temporal evolution of the engine speed and examines its dynamic behavior. The influence of parameters, including crank radius, heating temperature, and initial crank angle, on shaft power and brake thermal efficiency is thoroughly explored. To gain a profound understanding of the indicated work generation mechanism and the thermal-lag phenomenon, the discrete Fourier transform provides full spectra of the periodic quantities of interest generated by the CFDMD model. It is found that the numerical average engine speed at different heating temperatures and torques from the CFDMD model is in good agreement with the experimental data, with a maximum difference of 140 rpm. The shaft power decreases from 19.8 to 5.3 W, while brake thermal efficiency increases from 4.5 to 7.7% as the initial crank angle varies from −90° to 90°. The first-oscillation phase differences between pressure and volume are close to 180°, decrease as the heating capacity is raised, and dominate the swelling of the PV diagrams. Only the first harmonic oscillation of the heater temperature exhibits the thermal-lag phenomenon.

Junction temperature of CMOS electronics cooled by a Stirling cryocooler

This case study addresses the practical challenges associated with operating cryo-CMOS electronics. Previous research has been largely confined to impractical liquid cryogen cooling and the few under-reported examples of implementing electric cryocoolers. This study further clarifies the operation of cryo-CMOS devices cooled by electric cryocoolers. It develops and experimentally validates an analytical relation between the junction temperature and the parameters of a cryogenic system. To validate this relationship, we placed a printed circuit board in a thermal chamber, cooled it to 173 K using a Stirling cryocooler, and varied input clock frequencies. The measured values of the junction temperature were compared with calculations, which showed good prediction accuracy with a coefficient of variance of ±2%, a mean biased error of −2%, and a determination coefficient of 0.92. In addition, the measurements provided evidence for a multi-fold increase in the clock frequency of cryo-CMOS electronics. The results of this study help estimate accurately the steady-state junction temperature of cryo-CMOS devices in a wide range of cryogenic temperatures for different types of cryocoolers.

Linear generator design for concentrating solar power technologies: Optimization and prototype development

AbstractAn alternative way to generate electricity from solar energy is through the use of generators comprising Stirling engines with a parabolic collector. This study describes a parabolic collector with Stirling engine and investigates the design of a linear mobile generator for these systems. Especially for direct drive in free piston mechanisms proposed for these systems, linear generators are important due to their advantages such as compact structure, light weight, high efficiency and high power intensity. Although conventional one‐phase and three‐phase multi slot/pole models are widely available in the literature, three phase is considered for lower PM cost and to reduce the cost of power electronics and 3/2 slot/pole combination is considered for high speed and frequency. In this study, a new 3/2 slot/pole three‐phase tube‐type linear generator was designed and evaluated for performance and manufacturability. Objective functions were defined for power and efficiency increase to reduce generator power ripple, and the results of Mixed‐Integer Sequential Quadratic Programming, Multi‐Objective Genetic Algorithm (MOGA), and Screening methods were compared and examined. Performance of the two‐dimensional and optimized models was evaluated in terms of induced voltage, power density, efficiency, and output power. The best results were found with MOGA compared to other methods. When compared with the initial value, there was a 97.3% increase in power density for a 32.4% increase in moving weight. Moreover, the efficiency of the MOGA model is improved by 16.2% compared with the initial model. Additionally, significant improvements were shown in terms of thrust coefficient, voltage coefficient, power, and no‐load voltage. A prototype of the model was made and a motor‐driven crank mechanism system was used in place of the Stirling engine to undertake practical measurements.

Overcoming power-efficiency tradeoff in a micro heat engine by engineered system-bath interactions

All real heat engines, be it conventional macro engines or colloidal and atomic micro engines, inevitably tradeoff efficiency in their pursuit to maximize power. This basic postulate of finite-time thermodynamics has been the bane of all engine design for over two centuries and all optimal protocols implemented hitherto could at best minimize only the loss in the efficiency. The absence of a protocol that allows engines to overcome this limitation has prompted theoretical studies to suggest universality of the postulate in both passive and active engines. Here, we experimentally overcome the power-efficiency tradeoff in a colloidal Stirling engine by selectively reducing relaxation times over only the isochoric processes using system bath interactions generated by electrophoretic noise. Our approach opens a window of cycle times where the tradeoff is reversed and enables the engine to surpass even their quasistatic efficiency. Our strategies finally cut loose engine design from fundamental restrictions and pave way for the development of more efficient and powerful engines and devices.

Micro Combined Heat and Power (micro-CHP) Systems for Household Applications: Techno-economic and Risk Assessment of The Main Prime Movers

ABSTRACT To limit global warming and meet international environmental targets, more environmentally benign technologies must be used in the residential market. Combined Heat and Power (CHP) at the micro-scale (<50 kWe) is seen as one of the best solutions that offers simultaneous generation of both electricity and heat with high overall efficiencies using environmentally friendly fuels (e.g., biofuel, hydrogen, syngas). In this study, four major micro-CHP prime movers (i.e. Micro-gas turbine, Gas engine, Stirling engine, and Fuel cell) have been evaluated in terms of their performance and the currently available technologies and products. The most suitable options for household applications were identified using Political, Economic, Social, Technological, Legal, and Environmental (PESTLE) risk analysis and a Multi-Criteria Decision Analysis (MCDA). Fuel cells display the best environmentally friendly properties despite their price and high start-up time. Gas engines (Reciprocating engines) have the most developed technology with the fastest start-up time and high efficiencies but have vibration and noise issues. Micro-gas turbine micro-CHPs are emerging into the market with relatively cheaper prices, lower maintenance, and good start-up time. Stirling engines are fuel flexible with reasonable prices and available products. Micro-CHP systems, particularly when running on biofuels and hydrogen, provide excellent energy solutions for the future net zero buildings.

Optimization of a novel drive mechanism approaching the ideal cycle for beta type Stirling engines

Stirling engines are capable in the renewable energy gain and the recovery of waste heat, both of which are important issues today. Existing studies revealed that the piston trajectories convergent to the Stirling cycle provide more power than traditional crank mechanism. In this study, a hypocycloid drive mechanism not used in the literature was optimized on the ideal piston trajectories for beta type Stirling engines. Optimization was realized to improve the isochoric heating and cooling processes as well as the isothermal compression and expansion. The cyclic work was treated as another objective succession. The optimal results taking into account these five objectives were obtained with the gear ratios of 5 and 4 for displacer piston and power piston, respectively. The piston dwells provided the enough objective succession not only for the both isochoric processes but also for the isothermal compression process, while failed in the isothermal expansion. In comparison with crank mechanism, a cyclic work increment of 10% was achieved in the near-ideal piston trajectories, but the maximum increment of 14% eventuated in some out of ideal.

An Experimental Investigation of Fatigue Performance for A Flexure Spring

In order to prevent a free-piston Stirling cryocooler failure during mission life, the flexure spring's fatigue life is most critical and important. In the current research, an effort is made to examine fatigue life. The fatigue life testing of cryocooler flexures is frequently discussed in the literature. However, more information is needed about the economic approach and hardware needed for the testing. This research paper adopts the FEA approach of flexure spring and outlines the criteria for flexure fatigue testing. A moving coil linear motor was also designed and selected for the oscillating shaft and bending special-purpose flexure spring stack to validate stiffness criteria. The 108 no. of cycles required to validate the fatigue limit were also attained using the approach in a few days when testing at a relatively high frequency.

Multi-aspect exergo-economic/environmental study/optimization of an eco-friendly heat integration process for gas turbine modular helium reactors in integration with a Stirling engine

As a nuclear power source, gas turbine modular helium reactors (GT-MHR) can play a major role in the energy industry due to their high safety index and low operating costs. In light of the GT-MHR's high heat capacity, this study integrates a GT-MHR cycle with a Stirling engine to achieve a high-efficiency electricity production framework in an eco-friendly framework. Multi-aspect study from energy, exergy, economic and exergoenvironmental viewpoints are conducted. Detailed heat exchanger modeling is performed to accurately evaluate the system's performance conditions as part of the design process. The outcomes of this study reveal that in the optimum operating mode, the turbine's inlet temperature and expansion ratio are 1300 K and 2, respectively. The compressor compression ratio is 2.1, the Stirling engine inlet temperature is 850.6 K, and the regenerative effectiveness is 0.866. Consequently, the system's total electricity becomes 325.57 MW with 54.26 % energetic and 75.13 % exergetic efficiencies. The economic assessment of the system reveals that the unit electricity product cost is 8.9 $/GJ, and the total investment cost is 12737 $/h. Also, the optimal exergoenvironmental results show an exergoenvironmental index of 0.237, an environmental damage effectiveness index of 0.0031, and an exergetic stability factor of 0.24. Compared to GT-MHR, GT-MHR/KC, GT-MHR/APC, and GT-MHR/ORC, the newly designed system produces 44.57, 21.47, 29.74, and 25.17 MW more electricity and is more energetically efficient by 28.3 %, 24.45 %, 25.8 %, and 25.07 %, respectively.

Integration of a Gamma-Type Stirling Engine with LPG Cooking Stove for Micro-Scale Combined Heat and Power Generation

The Stirling engine is one of the most versatile micro-scale prime movers for combined heat and power applications, adaptable to different levels of heat sources. This study started a unique journey that included developing, experimenting, and analyzing the Gamma-type Stirling engine. Notably, the engine design ingeniously harnesses heat from a customized cooking stove burner powered by liquefied petroleum gas. The engine fabrication resulted in a compression ratio 2.014, accommodating a volumetric capacity of 181 cc. The Stirling engine test used air as the working gas, and the initial conditions were at atmospheric pressure. Stirling engine performance was analyzed using an ideal thermodynamic cycle model and burner efficiency using the water boiling method. The modified burner attains an average temperature of 699.5°C, producing a burner power output of 5.702 kW and a thermal efficiency of 32.7% or around 1.867 kW of heat for operating the engine and cooking activities. Simultaneously, tests of the Stirling engine revealed an average air temperature difference of 146.2°C between the expansion and compression phases. The flywheel rotation speed ranges from 158 to 369 rpm. During testing, the Stirling engine obtained an average thermal efficiency of 31.08%, accompanied by an ideal power spectrum ranging from 0.3 W to 42.6 W. The highlight of this study was the maximum pressure achieved at the end of the heat absorption stage, recorded at 296.1 kPa. Importantly, these findings underscore the promising potential of micro-Combined Heat and Power systems. Integrating the gamma-type Stirling engine with the LPG stove represents novelty and paves the way for further development and advancement in sustainable energy solutions.

Design and dynamic numerical modeling of a hybrid reverse osmosis/adsorption-based distillation system driven by solar dish Stirling engine for enhanced performance and waste heat recovery

In light of green solutions to address the shortage of freshwater shortage, energy crisis, and climate change. Tri-generation renewable systems are recently efficient approaches that can offer multiple energy outputs, such as heat, electricity, and distilled water. This paper presents a new hybrid freshwater production system driven by renewable energy for enhanced performance and waste heat recovery. This hybrid system proposes the utilization of reverse osmosis, an adsorption distillation system, and a solar dish Stirling engine (RO-ADS/SDSE). The cooperative concept is based on the re-desalination of the brine of the RO unit using an adsorption desalination system driven by the heat rejected from the SDSE, while the RO system is driven by the electrical energy produced by the Stirling engine. A mathematical model conducted in MATLAB coupled with EES package is developed to investigate the performance of the proposed RO-ADS/SDSE with and without pressure recovery (PR) mode at different salinity concentrations varying from 30000 to 50000 ppm at a constant feed pressure of 80 bar. The investigation is conducted based on the meteorological data of Tianjin, China. The results showed significant superiority in the performance of the proposed RO-ADS/SDSE system, as the annual freshwater production rate of the RO-ADS/SDSE with PR increased by up to 3000 m3 compared to 1700 m3 for the standalone RO system at a feed salinity of 45000 ppm and feed pressure of 80 bar. On a daily basis, the hourly permeating flow rate of the RO-ADS/SDSE with PR increased from 2.30 m3/h to 2.60 m3/h, and the permeate salinity decreased from 190 to 120 ppm. While the daily electricity consumption decreased from 5.20 to 4.40 kWh/m3. In conclusion, the proposed RO-ADS/SDSE system with PR can be considered a promising approach in maximizing the performance of desalination systems.

Dynamic response of a dual-opposed free-piston Stirling generator

As a type of distributed power generation device, free-piston Stirling generator (FPSG) is characterised by compactness, high efficiency, and long life. This work devotes to exploring the dynamic response of a low-vibration dual-opposed FPSG by employing the newly-developed time-domain acoustic-electrical analogy (TDAEA) method. Transient evolutions of self-sustained oscillations are first exhibited, which contains the start-up and steady stages. An exploration on steady-state performance is then conducted, which shows the maximum thermal-to-electric efficiency of 46.6% with an electric power of 3.92 kW, implying its enormous potential in distributed power generation for low-vibration applications due to its dual-opposed configuration. Dynamic response with jump in external load and gradual variation in heating temperature are subsequently investigated in detail. The results indicate that an acute drop in electric resistance leads to an oscillation attenuation, particularly, a short circuit results in an emergency shutdown. With the heating temperature declining gradually, the oscillation attenuates and even stops. Based on these results, some general operating regulation principles for FPSGs are further concluded to avoid the cylinder striking as well as frequent start and stop. This study gives a deeper understanding on the dynamic characteristics of FPSGs, as well as broadens the usage of the TDAEA method.

Influences of nonlinear parameters on the performance of a free‐piston Stirling engine

AbstractThis study presents a thermodynamic–dynamic model of a free‐piston Stirling engine (FPSE) with nonlinear coefficients of spring, load damping, and pressure for working spaces. The effects of the nonlinear coefficients of hardening and softening springs on the movement and operating frequency of a displacer and piston are investigated. Subsequently, the influences of the nonlinear pressure terms of the working space and buffer spaces, as well as the nonlinear coefficients of the load on the amplitudes of the two pistons, frequency, and output powers are discussed. The results indicate that the low nonlinear coefficients of the spring have a minor effect on the amplitudes and frequency. As the high nonlinear coefficients of the hardening spring increase, the amplitudes and output power decrease, while the frequency increases. As the nonlinear terms of the softening spring increase, the amplitudes and output power reach maximum values, but the frequency is reduced by 3.55%. The effects of the nonlinear pressure terms on the amplitudes and output power are not evident. However, an increase in the nonlinear load leads to a significant decrease in the FPSE's performance.

Quantum Stirling heat engine in two-coupled-qubit Heisenberg XYZ model

Quantum Stirling heat engine in two-coupled-qubit Heisenberg XYZ model

Numerical Modelling of Thermoacoustic Stirling Engines &amp; Refrigerators

Thermoacoustic machines depend on the complex relationship between thermodynamics and acoustics, and thus understanding it is vital in order to analyse the working principles and optimise parameters (i.e. geometrical or operational) to improve their performance. This paper investigates how numerical modelling can be used to explore this relationship and compares the accuracy of the performance predictions for different numerical simulation software. The software used included one designed for modelling Stirling machines called ‘Sage’ and one designed for modelling thermoacoustic machines called ‘DeltaEC’. To compare their results a model of both a thermoacoustic Stirling engine and refrigerator were developed from existing models in published papers, which contained experimental data to validate the numerical models. The results from the thermoacoustic Stirling engine model show that there is good agreement between the predictions from DeltaEC and the experimental data, as well as relatively good agreement between the Sage and DeltaEC predictions. However, due to Sage requiring a different approach to model the boundary conditions for the standing wave type machine (i.e. one end closed) the predictions varied slightly from those by DeltaEC. The results from the thermoacoustic Stirling refrigerator model, however, show improved agreement between the predictions from Sage and DeltaEC – potentially due to Sage and DeltaEC using a similar approach to model the boundary conditions for the travelling wave type (i.e. two open ends). Overall, it was found that although both can accurately model travelling wave thermoacoustic machines, the nature of Sage’s solving method makes it more complex to model the standing wave type compared to DeltaEC. A discussion on the use of numerical models as a tool for better understanding thermoacoustic machines, and the importance of the accuracy of the results to allow for optimisation and improvement in their design is presented.

Assessment of the integration of a switched nuclear isomer material with a kinematic Stirling engine

Presently, power systems are limited by the source and energy density of the chemical fuels that are currently in use. In response to this challenge, nuclear energy sources are being developed. Metastable nuclear states, or isomers, have an intrinsic energy density that is many orders of magnitude larger than chemical fuels. Through isomer depletion (or switching), these materials can be transformed from a long-lived half-life energy-storage state to a shorter-lived half-life energy-releasing state. The abundance of energy released by switched nuclear isomer (SNI) materials introduces unique engineering challenges for the power conversion system. Once an SNI material has been switched, its heat output decays with time. In contrast to hydrocarbon-based fuels, which can be throttled, generating constant power from an SNI system requires sufficient material based on duration of use and system efficiency. Despite these challenges, SNI materials could be used to replace chemical fuels for relatively high-power applications providing months of power from a compact energy source.The purpose of this work is to explore the coupling of an SNI heat source to a kinematic Stirling engine in order to assess the potential of such a system. First, the characteristics of a reference SNI material are discussed in the context of the conversion efficiency and power density of the power conversion system that it is coupled to. Second, the integration of an SNI with a kinematic Stirling engine is explored at two power levels, 6 kWe and 60 kWe, in order to explore the feasibility and efficiency of such a power system. In order to carry out this study, a second-order Stirling engine model (the Stirling Engine Trade-space Tool, or SETT) has been developed. The potential of the SETT tool to predict performance trends has been shown through comparison of the predicted to the measured performance for two baseline, well-documented Stirling engines: the GPU-3 (nominally 6 kW) and Mod II (nominally 60 kW) engines. The SETT model is then used to optimize the design of the integration heat exchanger that is used to thermally couple a Stirling engine to the SNI and evaluate the performance characteristics of this energy conversion system. The resulting model is also used to show that the kinematic Stirling engine has advantages related to its inherent ability to modulate its output through speed or inventory control, allowing it to effectively deal with the time varying rate of heat transfer that is inevitably provided by the SNI. This work shows that the kinematic Stirling engine is an attractive power conversion system to use with SNI material as it can achieve high conversion efficiency, high power density, and provides inherent modulation capability.

Evaluation of the operating conditions of the Stirling engine heat exchangers

The development of modern energy is inextricably linked with the further improvement of heat engines, which are still the only primary sources of mechanical energy on an industrial scale. Given the existing environmental problems, Stirling engines can be a good alternative to internal combustion engines and turbines. Of the many problems of creating such a sufficiently powerful and economical engine, it is customary to consider three main ones: 1) efficient transfer of large heat fluxes in the heater, regenerator and refrigerator; 2) creation of reliable and durable seals to hold the working fluid in the cylinder; 3) ensuring minimal friction in bearings and seals. And yet, of the three listed problems, the first should be given more attention, first of all, due to the unique conditions associated with the continuously changing thermo and mechanical loads. The instability of the loads is further complicated by the sharply different heat transfer coefficient on the external and internal heat transfer surfaces. That is, there are factors that contradict the requirements for the size of the heat transfer surface, the friction resistance of the working medium and the dead volume of the engine as a whole. In this regard, it is of interest to search for possible ways to reduce the effect of negative factors of non-stationary heat transfer in the calculation of heat exchangers. The current lack of suitable theoretical methods of calculation forces the use of semi-empirical methods. The aim of the paper is to assess the significance of the unique, characteristic of Stirling engines, the negative phenomenon of the delay in the flow of the working fluid in heat exchangers. For this purpose, a real kinematic diagram of the movement of the engine pistons with a rhombic drive by R. Meyer, created using the design program, is used. The analogue is the 4–235 type real engine of the Philips Company. In the calculations, a step-by-step procedure for calculating the time-varying parameters of the working fluid is used. The need for such a solution is due to the fact that in the rhombic drive of R. Meyer, the pistons movement and the change in the displaced volumes complexly depend on the angle of rotation of the engine shaft. In addition, the exact value of the real change in volume is possible only when the terms of the analytic equation of volume change under consideration are decomposed into a Fourier series.