This paper provides numerical simulation and thermodynamic analysis of SOLO 161 Solar Stirling engine. Some imperfect working conditions, pistons' dead volumes, and work losses are considered in the simulation process. Considering an imperfect regeneration, an isothermal model is developed to calculate heat transfer. Hot and cold pistons dead volumes are accounted in the work diagram calculations. Regenerator effectiveness, heater and cooler temperatures, working gas, phase difference, average engine pressure, and dead volumes are considered as effective parameters. By variations in the effective parameters, Stirling engine performance is estimated. Results of this study indicate that the increase in the heater and cooler temperature difference and the decrease in the dead volumes will lead to an increase in thermal efficiency. Moreover, net work has its maximum value when the angle between two pistons shaft equal to 90 degrees while efficiency is maximum in 110 degrees.
The urgent need to preserve fossil fuels and use renewable energies has led to the use of Stirling engines, which have excellent theoretical efficiency, equivalent to the related Carnot cycle. They can consume any source of thermal energy (combustion energy, solar energy, etc.) and they make less pollution than the traditional engines, [
A Stirling cycle machine is a device, which operates in a closed regenerative thermodynamic cycle. In the ideal cycle of the Stirling engine, the working fluid is compressed at constant temperature, heated at constant volume, expanded at constant temperature, and cooled at constant volume. The flow is regulated by volume changes so there is a net conversion of heat to work or vice versa.
The Stirling engines are frequently called by other names, including hot-air or hot-gas engines, or one of a number of designations reserved for particular engine arrangement.
Numerous applications of Stirling engines were raised during the 19th century and at the beginning of the 20th century. Stirling engine in the 19th century was confined largely by the metallurgical possibilities and problems of the time. By these reasons, the engine was finally pushed back by newly developed internal combustion engines. The Stirling engine was almost forgotten until the 1920s.
It was only in 1938, a small Stirling engine with an output of 200 W by Mr. N. V. Philips which stimulated interest in this engine type again. Development in material production technologies that took place in the 1950s opened new perspectives as well for the Stirling engine.
In the course of 1969-1970, Philips developed a drive unit with a rhombic mechanism for a municipal bus. A detailed calculation finally showed that, with a batch of 10000 pieces annually, the price would still be 2.5 times higher than that of a compression ignition engine of the same output due to the substantial complexity of the engine.
In the 1970s United Stirling worked hard on the development of a drive unit for passenger cars. One of the following V4X35 types was fitted in the Ford Taurus car in 1974. Despite the satisfactory results of a driving test covering 10,000 km, series production was never commenced due to the price of the drive unit.
High heat efficiency, low-noise operation, and ability of Stirling engines to use many fuels meet the demand of the effective use of energy and environmental security. Stirling engine-based units are considered best among the most effective low-power range solar thermal conversion units [
On the other side, the main disadvantages of Stirling engines are their large volume and weight, low compression ratio, and leakage of working fluid from the engine inner volume. To increase the specific power of Stirling engines, a number of methods have been developed, such as using hydrogen or helium as working fluid at high charge pressure, increasing the temperature difference between hot and cold sources, increasing the internal heat transfer coefficient and heat transfer surface and using simple mechanical arrangements, for example, a free piston Stirling engine [
The Stirling engine performance depends on geometrical and physical characteristics of the engine and on the working fluid gas properties such as regenerator efficiency and porosity, dead volume, swept volume, temperature of sources, pressure drop losses, and shuttle losses.
A machine that utilizes the Stirling cycle could function as an engine that converts heat energy from an appropriate heat source into kinetic energy, or, by employing the reverse cycle, as a refrigerator that can achieve low temperatures or provide a heat-absorbing effect by the injection of kinetic energy from an electric motor. The Stirling engine has also been proposed as a driver for electricity generators and heat pumps, and its practical applications have been realized. Conventionally, fossil fuels and solar energy have been considered as potential heat sources; more recently, however, the practical applications of engines that utilize biomass or the waste heat generated from diesel engines as fuel have been discovered [
The thermal limit for the operation of a Stirling engine depends on the material used for its construction. In most instances, the engines operate with a heater and cooler temperature of 923 and 338 K, respectively. Engine efficiency ranges from about 30 to 40% resulting from a typical temperature range of 923–1073 K, and normal operating speed range from 2000 to 4000 rpm [
Numerical investigations of Stirling engines have been performed by many researches. Thermodynamics analysis is the basic of almost all of these researches. Some of these researches focus on a specific part of engine, for example, regenerator [
Stirling engines are mechanical devices working theoretically on the Stirling cycle in which compressible fluids, such as air, hydrogen, helium, nitrogen, or even water vapor, are used as working fluids. The Stirling engine offers possibility for having high-efficiency engine with less exhaust emissions in comparison with the internal combustion engine. The Stirling engine is an external combustion engine. Therefore, most sources of heat can power it, including combustion of any combustive material, field waste, rice husk, or the like, biomass methane and solar energy. In principle, the Stirling engine is simple in design and construction and can be operated without difficulty [
The Stirling engine could be used in many applications and is suitable where multifueled characteristic is required; a very good cooling source is available; quiet operation is required; relatively low speed operation is permitted; constant power output operation is permitted; slow changing of engine power output is permitted; A long warmup period is permitted.
The engine cycle is represented on PV and TS diagrams in Figure
Stirling engine cyclic pistons’ arrangements [
To start with Stirling cycle, we assume that the compression space piston is at outer dead point (at extreme right side) and the expansion space piston is at inner dead point close to regenerator. The compression volume is at maximum and the pressure and temperature are at their minimum values represented by point 1 on PV and TS diagrams of Figure
Four processes of the Stirling cycle are [
During compression process from 1 to 2, compression piston moves towards regenerator while the expansion piston remains stationery. The working fluid is compressed in the compression space and the pressure increases from P1 to P2. The temperature is maintained constant due to heat flow from cold space to surrounding. Work is done on the working fluid equal in magnitude to the heat rejected from working gas. There is no change in internal energy and there is a decrease in entropy.
In the process 2-3, both pistons move simultaneously, that is, compression piston towards regenerator and expansion piston away from regenerator, so that the volume between pistons remains constant. The working fluid is transferred from compression volume to expansion volume through porous media regenerator. Temperature of working fluid increased from
In the expansion process 3-4, the expansion piston continues to move away from the regenerator towards outer dead piston while compression piston remains stationery at inner dead point adjacent to regenerator. As the expansion proceeds, the pressure decreases as volume increases. The temperature maintained constant by adding heat to the system from external source at
In the process 4-1, both pistons move simultaneously to transfer working fluid from expansion space to compression space through regenerator at constant volume. During the flow of working fluid through regenerator, the heat is transferred from the working fluid to the regenerator matrix reducing the temperature of working fluid to
The Stirling cycle is highly idealized thermodynamic cycle, which consists of two isothermal and two constant volume processes and the cycle is thermodynamically reversible. The first assumptions of isothermal working and heat exchange imply that the heat exchangers are required to be perfectly effective, and to do so, infinite rate of heat transfer is required between cylinder wall and working fluid. The second assumption requires zero heat transfer between walls and working fluid, both assumptions remain invalid in actual engine operation [
With respect to their mechanical arrangements, Stirling engines are classified into three groups: alpha, beta, and gamma. Each configuration has the same thermodynamic cycle but has different mechanical design characteristics, see Figure
The basic mechanical configurations for Stirling engine.
In the alpha-configuration, two pistons, called the hot and cold pistons, are used on either side of the heater, regenerator, and cooler. In the alpha type of mechanical arrangement, the thermodynamic cycle is performed by means of two pistons working in separate cylinders: one is held at the hot temperature and the other at the cold temperature.
In the beta-configuration, a displacer and a power piston are incorporated in the same cylinder. The displacer moves working fluid between the hot space and the cold space of the cylinder through the heater, regenerator, and cooler. The power piston, located at the cold space of the cylinder, compresses the working fluid when the working fluid is in the cold space and expands the working fluid when the working fluid is moved into the hot space.
The gamma-configuration uses separated cylinders for the displacer and the power pistons, with the power cylinder connected to the displacer cylinder. The displacer moves working fluid between the hot space and the cold space of the displacer cylinder through the heater, regenerator, and cooler. In this configuration, the power piston both compresses and expands the working fluid. The gamma-configuration with double acting piston arrangement has theoretically the highest possible mechanical efficiency. This configuration also shows good self-pressurization. However, the engine cylinder should be designed in vertical type rather than horizontal in order to reduce bushing friction [
In the ideal Stirling cycle, it is assumed that the total heat rejected during process 4-1 is absorbed by the regenerator and then released to the working fluid during the process 2-3. In reality, we cannot find the ideal regenerator and all of the regenerators due to their structure and used materials have deficiency. So, an imperfect regenerator cannot absorb the total heat released during process 4-1, and consequently cannot provide the total required heat of process 2-3. For this study, the temperatures of working fluid at exit of the imperfect regenerator are noted as
Total dead volume is defined as the sum of Stirling engine void volumes. The dead volumes are considered for regenerator, cold and hot pistons. It is evidenced that a real Stirling engine must have some unavoidable dead volume. In normal Stirling engine design practice, the total dead volume is approximately 58% of the total volume. Although many researchers have analyzed Stirling engines, there still remains room for further development. One can use the Schmidt equations to consider dead volumes on his/her analysis. However, ideal regeneration is assumed in the Schmidt analysis [
Stirling engine volumes contributions.
Another important issue which should be considered is the temperature of the regenerator. Correct estimating of regenerator temperature will have direct effects on the final results. In our approach, it is assumed that half of the regenerator dead volume is at
The basic assumptions for the Stirling engine are as follows [ Temperature in each gas space (cold and hot) is known and constant. There is no pressure difference between the gas spaces. Ideal gas law can be used for the working fluid. There is no leakage into or out of the working fluid space.
Effectiveness of a regenerator for hot and cold sides is defined as
For ideal regenerator, the effectiveness is equal to one. To consider the Stirling engines which do not use regenerator, the effectiveness should be set equal to zero. The temperatures at the exit of the regenerator are defined as
If we consider that
As it is discussed previously, estimation of the regenerator effective temperature is important. Three main approaches for estimation of regenerator effective temperature are as follows. Arithmetic mean approach:
Logarithmic mean approach:
Half hot space-half cold space approach:
By substitution of (
It is considered that pistons have simple harmonic movements. Therefore, volumes of pistons are defined as follows:
As it is shown in Figure
The dimensionless total dead volume is presented as follows:
It should be noted that the utilized algorithm is independent from volume variations definitions. However, in this study simple harmonic motion is used.
It is assumed that the working gas is an ideal gas and, therefore, the ideal gas state equation can be used for it. Total mass of the working fluid is sum of hot piston volume, hot piston dead volume, regenerator dead volume, cold piston dead volume, and cold piston volume. Therefore,
Each space follows the ideal gas state equation:
It is assumed that the pressure in these three spaces is equal. Substituting (
As it is mentioned previously, the Stirling cycle consists of four steps: isothermal compression process (based on Figure
Having the volumes and working gas mass, one can calculate the pressure using (
By changing the crank shaft angle from zero to 360 degree, it would be possible to obtain
Regenerator will provide required heat from 2-3′. Therefore, the remaining heat to warm the working gas from 3′-3 should be provided by heater. By considering that the process occurs in constant volume, the heat added during the isochoric heating process 3′-3 is given by
Thermal efficiency of Stirling engine including dead volumes and regenerator deficiency is given by
In what follows, the numerical solution algorithm is presented, see Figure
Values of basic engine parameters.
Hot piston sweep volume = 160.0 cm3 | Cold piston sweep volume = 160.0 cm3 |
Hot piston dead volume = 40 cm3 | Cold piston dead volume = 30.0 cm3 |
Regenerator dead volume = 30.0 cm3 | Total dead volume = 100.0 cm3 |
Regenerator effectiveness = 0.85 | Phase angle difference = 90.0 degree |
Heater temperature = 923 K | Cooler temperature = 300 K |
Average working pressure = 10 MPa | Frequency of engine = 1800 rpm |
Stirling engine performance estimation procedure diagram.
It should be noted that the conditions presented in Table
In Figure
Performance characteristics of the basic engine.
Working fluid | PAVG (MPa) | Net work ( |
Thermal efficiency | Total input heat ( |
---|---|---|---|---|
Helium | 10 | 887.993 | 26.9% | 3300.23 |
Theoretically, in the
Variations of total input heat against hot temperature for different regenerator effectiveness values.
Increase in hot temperature will result in increase of expansion work, the area below the line 3-4 in Figure
Variations of net work against hot temperature.
As it is stated before, increase in hot temperature will result in increase in total input heat and net work. When the difference between hot temperature and cold temperature is low, hot temperature from 400–600 K, in Figure
Variations of thermal efficiency against hot temperature for different regenerator effectiveness values.
In this study, we have fixed engine volumes and average pressure, and then the mass of working fluid is calculated. Because volumes and pressure are constant, net work is independent from working gas type. Therefore, thermal efficiency is just proportional to total input heat. Four different gases are considered, air, helium, hydrogen and nitrogen. As it is presented in Figure
Variations of total input heat against hot temperature for different working gas types.
Variations of thermal efficiency against hot temperature for different working gas types.
Compared to hot temperature effects, increase in cold temperature has reverse effects on total input heat, net work, and thermal efficiency. Although increase in cold temperature will result in decrease in total input heat which is desired, it will reduce net work which is undesired. The variations of total input heat and net work against cold temperature are presented in Figures
Variations of total input heat against cold temperature for different regenerator effectiveness values.
Variations of net work against cold temperature.
Because the decrease in net work is more than decrease amount of total input heat, thermal efficiency decreases with increase in cold temperature. The amount of decrease in higher values of regenerator effectiveness is more which is shown in Figure
Variations of thermal efficiency against cold temperature for different regenerator effectiveness values.
In lower values of cold temperature, helium needs lower amount of total input heat compared to air, hydrogen, and nitrogen, see Figure
Variations of total input heat against cold temperature for different working gas types.
Variations of thermal efficiency against cold temperature for different working gas types.
Effectiveness of regenerator indicates the required heat during process 3′-3 of Figure
Variations of total input heat against regenerator effectiveness for different working gas types.
It should be considered that, even in these cases, thermal efficiency of helium is much more than other considered working gases. By an increase in regenerator effectiveness, thermal efficiency of all working gases leads to a constant value, see Figure
Variations of thermal efficiency against regenerator effectiveness for different working gas types.
In the considered piston arrangement for Stirling engine, the difference angle between hot piston and cold piston is called phase angle difference. In this study, variations of volumes are considered harmonically based on the Simple Harmonic Motion theory [
As it is shown in Figure
Variations of normalized net work, efficiency, and total input heat against difference phase angle.
To find the optimized value for phase change difference, variations of thermal efficiency in different regenerator effectiveness are shown in Figure
Variations of thermal efficiency against difference phase angle for different regenerator effectiveness values.
Because the net work is independent from regenerator effectiveness, the optimum point of maximum net work, as stated in Table
Effects of difference phase angle variations on working parameters.
ALPH | W-Net | Thermal efficiency | Total input heat |
---|---|---|---|
60 | 788.545 | 0.217312 | 3628.62 |
70 | 848.686 | 0.238807 | 3553.86 |
80 | 881.95 | 0.256036 | 3444.63 |
90 | 887.993 | 0.26907 | 3300.23 |
100 | 867.31 | 0.277903 | 3120.91 |
110 | 821.141 | 0.282388 | 2907.84 |
120 | 751.371 | 0.282145 | 2663.07 |
130 | 660.43 | 0.276402 | 2389.38 |
140 | 551.199 | 0.2637 | 2090.25 |
150 | 426.92 | 0.241243 | 1769.67 |
In Figure
Variations of thermal efficiency against difference phase angle for different working gas types.
In Figure
Increase in regenerator effectiveness will reduce required heat during process 3′-3. Moreover, this required heat is linearly proportional to working gas mass. Therefore, an increase in average pressure will increase total input heat. These behaviors are presented in Figure
Variations of total input heat against average pressure for different regenerator effectiveness values.
In constant volumes and temperature conditions, average pressure has a linear relation with mass of working gas. Therefore, an increase in average pressure will lead to an increase in working fluid mass. Total input heat and net work is linearly proportional to mass of working gas. Therefore, increase in average pressure will result in increase of total input heat and net work which are presented in Figures
Variations of total input heat against average pressure for different working gas types.
Variations of net work against average pressure variations.
In Figure
Variations of total input heat against total dead volume ratio.
In Figures
Variations of net work against total dead volume ratio.
Variations of normalized working parameters against total dead volume ratio.
To compare rates of variations, normalized working parameters are shown in Figure
Pressure-Volume diagrams for different average pressure values.
In this paper, comprehensive parametric study is performed on the SOLO 161 Solar Stirling Unit. Hot temperature, cold temperature, regenerator effectiveness, working gas, average working pressure, phase change difference, and dead volume values are considered as variable parameters. Along with each parameter effects, interacting effects of these parameters are presented. As a result, the optimal working conditions and engine characteristics can be summarized as follows An increase in heater temperature will increase thermal efficiency and total input heat. Total input heat is increased linearly while the thermal efficiency leads to a limited value. The rate of increase is much more in engines with higher regenerator effectiveness. Reduction in cooler temperature will increase thermal efficiency and total input heat. In contrast to heater temperature variations, total input heat is increased exponentially. The rate of total input heat increase is much lower in engines with higher regenerator effectiveness. Engines with higher regenerator effectiveness need lower total input heat, and, therefore, have higher thermal efficiency. Among the considered working gases, helium has better working characteristics. Due to higher specific heat coefficient, helium needs lower total input heat. Therefore, engines which work with helium have higher efficiency. Phase angle difference between hot and cold piston shaft has direct effect on the engine performance. For the basic assumed engine, it is observed that around ALPH = 90 degree net work is maximum while the thermal efficiency is maximum in ALPH = 110 degree. Moreover, it is cleared that the optimum ALPH for maximum net work or efficiency is relative to regenerator effectiveness, and working gas type. Net work is linearly proportional to average pressure which increase in average pressure leads to an increase in net work. The rate of net work increase is equal to total input heat which means that thermal efficiency is not relative to average pressure. Consider that temperatures and volumes are held constant during these investigations. It is observed that an increase in dead volume will result in lower net work and higher total input heat, and, therefore, lower thermal efficiency.
Pressure
Volume
Temperature
Entropy
Gas Constant
Mass
Average Pressure
Regenerator Effectiveness
Heat
Work
Phase Angle Difference
Specific Heat at constant Volume
Efficiency.
Related to Hot Side
Related to Cold Side
Related to Regenerator
Hot Piston
Hot Piston Sweep Volume
Hot Piston Dead Volume
Cold Piston
Cold Piston Sweep Volume
Cold Piston Dead Volume
Total Volume
Regenerator Dead Volume
Total Dead Volume.