The ancient Stirling engine technology is more than 200 years old, and although it has significant advantages over internal combustion engines, it is only just beginning to be commercialized. Today, Stirling engines are the first choice in some special areas due to their unique advantages, such as low noise, high efficiency and long-term operation.
However, over the past few decades, despite technological advances, some challenges remain, such as heat transfer, lubrication, sealing, etc. To address these issues, the researchers simulated the thermodynamic and gas-dynamic behavior of the Stirling cycle to improve its understanding and optimize the engine.
This method produces detailed images of location and time, including gas pressure, mass flow, gas and metal temperatures, and more. Currently, researchers are developing a simplified method designed for the preliminary design and optimization of the Stirling engine.
The accuracy of this method has been tested on three well-documented Stirling engines and the results have been found to be fairly accurate. However, these engines are not efficient, as neither their construction nor efficiency can meet the requirements of modern high-performance Stirling engines.
As a result, researchers are investigating how more sophisticated models can be used to simulate high-powered Stirling engines to improve their efficiency.
Based on the engine configuration in the diagram, we can see that the best variables of this system are based on the ideal insulation method. In this system, W stands for work, V for volume, M for mass, T for temperature, Q for heat, and P for pressure.
The subscripts C, K, R, H, and E represent the compression space, the cooler or cold-side heat exchanger, the regenerator, the heater or hot-side heat exchanger, and the expansion space, respectively.
The double subscripts CK, KR, RH, and HE represent the four interfaces between the individual units, respectively. The engine is divided into five sections, corresponding to the compression space, the cooler, the regenerator, the heater, and the expansion space.
Compression and expansion of the space are assumed to be adiabatic. The energy is transferred at the interface between the individual units, through the mass flow rate and upstream temperature to the enthalpy transferred to and from the workspace.
The cooler and heater serve as ideal energy sinks and sources, respectively, that is, the temperature of the working gas in the heat exchanger is considered to be equal to the temperature of the heater and cooler.
Regeneration is considered ideal.
This graph depicts the temperature curve of an ideal adiabatic process. In the flow channel, the temperatures of the cooler, regenerator, and heater are fixed, while the variable volume temperature is derived from the energy equation.
These two equations are the equation of state (PV+MRT) and the differential form of the equation of state (3). CP and CV represent the specific heat capacity of the gas at a constant temperature and volume, respectively, I and O represent the inflow and outflow of gas, and M represents the mass flow rate.
The law of conservation of mass is used to connect these two equations. In this hypothesis, we consider quasi-steady-state flow, which means that the four mass flow variables remain constant in each integration interval with no acceleration effects.
Thus, we can reduce the problem to the solution of a system of seven ordinary differential equations. The easiest way to solve this system of ordinary differential equations is to formulate it as an initial value problem.
In this problem, the initial values of all the variables are known, and then from this initial state, the equations are integrated in a complete cycle, i.e. the crankshaft completes a complete rotation (360 degrees), returning the piston to its initial position.
It is important to note that the ideal static model of the engine is an initial value problem, and a boundary value problem is that we do not know the various initial values. However, by specifying arbitrary initial conditions for the seven variables to be integrated and integrating the equation through several complete cycles, we can reach a cyclic steady state, where the respective values at the beginning and end of the period are equal.
In addition, we can also account for non-ideal effects by extending the second-order formulation, including non-ideal regeneration, non-ideal heat exchangers, heat leakage from the regenerator walls, and pumping losses.
In an ideal case, the gas and wall temperatures are equal, but in a non-ideal case, the heat exchange will cause the gas and wall temperatures to be different. Taking the cooler wall and the heater wall as an example, by connecting the basic equations of heat transfer, we can get the effect of the heat transfer rate and the connection heat transfer coefficient on the temperature.
A non-ideal heat exchange will result in a different average temperature of the cooler and heater gases than the exchanger. Now that we take into account this non-ideal heat exchange situation, we can derive the heat loss from the basic equation of connected heat transfer.
This is known as the effectiveness of the regenerator and can be calculated from equation (5).
This text describes a second-order numerical recipe implemented using the Python open-source scripting language. A block diagram of the simulated process is shown, where GH and GK represent the gas temperatures of the heater and cooler, respectively.
Therefore, the variables TGH and TGK correspond to TH and TK, respectively, and are used to iteratively calculate the new gas temperature. In the initialization phase, the wall temperatures of the heater and cooler TWh and TWK are set to the temperatures of the input heater and cooler TH and TWK, respectively.
Then, by repeatedly performing an ideal adiabatic simulation and calculating a new gas temperature based on the simulation results, the simulation determines the gas temperatures in the heater and cooler, which are TH and TK, respectively.
Once the simulation reaches convergence, i.e., when the newly determined gas temperature is within a certain range of the gas temperature in the previous iteration, the enthalpy and heat leakage losses of the regeneration and the work losses of the pump can be determined.
With the help of a thermal workflow diagram, we can understand that there are two ways in which an engine is calculated. The first method determines engine performance and efficiency by calculating the losses of the regenerator and pump.
The second method achieves the same goal by reducing the losses of the regenerator and pump, but takes into account the heat generated by the friction of the gas during the calculation. Originally manufactured by General Motors Research Laboratories, the GPU-3 Stirling engine was designed with features such as increased clearance distance for pistons, calculated void volume for the regenerator, and inactive heat exchanger volume.
In the course of the simulation calculations, we found that the pressure-volume work in the gas cycle and the resulting pressure fluctuations were overestimated due to the interconnectedness of the various loss mechanisms.
In this case, the output power calculated using the first method was almost 15% and 54% higher than the actual value, while the second method produced better results.
Therefore, the second method is a more realistic way to simulate the average temperature of the expanded and compressed space more accurately.
Although the average temperature of the expansion and compression spaces is slightly less accurate, the overall effect is still satisfactory. Compared to the low-power baseline measurements listed in the table, the simulated expansion and compression space pressure fluctuations are slightly less accurate.
When estimating the heat input from the engine and the heat discharged from the cooler, the error is about the same as the error listed in the table.
This diagram shows that the working fluid is helium with an average pressure of about 42 megapascals, heating tube gas temperature of about 650 degrees Celsius, several variables related to engine speed.
We created these charts to verify the accuracy of the numerical formula as a function of various variables and engine speed.
The graph clearly shows the relationship between power output and efficiency and engine speed. The results show that the simulated output power and efficiency are significantly higher than the actual situation.
Although the actual power output starts to decline as the engine speed increases, the output power under both simulation methods still follows the same rate of decline. However, the simulated efficiency is about the same as the actual situation.
By using the third-order formula, we can see that the simplified momentum equation can be used for simulation until the peak power is reached. Even when the engine speed is well below peak power, we need to use the full momentum equation that includes the momentum flux and acceleration terms.
For helium, peak power occurs in the range of 2500 to 3000 rpm. From this moment on, there is an irregularity in the pressure of the expansion space.
We also found that in the simulation with air as the working fluid, there was a suffocation-type local pressure peak that had a negative impact on the output power.
These phenomena can only be simulated with more complex formulas than second-order simulations, so the simulated output power cannot keep up with the downtrend in the actual situation.
In the graph, you can see the actual heat input of the engine and the simulated gas heat input as a function of the engine speed. While the heat input of the gas will account for the majority of the engine's heat input, it will never exceed the total heat input, suggesting that the gas heat input simulated with alternative methods is more accurate.
Again, this method provides a more accurate determination of the heat output. This paper describes the implementation of a second-order equation for the Stirling engine** and improves it.
The accuracy of this equation has been verified by the real-world performance of the GPU-3 Stirling engine. Our aim was to build a tool that could be used for common engine pre-design and optimization.
In addition to output power and efficiency, our simulations also achieved a high level of overall accuracy with satisfactory results. Our alternative method can produce more accurate results than commonly used methods for estimating various other operating variables.
Although our second-order formula has some limitations, such as the inability to account for the intercorrelation of loss mechanisms, in general, the simulated trends and measurements are consistent.