Equivalent Simulation Study of Delta-Rotor Engine

Abstract

The mechanical structure, movement mode, and combustion process of a triangular rotor engine differ from those of a conventional engine with a crank connecting rod mechanism as the core. This makes it impossible to directly use existing simulation software to simulate the performance of the entire engine, rendering it difficult to conduct structural evaluation and performance prediction during the engine development stage. Therefore, using existing performance simulation software to establish a performance-equivalent model for a triangular rotor engine is crucial. To solve the above problems, this study proposes a performance-equivalent simulation modelling method for a triangular rotor engine based on a three-dimensional simulation model and the operating principles of the triangular rotor and four-stroke engines. Compared to various performance indicators obtained from the three-dimensional simulation model, the results show that the equivalent model established in this study has sufficient accuracy for key indicators such as the cylinder pressure, cylinder temperature, and in-cylinder quality under various working conditions. This can satisfy the requirements of the complete machine-performance simulation of a triangular rotor engine.

1. Introduction

Since the birth of the delta-rotor engine in the last century, its advantages such as high-power density, a simple structure, a small size, and a light weight have attracted the attention of countries worldwide (Kong and Haifeng, 2021) [1]. In the 21st century, with the increasing demand for various types of miniaturised, intelligent, and unmanned vehicles, rotary engines have become the best choice for various unmanned aerial vehicles (UAVs) and cruise missiles (Luo, 2021; Wu, 2022; Wang and Liu, 2023) [2,3,4]. However, owing to market demand, the most commonly used commercial software (2022.1) such as AVL’s FIRE, Ricardo’s WAVE, and GTI’s GT-SUITE do not have direct modelling and simulation capabilities for triangular rotor engines. This is because the mechanical structure, movement mode, intake and exhaust structure, and combustion process of the triangular rotor engine are different from those of conventional engines; therefore, it cannot be directly modelled, and other simulation software attempts for triangular rotor engines cannot meet current research and development needs, owing to their early development time (Li and Sun, 2023) [5].

Currently, two main methods are used to simulate the performance of the delta-rotor engine in the development stage: using special software for modelling a simulation, or treating the delta-rotor engine as equivalent to a conventional engine and modelling a simulation using commercial software. The former usually requires a large amount of experimental data as support for modelling analysis, which leads to poor versatility among different models. Moreover, newly developed software requires large investments, leading to an excessively long development phase (Dark, 1974; Faith, 1976; Hege, 2006) [6,7,8]. The latter has been proven to be able to achieve the purpose of modelling triangular rotor engines through some equivalent substitutions (Widener and Belvoir, 1995; Mitianiec, 2015; Peden, 2017; Wendeker et al., 2012; Tomlinson, 2016) [9,10,11,12,13]. Norman first simulated a rotary engine using non-commercial software (Norman, 1983) [14]. Tartakovsky et al. used GT-SUITE to establish a simulation model of a triangular rotor engine and studied its intake and exhaust processes (Tartakovsky et al., 2012) [15]. Dost et al. used equivalent methods to study the impact of multiple fuel gas mixtures on the combustion process of a rotary engine (Dost et al., 2020) [16]. Peden et al. used AVL BOOST’s module designed for the triangular rotor engine to establish a one-dimensional simulation model and comprehensively studied the impact of injection pressure and injector position on engine performance at different speeds (Peden et al., 2018) [17]. Cihan et al. used AVL BOOST to establish a one-dimensional simulation model and compared the fuel consumption and emission results of the simulation model and test at different speeds (Cihan et al., 2018) [18]. However, the aforementioned studies have shortcomings and do not comprehensively consider the unique movement mode, intake and exhaust structure, and combustion process of rotary engines. The established equivalent models were not compared or calibrated, and key indicators such as the cylinder temperature, cylinder pressure, and in-cylinder mass were not compared to verify the correctness and accuracy of the models. This brings difficulties to the establishment of the equivalent model of the triangular rotor engine, making it difficult to conduct structural evaluation and performance prediction of the engine during the development stage (Georgios, 2005; Danieli et al., 1978) [19,20].

To solve these problems, this study comprehensively considers the movement mode, intake and exhaust structures, and combustion process of a triangular rotor engine. Based on the relevant data of the three-dimensional simulation model verified through experiments, the common one-dimensional engine simulation software GT-SUITE 7.5 was used to establish an equivalent one-dimensional simulation model of the triangular rotor engine. The correctness and accuracy of the one-dimensional simulation model were verified by comparing key indicators such as cylinder temperature, cylinder pressure, and cylinder mass, and the universality of the model at different speeds was validated. This triangular rotor engine equivalent simulation method provides a structural evaluation and performance prediction method for the development and improvement of future rotary engines and provides a reference for the equivalent simulation-related research of other special-shaped engines. In the first and second sections, this study analysed the movement and combustion process of rotary engines and conventional engines based on theoretical and simulation results. Finally, in the third section, an equivalent simulation model was established, and the correctness of the model was verified.

2. Equivalence of Movement Modes

Unlike conventional engines that convert the reciprocating motion of the piston into the rotational motion of the crankshaft, the triangular rotor engine uses the expansion work generated by the combustion of the mixture in the combustion chamber to rotate the triangular rotor and then acts on the eccentric journal of the eccentric shaft to produce tangential force. With the support of the front and rear primary bearings, the main shaft rotates and performs work, thereby generating mechanical energy. The working process of a triangular rotor engine can be divided into four strokes: intake, compression, power, and exhaust. A comparison with a conventional piston engine is shown in Figure 1. The relationship between the internal and external gears of the triangular rotor engine makes its working cycle 1080° CA, and it also has the characteristics of a two-stroke engine. Based on the above analysis, this study treats the delta-rotor engine as an equivalent study to a four-stroke engine.

The movement of the engine is primarily reflected in the change in the cylinder volume with the crankshaft angle. Figure 2 shows a cross-sectional view of a triangular rotor engine; its single cylinder volume $V_1$ can be expressed as

$$V_1=V_f+V_{r1},$$

(1)

where $V_{r1}$ is the combustion–chamber volume with a fixed value. $V_f$ is the cylinder surface volume, which changes with the crankshaft angle and can be expressed as

$$V_f=\left[{\displaystyle \frac{\pi }{3}}+2\sqrt{K^2-9}-{\displaystyle \frac{3\sqrt{3}}{2}}K\mathit{sin}\left({\displaystyle \frac{2}{3}}\alpha +{\displaystyle \frac{\pi }{6}}\right)+\left({\displaystyle \frac{2}{9}}{K}^2+4\right){\mathit{sin}}^{-1}{\displaystyle \frac{3}{K}}\right]e^{2}B,$$

(2)

where B is the cylinder thickness, e is the eccentricity, and α is the crankshaft angle. The shape parameter, K, can be expressed as

$$K={\displaystyle \frac{R}{e}},$$

(3)

where R denotes the radius of creation. Then, we combined Equations (1)–(3) to discover

$$\begin{gathered} V_1=\left[\left({\displaystyle \frac{\pi }{3}}+2\sqrt{K^2-9}+\left({\displaystyle \frac{2}{9}}{K}^2+4\right){\mathit{sin}}^{-1}{\displaystyle \frac{3}{K}}\right)e^{2}B+V_r\right] \\ -{\displaystyle \frac{3\sqrt{3}}{2}}K{e}^{2}B\mathit{sin}\left({\displaystyle \frac{2}{3}}\alpha +{\displaystyle \frac{\pi }{6}}\right) \end{gathered}$$

(4)

where the first term is a constant, and the second term is a sine function of ${\displaystyle \frac{2}{3}}\alpha$. Equation (5) shows the curve of the cylinder volume $V_2$ of a four-stroke engine that changes with the crankshaft angle.

$$\begin{gathered} V_2=\left[\left(1-\mathit{cos}\alpha \right)+{\displaystyle \frac{r}{4l_1}}\left(1-\mathit{cos}2\alpha \right)\right]\pi r{\sigma }^2 \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+l_2\pi {\sigma }^2+V_{r2}, \end{gathered}$$

(5)

where r is the crank radius, $l_1$ is the connecting rod length, σ is the piston radius, $V_{r2}$ is the combustion chamber volume, and $l_2$ is the distance between the piston and the cylinder head when the piston moves to the top dead centre. We combined Equations (4) and (5) and converted the two motion periods to 720° CA to ensure the inner volume of the two cylinders were equal at all times:

$$\begin{gathered} \left({\displaystyle \frac{\pi }{3}}+2\sqrt{K^2-9}+\left({\displaystyle \frac{2}{9}}{K}^2+4\right){\mathit{sin}}^{-1}{\displaystyle \frac{3}{K}}\right)e^{2}B+V_{r1} \\ =\pi r{\sigma }^2+l_2\pi {\sigma }^2+V_{r2} \\ {\displaystyle \frac{3\sqrt{3}}{2}}K{e}^{2}B=\pi r{\sigma }^2 \\ \hspace{1em}\hspace{1em}{l}_1\to +\infty . \end{gathered}$$

(6)

In addition to the cylinder volume of the triangular rotor engine, the cylinder surface area is an important parameter that has a significant influence on the performance of the engine, such as heat dissipation loss and stress. The change in cylinder surface area $S_1$ with the crankshaft angle can be expressed as

$$\begin{gathered} S_1=2F+BL+B{L}^{\prime }+S_{r1} \\ F=\left[{\displaystyle \frac{\pi }{3}}+2\sqrt{K^2-9}-{\displaystyle \frac{3\sqrt{3}}{2}}K\mathit{sin}\left({\displaystyle \frac{2}{3}}\alpha +{\displaystyle \frac{\pi }{6}}\right)+\left({\displaystyle \frac{2}{9}}{K}^2+4\right){\mathit{sin}}^{-1}{\displaystyle \frac{3}{K}}\right]e^2 \\ L={\displaystyle \frac{\sqrt{3}\left[{\left(R+a\right)}^2-2\left(R+a\right)+4e^2\right]\left(R+a\right)\left(R+a-4e\right)}{2\left(R+a-4e\right)\left[{\left(R+a\right)}^2+4e^2-2\left(R+a-e\right)\right]}} \\ {L}^{\prime }=e\left(K+3\right){\int }_0^{{\displaystyle \frac{\alpha }{3}}}\left(\sqrt{1-k^2{\mathit{sin}}^2{\displaystyle \frac{\alpha }{3}}}\right)d{\displaystyle \frac{\alpha }{3}}+{\displaystyle \frac{3\alpha }{2}}{\int }_0^{{\displaystyle \frac{2\alpha }{3}}}\left({\displaystyle \frac{\frac{1}{27}{K}^2+1+\frac{4}{9}K\mathit{cos}\frac{2\alpha }{3}}{\frac{1}{9}{K}^2+1+\frac{2}{3}K\mathit{cos}\frac{2\alpha }{3}}}\right)d{\displaystyle \frac{2\alpha }{3}} \\ k={\displaystyle \frac{2\sqrt{K}}{\sqrt{3}{\left(\frac{1}{3}K+1\right)}^3}}, \end{gathered}$$

(7)

where $S_{r1}$ is the increased in-cylinder surface area after the addition of the combustion chamber and $a$ is the translation distance.

Equation (8) shows the curve of the change in the cylinder surface area $S_2$ of a four-stroke engine with the crankshaft angle.

$$S_2=2\pi {\sigma }^2+2\pi \sigma \left[\left(1-\mathit{cos}\alpha \right)+{\displaystyle \frac{r}{4l_1}}\left(1-\mathit{cos}2\alpha \right)\right]+{2\pi \sigma l}_2+S_{r2},$$

(8)

where $S_{r1}$ is the increased in-cylinder surface area after the addition of the combustion chamber. To simplify the calculation, the equivalent four-stroke engine combustion chamber is assumed to be a standard cylinder:

$$\begin{gathered} V_{r2}=l_3\pi {\rho }^2 \\ \hspace{1em}\hspace{1em}\hspace{1em}{S}_{r2}=2l_3\pi \rho , \end{gathered}$$

(9)

where $l_3$ is the height of the combustion chamber and ρ is the radius of the combustion chamber. To ensure that the inner surface areas of the two cylinders are equal at all times, Equation (7) must be equal to Equation (8); that is, $S_1=S_2$. This equation is transcendental. In Equations (6)–(9), there are five unknown quantities (crank radius r, piston radius σ, distance $l_2$ from the cylinder head when the piston moves to top dead centre, combustion chamber height $l_3$, and combustion chamber radius ρ). Taking Equation (6) as the constraint condition and Equation (10) as the fitness function Y, we use the NSGA-II algorithm to solve r, σ, and $l_2$. The NSGA-II is an optimisation method based on linear weighting and Pareto ranking that can effectively solve various multi-objective optimal solution problems (Yin et al., 2024; Liu et al., 2023; Wang et al., 2023) [21,22,23].

$$\begin{gathered} Y=0.5\times \left(S_1^0-S_2^0\right)+0.5\times {\displaystyle \frac{S_1^{\frac{3\pi }{2}}}{S_1^0}}\times \left(S_1^{{\displaystyle \frac{3\pi }{2}}}-S_2^{\pi }\right) \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} +0.1\times {\displaystyle \frac{S_1^{\frac{3\pi }{2}}}{2S_1^0}}\times {\displaystyle \frac{\sum _{i=1}^{179}\left(S_1^{\frac{i\pi }{120}}-S_2^{\frac{i\pi }{180}}\right)}{178}}, \end{gathered}$$

(10)

where $S_1^n$ and $S_2^n$, respectively, represent the in-cylinder area of the triangular rotor engine and the four-stroke engine at the equivalent working time when the crankshaft angle is n, respectively. By using the fitness function of Equation (10), the changes in the cylinder area at other angles can be made similar to those of the triangular rotor engine while ensuring that the maximum and minimum cylinder areas of the equivalent four-stroke engine are as equal as possible to those of the triangular rotor engine. Through the above calculation method, the equivalent basic structural parameters of the four-stroke engine can be obtained.

3. Combustion Process Equivalence

In conventional engine simulation modelling, to simplify the combustion model, the EngCylCylCombDIWiebe (direct injection diesel engine Wiebe model), or the EngCylCombSIWiebe (spark plug Wiebe combustion model), the Wiebe probability density function is often used to define the in-cylinder combustion process. The former is mainly used for simulating the combustion heat release rate of direct injection compression ignition engines, while the latter is mainly used for simulating the combustion heat release rate of intake of premixed ignition engines. This combustion model assumes that the mixed gas in the cylinder is burned as a percentage and that its heat release rate generally behaves as a uniform change curve, as shown in Figure 3a. However, triangular rotor engines are limited by their combustion chamber structure, and it is difficult to achieve uniform combustion. Figure 4 shows the flame distribution cloud diagram of the cylinder for different eccentric shaft rotation angles (Ye Y et al., 2020) [24]. It can be seen from the figure that after ignition starts, the fire core was formed from the ignition cavity and continued to develop into the combustion chamber, eventually forming a stable combustion flame surface. The flame spread slowly during the early development of the fire core. As the flame spread area increased, the flame spread speed also increased significantly. Because of the unidirectional flow field in the cylinder, it was difficult for the flame to propagate to the rear end of the combustion chamber, which resulted in an unburned area at the rear end. This combustion mode generated the typical heat release rate curve of a triangular rotor engine, as shown in Figure 3b. The heat release rate curve shows a decreasing trend at the end of combustion. The difference in the combustion process is the main reason for the inconsistency between the equivalent model and actual engine performance parameters, such as the cylinder temperature. To improve the simulation accuracy, the in-cylinder working process of a rotary engine was equated to two uniform combustions instead of one combustion. The heat release rate distributions of the two combustion processes were based on the three-parameter Weibull function shown in Equation (11). In the equation, $p_n$ is the shape parameter of the nth combustion, which determines the offset direction of the heat release rate distribution; ${\gamma }_n$ is the position parameter of the nth combustion, which determines the centre position of the heat release rate distribution; ${\beta }_n$ is the scale parameter of the nth combustion, which determines the steepness of the heat release rate distribution. The two combustions were independent of each other, and the total heat release rate in the cylinder was the sum of the heat release rates of the two combustions. Figure 5 shows the combustion heat release rate used in this study. It can be seen from the figure that the heat release rate curve is consistent with the common heat release rate curve trend of the rotary engine shown in Figure 3b. Suppose that the total heat releases of the two combustions are ${HR}_1$ and ${HR}_2$. Combined with relevant research and analysis, the ${\displaystyle \frac{{HR}_1}{{HR}_2}}$ of the triangular rotor engine under different working conditions is between 6.5 and 7.5, and the difference between the centre positions of the heat release rate distribution of the two combustions is between 65° CA and 90° CA. Figure 6 shows the calculation flowchart of equivalent heat release rate.

$$f_n\left(\alpha \mathrm{,}{p}_n\mathrm{,}{\beta }_n\mathrm{,}{\gamma }_n\right)={\displaystyle \frac{p_n}{{\beta }_n}}{\left({\displaystyle \frac{\alpha -{\gamma }_n}{{\beta }_n}}\right)}^{\left(p_n-1\right)}{e}^{-{\left({\displaystyle \frac{\alpha -{\gamma }_n}{{\beta }_n}}\right)}^{p_n}}n=\mathrm{1,2}.$$

(11)

To simplify the calculation, an SG (Savitzky–Golay) filter was first used to process and obtain a smoother heat release rate curve. The SG filter is a filtering algorithm based on the local polynomial least squares method, often used to smooth and denoise data. The greatest advantage is that it ensures that the shape of the data remains unchanged. The filter function is expressed by Equation (12), where P represents the original data, $P_j^{\prime }$ represents the filtered data, C is the SG filter coefficient, and N is the data length.

$$\begin{gathered} P_j^{\prime }={\displaystyle \frac{\sum _{i=-m}^{i=m}{C}_i{P}_{j+i}}{N}} \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}N=2m+1. \end{gathered}$$

(12)

After determining the crankshaft angle $T_{MAX}$ with the maximum heat release rate, the equivalent heat release rate curve for $T\le T_{MAX}$ was obtained first. The fitness function $Y_1$ at this stage is given as in Equation (13), where $T_1$ is the ignition time. After genetic algorithm optimisation, the best $p_1$, ${\beta }_1$, and ${\gamma }_1$ can be obtained.

$$Y_1={\displaystyle \frac{{\sum }_{i=T_1}^{i=T_{MAX}}{f}_1\left(i\mathrm{,}{p}_1\mathrm{,}{\beta }_1\mathrm{,}{\gamma }_1\right)-P_i^{\prime }}{T_{MAX}-T_1}}.$$

(13)

After determining the relevant parameters of the first combustion, the heat release rate curve of $T\ge T_{MAX}$ is considered to be equivalent. The fitness function $Y_2$ at this stage is as shown in Equation (14), where $T_2$ is the end time of the combustion. After optimisation by genetic algorithm, the best $p_2$, ${\beta }_2$, and ${\gamma }_2$ can be obtained. The relevant parameters of the equivalent heat release rate function can be determined using the aforementioned calculations (Zhang et al., 2023) [25].

$$Y_2={\displaystyle \frac{{\sum }_{i=T_{MAX}}^{i=T_2}{f}_1\left(i\mathrm{,}{p}_1\mathrm{,}{\beta }_1\mathrm{,}{\gamma }_1\right)+f_2\left(i\mathrm{,}{p}_2\mathrm{,}{\beta }_2\mathrm{,}{\gamma }_2\right)-P_i^{\prime }}{T_2-T_{MAX}}}.$$

(14)

4. Establishment and Verification of Equivalent Model

This study used MATLAB R2016b to build various mathematical models, GT-SUITE V7.5 to build equivalent simulation models, and CONVERGE 3.0 to build three-dimensional simulation models. The running environment of these software were Win10, and the workstation configuration was as follows: Intel C621 series chipset; Intel Xeon Gold 6145 processor, 40 cores and 80 threads, a main frequency of 2.0 GHz, a maximum turbo frequency of 3.7 GHz; 128 GB of memory, NVIDIA GT1030, and two GB graphics card.

In this study, a single-cylinder gasoline rotary engine was used for research and verification. The structural parameters of the rotary engine are listed in

Table 1. Rotor engine structure parameters.

Parameter	Value
Creation radius (mm)	60
Eccentricity (mm)	10
Rotor width (mm)	42
Translational distance (mm)	1.45
Speed (r/min)	8000
Single cylinder volume (cc)	133
Compression ratio	10
Ignition advance angle	30° BTDC
Ignition source	Single spark plug
Fuel type	Gasoline
Jet strategy	Premixed intake duct
Air–fuel ratio	1
Ignition diameter (mm)	8
Ignition energy (J)	0.08
Engine power (kw)	12.3
Torque (N.m)	15.66
Oil consumption (g/kw.h)	402.3

. This is a small rotary engine that uses gasoline as fuel, premixed with an intake port and ignited by a single spark plug. It is generally used as a power unit for long endurance unmanned aerial vehicles and cruise missiles, with an output power of 12.3 kw and a working speed of 8000 revolutions per minute. To verify the correctness and accuracy of the equivalent one-dimensional simulation model proposed in this study, CONVERGE was used to establish a three-dimensional simulation model in combination with the relevant boundary conditions, and the curves of the cylinder volume, temperature, pressure, and mass were extracted to verify the equivalent one-dimensional simulation model. The solid and computational fluid domain models established based on these parameters are shown in Figure 7. Based on the previous two sections, the equivalent simulation model established in combination with engine-related parameters is shown in Figure 8 (Lai et al., 2023; Liu et al., 2024; Xu et al., 2021) [26,27,28].

To verify the correctness and accuracy of the one-dimensional simulation model, we compared and analysed the in-cylinder volume, pressure, temperature, and mass curves of the one-dimensional and three-dimensional simulation models. The results are shown in Figure 9, Figure 10, Figure 11 and Figure 12, respectively, wherein Figure 9a, Figure 10a, Figure 11a and Figure 12a show comparative schematics of the cylinder volume, pressure, temperature, and mass curves of the equivalent engine (one-dimensional simulation model) and rotor engine (three-dimensional simulation model), respectively. Figure 9b, Figure 10b, Figure 11b and Figure 12b show the deviation values of the cylinder volume, pressure, temperature, and mass curves of the equivalent engine (one-dimensional simulation model) and the rotary engine (three-dimensional simulation model), respectively; Figure 9c, Figure 10c, Figure 11c and Figure 12c present the relative errors of the cylinder volume, pressure, temperature, and mass curves of the equivalent engine (one-dimensional simulation model) and the rotary engine (three-dimensional simulation model), respectively (Yang and He, 2015; Xu et al., 2021) [28,29]. To calculate the deviation value and percentage, the working cycle of the 3D simulation model was equivalent from 1080° to 720°, and the calculation was performed after difference processing. Figure 13 shows a comparison of the P–V plots of the equivalent engine (one-dimensional simulation model) and rotary engine (three-dimensional simulation model).

To further verify the accuracy of the one-dimensional equivalent simulation model established in this study, the cylinder volume, pressure, and temperature curves of the one-dimensional equivalent simulation model and three-dimensional simulation model as well as the average deviation, maximum deviation, maximum relative error, and average relative error of the main working stages of the in-cylinder mass curve from 5° CA before ignition to exhaust port opening (−35° CA to 135° CA) were calculated. The results are summarised in

Table 2. Deviation in the range of −35° CA to 135° CA.

Index	Volume Inside the Cylinder/m3	Cylinder Pressure/MPa	Cylinder Temperature/K	In-Cylinder Mass/kg
Maximum Deviation	7.0551 × 10−7	0.1503	68.3762	1.8849 × 10−6
Average Deviation	3.8096 × 10−7	0.0736	26.6056	1.3210 × 10−6
Maximum Relative Error	1.1398%	9.0314%	9.7714%	1.0964%
Average Relative Error	0.6037%	5.2542%	2.6841%	0.7708%

; the error of the equivalent model mainly occurs in the intake and exhaust stages (135° CA to 540° CA), the deviation in the main working stage (−35° CA to 135° CA) is small, and the maximum relative error is below 10%. The maximum deviation of the in-cylinder pressure is 0.1503 MPa (9.0314%), which occurs at −35° CA. the maximum deviation of the in-cylinder temperature is 68.3762 K (9.7714%), which occurs at −35° CA. The deviation of the maximum pressure in the cylinder is 0.0136 MPa (0.4572%), and the deviation of the crankshaft angle at the maximum pressure in the cylinder is 1.4004° CA. The average relative errors of the cylinder volume, cylinder pressure, cylinder temperature, and cylinder mass are less than 6%. The deviation of the maximum temperature in the cylinder was 2.8786 K (0.1381%), and the deviation of the crankshaft angle at the maximum temperature in the cylinder was 2.0058° CA. Based on these values, it can be concluded that the equivalent simulation model of the triangular rotor engine established in this study has high accuracy.

To verify the universality of the one-dimensional equivalent simulation model established in this study, the in-cylinder pressure and temperature of the one- and three-dimensional equivalent simulation models at 7500 r/min and 7000 r/min were compared. Figure 14 and Figure 15 show a comparison of the in-cylinder pressure and temperature curves of the equivalent engine (one-dimensional simulation model) and rotary engine (three-dimensional simulation model) at 7500 r/min and 7000 r/min speeds, respectively.

Table 3. Deviation in the range of −35 °CA to 135 °CA at different speeds.

Crank Speed	7500 r/min		7000 r/min
Index	Cylinder Pressure	Cylinder Temperature	Cylinder Pressure	Cylinder Temperature
Maximum relative error	9.4687%	10.2456%	9.8495%	10.0643%
Average relative error	5.6325%	2.8594%	5.8749%	3.1859%

lists the in-cylinder pressure of the one- and three-dimensional equivalent simulation models and the average deviation, maximum deviation, maximum relative error, and average relative error of the in-cylinder temperature curve from 5° CA before ignition to the opening of the exhaust port (−35° CA to 135° CA) in the main working stages. The equivalent simulation model of the triangular rotor engine established in this study has high accuracy under different working conditions and can satisfy the requirements of the performance simulation of the entire triangular rotor engine.

5. Conclusions

To address the problem of directly simulating the entire machine performance of a triangular rotor engine through simulation software, in this study, the mechanical structure, motion mode, and combustion process of the triangular rotor engine were investigated, and its motion mode and combustion process were equated using GT-SUITE to establish an equivalent one-dimensional simulation model. The key indicators of the triangular rotor engine, such as cylinder volume, cylinder temperature, cylinder pressure, and cylinder quality, were compared. The main conclusions of this study are as follows:

1. To equate the motion mode of the triangular rotor engine, this study used a genetic algorithm to simultaneously equate the changes in the cylinder volume and cylinder surface area during the motion of the triangular rotor engine. To ensure that the cylinder volume change of the equivalent four-stroke engine was consistent with that of the triangular rotor engine, the change in the surface area of the cylinder was considered similar to that of the triangular rotor engine. This ensured that the movement pattern, in-cylinder temperature, and in-cylinder mass of the equivalent engine were consistent with those of a triangular rotor engine;

2. For the equivalent combustion process of a triangular rotor engine, this study equated the in-cylinder working process of the rotary engine with two uniform combustions instead of one. The heat release rate distributions of the two combustion processes were based on two different three-parameter Wiebe functions, and a genetic algorithm was used to obtain piecewise equivalents for the two functions. This ensured that the in-cylinder pressure, in-cylinder temperature, and P–V plots of the equivalent engine were consistent with those of the triangular rotor engine;

3. To verify the correctness and accuracy of the equivalent simulation model established in this study, the in-cylinder volume, pressure, temperature, and mass curves obtained using a three-dimensional simulation model and a one-dimensional equivalent simulation model at rated speeds were compared, and the deviation in the main working process (−35° CA to 135° CA) was calculated. The results show that the deviation of the equivalent simulation model established in this study mainly occurs in the intake and exhaust stages (135° CA to 540° CA), the deviation in the main working stage (−35° CA to 135° CA) is small, the maximum relative error is below 10%, and the average relative error is below 6%; the deviation of the maximum pressure in the cylinder was 0.4572%, and the angle deviation was 1.4004° CA; the deviation of the maximum in-cylinder temperature was 0.1381%, and the angle deviation was 2.0058° CA. Moreover, the deviations in in-cylinder pressure and temperature at different speeds were minimal, which proves the correctness and accuracy of the equivalent simulation model established in this study. Thus, the equivalent simulation model can meet the needs of the performance simulation of a rotary engine. In subsequent research, we will expand the scope of research, conduct equivalent research on rotary engines with different structures and different fuels, and enhance the universality of the equivalent research method of triangular rotary engines.