Mathematical Modelling Comparison of a Reciprocating, a Szorenyi Rotary, and a Wankel Rotary Engine

Abstract This paper provides an explanation of the geometry, design, and operational principles for the three engines; having special emphasis in the Szorenyi rotary engine which has a deforming rhombus revolving inside a mathematically defined stator. A basic ideal mathematical simulation of those engines were performed, assuming the Otto cycle for the three engines. Also, it assumes the volumetric efficiency of 100%, a wide-open throttle (WOT), no knock nor any mechanical or thermal losses. This simulation focuses on how the fuel burns during combustion, creating pressure and thus, net work. A comparison in pressure traces and cycle performance is made. The study concludes analysing and comparing the ignition advance; finding the best advance for each engine thus the net work between the three engines during one working cycle. Finally, this paper analyses how the different volume change ratio for the combustion chamber of the Szorenyi, Wankel and the reciprocating engine have an effect in the pressure, net work and thermal efficiency generated inside the chamber during combustion for every working cycle.


Introduction
Rotary engines have seen many di erent designs over the years. The most successful of those, the Wankel engine, achieved production, but it had di erent problems such as high hydrocarbon emissions, increased oil and fuel consumption, more expensive manufacturing costs, and limited fuel exibility [1]. For all this reason, they have not been developed as much as reciprocating engines. In 2004, Peter Szorenyi got a patent approved with his partner Peter King of a hinged rotor internal combustion engine. The Szorenyi engine may o er some advantages over the Wankel rotary engine and over the reciprocating engine [2]. This paper explores the potential of the Szorenyi engine compared to the exiting Wankel rotary engine and the reciprocating engine. In consequence, the advantages, disadvantages of the Szorenyi engine against the reciprocating engine and Wankel rotary engine will be determined.
By running a series of basic fuel combustion simulations at di erent advance angles, and same angular speed. Their pressure, volumes, net-work and thermal efciency will be compared under the same working conditions. Also, it is assumed optimal conditions: not taking into account autoignition, thermal, mechanical and friction losses. In consequence, the di erences on how the volume change, will a ect the pressure change ratio and net work achieved from them during the cycle.

Background
Reciprocating engines consist of a piston inside a cylinder to create variable pressure and volume [3]. But there are not the only internal combustion engines. Rotary engines are a middle step between reciprocating engines and turbines. Rotary engines form 'a plurality of variable volume chambers' between a rotor and the outer part known as the stator [4]. Unfortunately, Rotary engines have several disadvantages as higher fuel consumption and emissions. Also, they are a less fuel exible engine, with sealing, lifespan, compression ratio disadvantages and longer ame travel path and other combustion e ects that makes the combustion unstable [5][6][7][8].

. Rotary Engines . . The Wankel Engine
The Wankel engine is a rotary engine where its most successful variety has a triangular shaped rotor. This rotor 'performs a planetary rotary movement relative to the outer body' [4]. This movement is generated using a planetary gearing, at the side plate to maintain the correct phase between the rotor and the eccentric shaft rotations but, it generates unbalanced forces. This forces can be cancelled by just adding balancing weights to balance the rotor [7,8]. It also forces the rotor to revolve at two thirds of the crank's angular velocity, meaning that , it produces three power stroke every two crankshaft revolution [5,[7][8][9]. Finally, breathing is through ports which can be in the stator face or at the sides of the housing [5,9,10].

. . . Wankel Engine Geometry
The Wankel engine has an epitrochoid stator in which the rotor rotates. (1) and (2) de ne the coordinates in X and Y axis of the stator shown in Fig. 1. Also, they were used to determine the volume of one chamber at any given time [7,11,12].

. . The Szorenyi Engine
The Szorenyi engine has a four-segment hinged rotor assembly which deforms and adapts continuously to the stator pro le during its rotation, changing from a square to a rhombus and back. Therefore, the Szorenyi is a rotary engine with four combustion chambers [2,13]. Since, there is no gearing, it has four power strokes per crankshaft revolution. Thus, one rotor of the Szorenyi engine generates the same number of power strokes as a reciprocating 4 stroke engine with eight cylinders [2,13]. Therefore, the Szorenyi engine should have higher power output at lower RPM having a lower mechanical complexity [13].
. . . Szorenyi Geometry The stator curve, patented as the Szorenyi curve, is determined by the right isosceles triangle, as shown in Fig. 2 [6]. It is 'generated by the base extremes A and B of an isosceles right angle, translating and simultaneously rotating triangle' [2]. In the present paper θ in Fig. 1 is the angle used to generate the x and y co-ordinates of the stator pro le. The crankshaft angle of the Szorenyi engine, as used in this paper, is de ned as the angle between the positive vertical axis and the centre of the rotor segment. Using the nota-  (4) de ne all points on the stator curve in Cartesian coordinates [13]. Then is possible to nd the volume at any given time applying the same basic concept than in the Wankel engine volume [7].

Method . Thermal Simulation
For the purposes of the present simulation, the total volume chamber for the three engines was de ned on 125cc.
The volumes on several stages were calculated and veri ed using CAD software. The simulation was executed with a spreadsheet to calculate the chamber volume, fuel fraction burned, pressure and the end-gas temperature at any time. The calculations used general and speci c parameters for each engine in order to have a net work output per chamber cycle of the three engines under equivalent conditions. These parameters are described in Table 1, for the modelling of the engines [9,10,[14][15][16][17][18][19][20]. Due to the limited scope of this paper, it should be noted that the present simulation does not consider pumping, heat, mechanical losses, knock nor their di erent efciencies. Note: The same compression ratio can be achieved using a di erent ω value by removing or adding material to the rotor face. The aim of the present work is not to determine the optimum value 'ω'. Although it will a ect the volume history of the combustion chamber. The scope of this paper does not extend to optimising the 'ω' value. By varying the 'ω' value, the volume history of the chamber could be optimised and this might lead to di erent results for the Szorenyi engine than those presented in the paper.

. Engines Combustion Chamber
For the Szorenyi and Wankel rotary engines, only one chamber was modelled for each engine. The engine modelling determined the volume of the combustion chamber at any time during the power stroke cycle. Then the points of the rotor segment in contact with the stator at any time were determined. After that, the volume between the rotor and stator at any time was calculated. The resulting area was multiplied by the rotor depth and a scale factor to determine the working volume [7]. For the Szorenyi rotary engine, in order to keep the simulation as simple as possible, the value 'ω' shown in (3) and (4) was modi ed until the desired compression ratio was achieved. In order to validate the geometry, the coordinates generated in the mathematical simulation were exported to a CAD software as seen in Fig. 3. Also, Fig. 3 illustrates the concept of calculating the projected area of the working chamber of the Szorenyi rotary engine.
For the Wankel rotary engine, in order to have the desired compression ratio, the rotor surface is further modied by a formation of a calculated rotor pocket, which reduces the volume in the chambers. Thus, increasing the compression ratio until the desired value [17].
As in the Szorenyi engine, careful consideration was taken in the Wankel rotary engine order to determine the area between the rotor and stator at any time as shown in Fig. 4 [7].
Likewise, the cylinder volume at any crank angle position was calculated for the reciprocating engine [10].

. . Fuel Fraction Burned -Wiebe Function
To create an accurate model of the engine which determines the temperature and pressure through the power stroke, it is important to know how much fuel is being burn inside the combustion chamber. The Wiebe function is used to determine this basic parameter as shown in (5) [9,10,[20][21][22][23].
All three engines rotate at the same speed but, their design forces them to have combustion cycles with di erent duration. In order to compare all three engines combustion in an equivalent way, the fuel mass fraction burned 'mfb', which is the fuel burned since the spark was generated, was compared to the percentage of the combustion period as it is shown in Fig. 5. It can be seen that the reciprocating and Wankel engines have a very similar mfb when it is compared to the percentage of combustion. And the Szorenyi engine has a slightly lower curve, which means that the fuel is burned slower at the beginning and faster at the end. This will have an e ect on how the pressure is generated in the combustion chamber.

. Pressure Change Determination
The incremental change in combustion chamber pressure is obtained based on the ideal gas law, the rst law of thermodynamics, Mayer law, ratio of speci c heats, and heat release equation [10,15,24]. The resultant pressure increment can be expressed in the form of (6) [15,23]. .

Work Determination and Thermal E ciency
The resulting data from the simulation, based on the P-V diagram, and assuming the friction value to be zero, makes it possible to determine the net work done per cycle [10]. Heat cannot be fully converted into work [24]. Therefore, thermal e ciency was calculated, in order to indicate how e ective, the engine is in converting heat input into a mechanical work [10].

Results and Discussion
The present research and modelling of combustion processes is based on the parameters presented previously at Table 1. Several simulations, using a Wiebe based thermodynamic model, were run to determine the best advance of ignition. The best advance of ignition is the one which provides the maximum net work for each engine.
The model only accounts for the positive net work generated by the engine. It does not take into account the negative work of intake and exhaust. It assumes a wide-open throttle.   Fig. 6 indicates that the Szorenyi engine has the slowest and better rate of volume change near TDC and the fastest near BDC. A slower rate of change near TDC is closer to the ideal Otto cycle because it brings a steadier volume for combustion. Therefore, a higher fuel mass fraction is burned.

. Net Work
The net peak work output with respect to the ignition timing is shown in Fig. 7. Consequently, it is observed in Fig. 7 the net work of the Szorenyi rotary engine is the greatest per working cycle, followed by the Wankel rotary engine and nally the reciprocating engine as summarised in Table 2. P-V diagrams of the three engines at their optimum ignition advance are shown in Fig. 8. This results in a highest maximum pressure for the Szorenyi engine, followed by the Wankel engine. Both maximum pressure and work are higher than the reciprocating engine.
The enclosed areas of the P-V diagram were computed with the result that the Wankel engine ideally produces 0.10% more net work per cycle than the reciprocating engine. Also, the Szorenyi engine ideally produces 0.462% more work per cycle than the equivalent reciprocating engine.
Furthermore, in Fig. 9, the top pressure of the Wankel engine is 4.58% higher than the reciprocating engine and the Szorenyi engine is 5.515% higher than the same reciprocating engine. This leads to a higher ideal net work per cycle for the Szorenyi engine followed by the Wankel engine and the least net work per cycle is produced by the reciprocating engine.  Table 3.
It can be seen in Table 3 that under the same working conditions, the Wankel engine has a thermal e ciency 0.079% higher than the reciprocating engine. Also, the Szorenyi engine has a thermal e ciency 0.462% higher than the same reciprocating engine.

. Power
Power is a function of time and torque, therefore, for one chamber of each engine, and the three engines rotating at the same speed at the crankshaft, 3000 RPM, will have  di erent chamber variation. Therefore, the ideal power output is stated in Table 4 for each engine at the same crankshaft speed.

Conclusions
As a result of mathematical modelling, it is established that: • The Szorenyi engine is more sensitive to changes in the advance of ignition respect to the crankshaft rotation angle than the reciprocating engine and it has a greater e ect on the net work produced. • The Szorenyi engine has a shorter rotation of the crankshaft during combustion than the reciprocating and Wankel engines. Therefore, it must have a smaller advance of ignition than the reciprocating and Wankel engine with respect to the crankshaft. • The net work per cycle produced by the Wankel engine is 0.079% higher than the reciprocating engine. Also, the Szorenyi rotary engine is 0.462% higher than the reciprocating engine for the same displaced volume. • The Szorenyi engine has the highest net work per cycle, and therefore the highest thermal e ciency, as shown in Table 2 and Table 3. The Szorenyi engine's net work and thermal e ciency is followed by the Wankel engine and, nally, the reciprocating engine has the lowest net work per cycle. • The Szorenyi engine higher power output is 100.90% higher than the reciprocating engine. Also, the Wankel engine is 33.43% higher than the reciprocating engine. These di erences are created by the net work, thermal e ciency and number of power strokes per crankshaft revolution of each engine when all engines are working at the same crankshaft speed.