Wankel engine

The Wankel engine (, ) is a type of internal combustion engine using an eccentric rotary design to convert pressure into rotating motion. The concept was proven by German engineer Felix Wankel, followed by a commercially feasible engine designed by German engineer Hanns-Dieter Paschke. The Wankel engine's rotor is similar in shape to a Reuleaux triangle, with the sides having less curvature. The rotor spins inside a figure-eight-like housing around a fixed gear. The midpoint of the rotor moves in a circle around the output shaft, rotating the shaft via a cam.

In its basic gasoline-fuelled form, the Wankel engine has lower thermal efficiency and higher exhaust emissions relative to the four-stroke reciprocating engine. This thermal inefficiency has restricted the Wankel engine to limited use since its introduction in the 1960s. However, many disadvantages have mainly been overcome over the succeeding decades following the development and production of road-going vehicles. The advantages of compact design, smoothness, lower weight, and fewer parts over reciprocating internal combustion engines make Wankel engines suited for applications such as , auxiliary power units (APUs), loitering munitions, aircraft, personal watercraft, , , Auto racing, and automotive range extenders.

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Concept The Wankel engine is a type of rotary piston engine and exists in two primary forms, the Drehkolbenmotor (DKM, "rotary piston engine"), designed by Felix Wankel, and the Kreiskolbenmotor (KKM, "circuitous piston engine"), designed by Hanns-Dieter Paschke,

(1971). 9780801955914, Chilton. ISBN 9780801955914

of which only the latter has left the prototype stage. Thus, all production Wankel engines are of the KKM type.

• In a DKM engine, there are two rotors: the inner triangular rotor, and the outer rotor, which has a circular outer shape, and an figure-eight inner shape. The center shaft is stationary, and torque is taken from the outer rotor, which is geared to the inner rotor.
  (2025). 9780786486588, McFarland. ISBN 9780786486588
• In a KKM engine, the outer rotor is part of the stationary housing, and is thus not a moving part. The inner shaft is a moving part with an eccentric lobe for the inner rotor to spin around. The rotor spins around the center of the lobe and around the axis of the eccentric shaft in a hula hoop-like fashion, resulting in the rotor making one complete revolution for every three revolutions of the eccentric shaft. Torque is taken from the eccentric shaft,
  (2025). 9780786486588, McFarland. ISBN 9780786486588
  making it a much simpler design to adapt to conventional powertrains.
  (2025). 9780786486588, McFarland. ISBN 9780786486588

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Development Felix Wankel designed a rotary compressor in the 1920s and received his first patent for a rotary type of engine in 1934.

(2025). 9780786486588, McFarland. ISBN 9780786486588

He realized that the triangular rotor of the rotary compressor could have intake and exhaust ports added, producing an internal combustion engine. Eventually, in 1951, Wankel began working at German firm NSU Motorenwerke to design a rotary compressor as a supercharger for NSU's motorcycle engines. Wankel conceived the design of a triangular rotor in the compressor.

(2025). 9780786486588, McFarland. ISBN 9780786486588

With the assistance of Professor from Stuttgart University of Applied Sciences, the concept was defined mathematically.

(2025). 9780786486588, McFarland. ISBN 9780786486588

The supercharger he designed was used for one of NSU's two-stroke single-cylinder engines. The engine produced a power output of at 12,000rpm.

(1975). 387943381X, Motorbuch Verlag Stuttgart. 387943381X

In 1954, NSU agreed to develop a rotary internal combustion engine with Wankel based upon his supercharger design. Since Wankel was known as a "difficult colleague", the development work for the DKM was carried out at Wankel's private Lindau design bureau. According to John B. Hege, Wankel received help from his friend Ernst Höppner, who was a "brilliant engineer".

The first working prototype, DKM 54, first ran on 1 February 1957 at NSU's Versuchsabteilung TX research and development facility. It produced . Soon after that, a second prototype of the DKM was built. It had a working chamber volume V_k of and also produced at 17,000rpm.

(1973). 9783540058861 ISBN 9783540058861

It could even reach speeds of up to 25,000rpm. However, these engine speeds distorted the outer rotor's shape, thus proving impractical.

(2025). 9780786486588, McFarland. ISBN 9780786486588

According to Mazda engineers and historians, four units of the DKM engine were built; the design is described to have a displacement V_h of 250 cm³ (equivalent to a working chamber volume V_k of 125 cc). The fourth unit built is said to have received several design changes, and eventually produced at 17,000 rpm; it could reach speeds up to 22,000 rpm. One of the four engines built has been on static display at the Deutsches Museum Bonn.

(2025). 9783658109011 ISBN 9783658109011

Due to its complicated design with a stationary center shaft, the DKM engine was deemed impractical. Wolf-Dieter Bensinger explicitly mentions that proper engine cooling cannot be achieved in a DKM engine, and argues that this is the reason why the DKM design had to be abandoned.

NSU development chief engineer Walter Froede solved this problem by using Hanns-Dieter Paschke's design and converting the DKM into what would later be known as the KKM. The KKM proved to be a much more practical engine, as it has easily accessible spark plugs, a simpler cooling design, and a conventional power take-off shaft. Wankel disliked Froede's KKM engine because of its inner rotor's eccentric motion, which was not a pure circular motion as Wankel had intended. He remarked that his "race horse" was turned into a "plough horse". Wankel also complained that more stresses would be placed on the KKM's apex seals due to the eccentric motion of the rotor. NSU could not afford to finance developing both the DKM and the KKM, and eventually decided to drop the DKM in favour of the KKM since the latter seemed to be the more practical design.

(2025). 9780786486588, McFarland. ISBN 9780786486588

Wankel obtained US patent 2,988,065 on the KKM engine on 13 June 1961.

(2025). 9780786486588, McFarland. ISBN 9780786486588

Throughout the design phase of the KKM, Froede's engineering team had to solve problems such as repeated bearing seizures, oil flow issues, and cooling issues.

(2025). 9780786486588, McFarland. ISBN 9780786486588

The first fully functioning KKM engine, the KKM 125, weighed in at only , displaced , and produced at 11,000rpm.

(2025). 9780786486588, McFarland. ISBN 9780786486588

Its first run was on 1 July 1958.

(1975). 9780812817195, Stein and Day. ISBN 9780812817195

In 1963, NSU produced the first series-production Wankel engine for a car, the KKM 502. It was used in the NSU Spider sports car, of which about 2,000 were made. Despite its "teething troubles", the KKM 502 was a powerful engine with decent potential, smooth operation, and low noise emissions at high engine speeds. It was a single-rotor peripheral port engine with a displacement of , a rated power of at 6,000rpm and a brake mean effective pressure (BMEP) of .

(1973). 9783540058861 ISBN 9783540058861

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Evolution Felix Wankel managed to overcome most of the problems that interfered with prior attempts to perfect the Wankel engine, by designing the apex seals with a tip radius equal to the amount of "oversize" of the rotor housing shape relative to the theoretical epitrochoid, to minimize radial apex seal motion, and cylindrical gas-loaded apex pins which strengthened the seals.

In the early days, unique, dedicated production machines had to be built for different housing dimensions. However, patented designs such as , G. J. Watt, 1974, for a "Wankel Engine Cylinder Generating Machine", , "Apparatus for machining and/or treatment of trochoidal surfaces" and , "Device for machining trochoidal inner walls", and others, solved such production issues.

Wankel engines have a problem not present in reciprocating piston four-stroke engines in that intake, compression, combustion, and exhaust occur at fixed locations, causing a very uneven thermal load on the rotor housing.

In contrast, four-stroke reciprocating engines perform these four strokes in one chamber, so that the extremes of the cold intake and hot exhaust are averaged out and shielded from working parts by a boundary layer. The University of Florida proposed the use of heat pipes in an air-cooled Wankel to overcome this uneven heating of the housing.SAE paper 2014-01-2160 Pre-heating of certain housing sections with exhaust gas improved performance and fuel economy, also reducing wear and emissions. Boundary layer shields and lubricant film act as thermal insulation, leading to a lower temperature of the film (approximately a maximum of on a water-cooled Wankel engine) and a more constant surface temperature. The temperature around the spark plug is about the same as in the combustion chamber of a reciprocating engine. With circumferential or axial flow cooling, the temperature difference remains tolerable.

Problems arose during research in the 1950s and 1960s as engineers were faced with what they called "chatter marks" and "devil's scratch" in the inner epitrochoid surface, resulting in chipping of the chrome coating on the trochoidal surfaces. They discovered that the cause was the apex seals reaching a resonating vibration, and the problem was solved by reducing the thickness and weight of the apex seals as well as using more suitable materials. Scratches disappeared after introducing more compatible materials for seals and housing coatings. Kenichi Yamamoto experimentally lightened apex seals with holes, identifying weight as the main cause and leading Mazda to use aluminum-impregnated carbon apex seals in their early production engines. NSU used carbon antimony-impregnated apex seals against a chrome housing surface; upon developing an "Elnisil" coating to production maturity, it returned to a metal sealing strip for the Ro 80. Mazda continued to use a chrome surface, but applied to a steel jacket in the aluminum housing. This allowed Mazda to return to the 3mm and later even 2mm thick metal apex seals.Yamamoto, Kenichi (1971). Rotary Engine. Toyo Kogyo. Page 60-61 Another early problem was the build-up of cracks in the stator surface near the plug hole, which was eliminated by installing the spark plugs in a separate conductive copper insert instead of screwing them directly into the block housing.

Toyota found that substituting Glow plug for leading-area spark plugs improved low-RPM partial-load specific fuel consumption by 7%, as well as emissions and idle performance.SAE paper 790435 A later alternative solution to spark plug boss cooling was a variable coolant velocity scheme for water-cooled rotaries, which was patented by Curtiss-Wright and saw widespread use., M. Bentele, C. Jones, F. P. Sollinger, 11/7/61 and , C. Jones, R. E. Mount, 4/29/63 and , C. Jones, 7/27/65 These approaches did not require a copper insert, but did not preclude its use. Ford tested a Wankel engine with the plugs placed in the side plates instead of the housing working surface (, 1978).

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Operation and design The Wankel engine has a spinning eccentric power take-off shaft with an eccentric lobe around which the rotor revolves. The rotor's crown gear has one and a half times the number of teeth as the gear that is fixed to the housing (a 2:3 gear ratio). The rotor and housing constantly form three moving working chambers.

(1973). 9783540058861 ISBN 9783540058861

The rotor does not make contact with its housing, so seals at the rotor's apices press against the housing's periphery to prevent pressure loss. The increase in pressure from combustion pushes against the rotor face, in turn transferring force to the eccentric part of the output shaft.

All practical Wankel engines are Otto cycle (i.e., four-stroke) engines, with each of the three rotor faces undergoing its own intake, compression, expansion, and exhaust cycles. The shape of the rotor between the fixed apices is to minimize the volume of the geometric combustion chamber and maximize the compression ratio, respectively.For a detailed calculation of the curvature of a circular arc approximating the optimal Wankel rotor shape, see In theory, two-cycle engines are possible, but they are impractical because the intake gas and the exhaust gas cannot be properly separated. As the Diesel cycle with its Diesel engine cannot be used in a practical Wankel engine,

Wankel engines typically have a high-voltage spark ignition system.

(1973). 9783540058861 ISBN 9783540058861

Wankel engines have a much lower degree of irregularity relative to a reciprocating engines, leading to much smoother operation. This is because the Wankel engine has a lower moment of inertia and more uniform torque delivery. For example, a two-rotor Wankel engine runs more than twice as smoothly as a four-cylinder piston engine.

The eccentric output shaft of a Wankel engine also lacks the stress-related contours of a reciprocating engine's crankshaft. The maximum engine speed of a Wankel engine is thus mainly limited by load on the synchronizing gears' teeth.Kenichi Yamamoto: Rotary Engine, 1981, 3. 3. 2, Fig. 3.17 page -25- Hardened steel gears are used for extended operation above 7,000 or 8,000rpm. In practice, automotive Wankel engines are not operated at much higher output shaft speeds than reciprocating piston engines of similar output. Wankel engines in auto racing are operated at speeds up to 10,000rpm, but so are four-stroke reciprocating piston engines with relatively small displacement per cylinder. In aircraft, they are used conservatively, reaching 6500 or 7500rpm.

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Torque delivery Wankel engines are capable of high-speed operation, meaning they do not necessarily need to produce high torque to produce high power. The positioning of the intake port and intake port closing greatly affect the engine's torque production. Early closing of the intake port increases low-end torque, but reduces high-end torque (and thus power). In contrast, late closing of the intake port reduces low-end torque while increasing torque at high engine speeds, thus resulting in more power at higher engine speeds.

(1973). 9783540058861 ISBN 9783540058861

A peripheral intake port results in the highest mean effective pressure throughout the RPM range (though moreso at high RPM and particularly if rectangularSAE Paper 950454 Page 7); however, side intake porting produces a more steady idle,Yamamoto, Kenichi. Rotary engine, fig 4.26 & 4.27, Mazda, 1981, p. 46. because it helps to prevent blow-back of burned gases into the intake ducts, which causes a "misfire" that manifests as alternating cycles of successful and unsuccessful mixture ignition. Peripheral porting is also linked to worse partial-load performance. Early work by Toyota led to the addition of a fresh air supply to the exhaust port. It also proved that a reed valve in the intake port or ductSAE paper 720466, Ford 1979 patent improved low-RPM partial-load performance of Wankel engines by preventing blow-back of exhaust gas into the intake at the cost of a slight loss of top-end power. Elasticity is improved with a greater rotor eccentricity, analogous to a longer stroke in a reciprocating engine.

Wankel engines operate better with a low-pressure exhaust system. Higher exhaust back pressure reduces mean effective pressure, especially in peripheral intake port engines. The Mazda RX-8's Renesis engine improved performance by doubling the exhaust port area relative to earlier designs, and there have been studies of the effect of intake and exhaust piping configuration on the performance of Wankel engines.Ming-June Hsieh et al. SAE papers Side intake ports, as used in the Renesis, were first proposed by Hanns-Dieter Paschke in the late 1950s. Paschke predicted that precisely calculated intake ports and intake manifolds could make a side port engine as powerful as a peripheral port engine.

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Materials As formerly described, the Wankel engine is affected by unequal thermal expansion due to the four cycles taking place in fixed places of the engine. While this puts great demands on the materials used, the simplicity of the Wankel makes it easier to use materials such as exotic alloys and . A commonplace method is, for engine housings made of aluminum, to use a spurted molybdenum layer on the engine housing for the combustion chamber area, and a spurted steel layer elsewhere. Engine housings cast from iron can be induction-brazed to make the material suitable for withstanding combustion heat stress.

(1973). 9783540058861 ISBN 9783540058861

Among the alloys cited for Wankel housing use are A-132, Inconel 625, and 356 treated to T6 hardness. Several materials have been used for plating the housing working surface, Nikasil being one. Citroën, Daimler-Benz, Ford, A P Grazen, and others applied for patents in this field. For the apex seals, the choice of materials has evolved along with the experience gained, from carbon alloys, to steel, ferritic stainless, ferrotitanium with carbon, and other materials.

The optimal combination of plating and seal materials was determined experimentally, to obtain the best duration of both the seals and housing. For the shaft, steel alloys with little deformation on load are preferred, such as maraging steel.

Leaded petrol was the predominant type of gasoline available in the first years of the Wankel engine's development. Lead is a solid lubricant, and leaded gasoline is designed to reduce the wearing of seals and housings. Early Wankel engines had an oil supply that only provided lubrication where leaded gasoline was insufficient. As leaded gasoline was being phased out, Wankel engines needed an increased mix of oil in the fuel to provide lubrication to critical engine parts. An SAE paper by David Garside extensively described Norton's choices of materials and cooling fins.

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Sealing Early engine designs had a high incidence of sealing loss, both between the rotor and the housing and also between the various pieces making up the housing. Also, in earlier Wankel engines, carbon particles could become trapped between the seal and the housing, jamming the engine and requiring a partial rebuild. It was common for very early Mazda engines to require rebuilding after . Further sealing problems arose from the uneven thermal distribution within the housing, causing distortion, loss of sealing, loss of compression, and uneven wear between the apex seal and the rotor housing, evident on higher mileage engines. Stressing the engine before it reached operating temperature would exacerbate these problems, which were eventually solved by Mazda. Current engines have nearly 100 seal-related parts.

The problem of clearance for hot rotor apices passing between the axially closer side housings in the cooler intake lobe areas was dealt with by using an axial rotor pilot radially inboard of the oil seals, plus improved inertia oil cooling of the rotor interior (C-W , C. Jones, 5/8/63, , M. Bentele, C. Jones. A.H. Raye. 7/2/62), and slightly "crowned" apex seals (with a different height in the center than the ends).Kenichi Yamamoto, Rotary Engine 1981, Page 50

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Fuel economy and emissions

Early Wankel engines had poor fuel economy due to the Wankel engine's combustion chamber shape and large surface area. The Wankel engine's design is, on the other hand, much less prone to engine knocking, which allows for the use of low-octane rating fuels without reducing compression. NSU tested low octane gasoline at the suggestion of Felix Wankel. On a trial basis, 40-octane gasoline was produced by BV Aral, which was used in the DKM 54 test engine with a compression ratio of 8:1; it ran without complaint. This upset the petrochemical industry in Europe, which had invested considerable sums of money in new plants for the production of higher quality gasoline.Dieter Korp, Protokoll einer Erfindung - Der Wankelmotor, Motorbuch Verlag Stuttgart 1975 p. 77-78'Rotary Engine', Kenichi Yamamoto; Toyo Kogyo, 1971, p. 104K. Yamamoto, T. Muroki, T. KobayakawaSAE Transactions, Vol. 81, SECTION 2: Papers 720197–720445 (1972), pp. 1296-1302 (7 pages) page 1297 test run down to 56 OktanRotary Engine and Fuel Kenichi Yamamoto 8th World Petroleum Congress Moskow 1971, Paper Number: WPC-14403

Direct injection stratified charge engines can be operated with fuels with particularly low octane numbers, such as diesel fuel, which only has an octane number of around 25.SAE Paper 2001-01-1844/4263 Direct injection stratified charge wankel enginesDirect Injection Stratified Charge Rotary Engine Zachary Steven Votaw .A., Wright State University, 2011 p. 6 As a result of worse efficiency, a Wankel engine with peripheral exhaust porting has a larger amount of unburnt Hydrocarbon (HC) released into the exhaust.

(1973). 9783540058861 ISBN 9783540058861

The exhaust is, however, relatively low in NOx (NOx) emissions, because combustion is slow and temperatures are lower than in other engines, and also because of the Wankel engine's good exhaust gas recirculation (EGR) behavior. Carbon monoxide (CO) emissions of Wankel and Otto engines are about the same.

The Wankel engine has a significantly higher (Δt_K>100 K) exhaust gas temperature than a reciprocating Otto engine, especially under low- and medium-load conditions. This is because of the higher combustion frequency and slower combustion. Exhaust gas temperatures can exceed under high load at engine speeds of 6000 rpm. To improve the exhaust gas behavior of the Wankel engine, an exhaust manifold reactor or catalytic converter may be used to reduce hydrocarbon and carbon monoxide emissions.

Mazda uses a dual ignition system with two spark plugs per chamber. This both increases power output and reduces HC emissions. At the same time, HC emissions can be lowered by reducing the pre-ignition of the T leading plug relative to the L trailing plug. This leads to internal afterburning and reduces HC emissions. On the other hand, the same ignition timing of the two plugs leads to higher energy conversion. Hydrocarbons adhering to the combustion chamber wall are expelled into the exhaust at the peripheral outlet.Rotary Engine', Kenichi Yamamoto; Toyo Kogyo, 1971 lower HC emisions with dual ignition with leading and trailing spark plug, p. 104Rotary Engine', Kenichi Yamamoto; Toyo Kogyo, 1971 lower HC emisions with dual ignition with leading and trailing spark plug, Fig.13.9 p. 141- Mazda used 3 spark plugs per chamber in their racing R26B engine. The third spark plug ignites the mixture in the trailing side before the "squish" is generated, causing the mixture to burn completely and also speeding up flame propagation, which improves fuel consumption.Mazda Motor Corp.: Ritsuharu Shimizu, Tomoo Tadokoro, Toru Nakanishi, and Junichi Funamoto Mazda 4-Rotor Rotary Engine for the Le Mans 24-Hour Endurance Race SAE Paper 920309 Page 7

According to Curtiss-Wright research, the factor that controls the amount of unburnt hydrocarbons in the exhaust is the rotor surface temperature, with higher temperatures resulting in fewer hydrocarbons in the exhaust. Curtiss-Wright widened the rotor, keeping the rest of the engine's architecture unchanged, thus reducing friction losses and increasing displacement and power output. The limiting factor for this widening was mechanical, particularly shaft deflection at high engine speeds.SAE paper 710582 Quenching is the dominant source of hydrocarbons at high speeds and leakage at low speeds. Using side porting, which allows the exhaust port to close around top dead centre, reduces intake and exhaust overlap and thus improves fuel consumption.

Mazda's RX-8 with the Renesis engine met the in 2004. This was mainly achieved by using side porting: The exhaust port, which in earlier Mazda Wankel engines was located in the rotor housing, was moved to the side of the combustion chamber. This approach allowed Mazda to eliminate overlap between intake and exhaust port openings while simultaneously increasing the exhaust port area. This design improved combustion stability in the low-speed and light load range, and reduced HC emissions by 35–50% compared to a peripheral exhaust port Wankel engine. However, the RX-8 was not improved to meet Euro 5 emission regulations, and it was discontinued in 2012. The new 8C engine in the Mazda MX-30 R-EV meets the Euro 6d-ISC-FCM emissions standard.

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Chamber volume

In a Wankel engine, the chamber volume $V\_k$ is equivalent to the product of the rotor surface $A\_k$ and the rotor path $s$. The rotor surface $A\_k$ is given by the rotor apices' path across the housing and determined by the generating radius $R$, the rotor width $B$, and the parallel transfers of the rotor and the inner housing $a$. Since the rotor has a trochoid (triangular) shape, the sine of 60 degrees describes the interval at which the rotor apices get closest to the housing. Therefore,

$A\_k=2\; \backslash cdot\; B\; \backslash cdot\; (R+a)\; \backslash cdot\; \backslash sin\; (60^\backslash circ)\; =\; \backslash sqrt\; 3\; \backslash cdot\; B\; \backslash cdot\; (R+a)$

(1973). 9783540058861 ISBN 9783540058861

The rotor path $s$ may be integrated via the eccentricity $e$ as follows:

$\backslash sum\; \backslash ,\; ds=\; \backslash int\_\{\backslash alpha=\; 0^\{\backslash circ\}\}^\{\backslash alpha=270^\{\backslash circ\}\}\; e\; \backslash cdot\; \backslash sin\; \backslash frac\; \{2\}\; 3\; \backslash alpha\; \backslash ,\; d\; \backslash alpha\; =\; 3e$

Therefore,

$V\_k=\; A\_k\; \backslash cdot\; s\; =\; \backslash sqrt\; 3\; \backslash cdot\; B\; \backslash cdot\; (R+a)\; \backslash cdot\; 3e$

(1973). 9783540058861 ISBN 9783540058861

For convenience, $a$ may be omitted because it is difficult to determine and small:

(1981). 9789997341174, Sankaido. ISBN 9789997341174

$V\_k=\; \backslash sqrt\; 3\; \backslash cdot\; B\; \backslash cdot\; R\; \backslash cdot\; 3e$

(1973). 9783034859745 ISBN 9783034859745

(2025). 9783662597149, Springer-Verlag. ISBN 9783662597149

(1971). 9781349011841, Macmillan. ISBN 9781349011841

A different approach to this is introducing $a\text{'}$ as the farthest, and $a$ as the shortest parallel transfer of the rotor and the inner housing and assuming that $R\_1=R+a$ and $R\_2=R+a\text{'}$. Then,

$V\_k=\; \backslash sqrt\; 3\; \backslash cdot\; B\; \backslash cdot\; (2\; \backslash cdot\; R\_1+R\_2)\; \backslash cdot\; e$

Including the parallel transfers of the rotor and the inner housing provides sufficient accuracy for determining chamber volume.

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Equivalent displacement and power output

Different approaches have been used over time to evaluate the total displacement of a Wankel engine in relation to a reciprocating engine, considering only one, two, or all three chambers. Part of this dispute was because of European Road tax being dependent on engine displacement, as reported by Karl Ludvigsen.

If $y$ is the number of chambers considered for each rotor and $i$ the number of rotors, then the total displacement is:

$V\_h=y\; \backslash cdot\; V\_k\; \backslash cdot\; i.$

If $p\_\{me\}$ is the mean effective pressure, $N$ the shaft rotational speed and $n\_c$ the number of shaft revolutions needed to complete a cycle ($N/n\_c$ is the frequency of the thermodynamic cycle), then the total power output is:

$P\; =\; p\_\{me\}\; \backslash cdot\; V\_h\; \backslash cdot\; \{N\; \backslash over\; n\_c\}\; =\; p\_\{me\}\; \backslash cdot\; y\; \backslash cdot\; V\_k\; \backslash cdot\; i\; \backslash cdot\; \{N\; \backslash over\; n\_c\}.$

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One chamber

Kenichi Yamamoto and Walter G. Froede placed $y\; =\; 1$ and $n\_c\; =\; 1$:

(1981). 9789997341174, Sankaido. ISBN 9789997341174

$P\; =\; p\_\{me\}\; \backslash cdot\; 1\; \backslash cdot\; V\_k\; \backslash cdot\; i\; \backslash cdot\; \{N\; \backslash over\; 1\}.$

With these values, a single-rotor Wankel engine produces the same average power as a $V\_h$ single-cylinder two-stroke engine, with the same average torque and the shaft running at the same speed, operating the unitary Otto cycle at triple the frequency.

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Two chambers

Richard Franz Ansdale, Wolf-Dieter Bensinger and Felix Wankel based their analogy on the number of cumulative expansion strokes per shaft revolution. In a Wankel engine, the eccentric shaft must make three full rotations (1080°) per combustion chamber to complete all four phases of a four-stroke engine. Since a Wankel engine has three combustion chambers, all four phases of a four-stroke engine are completed within one full rotation of the eccentric shaft (360°), and one power pulse is produced at each revolution of the shaft.
    This is different from a four-stroke piston engine, which needs to make two full rotations per combustion chamber to complete all four phases of a four-stroke engine. Thus, in a Wankel engine, according to Bensinger, displacement ($V\_h$) is:

(1973). 9783540058861 ISBN 9783540058861

$V\_h\; =\; 2\; V\_k\; \backslash cdot\; i$

If power is to be derived from BMEP, the four-stroke engine formula applies:

$P\; =\; \{p\_\backslash text\{me\}\; \backslash cdot\; V\_\backslash text\{h\}\; \backslash cdot\; \{N\; \backslash over\; 2\}\}$

With this values, a single-rotor Wankel engine produces the same average power as a $V\_h$ two-cylinder four-stroke engine, with the same average torque and the shaft running at the same speed, operating the unitary Otto cycles at 3/2 the frequency.

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Three chambers

Felix Heinrich Wankel in his early patent, Eugen Wilhelm Huber, and Karl-Heinz Küttner counted all the chambers, since each one has its own thermodynamic cycle. So $y\; =\; 3$ and $n\_c\; =\; 3$:, p. 16

(1993). 9783322940407, B. G. Teubner. ISBN 9783322940407

$P\; =\; p\_\{me\}\; \backslash cdot\; 3\; \backslash cdot\; V\_k\; \backslash cdot\; i\; \backslash cdot\; \{N\; \backslash over\; 3\}.$

With these values, a single-rotor Wankel engine produces the same average power as a $V\_h$ three-cylinder four-stroke engine, with 3/2 of the average torque and the shaft running at 2/3 the speed, operating the unitary Otto cycles at the same frequency:

$P\; =\; p\_\{me\}\; \backslash cdot\; 3\; \backslash cdot\; V\_k\; \backslash cdot$

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