Editing Wankel engine (section)

==Operation and design==
[[File:Wankel engine diagram.svg|249px|left|thumb|'''Figure 7.'''<br/>Schematic of the Wankel: {{ordered list
 |list_style=margin:0;
 |item_style=list-style-position:inside;
 |Intake
 |Exhaust
 |Stator housing
 |Chambers
 |Pinion
 |Rotor
 |Crown gear
 |Eccentric shaft
 |Spark plug
}}]]
[[File:Wankel Cycle (vector).svg|right|thumb|249px|'''Figure 8.'''<br/>The Rotary Cycle:{{ordered list
 |list_style=margin:0;
 |item_style=list-style-position:inside;
 |Intake (blue)
 |Compression (green)
 |Ignition (red)
 |Exhaust (yellow)
}}]]
[[File:Wankel3.ogv|left|258px|'''Figure 9.'''<br/>Video of a two rotor Wankel engine]]
[[File:Comparison between Wankel and reciprocating engines.svg|right|thumb|700px|'''Figure 10.'''<br/>Comparison between Wankel and reciprocating engines.<ref name="Yamamoto 1971 p. 109">{{cite book |last1=Yamamoto |first1=Kenichi |title=Rotary Engine |publisher=Toyo Kogyo |year=1971 |page=109 |quote=Fig 10.4}}</ref>
<span style="background-color: #AAF;">&nbsp;Intake&nbsp;</span><span style="background-color: #33D;color: white;">&nbsp;Compression&nbsp;</span><span style="background-color: #F00;">&nbsp;Expansion&nbsp;</span><span style="background-color: #500;color: white;">&nbsp;Exhaust&nbsp;</span>]]
[[File:Comparison between idealized Wankel and reciprocating engines.svg|right|thumb|900px|'''Figure 11.'''<br/>Each engine produces an average total power of 76.3&nbsp;kW at p<sub>mi</sub> = 11.1 bar and [[mean effective pressure|p<sub>me</sub>]] = 8.88 bar.
Chambers pressure, instantaneous unitary torque, instantaneous and average total torque plotted against shaft rotation angle. Instantaneous and average total power plotted against time.<ref name="Yamamoto 1971 p. 86">{{cite book |last1=Yamamoto |first1=Kenichi |title=Rotary Engine |publisher=Toyo Kogyo |year=1971 |page=86 |quote=Fig 7.7}}</ref>]]

The Wankel engine has a spinning eccentric power take-off shaft, with a rotary piston riding on [[eccentric (mechanism)|eccentrics]] on the shaft in a hula-hoop fashion, with the crown gear with one and a half times the number of teeth as on the eccentric shaft. Thus the Wankel is a 2:3 type of rotary engine, i.e., its housing's inner side resembles a two lobes oval-like [[epitrochoid]] (equivalent to a peritrochoid),.<ref name="Nash 1977">{{Citation|last1=Nash|first1=David H.|title=Rotary Engine Geometry|journal=Mathematics Magazine|volume=50|issue=2|pages=87–89|publisher=Taylor & Francis|date=1977-03-02|url= https://www.jstor.org/stable/2689731|doi=10.1080/0025570X.1977.11976621|jstor=2689731 |url-access=subscription}}</ref> In contrast, its rotary piston has a three vertices trochoid shape (similar to a [[Reuleaux triangle]]). Thus, the Wankel engine's rotor constantly forms three moving working chambers.<ref name="Bensinger 1973 p. 55">{{cite book |last1=Bensinger |first1=Wolf-Dieter |title=Rotationskolben-Verbrennungsmotoren |place=Berlin, Heidelberg, New York |date=1973 |isbn=978-3-540-05886-1 |oclc=251737493 |language=de |page=55}}</ref> The Wankel engine's basic geometry is depicted in figure 7. Seals at the rotor's apices seal against the housing's periphery.<ref name="Columbia"/> The rotor moves in its rotating motion guided by gears and the eccentric output shaft, not being guided by the external chamber. The rotor does not make contact with the external engine housing. The force of expanded gas pressure on the rotor exerts pressure on the center of the eccentric part of the output shaft.

All practical Wankel engines are four-cycle (i.e., four-stroke) engines. In theory, two-cycle engines are possible, but they are impractical because the intake gas and the exhaust gas cannot be properly separated.<ref name="Bensinger 1973 p. 54"/> The operating principle is similar to the Otto operating principle; the Diesel operating principle with its [[Compression-ignition engine|compression ignition]] cannot be used in a practical Wankel engine.<ref name="Bensinger 1973 p. 86">{{cite book |last1=Bensinger |first1=Wolf-Dieter |title=Rotationskolben-Verbrennungsmotoren |place=Berlin, Heidelberg, New York |date=1973 |isbn=978-3-540-05886-1 |oclc=251737493 |language=de |page=86}}</ref> Therefore, Wankel engines typically have a high-voltage [[Spark-ignition engine|spark ignition]] system.<ref name="Bensinger 1973 p. 124">{{cite book |last1=Bensinger |first1=Wolf-Dieter |title=Rotationskolben-Verbrennungsmotoren |place=Berlin, Heidelberg, New York |date=1973 |isbn=978-3-540-05886-1 |oclc=251737493 |language=de |page=124}}</ref>

In a Wankel engine, one side of the triangular rotor completes the four-stage [[Otto cycle]] of intake, compression, expansion, and exhaust each revolution of the rotor (equivalent to three shaft revolutions, see Figure 8.).<ref name="oGY5n">{{cite web|url= http://www.der-wankelmotor.de/Techniklexikon/techniklexikon.html |title=Techniklexikon|first1=Html; Kurt|last1=Heintz |website=der-wankelmotor.de}}</ref> The shape of the rotor between the fixed apexes is to minimize the volume of the geometric [[combustion chamber]] and maximize the [[compression ratio]], respectively.<ref name="Columbia">{{cite web|title=Internal-combustion engine |publisher=Columbia Electronic Encyclopedia |year=2008|url= http://www.encyclopedia.com/topic/internal-combustion_engine.aspx#1 |access-date=2011-01-04}}</ref><ref name="61ckr">For a detailed calculation of the curvature of a circular arc approximating the optimal Wankel rotor shape, see {{cite journal |last1=Badr |first1= O. |last2= Naik |first2= S. |last3=O'Callaghan |first3=P.W. |last4= Probert |first4=S.D. |doi=10.1016/0306-2619(91)90063-4 |issue=1 |journal=Applied Energy |pages=59–76 |title= Rotary Wankel engines as expansion devices in steam Rankine-cycle engines |volume=39 |year=1991|bibcode= 1991ApEn...39...59B }}</ref> As the rotor has three sides, this gives three power pulses per revolution of the rotor.

Wankel engines have a much lower degree of irregularity relative to a reciprocating piston engine, making the Wankel engine run much smoother. This is because the Wankel engine has a lower moment of inertia and less excess torque area due to its more uniform torque delivery. For example, a two-rotor Wankel engine runs more than twice as smoothly as a four-cylinder piston engine.<ref name="Bensinger 1973 p. 72">{{cite book |last1=Bensinger |first1=Wolf-Dieter |title=Rotationskolben-Verbrennungsmotoren |place=Berlin, Heidelberg, New York |date=1973 |isbn=978-3-540-05886-1 |oclc=251737493 |language=de |page=72}}</ref> The eccentric output shaft of a Wankel engine also does not have the stress-related contours of a reciprocating piston engine's crankshaft. The maximum revolutions of a Wankel engine are thus mainly limited by tooth load on the synchronizing gears.<ref name="YLcmp">Kenichi Yamamoto: Rotary Engine, 1981, 3. 3. 2, Fig. 3.17 page -25-</ref> Hardened steel gears are used for extended operation above 7,000 or 8,000{{nbsp}}rpm. In practice, automotive Wankel engines are not operated at much higher output shaft speeds than reciprocating piston engines of similar output power. Wankel engines in auto racing are operated at speeds up to 10,000{{nbsp}}rpm, but so are four-stroke reciprocating piston engines with relatively small displacement per cylinder. In aircraft, they are used conservatively, up to 6500 or 7500{{nbsp}}rpm.

===Chamber volume===
In a Wankel rotary engine, the chamber volume <math>V_k</math> is equivalent to the product of the rotor surface <math>A_k</math> and the rotor path <math>s</math>. The rotor surface <math>A_k</math> is given by the rotor tips' path across the rotor housing and determined by the generating radius <math>R</math>, the rotor width <math>B</math>, and the parallel transfers of the rotor and the inner housing <math>a</math>. Since the rotor has a trochoid ("triangular") shape, the sine of 60 degrees describes the interval at which the rotors get closest to the rotor housing. Therefore,

:<math>A_k=2 \cdot B \cdot (R+a) \cdot \sin (60^\circ) = \sqrt 3 \cdot B \cdot (R+a)</math><ref name="Bensinger 1973 p. 64">{{cite book |last1=Bensinger |first1=Wolf-Dieter |title=Rotationskolben-Verbrennungsmotoren |place=Berlin, Heidelberg, New York |date=1973 |isbn=978-3-540-05886-1 |oclc=251737493 |language=de |page=64}}</ref>

The rotor path <math>s</math> may be integrated via the eccentricity <math>e</math> as follows:

:<math>\sum \, ds= \int_{\alpha= 0^{\circ}}^{\alpha=270^{\circ}} e \cdot \sin \frac {2} 3 \alpha \, d \alpha = 3e</math>

Therefore,

:<math>V_k= A_k \cdot s = \sqrt 3 \cdot B \cdot (R+a) \cdot 3e</math><ref name="Bensinger 1973 p. 65"/>

For convenience, <math>a</math> may be omitted because it is difficult to determine and small:<ref name="Yamamoto 1981 p. 15"/>

:<math>V_k= \sqrt 3 \cdot B \cdot R \cdot 3e</math><ref name="Yamamoto 1981 p. 15"/><ref name="Corbat Pawlowski 1973 p. 8">{{cite book |last1=Corbat |first1=Jean Pierre |last2=Pawlowski |first2=Uwe L. |title=Kreiskolbenmotoren des Systems NSU-Wankel ihre Berechnung und Auslegung |place=Basel |date=1973 |isbn=978-3-0348-5974-5 |oclc=913700185 |language=de-CH |page=8 |quote=Formula 56 with k=R/e}}</ref><ref name="Bender Göhlich Springer-Verlag GmbH 2019 p. 126">{{cite book |last1=Bender |first1=Beate |last2=Göhlich |first2=Dietmar |publisher=Springer-Verlag |title=Dubbel Taschenbuch für den Maschinenbau Band 3. |place=Berlin |date=2019 |isbn=978-3-662-59714-9 |oclc=1105131471 |language=de |page=126}}</ref><ref name="Ansdale Keller 1971 p. 79">{{cite book |last1=Ansdale |first1=R.F. |last2=Keller |first2=H. |title=Der Wankelmotor: Konstruktion und Wirkungsweise |place=Stuttgart| publisher=Motorbuch-Verlag |year=1971 |language=de |page=79 formula 6.13}}</ref><ref name="v Manteuffel 1971 pp. 74">{{cite book |last1=v Manteuffel |first1=P. |title=Mechanical Prime Movers |chapter=Rotary Piston Engines |publisher=Macmillan |place=London |year=1971 |isbn=978-1-349-01184-1 |doi=10.1007/978-1-349-01182-7_6 |pages=74}}</ref>

A different approach to this is introducing <math>a'</math> as the farthest, and <math>a</math> as the shortest parallel transfer of the rotor and the inner housing and assuming that <math>R_1=R+a</math> and <math>R_2=R+a'</math>. Then,

:<math>V_k= \sqrt 3 \cdot B \cdot (2 \cdot R_1+R_2) \cdot e</math>

Including the parallel transfers of the rotor and the inner housing provides sufficient accuracy for determining chamber volume.<ref name="Yamamoto 1981 p. 15">{{cite book |last1=Yamamoto |first1=K. |title=Rotary Engine |publisher=Sankaido |year=1981 |isbn=978-99973-41-17-4 |page=15 |quote=Formula 2.27 and 2.30; Yamamoto uses V<sub>h</sub> for V<sub>k</sub>. In this article, V<sub>k</sub> is used for convenience}}</ref><ref name="Bensinger 1973 p. 65"/>

===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.<ref name="SAE J1220">{{Citation|title=Rotary-Trochoidal Engine Nomenclature and Terminology - SAE J1220|publisher=Society of Automotive Engineers|date=June 1978|url=https://www.sae.org/standards/content/j1220_197806/}}</ref>
Part of this dispute was because of Europe vehicle taxation being dependent on engine displacement, as reported by [[Karl Ludvigsen]].<ref name="Ludvigsen 2003">{{cite web|last1=Ludvigsen|first1=Karl|author1-link=Karl Ludvigsen|title=How Big Are Wankel Engines?|publisher=Bentley Publishers|date=2003|url=https://www.hemmings.com/stories/article/how-big-are-wankel-engines}}</ref>

If <math>y</math> is the number of chambers considered for each rotor and <math>i</math> the number of rotors, then the total displacement is:

:<math>V_h=y \cdot V_k \cdot i.</math>

If <math>p_{me}</math> is the [[mean effective pressure]], <math>N</math> the shaft [[rotational speed]] and <math>n_c</math> the number of shaft revolutions needed to complete a [[thermodynamic cycle|cycle]] (<math>N/n_c</math> is the frequency of the thermodynamic cycle), then the total power output is:

:<math>P = p_{me} \cdot V_h \cdot {N \over n_c} = p_{me} \cdot y \cdot V_k \cdot i \cdot {N \over n_c}.</math>

====Considering one chamber====

[[Kenichi Yamamoto (engineer)|Kenichi Yamamoto]] and [[:de:Walter Froede|Walter G. Froede]] placed <math>y = 1</math> and <math>n_c = 1</math>:<ref name="Yamamoto 1981 p. 37">{{cite book |last1=Yamamoto |first1=Kenichi |title=Rotary Engine |publisher=Sankaido |year=1981 |isbn=978-99973-41-17-4 |page=37 |quote=Table 4.1; Yamamoto uses Vh for Vk. In this article, Vk is used for convenience}}</ref><ref name="Froede 1961">{{cite journal|last1=Froede|first1=Walter G.|title=Kreiskolbenmotoren Bauart NSU-Wankel|journal=MTZ - Motortechnische Zeitschrift|volume=22|issue=1|year=1961|pages=1–10|language=de}}</ref>

:<math>P = p_{me} \cdot 1 \cdot V_k \cdot i \cdot {N \over 1}.</math>

With these values, a single-rotor Wankel engine produces the same average power as a <math>V_h</math> single-cylinder [[two-stroke engine]], with the same average torque, with the shaft running at the same speed, operating the Otto cycles at twice the frequency.

====Considering 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 rotary 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 rotary 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.<ref name="Bensinger 1973 p. 65">{{cite book |last1=Bensinger |first1=Wolf-Dieter |title=Rotationskolben-Verbrennungsmotoren |place=Berlin, Heidelberg, New York |date=1973 |isbn=978-3-540-05886-1 |oclc=251737493 |language=de |page=65}}</ref><ref name="Okimoto 2002 pp. 810">{{cite journal |last1=Okimoto |first1=Haruo |title=Der Rotationskolbenmotor Renesis |journal=MTZ - Motortechnische Zeitschrift |publisher=Springer |volume=63 |issue=10 |year=2002 |issn=0024-8525 |doi=10.1007/bf03226650 |pages=810 |language=de}}<br/>&nbsp;&nbsp;&nbsp;{{cite journal |last1=Okimoto |first1=Haruo |title=The Renesis rotary engine |journal=MTZ Worldwide |publisher=Springer |volume=63 |issue=10 |year=2002 |issn=2192-9114 |doi=10.1007/bf03227573 |pages=8}}</ref> 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 rotary engine, according to Bensinger, displacement (<math>V_h</math>) is:<ref name="Bensinger 1973 p. 66">{{cite book |last1=Bensinger |first1=Wolf-Dieter |title=Rotationskolben-Verbrennungsmotoren |place=Berlin, Heidelberg, New York |date=1973 |isbn=978-3-540-05886-1 |oclc=251737493 |language=de |page=66}}</ref><ref name="Ansdale Keller 1971 p. 82–83">{{cite book |last1=Ansdale |first1=R.F. |last2=Keller |first2=H. |title=Der Wankelmotor: Konstruktion und Wirkungsweise |place=Stuttgart| publisher=Motorbuch-Verlag |year=1971 |language=de |pages=82–83}}</ref><ref name="Wankel 1964">{{cite journal|last1=Wankel|first1=Felix|title=Die Anzahl der Zylinder und Kammern bei durchsatzgleichen Viertaktmotoren mit Hubkolben und mit Rotationskolben der Trochoidenbauart|journal=MTZ - Motortechnische Zeitschrift|volume=25|issue=12|year=1964|pages=489–494|language=de}}</ref>

:<math>V_h = 2 V_k \cdot i</math>

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

:<math>P = {p_\text{me} \cdot V_\text{h} \cdot {N \over 2}}</math>

====Considering three chambers====

Eugen Wilhelm Huber, and Karl-Heinz Küttner counted all the chambers, since each one operates its own thermodynamic cycle. So <math>y = 3</math> and <math>n_c = 3</math>:<ref name="Wankel 1958 p. 16">{{cite patent |inventor-last=Wankel |inventor-first=Felix |title=Rotary internal combustion engine |issue-date=1958-11-17 |patent-number=2988065 |country-code=US}}, p. 16</ref><ref name="Huber 1960">{{cite journal|last1=Huber|first1=Eugen Wilhelm|title=Thermodynamische Untersuchungen an der Kreiskolbenmaschine|journal=VDI-Berichte|volume=45|year=1960|pages=13–29|language=de}}</ref><ref name="Küttner 1993 p. 391">{{cite book|last1=Küttner|first1=Karl-Heinz|title=Kolbenmaschinen|publisher=B. G. Teubner|date=1993|isbn=978-3-322-94040-7|doi=10.1007/978-3-322-94040-7|language=de|page=391}}</ref>

:<math>P = p_{me} \cdot 3 \cdot V_k \cdot i \cdot {N \over 3}.</math>

With these values, a single-rotor Wankel engine produces the same average power as a <math>V_h</math> three-cylinder four-stroke engine, with 3/2 of the average torque, with the shaft running at 2/3 the speed, operating the Otto cycles at the same frequency:

:<math>P = p_{me} \cdot 3 \cdot V_k \cdot {{2 \over 3} N \over 2}.</math>

Applying a 2/3 [[gear set]] to the output shaft of the three-cylinder (or a 3/2 one to the Wankel), the two are analogous from the thermodynamic and mechanical output point of view, as pointed out by Huber.<ref name="Huber 1960"/>

====Examples (counting two chambers)====
;KKM 612 ([[NSU Ro80]])

* e=14 mm
* R=100 mm
* a=2 mm
* B=67 mm
* i=2

:<math>V_k = \sqrt 3 \cdot 67 \, mm \cdot (100 + 2 \, mm) \cdot 3 \cdot \, 14 \, mm \approx 498,000 \, mm^3 = 498 \, cm^3</math>

:<math>V_h = 2 \cdot 498 \, cm^3 \cdot 2 = 1,992 \ cm^3</math><ref name="Bensinger 1973 p. 133">{{cite book |last1=Bensinger |first1=Wolf-Dieter |title=Rotationskolben-Verbrennungsmotoren |place=Berlin, Heidelberg, New York |date=1973 |isbn=978-3-540-05886-1 |oclc=251737493 |language=de |page=133}}</ref><ref name="Dobler 2000 pp. 440–442"/>

;Mazda 13B-REW ([[Mazda RX-7]])

* e=15 mm
* R=103 mm
* a=2 mm
* B=80 mm
* i=2

:<math>V_k = \sqrt 3 \cdot 80 \, mm \cdot (103+2 \, mm) \cdot 3 \cdot \, 15 \, mm \approx 654,000 \, mm^3 = 654 \, cm^3</math>

:<math>V_h = 2 \cdot 654 \, cm^3 \cdot 2 = 2,616 \ cm^3</math><ref name="Dobler 2000 pp. 440–442">{{cite journal |last1=Dobler |first1=Helmut |title=Renesis — ein neuer Wankelmotor von Mazda |journal=MTZ - Motortechnische Zeitschrift |publisher=Springer |volume=61 |issue=7–8 |year=2000 |issn=0024-8525 |doi=10.1007/bf03226583 |pages=440–442 |language=de}}</ref>