Editing Wankel engine (section)

===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>