ABSTRACTAt present the market of Wankel engines is limited to somespecial applications. This fact explains absence ofcommercial software products specially developed for thisengine simulation and prediction of its performance.Conversely, there are available and widely used softwareproducts for simulation of reciprocating-piston enginesperformance. Some attempts are known in using this softwarefor prediction of Wankel engine performance. This paperdetails an approach used in these attempts. Main differencesbetween both types of engines are summarized and principlesof a virtual reciprocating-piston engine compilation aredeveloped. A method of virtual blowing was developed forassessment of discharge coefficients for intake and exhaustports. Comparison of simulation results with the measuredperformance of two UAV Wankel engines showed sufficientaccuracy of the suggested approach.

INTRODUCTIONNowadays there are many commercial software productsdealing with simulation of reciprocating-piston (RP) enginesperformance. The most popular of them are the BOOST andFIRE software of AVL (Austria), WAVE of Ricardo (UK),GT-SUITE of GTI (USA) etc. In contrast, there is nocommercially available software specially developed forsimulating Wankel engine performance. Some attempts ofcreating Wankel engine simulation software were made at theperiod between seventies and beginning of nineties[1,2,3,4,5,6,7,8,9,10,11,12,13]. However, none of them wasappropriately developed to contemporary commercial level.

Most probably, this results from the absence of significantdemand for such software since the present market of Wankelengines is very limited. The UAV application may bementioned which utilizes such well known benefits ofWankel engines compared to RP counterparts as higherpower-to-weight ratio, compactness and lower noise/vibrationlevels. Taking this into account, a prediction of Wankelengine performance using commercially available softwarefor RP engine might be of great interest.

Unfortunately, direct use of RP engine simulation software isimpossible for performance modeling of Wankel engines.Because of that two different approaches may be applied:• Development of specific software which is usually based ondetailed experimental data for some definite Wankel engine[1,2,3,4,5,6,7,8,9,10,11,12,13]. Taking into account the verylimited available experimental database on Wankel engines,such software is usually calibrated for only one engine typeand its applicability for other engines may be problematic. Ofcourse, development of new software requires verysignificant investment of time and resources.• Use of available RP engine software appropriately modifiedto account for unique features of a Wankel engine [14,15,16].In the work [14] the first attempt of such approach was madewhen some components of the RP engine code were replacedfor taking into account main peculiarities of a Wankel enginedesign and working cycle. The work was fulfilled on the baseof non-commercial RP software without detailed analysis ofRP and Wankel engines similarity criteria. In the paper [15]the modified commercial AVL BOOST software was usedfor simulation of Wankel engine performance where thesimulated Wankel engine was replaced with the virtual 3-

Simulation of Wankel Engine Performance UsingCommercial Software for Piston Engines

2012-32-009820129098Published

10/23/2012

Leonid Tartakovsky, Vladimir Baibikov, Marcel Gutman and Mark VeinblatTechnion Israel Inst. of Technology

Jonathan ReifElbit Systems Ltd

Copyright © 2012 SAE International

doi:10.4271/2012-32-0098

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cylinder, 4-stroke RP engine. The paper does not presentcriteria for choice of the virtual RP engine dimensions, heattransfer and heat release calculations. In the work [16] thecommercial software GT SUITE was used for prediction ofthe Wankel combustion chamber pressure as a function ofrotor angle. As in [15], the simulated Wankel engine wasreplaced with the virtual 3-cylinder, 4-stroke RP engine.Some assumptions that do not reflect peculiarities of Wankelengine geometry and working cycle were made in [15]: heatrelease and heat transfer calculation procedures were usedunmodified and did not take into account changes ofcombustion and heat transfer processes in Wankel enginecompared with RP counterpart, piston stroke of the virtual RPengine was taken equal to eccentric offset of the simulatedWankel engine, etc. These simplifications do not allowprediction of Wankel engine performance with an acceptableaccuracy.

The approach of [15 and 16] with use of the GT-POWERsoftware of the GT-SUITE package is discussed and furtherdeveloped in this paper. The GT-SUITE is an up-to-dateintegrated program for engines simulation and designanalysis. It contains the following components: GT-POWER- engine simulation for performance and acoustic analysis;GT-DRIVE - vehicle performance and cycle analysis for fueleconomy and emissions; GT-VTRAIN - valve trainkinematics, dynamics and tribology; GT-FUEL - injectionsystem pressure and flow analysis; GT-COOL - engine heatand cooling system analysis; GT-CRANK - crankshaftdynamics and torsional vibrations analysis. All sixcomponents of GT-SUITE are based on a common set ofmulti-physics libraries and thus have a large degree ofcommonality among them. The GT-POWER is the 2- and 4-stroke SI and diesel engine simulation tool widely usednowadays by engines makers. Among its advantages are easeof use and tight integration with the rest GT-SUITEcomponents. Its flexibility and usefulness are furtherenhanced by the possibility of integration with STAR-CD,Fluent, Simulink, MS/EXCEL and other general purposesoftware. Due to the large size of its user base, the GT-POWER has been thoroughly validated by real-lifeapplications. Use of this software which is focused on RPengines for simulation of Wankel engine performance hasrequired development of the algorithm for composing a 3-cylinder, 4-stroke virtual RP engine that would begeometrically similar to the considered Wankel engine.Development of special procedures that would take intoaccount differences between Wankel and RP engines incombustion and heat transfer, as well as assessment ofdischarge coefficients of intake and exhaust ports, wasrequired too.

MAIN DIFFERENCES BETWEENWANKEL AND PISTON ENGINESThe following main differences between Wankel and pistonengines that affect their performance can be mentioned:

• Difference in patterns of working chamber volume andsurface dependence on the angle of shaft rotation;

• Duration of the Wankel working cycle is 1.5 times longercompared with a 4-stroke piston engine (in terms of the angleof shaft rotation). Complete working cycle of a Wankelengine takes place in each working chamber per one rotorrevolution, or per three shaft revolutions. In other words, oneworking cycle in a Wankel engine occurs during 1080degrees of shaft rotation (or 360 degrees of rotor rotation),compared to 720 degrees of the crankshaft rotation in a 4-stroke piston engine;

• “Hot” and “cold” stator zones of the Wankel workingchamber surfaces are separated contrary to a piston engine,where the same working chamber surfaces are heated andcooled in-turn. This, in combination with charge rotationalmovement together with a working chamber, leads todifferences in the heat transfer conditions;

• Unfavorable shape of the Wankel working chamber (Figure1) leads to the higher surface-to-volume ratios and largerrelative value of crevice volumes, where flame quenchingtakes place. This causes differences in combustion patterns;

• Wankel working chamber has more complicated design ofseals (apexes and rotor side seals). This leads to thepossibility of higher charge leakage values;

• Wankel engine differs fundamentally from piston engine inits kinematic mechanism. This leads to differences in internalfriction power losses.

Figure1. Unfavorable shape of the working chamber in aWankel engine [17].

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SIMILARITY CRITERIAThe method of Wankel engine performance simulation byusing RP engine software which is discussed in this paperrequires compilation of some similarity criteria for theseengines and composing a virtual piston (VP) 4-stroke enginethat would allow simulation of Wankel performance.

The following similarity criteria of Wankel and the VPengines have been defined:

• Displacement equality;

• Compression ratio equality;

• Identical behavior of the working chamber volumedependence on angle of shaft rotation;

• Identical behavior of working chamber surface-to-volumeratio change vs. angle of shaft rotation;

• Identical behavior of intake and exhaust ports dischargecoefficients vs. angle of shaft rotation.

In addition to these similarity criteria which may besummarized as a geometric similarity, the followingimportant aspects should be taken into account:

• Instantaneous values of the heat transfer through thecombustion chamber walls vs. angle of shaft rotation have tobe calculated separately for non-firing (“cold”) and firing(“hot”) parts of the working cycle;

• Modeling the combustion should take into accountspecificities of the combustion process in Wankel engine.

GEOMETRIC SIMILARITYALGORITHMThe main purpose of the algorithm is to ensure that the VPengine would meet the criteria of geometric similarity listedin the previous section. It means that the VP engine has to beof the same displacement, compression ratio, behavior ofworking chamber volume and surface-to-volume ratio changeversus shaft angle as the modeled Wankel engine.

Equations characterizing geometric parameters of Wankel[18] and piston [19] engines are presented in Appendix A.Designation explanations are presented in Fig.2 (for aWankel engine) and in Fig.3 (for a piston engine). Joinedsolution of these equations for Wankel and piston enginesallows finding geometric parameters (bore, stroke andconnecting rod length) of a VP engine geometrically similarto the modeled Wankel engine. Unfortunately, an analyticsolution of these equations gives different results for variousangles of shaft rotation. Therefore, achievement of absolutegeometric similarity at each moment of the engines operationcycle is impossible and approximate iterative solution wasapplied with the following criterion of convergence. Valuesof the working chamber volume, surface and surface-to-

volume ratio for the VP engine should not differ from thecorresponding values of the modeled Wankel engine by morethen 1% at minimal working chamber volume values of theseengines.

A flow chart of the developed algorithm for calculation of thegeometric similarity parameters is presented in the AppendixB. Results of the computations are shown in Tables 1, 2 andFigures 4 - 6 for the example of the 802W Wankel engine. Itsmain parameters are:

• Naturally aspirated, spark ignition, single-rotor, combinedair-liquid cooling;

• Rated shaft speed - 8,000 rpm;

• Rated brake power - 52 kW;

• Displacement of each working chamber - 343.9 cc;

• Compression ratio - 7.4;

• Eccentricity - 11.6mm;

• Generating radius - 71.5mm;

• Rotor width - 80mm.

As can be seen from Table 2, a maximal deviation from thegeometric similarity at the top dead center (TDC) does notexceed 0.8% and at the bottom dead center (BDC) - 3.5%. Itis clear that from the viewpoint of combustion andperformance simulation, high accuracy of the geometricsimilarity at TDC is much more important then at BDC,where combustion does not take place.

Figure 2. Designations for Wankel engine parameters.

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Figure 3. Designations for piston engine parameters.

Table 1. Parameters of the virtual 4-stroke piston engine.

Table 2. Deviations from the geometric similarity of theVP engine.

Figure 4. Working chamber volume vs. angle of shaftrotation for 802W Wankel and virtual piston engines

Figure 5. Working chamber surface area vs. angle ofshaft rotation for 802W Wankel and virtual piston

engines.

Figure 6. Working chamber surface/volume ratio vs.angle of shaft rotation for 802W Wankel and virtual

piston engines.

As can be seen from Tables 1, 2 and Figures 4,5,6, thegeometrical similarity of the VP and the modeled Wankelengines is kept closely, especially in the vicinity of TDC.

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DISCHARGE COEFFICIENTS FORINTAKE AND EXHAUST PORTSValues of discharge coefficients (DC) for intake and exhaustports are very important because they significantly affectcalculations of gas exchange in the engine working chamber.It is clear that they greatly influence the accuracy of theengine performance simulation. Therefore, DC data used asthe inputs in a simulation approach must be found as accurateas possible.

Obviously, an accurate DC assessment would be based onexperimental measurements, when air is blown through thereal orifice. Unfortunately, these results are not alwaysavailable on early stages of engine development. Therefore amethod of flow simulation through the intake and exhaustports was developed. These calculations were performed forfew different positions of the rotor relative to thecorresponding port. Appropriate charts were prepared usingSolid Works software. Examples of this virtual blowing arepresented in Figures 7 (for intake port) and 8 (for exhaustport).

Gas flow rate values through the port were computed usingthe FLUENT software for each studied position of the rotor.Unburned mixture with air/fuel ratio of 14.5 (typical for SIWankel engines) and burned mixture of N2/CO2/H2O vapor(typical for complete combustion) were used for thesimulation of gas flow through the intake and exhaust port,respectively.

Figure 7. Simulation scheme for intake port blowing.

Molecular weight of each gas mixture was calculated byequation:

(1)where: Ci - mole fraction of the component “i” Mi -molecular weight of the component “i”.

The data that have been used for the calculation of gasmixtures molecular weight and calculation results for intakeand exhaust ports are given in Table 3.

DC values were calculated for each studied position of therotor using the isentropic gas flow rate equation for subsonicvelocity of outflow [20], where the real process effects weretaken into account to some extent by using a typical range ofthe polytrophic exponent n values:

(2)

Where: Q - gas mixture flow rate; DC - discharge coefficient;A - area of the port; n - polytropic exponent; ρo - upstreammixture density; p0 - upstream pressure; p - downstreampressure.

Figure 8. Simulation scheme for exhaust port blowing

Input data, such as pressure and temperature in the workingchamber, that are required for the calculation carrying out,have been assessed using a typical indicatory diagram of aWankel engine. Upstream data for the exhaust port anddownstream data for the intake port were spatially averagedfor the combustion chamber volume for each consideredopening area of the corresponding port. Downstream data forthe exhaust port and upstream data for the intake port weretaken as ambient conditions (1 bar and 20 degrees ofCelsius). Since pressure values in the working chamber thathave been assessed based on the typical indicatory diagramscannot be sufficiently accurate, the calculations were madefor several values of the pressure in the vicinity of thecalculated point. This allowed assessment of DC sensitivityto pressure value p in the working chamber. Fig. 9 illustratesexample of calculation results for 50% opening of exhaustport. As can be seen from this Figure, minimal DC value wasfound for the combination of the minimal pressure andpolytrophic exponent values. Accordingly, the maximal DCvalue anticipated for the highest values of pressure and n.Deviation of the computed DC values in the realistic range ofpressures and polytrophic exponents was found to be lessthan ±12%. In this work the DC values for each port openingwere assumed to be the average of those found by thecomputations for nine combinations of pressure value in theworking chamber and polytropic exponent in the range of

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1.5<p<2.5 bar (exhaust port), 0.7<p<0.9 bar (intake port) and1.2<n<1.4.

Table 3. Results of molecular weights calculation forunburned and burned mixtures.

Figure 9. Results of DC calculation for 50% opening ofexhaust port.

Results of computations that were performed by using thedescribed approach are shown in Figure 10 (for intake port)and Figure 11 (for exhaust port). As can be seen from Table4, maximal values of the DC obtained by using the suggestedapproach for the Wankel engine are substantially higher thanthose of reciprocating piston engines [21]. Obviously, thismay be explained by design simplicity of Wankel intake/exhaust ports in comparison with 2-stroke engines and,especially, with 4-stroke engines where intake/exhaust flowsare restricted by poppet-valves. Figures 10 and 11 contain DCvalues only for the process of ports opening. The same valuesof the discharge coefficients were assumed also for the portsclosing process. These values were used as an input data forthe engine simulation software.

Table 4. DC values calculated for Wankel engine andthose typical for reciprocating engines

Figure 10. Values of intake port DC at various portopenings.

Figure11. Values of exhaust port DC at various portopenings.

HEAT TRANSFER COEFFICIENTObviously, the heat transfer coefficient (HTC) calculated forthe virtual piston engine should reflect the heat transferconditions in the simulated Wankel engine. As describedearlier, the main factors leading to different conditions of heattransfer in Wankel engine compared with a piston counterpartare:

• Unfavorable shape of Wankel working chamber withsignificantly higher surface-to-volume ratio;

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• Non-firing (“cold”) and firing (“hot”) stator zones ofWankel working chamber surfaces are separated contrary topiston engine;

• Charge rotational movement together with a workingchamber.

These differences do not allow straightforward application ofthe HTC calculation procedure of GT-Power commercialsoftware [21] based on the Woschni correlation for an enginewithout swirl, as described in detail in Section 12.4.3 of [22].In this approach the first factor was taken into account bycomposing geometrically similar virtual piston engine withthe same surface-to-volume ratio of a working chamber, asdescribed earlier. Regarding the last two factors, it isimpossible to use standard computation routine of RP enginesoftware for assessment of the heat transfer coefficient.Therefore, in this work the approach used in [9, 13 and 14]was applied. This method is based on the classicalrelationship between the dimensionless groups nowadaysknown as Nusselt, Reynolds and Prandtl numbers that wasintroduced for the first time by W. Nusselt [23, 24] forturbulent flow in pipes or over flat plates:

(3)where: Nu = hL/k is Nusselt number; Re = vL/v is Reynoldsnumber; Pr = µcp/k is Prandtl number; K, a, b - constants; h -heat transfer coefficient; L - characteristic length; k - thermalconductivity of gas; v - characteristic velocity; v - gaskinematic viscosity; µ - gas dynamic viscosity; cp - specificheat at constant pressure.

In [9, 13 and 14] Prandtl number is assumed to be unity andthe flow pattern is assumed to be similar to that of the forcedconvection over a flat plate. The latter provides the followingvalues of the constants in equation (3), [14]: K = 0.087; a =0.8.

The characteristic length L for Wankel engine was assumedto be equal to the rotor width, i.e. Lw = H, and for the virtualpiston engine it was assumed to be equal to the cylinder bore,i.e. Lp = B. The characteristic velocity v was assessed indifferent ways for suction/compression and for combustion/expansion strokes. For the suction/compression stroke (non-firing):

(4)where: n - Wankel engine shaft speed; R - generating radius.

The instantaneous characteristic velocity for the combustion/expansion stroke (firing) was given by:

(5)

Values of the constants: C1 = 1; C2 = 0.00324 m/(s·K) weretaken from [14].

Substitution of all the data into the equation (3) and itssolution for h gives HTC values for each angle of thecrankshaft rotation. Fig. 12 presents Wankel HTC behaviorover crankshaft angle of the virtual piston engine operatedunder rated power, as calculated by using the describedalgorithm. This Figure contains also the HTC values for thesame operation regime of this engine that were computedusing the commercial software procedure. As can be seen, theHTC values computed by using the new algorithm for aWankel engine are larger by 10…15% than those assessedwith the aid of a commercial software procedure forreciprocating piston engines. The difference is a result of thepeculiarities of a Wankel engine, discussed above.

Figure 12. Heat transfer coefficient (HTC) patterns forthe virtual piston engine under rated brake power - 52

kW: 1 - computed by using commercial softwareprocedure; 2 - computed using the algorithm developed

for Wankel engine.

MODEL VALIDATIONThe validation of the developed approach was carried out bycomparison of the predicted 802W and 802 Wankel enginesperformance with the available experimental data for theseengines. The 802 engine differs from the 802W one by lesserrotor width (68mm compared with 80 mm). The calculations

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were carried out for the same ambient and cooling conditionsthat were observed under the engines tests. For eachcalculated regime piston surface temperature of the virtual RPengine was assumed equal to that one of the simulatedWankel rotor face, temperatures of cylinder head andcylinder surfaces were assumed equal to these of the Wankelhousing inner surface on the base of experimental data. HTCvalues were computed as described in the previous section.The engine tests and calculations were carried out overregimes of full load and propeller curves. The latter wasneeded since the simulated Wankel engines are intended forUAV propulsion, i.e. they are loaded by propeller where therelationship between the engine load P and speed n isdescribed by the equation P=kn3.

Figures 13,14,15 show satisfactory correspondence of thecalculated and measured values of brake power and fuelconsumption at various operation regimes of the 802W and802 engines.

Figure13. Wankel engines 802 and 802W: predicted(lines) and measured (dots) values of the brake power

over full load curve.

Figure 14. Wankel engines 802 and 802W: predicted(lines) and measured (dots) values of the fuel

consumption over full load curve.

Figure 15. Wankel engines 802 and 802W: predicted(lines) and measured (dots) values of the fuel

consumption over the propeller curve.

As can be seen from Figures 13, 14, 15, the maximaldiscrepancy of the measured and predicted results is about5% and 3-10% for the brake power and fuel consumption,respectively. This indicates the acceptability of the modelpresented here.

SUMMARY AND CONCLUSIONSIn contrast to piston engines, none of Wankel enginesimulation software was appropriately developed tocontemporary commercial level up to date. At present onlyAVL BOOST commercial software has additional block forWankel engine performance prediction by replacing it with avirtual RP engine. Most probably, this results from theabsence of significant demand for such software, since themarket of Wankel engines is very limited. So, a possibility ofWankel engine performance prediction using commerciallyavailable software for reciprocating-piston engine might be ofgreat interest.

Since straightforward use of the piston engine simulationsoftware is impossible, there are different approaches tosimulation of Wankel engine performance. Some of them arefocused on the development of new software for the specificWankel engine on the basis of its detailed experimental data.Another way is based on adaptation of available pistonengine commercial software products. The latter approach isdetailed in this paper. It required development of the Wankel-to- RP engine geometric similarity algorithm andcompilation of a virtual RP engine, which would allowassessment of Wankel engine performance. Peculiarities ofheat transfer and combustion processes in a Wankel enginewere taken into account as well.

The developed algorithm ensures meeting the geometricsimilarity criteria (equality of displacement, compressionratio, working chamber volume and surface-to-volume ratiovs. crankshaft angle). It allowed finding base dimensions ofthe virtual piston engine that can be used for performance

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simulation of a Wankel engine. The applied methodology ofthe heat transfer coefficient computing allowed accountingfor such features of Wankel engine as “hot” and “cold” statorzones separation, unfavorable shape of working chamber andthe charge rotational movement.

For the case when discharge coefficients cannot beexperimentally obtained (the common situation at earlyengine development stages), a method of blowing simulationthrough the intake and exhaust ports was developed. Thismethod includes compilation of charts for intake and exhaustport zones, when each of the charts has different rotorposition relative to the port (i.e. different rate of the portopening). Unburned mixture with air/fuel ratio of 14.5 andburned mixture of N2/CO2/vapor H2O were assumed forblowing through the intake and exhaust port respectively. Thegas flow rate through the port can be computed using anyavailable CFD software for different positions of the rotor,with appropriate calculation of discharge coefficients.

Combustion rate and heat release behavior were simulated byusing the conventional approach for piston engines - theWiebe equation. Values of the constants in the latter wereappropriately changed based on the available literature datafor Wankel engines.

This approach was demonstrated through the example of the802 and 802W engines brake power and fuel consumptionsimulation at various operation regimes. A comparison of thepredicted Wankel engines performance with the measureddata has shown satisfactory correlation.

The developed simulation approach can be used foroptimization of Wankel engine parameters, such as ignitionadvance map, intake and exhaust ports timing and geometry,air/fuel ratio, as well as assessment of the influence on engineperformance of air filter and exhaust muffler characteristics,flight altitude, etc.

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2. Horsfield, B., “Computer Modeling [Wankel RotaryCombustion Engine]” Electronics & Computing Monthly3(6): 56-7, 1983.

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7. Ramos, J.I., Shih, T.I-P., Schock, H.J., “Wankel enginemodeling”, Proceedings of the International Centre for Heatand Mass Transfer: 469-483, 1989.

8. Li, Z., Steinthorsson, E., Shih, T., and Nguyen, H.,“Modelling and Simulation of Wankel Engine Flow Fields,”SAE Technical Paper 900029, 1990, doi: 10.4271/900029.

9. Bartrand, T.A., and Willis, E.A., “Performance of aSupercharged Direct-Injection Stratified-Charge RotaryCombustion Engine”, NASA technical memorandum 103105,1990.

10. Kim, K.J., Chae, J.O., and Chung, T.Y., “A PerformanceSimulation for Spark Ignition Wankel Rotary Engine,” SAETechnical Paper 912479, 1991, doi: 10.4271/912479.

11. Raju, M., and Willis, E., “Three Dimensional Analysisand Modeling of a Wankel Engine,” SAE Technical Paper910701, 1991, doi: 10.4271/910701.

12. Raju, M., “Heat Transfer and PerformanceCharacteristics of a Dual-Ignition Wankel Engine,” SAETechnical Paper 920303, 1992, doi: 10.4271/920303.

13. Bartrand, T.A., and Willis, E.A., “Rotary engineperformance computer program. User's guide”, NASAContractor Report 191192, October, 1993.

14. Norman, T.J., “A performance model of a spark ignitionWankel engine: including the effects of crevice volumes, gasleakage and heat transfer”, M.S. thesis, MassachusettsInstitute of Technology, Massachusetts, June 1983.

15. Wendekerer, M., Grabovski, L., Pietrikovski, K. andMagryta, P., “Phenomenological model of a Wankel engine”,Silniki Spalinowe, 50(3), 2011.

16. Handschuh, R.F. and Owen, A.K., “Analysis of apex sealfriction power loss in rotary engines”, NASA/TM-2010-216353, September 2010.

17. Poojitganont, T., Izweik, H.T., and Berg, H.P., “Thesimulation of Flow Inside the Wankel CombustionChamber”, http://www.tsme.org/ME_NETT/ME_NETT20/article/pdf/tsf/TSF031.pdf, Nov. 2011.

18. Yamamoto, Kenichi, “Rotary engine”, Published byToyo Kogyo Co., LTD. 1969.

19. Automotive handbook, 8th edition, Robert Bosch GmbH,2011.

20. Mechanical engineering handbook, V.2, Moscow, 1961(in Russian).

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21. http://www.gtisoft.com/applications/a_Engine_Performance.php

22. Heywood, John B., “Internal combustion enginefundamentals”, McGraw -Hill Inc., 930 p., 1988.

23. Nusselt, W., “Das Grundgesetz des Wärmeüberganges”,Gesundheits-Ingenieur, 38 (42): 477-482 and (43): 490-496,1915.

24. Nusselt, W., “Der Wärmeübergang im Rohr”, Z. VDI,61, 1917.

CONTACT INFORMATIONDr. Leonid TartakovskyFaculty of Mechanical EngineeringTechnion - Israel Institute of TechnologyTechnion City, Haifa 32000, Israeltartak@technion.ac.il

ACKNOWLEDGMENTSThe authors are grateful to MAFAT, Israeli Ministry ofDefense for the financial support of this project.

DEFINITIONS/ABBREVIATIONSBDC - bottom dead centerDC - discharge coefficientHTC - heat transfer coefficientRP - reciprocating pistonSI - spark ignitionTDC - top dead centerUAV - unmanned aerial vehicleVP - virtual piston

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APPENDIX ARotor lobe surface area Ac is a constant value and was calculated by using the modified approximating formula from [14]:

(A1)

Where: Rx - radius of the circular arc approximation

(A2)

β is one half the angle subtended by the approximate circular arc:

APPENDIX

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(A3)

m is multiplier considering rotor lobe recess.

Following [14], in the equation (A3) values of R and e must be inserted in inches.

Instantaneous value of the housing surface area as function of the angle of rotor rotation was calculated analytically using thefollowing equation derived in this work:

(A4)

Instantaneous value of the side plate surface area consists of two parts - the first is the constant one Acompr (corresponds tocompression volume) and the second is the variable one Adispl (corresponds to displacement volume):

(A5)

Values of these parts were found using the following equations [16]:

(A6)

Where φmax = 3e/R.

(A7)

APPENDIX B

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Fig. B2. Designations of Wankel engine geometry elements.

Symbols designationAc - rotor lobe area;

As - area between rotor lobe arc and chord XY;

Vcr - volume of rotor recession;

Δα - angle step of calculations;

Fig. B1. Flow chart of virtual piston-to-Wankel engines geometric similarity algorithm. For designations see Fig.B2 and a listbelow.

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n - number of calculation steps;k - multiplier for conrod length iteration;α(i) - instantaneous value of rotor rotation angle;Ai - instantaneous value of area between arch Li and chord XY;

Li - instantaneous value of trochoidal arch;

Aw - instantaneous value of one working chamber surface area of Wankel engine;

Vw - instantaneous value of one working chamber volume of Wankel;

Vcw - compression volume of Wankel;

Vw - maximal value of one working chamber volume of Wankel;

Vhw - displacement of Wankel one working chamber;

B - virtual piston engine cylinder bore;R - crank radius;h(i) - instantaneous value of distance between piston and TDC;Ap(i) - instantaneous value of piston engine working chamber surface area;

Vhp(i) - instantaneous value of piston engine volume that corresponds to h(i);

Vcp - compression volume of piston engine;

Vp(i) - instantaneous value of piston engine working chamber volume.

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