Analytical functions approximating the burn rate in internal combustion engines are useful and cost-effective tools for engine cycle simulations. Most functions proposed to date are based on the law of normal distribution of a continuous random variable. The best known of these is the Wiebe function, which is used to predict the burn fraction and burn rate in internal combustion engines operating with different combustion systems and fuels. These include direct injection (DI) and indirect injection (IDI) diesel engines, classical spark ignition (SI) engines and gasoline direct injection (GDI) engines, engines with homogeneous charge compression ignition (HCCI) and premixed charge compression ignition (PCCI). This paper is a tribute to the lasting legacy of the Wiebe function and to the man behind it, Ivan Ivanovitch Wiebe. It includes a historical background to the development of the function in the mid 1950s in the Soviet Union, the controversy that surrounded its introduction, a description of the method used to arrive at the final formulation, and an overview of the many applications as prescriptive or predictive single-, double- and multi-function combustion models in engine research.
PuntigamW.HarmannT.SchierlB.HagerJ., ‘Thermal management simulations by compiling of different software packages to a comprehensive system.’SAE paper 2005-01-2061
2.
WiebeI. I., Progress in engine cycle analysis: Combustion rate and cycle processes Mashgiz, Ural-Siberia Branch
3.
WiebeI. I., ‘Semi-empirical expression for combustion rate in engines.’ In Proceedings of Conference on Piston engines, USSR
4.
FrickE.JanteA., ‘Das Wiebe-Brennengesetz-ein Fortschritt in der Thermodynamik der Kreisprozesse Verbrennungs Motoren.’Kraftfahrzengtechnik, Berlin9 (1960): 340–346
5.
VibeI. I., Brennverlaufund Kreisprozess yon Verbrennungsmotoren (: 1970): 286
6.
PischingerR.KrassnigG.TaucarG.SamsTh., Thermodynamik der verbrennungskraftmaschine (Thermodynamics of internal combustion engine)
7.
StieschG., Modeling engine spray and combustion processes (: Springer2003)
8.
MerkerG. P.SchwarzC.StiechG.OttoF., Simulating combustion: Simulation of combustion and pollutant formation for engine-development (: Springer2006)
9.
OppenheimA. K., Dynamics of combustion systems (: Springer-Verlag2006)
10.
HeywoodJ. B.HigginsJ. M.WattsP. A.TabaczynskiR. J., ‘Development and use of a cycle simulation to predict SI engine efficiency and NOx emissions.’SAE paper 790291
11.
ArkhangelskyV.KhovakhM.StepanovY.TrusovV.VikhertM.VoinovA., Motor vehicle engines
12.
LynW.-T., ‘Calculations of the effect of heat release on the shape of cylinder-pressure diagram and cycle efficiency.’Proc. Instn Mech. Engrs 1960—61 (1960): 34–46
13.
AustenA. E. W.LynW.-T., ‘The application of heat release analysis to engine combustion study.’ In Proceedings of the 6th International Congress on Combustion Engines, Copenhagen
14.
BowdenC. M.SamagatB. S.LynW.-T., ‘Rate of heat release in high-speed indirect injection diesel engines.’Proc. Instn Mech. Engrs 1969—70184 (1969): Part 3J
15.
SemenovN. N., Chemical kinetics and chain reactions (: Goskhimizdat1934) (in Russian). 1934.)
16.
SemenovN. N., Some problems of chemical kinetics and reactivity (: Academy of Sciences of the USSR1954) (in Russian). AH CCCP, 1954.)
17.
SemenovN. N., ‘Chain reactions and combustion.’Znanye (: 1957) Series VIII, No. 17 VIII, No. 17, 1957)
WiebeI. I.StavrovA. P., ‘The effect of some diesel engine operating conditions on the kinetics of the combustion process.’ In
21.
GhojelJ.HonneryD.Al-KhaleefiK., ‘Performance, emissions and heat release characteristics of direct injection diesel engine operating on diesel oil emulsion.’J. Appl. Thermal Engng26 (2006): 2132–2141
22.
WiebeI. I.FarafanovM. F., ‘Analysis of engine cycles using digital computers.’Tractors and Agricultural Machinery6 (1967): 232–250 (in Russian)
23.
RakopoulosC. D.GiakoumisE. G., ‘Second-law analyses applied to internal combustion engines operation.’Prog. Energy Combust. Sci.32 (2006): 2–47
24.
RamosJ. I., Internal combustion engine modelling 1st edition
25.
SitkeiG., Kraftstoffaufbereitung und Verbrennung bei Dieselmotoren (: Springer-Verlag1964)
26.
MiyamotoN.MurayamaT.FukazawaS., ‘Studies on low compression ratio diesel engines.’Bull. JSME15 (90) (1972): 1603–1616
27.
TodaN.KushiyamaT.OyamaT., ‘On a method of calculating characteristics of exhaust gas turbocharged two-stroke diesel engine.’Bul. JSME9 (35) (1966): 580
28.
IzumiS.YanoT.OmoteharaI.KushiyamaT., ‘Matching of exhaust turbo-chargers to two-cycle diesel engines.’ ASME paper 68-DGP-9
29.
ShipinskiJ.UyeharaO. A.MyersP. S., ‘Experimental correlation between rate of injection and rate of heat release is a diesel engine.’ ASME paper 68-DGP-11
30.
ChangK.BabajimopoulosA.LavoieG. A.FilipiZ. F.AssanisD. N., ‘Analysis of load and speed transitions in an HCCI engine using 1-D cycle simulation and thermal networks.’SAE technical paper 2006-01-1087
31.
WoschniG.AnisitsF., ‘Experimental investigation and mathematical presentation of rate of heat release in diesel engines dependent upon engine operating conditions.’SAE technical paper 740086
32.
LeeD. I., ‘Combustion simulation of direct injection diesel engine having ‘M— type combustion chamber.’ In Proceedings of COMODIA Symposium on Diagnostics and modeling of combustion in internal combustion engines, Tokyo, Japan, 4—6 September 1985
33.
KuoP. S., ‘Cylinder pressure in a spark-ignition engine: A computational model.’J. Undergrad. Sci.3 (1996): 141–145 (Harvard University)
34.
KooJ. M.BaeC. S., ‘Effect of breathing characteristics on the performance in spark-ignition engines.’ In Proceedings of FISITA World Automotive Congress, 12—15June 2000, Seoul, Korea, paper code F2000A027
35.
ChanS. H.ZhuJ., ‘Modelling of engine in-cylinder thermodynamics under high values of ignition retard.’Int. J. Thermal Sci.40 (2001): 94–103
36.
Abd AllaG. H.SolimanH. H.BadrO. A.Abd RabboM. F., ‘Combustion quasi-two zone model for dual fuel engines.’Energy Conversion Managmt42 (2001): 1477–1498
37.
ErikssonL. E.AnderssonI. K., ‘An analytic model for cylinder pressure in a four stroke SI engine.’SAE technical paper 2002-01-0371
38.
GrauJ. H., ‘Modelling methodology of a spark-ignition engine and experimental validation-Part I: Single-zone combustion model.’SAE technical paper 2002-01-2193
39.
ElmqvistC.LindströmF.ÅngströmH.-E., ‘Optimizing engine concepts by using a simple model for knock prediction.’SAE technical paper 2003-01-3123
40.
VitekO.MacekJ.PolasekM. J., ‘Simulation of pre-chambers in an engine combustion chamber using available software.’SAE technical paper 2003-01-0373
41.
CatonJ. A., ‘Effect of the compression ratio on nitric oxide emissions for a spark ignition engine: Results from a thermodynamic cycle simulation.’Int. J. Engine Res.4 (4) (2003): 249–268 DOI: 10.1243/146808703322743877
42.
CatonJ. A., ‘Effects of burn rate parameters on nitric oxide emissions for a spark ignition engine: Results from a three-zone, thermodynamic simulation.’SAE technical paper 2003-01-0720
43.
GingrichJ. W.CallahanT. J.DodgeL. G., ‘Humidity and temperature correction factors for NOx emissions from spark ignited engines.’ SwRI Project No. 03.10038, prepared for ENVIRON International Corporation
44.
SellnauM.RaskE., ‘Two-step variable valve actuation for fuel economy, emissions, and performance.’SAE technical paper 2003-01-0029
45.
ErikssonL., ‘VehProLib — Vehicle Propulsion Library development issues.’ In Proceedings of the 3rd International Modelica Conference, Linköping, 3—4November 2003
46.
RoelleM. J.ShaverG. M.GerdesJ. C., ‘Tackling the transition: A multi-mode combustion model of SI and HCCI for mode transition control.’ In Proceedings of IMECE'04 International Mechanical Engineering Conference and Exposition, Anaheim, California, USA, 13—19November 2004
47.
YelvingtonP. E.BernatiralloM.LiputS.TesterJ. W.GreenW. H.YangJ., ‘Prediction of performance maps for homogeneous-charge compression ignition engines.’Combust. Sci. Technol.176 (2004): 1243–1282
48.
CanovaM.GarcinR.Midlam-MohlerS.GuezennecY.RizzoniG., ‘A control-oriented model of combustion process in a HCCI diesel engine.’ In Proceedings of the American Control Conference, Portland, Oregon, USA, 8—10June 2005
49.
ShaverG. M., Physics-based modeling and control of residual-affected HCCI engines using variable valve actuation PhD Thesis, Department of Mechanical Engineering, Stanford University
50.
ShaverG. M.RoelleM. J.CatonP. A.KaahaainaN. B.RaviN.HathoutJ.-P.AhmedJ.KojicA.ParkS.EdwardsC. F.GerdesJ. C., ‘A physics-based approach to the control of homogeneous charge compression ignition engines with variable valve actuation.’Int. J. Engine Res.6 (4) (2005): 361–375 DOI: 10.1243/146808705X30512
51.
ShaverG. M.GerdesJ. C.RoelleM. J.CatonP. A.EdwardsC. F., ‘Dynamic modeling of residual-affected homogeneous charge compression ignition engines with variable valve actuation.’Trans. ASME, J. Dynamic Systems, Measmt, and Control127 (3) (2005): 374–381
52.
EdwardsK. D.WagnerR. M.TheissT. J.DawC. S., ‘An approach for investigating adaptive control strategies to improve combustion stability under dilute operating conditions.’ In Proceedings of the 2005 Fall Technical Conference of the Internal Combustion Engine Division of ASME, Ottawa, Canada, 11—14September 2005
53.
SoyluS., ‘Examination of combustion characteristics and phasing strategies of a natural gas HCCI engine.’Energy Conversion Mgmt46 (2005): 101–119
54.
ChangK.BabajimopoulosA.LavoieG. A.FilipiZ. S.AssanisD. N., ‘Analysis of load and speed transitions in an HCCI engine using 1-D cycle simulation and thermal networks.’SAE technical paper 2006-01-1087
55.
MansourC.BounifA.ArisA.GaillardF., ‘Gas—diesel (dual-fuel) modeling in diesel engine environment.’Int. J. Thermal Sci.40 (2001): 409–424
56.
SoyluS.GerpenJ. V., ‘Development of an autoignition submodel for natural gas engines.’Fuel82 (2003): 1699–1707
57.
SoyluS.GerpenJ. V., ‘Development of empirically based burn rate sub-models for a natural gas engine.’Energy Conversion Mgmt45 (2004): 467–481
58.
SoyluS., ‘Examination of combustion characteristics and phasing strategies of a natural gas HCCI engine.’Energy Conversion Mgmt46 (2005): 101–119
59.
KesginU., ‘Study on prediction of the effects of design and operating parameters on NOx emissions from a leanburn natural gas engine.’Energy Conversion Mgmt44 (2003): 907–921
60.
KesginU., ‘Genetic algorithm and artificial neural network for engine optimization of efficiency and NOx emission.’Fuel83 (2004): 885–895
61.
KesginU., ‘Efficiency improvement and NOx emission reduction potentials of two-stage turbocharged Miller cycle for stationary natural gas engines.’Int. J. Energy Res.29 (2005): 189–216
62.
KesginU., ‘Effect of turbocharging system on the performance of a natural gas engine.’Energy Conversion Mgmt46 (2005): 11–32
63.
PathakD., Use of a thermodynamic cycle simulation to determine the difference between a propane-fuelled engine and an iso-octane-fuelled engine Masters Thesis, Texas A&M University
64.
LammleC., Numerical and experimental study of flame propagation and knock in a compressed natural gas engine PhD Thesis, Swiss Federal Institute of Technology, Zurich
65.
OzcanH.SoeylemezM. S., ‘Effect of water addition on the exergy balances of an LPG fuelled spark ignition engine.’Int. J. Exergy2 (2) (2005): 194–206
66.
AtkinsonC. M.PetreanuS.ClarkN. N.AtkinsonR. J.McDanielT. I.NandkumarS.FamouriP., ‘Numerical simulation of a two-stroke linear engine-alternator combination.’SAE technical paper 1999-01-0921
67.
DahmW. J. A.NiJ.MijitK.MayorJ. R.QiaoG.DyerS. W.BenjaminA. G.GuY.LeiY.PapkeM. L., ‘Micro internal combustion swing engine (MICSE) for portable power generation systems.’ In Proceedings of the 40th AIAA Aerospace Sciences Meeting, Reno, Nevada, USA, 14—17January 2002, AIAA paper no. 2002-0722
68.
MacekJ.PolasekM., ‘Porous medium combustion in engines may contribute to lower NOx emissions.’ In Proceedings of the FISITA 2002 World Automotive Congress, Helsinki, 2002, paper code F02V147
69.
ShoukryE. F., Numerical simulation for parametric study of a two-stroke compression ignition direct injection linear engine PhD Thesis, The College of Engineering and Mineral Resources, West Virginia University
70.
HanssonJ.LeksellM.CarlssonF., ‘Minimizing power pulsations in a free piston energy converter.’ In Proceedings of the 2005 European Conference on Power electronics and applications, Dresden Germany, 11—14September 2005
71.
ShinY.ChangS.-H.KooS.-O., ‘Performance test and simulation of a reciprocating engine for long endurance miniature unmanned aerial vehicles.’Proc. IMechE, Part D: J. Automobile Engineering219 (D4) (2005): 573–581 DOI: 10.1243/095440705X11013
72.
BergmanM.GolovitchevV. I., ‘On transient temperature vs. equivalence ratio emission maps in conjunction with 3D CFD free piston engine modelling.’ In Proceedings of 17th International Multidimensional Engine Modeling User's Group Meeting at the SAE Congress, Detroit, Michigan, USA, 15 April 2007
73.
SherE.Bar-KohanyT., ‘Optimization of variable valve timing for maximizing performance of an unthrottled SI engine—a theoretical study.’Energy27 (2002): 757–775
74.
PayriF.MolinaS.MartinJ.ArmasO., ‘Influence of measurement errors and estimated parameters on combustion diagnosis.’Appl. Thermal Engng26 (2006): 226–236
75.
Cruz-PeragonF.Jimenez-EspadaforF. J., ‘A genetic algorithm for determining cylinder pressure in internal combustion engines.’Energy & Fuels21 (2007): 2600–2607
76.
ParkerG.RizzoniG., ‘Individual cylinder model of a general aviation aircraft engine.’J. Propulsion and Power15 (5) (1999): 681–688
77.
ChoH.LeeJ.YooJ., ‘A study of the adaptive control of spark timing using cylinder pressure in a spark ignition engine.’Proc. IMechE, Part D: J. Automobile Engineering213 (D5) (1999): 435–440 DOI: 10.1243/0954407991526991
78.
MinazoL.PengP., ‘Modeling and control of a variable valve timing engine.’ In Proceedings of the American Control Conference, Chicago, Illinois, June 2000
79.
RoelleM. J.ShaverG. M.GerdesJ. C., ‘Tackling the transition: A multi-mode combustion model of SI and HCCI for mode transition control.’ In Proceedings of IMECE'04 International Mechanical Engineering Conference and Exposition, Anaheim, California, USA, 13—19November 2004
80.
ÖbergP.ErikssonL., ‘Control oriented gas exchange models for CVCP engines and their transient sensitivity.’Oil Gas Sci. Technol. — Rev. IFP62 (4) (2007): 573–584
81.
di StasioS.MassoliP., ‘A dissymmetry ratio optical technique as applied to scattering pattern recognition of differently shaped soot aggregates.’Particle and Particle Systems Characterization15 (2) (1998): 90–99
82.
OppenheimA. K.KuhlA. L., ‘Life of fuel in engine cylinder.’SAE technical paper 980780
83.
KuhlA. L.FergusonR. E.OppenheimA. K., ‘Combustion of TNT products in a confined explosion.’ In Proceedings of the 17th International Colloquium on Dynamics of explosions and reactive systems, Heidelberg, Germany
84.
KuhlA. L.OppenheimA. K.FergusonR. E.ReichenbachH.NeuwaldP., ‘Effects of confinement on combustion of TNT explosion products in air.’ In Proceedings of the 28th International Symposium on Combustion, Edinburgh, Scotland, 30 July—4August 2000
85.
OppenheimA. K., Dynamics of combustion systems (: Springer-Verlag2006)
86.
ShawB.FischerG. D.HedrickJ. K., ‘A simplified coldstart catalyst thermal model to reduce hydrocarbon emissions.’ In Proceedings of 15th Triennial World Congress of the International Federation of Automatic Control, Barcelona, Spain
87.
PannangR.SanketiJ.Carlos ZavalaJ. K.HedrickM.WilcuttsT. K., ‘A simplified catalytic converter model for automotive coldstart applications with adaptive parameter fitting.’Proceedings of AVEC 06, the 8th International Symposium on Advanced vehicle control, Taipei, Taiwan, 20—24August 2006
88.
GoncharB. N., ‘Revised method for the computation and construction of engine indicator diagram. Collection of articles on the investigation of working processes in diesel engines,’: Central Diesel Research Institute
89.
SamsonYe. P., ‘Computaion of the indicator diagram of an engine with a combustion chamber in the cylinder.’: Central Diesel Research Institute
90.
DyechenkoN. Kh.MagidovichL. Ye.PugachiuvB. P., ‘Approximation of heat release characteristics in the cylinders of diesel engines (in Russian).’Energomashinostroyeniye310 (1969): 73 (LPI Publications)
91.
DyechenkoN. Kh.MagidovuchL. Ye.PugachiuvB. P., ‘Determination of main parameters of heat release characteristics during combustion in diesel engines (in Russian).’Energomashinostroyeniye316 (1970): 54 (LPI Publications)
92.
DyechenkoN. Kh., Theory of internal combustion engines (: 1974) (in Russian) (Mashinostroyeniye).
93.
GhojelJ. I., Investigation of cycle processes of direct injection diesel engines with cylindrical piston bowl PhD Thesis, Moscow Automobile and Road Institute (MADI) Moscow
94.
GhojelJ. I., ‘A study of combustion chamber arrangement and heat release in DI diesel engines.’SAE paper 821034, 1982, in Diesel Combustion and Emissions Research Report Sp. 525
95.
WatsonN.PilleyA. D.MarzoukM., ‘A combustion correlation for diesel engine simulation.’SAE paper 800029
96.
MiyamotoN.ChikahisaT.MurayamaT.SawyerR., ‘Description and analysis of diesel engine rate of combustion and performance using Wiebe's functions.’SAE paper 850107
97.
SierensR.Van HoveW.RiemslaghK., ‘Heat release analysis and power cycle calculation of the combustion in a medium-speed diesel engine.’ In
98.
BreuerC., ‘The influence of fuel properties on the heat release in DI-diesel engines.’Fuel74 (12) (1995): 1767–1771
99.
WittH.HassenforderM.GissingerG. L., ‘Modeling and identification of a diesel combustion process with the downhill gradient search method.’SAE technical paper 950854
100.
LiuZ.KarimG. A., ‘Simulation of combustion processes in gas-fuelled diesel engines.’Proc. IMechE, Part A: J. Power and Energy211 (A2) (1997): 159–169 DOI: 10.1243/0957650971537079
101.
EgnellR., ‘A simple approach to studying the relation between fuel rate, heat release rate and NO formation in diesel engines.’SAE paper 1999-01-3548
102.
Abd AllaG. H., ‘Computer simulation of a four stroke spark ignition engine.’Energy Conversion Mgmt43 (2002): 1043–1061
103.
PirouzpanahV.SarayR. K.SohrabiA.NiaeiA., ‘Comparison of thermal and radical effects of EGR gases on combustion process in dual fuel engines at part loads.’Energy Conversion Mgmt48 (2007): 1909–1918
104.
PetreanuS., Conceptual analysis of a four-stroke linear engine PhD Thesis, College of Engineering and Mineral Resources, West Virginia University
105.
ShoukryE. F., Numerical simulation for parametric study of a two-stroke compression ignition direct injection linear engine PhD Thesis, The College of Engineering and Mineral Resources, West Virginia University
106.
KumarR.ReaderG. T.ZhengM., ‘A preliminary study of ignition consistency and heat release analysis for a common-rail diesel engine.’SAE paper 2004-01-0932
107.
YasarH.SoyhanH. S.WalmsleyH.HeadB.SorusbayC., ‘Double-Wiebe function: An approach for single-zone HCCI engine modelling.’Appl. Thermal Engng28 (2008): 1284–1290
108.
GlewenW. J.WagnerR. M.EdwardsK. D. C.Stuart DawbC. S., ‘Analysis of cyclic variability in spark-assisted HCCI combustion using a double Wiebe function.’Proc. Combust. Inst.32 (2009): 2885–2892
109.
BilcanA.TazeroutM.Le CorreO.RameshA., ‘Ignition delay in dual-fuel engines: An extended correlation for gaseous fuels.’ In Proceedings of the Spring Technical Conference of the Internal Combustion Engine Division of ASME, Philadelphia, Pennsylvania, USA, 29 April—2May 2001
110.
CanovaM.GarcinR.Midlam-MohlerS.GuezennecY.RizzoniG., ‘A control-oriented model of combustion process in a HCCI diesel engine.’ In Proceedings of the 2005 American Control Conference, Portland, Oregon, 8—10June 2005
111.
GalindoJ.LujanJ. M.SerranoJ. R.HernandezL., ‘Combustion simulation of turbocharger HSDI diesel engines during transient operation using neural networks.’Appl. Thermal Engng25 (2005): 877–898
