Wiebe function
The combustion of the fuel in a internal combustion engine takes place so quickly that it is almost technically impossible to directly measure the amount of fuel that is burnt within a specific time period. However, temperature and pressure in the combustion chamber may be measured with sufficient accuracy. From these measurement values the progress of the combustion can be estimated to be used both as a prescriptive and a predictive combustion model. The nowadays predominant estimation function was developed in the early 1950s by Ivan Wiebe, a Russian engineer and scientist at the Chilyabinsk Politechnic Institute, currently known as the South Ural University. In contrast to former existing simple models his approach was based on the theory of chemical reaction kinetics and chain reaction. [1]
As his name might suggest he was of German descent. In 1960 it was for the first time that the Wiebe function became known outside of the USSR. It was published in a Journal in the German Democratic Republic. It took about another 20 years before it was adopted in the "Western" engineering world. Since then it is widely used due to its simplicity and versatility, sometimes overlaid with a second or further Wiebe functions with different parameters as a double function or multiple function. [1].
The combustion in an engine is interpreted by the heat release rate from the start of ignition up to the charge extinction. The Wiebe function estimates the fraction xb of burned mass of fuel mixture (mass burned fraction MBF) as a function of the crank angles and the Wiebe parameters a and m which determine the shape of the Wiebe function. The MFB is assumed to be proportional to the combustion pressure rise [2]. As the Wiebe function is an exponential function it asymptoticly reaches to the value 1 and the duration of combustion would be infinitely. The crank angle ΔΘ in which combustion (mainly) takes place therefore defines the time period in which 90% of the fuel gases have been combusted.
Examples for Wiebe parameters:
| fuel | rpm | cyl. bore [mm] | cyl. stroke [mm] | CR | a | m | timing advance BTDC Θ0 | burn duration ΔΘ | source | |
|---|---|---|---|---|---|---|---|---|---|---|
| petrol, lean mixture | 1200 | |||||||||
| petrol, rich mixture | 4000 | 2 | 6 | 50 | ||||||
| petrol | 900 | 8 | 2.30 | 2.75 | -10 | 30 | [6] | |||
| 80% petrol and 20% ethanol | 900 | 2.30 | 2.95 | -10 | 31 | [6] | ||||
| 60% petrol and 40% ethanol | 900 | 2.30 | 2.90 | -10 | 29 | [6] | ||||
| 26% petrol and 84% ethanol | 900 | 2.30 | 2.84 | -10 | 29 | [6] | ||||
| bio gas (19% H2; 30% CO; 5% CH4 ; rest N2 | 1800-2400 | 76.0 | 88.0 | 14 | [7] | |||||
| producer gas (generated from thermo-chemical conversion of bio-mass: 18-20% H2 and CO, 2% CH4, 12% CO2, balance N2) ** | 1500 | 102 | 120 | 10.5 | 2.4 | 0.7 | -24 | 40 | [8] |
** Lower calorific value (LCV) 5.00 MJ/kg; Air fuel ratio 1.35; laminar flame speed 0.50 m/s; Adiabatic flame temperature 1800 K. The high percentage of hydrogen enhances the flame speed due to a global increase in the reactivity and flame breakups into smaller cells causing an increase in the turbulent flame propagation speed. [8]
Bio gas, similar to natural gas is coupled with lower flame speed, narrow combustible range, higher auto-ignition temperature and requiring high ignition energy. Syn gas is a product of a thermochemical conversion of solid fuels that increases the hydrogen to carbon ratio of the feedstock. Syngases have moderate laminar flame speed and high self ignition temperatures. [7]
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[1] J I Ghojel, "Review of the development and application of the Wiebe function: A tribute to the contribution of Ivan Wiebe to engine research", International Journal of Engine Research Vol. 11, pages 297-312, DOI 10.1243/14680874JER06510.
[2] J.B. Heywood, "Internal Combustion Engine Fundamentals", New York, McGraw Hill International, 1988.
[6] Yeliana, C. Cooney, J. Worm, D. Michalek, J. Naber, "WIEBE FUNCTION PARAMETER DETERMINATION FOR MASS FRACTION BURN CALCULATION IN AN ETHANOL-GASOLINE FUELLED SI ENGINE", Journal of KONES Powertrain and Transport, Vol. 15, No. 3 2008.
[7] F. Hagos and A. Abd Aziz,"Mass Fraction Burn Investigation of Lean Burn Low BTU Gasification Gas in Direct-injection Spark-ignition Engine", SAE Technical Paper 2014-01-1336, 2014, DOI 10.4271/2014-01-1336.
[8]
Anand Shivapuji, S. Dasappa, "Experiments and zero D modeling studies using specific Wiebe coefficients for producer gas as fuel in spark-ignited engines" in ARCHIVE Proceedings of the Institution of Mechanical Engineers Part C Journal of Mechanical Engineering Science 1989-1996 (vols 203-210) 227(3):504-519 Follow journal DOI: 10.1177/0954406212463846