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Rassweiler - Withrow model
 The Rassweiler-Withrow method was originally presented in 1938 and many
still use the method for determining the mass fraction burned, due to its simplicity
and it being computationally efficient.
 In this model, the input to the method is a pressure trace p(θj) and the output is
the mass fraction burned trace xb(θj).
 A cornerstone for the method is the fact that pressure and volume data can be
represented by the polytropic relation:
pVn = constant
(n ~)
where the constant exponent n ∈ [1.25, 1.35] gives a good fit to
experimental data for both compression and expansion processes in an engine
[Lancaster, 1975]. An appropriate range γ for diesel heat release analysis is 1.3 to
1.35 [Heywood, 1988, p 510]. The exponent is termed the polytropic index.
Rassweiler - Withrow model
 The Rassweiler-Withrow method assume that the specific heat ratio is captured
by the (constant) polytropic index γ(T)= n
 Calculation of polytropic coefficient n (SAE Paper 2004-01-2973)
 In the Rassweiler-Withrow method, the actual pressure change ∆p = pj+1 − pj
during the interval ∆θ = θj+1 − θj , is assumed to be made up of a pressure rise due
to combustion ∆pc, and a pressure rise due to volume change ∆pv
∆p = ∆pc + ∆pv
Rassweiler - Withrow model
 By assuming that the pressure rise due to combustion in the interval ∆θ is
proportional to the mass of mixture that burns, the mass fraction burned at the end
of the j’th interval thus becomes
where M is the total number of crank angle intervals
 The result from a mass fraction burned
analysis by Rassweiler-Withrow method
Rassweiler - Withrow model
 There are several approximations made when using the Rassweiler-Withrow
method. The polytropic index n is constant. However, γ(T) varies from
compression to expansion and changes during the combustion process. It also
varies with engine operating conditions.
 Effect of gamma on thermodynamic analysis
From Brunt
• Gamma (γ) is the ratio of specific heats. A low value of gamma
produces heat release value that is too high and a heat release rate that is negative
after the completion of combustion.
• A temperature dependent equation for gamma is produced from
experimental data:
 = 1.338 – 6.0x10-5T + 10-8T2 – 0.01
• Gamma is also dependent on equivalence ratio, φ. The effect of
ignoring this term is an error of up to ±0.015 in gamma (0.8<φ<1.2)
Hoffman’s equations of gamma:
The main advantage of temperature dependent gamma is
that it adjusts to different engine operating conditions – higher values of gamma
would be used at low engine load.