1. 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.

2. 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 )
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

3. 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

4. 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:
 = – 6.0x10-5T T2 – 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.