An evaluation of beta PDF integration using the density-weighted PDF and the un-weighted PDF

In many turbulent combustion models, the mean values of scalars are often calculated by integrating the product of the scalar profile and the probability density function (PDF) over mixture fraction space. For the integration, researchers used the weighted and the un-weighted PDF interchangeably depending on their preference. For both PDFs, a β-function is normally presumed. However, a recent study by Liu et al. [Internat. J. Therm. Sci. 41 (2002) 763-772] showed that the predicted mean values of scalars might be significantly different depending on the employed PDF. This paper determines the reason for the difference and shows that the same result can be predicted by using the proper parameters in conjunction with the un-weighted PDF.