and 
:
In the non-adiabatic case, all properties are functions of the mixture fraction and enthalpy. Therefore, the averaged value of any scalar in a turbulent flow field is the instantaneous value integrated over the joint PPDF of and :
| (20) |
STAR-CCM+ makes a statistical independence assumption for 
so that:
| (21) |
and further assumes a Dirac delta function for 
. In other words, the mean value of enthalpy is used in the calculation of:
| (22) |
The integrals in Eqn. 10, Eqn. 18 and Eqn. 22 are pre-computed. When the value of an integral is needed during the calculation, a simple interpolation is all that is needed to provide the correct value.