and 
:
The application of the PPDF approach to the laminar flamelet model is very similar to that for the equilibrium model. The essential difference here is that, given Eqn. 14 for any state-space variable, we need to have a joint PDF of and :
| (15) |
In resolving the issue of the joint probability over 
and 
, note that in a one-dimensional laminar counterflow flame 
is a function only of the strain rate and the mixture fraction, i.e.:
| (16) |
where 
is proportional to the strain rate. We can integrate this equation with respect to 
and obtain the integrated scalar dissipation over the PDF of 
. This is then parametrized by the mean stoichiometric scalar dissipation, 
, which is related to the mean scalar dissipation rate, 
, obtained from the CFD analysis. The latter is calculated by applying a simple relaxation model for the evolution equation of the scalar dissipation:
 .
| (17) |
To complete the description, an independence hypothesis for the joint probability of 
and 
is used, expressed as:
| (18) |
Again, as in the equilibrium PPDF model, the mapping of mean scalar values to mean values of scalar dissipation rates and mixture fractions is accomplished through a pre-computed table.