Closures based on presumed conditional moments

For simple chemistry, a form for Q( x, t) can sometimes be found based on linear interpolation between two limiting cases. For example, for the one-step reaction discussed in Section 5.5, we have seen that the chemical source term can be rewritten in terms of a reaction-progress variable F and the mixture fraction f. By taking the conditional expectation of (5.176) and applying (5.287), the chemical source term for the conditional reaction-progress variable can be found to be 

The actual value of (F f) for finite k must lie somewhere between the two limits. Mathematically, this constraint can be expressed as 

Given (5.292) and the mixture-fraction PDF, the chemical source term in the Reynolds-averaged transport equation for (7) is closed  

if a presumed beta PDF is employed to represent / (f x, t), the finite-rate one-step reaction can be modeled in terms of transport equations for only three scalar moments (f), (f,2), and (7). The simple closure based on linear interpolation thus offers a highly efficient computational model as compared with other closures. 

Looking back over the steps required to derive (5.290), it is immediately apparent that the same method can be applied to treat any reaction scheme for which only one reaction rate function is finite. The method has thus been extended by Baldyga (1994) to treat competitive-consecutive (see (5.181)) and parallel (see (5.211)) reactions in the limiting case where k - oo.118 For both reaction systems, the conditional moments are formulated in terms of 72(X and can be written as 

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