### 7.14.2. Principal Variables and Transport Equation

The mixture is described as a three-stream system of fuel, oxidizer and residual material. The corresponding mass fractions neglecting any reaction sources (as for mixture fraction) are fuel tracer, , oxidizer tracer, , and residual tracer or EGR tracer, . The tracer variables obey the following obvious constraints, here written for the turbulent means (the same relations apply for the laminar/instantaneous quantities):

| (7–84)

where

The fuel tracer and mixture fraction are very closely related quantities. The difference is that the fuel tracer refers to the mass of fresh fuel, while the mixture fraction does additionally include a contribution from the residual materials. The fuel tracer is equal to mixture fraction if, and only if, the residual mass fraction is zero.

Without loss of generality the residual material is defined to be stoichiometric,

which establishes the following relation between the overall mixture fraction and the tracer variables:

| (7–85)

| (7–86)

Using Equation 7–84 and Equation 7–85 it is sufficient to solve for two of the four variables , , and . In addition to the mixture fraction equation discussed in section The Flamelet Model, a transport equation is solved for the Favre-averaged fuel tracer:

| (7–87)

The equation is identical to that for the mean mixture fraction, except for boundary and initial conditions.

When the residual material model is applied in combination with the weighted reaction progress model (for details, see Weighted Reaction Progress), the reaction progress is calculated based on the fuel tracer instead of mixture fraction. The relation between weighted reaction progress and reaction progress (Equation 7–44 and Equation 7–46) then becomes the following:

| (7–88)

where

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