The Adiabatic Equilibrium Model

In this model, the instantaneous value of any scalar at any spatial location and time is uniquely determined by the atomic concentrations and initial temperature at that location and time. Since, from mass conservation, the number of atoms in the reactor as whole does not change, the reaction state space can be characterized by a single conserved scalar. In other words, what is mixed has reacted and reached chemical equilibrium. This approach accommodates both forward and reverse reactions. A basic assumption of the model is that all species diffuse at the same rate, which is reasonable for turbulent flows where turbulent diffusivity is much greater than molecular diffusivity.

Under this assumption, if one defines a mixture fraction

(1)

where is the total mass of all atoms originating from the primary inlet (typically, fuel) at any spatial location, and is the total mass of all atoms originating from the secondary inlet (typically, oxidizer), it can be seen that is a conserved scalar. The equation for such a scalar is simple; is transported by convection and diffusion, and can accumulate locally, but there is no production. Therefore:

(2)

Note that for steady-state flows there is no accumulation, so the first term on the left-hand side of Eqn. 2 is zero. Boundary condition values are also needed for . After obtaining the latter, we can calculate the value of any conserved scalar at any spatial location according to the following formula:

(3)

where and are the values of the conserved scalar in the primary and secondary streams. In particular, if stands for the concentration of atoms of a particular element, then knowing automatically gives the concentration of that element's atoms at any point. Once the atomic concentrations are known, STAR-CCM+ feeds them and the initial temperature to an equilibrium routine that yields:

The only inputs needed for the equilibrium model are the boundary condition values for and the specification of the and streams in every respect, i.e. temperature, molecular weight, atomic composition, etc.