8.3.4 Using the Non-Premixed Model with the Inert Model

To model the effect of dilution on combustion without the expense of using two mixture fractions, ANSYS FLUENT allows the introduction of an inert stream into the domain. Unlike a secondary mixture fraction, the inert does not chemically equilibrate with the primary fuel and oxidizer - instead, its composition remains constant after mixing. However the inert stream does affect the solution due to its influence on enthalpy, specific heat, and density of the mixture. The equation for conservation of inert is written as:

where
	$Y_I$	=	inert stream tracer
	Sc $_t$	=	turbulent Schmidt number
	$\mu_t$	=	turbulent viscosity
	$\rho$	=	density

Equation  8.3-5 has no sources or sinks, because the problem is reduced to tracking a conserved scalar when it is assumed that the inert components have the same turbulent diffusivities.

Mixture Composition

The mixture properties are computed from the mean ( $\bar{f}$) and variance ( $\bar{f'}$) of the mixture fraction in the cell, the reaction progress variable ( $c$, when the partially premixed model is enabled), the cell enthalpy ( $H$, for non-adiabatic flows), and the inert tracer ( $Y_I$). The mixture is modeled as a blend of inert and active species, but the PDF tables need to be accessed with conditioned variables. Conditioning is necessary to take into account the volume taken up by the inert fraction, yet still be able to utilize previously built tables by straightforward lookup. The mean mixture fraction and mixture fraction variance used to access the PDF table is given by:

$$\bar{f_I} = \frac{\bar{f}}{1-Y_I}$$ (8.3-6)

$$\bar{f'^2_I} = \frac{\bar{f'^2}}{(1-Y_I)^2}$$ (8.3-7)

The reaction progress variable $c$ is not conditioned, however the cell enthalpy must be conditioned to account for the inert enthalpy. The inert enthalpy and active enthalpy are obtained from the following relationships:

where $H$ is the enthalpy of the cell at temperature $T$, $H^{pdf}$ is the enthalpy of the active mixture-fraction stream and $H^I$ is the enthalpy of the inert stream. Here it is assumed that the inert and the active streams have the same temperature, but different enthalpies. To calculate the temperature in the cell, Equation  8.3-8 is solved for the temperature and for $H^{pdf}$, which gives the partitioning of the energy between the inert and active streams.

The inert enthalpy is defined as

where $Y_i$ refers to the mass fraction of specie $i$ defined in the inert stream, $T_0$ is the reference temperature, $C_{p_i}$ the specific heat of specie $i$, and $N_I$ is the number of inert species.

The inert and PDF enthalpies are defined further in this equation in the separate User's Guide.

Property Evaluation

The specific heat of the mixture is evaluated by mixing the inert and active streams in the following way:

The density of the mixture is calculated by using a harmonic average of the densities of the active and inert streams, weighted by the inert tracer:

Here, the inert density ( $\rho^I$) is calculated from the ideal gas law.

For information on how to set up the inert model, see this section in the separate User's Guide.