Chemical Equilibrium

In some reacting systems, the reaction occurs almost instantaneously when the necessary species are present in the correct proportions. For these systems, the atomic concentrations and initial temperature at a location determine the instantaneous value of any scalar at that spatial location and time.

The Chemical Equilibrium model assumes that what is mixed has reacted and reached chemical equilibrium. The model assumes that all species diffuse at the same rate, which is reasonable for turbulent flows where turbulent diffusivity is much greater than molecular diffusivity.

In addition to tracking the mixture fraction and its variance, when the Chemical Equilibrium model is active, Simcenter STAR-CCM+ additionally solves for heat loss ratio.

Heat Loss Ratio

The heat loss ratio

γ

is used to indicate the amount of heat loss or gain.

For non-adiabatic systems, Simcenter STAR-CCM+ uses heat loss ratio $γ$ to account for the loss or gain of heat which arises due to situations such as wall heat transfer, radiation, spray evaporation, etc. The amount of heat loss or gain represents the departure of the system from its adiabatic state. The flamelet table is additionally parameterized by heat loss ratio—which accounts for the non-adiabatic effects in the CFD domain. Heat loss ratio in the flamelet table represents the normalized enthalpy difference between the cell enthalpy and its adiabatic state which is a function of mixture fraction

Figure 1. EQUATION_DISPLAY

γ = \frac{h_{a d} - h}{h_{s e n s}}

(3520)

where $h_{a d}$ is the adiabatic enthalpy, $h$ is the cell enthalpy and $h_{s e n s}$ is the sensible (or thermal) enthalpy.

The sensible enthalpy, $h_{s e n s}$ , that is used in Eqn. (3520) is defined as:

Figure 2. EQUATION_DISPLAY

h_{s e n s} = \sum_{k = 0}^{N_{s p e c i e s}} Y_{k} (\int_{T_{r e f}}^{T} c_{p, k} d T)

(3521)

where:

$Y_{k}$ is the mass fraction of the $k$ 'th species
$T$ is the temperature
$T_{r e f}$ is the reference temperature. This value is 200K by default. You can specify the appropriate value under Numerical Settings for the Chemical Equilibrium Table Generator. See Chemical Equilibrium Reference Table.
$c_{p, k}$ is the specific heat of the $k$ 'th species

The adiabatic enthalpy, $h_{a d}$ , is a linear function of the mixture fraction:

Figure 3. EQUATION_DISPLAY

h_{a d} = (1 - Z) h_{o x i d} + (Z) h_{f u e l}

(3522)

Temperature is stored in the table and then retrieved based on the table dimensions.

Tabulation for Chemical Equilibrium

The independent variables are:

Mixture Fraction $Z$
Mixture Fraction Variance $Z_{var}$
Heat Loss Ratio (when using the Non-Adiabatic model) $γ$

When the boundary condition values are given for

Z

and the streams are specified for

Z = 1

(fuel stream) and

Z = 0

(oxidizer stream), the value of any conserved scalar

ϕ

(concentration for a specific element) is calculated at any spatial location according to:

Figure 4. EQUATION_DISPLAY

ϕ = Z ϕ_{f} + (1 - Z) ϕ_{o x}

(3523)

where

ϕ_{f}

and

ϕ_{o x}

represent the values of the conserved scalars in the fuel and oxidizer streams respectively. Therefore, when

Z

is known, the concentration of any given element is also known at that location. When the elemental concentrations are known, Simcenter STAR-CCM+ passes them along with the initial temperature to an equilibrium routine which yields:

the mass fractions of all species
the density
the temperature at that point (using a Gibbs free-energy minimization technique)

Integration

The averaged value of any scalar in a turbulent flow field is the instantaneous value that is integrated over the joint PPDF of $Z$ and $h$ :

Figure 5. EQUATION_DISPLAY

ϕ_{m e a n} (Z_{m e a n}, h_{m e a n}) = \int_{}^{} ϕ (Z, h) P (Z, h) d Z d h

(3524)

Simcenter STAR-CCM+ makes a statistical independence assumption for $P (Z, h)$ so that:

Figure 6. EQUATION_DISPLAY

P (Z, h) = P (Z) P (h)

(3525)

and further assumes a Dirac delta function for $P (h)$ . In other words, the mean value of enthalpy is used in the calculation of:

Figure 7. EQUATION_DISPLAY

ϕ_{m e a n} (Z_{m e a n}, h_{m e a n}) = \int_{}^{} ϕ (Z, h_{m e a n}) P (Z) d Z

(3526)

The integrals in Eqn. (3524) and Eqn. (3526) are pre-computed. When the value of an integral is needed during the calculation, a simple interpolation is all that is needed to provide the correct value.

The non-adiabatic PPDF model can work with Lagrangian evaporation models. In that case, a source term resulting from the evaporation of the Lagrangian phase is added to the right-hand side of the transport equation for mixture fraction $Z$ (Eqn. (3494)).

See Beta Probability Density Function (PDF).