Basic Definitions

Some of the specific terms that are used throughout the reacting flow theory content are summarized here.

Basic Definitions

Cell Residence Time

The cell residence time

τ_{r e s}

is the time step over which chemistry in a cell is evaluated. For unsteady flows this is the specified time-step, and for steady flows, this is the time that a fluid element spends in the cell—calculated as the mass in the cell divided by the mass flux in the cell.

Figure 1. EQUATION_DISPLAY

τ_{r e s} = \frac{m}{\dot{m}}

(3328)

Combustion Progress Variable

Unnormalized Progress Variable: The unnormalized progress variable $y$ is used to track the progress of combustion. It is calculated by Simcenter STAR-CCM+ based either on chemical enthalpy or species weights $W_{i}$ , depending on the option that is chosen when generating the flamelet table. For species weights, $y$ is defined as:; Figure 2. EQUATION_DISPLAY

$y = Σ (W_{i} Y_{i})$

(3329)
Unnormalized Progress Variable Variance: The unnormalized progress variable variance $y_{var}$ is calculated using either a transport equation or by an algebraic relationship.; Figure 3. EQUATION_DISPLAY

$y_{var} = \bar{{(y - \bar{y})}^{2}}$
(3330)
Progress Variable: The progress variable is denoted as $c$ . When $c = 0$ , combustion has not yet commenced. When $c = 1$ , combustion is complete. Simcenter STAR-CCM+ calculates the progress variable by considering the proportions of products that are formed, such as $C O$ and $C O_{2}$ during hydrocarbon combustion, or $H_{2} O$ during hydrogen combustion.; Figure 4. EQUATION_DISPLAY

$c = \frac{y - y_{u}}{y_{b} - y_{u}}$

(3331)
Progress Variable Variance: The progress variable variance is defined as:

$c_{var} = \bar{{(c - \bar{c})}^{2}}$

(3332); which is calculated from the unnormalized progress variable and its variance.

Equivalence Ratio

The equivalence ratio

Φ

relates the mass fraction

Y

of fuel to oxidizer in an unburned mixture to that of a stoichiometric mixture. It is calculated from the amount of carbon, hydrogen and oxygen in the mixture:

Figure 5. EQUATION_DISPLAY

Φ = \frac{Y_{(C, H) u} / Y_{(O) u}}{{(Y_{(C, H) u} / Y_{(O) u})}_{s t o i c h}}

(3333)

Figure 6. EQUATION_DISPLAY

= \frac{m_{(C, H), u} / m_{(O), u}}{{(m_{(C, H), u} / m_{(O), u})}_{s t o i c h}}

(3334)

Figure 7. EQUATION_DISPLAY

= \frac{n_{(C, H), u} / n_{(O), u}}{{(n_{(C, H), u} / n_{(O), u})}_{s t o i c h}}

(3335)

where u denotes unburnt.

Flamelet

A flamelet refers to a thin laminar flame which can also be referred to in combustion simulations as a fine structure.

Mass Fraction

The mass fraction

Y_{i}

of species

i

is given by the fraction of the total mass of all molecules of that species in a mixture

m_{i}

to the total mass of all molecules of all species in the mixture,

m

Figure 8. EQUATION_DISPLAY

Y_{i} = \frac{m_{i}}{m}

(3336)

where:

Figure 9. EQUATION_DISPLAY

m = \underset{i}{Σ} m_{i}

(3337)

and

m_{i}

is calculated as:

Figure 10. EQUATION_DISPLAY

m_{i} = ρ_{i} V

(3338)

where

V

is the volume.

Mean Molecular Weight

The mean molecular weight of a mixture is defined as:

Figure 11. EQUATION_DISPLAY

{\bar{M}}_{w} = \frac{m}{n}

(3339)

Combining Eqn. (3342), Eqn. (3344), Eqn. (3336), and Eqn. (3339) gives an expression for the relationship between mole and mass fractions.

Figure 12. EQUATION_DISPLAY

Y_{i} = \frac{{M_{w}}_{i}}{{\bar{M}}_{w}} X_{i}

(3340)

Mixture Fraction

The mixture fraction

Z

is defined as the elemental composition that originated from the fuel stream. For example, consider a hydrocarbon molecule CnHm burning in air. The mixture fraction is then the elemental mass fraction of C and H, regardless of the state of reaction. That is, the C and H atoms can be in unburnt fuel, CnHm, or in burnt products of H₂O and CO₂.

The mixture fraction ranges from 0 for pure oxidizer streams, to 1 for pure fuel streams.

For more information, see Mixture Fraction.

Mole Fraction

The mole fraction

X_{i}

of species

i

, is given by the fraction of moles which that species occupies in the mixture

n_{i}

to the total number of moles of all species in the mixture,

n_{s}

Figure 14. EQUATION_DISPLAY

X_{i} = \frac{n_{i}}{n_{s}}

(3342)

where:

Figure 15. EQUATION_DISPLAY

n_{s} = \underset{i}{Σ} n_{i}

(3343)

Molecular Weight

The molecular weight of species

i

is given by the sum of the atomic weights

A

of all elements

e

in the species, weighted with the number of atoms of each element

n

in the given species.

Figure 16. EQUATION_DISPLAY

{M_{w}}_{i} = \underset{e}{Σ} n_{i e} A_{e}

(3344)

As an example, molecular hydrogen H2, which consists of two hydrogen atoms, has a molecular weight of twice the molecular weight of atomic hydrogen, H.

The molecular weight can also be calculated from a given mixture of species:

Figure 17. EQUATION_DISPLAY

{M_{w}}_{i} = \frac{m_{i}}{n_{i}}

(3345)

where

n_{i}

is the number of moles of species

i

and

m_{i}

is the mass of species

i

Reaction Front (Flame Front)

The location at which reactions occur in a flame is known as the flame front, or reaction front. To determine the position of the flame front, often the concentration of OH (an intermediate species) is monitored.

Scalar Dissipation Rate

The scalar dissipation rate,

χ

, is the rate at which fluctuations in scalar variables (that are caused by turbulence) are dissipated. The scalar dissipation rate is given by:

Figure 18. EQUATION_DISPLAY

χ = \frac{D_{f} Z_{var}}{τ_{t u r b}}

(3346)

where

D

is the effective diffusivity and

τ_{t u r b}

is the turbulent timescale.

Flame temperatures depend strongly on the scalar dissipation rate—the maximum flame temperature decreases with an increasing scalar dissipation rate. Therefore, high values of scalar dissipation rate can cause extinction. When simulating a turbulent non-premixed flame, the scalar dissipation rate is highest near the inlet nozzle of a combustor (where the velocity is high). Extinction of the flame near the nozzle can cause lift-off of turbulent flames.

Stoichiometric Coefficient

$υ_{i j}$ is the stoichiometric coefficient of species $i$ which defines how many moles of the species take part in reaction $j$ . For example, for a system of two reactions,

Figure 19. EQUATION_DISPLAY

C H_{4} + 2 O_{2} \to C O_{2} + 2 H_{2} O

(3347)

Figure 20. EQUATION_DISPLAY

C + O_{2} \to C O_{2}

(3348)

the stoichiometric coefficients are:


Species	Reaction 1	Reaction 2
$C H_{4}$	1	0
$O_{2}$	2	1
$C$	0	1
$C O_{2}$	1	1
$H_{2} O$	2	0

In the Simcenter STAR-CCM+ GUI, all stoichiometric coefficients are entered as positive numbers.

For example, reaction 1 (above) shows that for every one mole of $C H_{4}$ that is consumed, there are two moles of $H_{2} O$ produced.