Soot Two-Equation

The soot two-equation model developed here is based on the modeling framework provided by [785] and [799].

This model is a semi-empirical soot model where the overall soot emission has been modeled with four key physical processes:

Nucleation
Coagulation
Surface-growth
Oxidation

This model is particularly useful for non-premixed turbulent jet flames.

Although soot number density and mass density are the primary variables of interest, for numerical convenience, scaled variables are preferred for the transport equations. The transport equations for the soot two-equation model are given by:

Figure 1. EQUATION_DISPLAY

\frac{\partial (ρ Ø_{N})}{\partial t} + \nabla\cdot (ρ {v Ø}_{N} - D \nabla Ø_{N}) = {\dot{ω}}_{Ø_{N}}

(3651)

Figure 2. EQUATION_DISPLAY

\frac{\partial (ρ Ø_{M})}{\partial t} + \nabla\cdot (ρ {v Ø}_{M} - D \nabla Ø_{M}) = {\dot{ω}}_{Ø_{M}}

(3652)

Scaled soot number density $Ø_{N}$: The scaled soot number density is expressed in terms of soot number density $N$ [1/m³] as:; Figure 3. EQUATION_DISPLAY

$Ø_{N} = \frac{N}{ρ N_{A}}$

(3653); where $N$ is the soot number density and $N_{A}$ is Avogadro's constant.
Scaled soot mass density $Ø_{M}$: The scaled soot mass density is expressed in terms of soot mass density $M$ [kg/m³] as:; Figure 4. EQUATION_DISPLAY

$Ø_{M} = \frac{M}{ρ}$

(3654); where $M$ is the soot mass density.

Soot Two-Equation Source Terms

The Soot Two-Equation Source terms are expressed as:

For scaled soot number density rate:
Figure 5. EQUATION_DISPLAY

${\dot{ω}}_{Ø N} = \frac{d ρ Ø_{N}}{d t} = \frac{1}{N_{A}} \frac{d N}{d t}$

(3655)
For scaled soot mass density rate:
Figure 6. EQUATION_DISPLAY

${\dot{ω}}_{Ø M} = \frac{d ρ Ø_{M}}{d t} = \frac{d M}{d t}$

(3656)

Physical processes

The source terms

{\dot{ω}}_{Ø N}

and

{\dot{ω}}_{Ø M}

are split into four physical processes:

Nucleation

PAH

One nucleation submodel is based on polycyclic aromatic hydrocarbons (PAH), and may be expressed as:

Figure 7. EQUATION_DISPLAY

\begin{matrix} {(\frac{d N}{d t})}_{n u} = 8 \cdot c_{2} \frac{N_{A}}{M_{P}} [ρ^{2} {(\frac{Y_{C_{2} H_{2}}}{{M w}_{C_{2} H_{2}}})}^{2} \frac{Y_{C_{6} H_{5}} {M w}_{H_{2}}}{{M w}_{C_{6} H_{5}} Y_{H_{2}}}] e^{- \frac{E_{A}}{T}} \\ + 8 \cdot c_{3} \frac{N_{A}}{M_{P}} [ρ^{2} \frac{{Y_{C_{2} H_{2}} Y_{C_{6} H_{6}} Y}_{C_{6} H_{5}} {M w}_{H_{2}}}{{{M w}_{C_{2} H_{2}} {M w}_{C_{6} H_{6}} M w}_{C_{6} H_{5}} Y_{H_{2}}}] e^{- \frac{E_{A}}{T}} \end{matrix}

(3657)

where

E_{A}

is the activation energy, specified here.

c_{2}

and

c_{3}

are model constants with the following values:

$c_{2} = 127 \times 10^{8.88}$
$c_{3} = 178 \times 10^{9.50}$

See [799].

C2H2

Alternatively, an acetylene-based nucleation submodel can be used which requires the soot precursor C2H2. If C2H2 is specified in the chemical mechanism or species list—and also listed as a species for tabulation when using a flamelet model—C2H2 concentration is provided by the combustion model. When C2H2 is not provided by the combustion model, the C2H2 concentration is calculated empirically from the mixture fraction.

The nucleation rate for soot number density is expressed as:

Figure 8. EQUATION_DISPLAY

{(\frac{d N}{d t})}_{n u} = C_{1} N_{A} ρ \frac{Y_{C 2 H 2}}{{M w}_{C 2 H 2}} e^{\frac{- E_{A}}{T}}

(3658)

The nucleation rate for mass density is expressed as:

Figure 9. EQUATION_DISPLAY

{(\frac{d M}{d t})}_{n u} = \frac{M_{p}}{N_{A}} {(\frac{d N}{d t})}_{n u}

(3659)

where

M_{p} = 1200 \frac{k g}{k m o l}

and

C_{1} = 54 / s

.

Coagulation

Figure 10. EQUATION_DISPLAY

{(\frac{d N}{d t})}_{c o} = - {(\frac{24 R_{u}}{ρ_{s o o t} N_{A}})}^{1 / 2} {(\frac{6}{π ρ_{s o o t}})}^{1 / 6} T^{1 / 2} M^{1 / 6} N^{11 / 6}

(3660)

where

R_{u}

is the universal gas constant.

Figure 11. EQUATION_DISPLAY

{(\frac{d M}{d t})}_{c o} = 0

(3661)

Soot Surface Growth (requires C2H2)

Figure 12. EQUATION_DISPLAY

{(\frac{d N}{d t})}_{s g} = 0

(3662)

Figure 13. EQUATION_DISPLAY

{(\frac{d M}{d t})}_{s g} = C_{4} ρ \frac{Y_{C 2 H 2}}{M_{W, C 2 H 2}} e^{\frac{- E_{A}}{T}} [{(π N)}^{1 / 3} {(\frac{6 M}{ρ_{s o o t}})}^{2 / 3}]

(3663)

where

C_{4} = 9000.6 \frac{k g - m}{k m o l - s}

Soot Oxidation (requires O2 and OH)

Figure 14. EQUATION_DISPLAY

{(\frac{d N}{d t})}_{o x} = 0

(3664)

Figure 15. EQUATION_DISPLAY

{(\frac{d M}{d t})}_{o x} = - C_{5} ρ η \frac{Y_{O H}}{{M w}_{O H}} T^{\frac{1}{2}} [{(π N)}^{\frac{1}{3}} {(\frac{6 M}{ρ_{s o o t}})}^{\frac{2}{3}}] - C_{6} ρ \frac{Y_{O 2}}{{M w}_{O 2}} T^{\frac{1}{2}} e^{\frac{- E_{A}}{T}} [{(π N)}^{\frac{1}{3}} {(\frac{6 M}{ρ_{s o o t}})}^{\frac{2}{3}}]

(3665)

where

η = 0.13

,

C_{5} = 105.81 \frac{k g - m}{k m o l - K^{1 / 2} s}

, and

C_{6} = 8903.51 \frac{k g - m}{k m o l - K^{1 / 2} s}

.

Source Terms

The total source terms for number density and mass are:

Figure 16. EQUATION_DISPLAY

\frac{d N}{d t} = {(\frac{d N}{d t})}_{n u} + {(\frac{d N}{d t})}_{c o}

(3666)

Figure 17. EQUATION_DISPLAY

\frac{d M}{d t} = {(\frac{d M}{d t})}_{n u} + {(\frac{d M}{d t})}_{s g} + {(\frac{d M}{d t})}_{o x}

(3667)

Equations Eqn. (3666) and Eqn. (3667) are substituted in Eqn. (3655) and Eqn. (3656) to obtain the source terms for the transport equations.

User-Defined Source Terms: The two-equation model developed here allows you to specify your own rates for the four physical processes (nucleation, coagulation, surface growth, and oxidation).; The generalized framework for the computation of source term for $Ø = Ø_{N}, Ø_{M}$ is:
Figure 18. EQUATION_DISPLAY

${({\dot{ω}}_{Ø})}_{t o t a l} = A_{1} + A_{2} {\dot{ω}}_{Ø}$

(3668); where $A_{1}$ is the user-specified rate and $A_{2}$ is a scale factor that you specify for each of the four processes (nucleation, coagulation, surface growth, and oxidation).; If $A_{1}$ =0 and $A_{2}$ =1, the total source term ${({\dot{ω}}_{Ø})}_{t o t a l}$ is computed from equations Eqn. (3658) to Eqn. (3665).; By setting $A_{2}$ =0 (for nucleation, coagulation, surface growth, and oxidation), you can disable the Simcenter STAR-CCM+ built-in source computations. You can then implement user-specified source terms by defining the user-specified rates ( $A_{1}$ ) for each of the physical processes. When defining user-specified rates, an additional option to specify values for Jacobians is provided for better numerical stability.

Species Options

By default, the user-specified rate option is deactivated and all of the scaling factors are set to 1.0. The complete description of the physical processes requires specification of the mass fraction of a few species. These species may or may not be the part of the reaction set up. So choices are provided for you to compute the species concentrations. These options are available for O, OH, and C2H2.

Fluid Streams

Based on the fluid stream definition, an algebraic expression is used to compute the local mixture fraction with Bilger’s mixture fraction definition [783]:

Figure 19. EQUATION_DISPLAY

Z = \frac{\frac{2 (Y_{C} - Y_{C, o x})}{{M w}_{C}} + \frac{(Y_{H} - Y_{H, o x})}{{2 M w}_{H}} - \frac{(Y_{O} - Y_{O, o x})}{{M w}_{O}}}{\frac{2 (Y_{C, f} - Y_{C, o x})}{{M w}_{C}} + \frac{(Y_{H, f} - Y_{H, o x})}{{2 M w}_{H}} - \frac{(Y_{O, f} - Y_{O, o x})}{{M w}_{O}}}

(3669)

$Y$ refers to the elemental mass fraction. The $C$ , $H$ , and $O$ subscripts refer to elemental carbon, hydrogen, and oxygen. The $f$ and $o x$ subscripts refer to the fuel and oxidizer streams.

Activation Energy Values


Location of $E_{A}$	Value of $E_{A}$ 1/K
Eqn. (3657)	for $C_{2}$ = 4378 for $C_{3}$ = 6390
Eqn. (3658)	for $C_{1}$ = 21110
Eqn. (3663)	for $C_{4}$ = 12100
Eqn. (3665)	for $C_{6}$ = 19778