Soot Sections

The soot sectional method is based on a description of sections containing soot particles of equal volume, allowing a volume-based discretization of particle sizes together with conservation of the soot number density and mass.

The specified number of sections are transported in the section model. For each section, the soot mass fraction transport equation is solved, given by:

Figure 1. EQUATION_DISPLAY

\bar{ρ} \frac{\partial Y_{i, s o o t}}{\partial t} + \nabla\cdot [\bar{ρ} v_{j} Y_{i, s o o t} - \bar{ρ} \frac{μ_{t}}{{S c}_{s o o t}} \nabla Y_{i, s o o t}] = ρ_{s o o t} {\tilde{Ω}}_{i, s o o t}

(3715)

where

{S c}_{s o o t}

is the soot Schmidt number, and:

Figure 2. EQUATION_DISPLAY

Y_{i, s o o t} = \frac{m_{i, s o o t}}{m_{c e l l}}

(3716)

where

i

is the index of the section,

m_{i, s o o t}

is the soot mass in section

i

of a given cell, and

m_{c e l l}

is the gas mass of the cell.

ρ_{s o o t}

is the density of a soot particle that is used to determine the soot mass density

M_{s o o t}

in Eqn. (3750).

The soot source term

{\tilde{Ω}}_{i, s o o t}

is given by:

Figure 3. EQUATION_DISPLAY

{\tilde{Ω}}_{i, s o o t} = [{\tilde{Ω}}_{i, n u c} + {\tilde{Ω}}_{i, c o n d} + {\tilde{Ω}}_{i, s g} + {\tilde{Ω}}_{i, o x} + {\tilde{Ω}}_{i, c o a g}]

(3717)

where $n u c$ is nucleation, $c o n d$ is condensation, $s g$ is surface growth, $o x$ is oxidation, and $c o a g$ represents coagulation source terms.

To evaluate the soot mass in each section $i$ , the model describes all formation and oxidation stages ranging from gas phase to solid interactions.

The minimum

v_{1, \min}

and maximum

v_{1, \max}

volume of section 1 is given by:

Figure 4. EQUATION_DISPLAY

v_{1, \min} = \frac{3 v_{P A H}}{2}

(3718)

and:

Figure 5. EQUATION_DISPLAY

v_{1, \max} = v_{M I N} + v_{C 2}

(3719)

where

v_{P A H}

is the polyaromatic hydrocarbon (PAH) volume and

v_{C 2}

is the volume of two carbon atoms.

The maximum

v_{i, \max}

, mean

v_{i, m}

, and minimum

v_{i, \min}

volume of each section

i

, is given by:

Figure 6. EQUATION_DISPLAY

\begin{matrix} v_{i, \min} = v_{i - 1, \max} & f o r i > 1 \end{matrix}

(3720)

Figure 7. EQUATION_DISPLAY

v_{i, m} = \frac{v_{i, \min} + v_{i, \max}}{2}

(3721)

Figure 8. EQUATION_DISPLAY

v_{i, m a x} = (v_{M I N} + v_{C 2}) {(\frac{v_{M A X}}{v_{M I N} + v_{C 2}})}^{\frac{i - 1}{i_{\max} - 1}}

(3722)

where

v_{M I N}

and

v_{M A X}

are the absolute minimum and maximum volumes, respectively, across all sections, and

i_{\max}

is the index of the last section.

Figure 9. EQUATION_DISPLAY

Δ v_{i} = v_{i, \max} - v_{i, \min}

(3723)

A constant volume profile is assumed within each section thereby giving the soot volume distribution

q_{i} (v)

within section

i

Figure 10. EQUATION_DISPLAY

q_{i} (v) = \frac{ρ Y_{i}}{ρ_{s o o t} Δ v_{i}}

(3724)

The size repartition

f (i, s o o t)

of section

i

, and the total volume fraction

Q_{i}

are given by:

Figure 11. EQUATION_DISPLAY

f (i, s o o t) = \frac{ρ Y_{i}}{ρ_{s o o t} v_{i, m}}

(3725)

Figure 12. EQUATION_DISPLAY

Q_{i} = \int_{v_{i, \min}}^{v_{i, \max}} q_{i} (v) d v

(3726)

Nucleation: It is assumed that the collision of two PAH molecules forms a soot particle and that the volume of the first soot particle can be expressed as:
Figure 18. EQUATION_DISPLAY

${\tilde{Ω}}_{1, n u c} = v_{P A H} β_{P A H, P A H} N_{P A H}^{2}$

(3732)
where $β_{P A H, P A H}$ is the collision frequency of PAH molecules, and $N_{P A H}$ is the number of PAH molecules. The nucleation source term is only applied to the first section. You can use any of the three available nucleation options.
Condensation: The volume change in section $i$ due to condensation of PAH molecules can be expressed as:
Figure 19. EQUATION_DISPLAY

$Δ Q_{i, c o n d} = v_{P A H} N_{P A H} \int_{v_{i, \min}}^{v_{i, \max}} β_{P A H, P A H} n_{i} (v) d v$

(3733)
Figure 20. EQUATION_DISPLAY

$n_{i} (v) = \frac{q_{i} (v)}{v}$

(3734); Concentration of PAH is required to compute the condensation source. It is computed by solving the following quadratic equation:
Figure 21. EQUATION_DISPLAY

$R_{P A H} = 2 β_{P A H, P A H} N_{P A H}^{2} + Σ_{i}^{i_{\max}} N_{P A H} \int_{v_{i, \min}}^{v_{i, \max}} β_{P A H, P A H} n (v) d v$

(3735)
The net condensation rate for section $i$ , results from the balance of the condensation of particles from adjacent sections.
Figure 22. EQUATION_DISPLAY

$Δ Q_{i, c o n d} = Δ q_{i . c o n d}^{\to} - Δ q_{i . c o n d}^{↑}$

(3736)
$Δ q_{i . c o n d}^{↑}$ is the rate of soot particles leaving the section, while $Δ q_{i . c o n d}^{\to}$ is the rate of soot particles entering section $i$ .; Figure 23. EQUATION_DISPLAY

$Δ q_{i . c o n d}^{↑} = \frac{1}{\frac{(v_{i + 1, \max} - v_{i + 1, \min}) \ln (v_{i, \max} / v_{i, \min})}{(v_{i, \max} - v_{i, \min}) \ln (v_{i + 1, \max} / v_{i + 1, \min})} - 1} Δ Q_{i, c o n d}$

(3737)
Figure 24. EQUATION_DISPLAY

$Δ q_{i . c o n d}^{\to} = \frac{1}{\frac{(v_{i, \max} - v_{i, \min}) \ln (v_{i + 1, \max} / v_{i + 1, \min})}{(v_{i + 1, \max} - v_{i + 1, \min}) \ln (v_{i, \max} / v_{i, \min})}} Δ Q_{i, c o n d}$

(3738)
The condensation source terms for each section can be written as:
Figure 25. EQUATION_DISPLAY

$Ω_{1, c o n d} = - Δ q_{i, c o n d}^{↑}$

(3739)
Figure 26. EQUATION_DISPLAY

$Ω_{i, c o n d} = Δ q_{i - 1, c o n d}^{\to} - Δ q_{i, c o n d}^{↑}, i = 2, 3, ..., i_{\max} - 1$

(3740)
Figure 27. EQUATION_DISPLAY

$Ω_{i_{M A X}, c o n d} = Δ q_{i_{M A X} - 1, c o n d}^{\to}$

(3741)
Surface Growth and Oxidation: Soot surface growth depends on the soot surface reaction mechanism—HACA or HACA-RC (Hydrogen Abstraction Carbon Addition Ring Closure) mechanism. The soot volume change in section $i$ can be expressed as:
Figure 28. EQUATION_DISPLAY

$Δ Q_{i, s g} = α v_{C_{2}} k_{s g} \int_{v_{i, \min}}^{v_{i, \max}} {(\frac{v}{v_{C_{2}}})}^{\frac{θ}{3}} n_{i} (v) d v$

(3742)
Figure 29. EQUATION_DISPLAY

$Δ Q_{i, o x} = α v_{C_{2}} k_{o x} \int_{v_{i, \min}}^{v_{i, \max}} {(\frac{v}{v_{C_{2}}})}^{\frac{θ}{3}} n_{i} (v) d v$

(3743); $k_{s g}$ and $k_{o x}$ are the surface growth and oxidation rates, respectively, depending on the HACA or HACA-RC kinetics. The constant $θ$ is the soot surface fractal dimension which varies with the soot particle diameter. A lower fractal dimension value is set for particles smaller than 20nm, a higher value is set for particles greater than 60nm, and between these values the fractal dimension evolves linearly.; The net surface growth and oxidation for each section—based on equations similar to Eqn. (3737) and Eqn. (3738)—are given by:
Figure 30. EQUATION_DISPLAY

$Ω_{1, s g} = - Δ q_{i, s g}^{↑}$

(3744)
Figure 31. EQUATION_DISPLAY

$Ω_{i, s g} = Δ q_{i - 1, s g}^{\to} - Δ q_{i, s g}^{↑}, i = 2, 3, ..., i_{\max} - 1$

(3745)
Figure 32. EQUATION_DISPLAY

$Ω_{i_{M A X}, s g} = Δ q_{i_{M A X} - 1, s g}^{\to}$

(3746); and; Figure 33. EQUATION_DISPLAY

$Ω_{i, o x} = Δ q_{i - 1, o x}^{\to} - Δ q_{i, o x}^{↑}, i = 2, 3, ..., i_{\max} - 1$

(3747)

Figure 34. EQUATION_DISPLAY

$Ω_{i_{M A X}, o x} = Δ q_{i_{M A X} - 1, o x}^{\to}$

(3748)
Coagulation: The formulation for the coagulation source for each section is given by:
Figure 35. EQUATION_DISPLAY

$Ω_{i, c o a g} = \underset{v_{i, \min} < v_{k} + v_{j} < v_{i, \max}}{Σ} (v_{k} + v_{j}) N_{k} N_{j} β_{t r, c o a g} (v_{k}, v_{j}) + \underset{v_{k} + v_{i} < v_{i, \max}}{Σ} v_{k} N_{k} N_{j} β_{t r, c o a g} (v_{k}, v_{i}) - v_{i} N_{i} \underset{v_{i} + v_{j} < v_{i, \max}}{Σ} N_{j} β_{t r, c o a g} (v_{i}, v_{j}) - 2 v_{i} N_{i} N_{i} β_{t r, c o a g} (v_{i}, v_{j})$

(3749); $N_{i}$ is the number density of particle size $i$ and $β_{t r, c o a g}$ is the collision frequency for coagulation—which is a function of particle size, temperature, and pressure.

Soot Mass Density: Figure 36. EQUATION_DISPLAY

$M_{s o o t} = ρ_{s o o t} f_{v}$

(3750)
Soot Volume Fraction: Figure 37. EQUATION_DISPLAY

$f_{v} = Σ_{i = 1}^{i_{\max}} \frac{ρ Y_{i, s o o t}}{ρ_{s o o t}}$

(3751)
Soot Mean Diameter: Figure 38. EQUATION_DISPLAY

$d_{p} = {(\frac{6 f_{v}}{N_{t o t} \cdot π})}^{1 / 3}$

(3752)
Soot Number Density: Figure 39. EQUATION_DISPLAY

$N_{t o t} = Σ_{i = 1}^{i_{\max}} \frac{ρ Y_{i}}{ρ_{s o o t} (Δ v_{i})} \log (\frac{v_{i, \max}}{v_{i, \min}})$

(3753)
Particle Size Distribution Function: The particle size distribution function for section $i$ (where $i = 1 t o i_{\max}$ ), is given by:
Figure 40. EQUATION_DISPLAY

$\frac{d N}{d \log (d_{p})} (i) = \frac{N_{i}}{\frac{1}{3} \log_{10} (\frac{6 v_{i, \max}}{π}) - \frac{1}{3} \log_{10} (\frac{6 v_{i, \min}}{π})}$

(3754)