Soot Moments

Soot Moment Source

where $R_r$ , $G_r$ , and $W_r$ are the instantaneous rates of particle nucleation, coagulation, and heterogeneous surface reaction terms (that is, surface growth, oxidation, and PAH condensation), respectively.

Nucleation Source

Particle nucleation (or particle inception) is an important first step in process of soot formation. Particle nucleation results in generation of the smallest sized soot particles. Nucleation is usually modeled as the coalescence of two PAH species (soot precursors) into a soot particle.

Single PAH Species Nucleation

For the single PAH species nucleation option, the soot precursor for inception is assumed to be pyrene (A4). If A4 is absent from the mechanism, an isomer with same chemical composition (C16H10) is used, such as A3R5. The nucleation rate for the moments is given as described by Frenklach et al. [786]:

$$R_0=\epsilon \sqrt{\frac{4\pi k_{B}T}{m_C{n}_{C_{PAH}}}}{d}_{PAH}^2{N}_{PAH}^2$$

(3679)

$$R_r=2n_{C_{PAH}}{R}_{r-1}r=1,2,\mathrm{...}$$

(3680)

where:

• $n_{C_{PAH}}$ is the number of carbon atoms per PAH molecule. For pyrene precursor (A4), $n_{C_{PAH}}=32$ .
• $m_C$ is the mass of a carbon atom (12.0).
• $d_{PAH}$ is the diameter of the PAH molecule.
• $N_{PAH}$ is the number density of the PAH molecules.

$N_{PAH}$ is based on the molar concentration of the PAH species which is obtained from the following quadratic equation, assuming a quasi-steady approximation for the PAH species:

$${\dot{\omega }}_{PAH}=k_{pi}{\left[PAH\right]}^2+k_{cond}\left[PAH\right]$$

(3681)

where ${\dot{\omega }}_{PAH}$ is the rate of formation of PAH from the A4 precursor, $k_{pi}$ represents the collision rate between the PAH molecules that form the first soot particle, and $k_{cond}$ represents the collision rate for the condensation of a PAH molecule on a soot particle.

C2H2 Nucleation

If the chemical mechanism used does not include A4 species, Simcenter STAR-CCM+ allows acetylene (C₂H₂) to be used as a soot precursor for nucleation. The expression for nucleation when using C₂H₂ as a soot precursor is given by:

$$R_r=2\frac{N_A}{C_{\min }}{k}_N\left(T\right)\left[C_2{H}_2\right]r=0$$

(3682)

For $r=1,2,3$ :

$$R_r=2C_{\min }{R}_{r-1}$$

(3683)

where:

• $N_A$ is Avogadro's number.
• $C_{\min }$ is the number of carbon molecules in the smallest PAH which coagulates into a dimer to form the incipient soot particle—set as 10.

The expression for $k_N$ is given by:

$$k_N=0.63\times {10}^4\exp \left(\frac{-E_A}{T}\right)$$

(3684)

where $E_A=21100$ 1/K.

Multi PAH Species Nucleation

Nucleation from multiple PAH-based species is modeled as the collision and coalescence of various gas-phase PAH species to form initial dimers.

To reduce the efficiency of the collisions between small PAH molecules, a sticking coefficient $s_C$ is used which is assumed to scale with the mass of the PAH molecules to the fourth power.

The rate of PAH dimerization is expressed as:

$${\dot{\omega }}_{\dim er}={\displaystyle \sum _i{s}_C{\left(\frac{4\pi k_{B}T}{m_i}\right)}^{1/2}{\left(\frac{6m_i}{\pi {\rho }_{soot}}\right)}^{2/3}{\left[{PAH}_i\right]}^2}$$

(3685)

where, $m_i$ is the mass of species $i$ , $k_B$ is the Boltzmann constant, ${\rho }_{soot}$ is the soot density, $T$ is the gas temperature, and $\left[{PAH}_i\right]$ is the concentration of PAH species $i$ .

The sticking coefficients are as summarized below [784].

Species Name	Chemical Formula	Chemical Symbol	Molecular Mass	Sticking Coefficient
Naphthalene	C10H8	A2	128	0.002
Acenaphthylene	C12H8	A2R5	152	0.004
Biphenyl	C12H10	P2	154	0.0085
Phenathrene	C14H10	A3	178	0.015
Acephenanthrylene	C16H10	A3R5	202	0.025
Pyrene	C16H10	A4	202	0.025
Fluranthene	C16H10	FLTN	202	0.025
Cyclopenta[cd]pyrene	C18H10	A4R5	226	0.039

Removal of these dimers occurs by nucleation and condensation. Both the source terms for nucleation and condensation depend on the concentration of dimers which are assumed to be in quasi steady state. Thus, the dimer concentration is calculated as the solution of a quadratic equation given by:

$${\dot{\omega }}_{\dim er}={\beta }_N{\left[\dim er\right]}^2+{\displaystyle \sum _{i=0}^{\infty }{\beta }_{C_i}{N}_i}\left[\dim er\right]$$

(3686)

where $\left[\dim er\right]$ is the concentration of dimers, and $N_i$ is the number density of soot particles of class size $i$ .

${\beta }_N$ and ${\beta }_{C_i}$ are collision rates given by [796]:

$${\beta }_N=2.2\sqrt{\frac{16\pi k_{B}T}{m_D}}{d_D}^2$$

(3687)

$${\beta }_{C_i}=\sqrt{\frac{\pi k_{B}T}{2{\mu }_{Di}}}{(d_D+d_{c_i})}^2$$

(3688)

where $m_D$ and $d_D$ are the mass and diameter of a dimer, respectively. ${\mu }_{D_i}$ is the reduced mass of a dimer and a soot particle, and $d_{c_i}$ is the collision diameter of a soot particle.

The nucleation source term for the zeroth moment is then given by:

$$R_0=\frac{1}{2}{\beta }_N{\left[DIMER\right]}^2$$

(3689)

PAH Condensation Source

Some PAH molecules can condense on the soot particles. The source term due to PAH condensation is modeled as described in Frenklach and Wang [788].

$$W_r^{cond}=2.2\sqrt{\frac{\pi k_{b}T}{2m_C}}\underset{k=0}{\overset{r-1}{\mathrm{\Sigma }}}\left(\begin{array}{c}r\\ k\end{array}\right)(C_h^2{M}_{r-k+\frac{1}{2}}^{PAH}{M}_k+2C_h{C}_s{M}_{r-k}^{PAH}{M}_{k+\frac{1}{3}}+C_s^2{M}_{r-k-\frac{1}{2}}^{PAH}{M}_{k+\frac{2}{3}})$$

(3690)

$r=\mathrm{1...4}$

$k=\mathrm{0...}(1-r)$

where $d_a$ is the collision diameter of PAH. Since only one class of PAH is assumed to condense onto the soot surface, the PAH moments in the expression above can be reduced to:

$$M_r^{PAH}=m_{PAH}^r{N}_{PAH}$$

(3691)

in Eqn. (3691), $r$ can be replaced by either:

• $r-k$
• $r-k+\frac{1}{2}$
• $r-k-\frac{1}{2}$

Coagulation Source

$G_r$ is evaluated by a double interpolation after inserting a mass-equivalent expression of the frequency rate of soot particle collision $f$ . A general outline is presented here that is taken from Frenklach [787]. First, a grid function of order $l$ in general form is defined as:

$$f_l^{(x,y)}=\underset{i=1}{\overset{\infty }{\mathrm{\Sigma }}}\underset{j=1}{\overset{\infty }{\mathrm{\Sigma }}}{\left(m_i+m_j\right)}^l{m}_i^x{m}_j^y{(m_i^{\frac{1}{3}}+m_i^{\frac{1}{3}})}^2{N}_i{N}_{j}l=0,1,2,\mathrm{...}$$

(3692)

Then the $G_r$ can be represented in terms of grid functions as:

$$G_0=\frac{1}{2}{K}_{fm}^{\prime }{M}_0^2{f}_{\frac{1}{2}}^{(0,0)}$$

(3693)

$$G_r=\frac{1}{2}{K}_{fm}^{\prime }{M}_0^2\underset{k=1}{\overset{r-1}{\mathrm{\Sigma }}}\left(\begin{array}{c}r\\ k\end{array}\right)f_{\frac{1}{2}}^{(k,r-k)}r=2,3,\mathrm{...}$$

(3694)

Fractional-ordered grid functions can be obtained by Lagrange interpolation of integer ordered grid functions. For example:

$$f_{\frac{1}{2}}=f_0^{\frac{3}{8}}{f}_1^{\frac{3}{4}}{f}_2^{-\frac{1}{8}}$$

(3695)

$f_l(l=0,1,2,\mathrm{...})$ can be evaluated in terms of fractional-order moments through interpolation. For example, upon simplification:

$$f_1=2{\mu }_{\frac{13}{6}}{\mu }_{\frac{1}{2}}+4{\mu }_{\frac{11}{6}}{\mu }_{\frac{5}{6}}+2{\mu }_{\frac{3}{2}}{\mu }_{\frac{7}{6}}$$

(3696)

where ${\mu }_r$ and fractional-order reduced moment $\mu$ is obtained by interpolating whole-ordered moments. The coefficient $K_{fm}^{\prime }$ is given by:

$$K_{fm}^{\prime }=\epsilon \sqrt{\frac{6k_{B}T}{{\rho }_{soot}}}\left(\frac{3m_C}{4\pi {\rho }_{soot}}\right)$$

(3697)

Surface Growth and Oxidation

HACA

The HACA mechanism (Appel et al. [781]) is summarized below:

Reaction Number	Reaction
1	$C_{soot}H+H\rightleftharpoons C_{soot}•+H_2$
2	$C_{soot}H+OH\rightleftharpoons C_{soot}•+H_{2}O$
3	$C_{soot}•+H\to C_{soot}H$
4a	$C_{soot}•+C_2{H}_2\to C_{soot}H+H$
4b	$C_{soot}•+C_2{H}_2\to C_{soot}•+H$
5	$C_{soot}•+O_2\to 2CO+products$
6	$C_{soot}•+OH\to CO+products$

In these reactions, $C_{soot}H$ represents a reactive site on the surface of the soot particle, and $C_{soot}•$ denotes the corresponding radical. The reaction rates are computed as Eqn. (3365).

The soot kinetic model parameters are given as (Appel et al. [781]):

The reactions with the gas-phase species are modeled as:

$$W_r^{C_2{H}_2}=k_{4a}\left[C_2{H}_2\right]\alpha {\chi }_{Csoot}•\pi C_s^2{M}_0\underset{l=0}{\overset{r-1}{\mathrm{\Sigma }}}\left(\begin{array}{c}r\\ l\end{array}\right){\mu }_{l+\frac{2}{3}}{\left(2\right)}^{r-l}$$

(3698)

$$W_r^{O_2}=k_5\left[O_2\right]\alpha {\chi }_{Csoot}•\pi C_s^2{M}_0\underset{l=0}{\overset{r-1}{\mathrm{\Sigma }}}\left(\begin{array}{c}r\\ l\end{array}\right){\mu }_{l+\frac{2}{3}}{(-2)}^{r-l}$$

(3699)

$$W_r^{OH}={\gamma }_{OH}\left[OH\right]\sqrt{\frac{\pi k_{B}T}{2m_{OH}}}{N_{A}C}_s^2{M}_0\underset{l=0}{\overset{r-1}{\mathrm{\Sigma }}}\left(\begin{array}{c}r\\ l\end{array}\right){\mu }_{l+\frac{2}{3}}{(-1)}^{r-l}$$

(3700)

where:

• $\alpha$ is the steric factor—the fraction of the soot surface sites available for reaction.
• $m_{OH}$ is the mass of the OH radical.
• ${\gamma }_{OH}$ is the collision efficiency for the OH radical.

The rate of the surface reaction depends on the particle surface area and the number of active reaction (radical) sites available. The number density of active radical sites can be obtained from the total number density of all radical sites on the soot surface using a steady-state approximation. The total number density of all radical sites is estimated to be 2.5 × 10¹⁵ cm^-2.

$$C_s={\left(\frac{6m_C}{\pi {\rho }_{soot}}\right)}^{\frac{1}{3}}$$

(3701)

${\chi }_{{Csoot}^{•}}$ is the number density of the surface radicals, given by:

$${\chi }_{{Csoot}^{•}}={\chi }_{{Csoot}^{-H}}\frac{k_1\left[H\right]+k_2\left[OH\right]}{k_{-1}\left[H_2\right]+k_{-2}\left[H_{2}O\right]+k_3\left[H\right]+k_{4a}\left[C_2{H}_2\right]+k_5\left[O_2\right]}$$

(3702)

where ${\chi }_{{Csoot}^{•}}$ is the nominal number density of the soot radical sites, for which a value of 2.3×10¹⁹ m⁻² is prescribed. In the present work, the reaction pathway 4a from the table is considered, and 4b is ignored.

HACA-RC

The HACA-RC mechanism (Mauss et al. [794]) is summarized below:

Reaction Number	Reaction
1a	$C_{soot,i}H+H\stackrel{k_{1a}}{\rightleftharpoons }{C}_{soot,i}•+H_2$
1b	$C_{soot,i}H+OH\stackrel{k_{1b}}{\rightleftharpoons }{C}_{soot,i}•+H_2$
2	$C_{soot,i}•+H\stackrel{k_2}{\to }{C}_{soot,i}H$
3a	$C_{soot,i}•+C_2{H}_2\stackrel{k_{3a}}{\rightleftharpoons }{C}_{soot,i}{C}_2{H}_2•$
3b	$C_{soot,i}{C}_2{H}_2•\stackrel{k_{3b}}{\rightleftharpoons }{C}_{soot,i+1}H+H$
4a	$C_{soot,i}•+O_2\stackrel{k_{4a}}{\to }{C}_{soot,i-1}•+2CO$
4b	$C_{soot,i}{C}_2{H}_2•+O_2\stackrel{k_{4b}}{\to }{C}_{soot,i}+2CHO$
5	$C_{soot,i}H+OH\stackrel{k_5}{\to }{C}_{soot,i-1}•+CH+CHO$

In these reactions, $C_{soot,i}H$ represents a reactive site on the surface of the soot particle, and $C_{soot,i}•$ denotes the corresponding radical. $i$ and $i+1$ denote the size classes.

In the first step, an H radical is abstracted from the soot surface by either an H or OH radical forming an active radical site that can in turn be deactivated by the addition of an H radical. An addition of acetylene follows that leads to the formation of a new aromatic ring. This step is assumed to be reversible (3b). The next two reactions (4a) and (4b) describe the oxidation of a radical site by molecular oxygen while the last reaction in the scheme describes the oxidation of an active (non-radical) site by an OH radical. Introducing a steady-state assumption for the radical sites $C_{soot,i}•$ and $C_{soot,i}{C}_2{H}_2•$ leads to an algebraic equation for their respective concentrations:

$$[C_{soot}•]=\left(\frac{k_{1a,f}\left[H\right]+k_{1b,f}\left[H\right]+k_5\left[OH\right]}{k_{1a,b}\left[H_2\right]+k_{1b,b}\left[H_{2}O\right]+k_2\left[H\right]+k_{3a,f}\left[C_2{H}_2\right]f_{3,a}+k_4\left[O_2\right]}\right)\left[C_{soot}\right]$$

(3703)

$$[C_{soot}{C}_2{H}_2•]=\left(\frac{k_{3a,f}\left[C_2{H}_2\right]}{k_{3b,f}+k_{3a,b}+k_{4,b}\left[O_2\right]}\right)[C_{soot}•]$$

(3704)

where:

$$f_{3,a}=\frac{k_{3b,f}}{k_{3b,f}+k_{3a,b}+k_{4,b}\left[O_2\right]}$$

(3705)

The concentration of active sites can be calculated by:

$$\left[C_{soot}\right]=\underset{i=1}{\overset{\infty }{\mathrm{\Sigma }}}\alpha \frac{{\chi }_{soot}}{N_A}{S}_i{N}_i$$

(3706)

• $\alpha$ is the steric factor—the fraction of the soot surface sites available for reaction.
• ${\chi }_{soot}$ is the number of surface sites per unit area.
• $N_A$ is Avogadro's number.
• $S_i$ is the surface area of size $i$ .
• $N_i$ is the number of particles of size $i$ .

The rates of surface reactions for Eqn. (3703) to Eqn. (3706) are [798]:

$${\dot{M}}_{r,sg}=\alpha k_{3a,f}\left[C_2{H}_2\right]f_{3a}A\underset{k=0}{\overset{\mathrm{r–}1}{\mathrm{\Sigma }}}\left(\begin{array}{c}r\\ k\end{array}\right){}^s{M}_{k+2/3}\mathrm{\Delta }{m}^{\mathrm{r–}k},r=1,2,\mathrm{...}$$

(3707)

$${\dot{M}}_{r,ox}=\alpha (k_{4a}\left[O_2\right]A+k_5\left[OH\right])\underset{k=0}{\overset{\mathrm{r–}1}{\mathrm{\Sigma }}}\left(\begin{array}{c}r\\ k\end{array}\right){}^s{M}_{k+2/3}\mathrm{\Delta }{m}^{\mathrm{r–}k},r=1,2,\mathrm{...}$$

(3708)

The change of mass $\mathrm{\Delta }m$ takes on the following values:

• $\mathrm{\Delta }m=1$ for surface growth
• $\mathrm{\Delta }m=-1$ for oxidation

To describe the burnout of the smallest soot particles due to oxidation, the rate of the zero-th moment is corrected by:

$${\dot{M}}_{r,ox}=\alpha (k_{4a}\left[O_2\right]A+k_5\left[OH\right])M_{-1/3}$$

(3709)

The moment-independent pre-factors retrieved from the flamelet libraries are thus:

$$P_{sg}=\alpha k_{3a,f}\left[C_2{H}_2\right]f_{3a}A$$

(3710)

$$P_{ox}=\alpha (k_{4a}\left[O_2\right]A+k_5\left[OH\right])$$

(3711)

where:

$$A=\left(\frac{k_{1a,f}\left[H\right]+k_{1b,f}\left[H\right]+k_5\left[OH\right]}{k_{1a,b}\left[H_2\right]+k_{1b,b}\left[H_{2}O\right]+k_2\left[H\right]+k_{3a,f}\left[C_2{H}_2\right]f_{3,a}+k_4\left[O_2\right]}\right)$$

(3712)

In the HACA and HACA-RC mechanisms, the parameter $\alpha$ is difficult to quantify from a theoretical point of view and can have a marked influence on the soot levels. Steric factor $\alpha$ is defined as the fraction of the reactive sites ('armchair' sites) on the surface of the soot particle that are available for surface growth or oxidation reactions. The value of $\alpha$ ranges from zero to unity.

Various values and functional forms for $\alpha$ are found in literature. Simcenter STAR-CCM+ provides three ways to compute $\alpha$ : a constant value, a premixed temperature-based correlation, and as a user-defined scalar.

$a=12.65-0.00563T$

$b=-1.38+0.00068T$

and