3.2.13. Soot Generation Model for Multicomponent Combustion

Soot is an important contributor to radiative exchange within a fire and between a fire and its surroundings. Soot production, destruction and transport at flame scales are still active areas of research, with important chemical/physical processes not understood from a fundamental physics point of view. Basically, soot particles are carbon-rich solid particles generated in regions of excess pyrolyzate, such as on the rich side of a diffusion flame. Unagglomerated soot particles have characteristic dimensions in the range 0.01–0.05 $\mu$ ).

{it The main purpose of the soot model is for the calculation of the absorption coefficient} in the radiant energy transfer equation. For the current implementation we employ the soot model implemented in the KAMELEON code because it has been used for large turbulent fire calculations with participating media radiation. The model is discussed in Magnussen et al. [43] and Magnussen and Hjertager [62] It is a two-step formulation, first described by Tesner et al. [63]. The model for generation and combustion of soot can be summarized by three principal steps: 1) particle nucleation, where the first solid soot particles (often called radical nuclei) are created as a result of fuel oxidation and pyrolysis, 2) particle growth, whereby the soot particle size increases due to the addition of material which is primarily carbon (10–20% mole fraction hydrogen) through a series of reactions and coagulation, 3) particle oxidation, where soot particles are burned. Additional information is provided in the overview by Haynes and Wagner [64].

Since the soot model is primarily directed at closing emission/absorption terms in the radiative transfer equation, engineering approximations are made with respect to its inclusion in the Navier Stokes equations. Specifically, heats of reaction associated with formation and destruction are not accounted for in the heat balance, and the mass concentrations of soot and radical nuclei are not included in the species mass balances; they are treated as tracers. The model has a significant amount of empiricism associated with it, necessitated by the extreme length scale range of soot processes, its complexity, and the degree to which many processes have yet to be quantified from a first principles perspective. The model choice can be considered to be a pragmatic one based on its prior use in fire calculations.

The present model has been constructed to fit into the same framework as the conceptual model for turbulent combustion outlined in the theory section for the EDC model. In the following subsections, the basic mechanisms of soot formation and destruction are presented. These processes occur on a scale smaller than can be resolved numerically, therefore the following subsections present the basic approach to the subgrid modeling of the elementary mechanisms, suitable for use in a numerical model.

3.2.13.1. EDC Soot Model

It is important to note that the processes of turbulent soot formation and combustion occur on a scale smaller than can be resolved in a numerical approximation. Thus, the averaged governing equations to be solved numerically must be supplemented with subgrid models to account for these subgrid processes. The conceptual model for subgrid turbulent soot generation and combustion is consistent with the two-zone, turbulent, gas-phase, combustion model presented in the last section (see also Holen [52]). One zone is the flame zone (flame structure) and the other is the surrounding zone.

Soot reactions tend to be slower than gas phase hydrocarbon chemistry. Therefore, the infinitely fast chemistry limit used for the gas phase chemistry is not employed for soot. The current model assumes that the formation and combustion rates are long compared to turbulent mixing rates at flame scales. A steady-state, steady-flow assumption is used in the formulation between the production/destruction rates and the turbulent mixing rates to obtain the soot mass fraction in the flame zone in an algebraic manner (avoiding solution of stiff ordinary differential rate equations).

3.2.13.1.1. Criteria for Soot and Radical Nuclei Formation

To start, the first level criteria for formation of soot are

(3.431) $Y_{prod} > Y_{lim} \qquad {\rm and} \qquad \gamma \chi > 0 \qquad {\rm and} \qquad T^{\circ} > T_{lim} ,$

where $Y_{prod}$ denotes the mass fraction of products, $Y_{lim}$ and $T_{lim}$ are minimum values of product mass fraction and surrounding temperatures allowing soot generation, and $\gamma \chi$ is the volume fraction of the reaction zone of the current cell. If these conditions are met, then the first step is to determine how much carbon is available over and above what may potentially react with oxygen to produce ${\rm CO}_2$ , via the 2-step reaction postulated in the chemistry model (see EDC Turbulent Combustion Model). So, first form the elemental mass fraction of excess (over what may potentially form ${\rm CO}_2$ ) carbon in each species,

(3.432) $f_{c,i} = \max \left[ 0, \left( \hat{Y}^C_i - {1 \over 2} {{W_C} \over {W_O}} \hat{Y}^O_i \right) \right] ,$

where $\hat{Y}^C_i$ is the mass fraction of carbon in species $i$ , and $\hat{Y}^O_i$ the mass fraction of elemental oxygen in species $i$ . For example, for CO (carbon monoxide), $\hat{Y}^C_{CO} = 12 / \left( 12 + 16 \right)$ , etc. Also, for ${\rm CO}_2$ , the excess fraction $f_{c,co2}=0$ , while for any species containing oxygen but no carbon, the formula for the excess fraction is constructed to give zero. Hence, the fraction is non-zero only for species containing carbon but excluding carbon dioxide; i.e., the fuel and carbon monoxide species will have non-zero excess carbon fraction. With the 2-step reaction process being considered, the CO can be considered a fuel in the second reaction, in which CO and ${\rm H}_2$ are oxidized if enough oxygen in available after the first reaction. Thus, the computed carbon fraction, $f_{c,i}$ , is collectively the available carbon in the “fuel species”, comprised of the actual CHNO fuel and CO, and will be zero for other species (compounds). Note that this fraction excludes the carbon in the species that can potentially form ${\rm CO}_2$ via oxidation with the oxygen present in the species itself.

Now, the mass fraction of carbon potentially available to produce soot can be computed for the surrounding and flame zones from the following,

(3.433) $Y^{\circ}_{c \rightarrow s} = \sum_i f_{c,i} Y^{\circ}_i \qquad Y^{\ast}_{c \rightarrow s} = \sum_i f_{c,i} Y^{\ast}_i .$

Again, these mass fractions represent the potentially available carbon in the fuels, separated into flame zone and surroundings, for formation of soot. The average mass fraction of soot-producing-carbon is,

(3.434) $Y_{c \rightarrow s} = \left( \gamma \chi \right) Y^{\ast}_{c \rightarrow s} + \left( 1 - \gamma \chi \right) Y^{\circ}_{c \rightarrow s} .$

Now we must compare the amount of oxidant (not counting oxidant present in the fuel compound) actually available for burning these fuels to produce ${\rm CO}_2$ ; any excess carbon is available to produce additional soot and radical nuclei. The amount of oxygen required to react with {it all} of the available soot-producing-carbon ( $Y_{c \rightarrow s}$ , which already excludes the oxygen present in the fuel compound) to produce ${\rm CO}_2$ is

(3.435) $Y_{O_2,max} = 2 {{W_O} \over {W_C}} Y_{c \rightarrow s} .$

Now, if we can compare this to how much oxygen is actually available, we can decide how much excess carbon is available to produce soot and radical nuclei. Thus the fraction (molar ratio) of excess carbon for producing soot is determined by subtracting off the amount that will go to stoichiometrically react with the available oxygen to ultimately produce ${\rm CO}_2$ in the two-step reaction,

(3.436) $\xi_c = {{ Y_{c \rightarrow s} / W_c - Y_{O_2} / W_{O_2} } \over { Y_{c \rightarrow s} / W_c}} = 1 - {{Y_{O_2}} \over {Y_{O_2,max}}} \Rightarrow 1 - \min \left( 1, {{Y_{O_2}} \over {Y_{O_2,max}}} \right) ,$

where the last expression is the computational implementation, to take care of “lean” conditions where there is excess oxidizer, and which will result in zero mole fraction of carbon to produce soot.

In other words, it is assumed that for a given fraction of existing soot that gets mixed by turbulence into a flame zone, a fraction $\xi_c$ , will contribute to the growth of soot in the flame zone, while the balance, $(1 - \xi_c )$ will be consumed in the production of $\rm{CO}_2$ . Implicit in this assumption is that soot entering a flame will be consumed in proportion to the oxygen present. Therefore in fuel lean regions, soot entering flame zones will be preferentially destroyed.

Now we are in a position to determine whether soot and radical nuclei can be formed under present conditions. They will form if

(3.437) $Y_{c \rightarrow s} > Y_{soot} \qquad X_{c, soot} > 0 .$

The first inequality in (3.437) asserts that the available potential-soot-producing carbon in the fuel must exceed the present amount of soot before enabling generation of additional soot. The construction of $Y_{c \rightarrow s}$ sums the total potential soot-producing-carbon, without distinguishing whether the carbon exists as soot or fuel. The second requires enough carbon to exceed the requirements for the combustion reaction; i.e., soot will only be formed under fuel rich conditions.

3.2.13.1.2. Soot Formation and Termination Models

In general, soot may be considered to be generated in both the reaction zone and in the surrounding zone. This was the assumption invoked in KAMELEON-II (Holen, et al. [52]). As we shall see, in the present implementation for multicomponent species problems, formation/destruction is assumed to take place only in the surrounding fluid. The mass fraction of fuel in the reaction zone is assumed to be proportional to the mass fraction $\gamma^{\ast}$ , and the reacting fraction of the fuel in the reaction zone, $\chi$ . The total rate of radical nuclei formation and destruction is given by a volume averaged sum of the formation within the reaction zone and the surrounding zone.

Assuming the conditions in (3.437) are met, the rates of formation can be computed. The following models for soot formation and termination were originally described by Tesner et al. [63] and have been subsequently modified by Magnussen and co-workers. The elementary mechanisms (subgrid models for the fire code application) of formation and destruction of radical nuclei was described by Tesner et al. [63] in the form,

(3.438) $\dot{R}_n = n_0 + \left( f - g \right) n - g_0 N n \qquad \left[ {\rm particles/s-m}^3 \right] ,$

where $n_0$ is the spontaneous origination rate of radical nuclei in particles/(s- ${\rm m}^3$ ) (due to fuel oxidation and fuel pyrolysis), $f$ is the linear branching coefficient (whereby radical nuclei react to create additional radical nuclei), $g$ is the linear termination coefficient (where radical nuclei combine with existing radical nuclei), $n$ is the concentration of radical nuclei in particles/ ${\rm m}^3$ , $g_0$ is the linear coefficient of termination on soot particles (where radical nuclei combine with existing soot particles), and $N$ is the particle concentration of soot particles (assumed to be spherical with uniform diameter $d_p$ ) in particles/ ${\rm m}^3$ . The spontaneous origination rate of radical nuclei was given by Tesner as

(3.439) $n_0 = 1.08 a_0 \rho Y_{fuel} \exp \left( - {{E} \over {R T}} \right) .$

The rate of soot particle formation and destruction was given by Tesner et al. as,

(3.440) $\dot{R}_{N,f} = \left( a - bN \right) n \qquad \left[ {\rm particles/s-m}^3 \right] .$

The parameters appearing in the foregoing, as determined by Tesner et al. [63] and Holen, et al. [52], are given in Table 3.7 Tesner et al. [65] provide additional data for various hydrocarbons. In practice, the variables $a$ and $b$ are scaled (multiplied) by $10^{16}$ while $a_0$ is scaled (divided) by $10^{16}$ thereby effectively reducing the nuclei concentration by this amount.

Table 3.7 Soot model parameters (Tesner et al.(1971); Holen, et al.(1994))
$a$	$f-g$	$g_0$	$b$	$E/R$	$\rho_{soot}$	$a_0$	$d_p$
$\left[ {\rm 1/s} \right]$	$\left[ {\rm 1/s} \right]$	$\left[ {\rm cm}^3{\rm /part}-{\rm s} \right]$	$\left[ {\rm cm}^3{\rm /part}-{\rm s} \right]$	$\left[ {\rm K} \right]$	$\left[ {\rm g/}{\rm cm}^3 \right]$	$\left[ {\rm part/g}-{\rm s} \right]$	$\left[ {\rm cm} \right]$
$10^5$	$10^2$	$10^{-9}$	$8 \times 10^{-8}$	$9 \times 10^4$	$2.0$	$12.5 \times 10^{33}$	$17.85 \times 10^{-7}$

The elementary formation/destruction models of Tesner have been modified by Magnussen et al. (Holen, et al. [52]) for application to multicomponent fire simulation problems. First, for implementation into a computer program, transport equations for two field variables, radical nuclei and soot concentrations, are needed. For computational reasons, it is convenient to write all transport equations in a standard form,

(3.441) $\int {{\partial \rho \phi} \over {\partial t}} {\rm d}V + \int \rho u_j \phi n_j {\rm d}S = \int \Lambda {{\partial \phi} \over {\partial x_j}} n_j {\rm d}S + \rho S_{\phi} ,$

written for the arbitrary scalar field, $\phi$ , which will have units of intensity per unit mass (or be dimensionless, such as a mass fraction). Thus the computational variables for the soot model are, respectively, the radical nuclei concentration and soot mass fraction,

(3.442) $\beta = {n \over \rho} \qquad {\rm and} \qquad Y_{soot} = {{C_{soot}} \over \rho}$

where $C_{soot}$ denotes the mass concentration of soot (kg/ ${\rm m}^3$ ). In terms of these variables, the spontaneous origination of radical nuclei, as modified by Magnussen et al., is determined from,

(3.443) $\beta^{\circ}_0 = 1.08 a_0 \left( Y^{\circ}_{c \rightarrow s} - Y_{soot} \right) \exp \left( - {{E} \over {R T^{\circ}}} \right) ,$

(3.444) $\beta^{\ast}_0 = 1.08 a_0 \left( Y^{\ast}_{c \rightarrow s} - Y_{soot} \right) \exp \left( - {{E} \over {R T^{\ast}}} \right) ,$

in units of part/kg-sec, which, when compared to Tesner’s form, is seen to have been written in terms of the excess soot-producing carbon, rather than simply being proportional to the fuel concentration, of which only a fraction is available to produce radical nuclei and soot. Similarly, the linear branching and termination reactions for radical nuclei can be written in the form,

(3.445) ${{\dot{R}^{\circ}_{n,f-g,mod}} \over {\rho^{\circ}}} = \max \left(0, f^{\circ}_c \right) \left( f - g \right) \beta^{\circ} \equiv \max \left(0, f^{\circ}_c \right) {{\dot{R}^{\circ}_{n,f-g}} \over {\rho^{\circ}}} ,$

(3.446) ${{\dot{R}^{\ast}_{n,f-g,mod}} \over {\rho^{\ast}}} = \max \left(0, f^{\ast}_c \right) \left( f - g \right) \beta^{\ast} \equiv \max \left(0, f^{\ast}_c \right) {{\dot{R}^{\ast}_{n,f-g}} \over {\rho^{\ast}}} ,$

where the scale factors are defined by,

(3.447) $f^{\circ}_c = {{\left( Y^{\circ}_{c \rightarrow s} - Y_{soot} \right)} \over {Y^{\circ}_{c \rightarrow s}}} \qquad {\rm and} \qquad f^{\ast}_c = {{\left( Y^{\ast}_{c \rightarrow s} - Y_{soot} \right)} \over {Y^{\ast}_{c \rightarrow s}}} ,$

and represent the fraction of soot-producing carbon available in the surroundings and flame zone, respectively. The present formulation reduces the rates by the fraction of soot-producing carbon over and above that which is already present as soot, represented by the last terms in each equation. In contrast, the bilinear termination term for generation of soot is indirectly modified through the soot mass fraction, which is similarly modified (as will be shown shortly). Therefore, the termination term can simply be expressed in terms of the computational variables as,

(3.448) ${{\dot{R}^{\ast}_{n,g_0}} \over {\rho^{\ast}}} = g_0 {{\rho^{\ast} Y^{\ast}_{soot}} \over {m_p}} \beta^{\ast} \qquad {\rm and} \qquad {{\dot{R}^{\circ}_{n,g_0}} \over {\rho^{\circ}}} = g_0 {{\rho^{\circ} Y^{\circ}_{soot}} \over {m_p}} \beta^{\circ} ,$

in which the soot particle concentration has been expressed in terms of the soot mass fraction and an average mass of a soot particle, $m_p$ (kg),

(3.449) $N = {{\left( \rho Y_{soot} \right)} \over {m_p}}$

(3.450) $m_p = \rho_{soot} {4 \over 3} \pi \left( {{d_p} \over 2} \right)^3 ,$

(3.451) $m_p^{\circ} = {b \over a} \rho^{\circ} Y^{\circ}_{c \rightarrow s} ,$

(3.452) $m_p^{\ast} = {b \over a} \rho^{\ast} Y^{\ast}_{c \rightarrow s} ,$

See Table 3.7 for data used in these equations. The generation/destruction term for soot are also modified via the scale factors,

(3.453) ${{\dot{R}^{\ast}_{soot,form,mod}} \over {\rho^{\ast}}} = f^{\ast}_c m_p \left( a - b {{\left( \rho^{\ast} Y^{\ast}_{soot} \right)} \over {m_p}} \right) \beta^{\ast} \equiv f^{\ast}_c m_p {{\dot{R}^{\ast}_{soot,form}} \over {\rho^{\ast}}} ,$

(3.454) ${{\dot{R}^{\circ}_{soot,form,mod}} \over {\rho^{\circ}}} = f^{\circ}_c m_p \left( a - b {{\left( \rho^{\circ} Y^{\circ}_{soot} \right)} \over {m_p}} \right) \beta^{\circ} \equiv f^{\circ}_c m_p {{\dot{R}^{\circ}_{soot,form}} \over {\rho^{\circ}}} .$

to be used in the elementary source expression for the flame zone and surroundings.

The production/destruction of soot in the reaction zone should approach zero for $Y^{\ast}_{soot} \rightarrow Y^{\ast}_{c \rightarrow s}$ , since production should cease when the amount of soot equals the maximum available soot-producing-carbon in the reaction zone. This is easier to see by substituting this form into the production term,

(3.455) ${{\dot{R}^{\ast}_{soot,form,mod}} \over {\rho^{\ast}}} = f^{\ast}_c b \left( \rho^{\ast} Y^{\ast}_{c \rightarrow s} - \rho^{\ast} Y^{\ast}_{soot} \right) \beta^{\ast} .$

This term vanishes when the soot mass fraction equals the maximum carbon mass fraction, by virtue of its construction. However, this form is clearly not the form suggested by Tesner [63], the scaling factor notwithstanding.

3.2.13.1.3. Soot Combustion Model

The soot combustion model assumes that soot is destroyed in the flame zone based on two factors 1) the rate at which it is mixed into the flame zone, and 2) that there is sufficient oxygen to consume it. The mixing rate is the same as in (3.311) (in the gas phase combustion model section) where the species $Y_k$ are treated as follows: In the cell, the fraction of soot that will burn up in the flame zone is $(1 - \xi_c )Y_{soot}$ . In the flame zone, this mass is converted to ${\rm CO}_2$ , so its mass fraction in the flame zone is zero. The radical nuclei concentration is treated similarly. Therefore,

(3.456) ${\dot{R}_{n,comb} \over \bar{\rho}} = \left({{-\left( 1 - \xi_c\right) {n \over \rho}} \over \tau_{res}}\right) \left({{ \gamma \chi } \over { 1 - \gamma \chi}}\right) \chi_3 ,$

(3.457) ${\dot{R}_{soot,comb} \over \bar{\rho}} = \left({{-\left( 1 - \xi_c \right) Y_{soot}} \over \tau_{res}} \right) \left({{ \gamma \chi } \over { 1 - \gamma \chi}}\right) \chi_3 .$

It is convenient to define a new timescale,

(3.458) $\tau_h = {{\left( 1 - \gamma \chi \right) \tau_{res}} \over \chi_3}.$

3.2.13.1.4. Calculating Properties of the Reaction Zone

The foregoing models for soot and radical nuclei contain properties corresponding to the flame zone and surroundings. This section discusses the method employed by Magnussen et al. to compute these properties. The flame zone properties are computed by assuming local equilibrium mass transfer due to turbulent mixing between the reaction zone and surroundings. In other words, the production and combustion rates are sufficiently slow that the mass concentrations in the flame zone come to an equilibrium state with the surroundings via the turbulent mixing rate. This equilibrium rate is assumed to instantaneously adjust to the new cell conditions at every time step.

For this steady-state, steady flow approximation, a balance equation can be written for both nucleate particles and soot mass fraction for the flame zone. In words, the radical nuclei concentration (or soot mass fraction) mixed into the flame zone minus the radical nuclei concentration (or soot mass fraction) mixed out of the flame zone plus the production of radical nuclei (or soot) minus the combustion of radical nuclei (or soot) equals zero. Note that the combustion rates given above are equal to the mixing rates times the fraction of radical nuclei concentration (or soot mass fraction) able to combustion. So the difference in these terms is equal to the soot production rates or,

(3.459) ${{\left( \beta^{\ast} - \xi_c \beta \right)} \over \tau_h} = \beta^{\ast}_0 + { f^{\ast}_c{\dot{R}^{\ast}_{n,f-g}} \over {\rho^{\ast}}} - {{\dot{R}^{\ast}_{n,g_0}} \over {\rho^{\ast}}} ,$

(3.460) ${{\left( Y^{\ast}_{soot} - \xi_c Y_{soot} \right)} \over \tau_h} = f^{\ast}_c m^{\ast}_p {{\dot{R}^{\ast}_{soot,form}} \over {\rho^{\ast}}} .$

Solution of these two algebraic equations with two unknowns gives, $\beta^{\ast}$ and $Y^{\ast}_{soot}$ , the radical nuclei and soot concentrations in the flame zone, respectively. Note that the formation/destruction terms are of a bilinear form in the soot and radical nuclei concentrations. Thus, to compute the flame zone values of radical nuclei and soot mass fractions requires the simultaneous solution of this $2 \times 2$ system of equations. In particular, substituting for these terms from the formula given above, (3.460) can be solved for $Y^{\ast}_{soot}$ using (3.455). The result is that the mass fraction of soot in the flame zone in terms of the radical nuclei concentration.

(3.461) $Y^{\ast}_{soot} = {{\xi_c Y_{soot} + \tau_h f_c^{\ast} b \rho^{\ast} Y^{\ast}_{c \rightarrow s} \beta^{\ast}} \over {1 + \tau_h f^{\ast}_c b \rho^{\ast} \beta^{\ast}}} .$

(3.461) can be used in (3.459) to form a quadratic equation for $\beta^{\ast}$ ,

(3.462) $tilde{a}_s \left( \beta^{\ast} \right)^2 + \tilde{b}_s \beta^{\ast} + tilde{c}_s = 0 ,$

where,

(3.463) $\tilde{a}_s = f^{\ast}_c \tau_h \rho^{\ast} \left( \tilde{\alpha} b + \tau_h a g_0 \right) ,$

(3.464) $\tilde{b}_s = \tilde{\alpha} + \tau_h \rho^{\ast} \left( {{\xi_c Y_{soot} g_0} \over {\displaystyle{b \over a} \rho^{\ast} Y^{\ast}_{c \rightarrow s}}} - f^{\ast}_c b A \right) ,$

(3.465) $\tilde{c}_s = - A ,$

(3.466) $A = \xi_c \beta + \tau_h \beta^{\ast}_0 ,$

(3.467) $\tilde{\alpha} = 1 - \tau_h \left( f - g \right) f^{\ast}_c .$

The solution is the negative root of the quadratic, here written in a computationally appropriate form,

(3.468) $\beta^{\ast} = {{ - 2 \tilde{c}_s} \over {\tilde{b}_s + \sqrt{ \tilde{b}_s^2 - 4 \tilde{a}_s \tilde{c}_s}}}$

In the limit where

(3.469) $Y^{\ast}_{soot} \rightarrow Y^{\ast}_{c \rightarrow s}$

then the soot mass fraction becomes static and the radical nuclei concentration can be solved for directly. The result is

(3.470) $\beta^{\ast} = { {\xi_c \beta + \tau_h \beta^{\ast}_o } \over {1 + \tau_h \left( -f^{\ast}_c \left( f-g \right ) + {{a g_o} \over {b} }\right)} }.$

3.2.13.1.5. Calculating Properties of the Surroundings

Having computed the properties of the reaction zone, the properties for the surroundings are calculated from the definition of the cell (average) values,

(3.471) $\beta^{\circ} = {{\beta - \gamma \chi \beta^{\ast}} \over { 1 - \gamma \chi }} ,$

(3.472) $Y^{\circ}_{soot} = {{Y_{soot} - \gamma \chi Y^{\ast}_{soot}} \over { 1 - \gamma \chi }} .$

(3.473) $\beta^{\circ} = min \left(\beta^{\circ}, {{ {{a g_o} \over {b}}\times 10^{-6}} \over {\rho^{\circ}}} \right)$

Note that there is an upper bound to the number of nucleate particles based on a 50 percent dense mixture given they are monodisperse at the size given in Table 3.7 with mass given by

(3.474) $m = \rho_{soot} {4 \over 3} \pi \left( {{d_p} \over 2} \right)^3.$

Now we are in a position to specify the transport equations and source terms for the soot model.

3.2.13.2. Transport Equations and Source Terms

Two transport equations for radical nuclei and soot mass fractions need be solved,

(3.475) $\int {{\partial \rho \beta} \over {\partial t}} {\rm d}V + \int \rho \beta u_j n_j {\rm d}S = \int {{\mu_{eff}} \over {\sigma_Y}} {{\partial \beta} \over {\partial x_j}} n_j {\rm d}S + \int \rho S_{n} {\rm d}V ,$

(3.476) $\int {{\partial \rho Y_{soot}} \over {\partial t}} {\rm d}V + \int \rho Y_{soot} u_j n_j {\rm d}S = \int {{\mu_{eff}} \over {\sigma_Y}} {{\partial Y_{soot}} \over {\partial x_j}} n_j {\rm d}S + \int \rho S_{soot} {\rm d}V .$

In general, the source term, in particles/kg-sec, for radical nuclei is given by,

(3.477) $S_n = \gamma \chi \left({{\dot{R}^{\ast}_{n,form, mod}} \over {\rho^{\ast}}} - {{\dot{R}^{\ast}_{n,comb}} \over {\rho^{\ast}}} \right) + \left( 1 - \gamma \chi \right) {{\dot{R}^{\circ}_{n,form, mod}} \over {\rho^{\circ}}}$

where the form of the net formation/destruction source terms is,

(3.478) ${{\dot{R}^{\circ}_{n,form, mod}} \over {\rho}} = \beta^{\circ}_0 + \left( f - g \right) \beta^{\circ} \max \left(0, f^{\circ}_c \right) - g_0 {{a Y^{\circ}_{soot}} \over {b Y^{\circ}_{c \rightarrow s}}} \beta^{\circ} .$

For each of the reaction and surrounding zones, the (production destruction) of radical nuclei in the flame zone is given by the mixing balance, or

(3.479) $\left({{\dot{R}^{\ast}_{n,form}} \over {\rho}} - {{\dot{R}^{\ast}_{n,comb}} \over {\rho}} \right) = \left( {\beta^{\ast} - \beta } \over {\tau_h}\right).$

Substituting gives,

(3.480) $S_n = \gamma \chi \left( {\beta^{\ast} - \beta } \over {\tau_h} \right) + \left( 1 - \gamma \chi \right) \left( \beta^{\circ}_0 + \left( f - g \right) \beta^{\circ} \max \left(0, f^{\circ}_c \right) - g_0 {{a Y^{\circ}_{soot}} \over {b Y^{\circ}_{c \rightarrow s}}} \beta^{\circ} \right) ,$

The general source term for soot (1/sec) is given by

(3.481) $S_{soot} = m_p \gamma \chi \left({{\dot{R}^{\ast}_{soot,form, mod}} \over {\rho^{\ast}}} - {{\dot{R}^{\ast}_{soot,comb}} \over {\rho^{\ast}}} \right) + \left( 1 - \gamma \chi \right) m_p {{\dot{R}^{\circ}_{soot,form, mod}} \over {\rho^{\circ}}}$

The (production-destruction) of soot in the flame zone is likewise given by the mixing balance, or

(3.482) $m_p \left({{\dot{R}^{\ast}_{soot,form}} \over {\rho}} - {{\dot{R}^{\ast}_{soot,comb}} \over {\rho}} \right) = \left( {Y^{\ast}_{soot} - Y_{soot}} \over {\tau_h}\right).$

Substituting gives,

(3.483) $S_{soot} = \gamma \chi \left( {Y^{\ast}_{soot} - Y_{soot} } \over {\tau_h} \right) + \left( 1 - \gamma \chi \right) f^{\circ}_c b \rho^{\circ} \left( Y^{\circ}_{c \rightarrow s} - Y^{\circ}_{soot} \right) \beta^{\circ} ,$

which also follows the practice of using the scale factor and effective mass for a soot particle in the surroundings, $m_p = b \rho^{\circ} Y^{\circ}_{c \rightarrow s}/a$ .

The fact that the soot and radical nuclei concentrations are treated as tracers should be reemphasized. This means that their concentrations in the gas mixture are assumed insignificant such that they do not enter into calculations of density, or other properties of the mixture.