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20.3.2 Theory



The Single-Step Soot Formation Model


In the single-step Khan and Greeves model [ 179], FLUENT solves a single transport equation for the soot mass fraction:


 \frac{\partial}{\partial t} (\rho Y_{\rm soot}) + \nabla \cd... ..._{\rm soot}} \nabla Y_{\rm soot} \right) + {\cal R}_{\rm soot} (20.3-1)


where    
  $Y_{\rm soot}$ = soot mass fraction
  $\sigma_{\rm soot}$ = turbulent Prandtl number for soot transport
  ${\cal R}_{\rm soot}$ = net rate of soot generation (kg/m $^3$-s)

${\cal R}_{\rm soot}$, the net rate of soot generation, is the balance of soot formation, ${\cal R}_{\rm soot, form}$, and soot combustion, ${\cal R}_{\rm soot ,comb}$:


 {\cal R}_{\rm soot} = {\cal R}_{\rm soot, form} - {\cal R}_{\rm soot, comb} (20.3-2)

The rate of soot formation is given by a simple empirical rate expression:


 {\cal R}_{\rm soot, form} = C_s p_{\rm fuel} \phi^r e^{-E/RT} (20.3-3)


where    
  $C_s$ = soot formation constant (kg/N-m-s)
  $p_{\rm fuel}$ = fuel partial pressure (Pa)
  $\phi$ = equivalence ratio
  $r$ = equivalence ratio exponent
  $E/R$ = activation temperature (K)

The rate of soot combustion is the minimum of two rate expressions [ 229]:


 {\cal R}_{\rm soot, comb} = \min [{\cal R}_1, \; {\cal R}_2 ] (20.3-4)

The two rates are computed as


 {\cal R}_1 = A \rho Y_{\rm soot} \frac{\epsilon}{k} (20.3-5)

and


 {\cal R}_2 = A \rho \left(\frac{Y_{\rm ox}}{\nu_{\rm soot}}\... ...soot} + Y_{\rm fuel} \nu_{\rm fuel}}\right) \frac{\epsilon}{k} (20.3-6)


where    
  $A$ = constant in the Magnussen model
  $Y_{\rm ox}$, $Y_{\rm fuel}$ = mass fractions of oxidizer and fuel
  $\nu_{\rm soot}$, $\nu_{\rm fuel}$ = mass stoichiometries for soot and fuel combustion

The default constants for the single-step model are valid for a wide range of hydrocarbon fuels.



The Two-Step Soot Formation Model


The two-step Tesner model [ 373] predicts the generation of radical nuclei and then computes the formation of soot on these nuclei. FLUENT thus solves transport equations for two scalar quantities: the soot mass fraction (Equation  20.3-1) and the normalized radical nuclei concentration:


 \frac{\partial}{\partial t} (\rho b_{\rm nuc}^*) + \nabla \c... ...{\rm nuc}} \nabla b_{\rm nuc}^* \right) + {\cal R}_{\rm nuc}^* (20.3-7)


where    
  $b_{\rm nuc}^*$ = normalized radical nuclei concentration (particles $\times 10^{-15}$/kg)
  $\sigma_{\rm nuc}$ = turbulent Prandtl number for nuclei transport
  ${\cal R}_{\rm nuc}^*$ = normalized net rate of nuclei generation (particles $\times 10^{-15}$/m $^3$-s)

In these transport equations, the rates of nuclei and soot generation are the net rates, involving a balance between formation and combustion.

Soot Generation Rate

The two-step model computes the net rate of soot generation, ${\cal R}_{\rm soot}$, in the same way as the single-step model, as a balance of soot formation and soot combustion:


 {\cal R}_{\rm soot} = {\cal R}_{\rm soot, form} - {\cal R}_{\rm soot, comb} (20.3-8)

In the two-step model, however, the rate of soot formation, ${\cal R}_{\rm soot, form}$, depends on the concentration of radical nuclei, $c_{\rm nuc}$:


 {\cal R}_{\rm soot, form} = m_p (\alpha - \beta N_{\rm soot}) c_{\rm nuc} (20.3-9)


where    
  $m_p$ = mean mass of soot particle (kg/particle)
  $N_{\rm soot}$ = concentration of soot particles (particles/m $^3$)
  $c_{\rm nuc}$ = radical nuclei concentration = $\rho b_{\rm nuc}$ (particles/m $^3$)
  $\alpha$ = empirical constant (s $^{-1}$)
  $\beta$ = empirical constant (m $^3$/particle-s)

The rate of soot combustion, ${\cal R}_{\rm soot, comb}$, is computed in the same way as for the single-step model, using Equations  20.3-4- 20.3-6.

The default constants for the two-step model are for combustion of acetylene (C $_2$H $_2$). According to Ahmad et al. [ 5], these values should be modified for other fuels, since the sooting characteristics of acetylene are known to be different from those of saturated hydrocarbon fuels.

Nuclei Generation Rate

The net rate of nuclei generation in the two-step model is given by the balance of the nuclei formation rate and the nuclei combustion rate:


 {\cal R}_{\rm nuc}^* = {\cal R}_{\rm nuc, form}^* - {\cal R}_{\rm nuc, comb}^* (20.3-10)


where    
  ${\cal R}_{\rm nuc, form}^*$ = rate of nuclei formation (particles $\times 10^{-15}$/m $^3$-s)
  ${\cal R}_{\rm nuc, comb}^*$ = rate of nuclei combustion (particles $\times 10^{-15}$/m $^3$-s)

The rate of nuclei formation, ${\cal R}_{\rm nuc, form}^*$, depends on a spontaneous formation and branching process, described by


 {\cal R}_{\rm nuc, form}^* = \eta_0 + (f - g) c_{\rm nuc}^* - g_0 c_{\rm nuc}^* N_{\rm soot} (20.3-11)


 \eta_0 = a_0^* c_{\rm fuel} e^{-E/RT} (20.3-12)


where    
  $c_{\rm nuc}^*$ = normalized nuclei concentration ( $= \rho b_{\rm nuc}^*$)
  $a_0^*$ = $a_0/10^{15}$
  $a_0$ = pre-exponential rate constant (particles/kg-s)
  $c_{\rm fuel}$ = fuel concentration (kg/m $^3$)
  $f-g$ = linear branching $-$ termination coefficient (s $^{-1}$)
  $g_0$ = linear termination on soot particles (m $^3$/particle-s)

Note that the branching term, $(f-g) c_{\rm nuc}^*$, in Equation  20.3-11 is included only when the kinetic rate, $\eta_0$, is greater than the limiting formation rate (10 $^{5}$ particles/m $^3$-s, by default).

The rate of nuclei combustion is assumed to be proportional to the rate of soot combustion:


 {\cal R}_{\rm nuc, comb}^* = {\cal R}_{\rm soot, comb} \frac{b_{\rm nuc}^*}{Y_{\rm soot}} (20.3-13)

where the soot combustion rate, ${\cal R}_{\rm soot, comb}$, is given by Equation  20.3-4.



The Effect of Soot on the Radiation Absorption Coefficient


A description of the modeling of soot-radiation interaction is provided in Section  13.3.8.


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