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14.3.1 Theory



General Description


The relationships for calculating char particle burning rates are presented and discussed in detail by Smith [ 342]. The particle reaction rate, ${\cal R}$ (kg/m $^2$-s), can be expressed as


 {\cal R} = D_0 (C_g-C_s) = R_c(C_s)^N (14.3-1)

where


$D_0$ = bulk diffusion coefficient (m/s)
$C_g$ = mean reacting gas species concentration in the bulk (kg/m $^3$)
$C_s$ = mean reacting gas species concentration at the particle surface (kg/m $^3$)
$R_c$ = chemical reaction rate coefficient (units vary)
$N$ = apparent reaction order (dimensionless)

In Equation  14.3-1, the concentration at the particle surface, $C_s$, is not known, so it should be eliminated, and the expression is recast as follows:


 {\cal R} = R_c \left[C_g - \frac{{\cal R}}{D_0}\right]^N (14.3-2)

This equation has to be solved by an iterative procedure, with the exception of the cases when $N=1$ or $N=0$. When $N=1$, Equation  14.3-2 can be written as


 {\cal R} = \frac{C_g R_c D_0}{D_0+R_c} (14.3-3)

In the case of $N=0$, if there is a finite concentration of reactant at the particle surface, the solid depletion rate is equal to the chemical reaction rate. If there is no reactant at the surface, the solid depletion rate changes abruptly to the diffusion-controlled rate. In this case, however, FLUENT will always use the chemical reaction rate for stability reasons.



FLUENT Model Formulation


A particle undergoing an exothermic reaction in the gas phase is shown schematically in Figure  14.3.1. $T_p$ and $T_{\infty}$ are the temperatures in Equation  22.9-86.

Figure 14.3.1: A Reacting Particle in the Multiple Surface Reactions Model
figure

Based on the analysis above, FLUENT uses the following equation to describe the rate of reaction $r$ of a particle surface species $j$ with the gas phase species $n$. The reaction stoichiometry of reaction $r$ in this case is described by


\mbox{particle species} \; j \mbox{(s)} + \mbox{gas phase species} \; n \rightarrow \mbox{products}

and the rate of reaction is given as


 \overline{{\cal R}}_{j,r} = A_p \eta_r Y_j {\cal R}_{j,r} (14.3-4)


 {\cal R}_{j,r} = {\cal R}_{{\rm kin},r} \left(p_n- \frac{{\cal R}_{j,r}}{D_{0,r}}\right)^{N} (14.3-5)

where


$\overline{{\cal R}}_{j,r}$ = rate of particle surface species depletion (kg/s)
$A_p$ = particle surface area (m $^2$)
$Y_j$ = mass fraction of surface species $j$ in the particle
$\eta_r$ = effectiveness factor (dimensionless)
${\cal R}_{j,r}$ = rate of particle surface species reaction per unit area (kg/m $^2$-s)
$p_{n}$ = bulk partial pressure of the gas phase species (Pa)
$D_{0,r}$ = diffusion rate coefficient for reaction  $r$
${\cal R}_{{\rm kin},r}$ = kinetic rate of reaction $r$ (units vary)
$N_r$ = apparent order of reaction $r$

The effectiveness factor, $\eta_r$, is related to the surface area, and can be used in each reaction in the case of multiple reactions. $D_{0,r}$ is given by


 D_{0,r} = C_{1,r} \frac{\left [ (T_p + T_{\infty})/2 \right]^{0.75}}{d_p} (14.3-6)

The kinetic rate of reaction $r$ is defined as


 {\cal R}_{{\rm kin},r} = A_r T^{\beta_r} e^{-(E_r/RT)} (14.3-7)

The rate of the particle surface species depletion for reaction order $N_r =1$ is given by


 \overline{{\cal R}}_{j,r} = A_p \eta_r Y_j p_{n} \frac{{\cal R}_{{\rm kin},r} D_{0,r}}{ D_{0,r} + {\cal R}_{{\rm kin},r}} (14.3-8)

For reaction order $N_r =0$,


 \overline{{\cal R}}_{j,r} = A_p \eta_r Y_j {\cal R}_{{\rm kin},r} (14.3-9)



Extension for Stoichiometries with Multiple Gas Phase Reactants


When more than one gas phase reactant takes part in the reaction, the reaction stoichiometry must be extended to account for this case:

\mbox{particle species} \; j \mbox{(s)} + \mbox{gas phase species} \; 1 + \mbox{gas phase species} \; 2 + \; \dots


+ \; \mbox{gas phase species} \; n_{\rm max} \rightarrow \mbox{products}

To describe the rate of reaction $r$ of a particle surface species $j$ in the presence of $n_{\rm max}$ gas phase species $n$, it is necessary to define the diffusion-limited species for each solid particle reaction, i.e., the species for which the concentration gradient between the bulk and the particle surface is the largest. For the rest of the species, the surface and the bulk concentrations are assumed to be equal. The concentration of the diffusion-limited species is shown as $C_{d,b}$ and $C_{d,s}$ in Figure  14.3.1, and the concentrations of all other species are denoted as $C_k$. For stoichiometries with multiple gas phase reactants, the bulk partial pressure $p_n$ in Equations  14.3-4 and 14.3-8 is the bulk partial pressure of the diffusion-limited species, $p_{r,d}$ for reaction $r$.

The kinetic rate of reaction $r$ is then defined as


 {\cal R}_{{\rm kin},r} = \frac{A_r T^{\beta_r} e^{-(E_r/RT)}} {(p_{r,d})^{N_{r,d}}} \prod_{n=1}^{n_{\rm max}} p_n^{N_{r,n}} (14.3-10)

where


$p_n$ = bulk partial pressure of gas species $n$
$N_{r,n}$ = reaction order in species $n$

When this model is enabled, the constant $C_{1,r}$ (Equation  14.3-6) and the effectiveness factor $\eta_r$ (Equation  14.3-4) are entered in the Reactions panel, as described in Section  14.3.2.



Solid-Solid Reactions


Reactions involving only particle surface reactants can be modeled, provided that the particle surface reactants and products exist on the same particle.

\mbox{particle species} \; 1 \mbox{(s)} + \mbox{particle species} \; 2 \mbox{(s)} + \dots \rightarrow \mbox{products}

The reaction rate for this case is given by Equation  14.3-9.



Solid Decomposition Reactions


The decomposition reactions of particle surface species can be modeled.

\mbox{particle species} \; 1 \mbox{(s)} + \mbox{particle spec... ...+ \mbox{particle species} \; n_{\rm max} \mbox{(s)} \rightarrow


\mbox{gas species} \; j + \mbox{products}

The reaction rate for this case is given by Equations  14.3-4- 14.3-10, where the diffusion-limited species is now the gaseous product of the reaction. If there are more than one gaseous product species in the reaction, it is necessary to define the diffusion-limited species for the particle reaction as the species for which the concentration gradient between the bulk and the particle surface is the largest.



Solid Deposition Reactions


The deposition reaction of a solid species on a particle can be modeled with the following assumptions:

\mbox{gas species} \; 1 + \mbox{gas species} \; 2 + \; \dots ... ...ghtarrow \mbox{solid species} \; j \mbox{(s)} + \mbox{products}

The theoretical analysis and Equations  14.3-4- 14.3-10 are applied for the surface reaction rate calculation, with the mass fraction of the surface species set to unity in Equations  14.3-4, 14.3-8, and 14.3-9.

In FLUENT, for the particle surface species to be deposited on a particle, a finite mass of the species must already exist in the particle. This allows for activation of the deposition reaction selectively to particular injection particles. It follows that, to initiate the solid species deposition reaction on a particle, the particle must be defined in the Set Injection Properties panel (or Set Multiple Injection Properties panel) to contain a small mass fraction of the solid species to be deposited. See Section  14.3.3 for details on defining the particle surface species mass fractions.



Gaseous Solid Catalyzed Reactions on the Particle Surface


Reactions of gaseous species catalyzed on the particle surface can also be modeled following Equations  14.3-4- 14.3-10 for the surface reaction rate calculation, with the mass fraction of the surface species set to unity in Equations  14.3-4, 14.3-8, and 14.3-9.

The catalytic particle surface reaction option is enabled in FLUENT when Particle Surface is selected as the Reaction Type in the Reactions panel and there are no solid species in the reaction stoichiometry. The solid species acting as a catalyst for the reaction is defined in the Reactions panel. The catalytic particle surface reaction will proceed only on those particles containing the catalyst species. See Section  14.3.3 for details on defining the particle surface species mass fractions.


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