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Copy file name to clipboardExpand all lines: Manuals/FDS_Technical_Reference_Guide/Combustion_Chapter.tex
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@@ -342,16 +342,16 @@ \subsection{Finite-Rate Chemistry (Detailed Chemical Mechanism)}
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\label{detailed_chemistry_using_mechanism}
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A chemical mechanism represents different chemical pathways using multiple species and reactions. Typically, these mechanisms are available in Chemkin or Cantera YAML formats. Such a mechanism can be represented as:
Here, $X_j$ represents the chemical symbol of the $j$th species (i.e. $\mathrm{CH_4, H_2, O_2}$); ${\nu}_{ji}^{'}$ and ${\nu}_{ji}^{''}$ are the stoichiometric coefficients of the $j$th species in the $i$th reaction; ${N_{sp}}$ and ${N_{reac}}$ are the total number of species and total number of reactions, respectively.
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Here, $X_j$ represents the chemical symbol of the $j$th species (i.e. $\mathrm{CH_4, H_2, O_2}$); ${\nu}_{ji}^{'}$ and ${\nu}_{ji}^{''}$ are the stoichiometric coefficients of the $j$th species in the $i$th reaction; ${N_{\text{s}}}$ and ${N_\text{r}}$ are the total number of species and total number of reactions, respectively.
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The rate of change of the molar concentration \si{(kmol/m^3/s)} of each species can be represented using the following system of ordinary differential equations (ODEs):
The right-hand side of the above equation represents contributions from each reaction to the $j$th species. Here, $C_k$ is the molar concentration \si {(kmol/m^3)} of the $k$th species; $b_i$ is the reaction rate modification coefficient of the $i$th reaction due to third-body effects and pressure; $\nu_{ki} = {\nu}_{ki}^{''} - {\nu}_{ki}^{'}$; and $r_i$ is the reaction progress rate of the $i$th reaction.
Here, $k_{f,i}$ and $k_{r,i}$ are the forward and reverse reaction rate coefficients, respectively. The $k_{f,i}$ can be calculated using Eq.~(\ref{eq:rate_cons}), and the $k_{r,i}$ can be calculated by obtaining the equilibrium constant, as shown in Eq.~(\ref{eq:equilibrium_const}).
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\end{equation}
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For third-body reactions, presence of (or collission with) a third body (any other molecule) modify the reaction rate. If the third-body efficiency of species $j$ in reaction $i$ (denoted as $\alpha_{ij}$) is provided, then
If the third-body efficiency is not provided, the default value of $\alpha_{ij}$ is 1.0.
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To solve the reactive system, we assume a constant pressure reactor. For that, the concentration ODEs in the form of Eq.~(\ref{eq:chemistry_ode_system}) need to be solved along with a ODE of temperature given by:
Here, $\rho$ is the density $\mathrm{(kg/m^3)}$, $c_p$ is the specific heat of the mixture (J/kg/K), $h_k$ is the absolute enthalpy that includes enthalpy of formation (J/kg), $W_j$ is the molecular weight (kg/kmol) of species $j$, and $\dot{\omega}$ is the species production rate given by Eq.~(\ref{eq:chemistry_ode_system}).
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To solve the system of ODEs, we first convert the mass fraction of species to molar concentrations using $C_j=Y_j\rho/W_j$. Then, CVODE from Sundials \cite{cvodeDoc:2024} is used to solve the system of ODEs by supplying an analytical Jacobian. The details of analytical Jacobian formulation is provided in the Appendix \ref{chemistry_analytical_jacobian}.
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