You signed in with another tab or window. Reload to refresh your session.You signed out in another tab or window. Reload to refresh your session.You switched accounts on another tab or window. Reload to refresh your session.Dismiss alert
@@ -846,7 +846,8 @@ \chapter{Analytical Jacobian for Mixing and Detailed Chemistry}
846
846
847
847
The system of ODEs for concentration and temperature contains additional terms to account for turbulent mixing. Following the theory of Sec.~\ref{sec:subgrid_evironment}, we can derive these ODEs as follows.
848
848
849
-
\subsection*{Mass Balance}
849
+
\section{Mass Balance}
850
+
\label{mass_balance_mixing_chem}
850
851
851
852
Each cell in divided in two zones, unmixed and mixed. Chemical reaction is only allowed in the mixed zone. The total mass in the cell is $\rho V_c$ where $V_c$ is the cell volume. For the purposes of discussing the time integration for mixing and chemical kinetics, the variable $t$ here can be taken as zero at the start of an LES time step. Thus, we are concerned with integration of the state of mixing and reaction within the cell from $0\le t \le\delta t_{\mathrm{LES}}$. The mass in the unmixed portion of the cell is $U(t)$ and the mass in the mixed zone is $M(t)$:
Deatails of chemistry source term $\dot{{\omega}_k}$ can be found in Section \ref{chem_source_term}.
930
+
931
+
\section{Energy Balance}
932
+
\label{energy_balance_mixing_chem}
929
933
930
934
Now consider the total internal energy balance in the mixing zone:
931
935
@@ -972,8 +976,11 @@ \subsection*{Energy Balance}
972
976
973
977
Eqs.~(\ref{eq:jspeciemolarconcentration2}) and (\ref{eq:tempeqn}) will be used in CVODE as the ODE RHS.
974
978
979
+
Deatails of chemistry source term $\dot{{\omega}_j}$ can be found in Section \ref{chem_source_term}.
980
+
975
981
976
-
\subsection{Analytical Jacobian formulation}
982
+
\section{Analytical Jacobian formulation}
983
+
\label{jacobian_mixing_chem}
977
984
978
985
Providing an analytical Jacobian to the CVODE solver can significantly accelerate the chemistry calculations. The Jacobian for the given system can be written as:
For Eq.~(\ref{eq:chemistry_ode_system_appendix}), $C_j$ is the molar concentration \si{(kmol/m^3)} of the $j$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.
For Eq.~(\ref{eq:TemperatureDerivativeAppendix}), $\rho$ is the density \si{(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_appendix}).
Providing an analytical Jacobian to the CVODE solver can significantly accelerate the chemistry calculations. The Jacobian for the given system can be written as:
This Jacobian is the first part of equations \ref{eq:jspeciemolarconcentration3} and \ref{eq:tempeqn2} for the mixing and chemistry Jacobian matrix (\ref{eq:jacobian_mix_chem}).
1167
+
1168
+
\subsection*{Calculation of $\frac{\partial\mathbf{\dot{\omega}}}{\mathbf{\partial C}}$}
1169
+
By taking derivative of Eq.~(\ref{eq:chemistry_ode_system_appendix}) with respect to concentration, we can write:
Similar to $\frac{\partial b_i}{\partial C_l}$, $\frac{\partial b_i}{\partial T}$ can be calculated using Eqs.~(\ref{eq:reac_mod_coeff})-(\ref{eq:falloff_Fi}).
Here, $K_i$ is the concentration equilibrium constant obtained using Eq.~(\ref{eq:equilibrium_const}). Through mathematical derivation, it can be shown that:
Using the relations $\frac{\partial\rho}{\partial C_l} = W_l$ and $\frac{\partial c_p}{\partial C_l} = - \frac{W_l}{\rho}\left( c_p -c_{p,l} \right)$, we can rewrite Eq. ~(\ref{eq:Jac3rd}) as:
To derive the above equation, the relations $\frac{\partial\rho}{\partial T} = -\frac{\rho}{T}$ and $\frac{\partial h_j}{\partial T} = c_{p,j}$ are used. The term $\frac{\partial\dot{\omega}_j}{\partial T}$ can be obtained using Eq.~(\ref{eq:Jac2nd}).
\caption[The \ct{insulated\_steel} test cases]{Steady-state temperature profile within an insulated steel plate and pipe. Note that the pipe conduction case has been performed as part of a 3-D calculation (upper right) and a 2-D, axi-symmetric calculation (bottom).}
4771
+
\caption[The \ct{insulated_steel} test cases]{Steady-state temperature profile within an insulated steel plate and pipe. Note that the pipe conduction case has been performed as part of a 3-D calculation (upper right) and a 2-D, axi-symmetric calculation (bottom).}
0 commit comments