FDS Tech Guide: minor updates to equations of solid energy conservation

Manuals/FDS_Technical_Reference_Guide/Appendices.tex

@@ -2003,9 +2003,9 @@ \subsection{Solid Phase Heat of Reaction}
 
 For the first step, reference enthalpies for all the components in a reaction must be determined. FDS has a database for many common gases that includes predefined reference enthalpies. In a typical simulation any gases not in the database are likely fuels participating in combustion reactions where the reference enthalpy can be determined from the heat of combustion and the reference enthalpies of the combustion products. Therefore, gas reference enthalpies are treated as known quantities for the solid phase. A similar rationale applies to liquid particles produced by a solid phase reaction. Definition of a liquid particle includes specifying the gas it evaporates to and the heat of vaporization which means a reference enthalpy for the liquid can be determined. Reference enthalpies for solid materials are generally not known except for some pure substances. Even if available, the published values when placed in Eq.~\ref{eqn_solid_e_cons} may not exactly result in the specified heat of reaction. Reference enthalpies for any solid material participating as a reactant or product in a solid phase reaction must be determined. For solid materials not participating in reactions the assumption that the reference enthalpy is 0~kJ/kg at 0~K suffices as there is no exchange of mass (and its asssociated enthalpy) with the gas phase. Solid phase products for a solid phase reaction include both solid materials that remain part of the reacting surface and solid materials that leave the surface in the form of particles (i.e., a firebrand). Finding these enthalpies is done by solving a system of linear equations where there is one equation for every material reaction and one unknown for the reference enthalpy of every solid material participating in a material reaction as reactant or product. To build the matrix, Eq.~\ref{eqn_solid_e_cons} is applied to every material reaction. The enthalpy of material $\alpha$, $h_{\alpha}$, is given by~\cite{NIST_JANAF}:
 \be
-  h_{\alpha}(T) = h_{\alpha,ref} + \int_{T_{\rm ref}}^{T_{R}} c_{p,\alpha}(T) \, \mbox{d}T
+  h_{\alpha}(T_{\rm R}) = h_{\alpha,{\rm ref}} + \int_{T_{\rm ref}}^{T_{\rm R}} c_{p,\alpha}(T) \, \mbox{d}T
 \ee
-In this equation $h_{\alpha,ref}$ is the enthalpy at the temperature $T_{\rm ref}$. This is the value being solved for. In many tabulations of solid enthalpies the value of $T_{\rm ref}$ is 298.15~K; however, since it can be determined at any reference temperature, for convenience a temperature of 0~K is picked. The specific heat of a material, $c_{p,\alpha}(T)$, is defined in the input file. Evaluating the integral requires picking the upper bound temperature, T$_R$. This is taken as the reference temperature for the reaction. This may be user defined as part of the input. If not, it can be determined from the reaction $A_{\alpha\beta}$ and $E_{\alpha\beta}$ by performing a TGA simulation for material $\alpha$, reaction $\beta$. The value for $T_R$ is set to the temperature of the peak reaction rate. With $T_R$ defined, the various $\nu_{\alpha',\alpha \beta} h_{\alpha,ref}$ terms can be collected on the left hand side, and the heats of reaction and the various $\nu_{\alpha',\alpha \beta} \int_{T_{\rm ref}}^{T_{R}} c_{p,\alpha}(T) \, \mbox{d}T$ terms can be collected on the right hand side. The result is a system of linear equations $A \bar{x} = \bar{b}$ where the unknowns, $\bar{x}$, are the values of $h_{\alpha,ref}$; however, it is not guaranteed that there are an equal number of reactions and unknown material reference enthalpies. Solving the system of equations depends on the relative numbers of equations and unknowns as follows:
+In this equation $h_{\alpha,{\rm ref}}$ is the enthalpy at the temperature $T_{\rm ref}$. This is the value being solved for. In many tabulations of solid enthalpies the value of $T_{\rm ref}$ is 298.15~K; however, since it can be determined at any reference temperature, for convenience a temperature of 0~K is picked. The specific heat of a material, $c_{p,\alpha}(T)$, is defined in the input file. Evaluating the integral requires picking the upper bound temperature, $T_{\rm R}$. This is taken as the reference temperature for the reaction. This may be user defined as part of the input. If not, it can be determined from the reaction $A_{\alpha\beta}$ and $E_{\alpha\beta}$ by performing a TGA simulation for material $\alpha$, reaction $\beta$. The value for $T_{\rm R}$ is set to the temperature of the peak reaction rate. With $T_{\rm R}$ defined, the various $\nu_{\alpha',\alpha \beta} h_{\alpha',{\rm ref}}$ terms can be collected on the left hand side, and the heats of reaction and the various $\nu_{\alpha',\alpha \beta} \int_{T_{\rm ref}}^{T_{\rm R}} c_{p,\alpha'}(T) \, \mbox{d}T$ terms can be collected on the right hand side. The result is a system of linear equations $A \bar{x} = \bar{b}$ where the unknowns, $\bar{x}$, are the values of $h_{\alpha,ref}$; however, it is not guaranteed that there are an equal number of reactions and unknown material reference enthalpies. Solving the system of equations depends on the relative numbers of equations and unknowns as follows:
 
 \begin{description}
    \item[Reactions = unknowns]{Solution can be found by a simple solution of the linear system: $\bar{x} = A^{-1} \bar{b}$}
@@ -2016,13 +2016,13 @@ \subsection{Solid Phase Heat of Reaction}
 
 If there is a reaction where mass is destroyed (the LHS of Eq.~\eqref{eq:solid_nusum} is less than 1), this approach must be modified. An assumption must be made about the destroyed mass so that the reaction $\nu$ values balance. In this case, if a reaction looses mass, FDS assumes that the lost mass has the same temperature dependent specific heat as the original material. FDS then adds a virtual material for that product which is included in the solution vector. The virtual material is not tracked outside of determining reference enthalpies.
 
-Once the reference enthalpies are known, the heat of reaction at a specific temperature can be determined by applying Eq.~\ref{eqn_solid_e_cons} at that temperature and solving for the heat of reaction.
+Once the reference enthalpies are known, the heat of reaction at a specific temperature can be determined by applying Eq.~(\ref{eqn_solid_e_cons}) at that temperature and solving for the heat of reaction.
 
 \subsection{Accounting for Solid Cell and Gas Cell Temperature Differences}
 
 The second aspect of energy conservation during a solid phase reaction is accounting for the different temperatures at which reactants and products may exist. In FDS the solid phase reaction occurs at the temperature of the solid cell being evaluated. Therefore, if the reaction consumes a gas, that gas must first be brought to the solid temperature. If the gas is hotter or cooler, an energy source or sink term will exist in the solid phase. Reaction products are produced at the solid temperature. For products that are other solid materials, nothing needs to be done as their solid temperature remains the same. For a Lagrangian particle, particles are produced at the solid temperature as well. Again, no further adjustments are needed. Note, that since particles may not be created every time step, FDS keeps track of the particle mass and particle enthalpy created by material reactions. At the next particle injection time, the particle is created at a temperature that conserves the accumulated enthalpy. For a gas product, the reaction produces the gas at the solid temperature. In FDS, the gas is injected as if it is at the current gas cell temperature. Therefore, the opposite adjustment must be made for gases that are produced as it made for gases that are consumed. In the gas phase if the current gas cell temperature is hotter than the solid, an energy sink term must be generated. In both cases the term is the mass flow rate in kg/s times the difference in enthalpy of the produced or consumed material between the gas temperature and the solid temperature (kJ/kg).
 \be
-\dot{q}_{\alpha'}'''=r_{\alpha \beta}  \nu_{\alpha',\alpha \beta} \left( h_{\alpha'}(T_g) - h_{\alpha'}(T_s) \right)
+\dot{q}_{\alpha'}'''= \sum_\alpha \sum_\beta r_{\alpha \beta}  \nu_{\alpha',\alpha \beta} \left( h_{\alpha'}(T_{\rm g}) - h_{\alpha'}(T_{\rm s}) \right)
 \ee
 This term is added to the solid cell enthalpy for gases that are consumed ($\nu_{\alpha',\alpha \beta}<0$ when $\alpha'$ is a gas). This term is subtracted from the gas cell enthalpy for the gases that are produced ($\nu_{\alpha',\alpha \beta}>0$ when $\alpha'$ is a gas). When a lumped species is specified for a material reaction, FDS will attempt to determine what primitive sub-species is actually involved in the reaction. For example, consider a solid phase combustion reaction where the oxygen in the air lumped species plus a material is converted to products. Assuming the products species is appropriately defined, FDS can examine the difference in the masses of primitive species to determine that oxygen is consumed from air but the other components (e.g., ambient N$_2$, CO$_2$, and H$_2$O) are simply passed through to the products side. In this case the enthalpy correction term for the solid would only include the O$_2$ consumption, and the enthalpy correction term for the gas would only include those products in excess of the original air minus its oxygen.