Chemical Kinetics#
Pyrometheus generates code to evaluate chemical source terms. These appear in the conservation equations of reacting flows. Here, we lay out the corresponding equations. We focus on a homogeneous adiabatic reactor for simplicify. Yet, the systems explained here can easily be adapted to other configurations (e.g., isochoric or inhomogeneous reactors).
Species Conservation and Chemical Source Terms#
Our goal is to express, in as much detail, the chemical kinetics of reactive flows. We assume a homogeneous mixture of ideal gases evolves at constant pressure \(p\) and enthalpy \(h_{0}\). We characterize its chemical composition by the species mass fractions \(\boldsymbol{y} = \{ y_{i} \}_{i = 1}^{N}\). These evolve from an initial condition \(y_{i}(0) = y_{i}^{0}\) according to
where \(S_{i}\) is the chemical source term of the \(i^{\mathrm{th}}\) species (in \(\mathrm{s}^{-1}\)), \(W_{i}\) its molecular weight (in \(\mathrm{kg/kmol})\) and \(\dot{\omega}_{i}\) its molar production rate (in \(\mathrm{kmol/m^{3}-s}\)). The mixture density \(\rho\) (in \(\mathrm{kg/m^{3}}\)) is obtained from
where
is the mixture molecular weight, \(R\) the universal gas constant (in \(\mathrm{J/kmol-K}\)), and \(T\) the temperature (in \(\mathrm{K}\)). We explain how to obtain the mixture temperature in detail in Section 1.2.
To evaluate [species_conservation], we need the net production rates \(\dot{\boldsymbol{\omega}} = \{ \dot{\omega}_{i} \}_{i = 1}^{N}\). These represent changes in composition due to chemical reactions
where \(\nu_{ij}^{\prime}\) and \(\nu_{ij}^{\prime\prime}\) are the forward and reverse stoichiometric coefficients of species \(\mathcal{S}_{i}\) in the \(j^{\mathrm{th}}\) reaction. Per [reactions], species \(\mathcal{S}_{i}\) can only be produced (or destroyed) by an amount \(\nu_{ij}^{\prime\prime}\) (or \(\nu_{ij}^{\prime}\)) in the \(j^{\mathrm{th}}\) reaction. Thus, \(\{ \dot{\omega}_{i} \}_{i = 1}^{N}\) are linear combinations of the reaction rates of progress \(R_{j}\),
where \(\nu_{ij} = \nu_{ij}^{\prime\prime} - \nu_{ij}^{\prime}\) is the net stoichiometric coefficient of the \(i^{\mathrm{th}}\) species in the \(j^{\mathrm{j}}\) reaction. The rates of progress are given by the law of mass-action,
where \(k_{j}(T)\) is the rate coefficient of the \(j^{\mathrm{th}}\) reaction and \(K_{j}(T)\) its equilibrium constant. Depending on the reaction, the rate coefficient \(k_{j}(T)\) may take different forms (and even become a function of pressure). Its simplest form is the Arrhenius expression,
where \(A_{j}\) is the pre-exponential, \(b_{j}\) is the temperature exponent, and \(\theta_{a,j}\) is the activation temperature.
The equilibrium constant is evaluated through equilibrium thermodynamics
where \(p_{0} = 1\) \(\mathrm{atm}\) and
are the species Gibbs functions, with \(h_{i}\) and \(s_{i}\) the species enthalpies and entropies.
Species Thermodynamics#
Conservation of Energy#
To evaluate the rates of progress [reaction_rates], we need the temperature. Yet, we have defered any discussion on how to compute it from other state variables.