Reaction Models

This section briefly describes the reaction models available in PERFORM. For a comprehensive description of combustion mechanics well beyond the reaction models covered here (though also including them), we direct the reader to Combustion by Glassman and Yetter [GY08].

Finite-rate Mechanisms

Finite-rate reactions generally describe mixtures in chemical non-equilibrium, opposed to infinitely fast chemistry in which reactions are assumed to proceed to completion instantaneously. Finite-rate mechanisms must, as the name implies, compute finite reaction rates.

To preface this, we begin with the general form of the \(m\)th reaction (in a set of \(N_r\) reactions) between \(N_Y\) chemical species, given by

\[\sum_{l=1}^{N_Y} \nu_{l,m}' \chi_l \overset{k_{r,m}}{\underset{k_{f,m}}{\leftrightharpoons}} \sum_{l=1}^{N_Y} \nu_{l,m}'' \chi_l\]

where \(\chi_l\) is the chemical formula for the \(l\)th chemical species, and \(\nu_{l,m}'\) and \(\nu_{l,m}''\) are the stoichiometric coefficients of the reactants and products, respectively. These stoichiometric coefficients are input into PERFORM as the difference between the reactant coefficient and the product coefficient, i.e. \(\nu_{l,m} = \nu_{l,m}' - \nu_{l,m}''\), via nu in the chem_file.

The coefficients \(k_{f,m}\) and \(k_{r,m}\) are the forward and reverse reaction rates for the \(m\)th reaction. The forward reaction rate is computed as an Arrhenius rate, given by the formula

\[k_{f,m} = A_m T^{b_m} \text{exp} \left( \frac{-E_{a,m}}{R_u T} \right)\]

where the coefficients \(A_m\), \(b_m\), and \(E_{a,m}\) are tabulated constants given by the reaction mechanism, given by pre_exp_fact, temp_exp, and act_energy, respectively, in the chem_file.

The reaction source term \(\dot{\omega}_l\) introduced in Governing Equations is computed as a function of reaction “rates of progress” \(w_m\)

\[\dot{\omega}_l = W_l\sum_{m=1}^{N_r} (\nu_{l,m}'' - \nu_{l,m}') w_m\]

where \(W_l\) is the molecular weight of the \(l\)th species. The following methods are concerned with the calculation of these rates of progress.

Irreversible Mechanisms

The irreversible reaction mechanism model is activated by setting reaction_model = "fr_irrev" in the chem_file. An irreversible finite-rate mechanism assumes that reactions only proceed in the forward direction, i.e. converting reactants to products and neglecting the reverse reaction rate \(k_{r,m}\). The rate of progress for the \(m\)th reaction is given by

\[w_m = k_{f,m} \prod_{l=1}^{N_Y} [X_l]^{\tilde{\nu}_{l,m}}\]

where \([X_l]\) is the molar concentration of the \(l\)th species. Additionally, \(\tilde{\nu}_{l,m}\) are tabulated constants for each species and reaction which are input in the chem_file via nu_arr.

Irreversible reactions vastly simplify the calculation of the reaction source term, at the expense of accuracy. The exponential constants \(\tilde{\nu}_{l,m}\) are empirically-determined and may not be accurate under all flow and reaction regimes. The reduced cost of these mechanisms is often extremely attractive, and errors incurred by their approximations may be within acceptable limits.

Reversible Mechanisms

Coming soon!