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Chemistry - Finite-Rate Kinetics

Overview

FLINT computes mass source terms for species concentrations in chemically reacting flows, accounting for elementary, three-body, and pressure-dependent reactions. The finite-rate chemistry model captures non-equilibrium chemical kinetics essential for combustion, hypersonic flows, and plasma applications.

Mathematical Formulation

The rate of change of molar concentration \(c_i\) for species \(i\) is governed by contributions from all reactions:

\[ \frac{dc_i}{dt} = \sum_{r=1}^{N_r} \nu_{i,r} R_r \]

where:

  • \(c_i\) = molar concentration of species \(i\) (mol/m³)
  • \(\nu_{i,r} = \nu'_{i,r} - \nu''_{i,r}\) = net stoichiometric coefficient of species \(i\) in reaction \(r\)
  • \(\nu''_{i,r}\) = stoichiometric coefficient of species \(i\) as a reactant
  • \(\nu'_{i,r}\) = stoichiometric coefficient of species \(i\) as a product
  • \(R_r\) = rate of progress for reaction \(r\) (mol/(m³·s))
  • \(N_r\) = total number of reactions

For a general elementary reaction: \(\sum_{i=1}^{N_s} \nu''_{i,r} X_i \rightleftharpoons \sum_{i=1}^{N_s} \nu'_{i,r} X_i\)

Reaction Rate (Arrhenius)

The rate of progress for reaction \(r\) follows the modified Arrhenius form [1,2]:

\[ R_r = k_{f,r}(T) \prod_{j=1}^{N_s} c_j^{\nu''_{j,r}} - k_{b,r}(T) \prod_{j=1}^{N_s} c_j^{\nu'_{j,r}} \]

where the forward rate coefficient is:

\[ k_{f,r}(T) = A_r T^{b_r} \exp\left(-\frac{E_{a,r}}{R_u T}\right) \]

Parameters:

  • \(A_r\) = pre-exponential factor (units depend on reaction order)
  • \(b_r\) = temperature exponent (dimensionless)
  • \(E_{a,r}\) = activation energy (J/mol or cal/mol)
  • \(R_u\) = universal gas constant (8.314 J/(mol·K))
  • \(T\) = temperature (K)

The backward rate coefficient \(k_{b,r}(T)\) is computed from thermodynamic equilibrium:

\[ k_{b,r}(T) = \frac{k_{f,r}(T)}{K_{c,r}(T)} \]

where \(K_{c,r}(T)\) is the equilibrium constant in concentration units [3].

Pressure-Dependent Reactions

Three-Body Reactions

Three-body (or chaperon) reactions involve a collision partner \(M\) that stabilizes an excited intermediate without being consumed [4]:

\[ A + B + M \rightleftharpoons AB + M \]

The effective concentration of the third body is:

\[ [M] = \sum_{j=1}^{N_s} \alpha_{j,r} c_j \]

where \(\alpha_{j,r}\) is the third-body efficiency of species \(j\) in reaction \(r\). The reaction rate includes this effective concentration in the appropriate stoichiometric position.

Lindemann Reactions (Falloff Behavior)

Lindemann reactions describe the transition between low-pressure (termolecular) and high-pressure (bimolecular) kinetics [5,6]. These are also known as unimolecular dissociation/recombination reactions or pressure falloff reactions.

Physical mechanism: At low pressures, the reaction rate is limited by collisional activation/deactivation and is proportional to pressure (third-order kinetics). At high pressures, all molecules are in steady-state equilibrium with the excited state, and the reaction becomes pressure-independent (second-order kinetics) [7].

Rate expression: $$ k_r(T, P) = k_{\infty,r}(T) \left(\frac{P_r}{1 + P_r}\right) F_r $$

where:

  • \(k_{\infty,r}(T)\) = high-pressure limit rate coefficient
  • \(k_{0,r}(T)\) = low-pressure limit rate coefficient
  • \(P_r\) = reduced pressure (dimensionless): $$ P_r = \frac{k_{0,r}(T) [M]}{k_{\infty,r}(T)} $$
  • \([M]\) = effective third-body concentration
  • \(F_r\) = broadening factor (accounts for deviations from simple Lindemann theory)

Limiting behavior:

  • Low pressure (\(P_r \ll 1\)): \(k_r \approx k_{0,r}[M]\) (third-order)
  • High pressure (\(P_r \gg 1\)): \(k_r \approx k_{\infty,r}\) (second-order)

Troe Formulation

The Troe formulation provides an empirical broadening factor \(F_r\) that better matches experimental falloff curves than the simple Lindemann form [8,9]:

\[ \log_{10} F_r = \frac{\log_{10} F_{\text{cent},r}}{1 + \left[\frac{\log_{10} P_r + c}{n - d(\log_{10} P_r + c)}\right]^2} \]

where:

\[ F_{\text{cent},r} = (1-a) \exp(-T/T^{***}) + a \exp(-T/T^*) + \exp(-T^{**}/T) \]

and the parameters are:

  • \(a, T^*, T^{**}, T^{***}\) = Troe parameters (fitted to experimental data)
  • \(c = -0.4 - 0.67 \log_{10} F_{\text{cent},r}\)
  • \(n = 0.75 - 1.27 \log_{10} F_{\text{cent},r}\)
  • \(d = 0.14\)

Simplified Troe form (three-parameter): When \(T^{**} \to \infty\): \(F_{\text{cent},r} = (1-a) \exp(-T/T^{***}) + a \exp(-T/T^*)\)

Note: For simple Lindemann falloff (no Troe correction), set \(F_r = 1\).

References

[1] Arrhenius, S. "Über die Reaktionsgeschwindigkeit bei der Inversion von Rohrzucker durch Säuren." Zeitschrift für Physikalische Chemie, vol. 4, 1889, pp. 226-248.

[2] Kee, R. J., Coltrin, M. E., and Glarborg, P. Chemically Reacting Flow: Theory and Practice, 2nd edition. John Wiley & Sons, 2003.

[3] Smith, G. P., et al. "GRI-Mech 3.0." http://www.me.berkeley.edu/gri_mech/

[4] Baulch, D. L., et al. "Evaluated Kinetic Data for Combustion Modeling: Supplement II." Journal of Physical and Chemical Reference Data, vol. 34, no. 3, 2005, pp. 757-1397.

[5] Lindemann, F. A. "Discussion on 'The Radiation Theory of Chemical Action'." Transactions of the Faraday Society, vol. 17, 1922, pp. 598-599.

[6] Gilbert, R. G., Luther, K., and Troe, J. "Theory of Thermal Unimolecular Reactions in the Fall-off Range. II. Weak Collision Rate Constants." Berichte der Bunsengesellschaft für Physikalische Chemie, vol. 87, 1983, pp. 169-177.

[7] Gardiner, W. C. Gas-Phase Combustion Chemistry, 2nd edition. Springer, 2000.

[8] Troe, J. "Predictive Possibilities of Unimolecular Rate Theory." Journal of Physical Chemistry, vol. 83, no. 1, 1979, pp. 114-126.

[9] Gilbert, R. G., Smith, S. C., and Jordan, M. J. T. UNIMOL: Calculation of Fall-off Curves for Unimolecular and Recombination Reactions. Blackwell Scientific Publications, 1990.