Chemical Reaction Kinetics Fundamentals

Exactly. Chemical kinetics is the discipline that mathematically describes how fast fuel and oxidizer react. In combustion CFD, if this reaction rate is not modeled correctly, results for flame temperature and exhaust gas composition will be completely off the mark.

The core is the Arrhenius-type reaction rate equation. For each elementary reaction, the rate constant is expressed by three parameters: the pre-exponential factor $A$, the temperature exponent $n$, and the activation energy $E_a$.

The reaction rate constant $k$ can be written as follows.

Here, $A$ is the pre-exponential factor (frequency factor), $n$ is the temperature exponent, $E_a$ is the activation energy [J/mol], and $R$ is the universal gas constant 8.314 J/(mol K). For example, in the basic hydrogen reaction H$_2$ + O$_2$ system, $E_a$ is on the order of tens of kJ/mol.

Yes. The larger $E_a$ is, the smaller the exponential term $\exp(-E_a/RT)$ becomes, and at low temperatures, the reaction hardly proceeds. Conversely, as temperature increases, the rate increases exponentially. This directly relates to combustion ignition delay and flame stability.

The time rate of change of the mass fraction $Y_i$ of chemical species $i$ can be written by summing the contributions of all elementary reactions as follows.

Correct. $[X_j]$ is molar concentration, $M_{w,i}$ is molecular weight. Multiple elementary reactions intertwine to constitute the species production/consumption rate $\dot{\omega}_i$.

That's precisely the problem of Stiffness. For example, the detailed reaction mechanism for methane/air, GRI-Mech 3.0, contains 53 species and 325 elementary reactions, but the time constant for radical reactions is on the order of $10^{-9}$ seconds, while the main reactions are on the order of $10^{-3}$ seconds—a difference of over six orders of magnitude.

Exactly. Explicit Euler or Runge-Kutta methods would require extremely small time steps, making the computation time impractical. Therefore, implicit methods (BDF methods, Rosenbrock methods, etc.) or tabulation techniques like ISAT (In-Situ Adaptive Tabulation), which will be covered in detail in the next article, are used.

Reaction mechanisms have the following hierarchy.

Because the computational cost becomes enormous. Trying to solve a detailed mechanism with 100 species on a 3D mesh with 1 million cells would require solving a system of 100 ODEs in each cell every time step. Therefore, in practice, reduced mechanisms are used, or dimensionality is reduced using ISAT or FGM (Flamelet Generated Manifold).

Good. The details of numerical methods will be covered in the next article. If you make a mistake in setting the Arrhenius equation parameters, the ignition temperature can be off by hundreds of Kelvin, so always verify against experimental data.

Numerical integration of chemical reactions in combustion CFD is broadly divided into three approaches: (1) Direct Integration (DI), (2) Tabulation methods, and (3) Operator splitting methods.

This method solves the ODE system for chemical species in each CFD cell every time step. To handle stiff systems, implicit multi-step methods are used.

Let's compare typical implicit solvers.

Yes. In Ansys Fluent, a CVODE-based integrator is incorporated as the "Stiff Chemistry Solver." STAR-CCM+ also internally uses the SUNDIALS library. Evaluating the Jacobian matrix becomes a bottleneck, so automatic generation of analytical Jacobians is important.

A method proposed by Pope in 1997, it records the input-output of once-calculated chemical reactions in a binary tree structure table. When a new composition point arrives, if the answer can be returned via linear approximation from existing records, direct integration is skipped.

The tolerance $\epsilon_{\text{tol}}$ is set by the user. Fluent's default is around $10^{-4}$. ISAT can often reduce direct integration calls by over 90%. However, table size can strain memory, and in large-scale parallel computing, each process holds an independent table, making memory efficiency a challenge.

A method that solves fluid transport and chemical reactions in separate steps. Strang splitting is representative, alternately executing CFD transport steps and ODE integration for chemical reactions.