Species Transport Equation
Species Transport Equation: Theoretical Foundations
Overview
Professor, the species transport equation is the foundation of combustion CFD, right?
Exactly. The species transport equation describes how the mass fraction of each chemical component (CH4, O2, CO2, H2O, CO, NO, etc.) in a fluid changes. It is the foundation for all combustion CFD models (EDC, Flamelet, Species Transport + Finite Rate Chemistry, etc.).
Governing Equations
Please explain the species transport equation.
The transport equation for the mass fraction $Y_i$ of species $i$ can be written as follows.
The meaning of each term is as follows.
- Left-hand side, 1st term: Temporal change (unsteady term)
- Left-hand side, 2nd term: Convective transport
- $\mathbf{J}_i$: Diffusion flux (Fick's law: $\mathbf{J}_i = -\rho D_{i,m} \nabla Y_i$)
- $R_i$: Source/sink term due to chemical reaction
- $S_i$: Other sources (e.g., spray evaporation)
Reaction Source Term
How is the reaction source term $R_i$ expressed?
$R_i$ is the sum of contributions from all reactions and is given by the following equation.
Here, $k_{f,r}$ is the forward reaction rate constant of Arrhenius type, $\nu'$, $\nu''$ are the stoichiometric coefficients of reactants and products, and $[C_j]$ is the molar concentration of species $j$.
Do you consider the reverse reaction?
Reverse reactions are also important under conditions close to equilibrium. The reverse reaction rate constant $k_{b,r}$ is obtained from the equilibrium constant $K_{eq,r}$ as $k_{b,r} = k_{f,r}/K_{eq,r}$. In high-temperature combustion, the dissociation of CO2 ($\text{CO}_2 \rightleftharpoons \text{CO} + \frac{1}{2}\text{O}_2$) becomes important as a reverse reaction.
Diffusion Model
How do you model the diffusion flux?
There are three levels.
| Model | Computational Cost | Accuracy | Application |
|---|---|---|---|
| Fick's Law ($D_{i,m}$) | Low | Moderate | RANS Standard |
| Modified Fick's Method (Mass Conservation Correction) | Low | Good | Fluent Standard |
| Stefan-Maxwell Equations | High | Highest | When accuracy for light species (H2, He) is critical |
In turbulent flows, turbulent diffusion dominates over molecular diffusion, right?
Correct. The effective diffusion coefficient in a turbulent flow is $D_{\text{eff}} = D_{i,m} + D_t$, where $D_t = \mu_t/(\rho\,Sc_t)$ is the turbulent diffusion. $Sc_t \approx 0.7$ is a common value in combustion analysis. In high-Re turbulence, $D_t >> D_{i,m}$, so the choice of molecular diffusion model doesn't have much impact in RANS. However, in LES or laminar flames, the accuracy of molecular diffusion becomes important.
Mass Fraction Constraint
When there are $N_s$ species, solving $N_s - 1$ transport equations is sufficient. The last species is determined algebraically from the constraint $\sum Y_i = 1$. Typically, the component with the largest mass fraction (e.g., N2) is determined algebraically.
So the species transport equation is composed of three elements—convection, diffusion, and reaction—and all combustion CFD models are built upon this equation.
Exactly. Models like flamelet or PDF models are merely efficiency improvements that use table lookups instead of directly solving this equation. The essence lies in this transport equation.
Fick's Law of Diffusion—How a 1855 Law Became the Foundation of Combustion CFD
The diffusion law ($J = -D \nabla c$) published by Adolf Fick in 1855 was originally derived from experiments on salt dissolving in water. It wasn't until the early 20th century that it was found applicable to chemical species transport in gas mixtures. Extensions to multicomponent mixtures (Stefan-Maxwell model) and simplifications (Hirschfelder approximation) accumulated, leading to the modern species transport equation. To think that a 170-year-old saltwater experiment formula now operates as the fundamental equation for combustion simulations in gas turbines and furnaces today makes one appreciate anew the universality of physical laws.
Computational Methods for the Species Transport Equation
Details of Numerical Methods
Please tell me what to watch out for in the numerical solution of the species transport equation.
There are three main challenges in the numerical solution of species transport: (1) Suppression of numerical diffusion, (2) Guarantee of mass conservation, (3) Handling of stiff reaction source terms.
Spatial Discretization Schemes
What spatial discretization scheme do you recommend for species?
To resolve sharp fronts of species (like flame fronts), schemes with low numerical diffusion are necessary.
| Scheme | Accuracy | Numerical Diffusion | Stability | Recommendation |
|---|---|---|---|---|
| First Order Upwind | 1st order | Large | High | Initial convergence only |
| Second Order Upwind | 2nd order | Moderate | Good | RANS standard |
| QUICK | 3rd order | Small | Somewhat unstable | Use with caution |
| Central Difference | 2nd order | None | Unstable (oscillations) | For LES momentum equations |
| Bounded Central Difference | 2nd order | Small | Good | Recommended for LES |
So Bounded Central Difference is recommended for LES.
Correct. Central Difference has zero numerical diffusion but produces non-physical oscillations (Gibbs phenomenon). The bounded version suppresses oscillations with limiters while maintaining low diffusion. In Fluent, Bounded Central Differencing is available as an option.
Mass Conservation
What is the mass conservation problem?
When solving $N_s - 1$ transport equations and determining the last species by $Y_{N_s} = 1 - \sum_{i=1}^{N_s-1} Y_i$, numerical errors from each equation can accumulate, sometimes causing $Y_{N_s}$ to become negative.
Countermeasures include:
- Species Bounding: Clip each $Y_i$ to [0, 1] (Fluent standard)
- Flux-Corrected Transport (FCT): High-order, conservative scheme
- Calculate N2 last: Determine the component with the largest mass fraction algebraically (errors are less noticeable)
Operator Splitting
Please explain Operator Splitting for the reaction source term.
It's a technique that splits the species transport equation into "transport part" and "reaction part" and solves them alternately.
Strang splitting has 2nd-order accuracy, but splitting error becomes problematic when $\Delta t$ is large. Typical settings:
Related Topics
Experience the theory firsthand with the interactive simulator for this field
All Simulators| Parameter | RANS | LES |
|---|---|---|
| CFD Timestep | -- (Steady) | $10^{-5}$ - $10^{-6}$ s |
| ODE Substep | Automatic (CVODE internal) | Automatic |
| Splitting Method | Sequential (Fluent) | Strang (OpenFOAM) |