Combustion and Reactions Chapter 1: Chemical Reaction and Transport
This chapter builds the conceptual and mathematical framework for chemically reacting flows. It introduces reaction mechanisms, Arrhenius kinetics, equilibrium, thermodynamics of reacting mixtures, zero-dimensional reactors, ignition physics, transport equations, and key dimensionless numbers. It also establishes how chemistry is coupled to flow solvers through tools such as CHEMKIN.
This is the structural foundation of combustion modeling.
1. Why Reaction and Transport Must Be Studied Together
Chemical reactions do not occur in isolation. In real systems:
Species are transported by convection.
Species mix by molecular diffusion.
Heat is released.
Temperature changes modify density.
Density changes alter flow.
This tight coupling creates nonlinear feedback between chemistry, thermodynamics, and fluid mechanics.
In combustion, this coupling is the whole problem.
2. Chemical Reactions and Mechanisms
What a Chemical Reaction Really Is
A chemical reaction rearranges atoms. Molecules disappear and new ones form, but atomic elements are conserved.
The global reaction you learn in undergraduate chemistry (e.g., methane + oxygen → CO₂ + H₂O) is not how combustion actually proceeds. Real combustion involves:
Dozens to hundreds of species
Hundreds to thousands of elementary reactions
This ensemble is called the reaction mechanism.
Mechanism size determines:
Accuracy
Computational cost
Numerical stiffness
3. Reaction Rate Theory
Law of Mass Action
For an elementary reaction, the reaction rate depends on the product of reactant concentrations raised to stoichiometric powers .
Each reaction contributes to the production or destruction rate of every species.
In CFD, we typically work in mass units, converting molar rates to mass production rates for transport equations.
Arrhenius Law and Temperature Sensitivity
Reaction rate constants follow Arrhenius behavior:
A pre-exponential factor
A temperature power term
An exponential term depending on activation energy
The exponential dependence on temperature is crucial:
Small temperature changes → exponential increase in reaction rate.
This drives:
Ignition
Thermal runaway
Flame stabilization
It also introduces stiffness.
Reaction Order and Characteristic Time
For simple reactions:
First-order reactions exhibit exponential decay.
Second-order reactions depend on concentration squared.
The inverse of the rate constant behaves like a chemical time scale.
Different reactions in a mechanism have vastly different time scales. When these vary by orders of magnitude, the system becomes stiff.
Stiffness means:
Very small time steps needed for stability
Large computational cost
Specialized ODE solvers required
4. Chemical Equilibrium and Steady State
Partial Equilibrium
When forward and backward rates balance for a reaction, that reaction is in equilibrium.
The equilibrium constant depends only on thermodynamics, not kinetics.
This allows:
Computing backward rate constants from forward ones
Reducing computational effort
Global Chemical Equilibrium
At full chemical equilibrium:
All reaction rates balance
Species composition stops evolving
State depends only on pressure, temperature, and elemental composition
Equilibrium composition is typically computed by minimizing Gibbs free energy .
Equilibrium ignores kinetics. It represents the final state, not the path to reach it.
5. Accounting for Species
Species concentration can be expressed as:
Mass fraction
Mole fraction
Molar concentration
CFD prefers mass fractions, because transport equations are written in mass form.
Mixture molecular weight is computed from species composition.
6. Stoichiometry and Equivalence Ratio
Overall Stoichiometric Coefficient
For fuel–oxidizer systems, a global mass-based stoichiometric coefficient is defined.
It expresses how much oxidizer is required per unit fuel for complete reaction.
Equivalence Ratio
The equivalence ratio compares the actual fuel-to-oxidizer ratio to the stoichiometric one.
ϕ > 1 → fuel-rich
ϕ < 1 → lean
ϕ = 1 → stoichiometric
This single parameter characterizes mixture composition in premixed combustion.
7. Thermodynamics of Reacting Mixtures
Ideal Gas Mixture
Reacting gases are usually modeled as ideal gas mixtures:
Pressure depends on:
Density
Mixture gas constant
Temperature
Mixture gas constant depends on composition.
Species Enthalpy
Species enthalpy includes:
Sensible enthalpy (temperature-dependent)
Chemical enthalpy (formation enthalpy)
Formation enthalpy represents chemical energy stored in molecules.
Heat of Reaction
Heat of reaction equals the difference between formation enthalpies of products and reactants.
In combustion:
Chemical energy → sensible energy → temperature rise.
The energy equation reflects this through source terms.
Adiabatic Flame Temperature
Defined as the temperature reached when:
Reaction occurs completely
No heat losses
No mechanical energy change
It represents the theoretical maximum temperature.
It depends on:
Equivalence ratio
Pressure
Dissociation effects
8. Zero-Dimensional Reactors
These eliminate transport to isolate chemistry.
Closed Homogeneous Reactor
No mass exchange
No heat exchange
Internal energy constant
Used to:
Study ignition delay
Analyze reaction mechanisms
Perfectly Stirred Reactor (PSR)
Perfect mixing
Finite residence time
Inflow equals outflow
Species evolution depends on:
Chemical time scale
Residence time
If residence time is too short → extinction.
PSR is a fundamental building block in combustion modeling and is heavily used in CHEMKIN.
9. Ignition and Chain Reactions
Combustion proceeds via chain reactions:
Chain initiation
Chain propagation
Chain branching
Chain termination
Ignition occurs when heat generation exceeds heat losses, leading to thermal runaway.
Ignition delay strongly depends on:
Activation energy
Initial temperature
Pressure
10. Transport Equations for Reacting Flows
Reacting flows solve:
Continuity
Momentum
Species transport
Energy equation
Species equation includes:
Convection
Diffusion
Reaction source term
Energy equation includes:
Heat conduction
Chemical heat release
11. Molecular Transport
Key transport numbers:
Schmidt number (momentum vs mass diffusion)
Lewis number (thermal vs mass diffusion)
Prandtl number (momentum vs heat diffusion)
These determine:
Flame thickness
Stability
Diffusion behavior
12. Mixture Fraction and Conserved Scalars
Mixture fraction represents the local mixing state between fuel and oxidizer.
It behaves like a conserved scalar when chemistry is fast relative to mixing.
Used extensively in non-premixed combustion modeling.
13. Characteristic Time Scales and Damköhler Number
Important time scales:
Chemical time scale
Mixing time scale
Flow time scale
Damköhler number compares mixing and chemistry :
Large → mixing-limited
Small → chemistry-limited
This classification is central to combustion regime identification.
14. CHEMKIN as Chemistry Engine
CHEMKIN provides:
Reaction rate evaluation
Thermodynamic polynomials
Reactor simulations
Equilibrium calculations
In CFD coupling:
Flow solver handles transport
CHEMKIN provides source terms
Stiff ODE integration required
The separation between transport and chemistry is computationally essential.
Engineering Interpretation Framework
When analyzing a reacting system:
What controls the reaction rate?
Which time scale dominates?
Is the system mixing- or chemistry-limited?
Is equilibrium a valid approximation?
Which transport process is dominant?
This mental checklist applies to every combustion problem.
Study Priorities
Focus on:
Arrhenius temperature sensitivity
Reaction time scales and stiffness
Heat of reaction and flame temperature
Equivalence ratio and mixture fraction
Zero-dimensional reactors
Damköhler number interpretation
Key Takeaways
Reaction rate depends exponentially on temperature.
Multiple time scales create stiffness.
Heat release couples chemistry and flow.
Equilibrium depends only on thermodynamics.
Mixture fraction simplifies non-premixed modeling.
Damköhler number classifies combustion regimes.
CHEMKIN separates chemistry from transport numerically.