Soot Model
Theory and Physics
Overview
Professor, why is soot important in combustion CFD?
Soot is carbonaceous fine particles (particle size 10-100 nm) generated by incomplete combustion, and it is important for three reasons. (1) It is subject to exhaust gas regulations (PM: Particulate Matter), (2) It contributes significantly to radiative heat transfer (soot dominates flame radiation), (3) Health hazards (carcinogenicity). Soot prediction in diesel engines, aircraft engines, and industrial furnaces is an essential task.
Please explain the soot formation mechanism.
Soot formation progresses through a 4-stage process.
1. Nucleation: The initial soot nuclei are formed by the polymerization of PAHs (Polycyclic Aromatic Hydrocarbons). C2H2 (acetylene) is the main precursor for PAH growth.
2. Surface Growth: Carbon deposits on the soot particle surface via the HACA mechanism (H-Abstraction-C2H2-Addition).
3. Coagulation: Particles collide and coalesce, increasing in size.
4. Oxidation: Soot is combusted and eliminated by O2 or OH.
Moss-Brooke Model
Please explain the governing equations for the soot model used in CFD.
The Moss-Brooke 2-variable model solves two transport equations: for soot mass fraction $Y_s$ and soot number density $N$ (number of particles/kg).
The soot volume fraction $f_v$ is obtained by the following equation.
Here, $\rho_s \approx 1800$ kg/m$^3$ is the density of soot, and $d_p$ is the average particle diameter.
Key Parameters for Soot Formation
Under what conditions is more soot produced?
Let's summarize the main factors for soot formation.
| Factor | Direction for Increased Soot | Reason |
|---|---|---|
| Equivalence Ratio | Rich ($\phi > 1$) | Insufficient oxygen leads to incomplete combustion |
| Temperature | 1500-1800 K | Optimal temperature range for nucleation |
| Pressure | High Pressure | Increased collision frequency |
| Fuel Structure | Aromatic > Straight-chain | Easier generation of PAH precursors |
| Residence Time | Long | Ensures time for soot growth |
So 1500-1800 K is the temperature window for soot formation.
Correct. Below this temperature, nucleation rate is slow; above it, OH oxidation becomes dominant and soot burns out. This "soot formation window" is visualized as a $\phi$-T map. The $\phi$-T map for diesel combustion (Dec diagram) forms the basis for simultaneous reduction strategies of soot and NOx.
So the soot model is a fusion of chemical kinetics and particle dynamics.
Yes. Because it requires accurate description of both gas-phase PAH chemistry and soot particle dynamics, it is one of the most challenging fields in combustion modeling.
Soot is a Nano-sized Carbon Particle — The Challenge of Describing the Formation of Objects Below 1nm in Diameter with Equations
Soot formation is a process where carbon particles with diameters of 1–100 nm form, grow, and coagulate in just a few milliseconds during combustion. To represent this in CFD, the four processes — "nucleation → surface growth → coagulation → oxidation" — must be formulated mathematically. There are different schools of thought, such as the Fenimore-Jones two-equation model and Frenklach's detailed soot model, but describing the particle size distribution (PSD) is particularly difficult. Particles of 1 nm and 100 nm behave completely differently, yet CFD grids are on the order of millimeters — meaning we are dealing with a problem of different dimensions where the object size is one-millionth of the computational cell.
Physical Meaning of Each Term
- Temporal Term $\partial(\rho\phi)/\partial t$: Imagine the moment you turn on a faucet. At first, water comes out in an unstable, spluttering manner, but after a while, it becomes a steady flow, right? This "period of change" is described by the temporal term. The pulsation of blood flow from a heartbeat, or the flow fluctuation each time an engine valve opens and closes — all are unsteady phenomena. So what is steady-state analysis? It looks only at "after sufficient time has passed and the flow has settled down" — meaning this term is set to zero. Since computational cost drops significantly, solving first in steady-state is a basic CFD strategy.
- Convection Term $\nabla \cdot (\rho \mathbf{u} \phi)$: What happens if you drop a leaf into a river? It gets carried downstream by the flow, right? This is "convection" — the effect where fluid motion transports things. Warm air from a heater reaching the far corner of a room is also because the "carrier," air, transports heat via convection. Here's the interesting part — this term contains "velocity × velocity," making it nonlinear. That is, as flow speed increases, this term rapidly strengthens, making control difficult. This is the root cause of turbulence. A common misconception: "Convection and conduction are similar" → They are completely different! Convection is carried by flow, conduction is transmitted by molecules. There is an order of magnitude difference in efficiency.
- Diffusion Term $\nabla \cdot (\Gamma \nabla \phi)$: Have you ever put milk in coffee and left it? Even without stirring, after a while they naturally mix. That's molecular diffusion. Now a question — honey and water, which flows more easily? Obviously water, right? Honey has high viscosity ($\mu$), so it flows poorly. Higher viscosity strengthens the diffusion term, making the fluid move in a "thick" manner. In low Reynolds number flows (slow, viscous), diffusion dominates. Conversely, in high Re number flows, convection overwhelms and diffusion plays a supporting role.
- Pressure Term $-\nabla p$: When you push the plunger of a syringe, liquid shoots out forcefully from the needle tip, right? Why? Because the piston side is high pressure, the needle tip is low pressure — this pressure difference provides the force that pushes the fluid. Dam water discharge works on the same principle. On a weather map, where isobars are tightly packed? That's right, strong winds blow. "Flow is generated where there is a pressure difference" — this is the physical meaning of the pressure term in the Navier-Stokes equations. A point of confusion here: "Pressure" in CFD is often gauge pressure, not absolute pressure. If results go wrong immediately after switching to compressible analysis, mixing up absolute/gauge pressure might be the cause.
- Source Term $S_\phi$: Warmed air rises — why? Because it becomes lighter (lower density) than its surroundings and is pushed upward by buoyancy. This buoyancy is added to the equation as a source term. Other examples: chemical reaction heat generated by a gas stove flame, Lorentz force applied to molten metal by an electromagnetic pump in a factory... These are all actions that "inject energy or force into the fluid from the outside," expressed by source terms. What happens if you forget the source term? In natural convection analysis, forgetting buoyancy means the fluid doesn't move at all — a physically impossible result where warm air doesn't rise in a heated room in winter.
Assumptions and Applicability Limits
- Continuum Assumption: Valid for Knudsen number Kn < 0.01 (mean free path of molecules ≪ characteristic length)
- Newtonian Fluid Assumption: Shear stress and strain rate have a linear relationship (non-Newtonian fluids require viscosity models)
- Incompressibility Assumption (for Ma < 0.3): Density is treated as constant. For Mach numbers above 0.3, compressibility effects must be considered.
- Boussinesq Approximation (Natural Convection): Density variation is considered only in the buoyancy term; constant density is used in other terms.
- Non-applicable Cases: Rarefied gases (Kn > 0.1), supersonic/hypersonic flows (shock capturing required), free surface flows (VOF/Level Set, etc., required)
Dimensional Analysis and Unit Systems
| Variable | SI Unit | Notes / Conversion Memo |
|---|---|---|
| Velocity $u$ | m/s | When converting from volumetric flow rate for inlet conditions, pay attention to cross-sectional area units. |
| Pressure $p$ | Pa | Distinguish between gauge and absolute pressure. Use absolute pressure for compressible analysis. |
| Density $\rho$ | kg/m³ | Air: approx. 1.225 kg/m³ @20°C, Water: approx. 998 kg/m³ @20°C |
| Viscosity Coefficient $\mu$ | Pa·s | Be careful not to confuse with kinematic viscosity $\nu = \mu/\rho$ [m²/s] |
| Reynolds Number $Re$ | Dimensionless | $Re = \rho u L / \mu$. Indicator for laminar/turbulent transition. |
| CFL Number | Dimensionless | $CFL = u \Delta t / \Delta x$. Directly related to time step stability. |
Numerical Methods and Implementation
Details of Numerical Methods
Please explain the numerical implementation of the soot model.
Soot models used in CFD are broadly classified into three categories.
| Model | Accuracy | Computational Cost | Features |
|---|---|---|---|
| Empirical 2-Variable Model | Low-Medium | Low | Moss-Brooke, transports $Y_s$ and $N$ |
| Method of Moments (MoM) | Medium-High | Medium | Transports moments of particle size distribution |
| Sectional Method | High | High | Resolves particle size distribution with discrete sections |
Empirical 2-Variable Model
Please start with the simplest model.
The Moss-Brooke model (standard in Fluent) describes the particle population with two variables: $Y_s$ (soot mass fraction) and $N$ (number density). It uses Arrhenius-type rate expressions for each process: nucleation, surface growth, coagulation, and oxidation. It's simple, but particle size distribution information is limited to average values.
Method of Moments (MOMIC)
What is the Moment Method?
It is a method that solves transport equations for the moments $M_r = \int_0^\infty v^r n(v) dv$ of the particle size distribution function $n(v,t)$ ($v$ is particle volume). $M_0$ corresponds to number density, $M_1$ to volume fraction. MOMIC (Method of Moments with Interpolative Closure) proposed by Frenklach & Harris is available in Fluent 2020 and later.
Sectional Method
What are the advantages of the Sectional Method?
The particle size range is divided into discrete sections (bins), and the number density in each section is transported individually. It can represent arbitrary shapes of particle size distributions with the highest accuracy, but requires transport of 20-30 additional scalar variables, resulting in high computational cost. Available in STAR-CCM+ and CONVERGE.
Fluent Setup
Please explain the soot model setup procedure in Fluent.
1. Models > Species > Species Transport (assuming combustion model is already set)
2. Models > Soot > Moss-Brooke (simple) or MOMIC (recommended)
3. Setup of PAH precursor species (C2H2, C6H6, etc.) — must be included in the reaction mechanism
4. Coupling with radiation model — pass soot absorption coefficient to the radiation model
An important note: the soot model requires a reaction mechanism that includes PAH precursors (at least C2H2). Global one-step mechanisms do not include C2H2, so soot calculation is not possible. Use mechanisms like DRM-19 or higher.
Coupling of Soot and Radiation
How are soot and radiation coupled?
Soot particles emit and absorb radiation across a continuous spectrum. The absorption coefficient of soot is approximated by the following equation.
Here, $C_0$ and $C_2$ are optical constants. The continuous radiation from soot is added to gas radiation (band radiation from CO2, H2O), so in flames with high soot content, radiative loss increases significantly.
So the soot model is a triple coupling of combustion model + particle model + radiation model.
Yes. Balancing model complexity and computational cost is crucial. A practical approach is to first grasp trends with the Moss-Brooke 2-variable model, then move to MOMIC or the Sectional method as needed.
Numerical Methods for Soot Models — Choosing Between Transport Equation Methods and Sectional Methods
For soot analysis in CFD, there are mainly "2-equation models (Moss-Brookes: soot number density N + soot volume fraction f)" and "Sectional methods (tracking particle size distribution with discrete bins)." The 2-equation model has low computational cost and is standard in Fluent, but cannot provide detailed particle size distribution. The sectional method can resolve particle nucleation, coagulation, and surface growth by particle size, offering high prediction accuracy for light scattering/absorption properties. Recently, MoM (Method of Moments) is gaining attention as an intermediate approach, offering a good balance between computational cost and modeling accuracy. Practical implementation of sectional methods is accelerating for compliance with NVPM (Non-Volatile Particulate Matter) regulations for aircraft engines.
Upwind Scheme
First-order upwind: Large numerical diffusion but stable. Second-order upwind: Improved accuracy but risk of oscillations. Essential for high Reynolds number flows.
Central Differencing
Second-order accurate, but numerical oscillations occur for Pe number > 2. Suitable for low Reynolds number, diffusion-dominated flows.
TVD Schemes (MUSCL, QUICK, etc.)
Maintain high accuracy while suppressing numerical oscillations via limiter functions. Effective for capturing shocks and steep gradients.
Finite Volume Method vs Finite Element Method
FVM: Naturally satisfies conservation laws. Mainstream in CFD. FEM: Advantageous for complex shapes and multi-physics. Mesh-free methods like SPH are also developing.
CFL Condition (Courant Number)
Explicit methods: CFL ≤ 1 is the stability condition. Implicit methods: Stable even for CFL > 1, but affects accuracy and iteration count. LES: CFL ≈ 1 is recommended. Physical meaning: Information should not travel more than one cell per time step.
Residual Monitoring
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