Evaporation Model
Theory and Physics
Overview
Professor, what does a CFD evaporation model calculate?
It's a model that predicts the evaporation rate of droplets and liquid films. It handles heat and mass transfer with phase change, such as droplet evaporation after fuel injection, spray drying, cooling towers, and paint drying. The mainstream approach is to calculate evaporation for each droplet within Lagrangian particle tracking (DPM).
What is the physics behind droplet evaporation?
It's the process where vapor diffuses from the droplet surface into the surrounding gas. At the droplet surface, the concentration corresponds to the saturation vapor pressure, and the concentration difference with the lower concentration far away is the driving force. Simultaneously, the droplet temperature drops due to the absorption of latent heat of evaporation, making it a coupled problem of heat and mass transfer.
Governing Equations
Please tell me the equation for the evaporation rate.
In the classical Abramzon-Sirignano (1989) model, the droplet mass change rate is expressed as follows.
Here $d_p$ is the droplet diameter, $D_{AB}$ is the binary diffusion coefficient of the vapor, $Sh^*$ is the modified Sherwood number, and $B_M$ is the Spalding mass transfer number.
$Y_s$ is the vapor mass fraction at the droplet surface, right?
That's correct. $Y_s$ is obtained by finding the saturation vapor pressure at the droplet temperature from the Clapeyron-Clausius equation and converting from mole fraction to mass fraction. $Y_\infty$ is the vapor mass fraction far away (CFD cell average).
The droplet temperature change is determined from the heat balance.
$B_T$ is the Spalding heat transfer number, $Nu^*$ is the modified Nusselt number. The first term on the right side is convective heating, and the second term is evaporative cooling (latent heat absorption).
d-squared Law
I've heard of the d-squared law.
In steady evaporation, the square of the droplet diameter decreases proportionally with time.
$K$ is the evaporation rate constant, determined by the liquid type and ambient conditions. This linear decrease is the d-squared law, most commonly used as a validation metric for evaporation models.
D2 Law—The "Golden Rule" of Droplet Evaporation and Its Deviation from Reality
The classical "D2 law" governing droplet evaporation rate is a simple law stating that the square of the droplet diameter decreases proportionally with time. It is expressed as D^2 = D0^2 - K*t (K: evaporation constant), and this law originated from the fuel droplet combustion analysis published almost simultaneously by Godsave and Spalding in 1953. However, real droplets are influenced by internal circulation, Marangoni convection, gas-phase thermal radiation, etc., causing the D2 law to produce errors up to 30%. In dense spray environments with multiple droplets, the deviation becomes even larger due to the Group evaporation effect from vapor of adjacent droplets.
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 unstably, but after a while, the flow becomes steady, right? This "during the 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/closes—all are unsteady phenomena. So what is steady-state analysis? Looking only at "after sufficient time has passed and the flow has settled"—meaning setting this term to zero. Since computational cost drops significantly, solving first with a 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 end 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 the flow becomes faster, this term rapidly strengthens, making control difficult. This is the root cause of turbulence. A common misconception: "Convection and conduction are similar" → Completely different! Convection is carried by flow, conduction is transmitted by molecules. There's 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 it naturally mixes. That's molecular diffusion. Now a question—honey and water, which flows easier? Obviously water, right? Honey has high viscosity ($\mu$), so it flows poorly. When viscosity is high, the diffusion term becomes strong, and the fluid moves in a "thick" manner. In low Reynolds number flow (slow, viscous), diffusion is dominant. Conversely, in high Re number flow, convection overwhelmingly dominates, 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 plunger side is high pressure, the needle tip is low pressure—this pressure difference becomes the force pushing the fluid. Dam discharge works on the same principle. On a weather map, where isobars are tightly packed? That's right, strong winds blow. "Where there is a pressure difference, flow is generated"—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. When switching to compressible analysis, if results become strange, it might be due to mixing up absolute/gauge pressure.
- Source Term $S_\phi$: Heated air rises—why? Because it becomes lighter (lower density) than its surroundings, so it's pushed up 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 a factory electromagnetic pump... These are all actions that "inject energy or force into the fluid from the outside," expressed by the source term. What happens if you forget the source term? In natural convection analysis, forgetting to include buoyancy means the fluid doesn't move at all—a physically impossible result where warm air doesn't rise in a heated winter room.
Assumptions and Applicability Limits
- Continuum Assumption: Valid for Knudsen number Kn < 0.01 (molecular mean free path ≪ characteristic length)
- Newtonian Fluid Assumption: Linear relationship between shear stress and strain rate (non-Newtonian fluids require viscosity models)
- Incompressibility Assumption (for Ma < 0.3): Treat density as constant. For Mach number 0.3 and above, consider compressibility effects
- Boussinesq Approximation (Natural Convection): Consider density variation only in the buoyancy term, using constant density in other terms
- Non-applicable Cases: Rarefied gas (Kn > 0.1), supersonic/hypersonic flow (requires shock capturing), free surface flow (requires VOF/Level Set, etc.)
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 coefficient $\nu = \mu/\rho$ [m²/s] |
| Reynolds Number $Re$ | Dimensionless | $Re = \rho u L / \mu$. Criterion 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 tell me the numerical points of the evaporation model.
Within Lagrangian droplet tracking, the evaporation amount is calculated for each droplet at each time step. The mass lost by evaporation is reflected as a source term in the gas-phase species transport equation (two-way coupling).
The calculation flow is as follows.
1. Interpolate gas temperature and vapor concentration at the droplet position
2. Calculate saturation vapor pressure at the droplet surface (Antoine equation or Clausius-Clapeyron equation)
3. Calculate Spalding transfer numbers $B_M$, $B_T$
4. Calculate evaporation rate $\dot{m}$ and droplet temperature change rate
5. Update droplet mass and diameter
6. Reflect mass, momentum, and energy sources to the gas phase
Multi-Component Droplet Evaporation
How are droplets with mixed components, like gasoline, handled?
In multi-component evaporation models, the vapor pressure of each component is determined by Raoult's law.
$x_i$ is the mole fraction in the liquid phase, $\gamma_i$ is the activity coefficient, $p_i^{sat}$ is the saturation vapor pressure of the pure component. Light components evaporate first, and the droplet composition changes over time.
Is the temperature distribution inside the droplet considered?
The simple model (Uniform Temperature) assumes a uniform temperature inside the droplet. High-precision models (Diffusion Limit) solve for the radial distribution of temperature and composition inside the droplet. In Fluent, you can choose between "Infinite Diffusion" and "Diffusion-Limited".
Implementation by Tool
| Tool | Evaporation Model | Multi-Component | Features |
|---|---|---|---|
| Ansys Fluent | Convection/Diffusion Controlled | Raoult's Law | Coupled with Species Transport |
| STAR-CCM+ | Abramzon-Sirignano | Multi-component support | Lagrangian framework |
| OpenFOAM (sprayFoam) | Various evaporation models | Supported | Customizable |
| CONVERGE | Multi-component evaporation | Detailed chemistry coupling | AMR support |
In combustion analysis, coupling between the evaporation model and chemical reaction model is important. To accurately reproduce the process where vapor generated by droplet evaporation ignites and burns, the resolution of Species Transport and time step management are key.
Langmuir-Knudsen Model—Molecular Theoretical Basis for Thin-Film Evaporation
The Langmuir-Knudsen model, which describes the evaporation/condensation rate of molecules at a liquid surface from kinetic theory, handles the physics of the Knudsen layer where the mean free path of the gas-liquid interface becomes comparable to the film thickness. For thin-film evaporators in microdevices (MEMS) or evaporation of nano-droplets, the continuum assumption breaks down, making this model essential. However, the uncertainty of the "evaporation coefficient σ_e (ranging from 0.01 to 1 with almost 4 orders of magnitude uncertainty)" when implementing it in CFD is the biggest challenge, causing dramatically different calculation results even for the same water evaporation depending on literature values.
Upwind Scheme (Upwind)
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 (Central Differencing)
Second-order accurate, but numerical oscillations occur for Pe number > 2. Suitable for low Reynolds number diffusion-dominated flows.
TVD Scheme (MUSCL, QUICK, etc.)
Maintains high accuracy while suppressing numerical oscillations via limiter functions. Effective for capturing shock waves 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 method: CFL ≤ 1 is the stability condition. Implicit method: Stable even for CFL > 1, but affects accuracy and iteration count. LES: CFL ≈ 1 recommended. Physical meaning: Information should not travel more than one cell per time step.
Residual Monitoring
Convergence is judged when each residual for the Continuity Equation, momentum, and energy decreases by 3-4 orders of magnitude. The mass conservation residual is particularly important.
Relaxation Factor
Pressure: 0.2–0.3, Velocity: 0.5–0.7 are typical initial values. If diverging, lower the relaxation factor. After convergence, increase to accelerate.
Internal Iterations for Unsteady Calculations
Iterate within each time step until a steady solution converges. Internal iteration count: 5–20 times is a guideline. If residuals fluctuate between time steps, review the time step size.
Analogy for SIMPLE Method
The SIMPLE method is an "alternating adjustment" technique. First, velocity is tentatively determined (predictor step), then pressure is corrected so that mass conservation is satisfied with that velocity (corrector step), and velocity is revised with the corrected pressure—this back-and-forth is repeated to approach the correct solution. It resembles two people leveling a shelf: one adjusts the height, the other balances it, and they repeat this alternately.
Analogy for Upwind Scheme
The upwind scheme is a method that "stands in the river flow and prioritizes upstream information." A person in the river cannot tell the source of the water by looking downstream—it's a discretization method reflecting the physics that upstream information determines downstream. Although first-order accurate, it is highly stable because it correctly captures flow direction.
Practical Guide
Practical Guide
Please tell me the procedure for spray analysis involving evaporation.
Let's take the analysis of a spray dryer as an example.
1. Geometry Creation: Drying chamber, spray nozzle position, exhaust port
なった
詳しく
報告