Evaporation Model
Evaporation: Theoretical Foundations
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.
Computational Methods for Evaporation
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.
Evaporation in Practice
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
2. Mesh Generation: Fine mesh in spray region, coarse mesh in far field
3. Gas Phase Setup: CFD domain, inlet temperature/humidity, outlet pressure
4. Lagrangian Particle Injection: Droplet size distribution, initial velocity, mass flow rate
5. Enable Evaporation Model: Select appropriate model, set liquid properties
6. Species Transport: Set up species equation for vapor (two-way coupling)
7. Time Stepping: Choose time step to capture evaporation (typically smaller than steady analysis)
8. Run and Post-Process: Monitor droplet residence time, final size distribution, humidity increase