Modeling Particle Behavior
Learning Objectives
Students will be able to:
- Describe particle size distributions using lognormal and multimodal models
- Calculate mass and number concentrations from size distribution data
- Apply residence time concepts to predict indoor particle levels
- Use spreadsheet models to simulate particle dynamics
- Interpret model outputs and assess model limitations
The Big Question
"How do we mathematically represent the complex mixture of particle sizes in air, and how can models predict particle concentrations over time?"
Particle Size Distributions
Real aerosols contain particles spanning many orders of magnitude in size. We characterize this using size distributions.
Number Distribution
dN/dlog(dp)
Number of particles per unit volume in each size bin
Ultrafine particles dominate number concentration
Mass Distribution
dM/dlog(dp)
Mass of particles per unit volume in each size bin
Larger particles dominate mass concentration
The Lognormal Distribution
Particle sizes typically follow a lognormal distribution, characterized by geometric mean diameter (dg) and geometric standard deviation (sigmag):
n(d) = Nt / (sqrt(2 pi) * ln(sigmag) * d) * exp[-(ln(d) - ln(dg))2 / (2 * ln2(sigmag))]
| Parameter | Typical Values | Physical Meaning |
|---|---|---|
| dg | 0.01-10 um | Central tendency (median on log scale) |
| sigmag | 1.5-3.0 | Width of distribution (1 = monodisperse) |
| Nt | 103-106 /cm3 | Total number concentration |
Multimodal Distributions
Atmospheric aerosols typically have multiple modes, each from different sources:
Nucleation Mode
dp < 0.02 um
Freshly nucleated particles, high number, low mass
Accumulation Mode
0.1-1 um
Aged particles, dominates PM2.5 mass
Coarse Mode
dp > 1 um
Dust, pollen, mechanical generation
Residence Time and Loss Rate
The residence time (taur) is the average time a particle spends in a space before being removed:
taur = 1 / (lambdav + lambdad + lambdaf)
- lambdav: Ventilation loss rate (air changes per hour)
- lambdad: Deposition loss rate (settling, diffusion to surfaces)
- lambdaf: Filtration loss rate (air cleaners)
Total loss rate: lambdatotal = lambdav + lambdad + lambdaf
The Well-Mixed Box Model
The simplest model for indoor particle dynamics assumes the room is well-mixed:
Mass Balance Equation
V * dC/dt = E + Q * Cout * P - Q * C - lambdad * V * C - CADR * C
Where:
- V = room volume, C = indoor concentration
- E = emission rate, Q = ventilation flow rate
- Cout = outdoor concentration, P = penetration factor
- CADR = clean air delivery rate from filtration
Steady-State Solution
Css = (E/V + lambdav * Cout * P) / (lambdav + lambdad + lambdaf)
Activity: Spreadsheet Modeling
Build a Particle Dynamics Model
Create a spreadsheet model to simulate particle concentrations in a classroom:
Model Parameters
- Room volume: 200 m3
- Air exchange rate: 2 h-1
- Deposition rate: 0.2 h-1 (for 1 um particles)
- Outdoor PM2.5: 10 ug/m3
- Penetration factor: 0.7
Scenarios to Model
- Calculate steady-state indoor concentration with no sources
- Add an emission source (cooking activity: E = 1000 ug/min for 30 min). Model the rise and decay of concentration.
- Add a portable air cleaner (CADR = 200 m3/h). How does this change steady-state and decay rate?
- Compare time to return to background for scenarios with and without air cleaner
Numerical Implementation
C(t + dt) = C(t) + [E/V + lambdav*Cout*P - (lambdav + lambdad + lambdaf)*C(t)] * dt
Use dt = 1 minute for good accuracy
Key Takeaway
Mathematical models provide powerful tools for predicting particle behavior in indoor environments. The lognormal distribution captures the range of particle sizes present in real aerosols, while the well-mixed box model allows calculation of concentrations under various ventilation, filtration, and emission scenarios. These models, though simplified, provide essential insights for designing healthy indoor environments and evaluating the effectiveness of air quality interventions.