Math of Particulate Matter
Students apply differential equations, exponential functions, and steady-state analysis to mathematically model particulate matter dynamics in indoor environments, including emission rates, decay, ventilation, and air cleaner performance.
5
Lessons
5
Class Periods
Low
Materials Cost
3
NGSS Standards
Essential Question
How can we use mathematical models to predict and control indoor particle concentrations, and what equations govern the balance between emission, removal, and ventilation?
Lessons
-
1→Emission Rates and Concentrations
-
2→Exponential Decay Functions
-
3→Steady-State Calculations
-
4→CADR and Air Changes
-
5→Mathematical Modeling Project
Key Concepts
Mass Balance
- dC/dt = E/V + lambda_v*C_out - lambda_total*C
- Sources, sinks, and transport
- Well-mixed assumption
- Transient vs steady-state
Exponential Functions
- C(t) = C_0 * exp(-lambda*t)
- t_1/2 = ln(2)/lambda
- tau = 1/lambda (time constant)
- Integration and differentiation
Air Exchange
- ACH = Q/V (air changes per hour)
- Ventilation loss rate
- Penetration factors
- Recirculation effects
Air Cleaning
- CADR = Q_filter * efficiency
- Equivalent ACH = CADR/V
- Filter sizing calculations
- Cost-effectiveness analysis
Standards Alignment
| Standard | Description |
|---|---|
| HS-PS3-1 | Create a computational model to calculate the change in energy of one component in a system |
| HS-ETS1-4 | Use a computer simulation to model the impact of proposed solutions |
| HSF-LE.A.1 | Distinguish between situations that can be modeled with linear functions and exponential functions |