5
Evaluate
Mathematical Modeling Project
Duration
50 minutes
Type
Evaluate
Standards
HS-ETS1-4, HSF-LE.A.1
Project Overview
"Build a spreadsheet model to simulate particle dynamics in a real space and use it to optimize air quality interventions."
Project Requirements
Part 1: Model Development
- Choose a real space (your classroom, bedroom, or other)
- Measure or estimate room dimensions and calculate volume
- Research typical emission rates for a source in that space
- Implement the mass balance equation in a spreadsheet
- Use Euler's method with dt = 1 minute time steps
Part 2: Scenario Analysis
Model at least three scenarios:
- Baseline: Current ventilation, no air cleaning
- Improved ventilation: Double the ACH
- Air cleaner: Add portable HEPA with realistic CADR
Part 3: Optimization and Recommendations
Determine the most cost-effective approach to reduce average exposure by 50%
Spreadsheet Template Structure
Columns Required
| Column | Content | Formula Example |
|---|---|---|
| A | Time (minutes) | =A1+1 |
| B | Emission rate (ug/min) | Input or formula for time-varying |
| C | Concentration (ug/m3) | =C1+dC/dt*dt |
| D | dC/dt | =(E/V+lambda_v*C_out-lambda_tot*C) |
Deliverables
Technical Report
- Space description and parameters
- Model equations and assumptions
- Results graphs for each scenario
- Cost-effectiveness analysis
- Recommendations with justification
Spreadsheet Model
- Clearly labeled parameters section
- Time-series calculations
- Graphs showing concentration vs time
- Summary statistics (average, max, time above threshold)
Assessment Rubric
| Criterion | Excellent (4) | Proficient (3) | Developing (2) | Beginning (1) |
|---|---|---|---|---|
| Model Implementation | Correct equations, well-organized spreadsheet | Correct equations, minor organization issues | Some errors in equations | Major errors, model not functional |
| Scenario Analysis | Comprehensive comparison, insightful analysis | Good comparison, adequate analysis | Limited scenarios or analysis | Incomplete scenarios |
| Recommendations | Evidence-based, cost-effective, practical | Good recommendations with evidence | Recommendations lack full justification | No clear recommendations |
| Communication | Clear writing, professional graphs | Clear with minor issues | Unclear in places | Difficult to follow |
Unit Summary
This unit has developed the mathematical foundations for predicting and controlling indoor particle concentrations. From emission rates and mass balance to exponential decay and steady-state, these equations provide the quantitative tools for engineering better indoor air quality. The modeling project demonstrates how these principles apply to real spaces and real decisions about ventilation and filtration investments.