Aerosol Differential Decay Equations: The Mathematical Foundations of AHAM AC-1 CADR

Behind every Clean Air Delivery Rate (CADR) specification label lies rigorous differential calculus. In a sealed aerosol chamber, particle concentration behaves as a first-order rate-governed system dictated by exponential decay differential equations.

1. Mathematical Derivation of Particle Mass Balance

In a chamber of volume V (ft³), the rate of change of airborne particulate count C(t) is governed by the mass balance differential equation:

dC(t) / dt = - (k_n + k_e) × C(t)

Where k_n is the natural particle loss rate constant (due to gravitational settling, Brownian wall deposition, and thermophoretic losses) in units of min^-1, and k_e is the air cleaner removal rate constant:

k_e = CADR / V

Integrating this linear differential equation from time t = 0 to t yields the classic exponential decay solution:

C(t) = C_0 × e^(- (k_n + k_e) × t)

2. Linear Regression on Logarithmic Concentration Slopes

Taking the natural logarithm of both sides linearizes the decay data logged by optical particle spectrometers:

ln[C(t)] = ln[C_0] - k_total × t

Using ordinary least squares (OLS) linear regression over 1-minute intervals across the 20-minute test period, the regression slope yields k_total = k_n + k_e. The Clean Air Delivery Rate is calculated with high precision by subtracting the baseline natural slope:

CADR = V × (k_total - k_natural)

Under AHAM AC-1 protocol, the coefficient of determination (R²) for the linear regression must exceed 0.95; otherwise, the test run is invalidated and must be completely repeated.