Ideal source-free purge time

Estimate exponential dilution time to a user-chosen tracer concentration above supply background.

How this calculation works

This purge estimate applies only after tracer generation has stopped. Enter the well-mixed zone volume, replacement-air flow, initial tracer concentration, target concentration and concentration in the incoming air. The target must be below the initial value and above the supply background. The logarithm of the initial-to-target excess-concentration ratio gives the ideal number of volume replacements. Multiplying by volume divided by airflow gives elapsed time. Background is subtracted from both concentrations because the incoming air sets the asymptotic limit; that limit cannot be reached in finite time in this model. Constant flow and perfect mixing are assumed. Pockets, continuing releases and transfer from adjoining zones are excluded, so the result can underestimate real clearance time and does not establish safe entry or infection-control performance.

Inputs and units

Well-mixed zone volume (m³)
Replacement-air flow (L/s)
Initial tracer concentration (mg/m³)
Target tracer concentration (mg/m³)
Supply-air tracer concentration (mg/m³)

Method and formula

C(t)−Cb = (C0−Cb) exp(−Qt/V); t = (V/Q) ln[(C0−Cb)/(Ct−Cb)]. Q is m³/s, V is m³ and t is seconds.

Worked example

Example inputs

Well-mixed zone volume: 300 m³
Replacement-air flow: 500 L/s
Initial tracer concentration: 100 mg/m³
Target tracer concentration: 10 mg/m³
Supply-air tracer concentration: 0 mg/m³

Calculation steps

Replacement flow = 500 / 1000 = 0.5 m³/s. Initial excess = 100 − 0 = 100 mg/m³; target excess = 10 − 0 = 10 mg/m³.
Required ideal volume replacements = ln(100 / 10) ≈ 2.302585.
Time = (300 / 0.5) × 2.302585093 = 1381.551056 s ≈ 23.025851 min.

Example results

Ideal dilution time: 23.02585093 min
Required ideal volume replacements: 2.302585093 fraction

Example notes

Perfect mixing generally underestimates real clearance time. This does not establish safe entry or infection control.

Assumptions

Generation has completely stopped; perfectly mixed constant-volume zone and constant replacement flow.
Tracer is conserved except for removal by exhaust; supply background is constant.

Limitations

No continuing release, pockets, deposition, chemical reaction, filtering or neighboring-zone transfer.
The target is user supplied and is not judged safe or compliant.
Preliminary educational check; verify inputs and equipment data before design or operation.

Sources

CDC: Appendix B, Air: Table B.1 footnotes: exponential source-free purge equation and perfect-mixing limitation
NIST: Carbon dioxide ventilation metric: Single-zone mass balance, equations 1–3; constant generation, outdoor concentration and outdoor airflow