Steady-State Calculations
Learning Objectives
Students will be able to:
- Define steady-state and explain when it applies
- Derive steady-state concentration from mass balance equations
- Calculate equilibrium concentrations for various scenarios
- Analyze how changes in parameters affect steady-state
- Determine time to reach steady-state conditions
The Big Question
"When sources emit continuously, what concentration will the room eventually reach? How do we calculate this equilibrium?"
Deriving Steady-State Concentration
At steady state, concentration is constant, so dC/dt = 0:
Starting with: V dC/dt = E + Q*Cout*P - (Q + kd*V + CADR)*C
Setting dC/dt = 0 and solving for Css:
Css = (E/V + lambdav*Cout*P) / (lambdav + lambdad + lambdaf)
Special Cases
Indoor Source Only
Css = E / (V * lambdatotal)
When Cout = 0 or negligible
Outdoor Source Only
Css = Cout * P * lambdav / lambdatotal
When E = 0 (no indoor sources)
Time to Reach Steady State
The approach to steady state follows:
C(t) = Css - (Css - C0) * e-lambda*t
- After 1 time constant (tau = 1/lambda): 63% of the way to Css
- After 3 tau: 95% of the way
- After 5 tau: 99% of the way (practical equilibrium)
Activity: Steady-State Analysis
Problem Set
A 100 m3 classroom has: E = 200 ug/min (continuous source), ACH = 3 h-1, deposition rate = 0.5 h-1, outdoor PM2.5 = 10 ug/m3, penetration = 0.7
- Calculate lambdatotal and the steady-state concentration
- How long until the room reaches 95% of steady state?
- If you add a HEPA filter with CADR = 300 m3/h, what is the new steady-state?
- Calculate the percent reduction in steady-state concentration from adding the filter
Key Takeaway
Steady-state analysis allows us to predict the long-term equilibrium concentration that results from the balance between source emissions and removal processes. This is the concentration that occupants experience during prolonged activities. Understanding how to calculate and manipulate steady-state is crucial for designing ventilation and filtration systems that maintain acceptable air quality.