Fluid Dynamics of Air
Learning Objectives
Students will be able to:
- Define the Reynolds number and explain its physical significance
- Distinguish between laminar and turbulent flow regimes
- Apply Stokes drag law to particle motion in air
- Explain the concept of relaxation time and stopping distance
- Analyze how air flow patterns affect particle transport in buildings
The Big Question
"How do we characterize air flow mathematically, and how does this determine where particles travel in a room?"
The Reynolds Number
The Reynolds number (Re) is the most important dimensionless parameter in fluid dynamics. It represents the ratio of inertial forces to viscous forces:
Re = rho v L / eta = v L / nu
| Variable | Meaning | Units |
|---|---|---|
| rho | Fluid density | kg/m3 |
| v | Flow velocity | m/s |
| L | Characteristic length | m |
| eta | Dynamic viscosity | Pa-s |
| nu | Kinematic viscosity (eta/rho) | m2/s |
Laminar vs. Turbulent Flow
Laminar Flow (Re < 2300)
- Smooth, orderly flow in layers
- Streamlines do not cross
- Predictable particle trajectories
- Parabolic velocity profile in pipes
- Found in HVAC ducts at low velocities
Turbulent Flow (Re > 4000)
- Chaotic, irregular motion
- Eddies and vortices
- Enhanced mixing
- Flat velocity profile in pipes
- Most indoor air flow is turbulent
The transition region (2300 < Re < 4000) exhibits intermittent behavior between laminar and turbulent states.
Stokes Drag Law
For a spherical particle moving through a viscous fluid at low Reynolds number:
Fdrag = 6 pi eta r v
This linear relationship between drag force and velocity only holds for Rep < 1 (particle Reynolds number).
For air at 20 C with eta = 1.81 x 10-5 Pa-s:
| Particle Size | Velocity for Rep = 1 | Stokes' Law Valid? |
|---|---|---|
| 1 um | 15 m/s | Almost always |
| 10 um | 1.5 m/s | Usually |
| 100 um | 0.15 m/s | Often not (settling too fast) |
Particle Relaxation Time
The relaxation time (tau) is the characteristic time for a particle to respond to changes in fluid velocity:
tau = rhop d2 / (18 eta)
Stopping Distance
s = v0 * tau
Distance a particle travels before reaching fluid velocity (starting from v0)
Typical Values
- 0.1 um: tau ~ 10-7 s
- 1 um: tau ~ 10-5 s
- 10 um: tau ~ 10-3 s
Air Flow in Buildings
Factors Affecting Indoor Air Flow
- HVAC systems: Supply and return air create bulk flow patterns
- Thermal plumes: Warm objects (humans, equipment) generate buoyant flows
- Window/door flows: Natural ventilation and infiltration
- Human movement: Walking creates wakes and recirculation
- Furniture and obstructions: Create zones of stagnation and recirculation
Typical Indoor Air Velocities
| Location | Velocity (m/s) | Re (for L = 1 m) |
|---|---|---|
| Still room | 0.05-0.1 | 3,000-6,000 |
| Near HVAC diffuser | 0.5-2.0 | 30,000-130,000 |
| Human thermal plume | 0.2-0.5 | 13,000-33,000 |
| Open window | 0.5-5.0 | 33,000-330,000 |
Activity: Fluid Dynamics Problems
Problem Set
-
Reynolds number: Calculate Re for air flow in a circular duct (diameter 30 cm) at 5 m/s. Is the flow laminar or turbulent?
- Use: rhoair = 1.2 kg/m3, eta = 1.81 x 10-5 Pa-s
- Particle Reynolds number: Calculate Rep for a 20 um particle settling at its terminal velocity (vs ~ 1 cm/s). Is Stokes' law valid?
- Relaxation time: Calculate tau and stopping distance for a 5 um particle (density 2 g/cm3) initially moving at 1 m/s.
- Room analysis: In a classroom with ceiling-mounted HVAC, sketch expected air flow patterns. Where would you expect particles to accumulate?
Key Takeaway
Fluid dynamics provides the framework for understanding how particles move in air. The Reynolds number determines whether flow is laminar or turbulent, while Stokes drag law governs particle-air interactions at low Re. Small particles rapidly equilibrate with air flow (short relaxation times) while larger particles maintain their momentum longer. These principles are essential for predicting particle transport, designing ventilation systems, and understanding exposure patterns in indoor environments.