2.3 Fluid Mechanics and Airflow Fundamentals

Fluid Mechanics and Airflow Fundamentals

Local exhaust ventilation (LEV), dilution ventilation, air sampling manifolds, and particle transport lines operate according to the fundamental laws of fluid mechanics. Industrial hygienists must evaluate airflow quantities, duct velocities, static and dynamic pressure distributions, and energy losses to design, test, and troubleshoot engineering control systems. This section covers the continuity equation, Bernoulli's energy conservation, duct pressure relationships, the velocity pressure equation, Reynolds numbers, and friction loss calculations.

---

1. The Continuity Equation and Conservation of Mass

In fluid dynamics, the Law of Conservation of Mass states that mass can neither be created nor destroyed in a steady-state fluid system. For flow through a closed conduit or ventilation duct:

$\dot{m} = \rho_1 A_1 V_1 = \rho_2 A_2 V_2 = \text{constant}$

Where:

• m = mass flow rate (lb/min or kg/s)
• ρ = fluid density (lb/ft³ or kg/m³)
• A = cross-sectional area of duct (ft² or m²)
• V = average fluid velocity (ft/min or m/s)

Incompressibility Assumption in Industrial Ventilation

Under standard industrial ventilation conditions, pressure differentials rarely exceed ± 20 inches of water gauge (≈ 0.72 psi or 5 kPa), which represents less than a 5% variation from atmospheric pressure (14.7 psia). Air velocities in LEV systems are almost always well below 10,000 fpm (Mach number < 0.15).

Consequently, air can be treated as an incompressible fluid of constant density (ρ1 = ρ2), simplifying the continuity equation to volumetric flow rate (Q):

$\mathbf{Q = V \times A} \quad \Longleftrightarrow \quad \mathbf{Q = V_1 A_1 = V_2 A_2}$

Where:

• Q = Volumetric airflow rate in Cubic Feet per Minute (CFM) or m³/s
• V = Average air velocity in Feet per Minute (FPM) or m/s
• A = Cross-sectional duct area in Square Feet (extft²) or m²

Duct Geometry Formulas

---

2. Bernoulli's Principle & Total, Static, and Velocity Pressure

Bernoulli's Theorem expresses the conservation of mechanical energy along a streamline in steady, frictionless fluid flow:

$\frac{P_1}{\rho g} + \frac{V_1^2}{2g} + z_1 = \frac{P_2}{\rho g} + \frac{V_2^2}{2g} + z_2 + h_{\text{loss}}$

Because the density of air is low, elevation head changes (z1 - z2) across industrial duct runs are negligible and set to zero. Energy in a ventilation duct is therefore expressed in terms of fluid pressures, measured in inches of water gauge (in. w.g.) or Pascals (Pa).

The Fundamental Duct Pressure Relationship

At any point in a duct system, the Total Pressure (TP) is the algebraic sum of Static Pressure (SP) and Velocity Pressure (VP):

$\mathbf{TP = SP + VP}$

   +-------------------------------------------------------------+
   |                VENTILATION PRESSURE COMPONENTS              |
   +-------------------------------------------------------------+
   |                                                             |
   |  STATIC PRESSURE (SP)       +    VELOCITY PRESSURE (VP)     |
   |  - Potential energy              - Kinetic energy of motion |
   |  - Exerted in all directions     - Exerted only in flow dir |
   |  - Can be POSITIVE or NEGATIVE   - ALWAYS POSITIVE (> 0)    |
   |  - Collapses/bursts duct walls   - Directly yields velocity |
   |                                                             |
   |                              =                              |
   |                                                             |
   |                     TOTAL PRESSURE (TP)                     |
   |                     - Total energy in duct                  |
   |                     - Decreases downstream (losses)         |
   +-------------------------------------------------------------+

Detailed Characterization of Pressure Components

1. Static Pressure (SP):

   • Represents the potential energy of the air.
   • Acts perpendicularly and equally in all directions against the interior duct walls.
   • Sign Convention: On the suction side of the fan (upstream/inlet), SP is negative (below atmospheric pressure), acting to collapse the duct. On the discharge side of the fan (downstream/outlet), SP is positive (above atmospheric pressure), acting to expand the duct.
   • Measured using a static pressure tap flush with the duct wall.

2. Velocity Pressure (VP):

   • Represents the kinetic energy of the moving airstream.
   • Acts exclusively in the direction of airflow.
   • Sign Convention: Velocity pressure is always positive (VP > 0) in moving air, and zero in stationary air.
   • Measured using the differential between the impact tip and static side ports of a Standard Pitot-Static tube.

3. Total Pressure (TP):

   • Measures the total mechanical energy present in the fluid stream.
   • Because friction and turbulence convert mechanical energy into irreversible thermal energy, TP always decreases in the direction of airflow through passive ductwork, fittings, and hoods. Only the fan adds energy to create a sudden rise in TP.

---

3. Derivation of the Velocity Pressure Equation (V = 4005√(VP))

A cornerstone formula on the CIH exam is the equation converting measured velocity pressure (VP, in inches of water) to average linear air velocity (V, in feet per minute).

Step-by-Step Mathematical Derivation

From basic physics, kinetic energy head is h = V²/2g, or V = √(2gh).

1. Express fluid velocity in feet per second (v): $v = \sqrt{2 g h_{\text{air}}}$ Where g = 32.174 ft/s² and hair is fluid column height in feet of air.

2. Convert column height from inches of water (VP) to feet of air (hair):

   • Standard density of liquid water at 70°F: ρwater = 62.37 lb/ft³ ≈ 62.4 lb/ft³
   • Standard density of dry air at 70°F (21.1°C) and 29.92 in. Hg (760 mmHg): ρair = 0.07495 lb/ft³ ≈ 0.075 lb/ft³

   $h_{\text{air}} = \left(\frac{VP\text{ in. w.g.}}{12\text{ in/ft}}\right) \times \left(\frac{\rho_{\text{water}}}{\rho_{\text{air}}}\right) = \left(\frac{VP}{12}\right) \times \left(\frac{62.37}{0.07495}\right) = 69.345 \times VP$

3. Substitute into the velocity equation and convert velocity from ft/sec to ft/min (V = 60 × v): $v(\text{ft/s}) = \sqrt{2 \times 32.174 \times (69.345 \times VP)} = \sqrt{4462.6 \times VP} = 66.803 \times \sqrt{VP}$ $V(\text{ft/min}) = 60 \times v(\text{ft/s}) = 60 \times 66.803 \times \sqrt{VP} = \mathbf{4008.2 \times \sqrt{VP} \approx 4005\sqrt{VP}}$

Working Equations for Standard and Non-Standard Conditions

$\mathbf{V = 4005 \sqrt{VP}} \quad \Longleftrightarrow \quad \mathbf{VP = \left(\frac{V}{4005}\right)^2} \quad (\text{Standard Air: } 70^\circ\text{F}, 29.92\text{ in. Hg})$

For non-standard temperatures or elevations, density correction factor (d = ρactual / 0.075) is applied:

$V = 4005 \sqrt{\frac{VP}{d}} \quad \text{where } d = \left(\frac{530}{T(^\circ\text{F}) + 460}\right) \times \left(\frac{P_{\text{baro}}(\text{in. Hg})}{29.92}\right)$

---

4. Reynolds Number & Flow Regimes in Ducts and Sampling Lines

The Reynolds Number (Re) is a dimensionless parameter representing the ratio of inertial forces to viscous forces in fluid flow:

$Re = \frac{\rho V D}{\mu} = \frac{V D}{\nu}$

Where:

• ρ = fluid density (kg/m³ or lb/ft³)
• V = average velocity (m/s or ft/s)
• D = internal pipe/duct diameter (m or ft)
• µ = absolute dynamic viscosity (Pa·s or lb/(ft·s))
• ν = µ / ρ = kinematic viscosity (m²/s or ft²/s)

Flow Regimes in Circular Pipes and Ventilation Ducts

Key Principle in Particulate Transport: Industrial exhaust systems capturing dusts and fumes are intentionally designed for fully turbulent flow (Re >> 10,000, with transport velocities typically between 3,500 fpm and 4,500 fpm) to keep particles suspended and prevent duct clogging.

---

5. Duct Friction Losses & The Darcy-Weisbach Equation

As air moves through a duct, friction against the duct walls and internal turbulent eddies dissipate mechanical energy as heat. The fundamental equation for friction head loss (hf) is the Darcy-Weisbach Equation:

$h_f = f \left(\frac{L}{D}\right) \left(\frac{V^2}{2g}\right) = f \left(\frac{L}{D}\right) VP$

Where:

• hf = friction head loss in duct (in. w.g.)
• f = dimensionless Darcy friction factor (determined by pipe roughness and Re on the Moody Diagram)
• L = duct length (feet)
• D = duct diameter (feet)
• VP = velocity pressure (in. w.g.)

Duct Loss Coefficient Method (K)

For hood entries, elbows, branch entries, expansions, and contractions, pressure losses are expressed as a fraction of the velocity pressure:

$h_{\text{loss}} = K \times VP$

Where K is the specific fitting loss coefficient (e.g., K = 0.50 for a standard 90° flanged hood entry, K = 0.25 for a smooth rounded 5-piece duct elbow).

---

6. Worked Step-by-Step Calculation Examples

Worked Example 1.6: Continuity Equation and Duct Sizing for Particulate Transport

Problem: A local exhaust ventilation hood captures metal grinding dust from a buffing wheel at a required volumetric flow rate of Q = 1,400 CFM. To prevent heavy metallic dust particles from settling out and clogging the ductwork, the ACGIH Industrial Ventilation Manual recommends a minimum duct transport velocity of V = 4,000 fpm.

1. Calculate the required duct cross-sectional area in square feet.
2. Determine the exact duct diameter in inches.
3. If standard round spiral duct is available in whole-inch increments (e.g., 7-inch, 8-inch, 9-inch), select the correct size and calculate the resulting actual duct velocity.

Solution Steps:

1. Calculate required area using Q = V × A: $A = \frac{Q}{V} = \frac{1,400\text{ CFM}}{4,000\text{ fpm}} = 0.350\text{ ft}^2$

2. Calculate exact duct diameter (d in inches): $A = \frac{\pi d^2}{576} \implies d = \sqrt{\frac{576 \times A}{\pi}} = \sqrt{\frac{576 \times 0.350}{3.14159}} = \sqrt{64.17} = 8.01\text{ inches}$

3. Select standard duct size and verify transport velocity:

   • If a 9-inch duct were selected (A = 0.4418 ft²), actual velocity would be V = 1400 / 0.4418 = 3,169 fpm, which is below the 4,000 fpm minimum and would allow duct settling.
   • Therefore, select an 8-inch diameter duct: $A_{\text{actual}} = \frac{\pi (8)^2}{576} = \frac{201.06}{576} = 0.3491\text{ ft}^2$ $V_{\text{actual}} = \frac{Q}{A_{\text{actual}}} = \frac{1,400\text{ CFM}}{0.3491\text{ ft}^2} = 4,010\text{ fpm}$

Result: Select an 8-inch duct, providing an actual transport velocity of 4,010 fpm, safely satisfying transport requirements.

---

Worked Example 1.7: Pitot-Static Tube Velocity and Airflow Rate Calculation

Problem: A CIH performs a multi-point Pitot-tube traverse on a 12-inch diameter circular exhaust duct at standard air density. The average of the square roots of the velocity pressures across the traverse points is √(VP)-bar = 0.750 (in. w.g.)(1/2).

1. Calculate the average air velocity (V) in feet per minute.
2. Calculate the total volumetric flow rate (Q) in CFM.

Solution Steps:

1. Calculate average velocity using V = 4005 × √(VP)-bar:

   Measurement Rule: Always average the square roots of the velocity pressures, rather than averaging velocity pressures and taking the square root afterward (Vavg = 4005 × avg(√(VP)) ≠ 4005 × √(avg(VP))). $V = 4005 \times 0.750 = 3,003.75\text{ fpm} \approx 3,004\text{ fpm}$

2. Calculate duct area (A) for a 12-inch diameter duct: $A = \frac{\pi (12\text{ in})^2}{576} = \frac{3.14159 \times 144}{576} = 0.7854\text{ ft}^2$

3. Calculate volumetric flow rate (Q = V × A): $Q = 3,003.75\text{ fpm} \times 0.7854\text{ ft}^2 = 2,359.1\text{ CFM}$

Result: The duct air velocity is 3,004 fpm and the total airflow is 2,359 CFM.

---

Worked Example 1.8: Total, Static, and Velocity Pressure Balance Across a Hood

Problem: In an LEV branch duct located upstream of the exhaust fan, a technician measures a static pressure of SP = -2.75 in. w.g. and a velocity pressure of VP = 0.64 in. w.g..

1. Calculate the Total Pressure (TP) at this measurement plane.
2. Explain the physical significance of the sign of each pressure value.

Solution Steps:

1. Apply the total pressure equation (TP = SP + VP): $TP = SP + VP = -2.75\text{ in. w.g.} + 0.64\text{ in. w.g.} = -2.11\text{ in. w.g.}$

2. Physical Analysis:

   • SP = -2.75 in. w.g.: Negative potential energy because the duct is under suction from the fan inlet, drawing air into the hood against friction and entry losses.
   • VP = +0.64 in. w.g.: Positive kinetic energy corresponding to a duct velocity of V = 4005√(0.64) = 4005 × 0.80 = 3,204 fpm.
   • TP = -2.11 in. w.g.: Negative total energy relative to ambient room pressure (0.0 in. w.g.), reflecting the energy loss required to accelerate air into the hood and overcome upstream friction.