Fluid Mechanics for HVAC Engineers

Fluid Mechanics for HVAC Engineers

Fluid flow governs air and water distribution in HVAC systems. Accurate prediction of pressure drops, pump/fan power requirements, and flow velocities requires application of fluid mechanics principles. This guide provides the engineering equations for incompressible flow in ducts and pipes essential for sizing distribution systems.

Fundamental Fluid Properties

Density

Mass per unit volume determines momentum and energy content of flowing fluids.

Air (at sea level, 70°F): $$\rho_{air} = 0.075 \text{ lb/ft}^3 = 1.20 \text{ kg/m}^3$$

Water (at 60°F): $$\rho_{water} = 62.4 \text{ lb/ft}^3 = 1,000 \text{ kg/m}^3$$

Viscosity

Dynamic viscosity ($\mu$) quantifies internal fluid friction resistance:

Air (at 70°F): $$\mu_{air} = 1.22 \times 10^{-5} \text{ lb/(ft·s)} = 1.81 \times 10^{-5} \text{ Pa·s}$$

Water (at 60°F): $$\mu_{water} = 7.54 \times 10^{-4} \text{ lb/(ft·s)} = 1.12 \times 10^{-3} \text{ Pa·s}$$

Kinematic viscosity ($\nu$):

$$\nu = \frac{\mu}{\rho}$$

Units: ft²/s or m²/s

Bernoulli’s Equation

For steady, incompressible, frictionless flow along a streamline, total energy remains constant:

$$P_1 + \frac{1}{2}\rho v_1^2 + \rho g z_1 = P_2 + \frac{1}{2}\rho v_2^2 + \rho g z_2$$

Where:

• $P$ = static pressure (lb/ft² or Pa)
• $\rho$ = fluid density (lb/ft³ or kg/m³)
• $v$ = fluid velocity (ft/s or m/s)
• $g$ = gravitational acceleration = 32.17 ft/s² = 9.81 m/s²
• $z$ = elevation (ft or m)

In HVAC practice, expressing in pressure units (inches of water gauge, “w.g.):

$$P_{total} = P_{static} + P_{velocity}$$

Where velocity pressure:

$$P_{velocity} = \frac{1}{2}\rho v^2$$

For air at standard conditions (0.075 lb/ft³):

$$P_v = \left(\frac{v}{4005}\right)^2 \text{ in. w.g.}$$

Where $v$ is in ft/min (FPM).

graph LR
    A[Total Pressure<br/>P_total] --> B[Static Pressure<br/>P_static]
    A --> C[Velocity Pressure<br/>P_velocity = ½ρv²]

    style A fill:#667eea
    style B fill:#f8f9fa
    style C fill:#f8f9fa

Reynolds Number and Flow Regimes

Reynolds number determines whether flow is laminar or turbulent:

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

Where:

• $D$ = characteristic dimension (pipe diameter or duct hydraulic diameter, ft or m)
• $v$ = average velocity (ft/s or m/s)

Flow regime classification:

• $Re < 2,300$: Laminar flow (rare in HVAC except small pipes at low flow)
• $2,300 < Re < 4,000$: Transitional flow
• $Re > 4,000$: Turbulent flow (typical for HVAC systems)

Hydraulic diameter for rectangular ducts:

$$D_h = \frac{4A}{P} = \frac{4ab}{2(a+b)} = \frac{2ab}{a+b}$$

Where $a$ and $b$ are duct dimensions (ft or m).

Pressure Drop in Ducts and Pipes

Darcy-Weisbach Equation

Universal equation for pressure drop in straight sections:

$$\Delta P = f \frac{L}{D} \frac{\rho v^2}{2}$$

Where:

• $\Delta P$ = pressure drop (lb/ft² or Pa)
• $f$ = Darcy friction factor (dimensionless)
• $L$ = duct/pipe length (ft or m)
• $D$ = diameter or hydraulic diameter (ft or m)

For ductwork in “w.g. per 100 ft:

$$\Delta P = f \frac{L}{D} \frac{v^2}{(4005)^2 \times 12.96}$$

Where $v$ is in FPM.

Friction Factor Correlations

For turbulent flow in smooth ducts (sheet metal):

Colebrook equation (implicit):

$$\frac{1}{\sqrt{f}} = -2.0 \log_{10}\left(\frac{\epsilon/D}{3.7} + \frac{2.51}{Re\sqrt{f}}\right)$$

Where $\epsilon$ = absolute roughness (ft or m).

Swamee-Jain equation (explicit approximation):

$$f = \frac{0.25}{\left[\log_{10}\left(\frac{\epsilon/D}{3.7} + \frac{5.74}{Re^{0.9}}\right)\right]^2}$$

For smooth ducts ($\epsilon \approx 0$):

$$f \approx \frac{0.25}{[\log_{10}(Re/7)]^2}$$

Typical absolute roughness values:

Fitting Losses

Fittings (elbows, tees, dampers) cause local pressure drops:

$$\Delta P_{fitting} = C \frac{\rho v^2}{2}$$

Where $C$ = loss coefficient (dimensionless).

Equivalent length method expresses fitting losses as equivalent straight duct length:

$$L_{eq} = C \frac{D}{f}$$

Typical loss coefficients:

graph TD
    A[Duct System<br/>Pressure Drop] --> B[Straight Section Losses<br/>Darcy-Weisbach]
    A --> C[Fitting Losses<br/>Loss Coefficients]

    B --> D[f · L/D · ρv²/2]
    C --> E[Σ C · ρv²/2]

    F[Total Pressure Drop<br/>ΔP_total = ΔP_straight + ΔP_fittings]

    D --> F
    E --> F

    style A fill:#667eea,color:#fff
    style F fill:#764ba2,color:#fff

Pump and Fan Power

Mechanical power required to move fluid:

$$\dot{W} = \frac{Q \Delta P}{\eta}$$

Where:

• $\dot{W}$ = power (hp or kW)
• $Q$ = volumetric flow rate (CFM or m³/s)
• $\Delta P$ = total pressure rise (in. w.g. or Pa)
• $\eta$ = overall efficiency (fan or pump, dimensionless)

For air (in IP units):

$$hp = \frac{CFM \times \Delta P_{in.w.g.}}{6,356 \times \eta}$$

For water (in IP units):

$$hp = \frac{GPM \times \Delta P_{ft}}{3,960 \times \eta}$$

Where $\Delta P_{ft}$ is head in feet of water.

Typical efficiencies:

• Fans: 50-75% (total efficiency including motor and drive losses)
• Pumps: 60-85% (total efficiency including motor)

Pipe Sizing for Water Systems

Hazen-Williams Equation

Empirical equation widely used for water flow in pipes:

$$v = 1.318 C R^{0.63} S^{0.54}$$

Where:

• $v$ = velocity (ft/s)
• $C$ = Hazen-Williams coefficient (dimensionless)
• $R$ = hydraulic radius = $D/4$ for circular pipes (ft)
• $S$ = friction slope = $\Delta P / (\rho g L)$ (dimensionless)

For circular pipes:

$$\Delta P = \frac{4.52 L Q^{1.85}}{C^{1.85} D^{4.87}}$$

Where:

• $\Delta P$ = pressure drop (ft of water)
• $L$ = pipe length (ft)
• $Q$ = flow rate (GPM)
• $D$ = inside diameter (inches)

Hazen-Williams C-factors:

Velocity Limits

Design velocities for noise and erosion control:

Practical Applications

Duct System Design

1. Equal friction method: Size ducts for constant pressure drop per unit length (typically 0.08-0.15 “w.g./100 ft)
2. Static regain method: Size ducts to maintain constant static pressure at each branch takeoff
3. Velocity method: Size ducts for maximum allowable velocities (main ducts 1,200-2,000 FPM, branches 800-1,200 FPM)

Pipe System Design

1. Velocity-based sizing: Select pipe size to maintain velocities within limits (4-10 ft/s)
2. Pressure drop limit: Size pipes to limit pressure drop (typically 4 ft/100 ft or 1.73 psi/100 ft)
3. Economic optimization: Balance first cost (larger pipes) vs. operating cost (pump power)

Pump and Fan Selection

1. System curve: Plot pressure drop vs. flow rate using Darcy-Weisbach equation
2. Operating point: Intersection of system curve and fan/pump performance curve
3. Safety factors: Add 10-15% to calculated pressure drop for uncertainties and aging

Common Design Pitfalls

• Ignoring fitting losses: Fittings often contribute 30-50% of total pressure drop
• Using water velocities > 10 ft/s: Causes erosion, noise, and water hammer risk
• Undersizing ducts for noise: Velocities > 2,000 FPM generate objectionable noise
• Neglecting elevation changes: Static pressure changes 0.43 psi/ft for water, 0.036 “w.g./ft for air
• Using smooth pipe friction factors for aged systems: Corrosion and fouling increase friction significantly

Summary

Fluid mechanics principles govern HVAC distribution system design:

• Bernoulli’s equation relates pressure, velocity, and elevation energy
• Reynolds number determines flow regime (laminar vs. turbulent)
• Darcy-Weisbach equation calculates friction losses in straight sections
• Fitting loss coefficients quantify pressure drops in elbows, tees, and transitions
• Hazen-Williams equation provides empirical method for water pipe sizing
• Pump/fan power is proportional to flow rate and pressure rise, inversely to efficiency

Accurate pressure drop calculations enable proper fan and pump selection, duct and pipe sizing, and energy-efficient system design.

Related Technical Guides:

• Duct Design Fundamentals
• Hydronic System Fundamentals
• Pump Selection & Performance
• Fan Selection & Performance

References:

• ASHRAE Handbook of Fundamentals, Chapter 3: Fluid Flow
• ASHRAE Duct Fitting Database (DFDB)
• SMACNA HVAC Systems Duct Design, 4th Edition
• Crane Technical Paper No. 410: Flow of Fluids Through Valves, Fittings, and Pipe