Turbulent Flow

Turbulent flow represents the dominant flow regime in commercial and industrial HVAC piping systems, characterized by chaotic fluid motion, mixing eddies, and enhanced heat and momentum transfer. Understanding turbulent flow behavior is essential for accurate pressure drop predictions, pump sizing, and energy-efficient system design.

Turbulent Flow Characteristics

Reynolds Number Transition

The transition from laminar to turbulent flow occurs over a range of Reynolds numbers rather than at a discrete value. For smooth pipes, the critical Reynolds number is approximately 2,300, while turbulent flow is fully established above Re = 4,000. Between these values lies the transition zone, where flow exhibits intermittent turbulent bursts.

Reynolds Number Definition:

Re = ρVD/μ = VD/ν

Where:

• Re = Reynolds number (dimensionless)
• ρ = fluid density (lbm/ft³ or kg/m³)
• V = mean velocity (ft/s or m/s)
• D = pipe inside diameter (ft or m)
• μ = dynamic viscosity (lbm/ft·s or Pa·s)
• ν = kinematic viscosity (ft²/s or m²/s)

Typical Reynolds Numbers in HVAC Systems:

All typical HVAC piping operates well into the turbulent regime, with Reynolds numbers typically exceeding 10,000.

Turbulence Structure

Turbulent flow consists of superimposed velocity fluctuations on the mean flow. Instantaneous velocity at any point:

u(t) = ū + u’(t)

Where:

• u(t) = instantaneous velocity
• ū = time-averaged velocity
• u’(t) = fluctuating component

Turbulence Intensity:

I = u’_rms / ū

Where u’_rms is the root-mean-square of velocity fluctuations. For fully developed turbulent pipe flow, turbulence intensity typically ranges from 5% to 10% on the centerline, increasing toward the wall.

Eddy Formation and Energy Cascade

Turbulent kinetic energy cascades from large-scale eddies (comparable to pipe diameter) down to the smallest dissipative scales (Kolmogorov microscale). This energy cascade enables:

• Enhanced mixing and heat transfer
• Uniform velocity distribution across the pipe cross-section
• Increased wall shear stress and pressure drop
• Self-sustaining turbulent fluctuations

Velocity Profile in Turbulent Flow

Unlike the parabolic profile in laminar flow, turbulent velocity profiles exhibit a flattened core region with steep gradients near the wall. The profile shape depends on Reynolds number and wall roughness.

Logarithmic Law of the Wall

The universal logarithmic velocity profile applies in the inner layer (y+ > 30):

u+ = (1/κ) ln(y+) + B

Where:

• u+ = u/u* = dimensionless velocity
• y+ = yu*/ν = dimensionless wall distance
• u* = √(τ_w/ρ) = friction velocity (ft/s or m/s)
• τ_w = wall shear stress (lbf/ft² or Pa)
• κ = von Kármán constant ≈ 0.41
• B = constant ≈ 5.0 for smooth walls

Near-Wall Flow Regions:

Power Law Velocity Profile

For engineering calculations, the simpler power-law profile provides adequate accuracy:

u/u_max = (y/R)^(1/n)

Where:

• u = local velocity at distance y from wall
• u_max = centerline velocity
• y = distance from pipe wall
• R = pipe radius
• n = exponent depending on Reynolds number

Power Law Exponent:

For most HVAC applications with Re > 100,000, n = 7 (the “one-seventh power law”) provides sufficient accuracy.

Velocity Profile Ratio:

The ratio of average to maximum velocity:

V/u_max = 2n²/[(n+1)(2n+1)]

For n = 7: V/u_max = 0.817

This contrasts sharply with laminar flow where V/u_max = 0.5, demonstrating the much flatter turbulent profile.

Friction Factor Correlations

The Darcy friction factor (f) relates pressure drop to flow conditions through the Darcy-Weisbach equation. In turbulent flow, f depends on Reynolds number and relative roughness (ε/D).

Darcy-Weisbach Equation

The fundamental equation for pressure drop in pipe flow:

ΔP = f × (L/D) × (ρV²/2)

Or in head loss form:

h_f = f × (L/D) × (V²/2g)

Where:

• ΔP = pressure drop (lbf/ft² or Pa)
• h_f = head loss (ft or m)
• f = Darcy friction factor (dimensionless)
• L = pipe length (ft or m)
• D = inside diameter (ft or m)
• ρ = density (lbm/ft³ or kg/m³)
• V = mean velocity (ft/s or m/s)
• g = gravitational acceleration (32.2 ft/s² or 9.81 m/s²)

Note: The Darcy friction factor is 4 times the Fanning friction factor used in some chemical engineering texts.

Colebrook-White Equation

The most accurate implicit equation for turbulent friction factor, forming the basis of the Moody diagram:

1/√f = -2.0 log₁₀[(ε/D)/3.7 + 2.51/(Re√f)]

Where:

• ε = absolute roughness (ft or m)
• ε/D = relative roughness (dimensionless)
• Re = Reynolds number

This equation requires iterative solution, making it impractical for hand calculations but ideal for computer implementation. Convergence typically occurs within 3-5 iterations using initial guess:

f₀ = 0.25/[log₁₀(ε/3.7D + 5.74/Re^0.9)]²

Moody Diagram

The Moody diagram graphically represents the Colebrook-White equation, plotting friction factor versus Reynolds number with relative roughness as a parameter. Key features:

Laminar Region (Re < 2,300): f = 64/Re (straight line independent of roughness)

Transition Region (2,300 < Re < 4,000): Uncertain behavior, design conservatively

Turbulent Smooth Pipe: f depends only on Re, following Prandtl’s equation: 1/√f = 2.0 log₁₀(Re√f) - 0.8

Turbulent Rough Pipe (fully rough zone): f becomes independent of Re at high Reynolds numbers: 1/√f = 2.0 log₁₀(3.7D/ε)

Critical Observation: For typical HVAC systems operating at high Reynolds numbers (Re > 100,000), the friction factor becomes nearly constant and dependent primarily on relative roughness.

Swamee-Jain Equation

An explicit approximation to Colebrook-White with accuracy within 1% for:

• 5,000 < Re < 10⁸
• 10⁻⁶ < ε/D < 10⁻²

f = 0.25 / [log₁₀(ε/3.7D + 5.74/Re^0.9)]²

This equation enables direct calculation without iteration, making it ideal for spreadsheet calculations and hand calculations.

Example Calculation:

Given: 6-inch Schedule 40 steel pipe, chilled water at 45°F, V = 8 ft/s

Properties:

• D = 6.065 in = 0.5054 ft
• ε = 0.00015 ft (commercial steel)
• ε/D = 0.000297
• ν = 1.41 × 10⁻⁵ ft²/s (water at 45°F)
• Re = VD/ν = (8 × 0.5054)/(1.41 × 10⁻⁵) = 287,000

Swamee-Jain: f = 0.25 / [log₁₀(0.000297/3.7 + 5.74/287,000^0.9)]² f = 0.25 / [log₁₀(8.03 × 10⁻⁵ + 1.27 × 10⁻⁵)]² f = 0.25 / [log₁₀(9.30 × 10⁻⁵)]² f = 0.25 / [-4.031]² f = 0.0154

Haaland Equation

Another explicit approximation, slightly simpler:

1/√f = -1.8 log₁₀[(ε/D/3.7)^1.11 + 6.9/Re]

Valid for similar ranges as Swamee-Jain, with comparable accuracy.

Alternative Correlations

Blasius Equation (smooth pipes, Re < 100,000): f = 0.316/Re^0.25

Nikuradse Equation (smooth pipes, general): 1/√f = 2.0 log₁₀(Re√f) - 0.8

Karman-Nikuradse (fully rough): 1/√f = 2.0 log₁₀(D/ε) + 1.14

Pipe Roughness Values

Absolute roughness critically affects friction factor in turbulent flow. Surface condition varies with pipe material, manufacturing method, age, and service fluid.

Absolute Roughness for New Pipes

Aging and Fouling Effects

Pipe roughness increases with service time due to:

• Corrosion product buildup
• Scale formation in hard water
• Biological film growth
• Particulate deposition
• Chemical attack

Aging Multipliers for Steel Pipe:

For design purposes, conservative practice uses aged roughness values, particularly for systems expected to operate 20+ years.

Smooth vs. Rough Pipe Behavior

Hydraulically Smooth Flow

Flow behaves as hydraulically smooth when roughness elements are buried within the viscous sublayer (y+ < 5). This occurs when:

ε+ = εu*/ν < 5

Or equivalently: ε < 5ν/u* = 5ν/(V√(f/8))

For hydraulically smooth flow:

• Friction factor depends only on Reynolds number
• Roughness has negligible effect
• Common in small-diameter copper tubing at low velocities

Transition Regime

Most HVAC piping operates in the transition regime where:

5 < ε+ < 70

Both viscous effects and roughness influence friction factor. The Colebrook-White equation fully accounts for this interaction.

Fully Rough Regime

At high Reynolds numbers, roughness elements protrude through the viscous sublayer:

ε+ > 70

Friction factor becomes independent of Reynolds number, dependent only on relative roughness (ε/D). Typical for:

• Large-diameter condenser water mains
• Aged carbon steel piping
• Concrete cooling water piping

Rough Pipe Friction Factor:

f = [1.14 - 2.0 log₁₀(ε/D)]⁻²

Pressure Drop Calculations

Step-by-Step Procedure

1. Determine fluid properties at operating temperature:

   • Density (ρ)
   • Kinematic viscosity (ν)

2. Calculate flow velocity: V = Q/A = 4Q/(πD²)

3. Calculate Reynolds number: Re = VD/ν

4. Determine pipe roughness:

   • Select absolute roughness (ε) for pipe material
   • Calculate relative roughness (ε/D)

5. Calculate friction factor:

   • Use Swamee-Jain or Colebrook-White equation
   • Or read from Moody diagram

6. Calculate pressure drop: ΔP = f(L/D)(ρV²/2)

Worked Example

Given:

• 400 GPM chilled water at 45°F
• 4-inch Schedule 40 commercial steel pipe
• 200 ft equivalent length (including fittings)

Solution:

Fluid properties at 45°F:

• ρ = 62.4 lbm/ft³
• ν = 1.41 × 10⁻⁵ ft²/s

Pipe geometry:

• D = 4.026 in = 0.3355 ft
• A = π(0.3355)²/4 = 0.0884 ft²

Velocity:

• Q = 400 GPM × 0.002228 ft³/s/GPM = 0.891 ft³/s
• V = 0.891/0.0884 = 10.1 ft/s

Reynolds number:

• Re = (10.1 × 0.3355)/(1.41 × 10⁻⁵) = 240,000

Roughness:

• ε = 0.00015 ft
• ε/D = 0.00015/0.3355 = 4.47 × 10⁻⁴

Friction factor (Swamee-Jain):

• f = 0.25/[log₁₀(4.47 × 10⁻⁴/3.7 + 5.74/240,000^0.9)]²
• f = 0.0185

Pressure drop:

• ΔP = 0.0185 × (200/0.3355) × (62.4 × 10.1²/2)
• ΔP = 0.0185 × 596.1 × 3,184
• ΔP = 35,100 lbf/ft² = 244 psi

Or in head: h = 244 × 2.31 = 563 ft of water

Design Considerations

Velocity Limits

Recommended maximum velocities to limit noise, erosion, and pressure drop:

Minimum velocities ensure:

• Oil entrainment in refrigerant lines (typically 7 ft/s minimum in risers)
• Acceptable heat transfer in heat exchangers
• Prevention of sediment settlement

Economic Pipe Sizing

Optimal pipe diameter balances:

• Initial cost (pipe, fittings, installation)
• Pumping energy cost over system life
• Space constraints
• Noise considerations

Life-cycle cost analysis typically yields economically optimal velocities of 6-8 ft/s for chilled water systems.

Turbulence Enhancement

Intentional turbulence enhancement improves heat transfer in:

• Heat exchanger tubes (twisted tape inserts, rifling)
• Boiler tubes (dimpled surfaces)
• Cooling tower fill

Enhanced turbulence increases friction factor but may improve overall system efficiency through reduced heat exchanger size.

ASHRAE References

• ASHRAE Fundamentals Handbook, Chapter 3: Fluid Flow

  • Section 3.5: Flow in Pipes
  • Table 1: Absolute Roughness Values
  • Figure 6: Moody Chart for Friction Factor

• ASHRAE Fundamentals Handbook, Chapter 22: Pipe Sizing

  • Velocity limitations
  • Pressure drop calculation methods
  • Aging factors for piping materials

Code Requirements

Velocity Constraints

Building codes generally do not specify maximum velocities, but professional standards recommend:

• IMC (International Mechanical Code): References ASHRAE standards for pipe sizing
• UPC (Uniform Plumbing Code): Specifies friction loss methods for water distribution

Erosion Considerations

High-velocity turbulent flow causes erosion-corrosion in:

• Copper tubes above 8 ft/s (particularly with entrained air)
• Carbon steel above 15 ft/s
• Elbows and tees (velocity concentration)

Design practice maintains velocities below erosion thresholds, particularly at fittings where local velocities exceed bulk velocity.

Summary

Turbulent flow dominates HVAC piping system hydraulics, requiring accurate friction factor determination through the Colebrook-White equation or equivalent correlations. Key factors affecting pressure drop include:

• Reynolds number (fluid velocity, density, viscosity, pipe diameter)
• Relative roughness (pipe material, age, service conditions)
• System geometry (length, fittings, elevation changes)

Modern computational tools enable precise turbulent flow calculations, but fundamental understanding of velocity profiles, friction factors, and the transition from smooth to rough pipe behavior remains essential for effective HVAC system design.

Proper accounting for turbulent flow characteristics ensures:

• Accurate pump selection and sizing
• Energy-efficient operation
• Acceptable noise levels
• Equipment longevity through erosion prevention
• Life-cycle cost optimization