Flow in Pipes: Fundamentals & Losses

Laminar and turbulent flow, Reynolds number, friction factor, and calculation of head loss in pipes.

Overview

Pipe flow refers to the flow of a liquid in a closed conduit that completely fills the cross-section. The driving force is typically a pressure difference or gravity.

Flow Regimes

Reynolds Number

The nature of flow (laminar or turbulent) is determined by the dimensionless Reynolds Number (ReRe).

Reynolds Number

Laminar and turbulent flow, Reynolds number, friction factor, and calculation of head loss in pipes.

Re=ρVDμ=VDνRe = \frac{\rho V D}{\mu} = \frac{V D}{\nu}
  • Laminar Flow (Re<2000Re < 2000): Viscous forces dominate. Fluid moves in smooth, parallel layers. The velocity profile is parabolic.
  • Transitional Flow (2000<Re<40002000 < Re < 4000): Unstable flow, alternating between laminar and turbulent.
  • Turbulent Flow (Re>4000Re > 4000): Inertial forces dominate. Chaotic mixing and eddies. The velocity profile is flatter.
Reynolds Number Simulation: Experiment with fluid velocity, pipe diameter, and viscosity to see the flow regime change.

Reynolds Number Visualizer

Calculated Reynolds Number (Re)
50,000
Flow Regime: Turbulent

Particle visualization of flow lines. Note: visual speed is scaled.

Friction Losses (Major Losses)

Friction Loss

As fluid flows, energy is lost due to friction between fluid layers and against the pipe wall. This energy loss is expressed as head loss (hfh_f).

Darcy-Weisbach Equation

The most accurate and universally applicable formula for head loss.

Darcy-Weisbach Equation

<strong>Reynolds Number Simulation:</strong> Experiment with fluid velocity, pipe diameter, and viscosity to see the flow regime change.

hf=fLDV22gh_f = f \frac{L}{D} \frac{V^2}{2g}

Variables

SymbolDescriptionUnit
ffDarcy friction factordimensionless
LLPipe length-
DDPipe diameter-
VVAverage velocity-

Determining the Friction Factor (ff)

The value of ff depends on the flow regime and the pipe roughness (ϵ\epsilon).

Procedure

  1. Laminar Flow: ff depends only on ReRe.

    Determining the Friction Factor ($f$)

    The value of $f$ depends on the flow regime and the pipe roughness ($\epsilon$).

    f=64Ref = \frac{64}{Re}
  2. Turbulent Flow: ff depends on ReRe and Relative Roughness (ϵ/D\epsilon/D).
    • Colebrook-White Equation (Implicit):
      1f=−2log⁡(ϵ/D3.7+2.51Ref)\frac{1}{\sqrt{f}} = -2 \log \left( \frac{\epsilon/D}{3.7} + \frac{2.51}{Re \sqrt{f}} \right)
    • Haaland Equation (Explicit Approximation):
      1f≈−1.8log⁡[(ϵ/D3.7)1.11+6.9Re]\frac{1}{\sqrt{f}} \approx -1.8 \log \left[ \left( \frac{\epsilon/D}{3.7} \right)^{1.11} + \frac{6.9}{Re} \right]
    • Moody Chart: A graphical plot of ff vs ReRe for various ϵ/D\epsilon/D.

Moody Chart Calculator (Friction Factor)

10³10⁸
SmoothRough

Flow Regime

Turbulent

Friction Factor (f) = 0.02000

Calculated using Haaland approximation:

1f≈−1.8log⁡10[(ϵ/D3.7)1.11+6.9Re]\frac{1}{\sqrt{f}} \approx -1.8 \log_{10} \left[ \left( \frac{\epsilon/D}{3.7} \right)^{1.11} + \frac{6.9}{Re} \right]

Empirical Pipe Flow Formulas

Commonly used explicit formulas for water distribution systems.

Hazen-Williams Equation

Widely used in waterworks design because it is explicit and does not require calculating the friction factor (ff) iteratively. It is valid only for water at ordinary temperatures (5°C to 25°C) and for pipe diameters larger than 50 mm.

Hazen-Williams Equation

Commonly used explicit formulas for water distribution systems.

V=0.849ChwR0.63S0.54(SI Units)V = 0.849 C_{hw} R^{0.63} S^{0.54} \quad (\text{SI Units})

Variables

SymbolDescriptionUnit
VVMean velocity$m/s$
ChwC_{hw}Hazen-Williams roughness coefficiente.g., 140 for very smooth, 100 for old cast iron
RRHydraulic radius$D/4$ for full pipes
SSSlope of the energy grade line$h_f/L$

Manning Equation for Pipes

While more common in open channels, the Manning equation is also frequently used for full pipe flow, especially in sewer and storm drain design.

Manning Equation for Pipes

V=1nR2/3S1/2(SI Units)V = \frac{1}{n} R^{2/3} S^{1/2} \quad (\text{SI Units})

Variables

SymbolDescriptionUnit
nnManning's roughness coefficient-

Minor Losses

Minor Losses Overview

Energy losses occur at pipe components like valves, bends, expansions, and contractions due to flow separation and turbulence.

Minor Losses Overview

hm=KV22gh_m = K \frac{V^2}{2g}
  • KK = Loss coefficient (empirical).
    • Entrance: K≈0.5K \approx 0.5 (sharp), 0.040.04 (rounded).
    • Exit: K=1.0K = 1.0.
    • 90° Elbow: K≈0.9K \approx 0.9.
    • Globe Valve (Open): K≈10K \approx 10.
Key Takeaways
  • The Reynolds Number (ReRe) is a dimensionless parameter that dictates the flow regime by comparing inertial forces to viscous forces.
  • The Hazen-Williams Equation is an explicit, empirical formula often preferred in water distribution systems over the iterative Darcy-Weisbach equation.
  • Flow transitions from laminar (smooth, predictable) to turbulent (chaotic, highly mixed) as velocity or pipe diameter increases, or as fluid viscosity decreases.
  • The critical threshold for pipe flow is generally Re≈2000Re \approx 2000.
  • Major losses are due to friction over the length of the pipe and are best calculated using the Darcy-Weisbach Equation.
  • For laminar flow, the friction factor ff depends solely on the Reynolds number (f=64/Ref = 64/Re).
  • For turbulent flow, ff depends on both ReRe and the relative roughness of the pipe wall (ϵ/D\epsilon/D), often requiring the Colebrook-White equation or the Moody Chart.
  • Minor losses occur at localized points in a pipe system due to changes in geometry (valves, bends, expansions, entrances).
  • They are calculated as a fraction of the velocity head (KV22gK \frac{V^2}{2g}).
  • Despite the name "minor", in short, highly convoluted pipe systems, these losses can exceed the "major" friction losses.
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