Darcy-Weisbach Pipe Friction Head Loss
Evaluates the energy head loss due to friction in fluid flow through a circular pipe of constant diameter.
Primary Mathematical Expression
Nomenclature & Variables
| Symbol | Variable Name | Metric Unit | Imperial Unit | Description |
|---|---|---|---|---|
| h_f | Friction head loss | m | ft | Energy loss expressed as equivalent fluid column height. |
| f | Friction factor | dimensionless | dimensionless | Pipe roughness resistance coefficient (from Moody Chart). |
| L | Pipe length | m | ft | Total length of the pipe segment. |
| D | Inner diameter | m | ft | The inside diameter of the circular pipe. |
| v | Flow velocity | m/s | ft/s | Average velocity of fluid in the cross-section. |
| g | Gravity constant | m/s² | ft/s² | Acceleration due to gravity (9.81 m/s² or 32.2 ft/s²). |
Step-by-Step Derivation
- 1
Consider fluid flowing through a circular pipe of diameter D and length L. The shear stress \tau_o on the pipe wall resists flow.
- 2
Force balance: Pressure force change equals shear force: (P1 - P2) * \pi * D² / 4 = \tau_o * \pi * D * L.
- 3
This simplifies to head loss: h_f = (P1 - P2) / \gamma = 4 * \tau_o * L / (\gamma * D).
- 4
The wall shear stress \tau_o is related to dynamic pressure by friction factor f: \tau_o = f * \rho * v² / 8.
- 5
Substituting \tau_o and \gamma = \rho * g into the head loss equation gives the Darcy-Weisbach equation: h_f = f * L * v² / (D * 2g).
Worked Example Calculation
Water flows through a 50 m long pipe of diameter 0.1 m at a velocity of 2.0 m/s. The friction factor f is 0.02, and gravity g is 9.81 m/s². Determine the head loss due to friction.
- •Identify input parameters: L = 50 m, D = 0.1 m, v = 2.0 m/s, f = 0.02, g = 9.81 m/s².
- •Apply the Darcy-Weisbach Formula: h_f = f * L * v² / (D * 2g).
- •Compute the numerator: f * L * v² = 0.02 * 50 * 2² = 1.0 * 4 = 4.0.
- •Compute the denominator: D * 2g = 0.1 * 2 * 9.81 = 1.962.
- •Calculate head loss: h_f = 4.0 / 1.962 = 2.04 meters.
Engineering Assumptions
- •Steady, incompressible, one-dimensional fluid flow.
- •Fully developed flow inside a circular pipe of constant cross-section.
Design Limitations
- •Requires accurate friction factor f (e.g. from Moody chart or Colebrook-White equation depending on laminar or turbulent flow).
Academic References & Standards
Frank M. White, Fluid Mechanics, 8th Edition
Textbook detailing pipe flows and head loss formulations.