Pipe friction loss, also known as pressure drop, is a critical parameter in fluid dynamics and pipeline engineering. It represents the energy lost by a fluid as it flows through a pipe due to the friction between the fluid and the pipe walls, as well as internal friction within the fluid itself (viscosity). This energy loss manifests as a decrease in pressure along the length of the pipe.
Accurate calculation of friction loss is essential for designing efficient pumping systems, sizing pipes correctly, and ensuring that fluid delivery meets the required pressure and flow rate at the destination. Overestimating friction loss can lead to oversized (and thus more expensive) pumps and pipes, while underestimating it can result in system underperformance, cavitation, and inadequate flow.
The Darcy-Weisbach Equation
The most widely accepted and fundamental equation for calculating friction loss in pipes is the Darcy-Weisbach equation. It is applicable to both laminar and turbulent flow regimes. The equation is as follows:
h_f = f * (L/D) * (v²/2g)
Where:
h_f is the head loss due to friction (in meters of fluid head).
f is the Darcy friction factor (dimensionless).
L is the length of the pipe (in meters).
D is the inner diameter of the pipe (in meters).
v is the average velocity of the fluid (in m/s).
g is the acceleration due to gravity (approximately 9.81 m/s²).
To find the pressure loss (ΔP), you can convert head loss to pressure using:
ΔP = h_f * ρ * g
Where:
ΔP is the pressure loss (in Pascals, Pa).
ρ (rho) is the density of the fluid (in kg/m³).
Determining the Friction Factor (f)
The challenge in using the Darcy-Weisbach equation often lies in determining the friction factor, f. This factor depends on the flow regime (laminar or turbulent) and the relative roughness of the pipe.
1. Reynolds Number (Re):
The Reynolds number indicates whether the flow is laminar, transitional, or turbulent.
Re = (ρ * v * D) / μ
Where:
Re is the Reynolds number.
ρ is the fluid density (kg/m³).
v is the average fluid velocity (m/s).
D is the pipe inner diameter (m).
μ (mu) is the dynamic viscosity of the fluid (Pa·s).
2. Flow Regimes:
If Re < 2300: Flow is Laminar.
If 2300 < Re < 4000: Flow is Transitional.
If Re > 4000: Flow is Turbulent.
3. Calculating Friction Factor (f):
Laminar Flow (Re < 2300): The friction factor is independent of roughness and is given by: f = 64 / Re.
Turbulent Flow (Re > 4000): The friction factor depends on both the Reynolds number and the relative roughness of the pipe (ε/D, where ε is the absolute roughness). The Colebrook-White equation is commonly used for turbulent flow, but it is implicit and requires iterative solutions or approximations. A widely used explicit approximation is the Swamee-Jain equation:
f = 0.25 / [log10( (ε/D)/3.7 + 5.74/Re^0.9 )]²
Transitional Flow: This regime is complex. Often, engineers use interpolation or default to turbulent flow calculations for safety.
Calculator Logic
This calculator first calculates the fluid velocity from the given flow rate and pipe diameter. It then determines the Reynolds number to identify the flow regime. Based on the flow regime and pipe roughness, it calculates the Darcy friction factor using appropriate formulas (Swinney-Jain for turbulent, 64/Re for laminar). Finally, it computes the head loss using the Darcy-Weisbach equation and converts it to pressure loss.
Example Calculation
Consider pumping water (density ≈ 998 kg/m³, viscosity ≈ 0.001 Pa·s) through a 0.2 m diameter PVC pipe (roughness ≈ 0.000046 m) for a length of 100 m, at a flow rate of 0.1 m³/s.
Velocity (v): Area = π * (0.2/2)² ≈ 0.0314 m². Velocity = 0.1 m³/s / 0.0314 m² ≈ 3.18 m/s.
Reynolds Number (Re): Re = (998 * 3.18 * 0.2) / 0.001 ≈ 635,724. This indicates turbulent flow.