Darcy-Weisbach Calculator
Solve the Darcy-Weisbach equation for pipe head loss, pressure drop, or flow velocity. Covers laminar and turbulent flow with automatic friction factor calculation.
💧 What is the Darcy-Weisbach Equation?
The Darcy-Weisbach equation is the fundamental formula for calculating head loss due to friction in a pipe carrying a fluid: h_f = f x (L/D) x (V^2/2g). It links the geometry of the pipe (length L and diameter D), the flow condition (velocity V), gravity (g = 9.81 m/s^2), and the dimensionless Darcy friction factor f to the head loss h_f in metres. Head loss is the pressure drop expressed as equivalent fluid height, which engineers find convenient because it is independent of fluid density when dealing with incompressible flow.
The equation is used across a wide range of applications. Municipal water engineers use it to size distribution mains and balance loop networks. HVAC engineers calculate duct and chilled-water pipe sizes to keep fan and pump energy within budget. Oil and gas pipeline designers use it to determine required compressor and pump stages for cross-country pipelines. Process engineers apply it for every piping system inside a chemical plant or refinery.
The friction factor f is not a simple constant. In laminar flow (Reynolds number Re below 2300), the Hagen-Poiseuille result gives f = 64/Re exactly. In turbulent flow (Re above 4000), f depends on both Re and the pipe relative roughness (ε/D) through the implicit Colebrook-White equation. Because Colebrook-White requires iteration, this calculator uses the explicit Swamee-Jain approximation (1976), which matches Colebrook-White to within 3% for practical engineering ranges.
A common confusion is between the Darcy friction factor used here and the Fanning friction factor used in some chemical engineering texts. The Darcy factor is exactly four times the Fanning factor. Always check which factor a reference uses before substituting values into the head loss equation. This calculator uses the Darcy (also called Moody) friction factor throughout, consistent with the Moody chart and most civil and mechanical engineering textbooks.
📐 Formula
hf = f × (L ÷ D) × (V² ÷ 2g)
hf = friction head loss (m of fluid)
f = Darcy friction factor (dimensionless)
L = pipe length (m)
D = pipe inner diameter (m)
V = mean flow velocity (m/s)
g = gravitational acceleration = 9.81 m/s²
Pressure drop: ΔP = ρ × g × hf (Pa), where ρ = fluid density (kg/m³)
Laminar flow (Re below 2300): f = 64 ÷ Re
Turbulent flow (Re above 4000), Swamee-Jain: f = 0.25 ÷ [log10(ε/3.7D + 5.74/Re0.9)]²
Reynolds number: Re = V × D ÷ ν
ν = kinematic viscosity (m²/s); for water at 20°C: ν = 1.004 × 10−6 m²/s
Example: D = 0.1 m, L = 100 m, V = 2 m/s, ε = 0.046 mm: Re = 199,203 (turbulent), f = 0.0187, hf = 3.82 m, ΔP = 37.4 kPa
📖 How to Use This Calculator
Steps
1
Choose your calculation mode - Select Find Head Loss if you know the flow velocity and want the pressure drop, or Find Velocity if you have a head loss budget (such as a pump curve or gravity elevation difference) and need the corresponding flow rate.
2
Enter pipe geometry - Type the inner pipe diameter in mm (the inside bore, not the outside diameter or nominal size) and the pipe length in m. These determine the L/D ratio that amplifies friction losses over long runs.
3
Set flow velocity or head loss - In Find Head Loss mode, enter the mean flow velocity in m/s. In Find Velocity mode, enter the available head loss in metres (elevation difference for gravity systems, or pump head minus static head for pressurised systems).
4
Set roughness and viscosity - Enter the pipe roughness in mm (0.046 for new steel, 0.0015 for PVC) and the fluid kinematic viscosity in mm²/s (1.004 for water at 20°C). The calculator pre-loads water defaults.
5
Read the results - The calculator outputs head loss or velocity, pressure drop in kPa, flow rate in L/s and m³/h, Reynolds number, Darcy friction factor, and flow regime so you can verify whether your design is in turbulent or laminar territory.
💡 Example Calculations
Example 1 - Water Supply Pipe, 100 mm Diameter, 100 m Long
New commercial steel pipe (ε = 0.046 mm), water at 20°C, velocity 2 m/s
1
D = 0.1 m, L = 100 m, V = 2 m/s, ε = 0.046 mm, ν = 1.004 x 10^-6 m²/s. Reynolds number: Re = 2 x 0.1 / 1.004e-6 = 199,203 (turbulent).
2
Relative roughness: ε/D = 0.000046 / 0.1 = 0.00046. Swamee-Jain: f = 0.25 / [log10(0.00046/3.7 + 5.74/199203^0.9)]² = 0.0187.
3
h_f = 0.0187 x (100/0.1) x (4/19.62) = 0.0187 x 1000 x 0.2039 = 3.82 m. Pressure drop: ΔP = 998 x 9.81 x 3.82 = 37.4 kPa. Flow rate: Q = 2 x π/4 x 0.01 = 15.7 L/s.
hf = 3.82 m | ΔP = 37.4 kPa | Q = 15.7 L/s
Try this example →Example 2 - PVC Irrigation Pipe Finding Flow Rate from Elevation
PVC pipe (ε = 0.0015 mm), 50 mm diameter, 200 m long, gravity head = 15 m
1
Switch to Find Velocity mode. D = 50 mm, L = 200 m, h_f = 15 m, ε = 0.0015 mm, ν = 1.004 mm²/s.
2
Iterative solution: start f = 0.02, V = sqrt(2 x 9.81 x 0.05 x 15 / (0.02 x 200)) = sqrt(0.981) = 1.906 m/s. Re = 95,022, new f = 0.0181, new V = sqrt(0.981/0.905) = 2.005 m/s. Converges to V ≈ 2.00 m/s.
3
Flow rate: Q = 2.00 x π/4 x 0.0025 = 0.003927 m³/s = 3.93 L/s = 14.1 m³/h.
V = ~2.00 m/s | Q = ~3.93 L/s | Re = ~95,000 (turbulent)
Try this example →Example 3 - Cast Iron Water Main, 300 mm Diameter, 500 m Long at 1.5 m/s
Aged cast iron main (ε = 0.26 mm), D = 300 mm, L = 500 m, V = 1.5 m/s
1
Re = 1.5 x 0.3 / 1.004e-6 = 448,207 (turbulent). Relative roughness ε/D = 0.00026/0.3 = 0.000867.
2
f = 0.25 / [log10(0.000867/3.7 + 5.74/448207^0.9)]² = 0.25 / [log10(0.0002343 + 7.29e-5)]² = 0.25/(-3.515)² = 0.0202.
3
h_f = 0.0202 x (500/0.3) x (2.25/19.62) = 0.0202 x 1667 x 0.1147 = 3.86 m. Flow rate: Q = 1.5 x π/4 x 0.09 = 106 L/s = 382 m³/h.
hf = 3.86 m | Q = 106 L/s | f = 0.0202
Try this example →Example 4 - Laminar Flow: Glycerine in a Small Tube
Smooth tube (ε = 0 mm), D = 10 mm, L = 5 m, V = 0.05 m/s, glycerine ν = 700 mm²/s
1
Re = 0.05 x 0.01 / 700e-6 = 0.714. Well below 2300, so flow is laminar. f = 64/Re = 64/0.714 = 89.7.
2
h_f = 89.7 x (5/0.01) x (0.0025/19.62) = 89.7 x 500 x 0.0001275 = 5.72 m.
3
ΔP = ρ x g x h_f = 1260 x 9.81 x 5.72 = 70,700 Pa = 70.7 kPa (using glycerine density 1260 kg/m³; calculator shows water-equivalent value).
f = 89.7 | hf = 5.72 m | Re = 0.71 (Laminar)
Try this example →❓ Frequently Asked Questions
What is the Darcy-Weisbach equation used for?
The Darcy-Weisbach equation calculates head loss (pressure drop expressed as fluid height in metres) caused by friction as fluid moves through a pipe: h_f = f x (L/D) x (V2/2g). It is the standard formula for pipe flow design in water supply, HVAC, oil and gas, and chemical process systems. Engineers use it to size pipes, select pumps, and balance distribution networks.
What is the Darcy friction factor and how is it calculated?
The Darcy friction factor (f) quantifies resistance between the fluid and pipe wall. For laminar flow (Re less than 2300), f = 64/Re exactly. For turbulent flow, this calculator uses the Swamee-Jain explicit formula: f = 0.25 / [log10(ε/3.7D + 5.74/Re^0.9)]^2, which is accurate to within 3% of the Colebrook-White equation for Reynolds numbers between 5000 and 10^8.
What is the Reynolds number and what values indicate turbulent flow?
Reynolds number Re = VD/ν is the ratio of inertial to viscous forces. Re below 2300 means laminar flow with smooth, parallel streamlines and f = 64/Re. Re between 2300 and 4000 is the unstable transition zone. Re above 4000 is fully turbulent with chaotic mixing and a higher friction factor. Most water pipes in practice operate with Re from 50,000 to 500,000, well into the turbulent regime.
What pipe roughness value should I use?
Use these absolute roughness values (ε): PVC or drawn copper 0.0015 mm, new commercial steel 0.046 mm, galvanized steel 0.15 mm, asphalted cast iron 0.12 mm, cast iron 0.26 mm, concrete 0.3 to 3 mm. For aged pipes, double or triple these values to account for scale and corrosion. The Moody chart is the classical reference for relative roughness ε/D vs. friction factor.
How does pipe diameter affect head loss in a pipe?
Head loss is inversely proportional to D^5 when flow rate (not velocity) is held constant. Halving the pipe diameter while maintaining the same flow rate increases head loss by a factor of 2^5 = 32. This makes pipe sizing the single most impactful variable in pipe system design, with small diameter reductions causing very large increases in pumping energy and cost.
What is the difference between head loss and pressure drop?
Head loss (h_f, in metres) is pressure drop expressed as equivalent fluid column height. Pressure drop (ΔP, in Pascals) is the actual reduction in pressure. They relate by ΔP = ρgh_f. For water at 20°C, 1 m of head = 9.79 kPa = 1.42 psi. Head loss is preferred in hydraulics because it is independent of fluid density when comparing pipe sections with different fluid types.
What is the Swamee-Jain equation?
The Swamee-Jain equation (1976) is an explicit approximation for the Darcy friction factor in turbulent flow: f = 0.25 / [log10(ε/3.7D + 5.74/Re^0.9)]^2. It replaces the implicit Colebrook-White equation, which requires numerical iteration to solve. Swamee-Jain is accurate to within 3% of Colebrook-White for Reynolds numbers from 5000 to 10^8 and relative roughness ε/D from 10^-6 to 0.05.
How does this calculator find velocity from head loss?
Rearranging Darcy-Weisbach gives V = sqrt(2gDhf/fL), but f also depends on V through the Reynolds number. The calculator starts with an initial guess of f = 0.02, computes V, then computes Re and a new f, and repeats until convergence (typically 10 to 20 iterations). This iterative approach is standard in hydraulics software and pipe network analysis tools.
What kinematic viscosity should I use for water at different temperatures?
Water kinematic viscosity by temperature: 10°C = 1.307 mm²/s, 20°C = 1.004 mm²/s, 40°C = 0.658 mm²/s, 60°C = 0.474 mm²/s, 80°C = 0.365 mm²/s, 100°C = 0.294 mm²/s. Viscosity decreases significantly with temperature. Hot water systems have lower friction losses than cold water systems at the same flow rate, which matters for solar thermal and heating loop design.
What is the typical design head loss per kilometre for water pipes?
Common design guidelines for municipal water distribution recommend a maximum friction gradient of 4 to 10 metres per kilometre (4 to 10 m per 1000 m). Transmission mains typically target the lower end (4 m/km) to minimise pumping energy over long distances. Distribution networks may accept up to 10 m/km. Building internal plumbing often allows 1 to 2 m per 10 m length.
Is the Darcy-Weisbach equation more accurate than Hazen-Williams?
Yes. The Darcy-Weisbach equation is physically derived and valid for any incompressible fluid at any temperature and Reynolds number. The Hazen-Williams formula is empirical, calibrated only for water near 15°C in turbulent flow, and can underestimate head loss by up to 20% for very smooth or very rough pipes outside its calibration range. Modern hydraulic software and the ASCE standards recommend Darcy-Weisbach for all new designs.
Can I use this calculator for gases or compressible flow?
For low-velocity gas flow where density change along the pipe is small (Mach number below 0.3 and pressure drop below 10% of inlet pressure), the Darcy-Weisbach equation gives a useful approximation using inlet density and viscosity. For high-velocity or high-pressure-drop gas flow, compressibility effects become significant and require the full compressible flow equations. This calculator is designed for incompressible liquids and low-velocity gases only.