The Darcy-Weisbach Calculator estimates the change in pressure — the pressure drop — caused by friction in a pipe during fluid flow. The roughness on the inner surface of a pipe causes friction loss, resulting in a pressure drop, which is described through the Darcy friction factor. Pipes transport fluids across a wide array of domains, such as pumping a water supply, and the pressure drop is crucial for estimating the actual pressure output and for selecting the material and pipe dimensions during construction.
Read on to understand the workings of the Darcy-Weisbach equation and how to calculate the pressure drop across a section of pipe.
What is the Darcy-Weisbach equation?
The friction in a pipe due to surface roughness is related to the loss of pressure by means of an empirical equation. The equation also factors in the pipe dimensions — such as diameter and length — as well as fluid properties such as density and flow velocity, and material properties including the friction factor. Mathematically, the pressure loss ΔP for a pipe of diameter D and length L is written as:
ΔP = (f · L · V² · ρ) / (2 · D)
where:
- ρ — the fluid density;
- V — the flow velocity; and
- f — the Darcy friction factor.
The above relation is known as the Darcy-Weisbach equation, named in honor of the engineers Henry Darcy and Julius Weisbach. Alternatively, the pressure loss per unit length can be written as:
ΔP / L = (f · V² · ρ) / (2 · D)
The Darcy friction factor is the key part of the equation. It is based on the material's surface and is very difficult to calculate directly. One of the classic solutions uses the Moody diagram and the Colebrook-White equation. For a pipe with absolute surface roughness ε (sometimes written k), the Colebrook-White equation gives the friction factor implicitly:
1/√f = −2 · log10( ε/(3.7·D) + 2.51/(Re·√f) )
Because that form is implicit (f appears on both sides), this calculator uses the explicit Swamee-Jain approximation of the Colebrook-White equation, which is accurate to within about 1 % for turbulent flow:
f = 0.25 / [ log10( ε/(3.7·D) + 5.74/Re0.9 ) ]²
For laminar flow (Reynolds number Re < 2300) the friction factor does not depend on roughness at all and is simply f = 64 / Re. The Reynolds number itself is computed from Re = V · D / ν, where ν is the kinematic viscosity of the fluid.
How to calculate pressure drop
Calculating the pressure drop with the Darcy-Weisbach equation takes four steps:
- Find the Reynolds number Re = V · D / ν to determine whether the flow is laminar or turbulent.
- Determine the Darcy friction factor f — use f = 64/Re for laminar flow, or the Swamee-Jain / Colebrook-White equation for turbulent flow, using the relative roughness ε/D.
- Apply the Darcy-Weisbach equation ΔP = f · (L/D) · (ρV²/2) to obtain the pressure drop.
- Convert as needed — express the result as head loss hf = ΔP / (ρg), or as a pressure gradient ΔP/L per unit length of pipe.
US customary vs metric units
The calculator supports both unit systems and converts everything internally to SI:
- US customary: velocity in ft/s, diameter in inches, length in feet, roughness in inches, density in lb/ft³ — results in psi, kPa and feet.
- Metric (SI): velocity in m/s, diameter in mm, length in m, roughness in mm, density in kg/m³ — results in kPa, bar and meters.
Viscosity is entered in centistokes (cSt = mm²/s) in both systems. Built-in fluid presets (water at 20 °C and 10 °C, seawater, light oil, air) and pipe-material roughness presets (drawn tubing, PVC, copper, commercial steel, galvanized iron, cast iron, concrete, riveted steel) fill the density, viscosity and roughness fields automatically — or choose Custom to enter your own values.
Example — Using the Darcy-Weisbach Calculator
Suppose water at 20 °C flows at 2 m/s through a 100 mm diameter commercial-steel pipe that is 50 m long. The water density is ρ = 998.2 kg/m³, the kinematic viscosity is ν = 1.004 cSt, and the steel roughness is ε = 0.045 mm.
- Reynolds number: Re = (2 × 0.1) / (1.004 × 10−6) ≈ 199,200 — turbulent.
- Relative roughness: ε/D = 0.045 / 100 = 0.00045.
- Friction factor (Swamee-Jain): f ≈ 0.0185.
- Pressure drop: ΔP = 0.0185 × (50 / 0.1) × (998.2 × 2² / 2) ≈ 18.5 kPa (≈ 2.68 psi).
- Head loss: hf = ΔP / (ρg) ≈ 1.89 m of water.
Toggle the calculator to US customary and the same scenario is entered in ft/s, inches and feet, with the pressure drop reported in psi.
Frequently Asked Questions
What is the Darcy-Weisbach equation used for?
It is the most general and accurate way to calculate the pressure drop (or head loss) caused by friction when any fluid flows through a full pipe. Unlike the Hazen-Williams equation, which is restricted to water near room temperature, the Darcy-Weisbach equation works for any Newtonian fluid at any temperature, which is why it is the preferred method in mechanical and chemical engineering.
What is the Darcy friction factor?
The Darcy friction factor f is a dimensionless number that captures how much energy the flow loses to friction. It depends on the Reynolds number and the relative roughness ε/D of the pipe. For laminar flow f = 64/Re; for turbulent flow it is found from the Moody chart or the Colebrook-White equation. Note that the Darcy friction factor is four times larger than the Fanning friction factor.
How do I reduce the pressure drop in a pipe?
The most effective change is to increase the pipe diameter, because ΔP is inversely proportional to D and the velocity falls with the square of the diameter for a fixed flow rate. You can also lower the flow velocity, use a smoother pipe material (smaller ε), or shorten the pipe run.
What is the difference between pressure drop and head loss?
They describe the same friction loss in different units. Pressure drop ΔP is expressed in pressure units (Pa, kPa, bar, psi), while head loss hf is the equivalent height of a fluid column, in meters or feet. They are related by ΔP = ρ · g · hf.
🙋 Friction is always hard to model — its consequences, however, are fairly simple. Explore the related friction calculator and drag equation calculator to dig deeper.