Continuity, Darcy-Weisbach friction and total pump head
Give the calculator a flow (or a velocity) and a pipe, and it runs the full chain a pump selection actually needs: the other half of the continuity equation Q = A × V, the Reynolds number and flow regime, the Darcy friction factor (exact 64/Re when laminar, the Haaland explicit form of Colebrook when turbulent), friction loss by Darcy-Weisbach, minor losses from your fittings coefficient, elevation head, and the totals: total dynamic head, pressure drop, and the pump’s hydraulic power. Roughness values come from material presets you can override; nothing about your pipe is assumed silently.
Disclaimer: This calculator gives estimates for general information only. It is not financial, medical or other professional advice. Pipe sizing and pump selection for real systems belong with the governing codes and a qualified engineer; verify critical designs independently.
What this calculator does
One run connects the quantities that are usually scattered across three charts. Flow and velocity convert through the bore area. The Reynolds number (velocity × diameter / kinematic viscosity) selects the regime and the friction model, honestly: laminar flow uses the exact f = 64/Re; turbulent flow uses the Haaland equation, the standard explicit approximation of the Colebrook relation, fed by your roughness; the transition band is flagged and its factor labelled an estimate, because in that band it genuinely is one. Friction loss follows Darcy-Weisbach, minor losses follow the K method on the same velocity head, elevation adds directly, and the total is expressed as head, pressure and hydraulic power so it can be read against a pump curve without further conversion.
How to use it
- Inside diameter, always. A “50 mm” pipe’s bore varies by material and schedule, and loss scales roughly with diameter to the fifth power at fixed flow: a 5% bore error is about a 28% loss error. Measure or look up the real bore.
- Count fittings into K honestly. On short runs the fittings, not the pipe, own the loss: ten elbows and a check valve can out-cost a hundred metres of straight tube, which the split results expose line by line.
- Roughness is a condition, not just a material. The presets describe new pipe; scaling, corrosion and biofilm raise effective roughness over a system’s life, and prudent design re-runs the numbers at an aged roughness before committing a pump.
- Elevation is not loss. It appears in the total because the pump must lift the fluid, but unlike friction it is recoverable in a siphon leg; the breakdown keeps it separate precisely so it is never double-paid.
The formulas
- Continuity: Q = A × V with A = π/4 × D².
- Reynolds number: Re = V × D / ν. Below 2,300 laminar; above 4,000 turbulent; between, transition.
- Friction factor: laminar f = 64/Re; turbulent, Haaland: 1/√f = -1.8 × log10( ((ε/D)/3.7)^1.11 + 6.9/Re ).
- Friction loss: h_f = f × (L/D) × V²/(2g). Minor loss: h_m = K × V²/(2g). Total head = h_f + h_m + elevation rise. Pressure drop = ρ × g × total head; hydraulic power = ρ × g × Q × total head.
Worked examples
A steel line, fully traced. 50 mm bore, 100 m of commercial steel (roughness 0.045 mm), 5 L/s of water at 20°C, total K of 5, rising 10 m: velocity 2.55 m/s, Reynolds number 126,816.69 (turbulent), friction factor 0.0212. Friction loss 14.00 m, minor losses 1.65 m, elevation 10 m: total head 25.65 m, a pressure drop of 251.02 kPa, and 1,255.10 W of hydraulic power the pump must deliver to the water before its own efficiency is considered. The friction share exceeding the lift share is the typical signature of a long small line, and it is why the remedy is usually a bigger pipe, not a bigger pump.
The same physics from a velocity. Designing to 1.5 m/s in a 100 mm PVC line (roughness 0.0015 mm), 50 m long, K of 2, no rise: the flow is 11.78 L/s, Reynolds 149,402.39, friction factor 0.0165, friction loss 0.95 m, minor 0.23 m, total head 1.18 m, pressure drop 11.50 kPa. Gentle numbers, and a quiet illustration of the diameter law: double the bore of the first example transformed a punishing line into a cheap one.
Reading your results
Read the loss split before the total: friction-dominated systems respond to diameter and length; fitting-dominated ones to layout; elevation-dominated ones to route. Read velocity against service norms (roughly 1 to 3 m/s in pumped water lines: slower wastes pipe, faster wastes energy and invites water hammer). And read the friction factor’s regime: a laminar result in what you believed was a turbulent system means the flow, the viscosity or the diameter is not what you think it is.
Limitations
- Steady, incompressible, single-phase flow. Water hammer, cavitation, air locking, slurries and two-phase flow are outside the model; gases need density handling this page does not pretend to offer beyond the custom inputs.
- Transition band honesty. Between Reynolds 2,300 and 4,000 the factor is genuinely unstable in nature; the page estimates it at the band’s edge and labels it, rather than printing false precision.
- Fire hydrant testing is a different instrument. Hydrant flow tests use their own sourced formulas and unit conventions (the Hazen-Williams based test method with pitot readings) and are deliberately not folded into this Darcy-Weisbach page.
- Pump reality exceeds hydraulics. The power row is the water power; shaft power divides by pump efficiency and motor loading adds its own losses, none of which this page invents values for.
Frequently asked questions
Why Darcy-Weisbach and not Hazen-Williams?
Darcy-Weisbach is dimensionally sound for any Newtonian fluid at any temperature because viscosity enters it explicitly through the Reynolds number; Hazen-Williams is an empirical water-only formula whose coefficient quietly absorbs a narrow range of conditions and drifts outside them. Fire-service practice keeps Hazen-Williams for its own standardised tests (named in the limitations), but for a general pipe calculator the Darcy form is the defensible default, and it is the one engineering references from Crane to the standard fluid mechanics texts build on.
Why does one size up change the loss so much?
At fixed flow, velocity falls with the square of diameter while the friction term also divides by diameter once more and the velocity head falls again: the compound effect is loss proportional to roughly the fifth power of diameter, the steepest lever in the whole calculation. It is why the worked examples differ so violently, why bore honesty matters more than any other input, and why “just fit a bigger pump” is usually the expensive answer to a geometry problem.
What K values should I use for my fittings?
Start from the form’s hint values and the fitting makers’ published coefficients where they exist: entrances about 0.5 (sharp) to 0.05 (well rounded), exits 1.0, flanged regular 90° elbows about 0.3, threaded ones about 1.5, fully open gate valves about 0.15, globe valves near 10, swing check valves about 2.5. Sum every fitting on the run. Where the total is a guess, run the calculator at a low and a high K and design to the honest range the two answers bracket.
Can this size a pump?
It produces the duty point a pump must meet: the flow, and the total dynamic head at that flow. Selecting the pump means finding a published curve that passes through (or above) that point at acceptable efficiency, then checking shutoff head, NPSH and the system’s own curve across the operating range. The page deliberately stops at the duty point, because choosing the machine from a curve only its maker publishes is the step that keeps the calculation honest.
Related calculators
Method: continuity, Reynolds regime selection, Haaland friction for turbulent flow, Darcy-Weisbach friction and K-method minor losses as itemised in the formula section. Last reviewed: 14 October 2026.