Pressure drop is the bill friction sends the pump every secondPhoto: UnsplashDarcy-Weisbach Comes Straight from Bernoulli
Start with Bernoulli's equation — the bookkeeping of energy inside a pipe. Total head at any point in a steady, incompressible, inviscid flow is the sum of elevation head, pressure head, and velocity head:
z + P/(ρg) + v²/(2g) = constant
A real pipe is not inviscid. The fluid rubs against the wall and converts kinetic energy into heat; the amount lost between two sections is the friction head hf. For a horizontal pipe (z1 = z2) with no pump between the ends, Bernoulli's equation collapses to:
P₁/(ρg) + v₁²/(2g) = P₂/(ρg) + v₂²/(2g) + hf
In a pipe of constant diameter v1 = v2, so the velocity-head terms cancel. The pressure difference ΔP = P₁ − P₂ must be exactly the friction loss converted into pressure:
ΔP = f · (L/D) · ρ · v² / 2
Darcy-Weisbach — the universal pressure-drop formula
Multiply through by ρg and you get the friction-head form hf = f · (L/D) · v²/(2g). Same equation, different units. Either way, it says three things about pressure drop that nothing else disagrees with:
- Friction grows linearly with pipe length — double the run, double the loss.
- Friction falls inversely with diameter — double the pipe size and the loss drops by four times, because velocity and the L/D term both halve.
- Friction scales with velocity squared — double the speed and the loss quadruples. This is why every good pipe-sizing code publishes velocity ceilings.
The equation is attributed to Darcy and Weisbach working independently in the middle of the 19th century. It survives today because no engineering approximation — Hazen-Williams, Manning, Chezy — disagrees with it; each one is just a shortcut for a limited range of Reynolds number and roughness. If you know the friction factor f, Darcy-Weisbach is the answer. The hard part is finding f.
f Is Not a Constant — It Depends on Roughness and Reynolds
The friction factor f is a dimensionless number between roughly 0.008 and 0.1 that accounts for how much a given pipe surface and flow regime slow the fluid down. It depends on exactly two things:
- Roughness ratio ε/D — the absolute roughness of the pipe wall (ε, typically 0.001–3 mm) divided by the inside diameter D. A smooth PVC pipe (ε = 0.0015 mm) and a rusty cast-iron line (ε ≈ 0.5 mm) differ by more than two orders of magnitude.
- Reynolds number Re — ρvD/μ, comparing inertial forces to viscous forces. Below Re ≈ 2300 the flow is laminar (layers slide); above Re ≈ 4000 it is turbulent (churns and mixes); between is transitional.
Colebrook-White (1939) — the Implicit Equation
For turbulent flow (Re > 4000) the standard answer is the Colebrook-White equation. It is built to reduce to the laminar law (f = 64/Re) at low Re and to the fully-rough-wall regime (f independent of Re) at high Re:
1/√f = −2 · log₁₀( ε/(3.7·D) + 2.51 / (Re · √f) )
Why Colebrook is implicit (and why you do not care)
f appears on both sides — you cannot solve the equation with algebra. A Newton-Raphson iteration converges in 2–3 steps on any calculator; that is what every hydraulic engine does internally. The approximations below skip iteration and are accurate enough for engineering work.
Haaland (1983) — 3% max error
1/√f ≈ −1.8 · log₁₀( (ε/(3.7·D))^1.11 + 6.9 / Re )
Haaland replaced the 2.51/(Re·√f) term with a plain 6.9/Re and tuned the constants against the Moody chart. Maximum error vs Colebrook: 3%. Good enough for hand calculation.
Swamee-Jain (1976) — 2% max error
f ≈ 0.25 / [ log₁₀( ε/(3.7·D) + 5.74 / Re^0.9 ) ]²
Swamee and Jain fitted the entire Moody chart with a single expression. Maximum error vs Colebrook: 2%. This is what the live widget above uses — it recomputes on every keystroke without iterating.
Laminar flow — the exact answer
For Re < 2300, the friction factor is independent of roughness entirely. f = 64/Re (sometimes reported as f = 57/Re for very short lengths). Swamee-Jain gives almost exactly 64/Re in the laminar regime, so one formula covers both regimes cleanly.
Absolute Roughness ε for Common Materials
Roughness is a wall characteristic, not a material characteristic — the same material installed two ways (drawn tubing vs cast iron) can differ by two orders of magnitude. The numbers below are industry-standard estimates used by the calculator and by every major pipe-sizing handbook.
| Pipe Material | ε (mm) | ε (in) | Notes |
|---|---|---|---|
| Drawn tubing (PVC, copper, PEX) | 0.0015 | 0.00006 | Smooth, as-new |
| Commercial steel (new, mill scale) | 0.045 | 0.0018 | Mill-rolled, pickled, or light rust |
| Galvanized steel | 0.15 | 0.006 | Zinc coating adds significant roughness |
| Cast iron (new) | 0.26 | 0.010 | Sand-cast |
| Cast iron (used, 30 yr, tuberculated) | 0.5+ | 0.020+ | Rust tubercles & internal deposits dominate |
| Concrete (smooth form) | 0.3 | 0.012 | Steel formwork, well-finished |
| Concrete (rough, drawn, or rubble lined) | 3.0 | 0.12 | Poor workmanship, joint defects |
| Rubber hose (reinforced, new) | 0.0015 | 0.00006 | Smooth rubber liner |
Rule of thumb for aged cast iron. If you only know the pipe was installed pre-1960 and is on a mineral water line, estimate ε = 0.5 mm and add 20% to the total ΔP as a tuberculation margin. The last person who guessed "new cast iron" on a 1955 branch usually paid for a pump upgrade the following spring.
Fittings, Valves, and Entrances — K-Factors
Not all friction happens on straight pipe. Every time the flow turns, narrows, expands, or passes through a valve, momentum is lost in eddies and swirls. These losses are called local losses (or minor losses) and they are quantified with a dimensionless loss coefficient K:
hf,local = Σ( Ki · v² / (2g) )
Total head loss on any pipe run is the sum of the straight-pipe Darcy loss and all the local losses on the run:
hf,total = f · (L/D) · v²/(2g) + Σ( Ki · v²/(2g) )
Common K-Factors
| Component | K | Notes |
|---|---|---|
| 90° elbow (standard) | 0.75 | Threaded, standard |
| 90° elbow (long-radius) | 0.4 | Threaded or flanged long-radius |
| 45° elbow | 0.35 | Threaded |
| Gate valve (fully open) | 0.19 | Fully open, rising-stem |
| Globe valve (fully open) | 6 | Fully open — throttles by design |
| Ball valve (fully open) | 0.03 | Fully open, full-bore ball |
| Pipe entrance (sharp) | 0.5 | Square-edged entrance to a round pipe |
| Pipe exit | 1 | Velocity head is fully lost on exit |
When local losses are more than the straight pipe
Consider a 10-metre run of 1-inch copper (D = 25.4 mm) carrying 10 L/min. Straight-pipe loss with f ≈ 0.0205 is about 1.6 m. A gate valve (K = 0.19) + two 90° elbows (K = 0.75 × 2) + sharp entrance (K = 0.5) totals ΣK = 2.19. Local losses alone are K · v²/2g ≈ 2.19 × 0.198 ≈ 0.43 m — that is 21% of straight-pipe. Shorten the run to 5 m and the local losses exceed straight pipe. That is the rule: when L/D drops below 100, add up the K's — they dominate.
One note on pipe exit K = 1.0. When flow discharges from a pipe into a large reservoir, the exit velocity v is fully dissipated. The K = 1.0 is not an empirical number — it is exactly the velocity head v²/(2g) you already have. If you are using the Darcy form ΔP = f·(L/D)·ρv²/2 and your downstream pressure is truly reservoir static pressure, you already charged that velocity head; double-counting K = 1.0 is a common mistake on gravity drain and pump-suction calculations.
ΔP × Q = Hydraulic Power — Then Efficiency Takes Its Cut
Pressure drop ΔP is the force per unit area the pipe exerts against the fluid. Multiply by the volumetric flow rate Q and you get power — the number of joules per second the pipe wall converts into heat:
Phydraulic = ΔP · Q
SI: ΔP in Pa (N/m²), Q in m³/s → P in W (J/s)
A quick shortcut for water that everyone memorises after their first pump quote:
P(W) ≈ ΔP(kPa) × Q(L/min) / 60
The hydraulic power is the minimum the pump must deliver into the fluid. The motor at the wall draws more, because every pump and motor has losses (impeller slip, bearing friction, eddies in the volute, motor copper and iron losses). Brake horsepower at the motor shaft is:
PBHP = Phydraulic / η
Typical Efficiency Ranges
Residential (< 1 HP)
η = 60 – 75%
Well, booster, irrigation
Commercial (1 – 25 HP)
η = 75 – 85%
HVAC, process, fire pump
Industrial (> 25 HP)
η = 80 – 90%
Large centrifugal
Example — the shortcut in action
ΔP = 29.3 kPa, Q = 10 L/min, η = 70% → PBHP = 29.3 × 10 / 60 / 0.70 = 6.98 W. Same pipe, same flow, three elbows instead of one, η same: ΔP doubles to 58.6 kPa → PBHP = 14 W. The pump motor's nameplate is directly the friction budget of the pipe.
Compressible Fluids — Use the Operating Density
Everything above was derived for incompressible flow — liquids, mostly. Gases obey the ideal-gas law, so their density depends directly on their absolute pressure:
ρoperating = (Pabsolute · M) / (R · T)
Air at 600 kPa (6 bar g, 6 atm absolute) is roughly six times denser than air at atmospheric pressure. If you plugged in standard air density (1.2 kg/m³) when the line is at 6 bar, you would underestimate ΔP by a factor of six. Always compute ρ from the line's absolute pressure and temperature, then run the Darcy-Weisbach equation normally — the formula does not change.
For long gas lines (>100 pipe diameters), the pressure drop itself changes the density, which changes the velocity, which changes the Reynolds number and the friction factor. Break the line into 5–10 pipe-diameter segments and run the calculation on each one using that segment's local absolute pressure. The calculator on this site does this automatically when you choose a gas preset.
Step-by-Step — From Pipe to Pump
Example 1
100 m of 1-inch copper (ID = 25.4 mm), 10 L/min water
- // velocity
v = (10/60000 m³/s) / (π·0.0254²/4 m²) = 1.97 m/s - // Reynolds (water at 20 °C: ρ=998, μ=0.001)
Re = 998 × 1.97 × 0.0254 / 0.001 = 50,200 → turbulent - // roughness (drawn tubing)
ε/D = 0.0015 / 25.4 = 5.91 × 10⁻⁵ - // friction factor (Swamee-Jain)
f = 0.25 / log₁₀(5.91e-5/3.7 + 5.74/50200^0.9)² = 0.0205 - // straight-pipe head
hf = 0.0205 × (100/0.0254) × (1.97²)/(2×9.81) = 16.0 m - // pressure drop
ΔP = ρgh = 998 × 9.81 × 16.0 = 157 kPa (22.8 psi) - // pump power (η = 75%)
PBHP = 157 kPa × (10/60) L/s / 0.75 = 349 W ≈ 0.47 HP
Example 2
Short run with local losses — 5 m steel, 3 elbows, one gate valve
Schedule 40 steel, DN25 (ID = 26.67 mm), commercial steel (ε = 0.045 mm). Same 10 L/min, water, same roughness.
- v = 1.80 m/s, Re = 48,100 (turbulent)
- ε/D = 0.045 / 26.67 = 1.69 × 10⁻³, f ≈ 0.0248
- hf,straight = 0.0248 × (5/0.02667) × 1.80²/19.62 = 0.61 m
- ΣK = 3 × 0.75 (90° elbows) + 0.19 (gate valve) + 0.5 (entrance) = 2.94
- hf,local = 2.94 × 1.80²/19.62 = 0.485 m
- htotal = 0.61 + 0.485 = 1.10 m
- ΔP = 998 × 9.81 × 1.10 = 10.8 kPa (1.57 psi)
- // local losses are 44% of total on this short run!
Example 3
Compressed air at 6 bar g — density correction
50 m of DN80 (ID = 80.7 mm) Sch40 steel, ε = 0.045 mm. Q = 1000 Nm³/h (standard, 0 °C, 1 atm). Operating at 25 °C, 6 bar g (Pabs = 700 kPa).
- // operating density (ideal gas)
ρ = (700000 Pa × 0.029 kg/mol) / (8.314 × 298 K) = 8.18 kg/m³ (vs 1.2 kg/m³ at standard) - // actual volumetric flow at operating conditions
Qactual = 1000 Nm³/h × (101.3/700) × (298/273) = 159 m³/h = 0.0442 m³/s - v = 0.0442 / (π·0.0807²/4) = 8.64 m/s
- Re = ρvD/μ = 8.18 × 8.64 × 0.0807 / 1.8e-5 = 320,000 → fully turbulent
- ε/D = 0.045 / 80.7 = 5.58 × 10⁻⁴, f ≈ 0.0188
- ΔP = f·(L/D)·ρv²/2 = 0.0188 × (50/0.0807) × 8.18 × 8.64² / 2 = 35.8 kPa
- // If you had used standard air density (1.2), you would have got ΔP ≈ 5.2 kPa — six times too low. The operating density is non-negotiable.
Four Mistakes That Skew Every Pressure-Drop Calculation
PITFALL · Nominal vs true inside diameter
A "1-inch" label is not a measurement. Schedule 40 steel is 26.67 mm ID, Schedule 80 is 24.31 mm, Type L copper is 25.0 mm, PVC Sch 40 is 26.7 mm. Area scales with D², velocity with 1/D², pressure drop with 1/D⁴ — a 5% ID error becomes a 25–30% ΔP error. Always look up the schedule; the calculator's preset cards fill in the real value.
PITFALL · Forgetting local losses on short runs
On a 10-metre run with three elbows and a globe valve, local losses can double the straight-pipe ΔP. Run Example 2 above — L/D = 187 and local losses are still 44%. When L/D is below 100, add up the K's every time; below 30, start designing for them before the pump quote.
PITFALL · Using laminar f = 64/Re in turbulent flow
The laminar friction factor f = 64/Re gives a perfectly accurate answer at Re = 1000. At Re = 50,000 (a typical 10 L/min through 25 mm pipe), laminar f = 0.00128 while the real f is about 0.0205 — 16× lower. Underestimating by 2–3× is common when a spreadsheet uses the laminar shortcut past the Re = 2300 cutoff. Always switch at Re > 2300.
PITFALL · Kinematic vs dynamic viscosity in Reynolds number
The Reynolds number is Re = ρvD/μ, where μ is dynamic viscosity (Pa·s). Kinematic viscosity ν (m²/s) is μ/ρ. If you accidentally use ν in place of μ, your Reynolds number is wrong by a factor of roughly ρ ≈ 1000 for water. The result is a friction factor that is 1000× off — impossible to spot unless you cross-check against the calculator's output.
Common Pressure-Drop Questions
What is the difference between Darcy-Weisbach and Hazen-Williams?▼
Darcy-Weisbach (ΔP = f·L/D·ρv²/2) is the universal pressure-drop equation — it works for every Newtonian fluid, every pipe material, and every velocity regime. Hazen-Williams is an empirical shortcut tuned specifically for cold water at moderate velocities inside pressure-rated pipe; it fails on hot water, gases, high-viscosity fluids, or extreme velocities. Darcy-Weisbach always reduces to Hazen-Williams if you pick the right friction factor, never the other way around.
Why is Colebrook-White implicit — and does that matter?▼
The equation 1/√f = −2·log₁₀(ε/(3.7·D) + 2.51/(Re·√f)) puts f on both sides, so you cannot solve it in one shot. A Newton-Raphson loop converges in 2–3 iterations on any phone; that is what calculators do. Haaland and Swamee-Jain are explicit approximations good to 2–3% that skip iteration entirely — plenty of accuracy for engineering estimates, and what the live widget above uses.
How much do local losses (fittings, valves) change the answer?▼
On short pipe runs (less than 10 diameters) local losses can be 40–60% of the total head. On a 10-metre copper branch with a gate valve, two 90° elbows, and an entrance, the K-total is around 2.5 — which contributes roughly the same head as the straight pipe itself. On long runs (>100 diameters) local losses are usually less than 5% and can be rolled into the straight-pipe answer with a small margin.
Can I use the Darcy-Weisbach equation for gases?▼
Yes — but use the *local absolute pressure* to compute density ρ, not standard density. Air at 6 bar g is roughly seven times denser than air at atmospheric pressure, so ΔP scales linearly with that correction. Also, gas expands as it loses pressure, so velocity changes along the pipe; for long lines (>100 pipe diameters) break the line into segments and solve each one with the local density.
What inside diameter should I use — nominal or measured?▼
Always use the true inside diameter. A '1-inch' pipe (DN25) labels an outside dimension that drifted from reality a century ago: Schedule 40 steel is 26.67 mm ID, Schedule 80 is 24.31 mm, Type L copper is 25.0 mm, PVC Sch 40 is 26.7 mm. Diameter enters to the fifth power in a pressure-drop sensitivity, so a 5% ID error is a 25–30% ΔP error.
How do I convert pressure drop to pump power?▼
Hydraulic power P_hyd = ΔP·Q — with ΔP in Pa and Q in m³/s, the result is in watts. Brake power at the motor shaft is P_BHP = P_hyd / η where η is the pump + motor efficiency (60–75% for small residential, 75–85% for commercial, 80–90% for large industrial). A handy shortcut for water: P(W) ≈ ΔP(kPa) × Q(L/min) / (60·η).
Run Your Own Pressure-Drop Problem
Open the full calculator on the home page, pick your pipe size preset, set your material, and watch the ΔP, Re, and pump-power outputs update in real time — the same math this article derived.