Pipe Friction (Darcy–Weisbach)

The Darcy–Weisbach equation

When a real (viscous) fluid flows through a pipe, friction at the wall continuously converts mechanical energy into heat. The Darcy–Weisbach equation quantifies that loss as a "head" — an equivalent height of fluid column:

h_f = f · (L / D) · (v² / 2g)

where:
  h_f = head loss to friction   (m of fluid)
  f   = Darcy friction factor    (dimensionless)
  L   = pipe length              (m)
  D   = internal diameter        (m)
  v   = mean flow velocity       (m/s)
  g   = gravitational accel.     (9.81 m/s²)

Multiply through by ρg to get the pressure form, which is what a gauge actually reads:

Δp = f · (L / D) · (ρ v² / 2)

  ρ = fluid density (kg/m³)

The structure is worth memorizing. Loss is linear in length (twice the run, twice the loss), inverse in diameter (a fatter pipe loses less, and the v² term punishes small bores even harder because velocity climbs as 1/D² for a fixed flow rate), and quadratic in velocity in the turbulent regime. Everything that is messy and fluid-specific — viscosity, turbulence, wall texture — is bundled into the one dimensionless number f.

The friction factor and the Moody chart

The friction factor is not a constant; it is a function of two dimensionless groups: the Reynolds number Re = ρvD/μ and the relative roughness ε/D. The Moody chart is the canonical log–log plot of f against Re for a family of ε/D curves. Three regions matter:

• Laminar (Re < 2300): f = 64/Re exactly. Roughness does not appear — a smooth viscous core slides past the wall and the texture is buried inside the boundary layer. Friction falls as flow speeds up.
• Transitional (2300 < Re < ~4000): flow flickers between laminar and turbulent; f is poorly defined and design here is avoided.
• Turbulent (Re > ~4000): f depends on both Re and ε/D. At low Re the pipe behaves "hydraulically smooth"; at high Re each ε/D curve flattens into a constant — the fully rough regime, where f no longer depends on Re at all.

For turbulent flow the implicit Colebrook–White equation is the reference standard:

1/√f = −2 · log₁₀[ (ε / 3.7D) + 2.51 / (Re·√f) ]   (Colebrook–White)

Because f appears on both sides, it must be solved iteratively. For hand work the explicit Haaland approximation is within ~2% and needs no iteration:

1/√f = −1.8 · log₁₀[ (ε / 3.7D)^1.11 + 6.9 / Re ]   (Haaland)

Flow regimes and friction laws compared

	Laminar	Transitional	Turbulent — smooth	Turbulent — fully rough
Reynolds number Re	< 2300	2300–4000	4000 up to onset of roughness	high Re, large ε/D
Friction factor f	64/Re (exact)	ill-defined	Blasius: 0.316·Re⁻⁰·²⁵	constant, set by ε/D only
Roughness matters?	No	—	Weakly	Yes — dominant
h_f vs velocity	∝ v (linear)	—	∝ v^1.75	∝ v² (quadratic)
Velocity profile	Parabolic; v_max = 2·v_avg	Intermittent	Flat core, thin sublayer	Flat core, thick logarithmic layer
Typical example	Lubricant in a thin bore, blood in capillaries	Avoided in design	Drawn copper, new PVC at moderate flow	Old cast iron, concrete culvert at high flow

Worked example: head loss in a water main

Water at 20 °C flows at 2 m/s through a 100 m run of commercial steel pipe, internal diameter 100 mm. What is the friction head loss?

Fluid (water, 20 °C):  ρ = 998 kg/m³,  μ = 1.0 × 10⁻³ Pa·s
Pipe:  D = 0.10 m,  L = 100 m,  ε = 0.045 mm

Reynolds number:
  Re = ρ v D / μ
     = 998 × 2 × 0.10 / (1.0 × 10⁻³)
     = 1.996 × 10⁵        → turbulent

Relative roughness:
  ε/D = 0.045 / 100 = 4.5 × 10⁻⁴

Friction factor (Haaland):
  1/√f = −1.8 · log₁₀[ (4.5e-4 / 3.7)^1.11 + 6.9 / 1.996e5 ]
       ≈ 7.38
  f ≈ 0.0184

Head loss:
  h_f = f · (L/D) · (v²/2g)
      = 0.0184 × (100 / 0.10) × (2² / (2 × 9.81))
      = 0.0184 × 1000 × 0.2039
      ≈ 3.75 m of water

Roughly 3.75 m of head — about 37 kPa — is burned to friction over that 100 m. A pump must supply at least this much head on top of any static lift just to keep the water moving. Note how dominant the L/D ratio (1000) is: friction head scales directly with how many diameters of pipe the fluid must traverse.

Worked example: the payoff of one size up

Suppose the same 2 m/s flow rate (Q = v·A = 0.0157 m³/s) is pushed instead through a 125 mm pipe. Velocity drops because area rises:

New velocity:
  v = Q / A = 0.0157 / (π/4 × 0.125²) = 1.28 m/s

  Re = 998 × 1.28 × 0.125 / 1e-3 = 1.60 × 10⁵
  ε/D = 0.045/125 = 3.6 × 10⁻⁴
  f ≈ 0.0183 (Haaland)

  h_f = 0.0183 × (100 / 0.125) × (1.28² / (2 × 9.81))
      ≈ 0.0183 × 800 × 0.0835
      ≈ 1.22 m of water

Going up just one nominal size cut the friction loss from ~3.75 m to ~1.22 m — a 67% reduction — for the same delivered flow. Because head loss scales as roughly 1/D⁵ at fixed flow rate (v ∝ 1/D², and v² ∝ 1/D⁴, times the explicit 1/D), modestly bigger pipe pays back enormous pumping-energy savings over a system's life. This trade-off (capital cost of larger pipe versus lifetime pumping energy) is the heart of economic pipe sizing.

Equivalent roughness of common materials

Material	ε (mm)	Notes
Drawn tubing (copper, glass, brass)	0.0015	Hydraulically smooth at most flows
New PVC / PE plastic	0.0015–0.007	Very low; degrades little with age
Commercial / welded steel	0.045	The default "ε = 0.045 mm" of textbooks
Galvanized iron	0.15	Zinc coating adds texture
Cast iron (new)	0.26	Rises sharply as it tuberculates with age
Riveted / old corroded steel	0.9–9	Wide band; field-measured, not catalog
Concrete	0.3–3	Depends on finish and casting quality

Minor losses at fittings

The Darcy–Weisbach term only counts straight-pipe wall friction. Real systems lose additional head wherever the flow is forced to turn, contract, expand, or pass an obstruction — these are minor losses, even though in a fitting-dense plant they can exceed the straight-pipe loss. Each is written as a multiple of the velocity head:

h_m = K · (v² / 2g)

Typical loss coefficients K:
  Sharp pipe entrance      0.5
  90° standard elbow       0.3 – 0.9
  Tee (through flow)       0.2
  Gate valve (fully open)  0.15
  Globe valve (fully open) 10
  Sudden exit to tank      1.0

Total system head loss is the sum of the pipe-friction term and all the minor-loss terms:

h_loss = f·(L/D)·(v²/2g) + Σ K·(v²/2g)

An alternative bookkeeping is the equivalent length method: each fitting is assigned a length of straight pipe (L_e/D) that would dissipate the same head, and these are simply added to L. A globe valve might be worth 340 diameters of pipe — which is why throttling flow with a globe valve is hydraulically expensive.

Darcy–Weisbach versus Hazen–Williams

Water-distribution engineers often reach for the empirical Hazen–Williams formula, V = 0.849·C·R^0.63·S^0.54, which packs all roughness into a single coefficient C (≈140 for new PVC, ≈100 for old cast iron). It is convenient because C does not depend on velocity or Reynolds number — but that convenience is also its weakness. Hazen–Williams is calibrated for water near 16 °C in the turbulent range and drifts out of accuracy for viscous fluids, hot water, high velocities, or small bores. Darcy–Weisbach has no such restriction: because f explicitly carries the Re and ε/D dependence, the same equation handles crude oil at 60 °C, compressed air, or liquid sodium with only a change of fluid properties. For anything other than ambient water, Darcy–Weisbach is the defensible choice.

Failure modes and design traps

• Designing in the transition zone. Between Re ≈ 2300 and 4000 the friction factor is unstable and unpredictable; flow rate can hunt and head loss is non-repeatable. Size pipes to sit clearly in laminar or clearly in turbulent flow.
• Ignoring aging. Cast-iron and steel pipes tuberculate and corrode, so ε can grow by an order of magnitude over decades. A main sized with new-pipe roughness can lose 30–50% of its capacity in service — always apply an aging allowance.
• Velocity too high. Pushing water past ~3 m/s drives the v² loss term up steeply, invites erosion of the pipe wall, amplifies water-hammer surge pressures, and raises noise. Most codes cap design velocity around 2–3 m/s for water.
• Velocity too low. Below ~0.6 m/s, sediment and entrained solids settle out and accumulate, slowly throttling the bore — a self-reinforcing failure in sewers and slurry lines.
• Forgetting minor losses. In compact skids and pump rooms, fittings can dominate; ignoring them under-sizes the pump and leaves the system unable to reach its design flow.
• Suction-side losses causing cavitation. Friction on a pump's suction line lowers the available NPSH; too much suction head loss drops the inlet pressure below the fluid's vapor pressure and the pump cavitates, eroding the impeller.