Flow Through Pipes
Darcy-Weisbach Equation, Major & Minor Losses, Moody Chart — Complete Guide with Solved Problems
Last Updated: March 2026
📌 Key Takeaways
- Darcy-Weisbach equation: hf = f(L/D)(V²/2g) — the fundamental equation for friction head loss in pipes.
- Major losses: Friction along pipe length (Darcy-Weisbach). Minor losses: Fittings, bends, valves (hm = KV²/2g).
- Friction factor f: Laminar → f = 64/Re. Turbulent → Moody chart or Colebrook equation.
- Turbulent friction depends on both Re and relative roughness (ε/D).
- Pipes in series: Same flow rate, head losses add. Pipes in parallel: Same head loss, flow rates add.
- Pipe flow problems are the most frequently tested topic in GATE FM (2–4 marks typically).
1. Darcy-Weisbach Equation
Darcy-Weisbach Equation
hf = f × (L/D) × (V²/2g)
Where:
- hf = head loss due to friction (m of fluid)
- f = Darcy (Moody) friction factor (dimensionless)
- L = pipe length (m)
- D = pipe internal diameter (m)
- V = average flow velocity (m/s)
- g = 9.81 m/s²
The Darcy-Weisbach equation is universal — it works for both laminar and turbulent flow, for any fluid, and for any pipe material. The only variable that changes between flow regimes is how the friction factor f is determined.
In terms of pressure drop: ΔP = f(L/D)(ρV²/2)
In terms of flow rate: since V = 4Q/(πD²), the equation can be rewritten as: hf = f × 8LQ² / (gπ²D⁵)
2. Finding the Friction Factor
| Flow Regime | Friction Factor | Depends On |
|---|---|---|
| Laminar (Re < 2,300) | f = 64/Re | Re only (roughness has no effect) |
| Turbulent, smooth pipe | Blasius: f = 0.316/Re0.25 (4000 < Re < 10⁵) | Re only |
| Turbulent, rough pipe | Colebrook equation (implicit) | Re and ε/D |
| Fully rough turbulent | f depends on ε/D only (Re-independent) | ε/D only |
Colebrook Equation (Turbulent Flow)
1/√f = −2 log₁₀(ε/(3.7D) + 2.51/(Re√f))
This is implicit in f (f appears on both sides) — solve by iteration or use the Moody chart.
The Moody Chart
The Moody chart (Moody diagram) plots the Darcy friction factor f on the y-axis against Reynolds number Re on the x-axis, with separate curves for different values of relative roughness ε/D. It is the standard graphical tool for finding f in turbulent pipe flow. To use it: calculate Re and ε/D, then read f from the chart.
| Pipe Material | Roughness ε (mm) |
|---|---|
| Drawn tubing (brass, copper, glass) | 0.0015 |
| Commercial steel / Wrought iron | 0.046 |
| Galvanised iron | 0.15 |
| Cast iron | 0.26 |
| Concrete | 0.3–3.0 |
| Riveted steel | 0.9–9.0 |
| PVC / Plastic | 0.0015 |
3. Major Losses — Pipe Friction
Major losses occur along the entire length of straight pipe due to friction between the fluid and the pipe wall. They are calculated using the Darcy-Weisbach equation.
Key relationships for major losses:
- hf ∝ L — head loss is proportional to pipe length.
- hf ∝ V² (turbulent) or hf ∝ V (laminar) — velocity has a strong effect.
- hf ∝ 1/D (turbulent) or hf ∝ 1/D⁴ (laminar, for given Q) — larger pipes have much lower friction.
- hf increases with surface roughness in turbulent flow.
4. Minor Losses — Fittings & Valves
Minor losses occur at any point where the flow is disturbed: bends, valves, expansions, contractions, tee junctions, and entry/exit points.
Minor Loss Formula
hm = K × V²/(2g)
Where K = loss coefficient (dimensionless), V = velocity at the relevant section
| Fitting | K Value |
|---|---|
| Pipe entrance (sharp-edged) | 0.5 |
| Pipe entrance (well-rounded) | 0.04 |
| Pipe exit (to reservoir) | 1.0 |
| 90° elbow (standard) | 0.9 |
| 90° elbow (long radius) | 0.6 |
| 45° elbow | 0.4 |
| Gate valve (fully open) | 0.2 |
| Globe valve (fully open) | 10.0 |
| Check valve | 2.5 |
| Tee (through run) | 0.6 |
| Tee (through branch) | 1.8 |
Sudden Expansion
hexpansion = (V₁ − V₂)² / (2g)
Or: K = (1 − A₁/A₂)²
Sudden Contraction
hcontraction = K × V₂²/(2g)
K ≈ 0.5(1 − A₂/A₁) for sharp contraction
Equivalent length method: Minor losses can also be expressed as an equivalent length of pipe: Leq = KD/f. This allows adding minor losses to major losses using a single Darcy-Weisbach calculation with Ltotal = Lpipe + ΣLeq.
5. Total Head Loss — Modified Bernoulli
Total Head Loss
hL = hf + Σhm = f(L/D)(V²/2g) + ΣK(V²/2g)
This total loss is used in the modified Bernoulli equation:
P₁/(ρg) + V₁²/(2g) + z₁ = P₂/(ρg) + V₂²/(2g) + z₂ + hL
6. Pipes in Series & Parallel
| Configuration | Flow Rate | Head Loss |
|---|---|---|
| Series (one after another) | Q₁ = Q₂ = Q₃ (same flow through all) | hL,total = hL1 + hL2 + hL3 (losses add) |
| Parallel (side by side) | Qtotal = Q₁ + Q₂ + Q₃ (flows add) | hL1 = hL2 = hL3 (same loss across all) |
7. Worked Numerical Examples
Example 1: Friction Head Loss — Turbulent Flow
Problem: Water (ρ = 1000 kg/m³, ν = 10⁻⁶ m²/s) flows at 2 m/s through a 100 mm diameter, 200 m long commercial steel pipe (ε = 0.046 mm). Friction factor from Moody chart: f = 0.019. Find the head loss and pressure drop.
Solution
Verify: Re = VD/ν = 2 × 0.1 / 10⁻⁶ = 200,000 → Turbulent ✓
ε/D = 0.046/100 = 0.00046
hf = f(L/D)(V²/2g) = 0.019 × (200/0.1) × (2²/(2×9.81))
= 0.019 × 2000 × 0.2039
hf = 7.75 m of water
ΔP = ρghf = 1000 × 9.81 × 7.75 = 76,028 Pa ≈ 76 kPa
Example 2: System with Minor Losses
Problem: A pipe system has: 50 m of 150 mm pipe (f = 0.02), one sharp entrance (K = 0.5), two 90° elbows (K = 0.9 each), one gate valve (K = 0.2), and one exit (K = 1.0). Flow velocity is 3 m/s. Find total head loss.
Solution
Major loss: hf = 0.02 × (50/0.15) × (3²/19.62) = 0.02 × 333.3 × 0.4587 = 3.06 m
Minor losses: ΣK = 0.5 + 0.9 + 0.9 + 0.2 + 1.0 = 3.5
hm = 3.5 × 3²/(2 × 9.81) = 3.5 × 0.4587 = 1.61 m
hL,total = 3.06 + 1.61 = 4.67 m
Minor losses account for 34% of total losses — significant in this system.
8. Common Mistakes Students Make
- Confusing Darcy and Fanning friction factors: The Darcy (Moody) friction factor fD is 4 times the Fanning friction factor fF. Most engineering references and GATE use Darcy: hf = fD(L/D)(V²/2g). Using the wrong one gives answers off by a factor of 4.
- Using f = 64/Re for turbulent flow: This formula is only valid for laminar flow. Turbulent flow requires the Moody chart or Colebrook equation.
- Neglecting minor losses in short pipe systems: Despite the name "minor," these losses dominate in systems with many fittings and short pipe runs. Always check if minor losses are significant.
- Forgetting that hf ∝ Q² in turbulent flow: Doubling the flow rate quadruples the head loss (since hf ∝ V² and V ∝ Q). This has major implications for pump sizing.
- Using exit velocity for minor losses at contractions: For sudden contractions, use the velocity at the smaller (downstream) section. For expansions, the loss formula uses (V₁ − V₂)²/(2g).
9. Frequently Asked Questions
What is the Darcy-Weisbach equation?
The Darcy-Weisbach equation hf = f(L/D)(V²/2g) calculates the head loss due to friction in a pipe. It is universal — it works for any fluid, any pipe material, and both laminar and turbulent flow. The friction factor f is 64/Re for laminar flow and is found from the Moody chart or Colebrook equation for turbulent flow.
What is the difference between major and minor losses?
Major losses are the friction losses along the straight length of pipe. Minor losses are the additional losses at fittings, valves, bends, and other flow disturbances. Major losses use the Darcy-Weisbach equation; minor losses use hm = KV²/(2g) with a loss coefficient K specific to each fitting type.
How do you find the friction factor?
For laminar flow: f = 64/Re (exact). For turbulent flow: use the Moody chart — plot Re on the x-axis, find your relative roughness curve (ε/D), and read f from the y-axis. Alternatively, solve the Colebrook equation iteratively. For GATE, the friction factor is usually given in the problem or can be approximated from the Blasius equation (f = 0.316/Re0.25) for smooth pipes.