Choose a penstock diameter from the friction loss you can afford, using head loss in meters equal to 0.00165 × length × flow squared divided by diameter to the fifth power in SI units, and then rate the pipe for surge, not just static pressure. A steel pipe stopped suddenly from 2 m/s gains about 25 bar, five times the 4.9 bar static pressure of a 50 m head, while closing the turbine gate over 5 seconds on a 100 m pipe adds only about 0.8 bar.
Part of Micro-Hydro Turbine
The penstock is the pressurized pipe from the intake basin to the turbine nozzle; its diameter sets the friction loss, and its wall, joints and closing time set whether it survives a shutdown.
Two Different Failures Set Two Different Limits
Friction wastes head every second the plant runs. Surge threatens the pipe once, on a bad day. They are designed separately, in that order, because the pipe diameter that gives acceptable friction also fixes the water speed that drives the surge. The existing Penstock Design page covers routing and fittings; this page derives the numbers.
Static pressure is 1 bar per 10.2 m of head, so 50 m gives 4.905 bar. That is the easy part.
Friction: The Law That Decides The Diameter
Darcy-Weisbach, as that page uses it, is h_f = f × (L ÷ D) × (V² ÷ 2g), with f ≈ 0.02 for smooth pipe. Substitute V = 4Q ÷ (πD²) with Q in m³/s and D in meters:
h_f = 8 × f × L × Q² ÷ (π² × g × D⁵) = 0.00165 × L × Q² ÷ D⁵ (for f = 0.02, g = 9.81)
Two consequences follow from the fifth power. A 15% larger diameter cuts the loss to 1 ÷ 1.15⁵ = 0.50, half. And for a fixed loss budget, the diameter grows only as flow to the power 0.4: doubling the flow needs a pipe 2^0.4 = 1.32 times as wide.
Worked example: the carrier’s 5 m, 30 L/s, 200 m site
| Diameter | Speed | Loss over 200 m | Share of 5 m head |
|---|---|---|---|
| 150 mm | 1.70 m/s | 3.92 m | 78% |
| 200 mm | 0.95 m/s | 0.93 m | 19% |
| 250 mm | 0.61 m/s | 0.31 m | 6.1% |
| 300 mm | 0.42 m/s | 0.12 m | 2.4% |
For the 150 mm row: V = 0.030 ÷ (π × 0.075²) = 1.70 m/s; h_f = 0.02 × (200 ÷ 0.15) × (1.70² ÷ 19.62) = 0.02 × 1,333 × 0.147 = 3.92 m. The carrier’s rule that penstock speed should stay under 2-3 m/s says nothing against 150 mm here, yet that pipe loses 78% of the head. On low-head sites the velocity rule is not enough: use the loss budget. The pipe needs to be about 250 mm to keep losses near 5%.
The carrier’s table (75, 63, 50 and 40 mm for 5 L/s at 5, 10, 20 and 50 m head) states no pipe length, so it cannot be checked as written. A 75 mm pipe at 5 L/s runs at 1.13 m/s, and its loss is 5% of 5 m (0.25 m) only over about 14 m of pipe. The 40 mm row runs at 3.98 m/s, above the carrier’s own 3 m/s limit. Derive the diameter from your length.
Surge: Why The Closing Time Matters More Than The Wall
When flow stops faster than the pressure wave can travel the pipe and back, the pressure rise is Joukowsky’s: ΔP = ρ × a × ΔV, where a is the wave speed and ΔV the change in water speed. The Penstock Design page writes this with an extra factor of one half, but its own worked example (20 bar for a steel pipe at 2 m/s) uses the full relation without it, and so does this page.
The wave speed depends on how stiff the pipe is: a = √(K ÷ ρ) ÷ √(1 + K × D ÷ (E × e)), where K is the bulk modulus of water (about 2.2 GPa, recalled), E the pipe material’s modulus and e its wall thickness.
- Steel, 150 mm bore, 4 mm wall, E ≈ 200 GPa (recalled): a = 1,483 ÷ √(1 + 2.2×10⁹ × 0.15 ÷ (200×10⁹ × 0.004)) = 1,483 ÷ 1.19 = 1,248 m/s. ΔP = 1,000 × 1,248 × 2 = 2.50 MPa = 25.0 bar.
- HDPE, 110 mm outside diameter, 10 mm wall (DR11, outside diameter divided by wall), E ≈ 0.8 GPa (recalled; it falls with load time and temperature): a = 1,483 ÷ √(1 + 2.2×10⁹ × 0.11 ÷ (0.8×10⁹ × 0.010)) = 265 m/s. ΔP = 1,000 × 265 × 2 = 5.3 bar.
The Penstock Design page quotes 900-1,200 m/s for steel and 300-500 m/s for HDPE. My two values sit just outside those ranges, so treat the surge as uncertain by about ±20%. The robust conclusion is that a thin steel pipe takes roughly five times the surge of HDPE for the same flow change, and about five times its own static pressure.
What closure time buys
A closure slower than 2L ÷ a does not raise the full Joukowsky pressure. For L = 100 m, 2L ÷ a is 0.16 s for steel and 0.75 s for HDPE. For a closure time T longer than that, Michaud’s approximation (recalled, valid for a steady closing rate) is ΔP = 2 × ρ × L × ΔV ÷ T:
- T = 5 s: 2 × 1,000 × 100 × 2 ÷ 5 = 80,000 Pa = 0.8 bar.
- T = 10 s: 0.4 bar.
The carrier’s guidance to close over 5 to 10 seconds therefore matters more than any increase in wall thickness. A gate worked by a screw, which cannot be slammed, enforces it; a hand-quick valve does not. The last part of a gate’s travel changes the flow most, so close slowly at the end.
The design rule on the Penstock Design page, rate the pipe for 1.5 times static head, passes this check only with enforced slow closure: 1.5 × 4.9 = 7.4 bar against 4.9 + 0.8 = 5.7 bar. With a fast closure it fails by a factor of four.
Materials, Walls And Thrust
Steel. For a 200 mm pipe with a 3 mm wall at 10 bar, hoop stress is p × D ÷ (2e) = 1.0×10⁶ × 0.2 ÷ (2 × 0.003) = 33 MPa. Mild steel yields near 250 MPa (the figure used on the Generator Coupling page), a factor of 7.5. At a 25 bar surge the stress is 83 MPa, a factor of 3. Corrosion that thins the wall to 2 mm lifts these to 50 and 125 MPa, which is why external coating and annual inspection matter.
HDPE. The Penstock Design page rates DR11 at 12 to 17 bar by grade and recommends it for small hydro. A full stop at 100 m of head would give 4.9 + 5.3 = 10.2 bar. HDPE moves with temperature; HDPE Pipe gives about 0.2 mm per meter per °C, or 0.4 m over 100 m for a 20 °C swing.
PVC is limited to heads under 15 m on that page and fails brittle in the cold (PVC Pipe). Wrapped iron pipe is rated 3-5 bar at 25 mm (Metal Pipes), about 31-51 m of static head before any surge margin. A hollowed log carries low pressure only (carrier).
Thrust at bends. The unbalanced force at an elbow is 2 × p × A × sin(θ ÷ 2). For a 150 mm pipe (A = 0.01767 m²) at 5 bar, p × A = 8,836 N and a 90° bend gives 8,836 × 1.414 = 12,496 N, the weight of about 1,270 kg. At the 25 bar surge it is 62,478 N, about 6,370 kg. Size the thrust blocks for the surge pressure.
Air. Fit an air-release valve at every high point, as the Penstock Design page says. Draining a thin pipe without an air inlet can let outside air pressure collapse it, so open the high-point valves before draining.
Verify Before Filling
Fill the pipe slowly with the high-point valves open, as the Installation page requires. Hold the line at 1.5 times design static pressure with the turbine gate closed, as the Metal Pipes page does for new pipe, and watch the gauge fall: the 1 bar per 10.2 m relation tells you what it should read. A falling reading with everything shut means a leak.
Common Mistakes
- Designing from the velocity rule alone — on a 5 m site it leaves most of the head in the pipe.
- A quick-closing valve in a long steel line — it sees the full surge.
- Thrust blocks sized for static pressure — surge is five times larger.
Safety
A burst penstock throws water and pipe
At 25 bar a failed joint or wall sends a jet and fragments along the pipe and downhill. Keep people out of the line of the pipe during first fills and shutdowns. Close the turbine gate slowly; never use the intake gate to stop a full penstock suddenly. The slow gate is the stop. Pressure-test with water, never air.
What’s Next
- Learn next: Penstock Leak Repair for what to do when a joint weeps
- Combine with: Powerhouse Location, since the pipe’s end fixes the turbine’s position