Hydraulic Turbines Notes

Last reviewed 16 Sept 2026 · 8 min read

Hydroelectric power plant

Water from a reservoir (behind a dam) flows through penstocks to turbines in the power house, then leaves through the tailrace. The turbine shaft drives a generator. A surge tank protects long conduits against water hammer (see Water Hammer & Surge Tanks).

Gross head $H_{g}$ — difference between headrace (reservoir) and tailrace levels.
Net (effective) head $H = H_{g} - h_{f}$ (friction loss in the penstock); for a Pelton wheel $H$ is measured at the nozzle base.

Efficiencies

Efficiency	Definition
Hydraulic $η_{h}$	Power delivered to the runner ÷ water power supplied $= \frac{runner power}{ρ g Q H}$
Mechanical $η_{m}$	Shaft power ÷ runner power
Volumetric $η_{v}$	Water actually striking the runner ÷ water supplied
Overall $η_{o}$	Shaft power ÷ water power $= η_{h} \times η_{m}$ (× $η_{v}$ if considered)

Water power $= ρ g Q H$ (W); shaft power $P = η_{o} ρ g Q H$ .

Classification of turbines

Basis	Types
Energy at inlet	Impulse — all available head converted to kinetic energy in a nozzle; runner at atmospheric pressure (Pelton). Reaction — water enters with both pressure and kinetic energy; runner is enclosed and runs full (Francis, Kaplan)
Direction of flow through runner	Tangential (Pelton), radial (old inward-flow Francis), mixed (modern Francis), axial (Kaplan, propeller)
Head (typical, overlapping)	High head (about 250 m and above) — Pelton; medium head (about 60–250 m) — Francis; low head (below about 60 m) — Kaplan/propeller
Specific speed	Low — Pelton; medium — Francis; high — Kaplan

Pelton wheel (impulse turbine)

Main parts: nozzle with spear (flow regulation), runner with double-hemispherical (split) buckets — the central splitter divides the jet — casing (prevents splashing; no hydraulic function), deflector (breaks the jet on sudden load rejection), braking jet.

FormulaPelton wheel relations

Jet velocity: $V_{1} = C_{v} 2 g H$ , $C_{v}$ ≈ 0.97–0.99
Bucket velocity: $u = ϕ 2 g H$ with speed ratio $ϕ$ ≈ 0.43–0.47; $u = \frac{π D N}{60}$
Work done per second (bucket friction factor $k$ , outlet blade angle $β$ , i.e. deflection $180° - β$ ):

P_{r u nn er} = ρ a V_{1} (V_{1} - u) (1 + k cos β) u

Hydraulic efficiency (based on jet kinetic energy $V_{1}^{2} /2 g$ ; multiply by $C_{v}^{2}$ to refer it to the net head):

η_{h} = \frac{2 ( V _{1} - u ) ( 1 + k cos β ) u}{V _{1}^{2}}

Maximum hydraulic efficiency when $u = V_{1} /2$ : $η_{h, ma x} = \frac{1 + k cos β}{2}$ (→ 100% if $β$ = 0 and $k$ = 1)
Jet ratio $m = D / d$ (wheel pitch-circle diameter ÷ jet diameter), usually about 11–14 (not less than about 6)
Number of buckets: $Z = 15 + \frac{m}{2}$ (empirical)

Buckets deflect the jet through about 160–165° (not 180°, so the leaving water does not strike the next bucket).

Francis turbine (inward mixed-flow reaction turbine)

Main parts: spiral (scroll) casing — cross-section decreasing to keep velocity uniform; stay vanes; guide vanes (wicket gates) — adjustable, regulate flow and set the inlet angle; runner with fixed curved vanes; draft tube.

FormulaFrancis turbine relations

Work done per second per unit weight: $\frac{V _{w 1} u _{1} \pm V _{w 2} u _{2}}{g}$
For radial (or axial) discharge at outlet ( $V_{w 2}$ = 0, best design): $η_{h} = \frac{V _{w 1} u _{1}}{g H}$
Discharge: $Q = π D_{1} B_{1} V_{f 1} = π D_{2} B_{2} V_{f 2}$ (vane thickness neglected)
Typical ratios: speed ratio $u_{1} / 2 g H$ ≈ 0.6–0.9; flow ratio $V_{f 1} / 2 g H$ ≈ 0.15–0.30; $B_{1} / D_{1}$ ≈ 0.1–0.45

Kaplan and propeller turbines (axial-flow reaction)

Water flows parallel to the shaft; the runner has few (usually 3–8) blades on a large hub (boss).
Propeller turbine — fixed blades; efficiency drops sharply at part load.
Kaplan turbine — adjustable runner blades (and adjustable guide vanes: "double regulated") — high efficiency over a wide load range.
Flow area: $Q = \frac{π}{4} (D_{o}^{2} - D_{b}^{2}) V_{f}$ ( $D_{o}$ outer, $D_{b}$ hub diameter); peripheral velocity at inlet and outlet is equal at any radius.
Suited to low heads and large discharges (river-bed plants, barrages).

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Hydraulic Turbines