Hydraulic turbines

Mechanical Engineering (B.E./B.Tech) · Fluid mechanics and machinery · free study notes for exams, placements and interviews

In short

A hydraulic turbine converts the energy of water falling through a head into shaft rotation, which an alternator turns into electricity. The three workhorses are the Pelton wheel (high head, low flow), the Francis turbine (medium head) and the Kaplan turbine (low head, large flow), chosen almost entirely by head and specific speed. India has roughly 47 GW of large hydro plus about 5 GW of small hydro and a growing pumped-storage fleet (CEA publishes the current figures monthly), run by NHPC, SJVN, THDC, NTPC and the state hydel corporations, with turbines made by BHEL Bhopal, Andritz, Voith and Flovel. The topic is heavily examined in GATE (ME and CE), in SSC-JE/state-PSU papers, and in interviews for BHEL, NHPC, NTPC, L&T, Jyoti and Kirloskar — usually through velocity triangles, efficiency, specific speed and cavitation.

Explained

The plant, the heads and the efficiencies

Layout: reservoir → intake with trash rack → tunnel/headrace → surge tank (absorbs water-hammer pressure waves when load changes) → penstock (steel pipe on anchor blocks) → turbine inlet valve → turbine → draft tube (reaction turbines only) → tailrace.
Gross head Hg: difference between headwater and tailwater levels. Net (effective) head H = Hg − hf, where hf is friction and bend loss in tunnel and penstock. All turbine performance is quoted on net head.
Water power (hydraulic power available) = ρgQH; shaft power is always less.
Hydraulic efficiency ηh = runner power / water power (accounts for whirl, friction and shock losses in the water passages).
Volumetric efficiency ηv = flow doing work on the runner / flow supplied (leakage past seals and clearances).
Mechanical efficiency ηm = shaft power / runner power (bearing and seal friction).
Overall efficiency ηo = ηh × ηv × ηm = shaft power / ρgQH. Modern large machines reach 92-95% at best point.
Generator efficiency (96-98%) is separate; plant output = ρgQH × ηo × ηgen.

Classification

By action of water: impulse — all available head converted to kinetic energy in a nozzle, runner works at atmospheric pressure, no pressure drop across buckets (Pelton, Turgo, cross-flow). Reaction — water is under pressure throughout; pressure *and* velocity both fall across the runner, which runs full (Francis, Kaplan, propeller, bulb, Deriaz).
By direction of flow through the runner: tangential (Pelton), radial inward (old slow Francis), mixed flow (modern Francis: in radially, out axially), axial (Kaplan, propeller, bulb).
By head: high head above ~250 m (Pelton); medium head ~30-400 m (Francis); low head below ~60 m (Kaplan, bulb). Ranges overlap and manufacturers stretch them.
By specific speed (metric): low Ns 8-30 (Pelton), medium 60-300 (Francis), high 300-1000 (Kaplan). Values depend on the definition used, so always state the formula.
By shaft: horizontal (small units, 1-2 jet Pelton) or vertical (most large units; multi-jet Pelton and all large Francis/Kaplan).
Degree of reaction = pressure-energy drop in the runner ÷ total energy drop in the runner. Zero for impulse; roughly 0.4-0.6 for Francis, higher for Kaplan.

Impulse turbine: the Pelton wheel

Working: the penstock feeds a nozzle; a spear (needle) valve inside it varies the jet area. The free jet strikes double hemispherical buckets with a central splitter that divides the jet into two halves and turns each through about 160-170° (not 180°, or the returning water would strike the back of the following bucket).
Notch (cut-away) in the bucket lip lets the jet enter cleanly without being intercepted by the bucket tip.
Jet velocity V1 = Cv√(2gH), with coefficient of velocity Cv ≈ 0.98-0.99.
Speed (bucket) ratio φ = u/√(2gH) ≈ 0.43-0.48; theoretically maximum efficiency at u = V1/2.
Jet ratio m = D/d (wheel pitch diameter ÷ jet diameter) ≈ 11-14 for good efficiency; number of buckets Z ≈ 15 + m/2 (Taygun's rule).
Number of jets: up to 2 on a horizontal shaft, up to 6 on a vertical shaft. More jets at the same head and speed means more power from the same wheel and raises Ns.
Braking jet / deflector: on load rejection the spear cannot close fast (water hammer), so a deflector plate flicks the jet off the buckets instantly and the spear closes slowly behind it; a small braking jet on the back of the buckets stops the wheel.
Casing is a cover only — it prevents splashing and guides water to the tailrace; it carries no pressure.
Advantages: simple, no cavitation risk, tolerates silty water better, efficiency curve is flat at part load (good for varying flow), easy maintenance. Limits: only for high head; low specific speed means a large slow wheel for a given power; the head between the wheel and tailwater is wasted.
Turgo: impulse runner with the jet at about 20° to the runner plane, so water enters one side and leaves the other — handles more flow than a Pelton of the same diameter, used at 50-250 m. Cross-flow (Banki-Michell): water crosses a drum-shaped runner twice; cheap, used in micro-hydro (below a few hundred kW).

Reaction turbines: Francis, Kaplan and variants

Francis components (water path): spiral (scroll) casing of decreasing cross-section (so velocity stays nearly constant and flow is distributed uniformly) → stay ring/stay vanes (take the structural load) → adjustable guide vanes (wicket gates) operated by a regulating ring, which set the whirl Vw1 and control the discharge → runner (fixed, curved, mixed-flow blades) → draft tube.
Francis design values: flow ratio ψ = Vf1/√(2gH) ≈ 0.15-0.30; speed ratio φ = u1/√(2gH) ≈ 0.6-0.9; guide-vane angle α1 ≈ 10-35°; runner outlet is designed for no whirl (Vw2 ≈ 0) at the best operating point, so all whirl energy is used.
Francis is single-regulated — only the guide vanes move, so efficiency falls steeply below about 50% gate.
Kaplan: an axial propeller with 3-8 adjustable blades on a large hub, fed through a spiral casing and guide vanes. The blade pitch is linked to the gate opening by a cam so that both change together — double regulation — which keeps efficiency high from about 30% to 100% load.
Kaplan design values: flow ratio ψ ≈ 0.35-0.75; speed ratio φ ≈ 1.4-2.0 (much higher than Francis); boss/runner diameter ratio ≈ 0.3-0.5.
Propeller turbine: same geometry with fixed blades — cheaper, but efficiency collapses off design point; used where head and flow are nearly constant.
Bulb, tubular and S-type turbines: axial machines with the generator in a submerged bulb or outside via an S-shaped draft tube; nearly straight flow path, used for very low heads (2-15 m) on barrages, canal drops and tidal schemes.
Deriaz (diagonal) turbine: adjustable blades set diagonally, bridging Francis and Kaplan at 30-100 m; also used as a reversible pump-turbine.
Reversible pump-turbine: a Francis-type runner that generates when flowing inward and pumps when reversed — the heart of pumped storage, which India is building aggressively for solar balancing.
Advantages of reaction machines: can use low heads and very large flows; the draft tube recovers the exit kinetic energy and the head below the runner. Limits: cavitation, silt erosion of guide vanes and runner (a serious Himalayan problem), heavier and costlier civil works.

Draft tube, cavitation and runner setting

Functions of the draft tube: (1) it recovers most of the kinetic energy leaving the runner by gradual divergence, converting it to pressure, and (2) by running full and sealing the runner to the tailwater, it lets the runner be set *above* the tailrace without losing that head — pressure at the runner exit can be below atmospheric.
Types: straight conical divergent (best efficiency, needs depth, used in small vertical units); simple elbow; elbow with varying cross-section (circular at inlet to rectangular at outlet — standard for large Kaplan and Francis, keeps excavation shallow); Moody spreading (hydracone) draft tube with a central cone to reduce whirl. The divergence is kept gentle — included cone angle about 8° or less (half-angle ~4°) — to avoid flow separation.
Draft tube efficiency ηd = (actual pressure head recovered)/(kinetic head at inlet).
Cavitation: where local pressure falls to the vapour pressure, vapour bubbles form and then collapse violently in the higher-pressure region, pitting the runner (usually the blade back near the outlet and the draft-tube cone), causing noise, vibration and efficiency loss. Worst in high-Ns machines (Kaplan) and at off-design gate openings.
Thoma's cavitation factor σ compares the available suction margin with the head; cavitation is avoided if σ > σc (critical value, which rises with specific speed). Setting: Hs is the height of the runner centreline (Kaplan: blade axis) above tailwater — positive if above, and often *negative*, meaning the runner must be submerged.
Prevention/control: correct setting from σc, stainless-steel (13Cr-4Ni) or clad runners, polished profiles, avoiding prolonged part-load running, admitting air into the draft tube, periodic weld repair. For silt-laden Himalayan rivers, de-silting chambers plus hard coatings are used.

Specific speed, unit quantities, selection and performance curves

Specific speed Ns is the speed of a geometrically similar turbine producing unit power under unit head. It is the single number that fixes the *type* of runner — every similar machine has the same Ns regardless of size.
Unit quantities (unit speed, unit discharge, unit power) reduce a turbine's performance to a 1 m head so that behaviour at any other head can be predicted — the basis of model testing and of acceptance curves.
Selection sequence: net head and design flow → power → choose type from the head/Ns band → pick synchronous speed N = 120f/p (f = 50 Hz in India, p = number of poles) → compute Ns and check it lies in the band → size the runner → check σ and fix the setting → check runaway speed and the generator's flywheel effect.
Characteristic curves: main (constant-head) characteristics — plot unit power, unit discharge and efficiency against unit speed for several gate openings; operating characteristics — efficiency, discharge and gate against power at constant speed and head; Muschel (iso-efficiency, hill) curves — contours of equal efficiency on speed-power axes, used to define the operating envelope. Kaplan's curve is flat (double regulation), Pelton's fairly flat, Francis's peaky, propeller's very peaky.
Model testing follows IEC 60193 (model acceptance tests); field acceptance tests on the prototype follow IEC 60041. Prototype efficiency is higher than the model's — the difference is the scale (step-up) effect, computed by the formula in the code.

Governing, protection, codes and Indian practice

Why govern: the alternator frequency must stay at 50 Hz, so speed must be constant as load changes. The governor senses speed and moves a servomotor through a pilot valve using oil at pressure (an oil-pressure governor with its own sump, pump and accumulator); modern machines use digital PID/PLC governors.
What is moved: Pelton — spear valve plus deflector; Francis — guide vanes via the regulating ring; Kaplan — guide vanes *and* runner blades together (cam-linked).
Load rejection: the gates cannot close instantly or the penstock would suffer water hammer; hence the surge tank, relief valve or deflector, and a large rotating inertia (GD²) to limit speed rise. The design balances speed rise against pressure rise.
Runaway speed (gates open, load thrown off, governor failed) is roughly 1.8-1.9 × rated for Pelton, 2.0-2.2 × for Francis and 2.5-3.0 × for Kaplan/propeller; all rotating parts must survive it.
Indian standards and practice: IS 12800 (guidelines for selection of hydraulic turbines and preliminary dimensioning/layout of surface power houses — separate parts for small, medium and large plants) and IS 12837 (selection guidelines for medium and large power houses); CBIP manuals; MNRE/AHEC IIT Roorkee standards for small hydro. International: IEC 60193, IEC 60041, IEC 61116 and IEC 62006 (small hydro), IEC 60308 (governor/control-system testing). Exact clause numbers and editions are revised periodically — check the current BIS/IEC listing.
Regulatory frame: Electricity Act 2003 (CEA concurrence for hydro schemes above a notified cost threshold); hydro above 25 MW was classified as renewable in 2019 with a Hydro Purchase Obligation; small hydro is ≤ 25 MW and sits with MNRE (micro ≤ 100 kW, mini 101 kW-2 MW, small 2-25 MW). Thresholds and obligations have changed more than once, so verify before quoting.

Key points

Impulse turbine - pressure drop happens entirely in the nozzle; runner at atmospheric pressure; Pelton, Turgo, cross-flow.
Reaction turbine - runner always full and under pressure; pressure and velocity both drop across it; Francis, Kaplan, bulb.
Net head - gross head minus conveyance and penstock losses; all performance is referred to it.
Speed ratio φ - u/√(2gH): ≈0.46 for Pelton, 0.6-0.9 Francis, 1.4-2.0 Kaplan.
Flow ratio ψ - Vf/√(2gH): 0.15-0.30 Francis, 0.35-0.75 Kaplan.
Jet ratio m = D/d - 11-14 in a Pelton; buckets Z ≈ 15 + m/2.
Maximum Pelton efficiency - at u = V1/2, giving ηh,max = (1 + k cos β2)/2.
Euler's equation - work per unit weight = (Vw1u1 ± Vw2u2)/g; designing for Vw2 = 0 maximises work.
Guide vanes - control discharge and set inlet whirl; Francis is single-regulated, Kaplan double-regulated.
Spiral casing - area tapers so flow enters the runner uniformly all round at nearly constant velocity.
Draft tube - recovers exit kinetic energy and the height of the runner above tailwater; essential to reaction turbines, absent in Pelton.
Specific speed Ns = N√P/H^1.25 - fixes the runner type: ~8-30 Pelton, 60-300 Francis, 300-1000 Kaplan (metric units).
Unit quantities - Nu = N/√H, Qu = Q/√H, Pu = P/H^1.5; used to predict performance at a new head.
Thoma's σ - (Ha − Hv − Hs)/H; keep σ above σc to avoid cavitation; σc rises with Ns.
Runaway speed - highest no-load speed with gates open: ~1.8× (Pelton) to ~3× (Kaplan) rated.
Overall efficiency - ηh × ηv × ηm; 92-95% at best point for large modern units.
Surge tank and water hammer - limit the pressure rise when gates close; traded off against speed rise.
Pumped storage - reversible Francis pump-turbine; India's main grid-scale storage route for balancing solar.

Formulas and examples

Net head: H = Hg − hf, where Hg = gross head (m), hf = total losses (m)
Water power: Pw = ρgQH (W); ρ = 1000 kg/m³, g = 9.81 m/s², Q in m³/s, H in m
Overall efficiency: ηo = P_shaft /(ρgQH) = ηh × ηv × ηm
Runner (peripheral) speed: u = πDN/60 (m/s); D = diameter (m), N = rpm; N = 120f/p
Euler turbine equation: work/unit weight = (Vw1u1 − Vw2u2)/g (m); runner power = ρQ(Vw1u1 − Vw2u2)
Pelton jet: V1 = Cv√(2gH); Q = (π/4)d²V1 per jet; d = jet diameter (m)
Pelton force and power: F = ρQ(V1 − u)(1 + k cos β2); P = F·u; ηh = 2(V1 − u)(1 + k cos β2)u / V1²
Maximum Pelton efficiency: at u = V1/2, ηh,max = (1 + k cos β2)/2; k = relative-velocity friction factor (0.85-1), β2 = bucket outlet angle = 180° − jet deflection angle (15° for a 165° bucket)
Pelton geometry: speed ratio φ = u/√(2gH); jet ratio m = D/d; buckets Z ≈ 15 + m/2
Francis flow: Q = πD1B1Vf1 × (vane thickness factor); B1 = runner width at inlet (m)
Inlet blade angle: tan θ1 = Vf1/(Vw1 − u1); outlet (no whirl): tan θ2 = Vf2/u2
Kaplan flow: Q = (π/4)(Do² − Dh²)Vf; Do = runner tip dia, Dh = hub (boss) dia (m)
Specific speed (metric): Ns = N√P/H^1.25, with N in rpm, P in kW, H in m
Unit quantities: Nu = N/√H, Qu = Q/√H, Pu = P/H^1.5
Thoma's cavitation factor: σ = (Ha − Hv − Hs)/H; Ha = atmospheric head (≈10.33 m at sea level, falling ≈0.11 m per 100 m altitude), Hv = vapour-pressure head, Hs = runner height above tailwater (m, negative if submerged)
Draft-tube efficiency: ηd = [(V2² − V3²)/2g − hf]/(V2²/2g); V2, V3 = inlet and outlet velocities

Example 1 - Pelton wheel, full design check. Net head H = 400 m, Q = 0.8 m³/s, Cv = 0.98, φ = 0.46, N = 500 rpm, bucket deflects the jet through 165° (so β2 = 15°), k = 0.9.

√(2gH) = √(2 × 9.81 × 400) = √7848 = 88.59 m/s → V1 = 0.98 × 88.59 = 86.82 m/s; u = 0.46 × 88.59 = 40.75 m/s.
Wheel diameter D = 60u/(πN) = (60 × 40.75)/(π × 500) = 2445/1570.8 = 1.557 m.
Jet diameter: d = √[4Q/(πV1)] = √[3.2/272.7] = √0.011734 = 0.1083 m = 108 mm; jet ratio m = 1.557/0.1083 = 14.4; buckets Z ≈ 15 + 7.2 ≈ 22.
Force F = 1000 × 0.8 × (86.82 − 40.75) × (1 + 0.9 cos15°) = 800 × 46.07 × 1.8693 = 68.9 kN.
Runner power P = 68,895 × 40.75 = 2,807,000 W = 2,807 kW.
Water power = 1000 × 9.81 × 0.8 × 400 = 3,139 kW → ηh = 2807/3139 = 89.4%.
Ns = 500 × √2807/400^1.25 = 500 × 52.98/1788.9 = 14.8 — squarely in the single-jet Pelton band.

Example 2 - Francis runner from velocity triangles. H = 60 m, N = 600 rpm, Q = 2.5 m³/s, speed ratio 0.65, guide-vane angle α1 = 15°, ηh = 90%, no whirl at exit, ηm = 97%.

√(2gH) = √1177.2 = 34.31 m/s → u1 = 0.65 × 34.31 = 22.30 m/s; D1 = (60 × 22.30)/(π × 600) = 1338/1885 = 0.710 m.
With Vw2 = 0: ηh = Vw1u1/(gH) → Vw1 = (0.90 × 9.81 × 60)/22.30 = 529.74/22.30 = 23.75 m/s.
Vf1 = Vw1 tan 15° = 23.75 × 0.2679 = 6.36 m/s; flow ratio = 6.36/34.31 = 0.185 (normal for Francis).
Inlet blade angle: tan θ1 = 6.36/(23.75 − 22.30) = 6.36/1.45 = 4.39 → θ1 ≈ 77° (blade leans forward since Vw1 > u1).
Runner width: B1 = Q/(πD1Vf1) = 2.5/(π × 0.710 × 6.36) = 2.5/14.19 = 0.176 m.
Water power = 1000 × 9.81 × 2.5 × 60 = 1,471.5 kW; runner power = 0.90 × 1471.5 = 1,324 kW; shaft power = 0.97 × 1324 = 1,285 kW; ηo = 1285/1471.5 = 87.3%.
Ns = 600 × √1285/60^1.25 = 600 × 35.85/167.0 = 129 — a medium Francis.

Example 3 - Kaplan turbine sizing. Runner tip diameter 3.0 m, hub 1.2 m, H = 18 m, speed ratio 2.0, flow ratio 0.6, ηo = 88%.

√(2gH) = √353.16 = 18.79 m/s → u = 2.0 × 18.79 = 37.59 m/s; N = (60 × 37.59)/(π × 3.0) = 2255/9.425 = 239 rpm (nearest synchronous speed 230.8 rpm for a 26-pole machine, next step 250 rpm for 24 poles — the designer would then re-tune D or φ).
Vf = 0.6 × 18.79 = 11.28 m/s; Q = 0.7854 × (3.0² − 1.2²) × 11.28 = 0.7854 × 7.56 × 11.28 = 66.95 m³/s.
Water power = 1000 × 9.81 × 66.95 × 18 = 11.82 MW; shaft power = 0.88 × 11.82 = 10.40 MW.
Ns = 239 × √10,404/18^1.25 = 239 × 102.0/37.08 = 657 — high-Ns Kaplan, as expected.

Example 4 - Cavitation setting for that Kaplan. Site at 600 m altitude, water at 30 °C (vapour pressure 4.24 kPa).

Ha = 10.33 − (0.11 × 6) = 9.67 m; Hv = 4.24/9.81 = 0.43 m.
Using the standard empirical relation for Kaplan runners, σc = 0.28 + (1/7.5)(Ns/380.78)³ = 0.28 + (1/7.5)(657/380.78)³ = 0.28 + (1.726)³/7.5 = 0.28 + 5.14/7.5 = 0.965 (constants differ between texts and manufacturers use measured σ, so treat this as indicative).
Hs = Ha − Hv − σcH = 9.67 − 0.43 − (0.965 × 18) = 9.24 − 17.37 = −8.1 m, i.e. the runner axis must sit about 8 m below tailwater level — which is why low-head Kaplan power houses need deep excavation.

Example 5 - Unit quantities. A turbine gives 1,000 kW at 300 rpm with Q = 5 m³/s under H = 25 m. Find its behaviour at H = 16 m.

Nu = 300/√25 = 60; Qu = 5/5 = 1.0; Pu = 1000/25^1.5 = 1000/125 = 8.0.
At H = 16 (√16 = 4): N = 60 × 4 = 240 rpm; Q = 1.0 × 4 = 4 m³/s; P = 8 × 64 = 512 kW.
Check: ηo at 25 m = 1000/(9.81 × 5 × 25) = 81.5%; at 16 m = 512/(9.81 × 4 × 16) = 81.6% — efficiency is preserved, as similarity requires.

Feature	Pelton	Francis	Kaplan
Action	Impulse	Reaction (mixed flow)	Reaction (axial)
Head range	~250-1,800 m	~30-400 m	~2-60 m
Specific speed (metric)	8-30 (multi-jet to ~70)	60-300	300-1,000
Flow	Small	Medium	Very large
Speed ratio φ	≈0.46	0.6-0.9	1.4-2.0
Regulation	Spear + deflector	Guide vanes (single)	Vanes + blades (double)
Draft tube	None	Essential	Essential
Casing pressure	Atmospheric	Spiral, pressurised	Spiral, pressurised
Cavitation risk	Nil	Moderate	High
Part-load efficiency	Flat	Falls sharply	Flat
Indian example	Moolamattom, Idukki (6 × 130 MW)	Nathpa Jhakri (6 × 250 MW), Tehri	Barrage and canal-drop stations; bulb units on canals

Where it is used

Hydro generation and OEM engineering: BHEL Bhopal, Andritz Hydro (Faridabad/Prithla), Voith Hydro (Vadodara), Toshiba, Flovel and Jyoti design, model-test and supply runners, governors and inlet valves; roles include hydraulic design, CFD, mechanical design and site erection/commissioning.
Utilities and O&M: NHPC, SJVN, THDC, NEEPCO, NTPC and state boards (KSEB, MSEB/Mahagenco, KPCL, UJVNL, HPPCL) employ engineers for operation, runner-pitting repair, silt management, overhaul and renovation, modernisation and uprating (RMU) projects that re-blade old runners for more output.
Pumped storage: reversible Francis pump-turbines at Purulia (West Bengal), Kadamparai (Tamil Nadu), Ghatghar (Maharashtra), Srisailam left bank, the Sardar Sarovar riverbed power house and Tehri PSP — now a priority area with many private PSPs under development for solar balancing.
Small and micro hydro: cross-flow, Turgo and bulb/tubular units on canal falls, irrigation outlets and hill streams under MNRE schemes, with design standards from AHEC, IIT Roorkee; used for rural and industrial captive supply.
Industrial energy recovery: hydraulic turbines and reverse-running pumps (PATs) recovering energy from high-pressure let-down streams in water-supply mains, refineries and process plants.
Consultancy and civil works: WAPCOS, NHPC design wing, L&T and HCC size penstocks, surge tanks and power-house settings — the σ-based setting and water-hammer calculations above are exactly what is done there.

Interview questions

Q: What is the basic difference between an impulse and a reaction turbine? A: In an impulse turbine the entire head is converted to kinetic energy in the nozzle and the runner works at atmospheric pressure; in a reaction turbine the runner runs full and both pressure and velocity fall across it.
Q: Why is the Pelton bucket angle about 165° and not 180°? A: A full 180° reversal would send the outgoing water straight onto the back of the following bucket, braking the wheel; 160-170° deflects it clear at a small loss of force.
Q: Prove that maximum Pelton efficiency occurs at u = V1/2. A: Power = ρQ(V1 − u)(1 + k cos β2)u; differentiating with respect to u and equating to zero gives V1 − 2u = 0, so u = V1/2 and ηh,max = (1 + k cos β2)/2.
Q: Why does a Pelton turbine not need a draft tube? A: Its runner already works at atmospheric pressure and discharges a free jet, so there is no exit pressure energy to recover; the wheel must simply sit above maximum tailwater level.
Q: What are the two functions of a draft tube? A: It converts the kinetic energy leaving the runner into pressure energy by gradual divergence, and it allows the runner to be placed above tailwater without losing that height, since the tube runs full and can hold sub-atmospheric pressure.
Q: Why is the Kaplan turbine better than a propeller turbine at part load? A: Its blades are adjustable and cam-linked to the guide vanes (double regulation), so the inlet angle stays shock-free over a wide load range; the fixed-blade propeller suffers large incidence losses off design.
Q: Why does a Francis runner discharge with (nearly) zero whirl? A: Any residual whirl Vw2 carries kinetic energy out of the runner and reduces Euler work; designing Vw2 ≈ 0 maximises work and also avoids swirl-induced losses in the draft tube.
Q: What is specific speed and why does it matter? A: The speed of a geometrically similar machine producing unit power under unit head, Ns = N√P/H^1.25 — it is the single number that decides whether the runner should be Pelton, Francis or Kaplan.
Q: What is cavitation and where does it appear on a runner? A: Vapour bubbles form where pressure falls to the vapour pressure and collapse where pressure recovers, pitting the metal — typically on the suction (back) side of the blade near the outlet and in the draft-tube cone.
Q: How is Thoma's number used in practice? A: σ = (Ha − Hv − Hs)/H is computed for the proposed setting and compared with the critical σc for that specific speed; if σ < σc the runner is lowered (Hs made more negative) until there is margin.
Q: Why can't the governor close the guide vanes quickly on load rejection? A: Rapid closure would generate severe water hammer in the penstock; so closure is slowed and the speed rise is absorbed by rotating inertia, with a surge tank, relief valve or (in a Pelton) a deflector handling the transient.
Q: Why is silt a bigger problem for Francis than for Pelton machines in the Himalaya? A: The Francis runner and guide vanes see high-velocity pressurised flow over long blade passages and are prone to erosion plus erosion-assisted cavitation, whereas Pelton buckets are simpler, accessible and individually replaceable.
Q: What is runaway speed and why does a designer care? A: The maximum no-load speed reached with gates open and the governor failed — up to about 3× rated for a Kaplan; the shaft, runner and generator rotor must withstand the resulting centrifugal stresses.

Common mistakes

Using gross head instead of net head in ρgQH, or putting Q in litres/s and H in cm.
Confusing hydraulic efficiency (runner/water power) with overall efficiency (shaft/water power), and quoting generator efficiency as part of the turbine's.
Measuring the Pelton bucket outlet angle from the wrong reference — if the jet is deflected through 165°, use β2 = 15° in (1 + k cos β2).
Assuming the specific-speed number is unit-free: metric Ns (kW, m, rpm) and the non-dimensional or US forms give completely different values, so the bands differ too.
Forgetting the vane-thickness factor and the boss diameter — Kaplan flow area is (π/4)(Do² − Dh²), not (π/4)Do².
Treating Hs as always positive in Thoma's equation; for high-Ns Kaplan machines it is normally negative (runner submerged).
Claiming a Pelton turbine cavitates, or that a Kaplan can be run without a draft tube.
Taking the turbine's best-point efficiency as its efficiency at all loads — for Francis and propeller machines, part-load efficiency falls steeply.