Pelton turbine: Specific speed

Pelton turbine specific speed: speed, discharge and head in one number

Specific speed combines three quantities: how fast the machine turns, how much water it handles and from what height the water comes. Machines with the same specific speed have similar ratios of jet to runner, even if one is small and the other large. Full similarity needs more, for example the same shape of the buckets.

For engineers: n_q = n · √Q / H^0.75, with n in rpm, Q in m³/s, H in m.

Here we use the conventional, unit-dependent numerical value: n in rpm, Q in m³/s and H in m. It is not a dimensionless number. Values using other unit conventions or dimensionless definitions cannot be compared directly.

Formula as text: n_q = n · sqrt(Q) / H^0.75. Quantities: n_q specific speed (n in rpm, Q in m³/s, H in m) in rpm · (m³/s)^0.5 / m^0.75 (unit-dependent); n rotational speed in rpm; Q discharge of the turbine in m³/s; H head, referred to the high pressure reference section of IEC 60193, here the inlet of the distributor in m.

Source: H. Brekke, Hydraulic Turbines, NTNU 2015, section 9.3, p. 137.

Specific speed per Pelton nozzle: the number applies to the single jet

For the Pelton turbine the specific speed is calculated with the discharge of a single nozzle. The reason: it describes how heavily a single jet loads the buckets.

The specific speed per nozzle grows with the ratio of jet diameter to runner diameter. A high specific speed means a thick jet on a small runner.

For engineers: n_q = n · √Q_nozzle / H^0.75. With c₁ and u₁ from the head, n_q is proportional to d₀ / D₁. With the full jet and k_u = 0.47, n_q ≈ 74 · d₀ / D₁; D₁ / d₀ = 10 corresponds to n_q ≈ 7.4 (with u₁ = 0.48 about 7.6; Brekke's speed number Ω = 0.09 for one nozzle uses u₁ = 0.5 and corresponds to 7.9, since n_q = 89 · Ω). Brekke gives D / d = 10 as the smallest ratio for machines with six nozzles, and 9 for horizontal machines and for vertical machines with fewer than six nozzles.

Formulas as text: n_q = n · sqrt(Q_nozzle) / H^0.75; n_q ≈ 74 · d_0 / D_1, so n_q ≈ 7.4 at D_1 / d_0 = 10 (with full jet and k_u = 0.47; with u_1 = 0.48 ≈ 7.6; Brekke's Ω = 0.09 with u_1 = 0.5 ≈ 7.9). Quantities: n_q specific speed (n in rpm, Q in m³/s, H in m) in rpm · (m³/s)^0.5 / m^0.75 (unit-dependent); n rotational speed in rpm; Q_nozzle discharge per nozzle in m³/s; H head, referred to the high pressure reference section of IEC 60193, here the inlet of the distributor in m; d_0 jet diameter in m; D_1 pitch circle diameter of the runner in m.

Limit per Pelton jet: a jet that is too thick overloads the bucket

Above a certain ratio the buckets can no longer take in and release the water cleanly. The efficiency falls. This limit is the reason why there are machines with several nozzles.

The Pelton turbine calculator therefore sets no characteristic number as a limit but chooses the runner by the head: ten times as large as the jet is thick at 400 metres, fifteen times at 1400 metres, never below ten. The higher the head, the larger the runner must be in relation to the jet, otherwise the buckets suffer. The textbook by Brekke gives ten as the smallest ratio for machines with six nozzles and nine for horizontal machines and for vertical machines with fewer than six nozzles; below 9.5 water is lost according to the NTNU lecture. The characteristic number per nozzle is then a result of the design. A fixed limit such as “7 per nozzle” is no longer given on this page; it depends on the shape of the buckets and on the head and remains the manufacturer's business.

For engineers: D₁ / d₀ = 0.005 · H + 8, at least 10 (NTNU lecture, linear rule of thumb between 400 and 1400 m). Brekke only gives the lower limit.

Formula as text: D_1 / d_0 = 0.005 · H + 8, at least 10 (10 at 400 m, 15 at 1400 m). Quantities: D_1 pitch circle diameter of the runner in m; d_0 jet diameter in m; H head, referred to the high pressure reference section of IEC 60193, here the inlet of the distributor in m.

Source: H. Brekke, Hydraulic Turbines, NTNU 2015, sections 5.1 and 9.3, pp. 22 and 137–138; lecture slides “Design of Pelton turbines”, NTNU Trondheim (Kvaerner/GE Hydro), runner diameter; Josef Erlach, 3 October 2026.

Multi-nozzle Pelton turbine: higher speed at the same specific speed

If the water is split over several nozzles, each nozzle gets less. At the same speed the specific speed per nozzle falls. Or the margin is used to let the machine turn faster: then the runner becomes smaller, and so does the generator.

An example with a head of 400 metres and 3 cubic metres per second: with one nozzle the estimate gives 375 revolutions per minute and a runner of 2,121 millimetres. With six nozzles it is 750 revolutions and 1,060 millimetres. Horizontal Pelton turbine explains what this means for the type of machine.

For engineers: With z nozzles, n_q per nozzle falls by the factor √z at the same speed, or the speed rises by √z at the same n_q. The pitch circle diameter follows from D₁ = 60 · u₁ / (π · n) with u₁ = k_u · √(2 · g · H), usually with k_u around 0.47.

Formulas as text: n_q per nozzle = n_q with one nozzle / sqrt(z), at the same speed n; D_1 = 60 · u_1 / (pi · n); u_1 = k_u · sqrt(2 · g · H). Quantities: n_q specific speed (n in rpm, Q in m³/s, H in m) in rpm · (m³/s)^0.5 / m^0.75 (unit-dependent); z number of nozzles, no unit; n rotational speed in rpm; D_1 pitch circle diameter of the runner in m; u_1 peripheral velocity at the pitch circle in m/s; pi the number pi, no unit, equal to 3.1416; k_u share of the theoretical jet velocity at which the buckets run, no unit, typically 0.45 to 0.48; g gravitational acceleration in m/s², equal to 9.81; H head, referred to the high pressure reference section of IEC 60193, here the inlet of the distributor in m.