Pelton turbine: Design in 13 steps

From head and discharge to the machine in thirteen steps, by the textbook. The calculator works through the chain, six pages go deeper into single steps. Where the textbook ends, System Erlach begins.

This page holds for every Pelton turbine. Where a section is about System Erlach, the horizontal Pelton turbine with up to six nozzles designed by Josef Erlach, its heading says so. The difference is explained in the overview Pelton turbine System Erlach.

Designing a Pelton turbine: two numbers, three decisions, thirteen steps

At the start are two numbers the plant provides: the head and the discharge. Then come three decisions of the designer: the number of nozzles, the speed and the position of the shaft. Everything else follows in a fixed order, as taught in hydropower engineering courses and worked through here.

How nozzle, jet and bucket work together is explained under Pelton turbine working principle. For the same head and discharge there are always several solutions, as the table in the calculator shows.

Pelton design from A to Z: the thirteen steps

Head and discharge are given. The plant supplies both. The power follows. The head is referred to the inlet of the distributor, the reference section of IEC 60193. More under Pelton turbine nozzle.
Set jet and bucket velocity. For the first estimate the jet is calculated with the full velocity √(2gH); the roughly 2 per cent loss up to the jet stands in the loss balance as in Veselý. The buckets run at 47 per cent of it, just under half the jet velocity. More under Pelton turbine nozzle.
Choose the number of nozzles. More nozzles mean thinner jets, higher speed, a smaller machine. Conventional: horizontal up to two, vertical up to six; System Erlach: horizontal up to six. More under Horizontal Pelton turbine.
Jet diameter. From discharge per nozzle and jet velocity, by continuity. More under Pelton turbine nozzle.
Bucket width. About 3.1 times the jet diameter for one nozzle, 3.2 for two, 3.3 for four to five, more for six. Manufacturers deviate according to production and experience.
Runner diameter from the head. The higher the head, the larger the runner in relation to the jet: 10 times at 400 metres, 15 times at 1400 metres. Below 9.5 water is lost. More under Pelton turbine specific speed.
Speed. From peripheral velocity and diameter. More under Pelton turbine specific speed.
Generator pole pairs. The speed must match the grid frequency. The number of pole pairs is rounded to the nearest whole number; only if the runner would then fall below 10 times the jet is it rounded up.
Recalculate the speed. With the whole number of pole pairs the speed is fixed, slightly above or below the value of step 7.
Recalculate the diameter. The runner gets slightly smaller or larger accordingly. The specific speed per nozzle is now a result, not an input. More under Pelton turbine specific speed.
Number of buckets. At least 17, empirically. As an estimate 15 plus runner diameter over twice the jet diameter.
Housing. With a vertical shaft, runner plus 8 to 9 bucket widths in diameter. With a horizontal shaft about four bucket widths of pit width, and the splash water decides how many nozzles are possible. More under Horizontal Pelton turbine.
Heights. Runner to tailwater about 3.5 bucket widths, runner centre to ceiling 2 to 2.5 bucket widths.

Two things belong to the design and are not in the chain, because the textbook does not put them into numbers: the distributor, which should bring equal water to every nozzle, see Pelton turbine distributor, and the behaviour at low discharge, see Pelton turbine efficiency.

Source: lecture slides “Design of Pelton turbines”, NTNU Trondheim (Kvaerner/GE Hydro), steps 1 to 13 with the examples Jostedal and Khimti; H. Brekke, Hydraulic Turbines, NTNU 2015, pp. 40, 41, 106 and 137–138; jet and bucket velocity after Josef Erlach, 3 and 12 October 2026.

The Pelton calculator: the thirteen steps worked through

The calculator goes through the chain for the head, discharge and number of nozzles entered and shows each step with its number. The two examples come from the lecture; their results are recalculated every time the website is built. Below, the same calculation is shown for one to six nozzles side by side.

Examples from the NTNU lecture:

1. Head and discharge are givenH = 1,130 m, Q = 28.50 m³/s. From these the power P = ρ · g · Q · H · η = 284.3 MW (η = 0.90).
2. Set jet and bucket velocityJet c₁ = √(2gH) = 148.9 m/s, periphery u₁ = 0.47 · √(2gH) = 70.0 m/s.
3. Choose the number of nozzlesz = 5, per nozzle Q_nozzle = Q / z = 5.700 m³/s.
4. Jet diameter from continuityd₀ = √(4 · Q_nozzle / (π · c₁)) = 221 mm.
5. Bucket widthB = 3.30 · d₀ = 729 mm (factor by number of nozzles).
6. Runner diameter from the headD₁ / d₀ = 0,005 · H + 8 = 13.65, so D₁ = 3,014 mm.
7. Calculate the speedn = 60 · u₁ / (π · D₁) = 443.5 rpm.
8. Generator pole pairsp = 60 · f / n = 6.76, rounded p = 7 (14 poles, f = 50 Hz).
9. Recalculate the speedn = 60 · f / p = 428.6 rpm (slightly below step 7, the runner gets larger).
10. Recalculate the diameterD₁ = 60 · u₁ / (π · n) = 3,119 mm, D₁ / d₀ = 14.1, n_q per nozzle = 5.2.
11. Number of bucketsz_B = 22 (rule of thumb 15 + D₁ / (2 · d₀) = 22.1, at least 17).
12. HousingVertical: D_G = D₁ + K · B = 9.31 m (K = 8.5). Horizontal: pit width about 4 · B = 2.91 m; the splash water decides, see System Erlach.
13. HeightsRunner to tailwater about 3,5 · B = 2.55 m (vertical); runner centre to ceiling 2 to 2,5 · B = 1.46 to 1.82 m. Runaway speed, guide value: 771 to 814 rpm.

Calculated for the first estimate with the full jet velocity √(2gH), the losses up to the jet stand in the loss balance as in Veselý; after Josef Erlach the buckets run at 47 per cent of it. The lecture uses 48 per cent; the text output below shows what that changes.

The same head and discharge with one to six nozzles

Jet, runner, speed and number of buckets for one to six nozzles
Nozzles	Jet d₀ mm	Bucket B mm	Runner D₁ mm	Speed n rpm	n_q per nozzle	Buckets	Shaft conventional
1	494	1,530	6,683	200 (30 poles)	5.5	22	horizontal
2	349	1,117	4,901	272.7 (22 poles)	5.3	22	horizontal
3	285	926	4,010	333.3 (18 poles)	5.3	22	horizontal with narrow nozzle spacing, or vertical
4	247	815	3,564	375 (16 poles)	5.1	22	vertical
5	221	729	3,119	428.6 (14 poles)	5.2	22	vertical
6	202	675	2,673	500 (12 poles)	5.6	22	vertical

Result as text, with inputs, assumptions, formulas and the comparison with the lecture values

Pelton turbine: design in 13 steps. First estimate, not a design.
Calculator: https://www.erlach-erlach.com/en/pelton-turbine/design/

Inputs
- H = 1,130 m (head, referred to the high pressure reference section of IEC 60193, here the inlet of the distributor)
- Q = 28.50 m³/s (discharge of the turbine)
- f = 50 Hz (grid frequency)
- z = 5 (chosen number of nozzles)

Assumptions
- k_c = 1.00, k_u = 0.47 (shares of the theoretical jet velocity after Josef Erlach; the NTNU lecture uses 1.00 and 0.48)
- eta = 0.90 (for the power figure only), g = 9.81 m/s², rho = 1,000 kg/m³
- Synchronous generator without gearbox: n = 60 · f / p, the number of pole pairs p is rounded to the nearest whole number; if the runner would then fall below 10 times the jet, it is rounded up
- Not included: bucket shape, needle and nozzle tip, detailed distributor, strength, control, model test

The 13 steps
1. Given: H = 1,130 m, Q = 28.50 m³/s; P = rho · g · Q · H · eta = 284.3 MW
2. c_1 = k_c · sqrt(2 · g · H) = 148.9 m/s; u_1 = k_u · sqrt(2 · g · H) = 70.0 m/s
3. z = 5; Q_nozzle = Q / z = 5.700 m³/s
4. d_0 = sqrt(4 · Q_nozzle / (pi · c_1)) = 221 mm
5. B = 3.30 · d_0 = 729 mm (factor by number of nozzles: 3.1 for one, 3.2 for two, 3.3 for four to five, above 3.3 for six)
6. D / d_0 = 0,005 · H + 8 = 13.65 (at least 10); D = 3,014 mm
7. n = 60 · u_1 / (pi · D) = 443.5 rpm
8. p = 60 · f / n = 6.76, rounded p = 7 (14 poles)
9. n = 60 · f / p = 428.6 rpm
10. D = 60 · u_1 / (pi · n) = 3,119 mm; D / d_0 = 14.1; n_q per nozzle = n · sqrt(Q_nozzle) / H^0,75 = 5.2
11. z_B = 22 (rule of thumb 15 + D / (2 · d_0) = 22.1, at least 17)
12. D_G = D + K · B = 9.31 m with K = 8.5 (housing of the vertical machine); horizontal: pit width about 4 · B = 2.91 m
13. height runner to tailwater about 3,5 · B = 2.55 m (vertical); runner centre to ceiling 2 to 2,5 · B = 1.46 to 1.82 m
- runaway speed, guide value: 1,8 to 1,9 · n = 771 to 814 rpm; the value verified by the manufacturer governs

Comparison with the NTNU coefficients (k_c = 1.00, k_u = 0.48)
- d_0 = 221 mm; p = 7; n = 428.6 rpm; D = 3,185 mm; z_B = 22

Formulas
- P = rho · g · Q · H · eta
- c_1 = sqrt(2 · g · H)
- u_1 = k_u · sqrt(2 · g · H)
- Q_nozzle = Q / z
- d_0 = sqrt(4 · Q_nozzle / (pi · c_1))
- B = 3.1 · d_0 for one nozzle, 3.2 for two, 3.3 for four to five, above 3.3 for six
- D_1 / d_0 = 0.005 · H + 8, at least 10 (10 at 400 m, 15 at 1400 m)
- n = 60 · u_1 / (pi · D_1)
- p = 60 · f / n, rounded to the nearest whole number; rounded up if D_1 / d_0 would then fall below 10
- n = 60 · f / p
- D_1 = 60 · u_1 / (pi · n)
- n_q = n · sqrt(Q_nozzle) / H^0.75
- z_B ≈ 15 + D_1 / (2 · d_0), at least 17
- D_G = D_1 + K · B (vertical machine); horizontal machine: pit width ≈ 4 · B
- n_R ≈ 1.8 to 1.9 · n (guide value)

Quantities
- H head, referred to the high pressure reference section of IEC 60193, here the inlet of the distributor in m
- Q discharge of the turbine in m³/s
- Q_nozzle discharge per nozzle in m³/s
- z number of nozzles, no unit
- f grid frequency in Hz
- p number of generator pole pairs, no unit
- n rotational speed in rpm
- c_1 jet velocity in m/s
- u_1 peripheral velocity at the pitch circle in m/s
- d_0 jet diameter in m
- B bucket width in m
- D_1 pitch circle diameter of the runner in m
- 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_B number of buckets, no unit
- D_G housing diameter of the vertical machine in m
- K housing factor, 8 to 9, rising with the number of nozzles, no unit
- n_R runaway speed in rpm
- P power in W

Sources: lecture slides "Design of Pelton turbines", NTNU Trondheim (Kvaerner/GE Hydro); H. Brekke, Hydraulic Turbines, NTNU 2015, pp. 40, 41, 106, 137, 138; jet and bucket velocity after Josef Erlach.
System Erlach permits two to six nozzles on a horizontal shaft. This estimate does not check whether a machine is suitable for these inputs.

For engineers: First estimate, not a design. The number of pole pairs is rounded; the speed therefore deviates slightly from step 7, and the diameter follows in step 10. The two lecture examples are recalculated at every build; for Khimti the lecture rounds the jet to 0.15 m and chooses 22 buckets, the rule of thumb gives 20 to 21 depending on the coefficients. The rule of thumb, the rounding of pole pairs and the minimum of two pole pairs are assumptions of this page, not of the lecture. Not included: bucket shape, needle and nozzle tip, the distributor in detail, strength, control, model test.

Formulas as text: P = rho · g · Q · H · eta; c_1 = sqrt(2 · g · H); u_1 = k_u · sqrt(2 · g · H); Q_nozzle = Q / z; d_0 = sqrt(4 · Q_nozzle / (pi · c_1)); B = 3.1 · d_0 for one nozzle, 3.2 for two, 3.3 for four to five, above 3.3 for six; D_1 / d_0 = 0.005 · H + 8, at least 10 (10 at 400 m, 15 at 1400 m); n = 60 · u_1 / (pi · D_1); p = 60 · f / n, rounded to the nearest whole number; rounded up if D_1 / d_0 would then fall below 10; n = 60 · f / p; D_1 = 60 · u_1 / (pi · n); n_q = n · sqrt(Q_nozzle) / H^0.75; z_B ≈ 15 + D_1 / (2 · d_0), at least 17; D_G = D_1 + K · B (vertical machine); horizontal machine: pit width ≈ 4 · B; n_R ≈ 1.8 to 1.9 · n (guide value). Quantities: P power in W; rho density of water in kg/m³, equal to 1000; g gravitational acceleration in m/s², equal to 9.81; 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; eta turbine efficiency, no unit; c_1 jet velocity in m/s; u_1 peripheral velocity at the pitch circle in m/s; k_u share of the theoretical jet velocity at which the buckets run, no unit, typically 0.45 to 0.48; Q_nozzle discharge per nozzle in m³/s; z number of nozzles, no unit; d_0 jet diameter in m; pi the number pi, no unit, equal to 3.1416; B bucket width in m; D_1 pitch circle diameter of the runner in m; n rotational speed in rpm; p number of generator pole pairs, no unit; f grid frequency in Hz; 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_B number of buckets, no unit; D_G housing diameter of the vertical machine in m; K housing factor, 8 to 9, rising with the number of nozzles, no unit; n_R runaway speed in rpm.

Pelton turbine and System Erlach: where the textbook ends

The thirteen steps lead to the runner, the speed and the housing of a vertical machine. They leave three questions open, and these are exactly where the Pelton turbine System Erlach starts.

How does the water reach the nozzles? The textbook names the ring distributor. In System Erlach an axial distributor gives every nozzle the same inflow, see Pelton turbine distributor.
How many nozzles can a horizontal shaft take? Conventionally two, because otherwise the splash water falls back onto the runner. In System Erlach two guide rings lead the water away, up to six nozzles, see Horizontal Pelton turbine.
What happens at low discharge? Switch nozzles off instead of throttling all of them, with the same inflow for every nozzle, see Pelton turbine efficiency.

Source: J. M. Erlach, New Pelton Concept, Viennahydro 2012, Full Paper, sections 2.2 to 2.5; leaflet for Viennahydro 2012.

Pelton design, further quantities: jet deflector, water hammer, runaway speed

Jet deflector. If the generator suddenly loses its load, a deflector turns the jet away from the runner at once.
Water hammer. The needle may only close slowly. Otherwise water hammer occurs in the penstock. This is why the deflector is needed.
Runaway speed. If the load is lost and the jet is not deflected, the runner accelerates towards its runaway speed. As a guide value, 1.8 to 1.9 times the rated speed; the value verified by the manufacturer governs, up to about twice the rated speed according to Zhang. Runner and generator must be designed for it.

Formula as text: n_R ≈ 1.8 to 1.9 · n (guide value). Quantities: n_R runaway speed in rpm; n rotational speed in rpm.

Source: energie.ch, Peltonturbine (runaway speed); Josef Erlach, 3 October 2026 (guide value); Z. Zhang, Pelton Turbines, Springer 2016, chapter 17 (runaway speed up to about twice the rated speed).

Limits of the Pelton calculator: a preliminary design, not a machine

The chain works with textbook relations and usual values from experience. It gives the order of magnitude of a machine: jet, runner, speed, generator, number of buckets, housing. What makes a turbine is not in it: the shape of the buckets, needle and nozzle tip, the distributor in detail, strength, cavitation and abrasion, control, and the model test. The chain replaces neither the work of a design engineer nor the model test.