The animation shows a Pelton turbine wheel with nozzle, water jet and deflected jet. A schematic of the double bucket shows the velocity triangles at inlet and outlet, and a characteristic diagram shows the driving torque and the power over the speed ratio. Flow rate and load torque can be set; the rotational speed settles on its own.
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Description of the Animation
The animation is divided into three areas that show the same operating state from three points of view. Changing one slider makes all three react at once.
Left: the runner
Sixteen buckets are mounted on the runner. A horizontal water jet leaves the nozzle and strikes the buckets as they pass by. The jet is split into partial jets: each one ends exactly where it meets a bucket contour. This makes it visible how the jet travels from bucket to bucket during one revolution and is briefly shared between two buckets at the same time.
The deflected water leaves the bucket as a lighter jet and flies away on a projectile parabola. Its range depends on the relative velocity: the slower the wheel runs, the faster the water moves relative to the bucket and the further it is thrown. If the wheel runs almost as fast as the jet, the water merely dribbles out. The splashes along the jet are pure decoration and carry no physical meaning.
The on-screen rotation runs in slow motion (factor 1/100). At the real speed the wheel would only be recognisable as a disc. All displayed numbers, by contrast, are real-world values.
Middle: a single double bucket
In the middle you look down onto one bucket. Its characteristic shape is clearly visible: two cups separated by a central splitter. The splitter divides the incoming jet into two equal halves, which are turned back through 165° to either side. The lateral forces therefore cancel out and the shaft is not loaded to one side.
Drawn in are the velocity triangles at the inlet and at the outlet, together with the decomposition of the exit velocity into its components. The arrows are to scale: as the wheel speeds up, the green arrow grows and the orange one shrinks.
Right: the characteristic diagram
Two curves are plotted over the speed ratio \( u/c_1 \): the driving torque \( M \) (red, left axis) falls off in a straight line, while the mechanical power \( P \) (green, right axis) forms a parabola with a pronounced maximum. Two dots mark the current operating state. They travel along the curves whenever the rotational speed changes, which makes it possible to steer deliberately towards the best point.
Interactive Controls
- Flow rate Q (0–100 l/s): Operates the spear valve. A larger flow rate means a thicker jet, not a faster one.
- Load torque M (0–4000 Nm): The counter-torque the generator applies to the wheel. Together with the flow rate it determines which rotational speed is reached.
- Checkboxes in the value list: Show or hide individual velocity arrows. The components \( c_{u2} \) and \( c_{m2} \) are hidden at start-up.
- Reset: Closes the nozzle, brings the wheel to a standstill and returns the load torque to its initial value.
The rotational speed itself cannot be set directly. It follows from the driving torque, the load torque, the bearing friction and the inertia of the wheel – exactly as in a real machine.
How a Pelton Turbine Works
The Pelton turbine is an impulse turbine (free jet turbine). This means that the entire pressure energy of the water is already converted into velocity inside the nozzle. Inside the bucket the pressure is ambient everywhere – the runner turns in air, not in water. The turbine therefore does not extract a pressure difference from the jet, only its momentum.
The energy travels in three steps:
- Potential energy → velocity: The water falls from the reservoir through the penstock and is accelerated in the nozzle.
- Velocity → force: The bucket turns the jet back through almost 180°. This change of direction produces a force on the bucket.
- Force → power: The force acts on the lever arm of the pitch radius and drives the shaft.
Pelton turbines are used for large heads and small flow rates, typically in high mountain regions. For shallow rivers carrying a lot of water, Kaplan or Francis turbines are the right choice.
The Quantities in the Animation
The value list below the bucket schematic shows eight quantities. All velocities carry the notation customary in turbomachinery: index 1 refers to the inlet into the bucket, index 2 to the outlet.
| \( c_1 \) | Jet velocity (absolute velocity at the inlet), measured by a stationary observer |
| \( u \) | Blade velocity of the bucket on the pitch circle |
| \( w_1 \) | Relative velocity at the inlet: the speed at which the water arrives as seen from the moving bucket |
| \( w_2 \) | Relative velocity at the outlet, rotated by the deflection angle and somewhat smaller than \( w_1 \) because of friction |
| \( c_2 \) | Absolute velocity at the outlet: the motion the water still retains when it leaves the bucket (residual energy) |
| \( c_{u2} \) | Whirl component of \( c_2 \), in the direction of the blade velocity |
| \( c_{m2} \) | Flow (meridional) component of \( c_2 \), perpendicular to the peripheral direction |
| \( u/c_1 \) | Speed ratio: the ratio of blade velocity to jet velocity, the key parameter of the machine |
The relation between the three velocities is always the same. At the inlet:
\[ \vec c_1 = \vec u + \vec w_1 \]
Because jet and bucket point in the same direction here, this reduces to a simple subtraction: \( w_1 = c_1 – u \). The same relation holds at the outlet, except that \( \vec w_2 \) now points almost the opposite way:
\[ \vec c_2 = \vec u + \vec w_2 \]
The Jet: Head and Nozzle
The jet velocity follows from the energy balance between the water surface and the nozzle exit (Bernoulli). It equals the velocity of a body in free fall, reduced by the friction losses in the penstock and the nozzle:
\[ c_1 = c_v \sqrt{2 g H} \]
With a net head of \( H = 60 \) m and a nozzle coefficient of \( c_v = 0.98 \) this gives \( c_1 \approx 33.6 \) m/s, about 75 mph.
One point is essential for understanding the animation: the jet velocity does not depend on the flow rate. The spear valve only changes the cross-section of the opening, not the head. A half-closed slider therefore produces a thinner jet of exactly the same speed. The value \( c_1 \) in the list stays constant while the jet visibly becomes slimmer.
Force, Torque and Power
The bucket does not „see“ the whole jet, only the excess of its velocity over the bucket’s own motion. What matters is therefore \( w_1 = c_1 – u \). The mass flow turned per second is \( \dot m = \rho Q \). The momentum equation then gives the peripheral force:
\[ F = \rho Q \, w_1 \left( 1 + k \cos \beta_2 \right) \]
The bracket is the deflection factor. The 1 stands for stopping the incoming water, the second term for the additional reaction of throwing it back. For an ideal 180° reversal without friction it would be 2 – turning the jet around doubles the force compared with merely catching it. In the animation the deflection angle is 165°, so the remaining angle is \( \beta_2 = 15° \), and the velocity coefficient of the bucket is \( k = 0.90 \). This yields a factor of 1.87. The angle deliberately stays below 180° so that the outflowing water does not hit the following bucket.
The pitch radius \( r \) turns the force into a torque, and the angular velocity \( \omega = u/r \) turns the torque into power:
\[ M = F \, r \qquad P = M \, \omega = F \, u \]
Inserting the force puts the whole power into a single line:
\[ P = \rho Q \left( 1 + k \cos \beta_2 \right) \left( c_1 – u \right) u \]
Where Does the Maximum Power Lie?
The power equation contains two factors working against each other: \( (c_1 – u) \) and \( u \). Both limiting cases can be set directly in the animation.
- Blocked wheel (load torque to the far right, \( u = 0 \)): The force on the bucket is at its maximum, and so is the torque. But nothing moves, so no work is done. The power is zero.
- Runaway (load torque at 0, \( u \to c_1 \)): The bucket runs away almost as fast as the jet. The water barely catches up with it, so hardly any momentum is transferred. Force and torque approach zero – and so does the power. In the schematic the orange arrow \( w_1 \) visibly collapses.
Between two zeros there must be a maximum. The expression \( (c_1 – u)\,u \) is a downward parabola with the roots \( u = 0 \) and \( u = c_1 \); its apex lies exactly halfway:
\[ u_{\text{opt}} = \frac{c_1}{2} \qquad \text{so} \qquad \frac{u}{c_1} = 0.5 \]
A Pelton turbine delivers its greatest power when the buckets run at half the speed of the water jet. Substituting this back gives:
\[ P_{\max} = \frac{\rho Q \left( 1 + k \cos \beta_2 \right) c_1^{\,2}}{4} \]
The diagram shows this clearly: the torque drops off in a straight line from left to right, while the power – the product of torque and speed – forms an arc with its apex at \( u/c_1 = 0.5 \). Near the apex the curve is flat, so small deviations in speed cost hardly any power. That is why a Pelton turbine stays economical over a wide load range despite a rigidly fixed grid frequency.
The intuitive explanation
There is a second, energetic view of the same result. The turbine extracts exactly as much energy as the water has less at the outlet than at the inlet. Ideally, then, the water would drop straight down behind the bucket and stand still in the peripheral direction.
That is precisely what happens at half the jet velocity: as seen from the bucket the water arrives with \( w_1 = c_1 – c_1/2 = c_1/2 \) and is thrown back at the same speed. Viewed from outside, this backward motion just cancels the forward motion of the bucket. The water practically comes to rest and falls into the tailrace. This can be checked in the animation: set the speed ratio to 0.5 and the arrow \( c_2 \) becomes shortest (about 4.5 m/s compared with 33.6 m/s at the inlet), while the deflected jets at the wheel fall almost vertically.
Real machines run at a speed ratio of roughly 0.45 to 0.48. The difference arises from friction inside the bucket, windage losses of the runner spinning in air, and the fact that the deflection angle must stay below 180°.
Efficiency
The hydraulic power available in the water follows from head and flow rate. The efficiency is the ratio of delivered to available power:
\[ \eta = \frac{P}{\rho \, g \, Q \, H} \]
With the values of the animation (\( Q = 60 \) l/s, \( H = 60 \) m) 35.3 kW are available. At the best point the wheel delivers about 31.7 kW, so the efficiency is roughly 90 %. The missing 10 % are shared between the nozzle (about 4 %), friction inside the bucket (about 5 %) and the residual energy of the outflowing water (about 2 %). Large Pelton turbines exceed 90 %.
The role of the flow rate is worth noting: it enters the power linearly, but it does not shift the best point. Halve \( Q \) and torque and power are halved, yet the maximum stays at \( u/c_1 = 0.5 \). In the diagram both curves drop without moving sideways. This too is an advantage of the design: a Pelton turbine still works efficiently at part load.
How the Rotational Speed Settles
The animation does not prescribe the rotation but integrates it from the torque balance. The water drives the wheel, the selected load torque and the bearing friction brake it:
\[ J \frac{\mathrm{d}\omega}{\mathrm{d}t} = M_{\text{drive}} – M_{\text{load}} – d \, \omega \]
The moment of inertia \( J \) makes the wheel approach its new operating point gradually after every change. Equilibrium is reached when driving and load torque are equal. Because the driving torque falls as the speed rises, this state is inherently stable: if the wheel runs too fast, the driving torque drops and slows it down again.
One practical consequence follows: the speed is controlled through the load, not through the water. To hit the best point, both sliders have to be matched to each other. If a real power station suddenly loses its load, the turbine runs away – which is why Pelton turbines have a jet deflector that swings the jet clear of the runner within seconds while the spear valve closes slowly.
Design Data of the Animation
| Net head \( H \) | 60 m |
| Nozzle coefficient \( c_v \) | 0.98 |
| Jet velocity \( c_1 \) | 33.6 m/s |
| Pitch radius \( r \) | 0.60 m |
| Number of buckets | 16 |
| Deflection angle | 165° |
| Velocity coefficient \( k \) | 0.90 |
| Moment of inertia \( J \) | 400 kg m² |
| Flow rate \( Q \) | 0–100 l/s |
| Load torque | 0–4000 Nm |
| Speed at the best point | about 267 rpm |
Pelton Turbines in Practice
- High-head power stations: Pelton turbines work at heads from about 100 m to over 1800 m. The Bieudron station in Switzerland uses the world’s largest head at 1883 m.
- Multiple nozzles: Large machines have up to six nozzles arranged around the runner. At part load individual nozzles are shut off while the others keep running at their best point.
- Storage plants: Because the spear valve allows fast control, Pelton turbines are used to balance peak loads in the electricity grid.
- Micro hydro: Installations in the kilowatt range use the same design because it needs little water and is mechanically simple.
- Wear: Glacial water carrying sand acts like a sandblaster at 75 mph. Buckets and spear needles are therefore hardened or coated.
Historical Background
For centuries water wheels mainly used the weight of the water. Mining in the Sierra Nevada created a demand from about 1850 onwards to exploit water falling from great heights. The first jet wheels caught the jet centrally in flat cups – and the water splashing back slowed down the incoming jet.
Lester Allan Pelton solved this problem around 1878 with the divided double bucket: the central splitter cuts the jet, the water is led away sideways, and the interference with the jet disappears. He was granted a patent in 1880. According to the story, the idea came to him when a bucket on a test wheel worked loose and the jet accidentally struck its edge – whereupon the wheel ran faster.
The bucket shape used today, with the notch in its leading edge, goes back to William A. Doble (1899). It ensures that the jet enters the cup cleanly instead of first striking the back of the bucket. The underlying principle has remained unchanged ever since.
Further Resources
- Pelton wheel – Wikipedia – design, key figures and applications
- Water turbine – Wikipedia – overview of the types and their fields of use
- Euler’s turbine equation – Wikipedia – the general form of the momentum balance used here
Related Animations
Web Animations
Overview
| Title | Pelton Turbine |
| Audience | Teachers and lecturers |
| Features | full screen mode lossless magnification large screens and projectors supported |
| License | MIT |