Design the Francis turbine — the world's most widely used reaction turbine. Adjust the head, flow, efficiency, runner diameter and speed to see the output power, specific speed, speed ratio and shaft torque update in real time, and find the best operating point for medium-head dam hydropower.
Parameters
Effective head H
m
Effective drop between dam level and runner exit
Flow rate Q
m³/s
Volumetric flow of water through the runner
Turbine efficiency η
Ratio of shaft output to available hydraulic power
Runner diameter D
m
Outer diameter at the runner-blade inlet
Rotational speed N
rpm
Matched to the generator synchronous speed
Results
—
Output P (MW)
—
Available hydraulic (kW)
—
Specific speed Ns
—
Runner periph. speed u (m/s)
—
Speed ratio φ
—
Shaft torque T (kN·m)
—
Francis turbine cross-section — inward-swirl animation
Water spirals radially inward from the spiral casing, through the ring of guide vanes, onto the curved runner blades that rotate at the centre. The blue arcs in the centre are the runner blades.
Output vs effective head P(H)
Output vs flow rate P(Q)
Theory & Key Formulas
$$P=\eta\,\rho g Q H,\qquad N_s=\frac{N\sqrt{P}}{H^{5/4}}$$
Output power P and specific speed Ns. η: turbine efficiency, ρ: water density, g: gravity, Q: flow rate, H: effective head, N: speed. Specific speed is evaluated with power in kW.
$$\phi=\frac{u}{\sqrt{2gH}},\qquad u=\frac{\pi D N}{60}$$
Speed ratio φ and runner peripheral speed u. D: runner diameter. The Francis turbine suits medium head and runs at a medium specific speed.
What is the Francis Turbine Simulator?
🙋
There are several kinds of water turbine, right? What is special about the "Francis turbine"?
🎓
In short, the Francis turbine is "the most widely used water turbine in the world". It was perfected in the 19th century by an engineer named James B. Francis, and even today this type carries most of the world's installed hydropower capacity. It is a reaction turbine: water enters through a spiral-shaped casing, passes through a ring of vanes called guide vanes, and is directed radially inward onto the central runner. The water hands over its energy inside the curved runner blades and finally leaves axially down into a draft tube below.
🙋
A "reaction turbine" — is that different from a Pelton wheel? A Pelton is a turbine too.
🎓
Good point. A Pelton wheel is an impulse turbine: its nozzle converts all the water pressure into velocity, then throws the jet at buckets in open air. A Francis turbine is a reaction turbine: the water still carries pressure as it does work inside the runner. As it flows through the blade passages the pressure drops, and the runner spins on the "reaction" of that pressure difference. So the blades sit in a sealed space full of water. In the cross-section above you can see water spiralling inward and being drawn into the runner.
🙋
What sites is it good for? When I move the "effective head" slider the output changes a lot.
🎓
The Francis turbine is at home with medium heads — roughly 10 to 700 metres. That happens to be the most common head range for dam-based hydropower, which is exactly why it is used everywhere. The output is $P=\eta\rho g Q H$, proportional to both the head H and the flow Q. Four times the head means four times the output. That is in contrast to a Pelton wheel, where jet velocity only scales with the square root of head. Raise the head slider in the simulator and you will see the output card grow in a straight line.
🙋
There is also a "specific speed" card. What does that one represent?
🎓
Specific speed Ns is the index that decides "which type of turbine suits the site". It is $N_s=N\sqrt{P}/H^{5/4}$, computed from the speed, output and head. A Francis turbine has a specific speed of roughly 60-400. The high-head Pelton wheel sits lower, and the low-head Kaplan turbine sits higher. So the Francis turbine covers exactly the middle ground. Even within the Francis family, a high-head design uses a slender low-specific-speed runner while a low-head design uses a flat high-specific-speed runner — the blade shape changes.
🙋
The "speed ratio φ" is around 0.9. For a Pelton wheel the best value was 0.5. Why is it different?
🎓
The speed ratio φ is the runner peripheral speed u divided by √(2gH), the theoretical maximum velocity if all the head were turned into speed. A Pelton wheel is an impulse turbine, so the output peaks when the buckets run at half the jet velocity — φ ≈ 0.45. But a Francis turbine is a reaction turbine, and because the water does work while staying under pressure inside the blades, the runner can spin faster. φ is typically in the range 0.6-0.9. Change the speed N or the runner diameter D and φ moves, so use the simulator to find an operating point that stays inside that band.
Frequently Asked Questions
A Francis turbine is a reaction turbine in which water enters the runner radially inward through a spiral (volute) casing and a ring of guide vanes, gives up its energy to the curved runner blades, and leaves axially through a draft tube. It was developed by James B. Francis and is by far the most widely used water turbine in the world today. It suits medium heads — roughly 10 to 700 metres — which is the most common range for dam-based hydropower.
Specific speed Ns is the index that determines the turbine type. In the metric form with power in kW it is Ns = N·√P / H^1.25, where N is the rotational speed in rpm, P is the output power in kW and H is the effective head in metres. A Francis turbine has a specific speed of roughly 60-400, sitting between the high-head Pelton wheel (low Ns) and the low-head Kaplan turbine (high Ns). Once speed, head and power are fixed, Ns points to the best runner shape for the site.
The speed ratio (peripheral-velocity coefficient) φ is a dimensionless number equal to the runner peripheral speed u divided by the theoretical maximum velocity √(2gH): φ = u/√(2gH). For a Francis turbine φ is typically in the range 0.6-0.9 — smaller for low-specific-speed, high-head runners and larger for high-specific-speed, low-head runners. It is higher than the Pelton wheel's value of about 0.45 because, in a reaction turbine, the water does work while still under pressure inside the blade passages.
The choice depends on head and flow. High head and low flow call for a Pelton wheel (an impulse turbine), medium head and medium flow call for a Francis turbine (a reaction turbine), and low head with high flow calls for a Kaplan turbine (a propeller-type reaction turbine). The Francis turbine covers the widest range and carries most of the world's installed hydropower capacity. As a rough guide, Pelton is used above about 200 m of head and Kaplan below about 60 m, with the broad band in between belonging to the Francis turbine.
Real-World Applications
Large dam hydropower: The Francis turbine's biggest stage is the dam-based hydropower plants found all over the world. China's Three Gorges Dam houses many Francis turbines rated at roughly 700 MW each, and North America's Hoover and Grand Coulee dams also run mainly Francis turbines. Because the conditions — tens to hundreds of metres of head and hundreds of m³/s of flow, that is "medium head, medium flow" — fit it exactly, it is effectively the standard format for large units.
Pump-turbines for pumped storage: Many pumped-storage plants use a "reversible pump-turbine", a Francis machine that can also run in reverse. By day it generates as a turbine; at night, surplus electricity spins the same machine as a pump to lift water to the upper reservoir. A Francis-type runner has its blades in a sealed space, so reversing the rotation makes it work as a pump too, which is why it sits at the heart of this large-scale "battery".
Small hydro and re-development of existing dams: When generating equipment is retrofitted to an irrigation or flood-control dam — "dam re-development" — a Francis turbine is the choice for medium heads. It can be installed on either a horizontal or vertical shaft, and the wide flow adjustment offered by the guide-vane opening makes it well suited to rivers whose flow varies with the season.
A foundation for turbomachinery design: The Francis turbine's ideas — radial inflow with axial outflow, using guide vanes to set the flow direction, choosing the type by specific speed — are common to centrifugal pumps and centrifugal compressors as well. It is widely used in mechanical-engineering textbooks as a vehicle for learning Euler's turbine equation and velocity triangles.
Common Misconceptions and Pitfalls
The most common misconception is to think that "the Francis turbine is a universal turbine that works for any head". It does cover the widest range, but its sweet spot is still limited to roughly 10-700 m of head. Above this, the pressure at the runner exit drops too far and cavitation (the formation and collapse of vapour bubbles) becomes severe, so the Pelton wheel takes over. Below this, the blades become extremely flat to handle the large flow, and the Kaplan turbine with its adjustable blades wins on efficiency. Remember that the specific speed Ns is itself the map of "which type fits".
The next pitfall is assuming that output P scales with the square root of head H. That is the Pelton wheel's jet velocity; the Francis turbine's output is $P=\eta\rho g Q H$ — proportional to both head and flow. Double the head and you double the output. That is why the "Output vs effective head" chart in the simulator is a straight line. Specific speed $N_s=N\sqrt{P}/H^{5/4}$, on the other hand, is divided by head to the 1.25 power, so raising the head lowers Ns. Note that head affects output and specific speed in opposite directions.
Finally, be aware that this simulator shows the ideal performance at the design point, not the behaviour at part load. A real Francis turbine, when throttled to part load with the guide vanes, develops a swirling vortex-core oscillation in the draft tube known as "draft-tube surge", which causes power swings, noise and vibration fatigue of the machine. A Francis turbine runs most efficiently and quietly only in a narrow band near its rated point, and on grids with large demand swings the restricted operating range becomes an important design issue. What is calculated here is strictly the ideal value near the best-efficiency point.
How to Use
Set gross head (H) in meters using the headHNum input (typical range 20–250 m for Francis turbines); the headHRange slider provides real-time adjustment.
Enter flow rate (Q) in cubic meters per second via flowQNum; Francis designs typically operate 1–50 m³/s depending on installation scale.
Adjust runner diameter (D) in millimeters using runnerDNum to match your site constraints and target rotational speed (usually 50–600 rpm for grid-connected units).
Modify efficiency target (%) in efficNum to reflect runner blade optimization and hydraulic losses; industrial Francis turbines achieve 85–95% peak efficiency.
Observe real-time outputs: output power in MW, specific speed Ns (dimensionless), runner peripheral speed u (m/s), speed ratio φ, and shaft torque T (kN·m).
Worked Example
Design a Francis turbine for a 60 m head hydropower station with 8 m³/s available flow. Input H=60 m, Q=8 m³/s, D=1200 mm runner diameter, target efficiency 92%. Simulator calculates: output power ≈ 4.32 MW (accounting for 10% hydraulic and mechanical losses), specific speed Ns ≈ 80 (indicating medium-head Francis geometry), runner peripheral speed u ≈ 37.7 m/s (at 600 rpm), and shaft torque T ≈ 7.2 kN·m. Verify that Ns falls within 60–400 range; values outside this suggest runner redesign or speed adjustment.
Practical Notes
Specific speed Ns directly determines Francis runner blade shape: Ns <60 implies radial-flow geometry (steep heads >100 m); Ns 100–300 suits medium heads (40–100 m); Ns >300 requires mixed-flow design (low heads <40 m).
Runner peripheral speed u must remain below cavitation threshold (typically 40–50 m/s depending on barometric pressure and saturation conditions at your elevation).
For sites with variable flow, optimize runner diameter at average discharge; use speed ratio φ (ratio of blade speed to jet velocity) between 0.45–0.65 for peak efficiency zones.
Shaft torque output guides coupling and bearing selection; multiply by rotational speed (rad/s) to confirm mechanical power matches electrical generator rating.