A Model Turbine is a geometrically scaled-down hydraulic turbine built and instrumented in a laboratory to predict the performance of a full-size prototype before manufacture. Lester Pelton tested wooden model wheels in the 1870s before patenting his impulse runner in 1880, and the same approach now governs every Francis, Kaplan and Pelton design at firms like Voith and Andritz Hydro. The model runs under controlled head and flow so engineers can measure efficiency, cavitation onset and runaway speed, then scale results using similitude laws. The outcome — fewer surprises and millions saved on multi-MW prototype builds.
Model Turbine Hydraulic Similitude Interactive Calculator
Vary model and prototype size, head, and test speed to see equal-unit-speed turbine scaling and the resulting prototype speed and flow scale.
Equation Used
The calculator applies hydraulic similitude for a geometrically scaled turbine. Matching unit speed gives the prototype rpm, while matching unit discharge gives the flow scaling factor between model and prototype.
- Model and prototype are geometrically similar.
- Unit speed N11 and unit discharge Q11 are matched.
- Efficiency step-up and Reynolds corrections are not included.
- Heads are net turbine heads.
The Model Turbine in Action
A Model Turbine is a precise geometric copy of a proposed full-size runner — usually 1/10 to 1/25 scale — mounted on a closed-loop test rig with a calibrated pump, head tank, flow meter and dynamometer. You drive water through it at a controlled head, brake the shaft, and read torque, speed and discharge. From those four numbers — head H, flow Q, speed N, torque T — you build the full performance hill chart that predicts the prototype.
The physics works because of hydraulic similitude. If you keep the runner geometry identical and match the unit speed N11 = N × D / √H and unit discharge Q11 = Q / (D2 × √H), the model and prototype operate at hydraulically identical points. Efficiency does not scale 1:1 — the prototype is always a few points more efficient because boundary-layer losses shrink with size. Engineers correct for this with the Moody or IEC step-up formula, typically adding 2-4 efficiency points from a 1/15 model to a 200 MW prototype.
Where it goes wrong is tolerances. Blade profile must hold to within 0.1 mm on a 300 mm-diameter Francis model — not 0.2, not 0.5 — because a 0.3 mm leading-edge burr changes incidence angle by 1° and shifts the best-efficiency point by 5% flow. Surface roughness has to track the prototype's relative roughness, not the absolute. And if your test head fluctuates more than ±0.5%, the cavitation sigma curve smears and you cannot resolve the inception point. Most lab errors trace to one of three things: leaking shaft seals letting air in upstream of the suction cone, bearing friction the dynamometer never sees, or thermal drift in the flow meter over a 6-hour test sweep.
Key Components
- Scale Runner: The geometrically faithful copy of the prototype runner, typically machined from stainless steel or bronze on a 5-axis mill to ±0.05 mm on the blade profile. Diameter sits between 250 and 500 mm for most lab rigs — small enough to handle, large enough that Reynolds-number effects stay tractable.
- Spiral Case and Distributor: Scaled volute and guide-vane ring that feed water into the runner at the correct angle. Guide vane opening is the primary control variable on a Francis or Kaplan model, swept from 20% to 110% of nominal during a hill-chart run.
- Draft Tube: The diverging passage downstream of the runner that recovers kinetic energy as static pressure. Geometry must match the prototype exactly because draft-tube surge and vortex rope behaviour drive part-load instability — the model is the only way to see it before commissioning.
- Cradle Dynamometer: An electric or hydraulic brake mounted on trunnion bearings that measures shaft torque to within ±0.1%. Combined with a magnetic speed pickup it gives you mechanical power output Pm = T × ω directly.
- Electromagnetic Flow Meter: Calibrated to ±0.2% on the supply line, sized for full-scale flows of 0.3-1.5 m³/s. Calibration drift is the silent killer of model test campaigns — most labs re-calibrate against a weigh tank every 6 months.
- Head Measurement: Differential pressure transducers across the spiral inlet and draft tube exit, referenced to a still-well manometer for absolute calibration. Resolution must be 0.01 m water column on a 30 m test head.
Where the Model Turbine Is Used
Model turbines exist because nobody builds a 300 MW Francis runner on a hunch. Every major hydro project, every marine propeller, and every large pump-turbine passes through a scaled lab test before steel gets cut. The applications below are where the technique earns its keep, and the named labs are where most of this work actually happens — IIHR at Iowa, EPFL's LMH in Lausanne, the Voith hydraulic laboratory in Heidenheim, and Andritz Hydro's facility in Linz.
- Large Hydropower: Voith tested 1/12-scale Francis models for the Three Gorges 700 MW units at their Heidenheim hydraulic laboratory before committing to the prototype geometry.
- Pumped Storage: Andritz Hydro runs reversible pump-turbine model tests at their Linz lab for projects like the 250 MW Kops II plant in Austria, where the same runner must operate efficiently in both directions.
- Small Hydro and Refurbishment: Gilkes in Kendal UK tests scaled Pelton and Turgo models before re-runnering heritage stations like the 1920s Kinlochleven scheme, validating efficiency gains before committing to a full casting run.
- Marine Propulsion: MARIN in the Netherlands runs cavitation tunnel tests on scale propellers and Kort nozzle assemblies for ships like Maersk Triple-E class container vessels, treating the propeller as a model turbine in reverse.
- Tidal and In-Stream: OpenHydro and Orbital Marine ran scale rotor tests at the European Marine Energy Centre in Orkney before deploying full-size 2 MW tidal turbines off the Scottish coast.
- University Research: EPFL's LMH laboratory in Lausanne tests reduced-scale Kaplan and Francis models on its high-precision rig PF3 for industrial partners and PhD work on draft-tube vortex behaviour.
The Formula Behind the Model Turbine
The two most useful similitude expressions are unit speed N11 and unit discharge Q11. They let you take a measurement on a 300 mm model at 20 m head and predict performance of a 6 m prototype at 200 m head. At the low end of a typical test envelope — say 30% guide-vane opening — you are mapping part-load behaviour and looking for draft-tube surge. At nominal opening you find the best-efficiency point. At the high end, 110% opening, you are pushing for runaway speed and cavitation inception data. The sweet spot for hill-chart density is 60-90% opening, which is where prototypes spend 80% of their service life.
Variables
| Symbol | Meaning | Unit (SI) | Unit (Imperial) |
|---|---|---|---|
| N11 | Unit speed — speed of a 1 m runner under 1 m head | rpm·m1/2 | rpm·ft1/2 |
| Q11 | Unit discharge — flow of a 1 m runner under 1 m head | m³/s | ft³/s |
| N | Rotational speed measured on the model | rpm | rpm |
| D | Runner reference diameter | m | ft |
| H | Net head across the runner | m | ft |
| ηm, ηp | Model and prototype peak efficiency | dimensionless | dimensionless |
| Dm, Dp | Model and prototype runner diameters | m | ft |
Worked Example: Model Turbine in a heritage 12 MW Francis refurbishment in Norway
You are scaling a 1/14 Francis model test campaign at the EPFL LMH laboratory in Lausanne for the refurbishment of a 12 MW unit at the Tyssedal heritage hydropower station in Hardanger Norway. The prototype runner is 1.85 m in diameter, design net head is 220 m, and the original 1918 runner is being replaced with a modern profile. Your model is 132 mm reference diameter, tested at a lab head of 30 m. You measured peak efficiency ηm = 0.928 at N11 = 72 rpm·m1/2 and Q11 = 0.42 m³/s on the model.
Given
- Dm = 0.132 m
- Dp = 1.85 m
- Hp = 220 m
- Hm = 30 m
- N11 = 72 rpm·m1/2
- Q11 = 0.42 m³/s
- ηm = 0.928 —
Solution
Step 1 — solve for prototype speed at the nominal best-efficiency point using N11 = N × D / √H rearranged for N:
That sits cleanly on a 12-pole synchronous generator at 50 Hz (500 rpm) once you trim the runner diameter — the hill chart tells you which way to trim.
Step 2 — solve for prototype flow:
Step 3 — apply the Moody step-up to predict prototype efficiency:
Step 4 — at the low end of the operating range, 60% guide-vane opening, the model measured ηm = 0.89 which steps up to roughly ηp ≈ 0.94 on the prototype — perfectly usable for off-peak operation but with visible draft-tube vortex rope. At the high end, 110% opening, ηm dropped to 0.86 with cavitation pitting visible on the suction-side blade trailing edge after 2 hours of test running, predicting a prototype efficiency around 0.91 and a hard upper limit on continuous operation.
Step 5 — compute prototype hydraulic power at best-efficiency point:
That is shaft power before generator losses — well above the 12 MW unit rating, which means the runner is sized for a much smaller flow than full-Q11. The actual operating point lives at about 27% of full discharge on this hill chart, which is why the original 1918 runner was efficient on paper but unstable at part load.
Result
The prototype hits 577 rpm at the best-efficiency point with 21. 3 m³/s flow and a stepped-up efficiency of 96.6%. In practice that means the new Tyssedal runner will produce roughly 1.5 percentage points more output than the 1918 original at the same flow — about 180 kW of free power on a 12 MW unit, paying back the model test campaign inside one operating year. Across the operating range, expect 94% efficiency with a visible vortex rope at 60% gate, 96.6% clean running at nominal, and a hard cavitation ceiling around 110% gate where you cannot operate continuously. If your commissioning measurements come in 1-2 points below the predicted 96.6%, the usual suspects are: (1) prototype surface roughness exceeding the equivalent-sand-grain target Ks ≤ 8 µm specified for the cast finish, (2) a guide-vane closure gap above 0.3 mm leaking water past the closed position and biasing the flow measurement, or (3) draft-tube cone misalignment by more than 2 mm shifting the swirl recovery off-design.
Choosing the Model Turbine: Pros and Cons
Scale model testing is one of three ways to validate a turbine design before manufacture. Each has its place, and the cost-vs-confidence trade is the main decision driver for project owners.
| Property | Model Turbine Test | CFD Simulation | Full-Size Prototype Test |
|---|---|---|---|
| Capital cost (typical campaign) | €500k - €2M for 1/15 scale model + 6-week lab time | €50k - €300k for full hill-chart CFD | €20M+ for a 100 MW prototype |
| Efficiency prediction accuracy | ±0.3 points after Moody step-up | ±1.0 to ±1.5 points, depends on mesh and turbulence model | ±0.1 points (it is the prototype) |
| Cavitation inception detection | Direct visual + acoustic, gold standard | Approximate via vapour-pressure threshold, often optimistic | Direct, but you have already built the runner |
| Draft-tube surge / vortex rope | Captured directly with high-speed video | Possible with unsteady scale-resolving CFD, expensive | Captured but at full project risk |
| Time to result | 3-9 months including model build | 4-12 weeks | Months to years, plus refit risk |
| Best application fit | Multi-MW units where 1% efficiency = €100k/yr | Early concept screening, geometry optimisation | Final acceptance test only, never design validation |
| Reusability across projects | Model usable for similar head/speed family | Mesh and setup directly portable | None — site-specific |
Frequently Asked Questions About Model Turbine
The Moody formula assumes the prototype achieves the same relative surface roughness as the model — and that is rarely true on a first cast. A polished 132 mm model runner with Ra 0.4 µm has relative roughness around 3×10-6, while a typical sand-cast 1.85 m prototype with Ra 12 µm sits closer to 6×10-6. Twice the relative roughness costs you 0.5 to 1.0 efficiency points.
Check the prototype's blade surface against the spec — anything above Ks 8 µm equivalent sand grain on the suction side will show up as missing efficiency. Hand-finishing the trailing 30% of each blade usually recovers the gap.
The IEC 60193 standard sets the lower limit at 250 mm reference diameter for a Francis or Kaplan model, with a Reynolds number not below 4×106 at the test head. Below that, viscous scale effects start to dominate and the Moody step-up loses accuracy — you end up over-predicting prototype efficiency by 1-2 points.
For Pelton wheels the floor is higher, around 300 mm pitch diameter, because the jet-bucket interaction is sensitive to surface tension at small scale. If you are tempted to go smaller for cost reasons, save the money and run CFD instead — the answer will be more accurate than a too-small physical model.
Below about 10 MW the economics usually favour CFD plus a workshop string test on the actual prototype runner. A €1M model campaign cannot pay back on a project where 1% efficiency is worth maybe €40k/year. Above 20 MW the calculation flips and you almost always want a physical model.
The exception is anything with unusual head, unusual setting (high submergence variation), or known draft-tube instability — model tests catch part-load surge and cavitation that CFD still struggles to predict reliably. If your site has a wide head range or run-of-river flow swings, pay for the model.
This is almost always a draft-tube vortex rope that the model showed but nobody flagged. At 60-75% gate opening the runner exit swirl drives a precessing helical vortex in the draft-tube cone that resonates with the penstock or surge shaft on the full-size unit but not on the lab rig because the lab piping is short and stiff.
Check whether the model test report includes pressure pulsation traces in the cone — if pulsation amplitude exceeded 3% of head at the surge frequency, that was the warning. Fix on the prototype is usually an air admission valve at the cone or a fin-type vortex breaker, both of which can be retrofitted without re-runnering.
Yes, and you must — that is precisely why pump-storage projects justify the model cost. The runner has to operate efficiently as a turbine and as a pump with reversed rotation, and the design compromise is a single geometry that does both. Andritz Hydro and Voith both run four-quadrant test campaigns on reversible models, sweeping turbine, pump, runaway and reverse-pump regimes on the same rig.
What changes is the test rig plumbing — you need a four-quadrant capable circuit with reversible flow direction, which not every lab has. EPFL LMH, Voith Heidenheim and Andritz Linz all do, but smaller facilities typically only run unidirectional turbine tests.
Sigma at inception (σi) is dimensionless and transfers directly from model to prototype — that is the whole point of the parameter. You measure σi on the model by lowering test-rig pressure until you see or hear cavitation onset, then specify prototype setting depth so that the plant sigma σp = (Hatm − Hvapour − Hs) / H exceeds σi with a safety margin of typically 1.3-1.5×.
The trap is that σi is highly sensitive to operating point — it can rise sharply at part-load and high-load extremes. Use the σi at the worst operating condition the plant will see continuously, not the best-efficiency value. Many older plants were set on best-efficiency sigma and now suffer cavitation damage every time they run off-peak.
References & Further Reading
- Wikipedia contributors. Water turbine. Wikipedia
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