A hemispherical cup windmill is a drag-type vertical-axis wind rotor that uses two or more half-sphere cups mounted on horizontal arms around a vertical shaft. The cup is the working component — its concave face catches roughly 4× the drag of the convex back face, producing a net torque that spins the shaft regardless of wind direction. The geometry self-starts in light wind and needs no yaw mechanism, which is why it powers low-RPM water pumps, aeration shafts, and Robinson-style anemometers where reliability beats efficiency. Real installations run from 10 W classroom rigs to 2 kW off-grid pumping plants.
Hemispherical Cup Windmill Interactive Calculator
Vary wind speed, cup size, and drag coefficients to see the high-drag, low-drag, and peak cyclic cup loading.
Equation Used
The calculator applies the drag equation to the concave and convex sides of a hemispherical cup. The article's key comparison is the drag coefficient split, Cd about 1.42 for the concave face and about 0.38 for the convex back, giving roughly a 3.7:1 drag advantage that produces shaft torque.
- Standard air density rho = 1.225 kg/m3.
- Effective hemispherical cup area is used for the cyclic cup load estimate.
- Peak cyclic load uses a 1.1 allowance to match the article's approximate peak load statement.
- Cup drag is evaluated normal to the wind for the concave and convex faces.
The Hemispherical Cup Windmill in Action
The rotor works on a drag differential, not lift. When wind hits the concave face of a cup head-on, the drag coefficient CD sits around 1.42. When the same wind hits the convex back of the opposite cup, CD drops to roughly 0.38. That ~3.7:1 ratio is what produces net torque — every cup on the rotor is always either pushing the shaft or dragging on it, and the concave-side push always wins. Because the geometry is symmetric about the vertical axis, wind from any compass direction produces the same torque, so you do not need a tail vane, a yaw bearing, or any directional control hardware.
The key design constraint is tip speed ratio (TSR). A hemispherical cup rotor caps out around TSR = 0.3 to 0.4 — meaning the cup's linear velocity never exceeds about 40% of wind speed. Push it harder and the back-running cup starts producing negative drag faster than the forward cup can deliver useful work, and rotor power collapses. This is why cup windmills are slow-RPM, high-torque machines: a 2 m diameter rotor in a 10 m/s wind tops out near 50-60 RPM, not 500. If you over-gear the output (asking for too high a load speed), the rotor stalls in light wind and never restarts until the gust hits 8+ m/s.
Failure modes are mostly mechanical, not aerodynamic. Arm-to-hub welds crack first because the cups see cyclic drag loading every revolution — a 200 mm cup in 12 m/s wind pulls about 11 N peak, reversing direction relative to the arm's bending axis twice per turn. Cup deformation is the second issue: a dented concave face loses CD faster than people expect, and a 5 mm dent on a 200 mm cup measurably drops rotor torque. Bearing wear shows up as starting-wind-speed creep — a rotor that started at 2 m/s when new will need 3.5 m/s after the lower thrust bearing pits.
Key Components
- Hemispherical cup: The drag-generating element. Standard builds use a 150-300 mm diameter half-sphere in spun aluminium, glass-reinforced polyester, or deep-drawn stainless. Wall thickness 1.5-2.5 mm balances stiffness against rotor inertia. The lip must be rolled or hemmed — a raw cut edge generates flow separation that drops effective CD by 8-10%.
- Radial arm: Connects each cup to the central hub. Length sets the swept radius and torque arm. Typically a 12-20 mm OD steel or aluminium tube. The arm-to-cup joint must be a through-bolt with backing plate, not a single rivet — single-fastener joints fail in fatigue within a season at exposed sites.
- Vertical shaft: Transmits rotor torque down to the load. Sized for the peak wind torque, not average. For a 2 m diameter, 4-cup rotor in a 25 m/s storm gust, peak shaft torque can hit 90 N·m even though working torque is 8 N·m. Undersize the shaft and it will whip and crack the upper bearing race.
- Thrust bearing pair: Carries the rotor weight and any axial wind load. Tapered roller bearings sized for L10 life of 30,000 hours at rated load. The lower bearing must be sealed against grit and rain — open ball races last under 2 years on a coastal installation.
- Hub: Joins the arms to the shaft. Machined from billet aluminium or cast iron with a keyed bore matched to the shaft within 0.05 mm. The bore must be 25.0 mm — not 24.95, not 25.1 — or the hub fretting opens up and the rotor develops a wobble within months.
Where the Hemispherical Cup Windmill Is Used
Hemispherical cup rotors live in applications where you need wind-driven mechanical work but cannot tolerate the directional sensitivity, blade fragility, or governor complexity of a horizontal-axis mill. They are slow, draggy, and inefficient compared to a propeller turbine — peak power coefficient Cp hovers around 0.12 versus 0.45 for a modern HAWT — but they self-start in 2 m/s wind, run unattended for years, and don't care which way the gust comes from. That trade is exactly right for instrumentation, low-head water pumping, aeration, and educational rigs.
- Meteorological instrumentation: The classic 3-cup or 4-cup Robinson anemometer used at every NOAA ASOS weather station — typical cup diameter 50 mm, arm radius 65 mm, calibrated to read wind speed from the rotor RPM via a fixed factor near 0.33.
- Off-grid water supply: A small hemispherical-cup windmill driving a piston pump on a 30 m bore at a remote cattle station near Tennant Creek, Northern Territory, where horizontal-axis Aermotor mills are too maintenance-heavy for the road access.
- Aquaculture aeration: Wind-driven paddle aerators on shrimp ponds in Soc Trang province, Vietnam — a 1.8 m diameter 4-cup rotor on a 4 m mast spins a vertical shaft straight into the pond's paddle wheel, no gearbox.
- Educational and STEM kits: The Pitsco Hydro-Wind Power Bundle and similar classroom kits use 80-120 mm plastic hemispherical cups to demonstrate drag-driven rotors before introducing lift-based blade theory.
- Low-pressure air supply: A 2.5 m diameter cup rotor driving a diaphragm air pump for a constructed-wetland aeration project in rural Bihar, India — the rotor delivers around 30 L/min of air at 0.2 bar in a 6 m/s mean wind regime.
- Heritage and demonstration: Replica panemone-style cup mills at the Anholt Windmill open-air collection in Denmark, illustrating the 9th-century Persian vertical-axis wind machines that predate the European tower mill.
The Formula Behind the Hemispherical Cup Windmill
The shaft power of a hemispherical cup rotor follows directly from the drag differential between the concave and convex faces. What matters in practice is how the output scales with wind speed cubed and how the design TSR limits the usable wind range. At the low end of typical operating wind (3 m/s), output is barely enough to lift a piston — the rotor just keeps the load moving. At nominal site wind (6-8 m/s), the rotor delivers its design power. At the high end (12+ m/s) you actually need to spill power or the bearings and weld joints take a beating. The sweet spot is matching rotor swept area to a load that demands roughly the cube of nominal wind speed.
Variables
| Symbol | Meaning | Unit (SI) | Unit (Imperial) |
|---|---|---|---|
| P | Shaft power output | W | ft·lbf/s |
| ρ | Air density (≈1.225 at sea level) | kg/m³ | slug/ft³ |
| A | Rotor swept area (2 × radius × cup diameter for a 2-cup mill, or full disc area for multi-cup) | m² | ft² |
| v | Free-stream wind speed | m/s | ft/s |
| Cp | Power coefficient (0.08-0.15 for hemispherical cup rotors) | dimensionless | dimensionless |
Worked Example: Hemispherical Cup Windmill in an off-grid borehole pumping rig in rural Karoo, South Africa
An off-grid game farm in the Karoo near Beaufort West fits a 2.0 m diameter, 4-cup hemispherical rotor on a 6 m mast to drive a single-acting piston pump on a 25 m borehole. Cup diameter is 250 mm, arm radius 1.0 m. Design wind is 6 m/s mean, with a typical operating range of 3-12 m/s. Air density at the 1100 m site elevation is 1.10 kg/m³. The owner wants to know the shaft power at the low end, nominal, and high end of the operating range so the pump cylinder bore can be sized to match.
Given
- Drotor = 2.0 m
- Dcup = 0.25 m
- ρ = 1.10 kg/m³
- Cp = 0.12 dimensionless
- vnom = 6 m/s
Solution
Step 1 — compute the rotor swept area. For a 4-cup rotor we use the full disc area projected by the cup centres:
Step 2 — at nominal 6 m/s wind, compute shaft power:
That is enough to lift roughly 1.4 L/min from 25 m head — modest, but exactly what a livestock trough demands.
Step 3 — at the low end of the operating range, 3 m/s:
At 3 m/s the rotor barely overcomes pump seal friction and check-valve cracking pressure. You will see the rotor turn but the pump column will not lift water consistently — this is the stall threshold for this load.
Step 4 — at the high end, 12 m/s:
That is an 8× jump from nominal because power scales with v3. In reality you cannot use all of it — TSR limits cap rotor RPM, and the pump connecting rod will hit a peak rod load roughly 4× nominal, which is why a slip clutch or spring-loaded over-travel coupling is mandatory above about 10 m/s.
Result
Nominal shaft power is 5. 6 W at 6 m/s wind, which translates to a steady trickle pump output of about 1.4 L/min from the 25 m bore — enough for a small herd but nothing more. The range tells the real story: 0.70 W at 3 m/s (effectively dead — pump barely moves), 5.6 W at 6 m/s (design point), and 44.8 W theoretical at 12 m/s (which the rotor cannot fully convert because TSR caps tip speed). If you measure 3 W at a verified 6 m/s wind instead of the predicted 5.6 W, check three things in this order: (1) cup orientation — one cup mounted backwards drops output by ~50%, this is the single most common build error; (2) shaft bearing drag from over-tightened thrust nuts, which adds 1-2 W of parasitic load and is felt as a stiff hand-spin test on a calm day; (3) cup wall flexing on thin-gauge stock under 1.5 mm, which deforms the concave shape under load and cuts effective CD by 15-20%.
When to Use a Hemispherical Cup Windmill and When Not To
Cup-drag rotors compete with two well-known alternatives in the same niche: the Savonius rotor (also drag-type but with curved scoops instead of cups) and the small horizontal-axis wind turbine (HAWT) like an Aermotor or Bergey. The right choice depends on what you actually need — peak efficiency, self-starting torque, directional independence, or low build cost.
| Property | Hemispherical Cup Windmill | Savonius Rotor | Small HAWT (multi-blade Aermotor type) |
|---|---|---|---|
| Peak power coefficient Cp | 0.08-0.15 | 0.15-0.25 | 0.30-0.40 |
| Tip speed ratio (TSR) | 0.3-0.4 | 0.6-1.0 | 1.0-2.0 (multi-blade), 5-7 (3-blade) |
| Starting wind speed | 1.5-2.5 m/s | 2.0-3.0 m/s | 3.0-4.5 m/s |
| Yaw / direction control needed | No — omnidirectional | No — omnidirectional | Yes — tail vane or active yaw |
| Build complexity | Low — symmetrical, no airfoil shaping | Low-medium — requires curved scoops | High — airfoil blades, yaw, governor |
| Typical lifespan in coastal exposure | 8-15 years | 10-20 years | 20-30 years (Aermotor) |
| Best application fit | Anemometers, low-flow pumping, aeration | Low-RPM mechanical drive, ventilation | Bulk water pumping, electrical generation |
| Cost per rated watt (2024 typical) | $15-30/W | $10-25/W | $3-8/W |
Frequently Asked Questions About Hemispherical Cup Windmill
This catches almost everyone the first time. The 4-cup rotor has two cups always pushing and two always dragging — the trailing pair adds mass, parasitic drag, and a second negative-torque arm without adding swept area at the working radius. A 2-cup rotor sees a cleaner flow and produces 60-70% of the work for half the cup count.
The reason 4-cup builds still exist is rotational smoothness. A 2-cup rotor delivers torque pulses twice per revolution that hammer the load. A 4-cup spreads it out evenly, which matters for piston pumps and anemometers where pulsing is a nuisance. If raw power is the goal, build 2-cup. If smoothness or instrument linearity matters, build 3 or 4.
You were burned by the cube law. Mean wind speed is not the same as mean wind power. At a site with an 8 m/s mean and typical Weibull k≈2 distribution, the rotor sits below its starting threshold for hours every day — and the moments above 8 m/s contribute disproportionately because power scales with v3.
Size the rotor for the lower-quartile wind speed of your site, not the mean. For an 8 m/s mean site, that means designing the pump load to start cleanly at 4-5 m/s. You will harvest dramatically more total energy over a season even though peak power looks lower on paper.
The deciding factor is starting torque versus running efficiency. Cup rotors have very high starting torque per unit swept area because the CD ratio between concave and convex faces is nearly 4:1. Savonius scoops have a lower ratio (closer to 2.5:1) but a higher TSR ceiling, so they convert more energy once spinning.
Pick the cup rotor when the load demands break-away torque from a stop every time wind picks up — diaphragm pumps, piston pumps, paddle aerators with sediment loading. Pick Savonius when the load runs continuously once started — alternators, centrifugal aerators, vent fans. For a clean aeration pump that idles between breezes, the cup rotor wins on restart reliability.
This is almost always a TSR mismatch with the load. As wind speed climbs, rotor RPM tries to climb proportionally to hold its design TSR around 0.35. If the load (pump, generator, gearbox) cannot accept the higher RPM — because of valve timing, flow restriction, or generator back-EMF — the rotor gets pushed off its TSR curve and torque collapses.
Diagnostic check: disconnect the load and confirm the rotor freewheels smoothly to a high RPM in the same wind. If it does, your load is the limiter. The fix is either a slipping clutch that lets the rotor over-speed under gust, a load that scales with speed (centrifugal pump rather than positive-displacement), or a smaller swept area matched to the load's actual speed envelope.
No — and this is where backyard scale-ups break. Power scales with rotor area (radius squared), so 3× diameter gives 9× power. Shaft torque scales the same way. But cup drag force on each individual cup scales with cup area, not radius, so you also need to think about arm bending stress separately. Worse, rotational stresses on the arm-to-hub joint scale with the square of tip speed and the cube of radius if you keep the same TSR.
The practical rule: when you double diameter, more than triple your shaft diameter, double your bearing rating, and re-engineer the arm-to-hub joint with a gusseted or flanged connection. A 3 m rotor built with 1 m rotor structural details will fail at the welds within the first stiff breeze.
The most common cause on field-built anemometers is asymmetric cup mass. If the four cups vary by even 2-3 grams from each other (common with hand-spun aluminium), the rotor develops a slight pendulum bias and over-reads at moderate wind speeds.
Second-most-common: arm length mismatch. Industrial anemometers hold arm radius to ±0.2 mm. A homebuilt with arm-length variation of 1-2 mm will produce non-linear calibration error that shows up as a percentage offset rather than a fixed offset. Match your cups by weight on a scale before assembly, and machine-cut the arms from one length of stock rather than measuring each one.
References & Further Reading
- Wikipedia contributors. Anemometer. Wikipedia
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