A Paddle Wheel (form 2), also called a breast-shot wheel, is a vertical-axis water wheel where the stream strikes the paddles roughly at axle height rather than at the top or the bottom. John Smeaton's 1759 efficiency experiments at the Royal Society showed this mid-strike configuration extracts power from both the kinetic energy of the flow and the weight of water carried in the buckets. The wheel converts a head of 1.5 to 5 metres into rotational shaft power at 60-75% efficiency, driving mill stones, generators, or aeration paddles where head is too low for an overshot wheel.
Paddle Wheel (form 2) Interactive Calculator
Vary head, flow, efficiency, and wheel diameter to estimate hydraulic power, shaft power, speed, and torque for a breast-shot paddle wheel.
Equation Used
The calculator estimates available water power from head and flow, then applies the selected breast-shot wheel efficiency to obtain shaft power. Wheel speed is based on the article guidance that paddle tip speed should be about half the entry water velocity.
- Fresh water density is 1000 kg/m3.
- Efficiency represents total hydraulic-to-shaft efficiency.
- Best tip speed is approximated as half the entry water velocity.
- Flow is steady and the breast gap is properly controlled.
Inside the Paddle Wheel (form 2)
Water enters through a sluice or pentrough and hits the paddles between roughly 9 o'clock and 12 o'clock relative to the wheel axle — that's the defining feature of a breast-shot or form-2 paddle wheel. The flow does two jobs at once. The first is impulse: the moving water pushes the paddle face, transferring kinetic energy. The second is gravity: water trapped between the paddles and the curved breast (a close-fitting masonry or steel arc) rides down with the wheel and adds weight torque all the way to the tailrace. That dual loading is why a properly built breast-shot wheel hits 60-75% hydraulic efficiency, sitting between the undershot wheel (around 25-35%) and the overshot wheel (around 70-85%).
The geometry has to be right or you lose both halves of the energy budget. The breast — the curved wall hugging the wheel — must sit within 10-25 mm of the paddle tips. Open that gap to 50 mm and water short-circuits the paddle, slipping past instead of pushing it, and you can drop 15-20% efficiency overnight. Paddle count typically runs 24-48 for wheels in the 2-5 m diameter range; too few paddles and water spills between them, too many and the bucket volume shrinks until each paddle traps almost nothing. Tip speed wants to sit near half the entry water velocity for best energy transfer, which is why low-head wheel design starts with the available head and flow before anything else.
Failure modes are mechanical, not exotic. Wooden paddles waterlog and rot at the waterline first — replace them in pairs to keep the wheel balanced. Iron banding rusts through where it sees alternating wet and dry conditions. The tailrace silts up and raises the downstream water level, drowning the lower paddles and creating drag that can cost you another 10% of shaft power before you notice the wheel is running slower than last season.
Key Components
- Paddles (or Buckets): Flat or slightly cupped boards mounted radially around the rim. For a 3 m wheel, paddle width typically runs 600-1200 mm with a depth of 200-300 mm. Wood (oak, larch) or galvanised steel — wood is forgiving on small debris, steel lasts longer in continuous service.
- Shrouds (Side Rings): The two circular side plates that contain the water between paddles. Without proper shrouds the water spills sideways before the paddle reaches the bottom of its arc, and you lose the gravity component of torque. Tolerance to the breast: 5-15 mm side clearance.
- Breast (Apron): The curved masonry, concrete, or steel wall that hugs the lower-front quadrant of the wheel. Gap to paddle tips: 10-25 mm. This is the single most important tolerance on the whole machine — open the gap and you bleed efficiency directly.
- Sluice Gate and Pentrough: The controlled water inlet at roughly the 10 o'clock position. The sluice meters flow to match generator load; the pentrough shapes the jet so it hits the paddle face cleanly without splashing. A poorly aimed jet wastes 5-10% of the available head as turbulence.
- Axle and Bearings: Forged steel axle running in oil-bath bronze or self-aligning roller bearings. Wheel speeds are slow — 4 to 12 RPM is typical — so bearing life is measured in decades, not hours, provided the seals keep silty water out of the lubricant.
- Tailrace: The downstream channel that carries spent water away. Must be deep enough that the lowest paddle never plunges into standing water. If the tailrace level rises within 50 mm of the lowest paddle, you can measure the efficiency drop with a stopwatch and a tachometer.
Real-World Applications of the Paddle Wheel (form 2)
A breast-shot paddle wheel earns its place where you have moderate head (1.5-5 m) and steady flow, and where overshot construction is impractical because you can't get the water above the wheel. You see it on restored historic mills, on small off-grid hydro installations, on aeration ponds, and in industrial process equipment where the gentle, high-torque output is more useful than high RPM. The wheel runs slow, so direct-drive applications need geared output to reach generator speeds, and that gearing — usually a 1:50 to 1:200 ratio — drives a fair chunk of the install cost.
- Heritage Restoration: Mapledurham Watermill on the Thames runs an authentic breast-shot wheel restored in the 1980s, still grinding flour for sale and exhibiting the original Georgian-era hydraulics.
- Off-Grid Micro-Hydro: Powerhouse Hydro and Hydrowatt supply 3-5 m breast-shot wheels for rural sites in the UK and Germany, generating 2-15 kW continuous from creek-fed millponds.
- Aquaculture and Aeration: Paddle-wheel aerators on shrimp and tilapia farms across Southeast Asia — Taiwan-built units from companies like Fu Hua and Linn — use the same form-2 geometry to entrain air into pond water at 0.5-1.5 kW per wheel.
- Paper and Textile Heritage Industry: Cromford Mill in Derbyshire and Wookey Hole Paper Mill in Somerset still demonstrate breast-shot wheels driving line shafting for tourist operation and small-batch artisan paper.
- Brewery and Distillery: Several Scottish and Bavarian craft distilleries use restored breast-shot wheels for grain milling — the slow, heavy crush is preferred over high-speed roller mills for traditional malt processing.
- Education and Research: Engineering departments at universities like Lancaster and TU Delft maintain instrumented breast-shot wheels for fluid-mechanics teaching and low-head hydro research.
The Formula Behind the Paddle Wheel (form 2)
Shaft power output from a breast-shot wheel comes down to head, flow, and an efficiency factor. The formula tells you what's mechanically available before generator and gearbox losses. At the low end of typical operating conditions — say 1.5 m head with seasonal flow — you're working with a few hundred watts and you need to decide whether the install cost is worth it. At the nominal sweet spot (3 m head, full design flow) the wheel hits its 65% efficiency target and produces the rated power. Push the head past 5 m and you're better off with an overshot or a Pelton turbine; push the flow past the design point and water overspills the paddles, efficiency collapses, and the extra flow does nothing useful.
Variables
| Symbol | Meaning | Unit (SI) | Unit (Imperial) |
|---|---|---|---|
| P | Shaft power output | watts (W) | horsepower (hp) |
| η | Hydraulic efficiency (0.60-0.75 for breast-shot) | dimensionless | dimensionless |
| ρ | Water density (≈1000) | kg/m³ | lb/ft³ (≈62.4) |
| g | Gravitational acceleration (9.81) | m/s² | ft/s² (32.2) |
| Q | Volumetric flow rate | m³/s | ft³/s (cfs) |
| H | Effective head (vertical drop across wheel) | metres (m) | feet (ft) |
Worked Example: Paddle Wheel (form 2) in a 4 kW off-grid micro-hydro install
You are sizing a breast-shot wheel for a small cheese dairy in the Yorkshire Dales running off a millpond with 3.0 m of head and 0.20 m³/s of design flow. The dairy needs 3-4 kW of continuous shaft power to drive a refrigeration compressor and lighting through a small alternator and gearbox. You want to know what the wheel produces at low summer flow, design flow, and winter spate.
Given
- H = 3.0 m
- Qnom = 0.20 m³/s
- η = 0.65 —
- ρ = 1000 kg/m³
- g = 9.81 m/s²
Solution
Step 1 — at nominal design flow of 0.20 m³/s, calculate gross hydraulic power before efficiency:
Step 2 — apply the breast-shot efficiency factor of 0.65 to get nominal shaft power:
That hits the dairy's requirement with a small margin. The wheel turns at roughly 8 RPM, the gearbox steps up to 1500 RPM for the alternator, and you have steady output day and night.
Step 3 — at the low end of the operating range, summer flow drops to 0.08 m³/s:
That's enough to keep the lights on and the cold store ticking over but you'll need a battery buffer or grid backup to run the compressor. This is the reality of low-head hydro — your generation curve follows the catchment, not the load.
Step 4 — at the high end, winter spate brings flow up to 0.35 m³/s, but the sluice is set to pass only the design flow plus 10% headroom (0.22 m³/s) because anything above that overflows the paddles:
The extra spate water bypasses the wheel via the bywash channel. If you tried to force all 0.35 m³/s through the paddles, water would back up against the breast, the wheel would drown at the bottom, and efficiency would crash to maybe 0.40 — you'd actually generate less power than at design flow.
Result
Nominal shaft power is 3. 8 kW at 3.0 m head and 0.20 m³/s flow with 65% wheel efficiency — enough to run the dairy's compressor and lighting through a 1:185 step-up gearbox to a 1500 RPM alternator. At summer low flow you drop to 1.5 kW (cold store and lights only), at controlled spate flow you reach 4.2 kW, and the sweet spot for steady year-round operation sits right at the design flow. If you measure 2.5 kW instead of the predicted 3.8 kW at full flow, the usual culprits are: (1) breast-to-paddle gap opened up beyond 25 mm from worn timber paddle tips, letting water short-circuit the wheel; (2) the alternator running off its peak power point because the sluice is throttling flow to match a fixed-load instead of using a controller; or (3) sluice jet aimed too high or too low, hitting the paddle off-centre and dumping energy as splash rather than torque.
When to Use a Paddle Wheel (form 2) and When Not To
A breast-shot wheel is the right answer for a specific window of head and flow. Outside that window, other low-head and medium-head technologies do the job better. Here's how it stacks against the two alternatives you're most likely to be comparing it against in a real micro-hydro decision.
| Property | Breast-shot Paddle Wheel (form 2) | Overshot Water Wheel | Crossflow Turbine |
|---|---|---|---|
| Hydraulic efficiency | 60-75% | 70-85% | 75-85% |
| Useful head range | 1.5-5 m | 3-12 m | 2-200 m |
| Output shaft RPM | 4-12 RPM (heavy gearing needed) | 2-8 RPM (heavy gearing needed) | 300-2000 RPM (direct-drive possible) |
| Tolerance to debris and silt | High — paddles shrug off leaves and twigs | Medium — buckets can clog | Low — needs trash rack and clean water |
| Capital cost (per kW installed) | £3000-6000/kW | £4000-8000/kW | £2000-4000/kW |
| Service lifespan (civil works) | 80-150 years (well-built masonry) | 80-150 years | 30-60 years |
| Maintenance interval (paddles/runner) | Inspect annually, replace paddles 15-25 yrs | Replace buckets 20-40 yrs | Runner overhaul 8-15 yrs |
| Best application fit | Heritage sites, low-head off-grid, aeration | Small streams with vertical drop | Modern micro-hydro with reliable head |
Frequently Asked Questions About Paddle Wheel (form 2)
Check the tailrace level, not the inlet. Late-summer growth — algae, weed, and silt deposition — raises the downstream water level and starts drowning the lower paddles. Once the tailwater rises within 50 mm of the lowest paddle, that paddle is now pushing against standing water on its way out of the arc, and you've turned the bottom of the wheel from a power-producing region into a drag-producing one.
Walk the tailrace, clear weed and built-up gravel, and re-measure. A 100 mm drop in tailwater can recover 8-12% of shaft power on a typical 3 m wheel. This is one of the most common seasonal performance complaints and almost nobody thinks to look downstream first.
Both will work hydraulically. The decision comes down to three things: site context, debris load, and what you do with the output. If the site is heritage-sensitive or has visible amenity value, the wheel wins on planning permission alone. If the water is full of leaves and small debris and you don't want to maintain a fine trash rack, the wheel wins on tolerance. If you need 1500 RPM alternator output and want to minimise gearbox cost, the crossflow wins because it spins fast enough to direct-couple via a belt.
For a working dairy or off-grid cabin where output matters more than appearance, a crossflow is usually the more economic choice per kWh delivered. For a restored mill site or a smallholder who values the aesthetic and the slow, quiet running, the breast-shot wheel is worth the extra capital.
You're past the design flow of the wheel. Each paddle pocket has a fixed volume; once you exceed the rate at which the wheel can carry water down through its arc, the surplus overflows the paddle and bypasses the wheel entirely. The water you see spilling over is doing zero work.
This is why over-supplying flow doesn't increase power linearly — past the design point, efficiency drops because the surplus water adds turbulence and back-pressure without adding torque. The fix is to set your sluice to pass design flow plus around 10% headroom and route everything else down a bywash channel. You'll generate more total power passing 0.20 m³/s cleanly through the wheel than 0.30 m³/s sloppily.
For a 3 m wheel, 32-40 paddles is the practical range. The trade-off: too few paddles (under 24) and water spills between them before each paddle has carried it through the productive arc, costing efficiency. Too many paddles (over 48) and the volume between adjacent paddles shrinks so much that you carry less water per paddle, plus you've added cost and weight for no gain.
The classical rule from 19th-century millwright practice is one paddle every 250-300 mm of rim circumference, which lands you in that 32-40 range for a 3 m wheel. If you're restoring a historic wheel, count the original mounting holes in the shrouds before you order new paddles — the original builder usually got this right.
The 0.65 figure assumes the entire head H is converted between sluice and tailrace. In real installs you lose head to three places that aren't always obvious: friction in the headrace and pentrough (5-10% of H if the channel is rough or undersized), entry losses where the jet meets the paddle (another 3-5% if the sluice geometry is wrong), and exit losses if the wheel discharges into a tailrace that's too shallow or too narrow.
Add those up and the effective head doing useful work can be 80% of the measured static head, which knocks calculated power down by 20% before you even apply wheel efficiency. The fix is civil-engineering work, not wheel work — clean up the headrace, correctly shape the pentrough lip, and deepen the tailrace.
10-25 mm is the practical range. Tighter than 10 mm and you risk paddle tips striking the breast when the timber swells in winter or the wheel hub takes up bearing slack — a wheel turning at 8 RPM with a 600 mm-wide paddle hitting masonry will splinter the paddle and crack the breast.
Wider than 25 mm and water short-circuits past the paddle. At 50 mm gap you can lose 15-20% of shaft power directly. The classical millwright trick is to set the gap at 15 mm with new oak paddles, knowing the timber will swell roughly 3-5 mm in service, leaving you at a stable 10-12 mm running clearance. Steel paddles can sit at 10 mm permanently because they don't move with moisture.
This is almost always a drivetrain issue, not a wheel issue. Between the wheel shaft and the alternator terminals you have: gearbox losses (typically 5-8% per stage, so a two-stage 1:185 gearbox eats 10-15%), belt losses if there's a belt drive (3-5%), alternator efficiency itself (75-90% depending on load matching), and resistive losses in cabling.
Stack those up and a 4 kW wheel realistically delivers 2.8-3.2 kW at the terminals — which matches what you're seeing. If the number is below that range, check the alternator load matching first. Many off-grid alternators are most efficient at one specific RPM and load combination; running them off-design point can cost you another 10-15%. A controller that tracks the wheel's power point will recover most of it.
References & Further Reading
- Wikipedia contributors. Water wheel. Wikipedia
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