Bóbr · Pilchowice I and II · P = 9.81 · Q · H · η
From waterwheel to turbine on the Bóbr
Head has width. The same power can come from a tall, narrow Q × H stream or a low, wide one. Change set A and see which set B gives the same.
- A, useful power
- losses (1 − η)
- B, same power
| Quantity | A · you set it | B · calculated |
|---|---|---|
| Flow Q [m³/s] | 6.75 | 85.5 |
| Head H [m] | 38 | 3 |
| Product Q·H [m³/s·m] | 256.5 | 256.5 |
| Gross power 9.81·Q·H | 2.52 MW | 2.52 MW |
| Useful power (η = 85%) | 2.14 MW | 2.14 MW |
Set B (H = 3 m) needs 12.7× more water than A to give the same useful power (2.14 MW). The rectangle areas are equal: Q·H = 256.5 m³/s·m.
Head A: 38 m.
Flow A: 6.75 m³/s.
Set B: choose the head [m]
H = 3 m: head of Pilchowice II at Nielestno from the operator's table (per V2, to be verified). Q of set B is calculated from equal power; it is not operator data.
Load into set A
Pilchowice I, turbine unit no. 1: Q = 6.75 m³/s and H = 38 m are values from the operator's table (per V2, to be verified). Efficiency η is an assumption.
Pilchowice I · dam opened · working head 38 m, wall 62 m. Source per V2, to be verified
Pilchowice II at Nielestno · commissioned · head 3 m. Source per V2, to be verified
Power stays put, energy grows with time
The power P of set A from the chart above is the rate at which the water does work at a single moment. It does not depend on how long the water flows. Energy is that power times time: E = P · t.
Efficiency η is a single number for all the losses in the system. It is an assumption of the model, not a measurement of any machine.
The model assumes constant flow and constant head for the whole time t. It is not an annual operating series or a reading of a power station's output. t is a duration in hours, not an hour of the day.
How much of the water's power remains useful
useful power 85%losses (hatched) 15%
- Gross water power
- 2.52 MW
- Useful power, constant over time
- 2.14 MW
- Losses
- 377 kW
How much energy from this power over time t
After 24 h at 2.14 MW: 51.33 MWh
Bar: time t on a scale of 0–8760 h (a year). The 1 h row is narrower than a pixel, and 24 h is only 1–2 px; the difference shows only in the numbers.
- 1 h2139 kWh
- 24 h51.33 MWh
- 720 h (30 days)1.54 GWh
- 8760 h (year)18.74 GWh
A dam wall is not head. A wheel is not a turbine.
There is one power formula. What differs is how many metres of head the water actually gives up to the machine, and what the machine does with it. All the heights below are on a single scale of 0–70 m.
- Dam wall, Pilchowice Iheight of the structure, operator's table per V2, to be verified62 m
- Working head, Pilchowice Ioperator's table per V2, to be verified38 m
- Head, Pilchowice II at Nielestnooperator's table per V2, to be verified3 m
- Mill waterwheelteaching model, not data of any particular mill1.5 m
What the machine does with the water's power
Waterwheel
Water fills or pushes the blades and turns the wheel. The user gets the rotation of a shaft, which drives the mill's machinery. In the model: Q = 1.2 m³/s, H = 1.5 m, η = 45%. These are illustrative figures.
Turbine with generator
Water flows through a pipe into the casing and turns the runner, and the runner turns the generator. You get electricity that can be sent on. The operator lists six Francis turbines at Pilchowice I after the 2013–2014 modernisation (per V2). In the model, η = 85% is an assumption.
The diagram is conceptual. It does not show the actual machines at Pilchowice and is not a technical drawing.
Dates from the operator's material (per V2, to be verified)
- 1903Work on the water stage begins. The river had to be diverted before the main dam could be built.
- 1912Opening of the dam, with a ceremony on 16 November. The history of the structure goes back to the protection programme after the flood of 1897.
- 1927Pilchowice II at Nielestno was commissioned on 13 October. A run-of-river, diversion power station; head 3 m.
- 2013–2014Modernisation at Pilchowice I; the operator lists six Francis turbines.
Where to see it: from public places only
This list is separate from the calculation. It does not show the power station's current output or the water level and does not guarantee access. The descriptions come from V2 and need checking before publication and before you set out.
- Crest of the Pilchowice I dam. A pedestrian promenade on the crest with a view of the dam wall and the reservoir.
- Shore of Lake Pilchowice. A view over the reservoir basin from a public trail along the shore.
- Nielestno, Pilchowice II. A view from a public road of the water stage and the diversion canal.
Safety. Keep to publicly accessible trails, roads and the dam crest, where it is open. Do not enter weirs, technical structures, railway tracks or areas closed off by the operator. This page does not replace signage or instructions from officials on site.
Sources and limits of the model
Sources carried over from V2
- TAURON Ekoenergia, Pilchowice I: wall 62 m, working head 38 m, Q of turbine unit no. 1 = 6.75 m³/s, construction 1903–1912, modernisation 2013–2014, six Francis turbines.
- TAURON Ekoenergia, Pilchowice II: Nielestno, commissioned 13 October 1927, diversion type, head 3 m.
- Photo of the dam, Fallaner, Wikimedia Commons (licence CC BY 4.0, 2014). Link only; the photo is not embedded, so that the page works offline.
Status: the addresses and figures were carried over from V2. They were not reopened when V3 was built. Everything marked “per V2” needs checking by the editors.
Limits of the model
- Power: P = 9.81 · Q · H · η in kW, for Q in m³/s, H in m, at a water density of 1000 kg/m³. A single value of η covers all losses.
- For Nielestno the material gives only a head of 3 m. The flow of set B is calculated from equal power; it is not operator data, and it is not known whether the river carries it.
- The result is not a rated power or a power station's output. There are no flow series, heads varying over time or nameplates here.
- Values outside the range of the fields are not silently corrected: the field shows an error, and the results are replaced by a “—” sign.
What changed relative to V2
- Removed the notebook saved in the browser: it was not needed to understand the topic, and V3 saves nothing locally.
- Removed the embedded photo from the internet. It was replaced by a height scale of 0–70 m, and the photo became a link.
- Removed “Nielestno” with a preset Q = 6.75 m³/s. That was a hypothetical number that looked like data. Now Q of set B is calculated from equal power.
- Four controls at once were split up: Q and H on the chart, η and t in the second part.
- The balance bars were replaced by hatching of losses in the rectangle and a bar showing the split by η.
- Kept: the power formula, the values 6.75 m³/s, 38 m, 3 m and 62 m, the waterwheel model (1.2 m³/s, 1.5 m, 45%), the equivalent flow for 3 m, the safety rules and the links to the operator.