Suppose you have a stream that is 1 meter wide and 30 cm deep. The water flows at a speed of 3 sec/meter. How much water does it contain and how much can you pump up without loss?
We first make two assumptions to reduce the problem to its simplest form:
1) the stream has a flat bottom and perpendicular banks; its cross-section is therefore a rectangle with sides 1 m and 0.3 m and area 0.3 m²
2) the velocity is the same over the entire cross-section, at the bottom as well as at the surface; So 3 m/s.
Then 0.3 x 3 = 0.9 m³ of water flows past per second. So if you were to pump out 0.9 m³ per second, you would lead all the water through the pump and the stream downstream would be dry.
More pumps are not possible because there is simply no more water available.
The reality is less simple. The bed of a stream is not a rectangle, the banks have a slope. The cross-section is better approximated by a trapezoid.
In any case, the true cross-section, expressed in m², must be used.
The speed is also not the same everywhere, along the banks and against the bottom it is smaller than in the middle of the surface. You will have to calculate with an average speed.
The number of m³ that flows by per second is then : area of the cross-section in m² times the average speed in m/s.
And that will be smaller than the higher calculated value.
prof. French Cerulus
physics, especially classical theoretical mechanics, electromagnetism, quantum mechanics, history of physics .
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