Undershot water wheel

[Ahto]

Hi,

I was trying to estimate the power available from any given size paddle wheel floating on barges, but couldn't find any explanation on how the 'theoretical head' for such an undershot wheel is calculated. In all the examples the number is somehow assumed and then popped right into the usual P(W) = head(m) * volume(m3/s) * 9,81. It should come from mass and velocity somehow, right? Can anyone help me out with the math.

[Ungrounded Lightning Rod]

Sure.  Head comes from just velocity.

Figure out how high the water would have to be at a standing start to end up falling that fast, in free air (actually, in vacuum) at one gravity.  That's the head.  (Alternatively, imagine a frictionless curved surface turned the water from horizontal flow through ninety degrees to an upward vertical flow of the same velocity and figure out how high it fountains.)

h = v^2 / 2g  where g ~= 9.81 m/sec^2

Volume flow is volume flow.

[Ahto]

Oops, did I make a mistake in the above power formula? If volume is expressed in cubic meters, power should come out in kilowatts since we're now dealing with 1 ton of mass per unit of water.

So:

P(W) = head(m) * volume(l/s) * 9,81(m/s^2)

P(kW) = head(m) * volume(m3/s) * 9,81(m/s^2)

Correct?

There seems to be some confusion reagrding those formulas among different

sources on the web. Also most of them uses imperial units wich doesn't make any sense to me.

[ADMIN]

Hi Ahto --

Sorry, I only have this in US units. But the metric conversion should not be hard.

H: theoretical head in feet;

v: water velocity in feet per second;

Q: cubic feet of water actually impinging on a paddle per minute;

g: acceleration of gravity in feet per second per second

H = v^2 / 2g = v^2 / 64.4

So, gross horsepower = 0.00189 * Q * H

DAN F

[FoolAmI]

While the velocity method is valid in determining head it doesn't take into account the buildup of water behind the paddle and so it will usually underestimate the true power.       Also when calculating the efficiency of an undershot wheel remember that undershot wheels are only 10% to 20$ efficient @ converting the water's energy into power @ the axle so I generally use 15% in the formula for efficiency.