[Physics] How to calculate the flow rate necessary for a given cross sectional area, angle, and length

fluid dynamicsgravity

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I'm making a toy for my kids and this problem came up. I have a channel on a slight angle (angle is between ground and length of channel) and I'm pouring water into it. I want to know how quickly I need to pour water in to make it flow continuously at a given height.

So the water is (meant to be) .5m wide, .2m high, and 1m long, angled at 5 degrees. How fast is that water going to fall out, in other words, how fast do I have to pour it in?

I'm particularly looking for the technique to do this, not just an answer.

The part I find very confusing is that the water at the top of the box will accelerate a little, and have a low velocity, while the water at the bottom of the box will have accelerated for a while, and have a higher velocity. But intuition tells me the water will stay cohesive, for lack of a better word. Can someone explain what's going on here and how this can be calculated?

Best Answer

I was told by an engineer to use Manning's equation for open channel flow rate, as described here http://en.wikipedia.org/wiki/Manning_formula

Manning coefficient for some common materials: http://www.engineeringtoolbox.com/mannings-roughness-d_799.html - in my case it was acrylic sheet so 0.009 worked fine

Combining with discharge as stated in the wikipedia article means you can avoid calculating the velocity if you don't need it.

Q = cubic meters per second

A = .2 * .5 (cross sectional area in meters squared)

Rh = A / P, P is the wetted perimeter in this case .2 + .2 + .5

S = 0.09 (tan(5 degrees))

k = m^1/3/s

n = 0.009

So 0.08m^3/s, or in liters, 80 liters/second

Hmm, not sure if that is actually correct, but it's the right approach, and if correct, tells me I need to decrease the angle and decrease the depth in order to achieve a flow rate that I can find a cheap pump for!