Velocity of Fluid flow due to gravity

[Seth Vogt]

I am designing a system for an engineering project at school. It's essentially a clock to be powered by water turning a mechanical system of gears (not generating electricity).
Imagine a rectangular basin full of water placed at some height above the ground. There is a hole in the bottom, allowing water to flow out due to the effects of gravity. This water free falls (either through a pipe, or just through air) and hits a "water wheel", causing it to turn. The water wheel, in turn, causes other gears to turn, which go through a series of reduction processes to slow the turning down. If I need the gears to be turning at a certain rate to make the clock accurate, how can I find the speed at which the water strikes against the water wheel? Assume flow through the hole/pipe is constant. Is there a certain equation or set of equations I should be using to find this? Would I use Bernoulli's equation? Please tell me if I left something unclear.

 

[mfb]

Energy conservation is sufficient to get a good estimate - if some mass m leaves the pipe at a height h below the water surface of the tank, its kinetic energy is m*h*g, this allows to find the velocity.
Finding the speed of the wheel is trickier if you do not want to waste huge amounts of water. And you probably want some method to keep the height of the water in the tank constant.

 

[Seth Vogt]

Thank you for your input. For our purposes, in this project we are assuming the height in the tank is constant, as you said. This will ensure a constant pressure at the bottom of the tank, in theory keeping the flow of water constant. I originally thought I could use the kinematic equations of physics to solve for the velocity after free falling a certain distance "h". My instructor said to look in my Fluid Mechanics book to find another way, since water and other fluids do not behave the same way that solids do when in free fall. Also, the speed of the wheel could be related to linear velocity of the water, and then converted to angular velocity using v=w*r. Any other suggestions?

 

[mfb]

There are correction terms and you have friction of course, but I would expect that you can neglect those issues unless the pipe is very long.

 

[JakeBrodskyPE]

My instructor said to look in my Fluid Mechanics book to find another way, since water and other fluids do not behave the same way that solids do when in free fall. Also, the speed of the wheel could be related to linear velocity of the water, and then converted to angular velocity using v=w*r. Any other suggestions?

I think your instructor is expecting you to cite the Manning Equation, assuming you're driving a turbine from a dam via a penstock. It is an empirical estimate of flow based upon surface roughness, and boundary layer behavior. It can be used for both pipes and flumes. Also check whether flow is below the Froude number. The manning flow equation assumes gravity flow conditions. It specifically does not apply where the Froude number is greater than 1.

The Froude number is a dimensionless number, very similar to a Mach number, except that it applies to incompressible flow. You can observe the effects of a Froude number by noticing the hydraulic jump of the water when you open a faucet into a kitchen sink.

So, how complex do you want to go with this?

 

[Seth Vogt]

I had been looking into Manning's Equation and was planning on asking my instructor today about how it would relate. I have never been exposed to it until now, so I am a little confused about how to use it correctly. I understand there is a unit-less coefficient of roughness in the equation as well, so I will be researching the material I will use for the project. Regarding the complexity of the project, she wants us to relate it back to course work and show significant understanding of our project and the subjects we relate to it. There are a lot of assumptions that must be made to solve this particular problem, so although we plan to build a working prototype, it won't work exactly how we want because we assume so many conditions to be true (the constantly full basin of water, etc.). What do you think?

 

[JakeBrodskyPE]

This is one source of coefficients. Do note that the coefficients can change. For example, a clean sewer pipe has a different coefficient than an older sewer pipe that has a layer of solidified grease in it. When storms or jet rodding crews scour the pipe, the flow coefficient changes again.

The Manning Flow equation isn't a precise thing. If you're getting a single digit of accuracy, that's about as good as it gets. However it can be used to estimate upper and lower limits of gravity flow conditions in a partially full pipe or a flume.

If you want an example of these calculations, look here.