How Does Tank Shape Affect Water Stream Pressure and Trajectory?

[peleus]

Homework Statement

Hi all,

I'll outline the situation first, then I'll ask about what I need.

Tank (designed by me / other students) which has no closed in top or bottom. Can be any shape, i.e. cylinder / cone / square etc etc. Made from cardboard with plastic bag in the middle acting as waterproof liner.

This sits atop of a reservoir which has a hole in the top of it (bottom of tank) to drain the water. Once in the reservoir it has two bungs in it (parallel to the floor), a 5mm and 10mm. Once these bung's are pulled obviously it drains the tank (downwards) and a stream of water shoots out (sideways), which we have to try and catch in a vessel below and a set distance away. The only variable we control is the size and shape of the tank. It contains 500ml of water.

Homework Equations

You tell me :)

The Attempt at a Solution

First things first, I'm not meant to understand how to do this yet. It's something that will be taught to us over the coming weeks, however I'm trying to do some research into the principles behind it now so I have a better understand anyway.

My understanding of how it works. Assuming a cylinder for a moment, the wider and shorter the tank is, the less pressure per cm^2 we have. As a result we will have a lower pressure stream shooting out of the bung. A taller skinnier cylinder will result in a higher pressure stream (higher force per cm^2). Because this is a cylinder we will have a drop off of pressure over time, from when it is full to where it is empty at a fairly linear rate.

A cone would perhaps be a better shape, we want to keep the water stream resulting on exactly one spot (or as close as possible) which means changing the surface area to less as we have less water to get the same force per cm^2.

Now - I have no idea how to go about modelling the problem. This is what I SUSPECT will be involved.

Working out the force per unit of area, so we can see how much pressure is one the hole in the bottom.

From this we should be able to work out some type of flow rate, so we now what speed the water will be shooting out at.

From there it should be fairly simple math modelling the water as gravity effects it, ignoring air resistance (which may be a big mistake in my design, should it matter indoors?)

I'm not particularly looking for someone to do up an entire spreadsheet modelling the problem for me (of course I wouldn't say no :P ) but really just wondering if someone can point me in the right direction to some good resources to learn those things I suspect are involved, or tell me if any of my assumptions about what is involved is wrong.

Thanks folks.

 

[xxChrisxx]

Force on water is from pressure head (as you described). The thing you describe as 'high pressure' flow is high velocity flow they are not one and the same according to bernoulli's eq.

You want to keep the same velocity flow with a constantly reducing pressure head. You simply can't do that without a variable orifice. So the key is to calculate the best point to switch from the 10mm hole to the 5mm hole.

Now I am not 100% sure on this next point: a cone will put more pressure on the water in the bottom than a cylinder(but I am not sure how much or even how to calculate it). So the best shape of the reservoir will depend on how far you have to shoot the water. If it is further than you can possibly achieve with maximum pressure head with a cylinder then a cone is needed. Youll need to find out an equivilant pressure head.

Modelling the trajectory of water properly is tricky as it doesn't stay together like a solid. You can assume it acts like one though and get a rough arc the water will follow.

To do it you need the flow rate through the orifice, to get a volume of water. Treat this water as a little solid block. Then work out the acceleration and velociites on it until it hits the ground. That should give you a fairly good indication of the arc it wil follow.

 

[Mene]

I would recommend starting by researching the principles of fluid mechanics, specifically Bernoulli's principle and the equation of continuity. These concepts will help you understand the relationship between pressure, velocity, and flow rate in a system like the one you described. You can also look into the concept of laminar vs. turbulent flow, which may affect the accuracy of your model.

Once you have a better understanding of these principles, you can begin to create a mathematical model for your system. This will involve using equations to calculate the pressure at the hole in the bottom of the tank, as well as the flow rate and velocity of the water as it shoots out. You will also need to consider the effects of gravity and potential energy as the water flows downwards.

Additionally, it may be helpful to experiment with different tank shapes and sizes to see how they affect the stream of water. This will give you a better understanding of the relationship between shape and pressure/velocity.

Overall, it's great that you are taking the initiative to research and understand the principles behind this project. Keep exploring and learning, and don't hesitate to ask for help from your teacher or a qualified scientist if you need it. Good luck!