I am trying to solve a situation, where I'd like to know how much energy would be needed to push a mass of water out of a container. Here's an image to help understand:
The water tank has a height of 2 meters, length of 6 meters, and width of 4 meters. If a piston, much like a coffee press but watertight, pushes the water with only the hole on top of the container, as shown in the drawing, as an exit point, how much energy would be needed for that device to go all the way to the other side of the container? Assuming the hole is 1 square meter, if that can help.
I understand there are a lot of forces to take into consideration, such as friction, hydrostatic pressure, and more, but I'm trying to figure out if it would require a lot of energy or would it remain minimal? What is the force that will require the most energy? Hydrostatic pressure?
If there is a formula that would enable me to figure this out, that is all I'm asking, I can try and do the math myself, but I can't figure out where to start!
Thanks a lot for the help!
Best Answer
I don't think that the size of the hole has an influence on the energy required. Friction is a complicated matter here, not easy to calculate. Try to calculate the pressure imposed by the water on the piston. The force working against that pressure will require most energy.
Edit
The pressure on the piston isn't uniform along its height in this case. At any given height the pressure is $p=\rho g z$, where $z$ is the vertical coordinate. To get the force required to counteract the pressure imposed on the piston You need to integrate this formula across the piston surface:
$F=\int_Sda $ = $F=\rho g \int_0^w \int_0^h zdzdx$
Finally, after the integration:
$F_{piston}= \frac{1}{2} \rho g w h^2 $,
where $\rho$ is the density of the fluid, $g$ is the gravitational acceleration, $w$ is the tank's (and piston's) width and $h$ is the tank's (and piston's) height.
Having the force all You need to do is multiply it by tank's length to get the work (and energy) required.