What is the Volume of Water Leaving a Tank Through a Rectangular Opening?

[juanitotruan77]

Homework Statement

5. A rectangular opening on the side of a water tanks has a width of L. The upper part of the opening it's in a height of h1 underneath the water y and the bottom part it's in h2 . prove that the volume of water that leaves the tank is given by :
2/3(L)√(2g(h1³-h2³))

Homework Equations

Bernoulli's law

Continuity equation

Volumetric flux

both V and v are Speed.

The Attempt at a Solution

No idea so far.

Homework Statement

Homework Equations

The Attempt at a Solution

 

[tiny-tim]

hi juanitotruan77! welcome to pf!

imagine the opening is divided into tiny horizontal openings of height dh, and integrate from h₁ to h₂ …

what do you get? ​

 

[juanitotruan77]

I think i get it, i'll try it. But I'm not very familiar with using integrals. Can you explain me how?

thanks, btw.

 

[tiny-tim]

try first …

see how far you get ​

 

[juanitotruan77]

i'm trying, but i don't know how to get to torricelli's law from bernoulli's in a rectangular area.

 

[tiny-tim]

if dh is small enough, you can regard the pressure as constant

so find the speed v from Bernoulli's equation

 

[juanitotruan77]

i can't, i don't understand where i have to put the differential of h. I think i should ask a teacher tomorrow.

 

[tiny-tim]

the rectangle is length L, and starts at height h, and finishes at height h + dh

you can regard the pressure as being constant, = ρgh

find v, then multiply by the area (L*dh) to get the flow

 

[juanitotruan77]

oh, that sounds reasonable, thanks :D

 

[juanitotruan77]

well, i tried, but i couldn't make it work.

 

[tiny-tim]

(just got up :zzz:)

show us what you did

hint: 2/3(L)√(2g(h₁³-h₂³)) = (L)√(2g) times [(2/3)h₁^3/2 - (2/3)h₂^3/2]

 

[juanitotruan77]

Got it, bro, i already finished, i think i was cofused with the h terms. I used y terms instead. it's quite easy now that i did it. lol. Thanks.