# Bernoulli's Equation application to find velocity of a leak

I was trying to do my homework. I just want to understand how do I find a velocity of a leak when the area of a given hole should be considered. I can find it if the hole's area is omitted from by just using the torricelli's theorem which can be derived from bernoulli's equation. But not sure what to do when I know the area of the pipe that the water is going to leak

Bernoulli's Equation: $P_1+{1\over2}\rho v_1^2+\rho g h_1 = P_2+{1\over2}\rho v_2^2+\rho g h_2$

Torricelli's theorem: $v_2= \sqrt{2g\Delta h}$

Edit: I think I got it but I'm just writing this to be sure. If I apply bernoulli's equation to the pipe I can find its velocity. It is very simple but it was a long day sorry for bothering you

$\rho g h_2$ I can take $h_2$ as the diameter of the hole right? and from there I can get the velocity

## 2 Answers

By conservation of mass $$dM = \rho A_1 v_1 dt = \rho A_2 v_2 dt \Longrightarrow v_1 = \frac{A_2}{A_1} v_2$$ Where $A_1$ and $A_2$ are the areas of the pipe and the hole. Replace in Bernoulli's equation and solve for $v_2$.

You should note that Torricelli's theorem may not apply here as it assumes the approximation of a small hole compared to the size of a recipient (that's why the area doesn't feature in the formula, which is usually something relevant).

In cases like this thinking about the physics of the problem, like conservation laws, usually goes a long way.

I don't see why g is playing any role at all, the pipe could be inside a spaceship in deep space and the leak would be the same. Either way, you need to know the P in the pipe some other way (that part might involve g, but it depends on the rest of the problem). We know from covering the end of a hose that the area will affect the speed, but that's more of a conservation of mass issue than a Bernoulli theorem. If it's just a leak, we don't have conservation of mass either, we have a pressure at the hole and an area. It's not clear to me that the size of the hole matters to the speed if it is small-- it should just convert P into 1/2 rho v^2, should it not?

• I can easily find the pressure at pipe's level. However pipe has a chaning diameter that's why I'm trying to find the speed by using bernouille's equation. Nothing else is given in the question other than pipe's diameter and pressure of the fluid at that level – GGphys Jan 3 '17 at 22:11
• I think I have problem understanding the concept so I will re-read the chapter. – GGphys Jan 3 '17 at 22:23