explain about toricellis theorem-fluid dynamics?

Question

time taken for the water level to fall from height h1 to height h2

Answer 1

Torricelli's Theorem: The velocity out from a vented tank is equal to speed of a free body falling a distance h, where h is the height from the top of the tank to the location of the hole.

This is a specific use of Bernoulli Equation in Fluid Dynamics on the problem of water flowing out from a small orifice located near the bottom of a vented tank.

We start with Bernoulli equation:

(v^2/2) + gh + p/ρ = constant

where v = fluid velocity, p = pressure, ρ = fluid density, g = acceleration due to gravity, h = height

For the case of a vented tank as stated above, the equation can be rewritten as:

gh(1) + p(1)/ρ + (1/2)v(1)^2 = gh(2) + p(2)/ρ + (1/2)v(2)^2

where h(1) is height 1, h(2) is height 2, p(1) is pressure 1, p2 is pressure 2, v(1) is fluid velocity 1, and v(2) is fluid velocity 2.

Solving for v(2), we get:

v(2) = sqrt((2/(1-(A(2)/A(1))^2)(((p(1)-p(2))/ρ)+gh))

where v(1) = (A(2)/A(1))v(2) as given by the equation of continuity
and A(1) is the surface area of the open tank and A(2) is the surface area of the small hole, and h is height (1) - height(2).

In the case of a small hole, A(2) is much less than A(1). So we approximate (A(2)/A(1))^2 to be near 0, and also p(1) = p(2) because the tank is vented open and the pressure at the opening of the tank would be the same as the pressure at the small hole. Hence the equation becomes:

v(2) = sqrt(2gh), and this equation is Torricelli's Theorem.

Now to calculate the time it takes for the water level to drop from height(1) to height(2), or just height h. Well, Torricelli's Theorem tells us that the rate in which the water is leaking out is just simply v=sqrt(2gh). Hence,

Time for water to fall a height h = Height/Rate of Depletion, or

t = h/sqrt(2gh)

Answer 2

From ref. 1, "A liquid leaves a spigot at the bottom of a reservoir with the same speed that a freely falling object falling through the same height [has]. v1 = (2g(h2-h1))^0.5."

Per ref. 2, (approximately quoted) "Torricelli's theorem, also called Torricelli's law, Torricelli's principle , or Torricelli's equation: statement that the speed, v, of a liquid flowing under the force of gravity out of an opening in a tank is proportional jointly to the square root of the vertical distance, h, between the liquid surface and the centre of the opening and to the square root of twice the acceleration caused by gravity, 2g, or simply v = (2gh)^0.5." So if you know the area of the hole you can calculate the initial volume rate. Bernoulli's equation comes in when you want to obtain the time taken for the water level to fall from an initial height to a final height. See ref. 3.
EDIT: Answer 2 assumes the flow rate is constant, but actually it decreases as the height decreases. A similar error was made by self-styled genius Marilyn Vos Savant several years ago. Ref. 4 addresses this problem. The solution requires that you integrate rate over the height interval.