Consider a large tank that contains a fluid of density ρ at height h1 from bottom to the upper surface of fluid. The tank has an orifice at height h2 from its bottom. Thus h1 - h2 =h be the height of fluid from orifice to the upper surface of fluid.
Consider a large tank that contains a fluid of density ρ at height h1 from bottom to the upper surface of fluid. The tank has an orifice at height h2 from its bottom. Thus h1 - h2 =h be the height of fluid from orifice to the upper surface of fluid.
We assume that the fluid level falls so slowly that the liquid velocity at the upper level may be assumed to zero. Let v1 be the velocity of fluid at the upper surface and v2 be the velocity of the fluid at orifice which is called velocity of efflux. As both ends are open to the atmosphere, so the pressure on the upper and bottom surfaces is equal to the atmospheric pressure ‘P’ that is $$ P_{1}= P_{2} = P $$ According to Bernoulli’s equations $$ P_{1}+ \frac{1}{2} \rho v^2_{1} + \rho gh_{1} = P_{2}+ \frac{1}{2} \rho v^2_{2} + \rho gh_{2} $$ $$ P+ \frac{1}{2} \rho (0) + \rho gh_{1} = P + \frac{1}{2} \rho v^2_{2} + \rho gh_{2} $$ $$ \rho gh_{1} = \frac{1}{2} \rho v^2_{2} + \rho gh_{2} $$ $$ \frac{1}{2} \rho v^2_{2} = \rho gh_{1} - \rho gh_{2} $$ $$ \frac{1}{2} \rho v^2_{2} = \rho g (h_{1} – h_{2}) $$ $$ v^2_{2} = 2g (h_{1} – h_{2}) $$ $$ v_{2} = \sqrt{ 2g (h_{1} – h_{2})} $$ $$ v_{2} = \sqrt{2gh} $$ $$ In \; general $$ $$ v = \sqrt{2gh}$$ This equation is called Toricelli’s theorem which states that
“The velocity of efflux of the fluid through an orifice is directly proportional to the square root of the height of liquid from orifice to the upper surface of fluid”.
This theorem also shows that the velocity of efflux is independent of the nature of liquid, quantity of liquid in the tank and the area of orifice.