State torricelli's law for efflux speed …

Torricelli’s law

• Torricelli law states that the speed of flow of fluid from an orifice is equal to the speed that it would attain if falling freely for a distance equal to the height of the free surface of the liquid above the orifice.
• Consider any vessel which has an orifice (slit)filled with some fluid.
• The fluid will start flowing through the slit and according to Torricelli law the speed with which the fluid will flow is equal to the speed with which a freely falling bodyattains such that the height from which the body falls is equal to the height of the slit from the free surface of the fluid.
• Let the distance between the free surface and the slit = h
• Velocity with which the fluid flows is equal to the velocity with which a freely falling body attains if it is falling from a height h.

Derivation of the Law:-

• Let A₁= area of the slit (it is very small), v₁= Velocity with which fluid is flowing out.
  • A₂=Area of the free surface of the fluid,v₂=velocity of the fluid at the free surface.
• From Equation of Continuity, Av=constant.Therefore A₁v₁ = A₂v₂.
  • From the figure, A₂>>>A₁, This implies v₂<<v₁(This meansfluid is at rest on the free surface), Therefore v₂~ 0.
• Using Bernoulli’s equation,
  • P+ (1/2) ρ v²+ρgh = constant.
• Applying Bernoulli’s equation at the slit:
  • P_a+(1/2) ρ v₁²+ρgy₁(Equation 1) where P_a=atmospheric pressure,y₁=height of the slit from the base.
• Applying Bernoulli’s equation at the surface:
• P+ρgy₂(Equation 2) where as v₂=0 therefore (1/2) ρ v₁²=0, y₂=height of the free surface from the base.
• By equating(1) and (2),
• P_a+ (1/2) ρ v₁²+ ρgy₁= P+ρgy₂
• (1/2) ρ v₁² = (P-P_a) + ρg(y₂-y₁)
• =(P-P_a) ρgh (where h=(y₂-y₁))
• v₁²=2/ρ [(P-P_a) + ρgh]
• Therefore v1=√2/ρ [(P-P_a) + ρgh].This is the velocity by which the fluid will come out of the small slit.
• v₁ is known as Speed of Efflux. This means the speed of the fluid outflow.

Torricelli’s law. The speed of efflux, v1,from the side of the container is given bythe application of Bernoulli’s equation.

Case1:- The vessel is not closed it is opento atmosphere that means P=P_a.

• Therefore v₁=&radic;2gh.This is the speed of a freely falling body.
• This is accordance to Torricelli’s law which states that the speed by which the fluid is flowing out of a small slit of a container is same as the velocity of a freely falling body.

• Case2:-Tank is not open to atmosphere but P>>P_a.
  • Therefore 2gh is ignored as it is very very large, hence v₁= √2P/ρ.
  • The velocity with which the fluid will come out of the container is determined by the Pressure at the free surface of the fluid alone.