Torricelli’s theorem

Torricelli’s theorem . It is a Bernoulli application and studies the flow of a liquid contained in a container, through a small hole, under the action of gravity . From Torricelli’s theorem, the flow rate of a liquid through an orifice can be calculated. The speed of a liquid in an open vessel, through a hole, is what any body would have, freely falling in a vacuum from the level of the liquid to the center of gravity of the hole.

Summary

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  • 1 equation
  • 2 low approach speeds
  • 3 Discharged flow
  • 4 Sources

Equation

Vt = √2. g. (h + Vo2 / 2.g)

where:

  • Vt = theoretical velocityof the liquid at the exit of the fluid hole in the section considered
  • Vo = is the approach speed
  • h = distance from the liquid surface to the center of the hole.
  • g = acceleration of gravity

Low approach speeds

The previous expression is transformed into:

Vr = Cv √2.gh

Where:

  • Vr is the average real velocity of the liquid at the exit of the hole
  • Cv is the speed coefficient. For preliminary calculations in thin-walled openings, 0.95 can be accepted in the worst case.

taking Cv = 1

Vr = √2.gh

Experimentally it has been found that the average speed of a jet from a thin-walled hole is slightly less than ideal, due to the viscosity of the fluid and other factors such as surface tension , hence the meaning of this speed coefficient. .

Discharged flow

The flow rate or volume of the fluid that passes through the hole in time, Q, can be calculated as the product of Sc, the actual area of ​​the contracted section, by Vr, the average real velocity of the fluid passing through that section, and by Consequently, the following equation can be written:

Q = Sc. Vr = (S. Cc) Cv √2.gh

Q = Cd. S √2.gh

where

  • S√2.gh represents the ideal discharge that would have occurred if friction and contraction were not present.
  • Cc is the contraction coefficient of the fluid vein at the exit of the orifice. Its meaning lies in the abrupt change of direction that the particles of the inner wall next to the hole must carry out. It is the ratio between the contracted area and that of the hole. It is usually around 0.65.
  • Cd is the coefficient by which the ideal discharge value is multiplied to obtain the actual value, and is known as the discharge coefficient.

Numerically it is equal to the product of the other two coefficients.

Cd = Cc.Cv

The discharge coefficient will vary with the load and hole diameter. Its values ​​for water have been determined and tabulated by numerous experimenters. As a guideline, values ​​can be taken over 0.6. This shows the importance of using these coefficients to obtain acceptable flow results.

 

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