6.4.2 Torricelli's Theorem | Fluid Mechanics | Physics 11

Torricelli's theorem, also known as Torricelli's law, is a principle in fluid dynamics that relates the speed of a fluid flowing out of an orifice (an opening) to the height of the fluid above that opening. It states that the speed of efflux is the same as the speed an object would acquire by falling freely from the same height. This theorem is a specialized case of Bernoulli's Equation.

The formula for the speed of efflux ( $v$ ) is:

$v = 2 g h$

Energy Conversion

Torricelli's theorem is a statement of the conservation of energy. The potential energy of the fluid at the top surface is converted into kinetic energy as it exits the orifice.

Potential Energy: Stored in the fluid due to its height ( $h$ ) above the opening.
Kinetic Energy: The energy of motion of the fluid as it flows out of the hole.

The Setup

The theorem is typically applied to a scenario involving a large, open container or tank filled with a fluid, with a small hole near its base. The speed of the exiting fluid depends only on the vertical distance ( $h$ ) from the surface of the fluid to the center of the hole.

Derivation from Bernoulli's Equation

The theorem can be derived directly from Bernoulli's Equation, which describes the conservation of energy in a moving fluid.

$P_{1} + \frac{1}{2} ρ v_{1}^{2} + ρ g h_{1} = P_{2} + \frac{1}{2} ρ v_{2}^{2} + ρ g h_{2}$

Let's define two points for our analysis:

Point 1: The top surface of the fluid in the container.
Point 2: The orifice where the fluid exits.

Assumptions for Simplification

Atmospheric Pressure: The container is open to the atmosphere, so the pressure at the top surface ( $P_{1}$ ) and at the orifice ( $P_{2}$ ) are both equal to the atmospheric pressure ( $P_{a t m}$ ). Therefore, $P_{1} = P_{2}$ .
Large Reservoir: The cross-sectional area of the container ( $A_{1}$ ) is much larger than the area of the orifice ( $A_{2}$ ). From the Equation of Continuity ( $A_{1} v_{1} = A_{2} v_{2}$ ), this means the speed at which the top surface of the fluid falls ( $v_{1}$ ) is negligible compared to the exit speed ( $v_{2}$ ). We can approximate $v_{1} \approx 0$ .

Applying the Assumptions

Start with Bernoulli's equation: $P_{1} + \frac{1}{2} ρ v_{1}^{2} + ρ g h_{1} = P_{2} + \frac{1}{2} ρ v_{2}^{2} + ρ g h_{2}$
Apply the assumptions ( $P_{1} = P_{2}$ and $v_{1} = 0$ ): $P_{a t m} + 0 + ρ g h_{1} = P_{a t m} + \frac{1}{2} ρ v_{2}^{2} + ρ g h_{2}$
The atmospheric pressure terms ( $P_{a t m}$ ) on both sides cancel out: $ρ g h_{1} = \frac{1}{2} ρ v_{2}^{2} + ρ g h_{2}$
Rearrange the equation to solve for the kinetic energy term: $\frac{1}{2} ρ v_{2}^{2} = ρ g h_{1} - ρ g h_{2} = ρ g (h_{1} - h_{2})$
Let $h = (h_{1} - h_{2})$ , which is the vertical height of the fluid above the orifice. The equation becomes: $\frac{1}{2} ρ v_{2}^{2} = ρ g h$
Cancel the density ( $ρ$ ) from both sides and solve for the efflux velocity ( $v_{2}$ , which we can simply call $v$ ): $\frac{1}{2} v^{2} = g h ⟹ v = 2 g h$

This final expression is Torricelli's law.

Possible Questions and Answers

Q: Does the size of the hole affect the speed of the exiting fluid?

A: According to the ideal model of Torricelli's theorem, the speed of efflux depends only on the height ( $h$ ), not the size of the hole. However, in real-world scenarios, factors like fluid viscosity and the contraction of the fluid stream (vena contracta) mean that the actual exit speed is slightly less than the ideal value.

Q: Is Torricelli's law always accurate?

A: It is a very good approximation for non-viscous, incompressible fluids under ideal conditions. For real fluids, energy losses due to friction will result in a slightly lower exit velocity.

Energy Conversion

Torricelli's theorem is a statement of the conservation of energy. The potential energy of the fluid at the top surface is converted into kinetic energy as it exits the orifice.

Potential Energy: Stored in the fluid due to its height (

h

) above the opening.

Kinetic Energy: The energy of motion of the fluid as it flows out of the hole.

Derivation from Bernoulli's Equation

The theorem can be derived directly from Bernoulli's Equation, which describes the conservation of energy in a moving fluid.

P_{1} + \frac{1}{2} ρ v_{1}^{2} + ρ g h_{1} = P_{2} + \frac{1}{2} ρ v_{2}^{2} + ρ g h_{2}

Let's define two points for our analysis:

Point 1: The top surface of the fluid in the container.

Point 2: The orifice where the fluid exits.

Atmospheric Pressure: The container is open to the atmosphere, so the pressure at the top surface ( $P_{1}$ ) and at the orifice ( $P_{2}$ ) are both equal to the atmospheric pressure ( $P_{a t m}$ ). Therefore, $P_{1} = P_{2}$ .

Large Reservoir: The cross-sectional area of the container ( $A_{1}$ ) is much larger than the area of the orifice ( $A_{2}$ ). From the Equation of Continuity ( $A_{1} v_{1} = A_{2} v_{2}$ ), this means the speed at which the top surface of the fluid falls ( $v_{1}$ ) is negligible compared to the exit speed ( $v_{2}$ ). We can approximate $v_{1} \approx 0$ .

Start with Bernoulli's equation: $P_{1} + \frac{1}{2} ρ v_{1}^{2} + ρ g h_{1} = P_{2} + \frac{1}{2} ρ v_{2}^{2} + ρ g h_{2}$

Apply the assumptions ( $P_{1} = P_{2}$ and $v_{1} = 0$ ): $P_{a t m} + 0 + ρ g h_{1} = P_{a t m} + \frac{1}{2} ρ v_{2}^{2} + ρ g h_{2}$

The atmospheric pressure terms ( $P_{a t m}$ ) on both sides cancel out: $ρ g h_{1} = \frac{1}{2} ρ v_{2}^{2} + ρ g h_{2}$

Rearrange the equation to solve for the kinetic energy term: $\frac{1}{2} ρ v_{2}^{2} = ρ g h_{1} - ρ g h_{2} = ρ g (h_{1} - h_{2})$

Let $h = (h_{1} - h_{2})$ , which is the vertical height of the fluid above the orifice. The equation becomes: $\frac{1}{2} ρ v_{2}^{2} = ρ g h$

Cancel the density ( $ρ$ ) from both sides and solve for the efflux velocity ( $v_{2}$ , which we can simply call $v$ ): $\frac{1}{2} v^{2} = g h ⟹ v = 2 g h$

This final expression is Torricelli's law.

Possible Questions and Answers

Q: Does the size of the hole affect the speed of the exiting fluid?

A: According to the ideal model of Torricelli's theorem, the speed of efflux depends only on the height (

h

), not the size of the hole. However, in real-world scenarios, factors like fluid viscosity and the contraction of the fluid stream (vena contracta) mean that the actual exit speed is slightly less than the ideal value.

Q: Is Torricelli's law always accurate?

A: It is a very good approximation for non-viscous, incompressible fluids under ideal conditions. For real fluids, energy losses due to friction will result in a slightly lower exit velocity.