## Bernoulli's Equation

Bernoulli's Equation is an expression of the Conservation of Energy for fluids. At any point, the combination of the pressure, , the potential energy, , and the kinetic energy, , is the same as at any other point in an ideal fluid's path. Bernoulli's Equation equates these quantities at two points.

**Bernoulli's Equation**

Or:

**Solved Problems for Bernoulli's Equation**

A tank of water, 4 meters deep, has a small hole in the side, 1 meter from the bottom.
Assume the rate that the water lowers, at the top of the tank, is negligible. What is the velocity of the water as it exits the tank, through the small hole? |

Start with Bernoulli's Equation:

Both the top of the tank and the small hole are open to atmospheric pressure.

is the rate that the water goes down, at the top of the tank, which is negligible.

The density cancels out. The exit velocity is independent of the density of the fluid.

Solve for the final velocity:

Given , the equation: is like the equation for the velocity of an object in free fall. This is known as **Torricelli's Theorem**.

**2nd Problem for Bernoulli's Equation**

Water flows through a pipe as shown in the illustration. At , the gauge pressure is: The velocity of the water is: |

**Gauge pressure and velocity**

The velocity at is required in order to calculate the gauge pressure.

The velocity can be determined by using the **Continuity Equation**, discussed on the Flow Rate page:

The gauge pressure can be found by using **Bernoulli's Equation**:

The pressure is less than at , at this location of higher velocity. even while the location is lower by .

**Flow rate throughout the pipe**

Fluid flow rate is discussed on the Flow Rate page:

Using the location at :

0.27 cubic meters, or 270 liters, of water passes any point along the pipe, per second.

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