Bernoulli's equation enables you to express the relationship between a fluid substance's velocity, pressure and height at different points along its flow. It doesn't matter whether the fluid is air flowing through an air duct or water moving along a pipe.

In the Bernoulli equation

*P* + 1/2 *ρv*^{2} + *ρgh* = *C*

*P* is pressure, *ρ* represents the fluid's density and *v* equals its velocity. The letter *g* stands for the acceleration due to gravity and *h* is the fluid's elevation. *C*, the constant, lets you know that the sum of a fluid's static pressure and dynamic pressure, multiplied by the fluid's velocity squared, is constant at all points along the flow.

Here, the Bernoulli equation will be used to calculate the pressure and flow rate at one point in an air duct using the pressure and flow rate at another point.

Use the Bernoulli equation to solve other types of fluid flow problems.

For instance, to calculate the pressure at a point in a pipe where liquid flows, ensure that the liquid's density is known so it can be plugged it into the equation correctly. If one end of a pipe is higher than the other, don't remove

*ρgh*_{1}and*ρgh*_{2}from the equation because those represent the water's potential energy at different heights.The Bernoulli equation can also be arranged to compute a fluid's velocity at one point if the pressure at two points and the velocity at one of those points is known.

Write the following equations:

P_{1} + 1/2 ρ_v__{1}^{2} + ρ_gh__{1} = *C*

P_{2} + 1/2 ρ_v__{2}^{2} + ρ_gh__{2} = *C*

The first one defines fluid flow at one point where pressure is P_{1}, velocity is *v*_{1}, and height is *h*_{1}. The second equation defines the fluid flow at another point where pressure is P_{2}. Velocity and height at that point are *v*_{2} and *h*_{2}.

Because these equations equal the same constant, they can be combined to create one flow and pressure equation, as seen below:

*P*_{1} + 1/2 *ρv*_{1}^{2} + ρ_gh__{1} = P_{2} + 1/2 *ρv*_{2}^{2} + *ρgh*_{2}

Remove *ρgh*_{1} and *ρgh*_{2} from both sides of the equation because acceleration due to gravity and height do not change in this example. The flow and pressure equation appears as shown below after the adjustment:

P_{1} + 1/2 *ρv*_{1}^{2} = P_{2} + 1/2 *ρv*_{2}^{2}

Define the pressure and flow rate. Assume that the pressure *P*_{1} at one point is 1.2 × 10^{5} N/m^{2} and the air velocity at that point is 20 m/sec. Also, assume that the air velocity at a second point is 30 m/sec. The density of air, *ρ*, is 1.2 kg/m^{3}.

Rearrange the equation to solve for P_{2}, the unknown pressure, and the flow and pressure equation appears as shown:

P_{2} = P_{1} *−* 1/2 *ρ*(*v*_{2}^{2} *−* *v*_{1}^{2})

Replace the variables with actual values to get the following equation:

P_{2} = 1.2 × 10^{5} N/m^{2} *−* 1/2 × 1.2 kg/m^{3} × (900 m^{2}/sec^{2} - 400 m^{2}/sec^{2})

Simplify the equation to obtain the following:

P_{2} = 1.2 × 10^{5} N/m^{2} *−* 300 kg/m/sec^{2}

Because 1 N equals 1 kg per m/sec^{2}, update the equation as seen below:

P_{2} = 1.2 × 10^{5} N/m^{2} *−* 300 N/m^{2}

Solve the equation for *P*_{2} to get 1.197 × 10^{5} N/m^{2}.

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