How To Calculate Pressure From Flow Rate

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

\(P+1/2\rho v^2+\rho gh=C\)

​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.

Step 1

Write the following equations:

\(P_1+1/2\rho v_1^2+\rho gh_1=C\P_2+1/2\rho v_2^2+\rho gh_2=C\)

The first one defines fluid flow at one point where pressure is P₁, velocity is ​v​₁, and height is ​h​₁. The second equation defines the fluid flow at another point where pressure is P₂. 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:

\(P_1+1/2\rho v_1^2+\rho gh_1=P_2+1/2\rho v_2^2+\rho gh_2\)

Step 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:

\(P_1+1/2\rho v_1^2=P_2+1/2\rho v_2^2\)

Step 3

Define the pressure and flow rate. Assume that the pressure ​P​₁ at one point is 1.2 × 10⁵ N/m² 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/m³.

Rearrange the equation to solve for P₂, the unknown pressure, and the flow and pressure equation appears as shown:

\(P_2=P_1-1/2\rho (v_2^2-v_1^2)\)

Step 4

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

\(P_2=1.2\times 10^5-1/2(1.2) (900^2-400^2)\)

Step 5

Simplify the equation to obtain the following:

\(p_2=1.2\times 10^5-300=1.197\times 10^5\text{ N/m}^2\)

Solve the equation for ​P​₂ to get 1.197 × 10⁵ N/m².