Chapter 8: Problem 29

What happens to the pressure when the speed of air is increased?

Short Answer

Expert verified

Answer: When the speed of air (or any incompressible fluid) is increased, its pressure decreases, according to Bernoulli's Equation.

Step by step solution

Recall Bernoulli's Equation

Bernoulli's Equation states that for a flowing fluid along a streamline, the sum of pressure, kinetic, and potential energies per unit volume remains constant. Mathematically, Bernoulli's Equation can be expressed as: P + \frac{1}{2} \rho v^2 + \rho g h = constant Here, P is the pressure of the fluid, ρ (rho) is the fluid density, v is the fluid velocity (speed), g is the acceleration due to gravity, and h is the height above a reference point.

Assume constant height and incompressible fluid

For this problem, we will assume the height (h) of the fluid above the reference point remains constant, and the fluid is incompressible - meaning the density (ρ) does not change. With these assumptions, Bernoulli's Equation can be simplified to: P + \frac{1}{2} \rho v^2 = constant Now we are only considering the pressure (P) and the kinetic energy of the fluid (\frac{1}{2} \rho v^2).

Investigate the relationship between pressure and speed

From the simplified Bernoulli's Equation above, we can see that the sum of pressure and kinetic energy per unit volume remains constant. This implies: - As the speed (v) of the fluid increases, the kinetic energy term (\frac{1}{2} \rho v^2) also increases. - As the total sum of pressure and kinetic energy must remain constant, when the kinetic energy term increases, the pressure term (P) must decrease to maintain the balance.

Conclusion

Based on Bernoulli's Equation and our analysis, when the speed of the air (or any incompressible fluid) is increased, the pressure decreases. This principle is often used in fluid dynamics and aerodynamics to explain various phenomena such as the lift produced in airplane wings and the Venturi effect.

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What happens to the pressure when the speed of air is increased?

Short Answer

Step by step solution

Recall Bernoulli's Equation

Assume constant height and incompressible fluid

Investigate the relationship between pressure and speed

Conclusion

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