How Does Air Flow Affect Pressure and Velocity in a Venturi Tube?

In summary, the air is flowing from left to right in a horizontal pipe with different diameters. At point 1, the air has a velocity of 10 m/s and a pressure of 150000 Pa. A narrow U-shaped tube filled with water is attached to the pipe, and the pressure at point 2 is 150060 Pa with a velocity of 0.625 m/s. The height difference in the water is 0.6 cm. Assuming constant air density and negligible effects from transitions, we can use continuity and Bernoulli's equations to determine the velocity and pressure at point 2, as well as the height difference in the U-shaped tube.
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Air flows from left to right though a horizontal pipe that has different diameters in two different sections. the narrow section has a radius of 5 cm and the wider sectio nhas radius of 20 cm. At point 1 (r=5) the air is flowing with a velocity of 10 m/s and has an absolute pressure of 150000 Pa. A narrow u shaped tup has been attached to the air tube and this u tube is filled with water. note that no air actually flows into or out of the u shaped tube, but instead flows right across its openings. The pressures at points 1 and 2 are different so the water in the u shaped tube is not leve. Assume that the transitions between the sections have a negligible effect on the frictionless flow of air. and that the density of air is constant throughtout the pipe. Find the speed of the air and its pressure at point two and the differenece in heights of the water in the u shaped tube.

I have no clue where to start
 
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A device like this is called a Venturi tube. Since air is being treated as incompressible and inviscid, we can easily model this situation with a simple continuity equation and Bernoulli's equation. We will use the continuity equation to find the velocity of air at point two.
[tex]A_1 v_1 = A_2 v_2[/tex]
[tex]v_2 = \frac{A_1}{A_2} v_1 = 10 \frac{m}{s} \left( \frac{5 \ cm}{20 \ cm} \right)^2 = 0.625 \frac{m}{s}[/tex]
This is a logical result, in order for mass to be conserved, the section with the smaller cross-section area must have a higher velocity than the section with greater cross-section area. Now, we will use Bernoulli's equation in order to find the pressure at point two. When we analyze a Venturi tube we normally neglect the change in potential energy of the fluid, so we have
[tex]P_1 + \frac{1}{2} \rho v_1^2 = P_2 + \frac{1}{2} \rho v_2^2[/tex]
Notice we need the density of air, so we'll just use an average value of 1.2 kg/m3.
[tex]P_2 = P_1 + \frac{1}{2} \rho \left(v_1^2 - v_2^2 \right) = 150000 \ Pa + 0.5 \left( 1.2 \frac{kg}{m^3} \right) \left( \left(10 \frac{m}{s} \right)^2 - \left(0.625 \frac{m}{s} \right)^2 \right) = 150060 \ Pa[/tex]
According to the Bernoulli Effect, the section with the higher velocity will have a smaller pressure, so the results are coherent. Finally, in order to calculate the height difference in the manometer, we use the following equation
[tex]\Delta P = \rho g \Delta h[/tex]
This time, we need the density of water, so we'll just use ρ = 1000 kg/m3.
[tex]\Delta h = \frac{\Delta P}{\rho g} = \frac{60 \ Pa}{\left( 1000 \frac{kg}{m^3} \right) \left(9.8 \frac{m}{s^2} \right)} = 0.006 \ m = 0.6 \ cm[/tex]
 
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Related to How Does Air Flow Affect Pressure and Velocity in a Venturi Tube?

1. What is fluid dynamics?

Fluid dynamics is the study of the motion of fluids (liquids and gases) and the forces that act upon them. It involves understanding how fluids flow, how they interact with objects, and how they can be controlled and manipulated.

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Fluid dynamics has many practical applications in various fields such as aerospace engineering, automotive engineering, oceanography, meteorology, and even in everyday objects like pumps, pipes, and turbines. It is used to design and improve the efficiency of vehicles, study weather patterns, and analyze the behavior of fluids in different scenarios.

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Fluid dynamics is closely related to other scientific disciplines such as thermodynamics, mechanics, and electromagnetism. It also overlaps with other fields like chemistry, biology, and geology, as fluids play a crucial role in many natural processes.

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Some current research topics in fluid dynamics include turbulence, multiphase flows, flow control, and the study of biological and soft matter fluids. There is also ongoing research in the development of new computational methods and experimental techniques to better understand and model fluid behavior.

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