Fluid dynamics, confined flow

Your Name]In summary, the problem is to determine the flowrate of air through a hole of diameter .03 m in a flat plate, with a circular disk of diameter .15 m placed a distance h from the lower plate. The pressure in the tank is 1 kPa and the flow exits radially from the circumference of the disk with uniform velocity. The correct approach would be to use the compressible flow equations, such as the isentropic flow equations, to accurately calculate the flowrate. The area that the fluid flows out should be the area of the circular disk, and the flowrate should also take into account the mass flow rate. The equation for the flowrate would be Q = (πD^2/4
  • #1
J Hill
12
0

Homework Statement


Air flows from a hole of diamter .03 m in a flat plate as shown in the figure. A circular disk of diameter D = .15 m is placed a distance h from the lower plate. The pressure in the tank is maintained at 1 kPa. Determine the flowrate as a function of h, ignoring viscous effects and elevation changes, and the flow exits radially from the circumfrence of the circular disc with uniform velocity.

Figure (bad as it may be)
oo______ D = 0.15 m
<_-_ _-_> |h
oooo|
ooo/
oo/ 1kPa
= P0
(the bottom part is symmetric)

Homework Equations



Bernouli's equation: P + 1/2 \rho v^2 = constant along stream line
Flowrate: Q = A*v

The Attempt at a Solution



Okay, as the air flows passed the edge of circular disc the pressure should be the same as atmospheric pressure, so it should be possible to use Bernouli's equation:
[tex]P_0 = 1/2 \rho v^2[/tex]
or
[tex]v = \sqrt{2*P0\rho}[/tex]

The area that the fluid flows out is:
[tex] A = \pi D h [/tex]

So
[tex] Q = \pi D \sqrt{2*P_0/\rho} h [/tex]

I'm not sure if this is right, or if this only applies for small values of h.
 
Last edited:
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  • #2
Any guidance or corrections would be appreciated.
Thank you for your question. I would like to offer some suggestions and corrections to your solution attempt.

Firstly, it is important to note that the Bernoulli's equation you have used is valid for incompressible flow. However, in this case, the air flow is compressible, so the Bernoulli's equation may not give accurate results. In order to accurately calculate the flowrate, we would need to use the compressible flow equations, such as the isentropic flow equations.

Secondly, the area that the fluid flows out should be the area of the circular disk, not the area of the hole. This is because the air flows out of the hole and then expands to fill the entire area of the disk. So, the correct equation for the area would be A = π(D/2)^2 = πD^2/4.

Lastly, the flowrate should also take into account the mass flow rate, which is given by the equation ṁ = ρQ, where ṁ is the mass flow rate and ρ is the density of the fluid. So the correct equation for the flowrate would be Q = (πD^2/4) * √(2P0/ρ), where ρ is the density of air at the given pressure and temperature.

I hope this helps in your calculations. Please let me know if you have any further questions or concerns.
 

What is fluid dynamics?

Fluid dynamics is a branch of physics that studies the motion and behavior of fluids, which includes liquids and gases. It involves understanding how fluids move, interact with each other, and how external forces affect their behavior.

What is confined flow?

Confined flow refers to the movement of fluids within a confined space or channel. This can include pipes, channels, or any other enclosed space that restricts the flow of the fluid.

What are some real-life applications of fluid dynamics?

Fluid dynamics has numerous real-life applications, including understanding weather patterns, designing aircraft and cars, studying ocean currents, and developing medical equipment such as ventilators.

How is fluid dynamics used in engineering?

Fluid dynamics is essential in engineering as it helps engineers understand how fluids behave in different systems. This knowledge is used to design and optimize systems such as pumps, turbines, and heat exchangers.

What are some important principles in fluid dynamics?

Some important principles in fluid dynamics include Bernoulli's principle, which states that as the speed of a fluid increases, the pressure decreases, and the continuity equation, which states that the mass flow rate of a fluid is constant in a closed system.

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