Bernoulli's Equation and Fluid Mechanics?

In summary, the problem involves a flow of water through two pipes with different radii and a height difference of 0.60m. Using flow continuity and Bernoulli's principle, the gauge pressure in the second pipe can be found by setting the pressure at the first cross-section to atmospheric pressure and solving for the gauge pressure at the second cross-section. The height difference between the two pipes does not need to be set to 0, as it can be combined with the other terms in the equation.
  • #1
wmrunner24
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0
Water flows through a 0.30m radius pipe at the rate of 0.20m^2/s. The pressure in the pipe is atmospheric. The pipe slants downhill and feeds into a second pipe with a radius of 0.15m, positioned 0.60m lower. What is the gauge pressure in the second pipe?

So, what I've figured from the problem and what I've learned so far about flow continuity and Bernoulli's principle gave me this.

FR=A1v1
FR=A2v2

Flow rate = Area of a Cross-section * Velocity

So, I can find the velocity for Bernoulli's equation.

P1 + [tex]\rho[/tex]gh1 + 1/2[tex]\rho[/tex]v12 = P2 + [tex]\rho[/tex]gh2 + 1/2[tex]\rho[/tex]v22

And since there's no change in height at the first pipe cross-section, [tex]\rho[/tex]gh1= 0, right?

P1 + 1/2[tex]\rho[/tex]v12 = P2 + [tex]\rho[/tex]gh2 + 1/2[tex]\rho[/tex]v22

So, then I solve for P2, but since it wants gauge pressure, I have to subtract atmospheric pressure, which means that I can remove P1 from the equation, because the problem statement says it equals atmospheric pressure, right?

1/2[tex]\rho[/tex]v12 - 1/2[tex]\rho[/tex]v22 - [tex]\rho[/tex]gh2 = Pg

All I'm really looking for here is a logic check for this much of it. Am I right?
All help is greatly appreciated.
 
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  • #2
That looks correct. There is actually no reason to set h1=0 because the problem gave you the height difference between the two pipes.
 
  • #3
Oh? I was under the impression that whenever you wrote something like [tex]\rho[/tex]gh1, it was assumed to be the change in height, like in gravitational potential energy. I said h1 was zero because as you moved left to right, at the first cross-section, there was no change yet so it was 0. But, if I understand what you're saying, I factor to get h1 - h2 together and replace it with [tex]\Delta[/tex]h, right? Which actually seems to make more sense...
 

Related to Bernoulli's Equation and Fluid Mechanics?

1. What is Bernoulli's Equation?

Bernoulli's Equation is a fundamental equation in fluid mechanics that describes the relationship between the pressure, velocity, and height of a fluid in motion. It states that as the speed of a fluid increases, the pressure decreases, and vice versa.

2. How is Bernoulli's Equation derived?

Bernoulli's Equation is derived from the principle of conservation of energy. It takes into account the potential energy, kinetic energy, and pressure energy of a fluid in motion to determine the overall energy at any point in the fluid.

3. What are the applications of Bernoulli's Equation?

Bernoulli's Equation has many practical applications, including explaining the lift force on an airplane wing, the flow of fluids through pipes, and the operation of carburetors in engines. It is also used in the design of hydraulic machinery and ventilation systems.

4. What are the limitations of Bernoulli's Equation?

Bernoulli's Equation is based on several assumptions, such as the fluid being inviscid (having no internal friction) and incompressible (constant density). In reality, most fluids do have some viscosity and can be compressed, so the equation may not accurately predict their behavior in certain situations.

5. How is Bernoulli's Equation related to the Bernoulli's Principle?

Bernoulli's Principle states that as the speed of a fluid increases, the pressure decreases and vice versa. This is essentially the same relationship described by Bernoulli's Equation. However, Bernoulli's Principle is a simplified version that only applies to steady, incompressible fluids, while Bernoulli's Equation is more comprehensive and can be applied to a wider range of situations.

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