How Does Bernoulli's Principle Apply to Varying Pipe Diameters and Water Flow?

In summary, The problem involves a horizontal length of pipe with a tapered middle part and given gauge pressures. The goal is to find the flow speed ratio between the wider and narrower sections, the actual water flow speeds in both sections, and the volume flow rate through the pipe. Using the equations A=πr^2, Q=Av, and P+.5ρv^2=constant, we can find the ratio of the velocities to be 2:5. To solve for the actual flow speeds, we can substitute v1 and v2 with Q/A1 and Q/A2 respectively, where Q represents the volume flow rate and A represents the cross-sectional area. The gauge pressures given in the problem must be converted to absolute pressures by
  • #1
Matt Armstrong

Homework Statement



A horizontal length of pipe starts out with an inner diameter (not radius!) of 2.60 cm, but then has a tapered middle part which narrows to a diameter of 1.60 cm. When water flows through the pipe at a certain rate, the gauge pressure is 34 kPa in the first (wider) section and 25 kPa in the second (narrower) section.

(a) What is the ratio of the flow speed in the narrow section to the flow speed in the wider section?
(b) What are the actual water flow speeds in the wider section and in the narrower section? (Hint: you will need to set up and solve Bernoulli’s equation. Bernoulli’s equation determines what flow rate is produced by this particular pair of inlet and outlet pressures.)
(c) What is the volume flow rate through the pipe?[/B]

Homework Equations



r_1 = .013 m, r_2 = .008 m, P_1 = 34000 Pa, P_2 = 25000 Pa

A = pi*r^2

Q=Av

P + .5rho*v^2 = constant

P_1 + .5rho*(v_1)^2 = P_2 + .5rho*(v_2)^2[/B]

The Attempt at a Solution



For a, since the flow rate is Q = v*a, I made a ratio between the velocities and the Areas and got a ratio of 2 (for narrow) to 5 (for wider). Part b is where I have trouble. I know that Bernoulli's principle can be written between any singular point or any two points along the pipe, and I tried to solve for v_1 and v_2 using the second equation, but that wouldn't be possible because there are two unknown variables. I know that in certain applications where a difference in height delta-y can be used in a formula for velocity, but since no height or reference frame is identified, I left those out of my principle equations as well as neglected to use them. I feel like if I can understand part b, then I can do part c on my own. What do I need to do?

Thanks for any information you can provide
 
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  • #2
Hello,

You just need to substitute v1 and v2 with Q/A1 and Q/A2 respectively. Q (volume/time units) is the same because of the mass balance. So, the unknown variable will be one instead of two.

Beware: the problem gives you the gauge pressure, so you must add atmospheric pressure (101.325 kPa) in order to get absolute pressure and use it in the Bernoulli's equation.
 
  • #3
DoItForYourself said:
Beware: the problem gives you the gauge pressure, so you must add atmospheric pressure (101.325 kPa) in order to get absolute pressure and use it in the Bernoulli's equation.
Does it matter here, as long as the pressures are all from the same base?
 
  • #4
haruspex said:
Does it matter here, as long as the pressures are all from the same base?

You are right, it does not matter as long as pressure is added to each side of the equation and we solve for Q (ΔP remains the same).

But if we wanted to solve for P1 or P2, the result would have been wrong (in terms of units).
 

Related to How Does Bernoulli's Principle Apply to Varying Pipe Diameters and Water Flow?

1. What is Bernoulli's Principle?

Bernoulli's Principle states that as the speed of a fluid (such as air or water) increases, the pressure exerted by that fluid decreases.

2. Who is Bernoulli and how did he discover this principle?

Daniel Bernoulli was a Swiss mathematician and physicist who discovered this principle in the 18th century through his studies of fluid dynamics and kinetic theory.

3. How is Bernoulli's Principle applied in everyday life?

Bernoulli's Principle is applied in various fields such as aviation, aerodynamics, and hydrodynamics. It is used to explain the lift force on an airplane wing, the flow of air through a chimney, and the movement of water through pipes, among other things.

4. Is Bernoulli's Principle always true?

Bernoulli's Principle is an idealized concept and does not always hold true in real-life situations. Factors such as viscosity, turbulence, and compressibility of fluids may affect the accuracy of its application.

5. How does Bernoulli's Principle relate to the conservation of energy?

Bernoulli's Principle is based on the conservation of energy, specifically the conservation of mechanical energy. As the speed of a fluid increases, the kinetic energy of the fluid also increases, resulting in a decrease in its potential energy. This principle helps explain the transfer of energy between different forms in fluid flow systems.

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