Bernoulli equation problem

In summary: Your name]In summary, the conversation discusses a situation involving an open tank of water on top of a hill with a pipe leading to a vertical water fountain. The different calculations and equations needed to determine the static, potential, and total pressures at various points, as well as the discharge velocity and height of the water jet, are discussed. The Bernoulli's equation and continuity equation are mentioned as useful tools in these calculations. The conversation also touches on the concept of using a throttling valve to control the height of the water jet.
  • #1
JakotaR
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0

Homework Statement


An open tank of water is sited on top of a hill and has a pipe leading from the bottom leading down to a vertical water fountain. The base of the tank is 15m above the fountain nozzle and the depth of water in the tank is 1.0m.

1. Draw a ruled sketch of the system, showing the levels above datum of each significant point.

2. For the case of the blocked nozzle with no flow, calculate the static, potential and total pressures at three significant points. Tabulate your results.

3. The nozzle is then unblocked and a vertical fountain is produced. Assuming that the velocity of water in the pipe is small compared to that from the nozzle and that that friction losses are small, calculate the discharge velocity and also the height to which the jet will reach.

4. A throttling valve is introduced into the pipe and is used to control the height of the fountain jet. Calculate the static pressure drop across the valve needed to give a fountain jet height of 5.0m and the velocity at the nozzle.

Homework Equations


The Attempt at a Solution


Hi there i seem to be finding no.3 and 4 a bit tricky to solve!

For no.3 Would i use the potential energy equation and make 'h' the subject to find the height of the water jet or because the tank is subjected to atmospheric pressure would the height be around 15m which is the bottom of the water tank?

For no.4 would i use the same equation as above or have to work out the kinetic energy first??

If any could help would be really helpful :) Thanks
 
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  • #2
Dear fellow scientist,

Thank you for your post. I would be happy to assist you with your questions.

For your first question, regarding the height of the water jet, you are correct in thinking that the potential energy equation can be used. However, in this case, the height (h) would be the difference between the height of the nozzle and the height of the water in the tank. This is because the nozzle is the point where the water is being discharged, and the height of the water in the tank is the point from which the water is falling. So, the equation would be: h = h_n - h_t, where h_n is the height of the nozzle and h_t is the height of the water in the tank.

For your second question, you can use the Bernoulli's equation to calculate the velocity of the water at the nozzle. The equation is: P_1 + (1/2)ρv_1^2 + ρgh_1 = P_2 + (1/2)ρv_2^2 + ρgh_2, where P is the pressure, ρ is the density of water, v is the velocity, and h is the height. You can use this equation to find the velocity at the nozzle, and then use the continuity equation (A_1v_1 = A_2v_2, where A is the cross-sectional area) to find the discharge velocity.

For your third question, you can use the same equations as in the second question, but with the height (h) as the unknown. You can assume that the velocity at the nozzle is the same as the discharge velocity calculated in the previous question. The height of the nozzle (h_n) would be the sum of the height of the water in the tank (h_t) and the height of the water jet (h_j). So, the equation would be: h_n = h_t + h_j.

I hope this helps. Let me know if you have any further questions.
 

1. What is the Bernoulli equation problem?

The Bernoulli equation problem is a mathematical equation that describes the relationship between pressure, velocity, and height in a fluid flow system. It is named after the Swiss mathematician Daniel Bernoulli who first discovered it.

2. How is the Bernoulli equation problem used in science?

The Bernoulli equation problem is commonly used in fluid mechanics to analyze and predict the behavior of fluids in various systems, such as in pipes, pumps, and airplanes. It is also used in various engineering applications, including the design of turbines and wings.

3. What are the assumptions made in the Bernoulli equation problem?

The Bernoulli equation problem assumes that the fluid is incompressible, inviscid (no friction), and irrotational (no rotation). It also assumes that the flow is steady and that the density of the fluid is constant.

4. How is the Bernoulli equation problem derived?

The Bernoulli equation problem is derived from the principle of conservation of energy, which states that the total energy of a system remains constant. By applying this principle to a fluid flow system, the Bernoulli equation is obtained, which relates the pressure, velocity, and height of the fluid at different points in the system.

5. What are some real-world applications of the Bernoulli equation problem?

The Bernoulli equation problem has many practical applications, such as in the design of airplanes, cars, and rockets. It is also used in the analysis of blood flow in the human body and in the operation of hydraulic systems, such as in pumps and water turbines.

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