Problem related to the Bernoulli Equation

[Zahid Iftikhar]

Homework Statement

What gauge pressure is requried for water in a water supply line to reach a vertical height of 15m?

Homework Equations

Bernoulli Equation

The Attempt at a Solution

I have tried this numerical by taking velocity at ground level to be equal to velocity at top i.e. 15m height. The answer is correct as given in the textbook but I am not convinced by this technique. I wonder how can water have same speed at bottom and top? (Sorry I know no way to add image to PF, that is why I cannot add my work to show how I calculated it.)

 

[BvU]

Hi,

I think you can set v = 0 in the equation for this exercise.

Do show your working in more detail.

 

[Chestermiller]

If the water is considered incompressible, it has to have the same speed at the bottom as at the top.

 

[BvU]

Doesn't the exercise allow a wider pipe at the floor than at 15 m ?

 

[Zahid Iftikhar]

Thanks for the reply.
Here we assume water as ideal fluid, so obviously it is incompressible. But again the question of having same velocity at bottom and top is still there. As we want to calculate the minimum pressure, enough to reach the top level, so it is similar to a ball thrown upward with bare minimum velocity to reach the top, whree it will have zero velocity. I wonder if water has same velocity at the top when it may go further up.

Doesn't the exercise allow a wider pipe at the floor than at 15 m ?

There is no mention of pipe diameter, Even there may not be a pipe at all. Water just shoots out from a hole at the ground and reaches the top.

 

[Zahid Iftikhar]

If the water is considered incompressible, it has to have the same speed at the bottom as at the top.

Thanks for the reply.
Here we assume water as ideal fluid, so obviously it is incompressible. But again the question of having same velocity at bottom and top is still there. As we want to calculate the minimum pressure, enough to reach the top level, so it is similar to a ball thrown upward with bare minimum velocity to reach the top, whree it will have zero velocity. I wonder if water has same velocity at the top when it may go further up.

 

[Zahid Iftikhar]

Hi,

I think you can set v = 0 in the equation for this exercise.

Do show your working in more detail.

Would you please guide me how to show my work. I find no guidance from this page.

 

[Zahid Iftikhar]

Would you please guide me how to show my work. I find no guidance from this page.

Taking v=0 means we assume water to be stationary all along? How come Sir?

 

[Chestermiller]

Would you please guide me how to show my work. I find no guidance from this page.

You failed to tell us whether it is shooting up into the air like a jet from below, or whether it is flowing up a pipe of constant cross sectional area. Do you think this information might have enabled us to help you better? And let's see how you applied the Bernoulli equation to this problem.

 

[Zahid Iftikhar]

You failed to tell us whether it is shooting up into the air like a jet from below, or whether it is flowing up a pipe of constant cross sectional area. Do you think this information might have enabled us to help you better? And let's see how you applied the Bernoulli equation to this problem.

Thanks for the reply Sir
Actually I don't know the way to share my work on this page. How could I add the image file of my work on this page? Please guide.
Moreover, there is no mention of any pipe in the question. May we assume a virtual pipe of 15m vertical height and apply Bernoulli equation.
I try to show you how I did it.
I took both velocities , at bottom and top same, so they get eliminated the Bernoulli equation. Rest of relation is ΔP= ρgΔh, so taking Δh=15m, ρ=1000kgm-3 and g=9.8ms-2, we get ΔP= 14700Pa

 

[Zahid Iftikhar]

Thanks for the reply Sir
Actually I don't know the way to share my work on this page. How could I add the image file of my work on this page? Please guide.
Moreover, there is no mention of any pipe in the question. May we assume a virtual pipe of 15m vertical height and apply Bernoulli equation.
I try to show you how I did it.
I took both velocities , at bottom and top same, so they get eliminated the Bernoulli equation. Rest of relation is ΔP= ρgΔh, so taking Δh=15m, ρ=1000kgm-3 and g=9.8ms-2, we get ΔP= 14700Pa

I need help to understand how velocity of water is same all along. Water while rising up must slow down and stop at the maximum height of 15m.

 

[Chestermiller]

I need help to understand how velocity of water is same all along. Water while rising up must slow down and stop at the maximum height of 15m.

The picture I'm getting is a very large diameter horizontal pipe with a hole in it (at the top) and water jetting out of the hole upwards. As the water rises in the jet, its upward speed decreases, until the upward speed becomes zero at 15 m up. In applying the Bernoulli equation, you focus on two different locations, 1 and 2. For location 1 in this problem, I would focus on a location within the large pipe, far upstream of the jet hole. At this location, the gauge pressure is p, the elevation z is 0 m, and the speed of fluid flow is essentially zero (because of the very large diameter of the pipe). For location 2, I would focus on the location at the very to of the "jet." At this location, the gauge pressure is 0 (i.e., atmospheric), the elevation is 15 m, and the speed of flow is again zero (because the upward motion of the jet has stopped). So, your solution for the gauge pressure in the large pipe is correct.

To put a diagram into a post, you click on the UPLOAD button on the lower right of your response window. You then select a diagram from your files to upload. If you want it full size (rather than thumbnail), you click on that.

 

[Zahid Iftikhar]

Thanks dear Sir.
It looks bit complicate. It means question statement is not perfect. I got your point. It means it is similar to a water leak from a large diameter main pipeline with stream of water, rising to a height of 15m, and we are going to find the pressure at the point of leak.
If we solve it another way, do you think it is correct.
Assume a vitual cylinder of uniform cross-section of height of 15m, full of water and we want to find the pressure at the bottom. Now it is straight-forward to apply P=ρgh relation. We get the same answer. Please comment.

 

[Chestermiller]

Thanks dear Sir.
It looks bit complicate. It means question statement is not perfect. I got your point. It means it is similar to a water leak from a large diameter main pipeline with stream of water, rising to a height of 15m, and we are going to find the pressure at the point of leak.
If we solve it another way, do you think it is correct.
Assume a vitual cylinder of uniform cross-section of height of 15m, full of water and we want to find the pressure at the bottom. Now it is straight-forward to apply P=ρgh relation. We get the same answer. Please comment.

Yes. That’s OK too.

 

[BvU]

Yes. And it doesn't have to be a cylinder. It can be a jungle of piping as well.

 

[Zahid Iftikhar]

Yes. That’s OK too.

Thanks once again for your time.
I have one observation

Yes. That’s OK too.

Thanks once again for your time.
Still I have one observation about significance of size of the hole. Doesn't it matter?
I feel if there are different pipes of different szes all vertical and of same height , they should have different pressures of water at the top.
It is like leakage from main supply line with many holes of different diameter. Water jetting out from diffrent hole will attain different height.
How can, then, answer of this problem be independent of size of hole?

 

[Chestermiller]

Thanks once again for your time.
I have one observation

Thanks once again for your time.
Still I have one observation about significance of size of the hole. Doesn't it matter?
I feel if there are different pipes of different szes all vertical and of same height , they should have different pressures of water at the top.
It is like leakage from main supply line with many holes of different diameter. Water jetting out from diffrent hole will attain different height.
How can, then, answer of this problem be independent of size of hole?

It's independent of hole size as long as the hole diameter is small compared to the diameter of the main pipe. The only thing that will change by making the hole larger will the volumetric flow rate of water coming out of the hole. Its velocity at the hole will not change.