I've come across another fluid pressure problem I don't understand

[vdance]

Homework Statement

Personal interests aside, I've run into another fluid pressure problem I don't understand.

Relevant Equations

F=ρgsh，ρ=1000，g=10。

Neglect gravity other than that of water; neglect friction.
F1=ρgsh=1000*10*1*(1+4)=50000 N;
F1=ρgsh=1000*10*0.1*4=4000 N;
F3=50000-4000=46000 N?

 

[jbriggs444]

Homework Statement: Personal interests aside, I've run into another fluid pressure problem I don't understand.
Relevant Equations: F=ρgsh，ρ=1000，g=10。

View attachment 365841
Neglect gravity other than that of water; neglect friction.
F1=ρgsh=1000*10*1*(1+4)=50000 N;
F1=ρgsh=1000*10*0.1*4=4000 N;
F3=50000-4000=46000 N?

I agree that the downward force (suction) from the water on the first piston ($F_1$) is 50000 N.

I agree that the upward force (suction) from the water on the check second piston ($F_2$) is 4000 N.

Before we can calculate $F_3$ we need to decide what $F_3$ is. Is it the force of suction on the top piston alone? Is it the net force of water pressure on the two piston assembly? (I assume the latter).

What does the check valve do? Does it prevent air from entering but permit water to exit?

Importantly, what is the pressure in the sealed chamber between the lower piston and the check valve?

I find it easier to approach this problem by considering the external force that holds the piston assembly in place. What [range of] external force values would permit the system to remain in equilibrium?

Big hint: When I look at the drawing, I see a sealed chamber with a piston. That piston is not going to move unless the check valve is forced open or cavitation occurs.

 

[Lnewqban]

What is exactly what you don't understand?
Both represented forces should be pointing in the opposite directions: both should be pulling away from the liquid, which naturally tries to reach the level of the basin at the bottom.

 

[Steve4Physics]

Some general comments...

Relevant Equations: F=ρgsh，ρ=1000，g=10。

Units for ρ and g are essential. If the correct units aren’t clear, sometimes terrible things happen! E.g. see https://spacemath.gsfc.nasa.gov/weekly/6Page53.pdf

View attachment 365841
Neglect gravity other than that of water; neglect friction.
F1=ρgsh=1000*10*1*(1+4)=50000 N;
F1 [editg - you mean F2]=ρgsh=1000*10*0.1*4=4000 N;
F3=50000-4000=46000 N?

The intended meanings of F1, F2 and F3 are not clear. Are they meant to represent:
- the force of the water on the piston?
- the force of the piston on the water?
- the external force needed to keep the piston in equilibrium?
- something else?

Minior edits.

 

[haruspex]

I do not understand the function of the check valve here. It is drawn as though allowing flow upwards, which means admitting air if the upper piston is raised, it seems. If so, that air would rise to the top of that chamber. The net movement of water would therefore be from just above the position at which the lower piston starts to just above where the upper piston starts. That is an unknown distance, somewhat less than 1m.

 

[vdance]

I agree that the downward force (suction) from the water on the first piston ($F_1$) is 50000 N.

I agree that the upward force (suction) from the water on the check second piston ($F_2$) is 4000 N.

Before we can calculate $F_3$ we need to decide what $F_3$ is. Is it the force of suction on the top piston alone? Is it the net force of water pressure on the two piston assembly? (I assume the latter).

What does the check valve do? Does it prevent air from entering but permit water to exit?

Importantly, what is the pressure in the sealed chamber between the lower piston and the check valve?

I find it easier to approach this problem by considering the external force that holds the piston assembly in place. What [range of] external force values would permit the system to remain in equilibrium?

Big hint: When I look at the drawing, I see a sealed chamber with a piston. That piston is not going to move unless the check valve is forced open or cavitation occurs.

In Figure 3, water will inevitably flow out through the check valve. I added the check valve to prevent air from mixing with the water in the pipe cavity, ensuring the effect of M=0.1 square meters remains unchanged. Overall, water descending from the top will flow out through port M. The magnitude of the resultant force is likely the gravitational force of the water over this 1-meter height difference. What I want to understand is how to analyze and calculate this, and how to map the parameters in the formula to specific numerical values.

 

[vdance]

I do not understand the function of the check valve here. It is drawn as though allowing flow upwards, which means admitting air if the upper piston is raised, it seems. If so, that air would rise to the top of that chamber. The net movement of water would therefore be from just above the position at which the lower piston starts to just above where the upper piston starts. That is an unknown distance, somewhat less than 1m.

In Figure 3, water flows downward, so the water inside the pipe can only flow out through Port M and cannot be sucked back in through it. I added the check valve solely to prevent air from entering the system.

 

[vdance]

Some general comments...


Units for ρ and g are essential. If the correct units aren’t clear, sometimes terrible things happen! E.g. see https://spacemath.gsfc.nasa.gov/weekly/6Page53.pdf


The intended meanings of F1, F2 and F3 are not clear. Are they meant to represent:
- the force of the water on the piston?
- the force of the piston on the water?
- the external force needed to keep the piston in equilibrium?
- something else?

Minior edits.

Since the unit for cubic units couldn't be entered there, I didn't include the unit.
F₁ and F₂ represent the pressures exerted by water on the piston, respectively, while F₃ is the resultant force exerted by water on the piston.

 

[vdance]

What is exactly what you don't understand?
Both represented forces should be pointing in the opposite directions: both should be pulling away from the liquid, which naturally tries to reach the level of the basin at the bottom.

The arrow indicates the direction of water's pulling force on the piston.

 

[jbriggs444]

In Figure 3, water will inevitably flow out through the check valve.

If the pistons were motionless in equilibrium with some external force (as I had supposed), no water would flow.

I added the check valve to prevent air from mixing with the water in the pipe cavity

So the check valve permits a one way flow - allowing water out but not air in.

Overall, water descending from the top

You had not stated that the pistons move. It would be good to state that when posing the problem.

Do the pistons move slowly? So that the water emerging from the check valve has negligible kinetic energy and the the bottom of the right hand chamber is therefore sure to be at ambient atmospheric pressure. And so that we can continue to treat this as an exercise in hydrostatics.

We can assume that the check valve is ideal so that forward fluid flow results in zero pressure drop.

Note that if the pistons do not move slowly then we no longer have a problem in hydrostatics. We will need to consider that the water is accelerating at non-negligible rates. Pressure differences will no longer be given by $\rho g h$.

will flow out through port M. The magnitude of the resultant force is likely the gravitational force of the water over this 1-meter height difference. What I want to understand is how to analyze and calculate this, and how to map the parameters in the formula to specific numerical values.

As I suggested above. Start with the pressure at the bottom of the right hand chamber. Work from there to the pressure at the lower surface of the right hand piston. You can reason separately to obtain the pressure on the upper surface of that piston.

The hypothetical force of water pressure on the check valve's surface area of 0.1 m² should not enter into your calculation and will not be 4000 N in the situation illustrated in the third frame.

Edit: It now appears that the pistons do not descend slowly and that water is ejected through the check valve with a non-negligible velocity. The calculation scheme described above is not applicable to the intended scenario.

 

[jbriggs444]

Since the unit for cubic units couldn't be entered there, I didn't include the unit.
F₁ and F₂ represent the pressures exerted by water on the piston, respectively, while F₃ is the resultant force exerted by water on the piston.

Did you mean to say that $F_3$ is the resultant force exerted by water on the assembly that includes both pistons? That is to say that $F_3$ will be the sum of the forces from water pressure on each of the three piston surfaces.

It would have been cleaner phrasing if you said that "$F_1$ and $F_2$ represent the pressures exerted by water on the respective pistons". When you say "the piston" and then add "respectively" afterward, it sounds awkwardly like you are pairing a single piston with two forces.

 

[vdance]

Did you mean to say that $F_3$ is the resultant force exerted by water on the assembly that includes both pistons?

Yes, under natural free-fall conditions, F₃ represents the resultant force exerted by water on the two pistons. With the water level at the left side being 5m and at the M port being 4m, water will inevitably flow downward and exit through the M port. I want to know how to calculate the value of the resultant force F₃ at the exact moment the pistons begin their descent from the preset state.
During the piston's downward stroke, the water within the 4m section below the pipe remains stationary and does not flow. Does this portion of water affect the magnitude of the resultant force F₃?

 

[jbriggs444]

Yes, under natural free-fall conditions, F₃ represents the resultant force exerted by water on the two pistons. With the water level at the left side being 5m and at the M port being 4m, water will inevitably flow downward and exit through the M port. I want to know how to calculate the value of the resultant force F₃ at the exact moment the pistons begin their descent from the preset state.

You have not described the situation adequately.

Are we to assume that the piston assembly is held in place until a preset moment when it is released?

Edit: I think this is what you have in mind.

Your expectation is that the piston assembly will descend and that water will squirt out through the check valve with a non-zero velocity that may or may not increase over time. Is that correct?

During the piston's downward stroke, the water within the 4m section below the pipe remains stationary and does not flow.

I agree. The volume of water below and between the pistons does not change. So the water in the 4m section need not move.

Does this portion of water affect the magnitude of the resultant force F₃?

You tell us.

 

[vdance]

You have not described the situation adequately.

Are we to assume that the piston assembly is held in place until a preset moment when it is released?

Or is the piston assembly slowly winched down?


You tell us.

Assume Figure 3 depicts the initial static state, where the top of the left piston is secured by a rope. At the instant the rope is cut, what is the magnitude of the resultant force F₃ exerted on the piston by the freely falling water within the pipe? Although the 4-meter-deep water at the bottom remains stationary, it continuously exerts a steady pulling force on both pistons. Therefore, I believe its contribution to F₃ cannot be ignored.

 

[jbriggs444]

Assume Figure 3 depicts the initial static state, where the top of the left piston is secured by a rope. At the instant the rope is cut, what is the magnitude of the resultant force F₃ exerted on the piston by the freely falling water within the pipe?

What freely falling water? I see no freely falling water. I see water constrained by cylinder walls, a piston assembly, a cylinder bottom and a nozzle.

Although the 4-meter-deep water at the bottom remains stationary, it continuously exerts a steady pulling force on both pistons. Therefore, I believe its contribution to F₃ cannot be ignored.

It exerts an upward force on one piston and an equal downward force on the other. The water in the 4 meter tube does not move, even after the rope is cut. Why can it not be ignored?

How about we begin with a simple calculation. What is $F_3$ before the rope is cut? Assume that the [gauge] water pressure at the check valve is zero. So that the valve is just on the verge of allowing water to leak out. Meanwhile the motionless right hand piston is supporting the water between check valve and piston just like in the famous inverted water glass trick.

 

[vdance]

What freely falling water? I see no freely falling water. I see water constrained by cylinder walls, a piston assembly, a cylinder bottom and a nozzle.

It exerts an upward force on one piston and an equal downward force on the other. The water in the 4 meter tube does not move, even after the rope is cut. Why can it not be ignored?

Water is in a state where g = 10 N/kg, with no external forces other than gravity acting upon it.
According to your reasoning, you believe the 4m below can be neglected. Therefore, the effective height h for force F₃ is 1. This means the resultant force F₃ = ρgsh = 1000 × 10 × (1 - 0.1) × 1 = 9000 N. Is this the correct result?

 

[jbriggs444]

Water is in a state where g = 10 N/kg, with no external forces other than gravity acting upon it.

There are forces other than gravity acting on it. I listed a bunch of constraints on the water.

According to your reasoning, you believe the 4m below can be neglected. Therefore, the effective height h for force F₃ is 1. This means the resultant force F₃ = ρgsh = 1000 × 10 × (1 - 0.1) × 1 = 9000 N. Is this the correct result?

No. That is not the correct result. You will need to explain your reasoning in more detail. In particular, explain why you think that the 0.1 m² check valve area is relevant.

A useful way to proceed would be to calculate the force of water on each of the three relevant piston surfaces and sum them.

 

[vdance]

There are forces other than gravity acting on it. I listed a bunch of constraints on the water.

No. That is not the correct result. You will need to explain your reasoning in more detail. In particular, explain why you think that the 0.1 m² check valve area is relevant.

A useful way to proceed would be to calculate the force of water on each of the three relevant piston surfaces and sum them.

I explicitly stated in the question that the weight and friction of all components except water are negligible. Are there any other external forces acting on this system?
If the answer does not involve calculating the 46,000 N force generated at a height of 4m (50,000 - 4,000) nor neglecting the 4m height component F₃ = ρgsh = 1000 × 10 × (1 - 0.1) × 1 = 9000 N, I cannot understand how to proceed with the calculation.

It is equivalent to Figure 4, and it appears that F₃ = ρgsh = 1000 × 10 × (1 - 0.1) × 1 = 9000 N is correct.

 

[vdance]

I am seeking a method to draw water from the top of a pipe while minimizing gravitational changes in the water within the pipe. This concept is quite challenging, but I believe it is feasible.

 

[haruspex]

In Figure 3, water flows downward, so the water inside the pipe can only flow out through Port M and cannot be sucked back in through it. I added the check valve solely to prevent air from entering the system.

Then you drew the triangle representing the valve the wrong way up.

 

[vdance]

Then you drew the triangle representing the valve the wrong way up.

That was my oversight. Next time, I'll add a one-way water flow arrow internally.
I just thought of another new structural form that seems closer to my target requirements. I'm currently drafting the analysis diagram.

 

[256bits]

If the answer does not involve calculating the 46,000 N force generated at a height of 4m (50,000 - 4,000) nor neglecting the 4m height component F₃ = ρgsh = 1000 × 10 × (1 - 0.1) × 1 = 9000 N, I cannot understand how to proceed with the calculation

Originally, I thought the system was out in space, or on the international space station due to

Neglect gravity other than that of water;

If you did mot mean to neglect gravitational affects of the earth on the water column, then why the statement? ( Posting #1 )

Label all that needs to be labelled: L1 = 1m, L2 = 4,m Piston A, Piston B, etc to avoid confusion.

Label a datum level, where pressure is allocated to be 0.
Use a ∇ on that surface.
Then posts such as #3, from @Lnewqban, will not be necessary.

The calculation is incorrect because: ( From Post #1 )
Figure A calculates the static force F1 correctly om the piston with the one column of water.
Figure B calculates the static force F2 on the valve correctly with the one column of water.

Figure C is a s new system, with two columns. It must be analyzed separately in its own.
Is F3 the restraining force or the force of the pressure on th piston(s). Unclear.
A flapper valve should be added to the bottom piston, that allows water to enter the second chamber column when the piston ensemble is moving upwards.
1. The implied external static force F3 on piston(s) is different slowly moving up, than moving down.
Water ,moves slowly out the valve..
2. With the piston ensemble dynamically moving down without external restraining force, the water moves with a velocity out the valve. A pressure differential now exists across the valve.
3. Was a spring force of F2 added to the bottom valve M as surmised by the calculations. Where did this come from except by add hoc magic? ( Transferring over form Fig B is not allowed for this new system )
4. The lengths of the bottom piston within the 1m second column is not given.. As such, the pressures acting on the top not the bottom of this piston are not calculable. Add a n L3 or L4.
5. The stress within the connecting between the two pistons will need to be calculated. Is it a rope, in which case the top piston might not exert a force ( no compression, maybe tension ) on the bottom piston when moving downwards . If rigid, is it in compression, or tension?

Your system is more complicated than realized.

Somewhy, I am barred from accessing your profile to access previous posts when I tried to look that up.

 

[vdance]

Originally, I thought the system was out in space, or on the international space station due to

If you did mot mean to neglect gravitational affects of the earth on the water column, then why the statement? ( Posting #1 )

Label all that needs to be labelled: L1 = 1m, L2 = 4,m Piston A, Piston B, etc to avoid confusion.

Label a datum level, where pressure is allocated to be 0.
Use a ∇ on that surface.
Then posts such as #3, from @Lnewqban, will not be necessary.

The calculation is incorrect because: ( From Post #1 )
Figure A calculates the static force F1 correctly om the piston with the one column of water.
Figure B calculates the static force F2 on the valve correctly with the one column of water.

Figure C is a s new system, with two columns. It must be analyzed separately in its own.
Is F3 the restraining force or the force of the pressure on th piston(s). Unclear.
A flapper valve should be added to the bottom piston, that allows water to enter the second chamber column when the piston ensemble is moving upwards.
1. The implied external static force F3 on piston(s) is different slowly moving up, than moving down.
Water ,moves slowly out the valve..
2. With the piston ensemble dynamically moving down without external restraining force, the water moves with a velocity out the valve. A pressure differential now exists across the valve.
3. Was a spring force of F2 added to the bottom valve M as surmised by the calculations. Where did this come from except by add hoc magic? ( Transferring over form Fig B is not allowed for this new system )
4. The lengths of the bottom piston within the 1m second column is not given.. As such, the pressures acting on the top not the bottom of this piston are not calculable. Add a n L3 or L4.
5. The stress within the connecting between the two pistons will need to be calculated. Is it a rope, in which case the top piston might not exert a force ( no compression, maybe tension ) on the bottom piston when moving downwards . If rigid, is it in compression, or tension?

Your system is more complicated than realized.

Somewhy, I am barred from accessing your profile to access previous posts when I tried to look that up.

My objective is to calculate the downward resultant force exerted by water on the piston under Earth's gravitational conditions. As shown in Figure 1, the resultant force when an external force pulls the piston upward can be readily determined without analysis. To avoid interfering with the analysis, the check valve within the pipe is not labeled. To simplify calculations, the gravitational forces of all components except the water volume are neglected, and friction is also ignored. However, the resultant force calculation for this structure matches that in Figure 4. Based on the established objective, this model ultimately failed to meet my requirements.