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. That would be a good 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]

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.

 

[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?

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?