Hydrostatics and Stevin theorem 4

[dianad]

Hi,

In hydrostatics everyone learned that P=F/A
p=Pressure
F=force
A=area
From this relation we can determine for a cilinder(for example):

P=(m x g)/A = (mv x V x g)/A = (mv x A x h x g)/A = mv x h x g
m=mass (kg)
mv=specific heigth (kg/m3)
V=volume
h=height
g=gravity acceleration

So, from this late relation: P=mv.h.g , wich corresponds to STEVIN theorem.

My problem is that with Stein theorem, says that Pressure only depends on the depth and always is presented the image that i've attached.
But if we use for both images the relation P=F/A, instead of the Stevin relation, to determine the pressure at the bottom, we have the following:

m=10kg (both)
A1=2 m2
A2=4 m2

P1=10kg * 10m/s2 / 2m2 = 50 Pa
P2=10kg * 10m/s2 / 4mw = 20 Pa

If we used the relation of Stein, we would have the same pressure for both.

What is your comment about this?

Thanks

 

[desertfox]

Hi dianad

I am not sure were you got your area's of vessels from, I think if you have assumed one vessel to have a base with twice the area of another and then assumed that 10kg of a fluid in each vessel will sit at the same height may well be flawed.
Take 2 cylinders of different internal diameters and fill them to the same level, now the pressure exerted by each column of liquid on the base will be the same, however each cylinder will have a different amount or mass fluid.

regards

desertfox

 

[BigInch]

I think it works like this,

Hydrostatic pressure acts in all directions, up, down, to the left, to the right, and normally on all surfaces it comes in contact with. In the teacup, the normal pressures on the wall are transmitted downwards, while in the flask shaped vessel, the hydrostatic pressures acts normally to the wall and the vertical component of that is transmitted to the floor of the vessel as a tensile load, uplifting the edge of the floor. Also notice that the weight or vertical component of hydrostatic pressure acts directly on the floor of the flask shaped vessel near the pointy part and does not act directly on the floor of the teacup, but normally along its walls.

To get the weight of water from its hydrostatic pressure you have to integrate the normal hydrostatic pressures at all points and depths over the entire surface area in which the water is in contact.

Try it by making your shapes with an infinite number of tiny cylinders stacked one on top of each other. In the case of the teacup, working from large diameter top cylinder to smaller diameter lower cylinders, at each cylindrical element's interface to the next, calculate the weight of water in the larger portion of the top cylinder's extra differential radius and add it as an axial compressive load into the wall thickness of the next lower cylinder. When when you get to the bottom cylinder, you will find that the sumation of all larger cylinder water weights causes an edge load that must be placed on the edge of lowest floor. That, in combination with the weight of the water column above the lowest floor, now the integral of which must be taken around the edge of the lowest floor equals the weight of all the water contained in the teacup.

In the case of an inverted teacup, there will be a hydrostatic "roof" load from hydrostatic pressure in the up direction, that will tend to lift the higher cylinder off the next lower cylinder. make that a tensile load on the next lower cylinder. Carry them all sequentially down to the bottom cylinder.

When you get to the bottom cylinder, both will have equal total weights of water. Pressures are hydrostatic. In the corner of the flask, the hydrostatic pressure is rho * H * density, just as it is at the center of the flask. Not the same as the teacup. In the equivalent "corner", the upper corner of the teacup, the pressure is 0.

Average hydrostatic pressure can be used, but the average of a triangle pressure is at the 2/3 point. In the case of the teacup, 1/3 from the surface and, in the case of the flask, 2/3 from the surface. Therefore there are much higher average pressures at the centroid of the inclined surfaces of the flask than there are on the centroid of the same inclined surfaces of the teacup. Over half of its surface is at less than half the flask's average pressure on the topologicly equivalent surface.

As for pressure below the vessels, underneath the steel or glass bottom, I have more pressure under my foot than an elefant, so I don't see that as too unusual that they should be different.

Now think about what happens when the vessels have no bottom, like when you take a glass of water and turn it upside down on a perfectly smooth surface. Then you have to push down hard, or the water will lift the glass off the table and run out. Another good one is a plastic sheet glued at the edges to a perfectly smooth surface, then filled with air under pressure.

"If everything seems under control, you're just not moving fast enough."
- Mario Andretti- When asked about transient hydraulics

http://virtualpipeline.spaces.msn.com

 

[dianad]

No, no,

What i've assumed is that those two vessels have the same mass because one is inverted in relation to the other and i assumed also that the bottom area of one is twice of the other.
This is possible and it gives me different values!

Please comment.
Thanks

 

[hydtools]

The areas in fact are equal. You must account for all the area upon which the fluid acts. In the case of the narrow bottom example, the fluid acts on the sloped sides and the areas projected down add to the flat bottom area. To what biginch says, pressure acts in all directions.

You are neglecting all the area which supports the fluid in the narrow bottom case. You are neglecting the upward forces in the wide bottom case.

Also see:

http://www.ac.wwu.edu/~vawter/PhysicsNet/Topics/Pressure/HydroStatic.html

Ted

 

[BigInch]

I did this one with only two cylinders (like a town water supply tank), the largest diameter at the top. The floor load of the larger cylinder is transmitted to the walls of the smaller cylinder below. I have drawn the pressure distributions below and added them all up and checked it aginst two weight calculations. All are equal.

Now if you would please do the complementry problem with the smaller diameter cylinder on the top, not forgetting that instead of a floor load, you have a roof load on the area difference acting upward, so you will have tension in the walls of the lower cylinder and the resultants of that tension are reversed in direction for your case. Noting also that hydrostatic pressue on the floor of cylinder 2 are equal in both cases.


"If everything seems under control, you're just not moving fast enough."
- Mario Andretti- When asked about transient hydraulics

http://virtualpipeline.spaces.msn.com

 

[Compositepro]

Maybe I'm misunderstanding the problem but it appears very simple. The confusion probably has to do with word definitions which is why in science every word is very carefully defined. Pressure is a term used in fluids but is often misapplied to solids where the term stress should be used. The pressure on the bottom inside surfaces of both vessels is the same because the height of liquid is the same. The contact pressure (stress)on the outside bottom surface of the vessels with the supporting surface below is different for the two vessels. It is not a hydrostatic pressure. It is a force applied between two solid materials.

This is a very common misunderstanding I encounter with people trying to understand how a vacuum bag works in composite processing.

 

[BigInch]

I don't think the terms matter too much mathematically. Both are equal to P/A. I say pressure is a load and stress is the result of applying a load to a material, liquid, solid or gas. Liquids and solids both have internal stresses and both can have pressure applied to them or can apply pressure to each other. A rock at the bottom of the ocean is applying the same pressure to the surrounding salt water that the salt water is applying to the rock. Liquids can have stresses just as solids have stresses and deform in the same manner. There is a bulk modulus for water just as there is a bulk modulus for ice.

Two objects in contact can be in contact via pressure or point load. The only difference is how much area a "point" has. Point load or pressure on the head of a pin both mean pretty much the same thing, right?



"If everything seems under control, you're just not moving fast enough."
- Mario Andretti- When asked about transient hydraulics

http://virtualpipeline.spaces.msn.com

 

[dianad]

Thank you all guys, because with all this answers and links, i understood the issue perfectly!

The reason why i've putted this question in the forum, was because i had a real problem that putted me t think in this.

My problem is the same that is reported in the link gaved by HYDTOOLS, namely the example with Vessel B. I had a pump that had a pipr of 1 inch at outlet. The pump was working to extract water from a reservoir to another reservoir a few meters up. The pump was selected to overcome this total head and it worked fine a few days, but recently someone substituted a check valve at half way and throwed away the pipe upwards and put there a 1 and 1/2 inch pipe (bigger that the first). This modification caused that the pump wasn't able to pump the water upwards to the upper tank.
So, at the momment i've concluded that i had more mass to pump and the pump wasn't able to do the work. This made me reject P=mv*g*h, because of the problem caused.

I hope i made myself clear.
Thanks

 

[BigInch]

Well then you know its not the pipe or the check valve that caused the problem (unless the check valve was installed in the reverse direction). Is it possible that after a few days of pumping the liquid level in the higher tank finally reached the total discharge head of the pump causing backpressure on the pump to reach shut-off head? If so, liquid would no longer flow from the pump to the higer tank.

"If everything seems under control, you're just not moving fast enough."
- Mario Andretti- When asked about transient hydraulics

http://virtualpipeline.spaces.msn.com

 

[hydtools]

If the pump no longer worked immediately after the modification, I'd bet the check valve is in backwards.

If the check valve is spring loaded to close, it may require more pressure than the pump can deliver to open it.

Ted

 

[Compositepro]

Dianad, if you are suggesting that your pump stopped working because there was a larger diameter pipe put on the discharge and there is a greater mass of water in the larger diameter - you are very wrong and need to go back to understanding the basic principles.

BigInch, while pressure and stress are both in units of force per unit area they are not the same and are treated differently. Stress has direction while pressure does not.
The concept of pressure being the same in all directions is meaningless in a solid but is the basis of fluid mechanics. If you don't make a distinction, then the very simple problem in the original post becomes a very complicated one.
In fact the drawings of the two vessels in the original post are used in beginning physics books to teach the importance of making the distinction between pressure and stress.

 

[vzeos]

Stress has direction while pressure does not.

This statement is wrong. Actually stress and pressure are both tensors. A tensor is a vector product of (in this case) area and force resulting in 9 components; 3 normal stress and 6 shear stress components.

The concept of pressure being the same in all directions is meaningless in a solid but is the basis of fluid mechanics.

This concept is not the basis of fluid mechanics. Actually the flow of real fluids involves a good deal of shear stress.

Dianad, the best way to look at the original problem is to use the simple definition of pressure, P = F/A, where A is a unit area and F is the force exerted on the unit area by the liquid. So if your unit area is 1 square inch, you need to find the number of cubic inches of liquid above the unit area. The number of cubic inches multiplied by the weight density of the liquid will give you the force acting on one square inch, i.e., the pressure in psi in this case. Since this is a static liquid, the pressure will be the same in all directions.

 

[BigInch]

Compositpro, Slow down buddy. I know its been awhile since I've taken several advanced level machanics of materials classes in college, but I don't think there's been any changes in the basics as you are suggesting.

"BigInch, while pressure and stress are both in units of force per unit area they are not the same and are treated differently." I didn't say they were the same, just mathematically equivalent. Other than that, I don't treat them similiarly, or any differently either. I just remember that each can be the result of the other.

But you say, "Stress has direction while pressure does not."
Obviously Not correct. Gravity load from a concrete foundation on the soil below, believe me, it acts as a downward pressue and produces stress in the vertical direction in the soil. It also produces a shear stress in the soil at a different angle. Dynamic pressure equivalent of velocity head, acts in the same direction as the velocity of the fluid. Static fluid pressure acts in all directions. Look at how a pitot tube measures air velocity relative to static pressure in an airplane's environment. Pressure is just force per unit area and its been pretty well known for a long time that force has direction. It follows that pressure does too. In some cases, where a point is involved, pressure must be applied in all directions; because we have no other choice, since points have no area.

"The concept of pressure being the same in all directions is meaningless in a solid but is the basis of fluid mechanics." If force and stress can have direction, I don't see why pressure cannot, even if its all directions.

For studying the concept of pressures and stresses in solids, find some Mohr's circle example problems and note the directions at which pressures and shears are applied to a differential element and the directions of the resulting normal, shearing and principal stresses inside the differential element.

vzeos, thanks for heading that one off at the pass.

"If everything seems under control, you're just not moving fast enough."
- Mario Andretti- When asked about transient hydraulics

http://virtualpipeline.spaces.msn.com

 

[dianad]

Hi everyone,

This was my initial intention. To generate some discution about some issues that we've learneg a long time, and we think that they aren't useful until we have a problem like this in hands.

What was said here, shows my doubts and why i've posted this here. I've stuied also everything about stresses, pressure, etc. but because it was a long time ago, it would take me too much time to study it again, so i posted here my problem so that i have a good and understandeble answer.

Sincerely, i think that BigInch, along with Vzeos opinios is the most correct, following what i've studied in the past.

In relation to my problem, i must say that the check valve was installed correctly and was equal to the first one that was substituted. Th only change that was made was the pipe next to the valve. The upper tank does not make any head pressure to the water in the pipe, because is an open tank and the pipe leaves the water from the top.

I will study this issue more accuraly to complete YOUR'S opinions and as soon i made my opinion consistent, i'll post it here with most pleasure, since all that i've done until now was read you opinions with th bigger attention!

Thank you alll guys, and in respect with this post, i'll think it stays open for more opinions!!!

It's a funny issue and that's why i love this, because this makes gown and experient people to put their knowledge in queetion everytime.

 

[BigInch]

I just changed my "famous quote" from this one to the current. Perhaps I must change it back.

"What gets us into trouble is not what we don't know, its what we know for sure" - Mark Twain

Apt advice. And just to be sure I wasn't getting me or you into one of those troubled situations, I got out Excel first and worked it out top to bottom.

"If everything seems under control, you're just not moving fast enough."
- Mario Andretti- When asked about transient hydraulics

http://virtualpipeline.spaces.msn.com

 

[Compositepro]

I won't be pedantic about definitions because in engineering different field do use words differently but in the above discusion I've read someone come to the conclusion that the pressure at the bottom of a column of fluid is not proportional to its height and that the diameter is a factor. I also read that pressure has direction - it is in all directions. I suggest that it would be more illuminating to say that the force that pressure applies to a surface has direction.

I'm just saying that imprecise use of words does not help understanding. BigInch you say " I say pressure is a load and stress is the result of applying a load to a material, liquid, solid or gas". But isn't pressure in units of force per unit area and load is in units of force.

Obviously, there are a number of words that various posters are using differently and they only way around that for the speaker to define what he means or the audience to guess.

This thread has a title that sounds very technical and narrow and as a result not many people would be reading it. But we are discussing the princples of physics 1A. I'm very concerned with the conclusion that Dianad seems to have reached which is so fundamentally wrong. Is there anybody here that supports the conclusion that the pump stopped pumping because the discharge pipe was increased from 1" to 1.5" and the extra weight of water in the pipe(pipe length and height did not change) stopped the flow?

 

[BigInch]

"pressure (is) in units of force per unit area and load is in units of force."

Yes of course pressure is in units of force/unit area, but no, load is not always in units of force (alone). I learned that a load can be a "point load", a force located at a point, a load can be a distributed load, a load/unit length and a load can be a "pressure load", a force per unit area, now we can talk about moment loads if you like, but I'd really rather not.

At this point he should really start a new thread with his specific pump problems. This one began with a question about pressure that should really have been posted in the fluid mechanics forum and besides its been pretty much run into the dirt by now anyway.

"If everything seems under control, you're just not moving fast enough."
- Mario Andretti- When asked about transient hydraulics

http://virtualpipeline.spaces.msn.com

 

[dianad]

Sorry to all because of my question.
It seems that created a bad mood to everyone, but it wasn't my intention.

Sorry and thank you all for your ideas.

 

[BigInch]

No no. No bad moods here.

"If everything seems under control, you're just not moving fast enough."
- Mario Andretti- When asked about transient hydraulics

http://virtualpipeline.spaces.msn.com