What variables affect the height of a Heron's fountain?

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SUMMARY

The height of a Heron's fountain is influenced primarily by the pressure in "bottle b," which is determined by the hydrostatic pressure of the water in the system. Key variables affecting this pressure include the volume of liquid (V), the density of the liquid (ρ), and the height difference (Δh) between the water levels in the connected bottles. The relevant heights to consider are the top of the tube (tube d), the top surface of the water, and the bottom of the container (bottle c). Understanding these relationships allows for accurate predictions of fountain height based on the principles of fluid dynamics.

PREREQUISITES
  • Hydrostatic pressure principles
  • Fluid dynamics concepts
  • Understanding of pressure equations (p=F/A, ph=mgh)
  • Basic knowledge of viscosity and its effects on fluid flow
NEXT STEPS
  • Research the Bernoulli's principle and its application in fluid dynamics.
  • Study the effects of viscosity on fluid flow in tubes.
  • Explore the relationship between pressure and height in hydrostatic systems.
  • Learn about the design and construction of fluid systems, including Heron's fountains.
USEFUL FOR

Students studying physics, engineers designing fluid systems, and hobbyists interested in constructing hydraulic devices will benefit from this discussion.

  • #31
Next, what is the pressure inside tube d at A? Look at what connects this to B.
 
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  • #32
The tube is full of water, so I think the pressure in tube d at A would be
P = h1ρg+h3ρg ?
Of course the pressure changes there, but the initial pressure from the air in bottle B remains the same in the whole liquid no?
 
  • #33
Dami121 said:
The tube is full of water, so I think the pressure in tube d at A would be
P = h1ρg+h3ρg ?
Is the pressure inside the tube d at A more or less than the pressure at B?
Dami121 said:
the initial pressure from the air in bottle B remains the same in the whole liquid no?
Not sure what you mean.
Remember we started by supposing you have your finger on top of the tube d to inhibit the fountain so that everything is static. In that arrangement, there are two rules you can apply:
  • If two points are connected by a path passing only through air then the pressures are near enough equal;
  • If two points are connected by a path passing only through water then the difference in pressures will be ρwgh, where h is the height difference between the two points.
 
  • #34
Wow, I can't get my head around it now... I will go get some sleep and respond tommorow.

But let me try at least, since the height A is smaller than height B I guess the pressure in the tube d at A must be smaller than at B.
Then the pressure at A in the tube would be:

P = h3ρg-(h2-h1)ρg ?
 
  • #35
Dami121 said:
the height A is smaller than height B
To be clear, the way you are using h1 ... h3 they are depths from the top, not heights from the bottom. So you mean that the depth at A is less than the depth at B.
Dami121 said:
the pressure in the tube d at A must be smaller than at B.
Yes.
Dami121 said:
pressure at A in the tube would be:

P = h3ρg-(h2-h1)ρg
Yes.
So what is the pressure difference between inside the tube and outside the tube at A?
How does that pressure difference relate to the height of the fountain?
 
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Likes Dami121
  • #36
Thanks for that height-depth note, I was getting a bit tangled in this.

As for the pressure difference, the bigger the difference between in and outside the tube, the bigger the fountain would be.
And it can be expressed ΔP = h3ρg-(h2-h1)ρg - h1ρg = (h3-h2-2h1)ρg??
This looks so wrong... so I'll bet there must be something I missed in this calculation?
 
  • #37
Dami121 said:
h3ρg-(h2-h1)ρg - h1ρg = (h3-h2-2h1)ρg??
Try that step again.
 
  • #38
Oh, stupid mistake,
ΔP=(h3-h2)ρg

Is that it?
 
  • #39
Dami121 said:
Oh, stupid mistake,
ΔP=(h3-h2)ρg

Is that it?
Yes.
 
  • #40
Well, it's been a pleasure and honor sir!
I think I have all I need. It took me ages , but now I understand it very well.
Thanks a lot :)
 
  • #41
@Physicist1011 (who asked me about this thread): do you understand it down to finding the pressure difference at post #38?
 
  • #42
No I don't. I don't understand how the differences in heights of the water affect the water height and I also don't understand how you got that equation.

Edit: or does the heights of the water affect the water fountain not only the differences between the heights of the water in the containers?
 
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  • #43
Physicist1011 said:
No I don't. I don't understand how the differences in heights of the water affect the water height and I also don't understand how you got that equation.
First thing is to understand how the pressures in the different airspaces are related.
If two airspaces are connected by a tube of air, what can you say about the two pressures?
 
  • #44
The pressures are the same? but why?
 
  • #45
Physicist1011 said:
The pressures are the same? but why?
If they were different, what would the air do?
 
  • #46
Move towards lower air pressure.
 
  • #47
Physicist1011 said:
Move towards lower air pressure.
Right.
Early in this thread I advised Dami to consider the set-up with a finger on top of the top tube so that no fountain occurs. This is so that everything is static, which makes the analysis easier.
So we can assume no air is flowing, so the pressures are the same. (When the fountain is flowing there will be a small difference in the air pressures.)

Next, consider a tube that is filled with water, connecting two reservoirs. For simplicity, we can just think about the pressures at the surfaces of the reservoirs.
What can you say about those two pressures? How are they related?
 
  • #48
I am not sure.
 
  • #49
Physicist1011 said:
I am not sure.
If two reservoirs are connected by a filled pipe they are effectively a single reservoir. What relates the pressures at two points in a reservoir?
 
  • #50
Sorry I am really not sure as this is what I am confused about.
How does the pressure at d relate to the difference in heights since the pressure is what relates to the fountain's height right?
 
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  • #51
Physicist1011 said:
Sorry I am really not sure as this is what I am confused about.
How does the pressure at d relate to the difference in heights since the pressure is what relates to the fountain's height right?
Consider a tank of water. How does the pressure at height y1 from the bottom relate to the pressure at height y2 from the bottom?
 
  • #52
These 2 heights of water are connected in a closed space (except for the water surface at the top). Such that the pressure of the lower height will be larger than that of the higher water height.
 
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  • #53
Physicist1011 said:
These 2 heights of water are connected in a closed space (except for the water surface at the top). Such that the pressure of the lower height will be larger than that of the higher water height.
Yes, but by how much?
 
  • #54
Pressure difference = pgh2(higher water height) - pgh1(lower water height)

Edit: oh so change in pressure from upper height to lower height = change in height *pg
 
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  • #55
Physicist1011 said:
Pressure difference = pgh2(higher water height) - pgh1(lower water height)

Edit: oh so change in pressure from upper height to lower height = change in height *pg
Right.
There are four locations to consider: the surface of the water in each container, and the base of the fountain. (By "base of fountain" I mean the point inside the top tube that is the same height as the surface of the top reservoir.)
If you can move from one such location to another only moving through air then you know the pressures at those locations are the same; if you can move from one to another only moving through water then you can relate the pressures by the height differences.
And don't forget, you have a finger on top of the top tube.

What equations can you write relating these four pressures?
 
  • #56
Pressure difference between B and A = pg*height difference between water surface B and A
Pressure difference between B and C - same
Pressure difference between C and A = pg*height difference between C and A
I am getting a little confused - is this correct?
 
  • #57
Physicist1011 said:
Pressure difference between B and A = pg*height difference between water surface B and A
Is there a path between those two surfaces that is entirely in water?
 
  • #58
Yes there is.. that's why there is a pressure difference.
 
  • #59
Physicist1011 said:
Yes there is.. that's why there is a pressure difference.
I am not seeing it.
Tube f (water) connects A and C.
Tube e (air) connects B and C.
Tube d (water) connects fountain base with B.
 
  • #60
Oh the pressure at A is atmospheric pressure so the difference in pressure between B and A is PB=Patm
 

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