Height of a Heron's fountain

[Dami121]

Homework Statement

I have this homework on how a Heron's fountain works. The exact assigment is as follows: "Construct a Heron's fountain and explain what (which variables) influences the height of the fountain."

I've already constructed it and it works well, but I'm not so sure about the second part of the problem.

I understand that the water is making a fountain due to the increased pressure in the "bottle b" (see picture) which pushes the water up through the tube d. I also understand that the air is getting to the bottle b via tube e, because the air itself in "bottle c" is being pushed by the water that is coming down from "bottle a" (tube f).

But in the end I totally confused myself: so what does the pressure in bottle b depend on? Is it the hydrostatic force? Am I even asking the right question? I got into a loop and I need to get out of it, help me please!

Homework Equations

p=F/A
ph=mgh=Vρgh

The Attempt at a Solution

I suggest from these equations that the height of the heron's fountain directly depends on pressure in "bottle b" and that the relevant parameters are V (volume of the liquid) ρ (viscosity of the liquid) and Δh (difference in heights - the length of a tube connecting "bottle a" and "bottle c".

But I'm not very sure of anything right now, I would be very glad if someone gets me onto the right track.

 

[haruspex]

Assign unknowns to the various heights and write down expressions for the pressures. For this purpose, suppose you have your finger on the top of tube d so that nothing flows.

 

[Dami121]

I wrote this down and I'm not sure where are we heading.
At the top of the "bottle a" ph= h1*ρ*g = 0 (because the height is 0)
down the bottom of the "bottle a" ph = h2*ρ*g
at the bottom of "bottle c" [p][/h] = h3*ρ*g
...
I could go on (I have it on paper right next to me)

I feel stupid. I don't get how this could help me. (or where did I go wrong?)

 

[haruspex]

I wrote this down and I'm not sure where are we heading.
At the top of the "bottle a" ph= h1*ρ*g = 0 (because the height is 0)
down the bottom of the "bottle a" ph = h2*ρ*g
at the bottom of "bottle c" [p][/h] = h3*ρ*g
...
I could go on (I have it on paper right next to me)

I feel stupid. I don't get how this could help me. (or where did I go wrong?)

I cannot check that because you have not defined those heights. Can you post a diagram with heights or reference points marked?
Remember that the tubes containing air make some pressures equal. Your aim is to find the pressure near the top of tube d.

 

[Dami121]

I'm now not at home, sorry for not replying so long. I have it written on a paper, because it's almost impossible to write on my very old IPhone. I attached the photos here.

I think I must've made a mistake somewhere but I'm not sure where.

 

[haruspex]

I'm now not at home, sorry for not replying so long. I have it written on a paper, because it's almost impossible to write on my very old IPhone. I attached the photos here.

I think I must've made a mistake somewhere but I'm not sure where.

I can read the diagram, but I do not follow the working.
The first hurdle is to realize which heights are relevant. There are five types of level in the set up:

• Top of container
• Top surface of water
• Bottom of container
• Top of tube
• Bottom of tube

Clearly not all are relevant. Start with the type that is most clearly relevant and see how far you can get with that type only. Hint: very few of the types are relevant.

 

[Dami121]

Oh well, I tought you advised me to define various random heights, my bad!

I think that 3 heights are relevant in this problem.

The height at the top of the tube (tube d) - let's call it h(t)
The height at the top surface of the water - let it be h(0)
The height at the bottom of the container (bottle c) - h(1)

 

[haruspex]

define various random heights

Not random.

The height at the top of the tube (tube d)

If the top of the tube were made a little bit higher or lower, would that be likely to change the height the fountain reaches? How?

The height at the bottom of the container (bottle c) - h(1)

Imagine adding another water-filled section below the base bottle, c, then removing the existing base so that it becomes part of bottle c. Would that change any pressures or flows?

 

[Dami121]

Ohh...The height of the fountain reach wouldn't really change if we made the tube lower or higher.
The water would just stay inside the tube if we made the tube high enough, and if we made it lower it would seemingly make a higher fountain.

hmmmm... If we did this I suppose the pressures won't change because the actual gradient would remain the same.

Am I correct?

 

[haruspex]

Ohh...The height of the fountain reach wouldn't really change if we made the tube lower or higher.
The water would just stay inside the tube if we made the tube high enough, and if we made it lower it would seemingly make a higher fountain.

hmmmm... If we did this I suppose the pressures won't change because the actual gradient would remain the same.

Am I correct?

Correct.
What about my other question, making the bottom of the base bottle lower compared with everything else? Or making the top or bottom of any tube higher or lower, as long as it stays in the same region and medium (air versus water)?

 

[Dami121]

I think that is a similar problem, If we made the bottom lower it wouldn't really change anything since the gradient would stay the same.
Same applies to your second question.

Uh, this must imply that the only height that matters is the height of the top surface of the water relative to the tube d?

 

[haruspex]

the only height that matters is the height of the top surface of the water

Surfaces, plural.

relative to the tube d?

Haven't we established that the precise start and end of a tube is irrelevant?

 

[Dami121]

Surfaces, plural.

Haven't we established that the precise start and end of a tube is irrelevant?

Yes of course, I was tired and my brain was working slowly yesterday.
So the parameters that matter are the differences between the heights of water surfaces. Is that right?

 

[haruspex]

Yes of course, I was tired and my brain was working slowly yesterday.
So the parameters that matter are the differences between the heights of water surfaces. Is that right?

Yes.

 

[Dami121]

And anything else that matters?
Perhaps the viscosity or the width of the tube?

 

[haruspex]

And anything else that matters?
Perhaps the viscosity or the width of the tube?

Yes, those will matter of course.
Can you find the relationship between the surface heights and the height of the fountain?

 

[Dami121]

I was going over this and couldn't work it out.
It seems to me that if we made the surface height at the bottom bottle lower, the pressure would actually increase, and if we made the surface height lower in the middle bottle, the pressure would drop.
What to do now?

 

[haruspex]

I was going over this and couldn't work it out.
It seems to me that if we made the surface height at the bottom bottle lower, the pressure would actually increase, and if we made the surface height lower in the middle bottle, the pressure would drop.
What to do now?

Call the surface heights in the three bottles A, B, C. What is the pressure at C?

 

[Dami121]

I have literally no idea.
It's equal to the pressure force of air over the area of the water surface, but how big is that force... I really don't have clue how to calculate that

 

[haruspex]

I have literally no idea.
It's equal to the pressure force of air over the area of the water surface, but how big is that force... I really don't have clue how to calculate that

You know the pressure at A, and there is a continuous body of water from there, down through tube f, into bottle c, and back to surface C. In terms of those heights, what is the pressure at C?

 

[Dami121]

Is it like this?
p=hρg+1013,25 hPa
Hydrodynamics aren't really my thing, I hope that it's right.

 

[haruspex]

Is it like this?
p=hρg+1013,25 hPa
Hydrodynamics aren't really my thing, I hope that it's right.

Yes, but we do not need to fill in a number for atmospheric pressure. That is just a background value that will cancel out in the end. Call it P_a. So rewrite the equation using that and the variables we have defined for the three heights, A, B and C.

 

[Dami121]

Okay so we have:

Pressure at A -
P = h1ρg + Pa = 0+Pa
Pressure at B -
P = h2ρg + Pa
Pressure at C -
P = h3ρg + Pa

 

[haruspex]

Pressure at B -
P = h2ρg + Pa

How do you get that? What connects the contents of bottles B and C?

 

[Dami121]

Of course that's not right, sorry, I have no idea how I got to that.

The bottles are connected with a tube (full of air)
The problem is that I really don't know what is the pressure force applied on the air.

 

[haruspex]

don't know what is the pressure force applied on the air.

What will happen if the pressures at B and C differ?

 

[Dami121]

Oh I'm very sorry! I thought replied, and was wondering why you aren't writing back.
The pressures at B and C can't differ because that would mean the water would be pushed into the tube (with lower pressure).
Hence the pressure at B and C must be the same, that is P=h3ρg+Pa.
That is all, or no?

 

[haruspex]

The pressures at B and C can't differ because that would mean the water would be pushed

They are the same, but not for that reason. What tube connects B and C? What is in that tube?

 

[Dami121]

Well the bottles are connected with a tube full of air, so I guess the pascal law comes in, saying that pressure applied on the air must be same in every point in the given connected medium.

 

[haruspex]

Well the bottles are connected with a tube full of air, so I guess the pascal law comes in, saying that pressure applied on the air must be same in every point in the given connected medium.

There will be a very small pressure difference because of the weight of air in the tube, but we can ignore it.
If the tube were full of water then the pressures would be different (and the fountain would not work).

Edit: what I previously wrote in this next sentence was not what I meant to ask. See next post.

 

[haruspex]

Next, what is the pressure inside tube d at A? Look at what connects this to B.

 

[Dami121]

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?

 

[haruspex]

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?

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 ρ_wgh, where h is the height difference between the two points.

 

[Dami121]

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 ?

 

[haruspex]

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.

the pressure in the tube d at A must be smaller than at B.

Yes.

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?