Filling a long tube by submerging it in fluid

[JuliaH19]

Hi everyone,

I'm working on a first-year college assingment and need some help with it.
I need to figure out how deep does a tube, initially filled with air, need to be submerged into water in order get fully filled, meaning in order for the air to be fully displaced by water.
The tube is 1/2 inches, 10,000 ft long, and it's capped at the bottom.
The hole (full of water) where the tube is submerged is infinitely long and vertical.

I'm attaching a picture for clarity.

Questions:
1. I would appreciate some help picking the right model/equations to find "x": how deep does the inlet of the tube need to be for it to get fully filled with water?
2. I also wonder if some solubility (Henry's Law) comes into play.
3. The first variation of the assingment states that the temperature is higher at the bottom (350 F) yet remains 77 F at surface. It asks what could happen then. My guess is a gayser might happen, but what do you think?

Thank you!

 

[berkeman]

Welcome to the PF.

I need to figure out how deep does a tube, initially filled with air, need to be submerged into water in order get fully filled, meaning in order for the air to be fully displaced by water.

Why would the open top of the tube need to be submerged to some depth? As long as water can run into it, it seems like the tube will eventually fill up.

And the 350F temperature at the bottom does throw a curve ball in. What is the pressure of the water at that depth, and what is the boiling point of water at that temperature pressure?

EDIT -- fixed typo in the last sentence...

 

[sojsail]

The hole (full of water) where the tube is submerged is infinitely long and vertical.

Not sure what long and vertical means. So the water is infinitely deep. But I am wondering how broad the hole is. Would the distance from the water surface down to the top of the tube decrease as the tube fills? Your figure looks as if there are no sides to the hole. If so, my "distance from the water surface down to the top of the tube" would remain constant.

 

[DaveE]

I think we all need a better description of the problem. As is, the answer seems obvious to me, x=0 as berkeman said above. That makes me think I don't really understand your question.
Perhaps as a point of clarification: If the water fully fills the tube ("displaced") where does the air go? How (where) does the water get in?

 

[Comeback City]

And the 350F temperature at the bottom does throw a curve ball in. What is the pressure of the water at that depth, and what is the boiling point of water at that temperature pressure?

I don't think this would change the depth needed to fill the tube... even if this were to cause some of the water to boil out (depending on pressure), water from the surface would still fall in. It would be in a dynamic equilibrium. Or am I missing something obvious?

 

[berkeman]

I don't think this would change the depth needed to fill the tube... even if this were to cause some of the water to boil out (depending on pressure), water from the surface would still fall in. It would be in a dynamic equilibrium. Or am I missing something obvious?

The only way I could see that high temperature at the bottom of the tube is with an active heater. That boiling the water at the bottom of the tube would keep it from filling with water, I think.

But when I looked up the boiling point of water versus pressure, the 350F temperature at that depth is pretty close...

Spoiler

Hah hah, made you look! The OP needs to do the simple Google Images search for themself.

 

[JuliaH19]

berkeman:
Thanks for the welcome!

"Why would the open top of the tube need to be submerged to some depth? As long as water can run into it, it seems like the tube will eventually fill up."

My guess is there is a determined pressure (depth = head pressure) at which air gets FULLY displaced by water, different from surface, or close to surface pressure.

"And the 350F temperature at the bottom does throw a curve ball in. What is the pressure of the water at that depth, and what is the boiling point of water at that temperature pressure?"

The pressure of water at 10,000 ft (surrounding = outside the tube) would be 4440.84 psi. At this pressure the boiling temperature is about 760 F.

 

[JuliaH19]

sojsail:
As per the statement of the assignment the hole is infinitely long = deep. Just condirmed it is also infinitely wide too.
The tube is vertically submerged in the hole.

 

[berkeman]

My guess is there is a determined pressure (depth = head pressure) at which air gets FULLY displaced by water, different from surface, or close to surface pressure.

But if it only gets partially displaced because the top of the tube is not deep enough, what is the remaining air distribution in the tube? Is there a leftover bubble left somewhere in the tube that finally gets flushed out by a high enough pressure at the top of the tube?

 

[JuliaH19]

DaveE:
I think we all need a better description of the problem. As is, the answer seems obvious to me, x=0 as berkeman said above. That makes me think I don't really understand your question.
Perhaps as a point of clarification: If the water fully fills the tube ("displaced") where does the air go? How (where) does the water get in?

To your question, this is all the info we were given to solve this open-ended question!
Getting mor feedback from the class, the point of discussion is to tell how much water head-pressure is needed to fully displace the air in the tube, meaning up to which depth would the tube need to be submerged for the entire it to get filled with water. Considering the tube has a diameter of 0.5 in and is 10000 ft long.

Some thoughts I've been having are regarding a potential capillary effect, since the tube has a small diameter and is so long. Also, would I need to consider the solubility of air in water (Henry's Law) at all?

 

[berkeman]

@JuliaH19 -- to quote another post, use click-drag to select the text, and click "Reply" in the pop-up. That makes it a lot easier to see what the quote is versus your reply. Thanks.



ADD -- So you would get this for your quoted text:

I think we all need a better description of the problem. As is, the answer seems obvious to me, x=0 as berkeman said above. That makes me think I don't really understand your question.
Perhaps as a point of clarification: If the water fully fills the tube ("displaced") where does the air go? How (where) does the water get in?

 

[JuliaH19]

But if it only gets partially displaced because the top of the tube is not deep enough, what is the remaining air distribution in the tube? Is there a leftover bubble left somewhere in the tube that finally gets flushed out by a high enough pressure at the top of the tube?

Thanks :)
That's the assigment, to figure-out at which pressure does the last leftover bubble gets flushes out, if that's what happens and it doesn't get trapped in the tube, given the tube's geometry.

 

[berkeman]

Thanks :)
That's the assigment, to figure-out at which pressure does the last leftover bubble gets flushes out, if that's what happens and it doesn't get trapped in the tube, given the tube's geometry.

Yeah, but what do you think the physics is that would leave any air in the tube? So far, myself and other responders don't see that happening.

Have you studied some stability criteria for vertical fluid flows in very narrow aspect ratio tubes against air? Is there some wetting or capillary effect that you've studied that may apply? And the problem says nothing about the tube not being shaken or vibrated, so even if there were a stable condition where the air bubble resisted the flow of the fluid, that could be defeated with vibration of the tube (think of clearing air bubbles from IV lines...).

 

[Chestermiller]

I think that it is important to first articulate what is happening in this system mechanistically. I see the water entering the top of the tube and flowing downward around the periphery of the tube (annularly) while the air flows upward up the center of the tube. Is this what others envision?

 

[jbriggs444]

I think that it is important to first articulate what is happening in this system mechanistically. I see the water entering the top of the tube and flowing downward around the periphery of the tube (annularly) while the air flows upward up the center of the tube. Is this what others envision?

I agree in general terms. The picture I get is of bubbles rising, and water flowing downward around them. I would not expect a smooth flow.

1/2 inch diameter should be enough to allow air to bubble up the tube. One could perform that experiment with an ordinary test tube submerged in a kitchen sink. Unfortunately, I cannot find any test tubes around the house.

 

[haruspex]

I think that it is important to first articulate what is happening in this system mechanistically. I see the water entering the top of the tube and flowing downward around the periphery of the tube (annularly) while the air flows upward up the center of the tube. Is this what others envision?

Indeed, the question seems backwards.
With the surrounding water only just brimming over into the tube there should be no opportunity for bubbles to become trapped.
On the other hand, if the tube is thrust deep into the water immediately it maybe that bubbles can become trapped. Whether they would is a complicated(?) matter of surface tension and tube radius, and I'm not sure that increasing pressure from above would tend to dislodge them.