question revisited

[jagec]

Originally posted by: DrPizza
So... the new question becomes:

find a function for the air pressure at a distance r from the center of the earth inside a hole drilled through the earth.

(Oh no! if you puncture the earth, all the gravity will leak out!)

I'm fairly certain the highest pressure would be at the center.

yeah, it would.

Think of it this way...the differential element of air as we go down our tube experiences a pressure of P at the top, and exerts a pressure P+(w/Ac) (w is the WEIGHT of the element, will decrease as we near the center of the earth but always >0 for r>0). So the pressure will steadily increase until we reach the differential element at r=0, which contributes a pressure of zero. As we continue down the other side, the weight becomes negative (w<0 for r<0 according to our nomenclature..I know it's messy), and pressure decreases until it reaches P at the surface of the earth.

 

[FrankSchwab]

OK, so what will be the air pressure at the center of the earth?

Assuming adiabatic heating, what temperature will the air be?

What fraction of atmospheric air will have liquified by then due to the pressure?

Extra Credit: Calculate temperature by taking into account the heating due to the surrounding molten iron.


/frank

 

[jagec]

Originally posted by: FrankSchwab
OK, so what will be the air pressure at the center of the earth?

Assuming adiabatic heating, what temperature will the air be?

What fraction of atmospheric air will have liquified by then due to the pressure?

Extra Credit: Calculate temperature by taking into account the heating due to the surrounding molten iron.


/frank

Well, the good thing is that two of those questions can be dealt with right away if we assume steady state: the temperature of the air will be exactly the same as the temperature of the surroundings at that depth. Any heating effects caused by the compression of air as we drill the hole will have plenty of time to conduct heat away into the surroundings, so the only change in the situation will be the different thermal conductivity of the air vs. the stuff that was there before. If we assume the hole isn't very wide, that means the temperature will be pretty much that of the surroundings.

As for calculating the pressure, that gets to be tricky. I suspect the best way would be to use different assumptions at different depths: use the ideal gas law until the pressure starts to get near the critical pressure, than use more realistic correlations.

Actually, an even easier way would be to use experimental property tables for air (which I'm sure exist to a high enough pressure to apply in this situation), plot density versus pressure (and temperature if we're taking that into account), feed that into a computer and tell it to solve the differential equation.

 

[DrPizza]

no solution for the pressure? Everyone's as lazy as I am?

 

[jagec]

Originally posted by: DrPizza
no solution for the pressure? Everyone's as lazy as I am?

Here. Use that to look up density vs. pressure, and write a computer program to integrate.

 

[Armitage]

Originally posted by: DrPizza
no solution for the pressure? Everyone's as lazy as I am?

I have some notes on it but no time at the moment.

 

[onix]

Originally posted by: Armitage

Originally posted by: VTHodge
As far as air pressure is concerned, I believe there would be no pressure at the center of the earth. Fluid pressure acts on the surface between the fluid and the center of gravity.

eh? What's holding up all that air then? At the surface of the earth, the ground holds up about 14 lbs/in^2
If you have a hole all the way through, what's holding up that 14 psi? The 14PSI on the other side. Draw an FBD of the air in that hole ... there's nothing else to take that load, and the load will only increase the deeper you go because there is more air above you, even though the weight/mass of the air decreases the lower you go.

Armitage, you make an important point. (After posting I see others have caught on too.) You are absolutely correct that the earth's surface is providing the equal and opposite force to hold up the 14PSI of air pressure, however once you open up the hole there is no restoring force.

What I think happens now is that you have a long and narrow channel which if it has no air has pressure on both sides filling it up. In the absence of gravity, such as in the space shuttle crew area, the channel will fill up to the ambient pressure, e.g. 14PSI. Can't exactly say what will happen for the long earth channel, but there will be competing effects of bouyancy (gravity) and a channel driven by air pressure. Hard to imagine though the pressure at the center will be zero.

 

[onix]

I wonder if the gravitational rest position at the center of the earth might be metastable or even unstable. Is this obvious to anyone?

 

[Safeway]

Originally posted by: blahblah99
It would be a damped oscillation, with wind resistance acting as your source of friction. Eventually you'll just be floating in the center of the Earth. Not sure about the air pressure part though.

 

[DrPizza]

Originally posted by: onix

Originally posted by: Armitage

Originally posted by: VTHodge
As far as air pressure is concerned, I believe there would be no pressure at the center of the earth. Fluid pressure acts on the surface between the fluid and the center of gravity.

eh? What's holding up all that air then? At the surface of the earth, the ground holds up about 14 lbs/in^2
If you have a hole all the way through, what's holding up that 14 psi? The 14PSI on the other side. Draw an FBD of the air in that hole ... there's nothing else to take that load, and the load will only increase the deeper you go because there is more air above you, even though the weight/mass of the air decreases the lower you go.

Armitage, you make an important point. (After posting I see others have caught on too.) You are absolutely correct that the earth's surface is providing the equal and opposite force to hold up the 14PSI of air pressure, however once you open up the hole there is no restoring force.

What I think happens now is that you have a long and narrow channel which if it has no air has pressure on both sides filling it up. In the absence of gravity, such as in the space shuttle crew area, the channel will fill up to the ambient pressure, e.g. 14PSI. Can't exactly say what will happen for the long earth channel, but there will be competing effects of bouyancy (gravity) and a channel driven by air pressure. Hard to imagine though the pressure at the center will be zero.

There's plenty of gravity acting on objects in the space shuttle... if there weren't, it would continue in a straight line, instead of being accelerated in a circle (ellipse) around the earth. Using F of gravity = Gmm/r^2, r only increases from the radius of the earth to the radius of the earth plus a couple hundred kilometers.. I worked it out a month or so ago, but don't remember the exact value... the acceleration of gravity on the space shuttle is around 7 or 8 meters per second squared.