global warming napkin math

[metalhead]

If you look in my calc post, you can see that I ignored the energy cost to move the canister upward to that 100 km height (the potential energy was only 1 million joules).
A space elevator attached to a cable would indeed need only a little bit more than that 1 million joules.

But once you have your canister there, you still have to give it the escape velocity of the Earth to shoot it away from the Earth, which needs more than 60 million joules energy.

I believe the mechanics of the space elevator are such that you'd have the counter-weight at the end be roughly twice the height of geosynchronous orbit, and it would already be moving at or near escape velocity. The Earth's rotation is doing the work for you.

 

[Silurian]

If you look in my calc post, you can see that I ignored the energy cost to move the canister upward to that 100 km height (the potential energy was only 1 million joules).
A space elevator attached to a cable would indeed need only a little bit more than that 1 million joules.

But once you have your canister there, you still have to give it the escape velocity of the Earth to shoot it away from the Earth, which needs more than 60 million joules energy.

If the canister was released high enough it would have escape velocity.

 

[Hrothbern]

I believe the mechanics of the space elevator are such that you'd have the counter-weight at the end be roughly twice the height of geosynchronous orbit, and it would already be moving at or near escape velocity. The Earth's rotation is doing the work for you.

There is the Law of Conservation of Energy.
Within a system, in this case the Earth, the counterweight and the canister, the total energy must remain the same.
If what you say would work, it would mean that I can give a 1 kilo canister without energy cost 60 million joules energy !
So somewhere in the total equation of the system that energy has to come from.

That counterweight at twice the height of geosynchronous orbit has a certain orbital speed to be able to be there which is a certain kinetic energy content.
If I move a mass upward to that counterweight, the total mass of the now loaded counterweight goes up, while the total energy content of that now loaded counterweight has to stay the same.
The energy content of 0.5*mv2 has to stay the same.
Because the total mass went up, the velocity has to go down.

If the canister was released high enough it would have escape velocity.

yes, and from above argument, at the expense of the kinetic energy content of the counterweight, at the expense of the orbital speed of the counterweight.

EDIT

I assumed in above logic for that elevator a cable like construction. There you need to add all the time 60 million joules per kilo to compensate for the loss of orbital speed.

However if you would build a stif construction like a tower to enough height, you do not need to add energy to the counterweight, because that 60 million joules energy will now be supplied by the rotational energy of the Earth.
But because the Law of Conservation of Energy still applies, this will go at the expense of the rotational speed of the Earth.

 

[metalhead]

Since the counterweight is tethered to the Earth, and in tension, wouldn't the mass of Earth figure in as well, making that a trivial change in velocity?

 

[Silurian]

The Earth would slow down.

 

[Hrothbern]

Since the counterweight is tethered to the Earth, and in tension, wouldn't the mass of Earth figure in as well, making that a trivial change in velocity?

The rotational speed of the Earth has a certain amount of rotational energy, being the integral of the kinetic energy as function of the middlepoint to the surface. Both the mass and the rotational speed going up when closer to the surface.
Tapping into that energy slows the rotation. I give it a number crunch shot to see how much per gigaton of CO2.

As sidenote.
You could say that the moon is gravitational connected to the Earth with a similar effect. In the last 5 billion years, the rotation of the Earth has slowed down with around 5 hours per day because of that gravitational friction. The reciprocal tidal effect on the much smaller Moon has been that the Moon picked up that rotational energy as orbital speed, and is getting farther of the Earth as consequence and has a rotational speed tuned to always facing the Earth with the same side..
=> The Law of Conservation of Energy also applies to the Earth Moon system (whereby noted that part of that total energy, a share goes in friction of ocean water (the moving tides) and the fluid Earth core resulting in heat, and part goes in the rotational-orbital speed of the Earth and Moon).

As a way to easily imagine that moving mass from the Earth to a counterweight will reduce the rotation: an ice skating person making a pirouette, changing the rotational speed by keeping arms close to the centre of rotation or spreading them out lowering the rotation speed.

 

[metalhead]

The mass of the earth is roughly 6*10^24 kg. We need to remove (very) roughly 1*10^15 kg of CO2 from the atmosphere in order to return to pre-industrial concentration.

So even if you moved all of that mass to the end of the tether at one time, you're talking about a factor of 1/(6*10^9) of the mass of the Earth being taken out to the end of the tether, and in reality you would move thousands of smaller payloads, one at a time, making the difference in the 10^13 or 10^14 range for each trip.

So, sure, you might slow down the rotation of the Earth. But, by a perceptible amount? The ice skater is moving 1/10th or so of her body mass by extending her arms out. The difference in mass we're talking about is more like if the skater has her arms extended, and one of the hairs on her hand falls out.

 

[Hrothbern]

Number crunching the los of rotational energy per gigaton CO2 removed with a tower:

The rotational energy content of the Earth is according to Wikipedia 2.138*10^29 joules.

An example is the calculation of the rotational kinetic energy of the Earth. As the Earth has a period of about 23.93 hours, it has an angular velocity of 7.29×10−5 rad/s. The Earth has a moment of inertia, I = 8.04×1037 kg·m2.[1] Therefore, it has a rotational kinetic energy of 2.138×10^29 J.
https://en.wikipedia.org/wiki/Rotational_energy

Don't worry about that rotation period of 23.93 hours mentioned. That is the physical correct rotation time of the Earth seen from outside the Earth, and the right number for the energy calculation.
Because we are in orbit around the Sun, we need as Earth 23.93 hours to turn 360 degrees around our own axis and some minutes more to turn the same spot on Earth to the same angle with the sun again which takes slighly more time because of our (curved) orbit around the Sun.

1 gigaton CO2 is 10^12 kilo
As from my earlier post we need 62.72 million joules per kilo, or 6.272*10^6 joules
=> we need 6.272*10^18 joules per gigaton CO2

=> we reduce the rotational energy to (2.138*10^29 - 6.272*10^18) / (2.138*10^29) = 0.999999999971
and the rotational velocity, the square root of that energy, to 0.999999999985
The impact on lenghtening our day of 86,400 seconds is 86,400 / 0.999999999985*86,400 = 86,400.0000012673 seconds

If we would remove this way 1,000 gigatonnes of CO2 (that 1*10^15 kg you mention), our day would increase with roughly 1.3 milliseconds.
That's not much.
And considering that our natural rotation decrease by the gravitational friction with the Moon is adding 1.8 milliseconds per 100 years to our day, I see no real issue on our rotation for a temporary action.

 

[Timsup2nothin]

Don't worry about that rotation period of 23.93 hours mentioned. That is the physical correct rotation time of the Earth seen from outside the Earth, and the right number for the energy calculation.
Because we are in orbit around the Sun, we need as Earth 23.93 hours to turn 360 degrees around our own axis and some minutes more to turn the same spot on Earth to the same angle with the sun again which takes slighly more time because of our (curved) orbit around the Sun.

While a good approximation, this value is incorrect. It does not account for the motion of the sun.

 

[Hrothbern]

While a good approximation, this value is incorrect. It does not account for the motion of the sun.

yes
But it becomes so terribly complicated to use text to describe all that.

 

[Timsup2nothin]

yes
But it becomes so terribly complicated to use text to describe all that.

Beyond the scope of the napkin then?

 

[rah]

And is all that hair splitting really relevant to the broad sweeping discussion? But I'll admit, I've found it interesting.
But I keep imagining near earth space polluted with Co2 canisters.

 

[Timsup2nothin]

And is all that hair splitting really relevant to the broad sweeping discussion? But I'll admit, I've found it interesting.
But I keep imagining near earth space polluted with Co2 canisters.

I keep imagining interstellar traders opening a huge seltzer water production facility and laughing at us for giving them their raw materials for free.

 

[Hrothbern]

Beyond the scope of the napkin then?

yes

But within napkin is the number crunch how high that stiff tower must be to have at the top an orbital speed equal to the escape velocity of Earth.
No idea.... but could be higher as twice the geosynchronous orbit

 

[El_Machinae]

I am extremely in favor of a future space elevator. But seriously, you buy us a lot more time if you just reduce your footprint and can maybe convince a couple loved ones to do so as well

 

[metalhead]

And is all that hair splitting really relevant to the broad sweeping discussion? But I'll admit, I've found it interesting.
But I keep imagining near earth space polluted with Co2 canisters.

The nifty thing is that if you are producing fuel with the CO2 you capture, you could affix rockets to the canisters that run on the fuel perhaps, jettisoning them far enough from Earth that they just kind of float off into space. Or, alternately, you could launch them to slingshot around Earth and then shoot off in some other direction.

 

[Silurian]

I keep imagining interstellar traders opening a huge seltzer water production facility and laughing at us for giving them their raw materials for free.

I would aim them at Mars. Assuming we find no evidence of life there.