Energy savings with space elevator

[bksree]

Hi
I read in a paper that the energy saving in taking a payload to geostaionary orbit with a space elevator is (R/Rg)*(2-R/Rg) where R- radius of earth, Rg - radious of geostaionary orbit.

How is this obtained ?

TIA

 

[Nik_2213]

Uh, saved compared to what ??

Okay, an elevator is much, much more efficient than staged rocketry, much more efficient than even an air-breathing SSTO like Skylon, but you have the tether and counter-weight mining tech to launch 'up-front'. Sure, once all that investment is repaid, the Solar System is wide open...

 

[Vagn]

Erm, that also has the wrong units for energy, it could be the percentage of energy saved perhaps, but not the energy saved. Have you got a link to the paper mentioned?

 

[bksree]

Thanks for the replies. The paper is : P.K Aravind, 'The Physics of the space elevator', Am. J. Phys., 75(2), Feb 2007.
The eqn actually gives the percentage saving of energy w.r.t that required if rocket prpoulsion is used. I think it is related to the energy required to accelerate to escape velocity with rocket propulsion wheras the space elevator uses the centrifugal force ofthe rotating Earth to accelerate the satellite.

TIA

 

[ttmark]

I'm not sure how they plan to keep the space elevator from falling back to Earth. Even orbiting satellites must use boosters from time to time to maintain their velocity and orbit due to friction from particles. For the space elevator, it would have Earth's gravity acting along a large portion of it's tether.

 

[ttmark]

Thanks for the replies. The paper is : P.K Aravind, 'The Physics of the space elevator', Am. J. Phys., 75(2), Feb 2007.
The eqn actually gives the percentage saving of energy w.r.t that required if rocket prpoulsion is used. I think it is related to the energy required to accelerate to escape velocity with rocket propulsion wheras the space elevator uses the centrifugal force ofthe rotating Earth to accelerate the satellite.

TIA

For the Earth to transfer its centrifugal force to maintain the orbital velocity of the tether, the tether would have to be rigid, which can not be achieved over the distance in question.

 

[russ_watters]

Rigid or angled backwards.

 

[Ryan_m_b]

I'm not sure how they plan to keep the space elevator from falling back to Earth. Even orbiting satellites must use boosters from time to time to maintain their velocity and orbit due to friction from particles. For the space elevator, it would have Earth's gravity acting along a large portion of it's tether.

I was under the impression that this was what the counter weight was for.

 

[ttmark]

I was under the impression that this was what the counter weight was for.

It at first sounds as if it would work, but the moment we attach a payload the systems center of mass changes, the gravity effect increases, and the tether loses both velocity and orbital distance. If the system is able to generate enough energy from somewhere to lift the payload up to the counterweights orbit, then it would recoup its orbit distance and velocity plus some. This might be achieved at better efficiency than rocket propulsion because the payload on the tether has the advantage of slowly stepping its way to orbit at a slow speed. And could be done with electric motors either powering the tether or taking power off the tether which is being generated by its long length in Earths atmosphere.

 

[Nik_2213]

"..this was what the counter weight was for."

Exactly ! The 'elevator' is not a 'sky-scraper' which must some-how stand up on its own, but a 'cable' in tension thanks to the large counterweight. The latter 'soaks up' the minor orbital changes due to cargo riding the tether...

 

[ttmark]

"..this was what the counter weight was for."

Exactly ! The 'elevator' is not a 'sky-scraper' which must some-how stand up on its own, but a 'cable' in tension thanks to the large counterweight. The latter 'soaks up' the minor orbital changes due to cargo riding the tether...

I am not sure I understand... A large portion of the tether is not in space and thus does have gravity puling it back to Earth. The Counterweight will tend to make the tether maintain its orbital velocity in geosynchronous orbit but any payload added to the tether will cause the orbit distance to decrease. At this point we can expend the external energy needed to now lift the payload against gravity along the tether into space and as it gets there it should recover most of the lost orbital distance but the energy needed to lift the payload in the first place is still required to be applied into the system.

 

[Nik_2213]

Wiki has its issues --YMMV-- but it is a fair introduction.
http://en.wikipedia.org/wiki/Space_elevator

And here's one for the Moon. In passing, they mention tether & counterweight masses-- IIRC, the counterweight must be more massive than the tether which, in turn is much, much more massive than laden riders...
http://cpsx.uwo.ca/PS%20Seminar/PS%20Seminar/Papers/SpaceElevator.pdf

 

[David_Appell]

bksree: Here's a sketch of the derivation. But first note that your formula has a multiplication sign where there should be a division sign (see Aravind's paper, http://chaos.swarthmore.edu/courses/pdg07/ajp/ajp000125.pdf).

The energy savings from a space elevator is E(no SE) - E(SE), where SE=space elevator. The fractional energy savings is this difference divided by E(no SE).

Now, E(SE) is just the change in potential energy going to GEO: GM(1/R - 1/Rg).

Then, E(no SE) is the PE change + the change in kinetic energy. The change in KE is (KE in orbit - KE at launch). At launch KE=0, of course. The KE in orbit (at GEO) comes from the speed at GEO where centripetal force = force of gravity, i.e. mv^2/Rg = GMm/Rg^2.

Plug this velocity in and do the algebra and you get

fractional energy savings = p/(2-p) where p=R/Rg. This equals 8.2% for Earth, and 9.1% for Mars.

Hope this helps.

 

[bksree]

David Appell
Thanks for the reply

 

[Chronos]

The potential energy savings of a 'space elevator' would never justify the truly astronomical cost of constructing and maintaining such a device. Even assuming we had the material science, the base of the thing would have to be about the size of Switzerland.

 

[David_Appell]

Chronos: How so? A lightweight material like some carbon nanotube composite would only need a base width of less than a meter. The example Aravind gives in his paper at the end of his section IV has a base area of 1.5 square millimeters...

 

[phyzguy]

The potential energy savings of a 'space elevator' would never justify the truly astronomical cost of constructing and maintaining such a device. Even assuming we had the material science, the base of the thing would have to be about the size of Switzerland.

This isn't right. It's not a tower standing upwards, it's a cable hanging downwards. Imagine a large mass in geostationary orbit and unreeling a cable down to the Earth's surface. You don't even need to go all the way to the surface - in principle you could stop a few miles (or a few hundred miles) up. The problem, of course, is finding a material that is strong enough and light enough that it can support 22,000 miles of its own weight.

 

[Chronos]

Agreed, but, it still requires an enormous Earth based platform. Best guesses are a tower around 20 kilometers high and a sea based platform about the size of a major metropolitan airport. Not quite Switzerland, but, that puts it in perspective. It remains an unaffordably expensive proposition, even having solved the material science and engineering issues.

 

[Ryan_m_b]

Agreed, but, it still requires an enormous Earth based platform. Best guesses are a tower around 20 kilometers high and a sea based platform about the size of a major metropolitan airport. Not quite Switzerland, but, that puts it in perspective. It remains an unaffordably expensive proposition, even having solved the material science and engineering issues.

Perhaps it is my ignorance but it was my understanding that the cable/ribbon hangs from the geosynchronous weight in orbit. That being the case why would a huge anchor be needed?

 

[DrStupid]

Perhaps it is my ignorance but it was my understanding that the cable/ribbon hangs from the geosynchronous weight in orbit. That being the case why would a huge anchor be needed?

The mass of the anchor should be greater than the maximum payload to avoid a collapse of the cable. But I think nobody wants to lift up a major metropolitan airport.