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Space Elevator - PHYS 310
The Space Elevator
Katherine Burgess, Ross Gregorive, Mason Roberts, Kevin Thompson, Matt Uleman
As of today, and for the past 50 years, we use hundreds of tons of fuel to break the bounds of Earth's Gravity in order to reach space. 90% of a rocket's mass come from the fuel needed to lift it beyond the atmosphere, this amount of fuel adds up to about $10,000 per pound of material they put up there. In our project we are pursuing an alternative method for getting our payloads into space, and that's where our idea of a space elevator comes from. The concept behind a space elevator is that the gravity of the Earth, pulling the tower/cord down, is directly countered by the centripetal acceleration created by the Earth’s own rotation. Objects in low orbit must be going very fast in order for them to be traveling faster than they are falling back down to the Earth, but these objects do not remain stationary above a single point on Earth. These satellites no longer have any connection to the Earth other than by gravity. Objects farther out in orbit experience less of the Earth’s gravitational pull and also do not need to be going as fast as lower orbiting objects in order to maintain a stable orbit. These objects can sustain an orbit above a single point on the surface of the Earth, ideal for connecting a tower/cord between the two.
It is also important to consider the materials the structure will be built from, having to endure such stresses, and just where the structure will be located along the equator. In this research paper we will examine just what technologies, materials, innovations, policies and energy will need to go into the construction and maintained use of the Elevator.
The earliest designs for a space elevator, over 100 years ago, were based on towers. Towers encounter problems when they grow large due to compression. Currently no material is strong enough to support its own weight. Cord structures do not enough problems with compression, however, tension creates complications with material design. The cord structure can counteract gravity with centripetal acceleration. The tension will be greatest in the middle of the cord, so we will need to use a variable width for our cord, peaking in the middle.
The location of the elevator is very important because of this tensile design. Since we are using the earth’s rotation to produce centripetal acceleration, picking a site along the equator would maximize angular speed. The space elevator will be relatively fragile so earthquakes, asteroids and storms could be devastating. Therefore, we can mount the elevator to a large ocean vessel that can move linearly along the equator to dodge natural disasters. The tensile structure will not be very heavy because it uses centripetal force to offset its weight, so a large vessel will be able to support the elevator. The amount of material required to build a space elevator on earth is unrealistic for the near future. However, we could build an elevator on the moon using Kevlar. With two solar powered elevators, we could have a relatively carbon free trip to the moon and back.
In designing a space elevator it is important to understand just how high "up" it is going to extend. From there we can determine just what amount of forces are being exerted on the super structure and determine what materials and shape the thing will have. The point at which we want to reach is called geostationary orbit. At this point the velocity needed to sustain a stable orbit is the same angular velocity of the Earth's rotation and thus remains above a single point on the equator, ideal for what we are trying to accomplish.
A simple calculation of the variable gravity of Earth as you get further away,
and the variable angular acceleration pulling the tether away from Earth,
set equal to each other and solving for the radial distance yields,
This value for the distance from the surface of the Earth at the equator is around 36,000km! To give some perspective on how large this is, it would be 6 times the radius of the Earth, or one tenth the distance to the moon!
The elevator must extend past geostationary orbit to support its own weight. We plan to extend the tower 14,000 km past geostationary orbit and attach a 10 zg mass to the end. Below is a visual representation of the acceleration versus distance. The function crosses zero gravity right at the geostationary orbit of 36,000 km. The further past orbit we build the elevator, the less mass we will need on the end. However, carbon nanotubes are very expensive so we want to make the elevator as short as reasonably possible.
The material choice that allows for a material with high enough tensile strength is carbon nanotubes. Continuous strings of nanotubes would be able to carry the tensile load of the elevator at geostationary orbit where the stress is the greatest. The tensile stress at geostationary orbit is defined below,
where G is the gravitational constant, M is the mass of the earth, ρ is the density of the material h is the height of the elevator, and Rg is the height of the geostationary orbit. The carbon nanotube density is assumed to be 1500 kg/m^3, height is 50,000 km, and geostationary orbit is defined to be 36,000 km. The resulting maximum stress is 69 GPa. This strength requirement can be satisfied with carbon nanotubes. The tensile strength of current carbon nanotubes is around 130 GPa, which is a sufficient stress to not fail under the the stress at geostationary orbit.
The reality of obtaining 50,000km long nanotube without flaws is nearly impossible. However, since we are able to manufacture nanotubes with a strength of 130GPa, there is a possibility of making a material strong enough to handle the stresses generated. The resulting material would need to be a composite with discontinuous fibers of length 40mm as found in commercially available nanotubes. These fibers can be aligned and impregnated with a matrix material to distribute forces in the form of discontinuous fibers all aligned axially.
The matrix material must be processed in a special way to prevent the carbon from being absorbed in the material itself. This poses many problems with manufacturing The material being investigated for its high weight to strength ratio is Beryllium. Beryllium also is an isotropic material with a high modulus and corrosion resistant, giving favorable properties to be applied to the space elevator.
The minimum required length of carbon nanotubes for the discontinuous fiber composite to support the tensile loads is shown in the equation below. The length acquired from this equation gives the value in which the fiber will fail before the matrix fails. After calculating this value, the length should be extended even longer to increase the load characteristics of the of the composite. If the fiber length is greater than 5 times the critical fiber diameter, the composite will retain at least 90% of an continuous fiber composite.
Where L is the critical length, σ is the failure stress of the fiber, d is the diameter of the fiber and τ is the shear stress. Assuming there is perfect adhesion between the matrix and the fiber (yield strength of beryllium is 0.52GPa) and a fiber diameter of about 5nm, the critical length of the nanotubes was found to be 6.25x10^-7 m, which is significantly less than the commercial 0.04 m length nanotubes. From this information, the composite can be considered to act like a continuous fiber composite.
In this study, the critical length will be used to find the required volume percent of beryllium to determine the feasibility of making a composite that can support the stress. The volume percent value will reveal if the composite is possible. If the ratio was too high (greater than 60 vol%), the composite stops following the equations presented due to interactions between the fibers instead of the matrix material. The equation that describes this relationship with discontinuous fibers is found below
where σltu is the ultimate strength of the composite, σfu is the yield strength of the fiber, σm is the yield strength of the matrix, Lf is the length of the fiber and vf is the volume fraction of the fibers in the composite. The ultimate strength was set equal to the tensile stress at geostationary, length of fiber was found to be 40mm and the the yield strength of the matrix was 0.52GPa for beryllium. The resulting fiber content was found to be 52.5% fiber, which is a reasonable ratio. For the parameters used, it was found that there is a possibility for a composite to be made that will allow for the construction of the space elevator.
Earth’s gravity creates a downward force on the tether that is proportional to its length. Calculations done by individuals previously indicate that the tether material has to have a minimum breaking length of 5000 km at sea level, meaning that it can withstand 5000 km of it’s own weight without buckling, breaking, or deforming. For the space elevator, the material’s breaking length does not have to be the elevator’s full height because the force of gravity decreases with altitude. A principle issue with the feasibility of the space elevator is the availability of materials that meet these requirements. Most metallic alloys have breaking lengths of 30 km, while materials like Kevlar and carbon fiberglass have maximum breaking lengths of 400 km.
The best material for this structure was penis determined to be carbon nanotubing, a recently-developed fibrous material that is estimated to have a breaking length of 6400 kilometers, well above the necessary minimum. Details about this material are described above. Although this has only been constructed to a maximum length of 18 cm, innovators are hard at work expanding the properties and uses of this highly durable material. We hypothesize the in the future, the possibility of the extended use of carbon nanotubing can be realized. For the below calculation, we have assumed the use of carbon nanotubing for the tether. We have also assumed that the tether will be a cylinder, instead of wider at the middle than the top and bottom, so this calculation is for the upper limit on the amount of nanotubing needed.
This calculation turns out to be approximately 1.02 trillion kg, or 1.02 billion tons. This mass is only a small fraction of the mass of carbon contained on the Earth, so running out of materials for the tether would not be a major issue. The issue arises when you start thinking about the length of time needed to make all of this nanotubing. In order for this to be possible, we would have to drastically increase carbon production worldwide. This project of making the tether, in light of these limitations, is not feasible in ten years. It would take much longer to realize tether production.
At a tether length of 50,000 kilometers, we would need a 10 trillion kilogram counterweight.
We estimate that the elevator can safely make 50 trips a year while carrying 60,000 kg loads. If we assume a rate of $20,000 per kg the elevator can bring in 1.2 billion a trip. This totals 60 billion dollars a year. The profit margin will be largely determined by the cost of the elevator material. If we can acquire enough material cheaply, this can be a very profitable investment.
Before considering the environmental damage building this monstrosity will wreak, we first need to have a feasible investment. We calculated the cost of a sufficient amount of carbon nanotubes would be 1.05 trillion dollars. This large sum makes any other investment under 10 billion a negligible amount. After considering funding for a design team, shipping costs, the anchor and the solar power the project costs 1.1 trillion dollars.
An investment of this nature would be risk heavy and the return on investment is positive, but not a large enough to warrant such risk. However, a space elevator is an immense technological step forward. We believe that the United States would back a realistic proposal. If there was a theoretical investor that could loan us the 1.1 trillion dollars at an interest rate of three percent over thirty years, we would owe 54 billion annually. This yields a profit of 6 billion a year. While 6 billion is a very large amount of money, it is not a large enough yield to validate this from the reference frame of a private investor. If in the near future, the price for carbon nanotubes falls to a more reasonable amount, a space elevator return on investment would make it more viable project in the private industry.
A typical rocket launch can emit up to 17,000 tons of carbon dioxide. This equates to 2,000 americans annual emissions from driving. Rocket launches are also very heavy in ozone depleting substances and with increased frequency of launches, could be catastrophic to the ozone. After construction, our elevator would run on an electric motor powered by solar energy. This would be a very beneficial net change in emissions, however, construction is an issue. Just considering emissions from shipping the carbon nanotubing make this project seem unreasonable from an environmental perspective Shipping the elevators materials to the work site would emit over three million tons of carbon dioxide. The production of carbon nano tubing would add 41 million tons of carbon dioxide to the atmosphere. This put the carbon dioxide total at 44 million tons. It would take over 40 years of smooth space elevator operation to break even on emissions4.
The amount of materials required are not readily obtainable in the next ten years and trapping a 10 zg mass is unrealistic. The space elevator is a poor financial investment and would emit more carbon during construction than it would save in rocket launches over thirty years. We can not advocate construction of such a monstrosity at this time.
"Space Elevator." Wikipedia. Wikimedia Foundation, 13 June 2013. Web. 14 June 2013.
Aravind, P. K. "The Physics of the Space Elevator." American Journal of Physics 75.2 (2007): 125-30. Web. 12 June 2013.
Mallick, P. K. Fiber-reinforced Composites: Materials, Manufacturing, and Design. Boca Raton, FL: CRC, 2008. Print.
"Limits on the Space Launch Market Related to Stratospheric Ozone Depletion." Ross, Martin, Toohey, Darin, Peinemann,
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