Sizing Considerations for Electrodynamic-Tether-Driven Atmospheric Harvestor

G. Flanagan

Alna Space Program

October 11, 2011

This notebook presents a process for determining some of the factors that would go into the design of an atmospheric harvester. The focus is in the electrodynamic drive and power array. The study presents very little on the mechanics of the harvester itself. The altitude of the collector is determined by thermal considerations of the cable rather than the gas dynamics of collection. The goal is to give an idea of the scale of the total mass needed to achieve a mass collection goal. We have arbitrarily set a goal of 1000 kg/day of collected gas (total of oxygen plus nitrogen).

The basic idea of the haves tor is to drag a collection device through the upper atmosphere. An electrodynamic cable is used to balance the drag forces. A current flow through the cable interacts with the earth’s magnetic field to provide a propulsion force. The system is powered by a solar array at the upper end of the cable.

The drive cable will be made either entirely from aluminum alloy, or an aluminum steel mix. The material selection is based mostly on the good electrical conductivity of aluminum relative to it’s density. The disadvantage of aluminum is it’s relatively low operational temperature. One goal of the study is to demonstrate that a system can be designed even with modest temperature limits. In order to operate at lower altitudes, steel can be added to the cable (as separate strands, not an alloy). The composite mixture can have good conductivity as well as adequate strength as higher temperature.

For some quick numbers we use material properties for aircraft structures without regard for the material forms. In other words the properties may be for a plate form while the cable would be a drawn strand.

For aluminum, assume 7075 alloy and use the following graph for strength retention. Assume 10,000 hr exposure is close to infinite time.

Ultimate Tensile Strength Retention for 7075 Aluminum. (Fig 3.7.4.1.1(c) from Mil-Hdbk-5H).

Similarly, use a curve for modulus retention and construct a similar function.

Modulus Retention for 7075 Aluminum. (Fig 3.7.4.1.4 Mil-Hdbk-5H)

For steel, assume a constant strength. To cover the possible temperature range, the reference strength was multiplied by 0.9. For Aluminum, start with a reference strength of 70 ksi and multiple by the retention factor as a function of temperature.

The strength of the combination is computed by assuming both components have the same strain. In this approach, the aluminum is not allowed to “fail” in the sense of exceeding the ultimate strength. In reality, at very high temperatures the aluminum could plastically deform and completely unload into the steel. This behavior is a bit complicated to model, so a more conservative strength model is used instead.

The composite electrical resistance and density are computed by simple weighted sums.

The specific power for a large solar cell array (kW/kg) will determine the power unit mass. The value of 676 w/kg, from http://www.emcore.com/assets/photovoltaics/Paper_Navid_9-22-00.pdf. I’ve seen higher numbers (1003 kw/kg at http://www.orbital-power.com/home/thin-film-solar-cells/) for space solar power satellite studies. However, current space systems are much heavier. The specific power for the ISS solar arrays is only about 30 kg/kw (see http://www.shuttlepresskit.com/STS-97/payload81.htm)

The propulsion scheme uses current flowing one way through the cable. The current loop is completed by capturing electrons at one end, and ejecting electrons at the other end. There is some efficiency loss associated with this part of the current loop. There are some research papers on this topic, but for now use a made-up efficiency factor of 0.85. This is applied to the power needed for propulsion.

The coefficient of drag applies to the cable. Assume a value of 1. It is possible for the coefficient to be greater than one if the molecules bounce off the cable.

The solar power unit at the top of the system will be in the earth’s shadow for part of every orbit. We will assume that the drag is constant, but the momentum loss must be restored only during the daylight hours. Thus, the daytime power requirement is greater than would be needed if the drive operated continuously. We assume the system is massive enough relative to the drag loss that a cycle of turning off the propulsion during nighttime is acceptable.

An idea to be examined later: In order to keep the collector at a constant altitude during the nighttime momentum loss, consider a reel system that stores momentum by changing the free length of the cable.

Consider a cable 100 km long, and a collection altitude of 100 km.

Estimate the system center-of-mass (assuming the mass at the top and bottom nodes are equal), and the result orbital angular velocity.

From these, we can compute the velocity at the capture device.

If we assume that the device collects 1000 kg/day of gas, then the collection rate is

The collection rate times the velocity give the drag force for the collector.

Force times velocity gives the power needed to propel the capture device, not counting various system losses.

Assume that the altitude of the capture unit is determined by the maximum allowable temperature of the cable. The heat loads into the cable consists of aerodynamic heating, solar heating, and dissipated heat from the electrical current. The heat is carried away by radiation. We assume that because the cable is nearly vertical, there is no heat load from the earth.

The aerodynamic model uses H=1/2 ρ d, where ρ is the density of the atmosphere, and d is the cable diameter. The model assumes that all of the gas molecules deposit their kinetic energy onto the cable, which is probably very conservative. The solar heating model simply assumes that the cable is exposed to the solar energy load of 1387 , so the heat load into the cable is 1387 d. This assumes that the radiation absorption coefficient is 1.0, but this is compatible with the emissivity assumption of 1.0 used in the radiation model, unless there is a coating that selects for the radiation wavelength.

The radiation model simply uses the Stefan–Boltzmann law, H= 2 π d σ . The factor of 2π d accounts for the surface are of the cable. σ is the Stefan–Boltzmann constant, and T is the cable equilibrium temperature. This assumes that the cable is radiating to absolute zero in space. Not quite true, but should be an approximation.

Cable equilibrium temperature versus altitude.

The aluminum conductor has almost zero strength at 300 C. We can mitigate this loss of strength by adding some steel to the cable. If we target 200C for aerodynamic and solar heating alone, we are left with some room for electrical heating. Guess at an altitude of 115 km.

This will be the capture altitude.

Given an altitude governed by the maximum operating temperature, we can now compute the capture area. First, find the density of the gas the capture altitude.

The required area is then

The equivalent circular diameter is

Unfortunately, I have not studied the mechanism needed to capture and liquefy 1000 kg/day of hypersonic gas, and therefore I do not have any way to come up with a mass estimate for the collector. For the purpose of moving forward, I’ve assigned a mass of 5000 kg to the collector because that “feels” like the minimum that would be needed to construct such a device.

For the following, the cable diameter is adjusted by hand until the strength factor-of-safety (F.O.S) is at least 2. The increments for cable diameter trials was .05 cm. There is a table entry labeled “Top Mass Check”. This is a comparision of the calculated compatible top-node mass with the powerplant mass. The node mass must be greater than the powerplant mass. If not, the solution is increase the bottom node mass above the minumum assumed value for the collector mass. This forces the compatible top node mass to increase. The bottom-node mass was changed in 1000 kg increments, so again the results represent a very rough optimization.

Cable Length | 150 Kilo Meter |

Cable Dia | 1.05 Centi Meter |

Cable Drag | 132.725 Newton |

Collector Drag | 89.1496 Newton |

Power to Overcome Drag | 1.70899*10^^6 Watt |

Drive Voltage Drop | 37039.1 Volt |

Resistance Voltage Drop | 20110. Volt |

Current | 46.1402 Amp |

Total Power | 4.97866*10^^6 Watt |

Cable Mass | 61825.4 Gram Kilo |

Powerplant Mass | 7364.89 Gram Kilo |

Bottom Node Mass | 8000 Gram Kilo |

Top Node Mass | 7883.67 Gram Kilo |

Total System Mass | 77709. Gram Kilo |

Strength F.O.S | 2.0113 |

Top Mass Check | True |

Cable Max. Temp | 238.453 |

Capture Area |

Cable path with segment of earth surface

Cable angle relative to the local tangential direction.

Cable Length | 100 Kilo Meter |

Cable Dia | 0.7 Centi Meter |

Cable Drag | 88.4696 Newton |

Collector Drag | 89.661 Newton |

Power to Overcome Drag | 1.37993*10^^6 Watt |

Drive Voltage Drop | 25116.5 Volt |

Resistance Voltage Drop | 35918.7 Volt |

Current | 54.941 Amp |

Total Power | 6.18534*10^^6 Watt |

Cable Mass | 18318.6 Gram Kilo |

Powerplant Mass | 9149.91 Gram Kilo |

Bottom Node Mass | 9500 Gram Kilo |

Top Node Mass | 9256.45 Gram Kilo |

Total System Mass | 37075.1 Gram Kilo |

Strength F.O.S | 2.10494 |

Top Mass Check | True |

Cable Max. Temp | 260.375 |

Capture Area |

Cable path with segment of earth surface

Cable angle relative to the local tangential direction.

Cable Length | 75 Kilo Meter |

Cable Dia | 0.65 Centi Meter |

Cable Drag | 81.3209 Newton |

Collector Drag | 89.9186 Newton |

Power to Overcome Drag | 1.33035*10^^6 Watt |

Drive Voltage Drop | 18999.2 Volt |

Resistance Voltage Drop | 39818.6 Volt |

Current | 70.0215 Amp |

Total Power | 7.54796*10^^6 Watt |

Cable Mass | 11846.4 Gram Kilo |

Powerplant Mass | 11165.6 Gram Kilo |

Bottom Node Mass | 12000 Gram Kilo |

Top Node Mass | 11566.1 Gram Kilo |

Total System Mass | 35412.5 Gram Kilo |

Strength F.O.S | 2.12562 |

Top Mass Check | True |

Cable Max. Temp | 285.364 |

Capture Area |

Cable angle relative to the local tangential direction.

For this cable length, it was difficult to find a suitable design for the 60% aluminum combination. A 90% aluminum cable cable yielded a feasible design.

Cable Length | 50 Kilo Meter |

Cable Dia | 0.7 Centi Meter |

Cable Drag | 84.7648 Newton |

Collector Drag | 90.1774 Newton |

Power to Overcome Drag | 1.36303*10^^6 Watt |

Drive Voltage Drop | 12775.4 Volt |

Resistance Voltage Drop | 11422. Volt |

Current | 106.692 Amp |

Total Power | 4.93707*10^^6 Watt |

Cable Mass | 6186.38 Gram Kilo |

Powerplant Mass | 7303.36 Gram Kilo |

Bottom Node Mass | 8000 Gram Kilo |

Top Node Mass | 7349.09 Gram Kilo |

Total System Mass | 21535.5 Gram Kilo |

Strength F.O.S | 3.36712 |

Top Mass Check | True |

Cable Max. Temp | 266.396 |

Capture Area |

The aluminum conductor has almost zero strength at 300 C. We can mitigate this loss of strength by adding some steel to the cable. If we target 120C for aerodynamic and solar heating alone, we are left with some room for electrical heating. Guess at an altitude of 115 km.

This will be the capture altitude.

Given an altitude governed by the maximum operating temperature, we can now compute the capture area. First, find the density of the gas the capture altitude.

The required area is then

The equivalent circular diameter is

Keep the collector mass the same.

Perform a similar series of design iterations for the higher collection altitude. The same hand adjustments are made as for the previous case of a mixed steel/aluminum cable.

Cable Length | 150 Kilo Meter |

Cable Dia | 0.5 Centi Meter |

Cable Drag | 26.8612 Newton |

Collector Drag | 89.0834 Newton |

Power to Overcome Drag | 892403. Watt |

Drive Voltage Drop | 36843. Volt |

Resistance Voltage Drop | 4811.05 Volt |

Current | 24.2217 Amp |

Total Power | 1.97084*10^^6 Watt |

Cable Mass | 7952.16 Gram Kilo |

Powerplant Mass | 2915.44 Gram Kilo |

Bottom Node Mass | 5000 Gram Kilo |

Top Node Mass | 4799.22 Gram Kilo |

Total System Mass | 17751.4 Gram Kilo |

Strength F.O.S | 2.04352 |

Top Mass Check | True |

Cable Max. Temp | 126.565 |

Capture Area |

Cable angle relative to the local tangential direction.

Cable Length | 100 Kilo Meter |

Cable Dia | 0.45 Centi Meter |

Cable Drag | 23.942 Newton |

Collector Drag | 89.5937 Newton |

Power to Overcome Drag | 878868. Watt |

Drive Voltage Drop | 24982.9 Volt |

Resistance Voltage Drop | 5750.93 Volt |

Current | 35.1787 Amp |

Total Power | 2.11932*10^^6 Watt |

Cable Mass | 4294.16 Gram Kilo |

Powerplant Mass | 3135.09 Gram Kilo |

Bottom Node Mass | 5000 Gram Kilo |

Top Node Mass | 4709.61 Gram Kilo |

Total System Mass | 14003.8 Gram Kilo |

Strength F.O.S | 2.55919 |

Top Mass Check | True |

Cable Max. Temp | 132.879 |

Capture Area |

Cable angle relative to the local tangential direction.

Cable Length | 75 Kilo Meter |

Cable Dia | 0.4 Centi Meter |

Cable Drag | 20.814 Newton |

Collector Drag | 89.8506 Newton |

Power to Overcome Drag | 859100. Watt |

Drive Voltage Drop | 18897.9 Volt |

Resistance Voltage Drop | 7054.31 Volt |

Current | 45.46 Amp |

Total Power | 2.30069*10^^6 Watt |

Cable Mass | 2544.69 Gram Kilo |

Powerplant Mass | 3403.39 Gram Kilo |

Bottom Node Mass | 5000 Gram Kilo |

Top Node Mass | 4506.28 Gram Kilo |

Total System Mass | 12051. Gram Kilo |

Strength F.O.S | 2.47131 |

Top Mass Check | True |

Cable Max. Temp | 145.284 |

Capture Area |

Cable angle relative to the local tangential direction.

Cable Length | 50 Kilo Meter |

Cable Dia | 0.4 Centi Meter |

Cable Drag | 19.6222 Newton |

Collector Drag | 90.1088 Newton |

Power to Overcome Drag | 854300. Watt |

Drive Voltage Drop | 12707.1 Volt |

Resistance Voltage Drop | 6955.02 Volt |

Current | 67.2303 Amp |

Total Power | 2.56681*10^^6 Watt |

Cable Mass | 1696.46 Gram Kilo |

Powerplant Mass | 3797.05 Gram Kilo |

Bottom Node Mass | 5000 Gram Kilo |

Top Node Mass | 3944.36 Gram Kilo |

Total System Mass | 10640.8 Gram Kilo |

Strength F.O.S | 2.97874 |

Top Mass Check | True |

Cable Max. Temp | 167.722 |

Capture Area |

The table below summarizes the total mass results. There is a definite mass advantage to using a higher capture altitude. For the 150 km alum/steel mix it was hard to find a compatible design; the lower node mass had to be increased dramatically to support the mass of the power array at the upper node, resulting in a high system.

For the low-altitude system, the lower node mass had to be increased above the somewhat arbitrary minimum in order to balance the power array mass. If the collector mass is greater than our assumption, then there would be less penalty for using a lower altitude.

A reasonable system can be designed for a wide range of collection altitudes. This means that the selection will hinge more on the collection mechanism than on propulsion considerations. Lower altitudes may allow for continuum flow and use of ram compression. High altitude involves larger mean-free paths and atomic level collection. I have no insights on which may be preferred. This design study has gone into as much detail as is useful without further definition of the collector.

Cable Length | Alum/Steel | Alum |

150 | 77709. | 17751 |

100 | 37075 | 14004 |

75 | 35412 | 12051 |

50 | 21535 | 10641 |

The follow sketch summarizes one design case (75 km cable length, 125 km collection altitude) for quicker viewing.

Power to cover heat of vaporization of nitrogen and oxygen. N2=200 kJ/kg, O2 = 213 kJ/kg. Call the mix 206 kJ/kg so we don’t have to worry about the mixture ratio. For 1000 kg over 24 hrs, the power required is

which is small compare to the megawatts needed to drive the system.

Ratio of oxygen to total for 115 km capture

Ratio of oxygen to total for 125 km capture