Tuesday, October 27, 2015

Nuclear powered VASIMR and plasma propulsion doable now, Page 3: High efficiency conversion enables zero radiator mass.

Copyright 2015 Robert Clark

Low Radiator Mass at High Efficiency.
 Discussions on various forums have argued the point that actually low conversion efficiency is preferred to save on radiator mass. The idea behind this is that by the Stefan-Boltzmann Law, energy is radiated away according to the fourth power of the temperature. So high thermal energy conversion efficiency resulting in low output temperatures would require larger radiators than with low conversion efficiency with high output temperature. This is explained on this page:

Heat Radiators.
"As an example of the severity of this problem, let us examine the case of a
simple nuclear power plant whose energy conversion efficiency from thermal
to electric is approximately 10 percent. The plant is to generate 100 kW of
useful electricity. The reactor operates at approximately 800 K, and a
radiator with emissivity equal to 0.85 would weigh about 10 kg/m2. The
thermal power to be dissipated from the reactor would be about 1 MW. From
the Stefan Boltzmann Law, the area of the radiator would be about 50 m2 and
the mass approximately 500 kg. This seems quite reasonable.
However, we must assume that the electricity generated by the power plant,
which goes into life support systems and small-scale manufacturing, would
eventually have to be dissipated also, but at a much lower temperature
(around 300 K). Assuming an even better, aluminum radiator of about 5 kg/m2,
with again an emissivity of 0.85, in this case we find that the area of the
low temperature heat rejection component is 256 m2, with a mass approaching
1300 kg."

 This web site by the way provides a nice collection of the work that has been done on advanced space propulsion systems.

 An additional problem though in regards to low efficiency conversion is that you need higher mass for the reactor and larger amounts of radioactive material that needs to be launched to orbit. For example in the blog post:

"Nuclear powered VASIMR and plasma propulsion doable now, Page 2," ,

I suggested running the reactor at low power to extend the lifetime. In this mode the reactor weighing 2,200 kg would put out 8 megawatts using 200 kg of uranium fuel. If instead of the high conversion efficiency I was aiming for, we selected 10% conversion efficiency, we would need a 10 times larger reactor and 10 times more radioactive fuel. This would put the mass of the reactor at 22,000 kg and the uranium fuel at 2 tons.

 There might be some methods to reduce the heat that needed to be radiated away in the high efficiency scenario. For example Stirling engines can operate even at low temperatures. We could use these to make use of the heat at low temperatures coming from the turbines of  our generator.

 There is also research on lightweighting radiators such as by using carbon-carbon composites:

High Conductivity Carbon-Carbon Heat Pipes for Light Weight Space Power System Radiators.

 However an interesting possibility exists that we might not need any radiators at all for the high efficiency case

  In the high efficiency case at low output temperature, you can bring the output of the working-fluid down to cryogenic temperatures, as for example for a de Laval nozzle exhausting out to vacuum. If the electric generator equipment is at room temperature, or chilled to decrease resistance and therefore increase power output and specific electric power, then we can have the working-fluid output temperature be lower then the electric generator equipment. So it can used to remove the heat given to the electric generator due to small proportion of thermal not converted to electric power.

 Note this would not obtain for the low efficiency, high output temperature case. If the output is at, say, 800 K to 1,000 K, then by the Second Law of Thermodynamics (Clausius form) we could not use this to cool the generator operating at, say, 300 K or below.

 However, by exhausting down to cryogenic temperatures we could use the cold working-fluid to cool the generator. 

 We need also to cool the turbine. Turbines can be at 90%+ efficiency. The turbine however will be receiving the working fluid at the output temperature. We can though allow the working-fluid to expand further after exiting the turbine to drop to a lower temperature. This then could be used also to cool the turbine.

 An additional factor to provide cooling are the nozzle extensions used for high expansion ratio upper stage engines. For example the RL10-B2 nozzle extensions increase the area ratio to 285-to-1.

 The nozzle extension doubles the weight of the RL10 engine from 150 kg to 300 kg, so weighs 150 kg. The upper part of the nozzle is regeneratively cooled but the carbon-carbon extension is only radiatively cooled. The RL10-B2 engine puts out ca. 250 MW of power measured by its 110 kN thrust and 465.5 s Isp. 

 Because of the low mass of the spacecraft due to the short travel time the power level for the plasma propulsion can be less than 10 MW. Then, with regeneratively cooled upper nozzle section, the nozzle extension to bring the exhaust output down to cryogenic temperatures for this power level might only be 1/25th as large, so only 6 kg(!)

Correction to fuel replacement mass.
 In the comments section to the "Nuclear powered VASIMR and plasma propulsion doable now, Page 2" post, Rok Adamlje noted that my estimate for the fuel canister mass that needed to be replenished was too small. I was taking it to be just the uranium fuel mass when the actual total canister mass may be ten times higher.

 In the section there titled, "Long Life by Multiple Nuclear Fuel Canisters", I calculated the average specific power when using 468 fuel canisters at 200 kg each to last the 39 days of the flight. If we take the fuel canisters instead to be 2,000 kg then we can still get high average specific power by jettisoning the used canisters: 

 The rest of the engine mass, which will be fixed, is a small proportion of the total mass of the fuel canisters so we'll calculate the average specific power based on just the total fuel canister mass. The average specific power will be:

  545 MW       545 MW                  545 MW         545 MW
 ------------ +  --------------- + ... +  -------------  +  -------------
  468*2,000      467*2,000                2*2,000          1*2,000

545MW     1               1                    1            1
 --------- (-------- +    ------- + ... +  --------  +  -------)
  2,000       468          467                  2            1
=  ----------------------------------------------------------------------

 The summation in the numerator is the sum of the reciprocals of the natural numbers and is well approximated by the natural log function ln(n). So this will be:

(545,000,000/2,000) *(ln(468)/468) = 3,580 W/kg, still above the 1,000 W/kg level needed for the plasma propulsion.

 Note though the length of time the engine will need to be run is actually less than 39 days because of the small mission size due to the fast travel time. So we'll need fewer replacement fuel canisters and the actual average specific power will be greater.

 Also it could be much of the canister(s) can be reused, primarily replacing just the uranium fuel. This will also increase the average specific power.

        Bob Clark


  1. This comment has been removed by a blog administrator.

  2. Actually I don't agree to dump to vacuum is bad. Or more precisely, I do want to expand out to vacuum or nearly so: you don't want to expand exactly out to absolute zero, or want the exhaust to freeze to a solid. I'm undecided on whether you want to expand it out to liquify or not. Liquids are easier to compress to high pressure which you need for completing the cycle of injecting the working fluid back into the reactor under high pressure and heat conditions. Also liquids can get somewhat higher efficiency from the turbine. But then you might need a more complex thermodynamic cycle involving two phases, not just the Brayton cycle.

    Rocket nozzles are very good at converting heat to kinetic energy, getting very high efficiency. That is why I want to use them. The idea of instead going through all that trouble to create this great amount of energy and then just use 5% to 20% of it, literally throwing perhaps 90% away, offends my sense of scientific parsimony.

    Bob Clark

  3. This comment has been removed by a blog administrator.

    1. Rocket nozzle's have efficiencies in the upper 90's in converting heat to kinetic energy. For radiatively-cooled nozzles used on upper stage engines, as the name implies they use radiatively cooling to dispense the small percentage of heat not converted to kinetic.

      An image of the radiatively-cooled nozzle extension of the RL10-B2 glowing red hot during testing:

      An image of the Merlin Vacuum's nozzle glowing red hot during an actual flight:

      Note that because these nozzle's do get red hot means they specifically would be good at rejecting heat.

      MHD may be the best way to go since it has few moving parts. However, there has been much research on using the Brayton cycle for space nuclear power.

      Bob Clark

    2. This comment has been removed by a blog administrator.

    3. Comments deleted because they were abusive.

      Bob Clark

  4. Radiators will still be needed because inefficiencies cause the VASIMR to produce 30%-40% waste heat. Although without the reactor they may be smaller.

    Is there anyway to use the waste heat to boil off solid/liquid fuel?

    1. You could also use the working-fluid to cool the plasma drive. I thought of using the propulsive fuel, xenon, krypton, etc., to cool the drive, but remember the fuel flow rate is very small due to the small thrust of the electric propulsion methods so it wouldn't provide enough cooling.

      Bob Clark