Which Is Cheaper Per Unit of Energy, Thermonuclear Charges Or Chemical Fuels

For the Wang Bullet (Nuclear Space Launch Cannon) – How do thermonuclear charges compare in $ per megajoule (megawatt-second) to chemical propellants and fuels?

A guest post by Joseph Friedlander

Reader Ack_and_Eft wanted to know (in an article about Wang Bullet thermonuclear launch technology)

How do thermonuclear charges compare in $ per megajoule to chemical propellants?

Given the quantities involved, $ per gigajoule would be a better measure.

1 joule is a watt-second
1 megajoule is a megawatt-sec

1 kg coal equivalent = 7000 kilocalories =~ 29.3 MJ = ~ 8.141 kWh)

29.3 gJ/ton coal at $80 ton gives $2.73 a gigajoule coal

using our handy megatons to gigajoules calculator

4184000 gigajoules per megaton

given all that then lets see various sizes of device–

ton of TNT      t 4.184×10e9 J
kiloton of TNT kt 4.184×10e12 J
megaton of TNT Mt 4.184×10e15 J 

in Wikipedia in “TNT Equivalent”
and we need to know the cost of a given propellant device for the divisor–

I typically take the cost of a given thermonuclear device at $10 million. This is arbitrary. The biggest influences on cost appear to be numbers produced vs. overhead of the production complex. (Producing a single prototype will result in huge cost, producing many identical ones with economies of scales in much less cost.)

Cost is basically independent of yield up to the point of say 10s of megatons. After that it scales nearly linearly. (The cost of fusion fuel dominates after a certain point, but this is (or can be) much cheaper than the special nuclear materials that smaller devices demand such as U 235, Pu 239 and tritium. Deuterium can be as low as $500 a kilogram, for example. (1.2 tons needed for an efficient 100 MT burn. –$600000 total for the fuel.) So you can see that on the right side of the curve (in large quantities) device cost could easily be as low as a million each (a 1 MT device would use at these prices $6000 worth of D.)

…Deuterium produces energy through four reactions:
1. D + T -> He-4 + n + 17.588 MeV
2. D + D -> He-3 + n + 3.268 MeV
3. D + D -> T + p + 4.03 MeV
4. He-3 + D -> He-4 + p + 18.34 MeV

Since the reaction cross section of 1 is some 100 times higher than the combined value of 2 and 3 the tritium is burned as fast as it is produced, contributing most of the energy early in the reaction. Reaction 4, on the other hand, requires temperatures exceeding 200 million K before its cross section becomes large enough to contribute significantly. Whether sufficient temperatures are reached and quantities of He-3 are produced to make 4 a major contributor depends on the combustion efficiency (percentage of fuel burned).

If only reactions 1-3 contribute significantly, corresponding to the combustion of 25% of the deuterium fuel or less, then the energy output is 57 kT/kg. If reaction 4 contributes to the maximum extent, the output is 82.4 kT/kg. The maximum temperature generated by an efficient burn reaches 350 million K.

Militarized devices tend to use lithium 6 deuteride but if we ever have massive fusion use we will want to use the cheapest (envisionable) fusion fuel which is deuterium. This may be combined with some carrier chemical for easier logistics—or not.

So, 1 megaton device– $10 million–4184000 gigajoules per megaton
$2.39 per giajoule– about the same price as coal

29.3 gJ/ton coal at $80 ton gives $2.73 a gigajoule coal

10 megaton device– $10 million–41840000 gigajoules per megaton
0.239 per giajoule 1/10th the cost of coal

So, 100 megaton device– $10 million–418400000 gigajoules per megaton = ~1/100th the cost of coal

The takeaway is anything much below a megaton device is very

expensive relative to the larger sizes.

If the devices were free, the deuterium being the only cost, the price would be 1/1891 that of coal at $80 ton (given Deuterium at $500/kilogram)

All this is only if you can use the energy with equal efficiency to coal, otherwise discount by the wastage factor.

But the original question was,

How do thermonuclear charges compare in $ per MJ to chemical propellants?

We used coal at $80 a ton, you can multiply by a factor of 1.5 to 10 or 20 depending on fuel choice.

I would guess a fuel/oxidizer mix of around $350/ton for the whole fuel loading of the Space X Falcon series. This probably means a fuel loading of around $100,000 for a Falcon 9.
Wikipedia on the Spacex Merlin rocket engine.

Propellant LOX / RP-1(rocket grade kerosene)
The ratio would typically be 2.6 to 1 or so, so if LOX is (for illustration) $100 and RP-1 $1000 a ton, then the fuel oxidizer mix would be around $350/ton. This is between 4 and 5 times more expensive than coal per ton.

http://en.wikipedia.org/wiki/Energy_density

Storage material Energy per kilogram
Uranium 238
20 terajoules

Jet fuel, Kerosene
43 megajoules

If you count the fact that you need oxygen it drags the energy per kilogram of the fuel-oxidizer mix down to probably around one-millionth of nuclear fuel.

Given the above calculations, a 150 KT device as postulated in Brian’s article would be $15.93 per gigajoule (at $10 million) vs $2.73 a gigajoule coal(remember we are not calculating the cost of the oxidizer which is free atmospheric oxygen) The RP-1 price I have not seen quoted but on conclusion it should be a little less per gigajoule than a 150 kiloton thermonuclear device.

Remember though the cost of fuel in an ordinary (non solid fuel) space launch is only perhaps 1% of the launch cost. The vehicle is 99%. So a cheap Wang Bullet saves enormously on that 99%, even if say a 150kt device was say 2 times more expensive power than chemical propellants, the ship is enormously more robust and cheaper. So instead of thousands of dollars a kilo upward, tens of dollars a kilo should be possible ultimately.

FURTHER READING
The above article reviews the underground nuclear propulsion into space design that I had (along with detailed work with Joseph Friedlander) and which I have described a few times before

Videos of Underground Nuclear Test, Project Orion and the Quicklaunch Gas Gun Proposal

Amchitka 5 Megaton Test

Project Orion

Quicklaunch

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