Supercriticality

One of the most complex parts of a spacecraft is the power supply, which usually takes the form of a nuclear reactor.

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Cherenkov Radiation from a nuclear reactor core.

Nuclear fission is about 600,000 times more energy dense than the most energetic chemical power reactions, which very easily makes it the best power supply for space. In space, mass is a premium, so energy density is critical. Most capital ships use fission reactors to get their power.

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Schematic of a Radioistope Thermoelectric Generator.

Similarly, Radioactive Decay runs about 15,000 times more energy dense than chemical reactions, which is why Radioisotope Thermoelectric Generators (RTGs) are the next best thing. Most drones and missiles with significant power needs use RTGs. Anything with lesser power needs simply use small batteries.

Chemical energy requires tremendous amounts of mass produce reasonable amounts of power, after which this mass is ejected. This makes it not very viable for space except as rocket propellant.

earth_horizon_and_international_space_station_solar_panel_array_expedition_17_crew_august_2008
Solar panels in space. Beyond Earth, they rapidly lose effectiveness.

Solar power becomes more or less useless further away from the sun (at Jupiter, the irradiance is 30 times lower than that on Earth). They are also much harder to armor than radiators, making them a poor investment for space warships.

Beamed power suffers from the beam waist widening very quickly over distance, which limits its effectiveness to very low orbits around celestial bodies. This occurs even with a constellation of mirrors to extend the range. Beamed power similarly require receivers that can be damaged easily, and if they are to be used in close combat, they can’t be covered up, making beamed power fairly infeasible for warfare.

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A Z Machine, a device for researching Inertial Confinement Fusion (ICF).

Fusion Power is a power source that could make all other sources obsolete if it can be developed. In its current state, it is too far future of a technology. An additional consideration about Fusion Power is that it, along with Fission Power, are not limited by energy density, unlike the other aforementioned power sources. While the other power sources are limited by not producing enough power, Fission and Fusion Power both generate more power than modern systems could ever dream of using. Fission Power is not limited by how much nuclear fuel you have, it is limited primarily by how large your radiators are. This means any limitations on Fission Power currently will still remain even if Fusion Power replaces it.

All of this means that the main power supply in Children of a Dead Earth is nuclear fission.

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Cooling towers of a nuclear reactor. The actual reactor core is underground, and tiny compared to the machinery used to extract power from the core.

In Children of a Dead Earth, nuclear reactor cores are governed by the Six Factor Formula. This rather large and complex equation which determines the effective neutron multiplication factor. Given the masses of all the materials inside the core from the fuel to the moderator to the neutron poison to the coolant, the operational characteristics of the reactor can be determined. This means the rate at which a core achieves criticality or shuts down can be predicted, and it usually takes microseconds.

Given a working reactor core, the neutron flux of a reactor can be arbitrarily controlled. The neutron flux directly determines the amount of heat the reactor will produce, which means the reactor can arbitrarily control the heat produced, and by extension, the power produced.

These fission reactors are not limited by the Uranium-235 fuel (or Pu-238 or Am-242m, etc.) inside the core. In practice, the amount of nuclear fuel is minuscule compared to the rest of the reactor elements. This is why 3% enrichment of U-235 is actually quite reasonable for reactor fuel. Contrast this with nuclear weapons, where around 97% enrichment of U-235 is preferred.

The main power limit inside the core is how hot you can make it. And generally how much heat your reactor can withstand is dependent on how large you can make the reactor to reduce energy per unit area. Additionally, the design of your fuel makes somewhat of a difference, as fuels such as TRISO allow for a Pebble Bed Reactor design, which yields somewhat higher heat tolerances.

triso
Cutaway of a TRISO pellet, showing various ceramic layers over a U-235 pebble.

The reason temperature is the limiting factor of a nuclear reactor, rather than the actual fuel mass, is that the energy density is so high that a nuclear reactor will never come close to unlocking all of the power of even a small amount of fuel mass. Nuclear reactor design mostly boils down to how much energy can be extracted from a tiny amount of nuclear fuel without melting to slag.

Thus, how much nuclear fuel is needed has nothing to do with how much power you need, and everything to do with whether or not your reactor can achieve a critical mass.

candu_fuel_bundles
CANDU fuel bundles, another nuclear fuel design for heavy water reactors.

For this reason, using more or less energy dense nuclear fuels is basically pointless, as they all produce “too much”. The main difference between fuels is how they affect the neutron multiplication factor. Certain fuels, like Am-242m, can achieve a critical mass more easily, and thus, less nuclear fuel is needed. This means the reactor design might be made smaller for the same amount of power.

In practice, however, the mass of the radiators ultimately end up being the limiting factor on nuclear reactors.

Of course, simply creating an atomic pile and letting it spew neutrons and radioactive waste is not enough to produce power. Power needs to be extracted from the heat of these fast moving nuclear byproducts. A number of methods have been devised to do so.

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The exposed turbine of a turboelectric nuclear reactor.

Turboelectric Fission Reactors involves letting the neutrons heat up the coolant and using the hot vaporized coolant to turn a turbine. This technique is the most common, being used in just about every nuclear reactor in the world. It requires a large turbine to effectively extract the heat, as well as plenty of coolant and turbomachinery to run it properly. Due to the size, it can take a while to warm up. Due to the mass and complexity, it generally is not used for spacecrafts.

Turboelectric Fission Reactors are heat engines, meaning they can never exceed the efficiency of a Carnot Cycle:

\eta = 1 - \frac{T_c}{T_h}

Where \eta is the efficiency, T_c is the cold temperature of the coolant prior to passing through the reactor, and T_h is the hot temperature of the coolant after passing through the reactor.

As noted in Why does it look like that? (Part 3), the cold temperature needs to stay high because this is the temperature the radiators will cool the reactor at. Cold radiators function abysmally and require huge amounts of mass. Similarly, the hot temperature needs to stay low to prevent the systems from cracking from the thermal stress.

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A thermoelectric reactor which produces 13.5 MW. Does not include radiators.

The sample reactor has a radiator temperature of 1200 K and 1688 K as the hot temperature. Using the above equation, this means the efficiency of that reactor can never exceed 29%, if it were theoretically perfect. In practice, the actual reactor runs at 22% efficiency.

On Earth, turboelectric reactors often quoted at running at 60% efficiency. This is possible because the cold temperature can be brought much lower if you are using conduction or convection to cool off the reactor. In space, using only radiators forces the cold temperature to be rather hot (> 1000 K usually) in order to keep the radiators from getting too large.

snap_10a_space_nuclear_power_plant
The SNAP-10A reactor, a thermoelectric reactor developed by the USA in the 60’s.

Thermoelectric Fission Reactors are heat engines as well. However, instead of passing the hot coolant through a turbine, it instead passes against as a thermocouple heat exchanger. It involves much less turbomachinery, and thus can be produced in much smaller sizes with much lower masses. On the flip side, it is often much further from the theoretical Carnot Cycle efficiency than turboelectric reactors. In space, it is the primary go-to reactor design due to its low mass, simple design, and cheap cost.

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The TOPAZ nuclear reactor, a thermionic reactor developed by the Soviet Union and launched into space in the 80’s.

Thermionic Fission Reactors uses the concept that the flow of charge carriers (such as electrons) across a potential barrier can produce power. This flow of electrons is triggered thermally, and thus is set up very similarly to the way Thermoelectric Fission Reactors work. It is also a heat engine, and sees similar use in space as the thermoelectric fission reactor.

Fission Fragment Reactors do not use coolant at all, and instead extract power by decelerating the neutrons and radioactive waste products using a magnetohydrodynamic generator. This bypasses the Carnot Cycle entirely, allowing efficiencies estimated up to 90%, far greater than any of the previously mentioned heat engines. However, it is also the least developed of all the technologies, and to date, no working Fission Fragment Reactor has been produced.

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Another picture of the Cherenkov Radiation of a nuclear reactor. Just because.

Since Children of a Dead Earth restricts itself to functioning technologies, this means nuclear reactors are restricted to Heat Engine designs. And due to the Carnot Cycle, the efficiency is forever limited by the cold temperature, or the radiator temperature.

Due to this restriction, the most massive part of a nuclear reactor is the radiators that accompany them. The power extraction machinery tends to be the next most massive piece, with the reactor core itself generally being negligible in terms of mass. On the flip side, the tiny amount of reactor fuel tends to be one of the most expensive parts of the entire system.

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28 thoughts on “Supercriticality

    1. Nuclear Thermal Rockets (NTRs like NERVA) have their own reactor in the rocket thrust chamber. Doing so allows almost all of the waste heat to be vented via the exhaust, which means no massive radiators are required. This also allows NTRs to optimize their reactors for greater temperatures since they have no complex heat extraction machinery aside from a nozzle.

      Every other propulsion system uses the main power reactor, aside from Combustion Rockets, which supply their own reaction.

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        1. I actually experimented with implementing multi-modal NTRs and found they perform very badly, generally taking the worst of both worlds. Having a separate reactor and propulsion system(s) allows the different systems to vary significantly, essentially specializing to yield optimal results. Trying to press multiple designs into one causes all sorts of problems, and yields very bad results: lower exhaust velocity than ideal, lower thrust than what could be possible, very poor power efficiency.

          The issue is that certain reactor designs do better at producing power, and other reactor designs do better at producing thrust, and other designs do better still at producing delta-v. Trying to merge all design goals yields a reactor that does all three things poorly. Jack of all trades, master of none essentially.

          The only benefit is you save some on mass and cost, but the reactors are usually not the most massive or expensive part of a spacecraft. This is due to the fact that most reactors in game share more similarities with the MITEE NTR than the NERVA NTR, and are very small and compact as is.

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          1. Ah, Ok then. Are they still useful for anything, like outer system probes?

            Also, are you going to go into different types of nuclear reactor cores later on?

            Also also, I’d like to continue a line of questioning from Stealth in Space, as you don’t seem to have answered my question.
            I had two use cases presented for large banks of batteries:
            One, for Q ships that need to punch above their weight despite having civilian reactors and radiators. The need to disguise the vessel as civilian under close observation would seem to preclude having military reactors, as you would need correspondingly sized radiators to run them.
            A large bank of batteries could supplement a civilian reactor during battle, allowing a Q ship to (briefly) power weapons that would otherwise require a military reactor. For instance, a Q ship targeting a guarded fuel depot could mount a military grade coilgun, and destroy the target from range despite laser/kinetic point defense.

            Two, to allow smaller and cheaper reactors for military ships.
            Without temporary storage of power, each ship would have to have a reactor core(and attendant radiators) that could power all of the ship’s systems, all at once, continuously, even if these power peaks only happen occasionally. Furthermore, the reactor would have to be throttled up and down to match demand, with corresponding maintenance costs due to thermal delta.
            With a bank of batteries, this wouldn’t have to be the case-the reactor can be run at one power level during battle, charging the batteries between engagements, and getting supplemented by said batteries during engagements. This would also allow us to use smaller reactors and radiators-we don’t need a reactor sized to handle peak levels continuously.

            So I guess my question is: are large banks of batteries buildable ingame, and are they practical for the above cases?

            Also also also, unrelated question: Are non-nuclear EMP weapons a thing ingame? I’ve been thinking about this for long range anti missile/drone weapons.

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            1. Battery banks (no matter which technology) are very very very heavy and huge for the power they store.
              Flywheels: <= 0.5 MJ/kg. Superconducting magnetic energy storage: <= 0.04MJ/kg. Rechargeable Lithium-Ion: <= 0.875 MJ/kg. The experimental Lithium-Sulphur rechargeable battery (which is not yet long life capable): <= 1.44 MJ/kg.
              Hydrogen has 142 MJ/kg (does not include mass of oxygen and fuel cells and tankage). Uranium: ~76,000,000 MJ/kg. See https://xkcd.com/1162/ !

              In case of a warship: The weight and size and price of the reactor are likely unimportant. There are a lot more expensive, heavy, large things on a warship, remass will take space and lots of weight, and weapons and drones (usually with reactor!) tend to be fairly expensive.

              Weapons that need short term peak power (lasers, coil/railguns, …) will have a local short term energy storage anyway, smoothing out the power drawn from the reactor.

              Undersizing your reactor and relying on battery banks means you are carrying a *lot* of mass and and space, very likely more than a proper sized reactor and radiators. That is more expensive to build and to maintain and to fuel.

              If your weapon systems draw little power (drones, missiles, chemically propelled projectiles,…), upsizing the radiators a bit is cheaper and lighter than a slightly undersized reactor and a battery bank.

              If your weapon systems draw lots of power (lasers, rail/coilguns, …), undersizing your reactor and radiators will gain you much less weight and space than you need for your battery banks. You also limit the combat time of your vessel for no good reason. Additionally, if you lose just one radiator, will you be able to nurse the ship back home, with fairly empty batteries and a reactor neither able to recharge them nor having enough power for non-combat operations (like life support …)?

              Better than battery banks might be heat sinks and/or extendable, hidden radiators. Your "civilian radiators" can be run during cruise and when you need more power … there you go. Also you'll have some redundancy because radiators have been known to be shot off.

              Q-ships are basically traps to lure unsuspecting victims into a bad position.
              The closest analogue to what you want is a commando operation or a merchant cruiser.

              Q-ships were less successful than even normal minefields against submarines in WWI, and more than twice more Q-ships were sunk than they sank submarines; 4 in 5 attacked submarines survived. In WWII, they did accomplish nearly nothing at all.

              Another type of Q-ship was the "Uboot -Flakfalle", trying to kill planes by surprising them with 2 quad-2cm and one 3.7cm AA. And 'convoys' of U-boats through the Bay of Biscay to mass AA power. Not a successful experiment due to plane group tactics applied.

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            2. weissel covers the issues with batteries fairly well. Power density is very poor compared to nuclear or radioisotope sources.

              In the future, fiberoptics are expected to replace most electronics. While fiberoptics are not quite immune to EMPs, they fair far better than normal electronics. Given that additional shielding is fairly inexpensive (in the form of a faraday cage), I passed on EMP weapons.

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          2. Would Bimodal NTR be introduced? I can see the greater thrust produced while in “Afterburner” using LOX help with combat maneuvering while not using LOX provides higher exhaust velocity. Or would extra tankage required not provide any meaningful benefits?

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  1. I’ve heard that reactors on board of nuclear submarines or aircraft carriers use highly enriched uranium. Why ist that?

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    1. This is correct, higher enrichments allow much smaller reactor core sizes (the critical mass reduces significantly). Enrichments around 20% is quite common for nuclear aircraft carriers, and that is approximately the enrichments that most reactors run in Children of a Dead Earth (though you can customize this how you see fit if you design your own reactor in game).

      Nuclear submarines, on the other hand, often use weapons grade uranium (97% enriched). This is because reactor core noise reduces with enrichment, and keeping quiet is absolutely critical for submarines. Not really an issue in space though.

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    2. This is not true for all submarine designs, though; in particular French nuclear submarines use low-enrichment fuel. For example, the new Barracuda (Suffren class, to be nitpicky) attack submarines uses civilian-grade uranium at about 5% enrichment, which is even a decrease compared to the previous 7% of the Rubis class.
      A big advantage is being able to make fuel through the civilian industry, which allowed them to get rid of weapon-grade enrichment facilities entirely (nuclear weapons are using previous stockpiles).
      Essentially, they chose to have “good enough” submarines for a more reasonable price (better result/cost ratio), as opposed to the best possible submarines by sparing no expense (it should be noted that “good enough” is still “good enough to sink escorted aircraft carriers”, looking at naval exercises).

      Are there similar possible design compromises here?

      Getting rid of weapon-grade enrichment facilities also have political benefits through non-proliferation advocates, and possibly even security advantages by cutting out one possible source of weapon-grade uranium for hostile third-parties, but equivalents to that would probably be out of scope here.

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      1. Yes, you can vary the enrichment of your nuclear fuels arbitrarily. In practice, it’s a balance between cost and compactness of the core. Highly enriched cores require much smaller critical masses, so smaller reactors are possible.

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    1. MHD generators are probably too bulky and inefficient compared to the solid-state technologies discussed above. They’re useful for two-stage ground-based generation as first stage generators, though, is the impression I get.

      As for molten salt reactors, how small can those be made? As indicated above, you don’t need a lot of reactor to power a space ship.

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      1. MHD generators would still be better than thermocouples. Looking around has showed me thermocouples have an efficiency of 5-8%, necessating special materials. MHD generators are magnets and loops of wire, so no special materials constraints, and have much better efficiencies, up to a theoretical 60%.

        The only reason they don’t see much use on the ground is because they work best with high temp plasma that goes in and comes out hot. Useless if you’ve got literal tons of water to work with a large temperature gradient, but perfect for a spaceship trying to minimize radiator mass.

        Molten salt reactors have already been used on submarines, so they can be made small.

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        1. Let’s just consider the full system here. Reactor heats coolant, feeds into MHDG. Coolant comes out barely cooled at all, at high speed. And goes… where? Radiators? You might manage to get more energy out, but you’d need to run your radiators *really* hot to make it work, I think. Can’t really do the math here, but it seems like it might not be so feasible.

          As for the reactors, well, it’s given above that the system is cooling-constrained to a massive degree. I don’t know how much more interesting an MSR is over a pellet reactor or other design, other than slightly shift some parameters of a tiny portion of the system slightly. It might be one of those “Sure, but it won’t actually matter” additions.

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          1. Why would the coolant be unaffected? High MHD efficiencies generally translate into a much slower, colder coolant flow.

            Also, according to qswitched, the biggest offender in the mass department is the radiators. Radiator area is reduced by a factor ^4 as their operating temperature increases. For example, a 1300K radiator is 2.85 times smaller than a 1000K radiator, and a 2000K microtubule radiator running hot helium inside an array of corrosion-protected carbon tubules is 16 times smaller. An MHD generator that barely affects
            the coolant, and comes out with a low efficiency, will still end up with a lighter system.

            I mentioned an MSR because in the 400-1300K range, it is the best suited for coupling with an MHD generator. Large quantities of conductive fluid moving slowly is great for MHD.

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            1. I should point out that extracting power by decelerating coolant with a MHD generator is still a heat engine, and is still limited by the Carnot Cycle efficiency. The only way to get around this limitation is to not use coolant at all, and to directly extract energy from the radioactive waste products (like how the Fission Fragment Reactor works). Thus, using an MHD generator is limited in efficiency same as thermocouples, turbines, and thermionic generators.

              In the Carnot Cycle efficiency equation, the colder the cold temperature (aka the radiator temperature) is, the more efficient you can make the heat engine. And as you pointed out, hot radiators perform much better than cold ones. You either have enormous cold radiators and very efficient reactors, or small hot radiators and very inefficient reactors. In game, I found ~1200 K radiators to be a good middle ground between both extremes.

              The reason why turbines (and MHD generators) are usually quoted as having 60+% efficiency is because on earth, the cold temperature can be made very cold (i.e. you can cool them with room temperature sea water) without issue. But in space, you can’t use convection or conduction, and radiators need to stay hot to be effective.

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  2. One thing you missed : fusion power densities _can_ be much higher if the fusion reaction used produces mainly charged particles as an output. Proton-Boron-11 or Helium 3/deuterium are examples of such reaction.

    As you’ll probably point out, these reactions are still in early stages of development. No one has even achieved a fusion gain of enough to be usable in a practical system. Tri Alpha Energy claims to be close, but they are a secretive private lab and possibly an investment scam, though they have a lot of funding and credibility.

    Droplet radiators are also something you ignored but have the potential to boost power densities immensely as well. Basically, by direct energy conversion, from charged particles directly to electricity, and droplet radiators, 100 times or better performance is possible on paper than anything you have in the game. Fusion also makes for a drastically better engine.

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    1. There are a lot of ways to capture the neutrons emitted from ‘colder’ fusion reactions that produce less charged particles. They mainly revolve around inertial confinement fusion, where a fusion fuel pellet is detonated inside a sphere of neutron-absorbing material.

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      1. Every method to capture neutrons I know of

        a. Requires a big heavy something to absorb the neutrons at all
        b. Requires a heat engine to convert the heat of absorption back to electricity

        Electric charge capture grids can be very, very light. Literally a series of meshes. You can get your watts/kilogram down by an enormous factor.

        For such a system you don’t really even care about energy efficiency per say. You might design the reactor to just vent the (side reaction) neutrons to space. You care about specific power, since the energy per kilogram of fuel from fusion is so enormous that you only need to capture a small amount of it.

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    2. Droplet radiators have their drawbacks, though. Any acceleration can move the droplets out of the path to the collector. Apparently many designs want a small collector, spraying from a wide droplet generator, these will cause coolant loss even when aligned with the ship’s acceleration, even at very small amounts of acceleration. Which means you need to use more, smaller radiators, causing less heat to be rejected per radiator. (Adding iron or similar to the droplet liquid helps you a bit correcting problems. But there is only so much force you can apply.)

      You also have the problem of contaminating your vessel by the odd splashing droplet — or of course, any projectile passing through the droplet stream. (Whereas the coolant loss due to that is negligible.) Ordinarily you could use a transparent sheet around the droplet stream, but that would be impossible to armour and will not help with battle damage.

      The probably biggest problem however is that you need a very low vacuum pressure material (and non-corrosive, radiation resistant, chemically stable and unfazed by temperature changes are also important). There seem to be very few materials that fit the role. On the positive side, some of the materials work in the 200-300 K, giving you the choice of a lower “cold” temperature — though you still will need huge, though comparatively lightweight radiators, and the liquid going in must be pretty cool to begin with, or it will just boil away.
      On the negative side, it seems no materials are good for higher than 1000K.

      So even if 1000K sounds fine to you, you need a system to bridge the difference between the reactor temperature and the acceptable maximum droplet temperature. Which means more stuff can be broken and it is more maintenance intensive than a simple heat pipe from the reactor to a radiator.

      So you have a radiator where you may not spin, yaw or pitch in useful amounts in battle, and likely not even accelerate without losing coolant … when your reactor power is needed most and where moving the vessel needs to be done most.

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      1. These are all straightforward engineering problems and there are ways to handle what you are describing. 1000 K is a good thing, colder exit temperatures mean more efficient reactors. Even with the reduced temperature, it still outperforms any other radiator by an enormous margin – check the calculator here.
        http://www.5596.org/cgi-bin/dropletradiator.php

        On the other side of the coin, boosted performance would mean a radiator packing warship would outrange any ship not using radiators because it would have massively more power and thus much larger lasers and higher performance railguns. Better engine performance also means the commander of such a warship would have more options to choose from when to engage.

        So the reason why noone would use regular radiators for the main radiator in a world where droplet is available is that despite the drawbacks, nobody with 1/100 or so the energy supply is going to live long enough to get into weapons range.

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        1. Gerald, qswitched seems to be focused on full scale tested hardware with all the kinks and advantages known rather than theoretical or highly experimental technologies.

          For example:

          In-space nuclear reactors (Of the thermoelectric type, the ones described in this article) were tested in the SNAP program.
          NTRs were tested on the ground in the NERVA program.
          The Navy has working railguns.
          Lamp pumped lasers…have been around since the first laser(which was a ruby rod pumped with a flashbulb.)

          Liquid droplet radiators, on the other hand, are still being tested in the lab. Forget solving the maneuvering problem, we’re still at the ‘make it work at all’ stage.

          Another thing: All of the systems described so far seem to be mechanically very simple and hard to break-the most complex systems seems to be ammunition elevators, which are internal components and not easily hit by weapons fire. Solving the maneuvering problem for LDRs would seem to require a collecter boom moving on a track on the exterior of the craft-simply swiveling the boom in place would allow portions of the radiator fluid to escape anyway- which would be a high profile, mechanically complex, hard to armor, easy to cripple system on the surface of the hull.

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  3. Why not use supercritical CO2 turbines for thermal to electric conversion? It’s a lot more efficient than solid state converters, even though it wouldn’t be as efficient as down here on earth.

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