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      Design factors
     Using the Stefan Boltzmann equation, we can quickly see that a radiator with better emissivity, higher surface area and higher temperature removes more waste heat. 
  • On the left, 1100K radiators. On the right, 2700K radiators. The latter is actually handling three times as much waste heat. 


     On spaceships, it is important to use the lightest possible components for each task. A spaceship with lighter radiators will accelerate faster and have more deltaV, meaning it can go further and do more for less propellant. 
     If we want a lightweight radiator, we want it to have the highest emissivity. We can accomplish this by using naturally dark materials, such as graphite, or painting over shiny metals with black paint. 
     A larger radiator weighs more. We therefore want the smallest radiators possible. To compensate for lower surface area, we can increase the operating temperature. A small increase in temperature leads to a massive increase in waste heat removed. This means that hot radiators are massively lighter and smaller than cold radiators. 
     Further considerations

  • The ISS's EAC system


     A typical radiator accepts coolant from a hot component. The coolant's component exit temperature is the initial temperature at the radiator. The radiator serves as an interface that radiates away the coolant's heat, leading to a lower radiator exit temperature. The coolant is fed back to the component to complete the waste heat removal cycle.

  • Note how the maximum temperature the heat exchanger's maximum temperature, delivered to the steam, is the lowest temperature of the liquid sodium in the reactor core.


     Heat only flows from a hot object to a cooler object. A radiator can therefore only operate when the component's temperature is higher than the radiator's coolant exit temperature. For example, if a nuclear reactor operates at 2000K, the radiator must work at 2000K or less. 

  • A reactor from COADE. The reactor operates at 2907K but the radiator receives coolant at 2400K.


     The difference between the entry and exit temperatures in a radiator depends on many factors, but generally we want the largest difference possible. This difference in temperature is especially important for power generation. A large difference means more energy can be extracted from a heat source. It also means that less coolant is needed to cool a component.
     This creates problems with realistic designs.
     A general solution is to use two sets of radiators operating at different temperatures: one low-temperature circuit and one high temperature one. It works fine when your low temperature waste heat is a few kilowatts from life support and avionics. Other solutions have to be found for components that must be kept at low temperatures yet generate megawatts of waste heat, such as lasers.  

  • This design has three sets of radiators of decreasing area for different temperature components. Actually four sets if you count the habitat module radiators (Payload/Avionics Radiators)

     For low temperature high heat components, heat pumps must be used. They can move waste heat against a temperature gradient, allowing, for example, a a 1000K radiator to cool down a 500K component. However, this costs energy. Moving heat from 500K to 1000K costs 1 watt to the pump for every watt moved. A realistic pump will not be 100% efficient and will require more than 1 watt to move a watt of waste heat. 

Pump_power = (Waste_heat Tc / (Th - Tc)) / Pump_Efficiency

     Pump_power is how many watts the heat pumps consume. Waste_heat is how many watts must be removed from the component. Tc is the component's temperature. Th is the radiator's temperature, both in Kelvins. Pump_efficiency is a coefficient.
  • The refrigeration cycle is an example of a heat pump.


     A coolant must generally be kept liquid. This imposes a lower and upper limit to the coolant temperature; any colder and it will freeze and block the pipes, any hotter it boils and stops flowing. Water coolant, for example, can only be used between 273 and 373K. More importantly, it limits the temperature difference that can be obtained from a radiator.
     Large temperature differences require that the coolant spend a long time inside the radiator. This requires larger radiators or long, circuitous paths for the pipes. As the coolant becomes colder, it radiates at lower rates, meaning that the last 10 kelvin drop in temperature can take exponentially more time than the first 10 kelvin reduction. There are strong diminishing returns. 
     There are also structural concerns. Large temperature differences impose thermal stresses. These might be too great to handle. Lightweight, stressed radiators are prone to reacting badly to any sort of battle damage, making radiators a weak-spot for any sort of warship.

  • The ISS radiators' support spars. A spaceship under acceleration will need much more support.


     All in all, we must keep in mind that there is a restricted range of temperatures between the hot and cold ends of a radiator, and that its performance cannot simply be obtained by using the Stefan Boltzmann equation on the maximum temperature. We cannot use a simple average either, because the coolant loses heat at a quadratically declining rate as it moves from higher to lower temperatures. 
     Here is an example of 1 kg of sodium at 1000K being cooled by a 0.8 emissivity one-sided 1m^2 radiator panel:


     We can see that it takes 17 seconds for the sodium to cool down from 1000K to close to its melting point of 370K. Any cooler and it'll solidify in the pipes. If we average the radiated watts, we get a value close to 11.46kW. This corresponds to an average radiating temperature of 545K.  
     Finally, a radiator suffers stresses when a spaceship accelerates. Some types of radiator break or disperse under strong accelerations, so the spaceship's performance needs to be considered before selecting a design.
     Solid Radiators


     A straightforward design used today.
     It consists of a slab of metal run through with hollow tubes for a coolant to flow. The waste heat conducts out of the coolant and into the radiator material, which radiates it away from its exposed surfaces.  


     This design has a rather high mass per area and low temperature limits, making it one of the worst performing designs. The maximum temperature is whatever keeps the radiator materials both solid and strong, which is important as many metals rapidly lose strength as they approach their melting point. 
     The coolant must remain liquid throughout the cooling cycle, so this limits the temperature difference that can be achieved. Using metals such as tin or salts such as sodium allows for better temperature differences, but pumping them requires specialized, sometime non-reactive, sometimes power consuming equipment.

  • Multiple radiators will shine their heat into each other and lose efficiency.


     The arrangement of radiators around a spaceship must take into account inter-reflection, which is when one radiator's heat is intercepted and absorbed by another radiator. This reduces their efficiency. Anything more than two radiators per axis absorbs some of the heat of another radiator... at four radiators, only 70% of the heat escapes to space, at eight radiators, the efficiency falls to 38%. 
     NASA has studied solid radiators for use in its Nuclear Electric Propulsion concepts. It has specified 2kg/m^2 area density as a requirement for any thermal management system. The ISS's radiators mass 8 kg per square meter, or 2.75kg/m^2 if we only consider the exposed panels.
     So far, only bare carbon fibre radiators operating at 800-1000K have reached this area density. 

     An alternative design achieves better area density by removing the coolant loops and pumps. The heat pipe has a hot end and a cold end, separated by a vacuum.
  • Heat Pipe moving waste heat into a heatsink. 


     Solid coolant is boiled away and then condensed on the cold end, then re-circulated through capillary action or centrifugal acceleration. This method allows for high operating temperatures and does not require any pumps of moving parts, but high mass per area negates many of its advantages.

     On a warship, radiators are a weakpoint. Bright, exposed and hard to defend, they are easy to hit and once the are damaged, they can render a spaceship unable to function. They can mission-kill a warship without ever having to penetrate any armor. Redundant radiators impose a mass penalty. Covering the radiators in plates of armor massively decreases their thermal conductivity between coolant and exposed surfaces, which in turn reduces their efficiency. 
     Solutions for reducing the vulnerability of radiators include pointing them edge-on to the enemy, moving them to the back of the ship, or using retractable designs.
  • On the right, the radiators are exposed the enemy fire. On the left, the hull bulge protects the radiators from damage.


     If all radiators are retracted, the spaceship must rely on heat sinks for its cooling needs. A megawatt heat source can boil off a ton of water in less than seven minutes, so this will only work over very short time periods. 


     High temperature solid radiators run into issues, such as having to deal with the coolant boiling, or having to contain enormous pressures to keep fluids in a supercritical state. The solution is to use solid blocks of metal instead of coolant. Running these blocks like a train around tracks allows for robust radiators that can handle strong accelerations and temperatures up to the boiling points of the coolant blocks (4000K in some cases, if the tracks are actively cooled). The smaller the blocks, down to the size of pinballs, the faster they cool down and the shorter the track needs to be, leading to mass and area savings. 
     Moving radiators
     One of the biggest reasons why solid radiators are so massive is that they need coolant pipes, pumps and heat exchangers to move waste heat from equipment to exposed surfaces.
     To greatly reduce the area density, we can devise a radiator that does not require bulky coolant loops. Instead, we move the radiator.
     Moving radiators rely on the radiator material itself to move through a heat exchanger, out into space to radiate away the heat, then back in.
     Advantages include simpler construction, less fragile designs, less power consumed and very larger temperature differences between the hot and cold ends. This ends up giving them better kg/m^2 and kW/m^2 ratings. However, there are many more moving parts and the radiating surfaces are only a fraction of the volume the radiators take up. Unless very lightweight materials are used, the support structure will negate the mass advantage of such a radiator.

  • From High Frontier


     A disk-and-drum design has a heat exchanger shaped like a drum, rolling against a radiating disk. The hoola-hoop radiator is a large disk held at the tip by a drum heat exchanger. 

  • The belt loops are held edge-on to the sun. Angled loops would suffer less from re-absorption of radiated heat on the internal surfaces, which is more important at higher operating temperatures.


     If the wheel or loop is replaced by a flexible or track-linked belt, it can be made to follow various paths. A 'belt-loop radiator' could bring the radiator closer to the spaceship and reduce the structural strength required to survive accelerations or vibrations.


     A wire-loop configuration uses black carbon filaments as the radiating surface. They are flung out of the heat exchanger and held in place by centripetal force. Using high tensile strength materials allows for extremely lightweight loops.

  • From High Frontier. Uses carbon nanotube materials for the wires. 


     Rollers can guide the wires instead of centripetal force, thereby becoming an even lighter version of the belt-radiator. High tensile strength materials would be needed, as this allows the rollers and motors to hold the wires under tension to prevent them from sliding around or tangling.


     A rotating disk radiator is a moving radiator where the central component is a spinning disk. Coolant fluid is sprayed at the hub. The low vapour pressure liquid's surface tension causes it to spread into a thin, even film over the disk. As the disk rotates, centripetal force causes the film to flows as it cools to the collector troughs on the edges. This configuration does away with heavy heat pipes and radiator pumps, but requires the use of very low vapour pressure fluids. The disk can be angled inwards, outwards or canted to deal with spacecraft acceleration.


     Bubble-membrane radiators are a 3D version of the rotating disk radiator. Hot coolant is sprayed against an inflated membrane, causing it to spread out into a thin film that very effectively loses its heat. Spinning the membrane causes the liquid film to pool at the bubble's equator, where it is collected and recycled. 
     Advantages includes allowing the use of high vapor pressure coolants and very light construction. Disadvantages include having to contain high pressure vapours in a container that must remain light and transparent.


     Electric radiators
     The designs mentioned so far use physical structures to hold the radiators in place. This imposes some restrictions, such as having to stay within the temperature limits of the support structures, and larger radiators need heavy support to survive even light accelerations. 
     A solution would be to use magnetic forces to hold the radiators in place. Strong magnetic can replace physical support structures for significant mass savings. 


     Examples of such radiators include the flux-pinned radiator. Magnetic fields hold solid radiator components in place. Thermally conductive ribbons transport heat to the magnetic components.
     However, there are complications. Most metals lose their magnetic properties as they are heated, becoming completely insensitive to magnetic fields above their Curie point. Careful selection of the materials used and control of the temperatures is required. 


     A Curie point radiator operates around the temperature at which metallic dust particles lose their magnetism. Iron, for example, loses its ferromagnetism at 1043K.
     The Curie point radiator uses metal filings or even liquid droplets. They are heated to above the curie point temperature and ejected into space, away from the spacecraft. A magnetic field is in place, but they are not affected by it. Iron can be released at temperatures of up to 3134K and collected at 1043K, but Cobalt has a Curie temperature as high as 1388K, is naturally black and boils at 3400K, making it a better coolant. The small size of the particles or liquid droplets allows several megawatts of waste heat to be radiated away per square meter.


     Once the particles cool below the Curie point, they regain their ferromagnetism. They begin to be affected by the magnetic field and are drawn back to the spaceship to be collected.
     Magnetic radiators are excellent solutions for combat damage - at worst, the enemy will disrupt cooling for a few seconds. However, they consume a lot of power and require heavy equipment to generate strong magnetic fields. Any unexpected acceleration or jolt from the spaceship can disperse all the material held in place by magnetic fields. 
     An alternative electric radiator uses electrostatic forces to hold charged particles in place. One example is the ETHER charged dust radiator. Charged particles follow field lines and execute elliptical orbits between the heat exchanger and the collection point. Similar to a liquid droplet radiator, charged particles can be mechanically dispersed and collected efficiently at the other end by oppositely charged scoops. 


     The advantage of electrostatic radiators is that they consume less power, since creating a strong charge differential is easier than extending a strong magnetic field. The equipment is lighter and is less sensitive to temperature changes, since no superconducting or cryogenic equipment is used, and the charged particles can hold a charge across larger temperature differences than they can maintain their magnetic properties. 
     However, the charge carried by the particles can be nullified by natural solar wind or if they come into contact with a conductor. This means they need a clear, short path between heat exchanger and collection point. 
     Liquid droplet radiators
     Liquid droplet radiators do not use any radiating surfaces - they expose the coolant directly to the vacuum. The resultant droplets have incredible surface area for their mass, allowing for rapid cooling and extremely low area density.


     As the coolant does not need to be physically contained, it can be heated to very high temperatures and still cool down very quickly. There are no thermal stress constraints on liquids, so the temperature change can be as extreme or rapid as desired. They do not have to maintain magnetic properties or hold a charge either. This calculator can gives an approximation of an LDR's performance. At 1300K and using 50 micrometer droplets (a fine mist), area density can be as low as 0.047kg/m^2 with an effective performance of 57MW/m^2. This does not include the mass of the heat exchanger, droplet emitter and collector.


     Solutions have already been devised for issues such as the droplets being blown away by solar wind, colliding and merging into larger droplets or moving at different velocities within the droplet sheet. 
     Vapor pressure is still a concern - hot liquids in vacuum tend to evaporate quickly. Special low-vapor pressure coolants must be used, such as liquid gallium, aluminium or tin up to 1200K, lithium up to 1500K. Salting these liquids with a material such a graphite 'grit' or coating them with black ink is necessary to achieve high emissivity. Nano-fluids might allow even higher temperature liquids to be used. Reaching higher temperatures means accepting high coolant loss rates or enclosing the radiating volume in a membrane that condenses and collects vapors. The membrane has to be transparent at the radiating temperatures.  
      The droplets in a liquid droplet radiator need to be spaced evenly and by distances much larger than the droplet diameter — this is to prevent inter-reflection losses from becoming significant.
     Variations in liquid droplet radiators are mostly around how to contain and direct the coolant flow between ejection and collection points.  
      A rectangular LDR has droplet emitter and collector arms of equal length. The collector arm can be made wider than the emitter to catch droplets deviated out of their path by unexpected movements or errors in droplet formation. It might be possible to move the collector arm above and below the droplet plane to intercept droplets when the spaceship is accelerating, as this would cause the droplet sheet to bend away from the plane. 

  • An ICAN-II design with rectangular liquid droplet radiators.


     A triangular LDR saves mass by using a small collector dish instead of a long arm. However, it is less able to catch deviating droplets or compensate for spaceship acceleration.

  • Triangular LDR variants


     Some LDR designs dispose of the long arms and membranes and instead just spray the droplets into space. The momentum of the droplets makes them follow trajectories that land them right back at the collectors. A fountain LDR shoots droplets in front of an acceleration spaceship. They are scooped up once cool. This method of dispersing droplets produces the lightest possible designs, but there is a risk of droplet losses.

  • Droplets are dropped from the spacecraft's 'front' and fall into collector arms at the mid-section.


     It works best on spacecraft that gently accelerate over long periods of time, such as nuclear-electric craft on interplanetary trajectories. A shower LDR disperses droplets in front of the spacecraft and has the collectors simply collect them like a ram-scoop. It has less risk of dispersing the droplets than a fountain LDR but requires a long shower-head. 
     Pressure membranes can be an addition to any liquid droplet radiator. They enclose the volume the droplets traverse. Benefits include re-condensing vapours from too-hot droplets, catching stray droplets, allowing for faster droplet velocity and a greater tolerance for droplet sheet instabilities. However, they must remain transparent to all wavelengths the droplets are radiating at, and hold in the vapour gas pressure. These are competing requirements: low wavelength absorption is done with very thin membranes, while high pressure requires thick membranes.  
     Advanced radiators
     Magnetically pumped and focused LDR:

  • Magnetically focused by the collector nozzle.


     Ferrofluids at low temperatures and liquid metal at high temperatures can be used as coolant in liquid droplet radiators. They react to eddy currents and magnetic fields, allowing the coolant to be pumped without any moving parts through magneto-hydrodynamics. 

  • Schematic of the Mag-LDR patent, showing the dipole magnet.


     Magnetic fields can also be used to recover a droplet sheet. Cyclical fields can push and pull on a group of droplets over distances proportional to the field strength. High strength fields could allow droplet sheets to extend over several dozens of meters before being recovered. They would also allow the LDR to compensate for its vulnerability to droplet sheets being dispersed and lost when the spacecraft accelerates by holding the droplets in place.
     Together, an LDR can become extremely lightweight for the area is covers, as no physical support structure has to span its length. 
     Gas coolants: 
     We have looked at solid and liquids as coolants. Gasses can be used too.
     Gas coolants have been used in nuclear reactors already. Carbon dioxide and helium were selected as they are inert and support higher temperatures than water or sodium coolants. 
     In space, the principal advantage of a gas coolant is that it can operate at much higher temperatures than liquid or solid coolants. The same gas could be run from a nuclear reactor to a radiator's tubes and back. It also allows for inflatable structures for the radiators, which can be much lighter than rigid equivalents.

  • Inflatable fin radiators.
  • Multiple roll-out fin radiators.
  • Inflatable bags are simpler and more rugged than roll-out fins but have lower surface area.


     However, there are limitations and complications. Hot, pressurized gas can be very chemically reactive. While you can push a gas to 3000K+ temperatures, the walls of the pipes containing the gas must also survive these temperatures. Many of the mass savings that come from running a radiator at high temperatures are lost trying to contain and survive the gas coolant. Pumping gas requires much more power per kg moved than liquids, for example.
     Another difficulty is the very poor heat transfer rate between a heat exchanger and a gas. A hot, low density gas like heated helium might have a thermal conductivity hundreds of times lower than a liquid like molten sodium. This leads to difficulties both at the heat exchange interface and the radiating surface interface. 
     A lot of these issues can be solved by using a two-phase coolant loop, meaning it spends some of its time as a liquid and some of its time as a gas. Up to the heat exchanger, the coolant is in a liquid form. It flows through tubes using simple pumps. The heat exchanger is divided into many smaller tubes to increase the contact area between exchanger and coolant. 
     Past the exchanger, the coolant expands. The pressure drop allows it to boil into a gas. This gas travels through a volume enclosed by a hermetic membrane. Through a combination of expansion decompression and the Stefan-Boltzmann law, the gas quickly cools and condenses on the membrane walls. This forms a thin film in microgravity that can be directed towards collection points, where the liquid is pumped back to the heat exchanger. 
     Dusty Plasma radiator:
     This radiator uses conductive plasma, manipulated by magnetic fields, to move and manipulate dust particles. 


     The dust particles suspended in a plasma behave in fascinating ways, still being discovered by the dusty plasma field of research. Interesting behaviours include self-organising into quasi-crystalline structure, building DNA-strand-like bridges through plasma or collecting into disks with empty centres. This is all due to the self-repelling charges the dust particles gain inside the plasma.


     A better understanding of these behaviours can allow for a radiator to combine every advantageous characteristic: wide range of operating temperatures, very low mass per square meter, easily manipulated by electromagnetic and electrostatic forces, low vulnerability to damage and able to survive strong accelerations. 

  • A sheet of dust particles self-organizing into a 'plasma crystal'. 


     The plasma can be quite cold and still serve to manipulate the dust particles. Low-temperature plasma does is safe to manipulate and is quite transparent to the wavelengths the dust particles will be radiating at, meaning it won't heat up or be blown away by thermal expansion. 
     A simple dusty plasma radiator would have plasma trapped in magnetic loops, like coronal loops. Dust would travel along these plasma tubes. More advanced dusty plasma radiators would spray dust particles into a plasma and have it self-organize into thin planes for maximal radiating surface area. Simply changing the ionization state of the particles by running an electric current through the plasma would allow the dust to clump together and follow magnetic field lines straight back to a collector. 




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