Introduction

RocketCat sez

You might think that the problem with surviving in the "zero degree cold of space" is keeping from freezing to death. Nope, the problem is Heat.

Human bodies are little furnaces, which you can discover if you wrap your limbs and torso in plastic wrap and see how little time it takes to pass out from heat prostration (note to jackasses THIS IS AN ILLUSTRATION, DO NOT ACTUALLY TRY TO DO THIS!). In space, it's not like you can open the window for a cooling breeze, either. Your cosy little habitat module will turn into an oven.

If you stayed awake during Physics 101 class you'll know that the blasted laws of thermodynamics say there are only three ways of getting rid of waste heat. But only one of them will work in space: radiation.

So you'll need heat radiators or the crew is going to die horribly while sweating bullets.

And I am quite sure that you are going to make things infinitely worse by insisting on your precious nuclear power reactors and megawatt laser cannons. Human bodies only make enough waste heat to kill everybody, reactors and lasers can make the entire freaking ship glow white-hot and vaporize.

Ever see those titanic curved towers around nuclear power plants? Yep, cooling towers. You'll need something a bit more high-tech if you do not want your spacecraft's aesthetics spoiled by a 40 meter cooling tower or two.

Laser cannon are much worse. Rick Robinson described them as observatory telescopes with a jet engine at the eyepiece. Ken Burnside said they were blast furnaces that produced coherent light as a byproduct. Whatever you call them they are hot enough to make your ship go from solid directly to Solar-surface hot ionized gas without passing through the molten metal stage first.

But of course heat radiators are one of the major things conspicuous by their absence in science fiction TV shows and movies. Concept artists don't want their ultra-futuristic spacecraft decked with 17th century billowed sails. They even over-ruled Arthur C. Clarke for cryin' out loud! The only exception that comes to mind is the ISV Venture Star from the movie Avatar.

Power plants and some propulsion systems are going to require heat radiators or the ship will glow red then melt (NO, for the millionth time you CANNOT get rid of the heat by turning it into electricity!).

There are only three ways of getting rid of heat: convection, conduction, and radiation; and the first two do not work at all in the vacuum of space. So the ship designer is stuck with heat radiators, or what NASA calls Active Thermal Control Systems

Functionally they are not too different from the radiator on your automobile. Pipes full of radiator fluid are coiled around the cylinder heads and engine block, sucking up the heat so the engine doesn't turn into molten lava. The hot radiator fluid is moved by the coolant pump, carrying the heat into the engine coolant radiator (that flat box on the automobile's nose with all the scalloped holes). In the radiator, the heat is removed from the radiator fluid by conduction with the wind. The cool radiator fluid travels into the engine and the cycle begins anew.

Actually, in spaceships the heat radiators get rid of heat by … well … radiating, instead of conduction. Different design because there is no wind in space. But you get the idea.

See Thermophotovoltaic Energy Conversion in Space Nuclear Reactor Power Systems and HIGH TRADER for details.

Radiator Details

Now lets go in-depth on how these things work.

If you want to calculate this for yourself use the Stefan-Boltzmann law:

P = A * ε * σ * T4

A = P / (ε * σ * T4)

where

  • P = the power of waste heat the radiator can get rid of (watts)
  • σ = 5.670373×10-8 = Stefan-Boltzmann constant (W m-2K-4)
  • ε = emissivity of radiator (theoretical maximum is 1.0 for a perfect black body, real world radiator will be less. Should be at least 0.8 or above to be worth-while)
  • A = area of radiator (m2)
  • T = temperature of radiator, this assumes temperature of space is zero degrees (degrees K)
  • x4 = raise x to the fourth power, i.e, x * x * x * x

My source (Matthew DeBell) says that if P = 150 gigawatts, ε = 0.94, and T = 3000 K, A would be 34,941 m2. Actually it could be half that if you have a two-sided radiator, which would make the radiator 17,470 m2 (a square 132 meters on a side). Which is still freaking huge.

For estimating the mass of the radiator array, go here.

Ken Burnside says that if one examine the equation carefully one will notice that the radiator effectiveness goes up at the fourth power of the heat of the radiator. The higher the temperature, the lower the surface area can be, which lowers the required mass of radiator fins. This is why most radiator designs use liquid sodium or lithium (or things more exotic, still). 1600K radiators mean that you need a lot less mass than 273 K radiators.

Ken Burnside also noted that radiators are large, flimsy, and impossible to armor (except perhaps for the droplet radiator). A liability on a warship. However, Zane Mankowski (author of Children of a Dead Earth) makes a good case that heat radiators can indeed be armored. Mr. Mankowski says the thickness of the radiator material can be increased to provide armor-like protection for the working fluid tubes, with the price of reducing radiator efficiency.

Mr. Burnside has an entire essay about the problem of heat on combat spacecraft, entitled The Hot Equations: Thermodynamics and Military SF. Since thermodynamics is one of the most important (and most neglected in science fiction) factors in combat, the essay will repay careful study.

In the military the old bromide is that amateurs talk about battle tactics while professionals talk about logistics. In the real of spacecraft design, @AsteroidEnergy said "Amateurs discuss rockets, professionals discuss heat management."

But do realize that if the spacecraft does indeed have a nuclear propulsion system or something else dangerously radioactive, the radiators must be tapered to keep inside the radiation shadow shield. Or bad things happen.


TWO RADIATOR SYSTEMS NEEDED

I had initially thought that the heat from the life-system could be simply dumped by the same radiator system dealing with the multi-gigawatt waste heat from the propulsion system or power system. Richard Bell pointed out that I had not thought the problem through. Due to the difference in the temperatures of the waste heat from life-system and propulsion, unreasonably large amounts of energy will be required to get the low-level life-system heat into a radiator designed to handle high-level propulsion heat. The bottom line is that there will be two separate radiator systems.

Not only are you going to require two separate radiator systems, the one for the modest cooling required by the life-system is liable to have larger radiator surfaces than the one cooling the multi-gigawatt propulsion system. Radiator effectiveness goes up as the fourth power of the heat of the radiator, remember?


THERMAL MANAGEMENT IN SPACE

It should be pointed out that in a vacuum environment, convection is no longer available and the only mechanism of rejecting heat is radiation. Radiation follows the Stefan-Boltzmann Law

E = σT4

where
E = the energy rejected
σ = the Stefan-Boltzmann constant, = 5.67 W m-2 K-4
T = the temperature at which the heat is radiated

That is, the total amount of heat radiated is proportional to the surface area of the radiator. And the lower the radiation temperature, the larger the radiator area (and thus the radiator mass, for a given design) must be.

The radiator can only reject heat when the temperature is higher than that of the environment. In space, the optimum radiation efficiency is gained by aiming the radiator at free space. Radiating toward an illuminated surface is less effective, and the radiator must be shielded from direct sunlight.

The rejection of heat at low temperatures, such as would be the case in environmental control and in the thermal management of a materials processing unit, is particularly difficult.

Space-Based Power Generating Systems

Solar photovoltaic systems have a generating capability of up to several hundred kilowatts. The power output range of solar thermal systems is expected to be one hundred to perhaps several hundred kilowatts. While in principle these power systems can be expanded into the megawatt region, the prohibitive demands for collection area and lift capacity would appear to rule out such expansion. Megawatt and multimegawatt nuclear power reactors adapted for the space environment appear to offer a logical alternative.

Solar photovoltaics themselves will not burden the power generating system with a direct heat rejection requirement, since the low energy density of the system requires such a great collection area that it allows rejection of waste radiant energy. However, if these systems are to be employed in low Earth orbit or on a nonterrestrial surface, then a large amount of energy storage equipment will be required to ensure a continuous supply of power (as the devices do not collect energy at night). And the round-trip inefficiencies of even the best energy storage system today will require that a large fraction—perhaps 25 percent—of the electrical power generated must be dissipated as waste heat and at low temperatures.

Solar thermal systems, which include a solar concentrator and a dynamic energy conversion system, are presumed to operate at relatively high temperatures (between 1000 and 2000 K). The efficiencies of the energy conversion system will lie in the range of 15 to perhaps 30 percent. Therefore we must consider rejecting between 70 and 85 percent of the energy collected. In general, the lower the thermal efficiency, the higher the rejection temperature and the smaller the radiating area required. As with solar photovoltaic systems, the inefficiencies of the energy storage system will have to be faced by the heat rejection system, unless high temperature thermal storage is elected.

The current concepts for nuclear power generating systems involve reactors working with relatively low-efficiency energy conversion systems which reject virtually all of the usable heat of the reactor but at a relatively high temperature. Despite the burdens that this low efficiency places on nuclear fuel use, the energy density of nuclear systems is so high that the fuel use factor is not expected to be significant.

In all of these systems the output power used by the production system in environmental control and manufacturing (except for a small fraction which might be stored as endothermic heat in the manufactured product) will have to be rejected at temperatures approaching 300 K.


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.

Using the Stefan-Boltzmann Law,

E1 = 5.67×10-8 W m-2 K-4 (800 K)4
E1 = 5.67×10-8 W m-2 K-4 × 4096×108 K4
E1 = 5.67 W m-2 × 4.10×103
E1 = 23.3 kW m-2

900 kW / 23.3 kW m-2 = 38.6 m2
and 38.6 m2 / 0.85 = 45.4 m2

E2 = 5.67×10-8 W m-2 K-4 (300 K)4
E2 = 5.67×10-8 W m-2 K-4 × 81×108 K4
E2 = 5.67 W m-2 × 81
E2 = 459 W m-2

100 kW / 459 W m-2 = 0.2179×103 m2 = 218 m2
and 218 m2 / 0.85 = 256 m2

Therefore, we can see that the dominant heat rejection problem is not that of the primary power plant but that of the energy that is used in life support and manufacturing, which must be rejected at low temperatures. Using the waste heat from the nuclear power plant for processing may be effective. But, ironically, doing so will in turn require more radiator surface to radiate the lower temperature waste heat.

Heat Rejection Systems

In this section I will deal with systems designed to meet the heat rejection requirements of power generation and utilization. These heat rejection systems may be broadly classified as passive or active, armored or unarmored. Each is expected to play a role in future space systems.


Heat pipes: The first of these, called the “heat pipe,” is conventionally considered the base system against which all others are judged. It has the significant advantage of being completely passive, with no moving parts, which makes it exceptionally suitable for use in the space environment.

For the convenience of the reader, I will briefly describe the operational mechanism of the basic heat pipe. (See figure 36.) The heat pipe is a thin, hollow tube filled with a fluid specific to the temperature range at which it is to operate. At the hot end, the fluid is in the vapor phase and attempts to fill the tube, passing through the tube toward the cold end, where it gradually condenses into the liquid phase. The walls of the tube, or appropriate channels grooved into the tube, are filled with a wick-like material which returns the fluid by surface tension to the hot end, where it is revaporized and recirculated.

Essentially the system is a small vapor cycle which uses the temperature difference between the hot and cold ends of the tube as a pump to transport heat, taking full advantage of the heat of vaporization of the particular fluid.

The fluid must be carefully selected to match the temperature range of operation. For example, at very high temperatures a metallic substance with a relatively high vaporization temperature, such as sodium or potassium, may be used. However, this choice puts a constraint on the low temperature end since, if the fluid freezes into a solid at the low temperature end, operation would cease until the relatively inefficient conduction of heat along the walls could melt it. At low temperatures a fluid with a low vaporization temperature, such as ammonia, might well be used, with similar constraints. The temperature may not be so high as to dissociate the ammonia at the hot end or so low as to freeze the ammonia at the cold end.

With proper design, heat pipes are an appropriate and convenient tool for thermal management in space systems. For example, at modest temperatures, the heat pipe could be made of aluminum, because of its relatively low density and high strength. Fins could be added to the heat pipe to increase its heat dissipation area. The aluminum, in order to be useful, must be thin enough to reduce the mass carried into space yet thick enough to offer reasonable resistance to meteoroid strikes.

A very carefully designed solid surface radiator made out of aluminum has the following capabilities in principle: The mass is approximately 5 kg/m2 with an emissivity of 0.85; the usable temperature range is limited by the softening point of aluminum (about 700 K). At higher temperatures, where refractory metals are needed, it would be necessary to multiply the mass of the radiator per square meter by at least a factor of 3. Nevertheless, from 700 K up to perhaps 900 K, the heat pipe radiator is still a very efficient method of rejecting heat.

A further advantage is that each heat pipe unit is a self-contained machine. Thus, the puncture of one unit does not constitute a single-point failure that would affect the performance of the whole system. Failures tend to be slow and graceful, provided sufficient redundancy.


Pump loop system: The pump loop system has many of the same advantages and is bounded by many of the same limitations associated with the heat pipe radiator. Here heat is collected through a system of fluid loops and pumped into a radiator system similar to conventional radiators used on Earth. It should be pointed out that in the Earth environment the radiator actually radiates very little heat; it is designed to convect its heat. The best known examples of the pump loop system currently used in space are the heat rejection radiators used in the Shuttle. These are the inner structure of the clamshell doors which are deployed when the doors are opened (fig. 37).

Pump loop systems have a unique advantage in that the thermal control system can easily be integrated into a spacecraft or space factory. The heat is picked up by conventional heat exchangers within the spacecraft, the carrier fluid is pumped through a complex system of pipes (extended by fins when deemed effective), and finally the carrier is returned in liquid phase through the spacecraft. In the case of the Shuttle, where the missions are short, additional thermal control is obtained by deliberately dumping fluid.

Since the system is designed to operate at low temperatures, a low density fluid, such as ammonia, may on occasion, depending on heat loading, undergo a phase change. Boiling heat transfer in a low gravity environment is a complex phenomenon, which is not well understood at the present time. Because the system is subjected to meteoroid impact, the basic primary pump loops must be strongly protected.

Despite these drawbacks, pump loop systems will probably be used in conjunction with heat pipe systems as thermal control engineers create a viable space environment. These armored (closed) systems are rather highly developed and amenable to engineering analysis. They have already found application on Earth and in space. A strong technology base has been built up, and there exists a rich literature for the scientist-engineer to draw on in deriving new concepts.

Advanced Radiator Concepts

The very nature of the problems just discussed has led to increased efforts on the part of the thermal management community to examine innovative approaches which offer the potential of increased performance and, in many cases, relative invulnerability to meteoroid strikes. Although I cannot discuss all of these new approaches, I will briefly describe some of the approaches under study as examples of the direction of current thinking.

Improved conventional approaches: The continuing search for ways to improve the performance of heat pipes has already shown that significant improvements in the heat pumping capacity of the heat pipe can be made by clever modifications to the return wick loop. Looking further downline at the problem of deployability, people are exploring flexible heat pipes and using innovative thinking. For example, a recent design has the heat pipes collapsing into a sheet as they are rolled up, the same way a toothpaste tube does. Thus, the whole ensemble may be rolled up into a relatively tight bundle for storing and deploying. However, because the thin-walled pipes are relatively fragile and easily punctured by meteoroids, more redundancy must be provided. The same principles, of course, can be applied to a pump loop system and may be of particular importance when storage limits must be considered. These are only examples of the various approaches taken, and we may confidently expect a steady improvement in the capability of conventional thermal management systems.


The liquid droplet radiator: The basic concept of the liquid droplet radiator is to replace a solid surface radiator by a controlled stream of droplets. The droplets are sprayed across a region in which they radiate their heat; then they are recycled to the hotter part of the system. (See figure 38.)

It was demonstrated some time ago that liquid droplets with very small diameters (about 100 micrometers) are easily manufactured and offer a power-to-mass advantage over solid surface radiators of between 10 and 100. In effect, large, very thin radiator sheets can be produced by the proper dispersion of the droplets. This system offers the potential of being developed into an ultralightweight radiator that, since the liquid can be stored in bulk, is also very compact.

The potential advantages of the liquid droplet radiator can be seen if we consider again the problem that was discussed at the end of the section on heat pipe radiators. We found that a very good aluminum radiator would require 256 m2 and have a mass of nearly 1300 kg to radiate the low temperature waste heat from lunar processing. Using the properties of a liquid droplet radiator and a low density, low vapor pressure fluid such as Dow-Corning 705, a common vacuum oil, we find that, for the same area (which implies the same emissivity), the mass of the radiating fluid is only 24 kg.

Even allowing a factor of 4 for the ancillary equipment required to operate this system, the mass of the radiator is still less than 100 kg.

To achieve efficiency, the designer is required to frame the radiator in a lightweight deployable structure and to provide a means of aiming the droplets precisely so that they can be captured and returned to the system. However, present indications are that the droplet accuracies required (milliradians) are easily met by available technology. Recently, successful droplet capture in simulated 0 g conditions has been adequately demonstrated. An advantage of a liquid droplet radiator is that even a relatively large sheet of such droplets is essentially invulnerable to micrometeoroids, since a striking micrometeoroid can remove at most only a few drops.

The reader may be concerned that the very large surface area of the liquid will lead to immediate evaporation. However, liquids have recently been found that in the range of 300 to 900 K have a vapor pressure so low that the evaporation loss during the normal lifetime of a space system (possibly as long as 30 years) will be only a small fraction of the total mass of the radiator.

Thus, the liquid droplet radiator appears promising, particularly as a low temperature system where a large radiator is required.

Liquid droplet radiators for applications other than 0 g have been suggested. For example, in the lunar environment fluids with low vapor pressures can be used effectively as large area heat dissipation systems for relatively large-scale power plants. We may well imagine that such a system will take on the appearance of a decorative fountain, in which the fluid is sprayed upward and outward to cover as large an area as possible. It would be collected by a simple pool beneath and returned to the system. Such a system would be of particular advantage in the lunar environment if low mass, low vapor pressure fluids could be obtained from indigenous materials. Droplet control and aiming would no longer be as critical as in the space environment; however, the system would need to be shaded from the Sun when it is in operation. While this system is far less developed than the systems previously discussed, its promise is so high that it warrants serious consideration for future use, particularly in response to our growing needs for improved power management.


Belt radiator concepts: The belt radiator concept is a modification of the liquid droplet concept in which an ultrathin solid surface is coated with a very low vapor pressure liquid (see fig. 39). While the surface-to-volume ratio is not limited in the same fashion as for a cylindrical heat pipe, it does not quite match that of the liquid droplet radiator. However, this system avoids the problem of droplet capture by carrying the liquid along a continuous belt by surface tension. The liquid plays a double role in this system by acting not only as the radiator but also as the thermal contact which picks up the heat directly from a heat transfer drum. Variations on this scheme, in which the belt is replaced by a thin rotating disk, are also feasible but have yet to be fully assessed.

From THERMAL MANAGEMENT IN SPACE by Abe Hertzberg
Collected in Space Resources NASA SP-509 vol 2
MULTIPLE RADIATOR PANELS

You noted that having too many radiators distributed about an axis causes them to radiate into each other. It all boils down to what's known as a face factor, essentially how much of the radiation released by a surface is intercepted by another one. For two plates of equal length separated by an angle alpha (α), the face factor is:

F = 1 - sin(α/2)

So you can see right off the bat that for 2 radiators opposite each other, α = 180°, α/2 = 90° and the face factor is 0, no interception. But go up:

Multiple Radiator Panels
#Face
Factor
EmitEfficiency
10.0000000001.000000001.000000000
20.0000000002.000000001.000000000
30.1339745962.5980762110.866025404
40.2928932192.8284271250.707106781
50.4122147482.9389262610.587785252
60.5000000003.0000000000.500000000
70.5661162613.0371861740.433883739
80.6173165683.0614674590.382683432
90.6579798573.0781812900.342020143
100.6909830063.0901699440.309016994
110.7182674433.0990581250.281732557
120.7411809553.1058285410.258819045
  • # number of radiators spaced around the ship's long axis
  • Face Factor how much heat radiation from a radiator is wastefully intercepted by another radiator
  • Emit how much heat is effectively radiated by the total radiator array, in units of single radiator panels
  • Efficiency how efficient is this array at getting rid of heat, single panel = 1.00 or 100%

The second column is face factor, the third column is how much is emitted relative to a single surface, and the last is "efficiency", how much every individual panel is emitting relative to a single unlimited surface (efficiency is just 1 - F = sin(α/2), which is itself the face factor for the surface relative to its unobstructed surroundings). As you can see, it falls off very very quickly; the third radiator is only 60% as effective (goes up from 2.00 to 2.60), and the fourth adds to this only marginally (goes up from 2.60 to 2.80). Unless there really isn't room to simply stretch out the panels, it just doesn't seem worth it to pack more than 3 about an axis, and even 3 might be a stretch.

Neat thing is, using the face factor you can figure out the efficiency of radiators in weird geometries. My textbook has face factors for cylinders, enclosed spaces and plates of unequal size, if you so desire, which is to say I could tell you how much is going into your spaceship, or out an open dock.

Another neat side effect of face factors is that you can make a radiator more efficient per given mass by poking holes right through it, since the inner surface of the hole radiates at least partly out into its surroundings (the rest radiates back into itself, but that isn't really a problem). This reduces efficiency per unit area (though interestingly not by much for giant holes), and the panel is significantly weaker as a result (even more than you'd think, since the hole provides an area of stress concentration — it can reach multiples what it would normally there), but for very small holes that are very close together, you can get efficiencies per mass that are many times higher than they would be for a straight panel.

Take this arrangement: a square grid of side L, with a hole in the center of each square and one on each vertex, each hole being of radius 0.35L so that the holes at the vertices are nearly touching that in the middle. Say, for L varying from 0.1 to 1, 10 and 100 times thickness, relative mass efficiency goes up 4.342, 4.010, 1.828 and 1.094 times(by the way, because the relative size of the hole is the same, you need 4.660 times the area of panelling to get the same mass as a continuous radiator).

(ed note: the above was orginally erroneously writen as 1.199, 2.204, 4.184 and 4.609 times)

You can force even more holes into there if they're arranged hexagonally; take a hexagon of side L, with a hole at the center and one each vertex, you can reach a radius of up to 0.5L. Now, for L varying again from 0.1 to 1, 10 and 100 times the thickness, relative mass efficiency goes up 10.741, 9.610, 2.763 and 1.193 times over (in this case, you need 10.74 times the area of panelling to get the same mass as a continuous radiator)!

(ed note: the above was orginally erroneously writen as 1.486, 5.035, 9.816 and 10.644 times)

Given that "every gram counts", it's almost certainly worth the fragility. You could probably thicken the panels somewhat to make up for it and hit a sweetspot

From Zach Hajj (aka Zerraspace) (2015)
MULTIPLE RADIATOR PANELS 2

For one, I hoped to clarify some things in my former post, so I've provided images for the hole arrangements to make them easier to understand.

It turns out I made a little slip with the calculation — it's smaller holes that are close together that give higher efficiency, not large holes. It just means correcting a couple of sentences there — "very small" rather than "very large" holes, and for the figures, for a square grid: for L varying from 0.1 to 1, 10 and 100 times thickness, relative mass efficiency goes up 4.342, 4.010, 1.828 and 1.094 times, and for a hexagonal grid: for L varying from 0.1 to 1, 10 and 100 times thickness, relative mass efficiency goes up 10.741, 9.610, 2.763 and 1.193 times.

(ed note: I tried to make the changes specified in the above paragraph in the prior quote "Multiple Radiator Panels". They are marked in bold red letters.)


Now, for the new stuff. I’ve found out there’s a hard limit to the emission you can get by stuffing more panels about an axis. You see, mathematically, sine can be expressed by a Taylor series, so broken down:

E = n sin(a/2)
= n sin(2π/2n)
= n sin(π/n)
= n ((π/n) - (π/n)3/3! + (π/n)5/5! - (π/n)7/7!+…))
= π (1-(π/n)2/3! + (π/n)4/5! - (π/n)6/7!+…)

As n grows, π/n shrinks, till all terms but the first disappear, leaving us with π! You can think about this in geometric terms too — emission is basically the perimeter of the shape formed by the joining of the tips of each plate, so as n gets really huge, the shape transforms into a circle (in fact, this is how computers build circles, as n-gons with huge values of n). Instead of working to pump out every last drop possible within a confined space by throwing in essentially useless panels, you could replace them with one large cylindrical panel, giving you π emission, the maximum possible, but with 100% efficiency because there’s no self-interception! That being said, you’ll still lose out to two panels on opposite sides of the ship, seeing as those can get double the emission by radiating from both sides.

But don’t discount those extra panels yet. See, the formula I gave you makes one critical assumption — that the length of the plate so greatly outstrips width it might as well be infinite. This is not necessarily the case, so I decided to go and find a more expressive formula, here - http://www.thermalradiation.net/sectionc/C-16.html . It involves a very complex integral, so using a MATLAB code, I managed to solve it numerically, cross-referencing with the values given on the website to ensure accuracy. The results are presented in the attached Excel file; the values are only somewhat higher for triangular plates.

It seems that for small aspect ratios (width/length is less than ¼ or so), the value of the face factor is closely approximated by the simplistic formula I gave earlier. However, it falls off dramatically as the aspect ratio climbs, and for very large values (above 50 or so), the other panels barely present any obstruction at all!

The reason for this is that emission isn’t all perpendicular from the radiating surface — that’s simply the direction in which it is most intense. Some is emitted at an angle, so for huge aspect ratios, more can escape through the open sides. This is what makes belt radiators and heat pipes so effective, and lets us get any performance improvement out of poking holes in the panels. It also allows you to experiment with novel designs, as you can now have radiators that point right at each other without adjacent surfaces being rendered completely useless, so long as there’s some distance between them.

For one, you can further reduce face factor by swapping out flat panels. Cylindrical radiators fare spectacularly well: if my code is on the money, for length/radius greater than or equal to 5, face factor for up to 12 such tubes arranged radially is 2.45% or less, essentially negligible, and while I could not find a formula for cones, I can tell you performance will be intermediate between flat plats and cylinders. Attack Vector: Tactical’s radiator spikes would seem to be in the realm of possibility, though I wouldn’t vouch for their exact arrangement given in the illustrations (arranging spikes to the front and rear of one another blocks more and more of their surroundings, which kind of defeats the purpose of using them).

Even directly parallel panels can be worked with, as shown here: http://www.thermalradiation.net/sectionc/C-11.html

(Images for rectangular and circular panels are from Fundamentals of Mass and Heat Transfer, 6th Edition, by Frank Incropera, David Dewitt, Theodore Bergman and Adrienne Lavine; the last is from the online Catalog of Radiation Heat Transfer Configuration Factors, by John R. Howell at the University of Texas in Austin, available here: http://www.thermalradiation.net/tablecon.html )

Take two square panels of side X, separated by the same distance (X/L=1), then the face factor is only 0.2, not exactly murderous; you get nearly the same value if the panels are triangular (theta = 45, c/b = 1) or circular, with diameter equal to the separating distance (rj/L=0.5, L/ri = 2). If one dimension of a rectangular panel is much shorter than the distance between them, say a fifth of it or less (either X/L or Y/L < 0.2), then it doesn’t matter what the other dimension is, face factor is always below 0.1, and the same can be said for triangles. You could get some interesting geometries out of this — a homage to the TIE fighter, anyone?

Again, this analysis has to be taken with a grain of salt. None of these is a free lunch. You have to transport heat across that distance of paneling without temperature dropping off too much, or you’re only going to get the real heat disposal near the body of the craft, which just so happens to be where most of the interception takes place. This isn’t a problem when your panels are only a couple or even tens of meters across, but it does become a concern when they stretch kilometers away from the ship body. Moreover, they require more mass in the form of additional support and shadow shield coverage (if the ship needs such), and they present an even greater target for weapons’ fire. Whether or not this always pays off is a valid question.

From Zach Hajj (aka Zerraspace) (2015)

Open-Cycle Cooling

With certain kinds of rocket engines, you can cheat and avoid the need for heat radiators (and their ugly penalty weight). The dodge is called "open-cycle cooling", where the waste heat is carried away by the exhaust plume. In effect, the exhaust is their radiator, made out of rocket plasma instead of metal.

But it only works on certain kinds of engines.

Since the heat is carried by the rocket exhaust, you need plenty of exhaust. Which means each second of exhaust needs lots of propellant. Which means the engine needs a large propellant mass flow (called "" or "mdot"). This has consequences: raising the mdot will raise the thrust but drastically lower the exhaust velocity and specific impulse. Basically the engine will accelerate the spacecraft more quickly, but the gas mileage will fall into the toilet.

Some rocket engines (such as ion drives) have large exhaust velocities but low thrust. They cannot use open-cycle cooling because their ain't enough propellant in the exhaust plume to carry away all the heat.

Other engines (such as solid-core nuclear thermal rockets) have relatively low exhaust velocities but high thrust. They work splendidly with open-cycle cooling. So as a general rule, most NERVA type engines do not have heat radiators. If they do, this is because they are bimodal NTRs, and the radiator is only used when there is no rocket exhaust (when it is generating electricity instead of thrust).

The Glow

What color will the radiators glow? A practical one will only glow dull red. You can use the Blackbody Spectrum Viewer to see what temperature corresponds to what color. If it was glowing white hot, the temperature would be around 6000 Kelvin. This would be difficult for a solid radiator, since even diamond melts at 4300 degrees K.

Optimum Radiator Temperature

Here is some scary math about radiators from Dr. Tony Valle and Ray Robinson, along with some interesting conclusion. Remember that according to the radiator equation the hotter temperature the radiator is run at, the more waste heat it can dispose of.

RocketCat sez

Their "interesting conclusion" is that ya don't wanna design your heat radiator to run at 100% efficiency or the blasted thing will be huge, unwieldy, and bloated with penalty mass. Remember every gram counts!

For the sweet spot between maximum efficiency and minimum mass, design the radiator temperature to run between 3/5 and 3/4 of the power plant's hot end. But if you don't wanna take my word for it, feel free to dive into the following scary math.

It is surprising but there is an optimum temperature ratio at which to run a starship heat exchanger (or similar power source) to achieve maximum free power with a minimum of radiator area. The only assumptions necesary are that the power source obeys the laws of thermodynamics and that the starship may only get rid of waste heat by radiating.

Let us assume that we have a heat engine as a power source with a relative efficiency of η, and an absolute efficiency is η times the Carnot efficiency ε. We can write the available free power, F, as:

F = Qηε = Qη(1 - T1/T2)

where Q is the rate of heat flow into the exchanger and T1 and T2 are the temperatures of the cold and hot sides of the engine, respectively. The waste heat, H, released into the starship is Q - F, or:

H = Q(1 - η + ηT1/T2)
H = F (1 - η + ηT1/T2)/η(1 - T1/T2)

To simplify, we will measure temperature in units of T2 and let T1 be called just T. After dividing through by η the amount of waste heat associated with a given free power F is then:

H = F (η-1 - 1 + T) / (1 -T)

Now this waste heat must be radiated away from the ship. The power radiated by a black body at temperature T and with area A is given by the Stephan-Boltzmann Law:

P = AσT4

with σ a constant depending on the choice of units. Setting these equal to each other gives:

A = F (η-1 - 1 + T) / σ(T4 - T5)

Now we can ask what value of T will give the minimum radiator area. Taking the derivative of A with respect to T and setting it equal to zero gives:

(T4 - T5) - (4T3 - 5T4)(η-1 - 1 + T) = 0

Or, dividing by T3 and expanding:

T - T2 - 4η-1 - 4 + 4T + 5Tη-1 - 5T + 5T2 = 0

After collecting terms, we have:

4T2 + (5η-1 - 8)T + 4(η-1 - 1) = 0

or, dividing through by 4:

T2 + (5/4η-1 - 2)T + (η-1 - 1) = 0

We write η-1 as γ then the solution to the above quadratic can be written:

T = 1 - 5/8γ + 1/8 sqrt(25γ2 - 16γ)

In the special case where the exchanger runs at maximum theoretical efficiency, η = γ = 1 and the equation above gives T = 3/4. This means that the cold side of the heat engine is at 75% of the temperature of the hot side.

This is horribly inefficient as a Carnot heat engine goes, but if the radiator temperature drops, the surface area (and thus mass) must increase, because of the T4 behavior of the radiation law — colder radiators dump heat much less efficiently. This function is fairly flat — as η drops from 1 to a more plausible 0.1, T changes from 3/4 to 4/5

Dr. Tony Valle and Ray Robinson's commentary in Attack Vector: Tactical, "Science Behind The Rules: Power and Heat Generation" (2004)

Take a classic space opera warship. Onboard power is generated by one or more fusion reactors. If the overall power is 2 gigawatts, and the efficiency is 90% (a pretty generous estimate, since projections I've seen for MHD power generation are around 60%) then at full power, the reactors create 200MW of waste heat. At these sorts of power levels the waste heat of the crew, computers, coffee makers, etc. can be ignored. If there are energy weapons, assume they too are 90% efficient and use 500MW of power when fired, generating another 50MW waste heat. Lump in a lot of other minor systems and you get something like 300MW total waste heat that has to be gotten rid of at peak.

Where it gets complex, AFAIK, is the question of how hot you can allow the ship's interior to get. Let's assume that the environmental areas have heat pumps that allow them to stay a fair amount cooler than the engineering areas (since they're not generating the majority of the heat to begin with) and there are no low-temperature superconductors and so forth to worry about. If the engines and weapons can operate happily at 150 degrees Celsius, that's 423 degrees Kelvin. So that's our starting point — the coolant (probably liquid sodium or lithium at that temp) gets that hot before it's pumped through the radiators to cool off again, at which point:

Heat lost [watts] = area [m2] * emissivity * 5.67e-8[Stefan-Boltzmann constant] * T4 [degrees Kelvin].

(ed note: Stefan-Boltzmann law again)

If the radiators are perfectly black (emissivity of 1), and the coolant temperature is 423 degrees K, then in order to radiate away 275MW of heat, the radiator needs be about 150,000 square meters in area (of course it's double-sided, so the actual fin(s) only need to be 75,000 m2). That's a square 275 meters on a side, or roughly a large city block, simply to deal with the ship's own waste heat at full power. If the ship needs to radiate away additional heat due to taking in, say, 400MW of energy from an enemy ship's lasers, you'd probably have to double or triple that figure (and make darn sure to keep your fins edge-on to the enemy ship firing at you! :). Of course all this is very crude and assumes perfect efficiency of a number of things (some of which I'm probably unaware of :). In reality you might get 80% of that theoretical performance. Or perhaps less. And the first thing damaged in a battle would probably be the radiators (big, hard to protect).

The structural mass of a large radiator fin could be a substantial fraction of the entire ship's mass, and that slows down the acceleration of the ship, which needs more power for thrust, which gives off more waste heat, and so on... So the idea of using spray wands and droplet coolants is attractive.

OTOH, if you need to keep the whole ship at a comfy temperature like 20C, then it's almost hopeless. The radiating area required is so enormous that high acceleration isn't practical at all (something like half a million m2).

Another alternative is to design the ship to only radiate away normal, routine power levels, and to boil off propellant to deal with peak loads. But that goes through a lot of propellant pretty fast at high power levels. Dreadnaughts become like modern jet fighters — only good for a few minutes of intense combat before the fuel runs out. Once it's gone, you can't crash, but you have to surrender or be boiled...

(ed note: He is assuming that the radiator temperature is 423K. Some of the other estimates were for radiator temperatures of 1600K to 3000K, which would drastically lower the radiator surface area to 800 m2, if double sided it would be about 20 meters square.)

Radiator Types

Use the "Life Support" radiator data for life support and other low-waste-heat management. Use all the others for high-waste-heat management, such as fission/fusion reactors and weapons-grade lasers.


In each radiators Specific Area data table will be listed Heat Cap., Mass, and Op. Temp.

Heat Cap.: heat capacity in kWth/m2. This is how many kilowatts of waste heat each square meter of radiator can get rid of. Multiply the surface area of the entire radiator by the heat capacity to find the total amount of heat the radiator array can handle. kWth means "kilowatts of thermal energy" (i.e., waste heat) as opposed to kWe which means "kilowatts of electricity".

Mass: specific area mass of the radiator in kg/m2. This is the mass of each square meter of radiator in kilograms. Multiply the surface area of the entire radiator by the specific area mass to find the total mass of the radiator array.

Op. Temp.: the operating temperature of the radiator. You probably won't need this unless you want to fool around with the Stefan-Boltzmann equation. The higher the operating temperature, the higher the heat capacity. Which means the value listed for the heat capacity is only valid if the radiator operates at this temperature.


Use the "Specific Area" values in the tables to calculate the radiator mass.

  1. Decide how many kilowatts of waste heat the radiator will have to handle (from the engine, the power reactor, the laser cannon, etc.)
  2. Select which radiator type to use, and examine its Specific Area table.
  3. Divide the total waste head in kilowatts by the Heat Cap. entry of the table to get the square meters of radiator area required.
  4. Multiply the radiator area by the Mass entry to get the total mass of the radiator required.

or in other words:

radiatorMass = (wasteHeat / specificAreaHeat) * specificAreaMass

where:

  • radiatorMass = mass of radiator array (kg)
  • wasteHeat = amount of waste heat to dispose of (kWth)
  • specificAreaHeat = Heat Cap. from radiator table (kWth/m2)
  • specificAreaMass = Mass from radiator table (kg/m2)
Example

An Attack Vector: Tactical Medium Range Laser 2 has an input energy of 2 gigawatts (GW) and an efficiency of 12.5%. This means that 0.25 gigawatts become laser beam and 1.75 gigawatts turn into waste heat. About par for the course for lasers.

So the laser needs a radiator array that can deal with 1.75 gigawatts of thermal energy = 1,750,000 kWth. This is Step 1.

Looking at the list, by far the radiator with the largest heat capacity is the Molybdenum/Lithium Heat pipe: 469 kWth/m2. This is Step 2.

Divide the waste heat of the laser by the heat capacity of the Mo/Li Heat pipe and we get 1,750,000 / 469 = 3,731 square meters of radiator. This is Step 3.

The Mo/Li Heat pipe specific area mass is 150 kg/m2. Muliply the radiator area by the Mass entry and we get 3,731 * 150 = 559,650 kilograms = about 560 metric tons.

Which sounds like a lot, except if you used an ETHER Charged Dust radiator you'd need about 580 metric tons of radiator.

Note that Step 3 calculates the radiation surface of the radiator. If the radiator is layered flat on the ship's hull, the radiation surface is the same as the physical radiator size. However, if the radiator is attached edge on so it extends out as a fin or a wing, the physical radiator size will be one-half the radiation surface. This is because you can use both sides of the physical fin as radiator surface. Yes, even a liquid droplet radiator. This might not apply for some of the stranger radiator designs, but details are scarce.

Having said that, things are complicated for liquid drop radiators. The radiation surface is the surface area of the droplets. Figuring out the physical radiator size is compilcated, you can find the equations here. There is also Eric Rozier's online calculator.


Note, in the illustrations from the High Frontier game, it uses very strange game-specific terms. Each "mass unit" is equal to 40 tonnes, each thermometer is one "therm" and represents the radiator dealing with 120 megawatts of thermal waste heat (120,000 kWth). When a specific area value was missing I uesd the therm, mass points, and radiator area on the cards to calculate.


Here is a table of the various radiator types. Their area and mass has been calculated as if they were sized to handle 250 megawatts of waste heat.

The table is sorted by array mass, so the better ones are at the top. At least if you want the lowest mass radiator. If the radiation area was an issue you'd probably prefer a Mo/Li Heat Pipe instead.

The life support radiator was included even though it was not intended to handle waste heat over 100 kilowatts or so.

Radiator for 250,000 kilowatts waste heat
RadiatorSpecific area
heat
(Heat Cap.)
Specific area
mass
(Mass)
Radiation
area
Array
mass
Marangoni Flow293.04 kWth/m224.4 kg/m2853 m220,816 kg
Electrostatic Membrane51.3 kWth/m24.275 kg/m24,873 m220,833 kg
Hula-Hoop300 kWth/m233 kg/m2833 m227,500 kg
Buckytube Filament293.03 kWth/m248.839 kg/m2853 m241,667 kg
Curie Point212.75 kWth/m235.459 kg/m21,175 m241,667 kg
Tin Droplet38.49 kWth/m26.4154 kg/m26,495 m241,669 kg
Flux-Pinned Superthermal76 kWth/m217 kg/m23,289 m255,921 kg
Attack Vector: Tactical357 kWth/m2100 kg/m2700 m270,028 kg
Bubble Membrane21.01 kWth/m27.00 kg/m211,899 m283,294 kg
Mo/Li Heat Pipe453.54 kWth/m2151.18 kg/m2551 m283,333 kg
Microtube Array102.6 kWth/m234.2 kg/m22,437 m283,333 kg
ETHER212.75 kWth/m270.92 kg/m21,175 m283,337 kg
Ti/K Heat Pipe150.22 kWth/m2100.14 kg/m21,664 m2166,656 kg
SS/NaK Pumped90.83 kWth/m260.554 kg/m22,752 m2166,669 kg
Salt-Cooled Reflux tube75 kWth/m275 kg/m23,333 m2250,000 kg
Life Support0.19 kWth/m23.1 kg/m21,315,789 m24,078,947 kg
ALL THE RADIATORS

      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 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


     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.


     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. 


     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.  


     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.


     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.


     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.


     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.


     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.


     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.


     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. 


     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.


     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. 


     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.


     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.


     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:


     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. 


     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.


     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. 


     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. 

From TOUGH SF: ALL THE RADIATORS by Matter Beam (2017)

Life Support

Specific Area
Heat Cap.~0.19 kWth/m2
Mass~3.1 kg/m2
Op. Temp.? K

Technically you also need radiators to keep the life-system habitable. Human bodies produce an amazing amount of heat. Even so, the life-system radiator should be small enough to be placed over part of the hull, since life-support waste heat is quite tiny compared to nuclear reactor or gigawatt laser waste heat.

Use this radiator type for life-support and other modest waste heat management. Use the other radiators for gigantic waste-heat producers.

The life-system radiators on the Space Shuttle are inside the cargo bay doors, which is why the doors are always open while the shuttle is in space.

Troy Campbell pointed me at a fascinating NASA report about spacecraft design. In the sample design given in the report, the spacecraft habitat module carried six crew members, and needed life-system heat radiators capable of collecting and rejecting 15 kilowatts of heat (15 kW is the power consumption for all the systems included in the example habitat module). The radiator was one-sided (basically layered over the hull). It required a radiating surface area of 78 m2, had a mass of 243.8 kg, and a volume of 1.742 m3. It used 34.4 kg of propylene glycol/water coolant as a working fluid. In addition to the radiator proper, there was the internal and external plumbing. The Internal Temperature Control System (coldplates, heat exchangers, and plumbing located inside the habitat module) had a mass of 111 kg and a volume of 0.158 m3. The External Temperature Control System had a mass of 131 kg, a volume of 0.129 m3, and consumes 1.109 kilowatts.

What this boils down to is that the described system needs about 96 kilograms and 0.405 cubic meters of temperature regulating equipment per crew person. That's the total of the external radiator on the hull and the internal temperature control system.

Simple math tells me the radiator has a density of about 140 kg/m3, a specific area of 3.1 kg/m2 and needs a radiating surface area of about 5.2 m2 per per kilowatt of heat handled (1/5.2 = 0.19 kWth/m2). The entire system requires about 35 kg per kilowatt of heat handled, and 0.13 m3 per kilowatt of heat. But treat these numbers with suspicion, I am making the assumption that these things scale linearly.

Liquid Droplet

Liquid Droplet Radiators use sprays of hot droplets instead of tubes filled with hot liquid in the radiator. This drastically reduces the mass of the radiator, which is always a good thing. A NASA report suggested that for 200 kW worth of waste heat you'd need a 3,500 kg heat pipe radiator, but you could manage the same thermal load with a smaller 500 kg liquid droplet radiator.

The droplet generator typically has 100,000 to 1,000,000 orifices with diameters of 50 to 20 μm. They are a bit more susceptible to damage than the components of more conventional radiators.

A drawback is that the spray is in free fall. This means if the radiator is operating and the ship starts accelerating, the spray will start missing the collector and precious radiator working fluid will be lost into space. Brookhaven National Laboratory has patented a way to magnetically focus the droplet stream. Using a large radiator it will allow the spacecraft to maneuver at acceleration of up to 0.001 g (0.00981 m/s2) which is barely an improvement. The acceleration can be increased but only if the single radiator is replaced by numerous smaller radiators. Which of course makes the sum of the radiators have a larger mass than the single large radiator. Oh, and Brookhaven's patent expired in 1994.

Late breaking news, the Curie point type of liquid drop radiator is relatively immune to ship acceleration.

Many liquid droplet designs are well suited for warships, since they do not utilze large fragile panels vulnerable to hostile weapons fire. If a rail gun round or laser bolt passes through a spray of working fluid, it will just make a bit of fluid miss the collector. If weapons fire passes through a conventional panel it will wreck it.

temperature rangecoolant typeexample
250 K – 350 Ksilicone oils
siloxane
Trimethyl-Pentaphenyl-Trisiloxane
370 K – 650 Kliquid metal eutectics
500 K – 1000 Kliquid tin
TOWARDS ACCOUNTING FOR OCCLUSION IN DROPLET RADIATORS

Recently, +Matter Beam made an attempt to figure out how to compute the effect of droplet inter-reflection and -absorption in droplet radiators, concluding that the effect was generally small, especially for planar configurations. I was intrigued by the problem, and set out to solve it more accurately.

First, this is a difficult problem, and I had to completely start over at least twice while solving it, because of multiple screw-ups. But once you discover it, the correct math describing it is actually fairly simple. If you have trouble with the subscripts, Chrome on desktop seems to work okay.


Derivation and Listing of Formulae

There are two key effects here: inter-absorption and inter-reflection. In the first, light is absorbed, converted to heat, and re-radiated as thermal radiation. In the second, the light just reflects directly. Both are crucial, but the problem is complicated by the fact that when absorption happens, the energy gets routed through the Stefan–Boltzmann Law, which introduces a fourth power of temperature into an otherwise-simple geometric sum.

Happily, the problem has some neat symmetries, which we can exploit to work around the summation issue altogether.

First, we relate the thermal power coming out of a droplet (Φo, "o" for "out") to a radiometric quantity called "radiance" (Lo). Power is equal to integrating radiance over the hemisphere and then the sphere. You then solve that relation for radiance. I'll spare you the details and give you the answer ("r" is droplet radius):

	   Φo
Lo = ——————
	 4 π2 r4

By definition, this must be equal to the sum of emitted light (Le, "e" for "emitted") and reflected light (Lr, "r" for "reflected") from other droplets:

Lo = Le + Lr

The emitted light comes from the Stefan–Boltzmann Law (ε is the absorptivity/emissivity of the material, σ is a constant, "T" is temperature in degrees Kelvin). Note we're re-using the relationship between power Φ and radiance L given in the first equation (which holds for any coupled values of power and radiance on a sphere):

Φe = ε σ T4

	 ε σ T4
Le = ——————
	 4 π2 r4

Now, the reflected light is just the portion of the incoming light (Li) that is reflected:

Lr = (1-ε) Li

The key insight is that, while our droplet might radiate into another droplet, that other droplet is radiating back. Because every droplet is "average", both droplets have the same temperature, radiance, etc. In particular, the incoming radiance from an occluding droplet is the same as the outgoing radiance our droplet is sending back.

Call the fraction of occluded directions "f". In "f" of the directions, our droplet is occluded by another droplet emitting Lo. In "(1-f)" of the directions, we see "0" (assuming the radiance of space is zero, but you could calculate a tiny value from the CMBR if you want). Therefore, the incoming radiance to our droplet is on average:

Li = Lo*f + 0*(1-f)

Substituting and using the relation between radiance and power on a sphere (first equation):

Lo = Le + (1-ε) Lo f

	Φo      ε σ T4              Φo   
————————— = ———————— + (1-ε) ———————— f
 4 π2 r4    4 π2 r4          4 π2 r4

Φo - (1-ε) f Φo = ε σ T4

		ε σ T4
Φo = ————————————
	 1 - (1-ε) f

The net power (Φn, "n" for "net") of the droplet can now be computed (remember that "Li = Lo f", as above):

Φn = Φi - Φo
   = Φo f - Φo
   = Φo (f - 1)
		ε σ T4
   = ———————————— (f - 1)
	 1 - (1-ε) f

Thermal energy (J) is related to temperature by the specific heat capacity (c) of the droplet's material. This is considered to be constant (usually accurate as long as no phase change occurs). The relation is simply:

J = c T

We can combine this with the previous formula, since dJn/dt=Φn and integrate to get functions of time. Thus, we get the energy (or temperature) per time (and obviously we now need J(0) or T(0), the initial energy or temperature of each droplet):

							 1
		 ___________________________________________
J(t) =     _______________________________________
		  /  3 ε σ       1 - f              1
		3/   —————— * ———————————— * t + ————————
		√      c4     1 - (1-ε) f        (J(0))3

		
							 1
		 _____________________________________________
T(t) =     ________________________________________
		  /  3 ε σ       1 - f                1
		3/   —————— * ———————————— * t + ———————————
		√      c7     1 - (1-ε) f        (c T(0))3

Differentiating either of these will give you the power Φn as a function of time. The differentiation is trivial, so I'm not going to write it.

We can also compute the relative efficiency by dividing the equation for Φn when "f>0" by the same equation when "f=0" (the "ideal" case). The result is:

				 1 - f
Efficiency = ————————————
			 1 - (1-ε) f

Hopefully if I left anything interesting out, you should be able to derive it from one of the given formulae. The "f" factor (again, the fraction of occluded directions) needs to be user-provided, probably as the result of a simulation since I don't have any great ideas on how to compute a reasonable value in closed-form.


Some Examples

By my complete estimate, < 1% occlusion and emissivity 0.4–0.8 is the most plausible configuration. I'm not even sure 100% is possible without ridiculously packed droplets (and it's certainly not possible at all in planar arrangements).

0.001% occlusion, emissivity 0.8 => efficiency ~100.00%
0.01% occlusion, emissivity 0.8 => efficiency ~99.99%
0.1% occlusion, emissivity 0.8 => efficiency ~99.92%
1% occlusion, emissivity 0.8 => efficiency ~99.20%
10% occlusion, emissivity 0.8 => efficiency ~91.84%
50% occlusion, emissivity 0.8 => efficiency ~55.56%
90% occlusion, emissivity 0.8 => efficiency ~12.20%
100% occlusion, emissivity 0.8 => efficiency ~0.00%

0.001% occlusion, emissivity 0.4 => efficiency ~100.00%
0.01% occlusion, emissivity 0.4 => efficiency ~100.00%
0.1% occlusion, emissivity 0.4 => efficiency ~99.60%
1% occlusion, emissivity 0.4 => efficiency ~99.60%
10% occlusion, emissivity 0.4 => efficiency ~95.74%
50% occlusion, emissivity 0.4 => efficiency ~71.43%
90% occlusion, emissivity 0.4 => efficiency ~21.74%
100% occlusion, emissivity 0.4 => efficiency ~0.00%

Some Limitations

The main assumptions used in this analysis are that:

1: All droplets are Lambertian emitters (which means their radiance ("brightness") is the same in every direction). Most light sources and thermal emitters are nearly Lambertian.

2: All droplets are Lambertian reflectors (which means they reflect radiance equally in all directions). This is wrong, especially for metals. Using a more physically plausible BRDF would probably reduce inter-reflection effects even further in planar configurations (that is, decrease the attenuating effects considered here even more).

3: "f" is homogeneous. This is essentially true for the interior region of a radiator. Droplets near the edges have a lower "f" factor, so they radiate more efficiently (again, this decreases the attenuating effects).

Overall, these assumptions are extremely reasonable, and the only way to relax them further would be to do a full numerical simulation for the particular configuration of your radiator. The "right" way, in some sense, is to just simulate it with a path tracer. Unfortunately, I am currently rewriting my path tracer, so I can't show you what this would look like.

Rectangular LDR

Rectangular LDRs have collectors the same width at the droplet generator. The droplet density remains constant across the flight path. It is a simpler more robust design than a Triangular LDR, and has a larger radiating surface (twice the surface area).

However the triangular LDR is lighter (40% less massive) due to its smaller collector. As previously mentioned, the rectangular LDR's collector is a long bar the exact same width as the droplet generator bar. By way of contrast the triangular LDR's collector is a small bucket, which has about 40% less mass than a corresponding rectangular LDR collector. But it has drawbacks.

Triangular LDR

Triangular LDRs have a tiny collector a fraction of the width of the droplet generator, unlike rectangular LDRs. The droplet density increases across the flight path. It is 40% less massive compared to a comparable Rectangular LDR due to the smaller collector, reduced mass is always a plus.

However it is a more complicated design with more failure points, and it has only half the surface area of a same sized Rectangular LDR. Because the radiating surface is a triangle instead of a rectangle.

For reasons that have not been made clear to me, Triangular LDR is currently the focus of much of the research and development. NASA likes them better than Rectangular LDRs. I guess in NASA's eyes lower mass trumps all other considerations.

Eric Rozier has an online calculator for droplet radiators here, and for coolant systems in general here. He had this analysis:

Given that the main thing we want to determine is the surface area of the lithium droplets to calculate the heat it can radiate, I decided to build a model of the surface area.

Since no such radiator has been built we have to work with some plausible model data. To model the lithium drops themselves I dug into some meteorological data and found that raindrops typically range in size from 1mm to 3mm, sounds pretty reasonable. Assuming droplets are spherical (a reasonable assumption in zero gravity) then the surface area of any given droplet is of course 4*π*r2.

Working off the wedge based idea you cited here. We then model the full radiating body of the droplets as a triangle, reducing the emitter to a point source for simplification. I'm not sure how space out the droplets should be, but I figure if the distance between any two droplets is roughly twice the radius, the model is probably pretty conservative. Thus for an emitter with distance h from the emitter to the collector, and a collector plate of length h, we get the number of droplets suspended between them to be:

(0.5 * b * h)/(16r2)

We can then model the surface area of the lithium droplets as:

(0.5 * b * h)/(16r2) * 4*π*r2

If you want to modify the spacing of the drops, you can change the inter-droplet gap to q instead of r, rendering the following equation:

(0.5 * b * h)/(4r2 + 4r*q + q2) * 4*π*r2

Eric Rozier

So the equations are:

a = (0.5*b*h) / (16*r2) * 4*π*r2

a = (0.5*b*h) / (4*r2 + 4*r*q + q2) * 4*π*r2

where:

  • a = surface area of lithium droplets in radiator surface
  • b = length of base of radiator triangle
  • h = length of height of radiator triangle
  • r = radius of indiviual droplet
  • q = inter-droplet gap

Liquid-droplet radiators are also a possibility. There do exist liquids which have extremely low vapor pressure at high temperatures — certain organics up to ~600K, liquid metals (esp. lithium) to ~1500K. Using a carefully-designed nozzle to create a fan-shaped spray of fine droplets towards a linear collector results in a very efficient radiator, with minimal weight per unit radiating surface, high temperature, and high throughput.

The radiator would be essentially triangular when "deployed", with the spray nozzle at one vertex and the collector along the opposite side. If the nozzle-vertes is adjacent to the ship body, the collector "arm" will have to extend outwards. Alternately, the collector can be run along the side of the ship, and the spray nozzle extended on a boom and aimed inwards. A series of closely-spaced, narrow-angle nozzles would approximate a rectangular array.

There is always some loss of coolant due to evaporation in vacuum, hence use of liquids with extremely low vapor pressure. You also lose coolant if such a radiator is run under acceleration, unless the collector is over-long and aligned parallel to the thrust axis, which imposes a constraint on system geometry. You also lose coolant if the radiator "panel" is hit by enemy weapons fire; on the other hand there is no mechanical damage unless the much smaller nozzle or collector arms are hit. Bottom line — you'll need a small surplus of coolant, unless you are running a warship, in which case you'll need a large surplus.

If liquid metal is used as the coolant, MHD pumping can be used at the collector arms, resulting in a simplified design with no moving parts. Indeed, in such a case the coolant could also be used as the working fluid in an MHD generator, resulting in a single-fluid, single-cycle power system from primary energy generation to waste heat radiation. Again, a simple, efficient design with no moving parts.

Specific Area
Heat Cap.~38 kWth/m2
Mass6.4 kg/m2
Op. Temp.1030 K
Emissivity0.96

Tin droplet radiator

Atomization increases the surface area with which a fluid can lose heat. A hot working fluid sprayed into space as fine streams of sub-millimeter drops readily loses heat by radiation. The cooled droplets are recaptured and recycled back into the heat exchanger. If tin (Sn) is used as a working fluid, the kilos per power radiated is minimized, using a heat rejection temperature of 1030 K and a total power in the megawatt range (comparable to the game value of heat rejection of 120 MWth per therm). The low emissivity of liquid tin (0.043) is increased by mixing in carbon black, which distributes itself on the surface of the droplet. Evaporation losses are avoided by enclosing the radiator in a 1 μm plastic film, which transmits radiation in the 2 to 20 μm (IR) range. Such a film would continue to perform its function even if repeatedly punctured by micrometeoroids. The illustration shows a triangular liquid droplet geometry. The collector, located at the convergence point of the droplet sheet, employs centrifugal force to capture the droplets. The total specific area is 6.4 kg/m2.

K. Alan White, "Liquid Dropbt Radiator Devebpment Status," Lewis Research Center, 1987

From High Frontier by Philip Eklund

Curie Point LDR

There are certain magnetic materials that abruptly loose their magnetism if heated above a certain point. This is called the Curie Point for that material.

A team of scientists led by Mario D. Carelli used this property to create a species of liquid drop radiator.

Droplets or particles of magnetic material are used in the droplet radiator, heated above their Curie point so they lose their magnetism. They are sprayed in a stream, spreading out to maximize heat radiation. At some point they radiate enough heat to drop below the curie point and abruptly become magnetic. Whereupon they change course and make a bee-line for the magnetized droplet collector.


Just like the liquid droplet radiator, this design is relatively immune to meteor strikes and hostile weapons fire. Beyond NERVA points out another advantage. The Curie point radiator is somewhat immune to any maneuvering the spacecraft performs, which the LDR is not. The LDR drops move in a straight line to the collector, acceleration moves the position of the collector so the drops miss. But with the Curie point, the collector is magnetized, so the drops will automatically correct their course to enter the collector.

CURIE POINT RADIATOR 1

INTRODUCTION

     The Curie Point Radiator (CPR) is an advanced radiator concept that offers unique advantages in terms of possible improvements in space power system efficiencies, reduced radiator mass requirements and increased reliability for multimegawatt space power applications. The operating principle of the CPR is to transfer waste heat to small size ferromagnetic particles which are heated above their Curie point and released into space. As the particles lose heat by radiation, their temperature drops below the Curie point and thus they become ferromagnetic again allowing recapture by a magnetic field; the particles are then collected and cycled again through the heat exchanger.

     A previous paper (Carelli et al. 1986) has reported the results of an assessment of the design and development issues and of an analysis of particles characteristics and performance. This paper describes the design of two systems for different power applications.

SYSTEM DESIGNS

     Figure 1 provides a pictorial representation of the "self-contained" unit. The cold ferromagnetic particles are collected inside a central cone from which they are drawn into radial passages by the magnetic force exerted by the coil. The passages, which are of equal width slightly larger than 2 particle diameters, constitute the heating channels, where the particles receive heat from the channel walls by conduction and radiation. The particles move in an accelerated motion along the channel, since the magnetic force progressively Increases as the particle approaches the coil. When the temperature of the particles reaches and exceeds the Curie point, their motion becomes inertial and they proceed with essentially uniform velocity to the exit of the heat exchanger, which is contoured to impart the desired angular velocity component to the particles. Once the paramagnetic particles are released into space, they lose heat and as they become ferromagnetic again they are drawn back to the cone and the process Is repeated. As shown by Figure 1, two heat exchanger/collector/ejector units share the same magnet to reduce mass.

     The major advantage of the self-contained unit is Its compactness. The diameter of the heat exchanger and magnet is 2 m and the maximum height of the particles cloud Is approximately 1 m on both sides. The total heat rejection of the unit is in the 0.8-0.9 MW range and the specific mass is of the order of 0.4 kg/kW. A smaller unit, with a structures diameter of 1 m has a heat rejection capability of 0.5-0.6 MW and a specific mass of about 0.5 kg/kW. Several of these units in parallel provide SP-100 type heat rejection requirements, have a specific mass competitive with the best state-of-the-art radiator and offer minimal size, thus minimal armoring 1s required.

     For the much larger heat rejection requirements of multimegawatt applications, the self-contained unit concept Is not applicable, because the heat rejection capability Is limited by the relatively small size of the particles cloud. Increasing the cloud would require a significant increase In the magnet mass. Thus, a tandem unit was developed (see Figure 2.3) where four parallel clouds flow two each In opposite directions between two identical halves, which have the heat exchanger/collector/ejector functions. A magnet is located at each half and again can be shared by other units located back-to-back. The conceptual design utilizes state-of-the-art materials (e.g. the structures are of Nb-IZr). The heat exchanger provides a way of individually heating multiple sheets of particles as they pass through each curved leaf of the unit. A single tandem unit, with a separation distance of 20 m between the two halves has a heat rejection capability of the order of 8 MW with a specific mass slightly less than 0.2 kg/kW, which is a great Improvement over current radiators. The design also Includes special features for containing the particles securely during launch and deployment. A unique method to provide repeated startup/shutdown capability is featured.

     A detailed discussion of the design, analytical and experimental investigations supporting the theoretical feasibility of the CPR will be reported in the full length manuscript.

CONCLUSIONS

     The conceptual design and supporting theoretical analyses have confirmed that the CPR has: potential for highly reliable heat removal capability; flexibility and adaptability to a variety of operating conditions; extreme compactness; good maneuvering capability; no loss of particles Inventory; repeated startup/shutdown capability; and, most important, very low mass requirements.

ACKNOWLEDGEMENTS

     This work is being performed at the Westinghouse Advanced Energy Systems Division under DOE Contract DE-AC03-85SF15936.

REFERENCE

     M. D. Carelli, et al., "A Novel Heat Rejection System for Space Applications: The Curie Point Radiator," 21st lECEC, Vol. 3, pp 1875-1880, August 1986.

From THE CURIE POINT RADIATOR by Mario D. Carelli. Robert A. Markley. Bill L. Pierce and J. Ed Schmidt (1988)
CURIE POINT RADIATOR 2

The Curie Point Radiator (CPR), as shown in Figure 2.3, is similar to the LDR and transfers heat to a particle cloud which radiates directly to space as it passes exterior to the radiator's support and structural elements (Carelli 1989). The operating principle of the CPR is to transfer heat to small ferromagnetic particles which are heated above their Curie point and released into space. As the particles lose heat by radiation, their temperature drops below the Curie point, become ferromagnetic again, are recaptured by the magnetic field, collected and recycled through the heat exchanger.

The CPR offers several advantages as a space-based heat rejection system, including:

  • a particle cloud that is largely resistant to micrometeoroid degradation (or hostile weapons fire)
  • the system mass is relatively independent of temperature
  • the particle inventory losses can be readily controlled and nearly eliminated
  • the CPR is maneuverable
  • the use of solid particles eliminates vaporization losses
  • the CPR is completely passive
  • startup and shutdown is intrinsically simple

The CPR is able to provide a low system mass—in the 0.2 to 0.5 kg/MW range—and the flexibility to be used for both large and small thermal management systems.

CURIE POINT RADIATOR 3
Specific Area
Heat Cap.~300 kWth/m2
Mass35 kg/m2
Op. Temp.1200 K

Curie point radiator

A ferromagnetic material heated above its Curie point loses its magnetism. If molten droplets of such a substance are slung into space, they radiate heat and solidify. Once below their Curie temperature, they regain their magnetic properties and can be shepherded by a magnetic field into a collector and returned to the heat exchanger. A 120 MW system operating at 1200 K includes a 13 tonne magnetic heat exchanger and a rotating dust recovery electromagnet on a 25-meter boom, plus 7 tonnes of dust spread in a spiraling disk 27-meters in diameter (35 kg/m2). The usual medium is iron dust, which has a Curie point of 1043 K and is easily scavenged by magnetic beneficiation from regolith.

M.D. Carelli, 1989

From High Frontier by Philip Eklund

Enclosed Disk LDR

Specific Area
Heat Cap.51 kWth/m2
Mass4.3 kg/m2
Op. Temp.1000 K
Emissivity0.85

Electrostatic membrane radiator

This heat-rejection concept, also called a liquid-sheet radiator, encloses radiating liquid within a transparent envelope. It consists of a spinning membrane disk inflated by low gas pressure, with electrostatically-driven coolant circulating on its interior surfaces. The liquid coolant is only 300 μm thick and has an optical emissivity of 0.85 at a temperature of 1000 K. An electric field is used to lower the pressure under the film of coolant, so that leakage through a puncture in the membrane wall is avoided. The membrane has a specific area of 4.3 kg/m2 and 51 kWth/m2.

Shlomo Pfeiffer of Grumman, 1989

From High Frontier by Philip Eklund

ETHER

Specific Area
Heat Cap.213 kWth/m2
Mass71 kg/m2
Op. Temp.1200 K

ETHER charged dust radiator

To avoid the evaporation losses suffered by radiators that use liquid droplets in space, dust radiators use solid dust particles instead. If the particles are electrostatically charged, as in an electrostatic thermal radiator (ETHER), they are confined by the field lines between a charged generator and its collector. If the spacecraft is charged opposite to the charge on the particles, they execute an elliptical orbit, radiating at 1200 K with a specific area of 71 kg/m2 and 213 kWth/m2. The dust particles are charged to 10-14 coulombs to inhibit neutralization from the solar wind.

Prenger 1982

From High Frontier by Philip Eklund

Heat Pipe

Mo/Li Heat Pipe

Specific Area
Heat Cap.~469 kWth/m2
Mass150 kg/m2
Op. Temp.1450 K

Mo/Li heat pipe radiator

A heat pipe quickly transfers heat from one point to another. Inside the sealed pipe, at the hot interface a two-phase working fluid turns to vapor and the gas naturally flows and condenses on the cold interface. The liquid is moved by capillary action through a wick back to the hot interface to evaporate again and repeat the cycle. For high temperature applications, the working fluid is often lithium, the soft silver-white element that is the lightest known metal. Molybdenum heat pipes containing lithium can operate at the white-hot temperatures of 1450 K, and transfer heat energy at 240,000 kWth/m2, almost four times that of the surface of the sun. The specific area is 150 kg/m2.

David Poston, Institute for Space and Nuclear Power Studies at the University of New Mexico, 2000

From High Frontier by Philip Eklund

Ti/K Heat Pipe

Specific Area
Heat Cap.~153 kWth/m2
Mass100 kg/m2
Op. Temp.1100 K

Ti/K heat pipe radiator

A Rankine evaporation-condensation cycle heat pipe uses metal vapor as the coolant, which is liquefied as it passes through a heat exchanger connected to the radiator. A liquid metal near the liquid/vapor transition is able to radiate heat at a nearly constant temperature. The pipe is made from SiC-reinforced titanium (Ti) or superalloy operating at up to 1100 K, and the working fluid is potassium (K). The pipe is covered with a lightweight thermally-conductive carbon foam, which protects the pipe from space debris and transfers heat to the radiating fins. The total specific area is 100 kg/m2.

From High Frontier by Philip Eklund

Wickless Heat Pipe

WEIGHTLESS WICKLESS HEAT PIPE

Heat pipes are devices to keep critical equipment from overheating. They transfer heat from one point to another through an evaporation-condensation process and are used in everything from cell phones and laptops to air conditioners and spacecraft.

Normally, heat pipes contain porous metal wicks that return liquid to the heated end of the pipe where it evaporates. But engineers are working to develop wickless heat pipes that are lighter and more reliable. Researchers at Rensselaer Polytechnic Institute initiated the Constrained Vapor Bubble (CVB) project to study these wickless heat pipes for use in near-zero gravity environments for aerospace applications.

“Wick structures can be difficult to keep clean or intact over long periods of time. The problem is especially acute for applications, such as NASA’s Journey to Mars mission, that put a premium on reliability and minimal maintenance,” said Professor Joel Plawsky, who heads the Isermann Department of Chemical and Biological Engineering at Rensselaer.

Working with a NASA engineering team, the researchers are conducting CVB experiments at the International Space Station. Plawsky and postdoctoral research fellow Thao Nguyen recently wrote an article about the CVB project in Physics Today, published by the American Institute of Physics (AIP).

“The CVB project is designed to record, for the first time, the complete distribution of vapor and liquid in a heat pipe operating in microgravity. The results could lead to the development of more efficient cooling systems in microelectronics on Earth and in space,” Plawsky said.

A Familiar Technology in an Unfamiliar Environment

A heat pipe is partially filled with a working fluid, such as water, and then sealed. At the heat source, or evaporator, the liquid absorbs heat and vaporizes. The vapor travels along the heat pipe to the condenser, re-liquifies and releases its latent heat, eventually returning to the evaporator, without any moving parts.

In the CVB experiment, Plawsky’s team created a miniature heat pipe, using pentane (an organic liquid) in a glass cuvette with square corners. An electrical resistance heater was attached to the evaporator end. At the other end, a set of thermoelectric coolers kept the condenser temperature fixed. The transparent tube allowed the researchers to study the fluid dynamics in detail, and the sharp corners of the cuvette replaced the job of the wick.

Two main forces affect how a heat pipe performs: capillary and Marangoni forces. The capillary force is what drives the liquid back toward the evaporator. This is the same force that causes liquid to climb up a straw. The Marangoni force arises from a change in the fluid’s surface tension with temperature. This force opposes the capillary force and drives liquid from the evaporator to the condenser.

A Balancing Act

When the amount of liquid evaporating is larger than what can be pumped back by the capillary force, the evaporator end of the heat pipe begins to dry out. This “capillary limit” is the most common performance limitation of a heat pipe.

The researchers expected the same thing to happen in the CVB experiment. But, instead, the evaporator flooded with the liquid. That’s because the Marangoni and capillary forces were no longer fighting against gravity. As a result, the Marangoni force overpowered the capillary force, causing condensation at the evaporator end. However, the net effect was the same as if the heat pipe had dried up.

“As the flooded region grew, the pipe did a poorer job of evaporating liquid, just as would happen if the heater were drying out,” Plawsky said.

The researchers have countered this problem in the next stage of the CVB project by adding a small amount of isohexane to the pentane. Isohexane boils at a higher temperature and has a higher surface tension. This change in surface tension cancels out the temperature-driven Marangoni force, restoring the heat pipe’s performance.

“The School of Engineering at Rensselaer and NASA have had long-standing and productive collaborations on a number of important research projects," said Dean of Engineering Shekhar Garde. “Dr. Plawsky’s heat-pipe research is a great example of our work with NASA to help translate fundamental understanding of liquids into real-world applications here on Earth and in space.”

From WICKLESS HEAT PIPES: NEW DYNAMICS EXPOSED IN A NEAR-WEIGHTLESS ENVIRONMENT by School of Engineering, Rensselaer Polytechnic Institute (2018)

Bubble Membrane

Specific Area
Heat Cap.~21 kWth/m2
Mass7 kg/m2
Op. Temp.800 K

Bubble membrane radiator

This high-temperature concept uses a spinning bubble-shaped membrane to reject waste heat. A two-phase working fluid (hot liquid or gas) is centrifugally pumped and sprayed on the interior surface of the bubble. The fluid wets the inner surface of the sphere and is driven in the form of a liquid film by centrifugal force to the equatorial periphery of the sphere. As the liquid flows along the inner surface of the envelope it loses heat by thermal radiation from the outer surface of the balloon. The use of membranes woven from space-produced carbon nanotubes and cermet fabrics offers a specific area of 7 kg/m2, radiating from one side at 800 K. Liquid metal pumps return the liquid out of the sphere through rotated shaft seals to its source.

Koenig, 1985

From High Frontier by Philip Eklund

Buckytube Filament

Specific Area
Heat Cap.~300 kWth/m2
Mass~100 kg/m2
Op. Temp.1300 K

Buckytube filament radiator

Waste heat may be rejected by moving thousands of loops of thin (1 mm) flexible "Buckytubes" (carbon nanotubes), which radiate their thermal load prior to return to the heat exchanger. Cables constructed of Arm-chair type nanotubes are the strongest cables known, with design tensile strengths about 70% of the theoretical 100 GPa value. The moving filaments are heated by direct contact around a molybdenum drum filled with the heated working fluid, and then extended into space a distance of 70m by rotational inertia. Their speed is varied according to the temperature radiated (from 273 K to 1300 K). The loops are redundantly braided to prevent single point failures from micrometeoroids. Each element is heat treated at 3300 K to increase the thermal conductivity through graphitization to about 2500 W/mK.

Richard J, Flaherty, "Heat-transfer and Weight Aialysis Of a Moving-Belt Radiator System for Waste Rejection in Space", Lewis Research Center, Cleveland, Ohio, 1964.

From High Frontier by Philip Eklund

Flux-Pinned Superthermal

Specific Area
Heat Cap.76 kWth/m2
Mass17 kg/m2
Op. Temp.928 K
Emissivity0.9

Flux-pinned superthermal radiator

Variable configuration radiators take advantage of the surprising physics of high-temperature flux-pinning superconductors. These materials resist being moved within magnetic fields, allowing stable formations of elements. No power or active feedback control is necessary. The radiating elements fly in a flux-pinned formation, not physically touching, but connected by superthermal ribbon. Superthermal compounds hypothetically conduct heat as effortlessly as superconducting materials conduct electricity. The radiating surfaces are graphite foams, which have both a high emissivity (0.9) and a high thermal conductivity (1950 W/m°K) if the heat conducts in a direction parallel to the crystal layers. Operating at 928K, the superthermal radiator has a specific area of 17 kg/m2 and 76 kWth/m2.

Dr. Mason Peck, 2005

From High Frontier by Philip Eklund

Hula-Hoop

Specific Area
Heat Cap.300 kWth/m2
Mass33 kg/m2
Op. Temp.1300 K

Hula-Hoop radiator

By imparting heat to twin washer-shaped disks by direct conduction, the Hula-Hoop radiator avoids the diseconomies of scale that plague fluid radiators. Furthermore, they are robust against micrometeoroid strikes and hostile attack. The two hoop are 100-meters in diameter. They are made of braided cermets coated with graphite, and lubricated in a heat exchanger with tungsten disulfide (WS2). Radiating at 1300 K, each has a specific area of 33 kg/m2 and 300 kWth/m2.

This design is a Philip Eklund original, published here for the frst time.

From High Frontier by Philip Eklund

Marangoni Flow

Specific Area
Heat Cap.~300 kWth/m2
Mass24 kg/m2
Op. Temp.1300 K

Marangoni flow radiator

In zero-g, a surface tension gradient can create a heat pump with no moving parts, or drive micro-refining processes. This phenomena, called Marangoni flow, moves fluid from an area of high surface tension to one of low surface tension. Bubbles operating at 1300 K have a specific area of 24 kg/m2.

G. Harry Stine, "The Third Industrial Revolution," 1979

From High Frontier by Philip Eklund

Microtube Array

Specific Area
Heat Cap.~104 kWth/m2
Mass34 kg/m2
Op. Temp.1000 K

Microtube array radiator

Nanofacturing techniques can fabricate large, parallel arrays of microtubes for high performance radiators. The radiating surface comprises a heavily-oxidized, metal alloy with a 100 nm film of corrosion resistant, refractory platinum alloy deposited on it. The working fluid is hydrogen, which has low pumping losses and the highest specific heat of all materials. This fluid is circulated at 0.1 to 1 MPa through the microtubes, and the heat radiates through the thin (0.2 mm) walls. This allows a specific area of 34 kg/m2, including the hydrogen. The rejection temperature for titanium alloy tubes is from 200 K up to 1000 K, if a high temperature barrier against hydrogen diffusion is used. High speed leak detection capability and isolating valves under independent microprocessor control provide puncture survivability.

F. David Doty, Gregory Hosford and Jonathan B. Spitzmesser, "The Microtube-Strip Heat Exchanger," 1990

From High Frontier by Philip Eklund

Salt-Cooled Reflux

Specific Area
Heat Cap.~75 kWth/m2
Mass75 kg/m2
Op. Temp.1100 K

Salt-cooled reflux tube radiator

In contrast to a heat pipe, that uses capillary action to return the working fluid, a reflux tube uses centrifugal acceleration. This design is more survivable than heat pipes, especially when overwrapped with a high-temperature carbon-carbon composite fabric. Unlike metals, the strength of these composites increases up to temperatures of ~2300K. However, they degrade when subjected to high radiation levels. The working fluid is molten fluoride salts, the only coolant (other than noble gases) compatible with carbon-based materials. Radiating at 1100 K, this radiator has a specific area of 75 kg/m2.

Charles W, Forsberg, Oak Ridge National Laboratory, Proceedings of the Space Nuclear Conference 2005, San Diego, California, June 5-9, 2005.

From High Frontier by Philip Eklund

SS/NaK Pumped Loop

Specific Area
Heat Cap.~93 kWth/m2
Mass61 kg/m2
Op. Temp.970 K
Emissivity0.9

SS/NaK pumped loop radiator

A Rankine evaporation-condensation cycle exchanges heat using a liquid metal as a coolant, which is vaporized as it passes through a heat exchanger connected to the radiator. A liquid metal near the liquid/vapor transition is able to radiate heat at a nearly constant temperature. The usual medium is sodium (Na) or sodium-potassium (NaK), which has a saturation temperature of nearly 1200 K at 1.05 atm. The plumbing is stainless steel (SS) tubes operating at up to 970 K with an emissivity of 0.9. The tube wall is half a millimeter thick to guard against meteoroid-puncture, and each pipe is an independent element so that a single puncture does not cause overall system failure. Molecular beam cameras on long struts scan for meteoroid leaks, which are plugged with pop rivets installed by a tube crawler. Radiating at 970 K from both sides, this radiator has a specific area of 61 kg/m2, including fluid and heat exchanger.

J. Ca/ogeras, NASA/LaRC, 1990.

From High Frontier by Philip Eklund

Attack Vector: Tactical

This fictional radiaor is from the tabletop wargame Attack Vector: Tactical, which is why the description talks about weird units like "power points" and "heat points."

  • One game turn segment is 16 seconds.
  • One power point is 1000 megajoules delivered in 1 segment.
  • So a starship reactor that outputs 1 power point produces at a rate of 1000 MJ / 16 seconds = 62.5 megawatts.
  • 1 heat point is 250 megawatts.
  • 1 hull space holds 20 metric tons.)
SCIENCE BEHIND THE RULES: RADIATORS AND HEAT SINKS
Specific Area
Heat Cap.357 kWth/m2
Mass100 kg/m2
Op. Temp.1600 K
Emissivity0.9
(1 - 0.1)

Knowing that our reactors produce 62.5 MW as a base power unit, and using the proof at right, we get an efficiency of 4 J of waste heat per J of power generated. This tells us that wee need to radiate ~250 MW per point of power. The Stefan-Boltzman law states that the surface emits power at a rate of (1-A) * 5.67×108 Wm2 K4 * T4 where A is the albedo, and T the absolute temperature in Kelvins. With an albedo of 0.1, a temperature of 1600K, and 250 MW of output, we need 700 square meters of radiating surface. Extending as a fin, radiating from both sides, this is roughly 18 meters square. At roughly 0.3m thick, and flexible enough to be retracted and extended, we get something that's reasonably 70 tons, or a bit shy of 3 hull spaces. For the sake of game play, one hull space of radiators dissipates 100 MW, or 0.4 heat points.

A civilian (starship) reactor has a built in 16 meter by 16 meter radiator that dissipates its waste heat; this radiator is built into the hull of the ship. This is why civilian reactors require part of the hull of the ship to be unarmored.

Storing the heat before radiation requires a heat sink. A sodium heat sink is ~21.5 cubic meters of sodium, with a density of 0.968 tons per cubic meter. Sodium has a thermal capacity of 28.2 J/mole/K. A mole of sodium weighs 22.98 grams. One gram of sodium absorbs 2.82/22.98 = 1.22 J per K of heat increase. A heat sink of sodium weighs 20.82 tons, raising that mass by 1 K absorbs 25.4 MJ. Sodium melts at 372 K and vaporizes at 1252 K. Pressurized, it remains liquid to 1600 K, our radiator temperature. Assuming a working range of 1300K (room temperature to 1600 K), each heat sink stores 1300 * 25.4 MJ = 33.02 GJ, which is one heat point, assuming other inefficiencies.

Lithium's thermal capacity of 24.8 J/mole/K and molar weight of 6.94 allows 1 gram to absorb 3.57 J per K of heat increase, or 2.92× the heat capacity of sodium. By using 22 tons of lithium, we get triple the capacity of the sodium heat sink.

Water's thermal capacity is 4.186 J/gram/K, 3.426 times that of sodium. Room temperature to boiling is ~85 K, which limits its usefulness. Raising 1 ton of water by 85 K takes 355.88 MJ. One heat point is 33 GJ, and the amount of water needed to store one heat point is 33 GJ/350.88 MJ = 93 tons. Including the extra mass for plumbing, that's 5 hull spaces all told.

Vaporizing water takes 2256 J/g, 6.3× the energy needed to raise it by 85 K. Because the vaporization is not quite perfect, we treat it as 6 heat points removed when the heat sink is vented. The liquid metal heat sinks aren't vented, as vaporized metal would deposit on the sensors of the ship.

From ATTACK VECTOR: TACTICAL Core Rulebook by Burnside, Finley, and Valle (2004)

Discovery XD-1

The spherical pressure hull formed the head of a flimsy, arrow-shaped structure more than a hundred yards long. Discovery, like all vehicles intended for deep space penetration, was too fragile and unstreamlined ever to enter an atmosphere, or to defy the full gravitational field of any planet. She had been assembled in orbit around the Earth, tested on a translunar maiden flight, and finally checked out in orbit above the Moon.

She was a creature of pure space - and she looked it. Immediately behind the pressure hull was grouped a cluster of four large liquid hydrogen tanks - and beyond them, forming a long, slender V, were the radiating fins that dissipated the waste heat of the nuclear reactor. Veined with a delicate tracery of pipes for the cooling fluid, they looked like the wings of some vast dragonfly, and from certain angles gave Discovery a fleeting resemblance to an old-time sailing ship,

At the very end of the V, three hundred feet from the crew-compartment, was the shielded inferno of the reactor, and the complex of focusing electrodes through which emerged the incandescent star-stuff of the plasma drive. This had done its work weeks ago, forcing Discovery out of her parking orbit round the Moon. Now the reactor was merely ticking over as it generated electrical power for the ship's services, and the great radiating fins, that would glow cherry red when Discovery was accelerating under maximum thrust, were dark and cool.

From 2001 A Space Odyssey by Sir Arthur C. Clarke (1969)

The final decision was made on the basis of aesthetics rather than technology; we wanted Discovery to look strange yet plausible, futuristic but not fantastic. Eventually we settled on the plasma drive, though I must confess that there was a little cheating. Any nuclear-powered vehicle must have large radiating surfaces to get rid of the excess heat generated by the reactors — but this would make Discovery look somewhat odd. Our audiences already had enough to puzzle about; we didn’t want them to spend half the picture wondering why spaceships should have wings. So the radiators came off.

From Lost Worlds of 2001 by Sir Arthur C. Clarke (1972)

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