## Introduction

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.

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.

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?

If the spacecraft has one heat radiator plate, you do not have to worry about waste heat radiated from one plate shining on a second plate. Such shine is counter-productive, shined heat is re-absorbed by the second plate which is not helping matters.

However, a single plate has no redundancy. One meteor hole and the plate could be rendered in operative, forcing a shut-down of the nuclear reactor or whatever. Or one hole from hostile weapons fire, for that matter.

The simplest solution is two heat radiator plates that are oriented coplanar, mounted on opposite sides of the spacecraft. Since they are coplanar it is impossible for them to shine on each other. You can have as many coplanar heat radiators as you want, but this tends to force the spacecraft to be elongated. This increases the spacecraft's moment of inertia, so it turns more slowly.

If you want more than two radiators without a dachshund shaped ship, you will be forced to have radiators that are not coplanar. The heat shine penalty goes up with each extra radiator panel added.

## 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. And since the exhaust is not physically connected to the spacecraft, the "radiator" adds zero mass to the rocket structure.

But it only works on certain kinds of engines. And it doesn't work at all to cool most spacecraft weapons, with the exception of weird weapons like open-cycle chemical and bomb-pumped lasers.

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 will also 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.

Note that the blackbody spectrum does NOT go up the rainbow. Both go from red to orange to yellow. But the rainbow continues to green, blue, indigo, and violet. The blackbody spectrum instead continues to white, blueish-white, and light blue.

The force fields in E. E. "Doc" Smith's Skylark & Lensman series and the Langston Field in Larry Niven & Jerry Pournelle's The Mote in God's Eye go up the rainbow spectrum as enemy energy beams assault them. But that's space opera, not reality.

In the diagram above the blackbody spectrum is the curved black line labeled "Planckian Locus". Which as you can see passes through red, orange, yellow, white, blueish-white, and light blue. But it never gets close to green at all. Nor purple or magenta either.

This is also why there ain't no such thing as a green star.

Why does this happen? Well, mostly because the human eye is a most imperfect optical instrument.

A blackbody emission that has its peak in the green part of the rainbow spectrum is also emitting lots of light in the red and blue parts of the spectrum (note how the curves are not sharp peaks but rather sloping curves). To the human eye a mix of green, blue, and red light looks like white.

Above 16,000 K or so all stars look the same shade of blue. In reality the relative intensities of the shorter frequencies are quite different at various temperatures, but to the imperfect human eye they all look like blue. A spectroscope can see the differences quite easily.

The fact that the human eye can be fooled this way is the reason why computer monitors have pixels for red, green, and blue; but no pixels for yellow, orange, or violet. Since the imperfect human eye sees a mix of red and green light as yellow, why go to the expense of adding yellow pixels?

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.

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:

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

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
heat
(Heat Cap.)
Specific area
mass
(Mass)
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

### 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

#### 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:

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

#### 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.

### 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.)

The "Achilles Heel" of combat spacecraft are the heat radiators. Drives, power plants, and most weapons generate incredible amounts of waste heat. For unlimited operations, the heat has to be disposed of with radiators. This is not a problem for civilian non-combat spacecraft.

Warships are different. Since by their nature radiators are difficult or impossible to armor, radiators will probably be the first thing shot off by hostile weapons fire. Then you have about thirty seconds to scram the ship's reactor before the engineering section turns into a sea of molten metal. This is because shooting a hole in a spacecraft's radiator will have the same effect as shooting a hole in your automobile's radiator, except at a much higher temperature.

Droplet style heat radiators cannot be armored, but they are relatively immune to hostile weapons fire, since they are basically liquid sprays of coolant instead of physical panels. And before somebody mentions the "refrigerator laser" from David Brin's novel SUNDIVER, there appears to be certain theoretical reasons why it would not work. For one it probably violates the second law of thermodynamics.

And no, you cannot solve the problem by using a thermocouple to convert the heat into electricity.

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.

## Combat Heat Sinks

No, I'm not talking about those blocks of metal with fins attached to over-clocked CPUs. I mean "Phase Change Material (PCM) heat sink". For some annoying reason the cooling block on computer CPUs is called a heat sink when it is really a convection cooler. Heat sinks do not have heat dissipation fins dumping the heat into the air by conduction.

Here's the deal: combat spacecraft will need to get rid of huge amounts of waste heat from the drive and some of the weapons. A non-combat spacecraft does this with heat radiators. But combat spacecraft have to worry about hostile weapons fire trying to shred the radiators. Now while it is possible to armor the radiators this might not be possible in all cases. So what do you do?

What you do is have a large cold block of some material hidden in the core of your warship. If you cannot use radiators to dump the waste heat overboard, you pump the heat into the cold block. This is the heat sink. You use some phase change material that absorbs heat by melting, e.g., a chunk of ice.

Warships going into battle retract their radiators into armored cubbies. They then rely upon internal heat sinks to dispose of waste heat. The good thing is that the heat sinks are armored. The bad news is that they can only store a few minutes worth of heat. This puts a severe time limit on the length of combat. Naturally a battleship will have a larger heat sink than a destroyer, but it will also have a higher waste heat level to dissipate.

If one's heat sink fills up too soon, the only option is to "strike the colors" and signal surrender to the enemy by extending the vulnerable heat sinks (sort of like a dog in a dogfight surrendering by lying on its back and baring its throat). The alternative is being roasted alive as your ship melts.

Also note this is a very similar situation to the science-fictional Langston field. Specifically, if the enemy gives you enough time to cool down your heat sink, your ship becomes combat-ready again, and the battle starts anew. So the enemy is not going to allow that to happen. You will have to give the enemy a safe way to disable your ship, or they will be forced to destroy you. In case of the Langston field, the tradition is to allow the enemy to send a low-ranking ensign wearing a tactical nuclear weapon on a dead-man switch, who enters your ship, parks themself in the control room, and ensures you do not break your surrender.

Another combat use of heat sinks is in the ever-fruitless quest for stealth in space. The idea is that heat radiators are like giant neon advertising signs, while a heat sink is much harder to detect.

In THE CHILDREN'S HOUR by Jerry Pournelle and S. M. Stirling is a splendid technobabble heat sink. The sink material is Degenerate matter, i.e., white dwarf star matter.

I have my doubts. Degenerate matter apparently does not undergo phase changes so adding heat to the sink might be a problem. Further: the accursed stuff will be trying to explode due to electron degeneracy pressure. The density will be a problem, typically 10,000 kilograms per cubic centimeter. About the only advantage I see is a vastly reduced volume of the heat sink.

## Non-Combat Heat Sinks

There are uses for heat sinks that are not combat-related. They are not as sexy as the combat applications, but can come in handy.

## Atomic Rockets notices

This week's featured addition is SPIN POLARIZATION FOR FUSION PROPULSION

This week's featured addition is INsTAR

This week's featured addition is NTR ALTERNATIVES TO LIQUID HYDROGEN