How much energy can a space ship use internally without overheating?

Question

More thoughts on hard science possibilities for interstellar colonization and the request to check my understanding of physics.

Suppose you have a fully self contained colony ship. They have some giant reactor producing power and as long as the reactor has fuel they can produce everything they need for life support on board. This could be with green houses to grow plants for making oxygen and food or it could also be pure chemistry. According to my understanding of physics, as long as it is a closed cycle within the ship all energy will eventually be turned into heat. And as they are in space the only way to lose this heat is through thermal radiation. The Stefan–Boltzmann law now gives a way to compute the energy the ship loses through thermal radiation. It only depends on the surface area of the ship and the temperature.

It seems that for a fixed surface area and a fixed surface temperature this gives the exact amount of energy the ship needs to use to be in a stable equilibrium?

If they use more energy the ship will heat up (which will also increase the thermal radiation), if they use less the ship will cool down (which will also decrease the thermal radiation). As thermal radiation changes with the 4th power of absolute temperature, there is some room for maneuvering but if the energy use is off by an order of magnitude or two in either direction, they have a problem.

Edit: Some specific numbers. In my last question linked above I learned that you need the energy equivalent of the entire energy output of the sun for a few days to accelerate a large space ship to half the speed of light. This means that if the space ship uses the equivalent of all the solar energy hitting the earth (around $10^{16}$ Watts) for life support and other internal energy needs that is negligible in comparison even over a timespan of decades or centuries. But if we assume a spaceship as a 10km cube and a temperature of 300 Kelvin the total heat radiation by the Stefan-Boltzman law above is only $5.67*10^{-8}*300^4*(6*10^8) W = 2.75*10^{11} W$. Meaning by my naive interpretation of physics they would need to use energy of that order of magnitude to not overcook themselves, for $10^{16} W$ they would need a much much bigger or hotter spaceship.

Answer 1

Isolate your habitation section from your reactor section

Stefan–Boltzmann defines the energy blackbody emissions as:

P = AσT⁴

Or for those of us who actually like to know what our variables means:

$radiantEnergy = $surfaceArea * 5.67 * $temperature^4;

Common logic tells us that the correct course of action is to increase the surface area, and yes, this will help a bit, but look at that power of 4 on the temperature... that is a nice big exponent to exploit meaning there is a lot of room for exponential growth by just getting a little hotter.

The hotter you are, the faster you radiate heat; so, if you want to push a maximum amount of heat off into space, getting hotter is the way to go. Now because people live on your ship, getting hotter everywhere is a bad idea... and totally unnecessary. By separating your ship into capsules that are attached by thermal resistant materials, you can heat each part of your ship to its maximum threshold while keeping living quarters relatively cool.

For your habitat section you need to maintain a temperature of about 293K, but that is just what is good for humans to live in. A much larger part of your ship will be the greenhouse, but no-one says you have to live off of vascular plants. By farming Algea as you main food source you can crank this compartment up to 335K... but the next part will make this a trivial measure. Lastly is the reactor section. Now this is where establishing a true MAX has to digress a bit from "hard science" and into "science based" because we really don't know what power sources or materials we will be limited by in the future. However, if you were to construct a large capsule with thermal properties similar to tungsten, you could heat up your rear section to somewhere in the range of 2750K making it glow like a giant incandescent lightbulb... at which point it could passively heat your other 2 sections if you put it at the right distance.

So, let's say your ship is made of 3 equal sized capsules, all with a surface area of 1/3 the OP's proposed total surface area, this gives your habitat the ability to compensate for 9.17e10W, your farm section 1.57e11W, and your reactor section 7.12e14W. This is because your glowing hot reactor could offset ~7,750 times as much heat per surface area as a room temperature module.

This still only puts you at about 7% of your target... which is honestly not that bad. This still keeps you working in human time scales, but there are two things you can to to further boost this if you really want to hit that 1e16W benchmark.

First, there is no reason the modules should be the same size. Your farm will probably need to be much bigger (thus more surface area) than your habitat, and depending on choices you make as an author, your reactor/fuel/propulsion section could have much more surface area than the rest of your ship if you picture a fuel system that stores hydrogen for fusion with no oxygen to react with, you might as well store it as a super heated plasma as long as it does not get hot enough to melt your tungsten containment tanks. In this case, if you make your ship something like 5% habitat, 10% farms, and 85% reactor section, you could get yourself to about 17% of your target goal.

Secondly, you can go with higher heat. I chose 2750K as the operating temperature of a lightbulb... much hotter and tungsten become structurally unsound, but adding a meta-material into a space setting that can operate higher than this is not implausible. If you go with a ship that is 85% hot capsule and about 4280K, you should be able to reach 1e16W with that total surface area... but that would significantly exceed the melting point of any known element; so, if you are going for a more hard science universe, I would accept the slightly lower power output.

Answer 2

Ships can Control Surface Area and Temperature

A star or planet has a (relatively) static surface, and heat will always flow from hot to cold.

A ship is significantly more complex.

To heat up a ship, you can cover it with an isolating material. Heat flux from the covered areas will decrease, and the rest of the ship will warm up.

Conversely, to cool a ship you can transfer heat to certain exterior sections using a heat exchanger. Common household heat exchangers include Air Conditioning and Refrigeration - any case where you use movement of a working fluid to move heat between two locations.

Imagine a large set of wing-like radiators protruding from the ship. They have high surface area to radiate energy in every direction, and the ship can control their temperature (and therefore the amount of heat they radiate) by pumping fluids into the structure.

The key here is that the ship is not some static, uniform body, but a complex system that can impact its own emissions by expanding a small amount of energy.

Answer 3

Addressing some confusions in the original question

This means that if the space ship uses the equivalent of all the solar energy hitting the earth (around $10^{16}$ Watts) for life support ...

First at all clarify for yourself your naive interchangeability use of energy (as measured in joules) and power (as measured in Watts). A generation ship is not an totally opened system in which any energy that you use immediately transform in heat that needs to be evacuated immediately.

Then, what if they don't need the solar entire power that hits the Earth? It's not like 100 million people needs the power which sustain the entire life on Earth, from the depth of Mariana trench to the top of troposphere, including the formation of hurricanes, ocean currents and rain.

But if we assume a spaceship as a 10km cube and a temperature of 300 Kelvin the total heat radiation by the Stefan-Boltzman law above is only $5.67*10^{-8}*300^4*(6*10^8) W = 2.75*10^{11} W$. Meaning by my naive interpretation of physics they would need to use energy of that order of magnitude to not overcook themselves, for $10^{16} W$ they would need a much much bigger or hotter spaceship.

Not only the assumption that you need the Sun output just for 100M people is wrong, but the rate consideration are totally exaggerated - you simplified your model too much.

1. the ship is not a lump of steel or something. You have multiple energy sinks inside the ship - the very food that you grow will take light and heat to run those biochemical reactions and store them as bound energy. Which means the temperature of your ship isn't going to increase immediately (the way it happens with a lump of steel). True - eventually, you are going to have it transformed in heat (after you eat, digest and burn those nutrients), but there's be at a slower rate than if you just apply the same extra energy to a lump of steel (or amount of gass)

2. a generation ship will need to be masters of recycling or it won't last long. Heat is one thing that one can recycle to some extent. As an example, you have a large storage of water that you can afford to heat up using a thermal pump - and you will get to reuse part of the stored heat (the thermal gradient) sometime down the line. By doing it, you will create extra heat in operating that those heat pumps. But because a heat pump can reach 600% energy moving efficiency, you will only need theoretically to get rid of 1/6 extra heat. Do this in many other sophisticated ways and you address the rate in which you need to purge "waste heat" (heat in spatial configuration with such a low gradient inside the ship that you can't use to extract controllable power to run your processes. The equivalent of the "heat death of Universe" inside the microcosm of your ship)
   
   One on top of the other, in principle, you run your ship by recycling everything, with only the cost of extra energy required to recycle everything. And you can lower the amount of energy required to recycle everything to a much lower value than the cost requiring "brand new energy from the Sun in every moment and throwing the excess away."

3. All of the above would address the rate of the heat purge, but bottom line you will still need to purge it in the end. You can still control how you purge it into the lower temperature of the surrounding space. All your ship being thermally isolated with the exception of the radiators means that, by controlling the direction and the radiating area and the temperature of those radiators, you can use the thermal radiation for propulsion. Happened inadvertently with Pioneer 10/11 anisotropic radiation pressure slows them at a rate of 1 km/h over a period of ten years.

---

Bottom line: I don't know how or if a generation ship can be actually built, because it will be a matter of carefully trying to solve a great number of problems. What I can tell for sure is that you can't demonstrate the possibility or impossibility of a generation ship with computations on the back of a napkin using dumbed down models