Your differences from my number: A) you're working based on spacecraft average temperature and not realizing you're going to have a substantial thermal drop; B) you're assuming just one side of the surface radiates. They're on the same order of magnitude. Both of us are assuming that cooling systems, power systems, and other support systems make no heat.
You can pick a color that absorbs very little visible light but readily emits in infrared-- so being in the sun doesn't matter so much, and since planetshine is pulling you towards something less than room temperature, it's not too bad either.
None of these numbers make me think "oh, that's easy". You're proposing a structure that's a big fraction of the size of the ISS for one rack of GPUs.
I don't really think cooling in space is easy. The things I have to do to get rid of an intermittent load of 40W on a small satellite are very very annoying. The idea of getting rid of a constant load of tens of kilowatts, or more, makes me sweat.
Yes, I could make more optimistic calculations: use the steradians occupied by earth, find and use the thermal IR emissivities of solar panels place many thin layers of glass before the solar panel allowing energy generating photons through and forming a series of thermal IR black body radiators as a heat shield in thermal IR, the base also radiates heat outwards and at a higher temperature, use nonsquare base, target a somewhat higher but still acceptable temperature, etc... but all of those complicate the explanation, risking to lose readers in the details, readers that confuse the low net radiative heat transfer between similar temperature objects and room walls in the same room as if similar situation applies for radiative heat transfer when the counterbody is 4 K. Readers that half understand vacuum flasks / dewars: no or fewer gas particles in a vacuum means no or less energy those particles can collectively transport, that is correct but ignores the measures taken to prevent radiative heat loss. For example if the vacuum flask wasn't mirror coated but black-body coated then 100 deg C tea isolated from room temperature in a vacuum flask is roughly 400 K versus 300 K, but Stefan Boltzmann carries it to the fourth power (4/3) ^ 4 = 3.16 ! That vacuum flask would work very poorly if the heat radiated from the tea side to the room-temperature side was 3 times higher than the heat radiated by the room temperature side to the tea-side. The mirroring is critical in a vacuum flask. A lot of people think its just the vacuum effect and blindly generalize it to space. Just read the myriad of comments in these discussions. People seriously underestimate the capabilities of radiative cooling because the few situations they have encountered it, it was intentionally minimized or the heat flows were balanced by equilibrium, not representative for a system optimized to exploit radiative heat transfer.
Some small corrections:
>Both of us are assuming that cooling systems, power systems, and other support systems make no heat.
I do not make this assumption! all heat generated in the cooling, power and other support systems stem from electrical energy used to power them, and that energy came from the solar panels. The sum of the heat generated in the solar panel and the electrical energy liberated in the solar panel must equal the unreflected incident optical power. So we can ignore how efficient the solar panel is for the rest temperature calculation, any electrical energy will be transformed to heat and needs to be dissipated but by conservation of energy this sum total of heat and electrical energies turned into heat must simply equal the unreflected energy incident on the solar panel... The solar panel efficiencies do of course matter a lot for the final dimensions and mass of the satellite, but the rest temperature is dictated by the ratio of the height of the pyramid to the square base side length.
>You can pick a color that absorbs very little visible light but readily emits in infrared-- so being in the sun doesn't matter so much, and since planetshine is pulling you towards something less than room temperature, it's not too bad either.
emissivity (between 0 and 1) simultaneously dials how well it absorbs photons at that wavelength as well as how efficiently it sheds energy at that wavelength. A higher emissivity allows the solar panel to cool faster spontaneously, but at the cost of absorbing thermal photons from the sun more easily! Perhaps you are recollecting the optimization for the thermal IR window of our atmosphere, the reason that works is because it works comparatively to solar panels that don't exploit maximum emissivity in this small window. The atmospheric IR window location in the spectrum is irrelevant in space however.
> A) you're working based on spacecraft average temperature and not realizing you're going to have a substantial thermal drop;
of course I realize there will be a thermal gradient from base to apex of the pyramidal satellite, it is in fact good news: near the solar panel base the triangular sides have wider area and hotter temperature, so it sheds heat faster than assuming a homogenous temperature (since the shedding is proportional to the fourth power of temperature). When I ignore it it's not because I'm handwaving it away, it's because I don't wish to bore computer science audience with integral calculations, even if they bring better news. Before bringing the better news you need to bring the good news that its possible with similar order of magnitude areas for the radiator compared to the solar panels, without their insight that its feasible first, its impossible to make them understand the more complicated realistic and better news picture, especially if they want to not believe it... Without such proof many people would assume the surface of the radiator would need to be 10's to 100's of times the surface area of the solar panels...
> B) you're assuming just one side of the surface radiates.
No, I even explicitly state I only utilize 2 of the 4 side triangles of the pyramid (to sidestep criticisms that earth is also radiating heat onto the satellite). So I make a more pessimistic calculation and handicap my didactic example just to show you get non-extreme surface ratios even when handicapping the design. If you look at history of physics, you will often find that insights were obtained much earlier by prior individuals, but the community only accepted the new insights when the experimental design was simplified to such an extent that every criticism is implicitly encoded in the design by making it irrelevant in the setup, this is not explicitly visible in many of the designs.
Nah -- when we're talking about how much it takes to power 70kW of GPUs, we need to include some kind of power utilization efficiency number. If 70kW is really 100kW, then we need to make this ridiculously big design 40% larger.
> >You can pick a color that absorbs very little *visible light* but readily emits in *infrared*-
> how well it absorbs photons at that wavelength as well as how efficiently it sheds energy at that wavelength.
Yes. Planetshine is infrared, 290K-ish; sunshine is 5500K-ish and planetary albedo is close enough to this, with a very small portion of its light being infrared. You are being long winded and not even reading what you reply to.
So, for example, white silicate paint or aluminized FEP has a equilibrium temperature in full sun, with negligible heat conducted to or away from it, somewhere in the span of -70 to -40C depending upon your assumptions. It will happily net radiate away heat from above-room temperature components while facing the sun.
It will also happily net radiate away heat when facing the planet because the planet is under room temperature and the planet doesn't subtend a whole hemisphere even in LEO.
I don't really like argument from authority, but... I will point out that I am the PI for multiple satellite projects and have owned thermal design, and that the stuff I've flown in space has ended up at very close to predicted temperatures. I don't feel like this is an easy thermal problem.
I mean, it's easy in the sense of "it takes a radiator area about the same as the floor area of my house". It's not easy in the sense of "holy shit I need to launch a radiator that's bigger than my house and somehow conduct all that heat to it while keeping the source cool."
> of course I realize there will be a thermal gradient from base to apex of the pyramidal satellite
No, there will be a thermal gradient from the hot thing -- the GPU -- to the radiator surface. S-B analysis is OK for an exterior temperature, but it doesn't mean the stuff you want to keep cool will be that average temperature. This is why we end up with heat pipes, active cooling loops, etc, in spacecraft.
If this wasn't a concern, you could fly a big inflated-and-then-rigidized structure and getting lots of area wouldn't be scary. But since you need to think about circulating fluids and actively conducting heat this is much less pleasant.