A lot of the discussion refers to "the spacecraft" as if it were a single object. When considered thermally, "the spacecraft" is a collection of objects, each in a specific heat-transfer context. The outer skin, the inner skin, the enclosed air, the windows, the frame, the wiring, the coolant, and so forth are all objects that behave thermally in complex and interactive ways.
While basic heat transfer describes how heat transfers within and between objects, the overall picture of the actual thermal situation of anything as detailed as a spacecraft is not implied at all reliably from the simple equations. That's like saying that since "basic" physics describes one rock atop another, anyone with that knowledge can build Chartres cathedral.
In fact the thermal design of spacecraft is its own field. Clearly "basic" knowledge isn't enough to practice the field, otherwise the various texts, papers, and courses on the subject would be a waste of time and paper.
In practice, understanding heat transfer in a complex composition such as a spacecraft indeed involves basic constitutive relationships, but modeled on the small scale incorporating the actual geometry, relationships, and properties of the components. So while a basic equation can give you the equilibrium temperature of a simple object in a simple radiative environment, the thermal profile of a spacecraft has no closed form and requires iterative models that can be solved only by fast computers and a lot of patience.
"How was the spacecraft cooled?" implies it was naturally hot. A better question is, "How was the spacecraft temperature controlled?"
I can go on and on about that. First we should talk about heat rejection. In space you often reject heat by radiation. Something that's hot will radiate its heat away as electromagnetic energy. Slowly, but surely. So if you have a radiator (not an automotive radiator but a true one) you can face it away from the sun and let heat radiate away into the blackness of space. Often you need two radiators on opposite sides, so that if one is facing the sun and can't effectively radiate, the other can.
The service module had opposing radiators. The fuel cells had their own separate radiators.
So then you need to get the heat from where it is to where it can be rejected. That's best done by a liquid coolant that's pumped in a loop between heat sources and the radiator. Heat always flows from hot to cold, so if the coolant comes out of the radiator freezing cold, heat will naturally flow into it without much coaxing. Grab a metal faucet tap with the cold running and see how long you can hold onto it.
Heat might come from absorption from the sun. It might come from electronic equipment. It might also come from the astronauts' metabolism. If you can get it into the coolant, you can reject it to the radiator on the dark side of the ship.
In practice heat didn't come from the sun on Apollo. The reflectivity of aluminized Kapton varies from about 0.45 to about 0.80 depending on manufacture. Of aluminum itself, up to 0.9. The spacecraft just isn't going to absorb a lot of heat all over.
Electronic equipment was the biggest heat producer on Apollo. Old 1960s and 1970s electronic components, remember.
Humans are finicky. They need an air temperature within a fairly narrow band in order to remain healthy and productive. So for Apollo there was a special problem in keeping their air mass at an appropriate temperature. If you blow the air past an exchanger, you can either suck heat out of it (if the exchanger coils are cold) or put heat into it (if the coils are hot). For Apollo, the same coolant loop was used for cabin air temperature control as for sucking heat away from electronics. A thermostatic switch operated a solenoid valve that piped the cabin heat exchanger to the upstream leg (i.e., right after the radiator, before the electronics blocks) to cool the air, and to the downstream leg (i.e., after the electronics, before the radiator) to heat it.
Without the electronics operating, there was no way to add heat to the air by that mechanism.
But that's not the only way the air could be heated. The CM had five windows, at least one of which would probably be admitting sunlight. Light streams in through the windows and shines on cabin interior surfaces, where it warms them radiatively. Those surfaces in turn transfer heat convectively to the air. That's not a lot of heat compared to what's transfered by the ECS exchanger. But if that suddenly doesn't work, the second-order greenhouse effect may become the first-order effect.
When the Apollo 13 astronauts put shades over the window to try to sleep, the sun no longer shone in. The lit surfaces no longer were warmed by the sunlight, and no longer transferred heat to the air. Even with all the insulation and thermal design, you can't practically eliminate all the conduction paths from the inner structure to the outer skin. And the vast majority of the outer skin that doesn't have a view factor to the sun will radiate and cool. Heat from attached structure conducts to it. And heat from adjacent cabin air will convect/conduct to that structure. Thus at the air/structure interface, the heat transfer reversed. That's why the temperature dropped significantly in the cabin.
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