thermodynamics - How to combat the black-body temperature of an object?

I'm trying to model the temperature of a large spacecraft for a space colony simulation game I'm working on. In another question, I checked my calculations for the steady-state black-body temperature of an object, considering only insolation and radiation, and it appears I'm on the right track.

My understanding is that this black-body temperature formula works only for passive bodies with no active heating or cooling. Now I want to add active heating and cooling elements. But how?

For cooling, I think I can model radiators as simply increasing the surface area of the craft, with no significant change to insolation (since radiators are placed edge-on to the sun). Please correct me if I'm wrong on that.

For heating, I'm stumped. I can increase the amount of energy dumped into the system, by presuming a nuclear reactor or beamed power or some such, but when I try that, the effect is much smaller than I would expect. I end up having to dump many MW of power into a large craft just to raise it up to room temperature.

So I'm wondering: does it matter how the extra energy is used within the system? Is a kW poured into a big electrical space heater going to get things hotter than a kW spent twirling a beanie, and if so, how?

As a possibly related question, it's claimed that the greenhouse effect significantly raises the temperature of a planet -- for example, Venus's black-body temperature would be 330 K, but due to atmospheric warming, its actual surface temperature is 740 K (*). How is this possible? Isn't it Q_out = Q_in, no matter what? And however this works for Venus, can we do the same thing to warm our spacecraft?

Answer

OK, I think I've got it, thanks to your comments above as well as this link, which shows how to calculate the temperature of a solar oven. (My situation is very similar to a solar oven, except that the power dumped inside the craft is electrical -- but watts are watts, right?)

So, I believe that what I need to do is:

1. Calculate the steady-state temperature of the outside of the craft, as described here, but considering only insolation (no internal energy dissipation).


2. To calculate internal temperature, observe that P_out = P_in in the steady state, and then apply this critical formula: P_out = U A (T_in - T_out), which describes the power leaving as a function of U (the combined heat transfer coefficient of the walls), A (the area of the walls), and the temperature difference. Using P_out = P_in and solving for T_in, I get T_in = T_out + P_in / (U A).


3. Now I need only plug in the power dissipated inside the craft, and use the skin temperature found in step 1 for T_out, and I can find T_in.

This all makes sense to me, but I'm obviously no physicist. If anybody sees a mistake here, please let me know!