Given all of the above [now below], how did it avoid overheating?
Short answer: Not enough heat produced in the first place.
Long read: (as usual only basic physics needed).
(Standard Disclaimer: I hope I did remember all the formulas right, didn't add (many) calculation errors and most important picked the right English words.)
Heat in a device only builds up to a point where the energy put in (*1) put into than the device can is in equilibrium to the temperature needed to transfer exactly this amount of energy to the outside. It does not matter what kind of transport mechanism is used. Avection, convection, conduction or radiation, active or passive cooling is all the same and differ only by the resistance they provide - or not as we want to get it out without heating up much.
It's all about energy, and energy is counted in joule. Energy is power over time. Since it's sufficient to look at the BBC as static system (energy fluctuations are rather minor and equaled out over time), we can look at a fixed time interval and cancel out all time dependent calculation. Since one joule is one watt over one second, using one second is a good idea (*3), which now allows us to base further examination on power measured in watt (and equal watt with joule in all calculations). And number of watt is a value given to us by the manufacturer. Convenient, isn't it?
The BBC master's power supply was rated at (just) 35 watt (*4). This is what the PS was is supposed to deliver to the computer, its output. A PS is converting, and any conversion in this universe got losses. In case of a linear PS it's usually somewhere between 40 and 60%, while a switched delivers 80+%.
Assuming 50% efficiency (for the linear), we get about 70 watt to be dissipated as a first number. But let's get that number up to 100 watt. That not only simplifies calculations, but it also takes into account the many modified and all pimped up units.
The task is now to get rid of 100 watt of power.
Heat transfer is based like any transfer on a flow, the heat flow Q (*2) again measured in watt. So we just let 100 watt flow out, right, and temperature inside will stay the same as outside? Nope. Any flow needs a difference in height (or pressure) for water and in temperature difference here. Further this difference defines how much energy can be transferred through a given barrier within a specific area. In theory this allows to transfer any amount we need. Except, to keep our BBC working we can define a maximum temperature we do not want to exceed. Above ~150 °C solder joints may start to fail. Even more chips are usually only specified to work up to 70-85 °C. Assuming that the operating environment is usually below 35 °C we get a maximum usable difference of 40 °C.
The other important value is the surface the energy needs to pass. A BBC Master is roughly 40x35x6 cm^3 For the sake of simplicity we just use the top surface size, 40x35 or 0.14 m^2.
Bottom line, we want to get 100 watt through a surface of 1/7th of a square meter with not more than 40 degree temperature difference. Right?
The underlaying formula here is Q = A times delta-T times k. With A as size in square meter, T in Celsius and k as a material specific value. Flipping the formula toward k gives k = Q / (A x dT) with or 100 / (0,14 * 40) which equals ~18. Thus the material(s) used should have a thermal resistance of 18 watt per square meter per Kelvin or less. For real material this is described by a material specific constant called lambda and defined as W/m*°C. By dividing that constant by our value we will get therefore the maximum thickness.
So let's have a look at some materials and how thick a case made out of this can be to keep within the borders set above:
Material / lambda / thickness in cm
Silver / 429 / ~24,000
Steel / 80 / ~ 5,500
Concrete / 2.1 / ~ 12
Glass / 0.75 / ~ 4
ABS(*) / 0.18 / ~ 1
(*) Using ABS not only as middle of the road plastic, but also as it's one of the less thermal conductive ones (which makes it great to be used in 3D printers;). most others are better conductor - like PET with 0.24.
With this number our case may have steel walls of like 55 meter, 12 cm concrete or 1 cm of ABS.
There was a metal box around that part of the machine, but a plastic case around the metal.
The relation above already shows that the steel cover can be (almost) ignored as its conductivity is about 400 times higher than plastic. And unless the plastic is thicker than 1 cm it's all is good.
The BBC's case is less than 3 mm of plastic which again can be easy turned into the temperature difference due to its proportional nature. With 40 degree temperature difference (to transfer 100 W) through 10 mm (1 cm) plastic, using only 3 mm thereof will result in about 12 degree temperature difference (40/10*3)
Or the other way around, with that 3 mm cover even a power consumption of 333 watt (100*10/3) would not damage the machine ... only typing might become a hot experience :))
Result: Not enough heat produced to drive the internal temperature anywhere critical
I love physics (and SI units).
Now, with using exact measurements and so on we will come to a difference of way less than 10 degree Celsius between machine and environment, making the BBC good for environmental temperatures of 60-70 degree. Keep in mind all calculations used where always rounded toward the 'bad' side using a rather large margin. Also, only conductivity between fluids (air) and a solid material was used. No special cases like radiation and so on.
Admittedly it was used on an island where the ambient temperature is usually quite low
Now considering the last few weeks I say living on an island might not make much difference - and more serious, while the BBC was a big player in the UK, it sold as well ok-ish in France and Germany - despite the almost non-existent support.
*1 - Energy put into a computer, gets almost completely turned into heat as other forms like EM radiation/light are quite low, and no mechanical output produced.
*2 - Usually it's noted as uppercase Phi or Q with a dot above - here Q will do it. Similar for all other symbols.
*3 - A second is always handy when calculating in SI units as being one of the fundamental units in MKSA :))
*4 - Here is, BTW, a real nice writeup about how to give more power to a fully loaded system - and stay within the original thermal specs as 60 W with 80% efficiency is about the same as 35 W with 50%. Nice, isn't it?