# How cold for lack of heat?

I know that on the moon, while the daytime surface temperature can be above 270° F, in the shade but a few feet above the surface temperature can get quite low. Wondering whether a similar phenomenon can be observed if a vaccum is created around an object which is suspended in he middle of it.

Say for example, a steel ball is suspended in the middle of a plexiglass cube, by 3 plexiglass threads, and the interior volume of the cube is 1 cubic meter, and the ball has a diameter of 10cm, the box walls are 5cm thick. The cube has been evacuated to the practical limit of modern vacuum pumps. Let's say the steel ball is chilled to a temperature of -20 c (measured at the time the evacuation is complete). Let's say it's 70° F outside, and the box is sitting on a plexiglass tripod in the shade, shielded from direct sunlight.

What would be the temperature reading of the steel ball, approximately, after say 2 weeks? (i.e. at equilibrium).

(I don't mean to be overly prescriptive, but trying to be sufficiently specific to avoid answers like "it depends on how thick the walls are" and similar)

• Regarding your first question, remember the atmosphere of earth absorbs energy as well, the amount of energy your system would receive on the earth is not same as it would receive on the moon. And i don't understand how 'The box' comes into play ? It's not quite clear to me. Aug 19 '18 at 17:32
• the purpose of the box is to create the vacuum around the steel ball, to insulate, as much as reasonably possible the steel ball from the radiant heat of the earth and the surrounding atmosphere, just as on the moon, the lack of atmosphere insulates something above the surface from the very high temperatures on the surface. i.e. the heat can't travel from the wall of the box to the steel ball because there is no medium to conduct it. That's the question anyway. Aug 19 '18 at 17:36
• I think i got it, so you want to reduce the heat transfer by conduction and convection as much as it is possible, but you forget the transfer heat by radiation. I think your thermodynamic system would be in equilibrium in two weeks. Aug 19 '18 at 17:40
• Ah. Does heating by radiation require a medium of any kind? if you remove the difference between the earth and moon, i.e. an atmosphere, why the behavior seems to be different (or perhaps not?). If the box were shielded from visible light, infrared ultraviolet would radiant heating still occur? how does this happen? Aug 19 '18 at 17:49
• Everything tends to the same temperature over time - give enough tme... Aug 20 '18 at 5:51

Imagine the system below at equilibrium.

The energy balances are as follows:

At the ball ...

$$\epsilon_b T_b^4 =\epsilon_w T_w^4$$

At the wall ...

$$\epsilon_w A_w \sigma T_w^4 = h_a A_w (T_a - T_w) + \epsilon_a A_w \sigma T_a^4$$

In these balances, $\epsilon_j$ are emissivities, $A_j$ are areas, $\sigma$ is the Stefan-Boltzmann constant, $T_j$ are temperatures, and $h_a$ is the convection coefficient of air.

Choose all materials and sizes to specify all $A_j$ and $\epsilon_j$ values. Choose a still or windy day to specify $h_a$. Define the air temperature. This leaves two unknowns ($T_b$ and $T_w$) with two equations. The problem can be solved.

## Example

Take the case where the ball and walls are the same material so their emissivities are the same. We find that $T_b = T_w$. The radiation heat flows inside the box are balanced. Now assume the air has no radiation and the walls are perfect black bodies ($\epsilon_w = 1$) to obtain $\sigma T_w^4 = h_a (T_a - T_w)$. Take stagnate air with $h_a = 5$ W/m$^2$ K and $T_a = 275$ K. The wall and ball are at 238 K. In this case, the air is pumping heat to the walls by convection, and the (black body) walls are radiating back to the air. As the walls (and ball) go toward gray bodies (as $\epsilon_w \rightarrow 0$), the wall temperature increases. In the limit where the walls emit no radiation, we end with the simple case that $T_b = T_w = T_a$. No heat flows from anything at equilibrium.

• Thanks that's very helpful Jeffrey. I'm afraid I don't quite what the units and some of the letters are. (e.g. Aw, e, sigma?) Assuming no wind, can you give an example based on the setup described above? Aug 20 '18 at 16:55
• I updated with an explanation of the symbols and an example Aug 20 '18 at 21:50