# Surface temperature of a Sphere

1. Jul 18, 2012

### balulu

I am facing the following interesting problem.

A closed room\hall contains several identical machines in it they are fed by an electrical cable.
The machines can be turned on or off. When a machine is turned on, it consumes electrical energy and as a byproduct generates heat. The heat is conducted and radiated through the room walls to the outside air.
At present there are 2 machines in the room, but the designers consider placing more machines in the room. As more machines are added and become operational the structure outside temperature increases. The concern is that the outside surface area of the structure will become too hot. The outside skin is not conventional material but it is heat conductive, thin and radiating.
The radiation is to the outside air. The boundary (the sphere surface) is very thin and the only way to get rid of the heat generated inside the sphere is through its surface.

We don't know the inside temperature, but know the Thermal energy per unit of time generated by each machine when it is on. All machines are identical, for simplicity it can be assumed that they consume negligible volume/space and all of them are in the center of the sphere (so we don't have to deal with their specific locations).

It can be shown that the functional structure of the function that describes surface heat as a function of machines turned on is not dependent on the geometry of the room (the coefficients do). Using this assumption and in order to simplify the analysis and find the functional structure as a function of β, it can be assumed that the room is a sphere.
The Question is:
What is the function that describes the room surface temperature in steady state as a function of the number of machines that are turned on? Is it power of 2 or of 3 of β (the number of machines that are turned on)?

Possible approaches to handle this problem:

1. For a first approximation you can consider it as a straightforward linear relationship. If one machine of power P raises the surface temperature by say 5C then two machines would raise it 10C.

For example a typical power loss calculation for a sphere would be

Power = (Ts-Tair)/Thermal Resistance

Where Ts = Sphere air temperature
Tair = Outside air temperature
Power = Power loss through the walls. Under equilibrium conditions this is equal to the power going into the Sphere.
Thermal resistance = a constant.

I thought that a linear relationship holds, but after measuring the surface temperature in steady state for 1 machine versus 2 machines on, it is clear that the relationship is nonlinear (I had also no machines on as another reference point). My own interpretation is that Heat radiation is not a linear function.

It is inaccurate to base the functional relationships on two observations that are too close one to the other. The main objective is to move 20 to 50 machines into the structure.

2. A black body emits thermal radiation based on the fourth power of temperature according to the stephan-Boltzman equation. Other bodies have an emissivity factor to account for a non-black body surface and this factor can range from 0 to 1.Then there is absorbed energy, reflected energy and transmitted energy of the surface to factor in the model. Considering the surface to be a greybody then absorbed radiation equals emitted radiation of the surface.