Functional structure of Surface Heat

[balulu]

I am facing the following interesting question.

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 by product generates heat. The heat is radiated through the room walls to the outside air.
At present there are 2 machines in the room, but thed esigners consider adding mmore machines. The concern is that the outside surface area of the structure will become too hot.

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)?

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). In order to simplify the analysis and find the functional structure as a function of β, assume that the room is a ball.

 

[CWatters]

Since you haven't had a reply i'll have a go...

If we are talking about a real world building then for a first approximation I believe you can consider it 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 room would be

Power = (Troom-Tair)/Thermal Resistance

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

Typically the Thermal Tesistance would be made up several components in series. For example the thermal resistance of a wall would be the sum of the thermal resistances of it's component parts from the paint on the inside to the cladding or render on the outside.

The above applies to normal building materials. If we are talking exotic materials such as thin layers of polished aluminium foil and temperatures high enough for radaition to dominate conduction then then you need a better more detailed answer that I can't provide.

One word of caution...The outside surface temperature of some buildings is dominated by how sunny it is rather than what's inside generating heat.

 

[balulu]

Thank you CW,

Thank you for your reply.

The outside skeen is not conventional matrial but it is heat conductieve, thin and radiating.

I thoght on a linear relationship too, but than measured 1 machine versus 2 mcahines on, waited long enough to reach steady state (same outside temperature), but the relationship is clearly non linear (I had also no machines on as another referecne point). My own interpretation is that heat radiation has an effect. 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.

 

[256bits]

A black body emits thermal radiation based on the fourth power of temperature according to the stephan-Boltzmann 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.

If you consider the surface to be a greybody then absorbed radiation equals emitted radiation of the surface.

In your model, these and a few more parameters that need to be taken into account. Is the structure radiating into space on into another sphere greybody or balackbody. Are the machines to be considered as radiating spheres within the enclosur sphere and/or is conduction and convection to be considered.

for example, a sphere enclosed by another sphere and both exchanging energy by radiation is basic textbook. Adding more spheres should just be an extension of the problem.

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). In order to simplify the analysis and find the functional structure as a function of β, assume that the room is a ball.

For walls radiating to one another a shape factor from charts are available. If you choose to simpliy the problem by using spheres that would be an approximation of course, as the walls would not receive equal amounts of radiation.

Your model needs some more work done to it I think to figure out what more assumptions are necessary. Are you sure this isn't a homework question?

 

[balulu]

Thank you for your input, I appreciate your help.
I assure you that it is not a homework, it is a very serious and fundamental problem.


Yes, it can be assumed that the geometry is of a sphere, and the radiation is to the outside air. The boundery (the sphere surface) is very thin and the only way to get rid of the heat generated inside the sphere is through its surface.

I don't know the inside temperature, but I know the heat energy per unit of time generated by each machine when it is on. All machines are identical, for simplicity you can assume 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.

Another way to approach and simplify is to assume that the machines are uniformely distributed in the volume of the sphere.