RE: ATM Mirror cooling equation

["Arjan te Marvelde (EMN)" <Arjan.te.Marvelde@emn.ericsson.se>]

Nils Olof, Raphael,

: this is interesting stuff. 

Yes, i'm thinking about this for a while now; i believe there must be some model possible 
which gives a more scientific grip on the issue. This would give me a better idea of the 
problem than when i use my gut-feeling only. Part of the curriculum of my physics study 
was heat transport phenomena, only it's so long ago and also it did not have my full 
interest in these days. But i still have the books, so i'll be happily digging on!

: Been there, trying to sell vacuum cleaners ;-) At least if 
: your 1" thick mirror takes longer than 30 minutes to cool, 
: cooling isn't efficient enough.

So did you also try some modeling? Actually, i think that the equation i found is the 
ideal case only, and i don't know how much off you are in practise. Is it really 30 
minutes, or is 60 minutes more likely? I suppose the ideal case could be reached for 
example when the mirror is submerged in flowing water or so.


Raphael, thanks for your long mail giving a much simpler view on the problem. I think 
that your analysis is an approximation however, which may very well hold in practical 
mirror situations, but i don't know under which circumstances. Can you please detail on 
that?

What i see from your numerical example is that for a typical mirror the heat flux in the 
mirror is of the same order of magnitude as the heat flux to the environment (i.e. 22 vs. 
5-100 W/K.m^2), so cooling is not really governed by either conduction or conve
ction.

To understand the approximation made, the problem can be seen as an electrical circuit, 
where a capacitor is shorted with two series resistors. The resistors are the equivalent 
of 1/conductivity and 1/convection, the capacitor of the specific heat * mass of the 
mirror, where the voltage and currents are equivalents of heat flux and temperature drops 
respectively. So the mirror surface temperature is determined by the ratio of the two 
"resistors": if the conductivity is much lower than the convection, the surface 
temperature will be already quite low, and therefore (as you pointed out) a fan is not 
useful.

The problem with this approximation is, that the heat flux is in reality not constant 
throughout the mirror, because the temperature profile is not linear. There is always a 
point in the mirror where the temperature gradient is 0, while at the edges it is at 
maximum (represented by the cosine component in my equation). 

On the other hand, maybe you may as well use this linearisation. In that case, maybe you 
can detail on the conditions that allow this?

The equation i gave is the solution to the heat diffusion differential equation, where 
the static boundary condition is an infinitely efficient removal of heat from the mirror 
surface, i.e. the surface temperature is always at environment level; Tsur = Tenv.
A more realistic equation would be the solution to the differential equation, with a 
dynamic boundary condition like: Q = Hconv * (Tsur - Tenv). Tsur is then a function of 
time, like throughout the rest of the mirror. Convection is then treated as a linear 
conductivity component, as in the electric equivalent.

Do you (or anybody else) know the solution to that differential equation?

/Arjan te Marvelde