ATM Mirror cooling equation

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

Dear ATMs,

While digging through my old university books i found an equation which may be of use to 
ATMs. The equation is a solution to the heat diffusion differential equations. It gives 
the temperature distribution in an infinite plate of thickness 2L, for large times t. 
I think it may be a good estimate for the thin mirrors most people use. Although i belief 
it is valid, I can not guarantee the correctness of the following.


        T'(x,t) = (4/pi) * exp{-t*a*sqr(pi/2L)} * cos{(pi * x)/(2L)}

where 
        T' = (T - T1)/(T0 - T1)
        T0 is the mirror start temperature
        T1 is the environment temperature

        a  = l / (r*C)
        l is the heat conduction coefficient
        r is the material density
        C is the specific heat at constant pressure

        x  = 0 at the plate center
        x  = L at the plate surface

Boundary conditions are:
        t > 0.2 * sqr(L) / a
        the temperature of the plate surface is kept at T1

The second condition can be approached by keeping the boundary layer as thin as possible, 
i.e. by forced air movement. Convection alone will yield a higher actual surface 
temperature, and therefore slower cooling.

So what does this mean for e.g. a plate glass mirror of 250 x 20 mm?
Approximate values for
        l : 1 W/m.K
        r : 2500 kg/m^3
        C : 840 J/kg.K

T'(0,t) = 1.25 * exp( -0.01 * t);    (i.e. the center temperature)

According to this equation, to get the temperature down to 1% of the original difference 
would (in ideal circumstances) take approximately 480 seconds, or 8 minutes.

Of course we're not living in an ideal world: the surface temperature is most probably 
higher than the environment, and also usually the environment temperature is constantly 
dropping. A deviation that actually speeds up the cooling is the fact that a mirror is 
not an infinite plate, but also looses heat through the sides.
Apparently the sum of the deviations cause real cooling times to be much longer, but at 
least the equation gives an idea of where the end of the road is.

I hope this adds to our understanding of the thermic behavior of mirrors.

/Arjan te Marvelde