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Re: exponential cooling



Ludwik Kowalski asked:

Does the exponential law of cooling apply to the radiational mechanism,
for example, to a very hot brick suspended in very good vacuum?

No. The actual time dependence for the law of cooling in this case is very
complicated and different parts of the brick obey different cooling laws. To
properly solve this problem one needs to solve the diffusion (parabolic PDE)
equation for the temperature field on the interior of the brick with a
Stefan-Boltzmann radiative boundary condition on the surface of the brick.

There is no closed-form solution for this problem for the general case.

The *simplest* special case is when we assume that the thermal conductivity
of the brick is infinite. In this case we can assume that the temperature of
the brick remains uniform throughout and is the same as the surface
temperature. If we let: T= abs. temp. of brick, T_0 = ambient abs. temp. of EM
radiation in the vacuum environment, T_i = initial abs. temp. of the brick,
c = specific heat of the brick, m = mass of the brick, A = surface area of the
brick, e = emissivity of the brick surface, s= Stefan-Boltzmann constant, and
t = the time elapsed since the problem started, then the solution for T = T(t)
obeys the implicit equation:

arctanh(T_0/T) + arctan(T/T_0) = arctanh(T_0/T_i) + arctan(T_i/T_0) +
2*(T_0^3)*A*s*e*t/(m*c) .

Now since a brick is a poor conductor of heat (um, er I mean thermal energy)
the actual problem is significantly more complicated than described above,
since as the surface temp. drops a temp. gradient is established across the
interior of the brick. For short times after the brick has begun to radiate
(from an initial state of a uniform initial temperature profile) the above
formula is approximately true with a slight modification. The value of m is
not the mass of the whole brick but the mass of a thin near-surface layer of
the brick which is in intimate thermal contact with the surface. After a
relatively short while due to the finite value of the thermal conductivity
even this layer has a temperature difference across it with the surface side
cooler than the interior side. As the temp. gradient forms across the sample
each part of the brick cools at a different rate and has its own T(t)
function. After a longer time the surface temp. has reduced to just a little
above the ambient temp. and its value is determined by the radiation rate
needed to accomodate the dissipation of the heat (thermal energy) flow across
this gradient supplying the surface from the brick's interior. This conduction
process is the rate-determining step at longer times. Since the cooling rate
eventually becomes proportional to the temp. difference across this gradient
and since the surface temp. is close to the ambient temp. we can surmise that
the long-time asymptotic tail on the cooling curve will approach some sort of
an exponential curve if the near-steady-state behavior at long times has the
temp. difference across the internal gradient much larger than the temp.
difference between the surface and the ambient. If the long-time temp.
differences do not obey this strong inequality, then it is possible that the
cooling rate curve never approaches an exponential--even at long times.

This is an interesting problem. I think it would be quite instructive to
numerically solve it for a simple case (such as for a uniform spherical brick)
and monitor the time dependence of the temp. field as a function of radius.
It would be interesting to see how the time dependences vary as the relative
values of parameters in the problem are varied. A natural time-scale for the
problem (a suitable unit for denominating the time parameter) is given by
t_0 = c*m/(k*R) where R is the radius and k is the thermal conductivity. A
dimensionless ratio Z which measures the relative importance of surface
radiation effects compared to bulk conductivity effects is given by
Z = (s*e*R*T_0^3)/k . It is to be expected that the system behaves very
differently when the Z parameter is large compared to 1 than when the Z
parameter is very small compared to 1. When the time t is long compared to
both t_0 and t_0/Z then the long-time asymptotic behavior sets in. It would
also be interesting to explore the effect of various initial temperature
profiles across the sample.

Anyone want to simulate this problem and report back what the results are? If
I had the time I would do it myself. Maybe this would make for a nice
undergraduate "research" project. If one was ambitious, various shaped
samples could be investigated.

Since this problem is almost the same as the problem of the cooling of the
interior of a planet after its initial formation I suspect that it is worked
out somewhere in the planetary science literature. Lord Kelvin worked this
out for the case of the cooling of the earth before it was discovered that
naturally occuring radioactivity in the earth's interior invalidated the
calculation and provided a long time heat (thermal energy) source for the
earth's interior. Since a planet's composition is far from uniform with a
constant specific heat and thermal conductivity the planetary science
literature is probably overkill for this problem. Possibly a key paper or
two refer back to the literature of the simpler uniform sphere case. (Maybe
Carl Sagan knows the answer to this problem. :-) )

David Bowman
dbowman@gtc.georgetown.ky.us