Some Conclusions and Fourier's Analytical Theory of Heat

Some conclusions concerning Newton's law of cooling and the leaky vessel model

1.) Although we were initially skeptical about Newton's law the data collected, our evidence, indicates that it may account for most of the cooling but there is a discrepancy that grows over time. It looks like Newton's law reaches equilibrium too soon and the attempted fix it was flawed. More study may eventually resolve this issue although we are still left with doubts.

2.) The leaky vessel analogy indicates cooling is governed by a surface thermal barrier with heat corresponding to a quantity within the vessel and temperature to a thermal pressure. The actual nature of the barrier is still open to question. It appears to communicate heat two ways. Only a fraction of the of heat on either side of the barrier makes it across in a given period of time. We also note that radiant heat like light requires a medium for its propagation and can be both reflected by transmitted at a boundary between two media.

The Analytical Theory of Heat

So we will shelve our doubts about Newton's law of cooling for the time being and move on to Fourier's Analytical Theory of Heat (1822).

We return again to the flux tube in spherical coordinates. There is a relation between the flow rate through the tube and the rate of change of the temperature with distance known as Fourier's law, a constitutive equation. It describes the "resistive flow" of heat through through a porous medium and one can draw an analogy with fluid flow through long tubes of very small diameter where surface friction dominates and the resulting steady flow is determined by its equilibrium with the applied pressure. Starting with the equivalent of Ohm's law we can deduce Fourier's law.

Referring to the flux tube above and using Fourier's law we then derive Fourier's equation which governs the change in temperature within a solid spherical body.

Fourier then uses Newton's law of cooling to determine the boundary conditions at the surface of the sphere.

Supplemental (Mar 18): I noticed a few loose ends in the blog. First, dΩ is the solid angle for the radial flux tube section. Secondly, dS was pulled out of the conductance G=1/ρ making it the conductivity per unit surface area. Aslo, dQ_tube is the heat gained by the tube due to the difference between the change in heat at the ends. Finally, Ctube is the heat capacity of the flux tube section and c_sp is its specific heat capacity or heat capacity per unit mass.