# Thermal Conduction and Newton's Law of Cooling

Fourier's law of thermal conduction states that $$\mathbf{j}=-k\nabla T,$$ where $\mathbf{j}$ is the heat flux. Integrating both sides of this equation over a closed surface gives the equation $$\frac{dQ}{dt}=-k\int \nabla T \cdot d\mathbf A.$$

If there is a temperature discontinuity across this surface, then $\frac{dQ}{dt}$ diverges, in contradiction with Newton's law of cooling. Are Fourier's law of conduction and Newton's law of cooling mutually incompatible?

Chestermiller
Mentor
Fourier's law of thermal conduction states that $$\mathbf{j}=-k\nabla T,$$ where $\mathbf{j}$ is the heat flux. Integrating both sides of this equation over a closed surface gives the equation $$\frac{dQ}{dt}=-k\int \nabla T \cdot d\mathbf A.$$

If there is a temperature discontinuity across this surface, then $\frac{dQ}{dt}$ diverges, in contradiction with Newton's law of cooling. Are Fourier's law of conduction and Newton's law of cooling mutually incompatible?
What makes you think there can be a temperature discontinuity at the surface? There, of course, can be a discontinuity of the temperature gradient at the surface, but this equation applies inside the region bounded by the surface.

Chet

What makes you think there can be a temperature discontinuity at the surface? There, of course, can be a discontinuity of the temperature gradient at the surface, but this equation applies inside the region bounded by the surface.

Chet

Well, let's imagine that you put a warm bottle of beer in a refrigerator to cool it down. At the surface of the bottle there is (at least initially) a temperature discontinuity, because the beer and the air in the fridge are at different temperatures. Newton's law of cooling has no trouble handling this, but Fourier predicts (at least initially) an infinite rate of cooling.

Newton's law of cooling has no trouble handling this, but Fourier predicts (at least initially) an infinite rate of cooling.

It predicts an infinite rate of cooling of the infinitesimally thin layer of the can that is in contact with the cold air, which is probably approximately right.

Chestermiller
Mentor
Well, let's imagine that you put a warm bottle of beer in a refrigerator to cool it down. At the surface of the bottle there is (at least initially) a temperature discontinuity, because the beer and the air in the fridge are at different temperatures. Newton's law of cooling has no trouble handling this, but Fourier predicts (at least initially) an infinite rate of cooling.
Yes, this is true, but it only lasts an instant. And the cumulative amount of heat transferred at short times will be proportional to time to the 1/2 power. One can determine this by solving the transient heat conduction equation in the region near the boundary using a similarity solution (i.e. Boundary layer solution).

Chet