Understanding Newton's Law of Cooling and Its Derivation | dQ/dt = KA(dT/dx)"

[andyrk]

Newton's law of cooling is: dQ/dt = KA(θ - θ_o). Then where does the equation dQ/dt = KA(dT/dx) come from?

 

[Orodruin]

Newton's law of cooling is: dQ/dt = KA(θ - θo).

This generally applies to the surface interface between two materials.

Then where does the equation dQ/dt = KA(dT/dx) come from?

This is Fourier's law (or something reminiscent of it, you really should define what you mean by Q). It applies to the heat transfer within a material.

 

[andyrk]

This generally applies to the surface interface between two materials.
This is Fourier's law (or something reminiscent of it, you really should define what you mean by Q). It applies to the heat transfer within a material.

How can dQ/dt have two different dimensions?

 

[nasu]

K means different things in the two formulas.

 

[andyrk]

I would say that Q means different things too. Q in Newton's law of cooling is temperature whereas in Fourier's law it is heat.

 

[Orodruin]

I would say that Q means different things too. Q in Newton's law of cooling is temperature whereas in Fourier's law it is heat.

Well, Fourier's law is actually just a statement on the current. What appears in the left hand side is the heat transfer per unit time across a surface. This can be related to an actual change in temperature (or heat, they are related by heat capacity, volume, and density) through the continuity equation.