# What does 'k' in Newton's law of cooling represent?

1. Mar 24, 2013

### namitakn

Newton's law of cooling says:

Instantaneous rate of cooling = -k (Original temperature - Final temperature)

But what does this 'k' mean?

I know it depends on the nature of the surface; but what property does it correspond to?

Is 'k' related to specific heat capacity?
Does higher specific heat capacity mean higher k?

2. Mar 24, 2013

### Staff: Mentor

k includes everything about the setup, including heat capacity, thermal conductivity of materials, surface areas, the geometry and probably some things I did not think about.

3. Mar 24, 2013

### arildno

You ought to think of Newton's law of cooling in the same way as with Coulomb's law of friction:
They are both "hodge-podge" laws on the theoretical level in that they do not bother about the distinct mechanisms behind the overall complex phenomenon, but are nonetheless highly effective in predicting a variety of outcomes. Thus, they are archetypes on what constitute intelligent "engineering", rather than being helpful to probe the deeper secrets of the universe.

4. Mar 24, 2013

### dextercioby

To my (albeit modest) knowledge there's no derivation of Newton's law of cooling from the fundamental principles of non-equilibrium statistical mechanics, so that the "k" there is just a constant with dimensions of 1/time.

5. Mar 24, 2013

### AlephZero

The OP's statement of Newton's Law is too general to be of much predictive use. If you unpack the heat transfer into conduction, convection, and radiation, then conduction and radiation are fairly easy to relate to the underlying phyiscs, but convection is not.

I think "Newton's law of cooling" usually refers to the convective part, in the form $\dot Q = h A \Delta T$ where you might hope that the heat transfer coefficient $h$ was related to the underlying physics. In practice, $h$ is not a constant but can be modelled fairly well by (semi-empirical) functions of assorted non-dimensional fluid flow parameters, like the Reynolds, Prandtl, Grasshof, Rayleigh, etc, etc, numbers.

But given the current (lack of) understanding on how the Navier Stokes equations relate to the underlying physics, I'm not holding my breath waiting for an answer "real soon now".