# Thermal conductivity

## Intro

The law of cooling says that that rate of cooling is proportional to the temperature difference.

${dT \over dt}=k({T_2 \over T_1})$

This law assumes a constant heat capacity of materials which is true for a material over a small enough temperature range that is not on a phase boundary. The law is a consequence that the heat flow (watts/m^2) is also proportional to the temperature difference:

$Q=(J/L) \Delta T$

or in terms of differential quantities:

$Q=J {dT \over dL}$

Where
Q is the heat flow (e.g units watts per meater squared)
J is the thermal conductivity
L is the length
T is the temperature
DT/dL is the change in temperature with respect to length

## Related Properties

The thermal conductivity can be used to determine the aggregate heat flow properties thermal conductance, ${J A \over L}$ or it’s reciprocal thermal resistance.

It should be noted that the terms thermal conductance and thermal resistance, which refer to the total resistance to heat flow, are rather confusing names when compared to the names electric conductance and electric resistance. They are called by the same names not because they are the same thing but rather because they are mathematically analogous. Like electrical resistances, thermal resistances in series are additive. Thus, if you have three layers of material with thermal resistances R1, R2 and R3, then when piled on top of each other their total thermal resistance is:

Rtotal = R1 + R2 + R3 + ...

which is exactly the same equation you would use to describe adding resistances to an electric circuit.