Equation of conservation of energy

Without source :

Let's consider an homogeneous body (in fact, most of the time a liquid or a solid one) with its density, its thermic conductivity and its heat capacity.

Those quantities are supposed to be constants.

In a first time, let's suppose that there is no sources able to provide heat locally in the medium.

Finally, we keep working in a single dimension named .

Let's apply the first principle of thermodynamics to a small volume  :

Where is the variation of the internal energy of the volume and the conduction thermal flow.

The internal energy of the volume is, at time  :

So :

Furthermore, the conduction thermal flow is :

Or :

The first principle of thermodynamics finally results in :

Finally : (equation of conservation of energy without source)

With heat sources :

Let's suppose now that heat sources are present in the medium.

Let's note the algebric volumetric power of the sources.

Example (Joule heating) :

If an electrical current runs through the material, the small volume , with its electric resistance, with running through it, receives during the energy :

Hence the volumetric power due to Joule heating :

We can also write (see the lesson about electromagnetic energy and Local Ohm's law) :

With sources, the energy balance becomes :

That is to say :