Chapter 4

# Equation of conservation of energy

### Fondamental : One-dimensional local energy balance (with or without source)

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)  Conservation of energy

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 : ### Attention : One-dimensional local energy balance (with or without source)

Without source : With sources : Previous
Fouriers‘s law
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Equation of heat diffusion