Balance Equations — The Conservation Laws

Abstract

Let us consider an extensive variable defined by the volume integral (*)

$$$$

(2.1)

Assuming the boundary surface Ω at rest, we follow the time change of (2.1). This may be split into two parts:

$$$$

(2.2)

where d_eI denotes the variation due to the exchange with the external world, and d_lI the source of the quantity I inside the system. A conservation law for the variable I implies:

$$$$

(2.3)

As an example, let us consider the first law of thermodynamics, which expresses the conservation of energy (I = E).