Theoretical considerations often require budget equations of certain physical quantities such as mass, momentum and energy in its various forms. The purpose of this chapter is to derive the general form of a balance or budget equation which applies to all extensive quantities and those intensive quantities derived from them. Intensive variables such as p or T cannot be balanced.

Let us consider a velocity field v(r, t) in three-dimensional space. A volume V(t) whose volume elements dτ are moving with the velocity v(r, t) is called a fluid volume. The fluid volume may change its size and form with time since all surface elements dS of the imaginary surface S enclosing the volume are displaced with the velocity v(r, t) existing at their position.

Now we envision V(t) to consist of a certain number of particles also moving with v(r, t). Obviously, at all times the particles that are located on the surface of the volume remain there because the particle velocity and the velocity of the corresponding surface element dS coincide. Thus, no particle can leave the volume so that the number of particles within V(t) remains constant. Therefore, a fluid volume is also called a material volume.

In analogy to the fluid or material volume we define the fluid or material surface and the fluid or material line. As in the case of the material volume, a material surface and a material line consist of a constant number of particles.