Based on the continuum theory, conservation laws for mass, momentum, and energy for disperse multiphase flow can be derived using the Reynolds transport theorem, as illustrated by Gidaspow (1994). For multiphase flow systems, the only new concept in this approach is the introduction of phasic volume fraction, e,. For a single-phase system, e = 1, these equations must reduce themselves to the equations found in standard transport phenomena books, such as those of Bird et al. (2007). Here, we briefly show how these equations are derived, using a Lagrangian representation.

Assume that a system of constant mass goes through temporal and spatial changes as presented in Fig. 1.1.

The point (x^{0}, y^{0}, z^{0}) represents the spatial coordinates of the particle at some fixed time t^{0}. Then, the spatial coordinates of the particle at any time are given by functions of

In space, we define a property per unit volume 3(t,x), where t is time and x is the position vector such that (Aris 1962)

F(t) is the system variable quantity that can change with time. The balance made on F(t) gives the Reynolds transport theorem (Aris 1962)

H. Arastoopour et al., Computational Transport Phenomena of Fluid-Particle Systems, Mechanical Engineering Series, DOI 10.1007/978-3-319-45490-0_1

Fig. 1.1 Motion of a system of constant mass

In multiphase flow, the volume occupied by phase i cannot be occupied by other phases at the same position in the space at the same time. This distinction introduces the concept of the volume fraction of phase i, e_{i}. The volume of phase i, V_{i}, in a system of volume V, is (Gidaspow 1977, 1994)