Conservation Laws with No Particle Interaction and Collisions

By substituting for p mass, momentum, and energy, the corresponding conservation laws are easily obtained from Maxwell’s equation (2.30). For a case with no particle interaction/collisional contribution,

Conservation of Mass

Let p = m, since nm = e_{s}p_{s} = p

Conservation of Momentum

Let p = mc

Since P_{k} = p(CC) and (C) = 0

Conservation of Solid-Phase Fluctuating Energy

Let p = 1 mc^{2}

Note: q_{k} = |p C^{2}Cfc = 2nm(C^{2}C) and since в = C^{2})

It is easy to derive the conservation equation for the fluctuating energy from Eq. (2.38) as

where P_{k} = p(CC).

Similarly, we can obtain the equations for the stress tensor (CC). This equation is similar to the Reynolds stress equations in single-phase turbulent flow. However, in the Reynolds stress equation, the average is over a time interval. Here, the averaging is over the velocity space. These averages are not equal as experimentally shown by Tartan and Gidaspow (2004). If we include rotation (Goldshtein and Shapiro 1995) in addition to the translation presented here, we can obtain a balance for the rotational temperature, as in single-phase fluids (Condiff et al. 1965).