Consider an open system of mass, m_{h} that gains mass and thus energy at a rate d|r. The energy balance moving with phase i becomes (Gidaspow 1977, 1994)

where

and where the rate of heat transfer is related to the flux q_{i} by relations such as those used by Ishii (1975)

where A(t) is the area enclosing the volume of the system at any instant of time. The differential element of surface area of system i was taken to be simply e_{i}da, thus making no distinction between area and volume fraction.

The energy dissipation by means of friction is shown as the dissipation term D_{iss}

Now, by applying the Reynolds transport theorem to our original energy balance, we obtain

where h_{i}^{n} is defined as the net enthalpy per unit mass entering system i at possibly nonequilibrium conditions

The above energy balance could be written in terms of enthalpy as

Furthermore, the entropy form of the energy equation can be obtained by using the fact that the internal energy of phase i depends upon the entropy of phase i and upon the specific volume of phase i, as the following:

The second law of thermodynamics states that the entropy production for the mixture is zero for reversible processes and is positive for real irreversible processes. The energy equations in entropy form can be added to produce the entropy production for the mixture of i phases. As shown by Gidaspow (1994), some of the early multiphase energy equations violate the second law. Hence, it is necessary to check whether the equations programmed into the commercial computational fluid dynamics (CFD) codes satisfy the second law.

Expressions for entropy production are also needed for the design of energy efficient processes. For example, distillation column design is routinely done using availability analysis (Fitzmorris and Mah 1980), and vapor compression air-conditioning systems (ASHRAE 1977) are routinely designed by minimizing the entropy production for the vapor compression.