Density, Pressure, and Equation of State

Assume that the particles can be regarded as molecules of a second gaseous species. Hence, multiple or interactive collisions among particles, and forces between particles and gas molecules can be neglected. However, the finite volume of particles may be taken into account. It is shown later that with this type of treatment the mixture behaves as a van der Waals gas without correction for the interaction force between the molecules. 

Denote ap as the volume fraction of particles in a gas-solid mixture. The volume-averaged densities of both gas and particle phases are then given by 

It is convenient to express the particle loading by the particle mass fraction, which depends on the material densities of the phases as well as the volume fractions of the phases and is defined by 

Since the particles are treated as a second gas phase, it is important to consider their contribution to the pressure of the mixture. The molecular weight of this gas is 

the ratio of the pressure of the gas p to the pressure of the mixture pu becomes 

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