The model thus far deals only with the gas flow and gas volumes. The solids distribution between these phases hasn't been addressed.
In most models of this nature, all of the solids are assumed to be in the emulsion phase. The bubble phase consists of either individual bubbles or jets of gas, or both, that contain little solids. Under that assumption, the fraction f has an upper limit based on the minimum void fraction for the emulsion phase. Beyond that limit, there is not enough gas in the emulsion phase to allow fluidization.
The minimum void fraction for fluidization is near 0.4. With that assumption, the maximum values of f at two values of alpha are shown below.
The parameter estimation scheme to be discussed in the upcoming posts ignores this "constraint". If the data indicate the best model has an f higher than "possible", it just means that some of the solids must also be present in the bubble phase. Thus, a tracer study could still be used to validate the fully plug flow model by Dasila et al. mentioned in the previous post.
In practice, the bed volume expands as gas rate is increased (for Geldart Type A powders). At some flow rate, bubbles form. As the rate is increased further, the extra gas goes into the bubble phase. If the gas does not exchange greatly between the phases, high rates of gas beyond the initial bubble formation rate would be a waste of gas because it would go through the reactor unreacted. In contrast, if high rates of gas appear to increase the reaction rate, that is an indication of significant cross flow between the phases.