The determination of the entrainment requires knowledge of concentrations of each particle size in the bed. Figure 10 shows a general case and the various possible combinations (Geldart, 1986). For example:
- Without feed, 100% efficient cyclone; total recycle of product on cyclone;
- Continuous feed, cyclone's efficiency varies depending on dpi; partial recycle of the fines.
No matter the arrangement, the mass balances of each component and global must be reached. As an illustration, it is assumed that RE=RR=0, and F and Q are different from zero. The mass balance of the fraction considering dpi given by:
The global mass balance given by:
In addition, in a well-mixed bed: xQi = xBi
Substituting in the mass balance for each particle size and re-arranging:
This equation cannot be directly solved because RT depends on the values from xBi for each fraction. In practice a rapid convergence and an interaction error can be avoided by using RT=0 in the first attempt.
If the freeboard height is greater than 2m, and the column's diameter exceeds about 0.2 m, then for the fraction sizes will have Ut/U < 0,5, Isso simplifica e encurta os cálculos significativamente.