Underflows to-throughput ratio

Particle Re)aiolds number for the terminal velocity of the particle in the fluid Reynolds nvmiber for power-law fluids Underflow-to-throughput ratio Resistance of filter medium (1/L)... 

Note that the S-shaped grade efficiency curves do not necessarily start from the origin—in applications with a considerable underflow to throughput ratio (by volume) R, the grade efficiency curves tend to the value G x) = / f as X —> 0. This is a result of the splitting of the flow, or dead flux that carries even the finest solids into the underflow in proportion to the volumetric split of the feed. Section 3.4 discusses possible modifications to the efficiency definitions which account for the volumetric split and illustrate only the net separation effect. Such reduced efficiencies are widely used for hydrocyclones and nozzle-type disc centrifuges where large diluted underflows occur. 

Rf = UIQ is the underflow-to-throughput ratio (by volume) which is the minimum efficiency due to dead flux. 

The effect of flow splitting (or dead flux ) in applications with appreciable and dilute underflow, as is common with some separators, is to modify the shape of the grade efficiency curve to make it appear as if the performance of the separator were better than it actually is. As shown in Figure 3.15, the curve does not start from the origin (as it should for inertial separation) but has an intercept, the value of which is usually equal to the underflow-to-throughput ratio Rf. This is because the very fine particles simply follow the flow and are split between the underflow and the overflow in the same ratio as the fluid. The R ratio is defined as the fraction of the volumetric feed rate which turns up in the underflow, i.e. the underflow rate, divided by the feed rate. 

The downward current is partially counterbalanced by an upward flow in the core region, depending on the underflow-to-throughput ratio. There is a well defined locus of zero vertical velocities (LZVV) which follows the profile of the cyclone. 

Design variables are not the only ones to affect the cyclone number. While the effects of pressure, flow rate, viscosity and the densities of the fluid and particles are included, the strong influence of the feed concentration and of underflow orifice control are not. Most of the theories only apply to low solids concentrations, some (Holland-Batt, Schubert and Neesse) offer a correction for hindered settling and others consider to some extent the effect of the underflow-to-throughput ratio (Bradley) or the limited capacity of the underflow orifice (Schubert and Neesse). Once again, the effect of solids concentration for the purpose of scale-up is best described by dimensionless correlations derived from pilot tests and this is shown in some detail later. 

At low solids concentrations of below 1 or 2% by volume, the flow pattern in hydrocyclones is unaffected by the presence of particles, and particle-particle interaction is negligible. The volume of the particles that separate into the underflow is small and the underflow-to-throughput ratio, R, is usually assumed to have no effect on the cut size, xso, except for the effect of flow splitting which can be easily accounted for by using the reduced efficiency concept (section 3.4.1). Dimensional analysis coupled with the conclusion of most of the simple theories of separation in hydrocyclones... 

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