Pumping number evaluation

Three dimensionless numbers are used to characterize the pumping ability of a CSTR. Flow within the vessel is composed of a discharge flow, $q_d$ $-$ i.e. the flow ejected from the impeller $-$, an induced flow, $q_i$ $-$ i.e. the flow entrained by the discharge jet $-$, and a secondary circulation flow, $q_c$ $-$ i.e. the upward directed flow near the walls of the vessel. Dimensionless flow numbers are derived by dividing discharge flow, induced flow and circulation flow by $N d^3$, that is the proportionality constant for the volumetric flow rate. Each flow number is used to characterize a different aspect of the stirring ability of the CSTR. The discharge flow number is used to determine efficiency of the impeller. It is defined as

$$N_{qd}=\frac{q_d}{N d^3}$$ (10)

where $q_d$ is the flow exiting from a control volume around the impeller. In this work, a cylindrical surface around the impeller is considered. Integration of radial components of velocity is made on this surface from the bottom of the tank to the height at which the radial flow remains outward directed. High $q_d$ values correspond to high discharge efficiency impeller, i.e. jets of fluid ejected from the impeller region are highly energetic and surrounding fluid can be easily entrained, transferring motion from the impeller region to the rest of the vessel. The induced flow number measures efficiency of the interaction between the impeller jet and the surrounding fluid. The entrainment ability is dependent from the geometrical configuration of the vessel. The secondary circulation flow number is used to characterize the flow in the upward direction. It is calculated as

$$N_{qc}=\frac{q_c}{N d^3}$$ (11)

Circulation flow is computed integrating the vertical component of velocity over the portion of the section normal to the rotation axis where it is positive. This section is taken as close as possible to the impeller. Evaluation of secondary circulation flow number is particularly important for CSTRs used for solid suspension and to promote mixing within large volumes. Pumping efficiency is calculated as

$$\eta=\frac{N_{qc}}{N_p}$$ (12)

and is used to give a measure of effectiveness of the impeller for generating mean flow per unit of power input [Hockey and Nouri, 1996].