Flow-number

The values of the discharge flow number and the circulation flow number are calculated for the different working conditions. Discharge flow number is defined as

$$N_{qd}=\frac{q_d}{N d^3}$$ (4)

where $q_d$ is the flow exiting in the radial direction from a control volume around the impeller. The volume extends from the bottom of the vessel up to the height at which the radial flow reverts. Circulation flow number is defined as

$$N_{qc}=\frac{q_c}{N d^3}$$ (5)

where $q_c$ is the flow directed upward in a section normal to the rotation axis just above the impeller. Values of the discharge flow and of the circulation flow are gathered in Table 4.4. Fluid density and rotation speed are also recalled. Discharge and circulation flows are calculated as described in Appendix A. The circulation flow is larger than the discharge flow since fluid is entrained by the jet discharged from the impeller as the flow moves toward the wall of the vessel.

Table 4.4: Discharge flow and circulation flow for the different simulations.

$Re$	Discharge flow	Circulation flow	Density	RPM
$-$	$[kg/s]$	$[kg/s]$	$[kg/m^3]$	$-$
109	2.288	2.612	1200	84
165	7.405	7.864	1259	200
2040	3.781	3.829	1181	80
4592	8.784	8.927	1181	180
43200	3.123	3.182	1000	80
108000	7.711	7.847	1000	200

Values of discharge flow number and circulation flow number are plotted against Reynolds number in Figure 4.13.

Figure 4.13: Discharge Flow number and Circulation Flow number versus Reynolds number.

Power number values and corresponding discharge flow number are also used to calculate the pumping efficiency in the different operating conditions. The pumping efficiency is defined as:

$$\eta=\frac{N_{qd}}{Ne}$$ (6)

Figure 4.14 shows variation of pumping efficiency as a function of Reynolds number. Pumping efficiency stabilizes around 0.6 for Reynolds number larger than 2000.

Figure 4.14: Pumping efficiency versus Reynolds number.