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# Power Number

This is a continuation of the Process Intensifier - Optimization with CFD: Part 1 paper.

The power is computed directly from integrating the moment, or torque, of the surface tractions about the axis of revolution of the shaft for the impeller surfaces. These are directly influenced by the RPM of the shaft, and the fluid rheology properties. In this particular case, the flow regime is highly turbulent, given the impeller size, speed, and viscosity of the fluid (water). Therefore, the Spalart-Allmaras turbulence Model (SAM) with standard wall functions was used to appropriately account for the wall traction effects of the radial flow impellers. The Spalart-Allmaras Model does not require extremely fine boundary layer mesh resolution. Y+ values on the order of 100 or more are acceptable. The mesh around the axial impellers was much finer and so SAM was not needed. The resultant power number is computed using the standard relationships with power, density, impeller speed and diameter. Power numbers of many types of impellers can be found on the Impeller Page.

## Power Number of the Impellers in Standard Tank Geometries

To see just how good the CFD could predict the power numbers, we examined Case R (Fig. 6). It had a single 5" (127 mm) RP4 (4-bladed radial paddle) with h/D=0.2 (identical to the impellers used for Model 1) and ran at 360 RPM (approximately 5 Hp/1000 gallons = 1 kW/m3) in a 12.5" (317.5 mm) diameter vertical tank (D/T=0.4) equipped with 4 straight wall baffles, w/T=0.1 and filled to a liquid level equal to the tank diameter, Z/T=1. The impeller was placed at 1/3 the tank diameter off bottom. The CFD program predicted a Np = 2.985 (or 3.0). The power number should be Np = 3.4. The ratio of the CFD predicted Np and the literature Np-value is 0.88. Oldshue (Oldshue 1983) predicts the proximity factor of radial impellers (PF = ratio of powers based on the location of the impellers and other internals in a tank) for our verification model to be 0.87 (OB/D=0.83, COV/D=1.67). Hence the CFD is within 1% of established correlations based on experimental results with no tuning of any CFD parameters. This confirms the usage of the Spalart-Allmaras model to approximate the boundary layer near the radial impeller.

Case R

Click on the picture for a bigger picture

The power numbers derived by the CFD are in extremely good agreement with experimental data found in the literature. In both radial and axial cases the agreement was within 1%. There should be great confidence in the numbers predicted by the CFD for the same impellers but in a different environment.

Case A

Fig 7: Case A
PBT axial impeller)

## Power Numbers for the Process Intensifiers

The following power numbers were not experimentally verified, because experimental data for these configurations are not available. The impellers used for Model LTR are the same as those studied in Case R, which showed agreement to published results within 1%. The impellers of Model HGR are the same as for Model LTR except that the blades are wider. The impellers of Model LTA are the same design as the impellers used in Case A, except that the diameter was only 3.5" (88.9 mm). The geometry of the blades was the same, except that the hub diameter to impeller diameter was a little bit larger. The impellers for Model HGA were identical to the ones used in the Case A tests.

 Power Numbers (Np) Model LTR Model HGR Model LTA Model HGA Click on any picture for a bigger picture Lightnin Radials Hayward Gordon Radials Lightnin Axials Hayward Gordon Axials 0 GPM 2.3, 2.2, 4.5 3.0, 2.9, 5.9 0.69, 0.78, 1.47 0.65, 0.62, 1.27 650 GPM 2.0, 2.5, 4.5 2.4, 3.5, 5.9 0.76, 0.75, 1.51 0.65, 0.62, 1.27 1100 GPM 2.0, 2.7, 4.7 3.0, 3.7, 6.7 0.83, 0.80, 1.63 0.61, 0.68, 1.29 Table 1: Power Numbers (Np) for each model. The impellers rotational speed is 1750 RPM. The Power Numbers are listed in the following order: lower impeller, upper impeller, total.

The individual power numbers of Model LTR are lower than our verification in a "standard" mixing tank. This means that the proximity factors for this geometry are lower, about 0.59-0.79. The off-bottom of the lower impeller is 0.5T and the off-bottom of the upper impeller (to the orifice plate) is also 0.5T. The coverage of the lower impeller to the orifice plate is also 0.5T, whereas the upper impeller has a higher coverage, because of the T-pipe section.

The Oldshue proximity factor (PF) correlations are strictly reserved for fully baffled, vertical cylindrical tanks. For comparison purposes, such a tank with one impeller at 2.5" (63.5 mm) off-bottom and a coverage of 2.5" (63.5 mm) (to the orifice plate) results in a PF = 0.56 and therefore Np = 1.90. The value of 2.0 for the lower impeller is very close. Remember that in the Line Blender, 650 to 1100 GPM (148 -250 m3/hr) flow is flowing against the impeller, which is increasing the Np. For the upper impeller, the off-bottom is the same (to the orifice plate), but the coverage is now to the top of the T, or 7.87" (200 mm) (COV/D=1.57). According to Oldshue, PF = 0.77 or Np = 2.62. It is interesting and reassuring that the two Line Blender cases are within 4% of this value.

Power numbers of the wide-bladed radials from Hayward Gordon are not readily available. If we assume that the power of a radial impeller is proportional to the height of the blades (Nagata 1967), the power numbers of the Hayward Gordon (HG) impellers would have to be 0.3/0.2 = 1.5 times higher than the Lightnin impellers. Therefore, the lower HG radial should have a power number of about 3.0, which they do. The uppers would be approximately Np = 3.9, which is again very close to 1.5 times the Lightnin impellers.

The axial impellers are 3-bladed, pitched bladed impellers at 30 degrees but with a wider blade. The Hayward Gordon HGA had higher power numbers than our standard Case A, even though it is the same impeller. What is interesting is that the two impellers are opposing each other in flow direction. The upper impeller is a down-pumper, while the lower impeller is an up-pumper. The outlet flows of the impellers are impinging on each other. This is obviously driving up the power, with PF = 1.13 - 1.18. Furthermore, the spacing of the two impellers is 4" (102 mm) (or S/D=0.8. If the impinging zone is considered to be exactly in the middle between them, the distance to this plane is only S/D=0.4. It is a well-known fact that as axial impellers get closer to an impinging "wall", PF increases. It seems reasonable that the case of impinging flows would increase Np beyond the PF of impinging on a stationary wall.

The only difference between the LTA and HGA impellers is the diameter and the flow direction. It is interesting and unexpected that the power number of LTA is about 16% more than the HGA at 0 GPM and 25% more than HGA at 1100 GPM (250 m3/hr). Both of the LTA impellers are smaller and both are down pumping. This impeller configuration shows the greatest effect of flow rate on the power number. Maybe this is because the impellers are smaller in diameter and the cross flow has a greater impact on these impellers. Maybe the second horizontal baffle on the bottom of the pipe has a significant effect on hindering a swirl in the pipe, therefore driving up the power number. Since the horseshoe baffle is continuous around the bottom of the pipe, the flow through the pipe may be diverted into the outlet flow of the lower impeller causing the power to go up. Without flow, the upper impeller is pumping more than the lower impeller, most likely due to the larger entrance volume (T-section) above the impeller (see Fig. 3).

In all of the above cases except LTA, the side flow does not seem to have a great effect on the power number up to 650 GPM (250 m3/hr). Some of the models did see minor changes at 1100 GPM (148 m3/hr). HGR saw the greatest impact, as the impellers "appear" to be working harder.