5. Results of multiobjective optimization

A hybrid MOGA based on the response surface constructed from the RSA model is employed to obtain the global POSs by using a controlled elitist genetic algorithm (a variant of NSGA-II) for three objective functions. Figure 8 shows the three-dimensional POSs based on the three objective functions obtained using a hybrid MOGA combined with the RSA model. Here, the values of all the objective function are normalized according to the corresponding values in the reference design. Three-dimensional POSs are obviously the trade-off among the conflicting objective functions. As a result, a trade-off analysis shows an obvious correlation between the hydraulic efficiency and radial force sources. The arbitrary optimum design (AOD) is randomly extracted near the end of the POSs, which exhibits the best performance in terms of all objective functions, as shown in Figure 8. The AOD has objective function values that are remarkably improved relative to those in the reference design. Consequently, the value of each objective function in the AOD shows improvements of approximately 49%, 80%, and 4% in the sweep area (As) of the radial force during one revolution, the distance (Ds) of the mass center of the sweep, and the hydraulic efficiency (η), respectively, in comparison with the reference design. On the other hand, a relatively large error among the three objective functions is observed, especially for the distance of the mass center of the sweep. Nevertheless, the values obtained by the numerical analysis are better compared with the reference design.

To understand the optimization results, the trade-off of the POSs in each two-dimensional functional space is shown in Figure 9. As shown in Figure 9(a) and (b), the decrement in the distance of the mass center of the sweep clearly leads to deterioration in the other objective

Figure 8. Three-dimensional POSs based on three objective functions. (a) ds-as (b) ds-eff. (c) as-eff.

functions. Specifically, the reduced distance of the mass center of the sweep is obtained at a lower efficiency and higher sweep area of the radial force during one revolution. However, the efficiency and sweep area of the radial force during one revolution shows a positive relation, as shown in Figure 9(c). The trade-off analysis of the POSs therefore allows an engineering designer to choose any economic solution according to the required design conditions.

Figure 10 shows the isosurfaces having a low velocity of 2 m/s. As shown in Figure 10(a), an extremely low-velocity region is formed along the internal wall in the impeller flow path in the reference design, whereas a similar low velocity isosurface is reduced considerably in the arbitrary optimum model (Figure 10(b)). These results illustrate the enhancement of hydraulic efficiency in the arbitrary optimum model as a result of optimization.

Figure 11 shows the distributions of unsteady radial force sources, averaged at the boundary surface near the impeller outlet, during one revolution of the impeller for both the reference and AODs. Here, both values are normalized by the value of the maximum radial force in the reference design. The sweep area constructed from the unsteady radial force sources in the reference design leans slightly toward the four quadrant directions from the origin, whereas it is formed near the origin in the AOD. Furthermore, the sweep area in the AOD is remarkably decreased compared with that in the reference design. Consequentially, as discussed already, the sweep area and the distance of the mass center of the sweep in the AOD are decreased by 49% and 80%, respectively, compared with those in the reference design.

design. As shown in Figure 12, the amplitude values of the fluctuation of the net radial forces in the AOD decrease considerably for most theta angle positions, especially in the region where the value of theta is 100. In addition, its level is also less than the normalized value of 0.5 and mostly flat compared with the reference design. These phenomena clearly highlight the

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considerable decrease in the radial force sources as a result of optimization.

Figure 10. Isosurfaces having a low velocity of 2 m/s. (a) Reference design (b) Arbitrary optimum design.

Figure 9. POSs on two-dimensional functional space.

Figure 12 shows the unsteady fluctuations of the net radial forces for the reference and AODs during one revolution. Both values are also normalized by the maximum value in the reference State-of-the-Art Design Technique of a Single-Channel Pump for Wastewater Treatment http://dx.doi.org/10.5772/intechopen.75171 193

Figure 9. POSs on two-dimensional functional space.

functions. Specifically, the reduced distance of the mass center of the sweep is obtained at a lower efficiency and higher sweep area of the radial force during one revolution. However, the efficiency and sweep area of the radial force during one revolution shows a positive relation, as shown in Figure 9(c). The trade-off analysis of the POSs therefore allows an engineering

Figure 10 shows the isosurfaces having a low velocity of 2 m/s. As shown in Figure 10(a), an extremely low-velocity region is formed along the internal wall in the impeller flow path in the reference design, whereas a similar low velocity isosurface is reduced considerably in the arbitrary optimum model (Figure 10(b)). These results illustrate the enhancement of hydraulic

Figure 11 shows the distributions of unsteady radial force sources, averaged at the boundary surface near the impeller outlet, during one revolution of the impeller for both the reference and AODs. Here, both values are normalized by the value of the maximum radial force in the reference design. The sweep area constructed from the unsteady radial force sources in the reference design leans slightly toward the four quadrant directions from the origin, whereas it is formed near the origin in the AOD. Furthermore, the sweep area in the AOD is remarkably decreased compared with that in the reference design. Consequentially, as discussed already, the sweep area and the distance of the mass center of the sweep in the AOD are decreased by

Figure 12 shows the unsteady fluctuations of the net radial forces for the reference and AODs during one revolution. Both values are also normalized by the maximum value in the reference

designer to choose any economic solution according to the required design conditions.

Figure 8. Three-dimensional POSs based on three objective functions. (a) ds-as (b) ds-eff. (c) as-eff.

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efficiency in the arbitrary optimum model as a result of optimization.

49% and 80%, respectively, compared with those in the reference design.

Figure 10. Isosurfaces having a low velocity of 2 m/s. (a) Reference design (b) Arbitrary optimum design.

design. As shown in Figure 12, the amplitude values of the fluctuation of the net radial forces in the AOD decrease considerably for most theta angle positions, especially in the region where the value of theta is 100. In addition, its level is also less than the normalized value of 0.5 and mostly flat compared with the reference design. These phenomena clearly highlight the considerable decrease in the radial force sources as a result of optimization.

Figure 11. Unsteady radial force distributions during one revolution of the impeller.

Figure 12. Unsteady net radial force fluctuations during one revolution of the impeller.

Figure 13 shows the time history of instantaneous unsteady pressure contours at the boundary surface near the impeller outlet for both the reference and AODs. Here, both values are normalized by the maximum pressure value in the pressure contours. Both the instantaneous unsteady pressure contours are compared for one rotation τ of the single-channel pump impeller. This rotation is divided into six steps to clarify changes in flow structure with time during one revolution of the impeller, as shown in Figure 13. In the reference design, highpressure zones occur widely on the boundary surface near the impeller outlet, as shown in

Figure 13. Unsteady pressure contours during one revolution of the impeller. (a) T = 1/6τ. (b) T = 2/6τ. (c) T = 3/6τ. (d)

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T = 4/6τ. (e) T = 5/6τ. (f) T = 6/6τ.

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Figure 13. Unsteady pressure contours during one revolution of the impeller. (a) T = 1/6τ. (b) T = 2/6τ. (c) T = 3/6τ. (d) T = 4/6τ. (e) T = 5/6τ. (f) T = 6/6τ.

Figure 13 shows the time history of instantaneous unsteady pressure contours at the boundary surface near the impeller outlet for both the reference and AODs. Here, both values are normalized by the maximum pressure value in the pressure contours. Both the instantaneous unsteady pressure contours are compared for one rotation τ of the single-channel pump impeller. This rotation is divided into six steps to clarify changes in flow structure with time during one revolution of the impeller, as shown in Figure 13. In the reference design, highpressure zones occur widely on the boundary surface near the impeller outlet, as shown in

Figure 11. Unsteady radial force distributions during one revolution of the impeller.

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Figure 12. Unsteady net radial force fluctuations during one revolution of the impeller.

Figure 13(b), and a high-pressure zone caused by impeller-volute interactions becomes gradually larger. Consequently, this results in the unbalancing phenomena, along with the fluidinduced vibrations caused by unsteady radial forces, throughout the annulus passage area of the pump. Thus, the sweep area constructed from the unsteady radial force sources leans slightly toward the four quadrant directions from the origin, as shown in Figure 11. In the AOD, the pressure distribution is generally uniform; especially, at the same instantaneous time, the large high-pressure zone caused by impeller-volute interactions is obviously

suppressed, as shown in Figure 13(b). The AOD results in mostly stable flows throughout the annulus passage area of the pump. This explains the considerable decrease in the fluid-

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Figure 14 shows the spectra of the magnitude values at observation points on the casing wall for the reference and AODs. Here, the spectra are calculated based on the wall pressure fluctuation time history by using a fast Fourier transformation algorithm. Both magnitude values are normalized by the first blade passing frequency (BPF) in the reference design, and these values are also related to the vibration of pump. As shown in Figure 14, the BPF is approximately 30 Hz (BPF = Blade number rpm/60). The peak magnitude values are clearly seen at every harmonic BPF in steps of 30 Hz, which are due to the periodic motion of pump impeller rotation. In the AOD, a considerable decrease in the magnitude values at the first BPF is observed specifically, as well as at all observation points, especially for points B and C. It clearly shows that the large high-pressure zone caused by impeller-volute interactions is obviously suppressed, as shown in Figure 13(b). Consequently, the considerable decreases in

induced vibration caused by impeller-volute interaction owing to optimization.

these magnitude values reduce the vibration caused by impeller-volute interaction.

A state-of-the-art design technique was introduced for a single-channel pump for realizing both high efficiency and low-fluid-induced vibration. The technique is based on a theoretical approach and three-dimensional steady and unsteady numerical analyses. Furthermore, advanced multidisciplinary numerical design optimization techniques were discussed in detail to simultaneously improve hydraulic efficiency and reduce the flow-induced vibration caused by impeller-volute interaction in the single-channel pump. The CFD studies conducted in the last decades, along with an increase in computing power systems, have significantly contributed to the development of various turbomachines with a deep understanding of flow physics and mechanism. Of course, it was possible to suggest a state-of-the-art design technique for a single-channel pump because of the rapid increase in the computing power system and development of computational methods. The authors expect that the practical design technique introduced in this chapter will be useful for engineers designing various single-channel

This work was supported by the Demand-based-Manufacturing Technique Commercialization R&D Project of the Korea Institute of Industrial Technology (KITECH) (No. JB180001), which was funded by the Ministry of Science and ICT (MSIT). The authors wish to express gratefully our thanks to Ms. Bo-Min Cho (former master student in UST and currently CFD engineer in Anflux Co., Ltd., Korea) and Mr. Wang-Gi Song (currently master student in UST) for their

6. Conclusions

pumps in the near future.

Acknowledgements

cooperation in the numerical simulation.

Figure 14. Spectra of the magnitude values at observation points on the casing wall. (a) Location of the observation points (b) Point A. (c) Point B. (d) Point C. (e) Point D. (f) Point E.

suppressed, as shown in Figure 13(b). The AOD results in mostly stable flows throughout the annulus passage area of the pump. This explains the considerable decrease in the fluidinduced vibration caused by impeller-volute interaction owing to optimization.

Figure 14 shows the spectra of the magnitude values at observation points on the casing wall for the reference and AODs. Here, the spectra are calculated based on the wall pressure fluctuation time history by using a fast Fourier transformation algorithm. Both magnitude values are normalized by the first blade passing frequency (BPF) in the reference design, and these values are also related to the vibration of pump. As shown in Figure 14, the BPF is approximately 30 Hz (BPF = Blade number rpm/60). The peak magnitude values are clearly seen at every harmonic BPF in steps of 30 Hz, which are due to the periodic motion of pump impeller rotation. In the AOD, a considerable decrease in the magnitude values at the first BPF is observed specifically, as well as at all observation points, especially for points B and C. It clearly shows that the large high-pressure zone caused by impeller-volute interactions is obviously suppressed, as shown in Figure 13(b). Consequently, the considerable decreases in these magnitude values reduce the vibration caused by impeller-volute interaction.
