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(1. School of Energy and Power Engineering, Jiangsu University, Zhenjiang, Jiangsu 212013, China; 2. Zhenjiang Fire and Rescue Division, Zhenjiang, Jiangsu 212000, China; 3. Sitael S.P.A., Pisa 56121, Italy)

Abstract：A series of steady and unsteady numerical calculations of the internal flow in mixed-flow pumps with three different specific speeds were carried out based on the N-S equation coupled with the standard k-ε turbulence model under different operating conditions to investigate the relationship between the impeller specific speed and the pump performance as well as pressure pulsations. Meanwhile, the pump performance and pressure pulsations inside the mixed-flow pump with three different specific speeds were also analyzed and compared with the corresponding test data. From the results, the averaged deviations between the predicted and tested head among different impellers are below 5%, and with respect to the equivalent impeller specific speeds of 280 and 260, the values are 4.30% and 3.69%, respectively. For all the impeller schemes, the best efficiency point of the mixed-flow pump is found at the flow rate of 1.2Qd and the higher head deviation occurs at lower flow rates. Especially, it can be found that the specific speed has a slight effect on the pressure fluctuation in the impellers. Eventually, it is determined that the pump performance curves calculated by numerical simu-lations have good agreement with the relevant experimental results, which verifies that the numerical methods used in the present study are accurate to a certain extent. Furthermore, the results also provide some references to the pressure pulsation analysis and the performance improvement of the mixed-flow pump design.

Key words：mixed-flow pumps; impeller specific speed; hydraulic performance; pressure pulsation; numerical simulation; experiment

A mixed-flow pump has an intermediate specific speed with characteristics of both axial flow and centri-fugal pumps[1-3]. The pump specific speed is typically calculated considering the maximum hydraulic efficien-cy at the design point and can be used to identify or design the optimal blade shape since the shape of the blade passage varies as the pump specific speed (ns) changes[4-6].

Since the blade shape is closely related to the performance of the pump, the pump specific speed has a strong impact on the shape of the head-flow performan-ce curve, cavitation, unstable vibration, as well as undesired noises. For example, MIYABE et al.[7]found that a rotating stall occurred in a low-specific-speed mixed-flow pump at a 65% flow rate of the best efficiency point (BEP) together with periodical large-scale abrupt backflow generated from the vaned diffuser to the impeller outlet in their experimental tests. KIM et al.[8]designed a new mixed-flow pump using the trend of the design variables versus the specific speed and found that the specific speed was closely related to the shape and performance of the mixed-flow pump. They also indicated that the pump efficiency first increased and then decreased as the specific speed increased. ZOTOV[9]found that the higher the specific speed, the lower the load and the pressure drop, and the better the cavitation performance. More efforts are required to understand the relationship between the pump specific speed and the performance of mixed-flow pumps.

So far, many researchers have observed that complex flow phenomena can be involved in the blade tip clearance flow in a mixed-flow pump, and have focused on analyzing the influence of the tip clearance size on pressure fluctuation in a mixed-flow pump[10-12]. ZHANG et al.[10]found that the separation between the main leakage vortex and the secondary leakage vortex was strengthened with the increase in the tip clearance size. JI et al.[11]experimentally found more complicated frequency domain features occurring in the pump impeller due to the effect of the tip clea-rance flow. ZHANG et al.[12]found that the clearance variation had less effect on the pressure fluctuations in the guide vane. Therefore, it is necessary to investigate the dynamic characteristics at different specific speeds to reduce their power consumption and improve their efficiency.

In the present research, the hydraulic performan-ce of a mixed-flow pump at different flow rates was predicted using the ANSYS CFX software package[13]. The pressure fluctuation in the impeller outlet was further studied based on frequency domain analysis.

1 Modeling and numerical methods

1.1 Geometrical data

The mixed-flow pump in this study has been typically applied in South-to-North Water Diversion Projects in China[14].

As illustrated in Fig.1, the computational model of the mixed-flow pump included five-bladed impellers with different specific speeds, the ring, a pump casing, a guide vane, and the inlet and outlet pipes with their extended portions, respectively.

Fig.1 Three-dimensional view of model mixed-flow pump

The main dimensions and characteristics of the mixed-flow pump impellers with different specific speeds are as demonstrated in Tab.1, whereQdis the design flow rate,Hdis the design head,nis the rotational speed,Z1is the blade number of the impeller,Z2is the blade number of the guide vane,D1is the impeller inlet diameter, andD2is the impeller outlet diameter. The impellers in accordance with the three different equivalent specific speeds of 280, 270 and 260, are labeled as Impeller A, Impeller B and Impeller C, respectively.

Tab.1 Design and geometrical parameters of Impeller A, B, C

The big difference for these impellers lies in the design parameters such as the different design flow rates and design head.

1.2 Mesh generation and grid independence check

Structured hexahedron cells were applied as the numerical mesh using ANSYS ICEM CFD commercial software. In particular, five computational grids of the whole computational domain, as demonstrated in Tab.2, were chosen to analyze the impact of mesh size on the prediction of the hydraulic performance of the mixed-flow pump operating with Impeller A (ns=280) at the design point in detail based on the steady numerical calculations.

Tab.2 Details of mesh information for mixed-flow pump with Impeller A (ns=280)

As indicated in Fig.2, the prediction of the mixed-flow pump head coefficientHcfirst slowly changed with the increase in the number of grid elements, and then remained almost constant with the re-levant experimental data when the mesh number was larger than 6 million. The mesh with reasonably larger mesh elements (Mesh Case3) was chosen for the pre-sent simulations to save substantial CPU time and sto-rage, which has an acceptable influence on the calculation accuracy.

Fig.2 Head coefficient of mixed-flow pump calcula-ted by different mesh numbers

1.3 Governing equations and boundary condi-tions

Turbulent flow inside the mixed-flow pump opera-ting at different flow rates was numerically simulated using ANSYS CFX commercial software based on the Reynolds-averaged Navier-Stokes (RANS) equations combined with the standardk-εturbulence model. As for the inlet and outlet boundary conditions, the total pressure and mass flow rate were imposed in the pump inlet and pump outlet, respectively, for all of the impeller cases.

Considering that the accuracy of the transient results could be affected by the temporal resolution, such as the time steps selected during the unsteady numeri-cal simulation, one full rotation of the mixed-flow pump operating with Impeller C (ns=260) was there-fore discretized by 36 time steps in accordance with the angular change of 10° for each time step in the present case. Unsteady flow over 50 full rotor revolutions was numerically calculated for each computed flow rate, with the results corresponding to the last 10 full rotations exacted for further FFT analysis.

1.4 Experimental apparatus

Series of experimental tests were carried out in the laboratory of China Water Resources Beifang Investigation, Design and Research Co. Ltd. (BIDR, Tianjin, China), whose test facilities and measurement techniques are adequate for the specific requirements[15]. A picture of the simplified configuration for the hydraulic performance test is as displayed in Fig.3a.

In order to estimate the hydraulic performance of the mixed-flow pump in the present experiment, the rise in pressure between the pump inlet and pump outlet was measured by a pair of smart differential pressure transducers (Druck, model DPI832, 0-25 meter operating range, 0.05% precision class).

As for the measurement of the volumetric flow rate, a discharge electromagnetic flow meter (MF/C4011021200CR101, model LDG-500s, 0.21% precision class) was installed on the discharge line of the water loop to provide measurements of the outlet flow rate. The shaft rotating speed was measured by a smart rotational meter (model JCZL2-500, 0.1% precision class). Meanwhile, a series of hydraulic performance tests performed in water at different flow rates were carried out by adjusting the throttle valve at the discharge section.

(1)

whereρis the fluid density, kg/m3;Qis the volumetric flow rate, m3/h;His the head, m; andNis the shaft power, kW.

The uncertainties of the measured pump head and hydraulic efficiency were ±0.1% and ± 0.025%, respectively. The test results of the mixed-flow pumps with different impellers were illustrated for later comparison with the numerical simulations. The details of test parameters of the hydraulic performance test rigs are as follows: maximum test headHmax= 60 m; maximum test flow rateQmax=1 000 L/s; motor power (DC speed regulation)P= 160 kW; and test rotational speednT=0-1 500 r/min.

2 Results and discussion

2.1 Steady flow in mixed-flow pumps with diffe-rent specific speed impellers

Firstly, the hydraulic performance of the mixed-flow test pump was experimentally investigated to eva-luate the overall performance when operating with different specific speed impellers, and the corresponding data are as shown in Fig.4.

Fig.4 Comparison of hydraulic performances for various impellers operating at different specific speeds

Particularly, a series of steady numerical simulations were carried out for the prediction of the hydraulic performance of the model mixed-flow pump with Impeller A (ns=280) and Impeller C (ns=260) ope-rating at low, design and high flow conditions, namely at 70%, 80%, 90%, 100%, 110%, 120%, 130% and 140% of the design flow pointQd. Accordingly, the relevant experimental data obtained from the hydraulic performance test of the model mixed-flow pumps were finally analyzed in order to further confirm the relevant computational results.

Fig.4a and Fig.4b display the trend of the pump head and efficiency moving towards the specific speed at low, design and high flow rates. As expected, as the specific speed decreased, the head coefficient of the mixed-flow pump increased, and the pumping characteristicH-Qcurves started to overlap each other as the flow rate was higher than 1.6Qd, while the variation in the pump efficiency was opposite. The hydraulic efficiency characteristicη-Qcurves showed coincidence with each other until the flow rate was larger than 1.2Qd, where discrepancies started to occur and became even larger as the flow rate increased.

The predicted hydraulic performance was compa-red with the corresponding test data of the mixed-flow pumps operating at the specific speeds equal to 280 and 260 as shown in Fig.4c and Fig.4d, respectively. Additionally, the discrepancy in the hydraulic perfor-mance between the numerical and experimental data was also investigated. For Impeller A with a specific speed equal to 280, the predicted best efficiency point was found at a flow rate of 1.2Qd, and the large deviations were 8.35% and 8.28%, in accordance with the flow rates equivalent to 110% and 120% of the design value, respectively.

With respect to the specific speed of 260, the predicted best efficiency point occurred at a flow rate of 1.2Qdas well. Moreover, the prerotation formed in the inlet flow entering the mixed-flow pump impellers showed a more significant effect, especially at flow rates lower than 110% of the design value, where the head deviation values were as high as 7.36%.

Generally, the calculated performance curves of the mixed-flow pumps were consistent with the tested ones in a wide range of flow rates, which indicated that they were in good agreement with the experimental data and verified that the numerical methods used in the present study was accurate.

2.2 Unsteady flow in mixed-flow pump under different specific speed impellers

In order to further analyze the dynamic characte-ristics of the mixed-flow pump with equivalent specific speeds of 280 and 260, the unsteady flow in the mixed-flow pump was numerically simulated for the best efficiency point, which was at a flow rate of 120% of the design flow rate.

As displayed in Fig.5, the monitor points of the pressure fluctuation were placed in the rotating domain of the impeller outlet, and the monitor points located at the five-blade passages were numbered progressively along the anticlockwise direction, which was the same as the impeller rotation.

Fig.5 Location of monitor points at impeller outlet

In addition, the fast Fourier transformation of the unsteady pressure signals collected from the impeller outlet at the best efficiency point, which corresponded to 120% of the design flow rate, respectively, was as well obtained for comparison.

Moreover, the pressure coefficientCpwas applied with its expression shown as follows:

(2)

where Δpis the difference between the static monitoring pressure compared with the averaged pressure, Pa.

With respect to the frequency domain analysis of the pressure fluctuationAin the impeller outlet operating under equivalent specific speeds of 280 and 260, the distribution of the frequency domain at the same flow rate (1.2Qd) was chosen, and only the frequencies up to 220 Hz along thex-axis were investigated, as shown in Fig.6. As expected, several dominant peaks of the main frequency and the secondary main frequency pressure fluctuations were identified in the impeller outlet. Specifically, the same main and secondary main frequencies had almost the same amplitudes for both specific speeds, as investigated, with 24.21, 120.86 and 193.07 Hz, which were equal to the shaft rotational speed (fn= 24.2 Hz), the blade passing frequency (fb= 120.8 Hz) and 1.6 times higher than the blade passing frequency, respectively.

Fig.6 Frequency domain distribution of pressure fluctuation in impeller outlet at the best efficiency point

3 Conclusions

The present work described a comparison of the detailed results of the hydraulic performance of mixed-flow pumps operating under different specific speeds and with various flow rates, as obtained by numerical simulations and experiments. The frequency domain analysis of the unsteady flow within the impeller outlet was also carried out, at the optimal flow condition. The main conclusions from the work are as follows:

1) As the flow rate increased, the difference in pressure rise between the pump inlet and outlet became smaller for the investigated specific speeds, while the variation for the pump efficiency was obviously the opposite for higher flow rates. This could be the reason that at lower flow rates, the unstable flow, with respect to the case of the higher specific speed, was more intensive at a wide region of the impeller and the vane blade passages as well as the pump inlet pipe, resulting in higher hydraulic losses and reduction in hydraulic efficiency in comparison to the optimal operating point, which was in coincidence with the observation of the pressure pulsation in the impeller outlet with the high specific speed at lower flow rates.

2) Regarding the frequency domain analysis of the pressure fluctuation within the impeller outlet, the main frequency in the frequency domain was different for both the low and high specific speeds.

In summary, the predicted hydraulic performance of the model mixed-flow pump operating under different specific speeds agreed well with the corresponding test data over a wide range of flow rates, indicating that the proposed numerical methods adequately captured the inner flow in the mixed-flow pumps with different impeller schemes.