SCIENCE CHINA Technological Sciences • Article •
doi: 10.1007/s11431-014-5567-4
Effects of blade rotation angle deviations on mixed-flow pump hydraulic performance BING Hao* & CAO ShuLiang State Key Laboratory of Hydroscience and Engineering, Tsinghua University, Beijing 100084, China Received February 3, 2014; accepted April 14, 2014
By model test and numerical simulation, this paper analyzed the effects of different blades with varying rotation angle deviations on the hydraulic performance of a mixed-flow pump. It was found that when some blades had rotation angle deviations, the hydraulic performance curves of the mixed-flow pump would move. With a positive deviation, the curves moved towards the large flow rate; with a negative deviation, the curves moved towards the small flow rate. When some blades had rotation angle deviations, the symmetry and uniformity of the pressure distribution inside the mixed-flow pump flow passage both decreased; the larger the deviation, the greater the decrease. When a single blade had a large rotation angle deviation, a rather clear low pressure area was formed, lowering the cavitation performance. When two adjacent blades changed simultaneously, under the small flow rate condition, adverse pressure gradient and flow separation occurred in the flow field, and a hump appeared in the head curve and the operation stability of the mixed-flow pump dropped significantly. Near the best efficiency point (BEP), the simultaneous change of two alternate blades produced a more significant change of pressure in the flow passage, with an even larger area. Compared to the effect of two adjacent blades, two alternate blades, when changed simultaneously, made the mixed-flow pump slightly less efficient, but with a flatter efficiency curve and relatively wider high efficiency area. By fitting the test results, a functional relation among the BEP of the mixed-flow pump QBEP, the number of deviated blades N, and blade rotation angle deviation α was established, thus realizing an effective prediction of the BEP of the mixed-flow pump when blade rotation angles have deviations. mixed-flow pump, blade rotation angle, hydraulic performance, model test, numerical simulation Citation:
Bing H, Cao S L. Effects of blade rotation angle deviations on mixed-flow pump hydraulic performance. Sci China Tech Sci, 2014, doi: 10.1007/s11431-014-5567-4
1 Introduction The mixed-flow pump is a type of pump widely used in industrial and agricultural production, with a structure between the axial flow pump and the centrifugal pump. The pump has the advantages of an extensive application and a wide range of high efficiency operation. In recent years, based on tests and numerical simulations, many scholars have conducted in-depth researches in the three-dimensional design and optimization [1–3], flow measurement and anal*Corresponding author (email:
[email protected]) © Science China Press and Springer-Verlag Berlin Heidelberg 2014
ysis [4–6], hydraulic and cavitation performances prediction [7–11] and key parameters analysis [12–15] of mixed-flow pumps. To a mixed-flow pump with adjustable blades, during installation of its impeller, the blades’ rotation angles may have deviations, which significantly influence the hydraulic performance of the mixed-flow pump. Based on model tests, Bing et al. [16] carried out a qualitative research on the effect of a single blade’s rotation angle deviation on the head, efficiency and shaft power of a mixed-flow pump. However, except this one, few researches on this issue have been conducted in the academia. Based on previous tests and researches [16], this paper tech.scichina.com link.springer.com
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compared the prediction results from the numerical simulation with the test results, thus proving the credibility of the prediction by the numerical simulation. Then, based on the numerical simulation, by analyzing the hydraulic performance curves, static pressure distributions and relative streamline distributions of the mixed-flow pump, this paper carried out both qualitative and quantitative analysis about the effect pattern of both one blade’s rotation angle deviation and two blades’ rotation angles deviations on the mixed-flow pump hydraulic performance.
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Table 1
Major design parameters of the mixed-flow pump impeller
Parameters
Value 3
Design volume flow rate (m /s) Head (m)
0.54 17
Rotational speed (r/min)
1450
Specific speed
465
Blade number
5
Diffuser vane number
6
Maximum diameter of impeller (mm)
420
2 Model test A mixed-flow pump model with adjustable blade rotation angles was selected to carry out the test. With different blade rotation angles, the mixed-flow pump usually had varying best efficiency values. It was defined that when the mixed-flow pump reached maximum best efficiency, the blade rotation angle was 0°. Furthermore, when the blade rotated around its axis, if the impeller had a larger flow capacity, then the blade rotation angle was positive; if not, then negative. Table 1 shows major design parameters of the mixed-flow pump impeller. Figure 1 shows the mixedflow pump impeller. With the general test rig for hydraulic machinery model from Beifang Investigation, Design & Research Co. Ltd, the hydraulic performance test of the mixed-flow pump model was carried out. Figure 2 shows the major components of the test rig which has a random error Er≤±0.1%, and efficiency measurement composite error Eη≤±0.3%. According to Hydraulic turbines, storage pumps and pump-turbines — Model acceptance tests (IEC60193-1999) and Code for
Figure 2
Figure 1
Mixed-flow pump impeller.
model pump acceptance tests (SL140-2006), the hydraulic performance of the mixed-flow pump under two conditions were tested: when all the blades’ rotation angles were 4°, 2°, 0°, +2° or +4°; and when other blades’ rotation angles were 0°, and No. 1 blade’s rotation angle was 4°, 2°, 0°, +2° or +4° (shown in Figure 1). Figure 3 shows the mixed-flow pump efficiency curves
Diagram of the test rig.
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obtained through the model test. It can be found that the variation pattern of the best efficiency point (BEP) position of the mixed-flow pump when only No. 1 blade’s rotation angle changes is similar to the one when all the blades’ rotation angles change, but with a smaller variation range. It can be found that, when the blades rotate positively, the BEP moves towards the large flow rate; when the blades rotate negatively, the BEP moves towards the small flow rate.
3 Numerical simulation When simulating the internal flow inside the whole flow passage of the mixed-flow pump, the governing equations of the fluid flow were the three-dimensional steady incompressible Reynolds averaged Navier-Stokes equations and standard k-ε model was used for closing. The finite volume method was used to discretize the governing equations. The convection term was discretized by the second-order upwind scheme, and the other terms were discretized by central difference scheme. SIMPLEC algorithm was used to solve the governing equations. The inflow was set to be uniform at the inlet boundary and the velocity of the flow was calculated based on the flow rate and the area of the inlet section. The outlet boundary was set to support the fully developed flow. In addition, the solid wall was set to have no-slip condition and standard wall functions were employed to calibrate the turbulent model in the near-wall region. Based on the internal flow field of the mixed-flow pump impeller obtained by the numerical simulation, the mixedflow pump hydraulic efficiency was calculated through the following formula:
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gQH 100%, M
(1)
where ρ denotes the fluid density, Q denotes the flow rate, M is the moment of the fluid to the rotation axis, and H represents the actual head. With unstructured tetrahedral grids, whole flow passage computation grids of the mixed-flow pump were formed. To improve the precision of the calculation, local refinement was applied to the area of the blade surface boundary. The result of the independence validation on the grids is shown in Figure 4. When the grid number is relatively small, the head of the mixed-flow pump predicted by numerical simulation increases with the grid number increasing, but the increment goes down. When the grid number reaches to 2650000, any further adding of the grids can only increase the relative increment of head within 0.5%, indicating that the grid number no longer influences numerical calculation prediction significantly. Considering calculation precision and calculation efficiency, this paper chose a whole flow passage with 2650000 grids to conduct the calculation. Figure 5 shows the head curves and efficiency curves predicted by the numerical simulation when other blades’ rotation angles are 0°, and No. 1 blade’s rotation angle becomes 4°, 2°, 0°, +2° or +4°. The prediction results by the numerical simulation have the same variation pattern as the test results, and the two fit better near the BEP. Especially at the BEP, No. 1 blade with rotation angles of 4°, 2°, 0°, +2° and +4° has absolute errors between prediction and test results of only 0.3%, 0.4%, 1.0%, 1.0% and 1.3% respectively. This effectively verifies the accuracy of prediction about the mixed-flow pump hydraulic performance through numerical simulation, suggesting that the numerical simulation method is credible and thus can be used to predict and analyze the hydraulic performance and internal flow of the mixed-flow pump.
4 Analysis and discussion To further analyze the effect of blade rotation angle deviation on the mixed-flow pump performance, this paper used
Figure 3 Mixed-flow pump efficiency curves. (a) All the blades’ rotation angles change simultaneously; (b) only No. 1 blade’s rotation angle changes.
Figure 4 Mixed-flow pump head varying with the grid number under the design flow rate.
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Figure 5
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Verification of numerical simulation prediction under No. 1 blade with different rotation angles. (a) 4°; (b) 2°; (c) 0°; (d) +2°; (e) +4°
numerical simulation to predict the hydraulic performance curves of the mixed-flow pump under three kinds of conditions: when only one blade’s rotation angle changed; when two adjacent blades’ rotation angles changed simultaneously; when two alternate blades’ rotation angles changed simultaneously. The hydraulic performance curves of the mixed-flow pump were predicted by the numerical simulation, also, two sections were selected along the flow direction (Sections A and B shown in Figure 6) to analyze the internal flow in detail and the static pressure distribution of the mixed-flow pump. Under the first condition, with other blades’ rotation angles remaining at 0°, No. 1 blade’s rotation angle was adjusted to nine values successively including 0°, ±2°, ±4°, ±6° and ±8°. Under the second condition, with other blades’ rotation angles remaining at 0°, No. 1 and No. 2 blades’
Figure 6
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Meridional flow passage of the mixed-flow pump.
rotation angles were adjusted to five values successively including 0°, ±2° and ±4°. Under the third condition, with other blades’ rotation angles remaining at 0°, No. 1 and No. 3 blades’ rotation angles were adjusted to five values successively including 0°, ±2° and ±4°. 4.1
Single blade changes
Figure 7 shows the hydraulic performance curves of the mixed-flow pump in the condition that only No. 1 blade’s rotation angle changes. With other blades’ rotation angles staying at 0°, when No. 1 blade rotates positively from 8° to +8°, the BEP of the mixed-flow pump moves gradually towards the large flow rate. The best efficiency initially increases and then goes down. When all the blades’ rotation angles are adjusted to 0°, the best efficiency of the mixed-flow pump reaches its maximum. When No. 1 blade rotates positively or negatively, the best efficiency decreases. The larger the rotation angle becomes, the more significantly the best efficiency drops. Two factors have contributed to this: first, test results show that when all blades’ rotation angles are adjusted to 0°, the best efficiency of the mixed-flow pump reaches its maximum, which means any deviation from 0° will lead to a drop in the best efficiency; second, the more No. 1 blade’s rotation angle deviates from
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Figure 7 Mixed-flow pump performance curves with No. 1 blade’s rotation angle varying. (a) Efficiency curves; (b) head curves.
0°, the more it deviates from other blades, which leads to a decrease in the symmetry of the internal flow, thus inducing an even larger flow loss. With other blades’ rotation angles staying at 0°, when No. 1 blade rotates positively from 8° to +8°, the head of the mixed-flow pump under the same flow rate gradually increases, and the head curve moves towards the large flow rate. However, with the blade rotating further positively, the increment of the head under the same flow rate gradually decreases. When No. 1 blade’s rotation angle reaches +8°, the head increment is no longer significant; under some particular flow rates, the head even decreases. This is because with the blade rotating positively, the flow capacity of the flow passage of the mixed-flow pump impeller gradually rises, pushing the mixed-flow pump performance curves towards the large flow rate generally, thus leading to a larger head under the same flow rate. However, with the rotation angle increasing positively, No. 1 blade is further deviated from other blades, causing more flow loss, thus leading to a decreased increment of the head. Meanwhile, with the flow rate increasing, hydraulic friction loss also gets increased, which also causes the increment of the head to drop off. During the whole process of changing the rotation angle of No. 1 blade, no hump ever appears in the head curve of the mixed-flow pump. Figure 8 shows, at the BEP, the static pressure distributions inside Section A (shown in Figure 6) under different No. 1 blade’s rotation angles, with other blades’ rotation angles remaining at 0°. It can be seen that, when all the blades’ rotation angles are 0°, the static pressure distribution inside the section is well symmetrical and uniform. However, when No. 1 blade’s rotation angle deviates from 0°, the symmetry and uniformity of the pressure distribution de-
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creases; the larger the deviation, the more significant the decrease. When No. 1 blade rotates positively from 8°to +8°, on the position corresponding to No. 1 blade in Section A along the direction of flow, the static pressure gradually decreases, forming a clear low pressure area near the shroud. Especially when No. 1 blade’s rotation angle turns to +8°, a relatively large low pressure area is formed inside Section A before the impeller inlet, which greatly reduces the cavitation performance of the mixed-flow pump, making it easier to induce cavitation. This is because when No. 1 blade rotates positively from 8° to +8°, the flow passage formed by No. 1 blade and its adjacent blades (No. 2 and No. 5) gradually gains a larger flow capacity. As a result, the flow rate at the BEP gets increased, which directly leads to a decrease in the static pressure of the fluid. Figure 9 shows that at the BEP, with other blades’ rotation angles remaining at 0°, the static pressure distributions inside Section B (shown in Figure 6) under different No. 1 blade’s rotation angles. When all the blades’ rotation angles are kept at 0°, the static pressure distribution inside Section B is also well symmetrical as the situation inside Section A. The static pressure gradually increases along the relative motion direction of the fluid, with a smooth pressure isoline. When No. 1 blade rotates negatively from 0°, it gains a gradually increasing pressure on its suction surface and a decreasing pressure on its pressure surface near the leading edge of the blade. When No. 1 blade’s rotation angle reaches 8°, a clear low pressure area appears on its pressure surface near the leading edge of the blade, and the cavitation performance significantly drops off. When No. 1 blade rotates positively from 0°, the pressure on its suction surface gradually decreases, and the pressure on No. 2 blade’s pressure surface near the leading edge also decreases. When No. 1 blade’s rotation angle reaches +8°, a relatively large low pressure area appears on its suction surface near the hub, and another clear low pressure area is also found on No. 2 blade’s pressure surface near the leading edge. These two low pressure areas have similar minimum pressures and both have significantly reduced the cavitation performance. This suggests that when a single blade rotates too much, the pressure on its surface or its adjacent blade surface will decrease sharply, which hinders the cavitation performance improvement. 4.2
Two blades change simultaneously
Figure 10 shows the hydraulic performance curves of the mixed-flow pump when two blades change simultaneously. Under the same rotation angle, the mixed-flow pump with two adjacent blades (No. 1 and No. 2) changing simultaneously has the same BEP with the pump with two alternate blades (No. 1 and No. 3) changing simultaneously. However, compared to two alternate blades, the mixed-flow pump
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Figure 8
Static pressure distributions inside Section A under different No. 1 blade’s rotation angles (unit: kPa).
Figure 9
Static pressure distributions inside Section B under different No. 1 blade’s rotation angles (unit: kPa).
with two adjacent blades has a slightly higher best efficiency when they change simultaneously. When two blades rotate positively at the same time, the BEP of the mixed-flow pump moves towards the large flow rate; when rotate negatively, towards the small flow rate. Compared to only single blade changing, two blades changing simultaneously can make the BEP of the mixed flow pump move further. When two adjacent blades’ rotation angles change simultaneously, a hump appears in the mixed-flow pump head curve. Especially when the rotation angles of No. 1 and No. 2 blades are +2° or 2°, the hump area is significantly apparent, within which the mixed-flow pump operation will become much less stable. However, when two alternate blades’ rotation angles change, no hump appears in the head curve. Therefore, it can be concluded that, for the mixed-flow pump with five blades analyzed in this paper, a
single blade or two alternate and simultaneously changing blades, cannot lead to a hump in head curve of the mixed-flow pump; but two adjacent blades can. This suggests that two adjacent blades changing together will exert a more significant influence on the operation stability of the mixed-flow pump. Figure 11 shows, at the BEP, when other blades’ rotation angles are 0°, the static pressure distributions inside Section A (shown in Figure 6) with two adjacent or alternate blades changing simultaneously. When the blades’ rotation angle deviates from 0°, on the position corresponding to them in Section A along the direction of flow, the static pressure changes: when blades rotate positively, the pressure goes down; negatively, goes up. With the same blade rotation angle, the mixed-flow pump with two alternate blades changing simultaneously can induce a more significant static
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Figure 10 Mixed-flow pump performance curves when two blades change simultaneously. (a) Efficiency curves (No. 1 and No. 2 blades); (b) head curves (No. 1 and No. 2 blades); (c) efficiency curves (No. 1 and No. 3 blades); (d) head curves (No. 1 and No. 3 curves).
Figure 11
Static pressure distributions inside Section A when two blades change simultaneously (unit: kPa).
pressure variation on the position corresponding to No. 1 blade in Section A along the direction of flow, than the pump with two adjacent blades. Meanwhile, compared to the position corresponding to No. 2 blade of the impeller with two adjacent blades changing simultaneously, the position corresponding to No. 3 blade of the impeller with two alternate blades changing simultaneously has a larger static pressure variation. A more significant change in static pressure means a more dramatically drop in symmetry in the pressure field. To some extent this explains why the mixed-flow pump with two adjacent blades changing simultaneously can have slightly higher efficiency near the BEP.
Figure 12 shows, at the BEP, the static pressure distributions inside Section B (shown in Figure 6) of the impeller with two adjacent blades or two alternate blades changing simultaneously when other blades’ rotation angles are 0°. When the two adjacent or alternate blades rotate negatively from 0°, the pressure on the blades’ suction surfaces increases. Especially when the blades’ rotation angles reach 4°, No. 1 blade of the impeller with two adjacent blades changing, and No. 1 and No. 3 blades of the impeller with two alternate blades changing all have low pressure areas on their respective pressure surfaces near the leading edges. When the two adjacent or alternate blades rotate positively from 0°, the pressure on the blades suction surfaces
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Figure 12
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Static pressure distributions inside Section B when two blades change simultaneously (unit: kPa).
decreases gradually. Especially when the blades’ rotation angles reach +4°, No. 3 blade of the impeller with two adjacent blades changing, and No. 2 and No. 4 blades of the impeller with two alternate blades changing all have low pressure areas on their respective pressure surfaces near the leading edges. Generally, the two alternate blades, when changing simultaneously, can lead to a larger area of pressure variation. This can, to some extent, explain why the mixed-flow pump with two adjacent blades changing simultaneously can have slightly higher efficiency near the BEP. Figure 13 shows when No. 1 and No. 2 blades’ rotation angles are +2° and other blades’ rotation angles are 0°, the static pressure and relative streamline distributions inside
the Section x=0 under three flow rates in the hump range. It can be found that 0.413 m3/s is the flow rate at the hump peak. When the flow rate is 0.428 m3/s, the internal flow in the whole flow passage of the mixed-flow pump is uniform. When the flow rate drops to 0.413 m3/s, the relative streamline begins to band, in the diffuser zone and the outlet zone behind the diffuser. When the flow rate continues to decrease to 0.398 m3/s, clear adverse pressure gradient and flow separation appear (see Figure 13(a)) in the diffuser zone and the outlet zone behind the diffuser, the pressure distribution becomes less uniform, the flow becomes more in disorder, and hydraulic loss and energy dissipation also get larger. This explains why the head of the mixed-flow pump under this flow rate has a sudden drop.
Figure 13 Static pressure and relative streamline distributions inside the Section x=0 when two adjacent blades rotation angles are +2° (Unit: kPa). (a) Q=0.398 m3/s; (b) Q=0.413 m3/s; (c) Q=0.428 m3/s.
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Comprehensive performance analysis
Figure 14 shows the variation curves of the mixed-flow pump best efficiency under different blade rotation angle deviations. When blades rotate negatively, the mixed-flow pump with only a single rotating blade produces the highest best efficiency, while the pump with two alternate blades changing simultaneously produces the lowest. When blades rotate positively, the mixed-flow pump with two adjacent blades changing simultaneously generates the highest best efficiency, while the pump with two alternate blades changing simultaneously has the lowest. Generally, two alternate and simultaneously changing blades lead to a most significant reduction of the best efficiency, which is mainly because this change induces a more significant change in static pressure within a larger area (see Figures 11 and 12). Figure 15 shows the variation curves of the mixed-flow pump high efficiency range under different blade rotation angle deviations. High efficiency range is defined as the flow rate range within which the absolute value reduction of the best efficiency is less than 10%. It can be noticed that whether the blades rotate positively or negatively, the mixed-flow pump with two alternate blades changing simultaneously has the widest high efficiency range, while the pump with two adjacent blades changing simultaneously has the narrowest. This is mainly because when two alternate blades change simultaneously, the best efficiency is relatively lower and the efficiency curve is flatter (see Figure 10(c)), so the range, within which the absolute value
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reduction of the mixed-flow pump best efficiency is less than 10%, is accordingly getting much wider. However, when the two adjacent blades change simultaneously, a hump appears in the head curve. Operating the mixed-flow pump in the hump range leads to adverse pressure gradient and flow separation in the diffuser zone and outlet zone behind the diffuser (see Figure 13 (a)), which greatly reduces efficiency, thus making the high efficiency range narrower. 4.4
BEP prediction
Figure 3 shows the efficiency curves of the mixed-flow pump based on the tests of two cases: All blades change simultaneously and only No. 1 blade changes. From these curves, the BEPs with different blade rotation angle deviations α can be obtained. By five-point linear fitting, the prediction curve of BEP in the condition of all blades changing simultaneously is generated (shown in Figure 16): QBEP 0.019 0.424.
(2)
In addition, the prediction curve of BEP in the condition of only a single blade changing is also found (shown in Figure 16): QBEP 0.0049 0.427.
(3)
According to eqs. (2) and (3), by interpolation calculation, the prediction curve of the mixed-flow pump BEP when two blades change simultaneously can be obtained (shown in Figure 16): QBEP 0.0084 0.426.
(4)
Figure 16 shows the BEPs predicted by numerical simulation when a single blade changes, and when two blades change simultaneously. It can be noticed that the numerical simulation results comply well to the prediction curves obtained by fitting the test results: Combining eqs. (2), (3) and (4): QBEP k l. Figure 14 Mixed-flow pump best efficiency variation curves under different blade rotation angle deviations.
Figure 15 Mixed-flow pump high efficiency range variation curves under different blade rotation angle deviations.
(5)
Coefficients k and l change with the number of deviated blades N (shown in Table 2).
Figure 16
Prediction results and test results of mixed-flow pump BEP.
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Table 2
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Coefficients k and l with different numbers of deviated blades N 1 2 5
k 0.0049 0.0084 0.019
l 0.427 0.426 0.424
By linear fitting the coefficients k and l with the number of deviated blades N respectively, the expressions of k and l are detailed as following (see Figure 17): k 3.526 N 1.361 10 3 ,
(6)
l 0.730 N 427.6 103 ,
(7)
Combining eqs. (5), (6) and (7): QBEP 3.526 N 1.361 0.730 N 427.6 10 3. (8)
From eq. (8), it can be noticed that with the number of deviated blades N increasing, the influence of the blade rotation angle deviation α on BEP of the mixed-flow pump becomes more significant. Eq. (8) can be used to predict the BEP of the mixed-flow pump when blade rotation angle has deviation, so that the flow rate can be adjusted to keep the mixed-flow pump operating within the high efficiency range, thus improving its operation.
5 Conclusion Based on the model test and the numerical simulation, by analyzing the hydraulic performance curves, the internal flow field and the pressure field of the mixed-flow pump, this paper studied the effects of different blade rotation angle deviations on the mixed-flow pump performance to obtain the following conclusions. (1) When the rotation angle of a single blade was deviated, the hydraulic performance curves of the mixed-flow pump would move. They moved in the same direction as in the situation when all blades rotated simultaneously. That means, when a single blade rotated positively, the hydraulic performance curves moved towards the large flow rate; when a single blade rotated negatively, the curves moved towards the small flow rate. When a single blade rotation
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angle had deviation, the symmetry and uniformity of the pressure distribution inside the mixed-flow pump flow passage would decrease; the larger the deviation, the greater the decrease. When the deviation was relatively small, the best efficiency value of the mixed-flow pump did not drop significantly. Therefore, it was suggested that a relative small deviation could be used as an effective way to adjust BEP of the mixed-flow pump operation. When the deviation was large, the pressure on the blade surface or the adjacent blade surface would drop, forming a clear low pressure area, thus reducing the cavitation performance of the mixed-flow pump. (2) When two blades changed simultaneously, with same rotation angles, the mixed-flow pump with two adjacent blades changing simultaneously had the same BEP with the pump with two alternate blades changing simultaneously. After two adjacent blades changed simultaneously, under small flow rate, clear adverse pressure gradient and flow separation occurred in the diffuser zone and the outlet zone behind diffuser, and a hump appeared in the head curve of the mixed-flow pump, with greatly reduced operation stability. Near the BEP, two alternate blades changing simultaneously produced even more significant pressure variation inside the flow passage, and the area with pressure variation was getting larger. Compared to two adjacent blades, two alternate blades changing simultaneously led to a slightly lower efficiency of the mixed-flow pump, but with a flatter efficiency curve and a wider high efficiency range. (3) After the model test results were linear fitted, the fitting results were verified by numerical simulation results. Based on this, a functional relationship among BEP of the mixed-flow pump QBEP, the number of deviated blades N and blade rotation angle deviation α was established. By using this relationship, BEP of the mixed-flow pump was effectively predicted when there was blade rotation angle deviation, which was then utilized to adjust flow rate to keep the mixed-flow pump operating within the high efficiency range. This work was supported by the National Natural Science Foundation of China (Grant No. 51176088). 1
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3
4 Figure 17 Curves of coefficients k and l varying with the number of deviated blades.
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