SCIENCE CHINA Technological Sciences vs• RESEARCH PAPER •
January 2013 Vol.56 No.1: 1–13 doi: 10.1007/s11431-013-5308-0 doi: 10.1007/s11431-013-5308-0
Multi-parameter optimization design, numerical simulation and performance test of mixed-flow pump impeller BING Hao* & CAO ShuLiang State Key Laboratory of Hydroscience and Engineering, Tsinghua University, Beijing 100084, China Received March 19, 2013; accepted July 12, 2013
On the basis of the three-dimensional design platform of the mixed-flow pump impellers, an optimization design system was developed in this paper by improving the genetic algorithm with application of both strategies of keeping the optimal individual and employing the niche. This system took the highest efficiency of the impeller as the optimization objective and employed P, a0, h and t, which could directly affect the shape and the position of the blade, as optimization parameters. In addition, loss model was used to obtain fast and accurate prediction of the impeller efficiency. The optimization results illustrated that this system had advantages such as high accuracy and fine convergence, thus to effectively improve the design of the mixed-flow pump impellers. Numerical simulation was applied to determine the internal flow fields of the impeller obtained by optimization design, and to analyze both the relative velocity and the pressure distributions. The test results demonstrated that the mixed flow pump had the highest efficiency of 87.2%, the wide and flat high efficiency operation zone, the relatively wide range of blade angle adjustment, fine cavitation performance and satisfied stability. mixed-flow pump, impeller, optimization design, performance test, numerical simulation Citation:
Bing H, Cao S L. Multi-parameter optimization design, numerical simulation and performance test of mixed-flow pump impeller. Sci China Tech Sci, doi: 10.1007/s11431-013-5308-0
Nomenclature W V F Er
V Vm l x (Vr)0
relative velocity vector of fluid motion absolute velocity vector of fluid motion body force of a unit mass of fluid mechanical energy of relative motion of a unit mass of fluid blade wrap angle angular velocity of impeller rotation tangential component of absolute velocity meridional component of absolute velocity meridional streamline length of blade zone meridional streamline length velocity moment of leading edge
*Corresponding author (email:
[email protected]) © Science China Press and Springer-Verlag Berlin Heidelberg 2013
(Vr) incremental of velocity moment from leading edge to trailing edge m meridional streamline relative length relative coefficient of parameter a a0 r radial direction coordinate Z axial direction coordinate angle between OB and OA O angle between OB and OH h t1 angle between OB and OT1 t2 angle between OB and OT2 spherical radius of hub rh spherical radius of tip rt blade rotation angle on its own rotation axis flow rate coefficient head coefficient Q flow rate tech.scichina.com
www.springerlink.com
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H n D
actual head rotational speed of mixed-flow pump characteristic diameter fluid density hydraulic efficiency M moment of fluid to rotation axis NPSHc critical cavitation allowance pi static pressure absolute velocity ci Ai area of control unit on the inlet and outlet sections nin number of nodes on the inlet section of computational domain number of nodes on the outlet section of computanout tional domain Ng grid number
1 Introduction The mixed-flow pumps are widely applied in the fields of irrigation, drainage, flood protection and water treatment due to their various advantages, such as wide application range, wide high efficiency zone and low probability of cavitation. In recent years, a great number of scholars have carried out deep researches on the optimization design methods of the mixed-flow pumps, the prediction of hydraulic performance and the effects of parameters on the pump design and performance. Goto et al. [1, 2] developed an impeller design system of the mixed-flow pumps based on the three-dimensional inverse design method, CAD and CFD, and also built a pump diffuser design system of the mixed-flow pumps on the basis of three-dimensional inverse design method and CFD. Under the condition of fixed meridional flow passage, Kim et al. [3] optimized the impeller and diffuser of the mixedflow pump to improve the internal flow. In addition, Oh et al. [4] developed an optimization design method of the mixed-flow pump impellers with minimum fluid dynamic losses and geometric and fluid dynamic variables as the optimization objective and the optimization parameters, respectively. The performance curves of the mixed-flow pump predicted with CFD were highly close to the test results [5]. Esch et al. [6] processed unsteady numerical simulation to the mixed-flow pump impeller and predicted hydraulic losses and disk friction losses with loss models, which were also evaluated in the paper. Oh et al. [7] employed CFD to consider the hydraulic performances of the centrifugal blood pump [8], the axial-flow main coolant pump [9] and Francis hydraulic turbine [10], respectively. Based on this, the hydraulic and the cavitation performances of the mixed-flow pump were predicted with CFD and compared with the experiment results. Bonaiuti et al. [11] analyzed the effects of the blade loading distributions, the position of the leading edge and
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the shape of the meridional flow passage on the hydraulic and the cavitation performances of the mixed-flow pump, by inverse design, numerical simulation and experiment. Kim et al. [12] used the numerical simulation to discuss the influences of the straight vane length ratio and the diffusion area ratio of the diffuser zone on the mixed-flow pump efficiency. Oh et al. [13] compared the various parameters in the effects on the cavitation performance of the mixed-flow pump impeller. Bing et al. [14] developed the direct and inverse iterative design method and achieved three-dimensional design of the mixed-flow pump impeller. The numerical simulation was employed to analyze the effects of the blade parameters [15, 16] and the meridional flow passage parameters [17] on the impeller design of the mixed-flow pump, and also to analyze the influence of the diffuser vane on the mixed-flow pump performance [18]. Based on the parameter analysis, a loss model for the whole flow passage of the mixed-flow pump was accomplished and then verified by comparing prediction results with test data [19]. 3D design, loss model and genetic algorithm were integrated to achieve the ‘single objective and single parameter’ optimization with velocity moment distribution coefficient a as the optimization parameter [20]. Cai et al. [21] used the controllable velocity moment method to design the impeller of the mixed-flow pump, but the best efficiency point had an offset towards the low flow rate, due to not considering the blade thickness. Lu et al. [22] combined the inverse design and the neural network to optimize parameters of the meridional flow passage and the velocity moment distribution. This optimization led to improvements in both head and efficiency of the impeller. Jia et al. [23] employed NUMECA software to optimize the radial parameters of the blade of the mixed-flow pump and the optimized blade by this method had a more uniform pressure distribution on the surface. Xie et al. [24] also used NUMECA software to optimize the blade camber lines on both the hub and the tip. The obtained blade after this optimization had a lager curvature and, in addition, the flow separation inside the flow passage was effectively improved by this way. An optimization design system of the mixed-flow pump impellers has been developed in this paper based on the three-dimensional design platform of the mixed-flow pump impellers to enhance the operation capability and improve the internal flow. In this method, with the optimization objective of the highest hydraulic efficiency and the optimization parameters of the velocity movement distributions and the leading and the trailing edges positions of the blades, the loss model was used to predict the efficiency of the impeller and the genetic algorithm was improved to achieve the optimization of the mixed-flow pump impeller with single objective and multiple parameters. This platform can lead to the highly efficient and accurate optimization design of the mixed-flow pump impellers and effective improvement of
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the impeller design.
ity moment distribution given by experiences. The integral equation of the blade camber line was given as follows:
2 Three-dimensional design platform
(l )
In the two-dimensional flow assumption design method, the impeller is considered to consist of an infinite number of infinitely thin blades, so the effect of blade thickness on the flow field calculation will be ignored in this situation. The quasi-orthogonal method [25] is generally used to calculate the streamlines and the meridional velocity distributions according to the principle of uniform flow rate in each flow passage. However, the results of the flow field calculation can only satisfy the continuity equation of fluid. The three-dimensional design platform (TDDP), based on direct and inverse iterative design method, was developed to improve the design method of the mixed-flow pump impellers. This platform was considered by combining the direct calculation with the inverse design. Specifically, direct calculation offered the flow field information to inverse design and inverse design provided the impeller model to direct calculation. The impeller was not finally determined until that direct calculation and inverse design converged. The obtained blade shape of the impeller from TDDP satisfied both the continuity and motion equations of fluid. 2.1 Direct calculation Based on the theory of two kinds of relative stream surfaces [26], direct calculation of TDDP was achieved by iterative calculation on S1 and S2 stream surfaces to settle the following equations [27]: W 0, W V F Er ,
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(1)
l
r 2 V r
0
Vm r 2
dl ,
where was the blade wrap angle, was the angular velocity of impeller rotation, r was the meridional plane radius, V was the tangential component of absolute velocity, Vm was the meridional component of absolute velocity, l was the meridional streamline length of the blade zone. After drawing the blade camber lines one by one, the blade camber surface was completed by integrating all the obtained blade camber lines. In the conformal mapping plane, the blade camber surface was single-side thickened and the leading edge and the trailing edge of the blade were smoothed. Thus, the process of inverse design was finished. 2.3 Direct and inverse iterative design Based on direct calculation and inverse design, the direct and inverse iterative design was realized by building TDDP by programming. The steps [14] were described as follows (Figure 1). ① The initial impeller was designed based on the two-dimensional flow assumption design method. ② The direct calculation was finished to obtain the meridional flow field, which satisfied both continuity and motion equations of fluid. ③ Inverse design was finished by using the meridional flow field calculated by direct calculation, to obtain a new impeller. Steps ② and ③ above were repeated until the difference between the meridional transversals, which was obtained by two consecutive inverse design, met the set requirements. Figure 2 demonstrated the meridional velocity distributions in the flow passage of the mixed-flow pump. Where, x
where W was the relative velocity vector of fluid motion, V was the absolute velocity vector of fluid motion, F was the body force of a unit mass of fluid, Er was the mechanical energy of relative motion of a unit mass of fluid. In the process of iterative calculation, the velocity potential function equation of S1 stream surface was solved based on the finite element method and the meridional velocity gradient equation along the quasi-orthogonal of the S2 stream surface was settled based on the streamline curvature method. The blade shape and the flow field inside the impeller were not finally determined until the iterative calculation converged． 2.2
Inverse design
The shape of blade camber line was determined by integrating the differential of blade point by point, on the basis of the meridional velocity distributions along meridional streamlines obtained by the direct calculation and the veloc-
(2)
Figure 1
TDDP flowchart of the mixed-flow pumps.
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Figure 2 Meridional velocity distributions of the mixed-flow pump. (a) Initial impeller (2D flow assumption); (b) initial impeller (3D design platform); (c) final impeller (3D design platform).
was the length of the meridional streamline. It was found from this figure that, under the same design conditions, the meridional velocity distributions had significant differences between each other due to the differences in the design methods and design stages, although they had consistent overall trends (Figures 2(a)–2(c)). Compared with the meridional velocity distributions calculated based on the two-dimensional flow assumption (Figure 2(a)), the meridional velocity distributions obtained by TDDP (Figures 2(b) and 2(c)) had significant changes in the blade zone (dashed line in Figure 2) and the outlet zone (solid line in Figure 2). Specifically, the maximum meridional velocity in the blade zone slightly increased and the meridional velocity distribution in the outlet zone was significantly non-uniform. The reason was that TDDP had considered the effects of the blade thickness on the flow field calculation inside the impeller. Compared with the meridional velocity distributions of the initial impeller (Figure 2(b)), the meridional velocity distributions of the impeller designed by TDDP (Figure 2(c)) had significant changes in both the blade zone and the outlet zone. These changes were mainly due to the interaction between the blade shape and the meridional flow field during the process of direct and inverse iterative design. Specifically, the meridional flow field was taken as the basis of the blade drawing and the blade shape also had effects on the calculations of the meridional flow field. Therefore,
the obtained blade shape and meridional flow field after the convergence had precise corresponding relation, and could better fit the flow characteristics of fluid in the impeller. In other words, the precision of the blade could be effectively enhanced by employing TDDP based on direct and inverse iterative design method.
3 Optimization design system Based on TDDP, the optimization design system (ODS) of the mixed-flow pumps was structured by reasonably selecting the optimization parameters, the objective function and the optimization algorithm. 3.1
Optimization parameters
After determining the geometric shape of the flow passage, the leading and the trailing edges positions and the velocity moment distributions along the meridional streamlines had significant effects on the design of the mixed-flow pump impellers. The velocity moment distributions along the meridional streamlines directly affected the blade loading distributions [28] and the energy and cavitation performances. In the process of the impeller design based on TDDP, the velocity moment distribution along each meridional streamline was
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given according to the same pattern. The velocity moment distribution along any meridional streamline was determined as follows [29]: V r (V r )0 f (m)(V r ),
(3)
where (Vr)0 was the velocity moment of the leading edge, (Vr) was the incremental of velocity moment from the leading edge to the trailing edge, f(m) was the dimensionless distribution function, 0f(m)1, and calculated as f (m) am 4 bm3 cm 2 dm e,
(4)
where m was the meridional streamline relative length, a, b, c, d, e were parameters. There were five conditions for f(m) to have definite solutions: f(0)=0; f(1)=1; df(0)/dm=0; df(1)/ dm=P(P is the constant); df(m)/dm|m∈[0,1]0. Thus, these five parameters could be solved as follows: (m 1)(2m 1)a (3 1.5 P) x ( P 3), m [0,1], b 2a P 2, c a P 3, d 0, e 0.
(5)
Both parameters d and e equaled 0, and b and c were expressed with P and a. The value ranges of P and a were determined by the first inequality in eq. (5). Thus, the velocity moment distributions could be completely controlled by selecting the values of parameters P and a. The value ranges of P and a were then discussed in the following four cases: 3 1.5P m P 3 , m [0, 0.5), a m 1 2m 1 P 6, m 0.5, 3 1.5P m P 3 , m 0.5,1 , a m 1 2m 1 P 0, m 1.
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According to the definition of eq. (8), it was found that the value range of a0 was valued within [0, 1] for any value of P. The curve shape of the distribution function was determined directly by the derivative P of dimensionless distribution function f(m) of velocity moment at the trailing edge of the blade and the relative coefficient a0. Therefore, P and a0 were selected as optimization parameters of the velocity moment distribution. The positions of the leading edge and the trailing edge affected the work capacity and the internal flow of these locations. Particularly, the position of the leading edge had also influences on the cavitation performance. Reasonable selection of the positions of the leading edge and the trailing edge was significantly important to the impeller design. Figure 3 demonstrated the meridional projection of the flow passage of the mixed-flow pump impeller. In this figure, O indicated the centers of spherical surfaces of both the hub and the tip in the blade zone; OA indicated the rotation axis of the impeller; OB indicated the rotation axis of the blade; H indicated the intersection of the leading edge and the hub; T1 indicated the intersection of the leading edge and the tip; T2 indicated the intersection of the trailing edge and the tip; θO, ∆θh, ∆θt1 and ∆θt2 indicated the angles between OB and OA, OH, OT1 and OT2, respectively. In order to conveniently adjust the blade angle of the mixed-flow pump, spherical flow passages were employed for both the hub and the tip in the blade zone. OH rh , OT1 OT2 rt ,
(9)
where rh was the spherical radius of the hub, rt was the spherical radius of the tip. Generally, OT1 and OT2 were (6)
The value range of constant P was determined to be [0, 6] by eq. (6). A new function g(m) was defined as follows for more convenient discussion: g ( m)
(3 1.5 P)m ( P 3) . (m 1)(2m 1)
(7)
The maximum value a1 of g(m) in [0, 0.5) and the minimum value a2 of g(m) in (0.5, 1) were both determined according to eq. (7), so the value range of a was [a1, a2]. However, the value range of a changed as P changed. Therefore, in order to make the discussion convenient, the relative coefficient a0 of parameter a was defined as follows: a a1 a0 . (8) a2 a1
Figure 3 Schematic diagram of optimization parameters of the leading and the trailing edges.
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symmetrical around the rotation axis OB. t1 t 2 .
(10)
With condition of t1=t2=t, the coordinates of points H, T1 and T2 were determined as follows: rH rh sin O h , Z H Z O rh cos O h , rt1 rt sin O t , Z t1 Z O rt cos O t , rT2 rt sin O t , Z Z r cos . O t O t T2
(11)
In the process of impeller design based on TDDP, the positions of the leading edge (points H and T1) and the position of the trailing edge (point T2) on the tip were given by experiences and the position of the trailing edge on the hub was obtained by the calculation of impeller design. From eq. (11), it was found that these three points H, T1 and T2 were completely determined by parameters h and t after confirming rh, rt and the positions of the rotation axis of the blade. Therefore, the parameters h and t were selected as the optimization parameters of the positions of the leading edge and the trailing edge. Overall, four parameters P, a0, h and t, which mainly influenced the performance of the mixed-flow pump impeller, were selected as the optimization parameters. 3.2 Objective function
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the hydraulic machinery. Based on the basic principle of survival of the fittest, GA aims to find the global optimal solution by random genetic operation for the population in a given termination criterion. The basic procedure was introduced as follows. ① Select the optimization parameters and the fitness function, encode the optimization parameters and generate the initial group randomly. ② Calculate the individual fitness and judge whether the optimization results can meet the termination criterion according to the impeller optimization. If it can satisfy the termination criterion, end the optimization. If not, go to step ③. ③ Generate the new population by selecting individuals according to fitness, and operate crossover and mutation according to certain probability, and then repeat step ②. The impeller optimization was generally a multiple-peak problem [33]. In order to ensure that the final optimization results can converge to obtain the global optimal solution in the search space, the strategy of keeping the optimal individual was employed to ensure that the optimal individual of this generation was retained for the next generation and avoid changes caused by the mutation and the crossover operations. The niche strategy was used to maintain the diversity of individuals and avoid inbreeding within the population caused by greatly intensive individual distributions. The procedure of GA used in this paper was illustrated as Figure 4. In order to verify the accuracy of the optimization algorithm proposed in this paper, the global minimum value of Rastrigin function was calculated by GA optimization. This function was a multiple-peak function and the mathematical expression was shown as follows:
In order to simplify the optimization process, the highest efficiency of the mixed-flow pump impeller was selected as the only objective of the optimization design. As the impeller efficiency was always positive, it was determined as the fitness function of genetic algorithm. For an individual case, three-dimensional turbulent flow determination, based on solving the Navier-Stokes equations, had intensive computations. When the population size became large, the calculation would be time-consuming and therefore unable to meet the engineering requirements. Oh et al. [30–32] employed losses modeling to predict the performances of the centrifugal and the mixed-flow pumps and obtained comparatively satisfied results. In this paper, accurate and quick prediction of the impeller performance was achieved by combining the losses modeling of the mixedflow pump with the flow field calculation, based on the S1 and S2 stream surfaces iterative calculation [19]. 3.3
Optimization algorithm
Genetic algorithm (GA) is a kind of optimization algorithms to simulate the proliferation and the evolution of the natural species and has wide applications in the field of optimizing
Figure 4
Flowchart of genetic algorithm.
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R ( x) 4 A xi2 A cos 2πxi ,
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(12)
i 1
where A was generally a given constant and set to be 1 in this paper. The value range of parameter xi (i=1, 2, 3 ,4) was [5.12 , 5.12]. When xi=0, the global minimum value of R(x) is obtained. There are approximately 40 local minimum value points of R(x) within the definition domain. The population size was given as 10, the maximum evolution generation was given as 100. Thus, the minimum value of R(x) was solved by GA. Figure 5 demonstrated the change of the smallest and average values of the population during calculation process of the minimum value of R(x). It was found from this figure that the minimum value approached to 0 rapidly during the search process of R(x) value. The global minimum value was determined by GA as the population evolves to the 34th generation. This meant that the optimization results of GA were able to converge to the optimization objective within a small population and with a high convergence speed by improving the traditional GA. Therefore, the efficiency and accuracy of GA in this paper were proved. 3.4
Optimization design system development
Based on the TDDP of the mixed-flow pumps, the optimization design system (ODS) of the mixed-flow pump impellers was built by combining with GA and programming with FORTRAN language. The ODS flow chart was detailed in Figure 6. The procedure of optimization design of the mixed-flow pump impeller was introduced step by step as follows. ① Set the design parameters and generate the meridional flow passage. ② Encode the optimization parameters and generate the initial population randomly according to a given scale. ③ Determine the three-dimensional impeller and the internal flow field by the TDDP. ④ Calculate the individual fitness based on the loss model and the internal flow field. ⑤ Process the genetic operation and apply the strategies of keeping the optimal individual and employing the
Figure 5 GA.
Process of solving the minimum value of Rastrigin function by
Figure 6
Flowchart of ODS.
niche. ⑥ Repeat steps ③–⑤ until the termination criterion of GA was satisfied. 3.5
Case study of optimization design
The main design parameters of the mixed-flow pump impeller were selected as follows. The flow rate was 0.54 m3 s1. The head was 17 m. The rotational speed was 1450 r min1. The number of blades was 5. 40 S1 stream surfaces and 1 middle S2 stream surface were selected in the process of direct calculation. The shape of the meridional flow passage remained constant in the process of inverse design. In the procedure of optimizing the impeller parameters, the population size was set as 10 and the maximum number of generations was set as 100. The final three-dimensional figure of the impeller determined by optimization design was shown in Figure 7. The ODS (Figure 6) of the mixed-flow pump impeller built in this paper, which selected the highest efficiency of the impeller as the optimization objective and determined the blade parameters P, a0, h and t as the optimization
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parameters, realized the optimizations of the blade shape and position parameters. Figure 8 showed the curves of the supreme fitness and average fitness of the population during the optimization process. It was found that the highest efficiency of the mixed-flow pump impeller was obtained as 89.2% as the population evolves to the 33rd generation, and then this value would no longer change afterward. Figure 9
Structural design diagram of the mixed-flow pump.
4 Test verification The structural design of the mixed-flow pump (Figure 9) was developed after finishing the impeller optimization design in order to verify the practical effects of the ODS of the mixed-flow pump impellers introduced in this paper on improving the impeller design. Figure 10 showed the material object of the mixed-flow pump impeller and Figure 11 demonstrated the material object of the test equipment. Based on the structural design, the performance test was operated on the hydraulic performance test rig of the mixedflow pump (Figure 12) belonging to Beifang Investigation,
Figure 10
Material object of the mixed-flow pump.
Figure 11
Material object of test equipment.
Figure 7 3D figure of the mixed-flow pump impeller obtained by optimization design.
Figure 12
Figure 8
Curves of population fitness varied with evolution generation.
Schematic diagram of the mixed-flow pump test rig of BIDR.
Design & Research (BIDR) CO. LTD. The flow rate was measured by an electromagnetic flowmeter, which was installed horizontally. The straight pipes in the front of and in the back of the flowmeter had the lengths of more than five times the pipe diameter, and the measurement error of this
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flowmeter was in the range of ±0.2%. The head was measured by a differential pressure transmitter. The measurement points were allocated in the inlet and outlet straight pipes, and the measurement error of this transmitter was in the range of ±0.1%. The inlet absolute pressure was measured by a pressure transmitter. The measurement points were arranged in the inlet water tank, and the measurement error of this transmitter was in the range of ±0.1%. The torque and the rotational speed were measured by a torque speed sensor, the measurement error of which was in the range of ±0.1%. The temperature was measured by a digital thermometer and its measurement error was in the range of ±0.1%. The actual fluid levels of inlet and outlet water tanks were indicated by a flap level gauge and its measurement error was in the range of ±3 mm. The random error of this test rig was in the range of ±0.1% and the comprehensive error was in the range of ±0.3%, which reached the leading level. The performance test of the mixed-flow pump was carried out according to the standards of Hydraulic Turbines, Storage Pumps and Pump-Turbines Model Acceptance Tests (IEC60193-1999) and Code for Model Pump Acceptance Tests (SL140-2006). The mixed-flow pump impeller had adjustable blades. The blade angle α was defined as the blade rotation angle on its own rotation axis. If the blade rotation increased the flow capacity of the impeller, was positive, otherwise was negative. The flow rate coefficient and the head coefficient were defined below respectively.
Q , nD3
(13)
H . n D2
(14)
2
The performance test of the mixed-flow pump was operated to obtain the hydraulic performances and cavitation performances under 8 blade angles of 10°, 8°, 6°, 4°, 2°, 0°, +2° and +4°, respectively. Figure 13 demonstrated the hydraulic performance curves of the mixed-flow pump under different blade angles. The efficiency results showed that the highest efficiency of 87.2% was achieved as the blade angle of the mixed-flow pump was 6° and the variation range of the highest efficiency was within 1% as the blade angle changed from 10° to +4°. This demonstrated that the variation of the blade angle had quite small influence on the mixed-flow pump impeller introduced in this paper, so this impeller could be operated with high efficiency under a relatively wide blade angle adjustment range. In addition, the high efficiency operation zones of the efficiency curves were all wide and flat under different blade angles, which illustrated that the mixed-flow pump introduced in this paper could be highly efficiently operated with a relatively wide flow rate range and significantly improved the operation performance of the
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pump under the frequent change of the flow rate. Figure 14 showed the NPSHc curve with the change of the flow rate under the blade angel of 6°. It was seen that the mixed-flow pump had relatively good cavitation performance under the blade angle of the highest efficiency, thus to achieve the combination of highest efficiency performance and fine cavitation performance.
5 Numerical simulation and analysis When simulating the internal flow inside the whole flow passage of the mixed-flow pump obtained by ODS, the governing equations of the fluid flow were three-dimensional steady incompressible Reynolds averaged NavierStokes equations and employed standard k- model 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 employed to solve the governing equations.
Figure 13 Hydraulic performance curves of the mixed-flow pumps. (a) Head coefficient curves; (b) efficiency curves.
Figure 14
NPSHc curve under the blade angle of 6°.
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The inflow was set to be uniform at the inlet boundary and the velocity of the flow was calculated according to the design 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. The equation below was used to determine the hydraulic efficiency of the impeller, based on the internal flow field of the mixed-flow pump impeller obtained by numerical simulation. gQH 100%, (15) M where was the fluid density, Q was the flow rate, M was the moment of fluid on rotation axis, and H was the actual head defined as follows: nout p c2 H i i ci Ai i 1 g 2 g
c A i
i 1
2 o
i
out
nin
c A i 1
i
i
in
,
(16)
where pi was the static pressure, ci was the absolute velocity, Ai was the area of control unit on the inlet and outlet sections, nin was the number of nodes on the inlet section of computational domain, nout was the number of nodes on the outlet section of computational domain. The structured grid was used to structure the computation grids of the whole flow passage of the mixed-flow pump (Figure 15(a)), and local grid refinement was processed to the grids of the boundary layers on the blade surface to enhance the precision of the numerical computation (Figure 15(b)). Figure 16 demonstrated the validation results of grid independence. From this, it was found that after the number of computation grids of the whole flow passage reached 5.2 million, the absolute increment of the hydraulic efficiency of the mixed-flow pump was less than 0.1% as the number of grids continued to grow up. Therefore, 5.2 million grids were employed to compute for the whole flow passage. Figure 17 demonstrated the hydraulic performance curves of the mixed-flow pump with the blade angle of 0°. The head coefficients and the efficiencies, obtained by numerical simulation, basically match the test results. Particularly, around the design flow rate, the head coefficients and the efficiencies had errors of about 1% and 3% respectively, compared with the test results. This illustrated that the precision of the numerical simulation could meet the engineering requirements of predicting and analyzing the hydraulic performance of the mixed-flow pump. 5.1
design flow rate were analyzed by selecting six sections (Figure 18) from the inlet zone to the outlet zone. Figure 19 demonstrated the relative velocity distributions of six sections along the flow direction in the flow passage of the mixed-flow pump under the design flow rate. In all the sections, the relative velocity distributions were uniform and symmetric. In addition, the relative velocities had the
nout
p c i ci Ai i 1 g 2 g nin
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Relative velocity distribution
The relative velocity and static pressure distributions under
Figure 15
Overall and local amplified grids of the mixed-flow pump.
Figure 16 Hydraulic efficiencies of the mixed-flow pump with different grid numbers.
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Figure 17 Validation of the numerical simulation. (a) Head coefficient curves; (b) efficiency curves.
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higher static pressure of the fluid. In Section B, the lowest pressure appeared on the suction surface of the leading edge of the blade and this was also the minimum value in all the six sections, which illustrated that the leading edge of the blade was the most possible position to have cavitation. However, as this area was quite small, it could be concluded that the impeller inlet had relatively satisfied cavitation performance. In Section C, the pressure isoline increased gently and smoothly along the relative velocity direction (from suction surface to pressure surface). The minimum value appeared on the suction surface near the hub, and the maximum appeared on the pressure surface near the tip. In addition, for the same radial position, the pressure on the pressure surface was always higher than the suction surface. In Sections D and E, the pressures on the pressure and suction surfaces basically equaled to each other at the trailing edge of the blade. In Section F locating in back of the outlet, the pressure changed gently along the radial direction, but had very small change range.
6 Conclusions
Figure 18
Schematic diagram of section positions.
minimum values nearby the pressure surface in all the sections except the hub and the tip, but had no flow separation. In the sections of the blade zone (Sections B, C, D and E), the relative velocities on the pressure surface were significantly lower than the suction surface. In addition, the isolines of the relative velocities changed in a uniform and gentle way from the pressure surface near the hub to the suction surface near the tip. From this, it was clearly seen that the blades designed by ODS proposed in this paper had better control on the fluid flow, which illustrated that this system could lead to the blade in better accordance with the real flow, more stable flow inside the impeller and better hydraulic performance of the impeller. 5.2
Static pressure distribution
Figure 20 illustrated the static pressure distributions of six sections along the flow direction in the flow passage of the mixed-flow pump under the design flow rate. The pressure distributions were relatively uniform and symmetric for all the sections. However, the pressure at the same radial position of each section increased gradually from Section A in front of the impeller inlet to Section F in back of the impeller outlet, which reflected that the blade working led to
An optimization design system has been developed in this paper by improving the genetic algorithm with both strategies of keeping the optimal individual and employing the niche, based on the three-dimensional design platform. This system takes the highest efficiency of the impeller as the optimization objective and employs P, a0, ∆θh and ∆θt, which can directly determine the shape and the position of the blade, as optimization parameters. The three-dimensional design platform ensures the designed impeller in accordance with the real flow of the fluid and the application of loss model can lead to quick and accurate prediction of the impeller efficiency. The optimization results illustrate that this system can effectively improve the design of the mixed-flow pump impeller due to its advantages such as high accuracy and fine convergence. From the whole test procedure and results, it can be found that the mixed-flow pump designed in this paper has following characteristics. (1) The mixed-flow pump has the highest efficiency of 87.2%, and the wide and flat high efficiency operation zone to effectively improve the operation performance of the pump under the condition of frequent changes of the flow rate; (2) The highest efficiency of the mixed-flow pump changes within 1% as the blade rotates 14°, which illustrates that the pump is suitable for a relatively wide range of blade angle adjustment; (3) The designed mixed-flow pump has fine cavitation performance and steady operation performance. Numerical simulation is employed to calculate the internal flow fields of the impeller obtained by optimization design and analyze the relative velocity and the pressure distributions. In all the sections, the relative velocity distribu-
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Figure 19
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January (2013) Vol.56 No.1
Relative velocity distributions of various sections (unit: m s1).
Figure 20
Pressure distributions of various sections (unit: Pa).
tions are uniform and symmetric. The relative velocity increases gradually and the pressure isoline grows up gently and smoothly along the relative velocity direction. This can fully illustrate that the optimization design system can lead to the blade with better control of the fluid movement, the blade shape in better accordance with the real fluid, more stable flow and the impeller with better hydraulic performance.
This work was supported by the National Natural Science Foundation of China (Grant No. 51176088). 1
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