Quasi-three dimensional hydraulic design and performance calculation of high specific speed mixed-flow pump

According to the basic parameters of 211-80 high specific speed mixed-flow pump, based on the quasi-three dimensional flow theory, the hydraulic design of impeller and its matching spaced guide vanes for high specific speed mixed flow pump was completed, in which the iterative calculation of S1, S2 stream surfaces was employed to obtain meridional flow fields and the point-by-point integration method was employed to draw blade camber lines. Blades are thickened as well as blade leading edges are smoothed in the conformal mapping surface. Subsequently the internal fields of the whole flow passage of the designed pump were simulated by using RANS equations with RNG k-ε two-equation turbulent model. The results show that, compared with the 211-80 model, the hydraulic efficiency of the designed pump at the optimal flow rate increases 9.1%. The hydraulic efficiency of designed pump in low flow rate condition (78% designed flow rate) increases 6.46%. The hydraulic efficiency in high flow rate areas increases obviously and there is no bad phenomenon of suddenly decrease of hydraulic efficiency in model pump. From the distributions of velocity and pressure fields, it can be seen that the flow in impeller is uniform and the increase of pressure is gentle. There are no obvious impact phenomenon on impeller inlet and obvious wake shedding vortex phenomenon from impeller outlet to guide vanes inlet.


Introduction
With more attention of national economic development is paid on energy conservation and emission reduction, the high specific speed mixed-flow pumps with high hydraulic efficiency, strong cavitation erosion resistance, excellent running stability are required urgently. At the present stage, the pump design usually adopts similar conservation method or speed coefficient method based on the similarity theory, or hydraulic design method based on one dimensional or two dimensional theories, but these methods are empirical and design qualities largely depend on experience of designers and improvement after hydraulic performance tests [1]. It is not common for hydraulic design by using quasi-three dimensional or three dimensional methods. Based on the two families of stream surfaces theory, Binghao et al. put forward the direct and inverse problem iterative design method of mixedflow pump impeller and complete the design of impeller of mixed-flow pump which specific speed is 348 [2]. This method can effectively make up for the defect of traditional design method which the calculation of meridional flow only satisfies the fluid continuous equation. Otherwise, the influence of blade shape on the calculation of meridional flow field is also considered. However, there are relatively less researches on high specific speed mixed-flow pump based on quasi-three dimensional theory. On one hand, for the restriction of specific speed, the specific speed is generally from 200 to 500 in the traditional hydraulic design of mixed-flow pump model impellers. With the increase of specific speed, axial flow pumps are for most use and there are not good similar model pumps for reference. In addition, the development of high specific mixed-flow pumps becomes more and more difficult with the uncertainty of empirical parameters selection based on quasi-three dimensional hydraulic design method. On the other hand, due to the complex structure of high specific speed mixed-flow pump, there are obvious bend torsion characteristics of impeller blades. Otherwise, the shape of hub, the clearance between impeller and shroud as well as the clearance between impeller and guide vanes have obvious influences on the three-dimensional unsteady turbulent flow of high specific mixed-flow pump. Up to now, the cognition that people on the inner flow of high specific mixed-flow pump is limited [3,4].
According to the basic parameters of 211-80 high specific speed mixed-flow pump [5], based on the quasi-three dimensional theory, the self-complied program is adopted to complete the hydraulic design of new mixed-flow pump. The internal flow fields of the designed pump were simulated by using RANS equations with RNG k   two-equation turbulent model. The purpose of this paper is to explore a new rapid development method of high specific mixed-flow pumps.

Quasi-three dimensional hydraulic design
The quasi-three dimensional hydraulic design of designed pump is completed by calculation procedure (Program flow chart is shown in figure 1).

Meridional flow field calculation
The calculation of meridional flow field is to use the geometry and flow boundary conditions of whole flow domain formed by the flow passage of initial impeller and guide vanes to achieve. The orthogonal curvilinear is adopted for calculation (shown in figure 3), 1 q , 2 q and 3 q represent the direction of meridional streamlines, water section lines and peripheral direction respectively. The basic suppose of steady, incompressible, non-viscous, irrotational and only gravity is introduced. The flow of S1 stream surface is supposed irrational and the flow of S2 stream surface is axial symmetry. Where, 3 Hr  , 11 w   , 22 w   , 33 wr   . Supposing the absolute motion of the coming flow of impeller is irrational ( 0 v= ). There is the corresponding three dimensional potential function ( 0 v= ). So, the calculation of the flow of S1 stream surface transfers to the calculation of velocity potential function  . The conformal transformation [7] is introduced, so the potential function equation of S1 stream surface on conformal surface (x-y surface) can be expressed by the follow equation, Where, h represents the flow layer thickness of S1 stream surface. Ritz variation is introduced to solve potential function (1) by giving the inlet, outlet, surface and periodic boundary conditions. The variation function is as follow， The finite element method is adopted for numerical solution.
The velocity gradient equation in non-blade section is, where, The meaning of variable in the above mentioned equations can be seen from figure 4. Based on two-dimensional flow theory ( 0 u Ω  ) and iterative calculation of S1 and S2 stream surfaces, figure 5 and 6 represent the meridional velocity distribution from hub to shroud along meridional surface streamlines in the whole computational domain of impeller respectively. From the comparison of both, there is a little change of meridional surface velocity distribution in the inlet of domain and there is obvious change of meridional surface velocity distribution in blade area and outlet of impeller.   (4) There is 2 0 dq  on characteristic line 2 q const  . Besides, the velocity torque distribution of hub and shroud is given to draw blade camber lines, the wrapping angle of blade  can be obtained by equation (5). The quartic polynomial is adopted to show the velocity torque distribution [8] along meridional streamlines when the shape of meridional flow passage as well as the position of inlet and outlet of blade are given. Then, in order to reduce the dependence of experience on designed results, one of the parameters a is determined to control the distribution of velocity torque of blade through the theoretical analysis. The simplified quartic distribution function expressed by the follow equation, Where, the relative length of meridional streamlines in blade area can be expressed by The designed pump chooses p=0，a=1.6. The velocity torque distribution of hub in blade area is shown in figure 7.

Thickening and smoothing of blades
Blades are thickened as well as blade leading edges are smoothed in the conformal mapping surface [9]. The spaced guide vanes adopted the same designed method with impeller. Finally, the inlet and outlet of impeller and spaced guide vanes blade after thickening and smoothing can be obtained, the three dimensional model of impeller and spaced guide vanes can be seen in figure 8. Unstructured tetrahedral mesh discrete calculation domain is adopted under the comprehensive consideration of time and accuracy. The meshing result is shown in figure 10. Table 1 shows grid independence test results. The influence of grid number on numerical calculation results can be ignored when the correlation of head and efficiency with different grid number is lower than 1%. The finalized total grid number of designed pump is 242 million.  The RNG turbulent model and SIMPLE algorithm are applied to solve the RANS equations. The Multiple Reference Frame (MRF) model is applied to take into account the interaction between stationary parts and rotating impeller. For such calculations, standard wall functions based on the logarithmic law have been used. Standard Scheme has been used for pressure terms and second-order upwind discretization scheme has been used for convection terms. The boundary conditions are as follows: inlet is the velocity inlet and assumed as a uniform distribution, outflow is given as boundary condition at outlet, and the solid walls with non-slip condition such as blade surface, hub and shroud are given the moving wall. The flow rate-head curve and flow rate-efficiency curve of designed pump and model pump can be seen from figure 11. From the figure 11, it can be seen that, the experimental results and CFD simulation results is basically consistent. Thus the reliability of the numerical model was validated. In figure 11, the imaginary line A represents low flow rate condition ( 0.78 d Q ), the imaginary line B represents optimal flow rate condition, the imaginary line C represents high flow rate condition (1.16 d Q ). Compared with the simulation results of model pump, the head of designed pump increases 0.81m and the hydraulic efficiency of designed pump increases 9.1% at B flow rate condition. The hydraulic efficiency of designed pump increases 6.46% at A flow rate condition. In general, the designed pump can restrain the phenomenon of a sudden drop of hydraulic efficiency and the high efficiency areas of designed pump broaden significantly in high flow rate conditions. (m 3 /h)), the directions of velocity at blade inlet change obviously, the distribution of velocity is uninform. With the increase of flow rate, the distributions of velocity become more and more uniform, it is benefit to improve the hydraulic performance. And the velocity increases at same location of impeller blade with the increase of flow rate, the velocity distribution is still uniform, there is no obvious phenomenon of flow separation. 1118  (2) Compared with the 211-80 model, the high specific speed mixed-flow pump designed by the quasi-three dimensional theory has better Hydraulic performance. The hydraulic efficiency of the designed pump on the optimal flow rate condition increases 9.1%. The hydraulic efficiency of designed pump in low flow rate condition (78% designed flow rate) increases 6.46%. The hydraulic efficiency in high flow rate condition areas increases obviously and there is no bad phenomenon of suddenly decrease of hydraulic efficiency in model pump.

The performance curves
(3) The pressure distribution of impeller of designed pump is uniform. There is no obvious impact phenomenon on impeller inlet. The stream lines distribution of whole designed pump is gentle, and there is no obvious wake shedding vortex phenomenon from impeller outlet to guide vane inlet. ICPF2015 IOP Publishing IOP Conf. Series: Materials Science and Engineering 129 (2016) 012008 doi:10.1088/1757-899X/129/1/012008