Design and optimization of mixed flow pump impeller blades by varying semi-cone angle

The mixed flow pump is a cross between the axial and radial flow pump. These pumps are used in a large number of applications in modern fields. For the designing of these mixed flow pump impeller blades, a lot number of design parameters are needed to be considered which makes this a tedious task for which fundamentals of turbo-machinery and fluid mechanics are always prerequisites. The semi-cone angle of mixed flow pump impeller blade has a specified range of variations generally between 45o to 60o. From the literature review done related to this topic researchers have considered only a particular semi-cone angle and all the calculations are based on this very same semi-cone angle. By varying this semi-cone angle in the specified range, it can be verified if that affects the designing of the impeller blades for a mixed flow pump. Although a lot of methods are available for designing of mixed flow pump impeller blades like inverse time marching method, the pseudo-stream function method, Fourier expansion singularity method, free vortex method, mean stream line theory method etc. still the optimized design of the mixed flow pump impeller blade has been a cumbersome work. As stated above since all the available research works suggest or propose the blade designs with constant semi-cone angle, here the authors have designed the impeller blades by varying the semi-cone angle in a particular range with regular intervals for a Mixed-Flow pump. Henceforth several relevant impeller blade designs are obtained and optimization is carried out to obtain the optimized design (blade with optimal geometry) of impeller blade.


Introduction
Mixed flow pump is a combination of radial and axial flow pumps. It can be considered as a cross between radial and axial flow pumps. Impellers are most important components of the mixed flow pumps. It is always a needful to understand and study the flow passage of the impeller blades in order to understand the occurrence of low hydraulic performance of pump. Empirical calculations are used for incorporating the design methodology where the various occurrences that takes place in the flow passage are ignored. It always acts as a complicated task if any problem in the designing methodology occurs. Since the designer always has less idea about the happenings inside the impeller blade it is very difficult to identify the stages where the problems are found. Rotational-based designing methodology has to be implemented considering fundamentals of fluid mechanics and turbo machinery as prerequisites to resolve the issues related to the performance of pump. In the recent past, many research works have been carried out in this field. Wislicensus [1] was first to initiate this design methodology in 1965, where he focused on the design methodology considering impeller blades as helical surface. Since this designing methodology lacked logical approach and sufficient theoretical support to this concept, it was partially acknowledged. Later Myles [2] in 1965 related the Wislicensus [1] design methodology by superimposing the aerofoil in cascade in a centrifugal field/region assuming that shape of the blades could only be imposed to provide with actual flow field boundary conditions. Considering the effect of slip velocity Busemann 2 1234567890''""  [3] formulated the expression for slip velocity for radial flow pumps. However, its incorporation into a mixed flow pump was not a successful step, which led to the introduction of correction factor into the expression of slip velocity. Senoo, Nakase [4] and Inoue [5] developed a methodology for calculating the meridional stream functions for the swirl distribution. Analysis was carried out for the blade-toblade flow rate, which later was used for the determination of surface and meridional velocity distribution of the blade from hub to tip. In all the above calculations, they assumed the flow to be axisymmetric in nature. It was also concluded that with considering a large number of blades the approximations were found to be even better. For particular angle of blade angles variations Stepanhoff [6] provided the blade design procedure. Neumann [7] came out with a detailed design optimization with all the required steps in design methodology of a mixed flow pump. Gahlot and Nyiri [8] provided a detailed procedure for designing Radial, Francis and Mixed flow Pumps. Yumiko and Watanable [9] used the 3D inverse design methodology, CFD, DOE, RSM for designing of mixed flow pump impeller blades and used Multi Objective Genetic Algorithm for optimization. For the efficient design of mixed flow pumps Kim [10] came out with a procedural methodology. He considered Navier Stokes Equation with shear stress turbulent model and determined the efficiency. Sham Sunder [11] in the year 1981 visualized the distribution of stress in impeller blades using FEM. Ramamurthi and Balasubamanian [12] performed the steady state analysis of centrifugal pumps considering cyclic loading. Janker and Van Essen [13], Samir Lemeš and NerminaZaimović-Uzunović [14], Bhope and Padole [15], Hao et al. [16] and Mehta and Patel [17] summarized the results for analysis of highly complex blades in various pumps.

Nomenclature
Ns

Design Calculations
The adopted values for performing the calculations using the basic equations of fluid mechanics are Ns = 105.74 rpm, D1 = 250 mm, D2 = D2o= 324 mm, P = 20 kW.
From the above relations Dm2 = 278 mm and D2i = 224 mm. With a semi cone angle of 30 o specified layout of the blade was drawn. The blade height is found to be 58mm from the layout for semi cone angle of 30 o . In the same way from the above equations rm1and rm2 are found to be 102mm and 139mm. In order to satisfy the relations given by Myles [2] these values are modified and found to be 109mm and 132mm respectively. By performing the calculations the ratio of Cm1 and Cm2 is determined as 1.21 and the designed meridional profile gets changed to trapezoidal profile, which further is divided into ten equal sections from hub to tip names as A-

Sample Design Calculations for A-A Section of Blade
For section A-A of blade, From the relation (8) The semi-cone angle of mixed flow pump impeller blade has a specified range of variations generally between 45 o to 60 o (source: Pump model test codes and data book) [2]. Form the above sample equations the layout of the meridional plane is depicted in Fig. 2 Fig. 3. In the same fashion the velocity triangles for all the sections can be drawn. It is found that even of the dimensions of the pump layout gets varied the outline of pump layout still remains the same for various semi-cone angles.  It was a significant conclusion by researcher Stodola [18] that for any given tip speed and flow rate for the impeller, if the number of blades gets increased then it results in the reduction in eddy. Hence taking 8 blades into consideration and also the slip velocity in this particular case from the Carter's correlations; pitch, modified blade angle and actual chord lengths for each of the sections of the blade were also determined. In Table 1

Optimal Design of Impeller blades for Mixed Flow Pump
Stepanoff [6] and Stodola [18] performed the calculations considering a constant semi-cone angle of 30 o and varied the other factors those affect the design methodology like specific speed, inlet diameter and outlet diameter for several blade profiles. A comparative study of variation of blade parameters with varying semi-cone angle is tabulated in Table 2. Considering the boundary conditions and results, calculations were performed for obtaining the optimized blade profile. The adopted techniques signify the fact that deflection at the inlet section of blade shows a reduction in the thickness by 0.6mm.

Conclusion
In