International Journal of Fluid Machinery and Systems DOI: http://dx.doi.org/10.5293/IJFMS.2016.9.4.313 Vol. 9, No. 4, October-December 2016 ISSN (Online): 1882-9554 Original Paper

Centrifugal Impeller Blade Shape Optimization Through Numerical Modeling

Sayed Ahmed Imran Bellary1 and Abdus Samad2

1Department of Mechanical Engineering, ADCET Ashta, Shivaji University Kolhapur

Sangli-416 301, Maharashtra, India, ibellary@gmail.com 2Indian Institute of Technology Madras Chennai-36, India, samad@iitm.ac.in

Abstract

Surrogate model based shape optimization methodology to enhance performance of a centrifugal pump has been implemented in this work. Design variables, such as blade number and blade angles defining the pump impeller blade shape were selected and a three-level full factorial design approach was used for efficiency enhancement. A three-dimensional simulation using Reynolds-averaged Navier Stokes (RANS) equations for the performance analysis was carried out after designing the geometries of the impellers at the design points. Standard k-ε turbulence model was used for steady incompressible flow simulations. The optimized impeller incurred lower losses by shifting the trailing edge towards the impeller pressure side. It is observed that the surrogates are problem dependent and most accurate surrogate does not deliver the best design always.

Keywords: Centrifugal impeller, Optimal design, Blade number, Blade angles, Multiple Surrogate models, Hydraulic efficiency.

1. Introduction Centrifugal pump is used for fluid transportation in almost all industries and belongs to a class of turbomachinery. The centrifugal

action of impeller blade converts mechanical energy into hydraulic energy. Performance characteristics of pump depend on the impeller geometry. For better performance of pump, design parameters like impeller blade width, exit blade angle, blade height, number of blades, impeller diameter and curvature of blade plays key role.

The performance analysis of pump or impeller was studied with the help of computational fluid dynamics (CFD) simulations [1-2]. To obtain thorough performance prediction of a specific design on fluid behavior in the machine, numerical simulations is an alternate choice and CFD analysis predicts acceptable solutions within limited time [2-4].

Several authors [5-8] have reported the effect of number of blades on the centrifugal pump. Feng et al. [7] performed numerical investigation for different shaped vane diffuser pumps and reported that increase in the number of the impeller blades leads to an increase of pressure fluctuation on the impeller pressure side and on the whole diffuser vane surface. Rababa [8] performed both experiments and simulations on a centrifugal pump with different blade numbers and shape to investigate the flow field and found that increase in blade number augmented the head generation.

Sanda and Daniela [9] performed a numerical study on the radial impeller as a function of inlet blade angle and achieved an efficiency increment by 1-2%. Luo et al. [10] performed experimental test and numerical simulation to understand the effect of inlet geometry on centrifugal pump and reported that hydraulic performance of pump was increased. There are experimental or numerical results for different discharge angles for different centrifugal impellers [11-12] and they have shown an improvement in efficiency. Viscosity which can be defined as flow resistance reduces performance [13].

Surrogate approximation models are widely used in the design and optimization problems and it has been observed that the surrogate models perform differently for different applications [14]. Goel et al. [14] developed an averaged surrogate (PBA) models using response surface approximation (RSA), radial basis neural network (RBNN), and Kriging (KRG) models. Samad [15] implemented the method and concluded that the weighted average surrogate model was much reliable prediction method than an individual surrogate.

Recently Derakhshan et al. [16] used optimization method for the pump impeller based on the artificial neural network during optimization phase for application of artificial bee colony algorithms. Geometrical parameters like hub diameter, suction diameter, impeller diameter, impeller width and inlet and outlet blade angles were chosen as the optimization variables for the enhancement of efficiency. They reported 3.59% augmentation in efficiency for the optimized pump.

In the present problem; impeller blade number, inlet and exit angles were design variables and hydraulic efficiency was an objective function. Different surrogate models combined with a three-dimensional Navier-Stokes solver was used and flow properties were evaluated. Detailed description of the methodology and flow analysis has been presented.

Received March 5 2015; revised December 09 2015; accepted for publication August 5 2016: Review conducted by Young Seok Choi. (Paper number O15007K) Corresponding author: Abdus Samad, Associate Professor, samad@iitm.ac.in

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2. Numerical Formulation 2.1 CFD Modeling

The geometry and geometric parameters of impeller are shown in Fig 1 Table 1, respectively. For investigating the influence of blade number and blade angles on the performance of centrifugal impeller three different blade numbers 6, 7 and 8, inlet angles 10o, 15o and 20o and three outlet blade angles 30o, 40o and 50o are selected and optimization has been performed. Table 2 represents the three different fluids with density and dynamic viscosity used for analysis. All the simulations have been performed at 20o C fluid temperature.

Geometrical modeling and meshing of the flow domain was carried out by using Ansys-BladeGen and TurboGrid module. Ansys-

CFX 13.0 [17] was used for the flow simulations and the problem setting is given in Table 3. The mesh in the flow passage is with O-grid block and at the inlet and outlet it is with H-grid block. These grid blocks, help to account the wall shear stress and boundary layer losses accurately [18]. The mesh has been shown in Fig. 1(c).

Table 1 Features of impeller

Parameter Dimension Shaft diameter, Ds 40 mm Eye diameter, Do 182 mm Hub diameter, Dh 55 mm Inlet diameter, D1 160 mm

Inlet blade width, b1 54 mm Outlet blade width, b2 30 mm Inlet blade angle, β1 10o -20o

Outlet blade angle, β2 30o-50o Blade number, z 6-8

Blade thickness, y 5 mm Outlet diameter, D2 365

Table 2 Viscosity and density of the fluids

Fluid Viscosity [N-s/m2] Density [kg/m3]

Water 1.002E 997 Crude-oil 5.000E 835 Gasoline 5.000E 720

Table 3 Meshing and boundary conditions

Flow domain Single impeller

Interface Periodic Mesh/Nature Structural/Hexahedral Nodes/Elements 634,161/568,620 Fluid used Water, Crude oil and

Gasoline Turbulence model k-ε Inlet/Outlet Pressure/Mass flow rate Residual convergence 1x10-5 Residual type RMS Iteration steps 2000

Mass imbalance % 0.0001

314

cm2

D B β2

Wu2 u2

α2 C

A

W2 c2

cu2

(a) Inlet and exit velocity triangle

Y β1

u1 Wu1

X

W1 cm1= c1 α1 Z

Hub

Inlet

Blade

(b) I ll

(c) Flow domain and mesh

Periodic

Blade

Shroud

Outlet

Hub

Inlet

(d) Leading edge (e) Trailing edge

Fig. 1 Inlet and outlet velocity triangle, geometrical model and

flow domain

315

To account for Reynolds number (Re>105) standard k-ε turbulence model was employed. The k-ε turbulence model is a class of RANS models to which two partial differential equations are solved, one for turbulence kinetic energy, k, and another for turbulence kinetic energy dissipation rate, ε, to get length and time scale information needed to form local eddy viscosities [19]. The simulations were carried out on 3.4 GHz core i7-3370 processor with 8 GB ram. Time per iteration was within 29 seconds.

The blades were swept backwards (β2 < 90o) and used for pumps and blowers [20]. The blade exit angle exerts a decisive influence on the total head and efficiency of pump. As per Euler's equation head generated by impeller blade is given by:

gucucH uu 1122 −

= (1)

Normal entry at impeller inlet implies cu1=0 and from velocity triangle shown in Fig. 1(a) cu2 can be given as

2

222

cotDb

Qucu πb

−= (2)

From Fig. 1(a) it is clear that as β2 increases, the absolute exit velocity c2 also increases. Increase in c2 causes cu2 to increase which

results in increase in head. Hydraulic efficiency is given by,

th

th

th

a

HHH

HH ∆−

==h (3)

2.2 Optimization Methodology The objective of the problem was to enhance hydraulic efficiency (η) (eq. 11) and the design variables were blade number (z), inlet

blade angle (β1) and exit angle (β2). The authors [3, 7-10, 21] found that these are the most important parameters to increase performance of an impeller. Ranges of the variables (Table 4) are selected based on preliminary calculations and prior literature search [3, 21-22, 23-24]. A three-level full factorial design was used to select 27 design points within the variable ranges. These points were used to get efficiency values through CFD analysis and the values were used to construct the surrogate models.

Optimization methodology deals with the selection of the best alternative in the sense of the given designs. The procedure for surrogate based optimizations is presented in Fig. 2. In order to evaluate an objective function as a function of design variables, a lot of numerical simulations are required. For most of thermo-fluids problems, however, a single numerical simulation, such as 3D RANS analysis takes long time to complete. Surrogate models mitigate this burden by constructing approximation models that imitate the behavior of the simulation model as closely as possible while being computationally cheaper to evaluate. The surrogates are also problem dependent which creates difficulty in selecting any surrogate for a particular problem. The surrogates, such as RSA, RBNN, KRG and PBA were implemented by Samad and Kim [25] which are also used for this problem. A weighted average surrogate model was constructed and a search algorithm was used to find optimal points from the constructed surrogates using a sequential quadratic programming.

Problem statement Objective functions, design space and design variables

Design of experiment Allocation of design

points

Numerical analysis Calculation of the values of objective functions at each experimental point

Formulation of surrogates

RSA, RBNN, KRG or PBA

No

Design of experiment Allocation of design points

Yes Whether

optimal design point within design space?

Quest for optimal point Optimal point quest from constructed

surrogate using optimization algorithm

Fig. 2 Optimization procedure

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3. Results and Discussions 3.1 Grid Independency and Validation, Turbulence Models The flow rate, head and rotation of impeller meet the specifications given in [21]. Reasonably good agreement is seen between the

simulated values for water, as fluid, with that of the analytical solutions as in ref. [21]. The computations were carried out at different flow rates, i.e at design and off-design conditions. Figure 3 represents the grid independency test conducted for water at designed flow rate.

Fig. 3 Head variation with number of nodes at design flow rate (Q=0.1003 m3/s)

The pump head at design flow rate was taken as the parameter to evaluate grids and the influence of mesh size on solution was

calculated by setting convergence criteria of 10-5. Once the number of nodes reach a value of 550,000, the variation of head with the increase in number of nodes was not significant.

The results of CFD simulations were compared with the analytical results at design point when water was used as working fluid. The head and efficiency developed was over estimated by 7% and 2% as compared to that of the analytical result. The present result is similar to that of the others [26-27] and a difference of about 10% result between CFD and analytical is allowed [28]. Simulations were performed at 1470 rpm for design and off design conditions to study flow pattern.

The turbulence models are still on developing stage and it is required to test the models for different problems, which can produce more accurate result. Due to flow complexity, the choice of turbulence model is delicate issue to apply in flow simulations, for turbomachinery systems. Bradshaw [29] has reported that the selection of turbulence model is influenced by three criteria: a) physical nature of problem, b) quality of attended results and c) computing power. In spite of this, the traditional RANS models, like k-ε or k-ω are largely used and satisfactory results are achieved. The models k-ε, SST and k-ω were evaluated for impeller flow simulation, at design point and under similar conditions. Analytical result produced 40 m head at design flow rate, whereas by using k-ε, SST and k-ω turbulence models the values obtained are 43.11m, 43.95m and 45.15m, respectively. The least error (=7.21%) among the turbulence models was produced by k-ε and this model was chosen for simulations.

3.2 Optimization Results Initially parametric study was performed for the impeller and it was found that the impeller with z =7 has slightly better performance

as compare to z =6 and 8 designs, hence all further calculations were performed at z =7 with different β1 and β2 angles. Tables 5 contain the optimization results for the objective function with different surrogate models. It was found that the results show that the RSA gives lowest CV error among the other surrogates, and the RBNN performs the worst for the objective function. The low cross validation (CV) error for RSA is likely to indicate that the objective is modeled well as quadratic polynomials [30]. In PBA, the weight associated with RSA is the largest. Hence the equation for the PBA model takes a form:

Fpba= 0.388 Frsa+ 0.301 Frbnn+ 0.312 Fkrg (4)

42

44

46

2 3 4 5 6

H

Number of nodes (X10-5)

Table 4 Design space of the variables

Variables Lower limit Upper limit

β1 10 20 Β2 30 50 z 6 8

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Table 5 Optimal designs suggested by various surrogates and corresponding predicted RANS results for objective function Table 5 compares the predictions of the optimal values predicted by different surrogate models against the actual RANS calculation

at the corresponding points. Consistent with the aforementioned error estimate, RBNN performs the worst. RSA and PBA give better prediction of objective function value. Depending on the characteristics of the design space in each problem, certain surrogate models can perform more satisfactorily than others. However, it is difficult to tell a priori which surrogate model is more appropriate than the others. These are precisely the merits of adopting multiple surrogates and finding ways to form a more robust composite surrogate [30].

The Table 5 shows the CFD and surrogate predicted results. The equation (4) was used to find prediction by PBA model. The highest efficiency enhancement was found by RSA and PBA (=2.32%). It can also be noticed that the error in prediction (=FRANS-Fsur) is lower in the case of RSA and PBA at the optimal points predicted by the respective surrogates. Figure 4 shows the sensitivity of objective function with the 10% deviation of variables from optimal point. RSA model is used to evaluate objective function values. This figure shows that the outlet blade angle is more sensitive to change its value and a small change in value may lead to larger change in efficiency.

Fig. 4 Sensitivity of the variables

Fig. 5 Surrogate predicted response surface for RSA

Surrogate model RSA RBNN KRG PBA

Optimal design variables

β1 β2

16.02 47.52

16.07 43.91

15.38 42.00

15.73 43.96

Fsur 92.15 92.22 92.17 92.16 FRANS 92.09 92.05 92.06 92.09 FRANS - Fsur -0.06 -0.16 -0.18 -0.08 Fref 90.00 90.00 90.00 90.00 FRANS - Fref 2.09 2.06 2.06 2.09

Increase in efficiency 2.32 2.29 2.29 2.32

318

A response surface generated using the best surrogate RSA. Fig. 5 shows that the optimal values of the variables and the efficiency. Normalized values (ranged from 0 to 1) of the variables are presented in the figures. Peak value of the surface shows optimal objective function value.

The optimum values of β1 and β2 are 16.02 and 47.52, respectively. The optimal point having peak hydraulic efficiency (92.15) is located inside the boundary of design space as shown in Table 4. Here, one point can be noticed that the optimal objective function values are not hitting the boundary, which helped to accept the result in single optimization iteration. If improper design space was selected a reiteration was necessary to find the objective function value in the design space.

3.3 Performance Analysis The typical oil industry handles different viscosity fluids. In the present investigation, three fluids are considered for the performance

evaluations (Table 2). The reference and optimized geometry for water was used for the evaluation of pump performance for crude oil and gasoline. The performances were compared for all the fluids given in Table 2.

Performance curves for the fluids at design and off-design conditions are shown in Fig. 6. The hydraulic losses and theoretical head at design point differ to small extent from off design conditions. A measure of hydraulic efficiency is directly linked with head generated. As the head measurement is high for large exit blade angle, the corresponding hydraulic efficiency is more for the liquid with large discharge angle [24]. The optimized β2 has large exit blade angle value compare to designed blade. This results in an increase in head generation. It is due to increase in absolute flow velocity at exit (c2) and corresponding peripheral velocity (cu2) i.e. dynamic part of head increases more rapidly shown in Figure 1 (a). This is evident from Eq. (1) and (2).

Figures 7 demonstrate the contours of static pressure for design point (Q=0.1003 m3/s) at 50% span. It is clear from these figures that static pressure difference between impeller inlet and outlet increases with large exit blade angle. It is due to the fact that fluid flow velocity at impeller outlet decreases with increase in discharge angle. The diameter at the outlet is high; hence absolute velocity as well

Fig. 6 Characteristic curves for different fluids at design and optimum point

319

Fig. 7 Static pressure contours at 50% span for 0.1 m3/s for RSA

SS PS

OPT blade Rotational

direction

REF blade Water Crude oil Gasoline

Fig. 8 Meridional view of velocity contours (Q = 0.1 m3/s) for RSA

OPT blade

REF blade Water Crude oil Gasoline

320

as total pressure at the outlet is augmented. In addition, the increase in discharge angle increases the total pressure at the outlet. This enables the fluid velocity at the outlet to be decreased.

The pressure contours represents a smooth flow between the blades and its value increases continuously towards the exit of the computational domain. It is to be noted that the lowest static pressure was observed at the impeller inlet on suction side which is a location in centrifugal impeller where usually cavitation appears. Here it is above saturation pressure of liquid and the occurrence of cavitation is ruled out [27-28].

The highest static pressure occurs at the impeller exit and the kinetic energy of the flow reaches to its maximum extent. Key factor is the pressure which increases continuously as the mechanical energy given in the form of impeller rotation is converted into the pressure energy. Minimum pressure exists at the suction side and near the leading edge of the blade. The difference in pressure between outlet and inlet increases with increase in blade angle. Liquids at large exit blade angle (optimized blades) have more difference in pressure between inlet and outlet. The peripheral velocity (cu2) at the impeller outlet elevates for large discharge angle as depicted from Fig.1 (a). Reasonable good agreement is seen between the existing results and those of [23, 26-27]. Head being influenced by peripheral velocity at out let increases and in turn hydraulic efficiency as well.

The hydraulic efficiency for the handling crude oil and gasoline is lower than that for the handling water. The decrease in efficiency, while pumping the crude oil and gasoline is due to the viscosity which results in disc friction losses over the outsides of the impeller shroud and hub. These results are in good match with the analytical results of Srinivasan [20] and Gulich [28]. In addition to this, the head and discharge of the pump for which it is designed are at maximum efficiency point where hydraulic losses, profile losses and secondary losses are minimum. On the contrary at all other point of operation shock losses and secondary losses increases resulting into great hydraulic losses and decrease in hydraulic efficiency as well.

Figure 8 shows the meridional view for distributions of velocity contours for various fluids at reference point and optimum point for RSA/PBA. The optimized exit blade angle has large exit blade angle value compare to designed blade. This results in an increase in head generation. It is due to increase in absolute flow velocity at exit (c2) and corresponding peripheral velocity (cu2), i.e., dynamic part of head increases more rapidly. This is evident from equations (1) and (2). The highest velocity occurs at the impeller outlet and the kinetic energy of flow reaches to its maximum extent. The difference in absolute velocity from outlet to inlet increases with increase in blade angle. Liquids with optimized exit blade angle have more difference in velocity between inlet and outlet and greater velocity at the outlet. It is due to increase in peripheral velocity (cu2) at the impeller outlet for large discharge angle as reflected from Figure 1(a). This agrees the analytical results; as fluid flows from inlet to outlet of the impeller, the absolute velocity and pressure increases and for large exit blade angle (optimized angle) its magnitude is high.

Fig. 9 Total pressure (kPa) contours at LE on SS and at TE on PS for reference and optimized blade

OPT blade

REF blade LE Water Crude oil Gasoline

REF blade TE

OPT blade

321

Figure 9 shows the distributions of total pressure on suction surface at leading edge of designed and optimum blade. Pressure

changes along the leading edge on the suction surface of the blade. And, in the case of the efficiency optimized blade, it is noted that near tip of leading edge on suction surface the pressure shows an increased value. Therefore, it is found from this figure that enhancement of efficiency is achieved by modifying the flow angle near the tip of the blade. This is consistent with the results of Gulich [28], who suggested that the optimization of inlet blade angle has remarkable influence on efficiency. Figure 9 demonstrates the total pressure contour on pressure side at trailing edge of designed and optimized blade. It is observed that there is an increase in the total pressure along trailing edge of the blade. For optimized blade increase in total pressure along trailing edge is large compare to designed blade. This in turn increases the head and consequently increases the hydraulic efficiency.

Figure 10 depicts the magnitude and nature of peripheral velocity from inlet to outlet at reference and optimized blade for RSA. It is clear that the optimized blade for all the fluids have higher peripheral velocity component (cu2) at the outlet as compared to reference blades. It is because the absolute velocity and total pressure at the outlet is higher for large exit blade angle.

2. Conclusion The shape optimization of the blades for centrifugal pump impeller was performed for single objective function with three-

dimensional Navier-Stokes analysis. The conclusions are:

• The optimized blade shape produced higher hydraulic efficiency, and this is due to the reduction in flow loss and the disc friction losses over the impeller shroud and hub. The higher viscosity fluids have higher friction losses which hinders the impeller performance.

• At optimized point; hydraulic losses, profile losses and shock losses were found to be lease resulting into an increase in absolute flow velocity at the exit. This has facilitated to increase in dynamic part of head and consequently an augmentation in head has been observed.

• Increase in outlet blade angle by shape optimization has enabled to an increase in outlet cross section size, hence a reduction in fluid pressure drop is observed. This helps in a relatively high head generation and also in efficiency enhancement.

• RBNN gives the highest cross-validation error and performs the least for the objective function. RSA, with the less cross-validation error, shows the best reliability of the surrogate. The weighted surrogate PBA has produced satisfactory result which can be used reliably in the case of pump design modification and protects against choosing a poor surrogate if only one is chosen.

OPT blade

REF blade Water Crude oil Gasoline

Fig. 10 Peripheral velocity distributions. (Q = 0.1 m3/s) for RSA

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Acknowledgments The authors would like to acknowledge Indian Institute of Technology Madras for the NFSC grant (Grant code: OEC/10-

11/529/NFSC/ABDU) to conduct this research.

Nomenclature Ac Lx Rth Re

Cross-section area of micro-channel [m2] Length of heat sink [m] Thermal resistance [ oC/W] Reynolds number (=UbDh/ν)

T Ui

θ, φ ρ

Local mean temperature [K] Mean velocity components (i=1, 2, 3) Design variables, Wc/Hc and Ww/Hc Fluid Density

b CFD CV c D g H ∆H KRG k LE OPT P PBA PS p Q RANS RBNN REF RSA SS SST TE u W 1 2

Breadth, mm Computational fluid dynamics Cross validation Absolute fluid flow velocity, m/s Diameter, mm Acceleration due to gravity, m/s2 Head, m Hydraulic losses, m Kriging Turbulence kinetic energy, J Leading edge Optimized Power, W PRESS based averaging Pressure side Pressure, N/m2 Volume flow rate, m3/s Reynolds-averaged Navier Stokes Radial basis neural network Reference Response surface approximation Suction side Shear stress transport Trailing edge Peripheral velocity, m/s Relative fluid velocity, m/s Subscript Inlet Outlet

a design h krg m o pba RANS RBNN RSA s sur th u w α β ε η μ ρ ω

Actual Design flow Hub Kriging Meridional component Eye PRESS based averaging Reynolds-averaged Navier Stokes Radial basis neural network Response surface approximation Shaft Surrogate Theoretical Peripheral component Relative fluid velocity component, m/s Greek Symbols Flow angle, o Blade angle, o Rate of energy dissipation, J/s Hydraulic efficiency, % Dynamic viscosity, N-s/m2 Density of fluid, kg/m3 Angular velocity, rad/s

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