High-efficiency design of a mixed-flow pump using a surrogate model

Transcription

1 Journal of Mechanical Science and Technology 30 (2) (2016) 541~547 DOI /s High-efficiency design of a mixed-flow pump using a surrogate model Man-Woong Heo 1, Kwang-Yong Kim 1,*, Jin-Hyuk Kim 2 and Young Soek Choi 2 1 Department of Mechanical Engineering, Inha University, Incheon, , Korea 2 Thermal & Fluid System R&D Group, Korea Institute of Industrial Technology, Cheonan, , Korea (Manuscript Received June 4, 2015; Revised July 7, 2015; Accepted August 26, 2015) Abstract In the present work, the fluid flow characteristics of a mixed-flow pump have beenwere investigated numerically using threedimensional Reynolds-averaged Navier-Stokes equations. The shear stress transport turbulence model and hexahedral grid system were used to analyze the flow in the mixed-flow pump. The efficiency of the mixed-flow pump was evaluated using the variation of two geometric variables related to the inlet angle of the diffuser vane. The design optimization of the mixed-flow pump was performed to maximize the its efficiency at the prescribed specific speed using a surrogate model. Latin hypercube sampling was used to determine the training points for the design of the experiment, and the surrogate model was constructed using the objective function values at the training points. The results show that the efficiency of the mixed-flow pump at the prescribed specific speed is improved considerably by the design optimization. Keywords: Computational fluid dynamics; High-efficiency design; Mixed-flow pump; Optimization; Surrogate model Introduction A mixed-flow pump is a rotating machine in which flow and pressure are dynamically generated. Generally, mixedflow pumps have intermediate characteristics between axialflow and centrifugal-flow pumps, and are widely used in agricultural and sewage applications. Mixed-flow pumps exhibit a complex three-dimensional flow, which involves turbulence, secondary flows, and unsteadiness. This complex flow phenomenon directly affects the hydraulic performance of the mixed-flow pump, and many studies have been conducted to investigate its flow characteristics. Zangeneh et al. [1] demonstrated the suppression of meridional secondary flows in a mixed-flow pump impeller by controlling the blade pressure distribution with an inverse design method. Goto [2] conducted an incompressible Navier-Stokes analysis for a mixedflow pump impeller and validated the results by comparing them with the experimental data on the said impeller at four different tip clearances. Muggli et al. [3] reported the characteristics of a mixed-flow pump in a wide range of flow rates through Computational fluid dynamics (CFD) calculations. Kim et al. [4] designed an impeller and a diffuser using the designed experimental method and CFD to improve the total head and efficiency of a mixed-flow pump. In addition, Kim * Corresponding author. Tel.: , Fax.: address: kykim@inha.ac.kr This paper was presented at the ISFMFE 2014, Wuhan, China, October Recommended by Guest Editor Hyung Hee Cho and Yulin Wu KSME & Springer 2016 et al. [5] performed three-dimensional Reynolds-averaged Navier-Stokes (RANS) analysis to determine the effect of the diffuser s discharge area and vane length on the efficiency of a mixed-flow pump. Many studies have been conducted to investigate the performance characteristics of mixed-flow pumps in relation to various geometric parameters. Recently, design optimization using a surrogate model coupled with three-dimensional CFD has become an efficient tool for turbomachinery designs [6-11]. Kim et al. [6] applied a surrogate model to optimize the axial fan blades with six design variables related to the blade lean angle and blade profile. Kim and Kim [7] presented an optimization procedure using a radial basis neural network surrogate model for a vane diffuser design in a mixed-flow pump. Kim and Kim [8] also investigated the effects of the straight vane length ratio and the diffusion area ratio on the efficiency of a mixed-flow pump through three-dimensional RANS analysis. They reported the optimization of the diffuser vane using a surrogate model. In the present study, an investigation is performed on the effects of the geometry of the diffuser on the hydraulic performance of a mixed-flow pump, using three-dimensional RANS equations. In addition, the design optimization of the mixedflow pump is performed to maximize its efficiency at the prescribed specific speed with two design variables the hub and shroud inlet angles of the diffuser vane using a Response surface approximation (RSA) surrogate model [12].

2 542 M.-W. Heo et al. / Journal of Mechanical Science and Technology 30 (2) (2016) 541~547 Table 1. Design specifications of the mixed-flow pump. Rotational speed, RPM 1780 Total head, m 17.5 Flow coefficient 0.25 Number of rotor blade, EA 5 Number of stator vane, EA 7 2. Specification of the mixed-flow pump Fig. 1 shows the geometry of the mixed-flow pump used as the reference design in this work. The mixed-flow pump consists of five impeller blades and seven diffuser vanes and has a specific speed of Ns = n Q 0.5 /H 0.75 = 550 at the best efficiency point. The flow coefficient and total head at the reference design are 0.25 and 17.5 m, respectively. The threedimensional geometry of the mixed-flow pump with the impeller and vane diffuser is shown in Fig. 1. The detailed design specifications are listed in Table Numerical analysis (a) 3D geometry (b) Meridional geometry Fig. 1. Geometry of the reference mixed-flow pump and computational domain. In this study, the three-dimensional flow characteristics of a mixed-flow pump were analyzed by solving incompressible RANS equations with a Shear stress transport (SST) turbulence model. To solve the incompressible RANS equations, the commercial CFD code ANSYS-CFX 14.5 [13] was used by employing an unstructured grid system. Numerical analysis was performed through the finite volume method to discretize the incompressible RANS equations. Blade profile creation, boundary condition definitions, flow analysis, and post processing were conducted using Blade-Gen, Turbo-Grid, ANSYS CFX-Pre, CFX-Solver and CFX-Post, respectively. The SST model works by solving a turbulence/frequencybased model (k-ω) in the near-wall region and a k-ε model in the remaining regions. A blending function ensures a smooth transition between these two models. Bardian et al. [14] showed that the SST model more effectively captures flow separation under an adverse pressure gradient than other eddy viscosity models, and therefore thus precisely predicts nearwall turbulence. In the present work, the SST model was used as the turbulence closure. The total pressure and designed mass flow rate were set as the inlet and outlet boundary conditions of the computational domain, respectively. Water was used as the working fluid. The stage interface method was used for the connection between the impeller and diffuser stage. A hexahedra grid system was used to generate the mesh in the computational domain, and a grid dependency test was performed in a range of nodes. The test results determined as the optimal number of grids, which consists of 50000, , and nodes in the inlet, impeller, and diffuser domains, respectively. The design points were selected with the help of a Latin hypercube sampling as experimental design for the design of the experiment. The design optimization was performed using RSA, which is a method of fitting a polynomial function to the discrete responses obtained from the numerical calculations. It represents the association of the response function with the design variables. The constructed second-order polynomial response can be expressed as follows: N N N y( x) = b 0 + b x + b x + b x x 2 å j j å jj j å å ij i j (1) j = 1 j - 1 i ¹ j where N is the number of design variables, {x j } represents the design variables, and {β} represents the unknown coefficients in the polynomial. For the second-order polynomial model used in the current study, the number of regression coefficients is (N+1) (N+2)/2. The time scale affects the convergence of the numerical analysis for the steady-state problem. If the time scale is too large, then the resulting convergence behavior is bouncy. In this case, the designer should first reduce the time scale. If the time scale is too small, then the convergence is very slow. A steady-state calculation typically requires enough loop iterations to achieve convergence. Therefore, the results of the steady-state calculation are affected more by convergence than by the time scale. In the present study, the convergence was reasonable because the root-mean-square relative residual

3 M.-W. Heo et al. / Journal of Mechanical Science and Technology 30 (2) (2016) 541~ Table 2. Results of the design optimization. Design Reference Optimum Design β 1h ( ) variables β 1s ( ) Prediction, η (%) RANS, η (%) Increment, η (%) (a) Effects of the parameters on efficiency Fig. 3. Three-dimensional plot of the objective function. (b) Sensitivity of the objective function Fig. 2. Results of the 2 K factor experimental analysis. values of all the flow parameters were less than 1.0E-5, and the change in each parameter was less than 0.1% with at least 100 iterations. The physical time scale was set to 0.1/ω, where ω is the angular velocity of the blades. The solver completed a single simulation in approximately 1000 iterations. The numerical simulations were performed by an Intel Core I7 CPU at a clock speed of 2.94 GHz. Each calculation was subdivided into eight tasks. Parallel calculations by eight CPUs helped reduce the calculation time by as much as 35.6% compared with serial calculation. 4. Results and discussion In this work, eight geometric variables related to the diffuser vane the radius at inflection point (R 4h ), distance between the inlet and inflection point (Z 4h ), control point for the hub contour at inlet (CP 1h ), control point for the shroud contour at inlet (CP 1s ), vane inlet angle at hub (β 1h ), vane inlet angle at shroud (β 1s ), control point for the vane inlet angle at hub (β 1cph ), and control point for the vane inlet angle at shroud (β 1cps ) and their effects on the efficiency of the mixed-flow pump were estimated through the 2 K factor experimental method [15]. The 2 K factorial design is a major set of building blocks for many experimental designs. 2 K represents the designs with K factors, where each factor has two levels. These designs are generated to explore a great number of factors. Fig. 2 represents the results of the 2 K factor experiment. Among the tested geometric variables, the inlet angles of the diffuser vanes are the most important parameters that significantly affect the efficiency of mixed-flow pumps. Another observation is that the R 4h and Z 4h of the diffuser vane substantially affect the efficiency of the mixed-flow pump as the second most important parameters. However, the effect of the control points of the hub and shroud contours at the inlets (CP 1s and CP 1h,, respectively) on the efficiency of the mixedflow pump is negligible. Based on the results of the 2 K factor experimental analysis, the hub and shroud inlet angles (β 1h and β 1s ) of the diffuser vane were selected as the design variables for the optimization of the mixed-flow pump. The inlet angle exhibited a linear distribution from hub to shroud with changes in β 1h and β 1s. The design optimization was performed to enhance the efficiency of the mixed-flow pump with two design variables. The efficiency of the mixed-flow pump is defined as rghq h = (2) φ where ρ, g, H, Q and φ represent the density, acceleration of gravity, total head, volumetric flow rate, and power, respectively. Fig. 3 shows the surface of the RSA model. The axes (β 1h and β 1s ) of the base plane representing the design variables are normalized in the range [0, 1]. This figure evidently shows that the objective value is more sensitive to β 1s than to β 1h. The results of the design optimization are shown in Table 2.

4 544 M.-W. Heo et al. / Journal of Mechanical Science and Technology 30 (2) (2016) 541~547 (a) 20% span (a) Reference (b) 50% span (b) Optimum Fig. 4. Mid-span velocity contours. The efficiency of the reference shape was calculated to be 90.18% at the design flow coefficient. Through optimization, the optimum shape exhibited increased values of β 1h and β 1s compared with the reference design. The objective function value of the optimum design obtained by RANS analysis was enhanced by 1.36% compared with the reference design. In addition, the objective function value obtained by RANS analysis differed from the value predicted by the RSA model with an error of 0.001%. The mid-span velocity contours of the reference and optimum designs are plotted in Fig. 4. The low-velocity region occurred on the suction surface of the reference diffuser. However, the area of this low velocity region decreased, and the location was shifted downstream of the optimum vane. The static pressure distributions on the pressure and suction surfaces of the vane diffuser at the 20%, 50% and 80% spans are shown in Fig. 5. In the case of the 20% span, the static pressure on the pressure surface of the reference design showed partially low values in a range of streamwise location 10% 35%. This low-static pressure region was also observed in a streamwise location 25% 45% at the 50% span. However, these low-static pressure regions were improved through design optimization, and the static pressures on the pressure (c) 80% span Fig. 5. Static pressure distributions on the pressure and suction surfaces of the diffuser vane. surface of the optimum design were comparable to that of the reference design beyond 50% of the streamwise location. The optimum shape showed a higher value of static pressure beyond 15% streamwise location at the 80% span compared with the reference design. Fig. 6 shows the static pressure contours on the 50% span of the reference and optimum designs. The high-pressure region was widely distributed near the pressure surface of the optimum vane diffuser compared with the reference design. These explicitly show the reason for the enhancement of efficiency by optimization. The velocity contours and vectors on the 80% span of the reference and optimum designs are presented in Fig. 7. A recirculating flow near the pressure surface was observed in the reference vane diffuser but disappeared in the optimum vane diffuser with an increased inlet angle. In the reference design, the recirculating flow also appeared near the trailing edge of the suction surface of the diffuser vane. This flow was reduced

5 M.-W. Heo et al. / Journal of Mechanical Science and Technology 30 (2) (2016) 541~ (a) Reference shape (b) Optimum shape Fig. 6. Static pressure contours of the 50% span. (a) Reference shape (b) Optimum shape Fig. 7. Velocity contours and vectors on the 80% span. through the design optimization of the vane diffuser. Fig. 8 shows the flow inlet angle distributions along the span of the reference and optimum vane diffusers. The flow inlet angles of the reference and optimum designs generally increased along the span of the hub to the shroud. The rapid increase of the flow inlet angle was found near the shroud of both designs because of a rotating flow in the impeller. The reference and optimum designs exhibited flow inlet angles of about 150º and vane inlet angles of 145º and 150º, respectively, near the hub of the vane diffuser. The vane inlet angle of the optimum design is similar to the flow inlet angle because of the smooth pressure recovery in the passage of the vane diffuser as shown in Figs. 5 and 6. As a result, the recirculating flow observed in the passage of the reference vane diffuser was certainly diminished in the optimum vane diffuser through the pressure recovery shown in Fig. 7. Fig. 8. Flow inlet angle distributions along the span of the vane diffuser.

6 546 M.-W. Heo et al. / Journal of Mechanical Science and Technology 30 (2) (2016) 541~ Conclusion The vane diffuser in a mixed-flow pump was optimized using the RSA surrogate model and three-dimensional RANS analysis. First, the effects of eight geometric variables of the diffuser vane on the efficiency of the mixed-flow pump were investigated using 2 K factor experimental design. We found that the inlet angles of the diffuser vane had a stronger influence on the efficiency of the mixed-flow pump than the other tested geometric variables. Optimization was performed to enhance the efficiency of the mixed-flow pump with two design variables related to the inlet angle of the diffuser vane. The results of the design optimization showed that the efficiency of the optimum design was improved by 1.36% compared with the reference design. The relative error of the objective function predicted by the RSA surrogate model was only 0.001% compared with the RANS calculations. Acknowledgment This research was supported by the Korea Evaluation Institute of Industrial Technology (KEIT) grant funded by the Ministry of Science, ICT, and Future Planning (No ). Nomenclature CP 1h : Control point for the hub contour at the inlet CP 1s : Control point for the shroud contour at the inlet H : Total head N : Number of design variables n : Rotational speed of the impeller Ns : Specific speed P : Pressure Q : Volumetric flow rate R 4h : Radius at the inflection point RANS : Reynolds-averaged Navier-Stokes RSA : Response surface approximation SST : Shear stress transport x j : Design variables Z 4h : Distance between the inlet and inflection point β : Unknown coefficients in the polynomial β 1cph : Control point for the vane inlet angle at the hub β 1cps : Control point for the vane inlet angle at the shroud β 1h : Inlet angle of the vane at the hub β 1s : Inlet angle of vane at the shroud η : Efficiency ρ : Density φ : Power ω : Angular velocity of the blades References [1] M. Zangeneh, A. Goto and T. Takemura, Suppression of secondary flows in a mixed-flow pump impeller by application of three-dimensiona inverse design method: Part 1- Design and numerical validation, ASME Journal of Turbomachinery, 118 (3) (1996) [2] A. Goto, Study of internal flows in a mixed-flow pump impeller at various tip clearances using three-dimensional viscous flow computations, ASME Journal of Turbomachinery, 114 (2) (1992) [3] F. A. Muggli, P. Holbein and P. Dupont, CFD calculation of a mixed-flow pump characteristic from shutoff to maximum flow, ASME Journal of Fluids Engineering, 124 (3) (2002) [4] S. Kim, K. Y. Lee, J. H. Kim, J. H. Kim, U. H. Jung and Y. S. Choi, High performance hydraulic design techniques of mixed-flow pump impeller and diffuser, Journal of Mechanical Science and Technology, 29 (1) (2015) [5] J. H. Kim, H. J. Ahn and K. Y. Kim, High-efficiency design of a mixed-flow pump, Science in China Series E: Technological Sciences, 53 (1) (2010) [6] J. H. Kim, J. H. Choi, A. Husain and K. Y. Kim, Performance enhancement of axial fan blade through multi-objective optimization techniques, Journal of Mechanical Science and Technology, 24 (10) (2010) [7] J. H. Kim and K. Y. Kim, Analysis and optimization of a vaned diffuser in a mixed flow pump to improve hydrodynamic performance, Journal of Fluids Engineering, 134 (2012) Paper (10 Pages). [8] J. H. Kim and K. Y. Kim, Optimization of vane diffuser in a mixed-flow pump for high efficiency design, International Journal of Fluid Machinery and Systems, 4 (1) (2011) [9] K. S. Lee, K. Y. Kim and A. Samad, Design optimization of low-speed axial flow fan blade with three-dimensional RANS analysis, Journal of Mechanical Science and Technology, 22 (10) (2008) [10] S. Y. Lee and K. Y. Kim, Design optimization of axial flow compressor blades with three-dimensional navier-stokes solver, KSME International Journal, 14 (9) (2000) [11] C. M. Jang and K. Y. Kim, Optimization of a stator blade using response surface method in a single-stage transonic axial compressor, Proceedings of The Institution of Mechanical Engineers, Part A-Journal of Power and Energy, 219 (8) (2005) [12] R. H. Myers and D. C. Montgomery, Response surface methodology-process and product optimization using designed experiments, John Wiley & Sons Inc: New York (1995). [13] ANSYS CFX-14.5 Theory Guide of ANSYS CFX 14.5, ANSYS Inc. (2012). [14] J. E. Bardina, P. G. Huang and T. J. Coakley, Turbulence modeling validation, testing, and development, NASA TM (1997). [15] C. R. Hick and K. V. Turner, Fundamental concepts in the design of experiments, Fifth Edition, Oxford University Press, New York (1999)

7 M.-W. Heo et al. / Journal of Mechanical Science and Technology 30 (2) (2016) 541~ Man-Woong Heo received his bachelor s degree and master s degree from Inha University in 2009 and 2011, respectively. He is currently pursuing research in his Ph.D. degree on Thermodynamics and Fluid Mechanics at Inha University, Korea. His research interests include the design of turbomachinery, numerical analyses, and optimization techniques. Kwang-Yong Kim received his B.S. degree from Seoul National University in 1978, and his M.S. and Ph.D. degrees from the Korea Advanced Institute of Science and Technology (KAIST), Korea, in 1981 and 1987, respectively. He is currently an Inha Fellow Professor at the Inha University of Incheon, Korea. Professor Kim is also the current chairman of the Asian Fluid Machinery Committee, editor-in-chief of the International Journal of Fluid Machinery and Systems (IJFMS), and associate editor of ASME Journal of Fluids Engineering. He is also a Fellow of the American Society of Mechanical Engineers (ASME) and an Associate Fellow of the American Institute of Aeronautics and Astronautics (AIAA). Jin-Hyuk Kim received his Ph.D. in Thermodynamics and Fluid Mechanics at Inha University, Korea, on Aug He was a postdoctoral researcher in the Faculty of Engineering at Kyushu Institute of Technology, Japan, from Sep. to Nov Since Dec. 2013, he has been a Senior Researcher in the Thermal & Fluid System R&BD Group, at Korea Institute of Industrial Technology (KITECH), Korea. His research interests include turbomachinery designs, numerical analyses, optimization techniques, and experimental tests. Young-Seok Choi received his B.S. degree from Seoul National University in 1988, and his M.S. and Ph.D. in Mechanical Engineering at the same university in 1990 and 1996, respectively. He is currently a principal researcher in KITECH. His research interests include computational fluid dynamics and the design optimization of turbomachinery.