PARAMETRIC STUDY PERFORMANCE OF A CENTRIFUGAL PUMP BASED ON SIMPLE AND DOUBLE-ARC BLADE DESIGN METHODS

Transcription

1 3 rd International Conference on Experiments/Process/System Modeling/Simulation & Optimization 3 rd IC-EpsMsO Athens, 8- July, 2009 IC-EpsMsO PARAMETRIC STUDY PERFORMANCE OF A CENTRIFUGAL PUMP BASED ON SIMPLE AND DOUBLE-ARC BLADE DESIGN METHODS Spyridon D. Kyparissis, Eleni C. Douvi, Elias E. Panagiotopoulos, Dionissios P. Margaris, and Andronikos E. Filios * Fluid Mechanics Laboratory, Mechanical Engineering and Aeronautics Department University of Patras GR Patras, Greece kypariss@mech.upatras.gr * Fluid Mechanics & Turbomachinery Laboratory, Department of Mechanical Engineering Educators, School of Pedagogical and Technological Education, GR-42 Athens, Greece fmtulab@otenet.gr Keywords: Blade impeller, centrifugal pump, CFD numerical analysis, double-arc design, simple-arc design. Abstract. The development of computational fluid dynamics has successfully contributed to the prediction of the flow field through pumps and the enhancement of their design. The present analysis studies the influence of simple and double-arc blade design methods in the flow field and in the efficiency of a centrifugal pump. The one-dimensional Pfleiderer s approach based on empirical analytical equations is applied for the main geometrical characteristics of the pump impeller. The numerical solution of the discretized three-dimensional, incompressible Navier-Stokes equations over an unstructured grid is accomplished with a commercial CFD finite-volume code. The characteristic performance curves are resulted through the calculation of the internal flow field. For every blade design method, the pressure distribution of the pump is computed.. INTRODUCTION The complexity of any study of the flow in a turbomachinery is a real challenge and has led to much of the research work over recent decades. Referring to pumps, many studies have been carried out, but even nowadays some flow characteristics are still under study and far from being fully understood. With great advances in computational fluid dynamics (CFD) and rapid increasing of computing capacity, the numerical simulation gets to be used as a common tool of direct flow analysis in turbomachinery and has successfully contributed to the prediction of the flow through pumps and the enhancement of their design. Initially, the design of a centrifugal pump was based mainly on empirical correlation and engineering experience. Nowadays, the design demands a detailed understanding of the internal flow during design and off-design operating conditions. Various researchers have considerably contributed in revealing the flow mechanisms inside centrifugal pump impellers aiming to the design of high performance centrifugal pumps. The reported works by Johnson and Moore [], Borges [2], Lakshminarayana [3], Xu and Gu [4] and Zangeneh et al. [5] are a significant research on the computational and experimental verification of the flowfield in centrifugal pumps. Several research works have been published during the last decades, taking into account the prediction of the performance at various operating conditions. Byskov et al. [6] studied the flow field in the impeller of a centrifugal pump at design and off-design conditions, applying large-eddy simulations. Asuaje et al. [7] used inverse design method for centrifugal pump impellers and compared the results with numerical simulation tools. Peng [8] presented a practical combined computation method of mean through-flow for three-dimensional inverse design of hydraulic turbomachinery blades to consider the influence of interrelated hydraulic components. Majidi [9] used CFD analysis to solve the unsteady three-dimensional viscous flow in the entire impeller and volute casing of a centrifugal pump. Cao et al. [0] presented a combined approach of inverse method and direct flow analysis for the hydrodynamic design of gas-liquid two-phase flow rotodynamic pump impeller. Miyauchi et al. [] studied the optimization of the meridional shape design of pump impellers. Choi et al. [2] investigated the internal flow characteristics and its influence on the performance of a very low specific speed centrifugal pump. Conzález and Santolaria [3] also

2 studied the relationship between the global variables and the dynamic flow structure numerically obtained for a low specific speed centrifugal pump. Anagnostopoulos [4] developed a numerical model for the solution of the RANS equations in the impeller of a centrifugal pump and applied it for direct flow analysis and parametric investigation of the effect of some impeller design details on its hydrodynamic characteristics. Bacharoudis et al. [5] studied the influence of the outlet blade angle in the performance of a laboratory centrifugal pump. Anagnostopoulos [6] developed a numerical optimization algorithm based on the unconstrained gradient approach that was combined with the evaluation software in order to find the impeller geometry that maximizes the pump efficiency. The present analysis is concerned with the influence of simple and double-arc blade design methods in the flow field and in the efficiency of a centrifugal pump. The main geometrical characteristics of the pump impeller are estimated by Pfleiderer s one-dimensional analytical approach. Through the CFD numerical investigation of the flow field the design and off-design performance characteristic curves are resulted. In addition, the pressure distribution inside the pump is calculated for every blade design method. The computational fluid dynamics analysis is carried out with the commercial software package Fluent [7], which has been widely used in the field of turbomachinery. 2. BLADE DESIGN ANALYSIS The one-dimensional Pfleiderer s analytical method has been widely used for the design and the construction of centrifugal pump impellers on hydraulic machines. It is a simple and fast method with empirical coefficients and formulas [9]. The above analysis is applied for the estimation of a 6-blade centrifugal pump impeller main characteristics designed to operate at flow rate Q = 45 m 3 /h and total head H = 0 m, when the rotational speed is n = 925 rpm. These values are inserted to the Pfleiderer s algebraic algorithm [9] and the solution results are given in Table, taking into consideration the blade trailing edge angle β 2 = 20 o and a constant blade thickness s = 6.5 mm. A simple-arc and three different double-arc blade design methods are applied according to the Pfleiderer s theory [9] for the impeller blade design of a centrifugal pump. Design Impeller Characteristics Values Suction pipe diameter D s, mm 7 Diameter of the impeller at the suction side 40 D, mm Diameter of the impeller at the pressure side D 2, mm 280 Impeller width at the suction side b, mm 20 Impeller width at the pressure side b 2, mm 20 Blade leading edge angle β, deg 5.4 Number of blades z 6 Specific speed n q 8.4 Hydraulic efficiency η h 0.83 Table. Main characteristics of the impeller design applying the Pfleiderer s analytical method Simple-arc method (SAM) is based on the construction of one curve for the blade mean line, as shown in Figure. The mean line of the blade is the arc AC, E and R are the center of curvature and the radius of the arc, respectively. An auxiliary circle C a is drawn, concentrically to the suction and pressure side of the impeller. The diameter d of the circle C a is given from the following expression: d = D sin β () where D and β are taken from Table of the present analysis. The center of curvature E is defined at the tangent of the auxiliary circle C a from the start of the arc A at distance equal to the radius R of SAM curve that is formulated as:

3 2 2 2 D D 2 2 R = 2 D2 D cos β 2 2 cos β 2 (2) where D, D 2 are the diameters of the impeller at the suction and pressure side respectively and β, β 2 are the blade leading edge and trailing edge angles, respectively. The values of the magnitudes D, D 2 and β are taken from Table of the present analysis. Figure. Representation of the mean blade line using SAM on the plane xoy perpendicular to the shaft of the pump On the other hand, double-arc methods are based on the construction of two curves for the blade mean line. In order to examine the effect of the continuity between the two curves to the pump efficiency, Pfleiderer proposes three different blade design methods named DAM, DAM2 and DAM3 for the double-arc design approach. DAM is characterized as the poorest continuity of two curves to the connection point B, as shown in Figure 2. Figure 2. Representation of the mean blade line using DAM on the plane xoy perpendicular to the shaft of the pump The mean line of the blade is consisted of the arcs A B and BC, E and R are the center of curvature and the radius of the arc A B, respectively. Likewise, E 2 and R 2 are the center of curvature and the radius of the arc BC, respectively. The auxiliary circle C a is drawn, according to the definition formula (). The periphery of the suction side is divided to six parts, just as the number of blades z. The tangents of the auxiliary circle from the points A and A 2 are intersected to the point E. The point B is the end of the first arc and is defined at the extension of the line that connects the points E and A 2 at distance R, equal to the distance between the points E and A. The radius

4 R 2 of the second arc BC is defined as: 2 = 2 D2 cos β2 g R 2 2 D2 r 2 2 g r cos β g (3) where r g is the distance between the points O and B and β g is the angle between the OB and E B, as shown in Figure 2. DAM2 differs from DAM to the arc length of A B, keeping the initial point A fixed. This is represented by the formula: A B = 0. 75A B (4) DAM 2 DAM As a result, the center of curvature E 2 shifts downwards and the arc length of the blade becomes smaller while the magnitudes E, R and R 2 remain fixed. DAM3 analysis depends on the radius R to be about 20% bigger than the radius of DAM, as formulated in the following expression: R =. 2R (5),DAM 3,DAM This results the shift of the center of curvature E to the left and E 2 downwards, while the radius R 2 remains constant. It s obvious that the shift of E 2 is greater while comparing to the DAM2. Thus, the arc length of the blade DAM3 becomes smaller than the corresponding of DAM2. 3. FLOWFIELD NUMERICAL SIMULATION 3. Governing equations In the present analysis, the simulation of a three-dimensional centrifugal pump flow is based on the incompressible Navier-Stokes equations, which are expressed in polar coordinates for the rotating impeller and in Cartesian coordinates for the stationary parts. The governing equations for the rotating impeller flowfield procedure analysis are formulated in the following expressions, neglecting the energy equation and the gravity: Continuity equation Momentum equation u r = 0 (6) r r r r r r r r r p τ ur u r + 2ω u r +ω ( ω ) = + ρ ρ (7) where u r r is the vector fluid velocity in the rotating system, p and ρ are the static pressure and the density of the fluid, respectively, r ω is the angular rotation speed of the pump impeller and r τ is the stress tensor including both the viscous and the turbulence viscosity terms. The momentum equation (7) contains two additional acceleration terms: the Coriolis acceleration ( 2ω u r r r ω ω ). r r ) and the centripetal acceleration ( ( ) r For the simulation of the stationary parts of the centrifugal pump, the continuity equation (6) remains the same and the momentum equation is changed as the rotating terms in (7) are neglected: r r r p τ u u = + ρ ρ where u r is the local vector fluid velocity in the stationary frame of reference. The flow turbulence is modeled with the standard k-ε model that is rated as one of the most used models in a wide range of industrial flows showing satisfactory results. It is a semi-empirical model based on differential (8)

5 transport equations for the turbulence kinetic energy k and its dissipation rate ε as formulated below: r µ t ρku = µ + k +Gk ρε... σ k (9) 2 r µ t ε ε ρεu = µ+ ε +Cε Gk C2ε ρ... σ ε k k (0) In the above equations, turbulent viscosity µ t is computed by combining k and ε t µ 2 k µ =ρc () ε where µ is the laminar viscosity, σ k and σ ε are the turbulent Prandtl numbers, G k represents the generation of turbulent kinetic energy due to the mean velocity gradients and C ε, C 2ε, and C µ are k-ε model constants with values.44,.92 and 0.09, respectively. 3.2 Geometry and grid The computational flowfield solution of the centrifugal pump is accomplished with the commercial code Fluent [7] that utilizes the finite volume method for the solution of the steady three-dimensional incompressible Navier-Stokes equations. The geometry and the mesh of the computational pump domain were generated with Fluent s pre-processor, Gambit [8]. The main geometry of the pump is illustrated in Figure 3. Figure 3. Illustration of the centrifugal pump geometry The whole domain consists of three sub-domains or zones. All gaps between the impeller shroud and the pump casing are neglected in the present numerical simulation. The first zone represents the inlet pipe which diameter is D s = 7 mm, as shown in Table. The second zone that incorporates the pump impeller, is moving with the applied rotational speed of n = 925 rpm. The third zone is consisted of the volute and outlet pipe of the centrifugal pump. Both first and third zone are stationary. A multiple reference frame has been used, since the present work deals with a problem that involves both stationary and moving zones. The stationary reference frame involves the first and the third zone, while the moving frame of reference is involved by the second zone. An unstructured mesh with tetrahedral cells is used for the second and third (volute) zone. Moreover, unstructured wedges are generated in order to define the first and third (outlet pipe) zone. In the regions near the leading and trailing edge of the blades, the grid is refined, as well as in the region of the tongue to better resolves the flow details, as shown in Figure 4. The involved parameters regarding the turbulence k-ε model intensity and the hydraulic diameter are estimated with values of 5% and D s /2, respectively. The pressure-velocity coupling is performed through the SIMPLE algorithm. For convections terms of k-ε model, second order upwind discretization is used, while for diffusion terms central difference schemes. Although grid size is not adequate to investigate local boundary layer variables, global ones are well captured, using wall functions, based on the logarithmic law. For all design methods, a mesh of approximately one million cells is applied.

6 Figure 4. Refined mesh in the region of trailing edge and tongue For better simulation of the pumping circuit influence, the applied boundary conditions involve the extension of the computational domain by adding a reasonable length at the inlet and outlet pipes of the centrifugal pump. At the inlet zone, the axial velocity is a constant based on the through flow for the pump, while the absolute tangential velocity at the inlet is defined zero. In the rotating frame, the relative tangential velocity is -r ω r and the radial velocity is zero. At the exit of the outlet pipe, assuming a fully developed turbulent flow, a practically zero velocity gradient is set. The walls of the model are stationary with respect to their respective frame of reference and no slip condition is applied. Initially the simulations were executed with a low rotational speed and then the rotation is slowly increased up to the desired design level of n = 925 rpm, applying the law of similarity. According to this law, the quantity of the pump flow rate Q is changed proportionally to the rotational speed n. This process is used to control the large complex forces in the flow that may evoke less stable calculations as the speed of rotation and the magnitude of these forces increases. The number of iterations adjusted to reduce the scaled residual below the value of 0-4, which is the criterion of convergence. Depending on the blade design method, the convergence was achieved at different iterations. In order to achieve a smooth convergence, various runs were attempted by varying the underrelaxations factors. Initializing with low values for the first iterations steps and observing the progress of the residuals, their values were modified for accelerating the convergence. The working fluid for this analysis is the water with density ρ = kg/m 3 and viscosity µ = kg/ms. The flow rate is augmented per 30%, 45%, 50%, 55%, 75% and 00% and is assumed at the entrance of the inlet pipe. 4. NUMERICAL RESULTS In the present analysis CFD numerical investigation is applied for simple and double-arc blade design methods in order to study the hydrodynamic efficiency of a centrifugal pump. The CFD predicted nominal flow rate and total head for all blade design methods are computed Q n = 67.5 m 3 /h and H n = 9.33 m, respectively. The nominal flow rate is defined by the point in which the hydraulic efficiency of the pump reaches its maximum value. The above numerical analysis is given in Figure 5 presenting the total performance head curves for all blade design methods. In Figure 5 the ordinate is the dimensionless total head of the pump, while the abscissa is the dimensionless flow rate. It s observed that as the flow rate increases the total head decreases for double-arc design methods. For the SAM analysis at low flow rates, an increase of the total head is noticed. The hydraulic efficiency curves of each blade design method are depicted in Figure 6. For flow rates lower than nominal DAM3 is the most efficient method. On the other hand, for flow rates greater than the nominal, the pump is more efficient applying SAM. The curve of the hydraulic efficiency for Q < Q n is steeper for SAM and smoother for DAM. The opposite happens for Q > Q n, where the hydraulic efficiency decreases rapidly at DAM and DAM2 and slowly at SAM. At nominal flow rate the value of the hydraulic efficiency is ranged between 0.88 and for the examined blade design methods, which is in reasonable accuracy with the predicted value η h =0.83 (Table ) according to the one-dimensional Pfleiderer s approach. At this flow rate the maximum hydraulic efficiency is observed at SAM and DAM3 blade design and the minimum at DAM. For all design methods the maximum hydraulic efficiency is observed at nominal flow rate, except for DAM, where is noticed at lower flow rate. SAM is less efficient for the lowest examined flow rate, while DAM, which is characterized as the method with the worst curvature, for the higher ones. In addition, the pressure distribution inside the pump is studied for all the examined blade design methods. The static pressure variations in pump flowfield are shown in Figure 7 for three flow rates according to SAM and DAM2. At the suction region of the pump are prevailed low pressures, whereas at the volute and at the exit of the impeller there are regions with high pressures. At low flow rates, at the exit of the impeller higher pressure is

7 observed than at higher ones. This explains the high total head at low flow rates. Furthermore, at high flow rates, low pressure is noticed at the interaction of the leading edge of the blades. This phenomenon is more intense at DAM2, which explains the low efficiency at these flow rates. Figure 5. Predicted total head curves for the blade design methods Figure 6. Hydraulic efficiency curves for the blade design methods Figure 7. The static pressure distribution inside the pump at the middle span plane for SAM and DAM2

8 5. CONCLUSION Numerical simulation is accomplished in a 3D six-bladed centrifugal pump for the calculation of the flow field and the prediction of the pump characteristic performance curves. The one dimensional Pfleiderer s analytical approach is applied for the estimation of the pump impeller main geometrical characteristics. The results of the above numerical procedures show that when pump operates at nominal flow rate, SAM and DAM3 models are more efficient. Furthermore, at flow rates lower than nominal, DAM3 causes a significant improvement of the hydraulic efficiency than the other design methods. On the other hand, for flow rates greater than nominal, SAM analysis gives higher hydraulic efficiency. REFERENCES [] Johnson, M.W. and Moore, J. (980), The development of wake flow in a centrifugal impeller, ASME Journal of Engineering for Power, Vol. 02, pp [2] Borges, J.E. (990), A three-dimensional inverse method for turbomachinery: Part I Theory, ASME Journal of Turbomachinery, Vol. 2, pp [3] Lakshminarayana, B. (99), An assessment of computational fluid dynamic techniques in the analysis and design of turbomachinery, ASME Journal of Fluids Engineering, Vol. 3, pp [4] Xu, J.Z., Gu, C.W. (992), A numerical procedure of three-dimensional design problem in turbomachinery, ASME Journal of Turbomachinery, Vol. 4, pp [5] Zangeneh, M., Goto, A. and Takemura, T. (996), Suppression of secondary flows in a mixed-flow pump impeller by application of three-dimensional inverse design method: Part - Design and numerical validation, ASME Journal of Turbomachinery, Vol. 8, pp [6] Byskov, R.K., Jacobsen, C.B. and Pedersen, N. (2003), Flow in a centrifugal pump impeller at design and off-design conditions-part II: Large eddy simulations, ASME Journal of Fluids Engineering, Vol. 25, pp [7] Asuaje, M., Bakir, F., Kouidri, S. and Rey, R. (2004), Inverse design method for centrifugal impellers and comparison with numerical simulation tools, International Journal of Computational Fluid Dynamics, Vol. 8, pp [8] Peng, G. (2005), A practical combined computation method of mean through-flow for 3D inverse design of hydraulic turbomachinery blades, ASME Journal of Fluids Engineering, Vol. 27, pp [9] Majidi, K. (2005), Numerical study of unsteady flow in a centrifugal pump, ASME Journal of Turbomachinery, Vol. 27, pp [0] Cao, S., Peng, G. and Yu, Z. (2005), Hydrodynamic design of rotordynamic pump impeller for multiphase pumping by combined approach of inverse design and CFD analysis, ASME Journal of Fluids Engineering, Vol. 27, pp [] Miyauchi, S., Kasai, N. and Fukutomi, J. (2006), Optimization of meridional shape design of pump impeller, 23 rd IAHR Symposium on Hydraulic Machinery and Systems, Yokohama, Japan. [2] Choi, Y., Kurokawa, J. and Matsui, J. (2006), Performance and internal flow characteristics of a very low specific speed centrifugal pump, ASME Journal of Fluids Engineering, Vol. 28, pp [3] González, J. and Santolaria, C. (2006), Unsteady flow structure and global variables in a centrifugal pump, ASME Journal of Fluids Engineering, Vol. 28, pp [4] Anagnostopoulos, J. (2006), CFD analysis and design effects in a radial pump impeller, WSEAS Transactions on Fluid Mechanics, Vol., pp [5] Bacharoudis, E.C., Filios, A.E., Mentzos, M.D. and Margaris, D.P. (2008), Parametric study of a centrifugal pump impeller by varying the outlet blade angle, The Open Mechanical Engineering Journal, pp [6] Anagnostopoulos, J. (2008), A fast numerical method for flow analysis and blade design in centrifugal pump impellers, Computers & Fluids, Vol. 38, pp [7] Fluent Inc., Fluent 6.2 Documentation-User s Guide, [8] Computational Fluid Dynamics (CFD) Preprocessor, Gambit 2.3 Documentation-User s Guide, [9] Pfleiderer, C. (96), Die Kreiselpumpen für Flüssigkeiten und Gase, Springer-Verlag, Berlin.