Original Article
Direct and inverse iterative design method for centrifugal pump impellers
Proc IMechE Part A: J Power and Energy 226(6) 764–775 ! IMechE 2012 Reprints and permissions: sagepub.co.uk/journalsPermissions.nav DOI: 10.1177/0957650912451411 pia.sagepub.com
Lei Tan1, Shuliang Cao2, Yuming Wang1 and Baoshan Zhu2
Abstract A combined direct and inverse iterative design method was developed for the hydraulic design of centrifugal pump impellers. This method is based on the fluid continuity and motion equations and solves for the meridional velocity taking into account the effects of the blade shape on the flow. The blade shape is drawn by point-by-point integration with blade thickening and smoothing using conformal mapping. Two examples designed using the direct and inverse iterative design method are compared to results using the traditional design method with significantly different meridional velocity distributions and three-dimensional blade shapes. Numerical simulations and tests show that the highest pump efficiency is 2.2% higher with this design method than with the traditional design method. The numerical results agree well with the experiments with a smoother flow pattern than with the traditional design, especially in the volute. Keywords Centrifugal pump, direct and inverse iterative, design method, experiment, numerical simulation Date received: 10 November 2011; accepted: 24 April 2012
Introduction With respect to other pump typologies, centrifugal pumps are the most widely used pumps due to their simple structure and convenient installation. More importantly, they have great energy-saving potential considering the large amounts of energy consumed by the pumps. One of the most efficient ways to save energy for centrifugal pump is to improve the centrifugal pump hydraulic performance by optimizing the design method. The purpose of the hydraulic design is to determine the shape and dimension of the flow passage components, with good performance only when the flow passage component shapes accurately match the actual flow patterns. Therefore, the hydraulic design directly affects the efficiency and operation stability of the centrifugal pumps. In recent years, a number of scholars have presented new design methods for centrifugal pumps. Wang1 and Lu et al.2 proposed a design method based on the S2 stream surface. Goto et al.3 and Asuaje et al.4 presented a design method for centrifugal pumps including design, mesh generation and numerical simulation steps. Bonaiuti et al.5,6 analysed the influence of various parameters, especially the blade loading, on the
impeller performance. Computational fluid dynamics (CFD) is used to give a thorough understanding of the flow phenomenon in many centrifugal pump design methods.7–9
Hydraulic design method The major steps in the hydraulic design of centrifugal pump impellers are: (a) determining the general design parameters based on the requirements; (b) determining the meridional channel shape; (c) calculating the meridional velocity; (d) drawing the blade bone line shape; (e) thickening and smoothing the blade and (f) calculating the wooden pattern. The meridional channel shape is determined by comparing to similar high performance pumps with the 1
State Key Laboratory of Tribology, Tsinghua University, China State Key Laboratory of Hydroscience and Engineering, Tsinghua University, China 2
Corresponding author: Lei Tan, State Key Laboratory of Tribology, Tsinghua University, Beijing 100084, China. Email:
[email protected]
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curves smooth and fluent to guarantee good flow pattern. The traditional design method in which the meridional velocity is obtained by solving the fluid continuity equation based on one-dimensional (1D) or 2D assumptions, is not optimal since it does not satisfy the fluid motion equation and does not take the blade shape effects on the flow into consideration. The streamline curvature method is used to calculate the meridional velocity along the quasi-orthogonal curve in the traditional design method. The meridional velocity gradient equation along quasi-orthogonal lines is dV1 ¼ PðsÞV1 ð1Þ ds d @ 1 sin @ ln sin þ PðsÞ ¼ sin ds @l cos r @l ð2Þ where V1 is the meridional velocity, s the quasi-orthogonal line, l the meridional streamline, the angle between the meridional streamline and vertical, the angle between the meridional streamline and direction normal to the quasi-orthogonal curve, as shown in Figure 1, the blockage factor and r the calculation point radius. The general solution of equation (1) is Rs PðsÞds V1 ðsÞ ¼ V1c e sc
Figure 1. Orthogonal curvilinear coordinate system and quasi-orthogonal line.
the S1 stream surface is solved using the finite element method.10 The meridional velocity gradient equation of the S2 stream surface based on the quasi-orthogonal lines shown in Figure 1 is dV1 C ¼ AV1 þ B þ V1 ds The parameters A, B and C in the blade zone are A¼
ð3Þ
where V1c is the meridional velocity at the impeller hub and sc the quasi-orthogonal curve length at the impeller hub. The integration constant can be obtained from the continuity equation V1c ¼ R sb sc
Q Rs 2r e
sc
PðsÞds
ð4Þ cos ds
where Q is the flow rate and sb the quasi-orthogonal curve length at the impeller shroud. Equation (1) is solved based on the principle that the volume flow rates in each sub-flow channel are identical to calculate the meridional velocity. This method is superior to the traditional design method by calculating the meridional velocity using both the fluid continuity and motion equations and taking the effects of blade shape on the flow into consideration. Thus, the direct and inverse iterative design method possesses a stronger theoretical foundation. The meridional velocity is calculated in the direct and inverse iterative design method from two stream surfaces. The velocity potential function equation of
ð5Þ
1 1 @ sin 1 d þ þ 2 @l r cos ds @ 1þ r @l d @ 1 @ d @ 1 þ r2 sin sin þ r2 þ ds @l cos @l ds @l d @ 2 @ @ d 2 @ r r þ ds @l @l @l ds @l ð6Þ
2!r d @ sin cosð Þ B¼ 2 ds @l @ 1þ r @l C¼
1 dEr 2 ds @ 1þ r @l
ð7Þ
ð8Þ
where Er is the mechanical energy per unit mass. The parameters A, B and C in the non-blade zone are A¼
1 @ d sin sin sin cos @l ds r
B¼0
ð9Þ ð10Þ
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Figure 2. Process of the direct and inverse iteration design method.
C¼
r2 ! þ rV3 dðrV3 Þ dEr þ ds r2 ds
ð11Þ
where ! is the impeller angular velocity, the calculation point circle angle and V3 the circumferential component of the velocity. With the given blade angle distribution along the streamline, the blade bone line shape can be determined by integrating the blade differential equation pointby-point as Z ’¼ 0
l0
1 dl tan e r
ð12Þ
where ’ is the wrap angle, e the blade angle and l0 the total meridional streamline length. The blade angle distribution can be expressed by the quadratic function e ¼ ae l2 þ be l þ ce
ð13Þ
where ae, be and ce are coefficients. The coefficients be and ce can be found from the given e1 for the blade leading edge at l ¼ 0 and e2 for the blade trailing edge at l ¼ l0. Then, the blade angle distribution can be written as e ¼ e1 þ l ðe2 e1 Þ=lo þ ae l ðl lo Þ
ð14Þ
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Figure 3. Schematic diagram of centrifugal pump performance test apparatus: 1. inlet valve; 2. vacuum gauge; 3. centrifugal pump; 4. torque meter; 5. motor; 6. pressure gauge; 7. flow meter; and 8. outlet valve.
Table 1. Design parameters for pumps A1 and A2. Flow rate, Qd (m3/h) Head, H (m) Rotational speed, n (r/min) Number of blade, Z Impeller diameter, D2 (mm) Blade width at exit, b2 (mm)
25 7 1450 7 160 11
Therefore, the only unknown in the quadratic function for the blade angle distribution is the parameter ae, which can be given and optimized by the designer. The conformal mapping method is used to thicken and smooth the blade. The conformal mapping method converts the meridional streamline from the stream surface to a cylindrical surface using xc ¼ R0 ’ Z
l0
yc ¼ 0
R0 dl r
Figure 4. Meridional velocity distributions of the traditional design method.
ð15Þ ð16Þ
where xc and yc are the horizontal and vertical coordinates of the cylindrical surface and R0 the radius at the blade leading edge.
Figure 2 shows the procedure for the direct and inverse iterative design method. The major steps are: (a) an initial impeller is designed using the traditional design method; (b) the meridional velocity is calculated by iterations of two stream surfaces, which satisfies the fluid
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Figure 5. Meridional velocity distributions from the direct and inverse iterative design method for the two stream surfaces for the initial impeller.
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Figure 6. Final meridional velocity distribution from the direct and inverse iterative design method.
Figure 7. Impeller models for pumps (a) A1 and (b) A2.
continuity and motion equations and takes into account the effects of the blade shape on the flow; (c) the velocities determined by the direct calculation are used in the inverse design using point-by-point integration to draw the blade shape with the conformal mapping method to thicken and smooth the blade; and (d) once the updated impeller is designed, direct calculations are again used to obtain the meridional velocity.
The result gives the coordinate data for the 3D model to complete the hydraulic design of the centrifugal pump impeller.
Numerical and experimental verification The reliability of the design results was evaluated using numerical 3D turbulent flow simulations with actual tests of the hydraulic performance.
Numerical simulations Steps (c) and (d) are repeated until the blade shape change between two iterations falls below the designed tolerance. The present impeller designs used a tolerance of 0.0001.
The commercial CFD code Fluent was used to solve the incompressible continuity equation and Reynolds time averaged Navier–Stokes equation to simulate the flow through the pump passage, with the renormalization
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Figure 8. Computational grids for pump A2: (a) volute and (b) impeller.
Figure 9. Pressure coefficient distributions on the blade surfaces of pump A2: (a) Q ¼ 0.6Qd; (b) Q ¼ Qd; and (c) Q ¼ 1.3Qd.
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Figure 10. Pressure coefficient distributions in pump A2: (a) Q ¼ 0.6Qd and (b) Q ¼ Qd; and (c) Q ¼ 1.3Qd.
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Figure 11. Relative velocities between the blade of pump A2: (a) Q ¼ 0.6Qd; (b) Q ¼ Qd; and (c) Q ¼ 1.3Qd.
Figure 12. Hydraulic performance of pump A2.
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group k–" turbulence model.11 The governing equations above were solved using the SIMPLEC algorithm. The second order upwind scheme was used to discretize the convective term with the second order central
Table 2. Design parameters for pumps B1 and B2. Flow rate, Qd (m3/h) Head, H (m) Rotational speed, n (r/min) Number of blade, Z Impeller diameter, D2 (mm) Blade width at exit, b2 (mm)
220 32 1480 6 322 22
difference scheme used to discretize the other terms. The boundary conditions were: (a) constant flow velocity at the inlet; (b) constant pressure at the outlet and (c) no-slip conditions along the impeller blades, sidewalls, volute casing and the inlet and outlet pipe walls. The rotating impeller domain was coupled to the stationary volute domain using the multiple rotating reference frame method. The pressure coefficient Cp was defined as Cp ¼
p 0:5U22
ð17Þ
where p denotes the pressure, the density and U2 the peripheral velocity.
Figure 13. Impeller models for pumps (a) B1 and (b) B2.
Figure 14. Predicted and measured hydraulic performance characteristics for pumps (a) B1 and (b) B2.
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Figure 15. Relative velocity distributions in the impeller for pumps (a) B1 and (b) B2.
Experimental system
Pumps A1 and A2
The hydraulic performance was measured using the setup shown in Figure 3. The system consisted of a water supply section (parts 1–2), pump section (parts 3–5) and an exhaust section (parts 6–8). The centrifugal pump head, H, was calculated as
Pump A1 was designed using the traditional design method while pump A2 was designed using the direct and inverse iterative design method for the same design parameters given in Table 1. Figure 4 shows the meridional velocity distributions given by the traditional design method, where K ¼ 1 denotes the streamline at the impeller hub and K ¼ 14 denotes the streamline at the impeller shroud. Figure 5 shows the meridional velocity distributions given by the direct and inverse iterative design method after the two stream surfaces iteration converged for the initial impeller designed by the traditional method. Figure 6 shows the meridional velocities given by the direct and inverse iterative design method. The meridional velocity distributions given by the two design methods are quite different, especially in the blade zones. The meridional velocity distributions along each streamline for the direct and inverse iterative design method are not the same due to the 3D flow in the impeller. Figure 7 shows the two impeller models given by the two design methods. The blade twist of the direct and inverse design model is more serious than that of the traditional design to better match the meridional velocity distribution. Figure 8 shows the computation grid for pump A2 used for the numerical simulation. The grid had 1,020,000 elements, with yþ between 30 and 300. Figure 9 shows the pressure coefficient distribution on the blade surface for three flow rates. The pressure gradually increases from the impeller inlet to the outlet. The pressure on the blade pressure side (PS) is greater than on the suction side (SS) at the same radial position. In addition, the impact of the flow on the
H ¼ Zo Zi þ
po pi V2o V2i þ g 2g
ð18Þ
where Zo Zi is the height difference between the pressure gauge (part 6) and vacuum gauge (part 2) and pi, po, Vi and Vo denote the pressures and velocities at the pump inlet and outlet, respectively. The centrifugal pump shaft power, PW, was calculated as PW ¼
2nM 60
ð19Þ
where M is the centrifugal pump torque. The centrifugal pump efficiency, , was defined as ¼
gQH PW
ð20Þ
As uncertainty analysis of this test apparatus gave the error in the performance measurement of 0.35%.
Validation cases The hydraulic performance of the direct and inverse iterative design method for centrifugal pumps was evaluated for two cases.
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Figure 16. Relative velocity distributions in the volute for pumps (a) B1 and (b) B2.
surface causes a small low pressure zone on the SS at the blade leading edge for all three flow rates. Figure 10 shows the pressure coefficient distribution inside the centrifugal pump. The pressure gradually increases from the pump entrance to the outlet, and is not even between the blades due to the asymmetrical volute geometric structure and the interaction between the impeller and volute. The flow rate has a great impact on the pressure distribution with the best distribution at the design flow rate. Figure 11 shows the relative velocity between the blades. The relative velocity gradient is high near the blade surface. The relative velocity streamlines in the channel middle are similar to the blade bone line and the relative velocity is uniform without flow separation for the design flow rate. Thus, the blade designed by the direct and inverse iterative design method better controls the flow and is more suitable for the 3D flow. Figure 12 shows the hydraulic performance curves, with the head, H, efficiency, , shaft power PW in function of the flow rate, Q. At the optimum pump working condition, the highest efficiency was 74.8% for Q ¼ 22.6 m3/h and H ¼ 7.1 m. The high efficiency region is wide near the pump design point and the centrifugal pump is stable over the entire operating range. The head flow curve smoothly decreases.
as with pumps A1 and A2. The numerical models had 1,350,000 elements. The predicted hydraulic performances of the centrifugal pumps are compared with the test data in Figure 14. The optimum efficiency of pump B1 was 75.2% for Q ¼ 225.7 m3/h and H ¼ 30.5 m and for pump B2 was 77.4% for Q ¼ 225.8 m3/h and H ¼ 31.5 m. Thus, the efficiency of pump B2 was 2.2% higher than the highest efficiency of pump B1. The simulation results are in good agreement with the experimental data with a maximum difference of less than 5%. Figure 15 shows the relative velocity distributions in the impeller of pumps B1 and B2. The relative velocities in the impeller of pump B2 are more uniform than in B1, especially at the impeller exit. Figure 16 shows the relative velocity distributions in pumps B1 and B2 in the volute, where the fluid velocity gradually decreases along the flow direction as the kinetic energy changes into pressure energy. A vortex develops at the volute throat in pump B1, whereas the flow in this area in pump B2 is very smooth. This shows that the hydraulic performance of the impeller designed by the direct and inverse iterative design method is better than that of the traditional design method for the same design parameters.
Conclusions Pumps B1 and B2 Pump B1 was designed using the traditional design method while pump B2 was designed using the direct and inverse iterative design method for the same design parameters given in Table 2. Figure 13 shows the two impeller models given by the two methods with the blade shapes different
A direct and inverse iterative design method was developed for centrifugal pump impellers based on the fluid continuity and motion equations taking into account the influence of the blade shape on the flow. Two designs were evaluated numerically and in tests to validate the hydraulic performance by the direct and inverse iterative design method. The blade twist in the
Tan et al. direct and inverse iterative design is more serious than in the traditional design for both cases. The CFD analyses demonstrate that the flow patterns in the pumps designed by the direct and inverse iterative design method are better than in the pumps designed by the traditional design method, especially in the volute. The experimental results also verify that the pumps designed by the direct and inverse iterative design method have better hydraulic performance, with the highest efficiency 2.2% higher than in the pump designed by the traditional design method.
Funding This work was supported by the National Natural Science Foundation of China (grant nos. 51176088 and 51179090), National Basic Research Program of China (grant no. 2009CB724304) and Open Research Fund Program of State key Laboratory of Hydroscience and Engineering (sklhse-2012-E-02).
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