1 Introduction

The centrifugal pump is of wide application in industrial and agricultural productions, consuming a large amount of electric power with great potential in energy saving. Among all components in the centrifugal pump, the impeller is the most important flow passage because it is the only part that does work[1]. Therefore, new or modified design methods for the impeller have been proposed continually to improve its comprehensive performance especially the efficiency. WANG[2] and LU, et al[3] provided a design method for centrifugal pump impeller based on S2 stream surface

considering the slip factor on the blade. ZANGENEH[4] and GOTO, et al[5] designed the centrifugal pump impeller by governing the circulation distributions along streamlines.TAN, et al[6], proposed a direct and inverse iteration design method for centrifugal pump impeller. On the other hand,some researchers paid attention to the influences of design

parameters on pump characteristics. CAO, et al[7] and BING,et al[8] considered the effect of velocity torque distribution on the pump efficiency. BONAIUTI, et al[9–10], analyzed the influence of several parameters, especially the blade loading, on the impeller performance.

The blade wrap angle is defined as the one between the tangent lines at leading and trailing edges of the blade. An increase in blade wrap angle would lead to a longer flow passage between the blades and thus a significant rise in friction loss. On the contrary, a small blade angle will generate a short flow passage but result in a poor control on

the flow in impeller arousing separation loss probably. Therefore, the blade wrap angle is a key parameter for blade shape, flow pattern in impeller and performance of pump. For the same impeller diameter, YANG, et al[11], used a cubic curve to form blade shape and compared the efficiencies of those impellers with different blade wrap angles. CAO, et al[12], numerically simulated five impellers with different blade wrap angles and showed that the angle had great effect on the internal flow and hydraulic performance of pumps. ZHANG, et al[13], numerically analyzed the relationship between the blade wrap angle and the centrifugal pump performance.

In this paper, three centrifugal pump impellers were designed with different blade wrap angles using the direct and inverse iteration design method. The influence of the blade wrap angle on the centrifugal pump performance was investigated both numerically and experimentally.

2 Centrifugal Pump Impeller Design

2.1 Design method

The direct and inverse iteration design method for centrifugal pump impeller is based on the continuity and motion equations of the fluid with the effects of the blade shape on the flow being considered. This method is applied by a set of computer code written with FORTRAN language. The initial parameters at the beginning of the design program are as follows: (1) flow rate, head and rotation speed of centrifugal pump and the fluid properties;

(2) shape of hub and shroud, leading and trailing edges of the meridional flow passage; (3) blade thickness distribution, blade number and blade wrap angle.

2.1.1 Process of design

Fig. 1 shows the process of the direct and inverse iteration design method.

Fig. 1. Process of direct and inverse iteration design method First, according to the given design parameters of the centrifugal pump, the traditional design method is adopted to design the initial impeller configuration. Second, based on the continuity and motion equations of the fluid, the meridional velocity distribution is obtained by the iteration

between the S1 and S2 stream surfaces. Third, with the obtained meridional flow field, the blade shape is generated using point-by-point integrations. Through this step, a new impeller is obtained. Fourth, the iteration of the S1 and S2 stream surfaces is operated again for the new designed impeller to get an update meridional velocity distribution.

The third and fourth steps should be repeated until the differences of the meridional transversals of two successive designed impellers are within a prescribed tolerance. For the present impeller design, the tolerance is set to be 0.000 1.

2.1.2 Blade drawing

The relation between blade wrap angles and meridional streamlines can be described by the bone line integration equation as follows.

The blade angle distributions along meridional streamlines can be described as a quadratic function, which can be expressed as

where the parameter a is the only unknown variable, which can be given by the designer to determine the blade angle distribution.

The blade wrap angle is a key parameter for impeller designing as it is directly related to the ability of doing work, the flow pattern, the friction loss, the separation loss and so on. So the value of the blade wrap angle should be easily controlled in impeller design process[13]. In previous blade drawing method[14], the blade angle distribution was given by the parameter a, and then the blade shape was structured according to Eq. (1). It is easy to determine the distribution of the blade angle but hard to make the blade wrap angle equal to a constant in this method. Most

designers pay more attention on the value of the blade wrap angle rather than the distribution of the blade angle especially in the industry companies. To make the blade wrap angle satisfy the given value, a blade drawing method by controlling the blade wrap angle is given as shown in Fig. 2.

The main steps of this blade drawing method are given as follows: (1) setting the value of the blade wrap angle by designers; (2) generating a value of parameter a to form a blade angle distribution along the meridional streamline; (3) integrating the Eq. (1) to draw the blade shape and calculate the blade wrap angle; (4) thickening and smoothing the blade. Steps (2) and (3) are repeated until the blade wrap angle meets the given value.

2.2 Design parameters and results

2.2.1 Design parameters

Some design parameters for centrifugal pump impeller are summarized in Table 1. Three blade wrap angles are selected according to the specific speed to analyze the influence of the blade wrap angle on the centrifugal pump performance. The blade wrap angles are 100°, 110°, 120°for impellers A, B, C with all other parameters remaining the same.

2.2.2 Design results

Fig. 3 shows the blade angle distributions along the meridional streamlines for the impeller C with the blade wrap angle of 120°, where K=1 denotes the streamline at the impeller hub and K=14 denotes the streamline at the impeller shroud. Each distribution curve denotes a quadratic function corresponding to a different value of parameter a in Eq. (5). They are parabolic curves near the hub and linear lines near the shroud. Compared to the blade angle distribution of the traditional design method[14] on the

assumption of one-dimensional or two-dimensional flow theory, the results of the direct and inverse iteration design method are three-dimensional. The blade angle distributions obviously change from hub to shroud

corresponding to the twisty blade shape, because the fluid flow in the direct and inverse iteration design method is three-dimensional flow.

Three impellers with blade wrap angles of 100°, 110°and 120° are all designed by the direct and inverse iteration design method. Fig. 4 shows their 3D models. As can be seen in the figure, the curved surfaces are very smooth with a circular arc shape forming the blade head in order to improve the flow pattern at the impeller inlet. The blade length and the flow passage geometry vary as the blade wrap angles increase. Generally, the flow passage is larger with the larger blade wrap angles.

3 Numerical Simulation and Analysis

3.1 Computation domain and grid

The computation domain and grid of the centrifugal pump are shown in Fig. 5. Local refinement of the grid is applied near the blade surface and the volute tongue to guarantee the precision of the numerical simulation. The

comparisons of the computational results using different grid distributions and different strategies for grid refinement indicate that the meshes in the present research are all fine enough to obtain the grid-independent

computational results. When the number of the grid of the whole computation domain reaches 1 350 000, the value of y+ is between 30 and 300.

3.2 Governing equations and computation method The continuity and motion equation of the fluid for three-dimensional viscous flow in pump are as follows:

In this study, the commercial CFD code Fluent is used to solve the above governing equations. The boundary conditions are set as follows: a constant flow velocity is set at the inlet; the pressure outlet option is set at the outlet, the standard wall functions are imposed over the impeller blades

and sidewalls, the volute casing and the inlet and outlet pipe walls. The multiple rotating reference frame (MRF) is applied to couple the rotation and station domains.

3.3 Relative velocity distributions in impellers

The relative velocity distributions in three impellers are shown in Fig. 6. As can be seen, the relative velocity at the impeller inlet is small and then increases gradually in the impeller with the maximum value at the impeller outlet.The relative velocity gradient is comparatively large near

the blade surface. As the wrap blade angle becomes larger,the control ability of the blades on the flows in the impeller becomes stronger. Therefore, the relative velocity gradients

around the pressure surfaces become smaller, leading to a more uniform distribution of the relative velocity at the impeller outlet.

The centrifugal pump is of wide application in industrial and agricultural productions, consuming a large amount of electric power with great potential in energy saving. Among all components in the centrifugal pump, the impeller is the most important flow passage because it is the only part that does work[1]. Therefore, new or modified design methods for the impeller have been proposed continually to improve its comprehensive performance especially the efficiency. WANG[2] and LU, et al[3] provided a design method for centrifugal pump impeller based on S2 stream surface

considering the slip factor on the blade. ZANGENEH[4] and GOTO, et al[5] designed the centrifugal pump impeller by governing the circulation distributions along streamlines.TAN, et al[6], proposed a direct and inverse iteration design method for centrifugal pump impeller. On the other hand,some researchers paid attention to the influences of design

parameters on pump characteristics. CAO, et al[7] and BING,et al[8] considered the effect of velocity torque distribution on the pump efficiency. BONAIUTI, et al[9–10], analyzed the influence of several parameters, especially the blade loading, on the impeller performance.

The blade wrap angle is defined as the one between the tangent lines at leading and trailing edges of the blade. An increase in blade wrap angle would lead to a longer flow passage between the blades and thus a significant rise in friction loss. On the contrary, a small blade angle will generate a short flow passage but result in a poor control on

the flow in impeller arousing separation loss probably. Therefore, the blade wrap angle is a key parameter for blade shape, flow pattern in impeller and performance of pump. For the same impeller diameter, YANG, et al[11], used a cubic curve to form blade shape and compared the efficiencies of those impellers with different blade wrap angles. CAO, et al[12], numerically simulated five impellers with different blade wrap angles and showed that the angle had great effect on the internal flow and hydraulic performance of pumps. ZHANG, et al[13], numerically analyzed the relationship between the blade wrap angle and the centrifugal pump performance.

In this paper, three centrifugal pump impellers were designed with different blade wrap angles using the direct and inverse iteration design method. The influence of the blade wrap angle on the centrifugal pump performance was investigated both numerically and experimentally.

2 Centrifugal Pump Impeller Design

2.1 Design method

The direct and inverse iteration design method for centrifugal pump impeller is based on the continuity and motion equations of the fluid with the effects of the blade shape on the flow being considered. This method is applied by a set of computer code written with FORTRAN language. The initial parameters at the beginning of the design program are as follows: (1) flow rate, head and rotation speed of centrifugal pump and the fluid properties;

(2) shape of hub and shroud, leading and trailing edges of the meridional flow passage; (3) blade thickness distribution, blade number and blade wrap angle.

2.1.1 Process of design

Fig. 1 shows the process of the direct and inverse iteration design method.

Fig. 1. Process of direct and inverse iteration design method First, according to the given design parameters of the centrifugal pump, the traditional design method is adopted to design the initial impeller configuration. Second, based on the continuity and motion equations of the fluid, the meridional velocity distribution is obtained by the iteration

between the S1 and S2 stream surfaces. Third, with the obtained meridional flow field, the blade shape is generated using point-by-point integrations. Through this step, a new impeller is obtained. Fourth, the iteration of the S1 and S2 stream surfaces is operated again for the new designed impeller to get an update meridional velocity distribution.

The third and fourth steps should be repeated until the differences of the meridional transversals of two successive designed impellers are within a prescribed tolerance. For the present impeller design, the tolerance is set to be 0.000 1.

2.1.2 Blade drawing

The relation between blade wrap angles and meridional streamlines can be described by the bone line integration equation as follows.

The blade angle distributions along meridional streamlines can be described as a quadratic function, which can be expressed as

where the parameter a is the only unknown variable, which can be given by the designer to determine the blade angle distribution.

The blade wrap angle is a key parameter for impeller designing as it is directly related to the ability of doing work, the flow pattern, the friction loss, the separation loss and so on. So the value of the blade wrap angle should be easily controlled in impeller design process[13]. In previous blade drawing method[14], the blade angle distribution was given by the parameter a, and then the blade shape was structured according to Eq. (1). It is easy to determine the distribution of the blade angle but hard to make the blade wrap angle equal to a constant in this method. Most

designers pay more attention on the value of the blade wrap angle rather than the distribution of the blade angle especially in the industry companies. To make the blade wrap angle satisfy the given value, a blade drawing method by controlling the blade wrap angle is given as shown in Fig. 2.

The main steps of this blade drawing method are given as follows: (1) setting the value of the blade wrap angle by designers; (2) generating a value of parameter a to form a blade angle distribution along the meridional streamline; (3) integrating the Eq. (1) to draw the blade shape and calculate the blade wrap angle; (4) thickening and smoothing the blade. Steps (2) and (3) are repeated until the blade wrap angle meets the given value.

2.2 Design parameters and results

2.2.1 Design parameters

Some design parameters for centrifugal pump impeller are summarized in Table 1. Three blade wrap angles are selected according to the specific speed to analyze the influence of the blade wrap angle on the centrifugal pump performance. The blade wrap angles are 100°, 110°, 120°for impellers A, B, C with all other parameters remaining the same.

2.2.2 Design results

Fig. 3 shows the blade angle distributions along the meridional streamlines for the impeller C with the blade wrap angle of 120°, where K=1 denotes the streamline at the impeller hub and K=14 denotes the streamline at the impeller shroud. Each distribution curve denotes a quadratic function corresponding to a different value of parameter a in Eq. (5). They are parabolic curves near the hub and linear lines near the shroud. Compared to the blade angle distribution of the traditional design method[14] on the

assumption of one-dimensional or two-dimensional flow theory, the results of the direct and inverse iteration design method are three-dimensional. The blade angle distributions obviously change from hub to shroud

corresponding to the twisty blade shape, because the fluid flow in the direct and inverse iteration design method is three-dimensional flow.

Three impellers with blade wrap angles of 100°, 110°and 120° are all designed by the direct and inverse iteration design method. Fig. 4 shows their 3D models. As can be seen in the figure, the curved surfaces are very smooth with a circular arc shape forming the blade head in order to improve the flow pattern at the impeller inlet. The blade length and the flow passage geometry vary as the blade wrap angles increase. Generally, the flow passage is larger with the larger blade wrap angles.

3 Numerical Simulation and Analysis

3.1 Computation domain and grid

The computation domain and grid of the centrifugal pump are shown in Fig. 5. Local refinement of the grid is applied near the blade surface and the volute tongue to guarantee the precision of the numerical simulation. The

comparisons of the computational results using different grid distributions and different strategies for grid refinement indicate that the meshes in the present research are all fine enough to obtain the grid-independent

computational results. When the number of the grid of the whole computation domain reaches 1 350 000, the value of y+ is between 30 and 300.

3.2 Governing equations and computation method The continuity and motion equation of the fluid for three-dimensional viscous flow in pump are as follows:

In this study, the commercial CFD code Fluent is used to solve the above governing equations. The boundary conditions are set as follows: a constant flow velocity is set at the inlet; the pressure outlet option is set at the outlet, the standard wall functions are imposed over the impeller blades

and sidewalls, the volute casing and the inlet and outlet pipe walls. The multiple rotating reference frame (MRF) is applied to couple the rotation and station domains.

3.3 Relative velocity distributions in impellers

The relative velocity distributions in three impellers are shown in Fig. 6. As can be seen, the relative velocity at the impeller inlet is small and then increases gradually in the impeller with the maximum value at the impeller outlet.The relative velocity gradient is comparatively large near

the blade surface. As the wrap blade angle becomes larger,the control ability of the blades on the flows in the impeller becomes stronger. Therefore, the relative velocity gradients

around the pressure surfaces become smaller, leading to a more uniform distribution of the relative velocity at the impeller outlet.

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