*2.2. Meshing*

Commercial ANSYS-ICEM17.0 software was utilized to mesh various computational domains. As the structured meshes only contained quadrilaterals or hexahedrons in this case, their topological structure was equivalent to a uniformly orthogonal mesh within a rectangular domain. Accordingly, the nodes on each layer of the mesh lines can be effectively adjusted to ensure a high quality [20–24]. The overall computational domain was structured and meshed and the boundary layer of the meshes in the vicinity of the near wall of the blade was refined. The quality of meshes within all computational domains was above 0.35 (Figure 2).

(**b**) Computational domain and structure grid. 

**Figure 2.** Mesh of the pump.

The efficiency, head, and power of the rated point of the pump were considered indexes for mesh independence verification of the unslotted centrifugal pump model. The global maximum mesh size was used to control the mesh density of each computational domain. Local meshes within each computational domain were specifically refined to ensure the desired mesh quality. The meshing results are shown in Table 2.


**Table 2.** Analysis of the grid independent.

Figure 3 shows the effects of the number of meshes in different cases on the head, power, and efficiency of the simulated pump. A numerical calculation on a mesh with a global mesh size of 2 mm was conducted with both the computational cycle and computational accuracy taken into account.

**Figure 3.** Schematic diagram of the blade structure grid.

#### *2.3. Computational Cases and Boundary Conditions*

Numerical calculations were performed in ANSYS-CFX19.2 software (ANSYS CFX. 19.2" ANSYS CFX 19.2 Documentation. 2019). The turbulence model in this case was a standard *k-*ω turbulence model. The computational impeller domain was set to a rotational domain and all other computational domains were set to static. Data transfer at the interface between the static domains and the rotational domains was achieved by the frozen rotor method.

To consider the effects of the impeller cover plate on the flow, all other inner wall surfaces within the pump cavity excluding those in contact with the impeller outlet surface were set to a rotational wall surface. The roughness of each computational domain surface was set to 10 μm to observe the effects of the material on the internal flow characteristics of the pump. The boundary conditions were set to pressure inlet and mass outflow. The reference pressure was set to a standard atmospheric pressure, the wall surface was placed under a non-slip boundary condition, and a standard wall surface function was used with the convergence accuracy set to 10−4.

#### *2.4. Orthogonal Design of Blade Slots*

To explore the effects of blade-slotting on the medium specific speed pump systematically, four factors including the slotting position, slotting width, slotting depth, and slotting angles of the blades

were studied via the orthogonal design method. The orthogonal design method is a scientific design technique wherein test plans are reasonably arranged to determine the main factors that influence certain indexes within a brief testing time [25]. Many researchers have used orthogonal designs to study the performance of centrifugal pumps [26–29]. Considering the time and cost burdens of the test, the geometrical factors of the slots as they affect pump performance were observed in this study by combining CFD technology with an orthogonal design.

#### *2.5. Determining the Test Factors*

As discussed above, four sets of geometric parameters were taken as test factors: slot position *p* slot width *b*1, slot deflection angle β and slot depth *h* (Figure 4). Slot position *p* is the position of the slot on the blade. Based on the arc length of the blade profile, four uniform levels were taken from the inlet edge of the blade to the outlet edge of the blade: 20%*p*, 40%*p*, 60%*p*, and 80%*p*. Three levels of the slot width *b*1 were selected: 0.5 mm, 1 mm, and 1.5 mm. As shown in Figure 4, the angle β is the included angle between the slot and the tangent line of the blade in the slot position. The deflection angle of the slot relates to the effects of the slot jet on the liquid flow in the flow passage. Three angles were tested: 30◦, 45◦, and 60◦. The slot depth *h* is the axial distance from the inner surface of the front cover plate, relative to the blade outlet width *b*. Three depths were selected: 1/4 *b*, 2/4 *b*, and 3/4 *b*.

**Figure 4.** The schematic diagram of the gap geometry parameters.

As shown in Table 3, the L16 orthogonal table was selected for these four factors.


**Table 3.** Factor level table.

#### **3. Analysis of Numerical Simulation Results**

#### *3.1. Test Verification*

To validate the numerical simulation method used in this study, the original model was tested. As shown in Figure 5, the test rig is an open-type system, which is composed of two parts, namely, the data acquisition system and the water circulation system. The DN100 electromagnetic flowmeter whose maximum allowable error is ±0.5% was used to measure the flow rate *Q*. The valve of the pump inlet pipeline was fully opened during the test and the flow condition points were collected through the pump outlet pipeline valve. To secure a smooth external characteristic curve during the collection process, recording was performed at an interval of 5 m<sup>3</sup>/<sup>h</sup> from the shutoff point to the large flow condition point for a total of 17 operating points.

**Figure 5.** Schematic diagram of the test rig. 1. Motor; 2. torque meter; 3. centrifugal pump; 4. outlet pipeline pressure section; 5. DN100 electromagnetic flowmeter; 6. outlet pipeline valve; 7. water tank; 8. inlet pipeline valve and 9. inlet pipeline pressure section.

Table 4 shows the pump performance test results. As the rotational speed of the pump was not constant at 2850 r/min during actual operation, for an effective comparison against the numerical calculation results, the external characteristic data of the pump was converted to a rated speed of 2850 r/min according to the rules of similarity theory.


**Table 4.** Pump performance test results.

Figure 6 shows a comparison between the test-based and simulated pump performance indicators. To completely reflect the external characteristic variation curve from the shuto ff point to the maximum flow condition during the pump test, eight flow condition points were simulated from 0.1 *Q* to 1.4 *Q*. The numerical simulation results accurately predicted the external characteristic curve of the pump within the whole range of operating conditions as observed in the test. The relative errors in the head, efficiency, and power of the rated operating points were 4.4%, 2.95%, and 4%, respectively. All were smaller than 5%, which indicates that the numerical simulation method was accurate. Further, these results sugges<sup>t</sup> that the orthogonal design case accurately reflects slotting e ffects numerically.

**Figure 6.** Comparison of test and numerical results.

#### *3.2. Direct Analysis of Orthogonal Design Case*

Sixteen sets of orthogonal design cases were used in this study. Prototyping all of them would be costly and time-consuming, so considering the accuracy of the numerical calculation method, the full flow field numerical simulation method was selected as the research tool for this orthogonal design. To observe the e ffects of slotting on the performance of the medium specific speed centrifugal pump, full flow field numerical simulations were conducted at four operating condition points: 0.6*Q,* 0.8 *Q*, 1.0 *Q*, 1.2 *Q*, and 1.4 *Q* for 16 sets of slotted impellers in conjunction with the volutes.

This study centers on the e ffects of di fferent slotting cases on pump performance. Tables 5–7 show the numerical simulation results of 0.6 *Q*, 1.0 *Q*, and 1.4 *Q*, respectively. The orthogonal test data was processed to assess the main factors influencing the pump head and e fficiency in the slotted blade case [18]. The range analysis method was used to observe the e ffects of the levels of the factors at 0.6 *Q*, 1.0 *Q*, and 1.4 *Q* operating conditions on the pump's performance.


**Table 5.** 0.6*Q* numerical simulation results.

**Table 6.** 1.0*Q* numerical simulation results.



**Table 7.** 1.4*Q* numerical simulation results.

In the case of a greater range, different levels of a given factor lead to a larger amplitude of variations in the test indicators. To this effect, the factor corresponding to the maximum range was the most important factor. *K*i(*i* = 1,2,3,4) denotes the sum of the tests of the same level in any of the columns in Table 3, where *i* corresponds to different levels of the same factor, *ki* = *Ki*/*n* denotes the arithmetic mean value of different levels of the same factor, *n* denotes the number of occurrence of the same level in any of the columns in the table, and *R* = *max(k*1, *k*2, *k*3, *k*4) − *min(k*1, *k*2, *k*3, *k*4) denotes the range. A range analysis of 0.6Q are shown in Tables 8 and 9.


**Table 8.** 0.6*Q* head analysis.


**Table 9.** 0.6*Q* efficiency analysis.

The primary and secondary geometric parameters of slots influencing the pump performance at 0.6*Q* were obtained as shown in Table 10.

**Table 10.** The order of influence of the gap geometry parameters on pump performance at 0.6*Q*.


A range analysis of 1.0*Q* are shown in Tables 11 and 12.

**Table 11.** 1.0*Q* head analysis.



**Table 12.** 1.0 *Q* efficiency analysis.

The primary and secondary geometric parameters of slots influencing the pump performance at 1.0 *Q* were obtained as shown in Table 13.

**Table 13.** The order of influence of the gap geometry parameters on pump performance at 1.0 *Q*.


A range analysis of 1.4 *Q* are shown in Tables 14 and 15.


**Table 14.** 1.4 *Q* head analysis.

**Table 15.** 1.4 *Q* efficiency analysis.


The primary and secondary geometric parameters of slots influencing the pump performance at 1.4 *Q* were obtained as shown in Table 16.


**Table 16.** The order of influence of the gap geometry parameters on pump performance at 1.4 *Q*.

Range analyses of the operating condition points, 0.6 *Q*, 1.0 *Q*, and 1.4 *Q* indicated that the slot width and depth under the small flow conditions and rated conditions have the greatest e ffects on the pump head and e fficiency among the parameters tested. The slot position appeared to have little e ffect on the performance of the pump under small flow conditions. In the case of large flow conditions, however, the slot position had a greater e ffect on pump performance than any other parameter.

To analyze the e ffects of the changes in factor levels on the pump performance more intuitively, a trend variation chart was plotted with the head and e fficiency of the pump as indicators. As shown in Figure 7, the head *h*0.6Q was the largest when the blade slotting position *p* was close to the outlet side and the slot deflection angle β was the smallest under the small flow condition 0.6 *Q*. The head *h*0.6Q decreased progressively as slot width *b*1 and the slot depth *h* increased, and an inflection point emerged on the curve of the head *h*0.6Q as slot position *p* varied. Based on the steepness of the curve variation trend, the primary and secondary factors influencing the head *h*0.6Q were slot width *b*1, slot deflection angle β, slot depth *h*, and slot position *p*, respectively. This result was consistent with the range analysis results. The e fficiency η0.6Q also increased as *b1* and *h* decreased. The e fficiency η0.6Q curve trend also presented an inflection point with the changes in the slot position. Based on the trend graph, in order of intensity, the factors influencing the e fficiency η0.6Q were slot depth *h*, slot width *b*1, slot deflection angle β, and slot position *p*.

**Figure 7.** 0.6 *Q* head (**left**) and e fficiency (**right**)-factor relationship.

As shown in Figures 8 and 9, under the conditions of 1.0 *Q* and 1.4 *Q*, the heads of *h*1.0Q and *h*1.4Q were the largest when the blade slot position *p* was in the vicinity of the inlet edge of the blade. Like 0.6 *Q*, the heads of *h*1.0Q and *h*1.4Q and the e fficiencies of η1.0Q and η1.4Q decreased progressively as slot width *b*1 and the slot depth *h* increased. *h*1.0Q, *h*1.4Q, η1.0Q, and η1.4Q also increased progressively as slot deflection angle β increased.

**Figure 8.** 1.0*Q* head (**left**) and efficiency (**right**)-factor relationship.

**Figure 9.** 1.4*Q* head (**left**) and efficiency (**right**)-factor relationship.

#### *3.3. Analysis of Internal Flow Field*

The orthogonal test results sugges<sup>t</sup> that blade slotting improved the head at small flow condition points and the efficiency at large flow condition points, which is consistent with previously published results. Under the working condition of 0.6*Q*, the head of Case 1 was 39.3 m; in the original case the head was 38.5 m. The head and efficiency in Case 1 for the 1.4*Q* condition were 30.15 m and 77.86%, respectively, and in the original case were 30.05 m and 77.09%. To further explore the effects of the geometric slot parameters on pump performance, the distributions of performance curves of the original model and Case 1 were compared as shown in Figure 10.

**Figure 10.** Comparison of case 1 and original model.

Figures 11 and 12 show cloud diagrams of the static pressure distribution of the blade unfolding at the section of the pump impeller flow passage (the section value Span was 0.9) in the original model and Case 1 of slotted blades under the conditions of 0.6*Q* and 1.4*Q*, respectively. As shown in Figure 11, under the 0.6*Q* condition, the static pressure distributions of the blade unfolding in Case 1 and the original model differed significantly. The distribution of pressure in the impeller flow passage from the blade inlet to the outlet was characterized by a low-pressure region in the first half-section of the impeller flow passage and a high-pressure region in the second half-section of the passage. The pressure gradient in the second half-section of the impeller flow passage was large because the flow passage diffusion was severe, which might have created a secondary back flow at the outlet of the impeller under small flow conditions. This is also likely a cause of the low efficiency of the medium-low specific speed centrifugal pump at the small flow condition point.

**Figure 12.** Static pressure distributions on the cross section of the impeller channel at 1.4*Q*.

A significant low-pressure region was also observed in the position close to the inlet edge of the original model. Due to slotting in the position close to the inlet edge of the impeller, the distribution of blade unfolding static pressure disappears in the low-pressure region close to the inlet position of the blade in Case 1 and the pressure distribution is significantly more uniform than that in the original model. Vortexes and back flows are unlikely to form in the inlet position of the blade in this case, which is also one of the reasons why the head of the model in Case 1 is larger than that of the original model under the 0.6 *Q* condition.

Under the 1.4 *Q* large-flow condition, the diagram for the blade unfolding static pressure distribution in the case of the original model was similar to that in Case 1, however, the original model had a significant low-pressure region with considerable variations in the pressure gradient in the first half-section of the impeller flow passage inlet. This is mainly because the fluid flow angle of the incoming liquid increases with the flow rate while the inlet setting angle of the blade remains unchanged. As a result, the inlet setting angle is smaller than the liquid flow angle; a flow cuto ff forms at the working surface in the position of the blade inlet creating a low-pressure region. Similarly, the changes in pressure gradient in the static pressure distribution diagram of blade unfolding in Case 1 are smaller than those of the original model due to the fact that the blade is slotted near the inlet.

As shown in Figure 13, the pressure distribution is shown on the blade surface at the mean circumferential flow surface. Under 0.6 *Q*, the pressure distribution of the original model and the Case 1 model were quite di fferent (Figure 13). At the position near the inlet side, the pressure of the pressure surface and the suction surface of the Case 1 model were larger than the original model. This further illustrates that the Case 1 model improved the pressure distribution at the inlet edge of the blade due to slotting. The pressure near the inlet edge of the blade was higher than that of the original model under the 1.4Q large-flow condition.

**Figure 13.** Variation of the blade load with streamwise at 0.6 *Q* (**left**) and at 1.4 *Q* (**right**).

Figures 14 and 15 show the distribution of pressure clouds in the middle plane of the blade flow channel. The pressure gradient distribution of the Case 1 model is more uniform than the original model under the 0.6 *Q* condition (Figure 14); the original model shows a lower pressure than the Case 1 model near the blade inlet as well, which is consistent with the findings shown in Figures 11 and 12. The enlarged view in the figure shows where, due to the existence of a gap, the local low-pressure gradient distribution was more uniform in the original model. This gap jet made the streamline in the inlet low-pressure area closer to the profile of the blade airfoil, thereby improving the local flow field. Figure 15 shows that under the large flow rate of 1.4 *Q*, the impact of the gap on the local area was relatively small. The local enlarged view did not show similar phenomena to the small flow conditions. Generally speaking, the gap improved the local flow field under small flow conditions.

**Figure 14.** Static pressure distribution in the middle plane of the blade flow channel at 0.6*Q*.

**Figure 15.** Static pressure distribution in the middle plane of the blade flow channel at 1.4*Q*.

Figure 16 shows a diagram of the relative velocity distribution under the 1.4*Q* operating condition in the impeller calculation domain. This distribution was normal; the average velocity of the impeller calculation domain was basically 11 m/s. The velocity distribution amplitude of velocity in Case 1 was larger than that of the original model, which indicates that the velocity within the impeller was concentrated near the desired value and was uniform throughout the impeller calculation domain. This was also one of the reasons why the efficiency and head of Case 1 were larger than those of the original model.

**Figure 16.** Velocity distribution in the impeller calculation domain at 1.4*Q*.
