gence angle is changed. The specific value is shown in Table 2. The existence of the divergence angle makes the nozzle become a converging–diverging nozzle, and the outlet velocity increases significantly, as shown in Figure 6a. When the divergence angle γ = 0°, the outlet velocity of the water jet is 201.45 m/s. With an increase in the divergence angle, the outlet velocity of the water jet increases gradually. When the divergence angle γ = 30°, the outlet velocity of the water jet reaches 229.45 m/s, which is increased by 13.90% compared with that without the divergence angle. To better observe the change in the X-axis velocity of the water jet, the initial target distance starts The existence of the divergence angle makes the nozzle become a converging–diverging nozzle, and the outlet velocity increases significantly, as shown in Figure 6a. When the divergence angle γ = 0◦ , the outlet velocity of the water jet is 201.45 m/s. With an increase in the divergence angle, the outlet velocity of the water jet increases gradually. When the divergence angle γ = 30◦ , the outlet velocity of the water jet reaches 229.45 m/s, which is increased by 13.90% compared with that without the divergence angle. To better observe the change in the X-axis velocity of the water jet, the initial target distance starts from 20 mm, as shown in Figure 6b. With an increase in the divergence angle, the axis water jet velocity increases initially and decreases thereafter. In the whole flow field, the X-axis velocity of the water jet of the nozzle with divergence angle γ = 30◦ decays faster than that of other nozzles. Among the other three nozzles, the X-axis velocity of the nozzle with divergence angle γ = 10◦ is slightly higher. When the target distance is X = 100 mm, the sequence of velocity is as follows: γ = 10◦ > γ = 20◦ > γ = 0◦ > γ = 30◦ .

When there is an outlet divergence angle of the nozzle, the outlet diameter of the water jet increases, and the effective Y-axis extension distance of the water jet increases significantly. When the divergence angle γ = 30◦ , the effective Y-axis extension distance of the water jet reaches 3.24 mm, which is 33.88% more than that of the divergence angle γ = 0◦ . Moreover, the number of data points reaching the effective water jet velocity increases significantly, as shown in Figure 7.

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the sequence of velocity is as follows: γ = 10° > γ = 20° > γ = 0° > γ = 30°.

the sequence of velocity is as follows: γ = 10° > γ = 20° > γ = 0° > γ = 30°.

**Figure 6.** Effect of divergence angle on the X-axis velocity attenuation of the water jet. **Figure 6.** Effect of divergence angle on the X-axis velocity attenuation of the water jet. significantly, as shown in Figure 7.

from 20 mm, as shown in Figure 6b. With an increase in the divergence angle, the axis water jet velocity increases initially and decreases thereafter. In the whole flow field, the X-axis velocity of the water jet of the nozzle with divergence angle γ = 30° decays faster than that of other nozzles. Among the other three nozzles, the X-axis velocity of the nozzle with divergence angle γ = 10° is slightly higher. When the target distance is X = 100 mm,

from 20 mm, as shown in Figure 6b. With an increase in the divergence angle, the axis water jet velocity increases initially and decreases thereafter. In the whole flow field, the X-axis velocity of the water jet of the nozzle with divergence angle γ = 30° decays faster than that of other nozzles. Among the other three nozzles, the X-axis velocity of the nozzle with divergence angle γ = 10° is slightly higher. When the target distance is X = 100 mm,

**Figure 7.** Effect of divergence angle on the effective Y-axis extension distance of the water jet at X = 100 mm. **Figure 7.** Effect of divergence angle on the effective Y-axis extension distance of the water jet at X = 100 mm.

#### 3.2.3. Effect of the Length-to-Diameter Ratio on Water Jet Performance 3.2.3. Effect of the Length-to-Diameter Ratio on Water Jet Performance

Under the condition that the other parameters remain the same, the value of the length-to-diameter ratio is changed, and the specific value is shown in Table 2. Under the condition that the other parameters remain the same, the value of the length-to-diameter ratio is changed, and the specific value is shown in Table 2.

**Figure 7.** Effect of divergence angle on the effective Y-axis extension distance of the water jet at X = 100 mm. 3.2.3. Effect of the Length-to-Diameter Ratio on Water Jet Performance Under the condition that the other parameters remain the same, the value of the length-to-diameter ratio is changed, and the specific value is shown in Table 2. When the nozzle has no outlet cylindrical section, Figure 8 shows that the outlet and X-axis velocities of the water jet are significantly lower than those of the nozzle with an When the nozzle has no outlet cylindrical section, Figure 8 shows that the outlet and X-axis velocities of the water jet are significantly lower than those of the nozzle with an When the nozzle has no outlet cylindrical section, Figure 8 shows that the outlet and X-axis velocities of the water jet are significantly lower than those of the nozzle with an outlet cylinder. When the target distance exceeds 100 mm, the X-axis velocity of the water jet of the nozzle without an outlet cylinder decreases rapidly. When there is an outlet cylinder of the nozzle, outlet velocity of water jet increases initially and decreases thereafter with an increase in the length-to-diameter ratio. When *l*/*d* = 2.5, the outlet velocity of the water jet is the largest, X-axis velocity attenuation is the slowest, and water jet performance is the best. When *l*/*d* = 2 and *l*/*d* = 3, the X-axis velocity and attenuation of the water jet are the same. When the target distance is X = 100 mm, the X-axis velocity of the nozzle with outlet cylinder is considerably large, and maximum velocity is achieved when *l*/*d* = 2.5. This result indicates that when some parameters are constant, there is an optimal length-to-diameter ratio to optimize the water jet performance.

**Figure 8.** Effect of length-to-diameter ratio on X-axis velocity attenuation of water jet. **Figure 8.** Effect of length-to-diameter ratio on X-axis velocity attenuation of water jet.

an optimal length-to-diameter ratio to optimize the water jet performance.

an optimal length-to-diameter ratio to optimize the water jet performance.

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The non-zero value of the length-to-diameter ratio will increase the effective Y-axis extension distance of the water jet. When the length-to-diameter ratio is *l d*/ 2.5 = , the effective Y-axis extension distance is the maximum, which is increased by 8.3% compared with the minimum value, as shown in Figure 9. Note that the non-zero value of length-todiameter increases the effective area of the water jet and improves the effects of coal breaking and punching. The non-zero value of the length-to-diameter ratio will increase the effective Y-axis extension distance of the water jet. When the length-to-diameter ratio is *l*/*d* = 2.5, the effective Y-axis extension distance is the maximum, which is increased by 8.3% compared with the minimum value, as shown in Figure 9. Note that the non-zero value of lengthto-diameter increases the effective area of the water jet and improves the effects of coal breaking and punching. extension distance of the water jet. When the length-to-diameter ratio is *l d*/ 2.5 = , the effective Y-axis extension distance is the maximum, which is increased by 8.3% compared with the minimum value, as shown in Figure 9. Note that the non-zero value of length-todiameter increases the effective area of the water jet and improves the effects of coal breaking and punching.

The non-zero value of the length-to-diameter ratio will increase the effective Y-axis

outlet cylinder. When the target distance exceeds 100 mm, the X-axis velocity of the water jet of the nozzle without an outlet cylinder decreases rapidly. When there is an outlet cylinder of the nozzle, outlet velocity of water jet increases initially and decreases thereafter with an increase in the length-to-diameter ratio. When *l d*/ 2.5 = , the outlet velocity of the water jet is the largest, X-axis velocity attenuation is the slowest, and water jet performance is the best. When *l d*/ 2 = and *l d*/ 3 = , the X-axis velocity and attenuation of the water jet are the same. When the target distance is X = 100 mm, the X-axis velocity of the nozzle with outlet cylinder is considerably large, and maximum velocity is achieved when *l d*/ 2.5 = . This result indicates that when some parameters are constant, there is

outlet cylinder. When the target distance exceeds 100 mm, the X-axis velocity of the water jet of the nozzle without an outlet cylinder decreases rapidly. When there is an outlet cylinder of the nozzle, outlet velocity of water jet increases initially and decreases thereafter with an increase in the length-to-diameter ratio. When *l d*/ 2.5 = , the outlet velocity of the water jet is the largest, X-axis velocity attenuation is the slowest, and water jet performance is the best. When *l d*/ 2 = and *l d*/ 3 = , the X-axis velocity and attenuation of the water jet are the same. When the target distance is X = 100 mm, the X-axis velocity of the nozzle with outlet cylinder is considerably large, and maximum velocity is achieved when *l d*/ 2.5 = . This result indicates that when some parameters are constant, there is

**Figure 9.** Effect of length-to-diameter on the effective Y-axis extension distance of water jet at X = **Figure 9.** Effect of length-to-diameter on the effective Y-axis extension distance of water jet at X = **Figure 9.** Effect of length-to-diameter on the effective Y-axis extension distance of water jet at X = 100 mm.

#### 100 mm. 100 mm. *3.3. Analysis and Discussion*

In the process of jetting, the water jet will have a strong momentum exchange with the surrounding air, and increasing air will move forward with the water jet under the suction of the longitudinal vortex, making its axial speed continuously attenuated. With the involvement of air, jet flow will gradually increase and, in a certain range, will realize the increase in the effective Y-axis extension distance, thereby increasing the effective strike area. The change in the nozzle's key structural parameters will affect the air entrainment rate of the water jet, and affect the jet effect thereafter.

When the contraction angle increases, flow resistance decreases, nozzle exit velocity increases, and high-speed water jet will have a strong entrainment effect on the nearby air. When *t* = 0.02 ms, the water jet has just squirted out, and with an increase in contraction angle, the maximum vortex of the jet moves backward and the front of the jet fans out and spreads around, as shown in Figure 10a–c. When the contraction angle is 90◦ , the nozzle exit velocity is considerably large, air entrainment rate is the largest, and the air

volume fraction of the jet is the highest, thereby attaining the axial velocity attenuation fastest. Figure 10d shows that the air around the jet indicates a scattered strong volume of suction state, the consistency of the fluid moving forward is poor, and the front end of the jet is strongly disturbed. When *t* = 0.8 ms, the jet has been injected into the middle of the flow field, as shown in Figure 11. At this point, the velocity vector distribution of the flow field outside the nozzle of θ = 30◦ and θ = 50◦ is similar. When θ = 70◦ and θ = 90◦ , the air entrainment rate at the front end of the jet is substantially large. When *t* = 2 ms, the jet has been injected to the end of the flow field, as shown in Figure 12. That is, the larger the injection angle, the more evident the effect on air coiling. The air coiling rate of the entire flow field can be expressed by the volume fraction of air in the jet, as shown in Figure 13. fastest. Figure 10d shows that the air around the jet indicates a scattered strong volume of suction state, the consistency of the fluid moving forward is poor, and the front end of the jet is strongly disturbed. When *t* =0.8ms, the jet has been injected into the middle of the flow field, as shown in Figure 11. At this point, the velocity vector distribution of the flow field outside the nozzle of θ = 30° and θ = 50° is similar. When θ = 70° and θ = 90°, the air entrainment rate at the front end of the jet is substantially large. When *t* = 2ms, the jet has been injected to the end of the flow field, as shown in Figure 12. That is, the larger the injection angle, the more evident the effect on air coiling. The air coiling rate of the entire flow field can be expressed by the volume fraction of air in the jet, as shown in Figure 13.

In the process of jetting, the water jet will have a strong momentum exchange with the surrounding air, and increasing air will move forward with the water jet under the suction of the longitudinal vortex, making its axial speed continuously attenuated. With the involvement of air, jet flow will gradually increase and, in a certain range, will realize the increase in the effective Y-axis extension distance, thereby increasing the effective strike area. The change in the nozzle's key structural parameters will affect the air entrain-

When the contraction angle increases, flow resistance decreases, nozzle exit velocity increases, and high-speed water jet will have a strong entrainment effect on the nearby air. When *t* = 0.02ms, the water jet has just squirted out, and with an increase in contraction angle, the maximum vortex of the jet moves backward and the front of the jet fans out and spreads around, as shown in Figure 10a–c. When the contraction angle is 90°, the nozzle exit velocity is considerably large, air entrainment rate is the largest, and the air volume fraction of the jet is the highest, thereby attaining the axial velocity attenuation

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ment rate of the water jet, and affect the jet effect thereafter.

*3.3. Analysis and Discussion*

**Figure 10.** Velocity vector of the flow field when *t* = 0.02ms. **Figure 10.** Velocity vector of the flow field when *t* = 0.02 ms.

(**a**) θ = 30° (**b**) θ = 50°

(**c**) θ = 70° (**d**) θ = 90°

(**a**) θ = 30° (**b**) θ = 50°

**Figure 11.** Velocity vector of the flow field when *t* = 0.8ms.

(**c**) θ = 70° (**d**) θ = 90° **Figure 10.** Velocity vector of the flow field when *t* = 0.02ms.

(**c**) θ = 70° (**d**) θ = 90°

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**Figure 11.** Velocity vector of the flow field when *t* = 0.8ms. **Figure 11.** Velocity vector of the flow field when *t* = 0.8 ms. **Figure 11.** Velocity vector of the flow field when *t* = 0.8ms.

**Figure 13.** Effect of contraction angle on air volume fraction. **Figure 13.** Effect of contraction angle on air volume fraction. **Figure 13.** Effect of contraction angle on air volume fraction.

A comparison of Figures 4 and 13 shows that the axial velocity decay of the water jet

rapid the velocity decay. When the volume fraction of air is large, the Y-axis extension distance of the water jet will increase accordingly. For coal breaking and punching, only when the velocity reaches a certain level can an effective striking force be formed. This study calculates the effective Y-axis extension distance within a 15% decay of the Y-axis velocity. Figure 5 shows that an effective Y-axis extension distance does not increase constantly, although the larger the contraction angle, the higher the air roll absorption rate. When there is an outlet divergence angle of the nozzle, a converging–diverging-type nozzle will form, resulting in a certain cavitation effect and significantly increasing the exit velocity. Moreover, an increase in the divergence angle will also reduce the wall attachment effect of the water jet, thereby further enhancing the exit velocity and producing strong entrainment to the surrounding air, and changing the jet effect. When the divergence angle is changed, the volume fraction of air in the water jet is shown in Figure 14. A comparison of Figures 6, 7 and 14 indicates that the larger the divergence angle, the higher the air enrolling rate. Moreover, axial velocity attenuation increases. However, the presence of the divergence angle increases the nozzle exit velocity. Thus, the effective Yaxis extension distance remains considerably large when the target distance X = 100 mm.

rapid the velocity decay. When the volume fraction of air is large, the Y-axis extension distance of the water jet will increase accordingly. For coal breaking and punching, only when the velocity reaches a certain level can an effective striking force be formed. This study calculates the effective Y-axis extension distance within a 15% decay of the Y-axis velocity. Figure 5 shows that an effective Y-axis extension distance does not increase constantly, although the larger the contraction angle, the higher the air roll absorption rate. When there is an outlet divergence angle of the nozzle, a converging–diverging-type nozzle will form, resulting in a certain cavitation effect and significantly increasing the exit velocity. Moreover, an increase in the divergence angle will also reduce the wall attachment effect of the water jet, thereby further enhancing the exit velocity and producing strong entrainment to the surrounding air, and changing the jet effect. When the divergence angle is changed, the volume fraction of air in the water jet is shown in Figure 14. A comparison of Figures 6, 7 and 14 indicates that the larger the divergence angle, the higher the air enrolling rate. Moreover, axial velocity attenuation increases. However, the presence of the divergence angle increases the nozzle exit velocity. Thus, the effective Yaxis extension distance remains considerably large when the target distance X = 100 mm.

A comparison of Figures 4 and 13 shows that the axial velocity decay of the water jet

A comparison of Figures 4 and 13 shows that the axial velocity decay of the water jet is negatively related to the volume fraction of air in the water jet. The larger the volume fraction, the more momentum exchange between the water jet and the air, and the more rapid the velocity decay. When the volume fraction of air is large, the Y-axis extension distance of the water jet will increase accordingly. For coal breaking and punching, only when the velocity reaches a certain level can an effective striking force be formed. This study calculates the effective Y-axis extension distance within a 15% decay of the Y-axis velocity. Figure 5 shows that an effective Y-axis extension distance does not increase constantly, although the larger the contraction angle, the higher the air roll absorption rate.

When there is an outlet divergence angle of the nozzle, a converging–diverging-type nozzle will form, resulting in a certain cavitation effect and significantly increasing the exit velocity. Moreover, an increase in the divergence angle will also reduce the wall attachment effect of the water jet, thereby further enhancing the exit velocity and producing strong entrainment to the surrounding air, and changing the jet effect. When the divergence angle is changed, the volume fraction of air in the water jet is shown in Figure 14. A comparison of Figures 6, 7 and 14 indicates that the larger the divergence angle, the higher the air enrolling rate. Moreover, axial velocity attenuation increases. However, the presence of the divergence angle increases the nozzle exit velocity. Thus, the effective Y-axis extension distance remains considerably large when the target distance X = 100 mm. *Machines* **2022**, *10*, x FOR PEER REVIEW 13 of 21

**Figure 14.** Effect of divergence angle on air volume fraction. **Figure 14.** Effect of divergence angle on air volume fraction.

The presence of an outlet cylindrical section of the nozzle will play a certain stabilizing effect on the water jet, thereby reducing the turbulent flow pattern and improving the water jet cluster effect. Consequently, the axial velocity attenuation is reduced. However, when the cylindrical length is markedly long, flow resistance will increase, thereby reducing the exit velocity. When the length-to-diameter ratio is changed, the volume fraction of air in the water jet is shown in Figure 15. Figures 8 and 9 show an optimal length-to-di-The presence of an outlet cylindrical section of the nozzle will play a certain stabilizing effect on the water jet, thereby reducing the turbulent flow pattern and improving the water jet cluster effect. Consequently, the axial velocity attenuation is reduced. However, when the cylindrical length is markedly long, flow resistance will increase, thereby reducing the exit velocity. When the length-to-diameter ratio is changed, the volume fraction of air in the water jet is shown in Figure 15. Figures 8 and 9 show an optimal length-to-diameter ratio, which makes the water jet to air volume absorption rate considerably moderate, thereby obtaining superior axial velocity and an effective Y-axis extension distance.

ameter ratio, which makes the water jet to air volume absorption rate considerably moderate, thereby obtaining superior axial velocity and an effective Y-axis extension distance.

To obtain the optimal combination of nozzle structural parameters when the X-axis velocity and effective Y-axis extension distance of the water jet are the largest, a numerical simulation and an orthogonal experiment are performed to further study the relationship among key parameters and their effects on water jet performance. According to the preceding analysis results, three good levels are selected for each factor. Factors A, B, and C are the contraction angle, divergence angle, and length-to-diameter ratio, respectively.

**Figure 15.** Effect of length-to-diameter on air volume fraction.

**4. Orthogonal Experiment**

The specific values are shown in Table 3.

**Figure 14.** Effect of divergence angle on air volume fraction.

The presence of an outlet cylindrical section of the nozzle will play a certain stabilizing effect on the water jet, thereby reducing the turbulent flow pattern and improving the water jet cluster effect. Consequently, the axial velocity attenuation is reduced. However, when the cylindrical length is markedly long, flow resistance will increase, thereby reducing the exit velocity. When the length-to-diameter ratio is changed, the volume fraction of air in the water jet is shown in Figure 15. Figures 8 and 9 show an optimal length-to-diameter ratio, which makes the water jet to air volume absorption rate considerably moderate, thereby obtaining superior axial velocity and an effective Y-axis extension distance.

**Figure 15.** Effect of length-to-diameter on air volume fraction. **Figure 15.** Effect of length-to-diameter on air volume fraction.

#### **4. Orthogonal Experiment 4. Orthogonal Experiment**

To obtain the optimal combination of nozzle structural parameters when the X-axis velocity and effective Y-axis extension distance of the water jet are the largest, a numerical simulation and an orthogonal experiment are performed to further study the relationship among key parameters and their effects on water jet performance. According to the preceding analysis results, three good levels are selected for each factor. Factors A, B, and C are the contraction angle, divergence angle, and length-to-diameter ratio, respectively. To obtain the optimal combination of nozzle structural parameters when the X-axis velocity and effective Y-axis extension distance of the water jet are the largest, a numerical simulation and an orthogonal experiment are performed to further study the relationship among key parameters and their effects on water jet performance. According to the preceding analysis results, three good levels are selected for each factor. Factors A, B, and C are the contraction angle, divergence angle, and length-to-diameter ratio, respectively. The specific values are shown in Table 3.

The specific values are shown in Table 3. **Table 3.** Level table of each factor.

#### *4.1. Experiment Design*

In the study of the effect of nozzle key structural parameters on water jet performance, the effect of the interaction between various factors on the experiment index is fully considered. Combined with research rules and practical principles, the first-order interaction between factors is mainly investigated in this study. According to the number of selected factors, the orthogonal experiment table of L27 (313) was selected to make a reasonable arrangement for all factors in the experiment. The experimental data are shown in Table 4. In particular, single factors A, B, and C are placed in columns 1, 2, and 5, respectively. Moreover, A × B is in columns 3 and 4, A × C is in columns 6 and 7, and B × C is in columns 8 and 11. The interaction of the three factors A × B × C is placed in columns 9, 10, 12, and 13, and is taken as the error. When the target distance is X = 100 mm, the maximum X-axis velocity and effective Y-axis extension distance of the water jet are selected as inspection indexes. Given that the model is axisymmetric, the effective Y-axis extension distance is replaced by a half-value.

**Table 4.** Orthogonal experiment table L<sup>27</sup> (313) and experimental data.

## *4.2. Results and Discussion*

#### 4.2.1. Range Analysis of the Orthogonal Experiment

In the orthogonal experiment, the degree of effect of the different factors on the results is often compared through the intuitive analysis method–range method, and the range value *R* can be calculated using Formula (3) [38,39]. The greater the *R* value, the greater the effect of this factor on the results.

$$R = \max(T\_1, T\_2, T\_3) - \min(T\_1, T\_2, T\_3) \tag{3}$$

where, *R* is the range value and *T*<sup>i</sup> (i = 1, 2, 3) represents the sum of the 9 experimental data corresponding to the ith level in each column.

According to the preceding formula, range analysis was conducted on the factors corresponding to the maximum X-axis velocity and effective Y-axis extension distance at the target distance of 100 mm and their first-order interaction. The results are shown in Table 5.

**Table 5.** Range analysis of the water jet performance.

Range analysis indicates that the primary and secondary sequence of each factor and interaction is as follows for the maximum X-axis velocity:

$$\mathbf{A} > \mathbf{B} > \mathbf{A} \times \mathbf{B} > \mathbf{C} > \mathbf{A} \times \mathbf{C} > \mathbf{B} \times \mathbf{C}$$

For the effective Y-axis extension distance, primary and secondary sequences of each factor and interaction are as follows:

$$\mathbf{B} > \mathbf{C} > \mathbf{A} > \mathbf{A} > \mathbf{B} > \mathbf{B} \times \mathbf{C} > \mathbf{A} \times \mathbf{C}$$

#### 4.2.2. Variance Analysis of the Orthogonal Experiment

In the orthogonal experiment analysis, although the range analysis is simple, the effect of the experimental error on the results cannot be excluded, and the accuracy of the analysis cannot be verified. Therefore, the *F* function should be used to conduct variance analysis for each factor and the interaction between factors, further excluding the experimental error and exploring the effect of each factor on the index [40,41]. The *F* function is as follows:

$$F\_d(n\_1, n\_2) = F \tag{4}$$

where *a* is the significant level, *n<sup>1</sup>* is the degree of freedom corresponding to each factor, and *n<sup>2</sup>* is the sum of the degree of freedom of error.

When the significant level and degree of freedom are determined, the specific value of the *F* function can be obtained using the *F* distribution table [42]. In this experiment, *F*0.01 and *F*0.05 are taken as critical values. If *F* > *F*0.05 of a factor, then there is 95% confidence that this factor has a significant impact on the index value, which is statistically significant. If *F* > *F*0.01, then there is 99% confidence that this factor has a significant impact on the index value, which is highly statistically significant. In the variance analysis table of this paper, \* \* stands for highly significant and \* stands for significant.

Factors A × C and B × C are close to the sum of the squares of the error column A × B × C. Thus, they are combined into the experimental error column, and a new sum of squares and degree of freedom are recalculated thereafter. The variance analysis of the maximum X-axis velocity is shown in Table 6.

**Table 6.** Variance analysis of the maximum X-axis velocity.

Table 7 shows that the sum of the square of the factor A × C and error column A × B × C is close. Adding it to the error and the final results are as follows:

**Table 7.** Variance analysis of effective Y-axis extension distance.

#### *4.3. Comprehensive Analysis of Experimental Data Based on Orthogonal Experiment*

According to the results of the orthogonal experiment, when the X-axis velocity is maximum, optimal levels of each factor are A2, B1, A<sup>2</sup> × B1, C1, A<sup>2</sup> × C1, and B<sup>1</sup> × C1, comprehensively considering that A<sup>2</sup> × B<sup>1</sup> × C<sup>1</sup> is the optimal combination. A nozzle of this structure is shown in the orthogonal experiment table, and the maximum X-axis velocity is 200.484 m/s at 100 mm. When the effective Y-axis extension distance is the largest, the optimal levels of each factor are B3, C2, A3, A<sup>3</sup> × B3, B<sup>3</sup> × C2, and A<sup>3</sup> × C2. Hence, A<sup>3</sup> × B<sup>3</sup> × C<sup>2</sup> is the optimal combination. A nozzle of this structure is shown in the orthogonal table, and its effective Y-axis extension distance is 3.204 mm.

Further analysis of the data in Table 4 shows that the maximum X-axis velocity and maximum effective Y-axis extension distance do not appear simultaneously, but the two evaluation indexes should be considered comprehensively in the process of highpressure water jet breaking coal and punching. The effective Y-axis extension distance of the A<sup>2</sup> × B<sup>1</sup> × C<sup>1</sup> combination is 2.484 mm, which is below that of other combinations. The maximum X-axis velocity of the A<sup>3</sup> × B<sup>3</sup> × C<sup>2</sup> combination is 193.189 m/s, which is the lowest among the different combinations. Therefore, a comprehensive analysis of the results of the single factor method in Part 2 and orthogonal experiment method in Part 3 indicates that the combination of A<sup>2</sup> × B<sup>2</sup> × C<sup>2</sup> is the optimal parameter combination of the water jet performance. At this point, the maximum X-axis velocity is 198.277 m/s, and effective Y-axis extension is 3.044 mm, which are relatively superior in their respective index values. The improved nozzles processed with the three parameter combinations will be verified by field experiments and compared with the original nozzle. *Machines* **2022**, *10*, x FOR PEER REVIEW 17 of 21 B1 × C1 combination is 2.484 mm, which is below that of other combinations. The maximum X-axis velocity of the A3 × B3 × C2 combination is 193.189 m/s, which is the lowest among the different combinations. Therefore, a comprehensive analysis of the results of the single factor method in Part 2 and orthogonal experiment method in Part 3 indicates that the combination of A2 × B2 × C2 is the optimal parameter combination of the water jet performance. At this point, the maximum X-axis velocity is 198.277 m/s, and effective Y-axis extension is 3.044 mm, which are relatively superior in their respective index values. The improved nozzles processed with the three parameter combinations will be verified by field experiments and compared with the original nozzle.

#### **5. Field Experiments 5. Field Experiments**

To further verify the water jet performance after nozzle optimization, a water jet experiment was conducted in the Qianjiaying mining area of the Kailuan Group using the self-developed equipment of a high-pressure water jet. The schematic of the equipment and water jet performance are shown in Figures 16 and 17, respectively. To further verify the water jet performance after nozzle optimization, a water jet experiment was conducted in the Qianjiaying mining area of the Kailuan Group using the self-developed equipment of a high-pressure water jet. The schematic of the equipment and water jet performance are shown in Figures 16 and 17, respectively.

To eliminate the interference of other factors on the effect of coal breaking and punching and ensure the accuracy of the results, the firmness coefficient of the four wall surfaces selected in the punching experiment was measured. Taking wall 1 as an example, after removing the floating coal with a thickness of approximately 0.3 m on the surface, coal samples were collected in three locations: the upper left area of the coal wall, the area of the coal wall directly opposite of the high-pressure water jet equipment, and the upper right area of the coal wall. The coal samples are labeled A1, B1, and C1. According to the standard of the determination method of the coal firmness coefficient [43], the coal with

The three groups were measured in parallel at each position, and the arithmetic mean value of the 3 positions was taken as the final firmness coefficient of wall 1. Coal samples

**Figure 16.** Schematic of the self-developed equipment of high-pressure water jet. **Figure 16.** Schematic of the self-developed equipment of high-pressure water jet.

**Figure 17.** Performance of the water jet.

label A1 is sampled and measured.

*5.1. Determination of Coal Firmness Coefficient*

field experiments and compared with the original nozzle.

and water jet performance are shown in Figures 16 and 17, respectively.

**Figure 16.** Schematic of the self-developed equipment of high-pressure water jet.

**5. Field Experiments**

**Figure 17.** Performance of the water jet. **Figure 17.** Performance of the water jet.

#### *5.1. Determination of Coal Firmness Coefficient 5.1. Determination of Coal Firmness Coefficient*

To eliminate the interference of other factors on the effect of coal breaking and punching and ensure the accuracy of the results, the firmness coefficient of the four wall surfaces selected in the punching experiment was measured. Taking wall 1 as an example, after removing the floating coal with a thickness of approximately 0.3 m on the surface, coal samples were collected in three locations: the upper left area of the coal wall, the area of the coal wall directly opposite of the high-pressure water jet equipment, and the upper right area of the coal wall. The coal samples are labeled A1, B1, and C1. According to the standard of the determination method of the coal firmness coefficient [43], the coal with label A1 is sampled and measured. To eliminate the interference of other factors on the effect of coal breaking and punching and ensure the accuracy of the results, the firmness coefficient of the four wall surfaces selected in the punching experiment was measured. Taking wall 1 as an example, after removing the floating coal with a thickness of approximately 0.3 m on the surface, coal samples were collected in three locations: the upper left area of the coal wall, the area of the coal wall directly opposite of the high-pressure water jet equipment, and the upper right area of the coal wall. The coal samples are labeled A1, B1, and C1. According to the standard of the determination method of the coal firmness coefficient [43], the coal with label A<sup>1</sup> is sampled and measured.

B1 × C1 combination is 2.484 mm, which is below that of other combinations. The maximum X-axis velocity of the A3 × B3 × C2 combination is 193.189 m/s, which is the lowest among the different combinations. Therefore, a comprehensive analysis of the results of the single factor method in Part 2 and orthogonal experiment method in Part 3 indicates that the combination of A2 × B2 × C2 is the optimal parameter combination of the water jet performance. At this point, the maximum X-axis velocity is 198.277 m/s, and effective Y-axis extension is 3.044 mm, which are relatively superior in their respective index values. The improved nozzles processed with the three parameter combinations will be verified by

To further verify the water jet performance after nozzle optimization, a water jet experiment was conducted in the Qianjiaying mining area of the Kailuan Group using the self-developed equipment of a high-pressure water jet. The schematic of the equipment

The three groups were measured in parallel at each position, and the arithmetic mean value of the 3 positions was taken as the final firmness coefficient of wall 1. Coal samples The three groups were measured in parallel at each position, and the arithmetic mean value of the 3 positions was taken as the final firmness coefficient of wall 1. Coal samples in other areas were also measured strictly according to the preceding method. The obtained data are shown in Table 8. The difference in the firmness coefficient within 0.86% indicates that the firmness coefficient of the selected three wall surfaces is consistent. According to the orthogonal analysis experiment results, the initial nozzle, the improved 1, improved 2, and improved 3 nozzles were arranged to perform coal breaking and punching experiments at walls 1, 2, 3, and 4, respectively, to further reduce experimental error.

**Table 8.** Firmness coefficient of the coal wall.

**Coal Wall**

1

2

3

4

**Sample Position**

#### *5.2. Experimental Analysis 5.2. Experimental Analysis* Before the coal breaking and punching experiment, the pressure of the pump was

Before the coal breaking and punching experiment, the pressure of the pump was adjusted, and the nozzle inlet pressure was stabilized at 20 MPa according to the pressure gauge. The erosion time of each nozzle was 300 s. According to the analysis result in Part 3, three groups of parameters were selected to improve the nozzle. The field coal breaking and punching experiment is shown in Figure 18. The parameters of the nozzle and measured data are shown in Table 9. The punching depth is the maximum hole depth measured perpendicular to the punching end face. Given that all the holes are quasi-circular, the punching diameter is the arithmetic average of the maximum and minimum diameters. adjusted, and the nozzle inlet pressure was stabilized at 20 MPa according to the pressure gauge. The erosion time of each nozzle was 300 s. According to the analysis result in Part 3, three groups of parameters were selected to improve the nozzle. The field coal breaking and punching experiment is shown in Figure 18. The parameters of the nozzle and measured data are shown in Table 9. The punching depth is the maximum hole depth measured perpendicular to the punching end face. Given that all the holes are quasi-circular, the punching diameter is the arithmetic average of the maximum and minimum diameters.

*Machines* **2022**, *10*, x FOR PEER REVIEW 18 of 21

**Table 8.** Firmness coefficient of the coal wall.

A1 0.51 0.53 0.52

C1 0.52 0.52 0.54

A2 0.53 0.52 0.52

C2 0.54 0.51 0.52

A3 0.52 0.53 0.53

C3 0.52 0.51 0.54

A4 0.52 0.52 0.53

**Firmness Coefficient of Group 2**

B1 0.52 0.51 0.53 0.5222

B2 0.52 0.53 0.54 0.5256

B3 0.51 0.53 0.53 0.5244

B4 0.53 0.53 0.54 0.5267

**Firmness Coefficient of Group 1**

in other areas were also measured strictly according to the preceding method. The obtained data are shown in Table 8. The difference in the firmness coefficient within 0.86% indicates that the firmness coefficient of the selected three wall surfaces is consistent. According to the orthogonal analysis experiment results, the initial nozzle, the improved 1, improved 2, and improved 3 nozzles were arranged to perform coal breaking and punching experiments at walls 1, 2, 3, and 4, respectively, to further reduce experimental error.

> **Firmness Coefficient of Group 3**

**Average of Firmness Coefficient**

(**a**) Field experiments (**b**) The hole

**Figure 18.** Coal breaking and punching experiment. **Figure 18.** Coal breaking and punching experiment.

**Table 9.** Experimental results of coal breaking and punching before and after nozzle optimization.

According to the experimental data, compared with the initial nozzle, the punching depth and diameter of improved nozzles 1 and 2 has been significantly improved. However, the diameter and depth of punching cannot reach the maximum value simultaneously. The reason is that improved nozzle 1 is selected based on the maximum X-axis velocity, while improved nozzle 2 is selected based on the maximum effective Y-axis extension distance. This result also confirms that the two indexes represent the depth and area, respectively, of punching, which is consistent with the analysis result of the orthogonal experiment. Moreover, this result indicates the reliability of the orthogonal experiment. When the maximum X-axis velocity is optimal, punching depth is increased by 12.70% compared with the initial value. When the effective Y-axis extension distance is optimal, punching diameter is increased by 90.91% compared with the initial value. Note that improved nozzle 3 is worth focusing on, the maximum X-axis velocity and effective Y-axis extension distance of which are less than those of improved nozzles 1 and 2. However, the punching depth and punching diameter are optimal, which are 4.22% and 14.29% higher than the two nozzles on the punching depth and diameter, respectively. Both are improved by 17.46% and 118% compared with the initial nozzle. This result shows that punching depth and punching area are mutually affected. That is, when the punching depth deepens, it is easier to obtain a larger punching area. Similarly, when the punching area increases, it is easier to obtain a deeper punching depth. Therefore, to achieve the overall improvement of the water jet performance, the maximum X-axis velocity and effective Y-axis extension

distance must be considered simultaneously. In future research, multi-parameter and multi-objective collaborative optimization of the nozzle can be conducted to improve the water jet performance comprehensively.

## **6. Conclusions**

In this study, the effect of nozzle structure parameters on water jet performance is explored by CFD simulation and verified by field experiments. The research results provide a certain reference value for nozzle structure optimization. The major findings include:

(1) The key structural parameters of the nozzle have a significant impact on the performance of the water jet. Among the key parameters, the value of the contraction angle has the greatest impact on the maximum X-axis velocity, and the value of the divergence angle has the greatest impact on the effective Y-axis extension distance.

(2) Based on the comprehensive results of the orthogonal experiment, range analysis, and variance analysis, three optimal combinations of nozzle structural parameters are selected. Through the field experiment, the optimized nozzles can improve the coal breaking and punching ability of the water jet.

(3) The orthogonal experiment shows that interaction among multiple parameters will affect the water jet performance, and the X-axis velocity and effective Y-axis extension distance of nozzle do not reach the optimal value at the same time. According to the field experiment result in Part 4, the punching depth and area also affect each other. Therefore, in future research, multi-parameter and multi-objective collaborative optimization of the nozzle can be carried out to improve the water jet performance more comprehensively.

**Author Contributions:** Conceptualization, L.C.; Methodology, L.C.; Software, M.C.; Validation, L.G.; Formal analysis, Y.C.; Writing—original draft preparation, L.C. and M.C.; Writing—review and editing, L.C. and D.G.; Supervision, L.C. and D.G.; Funding acquisition, L.G. All authors have read and agreed to the published version of the manuscript.

**Funding:** This work was supported by the National Natural Science Foundation of China (Grant No.51874012).

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**

## *Article* **Research on the Influence of Tip Clearance of Axial-Flow Pump on Energy Characteristics under Pump and Turbine Conditions**

**Yanjun Li \*, Qixu Lin, Fan Meng, Yunhao Zheng and Xiaotian Xu**

Research Center of Fluid Machinery Engineering and Technology, Jiangsu University, Zhenjiang 212013, China<EMAIL_ADDRESS>(Q.L<EMAIL_ADDRESS>(F.M<EMAIL_ADDRESS>(Y.Z<EMAIL_ADDRESS>(X.X.)

**\*** Correspondence<EMAIL_ADDRESS>
**Abstract:** In order to study the influence of tip clearance on the performance and energy dissipation of the axial-flow pump and the axial-flow pump as a turbine, and find the location of high dissipation rate, this study took an axial-flow pump model as its research object and designed four tip radial clearance schemes (0, 0.2, 1 and 2 mm). The unsteady calculation simulation of each tip clearance scheme was carried out based on CFD technology. The calculated results were compared with the experimental results, and the simulation results were analyzed using entropy production analysis theory. The results showed that, under both an axial-flow pump and axial-flow pump as turbine operating conditions, increasing the blade tip clearance led to a decrease in hydraulic performance. Compared with the 0 mm clearance, the maximum decreases in pump efficiency, head and shaft power under 2 mm tip clearance were 15.3%, 25.7% and 12.3% under the pump condition, and 12.7%, 18.5% and 28.8% under the turbine condition, respectively. Under the axial-flow pump operating condition, the change in blade tip clearance had a great influence on the total dissipation of the impeller, guide vane and outlet passage, and the maximum variation under the flow rate of 1.0*Qdes* was 53.9%, 32.1% and 54.2%, respectively. Under the axial-flow pump as a turbine operating condition, the change in blade tip clearance had a great influence on the total dissipation of the impeller and outlet passage, the maximum variation under the flow rate of 1.0*Qdes* was 22.7% and 17.4%, respectively. Under the design flow rate condition, with the increase in tip clearance, the dissipation rate of the blade surface showed an increasing trend under both the axial-flow pump and axial-flow pump as turbine operating conditions, and areas of high dissipation rate were generated at the rim and clearance.

**Keywords:** axial-flow pump as turbine; tip clearance; entropy production; loss distribution; numerical simulation

## **1. Introduction**

The axial-flow pump is a kind of highly specific speed pump with a large flow rate and low head, which is widely used in irrigation, flood control, large-scale water diversion and industrial circulating water systems and other fields of national economic importance [1]. Some axial-flow pumps with good performance can be used as hydraulic turbines, which can convert the pressure energy of a high-pressure medium into the kinetic energy of a rotating machine to generate electricity for energy recovery [2].

At present, research on the pump and the pump as a turbine is mostly based on numerical simulation [3–5] and experimental methods [6–8]. With the development of modern computer technology and the improvement of simulation analysis theory, numerical computer simulation has been increasingly used in the research of pumps and turbines. Qian et al. [9] used numerical simulation to study the performance of the axial-flow pump both under pump and turbine conditions. In the study of hydraulic dissipation within the axial-flow pump and turbine, the spatial distribution of hydraulic dissipation in the axial-flow pump and turbine can be described by using entropy production theory for

**Citation:** Li, Y.; Lin, Q.; Meng, F.; Zheng, Y.; Xu, X. Research on the Influence of Tip Clearance of Axial-Flow Pump on Energy Characteristics under Pump and Turbine Conditions. *Machines* **2022**, *10*, 56. https://doi.org/10.3390/ machines10010056

Academic Editor: Davide Astolfi

Received: 17 December 2021 Accepted: 10 January 2022 Published: 12 January 2022

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

analysis; therefore, entropy production theory has an obvious advantage in hydraulic dissipation assessment, compared with the usual calculation method of total pressure loss [10]. Zhang et al. [11] used numerical simulation to obtain the value and spatial distribution of hydraulic dissipation inside a centrifugal pump by introducing the entropy production analysis theory, which proves the applicability of entropy production analysis theory in the study of centrifugal pump energy loss. Pei et al. [12] studied the influence of the distance between the impeller and guide vane on the performance of the bidirectional shaft tubular pump by using entropy production analysis theory; turbulent dissipation was found to be the main energy dissipation of the total dissipation in the bidirectional shaft tubular pump. Li et al. [13] introduced entropy production analysis theory for the numerical simulation of pump turbines in pump mode, analyzed the hydraulic dissipation and entropy production of pump turbines under different guide vanes, and obtained the hump characteristic of pump turbines in pump mode; the results show that the peak value of entropy production corresponds to the minimum value of efficiency in the external characteristic curve, and the entropy production of the pump turbine mainly occurs on the impeller and guide vane under the pump condition. Gong et al. [14] studied the distribution of turbine hydraulic dissipation based on entropy production analysis theory. It was found that the energy dissipation in the guide vane area accounted for nearly 25% of the total dissipation, which mainly occurred at the head and tail areas of the vanes, and the dissipation in the impeller accounted for more than half of the whole flow passage, which mainly occurred at the leading edge of the blade and the trailing edge of blade, and nearly 20% of the energy dissipation occurred at the elbow. The results show that entropy production theory is suitable for evaluating the performance of the turbine, and has the advantages of determining the amount and the location of energy dissipation.

There is a clearance between the impeller edge and the shell of the axial-flow pump, which can cause clearance leakage flow and result in cavitation, which will influence the operation stability and reduce the operation efficiency of the axial-flow pump under both pump and turbine conditions, and increase vibration and noise during operation [15,16]. Shi [17] and Shen et al. [18] studied the influence of tip clearance size on the internal flow and pressure pulsation performance of the axial-flow pump by using numerical calculation; the results show that the increase in tip clearance leads to the decrease in axial-flow pump efficiency and the increase in pressure pulsation. Chen et al. [19] established a simplified model of clearance flow and used it to evaluate the trajectory of tip leakage vortices. Li et al. [20] studied the influence of five different passive turbine tip clearance flow control methods on axial turbines based on a numerical calculation; the results show that by controlling the passive turbine tip clearance flow, the secondary flow loss including the passage vortex losses and the tip leakage losses near the outer casing can be reduced. Han et al. [21] studied the influence of tip clearance on the hydraulic and thrust performance of the contra-rotating axial-flow water-jet pumps by numerical calculation; the results show that with the increase in tip clearance, the tip clearance leakage of contra-rotating axial-flow water-jet pumps increases and the hydraulic performance decreases obviously.

Tip clearance has a great influence on the operation performance, energy characteristics and manufacturing difficulty of the axial-flow pump and axial-flow pump as a turbine; however, there are few works in the literature about the effect of tip clearance on the internal energy dissipation characteristics of axial flow pump as turbine based on entropy production theory, and a lack of research on the spatial distribution law of its internal hydraulic dissipation. This paper takes the axial-flow pump model as the research object; on the basis of numerical simulation, and the influence of tip clearance on hydraulic loss of the axial-flow pump under both pump and turbine conditions is analyzed using the method of entropy production analysis theory. It can obtain the specific location of high dissipation rate and solve the limitation of optimization design, providing a reference for coupling the optimization design of the axial-flow pump under pump and turbine conditions.

## **2. Calculation Model and Numerical Simulation Calculation**

#### *2.1. Three-Dimensional (3D) Modeling and Mesh Division*

In this paper, an axial-flow pump model with a specific speed of 1000 was selected as the prototype of the research object. Its components include the inlet passage (straight pipe in pump operation and elbow in turbine operation), impeller, guide vane and outlet passage (elbow in pump operation and straight pipe in turbine operation), as shown in Figure 1. The main hydraulic parameters of the model are as follows: pump design flow rate is *Qdes* = 0.39 m3/s; turbine design flow rate is *Q*<sup>0</sup> *des* = 0.47 <sup>m</sup>3/s; rotational speed is *n* = 1450*r*/min. The main structural parameters are as follows: the number of impeller blades is 3; the diameter of the impeller is 300 mm; the number of guide vane blades is 5; and the bending degree of the elbow is 60◦ . Four tip radial clearance schemes *R* were designed, namely, 0, 0.2, 1 and 2 mm, respectively.

**Figure 1.** Three-dimensional model of axial-flow pump. 1—inlet passage; 2—impeller; 3—guide vane; 4—outlet passage.

In order to ensure the computational accuracy and convergence of the grid model, the computational domain was divided into structural grids [22–24]. ICEM was used to generate the grid of the inlet passage, impeller and outlet passage, and TuboGrid was used to generate the grid of the guide vane, as shown in Figure 2. Local mesh encryption was carried out on the wall surface around the blade and the rim clearance area to ensure that the wall Y+ value met the solution requirements of the turbulence model [25]. In order to ensure the accuracy of the calculation, a total of 15 layers of grids were divided at the blade tip clearance. The number of grid nodes of the inlet passage, guide vane and outlet passage was 1 million, 1.2 million and 1.4 million, respectively, and the number of grid nodes of the impeller under the four tip clearance schemes was controlled at approximately 1.8 million.

#### *2.2. Setting of Boundary Conditions*

Based on Newton's continuity equation and momentum conservation equation for incompressible fluid, the SST k−ω [26,27] turbulence model was used to calculate the internal flow field and hydraulic performance of the axial-flow pump with different tip clearance schemes under axial-flow pump and turbine operating conditions. In both axialflow pump and turbine operating conditions, the inlet passage adopts a mass flow rate condition, and the outlet passage adopts a static pressure condition [28]. The boundary details adopt a non-slip and smooth wall. The mixing model with stationary components is composed of "None", and the mixing model with rotating and stationary interfaces is composed of a "Transient Rotor Stator". The residual type of convergence criterion was RMS, which was set as 10−<sup>5</sup> , and the time step was 0.000344828 s, that is, the time required for the impeller to rotate 3◦ . The calculation cycle was set as 10 laps, and the total time was 0.413793 s.

**Figure 2.** Axial-flow pump and impeller grid.

### *2.3. Comparative Analysis of Calculation Results and Experiments*

Figure 3 shows the test bench of axial-flow pump. In order to verify the feasibility of the model simulation, the external characteristics of the model pump simulation calculation were compared with the test results of the test bench of axial-flow pump. The flow rate was measured by a cologne intelligent electromagnetic flowmeter, the measurement uncertainty of which was ±0.2%; the head was measured by the YOKOGAWA EJA intelligent differential pressure transmitter, the measurement uncertainty of which was ±0.1%; torque speed was measured by the NJL2/500 Nm intelligent torque speed sensor, the measurement uncertainty of which was ±0.1%. Figure 4 shows the comparison between the experimental results and numerical calculation results of the model pump under different flow rates. The trend of CFD numerical calculation results is the same as that of the test results. The simulated operating efficiency of the axial-flow pump was slightly lower than the experimental value, and the error under the condition of a small flow rate was greater than that under the condition of a large flow rate, and the maximum relative error was 3.1%. The simulated head of the axial-flow pump was slightly lower than the test value under the condition of a small flow, and slightly higher than the test value under the condition of the design flow and large flow, and the maximum relative error was 2.1%. In conclusion, the error between the simulation results and the test results is small, and the reliability of the calculation results is good.

**Figure 3.** *Cont*.

**Figure 3.** Test bench of axial-flow pump. (**a**) Test instruments. (**b**) Layout of test bench.

**Figure 4.** Comparison of model pump external characteristics test and numerical calculation results.

## **3. Entropy Production Analysis of Axial-Flow Pump with Different Tip Clearances under Pump and Turbine Conditions**

## *3.1. Entropy Production Theory*

According to the second law of thermodynamics, the fluid in the system is always accompanied by energy dissipation in the process of flow. Reynolds stress and the viscosity of the fluid itself will lead to irreversible energy dissipation in the flow process of the fluid inside the pump, while the eddy phenomenon generated in the flow will increase the energy dissipation, which will lead to an increase in entropy generation and hydraulic

dissipation. The distribution of hydraulic loss in the axial-flow pump can be analyzed by entropy production theory.

Axial-flow pump internal entropy production Φ *T* is caused by direct entropy production *SPRO*,*<sup>D</sup>* and the turbulent entropy production *SPRO*,*D*<sup>0</sup> ; the calculation formula is as follows:

$$\overline{\left(\frac{\Phi}{T}\right)} = \mathcal{S}\_{\text{PRO},\overline{\mathcal{D}}} + \mathcal{S}\_{\text{PRO},\mathcal{D}'} \tag{1}$$

$$\mathbf{S}\_{\rm PRO,\overline{D}} = \frac{\mu}{\overline{T}} \left\{ 2 \left[ \left( \frac{\partial \overline{\mathbf{v}\_{1}}}{\partial \mathbf{x}} \right)^{2} + \left( \frac{\partial \overline{\mathbf{v}\_{2}}}{\partial \mathbf{y}} \right)^{2} + \left( \frac{\partial \overline{\mathbf{v}\_{3}}}{\partial \mathbf{z}} \right)^{2} \right] + \left( \frac{\partial \overline{\mathbf{v}\_{1}}}{\partial \mathbf{y}} + \frac{\partial \overline{\mathbf{v}\_{2}}}{\partial \mathbf{x}} \right)^{2} + \left( \frac{\partial \overline{\mathbf{v}\_{1}}}{\partial \mathbf{z}} + \frac{\partial \overline{\mathbf{v}\_{3}}}{\partial \mathbf{x}} \right)^{2} + \left( \frac{\partial \overline{\mathbf{v}\_{2}}}{\partial \mathbf{z}} + \frac{\partial \overline{\mathbf{v}\_{3}}}{\partial \mathbf{y}} \right)^{2} \right\} \tag{2}$$

$$\mathbf{S}\_{\rm PRO,D} = \frac{\mu}{\overline{T}} \cdot \left\{ 2 \left[ \left( \frac{\partial \mathbf{v}\_1'}{\partial \mathbf{x}} \right)^2 + \left( \frac{\partial \mathbf{v}\_2'}{\partial \mathbf{y}} \right)^2 + \left( \frac{\partial \mathbf{v}\_3'}{\partial \mathbf{z}} \right)^2 \right] + \left( \frac{\partial \mathbf{v}\_1'}{\partial \mathbf{y}} + \frac{\partial \mathbf{v}\_2'}{\partial \mathbf{x}} \right)^2 + \left( \frac{\partial \mathbf{v}\_2'}{\partial \mathbf{z}} + \frac{\partial \mathbf{v}\_3'}{\partial \mathbf{x}} \right)^2 + \left( \frac{\partial \mathbf{v}\_2'}{\partial \mathbf{z}} + \frac{\partial \mathbf{v}\_3'}{\partial \mathbf{y}} \right)^2 \right\} \tag{3}$$

Among them, *v*1, *v*2, *v*<sup>3</sup> are the components of the average time velocity on the coordinate *x*, *y*, *z* axes; *v* 0 1 , *v* 0 2 , *v* 0 3 are velocity pulsations on the *x*, *y*, *z* axes; µ is the dynamic viscosity; *T* is the time average temperature.

In order to solve the problem that *SPRO*,*D*<sup>0</sup> in the formula cannot be calculated by solving the Reynolds equation, Kock [29,30] and Herwig [31,32] proposed Equation (4),

$$S\_{PRO, D'} = \frac{\rho \varepsilon\_1}{\overline{T}} \tag{4}$$

where *ρ* is the fluid density and *ε*<sup>1</sup> is the dissipation rate of turbulent kinetic energy.

The direct dissipation rate Φ*<sup>D</sup>* and turbulent dissipation rate Φ*D*<sup>0</sup> can be calculated from Equations (5) and (6):

$$
\Phi\_{\overline{D}} = \overline{T} \cdot \mathcal{S}\_{PRO, \overline{D}} \tag{5}
$$

$$
\Phi\_{D'} = \overline{T} \cdot \mathbb{S}\_{PRO, D'} \tag{6}
$$

Thus, the direct dissipation and turbulent dissipation of each hydraulic component can be obtained by volume integration of Equations (5) and (6), as shown in Equations (7) and (8).

$$P\_{\overline{D}} = \int\_{V} \Phi\_{\overline{D}}dV\tag{7}$$

$$P\_{D'} = \int\_{V} \Phi\_{D'}dV\tag{8}$$

Therefore, the total dissipation rate Φ*<sup>T</sup>* and total dissipation *P<sup>T</sup>* can be calculated from Equations (9) and (10):

$$
\Phi\_T = \Phi\_{\overline{D}} + \Phi\_{D'} \tag{9}
$$

$$P\_T = P\_{\overline{D}} + P\_{D'} \tag{10}$$

#### *3.2. Analysis of Calculation Results*

3.2.1. Influence of Tip Clearance on External Characteristics

Figure 5 shows the comparison of the influences of different tip clearances on the external characteristic parameters of the axial-flow pump operating under pump conditions. Under the pump condition, both the head and shaft power of the axial-flow pump decreased with the increase in the flow rate. Under the same flow condition, both the head and shaft power decreased with the increase in tip clearance. Under the conditions of 0.8*Qdes*, 1.0*Qdes* and 1.2*Qdes*, the head decreased by 10.0%, 15.5% and 25.7%. At 0.8*Qdes*, 1.0*Qdes* and 1.2*Qdes*, the shaft power decreased by 3.2%, 6.3% and 12.3%. When the flow rate increased from 0.8*Qdes* to 1.0*Qdes*, the efficiency increased; when the flow rate increased from 1.0*Qdes* to 1.2*Qdes*, the efficiency decreased.

**Figure 5.** Comparison of (**a**) efficiency; (**b**) head; (**c**) shaft power under different tip clearances in pump condition; (**d**) Legend.

Figure 6 shows the comparison of the influences of different tip clearances on the external characteristic parameters of the axial-flow pump under turbine conditions. Under the turbine condition, both the head and shaft power showed a rising trend with the increase in flow rate. Under the same flow condition, both the head and shaft power decreased with the increase in tip clearance. The decreases in the head were 18.5%, 12.0% and 7.9% under 0.8*Qdes*, 1.0*Qdes* and 1.2*Qdes* conditions. At 0.8*Qdes*, 1.0*Qdes* and 1.2*Qdes*, the shaft power decreased by 28.8%, 15.9% and 11.5%. When the flow rate increased from 0.8*Qdes* to 1.0*Qdes*, the efficiency increased; when the flow rate increased from 1.0*Qdes* to 1.2*Qdes*, the efficiency decreased. According to the formula:

$$H = \frac{P\_{\rm out} - P\_{\rm in}}{\rho g} \tag{11}$$

where *H*—head, m; *Pout*—the total pressure of outlet, Pa; *Pin*—the total pressure of inlet, Pa; *ρ*—the fluid density, kg/m<sup>3</sup> ; *g*—the acceleration of gravity, m/s<sup>2</sup> ; *PS*—the shaft power, kW; *ηP*—the efficiency of axial-flow pump; *ηT*—the efficiency of axial-flow pump as a turbine; *Q*—the flow rate, kg/s; *M*—the torque of impeller, N·m; *ω*—the rotating speed, r/min.

It can be seen that the efficiency of the axial-flow pump was directly proportional to the head and inversely proportional to the shaft power, and the efficiency of axial-flow pump as a turbine was directly proportional to the shaft power and inversely proportional to the head. The efficiency of the axial-flow pump decreased greatly under the condition of a large flow rate, while the efficiency of the axial-flow pump as a turbine decreased slightly under the condition of a large flow rate. With the increase in tip clearance, the efficiency of the axial-flow pump showed a decreasing trend under both pump and turbine condition. In the pump condition, the maximum decrease was 7.1%, 9.8% and 15.3% under 0.8*Qdes*, 1.0*Qdes* and 1.2*Qdes* conditions, respectively. Under turbine conditions, the maximum decline was 12.7%, 4.5% and 3.9% under 0.8*Qdes*, 1.0*Qdes* and 1.2*Qdes* conditions, respectively.

**Figure 6.** Comparison of (**a**) efficiency; (**b**) head; (**c**) shaft power under different tip clearances in turbine condition; (**d**) Legend.

#### 3.2.2. Dissipation Distribution Based on Entropy Production Theory

Figure 7 shows the influence of tip clearance on total dissipation under the axial-flow pump condition. In all flow rate conditions, the total dissipation increased with the increase in tip clearance. With the increase in flow rate, the influence of tip clearance change on total dissipation decreased. Under the 0.8*Qdes* condition, the total dissipation increased by 47.7%; under the 1.0*Qdes* condition, the total dissipation increased by 45.0%; under the 1.2*Qdes* condition, the total dissipation increased by 11.2%.

**Figure 7.** Influence of tip clearance on total dissipation under pump condition.

Figure 8 shows the influence of tip clearance on total dissipation under the turbine condition of the axial-flow pump. Under all flow rate conditions, the total dissipation increased with the increase in tip clearance, and with the increase in flow rate, the influence caused by tip clearance changes also increased. Under the 0.8*Qdes* condition, the total dissipation increased by 7.9%; under the 1.0*Qdes* condition, the total dissipation increased by 10.5%; under the 1.2*Qdes* condition, the total dissipation increased by 14.9%.

**Figure 8.** Influence of tip clearance on total dissipation under turbine condition.

Figure 9 shows the influence of tip clearance on the distribution of total dissipation inside the pump under different flow rate conditions. Tip clearance had an obvious influence on the impeller, guide vane and outlet passage, but had little influence on the inlet passage. Under all three flow conditions, the total dissipation of the impeller and outlet passage increased with the increase in tip clearance, and the influence of tip clearance was more obvious with the increase in flow rate. Under 0.8*Qdes*,1.0*Qdes* and 1.2*Qdes*, the total dissipation of the impeller increased with the increase in tip clearance of 114.8%, 53.9% and 7.6%, respectively. The total dissipation of the outlet passage increased by 18.5%, 54.2% and 23.5% with the increase in tip clearance. For the guide vane, the total dissipation decreased with the increase in tip clearance at 0.8*Qdes*, with a maximum decrease of 8.4%. Under the condition of 1.0*Qdes*, the total dissipation increased with the increase in tip clearance, with a maximum increase of 32.1%. Under the condition of 1.2*Qdes*, the total dissipation of the guide vane decreased first and then increased with the increase in tip clearance.

Figure 10 shows the influence of tip clearance on the distribution of the internal dissipation of the pump turbine under different flow rate conditions. Tip clearance had an obvious influence on the impeller and outlet passage, but had little influence on the guide vane and inlet passage. Under the three flow rate conditions, the total dissipation in the impeller increased with the increase in tip clearance, and the smaller the flow rate was, the more obvious the influence of tip clearance was. The total dissipation increased by 4.9%, 22.7% and 42.2% with tip clearance at 0.8*Qdes*, 1.0*Qdes* and 1.2*Qdes*, respectively. For the outlet passage, the total dissipation increased with the increase in tip clearance at 0.8*Qdes*, and the maximum increase was 31.1%. However, under the conditions of 1.0*Qdes* and 1.2*Qdes*, the total dissipation decreased first and then increased with the increase in tip clearance.

**Figure 9.** Influence of tip clearance on total dissipation under pump condition; (**a**) 0.8*Qdes*; (**b**) 1.0*Qdes*; (**c**)1.2*Qdes*; (**d**) Legend.

**Figure 10.** Influence of tip clearance on total dissipation under turbine condition; (**a**) 0.8*Qdes*; (**b**) 1.0*Qdes*; (**c**) 1.2*Qdes*; (**d**) Legend.

Figure 11 shows the comparison of turbulence dissipation and direct dissipation caused by different tip clearance schemes in the pump and turbine conditions of the axialflow pump under the design flow rate condition. As can be seen from the figure, the total dissipation generated by the direct loss of the axial-flow pump under pump and turbine conditions was very small compared with the turbulence loss. The maximum direct dissipation of the pump and turbine conditions was only 3.1% of the total dissipation, and the maximum direct dissipation of the turbine condition was only 2.8% of the total dissipation. Therefore, the dissipation inside the pump was mainly affected by turbulence dissipation.

**Figure 11.** Comparison of direct dissipation and turbulence dissipation; (**a**) Pump condition; (**b**) Turbine condition.

Figure 12 shows the distribution of the blade surface dissipation rates with different tip clearances under the design flow rate of the pump condition. The dissipation rate from the hub to the rim on the blade surface increased gradually. A high dissipation area was generated at the rim on the pressure side. At the angle between the leading edge and the rim, a high dissipation area was formed on the suction side. On the suction side, the high dissipation area was significantly greater than the pressure side. This is due to the fact that the pressure side of the blade was mainly affected by the fluid circumferential velocity, and the suction side was mainly affected by fluid axial velocity; the fluid in the impeller had a great impact on the suction side of the blade. In addition, due to the increase in blade tip clearance, leakage and backflow phenomena at the clearance were intensified, leading to a gradual increase in the speed loss at the rim, and then the dissipation rate of the rim.

**Figure 12.** Dissipation rate distribution on blade surface under pump condition. (**a**) Pressure side. (**b**) Suction side.

In order to more intuitively explain the influence of tip clearance on the axial-flow pump, the impeller and guide vanes were expanded radially on a cylindrical surface, as shown in Figure 13. The radial coefficient of the expanded surface *r* ∗ can be calculated by Equation (12):

$$r^\* = \frac{(r - r\_h)}{(r\_t - r\_h)}\tag{12}$$

where *r* is the radius of the cylinder, mm; *r<sup>h</sup>* is the radius of the hub, mm; *r<sup>t</sup>* is the radius of the rim, mm.

**Figure 13.** Cylinder section of impeller and guide vane.

Figure 14 shows the cloud diagram of velocity distribution on the cylindrical section of the impeller under different tip clearance schemes and the pump design flow rate condition (*r* ∗ = 0.97). The flow velocity inside the impeller decreased from the impeller inlet to the impeller outlet, and as a result of the existence of the blade tip clearance, part of the fluid through the blade tip clearance produced return loss and leakage caused by impeller import flow velocity. The bigger the tip clearance, the greater the reflux and leakage phenomena; therefore, the increase in tip clearance speed reduced the impeller inlet from the front.

**Figure 14.** Cylindrical distribution of impeller internal velocity under pump condition (*r* ∗ = 0.97).

Figure 15 shows the distribution of the total dissipation rate on the cylindrical section of the impeller under different tip clearance schemes and the pump design flow rate condition (*r* ∗ = 0.97). The velocity loss caused by the backflow and leakage of some fluid in the impeller passing through the tip clearance led to an increase in the rate of dissipation; therefore, a high dissipation area was generated on the suction side of the impeller. With the increase in tip clearance, the dissipation rate on the suction side of the impeller increased gradually.

**Figure 15.** Cylinder distribution of internal dissipation rate of impeller under pump condition (*r* ∗ = 0.97).

Figure 16 shows the cloud diagram of velocity distribution on the cylindrical section of the guide vane under different tip clearance schemes and the pump design flow rate condition (*r* ∗ = 0.99). The kinetic energy of the fluid was transformed into pressure energy after passing the guide vane; therefore, the flow velocity inside the guide vane decreased from the inlet to the outlet; there was a large amount of backflow at the outlet of the guide vane, which led to a low-speed zone here, and a point with the lowest speed was formed at the guide vane inlet due to the mixture of backflow and mainstream. The impeller clearance also caused speed loss, resulting in a gradual decrease in the internal flow velocity of the guide vane with the increase in tip clearance.

**Figure 16.** Cylindrical distribution of guide vane internal velocity under pump conditions (*r* ∗ = 0.99).

Figure 17 shows the distribution of the total dissipation rate on the cylindrical section of the guide vane under different tip clearance schemes and the pump design flow rate condition (*r* ∗ = 0.99). The high dissipation area inside the guide vane was the suction side and the inlet of the guide vane, where the velocity gradient was large, leading to the generation of high dissipation. The high dissipation area at the suction side was caused by the impact of backflow; the velocity gradient near the inlet of guide vane was large, which also resulted in high entropy production.

**Figure 17.** Cylinder distribution of internal dissipation rate of guide vane under pump condition (*r* ∗ = 0.99).

Figure 18 shows the internal dissipation rate distribution of the inlet passage with different tip clearances under the pump design flow rate condition. Energy dissipation in the inlet passage mainly occurred at the inlet, the bend of the inner wall of the elbow and the junction of the rotating shaft and the outer wall. As the fluid in the inlet passed through the inlet transition section, the bend of the passage wall and the rotation around the rotating shaft, a certain degree of flow separation occurred, which resulted in energy dissipation.

**Figure 18.** Distribution of internal dissipation rate in outflow passage under pump condition.

Figure 19 shows the distribution of blade surface dissipation rates with different tip clearances under the design flow rate of the turbine condition. The surface dissipation rate of the blade pressure side increased gradually from the hub to the rim and produced a higher dissipation at the rim. The suction side produced a high dissipation area at the angle between the rim and the leading edge; the area of the high dissipation rate on the suction side was obviously larger than that on the pressure side. This shows that when the impeller runs under the turbine condition, the fluid cannot pass through the suction side properly, which has a great impact on it. In addition, with the increase in blade tip clearance, the leakage and backflow in the clearance were intensified, and as a result, the speed loss at the rim increased gradually, and then the dissipation rate of the rim increased gradually.

Figure 20 shows the cloud diagram of velocity distribution on the cylindrical section of the impeller under different tip clearance schemes and the turbine design flow rate condition (*r* ∗ = 0.97). The flow velocity on the suction side of the impeller was obviously greater than that on the pressure side; under the turbine condition, the fluid could not flow well over the suction side surface, and obvious flow separation occurred at the leading and trailing edges of the impeller; therefore, a low-speed zone was generated at the leading and trailing edges of the impeller.

**Figure 19.** Dissipation rate distribution on blade surface under turbine condition. (**a**) Pressure side. (**b**) Suction side.

**Figure 20.** Cylindrical distribution of impeller internal velocity under turbine condition (*r* ∗ = 0.97).

Figure 21 shows the distribution of the total dissipation rate on the cylindrical section of the impeller under different tip clearance schemes and the turbine design flow rate condition (*r* ∗ = 0.97). As the fluid near the blade suction side could not fit the blade surface effectively. This had a great impact and flow separation phenomenon and caused a large velocity gradient on the blade suction side. There was a high dissipation area on the suction side of the impeller. High energy dissipation also occurred at the blade outlet due to flow separation. With the increase in tip clearance, the dissipation rate of impeller working surface and outlet increased gradually.

**Figure 21.** Cylinder distribution of internal dissipation rate of impeller under turbine condition (*r* ∗ = 0.97).

Figure 22 shows the distribution of the dissipation rate of the outlet passage with different tip clearances under the turbine design flow rate condition. There was no rear guide vane when the axial-flow pump operated as a turbine. The fluid at the outlet of the impeller lacked the rectifier action of the guide vane. There was a high dissipation area at the inlet of the outlet passage. With the increase in tip clearance, the dissipation rate at the inlet of outlet passage increased. Moreover, due to the effect of the passage wall, the kinetic energy flowing out of the impeller was converted into pressure energy; therefore, the dissipation rate of the outlet passage decreased from the inlet to the outlet.

**Figure 22.** Distribution of internal dissipation rate in outflow passage under turbine condition.

## **4. Conclusions**

In this paper, a numerical simulation method was adopted to study the energy loss of the axial-flow pump under both pump and turbine conditions based on entropy production theory, and four different tip clearance schemes were analyzed. The following conclusions can be drawn:

Therefore, a smaller tip clearance should be selected when the pump condition is the main condition; however, a larger tip clearance can be selected when the turbine condition is the main condition. The study result shows the relationship between blade tip clearance and dissipation loss in the pump and its spatial distribution under both pump and turbine conditions. This provides suggestions for the optimization of the axial-flow pump as a turbine, to find suitable tip clearance in the axial-flow pump as a turbine, and provides a reference for solving the problems of large vibration and noise, low operating efficiency caused by clearance that is too large, and manufacturing difficulty caused by clearance that is too small.

**Author Contributions:** Conceptualization, Y.L. and Q.L.; methodology, Y.L. and F.M.; software, Q.L.; validation, Y.L. and Q.L.; formal analysis, Q.L.; investigation, Q.L.; resources, Q.L.; data curation, Y.L. and Q.L.; writing—original draft preparation, Q.L.; writing—review and editing, Q.L. and F.M.; visualization, Q.L.; supervision, Y.Z. and X.X.; project administration, Y.L.; funding acquisition, Y.L. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by National Natural Science Foundation of China, grant number 51809120 and Industry-University-Research Collaboration of Jiangsu Province, grant number BY2020373.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Conflicts of Interest:** The authors declare no conflict of interest.

### **Nomenclature**

#### **References**

## *Article* **Correlation between the Internal Flow Pattern and the Blade Load Distribution of the Centrifugal Impeller**

**Bo Chen \*, Xiaowu Chen, Zuchao Zhu and Xiaojun Li**

Key Laboratory of Fluid Transmission Technology of Zhejiang Province, Zhejiang Sci-Tech University, Hangzhou 310018, China<EMAIL_ADDRESS>(X.C<EMAIL_ADDRESS>(Z.Z<EMAIL_ADDRESS>(X.L.) **\*** Correspondence<EMAIL_ADDRESS>
**Abstract:** The blade load distributions reflect the working characteristics of centrifugal impellers, and the vortexes in the impeller channel affect the blade load distribution, but the mechanism of this phenomenon is still unclear. In this study, particle image velocimetry (PIV) was adopted to clarify the correlation between the internal flow pattern and the blade load distribution. The internal flow pattern and the blade load distribution were presented under different working conditions to study the influence of the internal flow pattern on the blade load. Results showed that the vortexes in the flow channel redistributed the blade load. The clockwise vortex made the position of the maximum blade load closer to the outlet, while the counterclockwise vortex had the opposite effect. Meanwhile, the vortexes caused the blade load distribution to be steeper, which reduced energy conversion efficiency. Moreover, the mean absolute flow angle was introduced to explain the mechanism of the effects of vortexes on blade load. The results can be used as a theoretical basis for the design of high-performance impellers.

 Introduction**

The centrifugal pump is an important energy conversion device. Many researchers adopted various methods to analyze the flow characteristics of the pump and its influence on the pump performance [1–3]. The blade load distribution is a main parameter that reflects the working characteristics of centrifugal impellers. The blade load has gradually gained people's attention [4–6]. Studying the correlation between internal flow pattern and blade load will deepen the understanding of impeller flow pattern and stable operation of the pump.

Many studies have been carried out to explain the flow pattern in pumps. Wang et al. [7] adopted the PIV experiment to measure the flow characteristics in the five-blade centrifugal impeller. The results showed that when the flow rate was reduced to the flow rate of 0.8 *Q*BEP, flow separation began to form on the suction side of the flow channel. Atkins et al. [8] applied the PIV technology to measure the velocity fields of the upstream, gap, and downstream areas around five side-by-side cylinders. Zhang et al. [9] used the PIV technology to measure the flow structure in a low-specific-speed centrifugal pump in detail. The results showed that when the flow velocity was lower than the nominal conditions, a typical structure of jet-wake was observed at the blade outlet. Zhang et al. [10] studied the internal flow characteristics and hydraulic performance of the twin-screw pump by CFD. Results showed that the screw pressure gradually increased from the inlet face to the outlet face along the screw axis under different heads. Bennacer et al. [11] applied the PIV technology to conduct an in-depth analysis of the internal flow. They found that the dominant mode accounted for more than half of the total kinetic energy. Guo et al. [12] used an experiment and numerical simulation to characterize the performance of pumps

**Citation:** Chen, B.; Chen, X.; Zhu, Z.; Li, X. Correlation between the Internal Flow Pattern and the Blade Load Distribution of the Centrifugal Impeller. *Machines* **2022**, *10*, 40. https://doi.org/10.3390/ machines10010040

Academic Editor: Davide Astolfi

Received: 22 November 2021 Accepted: 28 December 2021 Published: 5 January 2022

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

with variable pitch inducers at different speeds. Si et al. [13] adopted the PIV technology to study the influence of internal flow characteristics on the overall pump performance. Results showed that the main loss was occurred inside the vaneless part of the diffuser under the low flow rates. Wu et al. [14] used the PIV technology to measure the internal flow field under design flow working conditions, which revealed the distribution of the principal Reynolds normal stress and the principal Reynolds shear stress. Li et al. [15] studied the influence of internal flow patterns on hydraulic performance and energy conversion characteristics through visual experiments. Detailed studies showed that a counterclockwise vortex would increase energy losses, while a clockwise vortex would have a positive effect on the pump head. Westra et al. [16] used the PIV technology to study the internal velocity field distribution characteristics of centrifugal pumps, especially the secondary flow structure in the impeller flow channel. Yu et al. [17] used a laser doppler anemometer to obtain velocity measurements from a model of bio-centrifugal pump. Almost all the aforementioned studies have focused on the effects of flow rates on the flow pattern. In the present study, the effect of the rotational speeds on internal flow pattern were analyzed. Several researchers have aimed to investigate the internal flow pattern by analyzing the absolute flow angle. Liu et al. [18] applied PIV technology to study the unstable internal flow of a double-blade centrifugal pump impeller at a design flow rate. It focused on the analysis of the change trend of the mean absolute flow angle and the average dynamic pressure. Gaetani et al. [19] studied the pump performance and focused on computing spanwise mean flow angles and velocity distribution. Wang et al. [20] studied the formation mechanism of the reverse flow. Their results showed that the absolute flow angle at the blade inlet nearing the shroud was close to 180◦ because of the joint action of the leakage flow and blade inlet impact under low flow rates. Poujol et al. [21] tested the centrifugal compressor under the staggered angle of three inlet guide vanes. It showed that the absolute flow angle at the diffuser inlet decreases with the increase of IGV stagger angle near the surge point.

At the same time, many studies have been conducted to explain the effect of blade load on pump performance. Chalghoum et al. [22] analyzed the unsteady value of the strong interaction between the impeller blade and the volute tongue, then studied the correlation between the dynamic load and the unstable radial force. Zhang et al. [23] used a new method to design a new impeller based on the blade load. Li et al. [24] studied the effect of blade load distribution on the head, radial force, and pressure pulsation of a low-specific-speed centrifugal pump with cylindrical impeller blades. It showed that the front-loaded impeller produced a more uniform flow than the rear-loaded impeller. Li et al. [25] discussed the impact of monitoring points and blade load variations on the unstable behavior of the impeller through sensor experiment. Zhang et al. [26] studied the pressure fluctuation characteristics of a reversible pump turbine under various blade load conditions by using pressure signals. They found that pressure fluctuations were very obvious under the low load conditions. Sun et al. [27] revealed the optimal load distribution mechanism of the hub by different methods. Durante et al. [28] introduced a comparison for the analysis of the unsteady loads among different hydrodynamic models. It highlighted the ability to describe high frequency hydrodynamic loads. Li et al. [29] proposed the blade load distribution based on PIV technology to analyze the energy conversion mechanism in the centrifugal pump impeller. Few works have investigated the correlation between the blade load distribution and the flow pattern in the impeller. The current study aimed to clarify the effect of vortexes in the impeller channel on the blade load distribution. In this paper, the correlation between the flow pattern in the impeller and the blade load distribution was further discussed with the help of PIV.The current study aimed to clarify the mechanism of how vortexes in the impeller channel affect the blade load distribution. Results include the theory of blade load, the process of blade load data extraction, the size of vortexes in the flow channel, and the movement laws of the vortex core. Finally, the reasons for the special trend of some blade load distributions were analyzed.

#### **2. Implementation Process Based on PIV**

#### *2.1. Theoretical Basis of Blade Load*

The blade load distribution is an important factor to measure the hydraulic performance of the impeller in blade design. The blade load is defined as the pressure difference between the pressure side and the suction side in the length direction of the blade. By changing the distribution law of the blade load, the pressure difference distribution between the pressure side and the suction side of the blade can be effectively changed. Therefore, the reasonable change of the blade load distribution can improve the performance of the impeller.

As for the inverse design method, the distribution represents the change of the velocity loop *rV<sup>θ</sup>* along the streamline. *V<sup>θ</sup>* is the circumferential component of the absolute velocity. The distribution of the velocity circulation along the streamline reflects the distribution change of the blade load.

Based on this feature of impeller machinery, the distribution of *∂rVθ*/*∂m* along the streamline is used as the flow filed condition in the inverse design method. Therefore, the *F* (the value of blade load) can be rewritten as Equation (1):

$$F = p^{+} - p^{-} = \frac{2\pi}{B} \rho \mathcal{W} \frac{\partial (r \overline{V\_{\theta}})}{\partial m} \tag{1}$$

*p* <sup>+</sup> and *p* − are the pressure at the pressure side and suction side of the blade respectively, *B* is the number of blades, *W* is the relative velocity at the blade side, *ρ* is the density of water, *rV<sup>θ</sup>* is the velocity loop quantity, *m* is the relative axial side flow length.

The blades are represented by vorticity sheets, which depend on the specific distribution of circumferential mean angular momentum *rV<sup>θ</sup>* defined as follows:

$$r\overline{V\_{\theta}} = \frac{Z}{2\pi} \int\_{0}^{\frac{2\pi}{Z}} rV\_{\theta} d\theta \tag{2}$$

According to the velocity triangle, we have the following conclusions:

$$V\_{\theta} = \frac{Q}{A \tan \alpha} \tag{3}$$

$$A = 2\pi r b$$

where, *Q* is the flow rate through the impeller, *A* is the cross-sectional area, *b* is the width of the channel in the cross-section, and *α* (the absolute flow angle) is the angle between the circumferential direction and the absolute velocity.

In this paper, the blade load distribution was obtained by an experimental method. The data obtained from the PIV test were processed by MATLAB to obtain the relative velocity and absolute velocity in flow field, and then the blade load was obtained based on the blade load Equation (1). The following is an introduction to the experimental device.

#### *2.2. Methodology*

In this paper, PIV experimental method was used to extract blade load value after post-processing. Experiments of the test pump were conducted on a closed test rig as shown in Figure 1. The main parameters of the impeller are listed in Table 1. The valves were used to adjust the flow rate of the pump. The flow rate was measured by using an electromagnetic flowmeter. The water head was measured by two pressure sensors. The rotational speed was regulated by a frequency inverter.

**Table 1.** Main design parameters of the impeller. **Table 1.** Main design parameters of the impeller.

Specific speed *ns* 23.8

Blade number *Z* 5 The PIV measurement system was described in Li et al. [29]. The PIV system (TSI) consisted of a double-pulsed Nd: YAG laser (λ = 532 nm), a CCD camera (spatial resolution of 2048 px × 2048 px, time delay of 130 μs), a synchronizer, and a data-processing system (Insight 3 G). The image pair's acquisition was synchronized with the impeller rotation using an optical trigger which was located at the pump shaft. The spatial resolution of the acquired images was 102.38 μm/px. The seeding particles' (soild alumina particles) diameter was 20–60 μm, and their density was 1050 kg/m3. A multi-pass method with 50% overlap was chosen with a final interrogation region of 32 × 32 px2. According The PIV measurement system was described in Li et al. [29]. The PIV system (TSI) consisted of a double-pulsed Nd: YAG laser (λ = 532 nm), a CCD camera (spatial resolution of 2048 px × 2048 px, time delay of 130 µs), a synchronizer, and a data-processing system (Insight 3 G). The image pair's acquisition was synchronized with the impeller rotation using an optical trigger which was located at the pump shaft. The spatial resolution of the acquired images was 102.38 µm/px. The seeding particles' (soild alumina particles) diameter was 20–60 µm, and their density was 1050 kg/m<sup>3</sup> . A multi-pass method with 50% overlap was chosen with a final interrogation region of 32 <sup>×</sup> 32 px<sup>2</sup> . According to Li et al. [29], the uncertainty of the phase-averaged velocities was lower than 1–3% of themeasured velocity.

Before the PIV experiment, the performance characteristics of the test pump were firstly tested. The optimum flow rates (*Q*BEP) and the corresponding heads at different rotational speeds are listed in Table 2.

**Table 2.** Optimum flow rates and the corresponding heads for different rotational speeds.

Based on these optimum flow rates, the PIV experiments were conducted at flow rates of 1.2 *Q*BEP, 1.0 *Q*BEP, 0.8 *Q*BEP, 0.6 *Q*BEP, 0.4 *Q*BEP, and 0.2 *Q*BEP at different rotational speeds. During the measuring process, the images were acquired at the same section by using a triggering system. The velocity fields were obtained from 300 pairs of images which were obtained at the same blade position. The experimental data were then processed as follows: The velocities are a function of position *x*, *y* and time *t*, including phase average and pulsation, shown in Equation (5):

$$V(\mathbf{x}, y, t) = \nabla(\mathbf{x}, y) + V'(\mathbf{x}, y, t) \tag{5}$$

Then the velocity field corresponding to each working condition captured in the test was averaged to obtain the mean absolute speed field shown in the following Equation (6):

$$\overline{v}(\mathbf{x}, y) = \frac{1}{n} \sum\_{i=1}^{n} V\_i(\mathbf{x}, y, t\_0 + i\Delta t) (n = 300) \tag{6}$$

The phase-averaged relative velocity field was obtained by subtracting the local circumferential velocity of the impeller from the absolute velocity at each point:

$$\mathcal{W} = V - \Omega \times r\_a \tag{7}$$

*W* is the relative velocity of the particles. Ω is the -rotational speed of the impeller and *r*<sup>a</sup> is the local radius.

After the relative velocity field was obtained, the blade load value could be obtained by combining Equation (1).

#### *2.3. Results of Blade Load*

Figure 2a shows a schematic of the flow channel meridian. The *V<sup>θ</sup>* distribution on the same radius in the flow channel was averaged to obtain the mean *V<sup>θ</sup>* distribution along the meridian. The channel meridian was consistent with the blade profile. The blade load distribution and the mean absolute flow angle distribution were analyzed along this meridian. Figure 2b shows the side view of the impeller.

**Figure 2.** Schematic of flow channel meridian (**a**) and side view of the impeller (**b**); *r* is the distance from the vortex center to the origin of the impeller, *r*m is the distance from the impeller outlet to the origin of the impeller. **Figure 2.** Schematic of flow channel meridian (**a**) and side view of the impeller (**b**); *r* is the distance from the vortex center to the origin of the impeller, *r*m is the distance from the impeller outlet to the origin of the impeller. **Figure 2.** Schematic of flow channel meridian (**a**) and side view of the impeller (**b**); *r* is the distance from the vortex center to the origin of the impeller, *r*m is the distance from the impeller outlet to the origin of the impeller.

Figure 3 shows the distributions of the blade load for different rotational speeds. The blade load distribution curves behaved as an aft-loaded type at each flow rate. At different rotational speeds, the blade load distributions behaved inconsistently. The corresponding blade load value of each flow rate increased accordingly with the increase of rotational speed. Comparing the blade load distributions of the five rotational speeds, it can be found that the load peak area was distributed along *r*/*r*m = 0.75–0.85, and then the blade load started to decrease until the impeller outlet. Figure 3 shows the distributions of the blade load for different rotational speeds. The blade load distribution curves behaved as an aft-loaded type at each flow rate. At different rotational speeds, the blade load distributions behaved inconsistently. The corresponding blade load value of each flow rate increased accordingly with the increase of rotational speed. Comparing the blade load distributions of the five rotational speeds, it can be found that the load peak area was distributed along *r*/*r<sup>m</sup>* = 0.75–0.85, and then the blade load started to decrease until the impeller outlet. Figure 3 shows the distributions of the blade load for different rotational speeds. The blade load distribution curves behaved as an aft-loaded type at each flow rate. At different rotational speeds, the blade load distributions behaved inconsistently. The corresponding blade load value of each flow rate increased accordingly with the increase of rotational speed. Comparing the blade load distributions of the five rotational speeds, it can be found that the load peak area was distributed along *r*/*r*m = 0.75–0.85, and then the blade load started to decrease until the impeller outlet.

**Figure 3.** Blade load distribution curve under the five rotational speeds. **Figure 3.** Blade load distribution curve under the five rotational speeds. **Figure 3.** Blade load distribution curve under the five rotational speeds.

#### **3. Analysis of Internal Flow Patterns 3. Analysis of Internal Flow Patterns**

#### *3.1. Discussion on the Relative Velocity Field 3.1. Discussion on the Relative Velocity Field*

Li et al. [15,29] studied the flow field in the impeller channel at different rotational speeds. At different rotational speeds, their results showed that the developing trend of internal flow pattern with flow rate is consistent. The internal flow pattern behaved steadily near the optimum flow rates, while it vortexes appeared at low flow rates. However, at the same flow rates, the flow field behaved more stably with the increase in the rotational speed. Figure 4 shows the flow streamlines at the flow rate of 0.2 *Q*BEP for five rotational speeds. At the flow rate of 0.2 *Q*BEP, there were two vortexes in the channel for all the five rotational speeds. The clockwise vortex appeared on the suction side, whereas the counterclockwise vortex appeared on the pressure side. The two vortexes occupied almost the entire flow channel. When the rotational speed increased, the zone and intensity of the low-velocity area decreased with the increase of rotational speed. At the rotational speed of 600 r/min, the flow channel was almost completely covered by the low-velocity area. While at the rotational speed of 1200 r/min and 1400 r/min, a secondary vortex appeared at the inlet of suction side. Figure 4 shows that the primary vortex was accompanied by a secondary vortex, which also led to a variation in the radial scale and the position of the vortex center of the primary vortex. On the other hand, for the flow rate of 1.0 *Q*BEP, no separation vortexes were observed on the suction side or pressure side for all the five rotational speeds, and the flow pattern behaved in good condition. Li et al. [15,29] studied the flow field in the impeller channel at different rotational speeds. At different rotational speeds, their results showed that the developing trend of internal flow pattern with flow rate is consistent. The internal flow pattern behaved steadily near the optimum flow rates, while it vortexes appeared at low flow rates. However, at the same flow rates, the flow field behaved more stably with the increase in the rotational speed. Figure 4 shows the flow streamlines at the flow rate of 0.2 *Q*BEP for five rotational speeds. At the flow rate of 0.2 *Q*BEP, there were two vortexes in the channel for all the five rotational speeds. The clockwise vortex appeared on the suction side, whereas the counterclockwise vortex appeared on the pressure side. The two vortexes occupied almost the entire flow channel. When the rotational speed increased, the zone and intensity of the low-velocity area decreased with the increase of rotational speed. At the rotational speed of 600 r/min, the flow channel was almost completely covered by the low-velocity area. While at the rotational speed of 1200 r/min and 1400 r/min, a secondary vortex appeared at the inlet of suction side. Figure 4 shows that the primary vortex was accompanied by a secondary vortex, which also led to a variation in the radial scale and the position of the vortex center of the primary vortex. On the other hand, for the flow rate of 1.0 *Q*BEP, no separation vortexes were observed on the suction side or pressure side for all the five rotational speeds, and the flow pattern behaved in good condition.

**Figure 4.** Flow streamlines for the five rotational speeds at the flow rate of 0.2 *Q*BEP. **Figure 4.** Flow streamlines for the five rotational speeds at the flow rate of 0.2 *Q*BEP.

#### *3.2. Discussion on the Vortex in the Flow Channel 3.2. Discussion on the Vortex in the Flow Channel*

Figure 5 shows the position of the arcs over the vortex center and radial area of the vortex: arc 1 represents the position of the arc that through the center of the clockwise vortex, and arc 2 represents the position of the arc that through the center of the counterclockwise vortex. The radial scale is the length of the meridian between edge 1 and edge 2 [30]. Edge 1 and edge 2 represent the initial and end radial position of the vortexes, Figure 5 shows the position of the arcs over the vortex center and radial area of the vortex: arc 1 represents the position of the arc that through the center of the clockwise vortex, and arc 2 represents the position of the arc that through the center of the counterclockwise vortex. The radial scale is the length of the meridian between edge 1 and edge 2 [30]. Edge 1 and edge 2 represent the initial and end radial position of the vortexes, respectively.

respectively. Figure 6 shows the radial scales of the clockwise vortex and the counterclockwise vortex under the five rotational speeds. It can be seen that the clockwise vortex was larger in scale than the counterclockwise vortex in the radial direction. The clockwise vortex was caused by the increase of the incidence impulse angle, while the counterclockwise vortex was generated by the blocking effect of the clockwise vortex [7]. The clockwise vortex and counterclockwise vortex tended to increase in scale with the decrease of the flow rates. The

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scale of counterclockwise vortex behaved more consistently and increased slowly with the decrease of flow rates. *Machines* **2022**, *10*, x FOR PEER REVIEW 8 of 15

**Figure 5.** Radial scale of the vortex and the arc that through the vortex center. **Figure 5.** Radial scale of the vortex and the arc that through the vortex center. with the decrease of flow rates.

**Figure 6.** Radial scales of the clockwise vortex (**a**) and the counterclockwise vortex (**b**) at the five rotational speeds. **Figure 6.** Radial scales of the clockwise vortex (**a**) and the counterclockwise vortex (**b**) at the five rotational speeds.

**Figure 6.** Radial scales of the clockwise vortex (**a**) and the counterclockwise vortex (**b**) at the five rotational speeds. The relationship between the distance from the vortex center to the impeller rotation axis (*r*) under different flow rates and rotational speeds is shown in Figure 7. The diagram of *r* and *r*m is shown in Figure 2b. For the clockwise vortex, *r* decreased with the decrease The relationship between the distance from the vortex center to the impeller rotation axis (*r*) under different flow rates and rotational speeds is shown in Figure 7. The diagram of *r* and *r*<sup>m</sup> is shown in Figure 2b. For the clockwise vortex, *r* decreased with the decrease in the flow rate, and vice versa for the counterclockwise vortexes. At the same time, it can be seen that the position of clockwise vortex center showed a tendency to move toward the impeller inlet with the decrease of the flow rate at different rotational speeds. For the rotational speed of 600 r/min, 800 r/min, and 1000 r/min, it appears that the position of the clockwise vortex center was closer to the impeller outlet with the decrease of rotational

The relationship between the distance from the vortex center to the impeller rotation axis (*r*) under different flow rates and rotational speeds is shown in Figure 7. The diagram

speed. Compared with other rotational speeds, the position of the clockwise vortex center was closer to the impeller outlet under the rotational speed of 1200 r/min and 1400 r/min. This is due to the appearance of the secondary vortex at the rotational speed of 1200 r/min and 1400 r/min. The position of the counterclockwise vortex center gradually moved toward the impeller inlet with the increase of rotational speed. At the rotational speed of 600 r/min, the counterclockwise vortex center appeared much closer to the impeller outlet than that at other rotational speeds. speed. Compared with other rotational speeds, the position of the clockwise vortex center was closer to the impeller outlet under the rotational speed of 1200 r/min and 1400 r/min. This is due to the appearance of the secondary vortex at the rotational speed of 1200 r/min and 1400 r/min. The position of the counterclockwise vortex center gradually moved toward the impeller inlet with the increase of rotational speed. At the rotational speed of 600 r/min, the counterclockwise vortex center appeared much closer to the impeller outlet than that at other rotational speeds.

in the flow rate, and vice versa for the counterclockwise vortexes. At the same time, it can be seen that the position of clockwise vortex center showed a tendency to move toward the impeller inlet with the decrease of the flow rate at different rotational speeds. For the rotational speed of 600 r/min, 800 r/min, and 1000 r/min, it appears that the position of the clockwise vortex center was closer to the impeller outlet with the decrease of rotational

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In this paper, we focused on the effect of rotational speeds and flow rates on the internal flow pattern inside the impeller. Additionally, we concentrated on the analysis of the scale size of the vortex and position of the vortex center in the flow channel, and specifically analyzed how vortexes affect the blade load distribution. In this paper, we focused on the effect of rotational speeds and flow rates on the internal flow pattern inside the impeller. Additionally, we concentrated on the analysis of the scale size of the vortex and position of the vortex center in the flow channel, and specifically analyzed how vortexes affect the blade load distribution.

#### **4. Effect of Flow Pattern in the Flow Channel on the Blade Load Distribution 4. Effect of Flow Pattern in the Flow Channel on the Blade Load Distribution**

#### *4.1. Correlation of Internal Flow Pattern and Blade Load*

*4.1. Correlation of Internal Flow Pattern and Blade Load*  From the above analysis, it can be seen that the internal flow patterns were basically the same with the variation of flow rates for different rotational speeds. Therefore, the flow patterns at the rotational speed of 800 were selected as examples for analysis. The correlation between the internal flow pattern and the blade load distribution is shown in Figure 8. For the flow rates of 0.2 *Q*BEP, 0.4 *Q*BEP, and 0.6 *Q*BEP, the distribution curves can be divided into two sections: the first section of the blade load distribution curves was in the rising stage, which manifested as a slow increase, and then it started to reach the highest point linearly and rapidly. The second section decreased until the outlet. While, for the From the above analysis, it can be seen that the internal flow patterns were basically the same with the variation of flow rates for different rotational speeds. Therefore, the flow patterns at the rotational speed of 800 were selected as examples for analysis. The correlation between the internal flow pattern and the blade load distribution is shown in Figure 8. For the flow rates of 0.2 *Q*BEP, 0.4 *Q*BEP, and 0.6 *Q*BEP, the distribution curves can be divided into two sections: the first section of the blade load distribution curves was in the rising stage, which manifested as a slow increase, and then it started to reach the highest point linearly and rapidly. The second section decreased until the outlet. While, for the flow rates of 0.8 *Q*BEP to 1.2 *Q*BEP, the first section of the blade load curves increased slowly with a smaller slope, and the second section of the blade load curves decreased slightly until the outlet. This is also consistent with the conclusion mentioned in the paper by Li et al. [24]: the higher slope of the load curve, the lower pump efficiency. This indicates that the flow blockage and hydraulic losses in the impeller channel will increase when the slope of the blade load curve increases.

flow rates of 0.8 *Q*BEP to 1.2 *Q*BEP, the first section of the blade load curves increased slowly with a smaller slope, and the second section of the blade load curves decreased slightly until the outlet. This is also consistent with the conclusion mentioned in the paper by Li et al. [24]: the higher slope of the load curve, the lower pump efficiency. This indicates that the flow blockage and hydraulic losses in the impeller channel will increase when the

To analyze the correlation between the flow pattern inside the impeller and the blade load, the location of low-velocity-zone of pressure side, the vortex leading edge (the front point) and the vortex center (the latter point) are also shown in Figure 8. It can be seen that for the flow rates of 0.8 *Q*BEP to 1.2 *Q*BEP, the position where the blade load distribution curve began to increase rapidly coincided with the position where the low-velocity-zone started to appear on the pressure side. The peak position coincided with the end of the low-velocity zone. For the flow rates from 0.2 *Q*BEP to 0.6 *Q*BEP, the position of the blade load distribution curve began to increase rapidly, coincided with the initial position of the clockwise vortex on the suction side, and the peak position coincided with the position of core area of the clockwise vortex. The position where the blade load starts to decrease coincided with the initial position of the counterclockwise vortex on the pressure side. The blade load kept increasing when the clockwise vortex appeared. When the counterclockwise vortex appeared, the blade load increased a little, and at this time the strength of the clockwise vortex was greater than that of the counterclockwise vortex. When the strength of counterclockwise vortex was greater than that of the clockwise vortex, the blade load curve started to decrease rapidly at core area of the counterclockwise vortex. It is inferred that the vortex on the suction side increases the blade load, while the vortex

slope of the blade load curve increases.

on the pressure side decreases the blade load.

**Figure 8.** Blade load distribution and the position of the vortexes under different flow rates. For 0.2– 0.6 *Q*BEP: the front point is the position of the leading edge of vortex, the back point is the position of the vortex center. For 0.8–1.2 *Q*BEP: the two points are the position of the leading edge and back edge of low-velocity-zone in the pressure side, respectively. **Figure 8.** Blade load distribution and the position of the vortexes under different flow rates. For 0.2–0.6 *Q*BEP: the front point is the position of the leading edge of vortex, the back point is the position of the vortex center. For 0.8–1.2 *Q*BEP: the two points are the position of the leading edge and back edge of low-velocity-zone in the pressure side, respectively.

Combining Equations (1)–(4), the blade load is related with the absolute flow angle. Therefore, the absolute flow angle was used as an intermediate medium to correlate the flow pattern with the blade load. Figure 9 shows the mean absolute flow angle distribution along the meridian. It can be seen that the overall trends of the mean absolute flow angle distribution behaved similarly under different rotational speeds. When the rotational speed increased, the curve became steeper under the same flow rates. For the flow rates of 0.2 *Q*BEP to 0.6 *Q*BEP, the mean absolute flow angle curve dropped slowly from the To analyze the correlation between the flow pattern inside the impeller and the blade load, the location of low-velocity-zone of pressure side, the vortex leading edge (the front point) and the vortex center (the latter point) are also shown in Figure 8. It can be seen that for the flow rates of 0.8 *Q*BEP to 1.2 *Q*BEP, the position where the blade load distribution curve began to increase rapidly coincided with the position where the low-velocity-zone started to appear on the pressure side. The peak position coincided with the end of the low-velocity zone. For the flow rates from 0.2 *Q*BEP to 0.6 *Q*BEP, the position of the blade load distribution curve began to increase rapidly, coincided with the initial position of the clockwise vortex on the suction side, and the peak position coincided with the position of core area of the clockwise vortex. The position where the blade load starts to decrease coincided with the initial position of the counterclockwise vortex on the pressure side. The blade load kept increasing when the clockwise vortex appeared. When the counterclockwise vortex appeared, the blade load increased a little, and at this time the strength of the clockwise vortex was greater than that of the counterclockwise vortex. When the strength of counterclockwise vortex was greater than that of the clockwise vortex, the blade load curve started to decrease rapidly at core area of the counterclockwise vortex. It is inferred that the vortex on the suction side increases the blade load, while the vortex on the pressure side decreases the blade load.

Combining Equations (1)–(4), the blade load is related with the absolute flow angle. Therefore, the absolute flow angle was used as an intermediate medium to correlate the flow pattern with the blade load. Figure 9 shows the mean absolute flow angle distribution along the meridian. It can be seen that the overall trends of the mean absolute flow angle distribution behaved similarly under different rotational speeds. When the rotational speed increased, the curve became steeper under the same flow rates. For the flow rates of 0.2 *Q*BEP to 0.6 *Q*BEP, the mean absolute flow angle curve dropped slowly from the inlet to the outlet. After the position of *r*/*r<sup>m</sup>* = 0.8, the mean absolute flow angle's behavior was almost unchanged. For the flow rates from 0.8 *Q*BEP to 1.2 *Q*BEP, the mean absolute flow angle curve decreased more rapidly than that at the flow rates from 0.2 *Q*BEP to 0.6 *Q*BEP. In addition, combined with the analysis of the blade load curve, it can be seen that the blade load curve had an upward trend in the first half of the curve, which was opposite to the distribution trend of the mean absolute flow angle. This is because that as the decrease of absolute flow angle *α*, the *tanα* decreases, resulting in the increase of *V<sup>θ</sup>* , which coincides

with Equation (1). The mean absolute flow angle decreased slowly near the trailing edge of the blade. As a result, the mean absolute flow angle had little effect on the blade load near the impeller outlet. Therefore, the blade load was mainly affected by the variation of the blade profile near the impeller outlet. cides with Equation (1). The mean absolute flow angle decreased slowly near the trailing edge of the blade. As a result, the mean absolute flow angle had little effect on the blade load near the impeller outlet. Therefore, the blade load was mainly affected by the variation of the blade profile near the impeller outlet.

decreases, resulting in the increase of *V*

θ

, which coin-

inlet to the outlet. After the position of *r*/*r*m = 0.8, the mean absolute flow angle's behavior was almost unchanged. For the flow rates from 0.8 *Q*BEP to 1.2 *Q*BEP, the mean absolute flow angle curve decreased more rapidly than that at the flow rates from 0.2 *Q*BEP to 0.6 *Q*BEP. In addition, combined with the analysis of the blade load curve, it can be seen that the blade load curve had an upward trend in the first half of the curve, which was opposite to the distribution trend of the mean absolute flow angle. This is because that as the decrease of

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absolute flow angle

α

, the *tan*

α

**Figure 9.** The mean absolute flow angle distribution curve under the five rotational speeds. **Figure 9.** The mean absolute flow angle distribution curve under the five rotational speeds.

In addition, it was found that the blade load distribution trends for some flow rates and some rotational speeds were different from the overall distribution trend. At the rotational speeds of 600 r/min and 1400 r/min, the blade load distribution curves for the flow rate of 0.8 *Q*BEP were closer to the flow rate of 0.6 *Q*BEP compared with the situation at other rotational speeds. For the rotational speeds of 1000 r/min and 1200 r/min, at flow rate of 0.6 *Q*BEP, the development trend of the first half blade load distribution curves were consistent with that at the flow rate of 0.2 *Q*BEP and 0.4 *Q*BEP. However, the latter half of the curves began to move away from that at the flow rates of 0.2 *Q*BEP and 0.4 *Q*BEP, and the positions of the maximum values were significantly closer to the impeller outlet. The specific values are shown in Table 3. In addition, it was found that the blade load distribution trends for some flow rates and some rotational speeds were different from the overall distribution trend. At the rotational speeds of 600 r/min and 1400 r/min, the blade load distribution curves for the flow rate of 0.8 *Q*BEP were closer to the flow rate of 0.6 *Q*BEP compared with the situation at other rotational speeds. For the rotational speeds of 1000 r/min and 1200 r/min, at flow rate of 0.6 *Q*BEP, the development trend of the first half blade load distribution curves were consistent with that at the flow rate of 0.2 *Q*BEP and 0.4 *Q*BEP. However, the latter half of the curves began to move away from that at the flow rates of 0.2 *Q*BEP and 0.4 *Q*BEP, and the positions of the maximum values were significantly closer to the impeller outlet. The specific values are shown in Table 3.

#### *4.2. Reasons for the Difference in Blade Load*

The blade load is related with the absolute flow angle, which related to the flow pattern in turn. Therefore, it is necessary to describe how the vortexes affect the distribution of the absolute flow angle. Figure 10a shows the distributions of the absolute flow angle along arc 1 (across the position of the clockwise vortex center) at the flow rates of 0.2 *Q*BEP, 0.4 *Q*BEP, 0.6 *Q*BEP, and 1.0 *Q*BEP under the rotational speed of 1000 r/min, respectively. Arc 1 and arc 2 are shown in Figure 5. When the vortex appears on the suction side (at flow rates of 0.2 *Q*BEP, 0.4 *Q*BEP, 0.6 *Q*BEP), the absolute flow angle curves decreased slightly at first, then increased rapidly, and then decreased until the outlet. An interesting phenomenon was also found: the first half of the curve basically coincided with the spanwise area of the clockwise vortex. At the flow rate of 1.0 *Q*BEP, it can be seen that the absolute flow angle distribution curve was much flatter. In addition, when the vortex appeared, the sharp increase and decrease process occurred for the absolute flow angle. Although the maximum value of the absolute flow angle was larger than that without vortex, the growth gradient was so large that the average value was smaller than that without vortex. Therefore, the overall blade load was increased. Figure 10b shows the distribution of the absolute flow angle along arc 2 (across the position of the counterclockwise vortex center). When the counterclockwise vortex appeared (0.2 *Q*BEP, 0.4 *Q*BEP), the absolute flow angle fluctuated along the curve at first. Then, in the region corresponding to the counterclockwise vortex, it became negative, decreased sharply, and rose again after reaching the minimum value. At the flow rate of 0.6 *Q*BEP, only the clockwise vortex appeared, and the position of arc 2 could not be determined. For comparison, the positions of arc 2 at the flow rates of 0.2 *Q*BEP and 0.4 *Q*BEP were selected for the flow rate of 0.6 *Q*BEP. It can be seen that their distribution trend was consistent with that along arc 1 when the clockwise vortex appeared. When the flow rate was 1.0 *Q*BEP, the absolute flow angle distribution changed a little, and the value remained around zero. From the above analysis, it can be seen that the appearance of the counterclockwise vortex causes the absolute flow angle to become negative and make the overall blade load become negative, resulting in the unloading of the blade load.

Figure 11 shows the flow patterns of the flow rates of 0.4 *Q*BEP, 0.6 *Q*BEP, and 0.8 *Q*BEP at the rotational speeds of 600 r/min, 1000 r/min, 1200 r/min, and 1400 r/min. At the rotational speeds of 600 r/min and 1400 r/min, for the flow rate of 0.8 *Q*BEP, it can be seen that the clockwise vortex appeared, which caused the blade load distribution curve to be closer to the curve at flow rate of 0.6 *Q*BEP. At the rotational speeds of 1000 r/min and 1400 r/min, only the clockwise vortex appeared, while at the flow rate of 0.4 *Q*BEP, both vortexes appeared. For the flow rate of 0.6 *Q*BEP, the unloading effect of the counterclockwise vortex disappeared, which made the blade load curve smoother and the length of its growth section longer than that at the flow rate of 0.4 *Q*BEP.

According to the previous analysis, the vortex in the channel affected the absolute flow angle distribution and redistributed the blade load. The clockwise vortex made the position of maximum blade load move toward the blade outlet. It also made the value of the maximum blade load become larger. The counterclockwise vortex made the position of maximum blade load move toward the blade inlet, and caused the unloading of the blade load.

r/min. Suction side (**a**) and pressure side (**b**).

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**Figure 10.** Absolute flow angle distributions along arc 1 and arc 2 under the rotational speed of 1000 r/min. Suction side (**a**) and pressure side (**b**). **Figure 10.** Absolute flow angle distributions along arc 1 and arc 2 under the rotational speed of 1000 r/min. Suction side (**a**) and pressure side (**b**). **Figure 10.** Absolute flow angle distributions along arc 1 and arc 2 under the rotational speed of 1000

According to the previous analysis, the vortex in the channel affected the absolute flow angle distribution and redistributed the blade load. The clockwise vortex made the position of maximum blade load move toward the blade outlet. It also made the value of the maximum blade load become larger. The counterclockwise vortex made the position

According to the previous analysis, the vortex in the channel affected the absolute flow angle distribution and redistributed the blade load. The clockwise vortex made the position of maximum blade load move toward the blade outlet. It also made the value of the maximum blade load become larger. The counterclockwise vortex made the position

**Figure 11.** Flow streamlines. **Figure 11.** Flow streamlines.

#### **5. Conclusions**

In this paper, the blade load distribution in the pump was obtained through a PIV experiment. The characteristics of blade load were analyzed, and the specific correlation between blade load distribution and internal flow pattern were discussed.

**Author Contributions:** Conceptualization, B.C. and Z.Z.; methodology, X.L.; validation, B.C. and Z.Z.; investigation, X.C.; data curation, X.C.; writing—original draft preparation, X.C.; writing review and editing, B.C.; visualization, X.C.; funding acquisition, B.C., Z.Z. and X.L. All authors have read and agreed to the published version of the manuscript.

**Funding:** Project supported by the National Natural Science Foundation of China (Grant Nos. 51806197, 51976202), the Natural Science Foundation of Zhejiang Province (Grant No. LY20E060006) and the Key Research and Development Program of Zhejiang Province (Grant No. 2021C05006), Top-notch Talent Support Program of Zhejiang Province (2019R51002), the Fundamental Research Funds of Zhejiang Sci-Tech University (2021Q017).

**Informed Consent Statement:** Informed consent was obtained from all individual participants included in the study.

**Data Availability Statement:** The data used to support the findings of this study are included within the article.

**Conflicts of Interest:** The authors declare that they have no conflict of interest.

**Ethical Approval:** This article does not contain any studies with human participants or animals performed by any of the authors.

## **References**