Research and application of wide and efficient hydraulic design in centrifugal pump

Through the analysis of the wide and efficient mechanism of centrifugal pump, this paper finds that the shape of the blade inlet angle to the outlet angle distribution line of the centrifugal pump impeller within the blade wrapping angle is the key factor determining the width and efficiency. Two sets of blade thickening laws are adopted, namely the equal thickness type and the NACA1933 air-foil. Tests have shown that the hill-type is superior Linear scheme, NACA1933 air-foil blades are superior to equal-thickness blades. The hill-type solution achieves the performance indicators of wide and efficient centrifugal pumps. Firstly, the angle distribution line of the inlet angle to the outlet angle of the pump is designed, and the calculus equation is applied to solve the conformal transformation line in the square grid, so as to produce a wide and efficient centrifugal pump impeller profile. The design method is applied to different specific speed centrifugal pumps, and the test results show that this method achieves the purpose of wide and efficient centrifugal pump, reaches the international advanced level, and provides a new method for the hydraulic design of centrifugal pump.


Introduction
As we all know, vane -type centrifugal pump is the largest amount of water pump, and is widely used in various sectors of the national economy such as water supply and drainage and agricultural engineering, solid particles and liquid transportation, petroleum and chemical engineering, aerospace, navigation, energy and vehicle engineering [1].
In recent years, the rapid development of industrial technology has put forward higher requirements for centrifugal pumps.From the point of view of energy saving, pumps and fans have great potential for development.Pumps and fans are the most energy-consuming general machinery, such as the national power generation as 100%, the power consumption of pumps and fans accounted for about 30%; In the power plant, if the power generation of the power plant is 100%, the pump and fan consume more than 6.0% of the electricity; If the power consumption of the power plant is taken as 100%, the pump and fan will account for about 80% of the power consumption of the power plant, of which the pump accounts for 60% and the fan accounts for 20% [2] .
According to data released by the National Energy Administration on July 31, China surpassed the United States for the first time since 2013 to become the world's largest installed power generation equipment capacity.In 2022, the total installed capacity of China's power generation equipment exceeded 2.564 billion kilowatts, and in the same year, the installed capacity of the United States was 1.257 billion kilowatts, and the installed capacity of the United States was only 49.03% of China's, falling below the 50% mark.By the end of June 2023, the installed capacity of power generation equipment in China reached 2.71 billion kilowatts.
It can be seen that the research on the wide and efficient hydraulic design of centrifugal pumps will have great positive significance for the national energy strategy and the realization of the dual carbon goal of "Peak Carbon, Carbon Neutral".
Among them, the definition of the efficiency of centrifugal pump width and efficiency, the analysis of the mechanism of centrifugal pump width and efficiency and the design method of centrifugal pump wide and efficient are undoubtedly beneficial attempts for the research field of hydraulic design of centrifugal pump wide and high efficiency.

Centrifugal pump wide and efficient efficiency definition
In the context of the current energy shortage in the world, wide and efficient fluid machinery is a hot topic in the field of international fluid machinery research in recent years [3]- [9].On 26 June 2012, the European Union published Regulation "on Eco-design requirements for water pumps", No547/2012, to implement Directive 2009/125/EC of the European Parliament and of the Council (ERP Directive).The Minimum Efficiency requirements and information requirements for rotary power pumps are listed in Annex II of the Regulation, MEI (Minimum Efficiency Index).
The minimum energy efficiency index of a pump is independent of the size of the pump.A parameter used to measure the hydraulic efficiency of a pump at high efficiency points, low flow points (75% of high efficiency point flow) and high flow points (110% of high efficiency point flow).Pump minimum energy efficiency index regulations require that the impact of products on the ecological environment should be improved without excessive increase in costs, and the standards introduced this time are aimed at the hydraulic efficiency of pumps [10].
Specifically, the working range of the pump is required to be in high efficiency conditions, that is, based on the corresponding flow QBEP of the highest efficiency point of the pump, 0.75 times of the QBEP is the small flow point QPL, 1.1 times of the QBEP is the large flow point QOL, and the efficiency values of the above three points of the pump are assessed.The pump that requires its EU MEI standard value to reach or exceed 0.7 is the benchmark in the industry, that is, the wide and efficient product, and such a pump is in the top 30% of the industry's efficiency in the EU standard pump.
At present, there is no standard for wide and efficient centrifugal pumps in China, we refer to the minimum energy efficiency index of the EU MEI standard (Minimum Efficiency Index) of water pumps (see Figure 1, Figure 2), and propose the quantitative formula of the wide and high efficiency index of centrifugal pump as follows:   Specifically, if the measured efficiency of the above three points of the centrifugal pump meets the formula (1)(2)(3), the pump meets the requirements of wide and efficient centrifugal pump; If the measured average efficiency of the centrifugal pump meets formula (4), the pump meets the requirements of the centrifugal pump width, high efficiency and medium width.Of course, if the measured efficiency and average efficiency of the three points of the centrifugal pump meet the formulas (1) to (4) at the same time, the pump meets the requirements of the width and efficiency of the centrifugal pump.

Analysis of the wide and efficient mechanism of centrifugal pump
We think that the wide and high efficiency of centrifugal pump are both related and different concepts.The wide efficiency of the centrifugal pump refers to the ability of the centrifugal pump to adapt to the corresponding flow changes within its working range (0.75 times the rated working point Qn is the small flow point QPL, and 1.2 times the rated working point Qn is the large flow point QOL).
Specifically, the average efficiency value of its working range is within 3% of the efficiency of the rated working condition point of the pump (usually the highest efficiency point); High efficiency means that in its working range, the efficiency of the rated working condition point of the pump is higher than the national energy saving standard (GB 19762-2007 water centrifugal pump energy efficiency limit value and energy saving evaluation value), and the efficiency of each working condition point of the working range is different from the efficiency of the rated working condition point of the pump in the range of 6%.
How to be wide in centrifugal pumps wide and efficient?That is, the centrifugal pump is required to have the ability to adapt to the corresponding flow change within its working range.We know that the centrifugal pump does not have the ability to adapt to the corresponding flow change by adjusting the blade position like the axial flow slurry hydraulic machinery (Figure 3).We can see from the impeller speed triangle in Figure 4 and Figure 5.
We can adjust the inlet installation Angle of the centrifugal pump blade and its adjacent part of the blade to maximize the adaptation to the corresponding flow changes.Specifically, we can calculate the inlet installation Angle of 0.90 times Qn as a small flow design point and 1.1 times Qn as a large flow design point within the working range of the centrifugal pump.The inlet installation Angle of the centrifugal pump blade and its adjacent part of the blade shows a gradual rising distribution trend with the increase of flow rate.In this way, the inlet edge (section) of the blade can achieve the purpose of adapting to the corresponding flow change to the maximum extent.How to achieve high efficiency in wide and high efficiency of centrifugal pump?That is, the centrifugal pump is required to have the ability to adapt to the corresponding flow rate and high efficiency within its working range.We know that the high efficiency of the centrifugal pump can be achieved by the installation Angle of the inlet is greater than the installation Angle of the outlet, that is, the confluent transformation line is convex [11].
We can understand that the width of the centrifugal pump is determined by the maximum Angle of the inlet Angle of the impeller blade to the Angle distribution line (the corresponding concave section of the conformal transformation line); The high efficiency in the wide efficiency of the centrifugal pump is determined by the angular maximum of the distribution line to the outlet Angle (corresponding conformal transformation line kyphosis).
To sum up, in order to achieve the purpose of wide and efficient centrifugal pump, the shape of the distribution line from the inlet Angle to the outlet Angle of the centrifugal pump impeller in the range of blade envelope Angle is a parabola with maximum value (commonly known as hill type), and its corresponding conformal transformation line is a concave and kyphotic S-shaped curve (see Figure 6).

Wide and efficient design method of centrifugal pump
The design adopts the idea of blade corner zoning design, and the blade corner range is divided into three areas, namely non-control area, transition area and control area, and each area performs its own duties.Among them, the blade non-control area, that is, the completely unconstrained part between the two blades, is equal to 360 degrees divided by 2 times the number of blades Z, that is, 360/2Z; The blade transition zone is the area between the non-control area of the blade and the blade control area, which is equal to 360 degrees divided by 2 times the number of blades Z, that is, 360/2Z; The blade control area is the part of the complete constraint between the two blades, which is equal to the optimal wrapping angle of the blade [12] minus 360 degrees divided by the number of blades Z, that is, ΦBEP-360/Z; The calculation formula of the optimal wrapping angle ΦBEP of the blade is as follows (see Figure 6).ΦBEP=(0.8-0.9)L/tan(∠G) ∠G= (∠A+∠B)/2 (6) Formula: ∠G-chord angle; ∠A -blade inlet placement angle; ∠ B -blade outlet placement angle; L-Square grid inlet edge equinox length value In order to achieve the purpose of wide and efficient centrifugal pump, the shape of the blade inlet angle to the outlet angle distribution line of the centrifugal pump impeller within the blade wrapping angle is a parabola-like with maximum value, where the angular maximum value of the angular distribution line is what we call the turning angle, which requires it to fall within the blade transition zone.(see Figure 7) At the same time, by integrating the shape of the blade inlet angle to outlet angle distribution line of the centrifugal pump impeller within the blade wrapping angle range, the angle trigonometric tangent curve of the parabola-like angle with maximum value is calculated, and the conformal transformation line of the corresponding point is obtained so that the corresponding conformal transformation line is a anterior-concave and posterior-convex S-type curve.
Among them, the concave and kyphosphocon type S-type curve turning point of the conformal transformation line corresponds to the angular maximum value of its angular distribution line, that is, the turning angle position.
It should be emphasized here that this distribution line of the leaf inlet angle to the exit angle parabola with a maximum value is a function of the wrapping angle, and different wrapping angles will have different inlet angles to the outlet angle distribution line of the parabola-like line, but there is always a wrapping angle will correspond to an optimal shape of the inlet angle to the outlet angle distribution line, we call this wrapping angle the optimal wrapping angle of the blade.
The blade inlet angle to outlet angle distribution line can be constructed using Bernstein polynomial [13], although engineers prefer to use what is called the Bé zier curve [14], so that the higher-order Bé zier curve can ensure the smoothness of the blade inlet angle to outlet angle distribution line and its corresponding conformal transformation line.

Design example of wide and efficient centrifugal pump
The design method is applied to low specific speed centrifugal pump (Ns=67), medium specific speed centrifugal pump (Ns=120) and high specific speed centrifugal pump (Ns=172), the test results show that this impeller design method achieves the purpose of wide and efficient centrifugal pump and reaches the international advanced level.
In this test, a total of 6 impellers, namely impeller 0 to impeller 5.Among them, No. 0 impeller and No. 1 impeller are the prototype impellers of the pump, and the model drawings come from an internationally renowned hydraulic machinery research center, the difference between the two is that the casting of No. 0 impeller is stainless steel precision casting (see Figure 10), and the casting of No. 1 impeller is gray iron sand casting (see Figure 11).
Among them, the No. 2 impeller and No. 3 impeller are newly designed angular distribution lines are linear impellers, both of which are cast gray iron sand casting, the difference between the two is that the thickening law of the No. 2 impeller is of equal thickness, and the thickening law of the No. 3 impeller is the NACA1933 airfoil (see Figure 12).Among them, No. 4 impeller and No. 5 impeller are newly designed angular distribution lines of hill-type impellers, both casting are gray iron sand casting, the difference between the two impellers is that the No. 4 impeller thickening law is equal thickness, and the No. 5 impeller thickening law is NACA1933 airfoil type (see Figure 13).The pump test of the above 6 impellers has 5 common characteristics: the same pump body, the same motor, the same performance test bench, the same batch of testers, and the same time period.The measured results are shown in Table 1   From the summary table 1 and table 2 of the measured results, we can see that the difference between the prototype impeller 0 impeller and the No. 1 impeller of the pump is the material and casting method, which is specifically manifested in the surface roughness of the two, and the rated point efficiency difference is 1.89%, which is in line with the relevant literature.The impeller roughness affects the pump efficiency by about 2%.
The material and casting method of the No. 1 impeller to No. 5 impeller are exactly the same, so we can reasonably assume that if the impeller tested by the No. 1 impeller to No. 5 model adopts the same material and casting method as the factory product No. 0 impeller, its efficiency will be greatly improved, so its efficiency correction is shown in Table 2.It can be seen that the efficiency of No. 4 and No. 5 hill types (that is, parabolas with maximum values in the angular distribution line) is higher than that of No. 2 and No. 3 linear types (angular distribution lines are linear type): the efficiency of the thickened impellers of No. 3 and No. 5 airfoils is higher than that of their corresponding No. 2 and No. 4 thickened impellers.Because the thickened impellers of the No. 3 and No. 5 airfoils generate a certain lift force, that is, generate a positive additional torque, the efficiency is high.
The rated point efficiency of a total of 4 impellers in the two schemes designed this time is higher than the efficiency of the national energy-saving standard, and the rated point efficiency of the 4 impellers has reached the high efficiency index.Let's take a look at the efficiency of each working point in the pump's operating range (0.75 times the rated working condition point Qn is a small flow point QPL, and its 1.2 times rated working condition point Qn is a large flow point QOL), see Tab. 3 and Tab.4 of the measured results.
A summary of the measured results can be seen in Table 3 and Table 4.
The average efficiency of the linear No. 2 and No. 3 impellers in their working range is 85.70% and 86.36%, and the efficiency of the rated working point of the pump is 87.63% and 88.07%, and the difference ratio is within the range of 3%.The average efficiency of hill-type No. 4 and No. 5 impellers in their working range is 86.66% and 87.24%, and the efficiency of the rated working point of the pump is 88.10% and 89.46%, and the difference ratio is within the range of 3%.All 4 impellers meet the width requirements.
The efficiency of the linear No. 2 and No. 3 impellers at their rated working point Qn of 1.2 times, that is, the large flow point QOL, is 72.75% and 76.84%, and the efficiency of the rated working point of the pump is 87.63% and 88.07%, which is outside the range of 6%.It does not meet the requirements of high efficiency at each working point.
The efficiency of the hill-type No. 4 and No. 5 impellers at their rated working point Qn of 1.2 times, that is, the efficiency of the large flow point QOL is 84.40% and 87.09%, and the efficiency of the rated working point of the pump is 88.10% and 89.46%, and the difference ratio is within the range of 6%.Meet the requirements of high efficiency at various working points.
In summary, the hill type No. 4 and No. 5 impellers meet the requirements of centrifugal pump width and efficiency, and the linear type No. 2 and No. 3 impellers only meet the requirements of width but not high efficiency, it can be seen that the shape of the blade inlet angle to the outlet angle distribution line of the centrifugal pump impeller is the key factor determining the width and efficiency of the centrifugal pump.
Before the above test, we performed CFD analysis on the newly designed 4 impellers and calculated the results as follows (see Figures 14~17):     Efficiency of CFD sorting of visible centrifugal pump impeller: 4 > 3 > 5 > 2; measured efficiency order: 5 > 4 > 3 > 2. Although CFD and measured efficiency are slightly different, they are generally the same.
In addition, according to the relevant literature, we completed the schematic diagram of the impeller area ratio from No. 1 to No. 5 (see Figure 18, Figure 19).Among them, S0 -impeller suction port area, S1 -blade inlet area, S2 -blade outlet area, S3volute throat area.The upper limit, middle limit and lower limit are determined by the specific speed and are related to the limit line of cavitation, balance and efficiency, and the relevant ratio of pump impeller and volute throat is required within their constraints.
We can see from Figure 18 and Figure 19 that the area ratio of No. 4 and No. 5 impellers is better than the area ratio of No. 2 and No. 3 impellers, and the ratio of volute throat to impeller outlet area of No. 5 impeller is better than that of No. 4 impeller.

Conclusion
1.The shape of the blade inlet angle to the outlet angle distribution line of the centrifugal pump impeller is a key factor determining the width and efficiency of the centrifugal pump, in other words: the shape of the angle distribution line is the genetic code of the width and efficiency of the centrifugal pump; 2. The basic law of wide and efficient centrifugal pump is that the shape of the distribution line from the inlet angle to the outlet angle of the blade is a parabola-like with maximum value (we commonly known as the hill type), and its corresponding conformal transformation line is a concave and convex S-type curve; 3. The efficiency of thickening the impeller blade using NACA1933 airfoil is about 1% higher than that of its corresponding equal thickness thickening, which is caused by the additional lift generated by the blade using the NACA1933 airfoil, so the stronger the outer diameter of the impeller, the greater the torque, and the more significant the efficiency improvement.4. CFD analysis calculation and impeller area ratio schematic diagram can play a role in assisting in judging impeller performance; 5.The test shows that this design method provides a new method for the hydraulic design of wide and efficient centrifugal pump impeller, and opens up a new way.

Figure 4 .
Figure 4. Velocity triangle at any point in the impeller.Figure 5. Triangular vector diagram of impeller.

Figure 5 .
Figure 4. Velocity triangle at any point in the impeller.Figure 5. Triangular vector diagram of impeller.

Figure 6 .
Figure 6.Centrifugal pump wide and efficient angle distribution line and square grid blade conformal conversion line.

Figure 7 . 7 Figure 8 .
Figure 7. Partition diagram of wide and efficient angle distribution line of centrifugal pump.

Figure 10 .
Figure 10.The casting of the No. 0 impeller is stainless steel precision casting.

Figure 11 .
Figure 11.The casting of impellers No. 1 to 5 is gray iron sand casting.
C1: Hill + Airfoil ■ Note: The above 6 impellers have 5 common features: the same pump body, the same motor, the same sand-casting process, the same performance test bench, the same batch of testers, and the same time period.

Table 1 .
and Table2as below: Straight / Hill 1.0/5.0version models and prototype pump test results summary.

Table 2 .
Straight/Hill model and prototype pump test results.