ANALYSIS OF THE IMPACT OF IMPELLER OUTLET WIDTH ON THE STEEPNESS OF PRESSURE CHARACTERISTIC

V . M i l t y k h * E-mail: bohdanovych-viktoriya@yandex.ru M . S o t n y k Doctor of technical sciences, Associate Professor* E-mail: nsotnik@mail.ru *Department of Applied Hydroaeromechanics Sumy State University Rymskoho-Korsakova str., 2, Sumy, Ukraine, 40000 Досліджено робочий процес коліс двостороннього входу відцентрових насосів та виявлено залежність крутизни напірної характеристики від відносної ширини колеса. При числовому моделюванні змінювалася лише ширина робочого колеса на виході, а інші геометричні розміри колеса лишалися без зміни. Результати досліджень важливі при проектуванні робочих коліс двостороннього входу з наперед визначеною крутизною напірної характеристики Ключові слова: напірна характеристика, насосна станція, робоче колесо, ширина виходу, відцентровий насос


Introduction
In the design of pumping stations of industrial and drinking water supply, pumps are chosen based on the maximal water consumption considering its gradual increase in the future.The most common of the pumping units that work on such pumping stations are the pumps with double-entry impellers with estimated supply from 1250 m 3 /h to 6300 m 3 /h.At modern rates of slowing down production and, as a result, water consumption, when using inefficient methods of regulation, according to [1] about 60 % of the pumping stations work with an efficiency (Eff) of only 10-40 %.It should be noted that 75 % of the pumping systems have overestimated values of pressure higher than by 20 % [2].
In the efforts to increase the efficiency of the pumping stations (PS) they resort to the following methods of regulating the pumps performance in the system: using a variable-frequency drive (VFD), the use of the "on-off" method of the pumps working in parallel and their combinations.
For the majority of cases, functioning of the pump systems at variable operating modes of VFD is one of the most efficient ways to regulate the pump unit.However, the changes in the parameters of the pump can be obtained by replacing the impeller, designed for the new parameters of the functioning of the network and in this case it is necessary to take into account the change in the steepness of pressure characteristics of the pump.Also, when applying a particular method of regulating the supply system, there are some requirements for pressure characteristics of the pump.Thus, at VFD Q-H characteristic should have the largest possible steepness, and during the work of pumps in parallel, their characteristics should be maximally sloping [3].
In the well-known scientific literary sources [4,5] it is stated that the steepness of the pressure characteristic is one of the most important parameters of the pump, which must be taken into account when designing the impellers of pumps or PS control method.
The most important task is to increase efficiency of the PS performance, to reduce their energy consumption, provided they meet the requirements of consumers.Since, for the most part, this is due to a mismatch of the characteristics of a pump and the chosen way of regulating, the total efficiency of PS remains quite low, and actual issue is to define interrelations between geometric dimensions of the impeller of the centrifugal pumps and the parameters of their pressure characteristics.

Analysis of scientific literature and the problem statement
The impeller outlet width is one of the geometric parameters that defines the structure of the liquid flow in impeller, the magnitude of hydraulic losses, in particular, and the characteristics of impeller (H=f(Q), Eff=f(Q), P=f(Q)) as a whole.
According to the papers [6,7], a steepness is measured through tgφ H , the tangent of the angle to the pressure characteristic, which is determined by the following formula: where q р is the consumable option in the reference point: where u 2 is the circular velocity: where v m2 is the meridian component of the absolute velocity: where Q рк is the impeller supply.
Converting the equation ( 2) with ( 3) and ( 4) we have: As can be seen from ( 5) and ( 1), a consumable parameter, and hence the steepness of the pressure characteristic in the reference point, depends only on the external diameter of impeller D 2 , the impeller outlet width b 2 , the angle of the blade at the output of impeller β 2 at the constant frequency of rotations of the rotor n.Moreover, the largest impact is produced exactly by β 2 , b 2 .The parameters q р and β 2 , b 2 are inversely dependent: with increasing values of b 2 and β 2 , the value q р decreases and vice versa.That is, given the equation (1), at increasing β 2 or b 2 , the characteristic grows more sloping.
Thus, taking the equation of continuity as the basis, the following dependence was obtained: By supplementing dependence (6) with the results of statistical analysis, the aforementioned authors proposed to determine the impeller outlet width by the formula: where K is the coefficient obtained as a result of statistical analysis, H is the pressure.
The following dependencies were obtained entirely by statistical analysis of the data in the works [9][10][11], respectively: where k v is the coefficient obtained as a result of statistical analysis, where k v2 is the coefficient obtained as a result of statistical analysis, These authors offer to focus on the pressure H, the supply Q, even the speed of rotation of the rotor n.However, neither method of calculation mentioned the steepness of the pressure characteristic and the influence of the impeller outlet width.
Since 2000, following the development of numerical methods for solving hydrodynamic problems, the studies have been conducted to determine the impact of the change in the impeller outlet width on the steepness of the pressure characteristic.
Thus, the authors in the work [12] argue that the effect of changing the impeller outlet width on the characteristic of the pump is a very efficient method of regulating the supply of a pump unit.
The paper [13] demonstrated that the increase in an impeller outlet width leads to a decrease in losses to friction, which leads to increasing pressure and hydraulic efficiency of a centrifugal pump.
The results presented in the article [14] also demonstrated the increase in the efficiency of the pump by increasing the width of the impeller.
The paper [15] presented a study of six centrifugal pumps that have a specific speed n s =45…260 at variable values of the width of impeller.According to the results, the change in the impeller outlet width significantly influences the parameters of the working point, the nature of the flow and the steepness of a pressure characteristic under condition of the immutability of geometric dimensions of other elements of the flowing part of the pump.
The results of the studies of the impact of the change in the impeller outlet width on the characteristics of a centrifugal pump, described in [16], show that the change in its value significantly affects the efficiency of the working process of the pump at the supplies larger than the nominal.In this case the curve of dependence Q-η increases its curvature, indicating a decrease in the efficiency of the working process of a pumping unit.
According to [17], a pump pressure decreases with decreasing impeller outlet width in the pumps of high rapidity.However, in the pumps with low rapidity, the pressure hardly depends on the width of the impeller.
The authors in [18] presented results of a numerical study of the pump with four different values of the width, according to which the pumps with a wider impeller have the best energy efficiency.However, they noted that a too large value of the impeller outlet width leads to a decrease in efficiency of the pump.Similar results were obtained in [19].
All of the above-mentioned researchers in their papers agree on that the impeller outlet width is one of the main geometrical parameters, which significantly influences the characteristics of centrifugal pumps in general and its steepness in particular.However, they did not present a clear correlation between the value of the impeller outlet width and the steepness of its pressure characteristic.Conclusions about broad impellers providing a higher efficiency value of the pump operation while too wide impellers, on the contrary, leading to its decrease, cannot be used in the design of the pumps because they do not allow selecting the optimal value of the impeller outlet width.Therefore, it is important to conduct additional research to identify a clear mathematical dependence between the width of the output impeller and the steepness of its pressure characteristic.

The purpose and objectives of the study
The aim of this work is to determine a mathematical dependence between the width of impeller of double-entry impellers of centrifugal pumps with the specific speed n s =80…210 and the steepness of a pressure characteristic, for the design of them with a predetermined steepness of pressure characteristic adjusted for necessary changed parameters of water supply network.
To achieve this goal, the following problems were identified: -to perform numeric simulation of the working process of double-entry impellers with different values of the impeller outlet width; -to obtain a mathematical dependence between the width of the output impeller and the steepness of its pressure characteristic.

Materials and methods of numerical research into the working process of double-entry impellers of centrifugal pumps
We studied impeller of the following pumps: D 3200-75, D 6300-80-2, SЕ 2500-180-8, SЕ 2500-180-8 UHL 4Е, D 6300-27-2.We modeled their working process at the modes from 0,5Q ref to 1,2Q ref , where Q ref is the supply in the reference point.By the method of conducting research, at unchanging angles of installed blades at the input β 1 and the output β 2 and geometric sizes of the impeller D 2 , D 0 , d hub , only the width of impeller b 2 at the output changed, as shown in Fig. 1.
Thus, the values of the width of the impellers were selected in increments of 10 % of the value b 2 of the original impeller of the pump.The selected values of the impeller outlet width are displayed in Table 1.For conducting numerical study, by using SolidWorks software package, we created 3D models of impellers with different values of the impeller outlet width.In this the following assumptions were adopted: -internal flow is symmetric; -internal flow at the input to the estimated area is axisymmetric; -leaks through the impeller's seal are neglected.
In connection with the above-mentioned assumption, the estimated area is one channel of the half of the impeller without seals.The input and output borders of the estimated area are at such a distance from control sections that is sufficient for the flow to be stable.
To test the influence of the number of cells in the grid on the results of simulation, in the grid generator CFD we built unstructured grids of the estimated areas with the number of cells in a range from 140 000 to 5 million.Results of the analysis are shown in Table 2.An example of a constructed grid for the estimated area is presented in Fig. 2.

Fig. 2. Example of the constructed unstructured grid for the estimated area
Numeric simulation was performed in the Ansys CFX software package.
At the input of the estimated area and at its output we set the components of velocity and static pressure, accordingly.For numerical simulation we selected standard k-ε -turbulence model and scalable functions of the wall.The roughness of the surface of the walls is 6.3 μm.
The flow in the estimated area was adopted as three-dimensional, viscous, turbulent.As the working fluid we chose water at the temperature of 25 ºC, and the criterion of convergence was 10 -4 .The Reynolds number was 10 7 .
All calculations were performed in a stationary state.The input and output segments were fixed while the impeller was able to rotate.In addition, periodic borders were defined for the estimated area.

Results of the research into the influence of the impeller outlet width on its pressure characteristic
As a result of the conducted numerical research we obtained integrated pressure characteristics for impellers.Based on the obtained data about the pressure H, we built dependences of the steepness of the pressure characteristic K H on the relative width of impeller b 2 /D 2 for each group of impellers (Fig. 3).
The value of the steepness of a pressure characteristic is calculated by the standard formula: where H max is the pressure at 0,5 Q ref ; H ref is the pressure at Q ref .
Fig. 3. Chart of dependence of the steepness K H of a pressure characteristic on the relative width of impeller b 2 /D 2 , K H =f(b 2 /D 2 ) for the pumps impellers with -n s =80…110, -n s =110…150, -n s =150…210 Fig. 3 shows the points obtained in the result of the numerical simulation of the working process of impeller pumps with different values n s and the equation of their approximating curves.The equation that describes the dependence of the steepness of a pressure characteristic on the relative width of impeller, generally takes the form: where а and k are the obtained coefficients that have the following numerical values: а=0,009, k=1,264 -for impellers of pumps with n s =80…110; а=0,0102, k=1,435 -for impellers of pumps with n s =110…150; a=0,0713, k=0,953 -for impellers of pumps with n s =150…210.
Fig. 3 also displays the dependence, which shows that with the increase of the specific speed n s , the value of steepness K H increases as well.At decreasing b 2 , the graph K H =f(b 2 /D 2 ) is more sloping at n s =150…210 than at n s =80…150.
Using a dependence (2) and the data that were obtained as a result of a numerical experiment, we built the graphs of dependence (Fig. 4) of consumable parameter q р on the relative width of the impeller b 2 /D 2 for each group of impellers at D 2 =const, n=const, β 2 =const.
As we can see from Fig. 3 and Fig. 4, the nature of the dependences K H =f(b 2 /D 2 ) and q р =f(b 2 /D 2 ) is similar that allows making a conclusion about the identity of the influence of the width of the output impeller on q р and K H . Fig. 5 shows the graph of dependence of the steepness of a pressure characteristic on a consumable parameter in the reference point K H =f(q p ) for the groups of impellers.
It is evident from Fig. 5 that in a general case the dependence K H =f(q р ) takes the form: Fig. 4. Graph of dependence of consumable parameter q р on the relative width of impeller b 2 /D 2 for the pumps impellers with -n s =80…110, -n s =110…150, -n s =150…210 Fig. 5. Graph of dependence of the steepness of a pressure characteristic on a consumable parameter in the reference point K H =f(q p ) for the pumps impellers with -n s =80…110, -n s =110…150, -n s =150…210 By using the equation ( 5), we received: where E and F are the obtained coefficients that have the following numerical values: E=1,0459, F=-0,0852 -for impellers of pumps with n s =80…110; E=1,2699, F=-0,3107 -for impellers of pumps with n s =110…150; E=0,8666, F=0,0402 -for impellers of pumps with n s =150…210.

Discussion of the results of the research into dependence of impeller outlet width and the steepness of a pressure characteristic
As a result of the research into the working process of impeller with n s =80…210 of the pumps D 3200-75, D 6300-80-2, SE 2500-180-8, SE 2500-180-8 UHL 4E, D 6300-27-2, mathematical dependences between impeller outlet width and the steepness of a pressure characteristic were established for the first time.
The coefficients a, k, E and F from the equations ( 12) and ( 14) are different for impellers with different rapidity and depend on their design features.
Due to the fact that the studies were conducted only for the impellers of aforementioned pumps, to refer established dependences to other brands of pumps would not be justified.Therefore, the purpose of further research is to define the main geometric parameters of impeller, which essentially influence the coefficients a, k, E and F, to obtain universal dependences for centrifugal pumps with double-entry impellers for all types.
Received dependences are expedient to use when designing the adjustable impeller of the pumps D 3200-75, D 6300-80-2, SE 2500-180-8, SE 2500-180-8 UHL 4E, D 6300-27-2 during their modernization for new parameters of the supply Q and the pressure H with a predetermined steepness of a pressure characteristic.

Conclusions
1. Due to the use of numerical simulation in the software package ANSYS CFX, we received a possibility to research into the working process of 30 impellers with changing only one geometric parameter -the impeller outlet width b 2 , while keeping other geometric parameters unchanged, excluding their cumulative impact on the results.
2. For the double-entry impellers with the specific speed n s =80…210 of the pumps D 3200-75, D 6300-80-2, SE 2500-180-8, SE 2500-180-8 UHL 4E, D 6300-27-2, we established dependences between the impeller outlet width b 2 and the steepness of the pressure characteristic K H , and also the consumable parameter q р in the form of mathematical formulas ( E q F = ⋅ + ) and the constructed curves.The identity of the curves of these dependences indicate the similarity of the impact of the change in the impeller outlet width on K H and q р provided no other geometric dimensions change.The coefficients in the equations are variable and depend on the design features of the impeller.

Fig. 1 .
Fig. 1.Diagram of the meridian section of the impeller of centrifugal pump

Table 2 Parameters
of the grids of the studied impellers Variant of impeller geometry Number of nodes in the grid, thousand

Table 1
Variants of the impeller outlet width b 2 of the studied pumps impellers