Effect of impeller shroud trimming on the hydraulic performance of centrifugal pumps with low and medium specific speeds

ABSTRACT The effect of pump impeller modification by a new approach, pump impeller shroud trimming (PIST), on the flow field and pump performance is examined both numerically and experimentally. The primary purpose of this study is to investigate the hydraulic performance of single-phase centrifugal pumps, which are refined by the PIST method, with low and medium specific speeds. In this type of shroud trimming, different trimming sizes are applied only to the shroud plate of the closed impeller, while the geometries (diameters) of the hub and blades remain unchanged. This modification, which increases the clearance between the impeller and casing from the shroud side, causes desirable conditions for pumping fluid containing undissolved gas. Computational fluid dynamics software (ANSYS-CFX) is used to predict the hydraulic performance of centrifugal pumps. Two well-known turbulence models, the renormalization group (RNG) k-ϵ developed model and shear stress transport (SST) k-ω, are used to predict the flow patterns. The computational results are verified through the comparison of experimental and unsteady numerical simulations with steady numerical data. After ensuring the accuracy of the flow-field simulation approach, further numerical analysis is performed for both pumps by changing the impeller shroud diameter. The effects of the geometric changes on the performance curves, efficiency, flow field, pressure distribution inside the pump components and the radial force acting on both pump types are investigated comprehensively. The results show that the shroud trimming reduces the produced head and efficiency at the design points. Examination of the radial force, which is applied on the rotating parts, shows that shroud-trimmed impellers experience higher radial forces than closed impellers in both pump types owing to the lack of a uniform pressure distribution around the impeller outlet. Accordingly, the relevant information obtained is used to modify the existing coefficients to predict the radial forces.


Introduction
Impeller trimming is a special technique that reduces the diameter of a centrifugal pump impeller to make an impeller with a smaller diameter at an equal rotational speed. Under certain conditions, impeller trimming causes a reduction in the tangential tip speed at the impeller outlet, which results in providing the proper pump working condition based on the system's demand, which also may decrease excessive noise or vibration of the pumping system. The procedure of tip speed reduction leads to a decrease in the energy imparted to the passing fluid flow, which causes both the discharge flow and the head generated by the pump to drop (Karassik, 1989;Tsang, 1992;Hydraulic Institute and US Department of Energy, 2006). Considering the affinity laws, the trimming method must satisfy the geometric and kinematic similarities. Nevertheless, impeller trimming changes the nonlinearities of the affinity laws and, as yet, CONTACT Amir F. Najafi afnajafi@ut.ac.ir there are no quantitative relationships between impeller trimming and its adherence to these laws (Li, 2011). Figure 1 demonstrates different well-known typical trimming methods applied to a radial impeller (Gülich, 2007). This figure shows five trimmed impellers. In addition to changes in the imparted tangential momentum, trimming provides a greater clearance between the impeller tip and the casing.
The main objective of this study is to examine the condition in which trimming is applied only on the shroud side of a pump impeller with low and medium specific speeds. This specific type of impeller trimming, called pump impeller shroud trimming (PIST), does not make any considerable changes to the impeller meridional plane ( Figure 2).
Indeed, shroud side trimming modifies the regular meridional impeller passage by providing extra space for the fluid flow at the impeller outlet ( Figure 3). It alters  (Gülich, 2007, with permission). a) trims with shroud (volute pumps), b) trims without shroud (diffuser pumps), c) oblique trim, d) oblique trim, e) trim with shrouds. the space between the impeller shroud side and the pump volute, which consequently alters the hydraulic performance of the pump. Because of the differently shaped pumps in different impellers, with low [two-dimensional (2D) impeller profile] and medium [three-dimensional (3D) impeller profile] specific speeds, the effect of implemented shroud trimming will be different for each case.
The enlarged available space at the impeller exit may lead to a desirable condition for two-phase (liquid-gas) fluid flow pumping. As a prerequisite to the investigation of such a fluid flow domain, the single-phase flow for these new geometries should be studied to attain an appropriate understanding of the hydraulic performance of both pump types in these geometries. The probably beneficial impacts of implementing PIST on pumping the fluid containing undissolved gases will be investigated in future research.
As mentioned earlier, PIST causes changes in pump performance owing to changes in the fluid flow patterns passing through the pump, such as changes in the power and size of the vortices in the impeller passages and the casing volute. The undesirable vortices are responsible for head losses, flow pattern non-uniformity and slip. Concerning the numerous experimental and numerical studies on secondary flows in turbomachinery, here, some of the previous studies related to this case are reviewed. One of the most observed results of vorticities in centrifugal pumps is the 'jet-and-wake' phenomenon (Cheshire, 1945;Hamrick et al., 1954;Olivari & Salaspini, 1975). The outlet flow from the centrifugal impellers and nonuniformity of the velocity profile in this area are the main issues that were investigated by these researchers. Although the jet-and-wake phenomenon at the outlet of the impeller of centrifugal pumps is usually unavoidable, the presence of vorticities in the impeller passages exacerbates this phenomenon. Gruver et al. (1996) and Brun et al. (1994) observed strong secondary flows in mixed-flow pumps. In addition,  analogous secondary flows in other kinds of turbomachinery geometries were studied by some researchers. For instance, a hot wire probe was used to determine the secondary flow field in a rotating radial-flow passage by Moore (1973aMoore ( , 1973b. In this research, comparison of the obtained results with predictions from a potential flow approach showed an acceptable agreement. Several researchers studied the secondary flows in stationary circular and rectangular bends (Hawthorne, 1951;Kelleher et al., 1980;Sanz & Flack, 1986). Some analytical models were derived for streamwise vorticity and secondary flows in turbomachines (Hill, 1962;Horlock & Lakshminarayana, 1973;Lakshminarayan & Horlock, 1973;Smith, 1957;Wu et al., 1952). Later, these models were used to predict the secondary flows in a rotating bend (Johnson, 1978). Computational fluid dynamics (CFD) has facilitated the calculation of the flow field through the centrifugal pumps and the progression of their design. Several researchers have tried to figure out the flow field inside centrifugal pumps, to optimize the design of centrifugal turbomachines. For instance, Asuaje et al. (2005), Majidi (2005), Huang and Wu (2006) and Zhang et al. (1994) investigated the flow field in centrifugal pumps and carried out parametric studies using CFD methods.
Another important impact of shroud trimming is the change in the radial force, which plays a key role in how much load is exerted on the bearings and shaft, followed by shaft deviation resulting in vibration. When a pump operates outside its BEP, the peripheral distribution of static pressure at the impeller outlet is not uniform, which amplifies undesirable radial force. Numerous studies have been conducted to show how radial forces are affected by geometric parameters (Girdhar, 2005;Gülich, 2007). Stepanoff (1992) proposed an empirical relationship to estimate radial force, which has been extremely useful in the initial stages of pump design. Based on the examination of 16 pumps with low specific speeds, Agostinelli et al. (1960) suggested a relationship to predict the radial force, which was developed subsequently (Hydraulic Institute, 2009). The implemented approach was validated by comparing the obtained numerical steady results with unsteady numerical outcomes and experimental data that were obtained in simultaneous steps in this study.
The main purposes of this research are to analyze the values of head reduction due to impeller shroud trimming and to study changes in the efficiency and radial force in pumps with low and medium specific speeds.
In the present study, the fluid flow passing through each pump was simulated using steady-state numerical simulations. To ensure the accuracy of the results, an unsteady simulation for four cases (at BEP and lower flow rates than BEP) and experimental tests were conducted. The main focus of this study is on variations in the pump head and efficiency, the pressure applied on the rotating parts to calculate the radial force and modified correlations to predict the radial force in the pumps with trimmed shroud impellers. Another purpose of the research is to present the streamlines to analyze the intensity of vortices and flow separation in each case. In addition, the effects are compared in pumps with low and medium specific speeds, to gain a better understanding of the effectiveness of impeller shroud trimming in the hydraulic performance of the studied pumps.

Geometry
The main parameters of geometry related to the pump with N s = 10 and a fully closed impeller are shown in Figure 4. The design conditions and hydraulic parameters related to the studied pumps (N s = 10 and 24) are summarized in Table 1.
During the initial investigation, it was found that the head coefficient (ψ) = gH ω 2 D 2 2 and the flow coefficient must be presented. Here, D 2 , Q, ω and H represent the outer diameter of the impeller, the flow rate, the rotational speed of the pump and the pump head, respectively. By considering the special geometries in the present work (trimmed impeller shroud side), the average outlet diameter is calculated from Equation (1): Table 2 presents the percentages of shroud trimming for pumps with N s = 10 and 24. For example, and to avoid confusion, only some geometries of the pump's impellers (N s = 24), which have been changed by the PIST method, are shown in Figure 5.

Experimental set-up
The test rig diagram and its configuration are shown in The test rig consists of two valves, an installed pump, and pressure and flow measuring devices. The fullscale accuracy of the pressure transmitter is ±0.5%. To increase the measurement accuracy of the tests, experiments were performed several times at a certain flow rate and the average of the recorded data was reported. Using conventional methods for uncertainty estimation, the maximum flow rate uncertainty was ±0.2%. Additional information on the laboratory test circuit can be found in Appendix 1. In this test, the whole units of device packages were tested together to check the hydraulic or mechanical functioning. The purpose of this test was to check the performance of whole devices when working together. The vibration values for each piece of equipment must be within its tolerance range.

Governing equations
Since the turbomachinery flow field is known as one of the most complicated turbulent flows, choosing a proper model for turbulence modeling and evaluating Navier-Stokes equations are important. The governing equations, known as continuity and momentum equations [Reynolds-averaged Navier-Stokes (RANS)] for the  3D flow field and a description of the turbulence models, can be found in Appendix 2 (Alemi et al., 2015;Shukla et al., 2016;Yakhot & Orszag, 1986;Yakhot & Smith, 1992).

Boundary conditions
Boundary conditions of the fluid domains should be determined so that they represent the actual conditions. In the present study, the incompressible water flowed inside a rotating impeller at a temperature of 25°C. The equation representing the fluid flow in a particular region must be numerically constrained, which defines the boundary conditions. For the inlet part of the computational domain, the static pressure was set as the inlet condition. The mass flow rate was selected as the boundary condition of the outlet. In the flow field, there are two interfaces: (1) between the inlet duct and the impeller eye; and (2) between the impeller outlet and the volute. Therefore, by changing the outlet flow rate and the shroud  diameter, simulation of the flow field was performed in various conditions for the centrifugal pumps. The abovementioned boundary conditions were proposed by Caridad et al. (2008). All rotary parts rotated at a nominal speed of 1475 rpm and the other walls were stationary.
The walls were considered all smooth and the nonslipping condition was applied, which means that the roughness of the walls was not taken into account. The interface between the rotating and the stationary parts was set as a mixing plane for steady-state analysis, which was compared to experimental and unsteady harmonic balance results. Some steady-state methods (e.g. the frozen rotor and the mixing-plane models) simplify the natural periodic flow in a pump; hence, the time derivative term of the Navier-Stokes equation can be neglected. To evaluate the effect of inlet turbulent quantities on the solutions, turbulent intensity was varied from 5% to 10% (Najafi et al., 2005). There were no meaningful changes in the solutions (the predicted head changed by less than 0.01%). In actual conditions, the inlet flow can be considered fully turbulent. In this regard, the assumed turbulent intensity is reasonable for numerical simulation.

Mesh generation
For grid generation, 3D models of the impeller and volute were first generated and then exported to commercial software, ANSYS Meshing 2019. To achieve and implement the complete computational domain, first, the pumps were split into four components: inlet duct, impeller, volute and outlet duct. Straight ducts at the inlet and outlet were designed to prevent the effects of boundary conditions on simulation of the pump flow domain. The meshing processes of these separate sections was carried out simultaneously. The unstructured grid was used to generate the mesh for the whole computational domain. Figure 9 demonstrates the meshes generated for the rotary and stationary parts of both pump types.
Owing to the complexity of the geometries, special meshing techniques, such as face sizing, on the shared interfaces between the rotating and the stationary parts, were used. So, close to some important regions, such as the leading and trailing edges of the blades and the volute tongue, where flow separation may occur, mesh clustering was applied ( Figure 9). In addition, tetrahedral cells were used for meshing all parts of the geometries. Cells adjacent to the solid walls were modified by face sizing and boundary layer generation to ensure that the appropriate turbulence models could be applied.
Therefore, a boundary layer was formed perpendicular to the blade walls to ensure that the appropriate grid size was used. The mean Y + values on the impeller were about 5.3 and 5.7, while the maximum Y + values on the solid walls were 43.5 and 47.5, for pumps with specific speeds of 10 and 24, respectively. The renormalization group (RNG) k-turbulence model was then used to predict the turbulent fluctuations under the design and off-design conditions in all cases for both pumps. By considering the values of Y + obtained from the numerical simulation and the RNG k-turbulence model, the scalable wall function was used to capture the viscous sublayer in the flow (Alemi et al., 2015;Bel Hadj Taher et al., 2017).

Numerical simulation method and mesh independency
For numerical analysis, a commercial CFD code based on the finite volume method, ANSYS CFX 19, was used in the present study. A high-resolution scheme with second-order accuracy was used for the advection terms, and the second-order upwind scheme was applied for the space discretization. Equations were solved in a moving reference system. The convergence criteria for numerical simulation were set at maximum residuals of 1e-5.
For the steady simulation, impeller rotation was considered and the frame of reference was changed, but the relative orientation of components across the interface was fixed. The two frames of reference connect in such a way that each of them has a fixed relative position throughout the calculation. This is equivalent to taking a snapshot from the flow field in an instance of time. In other words, in both pump types, governing equations of the system were solved once in the rotating coordinate system for the impeller and once in the stationary coordinate system for the volute. Then, the solved equations in the rotational and stationary sections were coupled together through the mixing-plane interface capabilities, and the pressure and velocity data were exchanged between two domains.
The independence of the solution from the number of mesh elements was examined by solving a fluid flow domain for the basic geometries with a different number of elements for both pump types. Also, a fluid flow property (pump head) was chosen as the measure to check the mesh independency. Figure 10 summarizes the numerical values obtained from the computational grids with different cell numbers for both pump types, with low and medium specific speeds (N s = 10 and 24). As can be observed, by increasing the number of meshes from 2,500,000 to 3,421,000 and from 6,328,736 to 13,568,203, the changes in the predicted head are about 10% and 4.5% in pumps with N s = 10 and N s = 24, respectively. Therefore, all numerical results were obtained using meshing with 3,050,000 and 10,093,387 elements in pumps with N s = 10 and N s = 24, respectively. In the mesh generating process, the measuring qualities, such as vertical angle, aspect ratio and skewness, were examined to be in the desired range in accordance with the software guideline (CFX 20 solver). To obtain a steady-state solution, the mesh cells between the rotating impeller and the stationary volute were linked together through an interface.

Steady-state solution
The numerical investigations were conducted in the range of flow rates, which varied from 50% to 150% of the BEP discharge for both above-mentioned centrifugal pumps. So, to examine the reliability of the numerical simulation, the results obtained for the complete impeller, as the original model of pump with the specific speed of 10, and also the results for complete and trimmed shroud impellers of the pump with the specific speed of 24, were verified by experiments. Subsequently, because of non-uniform fluid flow properties in centrifugal pumps with low discharge rates, in some cases for each pump, the results obtained from the steady numerical simulations and the experiments were compared with the results of the transient solver obtained by the harmonic balance method. The comparisons proved the validity of the numerical solutions and the accuracy of the experimental set-up.
Using the measurement devices, the pump performance curves for the original impeller (complete impeller with no shroud trimming) of pumps with the specific speeds of 10 and 24 and Cases II and III of the pump (see Table 2) with the specific speed of 24 are provided at different flow rates. The comparison of these results, which is shown in Figure 11, shows relatively good agreement. In Figure 11(b) (Case I), a significant discrepancy can be seen in flow rates far greater than the value of BEP. This may be caused by the existence of a large amount of fluid friction losses at flow rates much greater than those at BEP and the elimination of surface roughness in numerical analysis. (The magnitude of friction loss depends on the roughness of the solid surface and the fluid velocity relative to the surface.) Considering the geometry of this pump (medium specific speed, N s = 24, with complete impeller shroud), other effective factors such as incidence losses may have contributed to this outcome. Furthermore, in Figure 11(a), (c) and (d) (Cases I, II and III, respectively), the differences between the numerical and the experimental results reached the maximum values at much less than the nominal flow rates. These differences also grew with the increases in the shroud trimming.
This behavior could be explained by the patterns of the streamlines in the rotating and stationary parts of the centrifugal pumps and the way in which the resulting vortices affect the mechanism of the pressure measuring devices at low flow rates. In addition, the smooth surface of the impellers and the negation of flow losses in the wear rings in the numerical simulation, as well as the probability of error (± 0.3 mm) in the impeller shroud trimming, which leads to the difference in blade thickness under the experimental tests and numerical simulations, may cause the differences between the outcomes of the experiments and the numerical analysis for the head coefficient.
Considering the complexity of the flow behavior passing through the pump and the special method applied to the original geometry due to impeller shroud trimming (PIST), numerical analysis was also performed using the shear stress transport (SST) k-ω turbulence model to capture the small vortices, flow separations and adverse pressure gradients in the vicinity of surface of the blades (Skerlavaj et al., 2011). The details of all turbulence models used in this study can be found in Appendix 2. Figure 12 depicts the streamlines at BEP in the pump with the specific speed of 10. As shown in Figure 12, the vortices first concentrate on the pressure side of the surface of the blades and gradually expand in the impeller passages, to approximately half of the length of the blades. Given the evaluated deviation percentages (Equation 2) in Table 3, the difference between the head coefficient values obtained by the RNG k-and the SST k-ω turbulence models relating to the pump with the specific speed of 10 are greater than the values in the pump with the specific speed of 24. These differences are mainly caused by fewer blades (and the wider impeller passages) and the lower capacity of the pump with the specific speed of 10. All of these factors change the flow field and lead to the creation of more vortices and stronger flow separation. (2) Considering the deviation percentages given in Table 3 and the resulting streamlines presented in Figure 12, to save the calculation expense, the RNG k-with the wall function treatment is precise enough to use in the numerical hydraulic simulation in the following steps of the research.

Transient solution
Using steady-state methods, such as frozen rotor and mixing plane, for analyzing the fluid flow passing through a pump may simplify the flow pattern conditions by neglecting the time derivative term in the Navier-Stokes equations. For instance, a lot of important phenomena, such as impeller/tongue interactions and wake/blade interactions, cannot be observed by a steady-state solver. This may be especially relevant at low flow rates where separation from the impeller blades may occur. Given that reaching a periodic steady-state solution using a transient solver is much more computationally expensive, to take advantage of a faster steady-state and more accurate transient solver, a harmonic balance method has been used (Cvijetic & Jasak, 2018;He & Ning, 1998).
Since the flow in a turbomachinery device is naturally periodic, this method uses a Fourier series, which contains harmonics of the main frequency (f = 2π /ω) to estimate the field variables. In this paper, a harmonic balance analysis was also performed and compared to experimental results for different low flow rates. The unsteady harmonic balance results are shown in Table 4. The explanation of this method can be found in Appendix 3.
Comparison of the numerical results of the steady and unsteady simulations with experiments shows there are no large differences between these results. Therefore, it can be expected that the steady simulation is accurate enough for the present study. In Table 4, the error percentages with respect to the experimental data, which are given in the last column, are calculated by Equation (3): Error percent considering the experimental data

Hydraulic performance analysis
Figure 13(a) and (b) shows the heads obtained from numerical simulations at various flow rates for the different cases of the two pumps (N s = 10 and 24). The results show that even with only shroud trimming, the head has been reduced. (The diameters of the impeller hub and the outer blades were kept unchanged with the PIST method.) The highest head values resulted from the impellers with an untrimmed shroud (original impellers) at full capacity. Figure 13(d) shows that the simultaneous reduction of the pump head and impeller average diameter (based on Equation 1) leads to a unique non-dimensional performance curve related to all six cases of the pump with the specific speed of 24. Regarding the pump with the specific speed of 10 (N s = 10) (Figure 13c), in the first three cases (based on Table 2), the non-dimensional performance curves are similar to each other, whereas for the last three cases, the curves show a completely different behavior. In this pump type, the 2D impeller profile and the smaller casing compared with the other pump type (N s = 24) cause the trimmed part of the shroud plate compensated with the inner surface  of the volute casing, which pushes the fluid flow to the hub side and prevents excessive head drop. So, a smaller reduction in the head compared to the square of the decreased average diameter (the main parameters in the head coefficient formula) pushes the non-dimensional curves up. Head reduction may result from the decrease in imparted tangential momentum and change in the desired flow field through the impeller passageways, which follow the discharge recirculation and the leakage from the region around the leading edge ( Figure 14). From Figure 14, the vortices in the discharge recirculation can be observed in both pump types. By impeller shroud trimming, the expanded vortices, which are created in the clearance of the impeller shroud side and the volute, cause the increased velocity at the impeller outlet that leads to a local pressure reduction. Also, Figures 12  and 14 (related to the pump with the specific speed of 10) prove that the impeller shroud trimming reduces the intensity of the vortices at the impeller passages at BEP.
The other point that should be considered as an effective parameter in head reduction is the slip. The slip is the deviation between the angle at which the fluid leaves the impeller and the angle of the impeller blade. During the fluid flow passing through the impeller, the centrifugal acceleration, which is produced by the impeller bend and a component of the Coriolis acceleration, generates a pressure difference between the shroud and the hub, which pushes the fluid from the suction side to the pressure side and causes the secondary flows. This tendency is opposed by the component of centrifugal acceleration, which is caused by impeller rotation (Figure 15).
Shroud trimming could change the secondary flows at the shroud side of the impeller outlet (the righthand picture in Figure 15). These changes will produce the undesired slip that generally causes the head reduction. As shown in Table 5, the different geometry (fewer blades and 2D curvature of the blades) causes a bigger slip, which leads to a greater head reduction in the pump with N s = 10. At BEP conditions, 20% impeller shroud trimming (Case VI for N s = 10 and Case V for    (Gülich, 2007). Note: a T is the torque that is applied to the impeller (N·m). The value of this parameter is obtained from the numerical solution. δ o is the absolute difference between the angle of the leaving streamlines and the impeller blade outlet angle.
N s = 24) results in head reductions of 25% and 21%, respectively. From Table 5, it can be seen that shroud trimming generally decreases the produced head, but for the pump with N s = 10, by increasing the shroud trimming to 13% of its main diameter (Case IV), the head is reduced; despite further trimming, the head remains approximately unchanged. Different behavior can be seen for the pump with N s = 24.
As mentioned before in this subsection, in different cases of the pump with the lower specific speed (N s = 10), by increasing the shroud trimming (i.e. more than 13% for this type), the trimmed part of the shroud is approximately compensated with the inner surface of the volute casing and prevents unexpected changes in streamlines and head reduction. This is because of geometric parameters such as small volute size and less clearance between the impeller and the inner pump casing side walls, as well as 2D curvature of the blades. The other point that should be noted is that shroud trimming gradually leads to a higher steepness of the head-capacity curves in the pump with N s = 24. Figure 16 indicates the corresponding hydraulic efficiency values for each impeller with a trimmed shroud at different flow rate coefficients. The hydraulic efficiency is calculated using Equation (4): From the data in Table 5 and Figure 16, the efficiency reduction of centrifugal pumps with trimmed shroud impellers can be attributed to the increased losses due to greater slip and vortex expansion in the volute casing (Figure 14), which have negative impacts on the efficiency.
Among all the studied shroud trimmed impellers at BEP, the highest efficiency is accomplished by the pumps with a complete impeller. Accordingly, for the pump with the specific speed of 10, the maximum drop in efficiency is 5.6% (Case IV), while it is 56% (Case VI) for the pump with the specific speed of 24. As expected, at flow rates far from the BEP, friction losses make up the largest portion of hydraulic losses. Reducing the shroud size causes decreasing fluid-solid interaction in the flow passages, which could be considered a fluid flow with lower friction losses. This caused a different efficiency variation pattern in these flow rates. So, the highest predicted efficiency value is achieved with the maximum shroud trimming.

Radial force prediction
Pressure contours in the pumps' impellers and volutes for the nominal flow rates related to both pumps, with low and medium specific speeds, are demonstrated in Figures 17 and 18. The results show that increasing the shroud trimming causes the fluid flow to leave the impeller, resulting in a pump volute with relatively low pressure. Accordingly, the lowest pressure gradients are related to the cases with the highest shroud trimmings. Also, a comparison of Figures 17 and 18 shows that the pressure gradients for the fluid flow passing through the pump are more sensitive to PIST for a pump with the higher specific speed.   Any changes in the flow conditions at the impeller outlet, which are expected to affect the hydraulic performance of the pump, will affect the static pressure distribution around the impeller outlet. This means that the magnitude of radial forces is susceptible to variations in pressure.
Considering the description in Figure 19(a) and the defined pressure coefficient (Equation 5), Figures 20 and 21 present the variations in the pressure coefficient around the impeller outlet at BEP for each pump case.
where P is the calculated pressure around the impeller outlet, ρ is the density of water, and U 2 is the circumferential velocity at the impeller outlet.
Considering the previous hydraulic analysis, the undesired flow fields produced by increasing slip and vortices as the result of impeller shroud trimming create an irregular velocity distribution around the impeller outlet and an imbalance in pressure. From Figures 20 and 21 and the pressure contours in Figures 17 and 18, the pressure distribution around the impeller outlet has less symmetry through the radial axis, which leads to the production of larger radial forces in comparison with those in a pump with a complete impeller. The curves in Figure 21 have a clear regularity, i.e. there are six distinct peaks in each pressure coefficient diagram related to the trimmed shroud impellers, which is the same as the number of impeller blades (z = 6).
Thus, the shroud trimming changes the appropriate distribution of pressure and subsequently produces undesired radial force. Radial force can then be obtained by an integration of the pressure and the shearing stress on the impeller surface. The shearing stress force is smaller, by about three orders of magnitude, than the pressure force (Benra et al., 2003), so the shearing stress force is negligible in this study, where the radial force of the impeller is calculated from the following equations (Equations 6-8). The parameters used in these equations have been introduced in Figure 19(a) and (b). Fundamentally, radial forces are generated when the circumferential distribution of the static pressure, p 3 , at the impeller outlet is non-uniform. Flow asymmetries in the volute, as well as a rotationally non-symmetric impeller inflow, can thus create radial forces. Components x and y of the radial force depend on the pressure distribution around the impeller. where radial force is: The ratios of the radial forces acting on the pump impeller with a trimmed shroud (at BEP) to the radial force applied to the pump rotor with a closed impeller (at BEP) are demonstrated in Figure 22. As can be seen, increasing the shroud tip trimming will result in higher radial forces compared to the radial forces acting on pumps with the original impellers. In both pump types, the minimum value of radial forces belongs to the pump with an untrimmed impeller. The effect of impeller shroud trimming on increasing the amount of radial force in the pump with a higher specific speed is much greater than in the pump with a lower specific speed.
In the following, an attempt is made to estimate the radial force deviation for different cases, relative to the untrimmed shroud. It is assumed that the radial force of a pump with a complete impeller is given by the manufacturer; however, with a good approximation, it can be also estimated via empirical relations (Agostinelli et al., 1960) as a function of specific speed. The radial force coefficient, k r , is defined as follows (Gülich, 2007;Stepanoff, 1992): Since the above coefficient is recommended for a complete impeller, a correction factor for this coefficient is defined based on the radial forces provided by the numerical simulations for different shroud trimmings of each pump separately ( Figure 23). The correction factor is calculated using Equation (10): For this purpose, first, the radial forces are calculated for different shroud trimming percentages. After calculating the radial force coefficient for the complete impeller, which shows a good agreement with the recommended value (Gülich, 2007;Stepanoff, 1992), the correction factor is suggested for different amounts of shroud trimming in both centrifugal pump types, with low and medium specific speeds. These results are illustrated in Figure 23. In this diagram, with an equal percentage of shroud trimming, D * (Equation 11), the radial force correction factor related to the pump with the higher specific speed  (N s = 24) is less than the value related to the other pump (N s = 10), while its relative radial force is greater. This issue is caused by the bigger radial force coefficient in the pump with the higher specific speed.

Conclusions
Pump impeller modification by a new approach for trimming (PIST) is investigated both numerically and experimentally in this study. Different shroud-trimmed impellers related to single-phase centrifugal pumps with low and medium specific speeds (N s = 10 and 24,   respectively) have been analyzed numerically (steady and unsteady) and experimentally. The comparison of numerical and experimental results presented reasonable compatibility. Flow analysis was performed using CFX software (ANSYS 2019), in which the Navier-Stokes equations were solved using the RANS and RNG kdeveloped turbulence model. The aim was to trim the side of the shroud, step by step, in six stages for each type of pump, while the geometry of the hub and blades remained unchanged.
• Owing to different parameters of geometries of the selected pumps, such as the number of impellers, 2D or 3D impeller blade profiles, related to pumps with specific speeds of 10 and 24, respectively, the outlet to inlet diameter ratio of each impeller, various blade thicknesses and different pump eye diameters, the same hydraulic behaviors are not observed as a result of impeller shroud trimming in the two pumps. • In pumps with trimmed shroud impellers, much lower discharges than the nominal values generally cause the differences in the head between the experimental and numerical results. These differences also increase with the increase in shroud trimming. This is caused by the undesired effects of vortices on the mechanism of the pressure measuring devices. Besides, the smooth surface of impellers in numerical modeling, the negation of the flow losses in the wear rings and the probability of error (± 0.3 mm) in the impeller shroud trimming lead to the differences between the head measured in the experimental tests and the head calculated from the numerical simulations. • The SST k-ω turbulence model is much more accurate than the RNG k-as it captures the small vortices in the viscose sublayer. Fewer blades, wider impeller passages and lower capacity in the pump with the specific speed of 10 cause different-scale vortices at the surface of the blades, which the SST k-ω turbulence model is more successful in capturing. In the pump with the specific speed of 24, at BEP, no large differences are observed between the results obtained by the SST k-ω and RNG k-turbulence models. • Considering the deviation percentages between the head coefficient which were predicted by the SST kω and RNG k-turbulence models, due to saving the calculation expense, the RNG k-with the wall function treatment was accurate enough to use in the numerical hydraulic simulation, instead of the SST k-ω turbulence model. • In both pumps with low and medium specific speeds, shroud trimming decreases the imparted tangential momentum to the fluid flow. Simultaneously, the slip increases as a cause of undesired flow field through the impeller passageways, which follows the discharge recirculation and leakage from the region around the leading edge. These issues result in a smaller pressure gradient, which leads to the head drop in both pumps. • In the pump with N s = 10, by increasing the shroud trimming up to 13% of its main diameter (Case IV), the head decreases. Despite further trimming, the head remains approximately unchanged. This is because of the geometric parameters (small volute size, less clearance between the impeller shroud side and the inner pump casing wall, and 2D curvature of the impeller blades), which lead to the fluid flow being pushed toward the hub side and the prevention of unexpected changes in streamlines and head reduction. • In both pump types, the highest efficiency is accomplished in the pumps with a complete impeller. In general, the greater the reduction in shroud diameter the greater the decline in hydraulic efficiency, which is clearer in the pump with the higher specific speed. • According to pressure diagrams and contours, investigated flow phenomena, such as vortices, have detrimental effects on the flow field. This creates an unbalanced pressure distribution around the impeller outlet. This issue leads to greater radial forces in the centrifugal pumps with trimmed shroud impellers. • Increasing the shroud tip trimming will result in higher radial forces, especially in a pump with a higher specific speed. • Applying the PIST method (due to enlarged space at the impeller outlet) may provide a desirable condition to pump two-phase fluid flow, which will be investigated in future research.

Disclosure statement
No potential conflict of interest was reported by the authors.
groups. The turbulence kinetic energy, k, and turbulence dissipation rate, ε, are given by the following transport equations: where α k = α ε = 1.393 and C 1ε = 1.42. The production term in Equation (4) is given by: where S is the modulus of the mean rate of the strain tensor: In addition, the effective viscosity is calculated by: where C μ = 0.0845 and C * 2ε = C 2ε − C μ η 3 1 − η η 0 1 + βη 3 ; η = (2S ij S ij ) 0.5 k ε η 0 = 4.38; β = 0.012; C 2ε = 1.68 (A7) The SST k-ω turbulence model, which accounts for crossdiffusion, was developed by Menter (1994) using the standard k-ω model and a transformed k-model. Applying the k-ω formulation in the inner sublayer, at the vicinity of solid surfaces, makes the model beneficial enough through the viscous sublayer. The SST k-ω formulation also switches to a k-behavior in the free stream and thereby avoids the common k-ω problem that the omega equation is too sensitive to the inlet's free-stream turbulence properties. The Reynolds stress models and the details of the SST k-ω turbulence model equations are as follows (Skerlavaj et al., 2011): The above-mentioned parameters are defined as follows: The constant parameters are defined as follow: k -ω closure → σ k1 = 0.85, σ ω1 = 0.65, β 1 = 0.075 k -ε closure → σ k2 = 0.001, σ ω2 = 0.856, β 2 = 0.0828 SST closure → β * = 0.09, α 1 = 0.31

Appendix 3: Harmonic balance method
The underlying concept of the harmonic balance method is based upon the assumption of periodicity. If domain quantities are periodic in time, then they can be represented by a Fourier series containing harmonics of the main frequency (f = 2π/ω).
(A i sin sin (iωt) + B i cos cos (iωt)) (A10) where F(t) is a continuous field variable and A and B are Fourier coefficients. Thinking of mass, momentum and energy conservation equations as transport equations with different conservative variables of the form below, a general formulation can be developed: In this equation, R contains convective, diffusive and transport terms. After substitution of series expansion, the above equation becomes: ωAF + R = 0 (A12) where F and R vectors contain Fourier coefficients, while A is a (2n + 1) * (2n + 1) matrix of coefficients. This equation is time independent, hence eliminating the problem of the original transport equation, which is time dependency, but it requires a frequency-domain representation of the R, which is not favorable. Therefore, He and Ning (1998) proposed taking a sequence of forward and backward Fourier transformations where E and E −1 are the discrete Fourier transform and inverse discrete Fourier transform, respectively. To sum up the procedure, we have replaced the time derivative term with a source term and changed one time-dependent equation into 2n + 1 coupled steady-state equations, which can be solved more quickly. A larger number of harmonics (n) is supposed to increase the accuracy of the solution, whereas sometimes it has an adverse effect. In other words, complicating the system of equations comes at the expense of an increase in computational cost. Although there is no time derivative in this equation, by introducing a pseudo-time term and using a conventional transient scheme, a periodic steady-state solution can be reached. ∂F ∂τ where τ is fictitious time. The time derivative term will eventually disappear as the variable reaches its final value (Cvijetic & Jasak, 2018;He & Ning, 1998).