Numerical Identification of Key Design Parameters Enhancing the Centrifugal Pump Performance: Impeller, Impeller-Volute, and Impeller-Diffuser

This paper presents the numerical investigation of the effects that the pertinent design parameters, including the blade height, the blade number, the outlet blade angle, the blade width, and the impeller diameter, have on the steady state liquid flow in a three-dimensional centrifugal pump. Three cases were considered for this study: impeller, combined impeller and volute, and combined impeller and diffuser. The continuity and Navier-Stokes equations with the k-ε turbulence model and the standard wall functions were used by means of ANSYS-CFX code. The results achieved reveal that the selected key design parameters have an impact on the centrifugal pump performance describing the pump head, the brake horsepower, and the overall efficiency. To valid the developed approach, the results of numerical simulation were compared with the experimental results considering the case of combined impeller and diffuser.


Introduction
At the present time, single and multistage centrifugal pumps are widely used in industrial and mining enterprises.One of the most important components of a centrifugal pump [1] is the impeller.The performance characteristics related to the pump comprising the head, the brake horsepower, and the overall efficiency rely a great deal on the impeller.To achieve better performance for a centrifugal pump, design parameters such as the number of blades for the impeller and the diffuser, the impeller blade angle, the blade height for the impeller and the diffuser, the impeller blade width, the impeller diameter, and the volute radius must be accurately determined, due to the complex liquid flow through a centrifugal pump.This liquid flow is three-dimensional and turbulent.It is therefore important to be aware of the liquid flow's behavior when traveling through an impeller.This can be done by accounting for the volute and/or the diffuser in the planning, design, and optimization phases at conditions of design and off-design.Many experimental and numerical studies have been carried out on the liquid flow through a centrifugal pump [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21], where the effects of the number of impeller blades on the pump's performance were examined experimentally in [11,12].The effects of the impeller outlet blade angle on the pump's performance were also investigated numerically [13,14], using a CFD code and experimentally in [15].In [16] the dynamic effects due to the impeller-volute interaction within a centrifugal pump were numerically investigated, whereas the effects of the volute on velocity and pressure fields were examined in [17,18].Additional experimental investigation carried out [19] consisted of measuring unsteady velocity, the pressure and flow angle at the centrifugal pump's impeller outlet, with and without volute casing.The liquid flow and head distribution within a centrifugal pump's volute were compared with the impeller's characteristics, without the volute casing.Moreover, two centrifugal pump impellers with different outlet diameters for the same volute were examined both experimentally and numerically [20], to evaluate the influence the radial gap between the impeller exit and the volute tongue had on the unsteady radial forces acting upon the impeller of a centrifugal pump with volute casing.Additionally, the effects of flow behavior in a centrifugal pump, whose diffuser was subjected to different radial gaps, were investigated numerically in [21] using a CFD code.The analysis of previous works clearly demonstrated that research results obtained are specific to the centrifugal pump design parameter values and thus cannot be generalized.In this work therefore a numerical study was performed using a finite volume method according to the CFX code [22] to gain further insight into the characteristics of the three-dimensional turbulent liquid flow through an impeller, a combined impeller and volute, and a combined impeller and diffuser, while also considering various flow conditions and pump design parameters: blade heights of 12 mm, 18 mm, and 24 mm; blade numbers of 5, 7, and 9 for the impeller and 5, 8, and 12 for the diffuser; outlet blade angles of 9 • , 28 • , and 60 • , blade widths of 4 mm, 10 mm, and 15 mm; impeller outer diameters of 285 mm and 320 mm.The reference dimensions selected for the impeller and diffuser were based on the existing impeller and diffuser [23].
Upon applying the continuity and Navier-Stokes equations, the liquid flow velocity and the liquid pressure distributions in an impeller, a combined impeller and volute, and a combined impeller and diffuser were determined, while accounting for boundary conditions.Since the rotating speed of the centrifugal pump under consideration was constant a valve installed on the pump's discharge side was used to regulate the volume flow rate.We accounted for suction pressure variation as a function of the valve volume flow rate in the numerical simulations being run.The pump head, brake horsepower, and efficiency were represented as a function of the volume flow rate, where the objective was to identify the values of selected key design parameter that might improve pump performance with respect to their value ranges.

Mathematical Formulation
Figure 1 shows a centrifugal pump consisting of three components, including an impeller, a diffuser, and a volute [24].The models selected for the liquid flow in an impeller, a combined impeller and volute, and a combined impeller and diffuser are depicted in Figure 2, placing greater emphasis on the fluid domain.
In the governing equations for liquid flow in the centrifugal pump components, the following assumptions were made: (i) a steady state, three-dimensional, and turbulence flow using the k-ε model; (ii) it was an incompressible liquid; (iii) it was a Newtonian liquid; (iv) the liquid's thermophysical properties were constant with temperature.
To account for these assumptions, the theoretical analysis of the liquid flow in an impeller, a combined impeller and volute, and a combined impeller and diffuser was based on the continuity and Navier-Stokes [22] equations.For the three-dimensional liquid flow through the components of a centrifugal pump as shown in Figure 2, the continuity equations are expressed by and the Navier-Stokes equations are given by where U = U(u(x, y, z), v(x, y, z), w(x, y, z)) is the liquid flow velocity vector, p is the pressure, ρ is the density, μ eff is the effective viscosity accounting for turbulence, ⊗ is a tensor product, and B is the source term.More particularly, for flows in an impeller rotating at a constant speed ω, the source term can be written as follows: where r is the location vector.
Triangle ABC : velocity triangle without slip Triangle ADE : velocity triangle with slip In addition, μ eff is defined as where μ is the dynamic viscosity and μ t is the turbulence viscosity.
Since the k-ε turbulence model is used in this work because convergence is better than with other turbulence models, μ t is linked to turbulence kinetic energy k, equation ( 6), and dissipation ε, equation (7), via the relationship where C μ is a constant.The values for k and ε come directly from the differential transport equations for turbulence kinetic energy and turbulence dissipation rates: where C ε1 , C ε2 , and σ ε are constants.p k is the turbulence production due to viscous and buoyancy forces, which is modeled using: where p kb can be neglected for the k-ε turbulence model.
Moreover, for the modeling of flow near the wall, the logarithmic wall function is used to model the viscous sublayer [22].
2.1.Impeller.Three velocity types are involved when considering the flow through a centrifugal pump impeller: the tangential velocity U = rω, the relative velocity W, and the absolute velocity V .The last is expressed in vector format as follows: Figure 3 shows the velocity triangles at the impeller inlet and outlet at the design conditions where the liquid enters and leaves the impeller at the blade angles β b1 and β b2 , respectively.The components of V and W in the direction of U are V u (swirl velocity) and W u , respectively, while those normal to U are V r and W r .Moreover, according to the Euler equation [1], the energy transfer per unit mass of flow for a centrifugal pump can be formulated as where H i is the ideal pump total head.Neglecting the swirl velocity at the impeller inlet, (10) can be expressed as follows: When accounting for the hydraulic efficiency, η h , the actual pump head rise is given by Also, the hydraulic efficiency can be calculated using the following empirical formula [1]: where Q is the volume flow rate in m 3 /s.It is given by Q = V r A with A as the flow passage area normal to the meridional direction.
Since in reality the flow through a centrifugal pump is turbulent and three-dimensional, the actual relative flow direction at the impeller exit is different from that of the blade angle.As depicted in Figure 4, the flow angle β f 2 is always less than the blade angle β b2 .This can lead to secondary flows in the flow passage, from the pump inlet to discharge [1].
As such, the slip factor μ s is used to take into account the difference between β b2 and β f 2 , which is formulated as where V u2 is the actual swirl flow velocity at the impeller exit and V u2 is the ideal swirl flow velocity at the impeller exit.
In addition, the slip velocity is given by

Frozen rotor interface
Frozen rotor interface  Taking into account the slip factor, ( 12) can be expressed as Moreover, to account for the leakage flow from the impeller, the volumetric efficiency is defined by where Q L is the leakage flow from the impeller exit back to the inlet through the clearance.
In addition, the pump's mechanical efficiency is formulated as follows: where P s is the brake horsepower and P imp the power delivered by the impeller to the fluid.P s is globally expressed by where C is the pump shaft torque, P h is the centrifugal pump horsepower.It is expressed as P f is the loss power due to the friction, which is given by P L is the loss power due to leakage, which is defined as: P m is the friction loss power in bearings and seals, and P df is the disk friction power due to impeller shrouds.P imp in (18) can be formulated as follows: Furthermore, (23) can be rewritten as Accounting for ( 23), ( 18) can be expressed as Thus, the overall efficiency of a centrifugal pump can be formulated as Finally, the overall efficiency can also be formulated in terms of the other efficiencies as 2.2.Volute Parameters. Figure 5 shows the parameters of a volute without diffuser defined by the radius of volute basic circle r 3 , the radius of volute cut water circle r v , the volute angle α v , the volute cross-sectional area A θ , which depends on the angle Θ, and the volute outlet cross-sectional area A t [1].The average flow velocity at the volute outlet is given by where the volute velocity constant K 3 is an empirical parameter correlated with the specific speed, as shown in Figure 6 along with other volute parameters such as the volute angle α v and the volute basic circle diameter D 3 [1].
In addition, the volute cross-sectional area A θ can be formulated as where r c is the centroid radius of the volute cross-sectional area, L is the angular momentum of flow at the impeller outlet, it is expressed as L = r 2 V u2 , and C ∼ = 0.95 to account for friction loss.

Diffuser Velocity and Pressure
Difference. Figure 7 shows the velocities at the inlet and the outlet of a vaned diffuser immediately downstream from the impeller.
The outlet velocity V 3 can be determined using [24] where where A 3 is the flow passage area normal to the meridional direction at the diffuser outlet.Finally, the pressure difference between the diffuser outlet and inlet is given by To solve ( 1) and ( 2) numerically while accounting for the boundary conditions and the turbulence model k-ε, the computational fluid dynamics ANSYS-CFX code, based on the finite volume method, was used to obtain the liquid flow velocity and the pressure distributions.In the cases examined involving the impeller, combined impeller and volute, and combined impeller and diffuser, the boundary conditions were formulated as follows: the static pressure provided was given at the inlet, while the flow rate provided was specified at the outlet.The frozen rotor condition was used for both the impeller-volute and the impeller-diffuser interfaces.A noslip condition was set for the flow at the wall boundaries.
Figure 8 shows the inlet, outlet, and interface domains for the selected centrifugal pump components.
Accounting for the fact that the pump rotating speed was constant, the volume flow rate was controlled by a regulator valve, which had an influence on the pressure at the pump inlet as indicated in Figure 9 [23].This was accounted for in the numerical simulations performed.
Furthermore, the ANSYS-CFX code comprises by geometry (DesignModeler), CFX-pre, CFX-solver, and CFXpost modules.According to the applied ANSYS-CFX code, Figure 10 depicts the steps specifically used to obtain the numerical simulation results from the geometry models to the numerical models for the impeller, the combined impeller and volute, and the combined impeller and diffuser.

Results and Discussion
Water was used as the working liquid for all simulations run and for use in this study considered to have the following reference values: temperature of 25 • C for water, density of ρ = 997 kg/m 3 , and dynamic viscosity of μ = 8.899 × 10 −4 Pa s.The main data for the reference impeller, volute, and diffuser are given in Tables 1, 2, and 3.

Case Studies.
Six key design parameters of a centrifugal pump were selected for an examination of their effects mainly on the pump performance: impeller blade height without volute, impeller blade width without volute, impeller blade angle without volute, impeller blade number with volute, impeller diameter with volute, and diffuser blade number with impeller.For the highest accuracy of numerical simulation results, mesh-independent solution

Effect of Impeller Outlet Blade Height.
To analyze the outlet blade height's effect on the pump head, the pump brake horsepower, and the overall pump efficiency, the values 0.012 m, 0.018 m, and 0.024 m were selected for outlet blade height, while keeping the other parameters constant.
Figure 11 shows the pump head as a function of the volume flow rate with the outlet blade height as a parameter.
There, it can be clearly observed that the pump head decreases with increasing volume flow rate due to decreasing liquid pressure.In addition, the pump head increases with increasing outlet blade height.This can be explained by the fact that, when the volume flow rate is kept constant, the increased outlet blade height leads to the decreasing meridional velocity, which increases the pump head since the outlet tangential velocity and the outlet blade angle remain constant.In other words, the liquid pressure drop in the impeller decreases as a function of the increase in outlet blade height.The curves expressing the pump brake horsepower as a function of the volume flow rate are shown in Figure 12, illustrating that the brake horsepower increases with increasing volume flow rate.This can be explained by the additional decrease in liquid pressure relative to the volume flow rate.Also, the brake horsepower increases relative to the impeller blade height due to the requested increase in pump shaft torque relative to the increased blade height.
As depicted in Figure 13, the curves representing overall pump efficiency as a function of volume flow rate illustrate that the overall efficiency for b 2 = 12 mm decreases rapidly to the right of the best efficiency point (BEP).The overall efficiency for b 2 = 18 is highest when the volume flow rate reaches 0.08 m 3 /s.Figures 14,15,and 16 show the corresponding contours for static pressure, liquid flow velocity vectors, and streamlined liquid flow velocities for Q = 0.065 m 3 /s.From these figures it can be observed that the static pressure is higher at the impeller outlet than at the impeller inlet.This is due to the decrease in liquid flow velocity at the impeller outlet.As such

Effect of Impeller Blade Width.
To investigate the effect that the impeller blade width has on the pump head, the pump brake horsepower, and the pump overall efficiency, the blade widths of 4 mm, 10 mm, and 15 mm were selected, while the other parameters were keep constant.Figure 17 shows the pump head as a function of the volume flow rate, illustrating that the pump head decreases with increased blade width.This is due to augmenting the liquid pressure drop with increasing blade width.Also, the required pump brake horsepower decreases when the blade width rises, as indicated in Figure 18.The corresponding overall efficiency  curves are shown in Figure 19, illustrating that the blade width's impact on the overall efficiency is more pronounced in at high volume flow rates.In other words, the overall efficiencies for the three blade widths decrease rapidly to the right side of the BEP and the lowest overall efficiency is obtained when e = 15 mm.

Effect of Impeller Outlet Blade
Angle.Three impeller outlet blade angle values of 9 • , 28 • , and 60 • were selected to investigate their influence on the pump head, the pump brake horsepower, and the pump's overall efficiency.Figure 20 depicts the distribution of the pump head as a 0 0.02 0.04 0.06 0.08 function of volume flow rate and with outlet blade angle as a parameter.This figure thus shows that the pump head increases with increasing outlet blade angle, which can be explained by the increased outlet cross-section size relative to the increased outlet blade angle, thus leading to diminution of liquid pressure drop in flow passage between blades.In addition, Figure 21 depicts the corresponding brake horsepower curves as a function of the volume flow rate, illustrating that the pump brake horsepower increases relative to the augmenting outlet blade angle.This is due to the increase in the requested shaft torque, along with the augmented outlet blade angle.Moreover, the efficiency curves shown in Figure 22 illustrate that the overall efficiency for β 2 = 9 • decreases rapidly to the right of the BEP.
Additionally, Figures 23 and 24 show the static pressure contour and the liquid flow velocity vector for Q = 0.065 m 3 /s.From these figures, it can be observed that the

Effect of Impeller Blade Number When Accounting for
Volute.To investigate the effect of the impeller blade number on the pump head, the pump brake horsepower and the overall pump efficiency, three impellers whose blade number were 5, 7, and 9 were selected, while the other parameters were kept constant.Figure 25 shows the pump head as a function of the volume flow rate, illustrating that the pump head increases with a greater blade number.This is explained by the decrease in the liquid pressure drop in the flow passage with an augmented impeller blade number, keeping the same total volume flow rate.Also, as shown in Figure 26, the pump brake horsepower increases relatively with the augmented blade number.This is due to the increase in the request pump shaft torque, as the pump blade number also increases.In addition, Figure 27 shows the overall efficiency curves, showing that the impeller having 5 blades has the lowest overall efficiency.
Moreover, Figures 28 and 29 depict the corresponding static pressure contour and liquid flow velocity vector for Q = 0.065 m 3 /s, respectively.These figures thus clearly show the increased static pressure difference between the volute outlet and the impeller inlet relative to the increasing blade number.This confirms the reduction in the liquid flow velocity at the impeller outlet relative to the greater blade number, as represented in Figure 29, where the average liquid flow velocities at the impeller outlet were 16.06 m/s, 15.40 m/s, and 12.53 m/s for 5 blades, 7 blades, and 9 blades, respectively.

Effect of Impeller Diameter When Accounting for Volute.
The impeller outlet diameter values of 285 mm and 320 mm were selected to investigate their effects on pump performance when keeping the other parameters constant.Figure 30 shows that the pump head increases with increasing impeller diameter, which can be explained by the fact that the liquid static pressure drop in impeller decreases with increasing impeller diameter.In other words, for a given volume flow rate, the pressure difference between the volute outlet and the impeller inlet is higher for an impeller with a greater diameter.In addition, Figure 31 shows that the brake horsepower increases relative to the increasing impeller diameter, due to the requested augmented impeller shaft torque relative to the size of the impeller diameter.
Moreover, the corresponding overall efficiency curves shown in Figure 32 indicate that the impeller having a great diameter has better overall efficiency with volume flow rates greater than 0.02 m 3 /s.

Effect of Diffuser Blade Number.
To analyze the effect the diffuser blade number has on the pump head, the pump brake horsepower, and the overall pump efficiency, three diffuser models with blade numbers of 5, 8, and 12 were selected, while the other parameters were kept constant.Figure 33 shows the pump head as a function of the volume flow rate, where it is observed that the impact of the diffuser number on the pump head is small, even if the pump head for the diffuser blade number of 8 is highest for the Q between 0.012 m 3 /s and 0.055 m 3 /s.As depicted in Figure 34, the variation in brake horsepower due to diffuser blade number is also small, even if the diffuser blade number of 12 corresponds to a lowest brake horsepower.
Furthermore, Figure 35 shows that, for the low and the high volume flow rates, the overall efficiency for the diffuser

Model Comparison.
The impeller and diffuser combination was selected to validate the numerical approach developed, since the experimental results of this case were available from Technosub Inc.When accounting for experimental boundary conditions for the numerical simulations run, Figures 38-40 show the comparison between the experimental and the numerical results for the pump head, the brake horsepower, and the overall efficiency.The discrepancies observed could be explained by the fact that lost mechanical power, power lost due to leakage, and the pump casing were not taken into account in the numerical simulations carried out.The horsepower for experimental pump brake was therefore higher than the numerical brake horsepower obtained, as illustrated in Figure 39.

Conclusion
In this study, a steady-state liquid flow in a three-dimensional centrifugal pump was numerically investigated.Models of impeller, combined impeller and volute, and combined impeller and diffuser were developed to analyze the effects the key design parameters, including the blade height, the outlet blade angle, the blade width, the blade number, and the impeller outer diameter, had on the pump head, the brake horse power, and the overall efficiency.The obtained results demonstrate, among others, that the pump head and the brake horsepower increase with increasing impeller blade number and impeller blade height, while they decrease with increasing impeller blade width.Also, the interaction between the impeller and the volute reveals that the decrease of the impeller outer diameter keeping the volute dimensions constant leads to the reduction of the pump head and the brake horsepower.The pump overall efficiency is also influenced by the selected key design parameter.A relatively good agreement was observed comparing the developed numerical approach with the experimental results for the case of the combined impeller and diffuser obtained from a pump manufacturer.Radial or perpendicular to the vector U s: Shaft or slip u: Direction of vector U v: Volumetric or volute w: Wall.

Figure 6 :
Figure 6: Volute velocity constant, volute angle, and diameter of volute basic circle versus specific speed.

Figure 7 :
Figure 7: Velocities at the diffuser inlet and outlet.

Figure 9 :
Figure 9: Pressure at the pump inlet versus valve volume flow rate regulation.

StartFigure 10 :Figure 11 :
Figure 10: Steps from 3D geometry model to 3D numerical model and to numerical simulation results.

Figure 40 :
Figure 40: Overall efficiency versus volume flow rate (parameter: numerical and experimental results).

Table 1 :
Main data of the reference impeller [23].
tests were conducted in each case study by varying the number of mesh elements.Table4indicates the required number of mesh elements to achieve mesh-independent results.

Table 4 :
Number of mesh elements.