Surrogate-Based Optimization of a Centrifugal Pump with Volute Casing for an Automotive Engine Cooling System

: This paper investigates the capability of a surrogate-based optimization technique for the advanced design of centrifugal pumps. The centrifugal pump considered in this work is designed for the automotive cooling system and consists of an impeller, a vaneless diffuser and a volute. A fully three-dimensional geometry parametrization based on Bézier surfaces is presented. The optimization procedure includes the following software packages: Scilab for the geometric parametrization, Ansys-CFX for the CFD simulations and DAKOTA for the optimization management. The initial geometry is deﬁned by a 0D code that provides a preliminary design of the pump, given the operating conditions, i.e., the volumetric ﬂow rate, the head and the rotating speed. In this work an operative point typical of high performance gasoline cars is considered.


Introduction
The role of the centrifugal pump in a car cooling circuit is to guarantee the liquid circulation, and, therefore, to control the engine temperature. The design of a centrifugal pump turns out to be a difficult task for the following reasons: (i) the working point varies over a wide range of flow rate and pressure head; (ii) automotive companies require ever strict constraints for pump dimensions and geometry coupling, possibly different for each car model. Moreover, traditional design approaches based on empirical correlations or trial and error methods (based on previous developed models) are characterized by low accuracy (the former) and by an excessive time/cost to reach the optimal design (the latter).
Optimization algorithms coupled with Computational fluid dynamic (CFD) have demonstrated to be a robust and automatic alternative in the design of centrifugal pumps [1], which can overcome the limits of the traditional approaches. However, the works available in literature are often characterized by a simple parametrization, a small number of design variables [2,3], and only some components are optimized, i.e., the impeller [2,[4][5][6][7][8], or the diffuser [9], or the volute [10].
Centrifugal pumps for the cooling system are characterized by a compact layout to meet the downsizing policies of car makers. The location of the suction pipes is usually prescribed by the engine, and, in general, the vaned diffuser is not adopted because it could decrease the pump performance in off-design conditions. The vaneless diffuser is often used only as a connection between the impeller and the volute. A typical pump geometry is made of an impeller, a small vaneless diffuser, and a volute. The objective of this work is to propose an automatic methodology based on the shape optimization to design from scratch a centrifugal pump for the automotive field. A preliminary work has been presented in [11], where some limitations are present: (i) a baseline design is not available in an automatic manner and, as a consequence, the range of the design space is difficult to set and can be too wide to be spanned with a genetic algorithm; (ii) the vaneless diffuser represents only the gap created by the parametrization of the impeller, and the volute, i.e., it cannot be controlled directly by the optimization process, and (iii) the volute cross section is free to change and its boundaries are parametrized with Bézier curves. The latter choice allows the optimization process to find cross sections with "original" shapes, which are often difficult and/or expensive to manufacture. For these reasons, the approach proposed in [11] has been extended. A baseline geometry is generated by a lumped parameter code and optimized with respect to the efficiency with a genetic algorithm. The optimized baseline geometry is parametrized and the 3D model of the pump is created. Finally, a surrogate based optimization strategy is used to optimize the 3D geometry (impeller, vaneless diffuser, and volute).

Problem Formulation
The operating conditions (see Table 1) of the pump to design are provided by the Italian company Industrie Saleri Italo S.p.A. (http://www.saleri.com/, accessed on November 2021), as a representative working point for a high performance gasoline car.
The objective is to design a pump that satisfies the operating conditions and maximize the hydraulic efficiency. In particular, the design must guarantee a prescribed total pressure rise ∆p t to ensure the proper flow rate in the cooling circuit. ∆p t is equal to p t4 − p t0 , where p t4 is the total pressure at the volute outlet and p t0 the total pressure at the suction pipe. The hydraulic efficiency is defined as follows: whereQ [m 3 /s] is the volumetric flow rate and W [W] the power at the impeller. The total pressure head value reported in Table 1 is constrained in the range ±5% during the optimization process. The geometrical constraints, in general, are stringent and can be ascribed to packaging limitations and to feasibility requirements. The packaging constraints vary with the working point and the application, e.g., the maximum impeller diameter varies in a range from 30 to 90 mm. Table 2 summarizes the constraints for this case. The feasibility requirements depend on the manufacturing technology. The molding, mostly used for the pump manufacture, requires a minimum blade thickness of 2 mm and a minimum outlet blade angle.

Geometry Parametrization
Automotive centrifugal pumps are made up of three main components, i.e., the impeller, the vaneless diffuser, and the volute, as shown in Figure 1. The main symbols adopted in this work are reported in Figure 2, where D 0 is the inlet pipe diameter, D 1,h and D 1,s are the leading edge impeller diameter at the hub and shroud, D 2 is the impeller outer diameter, b 2 is the blade height at the impeller outlet, D 3 and b 3 are the outlet diameter and width of the vaneless diffuser, β b1 and β b2 are the angles of the camber line at impeller inflow and outflow, γ is the stagger angle, and δ s is the relative twisting angle of the leading edge at shroud with respect to the hub. Impeller, vaneless diffuser and volute are parameterized through Bézier curves and surfaces with the open-source software Scilab [12], and 21 design variables (DVs) are adopted: 17 for the impeller (DV i ), 2 for the vaneless diffuser (DV d ), and 2 for the volute (DV v ). Table 3 summarizes the DVs adopted in this optimization methodology, which are described in detail in Sections 3.1 and 3.2. The DVs can include functions of the geometrical parameters chosen to drive the optimization process or explicitly their values.

Impeller
The impeller geometry considered in this parametrization has an axial inflow and a radial outflow. The parametrization is based on 17 DVs, which can be divided in four sub-sets: 3 DVs define the external dimensions of the impeller (DV i,a ); 9 DVs define the camber line at the hub and shroud (DV i,b ); 4 DVs define the meridional channel (DV i,c ); 1 DV defines the number of blades (DV i,d ).
DV i,a includes the impeller outer diameter (D 2 ), the inlet pipe diameter (D 0 ), and the blade height at the impeller outlet, (b 2 ). The bearing diameter is fixed equal to 6 mm. DV i,b includes the impeller diameter at the leading edge for the hub, D 1,h , and the shroud, D 1,s , the blade angle at the inlet for the hub, β b1,h , and the shroud, β b1,s , the blade angle at the outlet for the hub, β b2,h , and the shroud, β b2,s , the stagger angle for the hub, γ h , and the shroud, γ s , and the relative twisting angle of the leading edge for the shroud with respect to the hub, δ s . The blade profiles at hub and shroud are built from the camber line and a thickness function. The camber lines (their projection is described on a plane perpendicular to the rotation axis) are Bézier curves of fourth order (see Figure 3). The blade surface is defined as a linear interpolation between the profiles at hub and shroud.
The axial position of the control points 1, 2, 6, 7, i.e., z 1 , z 2 , z 6 and z 7 (see Figure 4) can be derived from DV i,c and are used to define the two Bézier curves adopted to represent the hub and the shroud for the meridional channel. The points 3, 4, 5 are fixed at z = 0 and points 8, 9, 10 at z = b 2 to obtain a radial outlet. The inlet channel before the blade is controlled by four control points: d and h correspond to the leading edge at the hub and shroud; points a, b, c and e, f, g are aligned in order to guarantee an axial inflow; furthermore, the axial position of points c and g is set to guarantee the continuity of the derivative of the meridional channel at the leading edge for both the hub and shroud.  After defining the control points, the blade camber surface is generated (see Figure 5) and a thickness function is added (see Figure 6): the leading edge is defined with a symmetrical 4-digit NACA profile (NACA0012 has been adopted for this work) with a maximum thickness of 2 mm; when the maximum thickness is reached, this value is kept constant along the rest of the blade. The blade ends with a blunt trailing edge. The outlet of the impeller region is placed at 1 mm from the outer diameter D 2 . DV i,d includes the number of impeller blades n b .

Vaneless Diffuser and Volute
As reported in [1], the vaneless diffuser dimensions, the baffle tongue angle, and the area of the cross section at the volute outflow strongly affect the volute performance. The parametrization is based on 4 DVs. In particular, from DV d , the outlet radius of the vaneless diffuser, R 3 , and the diffuser width, b 3 (see Figure 7) can be obtained. DV v includes the baffle tongue angle, α i , and the cross section area at the volute outflow, A 4 (see Figure 8).
The constraints on R 3 and b 3 always guarantee the presence of a vaneless diffuser, characterized by R 3 − R 2 = 1 mm and b 3 = b 2 . The volute geometry is built starting from the area of the cross section at the outflow. The shape of the cross section is the symmetrical horseshoe-shaped type, commonly adopted in commercial pumps [13] for its good performance. Finally, a linear interpolation is used to define the cross-section area from the outlet until the baffle tongue angle.

CFD Setup
The mesh generation and the flow-field have been computed using the 2020 R2 release of ANSYS [14]. The ANSYS CFX solver is used to solve the incompressible RANS (Reynolds Average Navier Stokes) equations coupled with the SST (Shear Stress Tensor) turbulence model [15].
The impeller inlet section has been extruded along the axial direction to avoid possible disturbance at the inflow due to the blade leading edge. The length of the inlet pipe is L = 3D 1 , where D 1 is the impeller inlet diameter. Only one blade passage is considered with periodic conditions. The mesh of the impeller is generated using the software ANSYS TurboGrid, while the mesh of the volute is generated using ANSYS FLUENT Meshing. The size of the elements adjacent to the solid walls is equal to an averaged non-dimensional distance y + ≈ 20 and y + ≈ 4 for the impeller and volute, respectively. CFX automatic wall treatment is applied to the blade wall, hub and shroud, i.e., CFX automatically switches from a wall function approach to a low Re approach, depending on the grid spacing near the wall.
At the domain inlet the volumetric flow rateQ = 7.33 × 10 −3 /m 3 /s, the turbulence intensity Tu 0 = 4%, and the ratio (ν t /ν) 0 = 10 are prescribed. The corresponding turbulent kinetic energy, k 0 , and the specific dissipation rate, ω 0 , are computed as At the outflow a static pressure p 4 = 0 Pa is set. The no-slip adiabatic condition is applied to the blade walls, hub and shroud.
Steady-state simulations are performed using the multiple reference frame (MRF) approach, which implies no relative mesh motion between the rotating and stationary parts. In the rotating reference frame, where the relative velocity is computed, the momentum equation is modified, adding Coriolis and centrifugal terms. The interface between moving and fixed domain is treated using the mixing plane approach. This approach, unlike the frozen rotor interface, avoids the convection through the pump of non-physical wakes created by the impeller blades and remove the impact of the relative position between impeller blade and volute. Moreover, Fracassi et al. [16] demonstrated how this approach provides results in good agreement with unsteady simulations.
High resolution schemes are used for the velocity and the turbulent quantities.

Mesh Convergence Study
A mesh convergence study has been performed for the baseline geometry, using three grids with the number of elements ranging from 0.83 M to 3.09 M. In particular, Figure 9 shows a detail of the coarse (top-left), medium (top-right) and fine (bottom) mesh, while Figure 10 shows details of the impeller and volute medium mesh for the baseline geometry. The grid convergence study is summarized in Table 4, where the predicted pressure rise and efficiency are reported. As suggested by the convergence study, the grid with 1.53 M elements (medium) ensures a good compromise between computing time and accuracy of the results, and, therefore, it is chosen for the optimization.

Optimization Strategy
Automotive companies require ever strict constraints in the design of centrifugal pumps, both in terms of size and coupling with the engine, which often lead to the manufacture of a new pump for each car model. As the design often starts from scratch, the design space becomes particularly large to explore, and, hence, a global optimization could require a huge computational cost, even if assisted by a surrogate model. As a consequence, the definition of a baseline design is of paramount importance. For this reason a lumped parameter (LP) code has been developed to provide the data to build the baseline 3D geometry, and to define the design space, which is used to perform a surrogate-based optimization (SBO). The SBO is based on a genetic algorithm assisted by a Kriging model. The flow chart of the proposed optimization strategy is reported in Figure 11.  Figure 11. Flow chart of the optimization strategy.

Optimization Based on Lumped Parameter Model
The LP code for the preliminary design of the centrifugal pump with a volute casing is an extension of the approach proposed by Peck [17] (the procedure is reported in Appendix A), and is based on the following input parameters: the flow rate,Q [m 3 Table 3. Notice that the geometry respects the packaging constraints. Otherwise, it is necessary to increase the angular velocity with respect the initial requirement.
The hydraulic efficiency of the LP model is defined as a function of some DVs of the geometric parametrization, i.e., η hyd = f (b 1 , b 2 , D 1 , D 2 , A 4 ), and it is used to drive a cheap optimization based on a genetic algorithm.
The main geometrical parameters provided by the LP code, which are used to build the 3D model of the baseline design, are reported in Table 5.

Surrogate-Based Optimization
In literature, the single-(SOGA [18]) or multi-objective (MOGA [19]) genetic algorithms are used for turbomachinery shape optimization, due to their easiness and robustness: objective functions derivatives are not requested and the probability to remain trapped in a local optimum is very low. To alleviate the computational effort requested by genetic algorithms due to the large number of evaluations, the use of a surrogate model to approximate and evaluate the objective functions during the optimization process is mandatory. Both Kriging (KRG) [20,21] and artificial neural network (ANN) are used for the pumps optimization, even if De Donno et al. [18] show that KRG performs better than ANN in this context.

The KRG model performs a global approximation involving interpolation and the objective function is defined as
i.e.,f is the sum of trend basis functions (here polynomials), which fits the DoE, and a stochastic function (x) with mean zero, which fixes the function to interpolate the experiments. g(x) is the vector of the basis functions, β is the vector of the generalized least squares estimates of the basis function coefficients and (x) is computed using a Gaussian correlation function.
The whole optimization strategy is managed by the Dakota [22] software and is summarized by the following steps: 1.
Computation of a Design of Experiments (DoE) to create a training points database. The DoE is generated using the Latin Hypercube Sampling (LHS) method, which allows to randomly and uniformly distribute the designs over the whole design space. The DoE consists of 10N training points, where N is the number of design variables, as proposed by Jin et al. [23].

2.
Training points evaluation. The points are evaluated using the CFD approach described in Section 4.

3.
Surrogate models generation based on Kriging for the approximation of pump efficiency and pressure head.

4.
Search for the maximum of the efficiency by means of the constrained SOGA applied to the surrogate model. Crossover rate and mutation rate are set equal to 0.8 and 0.1, respectively.

5.
Verification of the maximum through a CFD simulation. 6.
If the convergence criterion is not met, add the maximum to the training points database and return to step 3.
The iterative process described above improves continuously the surrogate accuracy and accelerates the optimization convergence. A global error is defined as where err = |f − f | is the error of the surrogate in evaluating the objective functionsf , i.e., the efficiency and the pressure head, with respect to the value f predicted by the CFD simulation. The convergence is considered to be reached when E < tol E (tol E = 0.05 in this work) for at least ten iteration, and the maximum efficiency is not changed.

Results and Discussion
The DoE consists of 210 training points, but 38 designs have been rejected after their evaluation because of errors in the mesh generation or CFD calculation phase. The efficiency and the pressure head of these design are set to 0.8 and 3.2 bar, respectively, to force the search of the optimum far away. The analysis of the optimization convergence (see Figure 12) shows the absence of further unfeasible design. Figure 12 shows the optimization convergence in terms of efficiency and pressure head. The SBO reaches the convergence after 68 iterations. Both values predicted by CFD and estimated by the Kriging model are reported. Figure 13 deeply investigates the optimization process in terms of the global error (see Equation (2)) and the achieved maximum efficiency. The global error shows a decreasing behaviour, while the efficiency reaches a value above 0.9 starting from 16-th iteration. The optimum design is characterized by η = 0.905, i.e., an improvement of 3% with respect to the baseline is achieved, and ∆p t = 3.35 bar.  The profiles of the blade camberline in a conformal mapping plane are compared in Figure 14 for the baseline and optimal geometries. The abscissa of the coordinate system is where x m = √ z 2 + r 2 is the distance in the meridional direction (the blade leading edge is taken as origin), and represents also the ordinate, r is the distance from the rotation axis, and ϕ is the wrap angle, defined as where β b is the blade angle, and L m the meridional distance between leading and trailing edge. Profiles at 0%, 50%, and 100% of the span are depicted. The baseline profile shows a constant distribution of β b1 and β b2 along the span, while a variation is appreciable for the optimized geometry. The optimal geometry is compared with the baseline in Figure 15, where the blade profiles of the impeller at hub and shroud (right), the meridional channel (left), and the volute shape (bottom) at the outlet are shown. Figures 16-22 compare the flow field between the optimized and the baseline geometries to investigate the effect of the geometrical changes on the efficiency. Notice that maximize the efficiency corresponds to minimize the power at the impeller W, as the ∆p t is set at less than a tolerance. Initially, the effect of the impeller blade geometry is investigated, comparing the pressure coefficient, c p , and the skin friction coefficient, c f , along the blade (see Figure 16) for the baseline and the optimal geometries. The coefficients are defined as follows: where p 0 and U 0 are the static pressure and the velocity magnitude at the inlet pipe and τ wall is the wall shear stress along the blade. The c p distribution on the suction side shows a reduction of the pressure drop near the leading edge, while it assumes lower values on the pressure side. The area defined by the pressure and suction side curves is lower for the optimal geometry, which corresponds to a decrease of the blade loading, and, as a consequence, of W. The difference in the friction coefficient c f at the leading edge between the suction and pressure side is reduced in the optimal configuration; this discrepancy is motivated by an erroneous blade angle at the leading edge for the baseline. On the suction side a lower c f distribution for the optimal geometry confirms a reduction of the losses. Figure 17 shows the static pressure at mid-span for the baseline (left) and optimized geometry (right). The pressure contours for the baseline are characterized by a low pressure area near the leading edge, which is completely removed in the optimal geometry, reducing also the cavitation problems. Figure 18 shows the velocity contours with streamlines at the plane z = 5 mm for the baseline (left) and optimized geometry (right). The small zone of acceleration near the leading edge of the suction side, probably motivated by a not correct blade inlet angle, is removed, and a smoother acceleration is visible in the blade channel. Finally, the wake near the trailing edge is reduced and the velocity distribution at the impeller outlet is more uniform. The flow in the meridional channel is analyzed in Figure 19. The velocity contours show a more uniform distribution along the span in the inlet pipe and inside the impeller for the optimal geometry. This is obtained with a modification of the hub and shroud profile, as shown in Figure 15. In particular, the reduction of the section at the end of the inlet pipe avoids the stall at the hub. Moreover, at the impeller outlet the velocity peak is reduced and the velocity distribution is more uniform along the blade height.
Finally, the flow field in the volute is analyzed. In particular, Figures 20 and 21 show the pressure and velocity contours with streamlines in a plane at z = 4 mm. The pressure contours are quite similar for both baseline and optimized geometries. However, the latter shows a reduction of the low pressure zone at the volute tongue. The velocity contours show smaller values in the optimized geometry for the increased cross-section area (see Figure 22). Moreover, the small recirculation zone after the volute tongue is removed in the optimal geometry.           The distribution of the velocity vectors for the baseline and optimized geometry is displayed in Figure 23 for a plane at z = 5 mm. A small recirculation zone is evident near the leading edge of the baseline geometry, while the optimized geometry is characterized by a smooth distribution of the vectors. A comparison of the baseline and the optimal design in terms of the static pressure rise, ∆p, total pressure rise, ∆p t and hydraulic efficiency is reported in Table 6. Moreover, also single components, i.e., impeller, vaneless diffuser and volute, are compared. The global efficiency is increased by 3%; this improvement has been obtained decreasing the power absorbed by the impeller, and slightly the total pressure rise. The impeller torque C (and the power at the impeller W) is reduced by 11% in the optimal design, and its efficiency, η imp =Q∆p t,imp /W, is increased by 3%. The reduction of C can be ascribed to the decrease of the blade loading, as shown in Figure 16 by the c P distribution, and of the losses. In particular, lower losses were achieved modifying the shape of the impeller blade and of the meridional channel, as shown in Figures 17-19. The vaneless diffuser of the optimal design provides an increase of the static pressure head about 5% together with an halving of the total pressure losses. The higher static pressure rise is achieved increasing the radial size of the vaneless diffuser due to the reduction of the impeller outlet diameter D 2 , and the increase of the diffuser outlet height, b 3 . There is also a reduction of the total pressure losses, which can be ascribed to the lower velocity of the flow entering the diffuser, as shown in Figure 19. The optimization of the volute leads to a slightly decrease of the total pressure losses. In fact, a higher volute area leads to a lower velocity but a wider wet surface, which have an opposite effect of the friction losses. The recover of the static pressure is decreased due to a slower flow at the exit of the vaneless diffuser. Table 6. Comparison of the baseline and the optimal design for the impeller, the vaneless diffuser and the volute. η imp is the impeller efficiency, C the torque absorbed by the pump shaft, ∆p the static pressure variation, ∆p tot the total pressure variation, η the pump hydraulic efficiency.

Conclusions
A procedure for the automated design of a centrifugal pump for the cooling of car engines is presented. It involves a shape optimization of the impeller, vaneless diffuser and volute, with the objective of maximize the efficiency of the pump, keeping constant the operating conditions. The baseline geometry is obtained with a lumped parameter code. The baseline geometry is enhanced with a 3D surrogate based optimization.
The proposed approach demonstrates its robustness, as it provides for the prescribed operating condition a 3D design from scratch, which was optimized to enhance the pump hydraulic efficiency (η is increased by 3% at the end of the optimization).
The optimal design presents a decrease of the blade loading, and a smoother acceleration along the pressure side. Moreover, the low pressure area at the leading edge is reduced, and, consequently, the related cavitation problems. The velocity field across the meridional channel is modified by changing the hub and shroud profile to obtain a more uniform distribution. In particular, the reduction of the section at the end of the inlet pipe avoids the stall at the hub. Moreover, at the impeller outlet the velocity peak is reduced and the velocity distribution is more uniform along the blade height. In the volute the pressure distribution is quite similar for both designs. However, the optimal design shows a reduction of the low pressure zone at the volute tongue, and smaller velocity values for the increased cross-section area. Moreover, the small recirculation zone after the volute tongue is removed in the optimal geometry.
The components that have the greater impact to improving efficiency are the impeller and the vaneless diffuser. Only a slight improvement is obtained in the volute, probably for the reduced number of DVs used for its geometrical parametrization.
Each simulation has been run in parallel on 32 cores, and 1 h is needed to complete the workflow (mesh generation and CFD simulation). DoE required 210 simulations, while 68 iterations were needed by the optimization procedure to converge to an optimum design. The computational cost of the whole optimization process was approximately 280 h (≈12 days) on 32 cores.
Ongoing work is devoted to enhance the parametrization of the volute, and to reduce the global number of DVs with an active subspace technique. The theoretical head, H th , can be related also to the pump head, H, with the Pfleiderer [24] slip coefficient, p, as where where K s is the impeller slip constant, and can be chosen in the ranges 1.7-1.8 and 1.5-1.6 for flat and fully backed off impeller outlet tips, respectively. R is the blade loading ratio and can be defined as where K A is the projected area coefficient and can be defined as Equating Equations (A5) and (A10), an equation of the form is obtained, where A, B, C, and F are known constants. This equation can be solved iteratively, assuming a tentative value for D 2 that neglects the effects of the vanes number, the outlet angle, and can be computed from the following equation: where K sp is a speed coefficient defined as Finally, the height of the blade, the area, the meridional velocity, and the blade velocity at the impeller outlet are given by: The following quantities at the impeller inlet are computed: the impeller diameter, D 1 , the blade height, b 1 , the inflow area, A 1 , the meridional velocity c m1 , the blade velocity, u 1 , and the blade angle, β b1 . From the eye velocity, v e = k e c m2 (1.3 < k e < 1.5), the eye diameter, D e , is computed as and D e = D 1 for this pump configuration. The inflow impeller angle, β b1 , can be computed as where c m1 is the meridional velocity at the impeller inlet, and is defined as where 1.3 < k 1 < 1.5, u 1 is the inlet blade velocity and is computed as The inlet whirl velocity, c u1 , is defined as where c 1 = K qm c m1 (1.1 < K qm < 1.2), and the height of the blade can be computed as Finally, the main dimensions of volute casing are computed, neglecting the vaneless diffuser. The whirl velocity at the impeller outlet is computed as The velocity at the volute throat is computed as where with K h = 27.8. Applying the free vortex theory, the radius and the area at the volute throat can be computed as To include the design procedure in an optimization loop, it is necessary to redefine η h as a function of the DVs used for the optimization. Impeller losses, l imp , can be defined as the sum of two contribution, l imp,m f (mixing and friction losses) and l imp,s (shock losses). The mixing/friction contribution can be defined as where c D is the dissipation coefficient where b * 2 = b 2 /D 2 , and c f is the friction coefficient where = 0.00005 m is the roughness, L b the blade length, which is approximated as L b ≈ D 2 − (D 1 − D r )/2, Re = w ave L b /ν the Reynolds number, ν the kinematic viscosity, and w ave is the average relative velocity across the impeller where a 1 and a 2 are the distance between vanes at the leading and trailing edge, respectively, and can be generally defined as The hydraulic diameter, D h , is defined as