The Influence of the Geometry of Grooves on the Operating Parameters of the Impeller in a Centrifugal Pump with Microgrooves

Abstract

Centrifugal pumps are one of the most widely used machines in all branches of industry. For this reason, their efficiency has a significant impact on energy consumption in global industry (about 20%). It is extremely difficult to design low-specific-speed centrifugal pumps at acceptable efficiency levels due to increasing losses in the impeller (hydraulic, volumetric, and mechanical). This article presents the influence of the geometric features of grooves (width, depth), and the number of them, on the operating parameters of the impeller in a centrifugal pump with microgrooves. The principle of operation of this solution involves the passive control of the flow in the boundary layer using rectangular grooves. The grooves are made inside the flow channels of the impeller and are intended to reduce hydraulic losses occurring during flow. This is mainly due to a reduction in losses in the boundary layer. In order to determine the influence of the grooves’ parameters on the energy characteristics of the impeller, numerical calculations and experimental measurements were performed on 15 sets of impellers, which were obtained using experimental planning methods. The CFD calculations were conducted in order to understand the flow phenomena that occur in the impeller with the microgeometry and their effect on a reduction in internal flow losses. Based on the research results, useful guidelines for designing impellers with low-speed characteristics were formulated.

1. Introduction

During a research study concerning centrifugal pump impellers that are characterized by ultra-low speed values [1], a concept for reducing hydraulic losses was developed and patented [2]. It assumes modification of the inter-blade channel of the impeller by adding microgeometry [3]. This solution involves making many small grooves on the front and rear walls inside the inter-blade channels of the impeller (Figure 1). Microgrooves were mainly developed to be used in pumps with classic centrifugal impellers that are characterized by low speeds (n_q < 15) and operate within the low-capacity range (Q < 10 m³/h) [4]. Such impellers have flow channels with a small width (b₁, b₂) and a considerable length that corresponds with a relatively large external diameter d₂ (large ratio d₂/d₁). The advantage of the proposed solution is the possibility of implementing it in existing units, which is due to the fact that it does not require significant interference in the pump flow system—changes are only introduced in the impeller. The base geometry is modified by adding appropriate microgeometry, while the main dimensions and angles of the impeller remain unchanged.

As such a solution is new and practically not tested or described in the literature, the main aim of this publication is to present the research results concerning the influence of the geometric features of the grooves on a pump’s operating parameters. The following were used to carry out the research: dimensional analysis, experimental planning, experimental methods, and numerical methods. The research results will allow for a better understanding of the flow phenomena in such impellers and also for the development of guidelines for their design.

2. Preparation of Experiment

2.1. Initial Assumptions

In order to identify the significance of the geometric parameters that affect the operating process of a centrifugal pump with microgrooves, a dimensional analysis was performed with the following simplifying assumptions:

• The geometry of the main flow system of the impeller is unchanged (in both cases, there are the same: d₁, d₂, β₁, β₂, b₁, b₂).
• The cross-section of a groove is rectangular, with width s_r and depth g_r.
• The groove’s depth is constant along its entire length.
• The camber line of a groove corresponds to the camber line of a blade.
• The grooves are evenly distributed on the internal walls of the impeller.
• The grooves are made symmetrically inside the area between the impeller’s blades (there are the same number of grooves on the front and rear walls).
• The compared impellers are made of the same material using the same technology, which allows the same roughness to be obtained—comparable losses of the rotating discs.

2.2. Dimensional Analysis

A dimensional analysis was performed while taking into account the total specific energy of the liquid. Specific energy can be presented as a function, which links the geometric and hydraulic parameters of the pump. This function, after adopting the assumptions presented above, can be written as follows:

Y = g · H = f(Q, P_w, ρ, ν, n, d₂, s_r, g_r, z_r).

(1)

Based on Buckingham’s theorem (Π theorem) [5,6,7,8,9,10,11], function (1) is presented in the form of a function of dimensionless modules (2):

π = f(π₁, π₂, π₃, π₄, π₅, π₆, π₇, π₈, π₉, π₁₀).

(2)

Based on literature [12,13,14], the following parameters were assumed as dimensionally independent variables: liquid density, ρ, rotational speed of the impeller, n, and external diameter of the impeller, d₂ (respectively, π₄, π₆, and π₇), which allowed for the determination of dimensionless products, written as Equation (3):

$$\frac{g\cdot H}{n^2\cdot d_2^2}=f\left(\frac{Q}{n\cdot d_2^3},\frac{P_w}{\rho \cdot n^3\cdot d_2^5},\frac{\upsilon }{n\cdot d_2^2},\frac{s_r}{d_2},\frac{g_r}{d_2},z_r\right).$$

(3)

Assuming steady pump operating conditions (i.e., for Q = const, P_w = const, and ν = const), Equation (3) can be simplified to the following:

$$\frac{g\cdot H}{n^2\cdot d_2^2}=f\left(\frac{s_r}{d_2},\frac{g_r}{d_2},z_r\right).$$

(4)

The obtained function (4) expresses the influence of the geometric dimensions of the groove on the value of the total specific energy that is related to the external diameter of the impeller. The function can be easily transformed to obtain a lifting height of H.

2.3. Experiment Plan

Based on the dimensional analysis performed, the parameters, the change in which would probably have the greatest impact on the characteristics of the pump, were determined. This allowed for the preparation of appropriate geometries of the impellers for the tests, which were conducted based on experiment planning techniques [15,16,17,18].

The scope of changes in the geometric parameters of a groove was estimated based on the technological possibilities of the adopted method of producing impellers (3D printing using SLS technology), as well as on the following assumptions: the maximum depth of the groove is 25% of the impeller’s width (b₂), while the width and thickness of the groove remain constant along the impeller’s channel (rectangular groove).

The range of changes in the values of the tested parameters is presented in Table 1. The number of grooves described in the table is related to the number located on one wall of one inter-blade channel of the impeller. In the calculations, the number of grooves was rounded to the nearest whole number. The basic geometry of the impeller and the characteristic dimensions of the groove are shown in Figure 2.

Table 1. Scope of changes in the examined parameters.

Parameter Name	Symbol	Minimal Value	Maximal Value	Value of the Change	Central Point
Thickness	gr, mm	0.10	0.5	0.13	0.30
Width	sr, mm	0.25	0.75	0.17	0.50
Number of grooves	zr, pcs.	3	7	1.33	5

Table 1. Scope of changes in the examined parameters.

Parameter Name	Symbol	Minimal Value	Maximal Value	Value of the Change	Central Point
Thickness	gr, mm	0.10	0.5	0.13	0.30
Width	sr, mm	0.25	0.75	0.17	0.50
Number of grooves	zr, pcs.	3	7	1.33	5

Figure 2. Characteristic dimensions of the tested geometry (dimensions in mm).

Figure 2. Characteristic dimensions of the tested geometry (dimensions in mm).

A multi-level planning method was used during the research. Based on the literature and the experience and conclusions obtained during similar research [4,19,20], rotatable planning was selected. This allows for the determination of linear quadratic models. Due to the number of analyzed parameters, the experiment was planned as a three-level one, which, with the number of independent variables being known, allowed for the calculation of the necessary experiments for the experiment conducted within the planning with a spherical distribution (5):

N = 2^S + 2·S + N₀ = 15,

(5)

where

S = 3—the number of independent variables;

N₀ = 1—the number of experiments in the core (such a low value for numerical experiments can be assumed due to very high repeatability—there is full control over the boundary conditions).

The value of the star arm for three variables is (6):

$$a=\sqrt[4]{2^S}=1.682 .$$

(6)

The set of impellers for the testing was obtained after multiplying the limit values by the matrix of the experimental plan. This in turn enabled 15 impellers, which differ in terms of the geometry of the grooves, to be obtained. The characteristic dimensions of each geometry and the total number of grooves are shown in

Table 2. Characteristic dimensions of the impellers.

Set Number	gr, mm	sr, mm	zr, pc. (Rounded up)	zrc, pc.
WIR_1	0.17	0.33	4	56
WIR_2	0.43	0.33	4	56
WIR_3	0.17	0.67	4	56
WIR_4	0.43	0.67	4	56
WIR_5	0.17	0.33	6	84
WIR_6	0.43	0.33	6	84
WIR_7	0.17	0.67	6	84
WIR_8	0.43	0.67	6	84
WIR_9	0.08	0.50	5	70
WIR_10	0.52	0.50	5	70
WIR_11	0.30	0.22	5	70
WIR_12	0.30	0.78	5	70
WIR_13	0.30	0.50	3	42
WIR_14	0.30	0.50	7	98
WIR_15	0.30	0.50	5	70

.

3. Research Methods

Numerical fluid mechanics was used as the main research method, whereas experimental methods were used to verify the numerical models. A detailed description of the measurement procedure and verification of the results are described in the following subsections.

3.1. Test Stand and Measurement Procedures

The experimental tests were carried out on a test stand, which is schematically shown in Figure 3. The stand included a tank (1) with a capacity of 1 m³, from which the model pump (2)—operating in a closed system—was powered. The pump’s suction pressure was measured using the FUJI FKP 01 sensor (Fuji Electric, Clermont-Ferrand, France) (4), and the discharge pressure was measured using the FUJI FKP 03 sensor (Fuji Electric, Clermont-Ferrand, France) (5). An ARKON MAGS1-ST electromagnetic flowmeter (Arkon Flow Systems, Brno, Czech Republic) (7) was used to record the volume flow in the system. The pump’s capacity was regulated by a MARS 88V valve (6), which was driven by an IntrOM OM-1 stepper motor (Introl, Katowice, Poland). In turn, the pump was driven by an asynchronous squirrel cage electric motor (3) connected to the pump shaft by a flexible coupling. The engine speed was regulated using a Danfoss VLT Micro Drive FC 51 (Danfoss, Graasten, Danmark) frequency converter. The METEST PPP730 active power converter (Metest, Zielona Góra, Poland) was responsible for measuring power, and the temperature of the liquid was measured by a temperature sensor located inside the tank.

The main element of the test stand was a model pump. The tested impellers were placed in an annular casing that had a rectangular cross-section (Figure 4). Its external diameter was d₃ = 175 mm, and the width was b₃ = 22 mm.

The test stand was fully automated. The measurement process was monitored by a computer and a control program, which allowed for very high repeatability of the test conditions. Measurements were performed in accordance with the PN—EN ISO 9906:2012 standard (Grade 1) [21]. A detailed description of the test rig, measurement procedure, and uncertainty of measuring instruments is provided in the publication [2].

3.2. Numerical Calculations

Numerical simulations were performed as stationary calculations using ANSYS CFX 19.2 software, which utilizes the finite volume method to iteratively solve the system of equations of mass, momentum, and energy conservation [22]. The MRF (Multiple Reference Frames) model was applied for three domains. The stationary domains were the liquid supply element (1) and the annular casing (3), while the impeller (3) moved at a set rotational speed. Detailed information about the boundary conditions is provided in

Table 3. Initial conditions for the stationary calculations.

No.	Initial Condition	Setting
1	Inlet	Gradientless inflow, constant static pressure
2	Outlet	Mass flow—in the case of constant density, it is equivalent to the velocity condition (rectangular velocity profile, Dirichlet condition)
3	Impeller’s domain	Rotation speed n = 2950 rotations/min
4	Liquid	Water with density ${\rho }_{H_{2}O}$ = 998.2 kg/m3 and viscosity ${\mu }_{H_{2}O}$ = 0.001003 Pa·s
5	Temperature	Isothermal calculations, T = 20 °C
6	Fixed walls	No-slip condition, zero speed on the wall
7	Movable walls	No-slip condition, zero speed on the wall, n = 2950 rotations/min
8	Convergence	Minimum level of convergence of the stationary calculations equal to 10−4
9	Advection Scheme	High Resolution
10	Turbulence Numerics	High Resolution

and Figure 5.

Based on our previous research and experience, the Shear Stress Transport (SST) turbulence model was used [4]. This model combines the k-ε turbulence model in the core flow with the k-ω model near the walls [23]. This allows them to preserve their advantages while eliminating their major limitations—maintaining stability in the flow core and being insensitive to mesh resolution in the boundary layer [24].

The discretization of the previously prepared “water” solids was performed in such a way as to ensure the possibility of appropriate mapping of the flow in a groove. Therefore, the size of the mesh mainly depended on the value of the y+ parameter and the “aspect ratio” [25]. According to the basic model assumptions, the required value of y+ should be less than y+ ≤ 2 near the walls [23]. This requirement was fulfilled by keeping y+ ≈ 1 for impeller walls and grooves. These values were constantly checked while the solver was operating (Figure 6).

The Grid Independence Test (GIT) was performed to obtain an optimal mesh size for stationary calculations. The number of mesh elements was around 30 million. Nevertheless, to keep the main parameters of the grid at an acceptable level, the mesh size had to be increased. The characteristic mesh parameters are presented in

Table 4. Main parameters of the numerical meshes.

Impeller	Number of Elements	Average Quality	Average Skewness	Aspect Ratio (Average)
WIR_0 (without grooves)	46,472,396	0.79715	0.20648	2.0746
WIR_1 (the smallest number of grooves)	72,213,142	0.8213	0.20785	2.0312
WIR_14 (the largest number of grooves)	106,056,646	0.81965	0.21698	1.9301

. The detail of the boundary layer mesh in the vicinity of the impeller walls is shown in Figure 7.

3.3. Verification of the Numerical Calculations

Validation of the obtained numerical results was carried out using the example of the base impeller, hereinafter referred to as WIR_0, and the impeller designated WIR_15, which is located in the central point of the experiment plan (for the flow range (0.56 ÷ 1.56)Q_n).

The considered impellers were printed (Figure 8) and tested on the test stand, and then the obtained actual and numerical characteristics were compared (Figure 9a,b).

As can be seen in the graph (Figure 9), the numerical calculations model the characteristics of the tested impellers very well. At the optimal operating point, the error regarding the lifting height between the actual values and those calculated using CFD does not exceed 3%. The largest differences can be observed for the smallest flows, where they amount to approximately 13%.

Based on the analysis of the obtained results, it can be concluded that the adopted assumptions and boundary conditions are correct and that the numerical model simulates the pump’s operation with sufficient accuracy. It can therefore be stated that CFD can be used as a reliable research tool when conducting analysis in accordance with the experimental plan.

4. Results

4.1. Numerical Calculation Results for Q_n

The results of the CFD calculations for the nominal flow Q_n, which were conducted according to the experiment plan, are presented in

Table 5. Calculated values of the lifting height and efficiency for Q_n.

Set Number	H, m	ΔH, m	δH, %	η, %	Δη, %	δη, %
WIR_0	15.03	-	-	35.16%	-	-
WIR_1	15.65	0.622	4.14%	35.40%	0.24%	0.68%
WIR_2	16.21	1.182	7.86%	37.90%	2.74%	7.79%
WIR_3	15.98	0.953	6.34%	37.53%	2.37%	6.75%
WIR_4	16.57	1.546	10.29%	37.54%	2.38%	6.76%
WIR_5	15.71	0.683	4.54%	37.74%	2.58%	7.33%
WIR_6	16.21	1.178	7.84%	37.90%	2.74%	7.78%
WIR_7	15.97	0.941	6.26%	37.49%	2.33%	6.63%
WIR_8	15.42	0.39	2.59%	35.45%	0.29%	0.82%
WIR_9	15.51	0.487	3.24%	35.77%	0.63%	1.78%
WIR_10	16.51	1.485	9.88%	38.13%	2.97%	8.44%
WIR_11	15.69	0.667	4.44%	35.87%	0.71%	2.01%
WIR_12	15.91	0.88	5.85%	36.80%	1.64%	4.68%
WIR_13	16.02	0.99	6.59%	37.19%	2.04%	5.79%
WIR_14	16.16	1.127	7.50%	38.91%	3.75%	10.67%
WIR_15	16.15	1.116	7.43%	37.75%	2.59%	7.38%

. The values ΔH and Δη determine the increases in the lifting height and efficiency compared to the base impeller (WIR_0), while δH and δη represent a relative increase in these values. It should be noted that the presented efficiency is understood as hydraulic efficiency, which is defined as the following (7):

$${\eta =\eta }_h=\frac{\rho gQH}{M\omega }.$$

(7)

When analyzing the calculated values of the lifting height, it can be noticed that its largest increase was observed for the WIR_04 and WIR_10 impellers. The increase in the lifting height was approximately 1.5 m, which gave a relative increase in relation to the base impeller of approximately 10%. All the impellers with microgrooves achieved higher lifting height values than the base impeller.

It can be seen that each of the impellers with grooves was characterized by having a higher efficiency than the impeller without them. The highest increase in efficiency was observed for the impeller marked WIR_14, and it amounted to 3.75%. This means that there was a relative increase with regards to the base impeller of over 10%. The WIR_1 impeller was characterized by having the smallest increase in efficiency. The obtained values did not differ significantly from those obtained for the base impeller.

For this reason, in the next part of the paper, three impellers will be analyzed: the base one (WIR_0), the impeller with the smallest efficiency increase (WIR_1), and the one with the highest efficiency increase (WIR_14). These impellers are marked in Table 5 using underlining.

4.2. Speed Distributions in the Numerical Calculations for Q_n

In order to determine the distribution of the relative speeds inside the inter-blade channel, three orthogonal lines were prepared (Figure 10a). They were located near the inlet edge of the impeller, in the middle of the inter-blade channel, and near the impeller’s outlet. The speeds were plotted on a graph, starting from the active side of the blade (t₀) towards the passive side (t₁).

The distributions were made on appropriate control planes that are perpendicular to the axis of the impeller (Figure 10b). For example, the speed distribution described as ORTO1 SPAN5 correlates with the results obtained on the first orthogonal line, which is located at 5% of the impeller’s width.

When comparing the distributions of the relative speeds made on the control plane that is closest to the front wall of the impeller (SPAN5) (Figure 11) and in the middle of the width of the impeller’s flow channel (SPAN50) (Figure 12), it can be seen that the speed distributions in the impeller with the microgrooves become more irregular (the more grooves, the greater the irregularity) the closer they are to the wall. This can be seen by comparing the distributions obtained for the grooves of the WIR_1-56 and WIR_14-98 grooves.

However, as the distance from the walls increases, the influence of the grooves on the speed field decreases, and for the middle plane SPAN50 (Figure 12), no significant differences between the impellers are visible.

Table 6. Averaged values of the relative speeds on the control planes.

Control Plane	Impeller Designation	Averaged Speed [m/s]
SPAN5	WIR_0	7.542
	WIR_1	7.309
	WIR_14	7.233
SPAN50	WIR_0	10.152
	WIR_1	9.962
	WIR_14	9.959

lists the average values of the relative speeds on the two control planes. This allows for the statement that the introduction of grooves reduces average speeds, which in turn results in a reduction in flow losses. For the WIR_14 impeller (by the wall), the speed is approximately 0.31 m/s lower than in the case of the base impeller. On the middle plane, these differences are smaller (approx. 0.19 m/s) but still noticeable.

It can be concluded that the disruption of the speed distributions, which is caused by the impact of the grooves, reduces the values of the average flow speeds. To precisely indicate how the grooves affect the speed field, the speed profiles determined on the ORTO2 line for three different distances from the wall (ORTO2 SPAN5, ORTO2 SPAN10, and ORTO2 SPAN50) were analyzed (Figure 13). This allowed the following conclusions to be drawn:

• The impact of a groove can be noticed in the distributions in the characteristic “break” of the speed profile—the maximum drop in speed is in the geometric center of the groove.
• The influence of the grooves on the flow is particularly visible near the wall near the active side of the blade (Figure 13a)—the speed profiles of the impellers with microgrooves are characterized by having low relative speed values.
• The graphs show the influence of the number of grooves on the speed distribution. In the case of the WIR_1 impeller, which has four grooves, four distinct depressions are visible on the speed profiles (Figure 13a), whereas in the case of the WIR_14 impeller, seven of them are visible (the impeller has seven grooves).
• The effect of groove depth can be seen by analyzing the depth of the speed profile’s “breaks”. It was observed that the higher the value of g_r, the greater the variability of the speed profile (for WIR_1 g_r = 0.17 mm and for WIR_14 g_r = 0.3 mm).
• The graphs show that the deeper the groove, the greater its impact on the speed field in the inter-blade channel, and this is especially visible when moving away from the wall. On the SPAN10 control plane, the peaks in the speed distribution, which were determined for WIR_14, are characterized by having a larger amplitude than those obtained for the impeller with a smaller depth of the grooves (Figure 13b).
• On the control plane located exactly in the middle of the impeller’s channel, it can be seen that the influence of the grooves is already very weak (Figure 13c). The speed profiles are very similar to each other, and only a slight distortion of the speed profiles is visible. However, the average speed values obtained for the impellers with microgrooves are lower than those obtained for the smooth impeller.

4.3. Analysis of the TKE and TED Distributions

In order to find the location of internal losses, the distributions of the turbulence kinetic energy (TKE) and turbulent dissipation rate (TED) can be used [26,27].

TKE is a measure of the kinetic energy of turbulent fluctuations per unit mass of liquid and is related to the generation of vortices in a turbulent flow. In turn, TED is a measure of the turbulent dissipation rate, and provides information on where the kinetic energy of turbulence is dissipated and converted into heat. For this reason, the analysis of both of these values allows for a better understanding of the flow phenomena and the location of the friction losses in the considered flow to be determined.

Figure 14 shows a comparison of the turbulence kinetic energy distributions made on the SPAN5 control plane. When analyzing the distributions, it can be immediately noticed that the smooth impeller (WIR_0) is characterized by having much higher values when compared to the impeller with microgrooves. This is especially visible around the inlet diameter and the middle part of the channel—near the passive part of the blade. The use of grooves has a positive effect on a reduction in TKE. The fluid areas located directly above the grooves are characterized by having much smaller values of TKE when compared to the smooth areas of the flow channel, as can be seen in the example of the WIR_1 impeller (Figure 14b). In the case of the WIR_14 impeller, which has the most grooves, this difference is most visible (Figure 14c). The most even distribution of TKE can be observed inside the inter-blade channel.

This has a direct impact on the distribution of turbulent dissipation rates because energy dissipation will occur in places with an increased TKE. By analyzing the TED distributions made on the SPAN5 control plane (Figure 15), key loss locations can be identified. For the smooth impeller, this is the area of the inlet to the inter-blade channel and the area of the passive side of the blade. The area of the occurrence of inlet losses was significantly reduced using grooves in both cases. A significant reduction in losses inside the channel was observed in the case of the WIR_14 impeller (Figure 15c), which is characterized by having the largest number of grooves. When looking at the distributions, it can be concluded that this impeller achieved the highest efficiency by reducing internal losses.

When analyzing the TED distributions made on the control plane located in the center of the impeller (SPAN50), it can be noticed that the impellers with microgrooves are characterized by having slightly higher inlet losses when compared to the base impeller (Figure 16). This may be caused by the secondary effect of the groove. However, the impellers with microgrooves have much lower losses inside the inter-blade channel.

In order to estimate the extent to which the use of microgeometry reduces losses in the boundary layer, the TED and TKE values were averaged on the control planes, which were then presented in graphs with regards to the position of the control plane.

Analysis of the average turbulence kinetic energy graph (Figure 17) allows for the statement that for all the analyzed impellers, the areas with high TKE values are located near the impellers’ walls. This is related to friction losses caused by large speed fluctuations in the boundary layer. The use of grooves significantly reduced these values. In the case of WIR_1, they were approximately 1.5 times lower, while in the case of the WIR_14 impeller, they were approximately 4 times lower when compared to the base impeller (WIR_0).

By comparing the turbulent dissipation rate graphs (Figure 18), similar conclusions can be drawn. The areas with the greatest losses are located near the walls. For the base impeller (WIR_0), these areas occupy approximately 40% of the width of the flow channel. The use of microgrooves allows this area to be reduced by approximately 20% in the case of the WIR_14 impeller, which has a positive effect on the reduction of hydraulic energy losses. This leads to the conclusion that the use of microgeometry significantly reduces the scope of the wall’s influence on the flow. The average values of the turbulent dissipation rate are approximately 1.6 times lower for WIR_1 and 2.8 times lower for WIR_14 than those obtained for the smooth impeller. The symmetrical nature of the graphs in relation to the middle plane indicates that there is nearly the same level of loss near both walls.

5. Guidelines for Designing Impellers with Microgrooves

Based on the performed numerical calculations and the analysis of the obtained results, it can be concluded that the most important geometric parameter of the groove, which will affect the energy parameters of the impeller, is its depth g_r and its width s_r. The graph (Figure 19) presents the influence of the s_r/g_r ratio on the increase in relative efficiency. Unfortunately, there is no strong relationship between these parameters. However, it can be seen that the largest cluster of points with the highest efficiency gain occurs within the range (0.8 ÷ 2)s_r/g_r.

The analysis of the dependence of the relative increase in the efficiency on the total cross-sectional area of all the grooves A_r = s_rg_rz_rc allows a certain correlation to be noticed (Figure 20). Based on the obtained data, the regression function (R² = 0.69) can be determined. It is a parabola described with the following equation:

$$\mathsf{\delta }\eta =-0.004A_r^2+0.0116A_r.$$

(8)

This function (8) has a maximum value of approximately A_ropt = 14 mm². In order to transfer the relationship to other impellers, the impeller’s outlet cross-section area needs to be calculated using Formula (9):

$$A_2=\frac{b_2\pi d_2}{{\varphi }_2}=665.7 {\mathrm{mm}}^2.$$

(9)

The optimal ratio of the impeller’s outlet cross-sectional area to the groove’s cross-sectional area can be expressed as follows (10):

$${\xi }_r=\frac{A_2}{A_{ropt}}=47.55.$$

(10)

Based on the calculated ξ_r and by taking the value of the groove dimension ratio from the range (0.8 ÷ 2)s_r/g_r, and also by assuming the number of blades, it is possible to determine the optimal groove dimensions for impellers with other dimensions. However, taking into account the observations from the previous point, it is expected that this relationship will work in the case of impellers, in which the size of the boundary layer will be significant in relation to the impeller’s width (small b₂).

6. Summary and Conclusions

The microgrooved impellers are a new concept in centrifugal pump design, operating at low and very low specific speeds. The concept uses microgeometry to modify the velocity pattern in the boundary layer to reduce hydraulic losses.

Based on the conducted research, the influence of the groove geometric features on the pump impeller head was determined. The main conclusions can be formulated as follows:

• Numerical fluid dynamics can be treated as a reliable research tool for modeling and analyzing flow phenomena that occur in pumps with microgrooves. The errors in estimating the lifting height near the optimal point do not exceed 3%.
• The use of microgeometry has a beneficial effect on a reduction in internal flow losses by reducing the area of impact of the boundary layer. The use of grooves allowed the area of impact of the boundary layer to be reduced by approximately 20%.
• It can be concluded that the influence of the groove’s depth g_r has the most significant impact on flow parameters. This is strongly related to the thickness of the boundary layer. In order for the use of microgeometry to make sense, the area of the negative influence of the boundary layer must constitute a significant part of the flow channel (losses in the boundary layer must constitute a significant part of the losses occurring in the inter-blade channel). For this reason, the possibilities of using grooves are limited to narrow impellers (n_q < 15 and Q < 10 m³/h).
• It was noticed that the influence of the groove on the speed distribution depends on the groove’s depth, and it decreases with decreasing depth.
• The optimal ratio of the impeller’s outlet cross-sectional area to the grooves’ cross-sectional area was determined.