In order to identify flow phenomena occurring during the liquid flow in the stator type and to determine the rational flow geometry (which has optimal operating parameters), numerical flow analyzes were made using CFD (Computational Fluid Dynamics). As a base liquid model of pump – geometry of an annular casing and a multi-piped impeller implemented in the test rig pump – was created as a 3D model in Catia software. Further, geometry of the impeller remained unchanged. However, the type and construction of stators cooperating with the impeller were altered. Geometry of basic annular casing was calculated and designed according to theory included in [3, 4].

In order to perform the calculations Ansys Fluent from Ansys Workbanch 18.2 package was used. The geometric model of the pump consisted of the following volumes: inlet, impeller, annular-type casing (and volute casing in late calculations), outlet. The simplified geometric model of the pump used for discretization was presented in Figure 6.

Figure 6 Simplified model of flow geometry of a multi-piped pump with an annular casing.

To determine an optimal size of a grid from accuracy and speed of calculation point of view the GIT (Grid Independence Test) was performed. The calculations were performed for six variants of grids, which differed in size of the element (about 30% each). In each case the grid was built from the tetrahedral elements and concentrated in the areas of walls of the impeller and stator (used function Inflation in Ansys Fluent). For further calculations, the smallest size of the calculation grid was assumed, for which the discrepancies in the results of the comparison parameters (in this case total head pump H_{u} and total moment on the impeller rotating surfaces M_{t}) with the next gradation of mesh were in the range: 2.8% for H_{u} and 0.9% for M_{t}. In chosen grid the minimum size of element was 0.2 mm and total quantity of tetrahedral elements was 29 million. The y + parameter on rotating surfaces does not exceed 1.

The all numerical calculations were performed as transient flow (PISO algorithm, second-order upwind for all equations, double precision Solver, convergence criterion for each equation was *ϵ* = 10^{−6}) in Ansys Fluent software. Rotational speed of the multi-piped impeller was constant and was n = 2870 rpm. For one rotation of an impeller 120 time steps were assumed, the value of one time step determined in accordance with [8] was t_{s} = 5.2·10^{−4} s. The calculations were performed by means of a turbulence model k-*ω* SST, which one was choose as optimal turbulence model by procedure described accordance with [5, 8]. Other – necessary for calculations – boundary conditions were defined compliant with Figure 5, as:

Inlet model – velocity of the liquid at the inlet corresponding the assumed efficiency and intensity of turbulence T_{i} = 4%.

Walls – velocity of the liquid in a perpendicular direction to the wall u_{x} = 0, zero pressure gradient dp/dn = 0.

Outlet model (diffuser) – the assumed pressure at the outlet p = 400 kPa, intensity of reverse flow turbulence T_{ibf} = 2%, constant liquid viscosity, constant mass flow.

To evaluate correctness of the assumptions for the numerical model the validation of numerical results was performed by comparing them with the experimental test results in the whole range of a pump characteristic. In the whole area of a pump with a multi-piped impeller and annular casing the difference between the experimental results and numerical ones does not exceed 3%. Numerical characteristics coincided with the experimental characteristics of model pump (Figure 5). The verified numerical model was applied in the tests on the influence of the type and construction of the stator cooperating with a multipiped impeller on the pump’s energetic parameters.

Main operating parameters of the tested centrifugal pump were determined in compliance with the following equations:

$$\begin{array}{r}H=({p}_{cout}-{p}_{cin})/\rho g\end{array}$$(1)$$\begin{array}{r}{P}_{w}={M}_{t}\omega \end{array}$$(2)$$\begin{array}{r}{\eta}_{h}={P}_{h}/{P}_{w}=\rho gQH/({M}_{t}\omega )\end{array}$$(3)$$\begin{array}{r}\eta ={\eta}_{h}{\eta}_{v}{\eta}_{m}\end{array}$$(4)The total efficiency of the pump (4) was determined assuming values of mechanical efficiency *η*_{m} = 0.90 and volumetric efficiency *η*_{v} = 0.92. Mechanical efficiency can be treated as a constant value, because it depends on the losses in bearings and sealing. Volumetric efficiency mainly depends on the pressure difference in an impeller gap sealing what it is unchangeable in our tests. Due to this fact the volumetric efficiency can be treated as constant as well. The hydraulics efficiency is the key factor especially for pumps working in ultra-low kinematic specific speed – it mainly affects the total pump efficiency.

The cooperation of the multi-piped impeller with two types of stators was analyzed in this paper: an annular casing and volute casing – with different configurations of

their geometric features. Outlet elements were designed in accordance with commonly used theoretical models based on one-dimensional flow theory of centrifugal pump (according to the constant and averaged velocity of liquid in the stator after impeller outlet).

The algorithm of numerical simulation of flow geometry rationalization was prepared to analyze the cooperation of a multi-piped impeller with an annular casing at various configurations of construction features. The following geometrical parameters of stator were the researched (Figure 7a)

Figure 7 Analysed features of flow geometry for both types of stator: a) annular casing, b) volute casing.

In first step – impact of the width b_{3kk} of the cross-section – basic value of width of model pump was 22.5 mm.

After finding a rational value of width, the next step was to research the outer diameter d_{4} of the channel – basic value of outer diameter of model pump was 180 mm.

After finding a rational value of outer diameter, the last step was to research the cross-section shape – basic was rectangular profile.

Then the algorithm of numerical simulation of flow geometry rationalization was prepared to analyze the cooperation of a multi-piped impeller with a volute casing at various configurations of its geometric features. The following parameters were tested (Figure 7b)

In first step – impact of the angle B_{cw} of the beginning of the volute tongue – basic value angle of theoretical spiral casing was 0^{∘}.

After finding a rational value of angle, the next step was to research the width b_{3sp} of spiral part of stator – basic value of width was 18.5 mm.

After finding a rational value of width, the last step was to research the cross-section shape – basic was rectangular profile.

The important information – the width of non-spiral part of stator b_{3} was constant in all numerical calculation and was equaled rational value of width b_{3kk} of annular casing (orange dimension b_{3} = 18.5 mm in Figure 7b)

The shown research changes of geometrical parameters of stators. Range of changes of value, base value of parameters (initial parameters of simulations) and rational value of them was contained. Base value of geometrical parameters of annular casing was taken from model pump (from test rig). Initial value of geometrical parameters of volute casing was calculated.

Table 2 Research changes of geometrical parameters of stator type.

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