The pump turbine model is used for performance investigation and internal flow analysis. The specific speed of the pump turbine is 31 [m-m³/s] and 95 [m-kW] at pump mode and turbine mode, which are calculated using Eqs. (1) and (2), respectively.

The pump turbine operates at the same rotational speed of 1800 min^-1 for pump and turbine modes, respectively. The head and flow rates for the pump mode and turbine mode are H_PM = 16.5 m, Q_PM = 0.020 m³/s and H_TM = 15 m, Q_TM = 0.026 m³/s, respectively.

Fig. 1 indicates the meridional and plane views of the pump turbine impeller. The diameter of the impeller is designed to be D₁ = 112 mm for the scaled-down model test.^[13] The β₁ and β₂ are impeller blade inlet and outlet angles for the pump turbine in pump mode, respectively. Fig. 2 shows the impeller blade angle distribution from the leading edge to the trailing edge in pump mode. The performance and internal flow of the pump turbine according to the β₁ and β₂ are examined. The blade outlet angle was fixed at β₂ = 18° and the inlet angle was varied. Similarly, for the change in the impeller outlet angle β₂, the inlet angle was fixed at β₁ = 15°, and the outlet angle was varied. The three kinds of low, medium, and high blade angle distributions for β₁ and β₂, respectively, are selected for the design of the pump turbine impeller.

Fig. 1. 
Meridional (left) and plane (right) views of pump turbine impeller in pump mode

Fig. 2. 
Blade angle distribution of impeller at hub in pump mode

The commercial code of ANSYS 2024R1^[14] is used for the computational analysis of pump turbines in pump and turbine modes. ANSYS CFX 2024R1^[14] uses Reynolds Averaged Navier Stokes (RANS) equations to solve the steady incompressible simulation with a shear stress transport (SST) turbulence model. The SST model combines the capabilities of the κ-ε model away from the walls and the robustness of the κ-ω turbulence model near the walls by blending functions. The SST model was designed to give highly accurate predictions of the onset and the amount of flow separation under adverse pressure gradients by transport effects into the formulation of the eddy viscosity. A large number of studies were carried out for the validation of the SST model.^[15] The SST model is appropriate for evaluating the flow at high rotation speed, especially in pumps and turbines.^[16] The hexahedral numerical grids are suitable for the proper and stable CFD analysis. ANSYS ICEM 2024R1^[14] is used to create the hexahedral numerical grids with a y⁺ < 10. The SST turbulence model with automatic wall treatment was applied for CFD analysis. The automatic wall treatment provides a better near-wall behavior of the turbulent quantities.^[17]

The SST with automatic near-wall treatment allows the smooth shift from a low Reynolds form to a wall function formulation. Therefore, the 1 < y⁺ < 100 range of y⁺ is preferable for SST with automatic near-wall treatment to conduct CFD analysis.

Fig. 3 shows the numerical grids of the pump turbine model. The mesh dependency test was conducted to obtain the suitable mesh number for CFD analysis. The mesh dependency test result is shown in Fig. 4. A 6.7 million node number is selected for the CFD analysis from the mesh dependency test. The detailed boundary conditions are shown in Table 1.

Fig. 3. 
Numerical grids for pump turbine model

Fig. 4. 
Mesh dependency test for medium β₁ pump turbine at Q/Q_BEP=1.0

The performance curves of the pump turbine in pump and turbine modes with leakage flow are shown in Figs. 5 and 6, respectively. The unit quantities are used to compare the performance curves of the pump turbine.

Fig. 5. 
Performance curves of pump turbine model in pump mode according to β₁ and β₂ variation

Fig. 6. 
Performance curves of pump turbine model in turbine mode according to β₁ and β₂ variation

The performance curves were evaluated in the pump mode of the pump turbine according to β₁ and β₂ variation. Fig. 5 shows the best efficiency of 84% matches the design point for low and high β₁ and β₂. At Q₁₁ =0.37, pump mode efficiencies are 83%, 81%, 84% and 83%, for low β₁, high β₁, low β₂ and high β₂, respectively. The best efficiencies of pump mode match the design point. At partial flow rate Q₁₁ = 0.20, the pump mode efficiencies are 65% and 68% for high β₁ and β₂, respectively. The pump turbine impeller with low β₁ and β₂ achieved 70% efficiency in pump mode. At the high flow rate, the performance of low β₁ is comparatively lower than others. At Q₁₁ = 0.20 and 0.90, the pump efficiency is decreased by 7% and 10% with a change in β₁ from low to high values, respectively. The pump mode efficiency decreased and increased by 2% and 9% with a change in β₂ from low to high values at Q₁₁ = 0.20 and 0.90, respectively.

Fig. 6 shows the performance curves of the pump turbine in turbine mode. The best efficiencies of low β₁ and high β₁ are 81% and 85% at Q₁₁ = 0.37 and 0.58, respectively. It indicates that turbine mode performance curves change drastically with β₁ variation. The high value of β₁ shows better performance in the turbine mode. The best efficiency in turbine mode is achieved at Q₁₁ = 0.42 for high and low β₂, respectively. The turbine mode efficiency for low and high β₂ is 87%.

The hydraulic losses in the turbine mode at partial flow rates induce the S-Curve characteristic in the pump turbine. The S-Curve occurs in the turbine mode when the speed-discharge curve has a positive slope. The S-Curves are prepared by changing the flow rate at the constant guide vane opening.

Fig. 7 shows the S-Curve region of the pump turbine with the variation of β₁ and β₂. The S-Curve is prominent when the pump turbine operates at Q/Q_BEP < 0.5. The S-Curve begins from Q₁₁ = 0.18 and 0.19, and N₁₁ = 62.85 and 64.17, and ends at Q₁₁ = 0.09 and 0.04, and N₁₁ = 61.03 and 62.13 for low and high β₁, respectively. The S-Curve starts at Q₁₁ = 0.19 and 0.18, and N₁₁ =65.01 and 63.93, and the endpoints are N₁₁ = 63.41 and 61.94, and Q₁₁ = 0.05 for low and high β₂ pump turbines.

Fig. 7. 
S-Curves of pump turbine according to β₁ and β₂

Fig. 8 shows the variation of P₁₁ vs N₁₁ according to the variations of β₁ and β₂. P₁₁ = 0 indicates zero power output condition in the pump turbine, which is a runaway speed condition. The runaway speeds for low and high β₁ are 62.81 and 65.12, and for low and high β₂ are 65.03 and 63.93, respectively.

Fig. 8. 
P₁₁ vs N₁₁ characteristic curves of pump turbine according to β₁ and β₂

Fig. 9 shows pressure contours with a medium β₁ in the S-Curve range at the turbine mode of the pump turbine. The pressure contours indicate that, at the Q/Q_BEP = 0.5 and 0.1, the intense pressure rise is observed at the blade inlet in turbine mode. With a decreasing flow rate, a sudden rise in pressure at the blade inlet in turbine mode is observed.

Fig. 9. 
Pressure contours in S-Curve range of pump turbine with medium β₁ at Q/Q_BEP = a) 1.0, b) 0.5 and c) 0.1 in turbine mode

The hydraulic losses of the pump turbine in turbine mode include losses in the casing, stay vane, guide vane, impeller, and draft tube. Eq. (6) shows the turbine mode equation for the pump turbine. The objective of the study is to analyze losses in impellers.

Fig. 10 shows the theoretical losses of the pump turbine impeller in turbine mode. As fluid passes through the impeller, it is subjected to hydraulic losses and leakage flow, and ideal energy transfer is not achieved. Shock and friction losses are dominant at a partial and high flow rate. Eq. (7) shows the main loss components in the pump turbine impeller in turbine mode, which includes shock and friction losses.^[19] The flow losses and throttling differences at various guide vane openings are considered for the pump turbine equation.^[19] The pump turbine equations using hydraulic losses are expressed by Eqs. (8)-(16).^[19]

Fig. 10. 
Pump turbine performance difference in turbine mode including hydraulic losses^[18]

ϕⁿ and kmn are values at the rated operating points.

The coefficient of the equation is dependent on the impeller blade angle, which is related to the loss components. The coefficients a, b, and c are related to shock and friction losses in the impeller. The d, e, and f coefficients are related to the throttling difference between guide vane openings. The study objective is to evaluate the impeller losses in the S-Curve region. S-Curves were prepared using CFD analysis. Figs. 11 and 12 show the comparison between theoretical and CFD analysis results for β₁ and β₂ variation at α = 17° and 21°, respectively. The curve-fitting regression methodology is applied to calculate Eq. (8) coefficients from CFD analysis. The losses in the casing, stay vane, guide vane, and draft tube are neglected, letting ζ_m = 0 and ζ_u = 1. The shock and friction loss coefficients were calculated by solving Eqs. (11)-(16).

Fig. 11. 
S-Curve according to β₁ by loss analysis at α=17

Fig. 12. 
S-Curve according to β₂ by loss analysis at α=21°

Fig. 13 shows the S-Curve comparison according to β₁ and β₂. S-Curve inclination is measured to evaluate the steepness of the S-Curve. The S-Curve inclination for low β₁ and high β₂ are 24° and 22°, respectively. It implies that β₁ variation has minimum influence in an S-Curve inclination. The S-Curve inclination decreased from 26° to 15° when β₂ increased from 14° to 20°. Fig. 14 shows the variation of the shock and friction loss coefficient according to β₁ and β₂ in the S-Curve region. The shock and friction loss coefficient decreases and increases with an increase in guide vane opening. At α =13°, the low β₁ and high β₂ showed the maximum and minimum shock loss coefficient. At α =13°, the shock loss coefficient is decreased by 14% and 15% with an increase in β₁ and β₂ from low to high value, respectively. When β₁ and β₂ increase from low to high values, the shock loss coefficient is reduced by 16% and 19%, respectively, at α = 21°. At α = 13°, the friction loss coefficient value is approximately the same for every β₁ and β₂ value. At α = 21°, the maximum friction coefficient value is 1.4 for high β₁. The friction coefficient value is increased and decreased by 20% and 8% with an increase in β₁ and β₂ from low to high value, respectively, at α = 21°. The loss analysis showed that the shock loss coefficient decreases with an increase in β₁ and β₂. The friction loss coefficient decreases with a decrease in β₁ and an increase in β₂. The loss analysis showed that high β₂ has comparatively a lower shock and friction loss coefficient value.

Fig. 13. 
S-Curve according to β₁ and β₂ by loss analysis at α=17°

Fig. 14. 
Loss coefficient in impeller according to β₁ and β₂ in S-Curve region