Introduction

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 105 [m-kW] at pump mode and turbine mode, which are calculated using Eqs. (1) and (2), respectively⁽¹⁹⁾.

where, n_sp specific speed of the pump turbine. N, Q, H, and P are rotational speed (min⁻¹), flow rate (m³/s), effective head (m) and output power (kW). The subscript P, and T represent the pump and turbine modes, respectively.

The pump turbine operates at the same rotational speed of 1800 min⁻¹ for both pump and turbine modes. The head and flow rate values for the pump and turbine modes are H_P = 16.5 m, Q_P = 0.020 m³/s, and H_T = 15 m, Q_T = 0.026 m³/s, respectively. The design of the pump turbine is accomplished with the background theory of gradient of flow variation to head variation⁽²⁰⁾. The meridional shape of the pump turbine is shown in Fig. 1, and the flow direction is according to the turbine mode. The meridional shape for a low specific speed pump turbine is shown in Fig. 1. The inlet and outlet diameters of the pump turbine are 198 mm and 112 mm, respectively, in turbine mode. The outlet width of the pump turbine impeller is 15.2 mm. The pump turbine diameter is considerably small, and leakage loss is prominent. The number of impeller blades is 5 and 14 guide and stay vanes each. The inlet and outlet locations are interchangeable with the pump and turbine modes in the pump turbine. β₁ and β₂ are inlet and outlet blade angles for the pump turbine in pump mode, respectively. The inlet and outlet blade angles are β₂ and β₁, respectively, in the turbine mode of a pump turbine.

Fig. 1 
a) Meridional shape and b) plane view of pump turbine in turbine mode

Fig. 2 shows the blade angle distribution from LE_TM to TE_TM at the hub and shroud in the pump turbine. The blade angle β₂ increases linearly from hub to shroud at LE_TM. The inlet angle is changed from 13° to 22°, 11° to 19° and 7° to 16° at the hub, mid span, and shroud, respectively, in turbine mode.

Fig. 2 
Blade angle distribution of impeller at hub, mid span and shroud in turbine mode

ANSYS 2022R2⁽²¹⁾ is used for CFD analysis of pump turbines in which Reynolds-averaged Navier-Stokes equations are used to solve steady incompressible simulations. The numerical grids play a vital role in CFD analysis. The hexahedral grids were generated using ANSYS 2022R2 ICEM⁽²¹⁾. The numerical grids for the casing, stay vanes, guide vanes, impeller, and clearance gap are shown in Fig. 3. CFD analysis is critically dependent on the numerical grids. The mesh dependency test was conducted to obtain the optimal grid numbers for the stable CFD analysis results. Grid convergence test is carried out with grid sizes G₁, G₂, and G₃ using grid convergence index (GCI). The efficiencies at pump and turbine modes are selected for the sensitivity analysis. GCI is calculated using Eq. (3)⁽²²⁾.

Fig. 3 
Numerical grids of pump turbine model a) casing, b) stay vane and guide vane, c) impeller and d) clearance gap

where G is grid number, η is efficiency, ϵ is the relative error, r is the grid refinement factor, p₀ is the apparent order of numerical solution, the subscript a and ext are approximate and extrapolated relative error values, respectively.

Table 1 shows the discretization error analysis for CFD analysis. GCI is less than 1% for both pump and turbine efficiencies. Fig. 4 shows the mesh dependency test conducted for turbine and pump modes. The mesh dependency test indicated that 6.7 million nodes are suitable for the stable CFD analysis results. Besides numerical grids, the proper boundary is required for accurate numerical analysis. The pump turbine operates in both turbine and pump modes and requires separate boundary conditions for each operation. For turbine mode, the inlet and outlet are total pressure and static pressure, respectively⁽²³⁾. In turbine mode, the guide vane opening controls the flow rate. Similarly, in pump mode, the inlet is static pressure, and the outlet is mass flow rate⁽²⁴⁾.

Fig. 4 
Grids dependency test for pump turbine analysis

Furthermore, the RANS equation with the shear stress transport (SST) model is used to solve simulations of the pump turbine in turbine and pump modes. The full-domain pump turbine was used to perform CFD analysis. The frozen rotor was selected for the interface between stationary and rotating components. The high-resolution scheme is chosen for the advection scheme and turbulence numeric. The residual root mean square value for the convergence criteria is 10^-6. The summarized boundary conditions for turbine and pump modes are shown in Table 2. For the analysis of the S-Curve region, the outlet of turbine mode is replaced from static pressure to mass flow rate. The S-Curve is obtained by decreasing the mass flow rate at the same guide vane opening. The mass flow rate at the turbine mode is reduced gradually to observe the S-Curve region at a partial flow rate. For reverse pump mode analysis, the inlet and outlet locations are reversed while keeping the impeller rotation direction the same as in turbine mode.

The performance curves for the pump turbine in pump and turbine modes with leakage flow were obtained u sing C FD a nalysis. Fig. 5 shows t he performance curves of the pump turbine in pump mode. The best efficiency of 84% and head of 15m is achieved in pump mode. Generally, the design of the pump turbine is considered with pump mode performance curves. The best efficiency point matches the design point in pump mode. Hence, the design of the pump turbine is acceptable. At partial flow rate, the high β₂ shows marginally better efficiency than the medium and low β₂ pump turbine impeller. However, at a high flow rate, the difference in efficiency and head of the pump turbine in pump mode is visible in Fig. 5. The efficiency and head of the pump turbine in pump mode are better in medium β₂ impeller than high and low β₂ impeller. The high β₂ impeller showed higher input power and low head. Hence, Fig. 5 shows that medium β₂ shows better performance of the pump turbine in pump mode.

Fig. 5 
Performance curves of pump turbine in pump mode

Fig. 6 shows the performance curves of the pump turbine in turbine mode. The performance curves of the pump turbine are approximately similar to β₂ variation in turbine mode. The best efficiency of the scale-down pump turbine model in turbine mode is 88% at the medium β₂. The constant head of 15m is reached in turbine mode. The performance curves of the pump turbine showed that the efficiency, head, and output power are approximately the same with the variation of impeller β₂.

Fig. 6 
Performance curves of pump turbine in turbine mode

Pump turbine design focused on the pump mode performance. Therefore, flow irregularities and secondary flow occurrences in the impeller flow passage are usual in turbine mode operation. Fig. 7 shows the blade loading of the pump turbine model in turbine mode. The pressure distribution at the pressure side increases with a rise in flow rate. The pressure at the trailing edge in turbine mode is higher than atmospheric pressure, which suggests that the possibility of cavitation is low in the turbine at various flow conditions. The crossover between the pressure and suction sides is avoided in the current design of the pump turbine in turbine mode. Hence, the design of the pump turbine in turbine mode is acceptable.

Fig. 7 
Blade loading at various flow rate in medium β₂ in turbine mode

Fig. 8 shows the blade loading at Q/Q_BEP=1.00 with various β₂ in turbine mode. The pressure distribution in the impeller blade with various β₂ is approximately the same from a normalized distance of 0.1-1.0. The magnified view of Fig. 8, at a normalized distance from 0.0-0.1, is shown in Fig. 9. The sudden rise in the pressure is observed at the leading edge (LE).

Fig. 8 
Blade loading at Q/Q_BEP=1.00 various β₂ in turbine mode

Fig. 9 
Magnify view of fig. 8 blade loading at Q/Q_BEP = 1.00 various β₂ in turbine mode

The high β₂ showed a comparatively high-pressure rise than medium and low β₂. At 0.02 normalized distance from LE, a sudden drop of pressure occurred. The pressure drop for high, medium, and low β₂ are 113 kPa, 122 kPa, and 142 kPa, respectively. These pressure drops at the LE cause shock loss in the pump turbine at turbine mode. The high β₂ reduces the pressure drop at LE, which eventually reduces the shock loss and improves the S-Curve region in the pump turbine at turbine mode.

Fig. 10 shows the pressure distribution from the hub to the shroud at LE of the impeller in turbine mode. The difference in the pressure distribution from hub to shroud is observed with low, medium, and high β₂ in turbine mode. The low β₂ impeller has a lower pressure value than the medium and high β₂ impeller in turbine mode. It implies that pressure at the inlet increases with β₂ in turbine mode. The pressure distribution from hub to shroud in medium and high β₂ is constant and linear, but with low β₂, the pressure distribution is non-linear. The pressure difference is observed at the hub and shroud in the impeller with low β₂, which leads to recirculation flow and shock loss. The higher value β₂ is preferable for constant and linear pressure distribution at LE from hub to shroud in turbine mode.

Fig. 10 
Pressure distribution from hub to shroud at Q/Q_BEP = 1.00 with various β₂ in turbine mode

The S-Curve region is prepared with a change in the flow rate in turbine mode. The unit discharge and unit speed are used to evaluate the S-Curve region.

where N, Q, H, and P are rotational speed (min^-1), flow rate (m³/s), effective head (m), and output power (kW), respectively. D is outlet diameter according to turbine mode.

Fig. 11 shows the variation in the S-Curve region according to β₂ in turbine mode. The S-Curve region occurred at Q₁₁ from 0.02-0.20 and N₁₁ f rom 60-65 . In turbine mode, the S-Curve region is influenced by head and flow rate. The β₂ variation shows the difference in the S-Curve region. The S-Curve begins at Q₁₁=0.19, 0.21, 0.18, and N₁₁=65.04, 63.81, 63.91 for low, medium, and high β₂, respectively. The S-Curve ends at Q₁₁=0.05, 0.05, 0.05, and N₁₁=63.41, 62.70, 61.92 for low, medium, and high β₂, respectively. The variation in Q₁₁ and N₁₁ changes the inclination of the S-Curve. The low β₂ causes the steeper S-Curve region compared to medium and high β₂. For medium and high β₂, the S-Curve region begins from the same point, but the S-Curve region for the high β₂ ends at a lower value of Q₁₁ and N₁₁. It implies that the steepness of the S-Curve for high β₂ is much steeper than medium β₂. However, the low β₂ has approximately the same ending point of the S-Curve region with medium β₂, but the beginning of the S-Curve region has a high Fig. 10 Pressure distribution from hub to shroud at Q/Q_BEP = value of Q₁ and N₁₁. 1.00 with various β₂ in turbine mode Hence, the S-Curve regions from Q₁₁=0.20-0.05, low and high β₂ are steeper than medium β₂. The steepness of the S-Curve region increases the vortexes and recirculation flow in the blade passage.

Fig. 11 
S-Curve region in the pump turbine with various β₂ impeller

Fig. 12 shows the pressure distribution from hub to shroud at Q/Q_BEP=0.10 in turbine mode inlet. The pressure distribution shows the inclination from hub to shroud in the S-Curve region compared to standard operation in turbine mode. The pressure difference from hub to shroud causes the recirculation flow and vortexes in the turbine mode inlet.

Fig. 12 
Pressure distribution from hub to shroud at Q/Q_BEP = 0.10 of S-Curve region with various β₂ impeller in turbine mode inlet

Fig. 13 shows the velocity vectors Q/Q_BEP=0.90 of the S-Curve region in turbine mode. The velocity vectors are smooth without any recirculation and eddies flow. The low, medium, and high β₂ have a smooth flow from the inlet to the outlet in the impeller flow passage in turbine mode. Figs. 14 and 15 show the velocity vectors at Q/Q_BEP =0.50 and 0.10 of the S-Curve region in turbine mode, respectively. The recirculation flow is clearly visible at the blade passage at Q/Q_BEP=0.50 and 0.10 of the S-Curve region in turbine mode. The recirculation flow in the blade passage at Q/Q_BEP=0.10 is significantly higher at high and low β₂ in turbine mode. The recirculation flow and eddies cause shock loss and flow blockage in blade passage.

Fig. 13 
Velocity vectors in impeller flow passage a) low β₂, b) medium β₂ and c) high β₂ at Q/Q_BEP = 0.90 of S-Curve region in turbine mode

Fig. 14 
Velocity vectors in impeller flow passage a) low β₂, b) medium β₂ and c) high β₂ at Q/Q_BEP = 0.10 of S-Curve region in turbine mode

Fig. 15 
Velocity vectors in impeller flow passage a) low β₂, b) medium β₂ and c) high β₂ at Q/Q_BEP = 0.50 of S-Curve region in turbine mode

Fig. 16 shows the velocity contours in the blade-to-blade flow passage in the S-Curve region of the medium β₂ impeller. The magnitude of velocity decreases with decreasing flow rates. At Q/Q_BEP=0.90, the velocity contours seem smooth without any rotating stall. At Q/Q_BEP=0.50 and 0.10, the small velocity magnitude is observed in the blade flow passage. The low-velocity region in the blade passage creates the rotating stall, which develops the blockage effect and reduces the efficiency of the pump turbine in turbine mode. In the S-Curve region, with a decrease in flow rate, the blockage effect is prominent and changes the slope of the speed-flow curves.

Fig. 16 
Velocity contours in medium β₂ impeller in S-Curve region at a) Q/Q_BEP = 0.90 turbine mode, b) Q/Q_BEP = 0.50 turbine mode and c) Q/Q_BEP = 0.10 reverse pump mode

Fig. 17 shows the radial velocity vectors in medium β₂ at various flow conditions. At Q/Q_BEP=0.90, the bulk of radial velocity is directed inward, and the recirculation flow is minimal. At Q/Q_BEP =0.90, the shock loss and recirculation flow are minimal. At Q/Q_BEP=0.50, the inward and outward radial velocities are observed. The difference between inward and outward radial velocities creates vortexes and shock loss in turbine mode.

Fig. 17 
Velocity vectors in medium β₂ impeller inlet in S-Curve region at a) Q/Q_BEP = 0.90 turbine mode, b) Q/Q_BEP = 0.50 turbine mode and c) Q/Q_BEP = 0.10 reverse pump mode

In reverse pump mode, the quantity and magnitude of outward radial velocity is higher than inward. The intensity of recirculation flow and vortexes is higher, which causes the flow blockage in the blade passage. Hence, vortexes, recirculation flow, and eddies are common occurrences in the S-Curve region.

Fig. 18 represents the velocity triangle in the pump turbine at various operating conditions. At Q/Q_BEP=0.90, the tangential velocity (v_u) and radial velocity (v_m) are comparatively higher. The tangential velocity contributes to the Euler head, and radial velocity directs the flow direction at the impeller inlet in turbine mode. When the flow rate is reduced, the tangential velocity decreases, and radial velocity starts to change the direction from inward to outward direction. Finally, in reverse pump mode, the radial velocity is radially outward.

Fig. 18 
Velocity triangle in S-Curve region at various operating conditions in pump turbine with medium β₂

Fig. 19 shows the velocity contours in the S-Curve region at Q/Q_BEP=0.50 with various β₂. The low velocity region is observed in each impeller flow passage at Q/Q_BEP=0.50 in the S-Curve region. At low β₂, the area of the low velocity region is comparatively higher than that of medium and high β₂ impellers. It implies that the blockage effect in low β₂ impeller is prominent, and the slope change in the speed-flow curve is more prominent than that of medium and high β₂ impellers.

Fig. 19 
Velocity contours in impeller a) low β₂, b) medium β₂ and c) high β₂ at Q/Q_BEP = 0.50 of S-Curve region in turbine mode

Fig. 20 shows the change in radial velocity vectors according to the β₂ at Q/Q_BEP=0.50 in the S-Curve region of turbine mode. The strength of inward radial velocity is higher for high β₂ compared to low β₂. Fig. 20 indicates that the quantity and magnitude of inward radial velocity are higher in high β₂ impeller. The outward radial velocity is higher for low β₂ at each inlet location in turbine mode. In high β₂ at Q/Q_BEP=0.5 0, t he m aximum f low is d irected inward compared to outward.

Fig. 20 
Velocity vectors in impeller inlet a) low β₂, b) medium β₂ and c) high β₂ at Q/Q_BEP = 0.50 of S-Curve region in turbine mode

It implies that high β₂ is suitable to reduce the outward radial flow in the S-Curve region. The velocity components have the same tendency with various β₂ at LE_TM. Finally, Fig. 21 shows the velocity triangle in the S-Curve region at Q/Q_BEP=0.50 in turbine mode. The tangential velocity magnitude is higher at a high inlet angle and lower at a low β₂ at Q/Q_BEP=0.5 0 in t urbine m ode. T he r adial velocity decreases with a decrease in β₂ in turbine mode. The magnitude of outward radial velocity increases with a reduction in β₂ in turbine mode. The high β₂ is better for reducing the outward radial velocity, and low β₂ is better for tangential velocity.

Fig. 21 
Velocity triangle at Q/Q_BEP = 0.50 of S-Curve region with various impeller inlet angle in turbine mode