The derivation of the principal equations of fluid dynamics is based on the fact that the dynamical behavior of a fluid is determined by the following conservation laws: the conservation of mass, momentum, and energy. The continuity, momentum, and energy equations in conservation form can be expressed in Equations (1)-(3), as follows:

where V→ is the flow velocity (m/s), ρ is the fluid density (kg/m³), u is the relative velocity (m/s), p is the pressure (Pa), f_x is the body force per unit mass acting on the fluid element in the x-direction (m/s²), τ is the shear force (N), e is the internal energy per unit mass (J/kg), μ is the viscosity (Pa · s), μ_t is the turbulent viscosity (Pa · s) and k is the thermal conductivity (W/mK).

Creating a high-quality mesh is one of the most critical factors that must be considered to ensure simulation accuracy, because the grid quality affects the rate of convergence, calculation precision, and time consumption. The simulation model was divided into four domains: suction, guide vane, runner, and draft tube domains. The grids of the fluid domains were generated with the meshing module, based on the hybrid mesh. The grids for the suction and draft tube were built with hexahedral elements, which are highly space-efficient, while the guide vane and runner meshes were tetrahedral. The refined mesh was implemented at the blade hub and shroud to obtain a higher mesh density, and a 10-layer inflation was applied to all the wall surfaces to increase the calculation accuracy for the flow field in the near-wall regions. Figure 4 illustrates the mesh formation of the guide vane and runner blade, and Table 3 contains the Y-plus value and mesh information for each domain.

Figure 4: 
Three-dimensional model of the runner blade

The grid-independent analysis is illustrated in Figure 5, where P is the output power of the turbine (in the case of 0° GV inclined angle, 6 vanes), and P1 is the output of the turbine with the smallest grid size. Based on the analysis result, Mesh 4 with 4.36 million elements was selected for simulations. When varying the GV number and inclined angle, a similar mesh generation setup was applied to the model.

Figure 5: 
Grid-independent analysis

Numerical analyses using a commercial CFD code, CFX, were performed to predict the turbine performance under different working conditions and various configurations of the equipped guide vane. The turbulence model is an important factor in determining the accuracy of the simulations. As mentioned in several documents, the shear stress transport (SST) turbulence model takes advantage of the combination of k–ε in the far-field, and k–ω in the viscous sublayer at the near-wall regions. The validation of the SST turbulence model can be found in researches on hydraulic turbine performance [7][9][10]. It was determined that SST has a high ability to estimate the vortex appearance and flow separation on the surfaces of complex geometry blades. With the merits mentioned above, the SST turbulence model was employed in the numerical analysis of the current study.

Table 4 lists the boundary conditions applied for the simulations in this study, and Figure 6 illustrates the dimensions and components of the simulation domains. A total pressure corresponding to a gross head of 2.5 m is set at the inlet of the suction domain. Meanwhile, the outlet applied various mass flow rate values to characterize the turbine performance under different working conditions. Water at 25 ℃ is chosen as the working fluid, and the simulation convergence criterion was set to 10^-5.

Figure 6: 
Dimensions and boundary conditions of the simulation domains

First, the performance of the turbine without a guide vane was evaluated at a series of working flow rates, which were alternated gradually to demonstrate diversified operation conditions. The calculation accuracy in the current study was validated by the previous research on a pico-hydro turbine performed by Tran [8]. The propeller turbine was tested numerically and experimentally, and the CFD predicted and experimental results were compared, to validate the numerical method applied in this study. The hydraulic power, output power, hydraulic efficiency, and effective head are calculated using Equations (4)-(8) as follows:

where P_exp is the expected hydraulic power (W), P_h is the actual hydraulic power (W), P_mec is the mechanical power (W), and ρ is the water density (kg/m³), ηh is the expected efficiency, η is the actual efficiency, Q is the flow rate (m³/s), H is the gross head (m), T is the torque on the runner blade (N.m), ω is the rotational speed (rad/s), H_eff is the effective head (m), and P_inlet and P_outlet are the pressures at the inlet and outlet (Pa).

Figure 7 illustrates the performance curves of the hydraulic efficiency and generating power according to the change in flow rate, plotted by numerical simulations. Two characteristic curves were investigated over a wide range of discharge, from 0.75Q_d to 1.25Q_d (Q_d is the rated flow). With an increase in the flow rate, the turbine extracted power increased progressively, reaching 3050 W at Q_d. A higher flow rate generated a higher torque acting on the runner blade, resulting in a higher generated power. Meanwhile, the hydraulic efficiency peaked at 0.79 for the best efficiency point, and dropped at partial or excess flow rates.

Figure 7: 
Performance curves of the propeller turbine

Figure 8 illustrates the distribution of the surface streamlines and pressure on the runner blade under three flow rates: 0.75Q_d, Q_d, and 1.25Q_d. The fluid patterns on the blade were the smoothest, and were evenly distributed in the design flow. The runner working with partial and excess flows witnessed slight flow separation at the leading edge near the blade tip, causing power loss because the radial flow does not contribute to power generation. This phenomenon explains the reason for the highest hydraulic efficiency of the turbine under the design conditions. To turn to the pressure contour displayed on the right-hand side, owing to the energy transfer process, a pressure reduction from the blade leading edge to the trailing edge was observed. There was also a significant increase in the pressure field on the runner when the discharge rate increased.

Figure 8: 
Distributions of the surface streamlines and pressure on the runner blade

The primary function of the guide vane (GV) in a hydro turbine is to direct the fluid flow to the turbine runner. By transporting the water at a proper attack angle, the inlet GV can adjust the blade torque, thereby altering the turbine power and efficiency. Therefore, the installation of GVs play an essential role in improving the turbine performance. To determine an appropriate GV number according to the turbine design working conditions, a series of GV configurations with different vane numbers and vane angles were built and tested along with the turbine runner.

Figure 9 illustrates 3D streamlines of the water pattern throughout the GV and runner, in four cases of GV number (with 30° inclined angle): 6, 8t, 10, and 12 vanes. With an increase in the GV number, its solidity increased, resulting in a change in the flow resistance and flow guidance effectiveness. The impact of the GV number on the turbine performance is summarized in Figure 10.

Figure 9: 
Water patterns for different numbers of guide vane

Figure 10: 
Effect of GV number on turbine performance

Figure 10(a) illustrates the turbine performance under a series of flow conditions. The turbine equipped with 8 GV achieved the highest efficiency, followed by the turbine with 6 GV and 10 GV, while the 12 GV-turbine exhibited the lowest efficiency. In Figure 10(b), the efficiency and generated power at the design point were investigated under the designed operating conditions. With more GV, the turbine output increased gradually, accounting for 3230 W in the case of 12 GV. Nevertheless, the turbine efficiency followed a different trend, peaking at 8 GV with a value of 0.853, compared with 0.851 for 6 GV and 0.846 for 10 GV. Although 12 GV gained the highest output power, it exhibited the lowest efficiency of 0.837, owing to the high flow resistance caused by the increased solidity. More equipped vanes induced a better-directed flow, but it also caused a stronger resistance force for the current in the pipe. Overall, the turbine runner operated most effectively, with 8 GV, because it obtained high efficiency and saved the manufacturing material, resulting in low-cost investment.

The GVs inclined angle is also an important factor affecting the turbine operation. The existence of a GV makes the flow no longer parallel to the turbine axis. Instead, its direction is altered, and the water current contacts the runner blade at a corresponding angle. Figure 11 illustrates the velocity vectors within the guide vane and runner domains, while Figure 12 illustrates the turbine efficiency and effective head in four cases of oblique GV from 10°–40°.The investigated range of angle was chosen as the above values, because if the GV angle is smaller than 10°or greater than 40°, the GV will no longer be helpful. As illustrated in Figure 12, a greater GV angle leads to a higher effective head, which means the turbine requires a higher head to operate at the designed rotational speed and generate the rated output power, which can be explained by the fact that the greater inclined angle resulted in a large GV cross-area perpendicular to the flow direction; therefore, the GV became an obstacle to the flow.

Figure 11: 
Velocity vectors for different angles of guide vane

Figure 12: 
Turbine efficiency and effective head with various GV angles at Q_d

Moreover, a greater inclined angle also caused a reversed flow behind the GV leading edge (marked by the white circle in Figure 11(d)), which negatively affected the main current. Owing to the separated flow, the head loss caused the turbine with 40°GV to obtain the lowest efficiency, solely comprising 0.768. The turbine with 30°GV performed the best, with an efficiency of 0.85, and an effective head of 2.45 m, satisfying the initial designed conditions. This was followed by the 20° GV and 10° GV with turbine efficiencies of 0.843 and 0.818, respectively. Concisely, the 30° GV exhibited better characteristics compared to the other cases, significantly improving the turbine efficiency and extracted power.