The blade geometry is designed using our laboratory software, which is programmed based on the blade element momentum theory and other additional calculations. The BEMT, which is used for analyzing the hydrodynamic performance of wind or tidal turbines, is employed in the design process of the turbine blade. The general BEMT is based on the combination of momentum and blade element theories. The momentum theory is used to derive the axial and tangential flow reduction factors with inclusions of tip loss factors to consider the finite number of rotor blades. The blade element theory is used to model the blade section drag and torque by dividing the rotor blade into a number of elements [7]. Figure 3 shows the components of velocity and force distribution in a blade element.

Figure 3: 
Velocities and forces acting on a blade element (source: Blade Element Theory, W. Froude, D. W. Taylor, S. Drzewiecki)

The repeated calculation of blade parameters is illustrated in Figure 4, where the output power, expected coefficient, mechanical efficiency, number of blades, and incoming water velocity are inputted as initial key factors. The blade diameter and tip speed ratio are determined by Equation (1) and Equation (2), respectively, as follows:

Figure 4: 
Flowchart of design process

where D is the blade diameter, P is the output power, C_p is the expected efficiency, η is the mechanical efficiency, λ is the tip speed ratio, ν₁ is the flow velocity, V_rel is the relative velocity, ω is the angular velocity, Chord_M is the blade mean chord, and ν is the kinematic viscosity.

In the first loop, the blade mean chord is estimated and utilized to calculate Reynolds number. Then, the hydrofoil dynamics characteristics at that Reynolds number are analyzed at several attack angles (α) using XFLR5 software to determine the value of α where Cl/Cd ratio is maximum. Using the initial value, a = 1/3, the tip loss factor is calculated for all blade sections by Equation (4). Thereafter the axial induction factor, -a, is recalculated Equation (5).

where R is the blade radius, r is the radius at a certain position, C is the chord length, B is the number of blades, f is the tip loss coefficient, a is the axial reduction factor, and Cl is the lift coefficient.

Equation (6) determines the chord length of each blade section based on the lift coefficient (C_l), tip loss factor (f), and recalculated axial reduction factor (a). Hence, the mean chord of the whole blade is calculated and compared to the initially assumed mean chord. This loop repeats several times until two values of the mean chord (estimated and calculated) are close to each other. At the end of the process, the twist angle for each section is specified as the difference between angles φ and α.

In this study, the airfoil covers 70% of the blade length, from r/R = 0.3 to r/R = 1. The blade is divided into 15 different sections; however, only the parameters for the eight main sections are shown below. Several calculations yielded geometry data for blade design, including twist angle and chord length for each blade section, from root to tip as listed in Table 1. Meanwhile, the tip loss coefficient is calculated and illustrated in Figure 5.

Figure 5: 
Tip loss coefficient

Table 2 summarizes the designed parameters of the 10W pico-hydro turbine. The turbine is designed to operate at a rated velocity of 1.2 m/s. From the calculations, the outer and hub diameters of 0.21 and 0.074 m, respectively, were obtained. Based on the results of another research conducted in our laboratory, the number of blades was three. The three-blade turbine exhibits its best performance at various incoming water flows and a number of rotational speeds. After the design parameters are obtained through several calculations, the Solidworks software is employed to model the 3D geometry of the turbine.

The casing, which is an indispensable component of the turbine apparatus, not only functions as a frame to protect the blade from damage caused by external forces. It also contains a small generator, which converts the turbine rotation to electricity and stores it inside an enclosed battery. Because the turbine is portable and can work flexibly underwater, the turbine blade cannot operate on its own; a front casing is required. The casing functions as an installation space as well as an intermediate component between the blade and connecting rod. The casing design must satisfy some of the following requirements: adequate strength to protect the blade from external impact; sufficient space for the blade and generator to fit; simple construction to reduce flow velocity reduction. Based on the aforementioned criteria, and considering the axial distance (between the casing frame and blade) and tip clearance (distance between blade tip and inner wall casing), the casing was built using the detailed parameters shown in Figure 6. The casing consisted of four supports with a 238 mm outer diameter, an 84.2 mm total casing width, and a 76 mm room for installing the generator. The front edges that are directly in contact with the flow are rounded in order to reduce flow resistance. The full three-dimensional model of the turbines (with and without casing) are displayed in Figure 7.

Figure 6: 
Design of the casing

Figure 7: 
3D Model of the turbine

In this study, the CFD simulation method is employed to analyze the performance of the pico-hydro turbine under different operating conditions. In the simulation analyses, the turbine performance was tested at a TSR range of 0.5 - 4 with a 0.5 step. Three incoming water velocities were applied: 1, 1.2, and 1.5 m/s. The turbine efficiency is evaluated by a power coefficient, which is calculated from the torque on the blade given by Equation (8).

where T is the torque on the blade and A is the sweep area.

The turbulence model is an essential factor; it indicates the accuracy of numerical simulations. The shear stress transport (SST) turbulence model is one that is preeminently built based on the k–ω model in the near wall region, but utilized in the k–ε formulation on the far field. It is designed to provide highly accurate initial predictions of the onset and amount of flow separation under adverse pressure gradients through the inclusion of transport effects into the eddy-viscosity formulation. This results in a major improvement in terms of flow separation predictions. Because of these improvements, it is commonly used in modeling wind and hydro turbines, as utilized in this study. In his research, M. Rahimian concluded that the simulation results that employed the SST turbulence model agreed better with the experimental results than those of other models [8]. The blade region is controlled by a y-plus value (Y+) of approximately 1 in order to ensure the accuracy of the near wall treatment at low Reynolds number computations provided by the SST turbulence model. The current numerical computation is performed with multiple frames of reference approach; the flow field and casing domains are in a stationary frame while the turbine blade domain is in a rotating frame.

The hybrid mesh is created independently for three fluid domains, including the flow field, rotor blade, and casing domains, using the commercial software Ansys-Workbench. The unstructured mesh is used by dint of the casing complexity and the leading and trailing edges in the rotor domain, whereas the flow field domain employs a structured mesh based on hexa-hedra type elements. A local refinement of the mesh is applied at some important regions, such as the leading and trailing edges of turbine blades, in order to capture the flow field structure precisely. The distribution of the y-plus value on the blade can be observed in Figure 8. The number of elements in each domain is carefully determined after conducting a mesh-independent study. To validate the accuracy of numerical computations, it is essential to conduct a mesh sensitivity study of computational domains. As for the theoretical aspect, the finer the mesh, the fewer grid-related errors occur [9]. Thereafter, the power coefficient at a rated TSR and the designed flow velocity are considered as standard parameters to evaluate five grids and determine the impact of mesh size on solution accuracy. The same turbulence model (the SST) is used to compare the power coefficient of each case.

Figure 8: 
Mesh formation and Y⁺ distribution on the blade

The mesh-independent study is conducted in five cases with different grid sizes from rough to fine meshes and extremely fine mesh. The number of elements for each grid and the influence of mesh on power coefficient are demonstrated in Figure 9. It can be clearly seen that the turbine obtained a higher power coefficient when a finer mesh was applied. However, the discrepancy among the power coefficients of the grid with 7.4, 8.9, and 9.5 million elements was negligible. Therefore, to ensure the balance between the simulation accuracy, computer resource, and time consumption, a grid with 7.4 million elements was considered as the most suitable to achieve precise simulations.

Figure 9: 
Impact on grid size on Cp

Figure 10 illustrates the flow field domain dimension, turbine installation position, and boundaries. In terms of the flow field, the domain size is 5R × 10R × 25R (where R is the blade radius), which is sufficiently large to capture the flow pattern and wake behind the turbine. The inlet boundary is set at a normal speed. A static pressure is imposed at the field domain outlet. The top and side faces are of the free flow boundary type where water can freely enter or exit the domain, whereas the casing and turbine blade are set up as a free slip wall boundary. In this numerical simulation, multiple frames of reference are involved. The rotor domain is in a rotational frame, whereas the two remaining domains are in stationary frames. The interfaces between rotational and stationary components are set as a frozen-rotor type, and those between two stationary domains are set as general connections. All simulations were performed in a steady state analysis and reached a convergence criterion of 1×10⁻⁵ for accuracy.

Figure 10: 
Dimensions and BCs of flow field domain

Figure 11 illustrates the wake pattern after water passed through the turbine when it worked at TSR 2 and an incoming velocity of 1.2 m/s. By dint of operating under a designed condition, the turbine experienced a good behavior. The streamline appeared smooth without any abnormality in its flow pattern. It is reasonable according to the basic characteristics of fluid particles that passed through the turbine rotor. In the case of installing a casing to work with the turbine blade, the generated wake was more complex considering that more vortices were formed than when the turbine was bare.

Figure 11: 
Wake pattern behind the turbine

Detailed streamlines on the blade surface are shown in Figure 12; the streamlines on the blade surface were distributed more evenly and reasonably by the increase of TSR (from TSR 1 to 3). In these computational analyses, the TSR values were changed by keeping the flow velocity constant and varying the rotational speed. At TSR 1, because of the low rotational speed, there were radial flows on the suction surface where water particles tended to move in a direction away from the root to the tip of the blade. This type of flow does not contribute to the energy transfer process between water and turbine. This phenomenon explained why the turbine extracted less energy when working at a low TSR value. At a higher rotational speed (for TSRs 2 and 3), the fluid within the turbine’s wake region flowed following the turbine’s rotational movement. The streamlines were smoother without disturbance and the radial flow ceased. It indicated that most of the water particles that passed through the turbine forced the blade to rotate and generate power.

Figure 12: 
Streamline on blade surface at different TSR

Figure 13 plots the velocity field around the blade and casing. The flow velocity variation ranged from 1 to 1.5 m/s. It can be observed that when the water flow came into contact with the casing and blade, some areas with high speed motion (such as in the casing edges and blade tips) were generated. This occurred because of the combination of flow velocity and rotor blade rotation; it made the velocity at the blade tip increase significantly and caused a well-known phenomenon - a blade tip vortex. It was observed that at a high inflow velocity, the low speed area behind the turbine was narrow. The torque on the blade was achieved from simulations; hence, the extracted power was calculated at different TSR values and flow velocities; results are compared and presented in the following section.

Figure 13: 
Velocity field at different working conditions

Figure 14 shows the output of this turbine at three water velocities: 1, 1.2, and 1.5 m/s. Evidently, higher flow velocities resulted in higher generated power. Under all circumstances, the maximum power output was obtained when the turbine rotated at a rated speed of TSR 2 followed by TSR 2.5, with a small reduction in the absorbed power. At a lower TSR (under 1) and a higher TSR (above 3), the turbine degenerated. This was indicated by the lower amount of energy obtained from water. In the case of the 1.5 m/s flow velocity, the power was considerably higher compared with those of the two remaining conditions. However, at a velocity of 1.2 m/s, the turbine was also capable of generating an amount of electricity, which satisfied the design requirement. The simulation results exhibited a good agreement with the estimated power calculated by a formula in which power is directly proportional to the cube of velocity.

Figure 14: 
Generated power

As for the turbine efficiency, Figure 15 shows the turbine power coefficient when working with and without casing. In general, under both operating conditions, according to the increase in the TSR, the turbine efficiency increased gradually. It achieved the highest value at the rated TSR before it decreased and thereafter increased to another higher value. Without the casing, the turbine performed best at TSR 2 and peaked at a Cp of 0.37. The presence of the casing resulted in a Cp reduction for the whole range of the surveyed rotational speed. As demonstrated by the 2.5% decrease in turbine efficiency, at a low TSR, the casing effect on turbine performance was low. On the other hand, when the TSR values were either rated or higher, the power coefficient was significantly influenced and accounted for 4.5% of the decrease. In others words, with the same flow velocity but at a higher rotational speed, the casing makes it more difficult for the turbine to obtain energy from water. Nevertheless, the casing was essential to protect the turbine blade from external forces as well as to house a small generator that converted the rotational movement into electricity.

Figure 15: 
Turbine efficiency

A series of hydro turbine efficiency curves produced by several researchers are plotted in Figure 16. Although each turbine had its own peak at a different TSR value, the efficiency curve patterns were the same. Among these, Coiro's turbine [10] had the best performance with a maximum Cp of 0.39, followed by the work reported by Bahaj [11]. The studies of Bryden [12] and Kinnas [13] had similar maximum power coefficients. On the other hand, the 10W pico-hydro turbine introduced in this study only had a Cp of 0.33 (indicated by the red line) in Figure 14. The results of previous studies provided valuable information and were a useful reference to validate the current study. The 10W pico-hydro portable turbine should be tested with a real model in experiments to confirm the reliability of the CFD results. This will be the next step in our research.

Figure 16: 
Comparison of power coefficient