Design of a ducted wind turbine for offshore floating platforms

Abstract

Within the Marine Energy Laboratory project, funded by the Italian Ministry for Education, University and Research, one of the first offshore wind plants on floating hulls, hosting ducted wind turbines, has been considered. The confinement of horizontal-axis wind turbines inside divergent ducts is reconsidered, in light of material innovation and direct drive coupling. Ducted wind turbines can take advantage of the flow rate increase due to the effect of the divergent shrouds. The conventional blade element momentum theory has been reformulated in order to deal with ducted turbines. Furthermore, computational fluid dynamics simulations have been carried out based on the solution of the steady two-dimensional Reynolds-averaged Navier–Stokes equations for axisymmetric swirling flows. In order to avoid any expensive mesh refinement near the actual rotor blades, the turbine effect on the flow field is taken into account by means of source terms for the momentum equations solved inside the domain swept by the rotor. This technique allowed us to optimize the geometry of the ducted wind turbine in an extremely effective way.

Keywords

Ducted wind turbine offshore computational fluid dynamics modified blade element momentum theory

Introduction

In the last decades, a real revolution in the relationship between man and sea occurred. Toward the end of the 20th century, an attempt has been made to exploit the energy from the sea. Several are the forms of renewable energy coming from the sea, among others: wave energy, kinetic energy of water currents, energy from temperature or salinity gradients, tidal energy, marine geothermal energy, and marine biomass energy (Pelc and Fujita, 2002). Currently, wind energy is becoming more and more a marine energy, in the sense that there is a strong trend toward the installation of wind turbines offshore where wind speed is much higher than on the terrain.

This research, carried out within the Marine Energy Laboratory (MEL) project funded by the Italian Ministry for Education, University and Research (MIUR), collects the most advanced technologies of naval maritime engineering and combines them with energy and turbomachinery technologies. The MEL project is innovative and has ambitious goals aiming at the development of the first offshore ducted wind turbine application on a floating hull. The presence of a divergent duct enables the interception of a greater air mass flow rate, allowing the reduction in rotor diameters at equal rated power, favoring the rotor rigidity, reducing the blade deformations, and increasing the rotational speed with respect to conventional turbines.

The size of the target turbine is less than currently installed multi-megawatt (MW) offshore turbines, but it can be mounted on the hull in a dry dock, reducing significantly installation costs, making possible a periodic transportation of the platform in dry dock for maintenance and hence guaranteeing a very long service life. It is also worth mentioning the lower costs for decommissioning, at the end of the useful life of these movable floating turbines with respect to conventional offshore wind turbines fixed on the seabed.

In order to design the blades of ducted wind turbines, an original analytical model has been developed, since the conventional blade element momentum (BEM) theories fail. Moreover, computational fluid dynamic (CFD) simulations have been carried out in order to evaluate the ducted wind turbine performance.

The model is based on the solution, by means of a robust commercial CFD code, of the steady two-dimensional Reynolds-averaged Navier–Stokes equations (2D RANS equations) for axisymmetric swirled flows, introducing source terms in the momentum equations in the domain zone swept by the turbine blades in order to take into account the turbine effect on the flow field. This approach allows us to avoid any expensive refinement of the grid near the rotor, to reduce dramatically the computational costs, with respect to conventional unsteady flow simulations, and to optimize the geometry of the ducted wind turbine in an extremely effective way.

Ducted wind turbine model

The schematic representation of the proposed ducted wind turbine is shown in Figure 1. The diffuser is actually composed by two concentric annular wings. The aerodynamic profile of each annular wing has a camber line designed according to a cubic Bézier’s curve, whereas its thickness follows the NACA0004 equation. The maximum diffuser diameter is equal to 4.2 m. The rotor has been designed for an optimal tip-speed ratio, λ_opt = 5. The blade cross-sectional profiles change from NACA6618 at root, to NACA5418 at midspan, to NACA1318 at tip. Along the rotor blade span, all the cross-sectional profiles can be represented by the generic NACAxy18 equation, where both x and y are computed according to a second-order interpolation from the corresponding values at root, midspan, and tip, respectively. The blade tip radius, R, is equal to 1.5 m. The number of turbine blades, B, is equal to 5.

Figure 1.

Schematic representation of the MEL ducted wind turbine.

Analytical model

When a conventional wind turbine is considered, a preliminary design can be carried out according to the BEM theory. Actually, the BEM theory equates two different methods in order to examine the interaction of the flow field with the wind turbine: the first one considers the momentum balance on a rotating annular streamtube passing through the turbine and the second one examines the aerodynamic forces acting on the blade profile, in terms of lift and drag coefficients, in each streamtube. Hence, these two methods give a series of equations that can be solved iteratively. However, in the case of a ducted turbine, the analysis becomes more complicated, since in the axial momentum balance, it is necessary to take into account the effect of the diffuser thrust.

Let us consider a blade element at radius r and the corresponding streamtube. Inside a streamtube, the free stream wind speed, V₀, is modified into the local absolute velocity, V, by the local axial induction factor $a = (1 - V / V_{0})$ (from which $b = V / V_{0} = 1 - a$ ) and the relative tangential velocity, Ωr, is modified by the angular induction factor, $a^{'}$ , defined as $a^{'} = ω / (2 Ω)$ , when approaching the rotor.

The turbine design is performed at constant angle of attack, α = 5°, approximately equal to the optimum one (at which the profile efficiency, E = L/D, is maximum). Once the angle, φ, between the plane of rotation and the relative velocity, W, is known, the blade angle, β, is then trivially obtained. The air flow gives rise to a lift force, L, and a drag force, D, whose resultant force can be resolved into its normal dF_n and tangential dF_t components (perpendicular and parallel to the plane of rotation, respectively). Neglecting, for the sake of simplicity, the drag coefficient with respect to the lift coefficient (C_D << C_L), it is possible to write

d F_{n} = \frac{1}{2} ρ V_{0}^{2} {(1 - a)}^{2} σ_{r} \frac{C_{L} c o s ϕ}{{(s i n ϕ)}^{2}} 2 π r d r

(1)

d F_{t} = \frac{1}{2} ρ V_{0}^{2} {(1 + a^{'})}^{2} λ_{r}^{2} σ_{r} \frac{C_{L} s i n ϕ}{{(c o s ϕ)}^{2}} 2 π r d r

(2)

Hence

t a n ϕ = \frac{(1 - a)}{(1 + a^{'})} \frac{1}{λ_{r}}

(3)

where $λ_{r} = Ω r / V_{0} = λ r / R$ is the local tangential speed ratio, and $σ_{r} = (B c) / (2 π r)$ is the local solidity.

Now, considering the same annular streamtube and applying the first law of thermodynamics from the free stream to just upstream the turbine rotor, and from just downstream the rotor to the far wake, and defining $V_{e} / V_{0} = (b / k)$ , one obtains

d F_{n} = \frac{1}{2} ρ V_{0}^{2} (1 - {(\frac{b}{k})}^{2}) 2 π r d r

(4)

Equating equations (1) and (4), one obtains

\frac{1}{b^{2}} - \frac{1}{k^{2}} = σ_{r} \frac{C_{L} c o s ϕ}{{(s i n ϕ)}^{2}}

(5)

Applying the momentum equation to the same annular streamtube from the free stream up to the far wake, we have to take into account for the thrust contribution due to the diffuser (e.g. by means of a diffuser thrust coefficient, C_Ds, supposed constant along the entire blade span)

d T - d F_{n} = \frac{1}{2} ρ V_{0}^{2} 2 (1 - a) (1 - (\frac{b}{k})) 2 π r d r - \frac{1}{2} ρ V_{0}^{2} (1 - {(\frac{b}{k})}^{2}) 2 π r d r = d T_{s} = C_{D s} \frac{1}{2} ρ V_{0}^{2} 2 π r d r \Leftrightarrow (1 - (\frac{b}{k})) (2 b - (1 + \frac{b}{k})) = C_{D s}

(6)

A similar analysis can be performed for the local torque contribution, dQ, from the local tangential force contribution, dF_t, and this is equated with the angular momentum in the same annular streamtube

d Q = \frac{1}{2} ρ V_{0}^{2} 4 a^{'} (1 - a) λ_{r} r 2 π r d r = r d F_{t}

(7)

From which, taking into account equation (3), one obtains

\frac{4 a^{'}}{(1 + a^{'})} = σ_{r} \frac{C_{L}}{c o s ϕ}

(8)

Finally, integrating the product of the torque contribution, dQ, and the angular speed, Ω, along the blade span, the expression of the power coefficient, C_p, can be obtained

C_{p} = \int Ω d Q = \frac{8}{λ^{2}} \int a^{'} (1 - a) λ_{r}^{3} d λ_{r}

(9)

Initialized the diffuser thrust coefficient, C_Ds, an iterative procedure can be set in order to define the blade geometry (in terms of blade chord length, $c = c (r)$ , and blade angle, $β = β (r)$ ), which maximize the product $a^{'} (1 - a)$ and satisfy both equations (5) and (6) at each radius. Actually, the blade span has been divided into 60 intervals, and, at each interval, using the non-linear Generalized Reduced Gradient (GRG2) optimization algorithm (implemented in Microsoft Excel), the optimal values of both the axial $(a)$ and rotational $(a^{'})$ induction factors were found. Since the turbine design is a function of the diffuser thrust coefficient, C_Ds, in order to define the optimal turbine design, different C_Ds have been considered, and each time, the corresponding geometry has been investigated by means of CFD simulations. The chosen configuration was the one with the highest power coefficient.

Simplified 2D CFD model

The CFD model is based on the solution of the steady, incompressible, 2D RANS equations for axisymmetric swirled flow, written in conservation law form and discretized according to a control volume approach. The velocity field is obtained from the momentum equations, whereas the pressure field is extracted by solving a pressure correction equation in order to enforce mass conservation (continuity) according to the SIMPLE (Semi Implicit Method for Pressure-Linked Equations) algorithm developed by Patankar (1980). Turbulence is modeled by means of the semi-empirical standard k–ϵ model (Launder and Spalding, 1972). Boussinesq’s hypothesis is applied in order to relate the Reynolds stresses to the mean velocity gradients. This turbulence model has been chosen in consideration of its robustness, economy, and reasonable accuracy for a wide range of turbulent flows, which explain its popularity in many industrial flows (Fluent 6.3 User’s Guide, 2006). All the viscous and convective terms are discretized by means of a first-order upwind scheme. The domain, swept by the rotor blades, is discretized using a single cell streamwise. The blade influence is accounted for by means of source terms in the momentum equations, activated only in this domain via user-defined functions (UDFs), written in C language and incorporated into the CFD code. This approach was first proposed by Rajagopalan and Fanucci (1985), who developed a 2D CFD code for straight-bladed Darrieus rotors and applied in a three-dimensional (3D) configuration, and integrated into a commercial CFD code by Torresi et al. (2013). The aerodynamic forces on each blade depend on the local velocity field in the cells that lie on the rotor blade path. Hence, an iterative approach is necessary to reach the solution. Once the local Reynolds number and angle of attack are evaluated in each cell swept by the rotor blades, lift and drag coefficients can be computed from the airfoil polars, and the source terms can be computed. In order to obtain all the airfoil polars, the QBlade software was used (Marten et al., 2013).

Computational domain

In order to perform the 2D CFD simulations, the hybrid mesh (190,000 cells) shown in Figure 2 has been used for the discretization of the computational domain (352 m × 45 m). The two annular wings, which constitute the diffuser, have been surrounded by a structured mesh in order to take into account the boundary layer (30 levels, first cell height = 0.1 mm, growth factor = 1.1). A structured domain, one-cell thick, has been used for the domain zone swept by the turbine blades. The area surrounding the ducted wind turbine has been discretized with a multiblock unstructured mesh. Finally, the remaining computational domain has been discretized by means of a regular structured mesh.

Figure 2.

Details of the 2D hybrid mesh around the MEL ducted wind turbine.

Results

The proposed model has been applied in order to design a ducted wind turbine with a tip radius R = 1.5 m. Since the ducted wind turbine will be designed with a direct drive permanent magnet synchronous generator set on an external annulus at the blade tips, no nacelle is necessary. For this reason, the rotor hub is slightly larger than the shaft radius (0.1 m). The turbine was designed at an optimal tip-speed ratio, λ = ωR/V₀, equal to 5. The design angle of attack is equal to 5° at each radius. Once the airfoil profile distribution is assigned (hence, all C_L and C_D), the method allowed us to obtain the optimal design of the ducted wind turbine, according to the procedure described in the section ‘Analytical model’. The diffuser thrust coefficient, C_Ds, turned out to be equal to 0.35.

In order to understand the influence of the diffuser on the blade design, Figure 3 shows a comparison of the blade angle, $β (r)$ , and the local solidity, $σ_{r}$ , for the optimized ducted wind turbine with respect to the case of a free stream turbine designed according to a conventional BEM theory.

Figure 3.

Blade angle (left) and solidity (right) versus radial position.

It is evident that the ducted wind turbine needs to have a lower solidity, in order to allow a greater mass flux inside the diffuser, and this determines the need of greater blade angles in order to preserve the desired constant angle of attack.

Figure 4 shows the streamline across the ducted turbine. Thanks to the optimal design, no flow separation occurs downstream the ducted wind turbine even though the diffuser significantly deflects the streamlines in the radial direction.

Figure 4.

Streamlines in the meridional plane.

With a free stream velocity V₀ = 11 m/s, the ducted turbine generates almost 6 kW with a power coefficient, C_p, equal to 1.037, defined on the rotor swept area. Furthermore, considering the power coefficient, ${C^{'}}_{p}$ , based on the maximum diffuser diameter, this is equal to 0.529 higher than the one obtainable with a conventional unducted turbine 0.44, demonstrating the effectiveness of this ducted configuration.

Conclusion

The purpose of this research was to develop an original mathematical model for the design of ducted wind turbines, to be installed on floating hulls, as an alternative to conventional offshore wind turbines, since the standard BEM theories are not suited for this application. Furthermore, CFD simulations have been carried out in order to verify the performance of the MEL ducted wind turbine. The optimized design showed a significant improvement in turbine performance. These encouraging preliminary results drive us to further investigate this solution.

Footnotes

Appendix 1 Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This research was funded by MIUR (Italian Ministry for Education, University and Research), within the PON Marine Energy Laboratory (MEL), project no. PON03PE_00012_1.

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