inventions

Article Aerodynamic Simulations for Floating Darrieus-Type Wind with Three-Stage Rotors

Mohamed Amine Dabachi 1,2,* , Abdellatif Rahmouni 1, Eugen Rusu 3 and Otmane Bouksour 1

1 Laboratory of Mechanics Production and Industrial Engineering (LMPGI), High School of Technology (ESTC), Hassan II University of Casablanca, Route d’El Jadida, Km 7, Oasis, Casablanca 8012, Morocco; [email protected] (A.R.); [email protected] (O.B.) 2 Doctoral Studies Center of National High School of and Mechanics (ENSEM), Hassan II University of Casablanca, Route d’El Jadida, Km 7, Oasis, Casablanca 8018, Morocco 3 Department of Mechanical Engineering, University Dunarea de Jos of Galati, 800008 Galati, Romania; [email protected] * Correspondence: [email protected]; Tel.: +212-645785786

 Received: 29 February 2020; Accepted: 27 April 2020; Published: 29 April 2020 

Abstract: Growing energy demand is causing a significant decrease in the world’s hydrocarbon stock in addition to the pollution of our ecosystem. Based on this observation, the search for alternative sorts of energy to fossil fuels is being increasingly explored and exploited. Wind energy is experiencing a very important development, and it offers a very profitable opportunity for exploitation since the wind is always available and inexhaustible. Several technical solutions exist to exploit wind energy, such as floating vertical axis wind turbines (F-VAWTs), which provide an attractive and cost-effective solution for exploiting higher resources of offshore wind in deep water areas. Recently, the use of the Darrieus vertical axis wind (VAWT) offshore has attracted increased interest because it offers significant advantages over horizontal axis wind turbines (HAWTs). In this context, this article presents a new concept of floating Darrieus-type straight-bladed turbine with three-stage rotors. A double-multiple stream tube (DMST) model is used for aerodynamic simulations to examine several critical parameters, including, solidity turbine, number of blades, rotor radius, aspect ratio, wind velocity, and rotor height. This study also allows to identify a low solidity turbine (σ = 0.3), offering the best aerodynamic performance, while a two-bladed design is recommended. Moreover, the results also indicate the interest of a variable radius rotor, as well as the variation of the height as a function of the wind speed on the aerodynamic efficiency.

Keywords: vertical axis; darrieus turbines; floating wind; double-multiple stream tube; three-stage rotors

1. Introduction In the last few years, the world has seen a growing interest in environmental conservation and the engagement of socio-economic development plans based on a global vision of sustainable development. In order to reconcile the energy needs of countries with the requirements of the preservation of the environment, the international energy strategy aims to increase the share of renewable energies. According to the International Agency (IRENA) [1], “the objective set for the year 2050 is to reach 17% of the global installed offshore wind capacity (6044 GW)”. To achieve this capacity, it is necessary to look for technological solutions for more powerful and efficient wind turbines [2–4]. The offshore is considered as one of many solutions, defined as a wind turbine exploiting the energy generated by the wind at sea [5].

Inventions 2020, 5, 18; doi:10.3390/inventions5020018 www.mdpi.com/journal/inventions Inventions 2020, 5, 18 2 of 18

Inventions 2020, 5, x 2 of 19 Consequently,the use of this type of wind turbine has significant advantages over onshore wind turbines. useIndeed, of this type at sea, of wind the winds turbine are has particularly significant strongadvantages and stable,over onshore which wind means turbines. more power Indeed, generated. at sea,Moreover, the winds theare particularly installation ofstrong offshore and windstable, turbines which means on large more water power bodies generated. significantly Moreover, reduces the visual installationand noise of pollutionoffshore wind for residents turbines on [6]. large There water are twobodies types significantly of offshore reduces wind visual turbines: and VAWTsnoise and pollutionHAWTs. for Theresidents vast majority[6]. There of are installed two types off shoreof offshore wind turbines: solutions VAWTs are HAWTs, and HAWTs. because The of their vastmaximum majority powerof installed coeffi offshorecient value, wind approximately power solutions up to are 50% HAWTs, of the Betz because limit (59.3%) of their [7 ].maximum In comparison, powerthe VAWTcoefficient procures value, 40%. approximately There are alsoup to other 50% theoretical of the Betz formulas limit (59.3%) that announce[7]. In comparison, a maximum thepower VAWTcoeffi procurescient value 40%. of aboutThere 61.7%are also or 64%other for theoretical Darrieus VAWTs.formulas This that resultannounce is due a tomaximum the fact that power the blades coefficient value of about 61.7% or 64% for Darrieus VAWTs. This result is due to the fact that the cross the airflow twice, first in the upstream phase of the rotation, and second in the downstream blades cross the airflow twice, first in the upstream phase of the rotation, and second in the phase [5]. downstream phase [5]. VAWTs can be classified into two main types based on the wind velocity, efficiency desired, VAWTs can be classified into two main types based on the wind velocity, efficiency desired, and and utilization. The first type is a (drag-driven family) that consists of two utilization. The first type is a Savonius wind turbine (drag-driven family) that consists of two or more or more simple semicircular blades. The design of the Savonius rotor makes them unsuitable for simple semicircular blades. The design of the Savonius rotor makes them unsuitable for large-scale large-scale offshore application, but it can be attractive for small-scale application as domestic wind offshore application, but it can be attractive for small-scale application as domestic wind turbines due turbines due to good starting characteristics, easy installation, and low cost [8,9]. The second type is to good starting characteristics, easy installation, and low cost [8,9]. The second type is Darrieus VAWTDarrieuss (lift-driven VAWTs family) (lift-driven that demonstrate family) that good demonstrate efficiency good compared efficiency to Savonius compared wind to Savoniusturbines, wind becauseturbines, of their because simple of theirblade simple design blade and designlower center and lower of gravity, center ofmaking gravity, them making more them attractive more attractivefor offshorefor off applicationshore applications.s. The first The patent first patentfor a modern for a modern Darrieus Darrieus VAWTVAWT was deposited was deposited by the byFrench the French inventorinventor Georges Georges Jean JeanMarie Marie Darrieus Darrieus first in first France in Francein 1925, in then 1925, in the then United in the States United in 1931. States The in 1931. pateThent covered patent covered two major two configurations: major configurations: curved curvednon-straight non-straight blades blades(non-SB) (non-SB) and straight and straight blades blades (SB)(SB),, and and this thisis illustrated is illustrated in Figure in Figure 1. There1. There are several are several variations variations of non of-SB, non-SB, such suchas cantilevered as cantilevered andand guy guy-wired-wired versions versions [10,11 [10].,11 These]. These types types of non of non-SB-SB VAWTs VAWTs minimize minimize the the bending bending moments moments in in the the bladesblades and are more prone to the the dynamic dynamic stall stall than than SB SB VAWTs VAWTs as as demonstrated demonstrated by by Scheurich Scheurich [12]. [12].In In addition, addition, SB SB has has several several variations:variations: H-rotor,H-rotor, helical H H-rotor,-rotor, tilted tilted H H-rotor,-rotor, articulatin articulatingg H H-rotor.- rotor.SB-VAWT SB-VAWT rests rest betters better than thannon-SB-VAWT non-SB-VAWT due to due self-regulation, to self-regulation, simple rotor simple geometry, rotor geometry, absence of guy absencewire of use, guy cost, wire etc. use, cost, etc.

Figure 1. Types of Darrieurs VAWT.

Offshore wind turbineFigure goes 1. to Types deep of Darrieurs water. VAWT. According to a study conducted by the National Renewable Energy Laboratory (NREL) [13], a floating foundation (tensioned-leg platform, sparOffshore floater, wind semi-submersible turbine goes platform)to deep water. is more According cost-effective to a thanstudy fixed conducted foundations by the (monopile, National tripod, Renewablejacket frames Energy... Laboratory) from a water (NREL) depth [ of13 about], a floating 60–100 m. foundation That is why (tensioned current- researchleg platform, on off shore spar wind floater, semi-submersible platform) is more cost-effective than fixed foundations (monopile, tripod, turbines is focused on F-VAWTs that have several advantages compared to F-HAWTs, such as: simple jacket frames…) from a water depth of about 60–100 meters. That is why current research on offshore mechanism, low center of gravity, no yaw control, withstand high turbulence wind, easier maintenance, wind turbines is focused on F-VAWTs that have several advantages compared to F-HAWTs, such as: cost effectiveness, etc. [14,15]. These benefits have motivated several countries such as the USA, France, simple mechanism, low center of gravity, no yaw control, withstand high turbulence wind, easier Sweden, and Japan to create F-VAWT design projects. Hiromichi et al. [16] developed a new concept maintenance, cost effectiveness, etc. [14,15]. These benefits have motivated several countries such as the USA, France, Sweden, and Japan to create F-VAWT design projects. Hiromichi et al. [16] Inventions 2020, 5, 18 3 of 18 of F-VAWT, called FAWT. This concept used a new mechanism named “bearing swivel rollers” to support the turbine axis and help transform the torque from the turbine to the electric generator. Nakamura et al.[17] has also developed the Savonius keel wind turbine Darrieus concept (SKWID), an improvement of the FAWT concept. In fact, it is a combination between a floating wind turbine and a Savonius water turbine. The aim of the Savonius water turbine is to provide a starting torque to overcome the inertia of the wind turbine, because the VAWT weakness is that it requires some external assistance to initiate the rotation, but Dominy et al. [18] and Hill et al. [19] indicated that self-starting is possible by using symmetrical under steady wind conditions for a two-blade H-rotor with fixed geometry. The French company Nenuphar proposed two wind turbines. The first one is VertiWind [20], which has a very high availability rate. The cost of its foundation can be reduced by optimizing the float architecture. The second one is Twinfloat [21], which is built based on two turbines of 2.5 MW, working in a counter-rotating mode, and creating a tunnel effect between them in order to increase electricity production. Other design projects are described in [22–24]. The aim of this paper is to propose a novel design of Darrieus-type straight-bladed F-VAWT with three-stage rotors. A DMST model was adopted in the aerodynamic modeling in order to analyze several critical parameters of VAWT, taking into consideration the turbine solidity, the number of blades, the rotor radius, and the aspect ratio. The remainder of the paper is organized as follows: the proposed design of F-VAWT with three-stage rotors is discussed in Section2. Section3 shows general mathematical expressions for the aerodynamic analysis of straight-bladed VAWT. Section4 is devoted to the DMST model, while Section5 presents the results and discussion. Lastly, Section6 concludes the paper.

2. Floating Darrieus-Type Wind Turbine with Three-Stage Rotors In this section, a new design of Darrieus F-VAWT with three-stage rotors is shown in Figure2. The rotors will have a straight-blade configuration, as it has been shown to demonstrate a higher aerodynamic performance and be self-regulating in all wind velocities in comparison to other possible VAWT configurations [25]. The concept allows having a higher height in order to benefit from the stronger winds. The three rotors of the wind turbine rotate independently around the fixed shaft, unlike conventional VAWTs where the rotor and shaft rotate at the same time, causing the increase of the inertia and the torque applied to the shaft [26], and create problems in the self-starting of the turbine. In addition, by reducing the number of bearings, the axial and radial forces exerted on the shaft are minimized, as well as the related manufacturing cost, compared to the rotating shaft, due to the simple geometric specifications and because our solution does not need much tolerance interval precision [27]. In this design, three permanent-magnet synchronous motors (PMSMs) can be used with different powers depending on the rotor radius, wind velocity, and the height where each rotor is located. The float used is a semi-submersible tri-float type. We are also considering the introduction of connection beams to support the VAWT and increase its stability. Moreover, by using aerodynamic airfoils for turbine supporting arms (or struts), we can avoid the problem of decreasing the aerodynamic efficiency of the turbine. Ahmadi-Baloutaki et al. [28] demonstrated that it is necessary to adopt an aerodynamic for struts, and when two struts per blade are used at intermediate location of 21% and 79% along the blade length, the bending stress distribution along the blade is reduced. Figure3 illustrates one rotor stage of F-VAWT. A mechanism consisting of a pivot (screw-nut) is introduced in the middle of the two blades, which is automatically actuated by a small electric motor and a lever in the shape of a parallelogram. The latter is attached to the end of both blades. When turning the screw, the lever moves with the help of the two sliding links and exerts a force on the two blades that allows us to increase the diameter. On the other hand, when the screw rotates in the opposite direction, the diameter will be reduced. This new mechanism aims to change the diameter of the wind turbine automatically according to the wind speed, and it solves the problem of starting a large VAWT rotor. Inventions 2020, 5, x 3 of 19

developed a new concept of F-VAWT, called FAWT. This concept used a new mechanism named “bearing swivel rollers” to support the turbine axis and help transform the torque from the turbine to the electric generator. Nakamura et al. [17] has also developed the Savonius keel wind turbine Darrieus concept (SKWID), an improvement of the FAWT concept. In fact, it is a combination between a and a Savonius water turbine. The aim of the Savonius water turbine is to provide a starting torque to overcome the inertia of the wind turbine, because the VAWT weakness is that it requires some external assistance to initiate the rotation, but Dominy et al. [18] and Hill et al. [19] indicated that self-starting is possible by using symmetrical airfoils under steady wind conditions for a two-blade H-rotor with fixed geometry. The French company Nenuphar proposed two wind turbines. The first one is VertiWind [20], which has a very high availability rate. The cost of its foundation can be reduced by optimizing the float architecture. The second one is Twinfloat [21], which is built based on two turbines of 2.5 MW, working in a counter-rotating mode, and creating a tunnel effect between them in order to increase electricity production. Other design projects are described in [22–24]. The aim of this paper is to propose a novel design of Darrieus-type straight-bladed F-VAWT with three-stage rotors. A DMST model was adopted in the aerodynamic modeling in order to analyze several critical parameters of VAWT, taking into consideration the turbine solidity, the number of blades, the rotor radius, and the aspect ratio. The remainder of the paper is organized as follows: the proposed design of F-VAWT with three-stage rotors is discussed in Section 2. Section 3 shows general mathematical expressions for the aerodynamic analysis of straight-bladed VAWT. Section 4 is devoted to the DMST model, while Section 5 presents the results and discussion. Lastly, Section 6 concludes the paper. Inventions 2020, 5, 18 4 of 18 2. Floating Darrieus-Type Wind Turbine with Three-Stage Rotors

Inventions 2020, 5, x 4 of 19

the shaft are minimized, as well as the related manufacturing cost, compared to the rotating shaft, due to the simple geometric specifications and because our solution does not need much tolerance interval precision [27]. In this design, three permanent-magnet synchronous motors (PMSMs) can be used with different powers depending on the rotor radius, wind velocity, and the height where each rotor is located. The float used is a semi-submersible tri-float type. We are also considering the introduction of connection beams to support the VAWT and increase its stability. Moreover, by using

aerodynamic airfoils for turbine supporting arms (or struts), we can avoid the problem of decreasing (b) the aerodynamic efficiency(a) of the turbine. Ahmadi-Baloutaki et al. [28] demonstrated that it is necessary to adopt an aerodynamic airfoil for struts, and when two struts per blade are used at FigureFigure 2. (a 2.) 3D(a) design 3D design of Floating of Floating Darrieus Darrieus-type-type wind turbine wind with turbine three with-stage three-stage rotors (b) Schematic rotors (b diagram) Schematic intermediate location of 21% and 79% along the blade length, the bending stress distribution along detailsdiagram details. the blade is reduced. In this section, a new design of Darrieus F-VAWT with three-stage rotors is shown in Figure 2. The rotors will have a straight-blade configuration, as it has been shown to demonstrate a higher aerodynamic performance and be self-regulating in all wind velocities in comparison to other possible VAWT configurations [25]. The concept allows having a higher height in order to benefit from the stronger winds. The three rotors of the wind turbine rotate independently around the fixed shaft, unlike conventional VAWTs where the rotor and shaft rotate at the same time, causing the increase of the inertia and the torque applied to the shaft [26], and create problems in the self-starting of the turbine. In addition, by reducing the number of bearings, the axial and radial forces exerted on

FigureFigure 3. 3Kinematic. Kinematic scheme scheme of of thethe diameterdiameter variation mechanism mechanism in in one one rotor rotor stage stage

3. GeneralFigure Mathematical 3 illustrates one Expressions rotor stage for of Aerodynamic F-VAWT. A mechanism Analysis ofconsisting SB-VAWT of a pivot (screw-nut) is Theintroduced general in mathematical the middle of expressions the two blades at a specific, which locationis automatically of the blade actuated and theby inputa small parameters electric usedmotor for theand aerodynamic a lever in the simulation shape of a areparallelogram described. below The latter [25,29 is, 30attached]. to the end of both blades. When turning the screw, the lever moves with the help of the two sliding links and exerts a force on 3.1.the Local two Relative blades that Velocity allows Variation us to increase the diameter. On the other hand, when the screw rotates in the opposite direction, the diameter will be reduced. This new mechanism aims to change the diameterThe wind of the flow wind passes turbine over automatically the turbine according blades which to the consist wind speed, of airfoil and cross-sectionit solves the problem profile to generateof starting useful a large torque VAWT and power. rotor. As in case of any airfoil, the chordal and normal velocity components (Vc and Vn) can be obtained as follows: 3. General Mathematical Expressions for Aerodynamic Analysis of SB-VAWT Vc = Rω + Va cos θ (1) The general mathematical expressions at a specific location of the blade and the input parameters used for the aerodynamic simulation are described below [25,29,30]. Vn = Va sin θ (2) where3.1. LocalVa is Relative the axial Velocity flow Variation velocity (i.e., induced velocity) through the rotor. A relative velocity componentThe VwindR can flow be obtained passes over from the the turbine cordial blades velocity which component consist and of airfoil the normal cross- velocitysection profile component to as cangenerate be seen useful from torque Figures and4 and power.5. As in case of any airfoil, the chordal and normal velocity

components (Vc and Vn) can be obtained as follows:

Vc = Rω + Va cos θ (1)

Vn = Va sin θ (2)

where Va is the axial flow velocity (i.e., induced velocity) through the rotor. A relative velocity component VR can be obtained from the cordial velocity component and the normal velocity component as can be seen from Figures 4 and 5. Inventions 2020, 5, 18 5 of 18

Inventions 2020, 5, x q q 5 of 19 2 2 2 2 VR = (Vn) + (Vc) = (Va sin θ) + (Rω + Va cos θ) (3) V = √(V )2 + (V )2 = √(V sin θ)2 + (Rω + V cos θ)2 (3) Tip speed ratio is definedR as:n c a a Tip speed ratio is defined as: Rω TSR = (4) RVω (4) TSR = V ∞ where R and ω are the radius and the rotational speed∞ of the rotor, respectively where R and ω are the radius and the rotational speed of the rotor, respectively Dividing Equation (3) by the stream velocity V , the adimensionalized expression of relative Dividing Equation (3) by the stream velocity V∞, the adimensionalized expression of relative velocity is: ∞ velocity is: q V R = ((1 a) sin θ)2 + (TSR + (1 a) cos θ)2 (5) VR (5) V = √((1 −−a) sin θ)2 + (TSR + (1 − a−) cos θ)2 V∞ where a is the axial induction∞ factor. where a is the axial induction factor.

FigureFigure 4. 4.Flow Flow velocitiesvelocities diagram of of Darrieus Darrieus VAWT VAWT airfoil. airfoil.

3.2.3.2 Variation. Variation of of Local Local Angle Angle of of Attack Attack A relationA relation between between the the angle angle of of attack attackα α(incidence (incidence angle),angle), thethe azimuth angle θ, induction factor factor a, anda, theand TSR the TSR was was defined defined from from the the velocity velocity triangle triangle analysis analysis as as follows: follows: V V sinθ  n   a  tanα = (Vn) = ( Vasin θ ) (6) tan α = Vc = Rω + Vacosθ (6) Vc Rω + Vacos θ By non-dimensionalizing Equation (5) results: By non-dimensionalizing Equation (5) results:V a sinθ V∞ tanα = ( V )  (7) Rω aVsina θ  +V cosθ  tan α = V∞ ∞V∞  (7)  Rω Va  (1+− a)sincosθ θ (8) α = tan−1 ( V V ) TSR∞+ (1 −∞ a)cosθ ! 1 (1 a)sin θ α = tan− − (8) 3.3. Normal and Tangential Force Variation TSR + (1 a)cos θ − 3.3. Normal and Tangential Force Variation The overall normal and tangential forces are given by:

ρ 2 Fn = 0.5Cn hcVR (9)

ρ 2 Ft = 0.5Ct hcVR (10) 3 where ρ = 1.225 Kg/m is the air density, h is blade height, c is the blade chord, Cn the coefficient of the normal force is the difference between the normal components of lift and drag forces

Figure 5. Force diagram of a blade airfoil. Inventions 2020, 5, x 5 of 19

2 2 2 2 VR = √(Vn) + (Vc) = √(Va sin θ) + (Rω + Va cos θ) (3) Tip speed ratio is defined as: Rω (4) TSR = V∞ where R and ω are the radius and the rotational speed of the rotor, respectively

Dividing Equation (3) by the stream velocity V∞, the adimensionalized expression of relative velocity is:

VR (5) = √((1 − a) sin θ)2 + (TSR + (1 − a) cos θ)2 V∞ where a is the axial induction factor.

Figure 4. Flow velocities diagram of Darrieus VAWT airfoil.

3.2. Variation of Local Angle of Attack A relation between the angle of attack α (incidence angle), the azimuth angle θ, induction factor a, and the TSR was defined from the velocity triangle analysis as follows:

Vn Vasinθ tanα = ( ) = ( ) (6) Vc Rω + Vacosθ By non-dimensionalizing Equation (5) results: V a sinθ V tanα = ( ∞ ) Rω V (7) Inventions 2020, 5, 18 + a cosθ 6 of 18 V∞ V∞ (1 − a)sinθ (8) α = tan−1 ( ) Cn = CLcos α + CDsin α, Ct the coefficient of theTSR tangential+ (1 − a) forcecosθ is the difference between the tangential components of lift and drag forces Ct = CLsin α CDcos α 3.3. Normal and Tangential Force Variation −

Figure 5 5.. Force diagram of a blade airfoil.

3.4. Total Torque and Power Output The normal and tangential forces given by Equations (8) and (9) are for the arbitrary azimuthal position and are regarded as a function of azimuth angle θ. Average tangential force (Fti) on one single airfoil at certain θ is: Z π 1 2 Fti = Ft(θ)dθ (11) 2π 0

Considering B blades, the total torque output (Qi) is obtained as follows:

Z π BR 2 Qi = BFtiR = Ft(θ)dθ (12) 2π 0

The total power output (Pi) is described by:

Z π BRω 2 Pi = Qi ω = Ft(θ)dθ (13) · 2π 0

3.5. Rotor Solidity σ Rotor solidity is the main parameter to define the geometry of the vertical axis wind turbines, and it is a function of the number of blades B, the chord length c, and the radius of the rotor R. It can be calculated as suggested by Strickland [31]:

Bc σ = (14) R

3.6. Rotor Aspect Ratio AR Rotor aspect ratio of VAWT defined as the ratio between the blade height and rotor radius:

h AR = (15) R

3.7. Power Coefficient Cp

Power coefficient CP is the ratio between the power absorbed by the rotor shaft P and the power available from the air stream flowing through the rotor swept area.

P Qω ωR CP = = = CmTSR (16) 0.5ρAV 3 0.5ρAV 3 V ∞ ∞ ∞ Inventions 2020, 5, 18 7 of 18

where P is the mechanical power, ρ is the air density, A is the projected area of the rotor, and Cm is a moment coefficient.

3.8. Reynolds Number Re Reynolds number is used to characterize the flow regime perceived by the blades. It represents the ratio between inertial forces and viscous forces: ρcV R = ∞ (17) e µ where µ = 1.647 10 5 Kg/m.s is the dynamic viscosity of the air. × − 4. Double Multiple Stream Tube Model In the literature [25], many aerodynamic models for VAWT have been proposed, such as double multiple stream tube (DMST), multiple stream tube (MST), vortex, cascade, and panel. Each of them has its advantages and disadvantages, and Table1 gives a comparison of the aerodynamic models according to important criteria such as complexity and computational time. Therefore, it is necessary to find a compromise between the validity of the results, the problem complexity, and the computational effort, depending on the objectives fixed. The model chosen for this work is the DMST developed by ParaschivoiuInventions 20 [2032, 5], x in 1981. This model combined the MST model with the double actuator disk7 of 19 theory (see Figure6a) to predict the performance of VAWT. The advantages of this model can be seen through ρcV∞ (17) the fact that it is relatively simple to implementR = and gives better correlation between calculated and e μ experimental results [25]. However, the major drawback of this model is to give over prediction of powerwhere for a휇 high= 1.647 solidity× 10− turbine,5 Kg/m. sand is the there dynamic appears viscosity to be of a convergencethe air. problem for the same type of turbine, especially in the downstream side and at the higher tip speed ratio [29]. 4. Double Multiple Stream Tube Model

(b) (a)

FigureFigure 6. ( 6a.) ( Shematica) Shematic of of two-actuator two-actuator disk in in tandem tandem (b ()b Diagrammatic) Diagrammatic of DMST of DMST model model..

In the literature [25], manyTable aerodynamic 1. Comparison models of aerodynamic for VAWT modelshave been [5]. proposed, such as double multiple stream tube (DMST), multiple stream tube (MST), vortex, cascade, and panel. Each of them BEM Model Cascade Model Vortex Model Panel Model has its advantages and disadvantages, and Table 1 gives a comparison of the aerodynamic models Complexityaccording to important criteria such as Low-Mediumcomplexity andLow-Medium computational time Medium-High. Therefore, it is necessary High Ease of implementation Easy-Medium Easy-Medium Medium Hard to find a compromise between the validity of the results, the problem complexity, and the Computational effort Low Low Medium-High Medium-High Restrictedcomputational to known effort airfoils, depending on theYes objectives fixed. Yes The model chosen Yes for this work is the No DMST Incorporatedeveloped unsteady by Paraschivoiu conditions [32] in 1981.Limited This modelLimited combined the MST Yes model with the Yes double Rotoractuator wake/ multipledisk theory rotor (see interactions Figure 6a) toNo predict/No the performance No/No of VAWT. Yes /TheLimited advantage Yess of/Yes this model can be seen through the fact that it is relatively simple to implement and gives better correlation between calculated and experimental results [25]. However, the major drawback of this modelAs shown is to ingive Figure over 6 predictionb, the rotor’s of power area foris divided a high solidity into a setturbine of adjacent, and there parallel appears stream to be atubes. The calculationconvergence isproblem performed for the separately same typefor of turbine,the upwind especially and downwindin the downstream half cycles, side and and at the the blade higher tip speed ratio [29].

Table 1. Comparison of aerodynamic models [5].

BEM model Cascade model Vortex model Panel model Complexity Low-Medium Low-Medium Medium-High High Ease of implementation Easy-Medium Easy-Medium Medium Hard Computational effort Low Low Medium-High Medium-High Restricted to known airfoils Yes Yes Yes No Incorporate unsteady conditions Limited Limited Yes Yes Rotor wake/multiple rotor interactions No/No No/No Yes/Limited Yes/Yes

As shown in Figure 6b, the rotor’s area is divided into a set of adjacent parallel stream tubes. The calculation is performed separately for the upwind and downwind half cycles, and the blade element theories and the momentum conservation are used for calculating the aerodynamic forces acting on the blades, which then makes it possible to calculate the torque and the power generated. π 3π For the upstream half-cycle ( ) ≤ θ ≤ ( ) 2 2

푢 푢 2 푢 2 VR = √(푉푎 sin θ) + (푉푎 cos θ + Rω) (18) Inventions 2020, 5, 18 8 of 18 element theories and the momentum conservation are used for calculating the aerodynamic forces acting on the blades, which then makes it possible to calculate the torque and the power generated.     For the upstream half-cycle π θ 3π 2 ≤ ≤ 2 q u ( u θ)2 ( u θ ω)2 VR = Va sin + Va cos + R (18)

u ! u 1 (1 a )sin θ α = tan− − (19) TSR + (1 au)cos θ −     For the downwind half-cycle 3π θ π 2 ≤ ≤ 2 r  2  2 d d θ d θ ω VR = Va sin + Va cos + R (20)

     1 ad sin θ  d 1 −  α = tan−   (21) TSR + (1 ad)cos θ  − (1 2a) Vd d = − − a where, at the downwind side, the induction factor a (1 2a) and the induced velocity − d  d Va = 1 a (1 2a)V . − − ∞ Once the new relative wind speed and angle of attack are determined to use the new induction factor, the torque coefficient, thrust coefficient, and power coefficient could be obtained. Equating the forces given by momentum equations to those defined by blade element theory:

fu(1 a) = πa (22) − where the upwind function fu is given by:

Z 3π/2  2 Bc cos θ sin θ VR fu = Cn Ct dθ (23) 8πR cos θ − sin θ Va π/2 | | | | Thus, the power coefficient for the upwind half of the turbine is obtained by:

Z 3π/2  2 Bch VR CP = Ct dθ (24) 2πA π/2 Va where A = 2hR is the turbine swept area The downwind part of the rotor is evaluated in the same manner, and finally, by the summation of the power coefficients of the two half-cycles, the total power coefficient of the rotor can be found.

4.1. QBlade Simulation Tool QBlade is used as an open source framework for the design and aerodynamic simulations of the wind turbines. In a recent paper by Nachtane et al. [33] QBlade solver was used to predict the hydrodynamic performance of a new hydrofoil named NTSXX20. QBlade utilizes the blade element momentum (BEM) method for the simulation of HAWTs, and a double multiple stream tube (DMST) model for the VAWTs performance. The XFOIL code is integrated seamlessly into QBlade based on the viscous-inviscid coupled panel method to generate 2D airfoil coordinates for blade design, airfoil lift, drag coefficients, and 360◦ polar extrapolation for turbine simulations [34]. Figure7 shows the di fferent modules and types of analysis (shown inside the dotted square) that can be performed by Qblade for aerodynamic simulation. Inventions 2020, 5, x 8 of 19

(1 − a푢)sinθ α푢 = tan−1 ( ) (19) TSR + (1 − a푢)cosθ 3휋 휋 For the downwind half-cycle ( ) ≤ 휃 ≤ ( ) 2 2

d d 2 d 2 VR = √(Va sin θ) + (Va cos θ + Rω) (20)

(1 − ad)sinθ αd = tan−1 ( ) (21) TSR + (1 − ad)cosθ

(1−2a)−Vd where, at the downwind side, the induction factor ad = a and the induced velocity Vd = (1 − (1−2a) a d a )(1 − 2a)V∞ Once the new relative wind speed and angle of attack are determined to use the new induction factor, the torque coefficient, thrust coefficient, and power coefficient could be obtained. Equating the forces given by momentum equations to those defined by blade element theory:

fu(1 − a) = πa (22)

where the upwind function fu is given by: 3π/2 2 Bc cos θ sin θ VR fu = ∫ (Cn − Ct ) ( ) dθ (23) 8πR π/2 |cos θ| |sin θ| Va Thus, the power coefficient for the upwind half of the turbine is obtained by:

3π/2 2 Bch VR CP = ∫ Ct ( ) dθ (24) 2πA π/2 Va where A = 2hR is the turbine swept area The downwind part of the rotor is evaluated in the same manner, and finally, by the summation of the power coefficients of the two half-cycles, the total power coefficient of the rotor can be found. Inventions 2020, 5, 18 9 of 18 4.1. QBlade Simulation Tool

FigureFigure 7. 7.Software Software modules inside inside QBlade. QBlade. 4.2. QBladeQBlade Validation is used as an open source framework for the design and aerodynamic simulations of the wind turbines. In a recent paper by Nachtane et al. [33] QBlade solver was used to predict the Inhydrodynamic order to show performance the validity of a ofnew this hydrofoil software, named the QbladeNTSXX20. was QBlade compared utilizes with the blade the experimental element and computationalmomentum (BEM) results method of Raciti for the Castelli simulation et al. of [HAWTs,35] with and various a double TSRs. multiple The comparison stream tube of(DMST) the power coeffimodelcient versusfor the TSRVAWTs results performance. from the The QBlade XFOIL with code experimental is integrated andseamlessly numerical into QBlade results based are shown on in Figurethe8. viscous It can be-inviscid observed coupled that panel the general method trend to generate of the 2D experimental airfoil coordinates plot was for capturedblade design, by the airfoil Qblade wherelift the, drag quantitative coefficients di, andfferences 360° polar (presented extrapolation in Table for 2turbine) are found simulations to be [ the34]. largestFigure 7 for shows the highestthe TSR equal to 2.21. The observed deviations may be explained by the uncertainties associated with the experimental data, e.g. boundary conditions, the blade surface roughness, the wake-blade interaction. Also, the effect of blade-spoke connection on the CP of the turbine is known to be more significant in higher TSRs [36]. As well as Qblade overestimates the CP at high TSRs due to the dynamic stall effects are not considered, which is difficult to predict because the angle of attack α change rapidly and the turbulence generated. Another source of error is from the extrapolation of the initial data obtained from XFOIL as the airfoil characteristics are required for 360◦. Regarding the extrapolation, Montgomerie [37,38] and Viterna [39] are the most widely used methods. The Montgomery method is founded on the assumption as a thin plate, whereas the Viterna method is formulated on the basis of the potential flow theory. Extrapolated values provide a good estimate if the initial limited data are obtained from experiments. However, in this case, the initial data is over predicted by Xfoil which can further extend the high lift/drag ratio error for other angles of attack. Furthermore, the Montgomery and Viterna methods are intended for static airfoils, and do not consider the unsteady effects. The airfoil characteristics calculated from the Xfoil assume that the flow is laminar, but in fact, VAWT airfoils face a turbulent and unstable incoming flow, and it becomes worse in the downstream side, where the blade operates entirely in wake for a high solidity turbine [40].

Table 2. Comparison of CP from experiment, CFD, Qblade and their relative deviation.

|∆C | [%] |∆C | [%] TSR C Experiment C Qblade C CFD P P P P P Qblade/Experiment CFD/Experiment 1.69224 0.0477033 0.0276889 0.26065 53.09408666 138.1186451 2.00088 0.127525 0.067902 0.412182 61.0181807 105.4857543 2.30071 0.249668 0.141027 0.523215 55.61422593 70.78613451 2.60494 0.310683 0.248951 0.563603 22.06156166 57.85749743 2.90035 0.297405 0.427411 0.554072 35.87282841 60.28747694 3.05908 0.285123 0.453624 0.546477 45.61805327 62.85569986 3.20899 0.267197 0.435643 0.530408 47.93295771 66.00033851

Table2 presents percent di fferences between numerical and experimental results of the power coefficient. The relative deviation between the numerical and experimental results is given by the equation: Inventions 2020, 5, x 9 of 19

different modules and types of analysis (shown inside the dotted square) that can be performed by Qblade for aerodynamic simulation.

4.2. QBlade Validation In order to show the validity of this software, the Qblade was compared with the experimental and computational results of Raciti Castelli et al. [35] with various TSRs. The comparison of the power coefficient versus TSR results from the QBlade with experimental and numerical results are shown in Figure 8. It can be observed that the general trend of the experimental plot was captured by the Qblade where the quantitative differences (presented in Table 2) are found to be the largest for the highest TSR equal to 2.21. The observed deviations may be explained by the uncertainties associated with the experimental data, e.g. boundary conditions, the blade surface roughness, the wake-blade

interaction. Also, the effect of blade-spoke connection on the CP of the turbine is known to be more significant in higher TSRs [36]. As well as Qblade overestimates the CP at high TSRs due to the dynamic stall effects are not considered, which is difficult to predict because the angle of attack α change rapidly and the turbulence generated. Another source of error is from the extrapolation of the initial data obtained from XFOIL as the airfoil characteristics are required for 360˚. Regarding the extrapolation, Montgomerie [37,38] and Viterna [39] are the most widely used methods. The Montgomery method is founded on the assumption as a thin plate, whereas the Viterna method is formulated on the basis of the potential flow theory. Extrapolated values provide a good estimate if the initial limited data Inventions 2020, 5, 18 10 of 18 are obtained from experiments. However, in this case, the initial data is over predicted by Xfoil which can further extend the high lift/drag ratio error for other angles of attack. Furthermore, the Montgomery and Viterna methods are intended for static airfoils, and do not consider the unsteady effects. The airfoil characteristics calculated from the Xfoil assume thatCp1 the Cflowp2 is laminar, but in fact, VAWT airfoils face a ∆CP = − 100 (25) turbulent and unstable incoming flow| , and| it becomesCp1+ Cworsep2 ×in the downstream side, where the blade operates entirely in wake for a high solidity turbine [402].

Inventions 2020, 5, x 10 of 19

Table 2. Comparison of CP from experiment, CFD, Qblade and their relative deviation

|∆퐂 | [%] |∆퐂 | [%] TSR C Experiment C Qblade C CFD 퐏 퐏 P P P Qblade/Experiment CFD/Experiment Figure 8. Comparison of power coefficient versus experiment and numerical results by Raciti Figure 8.1.69224 Comparison 0. 0477033of power coefficient0.0276889 versus 0 experiment.26065 and53 .numerical09408666 results138 by.1186451 Raciti Castelli Castelli etet al. al.[ [35352]..]00088. 0.127525 0.067902 0.412182 61.0181807 105.4857543 2.30071 0.249668 0.141027 0.523215 55.61422593 70.78613451 Table 22 .60494present s percent0.310683 differences 0.248951 between 0.563603 numerical and22.06156166 experimental 57results.85749743 of the power 4.3. Input Parameters2.90035 Used for0 the.297405 Aerodynamic 0.427411 Simulation0.554072 35.87282841 60.28747694 coefficient. 3The.05908 relative 0deviation.285123 between0.453624 the numerical0.546477 and45 experimental.61805327 results62.85569986 is given by the The mostequation: common 3.20899 profiles0.267197 used for0.435643 commercial 0.530408 Darrieus47.93295771 VAWTs are66.00033851 the symmetrical NACA profiles [10]. The blade airfoil will consist of a symmetrical four-digit NACA 0015 which has a maximum 4.3. Input Parameters Used for the Aerodynamic|Cp1 −SimulationCp2| (25) |∆CP| = × 100 thickness of t/c = 15% and a 0% maximum camber.Cp1 + Cp2 A study was conducted at the University of Windsor The most common profiles used for commercial2 Darrieus VAWTs are the symmetrical NACA in Canada analyzingprofiles [10]. the The performance blade airfoil will of consist several of a symmetrical airfoils for four Darrieus-digit NACA VAWT 0015 which[41]. Ofhas alla the airfoils tested, the NACA0015maximum thickness performed of t/c= 15% the and best a 0% performance. maximum camber. Figure A study9a was gives conducted a comparison at the University between the four aerodynamicof profilesWindsor in in Canada terms analyzing of coordinates. the performance To assessof several the airfoils aerodynamic for Darrieus VAWT efficiency [41]. Of of all each possible the airfoils tested, the NACA0015 performed the best performance. Figure 9a gives a comparison VAWT design,between the powerthe four aerodynamic coefficient profiles Equation in terms (16) of willcoordinates. be used. To assess Table the3 aerodynamic outlines the efficie inputncy parameters used for theof performance each possible VAWT curves design, for the di powerfferent coefficient aerodynamic Equation (16 profiles,) will be used and. Table Table 3 outline4 givess the the physical proprieties ofinput air usedparameters for aerodynamicused for the performance simulation. curves As for showndifferent inaerodynamic Figure9b, profiles aerodynamic, and Table profile4 NACA gives the physical proprieties of air used for aerodynamic simulation. As shown in Figure 9b, 0015 presentsaerodynamic the best profile aerodynamic NACA 0015 e ffipresentsciency the compared best aerodynamic to other efficiency aerodynamic compared to profiles. other Figure 10 6 depicts the liftaerodynamic and drag profiles. coeffi Figurecients 10 ofdepicts a NACA0015 the lift and drag airfoil coefficients at Re =of 10a NACA0015. airfoil at Re = In the following,106. we have used this profile for the parametric study of the VAWT rotor. In the following, we have used this profile for the parametric study of the VAWT rotor.

(a) (b)

Figure 9. (a)Figure Comparison 9. (a) Comparison between between the four the aerodynamic four aerodynamic profiles profiles in terms in terms of coordinatesof coordinates ( (bb) Performance Performance curves for different aerodynamic profiles. curves for different aerodynamic profiles. The objective of this study is to carry out a parametric study by the DMST method, in order to The objectiveshow the ofinterest this of study a stepped is torotor carry with variable out aparametric radius, and to determine study by the the aerodynamic DMST method,behavior in order to show the interestof the rotor of a during stepped the design rotor phase, with which variable then makes radius, it po andssible to to determineperform a structural the aerodynamic analysis of behavior the design.

Inventions 2020, 5, 18 11 of 18 of the rotor during the design phase, which then makes it possible to perform a structural analysis of the design. Inventions 2020, 5, x 12 of 19

(a) (b) Figure 10. (a) Lift coefficient versus angle of attack alpha of NACA 0015 airfoil at Re = 106; (b) Drag coefficient 6 Figureversus 10. (a angle) Lift of coeattackffi alphacient of versus NACA 0015 angle airfoil of at attack Re = 106 alpha. of NACA 0015 airfoil at Re = 10 ;(b) Drag 6 coefficient versus angle of attack alpha of NACA 0015 airfoil at Re = 10 . 5. Results and Discussion

5.1. Rotor TableSolidity 3. Influential geometric parameters for aerodynamic simulations.

The influence of alteringRotor the R [m]turbine solidity c [m] on h its [m] aerodynamic B V performance[m/s] σ has been studied using the DMST model as shown in Figure 11. The solidity 𝜎 was increased∞ from 0.1 to 0.4 by decreasing the turbine radius while the blades swept1. Aerodynamicarea was kept constant airfoils (see Table 3). The DMST model has been run many times for1 H-rotor 0.7with three 0.09 NACA 0015 0.45 blades 3 and Reynolds 10 0.38 number 106. It can be observed for 𝜎 = 0.3 that the CP grows rapidly compared to the other solidity. This is illustrated clearly 2. Rotor solidity in Figure 11 where the CP peak (0.342) expected at a low tip speed ratio (TSR = 3.51). 1 2 0.0667 0.1 2 1.5 0.1 0.2 0.45 3 10 3 1 0.1 0.3 4 0.7 0.0933 0.4 3. Number of blades 1 0.7 0.105 2 2 1 0.1 3 0.45 10 0.3 3 1.5 0.1125 4 4 2 0.12 5 4. Rotor radius 1 0.7 0.105 2 1 0.15 Figure 11. Performance curve0.45 for different2 solidit10y 0.3 3 1.5 0.225

As the turbine solidity4 decrease 2s (𝜎 = 0.30.1), the CP drops (0.183) at high TSR (6.51). Therefore, from a structural design standpoint, a low solidity turbine will experience greater centrifugal forces 5. Rotor aspect ratio on the blades and supporting arms because of its higher TSR operating conditions [25,29]. While these forces could be contained1 with the 0.7 use of 0.105modern composite 0.28 materials. It can be concluded that the 2 1 0.15 0.8 3 1 0.15 1 2 10 0.3 4 1.5 0.225 1.8 5 2 0.3 3.2 6. Wind speed/Rotor height 1 2.5 0.375 1 10 2 2 3.75 0.5625 1.5 12 0.3 3 5 0.75 2 14 Inventions 2020, 5, x 12 of 19

Inventions 2020, 5, 18 12 of 18

Table 4. Physical properties of air.

3 Re ρair [Kg/m ] µair [Kg/m.s] 106 1.225 1.647 10 5 · −

5. Results and Discussion

5.1. Rotor Solidity (a) (b) FigureThe 10. influence(a) Lift coefficient of altering versus the angle turbine of attack solidity alpha on of its NACA aerodynamic 0015 airfoil performance at Re = 106; (b has) Drag been coefficient studied versususing theangle DMST of attack model alpha as shownof NACA in 0015 Figure airfoil 11. Theat Re solidity= 106. σ was increased from 0.1 to 0.4 by decreasing the turbine radius while the blades swept area was kept constant (see Table3). The DMST model has 5.been Results run many and Discussion times for H-rotor with three NACA 0015 blades and Reynolds number 106. It can be observed for σ = 0.3 that the CP grows rapidly compared to the other solidity. This is illustrated clearly 5.1in Figure. Rotor Solidity11 where the CP peak (0.342) expected at a low tip speed ratio (TSR = 3.51). σ TheAs the influence turbine of solidity altering decreases the turbine ( = 0.1),solidity the ConP drops its aerodynamic (0.183) at high performance TSR (6.51). has Therefore, been studied from a usingstructural the DMST design model standpoint, as shown a low in Figure solidity 11. turbineThe solidity will experience𝜎 was increased greater from centrifugal 0.1 to 0.4 by forces decreasing on the theblades turbine and supportingradius while arms the blades because swept of its area higher was TSR kept operating constant conditions (see Table [ 253)., 29The]. WhileDMST these model forces has beencould run be containedmany times with for the H- userotor of with modern three composite NACA 0015 materials. blades Itand can Reynolds be concluded number that 10 the6. optimumIt can be solidity (σ = 0.3) was chosen for this VAWT design as it has achieved a higher aerodynamic efficiency, observed for 𝜎 = 0.3 that the CP grows rapidly compared to the other solidity. This is illustrated clearly the same optimum solidity has been found in other works [42,43]. in Figure 11 where the CP peak (0.342) expected at a low tip speed ratio (TSR = 3.51).

Figure 11. PerformancePerformance curve curve for different different solidit solidity.y

5.2. NumberAs the turbine of Blades solidity decreases (𝜎 = 0.1), the CP drops (0.183) at high TSR (6.51). Therefore, from a structural design standpoint, a low solidity turbine will experience greater centrifugal forces Increasing the number of blades is to increase the number of wakes and accentuate the interactions on the blades and supporting arms because of its higher TSR operating conditions [25,29]. While these of the blades with the wakes. Nevertheless, this has the advantage of smoothing the instantaneous forces could be contained with the use of modern composite materials. It can be concluded that the torque produced by the rotor and decreasing the radial component of the aerodynamic forces [44]. Furthermore, for a specific turbine solidity, the blade becomes leaner and subsequently has less resistance to bending when using more than two or three blades, which requires more struts, and this creates more parasitic drag. As a result, the aerodynamic performances of turbines decrease significantly, which is undesirable from a structural perspective. Therefore, the choice of the number of blades is a compromise between aerodynamic efficiency, the blade stiffness, and cost considerations [45]. The impact of increasing the number of blades on the turbine aerodynamic efficiency is given in Figure 12, the solidity was kept constant (Table3). It is found that the most identifiable change in rotor Inventions 2020, 5, 18 13 of 18

Inventions 2020, 5, x 13 of 19 efficiency is from two to three blades. We have chosen a two-bladed, as it allows minimizing both the optimumrotor weight, solidity cost ( and𝜎 = 0.3) the was problem chosen of startingfor this VAWT the turbine. design Also, as it withhas achieved the two-bladed a higher design, aerodynamic we can efficiencyintegrate the, the diameter same optimum variation solidity mechanism has been (Figure found3) thatin other allows works varying [42,43 the]. rotor radius according to the wind velocity available. 5.2. Number of Blades

Figure 12. EffectEffect of the number of blades on the turbine eefficiencyfficiency.

5.3. RotorIncreasing Radius the number of blades is to increase the number of wakes and accentuate the interactioIn thens literature, of the blades few research with the studies wakes. explore Nevertheless, the influence this ofhas the the rotor advantage radius on of the smoothing performance the instantaneousof VAWT. The torque variation produced in Darrieus by the rotor rotor radius and decreasing in solo Darrieus the radial and acomponent combined of Darrieus–Savonius the aerodynamic windforces turbine [44]. Furthermore, has an important for a especificffect on theturbine turbine solidity, performance. the blade Figure becomes 13a showsleaner theand variation subsequently of the has less resistance to bending when using more than two or three blades, which requires more struts, power coefficient CP with tip speed ratio for four different radii, although the solidity and number of and this creates more parasitic drag. As a result, the aerodynamic performances of turbines decrease blades were kept constant (Table3). The Cp attains its highest value of the tip speed ratio TSR = 4.96 significantly,for radius equal which to 0.7is undesirable m and it decreases from a structural significantly perspective. with the Therefore, increase of the the choice radius. of Wethe findnumber the ofsame blades result isfound a compromise by Sahim et al. between [46], and aerodynamic the numerical efficiency,results show the the bladeinterest stiffness of a variable, and radius cost considerations [45]. The impact of increasing the number of blades on the turbine aerodynamic rotor, whereby when we increase the rotor radius, obviously, the Cp decreases. In contrast, the power efficiency is given in Figure 12, the solidity was kept constant (Table 3). It is found that the most generatedInventions 20 by20 the, 5, x rotor increases when the rotor radius increases, as shown in Figure 13b. 14 of 19 identifiable change in rotor efficiency is from two to three blades. We have chosen a two-bladed, as it allows minimizing both the rotor weight, cost and the problem of starting the turbine. Also, with the two-bladed design, we can integrate the diameter variation mechanism (Figure 3) that allows varying the rotor radius according to the wind velocity available.

5.3. Rotor Radius In the literature, few research studies explore the influence of the rotor radius on the performance of VAWT. The variation in Darrieus rotor radius in solo Darrieus and a combined Darrieus–Savonius wind turbine has an important effect on the turbine performance. Figure 13a shows the variation of the power coefficient CP with tip speed ratio for four different radii, although the solidity and number of blades were kept constant (Table 3). The Cp attains its highest value of the tip speed ratio TSR = 4.96 for radius equal to 0.7 m and it decreases significantly with the increase of the radius. We find the same result found by Sahim et al. [46], and the numerical results show the interest of a variable radius rotor, whereby when we increase the rotor radius, obviously, the C p decreases. In contrast, the(a) power generated by the rotor increases when the(b) rotor radius increases, as shown in Figure 13b. FigureFigure 13. 13(.a ()a Power) Power coe coefficientfficient Cp Cp versus versus TSR TSR for for di differentfferent radius; radius (b; ()b Mechanical) Mechanical Power Power versus versus TSR TSR forfor di differentfferent radius. radius

5.4. Rotor Aspect Ratio As shown in Table 3, five H-Darrieus rotors with different radius, chord length, and height are considered to investigate the effect of altering the aspect ratio on the aerodynamic performance of the rotor. Figure 14 represents the obtained results of increasing the blade aspect ratio on the rotor's

power coefficient. It is clear that from aerodynamic point the peak of the Cp increases with high aspect ratio, and the maximum power coefficient value is 0.469 at TSR = 4.01. However, from a structural point, a rotor with high aspect ratio is subjected to high bending moments, so during the design phase, it should consider the aerodynamic performance aspects and the structural resistance aspects should be considered. Indeed, floating Darrieus-type wind turbines with three-stage rotors contain these two aspects because the rotor is subdivided into three stages with a different radius and height. This means a different AR which will allow us to vary the output performance. Therefore, this subdivision will also solve the problem of starting a single large turbine.

Figure 14. Effect of increasing the AR on the rotor’s power coefficient.

Inventions 2020, 5, x 14 of 19

(a) (b)

InventionsFigure2020 1,35. ,( 18a) Power coefficient Cp versus TSR for different radius; (b) Mechanical Power versus TSR14 of 18 for different radius

5.45.4.. Rotor Rotor Aspect Aspect Ratio Ratio AsAs shown in in Table Table 33,, fivefive H-DarrieusH-Darrieus rotorsrotors withwith didifferentfferent radius, chord length, and height are consideconsideredred to to investigate investigate the the effect effect of altering the aspect ratio on the aerodynamic performance of of thethe rotor rotor.. Figure Figure 1414 represents th thee obtained results of increasing the blade aspect ratio on the rotor’srotor's power coefficient. It is clear that from aerodynamic point the peak of the Cp increases with high aspect power coefficient. It is clear that from aerodynamic point the peak of the Cp increases with high aspectratio, and ratio the, and maximum the maximum power coepowerfficient coefficient value is value0.469 at is TSR0.469= at4.01. TSR However, = 4.01. However, from a structural from a structuralpoint, a rotor point with, a highrotor aspectwith high ratio aspect is subjected ratio is to subjected high bending to high moments, bending so moments, during the so design during phase, the designit should phase, consider it should the aerodynamic consider the performance aerodynamic aspects performance and the aspects structural and resistance the structural aspects resistance should aspectsbe considered. should Indeed,be considered floating. Indeed, Darrieus-type floating wind Darrieus turbines-type with wind three-stage turbines rotorswith three contain-stage these rotors two containaspects becausethese two the aspects rotor isbecause subdivided the rotor into is three sub stagesdivided with into a three different stages radius with and a different height. Thisradius means and heighta different. This AR means which a willdifferent allow AR us which to vary will the allow output us performance. to vary the output Therefore, performance. this subdivision Therefore, will thialsos subdivision solve the problem will also of solve starting the a problem single large of starting turbine. a single large turbine.

FigureFigure 14. 14. EffectEffect of of increasing increasing the the AR AR on on the the rotor’ rotor’ss power power coefficient coefficient.. 5.5. Wind Speed with Rotor Height

The rotor’s power efficiency was evaluated for a range of free stream velocity from 10 m/s to 15 m/s (the rotor high varied from 1 m to 1.5 m, respectively) at different tip speed ratio ranging from 1.5 to 7, with the influential geometric parameters are given in Table3. Considering Figure 15, which plots TSR as a function of the power output, it indicates that the power peak will augment without altering its optimal TSR when the free stream velocity increases. In addition, for large values of free stream velocity and rotor height, the power increases considerably compared to the low values of free stream velocity and rotor height. Hence, this showed the influence of the variation of free stream velocity (height rotor, respectively) on the power generated by the rotor. Inventions 2020, 5, x 15 of 19

Inventions 2020, 5, 18 15 of 18 5.5 Wind Speed with Rotor Height

Figure 15. EffectEffect of increasing wind velocity and height on the rotor’srotor’s power coecoefficientfficient..

6. Conclusions The rotor’s power efficiency was evaluated for a range of free stream velocity from 10 m/s to 15 m/s (theThis rotor paper high presents varied a newfrom design 1 m to of 1.5 F-VAWT m, respectively) with three-stage at different rotors. tip Three speed electrical ratio ranging generators from can1.5 to be 7, used with with the diinfffluentialerent powers geometric depending parameters on the are rotor given radius, in Table wind speed,3. Considering and the height Figure where 15, which each rotorplots isTSR located. as a function In fact, thisof the configuration power output, of three it indicates stage rotors that the helps power to optimize peak will power augment and solve without the problemaltering its of turbineoptimal self-starting. TSR when the The free result stream of the velocity performance increases. curve In ofadditio Qbladen, softwarefor large wasvalues validated of free bystream comparison velocity withand experimentalrotor height, the and power numerical increases results, considerably and it shows compared a good prediction to the low of values the rotor of performance.free stream velocity Moreover, and therotor aerodynamic height. Hence, profile this NACA showed 0015 the is influence employed of for the aerodynamic variation ofsimulations, free stream becausevelocity (height this profile rotor, gives respectively) the best aerodynamic on the power e generatedfficiency compared by the rotor. to other aerodynamic profiles, as well as being the most common profiles used for commercial Darrieus VAWTs. An investigation of6. Conclusions several influential geometric parameters of VAWT has been taken into account, including solidity, numberThis of paper blades, presents rotor radius, a new design aspect of ratio, F-VAWT wind velocity,with three and-stage rotor rotors. height. Three Thus, electrical numerical generators results obtainedcan be used by with the aerodynamic different powers simulations depending identify on the a rotor low solidity radius, turbinewind speed, (σ = 0.3),and the offering height the where best aerodynamiceach rotor is located. performance. In fact, this A configuration two-blade design of three is recommendedstage rotors helps to to minimize optimize the power rotor and weight, solve reducethe problem the problem of turbine of startingself-starting. the turbine,The result and of lower the performance the cost. In curve addition, of Qblade we can software integrate was a mechanismvalidated by that compar allowsison varying with experimental the rotor radius and numer accordingical results to the, windand it speedshows availablea good prediction by using of a two-bladedthe rotor performance. design. These Moreover findings, the also aerodynamic indicate the profile interest NACA of a variable0015 is employed AR and windfor aerodynamic velocity, as whensimulation theses parameters, because thisincreased, profile the gives aerodynamic the best performance aerodynamic will efficiency be improved compared as well. to other aerodynamic profiles, as well as being the most common profiles used for commercial Darrieus Author Contributions: Conceptualization and writing—original draft preparation, M.A.D. and A.R.; methodology, O.B.;VAWTs. software An investigation and validation, of M.A.D.; several investigation, influential formalgeometric analysis, parameters writing—review of VAWT and has editing, been visualization, taken into E.R.account, All authors including have readsolidity, and agreednumber to theof publishedblades, rotor version radius of the, aspect manuscript. ratio, wind velocity, and rotor Funding:height. Thus,This researchnumerical received results no externalobtained funding. by the aerodynamic simulations identify a low solidity Acknowledgments:turbine (𝜎 = 0.3), offeringThe work the best of the aerodynamic third author performance. was supported A bytwo the-blade project design “Excellence, is recommended performance to andminimize competitiveness the rotor weight in the Research,, reduce the Development problem of and starting Innovation the turbine activities, and at “Dunarealower the decost. Jos” In University addition, ofwe Galati”, can integrate acronym a “EXPERT”,mechanism financed that allows by the varying Romanian the rotor Ministry radius of Research according and to Innovation the wind speed in the framework of Programme 1—Development of the national research and development system, Sub-programme 1.2—Institutionalavailable by using Performance a two-bladed —Projects design for. The financingse findings excellence also indicate in Research, the interest Development of a variable and Innovation, AR and Contractwind velocity no. 14PFE, as /when17.10.2018. these parameters increased, the aerodynamic performance will be improved Conflictsas well. of Interest: The authors declare no conflict of interest. Author Contributions: Conceptualization and writing—original draft preparation, M.A.D and A.R.; methodology, O.B; software and validation, M.A.D; investigation, formal analysis, writing—review and editing, visualization, E.R.

Funding: This research received no external funding. Inventions 2020, 5, 18 16 of 18

Nomenclature

A turbine swept area, m2 (A = 2hR) au, ad upwind and downwind induction factors B number of blades c blade chord length, m CD drag coefficient CL lift coefficient Cn normal coefficient Cm moment coefficient Ct tangential coefficient CP power coefficient D blade drag force, N fu, fd upwind and downwind functions Fn,Ft normal and tangential force, N Fti average tangential force, N h height of the blade/turbine, m L blade lift force, N Pi total power output Qi total torque output R radius of the wind turbine, m Re local Reynold number Va upstream induced velocity, m/s u d Va ,Va upstream and downstream induced velocity, m/s Vc chordal velocity component, m/s Vn normal velocity component, m/s u VR relative velocity upstream, m/s d VR relative velocity downstream, m/s V free-stream velocity, m/s ∞ α blade angle of attack αu, αd upstream and downstream angle of attack ∆θ azimuth angle increment, correspond to one stream tube θ azimuth angle TSR tip speed ratio ρ density of the air (kg/m3) µ dynamic viscosity of the air (kg/m.s) σ solidity ω angular velocity of rotor, rad/s

Acronyms

DMST double multiple stream tube F-VAWTs floating vertical axis wind turbines F-HAWTs floating horizontal axis wind turbines HAWT horizontal axis wind turbine MST multiple stream tube SKWID Savonius keel wind turbine Darrieus SB-VAWT straight-bladed vertical axis wind turbine PMSMs permanent-magnet synchronous motors

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