On Optimization of Power Coefficient of HAWT

Transcription

1 Journal of Power and Energy Engineering, 14,, 198- Published Online Aril 14 in Scies htt://wwwscirorg/journal/jee htt://dxdoiorg/1436/jee1448 On Otimization of Power Coefficient of HAWT Marat Z Dosaev 1, yubov A Klimina 1, Boris Ya okshin 1, Yury D Selyutskiy 1, Shih-Shin Hwang 1 Institute of Mechanics, omonosov Moscow State University, Moscow, ussia Mechanical Engineering Deartment, Chien Hsin University of Science and Technologies, Zhongli City, Taiwan dosayev@imecmsuru eceived December 13 Abstract The horizontal axis wind turbine (HAWT blades rotation in the steady wind flow is considered We discuss the roblem of determining the blade twist which could guarantee the maximum value of the ower coefficient We define the blade twist as the technological turn of sections of blade around its axis This turn changes the effective itch angle of turbine blade along its length For descrition of aerodynamic load uon the blades we used the quasi-steady aroach Air velocities of centers of ressure of blade sections are reresented when taking into account comonents induced by flow and vortex We reduced the functional maximization roblem to find the maximum of non-dimensional function This function is given by iemann integral deending on section itch angle and ti seed ratio We suggested the algorithm for solving the roblem under consideration for a given blade shae Keywords HAWT Blade; Ti Seed atio; Maximum of Power Coefficient 1 Introduction The roblem of designing the new shae for wind turbine blade is very comlicated Along with taking into account vibration and strength roerties of blade, one must allow for necessity to utilize the maximum energy of wind Wide sreading of different tyes of wind turbine enables to construct engineering techniques for blade design For examle, in [1,] authors roosed the engineering method for choosing blades arameters, such as the aerodynamic rofile, blade width, itch angle, and etc for several sections along the flow On the other hand the wind turbine aerodynamics is comrehensively studied [3] Some authors use analytical aroaches for ower efficiency estimating of wind turbine The method based on using Goldstein functions is roosed in [4] for ideal turbine with the finite number of blades The simle enough algorithm of numerical estimation of the uer bound of the ower coefficient is suggested In resent aer, we roose the analytical-comutational aroach for calculating the distribution of twist blade angle along the blade length, which guarantee the maximum value of ower coefficient for given blade section rofile and give distribution of blade width (Figure 1 We use the mechanical-mathematical model of How to cite this aer: Dosaev, MZ, Klimina, A, okshin, BYa, Selyutskiy, YD and Hwang, S-S (14 On Otimization of Power Coefficient of HAWT Journal of Power and Energy Engineering,, 198- htt://dxdoiorg/1436/jee1448

2 Figure 1 Blade twist for several sections aerodynamic load uon turbine blades develoed in [5] This model is based on quasi-steady aroach In order to advance this model we introduce the comonents that related with induced velocities (see for examle, [5] Problem Formulation We consider the turbine with radius (where is a distance between axis of rotation and blade ti that has n blades (usually n = 3-6 and rotates with angular seed Ω in steady airflow The wind seed W is directed along the turbine axis The outut ower P of turbine looks as following: P C SW 3 = 5 ( λ ρ, where λ is TS (ti seed ratio of turbine, ρ is the air density, S = π is turbine swet area, C ( λ is the ower coefficient that characterizes the efficiency of wind turbine We assume that the blade section, which is located at distance r ( < r < from axis of rotation, reresents the given airfoil The shae of section is the same for all blade sections We formulate the following roblem: to determine the secified value λ and secified deendence ϑ = ϑ( r for twist angle of blade along its length, which ensure the maximum value of ower coefficient We use the flat cross-section hyothesis for needed characteristics obtaining We allocate the section element dr which is located at distance r from axis of rotation (Figure Deendence b = b(r is given The air velocity of oint of intersection of blade axis and blade section is a sum of flow seed W(1 - а and seed rω (1 + a of its rotation Here coefficients aa, to be determined are functions of r In Figure 3, the side view of this element is shown In this rojection, it is convenient to introduce all required values: itch angle θ, angle of attack α, velocity V with resect to the flow, angle φ between the turbine lane and V, and vectors D, comonents of the aerodynamic force (drag and lift, resectively acting uon the selected element By construction, we have: θ = φ α, tg φ = W(1 a / rω (1 + a W(1 a = Vsin φ, rω (1 + a = V cos φ, (1 V = W (1 a + ( rω (1 + a Aerodynamic forces are defined by the following conventional formulae: =, D 5 CD ( α ρv ds 5 C ( α ρv ds = ( where ds = b( r dr is the element area, and aerodynamic coefficients CD( α, C( α are known functions for a given airfoil [6] Note that such aroach to describing the aerodynamics was used in [7] The aerodynamic torque dm with resect to the rotation axis roduced on this element is given by the exression dm = r( cosφ Dsin φ or, taking into account (1 and (, ( ( ( α φ D( α φ ρ ( ρ ( α ( D( α ( ( ( dm = 5rb r C cos C sin V dr = 5rb r W 1 a C ( 1+ a C rω W 1 a + rω 1+ a W dr 199

3 Figure Blade element Denote: Figure 3 Side view of blade element Then the torque roduced by all blades can be reresented as an integral The ower P ω = rω / W = λr/ (3 ( ( ( ( ( ( ( D M = 5nρW rb r 1 a C α 1+ a C α ω 1 a + ω 1+ a dr = Ω M looks as follows: 3 1 ( ( ( ( D( ( ( P = 5nρW λ rb r 1 a C α 1+ a C α ω 1 a + ω 1+ a dr (4 Substituting r in (4 using the exression (3, we obtain Here the non-dimensional function is introduced P 3 = 5 nρw Φ ( λ (5 λ 1 ( ( 1 ( ( 1 ( ( 1 ( 1 b a C a CD a a d (6 Φ λ = λ ω α + α ω + ω + ω 3 Comaring (5 with the general formula P= 5C ρπ W for the ower roduced by the wind turbine, one can readily see that the following relation holds: C ( λ nπ ( λ = Φ, (7 and the initial otimization roblem is reduced to the roblem of maximum search for the function (6 3 Solution of the Problem Before solving the roblem of otimization of this coefficient (values to be varied are θ ( r and λ, blade chord b is a given function of r = ω/ λ, values a, a are also to be defined as functions of r = ω/ λ, consider the rocess of calculation of the integrand in (6 for fixed r, ω, λ, b, θ For that, following [3], we

4 f introduce the adjustment coefficient F = arccos( e / π, taking into account losses at the blade ti, where f = n( λ ω/ω, σ = nbλ( πω, CN = C( αcos φ + CD( αsinφ is the comonent of aerodynamic force along HAWT the axis, and CT = C( αsin φ CD( αcosφ is the comonent of aerodynamic force lying in the turbine lane Then induced seeds coefficients can be reresented as follows [3]: a = σcn( α(4fsin ϕ+ σcn( α, (31 a = σc ( α 4Fsinϕcos ϕ σc ( α T ( Angles α and ϕ can be exressed as follows (using (1 and (3: 1 1 =, ϕ arctg ( 1 a ω ( 1 a α ϕ θ ( T = + (3 elations (31-(3 make a system of four equations with resect to a, a, ϕα, Contrarily to the iterative method of solving these equations roosed in [3,5], consider another way for determining the sought values Substituting (31 into the second Equation (3, we obtain ( ω ϕ ϕ ϕ σ ( ω N T 4F sin sin cos + C + C = (33 From here, ϕ is exressed via α, ω, λ In articular, for F ( ω λ we have: ϕ = arctg [( λc CD / ( λcd + C ] After having solved (33, arameters a, a are also exressed from (31 via α, ω, λ, and then the angle θ is determined from the first Equation (3 as a function of the same arameters α, ω, λ Thus, the integrand in (6 reresents an exlicit function of α, ω, λ Such reresentation is more referable for solving the original roblem of maximization of the coefficient C Now return to the roblem of maximization of C, or of function Φ ( λ Taking into account that λ is a constant (though unknown, the integral (6 can be interreted as a functional defined on the class of differentiable functions αω ( Then the source roblem is deduced to the roblem of choice of a function α = α ( ω, λ that delivers maximum to the integral, and of subsequent choice of such a value λ that delivers maximum value to the function Φ ( λ In order to find α ( ω, we use the convenient method for solving variational roblems, we obtain the following necessary condition of the maximum of the integral (Euler-agrange equation: the full derivative of the integrand with resect to α is zero (this equation is not written here due to its comlexity In [5], results of numerical solution of Equations (31-(3 are given for several NACA airfoils, from which one can see that coefficients a, a almost do not change for ω > 1 Hence, we assume (as the first aroximation that Then the mentioned Euler-agrange equation looks as follows a = const, a = const (34 (1 a dc dα ω(1 + a dc dα = (35 From (35 it is ossible to find α as a function of ω : α= αω ( One of branches of this solution will contain the otimal deendence α = α ( ω, λ, and from (33 (or from (3 it is ossible to determine the intermediate variable ϕ, and then from (3 it is ossible to derive the exression for the desired blade twist containing the undetermined (so far arameter λ : (( a( a θ ωλ 1 ω 1 α ωλ D (, = arctg (, (36 Substituting the obtained exression for α ( ω into the integrand in (6, we obtain the function Φ ( λ of single argument λ, maximum value of which is determined analytically (if ossible or numerically Thus, the desired otimal value λ will be found, after which the otimal blade twist is determined from (36 as a function of ω, ( ω (, λ ] Note that ω = λ r/, ( r (, ] In rincile, the roblem is solved Taking into account deendences C( α, CD( α (eg, [6], one can show that the solution α= αω ( of Equation (35 exists as a monotonically decaying function If it delivers maximum to the functional (36, then uon having determined the otimal value of λ and returning back to the variable r, the otimal blade twist ( can also be decreasing function ( ( ( θ ( r = arctg λ r 1 a 1 + a α ( λ r/ 1

5 This monotony is qualitatively confirmed by the ractice of HAWT design 4 Conclusion In this aer we studied the roblem of maximization of ower coefficient for a HAWT by otimization of a blade twist and ti seed ratio We obtained an algorithm for determining of the otimal solution of this roblem in the frame of the quasi-steady model of aerodynamic action taking into account induced velocities Each ste of this algorithm deends on arameters of the model, so the final shae of the twist can be secified for each articular airfoil Still qualitative features of the otimal solution remain for the general case: for instance, there exists the otimal itch angle function that is monotone along the blade length Acknowledgements The work is artially suorted by FB, rojects NN , , and eferences [1] Perli, SB (1938 Windum and Windelectrical Units ONTI, Kharkov (in ussian [] Sabinin, GH (1931 Theory and Aerodynamic Calculation of Wind Engines Trudi TsAGI, Moscow (in ussian [3] Hansen, MO ( Aerodynamics of Wind Turbines James and James, ondon [4] Okulov, V and Sorensen, GN (8 Ideal Wind Turbine with Finite Number of Blades DAN, 4, (in ussian [5] Tenguria, N, Mittal, ND and Ahmed, S (11 Evaluation of Performance of Horizontal Axis Wind Turbine Blades Based on Otimal otor Theory Journal of Urban and Environmental Engineering, 5, 15-3 htt://dxdoiorg/149/juee11v5n1153 [6] Kashafutdinov, ST and ushin, VN (1994 Atlas of Aerodynamic Characteristics of Wing Airfoils SO ASHN, Novosibirsk (in ussian [7] Dosaev, MZ, Kobrin, AI, okshin, BYa, Samsonov, VA and Selyutskiy, YuD (7 Constructive Theory of Small-Scale Power Generators Part I, MSU Publishing, Moscow