Parametric Study and Comparison of Aerodynamics Momentum-Based Models for Straight-Bladed Vertical Axis Wind Turbines

Abstract

In this study, different momentum models for vertical axis wind turbines (VAWTs) are investigated and compared for different operating parameters. The performance over different rotor solidities and aspect ratios is studied to find the best configuration of the turbine. By comparing different models, the merit of double multiple streamtube model is established. Three different airfoils were investigated using the parameters of reference turbines from the literature. NACA0015 is found to be advantageous among NACA0012 and NACA0021. Also, the effect of different number of blades is studied and it is found that two-bladed rotor with NACA0015 airfoil provides excellent performance. By studying the turbine aspect ratio, it is found that a VAWT of 5.1 m height and 4.25 m diameter will generate a maximum power coefficient of 36.13% at TSR of 5.51 with good starting behavior.

Appendix A

The mathematical relations for momentum models are presented in this section, more details are available in [30].

1.1 A.1 General Relations

$$ V_{\text{c}} = R\omega + V_{a} \cos \theta $$

(A.1)

$$ V_{\text{n}} = V_{a} \sin \theta $$

(A.2)

$$ W = \sqrt {V_{\text{n}}^{2} + V_{\text{c}}^{2} = } \sqrt {\left( {V_{a} \sin \theta } \right)^{2} + \left( {R\omega + V_{a} \cos \theta } \right)^{2} } $$

(A.3)

Using free stream velocity, the relative velocity can be expressed in non-dimensional form ($a$ is the axial induction factor and $\alpha$ is the angle of attach:

$$ \frac{W}{{V_{\infty } }} = \sqrt {\left( {(1 - a)\sin \theta } \right)^{2} + \left( {\lambda + (1 - a)\cos \theta } \right)^{2} } $$

(A.4)

$$ \alpha = \tan^{ - 1} \left( {\frac{{V_{\text{n}} }}{{V_{\text{c}} }}} \right) = \tan^{ - 1} \left[ {\frac{{V_{a} \sin \theta }}{{R\omega + V_{a} \cos \theta }}} \right] $$

(A.5)

Recall that $V_{a} = V_{\infty } \left( {1 - a} \right)$ and the tip-speed ratio (TSR) $\lambda = {{R\omega } \mathord{\left/ {\vphantom {{R\omega } {V_{\infty } }}} \right. \kern-0pt} {V_{\infty } }}$ previous equation can be written as:

$$ \alpha = \tan^{ - 1} \left[ {\frac{(1 - a)\sin \theta }{\lambda + (1 - a)\cos \theta }} \right] $$

(A.6)

$$ C_{\text{t}} = C_{\text{l}} \sin \alpha - C_{\text{d}} \cos \alpha $$

(A.7)

$$ C_{\text{n}} = C_{\text{l}} \cos \alpha + C_{\text{d}} \sin \alpha $$

(A.8)

where $C_{\text{l}}$ and $C_{\text{d}}$ are the airfoil lift and drag coefficients, respectively, and $C_{\text{n}}$ is the normal coefficient.

$$ F_{\text{t}} = \frac{1}{2}C_{\text{t}} \rho cHW^{2} \quad {\text{and}}\quad F_{\text{n}} = \frac{1}{2}C_{\text{n}} \rho cHW^{2} $$

(A.9)

$F_{\text{t}}$ and $F_{\text{n}}$ are the tangential and normal forces, respectively. Different models were developed to calculate such forces.

$$ F_{\text{ta}} = \frac{1}{2\pi }\int\limits_{0}^{2\pi } {F_{\text{t}} (\theta ){\text{d}}\theta } $$

(A.10)

The total torque (Q) for the number of blades (N) is found as

$$ Q = NF_{\text{ta}} R $$

(A.11)

The total power (P) can be obtained as

$$ P = Q \times \omega $$

(A.12)

1.2 A.2 Single Streamtube Model

Owing to the rate of change of momentum, the streamwise drag force ($F_{\text{D}}$) is given by [19]:

$$ F_{\text{D}} = 2\rho AV_{a} (V_{\infty } - V_{a} ) $$

(A.13)

The rotor drag coefficient ($C_{\text{DD}}$) is defined as

$$ C_{\text{DD}} = \frac{{F_{\text{D}} }}{{\frac{1}{2}\rho AV_{a}^{2} }} = \frac{{4(V_{\infty } - V_{a} )}}{{V_{a} }} $$

(A.14)

$$ \frac{{V_{a} }}{{V_{\infty } }} = \left( {\frac{1}{{1 + {{C_{\text{DD}} } \mathord{\left/ {\vphantom {{C_{\text{DD}} } 4}} \right. \kern-0pt} 4}}}} \right) $$

(A.15)

$$ C_{\text{D}} = \frac{{F_{\text{D}} }}{{\frac{1}{2}\rho AV_{\infty }^{2} }} = C_{\text{DD}} \left( {\frac{{V_{\text{D}} }}{{V_{\infty } }}} \right) = \frac{{C_{\text{DD}} }}{{\left( {1 + \frac{1}{4}C_{\text{DD}} } \right)^{2} }} $$

(A.16)

The local relative dynamic pressure is given by:

$$ \frac{q}{{\frac{1}{2}\rho AV_{a}^{2} }} = \left( {\frac{R\omega }{{V_{a} }} + \cos \theta } \right)^{2} + \sin^{2} \theta = \left( {\frac{W}{{V_{a} }}} \right)^{2} $$

(A.17)

The total drag of a rotor having N blades read as:

$$ F_{\text{D}} = \frac{NcH}{2\pi }\int\limits_{0}^{2\pi } {q\left( {C_{\text{n}} \sin \theta - C_{\text{t}} \cos \theta } \right){\text{d}}\theta } $$

(A.18)

Thus, considering a straight-bladed rotor of height H and radius R, the area A = 2HR

$$ C_{\text{D}} = \frac{Nc}{R}\frac{1}{{2\pi \rho V_{a}^{2} }}\int\limits_{0}^{2\pi } {q\left( {C_{\text{n}} \sin \theta - C_{\text{t}} \cos \theta } \right){\text{d}}\theta } $$

(A.19)

The tip-speed ratio, (TSR) is calculated as:

$$ \lambda = \frac{R\omega }{{V_{\infty } }} = \frac{R\omega }{{V_{a} }}\left( {\frac{1}{{1 + \frac{1}{4}C_{\text{DD}} }}} \right) $$

(A.20)

The total torque of the rotor is given by:

$$ T = \frac{NcRH}{2\pi }\int\limits_{0}^{2\pi } {qC_{\text{t}} {\text{d}}\theta } $$

(A.21)

The power is given by:

$$ P = \omega \times T = \frac{NcRH\omega }{2\pi }\int\limits_{0}^{2\pi } {qC_{\text{t}} {\text{d}}\theta } $$

(A.22)

The maximum possible power, according to the Betz limit, is given by Paraschivoiu [19] and Mohammed et al. [30]:

$$ P_{\hbox{max} } = \frac{32}{27}\frac{1}{2}\rho V_{\infty }^{3} RH $$

(A.23)

The power coefficient is thus:

$$ C_{\text{P}} = \frac{P}{{P_{\hbox{max} } }} = \frac{27}{32}\frac{1}{2\pi }\frac{Nc}{R}\frac{R\omega }{{V_{a} }}\left( {\frac{{V_{a} }}{{V_{\infty } }}} \right)^{3} \int\limits_{0}^{2\pi } {\left( {\frac{q}{{\frac{1}{2}\rho V_{a}^{2} }}} \right)} C_{\text{t}} {\text{d}}\theta $$

(A.24)

1.3 A.3 Multiple Streamtube Model

The average streamwise force $\overline{{F_{x} }}$ applied by blade elements while passing the streamtube is found from [19]:

$$ \overline{{F_{x} }} = 2\rho AV_{a} \left( {V_{\infty } - V_{a} } \right) = NF_{x} \frac{\Delta \theta }{\pi } $$

(A.25)

$$ \frac{{NF_{x} }}{{2\pi \rho r\Delta h\sin \theta V_{\infty }^{2} }} = \frac{{V_{a} }}{{V_{\infty } }}\left( {1 - \frac{{V_{a} }}{{V_{\infty } }}} \right) = F_{x}^{*} $$

(A.26)

$$ F_{x} = - \left( {F_{\text{N}} \sin \theta + F_{\text{T}} \cos \theta } \right) $$

(A.27)

$$ F_{x}^{*} = \frac{{NF_{x} }}{{2\pi \rho r\Delta h\sin \theta V_{\infty }^{2} }} = \frac{NC}{4\pi r}\left( {\frac{W}{{V_{\infty } }}} \right)^{2} \left( {C_{\text{n}} - C_{\text{t}} \frac{\cos \theta }{\sin \theta }} \right) $$

(A.28)

$$ a = 1 - \frac{{V_{a} }}{{V_{\infty } }} $$

(A.29)

$$ a = F_{x}^{*} + a^{2} $$

(A.30)

This is a fundamental relation to solve the streamtube momentum equation iteratively.

The torque produced by a single blade is given by:

$$ T_{\text{B}} = \frac{1}{2}\rho cHW^{2} C_{\text{t}} R $$

(A.31)

The average total torque produced by the rotor is written as:

$$ T = \frac{N}{{N_{\theta } }}\sum\limits_{1}^{{N_{\theta } }} {T_{\text{B}} } $$

(A.32)

where $N_{\theta } = \frac{\pi }{\Delta \theta } + 1$ and $\Delta \theta$ is the size of streamtube.

The power coefficient is given as follows:

$$ C_{\text{P}} = \frac{T\omega }{{\tfrac{1}{2}\rho AV_{\infty }^{2} }} = \frac{{\sum\limits_{1}^{{N_{\theta } }} {\tfrac{NC}{2R}\lambda \left( {\tfrac{W}{{V_{\infty } }}} \right)^{2} C_{\text{t}} } }}{{N_{\theta } }} $$

(A.33)

1.4 A.4 Double Multiple Streamtube Model

As mentioned before, the calculation are performed twice for upwind half (${\pi \mathord{\left/ {\vphantom {\pi 2}} \right. \kern-0pt} 2} \le \theta \le {{3\pi } \mathord{\left/ {\vphantom {{3\pi } 2}} \right. \kern-0pt} 2}$) and for downwind half (${{3\pi } \mathord{\left/ {\vphantom {{3\pi } 2}} \right. \kern-0pt} 2} \le \theta \le {\pi \mathord{\left/ {\vphantom {\pi 2}} \right. \kern-0pt} 2}$) cycles. Equations (A.3) and (A.6) are used to find the relative velocity and angle of attach, respectively. Equating the forces given by momentum equations to those given by blade element theory [19, 31]:

$$ f_{\text{u}} (1 - a) = a $$

(A.34)

where the upwind function $f_{\text{u}}$ is given by:

$$ f_{\text{u}} = \frac{Nc}{8\pi R}\int\limits_{\pi /2}^{3\pi /2} {\left( {C_{\text{n}} \frac{\cos \theta }{{\left| {\cos \theta } \right|}} - C_{\text{t}} \frac{\sin \theta }{{\left| {\sin \theta } \right|}}} \right)} \left( {\frac{W}{{V_{a} }}} \right)^{2} {\text{d}}\theta $$

(A.35)

Thus, the power coefficient for the upwind half of the turbine is given by:

$$ C_{{{\text{P}}_{u} }} = \frac{NcH}{2\pi A}\int\limits_{\pi /2}^{3\pi /2} {C_{\text{t}} } \left( {\frac{W}{{V_{\infty } }}} \right)^{2} {\text{d}}\theta $$

(A.36)

The downwind part of the rotor is treated similarly and finally the power coefficients of the two half cycles is summed to find the overall power coefficient of the rotor.