Aerodynamic Design of ‘Box Blade’ and ‘Non-planar’ Wind Turbines

Abstract

In this paper the aerodynamic efficiency of wind turbines with horizontal axis is discussed and the so-called box blade concept, inspired by the Prandtl’s ‘Best Wing System’ is analysed; this wind turbine configuration is proved to be efficient than a conventional blade. Moreover, other non-planar blades, such as the winglet and C extension are analysed via vortex theory with the numerical method of Ribner and Foster for the optimum circulation and the recent model of Okulov and Sørensen for the performance evaluation; a generalization of the above mentioned models is also presented in this work. Finally, the box blades are verified by means of a commercial CFD software.

Appendix: Comparison Between Theodorsen and Okulov–Sørensen Theories

In this Appendix we show the limits of the theory developed by Theodorsen for the study of heavily loaded rotors when applied to a wind turbine. As pointed out in Sect. 4.1, the Goldstein analytical solution was derived for a lightly loaded rotor. Hence, Theodorsen tried to generalize it to the case of a heavily loaded propeller. When $\bar{w} \ll 1$ it is possible to neglect the slipstream contraction/expansion. The first case occurs when dealing with a propeller while the latter occurs when considering a wind turbine. The wake deformation is another source of non-linearity for the problem since the wake expansion ratio is not known a priori and it depends on the solution itself. Following the Theodorsen approach we need to derive some additional relationships to relate the ratio R _∞ ∕R of the wake radius far downstream over the rotor radius.

First of all the performance coefficients are evaluated in the Trefftz plane, by means of Goldstein circulation function. To this end the subsequent integral balances are carried out on the control volume shown in Fig. 36:

Fig. 36

Control volume for the integral balances of the Theodorsen theoretical model; figure taken from [21]

• mass balance;

• z-momentum balance;

• energy balance.

For the detailed calculations we refer to [21]. Here we just report the main results for the power and thrust coefficients:

$$\displaystyle{ T =\rho wAk\left [V - w\left (\frac{1} {2} + \frac{\varepsilon } {k}\right )\right ] \Rightarrow C_{T} = 2k\bar{w}\left [1 -\bar{ w}\left (\frac{1} {2} + \frac{\varepsilon } {k}\right )\right ]\;, }$$

(60)

$$\displaystyle{ P =\rho kwA\left (V - w\right )\left (V - w \frac{\varepsilon } {k}\right ) \Rightarrow C_{P} = 2k\bar{w}\left (1 -\bar{ w}\right )\left (1 -\bar{ w} \frac{\varepsilon } {k}\right )\;, }$$

(61)

where ɛ is the mass coefficient and k is the kinetic energy coefficient, both dependent on the dimensionless circulation and the wake pitch.

The dimensionless coefficients of power and thrust have been evaluated by dividing thrust and power by $\frac{1} {2}\rho \pi R_{\infty }^{2}V ^{2}$ and $\frac{1} {2}\rho \pi R_{\infty }^{2}V ^{3}$, respectively, where the radius R _∞ of the streamtube far downstream the rotor plane is chosen as reference. In this way we obtain the performance of the turbine if no wake expansion were present. To include the wake expansion we must divide by the rotor disk area:

$$\displaystyle{ C_{P_{r}} = C_{P}\left (\frac{R_{\infty }} {R} \right )^{2}\;, }$$

(62)

$$\displaystyle{ C_{T_{r}} = C_{T}\left (\frac{R_{\infty }} {R} \right )^{2}\;. }$$

(63)

The additional equations that are needed to evaluate R _∞ ∕R are derived by calculating thrust and torque by means of the blade element momentum theory. Let us consider the velocity triangle in the rotor plane. The magnitude of the relative velocity is the following:

$$\displaystyle{ u = V \frac{\left (1 -\bar{ a_{0}}\cos ^{2}\vartheta _{p}\right )^{2}} {\sin \vartheta _{p}} \;, }$$

(64)

where $\bar{a}_{0}$ and ϑ _p are the dimensionless wake advance velocity and the flow angle in the rotor plane, respectively. Now, with the Kutta–Joukowski theorem we can write

$$\displaystyle{ T =\rho B\omega \int _{0}^{R}\frac{1 -\bar{ a_{0}}\cos ^{2}\vartheta _{ p}} {1 -\bar{ a_{0}}} \Gamma (r)\,dr\;, }$$

(65)

whose dimensionless form is

$$\displaystyle{ \left (\frac{R_{\infty }} {R} \right )^{2}\,C_{ T} = \frac{4\bar{w}\left (1 -\bar{ w}\right )} {1 -\bar{ a_{0}}} \,\int _{0}^{1}\left (1 -\bar{ a_{ 0}}\cos ^{2}\vartheta _{ p}\right )K(x)xdx\;. }$$

(66)

Similarly the power coefficient is

$$\displaystyle{ \left (\frac{R_{\infty }} {R} \right )^{2}\,C_{ P} = 4\bar{w}\left (1 -\bar{ w}\right )\,\int _{0}^{1}\left (1 -\bar{ a_{ 0}}\cos ^{2}\vartheta _{ p}\right )K(x)xdx\;. }$$

(67)

By dividing member by member these equations we find the relationship between $\bar{a}_{0}$ and $\bar{w}$:

$$\displaystyle{ \bar{a}_{0} = \frac{\frac{1} {2}\bar{w} - \frac{\varepsilon } {k}\bar{w}^{2}} {1 -\bar{ w}\left (\frac{1} {2} + \frac{\varepsilon } {k}\right )}\;. }$$

(68)

By substituting Eq. (60) into Eq. (66) we can finally write the wake expansion:

$$\displaystyle{ \left (\frac{R_{\infty }} {R} \right )^{2} = \frac{\left (1 -\bar{ w}\right )\left (1 -\bar{ a}_{0}S\right )} {\left (1 -\bar{ a}_{0}\right )\left [1 -\bar{ w}\left (\frac{1} {2} + \frac{\varepsilon } {k}\right )\right ]}\;, }$$

(69)

in which:

$$\displaystyle{ S = \frac{2} {k}\int _{0}^{1}Kx\cos ^{2}\vartheta _{ p}\,dx = \frac{2} {k}\int _{0}^{1}K \frac{x^{3}} {x^{2} + \left (\frac{1-\bar{a}_{0}} {1-\bar{w}} \right )^{2}\left (\frac{R_{\infty }} {R} \right )^{2} \frac{1} {\lambda _{T}^{2}} } \,dx\;. }$$

(70)

To fully define the problem of the wind turbine blade design, we need to maximize the power coefficient with respect to $\bar{w}$:

$$\displaystyle{ \frac{d} {d\bar{w}}\left (C_{P}\left (\bar{w}\right )\left (\frac{R_{\infty }} {R} \left (\bar{w}\right )\right )^{2}\right ) = 0\;, }$$

(71)

in which it is necessary to show the explicit dependence of the wake expansion ratio from $\bar{w}$: this requires an iteration procedure, since the equations that relate R _∞ ∕R to S are not explicit. The numerical solution was done as follows: since S and $\bar{w}$ are comprised between 0 and 1, the vectors $\bar{S}_{j}$ and $\bar{w}_{k}$ are defined to discretize these quantities; then the function $\bar{a}_{0}(\bar{w_{k}})$ is derived in a discrete form by employing Eq. (68). Then the discrete function $S(\bar{w}_{k})$ is derived: for each element of the S vector the following error is evaluated:

$$\displaystyle{ \Delta = S_{j} -\sum \limits _{i=1}^{N_{v} }\Gamma _{i} \frac{x_{i}^{3}\Delta x} {x_{i}^{2} + \frac{\left (1 -\bar{ w_{k}} \frac{\varepsilon } {k}\right )\left (1 -\bar{ a}_{0k}S_{j}\right )} {\left [1 -\bar{ w}_{k}\left (\frac{1} {2} + \frac{\varepsilon } {k}\right )\right ]\lambda _{T}^{2}} }\;. }$$

(72)

The value of the function $S(\bar{w}_{k})$ is given by the value of the vector S _j which minimizes the difference in Eq. (72). At this point the wake expansion $(\frac{R_{\infty }} {R} (\bar{w}_{k}))^{2}$ can be derived and, thus, the power coefficient is known with respect to $\bar{w}_{k}$. Finally the maximum value of $C_{P_{T}}$ can be found.

Once we know the value of $\bar{w}$ which maximizes the power coefficient, the rotor tip speed ratio can be evaluated as well:

$$\displaystyle{ \lambda = (1 -\bar{ w})\lambda _{T}\;. }$$

(73)

In the present paper it is observed how this model provides unreasonable results for high values of λ _T . In fact when λ _T  → ∞ the helix angle approaches π∕2, so the wake is similar to a solenoid; thus the induced velocity is purely axial and so v _z  ≡ w, therefore the mass coefficient and the kinetic energy coefficient become

$$\displaystyle{ k \rightarrow 1\;,\quad \quad \varepsilon \rightarrow 1\;,\quad \quad S \rightarrow 1\;, }$$

(74)

thus the power coefficient becomes

$$\displaystyle{ C_{P_{r}} = 2\bar{w}\frac{\left (1 -\bar{ w}\right )^{3}} {1 -\frac{3} {2}\bar{w}} \;. }$$

(75)

Clearly this function does not have any maximum but it diverges when $\bar{w}$ approaches 2∕3, as we can see in Fig. 37. A likely explanation of this inconsistency provided in this paper is the following: in the momentum balance we have a term which is related to the wake overpressure with respect to the ambient static pressure (p − p ₀). This additional force is not present in the general momentum theory, so we expect a different result between the two models in the case of λ _T  → ∞. Hence the C _P does not approach the Betz limit, as shown in Fig. 38. This wake overpressure increases the power extracted by the wind, and together with the fact that the wake cross section increases when λ _T increases, the extracted power tends to diverge. The error due to the presence of this overpressure was underlined by Schouten [15, 16] for the propellers case. However, in this case the wake contracts and so this term does not produce significant errors, while in the case of a wind turbine the equations of Theodorsen tends to be singular.

Fig. 37

Divergence of the total power coefficient when $\bar{w}$ approaches 2∕3

Fig. 38

C _P −λ curves: the Theodorsen theory does not approach the Betz limit, unlike the Okulov and Sørensen does

We can state that the fixed pitch rigidly moving helicoidal model is fairly reasonable for the blade loading calculation, but for the performance calculations it needs some corrections. For example, the Okulov and Sørensen model neglects the wake expansion and thus no additional expression is required to evaluate the power coefficient. In fact they derive the performance of the rotor by applying the blade element momentum formulas directly on the rotor plane, without the integral balances that include the singular term of the overpressure. To relate the rotor induced velocities with the correspondent quantities in the Trefftz plane they simply show how these velocities are the half of the velocities far downstream. The result is that this model is fully consistent with the general momentum theory when the limit case of λ _T  → ∞ is analysed.