Problem Description

This article describes the performance of a novel vertical axis wind turbine (VAWT). The Zephyr VAWT has a patented stator cage that can increase the turbine’s performance through the creation of low-pressure vortices. The patented features of the turbine allow it to perform in both low-wind and high-turbulence conditions; however, a relatively low maximum efficiency is exhibited by the current prototype design. A problem arises when incoming air is diverted away from the convex sides of the rotors, reducing opposing forces to the direction of rotation; however, the solidity (blockage) is increased, reducing the overall flow through the turbine.

Figure 1: Geometrical variables of the Zephyr vertical axis wind turbine

Manufacturers of fluid machinery such as small wind turbines are interested in this because efficiency improvements often are needed to obtain more feasible designs for commercialization. This article uses dimensional analysis and numerical predictions to obtain a turbine-specific correlation for the ZVAWT. The correlation will predict the turbine’s power coefficient after changes to critical features of the VAWT and operating conditions, such as the rotor velocity and wind speed (see Figure 1). The new dimensionless correlation will provide a valuable design tool that can be extended to other types of turbines for the purpose of predicting performance over a range of operating conditions.

Solution Procedure by Buckingham Pi Dimensional Analysis

In regard to the Zephyr VAWT design, nine key variables are identified to represent the turbine power output (Ω ) for different stator geometries.  The variables are the rotor velocity (Φ), rotor radius (R), freestream velocity (V), air density (ρ), turbine height (H), turbine width (W), stator spacing (σ), and stator angle (θ). The power output depends on these variables as follows: F = f(r, V, R, H, W, s, q, W). The dimensions of these variables can be represented by the base dimensions of mass (M), length (L), and time (T), as follows: F ≡ M¹ ∙ L² ∙ T^-3, r ≡ M¹ ∙ L^-3 ∙ T⁰, V ≡ M⁰ ∙ L¹ ∙ T^-1, R ≡ M⁰ ∙ L¹ ∙ T⁰, H ≡ M⁰ ∙ L¹ ∙ T⁰, W ≡ M⁰ ∙ L¹ ∙ T⁰, s ≡ M⁰ ∙ L¹ ∙ T⁰, q ≡ M⁰ ∙ L⁰ ∙ T⁰, and W ≡ M⁰ ∙ L⁰ ∙ T^-1. From these nine variables and three dimensional units, the Buckingham pi theorem states that six dimensionless pi terms will be sufficient to represent and predict the performance of the turbine. The reference variables, ρ, , and R lead to the following three pi variables, as determined by the Buckingham pi theorem:

Combining these three terms yields the following power coefficient (Cp):

The fourth and fifth pi terms are additional terms for representing changes in the stator geometry:

The last pi term is represented in terms of R, Ω, and V. It is equivalent to the variable tip speed ratio (λ), a common dimensionless parameter for wind power analysis:

These terms are combined so that:

Which can be reduced to

where ø will be determined through numerical predictions with CFD (computational fluid dynamics). This expression will lead to correlations that provide useful insight into the turbine’s geometric characteristics. It also will provide key information for design and power output improvements.

Results and Discussion

The power correlation is developed from 16 numerically simulated geometrical configurations. Predicted values of C_p at varying magnitudes of П₄ are used for the dimensional analysis. Four power curves are generated to represent discrete values of θ, specifically 0.698, 1.57, 1.92, and 2.27 radians.

To collapse the four curves associated with each of the four cases into a single correlation, they are normalized in following  dimensionless plane:According to the function:where:

The normalization coefficient f(θ) can be represented as:where minimal variability is exhibited, compared with the numerical prediction (R² = 0.91).  The normalization coefficient g(θ) can be represented by:which shows minimal variability with the numerical prediction (R² = 0.98). After all power curves are collapsed onto a single normalized power curve, the general correlation becomes:where f(θ) and g(θ) are described by Equations  (9) and (10), respectively.

Additional numerical predictions have been obtained and compared with the correlation described by Equation (11). Figure 2 illustrates the ability of the model to predict results that lie outside the CFD simulated conditions. For example, three additional ZVAWT configurations are represented at θ = 2.09 radians and θ = 1.15 radians, with associated errors of 4.4%, 5.8%, and 2.9%, respectively. The correlation generated from the combination of a dimensional analysis and numerical simulations provides an excellent prediction of the turbine’s performance:

Figure 2: Illustration of the dimensionless power correlation

Conclusions

As a result of the development of a general correlation that predicts the ZVAWT’s performance, further design improvements can be undertaken without the need for time-consuming CFD predictions in each case. The correlation provides a useful design tool for adapting the turbine conditions and operating requirements specific to this drag-type VAWT. It becomes easier to predict quickly how changes to the VAWT’s design features will affect its performance with a reasonable degree of accuracy. The correlation also can be useful for developing an optimum turbine design while limiting the need for further CFD simulations and time-intensive mesh generation for each different turbine configuration.