^{1}

^{*}

^{2}

^{3}

^{4}

^{5}

^{6}

In our paper we demonstrate that the filtration equation used by Gorban’ et al. for determining the maximum efficiency of plane propellers of about 30 percent for free fluids plays no role in describing the flows in the atmospheric boundary layer (ABL) because the ABL is mainly governed by turbulent motions. We also demonstrate that the stream tube model customarily applied to derive the Rankine-Froude theorem must be corrected in the sense of Glauert to provide an appropriate value for the axial velocity at the rotor area. Including this correction leads to the Betz-Joukowsky limit, the maximum efficiency of 59.3 percent. Thus, Gorban’ <i>et al</i>.’s 30% value may be valid in water, but it has to be discarded for the atmosphere. We also show that Joukowsky’s constant circulation model leads to values of the maximum efficiency which are higher than the Betz-Jow-kowsky limit if the tip speed ratio is very low. Some of these values, however, have to be rejected for physical reasons. Based on Glauert’s optimum actuator disk, and the results of the blade-element analysis by Okulov and Sørensen we also illustrate that the maximum efficiency of propeller-type wind turbines depends on tip-speed ratio and the number of blades.

In 2001, Gorban’ et al. [

Here, P is the extracted (or consumed) power, and

According to Gorban’ et al. [

holds in an open domain

Following these authors,

where the filtration Equation (1.2) has been inserted. In accord with Equation (1.1), they obtained

Gorban’ et al. [

“The efficiency coefficient can be maximized by optimizing the resistance density. The optimal ratio between the streamlining current and the current passing through the turbines can also be obtained from this model. This parameter can be measured experimentally to determine how close a real turbine is to the theoretically optimal one.”

Obviously, the maximum of the efficiency coefficient deduced by Gorban’ et al. [

This Re value is far beyond the critical Reynolds number at which the transition from a laminar to a turbulent flow occurs.

In the following section, we will present the governing equations for macroscopic and turbulent systems relevant for wind power studies: (a) the local balance equations for momentum (also called the Navier-Stokes equation), (b) total mass (also called the equation of continuity), and (c) kinetic energy. It is shown that the Bernoulli equation for an incompressible flow can simply be derived from the local balance equation of kinetic energy. Furthermore, we will derive the simplified integral balance equations recently used by Sørensen [

In order to outline the generation of electricity by extracting kinetic energy from the wind field we consider the local balance equations for momentum (i.e., Newton’s 2^{nd} axiom), Equation (2.1), and total mass, Equation (2.2), for a macroscopic system given by (e.g., [

and

Here,

where ^{st} term of the left-hand side of Equation (2.1) describes the local temporal change of momentum, and the 2^{nd} term represents the exchange of momentum between the system under study and its surroundings, where ^{st} term on the right-hand side of this equation represents the gravity force, and the 2^{nd} one the Coriolis force. Equation (2.2) is the equation of continuity. In addition, local balance equations for various energy forms (i.e., internal energy, kinetic energy, potential energy, and total energy), various water phases (i.e., water vapor, liquid water, and ice), and gaseous and particulate atmospheric trace constituents exist. All these local balance equations can be derived from integral balance equations (e.g., [

where

Equation (2.4) may be written as

The curl of Equation (2.6) leads to the prognostic equation for the vorticity

As the curl of the gradient of a scalar field is equal to zero, Equation (2.7) can be written as

This equation plays an important role in the description of rotational flows as occurred in the wake of the wind turbine. If the friction effect is negligible and the density is considered as spatially constant like in case of an incompressible fluid we will obtain

To deduce the local balance equation for the kinetic energy of the flow, Equation (2.1) has to be scalarly multiplied by the velocity vector

and

leads to

The colon expresses the double-scalar product (also called the double dot product) of the tensor algebra. Furthermore,^{st} term of the left-hand-side of Equation (2.12) describes the local temporal change of kinetic energy, and the 2^{nd} term is the energy exchange of the system with its surroundings which is performed by the surrounding air on the boundary of the system. The 1^{st} term of the right-hand-side represents the conversion of potential energy into kinetic energy and vice versa, the 2^{nd} term describes the reversible work rate of expansion, ^{rd} term represents the irreversible work rate owing to viscous friction. This term represents the dissipation of kinetic energy into the reservoir of heat. The term of our primary interest reads

It describes the transport of kinetic energy by the flow, and it may be called the kinetic energy stream density, but it is also denoted as wind power density. Inserting the definition of the total pressure,

into Equation (2.12) yields

Since the ABL is mainly governed by turbulent motion, the use of the macroscopic balance Equations (2.1), (2.2), and (2.12) is rather impracticable. Therefore, these balance equations are customarily averaged in the sense of Reynolds [

As argued by various authors [

where

obvious that

Note that intensive quantities like the pressure, p, and the density,

In comparison with that of Reynolds, Hesselberg’s averaging calculus leads to several prominent advantages [

keeps its form, and (b) the mean value of kinetic energy can exactly be split into the kinetic energy of the mean motion and mean value of the kinetic energy of the eddying motion, according to

This advantage is especially important in the theoretical description of the extraction of the kinetic energy from the wind field for generating electricity. The use of density-weighted averages is the common way to define averages in studies of highly compressible turbulent flows (see also [

Hesselberg’s average procedure will be applied within the framework of this contribution. It can be related to that of Reynolds by (e.g., [

Here, the prime (

of a nearly incompressible fluid, the distinction between

In the averaged form, the local balance equation for momentum of the turbulent atmosphere reads (e.g., [

where

internal energy, kinetic energy, potential energy, and total energy), various water phases (i.e., water vapor, liquid water, and ice), and gaseous and particulate atmospheric trace constituents [

Averaging Equation (2.12) provides the corresponding local balance equation for the kinetic energy

Obviously, the local derivative with respect to time not only contains the MKE, but also the TKE as outlined by Equation (2.18). Assuming, for instance, steady-state condition leads to

This means that the total kinetic energy is time-invariant, but MKE can be converted into TKE. In the inertial range, for instance, the TKE is transferred from lower to higher wave numbers until the far-dissipation range is reached, where kinetic energy is converted into heat energy by direct dissipation,

tion,

great intensity of rotation and is characteristic for all turbulent flows. Except for the immediate vicinity of rigid walls, turbulent dissipation exceeds direct dissipation by several orders of magnitude depending on the Reynolds number (e.g., [

This equation describes the transfer of MKE and TKE by the mean wind field and the transfer of TKE by the eddying wind field. Ignoring the turbulent effects yields

i.e.,

where

Unfortunately, there is a notable inconsistency regarding the role of the turbulence intensity. According to de

Vries [

speed and

or

i.e. we have still to consider the fluctuations of all components in this coordinate frame. On the other hand, de

Vries [

Since

The term

for estimating the effect owing to turbulence this term is ignored which leads to

Ignoring the similar term in Equation (2.27) yields

where

To obtain the local balance equation of MKE, Equation (2.20) has to be scalarly multiplied by

or

with

The quantity H may be considered as the mean total pressure. Subtracting Equation (2.32) from Equation (2.21) yields

or

where

is a non-dimensional parameter characterizing the thermal stability of a turbulent flow. This stability parameter expresses the relative importance of the two TKE-terms. It may be interpreted as a generalized Richardson number. The difference between the well-known flux-Richardson number and the generalized Richardson number results from the parameterization of ^{nd}-order balance equation (2.35) is the only balance equation that additionally arises from averaging a macroscopic balance equation (e.g., [

The local balance equation for the mean total energy

This equation demonstrates that no production or destruction of mean total energy within any given fixed volume exists (e.g., [

From the perspective of the generation of electricity by extracting kinetic energy from the wind field, Equations (2.17), (2.20), and (2.32) play the dominant role. To obtain a tractable set of equations, effects caused by

molecular and turbulent friction,

approximated equations reads

and

Because of the condition of incompressibility,

Based on this condition, Bernoulli’s equation, which plays an important role in describing the conversion of wind energy, can simply be derived by considering this condition along a streamline. In accord with the natural coordinate frame for streamlines, the Nabla operator reads

Here, we consider a natural coordinate frame with the unit vectors

With respect to Equation (2.43), the condition (2.42) results in

This means that for any value of

is fulfilled along a streamline. Equation (2.45) is Bernoulli’s equation (e.g., [

so that Equation (2.45) results in (e.g., [

This approximation of Bernoulli’s equation customarily serves as the foundation of, and is used to derive the Rankine-Froude theorem.

The integration of Equations (2.39) to (2.41) over a time-independent control volume, encompassing the rotor of the wind turbine, yields [

and

In accord with Gauss’ integral theorem, Equation (2.47) and the left-hand side of Equation (2.48) can be written as

and

where

Here,

unit vector

for any wind speed smaller than the cut-out wind speed because

The second term of the left-hand side of this equation is usually ignored in the blade element momentum (BEM) theory. However, this term is not zero [

The velocity vector may be expressed by

drical polar coordinates, respectively; and

In accord with Gauss’ integral theorem, the left-hand side of Equation (2.49)reads

This term represents the power extracted by the rotor of the wind turbine. In case of a quasi-horizontal flow, the right-hand side of Equation (2.49) can be neglected because

Rearranging the left-hand side of Equation (2.49) yields

Because of

Obviously, the first term on the right-hand side of this equation is missing in that of Gorban’ et al. [

In the following, we assume a pure axial flow (one-dimensional problem), i.e., the undisturbed wind speed far

upstream of the wind turbine,

model sketched in

To derive the Rankine-Froude theorem we consider the variation of wind speed and pressure by approaching and leaving the rotor area as sketched in

Whereas the latter is given by

Here,

Assuming that

The thrust force acting on the rotor is then given by (the subscript x that occurs in Equations (2.52) and (2.53) is ignored in this section because a pure axial flow is presupposed so that

On the other hand, the thrust force experienced by the rotor can also be expressed by

According to

i.e., the mass flow rate through the wind turbine is

Equation (3.6)) may be written as

Thus, combining Equations (3.5) and (3.8) provides

Rearranging yields

or

i.e., the axial velocity at the rotor disk corresponds to the arithmetic mean of the axial velocities far upstream and far downstream of the wind turbine. Equation (3.11) is the Rankine-Froude theorem (e.g., [

According to Equation (2.55), the total wind power of the undisturbed wind field far upstream of the wind turbine is given by

and that of the undisturbed wind field far downstream of the wind turbine is given by

Again, we assume that

Inserting Equation (3.11) into Equation (3.14) yields

or

Defining the power efficiency by

where

second derivative test,

derivative becomes negative, i.e., for

According to Betz [

Sometimes, the axial interference factor, a, defined by (e.g., [

is inserted. Using this factor leads to

and

with

and

The latter may be used to define the thrust coefficient,

Thus, we have

The result of the Betz-Jowkowsky limit is based on simplified description of the flow field. Even though the flow field exhibits a pure axial behavior in front of the rotor, the exertion of a torque on the rotor disk by the air passing through it causes an equal, but opposite torque to be imposed on the air. Because of this reaction torque, the air starts to rotate in a direction opposite to that of the rotor; the air gains angular momentum and so in the wake of the rotor disk the air particles have a velocity component in a direction which is tangential to the rotation as well as having an axial velocity component [

To consider these rotational effects, Glauert [

As outlined in Appendix B, the general equations of the General Momentum Theory lead to (see Equation (B.24))

where

An exact solution of the general equations of the General Momentum Theory can be obtained when the flow in the slipstream is irrotational except along the axis [

Here, r is the radial distance of any annular element of the propeller disk. Equation (3.26) is the basis for Joukowsky’s constant circulation model [

On the basis of Equation (B.19) of Appendix B,

we can deduce that the axial velocity

of Appendix B),

and the conservation of angular momentum (see Equation (B.6) of Appendix B),

the following relationship [

In accord with Equation (3.30), Equation (3.25) becomes

If we assume again that

Using the definitions

and

where

is the tip speed ratio. Thus,

already derived by Wilson and Lissaman [

In accord with Equation (2.54), the torque,

where the area of the stream tube element is

Replacing

Thus, in contrast to the axial momentum theory, the wind power efficiency is given by [

Inserting

0.593 occurs around

we obtain

Glauert [

The condition of constant circulation k along the blade, which has been the basis of the preceding calculations, cannot be fully realized in practice since it implies that near the roots of the blades the angular velocity imparted to the air is greater than the angular velocity of the propeller itself. In any practical application of the analysis it is therefore necessary to assume that the effective part of the propeller blades commences at a radial

distance not less than

It implies that, near the roots of the blades, the angular velocity imparted to the air is greater than the angular velocity of the propeller itself [

From Equations (3.30) and (3.33) we can derive

or

The maximum values of the power efficiency for various tip speed ratios are also illustrated in

be adequate in case of no wind turbine and

radius of the wake,

For wind turbines, Glauert [

simple axial momentum theory, the axial velocity

of thrust becomes

Now, the torque experienced by this annular stream tube element is given by

Inserting

and

Since the related power is given by

Defining

Thus, the power efficiency is given by [

Alternatively, defining

This formula is equivalent to Equation (3.48). Obviously, the power efficiency strongly depends the tip-speed ratio, but weighted by the integral expression. Unfortunately, Equations (3.48) and (3.49) contain the two unknowns a and

The pressure increment at the propeller disk is given by [

From Equations (3.44) and (3.50) we obtain

To obtain the maximum power for a given tip-speed ratio

and

Thus, combining Equations (3.51) to (3.53) provides

and

The quantities

In case of finite-bladed rotor Equations (3.42) and (3.43) are imprecise. Based on the vortex theory, each of the rotor blades has to be replaced by a lifting line on which the radial distribution of bound vorticity is represented

by the circulation

Okulov and Sørensen [

efficiency is obtained when the distribution of circulation along the blades generates a rigidly helicoidal wake that moves in the direction of its axis with a constant velocity. Betz used a vortex model of the rotating blades based on the lifting-line technique of Prandtl in which the vortex strength varies along the wing-span (

Using the Kutta-Joukowsky-theorem

where

and

Here, we only discuss the torque. Since the related power is given by

Using the analytical solution to the induction of helical vortex filaments developed by Okulov [

Here,

and

where

The result of the second derivative test shows that Equation (3.63) characterizes the maximum of

agreement with

In the vortex theory of the Joukowsky rotor [

where

Here,

The result of the second derivative test shows that for this value of a characterizes the maximum of the power efficiency.

Since neither the axial interference factor a nor

and

In case of the Joukowsky rotor the tip-speed ratio can be expressed by [

were numerically determined for each of the seven wind turbines (see

Wind turbine | Hub height (m) | Swept area (m^{2}) | Cut-in wind speed (m∙s^{−1}) | Rated wind speed (m∙s^{−1}) | Cut-out wind speed (m∙s^{−1}) | Rated power (kW) | Wind Class |
---|---|---|---|---|---|---|---|

Enercon E-48 | 76 | 1810 | 2-3 | 13.5 | 25 | 800 | IEC IIa |

Suzlon S64 Mark II-1.25 MW | 74.5 | 3217 | 4 | 12.0 | 25 | 1250 | IIa |

General Electric 1.6 - 82.5 | 80 | 5345 | 3.5 | 11.5 | 25 | 1600 | IEC IIIb |

Senvion MM92 | 78 - 80 | 6720 | 3 | 12.5 | 24 | 2050 | IEC IIa |

Mitsubishi MWT95/2.4 | 80 | 7088 | 3 | 12.5 | 25 | 2400 | IEC IIa |

Enercon E-82 E4 | 78/84 | 5281 | 2-3 | 16 | 25 | 3000 | IEC IIa |

Siemens SWT-3.6 - 107 | 80 | 9000 | 3 | 14.0 | 25 | 3600 | IEC Ia |

Wind turbine | A | K | Q | B | M | u |
---|---|---|---|---|---|---|

Enercon E-48 | −24.9 | 811.2 | 0.54 | 1.0 | 10.9 | 2.3 |

Suzlon S64 Mark II-1.25 MW | −56.5 | 1250.6 | 3.88 | 2.0 | 9.6 | 4.5 |

General Electric 1.6 - 82.5 | −315.7 | 1601.3 | 1.66 | 2.0 | 9.8 | 7.2 |

Senvion MM92 | −267.6 | 2050.4 | 19.5 | 1.9 | 8.5 | 6.2 |

Mitsubishi MWT95/2.4 | −270.4 | 2403.3 | 12.2 | 1.5 | 8.8 | 4.9 |

Enercon E-82 E4 | −113.8 | 3038.8 | 1.49 | 0.6 | 10.6 | 1.7 |

Siemens SWT-3.6 - 107 | −414.3 | 3599.6 | 40.0 | 1.4 | 9.0 | 5.4 |

Joukowsky limit of

We demonstrated that the filtration equation used by Gorban’ et al. [

is relatively small in the undisturbed wind field over water surfaces. This effect may become more influential in case of aerodynamically rougher landscapes covered, for instance, with vegetation canopies. In case of wind farms the effect by turbulence may considerably increase inside the array of wind turbines. Based on Equation (2.57), we showed that the criticism of van Kuik et al. [

We also demonstrate that the stream tube model customarily applied to derive the Rankine-Froude theorem must be corrected in the sense of Glauert to provide an appropriate value for the axial velocity at the rotor area. Including this correction leads to the Betz-Joukowsky limit, namely of a maximum efficiency of 59.3 percent.

We also assessed Joukowsky’s constant circulation model that leads to values of the maximum efficiency exceeding the Betz-Jowkowsky limit for very low tip speed ratios. Some of these values, however, have to be rejected because of physical reasons.

Using Glauert’s [

Finally, we showed that the power efficiencies of seven wind turbines of different rated power are notably higher than 30-percent limit of Gorban’ et al. [

We would like to express much gratitude to the Alaska Department of Labor for funding Dr. Gary Sellhorst’s project work. We would like to extend gratitude to the National Science Foundation for funding the project work of Hannah K. Ross and John Cooney in summer 2012 through the Research Experience for Undergraduates (REU) Program, grant number AGS1005265. We also express our thanks to the Max Planck Institute for Chemistry for the current financial support for Dr. Dr. habil. Ralph Dlugi.

GerhardKramm,GarySellhorst,Hannah K.Ross,JohnCooney,RalphDlugi,NicoleMölders, (2016) On the Maximum of Wind Power Efficiency. Journal of Power and Energy Engineering,04,1-39. doi: 10.4236/jpee.2016.41001

Let us consider a natural coordinate frame with the unit vectors

and

are fulfilled. These unit vectors can be considered as either contravariant or covariant basis vectors. We may write

because the triple scalar product

The unit vector

where

The principal normal

This means that

We may also define a radius of curvature by

and

we may infer that

where the proportionality constant

The relations (A.7), (A.10), and (A.11) can be summarized as follows

These equations are the central equations in the theory of space curves customarily called the Serret-Frenet formulae (e.g., [

Equation (A.11) may also be written as

The vector

Obviously, the Darboux vector determines the new orientation (rotation) of the moving trihedron (see

The magnitude of the Darboux vector is the total curvature, sometimes also called the Lancret curvature (e.g., [

A trajectory is the actual path of an air particle, i.e., it characterizes the direction of the velocity that such an air particle is taking successively during a certain time interval (e.g., [

As sketched in

If we express the substantial derivative of the unit tangent in the Eulerian form, we will obtain

Here, the partial derivative of the unit tangent with respect to the arc length was replaced by

other hand, we have

Combining Equations (A.18) and (A.19) yields

In the two-dimensional case, we have

With respect to the unit vectors

where

Combining Equations (A.21) and (A.23) yields

The scalar form of the relation (A.24) is Blaton’s equation. In the case of steady-state conditions, the left-hand side terms of Equations (A.21) and (A.24) vanish and the curvatures of the trajectories and the streamlines are identical, i.e., the trajectories and the streamlines coincide.

If no tangential acceleration exists, i.e.,

nitude of Coriolis acceleration is small in comparison with those of the centripetal acceleration and the acceleration due to the pressure gradient, we will lead to the following conditions for a frictionless flow:

and

The solution of this equation set is then given by

The equation describes the cyclostrophic flow. Obviously,

is fulfilled. Both cyclonic and anti-cyclonic flows are possible. Assuming, for instance,

fulfills the condition

In this appendix, we only consider average values. Thus, all symbols that are characterizing average values are ignored here. Let be r the radial distance of any annular element of the propeller disk,

Generally, the torque is

If the force

i.e., the angular momentum is invariant with time (

where

Thus, the conservation of angular momentum provides

In accord with Equation (2.54), the torque is given by

The Bernoulli equation in its approximated form (2.46) yields

and

Thus, the difference of the total pressure heads,

In addition, the pressure difference

where the conservation of the angular momentum (see Equation (B.6)) has been used. Rearranging

leads to

Here,

In applying Bernoulli's equation to the flow relative to the propeller blades, we have to consider the relative angular velocity of the air that increases from

Thus, we obtain

The pressure gradient in the wake balances the centrifugal force on the fluid and is governed by

A balance between the pressure gradient force and the centrifugal force in the horizontal direction leads to the cyclostrophic flow well known in meteorology (e.g., [

or

where Equation (B.15) was used. Since

Equation (B.17) becomes [

Furthermore, the thrust is given by

The pressure increment at the propeller disk is given by [

Combining these two equations leads to

The equation of continuity provides

Finally, using

This equation already derived by Wilson and Lissaman [

Since Sharpe [

Using

This is identical with Sharpe’s Equation (4). Considering

that is identical with Sharpe’s Equation (7). Thus, our Equation (B.14) results in

This equation completely agrees with Sharpe’s Equation (8). Finally, by rearranging our Equation (B.19)we obtain

that is identical with Sharpe’s Equation (11). Thus, Equation (B.24) already derived by Wilson and Lissaman [

An exact solution of the general equations of the General Momentum Theory described before can be obtained when the flow in the slipstream is irrotational except along the axis [

Here, r the radial distance of any annular element of the propeller disk.

On the basis of the equation (see Equation (B.19) of Appendix B)

we can deduce that the axial velocity

and the conservation of angular momentum (see Equation (B.6) of Appendix B),

the following relationship [

or with respect to Sharpe’s [

where

is the azimuthal interference factor, and by analogy with that quantity,

considered for the fully developed wake. Thus, Equation (B.24) of Appendix B becomes

or

In accord with Glauert [

and

Inserting these definitions into Equation (C.9) yields

If we assume again, that

or

Inserting Equation (C.11) into this equation provides

Rearranging this equation yields

Since

This formula was already derived by Wilson and Lissaman [