The Betz limit and the corresponding thermodynamic limit

Abstract

The Betz’s limit for the maximum efficiency of an ideal wind turbine imposes a maximum value of about 60% on the conversion of the kinetic energy of an airflow into work. In this paper, we analyze the reason for this value because, from a thermodynamic point of view, it can be 100%. The present work explains the reason for this difference, since it appears to be relevant from a didactic point of view. However, from a practical point of view, the Betz’s limit does not affect in any way the more useful and widespread expression for calculating the ideal maximum power of a wind turbine, which is at the origin of the referred limit. Complementarily, two approaches for the calculation of the theoretical maximum efficiency, in line with thermodynamics, are also presented in this work.

Keywords

Betz efficiency limit wind turbine thermodynamic efficiency streamtube kinetic energy

Introduction

The second law of thermodynamics imposes a limit on the conversion of heat, disorganized energy, into work, organized energy, given by the well-known Carnot efficiency. However, for the conversion of a form of organized energy, macroscopic kinetic energy, for example, as an airflow has, into another form of organized energy, such as work, for example, there is no limit, except that imposed by the first law of thermodynamics and the inherent principle of conservation of energy, that is, 100% (Çengel and Boles, 2015).

Therefore, the aforementioned reality contrasts with the well-known Betz limit for the maximum efficiency of an ideal wind turbine (Betz, 2013), from which it follows that the conversion into work of the kinetic energy of an incompressible fluid flow is limited to 16/27 (59.3%). It is important to note that two other authors who lived at the same time as Albert Betz did the same research on their own (Van Kuik, 2007).

Other authors, for example, Greet (1980) and Allahverdyan and Khalafyan (2021), point out errors in the calculation of the Betz’s maximum theoretical efficiency, calling the method’s utility into question. However, the most useful formula, which is the one that allows us to figure out a wind turbine’s maximum ideal power, cannot be argued with because it does not depend on the efficiency mentioned above. As a result, it is still in use today (Villanueva and Feijóo, 2010).

In the work of Allahverdyan (2020), a new model is proposed for the efficiency of converting flow energy into work for compressible fluids. This model takes into account both isothermal and non-isothermal flows, that is, work extraction from flow kinetic energy and from flow enthalpy, respectively. For the isothermal case, which is also looked at in this work but for incompressible fluids, an implicit efficiency expression was found that, when simplified to the ideal case, that is, one-dimensional and isentropic flow, gives the same expression as the one found in this work.

The apparent discrepancy between the maximum efficiency allowed by thermodynamics and that proposed by Betz is investigated in this paper, which begins by reproducing the steps that led to the calculation of the referred Betz limit. As a result, a simple explanation for this discrepancy is provided, which appears to be pedagogically relevant. In addition, two methodologies for calculating the theoretical maximum efficiency in accordance with thermodynamics are described in this work.

In the present analysis, it is assumed that there are no heat exchanges with the fluid, that the flow is isothermal and isobaric (dissipative effects are absent), which entails that the fluid is incompressible and that its entropy remains constant, as the thermodynamic state of the fluid remains unchanged under these conditions. In this work, the idea that the flow has organized macroscopic kinetic energy means that the velocity at the entrance and exit of the streamtube is one-dimensional.

Deduction of Betz’s theoretical maximum efficiency

Figure 1 schematically shows a streamtube commonly used in the study of wind turbines. The pressure and temperature of the air remain constant, so the fluid is considered incompressible. By the conservation of mass, we have that

m = ρ A_{1} v_{1} = ρ A_{2} v_{2} = ρ A_{0} v_{0}

(1)

where $m$ is the mass flow rate that crosses the turbine as it flows through the streamtube, v_i is the average flow velocity in a generic cross section i of the streamtube, and A_i is the area of that cross section. Points i = 1 and i = 2 are upstream and downstream of the turbine, respectively. At the point i = 0, is the wind turbine, where A₀ is the circular area swept by the turbine and v₀ is the average velocity of the flow in this rotor disk.

Figure 1.

Schematic of a streamtube around a wind turbine.

Betz (2013) calculated the ideal power produced by a wind turbine, P_turb, of area A₀ subject to a free stream velocity, v₁. Ideally, this power is equal to that taken from the airflow. Two different approaches were used, first through the product between the flow velocity, v₀, and the axial force, F, exerted by the flow on the rotor disk of area A₀. Then, in a second way, through the product between the mass flow rate and the flow kinetic energy variation from points 1 to 2.

In the first approach, the force, F, is calculated through a momentum balance between points 1 and 2, equation (2).

F = m (v_{1} - v_{2})

(2)

Subsequently, the power is given by equation (3).

P_{turb} = v_{0} \cdot F = v_{0} \cdot m (v_{1} - v_{2})

(3)

In the second approach, the power is calculated through the following equation.

P_{turb} = m (v_{1}^{2} - v_{2}^{2}) / 2

(4)

By making equations (3) and (4) equal to each other, a value for v₀ was found, as shown in equation (5).

\begin{matrix} v_{0} \cdot m (v_{1} - v_{2}) = m (v_{1}^{2} - v_{2}^{2}) / 2 \Leftrightarrow v_{0} (v_{1} - v_{2}) = (v_{1} - v_{2}) (v_{1} + v_{2}) / 2 \\ \Leftrightarrow v_{0} = (v_{1} + v_{2}) / 2 \end{matrix}

(5)

Through an algebraic manipulation of the equations (3) or (4), and by replacing $m$ in them by, $ρ v_{1} A_{0} [1 + (v_{2} / v_{1})] / 2$ , see the following equation,

m = ρ v_{0} A_{0} = ρ (v_{1} + v_{2}) A_{0} / 2 = ρ v_{1} A_{0} [1 + (v_{2} / v_{1})] / 2

(6)

a third and more informative expression for the ideal power that a turbine can produce, P_turb, was obtained, equation (7). Equations (3), (4), or (7) all represent the same variable, P_turb.

P_{turb} = \frac{1}{4} ρ A_{0} v_{1}^{3} [1 + \frac{v_{2}}{v_{1}} - {(\frac{v_{2}}{v_{1}})}^{2} - {(\frac{v_{2}}{v_{1}})}^{3}]

(7)

Differentiating the previous expression with respect to $(v_{2} / v_{1})$ , keeping $A_{0}$ and $v_{1}$ constant, Betz (2013) found the maximum value for this ideal power, $P_{max}$ , which occurs for $v_{2} / v_{1} = 1 / 3$ , that is,

P_{max} = \frac{16}{27} \cdot \frac{1}{2} ρ A_{0} v_{1}^{3}

(8)

This maximum arises since, keeping v₁ and A₀ constant, when varying v₂, the mass flow rate will also vary, equation (1), due to the inherent variation of the areas since,

A_{1} = A_{0} [1 + (v_{2} / v_{1})] / 2

(9)

and

A_{2} = A_{0} [1 + 1 / (v_{2} / v_{1})] / 2

(10)

Since $m \propto [1 + (v_{2} / v_{1})]$ , see equation (6), this means that the mass flow rate increases linearly with the ratio $v_{2} / v_{1}$ . On the other hand, the variation of the kinetic energy of the flow decreases with the square of the ratio $v_{2} / v_{1}$ , because $(v_{1}^{2} - v_{2}^{2}) \propto [1 - {(v_{2} / v_{1})}^{2}]$ . In this way, the product of these two functions, equation (4), gives rise to this maximum ideal value, equation (8). As it turns out, P_max is independent of the turbine efficiency definition.

Subsequently, having defined the power available in the air stream upstream of the turbine, P_upst, based on the independent variables ρ, v₁, and A₀, as being

P_{upst} = \frac{1}{2} ρ A_{0} v_{1}^{3} = \frac{1}{2} (ρ A_{0} v_{1}) v_{1}^{2}

(11)

Betz established the ideal efficiency of this energy conversion, η, as the ratio between the ideal power transmitted to the turbine, equations (3), (4), or (7), and P_upst, that is, $η = P_{turb} / P_{upst}$ . Combining equation (7) with equation (11) gives rise to the following equation.

η = \frac{P_{turb}}{ρ A_{0} v_{1}^{3} / 2} = \frac{1}{2} [1 + \frac{v_{2}}{v_{1}} - {(\frac{v_{2}}{v_{1}})}^{2} - {(\frac{v_{2}}{v_{1}})}^{3}]

(12)

From equation (12), we can verify that η reaches the maximum value of 16/27 (59.3%) for $v_{2} / v_{1} = 1 / 3$ , also known as the Betz limit of the theoretical maximum efficiency.

Figure 2 shows how η changes with v₂/v₁, equation (12), which proves that there is a maximum value for this definition of efficiency when v₂/v₁ = 1/3.

Figure 2.

Turbine efficiency variation with v₂/v₁.

The work developed by Betz makes it possible to quickly find the maximum power that a given wind turbine can extract from an airflow by knowing the values of the independent variables v₁, A₀, and ρ, equation (8). It also gives a way to find the maximum efficiency of a wind turbine, equation (12), if equation (11) is used to represent the power available in the airflow, that is, the required input for the efficiency.

Approaches to efficiency that are consistent with thermodynamics

As mentioned earlier, thermodynamics allows us to state that the theoretical maximum limit for the conversion of macroscopic kinetic energy into work is 100%, so why is the Betz limit for the maximum theoretical efficiency significantly lower? The reason for this difference has to do with the fact that to calculate the maximum power available in the airflow upstream of the turbine, the expression given by equation (11) was chosen, which represents a value greater than the actual, which is equal to ${mv}_{1}^{2} / 2$ , since $m = ρ A_{1} v_{1} < ρ A_{0} v_{1}$ , because A₀ > A₁, see Figure 1, so $(ρ A_{0} v_{1}) v_{1}^{2} / 2 > m v_{1}^{2} / 2$ .

Equation (13), or equation (14), which is the same as the previous one but changed to best demonstrate the difference between this method and the one used by Betz (2013), can be used to find out how much power from the airflow is actually available to the turbine, $P'_{upst}$ .

P'_{upst} = \frac{1}{2} {mv}_{1}^{2} = \frac{1}{2} ρ A_{1} v_{1} v_{1}^{2} = \frac{1}{2} ρ A_{1} v_{1}^{3}

(13)

P'_{upst} = \frac{1}{2} ρ A_{1} v_{1} v_{1}^{2} = \frac{1}{2} ρ A_{0} v_{0} v_{1}^{2} = \frac{1}{2} ρ A_{0} \frac{(v_{1} + v_{2})}{2} v_{1}^{2} = \frac{1}{2} (ρ A_{0} v_{1}) v_{1}^{2} \frac{(1 + v_{2} / v_{1})}{2}

(14)

Comparing equations (11) and (14), it can easily be seen that the ratio between the available wind power used in the Betz approach, $P_{upst}$ , and the maximum available power that can effectively be used in the turbine, $P'_{upst}$ , $P_{upst} / {P'}_{upst}$ , is equal to $2 / (1 + v_{2} / v_{1})$ , varying this power ratio between the values of 2, for v₂ = 0, and 1, for v₂ = v₁. Since equation (11), which is the numerator of equation (12), represents a lot more energy than is actually available in the wind turbine, Betz’s efficiency cannot be higher than 16/27, which is less than the theoretical maximum of 100% that thermodynamics allows.

Defining an efficiency, $η'$ , based on $P'_{upst}$ , equations (13) or (14), instead of P_upst, equation (11), that is, $η' = P_{turb} / {P'}_{upst}$ , results in the following efficiency expression,

η' = \frac{\frac{1}{4} ρ A_{0} v_{1}^{3}}{\frac{1}{2} ρ A_{0} v_{1}^{3} \frac{(1 + v_{2} / v_{1})}{2}} [1 + \frac{v_{2}}{v_{1}} - {(\frac{v_{2}}{v_{1}})}^{2} - {(\frac{v_{2}}{v_{1}})}^{3}] = [1 - {(\frac{v_{2}}{v_{1}})}^{2}]

(15)

Figure 2 also shows the variation of $η'$ with v₂/v₁, and it can be seen that in this approach, for v₂ = 0, a situation in which all the available kinetic energy of the flow is converted into work, a new theoretical maximum limit value is reached, that is, $η' = 1$ , which is in line with the first and second laws of thermodynamics.

It should be noted that v₂ = 0 does not mean that there is no flow. By equation (6), we have in this case that $m = ρ v_{1} A_{0} / 2$ and by equation (10), we see that A₂ is inversely proportional to v₂ and vice versa, so $v_{2} = m / (ρ A_{2})$ tends towards zero as the area A₂ approaches infinity.

For a given v₁, when v₂ increases, the same happens to the kinetic energy loss downstream of the turbine, ${mv}_{2}^{2} / 2$ , and thus the efficiency $η^{'}$ drops continuously, see RHS of equation (15). When v₁ = v₂, there is no energy taken from the fluid and so P_turb = 0 and $η' = 0$ .

It is worth noting that the expression of Allahverdyan (2020), his equation (19), for the upper bound value of kinetic energy conversion efficiency, is obtained by replacing the velocities v_i in the RHS of equation (15) by the ratio m/(ρ·A_i).

The relationship between the two efficiencies, $η'$ and η, is given by the next equation.

η' = \frac{2}{(1 + v_{2} / v_{1})} η

(16)

For the maximum turbine power, that is, for v₂/v₁ = 1/3, the efficiency $η'$ has a value of 48/54.

In order to have an ideal efficiency, $η ″$ , that always has a top limit of 100% regardless of the ratio v₂/v₁, the power available in the airflow effectively transmitted to the wind turbine, equation (4) for example, must be the required input for this efficiency, as shown in the following equation,

η ″ = W_{turb} / [m (v_{1}^{2} - v_{2}^{2}) / 2] = W_{turb} / P_{turb}

(17)

where W_turb is the useful power generated by the turbine.

The efficiency given by equation (17) evaluates the performance of a wind turbine in converting the flow’s kinetic energy variation per unit of time, that is, $m (v_{1}^{2} - v_{2}^{2}) / 2$ , into useful power, W_turb, a conversion that ideally could be equal to 100% without contradicting the first or second laws of thermodynamics. Even though this ideal efficiency is always 100%, the useful power, W_turb, still depends on the ratio v₂/v₁, since the power transferred by the flow to the turbine, $m (v_{1}^{2} - v_{2}^{2}) / 2$ , is also a function of v₂/v₁, as shown in equation (7).

Conclusion

The fact that the Betz limit for the maximum theoretical efficiency is less than 100% is directly related to the reference value chosen for the power available in the airflow upstream of the wind turbine. This choice gives rise to a power value that is up to twice as high as that physically available in the wind turbine, which is why the limit is less than 100%.

The existence of the inherent maximum limit of 16/27 can easily convey the undidactic idea that the conversion of macroscopic kinetic energy into work cannot theoretically reach 100%, a possibility that thermodynamics does not exclude at all. This work also refers to two alternative approaches for calculating wind turbine efficiency that are both in accordance with the first and second laws of thermodynamics.

It is worth mentioning that the expression deduced by Betz for calculating the maximum power produced by a wind turbine, equation (8), is not affected by any efficiency definition, and is also quite practical, since, in addition to provide realistic values, it only depends on the independent variables v₁, A₀, and ρ, and therefore continues to be used nowadays.

Footnotes

Declaration of conflicting interests

The author declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Paulo Coelho

References

Allahverdyan

(2020) Work extraction from fluid flow: The analog of Carnot’s efficiency. Physical Review Letters 125(6): 5+10 pages.

Allahverdyan

Khalafyan

(2021) Reexamination of Betz’s limit for wind engines. Journal of Contemporary Physics (Armenian Academy of Sciences) 56(1): 38–46.

Betz

(2013) The maximum of the theoretically possible exploitation of wind by means of a wind motor (Translation of: Das Maximum der theoretisch möglichen Ausnützung des Windes durch Windmotoren, Zeitschrift für das gesamte Turbinenwesen, Heft 26, Seiten 307 bis 309, 1920, by Hamann H, Thayer J and Schaffarczyk AP, University of Applied Sciences Kiel, Germany). Wind Engineering 37(4): 441–446.

Çengel

Boles

(2015) Thermodynamics: An Engineering Approach, 8th edn. New York, NY: McGraw-Hill, pp. 58.

Greet

(1980) Maximum windmill efficiency. Journal of Applied Physics 51(9): 4680–4681.

Van Kuik

(2007) The Lanchester–Betz–Joukowsky limit. Wind Energy 10(3): 289–291.

Villanueva

Feijóo

(2010) Wind power distributions: A review of their applications. Renewable and Sustainable Energy 14(5): 1490–1495.