Proceedings of the 7th World Congress on Mechanical, Chemical, and Material Engineering (MCM'21)
Prague, Czech Republic Virtual Conference – August, 2021
Paper No. HTFF 136
DOI: 10.11159/htff21.136

A Re-examination of Power Coefficient as a Measure of Performancefor Horizontal Axis Wind Turbines
Thomas M. Adams, Benjamin E. Mertz

Rose-Hulman Institute of Technology
5500 Wabash Ave., Terre Haute, IN, USAadams1@rose-hulman.edu; mertzb@rose-hulman.edu
Abstract - Linear momentum theory as applied to horizontal axis wind turbines (HAWTs) provides perhaps the most useful basis forunderstanding their operation. In particular, the theoretically derived expression for power coefficient represents a convenient measureof performance, as well as provides insight into optimal operating conditions. The typical interpretation of power factor as an energy
conversion efficiency, however, especially in the context of converting the “power in the wind” to a power output, has several conceptual
difficulties. In this paper it is argued that the energy efficiency interpretation of power coefficient can be misleading, potentially leading
to misinterpretation of performance of different wind turbine designs. Instead, an interpretation of power coefficient as the “relative
capture area” of a wind turbine is suggested, analogous to the relative capture width parameter for ocean wave energy conversion devices.
Such an interpretation gives a more physically coherent picture of wind turbine performance and provides a more pragmatic measure of
performance, one that can also be applied to other wind machine designs.
Keywords: Power coefficient, Betz limit, capture width

1. Introduction: Linear Momentum Theory, Power Coefficient, and the Betz Limit

Linear momentum theory applied to the operation of horizontal axis wind turbines (HAWTs) dates back at least to Betz
[1] and provides one of the most accessible bases for conceptualizing the function of such devices. Given that many thoroughtreatments of the topic exist in the literature [2]-[4] we give only the most salient features here.
Figure 1 gives a side view of the analysed system, which consists of a diverging stream tube extending from thedownstream side of the wind turbine to the upstream side. The variables V and A refer to wind speed and cross-sectionalarea, respectively, whereas FT is the thrust force exerted on the hub by the wind and Ẇout is the power extracted by the turbine.
The subscripts indicate the planes in which the windspeed corresponds to its undisturbed, freestream value (i), the value atthe hub of the wind turbine (t), and the minimum downstream value before the wind reforms (e). Local static pressure varies
in the flow direction but takes on atmospheric values at locations (i) and (e). The hub is considered an “actuator disc” wherepressure changes discontinuously and across which all energy extraction occurs. The flow is modelled as incompressible
with density ρ, one-dimensional, isothermal, and frictionless.

Fig. 1: Diagram of stream tube used in the application of linear momentum theory.

HTFF 136-1

Macroscopic mass, momentum, and energy balances applied to the system in Fig. 1 result in a predicted value fordeveloped power given by𝑊𝑊̇𝑜𝑜𝑜𝑜𝑜𝑜 = 4𝑎𝑎(1 − 𝑎𝑎)2

1 3
𝜌𝜌𝑉𝑉 𝐴𝐴 ,2 𝑖𝑖 𝑡𝑡(1)

where a is the axial induction factor, a parameter representing the fractional decrease in windspeed as the freestream windapproaches the actuator disc [2],This is commonly rearranged as𝑎𝑎 =𝑉𝑉𝑖𝑖 − 𝑉𝑉𝑡𝑡.
𝑉𝑉𝑖𝑖

𝑊𝑊̇𝑜𝑜𝑜𝑜𝑜𝑜 = 4𝑎𝑎(1 − 𝑎𝑎)2

𝜌𝜌𝑉𝑉𝑖𝑖3 𝐴𝐴𝑡𝑡
= 𝐶𝐶𝑃𝑃 𝑃𝑃𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤 ,
2

(2)

(3)

where CP=4a(1-a)2 is the power coefficient and Pwind= ½ρVi3At is considered “the power in the wind.”
Differentiating the power coefficient with respect to axial induction factor and setting it equal to zero gives its maximum
value as 16/27, occurring at a value of a = 1/3:
𝐶𝐶𝑃𝑃,𝑚𝑚𝑚𝑚𝑚𝑚 =16, when 𝑎𝑎 = 1/3.
27

(4)Equation (4) represents the famous Betz limit for wind turbine performance, stating that a HAWT can deliver a maximumof just under 60% (59.3%) of the “power in the wind.” Figure 2 shows the relationship of CP to a.

Fig. 2: Relationship of power coefficient to axial induction factor according to linear momentum theory. The maximum value of CP is16/27 and occurs at a = 1/3.

Similarly, linear momentum theory gives an expression for the thrust the wind exerts on the actuator disc as

HTFF 136-2

𝐹𝐹𝑇𝑇 = 𝐶𝐶𝑇𝑇

1 2
𝜌𝜌𝑉𝑉 𝐴𝐴 ,2 𝑖𝑖 𝑡𝑡(5)

where CT=4a(1-a) is the thrust coefficient. The maximum value of thrust coefficient is CT,max = 1, occurring at a = 1/2. Of
note is that a = 1/2 corresponds to a wind speed of zero at (e) and that maximum power and maximum thrust are not realised
at the same the value of axial induction coefficient.

2. The Problem of Power Coefficient as Conversion Efficiency

Though the power coefficient likely serves as the most useful measure of performance for HAWTs, its interpretation as
an energy conversion efficiency comes with several inconsistencies. Principal among these is what best represents the various
inputs and outputs of energy. We suggest that an interpretation of power coefficient as a relative capture area bypasses such
ambiguities and gives a clearer physical picture of HAWT performance. To be clear, we are not arguing that power coefficientis a poor measure of performance, but that it is not a properly defined energy conversion efficiency.
2.1. Two Definitions of Efficiency
In defining a conversion efficiency, we typically track the inputs and outputs of energy of a device, the efficiency being
the ratio of the desired output to some input that represents the depletion of a resource. If the energy related quantities are
power, then the efficiency becomes𝜂𝜂 =𝑃𝑃𝑜𝑜𝑜𝑜𝑜𝑜.
𝑃𝑃𝑖𝑖𝑖𝑖

𝐶𝐶𝑃𝑃 =

𝑊𝑊̇𝑜𝑜𝑜𝑜𝑜𝑜.
𝑃𝑃𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤(6)

Implicit in employing such an efficiency is that part of the input power is not converted to the useful output, but rather tosome other unusable form and/or that it is rejected elsewhere. If we consider power coefficient to be such an efficiency, thenẆout corresponds to Pout and Pwind to Pin so that
(7)

Figure 2 above would suggest, then, that the maximum attainable value of efficiency for a HAWT is 16/27 and occurs at a
= 1/3.
In calculating power output, linear momentum theory makes use of conservation of energy applied to the stream tube ofFig. 1, the power output being the difference between the inlet and exit flows of kinetic energy:
𝑉𝑉𝑖𝑖2 𝑉𝑉𝑒𝑒2
𝑉𝑉𝑖𝑖2 𝑉𝑉𝑒𝑒2
̇
𝑊𝑊𝑜𝑜𝑜𝑜𝑜𝑜 = 𝑚𝑚̇ � − � = 𝜌𝜌𝐴𝐴𝑖𝑖 𝑉𝑉𝑖𝑖 � − �,
2
2
2
2

(8)

where ṁ=ρAiVi is the mass flowrate of wind through the turbine. Rearranging and making use of the axial induction factor,𝑊𝑊̇𝑜𝑜𝑜𝑜𝑜𝑜 =𝜌𝜌𝐴𝐴𝑖𝑖 𝑉𝑉𝑖𝑖3
𝑉𝑉𝑒𝑒2
𝜌𝜌𝐴𝐴𝑖𝑖 𝑉𝑉𝑖𝑖3
4𝑎𝑎(1 − 𝑎𝑎).
�1 − 2 � =
2
2
𝑉𝑉𝑖𝑖

(9)

If we consider the input flow of kinetic energy to the stream tube to be the input power, Eq. (9) cast in the form of a conversion
efficiency yields

HTFF 136-3𝜂𝜂 =𝑊𝑊̇𝑜𝑜𝑜𝑜𝑜𝑜𝑊𝑊̇𝑜𝑜𝑜𝑜𝑜𝑜== 4𝑎𝑎(1 − 𝑎𝑎).
𝑃𝑃𝑖𝑖𝑖𝑖
𝜌𝜌𝐴𝐴𝑖𝑖 𝑉𝑉𝑖𝑖3
2(10)

Figure 3 gives a visualisation the conversion efficiency of Eq. (10).

Fig. 3: Variation of energy conversion efficiency as given by Eq. (10).

Clearly Fig. 3 gives a much different picture of efficiency than Fig. 2. Rather than a maximum value of 16/27, Fig. 3
suggests that a HAWT can achieve an efficiency of one, and that that efficiency occurs at a=1/2, not at a=1/3. The reader
may also recognize that the expressions for η and the thrust coefficient CT are identical, implying that maximum efficiency
and maximum thrust force are indeed achieved simultaneously.
2.2. Ambiguity of Inputs
The discrepancy between the two definitions for efficiency arises from the different assumed inputs for each. In Eq.
(10), Pin corresponds to the power of the upstream wind for the actual mass flowrate passing through the actuator disc, ṁ =ρAiVi. On the other hand, Pwind is the power of a hypothetical mass flow of wind at speed Vi through the actuator area, At; thatis, ṁwind = ρAtVi. In other words, Pwind is the power that would be contained in a cross-sectional area of At if the turbine were
not present.
Figure 4 shows the air flow associated with the wind turbine and allows us to contrast the different flowrates. Since thespeed of the flow decreases as it approaches the actuator disc, the incompressibility of the fluid requires that the upstream
area Ai be less than that of the actuator disc, At. Consequently, the flow rate through the wind turbine ṁ must be less thanṁwind, becoming smaller still with increasing values of the axial induction factor. Tracking a quantity of ṁwind starting
upstream of the actuator disc, we find that (1-a)ṁwind makes its way through the actuator disc whereas aṁwind bypasses theactuator disc completely. Only in the case of zero power output, when a=0, does ṁwind = ṁ. We may therefore question
whether Pwind truly represents the power input to the turbine, since it is the power associated with ṁwind and not with ṁ.

HTFF 136-4

(a)

(b)

Fig. 4: (a) For any a > 0, a fraction of ṁwind is deflected around the actuator disc. (b) ṁ = ṁwind only when a = 0, which corresponds tozero power output.

Of the flow that does go through the actuator, a fraction η = 4a(1-a) of that is converted to mechanical power, theremainder exiting with the flow downstream through Ae. Hence, the power coefficient accounts for two different effects, thediversion of part of ṁwind around the actuator disc and the partial conversion of kinetic energy into power for the flow throughthe disc itself:
𝐶𝐶𝑃𝑃 = [fraction of 𝑚𝑚̇𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤 passing through 𝐴𝐴𝑡𝑡 ] ∙ �fraction of KE converted to 𝑊𝑊̇𝑜𝑜𝑜𝑜𝑜𝑜 �= [1 − 𝑎𝑎] ∙ [𝜂𝜂].

(11)Figure 5 shows the relative proportions of Pwind as a function of axial induction factor. The figure gives additional insight
into the power coefficient as shown in Fig. 2 in that the two effects of Eq. (11) are distinguishable. The maximum power
output is realized at a =1/3 even though the conversion efficiency of Eq. (10) gives a value of η = 8/9 at that point. As aincreases beyond 1/3, the power output decreases despite an increasing η due to smaller inputs of Pin delivered to the actuator
disc. The trend continues until the total output of the turbine drops to 50% of Pwind at a=1/2 although η = 1.

Deflected around At
Unconverted to ẆoutConverted to ẆoutFig. 5: Relative proportions of Pwind. The blue area is the fraction of Pwind that is deflected around the actuator. Of the power delivered
to the actuator, the yellow area is the fraction converted to power output whereas the grey area remains unconverted, ultimately exiting
through Ae.

HTFF 136-5

3. Reinterpretation of Power Coefficient as a Measure of Performance

Resolving the ambiguity of CP as an efficiency resides in its reinterpretation as a measure of performance. Though we
may loosely consider the power coefficient as an efficiency in that it compares power output to a conveniently calculablereference, with the result (usually) being less than unity, its interpretation as an energy conversion efficiency poses several
difficulties. In particular, the discussion above demonstrates that such an interpretation of CP would require us to make therather awkward modelling assumption that a flow of wind travelling around the turbine—a flow that avoids the actuator discaltogether—constitutes part of the inputs and outputs of the device. Furthermore, linear momentum theory invokes isothermal
and frictionless assumptions, which indicates reversible flow. Hence, there is no theoretical foundation from which to assertthat 100% of the kinetic energy of a flow cannot be extracted as power. Indeed, this is what Eqs. (8) and (10) along with Fig.
3 indicate, namely, that the rate of kinetic energy input Pin is completely converted to output power when the downstream
area is very large and the speed Ve becomes vanishingly small.
And so, if power coefficient does not constitute an energy conversion efficiency, what exactly does it measure?
3.1. Power Coefficient as Relative Capture Area
Part of the problem in defining an energy conversion efficiency for a wind turbine rests in characterizing the energy
resource itself. At the root of this difficulty is the fact that there is effectively no finite amount of power that the wind contains
per se, but rather what is referred to as a power density [5]. This power density is expressed as power per unit area,′′
𝑃𝑃𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤=

𝜌𝜌𝑉𝑉𝑖𝑖3.
2(12)

We can therefore characterize the output of a wind turbine in terms of how much of the wind power density it
captures expressed as an area,′′𝑊𝑊̇𝑜𝑜𝑜𝑜𝑜𝑜 = 𝑃𝑃𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤
𝐴𝐴𝑐𝑐𝑐𝑐𝑐𝑐 ,(13)

where we refer to Acap as the capture area. Equations (3), (12), and (13) are easily rearranged to show that Acap=CPAt, or
𝐶𝐶𝑃𝑃 =

𝐴𝐴𝑐𝑐𝑐𝑐𝑐𝑐
.
𝐴𝐴𝑡𝑡

(14)

Thus, the power coefficient is best described as the relative capture area, a parameter that measures the amount of wind
power density captured by a wind turbine relative to its size.
The idea of capture area is borrowed from ocean wave energy conversion. In wave energy devices, ocean waves arecharacterized via a power per unit width of wavefront, forthrightly acknowledging the difficulty in defining a discrete amountof power input to a conversion device. The equivalent width of wavefront corresponding to the power the device generates
is the capture width, with that width divided by the widest linear dimension of the device forming the relative capture width.
A minimum relative capture width of three or greater is cited as a guideline for the potential success of a design [2]. Also
pointed out in [2] is that since the relative capture width is often greater than one, it is not helpful to consider it an efficiency.
Similarly, both capture area and relative capture area serve as useful measures of performance for wind turbines. For
any turbine, we wish the capture area to be as large as possible, thereby maximizing power output. When considering relative
capture area, the cross-sectional area At is indicative of the turbine’s size, and thus, its total direct cost including both capitaland operational costs [6]. For two turbines with the same value of Acap, then, we prefer the one with the larger relative capturearea, as it offers us the more economical option for the same performance even if it operates at a smaller conversion
efficiency.
Such an interpretation offers several advantages. Primarily, it avoids the ambiguity of what constitutes a power input tothe device by more cleanly identifying the power in the wind as a density. We may also argue that it is the preferred measure

HTFF 136-6

of performance over an energy conversion efficiency in that it better aligns with the objective of the device. As an illustrationof the latter point, we may consider the contrast between fuel economy and thermal efficiency in motor vehicles. In internal
combustion engines, thermal efficiency is often highest at full throttle, indicating that the largest conversion of chemicalenergy into mechanical power occurs at the vehicle’s top speeds. Given that air drag scales with the square of vehicle speed,
however, operating at very high speeds leads to smaller distances travelled for the same amount of fuel consumed. Hence, if
our objective is to travel as long a distance as possible per unit energy consumed, we may be willing to forgo always operating
at the engine’s highest thermal efficiency [7].
Furthermore, since CP is not a conversion efficiency, we need not limit its value to between zero to one. This is an
important point when we consider wind turbine designs that seemingly surpass the Betz limit, such as HAWTs that include
shrouds as outlined in [8]-[10]. The shrouds, usually in the form of diffusers, increase pressure drop across the actuator discand thus increase mass flow. This results in power coefficients that are 2-5 times larger than traditional designs and therefore
values that sometimes surpass unity. In [11] it is suggested that although the cited power coefficients are legitimate, theycease to be efficiencies for shrouded turbines. A correction to the power coefficient that effectively adjusts Pwind to accountfor the increased mass flow is suggested, thereby reinstating the power coefficient’s alleged status as conversion efficiency,
and once again recognizing the Betz limit as its maximum value.
If we abandon the idea of power coefficient as conversion efficiency to begin with, however, such adjustments becomemoot—power coefficient as relative capture area becomes the common measure of performance for both shrouded andunshrouded designs, and for all other wind turbines as well, including vertical axis designs and drag machines. For a windturbine of a given size, maximizing the relative capture area will always maximize power output, whatever the energy
conversion mechanism it employs and the efficiency thereof.

4. Conclusion

In this paper we have shown that the power coefficient for horizontal axis wind turbines is best thought of as a relative
capture area, a parameter comparing the equivalent area of the wind power density captured by a turbine relative to theturbine’s size. The interpretation avoids the ambiguities associated with the power coefficient’s interpretation as an energy
conversion efficiency, better aligns with the objectives of operating wind turbines, and also serves as a common measure of
performance for all wind turbines regardless of construction and operating principle.

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HTFF 136-7