Abstract –  With reference to his 1927 paper, the aerodynamic concepts of Albert Betz are examined and contrasted with up-to-date understanding. In a finding contrary to the prevailing view, it is determined that Betz misconceived the process of lift and that Betz’s Limit holds no valid application for lift-powered wind turbines.

Sources – Figure 1 and the Animation are available at Douglas McLean’s article “Lift(force)”, at Wikipedia. The English translation of Betz’s 1927 paper, cited herein, is available as Technical Memorandum 474 of the National Advisory Committee for Aeronautics (NACA). This version of Betz’s 1927 paper also may be downloaded at Wind Works, H/T Paul Gipes.

Betz’s Limit, a principle familiar to students of wind power, is accepted throughout that field as a fundament of design. Albert Betz formulated his Limit in his 1920 paper, “Theoretical Limit for Best Utilization of Wind by Wind Motors”. In this paper Betz posited an ideal, maximally efficient “wind motor” and presented a mathematical argument that concluded that the ratio 16/27 set the upper bound of the fraction of kinetic energy of wind, available to a “wind motor” for “extraction” and conversion to mechanical energy. The limiting 16/27 ratio, known as Betz’s Coefficient, is applied in wind turbine design. This review finds that Betz misconceived the aerodynamics of lift and that Betz’s Limit therefore holds no meaning for lift-powered wind turbines.

Albert Betz’s aerodynamic concepts may be viewed in his 1927 paper “Windmills in the Light of Modern Research”, wherein he discusses lift-powered wind turbines of the horizontal-axis category. It should be noted that Betz’s phrase “… Light of Modern Research” refers to concepts of lift that obtained in the early years of aerodynamical studies. Such early concepts, in the main, have been overturned by advances in understanding gained since 1927. Herein, Betz’s 1927 paper is reviewed in light of observations and data acquired in modern wind tunnel tests.

Fig. 7b of his 1927 paper presents graphically Albert Betz’s aerodynamic concepts. It shows a build-up and then an abrupt drop in air pressure at a “windmill”, with pressure restored to normal in the wind turbine’s wake where velocity simultaneously diminishes. Present-day wind tunnel observations reveal Fig. 7b as representing misconceived pressure and airflow circumstances, relative to an airfoil turbine blade.

Companion text for Figure 7b is found on page 15 of Betz’s paper and an excerpt is quoted here, each sentence enumerated for reference: “ [i] Consequently, the air strikes the windmill with diminished velocity but with higher pressure. [ii] The energy extraction by the windmill first causes only a diminution of the pressure energy, so that the air that entered the windmill with increased pressure, leaves it with diminished pressure. [iii] The velocity itself cannot change in the thickness of the windmill. [iv] Only behind the windmill the velocity diminution again continues until the normal air pressure is restored.” 

In the excerpt, Betz presents his concepts of the aerodynamics of lift at a wind turbine. Instead of the word lift, Betz employs the phrase “energy extraction”, this phrase corresponding to the assumption that an airfoil turbine blade extracts kinetic energy of the wind. The excerpt is reviewed with reference to observations and data presented in Figure 1 & the Animation.

In [i], Betz mis-states the pressure/airflow circumstances at a turbine blade, ignoring the low pressure at the camber side, revealed in Figure 1, and the acceleration of air thereto, revealed in the Animation. In sentence [ii], Betz puts a causal relation between “energy extraction by the windmill” and “diminution of pressure energy”, with a sense that “pressure energy” is extracted from air by the turbine blade with a consequent drop in air pressure, which sense is depicted graphically in Betz’s Fig. 7b. When these concepts are contrasted with the air pressure data presented in Fig. 1, it becomes evident that Betz misapprehended the air pressure circumstances at an airfoil. Sentence [iii] is untrue, as the Animation shows that the velocity of air changes at each side of an airfoil. Regarding sentence [iv], the Animation shows that the postulated “velocity diminution” is not observed in the wake of an airfoil, during wind tunnel tests.

As now understood, an important aspect of lift is drag, the friction of air against an airfoil and the cause of the high pressure. The coincidence of high pressure and drag is confirmed in a comparison of Fig. 1 with the Animation. Betz failed to recognize the role of drag in the generation of lift, putting in pages  6 and 7 of his 1927 paper: “… the real resistance or “drag” determines the amount of kinetic energy consumed in the production of vortices… objects [airfoils] must be used of such shape that they will yield the greatest possible lift and the smallest possible drag.” In those pages, Betz regards drag solely as a source of inefficiency while, in fact, drag is essential to the generation of lift and, the more the drag, the higher the coincident pressure and the greater the force of lift.*  

 *That is, up to the angle of stall; the degree of drag is determined by the angle formed by the airfoil with incident airflow, as well as the velocity of airflow. Lift occurs if no more than ambient pressure obtains opposite the camber, in which case the low pressure at the camber, of itself, suffices to generate lift.

Discussion

Albert Betz conceived the operation of a lift-powered wind turbine as a process of converting the kinetic energy of wind into a corresponding mechanical energy.  Betz expressed this process as “energy extraction by a windmill”. Today, students of wind power commonly substitute the term “energy capture” or “energy uptake” for “energy extraction”. As conceived, the “energy extraction” depletes wind of kinetic energy, reducing its velocity, with depletion perforce limited to a fraction of wind’s total kinetic energy. These concepts may be stated as two assumptions: 1) the generation of lift at a wind turbine depletes wind of kinetic energy.  2) depletion cannot exceed a definite fraction of total kinetic energy. However, data from wind tunnel tests reveal that the generation of lift conserves, rather than depletes, the kinetic energy of air, which observation renders assumption #1 as false. It follows that assumption #2 is meaningless and, likewise, Betz’s Limit, regarding lift-powered wind turbines.

In Fig. 7b and the companion text, excerpted above, Betz apparently meant to demonstrate the generation of lift as a process of extraction and depletion of the kinetic energy of air. In fact, he put only a series of aerodynamic misconceptions, the whole of which is overturned by observations gained in wind tunnel tests. Those observations, incorporated into Figure 1 and the Animation, reveal the force of lift as a pressure differential across an airfoil, generated by a process that conserves the kinetic energy of air.

From Fig. 7B, it may be inferred that Betz was aware of the pressure differential across the turbine blade. If so aware, Betz should have recognized the consequent force but, in his paper, he expressed no such recognition. Rather, Betz put a “diminution of pressure energy”, represented in Fig. 7b as a precipitous drop in air pressure at the “windmill”, and signifying the postulated “energy extraction” at the blade, and thus Betz explained the process of lift. The curve v of Fig. 7b represents the velocity of air and signifies “velocity diminution” in the wake of the airfoil, or “behind the windmill”, as Betz put it. Such “velocity diminution” is not observed in wind tunnel tests.

As cited above, Betz misapprehended the effect of drag and regarded drag as kinetic energy lost to the process of lift. His 1927 paper offers much mathematical argument predicated on this particular misapprehension. However, drag is essential to the process of lift and thus the role of wind at a lift-powered wind turbine, in which role wind’s kinetic energy is conserved. 

It should be noted that drag at a rotating turbine blade is generated only in part through true wind, the balance being generated through motion of the blade. Therefore, lift at a turbine blade is not actually generated by wind, but by an apparent wind, this defined as the resultant of true wind and blade motion in vector combination. In his 1927 paper, in page 8 and Fig. 3, Betz recognized an apparent wind at a turbine blade, which he referred to as “relative velocity c”, but he otherwise failed to establish a distinction between the aerodynamics of true wind and the aerodynamics of apparent wind. The distinction is critical; the velocity of wind is a measure of its kinetic energy but such is not the case with an apparent wind, whose calculated velocity is no measure of its kinetic energy.

Betz’s 1927 paper, with its misapprehended pressure/airflow circumstances and misperception of the effect of drag, exemplifies the uncertainty of the early investigators in their attempts to account for the force of lift. That object now is long achieved but, remarkably, the antiquated and misconceived aerodynamics of 1920 persist, sustained by students of wind power through their adherence to Betz’s Limit.