Wind Engines in Light of Modern Research1

A. Betz, Göttingen

Introduction

In the postwar period, interest in wind engines has noticeably increased. This is explained chiefly by two reasons: on the one hand, the shortage of coal compels the search for new sources of energy; on the other hand, it has become evident that new aerodynamic investigations and the large body of experimental material accumulated in the construction of airplanes will make it possible to improve wind engines considerably. Indeed, the general advances of aerodynamics have naturally proved very fruitful for the development of windmill design, but, unfortunately, up to the present time these new investigations have not brought the expected substantial improvements on the economic side of the matter, and for now one can hardly place hopes upon them.

Here one observes a very widespread phenomenon, namely that modern science introduces only small improvements into those technical structures whose constructional experience has been accumulating since the most ancient times.

In particular, the new aerodynamic investigations have mainly clarified the nature of the processes occurring in wind engines and the means for achieving

of the desired effect, and also determined the principal limits of the power that can be expected from an engine.

The task of the wind engine consists in extracting the energy of the wind and converting it into a form convenient for use.

Water turbines pursue related aims; however, there is an important difference between them and wind engines. Water power stations usually have at their disposal a certain limited reserve of energy, accumulated by means of an expensive dam. In converting this energy into a form convenient for use, one usually takes care to obtain a high efficiency of the machine, since any loss in such conversion is equivalent to wasting valuable energy.

Here, naturally, the cost and depreciation of the entire installation play a large role.

By contrast, in the air ocean we possess such gigantic reserves of energy that our wind engines cannot in any way encompass them. Therefore it is quite immaterial what losses occur in the conversion of this energy. What is essential is only the cost of the structures required for such conversion.

The most economical wind engine is the one in whose operation each kilowatt-hour obtained costs the least. And since, in obtaining energy by means of wind engines, the expenses are reduced chiefly to depreciation and to interest on the amount spent in construction, the problem of designing a wind engine is reduced to obtaining the greatest possible power for a given cost of construction; in other words, to lowering the figure obtained by dividing the cost of construction by the amount of energy produced in a year.

This problem—obtaining energy by means of wind engines—is complicated by certain requirements, depending partly on the peculiarities of the wind as a source of energy, and partly on the distinctive qualities of the consumers of this ener-

WIND ENGINES

gines. The most significant difficulty is caused by the variability of wind speed.

During a considerable part of the year the wind may be so weak that it will not be able to turn the engine’s wheel; and even if this wheel does turn rather freely, the energy delivered in such cases will not cover even a small fraction of the required amount. At times, on the contrary, a storm may be so strong that it will be necessary, with great effort, to save the installation from the excess energy that may destroy it. It is hardly possible to build an engine that would work well both in the weakest wind and in a storm.

Usually one has to confine oneself to calculations for the wind speeds that occur most often (in the interior of the continent—from 3 to 10 m per second, in coastal regions—somewhat more).

In particular, one has to take care to provide a good device for regulating the engine, which would make it possible safely to divert excessively large flows of energy when the wind speed rises above a certain limit. In addition, the variability of wind speed is inconvenient in another respect as well: the energy obtained fluctuates so much in magnitude that one cannot adapt to these fluctuations when consuming the energy. Thus, for example, if a wind engine turns a dynamo serving for electric lighting, then the consumption of current must increase in the evenings. But, obviously, one cannot count on the wind reaching its greatest speed precisely at that time. Conversely, at another time, when the wind is especially strong, its energy may not be needed. Therefore it is necessary either to accumulate the energy obtained, or to seek such an application of the energy of a wind engine that would not depend on time (for example, to use wind engines at water-pumping stations for the needs of water supply or for irrigation purposes).

Devices for accumulating energy—for example, electric accumulators—are expensive, and therefore increase the cost of producing energy.

*

Further difficulties arise from the small number of revolutions per minute characteristic of wind motors. For reasons that will be discussed in more detail below, the ratio of the circumferential speed of the ends of the motor blades to the wind speed must lie within certain limits. But, on the other hand, in order to obtain any appreciable power, the diameter of the wind-motor wheel must be rather large. This is why this wheel has to be given a number of revolutions per minute considerably smaller than in all other machines.

In order to set some machine in motion by means of a wind motor, it is therefore necessary to introduce an intermediate transmission to increase the speed of rotation. And this always makes the installation more expensive and thereby increases the cost of the energy obtained. The difficulties here become all the more significant, the larger the wheel diameter, since as it increases the number of its revolutions falls and the transmitted power increases. It is therefore very important to attain the greatest possible number of wheel revolutions under the given external conditions, that is, in other words, to increase the ratio of the circumferential speed of the wheel to the wind speed.

Unfortunately, such high-speed wheels have other substantial shortcomings, which set a limit to the striving to increase speed.

In what follows we shall examine more closely two such problems. First of all—the main problem of how energy can be obtained by means of a wind motor, and what power one may hope to obtain by means of a wheel of given dimensions and at a given wind speed. The second problem is connected with the question of high speed.

Mechanism of Energy Extraction

First of all let us ask ourselves: how, generally speaking, can kinetic energy be extracted from moving air? We may imagine the following schematic device (Fig. 1): some body is placed in the wind, for example a flat plate with surface area $F$. The wind blowing on

WIND MOTORS

a plate with velocity \(v\), exerts upon it a force \(W\), proportional to the area \(F\), to the square of the velocity \(v\), and to the density of the air \(\rho\) (in the technical system of units, “\(m\)-kg-sec\({}^{2}\)”) for air at normal temperature and under normal pressure

\[ \rho=\frac{1}{8}\, kg\cdot sec\cdot m^{-4}. \]

They try to express this relation in the following way:

\[ W=c_w\cdot \frac{\rho}{2}\,F\cdot v^2 . \tag{1} \]

The coefficient of proportionality \(C_w\) is called the drag coefficient and depends on the shape of the given body.

If the body is allowed to move in the direction of the acting force, then this force will perform work which can be utilized (for example, to lift a weight (Fig. 1).

Fig. 1

Since in such motion the velocity of the wind relative to the body will decrease, the pressure it produces will also decrease.

If the velocity of motion of the body (plate) is equal to \(v'\), then the relative velocity of the wind will be \(v-v'\), and consequently the pressure force will prove to be equal to:

\[ W=c_w\frac{\rho}{2}F\left(v-v'\right)^2, \tag{2} \]

The work performed during one second (the power) will be:

\[ L=Wv'=c_w\frac{\rho}{2}F\left(v-v'\right)^2\cdot v'. \tag{3} \]

It is not difficult to show that this power, for a given value of the area \(F\), reaches a maximum when \(v'=\frac{1}{3}v\). Its maximum value proves to be equal to:

\[ L_{\max}=\frac{4}{27}\cdot c_w\cdot \frac{\rho}{2}F\cdot v^3. \tag{4} \]

Meanwhile, the available power of the wind is, obviously,

\[ L' = W \cdot v, \tag{5} \]

since the air, moving with velocity \(v\), acts with force \(W\) in the direction of its motion.

For \(v' = \frac{v}{3}\), the wind therefore gives up one third of the available power. The remaining two thirds are converted into the energy of vortices and, ultimately, into heat.

Although, as we have said, such a poor efficiency in no way excludes the suitability of the wind motor, it nevertheless gives rise to certain reflections; for an installation operating with small losses, with the same dimensions (and consequently the same cost), could develop greater power.

Fig. 2

The action of the device that has just been described was associated with an unavoidable dissipation of energy: each second, work equal to \(w(v - v')\) was expended there on the formation of vortices.

Is it possible, one may ask, to imagine such a mode of action of the wind in which such losses would be excluded?

It is not difficult to show that this is indeed possible.

In fact, let us choose such a shape of body that the pressure \(P\) exerted on it by the stream should not coincide with the direction of the relative velocity of this stream, but should make an acute angle with it (Fig. 2). Resolve the pressure force into two components: \(W\), directed along the stream, and \(A\), perpendicular to it. Then the energy going into the formation of vortices will be due only to the component \(W\), which expresses the so-called drag resistance to motion.

The work expended in overcoming it during one second will be equal to: \(L = Wv\), where \(v\) denotes

speed of the wind relative to the body. The second component—the “driving force”—causes no loss of energy, since it is perpendicular to the direction of motion. In this respect it is analogous to a centripetal force: it only changes the direction of motion of the air, but does not change its energy.

Therefore, in order to obtain a large amount of energy with small losses, one must choose such a shape of body as would give the greatest possible driving force and the smallest possible resistance force.

Such a body is called a wing. The ratio of the driving force to the resistance force, which characterizes the merit of the wing, is called the quality of the wing. In what follows, its reciprocal will be used; it bears the name of the reciprocal quality of the wing. We shall denote this reciprocal quantity by the single letter $\varepsilon$. Consequently:

\[ \varepsilon=\frac{W}{A}. \tag{6} \]

Just as the pressure on a body was computed above, one can now compute the driving force and the frontal resistance:

\[ A=C_a\cdot \frac{\rho}{2} Fv^2 \tag{7} \]

\[ W=C_w\cdot \frac{\rho}{2}Fv^2, \tag{8} \]

where $C_a$ and $C_w$ are coefficients of proportionality depending only on the shape and position of the wing. By the symbol $F$ here is usually meant the greatest projection of the wing.

For a rectangular wing this area is equal to the product of length by width: $F=l\cdot t$. For extracting energy from the wind we shall use only such wings.

Let the wind speed be equal to $v$. Place a wing in the wind and move it perpendicularly to the direction of the wind with speed $u$ (Fig. 3).

The air will then move relative to the wing with a speed $V$, equal to the geometric sum of $v$ and $u$ (Fig. 4)

\[ V=\sqrt{v^2+u^2}. \tag{9} \]

The direction of the velocity \(V\) makes with the direction \(v\) the angle \(\beta\) (Figs. 3 and 4), which, on the basis of Fig. 4, can be determined from the relation

\[ \operatorname{tg}\beta=\frac{u}{V}. \tag{10} \]

Since the “driving” force \(A\), acting on the wing in this kind of displacement, is directed perpendicular to \(V\), it makes the same angle \(\beta\) with the direction of motion \(u\) (Figs. 3 and 5).

Fig. 3

Fig. 4

The useful work will be produced by that component of the force \(A\) which coincides in direction with \(u\),

\[ T_1=A\cdot\cos\beta. \]

The resistance to the motion will be represented in the form of the component

\[ T_2=W\cdot\sin\beta. \]

In the direction of motion there will therefore act a force equal in all to:

\[ T=T_1-T_2=A\cos\beta-W\sin\beta =A\cos\beta\,(1-\varepsilon\,\operatorname{tg}\beta), \tag{11} \]

where \(\varepsilon=\dfrac{W}{A}\) denotes the above-mentioned inverse quality of the wing.

Fig. 5

The useful work proves to be equal to

\[ L_N=T\cdot u=A\,u\cos\beta\,(1-\varepsilon\,\operatorname{tg}\beta). \tag{12} \]

The components of the forces that coincide in direction with the wind velocity \(v\) will tend to displace the wing. The shearing force \(S\) will be:

\[ S = S_1 + S_2 = A\sin\beta + W\cos\beta = A\sin\beta(1+\varepsilon\cotg\beta). \tag{13} \]

The available energy of the wind is equal to

\[ L_w = S\cdot v = Av\cdot\sin\beta(1+\varepsilon\cotg\beta). \tag{14} \]

Taking into account that \(v\cdot\sin\beta = u\cdot\cos\beta\), from (12) one may obtain:

\[ L_x = L_w\cdot \frac{1-\varepsilon\tg\beta}{1+\varepsilon\cotg\beta} = L_w\cdot \frac{1-\varepsilon\dfrac{u}{v}}{1+\varepsilon\dfrac{v}{u}} \tag{15} \]

If the frontal resistance were equal to zero (as also the reciprocal quality of the wing), then the energy obtained by the wing in one second would be equal to the available energy of the wind. In this case there would be a conversion of energy not accompanied by any losses.

In reality, resistance to motion exists, and therefore losses also exist. The efficiency coefficient of the wing is determined, obviously, by the ratio:

\[ \eta_1 = \frac{1-\varepsilon\dfrac{u}{v}}{1+\varepsilon\dfrac{v}{u}}, \tag{16} \]

that is, by the ratio of the useful work to the available work.

The reciprocal quality \(\varepsilon\), generally speaking, is a very small number (approximately from 0.02 to 0.1). Therefore the efficiency coefficient is poor only when the ratio \(\dfrac{u}{v}\) of the velocity of motion of the wing to the wind velocity is either too large or too small. In the first case \(\varepsilon\dfrac{u}{v}\) (in the numerator), and in the second \(\varepsilon\dfrac{v}{u}\) (in the denominator), prove to be such that the efficiency coefficient decreases noticeably. Especially important is the case when the ratio many times exceeds

unity. This is why the upper limit for the speed becomes the striving toward a good coefficient of efficiency. Below we shall return once more to this question.

Let us now determine how large the area of the wing must be that is required to obtain a given power.

According to equation (7), the driving force is equal to

\[ A=C_a\,\frac{\rho}{2}\,FV^2, \]

where \(C_a\) denotes the lift coefficient (it is of the order of unity), while \(\rho\) and \(F\), as before, denote the density of the air and the area of the wing. The magnitude of the velocity \(V\) entering into this formula is determined from the relation

\[ V=\sqrt{v^2+u^2}=u\sqrt{1+\left(\frac{v}{u}\right)^2}. \]

This is the velocity of the wind relative to the wing (Figs. 3 and 4). If we substitute this expression into (12) and take into account that \(\cos\beta=\frac{v}{V}\) and \(\operatorname{tg}\beta=\frac{u}{v}\), then for the developed power we shall find:

\[ L_N=C_a\,\frac{\rho}{2}\,FV^2\cdot u\cdot \frac{v}{V}\left(1-\varepsilon\,\frac{u}{v}\right)= \]

\[ =C_a\cdot\frac{\rho}{2}\,F\left(\frac{u}{v}\right)^2 \sqrt{1+\left(\frac{v}{u}\right)^2}\left(1-\varepsilon\,\frac{u}{v}\right). \tag{17} \]

Comparing this expression with (4) and taking into account that both coefficients—both \(C_w\) and \(C_a\)—are of the order of unity, it is not difficult to convince oneself that now, with the same value of the wing area and the same wind speed, a considerably greater power is attained.

On the one hand, in the case of a wing, the coefficient \(\frac{4}{27}\), which entered formula (4), drops out, and therefore the power can increase still more as the ratio \(\frac{u}{v}\) increases.

Thus, for example, if \(\frac{u}{v}=3\), then the factor increasing the power is equal to

\[ \left(\frac{u}{v}\right)^2\cdot\sqrt{1+\left(\frac{v}{u}\right)^2}=3^2\cdot1.05=9.45. \]

Although, on the other hand, the last factor \(\left(1-\varepsilon \dfrac{u}{v}\right)\) causes a certain deterioration, it does not play a large role so long as the inverse quality of the wing is small and the ratio \(\dfrac{u}{v}\) is not too large.

For example, for \(\dfrac{u}{v}=3\) and \(\varepsilon=0.05\), this factor will be:

\[ \left(1-\varepsilon \frac{u}{v}\right)=(1-0.05\cdot 3)=0.85. \]

Assuming that the coefficients \(C_a\) and \(C_u\) in equations (17) and (4) are equal to one another, we find that by means of the wing a power can be obtained which is \( \dfrac{27}{4}\cdot 9.45\cdot 0.85\), that is, approximately 54 times greater than that obtained by means of the primitive device described earlier. If the ratio of velocities were 2, then the power would increase 27-fold, and even for \(\dfrac{u}{v}=1\) it would nevertheless increase approximately 9-fold.

Such a colossal difference in the power obtained certainly compels us to prefer the wing to the body upon which, in the first case, we made the wind act, despite the high cost of constructing the wing.

The Wind-Motor Wheel

Let us now turn to the question of how, in practice, to carry out the extraction of energy by means of moving wings. Up to this point we have considered an isolated wing, or an isolated body upon which the wind pressure acted, with both moving rectilinearly.

This kind of energy extraction is realized on a sailing vessel moving under the action of the wind.¹

¹ The quality of such a “wing” is very low, since, for reasons of stability, the sail cannot be made of sufficient length—as aerodynamic conditions require (see A. Betz, Theorie der Tragflügel. Naturwiss. 1918, p. 557).

In stationary constructions one has to make wings, or other bodies on which the wind acts, move along a closed path (Fig. 6). The simplest way is to make them rotate about some axis. There have been many attempts to use here the bodies that were discussed at the beginning, but on the basis of what has been said it is easy to be convinced of their unsuitability and of the great advantages of wings over such imperfect bodies. Therefore we shall confine ourselves to describing only the most commonly used forms of wings.

Usually the wheel is arranged as follows: a certain number of wings are fixed on an axis, about which the whole system can rotate when placed in the wind.

Fig. 6

If the angular velocity of rotation of the wind wheel is equal to $\omega$, then a certain point of the wing, at a distance $r$ from the axis, has the linear velocity $u = r \cdot \omega$, directed perpendicular to the wind. One might think that the same calculations as in the preceding paragraph would give here also the magnitudes of the forces acting on the wings, and the magnitude of the power developed by the wheel. The circumstance that the linear velocity on the wings varies from point to point appears to be only a certain complication in the calculations, but by no means an essential obstacle.

But it is not difficult to notice that here we are already overlooking the circumstance that until now we have assumed the power developed to be proportional to the area of the wings. Therefore, it would seem, one could obtain arbitrarily large power from a very small wheel—one would only have to increase the number of wings and make each wing wider—which contradicts all experience. Obviously, here the wings interfere with one another, so that their action on the wheel turns out to be different from the case of an isolated wing. We must, consequently, first of all turn our attention to this distortion of the picture, caused by a large number of wings.

In the preceding section we became acquainted with the forces with which the wind acts on a wing. The same forces, but with the opposite sign, act on the particles of air on the side of the wing. An especially important role is played by the component \(S\), coinciding in direction with the wind and reducing the angular velocity, whereas the component \(T\) causes only a deflection of the air current.

In the case where one wing runs after another, each of them works in a region where the air jets are more or less distorted by the preceding wings.1

An exact investigation of all distortions is extremely troublesome. For our purposes it is sufficient to solve a simplified problem—to try to eliminate the unknown disturbing effects, replacing them by a simple and reliable scheme that would make it possible approximately to calculate the maximum attainable power of the wheel.

The actual power of the wheel will be somewhat less than this theoretical value—depending on the perfection of the construction. To solve this problem we shall reason as follows.

The energy of the wind is kinetic energy. If \(v\) is the wind velocity, then some mass of air \(m\) possesses kinetic energy \(\frac{m}{2}v^2\). Since up to now we have denoted the density of air by \(\rho\), the energy of a unit volume of it is equal to \(\frac{\rho}{2}v^2\).

Since the wind wheel takes away part of this energy, behind the wheel the kinetic energy of the air, and consequently also its velocity, must be less than in front of the wheel.

Let us denote, for distinction, the velocity at a large dis—

…at a distance in front of the wheel by \(v_1\), and at a large distance behind the wheel by \(v_2\) (Fig. 7). Then, consequently, it will be: \(v_2 < v_1\).

We also know that the wings act on the air with a force \(S\), directed against the wind and reducing the speed of motion of the air.

The transition from a high speed to a low one occurs, of course, not suddenly: for the slow stream must have a larger cross-section than the fast stream (Fig. 7), and the streamlines require, for their curvature, some finite interval of time.

Fig. 7

The phenomenon proceeds as though, in front of the wheel, a certain backwater is formed in the air stream. Velocity is converted into pressure. Therefore the air enters the wheel with reduced velocity, but under increased pressure. The absorption of wind energy by the wheel is reflected in a lowering of the pressure energy, so that the air, having entered the wheel under increased pressure, leaves it under reduced pressure. If we imagine the wheel to be infinitely thin, then the velocity of the motion of the air itself may remain entirely unchanged while passing through the wheel. Only behind the wheel does the decrease in velocity continue, with the kinetic energy being converted into pressure energy until this pressure again reaches its normal value (Fig. 7, below).

The wings of the wind wheel do not fill the wheel uniformly over its entire surface: considerable gaps remain between them, and one might think that the retardation

of the air jets occurs only in those places where the wings happen to be located, and that in the intervals between them the air is not subjected to any action.

But in reality this is not so. Indeed, since the wings rotate about an axis, then, if not simultaneously, at least after short intervals of time, they must act upon all the air that flows through the circle swept out by the wings, or through the annulus (in the case where the wings do not reach the axis).

Owing to such periodic action a certain nonuniformity arises, but, for practical proportions of the wheel, it is quite insignificant. We may therefore, in our analysis, mentally replace the wheel by a certain disk capable of letting air pass through and causing only a slowing of the motion of the air (owing to the existence of a resistance \(S\)), leading to the extraction of energy by the wheel.

Here we may disregard the tangential forces \(T\), which cause only a deflection of the air jets. We might, for example, imagine behind the wheel a stationary apparatus which would deflect the jets back to their former direction.

If in each second a mass of air equal to \(m\) passes through the disk of the wind wheel, and if the velocity of the flow is thereby reduced from \(v_1\) to \(v_2\), then the power which the engine could extract is equal to:

\[ L=\frac{m}{2}\left(v_1^2-v_2^2\right). \tag{18} \]

This is the energy that could be obtained in the ideal case; in reality only part of it is extracted, owing to the existence of various losses, which were taken into account above in the derivation of the efficiency coefficient of the wing. Naturally, one must strive to obtain the greatest possible energy from the air flow. Therefore we may pose the question: how great is the power that can be obtained, in the best case, for a given diameter of the wheel \((D)\), that is to say

of its determined area \(F_0=\dfrac{\pi D^2}{4}\) and the given wind speed \(v_1\). At a cursory glance at equation (18) it might seem that the maximum power will be obtained when \(v_2=0\), that is, when the air, having passed through the wheel, gives up all its energy to it. But this is incorrect, since when \(v_2\) is decreased, the mass of air \(m\) flowing through the wheel also decreases. The more strongly the air stream is slowed, the less air flows through the engine. In this case, the air bypasses the obstacle placed in its path and flows around it on the sides, producing no useful work.

To determine the mass \(m\) flowing per second, it is necessary to know the speed \(v'\) with which the air passes through the wheel. This mass is expressed as follows:

\[ m=\rho\cdot F_0\cdot v' \]

The magnitude \(v'\) of the speed of passage through the wheel evidently lies between \(v_1\) and \(v_2\) (before and after the wheel). It can be shown that \(v'\) is precisely the arithmetic mean between \(v_1\) and \(v_2\):

\[ v'=\frac{v_1+v_2}{2}. \]

Indeed, when a mass of air equal to \(m\) flows through the wheel each second, and the wheel reduces the speed of this air from \(v_1\) to \(v_2\), the wheel thereby receives a power determined on the basis of equation (18). On the other hand, the same power can be expressed differently: the air flowing with speed \(v'\) encounters, on its way through the wheel, a resistance force \(S\), and therefore the power expended by it must be equal to:

\[ L=S\cdot v'. \tag{19} \]

Equating the expressions (18) and (19), we find:

\[ \frac{m}{2}(v_1^2-v_2^2)=Sv'. \]

But the force \(S\) must reduce the speed of the air from \(v_1\)

up to \(v_2\). Consequently, by the law of impulse of forces and quantity of motion:

\[ S = m(v_1 - v_2). \tag{20} \]

Substituting this expression into the preceding equation and taking into account that \(v_1^2 - v_2^2 = (v_1 - v_2)(v_1 + v_2)\), we find:

\[ \frac{m}{2}(v_1 - v_2)(v_1 + v_2) = m(v_1 - v_2)\cdot v \]

or

\[ \frac{v_1 + v_2}{2} = v'. \tag{21} \]

Now we have all the necessary material for determining the power \(L\) from the initial \(v_1\) and final \(v_2\) velocities. According to the preceding:

\[ m = \rho \cdot F_0 \cdot v' = \rho F_0 \cdot \frac{v_1 + v_2}{2} \tag{22} \]

and

\[ \begin{aligned} L &= \frac{m}{2}(v_1^2 - v_2^2) = \frac{\rho F_0}{4}(v_1 + v_2)(v_1^2 - v_2^2) \\ &= \frac{\rho F_0 \cdot v_1^3}{4} \left(1 + \frac{v_2}{v_1}\right) \left[1 - \left(\frac{v_2}{v_1}\right)^2\right]. \end{aligned} \tag{23} \]

The maximum power will be obtained if we attain the following ratio between the velocities:

\[ \frac{v_2}{v_1} = \frac{1}{3} \tag{24} \]

The magnitude of this maximum power will be:

\[ L_{\max} = \frac{16}{27}\,\frac{\rho}{2}\,F_0 \cdot v^3 \tag{25} \]

In reality, this velocity ratio \(\frac{v_2}{v_1} = \frac{1}{3}\) is difficult to achieve in a wind wheel; moreover, as we know, certain losses occur on the blades themselves. All this, of course, further reduces the useful power of the wheel as compared with the calculated figure.

The degree of perfection of the wheel is expressed by a certain coefficient \(\zeta\), equal to the ratio of the actual power to the maximum theoretical power:

\[ \zeta = \frac{L}{L_{\max}} \tag{26} \]

To express the results of experiments, a somewhat different number is often used—a certain power coefficient:

\[ C_l=\frac{L}{\frac{\rho}{2}F\cdot v^3}=\frac{16}{27}; \tag{27} \]

(cf. Fig. 9). It should be borne in mind that the theoretical limit for \(C_l\) is not unity, but \(\frac{16}{27}\).

Wing Dimensions and Engine Speed

We have now learned how to determine the efficiency coefficient of a wind wheel and to predict its maximum power. Next comes a new question: by what means can one achieve the most advantageous ratio of velocities,

\[ \frac{v_2}{v_1}=\frac{1}{3}. \]

Obviously, there are here certain known conditions that must be satisfied by the dimensions of the wings and by their number, since if the wings are arranged too closely the wheel will reduce the speed of the air stream too greatly, while if they are arranged too sparsely, too much energy will pass through the wheel without being utilized. But the decisive role here is played not only by the area of the wings, but also by the speed at which they rotate.

In the second section we calculated the power developed by a single wing, and found that this power depends not only on the area of the wing, but—to a large extent—also on the ratio \(\frac{u}{v}\), that is, on the speed of motion. Hence it is clear that, in order to attain the very same power, we may make do with the smaller wing area the greater the speed of its motion.

We cannot directly apply equation (17), found for an isolated wing, to the wings of a wind wheel, for, as we already know, these wings influence one another. But we now know this influence: it consists in the fact that the wind velocity acting on the wings is equal not to the original value \(v\), but to a smaller \(v'\), owing to-

as a result of which the power is correspondingly reduced. We can apply equation (17) to the blades of the wind wheel, substituting into it only, in place of \(v\), the quantity \(v'\) (the velocity of the flow through the wheel). From the preceding it is also known that we must try to make \(v_2=\dfrac{v}{3}\), and consequently there should be:

\[ v'=\frac{v+v_2}{2}=\frac{2v}{3}. \]

Therefore we may take, normally, that it should be:

\[ v'=\frac{2}{3}v. \]

Now we have everything necessary for calculating the required dimensions of the blades. A small complication, however, consists in the fact that the linear velocity \(u\) is not the same for different points of the wheel.

If \(\omega\) denotes the angular velocity of rotation of the wheel, then

\[ u=\omega r. \]

Therefore we must carry out our calculations separately, as applied to each value of the radius.

If we consider only a ring of thickness \(b\), cut out of the entire area of the wheel (Fig. 8), and if the mean radius of the ring is equal to \(r\), then the area of such an annular strip will be \(\Delta F=2\pi rb\).

To reduce threefold the velocity of the air flowing through the wheel in the amount \(m=2\pi rb\rho\cdot v'\) in one second, it is necessary that this strip act in the axial direction with a force equal to:¹

\[ \Delta S=m\,(v_1-v_2)=m\frac{2}{3}v=2\pi rb\rho\cdot\frac{4}{9}v^2. \]

If the wheel has \(n\) blades of width \(t\) and if

¹ This is not entirely exact: in reality, owing to compensating flows, the particles of air are also subject to certain axial interaction forces (cf. D. Toma, Foundations of the Simple Theory of the Screw. Zeitschr. f. Flugtechn. 1905, p. 206).

if the angular velocity of rotation of the wheel is equal to \(\omega\), then for the ring under consideration:

\[ u=r\omega \quad \text{and} \quad V=r\omega \sqrt{1+\left(\frac{V'}{r\omega}\right)^2} \]

On the basis of equation (17), and taking into account that \(\Delta L=\Delta S v\), we shall find, after some transformations:

\[ \Delta_1 S = C_a\frac{P}{2}nt\cdot b\cdot v^2 \left(\frac{\omega r}{v}\right)^2 \sqrt{1+\left(\frac{v'}{\omega r}\right)^2} \left(1-\varepsilon\frac{\omega r}{v}\right) = \]

\[ = C_a\frac{P}{2}tnbv^2 \left(\frac{\omega r}{v}\right)^2 \sqrt{1+\frac{4}{9}\left(\frac{V}{\omega r}\right)^2} \left(1-\frac{3}{2}\varepsilon\frac{\omega r}{v}\right) \]

Comparing this with the preceding equation, we obtain from this:

\[ C_a\frac{nt}{2\pi r} = \frac{16}{9}\left(\frac{v}{\omega r}\right)^2 \frac{1}{ \sqrt{1+\frac{4}{9}\left(\frac{V}{\omega r}\right)^2} \left[1-\frac{3}{2}\varepsilon\left(\frac{\omega r}{v}\right)\right] } \tag{28} \]

The expression \(\frac{nt}{2\pi r}\) represents the ratio of the wing width \(t\) to the distance between the wings, \(\frac{2\pi r}{n}\). It could be called the “frequency” of the wings. The expression standing in the denominator of the right-hand side differs little from unity; therefore the most important factor is \(\left(\frac{v}{\omega r}\right)^2\), that is, the square of the ratio of the wind velocity to the linear velocity of the points of the wings located at the given distance \(r\) from the axis of rotation.

The “frequency” of the wings will be precisely the smaller, the greater the linear velocity in the given annular zone of the wheel.

Owing to the existence of a certain freedom in the choice of the coefficient of driving force \(C_a\), we can adapt the width of the wings to constructional requirements. With a constant value of \(C_a\), the width of the wings would have to increase as one approaches the axis of rotation.

Usually, however, it is more convenient to leave the width of the wings approximately the same, as a result of which one must accept a decrease in the coefficient \(C_a\) as one approaches the periphery of the wheel. But in the immediate vicinity of the axis it is impossible to realize the dimensions required

by equation (28), since the coefficient \(C_a\) cannot be increased beyond a certain known limit and broad blades cannot be accommodated here. Moreover, the tangential force, which we have hitherto neglected, begins here to manifest itself disadvantageously. Therefore one usually dispenses with the central part of the wheel, which in any case would operate under unfavorable conditions.

Since the linear velocity \(\omega r\) at any point of a blade, distant from the axis by \(r\), is proportional to the velocity \(u_0\) of the blade tips \((u_0=\omega R;\ R\) is the radius of the wind wheel), the “frequency” of the blades may be characterized by the ratio of the circumferential velocity of the wheel to the wind velocity,

\[ \frac{u_0}{v}=\frac{R\omega}{v}. \]

This ratio expresses the speediness of the wheel. Thus we arrive at the conclusion:

The speedier the wheel must be, the smaller the frequency of the blades must be.

A high speediness of the wind wheel is desirable for two reasons: first, a smaller surface area of blades and wheel is required, and consequently it proves simpler. Secondly, as was already indicated in the introduction, this facilitates the construction of the drive to the machines served by the wind engine. Most machines make considerably more revolutions than a wind engine, and therefore there arises the need for a transmission increasing the number of revolutions. This transmission is evidently simplified when the wind wheel itself proves to be speedier. However, with a considerable increase in speediness, serious structural difficulties arise:

1. Back pressure. The factor \(\left(1-\varepsilon \frac{\omega r}{v}\right)\), which reduces the power, becomes appreciable as soon as the order of \(\frac{\omega r}{v}\) approaches \(\frac{1}{\varepsilon}\). Therefore speedy wheels have to be made extremely carefully, taking care to choose the blade profile correctly and to ensure their high aerodynamic quality (small back pressure). This, naturally, makes the wheel more expensive and reduces to naught the advantage of the speedy wheel—its simplicity.

1. Initial torque. In high-speed wheels, the force necessary for their rotation arises when their comparatively small blades move at high speed. But while the wheel is motionless, considerably smaller forces act on its blades, which are small in area, than on the large blades of low-speed wheels. Therefore, with the same friction, low-speed wheels begin to move in a weaker wind than high-speed wheels. And since, because of the inconstancy of wind speed, wheels may often stop, after which they must again be set in motion, hard-to-start high-speed engines often cease to operate in a weak wind, while low-speed ones are still operating.

2. The centrifugal forces that arise owing to inevitable inaccuracies in balancing the moving parts of the wheel act on its axle and the entire installation with a force that is the greater, the higher the rotational speed. Therefore one must either build a stronger tower and a more massive foundation, or center and balance the wheel more carefully. But both increase the cost of the engine’s construction.

Apparently, for these reasons, excessively high-speed engines have not gained acceptance. However, one can observe a certain tendency gradually to move toward more high-speed models.

The following values are very commonly used: \(\dfrac{u_0}{v}\), \(\dfrac{u_3}{v} = 1\) to 2, low-speed, with a large number of blades (wind turbines); \(\dfrac{u_0}{v} = 2\) to 3, transitional type, with increased speed (the old four-bladed mills also belong to it); \(\dfrac{u_0}{v} =\) up to 34, high-speed.

With a ratio \(\dfrac{u_0}{v}\) exceeding 4, operating engines are hardly ever encountered, although there have been isolated attempts to build them.

Results of Experiments

In the preceding paragraphs we have clarified the phenomena accompanying the extraction of energy from the wind, and the limits of attainable power. Therefore we can now judge the perfection of a given engine if its actual power is found experimentally. Unfortunately, there is very little experimental material on this question. There are two ways of determining power: 1) by means of experiments with technical types of engines intended for practical needs, 2) by means of experiments on models placed in an artificial air stream.

Both methods have their own substantial shortcomings. The first of them makes it possible without difficulty to measure the useful power developed by the engine (with ordinary technical instruments). But at the same time it is extremely difficult to judge the wind speed, on which the maximum attainable power depends. In measuring wind speeds the probability of errors is very great, and very great skill is required to obtain reliable results. And since power varies directly proportionally to the third power of the wind speed (equation 25), all errors in this speed measurement show themselves especially noticeably. Thus, if, for example, the speed is in reality 10% greater than the measured one, then as a result of the calculations the efficiency coefficient will prove to be overestimated by approximately 37%.

If the actual wind speed is twice the measured one, then the efficiency coefficient obtained will be 8 times too large. The very unsteadiness of the wind speed complicates the measurements: most instruments give, for example, approximately the mean wind speed over some interval of time, whereas for the power the decisive role is played by the mean value of the cubes of the speeds. Thus, if during equal intervals of time the wind speed was 8 and then 4 meters per second, then the mean speed was 6 (this is what the measuring instru-

choice). The cube of this magnitude is equal to \(216\ \mathrm{m}^3/\mathrm{sec}^3\). In reality, the power depends on another magnitude, namely on:

\[ v_m^3=\frac{1}{2}(4^3+8^3)=288\ \mathrm{m}^3/\mathrm{sec}^3. \]

Unfortunately, in various catalogs one can often encounter exaggerated values of efficiency coefficients, placed there on account of such systematic errors in measurements.

Comparatively good tests of new wind engines under natural conditions have recently been carried out by the Agricultural Institute of Oxford University.

The second path for investigating wind engines—experiments on models—makes it possible to eliminate the difficulties that arise when measuring wind speed under natural conditions: here an artificial air stream is used, the speed of which can be easily and accurately measured. However, here one must take into account the errors that are caused by using a small model instead of a real engine.

It is completely impossible to reproduce on a model all the smallest details of a real engine: all its bolts and rivets. But even if this could be done, the forces acting on the model would not give a true representation of the forces that occur in reality, since the law of similarity, on the basis of which the conversion is performed, is not fulfilled precisely for small details, owing to the presence of air between the air and the parts of the engine. However, the errors arising for this reason are not too large, since such details are avoided when constructing models. Figure 9 gives the results of several experiments carried out at the Aerodynamic Institute in Göttingen on models of wind wheels. One of them \((a)\) belonged to the class of slow-running ones, another \((c)\)—to the class of the fastest-running, and the third \((d)\) corresponded to a modern intermediate type.

Along the axis of abscissas are plotted the values \(\frac{u_0}{v}\)—the ratio of the circumference—

ratio of the peripheral speed of the wheel to the wind speed, and on the ordinate axis—the power coefficient

\[ c_l=\frac{L}{\frac{\rho}{2}v^3\frac{\pi D^2}{4}}. \]

From the preceding we know that this coefficient is limited by the upper theoretical bound \(\frac{16}{27}\). To give an idea of the behavior of the wheel during starting, the diagram shows by a dashed curve the change in the quantity:

Fig. 9

\[ c_d=\frac{M}{\frac{\rho}{2}v^2\frac{\pi D^2}{4}\cdot\frac{D}{2}} = c_l\cdot \frac{v}{u_0} \]

—the coefficient characterizing the torque \(M=\frac{L}{\omega}\), for various values of \(\frac{u_0}{v}\). From the diagram it is seen that the first wheel (a slow-running one) develops maximum power at \(\frac{u_0}{v}=0.9\); the second (intermediate type)—at \(\frac{u_0}{v}=1.5\), and, finally, the third (fast-running one)—at values of \(\frac{u_0}{v}\) from 3 to 5. On the other hand, the diagram shows that the initial torque \(\left(\text{at } \frac{u_0}{v}=0\right)\) of fast-running wheels is considerably smaller than that of slow-running ones.

Summary of the Exposition

After indicating the distinctive features of how the problem is posed in the case of wind engines, the question of obtaining energy from moving air was considered. It was found that the most suitable component for this purpose is the wing. It was then shown that the power of a wind wheel of a given diameter cannot be increased without limit by increasing the area of the wings. On the contrary, the dimensions of the wheel diameter already impose an upper limit on the power. For the surface of the wings (the solidity of the wings), there exist the most favorable values: the solidity of the wings should be the smaller, the higher the speed ratio the wheel is to have. For the speed ratio that is desirable in principle, there are limits determined chiefly by the energy losses that increase together with the speed ratio. In conclusion, several experimental results have been presented.

1. In the case of an isolated wing, the motion of the air in the immediate neighborhood of the wing is also distorted by the body of the wing. But its influence is already taken into account by the empirical coefficients \(u\), since in their determination in experiment there exist the same distortions of the jets. This must be borne in mind when changing the wing profile, when the conditions change. ↩↩