A. E. Krzhizhanovsky.

HYDRAULIC THEORY

OF THE FRANCIS TURBINE

OF THE HIGH-SPEED TYPE.

Calculation and Design of Blades.

Tomsk.

Steam Typo-Lithography of P. P. Makushin, Blagoveshchensky Lane, own house.

1907.

Notation

$Q$—water discharge.

$q$—discharge lost through the clearance.

$\alpha$—outlet angle from the guide apparatus.

$\beta$—angle of the inlet edge of the runner.

$\gamma$—outlet angle from the runner.

$H$—total available head.

$h$—head.

$W_1$ and $W_2$—relative velocities at entry into the runner and at discharge.

$U_1$ and $U_2$—linear rotational velocities at the inlet and inner circumferences.

$V_1$ and $V_2$—absolute velocities at the outer and inner circumferences of the runner.

$V_0$—velocity in the supply channel.

$V_a$—velocity in the draft channel.

$\omega$—angular velocity of rotation.

$D_0$—diameter of the suction pipe.

$D_1$—outer diameter of the runner, measured from the blade edge.

$b$—height of the runner blade at the inlet edge.

$b_1$—height of the guide-apparatus blades.

$N$—number of horsepower.

$n$—number of revolutions per minute.

$K_1$, $K_2$, $K_3$—lost heads.

$\zeta$ and $\zeta_2$—empirical coefficients, approximately equal to 0.13.

$r_2$ and $ra$—radii of the various circumferences of the turbine wheel.

$\varphi$—hydraulic efficiency coefficient.

$\eta$—overall efficiency coefficient (at the turbine shaft).

$L$—work of one kilogram of water.

$p_2$, $p_2$—water pressures in different parts of the turbine.

$t$—length of the blade pitch along the arc of the circumference.

$S$—blade thickness along the arc of the circumference.

$HP$—number of horsepower.

$H. B.$—natural size.

The Francis Turbine and Its Features

Despite its comparatively early appearance, the Francis turbine does not have a properly developed theory, and many phenomena occurring in it have not yet been sufficiently explained.

Especially complex motion occurs in the wheel of the newest, high-speed Francis turbine, a type that first appeared in America.

This type, which we call simply the American turbine, is characterized by the fact that water enters the turbine wheel along a cylindrical surface parallel to the axis, but leaves the wheel in jets directed at various angles to the axis; under such conditions of motion, the path of one particle of water differs considerably from that of its neighbor, and therefore the method generally applied in considering other types—namely, that the motion of all the liquid in the turbine channel is taken as identical with the motion of the mean jet—is here entirely unsuitable. Apparently, only a few years ago it became possible to find the path of theoretical investigation that provides satisfactory explanations for many phenomena in the turbine.

Runner of a high-speed Francis turbine.

This new method, based on the differential equation of a moving liquid particle and called hydrodynamic, in contrast to the previous, hydraulic one, arose chiefly through the works of Professor Prasil and is being developed by him.

and at the present time for the calculation of American turbines*).

However, the hydrodynamic method just described is rather complicated, requiring considerable information from higher analysis; moreover, it is still too new and little developed, and therefore has, chiefly, only theoretical significance.

At the turbine-building works of Western Europe, at the present time, an exclusively modified hydraulic method is used.

The modification consists in the following: the entire internal volume of the runner of the turbine is mentally divided into a series of elementary layers superposed one upon another; if we then imagine that these layers are separated by impermeable walls, we obtain a series of elementary turbines, superposed one upon another—abcd, cdef, efgh, ghkl—separated by surfaces of revolution (Fig. 1).

Fig. 1.

Considering each turbine separately, independently of the others, we note that in them, with considerable accuracy, the motion of the mean stream—shown in the drawing by a dotted line—may be taken as identical with the motion of all the others. The accuracy increases with the number of elementary turbines. For the mean stream of each little turbine its own design equation is adopted.

Such a calculation is only approximate: the motion in each elementary turbine is in fact closely connected with the motion in the adjacent ones, and therefore, in order to obtain good results, it is necessary, in a known manner, to coordinate the motion of the water in each elementary turbine with the motion in the others, as will be indicated below.

Despite all the inaccuracy of the method set forth, turbines calculated in this way possess a considerable coefficient of useful effi-

* Ueber Flüssigkeits-bewegungen in Rotationshohlräumen von Prof. Dr. Prasil in Zürich.

there is reliable evidence that the efficiency on the turbine shaft can reach 0.9. No reactive type of turbine has such a high efficiency. It should be noted that until now these turbines have not been calculated at all, but have been built purely empirically, as is still the case at most American works; their efficiency was considerably lower and they could not compete with other types. The theoretical backwardness, in this respect, of American designers explains to us the fact—strange at first glance—that the greatest and most important turbine installations in America, such as, for example, almost all those at Niagara Falls, were made by Swiss firms, despite the abundance of turbine-building works in the country.

The only type of reaction turbines being built at the present time in Western Europe and North America is the Francis turbine. The chief advantages of this type in comparison with others are the following:

1) High efficiency.

2) Convenient and rapid regulation (especially automatic regulation).

3) Ready applicability of suction pipes.

4) A large and, within wide limits, arbitrary number of revolutions (where this is necessary, e.g., for dynamo-machines).

5) Slight pressure on the heel bearing.

Guide wheel of a Francis turbine, with rotating blades. Top view. The upper rim, like the lower, consists of two rings, of which the outer one (lighter in the drawing) is fixed. By rotating this latter, rotation is imparted to the blades.

Fink Regulation and Its Features

The highest efficiency of a turbine is closely connected with the power developed by it. Usually, with a decrease in the normal power of a given turbine, it falls—sometimes very considerably, sometimes only slightly—depending on the method of regulation.

The most favorable results are given by Professor Fink’s method. In his arrangement all the vanes of the guide apparatus

Fig. 2.

\(1/6\) of the HPP runner: \(H = 400;\ H = 82;\ n = 435\).

have axes of rotation parallel to the axis of the turbine and can, by means of one device or another, all be turned simultaneously. Owing to this, the size of the outlet slot of the guide apparatus changes, and consequently also the quantity of working water and the power of the turbine. Fink regulation is now used without exception in all cases where it is necessary to economize on water consumption.* Drawing 2 shows, in plan, a section of part of the runner and guide apparatus of a Francis turbine. The blades of the guide wheel are of the Fink system; two of them are set in the extreme position for “maximum” operation, and two at “minimum,” when the outlet slots are completely closed.

The desire to keep the efficiency as nearly normal as possible at different magnitudes of operation led designers to the idea of calculating the turbine for only \(3/4\) of the greatest output. In this case the highest efficiency should be obtained. When the blades are turned in one direction or the other, the output increases or decreases. It is clear that the places for such an opening of the channels of the guide apparatus must be quite sufficient, at which the “maximum” of operation is obtained.

When the blades are turned in the direction of narrowing the outlet openings, the distance between the edges of the blades of the guide apparatus and of the runner increases. According to the old rules, efforts were made to reduce this distance as far as possible, on the grounds of a supposedly better direction of the jets; now, however, this rule has been completely abandoned and, on the contrary, it is considered useful to make the distance between the edges of the blades equal to 10–50 mm, depending on the size of the turbine.

Such turbines operate not at all worse than those of the old type; moreover, the possibility disappears of the wheel being braked by foreign bodies that have entered between the blades. It is self-evident that the clearance between the wheel rims must still be small (1–3 mm) in order to avoid leakage of water.

In drawings 3a and 3b two diagrams are presented, taken from almost identical turbines (in the laboratory of the Zurich Polytechnic), with drawing 3a showing the diagram of the change in efficiency for a Francis turbine of the new type, and drawing 3b—for the old type. In the latter, one always observes, beyond the “maximum” useful effect, a certain irregular bend of the curve, which occurs in all turbines with a very small distance

* Zodel regulation, which also works very economically, has at present been abandoned, chiefly because of the rapid loosening of the regulating parts.

between the edges. In turbines with a considerable distance the curve is entirely correct.

Fig. 3a.

Fig. 3b.

For a more detailed explanation of the difference between the first and second cases, let us take Fig. 4. Owing to the thickness of the blade edges, so-called dead angles are obtained (shaded horizontally), in which the water does not move. When the lower wheel is displaced, these angles act differently, generally impeding the correct motion. Increasing the distance between the edges is intended to remove these dead angles from the region in which the water acts on the runner blades.

To establish the fact that a considerable distance between the blade edges is useful, it remains to answer the question of what the motion of the water will be in the space between the edges. With respect to axial turbines this question does not give rise to any special doubts; with respect to Francis turbines, however, it is more complicated.

Experience shows that if the water has been brought to the circumference of the outlet edges of the guide apparatus along a logarithmic spiral (as occurs with the newest profiles of guide blades), then this motion will also be preserved in the clearance between the edges.

Fig. 4.

The theoretical proof of this fact is found in the work of Professor Prasil cited above; here, for brevity, we shall present our own.

Let us have two flat horizontal rings placed vertically one above the other (Fig. 5).

Let us further suppose that between these rings water moves from the outer periphery to the inner, as shown by the streamlines. If all the water streams approach the outer periphery at one and the same angle, then by the law of symmetry it must be assumed that their exit also occurs at the same angle for all of them. It is self-evident, of course, that such two rings, since there are no blades between them, will experience no rotational moment from the water (the friction of the water against the rings is neglected).

Fig. 5.

Let \(mn\) be one of the streams flowing between the rings, \(V\) its initial velocity, and \(V_1\) its final velocity. If the discharge per second of this stream is \(q\), then the moment of the force due to it is:

\[ m=\frac{q\cdot \gamma}{g}\left(V\cdot S - V_1\cdot S_1\right) \]

where \(S\) and \(S_1\) are the corresponding arms of the velocities. The moment due to all the streams is:

\[ \sum m = M = \frac{\sum q\cdot \gamma}{g}\left(V\cdot S - V_1\cdot S_1\right) \]

but \(\sum m\), according to the above, is \(=0\); therefore:

\[ V\cdot S = V_1\cdot S_1. \]

But if we decompose each of the velocities \(V\) and \(V_1\) into the radial velocities \(W\) and \(W_1\) and the tangential velocities \(U\) and \(U_1\), then it is easy to see that:

\[ V \cdot S = U \cdot r \quad \text{and} \quad V_1 \cdot S_1 = U_1 \cdot r_1, \]

and therefore

\[ U \cdot r = U_1 \cdot r_1 \ldots \ldots \ldots (a) \]

Moreover, since the motion with velocity \(W\) takes place along a radius perpendicular to a surface that decreases in proportion to the radius, then:

\[ W r = W_1 \cdot r_1 \ldots \ldots \ldots (b) \]

Combining equations \(a\) and \(b\), we see that:

\[ \frac{U}{W}=\frac{U_1}{W_1} \]

and this indicates that the angles between the directions \(V\) and \(U\), and also \(V_1\) and \(U_1\), are equal.

Thus, the angle of approach of the jets to circumferences of different radii is constant, which is the characteristic of a logarithmic spiral.

It is obvious that the rings taken by us may be regarded as parts of the rims of the Francis turbine wheels, covering the space between the edges. The proof is not impaired thereby.

General Equations for Calculation.

We shall now proceed to derive the general equation for calculating the Francis turbine, while, as before, we shall assume that the mean jet moves in the same way as all the others.

Let us denote by \(V\) the absolute velocity, by \(U\) the linear velocity of rotation, by \(W\) the relative velocity, and by \(h\) the head height; the corresponding subscripts will indicate the belonging of these quantities to one or another section.

By \(\alpha\) we shall denote the angle of the last element of the guide vane, by \(\beta\) the angle of the first element of the runner blade, and by \(\gamma\) the angle of the last element of the runner blade with the corresponding concentric circumferences (see Drawing 6).

Let the dotted line \(A, I, II, III\) represent the path of the mean jet.

The equation for the motion of water from point \(A\) to circumference \(I\) will be:

\[ \frac{V_0^2}{2g}+h_1+\frac{P_a}{\gamma} = \frac{V_1^2}{2g}+\frac{P_1}{\gamma}+K_1 \ldots \ldots (1) \]

Fig. 6.

Here \(K_1\) represents all possible losses along the path \(A\)—I.

For the path I—II we have:

\[ \frac{W_1^2}{2g}+\frac{P_1}{\gamma}+h_2 = \frac{W_2^2}{2g}+\frac{P_2}{\gamma} +\frac{U_1^2-U_2^2}{2g}+K_2 \ldots \ldots (2) \]

where \(K_2\) is all the losses along the path I—II.

And finally, for the path II—III:

\[ \frac{V_2^2}{2g}+\frac{P_2}{\gamma}+h_3 = \frac{V_a^2}{2g}+\frac{P_a}{\gamma}+K_3 \ldots \ldots (3) \]

where \(K_3\) is all the losses along the path II—III.

Adding all these three equations and carrying out reductions, we obtain:

\[ H-(K_1+K_2)-K_3+\frac{V_0^2}{2g}-\frac{V_a^2}{2g} +\frac{V_3^2}{2g} = \frac{W_2^2-W_1^2+U_1^2-U_2^2-V_3^2}{2g} \ldots (4) \]

In the case of water entering without shock, from the velocity triangle we have:

\[ W_1^2-U_1^2=V_1^2-2V_1\cdot U_1\cdot \cos\alpha \ldots (a) \]

In the case of normal outflow from the runner, we similarly have:

\[ W_2^2-U_2^2=V_2^2 \ldots \ldots (b) \]

Since both of these cases are present here, combining equations (4), (a), and (b), we have:

\[ H-(K_1+K_2)-K_3+\frac{V_0^2}{2g}-\frac{V_a^2}{2g} = \frac{V_1\cdot U_1\cdot \cos\alpha}{g} \ldots \ldots (5) \]

Let us denote the entire left-hand side of equation (5) by \(\varphi H\); then:

\[ \varphi \cdot H \cdot g=V_1\cdot U_1\cdot \cos\alpha \ldots \ldots (6) \]

From the velocity triangle,

\[ \frac{U_1}{V_1}=\frac{\sin(\beta-\alpha)}{\sin\beta}\ldots \ldots (7) \]

and therefore from (6) and (7) we obtain:

\[ V_1=\sqrt{\varphi}\, \sqrt{\,g\cdot H\cdot \frac{\sin\beta}{\sin(\beta-\alpha)\cdot \cos\alpha}\,} \ldots \ldots (8) \]

\[ U_1=\sqrt{\varphi}\, \sqrt{\,g\cdot H\cdot \frac{\sin(\beta-\alpha)}{\sin\beta\cdot \cos\alpha}\,} \ldots \ldots (9) \]

On the basis of the study of existing turbines, it may be assumed that \(\varphi = 0.8\text{–}0.85\); an exception is the case where there is a long pipe bringing the water, in which there may be a considerable loss; in this case \(\varphi\) is smaller. Taking, for a preliminary calculation, \(\varphi\) within the indicated limits, i.e. \(\sqrt{\varphi}=0.9\text{–}0.92\), and preliminarily choosing the angles \(\alpha\) and \(\beta\), from equations 8 and 9 we obtain \(U_1\) and \(V_1\).

Let us now consider how to calculate \(\varphi\) more accurately. For this purpose we return once more to equation (5) and consider in more detail the terms of the left-hand side:

\(H\)—the difference in elevations between the upper and lower levels—is a quantity always known.

\(V_0\)—the velocity of approach to the supply canal or pipe; in most cases this velocity is so small that the term

\[ \frac{V_0^2}{2g} \]

may be taken as \(=0\).

\(V_a\)—the velocity of the water in the tailrace canal; the term \(\frac{V_a^2}{2g}\) is also, in most cases, nearly equal to 0, but if there is no draft tube, then \(V_a=V_2\), i.e. the velocity with which the water leaves the runner. In this case the term

\[ \frac{V_a^2}{2g}=\frac{V_2^2}{2g} \]

has a considerable magnitude and must be included in the calculation.

\(K_1\)—represents all losses from the beginning of the supply canal to the inlet edge of the runner; \(K_1\) may be divided into two parts: \(K_1'\), the loss in the pipe or canal, and \(K_1''\), the loss during entry into and passage through the guide vanes and during passage through the annular space between the edges of the guide and runner blades.

The loss \(K_1'\) is determined by the formulas for the flow of water through pipes and canals. If the pipe and canal are short and the velocity in them is small, then \(K_1 \approx 0\).

To determine the loss \(K_1''\), let us once more turn to the drawing. Upon entering the guide apparatus, the water undergoes contraction. During the flow there occurs curvature of the streamlines and friction against the walls of the channel. To determine these losses exactly theoretically is rather difficult.

It is much simpler and more reliable, in the case of calculation for normal operation, to use the results obtained on the basis of experiments with the passages of turbines having stationary vanes.

It has been established by these experiments that the losses in this case are expressed as:

\[ \zeta \frac{V_a^2}{2g} \ldots \ldots (10), \]

where \(V_a\) is the velocity of the water at the outlet end of the guide vane, \(\zeta = 0.1\text{--}0.13\).

Entering the clearance, the water rapidly changes its cross-section to the thickness of the blade, acquiring the velocity \(V_a'\); if the blade pitch is \(t\), and the thickness, measured along the pitch, is \(S\) (see Fig. 7), then the resulting loss is expressed as:

\[ \frac{(V_a - V_a')^2}{2g} = \frac{V_a^2}{2g} \left( 1 - \frac{V_a'}{V_a} \right)^2 = \frac{V_a^2}{2g} \left( 1 - \frac{t - S}{t} \right)^2 = \frac{V_a^2}{2g} \left( \frac{S}{t} \right)^2 \ldots (11). \]

Fig. 7.

We shall neglect the friction of the water in the annular space; with sufficient sharpening of the edge of the runner, the loss upon entry into it is extremely small, and therefore we shall take it equal to 0.

The loss that cannot be neglected is the discharge of water into the gap between the rims. The result of this loss is a decrease in the velocity of the water after the gap, which must be taken into account in the calculation.

Up to now, the losses found by us are expressed by the sum of (10) and (11), i.e.

\[ \left[ \zeta + \left(\frac{S}{t}\right)^2 \right] \frac{V_a^2}{2g} \ldots (12), \]

but, since in the preliminary calculation \(V_1\) is known to us, and not \(V_a\), we shall replace \(V_a\) by \(V_1\) (Fig. 6).

According to what was proved earlier, the motion in the space between the blades takes place along a logarithmic spiral; therefore, for a constant water discharge:

\[ V_a' \cdot 2\pi \cdot r_a \cdot \sin \alpha = V_1 \cdot 2\pi \cdot r_1 \cdot \sin \alpha, \quad \text{that is} \quad \frac{V_a'}{V_1} = \frac{r_1}{r_a} \ldots (13) \]

However, if the entire discharge of water is \(Q\), and \(q\) is lost through the gap, then in fact \(V_1\) is smaller in the ratio \(\dfrac{Q-q}{Q}\), and therefore equation (13) must be corrected, namely:

\[ \frac{V'_a}{V_1}=\frac{r_1}{r_a}\cdot\frac{Q}{Q-q}\ldots\ldots (14) \]

In addition, from the drawing:

\[ \frac{V_a}{V'_a}=\frac{t}{t-s}\ldots\ldots (15) \]

From (15) and (14) we have

\[ V_a=V_1\cdot\frac{r_1}{r_a}\cdot\frac{t}{t-s}\cdot\frac{Q}{Q-q}\ldots\ldots (16) \]

Substituting in equation (12), \(V_a\) by the expression from (16), we finally obtain for the loss \(K=K_1'+K_1''\) the following expression:

\[ K_1=K_1'+\left[\,\xi+\left(\frac{s}{t}\right)^2\right]\cdot \left(\frac{t}{t-s}\right)^2\cdot \left(\frac{r_1}{r_a}\right)^2\cdot \left(\frac{Q}{Q-q}\right)^2\cdot \frac{V_1^2}{2g}\ldots\ldots (17) \]

This formula, derived by us, makes it possible to use the data obtained for turbines of older designs; furthermore, it immediately takes into account the effect of leakage on the velocity. It is practically suitable also for the case when the edges of the blades are almost in contact; at least, in calculation it gives an insignificant difference in comparison with the old ones.

Below we shall indicate how to determine \(q\) theoretically, but it should be noted that such a determination is very inaccurate, and here it is more reliable to use experimental data.

A good Francis turbine under normal operation rarely gives a leakage greater than \(3\%\). This value is usually adopted in calculations.

\(K_2\) is the loss in the runner blades. From experience this loss is:

\[ K_2=\xi_1\frac{W_2^2}{2g}\ldots\ldots (18)\quad \text{here } \xi_1=0.1-0.13 \]

\(W_2\) is always known from the preliminary calculation.

\(K_3\) is the loss in the draft tube, which arises because the water leaves the runner with a greater velocity than the flow in the tube itself; likewise, the water may leave the tube with a greater velocity than the flow in the tailrace; in both the first and second places an impact occurs. In addition, there is also friction against the walls of the tube.

These losses are easy to determine, knowing the velocities. Usually a good draft tube reduces the loss that would occur if there were no draft tube at all and the water left the runner, going out into the air with velocity \(V_2\). For simplicity, however, and also for reliability, it is often assumed that:

\[ K_3=\frac{V_2^2}{2g}. \]

Under this assumption, and also when:

\[ V_0=0 \quad \text{and} \quad V_a=0 \]

equation (5) will take the form:

\[ H-K_1-K_2-\frac{V_2^2}{2g} = \frac{U_1 \cdot V_1 \cdot \mathrm{cs}\,\alpha}{2g} \ldots\ldots (19) \]

which is suitable both for turbines with a draft tube and without one.

Here

\[ H-K_1-K_2-\frac{V_2^2}{2g}=\varphi H \ldots\ldots (20) \]

It remains for us to indicate the path for the theoretical calculation of the discharge of water through the clearance. Adding equations (1) and (3) and making the proper simplifications, we have:

\[ \frac{p_1-p_2}{\gamma} = h_1+h_3-\left(K_1+\frac{V_1^2}{2g}\right) \ldots\ldots (21) \]

but the left-hand side of this equality is that propelling head which forces the water to move through the slot. The corresponding velocity is:

\[ c=\sqrt{\frac{p_1-p_2}{\gamma}\,2g}. \]

Knowing the velocity in the slot and the size of the slot, it is no longer difficult to find the entire lost discharge \(q\):

\[ q=\mu \cdot f \cdot c = \mu \cdot f \cdot \sqrt{\frac{p_1-p_2}{\gamma}\,2g} \ldots\ldots (24) \]

where \(f\) is the area of the slot, and \(\mu\) is, unfortunately, an unknown coefficient of discharge; in any case it may be assumed to be less than \(0.5\).

To reduce this loss, it is useful to introduce as many resistances as possible along the path of the water moving through the slot, whether in the form of turns of the slot, or of sudden expansions and contractions.

Numerical example; practical data and geometrical considerations in constructing runner blades.

Let us calculate a high-speed Francis turbine of 16 horsepower under a head of 2 meters.

As was said earlier, the calculation is carried out for \(3/4\) of the maximum work, i.e. \(3/4 \cdot 16 = 12\) horsepower.

Assuming the overall efficiency \(\eta = 0.75\), we find the discharge:

\[ Q=\frac{12 \cdot 75}{2 \cdot 1000 \cdot 0.75}=0.6 \text{ cubic meters.} \]

The diameter of the draft tube \(D_1\) at the point where the water leaves the runner is taken such that the velocity of water in the tube is

\[ C_0 = 0.224 \text{ to } 0.3 \sqrt{2gH}. \]

For our case,

\[ C_0 = 0.225 \sqrt{2 \cdot 9.81 \cdot 2}=1.4 \text{ meters.} \]

The area of the transverse section of the tube is:

\[ F=\frac{0.60}{1.4}=0.425 \text{ sq. meters.} \]

The internal diameter of the tube is

\[ D_0=\sqrt{\frac{4F}{\pi}}=0.736 \text{ meters.} \]

The outer diameter \(D_1\) of the runner blade edge is taken as \(0.8\text{–}1.4D_0\); the first value corresponds to the highest-speed type, the second to an ordinary Francis turbine of the old type.

We shall take:

\[ D = 1.1D_0 = 0.810 \text{ meters.} \]

Now, by equations (8) and (9), we determine the velocities \(V_1\) and \(U_1\).

Taking \(\varphi = 0.81\), we obtain:

\[ V_1=0.9\sqrt{gH \cdot \frac{\sin \beta}{\sin(\beta-\alpha)\cdot \cos \alpha}} \quad \text{and} \quad U_1=0.9\sqrt{gH \cdot \frac{\sin(\beta-\alpha)}{\sin \beta \cdot \cos \alpha}}. \]

The angle \(\alpha = 18^\circ\text{–}22^\circ\), most often \(\alpha = 20^\circ\), \(\beta = 60^\circ\text{–}135^\circ\); the greater \(\beta\), the higher-speed the turbine.

We take \(\alpha = 20^\circ\), \(\beta = 90^\circ\); then:

\[ V_1 = 4.2 \text{ meters} \quad \text{and} \quad U_1 = 3.8 \text{ meters;} \]

the number of revolutions is

\[ n=\frac{60U_1}{\pi D}=90. \]

Knowing the water discharge \(Q\), the diameter \(D_1\) of the circumference along which the water enters the runner, and the angle \(\alpha\), it is not difficult to find the required height \(b\) of the runner from the equality:

\[ V_1 \cdot \pi \cdot D_1 \cdot \sin \alpha \cdot b = 0.6 \]

whence:

\[ b = 0.167 \text{ meter.} \]

We shall, however, take \(b = 180\) mm, since, first, the receiving edges of the runner may be insufficiently sharpened and therefore constrict the flow; secondly, if the turbine proves to be somewhat more powerful, the trouble will not be great, since it is regulated.

Fig. 8.

For the guide apparatus one may take \((b_1)\) several millimeters smaller.

Let us now take the schematic drawing of this turbine which we wish to construct (Fig. 8).

As was said, for purposes of calculation it is represented as if consisting of separate turbines, superposed one upon another: \(abcd\), \(cdef\), \(efgh\), \(ghik\). Each of these turbines has its own mean line, lying on the mean surface of revolution (shown in the drawing by solid lines).

The oblique outline of the outlet surface relative to the mean line may be replaced by a surface perpendicular to

of them, so that we shall obtain a row of turbines: \(a'b'c'd\), \(c''de'f\), \(e''fg'h\), etc.

Obviously, in the limit, if we were to take an infinitely large number of elementary turbines, the steps on the discharge surface would disappear. We shall now consider the entire turbine as though composed of a series of such stepped ones.

In each such imaginary turbine the admission of water takes place along a cylindrical surface, and the discharge along the surface of a truncated cone. Owing to the considerable narrowness of each turbine, the motion of all the liquid in it may be regarded as identical with the motion of the mean filament.

If the condition is imposed, as is done in all other turbines, that the absolute velocity of the water at exit be perpendicular to the circumference of the discharge edges of the runner, then it is obvious that, in this case, the absolute velocity of the departing water, referred to the mean filament, will lie in one plane with the axis of the turbine and, for each turbine, will be directed along a generatrix of a cone whose vertex is on the axis of the turbine and whose surface is tangent to the surface of revolution formed by the mean line of the turbine. For the first turbine the vertex of this cone is \(f\), and the generatrix \(fm\) is tangent to the mean line \(mn\) at the point \(m\); for another turbine the generatrix is \(pf'\) (tangent to \(pq\)) at the point \(p\), and the vertex is \(f'\), etc.

Each elementary turbine cannot be chosen entirely arbitrarily; but in order to have more grounds for assuming that one turbine will not disturb the motion of another, they are usually calculated so that, for all the turbines, the angles \(\alpha\) and \(\beta\), the velocity \(V_1\), and consequently also \(U_1\), are the same. In order that the efficiency remain the same for each, it is also necessary to take \(V_2\) the same.

Under this assumption each turbine will have its own, differing only in the velocity \(W_2\) and the angle \(\gamma\). It is not difficult to determine these quantities.

Indeed, we shall know the linear velocity of rotation on the outer circumference \(U_1\); from it, knowing the discharge profile of the blade, and consequently also the radius of the discharge circumference, it is easy to determine \(U_2\) from the relation:

\[ \frac{U_1}{U_2}=\frac{r_1}{r_2}. \]

Moreover, \(V_2\) will also be known to us; therefore it is easy to construct

velocity triangles for the discharge of water from the wheel (Fig. 9), whence \(W_2\) and \(\gamma\) will also be determined.

Here, incidentally, we note that the whole velocity triangle must be referred not to the end of the blade, but to the center of gravity of the outlet opening of the channel. This fact has been established both by experiments and by theory. In practice, however, the difference obtained is insignificant, and only in the sense of a somewhat greater passage of water; therefore the said triangle is usually referred to the end of the blade. This inaccuracy, however, is always easily corrected by lengthening the blades somewhat, without changing the calculation.

Fig. 9.

Thus, if (Fig. 10) the velocity \(W_2\) is referred to the end of the blade, then, by lengthening the blades so much that the midline of the transverse section of the lengthened channel lies exactly on the circumference passing through point \(b\), we shall obtain the same \(W_2\) referred to the center of gravity of the section, the blades being lengthened.

Fig. 10.

And so, all the elementary turbines have identical \(\alpha\), \(\beta\), \(V_1\), \(V_2\); moreover, they are taken so that each turbine passes one and the same quantity of water. This latter circumstance requires that all turbines have on the generating cylinder (the outer discharge surface) the same height, and that (Fig. 8) the surfaces of revolution \(a'c'\), \(c''e'\), \(e''g'\), etc. (the discharge surfaces) be equal.

When we have chosen the contour line of the discharge edge and the turbine has been divided into the said elementary turbines, we proceed to the geometrical operations, the purpose of which is to determine the shape of the blade; and if this shape is not satisfactory, the contour line of the edge has to be changed. A satisfactory shape is considered to be one in which

the entire surface of the blade is represented as smooth, without sudden bends and breaks, and such that the paths of all streams (at least the extreme ones) come out approximately of equal length.

And so let Fig. 11 represent to us the outline of the turbine, with the selected profile of the outlet edge of the blade \(acegi\), while

Fig. 11.

the turbine is divided into 4 equal parts. Let us note that we can construct the angle \(\gamma\) not only for the middle streams of the elementary turbine, but

and for any jet whatever, such as, for example, for the jets \(ab\) and \(ki\), moving along the rims of the wheels. Let us construct, for example, this angle for the jet on the surface of revolution \(ab\). The corresponding cone will have vertex \(f^0\) and generator \(f^0a\). As was noted earlier, the velocity \(V_2\) is directed at point \(a\) along the generator \(f^0a\), and the velocity \(U_2\) is perpendicular to the drawing, i.e., tangent to the surface of the cone. \(V_2\) and \(U_2\) are known to us; therefore the velocity triangle is completely determined. Over a small space one may assume, with sufficient accuracy, that the surface of the cone coincides with the velocity triangle and that the ends of the blades are constructed on this same surface of the cone.

We shall develop the surface of the cone onto the plane of the drawing, cutting it along the generator \(f^0a\); on the circumference of the base of the cone, in its shifted position, we shall construct the velocity triangle and the outline of the blade tips according to a logarithmic spiral, as is done in the drawing*). The pitch of the blades must be such that the jet receives the proper direction, i.e., the length of the blade tip outlined along the spiral must be sufficient for directing the jet.

Exactly the same constructions are performed for all the middle jets and for the outermost one on the surface of revolution \(ki\).

Let us now find the horizontal projection of the jets. First of all let us note that the curve chosen by us for the outline of the blade edge may be plane or spatial; suppose it is plane and lies in the axial plane; then on the horizontal plane its projection will be \(a' i'\). At the point \(a'\) let us construct the horizontal projection of the angle \(\gamma\); for this purpose we project the velocity triangle: the projection of the velocity \(V_2\) will be \(a' V'_2\), the velocity \(U_2\) is projected in its natural magnitude, and the angle between these projections remains, as in space, a right angle. Thus the projection of the velocity triangle is found, and the projection of the angle \(\gamma\), \(\gamma'\), likewise. The last element of the horizontal projection of the jet flowing over the surface \(ab\) must be outlined at the angle \(\gamma'\) (and here it is constructed according to a logarithmic spiral); all the remaining projection is chosen arbitrarily, so long as the curve is smooth. The same procedure is followed with the other jets. Thus we obtain a series of curves \(I, II, III, IV, V\), which represent the horizontal projections of the jets flowing over the surfaces of revolution.

Thus, the horizontal projections are known to us; from them one can find the vertical surfaces of the jets. However, since the jets

*) In practice, the arc of a logarithmic spiral is replaced by the arc of a suitably chosen circle.

are curves in space, and two projections are insufficient for determining their form, one proceeds otherwise: usually the vertical projection is not constructed, but the horizontal projection of the blade is cut by a series of vertical axial planes \(OM\), \(ON\), \(OP\), etc., and the points of intersection of these planes with the streamlines are found; these points are transferred to the vertical projection, but in doing so each point of intersection is conventionally turned about the axis until it coincides with the plane of the drawing.

In this way we obtain a series of curves \(MM'\), \(NN'\), \(PP'\), etc., lying in the plane of the drawing and representing sections of the blade surface taken by axial planes; if these curves are smooth and the distance between them varies with a certain regularity, then this indicates the smoothness of the blade surface. Consequently the outline curve of the blade, and the horizontal projection of the streams, have been taken correctly; otherwise corrections must be made. In the numerical example taken by us earlier, all that has been said will be explained better*).

Taking \(MN\) as the axis of the turbine (see drawing 12, separate table), draw to it the perpendicular \(NS\), and from point \(N\) lay off the radius of the draft tube

\[ r_0=\frac{D_0}{2}=NO=368\ \mathrm{mm} \]

and

\[ r_1=\frac{D_1}{2}=NP=405\ \mathrm{mm}. \]

Through point \(a\), situated \(5\ \mathrm{mm}\) from \(P\), draw a line parallel to the axis, and on it lay off \(aF=aN\); from point \(F\) downward lay off \(b=180\ \mathrm{mm}\), and divide this segment into 10 equal parts.

Divide the radius \(r_1=NO\) likewise into 10 such parts, so that the areas of the rings formed by the rotation of these parts about the axis \(NM\) would be equal.

Through the points of division of the turbine height draw horizontal straight lines, and through the points of division of \(NO\) draw vertical straight lines. Then connect the lines issuing from the corresponding vertical and horizontal divisions by arcs of circles whose centers lie on the vertical \(aF\). In this way we obtain 10 layers of revolution superposed one upon another. The outermost lines \(AF\) and \(OR\) shall be taken as the outlines of the rims of the wheel, and the curve \(AR\), chosen by us, as the outline of the profile of the outlet surface. Then the parts of the layers cut off by the surface of revolution of the curve \(AR\) may be regarded as elementary turbines.

Each two adjacent layers shall be considered as one turbine, and the surface separating them as the surface along which the mean stream of this turbine moves. We thus obtain five

*) All constructions in designing blades are carried out at full scale.

elementary turbines (in the drawing separated by dotted lines) \(a\beta\beta'\), \(g\delta\delta'\), etc.

Only the indicated method of composing elementary turbines belongs to Engineer Baashuus*) and gives very satisfactory approximate results, although the same can also be achieved by other means.

When the elementary turbines have been obtained in one way or another, it is necessary to make sure whether they all have equal discharge surfaces. For this purpose we draw, to the profiles of the surface of the mean streamline, the perpendicular straight lines \(a\beta\), \(g\delta\), \(g\zeta\), etc., and determine whether the surfaces described by these segments are equal to one another.

When Baashuus’s method is applied, the difference is usually extremely small, and the inaccuracy can be corrected by shifting the centers of the circles. Thus in our case it was necessary to shift the centers 3 to \(3'\), 4—\(4'\), and \(3/4\) to \(3/4'\)**). In this way turbines were obtained with completely equal inlet and outlet surfaces.

With regard to the choice of the curve outlining the outlet edge, which must lie on the surface of revolution of the curve \(AR\), it should be noted that it may be either plane or of double curvature.

In many first-class turbine-building works, for simplicity, it is always taken as plane, lying either in an axial plane (medium reaction) or in a plane parallel to the axis (strong reaction). The principal condition in choosing this curve is to obtain a smooth blade surface and, for all the turbine streamlines—or at least for the outermost ones—paths of almost equal length.

We earlier had the velocity in the draft tube \(C_0 = 1.4\) meters; the velocity \(V_2\), however, must be somewhat greater, since at the place of water discharge the blades obstruct the free space of the tube.

The obstruction of the draft tube by the blades amounts, for iron blades, to 15%, and for cast-iron blades to 20—25%.

It is convenient to take the number of blades from the following table.

	400	600	1000	1500	3000
Outer diameter of the runner \(D\), in mm =	400	600	1000	1500	3000
Number of runner blades	12	14	18	24	32
Number of guide-vane blades	14	16	20	26	34

*) Baashuus Zeitschrift D. V. D. Jn. 1901; Seite 1902).

**) Fractions denote the centers of the dotted paths.

Assuming the blades to be iron, we shall find approximately \(V_2 = 1.6\) meters. Since \(U_1\), and consequently also \(U_2\), are known to us for each point from the relation

\[ \frac{U_1}{U_2} = \frac{r_1}{r_2}, \]

we are now in a position to construct, for each stream filament, the exit velocity triangle.

Let us take the filament on the surface \(6A\); the corresponding cone mentioned above will have the generatrix \(AM\) and the vertex \(M\). \(AA_1\) is part of the developed base; on this development the velocity triangles are constructed and the ends of the blades are laid out.

A similar construction must be made for all five middle filaments and for the two extreme ones on the wheel rims.

We note that the lower elements of the turbine in our drawing have the middle filaments 33, 22, 11, and, lying in their lower part on a cylindrical surface, like the filaments \(oR\); therefore the ends of the filaments will have to be drawn on a cylindrical surface, i.e. as in an axial turbine.

To avoid loss of time, one should try, as far as possible, to construct the greater part of the blade, at least the whole part that lies on the cylindrical surface. In the event of an unsuccessful choice of the curve outlining the discharge edge, the blade profile may acquire such a form as will indicate quite clearly the need to change the curve, and thus further useless expenditure of labor will be reduced.

From the constructed angles of the discharge edges, their horizontal projections are laid out as described earlier, and the horizontal projections of the filaments themselves are constructed, the missing portions being drawn by hand. These projections are \(O'R'\), \(O'F'\), \(O'E\), \(OD'\), \(O'C'\), \(O'B'\), \(O'A'\). Then, on the horizontal projection, the cutting axial planes \(N'I\), \(N'II\), \(N'III\), \(N'IV\), \(N'V\) are taken.

The intersections of the blade surface with these planes are transferred onto the vertical projection by rotating these planes until they coincide with the plane of the drawing. The curves \(II\), \(IIII\), \(IIIIII\), \(IVIV\), \(VV\) are obtained; they are sufficiently smooth and are arranged with a certain regularity. If this were not so and the curves showed abrupt bends, the profile would have to be changed.

To manufacture the blade in practice, a model of it must be made. The most widespread method (though by no means the only one) consists in passing a number of secant planes perpendicular to the turbine axis, \(k, h, g, f, e, d\), etc. (see drawing 12, vertical projection); these planes are drawn through the lower part of the blade,

where the curvature is considerable, at a distance of 15 mm; in the upper part, at a distance of 30 mm; then the intersection of these planes with the surface of the blade is found.

To find the lines of intersection, the points of intersection of the streamlets and of lines I I, II II, III III, IV IV, V V with the planes taken are marked on the horizontal projection. By connecting the corresponding points, we obtain in plan a series of curves (these curves are denoted here by the same letters as the plane forming them, with the addition of a prime mark), which are nothing other than a series of topographic horizontals.

Now boards are taken of such thickness as the distance between the secant planes; on each board one horizontal is drawn and cut out along it with a saw; then one board is laid upon another in the order of succession of the horizontals and they are glued together.

To obtain a model of the blade surface, it remains only to cut off the projecting corners and, for accuracy, to check the angles of elementary turbines with templates cut from sheet iron.

Fig. 15.

Fig. 13 represents a model of the blade surface for demonstration. The wires are the streamlets; the sheets of cardboard are the secant horizontal planes; they are all cut along the lines of intersection with the blade surface.

The blades of the guide apparatus are usually constructed so that their outlet edges are outlined along a logarithmic spiral, by which the water is compelled to move...

along a logarithmic spiral; but we know that once such a motion has been established at one circumference of the ring, the water will have a natural tendency to preserve it throughout the ring; the character of this motion also extends to the whole vessel in which the guide apparatus stands. The blades work best if they do not hinder this natural motion; in this case they may be made comparatively short.

Experience shows that with such blades the loss obtained is smaller than with blades of the old systems, when it was supposed that the entry of water into the guide apparatus should be directed along the radius. Guide blades of the newest type with an axis of rotation are shown in drawing 12.

According to the preceding calculation, the angle of their inclination to the circumference of the outlet edges is \(\alpha = 20^\circ\).

Between the runner and the ends of the guide blades a considerable gap must be left, so as to have sufficient room to turn the blades when working at full power.

Verification of the Designed Turbine.

Thus, the turbine has been calculated and everything relating to the construction of the runner blades has been drawn out; for the calculation we set in advance the hydraulic coefficient of useful action \(\varphi\). It is now necessary to calculate this coefficient by more exact formulas and to make sure that it has not been taken too large. For this, take equality (20)

\[ \varphi H = H - \left(K_1 + K_2 + \frac{V_2^2}{2g}\right) \]

We shall find the right-hand side more exactly, determining separately all the terms entering into it:

In our case \(H = 2\) meters;

\[ K_1 = K_1' + K_1'' = 0 + K_1'' = \left(0.1 + \left(\frac{22.5}{172.5}\right)^2 \cdot \left(\frac{172.5}{150}\right)^2 \cdot \left(\frac{405}{440}\right)^2 \cdot \left(\frac{100}{97}\right)^2 \cdot \frac{4.2^2}{2 \cdot 9.81} \right) = 0.134 \]

meters (see equation 17).

To determine \(K_2\), the velocity \(W_2^2\) enters (equation 18), different for each elementary turbine; to obtain an average value, the values of \(W_2^2\) for the five middle jets should be added and divided by 5; then we obtain the mean value and introduce it into expression (18). All velocities \(W_2\) we take from the outlet velocity triangles for the middle jets. However, for greater reliability—

... the velocities may also be taken as the mean and the greatest of \(W_2 = 4\); we then obtain:

\[ K_2 = 0.1 \cdot \frac{4^2}{2 \cdot 9.81} = 0.08 \]

\[ \text{Loss } K_3 = \frac{V_2^2}{2g} = \frac{1.6^2}{2 \cdot 9.81} = 0.13 \]

and therefore

\[ \varphi \cdot H = 2 - (0.134 + 0.08 + 0.13) = 1.656, \]

whence

\[ \varphi = \frac{1.656}{H} = \frac{1.656}{2} = 0.828. \]

This value for \(\varphi\) has been derived with great reliability; we, however, assumed \(\varphi = 0.81\), i.e., still smaller, and therefore we may be confident of the good action of the turbine under its normal operation.

Action of the Turbine at Various Openings of the Guide Channels

We still have an unresolved question concerning the action of the turbine at various rotations of the blades.

Since the turbine must ordinarily operate at the same number of revolutions under different loads, I have investigated this question for the case of a constant, normal number of revolutions.

When the opening of the channels varies, the discharge obviously changes; \(W_2\) also changes in proportion to the change in discharge. If, however, the number of revolutions remains the same, the normality of the outflow from the runner is disturbed. Moreover, water entering at another angle and with another velocity \(V_1\) must meet an impact upon entering the runner.

All the quantities of which we speak here vary differently for every elementary turbine, and the equality of the outlet and discharge velocities for all elementary turbines, as existed earlier under normal action, is no longer present.

As a result of the abnormality of the outflow of water into the draft tube, a certain spiral-like motion is obtained, and the loss from this increases greatly. (In the opinion of some designers, draft tubes divided by axial partitions, or flat-shaped tubes, as is sometimes the case with concrete tubes, act better in this case.) This loss, however, may, with great reliability as before, be taken as

\[ = \frac{V_2^2}{2g}, \]

where \(V_2\) will, of course, already have its own value, specific to each case.

In the case of abnormal outflow and impact entry, the work \(L\) transmitted by one kilogram of water is expressed, as is known, by the formula

\[ L=\frac{1}{g}\left(U_1 \cdot V_1 \cdot \cos\alpha - U_2^2 + U_2 \cdot W_2 \cdot \cos\gamma\right). \ldots (23) \]

This work must be equal to the head after subtracting all losses. The losses consist of (Fig. 6):

1) The losses in the section \(AI\) (see the drawing), which we expressed by \(K_1\).

2) The loss due to impact upon entry into the runner; previously this loss was absent during normal operation of the turbine.

3) The loss in passing through the runner along the path \(I\,II\); we denote this loss by \(K_2\).

4) And the loss in the draft tube, according to the assumption,

\[ =\frac{V_2^2}{2g}. \]

For determining \(K_1\), expression (17) is used; the coefficient in the present case will be different than before. This coefficient chiefly characterizes the friction in the guide channel, which has its greatest value at the place of greatest velocity, i.e. at the outlet slot. With small changes of the channel it may be assumed that this coefficient varies directly in proportion to the quantity \(\frac{P}{F}\), where \(P\) is the perimeter of the outlet opening, and \(F\) is the outlet area. Since for each opening of the channel \(\frac{P}{F}\) can be found, \(\zeta\) is thereby determined. However, at very small openings of the guide vanes the channel, resembling a nozzle, should give considerably more favorable results, and moreover \(\zeta\) is very small, as in a conical nozzle.

The ratio \(\frac{q}{Q}\) is greater for small discharges than for large ones.

In fact: the velocity of flow through the slot depends on the propelling pressure determined by equation (21). If it is assumed that this propelling pressure remains constant for all openings of the channels (in reality they vary somewhat), then the discharge \(q\) must also be constant; but \(Q\) in this case of course changes, hence the ratio \(\frac{q}{Q}\) must increase at small discharges.

Using the considerations set forth, and taking the missing quantities from the drawing, we are able to determine, by expression (17), \(K_1\) for every opening of the guide channels.

To determine the shock upon entry into the runner, let us take (drawing 14). Let \(U_1\) be the linear velocity of rotation, always constant; \(V_1\) the absolute velocity of entry under normal operation; then the component \(W_1\) is directed along the blade. Let us then, instead of the velocity \(V_1\), have \(V_1'\); this velocity will be resolved into \(U_1\) and the relative \(\overline{U_1 V_1'}\); the relative \(\overline{U_1 V_1'}\), in turn, will be resolved into \(U_1N\) along the blade and \(\overline{NV_1'}\) perpendicular to it; this latter is lost by shock:

Drawing 14.

\[ \text{loss}=\frac{\overline{NV_1'}^{\,2}}{2}. \]

The segment \(\overline{NV_1'}\) can be determined by construction, and consequently this loss can also be found.

The loss \(K_2\) is determined as a function of \(W_2\), but if the new \(V_1\) is known to us, then the new \(Q\) is also known, and since the passage of the runner has not changed, \(W_2\) changes in proportion to \(Q\), i.e. it can always be determined. The coefficient \(\zeta_1\), as depending only on the form of the passage, remains unchanged.

Now we turn to the exit triangle (see drawing 15).

Drawing 15.

The velocity \(U_2\) is constant, \(W_2\) is the normal value of the velocity at which \(V_2\) is perpendicular to the circumference of the discharge edge. If now instead of \(W_2\) we obtain \(W_2'\), then the velocity \(V_2\) becomes \(\overline{U_2W_2'}\).

All the losses are thus represented as:

\[ K_1+K_2+\frac{\overline{NV_1'}^{\,2}}{2g}+\frac{\overline{U_2W_2'}^{\,2}}{2g} \]

and equation 23 will take the form:

\[ \zeta H = \frac{1}{g}\left(V_1\cdot U_1\cdot \cos\alpha - U_2^2 + U_2W_2\cdot \cos\gamma\right) = H-\left(K_1+K_2+ \frac{\overline{NV_1'}^{\,2}}{2g} + \frac{\overline{U_2W_2'}^{\,2}}{2g} \right) \ldots\ldots 24 \]

to this equation there must also be added an equation for finding the discharge:

\[ Q=\pi D_1 \cdot b \cdot V_1 \cdot \sin \alpha \ldots \tag{25} \]

Having determined from (24) the hydraulic efficiency, and from (25) the discharge \(Q\), it is easy to find the entire power output of the turbine. The difficulty in solving equation (24) consists in the fact that, in addition to the new unknown \(V_1\), it contains expressions representing explicit and implicit functions of \(V_1\), some of which we found only by construction. The solution must therefore be carried out by interpolation, through successive selection of \(V_1\).

Moreover, since, as noted earlier, all elementary turbines each have their own particular velocity \(V_1\), equation (24) would have to be applied to all elementary turbines, and from them the mean value for \(V_1\) taken. For simplicity, however, it is sufficient to restrict oneself to only the two extreme elementary turbines, the upper and the lower, and to take their mean value.

Numerical example.

As an example, let us determine the efficiency and the total power of the turbine at full opening of the blades, when the angle \(\alpha=30^\circ\) (drawing 12, separate plate). For the first approximation we take \(V_1=4\) meters; for this value we determine separately the expression:

\[ \frac{1}{g}\left(V_1 \cdot U_1 \cdot \cos \alpha - U_2^2 + U_2 \cdot W_2' \cdot \cos \right), \]

then the expression:

\[ H-\left(K_1+K_2+\frac{N V_1'^2}{2g}+\frac{U_2 W_2'^2}{2g}\right) = H-\sum \text{losses}; \]

if these quantities prove equal, then \(V_1\) has been chosen properly; otherwise another \(V_1\) must be taken.

Let us first take the upper turbine \(bA33'\).

Angle \(\alpha=30^\circ\), \(V_1=4\), (by equation 25), \(Q=920\) liters.

Since normally \(Q=600\) liters, then

\[ W_2'=W_2 \cdot \frac{920}{600}=1.53\,W_2. \]

Having constructed the outlet velocity triangles for the upper elementary turbine, we find \(W_2'=3.8\) meters, \(V_2=2.4\) meters.

Taking the inlet velocity triangle, similarly to how this was done in drawing 14, we determine the relative velocity lost by impact \(=0.5\) m; whence

\[ \frac{N V_1'^2}{2g}=0.02. \]

Taking then $\xi$, and the lost discharge $q$ in percent, for greater reliability as before, we obtain:

\[ K_1=\left[0,1+\left(\frac{12}{162}\right)^2\right]\left(\frac{162}{150}\right)^2\cdot \left(\frac{405}{412}\right)^2\cdot \left(\frac{100}{97}\right)^2\cdot \frac{V_1^2}{2g}= \]

\[ =0,12\frac{V_1^2}{2g}=0,1. \]

\[ K_2=0,1\cdot\frac{3,8^2}{2g}=0,075 \]

Loss in the suction tube:

\[ \frac{V_2'^2}{2g}=\frac{2,4^2}{2g}=0,325. \]

And therefore:

\[ H-\sum \text{losses}=H-(0,1+0,075+0,02+0,325)=H-0,520= \]

\[ =2-0,520=1,48. \]

The expression, however:

\[ \frac{1}{g}\left(V_1\cdot U_1\cdot cs\alpha-U^2+U_2W_2\,cs\gamma\right) = \frac{1}{9,81}\left(3,8.4.\,cs\,30^\circ-2^2+\right. \]

\[ \left.2.3,8.\,cs\,37^\circ\right)=1,6. \]

Since the two expressions are not equal, $V_1$ should be changed. Taking $V_1=3,8$ according to equation 25, we obtain $Q=875$ liters.

In this case

\[ H-\sum \text{losses}=1,52 \]

\[ \frac{1}{g}\left(V_1\cdot U_1\cdot cs\alpha-U^2+U_2W_2\,cs\gamma\right)=1,5 \]

Being satisfied with such an approximate equality, from equation (24) we have

\[ \xi H=1,5; \]

whence

\[ \xi=\frac{1,5}{H}=\frac{1,5}{2}=0,75 \]

and thus, for the upper turbine, at full opening we shall have:

\[ \xi=0,75 \quad \text{and} \quad V_1=3,8. \]

In a similar manner, for the lower turbine $[[unclear: symbols]]$ about $R$ we find, at full opening,

\[ \xi=0,77 \qquad V_1=3,6 \]

As average values for the whole turbine, we shall take:

\[ \varphi = 0.76 \qquad V_1 = 3.7. \]

The discharge in this case is \(Q = \pi D_1 \cdot b \cdot V_1 \cdot \sin 30^\circ = 0.854\) cubic meters.
For normal operation of the turbine we took, at full efficiency coefficient \(\eta = 0.75\), the hydraulic efficiency coefficient \(\varphi = 0.81\).

If now, with \(\varphi = 0.76\), we take \(\eta = 0.7\), then at a discharge of 854 liters we shall obtain work on the turbine shaft equal to

\[ \frac{854 \cdot 2}{75} \cdot 0.7 = \infty 16 \text{ h.p.} \]

Practical method for determining the opening of the blades at different turbine operations.

The method just described for determining turbine operation at different openings of the blades is somewhat laborious.

The labor can be greatly simplified if one uses data obtained by experiment. On the basis of experiments one may compile*) the following table for turbines calculated according to our type.

Water discharge through the turbine wheel.	maxim.	\(^{3}/_{4}\)	\(^{1}/_{2}\)	\(^{1}/_{4}\)
Full efficiency coefficient \(\eta\). .	0.7—0.75	0.8—0.75	0.7—0.75	0.65—0.7
Hydraulic efficiency coefficient \(\varphi\). .	0.75—0.8	0.85—0.8	0.8—0.75	0.75—0.7

We shall write equation 24 in the form:

\[ \varphi H = \frac{1}{g}\left(V_1 \cdot U_1 \cdot \cos \alpha - U_2^2 + U_2 \cdot W_2 \cdot \cos \gamma\right) \ldots (26) \]

here \(\varphi\) is chosen from the table, \(W_2\) is determined from the new discharge:

\[ Q = \frac{N \cdot 75}{H \cdot \eta} \ldots (27) \]

substituting the found \(W_2\) into equation (26), we find \((V_1 \cdot \cos \alpha)\); but on the other hand:

\[ Q = \pi D_1 \cdot b \cdot V_1 \cdot \sin \alpha \ldots (28) \]

whence we find \(V_1 \cdot \sin \alpha\);

*) (W. Müller. Die Francis-Turbinen. F. Prasil, Vergleichende Untersuchungen an Reaktions Niederdruckturbinen).

The ratio \(\dfrac{V_1 \cdot \operatorname{Sn}\alpha}{V_1 \cdot \operatorname{cs}\alpha} = \operatorname{tg}\alpha\) is the tangent of the angle at which the blade must be set for the \(\varphi\) we have adopted, or, according to the table, for the corresponding water discharge.

Numerical example.

Let us return once more to our example (Fig. 12).

With full opening \(\eta = 0.7\); \(\varphi = 0.75\); \(N = 16\) horsepower. By equation (27): \(Q = 854\) liters;

previously \(Q = 600\) liters; therefore \(W_2\) has now become greater by

\[ \frac{854}{600} = 1.43 \text{ times.} \]

For the upper elementary turbine we have:	For the lower elementary turbine.
\(U_1 = 3.8\)	\(U_1 = 3.8\)
\(U_2 = 2\)	\(U_2 = 3.4\)
\(\operatorname{Sin}\gamma = 0.8\)	\(\operatorname{Sn}\gamma = 0.9\)
\(W_2 = 2.5 \cdot 1.43 = 3.55\)	\(W_2 = 3.7 \cdot 1.43 = 5.25\)
from equation (26):	from eq. (26):
\(V_1 \cdot \operatorname{cs}\alpha = 3.6\)	\(V_1 \cdot \operatorname{cs}\alpha = 3.\)

From equation (28) \(V_1 \cdot \operatorname{Sn}\alpha = 1.87\).

For the first elementary turbine:

\[ \operatorname{tg}\alpha = 0.52 \]

\[ \alpha = 27^\circ 30', \]

For the second elementary turbine.

\[ \operatorname{tg}\alpha = 0.62 \]

\[ \alpha = 32^\circ \]

The mean value for \(\alpha\) is

\[ \alpha = \frac{32^\circ + 27^\circ 30'}{2} = 29^\circ 45'. \]

Elements influencing the high-speed performance of the turbine.

In concluding the present article, I consider it appropriate also to indicate the means of attaining maximum speed, which is very important for turbines operating under a small head:

1) Increasing the angle \(\beta\) entails an increase in the reaction of the turbine, and consequently an increase in speed.

Fig. 16.

\[ \frac{1}{8}\ \text{full size.} \]

\[ \mathrm{HP} = 700;\quad n = 500 \]

\[ H = 20\ \text{meters.} \]

The upper limit for existing Francis turbines is \(\beta = 135^\circ\). With an increase in the reaction of the turbine, the leakage through the clearance increases. Therefore, in this case it is especially necessary to ensure that the gap between the rims has enlargements in the form of grooves that reduce this loss.

Fig. 16 shows blades with a very large angle \(\beta\).

2) Another means is to reduce the outside diameter of the runner: it is taken up to \(0.8D_0\) (Fig. 16).

In this case the part of the lower rim of the turbine is made conical.

The water in this part moves along the radii of a sphere whose center is the apex of the cone \(O\). The velocities for all points of each spherical surface of one radius are equal.

For the design of such a turbine (Fig. 16), in order to divide the turbine into layers equal in cross-section, the part \(BC\) of the radius perpendicular to the axis is divided into segments as indicated above; in the same part where the motion takes place along the radii of a sphere, the surface of the sphere must be divided into belts with equal surface area, which is very easy if one recalls that the surfaces of belts of equal height are equal. If attention is paid to that velocity of the water,

Fig. 17.

\[ \mathrm{HP} = 110,\quad n = 180; \]

\[ H = 7.3\ \text{m}; \]

\[ \frac{1}{8}\ \text{HP}, \]

which lies in the axial plane, then, for the type just indicated, it is difficult to avoid an increase of this velocity on approaching the narrowest place, \(CB\), and then again its decrease.

In a turbine designed for the greatest efficiency, it is desirable that this velocity should steadily decrease along the entire path in the runner.

A double change in velocity inevitably entails a considerable loss. This loss, when the outside diameter is reduced too much, is unavoidable.

Drawing 18.

3) A reduction in the diameter of the draft tube, and consequently an increase of \(V_2\), also entails an increase in the number of revolutions, but at the same time a greater loss in the draft tube.

We see, therefore, that all three methods entail an increase in losses. Of course, in extreme cases all three are applied simultaneously; it must be remembered, however, that the efficiency will thereby be reduced.

According to the statement of the chief engineer at one of the outstanding factories in Switzerland, which tests turbines in its own laboratory, for the highest-speed types the efficiency should be taken as 7% less than the ordinary value.

Drawing 17 represents a turbine runner with a diameter ratio

\[ \frac{D_0}{D_1} = 1. \]

The only means that does not reduce the efficiency and increases the number of revolutions is to mount on the shaft, instead of one turbine, two, three, etc., with a power 2–3 times, etc., smaller, although it is clear that in this case the cost of the installation increases considerably.

Such designs are becoming ever more widespread; the scheme of such an arrangement is shown in Drawing 18.

It is evident that even the highest-speed types may be arranged according to the indicated scheme.

Fig. 12.

Annotations visible in the drawing:

• Inner edge of the blade on surface \(SB\).
• Inner edge of the blade on the surface.
• Outer edge of the blade on the surface.
• Inner edge of the blade on surface \(OR\).

Visible dimensional and symbolic markings include: \(b=108\), \(b=85\), \(b=178\), \(6a\), \(g=85\), \(h=36\), \(h=76\), \(h=30\), and point/curve labels \(A\), \(B\), \(C\), \(D\), \(E\), \(F\), \(G\), \(O\), \(R\), \(S\), \(U\), \(V\), \(W\), \(a\), \(b\), \(c\), \(d\), \(e\), \(f\), \(g\), \(h\).

Handwritten plate / design drawing.

Project of a pressure turbine, reversible,
\(H = 18\) meters; \(D_1 = 2\) meters; \(n = 90\); \(Q_{\max} = 850\) liters.
Leitner, V.M.V.

Labels visible on the drawing:

• blade 15
• blade 16
• development of the curved blades on surface 15
• development of the curved blades on surface 16
• development of the curved blades on surface \(2E\)
• development of the curved blades on surface \(3D\)
• [[unclear: note along upper right edge, partly cut off]]
• \(y_0 = 36^\circ\)
• \(558\)
• \(492\)
• \(472.5\)
• \(292\)
• \(300\)
• \(u_2 = 3.5\)
• \(v_2 = 1.6\)
• \(46\)
• \(u_3 = 3.5\)
• \(v_3 = 1.6\)
• \(u_2 = 3.5\)
• \(v_2 = 1.6\)
• \(105\)
• \(u_3 = 3\)
• \(v_3 = 1.6\)
• \(120\)
• \(u_2 = 3.5\)
• \(v_2 = 1.6\)
• \(90\)
• \(165\)