PROCEEDINGS

of the Tomsk Technological Institute

of Emperor Nicholas II

Vol. 18. 1910. No. 2.

VI.

A. A. Shutkov.

REVIEW OF THE WORKS BY MECHANICAL ENGINEER A. MILOVICH

“Designing Francis Turbine Blades by Prof. Pfarr’s Method” and “An Attempt at a Theory of the Draft Tube.”

1–6.

Review of the Works by Mechanical Engineer A. Milovich

“Designing Francis Turbine Blades by Prof. Pfarr’s Method” and “An Attempt at a Theory of the Draft Tube,” published in the Bulletins of the Polytechnic Society for 1906 and 1907

The Francis turbine, by virtue of its valuable qualities—high efficiency, universality, compactness, speed, and controllability—has brought about a revolution in turbine construction, displaced all types that preceded it, and made possible the wide spread of hydroelectric installations that has had to be noted in recent years. The Francis turbine owes its finished modern form and its dissemination chiefly to its famous designer, Pfarr, formerly employed at the first-class Voith works in Heidenheim, who was the first in Europe to begin designing them. At the same time, the theory of hydraulic motors that preceded the appearance of the Francis turbine could not be applied directly to its calculation, since the theory of the mean stream—and one, moreover, considered not along the entire path, but only at three points: the outlet from the guide wheel, the inlet to and the outlet from the runner—was clearly inapplicable because of the excessively sharp difference in the conditions of motion of the boundary streams. There was, moreover, no more rational theory for the calculation; theory usually does not precede, but merely completes, the development of a rational design. Therefore, calculation before the appearance of a more rational method still rests upon the mean stream, but is applied to elementary turbines obtained by dividing the entire space between the rims of the runner and satisfying the passage of a specified quantity of liquid under identical conditions of delivered work. Since all streams have identical conditions upon entering the runner, but different ones at exit, the velocity at the outlet from the runner is made the dominant velocity in the calculation; with respect to it various assumptions were made in calculation, and there also exists a series of theoretical investigations on this question. As to what profile should be given to the blades, this remained undeveloped; the old theory concerned itself with preserving the ang—

...the blades and elimination of jet contraction, outlining the ends by helical lines and by involutes, and prescribed only, for outlining, a smooth transition of angles and a short path in order to reduce friction. All this, in the end, could not satisfy a theory seeking a more rational foundation. Hydrodynamics, which had taken almost no part in the theory of hydraulic motors until the end of the nineteenth century, did serve as the basis for the modern rational theory of hydraulic motors—a theory not yet complete, but one that has already done much in this direction. Its creator is Professor Prášil of the Zurich Polytechnic. He applied cylindrical coordinates to the study of the motion of a particle relative to a rotating system, taking gravity, turning acceleration, and centrifugal acceleration as external forces.

Moreover, he began to seek not the law of motion from the shape of the wall, but, conversely, by prescribing one or another condition of the motion, to seek the form of the flow, the surface of the velocity potential, the surface of equal velocities, and the surface of levels. In his works, “Über Flüssigkeitsbewegungen in Rotationshohlräumen” and “Die Bestimmung der Kranzprofil und der Schaufelformen für Turbinen und Kreiselpumpen,” published in Schweizerische Bauzeitung for 1903 and 1906, a considerable number of examples are examined. In this same field, almost simultaneously, N. Lorenz also worked; in his articles he likewise used cylindrical coordinates, but considered not relative motion, but absolute motion, giving to the external forces the so-called forced acceleration from the blade profile, taking them as infinitely thin and their number as infinitely large, i.e. assuming the conditions on parallel circles to be identical.* In the same journal there appeared, in 1907, a polemic between N. Lorenz and Prášil, who, in the end, did not come to agreement. This indicates why Prášil’s theory is so poorly assimilated by his students, as Mr. Milovich mentions, and also by Mr. Krzhizhanovsky in his report on his foreign mission. From the studies of both authors, Prášil and N. Lorenz, one may arrive at identical conclusions, assuming a steady motion with a velocity potential that performs no work; and such motion of the flow leads to the determination either of the form of the draft tube, or of the ends of the blades, serving to impart to the jets identical directions with respect to the axis of rotation.

* N. Lorenz’s articles, “Neue Grundlagen der Turbinentheorie,” were published in Zeitschrift für das gesammte Turbinenwesen for 1905 and 1906.

At a time when in the West a revolution was taking place in views on the theory of hydraulic motors, and the type of the Francis turbine was being developed and brought to perfection, Russia continued to study and make use of the old theory of obsolete turbine types. In 1900, at the industrial exhibition in Riga, the Pirwitz firm, which by that time had built 343 turbines, exhibited—as exemplary—a Jonval turbine, in which the blades were directed at an angle to the radius, while Francis turbines were not mentioned. In 1901, when I visited many of the best German and Swiss turbine-building works, I encountered the old type of Jonval and Girard turbines only at the Queva works in Erfurt, manufactured specially for Russia. The same was true of the courses then being taught: much space was devoted to the study of obsolete types, and almost no information was given on Francis turbines. Recently the firm Voith has built, not far from Essentuki, a hydraulic installation with a headrace canal about 1 verst long. Two horizontal spiral Francis turbines have been installed, each of 500 horsepower, with automatic regulation by a hydraulic servomotor; the turbines are directly coupled to a dynamo machine. The installation is worthy of attention and should arouse interest and imitation.

The author of the work submitted to me for review by the Mechanical Department of the Tomsk Technological Institute, G. Milovich, is almost the first—if one does not count Professor Astrov’s atlas, which devoted much space to the designs and installations of the Francis turbine—to have published his works devoted to the Francis turbine and to the organ that has acquired particular interest in its application to it: the draft tube. The author attended lectures by such leading figures of modern turbine construction and its ideologists as Pfarr and Prášil, and is their direct follower. His first work, “Designing Francis Turbine Blades by Pfarr’s Method,” as is clear from the title itself, is only a communication filling a gap existing in the courses read in Russia; consequently, in it the author does not claim originality. Nevertheless, thanks to the introduction of Prášil’s method, the author illuminates the flow in the runner of a Francis turbine and provides a means for correcting a runner that departs from the simplest profile.

The author’s fundamental work is the theory of the draft tube, founded on the hydrodynamic theory of the flow of a liquid stream: the author transforms Euler’s equation of motion for a steady-state

states, the equation of continuity and incompressibility, and the components of the vortex from Cartesian coordinates into cylindrical ones. As the external force he takes a force having a potential. The author’s general equations differ from the general equations of Prášil and H. Lorenz, since in the authors named the external forces depend on the velocities and will be potential in the case when there exists a potential function of the velocities.

Thus, the equations with which the author operates differ from Prášil’s equations and include only what may be considered quite indisputable, taking into account the frictions of Messrs. Prášil and H. Lorenz.

Further on, the author considers irrotational motion, for which there exists a velocity potential; then the equation of continuity and incompressibility serves to determine the velocity-potential function, which serves to determine the velocities and, consequently, to determine the lines or streamlines of the flow and the surfaces of equal velocities; from Bernoulli’s equation, which applies here, the distributed pressure in the flow is determined—such is the scheme for using the differential equations of hydrodynamics. Making further assumptions, namely the absence of tangential velocity, and at the same time giving two of the simplest solutions of the differential equations of continuity, the author called the first solution—the motion of water through a cylindrical vertical tube—flow I; the second solution—the motion of a fluid, called flow II and considered by Prášil—is a very interesting case, in which the potential of the velocity function and the streamlines, the surfaces of equal velocity, and the level surfaces are simply expressed and do not coincide. This same case is considered in the author’s first article for outlining the rims of a Francis wheel, where it is used to elucidate the true phenomena in the motion of fluids between the rims of a Francis turbine.

H. Lorenz gives the outline of a Francis turbine likewise in accordance with this motion, but Prášil, from the hydrodynamic point of view, and V. Kaplan, from the design point of view, deny the rationality of such an outline. Further, the author considers, according to the same scheme, plane motion expressed in Cartesian coordinates, and obtains an analogous solution, similar to flow II, called by the author flow III. Then, in investigating these flows, the author determines the parameters as functions of the discharge, determines the geometric locus of the minimum of velocity, as well as the maximum of pressure, and gives a graphical construction of the distribution of pressures and of the energy expended along the vertical axis.

In applying the three flows he has examined to the actual design of draft tubes, the author also indicates the most rational method in each case for discharging the liquid into the basin. For a cylindrical tube the author considers it most rational to have a basin of infinite depth and surface; he relates to this case all the phenomena of the first flow, and examines the graphical distribution of pressure along the vertical axis, taking into account the friction of the liquid against the wall. For a draft tube, according to the second flow, in the so-called turbine of Professor Prášil, the author assumes that the discharge of water should take place into a basin of finite depth and infinite surface; he presents all the results of the investigation of the theory of the second flow, constructs the pressure distribution along the axis of the tube, devotes considerable attention to the transition of the liquid from the draft tube into the basin, and finds it entirely rational to recommend the use of a diffuser in this case. This same second flow the author also applies to the case when the liquid is discharged into a basin bounded on two sides by vertical walls meeting at an angle; this case is the most rational and was developed by the author independently, just as was the case of discharging water into a basin of finite depth with two or three plane vertical walls, based on the study of the third flow proposed by the author himself and proving to be the most rational in application to draft tubes.

The author then proceeds to the study of the flow under the condition that a rotational motion exists in the tube; here he touches upon two cases: the first—when there is no vortex component about the vertical axis; this case was also considered by Prášil, but the author analyzes it in somewhat greater detail; and the second case, that of the helical motion of the liquid, when the vortex lines coincide with the streamlines. The author considers the second case in Cartesian coordinates, under the condition that the velocity and the energy are independent of the coordinate of the vertical axis \(z\), and that the ratio of the vortex components to the corresponding velocities is constant; he obtains differential equations of the velocities as functions of the coordinates \(x\) and \(y\). Passing to cylindrical coordinates and taking the radial velocity to be zero, he arrives at two equations for determining the velocities along the tangent and the axis \(Z\) as functions of the radius \(r\), solvable with the aid of Bessel functions. On the basis of this investigation the author carries out an analysis of the actual flow in the draft tube and comes to the conclusion that the phenomenon of liquid flow in the draft tube does not

coincides with the established rotation, as Prášil assumes, but is closer to a screw motion; nevertheless, both cases require, in order to avoid the formation of reduced pressure near the axis in the tube, as a result of which a funnel is formed and air separation follows, that the axial space be better filled by some body of rotation.

In concluding this brief exposition of the contents of the article, I consider it my duty to point out that the plane motion, called by the author the third flow, and its application to the draft tube with the use of a diffuser, is, in my opinion, the most rational; and also that the screw motion, by its analogy with the phenomenon of motion in the draft tube, is an independent investigation by the author. In this, as in all the rest, the author has shown his ability to command mathematical analysis excellently, and his broad acquaintance with hydrodynamics and with its most difficult branch—the theory of vortices. As a reproach one must charge the author with his unpopular presentation, which has made his article, so interesting in content, little accessible among the broad circle of engineers.

A. A. Shutkov.

January 24, 1908.