V. N. Belyaev

On the Calculation of High-Pressure Turbines

Tomsk — 1929

V. N. Belyaev.

On the Calculation of High-Pressure Turbines.

General considerations on the calculation of turbines.

In modern power plants, especially those of large scale, the steam turbine occupies a quite firmly established position as a prime mover, owing to the considerable number of revolutions of its shaft and, consequently, to the possibility of direct coupling with a generator or another rotary machine. The great power of a working unit likewise brings it to the fore. In heat-power installations using exhaust heat, an even more important role is assigned to the turbine, owing to the exceptional purity of its exhaust steam. Its importance increases to an even greater degree at those stations where high-pressure steam is used for operation. In the latter case the economic efficiency of the turbine increases. Such a turbine, moreover, by its construction is adapted to high initial steam temperatures, whereby the moisture content of the steam extracted for heat-transport purposes, and the losses connected with this, are reduced.

The natural aim of steam-turbine designers is, under given thermal conditions, to extract as much valuable mechanical energy as possible and thereby to increase the economic effect of the heat-power installation. Along this path turbine builders have two designs at their disposal: the impulse turbine and the reaction turbine.

The impulse type—with large single- or two-stage disks—has a steam velocity of the order of 500–600 m/sec for the former and 700–1000 m/sec for the latter. The reaction turbine operates at low steam velocities and is always cut off in the high-pressure part, i.e., it operates starting from 1–2 at. abs., because its economical operation is possible only with a small clearance (in fractions of a millimeter) between the blade, the casing, and the drum, whereas, owing to uneven temperature deformations of the turbine, the magnitude of this clearance must be not less than 0.5–0.6 mm, and at a temperature of 300°C about 1.5–2 mm; consequently, the leakage of steam through such a clearance in the high-pressure part is large. Unfortunately, replacing the high-pressure part of a reaction turbine with the disks of an impulse turbine, despite the insignificance of the losses through clearances, does not give the desired effect, even at large powers, when the losses due to disk friction against steam and through clearances almost disappear.

As Engineer Lezelya, chief designer of the Brno works, succeeded in establishing, the reason for the low efficiency of these turbines lay somewhat deeper. Namely, in the curved channel between the blades, when steam passes through it at high velocity, so-called hydraulic losses occur, and these constitute a large part of the losses in the turbine.

Without eliminating these losses it did not seem possible to raise the internal efficiency of the turbine above 70%, nor to build a high-pressure turbine, since already at 20 at. abs. pressure, at high steam velocities in the impulse disks and, consequently, at a high peripheral speed, the work of friction of one wheel reaches hundreds of horsepower.

In order to eliminate hydraulic losses when steam passes through the curvilinear channel between the blades, Engineer Lesel carried out more than 300 experiments with various blade profiles. He absolutely denies the possibility of eliminating this undesirable phenomenon by an appropriate selection of the blade profile, and sees the only way out of the situation in reducing the steam velocity. Having conducted tests with a steam velocity varying within the range of \(500 \div 60\ \mathrm{m/sec.}\), he obtained, at a certain steam velocity, an almost complete disappearance of hydraulic losses.

The results of such a test may be represented by a graph (Fig. No. 1), where the steam velocities are plotted along the abscissa axis, and along the ordinate axis—

Fig. No. 1.

—the velocity coefficient \(\varphi\), expressing the magnitude of the ratio of the actual velocity of steam discharge from the channel to the theoretically possible one. As is seen from the drawing, the coefficient \(\varphi\) has two maxima, the second of which, within the velocity range \(c = 500 \div 700\ \mathrm{m/sec.}\), was known earlier and served as the basis for the calculation of impulse turbines with large disks. Lesel’s experiments were subsequently carried out at various initial steam pressures, and a displacement of the maximum of the coefficient \(\varphi\) along the abscissa axis was observed, depending on the change in the initial pressure. The turbine-building firm Brown-Boveri asserts that it succeeded in obtaining a good efficiency only by the appropriate selection of the blade profile, without, however, providing a sufficiently convincing substantiation of such success.

The author of the present article had the opportunity to hear reports by Engineer Lesel, Engineer Noack—chief designer of the Brown-Boveri firm—and Engineer Baumann—chief designer of the Metropolitan-Vickers firm—in Leningrad in 1925, as well as the discussion that followed them, with the participation of prominent specialists in heat engineering. From the exchange of opinions it became clear that the Brünn plant carries out the calculation of its turbines, depending on the initial steam pressure, approximately according to the following data:

\(p\) at. abs.	\(120—55\)	\(55—35\)	\(35—6\)	\(6\) and \(<\)
\(d\) mm.	300	400	600	800
\(H\) k. cal.	1,8	2,7	4,3	6,5

where \(d\) is the mean diameter of the disk at the blades; \(H\) is the partial heat drop of one stage, in calories; \(p\) is the initial steam pressure before the turbine, in at. abs.

The Vickers works uses stages of 5–6 cal., while the firm Br. Boveri adheres in this matter to a dual point of view, not giving particular preference to small disk drops at high pressure.

As is evident from the table given, its data pertain to very wide pressure limits and, without doubt, cannot claim any special accuracy. In this direction, the carrying out of research work on a broad scale is highly desirable.

Thus, in calculating a high-pressure turbine, it is first of all necessary to strive to eliminate hydraulic losses by a corresponding selection of the stage drop according to the initial steam pressure.

A turbine, in respect of the economy of its operation, is characterized sufficiently satisfactorily by the so-called quality coefficient

\[ Q=\frac{\Sigma u^2}{H^0}, \]

where \(u\) is the peripheral speed of the blades of the rotor, and \(H^0\) is the available heat drop. This coefficient can be increased in two ways: by increasing the number of terms \(\Sigma u^2\), i.e. the number of turbine stages, or the peripheral speed \(u\). The second, without doubt, is the cheaper method and has been applied by the designers of the firm Brown-Boveri; this, however, cannot be regarded as rational, since in this case, under the influence of a considerable centrifugal force

\[ \left(\frac{2\cdot m\cdot u^2}{d_{\mathrm{av}}}\right) \]

the steam, especially high-pressure steam of high specific weight, is thrown from the working passages to the periphery and is compacted, as in a centrifugal pump.

The losses in the effective coefficient of efficiency caused by this, according to the experiments of Lezel, reach, in %:

\(n\), rev./min.	3000	6000
\(\Delta \eta_e\), %	5	12

The table speaks for itself and needs no explanation.

Consequently, the only remaining possibility for increasing the quality coefficient is to increase the number of stages.

Despite its exceptional economy, a multistage high-pressure turbine would have no value if sufficient reliability of its operation were not provided. An important factor in the question of the reliability of turbine operation is the establishment of the magnitude of the clearance between the guide apparatus and the rotor blades.

Tests of the turbine at a steam pressure \(p=20\) at. abs. and a steam temperature \(t=400^\circ\text{C}\) were carried out by the Brno works with variation of this clearance within the limits \(0 \div 3\) mm by longitudinal displacement of the rotor. In the tests, the effective efficiency coefficient of the turbine was measured.

As can be seen from the diagram showing the results of these tests (Fig. No. 2), the increase in efficiency when the normal clearance is reduced from 2.5 mm to 0.2 mm reaches a maximum at a clearance of 0.6 mm; nevertheless, for the necessary reliability of turbine operation, in order to avoid the possible rubbing of the rotor blades against the blades of the guide apparatus, the turbine must operate with a clearance in the range of 1.5–2 mm, even at the cost of some reduction in efficiency.

Fig. No. 2.

To the same extent, the reliability of the turbine in operation is affected by the designer’s choice of the steam temperature before the turbine. Lezel boldly recommends choosing temperatures up to 500° C on the grounds that Martin steel withstands this temperature quite reliably.

Despite such a relatively high steam temperature, at a higher initial pressure, in the low-pressure part the steam acquires sufficient moisture, and in order to avoid erosion of the turbine blades by the impacts of the finest water droplets it is necessary either to use intermediate superheating, which complicates the control of the turbine and increases the cost of the equipment, or to use “Monel” metal (65% nickel and 35% copper) for the manufacture of the blades.

An entirely different point of view in the choice of the initial temperature is held by the Englishman Bauman, who warns against enthusiasm for high steam temperatures. On the basis of scientific experiments by Prof. Lee in Manchester and other experimenters on determining the strength of steel at various temperatures and under variations in the duration of loading, it has been established that steel, when the temperature is raised from 200° C to 300° C, becomes stronger, and then its strength falls to 0.5 of its normal strength at 400° C, to 0.25 at 450° C, and to 0.125 at 500° C; moreover, the material (steel, and especially cast iron) at high temperature exhibits an unpleasant property for the designer: the ability to elongate permanently with the passage of time (“to creep” and “to flow”)—rapidly at high temperature and the corresponding limiting stress, more slowly as the temperature is lowered; finally, the permanent deformation disappears almost completely at a temperature of 400° C. Consequently, the suitability of a material for operation at high temperature must be assessed from the standpoint of its ability to “flow” or “creep,” beginning at one temperature or another, if one does not wish the turbine constructed from it to have to be dismantled and adjusted within a short time from the beginning of its operation. Difficulties with such deformation of the metal make themselves especially felt in the design of the sealing bush on the high-pressure side, since the labyrinth clearances in the axial direction cannot be made as small as would be desirable from the point of view of reducing steam leakage through the gaps.

Thus, one should by no means go above a temperature of 400° C, and 375° C should be considered the normal temperature until new materials, more resistant at high temperature, enter turbine-building practice.

Raising the steam temperature above 400° C creates only a seeming theoretical economy in a condensing turbine; thus, for example, if instead of 400° C one designs and builds a turbine for a steam temperature of 425° C, then theoretically the saving in steam consumption will be about 4.1%; in practice, however, owing to deformation of the metal, the saving at first will fall to 1%, and after several months of operation even this last percent will disappear. In going from 25 at. to a steam pressure of 35 at., at the corresponding temperature, the gain in steam consumption theoretically reaches 4.9%, whereas the increase in losses from leakage and from disk friction against steam reduces the percentage of saving to 1%; moreover, the boiler plant becomes more expensive and the costs of its maintenance increase. Thus one must come to the conclusion that, apparently, for central power stations with a load factor below 50%, it is not advisable to exceed a pressure of 25–30 atm.

When choosing the initial steam pressure for turbines with back pressure, the question is decided in each case individually, and without doubt in favor of increasing the pressure.

Design Features of the Turbine

The standard number of revolutions of the turbine shaft (3000 revolutions per minute), with a low steam velocity and the correspondingly equally low peripheral velocity, gives a comparatively small size for the rotor disk—on average about 600 mm—which in turn greatly reduces the losses from disk friction against steam, which depend on the fifth power of its diameter.

As is known, in an impulse turbine, to which the high-pressure turbine also belongs, the leakage of steam through the clearances between stages is small, since it occurs only between the wall of the guide disk (diaphragm) and the shaft, and is especially small in a high-pressure turbine because of the small heat drop falling to a stage. At the same time, in an impulse turbine the length of the moving blade is always made 2–3 mm greater than the height of the guide passage. Usually the guide apparatuses consist of sheet-steel plates without special machining, cast into a cast-iron disk. Owing to unavoidable casting defects, the passages of the guide apparatuses thus acquire unequal height and radial position (Fig. No. 3); consequently, the moving blades must correspond to the greatest displacement of the edges of the passages from the designed dimension. In this case the steam jet does not fill the passage of the moving blades and acts like an ejector, i.e., it carries along neighboring particles of steam, as is seen in the sketch, and accelerates their motion. As a result there occurs the phenomenon of “suction” of steam from the clearance between the blade tips (the shroud band) and the casing rim and, consequently, a reduction of the velocity coefficient on the moving blades in comparison with the same coefficient for reaction blades.

To eliminate the defect just indicated in the high-pressure turbine, the guide apparatus is made of separate, carefully milled elements, accurately assembled on the guide disk (Drawing No. 4).

Drawing No. 4.

The velocity coefficient of such blades reaches the value

\[ \psi = 0.985 \]

at a steam velocity of about \(100 \ \frac{\mathrm{m}}{\mathrm{sec}}\) (according to experiments of the “Institution of Mechanical Engineers” in 1922–23 in London).

A slight excess of the length of the blade of the working disk over the height of the channel is permitted (one millimeter). The pitch of the guide blades is taken at about 18 mm. To eliminate “suction,” a small degree of reaction is introduced, whereby a slight pressure difference is created on the two sides of the working blade and a relative equilibrium is attained between the leakage through the clearance that is inevitable in this case (Drawing No. 5) and the “suction.” The degree of reaction, according to data of the Brown works, fluctuates from 0.1 to 0.3 in the high-pressure section, depending on the corresponding pressure and specific volume. An exact solution of the question of choosing the degree of reaction has not yet been found or published, and awaits its investigator*).

Drawing No. 5.

Taking into account the complexity of the phenomena occurring as steam flows through the channels of the working blades, and the impossibility of allowing in the calculation for the significance of each of them, one is satisfied with selecting the velocity coefficient depending only on the angle

* Unless one counts Stodola’s experiments with air, the results of which are not applicable to high-pressure steam.

turning along the curves proposed by Wagner (Fig. No. 6), namely along the upper one for reaction blades.

The small diameter of the turbine rotor disk makes it possible in most cases to construct the turbine with full admission of steam, while

\[ \gamma = 180 - (\beta_1 + \beta_2) \]

Fig. No. 6.

a large number of stages in it, owing to the small drop per stage, compels the designer to strive for the greatest possible shortening of it and thereby to utilize the exit velocity of the individual stages. The introduction of a small reaction and the utilization of the exit velocity, besides increasing the efficiency coefficient at the circumference,

\[ \eta_u = f\!\left(\frac{u}{c_1}\right) \]

give the efficiency curve a more extended character along the axis of abscissas at the maximum of its rise than for an ordinary impulse turbine (Fig. No. 7), which means a lower sensitivity of such a turbine, in terms of the specific steam consumption, to variable load conditions.

In designing the rotor blade, the question inevitably arises of choosing the corresponding profile for it. As has already been mentioned, opinions among turbine builders differ on this question: according to Lesel’s experiments, the blade profile has no—

Fig. No. 7.

whose value on the efficiency coefficient of the turbine, provided only that the conditions of shockless entry of the steam onto the blade are fulfilled, will give the corresponding design exit angle of the blade (i.e., the angles \(\beta_1\) and \(\beta_2\)), (Fig. No. 8).

Fig. No. 8.

According to the data of F. Brown-Boveri, on the contrary, the selection of the profile is of great importance in this question.

Since the data of F. Br. Boveri relate only to high steam velocities and comparatively low pressures, while the blades of the Brünn works have shown their good qualities in operation, it is evidently incumbent upon the designer to concentrate his attention on the simplest profile, produced by stamping, with subsequent slight machining of the blade edges by a milling cutter, until the advantage of a more complex profile produced by milling has been scientifically substantiated. The material for such blades is usually sheet stainless steel 1.5–2 mm thick. The blades are covered by a shroud of the same material. The width of the blades is \(< 20\) mm in the high-pressure part of the turbine; since the blade height at high steam pressure is small, the pitch is usually chosen at about 6 mm.

The working blades and guide apparatuses in that part of the turbine where the temperature is high must be enclosed not by a casing, as in ordinary turbines, but by special rings fastened in the casing with a certain degree of freedom (Fig. No. 9) and heated all around by steam. Such a ring expands under high superheat simultaneously with the guide apparatus, thereby avoiding dangerous tem—

Fig. No. 9.

temperature stresses, and, moreover, the magnitude of the clearances installed within it also changes little, which preserves the design advantages of the turbine.

The rotor carrying the blading of the turbine should, in the high-pressure part, be made from a solid-forged blank of ordinary open-hearth steel (Fig. No. 10).

Fig. No. 10.

In manufacturing the rotor it is necessary to anneal it in order to eliminate the residual local stresses in the material that arise during machining. At high temperatures of the order of 400° C its strength is the same as that of nickel steel, while its cost is lower. The working wheel with the sleeve, mounted on the shaft and subjected directly to the action of the hot steam, is heated more than the shaft, since the layer between the shaft and the sleeve, despite its minimal size, serves as thermal insulation (being a poor conductor of heat), as a result of which the disk warps. The designer must not overlook any possibility of shortening the length of the rotor between the bearing centers; otherwise, when calculating for the permissible critical speed of the rotor, the shaft diameter (a stiff shaft) becomes considerable, and the leakage of steam through the clearance of the interstage seal increases; in addition, the absolute elongation of the rotor assumes an undesirable magnitude. To eliminate this drawback, the designer has above all the possibility of installing the aforementioned relatively short (not more than 15 disks) solid-forged rotor without sleeves, of using such a design of sealing bush that would guarantee its insignificant length (for example, the seal of the Vickers works), of not lengthening the bearings without special necessity, of dividing the turbine into several cylinders, placing 15 stages in each, and so forth.

The turbine casing should preferably be made of cast (or even forged, for especially high pressures) steel. In manufacture, repeated annealing is provided after each machining operation up to rough turning.

Choice of the Principle of Regulation of the H.P. Turbine

In regulating a turbine, almost exclusively two methods are used: 1) by throttling the steam admitted to the turbine by means of a throttle valve, and 2) by filling, or changing the quantity of steam admitted to the turbine while its pressure remains unchanged.

In studying regulation from the standpoint of the economy of turbine operation under variable load, the method of Prof. Bér is the most convenient by virtue of its simplicity.

On the basis of his experimental investigations, Prof. Bér showed that, when regulating a steam turbine by throttling, the relation between the steam pressure before the guide apparatus and the quantity of steam flowing through the turbine (through each stage) per unit time may be expressed by a bundle of straight lines (the so-called “characteristics”) passing through the origin (Fig. No. 11). For the turbine under investigation, when the load is reduced, it is sufficient to find on the straight line belonging to the first stage the point corresponding to the steam pressure after the throttle valve, in order, at the points of intersection of the ordinate drawn through this point ($b_1$) with the other inclined lines, to determine those values of steam pressure which will be established before the corresponding stages (points: $b_2$, $b_3$, $b_4$, etc.); and at the point of intersection of this ordinate with the axis of abscissae, to read off the new steam consumption by the turbine per unit time.

Fig. No. 11.

When regulating by admission, the first stage of the turbine, besides its direct purpose of carrying a definite load, plays, with respect to the remaining stages, the role of a throttle valve; i.e., it lowers (when the load is reduced) the steam pressure before the second stage. Consequently, with respect to Bér’s bundle of straight lines, in this case as well, beginning with the second stage, the picture remains the same. The steam pressure before the first stage remains constant, but the quantity of steam is reduced by shutting several nozzles out of operation.

If now, for comparison, admission regulation is applied to the turbine instead of throttling regulation, and the quantity of steam flowing through the turbine per unit time is reduced to the value previously obtained by throttling, it may be observed that the pressure drop of the steam in the first stage (see Fig. No. 12) (segment $b_1 b_2$) under admission regulation is much greater than under throttling, since in the first case the steam pressure before the first stage is constant, whereas the pressure before the second stage in both cases falls equally. Thus it becomes clear that, when the power of the turbine is reduced by admission regulation, the load falling on the first stage increases in comparison with the normal load, owing to the increase of the pressure drop in this stage. From the diagram given it is seen that the pressure drop increases by the amount of the segment $k b_1$ in comparison with the drop of the same stage...

…during throttling by compression (segment \(k b_2\)), and the magnitude of the segment \(k b_1\), for the selected value of the steam consumption \(G\) (with a decrease in load), depends only on the angle of inclination of the “characteristic” of the first stage:

\[ k b_1=(G_n-G)\tan\beta, \]

whereas the magnitude of the pressure drop of this same stage at normal load will be expressed by the length of the segment \(a_1a_2\) and depends both on the angle \(\beta\) and on the angle of inclination of the “characteristic” of the second stage \(\alpha\).

The drop of the first stage of a high-pressure turbine, in comparison with that of an ordinary combined-type turbine, is considerably smaller; this, at one and the same initial pressure, will be expressed on the diagram (Fig. No. 13) by a larger value of the angle of inclination of the “characteristic” of the second stage \(\alpha\) (\(\alpha_v>\alpha_k\)). To determine the qualitative influence of regulation by admission on the first stage of a high-pressure turbine, let us find analytically how, in this case, the ratio noted above changes,

Fig. No. 12.

of the increase in drop to its drop at normal turbine load, if the latter is reduced.

For convenience, we shall put onto a separate drawing (Fig. No. 14) the “characteristic” of the first stage and the “characteristics” of the second stage of the high-pressure and combined turbines.

Let us denote by \(\Delta p_{nv}\) the pressure drop of the first stage of the high-pressure turbine and by \(\Delta p_{nk}\) the drop of the first stage of the combined turbine at normal load. It is evident that the drop \(\Delta p_{nk}\) is only a part of the drop \(\Delta p_{nv}\); therefore one may write

\[ \Delta p_{nk}=F\Delta p_{nv}. \tag{1} \]

where \(F\) is a proportionality factor depending on the number of stages of the high-pressure turbine that have replaced the first stage of the combined turbine.

Fig. No. 13.

bines, and always greater than unity \((F>1)\).

The increase in the drop of the first stage will be expressed on the diagram by the segment \(\Delta p_1\), of equal magnitude both in the case of a high-pressure turbine and in the case of a combined turbine. Denoting further the ratio of \(\Delta p_1\) to the drop \(\Delta p_{nv}\) by “B,” and the ratio of \(\Delta p_1\) to \(\Delta p_{nk}\) by \(K\):

\[ \frac{\Delta p_1}{\Delta p_{nv}} = B;\qquad \frac{\Delta p_1}{\Delta p_{nk}} = K, \]

we shall determine, by dividing both sides of these equalities by one another, which of the relative pressure increments is greater, “B” or “K”:

\[ \frac{B}{K} = \frac{\Delta p_1 \cdot \Delta p_{nk}}{\Delta p_{nv} \cdot \Delta p_1} = \frac{\Delta p_{nk}}{\Delta p_{nv}}. \]

Substituting here the value of \(\Delta p_{nk}\) from equality (1), we find:

\[ \frac{B}{K} = \frac{F \cdot \Delta p_{nv}}{\Delta p_{nv}} = F. \]

Since \(F>1\), then

\[ \frac{B}{K} > 1 \quad\text{or}\quad B>K. \]

Drawing No. 14.

And we arrive at the conclusion that the smaller the pressure drop utilized in the turbine at its first stage, the greater is the relative increase of this drop under admission control, as compared with throttling control.

In full correspondence with the change in the pressure drop, the heat head of the first stage will in this case also change. A considerable deviation of the heat head from its normal value will cause an equally considerable change in the velocity of the steam leaving the first nozzle apparatus and in its specific volume. As a result, it will turn out that the flow sections for the steam between the working blades of the high-pressure turbine, as well as the inlet and outlet edge angles of the blade, manufactured for the normal load, will be completely unsuited to the new velocity and specific volume of the steam. As a consequence, an impact action of the steam on the blade edges will arise, and the efficiency of the stage will fall to a considerable extent; this, in turn, will reduce to nothing the advantages of admission control. Thus, for a high-pressure turbine operating with a small stage drop, admission control should not be used if one wishes to avoid a purposeless increase in the cost of the turbine.

Thermal Calculations of a V. D. Turbine

The tendency of the designer, noted by us, to shorten the turbine rotor as far as possible (utilization of the exit velocity, a thin shaft, and steam leakages through the diaphragm clearance associated with its diameter) entails the necessity of dividing a turbine operating with a high initial steam pressure into several independent cylinders. Their number may range from 2 to 3, and sometimes even to 4, depending on the magnitude of the total available heat drop and on the total number of stages selected for the turbine being designed.

The number of turbine stages is selected on the assumption that the velocity of steam leaving the guide apparatus of an individual stage, \(c_1\), should not exceed \(180 \div 240\ \mathrm{m/sec}\), which means, depending on what degree of reaction is to be applied, the utilization in the stage of a heat head of \(4 \div 9\) calories. Smaller values of the velocity \(c_1\), although they increase the efficiency of the turbine, at the same time increase its cost.

To determine the initial state of the steam before the first stage, it is necessary to determine to what degree the steam approaching this stage will be throttled under the influence of hydraulic losses in the steam-distributing organs (stop and throttle valves and the steam-admission channel). In determining these losses one has to use, as a first approximation, resistance coefficients for hydraulic valves, since with respect to steam valves there are as yet no fully definite data. The pressure drop is calculated by generally known formulas and lies within the limits of \(0.5\text{--}1.0\ \mathrm{at}\).

After the steam has passed through the first guide apparatus, a considerable part of it enters the rotor wheel and performs work there, while another part branches off and passes outward through the sealing bush, or is led away for use in one direction or another (to an intermediate stage, for heating feed water, etc.). Thus, through the first guide apparatus and rotor wheel, different quantities of steam pass per unit time, which must not be overlooked in calculating the flow sections of the passages of the guide apparatus and rotor wheel, since the leakage of steam through the clearance at high steam pressure before the first rotor wheel may attain greater values than in ordinary cases, and it cannot be neglected. This leakage is calculated by the same method as is used in calculating labyrinth seals in general. It is only necessary to increase, compared with the usual number, the number of labyrinths, so as, as far as possible, to reduce the magnitude of this steam loss to permissible limits (\(2\% \div 3\%\), depending on the turbine output).

\[ G_{\mathrm{ut}} = f \sqrt{\frac{g}{z + 1.5}\,\frac{p_1}{v_1}} \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots (2) \]

where \(G_{\mathrm{ut}}\) is the quantity of steam flowing through the seal in one second, in kilograms \((\mathrm{kg/sec})\)

\(f\) is the cross-sectional area of the channel formed by the clearance, in square meters \((\mathrm{m}^2)\)

\(g\) is the acceleration of gravity in meters per second squared \((\mathrm{m/sec}^2)\).

\(z\) is the number of labyrinths.

\(p_1\) is the steam pressure before the seal, in kilograms per square meter \((\mathrm{kg/m}^2)\).

$v_1$ — the specific volume of the steam before the packing, in cubic meters per kilogram \(\left(\dfrac{\mathrm{m}^3}{\mathrm{kg}}\right)\).

This formula is applicable in the case where the steam pressure after the packing

\[ p_2 < p_1 \cdot \frac{0.85}{\sqrt{z_1 + 1.5}}, \]

where the right-hand side of the equality gives the magnitude of the critical pressure, which, obviously, will also occur in a high-pressure turbine.

In calculating by formula (2), either the number of labyrinths is specified for design considerations and the quantity of steam that has passed through the packing in 1 hour is determined, or, conversely, from the quantity of steam allotted to leakage, the number of labyrinths is determined.

Let us now turn to the choice of the most advantageous conditions of operation of the steam in a high-pressure turbine—a problem of extremely great importance in its calculation. These most advantageous conditions, as we know, are evaluated in every turbine by the magnitude of its internal efficiency.

In calculating a multistage turbine with utilization of the exit velocity, to which a high-pressure turbine belongs, it is convenient, as Stodola has shown, to use the concept of the so-called blade efficiency \(\eta_s\).

This efficiency is the ratio of the internal work to the adiabatic heat drop in the stage (without adding the energy of the steam at the outlet \(A\dfrac{c_2^2}{2g}\)). From it, for a preliminary calculation of the thermal process of the turbine, it is easy to compute with sufficient accuracy the internal efficiency \(\eta_{ig}\) from the expression:

\[ \eta_{ig} = \eta_s(1+\lambda) - \frac{c_2^2}{2g}\frac{A}{H_0}. \qquad \qquad \qquad \qquad \qquad (3) \]

where \(\lambda\) is the heat-return coefficient, \(\dfrac{c_2^2}{2g}\dfrac{A}{H_0}\) is the loss with the exit velocity of the last stage, referred to the total adiabatic drop of the turbine.**)

For the possibility of calculating the expression just given, however, a whole series of data is necessary. With respect to these data we shall outline the following order of calculation.

Considering as given the initial steam pressure and its temperature before the turbine and the steam pressure after the turbine in its exhaust branch pipe, we determine (or specify) the pressure drop due to throttling along the path of the steam from the stop valve to the first nozzle apparatus. We choose the number of turbine cylinders depending on the magnitude of the initial and final steam pressures, on the degree of desired economy, and divide the heat drop approximately within the following limits: first cylinder (high pressure)—the steam pressure changes from 35 to 15 at. abs.; second cylinder (medium pressure)—from 15 to 5 at. abs.; third cylinder (low pressure) from 5 to the specified vacuum. In this case the work of the steam in the turbine is divided, in its smaller part, almost equally between the HPC) and MPC), while in its greater part it falls on the LPC*).

) HPC—high-pressure cylinder. MPC—medium-pressure cylinder. LPC—low-pressure cylinder.
) Stodola. Die Dampf und Gasturbinen*. Berlin, 1924.

Since the utilization of the steam exit velocity in the following stage occurs in the best manner when the steam is admitted over the full circumference, the designer of a high-pressure turbine must be required to provide precisely this kind of admission. In choosing the degree of such utilization \((m)\), it must be borne in mind that, despite the careful machining of milled guide-vane apparatuses in the high-pressure part, one can expect here a smaller utilization of the exit velocity, since the thickness of the partitions (vanes) of the guide apparatus is relatively large, and one cannot count on an entirely shockless entry of the steam. The following limits may be recommended:

\[ \begin{aligned} \text{H.P.C.} \;—\; m &= 0.65—0.85; \qquad \text{M.P.C.} \;—\; m = 0.75—0.85 \\ \text{L.P.C.} \;—\; m &= 0.85—0.95 \end{aligned} \]

Having divided the heat drop among the cylinders, it should also be divided among the individual stages, as far as possible equally, specifying either their number or the drop of an individual stage. The following may serve as limits for selection: a drop of 3–5 cal. per stage in the high-pressure part; 3.5–5 cal. in the medium-pressure part; 6–9 cal. in the low-pressure part, provided that in the last case the vacuum is not especially deep; otherwise the L.P.C. part is designed as an ordinary reaction turbine. The number of stages should be chosen (in accordance with the drop) both for the high- and medium-pressure parts from 10 to 15; for the low-pressure part it will fall within the limits of 20–25 stages. The number of stages of a cylinder \(z\) is found from the expression

\[ z = \frac{H_0(1+\lambda)}{H} \ldots \ldots (4), \]

where \(H\) is the adopted drop of an individual stage.

In high-pressure turbines operating with good efficiency, the coefficient \(\lambda\) lies between 0.01 and 0.04 and depends on the number of stages per unit of available adiabatic drop of an individual cylinder, \(\frac{z}{H_0}\); namely, the larger this ratio, the larger the coefficient \(\lambda\). It is also the larger, the lower the pressure zone in which the process of the given cylinder takes place.

When the stage drop has been chosen, the diameter of the rotor disk (at the mean blade length) should be selected. In order that the ratio of the peripheral speed \((u)\) to the steam exit velocity from the guide apparatus \((c_1)\) be kept within the required limits and that the losses due to disk friction against the steam be reduced to a minimum, the diameter should be assigned the following value:

\[ \begin{aligned} \text{H.P.C.} &— 0.5—0.6 \text{ m;}\\ \text{M.P.C.} &— 0.6—0.7 \text{ m;}\\ \text{L.P.C.} &— 0.7—0.8 \text{ m.} \end{aligned} \]

The choice of the stage drop and of the disk diameter should be checked by the ratio \(\frac{u}{c_1}\)* from the standpoint of obtaining good efficiency \(\eta_s\), namely: for H.P.C. \(\frac{u}{c_1}\) should lie between 0.45 and 0.55;

for M.P.C.—it should lie between 0.50 and 0.60;

for L.P.C.—between 0.50 and 0.70,

provided that the degree of reaction \(\rho\) is chosen respectively as follows: for H.P.C.—0.1; M.P.C.—0.1–0.2; L.P.C.—0.3–0.5.

* The ratio of the peripheral speed to the steam exit velocity from the guide apparatus.

A noticeable influence on the efficiency coefficient of the stage \(\eta_i\) is exerted by the proper choice of the angle of steam exit from the nozzle apparatus \(\alpha_1\) and the angle of the outlet edge of the rotor blade \(\beta_2\). As design practice shows, the most advantageous values of these angles do not go beyond the limits

\[ \alpha_1 = 17^\circ \div 30^\circ;\qquad \beta_2 = 30^\circ \div 40^\circ. \]

Subsequently one may proceed to determining an approximate value for a separate cylinder, namely the high-pressure cylinder, of its internal efficiency \(\eta_{ig}\); for this purpose it is first necessary, from practical data, to assume its value within the limits \(\eta_{ig \ \mathrm{predv.}} = 0.83 \div 0.86\) (and higher, depending on the magnitude of the drop and the thoroughness of the turbine execution), and, according to the considerations given above, to choose the mean value of the heat drop of the stage, the disk diameter, the degree of reaction \(\rho\), and the mean values of the angles \(\alpha_1\) and \(\beta_2\).

Further, on the \(J S\) diagram (Fig. No. 15), from the initial point \(A\), characterized by the initial pressure \(p\) and temperature \(t\), one should draw the line of the throttling process \((J=\mathrm{const.})\) until it meets at point \(B\) the line of equal pressure \(p_1\), to which the drop has occurred on the path from the stop valve to the first nozzle apparatus (Fig. No. 15).

Fig. No. 15.

If from point \(B\) one descends along the adiabat to the intersection at point \(C\) with the line of constant pressure at the exhaust from the cylinder \(p_k\), then the segment \(BC\) will give the magnitude of the available drop of the cylinder \(H_0\). Multiplying \(H_0\) by the chosen value \(\eta_{ig \ \mathrm{predv.}}\), we find:

\[ H_0 \eta_{ig \ \mathrm{predv.}} = H_i \ \mathrm{predv.}, \]

where \(H_i \ \mathrm{predv.}\) is the (approximate) magnitude of the useful heat drop worked out inside the turbine or cylinder. If, then, the value \(H_i \ \mathrm{predv.}\) is laid off in the form of the segment \(BK\), and through point \(K\) a horizontal is drawn to its intersection at point \(D\) with the line \(p_k=\mathrm{const.}\), then, from the state of the steam in the points \(B\) and \(D\) thus found, one can find the value of the specific volume of steam \(v\), either directly from the diagram available to us or by calculation from the formula relating to the diagram. Knowing these volumes, one may take their mean value*) and determine, by Stodola’s known formula, the friction loss of the disk against the steam for one stage of the cylinder:

\[ N_f = \frac{1.46 \cdot D^2 \cdot u^3 \cdot 10^{-6}}{v_{\mathrm{avg.}}} \quad \text{hp,} \]

where \(D\) is the disk diameter along the mean

*) The mean value of the specific volume of steam is expressed as the half-sum of the steam volume at the beginning of the thermal process at point \(B\) and at its end at point \(D\):

\[ v_{\mathrm{avg.}} = \frac{v_B + v_D}{2}. \]

blade length in meters, and \(u\)—the peripheral velocity in meters per second; \(v_{\text{av.}}\)—the mean value of the specific volume of steam in cubic meters per kilogram. This loss, in calories per kilogram of steam passing through the disk blades in one second, \(G_{\text{sec}}\), will be expressed as:

\[ H_r=\frac{75}{427}\cdot \frac{1.46}{G_{\text{sec}}}\cdot \frac{D^2 u^3\cdot 10^{-6}}{v_{\text{av.}}} =2.564\cdot 10^{-7}\cdot \frac{D^2 u^3}{G_{\text{sec}}\cdot v_{\text{av.}}}. \tag{5} \]

For a possible calculation of the efficiency on the blades, it is also necessary to determine the choice of velocity coefficients when steam passes through the channels of the guide apparatus \(\zeta\) and of the moving blades \(\psi\). The results of experiments by the English “Institution of Mechanical Engineers” in 1922–24*), which confirmed Lesel’s data that the smallest losses in guide apparatuses occur at velocities considerably lower than the critical one (Drawing No. 16), are at present being disputed in some part by the American investigators Warren and Keenan. They showed that the magnitude of the coefficient \(\zeta\) depends to a considerable degree on the vortex motions of the steam arising at the exit from the guide apparatus, and, moreover, by applying a method of investigation by measuring reaction, they eliminated the deficiency of the method used in the English investigations. In the English experiments the investigation was carried out by measuring the pressure of the steam on a plate placed normal to its flow, which in itself carried the possibility of vortex formation and led to an understatement of the results. Indeed, if the vortex motions of the steam behind the guide apparatus are not eliminated, then the pattern of variation of the coefficient obtained is the same as that of the English, only its magnitude is somewhat greater (Drawing No. 17). According to the curves shown (Drawings No. 16 and No. 17), it is thus evident that, for high-pressure turbines, the choice of the coefficient should lie within the limits from 0.97 to 0.985, depending on the magnitude of the theoretical velocity of steam exit from the guide apparatus \(C_0\) \((150\div 250\ \mathrm{m/sec})\).

Drawing No. 16.

Drawing No. 17.

With regard to the velocity coefficient on the moving blades \(\psi\), it should be said that, in the absence in the literature of experimental data on its magnitude at low steam velocities, one has to use the well-known

*) “Engineering,” 1923, 1924, 1925.

Wagner curve for reaction blades, expressing the functional dependence of the coefficient \(\psi\) on the angle of deflection of the steam jet on the blades (Fig. No. 6).

The efficiency \(\eta_s\) will now be found from the expression:

\[ \eta_s=\frac{H_i}{H}=\frac{H-\sum H_{\text{п}}}{H}, \ldots \tag{6} \]

where \(H\) is the total adiabatic (available) drop of the stage from the initial state of the steam before the given stage to the steam pressure after it. \(\sum H_{\text{п}}\) is the sum of the heat losses of the given stage: in the guide apparatus, on the rotor blades, due to disk friction against steam, and part of the loss with the exit velocity.

Fig. No. 18.

The drop \(H\), selected earlier in determining the number of stages in a separate cylinder,*) must be divided, depending on the selected degree of reaction, into the active drop \(H_A\) for the guide apparatus and the reaction drop \(H_R\) for the rotor blades.

The heat process of an individual stage on the \(JS\) diagram is shown in (Fig. No. 18). The steam expands in the guide apparatus from a certain pressure \(p_n\) (point \(n\)) to the pressure \(p'_n\) (point \(n'\)), in accordance with which the steam velocity in the guide channel increases from the inlet velocity to the outlet velocity \(c_1\), where

\[ \left(\frac{c_1^2}{\varphi}\right)\frac{A}{2g}-m\frac{c_2^2}{2g}A=H_A \]

where \(m\frac{c_2^2}{2g}A=mH_a\) is the heat of utilization of the steam exit velocity from the preceding stage. Hence it follows:

\[ c_1=\varphi\cdot91.5\sqrt{\,H_A+\frac{m c_2^2}{2g}A\,} =\varphi\cdot91.5\sqrt{\,H_A+\frac{m c_2^2}{2\cdot9.81\cdot427}\,} \]

\[ =\varphi\cdot91.5\sqrt{\,H_A+1.194\cdot10^{-4}m c_2^2\,}\ldots \tag{7} \]

Since the magnitude of the velocity \(c_2\) is not known in advance, the problem has to be solved by trial selection, as will be shown in the calculation example. The loss in the guide apparatus is expressed as:

\[ H_d=(1-\varphi^2)\left(H_A+\frac{m c_2^2}{2g}A\right)\ldots \tag{8} \]

and the actual process of steam expansion will proceed along the line \(nn'\).

*) See p. 15.

From the velocity diagram (Fig. No. 19), with known \(c_1\) and \(u\), the value of the relative velocity of the steam entering the moving blades \(w_1\) and the angle \(\beta_1\) are determined. From the value found for the angle \(\beta_1\) and that adopted for \(\beta_2\), the value of the coefficient \(\psi\) is found from Wagner’s curve (Fig. No. 6).

Figure No. 19.

In the moving wheel, the adiabatic expansion of the steam will take place along the line \(n'n'\) down to the pressure before the next stage \(p_{n+1}\), whereby

\[ \frac{\left(\frac{w_2}{\psi}\right)^2}{2g} A - \frac{w_1^2}{2g} A = H_R \]

whence:

\[ w_2 = \psi \cdot 91.5 \sqrt{\frac{w_1^2}{2g} A + H_R} = \psi \cdot 91.5 \sqrt{H_R + 1.194 \cdot 10^{-4} w_1^2}. \tag{9} \]

The energy loss in the moving blades will be expressed as:

\[ H_s = (1-\psi^2)\left(H_R + \frac{w_1^2}{2g} A\right) = (1-\psi^2)\left(H_R + 1.194 \cdot 10^{-4} w_1^2\right). \tag{10} \]

The actual expansion, consequently, will take place along a certain polytrope \(n'n''\).

The loss due to disk friction against the steam is found from the expression (5) given above, and, together with the heat of the lost part of the exit velocity,

\[ (1-m)H_a = (1-m)\frac{c_2^2}{2g} A = 1.194 \cdot 10^{-4}(1-m)c_2^2 \ldots \tag{11} \]

gives the final point of the process \(n+1\). The sum of the losses is

\[ \Sigma H_n = H_d + H_s + H_r + (1-m)H_a. \]

Substitution of this value of \(\Sigma H_n\) into equality (6) gives the value of the coefficient \(\eta_s\).

Having now the value of this coefficient, we find, from expression (3), a more accurately approximated value of \(\eta_{ig}\)—the internal efficiency coefficient of the cylinder.

As before, we find (Fig. No. 15) the point \(D'\), which characterizes the state of the steam at the outlet, and consequently determine a new, more accurate value of the specific volume of the steam at the outlet from the cylinder.

It is evident that the method analyzed for determining the internal efficiency of an individual turbine cylinder is possible when the heat drops of its stages are equal.

If, as is sometimes done, a Curtis wheel or simply an impulse wheel with a large heat drop*) is introduced as the first stage, then the first stage must first be calculated in the usual manner, and the procedure just discussed should be applied only to the remaining stages.

Having then calculated the new mean value of the loss due to steam friction of the disk for one stage, we proceed to calculate a series of variants in order to find the most advantageous of them from the point of view of the blade efficiency coefficient \(\eta_s\) and certain considerations of a design character.

On the basis of the considerations given above concerning the choice of data for calculating a turbine with a small stage heat drop, the following scheme of calculation variations may be compiled (see the scheme on p. 21).

When using the scheme presented, the following must be kept in mind:

With respect to columns 1, 2, 3, 4, 5, and 6 of the scheme, in determining the possibilities for a preliminary calculation of the internal efficiency coefficient of the turbine \(\eta_{ig}\)**), sufficient considerations have been given for choosing particular values of the quantities entered in them. Beginning with column 6, both in the part for the medium pressure (MP) and in the part for the low pressure (LP), the calculation is carried out in two variants with different degrees of reaction. In column 7 the variants in all parts of the turbine branch, depending on the magnitude of the disk diameter in the HP and LP, into two subvariants, and in the MP into three. All subvariants, in turn, are divided according to columns 8 and 9 into groups of seven subvariants each, depending on the variable values of the angles \(\alpha_1\) and \(\beta_2\). Thus the total number of variants obtained is: for the HP—14, for the MP—42, and for the LP—28.

When varying the angles \(\alpha_1\) and \(\beta_2\) in magnitude within one group of subvariants, it is possible to select such a combination of them for which the coefficient \(\eta_s\) attains its maximum value. Comparison of the group maxima reveals the most advantageous operating conditions for each cylinder of the turbine. It is not difficult to show that, in order to find the maximum coefficient within one group, it is sufficient to make seven calculations.

Taking as given the mean (or any other, according to column 9) value of the angle \(\beta_2\) and denoting it by \(\beta_2''\), from the corresponding velocity diagrams, for three values of the angle \(\alpha_1\) (according to column 8), three values of the coefficient \(\eta_s\)***) may be calculated. If the values thus obtained are now plotted in the coordinate system \(\eta_s, \alpha_1\), then (Fig. No. 20), corresponding to the three values of the angle \(\alpha_1\): \(\alpha_1'\), \(\alpha_1''\), \(\alpha_1'''\), we obtain three points 1, 2, 3 characterizing the portion of the functional dependence of interest to us

\[ \eta_s=f(\alpha_1)/\beta=\text{const} \]

for the selected constant value of the angle \(\beta_2\).

The smooth curve drawn through the named points is evidently the section of the surface constructed according to the functional dependence

\[ \eta_s=\Phi(\alpha_1,\beta_2) \]

by the plane \(\beta_2=\beta_2''=\text{const}\), parallel to the plane \((\eta_s,\alpha_1)\).

The tangent to this curve, parallel to the \(\alpha_1\)-axis, gives at the point of tangency the maximum value of the coefficient \(\eta_s\) for the value of the angle \(\beta_2\) chosen for the investigation. In the case shown in Fig. No. 20, the point of tangency 2 has the coordinates \(\beta_2'', \alpha_1'', \eta_{s2}\). Now leaving the angle \(\alpha_1\) constant, namely setting \(\alpha_1=\alpha_1''=\text{const}\), one may, by the 2nd remaining—

*) For the possibility of controlling the turbine by admission.

**) See pp. 14, 15, and 16.

***) By the method given on pp. 17, 18, 19.

SCHEME OF VARIATIONS

Seq. no.	DESIGNATION OF THE CYINDER	I. H.P.C. (high-pressure cylinder)	II. M.P.C. (medium-pressure cylinder)	III. L.P.C. (low-pressure cylinder)
1	Pressure drop in one cylinder, atm. abs.	35—15	15—5	5—vacuum
2	Degree of utilization of steam velocity at exit from the preceding stage, in %	0.65—0.85	0.75—0.85	0.85—0.95
3	Heat drop of an individual stage, \(H\), in Cal.	3.0—5.0	3.5—6.0	6.0—9.0
4	Number of stages in one cylinder	15—10	15—10	20—25
5	Circumferential velocity at the exit of steam from the guide apparatus, \(\dfrac{u}{c}\)	0.45—0.55	0.50—0.60	0.50—0.60
6	Degree of reaction, \(\rho\).	0.1	0.1; 0.2	0.3; 0.5
7	Diameter of the disk along the mean line of the blades \(D\), in m.	for \(\rho=0.1\): 0.5; 0.6	for \(\rho=0.1\): 0.6; 0.7; 0.8 for \(\rho=0.2\): 0.6; 0.7; 0.8	for \(\rho=0.3\): 0.7; 0.8 for \(\rho=0.5\): 0.7; 0.8
8	Angle of steam exit from the guide apparatus \(\alpha_1\), in degrees.	17; 24; 30 | 17; 24; 30	17; 24; 30 | 17; 24; 30 | 17; 24; 30 | 17; 24; 30 | 17; 24; 30 | 17; 24; 30	17; 24; 30 | 17; 24; 30 | 17; 24; 30 | 17; 24; 30
9	Angle of exit edge of the working blade \(\beta_2\), in degrees.	30; 35; 40 | 30; 35; 40	30; 35; 40 | 30; 35; 40 | 30; 35; 40 | 30; 35; 40 | 30; 35; 40 | 30; 35; 40	30; 35; 40 | 30; 35; 40 | 30; 35; 40 | 30; 35; 40
10	Number of variants necessary for determining \(\eta_s^{\max}\)	14	42	28

using the unused values of the angle \(\beta_2\), compute two values of the coefficient \(\eta_s\), and obtain on the drawing the points 4 and 5, belonging to the curve 4, 2, 5 lying in a plane parallel to the plane \((\eta_s, \beta_2)\). In the preceding manner, the maximum ordinate of this curve will be found at the point \(5'\), with coordinates \(\alpha_1'', \beta_2^{\max.}, \eta_s'\). Further, with \(\beta_2 = \beta_2^{\max.} = \mathrm{const.}\), a change of the angle \(\alpha_1\) within narrower limits (the mean value between \(\alpha_1'\), \(\alpha_1''\) and \(\alpha_1'', \alpha_1'''\)) gives the points 6 and 7 of the new curve 6, \(5'\), 7. In a manner entirely analogous to the preceding one, its maximum is determined at the point \(7'\). Further repetition of the procedure proves superfluous, since the convexity of the surface

Drawing No. 20.

\(\eta_s = \Phi(\alpha_1,\beta_2)\) in the region of the maximum is very gentle, and further refinement of the maximum value of the coefficient \(\eta_s\) lies beyond the limits of accuracy of ordinary engineering calculations. Thus, the group maximum of the coefficient \(\eta_s\) is determined graphically as the ordinate \(\eta_s{}_{7'}\) of the point \(7'\).

With regard to the low-pressure part of the turbine, the following should be noted. The rotor of the low-pressure cylinder is most often made in the form of a drum with a shoulder of larger diameter on the exhaust side, in order to avoid an excessive length of the last blades. This circumstance should not be overlooked when calculating variants relating to the LPC.

Beyond any doubt, the scheme presented for varying the calculations of the coefficient \(\eta_{is}\) cannot lay claim to complete universality, and is regarded by the author as a first approximation to a possible generalization of the question. The consumer’s desire to obtain cheaper power-station equipment, even at the expense of its thermal efficiency—which may prove advantageous from the economic point of view when fuel is cheap—compels the designer to proceed toward such a reduction in cost and to make the turbine in the form of a two-cylinder turbine instead of a three-cylinder one, or even to utilize the entire heat drop within a single cylinder. The presence of a deep vacuum at a considerable turbine output may likewise compel the designer, in order to avoid excessively long blades in the low-pressure part, to reduce the number of turbine stages instead of adopting the more expensive method of branching the steam path. Obviously, in such cases the scheme given above can be used only partially, since the designer may have to abandon the principle of steam operation at low steam velocities, at least in the low-pressure part. Therefore the magnitudes of the drops given in the scheme, in relation to the corresponding steam pressures, cannot be regarded as strictly fixed. If a lower economy of the turbine is permitted, then the magnitude of the drops recommended by the scheme may be raised somewhat.

For the final choice of the design variant of one or another turbine cylinder, it is necessary to subject the group maxima of the coefficient \(\eta_{is}\) to criticism. The following should be taken as the basis for comparing them.

1. The calculated length of the first guide blade) with full steam admission should not be too small, since the smaller the blade dimension in the radial direction, the greater is the influence of the radial clearance on the unavoidable leakage, due to the impossibility of attaining complete equilibrium between suction and pressure on both sides of the blade by selecting the degree of reaction. Values of blade length below 4–5 mm*) should not be allowed.

2. The length of the last moving blade***) should not exceed \(\frac{1}{5} \div \frac{1}{4}\) of the diameter of the circumference drawn through the middle of the blade length. As is known, with an excessively long blade the ratio of the circumferential velocity to the velocity of steam issuing from the guide apparatus, \(\frac{u}{c_1}\), cannot be regarded as a constant quantity in the radial direction along the blade length; such nonconstancy leads to a deterioration of the operating conditions of the steam as it flows through the working passages.

3. In the structural design of the guide apparatus, the shape of its passage proves more rational (a short passage and its narrowing over a small distance at the outlet) in the case when the steam from the preceding stage approaches it at an angle \(\alpha_2\) close in value to \(90^\circ\).

4. The leakage of steam between the shaft and the diaphragm wall through the seal is the greater, the greater the active drop of the stage; that is, it depends both on the number of stages in the cylinder and on the selected degree of reaction.

) This consideration applies only to the HP cylinder.
) Only for especially high pressures, 50–100 atm. abs., is the use of a blade 2–3 mm long permissible.
**) A consideration that plays a role almost exclusively with respect to the LP cylinder.

Thus, for the HP cylinder, with equal values of the group maxima of the coefficient \(\eta_s\), preference should be given to the variant in which the blade of the first guide apparatus is longer; with respect to the LP cylinder, the variant with the shorter last moving blade is preferable; for all cylinders it is highly desirable to have the exit-velocity angle \(\alpha_2\) near \(90^\circ\), and it is more economical to prefer a greater degree of reaction and a small stage drop, provided only that this does not lead to an increase in losses through the radial clearance of the blade and to a considerable increase in the length of the shaft, and consequently also of its diameter and the associated greater leakage of steam through the diaphragm clearance.

Example of the thermal calculation of the high-pressure cylinder of a turbine operating at low steam velocities.

Assuming that the pressure drop of the steam along the path from the stop valve to the first guide apparatus of the turbine has already been taken into account, we take the following data as the specification:

\[ p = 30 \ \text{at. abs.} \]

— steam pressure before the first guide apparatus of the turbine.

\[ t = 370^\circ \text{C} \]

— steam temperature at the same place.

\[ p_k = 7 \ \text{at. abs.} \]

— steam pressure at discharge from the turbine.

\[ n = 3000 \ \text{rev/min.} \]

— number of rotor revolutions.

\[ N_{\mathrm{ef.}} = 3000 \ \text{k. w.} \]

— power at the terminals of the generator directly coupled to the turbine.

\[ \eta_g = 0.94 \]

— efficiency of the generator.

We determine the adiabatic heat drop from the \(J S\) diagram:*) \(H_0 = i - i_k = 754.9 - 671.4 = 83.5\) k. cal. (Fig. No. 21).

We divide this drop so that the high-pressure cylinder receives the drop

\[ H_0^{I} = i - i_k' = 754.9 - 715.0 = 39.9 \ \text{k. cal.} \]

at the exhaust pressure

\[ p_k' = 16 \ \text{at. abs.}, \]

and the medium-pressure cylinder receives

\[ H_0^{II} = i_k' - i_k = 715.0 - 671.4 = 43.6 \ \text{k. cal.} \]

Taking the coefficient of utilization of exit velocity \(m = 0.75\), we choose the number of stages in the HP cylinder, \(Z\), equal to 12.

With the heat-return coefficient \(\lambda = 0.02\), the drop of one stage \(H\) is found from expression (4):

\[ Z = \frac{H_0^{I}(1+\lambda)}{H}, \]

whence

\[ H = \frac{H_0^{I}(1+\lambda)}{Z} = \frac{39.9 \cdot 1.02}{12} = 3.391 \ \text{k. cal.} \]

Fig. No. 21.

) Tabellen und Diagramme für Wasserdampf* von O. Knoblauch. München und Berlin 1923.

The magnitude of the circumferential velocity \(u\) will be expressed as:

\[ \text{for mean diameter } D = 0.5\ \text{m.}\quad u=\frac{\pi D n}{60}=\frac{\pi \cdot 0.5 \cdot 3000}{60}=78.6\ \frac{\text{m}}{\text{sec}} \]

\[ \text{” ” ” } D = 0.6\ \text{m.}\quad u=\frac{\pi D n}{60}=\frac{\pi \cdot 0.6 \cdot 3000}{60}=94.3\ \frac{\text{m}}{\text{sec}} \]

With a degree of reaction \(\rho=0.1\): the impulse part of the heat drop is

\[ H_A=0.9H=0.9\cdot 3.391=3.052\ \text{k. cal.} \]

the reaction part:

\[ H_R=0.1H=0.1\cdot 3.391=0.339\ \text{k. cal.} \]

If the utilization of the exit velocity is preliminarily estimated as \(H_u=0.3\ \text{k. cal.}\), then, with velocity coefficient \(\varphi=0.97\),

\[ c_1=\varphi 91.5\sqrt{H_A+H_u} =0.97\cdot 91.5\sqrt{3.052+0.300} = \]

\[ 88.76\sqrt{3.352}=162.5\ \frac{\text{m}}{\text{sec}} \]

The value of \(\dfrac{u}{c_1}\) is written:

\[ \text{for } D=0.5\ \text{m.} \;—\; \frac{u}{c_1}=\frac{76.6}{162.5}=0.487 \]

\[ \text{” } D=0.6\ \text{m.} \;—\; \frac{u}{c_1}=\frac{94.3}{162.5}=0.580 \]

i.e., both values are within permissible limits.

Next, assigning a preliminary value of the internal efficiency common to the cylinder (and turbine), \(\eta_{ig.\,\text{prelim.}}=0.84^*)\), we find the approximate value of the usefully worked heat drop within the cylinder \(H_{i\,\text{prelim.}}\) (Fig. No. 15) from the expression:

\[ H_{i\,\text{prelim.}}=H_0\eta_{ig.\,\text{prelim.}}=39.9\cdot 0.84=33.52\ \text{k. cal.} \]

and thereby determine the position of point Д itself on the line \(p'_k=\text{const.}\) (in drawing No. 15 it corresponds to \(p_k=\text{const.}\)). Thus, in the example under consideration, point B is determined by the values \(i_B=i=754.90\ \text{k. cal.}\) and \(p_B=p=30.0\ \text{at. abs.}\), while for point Д we have—

\[ i_D=i_B-H_{i\,\text{prelim.}}=754.90-33.52=721.38\ \text{k. cal.} \]

\[ p_D=p'_k=16.0\ \text{at. abs.} \]

Using Knoblauch’s \(J P\) diagram, \(^{**}\) we find, from the determined values \(i_B\) and \(p_B\), \(i_D\) and \(p_D\), the specific volumes of the steam in its states at points B and Д:

\[ v_B=0.094\ \frac{\text{m}^3}{\text{kg}} \qquad v_D=0.150\ \frac{\text{m}^3}{\text{kg}} \]

The mean value of the specific volume will be expressed as:

\[ v_{\text{avg.}}=\frac{v_B+v_D}{2} =\frac{0.094+0.150}{2} =0.127\ \frac{\text{m}^3}{\text{kg}} \]

The loss due to disk friction against steam for one stage, according to expression (5):

\[ H_r=2.564\cdot 10^{-7}\frac{D^2u^3}{G_{\text{sec}}v_{\text{avg.}}} \]

\[ \text{* See p. 16.} \]

\[ \text{** Tabellen und Diagramme für Wasserdampf von O. Knoblauch. München und Berlin 1923.} \]

For the preliminary calculation we shall confine ourselves to only one value of the diameter, \(D = 0.5\ \mathrm{m}\).

The consumption of steam flowing through the cylinder per second, \(G_{\mathrm{sec}}\), is determined from

\[ G_{\mathrm{sec}}=\frac{D^{\mathrm{ef}}_{\mathrm{kw}}\cdot N^{\mathrm{ef}}}{3600}, \]

where \(D^{\mathrm{ef}}_{\mathrm{kw}}\) is the steam consumption in kilograms per actual kilowatt-hour at the generator terminals. Without taking into account steam leakage through the sealing glands, \(D^{\mathrm{ef}}_{\mathrm{kw}}\) is found from:

\[ D^{\mathrm{ef}}_{\mathrm{kw}}= \frac{760}{H_0\cdot \eta_{ig\ \mathrm{prelim.}}\cdot \eta_{\mathrm{m.}}\cdot \eta_s} \ldots \ldots \ldots \ldots \tag{12} \]

Considering the mechanical losses (without steam leakage) to be within the limits of \(1\%\)—\(2\%\), we take the mechanical efficiency of the turbine as \(\eta_{\mathrm{m.}}=0.985\).

Then

\[ D^{\mathrm{ef}}_{\mathrm{kw}} = \frac{860}{83.5\cdot 0.84\cdot 0.985\cdot 0.94} = 13.24\ \frac{\mathrm{kg}}{\mathrm{kw\ hour}}, \]

and

\[ G_{\mathrm{sec}} = \frac{13.24\cdot 3000}{3600} = 11.055\ \frac{\mathrm{kg}}{\mathrm{sec}}. \]

Thus,

\[ H_r = 2.564\cdot 10^{-7}\, \frac{0.5^2\cdot 78.6^3}{11.055\cdot 0.127} = 0.0221\ \mathrm{k.\ cal.} \]

Assigning mean values of the angles \(\alpha_1\) and \(\beta_2\): \(\alpha_1=24^\circ\), \(\beta_2=35^\circ\), we construct the velocity diagram (Fig. No. 19), for

\[ c_1=162.5\ \frac{\mathrm{m}}{\mathrm{sec}} \quad\text{and}\quad u=78.6\ \frac{\mathrm{m}}{\mathrm{sec}}. \]

From the diagram we have

\[ w_1=96\ \frac{\mathrm{m}}{\mathrm{sec}}, \quad \zeta_1=43^\circ 23', \quad \beta_1+\beta_2=43^\circ 23' + 35^\circ = 78^\circ 23'. \]

From the Wagner curve (Fig. No. 6), \(\psi=0.91\); consequently,

\[ w_2 = \psi\cdot 91.5 \sqrt{H_R+1.194\cdot 10^{-4}\cdot w_1^2} = \]

\[ = 0.91\cdot 91.5 \sqrt{0.339+1.194\cdot 10^{-4}\cdot 96^2} = 0.91\cdot 91.5\sqrt{1.439} \]

\[ w_2=99.8\ \frac{\mathrm{m}}{\mathrm{sec}}, \qquad \zeta_2=87^\circ. \]

Further, from the diagram,

\[ c_2=57.3\ \frac{\mathrm{m}}{\mathrm{sec}}. \]

\[ H_{\mathrm{i}} = m\,\frac{c_2^2}{2g}\,A = 0.75\cdot 1.194\cdot 10^{-4}\cdot 57.3^2 = 0.294\ \mathrm{k.\ cal.} \]

This differs only slightly from the adopted value.

Losses:

\[ H_a = (1-m)\,\frac{c_2^2}{2g}\,A = 1.194\cdot 10^{-4}(1-0.75)\cdot 57.3^2 = 0.098\ \mathrm{k.\ cal.} \]

\[ H_d = (1-\varphi^2)(H_A+H_{\mathrm{i}}) = (1-0.97^2)\cdot 3.352 = 0.0591\cdot 3.352 = 0.1980\ \mathrm{k.\ cal.} \]

\[ H_z = (1-\psi^2)\cdot \left( H_R-\frac{w_1^2}{2g}A \right) = (1-0.91^2)\cdot 1.439 = 0.1719\cdot 1.439 = \]

\[ = 0.2470\ \mathrm{k.\ cal.} \]

\[ H_r=0.0221\ \mathrm{k.\ cal.} \]

Summation of the losses gives:

\[ \begin{aligned} H_d&=0.1980\\ H_s&=0.2470\\ H_a&=0.0980\\ H_r&=0.0221\\ \hline H_n&=0.5651\ \text{k. cal.} \end{aligned} \]

\[ \eta_s=\frac{3.391-0.5651}{3.391}=\frac{2.8259}{3.391} \]

\[ \eta_s=0.8334 \simeq 83.3\%. \]

Determine

\[ \eta_{ig}=(1+\lambda)\eta_s-\frac{c_2^2 A}{2gH_0'}=1.02\cdot 0.833-\frac{0.392}{39.9}\simeq 0.84 \simeq 84.0\%. \]

where

\[ \frac{c_2^2\cdot A}{2gH_0'}= \frac{1.194\cdot 10^{-4}\cdot 57.3^2}{39.9} =\frac{0.392}{39.9} \]

The obtained value of \(\eta_{ig}\) does not differ from the preliminarily selected one, and recalculation of the average loss due to disk friction against steam is unnecessary.

We then proceed to the calculation of variants in order to choose the most advantageous of them:

Variant No. 1 \(\rho=0.1;\ D=0.5\ \text{m.};\ u=78.6\ \frac{\text{m}}{\text{sec}};\ \alpha_1=17^\circ;\)

\[ \beta_2=35^\circ;\ \varphi=0.97. \]

Assuming utilization of the exit velocity:

\[ H_{n\,\text{pr.}}=0.290\ \text{k. cal.} \]

\[ c_1=88.76\sqrt{3.052+0.290} =88.76\sqrt{3.342}=162.3\ \frac{\text{m}}{\text{sec}}. \]

According to the velocity diagram,

\[ w_1=90\ \frac{\text{m}}{\text{sec}};\ \beta_1=32^\circ; \]

\[ \beta_1+\beta_2=32^\circ+35^\circ=67^\circ. \]

According to Wagner’s curve, \(\psi=0.9\)

\[ w_2=0.9\cdot 91.5\sqrt{0.339+1.194\cdot 10^{-4}\cdot 90^2} =82.3\sqrt{0.339+0.967}= \]

\[ =82.3\sqrt{1.306}. \qquad w_2=94.2\ \frac{\text{m}}{\text{sec}} \]

\[ c_2=54\ \frac{\text{m}}{\text{sec}} \qquad \alpha_2=91.5^\circ \]

\[ H_n=0.75\cdot 1.194\cdot 10^{-4}\cdot 54^2 =8.955\cdot 10^{-5}\cdot 54^2 =0.261\ \text{k. cal.} \]

The difference from \(H_{n\,\text{pr.}}\) is

\[ (H_{n\,\text{pr.}}-H_n)=0.290-0.261=0.029\ \text{k. cal.} \]

We recalculate, assigning the new value

\[ H_{n\,\text{pr.}}=0.255\ \text{k. cal.} \]

\[ c_1=88.76\sqrt{3.052+0.255} =88.76\sqrt{3.307} =161.2\ \frac{\text{m}}{\text{sec}} \]

\[ w_1=89\ \frac{\text{m}}{\text{sec}};\ \beta_1\simeq 32^\circ;\ \beta_1+\beta_2=32^\circ+35^\circ=67^\circ;\ \psi=0.9 \]

\[ w_2 = 82.3 \sqrt{0.339 + 1.194 \cdot 10^{-4} \cdot 89^2} = 82.3 \sqrt{0.339 + 0.946} \]

\[ = 82.3 \sqrt{1.285} = 93.3 \ \frac{\mathrm{m}}{\mathrm{sec}} \]

\[ c_2 = 53.5 \ \frac{\mathrm{m}}{\mathrm{sec}};\quad \alpha_2 = 92^\circ;\quad \frac{u}{c_1} = \frac{78.6}{161.2} = 0.487 \]

\[ H_i = 8.955 \cdot 10^{-5} \cdot 53.5^2 = 0.243. \]
The difference from the preliminary value is small.

Losses:

\[ H_a = 0.25 \cdot 1.194 \cdot 10^{-4} \cdot 53.5^2 = 0.0810 \ \text{k. cal.} \]

\[ H_d = 0.0591 (H_a + H_i) = 0.0591 \cdot 3.307 = 0.1955 \ \text{k. cal.} \]

\[ H_s = (1 - 0.9^2) 1.285 = 0.19 \cdot 1.285 = 0.2440 \ \text{k. cal.} \]

\[ H_r = 0.0221 \ \text{k. cal.} \]

\[ \sum H_n = 0.5426 \ \text{k. cal.},\quad \eta_s = \frac{3.391 - 0.5426}{3.391} = \frac{2.8484}{3.391} \approx 0.84 = 84\%. \]

The second variant was calculated with a preliminary estimate of \(\eta_{is}\).

A presentation of the calculations for the subsequent variants would be of interest only as arithmetical computations, without giving anything new from the standpoint of the calculation method, and therefore we do not include it; the necessary characteristic quantities of the variants obtained in the calculation are given in the table placed below.

For convenience in the final selection of the variant, in the same table we shall enter the values of the blade heights of the first and last disks for all variants. These blade heights are calculated from the expressions

\[ l_1 = \frac{G_{\mathrm{sec.}}\, v_1\, t}{\pi \, D_{\mathrm{cp.}} \, c_{1a} \, t_1}; \quad l_n = \frac{G_{\mathrm{sec.}}\, v_n\, t}{\pi \, D_{\mathrm{cp.}} \, c_{2a} \, t_1}, \]

where the letter symbols denote:

\(l_1\) — height of the blade of the first guide apparatus, in m.

\(l_n\) — height of the blade of the last rotor wheel, in m.

\(G_{\mathrm{sec.}}\) — working quantity of steam flowing through the turbine per second, in \(\dfrac{\mathrm{kg}}{\mathrm{sec.}}\).

\(D_{\mathrm{cp.}}\) — mean blade diameter, in m.

\(v_1\) — specific volume of the steam at the exit from the first guide apparatus, in \(\dfrac{\mathrm{m}^3}{\mathrm{kg}}\); for a preliminary calculation it may be taken equal to the initial value.

\(v_n\) — the specific volume of the steam at the exit from the last blade passage, in \(\dfrac{\mathrm{m}^3}{\mathrm{kg}}\); for the preliminary calculation it is equal to the final volume found earlier from \(\eta_{ig\,\mathrm{prelim.}}\).

Fig. No. 22.

\(\dfrac{t}{t_1}\) — the degree of narrowing of the passage by the wall thickness, equal to the ratio of the pitch in the clear to the actual pitch (Fig. No. 22); for a preliminary calculation it may be taken as constant for all first guide apparatuses,

\[ \frac{t}{t_1} \approx 1.12, \]

and for all last rotor blades,

\[ \frac{t}{t_1} \approx 1.14. \]

\(c_{1a}\) — the projection of the velocity at the exit from the guide apparatus onto the direction of the turbine axis.

\(c_{2a}\) — the projection of the velocity at the exit from the rotor ring onto the same direction.

Variant	Angle \(z_1\), in degrees	Angle \(z_2\), in degrees	Angle \(z_3\), in degrees	Mean blade diameter in meters	Circumferential speed on blades in m/sec	Length of blades of first guide apparatus in mm	Length of last rotor blade in mm	Ratio \(D_{\mathrm{av}}/b\), in mm	Degree of reaction
1	17.0	35.0	92.0	0.5	83.99	15.34	22.46	83.4
2	24.0	—	87.0	—	83.34	10.93	21.14	—
3	30.0	—	81.5	—	82.74	8.83	19.58	—
4	17.0	40.0	95.5	—	82.76	15.13	19.46	—
5	—	30.0	87.5	—	82.55	15.27	25.83	—	0.1
\(1'\)	—	35.0	118.0	0.6	84.49	13.07	20.82	100.0
\(2'\)	24.0	—	109.0	—	83.86	9.31	18.62	—
\(3'\)	30.0	—	102.5	—	83.61	7.58	17.52	—
\(4'\)	17.0	40.0	116.5	—	83.66	12.88	17.92	—
\(5'\)	—	30.0	118.5	—	85.15	13.19	24.22	—

Examining this table of the results of calculating the variants, one may arrive at the following conclusion. Since in all variants the length of the last moving blade does not go beyond the ratio established by practice \((D_{\mathrm{avg}}/6)\), it therefore has no influence on the choice of the most advantageous variant. It remains to turn our attention to the length of the first guide blade. As is known, the longer such a blade is, the more its manufacture is simplified from the standpoint of relative accuracy, and the more accurately the distribution of drops over the individual stages will occur in accordance with the calculation assumptions.

The variants which, with a good efficiency, give a considerable length of the first blade are 1, \(1'\), and \(5'\).

Variant 1 has two advantages over \(1'\) and \(5'\): first, the exit angle of the steam from the moving blade \(\alpha_2\) in the first case is close to \(90^\circ\) \((92^\circ)\), whereas in the case of variants \(1'\) and \(5'\) it is \(118^\circ\)—\(118.5^\circ\); and second, the disk diameter of variant 1 is only \(0.5\ \mathrm{m}\). On the other hand, variants \(1'\) and \(5'\) give a higher efficiency. The designer must therefore coordinate his choice of variant with the fact that, in variant 1, the constructive execution of the guide apparatus (angle \(\alpha_2\)) is easier, and also that the forging of an integral forged rotor, with the smaller diameter, is more reliable from the standpoint of metal technology.

When one or another variant has been chosen, it remains to proceed to the calculation of the flow sections both of the guide apparatuses and of the moving rows of the given cylinder.

From the standpoint of reducing the cost of production, it is sometimes important to preserve one and the same blade profile for all turbine stages having the same mean diameter. To satisfy this requirement it is necessary to preserve one and the same velocity diagram for all such stages. The first stage of any turbine cylinder operates without utilization of the exit velocity; therefore, in order to preserve the velocity diagram, the drop of the first stage must be increased correspondingly by the amount of the utilization drop:

\[ H_u = m \frac{c_2^2}{2g}. \]

To determine the specific volumes of the steam required in calculating these sections, it is necessary to construct the working process. Under ordinary conditions such a process is constructed using a \(JS\) diagram at a scale of \(1\ \mathrm{k.\ cal.} = 1\ \mathrm{mm}\). In the case of calculating a high-pressure turbine, when the magnitude of the drop of an individual stage may reach \(2\)—\(3\ \mathrm{k.\ cal.}\), the reactive part of it is a small fraction \((0.1\)—\(0.2)\), and especially the individual losses may attain very small values, on the order of hundredths of a calorie, which means, at the above scale, hundredths of a millimeter. Obviously, the latter quantities cannot be taken into account in constructing the working process using ordinary drafting instruments. One must get out of this situation by preparing a large-scale \(JS\) diagram, namely \(1\ \mathrm{k.\ cal.} = 10\ \mathrm{mm}\). At such a scale, hundredths of a calorie on the process drawing are represented by tenths of a millimeter and can be laid off, using a proportional divider, without particular difficulty.

The working process is constructed, as is customary for impulse turbines, stage by stage, beginning with the first (Fig. No. 23); moreover, in order to avoid the accumulation of errors from the inevitable graphical inaccuracy of the constructions, the individual points of the process should be found as the result of the intersection of the line of constant heat content, previously calculated for the desired point, with the corresponding adiabat or isobar drawn from the preceding known point.

The calculation of the heat content of the individual points of the process must be carried out in a definite order. In full accordance with this order one may compile a table of calculated quantities; the use of such a table will, to a considerable extent, facilitate and systematize the construction of the thermal process.

As an example, let us use the thermal calculation of a high-pressure cylinder carried out for a turbine of 1000 k.w. capacity, with its rotor speed equal to 3000 per minute, at an initial steam pressure of 30 at. abs. and a temperature of 370° C. For this turbine the mean blade-circle diameter of the wheel was chosen as 0.5 m; the degree of reaction was taken as 0.1, and the degree of utilization of the exit velocity was taken as 66%. For the given exhaust pressure of 7 at. abs., the assumed heat drop is obtained as about 86 calories per kilogram of steam. The number of turbine stages was taken as 30, with an equal division of them between two cylinders, which gave for the impulse part of the drop:

\(H_A = 2.565\) k. cal., and for the reaction part:

\(H_R = 0.285\) k. cal.

For the calculation a variant was chosen with the angles:

\(\alpha_1 = 17^\circ\) and \(\beta_2 = 30^\circ\). The drop of the first stage in its impulse part is taken (after the corresponding selection) as \(H_A = 2.565 + 0.135 = 2.700\) k. cal. The velocity

Fig. No. 23.

the steam velocity at the outlet from the nozzle apparatus is found from \(c_1=\varphi \cdot 91.5\sqrt{H_a}\).

For \(\varphi=0.96\) we obtain:

\[ c_1=87.84\sqrt{2.700}=87.84\cdot 1.643 \]

\[ c_1=144.3\ \frac{\mathrm{m}}{\mathrm{sec.}} \]

According to the velocity diagram:

\[ w_1=73.0\ \frac{\mathrm{m}}{\mathrm{sec.}};\quad \operatorname{tg}\beta_1=0.709;\quad \beta_1=35^\circ 20' \]

\[ \beta_1+\beta_2=35^\circ 20'+30^\circ 00'=65^\circ 20' \]

According to chart No. 6, \(\psi \sim 0.9\), and consequently

\[ w_2=\psi\cdot 9.15\sqrt{\frac{A w_1^2}{2g}+H_k} =82.35\sqrt{1.194\cdot 10^{-4}\cdot 73^2+0.285} =79.06\ \frac{\mathrm{m}}{\mathrm{sec.}} \]

According to the velocity diagram:

\[ c_2=41.2\ \frac{\mathrm{m}}{\mathrm{sec.}} \]

The calculation of losses gives:

Total loss with exit velocity:

\[ H_a=1.194\cdot 10^{-4}\cdot 41.2^2=0.2027\ \text{k. cal.} \]

The drop used in the next stage is

\[ H_u=mH_a=0.67\cdot 0.2027=0.1352 \]

Actual loss with exit velocity

\[ (1-m)H_a=0.33\cdot 0.2027=0.0675\ \text{k. cal.} \]

Loss on the moving blades:

\[ H_s=\frac{A}{2g}\cdot \frac{1-\psi^2}{\psi^2}\cdot w_2^2 =1.194\cdot 10^{-4}\cdot \frac{1-0.9^2}{0.9^2}\cdot 79.06^2= \]

\[ =0.1750\ \text{k. cal.} \]

Loss in the nozzle apparatus:

\[ H_d=(1-\varphi^2)\cdot H_a=0.0784\cdot 2.700=0.2113\ \text{k. cal.} \]

The loss due to disk friction against steam, according to expression (5):

\[ H_r = 2.564 \cdot 10^{-7}\frac{D_{\mathrm{cp}}^{2}\cdot u^{3}}{G_{\mathrm{sec.}}\,v_{\mathrm{cp}}}{}^{*)} = 2.564 \cdot 10^{-7}\frac{0.5^{2}\cdot 78.6^{3}}{4.01\,v_{\mathrm{cp}}} = \]

\[ = \frac{7.78\cdot 10^{-3}}{v_{\mathrm{cp}}}, \]

where \(v_{\mathrm{cp}}\) is the mean value of the specific volume of the steam at the points of the thermal process corresponding to the state of the steam at the outlet from the guide apparatus and at the outlet from the working blades. With a very small, almost imperceptible error, \(v_{\mathrm{cp}}\) may be taken as equal to the specific volume of the steam at the outlet from the working blade. In order, when finding the value of this specific volume, to avoid solving by trial, the following approximate method may be used.

As was already noted, when determining the most advantageous variant, from the blade efficiency of one stage \((\eta_s)\) thus found one can quite accurately determine the overall internal efficiency of any turbine cylinder \(\eta_{ig}\). Moreover, the practice of calculating high-pressure turbines shows that, if a curve is drawn through the terminal points of the thermal process of the individual stages of the turbine cylinder, it almost completely coincides with the straight line drawn through the initial and terminal points of the thermal process of the entire cylinder. These considerations only simplify the problem of finding \(v_{\mathrm{cp}}\). It is necessary, from the overall internal efficiency of the cylinder, to determine the end of the thermal process, draw the mentioned straight line through the beginning and end of the thermal process (Fig. No. 24), divide it into equal segments according to the number of stages (in our case 15), and at each point of division determine, in this way, the heat content, pressure, and

Fig. No. 24.

*) For the present example the peripheral velocity of the working disk \(u\) has been computed from the expression

\[ u = \frac{\pi D_{\mathrm{cp}} n}{60}, \]

and the steam flow per second \(G_{\mathrm{sec.}}\) has also been determined in advance from the expression

\[ G_{\mathrm{sec.}}=\frac{D_{\mathrm{ef.}}N_{\mathrm{ef.}}}{3600}. \]

See p. 25, expression (12).

Fig. No. 25.

specific volume (from the \(Jp\) diagram) at the exit of the steam from each separate stage. If now, in the \(J, v\) coordinates, one plots points according to the values found for the heat content and the specific volume, then a smooth curve drawn through them will make it possible, graphically, to correct the errors inevitable in such a determination of the specific volume (Fig. No. 25). Thus, for the first stage we find:

\[ H_r=\frac{7.78\cdot 10^{-3}}{0.1014}= \]

\[ =0.0765\ \text{k. cal.} \]

For the following stages, all the same quantities enter into the calculation of the thermal process, with the exception of the loss due to friction of the disk against steam, which depends only on the variable value of the specific volume (if one neglects the insignificant, at a small heat drop, leakage of steam through the clearance between the stage diaphragm and the turbine shaft). The results of the thermal calculations*) for all stages are entered in the table mentioned above, and the order of the calculations becomes clear from a comparison of the conditional desig-

*) Omitted in the text so as not to encumber it.

... of the values of the table and of drawing No. 23, and requires no special explanation. It is only worth noting that the specific volumes entered in columns 17 and 18 serve for calculating the flow areas (blade height) of the guide apparatuses and the working passages of the turbine cylinder. The numerical values of these volumes are obtained by the same method from the \(Jp\) diagram and are carried out on the \(Jv\) diagram prepared for this purpose, as was done in determining the losses due to disk friction against steam.

From the values of the specific volumes of steam \(v_n'\) and \(v_{n+1}\) found and entered in the table (columns 17 and 18), the dimensions of the flow areas of the guide and working passages are calculated in the usual manner. The width of the guide blade is selected within the limits of 25–40 mm, with a pitch equal to 18–20 mm. The width of the working blade is kept within 15–20 mm, with a pitch of 6–8 mm.

List of References.

Loezl. The latest achievements in turbine construction. Teplo i Sila, 1926, No. 2.
Baer. Die Regelung von Dampfturbinen und ihr Einfluss auf die Leistungsentwicklung, in den einzelnen Druckstufen. Mitteil über Forschungsarbeiten; Herausgegeben von V. D. I. Heft. 86.
Christlein. Untersuchungen über das allgemeine Verhalten des Geschwindigkeitskoeffizienten von Leitvorrichtungen. Zeitschr. d. V D. I. 1911 S’. 2081.
Christlein. Zeitschr. f. d. ges. Turbinenwesen Bd. 9, (1912) S’. 149.
Flügel. Hydraulische Probleme, Vorträge auf Hydrauliktagung in Göttingen. S’. 139. ff.
Kräft. Neuer Versuch mit Dampfturbinendüsen, Z. Bd 68 (1924) S’. 182.
Some deductions from nozzle Experiments, Engineering Bd 130 (1926) S’ 303.
Stodola. Die Dampfund Gasturbinen C Aufl. Berlin 1924.
Wagner. Der Wirkungsgrad von Dampfturbinen schaufelungen. Berlin 1913.
Warren und Keenan. A machine for testing steam turbine nozzles by the reaction method. Mechanical Engineering Bd. 48 (1926) S’ 227.
Wieselsberger. Physikalische Zeitschrift Bd 22 (1921 und Bd 23 (1922).
Wewerka. Zeitschr. f. d. ges Turbinenwesen Bd 17 (1920) S 265.

1. Stage no.	2. \(i_n''-i_0''=H_0\)	3. \(i_n''\)	4. \(i_{n-1}''-i_n''=H_p\)	5. \(i_n'\)	6. \(i_n'-i_k''=H_k\)	7. \(i''\)	8. \(\dfrac{i_{n+1}''-i_n''}{H_n+H_{n+1}}=\dfrac{H_n}{H_n+H_{n+1}}(1-m)\)	9. \(i_{n+1}\)	10. \(p'\), by diagram at \(S\)	11. \(p_{n+1}'\), by diagram at \(S\)	12. \(p_n''\), corrected	13. \(p_{n+1}''\), corrected	14. \(v_{n+1}'\), by diagram at \(v\)	15. \(v_n''\), by diagram at \(p\)	16. \(v_{n-1}''\), by diagram at \(p\)	17. \(v_n''\), corrected	18. \(v_{n+1}'\), corrected
1	2,6950	751,5050	0,2113	751,7163	0,2850	751,4313	0,3190	754,2	30,00	30,00		30,00
\(1^\circ\)										28,80	28,65	28,80	28,65	0,1014	0,0983	0,0992	0,0980
\(1'\)
\(1''\)						751,4313	0,3190	751,7503
2	2,6650	749,1853	”	749,3966	”	749,1116	0,3173	751,7503	27,60	27,47	27,64	27,52	0,1040	0,1013	0,1022	0,1020	0,1025
\(2^\circ\)
\(2'\)
\(2''\)						749,1116	0,3173	749,4289
3	”	746,8639	-	747,0752	.	746,7902	0,3149	749,4289	26,44	26,32	26,44	26,34	0,1075	0,1056	0,1065	0,1070	0,1070
\(3^\circ\)
\(3'\)
\(3''\)						746,7902	0,3149	747,1051
4	”	744,5401	”	744,7514	.	744,4664	0,3123	747,1051	25,30		25,33			0,1096		0,1100
\(4^\circ\)
\(4'\)
\(4''\)

Row	1	2	3	4	5	6	7	8	9	10	11	12
5	742,2137	*	742,4250	*	742,1400	0,3093	744,7787	25,18	25,23	0,1115	0,1105	0,1110
5′									24,27	0,1140	0,1140
5″									24,25
5‴
6	739,8843	”	740,0956	”	739,8106	0,3076	742,4493	24,13	24,16	0,1164	0,1149	0,1160
6′									23,23	0,1182	0,1180
6″									23,20
6‴
7	737,5532	”	737,7645	”	737,4795	0,3055	740,1182	23,09	23,13	0,1195	0,1192	0,1192
7′									22,20	0,1228	0,1230
7″									22,19
7‴
8	735,2200	”	735,4313	”	735,1463	0,3037	737,7850	22,09	22,01	0,1235	0,1238	0,1238
8′									21,23	0,1275	0,1274
8″									21,20
8‴
9	732,8850	”	733,0963	*	732,8113	0,3012	735,4500	21,10	21,13	0,1272	0,1285	0,1282
9′									20,26	0,1325	0,1322
9″									20,23

1 Point no.	2 \(i_n-i_n=H_n'\)	3 \(i_n^0\)	4 \(i_n-i_n^0=H_{\dots}\)	5 \(i_n'\)	6 \(i_n-i_n'=H_k\)	7 \(i_n''\)	8 \(i_{n+1}-i_n''=H_s+H_i+(1-m)H_a\)	9 \(i_{n+1}\)	10 \(P_n\), according to diagram \(JS\)	11 \(P_{n+1}\), according to diagram \(JS\)	12 \(P_n\), corrected	13 \(P_{n+1}\), corrected	14 \(v_{n+1}\), according to diagram \(JV\)	15 \(v_n\), according to diagram \(JP\)	16 \(v_{n+1}\), according to diagram \(JP\)	17 \(V_n\), corrected	18 Mean \(v_{n+1}\), corrected
10	″	730,5475	″	730,7588	″	730,1738	0,2985	733,1125	19,32	20,13	19,37	20,17	0,1325	0,1376	0,1335	0,1377	0,1340
\(10'\)	″	730,5475	″	730,7588	″	730,1738	0,2985	733,1125	19,32	20,13	19,37	20,17	0,1325	0,1376	0,1335	0,1377	0,1340
\(10''\)	″	730,5475	″	730,7588	″	730,1738	0,2985	733,1125	19,32	20,13	19,37	20,17	0,1325	0,1376	0,1335	0,1377	0,1340
\(10'''\)	″	730,5475	″	730,7588	″	730,1738	0,2985	733,1125	19,32	20,13	19,37	20,17	0,1325	0,1376	0,1335	0,1377	0,1340
11	″	728,2073	″	728,4186	″	728,1336	0,2967	730,7723	18,44	19,23	18,50	19,27	0,1388	0,1430	0,1387	0,1433	0,1388
\(11'\)	″	728,2073	″	728,4186	″	728,1336	0,2967	730,7723	18,44	19,23	18,50	19,27	0,1388	0,1430	0,1387	0,1433	0,1388
\(11''\)	″	728,2073	″	728,4186	″	728,1336	0,2967	730,7723	18,44	19,23	18,50	19,27	0,1388	0,1430	0,1387	0,1433	0,1388
12	″	725,8653	″	726,0766	″	725,7916	0,2947	728,4303	17,60	18,35	17,68	18,40	0,1435	0,1483	0,1441	0,1490	0,1453
\(12'\)	″	725,8653	″	726,0766	″	725,7916	0,2947	728,4303	17,60	18,35	17,68	18,40	0,1435	0,1483	0,1441	0,1490	0,1453
\(12''\)	″	725,8653	″	726,0766	″	725,7916	0,2947	728,4303	17,60	18,35	17,68	18,40	0,1435	0,1483	0,1441	0,1490	0,1453
13	″	723,5213	″	723,7326	″	723,4476	0,2930	726,0863	16,81	17,51	16,85	17,58	0,1494	0,1548	0,1494	0,1533	0,1500
\(13'\)	″	723,5213	″	723,7326	″	723,4476	0,2930	726,0863	16,81	17,51	16,85	17,58	0,1494	0,1548	0,1494	0,1533	0,1500
\(13''\)	″	723,5213	″	723,7326	″	723,4476	0,2930	726,0863	16,81	17,51	16,85	17,58	0,1494	0,1548	0,1494	0,1533	0,1500
\(13^*\)	″	723,5213	″	723,7326	″	723,4476	0,2930	726,0863	16,81	17,51	16,85	17,58	0,1494	0,1548	0,1494	0,1533	0,1500

14
$14^\circ$	*	721,1756
$14'$	*	721,3869	.					16,03
$14''$			721,1019	0,2911	723,7406	16,72	16,04	16,75	0,1540	0,1560
15								15,94	0,1600	0,1621
$15^\circ$	*	718,8280			721,3930	15,95	15,27
$15'$	*	719,0393	.				15,27		0,1665	0,1671
$15''$			718,7543	0,4244*)	729,1787	15,19		15,19	0,1684	0,1670
16									0,1680

*) The loss with the exit velocity is calculated in full, since the 15th stage is the last in the high-pressure cylinder and the steam exit velocity is not used.