BBCBROWN BQVERI

Rotors orLarge Steam Turbines

Publication No. CH-T 060053 E

Rotors for Large Steam Turbines

A. Hohn

As the unit capacity of steam turbosets Increases, so toodoes the sfse of the rotor, and hence also the stressesapplied to it. The various designs ofrotor are discussed andresults of stress calculations given. Rotor materials areconsidered briefly, followed by comment on the futuredevelopment ofrotor design for large steam turbines.

Rotor Configurations

The designs current today are restricted to the formsshown in Fig. l:

-Diagram a shows two rotors, each produced from asingle forging.-Shrinking discs on to a central shai? which transmitsthe torque gives rise to the composite construction ofdiagram b.—In diagram c, separate discs have been welded togetherto form a drum. type rotor [I],

Each configuration has its own advantages and disad-vantages as regards production of the'teel, heattreatment, machining and testing, but these will not bedealt with specifically here. Distinctive differences in thematter of stresses are considered in the following'two sectlolls.

Statics of Rotors under the Influence ofSpeed, Disc Geometry and Temperature

The Disc under the influence ofRotation

introduction

Steam turbines today are remarkable particularly fortheir size: unit capacities of more than l000 MW are nowto be found both in conventional power stations withfossil-fuelled boilers and also in nuclear power plant. Fora number of reasons, unit capacities will rise even furtherin future, and it would bc premature at the moment tospeak of any limit. Machines of this size represent asubstantial financial commitment and in thc event offailure cause serious disruption of the power supply toboth domestic and industrial users. It is therefore under-standable that the manufacturer of such machines does asmuch as the latest state of the technology will allow inorder to ensure that these large machines are reliable inservice.This article is concerned with the heart of the machine,the rotor, and reference is made to the various rotordesigns and the difiercnces between them. Full treatmentof the subject would have to include thc static behaviourin steady-state operation and under transient conditions,and also the dynamics of the rotor under the influence ofthe flow of steam. This, however, would go beyond thcscope of an article, and therefore the main focus ofattention here is on steady-state operation which at allevents constitutes the basis of the mechanical design, andon which all other phenomena arc superimpose*

Cs ti cot (3 —.')o ~Ci~a

8

Disregarding any external tension for thc time being, thecurves of radial and tangential stress are found to be asfollows for:

a. a solid disc:

t. ~'(3+v) (..8

(2)

All designers of turbomachines use thc rotating disc inonc form or thc other as a basic component of the rotor.The following remarks on the rotating disc, which arc ofan elementary nature and can be pursued further in (2, 3,4] for example, are therefore applicable to all, withaccount taken of the boundary conditions particular to aspecific design.Ifthe equilibrium of forces in the radial direction is takenon a rotating disc element of constant thickness, al-lowance is made for the relationship between radial andtangential expansion in thc disc and Hooke's law forbiaxial stress is introduced, we obtain the diflerentialequation of the'rotating disc in terms of a< with thegeneral solution:

dand since crt = —(r rtr) + (2

r'o'l'2

cos (3 ~ «) / I + 3 «2rt ~ rr 'Z

8 ( 3+.b. a perforated disc:

(2 cos (3 + «) t I r Xr22trr r 2 q r 2 rr (4)s I. rz

ts cos (3 + «) I 2 , rt's' —; 3 «I."'+"'*+s I " 3+.

centre of the solid disc, i.e. with err ——clt = (lrs'co'3 + «)/8.Thc result can be seen in Fig. 2. To illustrate more clearlythe mutual infiuences of radial and tangential stress,Fig. 2 also includes the dimensionless comparative stressSv on thc assumption of constant work of deformation,thus:

;-y;*+;*-;.,utld Sv ~ 8 clv

(2 rs'o'3 + 2)

From Fig. 2 wc can draw a first conclusion:

In order to show equations (2) to (5) in general form theyare made dimensionless with the stress prevailing at the

For the same dimension (rs), the same material ((2) andthc same speed (co), the perforated disc will exhibit a

Fig. I - Dlirerent types of rotor construction Fig. 2 - Dimensionless radial, tangential and combined stresses ol'iscsofequal width

2,01,8'tOl 02 03 04 05 r2—~ 0,6 St126'rl

Stt Sv1,4

1,2

rs

1.0

0,9

0,8

070,6

0.5

0,4

0.3

0,2

O,I

O,l S

rtrs

0,2

0.4osr 06/ r

8 or——-Sr ~era er (3 +v)

8 ov-——Sv ~0 r22 ars (3 + r)

0.1 0,2 0,3 0,4 0.5 0,6 0.7 0,8 0.9 1,0

rs8 or

Ors er (3 +«)

higher loading than the solid disc. A measure of this isthe mean tangential stress.This result also remains essentially unchanged when theadditional loads caused by blade tension, steam pressureand shrinkage are superimposed on the rotationalS treSseS.

The considerations presented so far are suFIcient fordetermining the rotational stresses in the case of a soliddisc. For the perforated and shrunk-on disc of Fig. lb,however, deformation also has to be taken into account,owing to the diFerent stiffness of the central shaft and thedisc. Only then can onc define the required degree ofshrinkage, which in turn has an infiuence on the choice ofmaterial.

Deformation Affecting the Perforated Disc

Here we can again start from Eq. (I) and determine theintegration constants C< and Ct appropriate to theboundary conditions. With thc aid of thc calcuhtedstresses it is possible to determine the radial expansion,and hence also thc radial displacement U for any radius ofthe central shaA or of the shrunk-on disc. Of particularinterest are the relative displacements Ut of shaA and discat the point of attachment with radius r1. Thc result ofconsidering deformation in this way can bc read from thcTable. Thus, any expansion of disc orshaA is proportion-al to the forces. art and ars which cause it. There is asquare-law relationship between the expansion and rota-tion ru. Here it must bc noted that for different speeds thcexternal tension ars also varies as the square of the speed.The shrunkwn body has to satisfy the following condi-tions: at the point of contact between disc and shaA at

radius rt the sum of disc, expansion and shalt compres-sion must equal the degree of shrinkage du, i.e.

rtw—rts Utw + UtsWith this it is now possible to construct a "springdiagram" of the shrunk joint (Fig. 3), and within this therelative degree of shrinkage hrjr can bc determined foragiven geometry (rt, r1) and a desired shrinkage force o I~

8) B)CO

mru ii

(tt1) er1icu'(I -r)ri Juan, 4 a

Disc

riix (I —r)+—(I +r)r11

I'iiX

[—(S + r) +(I —rtrli

Figurc 3 shows these relauonships for standstill (co = 0),operating speed (co), overspeed (co ~ co') and liftofspeed(co' I 35co) for a disc of uniform width with a radiusratio ofrr/ri ~ 3. In this diagram thc elasticity propertiesof the disc and thc central shaft have been determined inaccordance with the Table. On the abscissa thc point oforigin is the desired degree of shrinkage (&/ri)o,which isselected according to the residual shrinkage (ordinate)desired at the overspeed condition. The individual com-ponents of the disc and shaft expansion due to rotationand external tension ore have also been taken from theTable. In order to establish the order of magnitude of thecompressive forces ori involved, and also the residualshrinkage, the diagram was compiled using realisticconditions such as occur in the case of I.p. rotors for half-speed steam turbines: n 1500 rev/min, equivalent toco = 157 s-', overspeed co' 12 co; blade tension erisbeing taken as 8 kgf/mrna at the normal operating speed.The residual shrinkage for any speeds can be obtaineddirectly from Fig. 3 by means of thc following conversionfrom the stationary shrinkage diagram. The basic prin-ciples of this are explained in [6].

oct

E

2,$ ~ IO 3

2,0

I,O

O,S

I 2

Uisps

3 4IUsw

rI

Us

ri pg

We have:

(9)

Since the residual shrinkage du at operating speed co isgiven by

co ~ 0

(10)

for the residual shrinkage we obtain

I—

co ~cu'I

I) Fig. 3 - Shrinkage diagram for shrunken discs under dlirereni operatingeondiiions

Thus it can be seen from Fig. 3 that an extremely largedegree of shrinkage (4 15 x 10-') is necessary to achievea lift-offspeed of co' I 35 co, taking into account theblade tension.If for the example in Fig. 3 it had been stipulated thatliiboffis to occur at 135% of operating speed withoutallowance for the external tension etre (i.e. without blad-ing), this would result in the standstill shrinkage diagramshown by the broken line in Fig.3, with a standstillshrinkage of2 6 x 10-s. In this case, however, the bladedrotor would lose its residual shrinkage even at smalloverspeeds (9% in this instance), owing to the bladetension, and some means such as keys would be neededto prevent the disc from slipping. The reserve of speed upto liftwifmentioned here is determined by thc residualshrinkage obtained with Eq. (11).

influence of Disc Geometry

The above statements are of a fundamental nature andaid one's understanding when comparing dificrentdesigns. But in practice the shrunk-on disc is not ofconstant width. The disc meridian will therefore beshaped in some way, it will be formed to yield a disc of

uniform strength or the perforated disc will be given ahyperbolic meridian similar to y = c/rn, in order to makethe best possible usc of the material. This then results in amore gentle disc characteristic than shown in Fig. 3, andhence in a reduction of the necessary shrinkage force. Buthere, too, a very tight shrink fit will still be needed for agreat variety ofdisc meridian shapes, which is one reasonwhy highly tempered materials are chosen for the discs.There are a number of methods (e.g. [2]) for calculatingthe stress in a disc of any technically feasible contour.The method of finite elements has recently come to beused for this purpose, even going to the extent of not onlydetermining the stress conditions in the individual parts(discs) of the rotor, but also of considering the rotor asan entity and taking into account the interactions be-tween neighbouring parts of the discs. A very goodoverall investigation of the rotor is always possible withthe method of finite elements, the fundamentals of whichcan bc found described in [6]. Detailed investigations,

FISA-GridforcnlcttlctinsttreticsR Inttnl ln n I.p. rotor hy the 5nhe dctncnt

methodl3 ~ to

l2

C IC'4 />" in

Injfuenee of Temperature

Under normal operating conditions the rotors of largesteam turbines arc in general exposed to a steady-statetemperature field: after start-up and settling down tonormal load an isothermal distribution becomes estab-lished in the respective rotors which varies only slightly inresponse to moderate load fluctuations.A knowledge of the isotherm distribution in the rotor isnecessary for two reasons:

—first, onc needs to know the local temperature in orderto compare thc local stress present with the characteristicof thc material (e.g. long-time strength) valid at this localtemperature,-second, the isothermal condition gives rise to a stressfield which it may be important to calculate for the totalloading on thc rotor.

This mises the question of how one determines theisotherm distribution in the rotor. Basically this is aproblem of thermal conduction P] in a rotationallysymmetrical body described by the Fourier equation

—~ arhTaTar

(12)

such as in the slots of blade fixings, need more refinedcalculation applied over a very fine grid, while the aid of .photoelastic techniques must bc enlisted for assessing thesurface stress in thc grooves. In this manner one canaccount for all the stress components involved.

where a'tr (l3)

-Thc isotherms in the rotor are found with the aid of anelectrical analogue model, in which case thc rotationalsymmetry of the rotor is accounted for by selectingsuitable resistances (perforations) on the twMimensionalmodel.The conduction of an electrical current through a body isdescribed by the equation

aU—~ —hUat c (l4)

and is thus analogous to the heat conduction equation(l2). Here, U is the applied voltage, C the electricalcapacitance and x thc electrical conductivity of thcmaterial. Lines ofequal voltage U, or equal potential, arcan analogue of the isotherms T ~ constant.

-Another possible way of determining thc temperaturedistribution in the rotor is to solve the heat conductionequation by numerical methods. This possibility hasgained greatly in significance in recent years with. the usc

Before setting about solving this equation one must knowthe boundary conditions, e.g. surface temperature, heatsupplied and removed.In practice, the rotor geometry does not follow a simpleshape and the temperature distribution at the surface iscomplex, owing the cooling eKect of the stcam. Con-sequently, one cannot expect a complete solution to theheat conduction equation. There are nevertheless twopractical ways of solving this problem:

Fig. S- Von Mlscs'combined ttrcssMd of tha toUd Ip. rotor shownla Fig. ta

Values 20 to 47 it//mm~.

-.~20.~

3t immi

1000

20'-''

I I47 40

of finite elements for calculating stress. One has thcadvantage that the results of calculating temperature inthis way lie on the same lattice as the subsequent stresscalculation, and thus can be used as a direct input forcomputing the termal stress.

Finally, as regards determining the isotherms it must besaid that without thc subsequent stress calculation it willalways be fragmentary and yield only moderately usefulinformation.

Practical Results of Stress CalculationsRorarlonal Stresses in Dig@rent LP Rotor Designs

The discussion in the previous section on stress calcula-tion in rotors of different constructions is now illustratedbelow with the aid of a few practical examples.Figure4 shows thc grid imposed on a I.p. rotor fordetermining the mechanical stresses by the finite elementmethod. All the basic designs depicted in Fig. I werccalculated in a similar manner.When computing the stresses, the speed and blade tensionwere kept constant for all types of rotor. Shape, dimen-sions, speed and blade tension correspond to values

'ound in practice.Figure 5 illustrates thc comparative stress field for a I.p.rotor machined from the solid as shown in Fig. Ia. Herethe comparative stress has been taken as according to vonMiscs:

av ~ ~ (ar—at)'+(at —az) +'(ar —ar) .'arr'15)

It will be seen that owing to the abrupt change of cross-section from the central shah portion to thc disc, stressconcentrations as high as 31 kgf/mme occur. Stress con-centrations of this kind are always to be found when theforce field is disturbed as a result of changes in cross-section. Fig. 5 also shows the stress level at thc innerbore, with a radius ratio of rt/rs ~015. At47kgf/mmthe stress herc reaches a very high value, although it isstill always below that of shrunken discs. Results ofcalculating the stresses in shrunk-on discs arc shown inFig. 6. Owing to the larger central bore for the shaft amuch higher stress of 68 kgf/mm's found here, other-wise the conditions are the same as in Fig.5. At thetransition from thc slim part of the disc to the broadouter shoulder one can again see a stress concentration inthe corner of the divergence, attaining local values of70 to80 kgf/mme and caused chiefly by disruption of the radialstress pattern.A technique often used in the past was to secure theshrunk-on discs with extra keys. This inevitably gives riseto stress concentrations in the keyway which in the mostfavourable case have a stress concentration factor ofabout three. What this means with the high basic stresslevel of a perforated disc is easy to appreciate: from thcstart a plastic zone will form round the slot which, ifthcproperties of the material are less than ideal, can lead tocracking and hence to failure of the disc when it isrotating. Sufficient instances of this have unfortunatelyoccurred in the past [9, IO). In order to meet thestandards, of reliability required in power stations, there-fore, it is essential that no keys of any kind should bcprovided as an extra means of securing the discs.As already explained in connection with Fig. 2, thc soliddisc will show the most favourable stress characteristics.

0-44 55 44 15 h 4) 1) l5

RS. 6- Coro biped Strett tidd of aLp. disc rotor ot showa la Fia. lb,Ia )$Sf/a)a)j

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Figure 7 illustrates the combined str«ss distribution (aAervon Mises) in a welded drum rotor of a type found inmachines of over 1000 MW. Tlie boundary conditions-outside diameter and bladntension- are comparable withthe designs shown in Fig. 5 and 6, the speed being takenas 1800 rcv/min in all the cases shown. It will bc noticedthat with a rotor of this kind, which is composed of soliddiscs, thc greatest stress is roughly between 40% (Fig. 5)and 60% (Fig. 6) lower than for rotors machined fromthe solid or for shrunken discs. This fact will again beimportant when considering the choice of material andthc bursting speed.

HP and IP Rotors, Including Temperature sects

Figurc 8 shows the isotherms in a welded h.p. rotor underconditions of full load. Here onc can see thc charactcris-

tic feature of steady-state operation that the isothermsrun almost perpendicular to thc axis of rotation, and onthe basis of the isotherm distribution one can predict thatthe thermal strcsscs will be very small compared to the.stresses caused by rotation. In this example they in factamount to only some 5 to 10% of the mechanical stresses.In contrast to the cold low-pressure section, thc mechani-cal design of rotors exposed to high temperatures in-cludes their behaviour in relation to time. Because ofcreep phenomena, which willbe discussed in more detailin the next section, the material ages in the course oftime. This ageing process is a function of the material,temperature and stress, as well as time, and therefore inorder to assess the suitability of a design one must knowall these parameters, i.e.

- the behaviour of the material as a function of loading,temperature and time,

Fig. 7 - Combined stress IIela welded drum rotor as shownin Fig. Ic

Values 20 to 28 ltgf/mme.I

20

I/

+2g

I000

20

26

- thc isotherm distribution in the rotor, and—the stresses in the rotor.

An example of a detailed study of a high-pressure bladefixing is shown in Fig. 9. Using photoelastic techniques,the edge stresses in the lateral grooves are determinedunder diFerent loads and added as supplementary in-formation to the results of a refined stress calculation(Fig.9). In this way, together with allowance for thebehaviour of the material and stringent production quali-ty control, it is possible to guarantee the performance ofthe rotor over many years.

The Rotor IVlaterialHigh-Pressure and Intermediate-Pressure Rotors

The rotors of modern large steam turbines are,all offcrritic material. This is related to the fact that forconventional plant the world over the live steam tempera-ture has become established at 538 'C. With this materialone can expect good long-time properties, no softening,

little creep, uniform heat treatment, adequate long-termductility, low notch sensitivity and good resistance toscale.Nuclear power stations at present do not raise anyproblems of temperature because the turbines run onsaturated steam, and even the high-temperature reactorsfor large power stations will not exceed the live steamtemperature of conventional plant, at least in the nearfuture.Figurc l0 shows two typical rotor steels [11] used for h.p.and i.p. turbines. To allow internationally consistentcomparisons, the long-time rupture values for l00000hours are taken as a basis for mechanical design pur-poses.The following remarks survey briefly the behaviour ofrotor materials under thc influence of temperature, stressand time.Ifa test bar is subjected to a load att and at thc same timea temperature Te, it will in time undergo plastic elonga-tion (creep) and finally break. For the same loading thebar will fail earlier with a higher test temperatureTt ) Te than with a lower temperature.

Fig.g - Isotherm distribution ina welded h.p. rotor

450

$ 00

5 I0 'C

460'00

I10

II ti Ia fig.

9- Combined stresses in thegrooves of h.p. blade Axing'the Agura denote the von Miscscombmcd stress m ltgtimms.

rI 3

500 'C 505 C 5IO C sls C

The creep process is illustrated in Fig. Il. We candistinguish three main phases of creep; primary (I),secondary (II) and tertiary (III), in which the bar rapidlyreaches breaking point. All high.pressure and interme-diate.pressure rotors operate within the secondary phase,and the designer has to make sure that his design has anadequate reserve with respect to the tertiary stage. In thesecondary phase the rate of creep i ~ de/dt is constant,

which in practice makes it easier to assess the rotor aftera long period in service. Every time the turbine isinspected, specially provided control diameters aremeasured and the results compared with measurements ofprevious years. Here,'however, account must bc takenof the fact that thc creep rate within a disc varieswidely from inside to outside owing to variations inthe stress.

- I.ong-timetion of steels 24

fig.loeomposi

fnptUre cnrvcsCrMov 55 and

eea in hgrtmms and chemical Lo~~pressure Rotors2l CrMov 5 I I7060

50

40pelt'0

25

20

IO9g1

6

5

4

IO

2I Cr Mo V5ll

ooo

iso.

24 Cr Mo V55

io'O«h

o ~

J ~

Whereas for high-temperature conditions the number ofdiHcrent rotor materials used by the various manufac-turers is limited, the selection of materials for low-pressure rotors is much wider. This is not all thatremarkable when one remembers the variety of l.p. rotordesigns, because the material is principally matched tothe dill'erent stress conditions of the individual types ofconstruction. Furthermore, because these rotors areessentially cool, the factors governing the choice of'material will only be the yield point, ultimate strength,elastic limitand notch toughness. Here it is assumed thatthe rotor operates in the upper part of the notch-toughness range, i.e. thc fracture appearance transitiontemperature is below thc operating temperature.Recently, and not the least of thc reasons being severalcases of explosive failure of solid and shrunkMisc rotors,which also extended to nuclear stations [IOJ, there hasbeen a tendency to base the choice of material onadditional criteria in order to avoid such instances ofbrittle fracture. For this, the rotor is considered from thestandpoint of fracture mechanics, the aim being to arriveat appropriate values of crack resistance and rate of

I05 propagation for subcritical crack growth Without goinginto the fundamentals of fracture mechanics —the subject

IO

rt)ro rt)ro

//

rtr const.0 /dl ) do

/z

ro

IOltN I

fig. 1 I - Creep curves for difFerent temperatures and loads (schematic)

failure of all control and safety systems. In this hypothet-ical situation, rejection of the electrical load would causethe rotor speed to run away, possibly resulting in ex-plosive failure.Our own studies have shown that h.p. and i.p. rotorshave a much higher bursting speed than I.p. rotors. Thereason for this is that the high and intermediate-pressurerotors stretch radially less than the low-pressure rotors,and while the material characteristic governing bursting isthe yield point, h.p. and i.p. rotors are generally designedto withstand long-time failure. Since the value for long-time failure is only a fraction of the corresponding yieldpoint. depending on the temperature, these rotors have alarger reserve with respect to the bursting speed than doI.p. rotors.In order to study the behaviour of diFerent disc designsin relation to the bursting speed wc again use the disc ofuniform width as a starting point. Grammel has shown[14) that the mean tangential stress in the disc is suitableas a measure of the resistance to explosive failure. Themean tangential stress trtM is given by

rs

crt dr(16)

is treated in (12) and [13), for example —it should bementioned that this aspect of mechanics was originallyevolved for high-strength, relatively brittle materials.However, it is only suitable for describing a crack whichalready exists, and takes no account of the actual forma-tion of the crack. At the same time it should not beforgotten that turbine rotors consist of ductile materialswhich have the ability, if need be, to fiow locally anddisperse stress peaks, thus preventing cracks from form-ing, or at least greatly delaying their onset.It can, of course, happen that there is some justificationf'r examining a rotor from a fracture mechanics view-point. This willalways be so if, because of the high levelof disc stresses, one has to resort to high-strengthmaterials or when, as in the case of solid low-pressurerotors, the large dimensions, make it very difficult todetect faults inside the forging. It may then be ofadvantage to assume a fault of a certain size in a certainposition and check to see what the consequences might bein the course of time.

crt×fs rt

and can be written in dimensionless form as follows:

n8 trt dr

g res tos (3 + t )StM

fs —rt(17)

(18)

'he

ratio of thc bursting speeds of perforated disc tosolid disc is then described by

For trt we then use Eq. (3) for a solid disc and Eq. (5) fora perforated disc. Ifwe now write the ratio of the mean,tangential stress Stwr, of the perforated disc to the meantangential stress Stlv of the solid disc, we have

Bursting Speed

2,0

l,ge sax, Setsasar Soir

l.g

1,2

V g

l.o

O,g

heal. I / 1( l+—"„+(

—,"j',6

0 0,2 0.4 0,6rtPs

Fig. lz - Ratios of mean tangential stress and berating speed lorperforated and solid discs

Figure 13 shows in qualitative terms the behaviour of twodiff'erent I.p. rotor constructions at elevated speed. Forthe same size. blade tension and operating speed, thctangential stress for the drum rotor will follow curve I.The same applies to the disc rotor, but at thc borediameter chosen this rotor shows a stress roughly doublethat of thc drum rotor. In the elastic region there isproportionality between the stress and the square of thespeed. If the speed is raised relative to the normal speedby a factor of I 4, for example, the stresses increase by afactor of 196 (curve 2). The inner portion of thc per-forated disc is then already beyond the yield point, andthe corresponding zone relaxes.Owing to plastic deformation, therefore, the elasticcurve 2 gives way to curve 2'nd thc parts of the discwhich are still elastic arc thus subjected to additional,stress. According to what has been said.so far, a measureof the rcscrve with respect to fracture is the ratio of the.yield point to the mean tangential stress, i.e.„essentially.the area in Fig. 13 contained between curves I and theyield point. The root of this area ratio represents the.relationship of the bursting speed of thc two designsshown in Fig.13. If one wishes to compensate thedisadvantage of the lower fracture speed of a perforateddisc by using more highly tempered material, the increase-in yieltI point required for a perforated disc can also be.found with Eq. (18) (Fig. 12). It can bc seen that with theradius ratios occurring in practice it is difficultto achievea perforated disc of such a quality that it is equivalent toa solid disc as regards its bursting speed. This wouldmean high-strength material has to be used, with theconsequent higher risk of brittle fracture.

Fig. ls - Behaviour of tteo types of I.p. rotor at «lerated speed

~tdidi

2l

dl

di Pa

Outlook

As mentioned earlier, the unit capacity of large steamturbosets will continue to risc in thc foreseeable future,and hence.influencc the demands made of the rotors. Adecisive, and to solne extent limiting, factor over the pastdecade was the final stage, which ifthe vacuum was goodhad to handle enormous flow volumes. Allmanufacturersof steam turbines therefore carefully developed longerfinal blades and introduced these to the market. Butlonger blades also means a larger rotor diameter, accom-panied by higher centrifugal loadings on both blades androtor. To keep stresses below the limit, the speed of themachines was halved. The technique employed in theUSA was to run thc high and intermediate-prcssuresections at thc full speed of 3600 rev/min, and combinethe low-pressure units with a 4-pole generator on asecond shaft string running at 1800 rev/min. Europe lateradopted the idea of the half-speed machine, although in

12

single-shaft form and only for nuclear plant. By halvingthe speed in this way. and at the same time doubling, thesize, the stresses in full-speed and half-speed machineswere kept the same, but the corresponding exhaust areaof the final blades increased fourfold.A feature of recent years has been a growing worldwideshortage of cooling water [15]. In the industrializedcountries, and these if only because of their. powerdistribution networks are the potential buyers of largemachines. it is becoming no longer possible to usc freshwater for cooling purposes. Future large power stationswill therefore be equipped mainly with wet or dry coolingtowers. which means the turbine vacuum willbe rclativclypoor and the steam exhaust volume correspondinglysmaller. It may thus well be that the final blade lengthsand I.p. rotor dimensions customary today will be ade-quate for some time to come, without being tied to half-speed I.p. sections because of thc stresses, even with largecapacities. It is likely that large machines for nuclearpower stations, with poor vacuum, will also be built forfull speed and still be able to cope with thc stresses in theblades and rotor.The possibility of making the I.p. rotor relatively smallalso improves the chances of the solid-rotor design tosome degree. Great advances in forging technology havebeen made over the past few years, and this has increasedconfidence in the use of forged one-piece shafts. Finishedweights of over 200 t have been achieved to date. Theserotors require an ingot weighing morc than 400t andwith the associated risks can be produced only in Japanand the United States. It is improbable that the steel-works will contemplate a further increase in rotor size.with the correspondingly heavy investment needed to dealwith larger ingots, because the market for these largeforgings is too restricted. The concept of the large one-piece rotor can therefore be extrapolated into the futureto only a limited extent.Thc situation is slightly different for the high-pressuresection. On the assumption that future nuclear powerstations will also operate with steam conditions such asare found today in conventional plant (I50 to 250 bar,538'C), the very size of the I.p. rotor could present astress problem. Overcoming this can be approached intwo different ways: the material and the design.There is no likelihood in the near future of finding adifferent material for h.p. rotors which has substantiallybetter long-term properties and does not forfeit theadvantages of the low alloy steels used at present. Muchmore probable is that stresses in the h.p. rotor can bekept in check through suitable design: the large steamturbine today is quite clearly following thc path takenmany years ago by the gas turbine towards cooling therotor by means of steam. The designer thus has at his

Symbols

F. = Modulus of elasticityL = Perforated discS< ——- Dimensionless radial stressSi = Dimensionless tangential stressSist =- Dimensionless mean tangential stressSv ~ Dimensionless equivalent voltageT = TempcraturcU = Radial displacementV = Solid disca = Thermal conductivityc = Specific heat of rotor materialnnn =. Bursting speedr = Considered disc radiusri = Inner radius of perforated discri ——Outer radius of dischr = Degree of shrinkagedu—~ Relative degree of shrinkagetir ~ Timee< ——Radial expansion

= Tangential expansion= Conductivity of rotor material~ Transverse contraction ratio= Specific mass of disc material

e< = Radial stresse

Bibliography

[I]A. LNhyr Some advantages of welding turbine rotors.Weld. J. June 1968.

[2] C.B. Bienzeno, R. Grammel: Technische Dynamik,vol. II, Springer 1953.

[3] W. Traupel: Thermische Turbomaschinen, vol. Il,Springer 1960.

[4] K. Lofter: Die Bcrechnung von rotierenden Scheibenund Schalen. Springer 1961.

[5] A. Bald: Besonderheiten grosser NassdampAurbo-sgtze. Mitt. Vereinig. Grosskesselbesitzer S2 1972 (4).

[6] 0.C. Zleriklewlczr The finite element method in struc-tural and continuum mechanics. McGraw-Hill, London1967.

[7] B. Bauler Die Mathematik des Naturforschers undIngenieurs, vol. IV. Hitzel 1952.

[S]H. Lelpholz: Festigkeitslehre fur den Konstrukteur.Springer 1969.

[9] H.D. Enunerr Investigation of large turbine spindlefailure. ASME Paper 55 - A 17?[IO] D. Calderonr Stcam turbine failure at Hinkley Point.Proc. Inst. mech. Engrs 186.

[II]Stghte fQr grQssere SchmiedestQcke (GQtevorschrift).Stahl-Eisen-Werkstoffblatt 550- S7.

[12] K. Hecke/: EinfQhrung in die technische Anwendungder Bruchmechanik. Hanser 1970.

[13] D. Radaj: Grundlegendc Beziehungen der lincar-elastischen Bruchmechanik. Schweissen u. Schneidcn 231971 (IO).

[14] R. Grammel: Die Erklarung des Problems der hohenSprengfestigkeit umlaufender Scheiben. Ingenieur-Archiv16 1947 (I).

[IS] H. Flohn, D. Hensehler, H. Schuller:" Der Wasser-

haushalt der Erde. Aus: Mensch und Umwelt. Tech.Rdsch. 64 1972 (47).

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