turbinas1

CHAPTER

FIVE TURBINES

5-1 INTRODUCTION

Usiog steam to provide mechanical work probably owes its birth to the need for pumping water from coa! mines. The very first successful attempt at this was a "pump-iog engioe" built by Thomas Savery (1650-1715) in England. In Savery's engine, steam at pressures betweeo 50 and 100 psig (4.5 to 8 bar) acted directly upon the surface nf water in a chamber to force it up through a pipe. A crude check valve xevented .eversc ftow. After the chamber was clea.ed of water, steam supply was manually cut off and cold water wus applied over the chamber, thus condensing the steam ioside and c.eating a vacuum that sucked in mo.e water. Tbe direct contact between steam and water caused large condensation losses, and the lack of safety valves was responsible for many explosions.

At about the same time, Denis Papio .(1647-1712), who invented the safety val ve, conceived nf the idea nf separating the steam and water by a piston, and Thomas Ncwcomen (1663-1729) designed and built an engioe with one. In it low-p.essure steam was admitted to a vertical cylinder, where it pushed the piston upwards. The steam lcft in the cylinder was theo condeosed by a jet of outside cold water, creating a vacuum in thc cylinder. Tbe outside atmospheric pressure pushed the pis ton back on thc working stroke, hcnce the name "atmosphcric engine." The pisto o was attached to onc end of a beam that had a fulcrum about midpoint. A piston in a separate pumping cylioder was attached to the other end. The pump piston was smaller in diameter than thc steam piston, so a g.eater water pressure Iban steam pressure was obtained. The arious valves of Newcomen's engine were operated manually at first. Automatic opcrarion, the first on record, was conceived by a smalllad who was hired to operate

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174 POWERPLANT TECHNOLOOY

the valves. Being smaller and lazier than the others, as the story goeso,, 1h~~elh:~:~ regular pattem of beam and valve operation and rigged up a string n allowed the beam to actuate the valves. Newcomen's engine used one-third per hph than Savery's.

It was not until sorne 60 years later that James Watt* developed the ideas "modem" reciprocating steam engine. Working asan instrument maker, he wsc~l;.. upon one day in 1764 to repair a Newcomen engine and noticed the waste condensed in the cylinder. In 1765 he conceived of the idea of a separate contden,.; and subsequently ideas such as the working stroke caused by steam expaosion double-acting cylinder, the Hyball throttling govemor, the conversion of ' to rotary motion (in 1781), and other importan! features. His now-famous engine a majar contributor to the industrial revolution. Watt's engine used 60 pen:ent coal than Newcomen's and 75 percent less than Savery's.

The next majar improvement was made by Corliss (1817-1888), who deveJ~, the quick-closing intake val ves that bear his name which reduce throttling dllri.. closing. The Corliss engine used about half as much coal as Watt's but still f~~-five times as much as modem steam-turbine powerplants. Next carne Stumpf (!86)..\ _ who developed the "uniftow engine," which was designed to further reduce coll

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11JIUIINES 175

Figure S 1 The aeolipile of Hero of Alexandria (from Aeolus, "god of the winds," and pila, a "ball") the first recorded steam turbine in h.istory.

thus causing it to rotate. This turbine operated on the impulse principie (Sec. 5-2). Later yet, in 1831, William Avery of the United States bui1t the first steam turbines, which were used commercially in sawmills and woodcutting shops, with at least one tried on a locomotive. The Avery turbine had similarities to Hero's in that it used a hollow shaft with two 2.5-ft-long hollow arms attached to it at right angles with a small opening at the end of each and each pointing in opposite directions. Steam supplied through the hollow shaft exited through the openings and caused the shaft to rotate. Avery's, like_ Hero's, was therefore a reaction turbine. The turbines, though claiming similar effidencies as contemporary reciprocating steam engines, were aban-doned because of high noise leve!, difficult regulation, and frequent breakdowns.

The turbines that were destined to take over from the reciprocating engine, how-ever, carne about late in the nineteenth century as a resu1t of the effons of a handfu1 of men, the most prominent of whom were Gustav de Lava!* of Sweden and Charles Parsonst of England. de Lava! first developed a small, high-speed (42,000 r/min)

Carl Gustav Patrick de Lava! (1845-1913), an engineering graduare of the University of Upsala, Sweden, was an inventor whose main income carne from a cream separator and was spent on various olher l.lnprofitable invenrions. The turbine he invented was intended for a cream separator. Also active in public

lffai~. he became a member of both houses of parliament and was honored repeatedly for his contributions to !tthnology.

~Sir Charles Algemon Parsons (1854-1931), an upper-class Englishman, was motivated by the need 10 find a steam drive for ships. He is credited with developing the reactionstage principl~.

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176 POWERPLANT TECHNOLOGY

reaction turbine but did not consider it practical-and so tumed to the deveiPnenl .,~ a reliable single-stage impulse turbine, whicb bears bis name until today. He is alsQ . credited witb being the first to employ a convergent-divergent nozzle for use in turbine. The first unit was tested in 1890 and the first commercial unit, 5 bp, into service in 1891. In 1892 he built a 15-hp turbinc with two whccls for ships: for forward and one for astem propulsion. Parsons dcveloped the multistage l'

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d ' l. ,.

" !J

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111RBINES 177

In more modern times, during World War II developers in Switzerland, a country solated by the war, developed the technology for power generation by gas turbines. ~ir Frank Whittle of England was one of many who recognized the applieability of as turbines for et propuls10n of aircraft. Such efforts led to the development of the ~et fighter and subsequently jet transport in many countries.

J The gas turbine is now used in the utility industry mainly as a peaking unit (to deliver excess power during periods of high demand), for powering isolated locations, on 0 pipeline routes, and more recently, in combined-cycle (gas and steam) pow-erplants (Chap. 8).

S-2 THE IMPULSE PRINCIPLE

Before discussing the impulse turbine, a review of the impulse principie may be useful. consider a horizontal fluid jet impinging in the + x direction on a fixed vertical flat plate (Fig. 5-2a). That fluid willspread out along the plate, its vel~ity in the direction of the jet reduced to zero, and wdl1mpart to 1t a honzontal force F m the + x duecuon. This force is called an impulse and is equal to the change in momentum of the jet in the + x direction

m F=-(V,-0) g,

where F = force or impulse, lb1 or N

m = mass-flow rate of the jet, lb..;s or kg!s V, = velocity in the horizontal direction, ft/s or m/s

(5-1)

g, = conversion factor, 32.2 lbm ft/(lb s2) or 1 kg rni(N s2) Now consider that the plate is free to move in the horizontal ~irection (Fig. 5-2b) with a velocity V8 V, - V8 will be the velocity of the jet relative to the plate. Now the fon:e on the plate is

(o)

m F =-(V, - Vs) g,

V

(b)

Figure S-2 The impulse of a fluid jet on (a) a fixed fl.at plate and (b) a moving flat plate.

(5-2)

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178 POWER!'L

m 2m F = -[V, - (2Vs - V,)] = -(V, - V8 ) 8c 8c

and the work per unit time is FV s or . m

W = 2-Vs(V, - Vs) g.

WRBINES 179

(5-7)

(5-8)

We shall now define a blade efficiency 11 as the ratio of the power, Eq. (5-8), to the inilial power of the jet, mV:!Zg., or

[v. (v)2] 17=4 V,- V, (5-9) and F, W, and 17 Eqs. (5-7) to (5-9), for the blade are twice the values for the tlat plate, Eqs. (5-2) to (5-4).

To find the optimum blade velocity that results in maximum power, again dif-ferentiate W with respect to V8 and equate to zero, also giving

and

V, v. =-

""' . 2

. mV: Wnwr. =-2g.

(5-10)

(5-11)

The optimum blade velocity is half the jet velocity, the same as for the tlat plate, but tbe maximum power is twice that for the tlat plate and equal to the total kinetic energy (per unit time) of the jet. In other words, the maximum b/ade efficiency as can be verified from Eq. (5-9) is

17.- = 100% (5-12) Because the blade moves away from the jet, continuous power can be obtained only if a series of blades were mounted on the circumference of a wheel so that as the wheel rotates they conlinually face the jet. A high-speed jet needs a nozzle which has physical dimensioos so that it is impossible to have the jet impinging on the blades in their direction of motion but at a shallow angle 8 (Fig. 5-4). Tbe blade-entrance angle also cannot be zero from horizontal because it should correspond nearly to the relative !luid direction. Tbe blade-exit angle also cannot be zero from horizontal or

1

Axis or rotation

~:>~):l-+x 1

1 Figure 5-4 Top vicw of a row of impulse blades on wheel.

i ~ i !!

!~ 1 : !i 1 i

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180 POWE:RPLVIT lCCliNOLOGY

- else the fluid would not be able to leave the row of su=ssive blades. The practica blade, therefore, is turned around an angle less !han 180".

The Velocity Dlagram To evaluate the work on the blade, which is in the direction of motion, one then need_, to construct a velocity vector diagram, shown for a single blade in Fig. 5-5. Fillllle 5-5a shows the velocity diagram in relation to the blade. Figure 5-5b and e shows simplified versions of it, called "extended" diagrams, with the blade shape removed.' In these diagrams

V" = absolute velocity of fluid leaving nozzle V8 = blade velocity V,1 = relative velocity of fluid (as seen by an observer riding on the blade) V ..:z = relative velocity of fluid leaving blade Va = absolute velocity of fluid leaving blade

O = nozzle angle 4> = blade entrance angle y = blade exit angle 8 = fluid exit angle (

1 The work on the blade may be obtained from impulse-momentum principies as t

above or from first-law principies. Both methods yield numerically identical results. From impulse momentum, the force, a vector quantity in the direction of motion

of the blade, is equal to the cbange in momentum of the ftuid in the direction of motion, or

m F = -(V" cos 8 - Va cos 8)

~ (5-13a)

The componen! of the steam velocity in the direction of blade motion is called thc velocity ofwhirl. Thus V,1 cos O is the entrance velocity of whirl Vw" and V,2 cos 3 is the exit velocity of whirl V w2 and the force may be written as Th

Th

Ao. whe

v, iniJ V, e

(a) (b) (e) ('-1( Flpre s.5 Velocity diagrams on a singJ~stage impulse blade. 1

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m F = -(V., - V.,) K e

TURBINES 181

(5-13b)

Th work per unit time (power) is equal to the product of force and distance traveled bY the blade in unit time or the product of force and velocity. Thus

. mvs W = ---"(V,1 cos fJ - V a cos S) K e (5-14)

Note that if both fJ and 6 are O, V,, = Vs - (V,, - Vs) in the + x direction (for frictiooless, nonexpansion or noncontraction flow), and Eq. (5-14) reverts to Eq. (5-8). Note also that cos 6 is positive if 6 is less than 90 (Fig. 5-5b) and negative if 6 is greater than 90o (Fig. 5-5c), so that the worlc is greater if 6 is greater.

The blade efficiency again is defined as the ratio of the blade work, Eq. (5-14), to the initial energy of the jet m~,l2gc or

11 = 2[(v) cos 8- (v)(v'') cos 6] Ysl Vst Yst (5-15) Optimum blade speed. By analogy with the 180 blade, the relative velocity of fluid entering blade in the +x direction is V, 1 cos fJ - Vs. With no friction, expansion, orconttaction, that is also the relativo velocity leaving the blade but in the - x direction. The absolute velocity of the fluid leaving the blade in the + x direction, V,2 cos 6, is therefore V8 - (V,1 cos 8 - Vs) = (2V8 - V1 cos 11). Equation (5-14) can thus be written in the form

. 2mV8 W = (V,1 cos 8 - Vs) (5-16) K e The optimum blade speed that yields maximum worlc is again obtained by differen-ating W with respect to V8 and equating the derivativo to zero, giving

Ysl COS 9 v ..... = 2 (5-17)

The maximum work is obtained by substituting Eq. (5-17) into Eq. (5-16), giving ,;, 2m

W...,. = ;;-(2

V, 1 cos 11)2 = -Vi . .,. (5-18) gc 8c

The maximum blade efficiency is obtained by dividing W...,. by mV;112gc, giving '7s.m.u = (cos 11)2 (5-19)

An examination of the velocity diagram sbows that in ideal flow wbere V" = V,. and when = y, the optimum blade velocity wbich results in maximum worlc also results in 8 = 90, or absoluto exit velocity siraight in the axial direction. ln that case V, cos 8, the exit velocity of wbirl, is zero. Equations (5-17) to (5-19) revert to Eqs. (5!0) to (5-13) for (J = O. .

From the jirst-law principies, with no change in potential energy and no heat

. -- ---. '---~--------~-

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transfer, the work is equal to the de crease in enthalpies and absolute kinetic energie, of the fluid

W = (H, H ) + . (v;, v;,) - 2 m---2gc 2gc (5-20) where H 1 and H 2 are the enthalpies of the fluid entering and leaving the blade

' respectively. H, - H2 is obtained by considering fluid flow relative to the blade (as t seen by an observer riding on the blade), where only relative velocities and no wor1 ! are observed. Thus

H,- Hz= (V~ V~1) 2gc - 2gc Combining with Eq. (5-20) gives

. ,;, W = -

2 [(V;, - v;,) - (V;, - V~)]

8c

(5-21)

(5-22)

This is a general equation for the work in any blade, i.e., including friction, expansion, or contraction of the fluid through the blade passage. In the case of apure impulse blade, where none of these effects is present, H1 = H,, V,1 = V,2 , and

m 2 2 W,.,.. ._ = ::"-

r

S

l

.. -. -. -~ --- --~ .-... ---- --

1URBINES 183

Figure 5--' T-s diagram for nozzle of s Example 5-l.

Ieaving nozzle if it were adiabatic reversible ata ares. = 1.5901 Btu/(lbm 'R) and by interpolation at 200 psia, h. = 1237.2 Btu!lbm.

Nozzle isentropic enthaipy drop = 1307.4 - 1237.2 = 70.2 Btu!lbm

Nozzle actual enthalpy drop = 0.9 X 70.2 = 63.18 Btu!lbm V,, = Y2 x 32.2 x 778.16 x 63.18 = 1779.4 ft/s

Refer to Fig. 5-5

where 8 = 20", V,1 cos 8 = 1672.1 ft/s

v ..... = V,1 cos IJ/2 = 836.05 ft/s

from which

from which

V,1 sin = V,1 sin 8 = 608.6 ft/s

V,1 cos = V,1 cos 8 - V8 = 836.05 ft/s

V,1 = 1034.1 ft/s and = 36.05"

V,.z = k,.V,1 = 1034.1 x 0.97 = 1003.1 ft/s V ,.z sin y = V,2 sin 8, y =

v,. cos y+ v,2 cos 8 = v. V,2 = 590.8 ft/s, 8 = 87.57"

Refer to Eq. (5-14) 1000 X 836.1

W = 32 (1672.1 - 25.1) = 4.28 X 107 ft 1b!S

.17

= 77,818 hp = 58.03 MW


Refer to Eq. (5-22) . 1000

W = 2 X 32.17 [(1779.42

- 590.82) + (1003.12 - 1034.12))

= 4.28 X 107 ft 1biS

which confinns Eq. (5-14). W 4.28 X 107 X 2 X 32.17

B1ade efficiency = (mv;,ng.) - 1000 x 1779.42 X 100 = 86.97%

W 4.28 X 107 Stage efficiency = . (h _ h = =::::-...:..:.:::':-7--""::=~ X 100

m 0 ) 1000 X 70.2 X 778.16

= 78.35%

S-3 IMPULSE TURBINES

Impulse turbines or turbine stages are simple, single-rotor or multirotor ( compounded) turbines to wbich impulse blades are attached. Impulse blades can be recognized by their shape. They are usually symmetrical and have entrance and exit ang1es, .p and 'Y respective! y, around 20. Because they are usually used in the entrance bigh-press~U> stages of a steam turbine, when the specific volume of steam is low and requires mucb smaller ftow areas than at 1ower pressures, the impulse blades are short and havc constant cross sections.

Impulse turbines are also characterized by the fact that most or all of the enthalpy, and hence the pressure, drop ol:curs in the nozzles (or fixed blades that actas nozzle.) and little or none in the moving blades. What pressure drop occurs in the moving blade is a result of friction that gives rise to the velocity coefficient k, discussed above. , Single-rotor and compounded impulse steam turbines will now be discussed.

The Single-Stage Impulse Turbine The single-stage impulse turbine, also called the. de Lava/ turbine after its invelllllo (see lntroduction, Sec. 5-1), consists of a single rotor to which impulse blades 111 attached. The steam is fed through one or severa! convergent-divergent nozzles whkl do not extend completely around the circumference of the rotor, so that only par! d ~ the b1ades are impinged upon by the steam at any one time. The nozzles also allot < goveming of the turbine by shutting off one or more of them.

The velocity diagram for a sing1e-stage impulse turbine has been shown in F'~J

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nJRBINES 185

Figure 5-7 Overall steam prcssure and absolute steam-velocity changes in an ideal single-stage impulse (deLaval) turbine.

!he steam occurs at nozzle exit and decreases from V,, to V,2 in the blades. The linear changes in pressure and velocity shown are only schematic and do not represen! the actual processes.

Compounded-lmpulse Turbines lt has been shown that the optimum blade speed in a single-stage impulse turbine is 0.5 V,, cos 8 or roughly one-half of the incoming absoluto steam velocity, 8 being small. Steam expanding from modero boiler conditions, say 2400 psia and !OOO"F to !he condenser pressure of 1 psia ( or e ven to atmospheric pressure) in a single nozzle stage, will have velocities of about 54QO ft/s (1645 mis), meaning a blade speed of 2700 ftls (820 mis). Such a speed is far beyond the maximum.allowable safety limits because of centrifuga! stresses on the rotor material. In addition, the large steam velocities result in large friction losses (proponional to the square of the velocity) and a reduction in turbine efficiency. The higb rotor speeds would also necessitate large and bulky reduction gearing to the electric generator. To overcome these difficulties, two methods ha ve been utilized, both called compounding or staging .- One is the velocity-compounded turbini:, and the other the pressure-compounded turbine.

The Velocity-Compounded Impulse Turbine The velocity-compounded turbine was first proposed by C. G. Curtis (see lntroduction) to solve the problems of a single-stage impulse turbine for use with high pressure and temperature steam. The Curts stage turbine, as it carne to be called, is composed of one stage of nozzles as the single-stage turbine, followed by two rows of moving blades instead of one. These two rows are separated by one row of fixed blades attached to lhe turbine stator, which has the function of redirecting the steam leaving the first

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186 POWERP!...AN"l' TECHNOLOGY

row of moving blades to the second row of moving blades. A Curtis stage impulse turbine is shown in Fig. 58 with schematic pressure and absolute steamvelocty changes through the stage. In the Curtis stage, the total enthalpy drop and be~~te . pressure drop occur ideally in the nozzles so that the pressure remains constant in al! three rows ofblades. The kinetic energy ofthe steam leaving the nozzle at V,., however is utilized in both rows of moving blades instead of a single row as before. The absol~ velocity of the steam decreases from V,, to V,2 in the first row of moving blades remains essentially constan! in the fixed blades, enters the second row ofmoving bl~ at V,,, and !caves at v, . Ideally V,z = V,,, but actually there is a loss as a result of friction in the fixed blades so that V,, < V,2 and they are related by a velocity coefficien k. similar to that of Eq. (524).

The velocity diagram for a Curtis stage, with friction in moving and fixed blades ' is shown in Fg. 5-9. The procedure for constructing that diagram is the same as~

for the single-stage impulse turbine (Fig. 55). In the diagram, because of friction

Vs3 < Ysz

v,.. - = ""' V.

v, - " - "v2 V,z

v .. - = ""' v,,

(526)

Using an analysis similar to that used for the singlestage impulse turbine, it is easy w write expressions for the worl< of the Curtis stage using either a momentumimpulse or first law analysis. The latter yields v

Moving t Stationary blades blades t

' Moving blades

a SI tl: di

ce

ki ce th fOI in

wh< exit

Flure 5-8 Ovcnill steam J>IO$SUie and absolure steam vclocil7 .!... t cbanges in an ideal vclocity-

1

1

i 1 ....._

Moving

. . . . .. -:.'-" -~'' . .. - - . . . . . ,. . .

TilRBINES 187

Figure 5-9 Vel


Figure 5-10 velocity diagram and blades for a velocity...compounded impulse turbine with three rows Of moving bladcs. .

The work ratio of the highest-to-lowest-pressure stages, in an ideal turbine, can be found to have the ratio 3:1 for a two-stage turbine (Curts), 5:3:1 for a three-stage turbine, 7:5:3:1 for a four-stage turbine, and so on. This points to one of the major drawbacks of ve1ocity compounding, name1y that the 1ower-pressure stages produce such little work that staging beyond two stages (Curts) is uneconomical. Another drawback is the still-high steam ve1ocities that resu1t in 1arge friction, especially in the high-pressure stages. -

If b1ade speeds must be reduced be1ow that afforded by two stages, the second method of compounding is resorted to .

The Pressure-Compounded Impulse Turbine To al1eviate the prob1em of high b1ade ve1ocity in the single-stage impulse turbine, the total enthalpy drop through the nozzles of that turbine are simply divided up, essential1y equa11y, among many sing1e-stage impulse turbines in series. Such a turbine is called a Rateau turbine, after its inventor. Thus the in1et stearn ve1ocities to eacb stage are essential1y equal and dueto a reduced Ah. From the nozzle equation, ignoring inlet ve1ocities to the nozzles

['tJ;: Vs1 = Vs2 = = vzgc~ (5-29)

where t.h,., is the total specilic enthalpy drop of the turbine and n the number of stages.

A two-stage pressure-compounded turbine is shown in Fig. 5-11. Note that al-though the enthalpy drops per stage are the sarne, the pressure drops are not. AA examination of a Mollier steam chart shows, for example, that if we were to divide

th< ap] pr< res

pul in 1

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Figure 5-11 A two-stage pressure-compounded impulse turbine (Rateau).

the total enthalpy drop from 1000 psia and IOOO'F to 1 psia in isentropic expansion, approximately 580 Btu/lbm, into four equal parts, approximately 145 Btu/lbm, the pressure drops in the first to fourth stages would roughly be 650, 260, 75, and 15 psi, respectively.

Figure 5-12 shows a velocity diagram of a three-stage pressure-compounded im-pulse turbine with friction so that V ..z < V,1, etc. The individual triangles are constructed in the identical manner of the single-stage impulse and the equations for that turbine

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f1cue 5-12 Velocity diagrains and nozzles and blades for a three-stage pressucc-compounded impulse Ularcau) turbine.

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192 POWERPI..ANJ' TECHNOLOOY

Flpre 5-14 Velocity diagrarn for a two-stage reac:tion turbinc.

from AJt,. A biade speed V8 is chosen, say, to correspond to optimum conditions equ.aJ to V,1 cos IJ (compared with V,1 cos 1!12 for the impulse turbine). V,1 is then found. Note that -y is nearly equal to IJ but is much less !han here.

The second half of the enthalpy drop Mm occurs in the moving bladc. This resulta in increasing the velocities relative to the blades. In othcr words, Mm in thc moving blades increases its relative velocity. Thus V,1 is increased to V,., (in thc impulse turbine V,., was equal to or Jess !han V,1 because of friction). For the sarne V8 , get the new V.a. which enters the next row of fixed blades to be increased to V,3 ,. and so on. Thus

(S-31)

and (S-32)

The work of a reaction stage can also be obtained from momentum-impulse or first. law principies. The change in momentum on the blade in the direction of motion + x is due to the change in the components of the relative velocities V,1 and V,.. in that direction. For one reaction stage in general

m F = --'(V,1 cos + V,., cos. -y)

8c

but as V, cos q, = V,1 cos IJ - V8 m

F = --'(V,1 cos IJ - V8 + V,., cos -y) 8c

The rate of work, or power, W = FV8 . v. w = m--(v,, cos IJ - v. + v,., cos 1

8c

(5-33)

(S-34)

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--- -"------------- ------- -- -

f!OID first-Iaw principies, Eq. (5-22), n:peated hen:, applies. . m w = ~2 (V:, - V:z> - M. - Vh))

gc

OptiJDUID Blade Speed

TURBINES 193

(5-22)

Tite optimum blade speed can be easily obtaioed for the case when: the fixed moving bJades are sinlar, so that 9 = y, and Eq. (5-34) is written in the form

or

and

Va W = tir--"(2V,. cos 9 - Va) gc

Again diffen:ntiating W with n:spect to Va and equating to zero dW dVa = 2V,. cos 9 - 2Va =o

Vs.opt = V., 1 cos 9 .

w...,. m m

= -(V,, cos 9)2 = -

194 POWE:JU>U.NT TECHNOLOGY

Lines P0 , P~o etc., represen! constan! presswe linos, which diverge to the right on the, Mollier chart. The actual expansion line 0-1-2-3-4 represents the actual condition of the steam in the two stages and is called a condition curve.

The jixed-blade, or nozzle, efficiency 1IN is the ratio of the kinetic energy chango to the adiabatic reversible (isentropic) energy change across the fixed blade. For the first fixed blade

(5-37)

The moving-blode efficiency TJs is the work of the blade, Eqs. (5-22) or (5-34), divided by the total energy available to that blade, which consists of the kinetic energy ofthe incoming steam at V,1 plus adiabatic reversible (isentropic) enthalpy drop across it. Note that the latter is greater Iban Mm because of the friction (irreversibility) in the blade, which causes an increase in entropy. Thus

. .

w w '1B = m(V,/2g.J + M,. = m[(v:,12g,) + (h, - h,ll

The stage efficiency TJ,,... of a reaction stage is the work of the moving blade in the stage divided by the adiabatic reversible (isentropic) enthalpy drop for the entire stage, including fixed and moving blades. Thus

.

w w TJ, .. ,, = m M, = m h, - h,, (ho - h,) + (hz - h,.,) > ho - h.,

The reaction turbine is an efficient machine that is suited for large capacities. For a given blade speed, limited by material centrifuga! stresses, the steam velocity in a reaction turbine is about half that in a pressure-compounded impulse turbine [compare Eqs. (5-35) and (5-17)], resulting in Jow-friction losses. On the other hand, its work, for the same Vs. is about half that of an impulse stage [Eqs. (5-36) and (5-18)].

Contrary to an impulse stage, a reaction stage has a pressure drop across thc

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TURB!NES 195

JII(IVDg blades. This malees it less suitable for work in !be high-pressurc stages where AP per unit enlhalpy drop is high, which results in steam leakagc around !be tips of !be blades. which in tum leads to lbrouling anda loss of availability. lt foUows lben dial impulse ~g is prefcrable in !be entrance stages of a turbine, when !be pressurcs ue bigh, steam specific volumes are low, and !be blade height is small so lbat steam veJocities would be correspondingly low. In !be low-pressurc stages, reaction stages are preferred because !be llP across !be moving blades is less; !be blades become progressively longer so lbat !be tip clearance becomes smaller relative to !be blade

. beighl. i.e., relative to !be steam volume. Wilb largc reaction blading, V8 is Iarger, negating !be disadvantage of lower power per stage Iban an impulse stage of !be same v .

Turbine rotors are subjected to an axial lbrust as a result of pressure drops across !be moving blades and changes in axial momentum of !be steam betwecn entrance and exit. This axial lbrust must be counteractea to keep !be rotor in place.

In impulse lllrbines, lbere is no pressurc drop across !be moving blades if !be turbine is ideal and liule pressure drop caused by friction in a real turbine. In addition lbere is an axial force on !be row because of !be change in !be axial compooeot of momentum of !be steam from entrance to exit. This is given by (sec Fig. 5-5)

F ~ .. = ~V,, sin q, - V,2 sin y) g, (5-40) 1bis axial lbrust results in no work. In !be case of pure symmelrical impulse blades, V,1 = V,., q, = 'Y and lbat lbrust is zero. The total axial lbrust oo an impulse turbine rotor is, in any case, small and poses no severe problems.

The case of !be reaction turbi.W is different. The axial components of the steam eorering and leaving a reaction turbine are nearly equal (sec Fig. 5-14), so lbat the axial lbrust due to !be change in axial momentum of the steam is, like an impulse turbine, essentially zero. There is, however, a large and continua) pressure drop across lhe moving blades. Although that pressurc drop decreases in !be low-pressure stages, lhe effect is counterbalanced by an increasing blade height and hence area. The resulting axial lbrust is quite large and must be coped with. In small turbines, Ibis is done by a lbrust bearing oo the rotor shaft or by one or more dummy pistons (di ses) inside the casing of sufficient area wilb high-pressure steam on one face only, the other face sealed 1>y a labyrinth packing.

In modem large utility steam turbines, the common solution is to have double-ftow turbines or turbine sections in which steam enters in the center, expands both left and right, and leaves to the next lowerpressure section or to the condenser at opposite ends. This gives the turbine an X shape (Fig. 5-16) with each side's axial lbrust canceling the other. lt also results in dividing up the blade heights and hence areas and axial lbrusts in two and reduces blade tip speeds.

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196 POWERPI..A.NT TECHNOLOGY

Fture 516 A double-ftow low-pressure turbine section rotor.

Twisted Blades Reaction blades are high, especially in the latter stages. Their height, often onethird '

' of the mean blade diameter, reaches 43 in. (about 1.1 m) in sorne cases (refer to Table S-2). The velocity diagrams constructed so far in this chapter assumed constant blade speeds V8 , given by

V8 = rrDN

-----Base ----Mean

.. - .. -----Tip

(5-41)

Figure 5-17 Effect of reaction bladc ~ height on entrance and exit angles. t. necessitating a warped radial shapt. Me ,, Drawn for same V,1 8~ and samcl:lil t-whirl. '

~ ~-\~

J --

-------~~~-"~- '- -'~~-

niRBINES 197

filare 5-11 33 1/2-in. reactioo blading showing twisted construction {38 J.

wbele D is the diameter of the blade and N the number of revolutions per unit time. AJthough N is constan!, D for a high blade obviously is not (high-pressure impulse blades, if used, are so short compared with the rotor shaft diameter that V8 for them

' can be considered constan!)_ Thus V8 increases with radios from base to tip of the blade, resulting in changes

in the shapes of the velocity diagrams along the blade length, as shown in Fig. 5-17, which is drawn for optimum conditions at midpoint. It can be seen that, assuming V,, and 8 do not vary in the radial direction, the blade entrance angle 4> increases, and exit angle "Y decreases, from base to tip, which necessitates giving the blade a twisted shape- lt can also be seen that the degree of reaction varies from base to tip with the bladc somewhat resembling an impulse blade at the base and having maximum reaction at the tip_ Sucb blades are called twisted, warped, or vonex blades. Figure 5-18 shows a turbinc wheel with 33!-in-long rcaction blades with Ibis characteristic [38]. See also Fig_ 5-2L

S-6 TIJRBINE LOSSES

Supersaturation

Wben steam expands rapid/y from a superheated state across the saturated vapor line (point 1, Fig_ 5-!9), a condition of metastab/e equilibrium exists in which the steam does not immediately condense upon crossing point l. lnstead there is no change in the character of the steam, which continues to follow the laws goveming superheated steam for sorne distance past point 1 until a certain lower pressure is reached. Al that point condensation suddenly takes place, and the condition of the system is once again in ~rmodynomic equilibrium, which has a quality dictated by the pressure and specific

;-

' ' t "

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Wi!son line

Entropy s

Flpre 5-19 SupcB8lUration conditi01 and the Wilsoo linc, showo on thc Mol-lier (hs) diagram.

volume (or entropy) at point 2. The phenomenon occws in hoth turbines and nozzles, where rapid expansion occws.

Steam in the region 1-2 is called supersarurated. or undercooled, steam. Tbe locus of points 2 at various pressures, really a band or a zone, is called the Wilson line (Fig. 1-19). It is ahout 60 Btu be1ow the saturated-vapor 1ine on the Mollier cban.

Initial condensation resu1ts in liquid drop1ets of very small diameters and thus large curvature (inverse1y proportinal to diarneter). The. vapor pressure of a highly curved surface is greater than that of a ftat or 1ess-curved surface at the same temperatwe because a mo1ecule on a highly curved surface is freer to leave that surface as it is restrained by fewer adjacent mo1ecules. Droplet diameters below which this effect is pronounced are believed to be around 10 (1 angstrom = w-so m). Convernly, for the same vapor pressure a small drop will be at a lower temperature than a larger one or than the saturation temperature corresponding to that pressure. Thus when expansion occurs rapidly to a given pressure and no condensation takes place, a lower temperature will be reached before the first droplets form. Once they fonn and grow, thennodynamic equilibrium retums to the system.

This phenomenon is further illustrated by the use of a modified Mollier chart (Fig. 5-20) that represents the region of most interest in steam-turbine supersaturation. In thennodynamic equilibrium, a two-phase mixture at a given temperature has one and . on1y one corresponding saturation pressure (e.g., 4.74 psia and 160"1'). In supena turation, the steam behaves somewhat like a gas and the temperature lines in lh< superheat region extend into the two-phase region, as shown by the dashed lines iD the figure; Af any given pressure then a supersaturated fluid such as at b has a lowcr " temperature, ahout 105'F, than if it were in thermodynamic equilibrium ( 160'F), ID

-

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---- ... - -- . ---- ---~-~~~--'-'-'--------=-

TIJRBINES 199

other words, the steam is undercooled. The ratio of actual pressure to the pressure corresponding to the 1ower temperature is cal1ed the degree of supersaturation, or degree of undercooling.

o

"' , ;

~ ~ ., : e

"'

Examp1e 5-2 Compare the final conditions and the steady-ftow work when su-perheated steam at 11.5 psia and 240"F expands (1) isentropical1y to 4.74 psia when expansion occurs to a supersaturated state or (2) s1ow1y and thermodynamic equi1ibrium is maintained. Assume supersaturated vapor obeys pyt.J2 = constant.

SoLUTION The initial conditions (point a, Fig. 5-20) from the steam tab1es are h, = 1165.0 Btu/lbm, v, = 35.88 ft'nbm, and s, = 1.8047 Btu/(lbm 'R).

This steam expands isentropically to point b. If it does so rapidly and becomes supersaturated, it will behave as a gas and obey Pvu2 = constan!. Thus

(P ) 111.32 ( 11 5) o. 1m

v = v, p: = 35.88 4_;4 = 70.22 ft3nbm (

p )(1.32 - 1)11.32 (4 74)0.2424 T.= T, p: = (240 - 460) 1 ~.5 = 564.6'R = 104.6"F From Tab1e 1-3

n Steady-flow work W.- =

1 (Pv - P,v,)

-n

1180

1160

1140

1120

1100

1080

1060 1.6 1.7

= 1.32 X 144 (4.74 1 - 1.32

= 60.1 Btu/lbm

1.8 Entropy_ s, Btu/(lb, 0 R)

X 70.22 - 11.5 X 35.88) 778.16

1.9

~ 5-lO Modified MoUier chart showing supersaturation (.rs, dashcd Unes) and thermodynamie equi-l.ibrium (te, solid lines).

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Therefore

h = h. - W. = 1165.0 - 60.1 = 1104.9 Btu/lb~ lf the steam expands slowly and maintains thennodynamic equilibrium, tbc

conditions at b are obtained from the steam tables at 4. 74 psia and = .r . ,. 1.8047 Btuilb~, giviog x. = 0.9728, h = 1102.9 Btullbm, and v = 75.1~ ft'ilbm. The steady-ftow work w. = h. - h = 1165.0 - ll02.9 = 62.1 Btuilbm. The solution is summarized in Table 5-l.

It can be seen that supersaturation results in a lower temperature, justifying the dual name "undercooliog," lower volume, and reduced work.

Wben expansion crosses the Wilson line, it reverts to thennodynamic equilibriwn ~ by sudden coodensation. This releases the eothalpy of vaporization of the condensed 1 vapor and results in a sudden pressure rise and reduction in specific volume and velocity. The phenomenon is called condensation shock, which is similar, though not identical, to normal shocks that occur in supersonic nozzles. lt is an irreversible procesg that results in further loss in availability,

Fluid Friction Fluid friction is the biggest cause of all turbine losses. lt occurs throughout the turbine. There is, to begin with, friction in the steam nozzles. Next there is blade friction, which we tried to minimize by decreasing steam velocities by compouoding, otc. Also there is turbulence in the blades wheo the blade shape does not possess the proper entrance angle for steam at other than design loads. There is also friction between the steam and the rotor discs that carry the blades. Here rotor design is importan! (Sec. 5-8). In addition, the rotor and blade rotation impar! a centrifuga! action on the steam, thus causing part of it to How tadially to the casing and be dragged along by the moving blades. In case of less-than-full steam adnssion to the moving blades, such as for an impulse stage, there is churning in the moving blades. This is called afanning loss.

Fluid friction los ses can amount to about 1 O percent of all the energy availablc to the turbine.

Table 51 Comparison of supersaturation and tbermodynanc equillbrium for the data from Example 5-2

Properties P. psia T, op S. Btul(lb.., o R) h. Btuilb. ll'tlb. w,, Btw1b.

lnitial ILS 240.0 1.8047 116S.O 3S.88 Final,ss 4.74 104.6 1.8047 1104.9 70.22 60.1 Final,te 4.74 160.0 1.8047 1102.9 7S.I8 62.1

ss = supersaturation, te = lhennodynamic equilibrium. /

_,

!ii-~~f~

~ k .-p ;:

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TtiltBINES 201

Leakage Steam Jeakage can occur within and to the outside of a turbine. Within the turbine team can leak between the tips of moving blades and the casing when there is a

;ressure drop across them, such as in a reaction turbine. This Jeakage is greater the greater the pressure drop, i.e., in the higher-pressure stages, and the greater the ratio of tip clearance to blade height. The leaking stearn is throttled and represents a loss of available energy. In a pressure-compounded (Rateau) impulse turbine, leakage occurs between the base of the stationary diaphragms that carry the nozzles and the sbaft.

Leakage can also occur to the outside of the turbine at the various shaft bearings. This is minimized by the use of proper seals or packings, such as a /abyrinth packing.

Leakage loss can account for about 1 percent of the total energy available to a turbine.

Moisture Loss Besides the losses encountered as a result of supersaturation in the two-phase region (above), the presence of liquid droplets causes further losses. These droplets have both a size and velocity distribution, not unlike !hose in a liquid nozzle spray. Sorne low-speed droplets splash against the moving blades, i.e., strike them at off-design angles, and thus reduce the mechanical work of the rotor. Others are accelerated by the stearn and remove sorne of its energy through momentum exchange. The result is that turbine sections that operate in the two-phase region are substantially less efficient than !hose that operate in the superheat region.

Turbines are usually designed to operare with exit-moisture canten! of no more than about 12 percent (88 percent minimum quality). Higher moisture canten! (afien coupled with high oxygen content in boiling-water reactors) cause blade erosion as a result of the impingement of water droplets on the blades, surface washing, and the so-called wire drawing caused by high-velocity water leaking through narrow passages. Oxygen, in addition, causes corrosion. If steam expansion causes higher moisture content than 12 percent, moisture extraction at certain stages in the turbine is resorted to keep the moisture canten! within this limit. This is the practice in boilingwater reactor turbines, for exarnple. The moisture extracted represents a mass-ftow, and hence work, loss to the turbine, though this effect can be minimized by combining it with bled stearn for feedwater heating. Moisture extraction can be accomplished by constructing the moving blades with grooves on the back side, where the drops are known to collect (Fig. 5-21). The drops are then thrown radially by the centrifuga! force of the rotating blades into a collecting charnber in the casing where they may be bled to a feedwater heater or the condenser [39]:

From an operational point of view, contaminants in the stearn besides oxygen, such as particulate matter and chemicals such as sodium and chlorine from water treatment operations, can cause stress-corrosion cracking ( caused by otherwise tolerable stressing, but in a corrosive atmosphere) and erosion. This calls for better water chemistry control, monitoring, and maintenance [40].

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20% POWERPLANT TECHNOLOGY

Figure 5-21 A grooved moisture-extracting turbine bladc [39].

Leaving Loss We have noted a velocity residual in individual turbine stages, both impulse and reaction. The corresponding kinetic energy is usually recovered in subsequent stages,

. except for that due to the last row in the turbine. The velocity leaving Ibis row, as a result of low-pressure steam of maximum specific volume, and the corresponding kinetic energy represen! a loss to the turbine. This velocity is approximately noi'Ollll to the plane of rotation near rated load but has a large forward componen! at ligbter loads. The magnitude of this veioeity can be changed by the designer by the proper combination oflast-blade height, speeds, and area ofthe exhaust ducts to the condensct. (We already noted that low-pressure turbine sections are double-ftow and have two

' e e r.

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. - ----~~------~----~-~-- .. -.- ..o .. . ____ ., ____ , __ - --' .-.-. _, ___ -- .. -.. _, ... ---' .. -. _;: _____ ,, ______ ..., __ ""---':~---

TURBINES 203

CJ

204 POWERPUUIT TI:CHNOLOGY

S-7 TURBINE EFFICIENCIES

We ha ve already seen tbat, because constan! pressure Unes diverge on a Mollier chart (true also for gases), the isentropic entbalpy drops charged toa turbine stage are I!Ieater: Iban !hose for multiple stages and for tbe entire turbine. It follows that tbe efficiency of a stage is less Iban tbat of a turbine section, etc. Tbis can be seen witb tbe help 0 Fig. 5-15 for a reaction turbine, though it applies to all turbines

~-h. h2-h4 Efficiency of 2 stages = > efficiency of 1 stage =-= ~-h.,. h2-h ...

Tbe ratio of the total individual isentropic enthalpy drops to tbe entbalpy drop 0 a turbine section or wbole turbine is called tbe reheat factor R . R. is obviously greater Iban 1.0, with values ranging between just above 1.0 to perhaps 1.065, depending upon tbe pressure range. For the two stages of Fig. 5-15

96

95

94

93

92

1""" 1100

so / ~

1 " // ;; 9 a 90

~ ~ 89 " : 88

~ ;- 87 ~ 86 11 a ss

84

83

82

8

80

1

'o 15 /: 10 '//

7 ~// S '// 11. 1 11

2 '/ 1 1.5 1

J 1 0.2

""" ...-: ~ ~ ~ p v ~ .-: V/ ~ v; ;- V V r/ VJ /

1 1/

o.s

R = (~ - h:z..) + (h2 - h.,) ~-h.,.

f-' ~ ~

::::

. . . . . . . --. -. ' ' . ~-~~~~-- . ' ..... ' ,, .. ....... ~;- ..

TURBINES 205

1f the designer wishes to ha ve equal work from the stages, dividing the isentropic enthalpy drop for the whole turbine in equal parts will not, because of the divergence of the pressure lines, result in equal actual work in each stage; that is, if 11, - hz.. = hz,. - h .. , it does not follow that 11, - h2 = h2 - h . To get the same actual work, the designer must take into accoimt this divergence.

We have also seen that turbine stages operating in superheated steam are more efficient than those operating in the two-phase region. The perfonnance and efficiency of stages or of a whole steam turbine are undoubtedly a complex function of many variables. Accurate knowledge of the condition curve of a turbine, which is affected by the individual stage rather than the whole turbine efficiency, is necessary, for example, todo the cycle analysis in reheat and moisture-extraction turbines.

Methods for predicting the perfonnances and efficiencies of various steam turbines are often manufacturer's proprietary infonnation, but sorne may be found in the lit--erature [41, 42). One such method [41] predicts the perfonnance of large turbines used in modero nuclear powerplants operating with low-superheat or saturated steam in Figs. 5-22 and 5-23 for the superbeat and two-phase regions, respectively. These figures give base efficiencies for a volumetric ftow (106 ft3/h, 7.867 m'is) and other dala. Corrections for the departure from these data are available in the original paper.

94 92

93 91 --.. )o 92 90 1.5

89~ '5 2 91 . 3 4 88 6

f.o ' 87 , zof-; 86 e-+--+-4'-'~-~~~-5~o~ 85 o 10 20 30 40 50 60

78 77 o~~-L-L~~s~oo~--L-~-L~Io~oo=-L-~-L-L~I~5

206 POWERPL\NT TECHNOLOOY

5-8 TURBINE ARRANGEMENTS

Combination Turbines In earlier times, turbines used to be built as pure impulse, of one type or another, or pure reaction. With the expiration of patents, manufacturen were free to use combi-nations, especially in medium and large sizes. One popular arrangement used to be a Curtis stage (two-row velocity-compounded impulse) followed by a series of Rateau (pressure-compounded impulse) stages. Another was a de Laval (single-stage impulse) followed by a Rateau or reaction turbine.

A more common arrangement is a combination of a Curtis stage followed by a large number of reaction stages. There are certain advantages of this arrangement. The impulse stage is more suited to the high pressure of admission than the reaction stage because there is virtually no pressure drop in the moving blades. Recall that for the sarne enthalpy drop, a much larger pressure drop occurs at high pressure. Also the clearance between the blade tip and the casing is greater for the shorter higb. pressure blades, which aggrevates the leakage problem should there be a pressure drop across the moving blades. After the impulse.stage, the pressure is sufficiently low that the more efficient reaction stages can now be used. They become progressively longer, the clearance proportionately less, and the pressure drop across their moving blades progressively less.

Partial admission to the Curtis stage, because of the limited number of nozzles around the periphery, is conveniently used for goveroing. The nozzles are arranged in groups, each receiving stearn through a valve that is actuated by the goveroor. The valves open in succession as demanded by the turbine load. Such a stage is called a governing stage or a control stage. _

A pressure drop naturally occurs in the goveroing stage, depending upon total stearn flow (load) and number of nozzles in effect. Usually the pressure drop is larger the lighter the load. ,

The goveroing stage has an additional peripheral advantage in that large pressure and temperature drops occur in the fixed nozzles, thus subjecting the turbine proper to greatly reduced pressures and temperatures, an importan! factor in modero turbines that use high-pressure and high-temperature stearn.

Turbine Configurations We have noted the necessity of double-flow turbine sections to cancel out the axial thrust (Sec. 5-5). In addition, modero large turbines, dictated by practical design and manufacturing considerations, are made of multiple sections, also called cylinders, in both tandem (on one axis) or cross-compound (on two parallel axes) arrangements (Table 5-2). The sections may be orie high-pressure (HP), one intermediate-pressure (IP), and two low-pressure (LP) sections, all in tandem, but with the two LP sections operating in parallel as far as stearn flow is conceroed. They may be one HP, three

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1

l

1URBINES 207

rabie 5-2 furbine-generator configurations

fossil Fossil Nuclear

c.2f LSB 26, 30 and 33 . .5 in TC-6F LSB 26. 30 and 33 . .5 in TC-4F LSB 38 and 43 in rwo c:asings 3600 rfmin Five casinp 3600 rlmin Three casings 1800 rfmin

P4 toi@] M ~fGl~~@] M~~@] Hl-IP LP HP HP LP LP ,...00 MW S.5~1000MW 450-1000 MW

TC-4f LSB 26, JO and 33 . .5 in TC-6F l.SB 30 and 33.5 in TC-6F l.SB 38 and 43 in 'fhreC casings 3600 r/min Five casings 3600 r/min (double Four casings 1800 r/min

reheal)

H ~~@] HP-IP LP LP H~IP ~fG{~~@] ~~~~@] 250-650 MW 4.50-723 MW 600-1100 MW

TC-4F LSB 26, 30 and 33.5 in CC-4F LSB 38 and 43 in Four casings 3600 rlmin Four casings -3600/1800 r/min

M~~~@] M l>

208 POWERPL\NT lCCHNOLOGY

(a) (b)

(e) (d)

F1pre S.24 Turbinc amngcmcnts as affcc:ted by differcnt steam paths: (a) stntight througb; (b) si.np n:hcat; (e) cxtraction; (d) induction.

Figures 5-25 and 5-26 show cross sectioos for a 3600-r/mio fossil-fueled turbine and a 1800-r/min nuclear-fueled turhine, respectively .

Turbine Rotors The rotor is !he heart of the turbine. Curren! designs are shown in Fig. 5-27. Figu~~ 5-27a and b shows two versioos of a rotor produced from a single forging. Figure; 27 e shows a composite consrruction produced by shrinking rotor discs on a ceulnl shaft. Figure 5-27 d shows a drum-rype rotor composed of separate rotor di ses that" welded together. This last design is receiving acceptance for !he very large unils beq built today, 500 to 1000 MW and larger, which would otherwise be extremely hea~ and uneconomical and would pose severe mechanical problems ..

Materials for such rotors are careful1y chosen to yield lasting resistance to softeui'l' and creep, receive uniform heat treatment, and ha ve lasting ductility and good resistalll' to scale. Besides materials, attention must be paid to manufacturing methods " operating stresses [43].

Water-cooled nuclear-reactor turhines pose fewer problems !han fossil-fuele

~

! 1 .S g

i ~ :a

- '--~- ---,-~~:.,;,.,""'~' - ~--~-- --

1 1 '

l!O

1 , 11

i 1

' ' ,,

-

' o

'

(a)

(e)

R n ~.---~:~~--;~~ *"""+- f----e-. ----__...___~0' y ;/'. & ~

t:l ' "" (b)

()

flpn!l 5-27 Different turbinc rotor designs.

S-9 GAS TURBINES

TURBINES211

Gas turbines for utility service are nonnally used for peak power production but sometimes also for intennediate and base-load duties when called upon during a majar plan! outage. Gas-turbine cycles will,be covered in Chap. 8.

There are two basic types of gas turbines: radial-ftow and axial-ftow. The radial jlow gas turbine is similar in appearance toa centrifuga! compressor, with the exception, of course, that gas ftow is radially inward instead of radially outward. Radial-ftow turbines are widelyused in small sizes. Theyfonn a compact rigid rotor when combined with centrifuga! compressors. A common use of such a combination is for turbo-chargers on stationary and marine diese! engines and, more recently, on both diese! and gasoline motor vehicle engines. Radial gas turbines, however, are not as suitable to the high-temperature. gases necessary for good !herma! efficiency (Chap. 8), and exccpt for small sizes, are notas efficient as axial gas turbines. Axial.iow gas turbines resentblc the stearit turbines discussed in this chapter. Because we are concemed with largcr sizes, the discussion thal follows pertains lo axial-ftow gas turbines.

Gas-turbine stages are similar lo !hose of steam turbines, except thal the fluid is either a purc gas, such as helium, which is proposed for use with high-temperature gas-

212 POWERI'LANf TECHNOLOGY

being about 6 to JO atm in fossil-fueled gas turbines and about 2 to 3 atm in heliu111 ... _ turbines. The number of stages in fossil-fueled turbines is small, usually one to thrte . but thc number is much larger in helium turbines. This can be shown by recalling ~ _ bladc vclocity V8 is a direct function of gas velocity V,: V8 .;,. = V, cos 8 [Eq. (S; 35)). The gas velocity can be obtained, ignoring inlct velocity to the tixed blades v ' from o,

v> ___!_ = ~- - h = e (T - T) 2J"0 11 po J g. (5-43)

where the subscripts o and s indicate entrance and exit of the fixed blades or noZl!es. . For a gas in ideal expansion

T, = (P,)'" - llik = r',} -

----,

. '

: ..... , ...

TURBINES lJJ

812. Figure 5-28 shows a cross section of a proposed design for a single-shaft, 600-MW helium turbine showing, from Jeft to right, 9-stage Jow-pressure, 10-stage me-diUJIIpressure, and 12-stage high-pressure axial compressors, and a double-How tur-bine with a total of 20 stages [ 44].

Gas-Turbine Blading In steam-turbine practice, relatively inexpensive straight blading, i.e., untwisted, is used and designed on conditions at mean diameter, except in the long-bladed low-pressure stages where the large change in blade velocity with radios necessitates the use of twisted or vortex blading; The gas-turbine (Brayton) cycle is not as efficient as the Rankine cycle, and the gas-turbine designer, striving to sqeeze improvements in efticiency from every stage, has used twisted blading throughout. Gas-turbine blad-ing is invariably ofthe reaction type, mearng, as in steam-turbine blading, part reaction and part impulse, but the degree of reaction increases from blade root to tip. Hence it has not been the practice of the gas-turbine designer to designate a degree of reaction, or even impulse and reaction blading, but rather to use the so-called vortex theory '45, 46].

A detailed discussion of vortex theory is beyond the scope of this book. We shall assume for simplicity that the degree of reaction is constan! along the blade and hence draw !he velocity diagrams for gas-turbine blades in the same manner* as steam-turbine reaction blades (Sec. 5-5). The velocities, however, are calculated from gas relation-ships. Helium, being a monatomic gas, has constan! specitic heats and is relatively easy to do. Combustion-gas properties can either be approximated by variable specitic heat air (the air-to-fuel ratios are usually high) or obtained by the use of gas tables that take into account variable specitic heats, the fuel-air mixture, and dissociation, App. l [48].

For the case of helium, or other gas with assumed constan! specitic heats, the enthalpy drop across the turbine t!.hr per unit mass-ftow is given with reference to Fig. 5-29 by

and

t!.hr = c,(T. - T,) T,. -= r

214 POWERPUNT TECHNOLOGY

T

\. ;''" ')tage \.~oood \tage

\;:id \' ..

Flgare S.29 Ts diagram for gas tnr. s bine expansion.

cp = specific heat at constan! pressure = 1.250 Btu!(lbm o 'R) or 5o233 kJ/(kg o K) for helium and Oo240 Btu/(lbm o 'R) or 1.005 kJ/(kg o K) for air at low temperatures

k = ratio of specific heats = 1.659 for helium and 1.40 for air at low temperatures

r p.T = pressure ratio across turbine = ratio of inlet pressure Po to exit pressure'P e

17r = turbine polytropic (adiabatic, or isentropic) turbine efficiency

For equal work by the stages, the total temperature difference T. - T, is divided equally (for constan! specific heat), and the stage exit temperatures, and pressures, are foundo For example, T2 and T4 are the exit temperatures of the first and second stages in a three-stage turbine (Figo 5-29)o The stage is now divided according to thc degree of reactiono For a 60 percent reaction in the second stage for example, T,, thc fixed-blade exit temperature, is found from (T2 - T3)/(T2 - T4) = Oo40o The fixcdo blade exit velocity, and the moving blade inlet velocity V.,, is obtained from the nozzle equation for gases

(5-49) wbere V,2, the inlet velocity to the fixed blades, is obtained from the velocity diagram of the previous stage, and J = 778016 ft o lbBtu, if English units are used. V,o. may be negligible, bowevero

-- ........ - .-.- -~- ---- - - -- .-.---- .. !"-.-~ .... ~-

111RBINES 215

The velocity diagrams are now constructed in the rnanner of Figure 5-14, using Eqs. (5-3 1) and (5-32) where here M = c,(T, - T3) and ilhm = c,(T3 - T.).

For the case of combustion gases with variable specific heats, products of com-bustion and dissociation, the gas tables [8] (see App. 1) are used. This is best illustrated bY an example.

Example 5-3 A gas turbine using 200 percent theoretical air receives combustion gases at 2460'R. The first stage has a pressure ratio of 2.0, an efficiency of 0.9, and a 60 percent reaction (assumed constan! along the blades). Referring to Figs. 5-14 and 5-29, take 9 = 20', V8 corresponding to optimum, and calculate (1) the stage exit temperature T2 , (2) the fixed-blade exit temperature T1 and velocity V,, (3) the moving-blade inlet and exit angles, and (4) the exit velocity for zero exit whirl. TI! e molecular weight of the gases is 28.88.

SoLUTION Using the gas tables for 200 percent theoretical air, App. 1:

60, 19168.6 T, = 24 R, h, = 28 .88

= 663.7 Btuilbm, and P, = 521.1.

Thus

Therefore

Thus

521.1 P, = -

2- = 260.55

ho- h, T,~,, = 0.9 = ho - h,. '

h, = 566.2 Btullbm

.T, = 2135.8'R

For the stage ilh = ho - h2 = 97.5 Btuilbm. For 60 percent reaction M = 0.4 X 97.5 = 39.00 Btu/lbm. h1 = ho - 39.00 = 624.7 Btullbm. Thus

T1 = 2331.3'R

Therefore

V, 1 - v'2 x 32.2 x .778.16 x 39.00 = 1398.0 ftls

v. = vb.p1 = v,, cos 9 = 1398.0 cos 20 = 1313.7 ft/s q,, = 90'


V,, = V,, sin 9 = 478.1 ftls

!J.hm = 0.6 X 97.5 = 58.50 Btu/lbm

- !J.hm = 58.50 X 778.16 = 45524.0 ft1bf

Therefore

V.,= Y2 x 32.2 x 45524.0 + 478.1 2 = 1777.7 ft/s For zero exit whir1 S = 90'. Thus

PROBLEMS

-y= cos-1 1313.7 - 4 o 1777.7 - 2 3

V,z - V,2 sin "Y = 1777.7 sin 42.3 = 1197.7 ftls

.5-1 A tlat plate mounted on wheels, Fig. 5-2a, receives a perpendicular jet of water from a S-em2 llOlZit at a velocity of 20 mis. calculate the nw:imum power, in watts, imparted to the plate and the velociry of the plate, in meters per second, corresponding to that maximum power. Take water density as 1000 kg/ml. S-% A large movable cylindrical blade, Fig. 5-2b, receives a jet of air of 1-in2 cross.sectional aiea. The air is at 2-atm pressure and lOOO"F. Fmd the necessary air velocity, in feet per second, to produce 1 maximum power to the bladc of 45 kW. 53 Steam enters a single-stage impulse (Del..aval) turbine at 900 psia and 900"F and leaves at 300 psia, Flow is adiabatic and reversible. The nozzle angle is 20. The blade speed corresponds to maximum bladc efficiency. The moving blade is symmettic. Determine the velocity diagram and nd (a) the velocity of tbe steam leaving the nozzle, in feet per secl:lnd, (b) the blade entrance angle, (e) the horsepower developcd for a steam ftow of 1 lb,/s, and (d) the blade efficiency. 54 A single-stage impulse turbine is requi.red to develop 50 MW of power. Steam enten the nozzles

saturated at 70 bars and !caves at 50 bars. The blades are symmetrical and have a velocity coefftcient ol 0.96. Calculate (a) the minimum steam f!ow, in kilograms per second, that would result in tbe requiml power, (b) the blade efficiency, and (e) the stage efficiency. 5-5 Steam expands ideally in a turbine from 2500 psia and lOOOOf to 1 psia. Compare the maximum steam velocities and the number of stages requiced by (a) a velocity-compounded impulse turbine, (b) a pressure-compounded impulse turbine, and (e) a 50 percent reaction turbine if the optimum blade velocity may 1111 exceed 885 ftls in any of them. Take all nozzle angles to be 25". S-6 1.08 x IQ6Jb,.,lh of steam entera Curtis stage witb an absolute velocity of 4000 ftls. The nozzle aag1e and discharge angle of stationary blades are both 20". The moving blades are.symmetric and rotate at 600 ftls. Assuming ideal steam llow in the nozzle and blades, detennine the velocity diagram and find (a) ~ power, in horsepower and rnegawans, developed in the stage, and (b) the blade"efficiency. 51 A Curtis stage receives 3.6 x IQ lb..,lh of steam at 2380 ftls and 20" angle. Tb.e blade spced is 5$0 ft/s. The velocity coefficients in rnoving and stationuy blades are 0.905 and 0.932, respectively. DetcrmiDe the velocity diagram and find (a) the total stage power in feet-pound force per second, horsepower, mi kilowatts and (b) the blade efficiency of the stage. 5-8 Steam enters a Curtis impulse stagc at 1000 psia and lOOOOf, and exits at 1 atm. The nozzle angle is 20" and its efficiency is 87 percent. The fixed blade exit angle is 25". The moving blades are symmetrical.

'

' l ,

11JRBINES 217

t\11 veloeity coeflicients are 0.97. For optimum woric, calculatc (a) the blade velocity, in feet per second, ) tbe work. done by each stage, in Btus per pound mass of steam, (e) the stage efficiency.

:, A Rateau turbinc operating between ltXlO psia and l()()(ff and 1 atm has synunetrical blades, nozzle anIcs of W, nozzle cfticiencies of 97 percent, and velocity coefficients of 0.97. (lc same data as that fot tbe cwtis stage in Prob. 58). Calculate (a) lhc number of stages necessary to limit the optimum blade

(ocity to thaf. of the Cunis stage (1009.6 ft/s), (b) the work of each stage, in Btus per pound mass, (e) : turbine efficiency, and (d) lhe percent error in steam inlet velocity to the second stage dueto ignoring tbC absOiute steam velocity leaving the fi~t stage. ;..lO A SO percent reaction turbine operates between 1000 psia and HXXJop and 1 psia (the same conditions

for tbc velocity-compound and presswe-compound impulse turbines of Probs. 5-8 and 5-9). AU steam :pansion efficiencies in fixed and moving blades are 87 percent. All steam absolute angles ( 6 and y) are 2Cf. 1be turbine has the same optimum blade speed as the impulse turbines of Probs. 5-8 and 5-9 (1009.6 ftls), assumed constant for all stages for simplicicy. (a) Find lhe number of stages, (b) detennine lhe steam veiocity dagram. (e) calculate lhe work. done by each stage, in BttiS per pound mass, and (d) calculate lhe fSt 5cage efficiency. 5-ll A two-nozzle- aeolipile similar to that of Hero of Alexandria contains saturated steam at 6 bus and eJthauSIS to 1 atm. The nozzles are 60 percent efficient, have exit areas of 20 cm2 each and lheir areas are 2 m apart. Calculate lhe torque, in joules, on lhc turbine shaft. 5-ll Consider one stage in a 50 percent reaction turbine. 1 lb,Js steam enters lhe stage at lOO psia and 40l1f and leaves at 40 psia. The adiabatic efficiency of the stage is 0.90. The blades have ex.it angles of 2(f. Thc blade-speed ratio (blade velocity to incoming steam velocity) is O. 8. Determine the velocicy diagram and nd (a) the pressure at the exit of the fixed blades, in psia, (b) the blade speed, in feet per second, and (e) tbe borsepower developed in the stage. 5-13 A 50 percent rection stage in a steam rurbine undergoes a total of 20 Btullb,. enthalpy drop. The oozzle efficiency and angle are 88 percent and 25, respectively. The blades move at 420 ft/s and ha ve v,. = 332 ft/s, Va = 386 ft/s, and y= 22. The steam ftow is 1.08 x IQlilb,,h. Find (a) the work. done by tbe stage in horsepower and megawatts, (b) the blade efficiency, (e) the stage (nozzle and bla~e) efficiency, and (d) tbe blade velocicy coaesponding to maximum efficiency, in feet per second. 5-14 A reaction turbine has 33-in-long blades that receive a constant steam velocity 800 ft/s along their cntile lengths. Tite blades are designed for optirnum conditions at midlength. They are attached to a 60-in-diameter rotor. Assuming ideal frictionless ftow, calculate the blade entrance and cxit angles (


if the overaJI pressure ratios are 2 for thc heliui turbine and 6 for thc air turbine. Take e, = 1.25 8tQ lb.,. "R for hclium and 0.243 Btu/lb.., "R for air. 5-19 A gas turbine composed of two reaction stages receives 1 lb..,/s of combustion gases (assUOled to be pure ait) at 80 psia and 1800"F. It exhausts at 15 psia. A1l stages are SO pcrccnt reaction. flow is COosidertd_ adiabatic and reversible. Thc blade exit angles are 20. The blade spccds correspond to maxim.um. efficiency Both stagcs produce equal powcr. Determine the vclocity diagram and lind the turbine powcr in horsepo~ and megawatts. For simplicity, assume V8 = constant, ande, = 0.24 Btuilb ... "R and le = 1.4. 520 An ideal helium gas turbine has six rows of 50 pcrcent reaction blades andan overall pres.sure ratio of 2.S. All stages have the same enthalpy drop. The maximum helium tempcrature is IOOO"F. The blade spccd corresponda to optimum work. AH blade en trance angles are 20. Considering lhc high-pressure stage determine the velocity diagram, and calculate (a) the helium ex.it tempcrature and (b) the horsepower aod mcgawatts developed in the stage for a helium ftow of 1 lb..,/s. 521 An ideal gas turbine receives combustion gases at 6 atm and 2000"F, and exhausts to 1 atm. It has two stages of SO percent reaction blading producing equal work per stage. "1 = 9 = 200, V,.1 = V1 Consider the turbine to be adiabatic and reversible. For thc high-pressure stage (a) draw the velocity diagram, assuDJing optimum blade speed, (b) find the horsepower for 1 lb..,/s air ftow, and (e) the blade efficiency. For simplicity assume the gases to have a constant cp = 0.24 Btu/lb.., oR k = 1.4. 5-22 An ideal-gas turbine composed of two reaction stages receives 1 lb,./s of combustion gases with 200 percent theoretical air at 80 psia and 1800F. It exhausts at 15 psia. AU stages are SO perccnt reaction. The blade exit angles are 20". The blade speeds correspond to maximum efficiency. 8oth stages produce eqllal power. Detennine the velocity diagram_and find the turbine power in kilowatts. For simplicity, assumc v, = constan!. The molecular weight of the gases is 28.88. Use the gas tables, App. l. 5-23 A reaction gas turbine stage with 59.16 percent degree of reaction (based on an isentropic enthalpy drop) receives 106 lb,/s of combustion gasses with 200 pcrcent theoretical air at 2560"R. The pressure ratio across the stage s 1.987. The fixed blades (nozzles) exit angle is 22 and their efficiency is 86.72 percent. The stage efficiency is 83.94 percent. The moving blade speeds are optimum. The molecular weigbt ofthe gases is 28.88. Using the gastables determine the velocity diagram and calculate (a) lhe gas velocity, in feet per second, and temperature, in degrees Rankine, entering the moving blades, (b) the gas velocity, in feet per second, and temperature, in degrees Rankine, leaving the stage, and the stage power, in megawatts. 524 A reaction gas turbine stage with S9.16 percent degree of_reaction receives combustion gases wilh 200 percent theoretical air at 2560R. The pressure ratio across the stage is 1.987. The fixed blade (nozzle) efficiency is 86.72 percent. The stage efficiency is 83.94 percent. Calculate (a) the gas velocity entering the moving blades, in feet per second, (b) the stage efficiency, and (e) lhe power developed, in megawatts, The molecular weight of the gases is 28.88.

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