VORTEX THEORY

In the previous section 'v\'e have discussed symmetrical and non.

symmetrical type of blading a~rangements. Another type of blading called free vortex blading is also t:sed which is designed to produce axially directed velocity leaving the rotating blades. In the earlier blade arrangements no consideration was given to the fact that velocity triangles change with the radius and these wer,, calculated at a particular radius only. The change in peripheral velocity with radius will modify the velocity triangles. The vortex design is an attempt to take into account the effect of change in radius of the blades.

Whenever the fluid has an angular velocity as well as a velo- city in the direction parallel to the axis of rotation it is said to ha•;e 'vorticity'. The flc.w through an axial flow compressor is vortex flow in nature. The rotating fluid is subjected to a centrifugal force and to balance this force a radial pressure gradient is necessary,

Consider a small element of fluid at radius r with pressures and velocities as shown in Fig. 5 21. The centrifugal force acting

0

Fig. 5·21, S!'•'-~ll elfment at radiu': r nn,l conditiol'. for r2dia! equilibrium.

on this e],,m::~nt is pr d;- de. C,}Jgr. Ev resolving the forces along the radial direction, the condition for ra:JiaI equilibrium is obtained as folkws :

[SEO. 5.·8] DYNAMIC 00MFRESS0RS 175,

dB Cw²

(P+dP)(r+dr)dB ._-P:r.dB-P.dr.2

.,=p

di" r dO

"'" gr

or (5·59)

The total enthalpy h at any radius r f,r the flowing fluid, which is assumed t<J be perlect, is given by

li=c1

,T+c

^2;2gJ

Since c2=ca2+cu;2

and CpT=__:i_.

JJ

.T=

_L

ppJ

y-1 J y-1

We have

y · P Ca² Cw² h y-1 pJ l-2g.1-l 2gJ

Differentiating the above equatioIJ. with respect to r, we get dh _OadCa ~ ^y l [ I __ dP P dp

J

^·5·60

dr - drgJ

+

drgJ

+

^y-1 gJ p dr - p² dr ( ) Assuming that for ~ small cliange in pressu~e dP across the annulus wall the flow is isentropic, i.e. it obeys the law P/pY=cons- tant, or in differential form

P p dP

di=

rP· dr.

Fronj

Eq.

(5·59) aµp (5,·60), we

g.et

Jj}i,

=OadOa +Ow dOw gy [··· 1 dP _ 1 dP

J

g ·· dr · dr dr

+

y-1 p dr py dr

=OadfJa +Ow dOw

+~ _!__.

Y-1. dP dr dr y~ 1 p y dr·

By putting the value of dP/dr, WI! get J d~ =O dOa+o dOw Cw"

gdr adr wdr+r (5·61)

This equation is the basic equation for all.vortex flow. In the design of axial flow compressurs it is assumed that the flow is free vortex flow in which the whirl velocity varies inversely \Yith radiuS:.fi3

We assume that. w0.rk input at all radii is equal to the total head tt'.mpcrature and hence the entha!py will remain conStfint for all radii. The axial \docity is also asmraed to be constant,

i.e.

dlifdr=O and

176 GAS TURBINE AND JET AND ROCKET l"ROPULSION

By putting these conditions in equation (5·61), we get dCw/ Cw= -C,,J r

which on integration gives

Cw X r= constant

[SEC. 5,·9]

i.e. the whirl velocity varies inversely as the radius, or in other words the condition of free vortex blading satisfies the radial equili- briu:n requirements.

The air angles needed can be calculated. In Fig. 5·22 are -shown the blade angles as they vary from root to tip for a 50 per

Fig. 5·22. Free vortex velocity diagram at three section.

cent reaction blading and a free vortex blading. It can be seen that free vortex blading has a considerable twist. So the Mach number will also change considerably during the passage of flow through the stage, and if the inlet Mach number is high shocks will occur. For this reason the blade and axial speeds are limited, resulting in bigger dimensions. Also, since the vortex blade is designed to produce an axially directed velocitv leaving the row.

for good efficiency, the air is actually accelerated in ·the fixed blades and- the pressure reduces instead of increasing in the fixed blades.

This gives less overall pressure rise per stage and the number of stages required is more than that for other types of blading.

The avoidance of vorticitv was earlier thought to be essential for attaining high efficiencies. However, the marked "wists in the rotor blades of a free-vortex design and the large frontal areas are the main disadvanta~es of this type of design. Moreover, with this design the best efficiency condition, i.e. 50 per cent reaction, can only be realized at one radius and not over the entire span of blades.

So now it is thought that free vortex de;ign is not esse1;.iial for good efficiency and stable flow ; and the constant reaction blading is used to maintain constant degree of reaction from root to tip of the blade for best efficiency and the condition of constant axial velocity is relaxed.

5·9. AEROFOIL THEORY

Jn order to achieve high pre,sure ratio with compressors with high efficiency it is necessary to minimise, as completely as possible, the

[SEC. 5·9] DYN .AMIC COMPRESSORS 177 losses occurring in the flow. Ivfini.mi.zation of the losse$ require.a , detailed knowledge of the flow process and for that the aerodynamic design of the flmv passage is necessary. One important aspect of the aerodynamic design is the study of blade profiles and the effect of the presence of other blades on the flow phenomena, i.e. cascading of the blades m a compressor.

Aerofoil geometry

Fig. 5·23 shows the gemnetry of an aerofoil. The mean tamber line is the line reprefenring the locus of all points midway

I I ^L

,'!',l_"'"e ciPcl,

I i .,.

' ' /O »-1t?dn ccnnber line

I [_.--· .

Upp@r .ruritoee

rY1___________ ^I

,j

Lower sur/acl?

1

~ - - - Choral

---.iio""i

Ar:_gft' of ct/lack

Fig. 5·23. Geometry of an aerofoil.

between the upper and lower surfaces of an aerofoil as measured perpendicular to t!1e mean line. Choid line is the line joining the ends of the mean camber line. Gamber is the maximum rise of the mean line from the chord line. Leading edge radius is the radius of a circle tangent to the upper and lower surfaces, ,with its centre located on the tangent to the mean camber line through the leading edge of this Hne. The angle of attack is the angle between the

178 GAS 'IURB1NE AND JET AND ROCKET PRO,PtJi..SllJN (SBC. S·9) relative air flow direction and the. reference line. The aerofoil is assumed .to be stationary,and the relative air f!ow direction is equal

•and OPFOSite to the velocity of the ae_rofoiJ./ Angle of incide.nce is the angle between the chord line arid-:t_b_e :~eforence measurement whereas the a,,gle of attack decides the des1-osl·.ion of tht: chord line of the 'aerofoil with respect to the air flo:.V. Another important ae:.o- foil dimemion is the · aspect ratio, i.e. the ratio of blade height to

blade chord: . . . .' ..

. .

-~t aqd Drag coefficient of an aerofoil

{~;· 'r.,:,There are two lypts of forces acting on an aerofoil moving througli the

fli.tW,

Fir:st are the shearing forces due to fluid fricti6n : (!f the air ·and.~~R..Surface of the ~erofo.il and

!

he second are t~e

<J J)r~ssure forc~s a'c1ftng o~ • the aerofo.tJ. Smee static. pressure . cann9t

'pro¢~·ce

a resuliiftforee on ^1). inpving aerofoil~ the" only reason

for ·

the de~~lopµienrof·the pressure ·(o~c~sis ~he dynamic pressure given by

1

pV~,,,

Tl'itig~,~{nium

fore~ ,wl1i~_h;m~f b,e_:;devi;;loped is. gi~·m by '.

wh)e }ii th:r.:J:J;:ni~l{Jt\,iii4;,dyt/messu~e

^6::

acting. HoWever,' the actua:l fo:,.ce ''develop'ed ,i,y, Jh~ ae1;1Jf?il is· much greater than that giv'~n by Eq. (5'63). ·Let· us considel"~a vane as shown---in~:Fig._j_·25 deflecting a How; ·· The force developed on the vane is given by : -

,[Z

,ti,.V :l·4f41,

Q .

Fig. 5·24. Deflection of flowby an a.:rofoi!;

-F=rate of change of momentum - dV ··

=•

&=pAVx Li.V

=1·414 p AV²

which is 2·828 times gre~ter than given by simple energy equa~ion.

The-aerofoil utilizes this aerodynamic mechanical advantage -obtained by deflecting-a flow and is capable of producing forces grea•er than those given by Eq. (5·63): The forces developed by the aerofoil, i.e. the ability of the aerofoil to deflect a stream of air depends on its angle of attack, the curvature. of the aerofoil (i.e • .camber), the thickness of the aerofoil, aud the velocity of air.

[s.,;c. 5· J OJ DYNAHIO C0MPRR8S0ES

The resultant forre acting on ^I he af'rofc,iI can 1)~ resolved into two firers-one dr,ig force D in !he direction of motion representing;

the frictional forces tending to retard the motion, and the other lift force L pt:rpendi.:::ular to cfoecti,m of motion. Lift is the basic force causing the aeroplane to maimain its flight.

The drag fo;ce is a resultant of the pressure distribution over the aerofoil and the skin friction and repres,,nts a loss, and the ratio or lift to drag L/ D is consi1..·kred as an indication of the efficiency of the aerofoil.

The lift and drag force can be given 'as

I ]

D=Qo - o V²A

' 'Lg • '' l

~nd L=cQr.

2g

p v2.4 _(5·66)

where Co and CL ,are the drag and lift coefficients respectively and A is the proje,:ted ll.rea of the aerofoil. These c0eifo:ients depend upon the shape o( !he profile.

. The variatfo~ of lift and drag coeffi~ients with angle of incidence zs shown in Fig, 5·25. If the angle of incidence is too much the flow

~reaks away from the profile. · ·

1-2

,---..---.--,---,----ir-,

Flg. 5·25. Varialion oi dog awl lift co;;ffkients v:i,:1 an;le of atta::k.

180 GAS TURBlNE A.ND JET AND ROCKET PROPUI.SION [SEC. 5·]0]

5·10. CASCADE THEORY

The work done on a rotating blade can be expressed in terms of lift and drag forces and hence the lift and drag forces can be expressed in terms of blade angles at inlet and exit.

Let us consider a blade moving with a velocity U, the air enter- ing with a relative velocity V1 at an angle ~1 with the axial direction and leaving the blade with a rdative velocity V2 at an angle ~_{2 •}

Ji\

and V2 have a mean velocity V m at an angle ~,,.. (See Fig, 5·26).

The resultant, of drag and lift forces can be resolved into two components Lw' and La', in rotational and axial directions respec- tively. The rotational component Lw' represents the force required to overcome the resistance of motion and La' represents the pres- sure force developed. The useful lift per unit length of blade in the direct.ion of rotation is given by

FJg. 5·26. Blade inlet an.:! outlet velocity diagrams.

511] DYNAMIC COMPRESSORS

l .

=-2^{-p Vm2 C} cos 13m{CL+Cn tan 13m)

(J

where area A =C X l, C being the chord.

Since C"

=

V,,, cos [3m

, _ p Ca² C { CL+Oo tan [3m )

- 2g \ COS 13,m .

\Vork done =Lw' XU o 0,,² Uc

2

^A (OL+Co tan [3m)

(J COS l"m

t~rom Eq. (5·68) the total work done per unlt mass flow is W =, - - ( UGa tan [31 - tan [32)

p

W'=mW=p O" sW 0,,2

u ,.

13

=o -·-_{\ IJ} s,ran · _\ ,-·tan _.ll.

181

(5·67}

(5·66}

where sis the distance between two bladez, Equating (5·68) and we get

or (5·71)

Neglecting C0 in Eq. (5·71), which is very small as compared to CL, we have

If dP 1s the loss of pressure in a cascade, this must be equal to the drag force in the direction of mean velocity. '\'Vriting these forces for unit blade length and equatbg, we get

·f-

_g ^p

^v;,.2

^{CD O=s cos 0m.dv {5·73)}

Putting this into equation (5·72), we get C ·- a d,. _ cos³ [3,,,_

D-7r 1 _-oV

₂ cos

r1

2g . 1

·. 182

t.~"1.'URBINE AND .,TET .. ^{> . .} urn ^. ROCKET.PROP0LH0N ^{. •} [SEC, 5·11]

' . . ^' .

, . '

Fig.

5:25

shdws the lift a119-\drag coefficients of a fixed g~o- metrical

form for

^differrnt

arigl~s

ofinciden,e .. Such data can ea~1ly be obtained frbm. wind tunnel tests and a. wi<le r,e.nge of geometrical forms of the case's.de c:an be investigated .. From 'these coefficients th~

blade angles a11d

hr

nee the efficiency of; the aerofoil can be found. The efficiency.of a cascade is dd'ini·d as t};P ralio of actu~l to theoretical pressure rise in the cascade. Thus the aerodynanHC investigations, can give. us a,. great insight into the desig:1 of the blades conducive to .hidktffid,1.q1cies at higher pr<'s,ure r.at10s.

l ' • ... --..--,:_ __ -~· ' .. " ... ,

5·:fl. SURGING IN

COMPRi'S's.OR§

ASD

rm

CONTROL All dynamic cornpressorA ha\'e certain charncterist;c insta)_,ility of flow over rbei,· operational range.. This instability is as1.;ociatecl with separation of flow from the blade surface and formatio:1 ·ot eddies and can be seen to be oftwQ types-stall an1 surge. Eoth.

these types of instabilities are associated with the positive slonc" 0!

the pressure mass flow curve of the compressor, i.e .. when the pre,~.ure rise increases with an increase in rrn,s flow .. :Stall, in ge:,;;r.(i, is characterised by reverse flow nt>ar the hlade tip which ur:,e ,,; the velocity distribution and hence ad,·ersdy aff,,cts the pe1r;,r,I11ar';'~e of the succeeding stages. Surge is a violent o;.;~eraU ir,stability ·

9f

^'10w

in which flow fluctuat1,:ms of lz.rge amplitudes occur thrbilg:lo::;t the compressor which can result in destructively severe v'bl'ations and noise ; in- comparison stall results in relatively mi'q. Iota! flow fluctuations. Both surge and stall impair the perfo:rman~e and limit the range of operation of the compressors. "'

The mechanism of the instability of flow can be understood as follows : Consider a typical cascade shown in Fig. 5·27. It is well known that a cascade blade at certain angles of attack .with the moving air develops a lift force. As the angle of attack incHases this lift force increases. However, further increase in the angle of attack, more than a critical value; results in a drastic decrease in the lift coefficient along with a rapid increa8e in. drag coefficient. (see Fig._5·25). This is beca,_ise an incrrnse in angle ~1 in Fig. 5·27 does not mcrease ~2 much bm raises the pressure coefficient. "When this increase is critical a separation of How occurs as th€ air finds it difficult to follow the convex surface of the blade. With still more increase in ~1 , the angle

!3

1 changes drastically and flow losses due to s~paration are high, therebv, reducing the lift coeffieient and increa~

srng the drag coefficient. Wiih reference to Fi:;. 5·25, it can be seen that this will reduce the component (L,,+D,,) which is responsible for pressure rise in the cascade. This phenomenon is called "stalli11,g".

Due to stalling ihe pre~sure rise will be reduced. The loss of pressure causes a recession of delivery, and intPrmittent back flow oc~urs. A simi1.ar thing happens when ar.gle ~1 is reduced below a certain value. Stall due to too high angle of attack is called "positive staU"

and that due to too low angle of attack is called "negative stall".

[SEC. 5·1 l] DYNAMIC COMPBESSOB,8 183

(11) N0rmal opedtion. (b) St~lled ooeration.

Fig. 5·27. Stalling in a cascade.

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

~ .. The stau does not o(cur on a11 blades at one time, not even on al\blades of a single stage, but mu ally occurs on a set ,)f '>l lde 1. .

ThiI stall inoves in a direction qpposite toJ that of blade rotation with a ,:e,loc~ty equal to half the rotational vdocity and is called rotating

s.tall.

Due to this rotation, which dissipates a very b.rge amount 9f.e,nerg.y' the flow, though having instable

flow

tendency, is affected only in ~he w.:iy that the rre,sure flow cu'fvejias a positiveslope an.d _re<;J,uces the performance of th·.~ co;npre~sor.

An

increase in. camber :md stagger angle3 increa~e theJendency ofthe blade to stall. The c11.sca,:le spdng and the thicknl!.:L_

of

the bl~de:1 also affc:ct the ten.:-

.J " ~ d . . -· ·· ;,.y,.., 1 . · . h l f

·,uc1cy to st:H,. ,., ecrease .. _i1~:ax_µ; .. ve ocity increases t e ang e o incide_nce and ;,talling tend~cellr' when the mass flow is reduced, i.~. at starring, closing, oC::,fpart load. · '

Surging h als0a:,~Gci 1ted·witll ~epuation of flow or ~talling but the main cause ,}!'· it is the dynamic instability of the system as a whole. Priip1gati(lg Stall may cause severe body. vibrat!ons of the vvhol•:! mass of airlin th~ cornoressor. One important characteristic of pr0pa<_~ating sta!Jci'$jhat th(!~xited forces developed by it'have the . sartl!=:frequeilC~'ii,Lthe"natural frequency of the blades_. This i~ ,·ery dangerous as 'ir<rniiv set severe vibrations and damage the blades.

: . : < -• • ,:. \ :.;:."".>\i :\:, ·.,

(;r

The Pl12~~Iq:i:¢nii•ofsnrgi.ng can l'>est be expl~ined wirh refer-

t

^{cnce tf)} the-· characteri~tic curve of Fig. 5·28. · For a const~:nt speed

ill ..

operation of the compress0r, starting from low flows fp,¢: pressure

1

t

rise increases with an increase in flow up to point B, where it reac,hes )the maximum value. This region has a positive slope and is unstable.

184 GAS TURBHirE AND JET AND :ROCKE'!' P.ROPULS!ON [SEC. 5·12]

AB ::: Stobie branch

lU)

=

lln.rlable

!JC ::: Reverse Flow

1::<'ig. s·28. Pressure volume flow characteristics at an axial compressor.

the comnressor openitcs stable

ope:rrtion in 1~n5tabk ·· iay at point if the will be an increase work and pressure ratio b,2t (he voL.ffile wi H decrease in the first In the second this flow has slower axial \'elocity due to th,, reduced mass flow

as iNZ'll as vo!nrne. This remlts i::: decrease in inlet angle

~ 1 and an in ¹he angle of attack and hence stalling will occur in the rest of the compressor ,tages, The effect of reducing flow is cumulative and the pressure fa!ls with reduced mass flow.

But if the presrnre in tht· receiver er pipe down ~t:ream does not fall so rapidly, flow wil! have a tendency to flow back to the compressor.

After some time the pressure in the downstream pipe reduces and forward fimv takes place. This forvvard and back\';ard flow of the air rf':sults in severe pulsations and the phenornemm is called surging. It should be noted that whi.\c stall pro<luces local sations tl1e surge causes pulsations in tLe whole system and the

~har,, ctcristics of the cnnnecting ~ystnn cr~·;it]y affect the onset, of surge.

For operation in stah!e region a,; incff,2,e k mass f1r.,w rc>duces the pre,sun: 1ir,<F.artd ultimateiy at C the mass fo .. '\'J .is n:,xi- mum but ratio is unity, i e. t!H: cffiFier:cy is zero. S:1d1 How

i11, called flouJ. Surge and cbc("-kiug decides tlie .1,""1,tver a

There is

no

^{abs6lute remedy} for surge. Usually the pressure ratio in a single unit is limited to about 7' to 9 and two

p:ression prcsrnre r,,.tios as high as 12 to 16 can be stage ble.eds, wnereby a portion of the compressor air flow

is tiscd to help:to ;t>dm:c instability at .low speed

'[SEC, 5'12] DYNAMIC COMPRESSORS 185 5·12. CHARACTERIST!C CURVES OF DYNAMIC COM~

PRESSORS

Figs. 5·29 and 5·30 show the characteristic curves of centrifugal and axial flow compressors respectively. Pressure ratio b:as been plotted for various mass fl.ow rates at different speeds. ho-efficiency curves have also been drawn. The whole of the compressor charac- teristics can be divided in.to stable and unstable regions separated by the surge line. The surge Lne shows the limit for the stable operation of the compressor-to its right is the stable region and to its left fa unstable region. It is a single parameter line, i.e. it is a fixed mass now rate which produces surge at a given pressure ratio.

3·0

t

'

I f

i i

__;__-_J

I)

.hg. 5·29. Typical crnrifogal compressor characteristics.

T~e efficiency ;;aries v..ith the mass How at a given speed and the maximum value 1s nearly same for all speeds. A curve repre- .senting the locus of points of.maximum efficiency has been drawn in