-
111 Chapter 4 .
I11 THE AERODYNAMIC DESIGN OF AXIAl-FlOWi
1I AND RADIAl-INFlOW TURBINES! I!
iI
By DR. A. DOUGLAS CARMICHAELADV ANCES in the performance of
axial-flow and (3= Relative angle measured from axial direction,
de-
I radial-inflow turbines have come from the application of
greesi basic fluid-dynamic principies. At present, the flow condi-
'"{= Isentropic exponent. tions in these machines are not
completely understood be-
-
W = Relative velocity, ft/sec W is the rate of output shaft work
per unit massz = Number of blades flowa = Absolute angle measured
from axial direction, e is the specific interna! energy of the
fluid
degrees P/pJ is the flow work associated with moving the fluid(3
= Relative angle l"Qeasured from axial direction, V2/2goJ is the
kinetic energy of the fluiddegre~s o' . A u~eful property of the
fluid termed enthalpy, h can be
'Y = Isentroplc exponent d f ' dA h h lo o elne-E = ngle between
t e tangent to t e stream Ines Inthe ~eridional plane and the
impeller axis of h == e + ~ (2)rotatlon pJ
~ = Slip factor for rotar . o .77 = Isentropic efficiency Th~
sum of klnetlc energy aijd enthalpy IS termed the stag-v =
Kinematic viscosity, ft2/sec natlon enthalpy, ho, where-
w = Angular speed, radians/sec ~
ho == h + - (3)Subscripts 2goJ
is = Isentropicm = Meridional The steady flow energy equation
can now be rewritten as-o = Stagnation o .
opt = Optimum Q - W = ho2 - ho! (4)S = Stator In a turbomachine
the heat transfer is small ti.e. the process
R = Rotor is nearly adiabatic) thus it is customary to neglect
the heate = Tangential transferred to the fluid, Q and
therefore-
0,1,2,3,4, = Stations through the turbineW = hO1 - hO2 (5)
Superscripts
* = Blade angle This equation indicates that shaft work
delivered by the
turbomachine isequalto the decrease in stagnation enthalpySign
Convention: Absolute and Relative angles and veloci- across the
machine in the absence of external friction
ties are positive in the direction of rota- torqueso .
tion. En trop y. A further useful property of the fluid
termed
entropy, s, is derived from the second law of thermodynam-BASIC
THERMODYNAMICS AND FLUID MECHANICS ics. For a reversible
process-
-A ~ummary ~f basic thermody~amic and fluid ~e~hanic - ( dQ) -,-
equatlons used In the study of axlal-flow and radlal-lnflow ds == -
(6)
turbines is presentedo No proofs or deriva!ions are given Tsince
these can be found in good texts on the subjectso Thissection is
similar, although not identical, to the basic therm- Entropy is
related to the other thermodynamic propertiesodynamics and fluid
mechanics given in the previous through the expression-chapter.U .
P ( 1) 1 nlts Tds = de + - d - = dh - - dP (7)
Engineering units are used throughout this chapter; all J p
pJforces are expressed in Ibf units, the unit of mass is the Ibmand
temperatures are in degrees R, The constant go in New- Ths
expression is valid for any process in the absence ofton 's second
law is used in all equations concerned with capillarity, chemical
reaction, electricity and magnetismokinetic energy and change
of~entum; go has the units PERFECT AND SEMI-PERFECT GASES, The
perfectft Ibm per Ibf sec2 o and semi-perfect gas have the equation
of state-
Thermodynamics PSTEADY FLOW ENERGY EQUATION. The steady - = RT
(8)
flow energy equation can be derived from the first law of
pthermodynamics and can be applied to any continuous flow R .
o o h . Th f where IS the gas constant.process such as expanslon
In a turbomac Ineo e sum o o owork done and heat supplied to the
fluid in a turbomachine Ispeclflc heat ds f~t Cdonstant pressure,
cp, and constant
o vo ume C are e Ine as-between statlons 1 and 2 can be express
as- , v'
Q-W= (e2 +~+ ~ ) -(e! +~+~ ) C ==/~\P2J 2goJ \ P.J 2goJ p
\aT/p(1) (9)
where (a )Q is the rate of heat transfer to the fluid per unit c
v == ~mass flow aT v
63
-
.For a perfect gas both cp and Cv are constant while for a POL
YTROPIC EFFICIENCY. Another concept of effi-semi-perfect gas both
cp and Cv are functions of tempera- ciency is the small stage or
polytropic efficiency. Polytropictu re and can be found from gas
tables. For a perfect gas the efficiency is the limiting value of
isentropic efficiency asfollowing relationship's may be definedor
derived: the pressure change in the expansion process
approaches
zero. Polytropic' efficiency can be determined for a perfectc
gas from equation 16 as-
'" - p1 = - (10) ToCv -d-
..," = To (17)"p
R 'Y-1c - c = - (11) ~ dPo)- P v J 1- 1+- 'Y
Po( 'Y ) R thus-
c = - - (12)p 'Y-1 J 'Y-1. ~ =(~) 11P1 (18)
The isentropic expansion of a gas is given by the following T 01
\POIexpression:
'Y Isentropic efficiency can be related to polytropic
efficiency
(!-.!.-\ = ( !!.2-)'Y = (!!-)-;y-:-1 (13) through the
expression-P2 J P2 T 2 'Y - 1
(P02 ) 1/P1 ISENTROPIC EFFICIENCY. As a consequence of the 1 -
Psecond law of thermodynamics it can be stated that the 11 = -- 01
- 1 (19)ideal adiabatic expansion occurs at constant entropy. This
. - (!:!!3-)..r.- permits the definition of isentropic efficiency,
1/, as- 1 P 'Y
01
Actual Work COMPRESSIBLE FLOW EQUATIONS. The following1/ = . W k
(14) one-dimensional equations can be determined for a perfect
- I sentroplc or -gas:
Mach Number-This may be written in terms of the change in
stagnationenthalpy across the expansion process from equation 5 as-
V V
M == - = (20)(hol -h02) a y "YgoRT
1/ = (15)(hol - h02,iS)Stagnation Temperature-An Entropy -
Enthalpy diagram illustrating the adiabatic
expansion is illustrated on Fig. 1. V2To == T + (21)
2goJcp
To 'Y-1, - = 1 + - M2 (22)T 2
Stagnation Pressure-this is'defined as th~ pressure of a mov,p
ing fluid when it is brought to rest isentropically-o,
~
;~ 'Yc ( - .. Po 'Y-1 2 1
Fig. 1. Entrophy- -= 1 + -- M ) 'Y- (23)Enthalpy Diagram P 2for
a TurbineEnltoP1 s
Contunuity Equation-
For a perfect gas-T011--TI 'Y+111 =
~(P )o~ ] (16) ~-y~ = M (~\1/2 ~ +~ M2);~)1 - ~ - 'Y PoA R ) \
2POI (24
64
-
These one-dimensional equations for compressible tIow are The
continuity equation is-
very useful in the design of turbomachinery and are often
tabulatedforair. a(p,Vr) a(p,vx)+ =0 (32)
a, axFluid Mechanics
CONTINUITY. Mass conservation for uniform, steady' BOUNDARY
LAYERS. It has been observed that the
flow can be expressed as- main effect of viscosity is restricted
to a region clase to
V - salid surfaces; this regan is termed the boundary layer.p A
cos (X - Constant (25) Outside the boundary layer the effect of
viscosity can be
o I . V neglected and the inviscid equations presented above can
bewhere (X iS the angle between the ve oclty and the normal used.
In general the boundary layer growth along any sur-
to the area A. . o face is governed by the Reynolds number
(Vx/v) and the
~ELOCITY ~RIANGL.ESo oVeloclty trlan.gle.s are used to gradient
of velocity (dVldx). At low Reynolds number, alsod.efloe the
veloclty an~ dlrectlon of the flUid in b~th rela- when the velocity
outside the boundary layer (the free-tlve and absolute coordlnate
systems as shown on Flg. 2. t I ot ) . odl IdV/d . " t O
ds ream ve OCI y Increases rapl y x IS pOSI Ive an' """,~:~
large) the boundary layer is laminar. At fligh Reynolds
~:I~~:~;. v, ~ number and also when the free-stream velocity
decreasesthe boundary layer becomes turbulento In regions where
the
free-stream velocity is decreasing it is possible that the- vx,
w, v, boundary layer may separate from the salid surface and
Rotational Velocily u, t ::=~.. ~I ~, give rise to large losses
in stagnation pressureo The boundaryv layers in turbo-machines are
not easy to analyze beca use of
.'
F. 2 V 1 it T o les the complicated three-dimensional nature of
the flow. How-Igo o e oc y rlangever, two-dimensional analyses give
a fair indication of true
WORK EOUA TION. The sum of the moments of .']11 the boundary
layer conditions in turbomachines and are nor-external forces is
equal to the rate of change of angular mally used to provide
approximate design limitations.
momentum of the system. Thus the torque delivered by a INEstream
tube element of a turbine rotar to change tangential AXIAL FLOW
TURB S
velocity from Vel to Ve2 between radii '1 and '2 respec- .tively
is- Turblne Performance
TURBINE CHARACTERISTICS. Fromdimensional anal-Torque = rl Vo 1
-'2 Ve2 (26) ysis considerations it can be shown that similar Mach
num-
Mass flow go bers and fluid flow directions are maintained in
the blading of
a turbine-usirig a particular perfect or near perfect gas at
The power delivered by the element of the rotar is there-
constant values of (m ~ /Po 1) and (NI..-f;;-; ). Similarfore-
boundary layer conditions occur when Reynolds numbers
are constant. In general the influence of Reynolds number
T J}" V is of secondary importance and performance
characteristics. w x orque u 1 "e 1 - U2 e2 .w = = (27)
1],and(poI/P02)arenormallydrawnlntermsof
J x Mass flow g oJ(m~/PoI )
This equation is sometimes termed the Euler Work Equa-
tion and may be rewritten from equation 5 as-
UI VOl - U2 Ve2 1ho 1 - h02 = (28) ~
gol g
EOUATIONS OF MOTION IN AXISYMMETRIC ~CYLINDRICAL CO-ORDINATES.
The equations for an f
. -" -inviscid fluid in axisymmetric cylindrical co-ordinates in
the .!
absence of body forces where , is the radial co-ordinate, e is
'
the angular co-ordinate, and x is the axial co-ordinate are
given by Horlock (1) as-
avr avr VO2 goap t 5V r - + V x - - - = - - - (29) ~ar ax r p a,
'--'.
~I~
a~ a~ V~ ~2V r --.! + V x ~ + ~ = o (30) ~.I~.a r a x r 10 20
3~
Fig. 3. An Example StaQna,;on P,...u'. Aa';., "o1.!P..of
Axial-Flow Tur-a V x a V x go ap bine Characteristics
Vr - + Vx - = - - - (31) for a Two-Stagea, ax p ax Turbine
65
-
and (N/ ~). A characteristic for a two stagQ turbine is for a
streamline at constant radius. If the axial velocity V xshown on
Fig. 3. In this example (my-r;;-;/PoI )and 1} are does not change
across the rotor then equation 33 may beplotted against stagnation
R!essure ratio with (N/~) as rewritten as-parameter. Other forms of
turbine characteristics are also u V (tan a - tan a )used; see Fig.
16 for a radial irtflow turbine. ho I - h02 = Mo = x 1 2 (34)
VELOCITY OIAGRAMS. The station numbers for a sin. golgle stage
turbine together with the velocity diagrams areillustrated on Fig.
4. and since tan {31 = tan al - u/V x and tan (32 = tan a2 -
u/V then ;Stalion Numbo,s 2 xo
uVX(tan{31 - tan{32)ilho = -- (35)
gol
ABSOLUTE ANO RELATIVE ENTHALPIES. Theenthalpy conditions in an
axial-flow turbine stage can bewritten as follows:
1. Upstream of the rotor blade rowa. absolute
NOZZLES vlC\ (\ hol =hl +- (36)~ ", 2gl)J
ROTaR h --j5J ~ J- -:;-;., Fig. 4. Geometry b. relative to the
rotor../-'" /-"" - w. u V2 v:x:, and VeloCity Tri-
1- - I ~ I_) angles for an Axial- W2 W2 - ~
--J ... Flow Turbine I 1 1w.,(-) hO1 R = h1 + -=hO1 + (37)
, 2goJ 2goJSeveral assumptions, associated with the blockage
effect
of the blade thickness, have been used in drawing thevelocity
triangles. Three such methods in current use are as 2. Oownstream
of the rotor blade rowf 11 a. absoluteo ows: -
1. The stations selected are between the blade rows
sothattheeffectoftrailingedgethicknessisneglected h =h + ~= - U(Vel
- Ve2) (38)except perhaps as a loss in stagnation pressure. 02 2 2g
J ho 1 g J
2. The stations selected are at the trailing edges of the o
oblade rows and the blockage effect of the trailing .edges is
included in the continuity equations. b. relatlve to the rotor
3. Separate velocity triangles are drawn for the exit of W~ u~ -
uone blade row and the entrance of the following h02,R = h2 + - =
ho l,R + = ho l,Rblade row. The blockage effect of the trailing
edges 2goJ 2goJand leading edges is included. (39)
There are discrepancies of the fluid flow angle when OEGREE OF
REACTION. Reaction, ~,is a useful termthese three methods are
compared which can be five to ten in turbines which describes the
form of velocity changes indegrees for blades having thick leading
and trailing edges. the blade rows. It is defined as the ratio of
the change inAlthough it is not possible to say which of the
methods static enthalpy to the change in stagnation enthalpy
acrossoutlined above is superior, experience has shown that some
the rotor or-difficulties are encountered with the first method
(wherethe velocity triangles are drawn between the blade rows) be-
hl - h2cause it is possible to draw velocity triangles which have
f{ = (40)subsonic velocity components while in the blading the hol
- h02blockage of the trailing edges results in sonic or
supersonicvelocity components. The exhaust duct conditions down.
For a rotor having constant radius equation 40 can be writ-stream
of the last turbine blade row would be based on ten in terms of the
rotor relative velocities and stagnationflow without trailing edge
blockage for all methods of enthalpy rise as-drawing the velocity
triangles.
WORK EQUATION. The drop in stagnation enthalpy W2 - W2across an
axial turbine (along a streamline) is obtained from f{ = 2 I
(41)equation 28 as- 2goJ ilho
h - h = U I Ve I - U2 V 02 Equation 41 shows that for a zero
reaction (or impulse con.01 02 g J dition) the exit relative
velocity, W2, is equal to the inlet
o relative velocity, W l. Experience has showri that turbine
V -:v ) performance deteriorates when the reaction is negative.=
u ( el 82 (33) For a turbine stage having V x unchanged across
the
gol rotor equation 40 can be manipulated to give-
66
-",
-
WOI + WO2If{ = - (42)
2u
and ha
$'V O + VO.,If{ = 1 - l - (43) R2 u . R
For a stage having zero tangential velocity at exit-P,
VOl hIf{ = 1 - - (44) I2 u .c h.,
WORK-SPEED PARAMETER. In addition to reaction > ho 2there is
another type of parameter which can be used to ~
cdefine the stage loading. Stewart (2) defined a work-speed x1-
h
parameter- ~ h2
U2>.. == (45)
gol ~ ho
Similar parameters serving the purpose of defining the
stageloading used by other workers are- ENTROPY .
Fig. 5. Entropy-Enthalpy Diagram for an Axial-Flow Turbine.gol
~ho 2goJ ~ho u and u-~' U2 . /- fA Lg J~h -/2- fA Lg J~h.
andtherotorlosscoefficientis-V 50J "" FIO V L50J "" FIO,;S
J(h2 - h2 . )For an axial turbine at constant radius the work
speed ~R = go + W2 ,IS (50)
be . 2parameter may wrltten as-- If reheat is-neglected (since
it is small) then the isentropic
"' u efficiency of a turbine stage can be expressed as-1\ =
(46)
(VOl - VO2)
for a stage having zero tangential velocity at exit- 11 = ~ + ~N
V? + ~R w~ J -1 (51)2u(VOI - VO2)
u>.. = -v;- (47) For a turbine stage with zero tan gen ti al
velocity the effi-
01 ciency can be written in terms of the loss coefficients,Th .
t . b .tt ' f . f work-speed parameter, and axial velocity ratio,
as-IS equa Ion can e rewrl en In terms o reactlon romequation 44
as- [ ] -1 2 1 VX ~N ~R>"
"' 1 ( 1 ) 11 = 1 + - >"(~N + ~R) ( - ) + - + - (52)1\ =""2 ~
(48) 2 u 2>.. 2
The value of work-speed parameter for an impulse (or zero
SODERBERG'S LOSS COEFFICIENTS. Soderberg'sreaction) blade element
is 0.5 and for a 50 per cent reaction correlations of loss
coefficients, presented by Stenning (3)stage >.. is 1.0 when the
leaving tangential velocity is zero. and Horlock (4) is in the form
of a basic loss coefficient
TURBINE EFFICIENCY. The efficiency of a turbine which
isafunctionoffluidturningangletogetherwithcor-can be predicted with
reasonable accuracy from a know- rections for Reynolds number,
aspect ratio, and tip clear-ledge of the geometry and velocity
diagrmas at the mean of ance.the hub and tip radius of the machine.
The various methods The basic loss coefficient, ~*, presented on
Fig. 6, wasof prediction are based on correlations of losses for
good evaluated for the following conditions:designs. 1. Optimum
solidity as given by Zweifel's rule, Fig. 9
The loss coefficients are defined in terms of the differ- 2.
Zero incidenceence in enthalpy of the actual and an isentropic
expansion 3. Reynolds number is 10sacross the blade row, Fig. 5.
The nozzle loss coefficient is- 4. Aspect ratio is 3:1
5. Tip clearance is zeroJ (h - h . ) Soderberg's correction
factors for aspect ratio, Reynoldsl l IS .
~N = go ' (49) Number, and tlp clearance can be expressed as-t
vt Aspect Ratio
67
-
( b ) AINLEY'S METHOD. Ainley and Mathieson (6) devel-~' = (1 +
~*) 0.975 + 0.075 - - 1 (53) oped an empirical method of predicting
the performance of
h turbine stages at off-design as well as at the design point..
;. . The method is somewhat more complicated than the Soder-
where b = blade axlal chord and h = blade helght berg method and
uses pressure loss coefficients rather thanReynolds Number enthalpy
loss coefficients. The procedure takes account of
deviation angle leaving the blade row, secondary losses,"-
( 10S ) 1/4 , losses due to trailing edge thickness, and
incidence effects.~ - & ~ (54) This method will not be
discussed here.
EFF ICIENCY PR EDICTIONS. The Soderberg methodwhere has been
used to predict the isentropic efficiency at design
point for a turbine stage having zero tangential velocity atRe =
DV I Iv, D == 2hs cos al/(h + s cos al) exit. The variation of
isentropic efficiency with work-speedf I d parameter and fluid
angle leaving the nozzles is shown on
or nozz es an F' 8 Th " f " h h ff ..' kIg.. IS Igure s ows t at
e Iclency Increases as worR - DW 1 D = 2h l a I I(h I .a 1 ) speed
parameter is increased and is maximum when thee - 1 V - S COS'-2 +
s COS .-2 fl .d I I . h . .
, UI ang e eavlng t e nozzles IS approxlmately 70 degrees.
for rotors. liIo.
ro .09>.
~ Re = 105o.. !! Tip clearance = O
. ~ 08 h/b = 30: 0'2 ~2 = Oi "~ ~I -uo ~ O,. ~I
~of O~o 1'0 1-2
Fig. 6. Soderberg's Work -speed Parameler, >-. Loss
Correlation
.{Stenning (3) Cour- Fig. 8. Predicted Isentropic Efficienc~
uSi~g Soderber~'s Methodt M I ' tit (Courtesy Northern Research and
Englneerlng -Corporatlon)esy ass. ns ute
o .. .0 lO lO '00 , ... of Technology)... O."..ti~. ,--
Smith (7) provided a correlation of many turbine-test re-TIP
CLEARANCE. Soderberg's correction for tip clear- sults, all
corrected to zero tip clearance. The correlation,
ance was to multiply the calculated efficiency by the ratio
shown on Fig. 9 indicates that stage efficincy is a functionof the
flow area minus the clearpnce area to the flow area. of work
coefficient, (gol Llho IU2) and flow coefficient,This method of tip
clearance correction is considered ade- (Vxlu),quate only for small
impulse turbines and a more plausible OFF-DESIGN PERFORMANCE. It
has already beencorrection factor from Cardes (5) is presented on
Fig" 7. suggested that Ainley's method can be used to predict
theThis correction takes account of reaction of the stage in
off-design performance of turbine stages. Whitney andaddition to
the tip clearance. Stewart (8) also developed a procedure for
calculating the
performance of turbine stages at off-design conditions. At100 an
angle of inciden~e, , it was suggested that there was an
~ additional enthalpy loss given by-o" 098~ W2sin2~ Incidence
loss = I (55)r. 0"96 2gol2~ 0.91, Despite the apparent simplicity
of this expression the pro->. cedure for calculating the
off-design performance of a tur-~ bine stage is time-consuming,
mainly beca use of the itera-:g 0,92 tion procedure to salve the
continuity equation at outletW from the rotar blade row.
0-90 LEAVING LOSSES. The kinetic energy of the fluid leav-ing
the turbine can be an appreciable portion of the avail-able
enthalpy drop and it is desirable to minimize this
o 0"4 08 1"2 1.6 2.0 24 28 H .. f fClearance Ralio, k/h per c.nt
energy loss. owever, the beneflclal ef ect on per ormanceof
reducing the leaving axial velocity is offset by an increase
Fi9: 7. Tip Clearance Correction Factor (Cordes (5) by
permission in the centrifu gal stresses in the rotar blades due to
theSprl nger- Verlag)longer blades. The turbine performance often
has to be
Soderberg's method can be used to predict the efficiency
compromised in order to obtain satisfactory stress levels.of a
turbine stage at design point. ~t is claimed that the pre- An
efficient diffuser downstream of the last rotor is adicted
efficiency is within two per cent of measured values. useful method
of enhancing turbine performance, if space
! 68I
-
permits, The influence of diffuser recovery factor, e on the The
radial equilibrium equations are derived from thetotal (or
stagnation) to static efficiency of a turbine can be equations of
motion* and basic thermodynamic relation-determined from the
following equation: ships and can be applied to any turbomachine
where the
-1 main direction of through flow is axial. This restriction to[
,N ~ + , W2 + (1 - e) V 2 + VO2] mainly axial flow appears because
equilibrium in the radial1 t; I t;R2 x2 2 (56) d '" ' dd ' f ' l '
11ts = + .. Irectlon IS consl !:!re at a series o axla
statlons,2(Vel - VO2) It is customary to make certain assumptions
about the
flow in arder to simplify the analysis, The flow is
generallyWhere the recovery factor e defines the proportion of the
assumed to be axisymmetric, that is, circumferential varia-kinetic
energy of the axial component of velocity recovered tions in fluid
conditions are neglected, A further assump-in the diffuser, tion
that is often used in axial turbines is to neglect the
SHROUDED ROTOR BLADES, It has been found that radial components
of veTocity. Procedures using the lattertip leakage is an important
source of loss in a turbine. Fig- assumption are termed simple
radial-equilibrium methods,l.ire 7 shows that there is between 1,8
and 3,5 per cent loss In many turbines the radial components of
velocity arein efficiency for one per cent of tip clearance,
depending on small and can be neglected so that the more
time-consum-reaction, One method of reducing the loss is to use
ing, but more accurate, analysis which includes the effectsshrouded
rotar blades in arder to reduce leakage flow by an of the radial
components of velocity is not often necessary,appreciable factor
without reducing the radial runningclearances, There is little
published information on the de- 3.0sign of rotar shrouds, .~
- 2,8REYNOLDS NUMBER EFFECTS, Soderberg's correc- ~tion for
Reynolds number given in equation 54 indicates ~ 2.2that the loss
coefficient vares asRe-%; Reynolds numbers " 1,8were based on
hydraulic mean diameter and velocities at ~ 1.'the throat, Ainley
(9) found that from turbine cascade tests ~ Fig. 9, Turbinethe
losses varied as Re-y., losses deduced from a four-stage ~ 1.0
Stage Efficiency
, ,'" ~ based on Smith'sturblne tended to conflrm thls result,
Alnley also stated .6.3,' ,5 ,6 ,7 ,8 ,9 1,0 1,1 1,2 1.3
Correlation (7),that there was little decrease in loss when the
Reynolds FI.w C.etticie.l. V,/Unumber exceeded 1 x 105 for the
four-stage turbine; thebasis fQr the Reynolds number was blade
chord and outlet RADIAL EOUILIBRIUM (INCLUDING STREAMLINEvelocity,
In a more recent paper by Holeski and Stewart CURVATURE), The
equation of motion in the radial direc-(10) the influence of
Reynolds number on the losses in sin- tion, equation 29, may be
combined with the entropy equa-gle stage turbines was studied,
Reynolds number was de- tion, equation 7, to eliminate the pressure
gradient term tofined as: m!.urm-where m is the mass flow and rm is
the give- -radius of the mean section of the blades, The main
observa- ( as ah) a V a V ~tions from the analysis were that below
approximately 2 x gol T - -- = Vr ~ + Vx -2- - ~ (57)105 there is a
variation of loss with Reynolds number and ar ar ar ax rabove that
Reynolds number there appeared to be verylittle variation of loss
with Reynolds number, There was Usng the definition of stagnation
enthalpy, given in equa-considerable scatter in the variation of
loss with Reynolds tion 3, the previous equation can be rewritten
as follows tonumber although Re- % appears to be a more accurate
rep- provide the radial equilibrium equation:resentation than Re-y.
'. Many of the turbines considered by ~. ) V 2Holeski and Stewart
were small impulse turbines which are gol T~ - ~ = V 3-r - V !!!-=
- Vo ~ - ~unrepresentative of gas turbine practice, ar ar x ax x ar
ar r
The variation of loss wit Reynolds number in turbine (58)blades
is complicated becau of the varying extents oflaminar boundary
layer, In hi h re ct 'on t rb ' l . The first term on the right
hand side of this equation isa I u Ines amlnar II h d ' ff ' I d .
b " 1boundary layers might extend ver considerable regions of
genera .v t e m?st I ICU t to eter,mlne e~ause It Invo ,vesthe
blade Surfaces e en at h ' h cid b d h ' h the radial veloclty, V
r' The evaluatlon of thls term requlresv Ig eyn s num ers an Ig h d
" f h ' , f h l " hlevels of turbulence because of t e accelerating
flow. Low t ,e pre Ictlon o t e,posltlons o t e stream Ines In t ~
tur-reaction blades would tend to hav mainly turbulent boun- bln~
and the calculatlon of the cur~atures (or second dlffer-dary
layers, at least on the suctio surfaces of the blades entlals) and
slopes of these streamllnes, .because of the adverse gradients ere,
It is concluded SIM:LE RADIAL
EOUILI,BRIUM,Theassump~lon~~attherefore, that it is impossible to
gene lile about the varia- the ra~lal5c80mponent of veloclty can be
neglected slmpllfles
, f I ' h R Id b ' h " l . equatlon to-tlon o oss Wlt eyno s num
er sin t IS IS strong y In-fI,uen~ed ,bY ~he types of blades and
the etails of the velo- ( ds dho) dV x d V e ~Clty dlstrlbutlons on
the blade surfaces, gol T - - - = - V x - - Ve - - -
\ dr dr dr dr rII (59)
Radial Equilibrium /" For design purposes it is often assumed
that the stagnationThe general term "radial equil~~ is used to
define enthalpy and entropy are constant radially so that
equation
the analytical method of calculating the radial variation in 59
reduces to-axial velocity at any axial station in the turbine, The
value :,., ,
f' " " *The absence af bady farces In the equatlans af matlan
Implles thatO axlal veloclty IS, of course, requlred as an
Important the analysis strictly applies ta regians between blade
raws and nat
component in the velocity triangles, within blade raws.
69
'.~._-
-
dV dVO Ve tion enthalpy downstream of the rotar and equation 59
is
V x -2- = - Vo - - - (60) used to determine the distribution of
axial velocity thus-d, d"
.' . dV dhThis differential equation is easy to salve when
simple varia- V x -2- = gol --.!!-. (69)tions in tangential
velocity with radius are specified. d, d,
FREE VORTEX DESIGNS. The free vortex distributionof tangential
velocity, defined as- and it can be shown that-
;,VO =constant (61) [ ( ) COS2a ]V 2 V 2 x2 - xl '1 =u.V.l--
(70) is very easy to salve using equation 60. The solution indi- 2
II 011 '1
cates that the axial velocity, V x' is constant at all
radii.
For a free vortex turbine stage where the inlet tangential The
variation in axial velocity at exit from the turbine with
velocity is given by- radius can be determined from equation 70,
It can be seen
from this equation that the variation in tangential velocity
v: = ~ (62) at ~xit from the tur.bine is presented i~ t~rms of
t~e inlet01 , radlus 'l. However, In most cases there IS Ilttle
loss In accu-
racy when the outlet radius '2 is used in this equation,
and the exit tangential velocity given by-
Turbine Blade Design
v: = !!.. The aerodynamic design of turbine blades is often
close-02 , (63) Iy linked with the mechanical design and also with
the de-
sign for manufacture. The aerodynamic designer therefore,
the stagnation enthalpy drop is therefore- does not often have
complete freedom to select the opti-
mum blade geometry. Despite this fact it is often possible
u(VO - VO)
-
2-. where
2- ar = cos-1 (;.)2-
1- 113; I = cos-1 (-;)J)- 1lO
~ O = throat width ~.3 1~ s. = blade pitch corrected for exit
blade thickness~ 12 An alternate definition of deviation is used
when the tur-CJI~ bine blades are drawn as airfoils by defining the
mean lines~ 10 and thickness distributions. Deviation for airfoil
blades is
'"
measured from the mean line at or near the exit of theo-
blade.
SUBSONIC DEVIATION. Experimental correlations ofdeviation data
for turbine blades have been given by Ainleyo- and Mathieson (6)
and Lee (12). In both correlations the
deviation angles are plotted against Mach number with exito
angle as para meter and in the regions of interest the two
50 70 80 correlations agree. The correlation of deviation angles
dueRelative Exi! Fluid Angle .-132 (deg) to Lee but replotted by
Cordes (5) is presented on F ig. 12.
Fig. 10. Optimum Spacing. Zweifel's Method
chord ratio for nozzle blades ({31 = O) and impulse blades
7.({31 = - 132) for a range of exit angles, and suggested thatthe
foss for other blades would vary as the square of the 6.fluid inlet
angle. The pitch-to-chord ratios for minimumloss were calculated
from these data by a curve fitting tech-nique for a wide range of
inlet and outlet angles and are 5,presented on Fig. 11-,- ~
Q13 .."O~ 4-0
..
o
1 2.0u"Ui..; Og~
~ ,. 1-0uCJIc 08..
o. O'"
0.4 0-5 0-7 o-e. oExit Mach Number
o Fig. 12- Subsonic Oeviation (Cardes (5), by permission
Springer-Verlqg)
O.SUPERSONIC DEVIATION. When the flow down-
stream of the throat becomes superconic there are patterns50 60
70 of shock waves and expansion waves which turn the average
Relaliv~ Exi! Fluid Angl. -13 (deg) . . . d .. D . h (13) d. 2
flow and glve rlse to supersonlc evlatlon. elc es-Fig. 11. OPtimu~
Spacing, Ainley's Correlation (Caurtesy Northern cribed an
experimental investigation of supersonic condi-Research and
Englneerlng Corporatlon) . d t f th bl d d f d th t th pertlons
owns ream o e a es an aun a e su -
DEVIATION D . t . . th t d t d .b h sonic deviation angle could
be predicted accurately pro-. evla Ion IS e erm use o escrl e te. .
. .
fl . d . h t k I d f h h f vlded the tralllng edge thlcknesses
were small- It can also be
UI turnlng tata es pace ownstream o t e t roat o . . .
th b I d D . . 11 d d h fl .d seen from hls results that the
devlatlon angle at low super-e a e. evlatlon genera y ten s to re
uce t e UI. . -t . . th bl d d - d f . d sonlc Mach numbers can be
predlcted very accurately uslngurnlng In e a e an IS e Ine as- -. -
. .
j the contlnulty equatlon and the assumptlon that the flow IS8
== ar - al for a nozzle blade row isentropic from the sonic throat
to the downstream condi-
and (72) tions. Supersonic deviation angles predicted using the
isen-8 == 132 - (3; for a rotor blade row tropic assumption are
presented on F ig. 13.
71
-
)-0 dV V- = - (73)dn rc
where n is measured in the normal direction and r c is theradius
of curvature considered positive when the centre ofcurvature is in
the direction of increasing n.2-0 This equation provides a method
for calculating the flowcondition along normal~ to the streamlines.
An iterative! solution is normally used for determining the flow
condi-
~ tion in a channel. The continuity equation is first used to..,
provide the positions of the streamlines for the calculation; of
the radius of curvature of the streamlines in equation 73.~ 1.0
This method can be used in compressible as well as incom--~
pressible flow and in addition it is not restricted to two-~
dimensional channels since allowance can be made in con-
tinuity calculation for any change in spacing of the
stream-lines in the axial planeo
An elegant solution of the cascade problem was devel-o oped by
Martensen (15). His process is essentially the solu-1-0 105 1.1
1-15 1.20 1-25 tion of an integral equation describing the
distribution of
Eail Nach Numb.r velocity on the blade surfaces. The method is
suitable forFig. 13. Supersonic Deviation Angles for Turbines
Blades (1 = 1.33, solution using a digital computer and can be
modified toR = 53.3 ft Ibf per Ibm deg R) salve compressible flow
problems, Payne (16).
The channel flow methods are relatively easy to salve.
THEORE~ICAL BLADE DE?IGN MET~ODS. The d~- using a digital computer
and much can be gained in terms
sl~n of turb.lne blade sha~es .uslng theoretlcal methods IS of
performance or perhaps confidence in performance bywld.ely used In
the gas. turblne Industry. Several methods a!e proper use of these
methods. The more elaborate methodavallable .for the solutlon of
the channel flow problem whlle of Martensen is somewhat more
difficult to salve using aothers can be used for solving the
cascade flow probler:n. digital computer but the results would be
more accurate,
In channel flow methods the problem of the flow In the
particularly in the regions of the leading and trailing
edges.cascade is simplified by considering only the flow betweenone
pair of blades. These metlteds are not accurate in the CASCADE TEST
RESULTS. An investigation of a re-regions of the leading and
trailing edges of the blades. The lated series of turbine-blade
profiles in cascade was con-principie of the channel methods is
illustrated on Fig. 14. ducted by Dunavant and Erwin (17). Low
speed tests were
Stanitz (14) described an analytical procedure for calcu-
carried out on profiles having a range of camber angles fromlating
the shape of the turbine channel having prescribed 65 degrees to
120 degrees. Solidities of 1.5 and 1.8 werevelocity distributions
on the blade surfaces. The method used and each section was tested
over a practical range ofcan be used for incompressible and
compressible flow in inlet angles. Some tests were also carried out
at high speedtwo dimensions. However, a relaxation method must be
conditions to determine the values of critical Mach number.used for
compressible flow and this is time consuming to Design incidence
was determined from the measurementsalve on a digital computer. A
good approximation using of pressure at a tapping located at the
leading edge of eachlinearized compressible flow is also provided
by Stanitz blade. When the pressure at the tapping registered the
stlg-which is very easy to salve using a digital computer. nation
pressure then this was defined as the design inci-
dence (Dunavant and Erwin used the term induced angle).These
tests showed that the design incidence was negativeand a function
of inlet angle and solidity; varying from -17degrees at zero inlet
angle, solidity 1.5 to -2 degrees at 60degrees inlet angle,
solidity 1.8.
,--( 5 ., uctlon, ,- 5 fI 010 I \ ur ace """ ,~ \\ I\\ ,\, \
", \", \ b
" ,
" "Fig. 14. Channel " "Flow between Tur- "bne Blades " "
"
Streamline curvative methods can be used to determinethe
velocity distribution of channels of prescribed geome- S'try. The
basic equation for this method is derived from the Fig. 15. Blade
De-condition for irrotational flow- sign Procedure S
72
-
BLADE DRAWING PROCEDURES. An examination of '9turbine profiles
used in curre,lt gas turbines shows that ,8there are three main
features of blades and channel shapes: '7
1. The blade turning is highest near the leading edge2. The
suction surface of the blade is nearly straight L~Q. .6
from the throat to the ti"ailing edge. ~ ,53. The geometric
throat of the blade is at exit from the ~
- -4blade channel or passage. ...These features are therefore
used in the preliminary design : ,3of the blade channels and
profiles together with any stress- ; 1'0ing requirements for
maximum thickness and trailing edge .thickness. The drawing
procedure is illustrated on Fig. 15.
THE RADIAL INFLOW TURBINE .,
Turbine PerformanceTURBINE CHARACTERISTICS. If dimensional
analysis .6
is used it can be demonstrated that dynamical similarity is
>-maintained in a turbine with varying inlet conditions pro-
~vided (m~ IPo 1), (u31 ~l) and Reynolds number re- :~ '4main
constant. Experiments show that the performance is ;less sensitive
to changes in Reynolds number than to .changes in the other two
parameters. The effect of changesin Reynolds number is therefore
considered as a separa tefactor and is not generally presented in
the turbine charac- ,2.4.'.8 1.0 1.2teristics. Although
characteristics for radial inflow turbines v.. ' A . / ,.- ..
ocII)' allo, u, -i&GeJan can be drawn in the same form as
axlal-flow turblne charac- oteristics Fig. 3 there is a tradition
for presenting them in a Fig. 16. An .Example of a Radial-lnflow
Turbine C~ara.cteristlc, , . (based on Hlett and Johnston (20),
Courtesy I nstltutlon ofdifferent way. This appears to have
developed from Impulse Mechanical Engineers)and steam practice
where the efficiency is plotted againstblade-jet speed ratio,
(u3/V2goJ),ho,i-r). In this expression meet the blades at the
leading edges at atangential velocity),ho,;-r is the isentropic
enthalpy drop from inlet stagnation given by slip conditions. At
this operating point there areto exit stagnation (or sometimes
static) conditions. An ex- no velocity peaks at the leading edge
and this suggests thatample of a turbine characteristic drawn in
this way is shown inlet losses to the rotar would be a minimum.
However, inon Fig. 16. The isentropie-efficiency characteristic for
this arder to obtain maximum efficiency, the tangential
velocityturbine is nearly unique for stagnation pressure ratios be-
leaving the rotar should be zero at this condition.tween 1.3 and
2.45. The stagnation-to-static efficiencycharacteristic varies
slightly with stagnation pressure ratio Station Numberover the same
range. The mass flow characteristic is also
. . Collectorshown on thls figure.
VELOCITY TRIANGLES. The components of the radialinflow turbine
together with the station numbers are illus- otrated on Fig. 17 and
the velocity triangles at inlet and exitof the rotar are shown on
Fig. 18. The velocity triangles at aexit of the rotar are often
drawn at a station corresponding Pto the trailing edge of the
blades so that the blockage of the 01trailing edge thickness is
accounted for in the continuity -equation.
WORK EQUATION. The stagnation enthalpy drop
Fi9.17.ComponentsofRadlal-lnflowTurbineacross a radial inflow
turbine along any streamline can be
~ ~~~~~~~~derived from equation 28 as-~3U3 VO3 - U4 VO4 IJ
(-)ft;:~ hO3 - hO4 = (74) 3 V3
gor W3 Vm3
In many applications it is desirable to minimize the kinetic I ~
ve3 --jenergy leaving the turbine so tht the tangential velocity I
"3 --1leaving the machine is arranged to be zero. Equation 74h d a
Inl.1 Triangl.t en re uces to- -
11'144(-) U3 "03 hO3 - hO4 = (75) ~4
golW4 Ym4 V4OPTIMUM INCIDENCE. An examination of the stream-
lines at the inlet to the rotar of a radial inflow turbine ~ "
.:J:!t-shows that there is a striking resemblance to ffow condi-
4tions in a centrifugal compressor; this is indicated on Fig. ~
Exil Triangl.19. The stagnation streamlines for the radial inflow
turbine Fig. 18. Velocity Trlangles for a Radial Infiow Turbine
73
-
.'.0.'ion
0.9
0.8 -~ Hielt and 10hnslonc. Based on Average:~ Slalic Density;;
0.7 . I I
Bas'ed on Inlel SI.gn.lio~Densily
0.&
0.5O 0.05 0.25 0.30
Sp&cific Sp&&d, Nsa COMPRESSOR b. TURBINE- - - Fig.
20. Correlation of Peak Efficlency with Specific Speed
Fig. 19. Comparislon of the Streamlines for the 1 mpeller of a
Cen-trifugal Compressor and for the Rotor of a Radial-lnflow
Turbine W. l. (21) f . f . I fl t b.IS Icenus trans ormatlon rom an
axla - ow ur Ine
The stagnation enthalpy drop at the optimum incidence ca~ca~e.
With these nozzles the solidity, defined as chord/. h . 11 ' I 't'
exlt pltch,wasselected as 1.33.Wlt zero tangentla eavlng ve OCI
YIS- SUBSONIC DEVIATION. Hiett and Johnston observed
( VfJ3 ) that there was very good agreement between the
measuredu~ - fluid angle downstream of the nozzles and the fluid
angle
Dho = U3 opt (76) predicted using the Wislicenus transformation.
They alsogol found that the approximate rule expressed as-
o* -1where a2 = cos - (80)
( VfJ3 ) = Vm3 (3* + r . . - - tan 3 ~ (77) gave reasonably good
agreement wlth the experimental re-U3 U3 sults. In this equation s
is the circumferential pitch.
Opt SUPERSONIC DEVIATION. In the design of radial-in-(33 is the
blade angte atTotor inlet flow turbines for very high pressure
ratios it is often neces-t is the slip factor for centrifugal
compressors given by sary to have supersonic conditions downstream
of the noz-
equations 81,82, or 83 of Chapter 3, of this Handbook. zle exit.
Although convergent-divergent nozzles could beFor a rotar with
radial blades at inlet (33 is zero and the used for this purpose it
has been found that a more accept-
stagnation enthalpy drop is- abre design solution in many cases
is to use convergent noz-tu2 zles and allow a supersonic expansion
downstream of the
~ho = -2 (78) nozzles. If sufficient radius is available a
shockless expan-gol sion can be provided by a vaneless space; but
there may be
large friction losses. An alternative design procedure is toand
the slip factor, t, is generally clase to 0.85. arrange that the
expansion from sonic conditions at the
SPECIFIC SPEED. Specific speed is a non dimensional throat to
supersonic conditions downstream of the nozzleparameter defined as-
occurs through a series of shocks and expansions at the
trailing edge of the nozzles. This method would be utilizedN ..Q
when a restriction is placed on the maximum radial dimen-
Ns == 601g 1 ~ h )% (79) sion. The shock losses associated with
the supersonic e.xpan-o o sion are very small when the leaving Mach
number IS low
. . 1.25). Supersonic deviation can be calculated assumingwhere
Q IS the 1~let volu":,e flow.. . . . the flow is isentropic from
the sonic throat. The method is
For conventlonal deslgns of turblnes, speclflc speed IS f 1I .f
h d . f as o ows.closely related to the ~eometry o t e rotar a~ IS
r.e~uent- At a supersonic Mach number equal to M2 at outletIy used
as a correlatlng para.meter f~r tu~blne e.fflclency. from the
nozzles the fluid angle is a2 and the blade angleThe performance.
of h~draullc machlnes In particular ha.s iven b cos-1 (o/s) is a;.
a2 and a; are related by thebeen correlated wlth thls parameter,
and the value of specl- 9 .y
f . d . . d . f W d (18) expresslon-IC spee IS a very Important
eslgn eature. 00 pre-sented a correlation of peak efficiencies for
radia! inflow (m~ )turbines as a function of specific speed. Wood's
correlation otogether with add itional data are presented on F ig.
20. cos a; P oA M = M 2
- = - (81)Component Design cos ~2 (m VT o )The terminology used
to define the components of a -~A M = 1.0radial inflow turbine is
given on Fig. 17.
NOZZLE BLADES. The design of nozzles for radial in- where (m
V-r:;;-/PoA) can be calculated for a perfect gasflow turbines was
investigated by M i?umachi (19) but the using equation 24.results
were inconclusive. Hiett and Johnston (20) designed VANEL.ESS
SPACE. Between the nozzles and the inletthe nozzles for angles of
60, 70, and 80 degrees using the to the rotar there is normally a
gap termed the vaneless
74
-
space. This gap may be utilized to provide an increase in
centrifugal compr8Ssors. These methods are generally quasi-Mach
number from sonic conditions at the throat. Normal- three
dimensional methods which use the assumption thatIy, however, the
gap is small and chosen for mechanical or the flow conditions can
be determined in two stages as fol-manufacturing considerations.
Flow in the vaneless space lows:can be analyzed by one-dimensional
methods allowing for 1. Axisymmetric flow obtained by rotating the
mea nwall friction in a way similartd the analysis of vaneless dif-
line of the blades around the rotational axis of the
fusers for centrifugal compressors. In many designs where
machine.the vaneless space is small the friction can be ignored and
2. Blade to blade flow obtained by using the blade load-the change
in flow conditions can be calculated assuming ing effects together
with the mea n conditions givenisentropic flow together with the
assumption that moment by the axisymmetric solutionof momentum,
,Ve, remains constant across the vaneless The equation defininp the
gradient in relative velocity
space. along normals to streamlines in the hub-to-shroud plane
wasROTOR. The rotar of a radial-inflow turbine is a compli- given
by Smith and Hamrick (24) for rotors with radial ele-
cated three-dimensional shape and it follows that the flow ment
blades as-
conditions relative to the rotar are difficult to analyze. For ~
= W - b (83)preliminary design purposes it is often assumed that
the dn a
meridional velocities at inlet and exit of the rotar do not
change from hub to shroud. where n is measured along each normal
and a and b are fac-NUMBER OF ROTOR BLADES. Using the assumption
tors which depend on the geometries of the streamlines and
that the flow at inlet to the rotar does not vary from hub the
mean blade surfaces.to shroud and the condition for absolute
irrotational flow The values of the factors a and b are given
by-Jamieson (22) obtained an expression for the minimumnumber of
blades to prevent reversed flow at the blades. 2 a . 2 aTh .. b f
bl d d . cos.. sin..
e mlnlmum num er o a es can be etermlned from- a = - - -
'c , cos e- 21T cos 2 (33 (84)Z - (82) sin (3 ( dWe )Vm3/U3 b =
- sin e cose - + 2(,,)
., ... . cose dxJamleson s analysls IS approxlmate but appears
to predlctblade numbers which are Glose to the optimum number for
.. .radial inflow turbines in some cases. Hiett and Johnston As In
oth~r streamllne curvatu!e methods the r~dlus of
(20 ) measured a sli g ht im p rovement ( one t) ' ff '-
curvature 'c IS normally the most Important factor In deter-per Gen
In el.. h . - . l . l .I h Iciency by doubling the number of blades
to conform to mlnlng t e v~rlatlon In re a~lve ve oclty a ong t e
norma ~.
Jamieson's mnimum value at a high nozzle angle setting. . Fo~
machlne~ ?f prescrlbed geQ~etry t~e flo~ con~l-Knoenschild( 23 )
discussed modifications to a low '- tlons In the meridional plane
are obtalned uslng an Iteratlvespecl d . h . h h . . f h . dtic
speed turbine to improve the distribution of rel t" proce ure In w
IC t e posltlons o t e streamllnes are a -
velocities on the blades of the rotors. This procedu~elvi~
justed until the continuity equation and equation 83 areillustrated
on Fig. 21. simultaneously satisfied. Smith and Hamrick (24)
devel-
oped a simple approximate method which was not iterativeImproved
but simply a step-by-step integration from known initial
c Original condtions. With this method the shape of an initial
stream-~ \ line and the variation in relative velocity along the
stream- - - - - - - - line are prescribed. The procedure is then to
work away ,5~ IJproved Shape from this initial streamline by
solving conditions across an~ incremental streamtube using the
basic flow equations. In-c this way a series of streamlines are
constructed from the
-- - - - 1'0. hub to the shroud. If the shape or velocity
distribution onDiamolor I Tip Oiamoler the final streamline is
unacceptable then a new distribution
of velocity along the initial streamline is assumed.The blade to
blade conditions can be determined by the
~ !la ..~ simple approximate method of Stanitz and Prian (25).
With '- u ~ 2 ~ ~ this method the concept of zero circulation is
used together:~mloan : : with simple assumptions relating the blade
surface velocity: ~: to the average velocity. At optimum incidence
the velocity. - ~
:: ~:: distributions near the leading edge of the rotar blades
canbe determined using the assumption that the flow at opti-
Mean Slreamline mum incidence is analogous to slip flow in the
impellers oflenglh centrifugal compressors. With this assumption
the flow in
a Original Shape b Improved Shape c Increa.e 0151ade Number the
region of the leading edge is assumed to follow the
Fig. 21. Procedure for Improving the Performance of a Radial-
direction given by-Inflow Turbine (Based on Knoernschild (23),
Courtesy of Mechan-cal Engineers) 2( ' ) ( ' )sin(3=E+F - + G -
(85)
THEORETICAL DESIGN METHODS. The flow condi-'3 '3tions in the
rotar of a radial-inflow turbine can be analyzedusing methods
developed for the design of the impellers of Where the constants E,
F, and G are selected to satisfy the
75
-
boundary conditions- ROTOR SCALLOPING. Rotors of radial-inflow
turbinesare often scalloped to lighten the wheel, also to
prevent
13 = 133 at, ='3 ;.; . cracking between blades. Penalties in
efficiency found by13 = blade angle at, = 'x H iett and Johnston
for such modifications depended on the
d(sin 13) = gradient of sin of blade angle at radius, extent of
the scallops. Maximum loss in efficiency was 4dr x per cent with
rather deep scallops. This loss in efficiency
The flow is assumed to follow the blades at a radius given was
reduced to 2 per cent by removing material at the backby- of the
disc to reduce the sudden change in cross section. It
was established by thes tests that there were penalties inIn ~ =
-O. 71 ~ sin E (86) efficiency resulting from removal of the disc
between
'3 z blades but the penalties were small if care was taken to
min-imize discontinuity in the flow path between the blades.
Losses and Efficiency OFF-DESIGN PERFORMANCE. The off-design
perfor-The isentropic efficiency of a radial inflow turbine can
mance of a radial inf~ow tu:bine can be calcula~ed from a
be written in terms of the actual stagnation enthalpy drop,
k.nowledge of. :he deslgn polnt losse~ tgether wlth correc-~ h
across the turbine and the "enthalpy" losses as- tlons for
addltlonal effects due to Incldence on the rotar
o, blades and tangential velocity in the exhaust duct.77 = ~ =
~ho (87) Futral and Wasserbauer (27) described a method for de-
~ hO.is ~ ho + ~ losses termining the off-design performance of
radial inflow tur-bines based on the method of Whitney and Stewart
(8) for
The overall "enthalpy" loss in the radia-inflow turbine
axial-flow turbines. With this method the loss coefficientscan be
separated into losses for the various components. were determined
in terms of the mean of the inlet and out-These component losses
can further be subdivided into jet kinetic energies.losses due to
various causes, such as skin friction, tip clear- Stator Loss up to
the Stator Exit-ance and corner losses. Useful information on the
losses inradial inflow turbines has been given by Jamieson (22) and
Vl> + vBalje (26). Ll = Kl 2gol (90)
The plot of efficiency against specific speed presentedon Fig.
20 provided another very simple method of estimat- Loss from
Turbine Inlet to Rotor Inlet-ing turbine efficiency.
Hiett and Johnstonj20) provide performance data for L =--K ~~
(91)such factors as nozzle angle, clearances, Reynolds number, s 2g
1and scalloped rotors. This information is summarized o
below. Rotor Loss-NOZZLE ANGLE. Tests were carried out with one
tur-
bine rotar at nominal nozzle angles of 60, 70, and 80 de- W2 +
W~grees. The efficiency was a maximum at a nozzle angle of LR = mK
3 (92)70 degrees and fell slightly at 80 degrees. However, the
'}golrotar was designed for a nozzle angle of 70 degrees and
hasbest matching at this condition. The results are somewhat The
stator loss up to the stator exit, L 1, is required ininconclusive
but it appears that both 70 and 80 degree noz- addition to the full
stator loss up to the rotar inlet, Ls, be-zle angles produced
higher efficiency than the 60 degree cause the moment of momentum
is required to determinenozzle angle. the rotar work. It can be
seen that the ratio of rotar loss
CLEARANCES. Basic performance tests on the turbines coefficient
to stator loss coeffi~nt, m, is used, rather thanwere carried out
with a clearance on the rotar between the assigning separate
unrelated loss coefficients. Conditionsblades and the casing of
approximately 0.01 in. (rotor were evaluated at the arithmetic mea
n of the hub and thediameter 5 in.). Tests with varying clearance
were carried shroud diameter at rotar exit.out on one turbine and
it was found that an increase in For calculating the loss due to
incidence it was assumed.clearance of one per cent of the exit
blade height reduced as in the case of axial flow turbines, that
the loss was equi-the efficiency by approximately one per cent.
Losses in valent to the kinetic energy of the velocity component
nor-efficiency due to nozzle clearances were found to be, to a mal
to the blade at rotar inlet. For a rotar with radialfirst order,
one per cent for each per cent of nozzle height. blades at inlet
the incidence loss was expressed as-
REYNOLDS NUMBER EFFECT. Tests were carried outon one radial
inflow turbine over a Reynolds number range W2 . 2 13of 0.5 x 105
to 2.2 X 105. Reynolds number was defined Lin = 3 sin 3 (93)as-
2gol
Re == ~ (88) It was assumed in this analysis that minimum loss
occurredv with zero incidence. A more satisfactory method might
be
. . to use the optimum conditions; equation 93 would then
bewhere b IS the top wldth of the rotor. rewritten as-
Hiett and Johnston found that the change in efficiencycould be
expressed in the form W~ sin2 (133 -133 opt)
Lin = (94)1-77cx:Re-o.16 (89) 2gol
76
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.Futral and Wasserbauer (27) analyzed the performance absolute
inlet angle, al , 64.7 degof a commercial turbine having a 4.5 in.
diameter rotar and inlet meridional velocity, V mi, 644
ft/seccencluded that the value of K I for the turbine nozzle was
exit meridional velocity, V m2, 550 ft/sec0.078,Khad a value of
0.1995 and m was 2.215. A feature inlet blade speed, U, 1664
ft/secof the turbine design analysed was the use of circular noz-
exit blade speed, U2, 1198 ft/seczles rather than vanes in
the"stator. The stator loss, Ls, was ex it absolute angle, a2 , O
degtherefore somewhat higher than one would expect for specific
heat, cp 0.272 Btu/lbm Rvaned nozzles because of the sudden
expansion after thecircular nozzles. Comparison between the
predicted and Answers:measured off-design performance was good over
wide al 64.7 deg; 131 -25.0 deg; 132 - 65.6;ranges of speed and
pressure ratio. ",Vel 1364 ft/sec; ve2 O ft/sec; Wel - 300
ft/sec;Acknowledgements We2 -1198ft/sec; VI 1510ft/sec; V2
550ft/sec;
The assistance of the staff of the Drawing Office at the ~ho 91
Btu/lbm; ~ T o 334 deg R.Mechanical Engineering Department of the
Imperial Collegeof Science and Technology, London, in preparing the
draw- 2. Derive an expression for the specific speed of a
radial-ings for this chapter is gratefully acknowledged. inflow
turbine in terms of the ratio of the inlet dia-
meter, di, to the outlet shroud diameter, d2, for thefollowing
conditions:
EXAMPLESrelative angle at inlet, 131 , O deg
Axial-Flow Turbine Examples relative angle at shroud exit, 132,
60 deg1. Construct velocity triangles (not to scale) for the hub
hub diameter/rotor tip diameter 0.25
or r.ot regi~~ of an axial-flow turbine having the fol- absolute
tangential veolcity atlowlng condltlons: exit, Ve2, O ft/sec
absolute angle, al , 70 deg A .nswer.axial velocity, V xl = V X
2, 500 ft/sec ~d2~3 1 ~d~blade speed, UI = U2, 800 ft/sec Ns =
0.214 - - - -
rotar deflection,.81 -132, 107 deg d J 16 dIspecific heat, cp
0.272 Btu/lbm R
Determine the drop in stagnation enthalpy and stag-nation
temperature for this turbin~ REFERENCES -Answers: . ..
TT Axlal Flow Turblnesa2 O deg; 131 49.0 deg; 132 -58 deg; "e 1
1375 ft/sec;TT Of / . W 760 f / . W 944 f / . 1. Axial Flow
Compressors, J.H. Hcirlock, Butterworths, London,"e2 tsec, 1 tsec,
2 tsec, 1958,p.7.~ho 44 Btu/lbm; ~ T o 162 deg R. 2. A Study of
Axial-Flow Turbine Efficiency Characteristics in
. Terms of Velocity Diagram Parameters, W. L. Stewart,2.
Calculate the absolute and relatlve Mach numbers for A . S . t f M
h . I E . ee P pe No 61 WAmerlcan ocle y o ec anlca ngln rs, a r .
- -the turbine root section described in Example 1 when 37,
1961.the inlet stagnation temperature T 01 = 1500 deg R, 'Y 3.
Design of Turbines for High-Energy-Fuel Low-Power-Output= 1.33, and
R= 53.0 ft Ibf/lbm R. Applications, A. H. Stenning, Massachusetts
Institute of Tech-Answers: nology, Cambridge, MIT Dynamic Analysis
and Control Labora-
tory Report No. 79, 1953.MI = 0.835; MI R = 0.435; M2 = 0.468;
M2R = 0.885 4. Losses and Efficiencies in Axial-Flow Turbines, J.
H. Horlock,3. Derive the expression for the specific speed of an
International Journal of Mechanical Science, Vol. 2, 1960, pp.
axial-flow turbine in terms of the hub-tip ratio of the 48-75.h.
A . I ( t . ) d. . 5. Stromungstechnik der Gasbeaufschlagten
Axialturbine, G.
mac Ine. ssume Impu se zero reac Ion con Itlons C d S V I B I.
or es, prlnger- er ag, er In.at the hubo 70 deg nozzle leavlng
angle at the hub, 6 A M th d f P f E t . t . f Ax . I FI T rb 'nes.
e o o er ormance s Ima Ion or la - ow u I ,uniform axial velocity,
uniform work at all radii and D. G. Ainley and G. C. R. Mathieson,
Aeronautical Researchzero tangential velocity at exit. Council,
London, ARC R and M No. 2974,1957.Answer: 7. A Simple Correlation
of Turbine Efficiency, S. F. Smith, Jour-
nal of the Royal Aeronautical Society, Vol 69, July, 1965
pp467-470.
~~ 8. Analytical Inyestigation of Performance of Two-Stage
Turbine 2, t ayer a Range of Ratios of Specific Heats From 1.2 to 1
2/3, W.
Ns = 0.143 V:...: \ - 1 J. Whitney and W. L. Stewart, National
Aeronautics and Space\ '1 Administration, Washington, NASA TN O
1288, 1962.9. Performance of Axial-Flow Turbines, D. G. Ainley,
Proceedings
Radiallnflow Turbine Examples of the Institution of Mechanical
Engineers, London, Vol. 159,1 C h l . . 1 ( t 1 ) . I 1948, pp.
230-244.
. onstruct t e ve oclty trlang es no to sca e at In et 10 S d f
NASA d NACA S. I S A . I FI T b. . tu y o an Ing e. tage xla ow ur
Ineand exit to the rotar at the shroud region of a radial
Performance as Related to Reynolds Number and Geometry, D.inflow
turbine and determine the drop in stagnation E. Holeski and W. L.
Stewart, Journal of Engineering for Power,enthalpy and stagnation
temperature for the follow- Trans. ASME Ser. A, Vol. 86,1964, pp.
296-298.ing conditions: 11. The Spacing of Turbo-Machine Blading
Especially with Large
77
-
~~,-~
Angular Deflection, O. Zweifel, The Brown Boveri Review, 20.
Experiments Concerning the Aerodynamic Performance in In-December,
1945, pp. 436-444. ward Flow Radial Turbines, G. F. Hiett and l. H.
Johnston,
12. Theory and Design of St.vam and Gas Turbines, J. F. Lee,
Paper 13, Thermodynamics and Fluid Mechanics Convention,McGraw-Hill
Book Company, Inc., New York, 1954. Cambridge, England, April 1964,
Institution of Mechanical
13. Flow of Gas Through Turbine Lattices, M. E. Deich, National
Engineers, 1964.Advisory Committee for Aeronautics, Washington,
NACA TM 21. Fluid Mechanics of Turbomachinery, G. F. Wislicenus,
McGraw-1393,1956. Hill Book Company, Inc., New York, 1947, p.
211.
14. Design of Two-Dimensional Channels with Prescribed Velocity
22. The Radial Turbine, A. W. H. Jamieson, Gas Turbine
PrincipiesDistributions Along the Channel Walls, J. D. Stanitz,
National and Practice, editor {>ir H. Roxbee-Cox, George Newnes,
Lon-Advisory Committee for Aeronautics, Washington, NACA Rep. don,
1955.1115, 1953. 23. The Radial Turbine, for Low Specific Speeds
and Low Velocity
15. Die Berechnung der Druckverteilung an Dicken Gitterprofilen
Factors, E. M. Knoernschild, Journal of Engineering for Powermit
Hilfe van Fredholmschen Integralgleichungen Zweiter Art, Trans
ASME, Ser. A, Vol. 83, 1961 pp 1-8.E. Martensen, Max-Planck I
nstitut fur Stromungsforschung, 24. A Rapid Approximate Method for
the Design of The Hub-Gottingen, Mitteilungen Nr. 23, 1959. Shroud
Profiles of Centrifugal Impellers of Given Blade Shape,
16. Isolated and Cascade Aerofoils, D. Payne, Mathematics M. Sc.
K. J. Smith and J. T. Hamrick, National Advisory CommitteeThesis,
University of London, 1964. for Aeronautics, Washington, NACA TN
3399, 1955.
17. Investigation of a Related Series of Turbine-Blade Profiles
in 25. A Rapid Approximate Method for Determining Velocity
Distri-Cascade, J. C. Dunavant and J. R. Erwin, National Advisory
bution of Impeller Blades of Centrifugal Compressors, J.
D.Committee for Aeronautics, Washington, NACA TN 3802, Stanitz and
V. D. Prian, National Advisory Committee for Aero-1956. nautics,
Washington, NACA TN 2421, 1951.
Radiallnflow Turbines 26. A Contribution to the Problem of
Designing Radial Turboma-18. Current Technology of Radial-lnflow
Turbines for Compressible chines, O. E. Balje, Transactions of the
American Society of
Fluids, H. J. Wood, Journal of Engineering for Power, Trans.
Mechanical Engineers, Vol. 74, 1952, p. 451.ASME, Ser. A, Vol.
85,1963, pp. 72-83. 27. Off-Design Performance Prediction with
Experimental Verifica-
19. A Study of Radial Gas Turbines, N. Mizumachi, University of
tion for a Radial-lnflow Turbine, S. M. Futral and C. A.
Wasser-Michigan, Ann Arbor, Industry Program of College of
Engineer- bauer, National Aeronautics and Space Administration,
Wash-ing Report No.1P-476, 1960. ington, NASA TN 02621,1965.
- A.DOUGLASCARMICHAELMassachusetts Institute of Technology
Cambridge, Mass.
Dr. Carmichael graduated from London University in 1949
andobtained his Doctor's degree at Cambridge University in
1958.
He began his industrial career at Bristol Aero Engines Ltd.
(IaterBristol Siddeley) and spent severa! years in the compressor
group onthe desing of axial flow compressors. On his return from
CambridgeUniversity in 1958 he beca me head of the aerodynamic
researchsection responsible for the advancement of the design of
both com-pressors and turbines.
In 1961 he joined Northern Research and Engineering Corpora-tion
of Cambridge, Mass. as senior project engineer and was respon-sible
for many programmes in the field of turbomachinery. He wasone of
the directors of the extensively sponsored research pro-grammes for
the design of centrifugal compressors and radial
inflowturbines.
In 1964 Dr. Carmichael became a Research Fellow in
Turbo-machinery at the Imperial College of Science and Technology
inLondon, and in 1968 he joined the English Electric Company
astechnical Consultant.
In 1970 he was appointed Professor of Power Engineering in
theDepartment of Ocean Engineering at the Massachusetts I nstitute
ofTechnology.
78
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