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Aerodynamic losses in partial admission turbines
Int. d. mech. Sc~. Pergamon Press. 1969. Vol. 11 ,pp. 417-431. Printed in Great Britain
AERODYNAMIC LOSSES IN P A R T I A L ADMISSION TURBINES S. M. Y z a r r A I n d i a n Institute of Technology, New Delhi, India
and M. D. C. DOYLE Fourah Bay College, University of Sierra Leone, Freetown, Sierra Leone (Received 30 J u l y 1968, a n d i n revised f o r m 2 J a n u a r y 1969)
S u m m a r y - - A theory for the determination of the aerodynamic losses in a flat-bladed rotor operating under partial admission conditions is taken from a previous paper. I t is adapted to obtain a formula for the loss coefficients for a real turbine rotor operating under partial admission conclitlons. A n approximate b u t much simphfied version of this formula is developed for certain cases. The loss formulae derived are verified using three turbine rotors of different blade pitch. The formulae are shown to be approximately correct and the effect of blade pitching is shown to be negligible. A method (numerical example) of predicting the stage efficiency of partial admission turbines is also given. NOTATION B = Ut 1, the admission sector length
C D J L iV P0
loss coefficients diameter mechanical equivalent of heat length of the rotor blade channel revolutions per minute of the rotor stagnation pressure in the settling chamber R e o isentropic Reynolds n u m b e r based on rotor blade chord and velocity (Co) T tension in the brake belt T o stagnation temperature in the settling chamber U peripheral velocity of the rotor blades at mean diameter V transient velocity at the exit of the rotor blade channel W relative velocity of the gas at the entry of the rotor blades b rotor blade chord c nozzle exit velocity co isentropic velocity at the nozzle exit e degree of admission f( ) function of g acceleration due to gravity h rotor and nozzle blade height or enthalpy 2W L m = - I - I - 2 lo / l + e k t l \ 4e~t~ kt 1 * k~ ge ~ - - ~ - - ) + kt,(1 + e~',) rh mass of gas flowing through the rotor per second 4 2 n=l-~ k t t ( 1 + e ktl)
kt 1
417
418
S.M. YAHYAand M. D. C. DOYLE s t t~ x
rotor blade pitch time or thickness of the brake belt total time eonmdered displacement of the rotor blade along the admission sector as shown m Fig. 2. BHP brake horse power SHP stage horse power Ah enthalpy drop = C2o/2g
~b'ubscripts 1 rotor blade entrance or tight side of the brake belt 2 rotor blade exit or slack side of the brake belt e corresponding to a degree of admission = e; or sudden expansion m mixing p partial admlssion ~s iscntroplc st stage N nozzle blade R rotor blade Superscmpts • air angle at the entry of the rotor blades Greek symbols All angles are from the peripheral direction as shown in Fig. 1 nozzle angle rotor blade angles total to static efficiency p density ~: blade aerodynamic loss coefficient =
h2 - h2,s
½W2
INTRODUCTION A P~EVIOUS paper, Y a h y a 1 gave the d e r i v a t i o n of a theoretical velocity profile for the flow of fluid t h r o u g h t u r b i n e blades which h a d just entered an active nozzle sector in a partial admission turbine, h a v i n g been inactive in the region of no admission. The t h e o r y was derived using a t u r b i n e r o t o r model with u n c a m b e r e d axial blades or " f l a t - p l a t e " blades. This t h e o r y was confirmed e x p e r i m e n t a l l y using a t u r b i n e with " f l a t - p l a t e " blades. I n this paper, the results derived a n d confirmed for the " f i a t - p l a t e " rotor are used to o b t a i n the m a g n i t u d e of the losses due to partial admission in an axial flow impulse turbine. The losses are expressed in t e r m s of a loss coefficient. The principal losses in partial admission turbines are due to (i) R o t o r blades p u m p i n g the gas in the inactive sector, (ii) Mixing of the active a n d inactive gas, (iii) L e a k a g e o f active gas t h r o u g h t h e inactive sector, (iv) S u d d e n expansion of the active gas into a r o t o r blade passage which is either entering or leaving the active admission sector. I n this p a p e r the losses due to r o t o r blade p u m p i n g only h a v e been determ i n e d experimentally. The analysis takes into a c c o u n t mixing, leakage a n d sudden expansion losses. O t h e r losses are regarded as negligibly small c o m p a r e d to these.
Aerodynamic losses in partial admission turbines
419
T h e t h e o r e t i c a l r e s u l t s , i n t h e f o r m o f loss c o e f f i c i e n t s , a r e c o m p a r e d w i t h the experimental values obtained from tests on three impulse turbine wheels w i t h d i f f e r e n t p i t c h i n g o f t h e b l a d e s . I n d e r i v i n g t h e l o s s c o e f f i c i e n t s i t is necessary to distinguish between the end of the sector where the rotor blades e n t e r t h e a c t i v e s e c t o r , w h i c h is c a l l e d t h e e n t e r i n g - e n d a n d t h e o t h e r e n d , w h i c h is c a l l e d t h e l e a v i n g - e n d . THEORETICAL
CONSIDERATIONS
The loss formulae are derived under the following assumptions: (i) The flow is frictionless and incompressible. (il) There is an infinite number of blades on the rotor and one-dimensional flow through the rotor blades. (iii) There is no pressure drop across the rotor. I t is assumed t h a t the leakage of active gas prevents a n y pressure build-up at entry to the rotor. (iv) The velocity profile due to mixing of active and inactive gas predicted b y Y a h y a 1 for the "flat-plate" rotor holds for the impulse turbine blades considered in the present section. Corrections are applied to allow for those aerodynamic losses which are inherent in turbines, such as profile loss and secondary loss.
1. Mixing loss coej~cient ignomng sudden expansion I f the sudden expansion b o t h a t the leaving-end and the entering-end of the admission sector is ignored, then mixing can be considered to continue until the blade channels leave the admission sector. I n such a case some of the m o m e n t u m of the gas, relative to the rotor at inlet, is lost in aeceleratmg inactive gas in the rotor blades and resulting in leakage of the gas past the rotor shroud. Thus the velocity of the gas entering the rotor is V, which is less t h a n W and depends on the distance moved b y the rotor blade into the aclmi~ion sector in time t. The direction of flow relative to the rotor will be a t an angle ~' to the tangential direction as in the full admission case. The above flow model is used to determine the blade work for partial and full admiasion conditions. Stage efficiencies in both cases are found b y taking into account the normal rotor blade aerodynamic losses. The difference between these efficiencies gives the value of mixing loss coefficient, C~. This gives the amount b y which the stage efficiency of a turbine is lowered due to mixing when the turbine is run under partial admission conditions. An expression for this loss coefficient is developed in Appendix 1.
2. Mixing and sudden expansion loss coe~cient When sudden expansion losses are considered, the effect cannot be separated from mixing losses at the entering-end, b u t a t the leaving-end and it m a y be assumed t h a t U
2/
U FIG. 1. Inlet and exit velocity triangles.
mixing losses are negligible. Thus partial admission work is best calculated set~rately for the entering- and leaving-ends. Therefore mixing is n o t assumed to occur in the length ( s - x) which is exposed a t a n y time. This is shown in Fig. 2.
420
S.M. YAHYA and M. D. C. DOYLE
The average length along the admission sector for which mixing is considered is thus given b y
if0, ( B + X - S ) d x
s
s
(1)
= B--~.
Thus the flow through length [B - (s/2)] of the nozzle sector can be considered to suffer mixing loss, while the flow through the remaining length 8/2 suffers sudden expansion. Furthermore, it is assumed that the gas entering the length ( s - x) expands to fill the whole blade passage. 2 Thus the exit velocity is W ( s - x)/s.
Ii//I
)))))))))))
FIO. 2. Length of the transition zone. A loss coefficient Cme b ~ e d on the above flow model is derived in Appendix 2. This loss coefficient gives ~he a m o u n t b y which the stage efficiency of a turbine is lowered due to mixing and sudden expansion when the turbine is r u n under partial admission conditions. INSTRUMENTATION
1. General layout A general layout of the test rig is shown in Fig. 3. I t consists of a Napier's Turbocharger supplying air, through a 3~ in. dia. British Standard orifice, to a settling cha~nber. The ratio of the cross-sectional areas of the settling chamber and the nozzle ring is approximately 14, which gives a comparatively low velocity on approaching the nozzles. SETTLING
--.....
C HAMBER
GAUZE
NOZZLE BOX
~
TURBINE WHEEL BI~,KE WHEEL
ELECTRIC MOTOR
EXHAUST 16 ANNULUS NOZZLES
FIG. 3. General layout of the test rig.
The settling chamber was provided with ten static pressure tappings and five thermometer pockets. The temperature and pressure as indicated at these points were practically the mime. Since the velocity of air approaching the nozzle was small, the temperature and pressure at the nozzle entry could be considered to represent the stagnation values.
Aerodynamic losses in partial admission turbines
421
The nozzle ring consisting of sixteen nozzles is attached at the end of the settling chamber, concentric with the air-supply pipe. Static pressure tappings were provided at the entry of each nozzle. They showed very nearly the same pressure indicating a uniform static pressure distribution at the entry of the complete nozzle annulus. Blanking segments of various lengths were used to blank a desired n u m b e r of nozzles for partial admission tests.
2. The brake The brake consists of a mild-steel flywheel ( 9 ~ in. outside dia.) keyed to the turbine wheel-shaft between the two pedestals as shown in Fig. 3. The braking torque was applied b y a ~ in. thick, 1 in. wide Ferodo belt which was fixed to a hydraulic pressimeter giving a manometer deflection, which was calibrated to determine the tension at one end and the other carried a weight pan. A ~ h.p. electric motor was used to t u r n the flywheel and the turbine rotor to determine the shaft and rotor blade pumping losses. THE
TURBINE
TESTS
These tests were performed to determine the brake horse power (BHP) and the i n p u t energy at various values of the total to static pressure ratio (Po/Pa) for full a n d partial admission cases. (Pa) was the atmospheric pressure and (P0) was taken as the nozzle box pressure. Most of the tests were performed at a turbine speed of 1500 r.p.m. The blade to gas speed ratio (U/co} varied due to the varying of the gas speed which resulted from the varying pressure ratio. After changing from one pressure ratio to another some time was allowed for the flow to become steady. Then the weights on the p a n were adjusted to bring the speed of the turbine to 1500 r.p.m. (or a n y other value as reported in the high-speed tests). The hydraulic pressimeter manometer was read for each load. The following readings were recorded during each test: (i) Orifice temperature, (ii) Orifice pressure, (iii) Static pressure drop across the orifice, (vi) Nozzle box temperature (To), (v) Nozzle box static pressure (P0), (vi) Weights in the weight pan, (vii) Hydraulic pressimeter manometer deflection, (viii) Turbine rotor speed in r.p.m. (N). At very low brake loads, the tension on the hydraulic pressimeter was very low and it was ineffective. The brake power at these low loads was determined b y t u r n i n g the turbine wheel at the test speed (with the same weights on the brake pan) b y the electric motor a n d measuring the torque, after the brake tests. MEASUREMENT
OF THE
ROTOR
BLADE
PUMPING
LOSS
To determine the pumping loss, when the turbine works with partial admission, loss due to "pumping" for no admission was measured as described below. Then for a given degree of admission (e) this loss was calculated, assuming it to be proportional to ( 1 - e ) . Thus, (pumping loss)~ = pumping loss for complete blade ring × ( 1 - e ) . The turbine wheel was rotated at its test speed b y the electric motor and the power supplied by it was measured b y weighing the torque on the motor. This power (say A) is equal to the sum of (i) Complete blade ring pumping, (ii) Disk friction of the turbine wheel, (iii) Flywheel disk friction, and (iv) Bearing friction.
422
~. M. YAHYA
and M. D. C. DOYLE
The s a m e test was t h e n p e r f o r m e d after b l a n k i n g t h e rotor blade ring b y pouring m o l t e n w a x into its blade channels a n d giving a s m o o t h finish to the ring. Thin t~me, the p o w e r r e q u i r e d to r o t a t e t h e wheel at t h e s a m e speed (i.e. t h e test speed) was less, due to t h e absence of blade p u m p i n g loss. The difference of powers m e a s u r e d in the two cases g a v e t h e loss due to p u m p i n g of t h e c o m p l e t e blade ring, as explained below. B l a n k i n g of t h e blade ring introduces additional surfaces (the two v e r t i c a l faces of the b l a n k e d blade ring) on which a e r o d y n a m i c d r a g acts and leads to an additional e x p e n d i t u r e of power. This is n o t t a k e n into account b y such tests and is considered to be neghg~bly small. The p o w e r absorbed in the second test (say B) m equal to the sum of (i) D r a g of t h e blade ring faces (as m e n t i o n e d above), (11) Disk friction of t h e turbine wheel, (iii) F l y w h e e l disk friction, and (iv) B e a r i n g friction. I f (i) is ignored in t h e above, t h e n the difference (A - B ) will give the " p u m p i n g loss" due to the c o m p l e t e r o t o r blade ring.
EVALUATION
OF
STAGE EFFICIENCIES COEFFICIENTS
AND
LOSS
1. I n p u t energy
F o r a g i v e n v a l u e of the pressure ratio (Po/Pa), t h e iscntropie v e l o c i t y at the nozzle e x i t is calculated f r o m c o = ~/(2gJAh,,) in ft/sec (2) and c o = 109.6 4{T0[1 - (Pa/Po) ° 286]}. (3) Therefore the ideal i n p u t energy to the stage is c0~/(550 × 2g) horse power.
(4)
The t e r m stage as used t h r o u g h o u t t h e t e x t comprises t h e nozzles a n d the rotor. The ideal i n p u t e n e r g y to t h e stage [given b y e q u a t i o n (4)] includes t h e stage horse p o w e r (SHP) a n d t h e nozzle and r o t o r a e r o d y n a m i c losses. 2. Stage p o w e r
T h e b r a k e power, as m e a s u r e d b y t h e b r a k e is g i v e n b y B H P - (TI - T~) K (D + t) N / a s 0 0
(5)
B t t P = 0.08084N(T 1 - T2) × 10 -3.
(6)
To o b t a i n t h e stage horse p o w e r (SttP), shaft losses should be a d d e d to the B H P . S h a f t losses are d e t e r m i n e d again b y r o t a t i n g t h e t u r b i n e wheel b y the electric m o t o r and m e a s u r i n g t h e p o w e r supplied. This p o w e r (say C) is equal to the stun of (i) C o m p l e t e r o t o r blade p u m p i n g , (il) D i s k friction of t h e t u r b i n e wheel, (iii) F l y w h e e l disk friction, a n d (iv) B e a r i n g friction. F r o m this, full and p a r t i a l admission shaft losses are calculated b y the following relations : F u l l admission shaft losses = C - ( A - B ) (7) P a r t i a l admission shaft losses = C - ( A - B) e.
(8)
T h u s t h e stage p o w e r is g i v e n b y S H P -- B H P + shaft losses.
(9)
Aerodynamic losses in partial admission turbines
423
3. Stage e;~ciency The total to static stage efficiency is defined b y the following relation: Stage efficiency = SHP/ideal i n p u t energy to the stage.
(10)
SHP Thus the experimental stage efficiency = 35,420--~-0o ~.
(11)
The quantities SHP, ~h, co and U/c o were obtained in each test on the turbine. Using these values in equation (11) stage efficiencies at various values of U/c o were calculated.
4. Partial admission loss coeJ~cient The value of the stage efficiency for different values of blade to gas speed ratio (U/co) (both at full and partial admission) were calculated as described in the preceding section. Thus, b y difference, the loss coefficient at a particular value of U/c o can be determined. Such values are the experimental loss coefficients, which are compared with the theoretical values given b y equations (24) and (30) derived in Appendices 1 and 2 respectively. DISCUSSION
OF THE
RESULTS
Theoretical and experimental full admission stage efficiencies of the wheels are shown in Fig. 4. The theoretical stage efficiency was calculated using Soderberg's correlation for losses given b y Horlock. a STAGE EFFICIENCY 11. EXPE.RIMENTAL WHEEL NO. *Ai •
-
'
J - - - - 1 --'lm"
A2
]i
THEORETICAL x-
--o..
g
O~--~-
I'O--
O~-
Oe-
O~
I 0.2
I 0-3
I I c~4 u / c o o.s
I o6
I 0.7
FzG. 4. T h e v a r i a t i o n o f t h e t h e o r e t i c a l a n d e x p e r i m e n t a l full a d m i s s i o n efficiencies.
S. M. YAHYA and M. D. C. DOYLE
424
Curves A1, A2 and A3 h a v e been o b t a i n e d f r o m a n u m b e r of tests on t~Lrbine wheels A1, A2 a n d A3 respectively, b y using e q u a t i o n (11). Specffications of these turbine wheels are given in Table l.
TABLE l.
"~Vheel AI A2 A3
S P E C I F I C A T I O N S OE T H E T U R B I N E W H E E L S
B l a d e height (h) (in.)
Pitch (s) (in.)
Axial chord (b) (m.)
P i t c h chord ratio
Aspect ratio
(s/b)
(h/b)
~ ~ ~
0.242 0.315 0.375
0.484 0.484 0.484
0.500 0.652 0.776
1.805 1.805 1.805
Nozzle angle to t h e plane of r o t a t i o n = a = 20.0 ° R o t o r blade angle at t h e e n t r y = fll = 33"5° R o t o r blade angle at t h e exit = f~2 -~ 26.5 D i a m e t e r of t h e wheels a t m i d - b l a d e h e i g h t ---- 12a-~ in. D i a m e t e r of t h e wheels a t t h e h u b --- 1 1 ~ in. D i a m e t e r of t h e wheels a t t h e tip = 1 3 ~ in. Nozzle ring h u b d i a m e t e r = 1 1 ~ in. Nozzle ring tip d i a m e t e r = 1a3-~ in. Nozzle annulus exit a r e a p e r p e n d i c u l a r to the flow -~ 0.066 ft 2 N u m b e r of nozzles --- 16 R o t o r blades in all t h e wheels axe of t h e s a m e shape a n d size. ) * These a p p l y to all the wheels A1, A2 and A3.
The m a i n reasons for the discrepancy b e t w e e n t h e theoretical and e x p e r i m e n t a l curves are(a) The nozzle used m t h e tests are n o t t y p i c a l of those used in the w o r k on which Soderberg's correlation is based, being of sheet m a t e r i a l a n d long axial length. (b) T h e r o t o r blades are designed for o p e r a t i o n a t low U/% and t h e theoretical a n d e x p e r i m e n t a l curves correspond m o s t closely in this region. A t high U/c o the loss corr e l a t i o n will n o t be accurate, as it is n o t i n t e n d e d to a p p l y to conditions far f r o m t h e design point. (c) The t h r e e rotors h a v e t h e s a m e blades w i t h different pitching. Only for wheel A1 is t h e blading m a t c h e d to t h e p i t c h i n g correctly. Stage effieieneies for p a r t i a l admission tests were also calculated as described before a n d t h e e x p e r i m e n t a l loss coefficients e v a l u a t e d b y comparison w i t h t h e full admission stage efficiencies. The theoretical values of these coefficients are p l o t t e d against the length of the admission sector (B/L) for blade to gas speed ratio (U/co) f r o m 0.1 to 0.8 in Fig. 5. The curves show little v a r i a t i o n in loss coefficients b e y o n d B/JL ~- 20. This suggested t h a t for values of B / L a b o v e 20, t h e " e n d of s e c t o r " losses for a g i v e n v a l u e of (U/%) m a y be considered i n v a r i a n t w i t h t h e l e n g t h of the admission sector. I n Fig. 6 loss curves o b t a i n e d (corresponding to U/c o = 0.4) b y t h e formulae of S t e n n i n g 2 a n d Surer a n d T r a u p e l 4 axe c o m p a r e d w i t h the present analysis and the test points. T h e comparison shows t h a t t h e f o r m u l a e of Stenning and Surer and Traupel u n d e r e s t i m a t e t h e losses due to p a r t i a l admission. I n Figs. 7, 8 a n d 9 t h e theoretical values of loss coefficients are c o m p a r e d w i t h some t e s t points o b t a i n e d f r o m tests on wheels A1, A2 and A3. The a g r e e m e n t b e t w e e n t h e o r y and e x p e r i m e n t is reasonable a t low values of U/% (below 0.4), b u t at higher values the d e v i a t i o n of test points f r o m t h e theoretical curves is large in some eases. This is a t t r i b u t e d to t h e prevalence of a low R e y n o l d s n u m b e r , corresponding to higher values of blade to gas speed ratio. This is due to t h e fact t h a t t h e blade speed was k e p t c o n s t a n t at 80.4 f.p.s. (corresponding to 1500 r.p.m.) in all t h e tests
Aerodynamic losses in partial admission turbines
425
0"25 C1~ 0.2
Oq5
ot
(>3 o.~
0,5
o,6
7 O'8
U/CO
(>OS I I 5
OL-, 0
~. I0
-
J t5
,,
I 20
I 25
B ~L "
..-l, 30
F I ~ . 5. The theoretical values of the mixing ]oss coefl~clents.
PRESENT A N A L Y S I S - - - - - S U T E R A N D TP-.AUPEI. - - - - STENNING x TESTS O N TURBINE WHEEL AI -' 'TESTS O N TURBINE WHEEL A,?. • TESTS O N TURBINE WHEEL A3
0"20,
0,15
~ O.IC 0 U
0 .a O.OS
O O
i
,I
S
I0
.--i
r IS
20
,,,
i 25
FIG. 6. Comparison of various formulae and test points.
30
B/L
0 ~. 0,!
O,I t O.~S
0t
0"2
B/S = 60
B/L:30,.
e=so*/o
0.2
B/L: 15'44 B/S'- 40
+~=;+S°/o
~2
0"3
0.3
0.3
I
0.4
0"4
0,4
I
I
O.S
0"5
O,S
EXPERIMENTAL.
:MMG} THEORETICAL
q,
,
@
0"6 U//C 0-7
0.6 ~'~0 0-7
•
0,41 , ~ 0
I
FIG. 7. The comparison b e t w e e n theoretical a n d e x p e r i m e n t a l loss coefficients for wheel A1.
0 J
0
~: 0.15 u
0 0,I
0.1 f O,Os
S:20
~
L--~.72
e: 12~'/=
•
------
0 0.1
OOS
04
0,45
3048
0.2
•
l~s--" 61.12
E/L=
e = 50°Io
O.2
Be/S: 30.56
0,4
O,S
~
/c o
0-6
0.4
0,5
0"6 U~
e
x
0 ' 6 UZ
. ~ ~ , ~ _ ~ - - ~ - - ~
0.3
0,3
B,/L : =Sdl4
*
O.S
• --25 °]e
"~"'- ) 0,4
"
Reo= 1+23X|05
~ 0,3
X
Clvle]( THEORETICAL CM J S • EXPERIMENTAL ReO=O'5XIO - -
0.2
B/s- ,s.~
e :12,s°/o B/L= 7.72
I
I
0-7
0.7
0,7
FIG. 8. T h e comparison beLween theorettcal and experimental loss coefficients for wheel A2.
,J
u u~005 0u
O.IS+
0 0"1
005
04
0'1 S
i
O
~J
>
t~
Aerodynamic losses in partial admission turbines
427
and the gas speed was varied in order to be able to change (U/co). Thus at U/co = 0"5, co -- 168"8 f.p.s, and Re 0 = 0"3 × l0 s. This shows that most of the tests were performed below the critical Reynolds number. But the present theory does not take into account the Reynolds n u m b e r effects; this explains the large deviation of test points from the theoretical curves at higher values of U/co (which correspond to low Reynolds number). __ - -
--
CM¢
CM j THEORETICAL • 0+2
O.IS
EXPERIMENTAL
• -.'I ~'.5 %
B/L = 7,72 B/S = 12..8
"
0.I o,o ,
o.I
0.2
I
0.3
0+4
0"5
0~6
U/~
0.7 0
0"25
0.2
¢ = 2S°/o
b-
B / t : 15.44
z O.lS (J
B/$ : 25.76
•
o.i us
o u 0.o5 o J
o 04
0.2 • = SO°/e
0,3
0'4
05
(>6
07 U/CQ
B/L = 30418
B/S: 5t.52
•
•
•
O,OS~ O/ 0.1 I~IG.
9.
The
-:--~-- . . . . 0"2
0"3
~-- ~-- ~ - - - - - I C~,
0'5
04
-
t
0*7
°/o0
comparison between theoretical and experimental loss coefficients for wheel A3.
To verify the effect of Reynolds n m n b e r on the losses, one test was performed at high Reynolds n u m b e r and losses calculated. The results are shown in Fig. 8 (e = 25 per cent), in which the agreement between theory and test points is much better for higher Reynolds number. Adams 5 reports some investigations on low Reynolds n u m b e r (less t h a n 105) in turbines with partial admission; partial admission losses are shown to be dependent on Reynolds number. The curves shown in a dashed line in these figures show the theoretical mixing a n d sudden expansion losses together (represented b y the loss coefficient Cm~) and the solid line curve only the mixing loss C~. The difference between C~+ a n d C~ is small, particularly at higher values of B/L. Theoretical calculations suggest that sudden expansion losses at higher values of B/_L (say above 5) can be ignored in comparison to mixing losses. I n some tests, the test points show a n irregular trend. This was firstly on account of the fluctuations in the blower delivery, which caused corresponding fluctuations in the nozzle box temperature and pressure. This led to inaccurate measurement of pressure, 29
428
S. M. YAHYA and M. D. C. DOYLE
temperature and mass flow. This discrepancy could not be ehmmated because the blower was driven by a steam turbine which received steam from a boiler having manual controls on its feed water and fuel systems. Secondly, the power developed by the test turbine was very low--powers measured during some tests were as low as 0.1 h.p. These low values of measured power limited the accuracy and increased the chances of casual errors. The theoretical values of Cm as given by equation (24) are only dependent on the pitching of the blades in so far as ~?st and ~:R are dependent on the pitching. Sudden expansmn effects will depend on the pitching but, in general, these are small anyway. The experimental results do not show any marked deviation from the theory due to change of pitch, except for some indication that deviations due to Reynolds number tend to occur at lower U/c o for the wider pitching. CONCLUSIONS
The main conclusions which can be drawn from these experiments a r e - 1. The theory based on simple assumptions predicts fairly accurately the magnitude of mixing and sudden expansion losses for a turbine operating under partial admission conditions. 2. Mixing, leakage and sudden expansion losses form the bulk of the end-ofsector losses. These losses for a sector of length greater than twenty rotor blade chords can be assumed invariant with length. 3. The effect of sudden expansion losses can be ignored for sector lengths greater than five chords and the equation (34) (derived in Appendix 3) will give a sufficiently accurate result. ~st can be calculated from a suitable loss correlation. 4. The effect of rotor blade pitching on partial admission losses appears to be negligible. 5. The formulae of Stenning and Surer and Traupel give lower values of the losses when compared with the test results and formula reported in this paper. Acknowledgement--The authors are indebted to Professor J. H. Horlock for his help and advice in the work reported in this paper. One of the authors, S. M. Yahya, gratefully acknowledges the financial grant awarded by the British Ship Research Association, for the period between September 1962 and October 1965. REFERENCES 1. S. M. YAHYA, Int. J. mech. Sci. 10, 65 (1968). 2. A. H. S ~ I N G , "Design of turbines for high energy fuel low power output applications." D.A.C.L. Rep. M.I.T. (Sept. 1953). 3. J. H. HORLOCK, Int. J. mech. Sci. 2, 48 (1960). 4. P. SUTER and W. T~AUP~.L, B.S.R.A. Translation No. 917 (Ref. No. 4 of Institute for Thermal Turbines, Zurich) (1960).
5. R . G. ADAMS, The Effect of Reynolds Number on the Performance of Partial Admission and Re-entry Axial Turbines. 65-GTP-3, ASME Journal, 1965. 6. M. D. C. D o ~ , Aerospace Sci. 29, 489 (May 1962). APPENDIX
1
Mixing loas coe~cient (C~) Full admission stage efficiency is given by U W(cos fl' + cos fl~)- ER(W'/2)
(12)
Aerodynamic losses in partial admission turbines
429
The work for a small increment of admission sector 8B is, UV(eosfl'+eosfl~) ph SB V sin fl"
(13)
B = Utl.
(14)
and Thus the total p a r t i a l admission work is given b y
f:'VV(cos fl" + cos f12) ph UV sin fl' dr.
(15)
The rotor aerodynamic losses are t'
/V2~
~R(-~-) ph UV sinfl' dr.
(16)
Thus the actual partial admission work is
U~(cos f~' + cos fi~) ph sin ~' ( J0t~V 2 dt - ½~R ph U sin[J" f:lva dr.
(17)
I t has been shown b y Y a h y a ~ t h a t V = W(e **- 1)/(e~*+ 1).
(18)
F r o m equation (18) the integrals in equation (17) reduce to
~lV~dt
nW~tl
(19)
and
f~
*V a dt = roW* t I •
(20)
F r o m equations (17), (19) and (20), partial admission work is rh,[(cos fl' +cosfl2)nUW-~_~ WS2]
(21)
and
rh~ = ph Ut 1 W sinfl'.
(22)
Therefore partial admission stage efficiency (cos fi" + cos fl~) n U W - ~_~m(W~/2)
(23)
Equations (12) and (23) yield
C,n = ~,t-~,~ = ( 1 - n ) ~ - ( n - m ) ~ . APPENDIX
~:R W~
(24)
2
Mixing and sudden expansion loss coeyficient (C,~) The mixing is now considered to t a k e place for a time t 1, for a n y rotor blade passage. This is given b y 8 (25) Ut 1 = B---~. E q u a t i o n (25) can now be used to amend the derivation given in Appendix 1. The p a r t i a l admiSsion work over the length [ B - (s/2)] is (26)
430
S.M. YAHYA and M. D. C. DOYLE
Considering sudden expansion at the leaving-end, the work which corresponds to the admission sector length 2s is
sJo
ks/
L •
8
= m.(~-~) WU(cosfl'+½cosf~,).
(27)
From equations (26) and (27) the total partial admission stage work is
rh.(1-~--B) [(cos[J'+cosfl,)nUW-m~R-W~]
Therefore the partial admission stage efficiency is ~st~ = ~ s1
(1-~B)[(cosfl'+cosfl,)nUW-m~R?] 1
s
Equations (12) and (29) yield
v,.. = h-~(cos~ +cosf~) 1-~-~ (1-~) 8
8
2Ah,~
Ah,s
APPENDIX
3
A simplified form of the lose coefficient At very small values of (s/2B), i.e. for very long admission sectors, equation (30) will give very neaxly the same result as equation (24), which can be obtained from equation (30) b y p u t t i n g 8/2B = 0 a n d using equation (12). Small values of s/2B correspond to large values of B / L as s and L are usually of the same order. As kt 1 = 2 W B / L U and W and U axe of the same order, kt 1 will be large for small values of s/2B. I n these circumstances the values of n and m can be approximated by 2
n = 1 - k~t
(31)
2.386 m = 1---
(32)
and
ktl
Equations (31), (32) and (24) give 2
0.386~RW ~ 2kt I Ah~,
(33)
B u t 0.386~RW2/2ktz Ah~swill be small compared with (2/ktz) 7/st and can often be neglected giving the simplified equation,
2 UL C,n = ~l~-:7-~~lst = ~-~'qst
(34)
This equation is suitable for long admission sectors and corresponds in form to an equation b y Doyle)
A e r o d y n a m i c losses in p a r t i a l a d m i s s i o n t u r b i n e s APPENDIX
431
4
Calculation of the stage efficiency of partial admission turbines using loss coefficients T h e following d a t a , w h i c h i n a c t u a l p r a c t i c e a r e k n o w n , a r e a s s u m e d for a t u r b i n e wheel: U/c o= 0"4, e = 50 p e r c e n t , B / L = 30.88, F u l l a d m i s s i o n s t a g e efficiency, ~,~ --- 68 p e r c e n t (from Fig. 4), M i x i n g loss coefficient, C~ -- 2 p e r c e n t (from Fig. 5), D i s k f r i c t i o n a n d p u m p i n g ( w i n d a g e losses), Lw = 6-87 ft/lbf. T h e s e c a n e i t h e r b e c a l c u l a t e d b y a m e t h o d s u g g e s t e d b y S u t e r a n d T r a u p e l or d e t e r m i n e d e x p e r i m e n t a l l y . I n t h e p r e s e n t case t h e c a l c u l a t e d v a l u e is n e a r l y t h e s a m e as t h a t measured. Mass flow t h r o u g h t h i s s t a g e rh = 24 l b m / m i n , I s e n t r o p i c v e l o c i t y a t t h e e x i t of t h e nozzles, c o = 205 f.p.s. W i n d a g e loss coefficient Cw =
Lw
mc~/2g 6 . 8 7 × 2 × 3 2 . 2 x 60
Cw =
24 x 205 × 205
x 100
and C~ = 2.63 p e r c e n t . S t a g e efficiency of t h e p a r t i a l a d m i s s i o n t u r b i n e is ~
= ~?~ = C ~ - C ~ .
S u d d e n e x p a n s i o n a n d o t h e r losses a r e c o n s i d e r e d as negligible for 50 p e r c e n t degree of a d m i s s i o n . T h u s ~,~ = 6 8 . 0 0 - 2 . 0 0 - 2.62 = 63.38 p e r c e n t .
AERODYNAMIC LOSSES IN P A R T I A L ADMISSION TURBINES S. M. Y z a r r A I n d i a n Institute of Technology, New Delhi, India
and M. D. C. DOYLE Fourah Bay College, University of Sierra Leone, Freetown, Sierra Leone (Received 30 J u l y 1968, a n d i n revised f o r m 2 J a n u a r y 1969)
S u m m a r y - - A theory for the determination of the aerodynamic losses in a flat-bladed rotor operating under partial admission conditions is taken from a previous paper. I t is adapted to obtain a formula for the loss coefficients for a real turbine rotor operating under partial admission conclitlons. A n approximate b u t much simphfied version of this formula is developed for certain cases. The loss formulae derived are verified using three turbine rotors of different blade pitch. The formulae are shown to be approximately correct and the effect of blade pitching is shown to be negligible. A method (numerical example) of predicting the stage efficiency of partial admission turbines is also given. NOTATION B = Ut 1, the admission sector length
C D J L iV P0
loss coefficients diameter mechanical equivalent of heat length of the rotor blade channel revolutions per minute of the rotor stagnation pressure in the settling chamber R e o isentropic Reynolds n u m b e r based on rotor blade chord and velocity (Co) T tension in the brake belt T o stagnation temperature in the settling chamber U peripheral velocity of the rotor blades at mean diameter V transient velocity at the exit of the rotor blade channel W relative velocity of the gas at the entry of the rotor blades b rotor blade chord c nozzle exit velocity co isentropic velocity at the nozzle exit e degree of admission f( ) function of g acceleration due to gravity h rotor and nozzle blade height or enthalpy 2W L m = - I - I - 2 lo / l + e k t l \ 4e~t~ kt 1 * k~ ge ~ - - ~ - - ) + kt,(1 + e~',) rh mass of gas flowing through the rotor per second 4 2 n=l-~ k t t ( 1 + e ktl)
kt 1
417
418
S.M. YAHYAand M. D. C. DOYLE s t t~ x
rotor blade pitch time or thickness of the brake belt total time eonmdered displacement of the rotor blade along the admission sector as shown m Fig. 2. BHP brake horse power SHP stage horse power Ah enthalpy drop = C2o/2g
~b'ubscripts 1 rotor blade entrance or tight side of the brake belt 2 rotor blade exit or slack side of the brake belt e corresponding to a degree of admission = e; or sudden expansion m mixing p partial admlssion ~s iscntroplc st stage N nozzle blade R rotor blade Superscmpts • air angle at the entry of the rotor blades Greek symbols All angles are from the peripheral direction as shown in Fig. 1 nozzle angle rotor blade angles total to static efficiency p density ~: blade aerodynamic loss coefficient =
h2 - h2,s
½W2
INTRODUCTION A P~EVIOUS paper, Y a h y a 1 gave the d e r i v a t i o n of a theoretical velocity profile for the flow of fluid t h r o u g h t u r b i n e blades which h a d just entered an active nozzle sector in a partial admission turbine, h a v i n g been inactive in the region of no admission. The t h e o r y was derived using a t u r b i n e r o t o r model with u n c a m b e r e d axial blades or " f l a t - p l a t e " blades. This t h e o r y was confirmed e x p e r i m e n t a l l y using a t u r b i n e with " f l a t - p l a t e " blades. I n this paper, the results derived a n d confirmed for the " f i a t - p l a t e " rotor are used to o b t a i n the m a g n i t u d e of the losses due to partial admission in an axial flow impulse turbine. The losses are expressed in t e r m s of a loss coefficient. The principal losses in partial admission turbines are due to (i) R o t o r blades p u m p i n g the gas in the inactive sector, (ii) Mixing of the active a n d inactive gas, (iii) L e a k a g e o f active gas t h r o u g h t h e inactive sector, (iv) S u d d e n expansion of the active gas into a r o t o r blade passage which is either entering or leaving the active admission sector. I n this p a p e r the losses due to r o t o r blade p u m p i n g only h a v e been determ i n e d experimentally. The analysis takes into a c c o u n t mixing, leakage a n d sudden expansion losses. O t h e r losses are regarded as negligibly small c o m p a r e d to these.
Aerodynamic losses in partial admission turbines
419
T h e t h e o r e t i c a l r e s u l t s , i n t h e f o r m o f loss c o e f f i c i e n t s , a r e c o m p a r e d w i t h the experimental values obtained from tests on three impulse turbine wheels w i t h d i f f e r e n t p i t c h i n g o f t h e b l a d e s . I n d e r i v i n g t h e l o s s c o e f f i c i e n t s i t is necessary to distinguish between the end of the sector where the rotor blades e n t e r t h e a c t i v e s e c t o r , w h i c h is c a l l e d t h e e n t e r i n g - e n d a n d t h e o t h e r e n d , w h i c h is c a l l e d t h e l e a v i n g - e n d . THEORETICAL
CONSIDERATIONS
The loss formulae are derived under the following assumptions: (i) The flow is frictionless and incompressible. (il) There is an infinite number of blades on the rotor and one-dimensional flow through the rotor blades. (iii) There is no pressure drop across the rotor. I t is assumed t h a t the leakage of active gas prevents a n y pressure build-up at entry to the rotor. (iv) The velocity profile due to mixing of active and inactive gas predicted b y Y a h y a 1 for the "flat-plate" rotor holds for the impulse turbine blades considered in the present section. Corrections are applied to allow for those aerodynamic losses which are inherent in turbines, such as profile loss and secondary loss.
1. Mixing loss coej~cient ignomng sudden expansion I f the sudden expansion b o t h a t the leaving-end and the entering-end of the admission sector is ignored, then mixing can be considered to continue until the blade channels leave the admission sector. I n such a case some of the m o m e n t u m of the gas, relative to the rotor at inlet, is lost in aeceleratmg inactive gas in the rotor blades and resulting in leakage of the gas past the rotor shroud. Thus the velocity of the gas entering the rotor is V, which is less t h a n W and depends on the distance moved b y the rotor blade into the aclmi~ion sector in time t. The direction of flow relative to the rotor will be a t an angle ~' to the tangential direction as in the full admission case. The above flow model is used to determine the blade work for partial and full admiasion conditions. Stage efficiencies in both cases are found b y taking into account the normal rotor blade aerodynamic losses. The difference between these efficiencies gives the value of mixing loss coefficient, C~. This gives the amount b y which the stage efficiency of a turbine is lowered due to mixing when the turbine is run under partial admission conditions. An expression for this loss coefficient is developed in Appendix 1.
2. Mixing and sudden expansion loss coe~cient When sudden expansion losses are considered, the effect cannot be separated from mixing losses at the entering-end, b u t a t the leaving-end and it m a y be assumed t h a t U
2/
U FIG. 1. Inlet and exit velocity triangles.
mixing losses are negligible. Thus partial admission work is best calculated set~rately for the entering- and leaving-ends. Therefore mixing is n o t assumed to occur in the length ( s - x) which is exposed a t a n y time. This is shown in Fig. 2.
420
S.M. YAHYA and M. D. C. DOYLE
The average length along the admission sector for which mixing is considered is thus given b y
if0, ( B + X - S ) d x
s
s
(1)
= B--~.
Thus the flow through length [B - (s/2)] of the nozzle sector can be considered to suffer mixing loss, while the flow through the remaining length 8/2 suffers sudden expansion. Furthermore, it is assumed that the gas entering the length ( s - x) expands to fill the whole blade passage. 2 Thus the exit velocity is W ( s - x)/s.
Ii//I
)))))))))))
FIO. 2. Length of the transition zone. A loss coefficient Cme b ~ e d on the above flow model is derived in Appendix 2. This loss coefficient gives ~he a m o u n t b y which the stage efficiency of a turbine is lowered due to mixing and sudden expansion when the turbine is r u n under partial admission conditions. INSTRUMENTATION
1. General layout A general layout of the test rig is shown in Fig. 3. I t consists of a Napier's Turbocharger supplying air, through a 3~ in. dia. British Standard orifice, to a settling cha~nber. The ratio of the cross-sectional areas of the settling chamber and the nozzle ring is approximately 14, which gives a comparatively low velocity on approaching the nozzles. SETTLING
--.....
C HAMBER
GAUZE
NOZZLE BOX
~
TURBINE WHEEL BI~,KE WHEEL
ELECTRIC MOTOR
EXHAUST 16 ANNULUS NOZZLES
FIG. 3. General layout of the test rig.
The settling chamber was provided with ten static pressure tappings and five thermometer pockets. The temperature and pressure as indicated at these points were practically the mime. Since the velocity of air approaching the nozzle was small, the temperature and pressure at the nozzle entry could be considered to represent the stagnation values.
Aerodynamic losses in partial admission turbines
421
The nozzle ring consisting of sixteen nozzles is attached at the end of the settling chamber, concentric with the air-supply pipe. Static pressure tappings were provided at the entry of each nozzle. They showed very nearly the same pressure indicating a uniform static pressure distribution at the entry of the complete nozzle annulus. Blanking segments of various lengths were used to blank a desired n u m b e r of nozzles for partial admission tests.
2. The brake The brake consists of a mild-steel flywheel ( 9 ~ in. outside dia.) keyed to the turbine wheel-shaft between the two pedestals as shown in Fig. 3. The braking torque was applied b y a ~ in. thick, 1 in. wide Ferodo belt which was fixed to a hydraulic pressimeter giving a manometer deflection, which was calibrated to determine the tension at one end and the other carried a weight pan. A ~ h.p. electric motor was used to t u r n the flywheel and the turbine rotor to determine the shaft and rotor blade pumping losses. THE
TURBINE
TESTS
These tests were performed to determine the brake horse power (BHP) and the i n p u t energy at various values of the total to static pressure ratio (Po/Pa) for full a n d partial admission cases. (Pa) was the atmospheric pressure and (P0) was taken as the nozzle box pressure. Most of the tests were performed at a turbine speed of 1500 r.p.m. The blade to gas speed ratio (U/co} varied due to the varying of the gas speed which resulted from the varying pressure ratio. After changing from one pressure ratio to another some time was allowed for the flow to become steady. Then the weights on the p a n were adjusted to bring the speed of the turbine to 1500 r.p.m. (or a n y other value as reported in the high-speed tests). The hydraulic pressimeter manometer was read for each load. The following readings were recorded during each test: (i) Orifice temperature, (ii) Orifice pressure, (iii) Static pressure drop across the orifice, (vi) Nozzle box temperature (To), (v) Nozzle box static pressure (P0), (vi) Weights in the weight pan, (vii) Hydraulic pressimeter manometer deflection, (viii) Turbine rotor speed in r.p.m. (N). At very low brake loads, the tension on the hydraulic pressimeter was very low and it was ineffective. The brake power at these low loads was determined b y t u r n i n g the turbine wheel at the test speed (with the same weights on the brake pan) b y the electric motor a n d measuring the torque, after the brake tests. MEASUREMENT
OF THE
ROTOR
BLADE
PUMPING
LOSS
To determine the pumping loss, when the turbine works with partial admission, loss due to "pumping" for no admission was measured as described below. Then for a given degree of admission (e) this loss was calculated, assuming it to be proportional to ( 1 - e ) . Thus, (pumping loss)~ = pumping loss for complete blade ring × ( 1 - e ) . The turbine wheel was rotated at its test speed b y the electric motor and the power supplied by it was measured b y weighing the torque on the motor. This power (say A) is equal to the sum of (i) Complete blade ring pumping, (ii) Disk friction of the turbine wheel, (iii) Flywheel disk friction, and (iv) Bearing friction.
422
~. M. YAHYA
and M. D. C. DOYLE
The s a m e test was t h e n p e r f o r m e d after b l a n k i n g t h e rotor blade ring b y pouring m o l t e n w a x into its blade channels a n d giving a s m o o t h finish to the ring. Thin t~me, the p o w e r r e q u i r e d to r o t a t e t h e wheel at t h e s a m e speed (i.e. t h e test speed) was less, due to t h e absence of blade p u m p i n g loss. The difference of powers m e a s u r e d in the two cases g a v e t h e loss due to p u m p i n g of t h e c o m p l e t e blade ring, as explained below. B l a n k i n g of t h e blade ring introduces additional surfaces (the two v e r t i c a l faces of the b l a n k e d blade ring) on which a e r o d y n a m i c d r a g acts and leads to an additional e x p e n d i t u r e of power. This is n o t t a k e n into account b y such tests and is considered to be neghg~bly small. The p o w e r absorbed in the second test (say B) m equal to the sum of (i) D r a g of t h e blade ring faces (as m e n t i o n e d above), (11) Disk friction of t h e turbine wheel, (iii) F l y w h e e l disk friction, and (iv) B e a r i n g friction. I f (i) is ignored in t h e above, t h e n the difference (A - B ) will give the " p u m p i n g loss" due to the c o m p l e t e r o t o r blade ring.
EVALUATION
OF
STAGE EFFICIENCIES COEFFICIENTS
AND
LOSS
1. I n p u t energy
F o r a g i v e n v a l u e of the pressure ratio (Po/Pa), t h e iscntropie v e l o c i t y at the nozzle e x i t is calculated f r o m c o = ~/(2gJAh,,) in ft/sec (2) and c o = 109.6 4{T0[1 - (Pa/Po) ° 286]}. (3) Therefore the ideal i n p u t energy to the stage is c0~/(550 × 2g) horse power.
(4)
The t e r m stage as used t h r o u g h o u t t h e t e x t comprises t h e nozzles a n d the rotor. The ideal i n p u t e n e r g y to t h e stage [given b y e q u a t i o n (4)] includes t h e stage horse p o w e r (SHP) a n d t h e nozzle and r o t o r a e r o d y n a m i c losses. 2. Stage p o w e r
T h e b r a k e power, as m e a s u r e d b y t h e b r a k e is g i v e n b y B H P - (TI - T~) K (D + t) N / a s 0 0
(5)
B t t P = 0.08084N(T 1 - T2) × 10 -3.
(6)
To o b t a i n t h e stage horse p o w e r (SttP), shaft losses should be a d d e d to the B H P . S h a f t losses are d e t e r m i n e d again b y r o t a t i n g t h e t u r b i n e wheel b y the electric m o t o r and m e a s u r i n g t h e p o w e r supplied. This p o w e r (say C) is equal to the stun of (i) C o m p l e t e r o t o r blade p u m p i n g , (il) D i s k friction of t h e t u r b i n e wheel, (iii) F l y w h e e l disk friction, a n d (iv) B e a r i n g friction. F r o m this, full and p a r t i a l admission shaft losses are calculated b y the following relations : F u l l admission shaft losses = C - ( A - B ) (7) P a r t i a l admission shaft losses = C - ( A - B) e.
(8)
T h u s t h e stage p o w e r is g i v e n b y S H P -- B H P + shaft losses.
(9)
Aerodynamic losses in partial admission turbines
423
3. Stage e;~ciency The total to static stage efficiency is defined b y the following relation: Stage efficiency = SHP/ideal i n p u t energy to the stage.
(10)
SHP Thus the experimental stage efficiency = 35,420--~-0o ~.
(11)
The quantities SHP, ~h, co and U/c o were obtained in each test on the turbine. Using these values in equation (11) stage efficiencies at various values of U/c o were calculated.
4. Partial admission loss coeJ~cient The value of the stage efficiency for different values of blade to gas speed ratio (U/co) (both at full and partial admission) were calculated as described in the preceding section. Thus, b y difference, the loss coefficient at a particular value of U/c o can be determined. Such values are the experimental loss coefficients, which are compared with the theoretical values given b y equations (24) and (30) derived in Appendices 1 and 2 respectively. DISCUSSION
OF THE
RESULTS
Theoretical and experimental full admission stage efficiencies of the wheels are shown in Fig. 4. The theoretical stage efficiency was calculated using Soderberg's correlation for losses given b y Horlock. a STAGE EFFICIENCY 11. EXPE.RIMENTAL WHEEL NO. *Ai •
-
'
J - - - - 1 --'lm"
A2
]i
THEORETICAL x-
--o..
g
O~--~-
I'O--
O~-
Oe-
O~
I 0.2
I 0-3
I I c~4 u / c o o.s
I o6
I 0.7
FzG. 4. T h e v a r i a t i o n o f t h e t h e o r e t i c a l a n d e x p e r i m e n t a l full a d m i s s i o n efficiencies.
S. M. YAHYA and M. D. C. DOYLE
424
Curves A1, A2 and A3 h a v e been o b t a i n e d f r o m a n u m b e r of tests on t~Lrbine wheels A1, A2 a n d A3 respectively, b y using e q u a t i o n (11). Specffications of these turbine wheels are given in Table l.
TABLE l.
"~Vheel AI A2 A3
S P E C I F I C A T I O N S OE T H E T U R B I N E W H E E L S
B l a d e height (h) (in.)
Pitch (s) (in.)
Axial chord (b) (m.)
P i t c h chord ratio
Aspect ratio
(s/b)
(h/b)
~ ~ ~
0.242 0.315 0.375
0.484 0.484 0.484
0.500 0.652 0.776
1.805 1.805 1.805
Nozzle angle to t h e plane of r o t a t i o n = a = 20.0 ° R o t o r blade angle at t h e e n t r y = fll = 33"5° R o t o r blade angle at t h e exit = f~2 -~ 26.5 D i a m e t e r of t h e wheels a t m i d - b l a d e h e i g h t ---- 12a-~ in. D i a m e t e r of t h e wheels a t t h e h u b --- 1 1 ~ in. D i a m e t e r of t h e wheels a t t h e tip = 1 3 ~ in. Nozzle ring h u b d i a m e t e r = 1 1 ~ in. Nozzle ring tip d i a m e t e r = 1a3-~ in. Nozzle annulus exit a r e a p e r p e n d i c u l a r to the flow -~ 0.066 ft 2 N u m b e r of nozzles --- 16 R o t o r blades in all t h e wheels axe of t h e s a m e shape a n d size. ) * These a p p l y to all the wheels A1, A2 and A3.
The m a i n reasons for the discrepancy b e t w e e n t h e theoretical and e x p e r i m e n t a l curves are(a) The nozzle used m t h e tests are n o t t y p i c a l of those used in the w o r k on which Soderberg's correlation is based, being of sheet m a t e r i a l a n d long axial length. (b) T h e r o t o r blades are designed for o p e r a t i o n a t low U/% and t h e theoretical a n d e x p e r i m e n t a l curves correspond m o s t closely in this region. A t high U/c o the loss corr e l a t i o n will n o t be accurate, as it is n o t i n t e n d e d to a p p l y to conditions far f r o m t h e design point. (c) The t h r e e rotors h a v e t h e s a m e blades w i t h different pitching. Only for wheel A1 is t h e blading m a t c h e d to t h e p i t c h i n g correctly. Stage effieieneies for p a r t i a l admission tests were also calculated as described before a n d t h e e x p e r i m e n t a l loss coefficients e v a l u a t e d b y comparison w i t h t h e full admission stage efficiencies. The theoretical values of these coefficients are p l o t t e d against the length of the admission sector (B/L) for blade to gas speed ratio (U/co) f r o m 0.1 to 0.8 in Fig. 5. The curves show little v a r i a t i o n in loss coefficients b e y o n d B/JL ~- 20. This suggested t h a t for values of B / L a b o v e 20, t h e " e n d of s e c t o r " losses for a g i v e n v a l u e of (U/%) m a y be considered i n v a r i a n t w i t h t h e l e n g t h of the admission sector. I n Fig. 6 loss curves o b t a i n e d (corresponding to U/c o = 0.4) b y t h e formulae of S t e n n i n g 2 a n d Surer a n d T r a u p e l 4 axe c o m p a r e d w i t h the present analysis and the test points. T h e comparison shows t h a t t h e f o r m u l a e of Stenning and Surer and Traupel u n d e r e s t i m a t e t h e losses due to p a r t i a l admission. I n Figs. 7, 8 a n d 9 t h e theoretical values of loss coefficients are c o m p a r e d w i t h some t e s t points o b t a i n e d f r o m tests on wheels A1, A2 and A3. The a g r e e m e n t b e t w e e n t h e o r y and e x p e r i m e n t is reasonable a t low values of U/% (below 0.4), b u t at higher values the d e v i a t i o n of test points f r o m t h e theoretical curves is large in some eases. This is a t t r i b u t e d to t h e prevalence of a low R e y n o l d s n u m b e r , corresponding to higher values of blade to gas speed ratio. This is due to t h e fact t h a t t h e blade speed was k e p t c o n s t a n t at 80.4 f.p.s. (corresponding to 1500 r.p.m.) in all t h e tests
Aerodynamic losses in partial admission turbines
425
0"25 C1~ 0.2
Oq5
ot
(>3 o.~
0,5
o,6
7 O'8
U/CO
(>OS I I 5
OL-, 0
~. I0
-
J t5
,,
I 20
I 25
B ~L "
..-l, 30
F I ~ . 5. The theoretical values of the mixing ]oss coefl~clents.
PRESENT A N A L Y S I S - - - - - S U T E R A N D TP-.AUPEI. - - - - STENNING x TESTS O N TURBINE WHEEL AI -' 'TESTS O N TURBINE WHEEL A,?. • TESTS O N TURBINE WHEEL A3
0"20,
0,15
~ O.IC 0 U
0 .a O.OS
O O
i
,I
S
I0
.--i
r IS
20
,,,
i 25
FIG. 6. Comparison of various formulae and test points.
30
B/L
0 ~. 0,!
O,I t O.~S
0t
0"2
B/S = 60
B/L:30,.
e=so*/o
0.2
B/L: 15'44 B/S'- 40
+~=;+S°/o
~2
0"3
0.3
0.3
I
0.4
0"4
0,4
I
I
O.S
0"5
O,S
EXPERIMENTAL.
:MMG} THEORETICAL
q,
,
@
0"6 U//C 0-7
0.6 ~'~0 0-7
•
0,41 , ~ 0
I
FIG. 7. The comparison b e t w e e n theoretical a n d e x p e r i m e n t a l loss coefficients for wheel A1.
0 J
0
~: 0.15 u
0 0,I
0.1 f O,Os
S:20
~
L--~.72
e: 12~'/=
•
------
0 0.1
OOS
04
0,45
3048
0.2
•
l~s--" 61.12
E/L=
e = 50°Io
O.2
Be/S: 30.56
0,4
O,S
~
/c o
0-6
0.4
0,5
0"6 U~
e
x
0 ' 6 UZ
. ~ ~ , ~ _ ~ - - ~ - - ~
0.3
0,3
B,/L : =Sdl4
*
O.S
• --25 °]e
"~"'- ) 0,4
"
Reo= 1+23X|05
~ 0,3
X
Clvle]( THEORETICAL CM J S • EXPERIMENTAL ReO=O'5XIO - -
0.2
B/s- ,s.~
e :12,s°/o B/L= 7.72
I
I
0-7
0.7
0,7
FIG. 8. T h e comparison beLween theorettcal and experimental loss coefficients for wheel A2.
,J
u u~005 0u
O.IS+
0 0"1
005
04
0'1 S
i
O
~J
>
t~
Aerodynamic losses in partial admission turbines
427
and the gas speed was varied in order to be able to change (U/co). Thus at U/co = 0"5, co -- 168"8 f.p.s, and Re 0 = 0"3 × l0 s. This shows that most of the tests were performed below the critical Reynolds number. But the present theory does not take into account the Reynolds n u m b e r effects; this explains the large deviation of test points from the theoretical curves at higher values of U/co (which correspond to low Reynolds number). __ - -
--
CM¢
CM j THEORETICAL • 0+2
O.IS
EXPERIMENTAL
• -.'I ~'.5 %
B/L = 7,72 B/S = 12..8
"
0.I o,o ,
o.I
0.2
I
0.3
0+4
0"5
0~6
U/~
0.7 0
0"25
0.2
¢ = 2S°/o
b-
B / t : 15.44
z O.lS (J
B/$ : 25.76
•
o.i us
o u 0.o5 o J
o 04
0.2 • = SO°/e
0,3
0'4
05
(>6
07 U/CQ
B/L = 30418
B/S: 5t.52
•
•
•
O,OS~ O/ 0.1 I~IG.
9.
The
-:--~-- . . . . 0"2
0"3
~-- ~-- ~ - - - - - I C~,
0'5
04
-
t
0*7
°/o0
comparison between theoretical and experimental loss coefficients for wheel A3.
To verify the effect of Reynolds n m n b e r on the losses, one test was performed at high Reynolds n u m b e r and losses calculated. The results are shown in Fig. 8 (e = 25 per cent), in which the agreement between theory and test points is much better for higher Reynolds number. Adams 5 reports some investigations on low Reynolds n u m b e r (less t h a n 105) in turbines with partial admission; partial admission losses are shown to be dependent on Reynolds number. The curves shown in a dashed line in these figures show the theoretical mixing a n d sudden expansion losses together (represented b y the loss coefficient Cm~) and the solid line curve only the mixing loss C~. The difference between C~+ a n d C~ is small, particularly at higher values of B/L. Theoretical calculations suggest that sudden expansion losses at higher values of B/_L (say above 5) can be ignored in comparison to mixing losses. I n some tests, the test points show a n irregular trend. This was firstly on account of the fluctuations in the blower delivery, which caused corresponding fluctuations in the nozzle box temperature and pressure. This led to inaccurate measurement of pressure, 29
428
S. M. YAHYA and M. D. C. DOYLE
temperature and mass flow. This discrepancy could not be ehmmated because the blower was driven by a steam turbine which received steam from a boiler having manual controls on its feed water and fuel systems. Secondly, the power developed by the test turbine was very low--powers measured during some tests were as low as 0.1 h.p. These low values of measured power limited the accuracy and increased the chances of casual errors. The theoretical values of Cm as given by equation (24) are only dependent on the pitching of the blades in so far as ~?st and ~:R are dependent on the pitching. Sudden expansmn effects will depend on the pitching but, in general, these are small anyway. The experimental results do not show any marked deviation from the theory due to change of pitch, except for some indication that deviations due to Reynolds number tend to occur at lower U/c o for the wider pitching. CONCLUSIONS
The main conclusions which can be drawn from these experiments a r e - 1. The theory based on simple assumptions predicts fairly accurately the magnitude of mixing and sudden expansion losses for a turbine operating under partial admission conditions. 2. Mixing, leakage and sudden expansion losses form the bulk of the end-ofsector losses. These losses for a sector of length greater than twenty rotor blade chords can be assumed invariant with length. 3. The effect of sudden expansion losses can be ignored for sector lengths greater than five chords and the equation (34) (derived in Appendix 3) will give a sufficiently accurate result. ~st can be calculated from a suitable loss correlation. 4. The effect of rotor blade pitching on partial admission losses appears to be negligible. 5. The formulae of Stenning and Surer and Traupel give lower values of the losses when compared with the test results and formula reported in this paper. Acknowledgement--The authors are indebted to Professor J. H. Horlock for his help and advice in the work reported in this paper. One of the authors, S. M. Yahya, gratefully acknowledges the financial grant awarded by the British Ship Research Association, for the period between September 1962 and October 1965. REFERENCES 1. S. M. YAHYA, Int. J. mech. Sci. 10, 65 (1968). 2. A. H. S ~ I N G , "Design of turbines for high energy fuel low power output applications." D.A.C.L. Rep. M.I.T. (Sept. 1953). 3. J. H. HORLOCK, Int. J. mech. Sci. 2, 48 (1960). 4. P. SUTER and W. T~AUP~.L, B.S.R.A. Translation No. 917 (Ref. No. 4 of Institute for Thermal Turbines, Zurich) (1960).
5. R . G. ADAMS, The Effect of Reynolds Number on the Performance of Partial Admission and Re-entry Axial Turbines. 65-GTP-3, ASME Journal, 1965. 6. M. D. C. D o ~ , Aerospace Sci. 29, 489 (May 1962). APPENDIX
1
Mixing loas coe~cient (C~) Full admission stage efficiency is given by U W(cos fl' + cos fl~)- ER(W'/2)
(12)
Aerodynamic losses in partial admission turbines
429
The work for a small increment of admission sector 8B is, UV(eosfl'+eosfl~) ph SB V sin fl"
(13)
B = Utl.
(14)
and Thus the total p a r t i a l admission work is given b y
f:'VV(cos fl" + cos f12) ph UV sin fl' dr.
(15)
The rotor aerodynamic losses are t'
/V2~
~R(-~-) ph UV sinfl' dr.
(16)
Thus the actual partial admission work is
U~(cos f~' + cos fi~) ph sin ~' ( J0t~V 2 dt - ½~R ph U sin[J" f:lva dr.
(17)
I t has been shown b y Y a h y a ~ t h a t V = W(e **- 1)/(e~*+ 1).
(18)
F r o m equation (18) the integrals in equation (17) reduce to
~lV~dt
nW~tl
(19)
and
f~
*V a dt = roW* t I •
(20)
F r o m equations (17), (19) and (20), partial admission work is rh,[(cos fl' +cosfl2)nUW-~_~ WS2]
(21)
and
rh~ = ph Ut 1 W sinfl'.
(22)
Therefore partial admission stage efficiency (cos fi" + cos fl~) n U W - ~_~m(W~/2)
(23)
Equations (12) and (23) yield
C,n = ~,t-~,~ = ( 1 - n ) ~ - ( n - m ) ~ . APPENDIX
~:R W~
(24)
2
Mixing and sudden expansion loss coeyficient (C,~) The mixing is now considered to t a k e place for a time t 1, for a n y rotor blade passage. This is given b y 8 (25) Ut 1 = B---~. E q u a t i o n (25) can now be used to amend the derivation given in Appendix 1. The p a r t i a l admiSsion work over the length [ B - (s/2)] is (26)
430
S.M. YAHYA and M. D. C. DOYLE
Considering sudden expansion at the leaving-end, the work which corresponds to the admission sector length 2s is
sJo
ks/
L •
8
= m.(~-~) WU(cosfl'+½cosf~,).
(27)
From equations (26) and (27) the total partial admission stage work is
rh.(1-~--B) [(cos[J'+cosfl,)nUW-m~R-W~]
Therefore the partial admission stage efficiency is ~st~ = ~ s1
(1-~B)[(cosfl'+cosfl,)nUW-m~R?] 1
s
Equations (12) and (29) yield
v,.. = h-~(cos~ +cosf~) 1-~-~ (1-~) 8
8
2Ah,~
Ah,s
APPENDIX
3
A simplified form of the lose coefficient At very small values of (s/2B), i.e. for very long admission sectors, equation (30) will give very neaxly the same result as equation (24), which can be obtained from equation (30) b y p u t t i n g 8/2B = 0 a n d using equation (12). Small values of s/2B correspond to large values of B / L as s and L are usually of the same order. As kt 1 = 2 W B / L U and W and U axe of the same order, kt 1 will be large for small values of s/2B. I n these circumstances the values of n and m can be approximated by 2
n = 1 - k~t
(31)
2.386 m = 1---
(32)
and
ktl
Equations (31), (32) and (24) give 2
0.386~RW ~ 2kt I Ah~,
(33)
B u t 0.386~RW2/2ktz Ah~swill be small compared with (2/ktz) 7/st and can often be neglected giving the simplified equation,
2 UL C,n = ~l~-:7-~~lst = ~-~'qst
(34)
This equation is suitable for long admission sectors and corresponds in form to an equation b y Doyle)
A e r o d y n a m i c losses in p a r t i a l a d m i s s i o n t u r b i n e s APPENDIX
431
4
Calculation of the stage efficiency of partial admission turbines using loss coefficients T h e following d a t a , w h i c h i n a c t u a l p r a c t i c e a r e k n o w n , a r e a s s u m e d for a t u r b i n e wheel: U/c o= 0"4, e = 50 p e r c e n t , B / L = 30.88, F u l l a d m i s s i o n s t a g e efficiency, ~,~ --- 68 p e r c e n t (from Fig. 4), M i x i n g loss coefficient, C~ -- 2 p e r c e n t (from Fig. 5), D i s k f r i c t i o n a n d p u m p i n g ( w i n d a g e losses), Lw = 6-87 ft/lbf. T h e s e c a n e i t h e r b e c a l c u l a t e d b y a m e t h o d s u g g e s t e d b y S u t e r a n d T r a u p e l or d e t e r m i n e d e x p e r i m e n t a l l y . I n t h e p r e s e n t case t h e c a l c u l a t e d v a l u e is n e a r l y t h e s a m e as t h a t measured. Mass flow t h r o u g h t h i s s t a g e rh = 24 l b m / m i n , I s e n t r o p i c v e l o c i t y a t t h e e x i t of t h e nozzles, c o = 205 f.p.s. W i n d a g e loss coefficient Cw =
Lw
mc~/2g 6 . 8 7 × 2 × 3 2 . 2 x 60
Cw =
24 x 205 × 205
x 100
and C~ = 2.63 p e r c e n t . S t a g e efficiency of t h e p a r t i a l a d m i s s i o n t u r b i n e is ~
= ~?~ = C ~ - C ~ .
S u d d e n e x p a n s i o n a n d o t h e r losses a r e c o n s i d e r e d as negligible for 50 p e r c e n t degree of a d m i s s i o n . T h u s ~,~ = 6 8 . 0 0 - 2 . 0 0 - 2.62 = 63.38 p e r c e n t .