NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS FOR

NATIONAL ADVISORY COMMITTEE
FOR AERONAUTICS
TECHNICAL NOTE 2421
A RAPID APPROXIMATE METHOD FOR DETERMINING VELOCITY
DISTRIBUTION ON IMPELLER BLADES OF
CENTRIFUGAL COMPRESSORS
B y John D. Stanitz and V a s i l y D. P r i a n
Lewis F l i g h t P r o p u l s i o n L a b o r a t o r y
Cleveland, Ohio
Washington
July 1951
BUSINESS, SCIENCE
& TECHNOLOGY DEP'T
1
NATI0IU.L ADVISORY COMMITTEE FOR AERONAUTICS
c
TECBNICAL NOTE 2 4 2 1
A RAPID APPROXIMATE
METROD FOR DETERMINING VELOCITY DISTRIBUTION
ON IMPELLER BLADES OF CENTRIFUGAL CCMPmSSORS
I\)
P
CD
By John D. S t a n i t z and Vasily D. Prian
P
SUMMARY
A r a p i d approximate method of a n a l y s i s w a s developed f o r both comp r e s s i b l e and incompressible, nonviscous flow through r a d i a l - or mixedf l o w c e n t r i f u g a l compressors with a r b i t r a r y hub and shroud contours and
with a r b i t r a r y blade shape. The method of a n a l y s i s i s used t o d e t e r mine approximately the v e l o c i t i e s everywhere along t h e blade surfaces,
b u t no information concerning t h e v a r i a t i o n i n v e l o c i t y across t h e
passage between blades i s given.
I n e i g h t numerical examples f o r two-dimensional flow, covering a
f a i r l y wide range of flow r a t e , impeller-tip speed, number of blades,
and blade curvature, t h e v e l o c i t y d i s t r i b u t i o n along t h e blade surfaces
w a s obtained by t h e approximate method of a n a l y s i s and compared w i t h t h e
v e l o c i t i e s obtained by r e l a x a t i o n methods. I n a l l cases t h e agreement
between t h e approximate solutions and t h e r e l a x a t i o n solutions was
s a t i s f a c t o r y except a t t h e impeller t i p where t h e v e l o c i t i e s obtained
by t h e approximate method d i d not, i n general, become equal on both
surfaces of t h e blade as required by t h e Joukowski condition.
INTRODUCTION
.
L
I n impellers of c e n t r i f u g a l compressors, p a r t of t h e viscous l o s s e s
and t h e phenomena of surge and choke a r e r e l a t e d t o t h e v e l o c i t y d i s t r i b u t i o n on t h e blade surfaces. Viscous l o s s e s i n impellers a r e
associated with t h e boundary l a y e r along t h e f l o w surfaces. The growth
of t h i s boundary l a y e r depends on the v e l o c i t y v a r i a t i o n along t h e f l o w
surfaces j u s t outside of t h e boundary l a y e r . In p a r t i c u l a r , i f t h e
v e l o c i t y d e c e l e r a t e s r a p i d l y along the blade surfaces, t h e boundary
l a y e r may separate causing large mixing l o s s e s . Also, if t h e v e l o c i t y
a t any p o i n t along t h e blade surface i s s u f f i c i e n t l y g r e a t e r than t h e
l o c a l speed of sound, shock l o s s e s w i l l r e s u l t . The choke phenomenon
occurs when t h e average v e l o c i t y between blades i s sonic. This average
sonic v e l o c i t y i s characterized by l o c a l supersonic v e l o c i t i e s along
p o r t i o n s of t h e suction surface of the blade. One possible cause of
NACA TN 2421
2
surge irl c e n t r i f u g a l impellers i s t h e formation of r e l a t i v e eddies on
t h e pressure surface of t h e blade (reference 1). These eddies a r e
characterized by negative v e l o c i t i e s , opposed t o t h e general flow
d i r e c t i o n , along p o r t i o n s of t h e p:reas-u-e surface. I n order t o analyze
t h e performance of c e n t r i f u g a l impellers it i s t h e r e f o r e necessary t o
determine t h e v e l o c i t y d i s t r i b u t i o n on impeller blades.
.)
-
Several methods of a n a l y s i s t h a t can be used t o determine t h e
v a r i a t i o n i n v e l o c i t y along blades with f i n i t e spacing have been
developed f o r two-dimensional incompressible flow (references 2 t o 5,
f o r example) and compressible flow (references 1 and 6 ) . All t h e s e
methods require considerable labor and t h e r e f o r e are not convenient
t o o l s f o r analyzing t h e performance of an a r b i t r a r y impeller design.
I n t h i s r e p o r t a r a p i d approximate method developed a t the NACA
Lewis laboratory i s presented f o r both compressible and incompressible,
nonviscous, two-dimensional flow between blades with f i n i t e spacing i n
r a d i a l - o r mixed-flow c e n t r i f u g a l compressors with a r b i t r a r y hub and
shroud contours and with a r b i t r a r y blade shape. The method of a n a l y s i s
can be used i n connection with an axial-symmetry s o l u t i o n t o determine
t h e v e l o c i t i e s everywhere along t h e blade surfaces, but no information
concerning t h e v a r i a t i o n i n v e l o c i t y a c r o s s t h e passage between blades
i s given.
Other approximate methods t h a t a r e l e s s r a p i d than t h e proposed
method f o r computing t h e v e l o c i t y d i s t r i b u t i o n on blade surfaces i n
impellers of c e n t r i f u g a l pumps and compressors a r e given i n r e f e r ences 6 t o 9. I n t h e s e c t i o n s SIMPLIFIED ANALYSIS of r e f e r ences 6 and 7 approximate methods a r e developed f o r computing t h e
t h e o r e t i c a l d i s t r i b u t i o n of v e l o c i t y across t h e passage along normals
t o t h e blade surfaces. The methods a r e l i m i t e d t o s t r a i g h t or
logarithmic-spiral blade shapes on r a d i a l or conic surfaces of revolut i o n and do not apply, because of assumptions, i n regions near t h e
impeller t i p and t h e impeller i n l e t . I n reference 8 methods a r e
developed f o r computing t h e d i s t r i b u t i o n of v e l o c i t y across t h e passage between blades i n t h e c i r c u m f e r e n t i a l d i r e c t i o n f o r incompressible
flow with a r b i t r a r y blade shapes and with a r b i t r a r y hub and shroud
contours. TBe methods do not apply, because of assumptions, i n regions
near t h e impeller t i p and t h e impeller i n l e t . I n reference 9 an
approximate method i s developed f o r computing t h e t h e o r e t i c a l v e l o c i t y
d i s t r i b u t i o n everywhere within t h e impeller. I n t h i s method t h e corr e c t i o n s required f o r compressibility and f o r blade unloading a t t h e
t i p a r e somewhat more complicated than t h o s e presented herein.
1
THEORY OF METHOD
4
The method of a n a l y s i s presented i n t h i s s e c t i o n determines t h e
v e l o c i t y d i s t r i b u t i o n along t h e p r o f i l e s of blade elements on s u r f a c e s
of revolution.
3
NACA TN 2421
Preliminary Considerations
1v
P
(D
P
Assumed nature of flow. - I n t h i s section c e r t a i n preliminary
assumptions a r e made concerning t h e three-dimensional flow of an i d e a l
compressible f l u i d through an a r b i t r a r y impeller passage between blades
such as shown i n f i g u r e 1. I n general, t h e f l u i d i s f r e e t o f o l l o w
whatever path t h e pressure and i n e r t i a f o r c e s r e q u i r e of it. If,
however, it i s assumed t h a t t h e number of blades i n t h e impeller
approaches i n f i n i t y , t h e space between blades approaches zero and t h e
path of t h e f l u i d i s r e s t r i c t e d t o t h e curved, mean surface of t h e
blade. (The blades become very t h i n s o t h a t t h e two s u r f a c e s of each
blade approach a mean surface.) Under t h i s assumption of axial
symmetry t h e f l u i d motion i s reduced from a general three-dimensional
motion t o a two-dimensional motion on t h e curved, m e a n blade surface.
The streamlines of t h i s two-dimensional motion can be p r o j e c t e d on t h e
meridional ( a x i a l - r a d i a l ) plane, as shown i n f i g u r e 2. &den ( r e f e r ence 10) has shown that, provided that t h e blades a r e not t o o widely
spaced, axial-symmetry s o l u t i o n s give a good p i c t u r e of t h e mean flow
between blades.
For f i n i t e blade spacing, flow conditions vary between blades i n
t h e c i r c u m f e r e n t i a l d i r e c t i o n about t h e a x i s of t h e impeller. I n order
t o i n v e s t i g a t e t h i s blade-to-blade v a r i a t i o n , it i s assumed t h a t t h e
motion of any f l u i d p a r t i c l e bounded by adjacent streamlines i n t h e
meridional plane ( f i g . 2 ) i s r e s t r i c t e d t o t h e annulus generated by
r o t a t i n g t h e s e adjacent streamlines about t h e a x i s of t h e impeller.
If t h e adjacent streamlines a r e s u f f i c i e n t l y close together, flow
,conditions i n t h e annulus can be considered uniform normal t o a mean
s u r f a c e of r e v o l u t i o n i n t h e m u l u s . Thus t h e f l u i d motion i s reduced
t o two-dimensional flow on t h e mean surface of revolution ( f i g . 3)
generated by r o t a t i n g t h e center l i n e between t h e adjacent streamlines
i n t h e meridional plane ( f i g . 2 ) about t h e a x i s of t h e impeller.
Blade-to-blade s o l u t i o n s of this type may be obtained f o r every
mean surface of revolution generated by t h e center l i n e s between
adjacent streamlines i n t h e meridional plane. Therefore, flow condit i o n s can be determined throughout t h e passage between blades. The
r e s u l t i n g quasi three-dimensional s o l u t i o n i s obtained by t h e combinat i o n of two types of two-dimensional solution, axial-symmetry solut i o n s i n t h e meridional plane and blade-to-blade s o l u t i o n s on s u r f a c e s
of revolution. Such a conibination of s o l u t i o n s p r o h i b i t s t h e p o s s i b i l i t y of a corkscrew path, which t h e f l u i d might follow i n an exact
three-dimensional solution, b u t i t can be expected t o give a b e t t e r
p i c t u r e of t h e flow than does any two-dimensional s o l u t i o n alone.
The method of a n a l y s i s j u s t described i s accomplished i n two
phases, axial-symmetry s o l u t i o n and blade-to-blade s o l u t i o n s . O n l y
t h e second phase, blade-to-blade solutions, will be considered i n t h i s
4
NACA TN 2 4 2 1
.r
r e p o r t . The shape and t h e d i s t r i b u t i o n of meridional streamlines i n t h e
a x i a l - r a d i a l plane a r e assumed t o be known from an axial-symmetry
solution (reference ll, f o r example). Thus, f o r a blade-to-blade
solution i n t h e annulus generated about t h e a x i s of t h e impeller by any
t w o adJijacerh nericliaKL s t r e d i n e s ( f i g . 21, t h e shape of t h e mean
surface of revolution ( f i g . 3) i s known from t b shape of t h e center
l i n e between the adjacent meridional streamlines, and t h e v a r i a t i o n i n
height of an elementary f l u i d p a r t i c l e ( f i g . 2 ) as it moves along t h e
m e a n surface of revolution from t h e impeller i n l e t t o t h e impeller t i p
i s known from t h e v a r i a t i o n i n spacing of t h e adjacent meridional
streamlines.
.
Coordinates. - The c y l i n d r i c a l coordinates R, 8, and Z are
shown i n f i g u r e 3. ( A l l symbols a r e defined i n t h e appendix.) These
coordinates a r e dimensionless, t h e l i n e a r coordinates R and Z having
been divided by t h e impeller-tip radius 'T.
The coordinate system i s
oriented with t h e Z - a x i s along t h e axis of t h e impeller. The coordin a t e s a r e f i x e d r e l a t i v e t o t h e impeller, which r o t a t e s with t h e
angular v e l o c i t y LU i n t h e p o s i t i v e d i r e c t i o n (right-hand r u l e ) about
t h e Z-axis, as shown i n figure 3.
An i n f i n i t e s i m a l distance dS i n t h e d i r e c t i o n ' o f flow ( t h a t i s ,
coinciding with t h e v e l o c i t y vector) has components dR, Rde, and dZ
( f i g . 3 ) . The p r o j e c t i o n of dS on t h e meridional plane i s given by
dM i n f i g u r e 3. The i n f i n i t e s i m a l distances dS and dM help t o
define two angles a and I3 where, from f i g u r e 3,
and
The angle a ( f i g . 3) i s determined by tangents t o t h e center l i n e ,
between adjacent meridional streamlines, t h a t generates t h e surface of
revolution. The angle p ( f i g . 3) i s t h e flow d i r e c t i o n on t h e surf a c e of revolution measured from a m e r i d i o n a l l i n e . From equation (2a)
because ds and 1-34 are always p o s i t i v e and f i n i t e .
t i o n s ( l a ) and (1k),
From equa-
4
NACA TN 2421
5
because f o r impellers of c e n t r i f u g a l compressors
considered p o s i t i v e (or zero).
dR and dZ w i l l be
F l u i d s t r i p . - A f l u i d s t r i p of i n f i n i t e s i m a l width dM l i e s on
t h e surface of r e v o l u t i o n and extends across t h e passage between blades
along a l i n e of constant R. A developed view of t h e f l u i d s t r i p i s
shown i n f i g u r e 4 . The f l u i d s t r i p has dimensions dM and RAG where
t h e angular width of passage between blades A6 i s defined by
i n which t h e s u b s c r i p t s d and t r e f e r t o t h e d r i v i n g and t r a i l i n g
f a c e s of t h e blades, r e s p e c t i v e l y ( l e f t and right walls of t h e channel
between blades in f i g . 4 ) . The height r a t i o H of t h e f l u i d s t r i p i s
defined as t h e r a t i o of t h e incremental height Ah { f i g . 2) of t h e
f l u i d s t r i p a t r a d i u s R t o t h e incremental height ( A ~ ) Tof t h e
f l u i d s t r i p a t R = 1.0. This height r a t i o i s completely determined
along a mean surface of revolution by t h e spacing between t h e adjacent
streamlines i n t h e meridional plane ( f i g . 2 ) .
Velocity components. - The r e l a t i v e v e l o c i t y Q on a surface of
r e v o l u t i o n has components % and Qe i n t h e dM and de d i r e c t i o n s ,
r e s p e c t i v e l y , ( f i g . 3). These v e l o c i t i e s a r e dimensionless, having
been divided by t h e absolute stagnation speed of sound co upstream of
impeller, where
eo2 = rgRT0
(4)
i n which R i s t h e gas constant, y i s t h e r a t i o of s p e c i f i c heats,
T i s t h e s t a t i c (stream) temperature and where t h e s u b s c r i p t o r e f e r s
t o stagnation conditions upstream of t h e impeller. The t i p speed of
t h e impeller i s likewise dimensionless and equal t o t h e i m p e l l e r - t i p
Mach number MT, w h i c h i s defined by
Thus, t h e t a n g e n t i a l v e l o c i t y of the i r p e l l e r a t any r a d i u s R i s
equal t o RMT and t h e absolute t a n g e n t i a l v e l o c i t y of t h e f l u i d i s
equal t o (q
+ Q ) . From figure 3
%
and
= Q
COS
N K A TN 2421
6
-
Fromthe general energy equation and
Thermodynamic r e l a t i o n s .
f r o m the i s e n t r o p i c r e l a t i o n between temperature and density, t h e
density r a t i o p/po i s r e l a t e d t o t h e r e l a t i v e v e l o c i t y Q by
where the subscript U r e f e r s t o conditions upstream of t h e impeller
and where h i s t h e whirl r a t i o (absolute moment of momentum divided
by rTco) given by
h
=
R ( W +
ri
a
d
N
(9)
Development of Method
-
Before outlining t h e method of a n a l y s i s it i s conAssumptions.
venient t o discuss t h e major assumptions. Consider t h e f l u i d s t r i p
i n f i g u r e 4. Along t h e i n f i n i t e s i m a l distances bounding t h e f l u i d
s t r i p a t t h e driving and t r a i l i n g blade surfaces, t h e v e l o c i t i e s may be
considered constant and equal t o Q and Qt, respectively, and t h e
flow d i r e c t i o n s may be considered constant and equal t o pd and B t .
Along the l i n e s o f constant R bounding t h e f l u i d s t r i p i n f i g u r e 4,
t h e v e l o c i t y v a r i e s i n some unknown manner f r o m Q t o Q t and t h e
flow d i r e c t i o n v a r i e s from pd t o p t . I n t h i s r e p o r t it i s assumed
that t h e average values of Q and B along l i n e s of constant R may
be u s e d t o s a t i s f y t h e conditions of c o n t i n u i t y and absolute i r r o t a t i o n a l motion. The average value of Q i s assumed t o be given by
Qa + Qt
Qav =
and, for
Also, f o r
R L
R,,
R, L R
2
t h e average value of
p
i s assumed t o be given by
5 1.0,
sin
pav
=
A
+
BR
+
CR2
.
.
NACA TN 2 4 2 1
7
where A , B, and C a r e c o e f f i c i e n t s t o be determined and where Rx
i s t h e l a r g e s t r a d i u s a t which t h e f l u i d i s considered t o be p e r f e c t l y
guided by t h e blades; that i s , t h e radius a t which t h e s i m p l i f i e d
analyses given i n references 6 and 7 break down. From f i g u r e 10 of
ref.erence 7 t h e value of R
, f o r A6 equal t o s r / l O and s i n a
equal t o 1.0 i s about 0.8. For other values of A6 and s i n a, t h e
value of R, can be estimated from
In Rx
In
---=-=
- 0.71
o*8
where a i s t h e average value over the i n t e r v a l RX k R 5 1.0. Equat i o n (13) i s based upon an extension of t h e work i n reference l where
f o r impellers with s t r a i g h t blades it i s shown t h a t t h e flow conditions
i n one impeller can be c o r r e l a t e d w i t h t h e f l o w conditions i n another
InR
impeller a t t h e same value of
(ne) s i n a' I n reference 1, t h e
i m p e l l e r - t i p Mach number and t h e compressor flow r a t e were found t o have
a n e g l i g i b l e e f f e c t on t h e value o f R,.
Outline of theory. - F l u i d s t r i p s such a s shown i n f i g u r e 4 exist
a t a l l r a d i i along t h e surface of revolution. From t h e assumptions of
t h i s a n a l y s i s t h e r e a r e t h r e e unknowns (Qd, q, and %,)
f o r each
f l u i d s t r i p . These unknowns can be determined by t h e simultaneous
s o l u t i o n of equation (10) and t h e equations of c o n t i n u i t y and zero
a b s o l u t e c i r c u l a t i o n f o r flow across t h e f l u i d s t r i p . Equations f o r
t h e d i s t r i b u t i o n of v e l o c i t y along the blade p r o f i l e on a s u r f a c e of
revolution w i l l be developed i n this r e p o r t .
-
Zero absolute c i r c u l a t i o n .
I n t h e absence of entropy gradients,
which r e s u l t from shock, viscous d i s s i p a t i o n , heat t r a n s f e r , and so
f o r t h , t h e absolute c i r c u l a t i o n around t h e f l u i d s t r i p i n f i g u r e 4 i s
zero so t h a t
r
1
where (Rl@ + @ ) a v i s t h e average absolute t a n g e n t i a l v e l o c i t y and
where from trigonometric considerations of t h e v e l o c i t y t r i a n g l e s
( f i g . 5) Q + % s i n p i s t h e absolute v e l o c i t y component along t h e
blade surface. From equations ( 7 ) and (14) and from t h e assumptions
t h a t Q and p equal &av and Pav,
r e s p e c t i v e l y , i n t h e passage
between blades,
NACA TN 2421
8
F i n a l l y , from equations (10) and (15)
and from equation (11)
If Qa,v and Bav a r e known,
equations (16) and ( 1 7 ) .
Q and Qt can be determined from
Average v e l o c i t y &av. - From c o n t i n u i t y considerations of t h e flow
across t h e f l u i d s t r i p i n f i g u r e 4,
from which
M
where t h e flow c o e f f i c i e n t Cp
i s defined by
i n which Aw i s t h e incremental flow r a t e through t h e passage between
two blades on t h e surface of revolution and ( h ) i~
s t h e incremental
flow area (between two blades) normal t o t h e d i r e c t i o n of €& a t t h e
impeller t i p
( h )= ~
~T(A~)T(A~)T
09a)
.
NACA TN 2421
9
The flow r a t e p e r u n i t flaw a r e a a t t h e impeller t i p A w / ( b ) ~ i s
known s o t h a t t h e flow c o e f f i c i e n t Cp can be determined by equat i o n (19). The density r a t i o pav/p,
i s given by equation (8) with
Q
equal t o
Qav
1
Therefore, t h e v e l o c i t y %v
can be determined by t h e simultaneous
s o l u t i o n of equations (18) and (20) provided t h a t t h e average flow
d i r e c t i o n BaV i s known.
Average flow d i r e c t i o n . - I n the passage between blades t h e
average flow d i r e c t i o n i s assumed equal t o t h e average blade d i r e c t i o n
(equation (11)) except near t h e blade t i p ( R x I R 5 1.0) where
s i n pav i s given by equation ( 1 2 ) . The exact v a r i a t i o n i n Bav
with R i n t h e i n t e r v a l Rx I. R 5 1.0 could be represented by an
i n f i n i t e s e r i e s . However, because the v a r i a t i o n i n sin Bav with R
w i l l not, i n general, contain an i n f l e c t i o n p o i n t , a p a r a b o l i c v a r i a t i o n i n s i n Bav with R has been assumed and only t h e f i r s t t h r e e
terms of t h e i n f i n i t e s e r i e s retained. The constants A, B, and C
i n equation (12) are determined from:
(1)
( s i n pav)x = A
(2)
("
(3)
(sin P a v ) ~ = A
+ BRx +
sz
pav) X = B
+
ZCR,
+
B
+
and
so t h a t
A = (sin
Pav)~- B
-C
C
CR, 2
NACA 'I" 2421
10
.
a t Rx, and where ( s i n pav)T i s determined f r o m t h e s l i p f a c t o r
which i s a e f i n e d b y (reference 6)
p,
rl
0,
rl
so t h a t
OJ
The s l i p f a c t o r p i s assumed t o be known, or can be estimated, a s a
r e s u l t of t h e work presented i n references 1 and 7, f o r example.
(Further discussion on Rx and t h e s l i p f a c t o r p i s given l a t e r i n
t h i s report.) The v e l o c i t y (Gv)r i n equation (22) i s obtained from
equations (18) and (20) with R and H equal t o 1.0 and with
(cos Pav)T replaced by
41 -
(sin
Bav)~2
where
(sin
Bav)~
i s given
by equation ( 2 2 ) :
Equation (23) i s solved f o r (&av)rby t r i a l and error. Therefore,
pav i s determined a s a function of R ( o r M) by equations (ll), ( 1 2 ) ,
and ( 2 1 ) . The v e l o c i t i e s Qd, Qt, and GV a r e determined as funct i o n s of M ( o r R) from equations (16), (17), and (18). (The l a s t
term of equation (16) i s determined from t h e slope of
(+ + &av s i n Bav) R (AQ) p l o t t e d a g a i n s t M.)
APPLICATION OF METHOD
The following o u t l i n e of t h e numerical procedure i s given f o r t h e
general case of a mixed-flow impeller with a r b i t r a r y hub and shroud
contours i n t h e meridional plane ( f i g . 2 ) and a r b i t r a r y blade shape
(curvature and thickness d i s t r i b u t i o n ) on s u r f a c e s of revolution. It
i s assumed that t h e surfaces of r e v o l u t i o n a r e known, that i s have been
generated by t h e c e n t e r l i n e s between adjacent meridional streamlines
11
NACA TN 2421
obtained from an axial symmetry solution (reference 11, f o r example).
The following o u t l i n e of t h e wmerical procedure r e f e r 9 t o any one of
t h e s e surfaces of revolution.
Specified conditions.
-
The following q u a n t i t i e s are specified:
(1)Flow c o e f f i c i e n t CP (defined by equation (19) i n which
c
Aw, and (h),
are known q u a n t i t i e s )
0’
%
( 2 ) Impeller-tip Mach number
( 3 ) Whirl r a t i o h,
(defined by equation ( 5 ) )
upstream of impeller (defined by equation ( 9 ) )
(4) Ratio of s p e c i f i c heats
y
(5) From t h e shape of t h e center l i n e between adjacent meridional
streamlines t h a t generate t h e surface of revolution,
R = R(M)
and
a = U(M)
where t h e distance M along a meridional l i n e on t h e surf a c e o f revolution i s a r b i t r a r i l y equal t o zero a t t h e impell e r t i p and decreases toward t h e impeller i n l e t
(6) From t h e spacing of t h e adjacent m e r i d i o n a l s t r e a n l i n e s ,
H
Variation i n s i n Pav.
the impeller t i p (R, <, R
5
=
H(M)
The v a r i a t i o n i n s i n pa,
with
i
s
determined
as
follows:
1.0)
R
near
(1)Compute t h e value of R,
by equation (13). If s i n a v a r i e s
i n t h e region R , S R 51.0, as it generally does, the
average value of a i n t h i s region i s used i n equation (13),
and because t h e average value of a v a r i e s with t h e value
of R,,
equation (13) must be solved by t r i a l and e r r o r .
However, because t h e value of a does not generally vary
g r e a t l y i n t h e region RxL R 51.0, a s a t i s f a c t o r y value
of Rx could b e obtained from equation (13) using t h e
average value of a obtained from an i n i t i a l l y assumed
value of R,.
Also, equation (13) w a s developed from
information (references 1 and 7 ) r e l a t i n g t o blades t h a t
a r e not designed t o unload a t t h e t i p . If t h e blades
being considered were designed t o unload a t t h e t i p , t h e
d i r e c t i o n of t h e mean f l o w p a t h near t h e impeller t i p
would deviate l e s s f r o m t h e mean blade d i r e c t i o n and t h e
12
NACA TN 2421
value of RX would be somewhat g r e a t e r than t h a t given by
equation (13). The value of Rx i s not e s p e c i a l l y c r i t i cal and i n t h e s e cases, with s u f f i c i e n t experience, it can
probably be estimated accurately enough from t h e r a d i u s a t
which an assumed path of %he mea= s t r e d i n e (sketched by
experience) deviates appreciably from t h e mean d i r e c t i o n
of t h e blade p r o f i l e on t h e surface of revolution.
( 2 ) Estimate t h e value of p, or obtain values from references 1
and 7. The values of p given i n references 1 and 7 were
obtained f o r blades t h a t are n o t designed t o unload a t t h e
t i p . If t h e blades being considered were designed t o unload
a t t h e t i p , t h e d i r e c t i o n of t h e mean flow p a t h a t t h e impell e r t i p would deviate less from t h e mean blade d i r e c t i o n
a t t h e t i p and t h e value of p would be somewhat greater
than t h a t indicated i n references 1 and 7; t h a t is, ( p a v ) ~
1
would be more n e a r l y equal t o 5 (pd -t- &)T.
The value
of p, l i k e t h e value of Rx, i s not e s p e a i a l l y c r i t i c a l
and i n t h e s e cases it can probably be estimated a c c u r a t e l y
enough f r o m the assumed shape of a m e a n streamline (sketched
from experience) between blade-element p r o f i l e s on t h e s u r f a c e of revolution,
f r o m equation ( 2 3 ) by trial and e r r o r .
(3) Compute
(€&)T
(4) Compute
( s i n fiav)T from equation (22).
(5) Compute c o e f f i c i e n t s
( 6 ) Compute
sin
Pa,
A,
B,
and
C
over t h e i n t e r v a l
from equations (21).
%S
R
5 1.0
by equa-
t i o n (12).
The v a r i a t i o n i n pav f o r R l e s s than % i s given by equat i o n (11). This equation i s assumed t o be v a l i d downstream t o the
impeller i n l e t . If t h e angle of a t t a c k a t t h e impeller i n l e t i s zero,
t h e assumption i s probably good. If the angle of a t t a c k i s small, t h e
e r r o r involved i s probably small and could be p a r t l y corrected by
estimating t h e path of t h e mean streamline between blades i n t h i s
region. For l a r g e angles of a t t a c k , t h e stagnation point on t h e blade
surface may e x i s t w e l l i n s i d e t h e impeller passage and t h e i d e a l flow
i s reversed along t h e blade surface downstream of t h i s point. Under
these conditions, t h e method of a n a l y s i s does not apply near t h e
impeller i n l e t but because of t h e high blade s o l i d i t y it does apply
elsewhere i n t h e impeller.
.
.
13
NACA TN 2421
-
Average v e l o c i t y &av.
The average v e l o c i t y €&
a t each value
of M ( o r R) i s determined by equation (18) i n which pav/po
i s given
by equation ( 2 0 ) . Because pav/po
a l s o contains Qv,
t h e simultsneous s o l u t i o n of equations (13) and (20) must be by trial and e r r o r .
A suggested procedure i s f i r s t t o compute pav/po
assuming t h a t Qav
i n equation ( 2 0 ) i s zero. T h i s value of pav/po
i s then used t o compute &av by equation (18). The process i s repeated each time using
t h e new value o f &av t o compuCe pav/po
u n t i l t h e value of &av
converges.
V e l o c i t i e s on blade surface6, Qd and %. - The v e l o c i t i e s on t h e
blade surfaces a t each value of M (or R ) a r e determined by equat i o n s (16) and ( 1 7 ) . The l a s t term i n equation (16) i s obtained from
t h e slope of (+
+ &av s i n 3j),
R ( M ) p l o t t e d a g a i n s t M.
F i n a l l y , t h e s t a t i c (stream) pressure p corresponding t o t h e
r e l a t i v e v e l o c i t y Q a t any r a d i u s R i s given by
The approximate a n a l y s i s method developed i n t h i s r e p o r t i s
a p p l i e d t o e i g h t examples f o r which r e l a x a t i o n s o l u t i o n s of t h e exact
d i f f e r e n t i a l equation for two-dimensional compressible flow i n impell e r s of c e n t r i f u g a l compressors a r e given i n references 1 and 7 .
Although these examples a r e f o r r a d i a l - o r conic-flow surfaces and not
f o r a r b i t r a r y surfaces of revolution, they cover a f a i r l y wide range
of design and operating v a r i a b l e s s o that a comparison of t h e v e l o c i t i e s (on t h e blade s u r f a c e s ) obtained by t h e r e l a x a t i o n s o l u t i o n s and
by t h e approximate a n a l y s i s method should serve as a check on t h e
v a l i d i t y of t h e approximate method.
Types of impeller. - The e i g h t numerical examples a r e f o r twodimensional radial-flow mailers f o r which a i s equal t o 90° and t h e
s u r f a c e s of r e v o l u t i o n are- ,lat planes normal t o t h e a x e s of the
impellers. The impellers ( f i g . 6) contain a f i n i t e nuniber of t h i n
s t r a i g h t (pa = B t = 0 ) o r logarithmic-spiral (ad = p t = constant)
blades, and t h e flow a r e a normal t o t h e d i r e c t i o n of % i s constant
s o that HR equals 1.0. Only t h e c r i t i c a l flow region toward t h e t i p
of t h e impeller w a s i n v e s t i g a t e d ( 0 . 7 0 IR 5 1.0). The d i f f u s e r vanes
( i f any) and t h e inducer vanes were assumed t o be far enough removed
n o t t o a f f e c t t h e flow i n t h i s region.
I
NACA TN 2421
14
-
Design and operating variables.
The following design and operat i n g variables were s p e c i f i e d f o r t h e eight examples:
Example
Cp
-
EG-r
0.5
1.5
1.5
1.5
2.0
1.5
1.5
1.5
1.5
.7
.9
.5
.5
.5
.5
.5
-
Fluid
A0
0
0
0
0
0
-0.5
-1.0
0
C ompre ss i b l e
Compressible
C ompress i b l e
C ompres s i b l e
Compressible
Compressible
Compressible
Incompressible
0.934
.937
.938
.935
.899
,834
.768
.892
The wZlirl r a t i o upstream of t h e impeller A,
w a s zero and f o r t h e comp r e s s i b l e f l u i d t h e r a t i o of s p e c i f i c h e a t s y was 1.4. The value
of t h e s l i p f a c t o r p given i n t h e t a b l e was obtained from t h e relaxat i o n solutions and was a l s o used t o compute ( s i n p a v ) ~ in equat i o n (22). For t h e incompressible example, t h e speed of sound co cont a i n e d i n t h e d e f i n i t i o n s of &, C P , and MT i s a f i c t i t i o u s quantity
(constant) considered equal t o t h e upstream stagnation speed of sound
of t h e compressible-flow examples with which t h e incompressible-flow
example i s compared.
-
Results.
The r e s u l t s of t h e comparison between the r e l a x a t i o n
solutions and t h e approximate method of a n a l y s i s a r e shown i n f i g u r e 7
f o r t h e eight examples. The v e l o c i t i e s Qd and Q,t a r e p l o t t e d
a g a i n s t R f o r t h e r e l a x a t i o n s o l u t i o n s and f o r t h e approximate solut i o n s . The average v e l o c i t y Gv used i n t h e approximate method of
a n a l y s i s t o obtain Q and Qt i s a l s o p l o t t e d .
The e f f e c t of increasing t h e flow r a t e (flow c o e f f i c i e n t ) on t h e
agreement between t h e r e l a x a t i o n s o l u t i o n and t h e approximate s o l u t i o n
i s shown i n f i g u r e s 7 ( a ) t o 7 ( c )
The agreement appears equally good
f o r a l l flow r a t e s . In view of the r e l a t i v e s i m p l i c i t y of t h e approximate method of a n a l y s i s , t h e agreement i s considered e n t i r e l y s a t i s factory'everywhere except i n t h e immediate v i c i n i t y of t h e impeller t i p
where t h e r e s u l t s of t h e approximate method of a n a l y s i s do not follow
t h e r a p i d unloading of t h e blades. This r a p i d unloading i s - c h a r a c t e r i s t i c of blades t h a t a r e not designed t o unload a t t h e t i p . I f t h e
blades were designed t o unload, t h e agreement between r e l a x a t i o n solut i o n s and approximate s o l u t i o n s should be b e t t e r . I n any event t h e
disagreement i s serious only over t h e l a s t 2 percent of i m p e l l e r - t i p
radius. The f a i l u r e t o unload a t t h e impeller t i p w i l l be observed i n
most of the remaining s o l u t i o n s .
.
rl
a
ri
N
15
NACA TN 2421
The following f i g u r q comparisons indicate: f i g u r e s 7 ( a ) and 7(d),
t h e e f f e c t of increasing FmpeUer-tip Mach number; f i g u r e s 7(a)
and 7 ( e ) , t h e e f f e c t o f increasing angular width of passage between
blades (A6); f i g u r e s 7 ( e ) , 7 ( f ) , and 7 ( g ) t h e e f f e c t of l a r g e r
negative blade angles B; and f i g u r e s 7 ( e ) and 7 ( h ) , t h e e f f e c t of
compressibility. In f i g u r e 7 ( h ) , t h e p e c u l i a r humps i n t h e v e l o c i t y
d i s t r i b u t i o n obtained by t h e approximate method of a n a l y s i s i n d i c a t e s
that f o r incompressible 'flow t h e blades start t o unload a t a lower
value f o r R, than t h a t given by equation (13).
,
N
P
CD
P
I n view of t h e r e l a t i v e s i m p l i c i t y of t h i s approximate, b u t rapid,
method of a n a l y s i s , t h e agreement between t h e r e l a x a t i o n s o l u t i o n s and
t h e approximate s o l u t i o n s i s considered good i n a l l cases investigated;
t h a t i s , over f a i r l y wide ranges o f flow r a t e , i m p e l l e r - t i p speed,
blade curvature, and number of blades.
SUMMARY OF RESULTS
A r a p i d approximate method of a n a l y s i s w a s developed f o r d e t e r mining t h e v e l o c i t y d i s t r i b u t i o n on impeller b l a d e s of c e n t r i f u g a l
compressors. I n e i g h t numerical examples t h e v e l o c i t i e s obtained by
t h e approximate method of a n a l y s i s were compared with t h e more n e a r l y
c o r r e c t values obtained by r e l a x a t i o n methods. I n a l l cases, t h a t i s ,
over a f a i r l y wide range of flow r a t e , i m p e l l e r - t i p speed, blade
curvature, and number of blades, t h e agreement between v e l o c i t i e s
obtained by t h e approximate method of a n a l y s i s and by r e l a x a t i o n
methods was considered good.
Lewis F l i g h t Propulsion Laboratory,
National Advisory C o d t t e e f o r Aeronautics,
Cleveland, Ohio, April 27, 1951.
NACA TN 2421
16
APPENDIX
1
- SYMBOLS
The following symbols a r e used i n t h i s r e p o r t :
A,B,C
c o e f f i c i e n t s defined by equation (21)
CO
stagnation speed of sound upstream of impeller, equation (4)
Q
a c c e l e r a t i o n due t o g r a v i t y
H
height r a t i o of f l u i d s t r i p normal t o surface of revolution,
W W T
M
distance along meridional l i n e on surface of revolution
(dimensionless, expressed as r a t i o of impeller t i p r a d i u s
( f i g . 3)
q)
M'l
impeller t i p Mach number, equation (5)
P
s t a t i c (stream) pressure
Q
r e l a t i v e v e l o c i t y on surface of revolution (dimensionless,
expressed i n units of t h e stagnation speed of sound upstream
of impeller c o ) ( f i g . 3)
R
c y l i n d r i c a l coordinate (dimensionless, expressed as r a t i o of
impeller-tip r a d i u s r T ) ( f i g . 3)
rT
impeller-tip r a d i u s
S
distance along streamline on surface of revolution (dimensionl e s s , expressed a s r a t i o of i m p e l l e r - t i p r a d i u s r T ) ( f i g . 3)
T
s t a t i c (stream) temperature
Z
c y l i n d r i c a l coordinate (dimemsionless, expressed a s r a t i o of
impeller-tip r a d i u s rT) ( f i g . 3)
U
slope o f surface of revolution i n d i r e c t i o n of
t i o n s ( l a ) and ( l b ) ( f i g . 3)
B
f l o w d i r e c t i o n on surface of revolution, equations (2a) and
( f i g . 3)
r
r a t i o of s p e c i f i c h e a t s
k,
equa-
( A E L ) ~incremental flow a r e a between two blades and normal t o the
d i r e c t i o n of % a t impeller t i p , equation (19a)
(a)
.
3
.
.
.
NACA TN 2421
17
Ah
incremental height of f l u i d s t r i p on surface of r e v o l u t i o n
Aw
incremental flow r a t e between two blades on surface of revolution
Ae
angular width of passage between blades, radians unless otherwise specified, equation (3)
e
c y l i n d r i c a l coordinate, radians unless otherwise s p e c i f i e d ,
( p o s i t i v e about Z-axis according t o right-hand rule) ( f i g . 3)
A
whirl r a t i o , equation ( 9 )
CL
s l i p f a c t o r , equation (22)
P
s t a t i c (stream) weight density of f l u i d
cp
flow c o e f f i c i e n t , equation (19)
CD
a
n
w
a
r v e l o c i t y of impeller ( i n d i r e c t i o n of p o s i t i v e
e)
Sub s c r i p t s :
ab s
component of absolute v e l o c i t y along blade surface
av
average
d
driving f a c e of blade (blade surface i n d i r e c t i o n of r o t a t i o n )
( f i g . 4)
M
component along meridional l i n e on surface of r e v o l u t i o n
0
absolute stagnation condition upstream of impeller
R,Q,Z
components i n p o s i t i v e
T
impeller t i p
t
t r a i l i n g f a c e of blade (blade surface opposed t o d i r e c t i o n of
rotation) (fig. 4)
U
upstream of impeller
X
p o s i t i o n along meridional l i n e on surface of r e v o l u t i o n a t which
t h e assumption of p e r f e c t guiding of f l u i d b y blades i s considered t o break down
R-,
e-,
Z-directions, r e s p e c t i v e l y
NACA TN 2421
18
1. Stanitz, John D., and Ellis, Gaylord 0.: Two-Dimensional Compressible Flow in Centrifugal Compressors with Straight Blades. NACA
Re?. 954, 1950.
2. Concordia, C., and Garter, G. K.: D-C Network-Analyzer Determination of Fluid-Flow Pattern in a Centrifugal Impeller. Jour.
Appl. Mech., vol. 14, no. 2, June 1947, pp. All3-Al18.
rl
3. SErensen, E.: Potential Flow Through Centrifugal Pumps and Turbines.
NACA TM 973, 1941.
4. Spannhake, W.:
Anwendung der konformen Abbildung auf die Berechnung
van Strtlmungen in Kreiselradern. Z.f.a.M.M., Bd. 5, Heft 6,
Dez. 1925, S. 481-484.
5. Busemann, A.: Das F8rderh8henverh8ltnis radialer Kreiselpmpen mit
logarithmisch-spiraligen Schaufeln. Z.f.a.M.M., Bd. 8, Heft 5,
October 1928, P. 372-384.
6. Stanitz, John D.: Two-Dimensional Compressible Flow in Turbomachines with Conic Flow Surfaces. NACA TIV 1744, 1948.
7. Ellis, Gaylord
O., and Stanitz, John D.:
Two-Dimensional Compressible Flow in Centrifugal Compressors with Logarithmic-Sptral
Blades. NACA TN 2255, 1951.
8. Spannhake, Wilhelm:
Centrifugal Pumps, Turbines and Propellers.
The Technology Press (Cambridge), 1934.
9. Sheets, H. E.:
The Flow Through Centrifugal Compressors and F'umps.
ASME Trans., v o l . 72, no. 7, Oct. 1950, pp. 1009-1015.
10. Ruden, P.: Investigation of Single Stage Axial Fans. NACA TM
1062, 1944.
11. Hamrick, Joseph T . , Ginsburg, Ambrose, and Osborn, Walter M.:
Method of Analysis for Compressible Flow through Mixed-Flow
Centrifugal Impellers of Arbitrary Design. NACA TN 2165, 1950.
cn
cu
rl
19
NACA TN 2421
.
Direction of r o t a t i o n
N
Ul
-4
N
Figure 1.
-
Passage between blades i n impeller of t y p i c a l centrifugal compressor.
NACA TN 2421
20
-
.
R T = 1.0
Mer i d i o n a l
/ / /
I m D e 1l e r
--------
I
I
~
C e n t e r l i n e between
streamlines t h a t
bound f l u i d p a r t i c l e
1
A x l a of i m peller
F i g u r e 2.
-
S t r e a m l i n e s i n m e r i d i o n a l p l a n e f o r axial-symmetry s o l u t i o n
of f l o w t h r o u g h i m p e l l e r o f f i g u r e 1.
NACA TN 2421
21
rl
a
N
Figure 3.
.
-
Surface of revolution with coordinates and velocity components.
-
22
NACA TN 2421
0)
Driving face
Figure 4.
-
Developed view of f l u i d s t r i p between blades on surface of revolution a t radius R .
Direction of
blade surface
0J2&
4’
/
I.
\
Q
0’ - x ’ /
/
/
\
\
/
/
- Velocity t r i a n g l e f o r computing component of
absolute velocity along blade surface.
F i g u r e 5.
.
NACA TN 2421
23
rl
II
I
ffi
”
d
////////
a
.
I
+
I
p:
Z
rl
NACA TN 2421
24
1.c
.9
N
t;
P
.a
-Relaxation
solution
Approximate method
.7
.6
Q
a 4
.‘1
.C
.I
-.I
I Rx
-.‘ .65
I
I
.?5
.70
I
.ao
R
.ti5 I
I
.90
I
.95
1.0
(a) Example (a): flow coefficient cp, 0 . 5 ; impeller-tip Mach number $, 1.5; constant flow
area (HR
1.0); angular width of passage AO, 12O; blade angle B, 0 ;
compressible flow ( y = 1.4).
I
Figure 7. - Varlation in velocity along blade surfaces as obtained by relaxation methods (references 1 and 7) and by approximate method.
4
NACA TN
2421
25
.
I
1
-Relaxation
aolution
Approximate method
c
-
.
-.,
1
1
I
I
.70
.?5
.80
(b) Example (b):
Figure 7.
-
I Rx
I I
R
.85
I
I
.90
.95
1
flow coefficient cp, 0.7; other parameters same as example (a).
Continued. Variation in velocity along blade surfaces as obtained by relaxation
methods (references 1 and 7) and by approximate method,
NACA TN 2421
26
1.
Q
.
0-Relaxation
solution
Approximate method
-
.75
.70
2
.85
.80
.90
.95
1.0
R
(c) Example (c):
Figure 7.
- Continued.
flow coefficient T, 0.9; other parameters same as example (a).
Variation in veloolty along blade surfaces as obtained by relaxation
methods (references 1 and 7) and by approximate method.
NACA TN 2421
27
I
-Relaxation
solutLon
Approximate method
' Q \
Q
(d) Example (d):
Figure 7.
-
impeller-tip Mach TJmber MT, 2.0; other parameters same as example (a).
Continued. Variation in velocity along blade surfaces as obtained by relaxation
methods (references 1 ani 7) and by approximate method.
NACA TN 2421
28
1
O-------Fielaxation
solution
Approximate method
Q .
-.
-
I
I
.70
.75
IRx
I
.00
R
Figure 7 .
- Continued.
V a r i a t i o n i n v e l o c i t y along
methods ( r e f e r e n c e s 1 and 7 ) and
.e5
I
.so
1
.95
1.0
b i a d e s u r f a c e s as o b t a i n e d by r e l a x a t i o n
by approximate method.
NACA TN 2421
29
1.
-Relaxation
solution
Approximate method
Q .
-
1
1
I
.10
.15
I Rx
I
.85
.80
I
.90
I
.95
R
(f) Example (f):
Figure 7.
-
blade angle 8, tan-’(-0.5);
other parameters same as example (e).
Continued. Variation in velocity along blade surfaces as obtained by relaxation methods (references 1 and 7) and by approximate method.
3
NASA TN 2421
30
-0
.
----
Relaxation solution
AprL uxlnlaie solution
0
Figure 7. - Continued. Varlatlon in velocity along blade surfaces as obtained by relaxation methods (references 1 and 7) and by approximate method.
.
NACA TN 2 4 2 1
31
1.0
.9
.e
.7
-Relaxation
solution
Approximate solution
.6
Qav
.5
0
-4
.3
.f
.1
C
.-1
.-‘
I
I
I
I
I
(h) Example (h): incompressible flow; other parameters same as example (e). Note that
for incompressible flow stagnation speed of sound co contained in definitions of Q, +,
and ‘P is a fictitious quantity which, if considered equal to co of example (e), enables
comparison of compressible (example ( e ) ) and incompressible (example(h) ) solutions for
same impeller-tip speed, weight-flow rate, and s o forth.
Figure 7.
Concluded. Variation in velocity along blade surfaces as obtained by relaxation
methods (references 1 and 7) and by approximate method.
-
NACA-Langley
- 7-19-51 - M