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A erodyna m ic and ge om etric op tim iza t ion for
the d es ign of ce ntr fug al com pressors
A . P e r d i c h i z z i a n d M . S a v i n i
Ma x im iz i n g e f f i c i e n cy is t h e ma in g o a l i n ce n t r i f u g a l co mp r e sso r d e s ig n . Th u s a
c o m p u t e r c o d e h a s b e e n d e v e l o p e d t o o p t i m i z e g e o m e t r i c a n d f l u i d d y n a m i c
va r i a b le s w i th r e sp e c t t o se ve r a l d e s ig n co n s t r a i n t s . Co mp u ta t i o n s a r e p e r fo r me d
w i t h a n a d i a b a ti c o n e - d i m e n s i o n a l a p p r o a c h u s i n g s t a t e - o f - t h e - a r t l os s a n d s li p
co r r e l a t io n s . Th e o p t im i za t i o n ta ke s in to a c co u n t me c h a n i ca l s t re ss l im i t s . R e su l ts
w i th d i f f e r e n t l o ss a n d s l i p co r r e l a t i o n s a r e co mp a r e d w i th th e a va i l a b le
e xp e r ime n ta l d a ta . Ch a n g e s i n o p t imu m e f f i c i e n cy a n d sp e c i f i c sp e e d d u e to
va r i a t i o n s o f ma ss f l o w r a te a n d p r e ssu r e r a t i o a r e a l so p r e se n te d a n d d i scu sse d
to g e th e r w i th th e t r e n d s o f t h e o p t imu m g e o me t r i c f e a tu r e s .
K e y w o r d s : c o m p r e s s or s , f l u i d d y n a m i c s , d e s i g n
Centrifugal compressors are widely used in small aero-
nautic and industrial gas turbines and in turbochargers
for internal combustion engines. The main demand is to
achieve the highest possible efficiency, especially for
highly loaded machines, ie at high compression ratios.
Numerous theoretical and experimental investigations
are aimed at resolving the fluid dynamic problems and
improving the design criteria.
The great number of geometric and fluid dynamic
variables and their interactions make it hard to choose the
best values for any given project using simple design
criteria. To overcome this obstacle, the authors have
developed a computer code which uses a mathematical
optimization algorithm for maximizing efficiency.
F l o w m o d e l
Since the results of the optimization process are strictly
dependent on the flow model accuracy, it is necessary to
take into account, even if in an approximate way, the
complex fluid dynamic phenomena, including boundary
layer growth, jet-wake formation, shock waves and
boundary layer interaction, etc, in a centrifugal compre-
ssor. One must also choose carefully the loss model,
because the available methods, by their very nature
always have an element of empiricism. Indeed, they are
based on a limited number of machines and problems may
arise when they are applied to others. To overcome this
drawback the authors tested several methods 1-5 to find
the most comprehensive for the medium-to-high com-
pression ratios.
Thermodynamic and fluid dynamic quantities are
computed in the one-dimensional approach at five sta-
tions according to the following scheme (Fig 1):
1 - rotor inlet
2 - rotor discharge
3 - vaned diffuser leading edge
4 - vaned diffuser throat
5 - vaned diffuser outlet
D i p a r t i m e n t o d i E n e rg e t ic a , P o l i t e c n i c o d i M i l a n o , P i a z za L e o n a r d o
d a V i n c i 3 2 , M i l a n o , I t a l y
R e c e i v e d 1 4 M a y 1 9 8 4 a n d a c c e p t e d fo r p u b l ic a t i o n o n 2 2 A u g u s t
1 9 8 4
R o t o r
Hub, mid-flow and tip quantities at the rotor inlet are
computed both outside and inside the blade row; trail-
ing edge thicknesses are considered and an optimum
incidence, ie minimum losses, is assumed. Station 2 is
solved evaluat ing dissipation in the flow field across the
entire impeller. Fluid dynamic losses may be estimated
in two alternative ways:
Nor thern Research m eth od 2
With the one dimensional limitation, this method carries
out a detailed analysis of the flow within the impeller.
Albeit roughly, the relative velocity distributions on the
suction and pressure sides of the blade are considered. On
the basis of these distributions, the various losses are
evaluated and the jet-wake development estimated as-
suming that separation occurs if
W Wmax
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A. Perd ich izz i and M. Sav in i
Fig 1 Geometr ic con f igura t ion o f the compressor
o t a t i o n
a
A
b
Cp
D
L
Leu
Kb
rh
M y
M w
n
N ,
o
P
r *
U
V
W
t~
T
X w
Z
B '
6
A h . t
Ahis
Ar
TC
S p e e d o f s o u n d
G e o m e t r i c a r e a
A x i al d e p t h
Pressu re recovery coef f i c ien t
D i a m e t e r
Di f fuser l eng th
Euler i an work
Bl o c k a g e f a c t o r
M a s s f l o w r a te
A b s o l u t e M a c h n u m b e r
Re l a t i v e Ma c h n u mb e r
Rota t iona l speed , r / s
vduh
a M echan ica l s t ress
ubscr ipts
ax Axial
D Di f fuser
h H u b
I Impel l e r
m f M e a n f lo w
O V O v e r a l l
r Rad ia l
ref Reference cond i t ions
t Tip
tg Tangen t i a l
VL Vane less di ffuser
VD Vaned d i f fuser
TS To ta l / s t a t i c
T T T o t a l / t o t a l
0 S tagn at ion cond i t ions
1 Ro to r in le t
2 Ro t o r o u t l e t
3 Van ed diffuser in let
4 Di f fuser th roa t
5 Diffuse r exi t
At /Be Blade loading losses
At/eL Clearance losses
At/oF Disc fr ict ion losses
Specif ic speed , n V i l / 2 / A h i s 3 / 4
Diffuser th roa t w id th
P r e s s u r e
Isen t rop ic degree o f reac t ion
Tangen t i a l ve loc i ty
Abso lu te ve loc i ty
Rela t ive ve loc i ty
N o r ma l b l a d e t h i c k n e s s
T e m p e r a t u r e
Volume f low ra t e
~o me ridion al leng th at spl i t ter
N u m b e r o f b la d e s
Abso lu te ve loc i ty ang le
Rela t ive ve loc i ty ang le
G eom etric blad e angle At/INC Incide nce losses
C l e a r a n c e b e t w e e n i mp e l l e r a n d s h r o u d
Ex terna l losses At/MiX M ixing losses
IsenLropic head
A t s v
Skin fr ict ion losses
Impel l e r rad ia l ex ten t At/VL Vaneless di ffuser losses
Flow coef f ic i en t,
V~ U2
At/vD Vaned diffuser losses
Pre ssur e rat io At /Ex Kine t ic energy discha rge losses
Di f fuser ha l f -d ivergence ang le Uni t s a re SI , ang les a re me asured f rom the mer id ion
Sl ip fac to r d i rec t ion
50 Vo l 6 , No 1 , M arch 1985
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Disc friction;
Backflow.
The mixing loss is included in the global loss while
the backflow
l o s s a l s o
includes the clearance leakage loss.
The slip correlations commonly quoted in the
litera ture and used here to compute the exit flow angle ~2
are the Wiesner formulationT:
= 1 - ( c o s / ~ ) o . s
z o . 7
and the Eckert formulation s as modified by the Northern
Research to consider the relative velocity deceleration
ratio:
i
p=
I f
D i s - + -D I I ' ~ "
) )
\ \
a n e l e s s d i f f u s e r
The equations of motion are solved at an appropriate
number of stations using the classical Stanitz approach
with regard to friction and heat exchange9. The aerody-
namic blockage due to the growth of the boundary layer
along the sidewalls and the joint losses are taken into
account.
V a n e d d i f f u s e r
The ent rance region is extremely important since the flow
is unsteady, often transonic, and the viscous effects are
significant: special attention has therefore been paid to its
modelling. The diffuser performance and the pressure
recovery coefficient C~ are strongly related to the boun-
dary layer growth up to the th roat since a boundary layer
that is well developed and near to separation may prevent
the diffusion process. Since it is impossible to predict
accurately the boundary layer in that region, a simplified
method proposed by Dean I allows one to estimate the
throat blockage once the pressure rise from the rotor
discharge is known. When the flow is supersonic at the
diffuser leading edge, it is assumed to become subsonic by
means of a normal shock wave: this makes it possible in
the one-dimensional representation to take into account
the shock wave-boundary layer interaction, in as much as
the shock pressure rise leads to a higher blockage. This is
substan tially confirmed by Kenny 's results 1o.
It is assumed that only straight channel diffusers
are used as this work refers mainly to high pressure ratio
machines. The design criteria and the performance eva-
luation are derived from Dean and Runstadler's Diffuser
Data Book ~1, assuming as independent variables the
throat Mach number M~,, the blockage factor Kh, and the
aspect ratio bU04.
Each set of these parameters is joined to the
optimum geometry (A 5/A4, e) suggested by the maps given
by Dean and Runstadler. The optimum is determined by
the highest Cpvdand the smallest area ratio able to warrant
stable operat ing conditions, that is far enough away from
the region of unsteady stall. In this way a certain range for
the off-design performance is ensured. The positive in-
fluence of the inlet swirl, compared with the uniform test
conditions, is treated in the manner suggested by Stevens
and Williams12. Once the diffuser geometry is defined, it is
possible to evaluate the diffuser performance by means of
two other methods:
O p t i m i z a t i o n o f c e n t r i fu g a l c o m p r e s s o r s
Northern Research method:
This divides the diffuser into two zones, before and after
the throat. The losses in the first part depend substantially
on the Mach number and rise, causing the latter to
approach unity; the other losses are due to throa t Mach
number, blockage and area ratio.
Galvas method:
This uses Dean and Runstandler's data in a simplified
mode, for instance without accounting for the aspect ratio
and for the divergence angle e.
O p t i m i z a t i o n s t r a t e g y
S o l v i n g t h e o p t i m i z a t i o n p r o b l e m r e q u i r e s:
Design specifications, namely mass flow rate, pressure
ratio, thermodynamic properties of the working fluid and
inlet conditions.
Project variables; eleven independent variables (Table 1)
allow one to fix the machine geometry and to compute the
efficiency. Blade thicknesses and the clearance gap are
determined as functions of the rotor outlet diameter, with
respect to technological limits.
Target unction; the function to be optimized is the overall
compressor efficiency. In some cases it may be useful to
optimize the total-to-static efficiency; in others the static-
to-static efficiency is optimised. The kinetic energy re-
covery factor DE is introduced therefore to cover both
definitions:
Ahi~ + OEV212
r /-
L~u + Ah~xt
where Leu is the Eulerian work and
A h e x
is the 'external'
losses.
Constraints; several geometric and fluid dynamic con-
straints limit the search field. These constraints must be
observed if physically meaningful and /or practically fea-
sible solution are to be found. In addition, mechanical
stress limitations within the rotor must be considered; the
critical regions are: the blade root at the outer diameter
owing to the bending stress; the blade root at about half
the radial extent owing to the sum of the bending and
centrifugal tensile stresses; and the bore because of
centrifugal force.
The stresses at these points are evaluated on the
basis of Osborne's simplified criteria 1 aild are compared
T a b l e 1 V a r i a b l e s a n d d e s i g n s p e c i f i c a t i o n s
Optimizat ion var iab/es
O l t , 0 1 h , 0 2 , fl ~ , b2, Zl, AJAr,
D31D2 b31b2
ZD, Xsp
Depent var iables
& = m a x ( 0 .3 m m , o r 0 2 / 4 0 0 )
tn lm in=max (0 .2 mm, o r 0 2 / 5 0 0 )
t .2 rn in=m ax (0 .6 mm, or D2 /2 00 )
tndt.h = 0 . 3 3
Design data
F l u i d t h e r m o d y n a m i c p r o p e rt ie s
I n le t c o n d i t i o n s
P r e s s u r e r a t i o
M a w w f l o w r at e
Whee l a l l oy
I n t . J . H e a t ~ t F l u i d F l o w 51
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A . P e r d i c h i z z i a n d M . S a v i n i
w i t h t h e y ie l d s t re s s w h i c h d e p e n d s u p o n t h e t e m p e r a t u r e
l e v e l a n d t h e a l l o y u se d . A l l t h e se c o n s t r a i n t s a r e sh o w n i n
T a b l e 2 .
Mathematical optimization;
t h e n u m e r i c a l p r o c e d u r e u s e d
i s a b l e t o o p t i m i z e a f u n c t i o n w i t h u p t o 3 0 v a r i a b l e s a n d
9 0 c o n s t r a i n t s ( 4 0 li n e a r i n th e i n d e p e n d e n t v a r i a b le s a n d
5 0 n o n - l i n e a r ) .
T h e s e a r c h r e q u i r e s n u m e r o u s a t t e m p t s ( a b o u t
4 0 0 0 ) w i t h d i f fe r e n t se t s o f v a r i a b l e s ; e a c h o n e p e r f o r m s a
f u ll d e s i g n a n d p e r f o r m a n c e e v a l u a t i o n o f t h e c o m p r e s s o r .
T h e t i m e t a k e n t o s o l v e a s a m p l e c a s e is a b o u t 3 0 0 s e c o n d s
o f C P U o n a U N I V A C 1 1 00 /8 0 c o m p u t e r .
R e s u l t s
T h e r e l i ab i l i ty o f t h e l o s s c o r r e l a t i o n s w a s c h e c k e d f o r
v a r i o u s c o m p r e s s o r s b y c o m p a r i n g p r e d i c t e d p e r f o r -
m a n c e s f o r th e a c t u a l g e o m e t r i e s w it h t h e e x p e r i m e n t a l
d a t a . T a b l e 3 is a n e x a m p l e o f th e s e c o m p a r i s o n s f o r t w o
m a c h i n e s , o n e b u il t b y F r a n c o T o s i S p A a n d t h e o t h e r b y
N u o v o P i g n o n e S p A , f o r w h ic h d e t a il e d m e a s u r e m e n t s
w e r e a v a i l a b le .
T h e N o r t h e r n R e s e ar c h m e t h o d , c o u p l e d w it h t h e
T a b l e 2 C o n s t r a i n t s o n o p t i m i z a t i o n
1 Geometric 2 Flui d dynamics
20* < ]~lt < 700 M w l t < 1.4 0 < Cpvt< 0 .4
0 . 4 < D 1 t / D 2 < 0 . 7
M wlm < 0.9 wake Kbax< 0.5
0 . 0 3 m < D 2 < 2 m
W2/Wlt>0 25
0 < f l 2 < 6 0 * 2 < 4 ( ~ 2 < 7 6 )
0.01