Aerodynamic and Geometric Optimization for the Design of Centrifugal Compressors

8/10/2019 Aerodynamic and Geometric Optimization for the Design of Centrifugal Compressors

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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


4/8

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

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Aerodynamic and Geometric Optimization for the Design of Centrifugal Compressors