centrifugal compressor map prediction and modification

JKAU: Eng. Sci., Vol. 24 No. 1, pp: 73-88 (2013 A.D. /1434 A.H.)

DOI: 10.4197 / Eng. 24-1.4

73

Centrifugal Compressor Map Prediction and Modification

N.N. Bayomi*,**

, R.M. Abdel-Maksoud**

and M.I.F. Rezk***

*

King Abdulaziz University, Jeddah, Saudi Arabia, and **

Mech. Power

Dept., Faculty of Eng., Mataria, Helwan University, and ***

Elsewedy for

wind energy generation-Elsewedy Electric, Cairo, Egypt

[email protected]

Abstract. Centrifugal compressors are utilized in various fields and are

used in vast applications. Their operational performance maps are

significant to be studied, modified and enhanced. Unfortunately, such

maps that describe experimental results do not cover each condition.

This is due to expenses as well as the uncovering operational zones.

Therefore, map prediction is important, however, it is very complex

because of its nonlinearity as well as unstable region that are not

easily to be assigned practically. Consequently, the present paper

introduced a methodology that predicts the centrifugal compressors

performance maps specified at stable and unstable conditions.

Enhancement and modification of the compressor performance map is

performed using the closed coupled valve and variable drive speed

where the later method was more preferable based on shifting of the

compressor map towards lower flow rate with less pressure drop.

Keywords: Centrifugal compressor, Compressor map, Rotating stall,

surge, Choke line.

1. Introduction

Centrifugal compressors have been used in vast industrial applications. Knowledge of their operational performance maps is significant; however such maps do not cover all the conditions due to expenses. Therefore, map prediction is important, however, it is very complex because of its nonlinearity as well as unstable regions that are not easily

to be assigned practically. From this insight, many researches adopte to predict its performance. As a result of these studies the empirical loss correlation method had been persistently developed by several

N.N. Bayomi et al. 74

researchers[1-2]

. On the other side, map performance is restricted by high flow rate limits denoted by choke line. Choke line was determined by Dixon

[3]. Furthermore, compressor performance is limited by small flow rates where operational instability occurs that are rotating stall and surge that are vastly studied by researchers[4-6]

. Compressor surge control was introduced by other researchers

[7-13].

The present work aimed to predict the stable and unstable compressor performance map and accounts for compressor losses. This is achieved by introducing a methodology. This is determined by pre-matching of the simulated actual with the experimental results to account for different losses represented by previous empirical formulas. Therefore, the uncovered zones in compressor map can be predicted. To estimate the choke line, the present model utilizes the formula of Dixon

[3] for chocking at the diffuser. In addition, the stall line and surge line are determined using the local stability method. The present work is extended to avoid compressor instability by close coupled valve and variable drive speed methods.

2. Methdology for Compressor Map Prediction

In this section, performance map prediction and modifications are demonstrated. Performance map prediction is determined by the actual Euler head at different speeds, the choke line and the instability lines. On the other side, performance modification is attained using two types of controllers that are the closed coupled valve and variable speed drive.

2.1 Performance Map Prediction

Foremost, theoretical Euler’s head should be determined. The theoretical Euler head can be written as:

th 2 u2 1 u1H U C UC= − (1)

The velocity triangles at the inlet and exit of typical centrifugal compressor impeller impeller is shown in Fig. 1.

Fig. 1. Velocity triangles for compressor impeller: a) Inlet velocity triangle, b) Outlet

velocity triangle.

Centrifugal Compressor Map Prediction and Modification 75

All symbol definitions in the different equations are listed in the nomenclature. The velocity triangles at the inlet and exit can be identified using the data of the impeller dimension, the rotational speed. Air density at the compressor inlet and exit is estimated using equation of state, the ambient conditions and the blade dimensions. The Slip factor used for velocity triangle calculation is specified by the following equation introduced by Stanitz

[14]:

( )0.631

z

⎛ ⎞× πσ = −⎜ ⎟

⎝ ⎠ (2)

Consequently, the theoretical Euler’s Head can be calculated. The actual Euler’s head at different condition can be specified and is given by the following equation:

act th lossH H L= − (3)

In order to assign actual Euler head, eight common different head losses, Lloss, will be estimated from the Table 1 by using the selection losses equations from Oh et al.[2]. The ranges of the coefficients of these equations are specified in Table 2. Substituting the certain values of these coefficients is accomplished using trial and error till matching between the actual Euler head and the experimental results will be performed. Consequently, the uncovered zones in compressor map can be assigned. Since the compressor operational condition is characterized principally

by the efficiency therefore, it is necessary to estimate the efficiency at

different conditions. The efficiency of the compressor can be defined by:

th

th loss

H

H Lη =

+ (4)

Using actual Euler’s head to get pressure ratio

( )1th tot

p 1

H L1

C T

γ

γ−⎡ ⎤⎛ ⎞−π = +⎢ ⎥⎜ ⎟⎜ ⎟η⎢ ⎥⎝ ⎠⎣ ⎦

(5)


Table 1. Losses description equations for centrifugal compressor.

Compressor losses Loss model

Blade Loading Loss

( )2

2 1

2

22 2

2

1 1 11

2 2 2

11

21 2

⎛ ⎞−⎜ ⎟⎜ ⎟

× − + ×⎜ ⎟⎡ ⎤⎡ ⎤⎜ ⎟− +⎢ ⎥⎢ ⎥⎜ ⎟⎣ ⎦⎣ ⎦⎝ ⎠

p

BL

t t

C T TK

W UU

W D DW z

U D Dπ

Incidence Loss 2

1

2

u

mc

WF

Impeller Disk Friction

Loss

3

52

2 2

2

0.2

2 2 2

2

0.0402

4

Ur

r

U r

m

ρ

ρ

μ

⎛ ⎞⎜ ⎟⎝ ⎠

⎛ ⎞⎜ ⎟⎝ ⎠

Skin Friction Loss

2

2

1 2

22

2

f

Dw

C WD D

⎛ ⎞⎛ ⎞+⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠+⎛ ⎞

⎜ ⎟⎝ ⎠

Clearance Loss

( )

2

2 2

1 1

2 2

22 22 1

1

40.6

1

t h

t

r rW W

b b Zr r

ε π

ρ

ρ

⎡ ⎤⎡ ⎤⎢ ⎥⎢ ⎥−⎢ ⎥⎢ ⎥

⎛ ⎞⎢ ⎥⎢ ⎥− −⎜ ⎟⎢ ⎥⎢ ⎥⎝ ⎠⎣ ⎦⎣ ⎦

Leakage Loss 2

2

cl clm U U

m

Recirculation Loss

( )2

2 1

2

22 2

2 2

1 1 11

2 2 2

0.02tan 1

1 2

⎛ ⎞−⎜ ⎟⎜ ⎟

− +⎜ ⎟⎡ ⎤⎡ ⎤⎜ ⎟− +⎢ ⎥⎢ ⎥⎜ ⎟⎣ ⎦⎣ ⎦⎝ ⎠

p

BL

t t

C T TK

W UU

W D DW z

U D D

α

π

Table 2. Coefficients used in present methodology.

Coefficient Value Unit

KBL 0.75 or 0.60 for conventional or splitter

impellers, respectively

-----

Fmc 0.7 -----

Cf 0.004 -----

ε 0.0038 m

lc 1 -----

The choking of mass flow can be expressed by the famous equation

of Dixon[3]

:


( )( )

( )

( )

2 1

2 12

2 1o1

o1 o1 12 2

22

o1

U1 1

am 2a

A 1U

1 1a

γ

γ−

γ+•

γ−

⎡ ⎤ησ+ γ −⎢ ⎥

⎛ ⎞⎣ ⎦= ρ ⎜ ⎟γ +⎝ ⎠⎡ ⎤σ+ γ −⎢ ⎥

⎣ ⎦

(6)

Since the above equation represents a theoretical relationship

between the choke line flow rate and the different parameters, a new

treatment is herein presented to suit the actual prediction. This is

performed where γ is replaced by the polytropic index that equals to 1.2

as a correction in order to be suitable for precise prediction.

In order to determine the stall and surge line local stability analysis is

used. This method is used extensively by previous researchers such as

Abed El-Maksoud[8]

. The local stability analysis method is herein used to

assess the system whether the system is stable (rotating stall) or unstable

(surge) at the left of the peak. Regarding the stability condition, stall

point is defined as stable condition, since the characteristics will be

asymptotically stable in low-flow small pressure rise region. Stall line

can be predicted just on the left the characteristic peaks. In case of surge,

the flow coefficient and pressure fluctuates with certain amplitudes and

such phenomenon is unstable. The local stability analysis method

depends on the roots of the Jacobean matrix of Moore and Greitzer

model[4]

system description state equations. The two state equations of

the Moore - Greitzer model are:

( )( )c

c

d 1

d l

ϕ= ψ ϕ −ψ

ζ and ( )2

c

d 1

d 4B l

ψ= ϕ − Γ ψ

ζ (7)

The following equation determines the compressor map that is

defined by a fifth order polynomial of the flow coefficient, ϕ:

( ) 4 3 2

c 1 2 3 4 5C C C C CΨ ϕ = ϕ + ϕ + ϕ + ϕ + (8)

Applying stability condition on Moore and Greitzer model, hence

system stalls or surges could be assigned. Stability analysis has been

implemented by several researchers to analyze the compression system.

The following equations present the treatment of stability analysis

method. Therefore, the Jacobean matrix of the Moore and Greitzer model

will be:


( )c

c c

2 2

c c

d1 1

l d l

1

4 l 2 l

⎛ ⎞Ψ ϕ−⎜ ⎟

ϕ⎜ ⎟⎜ ⎟Γ

−⎜ ⎟⎜ ⎟β β ψ⎝ ⎠

(9)

For stalling condition:

( )c

2

c c

d10

l d 2 l

⎛ ⎞Ψ ϕ Γ− <⎜ ⎟

ϕ β ψ⎝ ⎠ and

( )

( )

2

c

2

c c

c

2 22

c cc

d1

l d 2 l

d1 14

l d 4 l2 l

⎛ ⎞Ψ ϕ Γ−⎜ ⎟⎜ ⎟ϕ β ψ⎝ ⎠

⎛ ⎞⎛ ⎞Ψ ϕ Γ< − +⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟ϕ ββ ψ⎝ ⎠⎝ ⎠

(10)

Otherwise, the system surges.

2.2 Performance Map Modification

The first controller introduced herein is the closed coupled valve,

CCV. It is named so as the valve is installed very close and coupled to

the compressor. According to Gravdahl[6]

, close coupled valve is to be

understood that the distance between the compressor outlet and the valve

is so small that no significant mass storage can take place. The

assumption of no mass storage between the compressor and the valve

allows for the definition of an equivalent compressor. The closed coupled

valve is introduced into the compression system to achieve a pressure

drop on the compressor pressure rise. Thus, reduces the characteristic

peak and shifts it towers lower flow rates, thus avoiding falling into

surge. Consequently the second state equation of model Eq. 7 can be

rewritten as:

( )ccv2

c

d 1

d 4B l

ψ= ϕ − Γ ψ − Γ ψ

ζ (11)

The second controller introduced in this sub-section is the variable

speed drive. From the findings of Gravdahl[6]

, Variable speed drive is

herein used as a second method to achieve compression system

stabilization. This is approach is used as active surge control. The use of


the rotational speed as a control variable renders the equilibrium globally

exponentially stable and the use of the drive speed as control ensure

exponential convergence. The control manipulates the compressor map in

such a way that the compressor map is shifted to the left with lower flow

rates. The equation that describes this type of controller can be written as:

( ) ( ) ( ) ( ) ( ) ( )4 3 2

c 1 2 3 4 5N C N C N C N C N C NΨ = ϕ + ϕ + ϕ + ϕ + (12)

In the above equation the pressure coefficient, Ψc , is dependent on

the speed, N, that varies according to the onset of instability. The

controller is used to reduce the speed of the compressor so that the peak

of the performance characteristic is lowered and shifted towards the

lower flow rates similarly to the closed coupled valve. This behavior

avoids falling into surge. The following section deals with the results of

the present model and the assessment of the two controllers.

3. Results and Discussion

In order to determine the uncovered zones in the compressor map,

the present model results are matched to the experimental results. The

comparison is herein performed using three different ratios of Eckardt[7]

denoted in this finding by rotor A, B, and O and rotor of Bayomi[15]

.

Foremost, in order to conduct simulations, different variables of the

present model should be specified to estimate the different losses in the

present model. The ambient temperature and pressure are 287 K and 1

Bar, respectively. The specific heat at constant pressure, specific heat

ratio and gas constant for air are taken to be equal to 1005 J/kgK and

1.333 and 287 J/kgK, respectively. Using the experimental maps of these

rotors and their data of these rotors to matching these maps with the

present model to find out different uncovered zoned. Tables 3 and 4 show

the three Eckardt rotors and Bayomi rotor geometrical parameters,

respectively. Table 5 illustrates the performance experminal data of

Eckardt and Bayomi rotors.

Figure 2 demonstrates the results of model pre-matching with the

experimental results for Eckardt rotors A for different four speeds.

Consequently, the losses considered in Table 1 are valid at these speeds.

This makes the losses different rotational speeds could be determined.

Hence, the compressor stable operation at different speeds that is not

covered by the experimental results could be predicted.


Table 3. Eckardt rotors geometrical parameter.

Geometrical Parameter Eckardt

rotor A

Eckardt

rotor B

Eckardt

rotor O

Inlet Tip Diameter mm 280 280.3 280

Inlet Hub Diameter mm 120 191.8 90

Discharge Diameter mm 400 400 400

Discharge Width mm 26 26 26

Number of Blades 20 20 20

Length in axial direction mm 130 84.2 130

Blade Thickness mm 3 3 3

Inlet Blade Angle 30 30 30

Exit Blade Angle 30 40 90

Maximum Rotational Speed RPM 16000 16000 16000

Table 4. Bayomi rotor geometrical parameter.

Geometrical Parameters

Impeller outer diameter mm 160

Inducer tip diameter ratio 0.70

Inducer hub diameter ratio 0.2375

Inducer tip diameter mm 112

Inducer hub diameter mm 38

Exit width ratio 0.0766

Blade thickness ratio 0.0163

Impeller discharge width mm 12.256

Impeller blade thickness mm 2.608

Exit blade angle 60

Inducer tip angle 60

Inducer hub angle 40

Number of blades (7 splitter blades) 7

Design speed rpm 55000

After pre-matching is achieved, Fig. 3 illustrates Euler’s head and

total predicted losses for Eckardt rotor B at different operating speed in-

between stalling point the choking point. Total losses plotted in this

figure could be utilized to determine the compressor efficiency. As one

may observe that total losses increase with the rotational speed and mass

flow rate.

After estimating the different losses, it is principally necessary to

estimate the compressor efficiency at different speeds. Efficiency of

Bayomi rotor is assigned by plotting the simulated results in Fig. 4 using

Eq. 4. The global observation is that the efficiency values appear to be

higher with the increase of rotational speed. Furthermore, efficiency

increases with reduction of mass flow rate till or at least near the peak of

map characteristic.


Table 5. Data for Eckardt's Bayomi's experimental data.

m Pr m Pr m Pr m Pr

10,000 rpm 12,000 rpm 14,000 rpm 16,000 rpm

2.50 1.4376 3.00 1.665 3.5 1.94 4.2 2.305

3.10 1.4064 3.80 1.635 4.5 1.925 5.2 2.26

3.80 1.3908 4.40 1.59 5.3 1.88 6 2.2

4.60 1.3596 5.20 1.56 6.1 1.805 6.8 2.08 Eckardt

Impeller

A 5.00 1.328

10,000 rpm 12,000 rpm 14,000 rpm 16,000 rpm

2.315 1.359 2.869 1.531 3.304 1.750 3.695 2.031

2.675 1.359 3.391 1.531 3.913 1.750 4.260 2.000

3.135 1.328 3.913 1.484 4.521 1.718 4.826 1.984

3.675 1.281 4.521 1.421 5.086 1.656 5.391 1.938 Eckardt

Impeller

B 4.270 1.250

40,000 rpm 45,000 rpm 50,000 rpm 55,000 rpm

2.300 1.4687 3.086 1.7340 4.0000 2.0546 4.9130 2.5000

2.565 1.485 3.413 1.7500 4.2610 2.0781 5.1956 2.5312

2.782 1.500 3.739 1.7812 4.5650 2.1015 5.3913 2.5312

3.043 1.531 3.956 1.7960 4.7830 2.1250 5.6086 2.5468

3.261 1.547 4.26 1.7960 5.0000 2.1250 5.8695 2.531

3.521 1.547 4.521 1.7570 5.2610 2.1406 6.0434 2.531

3.739 1.500 4.739 1.7810 5.5220 2.1250 6.3913 2.516

4 1.484 4.956 1.7340 5.6960 2.0937 6.5652 2.500

4.261 1.469 5.152 1.7109 5.9130 2.0859 6.7608 2.469 Eckardt

Impeller

O 4.565 1.438 5.326 1.6875 6.0870 2.0781 6.9565 2.453

10,000 rpm 12,000 rpm 14,000 rpm 16,000 rpm

1.25204 1.278 1.34733 1.332 1.42378 1.385 1.49026 1.434

1.10135 1.675 1.14678 2.039 1.22434 2.46 1.38057 2.674

1.0105 1.768 1.03266 2.161 1.10689 2.557 1.22988 2.998

0.91299 1.859 0.90080 2.2 0.97947 2.576 1.16451 3.031

0.82103 1.9 0.77449 2.22 0.85094 2.585 1.05149 2.999

0.70026 1.917 0.66258 2.214 0.73793 2.567 0.98612 2.991

0.57505 1.923 0.48087 2.153 0.56397 2.542 0.90080 2.969

0.43544 1.935 0.39556 2.161 0.48641 2.527 0.58724 2.868 Bayomi

Impeller 0.32797 1.915


Fig. 2. The results of model pre-matching with the experimental results for Eckardt rotors

A at different speeds.

Fig. 3. Euler’s head and total losses for Eckardt rotor B at different speeds.

The experimental results of Eckardt rotors O and the simulated

results of Dixon equation is presented in Fig. 5. The results of

compressor mass flow rate, the compressor rotational speed is substituted

in Eq. 6. This plot illustrates good matching between experimental results

and mathematical results. Consequently, this equation can be used to

predict the choke line at different compressor speeds.


Fig. 4. Efficiency curves of Bayomi rotor at different rotational speeds.

Fig. 5. Comparison of the experimental choke line and that estimated mathematically for

Eckardt rotor O.

Surge line variation with different values of β is demonstrated in

Fig. 6. The stall line is always specified at the peak of the performance

map. Stall point is determined by performance peaks. This is the

traditional method mentioned by Gravdahl[6]

. It is obvious that the

increase of β shifts the surge line away from the peak. Consequently, the

parameter β has an effect on the system and specifies whether the system

surges or stalls. More details about the results of the present work can be

found in Rezk[16]

.


Fig. 6. Effect of β parameter variation on surge line location on performance map of

Eckardt rotor B.

The results of employing closed coupled valve and variable speed

drive on Eckardt rotor A are shown in Fig. 7 at 10000 rpm. To access the

two controllers the two controllers have to achieve certain specified flow

rate reduction with minimal pressure drop reduction. It is clearly revealed

that variable speed drive achieves lower drop in the pressure ratio

compared with closed coupled valve.

Fig. 7. Comparison between closed coupled valve and variable speed drive behavior for

Eckardt rotor A at 10000 rpm.


4. Conclusions

From this work, the following conclusions can be drawn:

1. A new methodology is herein introduced to predict and modify

the compressor performance map by pre-matching with the experimental

results. Consequently, the different conditions that are not covered by the

experimental map can be identified.

2. The present methodology can be used to determine the impeller

losses and its efficiency.

3. To estimate the choke line, the predicted data of the present

model is substituted in the formula of Dixon[3]

where the specific heat

ratio is replaced by polytropic index.

4. The stall line and surge line are specified by substituting of the

predicted compressor characteristic map of the present model in the

Moore - Greitzer model.

5. The closed coupled valve and variable drive speed methods are

herein used to extend the safety operating margin by shifting the

performance map to the left (i.e toward the low mass flow rate) on the

plenty of pressure ratio reduction. Such reduction appears to be less when

using variable drive speed.

Nomenclature

a Mach number (---)

b Impeller width (m)

Cp Specific heat at constant pressure (kJ/Kg K)

C1…C4 Polynomial coefficients that determine the performance map of the compressor

(---)

D Impeller diameter (m)

Hact Actual Euler head (m2/s2)

Hth Theoretical Euler head (m2/s2)

lc The compression system duct length (m)

Lloss Different Euler head losses (m2/s2)

m Mass flow rate (kg/s)

N Rotational speed (rpm)

Pr Pressure ratio (---)

r Impeller radius (m)

T Temperature (K)

W Relative velocity (m/s)

U Blade velocity (m/s)

Z Number of the blades (---)

w Impeller width (m)

α Absolute flow angle (Degree)


β Greitzer coefficient (---)

Γ Throttle valve coefficient (---)

ΓCCV Closed coupled valve coefficient (---)

γ Specific heat ratio (---)

η Impeller efficiency (---)

μ Dynamic viscosity of air (N.s/m2)

π Pressure ratio (---)

ρ Air density (kg/m3)

σ The slip factor (---)

ζ Non-dimensional time (---)

ϕ Flow coefficient (---)

Ψ Pressure coefficient (---)

Ψc Performance characteristic of the compressor (---)

subscript

1 Inlet

2 Outlet

h hub

cl Clearance

T Throttle

t Tip

Abbreviation

CCV Closed coupled valve

References

[1] Seleznev, K.P., Galerkin, Y.B. and Popova, E.Y., Simplified mathematical model of

losses in a centrifugal compressor stage. Chemical and Petroleum Engineering, 23(10):

471-476 (1987).

[2] Oh, H.W., Yoon, E.S. and Chung, M.K., An optimum set of loss models for performance

prediction of centrifugal compressors. Proceedings of the Institution of Mechanical

Engineers, Part A: Journal of Power and Energy; 211: 331-338 (1997).

[3] Dixon, S.L., Fluid Mechanics and Thermodynamics of Turbomachinery. Pergamon Press

Ltd, (1996).

[4] Moore, F.K. and Greitzer, E.M., A theory of post-stall transients in a axial compressor

systems. Journal of Engineering for Gas Turbines and Power, 108: 68-76 (1986).

[5] Erickson, C., Centrifugal Performance Modeling Development and Validation for a

Turbocharger Component Matching System. Kensas State University (2008).

[6] Gravdahl JT. Modeling and control of surge and rotating stall in compressors. PhD thesis

Dept. of Engineering Cybernetics, Norwegian University of Science and Technology,

1998.

[7] Eckardt, D., Flow field analysis of radial and backswept centrifugal compressor impellers.

ASME 25th Annual International Gas Turbine Conference and Exhibit and the 22nd Annual

Fluids Engineering Conference, March, Louisiana, USA (1980).

[8] Abd El-Maksoud, R.M., Modeling of rotating stall and surge in axial flow compressors.

Ph.D., Mech. Power Eng. Dept., Faculty of Eng., Mataria, Helwan University, Cairo, Egypt

(2004).

[9] Shehata, R.S., Abdullah, H.A. and Areed, F.F.G., Variable structure surge control for

constant speed centrifugal compressors. Control Engineering Practice, 17: 815-833 (2009).


[10] Galindo, J., Serrano, J.R., Climent, H. and Tiseira, A., Experiments and modeling of

surge in small centrifugal compressor for automotive engines. Experimental Thermal and

Fluid Science, 32: 818-826 (2008).

[11] Gravdahl, J.T., Egeland, O. and Vatland, S.O., Drive torque actuation in active surge

control of centrifugal compressors. Automatica, 38: 1881-1893 (2002).

[12] Gravdahl, J.T., Willems, F., de Jager, B. and Egeland, O., Modeling of surge in variable

speed centrifugal compressors: Experimental validation. AIAA Journal of Propulsion and

Power, 20 (5): 849-857 (2004).

[13] BØhagen, B. and Gravdahl, J.T., Active surge control of compression system using drive

torque. Automatica, 44: 1135 -1140(2008).

[14] Stanitz, J.D., One-dimensional compressible flow in vaneless diffuser of radial and mixed

flow centrifugal compressor, including effects of friction, heat transfer, and area change,

NACA, TN-2610 (1952).

[15] Bayomi, N.N., An investigation on the casing treatment of the radial compressors. Ph.D,

Mech. Power Eng. Dept., Faculty of Eng., Mataria, Helwan University, Cairo, Egypt

(1995).

[16] Rezk, M.I.F., Prediction of centrifugal compressor performance maps. MSc., Mech. Power

Eng. Dept., Faculty of Engineering, Mataria, Helwan University, Cairo, Egypt (2009).


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centrifugal compressor map prediction and modification

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