Centrifugal Compressor Map Prediction and Modification_new

8/10/2019 Centrifugal Compressor Map Prediction and Modification_new

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JKAU: Eng. Sci., Vol. 24No. 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**andM.I.F. Rezk****King Abdulaziz University, Jeddah, Saudi Arabia, and

**Mech. Power

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

Elsewedy forwind 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 compressorsperformance maps specified at stable and unstable conditions.

Enhancement and modification of the compressor performance map isperformed 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 complexbecause of its nonlinearity as well as unstable regions that are not easilyto be assigned practically. From this insight, many researches adopte to

predict its performance. As a result of these studies the empirical losscorrelation method had been persistently developed by several


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N.N. Bayomi et al.74

U2

C1 C1

researchers[1-2]

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

Dixon[3]

. Furthermore, compressor performance is limited by small flowrates where operational instability occurs that are rotating stall and surgethat 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 unstablecompressor performance map and accounts for compressor losses. This isachieved 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. Toestimate 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 aredetermined using the local stability method. The present work isextended to avoid compressor instability by close coupled valve andvariable drive speed methods.

2. Methdology for Compressor Map Prediction

In this section, performance map prediction and modifications aredemonstrated. Performance map prediction is determined by the actualEuler head at different speeds, the choke line and the instability lines. On

the other side, performance modification is attained using two types ofcontrollers that are the closed coupled valve and variable speed drive.

2.1 Performance Map Prediction

Foremost, theoretical Eulers head should be determined. Thetheoretical Euler head can be written as:

th 2 u2 1 u1H U C U C= (1)

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

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

C2

U1


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Centrifugal Compressor Map Prediction and Modification 75

All symbol definitions in the different equations are listed in thenomenclature. The velocity triangles at the inlet and exit can be identified

using the data of the impeller dimension, the rotational speed. Air densityat the compressor inlet and exit is estimated using equation of state, theambient conditions and the blade dimensions. The Slip factor used forvelocity triangle calculation is specified by the following equationintroduced by Stanitz

[14]:

( )0.631

z

=

(2)

Consequently, the theoretical Eulers Head can be calculated. Theactual Eulers head at different condition can be specified and is given bythe following equation:

act th lossH H L= (3)

In order to assign actual Euler head, eight common different headlosses, 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 thesecoefficients is accomplished using trial and error till matching betweenthe 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 principallyby the efficiency therefore, it is necessary to estimate the efficiency atdifferent conditions. The efficiency of the compressor can be defined by:

th

th loss

H

H L = + (4)

Using actual Eulers head to get pressure ratio

( )1th tot

p 1

H L1

C T

= +

(5)


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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 D

Incidence Loss2

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

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

Leakage Loss 2

2

cl cl m U U

m

Recirculation Loss

( )2

2 1

2

22 2

2 2

1 1 11

2 2 2

0.02 tan 1

1 2

+ +

p

BL

t t

C T TK

W UU

W D DW z

U D D

Table 2. Coefficients used in present methodology.

Coefficient Value UnitKBL 0.75 or 0.60 for conventional or splitter

impellers, respectively-----

Fmc 0.7 -----

Cf 0.004 -----

0.0038 m

lc 1 -----

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

of Dixon[3]:


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( )( )

( )

( )

2 1

2 12

2 1

o1

o1 o1 12 2

22

o1

U1 1

am 2a

A 1U

1 1a

+

+

= + +

(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 is replaced by the polytropic index that equals to 1.2as 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 andsuch 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:

( )( )cc

d 1

d l

=

and ( )2

c

d 1

d 4B l

=

(7)

The following equation determines the compressor map that isdefined by a fifth order polynomial of the flow coefficient, :

( ) 4 3 2c 1 2 3 4 5C C C C C = + + + + (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:


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( )cc c

2 2

c c

d1 1

l d l

1

4 l 2 l

(9)

For stalling condition:

( )c2

c c

d10

l d 2 l


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the rotational speed as a control variable renders the equilibrium globallyexponentially stable and the use of the drive speed as control ensure

exponential convergence. The control manipulates the compressor map insuch 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 2c 1 2 3 4 5N C N C N C N C N C N = + + + + (12)

In the above equation the pressure coefficient, c , is dependent onthe 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 thelower 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 287K 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 showthe 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 notcovered by the experimental results could be predicted.


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Table 3. Eckardt rotors geometrical parameter.

Geometrical ParameterEckardtrotor A

Eckardtrotor B

Eckardtrotor 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 60Inducer 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 Eulers 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 massflow 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.


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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 2.08EckardtImpeller

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 1.938Eckardt

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 2.469Eckardt

ImpellerO 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 rpm1.25204 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 2.868BayomiImpeller 0.32797 1.915


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

A at different speeds.

Fig. 3. Eulers 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 topredict the choke line at different compressor speeds.


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

Fig. 5. Comparison of the experimental choke line and that estimated mathematically forEckardt rotor O.

Surge line variation with different values of is demonstrated inFig. 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 shifts the surge line away from the peak. Consequently, the

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

surges or stalls. More details about the results of the present work can befound in Rezk

[16].


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Fig. 6. Effect of 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 ratiocompared with closed coupled valve.

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

Eckardt rotor A at 10000 rpm.


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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 heatratio 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)

C1C4 Polynomial coefficients that determine the performance map of the compressor(---)

D Impeller diameter (m)

Hact Actual Euler head (m2

/s2

)Hth Theoretical Euler head (m

2/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) Absolute flow angle (Degree)


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Greitzer coefficient (---)

Throttle valve coefficient (---)

CCV Closed coupled valve coefficient (---)

Specific heat ratio (---)

Impeller efficiency (---)

Dynamic viscosity of air (N.s/m2)

Pressure ratio (---)

Air density (kg/m3)

The slip factor (---)

Non-dimensional time (---)

Flow coefficient (---)

Pressure coefficient (---)

c Performance characteristic of the compressor (---)

subscript

1 Inlet2 Outlet

h hub

cl Clearance

T Throttlet Tip

Abbreviation

CCV Closed coupled valve

References

[1] Seleznev, K.P., Galerkin, Y.B. and Popova, E.Y., Simplified mathematical model oflosses in a centrifugal compressor stage. Chemical and Petroleum Engineering, 23(10):

471-476 (1987).

[2] Oh, H.W., Yoon, E.S. andChung, 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. andGreitzer, 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 25thAnnual International Gas Turbine Conference and Exhibit and the 22ndAnnual

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


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[10] Galindo, J., Serrano, J.R., Climent, H. and Tiseira, A., Experiments and modeling ofsurge 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. andEgeland, O.,Modeling of surge in variablespeed centrifugal compressors: Experimental validation. AIAA Journal of Propulsion and

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

[13] Bhagen, B. andGravdahl, 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_new