An analytical approach to predicting particle depositby fouling in the axial compressor of the industrialgas turbineT W Song1, J L Sohn1�, T S Kim2, J H Kim3, and S T Ro1

1School of Mechanical and Aerospace Engineering, Seoul National University, Seoul, Korea2Department of Mechanical Engineering, Inha University, Inchon, Korea3Turbomachinery Department, Korea Aerospace Research Institute, Daejon, Korea

The manuscript was received on 13 May 2004 and was accepted after revision for publication on 30 September 2004.

DOI: 10.1243/095765005X7547

Abstract: The gas turbine performance deteriorates with increased operating hours. Fouling in theaxial compressor is an important factor for the performance degradation of gas turbines. Airborneparticles entering the compressor with the air adhere to the blade surface and result in the changeof the blade shape, which directly influences the compressor performance. It is difficult to exactlyunderstand the mechanism of compressor fouling because of its slow growth and different lengthscales of compressor blades. In this study, an analytical method to predict the particle motion inthe axial compressor and the characteristics of particle deposition onto blade is proposed as anapproach to investigating physical phenomena of fouling in the axial compressor of industrialgas turbines. Calculated results using the proposed method and comparison with measureddata demonstrate the feasibility of the model. It was also found that design parameters of theaxial compressor such as chord length, solidity, and number of stages are closely related to thefouling phenomena. Likewise, the particle size and patterns of particle distributions are alsoimportant factors related to fouling phenomena in the axial compressor.

Keywords: fouling, gas turbine, axial compressor, collection efficiency

1 INTRODUCTION

Performance degradation of the gas turbine isdirectly related to the change of the blade profilesof compressor and turbine due to their deteriorationcaused by fouling, erosion, corrosion, foreign objectdamage (FOD), etc. Among these degradation-related factors, compressor fouling, defined as thedeposition process of airborne particles on compres-sor blades, is known to be the source of about 70–85per cent of performance degradation of industrial gasturbines [1]. The atmosphere includes various typesof airborne particles such as dirt, dust, salt, etc. thatare sources of compressor fouling. Although thefiltration system removes most of these particles,

unfiltered particles enter into the compressor andadhere to the blade surface as a mixture of moistureand lubricants. Severe fouling problems can beresolved by installing highly efficient filters at thecompressor inlet, but their use is restricted due toheavy cost and large pressure drop.

In the axial compressor of the industrial gasturbine, fouling results in the change of the shapeof leading edges of the blades and their surfaceroughness and, as a result, the airflow in the com-pressor cascade becomes distorted. This alters thecompressor characteristics. For example, as a resultof compressor fouling, after 100 h of continuousoperation, a low-pressure compressor experiencedabout 3–4 per cent drop in pressure ratio with 2–4per cent drop in efficiency and 10 percent drop inpressure ratio with 6–7 percent drop in efficiencyfor a high-pressure compressor [2]. The turbineoutput power decreases with the compressor

�Corresponding author: School of Mechanical and Aerospace

Engineering, Seoul National University, Seoul, Korea.

203

A06304 # IMechE 2005 Proc. IMechE Vol. 219 Part A: J. Power and Energy

performance degradation. This also results in thereduction in surge margin and dramatically unstableoperation of the whole gas turbine.

There are many previous studies for the predictionof the overall performance degradation of the gasturbine due to fouling phenomena. Saravanamuttooand Lakshminarasimha [3] studied the effect of foul-ing on the gas turbine performance by assuming onlyone stage (first or last stage) of the axial compressoraffected by fouling. Aker and Saravanamuttoo [4]insisted that fouling progresses linearly along com-pressor stages and, as a result, the degradation ofcompressor stage decreases linearly in the down-stream direction. Seddigh and Saravanamuttoo [5]quantified the effect of compressor fouling to thegas turbine performance degradation. Massardo [6]investigated the effect of fouling on pressure riseand efficiency of the axial compressor.

Detailed characteristics of particle depositions canbe investigated using advanced computational toolssuch as computational fluid dynamics [7]. Slateret al. [7] showed particle deposition characteristicson turbine blades in turbulent flow environment ina gas turbine using a computational fluid dynamicstechnique based on a fully Eulerian two-fluidapproach. Their results provide detailed inter-relatedmechanisms among particle deposition, geometriccharacteristics of the blade and particle inertia, etc.Tarabrin et al. [8] invented an analytical model ofthe fouling mechanism considering the motion offoulant particles by simplification of a compressorblade as a cylinder. Although this simplification isgood enough, real physical phenomena such asflow conditions around a blade cannot be reflected.In the present study, a newly modified analyticalmodel based on the cylinder model of Tarabrinet al. [8] is suggested. With this model, the para-metric studies of a blade profile and the flow con-ditions are performed and the effects of particlesize and particle distribution are also investigated.

2 CASCADE COLLECTION EFFICIENCY

Fouling is defined as the deposition process of air-borne particles on the solid surface. Figure 1 shows

the general mechanisms of the deposition of solidparticles on a cylinder in the flow field. A particlealong a streamline encounters diffusion by Brownianmotion, or is intercepted by the cylinder surface, orcollides due to its inertia. When a particle floats ina slow flow field, diffusion is the dominant physicalphenomenon and, if the particle size is not lessthan that of a cylinder, interception is dominant[9]. Impact occurs in a fast flow field due to inertiaif the cylinder diameter is very large compared withthe particle size. In this study, it is assumed thatfouling occurs only when particles hit on the cylinder[8]. In order to examine the fouling effect due toimpact, the collection efficiency, E, is defined as theratio of the number of particle impacts on the cylin-der to that of all particles entering perpendicularly tothe cylinder, as shown in Fig. 1.

E ¼

the number of particle impactson a cylinder

the number of particles enteringperpendicularly to the cylinder

¼H

L(1)

Here, ‘impacts to the cylinder’ means that, whenparticles enter perpendicularly to a cylinder, someparticles are separated from the flow field aroundcylinder by their inertia and collide with the cylinder.Equation (1) assumes that the entering particles areuniformly distributed and all collided particles donot bounce off but adhere. The collection efficiencydefined in the flow field around a cylinder can bemodified for the simplified cascade flow, as shownin Fig. 2, around a single axial compressor blade asfollows

E ¼H

L¼

H

c sin (bb � b1)(2)

Fig. 1 Characteristics of particle motion around a

cylinder

Fig. 2 Flow characteristics between two blades in the

axial compressor [8]

204 T W Song, J L Sohn, T S Kim, J H Kim, and S T Ro

Proc. IMechE Vol. 219 Part A: J. Power and Energy A06304 # IMechE 2005

If considering a blade row instead of a single blade,the cascade collection efficiency based on the aboveexpression can be defined as

Ec ¼ Ec

s

sin (bb � b1)

sinb1

(3)

The cascade collection efficiency means that ratioof the number of particle impacts on a blade to thatof particles entering between two blades. From theabove two equations, it is clear that the cascade col-lection efficiency can be determined by blade charac-teristics and the width of flow field containingparticles possible to collide with a blade is H. Tarabrinet al. [8] assumed a blade as a cylinder and Hwas obtained using two-dimensional potential flowaround a cylinder and derived the cascade collectionefficiency as follows

Ec ¼1þ 0:77

St

� ��1c

s

sin (b b � b1)

sinb1

(4)

where St denotes the Stokes number defined as

St ¼tw

Lc

¼{rpd2

p=(18m)}w

2c sin (bb � b1)(5)

The Stokes number is the ratio of a particle’s stop-ping distance to a characteristic length. Zero Stokesnumber means that all the particles move along withthe flow field and there are no particle impacts onthe cylinder (or a blade). When the Stokes numberincreases, the possibility may increase that a particlefalls out by impact on the cylinder [10]. As shown byequation (4), if the flow velocity at the blade inlet orits angle of attack to the blade increases, the cascadecollection efficiency also increases.

3 A NEW ANALYTICAL MODEL TO PREDICTCASCADE COLLECTION EFFICIENCY

The expression of the cascade collection efficiency,equation (4), derived by Tarabrin et al. [8], has alimitation when applied to the axial compressordue to the assumption of the flow over a cylinder.In this study, a new model for the cascade collectionefficiency is proposed by modelling a blade as a plateinstead of a cylinder. The governing equation of aparticle motion can be described as follows

tdv

dtþ v ¼ u (6)

where t is the relaxation time [9] defined as

t ¼1

18

d2prpCc

m(7)

Cc in the above equation is the slip correction factor,which corrects the change of a small particle’s slip inthe flow field [11]. The slip correction factor must betaken into account when micro scale particles aredealt with and can be expressed as

Cc ¼ 1þ Kn 2:514þ 0:8 exp�0:55

Kn

� �� �(8)

Since particles move in the flow field, the govern-ing equation for the particle motion [equation (6)]must be solved simultaneously with governingequations of the flow field, which requires CFD.Because of the physically complex flow field inthe compressor blade row, any advanced CFD toolcannot exactly capture all important physicalcharacteristics of the flow field. To derive a relativelysimple analytic model, it is assumed, in this study,that there exist infinite number of blades in an axialcompressor row, and the blade shape can be rep-resented by a plate, as shown in Fig. 3. The analysiscan be categorized by two cases according to theflow characteristics between blades, as describedbelow.

3.1 Model 1: constant velocity model

Assuming that the velocity between blades is notvariable, the right hand side of equation (6) becomesconstant. With the x-axis set to be parallel with theblades and the y-axis perpendicular to the x-axis asindicated in Fig. 3, scalar components of equation(6) can be expressed as follows

tdvx

dtþ vx ¼ w (9)

tdvy

dtþ vy ¼ 0 (10)

Fig. 3 Simplified cascade model assuming flat-plates

as compressor blades

Predicting particle deposit by fouling 205

A06304 # IMechE 2005 Proc. IMechE Vol. 219 Part A: J. Power and Energy

where

vx ¼dx

dt, vy ¼

dy

dt(11)

Analytic solutions of the above equations can beexpressed as

x ¼ wt þ t(vxi �w)(1� e�t=t ) (12)

y ¼ vyit(1� e�t=t) (13)

Here, vxi and vyi are x and y components of theparticle velocity at the blade row inlet, respectively.Combining the above two equations, an expressionfor the particle position between the blades isderived as

x ¼ �wt ln 1�y

vyit

� �þ (vxi �w)

y

vyi(14)

When a particle reaches the exit of a blade row, itsposition can be expressed as

x ¼ c þy

tanbb

(15)

From the above two equations, the y value of a par-ticle at the exit of a blade row can be calculated.With this value, the collection efficiency, E, and thecascade collection efficiency, Ec, are expressed as

E ¼H

L¼

y/sinbb

c � sin (bb � b1)/sinb1

(16)

Ec ¼ Ec

s

sin (bb � b1)

sinb1

¼y

s � sinbb

(17)

3.2 Model 2: variable velocity model

The main function of the compressor is to raisepressure by decreasing the flow velocity passingthrough blade rows. To account for the decrease inrelative flow velocity inside the blade rows, thismodel assumes that the inlet and outlet flowvelocities are satisfied with de Haller’s criterionw2=w1 . 0:72 and that the flow velocity along ablade decreases linearly. With these assumptions,the x-component particle motion can be expressedas below

tdvx

dtþ vx ¼ w0 1� 0:28

x � vyit(1� e�t=t)/tanbb

c

� �

(18)

where w0 is the relative flow velocity at the bladerow inlet.

The solutions of the above equation can becategorized into three cases:

Case I – D ¼ c � 1:12tw0 . 0

x ¼0:28tvyif1� exp (� t=t)g þ c tanbb

0:28 tanbb

þ t

ffiffiffiffic

D

r �c

0:28½l2 exp (l1t)� l1 exp (l2t)�

þ vxi �vyi

tanbb

� �½exp (l1t)� exp (l2t)�g (19)

where

l1 ¼ �1

2tþ

1

2t

ffiffiffiffiD

c

r(20)

l2 ¼ �1

2t�

1

2t

ffiffiffiffiD

c

r(21)

Case II – D ¼ c � 1:12tw0 ¼ 0

x¼0:28tvyif1� exp (� t=t)gþ c tanbb

0:28 tanbb

þ exp �t

2t

� ��

c

0:28þ vxi�

vyi

tanbb

�c

0:56t

� �t

� �

(22)

Case III – D ¼ c � 1:12tw0 , 0

x ¼ � exp �t

2t

� �c

0:28cos

1

2t

ffiffiffiffiffiffiffiffiffi�

D

c

rt

!(

� vxi �vyi

tanbb

�c

0:56t

� �sin�1=2t

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið�D=cÞ

pt�

1=2tffiffiffiffiffiffiffiffiffiffiffiffi�D=cp

)

þ0:28tvyif1� exp (� t=t)g þ c tanbb

0:28 tanbb

(23)

With these solutions, the collection efficiency E andthe cascade collection efficiency Ec can be computedby the same expressions as equations (16) and (17).Detailed derivations to obtain equations (19)–(23)are described in the Appendix 2.

4 RESULTS AND DISCUSSION

The feasibility study of the proposed analyticalmodel is conducted with the 12 stage axial compressorproposed by Iwamoto et al. [12], and results arecompared with those of Tarabrin et al. [8, 13]. Effectsof compressor design parameters such as chordlength and solidity on the fouling mechanism

206 T W Song, J L Sohn, T S Kim, J H Kim, and S T Ro

Proc. IMechE Vol. 219 Part A: J. Power and Energy A06304 # IMechE 2005

are examined. The role of particle sizes and theirdistributions on the fouling process are also inves-tigated. The geometric characteristics and designparameters of the compressor blade considered inthe present study are shown in Table 1.

4.1 Particle sizes

Figure 4 shows distributions of the cascade collec-tion efficiency with various particle sizes. Asexpected, the cascade collection efficiency increaseswith growing of particle size. The cascade collectionefficiencies predicted by the cylinder-based model ofTarabrin et al. [8] are larger than those by the plate-based model. This is caused by the increase in par-ticle impacts on the cylindrical geometry, whichhas larger curvature than flat-plates. The reasonwhy the collection efficiencies predicted by model 2are larger than model 1 is due to the effect of increasein inertia force of the particles caused by thedecrease in flow velocity between the two plates.

4.2 Blade chord and solidity

Figure 5 represents the distributions of the cascadecollection efficiencies with various chord lengths

and particle sizes when the solidity is fixed. Here,the reference chord length (Cref) is selected as thechord length of the compressor of Iwamoto et al.[12]. It is shown that the cascade collection efficien-cies decrease with the increase in chord length. Thismay be caused by the decrease in the probability ofparticle impacts on the blade due to the increase intime during which particles move in the flow fieldbetween the blades. Figure 6 shows that the cascadecollection efficiency increases linearly with theincrease in solidity when the chord length is fixed.This is not caused by the increase in particle impactson the blades but is due to the decrease in the numberof inlet particles by closer spacing of the blades whenthe solidity is low. These results correspond to thoseof Tarabrin et al. [8]. Those two figures also showthat the particle size is an important parameter inthis study as predicted by Fig. 4.

Table 1 Specification of blade and velocities of the

compressor of Iwamoto et al. [12]

Chord, c 7 cmMean radius, rm 18.7 cmSolidity, c/s 1.5Flow coefficient at design point, fd 0.547Degree of reaction 0.5Relative outlet angle from previous blade, a 158Stagger angle, bb 40.68Axial velocity 150 m s21

Blade tip speed (first stage) 350 m s21

Fig. 4 Distributions of cascade collection efficiencies

with various particle sizes

Fig. 5 Distribution of the cascade collection efficiency

over chord length and particle size (s ¼ 1.5)

Fig. 6 Distribution of the cascade collection efficiency

over solidity and particle size (c ¼ 7 cm)

Predicting particle deposit by fouling 207

A06304 # IMechE 2005 Proc. IMechE Vol. 219 Part A: J. Power and Energy

4.3 Flow rates

As expressed in equation (4), the cascade collectionefficiency is increased with the decrease in the inletflow angle (b1) or the increase in the relative flowvelocity (w) at blade row inlet. These two parametersare dependent on the magnitude of the flow rate.Figure 7 shows that the cascade collection effi-ciencies decrease with the increase in the flow coef-ficient. Here, the flow coefficient is normalized bythe flow coefficient at the design point of the com-pressor of Iwamoto et al. [12]. The decrease inthe flow coefficient means a reduced flow rate ofthe compressor at constant rotational speed. Theincrease in cascade collection efficiency with areduced flow coefficient is dominated by an increasein particle sizes. From these results, it can be con-cluded that compressor fouling will be severe whenthe inlet air contains relatively large fouling particlesentering the compressor at low velocity.

4.4 Inlet flow angles

The multi-stage axial flow compressor in heavy-dutygas turbines is generally equipped with variable inletguide vanes (VIGV) or variable stator vanes (VSV) fora stable start-up and the enhancement of exhaustheat recovery [14]. The closure of the guide vaneschanges the direction of incoming flow and bringsthe decrease of the inlet flow angle (b1). Figure 8shows the influence of the inlet flow angle on thecascade collection efficiency with a condition of fixedflow rate. The decrease of inlet flow angle enhancesthe possible interaction of incoming particles on theblade surface as shown in Fig. 2 and, therefore,increases the cascade collection efficiency. This resultcan also be clearly understood from equation (4).

4.5 Compressor stages

Assuming that particles entering a cascade are uni-formly distributed and specifications of all stages ofthe axial compressor are the same, when particlesenter into a cascade in proportion to s sinb1, someparticles deposit on the blade surface in proportionto Ec sin (bb � b1). Then, the number of particlescarried away through a stage is proportional tos sinb1� Ec sin (bb � b1).

As described in Figs 4–8, the amount of depositedparticles on the blade surface increases when theparticle size is large. This result shows that most ofthe large particles are deposited on the front stagesand, as a consequence, the number of particlesentering the downstream stages is decreased. Figure 9shows that the number of particles passing throughthe stages is decreased in the downstream stages. Thedecrease in the number of particles in the downstreamstages is due to the deposition of large particles on frontstages. This result is that the fouling is dominantespecially in the front stages of the axial compressor.

Fig. 8 Influence of the inlet flow angles on the cascade

collection efficiency with various particle sizes

Fig. 9 Distribution of the number of particles

captured by consecutive compressor stages for

the variation of the particle size

Fig. 7 Distribution of the cascade collection efficiency

over the flow coefficient for various particle

sizes

208 T W Song, J L Sohn, T S Kim, J H Kim, and S T Ro

Proc. IMechE Vol. 219 Part A: J. Power and Energy A06304 # IMechE 2005

4.6 Characteristics of particle distributions

In the above discussions, it was assumed that theflow entering the axial compressor contains thesame size of particles for each operating condition.However, in a real situation, there exist varioussizes of particles in the atmosphere and, therefore,it is also necessary to consider characteristic distri-butions of particle sizes contained in the inlet air.Since the distribution of particle sizes may changeaccording to the location and time, effects ofarbitrary particle distributions on the overall foulingmechanism are considered in this study.

In most industrial gas turbines, a filtration systemmust be installed in front of the inlet duct to protectthe blades from fouling and erosion by airborne par-ticles in the incoming air. The type of the filter deter-mines the maximum particle diameter. Filters can becategorized such as inertial filters, self-cleaning pulsefilters, high efficiency filters etc. Among these filters,the inertial filter is the simplest and cheapest one,which can remove particles with diameters over20 mm [1]. In the present study, the maximumallowable particle diameter is set as 10 or 20 mmdepending on the filter characteristics.

For the simplicity of the analysis, it is assumed thatthe stability time of the deposition layer, defined asthe time elapsed from the beginning of particle depo-sition on the blade to the instant when the depositionlayer does not grow any more, is the same for eachstage of the axial compressor. And, it is also assumedthat the rate of different particle sizes in each stage isthe same, as shown in Fig. 9. This tendency is pre-served regardless of the distribution of particle sizesof the first stage. The distributions of particle sizesof the first stage used in this study are shown inFig. 10. Here, P(dp) means the mass (mg) of a particlewith its diameter of dp (mm) per unit volume (m3) ofair. Two kinds of maximum particle diameters suchas 10 and 20 mm are chosen as described above.

The non-dimensional collection mass of particlesin each stage is defined as the collection of particlemasses from the smallest size (0.1 mm) up to largestone (for example 20 mm) in the stage normalizedby the particle mass in the first stage. The non-dimensional collection mass of particles in the jthstage, aj, can be expressed as follows

aj ¼

Pi (Ni,j �Ni,jþ1) � P(dp,i)Pi (Ni,1 �Ni,2) � P(dp,i)

(24)

where subscript i denotes ith particle size in eachstage.

Figure 11 shows non-dimensional collectionmasses of particles fallen out in each compressorstage for three different patterns of particledistributions with their maximum particle size of

20 mm. The predictions are compared with meas-ured data of the experimental compressor [13].Differences between measured and predicted dataespecially in downstream stages are due to thereduction in the adhesion force of particles to theblades by the evaporation of moisture containedin the air in the high temperature and pressureenvironment. In case of a dominant amount of

Fig. 10 Distribution patterns of particles adopted in

the present study. (a) Cases A–C; (b) cases

D–F; (c) cases A, D, H, and G

Predicting particle deposit by fouling 209

A06304 # IMechE 2005 Proc. IMechE Vol. 219 Part A: J. Power and Energy

large particles (case A), most of the large particlesadhere to the front stages and, as a result, non-dimensional collection masses of particles in down-stream stages are relatively small. On the otherhand, when the mass of small particles is dominant(case C), non-dimensional collection masses of par-ticles are increased in downstream stages becauseof the small portion of mass of large particles.Figure 12 represents similar results as Fig. 11 withthe maximum size of particles of 10 mm. Figure 13shows that the slope of the particle size distributiondoes not have a strong influence on the pattern ofparticle adhesion on the blades. Instead, comparingwith case D, it can be concluded that the maximumsize of particles, which is governed by the inletfilter, strongly influences the characteristics of par-ticle adhesion to each stage of the axial compressors.

5 CONCLUSIONS

An analytic method has been developed for theanalysis of fouling phenomena in the axial compres-sor of the industrial gas turbine based on a flat-plate-cascade model considering variable velocity betweenblades. Effects of geometric parameters and flowcharacteristics of the axial compressor on foulingmechanism are investigated as results of parametricstudies. The impact of the characteristics of foulingparticles, such as their size and distribution, on thefouling phenomena are also studied. Some impor-tant conclusions disclosed by the present studyare as follows:

(1) Fouling is closely related to the characteristics ofgeometry and flow of the compressor stage.Adhesion of particles to blades, defined as thecascade collection efficiency, is increased withthe decrease in chord length and the increasein solidity. Also, fouling is expedited with thedecrease in the flow rate and the inlet flow angle.

(2) A large particle size increases the cascade collec-tion coefficient. Deposition of large particles infront stages makes fouling in front stages domi-nant. Small particles pass through front stagesand influence downstream stages.

(3) Distribution of particle sizes is an importantparameter to determine the fouling level. Also,the maximum size of particles, which is governedby the inlet filter characteristics, plays an impor-tant role to the fouling characteristics.

ACKNOWLEDGEMENT

This work was supported by the Electric Engineeringand Science Research Institute of the Ministry ofCommerce, Industry and Energy in Korea.

Fig. 12 Effect of patterns of particle size distributions

on the non-dimensional collection mass of

particles with a maximum particle diameter

of 10 mm

Fig. 13 Effect of patterns of particle distributions

on the non-dimensional collection mass of

particles with different maximum particle

sizes

Fig. 11 Effect of patterns of particle distributions

on the non-dimensional collection mass of

particle with maximum particle diameter of

20 mm

210 T W Song, J L Sohn, T S Kim, J H Kim, and S T Ro

Proc. IMechE Vol. 219 Part A: J. Power and Energy A06304 # IMechE 2005

REFERENCES

1 Diakunchak, I. S. Performance deterioration in indus-trial gas turbines. ASME J. Engng Gas Turbines Power,1992, 114, 161–168.

2 Mezheritsky, A. D. and Sudarev, A. V. The mechanismof fouling and the cleaning technique in application toflow parts of the power generation plant compressors.ASME Paper 90-GT-103, 1998.

3 Saravanamuttoo, H. I. H. and Lakshminarasimha, A. N.A preliminary assessment of compressor fouling. ASMEPaper 85-GT-153, 1985.

4 Aker, G. F. and Saravanamuttoo, H. I. H. Predicting gasturbine performance degradation due to compressorfouling using computer simulation techniques. ASMEJ. Engng Gas Turbines Power, 1989, 111, 343–350.

5 Seddigh, F. and Saravanamuttoo, H. I. H. A proposedmethod for assessing the susceptibility of axial com-pressors to fouling. ASME J. Engng Gas TurbinesPower, 1991, 113, 595–601.

6 Massardo, A. F. Simulation of fouled axial multistagecompressors. Proceedings of IMechE Conference onTurbomachinery, 1991, Paper C423/048, 243–252.

7 Slater, S. A., Leeming, A. D., and Young, J. B. Particledeposition from two-dimensional turbulent gas flows.Int. J. Multiphase Flow, 2003, 29, 721–750.

8 Tarabrin, A. P., Schurovsky, V. A., Bodrov, A. I., andStalder, J. P. An analysis of axial compressor foulingand a blade cleaning method. ASME J. Turbomachin.,1998, 120, 256–261.

9 Heinsohn, R. J. and Kabel, R. L. Sources and Control ofAir Pollution. 1999, Prentice Hall, Engelwood Cliffs, NJ.

10 Hinds, W. C. Aerosol Technology: Properties, Behavior,and Measurement of Airborne Particles, 1982 (JohnWiley & Sons, New York).

11 Jenning, S. G. The mean free path. J. Aerosol Sci., 1988,19, 159–166.

12 Iwamoto, T., Ikesawa, K., Kanayama, T., Nagai, K.,Yukinari, A., and Nakagawa, T. Development of ahigh-pressure ratio axial flow compressor. Proceedingsof the 1991 Yokohama International Gas TurbineCongress, 1991, II, pp. 79–86.

13 Tarabrin, A. P., Schurovsky, V. A., Bodrov, A. I., andStalder, J. P. Influence of axial compressor fouling ongas turbine unit performance based on differentschemes and with different initial parameters. ASMEPaper 98-GT-416, 1998.

14 Kim, J. H., Song, T. W., Kim, T. S., and Ro, S. T., Modeldevelopment and simulation of transient behavior ofheavy duty gas turbines. ASME J. Engng Gas TurbinesPower, 2001, 123, pp. 589–594.

APPENDIX 1

Notation

aj dimensionless deposited mass,equation (24)

c chord, mCc slip correction coefficient, equation (8)

Cx axial velocity of fluid, m s21

dp particle diameter, mE collection efficiencyEc cascade collection efficiencyH width of particle groups impacting on the

cylinder, mKn Knudsen numberL radius of cylinder, mLc characteristic length, mN number of entering particlesP(dp) particle distribution function, mg m23

s pitch, mSt Stokes numbert time, su velocity vector of fluid, m s21

U tangential speed, m s21

v velocity vector of particle, m s21

vxi x-component of particle velocity at bladeinlet, m s21

vyi y-component of particle velocity at bladeinlet, m s21

w relative flow velocity in blade passage, m s21

a relative blade outlet angle of previousstage, deg

b1 flow or particle angle, degbb stagger angle, degw flow coefficient (¼Cx/U)m dynamic viscosity, kg m21 s21

rp particle density, kg m23

s solidity (¼c/s)t relaxation time, equation (7)

Subscripts

b bladep particle1 in2 out

APPENDIX 2

Detailed procedure for the solutions of thevariable velocity model

Equation (18) can be arranged in terms of x-component as follows

d2x

dt2þ

1

t

dx

dtþ

0:28w0

ctx ¼

w0

tþ

0:28w0vyi

c tanbb

(1� e�t=t)

(A1)

The procedure for solving linear second-orderordinary differential equation [equation (A1)] hasthree steps as follows.

Predicting particle deposit by fouling 211

A06304 # IMechE 2005 Proc. IMechE Vol. 219 Part A: J. Power and Energy

Step 1

Find the general solution, xh of the homogenous partof equation (A1)

d2x

dt2þ

1

t

dx

dtþ

0:28w0

ctx ¼ 0 (A2)

The general solution of the homogenous problemis classified into three according to the sign ofthe determinant coefficient of the characteristicequation. The characteristic equation and thedeterminant of equation (A2) are obtained as

l2 þ1

tlþ

0:28w0

ct¼ 0 (A3)

D ¼ c � 1:12tw0 (A4)

According the sign of D, the general solution ofequation (A2) can be solved as follows

(1) D . 0 (l ¼ l1, l2 [ R) xh(t) ¼ c1el1t þ c2el2t

(A5a)

(2) D ¼ 0 (l1 ¼ l2 [ R) xh(t) ¼ c1el1t þ c2tel1t

(A5b)

(2) D , 0 (l ¼ a + ib) xh(t) ¼ eat(c1 cos bt

þ c2 sin bt) (A5c)

where c1 and c2 are constants, which can bedetermined by the initial conditions.

Step 2

Find the particular solution of the non-homogenouspart of equation (A1). The particular solution isobtained by using the method of variation para-meters. Let the particular solution be

xp(t) ¼ c1(t)x1(t)þ c2(t)x2(t) (A6)

By inserting equation (A6) into equation (A1), c1(t)and c2(t) can be obtained as

c1(t) ¼

e�t(l1þ1=t)fe�t=tw0(1þ l1t)� 0:28w0vyit=

c tanbb½e�t=t(1þ l1t)� l1t�g

l1(l1 � l2)t(1þ l1t)

(A7a)

c2(t) ¼

e�t(l2þ1=t)fe�t=tw0(1þ l2t)� 0:28w0vyit=

c tanbb½e�t=t(1þ l2t)� l2t�g

l2(l2 � l1)t(1þ l2t)

(A7b)

Then, the particular solution becomes

xp(t) ¼ c1(t)el1t þ c2(t)el2t

¼0:28tvyif1� exp (� t=t)g þ c tanbb

0:28 tanbb

(A8)

Therefore, the general solution of the full non-homogenous problem is

x(t) ¼ xh(t)þ xp(t) ¼ c1x1(t)þ c2x2(t)

þ0:28tvyi{1� exp (� t=t)}þ c tanbb

0:28 tanbb

(A9)

Here, x1(t) and x2(t) can be determined with equation(A5) based upon the properties of the determinant ofthe characteristic equation.

Step 3

Find coefficients c1 and c2 of equation (A9) withinitial conditions; x0(0) ¼ vxi and x(0) ¼ 0. Accordingto the determinant of the characteristic equation,equation (A9) becomes equation (19), (22), or (23).

212 T W Song, J L Sohn, T S Kim, J H Kim, and S T Ro

Proc. IMechE Vol. 219 Part A: J. Power and Energy A06304 # IMechE 2005