vaneless diffuser

THE AMERICAN SOCIETY OF MECHANICAL ENGINEERS345 E. 47th St., New York, N.Y. 10017

The Society shall not be responsible for statements or opinions advanced inpapers or discussion at meetings of the Society or of its Divisions or Sections,or printed in its publications. Discussion is printed only if the paper is pub-lished in an ASME Journal. Papers are available from ASME for 15 monthsafter the meeting.

Printed in U.S.A.

93-GT-101

AERODYNAMIC DESIGN AND ANALYSIS OFVANELESS DIFFUSERS AND RETURN CHANNELS

Ronald H. AungierProduct Development

Elliott CompanyJeannette, Pennsylvania

ABSTRACTProcedures are presented for the aerodynamic design and

performance analysis of vaneless diffusers, crossover bends andreturn channels. The design of the crossover bend and returnchannel is formulated to permit a computerized interactive designsystem, which has provided dramatic improvements in bothquality of designs and engineering productivity. Mean streamlineperformance models are employed to fully support the interactivedesign system. These performance models are qualified againstexperimental data from several centrifugal compressor stage tests.A recent development program which used these procedures isreviewed to demonstrate their benefits.

NOMENCLATUREA - areaAR - area ratioa - chordwise location of the point of maximum camberb - passage hub-to-shroud widthC - velocityc - vane chord lengthcr - skin friction coefficientC o, - meridional velocity componentC, - tangential velocity componentD - passage divergence parameter, equations (5) and (7)d - minimum metal thickness of return channel, figure 8dH - hydraulic diameterE - diffusion efficiency, equations (9)h - gas enthalpylc - curvature loss term, equation (13)ID - diffusion loss term, equations (10) and (12)i - vane incidence angleK - blade loading parametersKB - area blockage factor (fraction unblocked)

L - passage lengthLC - total pressure loss coefficient (based on component

inlet velocity pressure)m - meridional coordinateM o - rotational Mach number (impeller tip speed divided by the

stage inlet total sound speed)n, - dimensionless specific speed (Babe, 1981)P - pressureR - mean streamline radius of curvature✓ - radial coordinatet - vane thickness✓ - gas specific volumew - mass flowx - distance along vane chord linez - number of vanesa - flow angle with respect to tangent• - vane inlet flow angle at minimum loss condition

- blade angle with respect to tangentO" - vane deviation angle at minimum loss condition0 - vane camber angleBo - equivalent divergence angle, equations (5) and (43).07 - vane trailing edge angle with axial directionq5 - angle of C o, relative to the axis of rotationa - vane equivalent solidity

Subscriptsav - average value or value at mid-chordc - crossover bend parameterh - parameter on the hub contourm - a maximum conditiono - exit turn bend parameters - parameter on the shroud contourt - total thermodynamic condition1 - passage entrance parameter

Presented at the International Gas Turbine and Aeroengine Congress and ExpositionCincinnati, Ohio — May 24-27, 1993

Copyright © 1993 by ASME

Downloaded From: http://proceedings.asmedigitalcollection.asme.org/ on 09/22/2015 Terms of Use: http://www.asme.org/about-asme/terms-of-use

h, = h + C22

4 - diffuser exit parameter5 - crossover exit parameter6 - return channel vane inlet parameter7 - return channel vane exit parameter8 - return channel passage discharge parameter

INTRODUCTIONThe aerodynamic performance of centrifugal compressor stages

is strongly influenced by the internal flow and losses in thestationary components. The return system stands out asundoubtedly the least understood component of the centrifugalcompressor stage. The presence of the upstream 180° annularcrossover bend, and the interaction of the flow through the bendwith the downstream vane row, present a formidable fluiddynamics problem. The literature offers little informationrelative to specific design procedures or performance predictionmodels for this component. Most published experimental andtheoretical investigations are directed toward developing a betterunderstanding of these complex flows (e.g., Davis, 1976, Fister,et. al., 1982, Japikse and Osborne, 1982, Nykorowytsch, 1983,Aungier, 1988a). Return system design has largely consisted ofapplying basic engineering judgement to control diffusion rates,passage curvatures and other design parameters, based primarilyupon empirical categorization of performance from pastdevelopment activity (e.g., Hohlweg, 1987).

The present paper presents aerodynamic performance analysistechniques for general vaneless annular passages and returnchannels, including comparison with experimental data. Specificaerodynamic design procedures for vaneless diffusers and returnsystems are presented.

VANELESS PASSAGE PERFORMANCEExperience gained while predicting vaneless diffuser and

crossover bend losses with the author's three-dimensional analysis(Aungier, 1988a) has provided insight into loss mechanisms topermit formulating an accurate mean-streamline performanceanalysis. The governing equations are

21crbCif / v = w (1)

d(rC )bC,,, dm" -

C -CinscP

r m dmdc cICC„, dip

b dm

Except for the last two terms in equation (3), this set of equationsis conventional for mean-streamline analysis (e.g., Johnston andDean, 1966). The additional terms address loss contributions dueto flow diffusion and passage curvature. Flow diffusion losses

are modeled by a classical diffuser analogy. The data of Reneauet. al. (1967) shows the low loss regime can be identified by thedivergence parameter

D = MAR - 1) 1 L = 2tane, (5)

where diffusion losses are low for values of D less thanDm = 0.4(b 1 / L)°35 (6)

Equation (6) is an empirical fit of the data in figure (8b) ofReneau et. al. (1967). The analogy for the vaneless annularpassage used is

b dC (7)C dm

D,,, = 0.4(b 1 1 L)°35sina (8)

The flow angle term in equation (8) is an empirical factor derivedfrom comparisons of predicted and measured loss data from over35 compressor stage tests. Based on this same comparison, anempirical diffusion efficiency model was formulated as

E=1;DO

E = 1 - 0.2(D / D,,)2 ; 0 DDm (9)

E = 1 D ; D

The diffusion term is given by

dip v dC= -2(P, - P)(1 - E)—C dm— (10)

dm

In addition to this streamwise diffusion loss term, an excessivemeridional gradient of the passage area can cause higher losses.Again, a diffuser analogy is used to check for this situation ateach computing station. The maximum, stall-free, local area isestimated by

(rb)m = (rb) 1 [1 + 0.16m / b 1]

which is equivalent to a diffuser divergence angle, 28 c , of 9°. Ifthe local area exceeds this value, a second estimate of thediffusion term is generated

Ip s.b .

= 0.65v(P, - P)(1r

r )

If this value exceeds the local value obtained by integratingequation (3), it replaces that lower value. The passage curvatureloss term is given by

/c. = v(P, - P)C,„ 1 (13RC)

(13)

Equation (13) was developed empirically from comparisons ofpredictions with test data for 35 different vaneless diffuser/returnsystem combinations. This term has negligible effect on vanelessdiffuser performance, but is always significant (and sometimesdominant) for crossover bends. Once the accuracy of the analysiswas established for vaneless diffusers, equation (13) wasdeveloped to extend the analysis to crossover bends. Its validityis further supported by the successful use of this same analysis in

vdpdm

(12)

2


overall stage performance prediction for other curved passagessuch as the stage inlet passage and the exit turn from returnchannels.

The area blockage factor and the skin friction coefficient arecomputed using a simple boundary layer growth model, based ona 1/7th power law for the boundary layer velocity profiles. Theanalysis requires specification of an inlet boundary layer thickness(typically supplied by an impeller performance analysis).Boundary layer growth is estimated from the change in angularmomentum predicted by integration of the governing equations.The boundary layer thickness is used to compute the blockagefactor. Skin friction is computed from an empirical correlationof classical fully-developed pipe flow friction factors, includinglaminar, transition and turbulent flow as well as surfaceroughness effects (e.g., Schlichting, 1979). Pipe flow frictionfactors are also valid for boundary layers if the pipe diameter isreplaced by twice the boundary layer thickness. Hence, thethickness of the two boundary layers is used in place of thepassage width to compute the friction factors. This leads tohigher values for cases where the boundary layers do not fill thepassage, which is quite significant for high specific speed stages.

This performance analysis is used for vaneless diffusers,crossover bends and other vaneless annular passages. Its validityhas been established for a broad range of stage specific speedsand operating conditions, including cases where D is far in excessof D„,. Use of velocity pressure (P, - P) rather then velocity headas a loss coefficient base, insures applicability to a broad rangeof Mach number levels. The analysis has been qualified againsttest data for Mach number levels up to 0.85 with no observableinfluence of Mach number level on prediction accuracy. But, itshould be noted that effects due to differences in inlet flow profiledistortion and turbulence intensity levels are not specificallymodeled by this analysis.

RETURN CHANNEL PERFORMANCE-The return channel vane passage performance shares many

common features with the author's vaned diffuser analysis(Aungier, 1990), both of which are adaptations of a meanstreamline impeller flow analysis. A significant difference is thetreatment of incidence losses, which are strongly influenced byflow distortion imposed by the upstream crossover bend. Twoestimates of aerodynamic area blockage factorsentrance are made, and the smaller value is used.

1-1 + g (12R)

KB = (r6b6)

Equation (14) is a simple inviscid flow estimate using the averageradius of curvature, R, of the crossover bend. Equation (15)estimates blockage due to stall, based upon equation (11). TheMinimum incidence loss is assumed to occur when the flow inlet

angle is equal to the flow angle in the vane throat. Since thereturn channel analysis does not use area blockage directly, theminimum loss incidence angle is corrected to account for theentrance blockage to yield

tana* = KBtan[sin-1(A, / A6)] (16)

where A, is the vane throat area. The incidence loss coefficientis given by

LCin, = 0.8[1 - Cii6 / (C6sina")]2 (17)

Skin friction loss is computed byLC D = (4Lcf cl,)(Cm, C6)2

(18)

where C., is the average of the discharge velocity and either theinlet or throat velocity (whichever is larger), d H is the average ofthe throat and discharge hydraulic diameters and c f is computedfrom the correlation discussed previously. The averageblade-to-blade velocity difference is computed from the vanecirculation

AC = 2n(r6C.6 - r7C.7) I (a)

(19)

and the blade loading loss coefficient is given byLCB, = [AC / (6C6)]2

(20)

The maximum vane surface velocity is estimated assuming itoccurs at mid-passage for a mid-loaded vane

= 0.5(C6 + C7) + AC (21)

and C m > C6 is required to include the more common case wherethe return channel vane maximum surface velocity is the inletvalue. When Cm > 2C7 , it is assumed that the boundary layerwill separate at a velocity, C„, = Cm/2. Otherwise, the velocitywhen the boundary layer separates from the blade is set to C7.

The velocity after wake mixing is estimated from

C o = V(C7A7 / A0)2 + C!7 (22)

where A7 includes the vane metal blockage, while A, does not.The wake mixing loss coefficient is given by

= [(Csp - Co) / C6]2 (23)

The loss coefficient due to the exit turn into the eye of the nextstage is given by

LCo = / C6)2

(24)

which is derived from equation (13) for a 90 ° turn, assuming aconstant velocity. Our analysis also includes the choking loss ofAungier, 1990. But, since the author has never seen choke in areturn channel, it will not be repeated here. The flow dischargeangle is computed from the transformed axial flow compressordeviation angle model reported in Aungier (1990). The positionof the point of maximum camber, the vane solidity and camberangle are estimated using the vane camberline vane angle atmid-chord, O.,.

at the vane

(14)

(15)

3


a 2 (13, - Pe)= Li

3(P7 - Po)]

z(r6 / r7 - 1)a -

2nsink,

° = P7 - P6

parameter defined in equation (43) and evaluated at the crossover(25) discharge. The return systems considered represent both fairly

modern good design practice and obsolete designs containing veryundesirable features. The multiple integrations required, andtheir use in the ratios of small differences to define the loss

(26) coefficients, result in considerable data scatter. However, theagreement between prediction and test data is consideredsatisfactory to support both design and application activity.

(27)

and the reference deviation angle is based on Howell (1947)6[0.92(a / c)2 + 0.02(90 - p 6)

8* (28)%FT - 0.026

Off-design incidence effects on flow deviation angle are includedby an empirical correlation of figure 177 of Johnsen and Bullock,1965.

di = exp[((1.5 - p, / 60)2 - 3.3)0]

(29 )

a77 = P7 8 - da (P4 - ar„)di _

The analysis is a simple iteration procedure, computing thelosses and fluid turning while balancing mass at the discharge,until convergence is achieved.

TABLE I: COMPRESSOR STAGE DESIGN DATA

STAGE n. M. b./r4 b./b4 b./R 20cs

A #1 0.776 0.70 0.081 1.448 0.922 5.20

A #2 0.776 0.70 0.072 1.767 1.995 15.41

B 0.854 0.88 0.058 1.444 1.453 8.09

C 0.272 0.50 0.011 3.064 0.717 8.74

0 0.699 0.70 0.049 2.338 1.747 17.65

E 1.110 0.70 0.097 1.294 0.913 3.78

COMPARISON WITH EXPERIMENTAL DATA

Validation of the performance models was accomplished forover 35 individual centrifugal compressor stage developmenttests. These tests include traverse probes at the impeller tip,vaneless diffuser exit and stage exit. These data are integratedand mass averaged to permit calculation of the component losscoefficients (where the crossover bend and the return channelvane section are treated as a single component). Figures 1 to 7are representative of the correlation between prediction and testdata for vaneless diffusers and return systems. As seen fromTable I, these cover a range of stage specific speeds androtational Mach numbers. The last column in Table I is a design

RADIAL VANELESS DIFFUSER DESIGNThe radial vaneless diffuser passage construction is constant

width, parallel wall style with a width adjustment in the first15-20% of the passage to blend to the impeller tip width.Specifications required are the inlet and discharge radii andwidths and the blend radius at the start of the constant widthsection. A second-order polynomial variation of width with

0 -

rnHo -

I-4C.)

44 •4100

0.-••

0 a

OO

O TEST DATA 0

0 - PREDICTION

010 15 20 25 30 35 40DIFFUSER EXIT FLOW ANGLE - degrees

FIGURE 1: VANELESS DIFFUSER PERFORMANCE - COMPRESSOR 0A41"

0 TEST DATA0 - PREDICTION

010 15 20 25 30 35 40DIFFUSER EXIT FLOW ANGLE - degrees

FIGURE 2: VANELESS DIFFUSER PERFORMANCE FOR COMPRESSOR "B"

4


0 0

•

O

- PREDICTION0 0 TEST DATA

050 60 70 80 90 100 110 120 130 140% DESIGN FLOW

FIGURE 3: VANELESS DIFFUSER PERFORMANCE FOR COMPRESSOR "C"

O TEST DATA- PREDICTION

0-12.0 -8.0 -4.0 0.0 4.0 8.0 12.0RETURN CHANNEL VANE INCIDENCE - degrees

FIGURE 5: RETURN SYSTEM PERFORMANCE FOR COMPRESSOR "B"


m.::;-6.0 0.0 4.0 8.0 12.0 16.0

RETURN CHANNEL VANE INCIDENCE - degreesFIGURE 6: RETURN SYSTEM PERFORMANCE FOR COMPRESSOR "A42"

O"

H 0zUU••••

ra,rW •47O'0

O r•W •0

0 TEST DATAN - PREDICTIONo-12.0 -6.0 -2.0 2.0 6.0 10.0

RETURN CHANNEL VANE INCIDENCE - degrees

FIGURE 4: RETURN SYSTEM PERFORMANCE FOR COMPRESSOR "A#1"

000

radius is used at the entrance to match the two specified widthsand the zero gradient condition at the blend radius. Normalpractice is to use a radial hub contour with all width adjustmentimposed on the shroud contour. This is believed to be moreeffective in improving the impeller tip flow profiles, which oftentend to deteriorate toward the shroud wall.

Design optimization involves selecting the discharge width andthe blend radius with the other specifications viewed as designconstraints. Loss curves similar to figures 1 and 2 are used toevaluate the alternate choices of these parameters. An interactivedesign program could be used for this purpose. But, direct useof a mean streamline centrifugal compressor performanceprogram is more convenient, since impeller discharge flowpredictions are provided directly.

N

0

zo OI-1

L .

▪

V)•

Cal0U

0 O0v

•0


0 -8.0 -2.0 2.0 6.0 10.0 14.0 18.0

RETURN CHANNEL VANE INCIDENCE- degrees

FIGURE 7: RETURN SYSTEM PERFORMANCE FOR COMPRESSOR "D"

5


Axial DistanceFIGURE 8: RETURN SYSTEM GEOMETRY

(33)

RETURN SYSTEM DESIGNFigure 8 illustrates the construction of the return system gas

path geometry. The 180° crossover bend is constructed with acircular-arc hub contour and an elliptical-arc shroud contour.The 90° exit turn uses a similar construction, but with the arctypes reversed for the hub and shroud contours. Straight linecontours are used for the return channel vane section, blended tothe crossover and exit turn contours by circular arcs with arcradii equal to the local bend contour elliptical-arc axial semi-axisor circular-arc radius. The vane camberline used is an adaptationof the author's vaned diffuser camberline (Aungier, 1988b).Leading and trailing edge blade loading parameters, 1( 6 and K7,are specified to define the vane camberline by

y r r6(30)

cote = - A - By - 2Cy 2 - 3Dy 3

0 = Aln(y) + B(y - 1) + C(y 2 - 1) + D(y 3

Y = r7, / r6

(cot0 6 - cot0 7)(k6 + - 2)D - (34)

3(Y - 1)3

(cotp 6 - coti3 7)(K7 - K6) 9C - + + I) (35)

4(Y - 1)2 4

B = K6(cot13 6 - cot(3 7) / (Y - 1) - 4C - 9D (36)

A = - cotp 6 - B - 2C - 3D (37)

These equations apply to radii greater than the vane dischargeshroud radius, r7s . Below r78 , a constant blade angle camberlineis used. The vane thickness distribution used is given as afunction of the distance, x, along the chord line between I -, andr7s . Below r7s, the vane thickness is held constant. Thethickness distribution is given by

t = to + - to)y e

(38)

to = rb + (t7 - t6)x / c (39)

y = x / xm ; for x >

(40)

y = (1 - x) 1 (1 - ; forx> x1, (41)

- + 0.05]e = [0.95(1 + /x1 (42)

The blade maximum thickness, t,,, and its location, x„,/c, arespecified, as are t 6 and t7 . Figure 9 shows a typical returnchannel vane designed with these equations.

With reference to figure 8, the design constraints imposed onthe return system design are r„„ b 4 , and the diffuser exit flowangle (from the vaned or vaneless diffuser design). To completedefinition of the crossover contours, the designer must specify thefollowing

1) either b 6 or the crossover exit flow angle2) either r4 or (A„,, - A 4)/(A5 - A 4)3) either R, or the average b/R over the bend

6


0 TEST DATA— PREDICTION

434,

sr6:7,414,

opppuse

0 20 40 60 80 100 120 140

t DESIGN FLOW

FIGURE 10: COMPONENT PERFORMANCE FOR COMPRESSOR "E"

0

E-■Z41ID'-

U,

6.0

0UM':M 0 °A

Cl "

0

0

where A., is the passage area mid-way through the bend and b/Ris the ratio of passage width to mean streamline radius ofcurvature. The alternate choices provide direct control over vaneincidence, the passage area distribution and the distribution ofb/R. Their availability greatly accelerates the design process.

With reference to figure 8, the designer specifies the returnchannel gas path contours by supplying r 6 , r8h, rss , bs/R0 , d, andthe area ratio across the 90° exit turn. The parameter, d, isuseful since it controls the minimum metal thickness in a castreturn channel. This parameter sets the minimum axial lengthachievable with the designer's other specifications. Design of thevane requires supplying the number of vanes, the vane trailingedge angle (0, of figure 8), t6, t7, t,,,, K6, K7, X„,/e, 136 and 137 .

Note that the entire gas path displayed on figure 8 is defined byonly 12 parameters, which provide direct control over the keyarea and curvature distributions. The vane requires 10 moreparameters which directly control the vane passage areadistribution and blade loading style. The vane and all contoursare defined analytically. This reduces the problem to a form wellsuited to a computerized interactive design system. Our designsystem allows the user to interactively modify the designparameters while supplying the following support functions ondemand

1) Screen displays of gas path geometry (figures 8 & 9).2) Screen displays of graphical and tabular summaries of

geometry and key area and curvature distributions.3) Aerodynamic performance analysis.4) Linearized blade-to-blade flow analysis Aungier (1988b).

Results to date from this design system have been impressive.Return channel design activity has been reduced from severaldays to a few hours. Figure 10 shows results from a recent highspecific speed stage development program. The present methods

were used to design the stationary components for this stage.Return system losses achieved are about 15% lower than ourstandard designs. This is equivalent to a gain in overall stageefficiency of over 1 percentage point. Within the usual test datascatter, both the vaneless diffuser and the return systemperformance are in good agreement with prediction.

DESIGN RECOMMENDATIONSVaneless diffuser design optimization normally leads to diffuser

discharge flow angles in the range of 30° to 35° , wherecomponent loss coefficients approach a minimum and a goodstable operating range can be expected (e.g., see figures 1 and 2).But, each choice of blend radius and discharge width produces adifferent loss curve. Only an investigation of alternatives canidentify the choices best suited to the design objectives. This isvery significant for low specific speed stages where loss curvesfor different passage widths can be dramatically different.

Crossover bend and exit turn design should control the ratio ofpassage width to mean streamline radius of curvature, b/R.Local values of b/R should be less than 1. An average over thebend less than 0.8 is preferred. Based on experimental loss datafor numerous return systems, the passage equivalent divergenceangles, 20,, should not exceed 9° , where

20c = 2tan - l [k(A / A4 - 1) / (m - m4) /2] (43)

Ideal diffusers show increased losses for 20c greater than about11 ° , so this lower value for a bend is not unexpected.

Return channel vane design should typically feature front loadedvanes to minimize discharge flow deviation. Since inlet Machnumbers are normally moderate, this loading style is a reasonablechoice. The vane loading parameters, K6 and K7 , define the localradial gradient of the cotangent of the blade angle as a fraction ofthe overall (average) gradient. This provides direct control of theblade loading style (Aungier, 19886). Values of 10 6 = 1.6 andK, = 0.4 have been found to be reasonable choices. The vanemaximum thickness, and its location along the chord line shouldbe used to control the vane passage area distribution to minimizelocal mean velocity gradients.

CONCLUSIONSMean streamline aerodynamic performance models have been

presented for vaneless annular passages and return channels.Predicted loss coefficients for vaneless diffusers and returnsystems show excellent agreement with experimental data. Theseperformance models are well suited to interactive aerodynamicdesign activity and can be incorporated into any mean linecentrifugal compressor performance analysis.

A systematic interactive design system for return system designhas been presented. It has dramatically reduced the engineeringtime required for this design activity. Test results from a recentreturn system design accomplished with this design system show

7


that the improved return system performance predicted by thisaerodynamic design and performance analysis system was actuallyachieved.

REFERENCESAungier, R.H., 1988a, "A Performance Analysis For The

Vaneless Components Of Centrifugal Compressors", Flows InNon-Rotating Turbomachinery Components, ASME FED-Vol 69,pp 35-43.

Aungier, R.H., 1988b, "A Systematic Procedure For TheAerodynamic Design of Vaned Diffusers", Flows In Non-RotatingTurbomachinery Components, ASME FED-Vol 69, pp 27-34.

Aungier, R.H., 1990, "Aerodynamic Performance Analysis OfVaned Diffusers", Fluid Machinery Components, ASMEFED-Vol 101, pp 27-44.

Balje, 0.E., 1981, Turbomachines, Wiley, New York,pp 34-37.

Davis, W.R., 1976, "Three-Dimensional Boundary LayerComputation on the Stationary End-Walls of CentrifugalTurbomachinery" Trans. ASME, J. of Fluids Eng., pp. 431-442.

Fister, W., Zahn, G. and Tasche, 1982, "Theoretical andExperimental Investigations About Vaneless Return Channels ofMulti-Stage Radial Flow Turbomachines", ASME Paper No.82-GT-209.

Hohlweg, W.C., 1987, "Correlation and Application ofCentrifugal Compressor Return System Losses" , Fluid Machineryfor the Petrochemical and Related Industries, Proceedings of theIMechE, pp 97-103.

Howell, A.R., 1947, "Development of the British Gas TurbineUnit", Lecture: Fluid Dynamics of Axial Compressors, ASMEReprint.

Japikse, D. and Osborne, C., 1982, Vaneless Drser, ReturnBend and Return Channel Investigation, Creare Inc., TN 346(Proprietary).

Johnsen, I.A. and Bullock, R.O., editors, 1965, Aerodynamic

Design of Axial Flow Compressors, NASA SP-36.Johnston, J.P. and Dean, R.C., 1966, "Losses in Vaneless

Diffusers of Centrifugal Compressors and Pumps", Trans. ASME,J. of Eng. for Power, pp. 49-62.

Nykorowytsch, P. , editor, 1983, Return Passages of Multi-StageTurbomachinery, ASME FED-Vol 3.

Reneau, L., Johnston, J. and Kline, S., 1967, "Performanceand Design of Straight Two-Dimensional Diffusers", Trans.ASME, J. of Basic Eng. , pp. 141-150.

Schlichting, H., 1979, Boundary Layer Theory, SeventhEdition, McGraw-Hill, New York, Chapter 20.

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