415

THERMODYNAMICS AND FLUID MECHANICS GROUP FLUID MACHINERY GROUP

SLIP FACTOR OF A CENTRIFUGAL COMPRESSOR AND ITS VARIATION WITH FLOW RATE

A. Whitfield, BSc, MSc, PhD (Graduate)*

To predict the complete performance map of turbocharger centrifugal compressors, it has been found essential to include a procedure for estimating the slip factor at off-design flow rates. The many correlations available for the slip factor only concern themselves with design-point operation and consequently only provide a single value for the slip factor. The approach presented here attempts, within the confines of a one-dimensional pro- cedure, to recognize the changing flow pattern at the impeller discharge in order to predict the variation of slip factor with flow rate. The procedure is based upon the well-established jet-wake flow model of Dean and the

overall slip factor calculated from the jet slip factor. Experimental and theoretical slip factors are presented for three impellers with 34, 30 and 12 radial blades, respectively. The complete performance map of the 12-bladed turbocharger compressor and the effect of using

a constant and varying slip factor on the predicted performance are presented.

1 INTRODUCTION THE SLIP FACTOR of the centrifugal compressor has re- ceived extremely detailed attention from numerous in- vestigators over many years. Wiesner (I)t reviewed the various methods that have been proposed for calculating slip factors and included the work of more than 20 investi- gators. This work was, however, restricted to the design point only, as most designers have confined themselves to design-point performance prediction. In more recent years, detailed attention has been given to predicting the compressor performance for both design-point and off- design operation (2) (3). This work has been aimed at matching turbomachinery components to internal com- bustion engines (4). This demands an accurate prediction of the compressor pressure ratio and efficiency for the complete operating range of the compressor.

The Euler turbomachinery equation leads to the ex- pression

Y K Y - 1)

p = [ 1 + 7 (:;” 17iuzc,,- U 1 C 4

for the compressor stagnation pressure ratio. Including the slip factor defined as

leads to

for the case where there is no inlet prewhirl.

This paper is intended for presentation at an Ordinary Meeting of the Thermodynamics and Fluid Mechanics Group and the Fluid Machinery Group in London on 30th October 1974. The MS. was received at the Institution on 13th October 1973 and accepted for publication on 18th June 1974. 33

* School of Engineering, University of Bath, Cloverton Down, Bath BA2 7AY.

t References are given in the Appendix.

@ IMechE 1974

The pressure ratio for any given speed is clearly a function of the efficiency and slip factor. The efficiency is a measure of the entropy gain attributable to the losses occurring within the stage. The treatment used to repre- sent these losses was presented in (2) and (3). The variation of the slip factor with the compressor flow rate has been found to be an important parameter in the performance prediction procedure for turbocharger compressors.

Stahler (5) presented the variation of slip factor with flow rate for a 19-bladed centrifugal compressor and con- cluded that the experimental results could be approxi- mately predicted by assuming that the relative leaving angle Fz of the impeller was constant for a given impeller, i.e. was independent of the gas flow rate and the rotational speed.

Sakai (6) also presented the change of slip factor with flow rate. By studying three different impellers with 17, 10 and 7 blades, Sakai found that the rate of change of the slip factor with the flow rate increased as the number of impeller blades was decreased. However, he did not in- clude this variation of slip factor with flow rate into his efforts to provide a theoretical correlation for the slip factor.

The published information on slip factors does not in- clude any attempt to consider the real fluid conditions that exist at the exit from the impeller. Indeed, the work of Wislicenus (7), which provided for a velocity distribu- tion in both the hub-shroud and the blade-to-blade plane, has largely been ignored as its inclusion generally leads to a poorer correlation between experimental and theoretical slip factors.

Dean (8) (9) has presented details of the flow at the im- peller exit. He has pointed out that the design compro- mise between the impeller and diffuser inevitably leads to an area ratio across the impeller which gives a diffusion rate that cannot be sustained. Consequently, most centri- fugal compressors operate with the impeller flow separated.

Proc lnstn Mech Engrs Vol 188 32/74

416 A. WHITFIELD

The model suggested by Dean then considers the flow pattern after the point of separation to be approximated by a high-velocity jet, which carries the bulk of the through-flow in a potential core, and a low-velocity wake. At the impeller tip, these jets and wakes rapidly mix giv- ing rise to the now well-established mixing loss (10). Dean gives a relative velocity diffusion ratio Wl,/ W, of 1.8 as a predicted ultimate that ought to be achieved with- out separation and considers that present state-of-the-art impellers achieve diffusion ratios of 1.3-1.4.

For the impellers presented in this paper, relative velo- city diffusion ratios in excess of 2.0 are attempted and consequently separation of the flow into a jet and wake is considered to occur. The flow through the jet is then considered to be similar to the potential flow theory of Stanitz (11) and consequently his derived slip factor can only be applied to the through flow jet.

Considering the wide range of pumps and compressors presented by Wiesner (I), some probably not separated at all and the others all with differing degrees of fluid separa- tion even at the design point, it can be considered somewhat remarkable that he was able to achieve any kind of correla- tion for the experimental data.

1.1 A

C c m D d h K ri2

P R T U W z Y 17 0 Y

P

U

P

V

Notation Flow area. Acoustic velocity. Absolute particle velocity. Torque coefficient. Diffusion ratio. Diameter. Impeller blade height. Non-uniform velocity distribution parameter. Mass flow rate. Pressure ratio. Radius and gas constant. Temperature (absolute). Impeller peripheral speed. Relative velocity in impeller. Number of impeller blades. Relative flow angle from meridional direction. Ratio of specific heats. Isentropic stagnation efficiency. Jet area angle at impeller tip (see Fig. 36). Slip factor. Kinematic viscosity. Density.

Subscripts J Jet. m Meridional direction. S Impeller inlet shroud. T Impeller tip. W Wake. 6 Tangential direction. 0

tion subscript). 0 Upstream of compressor. 1 Compressor impeller inlet. 2 Compressor impeller exit.

Stagnation state (with appropriate subsidiary loca-

2 EXPERIMENTAL SLIP FACTOR The experimental slip factors presented in Figs 1 and 2

Experimental shp factor -x--*--r

0 98 Jet slip foctor

------L- O 96)

Constant Stanitz i 0 9 4 b - 1 s l i p foctor L

I i ~~

090i 0 88 a

14 15 I6 17 16 19 20 21 7.2 Mass f low parameler. m G / P o ,

k q / s f l / b a r 0 96

i O 9 4 n Constont Stanitz s l i p f ac to r I

a

10 I I I2 ' I3 14 I S 16 17 16 Pass f l o w ~~ pararreter rn</e, ~~

kq/svfC/bor

a Compressor A. b Compressor B.

Fig. 1. Experimental slip factors

were calculated from the experimentally-measured stag- nation temperature rise across the impeller.

The Euler turbomachinery equation gives

With no prewhirl, C,, = 0 and with the slip factor de- fined in the conventional manner for radial compressors, Y = C,,/~2

U2"

To 1 a01 l + ( y - - 1 ) 7 p . . . (2) - _

The measured experimental temperature rise, however, exceeds that given by the Euler equation by an amount attributable to disc friction.

The contribution to the temperature rise due to disc friction was obtained using the correlation given by Daily and Nece (12)

where Cm is a torque coefficient given by 1/10

c, = 3'7( i ) Re1',

for Re < 3 x lo5

1/10

,.lo,( &) for Re > 3 x lo5

Rel15 and c, =

where s is the gap between the rotating disc and the sta- tionary casing and Re is the Reynolds number given by

Proc lnstn Mech Engrs Vol 188 32/74 0 IMechE 1974

0 9 0 -

088-

086- L

c 0 " .E 0 8 4 - a - - In

0 8 2 r

0 8 0 -

SLIP FACTOR OF A CENTRIFUGAL COMPRESSOR AND

a 0781 L Neqative

incidence

1 2 3 4 5 6 7 8 9 0 L 0 76 - _ I 1 ---.- - -

ITS VARIATION WITH FLOW RATE 417

0 9 * r I

0 9 0 t

Constant Stoni tz sl ip foctor

\ !

0 86 -x--x- Experimntol d i p factoi

\ -Jet slip foctor 0 86 ~

0781 C I

Mass f low parameter, m a / P o , kq/sfl/bar

a 50 000 revlmin. b 60 000 revlmin. c 70 000 revlmin.

Fig. 2. Experimental slip factors for compressor C

By subtracting the temperature rise due to disc friction from that obtained experimentally, the temperature rise attributable to the Euler work, To,-Tol, can be found. By substituting this into equation (2), the experimental slip factor can be found. The magnitude of the tempera- ture rise attributable to disc friction cannot be precisely assessed and using alternative correlations will yield dif- ferent slip factor magnitudes. For example, using the correlation given by Stahler (5) yields a slip factor of 0-8277 compared to the value of 0.8315 presented here. Assuming that there is no disc friction, a value of 0.8485 is obtained. The experimental magnitude of the slip factor clearly contains a degree of uncertainty.

3 THEORETICAL SLIP FACTOR

3.1 Calculating the through-flow jet The calculation of the impeller inlet conditions was de- scribed in detail in (2) (3). To compute the impeller exit conditions, the overall diffusion ratio D was specified (see section 5). The relative velocity can then be found from

With an assumed theoretical slip factor, generally cal- culated from the Stanitz correlation p = 1-0-63n/Z the tangential component of velocity C,, can be found. The exit velocity triangle in Fig. 3a is then completely defined and all velocity magnitudes and directions can be readily found.

The stagnation pressure ratio for the impeller is given by

The impeller stagnation efficiency 77 was calculated by considering the impeller losses as given in (2) and (3). Equation (3) then gives the impeller exit stagnation pres- sure and equation (2) the stagnation temperature. With the known absolute fluid velocity C,, the exit static conditions can be readily found. The jet flow area needed to pass the required mass flow rate was then found from the con- tinuity equation

A . = - m CrnaPz

and the jet angle 0, Fig. 3b, calculated from WI s w2 = 7

Q IMechE 1974 Proc lnstn Mech Engrs VoI 188 32/74

418 A. WHITFIELD

Without slip Assume diffusion ratio

assuming flow completely fills the exit

/-’- ---- Rotation

a Velocity triangle. b Jet wake.

Fig. 3. Impeller exit

The Stanitz correlation for the slip factor was then applied to the jet and the calculation repeated with the jet slip factor replacing the initially-assumed slip factor. If it were found that the area of the jet A , exceeded the total flow area available then the calculation was repeated assuming that the flow completely filled the passage, with a suitable allowance for boundary layer thickness, and the diffusion ratio D calculated. The calculation procedure is illustrated by the flow chart, Fig. 4.

This procedure then provided a means of calculating the magnitude of the through-flow jet to which the Stanitz correlation for slip factor can be applied.

3.2 Calculating the jet slip factor Equation (4) enabled the size of the jet to be calculated This is necessary to determine the jet slip factor. The jet flow, which is considered to carry the bulk of the through- flow in a potential core, should be amenable to analysis by the potential flow theory of Stanitz (11).

The Stanitz correlation for slip factor of a radial flow compressor

applies when the flow is assumed to fully fill the impeller passage. When applied to the jet of Fig. 3b, which only partially fills the flow passage, the Stanitz correlation reduces to

0 2

p+,h = 1-0.63- . . . . (6)

When applied to the jet width, the Stanitz procedure, however, yields a higher slip factor to that obtained experimentally.

Wislicenus (7) has shown that including a non-uniform velocity distribution in the through-flow jet leads to a

Proc lnstn Mech Engrs Vol 188 32/74

Czlculate impeller exit velocity triangle

and efficiency

I Calculate exit density and jet flow area

through the jet

Calculate slip factor based M jet size

Compare new slip factor with previous value

I

Calculate exit con- ditions assuming flow

completely fills the passage

diffusion ratio

To calculation of diffusion system

Fig. 4. Flow chart for calculation procedure

reduction in the slip factor. In particular, when the velo- city distribution in the hub-shroud plane at the impeller tip was assumed to be sinusoidal, Wislicenus showed that the slip factor was given by

n 2

wherc pth is the slip factor that would be obtained if the flow through the jet had a uniform velocity distribution, given by equation (6). It was considered unlikely that the sinusoidal velocity distribution would adequately describe the flow pattern in the jet and therefore equation (7) was generalized to

. (7) p . = I-- (1- 8 Pth) .

pj = 1--K(l-ptJ . . . (8) where K is a factor to allow for the non-uniform velocity distribution.

3.3 Jet-wake mixing theory for the slip factor The overall slip factor was obtained by combining the jet and the wake. Dean estimates that up to 20 per cent of the through-flow is carried by the wake. By considering the jet and wake, Fig. 3b, to mix instantaneously just outside

@ IMechE 1974

SLIP FACTOR OF A CENTRIFUGAL COMPRESSOR AND ITS VARIATION WITH FLOW RATE 419

the impeller, circumferential equilibrium can be applied across the mixing zone such that the change in the rate of the moment of momentum of the fluid is zero, leading to :

kjR,( U , + W, sin pi) +mwR,( Ut + Ww sin pw)

From the velocity triangle at the impeller tip, Fig. 3a, = d?,(U,+ Wt sin pi) (9)

Ce = Ut+ Wsinp

Equation (9) can be reduced to

mat,coi+m2,c,, = ri?c,, . . . (10)

Now from the jet slip factor

cej = pjUt

cot = put

and from the overall slip factor

Similarly a slip factor can be defined for the wake; how- ever, as the magnitude of the relative velocity is low the wake slip factor was assumed to be unity.

The overall slip factor is then given from equation (10) as

The overall slip factor, therefore3 deviates from the jet slip factor as the magnitude of the wake increases; this is illustrated in Fig. 5.

The degree of separation and hence the wake magnitude was considered to be a function of the overall diffusion rate W,,/W, attempted in the impeller, Dean (8) con- siders a diffusion ratio of 1.8 as the ultimate that can be achieved without separation; however, it is likely that the fluid separated from the impellers considered here at diffusion ratios below this figure of 1-8. The computed diffusion ratios presented in Fig. 6 show that the degree of separation and hence the magnitude of the wake flowni, can be expected to be large at low flow rates and reduce as the flow rate is increased.

From Fig. 5 it can be seen that the rate of change of the

I /

0 8 0 1 / - I .Ap, 080 082 084 086 088 090 092

J e t slip factor

Fig. 5. Deviation of overall slip factor from jet slip factor

~ 3ve-LIl dif'usion ratio

o--y- Diffusion ra t i o alter acceleration 4r 1 due to negative incidence

2 3

\ 22 \

\

2 1

~ c a s t l o w parameter , md<i,'<, ~~

kq/sJ@'bur _ ~ _ _ -

Fig. 6. Impeller diffusion ratio

overall slip factor p with the wake flow rate increases as the jet slip factor is reduced. Consequently, the rate of change of the overall slip factor with the flow rate will be greater when the jet slip factor is low (i.e. for impellers with a small number of blades) than when the jet slip factor is high (i.e. for impellers with a large number of blades). This jet-wake approach therefore qualitatively agrees with the experimental results presented by Sakai (6). This result is clearly dependent upon the assumed slip factor for the wake. If the wake slip factor were assumed to be equal to the jet slip factor, then the overall slip factor would also be equal to the jet slip factor.

By combining equations (8) and (11), the overall slip factor can be obtained

This equation enables the overall slip factor to be calcu- lated from the ideal slip factor of the through-flow jet, given by equation (6), provided that magnitudes can be specified for the parameter K and the wake flow rate proportion Istw/m.

4 PRESENTATION OF RESULTS The results presented were all obtained using an assumed diffusion ratio D of 1.5 to calculate the jet size. The flow rate through the wake was assumed to be zero and equa- tion (12) reduced to

where the parameter Kl allows for the velocity distribu- tion and the neglected flow rate through the wake.

Attempts were made to include the wake flow rate by assuming that m,/m varied linearly from zero at a diffu- sion ratio of 1.5 to 0.2 at a specified upper diffusion ratio limit, usually 2.5. This had the effect of increasing the magnitude of the ideal slip factor of the jet, as the jet size was reduced, when it carried only 80 per cent of the total

p = l--K1(1-,AtJ . . . (13)

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420 A. WHITFIELD

flow. Applying equation (1 1) then led to a further increase in the slip factor making it necessary to use large values of K (equation (8)), of the order of two or three, to obtain agreement between the theoretical and experimental slip factors. As introducing the wake flow rate only made minor changes in the rate of change of the slip factor with the flow rate, with the assumed maximum wake flow of 20 per cent of the total, and because of uncertainties associated with the magnitude of the wake flow rate, the complete procedure leading to equation (12) was aban- doned in favour of equation (13).

Figs l a and l b compare the calculated jet slip factors with those obtained experimentally for compressors A and B (34 and 30 radial blades at impeller exit, respectively). Both of these compressors had a vaned diffuser and the design-point operation was controlled by the incidence of the fluid on to the vaned diffuser. Consequently, both of the impellers operated with positive incidence on to the impeller and zero incidence never occurred.

In contrast, compressor C had a vaneless diffuser, the impeller had 12 radial vanes and operated through both the positive and negative incidence regimes. Figs 2a, 2b and 2c present the theoretical and experimental results for three impeller speeds. The experimental results differ from those obtained with compressors A and B in that a minimum slip factor occurred, followed by an increasing slip factor with increasing flow rate. This minimum in the slip factor coincided with the predicted onset of negative incidence on to the impeller inducer, a regime not en- countered with either compressor A or B. The calculation of the jet size and hence the jet slip factor used for com- pressors A and B had, therefore, to be modified and could not simply be extrapolated into the negative incidence regime.The calculation of the exit jet size in the positive incidence regime was based on an assumed overall diffu- sion ratio of 1.5, i.e. the exit relative velocity W, was given by

Fig. 6 shows that as the flow rate increases, the overall diffusion ratio attempted decreases and, when this reaches 1.5, the through-flow jet completely fills the exit channel. However, before this state is reached the negative inci- dence regime of the impeller is encountered and this causes a rapid increase of the relative velocity in the in- ducer (2) (3), i.e. the relative velocity after incidence W'ls exceeds W, and the actual rate of diffusion attempted in the impeller increases again as shown in Fig. 6. This leads to a reduction in the jet size and consequently to an in- crease in the jet slip factor.

The predicted jet slip factors do, however, differ greatly from those obtained experimentally and para- meter Kl of equation (13) must be introduced. For im- peller A, a value of Kl ranging between 2.5 and 2.8 was necessary to correlate the jet slip factor with the experi- mental slip factor; for compressor B a value of between 2-2 and 2.5 was needed, whilst for compressor C a value between 1.4 and 1.8 was needed.

Figs 2a, 2b and 2c also include the calculated slip factor based on a Kl value of 1.5 for all speeds. The variation with flow rate is accurately predicted, although the magni- tude deviates from the experimental values because a constant value of KL with speed was used. However, the

0 5 10 15 20 25 30 35 40 Number of impeller blades

Fig. 7. Variation of K1 with number of blades

0 +a Experimental

_-_._ Predicted constunt slip factor 0-044

Predicted variable s l ip factor -_,

Impeller neqative - .L r-de 2 3 4 5 6 7

Mass i i o w parameter rnJz/e, kq /sJK /ba r

\ \

Impeller neqotive incidence, ,

2 3 4 5 6 7 8 9 L - . . ~~

M o s s f low pornmeter, m G / F & kq/s.&/bar

Fig. 8. Performance map of compressor C at 60 000 rev/min

values predicted are always better than those predicted with the simple constant value given by the Stanitz correlation.

Fig. 7 presents a plot of the value of Kl against the number of impeller blades. This gives a guide to the value of Kl needed for impellers with different numbers of blades. This, however, can only be considered a tentative result, as impellers A and B are very similar and there is then a wide gap to compressor C; intermediate points are needed before definite conclusions can be drawn.

Finally, Fig. 8 compares the predicted performance of the compressor obtained assuming a constant slip factor with that obtained with the variable slip factor predicted in Fig. 2b. The experimentally-measured compressor performance is also included for comparison.

5 CONCLUSIONS The procedure presented to calculate the compressor slip factor is an integral part of a complete performance pre-

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SLIP FACTOR OF A CENTRIFUGAL COMPRESSOR AND ITS VARIATION WITH FLOW RATE 42 1

diction procedure and cannot be separated from the im- peller incidence model and the procedure used to account for the separation within the impeller. Until the complete three-dimensional flow through the impeller can be calculated to give the point of, and degree of, separation for both design-point and off-design operation, the diffu- sion ratio actually achieved by the impeller flow passage must be estimated. The magnitude of the diffusion ratio used (herein 1.5) governs the size of the through-flow jet and hence the calculated jet slip factor. Using a different diffusion ratio would clearly modify the results presented here. However, the jet size is not only dependent upon the estimated velocity, but also upon the calculated fluid density, which is a function of the estimated losses in- curred within the impeller. This work is part of a continu- ing research programme to develop a complete perform- ance prediction procedure for radial- and mixed-flow compressors. It is intended to study the effect of diffusion ratio, velocity distribution parameter and all the associated losses on the complete performance prediction rather than on the slip factor alone. The approach presented here, which accurately predicts the change of the slip factor with the flow rate, now provides a good foundation from which this study of performance prediction can be con- tinued.

6 ACKNOWLEDGEMENTS This work forms part of a Science Research Council- supported research project and I wish to acknowledge with thanks the award of an SRC research fellowship.

I am also grateful for the experimental data supplied by industry, the turbocharger unit supplied by Holset Engineering Co. Ltd., and to Mr R. C . Atkey for his assistance with the experimental work a t the University of Bath.

APPENDIX R E F E R E N C E S

(I) WIESNER, F. J. ‘A review of slip factors for centrifugal im- pellers’, J . Engng Pwr, Trans. Am. Sac. mech. Engrs 1967 89, 558.

‘A new approach to the problem of predicting the performance of centrifugal com- pressors’, 2nd Int. J S M E symp. Fluid Machinery and Fluid- ics 1972 225 (Japan Society of Mechanical Engineers, Tokyo).

‘Study of incidence loss models in radial and mixed flow turbomachinery’ Confer- ence Publication 3 1973, Paper C55j73 (Institution of Mechanical Engineers, London).

‘Matching of high-output diesel engines with associated turbomachinery’, Proc. Instn mech. Engrs 1973 187,535.

‘The slip factor of a radial bladed centri- fugal compressor’, J. Engng Pwr, Trans. Am. SOL. mech. Engrs 1965 87, 181.

‘On the slip factor of centrifugal and mixed flow impellers’, A.S.M.E. Paper No. 67-WA/GT-10.

Fluid mechanics of turbomachinery 1965 (Dover Publications, New York).

‘On the unresolved fluid dynamics of the centrifugal compressors’, A S M E Advanced Centrifugal Compressors 1971 (American Society of Mechanical En- gineers, New York).

‘The fluid dynamic design of advanced centri- fugal compressors’, Creare Inc. Technical Note 153, 1972.

‘Losses in vaneless dif- fusers of centrifugal compressors and pumps’,J. Engng Pwr, Trans. Am. SOL. mech. Engrs 1966 88, 49.

‘Some theoretical aerodynamic investiga- tions of impellers in radial and mixed flow centrifugal compressors’, Trans. Am. Soc. mech. Engrs 1952 74,473.

‘Chamber dimension effects on induced flow and frictional resistance of enclosed rotat- ing discs’J. bas. Engng, Trans. Am. Soc. mech. Engrs 1960 82, 217.

(2) WALLACE, F. J. and WHITFIELD, A.

(3) WHITFIELD, A. and WALLACE, F. J.

(4) WALLACE, F. J. and CAVE, P. R.

(5) STAHLER, A. F.

(6) SAKAI, J., WATANABE, I., FUJIE, I. and TAKAYANAGI, I.

(7) WISLICENUS, G. F.

(8) DEAN, R. C.

(9) DEAN, R. C.

(10) JOHNSTON, J. P. and DEAN, R. C.

(11) STANITZ, J. D.

(12) DAILY, J. W. and NECE, R. E.

@ IMechE 1974 Proc lnstn Mech Engrs Vol 188 32/74

Discussion on A. Whitfield: SLIP FACTOR OF A CENTRIFUGAL COMPRESSOR AND ITS VARIATION WITH FLOW RATE

Discussion

F. S. Bhinder Member

This is an interesting and important contribution on the phenomenon of slip in centrifugal compressors,

The slip factor is a convenient empirical coefficient introduced in one-dimensional calculations to take into account the variations in absolute velocity vector due to the three-dimensional nature of the flow at the impeller exit. A reliable estimate of the variation of slip factor with mass flow rate is important in performance prediction calculations. But to retain the simplicity of the one- dimensional procedure such an estimate must be made without resorting to a complete three-dimensional or quasi-three-dimensional analysis of flow in the impeller. The paper presents a very useful method by which the theoretical slip factor can be corrected for non-uniform velocity distribution and jet and wake mixing at varying flow rates.

My queries are on equations (1 2) and (13), and Fig. 2. Substituting for K in equation (12) its value (l-pj)/ ( 1 prh) from equation (8) gives:

But uiz, = riI r i z j , SO:

Since the assumption of uiz, = 0, i.e. rt i j = rti, leads to p = pj, equation (13) seems unnecessary, unless the parameters K and K , in equations (12) and (13) res- pectively are supposed to be different. If K , and K are the same, there can be no advantage in using equation (13) instead of the above relationship, particularly if the values of K , must be chosen quite arbitrarily for agree- ment between predicted and measured results. The author’s comments on this will be appreciated.

In Fig. 2a, b and c the calculated and measured values of jet and overall slip factors are compared. ,It would be interesting to know how the j,et slip factor was measured. In our work at Hatfield we found that near to the impeller exit temperature and pressure measure- ments are very sensitive to location of the sensing instrument, besides being affected by high-frequency variations of the flow parameters in the jet and wake regions. Details of the author’s measurement techniques and instrumentation would be useful.

The Stanitz slip factor for impeller C from equation (5) would be 0.8351. The value shown in Fig. 2a, b and c appears to be 0.845. Although it is a minor point, the author may like to take note of this for future work.

D. J. Myles Fellow

Papers on the estimation of fluid ‘slip’ in centrifugal impellers have been numerous since the mid 1930s and, as the equations advanced for calculationg slip co- efficient are almost as numerous, there is abundant scope for selection. This prompted Weisner (1) to compare them and, though slip can generally be accurately pre- dicted in impellers with low backward-curved blades at the design point, its variation with flow rate and the prediction of slip in radial-bladed impellers even at the design point remains enigmatical.

The author has approached these problems by recognizing that radial-bladed impellers can perform reasonably well even though the flow may be separated from the suction faces of the blades. Under these circum- stances the work capacity of the impeller is more adversely affected than current theories suggest, and the variation of whirl-to-blade ratio C,/U with flow rate is far from constant. I share these views though experiment- ally they are hard to prove. The author has obviously experienced this difficulty, since he has chosen to mea- sure the total temperature rise and estimate disc friction loss so as to calculate experimental C,/U values. What levels of accuracy does he put on these measurements and the calculated disc loss? Also, from what number and location of temperature readings was the mean assessed? These questions are particularly relevant, as extrapolations of the linear portion of the experimental slip factor curves for compressor B (Fig. lb) intercept the m = 0 line at above unity; a situation I have not previously met. There are also considerable variations in the curves of compressor C (Fig. 2) when reduced to a common speed (Fig. 9). The high-valued and variable coefficients needed to re- duce theoretical C,JU values to coincide with experi- mental results are of concern. Rewriting equation (13) as

implies that, when the curve is linear (Fig. l a and b) K, /Cm, is constant and the intercept on the p axis is unity. Curves of Fig. l a and those shown in (13) suggest this intercept is less than one and is in agreement with the well-known Busemann relative eddy theory which is not flow-dependent. If the relative eddy effect and that caused by boundary layer growth are regarded as separate but interdependent effects on the value of slip coefficient, the expression in (13) reduces to

c m 2

U2 K p = pCL,,--tan6

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D96 DISCUSSION ON A. WHITFIELD

0.86

?:0.84 V

3 0.82 2 3 0.80

m

L-

U

Q

0.90, 1 I I I I I I 1

-

-

-

-

1 .o-

0 8 -

L 0 $ 0.6- cp

Q

8 0 . 4 -

'c

.-

i x 50 000 rev/min 0 60000 revlmin 0 70 000 rev/min

0 0 xo

0 0

0'70< 0.76 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8

Mass flow paraJpara. a t zero incidence

Fig. 9. Experimental slip factors for compressor C plotted on a common speed basis

for radial blades, where tan 6 is the slope of the curve, which may or may not vary in the useful range of the impeller according to whether the incidence changes sign. pLth is the relative eddy effect corresponding to an effective blade number Z given by 2' = 2n/$. Using the modified Stanitz equation (6) and making the approxi- mation tan 6 = R {(27c/2)- B}/(R, ~ R for separation starting at the leading edge, p reduces to

The form of this equation implies that, as separation increases, 8 decreases and the relative eddy effect be- comes smaller owing to the reduced main flow volume. However, because of separation, the slope of the curve becomes more negative leading to a net reduction in p- a result to be expected logically.

As insufficient data are available in the paper to evaluate this expression could the author please make this comparison with his experimental values to see if better correlation obtains?

R E F E R E N C E

(13) MYLES, D. J. 'An analysis of impeller and volute losses in centri- fugal fans', Proc. Instn mech. Engrs. 1969-70 184 (Pt 1 ) (14). 253.

F. 0. J. Otway Fellow In Fig. 1 of the paper I assume the different lines are for different speeds, and request confirmation of this with the range of speeds covered. In my experience much better correlation of slip factor with speed comes from plotting against flow coefficient 6,, defined as:

c v2 X N D ; B ,

6, =

where C = constant (taken as 310.4), V, = volume flow at impeller outlet (ft3/min), N = rotational speed (rev/min), D, = impeller diameter (in) and B, = blade width at outlet (in). This coefficient is a constant times the ratio of radial velocity to blade tangential speed.

Fig. 10 illustrates this correlation with varying speed; both Figs. 10 and 11 are for radially bladed impellers with vaneless diffusers, for comparison with compressor C of the paper. Fig. 10 is based on results from the N.G.T.E. no. 13 impeller tested with a vaneless diffuser (14). The slip factor has been calculated assuming a power input factor of 1.04. Disc friction has been ignored. Note that it has a linear relationship with 6,, falling continuously over a range comparable with that shown for compressor C. This compressor impeller had 29 blades and a diameter of 20.7 in (52.6cm). While this compressor was associated with jet engine development so maximum pressure ratio was desirable. I believe many centrifugal compressor impellers have too many blades to achieve the highest efficiences. This particular impeller when operating with a vaned diffuser showed a maximum efficiency of 81 per cent while 83 per cent has been obtained with a 6.5 in (16.5 cm) diameter impeller having only 17 blades.

K E Y - "t

f t l scc .

1000 - I3 1 1 2 7 - 0 1 2 0 0 - x 1 2 9 8 - 0 1 4 1 2 - V 1 4 9 3 - Q

0.21

0 0.10 I 0 . 1 1 0.12 0 13 0 . 1 4 0.15 0 16 0.17 0 . 1 8 0 . 1 9 0 . 2 0

6 2

Fig. 10. Slip factor vs flow coefficient; N.G.T.E. no. 13 impeller in vaneless diffuser

Proc lnstn Mech Engrs Vol 188 32/74 0 IMechE 1976

SLIP FACTOR O F A CENTRIFIJGAI, COMPRESSOR A N D ITS VARIATION WITH FLOW RATE D97

1.0,

0 1 jI 0 4 I I 0 0 0 5 0 1 0 0 15 0 2 0

6 2

Fig. 11. Slip factor vs flow coefficient; gas circulator for research reactor

Fig. 1 1 shows the results for a relatively small high- speed and high-temperature gas circulator running on carbon dioxide. The impeller was 154 in (38.7 cm) diameter and had 23 vanes. Blade outlet width was 0.440 in ( 1 1.2 mm) and axial clearance 0.060 in (1.52 mm), which may be considered high, but this circulator oper- ated at 475°C. Power input factor was taken as 1.06 and again disc friction was neglected. The slip factor shows a levelling ofT above a flow coefficient 6, of 0.14. Possibly this too would have shown a rise at higher flows as obser- ved with compressor C of the paper. The author states that the work reported is part of a continuing research programme. However, from the information presented the methods propounded do not seem as yet to give sufficient accuracy for predicting centrifugal compressor performance direct from the drawing board. This still leaves the designer dependent on empirical formulae based on earlier test results. Does he agree with this statement? Has he plans to refine the methods described t o give more accurate prediction'!'

KP;FlCKIiNCl ;

(14) SIMONS, A. and S 1 A K M l : K . R. 'Tests on power jets no. 13: impcller fitted in thc diffuserless blower casing', N.G.T.E. report R12.

R. D. Pearson Bath

The paper presents some useful and interesting results. I confine my remarks to the non-linear variation of slip factor with mass flow shown in Fig. 2 for compressor C having a vaneless diffuser, and speculate upon .the possible effect of design of outlet scroll in such cases.

I have no results from compressor scrolls to present but, instead, summary curves showing peripheral flow distribution of inlet casings for radial flow turbines (see Figs. 12 and 13). The distributions for compressor scrolls designed by thr same rules ought to be very similar. In Fig. 12 the distribution measured for a standard 01 turbocharger casing is compared with a theoretical calculation based on the nominal geometrical form. This scroll was based on a design rule assuming linear varia- tion of cross-sectional area with azimuth angle. The measured distribution has a similar shape to that predic- ted and both are extremely non-uniform, having a peak- to-peak variation of almost 3:l.

Fig. 13 shows a distribution for a casing based on improved design rules which assume all cross-sections

are trapezoidal and that angular momentum is conserved. Theoretical distribution is uniform and that found experimentally is shown to be very nearly so. In Fig. 2c at a mass flow parameter of 7 the calculated slip factor is 0.82. On either side of 7 the slip factor is higher so, if a scroll having a non-uniform distribution were used, the average slip factor would be increased to about 0.85.

Flow distributions were measured by pitot-yawmeter and hot-wire anemometer axial traverses made at a radius equal to the rotor tip radius but with the rotor absent. In practice the revolving rotor would partly smooth out flow non-uniformities by propagation of pressure waves in the rotor channels, thus creating a non- uniform static pressure distribution at the rotor tips.

- Theoret ica l -0-e- Experimental

0.6

0.4

\ i O\ I

\

', I

\* d b--

o'2 I 0 1 " I ' " " I " ' 0 90 180 270 360

8 degrees

Fig. 12. Flow distribution in a standard volute

@j) IMechE 1976 Proc lnstn Mech Engrs Vol 188 32/74

D98 DISCUSSION ON A. WHITFIELD

1.6 -

LD 1.4-

1.2 - c m -

VO -'

0.8 -

0.6 -

0.4 -

0.2-

07

It is easily shown that a trapezoidal volute designed according to the rule described is specified for incom- pressible flow by the equation

P-. /I '0,

'\

P / \

\

/,a, / \

%+.' \

a\ \

'0-

Mean r a d i a l v e l o c i t y I dea l r a d i a l ve loc i t y c, =

by i n t e g r a t i o n o f a x i a l t raverses

I , 1 . 1 1 . . I 1 "

0 90 180 270

where RB = R,- WN/(2tanP) . . (15)

and for compressible flow by

( ( % / a O ) cos aT - _ - "B) tan a = ( ~ / a ~ ) cos a RT

Cos a can be assumed 0.99 for first approximation and a found for specified value of u/ao, whence

(uT/aO) cos @T r/RT = (u/ao) cos CI

If a linear variation of tan CI with r is assumed for small finite 0 increments between specified values O1 and O, , then for intermediate values of 0 we have

f I - 6 , = fn(?--) A + B r , l i A . (18) A + Br,

where t a n a = A + B r . . . (19)

A = . . (20) tan a, - ( r , / r , ) tan a,

1-(r1/r2)

(21) tan a,-tan a,

r2-rl and B = . . .

Fig. 13. Flow distribution in a trapezoidal volute

A correction is needed for the thickness of the tongue end by adding to the outer wall a radius increment Ar where

Ar = t , ("l-RB)@) ____ , . (22) RO-RB

This allows for both trapezoidal shape and conservation of angular momentum. The correction is blended to zero to meet the calculated scroll form over an azimuth angle of 30 degrees.

Notation a

B 0

R

RO Rl R B

t2 P a0

Y U

Gas flow angle at rotor tip radius relative to circumferential direction and based on W. Half-angle of trapezoidal cross-section measured from median plane. Azimuth angle at radius R measured from start of scroll. Inside wall radius of scroll outer wall measured from rotor axis. Value of R at 0 = 0. Value of R at 0 = 2n. Base radius, i.e. radius of intersection of sides of trapezoid. Tip radius of rotor. Tip width of rotor. Projected width of trapezoid at RT (less than Wto allow for circumferential corner radii). Tongue end thickness. Density ( p o is upstream stagnation density). Sonic velocity when u = 0. Absolute gas speed. Ratio of specific heats.

Subscript , Measured at rotor tip radius.

T. Sakai Tokyo

The paper gives a good explanation for the variation of slip factor with flow rate. Slip factor is normally defined under the assumption of inviscid flow. In the circum- stances studied however, flow within the impeller can be taken as inviscid when flow rate through the wake is assumed to be zero-because in this case we can consider the blade to be extended to occupy the whole area of the wake.

The assumption that the area of the potential core is a function of flow rate, and that the rate of change of the potential core with flow rate is greater for impellers with few blades, seems quite appropriate.

In our own experimental study of radially bladed impellers, the lowest value of slip factor was found to occur in the neighbourhood of zero incidence. This result coincides with the author's theory and experiments.

Determination of the parameter K , seems very diffi- cult because not only the diffusion factor but also impeller geometry may influence it.

The author's efforts to evaluate slip factor at off- design flow rates is indeed much appreciated.

Proc lnstn Mech Engrs Vol 188 32/74 0 IMechE 1976

SLIP FACTOR OF A CENTRIFUGAL COMPRESSOR A N D ITS VARIATION WITH FLOW RATE D99

A b s o l u t e 0.9 v e l o c i t y 0.8 C/CmCix

16 R e l a t i v e 15 e x i t

, I

Fig. 14. Flow at exit from impeller of a centrifugal pump

0 0 04 0 08 0 12 0 16 Flow coefficient

Fig. 15. Slip factor variation in a centrifugal pump

F low c o e f f i c t e r i t

Fig. 16. Relative exit angle variation in a centrifflgal pump

0 IMechE 1976

M. J. C. Swainston Member

This is a most interesting paper. Variation of slip factor at off-design conditions seems to have had too little attention in the past. This limiting of the field of study has doubtless contributed to the uncertainty still sur- rounding its evaluation at the design point.

Some years ago I initiated work on this topic in con- nection with centrifugal pumps. The pump studied was a double-entry machine of standard commercial type rated at 400 gal/min (1800 litre/min) and 40 ft (12.2 m) head rise at a rotational speed of 1410 rev/min. A inch (9.5 mm) diameter cylindrical yawmeter was fitted paral- lel to the machine axis at impeller exit and this was traver- sed across the full width of the volute.

Results at the design flow coefficient are given in Fig. 14, and show clear evidence of the boundary layer shed by the impeller hub section. The relative flow angle shows little variation across the section.

Fig. 15 shows slip coefficients calculated from

cp cot a2

= 1-cpcotp2 where 4 = C,,/U, and a, and p2 are respectiv~ely the absolute and nominal relative impeller exit angles measured from the circumferential direction. In this form the slip coefficient is defined as the ratio of the actual absolute whirl velocity to the whirl velocity based on the nominal impeller exit angle. For radial blades (p2 = 90’) this becomes identical with the author’s definition.

These results confirm the trend he found for compres- sor C, to the extent that the curve has a minimum. How- ever, his suggestion that the minimum slip coefficient relates to zero incidence for the inlet section is not sup- ported by my data.

Also shown in Fig. 15 is the slip coefficient obtained from relaxation solutions of the two-dimensional equa- tions for flow in the meridianal and blade-to-blade planes, after the Stanitz methods (11). This does not agree closely with the measured value. Note that the slip coefficient at the design condition corresponds closely to the value 0.691 recommended by Stepanoff (15) for pumps.

Fig. 16 shows the variation of relative exit angle over the flow range. It is far from constant and approaches the metal angle only at very high flow coefficients. This appears to show that Stahler’s claim (5)-that the relative exit angle is virtually constant at all flow con- ditions in centrifugal compressors- cannot be applied to pumps.

R E F E R E N C E

(15) STEPANOFF, A. J. Centrifugal and axial flow pumps, 1957 (John Wiley and Sons).

1. Watanabe Tokyo

This paper is interesting in that it adopts the concept of jet-wake region within the impeller channel in deriving the slip factor of centrifugal compressors. Recent experimental research on flow patterns within centrifu- gal impeller channels at Keio University has revealed that, for an impeller with straight radial blades, the wake domain or low-energy region exists at the shroud/ suction side of the impeller channel irrespective of

Proc lnstn Mech Engrs Vol 188 32/74

DlOO DISCUSSION ON A. WHITFIELD

impeller configurations (16). Thus, the jet-wake concept is applicable on the stream surface near the shroud, but not on that near the hub, because of the absence of the wake region so far as the domain near the impeller exit is concerned. So I suggest to the author that better correlation will probably (if not certainly) be obtained by adopting the above assumption.

Apart from this, I have derived (17) a forrriula for the slip factor of centrifugal impellers having straight radial blades as follows

where

tan c! rz -ro l+-

p being the slip factor, a the apex angle between shroud contour and hub contour of the blade on the meridional plane, z the number of blades, b, the blade height at impeller exit (subscript 2 denoting impeller exit con-

b,

dition), r2 the impeller exit radius, o the angular velocity of the impeller, V, the volume flow at impeller exit and ro the outer diameter of the inducer inlet.

In deriving this formula, I also used a concept of the wake region (relative flow velocity being zero) and the jet region, though the jet-wake concept was not so established in those days. In addition, I used the method adopted by A. Stodola in deriving his formula for slip factor. If the volume flow at impeller exit V, is not known, use of V,, the volume flow at impeller inlet, leads to negligible error. Would the author therefore, if possible, compare the above formula with his experimental results and with his own formula?

R E F E R E N C E S

(16) MIZUKI, S., ARIGA, I. and WATANABE, I. ‘Investigation concerning the blade loading of centrifugal impellers’, A.S.M.E. Paper

(17) WATANABE, 1. ‘On the type of impeller of a centrifugal blower with special references to the problem concerning the number of impeller blades’, Proc. Faculty Enging (Keiogijuku Univ.)

74-GT- 143.

July 1949 2 ( 5 ) , 49-71.

Proc lnstn Mech Engrs Vol 188 32/74 @ IMechE 1976