Relative-eddy Induced Slip in Centrifugal Impellers for Engineering Students

 

 

T.W. von Backström

Professor, MSAIMechE, Department of Mechanical and Mechatronic Engineering, University of Stellenbosch, South Africa. E-mail: twvb@sun.ac.za

 

 

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ABSTRACT

The paper presents a new method for deriving the relative-eddy induced slip factor in centrifugal impellers in an engineering teaching situation. The simple analytical method derives the slip velocity in terms of a single relative eddy (SRE) centered on the rotor axis instead of the usual multiple (one per blade passage) eddies. Features of the method are: the application of basic fluid dynamics to a consistent control volume and logical determination of empirical constants. The method shows correct limiting behaviour for zero blades and for 90° blade angle combined with unity radius ratio, and excellent agreement with the accurate analytical method of Busemann. The SRE method meets the main criteria for presenting slip factor in an engineering teaching situation. It is suggested as a replacement in the teaching situation for the commonly used methods of Stodola, Stanitz and Wiesner.

Additional Keywords: Engineering education, slip factor, centrifugal impeller, eddy-induced slip, pump.

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Nomenclature

Roman

c blade length

d rotor diameter

e eddy radius in Stodola derivation

F blade angle function

RR radius ratio

r rotor radius

s blade spacing, distance along integration path

U rotor speed

w circulation velocity

Z number of rotor blades

Greek

ßblade angle

Γcirculation

πcircle circumference to diameter ratio

σslip factor

Ω rotor angular velocity

Subscripts

e exit

i inlet

lim limiting value

p pressure side of blade

s suction side of blade

0, 1, 2, 3 blade angle function designations

 

1. Introduction

The rate at which centrifugal compressors and pumps do flow work is less than that calculated with the assumption that the relative flow at the exit of the rotor follows the blade trailing edges. The reduction in angular momentum imparted to the flow is determined by the slip factor. Engineering teachers and students need a reliable method for the calculation of slip factor in centrifugal impellers. It should be based on sound fluid dynamics, and be suitable for classroom derivation. The method should be widely applicable in terms of basic impeller geometry such as blade number and blade angle, and be relatively accurate for typical values of impeller radius ratio.

The main mechanism usually considered when predicting slip factor in radial flow impellers is the so-called relative eddy. This is an inviscid flow effect. A fluid element entering a radial flow impeller does not rotate around its own axis with an angular velocity equal to that of the rotor, but moves around the machine axis while maintaining a constant orientation relative to the machine casing. Relative to the rotor, however, the fluid element rotates at an angular velocity equal but opposite to the angular velocity of the rotor. The relative vorticity of the flow in the rotor will set up a recirculating flow pattern relative to the rotor. In centrifugal impellers it results in a change in circumferential velocity component relative to the rotor at the rotor exit plane, causing the flow to deviate from the blade direction at the trailing edges.

Directly or by implication, textbooks have generally treated the relative eddy as the major factor causing slip in radial flow turbomachines, for example Stodola¹, Eckert and Schnell², Ferguson³, Wislicenus⁴, Osborne⁵, Eck⁶, Watson and Janota⁷, Cumpsty⁸, Logan⁹, Dixon¹⁰, Johnson¹¹, Wilson and Korakianitis¹², Aungier¹³, and Saravanamuttoo et al.¹⁴. At least, they generally do not attempt to model the other contributing factors. Dean and Young¹⁵, Whitfield and Baines¹⁶ and Japikse and Baines¹⁷ do however consider the effect of the wake region in the blade passage, but jet-wake models still require a slip factor correlation in the jet flow region where viscous effects do not dominate.

 

2. Background

Busemann¹⁸ proposed aremarkable slip factor prediction method. He analytically solved the inviscid flow field through a series of two-dimensional impellers with logarithmic spiral blades. He generated maps of slip factor versus impeller radius ratio, with blade number as parameter, for various blade sweep angles for logarithmic spiral blades. Blade radius ratio (RR) is the radial distance of the blade leading edge from the axis divided by that of the blade trailing edge. Wislicenus⁴ and Wiesner¹⁹ reproduced these maps (for example figure 1). The Busemann maps indicated that slip factor depends on RR, but below a critical value of RR it is relatively constant, especially for high blade numbers. Figure 2 shows the Busemann slip factors for RR = 0 as dependent on blade angle, β. The Busemann values presented here have recently been recalculated by Hassenpflug²⁰.

 

 

 

 

Unfortunately the Busemann method¹⁸ is mathematically complex and not compact enough for inclusion in text books or derivation in the classroom, so various simplified approaches have been tried. The most popular of these are the equations of Stodola¹ and Stanitz²¹ and the curve fit by Wiesner¹⁹.

Stodola¹ presented a simplified and popular approximate derivation followed by many textbooks. He inserted a circular-shaped control volume between the blades, near the outer radius of the rotor (figure 3). The circle touches the suction side trailing edge of one blade and is tangent to the pressure surface of its neighbour. For a rotor with exit radius, r_e and number of blades, Z, the blade spacing is 2πr_e /Z and the eddy diameter is 2e ≈(2πr_e /Z) cos ß, with ß the blade exit angle measured from the radial direction. Stodola assumed the slip velocity caused by the relative eddy to be equal in magnitude to the speed of rotation of the circular eddy at its rim: ∆w = Ωe = Ωr_eπ(cos ß)/Z = U _eπ(cos ß)/Z. A recent example of such an approach is the paper of Paeng and Chung²².

 

 

The present study was started because the Stodola¹ assumption that the eddy rim velocity ∆w may be applied along the rotor perimeter (the edge of another control volume) as the so-called slip velocity was difficult to justify, especially in a teaching situation. The vaguely defined control volumes in the Stodola¹ approach leads to the unrealistic conclusions that blade length (or radius ratio) does not matter, and that in the limit of zero blades on the impeller, the slip factor is equal to minus infinity. Text books do not generally state the accuracy of the Stodola¹, Stanitz²¹ and Wiesner¹⁹ approaches compared to the Busemann¹⁸ exact inviscid flow solution. As a consequence students end up with a rather vague understanding of what causes slip factor, which method to apply where, and how well the popular approaches agree with Busemann¹⁸ or with experimental data.

 

3. Definition of Slip Factor

Before defining slip factor, the normalised slip velocity should be defined. In one common definition the slip is normalised by dividing the slip velocity by the rotor rim speed, and in another by the circumferential component of the ideal (slipless), absolute velocity at the rotor exit. The second one introduces the complication that the ideal circumferential fluid velocity component is dependent on the flow through the impeller, except in the case of radial blades (ß_e = 0) or zero flow through the impeller, when the two definitions are equivalent.

Slip factor is one minus the normalized slip velocity. For the sake of simplicity we shall follow Wiesner¹⁹ and use the first definition of normalised slip velocity. It is known that in practice slip factors are not independent of through-flow, but the present investigation will focus on eddy-induced slip, which is independent of flow.

 

4. Traditional Approaches

Stanitz²¹ presented the slip factor equation given below, based on inviscid flow numerical modelling:

Textbooks such as Dixon¹⁰ recommend the Stanitz²¹ equation for use when ß < 30°. When 60° < ß < 70° the Stodola¹ slip factor equation is recommended:

The two equations are equivalent when ß = 51°, and figure 4 shows that using the Stanitz equation for ß < 51° and the Stodola equation for ß > 51 ° compares reasonably well with Busemann in the recommended ranges, but only for rotors with many blades (Z > 8).

 

 

Wiesner¹⁹ proposed an empirical equation, valid for RR less than a limiting value, RR_lim, stating, in our angle notation, that:

Figure 5 compares the Wiesner prediction (for RR < RR_lim) with Busemann. The agreement is seen to be excellent for 16 blades when ß < 80°. It is also good for ß < 30° for 4 or more blades, but is difficult to find a contiguous region where the agreement is very good.

 

 

None of the methods described above models the Busemann data consistently well, and all predict the slip factor for the limiting case of zero blades to be -∞ instead of 0.

 

5. The Single Relative Eddy Approach

Von Backström²³ has recently presented a new, approximate method for the derivation of relative-eddy induced slip in centrifugal impellers. The most important assumptions are listed below:

□ Logarithmic spiral rotor blades.

□ The two-dimensional control volume consists of a curved sector bounded by five lines: two logarithmic spirals representing adjacent blades, two radial lines between the blade leading edges and the axis, and by the rotor perimeter between the trailing edges (figure 6).

 

 

□ The flow induced by the relative eddy causes no through-flow.

□ There is only one relative eddy in the whole rotor: it revolves around the axis and protrudes into the blade passages, and when it forms separate cells associated with each blade passage, these cells are included in the main cell centered on the rotor axis (figure 6).

With reference to its most distinguishing feature, it was called the Single Relative Eddy (SRE) method.

 

6. Derivation of SRE Equations

The complete derivation is given by Von Backström²³, but its fundamental principles are:

□ Each fluid particle in the rotor has a vorticity equal in magnitude to twice the rotor angular velocity, relative to the rotor.

□ There is a single average circulation velocity, Aw around the edges of the relative eddy.

□ The integral of the circulation velocity around the control volume divided by the control volume area is equal to the vorticity.

□ The integration path follows the suction surface from leading to trailing edge, then the rotor exit rim from the blade trailing edge to the next blade pressure side trailing edge, then to its leading edge, and then around its leading edge from pressure to suction side (figure 6).

The slip velocity as a fraction of the rotor rim speed is then given by:

F is the sum of the average circulation velocities along the blade suction and pressure surfaces, divided by the average circulation velocity along the exit boundary, ∆w_e:

The function F must be found to give good agreement with Busemann¹⁸ or with experimental data.

The normal definition of blade row solidity is the blade chord divided by the spacing, but to keep things simple, we shall replace the chord by the blade length (in a plane perpendicular to the rotor axis) and use the spacing at the radius, r_e of the blade trailing edges (rotor rim). The solidity is then:

It is instructive to point out in class that the slip is fundamentally determined by the ratio of the total blade length to the impeller circumferential length. It is also worth noting in retrospect that the derivation would have been possible without the logarithmic spiral blade assumption. This probably explains why Busemann's values have been successfully used for impeller blades with other shapes.

As defined above slip factor is one minus the normalized slip velocity:

Since the magnitude of the other factors affecting slip, like blade incidence angle, trailing edge pressure gradient relaxation and boundary layer blockage effect (including the existence of wakes) are also primarily dependent on solidity, solidity should correlate measured slip factors well, at worst with a different F for each family of impellers.

 

7. The Dependence of F on Blade Angle

The next step in finding F is to determine its dependence on Z, ß and RR. Stodola¹ and Stanitz²¹ ignore RR as a parameter, Wiesner¹⁹ brings it in only as a correction, and Busemann showed that at high blade numbers (Z > 8) slip factor is very weakly dependent on RR, especially for RR < 0.5 (figure 1). Figure 7 shows the SRE prediction of slip factor with radius ratio RR, and ß = 30° and for a provisional value of F = 4.6. The general trend of the lines of constant blade number is similar to that of figure 1. Both predict σ= 0 at RR = 1, with σincreasing as RR decreases, but the SRE prediction does not level off at low values of RR. Since most impellers have values of radius ratio between 0.4 and 0.6, a possible approach to finding a suitable expression for F would be to choose a value that would ensure good agreement between SRE with RR = 0.5, and Busemann with RR = 0, and then assuming that RR - 0.5 when RR < 0.5 in the SRE prediction. These assumptions enable us to calculate values of slip factor for all combinations of Zand β and compare them, on the same basis as the Stodola, Stanitz and Wiesner methods, to the corresponding Busemann values for RR = 0.

 

 

8. Comparison of SRE Predictions with Busemann

By finding the values of F that would give good agreement with the data of Busemann¹⁸ measured from the graphs in Wiesner¹⁹, von Backström²³ showed that the equation: F₁= F₀(cos β)^0.5, with F₀= 5.0, represented the trend well enough. Figure 8 correlates with the SRE slip factor for F₁with accurate Busemann values from Hassenpflug²⁰. Also shown are lines of constant solidity. It is apparent that the equation correlates with the Busemann data accurately when the solidity exceeds 0.5 and the blade angle is less than 70°. The agreement can be slightly improved, however if the exponent in the equation is changed from 0.5 to 0.45 (equation for F₂). The higher angle limit is then increased to 80° (figure 9).

 

 

 

 

When the required value of F for perfect agreement with Busemann is plotted against β for various blade numbers (figure 10) it appears that F₃ = 2 + 3 cos ß should lead to good agreement for 4 or more blades and blade angles up to 85°. The reason why F = 2 when ß = 90° is that blades oriented at 90° do not deflect the flow, so that the flow inside and outside the rotor remains stationary in the absolute frame. The relative velocities over the blade suction and pressure sides are then equal, and equal to the relative velocity along the rotor rim (the slip velocity) and:

 

 

Inserting F₃ into the SRE then leads to figure 11, which shows excellent agreement (within 0.005) with Busemann for all blade angles right up to 90°, for Z > 2 and c/s_e > 0.5, and good agreement (within 0.03) for Z > 1 and c/s_e > 0.25.

 

 

9. Discussion

The SRE method for the prediction of relative-eddy induced slip in inviscid flow in centrifugal impellers has been derived, based on the following:

□ Simple assumptions based on sound fluid dynamics

□ Consistent control volumes

□ Logical determination of empirical constants

□ Careful check against an accurate analytical method

The proposed equation for eddy-induced slip factor in

where rr is taken as 0.5 when rr < 0.5. As pointed out, the SRE slip factor automatically approaches 0 as rr approaches 1.0 (figure 7), without the need for a separate correction as in the Wiesner¹⁹ method. Application of the SRE equation above to the experimental data presented by Wiesner¹⁹ shows excellent agreement between the SRE predictions and the Busemann values plotted against the modified solidity F₃(c/s_e), for blade number, Z varying from 3 to 44, blade angle, ß from 0° to 82° and radius ratio, RR from 0.33 to 0.60 (figure 12).

 

 

The various SRE approaches represent the analytical method of Busemann extremely well. Von Backström²³ has shown that the SRE method also predicts a large range of experimental slip factors as accurately as the method of Wiesner¹⁹. Consequently students can use the SRE method for the prediction of relative-eddy induced slip with confidence.

 

10. Conclusions

The SRE method for the prediction of relative-eddy induced slip factor in centrifugal impellers represents the mathematically complex inviscid method of Busemann¹⁸ with sufficient accuracy for class room use. It is derived by applying basic fluid dynamics to a consistent control volume. It exhibits the correct limiting behaviour for zero blades and 90° blade angle combined with unity radius ratio, and is more accurate than the methods of Stodola, Stanitz and Wiesner. The SRE method meets the main criteria for teaching slip factor, and can replace the commonly used methods in an engineering teaching situation.

 

References

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2. Eckert B and Schnell E, Axial- und Radialkompressoren, Springer-Verlag, 1961, 345.

3. Ferguson TB, The Centrifugal Compressor Stage, Butterworths, London, 1963, 85 - 90.

4. Wislicenus GF, Fluid Mechanics of Turbomachinery, 2^nd ed., in two volumes, Volume One. Dover Publications, Inc., New York, 1965, 269.

5. Osborne WC, Fans, Pergamon Press, Bell and Bain Ltd., Glasgow, 1966, 129.

6. Eck B 1973, Fans, Pergamon Press, Germany, 37.

7. Watson N and Janota MS, Turbocharging the Internal Combustion Engine, MacMillan Education Ltd. London, 1986, 89.

8. Cumpsty NA, Compressor Aerodynamics, Longman Scientific & Technical, England, 1989, 245 - 249.

9. Logan E Jnr, Turbomachinery, Basic Theory and Applications, 2^nd ed. Revised and expanded, Marcel Dekker, Inc., New York, 1993, 167, 248.

10. Dixon SL, Fluid Mechanics, Thermodynamics of Turbo-machinery. Pergamon Press, 1998, 222 - 227.

11. Johnson RW, The Handbook of Fluid Dynamics, CRC Press, Springer, U.S.A., 1998, 41-12-41-14.

12. Wilson DG and Korakianitis T, The Design of High-Efficiency Turbomachinery and Gas Turbines, 2^nd ed., Prentice Hall, New Jersey, 1998, 240.

13. Aungier RH, Centrifugal Compressors - A Strategy for Aerodynamic Design and Analysis, ASME Press, New York, 2000, 55.

14. Saravanamuttoo HIH, Rogers GFC and Cohen H, Gas Turbine Theory, 5^th ed. Prentice Hall, Cornwall, 2001, 153, 155.

15. Dean RC and Young LR, The fluid dynamic design of advanced centrifugal compressors, Creare TN-244, 1976, 5-27.

16. Whitfield A and Baines NC, Design of radial turbo-machines, Longman Singapore Publishers (Pty) Ltd., Avon, UK, 1990, 220-231.

17. Japikse D and Baines NC, Introduction to Turbo-machinery, Concepts ETI and Oxford University Press, 1997, 4-6 -4-7.

18. Busemann A, Das Förderhöhenverhältniss radialer Kreiselpumpen mit logarithisch-spiraligen Schaufeln, Zeitschrift fur Angewandte Mathematik und Mechanik, 1928, 8, 371 - 84.

19. Wiesner FJ, A review of slip factors for centrifugal impellers, Trans. ASME Journal of Engineering for Power 1967, 89, 558 - 72.         [ Links ]

20. Hassenpflug WC, Personal communication, 2004.

21. Stanitz JD, Some Theoretical Aerodynamic Investigations of Impellers in Radial and Mixed-Flow Centrifugal Compressors, Cleveland, Ohio, Transactions of the ASME, 1952, 74, 473-476.         [ Links ]

22. Paeng KS and Chung MK, A new slip factor for centrifugal impellers, Proc. Instn. Mech. Engrs., 2001, 215, Part A, 645 - 649.         [ Links ]

23. Von Backström TW, A unified correlation for slip factor in centrifugal impellers, ASME Journal of Turbomachinery, January 2006, 128, 1-10.         [ Links ]