Advanced Compressor Loss Correlations , Part II : Experimental Verifications

Reliable efficiency calculation of high-subsonic and transonic compressor stages requires a 
detailed and accurate prediction of the flow field within these stages. Despite the tremendous 
progress in turbomachinery computational fluid mechanics, the compressor designer still 
uses different loss correlations to estimate the total pressure losses and thus the efficiency of 
the compressor stage. The new shock loss model and the modified diffusion factor, developed 
in Part I, were implemented into a loss calculation procedure. In this part, correlations 
for total pressure loss, profile loss, and secondary loss coefficients are presented, using the 
available experimental data. Based on the profile loss coefficients, correlations were also 
established for boundary layer momentum thickness. These correlations allow the compressor 
designer to accurately estimate the blade losses and therefore the stage efficiency.


INTRODUCTION
The theoretical background and discussion presented in Part ! of this paper showed a direct correlation between the profile losses and the boundary layer quantities, particularly the boundary layer momentum thickness.Investigations by NACA, summarized in NASA SP-36 [1976] and briefly reviewed in Part I,   showed that measuring the total pressure losses can experimentally determine the momentum thickness.Further investigations by Gostelow and Krabacher  [1967], Gostelow [1971], Seylor and Smith [1967], Seylor and Gostelow [1968], Gostelo et al. [1968], Krabacher and Gostelow [1976a,b], and Monsarrat et  al. [1969] deal with the spanwise distribution of the total pressure and the total pressure loss coefficient.For the aerodynamic design of a single stage com- pressor, Monsarrat et al. [1969] presented correla- tions between the profile loss parameter and the dif- fusion factor using the experimental data by Sulam et  al. [1970].The loss correlations by Monsarrat et al. [1969] are frequently used as a guideline for design- ing compressor stages with the profile similar to that described by Monsarrat et al. [1969].Gostelow et al.  [1968] performed systematic and detailed experimen- tal investigations on four different rotors to determine *Corresponding author.Tel." (409) 845-0819  the optimum blade camber line shape.Although the experimental data revealed certain systematic tenden- cies, no attempt was made to develop a correlation to describe the loss situation in a systematic manner.These facts gave impetus to consider the above ex- perimental data in the present analysis.

ANALYSIS OF EXPERIMENTAL DATA
To establish the loss correlations, the existing avail- able experimental data were reevaluated, particularly those in Gostelow et al. [1968] and Krabacher and  Gostelow [1976a,b], which used four single stage compressors with multi-circular-arc profiles.A de- tailed description of the compressor facility and the stages are found in their reports.The data analysis used the following information: (1) the total pressure losses as a function of diffusion factor in the spanwise direction, (2) inlet, exit, and incidence angles, (3) Mach numbers, (4) velocities, and (5) geometry.To consider the compressibility effect discussed in Part ! of this paper, the modified diffusion factor D was obtained using the information from Gostellow's report mentioned previously.Regression analysis was used for a systematic evaluation of the loss parameters.Figures (la) to (ld) show the results.Starting from an immersion ratio H (R R)/(R Rh) 10%, Fig. (la) shows the loss parameter as a function of a modified diffusion factor for the investigated ro- tors 1B, 2B, and 2D in the reports by Gostelow and  Krabacher [1967], Gostelow [1971], Seylor and   Smith [1967], Seylor and Gostelo [1968], Gostelo et  al. [1968], Krabacher and Gostelow [1976a,b].For the sake of clarity, all the measured data are shown uniformly with filled squares.As shown in Fig (la) and in all subsequent figures, a systematic dependency is clearly visible and the individual rotors do not exhibit any deviatory characteristics.The intro- duction of the modified diffusion factor with com- pressibility effect has caused a shift of D to higher values.Figures (lb) to (ld) exhibit a similar depen- dency for H 50, 70 and 90%, where the smallest losses are encountered at H 60-70. Moving fur- ther toward the hub at H 90%, the total pressure losses experience a continuous increase attributed to higher friction losses and the secondary flow caused by the secondary vortices at the hub.A comparison of the loss parameters plotted in Figs.(la)-(ld) shows that the total pressure losses in the tip region (H 10%) are much higher than those in the hub region (H 90%).The higher losses at the tip are due to the existing shock losses and the secondary flow ef- fect due to the tip clearance vortices.

CORRELATIONS FOR TOTAL LOSSES, PROFILE LOSSES AND REST LOSSES
Correlations are derived for total loss coefficients, profile loss coefficients, and secondary flow loss co- efficients from the available experimental data, particularly those discussed previously.These correla- tions express the direct dependency of the above losses as a function of modified diffusion factor defined in Part I. Furthermore, correlations for momen- tum thickness and the diffusion factors are given from the profile loss parameter.To obtain a correlation for the profile loss coefficient (see definition in Part I, Section 2), the shock loss coefficients are calculated and subtracted from the total loss coefficients.By definition, this so called profile loss coefficient not only includes the primary losses (see definition in Part I) but also contains the losses due to the second- ary flow.As pointed out in Part I, the secondary flow losses assume higher values by approaching the hub/ tip regions, respectively.The profile losses corre- spond to primary losses only at the design section, which includes only the total losses due to the profile friction.Figure (2) shows the total pressure loss pa- rameter as a function of modified diffusion factor with the immersion ratio as a parameter.The highest loss parameter is encountered near the tip.The losses continuously decrease by moving toward the blade midsection up to H 0.6.The total pressure loss coefficient assumes a minimum at Hr 0.7.At this radius, the secondary flow effect apparently disap- pears completely, so that the total pressure loss coef- ficient corresponds to the primary loss coefficient.For immersion ratios greater than H 0.7, the losses start increasing again, which indicates the strong ef- fect of the secondary flow.As previously mentioned, subtracting the shock loss coefficients from the total loss coefficients obtains profile loss coefficients.The resulting profile loss coefficients plotted in Fig. (3) are approximately 30% smaller than the total pressure 70% and (d) H 90%, from the tip.Experiments from NASA-CR-45481-45485, rotors" 1B, 2B, 2D.
loss coefficient shown in Fig.
(2).The fact that the total pressure loss coefficients exhibit a minimum at H 0.7, where the secondary flow effect diminishes, enables the compressor designer to estimate the rest losses.To determine the distribution of the rest losses, we start from the profile loss distributions at different spanwise locations and subtract the losses at H 0.7.As a result, these rest losses include the effect of the secondary flows associated with wall boundary layer development and clearance vortices.Figure (4) shows the distribution of the rest loss parameter as a function of the modified diffusion factor.Figure (4) exhibits a linear dependency of the rest loss parameter as a function of the modified diffusion factor with the immersion ratio as a parameter.Since the diffusion factor is directly related to the lift force and thus to the lift coefficient (Cads) as a linear func- tion, one may conclude that the rest losses are lin- early proportional to (CLC/S).This is in agreement with the measurements by Grieb et al. [1975] and in contrast to the correlation proposed earlier by Carter [1948] that includes the term (CLC/S) 2 and is adopted by many other researchers.

CORRELATIONS FOR BOUNDARY LAYER MOMENTUM THICKNESS
As shown in Part I, the profile loss coefficient and the boundary layer momentum thickness are interrelated by: (0) (sin12F(H12, H32, O, or, c) (1) po-7 \sin[32/ The momentum thickness in Eq. ( 1) is the projection of the suction surface and pressure surface momen- tum thicknesses given by" where the subscripts S and P refer to the suction and pressure surface, respectively, and the function F is given by: + H32 with H12, H32 as the displacement and energy form factors, 0 as the boundary layer momentum thickness, o-as the solidity, and c as the blade chord.In the literature (see also Hirsch [1978] and Swan [1961]), the function F is frequently approximated as a con- stant with the value F 2. For a realistic velocity distribution, Schobeiri [1987] showed that the value of F may differ from 2. To arrive at a better estima- tion for F, the boundary layer velocity profile is ap- proximated by several simple functions such as a lin- ear function, a power law, a sine function, and an exponential function.A close examination of the re- suits and their comparison with the experiments showed that the velocity approximation by a power function yields better results.However, the exponen- tial approximation would be more appropriate for those profiles that are dose to separation.Using the law function approximation, we arrive at: HI2 + H32 3H12-(4) Introducing Eq. ( 4) into Eq.( 3) and the results into Eq.( 1) leads to" P sin[31 ) completely, so that the momentum thickness corre- sponds to the one generated by the blade surface friction only.For immersion ratios greater that H 0.7, the momentum thickness starts increasing again, which indicates the strong effect of the secondary flOW.
With Eq. ( 5) and (3), the momentum thickness is de-  5) with the immersion ratio as a parameter.The highest value for the momentum thickness is encountered in the vicinity of the tip that includes the viscosity effects as well as the sec- ondary flow effects.Similar to the profile losses, the momentum thickness continuously decreases by moving toward the blade midsection up to H 0.6.
It assumes a minimum at H 0.7.At this radius, the secondary flow effect apparently diminishes The correlations presented above are based on exper- imental results performed on typical high perfor- mance compressors with specific flow characteristics and blade geometries similar to those discussed previously.These correlations may be applied to other compressors with similar geometries but different flow conditions by considering the effect of the fol- lowing individual parameters.

MACH NUMBER EFFECT
Estimating the Mach number effect requires calculat- ing the critical Mach number.When the Mach num- ber reaches unity locally in a compressor cascade, the corresponding inlet Mach number is said to have reached its critical value.Jansen and Moffat [1967] made the assumptions that below the critical Mach number, the total pressure losses and the turning an- gle are essentially constant.The pressure losses in- crease rapidly beyond this value.Using the gas dy- namics relations, Jansen and Moffat [1967] deter- mined the local critical Mach number by the following implicit relation: To estimate the critical Mach number directly, Davis 1971 suggested the following explicit relation: M1 2.925 2.948 \ V1 -t-1.17 \v/ 0.1614\ V1 As seen in Part I, the velocity ratio grnax/Vl is directly related to the circulation function and thus the diffu- sion factor.With the critical Mach number from Eq. ( 7) or (8), the profile loss coefficient can be corrected as." ,cor p [A(M1 Ml'r) + 1.0] ( with A 1.8-2.0(see Moffat [1967] and Davis  [1971]).For DCA-profiles, Dettmering and Grahl   [1971] found that Eq. ( 9) underestimates the correc- tion and suggested the following modified approxi- mation: {,.o p {14.0 [M, (M,, 0.4)3] + 1.0} (10)   its changes do not affect the profile losses.The fol- lowing profile loss correction is suggested for Reynolds number ranges Re < 2.5 105: Re 0.2 pcor p \Recor (11)

BLADE THICKNESS EFFECT
To consider the effect of the thickness ratio t/c, the boundary layer momentum thickness may be cor- rected using the correlation by Fottner [1979].

CONCLUSION
In the second part of this paper, correlations were established for the total pressure loss coefficient, pro- file loss coefficients, and the secondary loss coeffi- cients.Based on the profile loss coefficients, correla- tions were also established for boundary layer mo- mentum thickness.The modified diffusion factor, discussed in Part I, was a major variable of the cor- relations with the immersion ratio as the parameter.Different loss parameters as well as boundary layer momentum thickness can be calculated for the com- pressor stages with similar blade profiles using the stage velocity diagram.These correlations allow the compressor designer to accurately estimate the blade losses and therefore the stage efficiency.Some cor- rections are recommended for inlet flow conditions different from those for which the above correlations were established.

REYNOLDS NUMBER EFFECT
This effect is only at lower Reynolds number ranges of practical significance.For high performance com- pressors, the Reynolds number is high enough so that FIGURE a, b Total pressure loss parameter as a function of modified diffusion factor with immersion ratio H as parameter for (a) Hr 10% and (b) H50% from the tip.Experiments from NASA-CR-45481-45485, rotors: 1B, 2B, 2D.
FIGURE c, d Total pressure loss parameter as a function of modified diffusion factor with immersion ratio H as parameter for (c) H losses as previously discussed, the correlation for the momentum thickness as a function of modified equivalent diffusion factor (see Part I) are plotted in Fig.( , R Cascade, stator, rotor E EN NE ER RG GY Y M MA AT TE ER RI IA AL LS S Materials Science & Engineering for Energy Systems Economic and environmental factors are creating ever greater pressures for the efficient generation, transmission and use of energy.Materials developments are crucial to progress in all these areas: to innovation in design; to extending lifetime and maintenance intervals; and to successful operation in more demanding environments.Drawing together the broad community with interests in these areas, Energy Materials addresses materials needs in future energy generation, transmission, utilisation, conservation and storage.The journal covers thermal generation and gas turbines; renewable power (wind, wave, tidal, hydro, solar and geothermal); fuel cells (low and high temperature); materials issues relevant to biomass and biotechnology; nuclear power generation (fission and fusion); hydrogen generation and storage in the context of the 'hydrogen economy'; and the transmission and storage of the energy produced.As well as publishing high-quality peer-reviewed research, Energy Materials promotes discussion of issues common to all sectors, through commissioned reviews and commentaries.The journal includes coverage of energy economics and policy, and broader social issues, since the political and legislative context influence research and investment decisions.S SU UB BS SC CR RI IP PT TI IO ON N I IN NF FO OR RM MA AT TI IO ON N Volume 1 (2006), 4 issues per year Print ISSN: 1748-9237 Online ISSN: 1748-9245 Individual rate: £76.00/US$141.00Institutional rate: £235.00/US$435.00Online-only institutional rate: £199.00/US$367.00For special IOM 3 member rates please email s su ub bs sc cr ri ip pt ti io on ns s@ @m ma an ne ey y. .cco o. .uuk k E ED DI IT TO OR RS S D Dr r F Fu uj ji io o A Ab be e NIMS, Japan D Dr r J Jo oh hn n H Ha al ld d, IPL-MPT, Technical University of Denmark, Denmark D Dr r R R V Vi is sw wa an na at th ha an n, EPRI, USA F Fo or r f fu ur rt th he er r i in nf fo or rm ma at ti io on n p pl le ea as se e c co on nt ta ac ct t: : Maney Publishing UK Tel: +44 (0)113 249 7481 Fax: +44 (0)113 248 6983 Email: subscriptions@maney.co.uk or Maney Publishing North America Tel (toll free): 866 297 5154 Fax: 617 354 6875 Email: maney@maneyusa.comFor further information or to subscribe online please visit w ww ww w. .mma an ne ey y. .cco o. .uuk k C CA AL LL L F FO OR R P PA AP PE ER RS S Contributions to the journal should be submitted online at http://ema.edmgr.comTo view the Notes for Contributors please visit: www.maney.co.uk/journals/notes/emaUpon publication in 2006, this journal will be available via the Ingenta Connect journals service.To view free sample content online visit: w ww ww w. .i in ng ge en nt ta ac co on nn ne ec ct t. .cco om m/ /c co on nt te en nt t/ /m ma an ne ey y Correlations for total pressure loss parameter as a function of modified diffusion factor with the immersion ratio H,.
as parameter.
Correlations for profile loss parameter as a function of modified diffusion factor with the immersion ratio H as parameter.
Correlations for rest loss parameter as a function of modified diffusion factor with the immersion ratio Hr as parameter.Correlations for dimensionless boundary layer momentum thickness as a function of modified diffusion factor with the immersion ratio H as parameter.