Small Radial Compressors : Aerodynamic Design and Analysis

This paper presents a computational procedure for the analysis of steady one-dimensional centrifugal compressor. The numerical model is based on the conservation principles of mass, momentum and energy, and has been utilized to predict the operational and aerodynamic characteristics of a small centrifugal compressor as well as determining the performance and geometry of compressor blades, both straight and curved.


INTRODUCTION
The advantages of small centrifugal compressor in comparison to the axial flow type were demon- strated at the end of the 1950s.Nevertheless few models were developed intentionally for aeronau- tical applications.Independent research programs appeared in the 1960s and 1970s as Pratt & Whitney, Canada (Kenny, 1984) and National Gas Turbine Establishment (Came, 1978).The real flow field in the compressor is far too complex to be treated analytically.For this reason many simpli- fications, empirical formulations and other simplifying assumptions are usually adopted to design or predict their performance.Clements and Artt (1987) analysed the influence of the diffuser geometry on the efficiency and operational range of the compressor, while Bammert et al. (1979) illustrated the importance of coupling the rotor and diffuser in the calculation of the rotor losses.Senoo (1987) analysed the effect of tip clearance on the compressor flow field and its performance.Stow (1989) revised models and advanced computational methods of potential use in blade design while Kenny (1984) reported   progress in the numerical techniques, viscous solutions, flow measuring techniques adaptable to centrifugal compressor testing and new developments in materials.Takeda (1987) realized a comparative study of three computer codes for the design and performance prediction of a centrifugal compressor for small gas turbine unit.
Given the shortage of technical information, experience and know how in this area in Brazil, a small research program devoted to this area was started with the objective of designing a small Fax: 55 (019) 239-3722.E-mail: kamal@fem.unicamp.br.modular gas turbine unit for emergency and similar applications.The first part of the project handles the centrifugal compressor (Rosolen, 1994) and the associated centrifugal gas turbine (Benevenuto, 1996).The analysis is based upon using refined one-dimensional formulation eliminating the use of empirical factors and equations unless absolutely necessary.As a result the flow, t6mperature and pressure curves can be evaluated along the stream lines in the rotor, diffuser and spiral duct which help in identifying and pointing corrections to the calculated geometry.Also the proposed model allows the analytical design of straight and curved blades, systematic adjustments when necessary and final evaluation of the performance of the unit as well as the determination of the local flows properties.

FORMULATION
Applying the first law of thermodynamics to the one-dimensional steady fluid flow in the com- pressor one can compute the power used in the compression process Pc of the total mass flow Fht: Pc thtCp T03 TOl )" (1) A sketch of the centrifugal compressor with straight profile and diffuser is presented in Fig. 1.
The compression ratio Rc can be calculated in terms of the isentropic efficiency r/c, the stagnation temperatures at the inlet T01 and the outlet T03 of compressor and the ratio "7 Cp/Cv: Rc ---Po3/Po1 --[1 + /c(T03-Zol)/Zol] 7/(7-1). ( 2 Due to the inertia and viscous effects the fluid does not follow the rotor outlet tangential speed U2 and a slip factor 0-2 is defined as 0-2 1--Vescor/U2. (3) The slip factor is empirical and depends upon the number nr and geometry/32 of the blades and is usually given by Wiesner's formula (Came, 1978),  Applying the momentum conservation equation between the entry and exit sections of the rotor with rotational velocity co (rad/s) ou N (rpm) one can write Pc thtco(r2Cw2 rl Cwl).( 7) Combining Eqs. ( 1) and ( 7) and considering that the flow is adiabatic one obtains To2 T01 co(r2Cw2 rlCwl)/Cp.( Hence the rotor outlet tangential speed, can be calculated as 02-[p(T02-T01)/2] 1/2. (9) The continuity equation is used to calculate the areas and the local flow properties both in the rotor and the diffuser.At the diffuser entrance the tan- gential velocity component is obtained by consid- ering that the flow in the vaneless space is free vortex type while the other two components are obtained from the mass conservation principle.
The flow velocity is assumed constant from entry to the diffuser throat and its direction is assumed constant until the diffuser exit.The fluid is conducted from the throat through diffusing channels to the through spiral duct which is calculated assuming free vortex flow.
In order to determine the total power necessary for running the compressor additional equations are used to estimate the losses.The total power P is the sum of the useful power, Putil, the volumetric losses, Pvol, the aerodynamic losses, Pa, and the mechanical losses (Pr + Pm): P-(Putil + Pvo + Pa) q-(Pr + Pm) ( The useful power is the power used in the real compression process of the net mass flow rate, rh, and is written as Putil thCp (To3 r01 ). (12) The volumetric efficiency T]vol is given by vol (tht-/kth)/tht--th/tht, (13 where Arh is the fluid loss in the compressor.The bearing efficiency r/m has been estimated while the disc friction power losses, Pr, as well as the aero- dynamic losses are calculated in a conventional manner. The global efficiency r/has been estimated using as a definition the ratio of isentropic compression power, Pis, to the total shaft power, l Pis / P flcPutil / P. (14) The above system of equations is used to calculate and refine the compressor geometry, calculate the local flow properties and finally evaluate the losses and determine the operation parameters of the compressor shown below.

THE CALCULATION PROCEDURE
The overall operational conditions of the com- pressor as well as the global geometrical character- istics of the rotor with straight radial blades can be defined using the equations presented.In order to determine the local flow properties and refine the aerodynamic aspect of the blades the rotor is subdivided into three regions in the meridional plane and along the flow direction (see Fig. 3).In the first region, the leading edge of the rotor cascade is tangent to the direction of the flow velocity relative to the rotor.At the end of this region, the rotor cascade is tangent to the axial direction as it is admitted that the flow tangential velocity is compatible with that of the rotor.
Between these two sections a cylindrical surface of consistent radius "T,g is adopted for the cascade profile, assuming that the axial velocity is constant while the other properties are calculated.In the second, or intermediate, region the flow direction changes to radial with no slip.In the third region, or exit region, the axial component is considered zero, slip occurs and an expression similar to Eq. ( 4) is used to calculate the local slip factor.
Expressions similar to Eqs. ( 6) and ( 8) are used to calculate the rise in stagnation temperature and pressure between the sections y and x defined by the radius of their mean points, in terms of the politropic efficiency of impeller r/r, as below: Tox-roy--Co2(Oxr2-Oyr2y)/Cp POx/POy) (Tox/Toy) rrT/(7-1).
In this manner the stagnation properties at a transversal the mean point of each section are obtained while the other properties are calculated from the thermodynamic relations and the conser- vation equations.

SMALL RADIAL COMPRESSORS 193
To improve the accuracy of the calculation procedure the flow passage is subdivided into a number of stream tubes (varying from to 10).The accuracy and the computing time are observed.The procedure for the calculations for each stream tube is practically the same.
Using the one-dimensional basic model it is possible to establish curved blades with a specific exit angle and hence analytically, avoid empirically based curved blades.To achieve this objective the meridional geometry determined for straight radial blades was adopted, together with the rotational velocity and discharge rate.The new operational conditions of the compressor and the flow properties at the rotor exit can be determined.For the intermediate sections of the exit region the radial velocity profile Cr is evaluated from the rotor calculations admitting a hypothetical constant slip factor equal to that of the exit section.Based upon this one-dimensional formulation and the proposed calculation scheme a computational code was elaborated to help in the design of centrifugal compressors.This code was then used to determine the effects of the design parameters on the performance of the compressor rotor.The effect of the parameters Rc, N and Pc while keeping the rest of the variables as constants is shown in Table I.
Table II shows the values of (T02-T01) and U2 defined by Rc for rotors of straight profiles.The use of conventional materials limits the value of Rc to about 4, since U2 must not be higher than 460 m/s due to the stresses in the material (Cohen et al.,  1987).
Tables III and IV present respectively the cor- responding values of r2, N, rht and Pc for the pressure ratios Rc used here.b2 [10 -3 m]   5.5 3.6 5.8 8.0 6.1 6.9 5 The overall values calculated for the compres-   Figures 4-9 present the internal geometry of the rotor calculated based upon subdividing the flow passage into ten stream tubes while the flow properties are presented only for the extreme stream tubes, that is, and 10.
One can observe from Fig. 4 that the meridional geometry results in discontinuities in the transition sections between the rotor regions because of the different formulations adopted in each region.The correction of the geometry is done by tracing two tangents to the blade root profile; one passing by the inner point of the entry section and the other passing by the point of greatest axial coordinate while maintaining the same radial direction until the rotor exit.The local depth of the sections in- creases and the local flow properties are recalcu- lated.The maximum variation obtained in the velocity is 1.5%.The recommendation of positioning the entry section at small radii to obtain deeper rotor passages is illustrated in Fig. 5 which illustrates the variations of rotor C of Table V.
Figure 6 shows the influence of the rotor velocity.One can observe that the effect of the increase of N is to decrease r2 in order to maintain the peripheral velocity U2 defined by Rc as in Tables II and III.
Keeping the pressure ratio and the rotor rota- tional speed as constants, the increase of the value of Pc leads to corresponding increase in the depth of the flow passages of the rotor as in Fig. 7.
Figure 8 shows the effect of the pressure ratio on the compressor geometry.This was achieved by keeping the mass flow rate and the rotor rotational velocity as constants while increasing the value of Rc.Then rotor flow passages are found to increase radially.

Rotor with Straight Blades
Although the calculations were realized for pres- sure ratio up to 4, we chose a minor value of 3 to keep the peripheral velocity well below its limiting value.For the intermediate velocity of 30 000 rpm, one can observe from Fig. 7 that the depth of the flow passages is very small for Pc 50 kW, case B, while for Pc= 150kW, as can be verified from Table VII, the compressor D is also adequate.Hence the option D, whose blade profile is shown in Fig. 9, is chosen to present the results illustrating the proposed calculation methodology.

Rotor with Curved Blades
For the calculation of the compressor with curved blades, we used the same meridional geometry obtained for the case of straight blade of Fig. 9.By keeping the same rotor rotational velocity N, one recalculates the operational conditions for the angle /o02 30 , as suggested by Takeda (1987).
Table VI shows the data values obtained for both compressors, rotors with straight and curved blades.One can also observe that for the com- pressor with curved blades the pressure ratio is smaller, also the compression power is equally smaller while the angles of the blade exit/oo2 and the flow exit angle/2 are very close.
In case of the rotor D with curved blades the pressure and temperature distributions along the   13(a) and (b) indicating discontin- uities due to the adopted assumptions for the slip factor in the intermediate and exit regions of the rotor.This factor changes abruptly from a unit constant value in the intermediate region to an assumed low value at exit.The correction is based upon using smooth and continuous variation of the slip factor.Hence in evaluating Cr the slip factor instead of being adopted as constant and equal to #2, is compared to the minimum value necessary to keep the stagnation temperature higher or equal to that of the previous section.Also in the cal- culation of the blade inclination/3 in the radial plane of the exit region, the slip factor is assumed in the start such that the variation of the stagnation temperature is the same for the three initial sections and starting from the last of these sections, Eq. ( 4) is used for the slip factor calculations.The properties distribution after the corrections of the slip factor are shown in Fig. 14.As can be noticed the discontinuities near the rotor exit remained and the blade profile in the radial plane does not tend the value /c2-30-In the calcula- tion of the Cr profile, the variation of the tan- gentially projected thickness is not included together with the variation of the slip factor.Hence the correction of Cr is based basically upon  FIGURE 14 (a) Variation of the local stagnation pressure in the rotor with curved blades without correction in Cr.
(b) Variation of the local stagnation temperature in the rotor with curved blades without correction in Cr.
redistribution of the slip factor and considering the projected thickness effect.Figures 15 and 16 present the distributions of the static and stagnation properties (pressure and temperature) while Fig. 17 presents the rotor geometry with curved blades.Figure 18 shows a comparison with the rotor with circular arc blade profile.
Following the same method used for the compressor option D, the details of the options A to G are calculated, and presented in Table VII.
One must notice that the compressor option D presents higher overall efficiency than that of compressor option C.

CONCLUSION
The most important conclusion of the present study is that the proposed model is able to predict the overall working parameters, the local flow properties, the compressor performance characteristics and finally enables refining the rotor geometry and predict its new performance.

FIGURE 3
FIGURE 3 Regions of the rotor.
FIGURE 9 Meridional geometry of rotor D.

FIGURE 12
FIGURE 12 Variation of the local static and stagnation temperatures (rotor D with straight radial blades).
FIGURE 13 (a) Variation of the local stagnation pressure in the rotor with curved blades and without corrections in 92 and Cr. (b) Variation of the local stagnation temperature in the rotor with curved blades and without corrections in 2 and C.
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TABLE III
Values of r2 specified in terms of Rc and N

TABLE IV
Values of rht specified in terms of Rc and Pc

TABLE V
sors denominated A to G with rotors of straight geometry are presented in TableV.
FIGURE 5 Effect of re on the rotor geometry.