A Simple Calculation Method for Ratio of Relative Velocity Within Centrifugal Impeller Channel

Transcription

1 E THE AMERICAN SOCIETY OF MECHANICAL ENGINEERS 345 E. 47 St., New York, N.Y s The Society shall not be responsible for statements or opinions advanced in papers or in discussion at meetings of the Society or of its Divisions or Sections, or printed in its publications. Discussion is printed only if the paper is published in an ASME Journal. Papers are available ]^! from ASME for fifteen months after the meeting. Printed in USA. Copyright 1986 by ASME 86-GT-25 A Simple Calculation Method for Ratio of Relative Velocity Within Centrifugal Impeller Channel ABSTR1:CT A simple but accurate method of calculating ratio of relative velocities within centrifugal impeller channels is proposed using a one-dimensional flow model, whose major parameters are specific speed, non-dimensional root- mean- square radius of the inducer inlet, slip factor, flow coefficient and flow angle at impeller exit. After the non dimensional relative velocity at inducer inlet and that at impeller exit are derived, the ratio of relative velocity at impeller exit to that at inducer inlet is obtained. In addition to this, the ratio is divided into two parts: one ratio for the inducer portion and another ratio for the radial portion of the impeller channel. The computations are conducted both for adiabatic inviscid flow and for two conditions assumed for viscous flow, in which one used an empirical relationship between the total pressure ratio and the peripheral speed of impeller and the other used experimental values for the total pressure ratio as a funtion of the flow rate. By the present simple method, the non-dimensional relative velocities as well as the ratios of the relative velocities for the inlets and the exits of an inducer and an impeller channel are calculated accurately. INTRODUCTION In the design of a centrifugal impeller, the performance characteristics are usually estimated initially using one dimensional flow calculations with empirical factors and formulae for the losses assumed within the channel(1-3.. Following this initial estimation of performance characteristics, more detailed flow patterns within the impeller channel are examined by quasi-three or three- dimensional flow analyses (4-6) to evolve a configuration with maximum efficiency. Because of the existence of flow separation, boundary layers, and tip leakage flow within the channel, it is still difficult, however, to estimate the flow patterns and the performance characteristics with sufficient accuracy. Time consuming iterative procedures in these calculations are necessary to obtain a final configuration which satisfy the operating conditions. The present authors previously investigated a SHIMPEI MIZUKI Professor Hosei University Koganei, Tokyo Japan ICHIRO WATANABE Professor Emeritus Keio University Yokohama, Kanagawa Japan Presented at the International Gas Turbine Conference and Exhibit Dusseldorf, West Germany June 8-12, 1986 method for estimating performance characteristics of centrifugal compressors using inducer incidence loss, blade loading loss, skin friction loss, mixing loss, recirculation loss, disk friction loss, vaneless diffuser loss and vaned diffuser loss (7). In related work internal flow patterns within impeller channels were measured (8). This work showed that the ratio of relative velocities at inducer inlet to that at impeller exit was one of the most important factors which governed the performance characteristics of centrifugal impellers. Numerous other reports exist on this subject (9-11), but the methods presented usually require long computer time to calculate the relative velocities. Therefore, it would be useful to accurately calculate the relative velocity ratios using a simple method prior to the main part of impeller design. With this point of view, a simple method to calculate relative velocities within centrifugal impeller channels was proposed using specific speed, slip factor, non-dimensional root-mean-square radius at inducer inlet, flow coefficient and flow angle at impeller exit as parameters. The computations were made initially for adiabatic flow and then, extended to actual flow conditions with viscous effects by employing two empirical assumptions. One assumption used an empirical relationship between total pressure ratio and peripheral speed at the impeller tip excluding the variation of total pressure ratio with flow rate and the other empirical assumption was directly given by experimental results of total pressure ratio as a function of flow rate. In addition to these two assumptions, the ratio of relative velocities at inducer exit to that at inducer inlet was estimated. Thus, the ratios of relative velocities at the inducer portion and at the radial portion of an impeller channel were obtained separately. NOMENCLATURE A =area (m2 ) a, =coefficient of polynominal Ct =lift coefficient - i,z ) Cp =specific heat (=m- s' K ) 2Pf` D =diameter (m) G =weight flow rate (Kg/s) Had =adiabatic head (fn)

2 L =lift (Kg) and no pre-whirl will be considered hereafter in the M =Mach number present study. The specific speed is given by N =rotational speed (rpm) Na =specific speed P =pressure (KPa^ Ns= N (3) Ha(P 75 Q =volume flow rate (ms /s) R =gas constant (K y' w 'Kg 'K ) where the slip factor is defined as the ratio of S =area of wing (m circumferential velocities at impeller exit with a T =temperature ( K finite number of blades to that with an infinite number t =blade pitch of blades. U =circumferential velocity of blade (m/s) By introducing Eqs. (2) and (3) into Eq. (1), the V =absolute velocity (m's) following equation is obtained, W =relative velocity (m 's) Z =numnber of blades a =flow angle measured from radial direction 112=( p \o s (4) (deg) (3 =flow angle measured from circumferential where direction (deg) y =specific weight (Kg/m 3 1 F =circulation of inducer blade C = p 1 s. Ns2. y 1 5 (5) 4 =difference of value from former iteration e =non-dimensional radius ratio at inducer inlet (=D1 rms / D2 The above formula can be shown in a different way e' =non-dimens onal rad ius ratio at inducer as follows: inlet (_^I-DY11/D2 ) g =flow coefficient (=VjZ/L'z s _ VA = B. ' Fe!7n (6) e =specific heat ratio x= 02 ( Ns p ' 5 s =total pressure ratio p =density (Kg s 2/m4 ) where B is x =non-dimensional circumferential velocity at impeller exit B= q x (= V2i / Uz ) - p =slip factor 2405 x 5 3 Subscripts In this way, the non--dimensional relative velocity 0 =inlet condition at inducer inlet is obtained as a function of flow I =at inducer inlet (Fig.l) coefficient and non-dimensional root-mean-square radius 1' =at inlet to inducer exit (Fig.1) at inducer inlet by Eq.(1) or related to the parameters 1H =at inducer hub of flow coefficient, slip factor, specific speed, IT =at inducer tip non-dimensional circumferential velocity at the 2 =at impeller exit impeller exit and non-dimensional root-mean-square 2' =at diffuser exit (Fig.1) radius at the inducer inlet by Eq.(6). When Eq.(1) is 5 =at compressor exit employed, relative velocity will be calculated ad =adiabatic implicitly if the radius is given. However, when Eq.(6) Av. =averaged is used, the parameters will not be able to vary b =blade independently with each other, because the Max. =maximum value obtained by the method in non-dimensional circumferential velocity at impeller Ref.7 or maximum value in Fig.3 exit cannot be decided without taking into account the Min. =minimum value obtained by the method in flow condition at impeller exit. Thus, the values of Ref.7 or minimum value in Fig.3 the parameters will be chosen to satisfy the continuity rel =relative equation, the equation of the state and the adiabatic rms =root-mean--square condition or the given empirical conditions in s =static conjunction with the non-dimensional relative velocity t =total at impeller exit as will be shown in Eq.(7). =circumferential component =axial component 0NE-DIMENSIONAL CALCULATION METHOD (1) Non-Dimensional Relative Velocity at Inducer Inlet The non-dimensional form of relative velocity at an inducer inlet based on the impeller tip speed can be expressed as a function of inlet flow coefficient and overall radius ratio as follows: W1=(AI rz)o.5 Uz When the pre--whirl is absent, the adiabatic head given by an impeller is Had=µ U2 la (2)Non-Dimensional Relatite Velocity at Impeller Exit The non-dimensional relative velocity at impeller exit is expressed by the following equation based on a simple investigation of the velocity triangle at impeller exit. Uz=Ex2cot2a2+ (x_ 1 )2]O.5 (7) (3)Rao Rat i of Relative Velocity at Impeller Exit to That at Inducer Inlet This ratio can be shown by the ratio of Eq.(4) and E q. (7). z 2 z Wz x cot az+ (x-1) J 2 (8 )0.5 W ^ (Pt+ (4) Ratio of Relatiix' Velocity at Inducer Exit to That at Inducer Inlet Vz" (2) The ratio of relative velocity at inducer inlet to 9 that at inducer exit is considered here in addition to the ratio of relative velocities corresponding to the 2

3 inducer inlet and the impeller exit. The velocity triangle within an inducer channel was considered as in Fig.1. The channel was assumed to be consisted by the inlet to inducer and the relative diffuser parts. The knee + exducer indicates the radial portion of impeller channel. Because the relative velocity in the inducer channel varies rapidly at the inlet to inducer portion formed by the curved blades and after that, varies slightly at the relative diffuser portion, the above-mentioned distinction was made. The mean-camber-line of inducer was approximated by a circular arc. As the pressure rise at the circular arc section is expressed by the following formula, PP1=p t. (U1- d2 ) (9) that at the diffuser part becomes as follows: P2'-Pi S =2(1tY 4 -V02 ) (10) In this way. the static pressure rise within the inducer channel is shown as follows: P2 S-Pis=2([x122- V02) +p t (Ut - ) (11) where the lift coefficient CL is given by experimental data. In the present study, the data of NACA 2412 aerofoil were employed. Based on Eq.(11), the relative velocities at the inducer inlet and the exit can be obtained. COMPUTATIONAL PROCEDURE (1) Input and Initial Values The flow chart of the computational procedure is shown in Fig.2. The criteria for the maximum and the minimum values of parameters were decided artificially Inducer Knee * Relative Inlet to - Exduce r Diffuser Inducer Circular Arc Blade I I 2 31' 1 Fig.1(a) Model of Inducer Channel INPUT 1. Configuration of Impeller (Table.2) 2. Input Data (Table.1) 3. Inlet Condition COMPUTATION OF FLOW COEFFICIENT 1. Maximum and Minimum Flow Coefficient 2. Flow coefficients at which calculations are made COMPUTATIONS OF INITIAL VALUES 1. Slip Factor 2. Specific Speed 3. Specific Weight at Impeller Exit (1.2) 4. Flow Angle at Impeller Exit COMPUTATIONS OF RELATIVE VELOCITIES INDUCER CHANNEL 1. Non-dimensional Relative Velocity at Inducer Inlet (Eq.(1)) 2. Specific Weight (Eq.(12)) 3. Absolute and Relative Velocities at Inducer Inlet (Fig.1) 4. Lift Coefficient 5. Static Pressure, Circulation and Velocities (Eqs.(9) and (10)) 6. Mach Number. Incidence Angle and Ratio of Relative Velocity at Inducer Inlet to That at Relative Diffuser Exit (Fig.1) IMPELLER CHANNEL 1. Non-Dimesional Circumferential Velocity at Impeller Exit (Eq.(6)) 2. Check Velocity Triangle at Impeller Exit (Modify values of V2 r,v2 and Y2' if they varied too much from those by former iteration.) If V2r>0.7xU2 V=0.5xU2 If Y2>1.5 tp If Y2<7l ; Y2=1.01 xyt If V2<0. 0 V=V2. tana2 If Ns>2.0xNs, Ns=1.iVsmax If Ns<0.5xNsm,,, ; Ns=Ns.i n 3. New Velocity Tiangle 4. Check Number of Iterations 5. If V2 > U2, Decrease Value of Specie Speed and Go to Total, Dynamic and Static Pressure 7. Check Total or Static Pressure ;Total for Adiabatic Flow (Modify values if they exceed given limits.) If P2s<Pk ; P2S=Pls 8. Check Convergence of Specific Speed If Ns>2.0xNsmax ; Ns=1.5NSm, T If Ns<0.5x Vsm i n ; Ns=NSmin 9. Check Number of Iterations 10. If Specific Speed Does Not Converge, Go to Check Convergences of Radial Velocity, Flow Angle at Impeller Exit, Specific Speed, Specific Weight and Weight Flow Rate If V2r>0.5U V2r=0.502 If 02>90.0 a2=89.0 If i>1.0 ; µ=1.0 If p<0.7 ; Ns=1.O1 x Ns If d G2 ig=0.05 V2r=G1/G2VVT If d G2/G=0.05 ; Y2=G,/G?Y^ 12. If Slip Factor, Specific Speed and Radial Velocity Do Not Converge, Go to Ratios of Relative Velocities at Inducer and Impeller Channels END Fig.1(b) Velocity Triangle at Inducer Inlet Fig.2 Flow Chart 3

4 by examining the computed values in order to get converged values of them simultaneously. However, sometimes the computed results failed to converge as will be discussed in the next section. It would be necessary to find out more meaningful criteria for the limitations of the maximum and the minimum values which will be able to make more reasonable and efficient convergences. All of the experimental and the input data were taken from Ref.(7) and results computed by the present method were compared with those by the method in Ref.(7). The compressors used in the present study are shown again in Tables 1 and 2. The maximum and the minimum flow rates were obtained from the input data in Table 1 and computations were made in every 0.01 increment of flow coefficient between them. The initial values of flow angle at impeller exit and specific speed for the first iteration were given by the estimated values of input data, which were assumed to vary in proportion to the variations with flow coefficient. The slip factor was given by the formula of F.J.Wiesner (12) and the specific weight at impeller exit was initialized to 1.2. Three different empirical relationships between total pressure ratio and impeller tip speed were considered first. The maximum, the averaged and the minimum curves of total pressure ratio versus impeller tip speed as in Fig.3 were used. A twentieth order polynominal, Ea,U2', was employed to get the curves obtained by the method of least square. Although the deviation of the minimum curve from the plotted data in Fig.2 occured, this polynominal gave the most smooth curves for the maximum and the averaged ones compared with the other eleven kinds of polynominals examined to get smooth curves. Because of the employment of compressor stage total pressure ratio, 7t1_5, in Fig.3, the maximum curve seems to give the most appropriate Table.1 Operating Condition A B C D E W I G C Tye El E2 E3 F G H1 H2 Il N rpn U MU n mn i G G mac Type J1 J2 J3 K N rpn U Mug fimin m^ II G Giax Table.2 Major Dimension of Compressor Ref. 3 (11) ) (14 ) Type A B C D E1 E2 E3 F G H1 H2 I1 12 J1 J2 J3 K Symbol O O O G 4 d D , J D ka2 7A ] _0_599_8b_4_0TY ' O , ^(b

5 values of 7r9 at the impeller exit. The averaged and the minimum curves were, however, tried in order to examine whether they also could converge to certain results or not. When the experimental values were used directly for the total pressure ratio, two cases were examined. First, the total pressure ratio was assumd to increase linearly with flow rate from the minimum to the maximum total pressure ratio in Table 1. Secondly, the opposite variation with flow rate was assumed. They were examined whether the variation of total pressure ratio with flow rate could be applied for the present method or not. It would be possible to use more realistic assumptions if the generalized relationship among total pressure ratio, flow rate and impeller tip speed exists The inlet conditions were given based on the standard condition (101.3KPa, < ). (2) Inducer Channel The velocity triangle at inducer inlet is easily computed using the following equation for the specific weight. Yi= Pit 1 R (12) R Ti^l 2 t ( J - Cp ) Because the inlet conditions and the flow coefficient were given, the axial and the circumferential components of relative velocities, the incidence angle, the absolute velocity, the static pressure, the static temperature, the Mach number and all of the other information were obtained. Based on the results, the lift coefficient CL was interpolated from the data of NACA 2412 aerofoil by the spline fit interpolation. Thus, the relative velocity within an inducer channel was computed by Eq.(11) , (3) Impeller Chalurcl The impeller exit total and static pressures were calculated using three different procedures: one for adiabatic flow and two for viscous flows. For adiabatic flows, the static pressure at impeller exit was obtained by the following formula, Ps _const. (13) Y K and the total pressure was obtained from the absolute velocity and specific weight. On the other hand, when the total pressure at impeller exit was given first, the static pressure was computed from that. By using the initial values or those given by the former iteration for specific speed, flow angle at impeller exit, slip factor and specific weight at impeller exit, the non dimensional circumferential velocity at impeller exit was obtained by Eq.(6). Based on this velocity, a new velocity triangle at impeller exit was given. The slip factor was also obtained. If the circumferential velocity at impeller exit exceeded the impeller tip speed, the value of specific speed was increased and the computation was sent back to the beginning. Without this iteration loop, the divergence of the specific speed sometimes occured and all of the other values gave unreasonable results. This divergence might be caused by the lack of some conditions which assure the convergences of all of the values of the parameters into their respective values. By this iteration loop, the values approached to similar ones irrespective of the employments of the different conditions at impeller exit even if the convergences were not enough. After the initial estimation of velocity triangle at impeller exit was established, static pressure was given by adiabatic flow or total pressure was given by the empirical formulae and the input data. Because all of the velocity components were known, the informations about pressure, temperature and specific weight could be obtained by using equation of continuity and equation of state. A new value of the specific speed was also obtained. Here, the computation was sent back to the beginning again if specific speed, specific weight and slip factor did not converge to certain values. When the above-mentioned two iteration loops gave the converged values, velocity triangle, pressure, temperature, slip factor, specific weight, weight flow rate and specific speed were computed again. If the specific speed, the slip factor, the radial component of relative velocity and the flow angle at impeller exit did not converge to their respective values, the iteration loop was sent back to the beginning. (4)Criteria for Coniergences The criteria for the convergences during the iterations for the specific speed were 0.1, for the slip factor , for the radial component of relative velocity 0.5m/s and for the specific weight , respectively. The maximum number of iterations was set to 300. The total of the computation times for 46 operating conditions was eleven minutes by the FACOM M360P in the computer center of the Hosei University. The total memory area was about 3MB. RESULTS AND DISCUSSIONS U2 (m/sec) Fig.3 Relationship between Total Pressure Ratio and Impeller Tip Speed (Ref.7) (1) Non-Dimensional Relative Velocity at Inducer Inlet The results of the non-dimensional relative velocities at inducer inlet given by Eq.(1) gave exactly the same results as in Fig.12 of Ref.7, which is shown again in Fig.4. In this figure, the predicted results in Ref.7 were valid only for the impellers with k,

6 straight radial blades, the slip factor of 0.90 and the specific speed of 100, 80 and 50. This means that the plotted symbols are the present results. Moreover, the straight lines which indicate the values of the specific speed given in Ref.7 were also valid for the present results obtained after the computations of Eqs. (4), (5) and (7). (2) Comparison of Resul ts betunen Adiabatic Flow and Assumed Viscous Flow Conditions at Impeller Exit The comparisons of the variations of non-dimensional relative velocities at impeller exit given by adiabatic flow, the three relationships between total pressuse ratio and impeller tip speed and the two experimental values of total pressure ratio for 04 Predicted Eq.(16), Ns^50 (/,I=0.90 ) Ns ,,., Tt Fig.4 Non-dimensional Relative Velocity at Inducer Inlet (Ref.7) the variation with flow rate are shown in Fig.5. It is apparant that the relationship between the maximum value of total pressure ratio and impeller tip speed gave the best result. All of the other results showed the less converged values with the fluctuations or the deviations from the values given by the method in Ref.(7). The adiabatic flow apparantly did not result in converged values with the spikes and the fluctuations in Fig.S. Because only the static pressure at impeller exit was given by Eq.(13), the iterative computations were necessary to get converged values among specific weight, relative as well as absolute velocitiy components and total pressure. which also gave influences on the values of slip factor, flow angle at impeller exit and specific speed in the outermost iteration loop. Thus,they could not reach to the converged values. On the other hand, when the total pressure was given at the exit, the outermost loop was used merely to find out the relationships among the values of the parameters which already approached to the converged values by the inner two iteration loops. If there did not exist the values satisfying the empirical condition at impeller exit, they fluctuated between those given by Eqs. (6) and (7). In addition, another modification was applied in the computations. One criterion for the slip factor which limited the value larger than unity was removed. By this modification, the results which did not converge were clearly found. The above mentioned maximum curve in Fig.3 gave only the values less than unity except for few flow rates. Moreover, the maximum curve gave the similar values as given by the formula of F. J. Wiesner (12) as in Fig.6. No limitation to keep the values at those by F. J. Wiesner were given during the computations. The maximum relationship between total pressure and impeller tip speed gave the reasonable values, and, thus, in the following, the results only by this relationship will be discussed. Although it will be \ I^ 3 04 Ref Adiabatic ---- Max. --- Av. in Fig.3 Min. itmax at 9min 7tmin at Tlmax r 0.2`` T Fig.5 04 Comparison of Non-dimensional Relative Velocity at Impeller Exit for Difference of Assumption of Total Pressure

7 better to use a modified relationship between them by introducing the effect of the variation with flow rate, this was not examined in the present study because such a relationship could not be found for the present compressors. (3) Non-Dimensional RelatiLe Velocity at Impeller Exit The variations of non-dimensional relative velocities at impeller exit for the corresponding flow rate are shown in Fig.7. The maximum and the minimum values obtained by the present method and the method in Ref.7 are shown by the symbols, respectively. The results of six impellers showed good agreements. Similar agreements could be also obtained for all of the other impellers. Thus, it was clear that the non-dimensional relative velocity at impeller exit was correlated to that at inducer inlet through the flow coefficient Wiesner Present Method ; rpm ---- A ; rpm 0.9 -" B ; 27920rpm K ; 33680rpm T, Fig.6 Comparison of Slip Factor between Present Results and Those Obtained by formula of F. J. Wiesner (12) N rpm o' A o2 A Present e C Method o D Ref.7 o' El El E A I A In order to investigate the variation of this non-dimensional relative velocity with flow angle at impeller exit, the same relationship as in Fig.14 of Ref. 7, which is shown in Fig.8(a) here, is compared with the present results in Fig.8(b). The resemblances between them can be seen. They also indicated that the velocity triangles were computed reasonably well by the N S Q Predicted (18)(7) I / 1= AQ Some of the Data were at the Surge All of the Data were at the Surge o(2 (Degree) Fig.8(a) Variation of Relative Velocity at Impeller Exit for That of Flow Angle at Impeller Exit (Ref.7) N 0.4 ^ fa^2 n7 //i Q1 0.2 P, Q3 Fig.7 Variation of Non-dimensional Velocity at Impeller Exit for That of Flow Coefficient G2 (Degree) Fig.8(b) Variation of Relative Velocity at Impeller Exit for That of Flow Angle at Impeller Exit (Present Method)

8 present method. Furthermore, the weak effects on the variations of losses for the variations of the flow rate could be estimated. The deviations of three results from the others showed that the convergences in the calculations were not enough and slip factors exceeded unity. In Fig.9, the relationship between specific speed and flow coefficient is shown. The present results were compared with those in Ref.7 by the same manner as in Fig.7. They exhibited similar values. The relationship between specific speed and Mach number of the peripheral component of the absolute velocity at impeller exit also showed good agreements between them (Fig.10). Thus, all of the computed values by this method resulted in quite the same ones given by the method in Ref.7. By the above mentioned comparisons, it became apparant that the non-dimensional relative velocity at impeller exit was given by the parameters of slip factor, specific speed, flow angle at impeller exit and flow coefficient. N rpm 200 o A Present Method o B Ref.7 o D v E V K o N rpm A A B C El F G o I Q2 Q3 0.4 SP Fig.11(a) Ratio of Relative Velocity at Inducer Exit to That at Inducer Inlet " N rpm Fig.9 Relationship between Specific Speed and Flow Coefficient 12 Q^ da o A e C N rpm 1.0 a El o A Present v F o B Method e G Q D Ref.7 o I F v K ,120 z e--e e 80 Q M^2 Q Fig.10 Relationship between Specific Speed and Mach number of Circumferential Component at Impeller Fig.11(b) Ratio of Relative Velocity Exit at Impeller Exit to That at Inducer Exit 8

9 (4) Ratio of Relatiue Velocity at Inducer Exit to That at Inducer Inlet The ratios of the relative velocity at inducer exit to that at inlet and that at impeller exit to that at inducer exit were compared in Figs. 11(a) and (b), respectively. Because the length of the inducer camberlines and the blade angles were mostly estimated ones from the drawings, they might involve some errors in their values. Notwithstanding, some qualitative considerations were found possible. Examination of Fig.11(a) revealed that the ratio at the inducer portion increased as the increase with flow rate and the tendencies of the curves for each impellers were similar. Because the incidence angle is the most important factor for this ratio, these tendencies can be realized by examining the velocity vectors in Fig.1. The variation of the circumferential component of relative velocity played an important role for that of the relative velocity within the inducer channel. On the other hand, the ratio at the radial portion of impellers were maintained nearly constant for the variations with flow rates as in Fig.11(b). The differences of the slopes in Figs.4 and 7 could be explained by this. Blade Loading of Centrifugal Impeller", ASME Paper 74-GT-143, (1974) 9. Mishina, H., "Effect of Relative Velocity Distribution on Impeller Efficiency", Trans. JSME, Vol , (1979) 10. Rodgers, C., "Specific Speed and Efficiency of Centrifugal Impeller", Performance Prediction of Centrifugal Pumps and Compressors", ASME, New York, (1980) 11. Bammert, K., "The Influence of the Meridional Impeller Shape on the Energy Transfer in Centrifugal Compressors", ASME Paper, 80-GT-48, (1980) 12. Wiesner, F. J., "A Review of Slip Factor for Centrifugal Impellers`, Trans. ASME, Ser.A. Vol.89-4, (1967) CONCLUSIONS The following conclusions were derived from the present study. (1) A simple method for the calculations of non-dimensional relative velocity at inlet and exit of both an inducer and an impeller was proposed. By employing a proper assumption about the total pressure at impeller exit. the present method clarified that the specific speed, the slip factor, the flow angle at impeller exit, the non dimensional radius at inducer inlet and the flow coefficient are the predominant parameters which govern the relative velocities within inducer and impeller channels. (2)Relative velocity varied greatly with flow rate within inducer channels. While the ratio of relative velocity at impeller exit to that at inducer exit is almost constant irrespective of the variation of the flow rate. (3) The computed results of the slip factor gave quite the same values as those obtained by the formula of F. J. Wiesner. REFERENCES 1. Dallenbach. F. et al., "Study of Supersonic Radial Compressors for Refrigeration and Pressurization Systems", WAD TR , (1956) 2. Boyce, M. P., "New Developments in Compressors", Proceedings of Texas A&M University, (1972) 3. Galvas, M. R., "Fortran Program for Predicting Off-Design Performance of Centrifugal Compressors", NASA TN D-7487,(1972) 4. Wu, C. H., "A General Theory of Three-Dimensional Flow in Subsonic and Supersonic Turbomachines of Axial-, Radial- and Mixed Flow Types", NACA TN 2604, (1952) 5. Marsh, H., "A Digital Computer Program for the Through-flow Fluid Mechanics in an Arbitrary Turbomachines Using a Matrix Method", ARC R&M 3509, (1968) 6. "Computational Methods in Turbomachinery", I Mech E Conference Publications, (1984) 7. Mizuki, S. et al, "A Study Concerning Performance Characteristics of Centrifugal Compressors", ASME Paper 85-GT-97, (1985) 8. Mizuki, S. et al, "Investigations Concerning the 9