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  • 251

    Non-dimensional aerodynamic design of a centrifugal compressor impeller

    A Whitfield, BSc, PhD, CEng, MIMechE School of Mechanical Engineering, University of Bath

    A fully non-dimensional preliminary design procedure for a centrfugal compressor is presented. The procedure can be applied for any desired pressure ratio to develop an initial non-dimensional skeleton design. The procedure is applied to compressor design for pressure ratios of 2, 6.5 and 8, and illustrates how the initial design can be developed without recourse to empirical loss models and the associated uncertainties.

    a A b C p i p

    f h m M M' M"

    ns N P PR Q r R Re T TR U W a

    B

    A h O S

    Y

    : 1 P

    P

    Y

    V

    6 0

    NOTATION

    speed of sound area blade or passage height absolute velocity slip velocity diameter function of enthalpy mass flowrate Mach number relative Mach number non-dimensional tip speed of the impeller =

    specific speed rotational speed (r/min) pressure pressure ratio = P,,/P,, volume flowrate radius gas constant Reynolds number = rit/(,ud2) temperature temperature ratio = TO2/To1 blade velocity relative velocity absolute flow angle, positive in direction of impeller rotation relative flow angle, positive in direction of impeller rotation isentropic stagnation enthalpy change ratio of specific heat stage efficiency non-dimensional mass flowrate = m/(polaoinr~) work input factor slip factor impeller inducer hub-shroud radius ratio density flow coefficient = m/bOl U, mi) head or enthalpy coefficient rotational speed (rad/s)

    UzIa01

    The MS was received on 23 October 19W and was accepted for publication on 12 June 1991.

    Subscripts B blade s shroud tip position 8 tangential component 0 stagnation condition 1 2

    station position at stage inlet station position at impeller outer diameter

    1 INTRODUCTION

    The initial preliminary design of a centrifugal compres- sor impeller provides the framework upon which the detailed passage shape must be built. This preliminary design provides the leading dimensions of the impeller together with the inlet and discharge blade angles. While it is always possible to modify the initial frame- work as the design develops it is important to minimize the iterative loops involved. To this end a number of papers have been published with a view to optimizing the initial design using detailed numerical optimization techniques [see Bhinder et al. (1) and Wang Qinghuan and Sun Zhiqin (2)]. These techniques search for the optimum dimensions using numerical techniques and empirical loss models to calculate the efficiency. Non- dimensional techniques are not usually adopted; instead the operating requirements, with the exception of the desired pressure ratio, are specified in absolute units and the impeller dimensions derived [see Osborne et al. (3) and Came et al. (4)]. Balje (5, 6) provides an excep- tion to this general approach and presents a non- dimensional design procedure based on the parameters of specific speed and specific diameter. This, however, seems to receive a mixed reception as the non- dimensional groups are composite parameters built up from the basic groups used to describe compressor per- formance, and they do not lend themselves to im- mediate comprehension-even the names are unhelpful descriptions. While basic non-dimensional groups are used to describe and assess the performance of the com- pressor, either different groupings are constructed for design purposes or the non-dimensional approach is abandoned. Here the non-dimensional groups widely adopted to describe compressor performance are also used to develop the initial conceptual design.

    A05690 Q IMcchE 1991 0957-6Sl9/91 $200 + .05 Roc Imtn Mcch Engrs Vol205

  • 258 A WHITFIELD

    The initial design procedure cannot take full account of the complex three-dimensional separated flow that occurs in the impeller. It is, however, important to have these complex flow phenomena in mind and carry out the initial design with a view to minimizing the poten- tial for flow separation. The later stages of the design process will then consider the detailed flow phenomena more fully.

    2 NON-DIMENSIONAL ANALYSIS

    The classical application of dimensional analysis to the basic parameters that influence the behaviour of a turbo- machine reduces the number of parameters involved from ten to six and leads to a functional relationship between the nondimensional groups as [see Whitfield and Baines (711

    f(PR, TR, e, Mu, Re, Y) = o (1) These basic non-dimensional groups are simple ratios of a parameter to a reference parameter, usually based on the inlet stagnation conditions; the Reynolds number is a simple ratio of forces. These basic groups are often combined to yield alternative parameters, for example the temperature ratio is usually combined with the pressure ratio and replaced by the more useful total-to- total isentropic efficiency through the relationship

    Similarly the pressure ratio is sometimes replaced by a head, or enthalpy, Coefficient defined as

    (3)

    It can be shown that this head coefficient is a com- bination of the pressure ratio and non-dimensional impeller speed through the relationship

    ? 1

    (4)

    In addition, the non-dimensional groups are often sim- plified for convenience by presenting them in a dimen- sional form. This practice will not be followed here; non-dimensional groups will be strictly adhered to.

    If the application is restricted to a single working fluid the ratio of specific heats ceases to be a variable and can be disregarded as a separate non-dimensional group. The flow Reynolds number may be considered a secondary variable provided it is high enough such that changes in magnitude have little effect on performance. The influence of the Reynolds number on compressor performance has received considerable attention (8-1 1) and should not be generally neglected; however, for an initial design study it can be considered to be a second- ary influence and equation (1) reduced to

    (5 ) These parameters are widely adopted for the presen- tation of compressor performance, with pressure ratio and efficiency presented as functions of non-dimen- sional mass flowrate and impeller speed.

    f (PR, qs, e, Mu) =

    Part A: Journal of Power and Energy

    This classical non-dimensional analysis employs the impeller outer diameter as a characteristic dimension and can be applied to a range of geometrically similar machines. When, however, design is considered the design options will not be geometrically similar and the major dimensions of the impeller must be considered in addition to the overall size. The inclusion of these dimensions transforms equation (5) to

    Increased efficiency, through improved design, leads to a reduction in impeller speed, Mu, for any desired press- ure ratio, or, if stress levels will permit, the ability to adopt increased blade backsweep. For design purposes the pressure ratio need not be treated as a variable as a single specific magnitude is required. For any design pressure ratio equation (6) can be rearranged to give

    (7)

    The application of dimensional analysis reduces the number of variables to be considered, but the further simplifying step of geometric similarity is not available. The designer must find the combination of non- dimensional parameters which will either maximize the efficiency or provide a satisfactory compromise with any other restraints, such as impeller size and speed limitations.

    When compressor design is considered some designers make extensive use of the alternative par- ameters of specific speed and specific diameter (5, 6, 12, 13) whilst others are either critical of this approach (14) or adopt a direct dimensional design procedure (3, 15, 16).

    Specific speed has been adopted from the application to incompressible flow machines and is usually defined as

    n, =

    Specific speed, like efficiency and head coefficient, is a composite parameter, being constructed through a com- bination of the basic non-dimensional parameters of equation (5 ) and as such is not a simple ratio of a parameter to a reference condition. It can be shown that specific speed is given by

    (9)

    where BjM, = #J = m/bOl U 2 nrl) is a commonly applied flow coefficient used as an alternative to 8. It should also be noted that the volume flowrate Q is defined at the inlet stagnation conditions, that is Q = m/pol. A machine with a high specific speed will have a large flowrate and a low head coefficient or enthalpy rise-usually an axial flow machine. A machine with a low specific speed will have a low flowrate and a large head coefficient, and a radial machine is usually

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  • NON-DIMENSIONAL AERODYNAMIC DESIGN OF A CENTRIFUGAL COMPRESSOR IMPELLER 259

    required. For design purposes the pressure ratio is fixed and the non-dimensional speed, Mu, is often limited and thereby fixed through stress considerations. Specifi- cation of the specific speed is then tantamount to specifying the non-dimensional flowrate, and as this gives the mass flowrate per unit frontal area directly it is to be preferred to the indirect use of specific speed.

    3 BASIC DESIGN APPROACH

    Generally a centrifugal compressor is required to produce a specified pressure ratio at maximum effi- ciency. The desire to achieve maximum efficiency may be compromised in order to meet other design restraints such as minimizing overall size and weight and maxi- mizing the flowrate between surge and choke. From equation (5) there will exist a non-dimensional flowrate 0 and non-dimensional speed Mu at which maximum efficiency will be achieved. The designer must establish these parameters and then develop the overall geometry of the impeller. From the resultant non-dimensional design a range of geometrically similar designs can be constructed to handle a range of flowrates which, in theory, should all yield the same performance, provided that Reynolds number effects are not significant.

    The design procedure will be developed from the desired pressure ratio, with specified target efficiencies for the impeller and complete stage. The impeller efi- ciency is required to effectively transform the desired stage pressure ratio to that required from the impeller only; however, the desire to maximize the efficiency will be maintained. In addition to establishing the optimum non-dimensional mass flowrate and impeller speed the non-dimensional geometry of the impeller will be devel- oped in terms of the radius ratio, r l s / rz , the discharge blade height, b2/rz , and the inlet and discharge blade angles.

    In order to maximize the efficiency it is necessary to minimize the losses. Whilst the losses are not explicitly calculated through the application of loss models it is essential to assess the consequences of any design choices on the loss-generating processes. For the impeller the losses commonly considered and the steps required to minimize them are as follows:

    Incidence loss. This is a function of the magnitude and direction of the relative Mach number at the impeller inlet. It is necessary to minimize the relative Mach number and also to select the blade leading edge angle to accept the flow with minimum inci- dence effects. Friction loss. Here it is necessary to ensure that the gas velocity relative to the metal surface is no larger than necessary; that is minimize the relative Mach numbers. It is also necessary to consider the flow path length and the magnitude of the effective hydraulic diameter of the passage. Diflusion or blade loading loss. Ensure that the overall diffusion across the impeller is not excessive ; that is assess the magnitude of the relative velocity ratio [see Rodgers (171.

    4. Clearance loss This 6 a function of the essential clearance gap between the tips of the rotating blades and the stationary shroud. A minimum clearance will be necessary, and the proportion of the flow passage

    @ IMechE 1991

    occupied by the clearance gap will increase as the actual blade height is reduced. It is necessary, there- fore, to assess the magnitude of the non-dimensional blade height, bJr2.

    5 . Diffuser system loss. The diffuser is not considered explicitly but it is essential to bear in mind the fact that the high Mach number discharged from the impeller must be diffused and consequently it is important to ensure that the discharge Mach number is no higher than necessary. Also if the flow angle from the radial direction is large the flow path through a vaneless diffuser will be long, and in addi- tion to high friction losses stall and flow reversals back into the impeller may occur, followed by violent surge.

    The essential objective of the design procedure is to establish the optimum velocity triangles at the inlet to and discharge from the impeller. This is carried out by specifying and systematically varying the absolute and relative flow angles. The case of swirl-free flow at the inlet is considered here and the absolute flow angle at the impeller inlet is then zero. For the absolute flow angle at impeller discharge Johnston and Dean (18) showed that an optimum flow angle, for design pur- poses, lies between 63" and 68". Similarly, Rodgers and Sapiro (19) considered the optimum flow angle to lie between 60" and 70". Osborne et al. (3) used a magni- tude of 70" in the design of an 8 : 1 pressure ratio com- pressor, whilst for a 6.5 : 1 pressure ratio compressor Came (16) indirectly used a magnitude of 75". For the illustrative examples used here an absolute flow angle of 65" will generally be adopted, with an assessment made of the application of alternative magnitudes. The rela- tive flow angles at the inlet and discharge are systemati- cally varied and assessed. Once desirable magnitudes have been identified the inlet and discharge blade angles must be derived through a knowledge of the optimum incidence angle and flow deviation respectively. Through the desired stage pressure ratio and target effi- ciency the non-dimensional speed of the impeller, U2/aol , can be found and the velocity -triangles estab- lished. The impeller radius ratio, r lJr2 , then follows from the derived blade speeds at impeller inlet and dis- charge. The impeller target efficiency must then be introduced to calculate the impeller discharge blade height, b2/rz .

    3.1 Impeller discbarge conditions As indicated above, the impeller discharge velocity tri- angle can be established through the specification of the relative flow angle. However, the main parameter to be established at the impeller discharge is the magnitude of the blade backsweep to employ. It is well established that backward swept blades are beneficial both from the point of view of maximizing efficiency and extending the stable operating range (4). Blade backward sweep is limited by the stressing imposed by the use of non- radial blade fibres, and it may not be possible to have that which is most aerodynamically desirable. As the blade angle is quoted in many published design ex- amples (3, 16) and is limited by stressing, it is specified rather than the relative flow angle, and the effect of pro- gressively increasing it assessed. The magnitude of the

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  • 260 A WHITFIELD

    relative flow angle is, however, a function of the devi- ation of the flow from the blade direction. The magni- tude of the flow deviation is usually derived through the specScation of the slip factor defined as, see Fig. 1,

    It can then be shown that the relative flow angle is related to the blade angle through [see Whitfield and -es (711

    tan Bs2 1 - P tan 012 tan B2 = - - - P P

    This expression is shown graphically in Fig. 2 for a range of slip factors and two assumed absolute flow angles. Figure 2 provides a means of relating the speci- fied blade angle to the actual relative flow angle and assessing the effect of alternative assumptions. It is interesting to observe that the natural phenomenon of slip is beneficial insomuch as it increases the effective backsweep for any specified blade angle. Consequently, the number of blades at the impeller discharge should not be chosen in order to minimize the slip, but to ensure a satisfactory blade loading and to minimize the effect of flow separation.

    Through the specification of the discharge blade angle and slip factor together with the absolute flow angle and a target efficiency it can be shown that the stage pressure ratio is given by

    (12) (PJ- ' ) ' l= 1 + (7 - l)qsm:

    A=---= ce2 P

    where

    U2 1 - tan &/tan a2 For any desired pressure ratio the non-dimensional speed of the impeller can be derived through equation (12). The speed, however, is often limited by stress con-

    G i p

    H

    Backward swept blade

    Fig. 1 Impeller discharge velocity triangles

    - a2 = 65 - - - a z = l 5

    0 / ;0.8

    0 1 0 / 0 I / ' /' 0.88

    0.92 O.%

    1 .o

    0 0 -10 -20 -30 -40 -50

    Blade angle Pe2 Fig. 2 Relative flow angle as a function of blade angle and

    slip

    siderations and it may be necessary to reduce the mag- nitude of blade backsweep if the desired pressure ratio is to be achieved.

    3.2 Impeller inlet conditions The inlet velocity triangle, at the inducer tip, can be established through the specification of the relative flow angle and one other parameter. The inducer design should be such that it passes the desired mass flowrate at the minimum possible inlet relative Mach number. Stanitz (20) described a procedure to maximize the flow- rate per unit frontal area for any desired inlet relative Mach number. He showed that the relative flow angle is then a function of the relative Mach number

    3 + y M 2 2M;', cos2 PIS =

    This expression provides a relative flow angle ranging from -56" to -62.5" at relative Mach numbers of 0.5 and 1.2 respectively [see Whitfield and Baines (7)]. It is, of course, possible to specify relative flow angles that do not satis@ the above expression; however, it is necess- ary to assess the consequences of doing this. To complete the velocity triangle one other par-

    ameter must be specified. For the procedure used here the impeller radius ratio, riJr2, was specified and sys- tematically varied. This leads to the magnitude of the non-dimensional inlet blade velocity to complete the velocity triangle. As alternatives, the inlet relative Mach number, absolute Mach number or non-dimensional flowrate, 6, could be specified.

    Q IMcchE 1991 Part A: Journal of Power and Energy

  • NON-DIMENSIONAL AERODYNAMIC DESIGN OF A CENTRIFUGAL COMPRESSOR IMPELLER

    1.0 J

    5 0.9- 8

    5

    ,M

    a

    2 5 0.8 - m

    m c

    0.7 -

    261

    4 DESIGN ASSESSMENT

    The design procedure was applied to pressure ratios of 2,6.5 and 8. The pressure ratio of 2 is a moderate press- ure ratio perhaps typical of turbocharger compressors (although current requirements are for higher pressure ratios). The other pressure ratios are more typical of those required for small gas gas turbines and were selec- ted for comparison purposes as detailed design data are available in the published literature.

    4.1 Design pressure ratio of 2 0 The non-dimensional speed of the impeller for a range of blade backsweep angles together with the discharge Mach number is shown in Fig. 3a. The clear advantage of backward swept blades is illustrated through the reduction in the Mach number of the fluid at discharge. This, however, is accompanied by an increase in the non-dimensional speed of the impeller, which in addi- tion to increasing the stress levels will also increase the blade speed at the impeller inlet. The non-dimensional speed of the impeller can be selected for any desired magnitude of blade backsweep subject to the uncer- tainties associated with the specification of the slip factor and stage eficiency.

    The inlet flow conditions are shown in Fig. 3b for a range of assumed impeller radius ratios and an inlet relative flow angle of - 60". Increasing blade backsweep and the accompanying increase in non-dimensional impeller speed leads to an increase in the inlet relative Mach number for any specified radius ratio. Whitfield and Baines (7) showed that the inlet absolute and rela- tive Mach numbers are related to the mass flowrate per unit frontal area through

    MfO 1 1 - v z M,,

    M ; : = M ; ~ + - -

    (3y- l)/Z(r - 1)

    x ( l+ - y ; M:.) (14) This is illustrated in Figs 4 and 5 where contours of non-dimensional mass flowrate and inlet relative flow angle are presented as a function of the inducer Mach numbers for discharge blade angles of 0" and -40". These results are presented for an inducer hub-tip radius ratio of 0.4. Aerodynamically the optimum hub-tip radius ratio is the smallest possible as this pre- sents minimum blockage to the flow. In practice the minimum hub size needed is usually fixed through con- siderations of either the minimum cross-sectional area to transmit the required torque and avoid critical vibra- tion problems or the space required to accommodate the required number of blades around the hub periph- ery. Clearly any desired mass flow per unit frontal area, 8, can be passed at a minimum inlet relative Mach number and as a consequence the inlet relative flow angle follows [see also equation (1311.

    Also included on Figs 4 and 5 are contours of impeller radius ratio. These were derived from the defi- nition of the non-dimensional mass flowrate as

    (15) m = r:. (1 - $) EL - CIS 8 =

    ~4POlaOl 4 Po1 a01 Q IMechE 1991

    a2 = 65 ,u = 0.85

    qs = 0.8 PR = 2.0

    Y -20

    I /\ I

    I I I 1 0.9 I .o 1.1 I .2

    Non-dimensional impeller speed Mu

    (a) Discharge Mach number

    I PI, = -60 0.7

    0.65

    0.6

    I Jr2 T

    I I I I 1

    0.9 1 .o 1.1 I .2 Non-dimensional impeller speed Mu

    (b) Inlet relative Mach number

    Fig. 3 Impeller Mach numbers for P , = 2.0

    The data provided in Figs 3, 4 and 5 can be used to provide the initial impeller skeleton design. The con- tours of non-dimensional mass flowrate shown in Figs 4 and 5 are also contours of specific speed and non- dimensional blade height, bJrz - The corresponding magnitudes are also tabulated in the figures.

    In addition, the relative Mach number ratio across the impeller can be represented by horizontal contours on Figs4 and 5. These are shown as the secondary ordinate. Contours of relative velocity ratio, W,$W,, could also be included on Figs 4 and 5, but for simpli- city have been excluded.

    Selection of the non-dimensional mass flowrate together with the minimum relative Mach number con- dition leads to the basic impeller dimensions rl$rz and

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  • 262

    1.0

    0.9 -

    0.8 -

    0.7 -

    0.6 -

    A WHITFIELD

    % .s E - RI 0.2 1.12 2

    - 2.4

    - 2.2

    - 2.0 0.c - 1.8

    - 1.6 I I I I I I I

    0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

    P, = 2.0 p = 0.85 U2/a,, = 0.897

    Be2 = 0

    r~, = 0.9

    v = 0.4 qs = 0.8 (12 =65

    1.0 -

    0.9 -

    0.8 -

    0.7 -

    0.6 -

    gs g .s - ,e B .- -

    - 2.2

    - 2.0

    - 1.8

    - 1.6

    bzlrz

    0.22

    0.167

    0.111

    bJr2 for any selected discharge blade angle. The adop- tion of blade backsweep, increased from 0" to -40" for the examples shown, leads to a reduction in the dis- charge Mach number, a reduction of the relative Mach number ratio across the impeller and an increase in the inlet relative Mach number. Whether there is a point

    where further increases in the degree of backsweep will lead to a reduction in performance due to the detrimen- tal effects of the increasing inlet relative Mach number offsetting the other benefits cannot be easily assessed. However, the degree of backsweep is probably limited in most cases by stressing considerations.

    P, = 2.0

    a2 = 65 PB2 = dm

    p = 0.85

    qS = 0.8 Y = 0.4

    tYz/aoI = 1.058 r ~ 1 = 0.9

    -70 -65 -60 -55 P I S 1

    e 0.2

    0.15

    0. I 0.75 r d r 2

    I I I I I I I I

    0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

    Inlet Mach number

    Fig. 5 Impeller radius ratio as a function of inducer conditions Part A: Journal of Power and Energy

    "s b2lr2

    1.32 0.24

    1.14 0.18

    0.93 0.12

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  • NON-DIMENSIONAL AERODYNAMIC DESIGN OF A CENTRIFUGAL COMPRESSOR IMPELLER 263

    For any degree of backsweep, increasing the non- dimensional mass flowrate leads to an increase in the inlet relative Mach number, the magnitude of diffusion attempted across the impeller, represented by M l J M 2 or W1,/W2. This leads to an increase in the impeller radius ratio rls/r2 and non-dimensional blade height

    For a non-dimensional flowrate of 0.2 an impeller radius ratio of approximately 0.8 is necessary for a radi- ally bladed impeller (Fig. 4), and for an impeller with a discharge blade angle of -40" the radius ratio at the minimum relative Mach number is approximately 0.75 (Fig. 5). Radius ratios in excess of 0.75 are probably impractical as the sharp turn from the axial inlet to the radial discharge will lead to increased losses. Clearly non-dimensional flowrates of the order of 0.2 cannot be achieved efficiently with a radial machine due to the required high impeller radius ratio and inlet relative Mach numbers. For high non-dimensional flowrates an axial flow machine is clearly required and it is not necessary to resort to specific speed to make this judge- ment. Low non-dimensional flowrates, for example below the 0.1 value shown, will lead to impellers with a small radius ratio and small non-dimensional blade height. This could give increased clearance losses and increased friction losses due to the long narrow flow channel, despite the reduced relative Mach numbers. Derivation of the optimum non-dimensional flowrate requires a detailed assessment of the losses, such as that carried out by Galvas (21), and it is doubtful whether empirical loss correlations will be sufficiently accurate to discriminate between the narrow options presented through Figs 4 and 5. The non-dimensional mass flow- rate for turbocharger compressors lies typically between 0.1 and 0.1 5, and this corresponds to the accepted range for specific speed. With increasing blade discharge back- sweep it is necessary to reduce the non-dimensional flowrate in order to avoid increased relative Mach numbers at the inlet.

    The above results have all been obtained with an assumed discharge flow angle of 65". The effect of modifying this assumption on the discharge Mach number is illustrated in Fig. 6 for a range of discharge blade angles. As the blade backsweep is increased the minimum Mach number moves to reduced flow angles. At a blade angle of -30" the minimum Mach number occurs at approximately 75". This large flow angle is probably unacceptable as it will lead to a long flow path through the vaneless diffuser. As the degree of backsweep is increased, however, the absolute flow angle decreases. This indicates that if increased blade backsweep becomes possible (through the use of improved materials, for example) it may be beneficial to design at discharge flow angles below 60". This will reduce the Mach number and shorten the flow path through the diffuser. Such possibilities could be readily assessed further.

    bzlr,.

    4.2 Design for high-pressure ratio The high-pressure ratio compressor designs presented in some detail by Osborne et al. (3) and Came et al. (4) have been used to assess the non-dimensional design procedure presented here. In both cases the impeller Q IMcchE 1991

    P, = 2.0 = 0.85 qr = 0.8

    Minimum Mach number

    i.

    2 $ L:

    - 70

    I 50 60 70 80

    Absolute flow angle

    Fig. 6 Discharge Mach number as a function of flow angles

    discharge blade angle was selected through mechanical considerations and in both cases a magnitude of -30" was selected. Through the specification of a head coeffi- cient, Y, of 1.6, Came et al. (4) indirectly specified an impeller discharge flow angle, u 2 , of 75.3", while Osborne et al. (3) specified a magnitude of 70". For each design, the stage and impeller efficiency and slip factor are taken as given by the respective authors.

    4.2.1 Design pressure ratio of 6.5 The impeller Mach numbers for a range of discharge blade angles and impeller radius ratios are shown in Fig. 7. In addition the effect of modifying the selected absolute flow angle at discharge from 75.3" to 70" and 65" is also shown. It can be seen that reducing the absolute flow angle leads to a significant increase in the discharge Mach number for a radial bladed impeller, but this effect becomes less significant as the blade backsweep is increased. As shown in Fig. 6 at high degrees of blade backsweep, a flow angle exists at which the Mach number is a minimum.

    Selection of the appropriate impeller radius ratio can be made through consideration of the non-dimensional mass flowrate together with the consequent magnitudes for the discharge non-dimensional blade height, the relative Mach number ratio and the relative velocity ratio. Contours of non-dimensional mass flowrate are shown in Fig. 8 where the magnitude of 0.0732 corre- sponds to that used by Came et al. (4). Also included in this figure is a contour for a discharge flow angle of 70". Selection of the minimum relative Mach number condi- tion leads to the inlet relative flow angle and it is then necessary to select the appropriate non-dimensional flowrate. Also included on the non-dimensional flow contours are the associated magnitudes of non- dimensional blade height and specific speed. Clearly selection of low non-dimensional flowrates will lead to a long narrow flow channel through low magnitudes for the impeller radius ratio and blade height. High magni- tudes, 0.1 and above in Fig. 8, will lead to high inlet

    Proc Instn Mcch Engrs Vol 205

  • 264 A WHITFIELD

    P, = 6.5 a2 = 75.32 qI = 0.87 P I S = -60.81 p = 0.92 q s = 0.78 Y = 0.45

    1

    P B 2 , -10

    \ x -20

    0.9 I I I I 1.4 1.5 1.6 1.7

    Non-dimensional impeller speed Mu (a) Discharge Mach number

    0.8 1 I

    I I I

    1.4 1.5 1.6 1.7

    Non-dimensional impeller speed Mu

    (b) Inlet relative Mach number

    Fig. 7 Impeller Mach numbers for P , = 6.5

    relative Mach numbers, in excess of unity, but no clear additional disadvantages. The modification of the dis- charge flow angle to 70" has a small effect on the design choices available. The non-dimensional blade height and specified speed are only slightly modified to 0.034 and 0.54 respectively.

    The impeller relative velocity ratio is shown in Fig. 9 and this can be used in the selection of the most appro- priate non-dimensional flowrate. The magnitude corre- sponding to the Came et al. design (4) is shown to be approximately 1.9. This does not correspond with the value of 1.6 quoted by Came et al. (4); however the author has not investigated the reason for this difference as it could be a function of the additional design con- straints imposed and/or the nature of the more detailed flow model used. Also shown is the consequence of modifying the discharge flow angle to 70" where a clear reduction in the relative velocity ratio is shown. Part A: Journal of Power and Energy

    For the non-dimensional design presented the overall dimensions can be selected through the specification of the minimum inlet relative Mach number condition, the absolute flow angle at discharge and the relative vel- ocity ratio, the discharge blade angle having been set at - 30" due to the mechanical limitations.

    4.2.2 Design pressure ratio of8 In this case the designers made the deliberate choice to design for a supersonic inlet relative Mach number of 1.2. The impeller Mach numbers for a range of dis- charge blade angles and impeller radius ratios are shown in Fig. 10, and this exhibits the same pattern as that for the previous designs. The design selected can be identified through the discharge blade angle of - 30" and the inlet relative Mach number of 1.2. The non- dimensional flow contours are shown on Fig. 11, together with the corresponding non-dimensional blade heights and specific speed. The contour of 0 = 0.109 is that indirectly used by Osborne et al. (3). Again the design selection has been based on the minimum inlet relative Mach number condition, and the non- dimensional flowrate, and hence specific speed, is based on the selection of an inlet relative Mach number of 1.2. At this pressure ratio selection of a subsonic inlet rela- tive Mach number will lead to a non-dimensional blade height below 0.03 and an impeller radius ratio of the order of 0.46. The relative velocity ratio across the impeller is shown in Fig. 12. In contrast to the 6.5 press- ure ratio design described above the diffusion ratios are much lower due to the selection of a discharge absolute flow angle of 70".

    Figures 11 and 12 also illustrate the effect of an assumed alternative magnitude for the inducer hub to shroud radius ratio, v . Osborne et al. (3) considered a range of magnitudes between 0.3 and 0.5, and selected the value of 0.5 in order to meet the imposed inlet rela- tive Mach number of 1.2 at a desired impeller rotational speed. As can be seen, a reduction in v to 0.3 makes it possible to meet the design requirements with a reduced inlet relative Mach number and a reduced impeller rela- tive velocity ratio. If the smaller magnitude is mechani- cally acceptable it is preferable to designing at the imposed inlet relative Mach number of 1.2.

    For the non-dimensional design procedure the impo- sition of a magnitude to the inlet relative Mach number locates the design point on Fig. 11 if the minimum inlet relative Mach number condition is accepted. As can be seen, this coincides closely with the flow angle adopted by Osborne et al. (3).

    5 DERIVATION AND ASSESSMENT OF THE ABSOLUTE DESIGN

    The non-dimensional design can be transformed into absolute dimensions through the specification of the inlet stagnation conditions and the required mass flow- rate. If further constraints are imposed, such as the spe- cification of the rotational speed of the impeller, then it may be necessary to compromise the non-dimensional design. As the non-dimensional speed of the impeller follows once the degree of discharge blade backsweep is decided the impeller diameter can be derived if the rota- tional speed is imposed together with the inlet stagna-

    Q IMechE 1991

  • NON-DIMENSIONAL AERODYNAMIC DESIGN OF A CENTRIFUGAL COMPRESSOR IMPELLER 265

    1.2

    1.1

    1 .o

    0.9

    0.8

    0.7

    0.6

    - a2 = 75.32 -X-X- a2 = 70

    0.031 1 0.0455 0.0626 b2/r2 0.435 0.526 0.616 n.

    I 1 I I I I I 0.1 0.2 0.3 0.4 0.5 0.6 0.7

    Inlet Mach number

    Fig. 8 Impeller radius ratio as a function of inducer conditions for P, = 6.5

    2.2

    2.0 g h .o 1.8 ti b '8 7 1.6 0 .- CI - 2 1.4

    1.2

    / 0.6 - - \

    / 0.42 0.5 0.46 - rd'2

    For a2 = 70

    I 1 I I 1 .o 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Inlet Mach number

    Fig. 9 Impeller diffusion ratio for P, = 6.5 @ IMcchE 1991 Proc Instn Mech Engrs Vol205

  • 266

    1.3 -

    s? 1.2 - B g

    3 .- 3 n

    2z M z 1.1 -

    A WHITFIELD

    The non-dimensional design effectively deals with flow conditions, that is flow areas and flow angles. To derive the desired geometry it is necessary to apply appropriate blockage factors to allow for boundary layer effects and non-uniform flows, as these have not been applied directly in the non-dimensional design. To convert the actual flow angles to blade angles it is necessary to consider the optimum incidence angle at blade inlet and flow deviation at blade discharge. By applying the slip factor directly in the design procedure the actual blade angle has been considered at the impeller discharge. The inlet blade angle follows from a specification of the optimum incidence angle. Came et al. (4) selected an optimum incidence angle of 7" to give a blade angle of 53.81". This selection was based on a requirement to achieve a specified flow margin (design flow/inducer choking flow). Osborne et al. (3) selected an incidence angle of 4" so that the change in incidence angle between the design point and surge was 5.25".

    The non-dimensional or absolute design can be further assessed by predicting a complete performance map of the proposed design using suitable empirical flow and loss models [see Whitfield and Baines (711. If the predicted performance does not meet the require- ments it may be necessary to return to the preliminary design procedure described above in order to reassess the design together with any constraints that were orig- inally imposed.

    P , = 8.0 a2 = 70

    /Ils = -60.3 p = 0.88

    q, = 0.865 q s = 0.825

    v = 0.5

    "O 1

    092 0

    - 30

    All rla'rz

    1.6 1 .7 1.8

    Non-dimensional impeller speed Mu

    (a) Discharge Mach number

    092 - 20

    1.3

    %- 1.2 j E

    f s 1.1 .- Y 3 - 2

    T2 d - 1.0

    - 30

    /

    /

    /

    I I I 1.6 1.7 1.8

    Non-dimensional impeller speed M,

    (b) Inlet relative Mach number

    Fig. 10 Impeller Mach numbers for P , = 8

    tion conditions. Consequently the non-dimensional flowrate follows from the specified mass flowrate and the design can be located on the inducer Mach number diagram, for example Fig. 11. Once the absolute dimen- sions have been determined it will be possible to see that the hub size is sufficient to carry the desired number of blades, and the original assumption with respect to the magnitude of the slip factor can be checked. Part A: Journal of Power and Energy

    6 CONCLUSIONS

    By applying a full non-dimensional approach to the preliminary design of a compressor impeller it is poss- ible to reduce the number of variables involved to man- ageable proportions such that the design options can be readily represented graphically. The non-dimensional design procedure has been applied to compressor designs with pressure ratios of 2, 6.5 and 8. The blade discharge angle is usually selected through stress con- siderations, and the inlet relative flow angle from the desire to minimize the inlet relative Mach number. With these blade angles readily fixed the remaining parameter to be specified is the non-dimensional mass flowrate, 0. This has usually been done indirectly; Osborne et al. (3) specified an inlet relative Mach number of 1.2 and Came et al. (4) specified a specific speed of 0.527. Such specifications effectively fix the non-dimensional design of the impeller, and the further specification of the required mass flowrate and inlet stagnation conditions leads to the absolute dirnensions. Nonetheless both authors carried out a detailed loss assessment in order to arrive at the 'optimum design'. This in effect only calculated the optimum inlet relative flow angle [Came et al. (4) gave a magnitude to two decimal places] and showed it to be very close to that which gives the minimum inlet relative Mach number ; the small differ- ence is likely to be due to the uncertainties associated with the loss models used.

    Selection of an appropriate non-dimensional flowrate is equivalent to the selection of the specific speed, and clearly different magnitudes are required for any desired pressure ratio and selected blade backsweep. Specific speed has no special qualities relative to the basic non- dimensional flowrate, which is widely used in many dis- guises for performance presentation. In any case design

    Q IMechE 1991

  • NON-DIMENSIONAL AERODYNAMIC DESIGN OF A CENTRIFUGAL COMPRESSOR IMPELLER

    P, = 8 PBZ = - 3 0 +-x- v = O . 3

    1 .3

    1.2

    1.1

    1.0

    0.9

    0.8

    0.7

    261

    -

    -

    -

    -

    -

    -

    -

    I I I I I I I

    -X- -X- v = 0.3

    t 1 .o

    0 0.1 0.2 0 . 3 0.4 0.5 0.6 0.7

    Inlet Mach number

    Fig. 12 Impeller diffusion ratio for P, = 8 Proc Insta Mcch Engrs Vol205 0 IMcchE 1991

  • 268 A WHITFIELD

    selection is more appropriately based on the choice of non-dimensional flowrate (flowrate per unit frontal area), inlet relative Mach number and/or the associated impeller diffusion ratio represented by the relative Mach number or velocity ratio. Given the additional constraints that are inevitably imposed on a compressor design (such as a specified rotational speed or a desire to maximize the flow range or minimize the inertia of the rotor), the designer may have little scope in selecting the optimum inlet relative Mach number through the application of loss models. It may well be found, however, that some of the design restraints may force the designer to select a design point away from the minimum inlet relative Mach number condition; for example selecting an inlet relative flow angle in excess of - 60 (for example - 65) will lead to an increase in the impeller radius ratio and a reduction in the absolute diameter with a consequent reduction in inertia. The non-dimensional design procedure described forms a sound basis from which the full design, with all the imposed constraints, can be developed.

    REFERENCES 1 Bhinder, F. S., Mashimo, T. and Jamad, S. N. On the use of

    numerical optimization in designing the radial impellers for cen- trifugal compressors. Tokyo International Gas Turbine Congress, 1987, paper 87-TOKYO-IGTC-7.

    2 Wang Quinghuan and Sun Zhiqin. Optimization design of the over-all dimensions of a centrifugal compressor stage. Gas Turbine Congress, Toronto, 1988, ASME paper 8843-134.

    3 Osborne, C., Runstadler, P. W. and Stacy, W. D. Aerodynamic and mechanical design of an 8 : 1 pressure ratio centrifugal compres- sor. Report NASA CR-134782, 1975.

    4 Came, P. M, McKenzie, I. R. I. and Dadson, C. The performance of a 6.5 pressure ratio compressor having an impeller with swept- back blades. NGTE memorandum 79013, Pyestock, UK, 1979.

    5 Balje, 0. E. A study on design criteria and matching of turbo- machines : Part A-similarity relations and design criteria of turbo- machines. Trans. ASME, J Engngfor Power, January 1962, 83.

    6 Balje, 0. E. Turbomachines: a guide to design selection and theory, 1981 (John Wiley).

    7 Whi~eld, A. and Baines, N. C. Design and radial turbomhines, 1990 (Longman Scientific and Tcchnical) (in USA, John Wiley).

    8 Wiesner, F. J. A new appraisal of Reynolds number effects on cen- trifugal compressor performance. Trans. ASME, J . Engng for Power, 1979, 101, 384.

    9 Casey, M. V. The effects of Reynolds number on the efficiency of centrifugal compressors. Trans. ASME, J. Engng Gas Turbines and Power, April 1985,107,541.

    10 Sbub, R A. et al. Influence of the Reynolds number on the per- formance of centrifugal compressors. Trans. ASME, J. of Turbo- machines, October 1987,109,541.

    11 Wright, T. Comments on compressor efficiency scaling with Rey- nolds number and relative roughness. ASME paper 89-GT-31, 1989.

    12 Rodgew C. and Langworthy, R. A. Design and test of a small two stage high pressure ratio centrifugal compressor. ASME paper

    13 Rodgers, C. Efficiency of centrifugal compressor impellers. AGARD report CPP 282, Brussels, May 1980.

    14 Vavra, M. H. Basic elements for advanced designs of radial flow compressors. AGARD lecture series 39, 1970.

    15 Jones, M. G. Impeller computer design package, Part I. A prelimi- nary design program. NGTE note NT 1002,1976.

    16 Came, P. M. The development, application and experimental evaluation of a design procedure for centrifugal compressors. Proc. lnstn Mech. Engrs, 1978,192,49.

    17 Rodgers, C. A diffusion factor correlation for centrifugal impeller stalling. ASME paper 78-GT-61, 1978.

    18 Johnston, J. P. and Dean, R. C. Losses in vaneless diffusers of centrifugal compressors and pumps. Trans. ASME, J. Engng for Power, 1966,49.

    19 Rodgers, C. and Sapiro, L. Design considerations for high pressure ratio centrifugal compressors. ASME paper 72-GT-91, 1972.

    20 Stanitz, J. D. Design considerations for mixed flow compressors with high flow rates per unit frontal area. NACA report RM E53A15, 1953.

    21 Calves, M. R. Analytical correlation of centrifugal compressor design geometry for maximum efficiency with spefific speed. NASA technical note TN D-6729, March 1972.

    74-GT-137,1974.

    Part A : Journal of Power and Energy Q IMechE 1991