# Non-dimensional Aerodynamic Design of a Centrifugal Compressor Impeller

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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

Q IMechE 1991

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 ov

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