The Design Space Boundaries for High Flow Capacity ... · PDF fileTHE DESIGN SPACE BOUNDARIES...

1 Copyright © 2012 by ASME

Proceedings of ASME Turbo Expo 2012

GT2012 June 11-15, 2012, Copenhagen, Denmark

GT2012-68105

THE DESIGN SPACE BOUNDARIES FOR HIGH FLOW CAPACITY CENTRIFUGAL COMPRESSORS

Daniel Rusch Compressor Development (Dept. ZTE)

ABB Turbo Systems Ltd. Bruggerstrasse 71a

CH-5401 Baden, Switzerland

Michael Casey Institute of Thermal Turbomachinery (ITSM)

University of Stuttgart, Germany and PCA Engineers Limited,

Lincoln, England

ABSTRACT A methodology has been derived allowing a fast

preliminary assessment of the design of centrifugal compressors specified for high specific swallowing capacity. The method is based on one-dimensional (1D) design point values using classical turbomachinery analysis to determine the inlet geometry for the maximum mass flow function. The key results are then expressed in a series of diagrams which draw out the nature of the conflicting boundary conditions of the design. In particular it is shown how the inlet casing relative Mach number causes the design flow coefficient to decrease with the total pressure ratio and determines the inlet eye diameter. Physically-based boundaries of operation are added to the diagrams giving guidelines for the proper choice of specification values to the designer. In addition, links are given to some well-known impeller efficiency correlations, so that a preliminary estimate of the performance can be made. Comparisons are made with a range of compressor data which supports the approach. The derived methodology allows any given specifications to be checked rapidly for feasibility and development risk or can be used to define a challenging specification for the design of a new product.

NOMENCLATURE 1A = area of compressor eye (m2) a = speed of sound (m/s) c = absolute flow velocity (m/s) pc = specific heat at constant pressure (J/kg/K) D = diameter (m) h = specific enthalpy (J/kg)

k = impeller inlet shape factor, 2111 sh rrk (-)

1cM = absolute Mach number at impeller inlet,

111 RTcM mc (-) 1wM = relative Mach number at impeller inlet casing

111 RTwM w (-)

2uM = tip-speed Mach number, 122 tu RTuM (-) m = mass flow rate (kg/s) p = pressure (Pa) R = gas constant (J/kg/K) r = radial location (m) s = specific entropy (J/kg/K) T = temperature (K) u = impeller blade speed (m/s) 1tV = volume flow rate, 11 tt mV (m3/s)

refTV = corrected volume flow rate,

11 treftT TTVVref

(m3/s) w = relative flow velocity (m/s) Greek Symbols s1 = relative flow angle (°) at casing of impeller inlet, relative to axial direction = isentropic exponent (-) = work input or enthalpy rise coefficient,

22uht (-)

= isentropic efficiency, total to total, tts hh (-)


p = polytropic efficiency, (-)

1t = global flow coefficient, 22211 Dum tt (-)

= mass flow function (equation (7)) = modified mass flow function (equation (17)) = stage total pressure ratio (-) = density (kg/m3) s = specific speed (-) = angular velocity (rad/s) Subscripts t = total conditions t1 = inlet total conditions 1 = inlet and inlet conditions 1h = inlet hub 1m = inlet meridional 1s = inlet casing (shroud) 2 = impeller outlet and outlet conditions ref = reference value INTRODUCTION

A common trend in centrifugal compressor development is to strive for more compact products and hence for stages with a high specific swallowing capacity. More compact stages have a lower weight and inertia, a smaller frontal area and a reduced cost. On the other hand, the compactness conflicts with the requirements for high pressure ratio and efficiency and with the mechanical limits of the impeller material. A suitable compromise has to be found in order to balance all requirements and to end up with a realizable specification.

In turbochargers and industrial compressors, a smaller size results in lower material and machining cost and lower weight of the product. Lower weight and dimensions are also favorable for the overall cost of the machine and the supporting structure and also for the natural frequencies of the machine, provided they do not compromise fuel consumption. The corresponding lower rotor inertia is beneficial for fast acceleration and response of the compressor which is a requirement during engine load increase in turbocharger applications. In aircraft engine applications, a high swallowing capacity of the centrifugal compressor also results in a frontal area reduction. This helps to decrease the aerodynamic drag on the nacelle and hence reduces fuel consumption. In addition, the lower engine weight directly reduces the fuel consumption as the required aircraft lift decreases.

The swallowing capacity also strongly influences the thermodynamic performance of the compressor. Rodgers [1] has presented various correlations of the efficiency for stages tested in air as a function of specific speed, defined as follows

431

431 2 t

ts

ts h

V with

(1)

22211 uDV tt ,

2

22Du

and 22uhh tts

Most of Rodgers’ published correlations are based on impeller efficiency, but an example of one presented in terms of stage efficiency is given in Figure 1, taken from Rodgers [2]. This correlation suggests that an optimum specific speed exists for a given stage pressure ratio and shows, by the dashed lines, that at higher pressure ratio a high inducer Mach number is required. The background to this will be examined in more detail in this paper.

Figure 1: Dependence of stage adiabatic efficiency on the dimensionless specific speed ( ss N ), pressure ratio and impeller inlet casing relative Mach number ( sw MM 11 ) given by Rodgers, [2] for high pressure ratio stages.

Figure 2: Dependence of compressor polytropic efficiency on the flow coefficient and tip-speed Mach number ( MuM u2 ) derived by Casey and Robinson, [3].


In more recent work, Casey and Robinson [3] and Robinson et al. [4] reformulated the Rodgers’ correlations, together with those from other sources, in terms of the polytropic compressor efficiency as a function of the compressor inlet flow coefficient and the tip-speed Mach number as shown in Figure 2. The background and equations to this correlation are given in the appendix. The inlet flow coefficient is sometimes known as swallowing capacity coefficient or “gulp factor” but will be referred to here as simply flow coefficient, and the flow coefficient and tip-speed Mach number are defined as

22212

22

11 uD

muD

V

t

tt (2)

1

22

tu

RTuM (3)

This correlation shows that an optimum flow coefficient can be found for a given tip-speed Mach number and that the optimum swallowing capacity reduces for a higher tip-speed Mach number level. The differences to the Rodgers’ correlation are discussed below.

Figure 3: Dependency of isentropic efficiency on the polytropic efficiency and total pressure ratio according to equation (4).

The measure of efficiency predicted by this correlation is the polytropic efficiency, as defined in equation (4) below, as it is a more suitable measure of the aero-thermodynamic quality of the design than the isentropic efficiency, see Dixon [5]. As a result of this, the effect of tip-speed Mach number on efficiency appears to be much weaker in Figure 2 than the effect of pressure ratio in Figure 1. This thermodynamic effect arises because for a constant value of the small-scale polytropic efficiency the isentropic efficiency falls with the pressure ratio as shown in Figure 3 and according to, see [6]:

1

11

1

p

(4)

The efficiency is correlated as a function of the tip-speed Mach number, rather than the pressure ratio, as it then becomes thermodynamically correct for different gases. Different pressure ratios occur at the same Mach number levels if the gas is changed, as the pressure ratio in an ideal gas is given by

)1/(22)1(1 uM (5)

In addition the specific speed in Rodgers’ correlation, which includes flow capacity and head rise in its definition, see equation (1), is replaced by the flow coefficient as a measure of the non-dimensional swallowing capacity. The justification for this is that two stages with the same non-dimensional flow capacity, but a different non-dimensional head rise, would have different values of the specific speed, so the specific speed is clearly not a good measure of swallowing capacity alone. In fact the head rise is determined mainly by the design of the impeller outlet (backsweep) and the swallowing capacity by the impeller inlet (throat, inlet eye diameter) so the specific speed confuses these independent features. In order to avoid this confusion the specific speed needs to be used together with a similar non-dimensional parameter, known as the specific diameter, and then its use would be acceptable. Unfortunately most turbomachinery work does not specify both of these parameters.

Other disadvantages of specific speed are its complexity, in that it includes the square root of the flow capacity, as this also means the value of the specific speed has no direct link to the continuity equation. Despite its disadvantages, many publications on centrifugal compressors make use of the specific speed rather than the inlet flow coefficient to characterize the non-dimensional swallowing capacity of a centrifugal compressor. It should be noted however that the use of a specific speed is only reasonable for comparing swallowing capacity in those cases, like centrifugal compressors, where the non-dimensional head coefficient remains sensibly constant over the range of designs being considered.

The efficiency correlations described here, and also others such as that of Aungier [7], show that there is an optimum non-dimensional swallowing capacity, and that the efficiency decreases towards higher or lower values. In addition, the optimum non-dimensional swallowing capacity is reduced with tip-speed Mach number or pressure ratio. This behavior is the subject of this paper.

The underlying physical mechanisms for this increase in losses at high flow capacity and tip-speed Mach number are manifold, but are fundamentally related to the need for a higher inlet area with a larger impeller eye diameter for higher non-dimensional flow capacity. Firstly, the use of a higher


impeller inlet eye diameter at high flows, or higher trim, leads to less centrifugal effect along the casing streamline of the impeller and more diffusion is then needed to achieve the same pressure rise with a potential for higher losses. Secondly, the higher trim gives an increasingly sharp turning of the axial inflow towards the radial direction in the impeller and this can cause higher hub to shroud loading with an associated increase in secondary flow losses. Thirdly, limiting the eye diameter to avoid these problems causes higher axial flow velocities in the impeller inlet with higher frictional flow losses and kinetic energy leaving losses at high swallowing capacity. Finally, the increased impeller inlet casing relative Mach number associated with a high eye diameter or a high axial inlet velocity causes possible shock losses and a requirement for a higher diffusion level in the impeller. Hence, striving for more compact centrifugal compressors by increasing the swallowing capacity can result in insufficient stage performance. The design of a centrifugal compressor for high flow capacity inevitably requires a careful balance between the compactness and efficiency, especially in high pressure ratio stages with high Mach number levels.

OBJECTIVE

A method for a rapid assessment of thermodynamic performance and compactness requirements is presented in this paper. A one-dimensional steady (1D) analysis is used and combines stage pressure ratio, swallowing capacity, impeller inlet casing relative Mach number, tip-speed Mach number and the dimensionless impeller eye diameter. It is shown how these dimensionless numbers are related and can be displayed in one diagram.

The new method is based on the mass flow function, see Dixon [5], which is derived in the next chapter. Based on the mass flow function, the swallowing capacity and the dimensionless impeller eye diameter are expressed as a functions of gas and impeller inlet casing flow properties, stage efficiency, work input parameter, impeller inlet shape factor and total pressure ratio. A method is then described to provide a single plot for design parameter assessment. A new feature of this work is that this plot links the impeller inlet eye dimensions to the overall impeller dimensions and the overall pressure ratio. The plot can be directly linked with efficiency correlations both on the basis of specific speed and flow coefficient.

THE MASS FLOW FUNCTION For highly loaded centrifugal compressors with high

swallowing capacity it is necessary to limit the inlet relative Mach number at the casing inlet. Stages designed for other applications may require other design strategies. In transonic inducers, the entropy generation due to the shocks has to be controlled for high efficiency stages. The loss directly related to the entropy production across the shock is small but

increases steeply with the upstream Mach number roughly as [8]:

32 1Ms (6) More importantly, the pressure rise at the shock can cause

boundary layer separation and be a cause of further losses. According to Bölcs, [9] separation due to shock interaction in an axial compressor cascade occurs if the shock upstream Mach number exceeds approximately 1.35. The separated boundary layer itself causes additional losses due to the related secondary flow and mixing losses. Careful design of the impeller blade profile at inlet is required for higher relative supersonic inlet Mach numbers, see Lohmberg et al. [10].

It is well known that an optimum inlet blade metal angle at the impeller inlet casing exists which minimizes the relative flow Mach number at the impeller inlet casing for a given mass flow, rotational speed and inlet total conditions. A derivation of the governing equations can be found in Dixon, [5] and a useful discussion can be found in Lohmberg et al. [10]. Additional relevant and similar derivations can be found in Whitfield and Baines [11] and in Stanitz [12]. The set of equations for zero inlet swirl is derived here in a slightly different form using a clearer definition of the relevant non-dimensional parameters. We define the mass flow function of the stage as the mass flow relative to that which can pass through an area of 2

2D with a gas velocity equal to the inlet total speed of sound with the density at inlet total conditions as

211

22221

2211

uttttt

Mau

Dum

Dam (7)

Figure 4: Velocity triangle at impeller inlet casing.

This formulation of the mass flow function differs to that used by Dixon. This is consistent with the conventional definition of reduced mass flow and replacing the density from the ideal gas relation and speed of sound by the usual ideal gas equation leads to the usual definition of the non-dimensional mass flow function as )/( 1

221 tt pDRTm . In this form, however, the

equation immediately highlights the physics that a high mass

mc1

sw1 su1

s1

Axis of rotation


flow function is related to a combination of both a high flow coefficient and a high tip-speed Mach number. Note also that the correlation of Casey and Robinson [3] given in Figure 2 and the appendix uses the product of flow coefficient and tip-speed Mach number for the effect of Mach number on efficiency. If we substitute the mass flow at the impeller inlet from the continuity equation as mcAm 111 the mass flow function can be expressed as follows:

2211

111

DacA

tt

m (8)

Following Dixon, the inlet area of the compressor eye is expressed using an impeller inlet shape factor k which represents the hub blockage area:

21

21

211 )4/()( shs DkrrA with

2111 sh rrk and 2121 // uuDD ss

(9)

For uniform axial inflow, the velocity triangle yields ssm wc 111 cos and sss wu 111 sin (10)

where mc1 denotes the inlet axial absolute velocity, sw1 the relative flow velocity and s1 the angle between the relative and absolute flow at the impeller inlet casing, see Figure 4. The axial velocity is assumed to be uniform over the inlet area. Assuming zero incidence angles, which is realistic at high transonic Mach numbers, the angle s1 corresponds to the blade metal angle at the impeller inlet casing. With the above equations, the mass flow function can now be expressed as follows:

ssts

tt

sst

s

t

ua

aw

aak

uaw

k

112

22

21

31

31

31

31

1

1

112

221

31

1

1

cossin4

cossin4

(11)

The definitions of the relative and absolute inlet Mach numbers

swssmc MawacM 11111111 coscos with

111 aMw ws (12)

are used to rewrite the mass flow function as:

ssu

w

tt MM

aak 11

222

31

31

31

1

1 cossin4

(13)

In the next steps, the static flow quantities at the impeller inlet are replaced by the total ones using the following expressions

)1/(1212

111 1 ct M

2/1212

111 1 ct Maa

(14)

The absolute inlet Mach number is replaced by the relative Mach number as follows:

swc MM 111 cos (15)

to relate the relative Mach number at the impeller inlet casing and the relative flow angle to the mass flow function. The final result is:

23)1/(1

122

121

112

22

31

cos1

cossin4

sw

ss

u

w

MMM

k (16)

And this can be reformulated to yield the modified mass flow function as:

23)1/(1

122

121

1123

122

cos1

cossin4

sw

sswu

M

MkM

(17)

which is the same expression as derived by Dixon [5] for his modified form of mass flow function, and similar to that given by Whitfield and Baines [11].

The modified mass flow function swM 11, is plotted over the relative inlet flow angle s1 for a series of impeller inlet relative Mach numbers and two different values of the isentropic exponent in Figure 5. These represent air (1.4) and a high molecular weight refrigerant gas (1.2). The important feature of this curve is that for a given inlet mass flow function there is a certain inlet angle leading to the minimum relative inlet Mach number. This can be explained physically by the fact that a smaller inlet flow angle is accompanied by a low inlet casing diameter and a higher inlet Mach number due to the small inlet flow area. A large inlet angle is accompanied by a large casing radius and in this case the inlet Mach number rises due to the increased inducer casing blade speed. Between these extremes there is an optimum flow angle that leads to a maximum inlet relative Mach number.

Figure 5: Modified mass flow function for a centrifugal compressor with zero inlet swirl and for two different values of the isentropic exponent based on equation (17).


A curve connecting the maxima of the mass flow functions for different impeller inlet casing relative Mach numbers is included. This curve can be intersected with a given design mass flow function value to yield the optimum design location with respect to inducer shock related losses as it represents the lowest impeller inlet relative Mach number possible for this design value. An analytical expression for the optimum inlet angle has previously been given by Whitfield and Baines [11], but from equation (17) a more simple expression for the optimum relative flow angle can be derived as:

1

1211

21

,1 22323

cosw

wwwwopts M

MMMM (18)

As can be seen, at low absolute inlet Mach numbers, swc MM 111 cos , the term on the bottom line of equation

(17) becomes unity so that the maximum mass flow function then occurs at the peak value of ss 11

2 cossin which is

2tan 12

s or 74.541s . The optimum angle increases with increasing Mach number and is about 60° for a typical relative inlet Mach number of unity at the impeller inlet casing. For a given relative Mach number the dependence of the mass flow function on the choice of angle is weak so that other angles within ±5° from the optimum do not substantially change the mass flow function. The throat area selected is considered to be an aspect of the detailed design where other tradeoffs may be appropriate, but these are only possible when the correct inlet Mach number and inlet angles have already been selected to ensure low relative inlet Mach number. The designer may choose slightly lower values than the optimum to increase the throat if he is concerned with choke, or higher values if he is interested in decreasing incidence for higher surge margin.

It has to be stated that in the derivation of equation (16) effects due to metal blockage, boundary layer blockage, incidence angle and non-uniform span-wise distribution of axial velocity have been neglected. The effects of flow non-uniformity and blockage can be included by a small modification of the shape factor k.

FLOW COEFFICIENT AND DIAMETER RATIO To derive an expression for the swallowing capacity,

equation (16) is further processed. Firstly we use equation (7) and (17) to derive an expression for the flow coefficient, as follows:

32

1 4 ut M

k (19)

This is plotted in Figure 6 which shows the variation of tip-speed Mach number with flow coefficient and impeller inlet casing relative Mach number for the optimum designs at the peak mass flow function in Figure 5. For a design at a given impeller inlet relative Mach number, the flow coefficient

decreases as the tip-speed Mach number is increased. Designs for high flow coefficient and high tip-speed Mach number are penalized by a higher inlet relative Mach number. The solution to this problem is also shown in the diagram in that designs for higher pressure ratio automatically require a lower flow coefficient (or a lower trim) to avoid high impeller inlet relative Mach numbers.

Figure 6: Dependence of 2uM on kt1 and different values of according to equation (19).

In practical applications it is useful to express the tip-speed Mach number in terms of the pressure ratio, leading from equation (5) to:

)1()1(

1

22uM (20)

so that finally the flow coefficient can be expressed as 2/3

11

1

)1(4

kt (21)

The actual impeller inlet relative Mach number required in equation (16) and (17) for the modified mass flow function is however a function of the impeller inlet casing diameter, so that we also need to derive an expression for this in terms of the other parameters. From the velocity triangle at the impeller inlet, shown in Figure 4, an expression for the dimensionless impeller inlet diameter can be derived by expressing the inducer tip-speed as a function of inlet total condition, relative Mach number and relative flow angle. The relations

)cos( 111 swc MM 12

121

11 1 ct MTT

11 RTa , 1111 cm Macc , 111 wMaw )sin( 1111 sws Mau

(22)


are combined to express the inducer casing velocity as a function of relative Mach number, flow angle and total inlet temperature:

11

2212

11111 )(cos1)sin( swtsws MRTMu (23)

Using the definition of the angular velocity

2

2

1

1

2 Du

Du

s

s (24)

and combining this with equations (5) rewritten as 12

2 11

tpTcu (25)

and (21) finally yields the expression for the non-dimensional impeller inlet casing diameter:

21

122

121

112

1

1)(cos1

1)sin(1

sw

sws

MM

DD (26)

Equation (21) is visualized in Figure 7 for a given parameter set of the total-to-total stage efficiency, the work input parameter and the inlet shape factor and for a total pressure ratio of 4.5 in air. The overall shape of the curves is the same as for the modified mass flow function shown in Figure 5. This can be confirmed by examining equation (19) as the inlet flow coefficient scales linearly with the mass flow function for a fixed pressure ratio. A minimum impeller inlet relative Mach number can be found for any specified inlet flow coefficient. The corresponding points in the figure are the optimum impellers with a maximum mass flow function which lie on the maximum of each curve indicated with the circle symbol.

Figure 7: Flow coefficient according to equation (21). The diameter ratio values are given for an example design at an inlet flow coefficient of 09.01t in air.

The diameter ratios according to equation (26) are also given in Figure 7 for a typical value of flow coefficient of 09.01t . With this value, the minimum impeller inlet casing relative Mach number is about 1.1 for a stage with design total-to-total pressure ratio of 4.5, a flow coefficient of 0.09 and the given parameter set according to the presented 1D theory. The corresponding diameter ratio is roughly 0.65 and the relative flow angle at the shroud is about 61.6°. Designers of radial compressor impellers will recognize these values as being typical for the most common styles of design.

Before continuing it is worthwhile clarifying the utility of these equations and summarizing the results so far. Firstly the analysis has been derived in a new form which gives clearer indication of the relevance of the terms and the relationship of the mass flow function to the swallowing capacity and tip-speed Mach number. For a given impeller inlet casing relative Mach number, which may be considered to represent the difficulty of the impeller inlet design or the technology level of the design, we can select an appropriate blade inlet angle (near to 60°) which minimizes the impeller inlet relative Mach number and calculate the modified mass flow function . The new analysis then shows that from equation (21) the inlet flow coefficient of the design is related to the required pressure ratio and the scale factor k . A higher pressure ratio or a higher tip-speed Mach number requires a lower flow coefficient. So this equation provides us with a way of examining how the non-dimensional swallowing capacity of the stage varies with Mach number, trim shape factor, and pressure ratio. Equations (21) and (26) show that both the swallowing capacity as well as the impeller diameter ratio are independent of the inlet total conditions and can be expressed as functions of the inlet shape factor k , the isentropic exponent (gas property), the stage isentropic efficiency , the work input factor , the stage pressure ratio and of the impeller inlet relative Mach number 1wM and flow angle s1 , which can be expressed functionally with the new equations as:

swt MkF 1111 ,,,,,,

sws MFDD 11221 ,,,,, (27)

ALTERNATIVE FLOW CAPACITY EXPRESSIONS For turbocharger centrifugal compressor maps, the mass

flow rate can be corrected for total inlet temperature using a corrected volume flow rate,

refTV . The mass flow is then expressed as:

ref

ttT T

TVmref

11 (28)

The conversions of the different flow rate definitions are as follows:


2

1

222

1

2222

22

11

11

uref

tT

ref

tT

tt MT

TuD

V

TT

VuDuD

V ref

ref (29)

With the equations (21), (25) and (29) an expression for the specific flow rate can be derived as:

kRT

D

V refTref

14

)1(1

2/1

22

(30)

This definition of specific flow rate has the dimension of a velocity and is often used in matching calculations of the turbocharger compressor with the turbine. Another common alternative formulation of the swallowing capacity would be in terms of the specific speed in which case equation (21) and equation (1) would need to be combined.

PRESSURE RATIO VERSUS SWALLOWING CAPACITY

In this section, charts showing the stage total pressure ratio against the stage swallowing capacity are elaborated. To come up with a figure to represent the total-to-total pressure ratio versus flow coefficient, it is assumed that the optimum point for centrifugal stage design lies on the maxima-curve in the swt MkF 1111 ,,,,,, diagram as this minimizes the impeller inlet relative Mach number and its associated losses. In the first instance, the values of the shape factor k , and the efficiency and the work coefficient ( and ) are kept constant.

Figure 8: Stage total pressure ratio vs. swallowing capacity expressed as inlet flow coefficient, assuming constant efficiency.

Using these assumptions, the optimum points plotted in Figure 7 using circular symbols can be determined for a range of pressure ratio and plotted against the flow coefficient 1t as shown in Figure 8 to yield the iso-lines of impeller inlet relative Mach number (colored lines). As for each point 1wM and s1 are known, the diameter ratio can be computed according to equation (26) and also plotted as solid iso-lines. The lines of constant tip-speed Mach number are evaluated according to equation (5) and indicated with dashed lines. The exemplary design point used in Figure 7 is highlighted in Figure 8 with the red dot and labeled as DP.

The plot can be translated to the dimensionless specific speed formulation presented by Rodgers, [1] using equation (1), and as shown in Figure 9, as follows:

143

2ts (31)

This diagram has the disadvantage over Figure 8 that the band of curves shifts horizontally when the work coefficient is varied. This identifies the clear advantage of flow coefficient over specific speed as a measure of flow capacity.

Figure 9: Stage total pressure ratio vs. swallowing capacity expressed as specific speed, assuming constant efficiency.

Instead of assuming constant efficiency over the entire

,1t space, the efficiency correlation shown in Figure 2 can be used. In this case a parameter set of ,, k and initial values for ),( 1t have to be specified. ),( 12 tuM is then computed with equation (5). Knowing the tip-speed Mach number, a new estimate for the isentropic total-to-total efficiency ),( 1t can be derived from Figure 2 and equation (4). The iteration is performed until convergence in ),( 1t


is reached. Afterwards, ),( 1t is computed from equation (19) which in turn is used to derive ),( 11 twM and

),( 11 ts from equation (17) by intersecting the optimum line, shown in Figure 5, at the given value of ),( 1t . The diameter ratio is then computed using equation (26). An example of such a result is given in Figure 10. Note that the lines of constant tip-speed Mach number are now no longer lines of constant pressure ratio, but become lower at high and low flow coefficients, see equation (5).

DISCUSSION As a result of the assumptions made during the derivation

of the pressure ratio versus swallowing capacity chart, Figure 10, each point in the chart represents a compressor geometry specially designed for this point. The chart should not be confused with a performance map for a given impeller geometry as the off-design effects of flow on efficiency and work are not included, which is not a realistic assumption for a whole compressor map. Limitations of the method

The underlying assumption, for the derivation of the swallowing capacity plots, is that the optimum design point is characterized as the point of minimum inlet casing relative Mach number. Clearly, the method is limited to designs that are carried out with this objective, but as this is usually done in high flow capacity compressors this is not a serious limitation. In addition, the following parameters need to be specified:

impeller shape factor, k work input coefficient, isentropic stage efficiency,

In Figure 8 and Figure 9 the efficiency is specified as a constant value for the whole map, which is not entirely realistic at very low or high flow coefficients as viscous or Mach number effects will contribute to an efficiency deficit. Figure 10 takes the efficiency to be that of the correlation shown in Figure 2 and the efficiency then falls to high and low flow coefficients, as expected. It is worthwhile to note that no information about total inlet flow conditions needs to be specified.

Nevertheless Figure 8 and Figure 9 remain useful, as the expected achievable isentropic efficiency can be used to represent the design point, so the diagram is correct in the neighborhood of this point. For the examples given in Figure 8 and Figure 9 with 5.4 the design point isentropic efficiency is computed as about 83%. This value is consistent with one chosen to plot the swallowing capacity diagram.

The efficiency values given in Figure 2 are values for compressors with an impeller diameter of 450mm and low clearances. More detailed information can be found in the appendix. Depending on the investigation, the values have to be corrected for Reynolds number effects due to different

impeller sizes and other effects using correlations in the open literature. Instead of using the inlet flow coefficient, the investigation can also be performed in terms of specific speed by comparing Figure 9 and the efficiency correlations presented in Figure 1.

Figure 10: Stage total pressure ratio vs. swallowing capacity expressed as inlet flow coefficient, but using the efficiency correlation according to Figure 2.

A further limitation of the method is that it is one-

dimensional and it assumes a constant meridional velocity across the span. In fact the curvature of the impeller causes an acceleration of the flow at the casing of the inlet which is stronger for short stages with a higher curvature. To take this into account the parameter k in equation (9) needs to be adjusted by an additional acceleration factor for the effect of the curvature, which needs to vary for stages of different axial lengths. Physical limits in compressor design

Before discussing the physical limits it is useful to examine the value of the flow coefficient and the modified mass flow function for a case with a few simple approximations. The modified mass flow function, equation (17), has a peak close to 60° (see Figure 5), so this angular value is used in the approximation. In addition, the correction for the change from total to static conditions in the term on the bottom line of equation (26) is close to unity for low inlet Mach number 1cM (see equation (15)). Equations (17) and (26) then simplify to

sswu M

kM

1123

1

22 cossin4 (32)


21

112

1

1

1)sin(1sw

s MDD (33)

Combining equations (21) and (33) and substituting by equation (32) gives

)sin()cos(

4)sin(4 1

1

3

2

13

11

3

2

11

s

ss

sw

st D

DkMD

Dk (34)

Approximating 601s and applying equation (7) leads to 3

2

11

34 DDk s

t (35)

2

3

2

1

34u

s MDDk (36)

Two distinct physical swallowing capacity limits can be identified from this and are shown in Figure 11. For low design total pressure ratios, the swallowing capacity is limited by the impeller diameter ratio and not by the impeller inlet casing relative Mach number, indicated with the bluish zone in Figure 11 ( 75.021 DD s ). Turbocharger stages and industrial process compressor stages with pressure ratios below 3 (or a tip-speed Mach number below 1.2) tend to be limited in flow capacity by the impeller diameter ratio. The impeller inlet casing relative Mach number is low and the increase in losses with high flow coefficient effectively determines the design flow capacity limit. This mechanism can also be identified in Figure 2 following a curve of constant 2uM towards high inlet flow coefficient. As the diameter ratio approaches 1, the geometry would transit towards a mixed flow design or an axial compressor. A typical upper limit for low pressure ratio process centrifugal compressors is 725.021 DD s and with a large hub diameter equation (35) leads to a maximum flow coefficient of around 0.13, which is the typical maximum flow coefficient found in such machines. Higher values might be acceptable if a tradeoff between swallowing capacity and stage efficiency can be found for a given application.

For high design total pressure ratio, the swallowing capacity is limited by the impeller inlet relative Mach number 1wM , indicated with the reddish region in Figure 11 ( 2.11wM ). In this region it is the mass flow function that is limited by the value of the impeller inlet relative Mach number. The physical background for the maximum impeller inlet relative Mach number is the shock boundary layer interaction, as discussed in the introduction. High values of the impeller inlet relative Mach number are inconsistent with highly efficient centrifugal compressors, as 1wM represents an averaged value just upstream of the inducer resulting from 1D correlations. In reality, the flow will further accelerate in the inducer at the suction side of the blade due to blade metal blockage, streamline curvature and uncovered passage turning

resulting in even higher pre-shock Mach numbers. An impeller inlet casing relative Mach number smaller than 1.35 is suggested for good performance. If we set this limit it effectively limits the mass flow function and Figure 11 then shows that an increase in pressure ratio (or tip-speed Mach number) can only be achieved by decreasing the impeller diameter ratio. If the specification requires a higher impeller inlet relative Mach number then the diagram suggests the degree of difficulty of the design, and the designer may then make use of special transonic flow design features to combat this, such as extremely thin blades with reduced suction surface curvature, and leading edge sweepback, as described by Rodgers [13].

Figure 11: Physical limits of the centrifugal compressor design space. The shading is explained in the text and indicates the regions where designs become more difficult.

A third physical border which can be identified from

Figure 11 is the achievable total pressure ratio due to material limitations. As shown in equation (5) the total pressure ratio is linked to the tip-speed Mach number, 2uM and hence to mechanical stresses within the impeller. This region is schematically indicated with the greenish area in Figure 11 and depends on the available impeller material and geometry. The geometry combined with the required rotational speed define the stress levels in the compressor which in turn have to be within a safety margin compared to the maximum allowable stress defined by the impeller material choice.

A further physical border is given by blade vibrations issues: Designing for no interference up to a given harmonic requires sufficient stiffness of the blades which can be demanding for high 21 DD s and k values. Increasing the natural frequency of the blades by increasing blade thickness


increases blockage and hence reduces both efficiency and swallowing capacity by trend. A final physical border is the drop off in efficiency to low flow coefficients, and this is identified clearly in Figure 10. This is not the subject of this paper and is not considered further here.

The presented values for the different physical limits are not sharp boundaries but just rough indications. During the design process a tradeoff has to be found between swallowing capacity and efficiency for a given application. The range of achievable total pressure ratio can be extended towards higher values by changing the impeller material. Towards high pressure ratio stages, not only the impeller shock related losses but also the diffuser losses due to high diffuser inlet Mach numbers increase which in turn reduce the overall achievable isentropic stage efficiency.

The discussed limits can also be identified in Figure 10 where the correlation from Figure 2 is used instead of the assumption of constant efficiency. Peak efficiency values are obtained roughly along the constant diameter ratio line of

6.021 DD s for the example with a work input coefficient of 0.75, an isentropic exponent of 1.4 and a shape factor of 0.9. For low pressure ratios, the iso-efficiency zones are more aligned to the 21 DD s iso-lines whereas for high pressure ratios, they tend to align with the iso-relative-Mach curves in the region of high efficiency.

The different limits given in the figures presented above are identified by this analysis in a clear and logical way. It is interesting to note that a design chart of a similar nature can also be found in the book by Eckert and Schnell [14].

DEMONSTRATION OF THE METHODOLOGY FOR COMPRESSOR DESIGN

The equations and methodology given here have been incorporated into a preliminary design and assessment system for radial compressors. A 1D design flow diagram of this is presented in Figure 12. It starts with two specification blocks, an inner specification part A and an outer specification block, called part B. In the inner specification block, part A, only dimensionless properties are specified whereas the specification part B represents the total inflow condition of the compressor.

The design pressure ratio, the target efficiency, the fluid property and the material choice have to be taken in the specification part A. The work input coefficient has to be specified due to matching restrictions or material strength considerations. In addition, the impeller shape factor is used as an input, which can be chosen from experience on other available designs.

Based on the specification part A, the swallowing capacity diagram can be computed. For the design total-to-total pressure ratio, a suitable trade of between the design flow coefficient, the impeller inlet casing relative Mach number and diameter ratio has to be found based on experience.

Figure 12: 1D design flow chart.

Specification part A: design pressure ratio, target efficiency,

Fluid: isentropic exponent,

Material choice: required work input factor,

Empirical data: Impeller shape factor, k

swt MkF 1111 ,,,,,,

sws MFDD 11221 ,,,,,

Tradeoff for given : flow coefficient, 1t

relative Mach number, ,1wM relative flow angle, s1 diameter ratio, 21 DD s

Effic

ienc

y pl

ausi

bilit

y ch

eck

Case 1: known hr1 from known k : sr1 and sD1 from known 21 DD s : 2D

Case 2: known 2D form known 21 DD s : sD1 and sr1 from known k : hr1

Specification part B: volume flow rate, 1tV total inlet temperature, 1tT

Check flow rate and material stress blade tip speed, 2u

volume flow rate, 1tV


The relative flow angle at the inducer casing is a result of the chosen design point. Using equation (5) the tip-speed Mach number can be computed which is used together with the flow coefficient to check the target efficiency for plausibility based on efficiency correlations such as given in Figure 2.

In case of a mismatch, the target efficiency or the tradeoff parameters have to be changed until convergence is reached in terms of efficiency values. Alternatively, Figure 10 can be directly plotted for a known set of ,, k which includes the inner efficiency loop.

In the next step, geometry information is required: In case 1, the inducer hub radius is specified from which the inducer casing radius can be computed as the impeller shape factor is known. The impeller outer diameter follows from the known diameter ratio. In case 2, the impeller outlet diameter is known and the inducer casing radius is computed from the known diameter ratio. The inducer hub radius follows form the known shape factor.

From the specified total inlet temperature (specification part B), the blade tip velocity can be computed from the definition of isentropic total-to-total stage efficiency and work input coefficient according to equation (25). Knowing the inlet flow coefficient this information is used to both check the material stress levels and the specified volume flow rate according equation (29).

Validation

It is difficult to provide exact validation for the design methodology given here, as it would require detailed commercially sensitive information of stage types, specifications and design performance. Figure 13 shows an attempt to do this for a range of commercially insensitive stage designs available to the authors, covering various process compressor, gas turbine, automotive turbocharger and large turbocharger applications, including data from various published sources. The points in Figure 13 show the design point pressure ratio and the flow coefficient of some successful designs from a range of sources, and the background of this figure is based on Figure 10. Firstly it can be seen that the spread of the data points overlaps with the peak efficiency region of Figure 10, demonstrating that most designs fall within the range of the approach suggested here. Not all of these stages in the figure have been designed for maximum possible flow capacity as some are clearly design for maximum efficiency. In addition many of the stages have a lower work coefficient than 0.75, the basis of Figure 10. Nevertheless the stages on the right hand of this diagram demonstrate a good agreement with the design limits for high flow capacity suggested in this paper. In addition there is evidence from this that the low pressure ratio stages are limited by the ratio of the inlet eye to impeller diameter and the high pressure ratio stages by the inlet casing relative Mach number, as discussed above.

Figure 13: Stage total pressure ratio vs. swallowing capacity for a range of compressor stages in comparison to Figure 10.

Specification assessment

The method presented can be used for the design of new stages, but can be also used to quickly analyze existing stages or assess new specification data. The discussed procedure allows the design point to be judged and the associated development risks to be assessed by deriving Figure 8 for the specification which can be compared against the efficiency correlations. The inclusion of sound physically-based limits not only helps to judge the specification but also helps to explain the risks to stake holders and iterate the specifications to come up with realistic and feasible values.

Figure 10 can also be used for specification assessment. Instead of using absolute values it might be more useful to compare different designs based on efficiency differences. Hence, based on a measured design, the trend on efficiency of a new design can be estimated and judged.

CONCLUSIONS A method has been presented allowing a fast preliminary

assessment of the design of centrifugal compressors with high swallowing capacity. A diagram of total-to-total stage pressure ratio versus inlet flow coefficient is derived and this includes physical boundaries that provide useful guidelines for the designer. The diagram is expressed in non-dimensional terms and both the inlet flow coefficient and the impeller diameter ratio are independent of total inlet conditions.

Physical borders which limit the maximum specific flow rate of centrifugal compressors are given in the diagram. At low total-to-total design pressure ratios, the impeller inlet eye to outlet diameter ratio limits the swallowing capacity whereas at high pressure ratios, the impeller entry relative Mach number at the blade casing is critical. The maximum


achievable pressure ratio is also limited by the impeller geometry and material.

A guideline including a flow chart is given how to use the derived equations for preliminary 1D centrifugal compressor design and assessment. It shows that the specification can be split in to dimensionless and non-dimensionless numbers and how they interact. Experimental data from a range of compressor designs support the use of this guideline for design.

An additional and important conclusion of the work is that it shows clearly that the use of specific speed as a measure of swallowing capacity cannot be recommended.

ACKNOWLEDGMENTS The first author thanks ABB Turbocharging for

permission to publish this paper. The many valuable discussions with Gerd Mundinger of ABB Turbo Systems Ltd, Baden, Switzerland, are gratefully acknowledged. Useful comments on an early draft of this paper from Colin Rodgers, Peter Came, Chris Robinson and the reviewers were very helpful.

REFERENCES [1] Rodgers, C., (1991), “The Efficiencies of Single-Stage Centrifugal Compressors for Aircraft Applications”, ASME 91-GT-77, Orlando, USA. [2] Rodgers, C., (1992), “Centrifugal compressor design: state of the art performance”, Cranfield University short course on centrifugal compressors, March, Cranfield University. [3] Casey M.V., Robinson C.J. (2006), “A guide to turbocharger compressor characteristics”, in “Dieselmotoren-technik”, 10th Symposium, 30-31 March, 2006, Ostfildern, Ed. M. Bargende, , TAE Esslingen, ISBN 3-924813-65-5. [4] Robinson, C. J., Casey, M. V., Woods, I. (2011) “An integrated approach to the aero-mechanical optimisation of turbo compressors”, published in 2011 – Current Trends in Design and Computation of Turbomachinery”, conference organised by KD Nové Energo & TechSoft Engineering in Prague, Czech Republic, May 2011. [5] Dixon, S. L., (1997), Thermodynamics of Turbomachinery, 3rd Edition, Butterworth-Heinemann, Oxford, England. [6] Hill, P. and Peterson, C., (1992), Mechanics and Thermodynamics of Propulsion, 2nd Edition, Addison-Wesley Publishing Company, Reading, Massachusetts, USA. [7] Aungier, R. H., (2000), Centrifugal compressors – a strategy for aerodynamic design and analysis, ASME Press, New York, USA. [8] Denton, J. D. (1993), “Loss mechanisms in Turbomachines”, Trans. ASME Journal of Turbomachinery October 1993, Volume: 115, pp. 621 – 656. [9] Bölcs, A., (1986), Transsonische Turbomaschinen, Braun, Karlsruhe, Germany.

[10] Lohmberg, A., Casey, M., Ammann S., (2003), “Transonic Radial Compressor Inlet Design”, Proc. Instn. Mech. Engrs., Vol. 217, J. Power and Energy, 2003. [11] Whitfield, A. and Baines, N.C., (1990), Design of radial turbomachines, Longman Scientific and Technical, UK. [12] Stanitz, J. D., (1953), “Design considerations for mixed flow compressors with high flow rates per unit frontal area”, NACA RM E53A15. [13] Rodgers, C. (2003) “ High specific speed, high inducer tip Mach number centrifugal compressor,” Paper GT 2003-38949, ASME TurboExpo 2003. [14] Eckert, B. and Schnell, E. (1961), Axial- und Radialkompressoren, Springer, Berlin. [15] Casey, M.V., and Marty, F., (1985), "Centrifugal compressors - performance at design and off-design conditions", Proceedings of the Institute of Refrigeration, Vol. 82, 1985-1986, pages 71-80.

APPENDIX The correlations of in Figure 2 are discussed in Casey and Robinson [3] and Robinson et al [4] and are based on experimental data published by Rodgers [1] and Casey and Marty [15]. The equations have not been previously published and are given below. The performance level is considered to be typical of that which a good designer with good design tools can be expected to achieve, and may even be a bit conservative up to 2 % points. The values predicted are generally in line with other correlations of experimental data such as those of Aungier [7]. Levels of performance higher than these can be expected if a thorough development program with CFD analysis and an experimental testing program is undertaken to evaluate different design choices. The correlations are for typical large turbocharger stages with “state of the art” back-swept open impellers (circa 40° backsweep), impeller diameter of D2 = 450mm with a small tip clearance of say 0.5 mm in operation and roughness typical of milled components, and with a well-designed and well-matched vaned diffuser. Reductions in efficiency for the following effects are needed:

Increased tip clearance Shroud friction in shrouded stages Vaneless diffuser Smaller size Lower Reynolds number Increased relative roughness Less back-sweep Fewer blades

The individual corrections require specific correlations for each effect and lead to much lower efficiencies of 75 to 80% for typical turbocharger automotive stages. The equations for the variation in efficiency at a tip-speed Mach number of 0.8 and below are:


10,5000,27,08.0,86.0

1:08.0

1:08.0

321maxmax

2max13max1

41max2

21max1max1

kkk

k

kk

tpt

ttpt

(37)

At Mach numbers above a tip-speed Mach number of 0.8 an efficiency deficit is imposed as:

)8.0(,3,05.0

:8.0

0.0:8.0

2154

2542

2

ut

pu

pu

MPkkPkPkM

M

(38)

Note that the strength of the deficit is based on the product of tip-speed Mach number and flow coefficient consistent with the findings of this paper and equation (5).

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