Lewis R.I. Turbomachinery Performance Analysis

94

Performance analysis for axial compressors and fans

stage using this cascade. Compare the pitch/chord ratios and ideal lift coefficients

CLoo i

(ignoring drag) for diffusion factors DF = 0.5 or 0.6.

Solution

From Eqn (4.22)

~b = 1/(tan

fll d-

tan f12)

-- 0.5

From Eqn (4.21a)

r = 2~b tan

fll --

1 = 0.406 62

We note that this lies within the

Omax

limit suggested by Eqn (4.19). From Fig. 4.4

the expected stage efficiency is about 90.3%.

We may now evaluate Eqn (4.27) to give the recommended pitch/chord ratio for

DF = 0.5 and 0.6. Then we may evaluate the ideal lift coefficient from Eqn (4.23)

ignoring the drag coefficient CDo~, namely

(t)( ) 423a,

CLoo i -"

2 7

V'4q~ 2 +

1

The outcome is as follows"

DF

t/l

CLooi

0.5 0.731 0.849 72

0.6 1.162 1.33641

It is clear from this, as we might expect, that a reduction in pitch/chord ratio

t/l

will

result in a reduction in aerodynamic blade loading as revealed by both the diffusion

factor and the lift coefficient.

Example 4.2

Problem

A compressor is to receive atmospheric air at 15 ~ and raise its pressure from 1.0 bar

to 3.0 bar. The mean radius is to be 0.5 m and the rotational speed 5000 rev min -1.

Select a suitable stage duty (~b, q0 and choose a suitable number of stages. Calculate

the cascade inflow and outflow angles

fll

and/32 assuming 50% reaction and find the

required pitch/chord ratio for a diffusion factor DF = 0.5.

Solution

From Fig. 4.4 we will opt for ~b = 0.5, q~ = 0.35, for which r/Tr = 91.5%. Next we

can calculate the specific enthalpy rise per stage Ah = qjU2:

U = rmI~ = 0.5 x 27r x 5000/60 = 261.8

m s -1

Therefore

Ah = 0.35 x 261.82

x 10 -3 =

23.99 kJ kg -1

4.5 Compressor stages with arbitrary reaction

95

Now the machine overall specific enthalpy rise is given by

ho2- hol = "r/Tr(ho2 s -- hol )

= 0.915 x 1.005 x

288.2{30"4/1"4- 1}

= 97.72 kJ kg-1

Therefore the number of stages = 97.72/23.99 = 4.0734. Let us therefore select four

stages. We now need to recalculate ff from

97.72x 1000 1

~p = 261.82 x ~ = 0.356 44

Note that the total-to-total efficiency will be slightly less than our original estimate,

Fig. 4.4, but we will ignore this. From Eqns (4.21) and (4.22) the cascade inflow and

outflow angles are

1 + 0.356 44 }

~1 --

arc tan 2 x 0.5 = 53"60~

{1-0.35644}

/32 = arc tan 2 x 0.5 = 32"76~

Substituting for ~b = 0.5, ~= 0.35644, DF = 0.5 into Eqn (4.27), the required

pitch/chord ratio for stable efficient flow is

t/l

= 0.9723.

Suppose we now decide to reduce the number of stages from four to three. The

work coefficient will go up by 4/3 to ~= 0.475 25. For the same diffusion factor

DF = 0.5 and flow coefficient ~b = 0.5, the required pitch/chord ratio, Eqn (4.27),

now becomes

t/l

= 0.501 214. Thus we would need almost double the number of

blades to achieve the increased flow turning, in this case from 131 = 55.87 ~ to

f12 =

27.69~

4.5 Compressor stages with arbitrary reaction

So far in this chapter we have considered only 50% reaction axial compressor stages.

Let us now extend the analysis to deal with repeating stages of arbitrary reaction

R. The velocity triangles for these, shown in Fig. 4.6, are similar to those of the

repeating 50% reaction stages illustrated in Fig. 4.2 in all but one respect, namely

that they are no longer symmetrical. The dimensionless velocity triangles, Fig. 4.6(b),

are now dependent upon the three variables ~b, ~ and R where the reaction R

determines the degree of asymmetry.

Let us begin by reconsidering the definition of reaction R given previously in Eqn

(4.5):

R = Specific enthalpy rise across the rotor

Specific enthalpy rise across the stage

h 2 - h 1

= h3- hi [4.5]

96

Performance analysis for axial compressors and fans

(o)

I_.. w02 .._I_.. Ac0 = c02-c01

..._I_..

COLI

1 =C3

u

(b)

IA R-~,/2 j~.. ~, ~jl-R-~/~

r" -r" -I" -I

cI/U = c3/U

~l

,x2

/$2 ~ -- ~3

Fig. 4.6 (a) Velocity triangles and (b) dimensionless velocity triangles for repeating axial compressor

stages

of arbitrary reaction

Since we are limiting the analysis to repeating stages for which entry and

leaving velocities are identical, c3 = Cl, the denominator of Eqn (4.5) may be

simplified to

h3-hl= ( ho3- ~) - ( hol- ~) =ho3-hol= Aho

(4.28)

where Aho is the stage stagnation enthalpy rise. The numerator of Eqn (4.5) may

be rewritten

(4.29)

But since there is no work or heat input through the stator, ho2

= ho3

and thus

ho2-hol = Aho. Also since the axial velocity is assumed to be constant,

c~ - ci = (Cx ~ + c~) - (Cx ~ +

&)

= C 2 __ C21

= (co2- co~)(co2 + col)

From the Euler pump equation (4.3)

Aho

co2 - col = ~ = $U

U

(4.30)

Introducing these results into Eqn (4.29) we have

Ah0

h2- hi = Aho- ~(Co2

+ col)

(4.29a)

4.5 Compressor stages with arbitrary reaction

97

so that the reaction R, Eqn (4.5) above, becomes

1

R = 1 - ~ (co2

+ col)

providing a second equation for

Col

and c02, namely

Cot 2 + CO1 "-

2U(1 - R)

(4.31)

Solving simultaneous equations (4.30) and (4.31) we have the dimensionless swirl

velocities

U

C02

1

c~ 1 -R-~

1

~=I-R+~

U

1

Wol = 1 C~ = R +

v --U-

Wo2 Co2 =R_I

U = 1 - --~- ~,

(4.32)

and the remaining velocities also expressed in dimensionless form are thus

Wl= ;$2+ (R+~) 2 Rotor relative inflow velocity

--ff

w2 ;th2+ (R_~) 2 Rotor relative outflow velocity

-6-

2 (4.33)

Cl ;~b2+ (a_R_ 2~__ ) Rotor entry and stage exit velocity I

.~.

m=

uC2 qb 2 + (1 - R + ~) 2 Stator inflow velocity J

All angles of the velocity triangles can now also be expressed as functions of ~b, ~,

and R as follows: for the rotor,

/31 = tan-1 R + ~ q~ Relative inflow angle

/32 = tan- 1 R - ~ 0 Relative outflow angle

f ~ 1

ea = tan-1 4 ~2 + R2 - ~qpl Fluid deflection

(4.34)

98

Performance analysis for axial compressors and fans

and for the stator,

ae=tan_l{l( 21 )}

-~ 1 - R + qJ Inflow angle

{1( 1 )}

% = al = tan-1 -~ 1 - R - ~ ~ Outflow angle

1 Fluid deflection

es = tan-1 4 ~2

+ (1 - R)2 _ 4 02

(4.35)

4.5.1 Stage losses and efficiency

The loss coefficients for rotor

~'R

and stator ~'s have been defined by Eqn (4.9) and

the total stage loss (Apo)los s follows from Eqn (4.17a). Introducing

Wl/U

and

c2/U

into this we have

(Apo)lo s

pU 2

1

-6

1 2

(4.17a)

+ ~ ~'s + 1 - g + (4.36)

The total-to-total efficiency then follows by substitution into Eqn (4.16), namely

1 2 q~ 2

In Section 4.1 we were able to show from dimensional analysis using the ~r-theorem

that rla--r is dependent upon the five groups ~b, qJ, R, ~r R and ~'s in the general form

expressed through Eqn (4.13). We have now shown that this relationship can be

converted into the explicit analytical form given by Eqn (4.37) as a result of linking

stage duty (~b, qJ) and reaction R to velocity triangles and thus to stage aerodynamics

and thermodynamics. This is a most useful and extremely powerful reduction.

4.5.2 Diffusion factors and selection of pitch/chord ratio

Since the velocity triangles are no longer symmetrical for values of reaction R other

than 0.5, the rotor and stator blade rows will have quite different profile geometry.

In order to select suitable values of pitch/chord ratio

t/l

to control aerodynamic

loading, it is again helpful to express the diffusion factor in terms of ~b, 0 and R for

each blade row. Introducing Eqns (4.34) and (4.35) into the expression for DF, Eqn

(2.28), we thus have for the rotor

cos l(/)

DFR = 1

cos ~1 ~.

(tan

fll --

tan f12)

COS f12

2

R

4.6

Selection of optimum blade profile geometry

99

= 1

-

6 2 + (R + qd2) 2 + 2

R

V'6 2 --b (R + qd2) 2

and for the stator

DFs 1

COS Or2

c~ I

= ~- (tan

O~ 2 --

tan o~3)

COSO~ 3

2 \/7 s

=1-

] 439.

t~ 2 "k-

(1 - g

+

~12) 2

+ 1

S

"~V/t~ 2 -k- (1- g

+ ~12) 2

Inverting these we may obtain expressions for the pitch/chord ratios"

(/)R = -~~/th2 + (R-~)2_ ~ (1- DFR) ~/th2 + (R+ _~)2

...... (4.40)

(/)s _~~r + (I-R-_~)2_ _~ (1- DFs) ~/~2+ (l-R+ 2~---) 2

4.6

Selection of optimum blade profile geometry using

theoretical cascade flow analysis

So far in this chapter we have considered only the selection of optimum duty

coefficients th, q~ for an axial compressor stage and the influence of these, together

with the reaction R, upon the velocity triangles. A simple design sequence was

presented in the introduction to this chapter, the final stage of which involves selection

of the cascade blade profile to achieve the following:

(1) The correct fluid deflection and efflux angle

fl2-

(2) Acceptable aerodynamic loading.

(3) Smooth entry flow.

The second of these design requirements has already been catered for by specifying

the diffusion factor DF as considered in Section 4.4, resulting in expressions for the

cascade pitch/chord ratio, Eqns (4.27) and (4.40). Requirements (1) and (3) may then

be met by use of cascade analytical design techniques such as that offered by the

computer program CASCADE. The design objective is to select the stagger angle

A and camber angle 0 needed to achieve the correct outlet angle

~2

with 'shock-free'

inlet flow. A base profile thickness (e.g. C4 profile) and type of camber line (e.g.

circular arc or parabola) must also be specified.

Figure 4.7 shows curves of/32 versus/31 for 'shock-free' entry flow cascades made

up by superimposing the C4 profile upon a circular arc camber line. (A, 0) plots are

shown for three pitch/chord ratios. The curves were derived from multiple runs of

the program CASCADE which enables the user to obtain the shock-free inlet angle

/31 and its associated outlet angle/32 for a specified stagger A and camber 0. A set

of such data may be obtained very rapidly with the help of CASCADE and provides

a quick way forward for optimum cascade selection as illustrated by the following

numerical example.

100 Performance analysis for axial compressors and fans

70

60

50

40

30

20

I ,'08 I' 1

-

~ ~-Stagger ,

[ [ angles Z, 60 ~

..

1 ] I .A" "-~

Camber 6( 70~//

angles 0

80. ~ •

50

55 60 65 70 75 80 85

90

i

60

1

Stagger

50 ~ .

40

30O40o

20

Camber 70o',~

10 angles 0 80 o

50 55 60

IU v

65 70 75 80 85

tJl

70

60

50

t~2

40

30-

2O

10

45

I

,0o

30 o~

40 ~ .~-~

i- 50*1

Camber

1

Stagger 600./

angles k~

'70orN

80 ~

angles 0 I

50 55 60 65 70

I~J

75 80 85

Fig. 4.7 Compressor cascades designed for 'shock-free' entry flow using computer

program

CASCADE

Example 4.3

Problem

For the compressor considered in Example 4.2, select a suitable blade geometry for

the mean radius section. Summary of given data:

Inlet angle

fll =

53.60~

Outlet angle /32 = 32.76 ~

Diffusion factor DF= 0.5

Pitch/chord ratio

t/l

= 0.9723

Solution

From Fig. 4.7 we can select the stagger A and camber 0 to give smooth entry flow

at the leading edge. Thus from the graph for

t/l

= 1.0 (which is close enough to the

specified value of

t/l

= 0.9723 to suffice), we read off

A ~ 41.5 ~

0 ~ 35 ~

To confirm or adjust this choice we can now run the program CASCADE with this

data, using the default profile thickness C4 and circular arc camber (see Appendix

4.6 Selection of optimum blade profile geometry 101

Fig. 4.8 Predicted surface pressure distribution

for compressor cascade - Example 4.3

II for instructions in the use of CASCADE). We find that the predicted outlet angle

is then/32 = 32.8 ~ which is close enough to our design requirements, and the predicted

'shock-free' inlet angle is 54.07 ~ which is also near enough to the design value to

suffice. As proof of this we may check the surface pressure distribution, shown here

as Fig. 4.8.

From this we see that the stagnation point (for which Cpl = 1.0, Eqn (2.14)) is

located directly on the profile leading edge, resulting in smooth entry flow as

described in Section 2.6 and illustrated by Fig. 2.11. The pressure on the upper convex

surface falls rapidly to a minimum value

of (Cpl)min =

-0.572 at about x/l = 0.16 and

then diffuses steadily towards the exit value of

(t.,)/f'plXexit -- P21 -- 2Pl

~pCl

1 (cos.1)2

COS ~2

= 0.502

[2.141

Note that the theoretical analysis, being based upon potential flow modelling,

predicts the trailing edge pressure distribution badly for

x/l > 0.96 since there is also

an implied stagnation point at the trailing edge associated with the Kutta-Joukowski

condition which has to be imposed at that point. Despite this minor setback, Fig.

4.8 gives us a firm indication that we have selected an aerodynamically acceptable

blade cascade geometry. Furthermore, we may use the predicted

Cpl

plot to check

the upper surface stability against the later version of diffusion factor Dloc due to

Lieblein (1956) which was defined in Section 2.7.1, Eqn (2.23). Thus

COS ~1

1

Dlo c =

1

COS~2 ~//1 -- (Cpl)min

[2.23]

= 0.436

102

Performance analysis for axial compressors and fans

/ Wl~~c 1

=

C x

U

/~ofor f

v

U

c3

= cl t~

S

f "--U

Stotor

Rotor

1.0

Fig.

4.9 Velocity triangles at the mean radius for an axial fan stage

According to Lieblein's experimental studies Dloc should be no greater than 0.5 which

confirms that this cascade is conservatively designed as intended in Example 4.2.

4.7 High reaction axial fan stages

As discussed in Section 2.2 it can be argued, by considering its role as a diffusing

device, that the optimum reaction of an axial compressor is in the region of 50%.

Axial fans such as that illustrated in Fig. 4.9 are, on the other hand, of much higher

reaction. The main aim is usually to move large volumes of gas with less emphasis

on the demand for pressure rise. Many fans achieve this by means of a single rotor

with no stator blade rows at all, for example for ventilation applications. Bearing

in mind the definition of stage reaction for a compressor or fan, Eqn (4.5), such fans

are clearly operating at 100% reaction. If the entry velocity Cl is axial there must

of course always be some swirl velocity c02 downstream of the rotor, Fig. 4.9. If the

level of this is sufficiently high then it may be advantageous to include also a

downstream stator, as illustrated here, to return the flow at exit from the fan stage

to the axial direction c3 = Cl. We will adopt this arrangement as our model fan stage

and derive next the related equations for its design and performance analysis.

The equations derived in Section 4.5 for compressor stages of arbitrary reaction

are of course still applicable to our fan stage but with one restriction. There is to

be zero swirl at entry, that is

Col

= 0. From Fig. 4.6 and/or Eqn (4.32a) we observe

that this determines the reaction R as a function of work coefficient ~, namely

R=I 0

2 (4.41)

Thus for a fan comprising rotor plus stator without inlet guide vanes the reaction

is no longer an independent variable once the design operating duty (~b, ~) has been

chosen. For example, if we selected the design duty 4} = 0.5, ~ = 0.3 the fan stage

reaction would be R = 0.85. From the 'Smith' charts for experimental test axial

4.7 High reaction axial fan stages

103

compressors, Fig. 4.4(c), the nearest reaction to compare with is 90% for which the

proposed duty for our fan is in fact very close to optimum. The velocity triangles

shown in Fig. 4.9 have been chosen to suit this prescribed duty. The blade profiles

and cascade geometry shown in Fig. 4.9 have been selected by the procedures outlined

in Section 4.6 for ensuring that the velocity triangles are actually delivered and with

optimum aerodynamic performance setting the diffusion factor level at DF = 0.5.

Introducing Eqn (4.41) into Eqns (4.32) to (4.40) of Section 4.5 we can now express

all velocity triangle or other useful design data as functions of 4) and 0 as follows.

Dimensionless velocities are given by

W---! -- V't~2 -+ - 1

U

W'-"~2 = "V"~ 2 +

(1

- I//) 2

U

Cl C__~3

-U- u-6

(4.42)

~2= v~+

U

Flow angles are given by

O~ 1 = Or3 = 0

a 2 =

tan-

1 (@/6)

~1

--

tan-

1 (1/(f))

f12= tan- 1( l-q' )4)

The total-to-total efficiency becomes

1

r/Tr =

1 - ~-{

SrR(1 + 4) 2) + ~'S(~ 2 + I//2) }

The diffusion factors, Eqns (4.38) and (4.39), become for the rotor

j 2(/)( )

4, 2 + (1 ~,)2 +

D FR

1

4)2+1 l R V'4~2+1

and for the stator

(4.43)

(4.44)

(4.45)

OFs=' ' ) ,446,

V'~ 2 + I/12 + 2 S "V'~ 2 + I/12

from which we may express the pitch/chord ratios as functions of 4), q~ and DF,

namely

2

-- --~-" [V'~ 2 +

(1 - q,)2_ (1 - DFR)V'4~ 2 + 11

2

= ~-[4) - (1 - DFs) V'4}2 + q,21

(4.47)

Lewis R.I. Turbomachinery Performance Analysis

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