Lewis R.I. Turbomachinery Performance Analysis

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Radial section

//~~~~~

Inducer s

184

Mixed-flow and radial turbomachines

Fig. 7.5 Centrifugal compressor rotor showing

inducer and radial sections

of the particle will be 2k1~, resulting in the Coriolis force normal to r of

F = 2m~l~. The shaft power input is thus

dW = FrO = 2mr dr

~"~2

(7.4)

Integrating between r 1 and r 2 the specific work done on the particle is thus

ff,=W i rE

-- = 21~2r dr = 1~2(r 2 - r 2) (7.5)

which is identical to the second term on the right-hand side of Eqn (7.3).

This observation is of considerable importance. Apparently energy may be trans-

ferred from rotor to fluid in large measure simply by imposing meridional streamline

radially outward shift. In other words the annulus design for mixed-flow turbo-

machines becomes as crucial to the specification of work input and hence enthalpy

or head rise as the selection of blade profile shapes is for axial machines, Chapter

2. Furthermore it is obvious from Eqn (7.3) that a very large head rise will result

from the specification of a large radius ratio

r2/r 1.

Another appropriate design technique might be to maximise the work input and

consequent head rise due to aerodynamic forces by attempting to set WOE = 0 by

adopting radial blading.

The typical centrifugal compressor rotor depicted in Fig. 7.5 meets all of these

objectives. First of all the flow enters an inducer section which in many respects is

like an axial compressor blade row. The task of the inducer is to turn the flow from

its relative inlet angle/31 towards the axial direction before it begins its large radius

change from rl to r E. The fluid then enters the main radial section of the impeller

where it is first turned from the axial direction to the radially outward direction. All

of the relative swirl

Wol

at inlet has now been absorbed and the aim is to eject the

fluid at the blade tips with zero relative swirl velocity wOE = 0. In practice this proves

impossible with purely radial blades at exit as illustrated in Fig. 7.5 due to 'relative

eddy' and its related 'slip', matters to which we will return later in Section 7.5.

7.2 Stagnation enthalpy relative to a rotor and

rothalpy 185

7.2 Stagnation enthalpy relative to a rotor and rothalpy

As shown in Section 1.1.2 the steady flow energy equation for a turbomachine may

be expressed as

0 - I,V = (h2 + 89 - (hi + 89

= ho2- hol

[1.5]

where stagnation enthalpy is defined as

C 2

ho = h +-~- (7.6)

The heat transferred into the system per unit mass Q will normally be negligible.

For a

stator

the specific work input W will also be zero so that

ho2 = hol ~

for a stator

(7.7)

ho = constant

Thus stagnation enthalpy is conserved through a stationary blade row, even if there

are frictional losses.

For a

rotor,

on the other hand, the specific work W is non-zero so that the steady

flow energy equation becomes

ho2-

hol = - W (7.8)

Suppose, however, that we were to sit on the rotor and observe the relative flow.

In such a rotating laboratory the turbomachine rotor would now appear to us to be

stationary and thus we might justifiably expect the stagnation enthalpy measured

relative to our observer's framework to be conserved. To test this let us define relative

stagnation enthalpy as we did for the non-rotating reference frame, Eqn (7.6),

namely

W 2

(ho)re I

h + --~ (7.9)

where h is the specific enthalpy and w is the fluid velocity which we observe relative

to our rotating framework. Adopting cylindrical polar coordinates

(x,r,O)

the

absolute velocity components

Cx, Cr

and

at radius r may be related to the velocity

components viewed relative to the rotor

Wx, Wr

and

through

CX ~ WX 1

C r = W r

co= rl) + Wo

(7.10)

Rotation obviously has no effect upon axial and radial velocity components but only

upon the 0 component as already stated by Eqn (7.2). The stagnation enthalpy ho

may now be related to (ho)rel at the point P within the mixed-flow fan or pump rotor

186

Mixed-flow and radial turbomachines

Fig. 7.6 Meridional flow through mixed-flow turbomachines: (a) mixed-flow pump or fan rotor; (b)

Francis turbine

depicted in Fig. 7.6, as follows:

12 12 12

ho = h + ~Cx + ~Cr + ~Co

1 2

= h + lw2 +

~W r

+ 89

+ Wo) 2

1 2 ~r2~2

= h + lw2 +

~W r + 1W2

+- -1- rl)wo

U 2

= (ho)re I +

rOwo + -~

(7.11)

At rotor inlet

hol = (hol)rel + r 1[~W01

-I-

2 (7.12)

If these equations are subtracted, then at any point P within the rotor

h o - hol = (ho)re I - (hol)rel +

~(rw O - r 1Wol) q

u 2 u 2

2 2

But for a location such as that of P on a meridional streamline within the rotor, the

Euler pump equation (7.3) may be adapted to give the local value of ho, namely

h o - hol =

l)(rwo-

r 1 wol ) + U 2- U?

(7.3a)

Subtracting the last two equations we have finally the steady flow energy equation

expressed relative to the rotor, whether it be a turbine, pump, compressor

or fan:

v v? t

(h~

= (h~

(7.13)

= I = constant along a meridional streamline

The quantity I

= (ho)re I -

U2/2

is called the

rothalpy

and the energy conservation law

7.3 Dimensionless performance parameters for mixed-flow fans and pumps

187

applicable to rotating fluid systems is that the rothalpy I remains constant along a

meridional streamline passing through the rotor. We note also that in non-rotating

systems such as stators, since U = 0 the rothalpy is equal in value to the stagnation

enthalpy, I = ho.

It follows immediately from the above that turbomachines designed to operate in

a non-rotating framework may move off design performance if located in a spinning

environment. Thus even in earth-bound laboratories the stagnation enthalpy will not

be conserved precisely due to the earth's rotation, although the errors involved in

assuming this are trivial since the variation of U = rI~ from bottom to top of the

turbomachine is negligible. The same may not be true for turbomachines installed

on space platforms spun to generate artificial gravity, which could be subject to more

significant departure from intended duty if such rotational effects were not carefully

considered at the design stage.

7,3

Dimensionless performance parameters for mixed-flow

fans and pumps

Following similar arguments to those which led to a rational performance analysis

for axial turbines (Chapter 3) and axial compressors and fans (Chapter 4), we would

expect the general performance of a mixed-flow fan or pump to depend upon a

number of leading geometrical or fluid-dynamic design variables, some independent

and some dependent. Thus we might anticipate that the total-to-total efficiency */an:

would be influenced by the following minimal selection of 11 likely design

variables"

tit r = f(Cs2 , Woo,

U2, rl,

r2,A1,A2,

Aho, (Apoi01oss , p,/.t)

~, .; k .; k 2

Y Y ~f

Velocity Annulus Head Fluid

triangles geometry rise physical

properties

(7.14)

Velocity triangles and annulus geometrical data are shown in Fig. 7.7 where

attention is focused on the mean meridional streamline q~o as representative of the

average stage performance from hub to casing.

Cs2

and U2 are the meridional and blade

velocities applicable at the rotor trailing edge flow exit point P2. w~o is the vector mean

velocity relative to the fan rotor, that is the vector average of W l and w2, typifying

velocity levels relative to the rotor blades, rl and r2 are as defined in Fig. 7.7(a)

Cs2

i//o\\\\~~-~, r2 f12 w2 b~,~i 2- ...... ]

;1 -""'~r ~1 ~ "x~l~2~

U2 r~

(a) (b)

Fig. 7,7 (a) Meridional annulus geometry and (b) veloci~/triangles for a mixed-flow fan

188

Mixed-flow and radial turbomachines

and A1 and A2 are the annulus areas at the leading and trailing edge planes viewed

along the directions of

Csl

and c~2, which may be approximated as follows:

A1 = 7r(r21 - r21)/cos

X1 ]

A2 = 7r(r22 - r22)/cos X2

(7.15)

Here gl and X2 are the leading and trailing edge 'lean' angles as defined in Fig. 7.7(a)

and

(rhl,rtl)

and

(rh2,rt2) are

the hub and tip radii of the rotor leading and trailing

edges. Selecting suitable dimensionless groups, r/Tr may be expressed alternatively as

follows:

r/TT = f(q~, @, r2rl, ~,A1A2 ~,

Re2 ) (7.16)

with the following definitions

~p- Cs2/U2

= Aho/U 2

rE/rl

A 2/A1

(ApoR)loss

"~176 1 2

~pwoo

2r2 U2

Re2 =

Flow coefficient

Work input or head coefficient

Annulus radius ratio

Annulus area ratio

Rotor loss coefficient

Machine Reynolds number

(7.17)

Although Eqn (7.16) provides immediate guidelines for data reduction and selection

of suitable correlation formats for systematic model tests (e.g. Oh, qJ) charts for specified

ranges of

r2/rl

and

A2/A

1), it is in parametric form only and gives no indication of the

likely dependency of r/Tr on the six dimensionless variables. It is possible to move

towards a more rational analytical equivalent equation to do just that with the help of

the velocity triangles, Fig. 7.7(b). Let us consider the case of an incompressible fluid

for which r/a~r is defined as

Actual stagnation pressure rise

flair = Frictionless ideal stagnation pressure rise

Apo -- (Apo)loss

Apo

1 2

(7.18)

From velocity triangles, Fig. 7.7(b),

w (c l+Cs2) +(W l+WO)2

4{( A2)2 }

2 2 " = ~2 1 + ~11

(vl +

U2 -- r

7.3

Dimensionless performance parameters for mixed-flow fans and pumps 189

6.00

4.00

2.00

0.00~

0.20

= 0.2

(a) r2/rl = 1.0

(Axial)

0.5

0.7

0.40 0.60 0.80 1.00 ~ 1.20 1.40

6.00

4.00

2.00

6.00

4.00

0.00

0.4

0.5 f

2.00

0.7

0.00

0.20 0.40 0.60 0.80 1.00 ~ 1.20 1.40 0.20 0.40 0.60 0.80 1.00 ~ 1.20 1.40

0.4

0.5

0.7

Fig.

7.8 Loss

weighting coefficients for mixed-flow fans for three different radius ratios

r2/r I

with

constant annulus area ratio

A2/A 1

1.0

But from the definition of flow coefficient, Cs2

= ~)U2,

and from the Euler pump

equation (1.9a), with zero inlet swirl c01 = 0, Aho = U2c02 = q~U 2 and thus c02 = qJU2.

The previous equation then becomes

(7.19)

Finally the total-to-total efficiency of the fan or pump may be expressed in the

following dosed analytical form"

r/TT=a - ~'= 4) 2 1+~- 7 +l--q,

= 1 -- f(4~, ~, q/r2, A21A 1) ~'~

(7.20)

The losses are thus the product of the profile loss coefficient ~'= and the loss weighting

coefficient f.

To summarise:

(1) The loss coefficient ~'~ is dependent upon the blade profile shape and the

consequent boundary layer development. It Will also be dependent upon the

190

Mixed-flow and radial turbomachines

Reynolds number

Re2 ,

accounting for the last dimensionless variable listed in

Eqn (7.17).

(2) The magnitude of the weighting coefficient f is a unique analytical function of

the selected duty (~b, ~) and the annulus geometry through

(rl/r2)

and (A

1/A2),

namely

(

A l) +(rl

f = -~-~ 1+ --+1-~

(7.21)

This powerful method of analysis thus brings out quite clearly the primary influence

which the dimensionless groups will exercise over the expected efficiency of a

mixed-flow fan. Irrespective of the detailed profile aerodynamic characteristics as

represented here by ~, the rotor losses will depend directly upon the weighting

coefficientfwhich is itself governed by the selected duty loading (r ~) and the annulus

geometry

(r2/rl, A2/A1).

To illustrate these dependencies fhas been evaluated in Fig.

7.8 for a wide range of (~b, ~) duties and for

r2/rl

= 1.0, 1.5 and 2.0 with constant

annulus

area

A2/A 1

1.0. Figure 7.8(a) represents the limiting case of an~axial

machine

(r2/rl

= 1) for which the loss weighting coefficients exhibit the highest values.

As might be expected the duty coefficients ~b and ~ exercise the strongest influence

over f, but for a typical (~b, ~) duty the radius ratio

r2/rl

will produce significant changes

in predicted efficiency as illustrated by the following example.

Example 7.1

Problem

Assuming a rotor loss coefficient ~'~ = 0.08, estimate and compare the mean streamline

efficiency of mixed-flow fans of varying

r2/rl

for a prescribed duty of ~b = 0.6,

= 0.35.

Solution

Making use of Eqns (7.20) and (7.21), the data tabulated in Table 7.3 may be obtained.

According to this simple example, gains in rotor efficiency are indicated for increasing

radius ratio

r2/rl,

with the important reservation that we have assumed the same loss

coefficient ~ for all four designs. Bearing in mind, however, that the (~b, ~) duty

loading is the same for all four machines, this may not be too unreasonable an

assumption for this simple study aimed merely at revealing trends.

To conclude this section the weighting coefficients are shown in Fig. 7.9 for two

mixed-flow fans of radius ratio

r2/rl

= 2.0, but with different area ratios, one diffusing

Table 7.3 Predicted efficiency of mixed-flow fans as a function of radius ratio

r2/r 1

r2/n f ~ nTr = 1 - f~

(%)

1.0 1.43646 0.08 88.51

1.5 1.083 29 0.08 91.33

2.0 0.936 46 0.08 92.51

2.5 0.857 89 0.08 93.14

7.3 Dimensionless performance parameters for mixed-flow fans and pumps 191

6.00

4.00:

2.00=

0.00 ~t~

0.20

(a) A2/A 1 = 0.75

~b = 0.2

0.4

0.5

0.7

6.00

4.00

2.00

0.00

(b) A2/A 1 = 1.25 ik = 0~-~

0.3

0.5

0.7

0.40 0.60 0.80 1.00 1.20 1.40 0.20 0.40 0.60 0.80 1.00 1.20 1.40

Fig,

"/'.9 Effect of annulus area ratio Az/A1 upon loss weighting coefficients for a mixed-flow fan with

radius ratio r'2/rl = 2.0

dl-"

w b

Cs2

U2 /U 2

a 1_.. rl/r2

I-"

Fig. 7,10 Dimensionless velocity triangles for a mixed-flow fan

(A2/A1 = 1.25) and the other accelerating (A2[A 1 = 0.75). There is clearly a gain in

rotor efficiency offered by the accelerating annulus, although of course this would

impose additional demands on the downstream diffusing requirements of the exit

stator or volute.

7.3.1 Dimensionless velocity triangle relationships

Dimensionless velocity triangles may be constructed if all velocities in Fig. 7.7(b) are

divided by the rotor exit blade speed U2. The outcome of this is shown in Fig. 7.10.

The various flow angles can now be expressed in terms of the chosen dimensionless

groups as follows. The absolute exit swirl angle is given by

a2 = tan-l(qdth) (7.22)

192

Mixed-flow and radial turbomachines

and the relative swirl angles are given by

tan (Z /1

f12 ="

tan-a{+ ( 1 -O)}

l-g,+ --

/3oo = tan-

The fluid deflection relative to the rotor then follows from

8r= ~1--~2

rlAl

~b r~ 2 l+g,

tan-1 --- (rlAi

~-2- (1 - ~) \

r2A2

(7.23)

(7.24)

Table 7.4 Velocity triangle data for a range of mixed-flow fans

Prescribed duty

A2/A1

= 1.0

~b = 0.6, q~ = 0.35

A2[A 1

=0.75

r2/rl a2 f12 [31 [3oo 8 R [31 [3oo E R

1.00 30.256 47.291 59.036 53.973 11.746 65.772 57.529 18.482

1.25 30.256 47.291 53.130 50.389 5.839 60.642 54.090 13.352

1.50 30.256 47.291 48.013 47.654 0.722 55.981 51.429 8.690

1.75 30.256 47.291 43.603 45.507 -3.688 51.780 49.316 4.489

2.00 30.256 47.291 39.806 43.781 -7.485 48.013 47.603 0.722

The above equations have been evaluated for a family of mixed-flow fans to illustrate

the influence of

r2[r 1

and

A2/A

1 upon the velocity triangles, Table 7.4. The following

observations may be made from these data:

(1)

Exit flow.

As can be seen from Eqns (7.22) and (7.23b) the absolute and

relative exit swirl angles a2 and/32 are in fact dependent only upon the fan duty

(~b, qJ) and are thus the same for all of the fans considered here.

(2)

A2/AI

= 1.0. It is of special interest to note that the rotor deflection angle eR

decreases as the design value for

r2/rx

is increased. Indeed, if

r2/rl

> 1.538, then

/32 >/31 and the blade profiles must be those of an accelerating (turbine) cascade

rather than those of a diffuser. Surprisingly, eR is then negative.

7.3 Dimensionless performance parameters for mixed-flow fans and pumps

193

(3)

A2/A1

0.75. With this modest contraction of the annulus area, significantly

greater fluid deflections e R are required to deliver the same duty ~b and e a then

remains positive (i.e. diffusing) for all radius ratios

r2/r 1.

Item (2) illustrates rather dramatically the discussion already presented in Section 7.1

regarding the two origins of specific work and hence work coefficient qJ in mixed-flow

turbomachines, namely

(a) Coriolis forces, and

(b) aerodynamic (Newtonian reaction) forces.

For large radius ratios the specific work input due to Coriolis forces is simply too great

and therefore negative specific work must be supplied by the aerodynamic forces for

delivery of the specified work coefficient q~. To illustrate this further we could express

the work coefficient as the sum of these two contributions thus:

I/t "- I/tAero" + I/tCor.

(7.25)

From the Euler pump equation (7.3), dividing through by U 2 we then have

~Cor.=l-- (rl) 2-r2

I/tAero.=l/t--l+ (r~-~) 2

(7.26)

From the second of these equations it follows that the blade profile aerodynamics will

be those of a turbine rather than a compressor unless

r 2 1 Limits of

r2/r I

for positive

< ~ aerodynamic specific work input (7.27)

X/1 -

However, some caution is needed at this point, since this limit does not actually

coincide with that of zero deflection. Thus from Eqn (7.24) we see that eR is positive

only if

-- r ra A (11 de,ection imits o,r ,ra,orpositive R

The explanation for this is subtle and not of easy perception. From the Euler pump

equation (7.3) we see that the specific work input from blade profile aerodynamics may

be expressed as

WAero. =

~(r2w02- rl wol)

(7.29)

For zero deflection the relative swirl velocities are equal, namely

wol

W02 , whereas

in Eqn (7.29) for zero specific work we must have the condition of equal relative

angular momentum rl

wol = re wOE.

Thus the radial shift rl to r2 does in fact influence

also the aerodynamic work input and it would seem that a small measure of negative

rotor deflection can still produce some pumping effect.