Yamaguchi H. Engineering Fluid Mechanics (Fluid Mechanics and Its Applications)

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182 4 Perfect Flow

analysis and performance laws, probably the widest comprehension of the

general behavior of all turbomachines, are presented prior to the discus-

sions on more general treatment of the similitude and dimensional analysis

in later chapter. Methods of analyzing the detailed flow process and the

performance characteristics differ largely, depending upon the geometrical

configuration of the turbomachines, and in this text no detailed behavior of

individual types in actual operation is considered, but more general theo-

ries are investigated.

Fig. 4.34 Various types of rotors (impeller vane)

in turbomachine

In order to achieve the energy conversion between vanes and fluid, it is

very necessary for both vanes and fluid to gain force in a rotational direc-

tion. For fluid passing through vanes, the change of velocity component in

rotational direction occurs by force, so that the torque or the magnitude of

energy conversion can be readily estimated by Euler’s pump (or turbine)

law verified in Eqs. (4.1.51) and (4.1.53). We need to know, however, the

flow fields at inlet and outlet to the vanes to obtain the energy conversion a

priori. For a turbomachine with a greater number of vanes and with larger

contact between the vane surface and the fluid flow, the moment of mo-

mentum theory (the Euler’s pump or turbine law) would be more appropri-

ate to apply. On the other hand, when the number of vanes are less and

have smaller contact surface area with fluid flow, the airfoil theory may be

more appropriate to apply. Both approaches are useful for designing

4.2 General Theories of Turbomachinery 183

4.2.1 Moment of Momentum Theory

Figure 4.35 shows schematics for an idealized radial flow pump impeller,

in which (a) is a control volume from the frontal view together with its me-

ridian cross section. In an idealized impeller, the thickness of each vane is

to be neglected so that, as seen in Fig. 4.35(a), the flow enters to the con-

trol volume radially, along vanes with a relative velocity

w , and thus

leaves with a relative velocity

, while in the meridian plane the axial

flow is diverted into a radial direction as the impeller rotates with an angu-

lar velocity

The representative geometry of the impeller is that

r is the inlet radius,

r the outlet radius,

b and

b the width of the impeller at inlet and outlet

respectively. The velocity diagram at the inlet with suffix 1 and the outlet

with suffix 2 is illustrated in Fig. 4.35(b). u is the circumferential velocity,

i.e.

, and c is the absolute velocity to the inertial reference frame. It

is worth mentioning that the image of the flow configuration in Fig.

4.35(b) is derived from the general concept of turbomachine, as displayed

in Figs. 4.5, 4.6 and 4.7. The angle

is vane angle, through which the

flow enters and leaves the impeller in the control volume with relative ve-

locity

w and

w respectively. The radial and circumferential component

of c are respectively denoted by

c and

at locations of 1 and 2.

the angle between the absolute velocity vector c and the circumferential

velocity vector u at locations 1 and 2.

According to the theory of the moment of momentum and the Euler’s

pump (or turbine) equation, the shaft torque

T and power

P transmitted

to a fluid from an impeller can be conventionally written as



11222

112

coscos

DDU

crcrQ

crcrmT





(4.2.1)

and



cucuQ

1122





(4.2.2)

nd mixed flow turbomachines with

the airfoil theory. Figure 4.34 shows the typical geometric configurations

of turbomachines based on the flow direction through the machines.

turbomachines, particularly for radial a

the moment of the momentum theory, and for axial flow turbomachine with

184 4 Perfect Flow

where

n and

are conventionally replaced by notions of

T and

that are the torque and angular velocity respectively in Eqs. (4.1.51) and

(4.1.53). It is noted again that in the case of

T and 0!

P , for the pump,

and in the case of

T and 0

P , for the turbine.

Fig. 4.35 Idealized radial flow pump

Noting that the fluid is a perfect fluid, which is satisfactorily justified

for actual turbomachines through which flows have large Reynolds num-

ber and can be regarded as the inviscid flow. The theoretical power trans-

mission of

L per unit mass, neglecting energy loss, is expressed as

4.2 General Theories of Turbomachinery 185

cucu

1122



(4.2.3)

L is often called the theoretical specific energy, and furthermore the

theoretical pressure head (Euler head) that rises across pump

H is writ-

ten as



cucu

1122



(4.2.4)

With the aid of the velocity diagram at points 1 and 2 in Fig. 4.35(b), we

have

uccu

uccuw

cos2

222



(4.2.5)

Combining Eqs. (4.2.5) and (4.2.4), we can reduce

H to write

 

222

wwuu

ccwwuu













ggg

(4.2.6)

When we consider the equation given in Eq. (4.1.41) where the rotating

reference frame at points 1 and 2, we can write Eq. (4.2.6) as follows

ppcc







(4.2.7)

Equation (4.2.7) indicates that for a pump, i.e. 0t

H , the energy given

by rotating the impeller to a fluid results in the rise of kinetic energy (the

first term of Eq. (4.2.7)) and the pressure rises (the second term of Eq.

(4.2.7)). Similarly for turbines, it becomes the opposite, i.e. 0d

H . The

actual delivered head

, measured as the head difference between the

inlet and outlet flanges of the pumps, as sometime called the manometric

head, is less than the theoretical head

H , due to a loss in head

h , as this

can be written as

lth

hHH 

(4.2.8)

For hydraulic turbines, the impeller can be regarded as a runner and

water flows in opposite direction to the pump. For example, in the Francis

turbines, the fluid power input to the turbine would be given as

186 4 Perfect Flow

thtw

QHP

)(

(4.2.9)

in which

tht

)(

is the theoretical head drop across the turbine. The actual

head drop



may be



lthtt

hHH 

)(

(4.2.10)

where the loss of head

h will be added to the theoretical head

tht

)(

Returning to the pump as our representative objective, the maximum

theoretical head

H in Eq. (4.2.4) can be achieved if

becomes zero.

This is the case when the inlet flows to an impeller and meets a condition,

, i.e. 0

in Fig. 4.35(b), and namely this is the case when the

flow has no swirling motion at the inlet and enters to an impeller along

vanes normal to rotational direction. So that ideally we have

uth

cuH

(4.2.11)

and from the velocity diagram at 2 in Fig. 4.35(b), we have

222

cot

EZZ





cuuH

mth

(4.2.12)

where

and

are kept constant when geometric parameters and the ro-

tational speed are fixed. It should be noted that in deriving Eq. (4.2.12) the

following relations are used

222222

cotcos

cucc



(4.2.13a)

2 br

(4.2.13b)

(4.2.14)

Equation (4.2.13b) is obtained by applying the mass continuity equation to

the outlet (point 2) of the impeller. In an actual design of a radial flow tur-

bomachine, there are vanes of finite thickness (conventionally 6~8 vanes

in a pump impeller and 15~17 vanes in a turbine runner), and particularly

4.2 General Theories of Turbomachinery 187

in a pump where there is a slip of flow at the outlet of an impeller, as illus-

trated in Fig. 4.36(a). As shown in Fig. 4.36(a), the relative velocity

with the vane angle

would be shifted to

and

respectively. This

is largely caused by the fact that the flow is affected through the impeller

of a pump, as illustrated in Fig. 4.36(b), due to the circulatory flow which

is induced to have a rotation opposite to that of the impeller and thus modi-

fies the outlet velocity diagram from an ideal pump. Therefore, there be-

comes a decelerating flow where the flow is quite uneven between the

vanes. As a result, we have the difference from

fth

H to be written as



uuthth

ccuHH

222

'  

(4.2.15)

and

fff



thcthth

HPHH

(4.2.16)

Fig. 4.36 Slip flow at outlet of impeller

Equation (4.2.16) compares the relative head difference between the

ideal head

fth

H and the actual (after considering slip) head

H , giving

the ratio between the tangential components of absolute velocity corre-

sponding to the angles

and

respectively. This is indicated in Fig.

4.36(a). Equation (4.2.16) can be further reduced to a formula

188 4 Perfect Flow



(4.2.17)

where

is called a slip factor, and

P is a correlation constant obtained

by experience in an actual pump design.

P may be written, as a result of

many attempts in predicting the amount of slip from impellers, Pflei-

dener(1961)

(

4.2.18)

Note that in Eq. (4.2.18),

q is an experimental constant and usually given

by the following correlational formula for a radial flow impeller (10–30%

lower value for pumps with outlet guide vanes)



sin6.068.055.0

 ᨺ

(4.2.19)

and for impeller vanes with a three dimensional curved surface

 



sin12101

.. ᨺ

(4.2.19)

In Eq. (4.2.18),

is the first moment of an area for the axis of rotation

along the center line of the meridian cross-section, as shown in Fig. 4.35(a)

and defined for the radial flow pump as





rrrdrI

(4.2.20)

s is the number of vanes in an impeller, where for pump

(4.2.21)

The information given from the slip factor given in Eq. (4.2.17) is of vi-

tal importance, particularly to the compressor designer, where accurate

knowledge of it appears to preclude the possibility of achieving a higher

energy conversion rate between the impeller and fluid.

4.2.2 Airfoil Theory

As mentioned in the previous section, the airfoil theory is more appropriate

to apply for designing turbomachines, whose numbers of vanes are smaller

4.2 General Theories of Turbomachinery 189

and with a less contact surface area than with fluid flow. This is often en-

counters in axial flow turbomachines, where the flow to and from a turbo-

machine is in the axial direction of the rotation (shaft rotation), as illus-

trated in Fig. 4.37(a) and (b). In order to figure out the axial flow

turbomachine, (a) schematics of an axial flow pump and (b) an axial flow

turbine (Kaplan turbine) are displayed in Fig. 4.37. Usually in the actual

axial flow turbomachines, there are inlet guide vanes (which are not re-

garded as part of a control volume in the case of a pump. It usually func-

tions in directing flow away from the axial direction), rotor vanes (impeller

vane in pump and runner in turbine) and stationary guide vanes in pumps.

In axial flow turbomachines, a number of identical vanes, which are

equally spaced and paralleled to one another, are arranged so that they

form a cascade geometry. For the axial flow turbomachines of a high hub-

tip ratio, radial velocities are negligible and, to a close approximation, the

flow may be regarded as two dimensional. To obtain a truly two dimen-

sional flow, however, there would require a cascade of infinite extent so

that there would not be any flow interaction between each neighboring

vane. In Fig. 4.38(a), the control volume of an impeller vane is illustrated,

and with it the energy input through the shaft rotation is converted into

flow energy.

The flow approaches the cascade, as illustrates in Fig. 4.38(b), with an

absolute velocity

c at an angle

and leaves downstream of the cascade

with the absolute velocity

at an angle

. It is denoted that in the fol-

lowing analysis the flow is assumed to be incompressible



30.

and

steady in the control volume with the rotational speed

u . The assumption

of the steady flow is valid for an isolated cascade, where for the control

volume the number of vanes is low. The velocity diagrams at the inlet and

the outlet can be composed as shown in Fig. 4.38(b). The absolute veloci-

ties

c and

c are obtained by the vector sum of the rotational speed u

and the relative velocity w at the inlet and outlet respectively. In the airfoil

theory, approaching the axial flow turbomachines, the averaged velocities

to a representative airfoil are composed by taking the averaged velocity

diagram with reference to Fig. 4.38(c), where

w and c are the average

relative velocity and absolute velocity respectively with the average angle

and

. In Fig. 4.38(c), c is composed with u and the arithmetic

mean of

and

, whose circumferential component is



21 uu

www 

190 4 Perfect Flow

Fig. 4.37 Axial flow turbomachines and their nomenclature

4.2 General Theories of Turbomachinery 191

Fig. 4.38 Idealized axial flow pump

locity and force