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

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202 4 Perfect Flow

 



max

(4.2.69)

where

n is called the specific speed (pump) and

n is the power specific

speed (turbine). It is noted that in Eqs. (4.2.68) and (4.2.69),

g and

are

often eliminated from both

and

in common industrial practices,

through which they become dimensional parameter. They are called the

homologous specific speed, and in the case of using the homologous spe-

cific speed one should be aware of a unit to be used. From industrial ex-

perience, in selecting the type of pump based on an impeller (as typically

displayed in Fig. 4.34(a)), the specific speed

n is correlated as follows

1

n ; radial flow pump

41 

; mixed flow pump

n ; axial flow pump

(4.2.70)

For further reference, in selecting the type of hydraulic turbine (as typi-

cally displayed in Fig. 4.34(b)), the specific speed

n can be correlated an

follows

10 

n ; impulse turbine (e.g. Pelton type)

5.31 

n ; radial turbine (e.g. Francis type)

0.75.3 

n ; mixed flow turbine

0.140.7 

; axial flow turbine

(4.2.71)

where for Eqs. (4.2.70) and (4.2.71) the units are in n [rad/s= 30rpm

u ],

Q [m

/s], W [W],

[kg/m

] and g [m/s

]. It is noticeable that in general a

higher specific speed implies a larger machine, because of a higher volu-

metric flow rate with a low head (pump) and a higher output power and

low head drop (turbine). For economic reasons in developing a turbo-

machine, it is recommended to select the highest possible specific speed

consistent with high efficiency for a given duty. It is also worth noting that

larger turbomachines are more efficient than smaller ones of the same

geometric family, known as the scale effect. Among many proposed corre-

lations, the following formula is often used in practice, relating efficiencies

to size, and according to L. F. Moody (ref. Kittredge, 1968)

4.2 General Theories of Turbomachinery 203





(4.2.72)

where

can be used for both pumps and turbines.

4.2.4 Cavitation

Cavitation is the local formation of vapor bubbles in a liquid due to a pres-

sure reduction below the vapor pressure caused by a dynamic action of the

liquid, rather than an increase of temperature. The cavitation is largely due

to a local boiling in a hydraulic turbomachine (and more generally in an

element of a flow channel), through which causes the deterioration of a

machine’s performance, giving an actual limitation of machine design. Par-

ticularly, in selecting a type of turbomachine for a given head

and flow

rate

, it is desirable to choose the highest possible specific speed because

of the resulting reduction in size, weight and cost of production. However,

this would require the increase in fluid velocities, resulting in a local pres-

sure reduction, and thus causing the cavitation. The lower limit of size in

designing a turbomachine is therefore dictated by the cavitation. Figure

4.44 shows a typical performance characteristic, where the

–

relation-

ship represented in the

–

plot of a radial flow pump, showing the dete-

rioration in performance due to a fully developed cavitation. In the fully

developed cavitation state, pockets of vapor are formed, as illustrated in

Fig. 4.45 referring to the pressure state diagram Fig. 4.36(b), affecting the

whole flow field in vanes.

The cavitation also leads to consequences of structural damage of near

solid surfaces, known as cavitation erosion. The cavitation erosion is

caused by the following process. When the local pressure in a turbo-

machine is approached toward a critical pressure, depending upon flow

conditions, cavitation occurs. This is called cavitation inception. Bubbles

collapse, when they are swept into higher pressure regions, and a pressure

wave is produced by a sudden bubble collapse that propagates and hits a

solid surface. The repeated action of the bubbles collapsing near the solid

surface causes the cavitation erosion over a long time period, and results in

a functional failure of turbomachines. In designing turbomachines, the

avoidance of cavitation inception is one of the major tasks of design engi-

neers. The cavitation may also cause vibration and noise in turbomachines.

204 4 Perfect Flow

Fig. 4.44 Performance characteristic of a radial flow pump and cavita-

tion

Fig. 4.45 Cavitation region

A criterion for the formation of vapor bubbles is obtained in considera-

tion of the pressure difference between the absolute pressure and the vapor

pressure being equal to that of the dynamic pressure of the liquid. In Fig.

4.46, a suction pipe connected to the inlet of a pump (or draft tube con-

nected to discharge of a hydraulic turbine) is illustrated, where

H is the

height (head) of the pump inlet

(the turbine discharge).Also,

p is the

absolute pressure,

c is the absolute velocity and

w is the relative velocity

4.2 General Theories of Turbomachinery 205

at s .

p is the absolute pressure at point e in the vane rotor, whereas

hr is the loss head in the pipe and

p is the atmospheric pressure at the

liquid level. The energy balance between 0 and

can be written in terms

of head as

ggg 2

lsv

 r

(4.2.73)

Fig. 4.46 Suction (or draft) configuration

Noting that





]

is the dynamic head loss between

and e , and by

defining

]

as the loss coefficient, we have

ggg 2

ses

wpp

]



(4.2.74)

where the constant

]

is determined by a channel configuration. Combin-

ing Eq. (4.2.73) with Eq. (4.2.74), we have

ggggg

UUUU

esse

lsv

pcpp

 r

gg 22

]



(4.2.75)

206 4 Perfect Flow

sure

p where cavitations tend to occur. The condition without cavitations

thus can be written as follows

ggggggg 222

222

ssvssv

wcpcpp

]

UUUU

t r

(4.2.76)

where we set the total dynamic head as

gg 22

max

]



(4.2.77)

It will prove useful in practical engineering to define the cavitation limit, at

which the cavitation occurs, as

svsv

limit

(4.2.78)

It is mentioned that

H depends upon the operating condition of a pump

(in the case of pump operation), but

limitsv

H is to be determined uniquely

for each individual pump and gives a criterion of cavitation.

limitsv

H is

called the net positive suction head (NPSH), through which the critical

cavitation number is defined by the following formula

pcp

vss

limit



ggg

(4.2.79)

where

is the actual head for a given flow rate Q at a speed of revolu-

tion n . In the case of a hydraulic turbine, the critical cavitation number is

defined as

limit



(4.2.80)

where

lsvs

hHH 

. Therefore, the condition to avoid the cavitation is

given by the pump

Hı

vss

t

ggg 2

(4.2.81)

and for the turbine

Now cases are examined when

p reaches the saturation vapor pres-

4.2 General Theories of Turbomachinery 207

Hı

t

(4.2.82)

In operating turbomachines, the performance laws must include the

additional variable

, taking into account the effects of cavitation. Con-

sidering the similitude at an inlet of the pump (or discharge of hydraulic

turbine), we can derive the suction specific speeds, along with Eq. (4.2.68),

by replacing

H as follows



(4.2.83)

and for the critical cavitation number given in Eq. (4.2.79) as

HıH

limit

(4.2.84)

More conveniently, in practice it is useful to relate

n and

n , where we

find a relation by substituting

limitsv

H to

H as in Eq. (4.2.83) with the

aid of the definition in Eq. (4.2.68) as follows

/ınn

(4.2.85)

From past experiences based on experimental results, the cavitation incep-

tion occurs almost at the same conditions, satisfying the following rela-

tionship.

nı v

(4.2.86)

Equations (4.2.85) and (4.2.86) give the result that values of

for all

pumps (also for all hydraulic turbines) are kept constant as to designate a

resistance to cavitation. They are approximately

Pumps ; 952

(4.2.87)

Turbines ; 963

(4.2.88)

where

n is in

30rpm

u ,

is in

and svHg is in

sm.

208 4 Perfect Flow

Exercise

Exercise 4.2.1 The Pelton Wheel

Turbines (hydraulic steam or gas), with stationary guide vanes and runner

(as studied in Section 4.2.2) have a continuous pressure drop when a fluid

flows through the passages, where a part of the available head is converted

into kinetic energy of flow. These turbines are categorized as the reaction

turbine. Reaction turbines usually have sophisticated functions and can run

with lower head installations. However, when a very high head is available,

it is often more suitable to adopt the impulse type of turbine (particularly

in the case of hydraulic power generation). The Pelton wheel is one which

is extensively used for high head installations, with its simple construction

and thus is economically competitive against reaction turbines.

Considering the theory developed in Exercise 4.1.8, explain the func-

tion of a Pelton wheel and discuss the energy conversion characteristics.

Ans.

A Pelton wheel primarily consists of a stationary inlet nozzle, a runner

and a casing. The runner has multiple buckets mounted on the periphery of

its rotating wheel, as shown in Fig. 4.47(a). The runner is driven by water

jets discharged by a nozzle with discharge rate

Q . As the jet impacts the

rotating buckets with an absolute velocity

v , the kinetic energy of the jet

is converted into a power to drive the runner. Each bucket has a splitter

that divides the jet into two equal streams and each stream leaves the

bucket with its relative velocity vector in the horizontal plane. It is noted

that in practice, the ideal angle of

q180 is limited to a value between q160

and

q168 so that the water leaving a bucket may stay free of the trailing

buckets.

In order to estimate the net power output, we consider a case where the

peripheral speed of a bucket, i.e. the circumferential speed, is assumed to

, and the angle of the leaving jet streams is

, as indicated in Fig.

4.47(b). According to the relationship obtain in Exercise 4.1.8, the torque

given to the runner by the water jet is

 

cos1



 uvQrT

(1)

where

is the runner radius as shown in Fig. 4.47 (a). If the runner rotates

with an angular velocity

, the ideal power output



PW can be writ-

ten as

Exercise 209

 

cos1 uvQrW

 

cos1

 uvuQ

(2)

Fig. 4.47 Impulse turbine; the Pelton wheel

Introducing the volumetric efficiency

and the mechanical efficiency

, with reference to Eqs. (4.2.42) and (4.2.43), the net power output

is obtained from the relation

210 4 Perfect Flow

mvi

(3)

On the other hand, the net power output

W is also estimated by the

head drop

as follows



mvh

HQW

(4)

and

HQW g

(5)

where

is the hydraulic efficiency defined in Eq. (4.2.41) and

is the

total efficiency defined in Eq. (4.2.44).

From Eqs. (3) and (4), the hydrodynamic coefficient

is written

where

 

cos1

 uvu

HHQ

(6)

The water jet velocity

v can be given where the head drop is

HCv

(7)

C is the velocity coefficient. Therefore, with Eqs. (6) and (7), we can ob-

tain an expression for

where

 

cos1

 uvu

(8)

and



cos12

cos1

222

ENN





pvp

ggg

(9)

is called the speed factor. According to Eq. (8),

reaches

the maximum when

vu , so that it follows that



max

cos1



(10)

Exercise 211

For plotting

vs.

in Eq. (9), we have a diagram in Fig. 4.48. If we

take representative values (typically) 940

C and q 168

, we have

the maximum hydraulic efficiency 8310

max



. However, in actual op-

eration,

max,h

will be reduced due to the reduction of relative velocity

over the bucket via friction and non-uniform velocity splitting, which re-

sultantly alters the Eq. (10) to



max

cos1

EHK



(11)

is the friction factor and typically 90.

in an actual operation.

Fig. 4.48

Exercise 4.2.2 Radial Flow Impeller for Turbo-Pump

A centrifugal pump with idealized radial flow impeller rotates with

1500

n rpm, as shown in Fig. 4.49. Water enters the impeller axially

through the eye, flows radially along the absolute pass line in the blades,

and is discharged radially. Denoting the inlet location 1 and the outlet loca-

tion 2, the idealized velocity diagrams are drawn in Fig. 4.49, using the

sign notation of Fig. 4.35(b), where

2rD

and

2rD

in Fig. 4.49

and their basic dimensions of the impeller are mm200

D , mm400

D ,

mm100

b , and mm50

b . If the volumetric flow rate is /sm250



Q ,

obtain the theoretical torque

T and Head

H . Also calculate

T and

when the discharge rate

Q is doubled. The meridional component of ve-

locity

c is assumed to be maintained constant as fluid flows through the