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

212 4 Perfect Flow

impeller, in which the fluid enters the impeller with a negligible tangential

velocity component. It is also assumed that the outlet vane angle

2

E

is

such that

$

22

2

E

. Show the specific speed

s

n

for both operation of Q

and

Q2 as well.

Fig. 4.49 Radial flow impeller of turbo-pump

Ans.

With reference to Fig. 4.49, since the tangential component

111

cos

D

cc

u

is assumed to be zero,

th

T can be estimated from Eq. (4.2.1)

as



2

22

/DQcT

uth

U

(1)

and from the velocity diagram at the outlet 2 ,

u

c

2

will be derived as

2222

cot

E

mu

cuc 

(2)

where

222

b

D

Qc

m

S

/ and 60

22

/nDu

S

. For the given values of these

parameters, we can obtain



>@

m/s98.305040250

2

uu ../.

S

m

c

>@

m/s4.3160150040

2

uu /.

S

u

Exercise 213

>@

m/s52122cot98.34.31

2

.c

u

q

Therefore, from Eq. (1), we have



>@

mN10755212402501000

22

 uuu ...

uth

cQrT

U

and also from Eq. (4.2.4),

th

H is

>@

m868819521431

22

....

u g

uth

cuH

When the discharge flow rate Q is doubled

>

@

sm502502

3

.. u Q

so that



>

@

m/s9670504050

2

.../. uu

S

m

c

Thus

>@

m/s71122cot967431cot

2222

... q 

E

mu

cuc

Therefore, in similar manner we have



>@

mN1170711240501000  uuu ./..

th

T

>@

m437819711431 ../.. u

th

H

The specific speeds for Q and Q2 are respectively calculated in Eq.

(4.2.68)



5900

868819

250

30

1500

43

21

43

21

.

..

.

/

u

¸

¹

·

¨

©

§

u

S

H

Q

nn

s

g



321

437819

50

30

1500

43

21

.

..

.

/

u

¸

¹

·

¨

©

§

u

c

S

s

n

The results show that for Q2, 1!

c

s

n , indicating a mixed flow pump may

be more appropriate.

Exercise 4.2.3 Power Output from Axial Flow Hydraulic Turbine

For a low head and high specific speed, axial flow hydraulic turbines are

often appropriate because of a higher meridional velocity, running faster

with higher efficiencies. The propeller turbine with adjustable blades is

214 4 Perfect Flow

known as the Kaplan turbine, which can run with a wide output range by

setting the runner blades and guide vanes simultaneously. Referring to Fig.

4.50 (a) representative dimensions, (b) velocity diagrams, (c) draft tube

configuration, consider the following problems:

The mean radius

R

to the runner is

m5.0 R

, and the width

B

of it is

m10. B

. The stream of water enters to the runner with an angle of

$

35

1

D

, and is discharged at atmospheric pressure with an angle of

$

25

2

E

from the outlet of the runner. The head difference

1

H between

the upper level of the water surface and the outlet of runner is

m12

1

H

.

Before entering the runner, the stream turns to a right angle of the runner,

so that the absolute velocity

2

c at the outlet of the runner is perpendicular

to the circumferential speed

u , i.e.

$

90

2

D

. It is assumed that there is no

radial velocity when water passes through the vanes, and that the axial ve-

locity at the inlet to the runner and the outlet is kept constant.

(i) Estimate the power transmitted to the runner and the revolutional speed

of the runner, ignoring hydraulic losses.

(ii) As illustrated in Fig. 4.50(c), if a draft tube of a diffuser type is at-

tached between the outlet of runner to the lower discharged water sur-

face, discuss the change of the power transmitted and revolutional

speed compared to problem (i). Set m4

s

H for problem (ii).

Fig. 4.50 Axial flow propeller turbine; Kaplan turbine

Exercise 215

Ans.

(i) Denote the actual head

H

to the runner is given where

g2

2

21

cHH 

(1)

With the given condition

u

c

2

assumed to be zero, so that from Eq. (4.2.4),

the theoretical head is to be written as

g

u

ucH

1

(2)

By the velocity diagram,

u

c

1

can be obtained by knowing that

21

cc

m

121

cot

D

cc

u

(3)

The discharge flow rate

Q is also estimated as

m

RBcQ

1

2

S

2

2 RBc

S

(4)

Knowing the condition at the outlet with reference to Fig. 4.50 (b), the ab-

solute velocity

2

c is written as

2

cot

E

u

c

(5)

The key parameters of

Q ,

H

,

2

c ,

u

c

1

, and u in Eqs. (1–5) are thus calcu-

lated by the known quantities,

R

,

B

,

1

D

,

2

E

, and

1

H

as follows

12

2

21222

cotcot

1

cotcot

1

DEDE

cccH

gg

(6)

and with Eqs. (1), (2) and (6), we can obtain the relationship

12

2

1

cotcot

1

2

DE

c

H

gg



(7)

so that

2

c will become

1cotcot2

2

12

1

2



DE

H

c

g

(8)

Once

2

c is obtained, therefore, the rest of unknown are readily calculated,

using each given value as follows

216 4 Perfect Flow

>@

sm755

14311422

128192

2

.

..

.

uu

uu

c

>@

m3110

8192

755

12

2

.

u

 H

>

@

sm81175510501432

3

..... uuuu Q

As a result, the power (work)

W

transmitted to the runner is obtained to

give

  

>@

kW18331108191000811 u

u

u ...HQW g

U

Also with the relation

Z

Ru , we are able to estimate the revolutional

speed of the runner from the angular velocity

Z

>@

srad724

50

142755cot

22

.

..

u

R

c

R

u

E

Z

so that we can obtain the revolutional speed with

>@

rpm235

2

60724

2

60

u

SS

Z

.

n

(ii) The installation of the draft tube adds the extra head to

1

H , as illus-

trated in Fig. 4.50(c), i.e.

s

HHH 

1

(9)

With the same manner as verified in problem (i), we should be able to cal-

culate

Q and

H

after obtaining

2

c likewise know that

12

2

21

cotcot

1

DE

cHH

s

g



(10)

so that



>@

sm16.7

43.114.2

41281.9

cotcot

12

1

2

u

u



DE

s

HH

c

g

Exercise 217

>@



>@

rpm293

2

60503515

sm3515142167

kW353412252819

sm25216710502

3

u

uu

uuu

S

..

...

..

....

n

u

W

Q

There is certainly an improve of the output characteristic of the tur-

bines by adding the draft tube.

Exercise 4.2.4 Maximum Suction Head for Cavitation Limit

A single-suction radial flow pump is to operate for the actual head

m50 H , the volumetric flow rate /sm0250

3

. Q and the revolutional

speed rpm2900 n . Determine the maximum suction head

maxsv

H , below

which the pump can be operated without the cavitation. It is assumed that

water temperature in the suction pipe is

ഒ60 and the atmospheric pres-

sure of the water surface at suction is

kPa3101.

a

p . It is further assumed

that the loss head in the section pipe is m51.

l

h . The constant for the

critical cavitation number

c

V

in Eq. (4.2.86) would be obtained from

2360.

s

k , where

s

k is defined as

34

ssc

nk

V

.

Ans.

The specific speed

s

n of the pump is



4590

50819

0250

2

60

2900

4

3

2

1

.

u

¸

¹

·

¨

©

§

u

S

H

Q

nn

s

g

(1)

From Eq. (4.2.86), knowing that the constant is

s

k , the critical cavitation

number

c

V

is thus given where



084045902360 ... u

c

V

(2)

NPSH is then

>@

m18.450084.0

limit/

u HH

csv

V

(3)

2

1

4

3

4

Thus,

218 4 Perfect Flow

The water density is

3

kg/m2983.

U

and the saturation vapor pressure is

kPa919

.

v

p at the temperature ഒ60 , which are obtained from a steam

table. Therefore,

maxsv

H will be calculated via Eq. (4.2.82), yielding

>@

m772

18451

892983

19900

892983

101300

limitmax

.

..

....

/



u



u



svl

va

sv

Hh

pp

H

gg

UU

Thus, the suction pipe must be installed as m772. high above the water’s

surface, so that there would not be cavitation.

Problems

4.2-1 A centrifugal water pump runs at rpm1450 n . The impeller has the

following dimensions: mm300

2

D , mm20

2

b and q 45

2

E

. If

the power consumption of the pump is kW40 W , calculate the

theoretical discharge

th

Q and obtain the theoretical head

th

H . There

would be no-slip flow in vanes due to internal circulation flow, so

that only a (Euler) theoretical head

th

H is to be considered. Assume

that the hydraulic efficiency is 850

.

h

K

, and the overall efficiency

is 750

.

K

. The fluid enters the impeller with a negligible tangential

velocity component. Also calculate the actual head.

Ans.



»

¼

º

«

¬

ª

»

¼

º

«

¬

ª

¸

¹

·

¨

©

§



m36.04

m442

m08480

cot

W

3

2

22

H

sQ

bD

Q

uu

Q

th

h

th

.

th

H

gH

Hg

E

S

KUK

4.2-2 A model of a centrifugal pump with the impeller diameter of

mm220

M

D

is tested at the revolutional speed of

rpm1500

M

n

.

Results of the test run were such that: the actual head was

m81

.

M

H , the actual discharge was minm751

3

.

M

Q and the

Problems 219

mechanical efficiency was 8.0

m

K

. Estimate the head

P

H , the

discharge

P

Q and the shaft input power

P

W , when the prototype of

a geometrically similar pump is in operation at the revolutional

speed of rpm800

P

n . The diameter of the prototype is

mm1000

P

D .

Ans.

»

¼

º

«

¬

ª

kW189

minm787

m610

3

P

W

Q

H

.

4.2-3 A centrifugal pump is to be operated with an actual head of

5.0 m1 H . The critical cavitations number is 080

.

c

V

at the

operating condition. The atmospheric pressure on the suction water

surface is kPa3101

.

a

p . The saturation vapor pressure at the op-

eration condition is kPa03

.

v

p and the head loss along the suc-

tion pipe is m61

.

l

h . Referring Fig. 4.46, obtain the maximum

allowable height between the suction water surface to the pump

inlet.

Ans.

>@

m27

max

.

sv

H

4.2-4 A Pelton wheel with a runner diameter of mm200 D is generating

a power output for a head drop of m100 H . In order to verify the

net power output a braking test was carried out for an ideal nozzle

being fully opened. The torque required to hold the wheel was

mN1500 

r

T . Denoting the angle of leaving the jet stream flow

from a bucket is

q 12

2

E

and the overall efficiency of the wheel is

850

.

K

, obtain the revolutional speed of the runner and estimate

the net power output.

Ans.

»

¼

º

«

¬

ª

kW142W

rpm713n

4.2-5 An axial flow hydraulic turbine is required to produce the power of

kW8800 W with a given head of m20 H . The overall effi-

ciency is 880

.

K

, and the hydraulic efficiency is 930.

h

K

for the

runner of a vane tip diameter of m4

0

D and a hob diameter of

m751.

n

D . Calculate the inlet vane angle

1

E

and the outlet vane

angle

2

E

of the runner. Assume there is no draft tube and the flow is

discharged to the atmosphere.

220 4 Perfect Flow

Ans.



»

¼

º

«

¬

ª



o

2

o

1

2

1

111

5311

918

tan

.

E

m

u

m

uc

uuc

Nomenclature

A

area

a

sound speed

c

absolute velocity

c

C

contraction coefficient

d

C

discharge coefficient

D

C

drag coefficient

L

C

lift coefficient

M

C

moment coefficient

Q

C

volumetric flow rate coefficient

v

C

velocity coefficient

W

C

power coefficient

n

CCCC ...,,,,

210

complex constants

D

drag

i

FF,

force, i designated forces

H

total head

th

H

theoretical head (Euler head)

k

specific heat ratio

s

k

constant

f

K

flow coefficient

L

lift

c

l

chord length

M

Mach number

m



mass flow rate

c

n

critical cavitation number

i

n

toque component (i =1,2,3)

s

n

specific speed (pump)

t

n

power specific speed (turbine)

n

ˆ

normal unit vector

P

pressure function

c

P

correlation constant

Nomenclature 221

w

P

power transmitted to (or from) fluid

atm

p

(

a

p

)

surrounding pressure

Q

flow rate

c

q

experimental constant

Re

Reynolds number

r

position vector

r

radius

c

s

number of vanes

r

T

torque

c

t

pitch of vanes

f

uU ,

real constants, average upstream velocity

vu,

velocity vector

r

v

relative velocity vector

W

work transfer, complex potential

W

( z )

velocity potential

w

complex velocity, relative velocity

z

complex number iy

x

z 

J

D

,

angle

E

angle, diameter ratio

*

circulation (potential vortex)

M

H

(Mach number) correction coefficient

]

pressure (head) loss coefficient

K

efficiency, total (overall) efficiency

d

K

diffuser efficiency

0

K

viscosity

p

K

propeller efficiency

p

N

speed factor

c

V

critical cavitation number

s

V

slip factor

ij

W

stress tensor

X

kinematic viscosity

[

correction factor

scalar potential

I

potential function, velocity potential

M

head coefficient

)

\

stream function

Z

angular velocity

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

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