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FLUID MECHANICS IIBMCF 3233

DIMENSIONAL ANALYSIS ANDSIMILARITY

Semester 1 Session 2013/2014

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DIMENSIONAL ANALYSIS &

SIMILITUDE FOR

TURBOMACHINERY

The relationship gravity times net head (gH) andpump properties such as flow rate (Q), diameter ofthe impeller blades (D), blade surface roughnessheight (), impeller rotational speed (), fluiddensity () fluid viscosity () yields

D

D

D

Qf

D

gH

DQfgH

,,

),,,,,(

2

322

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Cont

A simillar analysis with input brake horse power(bhp) as a function of the same variables results in

D

D

D

Qf

D

bhp

DQfbhp

,,

),,,,,(

2

353

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Application of the techniques of dimensionalanalysis to fans and pumps yields relationships ofthe form

)Hp(

tcoefficienhead

numberReynoldsRe

tcoefficienrateflow

tcoefficienpressure

tcoefficienpower

22

2

3

22

53

D

gHC

D

D

QC

D

pC

D

WC

H

Q

P

W

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Cont

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where W is shaft power, Q is volume flowrate, p is the pressure rise across the unitrotating at speed , and of diameter D. H

represents either the pressure head rise inthe case of pump, or the pressure headdrop for a turbine. The fluid type isdefined by density and viscosity .

Thus, for model testing to be valid, eachof these groups should have identicalvalues for the model and the prototype.

Cont

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Another dimensionless parameter for turbomachinery;

tp

41

2

1

1

2

byreplacing

byesfor turbinusedbealsocanrelationThis

1

1

is;sizetoesefficiencirelating1957),(Stepanoff

ncorrelatioempiricalacceptableAn

D

D

QH

W

CC

C

W

QH

C

CC

p

p

HQ

W

t

W

HQ

p

Cont

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Dimensionless number that characterizes optimum

condition of pump;

pumpflow-axial4

pumpflow-mixed41

pumpflow-radial1

speedspecificon thebased

selectionpumpypreliminarA

43

21

43

21

p

p

p

PH

Q

p

gH

Q

C

C

Cont

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Dimensionless number that characterizesoptimum condition of pump;

Turbines also can be classified according to

turbine specific

45

21

45

21/

t

t

H

Wt

gH

W

C

C

Cont

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EXAMPLE 1Select a pump to deliver 500 gal/minof water with a pressure rise of 65

psi. Assume a rotational speed not toexceed 3600 rpm.

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59.0C

QD

diameter;theDetermine

36003200

30

335N

/335

asestimatedisspeedthe

pump)flow(radial0.69gH

Q

speedspecifictheFind

/s1.11ft607.48

500Q

ft150g

p

/37730

3600

31

Q

43

P

43

P

p

3

ft

rpmrpm

sradQ

gH

HgHp

srad

P

PP

Cont

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EXAMPLE 2

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EXAMPLE 3

A one-fifth scale model of a water turbine is tested

in a laboratory at T = 20C. The diameter of the

model is 8.0 cm, its volume flow rate is 17.0 m3/h,

it spins at 1500 RPM, and it operates with a net

head of 15 m. At its best efficiency point, it delivers

450 W of shaft power. Determine the appropriate

rpm and volume flow rate for best efficiency.

Predict the brake horsepower output of the

prototype turbine, assuming exact geometric

similarity.

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