lec 3 -dim analysis
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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.
BMCF3233
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