Fluid Mechanics
Fluid Mechanics
8th Edition
ISBN: 9780073398273
Author: Frank M. White
Publisher: McGraw-Hill Education
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Chapter 5, Problem 5.73P
To determine

i.

The relationship in dimensionless form.

Expert Solution
Check Mark

Answer to Problem 5.73P

The Pi terms are f(n,ΩDv)=PD2ρv3

Explanation of Solution

Given:

Windmill parameter:

Diameter = 50 cm

Power developed = 2.7 kW

Speed of rotation = 4800 rpm

Concept Used:

P=f(D,ρ,v,Ω,n)k=5j=3(M,L,T)Numberofterms=kj

Calculation:

P=f(D,ρ,v,Ω,n)k=5j=3(M,L,T)Numberofterms=kjNumberofterms=53Numberofterms=2

The dimensions of each variable are,

{P}={ML2T3}{D}={L}{ρ}={ML3}{v}={LT1}{Ω}={T1}{n}=1Nowtaking(D,ρ,v)asrepeatingvariables,wegetf(n, ΩDv)=PD2ρv3

Conclusion:

The Pi terms are f(n,ΩDv)=PD2ρv3.

To determine

ii.

The power developed by a geometrical and dynamically similar prototype with a diameter of 5 m.

Expert Solution
Check Mark

Answer to Problem 5.73P

The power developed is 5.989 kW

Explanation of Solution

Given:

Geometric parameter of the prototype:

Diameter = 5 m

Wind speed = 12 m/s

Altitude = 2000 m

Concept Used:

f(n,ΩDv)=PD2ρv3=constant

Calculation:

For geometric similarity  n( number of blades) = constant

For Dynamic similarity:

( ΩDv)=PD2ρv3=constantP1D12ρ1v13=P2D22ρ2v23P2=P1*D22ρ2v23D12ρ1v13Thedensityofairat2000maltitudeis1.0067kg/m3Thedensityofairatsealevelaltitudeis1.2255kg/m3P2=2700*52*1.0067* 123 0.52*1.2255* 403P2=5989W5.989kW

Conclusion:

The power developed is 5.989 kW.

To determine

iii.

The rotation rate by a geometrical and dynamically similar prototype with a diameter of 5 m.

Expert Solution
Check Mark

Answer to Problem 5.73P

The rotation rate is 144 rev/min

Explanation of Solution

Given:

Geometric parameter of the prototype:

Diameter = 5 m

Wind speed = 12 m/s

Altitude = 2000 m

Concept Used:

f(n,ΩDv)=PD2ρv3=constant

Calculation:

For geometric similarity  n( number of blades) = constant

For Dynamic similarity:

( ΩDv)=constantΩ1D1v1=Ω2D2v2Ω2=Ω1D1v1*v2D2Ω2=4800*0.540*125Ω2=144rev/min

Conclusion:

The rotation rate is 144 rev/min.

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Chapter 5 Solutions

Fluid Mechanics

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