Fluid Mechanics
Fluid Mechanics
8th Edition
ISBN: 9780073398273
Author: Frank M. White
Publisher: McGraw-Hill Education
Question
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Chapter 11, Problem 11.95P
To determine

(a)

To prove:

The maximum power, hf=H3 for an impulse turbine.

Expert Solution
Check Mark

Answer to Problem 11.95P

The maximum power generated by an impulse turbine is hf=H3.

Explanation of Solution

Given:

Referencing the below diagram:

Fluid Mechanics, Chapter 11, Problem 11.95P , additional homework tip  1

The equation given below is also taking into consideration:

Fluid Mechanics, Chapter 11, Problem 11.95P , additional homework tip  2

Concept Used:

The maximum power for an impulse turbine can be obtained when, u=Vj2

Where,

Vj = jet velocity and u = vane velocity

Calculation:

Now, for determining maximum power:

P=ρQu(Vju)(1cosβ)P=ρQVj2(VjVj2)(1cosβ)P=ρQVj24(1cosβ)

Where,

ρ

  • = density of fluid
  • Q = flow rate of turbine
  • β
  • = exit angle
  • Jet area has an equation of,

    Aj=QVj

So the maximum power of turbine is reduced to,

P=ρA4(1cosβ)Vj3.....(1)P=CVj2Vj where C = ρA4(1cosβ) is constant

For the reservoir and the outlet jet, applying steady flow energy equation:

H=fLDVpipe22g+Vj22g....(2)Vj2=2gHfLVpipe2D

Where:

  • H = head of turbine
  • f = friction factor
  • L = length of pipe
  • D = diameter of pipe
  • Vpipe
  • = velocity of pipe
  • Now, using continuity equation:

AjVj=ApipeVpipeπDj24Vj=πD24VpipeVpipe=Dj2D2Vj...(3)

In equation (2), let us put the value of Vpipe=Dj2D2Vj from equation (3).

Vj2=2gHfLDj4D5Vj2

Lastly, putting the value of Vj2=2gHfLDj4D5Vj2 in equation (1),

P=C[2gHVjfLDj4D5Vj3 ]

To determine maximum power, below is the condition:

dPdVj=0Cd[2gHVjfL D j 4 D 5 Vj3 ]dVj=02gH3fLDj4D5Vj2=02gH3fLDVpipe2=0...(4)

But the friction head loss of pipe:

hf=fLDVpipe22g

Therefore, the maximum power of an impulse turbine is hf=H3.

Conclusion:

The maximum power generated by an impulse turbine is hf=H3.

To determine

(b)

To prove:

The optimum velocity is Vj=43gH for given impulse turbine.

Expert Solution
Check Mark

Answer to Problem 11.95P

The optimum velocity of an impulse turbine is Vj=43gH for given impulse turbine.

Explanation of Solution

Given:

Referencing the below diagram:

Fluid Mechanics, Chapter 11, Problem 11.95P , additional homework tip  3

The equation given below is also taking into consideration:

Fluid Mechanics, Chapter 11, Problem 11.95P , additional homework tip  4

Concept Used:

Jet area has an equation of,

Aj=QVj

So, the maximum power of turbine is reduced to:

P=ρA4(1cosβ)Vj3.....(1)P=CVj2Vj where C = ρA4(1cosβ) is constant

For the reservoir and the outlet jet, applying steady flow energy equation.

H=fLDVpipe22g+Vj22g....(2)Vj2=2gHfLVpipe2D

Where,

  • H = head of turbine
  • f = friction factor
  • L = length of pipe
  • D = diameter of pipe
  • Vpipe
  • = velocity of pipe
  • Calculation:

For getting optimum velocity, we have equation (2):

Vj2=2gHfLVpipe2DVj=2gH2ghfVj=2gH2gH3Vj=43gH

The optimum velocity of an impulse turbine is

Vj=43gH proved.

Conclusion:

The optimum velocity of an impulse turbine is Vj=43gH for given impulse turbine.

To determine

(c)

The best nozzle diameter is Dj=[D52fL]14 should be proved for given impulse turbine.

Expert Solution
Check Mark

Answer to Problem 11.95P

The best nozzle diameter is Dj=[D52fL]14 for given impulse turbine is proved.

Explanation of Solution

Given:

Referencing the below diagram:

Fluid Mechanics, Chapter 11, Problem 11.95P , additional homework tip  5

The equation given below is also taking into consideration:

Fluid Mechanics, Chapter 11, Problem 11.95P , additional homework tip  6

Concept Used:

The continuity equation:

AjVj=ApipeVpipeπDj24Vj=πD24VpipeVpipe=Dj2D2Vj...(3)

The friction head loss of pipe:

hf=fLDVpipe22g

hf=H3

Calculation:

For determining nozzle diameter,

hf=fLDVpipe22gH3=fLDVpipe22gVpipe2=2gDH3fL...(5)

We already have equation (3) with value of Vpipe=Dj2D2Vj...(3)

2gDH3fL=Dj2D2Vj22gDH3fL=Dj2D243gH where(Vj= 4 3gH)Dj=[ D 52fL]14

Conclusion:

The best nozzle diameter is

Dj=[D52fL]14 for given impulse turbine is proved.

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Chapter 11 Solutions

Fluid Mechanics

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