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Concept explainers
(a)
The volume of air passes through the circular area swept out by the blades.
(a)
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Answer to Problem 97P
The volume of air passes through the circular area swept out by the blades is
Explanation of Solution
Given that the length of the blade is
Write the expression for the volume of air swept out by the blade.
Here,
Since the blade sweeps out circular area, the radius of the circle is equal to the length of the blade.
Write the expression for the area swept out by the blade.
Here,
Write the expression for the distance travelled by the air in the given time.
Here,
Use equation (II) and (III) in (I).
Conclusion:
Substitute
Therefore, the volume of air passes through the circular area swept out by the blades is
(b)
The mass of
(b)
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Answer to Problem 97P
The mass of
Explanation of Solution
The density of air is
Write the expression for the mass of air in terms of its density and volume.
Here,
Conclusion:
Substitute
Therefore, the mass of
(c)
The translational kinetic energy of the air.
(c)
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Answer to Problem 97P
The translational kinetic energy of the air is
Explanation of Solution
Given that the mass of air is
Write the expression for the translational kinetic energy.
Here,
Conclusion:
Substitute
Therefore, the translational kinetic energy of the air is
(d)
The electric power output of the turbine.
(d)
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Answer to Problem 97P
The electric power output of the turbine is
Explanation of Solution
Given that the turbine can convert
Write the expression for the power output of the turbine (it is the energy converted per unit time).
Here,
Since the turbine converts
Thus, the equation (VII) becomes,
Conclusion:
Substitute
Therefore, the electric power output of the turbine is
(e)
The power output of the turbine if the speed of the wind is decreases to half of its initial value and the conclusion about the electric power production by wind turbines.
(e)
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Answer to Problem 97P
If the speed of the wind is decreases to half of its initial value, the power output of the turbine decrease to
Explanation of Solution
Equation (VII) gives the power output of the wind turbine.
Assume that the entire kinetic energy of the wind is converted to electrical energy. Then the equation (VII) can be modified as,
Use equation (V) in (IX).
Use equation (IV) in (X).
Conclusion:
From the expression (XI) it is clear that the power output of the turbine is proportional to the cube of the wind speed.
Let
Take the ratio of
This indicates that the power production of an individual wind turbine is inconsistent, since modest changes in the wind speed produce large changes in power output.
Therefore, If the speed of the wind is decreases to half of its initial value, the power output of the turbine decrease to
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