Chapter 6, Problem 96P

(a)

To determine

The volume of air passes through circular area swept out by the blades.

Answer to Problem 96P

The volume is $\underset{\_}{500{\text{m}}^3}$.

Explanation of Solution

Write the expression for volume of air swept out by the blade.

    $V=Ad$        (I)

Here $V$ is the volume, $A$ is the area and $d$ is the distance moved by the air.

Write the equation for circle as the blade swept out circular area.

    $A=\pi L^2$        (II)

Here $L$ is the length of the blade.

Write the expression for the distance travelled by air in the given time.

    $d=v\Delta t$        (III)

Here $v$ is the speed of air and $\Delta t$ is the time interval.

Substitute (II) and (III) in (I)

    $V=\pi L^{2}v\Delta t$        (IV)

Conclusion

Substitute $4\text{m}$ for $L$, $\text{10}\text{m}/\text{s}$ for $v$ and $1\text{s}$ for $\Delta t$ in (IV)

    $\begin{array}{c}V=\pi {\left(4\text{m}\right)}^2\left(\text{10}\text{m}/\text{s}\right)\left(1\text{s}\right)\\ =502{\text{m}}^3\\ \approx 500{\text{m}}^3\end{array}$

The volume is $\underset{\_}{500{\text{m}}^3}$.

(b)

To determine

The mass of air.

Answer to Problem 96P

The mass of air is $\underset{\_}{600\text{kg}}$.

Explanation of Solution

Write the expression for mass in terms of density and volume.

    $m=V\rho$        (V)

Here $m$ is the mass, $V$ is the volume and $\rho$ is the density.

Conclusion

Substitute $500{\text{m}}^3$ for $V$ and $1.2\text{kg}/{\text{m}}^{\text{3}}$ for $\rho$ in (V)

    $\begin{array}{c}m=\left(500{\text{m}}^3\right)\left(1.2\text{kg}/{\text{m}}^{\text{3}}\right)\\ =600\text{kg}\end{array}$

The mass of air is $\underset{\_}{600\text{kg}}$.

(c)

To determine

Translational kinetic energy of the air.

Answer to Problem 96P

The translational kinetic energy of the air is $\underset{\_}{30\text{kJ}}$.

Explanation of Solution

Write the expression for translational kinetic energy.

    $K=\frac{1}{2}m{v}^2$        (VI)

Conclusion

Substitute $600\text{kg}$ for $m$ and $\text{10}\text{m}/\text{s}$ for $v$ in (VI)

    $\begin{array}{c}K=\frac{1}{2}\left(600\text{kg}\right){\left(\text{10}\text{m}/\text{s}\right)}^2\\ =30,000\text{J}\\ \text{=30}\text{kJ}\end{array}$

The translational kinetic energy of the air is $\underset{\_}{30\text{kJ}}$.

(d)

To determine

The electric power output of the turbine.

Answer to Problem 96P

The electric power output of the turbine $\underset{\_}{12\text{kW}}$.

Explanation of Solution

Write the expression for the power output of the turbine

    $P=\frac{\Delta E}{\Delta t}$        (VII)

Here $P$ is the power output, $\Delta E$ is the energy converted and $\Delta t$ is the tie interval.

As the turbine converts $40\%$ of the kinetic energy to the electrical energy,

    $\Delta E=0.4K$.

Then (VII) becomes,

    $P=\frac{0.4K}{\Delta t}$        (VIII)

Conclusion

Substitute $30\text{kJ}$ for $K$ and $1\text{s}$ for $\Delta t$ in (VIII)

    $\begin{array}{c}P=\frac{0.4\left(30\text{kJ}\right)}{1\text{s}}\\ =12\text{kW}\end{array}$

The electric power output of the turbine $\underset{\_}{12\text{kW}}$.

(e)

To determine

The power output of the turbine if the speed of the wind is decreased to half its initial value and the conclusion about the electric power production by wind turbines.

Answer to Problem 96P

The output power of the turbine decreases to $1/8$ of the previous value. It can be concluded that the power production of an individual wind turbine is inconsistent, since modest changes in the wind speed produce large changes in power output.

Explanation of Solution

Refer subpart (d) and write the equation for power output of the wind turbine.

    $P=\frac{\Delta E}{\Delta t}$

Assume that the entire kinetic energy of the wind is converted into electrical energy.

Then,

    $P=\frac{\frac{1}{2}m{v}^2}{\Delta t}$        (IX)

Substitute (V) in (IX)

    $P=\frac{\frac{1}{2}V\rho v^2}{\Delta t}$        (X)

Substitute (IV) in (X)

    $\begin{array}{c}P=\frac{\frac{1}{2}\left(\pi L^{2}v\Delta t\right)\rho v^2}{\Delta t}\\ P\propto v^3\end{array}$        (XI)

From (XI) it is clear that the power output of the turbine is proportional to the cube of wind speed.

Let $P_v$ and $P_{\frac{v}{2}}$ be the power output of the turbine when wind speed id $v$ and $\raisebox{1ex}{$v$}\!\left/ \!\raisebox{-1ex}{$2$}\right.$

Take the ratio

    $\begin{array}{c}\frac{P_{\frac{v}{2}}}{P_v}=\frac{{\left(\raisebox{1ex}{$v$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\right)}^3}{v^3}\\ =\frac{1}{8}\end{array}$

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.

Conclusion

Therefore, the output power of the turbine decreases to $1/8$ of the previous value. It can be concluded that the power production of an individual wind turbine is inconsistent, since modest changes in the wind speed produce large changes in power output.