Chapter 2.8, Problem 69P

To determine

The actual electric power generated by the wind turbine.

The amount of electric energy and the revenue generated per day.

Answer to Problem 69P

The actual electric power generated by the wind turbine is $\text{804}\text{.2}\text{kW}$

The amount of electric energy and the revenue generated per day are $19,300\text{kWh}$ and $\$1737$ respectively.

Explanation of Solution

Write the formula to calculate mechanical energy of air per unit mass $\left(e_{\text{mech}}\right)$.

$\begin{array}{c}{e}_{\text{mech}}=ke\\ =\frac{V^2}{2}\end{array}$ (I)

Here, kinetic energy is ke and velocity of air is V.

Write the formula to calculate the mass flow rate of air $\left(\dot{m}\right)$.

$\begin{array}{c}\dot{m}=\rho VA\\ =\rho V\frac{\pi D^2}{4}\end{array}$ (II)

Here, density of air is $\rho$, area of the wind turbine is A and diameter of a wind turbine blades is D.

Write the formula to calculate power generation potential $\left({\dot{W}}_{\max }\right)$.

$\begin{array}{c}{\dot{W}}_{\max }={\dot{E}}_{\text{mech}}\\ =\dot{m}{e}_{\text{mech}}\end{array}$ (III)

Here, rate of total mechanical energy of air is ${\dot{E}}_{\text{mech}}$.

Write the formula to calculate actual electric power generation.

${\dot{W}}_{\text{elect}}={\eta }_{\text{wind}\text{turbine}}{\dot{W}}_{\max }$ (IV)

Here, efficiency of wind turbine is ${\eta }_{\text{wind}\text{turbine}}$.

Write the formula to calculate amount of electricity generated per day.

$\text{Amount}\text{of}\text{electricity}={\dot{W}}_{\text{elect}}\times \text{Operating}\text{hours}$ (V)

Here, wind power is ${\dot{W}}_{\text{elect}}$.

Write the formula to calculate revenue generated per day.

$\text{Revenues}=\left(\text{Amount}\text{of}\text{electricity}\right)\left(\text{Unit}\text{price}\right)$ (VI)

Conclusion:

Substitute $8\text{m}/\text{s}$ for V in Equation (I).

$\begin{array}{c}{e}_{\text{mech}}=\frac{{\left(8\text{m}/\text{s}\right)}^2}{2}\\ =\frac{64{\text{m}}^2/{\text{s}}^2}{2}\left(\frac{1\text{kJ}/\text{kg}}{1000{\text{m}}^2/{\text{s}}^2}\right)\\ =0.032\text{kJ}/\text{kg}\end{array}$

Substitute $1.25\text{kg}/{\text{m}}^3$ for $\rho$, $8\text{m}/\text{s}$ for V, and $100\text{m}$ for D in Equation (II).

$\begin{array}{c}\dot{m}=\left(1.25\text{kg}/{\text{m}}^3\right)\left(8\text{m}/\text{s}\right)\frac{\pi {\left(100\text{m}\right)}^2}{4}\\ =78,540\text{kg}/\text{s}\end{array}$

Substitute $78,540\text{kg}/\text{s}$ for $\dot{m}$ and 0.032 kJ/kg for $e_{\text{mech}}$ in Equation (III).

$\begin{array}{c}{\dot{W}}_{\max }=\left(78,540\text{kg}/\text{s}\right)\left(\text{0}\text{.032}\text{kJ}/\text{kg}\right)\\ =2513\text{kJ}/\text{s}\\ =2513\text{kJ}/\text{s}\left(\frac{1\text{kW}}{1\text{kJ}/\text{s}}\right)\\ =2513\text{kW}\end{array}$

Substitute $2513\text{kW}$ for ${\dot{W}}_{\max }$ and 0.32 for ${\eta }_{\text{wind}\text{turbine}}$ in Equation (IV).

$\begin{array}{c}{\dot{W}}_{\text{elect}}=\left(0.32\right)\left(2513\text{kW}\right)\\ =804.2\text{kW}\end{array}$

Thus, the actual electric power generated by the wind turbine is $\text{804}\text{.2}\text{kW}$

Substitute $\text{804}\text{.2}\text{kW}$ for ${\dot{W}}_{\text{elect}}$ and $\text{24}\text{h}$ for operating hours in Equation (V).

$\begin{array}{c}\text{Amount}\text{of}\text{electricity}=\left(804.2\text{kW}\right)\left(\text{24}\text{h}\right)\\ =19,300\text{kWh}\end{array}$

Thus, the amount of electric energy generated per day is $19,300\text{kWh}$.

Substitute $19,300\text{kWh}$ for amount of electricity and $\text{\$0}\text{.09}/\text{kWh}$ for Unit price in Equation (VI).

$\begin{array}{c}\text{Revenues}=\left(\text{19,300}\text{kWh}\right)\left(\text{\$0}\text{.09}/\text{kWh}\right)\\ =\$1737\end{array}$

Thus, the revenue generated per day is $\$1737$.