Respuesta :
Answer:
The wind turbine generates [tex]19297.222[/tex] kilowatt-hours of electricity daily.
The wind turbine makes a daily revenue of 1736.75 US dollars.
Explanation:
First, we have to determine the stored energy of wind ([tex]E_{wind}[/tex]), measured in Joules, by means of definition of Kinetic Energy:
[tex]E_{wind} = \frac{1}{2}\cdot \dot m_{wind}\cdot \Delta t \cdot v_{wind}^{2}[/tex] (Eq. 1)
Where:
[tex]\dot m_{wind}[/tex] - Mass flow of wind, measured in kilograms per second.
[tex]\Delta t[/tex] - Time in which wind acts in a day, measured in seconds.
[tex]v_{wind}[/tex] - Steady wind speed, measured in meters per second.
By assuming constant mass flow and volume flows and using definitions of mass and volume flows, we expand the expression above:
[tex]E_{wind} = \frac{1}{2}\cdot \rho_{air}\cdot \dot V_{air} \cdot \Delta t \cdot v_{wind}^{2}[/tex] (Eq. 1b)
Where:
[tex]\rho_{air}[/tex] - Density of air, measured in kilograms per cubic meter.
[tex]\dot V_{air}[/tex] - Volume flow of air through wind turbine, measured in cubic meters per second.
[tex]E_{wind} = \frac{1}{2}\cdot \rho_{air}\cdot A_{c}\cdot \Delta t\cdot v_{wind}^{3}[/tex] (Eq. 2)
Where [tex]A_{c}[/tex] is the area of the wind flow crossing the turbine, measured in square meters. This area is determined by the following equation:
[tex]A_{c} = \frac{\pi}{4}\cdot D^{2}[/tex] (Eq. 3)
Where [tex]D[/tex] is the diameter of the wind turbine blade, measured in meters.
If we know that [tex]\rho_{air} = 1.25\,\frac{kg}{m^{3}}[/tex], [tex]D = 100\,m[/tex], [tex]\Delta t = 86400\,s[/tex] and [tex]v_{wind} = 8\,\frac{m}{s}[/tex], the stored energy of the wind in a day is:
[tex]A_{c} = \frac{\pi}{4}\cdot (100\,m)^{2}[/tex]
[tex]A_{c} \approx 7853.982\,m^{2}[/tex]
[tex]E_{wind} = \frac{1}{2}\cdot \left(1.25\,\frac{kg}{m^{3}} \right) \cdot (7853.982\,m^{2})\cdot (86400\,s)\cdot \left(8\,\frac{m}{s} \right)^{3}[/tex]
[tex]E_{wind} = 2.171\times 10^{11}\,J[/tex]
Now, we proceed to determine the quantity of energy from wind being used by the wind turbine in a day ([tex]E_{turbine}[/tex]), measured in joules, with the help of the definition of efficiency:
[tex]E_{turbine} = \eta\cdot E_{wind}[/tex] (Eq. 4)
Where [tex]\eta[/tex] is the overall efficiency of the wind turbine, dimensionless.
If we get that [tex]E_{wind} = 2.171\times 10^{11}\,J[/tex] and [tex]\eta = 0.32[/tex], then the energy is:
[tex]E_{turbine} = 0.32\cdot (2.171\times 10^{11}\,J)[/tex]
[tex]E_{turbine} = 6.947\times 10^{10}\,J[/tex]
The wind turbine generates [tex]6.947\times 10^{10}[/tex] joules of electricity daily.
A kilowatt-hours equals 3.6 million joules. We calculate the equivalent amount of energy generated by wind turbine in kilowatt-hours:
[tex]E_{turbine} = 6.947\times 10^{10}\,J\times\frac{1\,kWh}{3.6\times 10^{6}\,J}[/tex]
[tex]E_{turbine} = 19297.222\,kWh[/tex]
The wind turbine generates [tex]19297.222[/tex] kilowatt-hours of electricity daily.
Lastly, the revenue generated per day can be found by employing the following:
[tex]C_{rev} = c\cdot E_{turbine}[/tex] (Eq. 5)
Where:
[tex]c[/tex] - Unit price, measured in US dollars per kilowatt-hour.
[tex]C_{rev}[/tex] - Revenue generated by the wind turbine in a day, measured in US dollars.
If we know that [tex]c = 0.09\,\frac{USD}{kWh}[/tex] and [tex]E_{turbine} = 19297.222\,kWh[/tex], then the revenue is:
[tex]C_{rev} = \left(0.09\,\frac{USD}{kWh} \right)\cdot (19297.222\,kWh)[/tex]
[tex]C_{rev} = 1736.75\,USD[/tex]
The wind turbine makes a daily revenue of 1736.75 US dollars.
