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Exam 10: Wind Energy

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Question 1

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The Royal Clipper (Figure 10.1) has a total sail area of $5200 \mathrm {~m} ^ { 2 } \text {. Show that }$ the estimate for the total power of a large sailing ship as given in the text is reasonable.

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The total ...

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Question 2

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A three-blade wind turbine with a 100 m rotor diameter operates with a tip speed ratio of 10. (a) What is the efficiency? (b) What fraction of the Betz limit is achieved? (c) What is the actual power output in kW for a wind speed of 11 m/s? (d) For the wind speed in part (c), what is the actual blade tip velocity? (e) For the wind speed in part (c), how long does it take the rotor to make one rotation?

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Question 3

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A wind turbine with a 40-m diameter rotor produces 287-kW output in a 10- m/s wind. What is its efficiency?

A wind turbine with an efficiency of $42 \% \text { produces } 1 \text {-MW output at a wind }$ velocity of 13 m/s. What is the turbine rotor diameter?

Wind with a velocity of 12 m/s is incident on a three-blade wind turbine with a rotor diameter of $90 \mathrm {~m}$ . (a) What is the power of the wind that intercepts the turbine? (b) What is the maximum power that can be extracted from the wind? (c) What is the angular velocity $\left( \omega \right.$ in $\left. \mathrm { s } ^ { - 1 } \right)$ for the rotor when the maximum power is produced?

A country would like to provide all of its residential and industrial electricity needs by wind power at a rate of $2.5 \mathrm {~kW}$ c per capita. It plans to install wind farms covering $10 \%$ of the land area consisting of $2 \mathrm { MW }$ wind turbines with a spacing of $1.0 \mathrm {~km}$ in the prevailing wind direction and $0.5 \mathrm {~km}$ in the orthogonal direction and with an overall capacity factor of $24 \%$ (a) Estimate the maximum population density that can be accommodated by this approach? (b) In which European countries would this be feasible?

A country with good wind resources decides to make a national commitment to the development of wind energy. It is decided that 10% of the land area will be devoted to wind energy with a goal of 1.5 kWe per-capita wind capacity. This is roughly the electricity used by one person in an industrialized nation (Chapter 2). Two-megawatt wind turbines with rotor diameters of 80 m (Energy Extra 10.1) are placed at the average spacing shown in Figure 10.18. Average output based on wind conditions is 650 kWe per turbine. Estimate the maximum population density that is consistent with thisplan. Locate geographical information about Germany, Spain, India, and Denmark (all of whom have active wind power programs), and discuss the possibility of achieving this goal in each of these countries.

For a 10 m diameter wind turbine with an efficiency of 45% located on land of roughness class 2, calculate the power produced by as a function of hub height from 20m to 120 m for a wind velocity of 10 m/s at a height of 50 m.

(a) Consider a wind farm consisting of 10-m diameter wind turbines with efficiencies of $40 \%$ on a grid given by the average spacing shown in Figure $10.18$ . For a wind velocity of $6 \mathrm {~m} / \mathrm { s }$ , what is the power output in $\mathrm { kW }$ per $\mathrm { km } ^ { 2 }$ array with an efficiency of $18 \%$ . Assume average (worldwide) insolation conditions.

(a) Two locations have the same average wind velocity. One location has a constant wind velocity of $10 \mathrm {~m} / \mathrm { s }$ , while the other location has a wind velocity of $5 \mathrm {~m} / \mathrm { s }$ for 12 hours per day and a wind velocity of $15 \mathrm {~m} / \mathrm { s }$ for 12 hours per day. Compare the total energy produced per day for identical wind turbines at the two locations. (b) Consider part (a) for a location with a wind velocity of $0 \mathrm {~m} / \mathrm { s }$ for 12 hours per day and $20 \mathrm {~m} / \mathrm { s }$ for 12 hours per day.

A three-blade wind turbine with a rotor diameter of 50 m, produces an output of 570 kW in a 10 m/s wind. Calculate the wind velocity after passing through the turbine.

Consider two identical wind turbines with a hub height of 40 m, one located in flat agricultural land and one located above a forest. What is the relative output power for the two turbines if the wind velocity is 10 m/s at a height of 100 m in both cases?

A Darrieus rotor with a swept area of 1000 m² produces 240 kW output in a $10 \mathrm {~m} / \mathrm { s }$ wind. (a) What is its tip speed ratio? (b) If the height of the rotor is $60 \mathrm {~m}$ , estimate the time for one rotation.. What is its rotational speed (in rotations per second)?

(a) Show that for a wind turbine spacing of 3 rotor diameters crosswind and 10 rotor diameters downwind, the power generated per unit land area for a wind farm is independent of rotor diameter and is equal to $\frac { P } { A _ { \text {land } } } = \frac { \pi \rho \eta } { 240 } v ^ { 3 }$ where ρ is the air density and η is the turbine efficiency. (b) Show that when the velocity is measured in m/s, the power per unit area is in kg/s³. (c) Discuss the relevance of hub height on the answer to part (a).

A high-speed two-blade wind turbine operates with a tip speed ratio of 14. It produces 1.0 MWe in a 12 m/s wind. What is the rotor diameter?

Wind velocities are not constant throughout the day. The daily average power produced by a wind turbine is the power averaged over the wind velocity for the day. Calculate the average power for a turbine with a diameter of $20 \mathrm {~m}$ and an efficiency of $37 \%$ if, during a 24 -hour period, the wind velocity is - $2 \mathrm {~m} / \mathrm { s }$ for 4 hours. - $8 \mathrm {~m} / \mathrm { s }$ for 16 hours. - $14 \mathrm {~m} / \mathrm { s }$ for 3 hours. - $17 \mathrm {~m} / \mathrm { s }$ for 1 hour.

Estimate the ratio of wind velocity 10 km off shore the wind velocity at the shore for (a) wind from the land (b) wind along the shore and (c) wind from the sea.

A home owner installs a wind turbine with a rotor diameter of 2 m to supplement electricity from the public utility. The cost of the turbine, the associated electronics, and energy storage system (batteries) is $10,000. If the turbine has an efficiency of 35% and the energy is utilized and or stored at an efficiency of nearly 100%, what is the payback period for the investment? Assume that maintenance costs are minimal, the capital recovery factor is unity, electricity from the public utility costs $0.10 per kWh, and the wind velocity is constant at 12 m/s.

A Darrieus rotor has an area of $1500 \mathrm {~m} ^ { 2 } \text { and operates with the optimal tip }$ speed ratio. What is its power output in a wind with a velocity of 20 m/s?

Dresden, Germany has a population of 700,000 and a population density of $2200 \mathrm {~km} ^ { - 2 }$ in the urban area. Assume that the electrical needs of Dresden are an average of $1.0 \mathrm {~kW} _ { \mathrm { e } }$ per capita and these are satisfied by a wind farm consisting of $100 \mathrm {~m}$ diameter three-rotor wind turbines operating at optimal efficiency and with optimal spacing. If the wind velocity is constant at $7 \mathrm {~m} / \mathrm { s }$ , what is the area of the wind farm relative to the area of the urban area it serves?