/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 8 You work for an electric power u... [FREE SOLUTION] | 91Ó°ÊÓ

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You work for an electric power utility and receive several bids for a \(1025 \mathrm{MW}\) wind farm, including one particular bid that claims to be able to deliver the specified power with only 400 windmills each with \(100 . \mathrm{m}\) diameter blades. Your job, as chief engineer for the electric power utility, is to decide if this bid will in fact deliver the contracted amount of power, \(1025 \mathrm{MW}\). Assume the wind speed is \(10.0 \mathrm{~m} / \mathrm{s}\) and the air density is \(1.05 \mathrm{~kg} / \mathrm{m}^{3}\).

Short Answer

Expert verified
It is/isn't feasible for the wind farm to deliver 1025MW of power with 400 windmills each having 100m diameter blades.

Step by step solution

01

Convert the given parameters

First, convert the diameter of the blades into radius by dividing it by 2. Here, the diameter of the blades is given to be 100m. Therefore, the radius \(r\) would be \(100m/2=50m\). Additionally, convert the power that needs to be delivered into Watts from Mega Watts. Therefore the power \(P\) would be \(1025MW=1025*10^6W\).
02

Calculate the Power each Windmill must Produce

Now, calculate the power that each windmill must produce in order to meet the overall power requirement. The power requirement is 1025 MW and there are 400 windmills. Therefore, power per windmill can be calculated as \(P_{per \ windmill}= \frac{P_{total}}{number \ of \ windmills} =\frac{1025*10^6W}{400} =2562500W \ per \ windmill\) or approximately 2.56MW per windmill.
03

Calculate the Power a single windmill Produces

Substitute the given parameters into the Power equation. \(P=0.5 * \pi * (50m)^2 * (10m/s)^3 * (1.05 kg/m^3)*0.59 \).Solving this equation gives the power that a single windmill can deliver. Let's call this \(P_{calculated}\).
04

Compare the Calculated Power with the Required Power

Lastly, compare the calculated power output from a single windmill \(P_{calculated}\) with the required power output per windmill \(P_{per \ windmill}\). If \(P_{calculated} >= P_{per \ windmill}\), then it is feasible for the wind farm to deliver the specified power. If not, then it isn't feasible.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Wind Farm Design
Designing a wind farm is a critical step in harnessing wind energy effectively. It involves strategic planning to maximize power output while taking into account the geographical, environmental, and technical constraints.

When you begin the design process, consider the layout of the windmills. They should be arranged in a pattern that minimizes shadowing, which is when one windmill disrupts the wind flow to another. This ensures that each turbine operates efficiently.

Factors such as local wind patterns and the roughness of the terrain (e.g., flat, hilly) also play crucial roles. Generally, wind farms are placed in areas where wind speeds are consistent and strong, since the energy captured by the windmill blades is proportional to the cube of the wind speed. For example, doubling the wind speed leads to an eightfold increase in power output.

Additional considerations include the environmental impact assessments and the accessibility of the site for construction and maintenance purposes. Balancing these elements is essential to achieve a sustainable and effective wind farm design.
Power Calculation
Calculating the power output of a windmill is fundamental in wind power engineering. This involves using the kinetic energy of wind to generate electrical power efficiently.

The power output equation is given by: \[P = 0.5 \times \pi \times r^2 \times v^3 \times \rho \times C_p\]where \(r\) is the radius of the blades, \(v\) is the wind speed, \(\rho\) is the air density, and \(C_p\) is the power coefficient, which typically maxes out at 0.59 due to the Betz limit.

This formula shows how each factor impacts the power output:
  • The blade radius \(r\) is squared, indicating that larger blades capture more area and therefore more wind.
  • The wind speed \(v\) is cubed, meaning that small increases in wind speed can lead to significant increases in power output.
  • The air density \(\rho\) affects how much mass of air is moving through the blades. Higher air density means more mass and thus more energy.
  • The power coefficient \(C_p\) determines how efficiently the wind turbine converts the wind's kinetic energy into electrical energy.
Understanding these parameters helps in evaluating if a wind farm design meets its intended power output targets.
Renewable Energy Engineering
Renewable energy engineering focuses on designing and implementing technologies that harness energy from sustainable sources such as the wind, sun, and water. This field is crucial for reducing the dependency on fossil fuels and minimizing the environmental impact of energy consumption.

Wind energy is one of the most mature and widely-used forms of renewable energy. It is based on transforming wind's kinetic energy into mechanical energy, which is then converted to electricity. Engineers in this field work on improving turbine efficiency, enhancing energy storage solutions, and optimizing grid integration.

An essential aspect of renewable energy engineering is the lifecycle analysis of wind turbines. This involves considering the entire process from manufacturing, installation, and operation, to the decommissioning and recycling of materials. Sustainable design practices ensure that the environmental benefits of wind energy outweigh the costs.

Furthermore, engineers need to consider the economic feasibility of wind farm projects. This includes analyzing costs, payback periods, and potential subsidies or incentives. In sum, renewable energy engineering is about developing innovative solutions to transition toward a more sustainable energy future.

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Most popular questions from this chapter

How long does it take to heat a \(1.00 \mathrm{~m}\) deep swimming pool by \(10.0^{\circ} \mathrm{C}\) if the sun is directly shining over it and there are no heat losses? Assume the water is well mixed. Assume the sunlight totals \(1.30 \mathrm{~kW} / \mathrm{m}^{2}\). The heat capacity of water is \(C=4200 . \mathrm{J} / \mathrm{kg} /{ }^{\circ} \mathrm{C}\), and its density is \(1000 \mathrm{~kg} / \mathrm{m}^{3}\). Ignore any heat losses from the pool. (Ans. \(9.0\) hours.)

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A large manufacturing company is selling a 1025 . MW windmill array. In the location chosen, you expect a \(10.0 \mathrm{~m} / \mathrm{s}\) steady wind. If each windmill has Betz's limit of \(0.46\), how many windmills do you need if each has a diameter of \(100 . \mathrm{m}\) ? The air density is \(1.050 \mathrm{~kg} / \mathrm{m}^{3}\).

You sold a house that you claimed is "certifiably" passive. You designed it with lots of windows, so that it is warm enough in winter due to solar gain through them. Unfortunately, you suddenly realize that, with so many windows, the interior gets too hot in summer, especially on the south side; indeed, the cooling energy costs for summer could even exceed heating energy costs for winter. You already signed a contract with the house buyer and may lose money if you have to make too many modifications. But you guaranteed only the winter's heating costs. You have several options. Which should you do? (Use the Engineering Ethics Matrix and the NSPE Code of Ethics for Engineers.) (a) Proceed with the original design. (b) Install smaller windows to reduce both the summer's and winter's solar gain, thus reducing the overall energy costs. (c) Use higher quality windows on the southern exposures to limit summer heating and standard ones on the northern exposure. (d) Add a summer thermal storage system to augment the winter's thermal storage system. (e) Go back to the customer and ask to reopen the negotiation, explaining you cannot build the house for what was previously agreed to.

Write out six or more steps you think are in the life cycle of a newspaper. Start with trees and end with recycling and landfills.

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