Question

Traveling down the road, you carefully watch a three-bladed wind turbine, determining that it takes two seconds to make a full revolution. Assuming it's operating near the peak of its efficiency curve \(^{41}\) according to Figure \(12.4\), how fast do you infer the wind speed to be if the blade length \(^{42}\) appears to be \(15 \mathrm{~m}\) long?

Short answer

Answer: The wind speed of the wind turbine when operating at peak efficiency is approximately 6.74 m/s.

Step-by-step solution

Find Angular Velocity and Tip-Speed Ratio

We can find the angular velocity of the blades using the full revolution time: Angular Velocity (ω) = \(2 * \pi / \) time_period where time_period = 2 seconds. Once we find the angular velocity, we can find the tip-speed ratio: TSR = \(ω * R / V\) where R represents the blade length (radius) and V represents wind speed. TSR is known at this point from Figure 12.4 for peak efficiency. So we have an equation relating the unknown wind speed.

Find the Wind Speed

Now we have an equation with one unknown variable – the wind speed (V). We will rearrange the equation and find the wind speed for the given peak efficiency point. So, V = \(ω * R / \) TSR

Calculate the Wind Speed

Now we have all the variables necessary to find the wind speed: 1. Time period = 2 seconds 2. Blade length (radius) = 15m 3. TSR from Figure 12.4 for peak efficiency = 7 (assuming that #41 is the given peak TSR) First, find Angular Velocity (ω), ω = \(2 * \pi / 2\) = \(\pi \mathrm{~rad/s}\) Next, use the TSR equation to find the wind speed: V = \((\pi * 15) / 7 \approx 6.74 \mathrm{~m/s}\) So the wind speed is approximately 6.74 m/s.

Key concepts

Understanding Angular Velocity

Imagine seeing the blades of a wind turbine spinning in a gentle breeze. To understand the speed at which the blades are rotating, we can calculate the angular velocity. It's a fancy term to describe the speed of rotation and it's measured in radians per second, which is the standard unit of angular measurement in physics. Think of it as how many 'angles' (measured in radians) the turbine covers per second.

To find the angular velocity, we need nothing more than the time it takes for one full turn of the blades. If you've clocked that single turn at two seconds, like a watch's second hand ticking two ticks, you can find the angular velocity by dividing 2π (that's a full circle in radians) by the time period. Remembering that π is roughly 3.14159, we can see that our blades are spinning at a speed of π radians per second when the time period is two seconds.

Deciphering the Tip-Speed Ratio (TSR)

Now, with angular velocity in hand, we can delve into the tip-speed ratio (TSR). The TSR is essentially how fast the very tip of the wind turbine blade is moving compared to the wind itself. It's a bit like comparing the speed of a race car on a track to the speed of the wind it's driving into. A higher TSR means the blade tip is moving significantly faster than the wind.

Here's how it works: multiply the angular velocity by the blade's length, which at 15 meters is pretty tall, taller than four average adults stacked on each other's shoulders! Then, divide by the wind speed to get TSR. By knowing TSR from a chart in our textbook, which tells us the turbine's peak efficiency, we have a critical piece of the puzzle to figure out the actual wind speed.

Maximizing Wind Turbine Efficiency

When it comes to harnessing the power of the wind, we want to be as efficient as possible, right? This is where understanding wind turbine efficiency becomes key. It's about striking that perfect balance between extracting as much energy from the wind as possible while not overworking or underutilizing our turbine.

Our turbine operates at peak efficiency at a specific TSR; too high or too low, and we're not getting the most out of the windy conditions. By knowing the TSR for peak efficiency is 7 (which, let's be honest, sounds like a lucky number), and using the formula from earlier, we can work out the wind speed that will give us the best energy output for the day. It's fascinating to think that this careful dance between blade speed and wind speed can mean the difference between a good and a great day for renewable energy production!

The Importance of Renewable Energy Education

Embarking on a journey of renewable energy education can be like unlocking the mysteries of a cleaner, more sustainable world. By understanding concepts like wind turbine efficiency, angular velocity, or TSR, we empower ourselves to make informed decisions about our energy future.

Education in this field goes beyond calculations; it's about being aware of the impact renewable energy can have on our planet. It promotes critical thinking and innovation, sparking ideas for improved technology and methods for harnessing energy from the elements around us. So, whether you're a student or a curious mind, remember that every bit of knowledge you gain contributes to a greener, brighter future for all.