Question

Efficiency of wind turbines A wind turbine converts wind energy into electrical power. Let \(v_{1}\) equal the upstream velocity of the wind before it encounters the wind turbine, and let \(v_{2}\) equal the downstream velocity of the wind after it passes through the area swept out by the turbine blades. a. Assuming that \(v_{1}>0,\) give a physical explanation to show that \(0 \leq \frac{v_{2}}{v_{1}} \leq 1\) b. The amount of power extracted from the wind depends on the ratio \(r=\frac{v_{2}}{v_{1}},\) the ratio of the downstream velocity to upstream velocity. Let \(R(r)\) equal the fraction of power that is extracted from the total available power in the wind stream, for a given value of \(r .\) In about \(1920,\) the German physicist Albert Betz showed that \(R(r)=\frac{1}{2}(1+r)\left(1-r^{2}\right),\) where \(0 \leq r \leq 1\) (a derivation of \(R\) is outlined in Exercise 70 ). Calculate \(R(1)\) and explain how you could have arrived at this value without using the formula for \(R\). Give a physical explanation of why it is unlikely or impossible for it to be the case that \(r=1\) c. Calculate \(R(0)\) and give a physical explanation of why it is unlikely or impossible for it to be the case that \(r=0\) d. The maximum value of \(R\) is called the Betz limit. It represents the theoretical maximum amount of power that can be extracted from the wind. Find this value and explain its physical meaning. (IMAGE CAN'T COPY)

Short answer

In conclusion, the analysis of the efficiency of wind turbines in converting wind energy into electrical power involves exploring the ratio of downstream to upstream velocity and calculating the fraction of power extracted from the wind according to Betz's formula. It was found that the maximum value of the power extraction coefficient (known as the Betz limit) is approximately 0.593, which represents the theoretical upper limit for the efficiency of a wind turbine. In practice, the actual efficiency is lower due to various factors such as friction and non-uniform wind flow.

Step-by-step solution

a. Bounds of the downstream to upstream velocity ratio

Given that the wind starts with an upstream velocity (\(v_1\)), the wind turbine slows it down to a downstream velocity (\(v_2\)) after extracting some energy. Hence, \(0 \leq v_2 \leq v_1\). Now, we can write the downstream to upstream velocity ratio as \(0 \leq \frac{v_2}{v_1}\leq 1\).

b. Calculate R(1) and provide a physical explanation

To calculate \(R(1)\), plug in r=1 into the given formula: \(R(r)=\frac{1}{2}(1+r)(1-r^2)\). \(R(1) = \frac{1}{2}(1+1)(1-1^2) = \frac{1}{2}(2)(0) = 0\) Alternatively, if \(r=1\), it means that the downstream velocity is equal to the upstream velocity (\(v_2=v_1\)). In this scenario, the wind turbine is not extracting any energy from the wind because the wind speed remains unchanged. So, without using the formula, we could deduce that \(R(1)=0\). Physically, it is unlikely or impossible for \(r=1\) because a wind turbine always extracts some energy from the wind, leading to a decrease in the wind's downstream velocity.

c. Calculate R(0) and provide a physical explanation

To calculate \(R(0)\), plug in r=0 into the given formula: \(R(r)=\frac{1}{2}(1+r)(1-r^2)\). \(R(0) = \frac{1}{2}(1+0)(1-0^2) = \frac{1}{2}(1)(1) = \frac{1}{2}\) Physically, if \(r=0\), it means that the downstream velocity is zero (\(v_2=0\)). In this scenario, the wind turbine has extracted all the available energy from the wind, bringing the wind to a halt. However, it is unlikely or impossible for \(r=0\) because completely stopping the wind would require an enormous amount of energy extraction, and it would also significantly impede the flow of air, decreasing the efficiency of the wind turbine.

d. Find the maximum value of R and explain its physical meaning

To find the maximum value of \(R\), let's take the derivative of \(R\) with respect to \(r\) and set it equal to zero. Let \(R(r)=\frac{1}{2}(1+r)(1-r^2)\). Then, we have: \(\frac{dR}{dr} = \frac{1}{2}[(1-r^2) + (1+r)(-2r)]\) Setting \(\frac{dR}{dr} = 0\), we have: \((1-r^2) - 2r(1+r) = 0\) Simplifying this equation and using the quadratic formula, we find that \(r= \dfrac{1}{3}\). Now plug in the value of \(r\) back into the equation for \(R(r)\): \(R\left(\frac{1}{3}\right) = \frac{1}{2}\left(1+\frac{1}{3}\right)\left(1-\left(\frac{1}{3}\right)^2\right) \approx 0.593\) So the maximum value of \(R\), known as the Betz limit, is approximately 0.593. This limit represents the theoretical maximum amount of power that can be extracted from the wind by a wind turbine. In reality, wind turbines cannot attain this limit due to various factors such as loss of energy through friction and non-uniform wind flow.

Key concepts

Wind Turbine Efficiency

Wind turbines are marvels of modern engineering designed to convert wind energy into electrical power efficiently. The efficiency of a wind turbine is a measure of how well it can perform this conversion. To understand this efficiency, we must look at the critical relationship between the speed of the wind before and after it interacts with the turbine, known as the upstream velocity (\(v_{1}\)) and the downstream velocity (\(v_{2}\)), respectively.

The downstream to upstream velocity ratio is expressed as \( r = \frac{v_{2}}{v_{1}} \). This ratio is crucial as it directly impacts the efficiency of energy extraction. It's physically intuitive that the downstream velocity cannot be greater than the upstream velocity, hence the ratio \( r \) will range between 0 and 1, signifying the proportion of wind speed retained after energy extraction.

Efficiency, in its purest form, can be seen as the optimal use of incoming wind to generate power without wastage. However, due to practical constraints such as aerodynamic losses, mechanical limitations, and the need to maintain a consistent air flow beyond the turbine, no turbine can convert 100% of the wind's kinetic energy into electrical power. This leads to a fascinating blend of physics and engineering to maximise the output within the given constraints, ensuring the turbines are both effective and durable.

Downstream to Upstream Velocity Ratio

The downstream to upstream velocity ratio provides insight into how the wind's interaction with the turbine blades affects its speed. With \( r = \frac{v_{2}}{v_{1}} \) defined, two extremes can be considered. When \( r = 1 \) the downstream velocity is equal to the upstream velocity, indicating no energy extraction, rendering a turbine essentially useless; hence, we expect this scenario to be improbable and, in practice, impossible. On the other hand, an \( r = 0 \) scenario would imply the wind is completely halted after passing the turbine, suggesting maximum energy extraction.

However, total conversion would defy the principles of energy conservation and mechanical efficiency. In practice, a significant slowdown would occur, causing back pressure, turbulence, and possibly structural damage or lowered operational integrity. Thus, turbines are designed to avoid either extreme, operating within a range where \( r \) is neither 0 nor 1, maintaining an equilibrium that offers effective energy conversion while allowing for the continuous flow of wind.

Maximum Power Extraction from Wind

Deriving the maximum theoretical power that can be extracted from wind is an essential exercise in optimizing wind turbine performance. This task entails examining factors like wind speed, air density, and the practical abilities of a turbine to harness the wind's kinetic energy. Here is where the Betz limit comes into play, named after German physicist Albert Betz, who proposed that no turbine can convert more than approximately 59.3% of the kinetic energy of wind into mechanical energy.

This value, known as the Betz limit, is found by examining the function \( R(r) \) for different ratios of downstream to upstream velocity, identifying the optimal point where the extraction of power is at its peak. The efficiency function \( R(r) = \frac{1}{2}(1+r)(1-r^{2}) \) peaks when \( r = \frac{1}{3} \) leading to this maximum efficiency. While turbines cannot achieve this limit due to real-world inefficiencies, such as friction and suboptimal wind conditions, the Betz limit represents a benchmark for wind turbine design. Engineers aim to reach as close to this theoretical threshold as possible, continuously striving for innovations that could push the limits of wind energy conversion.