Wind Power

How can we harness a substantial amount of power from the wind?

Wind energy is the kinetic energy of the air flow; its ultimate source is the solar energy, which drives convection in the Earth’s atmosphere. Wind power has been harnessed for possibly two millenia[1]Dodge, Darrel. Part 1 – Early History Through 1875: Wind Power’s Beginnings (online). Illustrated History of Wind Power Development. http://www.telosnet.com/wind/early.html [2019-09-20]. , but only now is it starting to become a significant part of the total power needs of industrialized nations[2] The Guardian windpower page, https://www.theguardian.com/environment/windpower [2019-09-20]..

To calculate the rate at which kinetic energy, $KE = \frac12mv^2$, can be extracted from an air stream, we will need the rate at which air passes through the rotor of the wind turbine. The rotor is made up of the blades, and the total swept area $A$ of these blades can be approximated by a circle whose radius is the blade length $r$, i.e. $A = \pi r^2$. Now we can calculate the mass $m$ of air, density $\rho$, velocity $v$, passing through this circle in time Δt (Fig.1).

Fig.1. Wolfe Island wind farm (author photo).

From this, we can see that the mass is (Eqn.1):

$$\begin{eqnarray} m = \rho A v \Delta t \tag{1} \end{eqnarray}$$

Note that the area $A$ is the area swept by the blades, not the area of the blades. This is because the blade moves much faster than the air and so each particle of air is affected by the blade. Therefore, the kinetic energy $KE$ of this mass of air is as follows (Eqn.2):

$$\begin{eqnarray} KE &=& \dfrac{1}{2} m v^2 \tag{2} \\ &=& \dfrac{1}{2} \rho A t v^3 \nonumber \end{eqnarray}$$

Thus the power $P$ of the wind passing through this circle is (Eqn.3):

$$\begin{eqnarray} P &=& \dfrac {\dfrac{1}{2} \rho A t v^3}{t} \tag{3} \\ &=& \dfrac{1}{2} \rho A v^3 \nonumber \end{eqnarray}$$

The power that can be extracted from this air stream is less than $P$ because the air cannot stop dead on the other side the blades. Some kinetic energy has to remain in the air stream so it can continue to flow. The theoretical “Betz” limit is about 60%[3]Betz’ Law http://en.wikipedia.org/wiki/Betz’_law [2019-09-20]. . The limit is reached when the wind velocity downstream of the turbine is 1/3 of the incident velocity. To understand this result, you have to recognize that the area of the incoming air stream is less than that of the outgoing, which is a simple result of mass conservation.

In addition, there are inevitable losses due to turbulence and inefficiencies in converting energy in the blade rotation to useful electrical energy.

The $v^3$ term emphasizes the need to have a high wind speed in order to capture a useful amount of power. Doubling the wind speed increases the power eight-fold.

An experiment with wind power in reverse: measuring the efficiency of an office fan

When you’re out shopping for an electric fan, it might be a good idea to look for one that not only keeps you cool, but keeps your energy consumption low, too. Using an anemometer, Kill-A-Watt meter and a little bit of math, you can figure out how efficient your fan really is.

What is Efficiency?

In the broadest sense, efficiency can be defined as useful output over total input in Eqn.1.

$\textnormal{Efficiency}=\dfrac{\textnormal{useful output}}{\textnormal{total input}}\tag{1}$
In our case, our useful output is wind power, and our input is electrical power. So we can express the efficiency of the fan in Eqn.2.

$\textnormal{Efficiency}=\dfrac{\textnormal{wind power}}{\textnormal{electrical power}}\tag{2}$
Measuring Power Input and Output

We can measure the electrical power that our fan draws using a Kill-A-Watt meter [4] Kill-a-Watt Meter, http://en.wikipedia.org/wiki/Kill_A_Watt [2019-10-15]. in the circuit between the fan and outlet. But what about the wind power? First, let’s try to find the power generated by the fan in an area where the wind speed is constant. Wind power is the rate at which kinetic energy is added to our air stream. Given a volume of air in the stream, wind power is the kinetic energy in that volume divided by the time it takes for that volume to pass a given plane (Eqn.3):

$\textnormal{Wind Power}=\dfrac{\textnormal{kinetic energy in volume}}{\textnormal{time taken to pass plane}}=\dfrac{\Delta (\frac{1}{2}mv^2)}{\Delta t}=\frac{1}{2}v^2\times\frac{\Delta m}{\Delta t}\tag{3}$
The expression $\frac{\Delta m}{\Delta t}$ is the mass flow rate of the air, or the mass of the air that passes through the area in a given time. Picture a plane, perpendicular to the air flow, of area $A$, as in Fig.1. You can obtain the mass flow rate through the plane by measuring the volume of air that travels past the plane in the time interval $\Delta t$, and multiplying it by the density of air (1.225 kg/m3 at room temperature, sea level), $\rho$. Eqn.4 gives the mass flow rate.

Fig.1. Air flowing through a plane $A$ at a speed $v_o$ for a time $t$.

$\textnormal{mass flow rate}=\dfrac{\Delta m}{\Delta t}=\rho \times \dfrac{Av \Delta t}{\Delta t}=\rho A v\tag{4}$
Putting it all together, we get an equation for wind power $P$ at this plane, in terms of area and wind speed (Eqn.5):

$\textnormal{P}=\frac{1}{2}v^2\times\frac{m}{t}=\frac{1}{2}v^2\times\rho A v=\frac{1}{2}\rho v^3 A\tag{5}$
To check our formula, we can look at its units in Eqn.6:

$\textnormal{Units of Power}=\dfrac{\textnormal{kg}}{\textnormal{m}^3}\dfrac{\textnormal{m}^3}{\textnormal{s}^3}\textnormal{m}^2=\dfrac{\textnormal{kgm}^2}{\textnormal{s}^3}=\dfrac{\textnormal{Joules}}{\textnormal{second}}=\textnormal{Watts}\tag{6}$
So now we know how to find the power generated in an area of constant air speed, but how can we apply this to our fan, where the wind speed is variable? We can measure the wind speed at many different points, so that each area of the air flow is small enough that the speed can be considered constant. Then we can calculate the power generated by the fan in each of those areas and sum them all up to approximate the power generated by the whole fan. The smaller each constant speed area is, the more accurate our estimation will be. As you can imagine, this can get very, very tedious — luckily for us, we can take the data quickly with a digital anemometer and record it directly into a spreadsheet. All we need to do is figure out how to measure the wind speed at many regular points in front of the fan.

Setting Up The Experiment

We set the fan to blow at its highest speed setting, and used a digital anemometer[5] Digital Anemometer, https://en.wikipedia.org/wiki/Anemometer#/media/File:Digital_Handheld_Anemometer.jpg [2019-10-15]. to measure air speed at a distance of 20 cm from the front grate of the fan, which was close enough so that the airflow would not have dissipated too much, but not so close that the grate would have an effect on the air flow. In order to get as accurate readings as possible, we wanted to minimize the number of solid objects in front of the fan that might interrupt air flow, so we placed the anemometer on a stand instead of holding it up by hand. The stand could be adjusted vertically and moved horizontally, which allowed us to divide the area in front of the fan into a a 10 x 10 grid and take measurements where the grid lines intersect, shown below in Figs. 2 and 3.

Fig.2. Measurement grid for an office fan.

Fig.3. Measurement distance from fan.

Analyzing The Data

Now that we have collected all of our data, there are two ways we could go about finding efficiency. The simplest thing to do would be to just calculate the power generated by the fan in each square, and sum the squares up. A more sophisticated method would be to fit our data to a guessed function, and then calculate the power generated there — the basic principle is still the same (adding up the power on many different squares), but by using a function we can make each square much smaller, thus improving the accuracy of our efficiency estimate.

Finding Efficiency: Simple Method

In order to plot our data on a 2-D graph, we note that the wind speed distribution of the fan is approximately symmetric, and plot velocity versus R, as shown in Fig.4.

Fig.4. Velocity data for the office fan.

At this point, we can use this velocity data to find the power generated by the fan in each 5cm x 5cm area, and sum those to get an estimate of the total power generated by our fan. In doing this, we found that the fan generates approximately 6.2 W of power. Measurements with the Kill-A-Watt meter showed 41.6 W of electrical power input, allowing us to calculate efficiency (Eqn.7):

$\textnormal{Efficiency}=\dfrac{\textnormal{Wind power}}{\textnormal{Electrical power}}= \dfrac{6.2 \textnormal{W}}{41.6 \textnormal{W}} = 15\%\tag{7}$
This method of summing up each square is straightforward and gives us a good order of magnitude estimate, but it also assumes large areas (25 cm2) of constant wind speed. For those who are mathematically inclined and wish to go further, we can try to refine our estimate by fitting our data points to a trial function.

Finding Efficiency: Curve Fitting Method

Since, as we noted above, the wind speed distribution of the fan is approximately symmetric, we end up with multiple wind speeds for the same $R$ distance from the fan’s axis. We can plot the average wind speed for each $R$ and add error bars showing the standard deviation of each. Looking at our new plot (below, left), we noticed that the plot looks similar to two Gaussian functions, one positive and one negative, overlaid on top of each other. This gives us the function (Eqn.8):

$v(R)=Ae^{\frac{-R^2}{2(\sigma_1)^2}} – Be^{\frac{-R^2}{2(\sigma_2)^2}}\tag{8}$
where $e \approx 2.718$. In order to find the constants $A$, $B$, $\sigma_1$ and $\sigma_2$ that best fit our data points, we can either use MATLAB’s curve fitting toolbox, or perform a chi-square analysis using the solver in Excel. We used Excel to determine the following values: A= 226.5 m/s B= 234.4 m/s $\sigma_1$= 0.0584 m $\sigma_2$= 0.0553 m

Fig.5. Fits to the fan velocity data with a trial function. Note: this method – where there is not a priori reason expect that particular function – is only as good as the fit.

Note: Using these coefficients for our function, a region of negative wind speed (i.e. wind blowing in the opposite direction) appears around $R$ = 0. Since in this case this is not feasible, we have taken all of the negative values and set them to be a wind speed of 0 m/s. By using a continuous, smooth function instead of discrete points, we can make the areas of constant wind speed much smaller, limited only by the amount of computing power we want to devote to this problem. We use MATLAB to sum everything up, and found that the bladed fan output approximately 5.5 W of wind power (Eqn.9).

$\textnormal{Efficiency}=\dfrac{\textnormal{Wind power}}{\textnormal{Electrical power}}= \dfrac{5.5 \textnormal{W}}{41.6 \textnormal{W}} = (13 \pm 4)\%\tag{9}$
This result is consistent with the efficiency we found by simply summing the data points, however the fit allows to estimate the uncertainty based on the scatter of the points.

Fig.6. Measure the efficiency of other types of fan, like this Dyson.

Now try this method with other types of fan in your school or home.

Updated (CEW) 2019-10-15

Footnotes

↑1 Dodge, Darrel. Part 1 – Early History Through 1875: Wind Power’s Beginnings (online). Illustrated History of Wind Power Development. http://www.telosnet.com/wind/early.html [2019-09-20]. The Guardian windpower page, https://www.theguardian.com/environment/windpower [2019-09-20]. Betz’ Law http://en.wikipedia.org/wiki/Betz’_law [2019-09-20]. Kill-a-Watt Meter, http://en.wikipedia.org/wiki/Kill_A_Watt [2019-10-15]. Digital Anemometer, https://en.wikipedia.org/wiki/Anemometer#/media/File:Digital_Handheld_Anemometer.jpg [2019-10-15].