Fan Efficiency

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How can we measure the efficiency of an electric fan?

Big Ideas: 
  • Efficiency is useful output over total input.
  • Power is the rate at which energy is changed from one form into another.

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:

$ \textnormal{Efficiency}=\dfrac{\textnormal{useful output}}{\textnormal{total input}} $

In our case, our useful output is wind power, and our input is electrical power. So we can express the efficiency of the fan as:

 

$ \textnormal{Efficiency}=\dfrac{\textnormal{wind power}}{\textnormal{electrical power}} $

 

Measuring Power Input and Output

We can measure the electrical power that our fan draws using a Kill-A-Watt meter plugged 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:

$ \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} $

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. 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 $.

$ \textnormal{mass flow rate}=\dfrac{\Delta m}{\Delta t}=\rho \times \dfrac{Av \Delta t}{\Delta t}=\rho A v $

Putting it all together, we get an equation for wind power P at this plane, in terms of area and wind speed:

$ \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 $

To check our formula, we can look at its units:

$ \textnormal{Units of P}=\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} $

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 to measure air speed at a distance of 20cm 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:

       

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 below:

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:

$ \textnormal{Efficiency}=\dfrac{\textnormal{Wind power}}{\textnormal{Electrical power}}= \dfrac{6.2 \textnormal{W}}{41.6 \textnormal{W}} = 15\% $

This method of summing up each square is straightfoward 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 guessed 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:

$ v(R)=Ae^{\frac{-R^2}{2(\sigma_1)^2}} - Be^{\frac{-R^2}{2(\sigma_2)^2}} $

where $ e $ is Euler's number (approximately 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

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.

$ \textnormal{Efficiency}=\dfrac{\textnormal{Wind power}}{\textnormal{Electrical power}}= \dfrac{5.5 \textnormal{W}}{41.6 \textnormal{W}} = (13 \pm 4)\% $

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.

          

This technique of using an anemometer and Kill-A-Watt meter can be used to determine the efficiency of any electric fan. Try the measurement yourself with different styles of fans like the ones pictured above! For a look at the relation between wind and power from a different perspective, check out our articles on wind turbines, linked below:

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