Making efficiency modifications?  Calculate the coefficients of Rolling Resistance and Aerodynamic Drag for your vehicle.

It takes effort to move.  Then once you’re moving, it takes effort to stay in motion.  There are several forces that act on a moving vehicle to slow it down, and the constant fight against them costs you fuel.  If you are considering making modifications for fuel efficiency, this article will help you quantity the forces against you to see if your modifications were successful.

Rolling Resistance

This is the collective term given to several forces that slow down a wheel rolling on a surface.  Tyres deform, and to a lesser extent so does the road.  Tyres will stick to the road, which is great for grip but does slow you down.  The bearings and gears in the drivetrain of the vehicle also add their own resistance.
The rolling resistance will vary with the mass of the vehicle and the force of gravity.  You can’t do much about gravity, but you can keep weight to a minimum.
The most common way of reducing rolling resistance is to purchase low rolling resistance tyres.  If you are considering an electric conversion though, you might decide to be a bit more drastic and remove the gearbox!

Aerodynamic Drag

Again, this is a collective term given to several forces, all of which are related to the air your vehicle is pushing through.  There is simple resistance between the air molecules and the surface of the vehicle, but drag quickly becomes complex when you factor in air compression and turbulence.
Drag depends on the frontal area of your vehicle (the bigger it is, the more drag), as well as the density of the air.  You probably don’t have much control over either of these factors, but you can change the shape and texture of your vehicle.  To reduce drag you might like to try closing windows, filling in the grille, removing or adding airfoils or mudflaps.  Underbody covers can also be added to reduce turbulence.

The two forces have an important distinction that helps us to calculate them from experimental observation.  Rolling Resistance is mostly not dependent on speed.  It can be taken as a constant.  Aerodynamic drag, however, does vary with speed.  The faster a vehicle moves, the more drag is experienced. 

The Experiment

The test that we are going to perform comes from Rob Steves, via this Instructables article: Measure the Drag Coefficient of your Car.  The steps are:

  1. Find a straight, flat piece of road that doesn’t have too much traffic.  I’d suggest that it should be at least 1 km long.  You might decide that a particular time is best, perhaps a Sunday morning, so that you don’t get in the way of other drivers.  Strong winds are also best avoided, although we do attempt to compensate for these.
  2. Nominate a speed to start your test (I did 80 km/h), and another speed slightly above (85 km/h).  Drive your vehicle at the higher speed towards a certain point on the road.  Shift to neutral when you pass the point.
  3. Let the vehicle coast down in speed.  Start a timer when you reach your test speed (80 km/h) and record the speed from the speedometer every 10 seconds after that.  Keep recording for up to 70 seconds, and ideally you should be close to stopping after that time.
  4. Repeat the test in the opposite direction.  This should help cancel out any slight hills or wind effects.
  5. Repeat the whole thing twice more.  You should have three tests in each direction, for a total of six coast-down tests.
  • If you find that you are still going quite fast at 70 seconds, you might want to choose a slower starting speed.
  • If you have run out of road before the 70 seconds, again you might want a slower starting speed.
  • Try to choose your starting point in both directions so that the test covers the same section of road.  Your end point in one direction should match up with your start point in the other direction.  This helps to average out results, in case the road surface is different or there are slight hills or dips.  You might like to take a marker with you to mark the place to start your test.
  • Using a video camera to record the speedo is a good way to avoid distractions while performing the test.  It can be difficult to set up though, so you might prefer to have a passenger to take down the readings.  The passenger should have a timer and a clipboard, to write down the speed at the 10 second intervals.

The Spreadsheet

Rob Steves created a spreadsheet to calculate the forces from the experimental data.  The spreadsheet uses the ‘Solver’ function in Excel.  He notes that the Solver function for the free alternative to Excel, Libre Office Calc, does not handle non-linear systems such as this one, and so you must have Excel to use the spreadsheet.  This is fine for people who can afford Excel, but for some of us this is a step too far!  Since Rob wrote his article, Google Sheets has emerged as an alternative free spreadsheet program, and an appropriate Solver function is available.  I’ve taken Rob’s original spreadsheet and modified it for use in Google Sheets.
You can find Rob’s original Excel spreadsheet here.
My Google Sheets spreadsheet is here.  You’ll need a Google account to copy the spreadsheet (File -> Make a copy) for your own use.  The Solver function is an ‘Add-on’.  To get it, choose Add-ons -> Get add-ons.  Search for ‘Solver’:

Enter the data from your coast-down tests at the top of the spreadsheet.  It doesn’t matter if you don’t have data for the full 70 seconds, and it doesn’t matter if you didn’t do the full six replicates.  But since the speed is averaged across the table, it is important that you even it up.  Don’t include more replicates from one direction than the other direction or your average would be skewed in one direction.

Stop Right Here!

You already have the data you need to see if a modification has affected your total drag (rolling resistance + aerodynamic drag).  Perform the test before and after your modification, entering the data into two copies of the spreadsheet.  Now look at the average vehicle speed after 70 seconds (or choose another time if 70 doesn’t work for you).  If the speed after your modification is greater than before your modification then you have decreased the drag!
You only need to fill in the remainder of the spreadsheet if you really want to break out the rolling resistance and aerodynamic drag into two separate coefficients.

The Constants

To calculate the rolling resistance and aerodynamic drag we need a few vital statistics about your vehicle and the test conditions.

Air density (rho) changes with air pressure, and air pressure varies from day to day.  To find this for your location:

  1. Find the local air pressure for the site where you performed the test.  In Australia, visit the Bureau of Meteorology website and look for ‘observations’ for a weather station in your area.  The nearest station might not be that close to your test site, but air pressure typically doesn’t change much over a reasonably wide area (10s of km).  Find a reading reasonably close to the time of the day that you performed the test.  It will be something like 1014 hPa.
  2. Air pressure decreases with increasing height, and the value from the weather station will be reported as a sea level value.  This is so that air pressures from different station altitudes can be compared easily.  To calculate the air pressure at the site of your test, go to an air pressure calculator website such as this one: Air Pressure at Altitude Calculator.  Enter in the air pressure from the weather station, the approximate temperature and altitude.  You’ll receive a new air pressure which will be lower than the original (assuming your test was above sea level).
  3. To calculate the air density from the air pressure, go to a website calculator such as this one: Omni Calculator.  Enter your calculated air pressure, temperature and humidity.  You can guess the temperature and humidity, or if the weather station you chose in the step above is close enough, you could use that data.
  4. Enter the air density value into the spreadsheet.  If you do the test again on a different day then you need to go through this process again.

The gravitational constant (g) will not vary at your site, and is likely to be the same within a few hundred km of your site.  So you can leave this at the default value and don’t need to change it from one test to the next.  If you’re a pedant (and there is a good chance that you are if you’ve made it this far through the article!), then look up your local value for g from a website such as this one: SensorsONE.  It will ask for your latitude and longitude, which you can find from Google Maps if you right click click on an area.

The frontal area (A) of your vehicle can be found by measuring the width and height in metres, then multiplying them together.  You may find a more accurate value in your vehicle specifications.

The mass (M) of your vehicle needs to include all passengers who were in the vehicle for the test.  For standard vehicles you can look up the mass online, then add the mass of the passengers.  For non-standard vehicles, such as converted vehicles, you’ll need to weigh your vehicle at a weighbridge.  This is often possible at your local rubbish dump, and there may be a fee.

Calculating the results

The drag coefficient and coefficient of rolling resistance are shown down the bottom of the spreadsheet, but are not calculated automatically.  To perform the calculation use the solver function.  Instructions here are for Google Sheets, but Excel is similar:

  1. Click on cell F39
  2. Choose Add-ons -> Solver -> Start (if the Solver option is not there, choose Add-ons -> Get add-ons and search for ‘Solver’).
  3. Choose options as shown in the screenshot below.  You are asking for F39 to be minimised by adjusting F41 & F42.

4. Click ‘Solve’ and wait for it to do its thing.  You should see the values in F39, F41 & F42 change, and the Solver pops up a little window saying that it is successful.

Caveats

The spreadsheet pulls out the retarding forces from your experimental data, and divides them into forces that are constant, and forces that increase with speed.  It labels the constant forces ‘rolling resistance’ (Crr), and the varying forces ‘drag coefficient’ (Cd).
You may have gathered from the heading above that this is not quite accurate.  To get accurate figures you’d need to test the values separately, for example using a wind tunnel for aerodynamic drag or a drum test for rolling resistance.  We’ve assumed that rolling resistance does not vary with speed, but actually there are some rolling resistance forces that do.  In particular, a car manufacturer may design an aerodynamic shape to increase the apparent weight of the vehicle at speed to improve handling.  Therefore the values obtained will never be exactly correct, and will differ for different vehicle designs.  They are still fun to find though.
The coast-down speed is probably the most useful value for someone intending to modify their vehicle.  This is great for testing if your modification had an effect, although it is not great to compare different vehicles.

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