Dot Physics

Which wastes more fuel? (and thus produces more carbon dioxide). This is a difficult to question to answer for a variety of reasons. The main reason is that a speed change from 71 mph to 70 mph is different than a reduction from 56 to 55 mph.

First, let me be clear that the question of how much fuel is wasted using daytime running lights (or DRL as they are called) has already been addressed. The first source I found was howstuffworks.com

Assumptions

  • The daytime running lights on a car run at about 100 watts (for the pair)
  • The energy density of gasoline is 1.21 x 108 Joules/gallon.
  • A car is 20% efficient at converting this energy to mechanical energy.
  • The alternator is 70% efficient at converting mechanical energy into electrical.
  • At highway speeds, air resistance is the dominating factor in fuel efficiency (this might be wrong)
  • The air resistance can be modeled as Fair = (1/2)rhoCAv2
  • I will assume an “average” car that has combined CdA of 9 ft2 or 0.84 m2 (where Cd is the coefficient of drag and A is the cross sectional area. Also rho is the density of air, about 1.2 kg/m2)
  • An average trip of 50 miles (I completely made this up).
  • My mythical “average” car gets 25 mpg when going 70 mph

Now to Calculate

First, for this 50 mile trip going at 70 mph, how much gas is “spent” on daytime running lights? Well, a 50 mile trip at 70 mph would take:

i-b609560d49708ed3880935b818bd8fd0-2009-12-23_gascalc_1.jpg

During this time, the lights are running at 100 watts, this is the same as:

i-d5f164091ec7a7fee9dc48de8f9be060-2009-12-23_drl_calc_2.jpg

How much gas does this require? Using the information from the above, I can write an expression for how much energy a certain number of gallons of gas would produce using the alternator. Note in this expression, the 0.2 and 0.7 are the efficiencies of the engine and alternator.

i-4c4c0f09219dfdc314d1d9f922b924c7-2009-12-23_drl_3.jpg

Solving this for “gallons of gas”

i-87911d5f513a03a200fdb01e4c129007-2009-12-23_drl_4.jpg

Note: this wouldn’t change much if driving 71 mph, the time would be about the same.

Now for the slowing down part.

How much energy would be saved by driving 70 mph instead of 71 mph? What is the energy lost due to air resistance? To calculate this, I can calculate the work done by the air resistance (which would be equivalent to the energy lost). Remember that work is

i-7a11ec0ff94a419ca182b28279f74aaa-2009-12-23_drlwork.jpg

In the case of air resistance, the force of air and the direction of motion are in the opposite directions. This means that the work done by friction is:

i-7ad0febb8c9a7e1e38887f95cef24cc2-2009-12-23_drlworkfriction.jpg

where these are the magnitudes of the vectors (d is the distance traveled).
Since I am not really concerned with the work done by air resistance, but rather the CHANGE in work done in going 71 mph vs. 70 mph, I can just calculate:

i-1e64de1151fbd12bd1ea869f8abd22ba-2009-12-23_drlcalc_5.jpg

Now to convert this into gallons of gas. Note: I only need to include the 20% efficiency of the engine and not the 70% efficiency of the alternator.

i-30acbe3feaf67bc2240dad9fc22a5029-2009-12-23_drleff.jpg

The Answer

So, there you have it. Slowing down from 71 mph to 70 mph saves more gas than turning off your daytime running lights. There are people that say daytime running lights are a waste of gas, but they don’t seem to be that big of a problem. If you are really concerned, you could slow down. That will have a much bigger impact on fuel usage. (Just imagine how much you would save if you slowed down to 55 mph). Plus, daytime running lights are likely safer (that is what they claim, and it makes sense).

Comments

  1. #1 Antoni Jaume
    December 24, 2009

    Would it not be easier to look at the increment of power needed to go at the different speeds?
    P=F * v, if F = (1/2)rhoCAv**2 then one has only to look at (1/2)rhoCAv2**3-(1/2)rhoCAv1**3, v1, v2 being the two speeds in consideration. If it’s bigger than 100W/0.7, then reducing speed would save fuel.

  2. #2 TGAP Dad
    December 24, 2009

    I am old enough to have just begun driver’s education when the federal government lowered the national speed limit to 55MPH in response to the oil embargo. As a side benefit, it also saved a significant number of lives as well. What a difference 30 years makes. The national response went from “slow down and save” to “drill baby, drill.”

    FWIW, Rhett, I love your blog, which is now among my daily must-reads. Though I am more of a lurker than a participant, I thought you might like to know that. Cheers and happy holidays.

  3. #3 ex-stoner
    December 24, 2009

    re: daytime running lights, hippies had it right: light up and live.

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