The Math of Fuel Mileage

I guess when you have people feeding you all the numbers you need through your earpiece, you think they’re easy to come by.  That’s the only explanation I can figure out for the snarky comments by television commentators about crews not being “smart enough” to figure out how much gas to put in the car so that it doesn’t run out before the end of the race.  There have been a lot of fuel mileage races the last few weeks.  Pocono is traditionally also highly likely to be a fuel mileage race, so let’s clarify how easy (or hard) it is to not run out of fuel.

Average mileage under green is about 4 miles per gallon.  At a one-mile track, than means one lap (one mile) requires one quarter of a gallon, which is one quart.  A car running out of gas coming out of turn four is short by probably a cup of fuel.   On the one hand, it’s amazing that it takes a whole quart of gas to do one lap.  On the other hand, the fuel cell holds 18 or 19 gallons.  Let’s say they get 18.5 gallons in the fuel cell – that’s 74 quarts, so you’re talking being off by 1/74th of a tank, which is a pretty narrow margin of error.

For comparison, a passenger car getting 32 mpg would need only a half a cup of gas to do a lap at Phoenix.  Although much more fuel efficient, the television ratings would likely be much lower.

There are some other considerations.  Here are two that are hard to quantify:

• The pickups on the fuel cell can’t pull all the gas out of the tank, no matter how much swerving the driver does.  There’s likely to be some fuel in the fuel cell that just doesn’t make it to the engine.  It is a small fraction of the fuel cell, but  if we’re talking about 8 ounces of fuel being the difference between making it and not, small amounts matter a lot.
• The driver’s ability to save fuel varies, depending on the driver and if he’s racing hard or if he’s able to set his own pace.  If he’s racing hard with another driver, he’ll likely get less than the expected fuel mileage.  If he’s skilled (getting off the throttle earlier going into the corner and getting onto the throttle later coming out of the corner), he might save a lap or two or three worth of gas.  It’s the same principle as you and I not stomping on the gas or the brake to be more fuel efficient.  When the crew chief asks the driver how much gas he’s saved, the only thing the driver can do is guess.  The more experienced the driver, the better feel he is likely to have for how much gas he saved.

One of the biggest challenges for the crew chief is calculating the actual gas mileage.  Let’s say you or I are calculating the fuel mileage of our car.  We go to the gas station and fill up the car.   The next time we stop for gas, we figure out how many gallons it takes to fill the tank back up and how far we drove.  For example:

I fill up my tank.  300 miles later, I stop for gas again and find that I need 10 gallons to fill up the tank.  It took me 10 gallons to drive 300 miles, which means my gas mileage is 30 miles per gallon.

OK, that’s not perfectly accurate because what does “fill up” mean?  Some people top off the tank and others stop as soon as they sense it is close to full.  There’s some variation in the fuel pumps as to where the pump shuts off automatically.  300 miles on the expressway is different than 300 miles in town.  If you want a meaningful number that characterizes your own gas mileage, you need to measure it consistently over a period of time and use an average.  Of course, that’s not possible in NASCAR.

But at least you and I get a decent measurement of how much gas we put in the car.  NASCAR teams don’t get to measure how many gallons of fuel goes into the car: They get to measure how many pounds of fuel went into the car.

A NASCAR fuel can holds about 12 gallons of fuel.  Gas weighs about 6 lbs per gallon, so the full gas can holds 76 lbs of gas.  The can itself is about 20-25 lbs, so round numbers, 95-100 lbs total. (Thanks to the NASCAR Insiders for the numbers.  I am writing this from a neuroscience retreat and don’t have my notes handy.)

Before each pit stop, the team weighs each one of the gas cans.  Let’s say one of them weighs 96 lbs.  The car comes in to pit, they add fuel and then weigh each gas can again.  Let’s say that the can weighs 36 lbs after a stop.  The change in weight is 96 lbs – 36 lbs = 60 lbs.  At 6 lbs per gallon, you can infer that the can is missing 10 gallons.

Note that I very carefully said ‘the gas can is missing 10 gallons’ because we have no assurance that all 10 gallons went into the car.  You’ve seen gasoline spill out everywhere when the gasman pulls the dry break away from the fuel cell inlet.  That happens even more with the new dry breaks because they are a little trickier to put in place and pull out than the old gas cans were.

The crew chief looks down and makes a mental estimate of how much fuel is spilled, converts the masses from the cans into gallons and comes up with a number for how much fuel he thinks is in the car.  From that, he estimates how many laps they can run.  If you want to see a frustrated crew chief, look for the gas man with the raised eyebrows and the shrugging shoulders.  He thinks he got it full… but he’s not sure.  That’s actually sometimes worse than the one who knows he didn’t get it full.  Sometimes it’s better to know the answer, even if it’s bad, than to be unsure.  The scales in the pits have at least one decimal place, and my friend Josh (a member of the ex-Elliott crew chief club) suggests that the better teams have almost certainly moved to scales with two decimal places.

Do the decimal places really matter?  Turns out they do.  Sunoco provides NASCAR teams with the exact density of the gas on race day, and they provide it to two decimal places.  So instead of 6.00 lbs/gallon, they’ll tell you 5.94 or 6.06 lbs/gallon.  If you weigh 60 lbs of gas, that’s 1o gallons @6.00 lbs/gal vs. 10.6 gallons @6.06 lbs/gal.  Remember that on a one mile track, one lap requires 0.25 gallons.  That 0.6 gallons difference is more than two laps on a one-mile track.

One more thing that’s different this year.  Here’s your word to impress people with this week:  Hygroscopic (hi-grow-skop-ick).  It means very attractive to water.  Ethanol – and 15% of the NASCAR fuel is ethanol – is highly hygroscopic.  If you turn your back on ethanol for even a moment, you turn back and there it is sucking up water.  We use ethanol in the lab to clean things and we actually have to use acetone afterward to get rid of the water the ethanol leaves.

Two issues with hygroscopicity:  First, you’re getting water in the fuel and water isn’t combustible.  You put the same volume of liquid in the cylinder and you get less power because some of the molecules turn into steam instead of combusting.  So you need more rotations to get the same power and thus you’re using fuel at a different rate.

Second, water has a different density than the hydrocarbon fuel molecules (or the ethanol), so the amount of gas you’re getting in the car is different that what you think.   Density changes with temperature, so if you think about a race like Kansas, where it was really hot, or like Charlotte, when the temperature varied quite a bit from start to finish, you might experience meaningful changes in the density over the course of a race.  Even if you did all the calculations successfully, you might still be surprised because one of the inputs was off. Also, when the temperature rises, more water can be absorbed by the ethanol.  The water molecules hang out in the gas, pretending they belong there.  But when it cools down, the water can separate from the fuel, so it’s possible to have liquid in the tank, but not have a lot of fuel.  This is a tremendous unknown that the teams have no experience with and it may account for why there have been so many fuel mileage surprises.

A lot of factors go into correctly calculating fuel mileage.  I think if you really want to get it right, you’d want to use a model that involves calculus.  And I bet there are at least a couple teams doing that.  You can make little widgets for things like fuel consumption or gear ratios and rpm using something as simple as Excel.  I know NASCAR likes to portray itself as simple, but let’s give the folks sitting with all the computers up on the pit box their due.

A few misc notes:

• Happy to hear that Chad Johnston is getting a shot at crew chief for the 56 team.  Chad was the engineer for Elliott Sadler’s team when I was following them around for the Physics of NASCAR book.  Chad is a talented guy who reminds me a little of Rodney Childers – not self-promoting, doesn’t talk when he doesn’t have anything to say, but when he has something to say, make sure you listen.
• I wish the story about what happened to the Second Chance Motorsports Nationwide crew at Chicago got just a small fraction of the attention Richard Childress/Kyle Busch did.  It’s sad, but there are so many people trying to get into NASCAR that there will always be some people who will work for someone who doesn’t have a history of treating people right.
• BTW – I’m tired of hearing about RC/KyBu… you can stop now.
• Here are a couple neuroscience tidbits I learned this week.  Perhaps the most useful thing was that if you get eight hours of sleep, but it’s not continuous (think new moms), your reflexes and ability to think are comparable to someone seriously sleep deprived.  The least useful (but perhaps most interesting) piece of information was that rodents lack the ability to vomit.  If you want to test whether a drug induces nausea, you use ferrets because they barf pretty readily.  Moral of the story:  If you’re going out drinking, take the rat as your bar buddy and let the ferret be the designated driver.  (The second moral is that if you went into physics because you have a queasy stomach, watching that talk right before lunch was maybe not the best thing to do.)
• Where have I been?  Well, the last year or so I’ve been dealing with some really, really serious medical issues and it’s been all I can do to get through the day.  Blogging was one of the many things in my life that just seemed to require too much energy to manage.  I’m starting to feel better now – sometimes I would go so far as to say “inspired” – so I’m hoping my comeback will keep.  Thanks to the many online buddies who have kept me in their thoughts and brightened my days.  You don’t know how much you have been appreciated.