“Be kind, for everyone you meet is fighting a hard battle” - Often attributed to Plato but likely from Ian McLaren (pseudonym of Reverend John Watson)

## Saturday, November 24, 2007

### Hills

The only way to my house is to select one of two hills to climb. As I make my choice and climb, I watch the average mileage for that tank full (the LR3 clears the average mileage at fill up when the trip odometer is reset) decrease. And quite a few web sites that discuss gas mileage state that, when possible, use the least hilly route available. This got me to wondering what the effects of hills actually are so, as usual, I decided to do some calculating.

To start, I found the elevation at the bottom and top of the hill I usually climb by getting the latitude and longitude from Google Earth and then plugging the coordinates into the height/elevation tool of EarthTool: Webservices. I determined that I climb 123 meters. Doing this in a vehicle whose mass is, on average, 2,673 kilograms means that I add 3,222,000 joules of potential energy to the vehicle in climbing the hill. This energy comes from burning gasoline, but since I'm only able to use about 25% of the heat of the combustion of fuel, I need four times this amount, or about 12,890,000 joules of heat energy from gasoline. This is the amount in about 0.1 gallons. This is in addition to the fuel I burn just to drive the 2108 meters of road (as measured by Google Earth) to climb the hill.

Since I typically drive this hill at about the speed limit of 35 m.p.h., on level ground I'd get something like 21 m.p.g. and use about 0.062 gallons. Thus, I use much more fuel to climb the hill than I do to drive the distance. Adding the two numbers, I use 0.062 + 0.1 gallons to drive 1.31 miles for a gas mileage number of about 8.1 m.p.g. This squares nicely with what the readout on the panel display says.

BUT... When I go down the hill, I turn my engine off and coast to the bottom of the hill. The distance down is the same as the distance up, so if I drove it on level ground, I'd use the same 0.062 gallons. Instead, I use none. So driving up the hill and coasting down uses 0.162 gallons, driving the same distance on level ground would use about 0.124 gallons. Thus, the necessity of climbing the hill requires the expenditure of 0.038 gallons of fuel. Since I typically do this about five times per fill up, I put something like 0.19 extra gallons of fuel in the tank because I live at the top of a hill.

So how does this affect my fuel economy? Well, let's say I fill up at 340 miles and put in 17 gallons. This is fairly representative, and equates to exactly 20 m.p.g. The extra 0.17 gallons would reduce my mileage to about 19.8 m.p.g. I'm kind of surprised by this result, as I intuitively expect that I'll convert the potential energy going down the hill. If I didn't have to brake going down the hill and if I could hit the bottom, turn the corner and coast down to cruising speed I'd be able to recover a lot more of it, but these actions aren't possible. This means that I end up using the stored potential energy gained by burning fossil fuel to heat the metal in my brakes.

The hill seems pretty steep. Running the trigonometry, it turns out to average 3.34 degrees. It turns out that going up the hill and down adds eight meters to the distance I would travel if there were no elevation change from the location of the beginning of the hill to my house. While I will always turn off my engine and coast the last eight meters into a parking space where possible, it isn't saving much. The extra eight meters driven five times for a fill up use about a thousandth of a gallon. Anyway, traveling up and down hills clearly does not help fuel economy. I guess I should move downhill. Better yet, move next door to my job.