Science

I Think I Need a Better Mileage Model

Previously, I estimated the fuel efficiency of a car as a function of speed. This is essentially what I had. Where m is the gas mileage, K and B are constants. In this model, I assumed the constant frictional forces were small enough to ignore. Maybe it was a dumb idea, but it let me […]

Previously, I estimated the fuel efficiency of a car as a function of speed. This is essentially what I had.

Mileagemodel 1

Where m is the gas mileage, K and B are constants. In this model, I assumed the constant frictional forces were small enough to ignore. Maybe it was a dumb idea, but it let me estimate the fuel efficiency for different speeds if I knew the efficiency at a particular speed.

So, what is the problem? Consumer Reports is the problem. I happened to pick up a new car buying guide. In this issue, there was a short segment about improving gas mileage. They found that with two cars (a Toyota Camry and a Ford Explorer) the cars got better mileage at a slower speed. Fine, but here are there values.

For the Camry:

  • 40 mpg at 55 mph.
  • 35 mpg at 65 mph.
  • 30 mpg at 75 mph.

For the Explorer:

  • 24 mpg at 55 mph.
  • 21 mpg at 65 mph.
  • 18 mpg at 75 mph.

Here is a plot of my (obviously wrong) model and the data from Consumer reports. I used the 65 mph speed and mileage to calculate the constant K/B. The blue data are for the Camry and the green is for the Explorer.

Figure 1

Can I figure out what went wrong? Or more importantly, could I make a new model?

How about I 'undo' one of my assumptions. I had determined the following model for fuel efficiency:

La te xi t 1

The assumption was that the frictional force (Ff) was small enough to ignore. Let me rewrite the expression above:

La te xi t 1 1

According to this, if I plot 1/m vs. the velocity squared, it should be linear. The slope should be B/K and the intercept should be Ff/K.

Here is that plot for the Camry data:

Figure 1 1

A linear regression (using the polyfit function) gives the slope and intercept such that:

La te xi t 1 2

Problem: now either K or Ff is negative.

Starting with a new model

Now, I will just use the data from the Camry and Explorer to see if I can find a new model. The data looks quite linear, so if I fit a function to both of the sets I get:

La te xi t 1 3

Here, the e subscript is for the Explorer and the c is for the Camry. Oh, also, I left off the units because I am lazy. This model seems to be fine. However, the problem is that I would need two data points (mileage and speed) for each car to create this model. It would be nice if I could get a model from just one data point.

I guess there is a chance to estimate the air drag coefficient and stuff in front of the velocity squared term (which I called B). Wikipedia does list lots of data for different vehicles in terms of drag coefficients. Maybe I will try that sometime.

Let me just leave you with the data I used a long time ago. This is from the DOE Transportation Energy Data book from a few years ago.

Dd 1

Oh snap. It seems there is more recent data. The 2010 edition of the Transportation Energy Data Book has data from 2008 vehicles. (Here is Chapter 4 of TEDB

- warning, that is a pdf file.) In it, there is this table:

table

Ok, I need to think about this a little more.