“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, April 12, 2014

### Drag and weight as parameters of fuel economy in passenger cars

 Image credit: www.modified.com
I've published previously that, for my car of that time that, below about 50 m.p.h., rolling resistance is the greater contributor to my need to burn fuel and above, it's aerodynamic drag. That vehicle was a Land Rover LR3 HSE, a much larger, heavier, draggier vehicle than my current chariot (a Lexus CT200h). Going through the same calculations, I find the crossover point to be about 38 m.p.h. That is (at steady speeds), below 38 m.p.h, rolling resistance provides the greater force to be overcome by burning fuel (or running electrons from high potential to low), above 38 m.p.h., it's aerodynamic drag. Below is a graphic taking these fractions from 0 to 40 m/s (about 89 m.p.h., far above my maximum). It should be noted that, in all of this, I only consider the external forces being overcome.

At my highway speed of 55 m.p.h., about 68% of my fuel is burned to overcome aerodynamic drag. And, since something like 70% of the miles I drive are on the freeway at my typical freeway speed, it's clear that drag represents a large portion of my fuel expenditures.

So let's take a look at how fuel economy in miles per gallon varies with the coefficient of aerodynamic drag (Cd). Below is a graphic showing an estimate of fuel economy as a function of Cd at 60 m.p.h. for a Toyota Camry-like vehicle (note that axes are not zero scaled). While the curve in this range looks to be close to linear, over larger ranges it's not, since fuel economy is inversely proportional to Cd and thus the graph is that of a hyperbola.

So what can be accomplished by reducing Cd from, say, 0.32 to 0.29? At 60 m.p.h. (and using my very simple model), this would result (for the Camry-like vehicle) in an increase from about 47.5 m.p.g. to 50.7 m.p.g. In a typical 12,000 mile year with 50% of the miles driven at highway speed, this would save some 8 gallons of fuel that might cost \$32. Meh.

I attended a conference sponsored by the American Physical Society entitled "Physics of Sustainable Energy" (this was the third triennial such conference, I attended the second as well) at UC Berkeley. Amory Lovins of the Rocky Mountain Institute was the banquet speaker and made a presentation during the course sequence as well. Mr. (though Lovins has several honorary doctorates, I'm not sure that the "Dr." honorific is appropriate) Lovins has a huge portfolio of concepts that he claims, if implemented, would result in massive reductions in energy use in buildings (industrial, commercial, institutional, residential), transportation, and manufacturing. As time allows, I'll look into some of these.

But with respect to the topic of this post, Mr. Lovins stated that "two thirds of the energy used in a personal car is mass dependent." It seemed high when I heard it, let's consider. Energy is used in a car to accelerate (very much mass dependent but, in a hybrid, some of the kinetic energy imparted by accelerating a car's mass can be recovered by regenerative braking during deceleration), climb hills (very much mass dependent but descending hills can recover some, and in the case of hybrids with regenerative braking, much of the energy used in climbing), overcoming rolling resistance (mass dependent), overcoming aerodynamic drag (not mass dependent), and overcoming drive line friction and inertia moments of the rotating masses (both indirectly mass dependent in that lighter cars will need smaller, less powerful engines and, hence, lighter drive line components).

So, a lot of the energy is mass dependent. Is two-thirds a reasonable estimation? This is a complex question and will vary by car and by driver, but surely we can approach it. I'll assume that the car is not a hybrid. In addition to the usual coefficient of rolling resistance (Crr) assumptions, a number of others are required, among them: fraction of city vs. highway miles (I assumed 0.4 and 0.6); stops and starts per mile for both city and highway (I assumed 4 highway accelerations per 25 miles and 4 per mile in the city), engine efficiency (I assumed 22%). I ignored hill climbing (this would sway the fraction we're seeking higher). Should my readership clamor for it, I can elaborate on the process I used to calculate. In the end though, my estimate of the fraction of energy used in mass dependent aspects of fuel economy in this personal vehicle is 37%.

It's actually more complex than this since, at very low speeds, a large proportion of the energy used is devoted to keeping the engine going. In the extreme, stopped at a light, all of it is (though in my hybrid, the engine shuts off at a stop and I used to turn off the engine in my LR3 to eliminate this). And the amount of energy devoted to keeping the engine turning is dependent on the size of the engine and, thus, on the mass of the car. This argues for increasing the estimate of the mass dependent portion of energy used. But for the car I'm considering, it's hard for me to imagine that that portion exceeds half.

It's clear though that changes in the assumptions will have a large effect on this calculation. For example, reducing the highway portion would increase mass dependent energy; decreasing the average stop/accelerate cycles per mile in city driving would decrease it. In any event, it's clear that mass reduction in a vehicle will significantly enhance fuel economy. This post is already pretty long, so I'll elaborate in a future post.