The Car Equation

The physics of automotive motion is complicated, but the basics are clear from a relatively simple equation. It’s worth posting, as it illustrates some of the main challenges faced by design engineers and drivers who aim to increase fuel economy. 

So, ignoring secondary effects, here is what might be termed the fundamental equation of automotive force: 

F = mgC_rr + ½ρC_DAv² + ma + mgsin(θ)

where 

F = force required at the wheels of the vehicle
m = mass of the vehicle
C_rr = coefficient of rolling resistance between tires and road surface
ρ = density of the ambient air
C_D = coefficient of drag of the vehicle in the direction of travel
A = cross-sectional area of the vehicle
v = speed in the direction of travel
a = acceleration of the vehicle
g = local acceleration of gravity
θ = angle (relative to horizontal) of the road surface

To increase fuel economy, engineers work to increase the efficiency of the drive train that delivers the force that’s required at the wheels. This is complicated work, and they have been at it for decades. But only recently has attention focused on the capture, storage, and re-use of kinetic energy that is normally lost when vehicles slow down. Electric hybrids are just the beginning. 

On the other side of the equation, engineers work to decrease the amount of force that’s required. From that side, you can see why they try to do three things based on variables that are primarily under their control (vs. the driver’s control): 

• reduce the vehicle mass (particularly important, since there are three terms in the force equation that are proportional to the mass, and because vehicles today are made from heavy materials) 
• reduce the rolling resistance (tires) 
• improve the aerodynamics (reduce cross section, reduce coefficient of drag)

Of course, there are economic and other tradeoffs involved. Reduce mass, but don’t give up too much safety; reduce rolling resistance, but don’t give up too much wear or grip; improve aerodynamics, but don’t give up too much comfort, convenience, and carrying capacity (and don’t make it ugly!). 

While these variables are primarily under the design engineer’s control, the driver has some control as well. If you load up the car, the mass increases. If you put stuff on the roof, the aerodynamic drag increases (likewise if you open all the windows). If you choose the wrong replacement tires or don’t keep the tires properly inflated, the rolling resistance increases. 

On the other hand, the equation shows the importance of variables that are primarily under the driver’s control: 

• acceleration - how often and how strongly you accelerate 
• speed - particularly important given that the second term is proportional to the square of your speed (drive a little faster, and there’s considerably more drag) 
• road surface – where you drive affects rolling resistance 
• hills - how often you go up them and how steep they are; (the sign of the last term becomes negative when you go back down, but you never get back all you lose going up. Although there are many variables involved - speed, cornering, braking, regeneration, etc. – at best you can recover most of the potential energy gained by climbing the hill. However, the energy spent going uphill is considerably greater than that potential energy gain since engines and drivetrains are not 100% efficient. The difference is lost forever.)

As mentioned, these variables are primarily under the driver’s control; but they are also affected by vehicle design. For example, drive trains may be most efficient at a particular speed (so driving very slowly may not yield the best fuel economy). And, obviously, you can’t accelerate or drive faster than the vehicle’s capabilities (which may be electronically limited). Similarly, unless you’re an off-roader, your choices of road surface and terrain are limited strongly by those in charge of road construction and maintenance.

Finally, note that there’s an important effect that’s beyond the scope of the equation above, namely that fuel economy is reduced by diverting energy for purposes other than forward motion – in particular for comfort and convenience features, including climate control, on-board electronics, etc. Here, the engineer is responsible for how energy-efficiency the features are, while the driver is responsible for how much they are used.

Note: post revised on 12 November, 2007; see comments; js

Comments

[Chris Ellis]

This is a great summary, apart from the phrase 'but you never get back what you lose going up'. On most downgrades most drivers seldom use their brakes, because the potential energy their vehicle gained going up, five minutes before or a hundred miles away, is now being used to overcome air and tire resistance, etc. Of course, with a hybrid, any gentle braking required will result in energy being recovered for later use. Only going down a steep mountain pass will some of the potential energy gained in reaching altitude be lost, and then not all of it. But then I'm still young enough to accelerate out of downhill hairpin bends... Most vehicles spend less than one mile in a thousand going down really steep, long hills, so clearly the phrase is wrong and should be dropped. Then the 'The Car Equation' may become the definitive primer. I would't have bothered with this blog if I didn't think it could....

[Ken Fry]

This is an excellent summary, John.

At the risk of over-complicating things, I’ll reinforce that what is being calculated is a force: tractive force. The equation applies to one instant for a particular car, on a particular grade at a particular speed and acceleration.

To get from force to power, we need throw v (velocity) into the equation again. From power, we can say how much energy a car will consume per minute.

Which leads us to the other equation: given that one needs, let’s say, 10 kW to move a particular vehicle at 60 mph along a flat road, then how to best supply that? Generate electricity at 38% efficiency (although the DOE says 33%) cranking out a lot of carbon in the process… or generate gasoline at 82% efficiency? Once the energy source has reached the vehicle, do you drive on electricity at 85% efficiency (including battery charging, through the controller, and through the motor to the motor output)… or do you drive at the 38% (peak) efficiency of a Prius engine (ignoring the hybridization factors which improve the vehicular efficiency). Using the DOE figure, the Prius is more efficient than the electric car: 82% x 38% > 85% x 33%. Using the 38% figure for overall electrical generation sometimes heard, the tables turn… but in practical terms, the > or < symbols might just as well be =.

But then 49% of our power is generated with coal, which creates about 50% more carbon per million BTU than gasoline. So today, in the real world, an efficient gasoline-powered car is perhaps less egregious, environmentally, than an electric car. The DOE predicts that the percentage of electricity generated by coal will increase to 57% by 2030. The use of renewables will increase in kW terms but not as a percentage of the total. See: http://www.eia.doe.gov/oiaf/aeo/electricity.html

Is that a dismal prospect, or what??

Having written all this, however, I’d have to say that the Tesla, especially, will help to change things in a fundamental way. Electric vehicles are clearly the way to go in the long term, and great credit should go to Tesla for making the technology attractive, today. While either my own Pod One microcar or an Aptera can squeeze about twice as many miles out of a chunk of coal as a Tesla, it’s the Tesla that says, today, that driving an electric car can be a blast – and that an electric car can compete (in coolness) with a Porsche, head to head, dollar for dollar. In the not too distant future, all cars will be full electric. As we, as a country and as a world, actually begin to think, really think, about how we use energy, the move to renewables, and especially solar, will come. People will start to think about how profoundly foolish it is use up all the resources we have so little of while ignoring the one we have so much of. A single year’s solar input is far greater than all the fossil fuels we can ever use. But this graph of energy usage has to be expanded in scale twice for solar voltaic usage to even become visible: .04% of total energy usage.
See: http://en.wikipedia.org/wiki/Image:World_energy_usage_width_chart.svg

Even today, solar voltaics are competitive economically with grid electricity in many parts of the US. In the future (probably near future), that will only improve. My hope is that the Automotive X Prize will stimulate enough real thought to speed that transition to solar power. In the meantime, PHEVs are a great stepping stone.

[John Shore]

Chris Ellis and I had a side conversation after his comment on "The Car Equation", above. I believe I convinced him that "you never get back what you lose going up", but I may have oversimplified by not mentioning the many relevant variables (speed, cornering, braking, regeneration, etc.), and I realized that the basic reason for energy loss may not be obvious. Consequently, I edited and then republished the post, in particular replacing this:

• hills - how often you go up them and how steep they are (what goes up, must come down, and the sign of the last term becomes negative when you go down; but you never get back all you lose going up)

with this:

• hills - how often you go up them and how steep they are; (the sign of the last term becomes negative when you go back down, but you never get back all you lose going up. Although there are many variables involved - speed, cornering, braking, regeneration, etc. – at best you can recover most of the potential energy gained by climbing the hill. However, the energy spent going uphill is considerably greater than that potential energy gained since engines and drivetrains are not 100% efficient. The difference is lost forever.)

js

[John Prikkel PE]

I am pleased to see that weight reduction is a prominent factor to be considered. liteflex LLC has been designing and manufacturing light weight springs (leaf and coil) for automotive (Corvette 25+ years) and truck applications (Peterbuilt, Freightliner etc). Weight is reduced 60 to 75%) while performance is improved (ride, handeling, durability etc).
Concerning reducing rolling resistance, Liteflex has developed a low cost alternative to run flat tires. better rolling resistance and no need for a heavy spare tire. Fell free to contact me if interested cell 937-469-3962. John P

[Bruce McHenry]

Regarding Chris Ellis' comment above:
"Generate electricity at 38% efficiency (although the DOE says 33%) cranking out a lot of carbon in the process… or generate gasoline at 82% efficiency? Once the energy source has reached the vehicle, do you drive on electricity at 85% efficiency (including battery charging, through the controller, and through the motor to the motor output)… or do you drive at the 38% (peak) efficiency of a Prius engine (ignoring the hybridization factors which improve the vehicular efficiency). Using the DOE figure, the Prius is more efficient than the electric car: 82% x 38% > 85% x 33%. Using the 38% figure for overall electrical generation sometimes heard, the tables turn… but in practical terms, the > or 85% (PEV storage) x 33% (Electric generation + distribution):

The quoted 38% Prius efficiency quoted must be for the peak efficiency of the engine. The engine is not always operated at peak efficiency which is also substantially reduced when the output is stored and retrieved from the battery. My guess is that that round-trip loss averages more like 40% than 15%. Can you find any data on that? Overall, I'd expect average Prius' efficiency around 25%, not 38%.

When electric utilities use natural gas in new turbines, efficiency is 55% or better, presumably because combustion inside the turbine is more efficent. Higher efficiency should also be expected for atomized liquid fuels such as gasoline and oil. I assume the 33% DoE figure you quote includes transmission losses. What are they? If 15%, the fuel equivalent utility power is .55 * .85 = .47 efficient.
Revising your equation yields .82 * .25 = .21 for Prius versus .47 * .85 = .40 for electric, twice as good. Say this goes down to .33 * .85 = .28 for coal. If your 1.5X carbon from coal is correct, coal power is also carbon equivalent.

Not factored into the calculation above is the weight of the on-board power plant which results in higher power needs. Going all-electric seems to reduce weight, but how much at a given range?

Going all-electric with high speed, long range power coming from the directly from the grid would allow most of the on-board batteries to be eliminated which will probably reduce weight relative to traditional vehicles by about 40%. Such vehicles would weigh much less than the equivalent Prius, PHEV and PEV.

How can vehicles be powered while cruising down the arterials? How do you power them when off-grid? See http://roadtrains.us for some ideas in development.

[roger]

hi all
sorry, but overall prius eff. is much lower than 38%, even a lot lower than 25%. the avg. prius driver uses 5 liters premium for 100km. that's 5 x 8900Wh/l or 44,5 kWh. let's say we're doing an avg. of 60 km/h. drag, rolling res. & friction losses will sum up to about 6kW power demand at that speed. therefore, total eff. is somewhere around 14%. it's still a clever thing, though, at least in terms of marketing.

[Jim Bullis]

I need to reiterate my concern about the MPGe method since I read your plans for the competition. On a separate issue, I appreciate your inclination to allow tandem seating.

By promoting the MPGe term, harm is done to the overall intention to limit carbon dioxide and its equivalent compounds. The harm is that the public sees the MPG part and is led to think that we will make great progress against global warming by simply changing to electric power systems.

I offer evidence in the fact that, today, Tesla claims on the headline page of their website at www.teslamotors.com that their roadster gets 135 miles per gallon equivalent.

Since last summer when I first objected to this criterion I have looked at "Mechanics, Heat and Sound by Sears, 1950 (probably more engineers learned freshman physics from his very clear writings than from any other). Hopefully my argument is now more refined.

Obviously electricity does not come in gallons, so the desire to make some sort of equivalent is understandable. An incorrect formulation can result in statements that mislead by a factor of 3 or more. The "miles per gallon equivalent" measure is only valid for electricity that falls from the sky, meaning hydro-electric, solar, (lets include) wind and geothermal, where the heat engine process is not required. However, by far the most electrical energy comes by making heat from some sort of fuel and running it through a heat engine of some kind. The second law of thermodynamics now stands in the way. If we are talking about fossil fuels, which is the only way to get electricity that is not already spoken for, then the USA efficiency in 2006 for electrical energy produced from fossil fuel is 34%. Without worrying about distribution losses, this means that the Tesla roadster example of which is claimed 135 mpg(equivalent) has to be divided by about three. It might be claimed that in California, the mix of electric sources allows a divisor of about two. So taking these as a range of reasonable, the Tesla gets between 45 and 68 mpg(equivalent). So I think it is clear that this is marketing over-done. The shame is that the Tesla is quite a remarkable achievement, where sports car performance is obtained, while being more economical than a Prius.

The outcome of this is that there is an unwarranted sense in the media and hence, the public, that such cars put us on track to solve global warming, and since they are so attractive, no further effort is called for.

The good side of this exercise in outrage is that it has made me realize that, not only are we excessively wasteful due to operation of inappropriate cars, we are similarly wasteful in electric power generation. The system of central power plants where heat rejected by the heat engine is thrown away is all wrong. Rather, a distributed power generation system, where a cogeneration process is carried out at our respective households would allow full use of that otherwise wasted heat. If this were to be the way we make the electricity, then MPGe is actually a valid unit of measurement.

A key requirement for such cogeneration is that the motor-generator be sized appropriately for the household such that all heat can be used effectively.(PGE rebate rules make a looser but similar statement).

If high efficiency cars include motor-generator systems, then distributed cogeneration of power, using those motor-generators in those cars can be implemented with very little additional cost. Conduits for heat transfer to the household from the car and piping to supply natural gas from the household to the car are anticipated.

It is amazingly fortuitous that this combination could put us on the 7% Kyoto CO2 reduction track much faster than previously thought possible.

I realize this goes beyond your immediate concerns, but I am interested in solving the whole problem. I think this warrants X-prize involvement in some form.

Best regard, Jim Bullis

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