Modeling and Measuring the Efficiency of Your Car

If you read automotive blogs, news sites, press releases from manufacturers, or mainstream news outlets when gas prices go up, you have undoubtedly seen the word "efficiency" thrown around. This is usually brought up when a manufacturer wants to tout a new engine with high thermal efficiency—e.g. Nissan’s variable-displacement engine a few years back, or Toyota's current "Dynamic Force" hybrid engines—or when people want to argue the merits of battery electric vehicles (BEV) versus internal combustion engine vehicles (ICEV), or when gas prices shoot up.

 
But what do we mean when we talk about "efficiency"? What is thermal efficiency, and why is it important? Are there other ways to characterize efficiency in cars that may be more useful? Can you measure the efficiency of your own car? And finally, is efficiency different from economy?

This XL1 was reported to get more than 100 MPG on the highway (the car was never US-certified and has no EPA rating). Does this tell us how efficient it is? Yes and no.

Thermal Efficiency
 
Don't be confused by the terminology here. In physics and engineering, we use "thermal" to mean internal energy while "heat" is an energy transfer or process, the movement of energy from one place to another. An object or system cannot "have" or "store" heat; it has internal (or thermal) energy. Thermal efficiency here simply means the efficiency of a thermodynamic cycle. These cycles move energy from one place to another as heat and hence are known as "heat engines." Standard formulation of the First Law of Thermodynamics shows this; on the left are energy movements in or out of the system (heat and work) while on the right are the changes in state (thermal, potential, and kinetic energy):

In any heat engine model, we have to put in energy (in the form of heat) to get energy out (usually in the form of work, but sometimes as kinetic energy in an exhaust stream). In a perfect or "ideal" engine, we would get out as much energy as we put in, in a useful form like work; in a real engine, energy is made unavailable due to inefficiencies in the real process—things like incomplete combustion or friction between the piston and cylinder walls. The ratio of energy out to energy in gives us the thermal efficiency of the cycle: in the case of a car, how much shaft power we get out of an engine divided by the amount of energy available in the fuel. Since power is a rate, to ensure the units cancel here we must divide by the fuel heating value (energy per unit mass) multiplied by the fuel flow rate (mass of fuel per unit time), giving us energy per time (i.e. power) in the numerator and denominator:

(That's the lowercase Greek letter "eta"). This is the number you will see manufacturers trumpet when they improve their engines, and the best engines now have peak thermal efficiencies of 40% or more. But this is only part of the story when we talk about vehicle efficiency. Thermal efficiency tells us how much work we get at the output shaft of the engine for energy input from fuel; the other part is how much of that work makes it to the road.
 
Propulsive Efficiency
 
The ratio of work done on the car to work available at the engine output shaft is called "propulsive" efficiency. There are numerous losses between the crankshaft and tire treads: friction and drag in bearings, material strain in all drivetrain components (but especially tires, which deform significantly), brake drag, etc. Each of these loses a little of the initially available energy, which results in an increase in internal energy in the air flowing past these components, in the air trapped in the tires, in other fluids such as brake hydraulic fluid or bearing grease, in solid components such as axles and wheels, and in other measures of entropy or the dispersion/"spread" of energy throughout the system. Additionally, some of the work at the engine output shaft is deliberately siphoned away to power other essential things—like the electrical system of the car or the water pump—and nonessential things, like the AC compressor.
 
The rate of work used to actually move the car is modeled as the propulsive force, F, times vehicle speed, u (and this quantity will always be less than the work available at the engine output shaft), giving propulsive efficiency as the ratio,

Overall Efficiency
 
Now that we've defined (and distinguished between) thermal and propulsive efficiency, we need to combine them to determine the overall efficiency of a car. This is the number that matters! Thermal efficiency tells us how well the energy in the fuel source is converted to work output by the engine and propulsive efficiency tells us how much of that shaft work reaches the road. But it is the combination of the two in overall efficiency that measures how much of the energy in the fuel actually does useful work propelling the car.

Thermal efficiency is some percentage of fuel energy turned into shaft work (the rest lost to incomplete combustion, kinetic energy in the exhaust gases, heat, friction, and other dissipative effects); then, propulsive efficiency is some percentage of that shaft work turned into work on the vehicle (the rest lost to power offtake, heat, friction, and other dissipative effects—I am not repeating myself, I am not repeating myself; oh God, I'm repeating myself).
 
Overall efficiency is simply the product of thermal and propulsive efficiency; that is, multiply them together. Since both thermal and propulsive efficiency must be less than 1, this means overall efficiency is, as we would expect from intuition, smaller than either thermal or propulsive efficiency (the losses compound—they can't go in the other direction).

Write out the equations we derived above, cancel terms, and we arrive at the relationship,

where F is the total force required to overcome resistance, u is vehicle speed, ṁ_f is fuel flow rate, and Q_R is the fuel heating value. Furthermore, this relationship applies not just to cars but to any vehicle—for example, jet aircraft, trains, cargo ships, or bicycles.
 
Measuring Overall Efficiency of Your Own Car
 
That's all well and great, but is there a way to measure the efficiency of our personal vehicles? Yes, assuming you can find or estimate some information. Let's go through step by step.
 
First, you will need an estimate of the drag area of your car. If you have an unmodified car and are lucky enough that the manufacturer publishes its drag coefficient, this is straightforward. Otherwise, you'll have to estimate this from testing. Then, record ambient temperature and pressure at the time and location of testing to find air density by the Ideal Gas Law.
 
Next, you will need to estimate the rolling resistance of your car. Use the test I developed to estimate static C_RR and a truck scale or automotive race scales to find its weight.
 
Finally, you will need to know the fuel heating value of whatever energy carrier your car uses. For most of us in the US, this is E10, a blend of gasoline and up to 10% ethanol. The fuel heating value of E10 is, according to the US Department of Energy, 118,200 BTU/gal (this is the average of the range of "lower" and "higher" heating values. The difference between these is in the state of the water in the combustion products; lower heating values assume it is released entirely as steam while higher heating values assume water in the exhaust condenses, leaving the energy that otherwise would escape with the vapor free for cycle output).
 
Now, on to some road testing. Using an OBD reader and the flattest section of road you can find on a calm and temperate day, drive your car at various speeds and record the average fuel flow rate (usually reported in gallons per hour, GPH, for American passenger cars). Check the OBD reader for the car's actual speed, which will be slightly below the value displayed on the speedometer (usually by 2 mph or so). Alternatively, if your car cannot display average fuel flow rate, you can back it out from average MPG measured at various speeds over the test interval, as

You should get data that look like this (recorded on a 58°F day with light winds over a 10-mile test section, averaged in both directions and corrected for the optimistic onboard fuel consumption display):
 

Indicated Speed (mph)	Fuel Economy (MPG)	Fuel Flow Rate (GPH)
45	68.3	0.63
55	61.9	0.86
65	50.8	1.24

Now, using the equation for aerodynamic drag (which varies with the square of speed) and rolling drag (which, in its simplest model, is constant—but you can use a more refined model in which there is a speed-proportionate term if you want, as I have here), known speed and fuel heating value, measured fuel flow rate, and converting everything to like units, you can calculate your car's overall efficiency as a function of its speed using the equation above:

Indicated Speed (mph)	Vehicle Power (kW)	Fuel Power (kW)	Overall Efficiency ηo
45	6.6	21.8	30%
55	9.7	29.7	33%
65	13.9	43.0	32%

You can see that my car's efficiency varies with speed, improving slightly as speed goes up. This tells us that the engine is probably running at less than its peak thermal efficiency at 45 mph but perhaps closer to it at 55 mph. These numbers are in the expected range; the 2ZR-FXE engine in the car has a maximum thermal efficiency of 38% according to the manufacturer, so overall efficiency must be a smaller number due to losses after the engine output shaft, in the transmission and drivetrain. But you can see that we don't actually need to know the thermal efficiency of the engine here to determine overall efficiency; bumping up thermal efficiency, as manufacturers try to do with each successive generation of engine, will be reflected in an increase in overall efficiency.
 
Notice also that efficiency is a proportion of the total power required for the vehicle to move at a certain speed to the total power available in the fuel flow; as such, it is relative. It is entirely possible, for example, for a small, light car with low aerodynamic and rolling drag to have a lower efficiency than one with higher aerodynamic and rolling drag—if that second vehicle uses the available energy in its fuel (or "energy carrier") more efficiently.

High fuel economy cars. High efficiency? Maybe, maybe not.

This is an important distinction: you might have wondered at the outset of this article, "Isn't MPG a direct measure of efficiency?" Yes, to a point: when comparing two vehicles that have essentially the same resistance force, for instance the Honda CR-V and CR-V Hybrid. There, the vehicle with greater MPG is more efficient. Comparing different cars, however, fuel economy doesn't tell us how efficient the vehicle is with regard to how much force it must overcome and how well it uses the energy in its fuel to do that. An 18-wheeler, for example, is very efficient at converting fuel energy to work at the tires, but since it is moving a lot of mass it requires that much more energy input. On a per pound-mile basis, the 18-wheeler may use less energy than other forms of transport because of its high efficiency. Some quick back-of-napkin math suggests that a typical tractor-trailer can move one pound of cargo ~2 times further than my Prius can for the same fuel power input, for instance.
 
Improving Fuel Consumption
 
All that said, there are two ways to improve the fuel consumption of your car. First—and this is what most people focus on, since it is the most obvious—you can reduce the power required for it to move by reducing aerodynamic drag, rolling drag, or speed; assuming its overall efficiency remains constant (in reality, it will vary depending on a number of factors), this will reduce the fuel required commensurately. But second, you can improve the overall efficiency of your car by figuring out ways to get more of the available energy in the fuel to the road in the form of useful work. This second approach is one of the inherent advantages of BEV: they carry a lot less energy onboard than gas-powered cars but a greater proportion of that energy is turned into work.
 
How much? I asked my brother to do some testing on his Model 3, recording power draw from the battery (not power produced by the motor—an important distinction here!) at various speeds. Efficiency plotted against my Prius shows,

Despite having about the same total drag as my car at each speed, the electric car is much more efficient; a significantly larger portion of the energy carried onboard the car is turned into work.
 
An easy way to improve efficiency, regardless of fuel source: turn off as many power-consuming peripherals as possible. The increase of battery electric vehicles in the new car market has unfortunately coincided with an explosion in electric "tech," in the form of computers, screens, sensors, and unnecessarily powered devices like door handles and ambient lighting. Those all siphon energy away that could otherwise have been used to move the car, reducing η_p and consequently η_o.

Door handles were a solved problem, automakers; stop trying to reinvent the wheel. These are 35 years old, don't use electricity, and still function perfectly.

 
Economy vs. Efficiency

Before taking a propulsion class my last semester in school, I wasn't really clear on the difference between economy and efficiency, using the two terms interchangeably. However, while they are related, they really mean different things. "Fuel economy" is a measure of how far your car can travel on a specified amount of energy, while "efficiency" is a dimensionless measure of how well your car converts fuel energy into work. This is something that can be confusing or even deliberately misleading. For example, this article on the recent rise in gas prices quotes a Yale professor who says:

It may indeed be true, as I found on my car, that efficiency is highest between 50-60 mph but this does not mean that fuel economy is highest in that range. As we saw above, my car gets much better gas mileage at 45 mph than it does at 55 or 65 despite having a lower efficiency at that speed, and yours probably does too.

Efficiency can also be a useful measure for evaluation of outlandish fuel economy claims. If you can find the maximum thermal efficiency of the engine in question (which manufacturers sometimes publish for economy cars), and knowing that overall efficiency must be smaller than this number, you can determine how much the resistance force of the car would have to be reduced to achieve some claimed flat-ground constant-speed fuel economy. You will find that all those social media "I did some untested aeromods and now my car gets 90 MPG!" claims are complete fantasy that would require drag reduction to levels not seen outside of a handful of research cars. For example, in the same test conditions as above and assuming no change in overall efficiency, in order to improve fuel economy to 90 MPG at 55 mph my car would need its drag coefficient reduced to:

A drag coefficient that low on a real, road-going car is simply not possible. When you see people making these claims, something else is going on: hypermiling techniques (which can massively inflate fuel economy through a number of mechanisms and driving tricks), inaccurate data collection and record keeping, unaccounted testing errors, or straight up deception and exaggeration. Cars are not magic.

Now you also know why the 100 MPG reported for the XL1 on the highway doesn't quite tell us its efficiency. That car was designed for extremely low aerodynamic and rolling drag (so the numerator of η_o is smaller than a normal car) and ran on diesel (so the fuel heating value in the denominator is larger than a normal car). Its overall efficiency may be no better or worse than a typical car while its fuel economy is far higher. Rearrange the equation for overall efficiency to show the relationship (assuming like units throughout):

…where F is a function of speed. Reduce F for any given speed and fuel economy goes up; increase Q_R (for example, by running the car on diesel instead of gasoline) and fuel economy goes up. Or increase η_o to improve fuel economy; as we saw with the EV above, overall efficiency can be significantly higher than a car that uses a combusted energy carrier.

Don't forget that aerodynamic drag is not the only resistance force your car must overcome. Getting rid of weight—for instance, by replacing the rear seat with a lightweight platform (poplar frame and birch ply here, saving just over 40 lb)—also reduces F and increases fuel economy.

Try measuring efficiency on your own car!