How much energy does a vehicle require?
To get an idea of the size of the challenge, a $1000 in batteries for an electric car contain the equivalent energy of 2/3 of a gallon of gasoline. Going 63 miles each trip, with a recharging cycle in between, means the car is sleek and lightweight enough to approach a 95 mpg performance level. This is doable; consider the "Volkswagen One Liter". But while doable, it may be difficult to achieve this when starting with a big, heavy American vehicle. As Brant notes, "converting a big, heavy, gas-guzzling American car gives you a big, heavy, electricity-guzzling car".
To reach the end goal, we need to find, clarify, and organize the required physical expressions (physics equations) which will predict the performance of any car design, defering as long as possible the choice of engine type. Thus the following sections concentrate on quantifying the forces acting on any vehicle. Almost all of the "forces" mentioned here are, to be accurate, "retarding forces". It should be clear from context which is which.
The emphasis on forces is related to two key physics relationships:
Force x Distance = Energy (more) and Force = Mass x Acceleration (more)A late step in the following process will be to convert forces to whpm which sets up nicely for our all-important range calculation. Forces can be measured in lots of units but simple "pounds" will work nicely. To find out what makes the various forces, we'll pull needed material from Bob Brant's excellent book, wikipedia.org, EV-related websites, some university class notes, and a government study or two.
To get any equation to put out correct numbers, all the supplies input values must be in the proper units. Sometimes the units are specified and sometimes the reader is supposed to assume that one of the international systems of units can be used. As long as all the terms are put into the units of such a system, the result will also be from that system of units. See glossary.
This note, however, will follow Brant's lead and ignore the internationally accepted sets of unit. Instead we'll use familiar units of feet, mph, and pounds of force. This will cause some of the equations to differ slightly from the internationally recognized form.
The effort to "clarify and organize equations" is also a necessary step in writing a program which uses these equations to calculate values. A program, named 'ev.c', written in C, is being constructed in parallel. As its name implies, it allows itself an openly pro-EV bias. It's results are written in HTML, a format compatible with web browsers. Each table from ev.c focuses on one car, characterized by its weight, tire choice, desired range and speed, etc. Each line of the table looks at a particular battery type trying to fit that car's speed and distance needs. The program calculates the forces, voltages, currents, and range for that combination of components.
To defer the choice of engine type in this note, the early discussion will focus only on rear wheel horsepower, "RW HP", temporarily disregarding what had to produce that power.
By assuming that the retarding forces exact balance the driving force, the imagined car will neither speed up nor slow down. The balanced forces means there's zero force remaining to change the speed.
To test my understanding, the results calculated by 'ev.c' will be compared to web-based performance calculators, most noteably that at EvConvert.com. I can already see that there are disagreements between the calculators. Reconciling the differences will be educational!
We all know that very strong winds can knock down trees. In physics terms, the wind 'applies a force to the tree' to knock it over. And a strong gust can also move a person around. This again is a force. Likewise, a car battles the force of a 'wind' as it moves down the freeway. The amount of force is not the same for all cars; the force is larger if the car is larger, the speed is higher, or the car's shape is not streamlined.
People who have quantified these notions learned to separate size from shape. Thus there is a number, 'Cd', representing the drag due to the sleekness of a shape and there is a related number, 'A', representing the size of the studied object. 'Cd' is called the coefficient of drag and 'A' has several names:
Keep in mind that a 20% reduction in width has the same effect on CdA as a 20% reduction in Cd. Likewise, reducing a car's height 20% is equally effective.
In 2003, Car and Driver Magazine complained that 'Cd' had become a marketing term and was only partially understood by the public. Their point, I believe, is that it doesn't do much good to have a low Cd if the area 'A' just goes up. They suggested that people just focus on CdA and ignore the individual terms. "Publish CdA, not Cd" etc. However, you'll find that even now in 2008, we still have a mix of published tables of Cd's (but not A's !) and sometimes you can find a published table of CdA's. See link.
The resistive force of aerodynamic drag can be calculated with the following expression (ev.c uses the following):
It is immensely important to note that the drag force increases with the square of the speed ('V'); that is, going twice as fast produces four times the aerodynamic force. Going 20% faster (1.2 times) produces 1.2 x 1.2 or 1.44 times the force (a 44% increase). This effect is what causes the U.S. to lower freeway speed limits to 55mph years ago.
Since the oil embargo of the early 70s, tires have made many reductions in Fr and now publish a value called 'RRC', Rolling Resistance Coefficient, with each model tire.
Thus a 3000 lb car, riding on tires with an RRC of 0.015, will have a Rolling Resistance of 45 lbs pulling in the opposite direction. The same vehicle with 0.010 tires would have 30 lbs retarding its progress.
According to my first sources, rolling resistance (force), 'Fr', is basically constant with speed. Brant, however, suggests that it increases linearly with speed and has doubled when your speed reaches 100 mph (that'll be fun to test...). And even that's not the whole story because the resistance decreases as the tire softens from heating.
Amongst all this detail, realize that at very low speeds Fr is almost all the drag on the vehicle (because aerodynamic drag is nearly zero). Around 50-60 mph, however, the aerodynamic drag is often equal to the Fr. You can slow down to reduce aero drag but Fr is always with you.
RRC decreases by 25% over the 'wear life' of the tire; thus, buying used tires can actually increase your efficiency !
Tire manufacturers improve the RRC nearly every year. Though the very lowest RRC tires have a value around 0.0062, for ev.c programming purposes a more available, realistic value might be 0.009(-). Goodyear Invicta GLR, Invicta GL (.0087), and Bridgestone/Firestone B381 (.0062) were near the top of the list I obtained. Also 'replacement tires' aren't generally as good as 'OE' (Original Equipment) tires. The difference can be 25% so ask for the RRC. I'm not sure how you obtain OE tires...
Many published values assume common rim sizes but RRC decreases further as the diameter increases and as the tire gets narrower. While all the RRC literature encourages larger wheel diameters, there's an aerodynamic cost to taller wheels. I believe it noteworthy that the MIT solar car racer uses special forms of small Moped tires, pumped to a rock-hard 120 psi. Perhaps the larger tire advice is best suited to low speed NEV vehicles.
The cited literature strongly suggests that it's not possible to predict RRC from tire measurements but Brant offers one, in case you're curious.
The RRC pertains to perfect conditions which include tire pressure. A reduction in tire pressure from 29psi to 24psi will increase Fr by 10%. For tires in the 36psi range a drop of 1psi leads to a 1.4% increase in Fr. Yes, this provides an excuse to buy that compressor!
Another surprise was that a soft surface, such as sand, can increase the rolling resistance by about four times! If given a choice, take the harder surface.
Older EV literature talks of 'finding LRR tires' ("Low Rolling Resistance") but nowadays, in 2008, nearly all tires try to address this issue; just ask for the RRC. "Just asking for the RRC" also lets the tire industry know that people are becoming knowledgeable about this connection to fuel economy.
Brant gave the following efficiency values in percentages. Implicitly (ICE) engine HP is the 100% starting point.
Added altogether, the above string of components would have an efficiency of 90% meaning 10% of your engine's HP went to making heat in the above parts. Note that certain cars, like front wheel drive cars, don't have a driveshaft therefore you have to taylor the list to the specific car. Also, it's said that changing the internal lubricants to lightweight variants can "save 1%".
This becomes a relatively minor issue for EVs, especially for conversion projects. A related EV issue is whether to use direct drive or to use the existing drivetrain. Direct drive is awkward because the design is stuck with a single gear ratio. There's also the issue of whether to have 1 or 2 drive wheels. If two, how will the turn-a-corner function of the differential be accomplished? Most people just reuse the drivetrain.
Brant's calculation of Fd friction value shows a dependence on speed and rpm while other books treat it as a constant.
A problem with citing a value like Fd as a percentage is that, when the engine changes, presumably the Fd would change. Obviously that's not necessarily the case so ev.c has to work in absolute values, not ones which changes with engine choice.
A road wet from rain undoubtedly offers more resistance than a dry road. Low temperatures prevent a battery from acting efficiently. Some batteries specify a low temperature limit at which the battery should no longer be used. Some EVs have battery heaters. High temperatures can also be a problem. The Ford Ranger EV uses a radiator to keep its batteries from becoming too warm.
To be useful, the forces found in the last sections have to be converted into energy effects. As mentioned early, the initial energy unit will be whpm, Watt-Hours Per Mile. For an EV, once whpm is known, the range falls out quickly with:
range in miles = (usable_pack_energy in watt-hours) / whpm
Note that whpm is energy per unit distance while our familiar 'mpg' is distance per energy (gallon). So whpm and mpg are different, related ways of describing the same thing. So we can still use whpm even if we later decide to use a gas-powered car (Europeans already rate their gas engines in watts, not horsepower). See the glossary for details of whpm-mpg conversions. To convert the studied forces to whpm energy values, we can use an old equation:
and it shows that by multiplying our forces by distance, energy can be calculated. In fact, since watt-hours/mile is the target, the distance can be 1 mile.
(Fa + Fr + Fd + Fslope ...) x 1.98 = whpm (derivation)
Lastly, if calculations show your proposed EV just barely making the requested range, you're in trouble!
You can, for instance, measure the forces which slow your car.