Colleen and I spent a fair amount of time trying to understand how to talk about energy issues. 1,000 miles per gallon on a bicycle gets attention as does telling someone for every four gallons of gasoline they use only one moves the car and the other three are mostly used to heat the air. The notion that energy can't be created or destroyed, but can change form leads is the basis of how work is done. Chemical energy in gasoline is converted to mechanical energy and heat when you drive a car. The sums all work out, but some forms of energy - like the heat from an engine - are difficult for us to use, so we tend to think of them as waste.

We also learned that people tend to think in terms of power - the rate at which energy is used. You buy some amount of energy, ten gallons of gasoline for example, but how you drive determines how quickly the tank is emptied. You eat a meal but it lasts for less time if you're running a marathon rather than watching tv. We associate power with activity and we can control activity.

To understand what it takes to make a car more efficient and where new forms of power may first become practical, it is useful to build a simple model of the power use. A bit of physics 101 is used to build this model, but I'll try to talk about the results in the main body of the post and reference two sheets that describe what goes into the model for those that want deeper detail.

1

2

The energy that goes into your car is used to do roughly five kinds of work. An easy way to build the model is to use consider power rather than energy and to consider each of the terms separately as most are independent. Calculating total energy use is simple - at the end, when the model is complete, just integrate over time.

power is used to:

speed up and slow down

overcome air resistance

deal with the inefficiency of the power train (mostly waste heat)

overcome rolling resistance

run the car's heater, air conditioner and electronics

You accelerate your car to speed increasing the its kinetic energy. To slow down the brakes are used to turn the kinetic energy into heat (equation 1). When it is moving the car must overcome air resistance. You can think of this as causing the air to move and swirl. The exact calculation of this is difficult, but you can get a good enough estimate by looking at the kinetic energy the air receives and knowing how aerodynamic the car is. (equation 2 - an earlier post had a more detailed explanation)

The simplest model of a car's power use is the sum of these two terms (equation 3). There are more terms to consider, but this much usually dominates and there are some interesting observations that can be made. The power used goes as the cube of velocity - driving fast demands a lot of power. Stop and go low speed driving is dominated by the first term and high speed highway driving by the second. The first term has distance associated with it - the average distance between braking events. Using this it is possible calculate a special distance below which the stop and go term is more important and above which the highway driving term dominates (equation 4).

The most important characteristics for stop and go driving are made clear in equation 1 - low mass and low velocity. At highway speeds the dominant term is velocity. Note that mass isn't a concern. Once you get something like a two ton car up to speed you are no longer accelerating and are only moving swirling the air away.^{1}

Why not plug in a few numbers to get a sense of what's going on at this point? The special distance that is the boundary between stop and go and highway driving is about 1,150 meters for an average family car.^{2} That goes up with mass, so a heavy SUV will be longer. Being able to shut off the engine when the car stopped is useful and regenerative braking, which recovers some of the energy that would normally go into waste heat during braking, can be very useful.

At higher speeds without having to constantly speed up and slow down you have two main options. Make the car slip through the air more efficiently by reducing its frontal area and/or its drag coefficient, and lowering your speed. Making the car slipperier is difficult if you want a car that people will buy. Frontal areas don't change much within a category of car and drag coefficients are usually in the 0.25 to 0.35 range. Moving much lower than 0.20 dictates a shape many people find too strange.

To get some realistic numbers for power we need to consider the efficiency of the power train. The average efficiency of an internal combustion engine power train is about 25%. An electric vehicle can be over 80% efficient. Using equation 3 and factoring in the power train efficiency (the first equation on the second page - I forgot to number it) we find a family car with average aerodynamics moving at 70 mph (about 110 km/h) uses about 80 kilowatts of power. An electric vehicle with similar aerodynamics would use about 25 kw. For a sanity check note that a gallon of gasoline has about 33 kilowatt-hours of energy. If you run the car at speed for an hour you use 80 kWh of energy and travel 70 miles. This is about 2.5 gallons of gasoline in 70 miles or 28 mpg.. not too bad for such a simple model.

If you dropped your speed by half to 35 mph your power use would be reduced by a factor of eight and the total time it would take you to make the trip would double. This turns out to not be quite the real world case as internal combustion engines have very narrow ranges where they operate at peak efficiency. It is necessary to have a transmission with multiple gears that cover a wide range of speeds. Some cars have eight or nine speed transmissions and some have moved to continuously variable transmissions.^{4} Most cars are designed to be most efficient at constant speeds in the 45 to 60 mph range. Exceeding that by much requires a lot of power to break through the air and the engine is usually out of its most efficient operating range.

If you've ever tried to ride a bike with too little air pressure in the tires, you've thought about rolling resistance. Equation 5 shows resistance increases with the car's mass and velocity. For an average family car with is about 4,500 watts at 70 mph so the total power needed to cruise at this speed is a bit less than 85 kW.^{5} Keeping your tires inflated to their maximum safe pressure is important for your fuel economy and the car's safety. Under-inflation can double the rolling resistance dropping your fuel economy by about five percent and reducing your safety.

Finally power is needed to heat and cool the car and run the car's electronics. An IC car gets heat "free" as a bit of waste heat from the engine is diverted into the passenger area. Air conditioning is a load that can add a few kilowatts to the requirement depending on temperature and passenger space.^{6} Electronic loads like entertainment systems, computers, gps, and lights are another load that vary a lot from vehicle to vehicle - one or two kW is common.

I'm already over time and should probably detail where exactly electrics will begin to dominate in yet another post, but some of the basics hopefully are now a bit more clear. Electrics are fundamentally more efficient, but batteries are expensive and heavy. To do the calculus you need to look at use cases and ponder technology curves. Also there is a hard limit increasing the effiency of an IC engine. Doubling the efficiency is extremely unlikely, so improvments have to come from radical and departure in design and use. At this point there are cost and social issues that make big improvements difficult.

but ... the first two terms of the model dominate. You can optimize a vehicle for the highway or for stop and go use, but if you want both there is compromise. Acceptable design for the job we want a car to do (safety, moving stuff, style,..) require further compromise. If you just move yourself over short distances you can't beat a bicycle using yourself as a power source.

To be continued...

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^{1} This is a overly simplistic model at this point! The rolling resistance and other terms have a velocity dependent component, but we're staying simple at this point.

^{2} mass = 1,500 kg, C_{d} = 0.33, frontal area = 3 m^{2}

^{3} Here C_{d} is assumed to be 0.33 and frontal area 3 m^{2} as before, Efficiency is 0.25.

^{4} Electric motors have much different power curves than internal combustion engines, some getting by without a multi-speed transmission and others just using two or three speeds. This is much less expensive and lower the weight of the power train dramatically.

^{5} Cr = 0.01, mass = 1,500 kg and velocity is 31 m/s (70 mph). Sorry for the ugly switching of units .. most people who read this are more comfortable with miles per hour and such.

^{6} Opening windows at high speeds increases a car's drag coefficient. The break-even varies from car to car, but above 45 mph it is usually more efficient have the windows up and use the AC. Of course it is even more efficient to leave the windows up and not use the AC, but ...

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Recipe corner ..

Not much really other than go out and get some fresh produce and try grilling it with a bit of olive oil. There are grilling baskets that can be useful for the small stuff. Grilled peppers are a revelation that can be used in many dishes. Even carrots are amazing.

The Maillard reaction rules.

Excellent post. I recently wondered what Toyota is up to with the next generation of Prius and it seems that they are targeting the engine and drive train efficiency issues. They recently had a press conference where they made the claim that their new small engines will reach 37-38% engine efficiencies (source: https://www.youtube.com/watch?v=OCtyDzv_UMo ). They also recently announced work at making electric modulation more efficient by going with SiC (http://newsroom.toyota.co.jp/en/detail/2656842). They seem like big leaps forward though still within sustaining innovation. What really surprised me was this: http://www.greencarreports.com/news/1091794_toyota-describes-combustion-engine-that-generates-electricity-directly

If they can get a linear engine to work they can efficiently convert the chemical energy in gasoline into electricity for use within a short period of time (acceleration), limiting the need for large costly batteries. It should also be a much simpler engine than what we are used to and I guess it more or less scales with car size. If you need more power add 2 or 4 cylinders. This suggests much lower initial and lifetime costs. Of course we still need gasoline but if demands are much lower it might make sense for society to switch to more expensive "renewable fuels" from algae or some such. No need to put in new infrastructure either. Of course they might never get it to work or the issues in battery chemistry might get solved first.

Posted by: David | 06/03/2014 at 11:49 AM