While I procrastinate in writing about more important things...

Physicists (one of which I am not) are quite concerned about units and dimensions and use dimensional analysis for a variety of purposes. And if the units in an equation don't match in all terms and across the equals sign, you've erred.

But sometimes this can lead to confusion. For example, torque (exerting a force about an axis) is measured in dimensions of [force]*[length]. It could be pound feet or newton meters or, for the matter of that, dyne centimeters or ton furlongs. But work and energy are also measured in such units. A joule is a newton meter.

Thus, in thinking about my fuel economy as a U.S. resident, I'm accustomed to thinking of miles/gallon but in countries who've adopted the SI (metric) system, people make a very sensible inversion of this and, rather than distance/volume, they use volume/distance, typically liters/100 kilometers. While this isn't an SI unit, it is metric.

But it's also a volume divided by a length or [length]^3/[length] which is [length]^2 or an area. So I converted my 50 m.p.g. to an area to note that my fuel economy is 4.704*10*10^(-8) m^2 or 0.04704 mm^2. Next time someone asks about my fuel economy, I'm going to say "a bit over 47 thousandths of a square millimeter."

It's not surprising to note that Randall Munroe*, of xkcd fame, has beat me to it. I will state, for the record, that I noted his post after composing all but this portion of this one.



*This is the first time I've noticed a photo of Munroe. It always amazes me how little resemblance there is between what I picture someone to look like in my mind and what they actually look like when I see them or their photo. And I always picture them.

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 (C_d). Below is a graphic showing an estimate of fuel economy as a function of C_d 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 C_d and thus the graph is that of a hyperbola.

So what can be accomplished by reducing C_d 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 (C_rr) 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.

Looking into the CT200H mileage trends

What with work and family, sometimes I don't have as much time as I'd like to devote to authoring posts. And some of them take a significant amount of time. In this case, I've made a couple of posts regarding the the possibility of sequestering CO₂ from power plants in carbonate rocks, pavers, bricks, etc. I still owe my audience an analysis of this process in terms of energetics and economics. Those are taking some time.

In the mean time, I want to take a look at some of the data from my records of mileage in the Lexus CT200h that's my daily driver. To the left is a plot of the mileage at each fill-up since my acquisition of the car. It raises some questions.

Of course, the very low and very high numbers are related to the profile of driving done during the applicable tank. The lowest, for example, involved climbing into the mountains above Los Angeles for an outing with my son.

But I noted a trend, beginning at the end of the third quarter of 2012 and extending to the end of the first quarter of 2013, of declining mileage. I took the vehicle in for scheduled service in mid-March and told the service crew about the declining fuel economy. When they returned the car, they said they'd checked all applicable parameters in the fuel delivery system, the engine control unit, etc. and found no anomalies and adjusted nothing. But the mileage increased noticeably and is still appearing to be on that upward trend.

Now, to the best of my ability, I always drive in the same way (much to the frustration of my passengers - those who will still ride with me anyway - and the vehicles that share that road with me). And, to the extent possible, I try to always purchase gasoline from the same station. Assuming that it's not related to my route or driving techniques, what could explain it?

One possibility is "winter blend" vs. "summer blend" gasoline. In summer, particularly in California, refiners must use gasoline blends with lower vapor pressure to minimize vaporization due to warmer temperatures (and more driving). In winter, refiners add butane to blends because it's cheaper (thereby partially explaining winter's lower gas prices) and the higher vapor pressure of butane containing blends isn't as harmful due to the lower temperatures and lesser total vehicle miles driven. And butane has a lower specific energy content, thus possibly explaining my reduced fuel economy.

Is there such an annual "signal" in my fuel economy data? I've got 90 data points, and so ran a Fast Fourier Transform of the data. Such a process is used to transform data from the "time domain" (as in a time series) to the frequency domain (showing periodic components in the data). If the winter/summer blend switch, which happens annually, is a significant part of the fuel economy changes I've noted, my theory is that such periodicity should appear in the frequency domain. If you squint, you can even convince yourself that it's there - lower in winter and higher in summer.


Alas, there's no such peak apparent in the Fourier Transform. It's back to the drawing board. I can't imagine that the dealer fixed  or adjusted something and didn't charge me for it!

The (probably) last post on regenerative braking

I've posted a couple of times on regenerative braking in my CT200h. This will, I expect, be the last. In the previous post I estimated that regenerative braking on a trip saved me about 5.9% of the gasoline I'd have used without it. I decided that a better test would be a full tank, so I monitored all of the regenerated watt hours for my most recent tank. Since it's kind of a pain in the rear, I'm not going to keep it up.

Calculating in a more efficient way than the very detailed way in the previous post, the results are as follows:

• The measured economy by miles divided by gallons at fill-up: 50.60
• The calculated economy without regenerative braking: 47.55
• Gallons per 100 miles: 1.976
• Gallons per 100 miles without regenerative braking: 2.103
• Per cent fuel savings: 6.04%

Not much different, so I think that it's safe to say that regenerative braking saves about 6% of the fuel I'd otherwise use.

I'm a bit surprised that the number is that low. In this post I discussed some of the factors that make hybrids so much more fuel efficient than their non-hybrid cousins and the regenerative braking was one of the factors I considered most important.

There is no non-hybrid CT with which to compare the fuel economy. I went to the DOE fuel economy site for the Camry (the four cylinder version)  and for the Camry hybrid. Using the combined highway and city estimates for each (28 m.p.g. and 41 m.p.g. respectively) it looks like the hybrid, per the government's test protocol, will use about 31.7% less fuel over any distance. It's reasonable to infer that, while the regenerative braking is a significant fuel saver, other factors (operating more frequently on more efficient areas of the engine map, capturing energy while coasting, automatic engine shut-off where appropriate, etc.) are at least as important.

More on fuel saved by regenerative braking

I published a post regarding how much energy is captured in the regenerative braking system in my Lexus CT200h hybrid. After some discussion with commenter Gabriel Grosskopf, I estimated that about 59% of the energy available (after subtracting the energy used to overcome aerodynamic drag, rolling resistance, and internal friction) was recaptured and used to charge the battery.

Since I (and others) have represented that the regenerative braking system is among the key reasons that hybrids achieve superior fuel economy, I decided to check the actual impact.

My round trip commute, generally downhill in the morning and uphill in the evening, is 62.46 miles and, for the last 10 fill ups, my average m.p.g. has been 52.47. So, to make my commute, I use, on average 62.46/52.47=1.190 gallons of gasoline. My display showed me today that my regenerative braking system added 700 watt hours or 2,520,000 joules to my battery that I could use for accelerating, hill climbing, etc. If I assume my electric motor is 90% efficient, I put 2,268,000 of these joules to work.

A gallon of gasoline (reformulated blend in this case) has an energy upon oxidation of 111,836 btu or 117,993,000 joules. I estimate that my internal combustion engine is about 25% efficient, so I put about 29,498,000 of these joules to work. My 1.19 gallons thus provide 35,103,000 joules that propel my vehicle (the remainder being lost as waste heat in myriad ways).

If I assume that I used all of the energy my brakes provided, then 35,103,000 + 2,268,000 = 37,371,000 joules of work were done to propel my car. Then, dividing by 0.25, I can estimate that 149,484,000 joules of oxidized gasoline would have been necessary to do this work. This is the energy in 1.267 gallons. Dividing this into 62.46, I find that the fuel economy without the regenerative braking would have been about 49.30 m.p.g. The regenerative braking thus upped my m.p.g. by 3.17.

As I've often said, it's much more intuitively informative to discuss gallons per mile, or gallons per 100 miles. So, the regenerative braking took me from 2.03 gallons per 100 miles to 1.91 gallons per 100 miles. So it takes me 5.9% less fuel to go a given distance, ceteris parabus.

There's no question that I'm carrying a lot more significant figures (apologies to John Denker) than are warranted by the precision of my data, but I think that the figure I've determined is probably in the ballpark.

Gadgetman Groove update

Back in February of 2011, I wrote a post on the Gadgetman Groove, a modification purported to provide spectacular gains in fuel economy and power. And when I say "spectacular," I'm talking about double the m.p.g. and more. As it happens, my Groove page turns up on the most visited statistic fairly frequently. I suspect the visitors may not read what they'd hoped to but, as the Apostle Paul wrote in his first letter to the Corinthians, the time comes when we must "put the ways of childhood behind" (New International Version 1 Corinthians 13:11).

In any case, seeing that post come up as one of my most frequently visited piqued my curiosity and I paid Ron Hatton's (the inventor of the Groove) web site another visit, where I found the following quote: "The EPA tells us more than 60% of the power in your fuel is wasted in the exhaust." Does the EPA actually say such a thing? Well, sort of...

The EPA (and anyone else with a degree of knowledge of engineering thermodynamics) will

tell you that much of the chemical potential energy released in the burning of fuel in the cylinders of an internal combustions engine exits the engine as low grade waste heat in the exhaust, through the radiator, radiantly from the engine, and elsewhere, and 60% is really a very low number for that "waste" in an otto cycle engine. Such losses in a heat engine (or any engine) are an inevitable consequence of the second law of thermodynamics.

Ron Hatton, though, implies that this waste is fuel that isn't burned in the engine. Or, perhaps is burned in such a way as to not produce motive power - I'm not really sure. He has a fourteen minute explanation of its working principles (as he understands them) here. At one point, he mentions that the "ball" of high pressure air created by the groove is a million times more dense than ambient air. That would be ~1.22*10^6kg/m^3~. Yes, the air is so dense that a cubic centimeter of it weighs (ok, has a mass of) 1.22 kilograms or weighs (here at the Earth's surface) 2.7 pounds. This is about 90 times as much as a cubic centimeter of mercury weighs! If it's an ideal gas (it's not, but we're talking order of magnitude here) its pressure is on the order of ~3*10^{13} Pa~ or ~4*10^9 pounds/inch^2~ (where I've speculated that the temperature is around 470 K).

I actually watched an online "talk show" called "Talk For Food" wherein Adam Abraham holds forth on a variety of rather outré subjects. In the subject episode, Abraham interviewed Gadgetman Ron Hatton and had the groove installed in his 1993 Lexus. In the course of the interview, Hatton claimed as much as 90% of the fuel going into a typical internal combustion engine is not burned in the cylinders to produce motive power. Rather, it's burned in the catalytic converter or exhausted unburnt.

This is irrational. A gallon of gasoline releases about 132 MJ (megajoules)/US gallon upon complete oxidation. Let's assume that 13.2 (10%) of those potential MJ are actually released in consuming a gallon, and the vehicle goes, say, 18 miles on that gallon at 55 m.p.h. Let's further, generously, assume that the internal combustion engine (ICE) can utilize 30% of these 13.2 MJ (i.e., the ICE is thermodynamically 30% efficient) then we can calculate that abut 4.5 horsepower is what's required to push this vehicle down the road. Sorry, that dog don't hunt.

Hatton had an analyzer hooked to Abraham's exhaust for a before and after test. The footage showed somewhere around 3,900 ppm (parts per million) for hydrocarbons in the exhaust before installation of the groove and 0 (yes, zero) after. Now, I've not seen any independent testing of the exhaust stream, so I can't say that these results have been replicated (nor that replication has been attempted and failed).

I don't claim that the Groove doesn't work, or that it provides no benefits. I simply state that all the "I think I'm getting about 28 m.p.g. and I used to get 12 m.p.g. and it sure does run smooth now" anecdotes on youtube provide no evidence that it does provide benefits.

But, at a broader level, I ask you: If this simple modification could be so effective, why aren't all the vehicle manufacturers beating a path to Hatton's door to license the (patented) technology? Can the oil companies really afford to pay them off? Imagine that Chevy could announce a Chevy Cruze that achieved 45 m.p.g. or even 60 m.p.g. and that car cost not a penny more to manufacture (the groove would, after all, not be installed by auto workers with Dremel tools as Hatton does it).

Anyway, I'll answer a question that comes up when I write about such matters and then make a comment on a comment I've seen on my blog posts and elsewhere. The question: "why do I care? The customers are satisfied, Ron Hatton seems like a nice guy." I care for a couple of reasons. First, the rising (worldwide, if not in the US) demand for petroleum based transportation fuel coupled with our stagnant ability to provide it makes critical analysis of possible efficiencies crucial and the discounting of pixie dust pivotal. Second, the gullibility and inability to think critically of the US public is disturbing and each example troubles me.

As to the comments, I've frequently seen (at PESN, on my blog, and elsewhere) a troubling retort to physics based debunking of alleged miracle fuel saving devices, miracle cures, etc. The retort is along the lines of "I'm sure glad I never took physics so that my view isn't limited by the dogma of traditional physics. I can be open to new ideas." You'll see such a comment on my original Gadgetman post. It's sad, so very much is possible within what we know and, though we certainly don't know everything, we know a lot more than nothing. And knowing what is and is not possible, the "man will never fly" and "aerodynamics says bumblebees can't fly, yet they do" tropes aside, enables efforts to be directed at things that have, at least, the possibility of paying off.