Why I (mostly) drive below the speed limit

(Updated 2/19/13 - see below)

Although it's not at all obvious to most, the speed you drive has a highly significant effect on the fuel consumption of your vehicle.  The key is the drag equation, which is an approximation of the drag force on an object moving at high speed (relative to the characteristics of the fluid in which it's moving - more on that later).

The drag equation is

where F represents the force, rho is the density of the fluid, v is the velocity of the fluid with respect to the object (in the case of a car, it's equivalent for us to consider the car in motion and the fluid stationary), and A is the cross-sectional area of the surface relative to the fluid - for us, the "face" of the car moving into the "wind."  The C term is the drag coefficient, which is a "fudge factor" that's usually determined empirically and depends on the properties of the surface, its shape, and the properties of the fluid*.

The key to take away from the drag equation is that the drag force depends on the square of velocity.  However,  if we want to talk about fuel consumption (call it gasoline in gal/min), we're going to need to move beyond just the drag force.  We can compare gasoline consumption at different speeds by using the energy requirement of your car, in, say, horsepower.  Horsepower is a power term (in case the name didn't make it obvious), meaning it expresses some amount of energy expended over some period of time**.

We can relate the force to the power required to overcome it by using the definition of work (a force exerted over a distance) and diving by the total time.  Since the distance over which you travel divided by the time over which you travel is your average velocity, this gives the power equation above***.  Note now that the power is proportional to the cube of velocity.

So now, onto why I drive below the speed limit.  One fine day I was driving from Ithaca, NY to Syracuse, NY, on the I-81N out of Cortland.  The road is pretty boring except for the occasional deer, so I did some mental math.  Because the fuel consumption (power) required from my car to overcome drag is proportional to the cube of velocity, the ratio of fuel consumption at, say, 75 mph vs at 65 mph is simply (75^3)/(65^3). 

The ratio turns out to be about 1.54.  This means that to a first approximation, at 75 mph I was consuming fifty percent more fuel than I was at 65 mph! Ouch.  I knew that on country roads, my Subaru got about 23 mpg at 65 mph, which means that over a 60 mile drive at around 65 mph it'd consume roughly 3 gallons.  At 75, it'd be consuming 1.5 more gallons. But wait - I was saving time by driving faster. Was my time worth it?

This calculation was easier - between 75 and 65, over the course of an hour it'd take (60/75) hours (0.8 hours, or 48 minutes) to go at 75 mph versus 60/65 hours (0.92 hours, or 55 minutes and about 20 seconds) to go at 65 mph.  How much was 7 minutes of my time worth?  At the wages I was working at that point (about $15 an hour), $1.75.  How much was 1.5 gallons of gas worth? At that time, roughly $6.

I decided to drive slower.  However, on the way back, I had my girlfriend, which meant I, ahem, valued the time saved from driving faster more.  I drove faster on the way back.

To this day I've yet to make enough money in hourly terms to justify regularly driving 75 mph over driving 65 mph on the highway.   At the current average gasoline price in New York state ($3.92/gal), I'd have to be making $50.40 an hour to justify it, which for a standard 8-hour work week is $104,832 per year.

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This analysis is relatively simple.  There are a couple of (to my mind, anyway) interesting little wrinkles to this calculation, the magnitude of which I can't calculate off the top of my head but I think on balance points to even more fuel consumption at higher speed.

The first wrinkle occurs when we relax the assumption that the drag coefficient is the same at different velocities.  Earlier, we assumed that the drag coefficient is the same at 75 mph and 65 mph - at least, I didn't call attention to the variation, which means I assumed the drag coefficient was the same.  If we relax that assumption, it turns out that there's an additional velocity dependence in the drag equation.  The drag coefficient is proportional to some power of the Reynolds number, which for an object moving through fluid is proportional to velocity.  Usually the expression is something along the lines of C_d = k*Re^a, where k is some constant that's empirically determined and a is some number below 1, also empirically fit.  So the power equation is actually proportional to velocity to the (3+a) power.  So power required to overcome drag force is actually somewhat higher because of this effect.

The second wrinkle occurs when you consider that an engine doesn't linearly deliver horsepower with increasing fuel consumption.  Engines have power curves that describe their performance given the acceleration they must deliver and the rpm they are already moving at.  A power curve for the motor on a Kia Alto (a small car) is shown below:

 

Fuel consumption is the bottom curve (PS is a German abbreviation, common in the car industry, for metric horsepower).  Most cars will operate in the downward sloping part of the curve (we usually change gear if we're not), which means that if we increase speed and don't change gears, fuel consumption per amount of energy delivered actually goes down.  However, we're still delivering a much higher amount of energy, and the ratio varies (in the region we're likely to see) from about 4.5 to maybe 3.25.  In reality the difference in engine RPM is likely to just be the ratio of the velocities, 75/65 or about 1.15, while the fuel consumption ratio occurs over a much larger range (about 3.5).  Still, it means that the fuel consumption at higher velocities, all things being equal, is lower than we might expect because fuel consumption increases less than linearly.

Finally, we have to consider that engine work is also used to overcome rolling resistance in addition to drag.  Rolling resistance is a subject that I don't have any experience in, but I know that for a car it generally increases at a much higher than one-to-one rate with applied torque.  This has much more to do with the way the wheels interact with the pavement then the way bearings behave; at high speeds there's some slippage and the like that cause inefficiencies.  If it were the bearings alone, we'd actually expect less resistance as speed increases because bearing lubricants tend to be shear thinning.  But all in all, this causes required power and hence, fuel consumption to increase at higher velocity more than we would expect from drag alone.

edit: Thanks to a lively discussion with my friend Tom, here's some more interesting stuff:

Tom calculated that the optimal speed at my then wage of $15 by minimizing the total cost of a 60 mile journey:

Cost(v) = Gas cost + Time Cost
C(v) = (60/23) gal*$4/gal*(v^3/(65mph)^3) + $15/hr* 60 miles/v
C(v) = $10.43 * v^3/(65 mph)^3 + $900 mi/hr / v

The local minimum for this can be calculated with some basic calculus, but we were lazy and used Wolfram Alpha:


This means that my optimal speed at that wage, assuming that the drag force dominated my costs, was 53 mph.  It turns out that's a pretty darn good assumption.

Thanks to some quick googling, I found the following diagram (which is given for a parcel truck):

Rolling resistance increases with a relatively low order with respect to velocity while aerodynamic drag has (as we've discussed) a very high order with respect to velocity.  Thus at high speeds it's a bit tricky to discuss aerodynamic drag as the only force, but it isn't an overwhelmingly bad assumption. Note that on the above graph, the curves stop at about 120 kph which is about 75 mph.

All in all, a fun physics problem.


* The drag coefficient for most objects is close to 1.  It's a measure of how close the object is to representing an ideal object for modeling, usually something like a cylinder of infinite height or some such.  Most objects don't deviate much from that - hence, the coefficient doesn't modify the ideal equation much.  For cars, though, a considerable amount of R&D goes into making that drag coefficient lower.  In terms of fluid properties, however, it's also related to the Reynolds number, a dimensionless quantity that is related to the inertia of a fluid (usually involving the velocity) and the viscosity.

** It's easy to get confused about what quantity to look for when talking about fuel consumption, but we can review the units to double check: fuel burned releases energy (in joules) over time (min), while force (Newtons) must be multiplied by distance (Newton*meter = Joule) and divided by time (min) to produce an equivalent quantity.

*** Strictly speaking, if we wanted to be precise then the power equation should be given by d/dt(F_d \dot x), where F_d and x (the distance) are both vector quantities and d/dt is the differential operator with respect to time.  This gives the instantaneous power consumption, which given the data for a whole trip can be averaged out taking into account all acceleration and deceleration and the like.  However, for a rough comparison of fuel consumption at constant speeds it's not necessary.

Arguments that annoy me, part 1

I love debate.  Once upon a time, I liked competitive debate, too, and thought that I liked pounding other people's arguments into the dust.  However, as time went on I realized that what I enjoyed was not winning per se, but having conversations with intelligent people.  People with poor argumentation and poor logic not only made for poor debate, but simply annoyed me.

To this day, there are few things that annoy me more.  So I'm going to blog about them. Hah.

1.  "It is better to be consistent than correct"

Hypocrisy is an irrelevant ad homoniem, but I see this all the time.  Essentially, someone argues that because person/country/organization A has done something before, but subsequently advocates against it, the argument is invalid.  A is hypocritical!  Common examples:

• The US has violated someone's sovereignty before, therefore its advocacy against a violation of other people's sovereignty is invalid.
• <insert political party here> did something in the past, therefore its current stand against the practice is invalid.
• You once believed this, but now you've flip-flopped and your argument is weaker because of it.

I could go on and on.  This is a poor argument.  Accusations of hypocrisy are often valid, but the only thing that they should do is cast aspersions on the character of the advocating party - they never have any relevance to an argument itself, unless that argument is about a person or organization's character.

When presented in simple terms as above, it's easy to spot how fallacious an accusation of hypocrisy is.  I more frequently see it presented in forms more difficult to recognize.  For example, if I were to hypothetically accuse the Republican party of being obstinate in their use of the filibuster in the Senate of the US, I might receive the following reply:

"But when the Democrats were in the minority during the Bush years, they also heavily used the filibuster and justified it by claiming that they were doing the job of the minority in protecting their interests, and you didn't complain then!"

Sound familiar?  Doesn't address the issue at hand, doesn't address anything but the parties to the argument (the Democrats and the proposer), allegation of hypocrisy.  While pointing out how hypocritical a group or person is might cast aspersions on their character, it has zero to do with the argument.  Instead, it's a statement that amounts to "it is better to be consistent than correct," or, alternatively, "consistency is a prerequisite for having a valid argument."

Top places I see that argument:

• Anti-colonial related rants
• Chinese uberpatriots
• American political debate

Next in this series: "It is preferable to be understood than to be correct"

What's this process configuration called?

There's a relatively common process configuration for pilot and demonstration plants that operate at least in part continuously, where process steps are isolated in a modular fashion and tankage is placed between them to store intermediates/effluent.  It's a method of design that's focused on being a modular platform for testing and investigation rather than for a manufacturing campaign, such that individual process steps/units can be devoted to different teams for testing and investigation, but it can be integrated to achieve pseudo-steady state operation if need be. It looks approximately like this:

I've seen it in pilot and demonstration plants all over the place, but what is it called?  I want to not look like an idiot when I speak at a conference.

Price Controls Have Consequences

Two recent events have brought the topic of energy price controls to my mind. The first is in Brazil, where government pressure on Petrobras, the state-owned energy producer, has caused it to continue to hold down prices on gasoline in Brazil for the foreseeable future.  Thanks to governmental pressure, Petrobras has subsidized its gasoline, selling below fair market value, since 2005.  Brazil's government has a strong interest in cooling down inflation and part of that is attempts to hold down the value of the currency, which is already overvalued.  But trying to push against a gas price rise has led to some unintended consequences (gated, unfortunately).

Essentially, cane sugar mills that might otherwise be producing high-value ethanol to sell into the fuels market are now facing a situation where rising fundamental input prices are pushing them out of the market. The price levels of their fundamentals - labor, raw material inputs, and the like - are continuing to rise with inflation, because there is no universe in which government price controls can change the underlying market forces.

In order to remain competitive, ethanol must be sold at a significant discount by volume to gasoline, which has a higher energy content.  Price controls on gasoline have historically cut the profitability of ethanol, and it is now reaching the point where cane ethanol producers cannot sell into their own domestic market because of the artificially low price of the competing product, gasoline.  Perversely, this has caused the Brazilian ethanol market to preferentially export ethanol to the US once again (aided and abetted by the US's renewable fuels standard, and reversing the trend of the last few years) and to overproduce sugar, the price of which isn't being held down by the Brazilian government.

The second bit of news was brought to my attention by the always-informative Geoffrey Styles at his blog Energy Outlook.  Mr. Styles' discussion is more detailed than my own, so I would definitely recommend reading his post.

After the recent election Senator Ron Wyden (D-OR) is now likely to chair the Senate Committee on Energy and Natural Resources.  His views on shale gas exports show a dangerous tendency to protectionism.  Essentially, Senator Wyden is worried that raising the natural gas price in the US by selling LNG onto the world market will harm America more than it will help it.

Senator Wyden's protectionist reasoning is dependent on two assumptions: first, that shale gas production in the US will continue to provide energy at the same cheap price levels we see today, and second, that a low gas price is unambiguously good for America now and in the future.

The first assumption is unambiguously wrong. The shale gas price level is so low that drilling activity has drastically decreased and continuing production is depending on co-production of natural gas liquids - so-called "wet" gas wells.  "Dry" gas fields with little-to-no natural gas liquids have largely remained unexploited this year.  Additionally, Mr. Styles points out one reason that the pricing mechanisms that govern gas production might currently be difficult to discern: that much of the continuing production is due to contractual obligations that require the exploitation of reserves at a set pace independent of market signals.

However, this lack of a price incentive to drill is keenly felt in industry. There is a virtual consensus that price levels must go up to encourage more drilling.  If exports are blocked and contribute significantly to the price remaining low (there is some controversy as to whether exports will play a big part of this), then Senator Wyden will see his policies causing the shale gas revolution to peter out.

Senator Wyden is also wrong to think that low gas pricing is unambiguously good for America in both the short and long term.  In the short term, it is worth remembering that high energy prices are good for the (US-based) producers as well as for consumers. At minimum, keeping the shale gas price artificially low by minimizing exports will amount to a redistribution from resource extraction companies to consumers of those resources, with the results unclear.

In the long term, Senator Wyden may also find that keeping the gas price low hurts some of the very sectors he may be inclined to support.  Renewable energy hopefuls for both electricity and vehicular fuel must now compete against a very cheap, high quality and clean burning substitute that was literally not in contention three years ago.  Senator Wyden will undoubtedly find that holding down the price of natural gas will sound the death knell for a whole host of renewable electricity and renewable fuels projects - maybe even including some in his home state.  We may venture that holding down gas prices now might well lead to a very uphill battle when the gas begins to run out and we have no renewable infrastructure to fall back on.

Price controls create perverse unintended consequences, in the energy world and out.

I don't want to live in the Zuiderzee

Geoengineering is a term for technical measures to change the way the climate works. Most commonly, it's applied to the problem of AGW, with the general hope that all of this expensive transition to low-carbon fuel can be averted for pennies on the gallon of gas to pay for carbon sequestration, sulfur aerosols, water vapor boosters, and the like.

I've never been much of a fan of geoengineering schemes. Most of the time I can dismiss geoengineering for a variety of technical reasons.  For example, most schemes for sequestering carbon once it's been dispersed from the atmosphere are, I think, dead on arrival, because they are far too expensive to implement due to the mixing energy barrier (in the case of atmospheric scrubbers - it's "simply another chemical engineering process" but the economics will never work out compared to CCS at the source) or due to natural homeostatic mechanisms (for those who think fertilization and bio-sequestration in plants is the answer). They also suck up carbon dioxide or other greenhouse gases independently of the source, which requires an external entity - probably the government - to pay them for their services out of a carbon tax, all of which opens up problems of regulatory capture and government inefficiency. These reasons provide a pretty good blanket justification against all geoengineering ideas of this type.

However, for those schemes that directly mitigate AGW effects, e.g. sulfur aerosols dumped into the air as popularized by Steven Levitt, no matter how many technical justifications you offer it can't change the fact that there are possible technical solutions out there that might offer us a way out through active intervention in the climate. Unfortunately, there are good reasons to go against that approach as well.

Distiller's Grains on the Brain

The always informative Energy Collective has posted this article by Professor Simon Donner of UBC trying to shed light on the media circus around the US corn crop and the food-fuel substitution argument. I like that someone is trying to dispel the myths around this, but unfortunately he falls into the same fallacy of claiming that feed uses for corn are going away that most analysts with little understanding of the corn ethanol value chain continue to emphasize. To support it, he posts this graph:

This is originally derived from another work, I believe, but all data ultimately stems from the USDA feed grains yearbook.  At first glance this appears to tell a compelling story of how the amount of animal feed in the US has declined, while the amount going to sweetener, chemical or direct food use ("other") has remained fairly constant. Since I've repeatedly asserted this but never before posted data on it, consider this the moment where I finally try to prove my claims instead of simply referring obliquely to the data sources.

The unfortunate part of this analysis is that without an understanding of the full ethanol value chain, you cannot tell the whole story. The whole story is that for every bushel of corn consumed, 17/52 by weight becomes distiller's grains, a food supplement for cattle, hogs and chickens. Cattle can consume it exclusively because they're ruminants, and hogs and chickens can consume up to 15% by weight of it in their diet. If you include distiller's grains in the mix, the story starts to support my assertion that the feed picture hasn't changed, at least in terms of quantity.

So let's look at the US feed consumption of corn.

Note that the USDA uses a slightly different calendar system than the financial world due to the growth/harvest cycle; Q1 06/07 corresponds roughly to September - November 2006. It's immediately clear that the feed business has a very cyclical component, with feeding hitting a peak around Sep-Nov each year and hitting a trough in Jun-Aug the following year. I don't know why this happens because I don't know the animal feeding business well enough. What is clear, however, is that each year since 2006 the peaks have been getting lower and the troughs have as well.

This picture changes when you start to include the mass of DDG produced each quarter, which goes exclusively into feed use:

Total Corn and Corn Derivatives Going Into Feed Uses; DDG is included on an equivalent mass basis to corn

Suddenly, the picture has become less clear, though a slight decrease is perhaps detectable. But wait! DDG is more nutritious than corn is, and the USDA last year acknowledged that a pound of DDG can be substituted for 1.22 pounds of corn in most feeding situations. If we take that into account:

Total Corn and Corn Derivatives Going Into Feed Uses; DDG included on an equivalent nutritional basis to corn

Suddenly, there are no discernible trends. At least in the sense of quantity used for feed and food, the food-fuel substitution crowd doesn't have an argument.

However, this is not where the food-fuel substitution argument against corn ethanol is strong. Where it is strong, and why I think it's significant, is in the realm of pricing.


Sources: FGYearbook Table01, Table 04, Table31, link to USDA feed grains yearbook above.