Friday, April 20, 2012

Will the Chevrolet Volt Save You Money?

EDIT 4/23/12: I made a pretty huge mistake here in not taking the relative efficiencies of gas vs. electric car motors into account and just assuming they were roughly equal.  In reality electric motors are about a factor of four better.  That completely changes the conclusion of the post, which originally had the Volt costing about as much as a regular car to drive around.  I've edited heavily to take all that into account.  

The Chevrolet Volt, and other plug-in hybrid vehicles like it, are marvels of engineering that should have all self-respecting nerds salivating for them to come down to normal-people prices.  Their genius is in the way they get around what's always been the main issue with electric cars: the energy density (the amount of energy you can store in a given weight or volume) of batteries, while it sucks less than it used to, has never been able to match the energy density of gasoline.  As a result, electric cars either need to carry several times the mass of a full gas tank in storage batteries (almost always impractical) or have their range drastically compromised.  The Volt and its ilk work around this by adding a gasoline engine to basically act as a battery charger on the road; it doesn't do much more than burn (high-energy-density) gasoline to keep the (low-energy-density) battery topped up, while the battery powers the drivetrain via electric motor.  This is different from parallel-hybrids like the Prius, which freely switch between using their gas and electric motors to power the drivetrain, and in principle should be able to seriously increase gas mileage to well above what even current hybrids are capable of.  It does in fact succeed in doing this, as this photo from James Fallows' blog shows:

For reference, that's approximately Philadelphia to Denver on a single tank of gas.

That's bananas, right?  Unfortunately it's not that simple.  The Volt ain't magic; it still takes the same amount of energy to push it that 1389.4 miles as it does a regular car of the same size.  The difference is that a lot of that energy is coming from electricity (via charge-ups between drives) rather than gasoline.  Since electricity isn't generated from nowhere and definitely isn't free, miles per gallon of gasoline is an extremely deceiving metric to use when evaluating the efficiency of plug-in hybrids; we need to think of a way to take that generated electricity into account too.  I decided to figure out a way to do this and, in the process, find out (roughly) how much money you'd actually save by driving a Volt vs. a regular, decently-efficient car.

I made a couple of simplifying assumptions in figuring all this out:
  •  I assumed the car usage summarized in the photo above was typical.  From the numbers I've heard thrown around for plug-in hybrids, it's probably close.
  • My "normal" car was assumed to have an average gas mileage of 30 mpg over the same 1389.4 miles that Volt drove.  That's a reasonable assumption for a well-built traditional car of the Volt's size.  I also assumed that it had roughly the same size, weight, drag coefficient, etc. as the Volt, so the energy used to move both of them would be close to identical.
  • I assumed that the efficiency of charging the Volt's battery with a gasoline motor is the same as the efficiency of moving the regular car with a gasoline motor. The Volt is probably slightly more efficient for various reasons. 
  • I assumed that all the electric power used to push the Volt came from charging it off a residential power grid.  In real life some of it will come from regenerative braking, but probably not enough to throw off our calculations by more than a couple of percentage points.  
We need to know a few numbers before we can get started:
  • The energy you get from burning gasoline is, according to Wikipedia, about 34 MJ/liter.  In American units, that converts to about 35.75 kW-hr/gallon
  • The average US price of a gallon of gas, at the time of this writing, was about $3.88.  We can calculate the energy cost of burning gasoline to be about 10.9 cents per kW-hr, using the previous number. 
  • Likewise, the average national cost of 1 kW-hr of electricity in 2010 (the most recent data I could find with a quick Google search) was about 11.5 cents.
  • The energy efficiency of a good internal combustion engine is about 20%; electric motors run closer to 80%.  In other words, you can get an electric motor to do the same amount of work as a gasoline motor for 1/4 the input energy.  

First we need to calculate the total energy used to move the car over our representative 1389.4 miles.  If we assume it's the same for both our Volt and normal car, and that our normal car can average 30 mpg efficiency, then we know the normal car will need 46.313 gallons of gas to go that distance.  Since we know how much energy is contained in a gallon of gas now, we can work out that the gas-powered car will use about 1656.16 kW-hrs of energy during the drive.  Since the engine is only 20% efficient, only about 331 kW-hrs of that were actually needed to move the car; the rest gets lost as heat, noise, etc.

All of that energy came from burning gasoline for the traditional car, so we can pretty easily calculate the total cost of the drive by using the price of gas and the energy density of gas: about $180.52.

For the Volt, things are slightly more complicated.  The electric motor powering the drivetrain is about 80% efficient, so dividing 331 kW-hrs by that gives us 414 kW-hrs, the input energy needed to move the car.  Since we know we burned 10.4 gal of gasoline during the trip, we can calculate that about 372 kW-hrs was used by the (20% efficient) gasoline engine.  If we assume all of that was used to charge the battery (in reality some would have been used to power the drivetrain, but we'll ignore that for simplicity) that's about 75 kW-hrs of battery charge from burning gasoline.  The rest of the battery's charge would have come from the power grid;; we can calculate that by subtracting the gas engine's contribution to charging the battery from the total 414 kW-hr charge.  The total energy cost can then be calculated as follows, by adding the battery-charging contributions of the gasoline engine and the power grid:

Cost = (372 kW-hrs)*($0.109/kW-hr) + (414-75 kW-hrs)*($0.115/kW-hr)

The total cost comes out to about $79.50, more than a factor of two less than the conventional car

So that's about 13 cents/mile to drive the conventional car, vs. 5.6 cents/mile for the Volt.  That's a pretty big difference; even with the money you're paying to charge the car, it's still costing you about half as much to drive your Volt around as it would a similarly-sized conventional car.  To put that in perspective, if you drive 20,000 miles in a year, the Volt will save you almost $1500 annually.  That's a long way from "paying for itself," at least at current prices (a Volt will run you almost twice as much as a similar conventional car), but if plug-in hybrids like the Volt are anything like the current generation of hybrids they should fall in price pretty quickly over the next few years. 

So yes, the Volt will save you quite a bit of gas money, even though it's far from the free lunch that the mpg numbers being thrown around make it look like.  From a cost standpoint, you could assume it's equivalent to a hypothetical gas car that got around 70 mpg; that's not exactly Philly to Denver on a tank of gas, but it's pretty good.   The cost savings will probably only get bigger, as gas prices continue to rise faster than electricity prices, and in places like Europe where gas isn't artificially cheap the Volt is already close to paying for itself in a couple of years.  

It's worth mentioning that this is only looking at raw fuel cost; there are a lot of other advantages (less pollution and reduced dependence on foreign oil are two big ones) to using centrally-generated electric power vs. burning gasoline to power our cars.  It's far from a perfect solution to the transportation problem the US is going to have to deal with when our current era of cheap gas ends, but it's one of the best things anybody's come up with so far.

Tuesday, April 10, 2012

Why Is The Earth's Crust Mostly Silicon?

A couple posts ago, I mused that it was a pretty goddamn convenient coincidence that most of the crust of the planet we live on was made of the one element that's absolutely essential to all modern technology.  Being a generally lazy person, I was ready to just shrug and say "eh god did it" until I remembered that I'm, at best, an agnostic and not supposed to be doing that.  So I went with my backup plan-- shrugging and saying "eh, astronomy/geology did it."  It wasn't really germane to the earlier post's topic anyway, but the whole point of this blog is to actually try to find out the answers to all the things in life I usually just shrug and accept.  Plus thinking about it got me curious about two things I know next to nothing about: how heavy elements are formed and how the earth was formed.

The answer to this one goes all the way back to the beginning of the universe, when all the matter in existence (your desk, my computer, Andy Reid, etc) was created in the first couple of hundred thousand years after the Big Bang.  Problematically, that matter at the time consisted almost entirely of hydrogen and helium, since a rapidly cooling quark-gluon-lepton plasma (the mess left by the Big Bang) is going to relax into the least energetic state possible. In this case, that means lots of individual or double protons that were eventually able to capture an equivalent number of electrons as the universe continued to cool.

Hydrogen is great for making water and explosions and everyone loves balloons, but as you've probably noticed almost everything solid in the universe is made up of heavier elements like carbon, silicon, and iron.  So how did we go from "shit-tons of hydrogen, helium, and not much else" to the clusterfuck of 100+ elements that makes up the periodic table?

Short answer: explosions, and lots of 'em.  All those clouds of hydrogen and helium in the early universe would eventually (~1 billion years) coalesce into discrete masses, aided by gravity.  Eventually these masses got dense enough that the hydrogen and helium at the cores was under enough pressure to undergo fusion.  The result was lots and lots of gigantic primordial stars. 
These primordial stars, being much purer hydrogen-helium blobs than most of our current crop, were able to burn a lot hotter and, as a result, could get quite a bit bigger.  More mass means more core pressure means way more fusion than we see in most "modern" stable stars, which mostly just make helium; large numbers of protons could be fused into heavy elements. Every element from carbon through iron is/was formed via extreme stellar fusion this way.

Conveniently, the stellar mass that's necessary for this kind of higher-order fusion to occur also tends to make a giant star (superstar?) extremely unstable, so after creating heavier elements in its core for awhile it generally goes boom in a supernova/hypernova event, spreading those elements out through the universe.  As a result, the universe's supply of heavy elements consists overwhelmingly of the stuff between carbon and iron on the periodic table.  The elements heavier than iron, created from less-common non-fusion processes in large stars (physical limits on stellar mass mean iron is about the heaviest thing you can make with pure stellar fusion), are quite a bit rarer and get even more so as their atomic number goes up.

Relative abundance of the elements in our solar system (and, by extension, the galaxy/universe).  The weird sawtooth pattern is due to the fact that elements with even atomic numbers have a higher binding energy than odd-numbered ones.  Note that the y-axis is log scale, so differences are bigger than they look.  (thx Wikipedia)

So a lot of the early history of the universe was just giant stars forming and exploding, making lots of heavy elements in the process (it's worth mentioning that this is still going on, although less frequently).  At the same time this supernova-fest was happening, more reasonably-sized stars that didn't explode all the damn time were also getting formed and coalesced into clusters, galaxies, etc, eventually giving us approximately the universe we know and love today.  Once there were stable stars, the whole process of gravitational capture of heavy elements and planetary accretion started creating solar systems, including ours. 

So at the end of the day (or couple billion years or whatever), the top ten most common heavy elements in the galaxy (in order of abundance) are oxygen, carbon, neon, iron, nitrogen, silicon, magnesium, sulfur, argon, and calcium.  It's a pretty safe bet that most of these are going to have a lot to do with Earth's composition.  We can rule neon and argon out almost immediately though; they're noble gasess and aren't going to form anything solid without lots of coercion.  Of the others, oxygen has a tremendous advantage: it can form stable, solid compounds with everything else on the list except the carbon and nitrogen, and lots of other elements too.  More importantly, it's the only top-ten element that's capable of doing this.  So it's pretty much a given that the crust is going to be made up of mostly "rock-like" (solid at planetary temperatures) oxides of abundant elements.  Oxygen, ergo, is pretty much a lock for most common crustal element, and indeed wins by more than a factor of two over the first runner-up.

So now that we're battling for second place, the question now becomes "which oxides?"  You can roughly work this out by looking at all the rock-like oxides, rating them by the galactic abundance (or lack thereof) of the other element involved, and then accounting for each oxide's molecular weight.  The weight matters because Earth was basically a liquid during its formation; heavier elements/compounds had a tendency to sink down toward the core, while the lighter ones floated around in what would become the crust.  So while you'd expect iron oxide to be the most common compound in the crust, its relatively high molecular weight causes it to place a distant fifth, after the silicon, aluminum, calcium, and magnesium oxides.  Same deal with magnesium, to a lesser extent; the less common, but much lighter aluminum and calcium oxides end up beating it out even though aluminum isn't even in the top ten of galactic abundance.

Relative abundance of elements in the Earth's crust.  Note that the green blob (elements that form rocky oxides) is kicking everything else's ass. (thx Wikipedia)
Silicon, though, is the best of all worlds: not only is it the second most common rock-like-oxide forming element in the galaxy (after iron), but the oxide it forms is also pretty light as these things go. Result: lots of silicon oxide in the overall composition of the earth, and nearly all of it floating at the top in what would eventually cool down and become the crust.  The only other oxide that even comes close is aluminum, and even it still lags more than a factor of three behind silicon oxide in crustal abundance.

So as usual, there's a perfectly reasonable, if somewhat long and complicated, explanation for why the most common element in the crust of our home planet is also one of the most useful.  Yes, it's a complete coincidence that silicon happens to also be a semiconductor as far as I can tell, but at least now we know why there's so much of it around.  Still, if silicon didn't semiconduct we'd be pretty SOL; the next most common Si-like elemental semiconductor is germanium, which is about six orders of magnitude less abundant than silicon.  (Slight caveat for the pedantic: carbon, in diamond form, will semiconduct, but not in ways that are very conducive to the low-power digital electronics we like so much.  Still, we might've made it work if we had to, we're clever like that.) 

An interesting, largely unrelated fact I learned while looking all this up is that the galaxy (and by extension probably the universe) is, even now, still more than 99% composed of hydrogen and helium.  All the rest of the other elements put together barely comprise enough matter to even rate as a contaminant.  Even weirder, that contaminated field of hydrogen-helium only comprises about 5% of the universe; the rest is apparently dark matter and dark energy.  And that's where I'll stop, because I really don't want to have to go there.  

Thanks to Wikipedia for most of the basics of this one, and the blog's astrophysicist pal for some fact-checking of the parts with stars in them.

Wednesday, April 4, 2012

Why Are Manhole Covers Round?

One of the things they (or I guess "we" at this point, since I routinely fail to mention this at science-outreach events) never tell you about science is that most of it is waiting around.  A good percentage of my work time is spent waiting for processes to finish, waiting for vacuum systems to pump down, and waiting for machines to stabilize.  To make matters even more boring, most of that waiting happens in a clean room, where I'm wearing a bunny suit and forbidden from bringing in outside objects; it's not like I'm going to spend that time catching up on paperwork or reading a book.  The point I was gradually getting to: the other morning I was waiting for something science-y or other to happen and  killing time by repeatedly clicking the "random" button on  While doing so, I ran across this:

(this is where you take a break and go read the hard-to-embed comic)

I realize that this is sort of missing the point, but the manholes question bugged the crap out of me for a solid day afterward.  The best answer I could come up with was "because the pipes they cap off are also round," but that's the worst kind of kick-the-can answer and just begs the question "well then why are the pipes round genius?"  So that's less than helpful.  All I accomplished from trying to come up with a better answer was getting the Teenage Mutant Ninja Turtles cartoon theme stuck in my head, so I decided to go to the Wikipedia.

Apparently this is an actual, somewhat famous question Microsoft likes to ask at interviews.  So for probably the 20th time today (and it's only a bit after noon), I'm extremely grateful to have a job despite quite clearly not deserving one.  Anyway, the reason Microsoft likes it so much is that there are a lot of good answers, and apparently which one you pick gives some unique insight into your personality (all my answer tells you is "MIT's Ph.D program has really gone downhill," so SUCK IT MICROSOFT).

Japanese manhole cover, which looks exactly how I'd expect it to.

The most important reason for having round sewer caps involves spatial geometry, so as someone who once got tagged "possibly learning disabled" in high school math I really never had a prayer here.  Basically, there's no orientation of a round sewer cap that can make it fall through a round hole of the same size.  A square/rectangular sewer cap, on the other hand, can easily be dropped down its corresponding hole if the angle is right. As an abortive childhood adventure inspired by either Goonies or Ninja Turtles taught me, sewer caps weigh like a quarter-ton; I can respect that dudes working underground, waist-deep in human excrement, probably have enough problems without the threat of one falling on their head.

The fact that sewer caps are so heavy also leads to a few other good reasons for them being round: they'll fit on the sewer in any orientation (no need to rotate to the correct angle), and they can be easily moved around via rolling.  Similarly, if there's a sewer cap in the road that's not correctly seated, it's a lot less likely to shred your car tires if it's got a round, vs. a sharply-cornered, edge sticking out.

Perhaps somewhat more practically, circular tunnels are both easier to dig and more stable than any other shape, so if you don't want your sewer system to collapse they're an excellent choice.  And finally, there's simple economics to consider: there's only a couple of companies that make sewer caps, and they all make exclusively round ones for all of the above reasons.  If you want any other shape, it's going to cost significantly more to have it custom-made.

You do occasionally see square sewer/utility hatches, usually in places that are a) not in a road, and b) leading down to electrical conduits, storm drains, or other things that don't run very deep.  The rugged individualists of Nashua, NH apparently use triangular sewer caps because LIVE FREE OR DIE or something.  I really hope their libertarian utopia is willing to make an exception and give sewer workers some decent health insurance. (addendum: there's no way a triangular cap can fall through a hole of the same size either.  See earlier comments on spatial geometry)


So that's about six different, very good reasons for making sewer caps round, none of which I was able to come up with on my own.  That's disappointing, but I'm heartened by the fact that judging by such bestselling products as the Zune and Windows Phone 7, the question doesn't seem to be working out all that well for talent-screening at Microsoft.  Still, now you can go get a job there if you want, assuming they're dense enough to not change their interview questions after they turn up on both Wikipedia and random webcomics.  They'll probably ask you something like "why do we have a 120V/60Hz AC power grid" now, in which case you're welcome and you can repay me by doing something about the idiotic User Access Control in Windows.