Prime trivia

April 24, 2025 at 1:12 PM by Dr. Drang

I was at a trivia contest last night, and one of the questions was: What is the largest three-digit prime number? One of my teammates and I both guessed 997 and went about trying to prove or disprove it before the next question came up.

We agreed that 997 wasn’t a multiple of 7. My thinking was that since

997-700=297

and

297-280=17

and 17 isn’t a multiple of 7, 997 isn’t a multiple of 7. Similarly, since

997-990=7

997 isn’t a multiple of 11, either. I was trying to work out my reasoning for 13, starting with

997-390×2=217

when we decided to just go with 997 as our answer because time was running out. Later I realized that I should have used a different multiple of 13 and done the subtraction in the other direction:

520×2-997=43

It’s more obvious—to me, anyway—that 43 isn’t a multiple of 13 than that 217 isn’t.

Despite our failure to check if it was a multiple of 13 (or any higher prime), we got the answer right.

If you’re feeling an itch to tell me some rules about checking divisibility, don’t bother. As Matt Parker said in this video, there are an endless number of them, and I just don’t see myself committing any of them—other than the rule for 3, which I’ve known since I was a kid—to memory. Integer arithmetic doesn’t show up much in structural or mechanical engineering and has never seemed natural

1 to me.

This morning, I decided to look into how Mathematica handles primes. One function, Prime[n], returns the nth prime number, and another, PrimePi[x] gives the number of primes less than or equal to x. It gets its name from the prime counting function, π(x)

I put these functions together like this,

Prime[PrimePi[999]]

to get 997, which is a reasonably convenient way to get the largest prime less than or equal to a number. But what if I wanted to get the five largest three-digit primes?

I could work my way down the ladder.

Prime[PrimePi[996]]

returns 991, and

Prime[PrimePi[990]]

returns 983. But this is tedious, and there should be a way to get them all at once. One way is to use the Table[] function to get a list of all the three-digit primes,

Table[Prime[n], {n, PrimePi[999]}]

and then pull out just the last five:

Table[Prime[n], {n, PrimePi[999]}][[-5;;]]

This returns a list comprising 971, 977, 983, 991, and 997. I find Mathematica’s list indexing notation hard to remember, mainly because everything is doubled. The brackets have to be doubled because Mathematica uses single brackets to enclose function arguments. And the double semicolons are a single term; the expression

[[-5;;]]

means “start 5 items from the end and go to the end.” It’s like

[-5:]

in Python, only harder to read.

Although it’s not obvious from the documentation, Prime[] can take a list of integers as its argument and will return the corresponding list of primes. So

Prime[{1, 2, 3, 4, 5, 6}]

returns the list comprising 2, 3, 5, 7, 11, and 13. We can use this and the Range[n] function to simplify our expression for the five largest three-digit primes:

Prime[Range[PrimePi[999]]][[-5;;]]

OK, it’s not that much simpler. I often think Mathematica and Perl are too heavily influenced by TMTOWTDI.

  1. Yes, that’s an intentional pun. ↩

Plastic bottles and pressure vessels

April 19, 2025 at 9:52 PM by Dr. Drang

Bill Hammack (The Engineer Guy on YouTube) had a nice video last week on the engineering behind plastic soda bottles. It’s well worth your 13 minutes.

Prof. Hammack is a chemical engineer, so you should pay particular attention when his topic is plastic. About 3:30 into the video, he talks about how, during the expansion of the bottle from its original size and shape (the preform), the long chain PET molecules tend to untangle and align themselves in the axial and hoop directions. This strengthens the bottle in those directions and makes it better able to handle the internal pressure.

Video frame showing polymer alignment

Why do these directions matter? Because they’re the directions of principal stress in the bottle’s wall, and the hoop direction is where you’ll find the maximum tension. This is one of the elementary bits of stress analysis you’ll find in an undergraduate mechanics of materials class. Not specifically for soda bottles, but for pressure vessels in general.

Here’s a typical cylindrical pressure vessel. The end caps—which I’ve made ellipsoidal because that’s relatively easy to do in Mathematica—don’t match what we see in soda bottles, but that’s OK. When you’re interested in the stresses in the vessel’s midsection, the shapes of the end caps don’t matter.

1

Pressure vessel

Something that does matter is the thickness of the wall relative to the radius. The analysis below is for thin-walled pressure vessels, which means that t≪r, where t and r are the wall thickness and cylinder radius, respectively.

You might be wondering “which radius?” There’s an internal radius and an external radius. Which one is r? For all practical purposes, it doesn’t matter because the internal and external radiuses are nearly the same, since

ro=ri+t=ri(1+ t ri ) ≈ ri

If the ambiguity bothers you, you can say r is the average radius.

We’re also going to assume that the pressure in the vessel is much larger than the hydrostatic pressure due to the weight of the fluid alone. That way, we don’t have to worry about the pressure being higher at the bottom of the vessel than at the top. This is certainly true in a soda bottle. Prof. Hammack says the pressure has to be about 70 psi to dissolve the CO2 in the liquid. The additional pressure at the bottom of the bottle due to the weight of the soda is only about 0.4 psi, so it’s reasonable to ignore it.

2

One last assumption: the stresses in the wall are constant from the inner surface to the outer surface. You might think of this as the wall being too thin for the stresses to have room to change. If you’re skeptical, I can tell you that more advanced analysis shows that the stress is indeed nearly constant across the wall thickness.

Let’s get to analyzin’. We’ll start by passing a plane through the bottle at right angles to its axis, and consider the free body diagram of the vessel and its contents on one side of that plane.

Axial stress FBD

The red arrows represent the axial tension, σa, in the wall of the vessel, and the blue arrows represent the pressure, p, in the bottle. There are no shear stresses in the fluid because fluid’s don’t carry shear stresses when they’re not flowing, and there are no shear stresses in the wall because of the vessel’s symmetry.

Balancing the forces in the axial direction gives us

σa(2πrt)=p(πr2)

which simplifies to

σa= pr 2t

Now we make a new free body diagram by passing two parallel planes a distance w apart through the vessel perpendicular to its axis. Then we pass another plane that runs along the axis and consider the wall and fluid on one side of that plane.

Hoop stress FBD

Because we’ve already analyzed the forces in the axial direction, and because they don’t affect equilibrium at right angles to them, I’ve left them out of this drawing. The balance of forces here gives

σh (2tw) = p(2rw)

which simplifies to

σh= pr t

So the tension in the hoop direction is twice the tension in the axial direction. But note that because t≪r, both stresses are much larger than the pressure in the vessel.

I don’t have any 2-liter bottles around the house, but I cut open a can of LaCroix (also a pressure vessel), and measured its diameter

3 at 66 mm and its wall thickness near the center at 0.10 mm. That means r/t=330. So the hoop stress in that can would be more than two orders of magnitude greater than the pressure in the fluid.

If the material the vessel is made from has the same strength properties in both directions, the hoop stress is the one you have to worry about. In plastic soda bottles, though, you probably have to worry about both stresses because the extent of molecule alignment is not necessarily the same in both directions. If that’s the case, the strengths in the two directions will be different, and you have to compare each stress with its corresponding strength.

  1. This is due to a phenomenon known as Saint-Venant’s principle, and it’s a good thing. If we had to account for the shapes of the end caps—which we do when calculating stresses in and near the end caps—the analysis would be considerably more complicated and we wouldn’t be teaching it at the undergraduate level. ↩

  2. He doesn’t say it, but I’m pretty sure that 70 psi is the absolute pressure. The gage pressure—the difference between the pressures inside and outside the bottle—is therefore about 55 psi. Still very large compared to 0.4 psi. In the analysis below, p is the gage pressure. ↩

  3. Nobody measures radius directly. You measure diameter and divide by two. ↩

Concrete on Rare Earth

April 17, 2025 at 7:19 AM by Dr. Drang

A recent episode of the BBC Radio 4 science and environment show, Rare Earth, discussed concrete. I approached the show with some trepidation. Concrete has become an environmental whipping boy because of the great quantities of carbon dioxide released during the manufacture of cement. But the show was fair. It didn’t shy away from concrete’s problems, but it also discussed how valuable concrete is to society. It’s well worth a listen.

Like most civil/structural engineers, I have a fondness for concrete. It’s a great building material, and it’s hard to imagine the modern world without it. In the show, they talk about it mostly in the context of buildings, but roads, bridges, harbors, tunnels, dams, and wastewater systems all depend on concrete.

But its carbon footprint is undeniably huge, primarily because of cement. Cement is the powder that reacts with water to form the hard material that binds together sand and gravel into concrete. It’s made by burning limestone, clay, and a few other materials in giant kilns, and that’s where the environmental trouble lies.

Here’s a fun image from the first edition (1956) of Troxell and Davis’s Composition and Properties of Concrete, showing the various parts of cement production in pictorial form. The kiln is the long tube near the center. It’s set on an incline and rotates slowly, with the raw materials turning into cement clinker (nodules of cement that get ground into powder) through a series of reactions as they make their way down the slope.

Figure from Troxell and Davis

Several of these reactions, called calcination, produce carbon dioxide. The simplest is

CaCO3 →CaO+CO2

where heat drives off carbon dioxide from limestone (calcium carbonate) to produce quicklime (calcium oxide), which is used in later reactions to produce cement itself. It’s this chemistry that makes cement production such a profligate generator of CO2. Even if fossil fuels were eliminated from the heating, grinding, and mixing processes, there’d still be carbon dioxide coming out of cement plants.

The show discusses ways of reducing the CO2 produced by cement, mainly by reducing the amount of cement needed in concrete. Materials like fly ash and granulated blast furnace slag are used to replace some of the cement in concrete. And because these are the waste products of other industries—coal-fired power plants and steel mills, respectively—they have an additional environmental benefit.

Like many Radio 4 shows, Rare Earth is available as a podcast (that’s a link to its RSS feed). After listening to this episode, I subscribed.

Northern lights journalism

April 16, 2025 at 10:39 AM by Dr. Drang

I read a story on Apple News this morning, Northern lights may be viewable in some US states this week: Where and when to see it, and I want to complain about it. Unlike my previous two posts, this will be short.

The article is from USA Today, and you can read it on their site in your browser, which you may prefer, as that gives you the opportunity to skip the ads via Reader Mode. But my complaint isn’t about ads in Apple News, it’s about the article itself.

You might think from the article’s title that its main topic is where you’ll be able to see tonight’s northern lights. And it does get to that, but only after 400 words in a 600-word article, a frustrating inversion of the inverted pyramid. The 400 words of throat clearing explains what the northern lights are, what causes them, that they’re also called the aurora borealis, that there’s a similar phenomenon in the southern hemisphere called the aurora australis, and how NASA uses satellites to predict aurorae. These are all useful bits of information to people who don’t already know them, but they don’t belong before the list of states where you have a decent chance of seeing the northern lights.

(In some ways, of course, this is a complaint about ads. The inverted inverted pyramid structure is there to force you to scroll through the ads that Apple and USA Today want you to see.)

BBEdit says I’m now about 250 words into this post, so let me give you the list of states:

That’s the same order as in the article, and if you can figure out the rationale behind it, you’re doing better than I am. I thought at first that it put the northernmost states first, but that doesn’t explain Vermont and New Hampshire coming after Illinois. And you may be wondering how someone could lead off with North Dakota and include Nebraska but somehow omit South Dakota. 🤷‍

By the way, the article was written yesterday afternoon. As of 10:30 this morning (CDT), the NOAA Space Weather Prediction Center says the likelihood of seeing northern lights in the southernmost of these states isn’t very good.

NOAA aurora prediction for tonight

If you’re like me, the red-on-black text at the bottom of the image is hard to read. It says

View Line Indicates The Southern Extent of Where Aurora Might Be Seen on the Northern Horizon

So the southern extent is now just north of the Illinois/Wisconsin border. I’ll probably go out tonight to take a quick look anyway.

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