Beet the Traffic

We watch the local news every morning, and this week a story caught my interest: Alternatives to using road salt to avoid harmful risks. The key point of the story is that runoff from salting the roads can damage the surrounding environment, but adding beet juice to the solution can make it stick to the roads better. Reading up on the idea, I've found it's still debated whether this idea is really better for the environment, but the bit I was curious about is the ability to stay on the road.

Looking into ways I could model this, I found a paper discussing how droplets spread over time based on their surface tension. Their model was a bit more involved than I wanted to get, so I made some simplifications: The droplet takes the form of an ellipsoid with constant volume and circular base. This constrains the relationship between the height and the radius. The paper defines a value they call h* where gravity and surface tension balance. After my simplifications, it takes the form

where ρ is the density of the fluid, g is the acceleration due to gravity, and γ is the surface tension. For a given fluid, we can look up the surface tension and density. I decided to try a salt-water solution (γ, ρ), a sugar-water solution (γ, ρ), and molasses (γ, ρ).

The paper gives a t^1/5 form for the height of the drop, so we can start each of these fluids as a hemisphere and see how they spread as they approach their respective h* values:

This shows the sugar and salt spreading at roughly the same rate, contrary to the idea given in the report, so I expect my model is not capturing all their qualities. What I find really interesting though is that the thicker molasses actually spreads faster, because it's more dense that the other two, so gravity exerts a stronger force. Naturally, this brought to mind a bit of history from my home state, when a flood of molasses from a ruptured tank cut a swath of destruction through Boston!

Tactical Sailing

Marika is an avid sailor, and former Fleet Captain of the University of Michigan Sailing Club. In hearing about her exploits on the high seas (or Huron River, as the case may be), I've often thought about the mechanics of sailing. Specifically, I was curious if I could model the forces involved in maneuvers like tacking and jibing.

The model I came up with was to consider two connected forces on the boat: wind hitting the sail, and drag from the water. When wind hits the sail, the air bounces off, imparting momentum in the direction of the surface of the sail. The magnitude of the momentum transferred will be proportional to the dot product of the sail direction and the wind direction – maximum is when the wind hits the full face and zero when it's perpendicular. The drag from the water will be similar to the wind hitting the sail, but this time we're looking at the shape of the hull for each bit of water that hits the boat. Both these forces depend on and change the velocity of the boat, so we iterate the calculation until we find the equilibrium. To do that, we need to come up with a model for the hull of the boat.

I couldn't find any existing mathematical models for hull shape, but looking at common designs, I came up with this concept: Take two halves of an ellipse with semi-axes a and b, and put them a distance W apart. As long as W is less than 2a, the halves will intersect at each end. Now measure L from one end, and cut the remainder off. Using numbers for the width and length from here, and picking a and b that seemed reasonable, here's what I came up with:

Now to the simulation: We consider a wind going from left to right and try different orientations of the boat and sail. For each case, we start with zero velocity and calculate forces from wind and drag. Those forces give a new velocity, and we repeat until there's little change. The final velocity is shown by the arrow. Now, I didn't want to spend a lot of time figuring out the exact momentum transfer for the wind and drag, so I fudged the scaling to get reasonable results. Even with this super simple model though, we can see cases where it's possible to sail into the wind! (Look for the arrow pointing left.)

As with all of my posts, I don't recommend trying this yourself based on my calculations, but I'm hoping Marika will give me some experimental experience in the future!

On the rEvolution of Doorways

Dipping once again this week into my list of topics, I chose a particularly old one: revolving doors. Ever since my stay at Mass. General Hospital, I've been curious how efficient revolving doors are at keeping heat in or out, compared with sliding doors. Last time, aside from the disadvantage of chemo effects, I attempted to do a detailed simulation of the motion of particles through the doorway, which never panned out. This time, I took a much simpler approach using an approximation for the rate of heat flow between two reservoirs:

where Q is the heat energy transferred in a time Δt through a surface of area A. The temperature difference ΔT between the two reservoirs is spread over a distance Δx, and the thermal conductivity k is a property of the air, which we can look up. The idea with a revolving door is that the inside and outside are never in direct contact: The air moves from outside, to a segment of the door, then to the inside. I wasn't sure how to get the distance, since it will change with time as the air moves, but I went with a guess of 20 cm. For the dimensions of the door, I found an architecture page that gave some example measurements. I went for a 3-segment door. For each person who enters, the door will turn 120°, so we can try a few different rates.

Along with the heat transferred, we need to know the current temperature in each section of the door. That will be a simple scale factor, the heat capacity. Since the change in energy/temperature is proportional to the current energy/temperature, we'll get an exponential relationship in time, where the door section will approach the in/outdoor temperature, but never quite make it.

I put together a simulation with some values more suited to my current Florida environment: 20°C (68°F) inside and 35°C (95°F) outside. First, we can look at the temperature in each section of the door as it rotates with 1 person every 30 seconds:

I was surprised how consistent the temperature stayed – In total it's only about half a degree C in variation. To get a visualization of what was going on, I made an animation from the same run:

I find it really interesting that the oscillations are consistent enough that the door segments return to their original uniform temperature about every 1.5 minutes. I did not expect such a clean result.

Turning to the comparison with the sliding door, we can consider having direct contact between the inside and outside temperatures for the same time it takes for a person to go through the revolving door. We can add up the energy transfer over time for different rates of entry for both the sliding and revolving door. Below I've plotted the results on a log-log scale:

According to this model, the revolving door is only more efficient if more than 120 people per hour are going in/out. However, I suspect this could change drastically with the choice of the temperature spreading distance I mentioned earlier. A more reliable result may be that all the revolving door cases lose nearly the same amount of energy. I've often scoffed at airports that have motorized continuously rotating doors, on the belief that they were wasting power pumping air between the inside and outside, but perhaps I owe those architects an apology!

You Can't Take the Sky from Me

[Yes, I recently rewatched Firefly.]

With social distancing on my mind this week, I started thinking about taking it to an extreme: interstellar travel. Specifically, I wanted to try some calculations with the Tsiolkovsky rocket equation, which applies to any space ship that moves by using conventional rockets.

The idea is this: To get moving, you need to push off of something. On the ground, you can push off the ground or the air, but in space your options are more limited. Rockets accelerate fuel out the back, which pushes the payload forward:

via Wikipedia

The problem is that the more mass you carry, the more fuel you need, but more fuel means more mass. If you work out the particulars, the equation you (or rather, Tsiolkovsky) get is:

This says that the more change in velocity you want, the more fuel you need, but there are diminishing returns.

When my parents were shopping for a camper van, they described their ideal as a "Russian space capsule", so let's take them at their word and imagine attaching rockets to their 2000 Chinook Premier. The manual gives the loaded weight as 10,700 lbs, or 4,853 kg. This is m in the equation above. For the exhaust velocity, ve, we can use the numbers for the (now retired) space shuttle: 4,447 m/s. If our target is the edge of the Solar System, we can relate the travel time to the fuel required:

The dashed line shows the mass of the camper, so to get there in under ~2 months, we need more fuel than cargo.

Of course, there's not much to do at the edge of the Solar System, so how long does it take to get to our nearest neighbor, Proxima Centauri, 4.2 light years away? If we want to get there within a reasonable time, we need to go pretty fast, so we'll use the Special Relativity version of the rocket equation:

Relativity says that as you get closer to light speed, it takes more energy to go any faster. Using the same exhaust velocity from before, the curve looks like this:

For reference, the mass of the Earth is around 6 x 10^24 kg, so burning the entire planet could get our camper there in under 10,000 years!

This is the reason other propulsion techniques have been proposed, like the LightSail I mentioned a couple weeks ago, or Project Orion, which involved accelerating the spacecraft by exploding nuclear bombs behind it. I think I prefer to associate my name with the new shuttle replacement, rather than something that leaves a trail of fallout...

Cents and Cents-Ability

A late post this week, since I'm currently traveling in Italy with my parents! Going between EU countries and using cash more often than usual has reminded me of how inept I feel making change in euros after 30 years of using the US denominations. I thought I'd take a look at the relationship between the different coin/bill values and the ways to get a certain amount of money.

Here are the US denominations less than $5 (not including $1):

Wikipedia

And the EU denominations less than 5€:

Wikipedia

We're looking for ways to get a total value using some number of each of these coins. We can write this as an equation, for example

where p, n, d, q, and D are the numbers of pennies, nickels, dimes, quarters, and dollars. Since we no longer cut coins to make change, all these values must be integers, which makes this a Diophantine equation. These can be difficult to solve; the Python package SymPy has tools for it, but I couldn't find a way to restrict it to only positive integers (handling anti-pennies is far too dangerous). I was able to make my own though, which uses a simple brute-force technique.

Since it slows down exponentially as the total value gets larger, I was only able to run it for values up to 4 dollars/euros. The results are still interesting though. Here's the minimum number of coins needed to make a total:

As each larger denomination becomes available, there's a sudden drop in the number of coins needed. Since euros include many more denominations, they're able to use fewer.

We won't always get the minimum in change though, so it's useful to look at the different ways to make a value, and find out what the average number of coins is:

These are going to include many sets with mostly pennies though, so we can also look at the median:

At the very beginning of each of these, you can see there are a few cases where the US uses more coins, probably due to the euro having a 2 cent coin, but in the end, more denominations means more coins on average.

There are many things I'll need to unlearn when I return to the US next month (like saying "merci" instead of "thanks"), but since I never really got the hang of euros I suppose making change won't be one of them.

Lean on Me

For some reason, my bus the past week has been especially full, forcing me to stand for most of my commute. I'm always surprised by how difficult it is to keep my balance going around curves, so I thought I'd try putting some hard numbers to the issue.

I've talked about the strange, non-inertial effects of being in a turning vehicle before, but this situation is much simpler. When the bus goes around a turn, it drags my feet along with a centripetal acceleration. Balance is all about keeping the total forces on you over your feet:

Here, ar is the centripetal acceleration of the bus, and g is the acceleration from gravity. Using geometry,

When something moves in a circle with radius r and velocity v, the centripetal acceleration is

Putting these together and solving for v gives

The turning radius for a standard city bus is about 21.5 feet, and the maximum angle a person can lean is around 15°. Putting these together gives a speed of just 9 mph! As a non-driver, I don't really have a feel for typical turning speeds, but the National Association of City Transportation Officials suggest a maximum of 15 mph, so our limit is well below this. Of course, when you're riding a bus you can lean into the curve, and hold on to the railings, neither of which we've accounted for here. Still, I feel a little justified in lurching around awkwardly during my commute.

Zoned Out

This past week, I flew back to the US with Marika and Lorna to visit family. I'm back in France now, but the jet lag I've been suffering from has put timezones on my mind. The idea behind timezones is pretty simple: We use time (meaning the value read on a clock, not the duration between two events) to measure the progress of a day, which is connected to the rotation of the Earth. Every point on the Earth has a different angle with relation to the Sun, so in the most precise sense, every point has a different time. That would make things difficult to keep track of though, so instead we divide the planet into zones, usually every hour.

It would be inconvenient to simply draw straight lines though, since that might put a small corner of one state or country in a different zone from the rest, so instead we define the zones to follow borders:

Wikipedia

Some of these are pretty misshapen, and it made me wonder about the extremes in the difference between position on the planet, and designated local time. I found there are a number of handy Python packages for dealing with timezones, so I put together a script to get some statistics.

The main thing I wondered was in which timezone was it possible to get the farthest from the "true" time, determined by the rotation of the Earth. This is equivalent to the zone that covers the greatest longitude. When I tried to get that with my script, it gave me Antarctica, which has a timezone covering the full 360°! That's not really an answer to my question though, so I had it exclude points in the Arctic/Antarctic circles, where there can be 24 hour days/nights. With that done, I found the widest timezone was UTC+1, covering Ireland, parts of Africa, and importantly, Iceland, which gives it the extra width to beat China, at least measuring by angle. The northern latitude of Iceland means there's less distance in every degree of longitude. This timezone is nearly 45°, or 3 hours wide.

The narrowest zone, on the other hand, is UTC+6.5, covering the Cocos Islands. If you zoom in on the map above, you can see it just above the label for the Indian Ocean. That zone spans just half a degree, or about 2 minutes. It made me curious though about timezones that don't fall on an hour. The full list is America/St_Johns, Asia/Colombo, Asia/Kabul, Asia/Kathmandu, Australia/Broken_Hill, Australia/Eucla, Australia/Lord_Howe, Indian/Cocos, and Pacific/Chatham. I was surprised not to see North Korea on this list, since I thought I remembered they had a 30-minute offset, but it turns out they just got rid of that last year.

I'm still a bit jet-lagged, so I'll stop here, but if you take a look at the code you can see an attempt to find where you can get the biggest time change in the shortest distance (and maybe fix it yourself).

I Case My Rest

Since moving to France, I've noticed many small cultural differences. Among the more unexpected is the shape of pillows here – They're square, rather than rectangles. It might seem like this is irrelevant to getting a good night's sleep, but it actually changes the way the pillow inflates, and initially I found it incredibly uncomfortable. I was curious if I could find a way to model how the different dimensions lead to different pillow shapes.

It turns out this is a well-studied problem, usually called the paper bag or teabag problem. Unfortunately, it's a problem with no simple solution. There are approximations of the total volume contained, but I was more interested in the shape of the pillow. That can be modeled with a computer program, but I failed in my attempts to get that one to work.

However, the original post on that page suggested a different way to make the comparison – Inflate some paper bags myself! The French pillowcases are 65 x 65 cm, and according to this site, American ones are 20 x 26 in or about 51 x 66 cm. I cut out 1/5 scale replicas of the two sizes, and blew them up as best I could:

The area for the French pillows is only about 25% larger, but this leads to much thicker pillows! It seems after a few months here, my spine has reshaped itself, and I'm sleeping much better now. Still a funny bit of culture shock though.

Zap!

Happy Diagnosis Day! It's been 8 years since that fateful day when doctors discovered a 3 cm mass perched on my pineal gland. In that time, I graduated from college, went off to graduate school, married an amazing woman, got my PhD, and moved to France for a post-doc! In honor of the occasion, today I wanted to discuss radiation.

Part of my treatment included proton radiation, which I've talked about before (long before). While that exposure was medicinal, it's still best to limit overall radiation exposure to skin to about 50 millisievert (mSv) per year. For a couple years after my treatment, I avoided the TSA body scanners, opting for the manual screening. However, during a recent visit (to Michigan, not France) my friend Kevin wondered, How much radiation exposure do you get from flying in the upper atmosphere, compared to the security scans?

It turns out the TSA has actually used two different types of scanner in airports: x-ray backscatter, and millimeter wave. Up until 2012, the TSA mainly used x-ray backscatter machines, but switched to millimeter wave after manufacturer disputes. The EU actually banned x-ray backscatter in 2011 due to health risks.

Normal x-ray machines work by measuring the fraction of x-rays that pass through a material. Denser substances, like bone, absorb more of the x-rays and appear as white areas on the resulting photographic negative. Backscatter machines instead measure the rays that are reflected from an object, giving an image of the outer surfaces. The trouble with this is, x-rays are a form of ionizing radiation:

By Spazturtle - Own work, CC BY-SA 4.0, Link

Ionizing radiation means that it carries enough energy to damage cells, potentially resulting in cancer. This is why (in the EU anyway) these machines are no longer in use.

The alternative full-body scanning device is the millimeter-wave scanner, which use waves with a frequency of

On the chart above, this is just below the visible range, well into the non-ionizing region. Like the x-ray backscatter, these machines form an image by bouncing the waves off your body, and interpreting the reflected signal.

So, on to the exposure issue: The point Kevin brought up is that our sun puts out damaging ultraviolet light, and there are even more powerful cosmic rays coming from elsewhere in the universe. Typically, the Earth's atmosphere protects us from those sources, but by going high in the air, we strip away some of those protections.

The American Association of Physicists in Medicine released a report in 2013 comparing the exposure levels for a person of my size (5'10" and 160 lbs) on the ground, in the air, and in a scanner. The way they present their results though is a bit confusing (2.84 hour flight?). Instead, let's do some unit conversions to find how long, on the ground and in the air, it takes to equal the 11.1 nanosieverts they measured from an x-ray backscatter machine (which, remember, is no longer in use). On the ground, a person my size gets about 3.11 millisieverts per year, which means it takes 113 seconds to equal one scan. In the air, it's even less than that, 12.1 seconds!

There are lots of other issues with airport scanning machines, including privacy, and other possible health risks associated with radiation exposure. From the perspective of standard exposure limits though, you're a lot better off staying in the machine than going in the plane!

Dimensional Dilemma

Just a short post, since I got an apartment earlier this week, and there's still lots of set up to do!

While planning for this move to France, I had to figure out how to fit everything I want to have the next 2 years into airline-approved bags, and I found their size limit for checked luggage interesting. I would have assumed the limit would be volume:

where x, y, and z are the length, width, and height. The trouble with this though, is that it would allow narrow objects with enormous length, like a long pole (and you don't want to get a physicist started on fitting poles into things). Instead, airlines place a limit on the sum of the dimensions:

What's interesting is that this also places a limit on the volume of the object. We get the maximum volume when x = y = z, so

It seems like we're getting extra information here: With only one expression we can limit both the volume and the maximum size. It's not entirely clear to me where this information comes from, but my guess is that it's due to the fact that the linear limit defines a coordinate system, in this case Cartesian. We could imagine a different limit, where items were required to fit inside a sphere of radius r. This would also limit the volume:

Depending on how you choose the linear measurement, you can get different maximum volumes. Makes me wonder whether airlines initially used the volume limit, only to be inundated with pole-vaulters...