"We are more afraid of excellence than of failure." -Marianne Williamson, A Return To Love: Reflections on the Principles of A Course in Miracles
Question #88974 posted on 04/20/2017 12:03 a.m.
Q:

Dear Frère Rubik,

I was thinking about Board Question #85880 at 1:17 am on a Friday morning for some reason, the question from like a year ago about the heat escaping the house on Pluto. I know, I know, everyone else was too. Anyways, I thought of something that might impact the answer but don't know enough about physics to make a correction. Or even really know enough to know if this changes anything at all.

I have heard that air has a lot to do with how fast heat is transferred on earth because air is a fluid that allows heat to transfer by convection. I have read that in space, heat can only be lost via infrared radiation. (For example, apparently you could definitely stay conscious for like 10 seconds in space without a space suit!) according to this How much is Pluto like regular space? Assuming the walls of this "house" are infinitely strong and the atmosphere on the inside is the same density as Earth and the density of the atmosphere outside is the same density as Pluto, would that R value still make sense? Maybe this was some of the stuff you were talking about at the end of your original question that is a bit outside my understanding. If so, I apologize, and feel free to humiliate me (anonymously) in public.

If you have better things to do than answer this question that has clearly been on everybody's mind, I will be very angry and sad but I will understand even if I resent you forever.

-Sheebs

PS - Thanks for the trapdoor with a portal to Earth, that sounds like a totally groovy addition to a house that already has an amazingly vaulted ceiling

A:

Dearest Sheebs,

You're absolutely right! The atmosphere on Pluto is 100,000 times less dense than the atmosphere on Earth, and that makes a huge difference in heat transfer. There are three ways that energy can be exchanged between media: Conduction, Convection, and Radiation. Conduction happens when different substances are touching each other; convection happens when they're separated by a fluid (or when a substance is in contact with a fluid or when two fluids are in contact with each other), and radiation just kind of happens spontaneously: everything that's above absolute zero emits radiation. That's right, Sheebs: you're radioactive! I'm radioactive! Every member of Imagine Dragons is radioactive when they sing "Radioactive!" 

Now, in a vacuum, there is no fluid, so convection can't take place. If two objects in a vacuum are touching each other, they can still transfer energy via conduction (and, if they're both made out of the same stuff, like two iron bars, they'll stick together and become one object via a process called "cold welding," but that's another story), but if they're separated then the only way they can share energy is by radiating at each other. This is the idea behind those super-cool (ha!) insulated water bottles that I love. They don't have a perfect vacuum inside of them, but they do have a very, very low-pressure zone in between the inner surface and outer surface, which reduces how much heat can be gained/lost through convection, allowing the stuff inside to stay colder/hotter for longer.

Now that that's established, the next question we have to ask ourselves is this: how close is Pluto's atmosphere to a vacuum? Turns out, pretty close: Pluto's Wikipedia page lists the surface pressure of its atmosphere as 1 Pascal. For comparison, standard atmospheric pressure on earth is 101,325 Pascals. So the atmosphere on Pluto is 1/100,000th as dense as Earth's. A vacuum exists at 0 Pascals, so I'd say we're fairly justified in saying Pluto's atmosphere is like a vacuum.

Our house on Pluto can still lose heat to the ground via conduction, but I've chosen to ignore that for this particular model. We're interested in what the heat loss due to radiation looks like, and if we threw conduction in there it would quickly dominate the process and we wouldn't learn anything.

With all that in mind, I proposed this situation as a subject for a final problem in one of my computational physics classes, and it was approved! I then spent the next four weeks writing a MATLAB script that would make a model of our little Plutonian house and measure how it lost heat. In this model, our "house" is actually a 2D square; 3D was a bit too ambitious for the amount of time I had to work with (so it's more like a cross-section of our house, but it's still a pretty good approximation). The "house" is 5 meters x 5 meters, and it has walls that are 0.25 meters thick. The inside of the house is filled with normal earth air. My program divides the house into a 100 x 100 grid (with each grid square being 5 centimeters x 5 centimeters big) and records the temperature for each square as it goes.

To add some depth and breadth to my data, I wrote a feature in the program that let you choose what kind of material you wanted the walls of the house to be made of. Here are the options I went with:

1) Aluminum
2) Copper
3) Manganese
4) Diamond
5) Brick

I picked Aluminum and Copper because they seemed like pretty standard, average metals. I picked Manganese because it has the lowest thermal conductivity of any pure metal (thermal conductivity and volumetric heat capacity are used to determine a property called diffusivity, which is similar to the R-value we used in the previous answer). I picked Diamond because diamonds are a Sheebs' best friend (and also because diamond has a crazy high thermal conductivity). Finally, I chose brick because I wanted to have one substance that could stand in for an average house wall, and I was having a hard time finding numbers for the thermal conductivity of regular house walls.

Finally, so we could have a metric to measure all the values against, I decided to time how long it would take for the temperature in the room to fall from 300 Kelvin (about 80°F) to 280 Kelvin (about 45°F). I did this for three different spots in the room that seemed to change at different rates based on some early trial runs: the middle of the room, against one of the walls, and right in the corner where two walls meet. For the purposes of this answer, we'll focus mostly on the numbers from the middle of the room.

Alright, now that all the setup's out of the way, let's get into the fun stuff: the results. Before we do, though, take a second to look at the options and think about which one you think will work best and why.

All done? Then let's go!

Here are the times that it took for each of the houses to cool from 300K to 280K, arranged from quickest to longest:

1) Aluminum: 24.15 hours
2) Brick: 24.47 hours
3) Copper: 28.48 hours
4) Manganese: 29.64 hours 
5) Diamond: 33.11 hours

Right off the bat, we can see that the times here are waaaaay longer than I originally estimated a year ago; back then I was talking about minutes, whereas here we find that all of the materials tested take at least a day to cool down to the desired temperature. Pretty crazy, no? 

Now, the moment of truth: did your earlier pick end up being the longest-lasting material? If you're me, then definitely not. Usually, when something has a high diffusivity, we think that it will transfer heat really quickly, and this is definitely true for conduction. By that logic, the diamond house should have lost its heat the quickest, and brick the slowest, since they have the highest and lowest diffusivities, respectively. However, that's not what we see here: the diamond house lasted the longest of the five, and the brick house was almost the fastest in dumping its heat.

Another oddity: Whether or not the higher-diffusivity materials were faster or slower, I would have expected the results to line up according to the diffusivities of the materials. The order I expected would have looked something like 1) Brick 2) Manganese 3) Aluminum 4) Copper 5) Diamond or the reverse, since that's the order their diffusivities go in. However, as we see, it doesn't really follow that order at all; it goes 3-1-4-2-5. For the earlier regularity, I chalked it up to learning something new about space (that apparently high-diffusivity materials are really good insulators when all you have to worry about is radiating heat away). For this one, though, I have no idea what to think. Part of me wants to say "Well, maybe there's some other property of these materials that's making them act differently than you expect," but the problem with that argument is that the program only knows what I told it about these materials; it doesn't know that diamond is clear or brick is rough or that manganese can act as a neurotoxin if present in large amounts in the human body (though it does know all of their nicknames, which are secret).

The final surprising thing I found was this: After a certain amount of time, the rate of heat loss in the system goes down. The house never stops losing heat, but as time goes on, it loses less and less. Allow me to illustrate with some graphical representations (or, in layman's terms, look at these graphs):

amr.jpg

acr.jpg

(Sorry they're not labeled)

Both graphs are looking at the rate of heat loss as a function of time for the aluminum house. The top graph was taken from looking at a point in the middle of the room, and the bottom graph was taken from one of the corners. In the top graph, the rate of heat loss gets bigger and bigger (negative numbers on the graph = higher rate of loss), but toward the end the rate at which it increases starts to slow. In the bottom, we see it very quickly hit a maximum value, and then it gradually starts getting slower and slower. This is another phenomenon that I have no explanation for, but it's consistent across all the materials. 

As before, I'm not 100% confident in these models and findings. I'm a great deal more certain than I was before, especially since my professor helped me create this simulation, but there's still enough oddness going on in there that I can't say there's definitely nothing that I missed.

Finally, to wrap things up, here's a graph of the temperature distribution from the diamond graph once the middle has hit 280K. The x and y dimensions represent the dimensions of the house, whereas the z direction represents temperature: higher points imply higher temperature, as indicated by the scale on the right:

dmd.jpg

From the graph we can see that the temperature in the walls falls much more quickly than the temperature of the air inside, which is basically how insulation works. Pretty cool, eh? I also have a version of the program that makes an animation of the temperature falling. It's really cool to watch, but really tricky to export. If there are any MATLAB-savvy readers out there, they can download my script here (saved as a text file because I didn't know how nicely the Board system would play with a .m file).

That's all for now! Thanks for the question! It made for a really fun project.

-Frère Rubik