Dear 100 Hour Board,
If you had a well insulated house and you put it on Pluto how long would it stay warm?
-My Name Here
Dear Martian New Housing (the finest real estate in the Milky Way!),
It depends on how you define "well insulated." Also "warm," since my Californian friends start shivering at 60 degrees Fahrenheit. Such is the life of a scientist, always having to qualify things.
Do you know what else is the life of a scientist? Making assumptions. For example, here I'm going to assume that your "house" is actually a 10m x 10m x 10m box that we've somehow just dropped onto the surface of Pluto (that's a pretty big house, if you think about it). It has no doors nor windows, but it does have a portal back to Earth in the floor (said portal has nothing to do with our calculations here, but I just felt bad about leaving you in a box on Pluto with no way to get home).
To calculate the rate of heat loss, we'll use the equation found here on Hyperphysics. However, while the basic equation is sound, the units are not; English units are silly and nonsensical and not scientific at all. I went ahead and did the conversions; we just have to multiply our R-value by 0.316835 in order to get our units to meters-squared per Joules per seconds. We also have to change our temperatures to Kelvin, but we also have to figure out what our temperatures are. Space.com says that at Pluto's coldest, it's around 40 Kelvin, and at its hottest it's about 50 Kelvin, so we'll take an average and say 45 Kelvin (that's -378.67 degrees Fahrenheit; think about that the next time you walk up to campus on a cold day).
How warm is your house? Well, room temperature is about 73 degrees Fahrenheit, or 295 Kelvin (if you're following along at home by doing the conversions on your own and see any errors in my math, know that I'm rounding very generously and that you should probably find better ways to spend your time). So, the temperature difference is around 250 Kelvin.
Lastly, we need an R value. According to Hyperphysics, the recommended R-value is 19, so we'll just multiply that by our calculated value and get a metric R-value of 6.019865.
So, then, our heat-loss rate of Joules per second is given by (500m^2)*(250 K)/(6.019865) = 20,764.585 Joules per second.
That number is fine and dandy (it wears a monocle and greeted me with a "What ho, good chap?"), but what does it mean? (Especially with the "What ho" part. I'm not sure I like what he's insinuating...)
Well, what it means is that 20,764.585 Joules of energy are leaving your house every second. To put it in perspective, a change in temperature from room temperature to 60 degrees Fahrenheit represents a drop in energy of 581.408 Joules. On Pluto, that drop would take 0.028 seconds. Let's say you're the sturdy, alpine type that doesn't consider it to be "cold" until the temperature hits freezing. I have good news for you: it will take a full 0.088 seconds until you start feeling a bit chilly.
Now, here's where our subjectivity comes in. We've defined "well-insulated" to be well-insulated for Earth, where temperatures are roughly six times as high as they are on Pluto. We could up the R-value by a factor of six; that would drop the rate down to 3460.764 J/s, and it would take roughly ten times as long to drop from room temperature to 60 degrees Fahrenheit (0.167 seconds). That's still pretty fast, and it's mainly due to the fact that the temperature difference is still 250 Kelvin. Put in perspective, the difference in temperature from where water freezes on Earth to absolute zero (the coldest any thing can possibly be anywhere in the universe) is 273.15 Kelvin. Since temperature differences on Earth usually fall within 100 K of each other, I'd say we'd be justified in beefing up the insulation of our walls. If we want it to take 10 minutes for the temperature to fall to 60 degrees Fahrenheit (enough time to grab a warm coat, decide this was a terrible idea, activate the portal to Earth and jump through), we'd need our walls to have an R-value of about 128997.1931.
Now, obviously this is a very rough estimate and there's a lot of other things to consider. For one thing, our Pluto house is just an ordinary box with no other features. Windows would be a big factor, as they let heat out a lot more quickly than walls. Also, the atmosphere on Pluto is about 1/100,000th the density of Earth's, so whatever walls (or windows) we construct there are going to have to be pretty darn strong if we want to be able to breathe in our little home-way-far-away-from-home. If I had the time, I could maybe come up with estimates for these numbers. But, in (more or less) the words of the great Kimberly "Sweet Brown" Wilkins,
P.S. All of this math was done in the middle of midterms with the help of sketchy websites. While I've already asserted that these estimations are rough, they could also be completely off base, for all I know. If an enterprising reader with gumption and know-how (and enough time) would like to improve on my work, I wholeheartedly invite them to with a correction below.