## Scaling Problem: House Size and Heating Bills

December 25th, 2014 by PotatoThere was an article in the Globe & Mail a while ago claiming that it’s best to go with a smaller house because the bigger the house, the bigger the associated bills. Ok, that makes perfect sense.

But then it went on to claim that “it would seem reasonable to assume that it would cost twice as much to heat (or air condition) a 3,200 square foot home than it would one that is 1,600 square feet. But, as reasonable as this seems, it’s incorrect; it actually costs *more* than twice as much. […] Circumstances vary, but it can cost up to three times as much or more to heat and cool a home that is only twice as big.”

Now that just doesn’t make physical sense to me. We all know how scaling laws work: assume you have a spherical house, then the surface area will scale by r^2, while the volume will scale by r^3.

Ok, we don’t live in spherical houses, but still, this guy’s math must be way off. So I thought about it, and scaling with houses is actually a problem without any clear answer. Let’s set aside the complications like your own body heat or the waste heat of your home server farm (everyone has that, right?) and just talk about heat loss through the outside walls: even narrowed down with all that *ceteris paribus* it’s still a tricky question because houses are not spherical.

The simplest case I can think of is to take a cubical house. It has 6 unit surfaces: the roof, floor, and 4 walls. Now if you make that house twice as big by adding a second storey, the roof and ground floor are the same, and you’ve doubled the size of your walls (8 unit-walls). So doubling your floor space was less than doubling in your heat transfer area: only 1.67 times as much.

There are other ways to double the size of a house. You could go longer: expanding your floor plan from a unit square to a 2×1 rectangle. You only save on one shared wall between the unit squares in that case, so you do nearly double the outside area: 6 unit walls facing the outside, 2 floors, 2 roofs… but that’s again a 1.67 times increase (though more roof and floor with fewer walls added). Oh yeah, that’s just the first case turned sideways.

If you want to go crazy with shapes you could try find a way to get really inefficient. If you built a really long house (or made a C-shaped house to fit it on the lot — same difference for walls) that was 5 times as big as our unit square house, then it would be 3.67 times as costly to heat… wait that’s still going in the way I thought it would, with bigger houses being more costly, but scaling less than the increase in space.

In fact, the only way the author’s math works out is if you do non-apples-to-apples comparisons, like one house at 1,600 sq.ft. with 8’ ceilings and one at 3,200 sq.ft. with 16’ ceilings to drive the volume up but not the livable space measured in square feet. Or maybe it comes down to one of the complications I ignored, like floors and walls being roughly equivalent in terms of heat loss… but I doubt it.

He does mention more windows and doors just after the part I quoted, but again that doesn’t make sense to me. Yes, I lose more heat through my door than through a solid wall, but my house has two doors. A slightly bigger house would still have two doors. My parents’ house, which is maybe 2.5-3 times the size of our house, does have four doors, and my friend’s parents’ house, which is in-between, has three. But again, the number of doors are not scaling up *faster *than the increase in the size of the house. And the portion of the walls that are windows is not really any different with the bigger house.

So I will conclude for now that yes, a larger house *will *cost more to heat and cool, but it’s likely to scale *less* than the difference in size, because math. Fortunately, the massive building boom of recent times means that somewhere out there are a few developments with good test houses, ones built with the same insulation and materials and styles, but to different sizes. If anyone has some experimental data to back up (or refute) the spherical house reasoning, I’d love to hear it.