The Thermodynamics of Heating a House

Follow along as I explore trying to make my house a little more energy efficient, look at my energy usage history, and do a little thermodynamics. There will be some small equations, but I'll explain them in words as well. At the end, I discover something about the insulation efficiency of my house.

I live in a region which is heating-challenged. Every house or apartment I've lived in has had issues with heating in the winter and cooling in the summer. Builders in this area just don't seem to get that a little insulation goes a long way. For my current house, the homeowner's association also decided in its infinite wisdom that it would rip out all the old oil-fired boilers because they were too expensive, and replace them with electric baseboard heaters, because they are more economical. Whatever insane reasoning that led to that decision is now negated, especially since electricity has doubled in price over the past two years here. While it is true that electric heaters themselves are 100% efficient, the power plant and transmission lines are not. Furthermore, baseboard heaters tend to be mounted on outer walls below windows, so much of the heat can be conducted through the wall and escape the house.

I'm trying a few new strategies to try to make my house more comfortable, given its current limitations. First, I added transparent window films to almost all of the windows. The idea is that they hold an still pocket of air against the window, which adds an extra insulation factor. They also can contain small drafts so that cold air can't get in. It does take some work to install them, which basically involves stretching a huge sheet of saran wrap onto double stick tape mounted on each window frame, but eventually I developed a pretty efficient method (especially for smoothing the wrinkles).



A second thing I did was install curtains in the living room doorways, in order to keep the heat from escaping to colder parts of the house from the room I use most. These are cheap but heavy curtains I got on sale at Wal-Mart, hung from an expandable shower curtain rod across two doorways. Finally, I put some foam-board over my back door. It's a thin wooden door that conducts a lot of heat out.

I think these efforts have helped in a very qualitative sense. The living room is much less drafty, especially near the windows. Before installing the films, a cold down-draft from the windows would collide from an up-draft from the heaters to make chilly turbulent zone right where I was sitting. These drafts are gone now. The curtains also definitely help keep the heat where I appreciate it most.

Comfort is good, but I'd also like to know if this is saving energy and money.

PEPCO kindly puts my energy usage history on each bill, so it was a matter of collecting a few old bills and entering them in the computer. That's shown in black below (click for larger image).



The plot shows the number of kiloWatt-hours I use each month (ignore the red and blue curves for the moment). Unfortunately, I don't have data yet for December, the first month that I installed the window films or curtains, so I have to put the efficiency question on hold for now.

I decided to check out this plot a little more carefully. I use the greatest energy in the winter, obviously for heating. There are also small bumps in the summer, corresponding to cooling. Up until recently, I had a very old air conditioner which I rarely used, so my cooling expenses have never been large.

What to compare this with? Well, the there is a nifty number called a heating degree day used for heating calculations. Basically, any day that the mean temperature dips below 65 degrees Fahrenheit is considered a "heating day," and for that matter when the mean temperature is above 65 it is a "cooling day." The number of heating degree days is the number of degrees the mean temperature is below 65. The US National Climatic Data Center (not to be confused with the Climactic Data Center! Ooo la lah!) provides tabulated historical heating and cooling degree day data. The monthly total heating and cooling degree days are shown in the above plot (red=heating; blue=cooling; averaged over Maryland & Washington DC).

It's no big surprise that the heating and cooling curves match up with my energy usage pretty well. It's physics after all.

In fact, the Mr. Quantitative in me wants to do more. I decided to perform a linear regression between these quantities, with energy usage per day as the dependent variable, and heating/cooling degree days per day as the two independent variables. The simple function I tried was:

E = Constant + H(T_h) + C(T_c)

where H(T_h) is some function of heating degree days (per day), and C(T_c) is another function of cooling degree days (per day), both of which describe power usage versus temperature. This equation has the interpretation that I use some constant electric power all the time (for lights, water heater, etc.), plus the amount I use for heating and cooling, which depend on temperature.

The obvious choice is to make the two electric heating functions, H and C, proportional to temperature. However, I found that wasn't a good fit, as you will see below. Instead, there is an activation threshold. For small temperature excursions, no heating or cooling is required, and I don't use energy. This would be my comfort zone, the temperature range I'm willing to tolerate. I imagine I have a larger comfort zone than many people. As the outside temperature gets more extreme, then I use energy to maintain the inside house temperature within the comfort range. This function can be written as a constant when the heating/cooling temperature is within the comfort threshold, and a linear function outside of that. The linear coefficient of the function describes the number of kiloWatt-hours per day needed to heat (or cool) the house one extra degree Fahrenheit.

The fit works quite well, and here is how the results look. On the heating side, the function H(T) looks like this:

This means that I am willing to tolerate mean outside temperature drops of about 6.5 degrees (F) below the baseline temperature of 65 degrees before turning on the heat, and then I use about 0.84 kWh of energy per day for each degree (F) that it gets colder. At the current PEPCO price of 10.96 cents/per kWh, I pay an extra 10 cents per day for each degree colder that the outside temperature goes below about 59 degrees.

On the cooling side, the curve looks like this:

I'm apparently willing to tolerate large excursions before turning on the air conditioner (up to 10 degrees above the 65 degree baseline), and then I use 1.31 kWh of energy per day for each degree above that (for a cost of about 14 cents per day for each degree).

Finally, it's worth noting that I use 9.7 kWh of energy every day, no matter what the outside temperature is, just keeping the house going. I know for a fact that my refrigerator uses about 3.8 kWh every day on average, or about 40% of the total. It's a very old refrigerator from 1982 (!) which needs to be replaced. I used my handy Kill-a-Watt energy meter to measure this and other devices in the house. The refrigerator is by far the largest constant energy user.

Interestingly, last winter I changed from incandescent and halogen lamps to compact fluorescent bulbs. I predict this should save me between 1-2 kWh per day. A change such as this is barely detectable on the graphs, given the season and monthly fluctuations.

As one final exercise, I can estimate the overall efficiency my house, the effective "R-value". This quantity is defined as the reciprocal of the amount of heat lost per unit time per exposed area per degree temperature change, and has units of ft² per (BTU/hour/Fahrenheit). I already know the second quantity, since it's the linear heating coefficient I found above (0.845 kWh/day/F = 120 BTU/hour/F). The exposed area of my house is about 2000 ft², giving an effective R-value of 17. (NOTE 14 Jan: my original value of R-0.7 was had a unit conversion error and was incorrect).

An overall insulation efficiency of R-17 is okay but not great. As pointed out here, a house in my region (zone 2) demands an R value in the range of 18 (walls) to 49 (attic). However, as one of my commenters notes, there are other factors to consider, like how much air circulates through the building.

Remember that this data is all based on my house before I made the few changes above. Neither my usage data nor the climate data for the winter heating season are available yet. I hope to see improved efficiency!

Update (14 Jan): Oops! I made a unit error when converting from kWh/day to BTU/hr (missed a factor of 24). After the correction, the overall insulation efficiency of R-17 is more reasonable.

When do you throw in the towel?

Since 1999, I have been the proud owner of a Toyota Camry, which my parents graciously gave it to me. This Camry was born in 1992 and has had a string of owners, starting with as corporate fleet, then to some other wanker, then my parents, and now me. It startles me to think that this car pre-dates the beginning of The X-files!

Well, gradually over the years, this car has been becoming more difficult to maintain. I try to do the standard up-keep, but more and more things are starting to fail.

The most recent failure is the passenger side window motor. Unfortunately for the current season (winter), the window has failed in the most awkward state (open), which can make for uncomfortable driving (breezy). The driver-side window failed a few years ago, so it's likely that the other window motors will go bad at some stage. It also means that I can't keep anything valuable inside the car, and it certainly isn't waterproof when it rains, like it did last night. On the positive side, I don't have to preoccupy myself with locking up!

Surely, as this is a very old car, there are many components that are near their failure threshold. And, as any rider of my car knows, I also just live with some of its (er) shortcomings. It's just a matter of time before more things go bad. The question is, how long do I keep dumping money into it for repairs?

Over the past few years, I've become more conscious of this problem. I've justified it to myself on the basis of, this $X repair will keep me going for another year. Last summer I spent a considerable amount on some repairs, with a certain amount of resolve that it would be the last time. So here I am, less than a year later, and something new has come up.

My friends and colleagues all say, "Dude, just get a new car!" Good point, but I'm way too analytical to just do that.

Unlike my first car, a Nissan Sentra (also a gift from my parents! Thanks again!), I don't have as much of an emotional attachment to the Camry. So from an emotional standpoint, I'm probably willing to let go.

Economically, the amount of repair work required to get the car into a "good" condition is approximately equal to its re-sale value if it were in good condition. So from that perspective, its probably worthless to me.

From a timing standpoint, it's good in a way. I have some travel coming up, where I wouldn't need a car anyway. On the other hand, it would have been "nicer" for this to happen in December. I could have donated the car to charity, taken a tax break for 2007, and bought a car at the end of the calendar year when it might have been easier to get a good deal.

From a personal convenience standpoint, it's not so great. I'm the kind of person that takes forever to buy a can of shaving cream, much less deciding on ditching a car and buying a new one. I don't relish spending my time visiting car dealerships, or going through the hassle of buying a used car.

There is no magic conclusion here, but my gut is telling me it's time to trow in the towel.

No Longer A Furnishing Virgin

This is the first new piece of furniture that I've ever owned.

Red is not my favorite color, but I've decided that it's a color that goes well with my house, and the greens of the forest outside my window. It replaces an absolutely dreadful sofa, that was collapsing... literally.

A Whole Lotta Libido Goin' On

Would you run for president? I probably wouldn't, but if I did, it would probably be for reasons like saving the world, or some other altruistic B.S. like that. But not Lee L. Mercer, Jr.. No, he has much bigger fish to fry. Here is one of the many reasons he is running for president:

49. To Prove the United States Government killed my sex life, my wife sex life, my daughter-in -laws sex life both may sons and other of my family members sex life with Espionage Experimentation and Espionage Exploitation sex killing. (sic)

I'm glad that someone has the courage to use the presidential pulpit to air his dirty laundry about his dirty dancing. And his wife's. And his sons'. And their wifes'. (How he has all of this in-depth knowledge about his family's intimate lives... is not something I want to know).

I suspect Mr. Mercer's problems run deeper than his sex life, but it's clear he's running for almost purely personal reasons. Which, strangely, makes me think of our current president.

Thanks to Swing State Project for the link.

Restless Blackjack Syndrome

I can't imagine how difficult it must be, to be afflicted by the horrible condition known as Restless Leg Syndrome. Suffers have symptoms like, "an urge to move, usually due to uncomfortable sensations that occur primarily in the legs." When I was a kid, we called that being antsy. But from the nifty little article on Wikipedia I gather that at least some people have real problems with it.

Thankfully, Mirapex is ready to heal your restlessness (with a doctor's prescription of course). I experienced a Mirapex advertisement today, and one phrase blew my mind:

... if you experience increased gambling, sexual, or other intense urges, then see your physician...

Come again? Should we be calling it Viagrapex? That must be some potent stuff indeed.

I kind of thought of Restless Leg Syndrome as a great alibi when you kick your partner in bed. Sorry honey, I guess it's the RLS acting up again! (wink, wink) But it sounds like the side effects are even worse. After all, it's easy to blame an innocent kick on RLS, but it's quite a bit harder to explain how you got caught binging on Butterfingers at a Texas Hold'em poker orgy. I guess it was the Mirapex, sweety!

World's Highest Product Placement

Continuing on the theme of the Everest show, I started looking a little more closely at the climbers' jackets, and I started seeing things. Well actually, I started seeing corporate logos. In particular, the best climber, Mogens Jensen, has a bunch of "gsk" logos stuck to his jacket and his hat. Could that be the drug conglomerate Glaxo-Smith-Kline? Yep.

And if I wasn't mistaken I also saw logos for Motorola and (maybe?) ask.com.

Now I seriously doubt that Glaxo-Smith-Kline manufactures mountaineering parkas or hats, nor does Motorola.

Of course not. These climbers are advertising for corporate sponsors. What I don't know is whether they found their own personal sponsor, or if the producers provided a certain number of "product placement" emblems throughout the show. On the one hand, it's clear that "gsk" has been Mogens Jensen's personal patron for at least two years (he's an asthmatic, and I'm sure they have asthma medicines for sale). On the other hand, product placements are sure money-makers for reality shows. Other shows like Top Chef have so many product placments that it's hard to draw the line between entertainment and advertisement anymore.

I predicted that by the third season, we would be seeing aspiring young actors auditioning for the show. Now I wonder if we will instead be watching a climber drinking his Evian springwater while warming his feet on a Coleman Stove and watching CNN on his Motorola mini-TV...

Beyond the Limit

The Discovery Channel is airing a new season of its reality program, Everest, Beyond the Limit. I find it fascinating that the show is sponsored by Mastercard. The producers are already working on the new hit series, Your Credit, Beyond the Limit. I mean really, did Mastercard even look at the title before they sponsored it?

The program itself is quite fascinating. It's a reality-info-tainment documentary of climbers that pay a guide service for the privilege of climbing Mount Everest. I assume that the producers are paying for the same privilege for their camera and sound men to trail along and catch all of the grueling details. And to attach small cameras to the Sherpas that blaze trails and help climbers. The so-called SherpaCams™ can go places that normal mortal camera dudes can't.

The danger of death is very real, much more real than Deadliest Catch, where Alaskan crab fishermen lounge around at sea level and breathe luxurious 21% oxygen. Climbers have to deal with high altitudes, icy and windy conditions, and bad weather. The casualty rate is usually not measured by whether somebody died in the season, but how many died. And even those who don't die are susceptible to losing their fingers or toes (or nose!) due to frostbite. But this is not a huge surprise, to anybody who has read the book Into Thin Air by Jon Krakauer.

The really interesting thing is that the conditions are so extreme, and the location is so remote, the documentary crew are not really impartial observers any more. Last season, a climber David Sharp died right on the trail as other climbers hiked by. There is some controversy over whether Sharp was seen by the climbers early in the climb on the way up, or later on their way down. Still, the expedition leader David Brice made the decision to continue his expedition and provide no rescue assistance. Understanding that the camera crew, Sherpas, and Brice himself were being paid by Discovery channel and the other climbers, one can imagine that the urge to continue the expedition was quite strong. We have a TV show to make here. Interestingly, the portion of the video where Sherpas encountered Sharp was editted out of the show. It's not clear if there was a "right" decision that could have been made, but in that kind of situation the decisions of the filmmakers themselves could have life-or-death consequences. (Another example of this is the documentary Black Tar Heroin which followed several heroin users for a year in San Francisco.)

This is the second season of the Everest show. I can already see that the producers have "selected" the climbers for the greatest entertainment value. So far we have been introduced to the Biker Dude, the L.A. Reporter, the British Pub Mate, and the Asthmatic Athlete. I'm surprised that none of them is an aspiring actor trying to get onto Desperate Housewives. It's so hard to get onto reality television these days, don't you know?