The mysterious number 2.6

When performing the accumulated Cooling Degree Days analysis, which the mindful reader will recall incorporated the memory (a.k.a. hysteresis) of the system to try to get a better handle on the effectiveness of the improvements, we ran across an interesting, nay fascinating result.
It turned out that a pretty good linear model for the response of energy usage to varying cooling degree days (CDD's) took into account the temperature of the previous month as well as the temperature of the current month. The fascinating part of this was that, when determining how much energy my house was going to use, the temperature of the previous month mattered 2.6 times more than the current outside temperature.
This flies in the face of everything that I had learned about homes and insulation theory, at least before I started down the path of Energy Efficiency enlightenment. How is that? Well, like many of you out there, I had learned that the way to keep your buildings cool was to keep that hot air out, and keep a good thermal wall between the cool inside air and the hot outside air, in the form of things like double-pane windows, thick walls, and thick insulation. This would slow the flow of heat from the hot outside air into the interior airspace. And to be sure, my house is now well sealed and has decent windows. However, if that were the 100% correct approach, then why does my real world data show that some factor other than the current outside air temperature matters more? In other words, if the problem is really hot outside air, why doesn't the current outside air temperature dominate the model? If the traditional insulation theory is valid in my case, why does the temperature of air from last month that's not even around any more account for some 72% of the correlation, while the supposed main problem, hot air currently surrounding the house, accounts for only 28%?

Something is definitely amiss with our theory.

Ruminating on this problem leads us to an inescapable conclusion: in my area, during my cooling season, the main heat transfer mechanism into my house must not be direct convection from hot outside air. The traditional "double-pane window, lots of insulation" approach which attacks the convection problem is quite literally missing 72% of the target.
So how is most of the heat entering my house? Well, we have exactly one clue to help us find the culprit. Since we've seen that greatest determinant of the energy usage in June, for example, is the temperature in May, it must have to do with the heat being stored over time in someplace other than the outside air. Possible culprits:

1. Heat stored in the ground -- During the course of the typical dry summer here in central Texas, the ground becomes parched and unable to cool itself through the mechanism of evaporation, since there is no water to evaporate. Much of the vegetation also goes dormant to survive, not performing its usual transpiration which might also effect some cooling. The ground is in thermal contact with my house's foundation, and heat could certainly conduct into the concrete foundation, and once there, into the interior airspace via conduction or radiation.
   
2. Heat stored in all the thermal masses around my house -- Neighboring houses (all of which have nice heat-storing brick sides), sidewalks, and streets all store heat during the day, and radiate so much of it that it is noticeable even to the casual observer walking by after sunset. That radiated energy will be coming into my house from low angles and thus will unfortunately avoid my under-the-rafters radiant barrier.
   
3. Heat stored in the thermal mass of my own house -- The bricks on the outside walls certainly store a lot of heat, and that heat can conduct or radiate into the interior of the house. The attic itself consists of a lot of wood and some metal as well as insulation, all of which can and do heat up and store that heat over time, again conducting or radiating to become a problem.
   

As far as which of these is the most significant, it is difficult to say. I have not seen any indication from other web sources that heat conduction from the ground into the slab is a big problem; it may well be that down at the bottom of my slab, several feet underground, the ground is not particularly hot. (Some data on soil temperature variation with depth in my area would be nice, if anyone can point me to it). However, all that concrete would certainly store a LOT of heat. Unfortunately, now that the house is built, I don't have any good way to insulate between the foundation and the ground, or between the foundation and the interior space. I have heard of foundation insulation during construction, but all the examples I have seen so far are in much colder climates, attempting to reduce heat flow out of the house rather than into it. Furthermore, I have no low-energy means to cool the foundation itself.
To address #2 and #3, having a radiant barrier in all the walls and windows would go a long ways towards reducing the heat gain there. Even my own bricks could be separated via a radiant barrier from the interior space, keeping their significant heat gain from being a problem on the inside of the house. I would be interested to learn how much heat gain is radiating in through my windows, and how much through the walls. There are "low-e" (emissivity) coatings that can be added to existing windows, and I suspect those could be a significant help to me.
Unfortunately for #3 (the heat stored in the thermal mass of my own house) since my house is already built, it would be prohibitively difficult and expensive to rip open the walls and install foil barrier there.
So with this analysis we have identified 3 possible culprits for the main predictor of the heat load coming into the house, and only one of them can be partially addressed as a retrofit project: the radiation of heat in through my windows from nearby sources. The others needed to be addressed in construction, and it is now too late for that.
Well, now that the attic is pretty well taken care of, our analysis of that mysterious 2.6 factor seems to be pointing us in the direction of windows. And so let us begin to follow this new trail; let us examine and look into the efficiency of windows, and how to improve it, soon...

Take 2: Characterizing Hysteresis

In my last post, one of the caveats about my results in determining cooling effectiveness was that the real-world data shows hysteresis, or memory, which makes determining the effectiveness of cooling difficult if it the calculation is based only on the temperature outside.
I have attempted to take into account some of this memory by replotting the energy use (kWh) vs. Cooling Degree Days (CDD) data, but with the following change: the CDD numbers on the X axis are the sum of the current CDD value added to a factor times the previous month's CDD value. For for a given month on this plot,

Accumulated CDD = (Factor * previous CDD) + CDD for this month

I utilized the Standard Error function in Excel, STEYX(), to assist me in determining the value of "Factor" by trial and error. I chose a value for "Factor" that minimzed the total standard error of the function; i.e. the sum of the errors of Y as they can be predicted by X. In another intriguiging twist, this yielded a value of greater than 1; in fact, a value around 2.6 actually worked best.
What does this mean? This means that last month's weather matters a lot more (2.6 times more) than this month's weather in determining how much heat is going to entering my home. Another fascinating result, suggesting that once again, simple air convection or conduction (represented by this month's CDD) through the exterior of the house has a lot less to do with cooling load than radiation of the hot environment around me (represented by last month's CDD). Another argument for full-house radiant barriers, if anyone is listening! In fact, the low-emissivity ("low-e") coating on modern windows helps perform exactly that radiant barrier function, but on the sides of the house, somewhere that my attic-based radiant barrier doesn't cover, and something I hope to learn more about in the future. So far, replacing windows has been off of my list due to cost constraints (my entire radiant barrier cost a lot less than 1 window), but there are efficiencies to be gained there as well. But I digress...
I actually also tried this model with a second factor multiplied by the CDD from 2 months ago, but trial and error yielded almost no impact: the factor was less than 0.1. This indicates that the main factors are (in order of importance) last month's temperature, then this month's temperature.



You'll notice a few things right off the bat:

1. The "best fit" lines look a little high. The reason: I made them fit starting at 500 accumulated CDD; I considered everything to the left of that as noise. So the slopes should match the hottest (right hand) part of the curve well, even if their level looks high. The slope is what I'm after; we're going to ignore the intercept (height) of the lines.
2. If you compare to the previous 2005 graph, you'll see that the effect of using accumulated CDDs has been to turn the large loop of 2005 data into a double loop, with a crossing in the middle. Intuitively that makes sense: that minimizes the error between a line drawn through the middle of the loop and the loop itself. The standard error for the raw 2005 CDD data was 564; the standard error for accumulated CDD data is 473, a significant improvement, meaning that a line fits this data better (although clearly still not great!)
   
3. If you compare to the previous 2009 graph, you'll see that the effect of using accumulated CDDs has been to turn the small loop of 2009 data into almost an exact line. The fact that it's a nice fit for the line means that the house's response to temperature has become much more linear; the "loopy" nonlinear part only appears far to the right in the hottest areas. The standard error for the raw CDD data for 2009 was 141; for the accumulated data, it drops to 95. Looking at the graph, you can see that the best-fit line matches quite well.

So now that we've got a better model, particularly for the 2009 data, how do the slopes compare? We've got a slope of:
2005: 0.99 kWh / accumulated CDD
2009: 0.27 kWh / accumulated CDD

Taking reciprocals to convert to effectiveness:
2005: 1.01 accumulated CDD cooled per kWh spent
2009: 3.73 accumulated CDD cooled per kWh spent
Effectiveness ratio: 3.7 to 1
Conclusion: when taking the hysteresis of the system into account, our home improvements look even better. Rather than a 70% improvement in effectiveness based on the immediate CDD method, I may well be looking at a 73% improvement based on a more accurate model.
While not a massive change, this analysis makes me feel a bit better in that I've now accounted for the rather obvious hysteresis in the data, and come out with pretty similar numbers.
There remains the lingering question of the nonlinearity all through the 2005 data and at the very far right of the 2009 data. It would appear that when things get hotter than the house can handle (which seems to happen immediately in 2005, but not until about x=2000 accumulated CDDs in 2009), energy use gets bumped up nonlinearly. There are probably good physics reasons for this nonlinearity, but I will leave it to my readers to write in a let me know what they might be and how I might model them - or even better, prevent them!
In the meantime, this diversion into analysis was fun, but we still have a little bit more story to tell about the home energy efficiency projects completed in 2009. Stay tuned!

Analysis: Effectiveness of Ventilation + Barrier

I will continue the analysis of the cooling effectiveness of my home, skipping forward to 2009. The improvements in 2007 (related to ventilation) have been enhanced by the addition of a foil radiant barrier and a 14 SEER air conditioner (to replace the failed 11 SEER unit).
Here is a graph of energy usage vs. Cooling Degree Days (CDD's, see previous posts for explanation) for 2005 and 2009. Since 2009 isn't over yet, I have substituted 2008 data for both November and December, but neither of those months is real significant for cooling costs.

1. Again, as in the previous post, the hysteresis of the system is quite apparent.
   
2. As expected from personal experience, the 2009 loop extends almost 100 CDDs further to the right than the 2005 loop, indicating the record breaking HOT summer here.
3. The curve for 2009, while showing some positive slope, looks almost unfazed by the increasing heat.

So at a glance, it appears that adding the radiant barrier and more efficient air conditioner to the already improved 2007 system has helped. But how much? Let's perform our linear best fit again and look at the results:


Including the numbers from our last post covering 2007, here are the final results of this analysis method:
2005 Slope: 2.8 kWh/CDD
2007 Slope: 1.1 kWh/CDD
2009 Slope: 0.84 kWh / CDD
So, by 2009, the effectiveness of my cooling system has increased again. Stated as "kWh of energy to cool 1 CDD", the effectiveness has gone up from:
(1/2.8)= 0.35 CDD per kWh expended to
(1/0.84)=1.19 CDD per kWh expended.
Finally: something meaningful to compare. Stated as simply as I can, based on linear best-fit analysis to remove weather dependencies:

The energy cost of cooling my home 1 degree has dropped 70% due to my efforts.