Summer 2009: Vaulted Ceilings Saga continues

So the weekend before the roofing crew was to show up, I needed to create roughly 16 pieces of 4'x8' decking, covered with radiant barrier foil and properly spaces baffles, to be ready for the job the following week.
As luck would have it, illness struck that weekend, but even in those early days, I knew that Energy Efficiency would wait for no man(!), and I toughed out the job of moving those rather awkward pieces of decking around and working on them. The job was made easier by the fact that the radiant barrier was already 4' wide, making it a matter of a single measure and cut operation to get the barrier on each piece. To cover 8' of length with baffles required 4 baffles per decking piece, but again, that was fairly simple and required mostly eyeballing with a little measuring.
Finally, it was the next week, and the crew arrived. I couldn't sit inside and just hope that the new decking was installed properly; I had to go out and see it. Although they wouldn't let me up on the roof, I was able to assist with a few things like cutting decking for odd-shaped areas near the corners of the roof, and creating 2 or 3 more pieces of radiant barrier decking since the original estimate had been a little low.

Preparing to staple baffles to a new piece of decking

The roofing crew was quite competent, and despite the July heat they completed the job in a single day. My hat is off to them for their professionalism in dealing with a job that had a few more components to it than usual. Here is a shot of one of the last pieces going on to the back roof. The baffles and radiant barrier are facing down, of course, but they are there. You might be able to tell that we had to go two 4' widths in from the outside wall to make certain we had enough length to cover the vaulted ceiling portions, as well as to get above the level of insulation in the attic so that the open end of the baffles inside the attic would be clear.

Later that night, it was clear that the new barrier had helped, although the change was not as dramatic as adding the initial barrier; after all, we were adding roughly 600 square feet to an already existing 1500 or so square feet. Significant, but not as much as going from 0 to 1500 square feet the first time.
This improvement yielded us the following benefits:

1. 600 square feet of radiant barrier
2. Far better barrier coverage over the master bedroom (from 70% to 100%)
   
3. Significant improvements to the intake ventilation
   
4. 600 square feet of new decking to replace worn (admittedly still functional) decking
   

The overall cost was around $700, $540 for the plywood decking and around $150 in materials (baffles and foil barrier) making the roughly $1.20/square foot cost of this improvement the most expensive per-square-foot change that I'd made. Again, I balked at doing it initially due to the cost, but I thought that I would regret it over the next 15? years until we got our next new roof if I didn't do it. The improved airflow and ventilation coverage seemed worth it.
The benefits from this job are already pretty well accounted for in the analysis of the 2009 energy data here since this change occurred in July of 2009. I am considering trying to examine 2009 May and June, before the change, with July-September, after the change, but that might not prove very conclusive given the short timescales.

In the meantime, the number 2.6 has been on my mind, and its significance might require some more discussion. Until next time!

Summer 2009: Finally addressing vaulted ceilings!

After a few posts on analysis, we are returning to the narrative of what changes I made to my inefficient house to achieve dramatic savings and comfort.
After the early spring completion of the radiant barrier and the subsequent installation of a good chunk of blow-in insulation, my attic efforts were largely complete. The only remaining concern nagging at me was the fact that around the border of roughly half of the house, I had vaulted ceilings that were inaccessible from the attic as well as from the outside. There was no way, short of removing roof shingles and decking and attacking from the top, to install radiant barrier in these areas.
This was somewhat unfortunate, because this was a good 4 feet of ceiling along the outside wall that I could not cover, extending around half the house. Furthermore, two runs of vaulted ceiling intersected over the master bedroom, making that room at best 70% coverable with radiant barrier, unless one were to do something dramatic like replace the roof.
Well! As it happened, a (rather convenient) massive, record-breaking hailstorm occurred in March. The damage from that particular storm damaged roofs in my area severely enough that crews are _still_ in our neighborhood replacing roofs, lo these 7 months later. In fact, they are all over this part of town. (And, in a development near and dear to all of our Energy Efficient hearts, a lot of the new roofs I see are getting ridge vents installed. I do not know if they have investigated clearing their soffit vents or not: perhaps some of them will read this blog and know what to do.)
So, although the insurance deductible payment was not trivial, this turned out to be the year to get a new roof. And of course, since this roof replacement event happens less than once a decade, I wanted to take the opportunity to try to make my changes to the decking over the vaulted ceiling areas.
From the beginning I learned that this was unlikely to be as cost-effective as the other improvements that I had made, mainly because I would be forced to replace existing material in the house rather than simply add to it as I had until this point. The material I had to replace was roof decking; there was no practical way to bring up the old decking without tearing it up. In effect, the extra cost I was paying was because I was giving up decking that probably had another 10 or 15 years of life on it.
The deal I reached with the roofer, who was quite helpful, was this: The new decking was going to cost me roughly $1/square foot. Since he couldn't find radiant barrier decking in the exact thickness to match my existing decking, he was going to supply me with the new decking the weekend before the job. Since I still had plenty of radiant barrier left over, I would construct homemade radiant barrier decking by stapling sheets of radiant barrier along one side of the 4' x 8' pieces of decking. Then the crew would remove the existing non-radiant-barrier decking from the parts of the house that I showed them, and replace it with the new homemade radiant barrier decking. The only cost to me would be the cost of the new decking, which I felt was more than fair.
Another helpful thing about replacing the roof decking is that it would help me resolve another problem that had dogged my attic from the beginning: lack of intake ventilation. As you may recall, I had unclogged all the soffit vents that I could reach from the attic. However, there were many places, all of them over vaulted ceiling, where the vents were still blocked due to the impossibility of getting my body in front of the area to work. Replacing the old decking gave me access from outside, allowing a chance to install baffles on the bottom of the new decking to create airflow where there had been none before.

4'x8' decking with baffles and foil barrier

You can see from the picture above that the combination of radiant barrier and baffles works quite well! I had first stapled radiant barrier to each piece of decking (except the last couple of inches on either end, which were going to be nailed to rafters anyway), then stapled baffles over the top of these, spacing the baffles to where the parts flush to the decking would coincide with the rafters.
To review our radiant barrier theory, the radiant barrier needs an airspace adjoining it to work; having it touch insulation or anything else destroys its reflective properties (recommended is at least a 3/4" space; these baffles gave about 1"). The baffle serves to create that needed airspace. In addition, the airspace is creating a channel to allow outside airflow into the attic from the soffit vents, cooling the attic during the evening hours when the barrier isn't helping. Thus, these cheap Styrofoam baffles perform double duty, allowing the barrier to reflect heat out before it causes trouble, and cooling the heat that does get through by enhancing attic airflow. They are truly an excellent value to the discriminating Efficiency Enthusiast!
The only downside is apparent from the photo: each baffle has a couple of inches running down the edge and through the middle that touch the barrier. These are necessary to give you a place to staple the baffle as well as to provide structural support, but along these runs, the radiant barrier will not work due to lack of adjoining airspace. So for each of these pieces of decking with barrier and baffles, I am probably getting only about 80% of the reflective power of the barrier. Still quite worth doing. I even considered trying to install small strips of radiant barrier along the top of the problematic portions of the baffle, but that would have been a good bit of work for very little square footage of reflection, plus, depending on how much the insulation bent under the pressure of the baffle, these sections might have contacted insulation anyway after installation, making them nonreflective. So I left the decking as you see it in the photos.

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!

Cooling effectiveness: check!

I will repeat from my last post:

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

Energy efficiency enthusiasts, be enthusiastic! This is an incredible number. Every kWh of energy I run through the air conditioner cools me 3.4 times more effectively than it did in 2005 when I embarked on this journey. And, in a big Energy Efficiency Man plus, almost none of the improvements that brought me these savings will break, wear out, or require maintenance. Ridge vent? No moving parts. Baffles and soffit ventilation improvements? No moving parts. Additional insulation? Will last the lifetime of the building barring roof leaks. Radiant barrier? It's foil hanging from the rafters, folks. The business end is the reflective side facing down, and it won't even get dusty after decades up there.
Yes, the more efficient A/C unit will wear out, but I have little choice but to have an A/C unit of some type here. It is definitely the weak link, and will undoubtedly give me trouble, but it's awfully nice to have when it's 100 degrees in September.

Caveats:

1. This estimate is based on a linear analysis with a simple "slope-intercept" fit. A better analysis would take the hysteresis of the system into account.
2. Your mileage may vary: these improvements work quite well in my 5-month cooling season-dominated climate. Effectiveness of all improvements depend on your climate and house situation.

The above notwithstanding, the data (and energy bills!) clearly show much less dependence of the house on the outside temperature as was true only 4 years ago. Truly, the house is cooler to live in as well as easier to cool through most of the year.

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.

The Analysis Continues: Effectiveness of Ventilation

So given that I have energy usage data, and "cooling degree day" (CDD) data for the summers since 2005, I should be able to determine a correlation between the two, and figure out how much my improvements have helped me while eliminating the variations in the weather. I want to know: how much of my energy savings in 2007 was due to the cooler weather, and how much was due to fixing the ductwork and improving the ventilation?
To answer that question, let's take a look at a graph that plots what I'm interested in (energy usage), vs. the thing that I thing most influences it (CDD's) for both my starting year 2005, and the year we're analyzing, 2007. The graph below has been smoothed by Excel to make the curves easier to follow.

What a fascinating result! The larger, blue "loop" is the 2005 data, and the smaller "green" loop below it is the 2007 data. A few things jump out immediately:

1. The relationship between kWh and CDDs is not as simple as might be hoped
   
2. As expected, the green loop does not extend as far to the right, since a cooler summer caused there to be fewer CDDs in 2007
3. As expected, the green loop does not extend as high as the blue loop, since we know we used less energy in 2007
4. The graph indicates a hysteresis, or memory, is at work in our system. We know this because for any given X value, there tends to be more than 1 Y value, and you would need to know more about the system to determine which Y value to use.

In fact, intrepid readers, we have already learned something about my house as a physical system: it has a large amount of memory/hysteresis with regards to outside temperature and energy use. The reason for this seems pretty simple: the house itself, along with everything around it, has heated up more at the end of the year than at the beginning. Thus, even for the same number the CDDs, I must expend more energy to remove all the heat from the house at the end of the summer than at the beginning. Restated, it costs me a lot more to cool the house on a 100 degree day in August than it does on that same 100 degree day in May.
We learned all that from simply looking at the graph! Truly, a picture is worth at least 1000 words.
To verify that the hysteresis is caused by what I think it is, I should be able to look at the raw data and determine that for each year, the lower half of the curve occurs in January-June, and the upper half in July-December. Wondrous to see, I have looked, and it does indeed. If you follow the data points making up each year, you start at the left in January, move across the bottom of the loop going rightward until about the peak is reached in July or August, then move back across the top of the loop during the last few months of the year.
One thing that this complex graph tells me is that it won't be fully accurate to just estimate a slope for each of those loops and compare those slopes to determine the effectiveness of my home improvements to 2007. Nevertheless, I can do it, so I will. Again using simple features in Excel, I come up with the following "best fit" lines for each loop.

Looking at the slope of each of these "best fit" lines will give me a rough idea how well my improvements in 2007 had improved my situation regardless of the weather changes. The numbers?
2005 Slope: 2.8 kWh/CDD
2007 Slope: 1.1 kWh/CDD

This is yet another fascinating result! To the extent that it is accurate, I had already increased the energy-effectiveness of the cooling of my house by some 60% in 2007, yet my energy usage had dropped only some 43% over the same time. Perhaps that makes some sense: I had only improved the cooling-related energy use, and not any other aspect of energy use. Since the cooling-related energy use was reduced due to the cloudy summer in 2007, and that's the only part that I optimized, perhaps the overall savings should be less than 60%.

However, this analysis remains unsatisfying. Looking at the 2nd graph above, the linear fit is just a really poor estimate of the actual function we're graphing. Clearly, the memory at play in the system is significant, and perhaps I can find a way to account for it.

In the meantime, I need to finish the analysis to account for the radiant barrier added later in 2007 and 2008. Read on to fulfill your curiosity!

Analyzing Usage and Weather

I located some logged online Cooling Degree Day data nicely packaged by these folks.
To review, one Cooling Degree Day (CDD) recorded at a base of 65 degrees, for example, is a day where the temperature averaged over a day was 66 degrees. If the temperature for that day averaged 85 degrees, the CDD would be 20 for that day. Negative CDDs are ignored - actually, they would be counted as Heating Degree Days, which I am currently not using in my analysis (I may at some point in the future when I look at my natural gas usage, which heats my house).
The first question was which weather station to use; there are many in my area, and their numbers are all different. Not _too_ different, but somewhat. I decided to use the airport data since that data has the fewest gaps, even though the airport is a good half-hour drive away, and is well outside the urban heat island that I live on. One of the issues in dealing with long term archived data is dealing with the gaps. In my case, only a few months were missing, and I filled those in with data from another source.
I have the vague idea that my energy usage is mostly driven by air conditioner usage, supported by the fact well known to many Texans that the electric bills are largest in July and August, and are still not fun in June and September. Of course, I'm also using energy for other things such as lighting, computers, televisions, refrigerator, etc. but those loads should not vary as much seasonally, although, in yet another complication, lighting usage varies seasonally as the days get shorter.
So we have some known issues with our analysis, which we can hope will not matter too much:

1. Location difference between airport (out of town) and house (in town, downwind from downtown in the "heat island")
2. Small gap in the CDD data filled by data from a different source
3. Many things sum to make energy usage; air conditioning is only one, albeit a big one
4. CDD's measurements themselves can be taken different ways. Measuring the temperature every hour, and summing those results over a day, yields a different number than just looking at the (highest - lowest)/2 value that some data providers might use.
5. CDD's at base 65 might not be the wrong "baseline" for my house. Perhaps my air conditioner does not kick on until the daily average is over 70, for example.
6. CDD's do not consider sunlight, which delivers far more heat than convected air, particularly on cooler days. It would not surprise me to see some air conditioners running on a day with 0 CDD's but is sunny, and not run on a day with a few CDD's but is cloudy. However, installing a radiant barrier should have reduced this problem for me; I have far less sunny heat gain than before.
7. CDD's do not consider the non-air environment, except as it affects air temperature. What do I mean by this? Well, a lot of us who live here have seen 100 degree days in May. Although unwelcome, those days never seem so bad as 100 degree days in August. Why? For one thing, there is still moisture in the soil in May. Grass is green and growing, plants are moist and lush, and the ground hasn't been baked for months on end to a nice shade of brown. All of these things will reduce the heat radiating around the area and hitting me and my house, even in high air temperature. In August, on the other hand, the grass is dormant and not evaporating water, cooling the ground. The streets, sidewalks, and bricks in the houses are storing a lot of heat built up over the summer that they didn't have in May. All of that heat is radiated and hits me and my house in August, causing more cooling load, even though very little of it affects the air temperature (particularly the air temperature at the airport, which is out of town).

One way that I hope to discover whether some of these factors matter, and perhaps how much they matter, is to simply graphically look at the data. Do the data make sense? If I graph my monthly consumption in kWh vs. the CDD's for that month, a relationship should emerge if there is one. In a nice, pretty world, it would be a linear relationship, with a slope showing how much energy I need to expend to handle one CDD, but we'll see if that is the case.

Here is a look at the Cooling Degree Days at the airport since 2005. You can clearly see the seasonality reflected, and the amazingly cool summer of 2007 right in the middle of the graph. You can also see how our January's have been getting warmer every year fairly consistently, even while the summers fluctuate, and you can see that the summer of 2009 was all-record-breaking in terms of heat. The graph has been 5-point smoothed (each point has been averaged with the 2 before it and the 2 after it) to make it look nicer.