Refining Residence Time Comparison

I've been thinking about how to more accurately compare the Global Warming Impact (GWI) of the HFC released during our closed cell foam insulation job against the amount of carbon that I estimate will be saved from the insulation. My original estimate, in this post, was that it would take 214 years before the estimated amount of carbon saved would equal the CO2 equivalent of the HFC emitted by the insulation job, since HFC has about 1000x the GWI power of carbon dioxide comparing by weight. But that calculation didn't take into account the residence time. The residence time of HFC in the atmosphere is just 10 years while that for carbon dioxide is around 500. Because it hangs around in the atmosphere 50 times longer than HFC, carbon dioxide causes a much larger impact over time. In this post, using a fairly simple measure that incorporated residence time, I calculated that the carbon payback time would be around 4.3 years. That calculation seemed too simplistic to me, so I decided to try to develop  a mathematical model of the GWI for both HFC and carbon dioxide. In this post, I'll explain the model and why I think it is the right way to think about comparative GWI for HFC and carbon dioxide. I'll try to keep the math simple, but there will be some as it is unavoidable given the topic. If you suffer from a math allergy, I'd suggest skipping to the end of this post, where a couple of graphs give a nice pictorial representation of the model.

First, let's review the basic facts about the comparison. The HFC from our insulation job resulted in the release of around 128 metric tons of CO2 equivalent (CO2e) HFC. Now, in actual fact, only part of that was released immediately. Some will leak out over time as the foam ages, some will remain trapped in the foam and only be released when the house  is demolished. But for purposes of comparison (and buying carbon offsets, which is the important point) we can assume that the insulation job resulted in an instantaneous release of 128 metric tons of CO2e  when the foam was installed, because we have even a less precise model of how these other processes work than for an instantaneous release. I've estimated that the insulation will decrease heating gas consumption in the winter around 30%, which will result in saving around 0.5 metric tons of CO2 per year. So these are the two numbers we have to work with: 128 metric tons of CO2e released in a single pulse this year  and 0.5 metric tons per year of CO2 saved as long as the house stands and we use gas heat.

Both CO2 and HFC decay in the atmosphere over time. Taking the case of HFC, we have a pulse of the compound emitted at the beginning of the time period, then after that, the rate the concentration decreases depends on the rate that is still there. By year 10, the concentration will be zero. We can model this with a rate equation like the following

Where R(t) on the left indicates the rate at which the concentration changes at year t, C(t) on the right indicates the concentration at year t, and 1/N is the time constant of the decay, with N being the year at which the concentration reaches zero (10 in the case of HFC).

Now this equation doesn't give the actual concentration of C, it just gives the rate at which it decays. But as it turns out, this equation can be analyzed and turned into an equation in t and C which will let us calcuate, for any year between 0 and N, what the concentration of C is. The procedure is well known and many natural processes exhibit the same general form. It's called exponential decay.

While this equation works for the case of the HFC emission because we're assuming all the gas is emitted at once, it won't work for the CO2 we save from not having to heat the house. That is released over time. In fact, as mentioned above, 0.5 metric tons are released every year. We need a new rate equation, one that takes the difference between the rate at which the CO2 is being released and the rate at which it is decaying. Because the rate at which it decays is small (500 years is a long time, after all), the contribution of the decay is not big until the house stops being heated, which amounts to the time at which the house is torn down. The rate equation in this case is:

A here is the amount that is emitted every year and 1/N * C(t) is the rate at which the amount that is there is decaying. Note that the equation is only valid up until the year T in which the house is torn down, then the first equation becomes valid, because the house isn't emitting any more CO2, but the amount that was emitted still needs to decay. It will take 500 + T years for all of it to be gone from the atmosphere. This equation isn't as easy to analyze as the above one, but with the help of a computer it is possible to numerically calculate a series of numbers giving the estimated concentrations.

Enough math. The following graph compares the concentration path for three different cases: the HFC, CO2 from heating if the house stands for 30 years, and CO2 from heating if it stands for just 6 years:

The blue shows the HFC concentration, the red shows the CO2 concentration if the house continues to emit CO2 for 30 years and the green shows the concentration if the house emits CO2 for 6 years. The graph runs out to 530 years (sorry, Excel won't put any numbers on the far end).

Just looking at this graph, the emissions from the house even if they  last 30 years don't look so bad. Sure, they go on for a long time, but they never get as large as the HFC. However, if you instead look at the cumulative effect over time, estimated by adding up the amount of carbon in the atmosphere every year until the concentration goes to zero, the emissions from the HFC look fairly harmless in comparison to the house:

This graph shows the cumulative GHI for 5 different scenerios: the three shown above, plus two cases where the house stands for 4 years or 5 years.  The GHI impact of the HFC effectively ends in year 10, while that for the 4 heating scenerios goes on for up to 530 years. If the house is assumed to stand for 30 years, then the cumulative GHI is more than 6 times that of the HFC release scenerio!

This graph is, to me, truly frightening. Turning my thermostat up essentially contributes incrementally to modifying the climate 400 years from now. The graph also tells me that, despite the GHG emissions from HFC, putting closed cell foam into the house was the right decision, from an environmental standpoint.

The three cases where the house is assumed to stand for 4, 5, and 6 years help figure out what the carbon payback time is, taking residence time into account. For 4 and 5 years, the cumulative GHI is less than that of the HFC, while for 6 years it is slightly greater. This says that a cumulative GHI of 6 years is about right for figuring out carbon offsets. After 6 years, the total amount of CO2 emitted is 3 metric tons. The carbon offsets, at $10 per metric ton, would therefore be $30.