How to Account for Residence Time?

If you recall from my last post, I estimated a carbon payback time for our closed cell foam insulation at around 214 years, based on an estimate of 120.38 kg (0.12038 metric tons) HFC emitted and an estimated savings of 0.562 metric tons (mt) of carbon per year (30% reduction in annual natural gas use). The calculation assumes that HFC245fa has an instantaneous global warming impact (GWI) of 1000x carbon dioxide.

There is one problem with this way of measuring the impact HFC245fa relative to CO2. It does not take into account that the residence time of HFC245fa in the atmosphere is around 7-10 years while that of carbon dioxide is around 450-500 years, and when the HFC breaks down, there are no long lasting products remaining. So this means a kilogram of HFC245fa put into the atmosphere now will cause 1000x the global warming of a kilogram of carbon dioxide for 10 years, but then it will be gone. A kilogram of carbon dioxide, on the other hand, will continue to warm the planet for another 490 years before it disappears. Simply comparing the two gases based on their instantaneous GWI therefore doesn't seem quite right.

This is a problem system engineers face all the time: how to calculate a number that lets you compare the impact of two different technologies/treatments/etc. on a system. The number is called a figure of merit. In effect, it lets you compare apples and oranges along some common metric, like for example the percent energy in the pure red color light (at 700–635 nm wavelength) reflected by their skins, of value in judging their relative contribution to solving some problem. In our case, we want a formula to calculate the weight of carbon dioxide equivalent (CO2e) for the HFC that takes into account both the instantaneous GWI relative to CO2 and the residence time relative to CO2.

My previous post calculated the CO2e for the HFC like this:

       wt. CO2e   = wt. HFC * instantaneous  GWI

But, as mentioned, this doesn't account for the residence time.

So we need to adjust the right hand side so that the reduced residence time for HFC over the equivalent amount of CO2 is accounted for. This means that we need to multiply the instantaneous GWI by a number that will decrease when the residence time of the gas for which the GWI is desired decreases relative to CO2 (or correspondingly increases if the gas has a longer residence time). One way to do that is to divide the residence time of the gas by the residence time of CO2 and multiply the instantaenous GWI by that to obtain a time adjusted GWI:

       time adjusted GWI = instantaneous GWI * (residence time of HFC / residence time of CO2)

If the residence time of the HFC increases relative to CO2, the factor on the right side will increase, causing the time adjusted GWI to increase, and vice versa, which is what we want.

Now we can use the time adjusted GWI for the CO2e calculation:

      wt. CO2e = wt. HFC * time adjusted GWI

In the case of HRC235fa, we have:

     instantaneous GWI      = 1000
     residence time of HFC = 10
     residence time of CO2 = 500
     time adjusted GWI       = 1000 * (10 / 500) = 1000 * 0.02 = 20

Plugging the time adjusted GWI into the equation for CO2e for our case gives:

      CO2e = 0.12038 mt * 20 = 2.4076 mt

At 0.562 mt of carbon eliminated per year, this gives a payback time of:

    payback time = 2.4076 mt CO2e / 0.562 mt CO2 per year = 4.284 years (!)

This is considerably better than the previous figure of 214 years. The carbon offsets for 2.4076 mt CO2 come to only $24 rather than $1203.80 as calculated in the previous post.

People who are not used to making engineering tradeoffs look at this kind of calculation and scratch their heads. Which number is right? Are we just playing with numbers or is there some kernel of truth  in all this arithmetic? I suppose you will have to answer that question for yourself. But if we are ever to come up with sustainable technical solutions to the systems in our society upon which we depend, we are going to have to come up with ways that fairly and accurately compare different kinds of solutions that have differing impact. Having a single number that summarizes this impact, like the time adjusted GWI, makes such comparisons a lot easier.