Again, it's
not about the net present value of future energy use savings against the installed cost of the heat exchagner. Rather, it's about gaining the showering capacity without taking up more space or throwing away a perfectly good hot water tank for a bigger one. Compared to the installed cost of a tankless big enough to make up the difference (the original question) it's comparable (or cheaper if done as a DIY.)
There's also no "payback" on raising the storage temperature which was my original suggestion.
Nor is there "payback" on installing a second tank in parallel. Though the initial cost of a second tank may be lower than a drainwater heat exchanger, it takes more space, and you'll go through three or four of those before the heat exchanger would crap out.
Oh yeah- AND it would use more energy. From a lifecycle cost point of view compare the installed cost of 2-3 replacement tanks plus the difference in energy use cost and it's not hard to come up with discount rates and energy price inflation rates that would make it financially rational, even if purely on the energy savings alone it doesn't "pay".
If you're only planning to live there another year, find a low-cost gym membership and shower there!
If you plan to spend a couple decades there, do what's rational for the longer term. The "install it and forget it" aspect of the long lifecycle is itself appealing.
The embodied CO2 of the heat exchanger is roughly 2-6x that of it's copper weight.
(source1,
source 2 ) The shipping weight of a
3" x 60" or
4" x 48" is 25-35lbs, so pessimistically it's 35lbs x 6= 210 lbs of CO2.
Burning natural gas is worth about 11.7lbs of CO2 emissions per therm, so the lifecycle embodied carbon emissions of the heat exchanger it's no worse than lighting off (210lbs/11.7=) 18 therms of natural gas.
If heating hot water with natural gas it'll save a typical showing family of four about 1.5-3x that amount of fuel every year. But if you're heating hot water with all hydro grid power in the Pacific Northwest it may have a net-lifecycle carbon footprint, but not a huge one. Assuming a 40 year lifecycle that's about 5lbs of carbon emissions per year, which is the same order of magnitude of climate damage of the average human's flatulence-methane emissions per year (about 2lbs of CO2-equivalent.)