Edit 11/10/2023: On re-reading this, I think it’s wrong. The estimate for the energy radiating into the opening of the mason jar seems right, but the analysis fails to account for the energy radiating out of the opening of the jar. The net radiant energy input is much smaller than what I calculate below.
So now, the excellent performance of the mason-jar cloche is a bit of a mystery.
Sure, it works in practice. But does it work in theory?
My prior gardening post demonstrated that a standard Ball jar (mason jar) provides excellent frost protection, if used in a garden bed with relatively warm soil below the surface.
Now, let’s use a physics law, an insulating R-value, and a bit of math, and show that this works in theory.
The Stefan–Boltzmann law shows the total amount of power radiated by a perfect “black body”, for any given temperature. Let me assume that the soil under my mason jar is a black body at 50 degrees F. And that my glass jar is going to capture all of that power.
How hot will the interior of the jar get, before heat losses through the glass walls bring it into equilibrium with the cold air outside the jar?
Answer: Assuming the deep soil of the garden bed is somewhere around 50 degrees Fahrenheit, in theory, the inside of the Ball Jar could end up 9.4 degrees warmer than the outside.
I think that’s a pretty remarkable correspondence with the actual observed values (prior post), all things considered. There are some uncertainties here, and my temperature recorders only read temperatures to the nearest 2 degrees F.
You need to know a few more things to be able to have some faith in that calculation.
First, thicker glass has almost the same R-value (insulating properties) as thinner glass. You can infer that from the table of U-values at this reference. (U = 1/R). So the fact that a Ball jar is thicker than a pane of window glass is more-or-less irrelevant. The R-value (resistance to heat flow) will still be about 1.0 in U.S. units.
(I keep adding that phrase “in U.S. units” because, confusingly enough, countries that use the metric system (i.e., everybody but the U.S.) also refer to R and U values for insulation. But the actual values attached to materials are vastly different from U.S. R and U values. In S.I. units, R-values are about one-sixth what they are in inch-pound units (reference).
Second, ordinary window glass doesn’t transmit long-wave radiation (infrared), nor does it reflect it very much. Mostly, it absorbs it. The figures given in this reference say that it absorbs 86% of long-wave infrared, and reflects 14%.
This is why I more-or-less ignored energy losses through the opening of the jar itself. Dry, mulch-covered soil is a pretty good insulator, so I wasn’t going to lose much heat via condition through the jar opening embedded in the soil. (At least, not compared to the heat losses through all that above-ground glass area.) Although, in theory, I ought to net out the 14% that is reflected, if it ends up being reflected back down into the soil. But I think this is close enough as-is.
The upshot is that the long-wave radiation being given off by the ground is like having a 1.6 watt old-fashioned incandescent bulb lit inside the mason jar. That steady flow of 1.6 watts is in a form of energy that is almost entirely absorbed by the glass. That heats the glass, which then in turn heats the inside of the jar.
We knew it worked in practice. Now I know it works in theory. I think.
Now that I realize this works by heating the glass — because glass is an excellent absorber of long wave infrared — I have to try this with a layer of insulation around the glass. Plausibly, a mason jar and some bubble wrap might keep plants from freezing down 15F or so.
And that knowledge would be a great thing to have in your back pocket, just in case. I think that will be tonight’s experiment, if I can scrounge up some bubble wrap.
