About two weeks ago, my Vermonting friend Shane asked me a question that I thought would be pretty easy to answer: about how much alcohol is present in a batch of cider when honey is added after the original gravity measurement? Estimating the sugar increase is a pretty easy process. Figuring out the change in volume, which affects the density that affects the alcohol content, is the tough part. Thankfully, my recipe spreadsheets are capable of doing that. Unfortunately, the numbers they returned didn't agree with a simple sanity check I performed to make sure I was doing things correctly. The problem, I eventually learned, was caused by several errors:
1. The formula I'd been using to calculate extract, aka the weight of dissolved sugar, didn't agree with the definition of degrees Plato (extract as a percentage of total solution mass). I'm bummed out by this because the formula's source is the wonderful book "Brewing" by Michael Lewis and Tom Young. If you're dying to find a practical use for an outdated formula, the calculation is Extract = (258.7+degrees Plato)*degrees Plato*barrels/100. If you've wiped your nose with this stuff before, you may have noticed that 258.7 is the density of water, in lbs/barrel, at 39.2 degf. The American Society of Brewing Chemists (ASBC) defines degrees Plato at 68 degf. Reaching for that low-lying fruit by changing the value in the formula was one of the first things I tried. It didn't work.
2. Homebrewers typically measure gravity in Specific Gravity instead of degrees Plato. Additionally, disowning the degrees Plato -> Extract formula requires you to know specific gravity to determine extract. My prior conversions between degrees Plato and specific gravity were third-order polynomials fitted to an ASBC table, but the table is only valid for specific gravities of 1.083 (20.007 degrees Plato) and lower. Not terribly helpful when you're brewing a barleywine, and downright inaccurate when you're trying to determine how adjuncts like honey (gravity of 82.1 degrees Plato for the Wikipedian varietal) affect final liquid properties.
3. I was calculating all of my adjunct additions based on a parameter called "total wort volume", which was essentially the post-boil volume plus the volumes of any adjuncts added afterward (i.e. a yeast starter). Another way of thinking about it is that total wort volume = initial fermentation volume plus kettle wort losses. It's a made-up variable that never physically exists, but I thought it was a clever way to manage the interactions of several adjuncts. The problem? When you add an adjunct to the boil kettle, you lose some of it. When you add it directly to the fermenter, you don't. Since I typically specify adjunct additions as percentages of total extract, equating them to percentages of total wort volume is invalid.
I solved issues #1 and #3 by calculating the degrees Plato, specific gravity, volume, mass and extract for every stage of the brewing process and using those values where applicable instead of using total wort volume everywhere. To do so, I needed to "fix" problem #2 by adding an endpoint to the ASBC table. Since sucrose has a density of 1.587 g/mL and water has a density of 0.9982 g/mL at the ASBC reference temperature of 20 degc, the specific gravity of sucrose is 1.587 / 0.9982 = 1.589. Since degrees Plato is defined as % sucrose by weight, 100 degrees Plato should equal 1.589. One of the curve fits is shown below:
I say "fix" because I don't actually know what the data looks like between 20 degrees Plato and 100 degrees Plato. At least the curve converges on a reasonable endpoint, which is good enough for government work. Here's how the process works for Shane's cider example (5.0 gallons of cider at 1.052, 4.8 lbs of honey with an assumed gravity of 82.1 degrees Plato, final gravity of 1.004):
1. Water density at 68 degf = 0.9982 kg/L = 8.3316 lbs/gal.
2. Initial cider mass = density*volume = (SG*Dwater)*volume = (1.052*8.3316)*5 = 43.8 lbs. Leave me alone about using lbs as a unit of mass; it's a lot easier than slugs.
3. Initial cider gravity = ((116.716*SG-569.851)*SG+1048.046)*SG-594.914 = 12.9 degrees Plato.
4. Initial cider extract = (degrees Plato/100)*mass = (12.9/100)*43.8 = 5.6 lbs.
5. Honey extract = (82.1/100)*4.8 = 3.9 lbs.
6. Total mass = cider mass + honey mass = 43.8 + 4.8 = 48.6 lbs.
7. Total extract = cider extract + honey extract = 5.6 + 3.9 = 9.6 lbs (5.64 + 3.94 = 9.58, which gets rounded up).
8. Original gravity = 100*extract/mass = 100*9.6/48.6 = 19.7 degrees Plato.
9. Final gravity = ((116.716*SG-569.851)*SG+1048.046)*SG-594.914 = 1.0 degrees plato.
10. Alcohol by volume = 0.516*(original gravity - final gravity) = 0.516*(19.7-1.0) = 9.6%. It should be noted that calculating alcohol content is always approximate; to determine the exact value, you need to perform a distillation test on a physical sample.
Thanks for the mental workout, Shane. Updated recipe and brewlog spreadsheets can be found in the file cabinet.