The Man burns in 11 days.

Playa Dust Pale Ale recipe (PDF)

What was really needed was a real, independent three-dimensional surface fit. Fortunately, I found a truly excellent site that did the heavy lifting for me: ZunZun.com. This is one of those transformative “cloud apps” that journalists keep telling us are so revolutionary, and you really ought to check it out. Anyway, equipped with data from twelve FG readings and a little free time today, I decided to see if I could get a reasonable equation together. The data I’ve collected so far is, I think, pretty representative: OGs range from 1.036 to 1.106, FGs from 1.007 to 1.022, and ADFs from 73% to 91%. It turns out there is actually a very good (R² ≅ 0.98) fit, using a full cubic expansion:

FG = 1.0929176 – 0.0956887RI_i + 0.160699RI_f + 0.0103753RI_i² – 0.00449931RI_f² + 0.000585957RI_i³ – 0.00911434RI_f³ – 0.0165360RI_iRI_f – 0.00538394RI_i²RI_f + 0.0128988RI_iRI_f²

Where RI_i and RI_f are the initial and final refractive indices, respectively, in wort-corrected degrees Brix. The concern, of course, is that mapping 12 data points to 10 coefficients will result in substantial over-fitting. For the time being, I’m sticking with a simplified cubic form:

FG = 1.0111958 – 0.00813003RI_i + 0.0144032RI_f + 0.000523555RI_i² – 0.00166862RI_f² – 0.0000125754RI_i³ + 0.0000812663RI_f³

The mean deviation is, essentially, zero (10^-15), with a maximum of 2.1 points, and the standard deviation is reduced to 0.98 points – still not as good as a quality hydrometer, but quite possibly acceptable to most brewers. In fact, you can reduce the polynomial quite a bit and retain accuracy; I’m reporting the simplified cubic because it can be quickly inserted into existing software by changing the coefficients. A basic linear fit has an SD of 1.1 points, with a maximum deviation of 2.0, and the added advantage of being able to be (quickly) worked out on paper:

FG = 1.00358522 – 0.00123861*RI_i + 0.00380186*RI_f

It’s worth noting that even this highly simplified equation is superior to the default correlation, although again, this is based on a set of only twelve points. Caveat calculator.

Finally, we come to what I thought was a pretty neat visualization of all this razzmatazz. I plotted the old-school (red), logarithmically corrected (yellow), new cubic fit (green), and simplified linear (blue) correlations against the expected FG values. A perfect fit would result in all the data points lying on the dotted line. As you can see, anything other than the stock correlation provides reasonably good results.

My FG/Attenuation/ABV spreadsheet has been updated to utilize the new correlation, and add the new visuals. If you find it useful and/or accurate, please drop me a line. I’d love to hear more brewers’ experiences with using refractometers to estimate FGs.
 
 
Update: 06 Aug 2010

I made a minor adjustment to the spreadsheet, changing the default wort correction factor to 1.02, which is what my own refractometer data support. The current version is now 2.1.
 
 
Spreadsheet download:
fg_calculator_v2.1.ods | fg_calculator_v2.1.xls

Graphics courtesy the Beer Labelizer.

Seriously, it was pretty nasty yesterday, but I had to get in my Burning Man brew session; I’d already put it off at least a month longer than I should have. From last year’s burn I learned two important facts:

1. Having ice cold beer on the playa is essential.
2. Keeping a keg ice cold on the playa is foolhardy.

So this year I’ve decided to bottle-condition – in plastic bottles, no less. I’m leaving August 9, and I’m sure that three weeks rattling around in a hot car trunk will be plenty of time for the beer to carbonate, even if it isn’t a great idea otherwise. But that means that I have less than five weeks to take a 7.2% ABV lager from brewing to bottling. I have some reservations, but what are you going to do?

Anyway, the brewday went off without a hitch. I under-estimated my efficiency slightly, and ended up with an OG of 16.7°P (1.069). I had to chill the wort in my kegerator overnight to get it down to pitching temps, then this morning I aerated, pitched, and took a sample to conduct a fast ferment test. Normally I don’t bother, but in this case a few extra days of lagering could really make a difference, so I need to get the beer out of primary as soon as it’s ready – hopefully no more than about 10 days from now. I’ll do a short diacetyl rest, then lager it at 0°C for as long as possible.

I also racked my properly fermented Oktoberfest into a keg to make room, and man oh man is that tasting good.

Black Rock Maibock recipe (PDF)

The penultimate attempt, IPA #17, at right.

The recipe for Two Hearted is actually pretty straightforward. The base is regular domestic two-row pale malt, with a fairly significant dose of light Munich, and a small addition of light crystal malt. Some allegedly authentic sources suggest that Bell’s actually uses Vienna malt, but the Munich I use is rated 4-8 Lovibond, overlapping the typical range for Vienna considerably. Consult your maltster’s specifications and act accordingly. Hopping is 100% Centennial, of course, with more or less equal quantities for bittering, aroma, and dry-hop additions, doubled for the flavor addition. I like to split that between 20 and 10 minute doses, rather than a single addition at 15 minutes, because I feel that it gives a smoother and more complex hop flavor, at least in theory. Whether or not the difference is actually detectable, or which technique Bell’s uses, I couldn’t say.

In order to be 100% accurate, you’ll need to culture the yeast from a bottle; any of Bell’s American ales will use their house strain, so Oberon is probably the easiest source. Having brewed with both, though, I’ve found that I actually prefer Wyeast 1272. The differences are subtle, but its slightly fruitier esters seem to prolong the Centennial hop flavor, which is a plus if you won’t be finishing the keg for a couple months. I pitch at the standard 0.75 billion/L-°P, starting at 17°C/63°F, then let the beer rise on its own to 20°C/68°F and hold it there for the remainder of fermentation. Dry-hopping is conducted for 10 days at the same temperature.

I have to dilute my tap water 2:1 with reverse osmosis water to brew a beer this pale, then add gypsum and calcium chloride. The combined effect is to reduce the residual alkalinity to about 20 ppm CaCO₃ and brings the pH to a perfect 5.3.

• Ca²⁺: 103 ppm
• Mg²⁺: 9 ppm
• Na⁺: 7 ppm
• SO₄^2-: 144 ppm
• Cl^-: 29 ppm
• HCO₃^-: 119 ppm

The last time I tasted the two beers side by side (pictured), they were extremely close, but I could still differentiate them, even tasting blind. The clone was a hair too dark, and also a little too malty. For this last revision, I reduced the Munich malt and increased the bittering charge slightly, and the two are now basically indistinguishable. About the only way I can reliably tell the difference is to finish a glass; the homebrew has better lacing.

IPA #18 recipe (PDF)
 

The second, more intractable problem with using a refractometer to determine specific gravity is that once fermentation begins, the beer becomes a three-part solution: sugars, water, and alcohol. There is no longer fidelity of measurement – that is to say, there can be more than one specific gravity that will correlate to the same refractive index. Generally speaking, however, only one of the potential data points will be sensible for a real beer. Making that assumption, it should be possible to develop a correlation between the measured refractive index and the actual gravity of the beer, as long as the alcohol content can be estimated. This means that if both pre- and post-fermentation readings are taken, the FG can be predicted. Various software packages and websites incorporate tools to do just that, all of which seem to use the same correlation:

FG = 1.001843 – 0.002318474*RI_i – 0.000007775*RI_i² – 0.000000034*RI_i³ + 0.00574*RI_f + 0.00003344*RI_f² + 0.000000086*RI_f³

Where RI_i and RI_f are the initial and final refractive indices, respectively, in wort-corrected degrees Brix.

I took pre- and post-fermentation readings of ten beers, with OGs ranging from 1.036 to 1.103, using both a refractometer and hydrometer. In every case the refractometer correlation provided an FG that was lower than the hydrometer reading, by anywhere from 0.5 to 8.5 “gravity points” (1000*(SG-1)). The mean discrepancy is 5.1 points. The main variable of concern seems to be the attenuation of the beer; the greater the attenuation, the larger the discrepancy. The results are plotted below.

Note that the discrepancy is zero at about 58% attenuation (71% apparent attenuation). I have no information on who originally developed the correlation, but my supposition is that they only tested worts with about this degree of fermentability. A logarithmic curvefit provides a reasonably good (R² ≅ 0.7) approximation for the offset that is needed; by adding this correction factor to the standard correlation, the maximum discrepancy for this dataset is reduced to only 2.1 points, and the average to 0.1 points. Unfortunately, the resulting equation is a bit unwieldy:

FG = (1.001843 – 0.002318474*RI_i – 0.000007775*RI_i² – 0.000000034*RI_i³ + 0.00574*RI_f + 0.00003344*RI_f² + 0.000000086*RI_f³) + 0.0216*LN(1 – (0.1808*(668.72*(1.000898 + 0.003859118*RI_i + 0.00001370735*RI_i² + 0.00000003742517*RI_i³) – 463.37 – 205.347*(1.000898 + 0.003859118*RI_i + 0.00001370735*RI_i² + 0.00000003742517*RI_i³)²) + 0.8192*(668.72*(1.001843 – 0.002318474*RI_i – 0.000007775*RI_i² – 0.000000034*RI_i³ + 0.00574*RI_f + 0.00003344*RI_f² + 0.000000086*RI_f³) – 463.37 – 205.347*(1.001843 – 0.002318474*RI_i – 0.000007775*RI_i² – 0.000000034*RI_i³ + 0.00574*RI_f + 0.00003344*RI_f² + 0.000000086*RI_f³)²))/(668.72*(1.000898 + 0.003859118*RI_i + 0.00001370735*RI_i² + 0.00000003742517*RI_i³) – 463.37 – 205.347*(1.000898 + 0.003859118*RI_i + 0.00001370735*RI_i² + 0.00000003742517*RI_i³)²)) + 0.0116

In order to spare anyone who might be interested some trouble, I’ve put together a simple spreadsheet that will calculate FG using both the old and new correlations, in addition to attenuation and ABV. If you end up using it for a significant number of batches, please share your results.
 
 
Update: 20 July 2010

I’ve since refined the FG correlation, using a more mathematically rigorous method. I leave the original post up for transparency’s sake, but if you’re looking for an FG calculator, please check out the new post.
 
 
Spreadsheet download:
fg_calculator.ods | fg_calculator.xls

• My newest toy – a NIST-traceable waterproof digital probe model
• A TruTemp-branded compact digital that set me back about $9
• An AcuRite-branded digital “meat” thermometer
• A no-brand kitchen probe thermometer from Walmart
• An ordinary floating alcohol thermometer
• Several LCD strip thermometers
• The bulkhead thermometer in my boil kettle

Since the kettle thermometer requires about three gallons, and the strips five gallons, of liquid in order to be submerged, it didn’t really seem practical to check them against the others. All the others read to 0.1°C, except for the AcuRite (1°C) and the Walmart thermometer, which only does Fahrenheit and which I know not to be accurate anyway.

The other five thermometers were checked at four temperatures: 0.0°C (ice water), ~23°C (room temperature), ~59°C (mixture of ice-cold and boiling water), and 99.2°C (boiling water). That gives me, roughly speaking, fermentation and mash temperature calibrations for each, plus a couple of outliers for pretty curve fits. For the two middle values, the traceable thermometer’s readings were assumed to be correct, which given the readings at freezing and boiling seems justifiable.

Generally speaking, all the thermometers did well. Throwing out the no-name Walmart model and the surprisingly inaccurate alcohol thermometer, the maximum variation was 1.3°C, which isn’t great for lab equipment but is probably fine for brewing. Surprisingly, the AcuRite kitchen thermometer is every bit as accurate as the ISO 17025-calibrated lab thermometer, landing all four readings within its (admittedly larger) resolution. The TruTemp model turned out to be less accurate, albeit with a highly linear discrepancy. This is interesting because it could point to a design problem – assuming a linear voltage response for the thermocouple rather than doing a multi-point calibration. More to the point, it means that it can still be useful for brewing once the proper offset is applied.