

08242011, 09:41 AM

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Hydrometer corrections: a black art?


Mostly for my own amusement, and as an exercise to maintain my rusty programming skills, I recently wrote one of those programs to predict S.G. as a function of the grain bill. It was when I attempted to incorporate hydrometer temperature corrections that I ran into trouble. Now, I dare say I'm no babe in the woods when it comes to physical chemistry  I have degrees in both chemistry and the biological sciences, yet, I can discern no consistent manner for correcting hydrometer readings from authority to authority. Papazian recommends simple, acrosstheboard additive corrections, independent of the SG of the wort. Dave Line's book, on the other hand, describes a multiplicative correction, that therefore produces a higher degree of correction for stronger worts. At higher temperatures and specific gravities, this leads to wildly different calculated gravities from those given by Papazian's simple correction chart.
Of the online calculators, only that at HBD.org reveals their methodology: this calculator appears to be a Javascript implementation of a formula derived by C. Lyons and based on values in the CRC handbook. However, those original values are based on the SG of pure water at various temperatures, and thus do not reflect any contribution  which surely must exist  by dissolved sugars. Finally, there is the Fermsoft.com calculator. They do not actually describe their methodology, but comparing results obtained by plugging in SG values for worts of 1.010, 1.050, 1.070, and 1.100 at different temperatures from 10 to 90C shows that their method apparently does indeed incorporate some sort of correction factor for worts of different densities.
The bottom line here is that if anybody knows of any definitive discussion concerning hydrometer corrections vs. temperature vs. wort gravities, I'd be delighted to learn about it.



08242011, 01:21 PM

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The basic theory that most correction schemes use is that the variation in density with temperature of a sucrose solution of the strength brewers usually deal with is essentially the same as that of water. For example, the density of water at 20 °C is 0.998203. The density of water at 90 °C is 0.965304 which is 0.97794 times the density at 20 °C. For a 12 °P sucrose solution the density at 20 °C is 1.04431 while at 90 °C is it 1.00963 which is 0.96679 times that at 20 °C. Thus you can divide a reading at 90 °C by 0.9653 and be pretty close. But 0.9653 is 1  0.03296 so you can multiply the high temperature reading by 0.03296 and add it to the high temperature reading. In this example 1.00963*0.03296 = 0.033277 which added to 1.00963 gives 1.04291 compared to the actual 1.04431. So it's not terribly accurate but then you usually aren't concerned with 90 °C.
If you want more accuracy then you would have to program up the ICUMSA polynomial and do your calculations based on that. They track the ASBC official data very closely at 20 °C (which is the only temperature for which the ASBC publishes data which I believe is derived from AOAC or NIST data).
In a case like this it is, IMO, always best to use the method of an accepted standards body which, in North America, is the ASBC. Their correction table can be fit by the polynomial
0.000383644+0.0496253*T6.35525e05*P+0.00110774*T*T+0.00091996*T*P+1.36646e05*P*P
The 90/20 apparent specific gravity of 12 °P sucrose solution is 1.00963/0.998203 = 1.01145 so a Plato hydrometer would read 2.92832 and inserted in the correction polynomial would call for a correction of 9.09 °P. Added to 2.92 °P this gives 12.02 which is pretty accurate.
There are lots of ways to skin this cat. I recommend the ASBC method because of its "official" standing.
[Edit] In rereading your post I see you are not in North America. I'm sure EBC has a correction table similar to ASBC's  it may even be the same.
[Edit] Note that T here is the difference between the temperature of the wort and 20 °C. Thus for 90 °C wort temperature as in the example T = 90  20 = 70



08242011, 05:26 PM

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Quote:
Originally Posted by ajdelange
Their correction table can be fit by the polynomial
0.000383644+0.0496253*T6.35525e05*P+0.00110774*T*T+0.00091996*T*P+1.36646e05*P*P

Thank you for both the thorough explanation of principles involved and the mathematical rigor which were missing in other sources I'd consulted. I take it that in the above quadratic (apparently the result of a leastsquares fit of actual data, to judge by all those decimal places), T is temperature in degrees Celsius, and P gravity in degrees Plato?
My immediate reaction is that there's most likely no significant difference between EBC and ASBC standards, with the possible exception of the units employed, since the basic physical properties of sugar solutions are going to be the same everywhere. (And at any rate, I'm not a European, I just live there.)
Again, thanks for taking the time to provide this detailed response.



08242011, 05:37 PM

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Yes, correct on all the assumptions (least squares fit to ASBC table; T~ °C; P ~ °P).



09052011, 01:52 AM

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Dave Line messed up his temperature correction table in his "Big Book Of Brewing".
If you are going to implement a polynomial, you might as well use a polynomial that gives the density of water at different temperatures and then calculate the correction directly. This gives total flexibility about hydrometer reference temperature.
The water density polynomial that I use is:
1(T+288.9414)/(508929.2*(T+68.12963))*(T3.9863)^2
Where T is temperature in °C.
The simplest way (that is ignoring S.G.) of deriving a correction figure is:
(Density_of_water@reference_temp / Density_of_water@sample_temp) 1
Basically divide one by t'other and subtract 1
Where reference temp is the temperature written on the hydrometer and sample temp is the temperature of your wort.
That polynomial will give:
0.9982337 @ 20°c
0.9956783 @30°C
Therefore the correction for a 20°C hydrometer, at 30°C wort temperature will be (0.9982337/ 0.9956783)1 = 0.002566.
Depending upon how you want the figure expressed you may need to multiply it by 1000 giving 2.6.
As beer is mostly water, it is considered accurate enough by many and is the implementation used in simple correction tables.
If you want to bring SG into it, the drill is:
(SG*density@reference_temp/density@sample_temp)SG
The brackets are superfluous.
The SG probably needs to be in the 1.050style form, which is how I use it.
There is a gotcha here as you need an SG to begin with. No good for a general purpose table for instance. Obviously you will be computerising this, as you are using a polynomial, so you should put in a test in to ensure that SG is set to 1.000 (defaults to 1.000) in the absence of the true SG (if that makes sense). Then it will be at least as accurate as the previous formula.
Reason: Missing bracket



09052011, 02:06 AM

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Hooray! A thread that's not about water salts!
Enjoyed reading this, though I think I'll need to read it again, as now, it is well past my bedtime.



09052011, 05:05 AM

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If you want to use the density of water directly the following polynomial gives rms residual of 3E7 with respect to the data in the table from
Bettin, H.; Spieweck,F.: "Die Dichte des Wassers als Funktion der Temperatur nach Einführung der Internationalen Temperaturskala von 1990. PTBMitt. 100 (1990) pg 195196
density = 0.99984+T*(6.7715e05T*(+9.0735e06T*(1.015e07T*(+1.3356e09T*(1.4421e11T*(+1.0896e13T*(4.9038e169.7531e19*T))))))) gram/cc.
The density of water at 20 °C is 0.998203 g/cc and at 30 °C is 0.995645 according to the table. The polynomial returns 0.99820298 and 0.995645047 respectively.
Not that this level of accuracy is required for correcting a hydrometer reading made to 3 decimal places but if you are going to paste a polynomial into a spread sheet it is as easy to paste this one as a less accurate one.



09052011, 08:36 AM

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Yes, one might as well use an accurate formula as a less accurate one.
However, one has to be a bit careful here. Some of these tables give density measured in a vacuum, which is not how we use a hydrometer, of course. I know that it caught me out for while. I couldn't understand why there were such variations in density tables depending upon where one looks. There still are variations, but not so great when one realises why.
It was a long time ago that I 'found' that polynomial, but I am sure that it fitted a particular table well enough. Perhaps I will look into it again after I've done a day's work.
Of course if normal variations in atmospheric pressure make a significant difference to the last few decimal places, there isn't a busting lot of point.
Reason: Clarity



09052011, 11:59 AM

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I was already familiar with the "pure water" data (see paragraph 2 of my original post). I agree that the thermal expansion (and density curve) of wort, especially a lighter one, will most likely be quite close to that of water, I don't believe it'll be exactly that of water, and almost certainly not out to the five or six decimal places given by polynomials based on published data for pure water. For that reason, the formula given in reply 2 by ajdelange seems the most reasonable one to use in the interest of achieving the greatest accuracy  particularly if it's going into a spreadsheet or program where all those digits don't have to be keyed in each time.
Dave Line made a related logical error in chapter 19 in his discussion of "Degrees of Extract". Where he speaks of adding a weight of sugar to a volume of water, he appears to tacitly assume that the volume of liquid remains constant, and that only the weight changes. This is certainly not the case, and unfortunately, as a result, many of the conclusions put forth in this entire chapter are questionable. Similarly, it strikes me as courting error to assume  in the absence of much in the way of supporting data  that a weight of sugar dissolved in a volume of water doesn't significantly alter that solution's thermal expansion properties.



09052011, 01:30 PM

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Quote:
Originally Posted by GWheeler
However, one has to be a bit careful here. Some of these tables give density measured in a vacuum, which is not how we use a hydrometer, of course. I know that it caught me out for while. I couldn't understand why there were such variations in density tables depending upon where one looks. There still are variations, but not so great when one realises why.

The ASBC tables use apparent specific gravities. When using a hydrometer the differences between apparent and true are less than the accuracy to which the instrument can be read. For example 12 °P wort has true (in vacuuo) SG = 1.04839 and apparent SG 1.04833. The Bettin and Spieweck data are for density which is always in vacuuo (though of course it isn't measured that way). Note that this data is on the ITS 90 temperature scale and I think most of the differences between older fits and this one are from that and fewer terms in the polynomial. I took as many terms as were necessary to make the residual noiselike.
Quote:
Originally Posted by GWheeler
Of course if normal variations in atmospheric pressure make a significant difference to the last few decimal places, there isn't a busting lot of point.

They don't. As you can see from the example the difference at typical wort strength are in the fifth decimal place. The fanciest density meter you can buy (that I know of and the one used by breweries, distilleries and, most importantly, the TTB) only reads to 5.3 decimal places so it's generally moot in terms of actual observed values. What is important is that there is a fixed mapping, based on an assumed fixed air density between apparent specific gravity and density.
More to the point, I think, is that
P_corr = P + 0.000383644+0.0496253*T6.35525e05*P+0.00110774*T*T+0.00091996*T*P+1.36646e05*P*P
is pretty darn simple and gives a result more accurate than the water density ratio method. For example, 12 °P wort has density 1.04644 at 20 °C and density 1.03509 at 50 °C. A hydrometer (or density meter) is calibrated to read 9.266 in 12 °P wort at 50 °C (you can't read the hydrometer that accurately, of course but we continue assuming you could). Stuffing 9.266 and 50 into the formula above gives 12.008 °P for the reading as corrected to 20 °C. Using the ICUMSA polynomial to find the density corresponding to 9.266 at 20 °C to find the density the hydrometer is experiencing (1.03509) and then using that with the ICUMSA polynomial again at 50 °C to find what strength sucrose solution gives that density at 50 yields 11.9993 °P as the corrected value so obviously that is the way to go for best accuracy. Taking the ratio of water densities at 20 and 50 °C and multiplying that by the density the hydrometer sees, i.e. 1.03509 to estimate the density at 20 °C and inserting that into the ICUMSA polynomial at 20 °C gives 11.8334 °P. Now that is less accurate than the ASBC correction but is still only 0.17 °P in error which is pretty small. A good subrange hydrometer can be read to perhaps a bit better than 0.1 P (0.0004SG) but one clearly cannot expect that level of accuracy from the plastic hydrometers included with most homebrewing kits.





