Converting SG(15/15) to SG(20/20)

[ljm]

None of the SG hydrometers I have found are calibrated to 20C. I have been thinking about a method or methods to do this conversion, but can't try them without, for example, both gravities for a 10P solution.

The more complicated nerdy method starts with equality of mass of extract (wex):

wex = d15 * SG15 * P * V15 = d20 * SG20 * P * V20

where
d15 = density of water at 15C
V15 = volume of wort at 15C
SG 15 = specific gravity of wort at 15C = SG(15/15)
etc.

Then a little algebra gives

SG20 = d15/d20 * SG15 * V15/V20

To get the volume ratio I could use the cubic coefficient of expansion for water (k) (or better yet for wort). And this the first problem - for water at 20C k = 2.07E-4; at 15C k=1.51E-4

Anyway,substituting the volume ratio with V15/V20 = exp(k*(20-15)) ought to give me an answer for SG(20/20). Except (1) I can't find any SG(15/15) and SG(20/20) for the same Plato and (2) k is clearly not even close to constant which is important since the none of the multiplicative factors in the SG20 equation are very far from 1.

I also thought perhaps if there were reliable SG20 and SG15 data the equation could be used to find an optimal k to force the conversion to work for other solution concentrations.

The easiest way might be to multiply SG(15) by the density ratio of water at the two temperatures. But what fun is that? And I still need some data for comparison.

Any help?

 

[ajdelange]

At first blush the density of a sucrose solutiom changes with temperature in pretty much the same way that the density of water does. Therefore, SG 15/15 should be close to SG 20/20. Let's check that.
The density of a 10 °P sucrose solution at 20 °C is 1.03811 g/cc and that of water 0.998203 g/cc. Thus the true 20/20 SG of 10 ° wort is 1.03811/0.998203 = 1.03998. At 15 °C the density of 10 °P sucrose solution is 1.03926 and that of water 0.916796. The ratio of those gives a true SG of 1.04020. The magnitude of the difference is 0.00021 which should be good enough for most home brewing. For 20 °P wort the difference is 0.00043 and for 30 it is 0.00065 (i.e. it's pretty linear with the solution strength).

For more accurate results you will need data on sugar solution density beyond what is given in the ASBC/EBC table. The usual source for this data is the ICUMSA polynomial. But note that the error in equating 15/15 SG to 20/20 SG is about the same as the difference between apparent and true SG and you can only measure apparent with a hydrometer.

 

[ljm]

So assuming we are measuring the same solution with two differently calibrated hydrometers at their respective calibration temperatures, the volume ratio is effectively the inverse of the water density ratio at the two temperatures and the equation in #1 becomes SG(T1/T1) = SG(T2/T2) = SG(T3/T3) = ... at least as long as the temperatures are not too far apart.

I suppose 15C/60F/59F are just legacy temperatures from early Baume/Brix days and no SG manufacturer is interested in calibrating at yet another temperature...

Thanks again.