Thermal expansion/contraction of wort formula?

General Homebrew Discussion
1

I’ve been researching this for a few weeks now and have only been able to come with online or in hobby-level homebrew books is anecdotal evidence. I’m trying to find a formula which will output wort volume at a specific temperature based on a known wort volume and temp. For example,

5.0 gallons wort at 68 degrees F = X gallons of wort at 212 degrees F

I’ve found many references to using .96 or .97 as a factor depending on a wort temp of 212F or 160F, respectively. For example,

5.0 gallons / .96 = 5.2 gallons at boiling

I believe people are basing this factor on the thermal expansion of water formula (Wikipedia: Thermal Expansion):

207 * 10^-6 * (temp in C) * (volume) = expansion volume

So,

207 * 10^-6 * 100 Celsius * 5.0 gallons = 0.10 gallons
0.10 gallons + 5.0 gallons = 5.10 gallons at boiling

Regardless of which one I’ve used, they both appear to be way off based on my own experience. Over the course of five different brews, I’ve recorded the following data:

Brew #1
5.25 gal @ boiling, 4.75 gal @ 68F → 4.75 / 5.25 = .91

Brew #2
5.75 gal @ boiling, 5.25 gal @ 66F → 5.25 / 5.75 = .91

Brew #3
5.60 gal @ boiling, 5.00 gal @ 76F → 5.00 / 5.60 = .89

Brew #4
5.65 gal @ boiling, 5.10 gal @ 68F → 5.10 / 5.65 = .90

Brew #5
5.50 gal @ boiling, 5.00 gal @ 90F → 5.00 / 5.50 = .91

As you can see, my own measurements show an expansion/contraction rate of about 10% between 68F and boiling which doesn’t at all agree with the expansion of water or the .96 (4%) factor thrown around so often. It seems like there must be a thorough formula for wort expansion which has been lab produced but I can’t find it. Wouldn’t this be covered in one of the professional brewing science texts (which I don’t have access to)? Since I’m working on building an online calculator for this, I’d prefer to have a real formula to work with and not just anecdotal evidence. Anyone have any ideas? This would’ve been a great question for John Palmer I think but I missed out on that one; maybe I should submit it to Rand Moser for the next “Ask the Experts” session?

2

I believe we should be able to determine the correct expansion factor from the temperature correction formulas for wort sg. After all, the mass of the wort stays the same and:

sg = mass / volume

Let me see what I can find.

Kai

3

Using this calculator: http://hbd.org/cgi-bin/recipator/recipator/hydrometer.html?13082506

I get this:
T = 212 F
sg = 1.010

T = 68 F
sg = 1.050

Assuming a mass of 1 kg I get these volumes:

T = 212 F : V = 1/1.010 = 0.9901 l
T = 68 F  : V = 1/1.050 = 0.9524 l

This is a shrinkage by 4%

Now it is possible that even the temperature correction formulas, which we are using, are incorrect.

One other place I like to turn to with these questions is A.J. deLange’s work. In particular this paper: http://ajdel.wetnewf.org:81/Brewing_articles/Sugar_Gravity.pdf. While it is very difficult to follow, he has a section that discussed temperature dependent sg changes and the gravities listed there match up with the example given above.

foobrew, could there be some systematic measurement error that is at fault here?

Kai

4

Thanks for the response, Kai.

It’s possible, of course. I’m reviewing my measurement procedures now to see if I missed something.

5

I use 4% shrinkage to calculate my 68°F volume from my volume at boil and it has always been accurately predictive.

6

I’ve also always seen a 4% contraction.

7

I discovered there were some factors with my measurements which I wasn’t accounting for, the most notable of which was the significant inaccuracy in my kettle. The sight glass on the 10 gal Blichmann Boiler Maker is not accurate, but it is consistent. The markings start at 2.0 gallons on the sight glass and this is accurate. For every gallon past 2.0, however, the marks overestimate 7 ounces. So, if you have a reading of 6.0 gallons, you really have: 6.0 + (6.0 - 2.0) * 7 / 128 = 6.22 gallons. I’m guessing the sight glass is calibrated for one of their larger kettles and just re-used for all the others. A little disappointing for a $450 kettle.

8

I’m glad you figured out what’s wrong.

That is a defect for which I would request another sight glass or the other kettle. I agree that for a $450 kettle thar things should not happen. Maybe they just installed the wrong sight glass.

I have a fementation bucket where the markings are off significantly. But that was only $20.

Kai

9

Kai - the link off the HBD page has always proven to be accurate for SG temp correction in my brewing experience
http://hbd.org/brewery/library/HydromCorr0992.html

For anyone who cares, this should prove interesting and will explain the 4%

10

You are a master of understatement.  I would be livid with rage.