FORUM ON BEER, HOMEBREWING, AND RELATED ISSUES
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Contents:
Carbonation levels at bottling time (Fred L Johnson)
CO2 Volumes ("A. J. deLange")
CO2 Volumes - Forget Polynomials ("A.J deLange")
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Date: Sat, 1 Aug 2009 12:27:11 -0400
From: Fred L Johnson <FLJohnson52 at nc.rr.com>
Subject: Carbonation levels at bottling time
Following a fermentation in an unpressurized container (carboy,
bucket, etc.), the carbon dioxide levels in the beer can be
calculated based on the temperature of the beer. For example, at 65
degrees F and with the headspace filled with CO2, one can assume that
the beer has about 0.9 volumes of CO2 in it. If one intends to bottle
this beer and prime the bottle to achieve a final CO2 level of 2.5
volumes, one should add only enough sugar to provide 1.6 volumes of
CO2 in addition to the 0.9 volumes already in the beer. This all
assume that no CO2 is lost during the transfer(s), which would be one
in one bottles straight from the fermentor or two if one racks it to
a bottling vessel before transferring to the bottle. If the latter,
batch priming the entire batch would certainly lead to additional CO2
loss. The question is how much?
Does anyone have knowledge of how much CO2 is lost from the beer by
simply racking the beer directly to a bottle? I'm sure that
commercial bottling lines take this into consideration.
Fred L Johnson
Apex, North Carolina, USA
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Date: Sat, 01 Aug 2009 21:20:47 -0400
From: "A. J. deLange" <ajdel at cox.net>
Subject: CO2 Volumes
A day or 2 ago I posted some polynomial fits to what I said were the
ASBC tables. Fred saw some disagreement there and so I went back and
looked and looked and couldn't find anything on my disks pertaining to
a fit to the ASBC tables but what I did find was some fits to
McDantim's curves and so I think that must be where those polynomials
came from. The McDantim curves are straight lines on volume vs.
pressure charts at a given temperature. The ASBC constant temperature
curves are not straight lines. Thus the McDantim curves are much
easier to fit and the residuals are smaller but the numbers don't
agree with the ASBC tables or fits to them.
Here is a simple fit to the portion of the ASBC table up to 30 psi
(minimum pressure in the table is 5 psig) and 86 F (min. temp is 32
F). Does anyone carbonate at 86F? No, didn't think so. Reasoning this
way I took the approach of fitting best in the portion of the table
near 38 F and 10 psi (there is as much art in fitting as science).
This is around 2.4 volumes which I think is probably typical of what
we do. Error is less than 0.05 Vol between 36 and 46 F for any
pressure between 5 and 30 psig.: Error climbs to as much as 0.1 vol at
32F and 51 F (i.e. it's less than 0.1 from 32 - 50 F).
V = 3.0355 -0.040798*T + 0.10122*P - 0.00028923*P*T
Including squared P and T terms improves things noticeably and extends
the range:
V = 3.4361 - 0.070962*T + 0.14049*P + 0.00047273*T*T - 0.0011049*T*P
- 0.00022153*P*P
Here disagreement with the tables in less than 0.04 Volumes for any
pressure between 5 and 30 for temperatures between 32 and 60 F and
less than 0.01 from 36 to 41F (which I'm supposing covers the usual
range of storage temperatures), again at any pressure between 5 and 30
psig. Agreement with the table is better than 0.05 up to 72 F, and
0.1 up to 80F.
If you are going to use these in a spreadsheet it doesn't really
matter how many coefficients you have (once you have them typed in and
you can cut and paste from the e-mail to do that) so here are the
coefficients for a 3rd order fit. I'm just going to list them.
K0 =3.3518
K1 =-0.074649
K2 =0.16466
K3 =0.00072579
K4 =-0.0020645
K5 =-0.00015
K6 =-2.5727e-06
K7 =7.6321e-06
K8 =9.3369e-07
K9 =1.3224e-06
The coefficients progress in the order of the power of the terms. Thus
K0 is the constant (0 power) term, K1 and K2 are the coefficients for,
respectively, T and P, K3, K4 and K5 are for the second order terms,
respectively T*T, T*P and P*P and K6 through K9 are for the 3rd order
terms : T*T*T, then T*T*P, T*P*P and P*P*P in that order. No weighting
was applied in doing this fit and the result is agreement to better
than 0.03 volumes between 5 and 30 psig and 32 to 86 F. The rms
disagreement with the table is 0.005 Vol. Going to higher order fits
does not improve things and in fact results in singularities in the
fitting matrix. Thus in reality these data are not that well suited to
polynomial fitting.
Just as a reminder: here 1 Vol means that a liter of beer contains
gas which has a volume of 1 liter at 0 C and 760 mm Hg (1013.25 mb).
------------------------------
Date: Sun, 2 Aug 2009 15:58:24 -0400
From: "A.J deLange" <ajdel at cox.net>
Subject: CO2 Volumes - Forget Polynomials
Based on the comments I made in my last post about the ASBC volume
table not really being well suited for polynomial fitting I thought
more about the problem and decided that a better approach might be to
calculate the effective Henry coefficient (ratio of dissolved gas to
gas pressure) from the tabulated data, see if it is relatively
constant with pressure (it is except at low temperature) and see if I
could fit the ratio to temperature. When I sat down at my machine to
have a look at this I found an e-mail from Fred with a formula using
just such an approach that he had found on Kai's site (www.braumeister.com
). Fred asked how it compared to the ASBC table. The answer is that it
compares reasonably well at low temperature but not so well at the
high temperatures (at which no one in his right mind would be
carbonating beer anyway). I went ahead and did what I intended to do
and came up with
Vols = (P+ 14.695)*(0.018061 + 0.090166*exp(-(T-32)/43.157))
which is certainly a lot simpler than the polynomials and more
accurate - over the entire span of the ASBC table. It is in English
units, i.e. P is in psig and T in Fahrenheit and the result is in
volumes. Kai's similar formula (which you can find at his site as
given above) takes pressure in atmospheres, temperature in Celsius and
returns CO2 quantity in grams per liter. This formula compares to the
ASBC table slightly better than Kai's formula in the region of
interest (temperatures below 40 F) and appreciably better above 80 F
(but who cares about above 80F?). The reason neither of these formulae
are accurate at the lower temperatures is that the "Henry coefficient"
has a weak dependence on pressure at these lower temperatures (at
least according to the data in the ASBC table). The ASBC table
contains 1371 entries. The formula is in agreement with 95% of them to
better than 0.02 volumes and to 75% of them to less than 0.01 volume.
At temperatures between 38 and 41 F and pressures between 17 and 21
psig errors of a much as 0.045 volumes can be incurred.
A.J.
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End of HOMEBREW Digest #5586, 08/02/09
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