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	FORUM ON BEER, HOMEBREWING, AND RELATED ISSUES
		Digest Janitor: pbabcock at hbd.org


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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).



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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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