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 Home Brew Forums > Step Feeding and Dilution Equation

10-20-2012, 02:32 PM   #1
seth8530
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 Step Feeding and Dilution Equation

Hey guys, I made a reference to this equation earlier that I derived but I have decided to clean it up making it easier to read and also include an excel sheet that will do the calculation for us.

First off I am going to derive the equation, just to prove I am not pulling this out of thin air. If you do not want to see the derivation just skip to the bottom of the post.

This formula is how you correctly calculate ABV when you step feed or dilute your beer/wine/mead with some kind of liquid.

First off we will define what ABV is

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Next we will define how we calculate ABV gain per step via SG

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From this we can extract how much PURE alcohol we gain per step based on Volume and the definition of ABV

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This can be simplified to be a little prettier as

Thus to get the Total Pure alcohol we would need to sum up each step

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Which can me simplified to be

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Now Substitution into the definition of ABV,

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We come up with this as our final equation for ABV

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Volume1=Original volume.. Volume2=Original volume+Volume of dillutint1 Volume 3= Volume2+Dillutint2 and so on and so forth. Total Volume is equal to the last volume in the series. Ie your final volume.

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Last edited by seth8530; 10-20-2012 at 02:46 PM. Reason: Clarification

10-20-2012, 02:43 PM   #2
seth8530
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Alright, all this math is great.. But how about an example?

Lets say that I make 5 gallons of alcohol with an OG of 1.100 I then ferment it dry down to a FG of 1.00.. I then decide to step feed my beer with 1 gallon of concentrated wort.

The one gallon of wort brings the SG of my beer up to 1.030. It then Ferments dry back down to 1.00.

What is my ABV?

Well, you might be tempted to say well, the sg dropped from 1.100 down to 1.000. Then from 1.030 to 1.000 thus the sg dropped 1.130 points. Then looking at the hydromoter or a SG to ABV chart you might be tempted to say your abv is 17.7 percent. Hooray for rocket fuel and all hail Bacchus and the step feeding gods!

Unfortunately, things do not quite work out that way. We need to somehow account for the to the ABV that adding the wort in contributed. Thus we use the handy dandy equation that we have built to find out what our ABV actually is.

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Hmm, 14 Percent ABV is a little more modest than say 17%

Lets calculate the percent error for laughs.

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So, by not accounting for the dilution from step feeding we would of been off by 21% on our ABV. Pretty significant.

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10-20-2012, 03:27 PM   #3
seth8530
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Ok and now its time for an excel spread sheet. Remember, to use all the same units of volume or god knows what this spreadsheet will spit out.

All fields start out with 0's in them. Replace the 0's as needed to account for each step.

To make sure you are using the Excel sheet properly check it against my above example.

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10-20-2012, 04:38 PM   #4
seth8530
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So being able to look back at what we have done is all nice and great.. But what use is that when you are trying to plan a beer?

So I am going to go through an example of using the excel file I have uploaded to create a Utopia Clone.

Lets start out with 5 gallons of wart with 3 additional additions of 1 gallon each.

We start our wort out at 1.070 and lets say it drops to 1.020. This will give us an initial potential of 6.1% We are shooting for 27%

So for step 2 lets add in 1 additional gallon of wort that is strong enough to raise the gravity back up to 1.1 ( use a person square) Let it ferment down to 1.025.
So in the spread sheet let OG1=1.07 FG1=1.02 OG2=1.1 FG2=1.025 Volume 1=5 Volume 2=6 And let Total volume = 6

This gives us an ABV of 14.58% We need another addition
Ok lets repeat step 2 for step 3 But let FG3=1.030 Vol3=7 total vol=7

We find our abv to be

21.25%.. We are getting close.

For the final step lets go from 1.08 to 1.035 with a 1 gallon addition.
Vol4=8 total vol=8

This gives us a final ABV of 24% Very nice!

Just for comparison lets compare this against not accounting for dilution.
We would get a final ABV of 26.25% if you did not account for the dilution.

That is a 9% error on the ABV. And that is using an extremely concentrated wort. The error would be much greater if say the wort you used to raise the gravity up with was say at 1.100 instead of being strong enough to raise the overall wort up to 1.100

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10-20-2012, 04:45 PM   #5
jlangfo5
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Very interested in trying this out to design my first super high gravity beer! You should try and submit it as write up!

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10-20-2012, 07:13 PM   #6
ajdelange
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Lets look at your example problem

SG 1.100 implies OE of 23.77 °P i.e. 100 g of wort contained 23.77 g sugar and 76.23 grams of water. At this OE the apparent extract Balling factor is 0.433. If the AE falls to 0 (1.000 DG) the ABW is 0.443*(23.77 - 1) = 10.29 grams/100 grams. The true extract balling factor for this OE is 0.551 and that allows us to compute the TE as TE = OE - ABW/0.551 = 5.09 °P. Thus 100 grams of this beer must contain 100 - 5.09 - 10.29 = 84.62 grams of water. But we only had 76.23 grams of water to begin with so we must have 100*76.23/84.62 = 90.09 grams of beer from the original 100. The mass loss is to CO2 and yeast.

The true extract of this beer is estimated at 5.09 °P. This corresponds to specific gravity of 1.020. Thus the alcohol suppressed specific gravity by 20 'points'. Applying Tabarie's principal to the beer after addition of the wort it's specific gravity would be 1.030 + 0.020 = 1.050 corresponding to 12.39 °P. Since you added 3.785 L of wort to 18.9 L of beer you have 22.71 L of wort of effective density 1.050*.998203 at 12.39 °P which contains 22.71*1.05*998.203*12.39/100= 2949.15 grams of extract. Five gallons of the beer (before the addition) contained 5*998.203*1.02*3.785*5.09/100 = 980.78 grams of extract. Thus you have added 2949.15 - 980.783 = 1968.37 grams of extract which, from one gallon, is awfully strong wort i.e. 43.48 °P ~ 1.197 SG. All this would be so much simpler if you said you added a gallon of wort at 43.48 °P. I would not have had to do any of the stuff I've done so far except calculate the OE and I wouldn't have to mess with Tabarie's principal which I have always been a bit suspicious of. Be sure you understand that where you use the factor 125 you are relying on Tabarie's principal as well.

So now that we know the extract of the addition and the original let's start over again and make it easy.

18.9L of wort and apparent SG 1.100 ~ 23.77 °P contains 4932.9 gram extract.

Add 3.785 gal of concentrated wort at 43.48 °P ~ 1.197 SG containing 3.785*998.203*1.197*43.48/100 = 1968 grams of extract
4933+1968 = 6901 grams equals the effective total original extract.

Ferment to 1.000 and assume no losses i.e. you have 6 gallons (22.71L) at 1.000. The estimated original extract is then 6901/22.71 = 303.9 g/L. This corresponds to an effective original gravity of 1.1163 or 27.28°P.

For this effective OE the Balling factor is .451 so ABW = 0.451*(27.28 - 0) = 12.305
ABV = ABW/0.797 = 15.44%

Now lets say the final volume was really 5.5 gal i.e. that you lost half a gal to evaporation. You'd then have 20.82L and use 6901/20.82 = 331.5 g/L as your effective total extract. This corresponds to an OG of 1.177 and an OE of 29.47. The Balling factor for this OE is 0.456 (not much change) so ABV is
0.456*(29.47-0)/0.797 = 16.86%.

This is as expected. You've put the same amount of extract into less beer so that the alcohol content is higher.

So it's not quite as simple as your formula but, if you keep track of the strength of each addition and the final volume it isn't much more complicated. You do need to be able to compute effective OG from grams/gallon but that's not too hard to do.

As noted in the other thread a more robust way of doing all this is to measure the True Extract of the finished beer. This is a bit of a pain but if you are really after an accurate estimate you should be willing to do it. With TE you can calculate the total extract put into the brew and the total amount present at the end. Each 2.0665 grams consumed yields 1 gram of alcohol. That's still an approximation but better than an approximation plus the errors from Tabarie.

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10-20-2012, 08:21 PM   #7
seth8530
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That is an interesting post you have made. Personally, my primary use for my equation is as an EASY better way to get the abv of something that I have step fermented. I realize that it is not a perfect prediction tool because it does rely on 125(Deltasg)=abv which in itself is not perfect. And it is difficult to predict volume change to blow off. However, I feel like if you give my equation the DeltaSg it needs and the correct volumes it needs it will yield rather accurate results. I have not had the opportunity to test if this yields perfect results because I do not have the lab equipment to do so.. However, if some one here has the means to test it I would be thrilled! As a heads up I am a wine and meadmaker first ( a land where things do not get boiled) and I have not had the chance to make much beer yet..

Also, I do realize that in the example I gave that you would need to make an extremely rich extract. But, I meant it to be an easy to follow example not a recipe.

My goal in creating this equation was to derive a formula that gives as accurate as possible of an ABV based on SG. I would love to see it tested out.

Thanks

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10-20-2012, 09:17 PM   #8
ajdelange
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Given that you would probably be disappointed in the accuracy of the Balling formula working with true extract and a simple (no supplementation) brew taking the Balling factor into account I expect you would be even more disappointed in how good the apparent extract formula is when the variability in the Balling factor is not taken into account. Then the additional approximations you are making would add even more uncertainty. Part of the problem in any of these methods is that you really need to know the initial volume and initial extract. Volume is a bit tough to measure especially as temperature is a factor as is evaporation. What really was your OG? I have given up on trying to get a good measure of it and thus, for purposes of calculating efficiency, use the effective OG estimated from the alcohol content (which I do measure). I've noticed that the Anton Paar Alcolyzer does exactly the same thing. I don't know if that is because it's the way mega brewers determine OG or if it's just on their machine because it's easy to calculate once you have TE and alcohol content.

Couple this with the accuracy of hydrometer reading made with a \$3.00 plastic hydrometer and it is not hard to see that ±1% would be pretty good accuracy for an alcohol estimate.

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10-20-2012, 09:34 PM   #9
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Right, im not saying my method is perfect because of it reliance on the balling formula, but I do believe that the only error that is inherent in my method comes from the balling formula. Their are no other additional approximations or assumptions made other than the balling formula gives accurate ABV. Initial volume can be easily measured with a ruler and a calculator and evaporation does not matter when taking sg. If you take temp corrected sg and properly measure your volume im pretty sure this should model your abv really well.

Like I said earlier I wish I had the means to perform an experiment to see how accurate this is. BTW, do you have a link or something about the manner of the inaccuracy of the balling formula? Im interested in reading up on it.

One thing in engineering that we like to talk about is alot of times is how accurate is accurate enough? Is this model good enough? Is it worth the time and resources to pursue it further? For instance say if my method gives accuracy within 1% to .5% vs being off by 4% to 5% by not taking dilution into account I would say that my method is an improvement. Especially considering how easy it is for anyone to use.
BWT another good post.

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10-20-2012, 11:00 PM   #10
ajdelange
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Quote:
 Originally Posted by seth8530 Right, im not saying my method is perfect because of it reliance on the balling formula, but I do believe that the only error that is inherent in my method comes from the balling formula. Their are no other additional approximations or assumptions made other than the balling formula gives accurate ABV.
You are making a couple of assumptions beyond that the Balling formula is not perfect. You are assuming that the Balling formula is independent of OG. It isn't. For the case of 1.100 wort fermented down to 1.000 gives, using your constant factor formula 125*(1.100 - 1.000) = 12.5% for the ABV. Balling's formula gives 0.443*(23.77 - 0)/0.797 = 13.2%. For a 12°P beer (1.048) fermented to 3°P (1.012) the constant formula would give 125*(1.048-1.012) = 4.5% whereas the Balling formula would give 0.419*(12-3)/0.797 = 4.73%. Much closer but still a bit different. Also Balling's formula is based on extract - not specific gravity. Extract is very nearly a linear function of specific gravity but is not exactly a linear function. Part of the 'error' in the above comparisons is due to that. Also you are assuming that adding extract does not change the volume of the beer/wort. It does. In addition when you use apparent extract in the Balling formula you are accepting not only the errors inherent in the Balling formula but the deviation of the real world from the statement of Tabarie's principle. Each of these is small but cumulatively they add up. It is easy to deal with each of them.

1. Recognize the dependence of the mulitiplier on OE. This is quite simple. The multiplier is

0.48394 + 0.0024688*OE + 0.00001561*OE*OE

for true extract (°P) and

0.39661 + 0.001709*OE + 0.000010788*OE*OE

for apparent extract (°P).

2. Use extract, not specific gravity (the Balling factor formulas require this anyway).

3. Use true extract - this eliminates the additional innacuracy due to Tabarie

4. Look at total extract and final volume.

Quote:
 Originally Posted by seth8530 Initial volume can be easily measured with a ruler and a calculator
Can it? I can't do that. I can get approximate wort volume measurements by measuring down from the rim of the vessel but just assuming the vessel is a cylinder of known diameter doesn't give a terribly accurate answer. I've gotten around this by weighing the tared vessel at various depths and that helps but the temperature corrections are based on water (not wort) and while this is another small source of uncertainty is is, nevertheless, an uncertainty.

Quote:
 Originally Posted by seth8530 and evaporation does not matter when taking sg.
Yes it does because it isn't the specific gravity in the fermenter that counts. It's the effective specific gravity. If you brew a beer and get it into the fermentor such that there are 12 pounds of extract in 120 pounds of wort that's 10 °P. Now suppose that the ferment is really vigorous and 5 pounds of water go off with the CO2. Now you have 12 pounds of extract in 115 pounds of wort and that's 10.43 °P and your beer will be stronger as a result. Then what about starter's? What about volume hung up in the chiller? Did you push that through with water? We've listed quite a few error sources here and they do mount up.

Quote:
 Originally Posted by seth8530 BTW, do you have a link or something about the manner of the inaccuracy of the balling formula? Im interested in reading up on it.
It's based on the assumption that 2.0665 grams of extract produce 1 gram of ethanol, 0.9565 grams of CO2 and 0.1100 grams of yeast biomass. Obviously they don't always do exactly that as some strains will produce more or less biomass and they also produce things like esters and acids.

Somewhere at home I have data on the comparisons between alcohol levels calculated from OG as best I can measure it and what the actual alcohol levels turned out to be. I'll look into that when I get home next week (if I can remember a whole week back).

You might want to have a look at http://en.wikipedia.org/wiki/Gravity_%28alcoholic_beverage%29 and the reference to DeClerck which is mentioned there if you have it.

Quote:
 Originally Posted by seth8530 One thing in engineering that we like to talk about is alot of times is how accurate is accurate enough?
Sometimes it's clear. The accuracy of the terminal guidance system at an airport is an example of that. Here it isn't so clear. It is whatever it takes to make you comfortable. For some reason late in my career I started chasing errors, doing error propagation, error budgets etc. and it became kind of a campaign for me as I found too few of my colleagues ever gave it a thought. If the meter said 12 dBm it must be 12 dBm.

Quote:
 Originally Posted by seth8530 Is this model good enough? Is it worth the time and resources to pursue it further? For instance say if my method gives accuracy within 1% to .5% vs being off by 4% to 5% by not taking dilution into account I would say that my method is an improvement.
You certainly took a step in the right direction by considering volume. As to whether the model is accurate enough that is really up to you to decide. Whenever this comes up I always go through the extract based method and always push people writing brewing software etc. to use it but I don't usually try to push it on the casual home brewer. For most purposes I think points per pound per gallon is accurate enough but problems where the wort/beer is subsequently diluted or where extract is added really, IMO, need to consider the things we've discussed. It is, ultimately, simple conservation of mass.

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