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The Mathematics of Brewing Tom Aydlett Jay Martin Alison Schubert

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Mashing In Regardless of style all beer starts with making a wort by steeping malted grain in hot (153-155° F ) water for about 30 min This allows the enzymes to break down the complex sugars.

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How Much Grain To use? The amount of sugar you extract into your wort is your efficiency. Most recipes will also list their presumed efficiency. So what do you do when they are different?

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Scaling the Grain

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The Boil After the grains are removed the liquid is left to boil for 60min. During this process the hops are added to adjust the bitterness and aroma of the finished beer. At the end of the boil we are left with a sugary hoppy wort just too hot for our yeast!

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Yeast In short yeast is a microbe that converts sugars to alcohol and carbon dioxide. Since it is a living organism it cannot survive at extreme temperatures. Most brewers ale yeast strains require temperatures between 65°F to 72°F.

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Cooling the Wort For most home brewers this is a two step process: – Cool the 2.5gal of wort partially by surrounding it with an ice water bath – Add enough tap water to drop the temperature to 70°F and raise our volume to 5.5 gallons of wort

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Cooling the Boiling Wort What temperature should the 2.5 gallons of wort reach to mix with 3.0 gallons of tap water at 48 0 F in order to obtain 5.5 gallons of wort mixture at 70 0 F? 2.5gal * T 0 + 3 gal * 48 0 = 5.5 gal * 70 0 which results T = 96.4 0 F

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Cooling the Boiling Wort Tap water temperature changes depending on the time of year. Also, the gallons of wort after the boiling will be different each time, So, what is the temperature needed in terms of tap water temp and wort volume? A gal * T 0 + (5.5 – A) gal * t 0 = 5.5 gal * 70 0 T = (385 – A * t) / (5.5 – A) 0 F.

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Modeling Temperature Data How long will it take to cool a boiling wort to 96.4 0 F? We need a model for the wort temp. Time Elapsed Time Temperature of the Wort Temperature of the Water 10:29019034 10:31215240 10:33413642 10:35612050 10:37811453 10:391010558

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Modeling Temperature Data

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Elapsed Time Temperature of the Wort (Actual) Temperature of the Water (Model) Difference 019034.238155.762 215239.0094112.9906 413643.780892.2192 612048.552271.4478 811453.323660.6764 1010558.09546.905

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Modeling Temperature Data Add the linear function to this exponential model to shift it up to the original data

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Finding The Cooling Time Set this function = 96.4 and solving results in a cooling time of 14.5 minutes.

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Fermentation Now that your beer is cooled it is time to pitch the yeast. Once added the yeast goes to work, digesting the sugars, and replicating itself, creating alcohol and carbon dioxide in the process.

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Chemical Reaction For each molecule of sugar, how many molecules of ethanol and how many molecules of carbon dioxide are being created by the reaction from the yeast?

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Chemical Reaction The molecular makeup of Glucose sugar is C 6 H 12 O 6, ethanol is C 2 H 6 O and carbon dioxide is CO 2. The Chemical Equation to Balance: Sugar + Water = Ethanol + Carbon Dioxide C 6 H 12 O 6 + H 2 O C 2 H 6 O + CO 2

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Chemical Reaction

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Thus water should be present on both sides of the equation and does not contribute molecules to alcohol nor carbon dioxide. Therefore w = 2, x = 1, and z = 2, which states that for each molecule of glucose, two molecules of ethanol and two molecules of carbon dioxide are being created during fermentation.

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Specific Gravity Brewers measure alcohol content by measuring the reduction in weight, called specific gravity. Specific Gravity is a relative density of the wort to water of the same temperature. Original gravity of 1.042 means the wort is 4.2% more dense than water.

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Fermentation There are three main phases of yeast activity during fermentation: – Lag Phase (0-15 hours) where yeast absorbs the nutrients it needs to replicate – Exponential Growth Phase (1-4 days) where it rapidly replicates producing alcohol – Stationary Phase (3-14 days) where flavors mature and the yeast settles out

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Data Time (hrs) Specific Gravity 0.0001.042 10.2331.042 21.7501.025 28.0001.021 45.9671.018 68.5001.015 106.5001.015 165.0001.014 Approximate start of exponential growth phase Approximate start of stationary phase

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Model Time Fermenting Adjusted Specific Gravity 0.0000.028 11.5170.011 17.7670.007 35.7330.004 58.2670.001

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Calculating Your Efficiency While youre waiting for the beer to ferment you can use the original gravity to calculate your own efficiency. In addition to the original gravity of your beer you also need to know the potential yield, in points per pound of the grain you used.

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Calculating Your Efficiency Lets assume the recipe you followed used: – 9.3 lb extract (35 points/lb/gallon) – 1.5 lbs caramel malt (33 ppg) – 0.75 lb chocolate malt (28 ppg) – 2 lb sugar (46 ppg) And yielded 5 gallons of wort with an original gravity of 1.072

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Calculating Your Efficiency

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Measuring Alcohol

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The difference between the final and original gravity shows how much CO 2 has been released. From molecular weights about 1.05 grams of CO 2 are released for each gram of ethanol formed. Divide by the final gravity to get a percent alcohol by weight. Divide by the density of ethanol, 0.79 g/mL, to obtain the percent alcohol by volume.

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Measuring Alcohol

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Transferring After about 2 weeks most of the sugars are converted into alcohol and the yeast become inactive and fermentation slows. By now most particles have settled out and it is time to move your beer to secondary fermentation.

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Torricellis Law

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Transferring Time

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Data So how does this theoretical time compare to the actual?

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Improving The Model

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Data Using the 1 st & 3 rd points to determine k so how does this theoretical time compare to the actual?

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Contact Info Tom Aydlett – tjaydlett@waketech.edu Jay Martin – jemartin@waketech.edu Alison Schubert – alison.schubert@waketech.edu Presentation – http://sdrv.ms/1g6GFXz

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