# The Mathematics of Brewing

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The Mathematics of Brewing
Tom Aydlett Jay Martin Alison Schubert

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

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?

Scaling the Grain If a recipe calls for “2.00 lb English Chocolate” and specifies a “Brewhouse Efficiency: 75%” but your efficiency is only 70% how much grain do you really need? At 75% efficiency we would extract =1.5lbs of usable sugar So at 70% we would need =2.14lbs

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!

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 brewer’s ale yeast strains require temperatures between 65°F to 72°F.

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

Cooling the Boiling Wort
What temperature should the 2.5 gallons of wort reach to mix with 3.0 gallons of tap water at 480F in order to obtain 5.5 gallons of wort mixture at 700F? 2.5gal * T gal * 480 = 5.5 gal * 700 which results T = 96.40F

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 * T0 + (5.5 – A) gal * t0 = 5.5 gal * 700 T = (385 – A * t) / (5.5 – A) 0 F.

Modeling Temperature Data
How long will it take to cool a boiling wort to 96.40F? We need a model for the wort temp. Time Elapsed Time Temperature of the Wort Temperature of the Water 10:29 190 34 10:31 2 152 40 10:33 4 136 42 10:35 6 120 50 10:37 8 114 53 10:39 10 105 58

Modeling Temperature Data

Modeling Temperature Data
Elapsed Time Temperature of the Wort (Actual) Temperature of the Water (Model) Difference 190 34.238 2 152 4 136 6 120 8 114 10 105 58.095 46.905

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

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

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.

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?

Chemical Reaction The molecular makeup of Glucose sugar is C6H12O6 , ethanol is C2H6O and carbon dioxide is CO2 . The Chemical Equation to Balance: Sugar + Water = Ethanol + Carbon Dioxide C6H12O6 + H2O → C2H6O + CO2

Chemical Reaction 6𝑥+0𝑦=2𝑧+1𝑤 12𝑥+2𝑦=6𝑧+0𝑤 6𝑥+1𝑦=1𝑧+2𝑤
No parts of each molecule can disappear or be added to balance C6H12O6 + H2O → C2H6O + CO2 6𝑥+0𝑦=2𝑧+1𝑤 12𝑥+2𝑦=6𝑧+0𝑤 6𝑥+1𝑦=1𝑧+2𝑤 where x=# Glucose, y=# Water, z=# Ethanol, and w=# Carbon Dioxide

Chemical Reaction 1 0 0 − 1 2 0 0 1 0 0 0 0 0 1 −1 0 → 𝑥= 𝑤 2 𝑦=0 𝑧=𝑤
6𝑥+0𝑦=2𝑧+1𝑤 12𝑥+2𝑦=6𝑧+0𝑤 6𝑥+1𝑦=1𝑧+2𝑤 ⟶ 6 0 −2 − − −1 −2 0 1 0 0 − −1 0 → 𝑥= 𝑤 2 𝑦=0 𝑧=𝑤 The smallest integer solution is x=1, y=0, z=2, and w=2

Chemical Reaction 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.

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 means the wort is 4.2% more dense than water.

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

Data Time (hrs) Specific Gravity 0.000 1.042 10.233 21.750 1.025
28.000 1.021 45.967 1.018 68.500 1.015 1.014 Approximate start of exponential growth phase Approximate start of stationary phase

Model Time Fermenting Adjusted Specific Gravity 0.000 0.028 11.517 0.011 17.767 0.007 35.733 0.004 58.267 0.001

While you’re 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.

Let’s 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

As before we first calculate the potential yield: (46) 5 =97.6 points With a specific gravity of we have 72 points in our wort. Comparing that to the recipe of 97.6 points ⋅100=73.8%

Measuring Alcohol Brewers measure alcohol content by calculating the reduction in specific gravity after fermentation, since alcohol is less dense than water. Alcohol content is commonly expressed as percent alcohol by volume (ABV), which can be calculated from the original and final gravities: 𝐴𝐵𝑉= 1.05 𝑂𝐺−𝐹𝐺 0.79𝐹𝐺

Measuring Alcohol The difference between the final and original gravity shows how much CO2 has been released. From molecular weights about 1.05 grams of CO2 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.

Measuring Alcohol If OG = and FG = , we have 𝐴𝐵𝑉= − =0.071 So our beer is about 7.1% alcohol by volume.

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.

Torricelli’s Law As with any liquid the rate of flow is governed by Torricelli's law: 𝑑𝑉 𝑑𝑡 =−𝑎 2𝑔ℎ 𝜋 𝑅 2 𝑑ℎ 𝑑𝑡 =−𝑎 2𝑔ℎ Where 𝑅 is the radius of the tank, 𝑎 is the area of the hose, 𝑔 is the gravity constant, and ℎ is the height from the surface of the wort to the end of the siphon.

Transferring Time Since this is a separable differential equation we have the general solution: ℎ= − 𝑔 𝑟 2 𝑅 2 𝑡+𝐶 2 For Jay’s tank with a radius of 𝑅=6.75in, hose of 𝑟=4.5mm in radius and an initial height of 13in this means it will take about 6.14 min to transfer.

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

Improving The Model To account for viscosity and friction we can use the more general model 𝑑ℎ 𝑑𝑡 =𝑘 ℎ Where 𝑘 is the proportionality constant Solving this differential equation we have the general solution: ℎ= 𝑘𝑡+𝐶 2

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

Contact Info Tom Aydlett Jay Martin Alison Schubert Presentation