Presentation on theme: "The Mathematics of Brewing"— Presentation transcript:
1 The Mathematics of Brewing Tom AydlettJay MartinAlison Schubert
2 Mashing InRegardless of style all beer starts with making a wort by steeping malted grain in hot ( ° F ) water for about 30 minThis allows the enzymes to break down the complex sugars.
3 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?
4 Scaling the GrainIf 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 sugarSo at 70% we would need =2.14lbs
5 The BoilAfter 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!
6 YeastIn 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.
7 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 bathAdd enough tap water to drop the temperature to 70°F and raise our volume to 5.5 gallons of wort
8 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
9 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 * 700T = (385 – A * t) / (5.5 – A) 0 F.
10 Modeling Temperature Data How long will it take to cool a boiling wort to 96.40F?We need a model for the wort temp.TimeElapsed TimeTemperature of the WortTemperature of the Water10:291903410:3121524010:3341364210:3561205010:3781145310:391010558
12 Modeling Temperature Data Elapsed TimeTemperature of the Wort (Actual)Temperature of the Water (Model)Difference19034.23821524136612081141010558.09546.905
13 Modeling Temperature Data Add the linear function to this exponential model to shift it up to the original data
14 Finding The Cooling Time Set this function = 96.4 and solving results in a cooling time of 14.5 minutes.
15 FermentationNow 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.
16 Chemical ReactionFor 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?
17 Chemical ReactionThe molecular makeup of Glucose sugar is C6H12O6 , ethanol is C2H6O and carbon dioxide is CO2 .The Chemical Equation to Balance:Sugar + Water = Ethanol + Carbon DioxideC6H12O6 + H2O → C2H6O + CO2
18 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 balanceC6H12O6 + H2O → C2H6O + CO26𝑥+0𝑦=2𝑧+1𝑤 12𝑥+2𝑦=6𝑧+0𝑤 6𝑥+1𝑦=1𝑧+2𝑤where x=# Glucose, y=# Water, z=# Ethanol, and w=# Carbon Dioxide
20 Chemical ReactionThus 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.
21 Specific GravityBrewers 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.
22 FermentationThere are three main phases of yeast activity during fermentation:Lag Phase (0-15 hours) where yeast absorbs the nutrients it needs to replicateExponential Growth Phase (1-4 days) where it rapidly replicates producing alcoholStationary Phase (3-14 days) where flavors mature and the yeast settles out
23 Data Time (hrs) Specific Gravity 0.000 1.042 10.233 21.750 1.025 28.0001.02145.9671.01868.5001.0151.014Approximate start of exponential growth phaseApproximate start of stationary phase
24 Adjusted Specific Gravity ModelTime FermentingAdjusted Specific Gravity0.0000.02811.5170.01117.7670.00735.7330.00458.2670.001
25 Calculating Your Efficiency 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.
26 Calculating Your Efficiency 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
27 Calculating Your Efficiency As before we first calculate the potential yield: (46) 5 =97.6 pointsWith a specific gravity of we have 72 points in our wort.Comparing that to the recipe of 97.6 points ⋅100=73.8%
28 Measuring AlcoholBrewers 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
29 Measuring AlcoholThe 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.
30 Measuring AlcoholIf OG = and FG = , we have 𝐴𝐵𝑉= − =0.071So our beer is about 7.1% alcohol by volume.
31 TransferringAfter 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.
32 Torricelli’s LawAs with any liquid the rate of flow is governed by Torricelli's law: 𝑑𝑉 𝑑𝑡 =−𝑎 2𝑔ℎ 𝜋 𝑅 2 𝑑ℎ 𝑑𝑡 =−𝑎 2𝑔ℎWhere 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.
33 Transferring TimeSince this is a separable differential equation we have the general solution: ℎ= − 𝑔 𝑟 2 𝑅 2 𝑡+𝐶 2For 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.
34 DataSo how does this theoretical time compare to the actual?
35 Improving The ModelTo account for viscosity and friction we can use the more general model 𝑑ℎ 𝑑𝑡 =𝑘 ℎ Where 𝑘 is the proportionality constantSolving this differential equation we have the general solution: ℎ= 𝑘𝑡+𝐶 2
36 DataUsing the 1st & 3rd points to determine k so how does this theoretical time compare to the actual?