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The Mathematics of Brewing
Tom AydlettJay Martin
Alison Schubert
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.
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 lbs of usable sugar
• So at 70% we would need lbs
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 * T0 + 3 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 0 190 3410:31 2 152 4010:33 4 136 4210:35 6 120 5010:37 8 114 5310:39 10 105 58
Modeling Temperature Data
0 2 4 6 8 10 120
20406080
100120140160180200
f(x) = 2.38571428571429 x + 34.2380952380952R² = 0.984143974592719
Wort & Ice Bath Temperature
Time after 10:29 (min)
Tem
p
Modeling Temperature Data
Elapsed Time
Temperature of the Wort
(Actual)
Temperature of the Water
(Model)Difference
0 190 34.238 155.7622 152 39.0094 112.99064 136 43.7808 92.21926 120 48.5522 71.44788 114 53.3236 60.6764
10 105 58.095 46.905
0 2 4 6 8 10 120
50
100
150
200
f(x) = 148.48938028175 exp( − 0.116021075142595 x )R² = 0.993096615743465
Difference in Temperature
Time after 10:29 (min)
Tem
p
Modeling Temperature Data
Add the linear function to this exponential model to shift it up to the original data
0 2 4 6 8 10 120
20
40
60
80
100
120
140
160
180
200
Final Model: T=148e-.116x+2.39x+34.2
Series1Series3
Time (min)
Tem
pera
ture
(Deg
F)
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
• No parts of each molecule can disappear or be added to balance
C6H12O6 + H2O → C2H6O + CO2
• where x=# Glucose, y=# Water, z=# Ethanol, and w=# Carbon Dioxide
•
• The smallest integer solution is x=1, y=0, z=2, and w=2
Chemical Reaction
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 1.042 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
DataTime (hrs)
Specific Gravity
0.000 1.04210.233 1.04221.750 1.02528.000 1.02145.967 1.01868.500 1.015
106.500 1.015165.000 1.014
Approximate start of exponential growth phase
Approximate start of stationary phase
0.000 20.000 40.000 60.000 80.0000
0.01
0.02
0.03f(x) = 0.028 exp( − 0.058412203054748 x )R² = 0.987845723829761
Fermentation Rate
Time after 10.233 hours
Adju
sted
SG
ModelTime
FermentingAdjusted Specific Gravity
0.000 0.02811.517 0.01117.767 0.00735.733 0.00458.267 0.001
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.
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
Calculating Your Efficiency
• As before we first calculate the potential yield:
• With a specific gravity of 1.072 we have 72 points in our wort.
• Comparing that to the recipe of 97.6 points
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:
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 = 1.072 and FG = 1.018,we have
• 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:
• Where is the radius of the tank, is the area of the hose, is the gravity constant, and 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:
• For Jay’s tank with a radius of in, hose of mm 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?
0 50 100 150 200 250 300 3500
0.20.40.60.8
11.2
Height of Wort
Measured Toricelli's
Time (sec)
Hei
ght (
ft)
Improving The Model
• To account for viscosity and friction we can use the more general model
Where is the proportionality constant
• Solving this differential equation we have the general solution:
DataUsing the 1st & 3rd points to determine k so how does this theoretical time compare to the actual?
0 50 100 150 200 250 300 3500
0.20.40.60.8
11.2
Height of Wort
Measured
Predicted
Time (sec)
Hei
ght (
ft)
Contact Info
• Tom Aydlett – [email protected]
• Jay Martin – [email protected]
• Alison Schubert– [email protected]
• Presentation– http://sdrv.ms/1g6GFXz
