1. The Mathematics of Brewing
Tom Aydlett
Jay Martin
Alison Schubert

2. 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.

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 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

5. 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!

6. 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.

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 bath
Add 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 * 700
T = (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.
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

11. Modeling Temperature Data

12. 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

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. 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.

16. 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?

17. 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

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 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

19. 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

20. 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.

21. 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.

22. 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

23. 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

24. Adjusted Specific Gravity
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

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 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%

28. 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

29. 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.

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

31. 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.

32. 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.

33. 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.

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

35. 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

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

37. Contact Info
Tom Aydlett
Jay Martin
Alison Schubert
Presentation