Calculating The Hop Amount

Here we will calculate the amount of hops needed for our Bourbon County Stout clone recipe from the previous post. This calculation is greatly simplified by the fact that we have only a single hop addition and we know the final IBU. As stated in the previous post, we will be using the Tinseth method for IBU calculations. Let's take a look at that equation (Information from HowTo Brew):

IBU = (AAU * U * 75) / Volume of Recipe

• AAU = Alpha Acid Units = (weight of hops in oz. * % Alpha Acid of hops)
• U = Utilization factor, which is a factor of the wort gravity and boil time
• Volume of Recipe = the batch size = volume in fermenter (V_fermenter)

Now let's take a look at the utilization factor. We said it is a function of both gravity and time. Therefore we can express the equation for utilization as:

U = f(G) * f(t)

I won't go into an in-depth explanation of why these functions are relevant except to say that the bigger the beer (higher gravity) the less effect the hops will have (the utilization factor decreases). The amount of time the hops are boiled also affects the IBU's.  The longer the boil, the more the alpha acids are isomerized (the utilization factor increases). The equation for each function is:

f(G) = 1.65 * 0.000125^(Gboil - 1)

f(t) = [1 - e^(-0.04 * t)] / 4.15

• Gboil = pre-boil specific gravity or specific gravity at time of hop addition if added later in the boil
• t = time in minutes that the hops are in the boil

We already know the IBU of the recipe,but we need to calculate the hop amount. Therefore we rearrange our IBU calculation to solve for the AAU:

AAU = (IBU * Volume of Recipe) / (U * 75)

Also, remember that

AAU = Weight of hops in ounces * %AA of hops

So far we have all the information needed to solve the equations except for the gravity of the boil (Gboil). Actually, Gboil should be the gravity at the time of the hop additions. In our recipe specifics from the previous post, we know that we are boiling for 80 minutes, and adding the hops with 60 minutes remaining in the boil. From the post on water amounts, we know that I lose 0.117 qt/min. For 20 minutes of boil time (counting down from 80 minutes to 60 minutes), we will lose about 2.34 quarts of wort. The pre-boil volume calculated in the previous post was 6.5 gal or 26 quarts. This means we will have a volume of 26 – 2.34 = 23.66 quarts (about 6 gal.) at the time we add the hops.

We now know our wort volume at the time of the hop addition, but we need to know the gravity at this time. For this, we use the fact that the sugar in the wort remains constant– it is the concentration that changes due to volume changes. This means that the gravity points multiplied by the volume at point 1 is the same as the gravity points multiplied by the volume at point 2. The equation looks like this:

G1 * V1 = G2 * V2

We know the gravity and volume in the fermenter, and we know the volume at the time of the hop addition. Therefore, we rearrange the equation to solve for G2 (point 1 represents the fermenter and point 2 represents the boil):

Gboil = (Gfermenter * Vfermenter) / Vboil

Plugging in the numbers:

Gboil = (130 * 3.25) / 6.0 = 70.4

Represented as specific gravity: 1.0704

In this manner, you can calculate the expected specific gravity values for various points in the process (namely pre-boil) and augment the recipe with DME as necessary in order to hit the expected original gravity. That exercise is for a future post. In the meantime, we now have all the data needed to plug into the IBU equation and find our needed hop amounts. Let's summarize the data we have:

Variable	Value
IBU	60
Volume of recipe (Vfermenter)	3.25 gal
Time in minutes (t)	60 min
Gboil	1.0704

Let's start plugging in these numbers.

Utilization Equations

f(G) = 1.65 * 0.000125^(1.0704 - 1)
f(G) = 0.876

f(t) = [1 - e^(-0.04 * 60)] / 4.15
f(t) = 0.219

U = 0.876 * 0.219 = 0.192

IBU Equation

AAU = (60 * 3.25) / (0.192 * 75)
AAU = 13.54

Hop Amount

Recall that AAU can be represented with the equation:

AAU = weight of hops in ounces * %AA of hops

Solving for the weight in ounces:

weight of hops in ounces = AAU / %AA of hops

Our recipe calls for Willamette hops, which have a %AA in the range of 3.5 to 6.0. For this example, we will assume that the %AA = 4.75 (average value). When I purchase the hops, I will use the %AA on the package and re-calculate the number of ounces needed. For now, we get a good ballpark value:

weight of hops in ounces = 13.54 / 4.75

weight of hops in ounces = 2.85

Summary

We calculated our utilization number,but we could have just used a simple lookup table found here. You will see that we are right in line with the table values. I do question whether or not the volume in the fermenter should be used, or if we should have used the volume after boiling. I'm not sure it matters since these equations are nothing more than estimates. It is more important that we solve for the equations in the same manner each time, and that we use the same equation each time (Tinseth as opposed to Daniels or Rager or Mosher or....). Through the consistent use of equations, we build up correlations between the IBU value and the hop profile sensed in the beer.

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Recipe Formulation with Water Calculations

In this post, we will formulate a new recipe and calculate the needed water amounts at the same time. This is not a big deal since the only real water change is the amount of water absorbed by the grains. Because we do not yet know the amount of grains we will use, we calculate the water from the target batch size back until we get to the water absorbed by the grains. At that point, we are able to calculate our grain amounts and then complete the water calculations. Rather than just talk about this process, I want to demonstrate the steps needed so that it serves as a complete example for reference.

Before moving on, I feel I need to make something clear: The Original Gravity (OG) and Final Gravity (FG) measurements are both taken from my fermenter. The OG is measured prior to adding the yeast for fermentation, and the FG is measured after fermentation has completed. This is important, and I did not make this distinction previously which could lead to some confusion when talking water amounts used in calculations.

OK, I am excited about this recipe because it is a first attempt at cloning my favorite beer: Goose Island's Bourbon County Stout. In this post, I do not want to get into my feelings towards the InBev purchase of Goose Island, so I will just summarize that my initial optimism and acceptance was turned sour by the discovery/learning of the unscrupulous InBev corporate views of the craft beer industry and their unethical business practices. So, all the better if the beer can be replicated at home. On to the recipe....

As I mentioned, I am excited to try and brew this beer, but the recipe guidelines are not from me. The grains and hops used are from the Goose Island description linked above; The gravity values used are from a post in the HomeBrewTalk forums (using the 2007 version of the beer); The grain percentages used are also from a post in HomeBrewTalk. Here are the specifics we are using:

Recipe Parameters

Parameter	Value
Batch Size (fermenter volume)	3.25 gal.
Target Original Gravity	1.13
Target Final Gravity	1.042
Color	Black ~100 SRM
IBU	60 (assume Tinseth method)
Length of Boil	80 min.
Mash Steps	Single infusion 152°F for 60 min.
Brewhouse Efficiency	70%

Grains

Grain	% Used	Max Extraction	Color (°L)
2-Row	34	37	1.8
Munich	30	34	10
Chocolate	14	34	350
Crystal 60	10	34	60
Roast Barley	10	25	300
Debittered Black (Briess Blackprinz)	2	25	500

Hops

Willamette 3.5% to 6.0% AA @60 min. ~4 oz. (TBD)

Yeast

Safale US-05 Dry Yeast 3 packets

STEP 1: Calculate Initial Water Amounts

Remember that the constants used in finding water amounts are specific to my system. Your values may be different. In some cases they are just guesstimates waiting for me to collect more accurate information

• V_final = 3.0 gal (amount in bottles or keg)
• bottling losses: ~ 0.25 gal.

• V_fermenter = 3.25 gal.
• Transfer and trub losses: ~0.75 gal.

• V_post-boil = 4.0 gal.
• Temperature shrinkage: 4% ~ 0.2 gal.
• 80 minute boiloff: 2.34 gal.

• V_pre-boil = 6.5 gal.

This is as far as we can get without knowing the grain amounts.

STEP 2: Estimate Grain Amounts

To summarize the values of interest here:

• Original Gravity: 1.13 (130 Gravity Points)
• V_fermenter : 3.25 gal.
• Brewhouse Efficiency: 70%

     ► Total Gravity (GT)

GT = Gravity Points * Volume

GT = 130 * 3.25

GT = 422.5

This is the total number of gravity points in our fermenter that we need to be contributed by the grains.

     ► Individual Grain Contributions to Total Gravity

Here we use the percentage of each grain used to determine the amount of gravity points each individual grain contributes.

Grain	Percent Calculation	Grain Contribution
2-Row	0.34 * 422.5	143.6
Munich	0.30 * 422.5	126.7
Chocolate	0.14 * 422.5	59.2
Crystal 60	0.10 * 422.5	42.25
Roasted Barley	0.10 * 422.5	42.25
Blackprinz	0.02 * 422.5	8.5

     ► Grain Amounts Needed to Achieve Gravity

Now we use the individual gravity contributions, the maximum gravity that can be extracted from the grain and our brewhouse efficiency to find the grain amount. Here is the equation used:

Grain Gravity Contribution / (Brewhouse Efficiency * Max Extract Points)

Grain	Formula	Value (lbs)	Value (lbs And oz)
2-Row	143.6 / (0.7 * 37)	5.5 lbs.	5 lbs. 8 oz.
Munich	126.7 / (0.7 * 34)	5.3 lbs.	5 lbs. 5 oz.
Chocolate	59.2 / (0.7 * 34)	2.5 lbs.	2 lbs. 8 oz.
Crystal 60	42.25 / (0.7 * 34)	1.8 lbs.	1 lbs. 13 oz.
Roasted Barley	42.25 / (0.7 * 25)	2.4 lbs.	2 lbs. 6 oz.
Blackprinz	8.5 / (0.7 * 25)	0.5 lbs.	0 lbs. 8 oz.
TOTAL:		18 lbs.	18 lbs. 0 oz.

STEP 3: Calculate Remaining Water Amounts

We now continue where we left off calculating our water amounts. Our last calculation was for the pre-boil volume, so we start there:

• V_pre-boil = 6.5 gal.
• Transfer loss: ~0.25 gal.
• Grain Absorption: 18.0 lbs. * 0.2 gal./lb. = 3.6 gal.

• V_total = 10.35 gal. ≈ 10 gal. 1.5 qt.
  
  

STEP 4: Calculate Mash and Sparge Volumes

For my mash, I am going to use 1.3 quarts of water per pound of grain. Also, remember that I have a full 4 quarts of space under my false bottom in my mash tun, so that needs to be added to the mash water.

V_mash = 1.3 qt/lb * 18.0 lbs + 4 qt

V_mash = 27.4 qt ≈ 6.85 gal. = 6 gal. 3.5 qt.

The sparge volume is now just a matter of subtracting the mash volume from the total volume.

V_sparge = V_total – V_mash

V_sparge = 10.35 gal – 6.85 gal.

V_sparge = 3.5 gal. = 3 gal. 2 qt.

Since I doubt I am 100% correct, I will make sure I have probably 5.0 gallons of sparge water available. I will just need to make sure I hit the proper volume in my boil kettle while watching the gravity readings of my runnings.

Conclusion and Remaining Information Needed

So that is it for calculating our grains and water. Of course, there is some information missing in regards to brewing the recipe:

1. We really didn't do anything with the hops. We know the IBU we need to hit, and I mentioned that we are using the Tinseth method for IBU calculations. Through that, we can estimate our hop amounts (it also depends on the actual %AA of the hops you purchase). Maybe this will be my next post.
2. Based on the gravity and volume amounts, I believe the 3 packages of dry yeast will be sufficient for this big beer. One thing we did not cover is the fermentation schedule (times and temps). This will definitely be racked to a secondary fermentation where we will allow the beer to sit on bourbon-soaked oak cubes for probably 6 months. I am not 100% sure, but I will probably let this sit in primary for two weeks; with week one being at a cooler temperature of maybe 60 degrees or so, and the second week at around 68 degrees. The secondary will be 6 months, and that temperature will probably be the same 68 degrees (not a lot of options for my basement).
3. Speaking of bourbon-soaked oak cubes, how much bourbon and how many cubes? Since we only have 3 gallons, I am assuming one 1 ounce cube. As far as the bourbon, I am not sure yet (amount or brand). I used Makers Mark and oak chips for my last beer and was less than happy, so this time I am moving to cubes and some other bourbon brand.

So these are questions that still need to be answered, but we are well on our way to making a first attempt at our clone. If this recipe doesn't work out, we get to drink what will probably still be a great beer and then try again. If it does work, we make more. Truly a win-win.

Math Equations Test

Here are some examples of entering math equations (normal text representations added for readers that strip JavaScript):

• SimpleFraction:
• 23/6.75

$$\frac{23} {6.75}$$

• Test:
• cos(x) = (e^(ix) + e^(-ix)) / 2

$$\cos x = \frac{e^{ix}+e^{-ix}}{2}$$

Hopefully we see equations above and not gibberish. I say this because the preview of the page is not rendering correctly, but someone stated that the published version is fine. So, here goes....

Woo Hoo, it worked! Thanks to this post. I still need to test this on mobile readers (and resize my images too), and then it is back to beer again.

Edit: So it does not work in my iPad blog reader (Blogshelf II - an awesome reader that apparently simplifies the blog pages for easier reading). This is where I also have an issue with the image sizes. If I view the blog with a web browser, all is fine. What I would like to find is a way for the equations to default to some text if the MathJax JavaScript is not present (kind of like displaying text instead of a SVG graphic if it is not supported by the browser). I will continue to look into this, but will still get to the next post as well (in case anyone cares). Since the equations are nothing fancy, I really can represent them without an equation renderer; so I will change to the standard characters if a solution is not found)