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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