# Hop Bittering Calculations

For those of you who dislike math, I will make this as straightforward as possible. We will use the following example:

 Joe Ale6 lbs. of Amber DME1.5 oz of 6.4% AA Perle hops (60 minutes)1 oz of 4.6% AA Liberty hops (15 minutes)For a 5 gallon recipe, we will boil 1.5 oz of Perle hops for 60 minutes for Bittering and 1 oz of Liberty for 15 minutes for Finishing. The recipe calls for 6 lbs. of dry malt extract and it will be boiled in 3 gallons of water because of the pot size. The remaining water will be added in the fermenter.

The first step is to calculate the Alpha Acid Units (AAUs).

AAU = Weight (oz) x % Alpha Acids (whole number)

AAU (60) = 1.5 oz x 6.4 = 9.6 AAUs of Perle and AAU (15) = 1 oz x 4.6 = 4.6 AAUs of Liberty

Whenever a brewer is using AAUs in a recipe to describe the quantity of hops, it is important to specify how long each addition is boiled. The boiling time has the largest influence on how bitter a hop addition makes the beer. If no times are specified, then the rule of thumb is that bittering hops are boiled for an hour and finishing hops are boiled for the last 10-15 minutes. Many brewers add hops at 15 or 20 minute intervals and usually in multiples of a half ounce (for ease of measurement).

To calculate how much bitterness the final beer will have from these hop additions, we apply factors for the recipe volume (V), gravity of the boil and the boil time. The time and gravity of the boil are expressed as the utilization (U). The equation for IBUs is:

IBU = AAU x U x 75 / Vrecipe

75 is a constant for the conversion of English units to Metric. The proper units for IBUs are milligrams per liter, so to convert from ounces per gallon a conversion factor of 75 (74.89) is needed. For the metric world, using grams and liters, the factor is 10. (For those of you paying attention to the units, the missing factor of 100 was taken up by the % in the AAU calculation.)

Gravity of the Boil
The recipe volume is 5 gallons. The gravity is figured by examining the amount and concentration of malt being used. As noted in the previous chapter, dry malt extract typically yields about 40 pts/lb./gal. Since this recipe calls for 6 lbs. of extract to be used in 5 gallons, the calculated OG = 6 x 40 / 5 = 48 or 1.048

But, since we are only boiling 3 of the 5 gallons due to of the size of the pot, we need to take into account the higher gravity of the boil. The boil gravity becomes 6 x 40 / 3 = 80 or 1.080

It is the gravity of the boil (1.080) that is used in figuring the Utilization. As you will see in the next section, hop utilization decreases with increasing wort gravity. The higher concentration of sugars makes it more difficult for the isomerized alpha acids to dissolve. I use the initial boil gravity in my utilization calculation; others have suggested that the average boil gravity should be used. (The average being a function of how much volume will be boiled away during the boiling time.) This gets rather complicated with multiple additions, so I just use the initial boil gravity to be conservative. The difference is small—overestimating the total bitterness by 1-3 IBUs.

Utilization
The utilization is the most important factor. This number describes the efficiency of the isomerization of the alpha acids as a function of time. This is where a lot of experimentation is being conducted to get a better idea of how much of the hops are actually being isomerized during the boil. The utilization numbers that Tinseth published are shown in Table 7. To find the utilizations for boil gravities in-between the values given, simply interpolate the value based on the numbers for the bounding gravities at the given time.

For example, to calculate the utilization for a boil gravity of 1.057 at 30 minutes, look at the utilization values for 1.050 and 1.060. These are .177 and .162, respectively. There is a difference of 15 between the two, and 7/10ths of the difference is about 11, so the adjusted utilization for 1.057 would be .177 - .011 = 0.166.

The Utilizations for 60 minutes and 15 minutes at a Boil Gravity of 1.080 are 0.176 and .087, respectively. Inserting these values into the IBU equations gives:

IBU(60) = 9.6 x .176 x 75 / 5 = 25 (rounded to nearest whole number) and
IBU(15) = 4.6 x .087 x 75 / 5 = 6
Giving a grand total of 31 IBUs.

Table 7 - Utilization as a function of Boil Gravity and Time

 Gravity vs. Time 1.03 1.04 1.05 1.06 1.07 1.08 1.09 1.1 1.11 1.12 0 0 0 0 0 0 0 0 0 0 0 5 0.055 0.05 0.046 0.042 0.038 0.035 0.032 0.029 0.027 0.025 10 0.1 0.091 0.084 0.076 0.07 0.064 0.058 0.053 0.049 0.045 15 0.137 0.125 0.114 0.105 0.096 0.087 0.08 0.073 0.067 0.061 20 0.167 0.153 0.14 0.128 0.117 0.107 0.098 0.089 0.081 0.074 25 0.192 0.175 0.16 0.147 0.134 0.122 0.112 0.102 0.094 0.085 30 0.212 0.194 0.177 0.162 0.148 0.135 0.124 0.113 0.103 0.094 35 0.229 0.209 0.191 0.175 0.16 0.146 0.133 0.122 0.111 0.102 40 0.242 0.221 0.202 0.185 0.169 0.155 0.141 0.129 0.118 0.108 45 0.253 0.232 0.212 0.194 0.177 0.162 0.148 0.135 0.123 0.113 50 0.263 0.24 0.219 0.2 0.183 0.168 0.153 0.14 0.128 0.117 55 0.27 0.247 0.226 0.206 0.188 0.172 0.157 0.144 0.132 0.12 60 0.276 0.252 0.231 0.211 0.193 0.176 0.161 0.147 0.135 0.123 70 0.285 0.261 0.238 0.218 0.199 0.182 0.166 0.152 0.139 0.127 80 0.291 0.266 0.243 0.222 0.203 0.186 0.17 0.155 0.142 0.13 90 0.295 0.27 0.247 0.226 0.206 0.188 0.172 0.157 0.144 0.132 100 0.298 0.272 0.249 0.228 0.208 0.19 0.174 0.159 0.145 0.133 110 0.3 0.274 0.251 0.229 0.209 0.191 0.175 0.16 0.146 0.134 120 0.301 0.275 0.252 0.23 0.21 0.192 0.176 0.161 0.147 0.134

Utilization numbers are really an approximation. Each brew is unique; the variables for individual conditions, i.e. vigor of the boil, wort chemistry, or for losses during fermentation, are just too hard to get a handle on from the meager amount of published data available. Then why do we bother, you ask? Because if we are all working from the same model and using roughly the same numbers, then we will all be in the same ballpark and can compare our beers without too much error. Plus, when the actual IBUs are measured in the lab, these models are shown to be pretty close. Click Above for Information on the Homebrewing Recipe Calculator

Hop Utilization Equation Details
For those of you who are comfortable with the math, the following equations were generated by Tinseth from curve fitting a lot of test data and were used to generate Table 7. The degree of utilization is composed of a Gravity Factor and a Time Factor. The gravity factor accounts for reduced utilization due to higher wort gravities. The boil time factor accounts for the change in utilization due to boil time:

Utilization = f(G) x f(T)
where:
f(G) = 1.65 x 0.000125^(Gb - 1)
f(T) = [1 - e^(-0.04 x T)] / 4.15

The numbers 1.65 and 0.00125 in f(G) were empirically derived to fit the boil gravity (Gb) analysis data. In the f(T) equation, the number -0.04 controls the shape of the utilization vs. time curve. The factor 4.15 controls the maximum utilization value. This number may be adjusted to customize the curves to your own system. If you feel that you are having a very vigorous boil or generally get more utilization out of a given boil time for whatever reason, you can reduce the number a small amount to 4 or 3.9. Likewise if you think that you are getting less, then you can increase it by 1 or 2 tenths. Doing so will increase or decrease the utilization value for each time and gravity in Table 7.

Calculating the IBUs for each hop addition will help you to design your own beer recipes. You will not be a slave to any recipe book but will be able to take any beer style, any combination of malts, and plan the amount of hops to make it a beer you know you will like.

References
Garetz, M., Using Hops: The Complete Guide to Hops for the Craft Brewer (HopTech, Danville, California, 1994).

Pyle, N., Ed., The Hop FAQ, 1994.

Tinseth, G., The Hop Page, 1995.

Tinseth, G., personal communication, 1995.