There is a basic formula that can be used to calculate the weight of hops added to wort during boiling, which is: Hop weight (grams) = ((liters wort) * (IBU contribution of addition) * (0.001)) ÷ ((% utilization) * (% alpha acids of hop addition))
There are certain calculations that simply do not make sense converting to English units, and hop calculations are one example. The IBU (International Bitterness Unit) is expressed as milligrams iso-alpha acids/liter beer, so the calculations using IBUs are metric by nature. When crunching numbers with this equation it is important to convert the utilization and alpha acid percentages to decimals; for example, 30% utilization should be input as 0.30, not 30.
Hop utilization is the key to bitterness estimation through hop calculations. This value numerically defines the amount of iso-alpha acids in the finished beer compared to the amount of alpha acids added to the wort during boiling. If you have a beer with 20 IBUs that was made by adding only one hop addition during boiling and the hop utilization was determined to be 25%, you can determine that 80 mg of alpha acids/liter of wort (post boil volume) must have been added.
Hop utilization is not an easy value to know unless you analyze your finished beer and back-calculate utilization based on the brew. Most small craft brewers do not have the labs required for this analysis and those who do actually measure beer bitterness use outside labs. Since this analysis is not exactly cheap, most small craft brewers, and the majority of homebrewers, do not really know their hop utilizations for the various beers brewed.
Most small brewers rely on tables and equations developed by others to estimate utilization. In general terms utilization increases as wort gravity goes down and increases as exposure to hot wort increases. The hop preparation also influences utilization, with pellets usually yielding higher values than cones. And pH, kettle design, fermenter design, yeast strain and clarification method can also affect utilization.
In my experience, utilization varies between 2–35% for pellet hops added to 12 °Plato during the boil. The maximum utilization of this range occurs after about 60 minutes of boiling and the minimum is obtained when hops are added to the whirlpool after the boil. The increase in utilization with boil time is linear over the normal range of what most brewers do in the brewhouse. This means that you could approximate the results of “continuous” hopping one of two ways.
The easiest method is also the most realistic, and that is to approach the problem as a bunch of discrete additions over the course of the boil since you probably will be using pellets and not a liquid hop preparation.
For the sake of discussion purposes, I am going to assume that a batch of wort is prepared by boiling for 60 minutes, during which time 120 individual hop pellets are added to the kettle. Furthermore, let’s assume that the brewer has decided on a form of self-abuse for the brew and has decided to use a timer as a reminder to add 1 hop pellet every 30 seconds for the hour boil. The math is pretty simple. Step one is to develop a regression equation based on data from others (Malowicki and Shellhammer, for example). You can easily write a formula to provide the expected utilization for each of the 120 additions. After this you calculate the IBU contribution of each addition, sum the 120 additions and now you have an educated guess about the predicted bitterness of the finished beer.
I wish I knew more about what I do today when I took calculus during my freshman and sophomore years of college. I don’t remember how to use calculus, but do know its applications and there are many to be found in a brewery. For all the math whizzes out there in BYO-land, now is your chance to apply your integration skills to brewing. If you go all out and solve this problem by integration you really need to use a liquid hop extract preparation and design a system to deliver a constant flow rate of hop extract to the kettle. And now the problem becomes fun! You cannot simply put the extract in a burette, for example, and open the outlet valve to a set point because the flow of extract will change as the container is emptied (variable static head). I am imagining some pretty cool projects coming from this discussion.