### by Rob | March 6th, 2009

I spent a good amount of time the last couple of days reading up on the strategies homebrewers can take to estimate the International Bittering Units (IBU) of their beer. If you’re not already familiar with what IBUs are, or how to calculate them, the internet is full of interesting articles on the subject. I’d start at Norm Pyle’s Hops FAQ. I’ve recently become interested in compiling a collection of all the useful brewing related equations. I know I could buy something like ProMash or BeerSmith and I’m sure they do the job fantastically, but part of the enjoyment of home brew is actually having to work for it. And, plus punching in numbers into a calculator wouldn’t satisfy my curiosity. So, as time permits, I am hoping to put together some software or perhaps Maple Worksheets to assist me in my brews.

If you take a look at Norm Pyle’s Hops FAQ he lays how to calculate Jackie Rager’s method for estimating IBUs. One of steps in this method is to calculate the hop utilization, which is a function of how long the hops are boiled. Here is the table of data points provided:

Boiling Time (minutes) %Utilization ----------------------------------- 0 - 5 5.0 6 - 10 6.0 11 - 15 8.0 16 - 20 10.1 21 - 25 12.1 26 - 30 15.3 31 - 35 18.8 36 - 40 22.8 41 - 45 26.9

As an alternative to the table a function was given to approximate the hop utilization. Note that *x* is the number of minutes the hops are exposed to a boiling wort.

I wasn’t able to find the original article (”Calculating Hop Bitterness in Beer”, Zymurgy Special 1990) by Rager, so I’m not entirely sure where the function came from or if the data points referenced are exactly as I have them here. In any case, I graphed the function above against the plotted points to see how well they matched up. I found that the curve deviates from the data as much as 10%. I wanted to find a curve with a more accurate representation of the data points. Specifically a curve that actually hit each point in the data set.

There are a few different methods of finding a best-fit line. The first method I tried was polynomial interpolation. The result wasn’t horrible but the function shot off wildly when *x* went beyond the scope of what was defined by Rager’s data set. The piecewise method of interpolation, cubic spline, ended up giving me the nicest looking curve. The graph below tracks 4 pieces of information; (1) Rager’s data points, (2) Rager’s data points on 5 minute intervals, (3) Rager’s utilization function, and finally (4) the cubic spline interpolation of Rager’s data points.

The nice thing about the interpolation is the utilization percentage at a specific boiling time is extremely close to the actual data set. You don’t get that with Rager’s utilization function. Unfortunately the function defining the cubic spline interpolation is definitely not as easy on the eyes as its curve is! I mentioned earlier the cubic spline method of finding a best-fit line is done piecewise, so there is a polynomial defined between each interval in the data set. As you’ll see in the function below I am using Rager’s function when *x* > 45; this is because I don’t have data for hop utilization beyond 45 minutes. It’s not clear what the function is suppose to look like after *x* > 45, so it seemed reasonable enough to just do exactly what Rager’s utilization function did.

I’m not entirely sure interpolation is any how “better” than Rager’s utilization function, or even just using the table directly. The 10% deviation I mentioned is probably not very noticable. Until the day I have a chemist’s lab in my house and am able to measure IBUs I’ll be stuck with the art of estimation. I’ve heard it’s best to stick with one particular method, and regardless of actual IBUs you’ll at least be able to consistently hop the beer. Makes sense to me! For those interested here is the PDF of my Maple Worksheet: Interpolation of Rager’s Data Points. Cheers!