From the HBD Archive
From: Mike Lemons <mikel@cts.com>
Subject: Hop Utilization Mathematics
Date: 1995-03-05 19:52:15 GMT

Background: I wrote a program a couple of years ago that tells me how
much hops to add to hit a target IBU. I've been satisfied with the
results. Then I bought Mark Garetz's book last week and changed the
program all around to match what's in his book. The next day, I helped
my uncle brew an SNPA clone. (For the first time, I knew how to
calculate utilization for partial boils.) The amount of hops that we
added seemed really high, so I FTP'ed a month's worth of HBD's to see if
"Boy, am I stupid."

I found some hard figures in HBD #1657 from "George A. Dietrich"
<74543.310@compuserve.com>. Here's what happened when I ran his figures
through my equations:

25.6 IBU Pilsner:
Garetz: 19.25 IBU Rager: 28.73 IBU

35.3 IBU Steam:
Garetz: 20.90 IBU Rager: 35.30 IBU

The Rager equation was so close on that second beer it's scary! I can
explain the error on the first beer as being a result of using a low
alpha-acid hop for bittering. If we assume that the hops were 3.5%
instead of 3.6%, the Rager equation predicts 19.48 IBU.

Technical Details
=================
I left all of Mark Garetz's modifiers in the program and switched back
and forth between the two utilization equations that follow. However,
the only Garetz modifier that isn't at unity for this batch is the
"Hopping Rate Factor":

utilization /= 1.0 + (ibu / 260.0);

(I put in the lab results for ibu.) This appears to be the reason why
my results differ from other people's results. The HF factor decreases
predicted hop utilization by 13.6% for the steam beer. Does anyone know
where Mark got this equation? Is it made up or does it come from sound
research? It sure makes the numbers come out right. We need more data,
though.

Here are my other equations:

total_ibu += 2.835 * alpha_acid *
calc_util(minutes, gravity) *
ounces / (gallons * 3.7854);

The Garetz equation:
if(minutes < 11.0) return(0.0);
utilization = 7.2994 + 15.0746 * tanh((minutes - 21.86) / 24.71);

The Rager equation:
utilization = 18.109069 + 13.862039 * tanh((minutes - 31.322749) / 18.267743);

(tanh is the hyperbolic tangent)

Someone else figured out the coefficients of the Rager equation a long
time ago. I think that he used some technique that actually requires
thought, while I just plugged the Garetz equation into a computer, told
it to minimize the total error, and let it run for a couple of days.

Here are the results of the equations plotted against the authors' tables:

Garetz Rager
Time Equation Table Time Equation Table
13.0 2.08 2.00 4.0 5.57 5.00
18.0 4.94 5.00 8.0 6.25 6.00
23.0 8.00 8.00 13.0 7.53 8.00
28.0 11.00 11.00 18.0 9.48 10.10
33.0 13.72 14.00 23.0 12.20 12.10
38.0 16.00 16.00 28.0 15.62 15.30
43.0 17.81 18.00 33.0 19.38 18.80
48.0 19.18 19.00 38.0 22.96 22.80
55.5 20.55 20.00 43.0 25.93 26.90
65.5 21.54 21.00 48.0 28.12 28.10
75.5 22.00 22.00 55.5 30.14 30.00
85.5 22.21 23.00 Total Error = 4.19
Total Error = 2.67

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