Tag Archives: IBU

mIBU Experiments #1 and #3

Abstract
This post summarizes two of the three experiments I conducted in order to evaluate the accuracy of the mIBU approach described earlier, specifically Experiments 1 and 3. (The second experiment is described in a separate post, “An Analysis of Sub-Boiling Hop Utilization“.)  The results from the current two experiments show that when estimating IBUs, it’s important to have good estimates of (a) the alpha-acid rating of the hops, (b) storage conditions of the hops, (c) alpha-acid concentration in the wort, and (d) age of the beer.  If these factors are accounted for, the IBU estimates in these experiments are fairly close to measured IBU values.  When the wort is allowed to cool naturally after flameout for (in this case) 15 minutes, the use of the mIBU approach yields much better estimates for hop additions at flameout and with short boil times.

Introduction
For the first experiment, I brewed four batches of beer with hops added at different times during the boil and with forced cooling at flameout, in order to calibrate my brewing setup and resulting measured IBU values with the Tinseth IBU formula.  For the third experiment, I brewed five batches, each with 15 minutes of post-flameout natural cooling, to compare the measured IBU values with values predicted by the Tinseth formula and the mIBU approach.

In both of these experiments, IBU values were measured by Analysis Laboratory.  Scott Bruslind from Analysis Laboratory was very responsive and encouraging, providing a full set of measurements (including gravity, pH, and attenuation, in addition to IBUs) as well as alpha-acid measurement of hops.

Experiment #1
The first experiment calibrated measured IBUs obtained from my brewing setup with the standard Tinseth IBU formula.  As a result of this experiment, I got some idea of how much variation to expect in IBU measurements, and I found that several factors inadvertently impacted both measured and modeled values.

Experiment #1: Methods
In this experiment, four batches of beer were brewed with forced cooling at flameout.  Each batch was brewed separately: 2.0 lbs (0.91 kg) of Briess dry malt extract in 2 G (7.6 liters) of water, with 0.60 oz (17.0 g) of Cascade hop cones (in a loose mesh bag) and a slurry of 0.08 oz (2.3 g) of Safeale US-05 yeast.  The boil time of the wort for all conditions was 60 minutes.  The hops were added at 60 minutes (condition A), 40 minutes (condition B), 20 minutes (condition C), and 10 minutes (condition D) prior to flameout.  All batches had the following targets: pre-boil volume of 2.15 G, pre-boil specific gravity of 1.043, post-boil volume of 1.45 G, and (post-boil) original gravity (OG) of 1.060.  The wort was quickly force-cooled and the hops were removed immediately at flameout.  The wort was left to sit, covered, for several minutes, and then 3½ quarts were decanted into a 1 G (4 liter) container.  After 90 seconds of aeration (a.k.a. vigorous shaking), the yeast was pitched.  Fermentation and conditioning proceeded for 19 days.  The beers were bottled (with 0.46 oz (13 g) of sucrose per condition as priming sugar) and left to bottle condition for an additional 8½ weeks before IBU values were measured.

The Cascade hops, purchased in June, had an alpha-acid (AA) rating on the package of 8.0%.  I had the alpha acids measured close to the time of the experiment by both Analysis Laboratories (AL) and subsequently by KAR Labs (KAR).  The AL alpha-acid rating was 6.25% (with 7.25% beta acids and a Hop Storage Index (HSI) of 0.45), and the KAR rating was 4.11% (with 5.40% beta acids).  An HSI of 0.45 indicates 28% loss or 72% AA remaining, which translates into an AA rating on brew day of 5.76% if the harvest AA rating was 8.0%, or a harvest AA rating of 8.7% if the level was 6.25% at the time of the experiment.  From the AL numbers, the alpha/beta ratio is 0.862 and the from the KAR numbers, the alpha/beta ratio is 0.761, both on the low side for Cascade.  From these various numbers, two things are clear: (1) the actual AA rating at the time of brewing could easily have been anywhere from about 4% to 6.25%, which is a pretty wide variation, and (2) I had inadvertently used hops that had been improperly stored.  Afterwards, I had a nice chat with my LHBS, and they confirmed that while the hops were stored in very good mylar bags, they spent at least part of the year in an air-conditioned room at the back of the store.  I’ve since become much more concerned and proactive about the storage conditions of my hops.  At any rate, Glenn Tinseth recommends, if needed, adjusting the linear scaling factor (4.15) in his equation to fit the current conditions, so we can pick our best guess of the AA rating and adjust the scaling factor to fit the data.  Equivalently, we can pick one scaling factor (e.g. the recommended 4.15) and adjust the AA rating to fit the data.

Experiment #1: Results
Table 1 (below) shows measured and modeled IBU values for each of the conditions in Experiment 1, along with a variety of other measured parameters (e.g. original gravity).  The observed and modeled IBU values are plotted below in Figure 1.

Determining the post-boil volume was a little tricky… if the hops are in the wort they will increase the measured volume by displacement, and if they are removed from the wort they will decrease the volume by soaking up wort.  In the end, I took the ratio of pre-boil gravity points divided by post-boil gravity points, and multiplied that by the initial volume.  The post-boil specific gravity (i.e. the OG) measured by Analysis Laboratory was determined from the original extract reading in degrees Plato.

The average alpha acid concentration of about 210 ppm for all conditions is less than the threshold of 230 ppm that seems to be the cutoff for a linear increase in IBU values with alpha-acid concentration.  Therefore, the Tinseth equation should still yield good results at this concentration.

For IBU values from the Tinseth equation, I used the recommended scaling factor of 4.15 and the average specific gravity of the start and end of the boil, as recommended by Tinseth, and adjusted the AA rating to minimize the error.  This yielded an AA rating of 5.79%, about the middle of the range between 4.00% and 6.25%, and a root-mean-squared (RMS) IBU error of 4.32 IBUs.  How good (or bad) is this error?  It’s hard to say, but it’s within the reported perceptual threshold of 5 IBUs, with one condition having a difference of about 7 IBUs.  The problem in getting a better fit is that the modeled IBU value at 60 minutes is higher than the measured IBU, and the modeled IBU at 10 minutes is lower than measured; a linear scaling factor can’t fix that.  These differences at high and low steeping times may be due to the large amounts of oxidized alpha and beta acids in the poorly-stored hops that I used.

In a separate blog post, I present a more detailed model of IBUs; the values obtained from that model for this experiment are also given in Table 1.  This more detailed model takes into account factors such as original gravity, hopping rate, age and storage conditions of the hops, alpha/beta ratio, age of the beer, and form of the hops.  Using this model, the estimated AA rating at harvest was 8.0% (the same as the value on the package) and the estimated degradation factor was 0.82 (somewhat higher than the HSI-based factor of 72%), yielding an AA rating on brew day of 6.6%.  An AA rating of 6.6% is higher than the AA rating estimated from the Tinseth equation (5.79%) but close to the value from AL (6.25%).  The estimated alpha/beta ratio was 0.75, somewhat lower than the value from AL (0.86) but very close to the KAR value (0.761).  The RMS error from this model was 2.93 IBUs (about two-thirds the error of the Tinseth model), with a maximum difference of 3.7 IBUs.  According to this model, isomerized alpha acids contributed 67%, 61%, 48%, and 36% to the IBU values of conditions A through D, respectively.  The low percentage for even the 60-minute boil is due to the age, poor storage conditions, and low alpha/beta ratio of the hops.  I used the average boil gravity and average volume over the other four conditions to estimate 4.9 IBUs at a boil time of 0 minutes (0% from isomerized alpha acids); this value is higher than it would typically be, because of the poor storage conditions of the hops.

condition
A
condition
B
condition
C
condition
D
pre-boil SG (from hydrometer)
1.042 1.0425 1.042 1.042
pre-boil volume
2.11 G / 7.99 l 2.13 G / 8.06 l 2.15 G / 8.14 l 2.15 G / 8.14 l
time of hops addition
60 min 40 min 20 min 10 min
post-boil SG (from hydrometer)
1.059 1.058 1.061 1.063
post-boil SG (measured by AL)
1.05986 1.05891 1.06337 1.06417
post-boil volume 1.49 G / 5.64 l 1.54 G / 5.83 l 1.44 G / 5.45 l 1.42 G / 5.38 l
FG (measured by AL)
1.01134 1.00863 1.00928 1.00950
measured IBUs (from AL)
35.7 34.3 27.1 22.0
IBUs from Tinseth
40.0 34.0 24.7 14.9
IBUs from detailed model
38.9 32.1 24.9 18.2

Table 1. Measured and modeled values of the four conditions in the first experiment.  Results provided by Analysis Laboratories are indicated by “AL”.

mIBU-exp1

Figure 1. Measured IBU values (red line), IBU values from the Tinseth model (blue line), and IBU values from the detailed model (green line). The model values were fit to the measured values by minimizing the error, which was necessary because the AA rating at brew day was basically unknown.

Experiment #1: Conclusion
A number of issues came up when analyzing the data from this experiment.  The point of this first experiment was, in some sense, to discover such issues and be able to address them in subsequent experiments.   (Regardless of the numerical results, all of these experiments have been a wonderful learning opportunity.)  Here’s a list of bigger issues with the first experiment: (1) I don’t have a reliable estimate of the AA rating of the hops on brew day, which obviously impacts any modeled IBU value; (2) the hops were improperly stored, which drastically decreased the amount of alpha acids and increased the amount of oxidized alpha and beta acids, impacting the measured IBU values; (3) I used a digital kitchen scale to measure 0.60 oz of hops, which was OK but not ideal… I’ve since upgraded to a more precise jewelry scale; and (4) boiling a small amount of wort for 1 hour yields a large change in specific gravity and an evaporation rate that is very difficult to control, leading to unwanted variability.

Despite these issues, fitting the AA rating to the IBU data yielded a not-terrible fit to the Tinseth model (with an RMS error of 4.32 IBUs).

Experiment #3
The third experiment was similar to the first, except that the wort was left to sit and cool naturally for 15 minutes after flameout.  The purpose of this experiment was to compare measured IBU values with IBU values predicted by the Tinseth formula and the mIBU approach.

Experiment #3: Methods
In this experiment, five batches of beer were brewed with 15 minutes of natural cooling at flameout, and forced cooling when the 15-minute mark was reached. This time, I made one batch of wort and divided it into equal portions for each condition.  In this case, 9.25 lbs (4.20 kg) of Briess dry malt extract was added to 7.0 G (26.5 liters) of water to yield 7.75 G (29.34 liters) of wort, with a specific gravity of 1.057.  This wort was boiled for 30 minutes and left to cool with the lid on. The specific gravity of the wort after the 30-minute boil was 1.062, with a volume of about 7 G (26.5 liters).  The wort for each condition was taken from this larger pool of wort, to guarantee the same specific gravity at the start of the boil.  The hops were boiled for 60 minutes (condition A), 30 minutes (condition B), 15 minutes (condition C), 7½ minutes (condition D), and 0 minutes (condition E).

For each condition, 1.3 G (4.92 liters) was heated to boiling.   When the wort reached boiling, 0.80 oz (22.7 g) of Cascade hops were added.  The wort was boiled for the amount of time specified for each condition, and the boil was conducted with the lid on, in order to minimize evaporation losses and keep the boil gravity from increasing too much.  At flameout, the lid was removed (to make it easier to measure the change in temperature over time) and the hops remained in the wort.  At 15 minutes after flameout, the hops were removed and the wort was quickly cooled.  The wort was left to sit, covered, for several minutes, and then 3½ quarts (3.31 liters) were decanted into a 1 G (4 liter) container.  After 90 seconds of aeration (a.k.a. vigorous shaking), a slurry with 1.5 oz (42.5 g) of Safeale US-05 yeast was pitched into each condition.  Fermentation and conditioning proceeded for 21 days.  The beers were bottled (with 0.45 oz (12.75 g) of sucrose per condition as priming sugar) and left to bottle condition for an additional 5 weeks before IBU values were measured.

In order to have better control over the hops in this experiment, I used some of my precious home-grown Cascade.  The AA rating at harvest, measured by KAR Labs, was 6.64% (with a beta acid percentage of 5.38%).  While they were nearly 8 months old at the time of the experiment, I had stored them in vacuum-sealed bags in a freezer at  -6°F (-21°C).  Around the time of the experiment, I sent samples to both KAR Labs and Alpha Analytics.  This time, KAR Labs reported an AA rating of 6.66% and beta acid level of 5.51%; Alpha Analytics reported an AA rating of 7.70% and beta acid level of 6.80%.  The HSI value from Alpha Analytics was 0.22, indicating no significant degradation over the 8 months.  Once again, there was a surprising lack of clarity in the AA rating from the laboratory-measured values… it could be anywhere from 6.6% to 7.7%, or even outside this range.  The alpha/beta ratio was approximately 1.1 to 1.2.  Fortunately, the data from both KAR Labs and Alpha Analytics indicate that the hops were well preserved, so the hop degradation factor should be close to 1.

Experiment #3: Results
Table 2 provides measured and modeled IBU values for each of the conditions in Experiment 3, along with a variety of other measured parameters. The observed and modeled IBU values are plotted below in Figure 2. The post-boil volume and specific gravity were determined using the same methods as in Experiment 1.

I thought that by keeping the lid on the kettle during the boil, there would be almost no evaporation and therefore almost no change in specific gravity between conditions.  Instead, I found a fairly large change in original gravity between the different conditions, probably because I did take off the lid occasionally to stir the wort.  In the future, I’ll have to take this source of variability into account.

In this experiment, the alpha-acid concentration of about 345 ppm was (unfortunately) well above the estimated threshold of 230 to 260 ppm.  (The alpha-acid concentration can be computed as AA × W × 1000 / V, where AA is the alpha-acid rating of the hops (on a scale from 0 to 1), W is the weight of the hops (in grams), and V is the volume of the wort (in liters).  Therefore, the Tinseth equation will predict IBU values higher than measured IBU values, unless this concentration is taken into account.

I kept a minute-by-minute record of the decrease in temperature after flameout for each condition.  Since the volume of each condition was similar, the temperature decay was also similar for each condition.  I used a single temperature-decay function, fit to the temperatures from all five conditions, to model post-flameout temperature decay in this experiment:  temp = 0.1065t2 – 5.1294t + 211.682, with temperature temp measured in Fahrenheit and time t measured in minutes.  (While larger volumes seem to fit well with a straight line, these small volumes had a temperature decay that fit much better with a quadratic function.)

The recommended scaling factor of 4.15 in the Tinseth model did, in fact, yield predicted IBU values that were much higher than measured values.  In the first experiment, it seems that the default value worked well as a compromise between the age of the beer (which, unaccounted for in the Tinseth model, would have yielded larger predicted values than measured values) and the degradation of the hops (which, given the storage conditions and alpha/beta ratio less than 1, would have yielded smaller predicted values than measured values).  In this third experiment, the storage conditions and alpha/beta ratio are probably closer to what Tinseth used when he developed his model, and so the combination of hopping rate and age of the beer yielded predicted values much greater than measured values when using the default scaling factor.  The purpose of this experiment is to compare the Tinseth and mIBU models, and so we can adjust the scaling factor in both models to fit the data, and see which model produces values closer to the measured values given the best scaling factor.  In this case, a scaling factor of 6.15 with the AA rating estimated by the detailed model (6.0%, as described below) provided the best fit of the Tinseth model to the measured IBU values.  With this scaling factor, there is an RMS error of 8.33 IBUs and a maximum difference of 16.1 IBUs (at the 0-minute condition).  (If a different AA rating is used, the same error is obtained with a different scaling factor.)

Another option for fitting the data is to explicitly account for the hopping rate and age of the beer, and use the recommended scaling factor of 4.15 in both the Tinseth and mIBU models.  We can approximate the alpha-acid solubility limit by simply limiting the alpha-acid concentration in the Tinseth equation to 260 ppm.  (Computationally, we can adjust the weight of the hops to an “effective” weight that limits the alpha-acid concentration to no more than 260 ppm at the beginning of the boil.)  We can estimate the impact of age on IBUs using an adjustment factor developed in a separate blog post: 0.32 × e0.08 ageweeks + 0.68, where ageweeks is the age of the beer in weeks.  With these modifications to the Tinseth formula and the recommended scaling factor of 4.15, there is an RMS error of 8.24 IBUs and a maximum difference of 16.1 IBUs (at the 0-minute condition).

For the mIBU model, a scaling factor of 6.60 provides the best fit to the data when not accounting for alpha acid concentration or age of the beer.  In this case, there is an RMS error of 1.92 IBUs, with a maximum difference of 3.41 IBUs (at the 0-minute condition).   When accounting for these two factors and using a scaling factor of 4.15, there is an RMS error of 1.89 IBUs, with a maximum difference of 2.74 IBUs (at the 30-minute condition).

For the more detailed model, the best fit was obtained by adjusting the AA rating, alpha/beta ratio, and decay factor to fit the data.  An AA rating of 6.0% (somewhat lower than the value of 6.64% reported by KAR), an alpha/beta ratio of 1.6 (higher than the value of 1.21 reported by KAR), and a decay factor of 0.95 provided the best fit to the data.  With these values, there is an RMS error of 1.36 IBUs and a maximum difference of 2.5 IBUs (for the 60-minute condition).  According to this model, isomerized alpha acids contributed 75%, 67%, 56%, 44%, and 23% to the IBU values of conditions A through E, respectively. Given the good storage conditions of the hops, the fairly low iso-alpha percentage for even the 60-minute boil is, in this case, due to the alpha-acid concentration above the solubility limit.

condition
A
condition
B
condition
C
condition
D
condition
E
pre-boil SG (from hydrometer)
1.062 1.062 1.062 1.062 1.062
pre-boil volume
1.30 G / 4.92 l 1.30 G / 4.92 l 1.30 G / 4.92 l 1.30 G / 4.92 l 1.30 G / 4.92 l
time of hops additions
60 min 30 min 15 min 7.5 min 0 min
post-boil SG (from hydrometer)
1.075 1.069 1.067 1.069 1.065
post-boil SG (measured by AL)
1.0760 1.0720 1.0685 1.0689 1.0658
post-boil volume 1.06 G / 4.01 l 1.12 G / 4.42 l 1.18 G / 4.47 l 1.17 G / 4.43 l 1.22 / 4.62 l
FG (measured by AL)
1.01190 1.01114 1.01008 1.01016 1.00944
measured IBUs (from AL)
46.4 35.4 26.1 21.2 16.1
IBUs from Tinseth, scale 6.15
49.2 36.6 22.6 13.0 0.0
IBUs from Tinseth, scale 4.15
44.6 35.0 22.8 13.0 0.0
IBUs from mIBU model, scale 6.60
46.8 37.1 26.3 19.3 12.7
IBUs from mIBU model, scale 4.15
45.5 38.1 28.5 20.7 14.2
IBUs from detailed model
48.9 35.8 26.1 20.6 14.6

Table 2. Measured and modeled values of the five conditions in the third experiment.  Results provided by Analysis Laboratories are indicated by “AL”.

mIBU-exp3-new

Figure 2. Measured IBU values (red line), IBU values from the Tinseth model (blue line), IBU values from the mIBU model (black line), and IBU values from the detailed model (green line).  The Tinseth, mIBU, and detailed-model values take into account the initial alpha-acid concentration and the age of the beer.

Experiment #3: Conclusion
Results obtained (a) by adjusting the scaling factor to fit the data, or (b) by using the default scaling factor and incorporating modifications to the Tinseth formula to account for alpha-acid concentration and age of the beer, were similar.  In both cases, the mIBU approach showed an improved estimate, especially at the 0-minute and 7½-minute conditions.  In these two cases, the differences between the two models (14.2 and 7.7 IBUs, respectively) seem to be outside the range of typical random variation, with the mIBU results much closer to measured IBU values.

The detailed model also showed a good fit to the observed data.  I find it interesting that a complicated model with many parameters performed about as well, in this case, as the simpler mIBU model, after accounting for alpha-acid concentration and age of the beer.

Overall Summary
Analysis of the results indicates: (1) In the first experiment, the poor storage conditions of the hops, the low alpha/beta ratio, and the age of the beers probably caused the values predicted by the Tinseth formula (with the recommended scaling factor) to be somewhat different from the measured IBU values, but an inability to get a good value for the alpha-acid rating of the hops on brew day prevents more specific conclusions; (2) Accounting for the hopping rate, storage conditions of the hops, alpha/beta ratio, age of the beer, and other parameters in a much more detailed model of IBUs provided a better fit to the data; (3) In the third experiment, the mIBU method provided good estimates with the recommended scaling factor of 4.15, after taking into account the alpha-acid concentration and age of the beer (and with the use of well-preserved hops); and (4) Results from the third experiment show the expected increase in IBUs caused by post-flameout utilization, and that this increase was modeled well by the mIBU method.

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A Summary of Factors Affecting IBUs

This blog post is excessively long.  In order to make it somewhat more manageable, here are links to the various sections:
1. Introduction
2. Definitions of IBUs
xxxxx2.1 IBU Definition from the American Society of Brewing Chemists (ASBC)
xxxxx2.2 IBU Definition from Val Peacock
3. A General Description of Factors Affecting IBUs
xxxxx3.1 Concentration of Isomerized Alpha Acids (IAA) Under Ideal Conditions
xxxxx3.2 Accounting for Post-Boil Utilization
xxxxx3.3 Adjustments to the Concentration of Isomerized Alpha Acids
xxxxx3.4 A Revised IBU Formula for nonIAA Components
xxxxxxxxxx3.4.1 Oxidized Alpha Acids
xxxxxxxxxx3.4.2 Oxidized Beta Acids
xxxxxxxxxx3.4.3 Polyphenols
xxxxxxxxxx3.4.4 Solubility of nonIAA Components
4. Available Data, Parameter Estimation, and Results
xxxxx4.1 Overview
xxxxx4.2 Sources of IBU Data
xxxxxxxxxx4.2.1 Tinseth Utilization
xxxxxxxxxx4.2.2 Peacock Hop-Storage Conditions
xxxxxxxxxx4.2.3 Personal Experiments
xxxxx4.3 Parameter Estimation and Results
5. Discussion of Results
6. Summary
References

1. Introduction
This blog post presents a summary of some of the factors that affect the measurement of International Bitterness Units (IBUs) (or simply “Bitterness Units” (BU) if you’re already international).  In order to provide as cohesive a summary as possible, I provide both qualitative and quantitative descriptions of these factors.  The purpose of the quantitative model is descriptive, not predictive.  In other words, the information here may be helpful in understanding how certain factors affect IBU values, but it may not be sufficient to predict the IBU level of your beer much better than existing predictive formulas (e.g. the Tinseth formula).  With so many interrelated factors and guesses of appropriate values for many factors, there is a very good chance that IBU values predicted from this quantitative description will not be the same as measured IBU values.  If, however, you simply want to get a better understanding of what components contribute to an IBU value, how the storbuage conditions and amount of hops used may impact IBUs, or how late hopping may decrease the relative proportion of isomerized alpha acids, then this might be the blog post for you.

The more I learn about hops, the more complex the topic becomes, with a seemingly never-ending level of detail. If you’re familiar with Alice In Wonderland, then this blog post goes only one level down the IBU rabbit hole, and it looks briefly through a number of open doors at that level without going through them.  In other words, there’s a lot of research, chemistry, opinions, known unknowns, unknown unknowns, and contradictions that I’m not going to touch on.  If you’re not familiar with Alice In Wonderland, then think of this post as an impressionist painting: if you stand back far enough, you should be able to see a complete picture.  If you look too closely and focus too much on the details, however, things that make sense in their relationship to other things may become, when isolated from the larger context, meaningless splotches.

The IBU measurement itself is not always highly regarded.  While it is often reported to be correlated with the bitterness of beer (e.g. [Priest and Stewart, p. 266]), the perception of bitterness is not linear (especially at high bitterness levels [Hieronymus, p. 184]), bitterness may have different qualities not captured by the IBU measurement [Peacock, p. 163], and the correlation between IBU levels and bitterness doesn’t hold up under every circumstance (e.g. with dry-hopping [Maye et al., p. 25]).  On the other hand, it is a universally-known and (sometimes grudgingly) accepted quantitative measurement.  This post doesn’t touch on the pros and cons of the IBU, but, accepting it at face value, tries to break it down into various components and relationships.

This post provides a summary of a large number of sources, including Val Peacock’s article “The International Bitterness Unit, its Creation and What it Measures” in Hop Flavor and Aroma (ed. Shellhammer); Mark G. Malowicki’s Masters thesis, Hop Bitter Acid Isomerization and Degradation Kinetics in a Model Wort-Boiling System; Michael L. Hall’s article “What’s Your IBU” in Zymurgy (1997); Michael J. Lewis and Tom W. Young’s chapter “Hop Chemistry and Wort Boiling” in Brewing; Mark Garetz’ article “Hop Storage: How to Get – and Keep – Your Hops’ Optimum Value” in Brewing Techniques, and his book Using Hops; Stan Hieronymus’ book For the Love of Hops; J. S. Hough et al.’s Malting and Brewing Science (volume 2); and many other theses, print, and internet sources.  If you look at the bibliography, you’ll see many publications produced under the guidance of Thomas Shellhammer.  I’ve tried to cite appropriately, and I’ve put the full bibliography at the bottom of this post.  I’ve omitted a lot of interesting details from these sources, in order to maintain a more focused narrative.

2. Definitions of IBUs
2.1 IBU Definition from the American Society of Brewing Chemists (ASBC)
Because of the complexity of hops and IBUs, it’s probably a good idea to start at a high level of description, which is deceptively simple but not very informative: An IBU is a measurement of the amount of absorption of light at 275 nm (abbreviated as A275nm) in a liquid, multiplied by 50.  The liquid in this case is not just any liquid, but beer that has been combined with twice as much iso-octane (TMP) and also diluted in octyl alcohol and hydrochloric acid [American Society of Brewing Chemists], i.e. “acidified beer.”  In mathematical form, we can say:

IBU = A275nm(beer) × 50 [1]

where IBU is the resulting IBU value, “beer” indicates the substance being analyzed (after proper acidification), and A275nm(beer) is the amount of light absorbed at 275 nm from a sample of acidified beer [Peacock, p. 158].

This measurement has been found to correlate well with the perception of bitterness in beer.  As Lewis and Young state, “the value for [IBU] is a good representation of the sensory bitterness of beer” [Lewis and Young, p. 266].  Why does this correlation exist?  There are three intertwining factors: (1) the absorption of light at a particular (ultraviolet) frequency (275 nm) through a sample, (2) the concentration of certain substances in this acidified beer that absorb light at this frequency, and (3) the perception of bitterness that is associated with these substances.  This blog post pretty much ignores the first and third factors, assuming that it is predominately those substances that absorb more light at this frequency that have a bitter taste in beer.  What this post focuses on, then, is the second factor: the concentration of substances in acidified beer that absorb light at 275 nm.  In the development of the IBU measurement, there was a deliberate decision to include not only the bitter isomerized alpha acids (abbreviated here as IAA) that are produced during the boiling of hops in wort, but also other “bittering substances” that contribute to the perception of bitterness [Peacock, p. 159], and which happen (by happy circumstance) to absorb light at 275 nm (as isomerized alpha acids do).

The amount of absorption of light at 275 nm by a sample of acidified beer, multiplied by 50 (or, more precisely, 51.2), was found to provide a good approximation to the concentration of isomerized alpha acids in typical beer of the 1960s (when the IBU measurement was developed) [Peacock, p. 161].   So, we can say:

[IAA]beer1960s = A275nm(beer1960s) × 51.2 [2]

where [IAA]beer1960s is the concentration of isomerized alpha acids in the 1960s beer (in mg of isomerized alpha acid per kg of beer, or parts per million (ppm)), and “beer1960s” on the right-hand side of the equation indicates that we’re measuring the absorption of a certain type of beer.  (Note that beer contains a number of types of substances that absorb light at 275 nm; IAA is the usually predominant, but not only, substance [Peacock, p. 159].)  The IBU value can approximately equal the concentration of IAA (i.e. Equations [1] and [2] can be approximately equal), but generally only for hops and boiling times typical of the 1960s, because of the relative concentrations of other bittering substances.

If one has a solution that contains only isomerized alpha acids and no other substances that absorb light at 275 nm, the concentration of IAA can be estimated with the following equation [Peacock, p. 161]:

[IAA]IAAsolution = A275nm(IAAsolution) × 69.68 [3]

where [IAA]IAAsolution is the concentration of isomerized alpha acids in this solution, and “IAAsolution” on the right-hand side of the equation indicates that the solution being analyzed contains only isomerized alpha acids as the relevant (light-absorbing) substance.

Figure 1(a) shows hypothetical (i.e. completely made up) data that represent absorption of light at 275 nm on the horizontal axis and the measured concentration of a substance X on the vertical axis.  (The data are fake, but the figure will hopefully be useful to illustrate some concepts.)  In this case, a line can be fit through the data to predict concentration given absorption: concentration = (69.68 × absorption) + 0.  The offset of this line is 0 (meaning that the predicted value for an absorption of 0 is a concentration of 0), and so we can ignore the offset, characterizing the relationship with a single multiplication factor (69.68).

absorptionVsConcentrationALL

Figure 1. Concentration as a function of light absorption for various circumstances. (a) Concentration of X is approximated by light absorption multiplied by 69.68.  (b) Both X and Y can have their concentration predicted by multiplying absorption by 69.68.  (c) The concentration of substance Z is predicted by light absorption multiplied by 696.8 (10 × X).  We can model the concentration of Z multiplied by a scaling factor (0.10) as a function of absorption, which allows us to treat it like substances X and Y (with a multiplication factor of 69.68).

2.2 IBU Definition from Val Peacock
This background leads us to a second high-level description of IBUs:  An IBU is an estimation of the concentration of isomerized alpha acids in typical 1960s beer, based on the combined concentration of isomerized alpha acids and other bittering substances in beer, multiplied by 5/7 [Peacock, p. 161].  In mathematical notation:

[IAA]beer1960sIBU = 5/7 × ([IAA]beer + [nonIAA]beer) [4]

where [IAA]beer1960s is the concentration of isomerized alpha acids in 1960s beer, [IAA]beer is the concentration of IAA in the beer being analyzed, “nonIAA” are “other bittering substances that aren’t isomerized alpha acids” in beer (which is not the same as “non-isomerized alpha acids,” despite the abbreviation), and [nonIAA]beer is the concentration of these substances in the beer being analyzed.

Why is there the multiplication by 5/7 in Equation [4]?  We can derive it from Equations [2] and [3] if we make three assumptions: (1) For substance X in an appropriate solution, if the absorption of light at 275 nm is zero (A275nm(X) = 0), then the concentration of X is zero ([X] = 0).  (2) There is a linear relationship between the absorption of light at 275 nm and the concentration of relevant bittering substances in beer, at least within the range of interest.  (3) The linear relationship between absorption and concentration is the same for all relevant substances in beer, namely 69.68.  The first two assumptions were made by the 1967 Analysis Committee of the European Brewery Convention that developed the unit that became the IBU [Peacock, p. 160-161], so they seem reasonable.  The third assumption is not necessarily true, but we can modify it for those cases where it isn’t true, so let’s assume it’s true for now.

Let’s start by looking at two beers that have the same amount of absorption of light at 275 nm (i.e. the same level of bitterness): one beer is a (cryogenically preserved) 1960s beer with this bitterness level, and the other beer is something you just brewed:

A275nm(beer1960s) = A275nm(beer) [5]

where beer1960s is our 1960s beer, and beer is the one just brewed.

We can then multiply the numerator and denominator of the left-hand side by 51.2, and multiply the numerator and denominator of the right-hand side by 69.89, and the relationship still holds:

(A275nm(beer1960s) × 51.2) / 51.2 = (A275nm(beer) × 69.68) / 69.68 [6]

The relevant bittering substances in beer are IAA and nonIAA (by definition), so we can replace beer in Equation [6] with (IAA + nonIAA):

(A275nm(beer1960s) × 51.2) / 51.2 = (A275nm(IAA + nonIAA) × 69.68) / 69.68 [7]

From Equation [3], we can multiply absorption of light at 275 nm by 69.68 to predict the concentration of IAA in a solution that contains only IAA as the relevant substance.  From our third assumption, nonIAA substances have the same relationship between absorption and concentration, so we can also multiply the absorption of light at 275 nm by 69.68 to predict the concentration of nonIAA in a solution that contains only nonIAA as the relevant substance.  This is illustrated in Figure 1(b), showing two different substances that have the same mapping between absorption and concentration.  Since the relevant bittering substances in beer are IAA and nonIAA, we can predict the combined concentration of (IAA + nonIAA) from the absorption of light at 275 nm in a solution containing both substances.  (For example, if we have 30 mg of IAA in 1 kg of solution, we have 30 ppm and light absorption of 0.43.  Likewise, if we have 21 mg of IAA and 9 mg of nonIAA in 1 kg of solution, we have a total of 30 mg of (IAA + nonIAA), or 30 ppm.  That 30 ppm will also have a light absorption of 0.43.)  Now we can map from absorption to concentration, using Equation [2] for the left-hand side and the third assumption for the right-hand side:

[IAA]beer1960s / 51.2 = [IAA + nonIAA]beer / 69.68 [8]

We can then bring the 51.2 from the left to the right by multiplying both sides by 51.2, and note that the combined concentration of both IAA and nonIAA in beer ([IAA + nonIAA]beer) is equal to the sum of the concentrations of the individual substances ([IAA]beer + [nonIAA]beer) :

[IAA]beer1960s = (51.2 / 69.68) × ([IAA]beer + [nonIAA]beer) [9]

Next, we can simplify 51.2/69.68 to 5/7, and note that then the right-hand side equals Peacock’s definition of an IBU, and the left-hand side indicates that this is approximately equal to the concentration of IAA in the 1960s beer:

[IAA]beer1960sIBU = 5/7 × ([IAA]beer + [nonIAA]beer) [4] = [10]

Let’s look at a quick example… say we brew a beer with pure isomerized alpha acids, and we end up with [IAA]beer equal to 10 ppm.  In this case, [nonIAA]beer is zero, and the measured IBU value will be 7.  A beer with the same bitterness level brewed in the 1960s would have had, typically, 7 ppm of IAA and (the equivalent of) 3 ppm of nonIAA, with the same net concentration of bittering substances (10 ppm).  As another example, let’s say we brew a beer with poorly-stored hops, and we end up with equal concentrations of IAA and nonIAA, at 10 ppm each.  Now our beer will have an IBU value of 14.  A typical beer with the same bitterness level brewed in the 1960s would have had an IAA level of 14 ppm and a nonIAA level of 6 ppm.

Now let’s revisit the assumption that the concentration of nonIAA substances can be predicted from light absorption with a scaling factor of 69.68.  For the sake of explanation, let’s consider a hypothetical case where nonIAA substances have a scaling factor of 696.8, ten times that of IAA, as illustrated in Figure 1(c).  We can then plot the concentration of nonIAA substances divided by 10 (i.e. [nonIAA]/10) as a function of light absorption (Figure 1(d)), and return to our desired IAA scaling factor of 69.68.  We then just need to note in our equation that we’re no longer modeling the actual concentration of nonIAA, but the scaled concentration [nonIAA]beer × scalenonIAA:

[IAA]beer1960sIBU = 5/7 × ([IAA]beer + ([nonIAA]beer × scalenonIAA)) [11]

where scalenonIAA is the scaling factor needed to convert the absorption-to-concentration relationship of nonIAA (696.8 in our example) to the absorption-to-concentration relationship of IAA (69.68).  In our example, scalenonIAA is 0.10.  In a similar way, we can consider nonIAA as a group of substances, each with its own scaling factor.  If nonIAA consists of three different substances, nonIAA1, nonIAA2, and nonIAA3, we can write the relationship like this:

[IAA]beer1960sIBU = 5/7 × ([IAA]beer + (([nonIAA1]beer × scalenonIAA1) + ([nonIAA2]beer × scalenonIAA2) + ([nonIAA3]beer × scalenonIAA3))) [12]

where scalenonIAA1 is the scaling factor for the first nonIAA substance, scalenonIAA2 is the scaling factor for the second nonIAA substance, and scalenonIAA3 is the scaling factor for the third nonIAA substance.

The IBU value was designed to be approximately equal to the concentration of isomerized alpha acids (in ppm), given the boiling time, alpha acid levels, and storage conditions of 1960s beer and hops [Peacock, p. 161].  At that time, hops seem to have been stored for long periods of time at cellar or room temperature without special packaging [Peacock, p. 160 and 162].  As Peacock explains, for a typical beer made from typical hops with a typical age and duration of hop boiling, the relative concentration of IAA to all bittering substances (IAA + nonIAA) was about 5/7, or about 71%.  In more recent times, it is much more likely that hops are stored at freezing temperatures with less oxygen for less time, which makes the relative concentration of IAA (with a typical 1960s hop boiling time) much higher.  So, an IAA concentration of 14 ppm from a 60-minute boil might yield an IBU value closer to 12.  On the other hand, it is also common now to add a lot more hops much closer to flameout, which increases the relative concentration of nonIAA components in the beer (compared with longer boiling times), as discussed below.

3. A General Description of Factors Affecting IBUs
The preceding descriptions of IBUs actually helped us.  Now we know that there are only three things we need to worry about when modeling IBUs: the concentration of isomerized alpha acids (IAA), the concentrations of other bittering substances (nonIAA), and the scaling factors for the nonIAA substances.  Thanks to Peacock’s formulation, we’ve moved from the absorption of light at 275 nm (which is very difficult for a homebrewer to  predict) to the concentrations of different substances (which we can approximate).  This section looks at these three items in more detail.

Before getting too far into this section, this might be a good place to define some terms related to alpha acids and beta acids.  Alpha acids are part of the soft resins in the hop lupulin gland, and the alpha acids contain humulone, cohumulone, and adhumulone  [Oliver, p. 34].   Older work may refer to all alpha acids as humulones [Oliver, p. 462].  The oxidized alpha acids contain humulinone as their most important component [Algazzali, p. 13].  The soft resins also contain beta acids, which are also called lupulones [Oliver, p. 462].  The beta acids are composed of colupulone, adlupulone, lupulone, and prelupulone [Oliver, p. 260].  The oxidized beta acids contain mostly hulupones [Algazzali, p. 15].  The oxidized form of colupulone is called cohulupone [Stevens and Wright, p. 496].

3.1 Concentration of Isomerized Alpha Acids (IAA) Under Ideal Conditions
A lot of research has been conducted on modeling isomerized alpha acids.  We can use this work to estimate the IAA concentration that we need to model IBUs.  Mark Malowicki [Malowicki] provides a model for both the conversion of alpha acids into isomerized alpha acids and the subsequent conversion of isomerized alpha acids into other “uncharacterized degradation products”, as functions of time and temperature, under fairly ideal laboratory conditions (with pH 5.2 and an alpha-acid concentration of 80 ppm).  (These degradation products include humulinic acid, isobutyraldehyde, and iso-hexenoic acid [Hough et al., p. 480].)  Malowicki describes the conversion of alpha acids into isomerized alpha acids as a first-order reaction following an Arrhenius equation with a temperature-dependent rate constant k1:

k1(T) = 7.9×1011 e-11858/T [13]

where k1(T) is the rate constant for the conversion of alpha acids into isomerized alpha acids and T is the temperature in degrees Kelvin.  A first-order reaction is of the form [X] = [X]0ekt (where [X] is the concentration of substance X at time t, [X]0 is the initial concentration of X (at time 0), and e is the constant 2.71828…), so we can describe the reduction of alpha acids (due to their conversion into isomerized alpha acids) as:

[AA]wort = [AA]0 ek1(T)t [14]

where [AA]wort is the resulting concentration of alpha acids in the wort at time t (in minutes), [AA]0 is the initial concentration of alpha acids (at the start of the boil), and k1(T) is the rate constant from Equation [13].  We can assume that the reduction in alpha acids is mirrored by a corresponding increase in isomerized alpha acids (see [Malowicki p. 6]).  Second, Malowicki describes the subsequent conversion of isomerized alpha acids into degradation products, also as a first-order reaction with a temperature-dependent rate constant:

k2(T) = 4.1×1012 e-12994/T [15]

where k2(T) is the rate constant for the conversion of isomerized alpha acids into other products (and T is still in degrees Kelvin).

Both Malowicki [Malowicki, p. 27] and Yarong Huang et al. [Huang 2013] show how to combine these equations into a single model of the cumulative concentration of isomerized alpha acids as a function of time and temperature:

[IAA]wort = [AA]0 (k1(T)/(k2(T)-k1(T))) (ek1(T)t-ek2(T)t) [16]

where [IAA]wort is the concentration of isomerized alpha acids in the wort at time t and temperature T.  We can plot this equation in Figure 2, with time on the horizontal axis, relative concentration of isomerized alpha acids (compared with the initial concentration of alpha acids) on the vertical axis, and a few different steeping temperatures represented with different colors:

isoAlphaAcidConcentraion

Figure 2.  Theoretical relative concentration of isomerized alpha acids in water, as a function of time and temperature.

This plot at 100°C (212°F) looks reassuringly similar to the utilization of alpha acids in the Tinseth equation for predicting IBUs [Tinseth]; the scale is different, and the shape is somewhat different, but the general trend at boiling is similar.

Equation [16] relies on the initial concentration of alpha acids at the beginning of the boil, which we can determine from the volume of wort (in liters), the weight of hops added (in grams), and either (a) the measured percentage of alpha acids at the time of the boil or (b) the measured percentage of alpha acids at the time of harvest and the degradation of alpha acids over time.  These values will give us the concentration of alpha acids in wort (in ppm):

[AA]0 = AA × W × 1000 / V [17]

where AA is the alpha acid rating of our hops, scaled to the range 0 to 1 (i.e. AA is the proportion of the hop (cone, pellet, or extract) that is alpha acids, from 0 to 1; e.g. an alpha acid rating of 7% becomes 0.07), W is the weight of the hops in grams, the factor of 1000 converts from grams to milligrams, and V is the volume of the wort in liters.  These units combine to give us milligrams of alpha acids per kilogram of wort (since 1 liter of water equals 1 kg; we’ll ignore the extra weight of the extract), or approximately parts per million.

Is V the volume at the beginning, middle, or end of the boil?  While [AA]0 indicates the initial level of alpha acids (at the beginning of the boil), we don’t have a factor that adjusts for volume changes between the beginning and end of the boil.  If we did have such a factor, it would describe the difference between the pre-boil volume and the post-boil volume, since the final concentration of isomerized alpha acids is determined by the post-boil volume (before racking losses that reduce the volume but don’t change the concentration).  Instead of having a separate factor and applying it explicitly, we can specify that V is the post-boil volume, and the numbers will come out the same as if we started with pre-boil volume and then accounted for evaporation.  In short: V should be post-boil wort volume (not including the volume of hops or trub).

If we don’t know the alpha acid rating of the hops when we brew our beer, we can use the initial (harvest) estimate with a model of how alpha acids degrade over time, developed by Mark Garetz [Garetz article] to estimate the alpha acid rating for hop cones:

AAAAharvest × AAdecayfactor = AAharvest × 1/ek×TF×SF×D [18]

where AAharvest is the alpha acid rating of the hops after harvest and drying, AAdecayfactor is a multiplication factor for how much the AA level has decayed over time (1.0 for fresh hops), k is a value that depends on the percent of alpha acids lost after 6 months at room temperature (which in turn depends on the variety of hops), TF is the temperature factor that describes how degradation is affected by temperature, SF is the storage factor that describes how degradation is affected by storage conditions, and D is the age of the hops, in days.  The full definition of all terms is provided in Garetz’s article [Garetz article].  For hop pellets, the rate of deterioration is much slower.  Hieronymus says that while whole hops can lose up to 100% of their alpha acids when stored at 68°F (20°C) for one year, pellets lose only 10% to 20% under the same conditions [Hieronymus, p. 230].  If you use pellets that were made immediately after harvest, and they’ve been stored in the refrigerator or freezer, it’s probably safe to assume that losses are somewhere between 5% and negligible, yielding a correction factor between 0.95 and 1.0.  If you don’t know how long the hops in your pellets were in whole-cone form, or what the storage conditions were, predicting the losses becomes quite difficult.

3.2 Accounting for Post-Boil Utilization
It’s clear that at flameout, the wort (unfortunately) does not instantaneously cool to pitching temperature.  According to Equation [16], there can still be measurable isomerization even at 158°F (70°C).  Therefore, as the wort cools after flameout, there can be a significant increase in the concentration of isomerized alpha acids.  I’ve suggested in a previous blog post that we can model this post-flameout increase in IBUs by multiplying the change in IAA concentration at time t by a temperature-dependent factor at t (with a factor of 1.0 for boiling), and then integrating the instantaneous values over time to arrive at a cumulative IAA concentration that reflects post-flameout temperature changes. In the current framework, we have a function (Equation [16]) that is already dependent on temperature, so we can take the derivative with respect to time, compute the instantaneous change in concentration at time t and temperature T, and then integrate over time t to arrive back at total concentration of IAA.  While the temperature is boiling, we will arrive at the same answer as if we didn’t take the derivative and then integrate.  As the kettle cools after flameout, we change the rate constants to reflect the lower rate of isomerization.  This can be implemented in less than 20 lines of programming code, and I’ve since noticed that Malowicki suggests this very approach, saying “for conditions in which the rate constants change with a changing temperature profile, the concentrations of iso-humulones formed during kettle boiling can be calculated using [equations] which define the differential change in alpha-acid, iso-alpha-acid, and degradation product concentrations with respect to time” [Malowicki, p. 27].

We  can take the derivative of Equation [16] in order to compute the change in IAA concentration at time t:

d([IAA]wort)/dt = [AA]0 (k1/(k2k1)) (k2e-k2tk1e-k1t) [19]

where d([IAA]wort)/dt is the rate at which the IAA concentration changes, in ppm per minute.  However, instead of using [AA]0 to compute the change at any time t, we can use the equations defined by Malowicki [p. 27] to compute the change in concentration at time t using current concentration levels:

d([AA]wort)/dt = –k1 [AA]wort [20]
d([IAA]wort)/dt = k1 [AA]wortk2 [IAA]wort [21]

By using these equations, we only need to know the total concentration of these substances at the previous time step in order to compute the concentrations at time t.  Only at t=0 do we need to know the initial concentration of alpha acids.  Since we’re integrating the instantaneous values anyway, Malowicki’s formulation of the derivatives ends up being just as easy to program and incredibly more efficient at dealing with multiple hop additions.

A model of how temperature decreases after flameout can be obtained by bringing the desired volume of water to a boil, turning off the heat, measuring the temperature at one-minute intervals, and then fitting a line or polynomial to the data.  I’ve found that the temperature decrease of a 6-gallon (23-liter) volume (no lid on the kettle) can be modeled fairly well with a straight line, at least for the first 20 minutes or so:

TF(tf) = -1.344 tf + 210.64          (for temperature in Fahrenheit) [22a]
TC(tf) = -0.74667 tf + 99.244    (for temperature in Celsius) [22b]
TK(tf) = -0.74667 tf + 372.394  (for temperature in Kelvin) [22c]

where TF is the estimated temperature in Fahrenheit, -1.344 is the rate of change (°F per minute), tf is time after flameout (in minutes), and 210.64 is the approximate temperature at flameout (when tf = 0, in °F). Likewise, TC is the estimated temperature in Celsius, -0.7466 is the range of change (°C per minute), and 99.244 is the approximate temperature at flameout (in °C); TK is temperature in Kelvin modeled with -0.74667 degrees Kelvin per minute and a flameout temperature of 372.394 Kelvin.  (Note that this formula will only yield reasonable results for a typical home-brewing system with a 6-gallon (23-liter) volume and an uncovered kettle, and even these “reasonable” results will be affected by factors such as kettle material and size.  To maximize accuracy, one should measure the temperature decay of their own system and determine a formula based on system-specific data.  Fortunately, the data I’ve collected so far indicates that this function is not significantly dependent on ambient temperature or relative humidity, so this function only needs to be constructed once per brewing system.)

We can model total concentration of IAA by integrating the change in [IAA] at each instant, where this amount of change is dependent on the steep time and temperature of the wort.  Rather than expressing this as a formula, I think a short amount of pseudo-code will be easier to understand (referred to as Code [1]), even if you’re not a programmer:

integrationTime = 0.001;
AA = AA0;
IAA = 0.0;
time = 0.0;
while (time <= totalTime) {
    if (time <= boilTime) 
        temp = 373.15; 
    else 
        temp = (-0.74667 * (time - boilTime)) + 372.394;
    k1 = 7.9 * pow(10,11) * exp(-11858.0/temp);
    k2 = 4.1 * pow(10,12) * exp(-12994.0/temp);
    dAA = -1.0 * k1 * AA;
    AA = AA + (dAA * integrationTime);
    dIAA = (k1 * AA) - (k2 * IAA);
    IAA = IAA + (dIAA * integrationTime);
    time = time + integrationTime;
}

where the integration time of 0.001 (called integrationTime) is sufficient for accuracy to at least two places past the decimal point.  The variable AA0 is the initial concentration of alpha acids, in ppm (see Equations [17]).  Here, AA is the total concentration of AA, or [AA], after time time (in minutes).  Likewise, IAA is the total concentration of IAA, or [IAA], after time time.  The value of totalTime is the length of the boil in minutes (boilTime) plus any time after the boil when isomerization might be happening (postBoilTime).  A loop is set up to evaluate (and integrate) all time points from 0.0 to totalTime in increments of 0.001 minutes, with time representing the current time instant.  The temp variable is temperature at the current time, in Kelvin.  The k1 and k2 variables are the rate constants from Equations [13] and [15].  The variable dAA is the derivative of [AA], or change in [AA] per minute, as defined in Equation [20].  Likewise, the variable dIAA is the derivative of [IAA], or change in [IAA] per minute, as defined in Equation [21].  The pow() function raises the first argument to the power of the second argument; the exp() function computes the exponent of its argument.  After finishing the loop, IAA will equal the total concentration of isomerized alpha acids, accounting for both time and (post-flameout) temperature.

3.3 Adjustments to the Concentration of Isomerized Alpha Acids
Now we know how to measure the concentration of IAA in wort during the boil under ideal conditions.  We can use this as the basis for a quantitative model of IBUs.  What we need next is a way to describe the differences between ideal laboratory conditions and (home) brewery conditions.  Many factors affect the rate or amount of conversion from alpha acids to isomerized alpha acids: temperature (e.g. boiling at high altitudes), pH of the wort, form of the hops (e.g. extract, pellet or cones; loose or bagged), and alpha-acid concentration in the wort.  Other factors can be described as losses of IAA that are produced in the boiling wort but never make it into the pint glass: losses due to high wort gravity, other losses during the boil, and losses due to fermentation, filtration, and aging.  We’ll look at each of these briefly in this section.  In general, if we have a loss of x%, the loss factor will be (1 – (x%/100)); for example, a loss of 10% will become a loss factor of 0.90.

Before getting into too much detail, this is a good place to define a high-level term, “utilization.”  Hop utilization, U, is the ratio of the amount of isomerized alpha acids in finished beer, divided by the amount of alpha acids added to the kettle, and then multiplied by 100 to convert to percent [e.g. Lewis and Young, p. 266]:

U = 100 × (isomerized alpha acids in beer) / (alpha acids added to kettle) [23]

It should be noted that utilization refers only to the creation and loss of isomerized alpha acids, not to IBUs.

Isomerization as a Function of Temperature and pH: According to Malowicki’s equations (above), a decrease in temperature (e.g. below 100°C) will decrease utilization.  If you live at a high enough altitude, your wort will boil at less than 100°C, in which case you might want to adjust k2 and k2 in Equations [13] and [15], or include a temperature-dependent rate factor, RFtemp(T).  Post-flameout temperature dependencies are discussed above.  (Lewis and Young, Palmer, Hieronymus, and others note that the intensity of the boil affects utilization [Lewis and Young, p. 266; Palmer p. 55; Hieronymus, p. 188], which is presumably related to wort temperature.)

It is generally accepted that an increased wort pH will increase utilization [e.g. Lewis and Young, p. 266].   This was demonstrated by Kappler (at pH levels of 4.0, 5.0, and 6.0) and by Huang (at pH levels of 4.5, 5.2, 5.5, and 6.5), looking at the degradation of isomerized alpha acids added to the boil [Kappler, p. 334; Huang, p. 50].  However, Malowicki looked at isomerized alpha acids produced and degraded during boiling at pH values of 4.8, 5.2, 5.6, and 6.0, and found that “the dataset did not show a significant effect of pH on rate of iso-alpha-acids produced” [Malowicki, p. 38].  The typical homebrewer should aim for a mash pH in the ballpark of 5.2 to 5.4 [Palmer and Kaminski, p. 60; Noonan, p 144; Fix, p 49; Troester citing Kunze (2007) and Narziss (2005)], although a pH as high as 5.8 is still acceptable [Troester].  Since Malowicki’s work looked at both the production and degradation of isomerized alpha acids, and since this work didn’t show a significant effect of pH in the range of interest, the model of IBUs that is built in this post does not have a dependency on pH.  Therefore, the rate factor for wort pH, RFpH(pH), is set to be 1 for all values of pH.

Isomerization as a Function of Form of the Hops: It is often said that whole hops do not provide as much utilization as hop pellets [e.g. Daniels p. 78].  According to Lewis and Young, “the alpha acids dissolve most easily from extracts, less easily from pellets …, and least with whole hops” [Lewis and Young, p. 266].  The higher rate at which alpha acids from pellets dissolve, compared with whole cones, is because “the pelletization process ruptures the lupulin glands and spreads the resins over the hop particles, giving a larger surface area for isomerization” [Hall, p. 58].  Noonan says that “with pelletized hops, ruptured and better-exposed lupulin glands give greater utilization” [Noonan, p. 154].  Garetz indicates that pellets have better utilization up to a boil time of 30 minutes (after which utilization is the same), because after 30 minutes all of the alpha acids have been dissolved, regardless of whether they come from cones or pellets [Garetz book, p. 131].

Hough et al. say that alpha-acid extracts are actually much less efficient than whole or pelletized hops: “the solubility of humulone was the limiting factor in its utilization.  … In trials using pure humulone, only 50-60% of the resin added was isomerized during [the] 1.5 h boil.  In contrast, 65-75% of the alpha acids present in hops are isomerized in the same period, which supports the view that the isomerization of humulone is catalyzed by the presence of hop cones, break, or even an inert surface such as Celite.” [Hough et al., p. 489, citing Maule, p. 288].  Since Malowicki used alpha-acid extract in his experiment (with no added surfaces to serve as a catalyst), the correction factor for the form of the hops in our quantitative description is 1.0 for extracts and about 1.27 (70%/55%) for non-extract forms.  (Note that there is 57% isomerization of alpha acids at 90 minutes according to Equation [16], which is very much in line with the statement by Hough et al.)

Expressing whole hops as less efficient than pellets, Noonan provides a whole-hop correction factor (in table form) that varies from 0.66 to 1.0, based on boil time and gravity [Noonan, p. 215].  Garetz recommends a correction factor of 0.90 for boil times up to 30 minutes, otherwise a correction factor of 1.0 [Garetz book, p. 141].  Hieronymus says that hop pellets are 10% to 15% more efficient than cones [Hieronymus, p. 188], translating into a correction factor between 0.87 and 0.91 when using whole cones.  According to Michael Hall, Mosher specifies a correction factor of 0.75 [Hall, p. 62].  This leaves a wide range of possible correction factors for the use of whole hops compared with pellets (from 0.66 to 1.0), with a median factor of 0.91.  For the model of IBUs being built, I’ll assume a factor of 0.91.  This whole-hop vs. pellet correction factor is in addition to (i.e. multiplied by) the correction factor for non-extracts, 1.27.  Therefore, pellets have a correction factor of about 1.27 and whole hops have a correction factor of about 1.16.

Garetz also says that hops kept in a mesh bag during the boil have lower utilization than loose hops, with a correction factor of 0.91 for loosely-stuffed hops and 0.83 for a full bag. Whole hops in a loosely-packed mesh bag will then have a combined correction factor of 1.05 (1.27 × 0.91 × 0.91) [Gartez book, p. 141].  I recently looked at the effect of a mesh bag vs. loose hops on measured IBUs, and found no significant difference (blog post to come in the future).  Marshall Schott at Brülosophy also looked at bagged vs. loose hops, and found 25 IBUs for the bagged hops and 27 IBUs for the loose hops [Schott].  While this difference is not significant, this ratio of 0.926 (25/27) is close to Garetz’s correction factor of 0.91.  In short, it’s not clear if hop cones in a mesh bag really do have lower utilization, but I’ll keep the suggested correction factor of 0.91.  For the model being developed, I’ll assume that bagged hops are always loosely bagged, for a “bagging” correction factor of 0.91.

Isomerization as a Function of Alpha-Acid Concentration: Along with the form of the hops, the relative amount of hops (and therefore also the relative amount of alpha acids) in the wort affects utilization.  As Lewis and Young say, “a high hopping rate reduces extraction efficiency” [Lewis and Young, p. 267].  Daniels phrases this as “simply adding more and more hops does not produce a linear increase in the amount of bitterness produced” [Daniels, p. 85].  Fix also notes that the utilization rate is affected by hop concentration [Fix, p. 47].  Hough et al. say that “hops are utilized more efficiently at low rates” [Hough et al., p. 489].  Maule determined that reduced utilization at higher hop rates can only be accounted for by the “difficulty with which [isomerized alpha acid] enter[s] solution when wort [is] boiled with large amounts of [alpha acid]” [Maule, p. 290], and that “only a small portion of the resin present on the hot break … can be claimed to be truly adsorbed” [Maule, p. 289].

Garetz provides the only source I’m aware of with a quantitative model of the relationship between amount of hops and utilization.  He proposes a hop-rate correction factor (also described by Hall and Daniels) that depends on volume and “desired IBU” to determine the weight of hops needed [Garetz book, p. 137; Hall, p. 63; Daniels, p. 86].  When I was initially developing this blog post, I used a modified form of his equation to estimate a correction factor based on volume, weight of hops, and alpha acid rating of the hops, since we don’t know the desired IBU when trying to predict an IBU value.  However, after some difficulty fitting the IBU model developed in this post to available data, and after further experimentation, I concluded that Garetz’s correction factor underestimates the effect of alpha acid concentration on utilization.

A better fit to the data available to me can be obtained by either (a) simply limiting the alpha acids available for conversion to about 260 ppm, or (b) limiting the alpha acids in solution to 230 ppm, increasing Malowicki’s rate constant k1(T) (Equation [13]) by a factor of 3.9, and increasing the rate constant k2(T) (Equation [15]) to some very large value, e.g. 1000.  (Increasing k1(T) means that alpha acids not yet in solution are converted more quickly into IAA, but increasing k2(T) means that those IAA that are created while not in solution are nearly instantly degraded.)  (I will provide more detail about the second suggested limit on alpha acid solubility in a future blog post.)  These limits are greater than the solubility of alpha acids at room temperature (around 90 ppm [Malowicki, Appendix A, pp. 51-54]), but Spetsig’s paper on the solubility of humulone indicates an approximate limit of around 300 ppm at boiling and pH 5.2 [Spetsig 1955, p. 1423-1424].  Using this approach, utilization increases linearly with alpha-acid concentration until the solubility limit is reached (e.g. 230 ppm or 260 ppm); at higher concentrations utilization increases either much less or not at all.  The values for the solubility limit and the increase in k1(T) were obtained using the model described in this blog post, so these are two unknown parameters in the model development; they will be referred to as [AA]limit and k1_scale.

Instead of providing a formula to show how the model of alpha-acid isomerization is affected by alpha-acid solubility, I think that modifications to Code [1] that set the solubility limit to 230 ppm and include changes to the rate constants will be more helpful in understanding how the modified model works.  This new code will be referred to as Code [2]:

integrationTime = 0.001;
AA_limit = 230;
k1_scale = 3.90;
if (AA0 > AA_limit) {
    AA_sol = AA_limit;
    AA_unsol = AA0 - AA_limit;
} else {
    AA_sol = AA0;
    AA_unsol = 0.0;
}
IAA = 0.0;
time = 0.0;
while (time <= totalTime) {
    if (time <= boilTime) 
        temp = 373.15; 
    else 
        temp = (-0.74667 * (time - boilTime)) + 372.394;
    k1_sol = 7.9 * pow(10,11) * exp(-11858.0/temp);
    k2_sol = 4.1 * pow(10,12) * exp(-12994.0/temp);
    dAA_sol = -1.0 * k1_sol * AA_sol;
    AA_sol = AA_sol + (dAA_sol * integrationTime);
    if (AA_sol < AA_limit AND AA_unsol > 0) {
        AA_sol = AA_sol - (dAA_sol * integrationTime);
        AA_unsol = AA_unsol + (dAA_sol * integrationTime);
    }
    dIAA_sol = (k1_sol * AA_sol) - (k2_sol * IAA_sol);
    IAA_sol = IAA_sol + (dIAA_sol * integrationTime);
    k1_unsol = k1_sol * k1_scale;
    k2_unsol = k2_sol * 1000.0;
    if (AA_unsol > 0.0) {
        dAA_unsol = -1.0 * k1_unsol * AA_unsol;
        AA_unsol = AA_unsol + (dAA_unsol * integrationTime);
        dIAA_unsol = (k1_unsol * AA_unsol) - (k2_unsol * IAA_unsol);
        IAA_unsol = IAA_unsol + (dIAA_unsol * integrationTime);
        if (IAA_unsol < 0) IAA_unsol = 0.0;
    } else {
        AA_unsol = 0.0;
        dAA_unsol = 0.0;
        dIAA_unsol = 0.0;
        IAA_unsol = 0.0;
    }
    time = time + integrationTime;
}

where the variables are the same as in Code [1], but dAA, AA, dIAA, IAA, k1, and k2 now have two parts: the part that is related to alpha-acids that have been dissolved in the wort (dAA_sol, AA_sol, dIAA_sol, IAA_sol, k1_sol, and k2_sol) and the part that is related to the alpha-acids that are not yet dissolved in the wort (dAA_unsol, AA_unsol, dIAA_unsol, IAA_unsol, k1_unsol, and k2_unsol).  If we start with an alpha-acid concentration above the solubility limit [AA]limit, the part that is less than the limit (dissolved in wort) isomerizes and is destroyed at the same rates as described by Malowicki.  As dissolved alpha acids are converted into isomerized alpha acids, lowering the concentration of dissolved alpha acids, some of the alpha acids not yet dissolved go into the wort solution to maintain the solubility limit.  The not-yet-dissolved alpha acids are converted into isomerized alpha acids at a higher rate (k1(T) × k1_scale), and the not-yet-dissolved isomerized alpha acids are quickly destroyed and don’t contribute to the final concentration of isomerized alpha acids.

Isomerization as a Function of Kettle Size and/or Geometry: The kettle size and/or kettle geometry may also impact utilization [Daniels, p. 78; Fix, p. 47].  As Hieronymus says, “larger kettles are more efficient, and the difference between a five-gallon homebrew system and even a 10-barrel (310-gallon) commercial brewery is startling” [Hieronymus, p. 188].  There are other claims, however, that recipes should scale linearly with kettle size, indicating no impact on utilization [e.g. Spencer].  If there is an impact, the reason for the change in utilization is not clear to me, especially since Malowicki used only tiny volumes of wort (12 ml) [Malowicki, p. 19] and obtained high utilization rates at boiling (see Figure 2).  The only quantitative description I’ve seen of this impact on utilization is in an article on BeerSmith, which says that “Hop utilization is much higher at craft brewing scales, because large boils simply extract more bitterness. … The Hop Utilization Factor … can easily be 125%, 150% or possibly more for a multi-barrel brewing system” [Smith].  It may be that the observed increase in utilization with kettle size is a reflection of longer times between flameout and cooled wort, which is already accounted for in the current model with post-boil utilization.  In short, kettle size (or wort volume) may (or may not) have an impact on utilization, with a scaling factor ranging from 1.0 (no impact) to 1.5 (large impact).  Because of the difficulty of reconciling Malowicki’s use of tiny volumes and resulting high utilization, I assume that kettle size has no impact on utilization.  Therefore, the rate factor for kettle size, RFsize(V), is assumed to be 1.

Losses Due to Wort Gravity: Utilization decreases with increasing wort gravity, at least at higher gravities [e.g. Lewis and Young, p. 266; Hieronymus, p. 188; Hall, p. 62; Daniels, p. 78; Palmer, p. 55; Malowicki, p. 44; Garetz book, p. 130; Hough et al., p. 489; Kappler, p. 334].  As Lewis and Young state, “iso alpha acids react with proteins of wort whence they are partially removed as trub or hot break” [Lewis and Young, p. 266].  As the gravity increases, the concentration of wort proteins increases, implying a greater loss of isomerized alpha acids with increasing gravity.  Kappler found losses of isomerized alpha acids at higher gravities when adding (already) isomerized alpha acids to the boil [Kappler, p. 335], indicating that higher gravity causes more isomerized alpha acids to bind with trub and settle out of solution (as opposed to slowing the rate of conversion from alpha acids to isomerized alpha acids).  Malowicki did not find a significant change in utilization at specific gravities of 1.000 and 1.040 [Malowicki, p. 39], and Garetz indicates that the lower limit for this effect is a specific gravity of 1.050 [Garetz book, p. 130].  Greg Noonan [Noonan, p. 215] provides a table of utilization as a function of boil time, original gravity, and form of the hops.  (His table simply lists “wort density” and “specific gravity”, but he defines wort density as original gravity [Noonan, p. 204].) The original gravity in his table seems to be an independent scaling factor of the other two parameters, with scaling factors of about 1.0, 0.921, 0.865, 0.842, and 0.774 at averaged gravities of 1.040, 1.058, 1.070, 1.080, and 1.090, respectively.  A line can be fit through these points to determine an original-gravity correction factor as a function of original gravity:

LFOGN(OG) = (-4.944 × OG) + 6.166    if OG > 1.045, else 1.0 [24]

where LFOGN(OG) is Noonan’s gravity loss-correction factor (expressed as an equation instead of the original table form) and OG is the original gravity.  If OG is less than or equal to 1.045, LFOGN(OG) is defined as 1.  Glenn Tinseth models the  gravity correction factor as LFWGT(WG) = 1.65 × 0.000125(WG − 1), with a scaling factor of 1.0 at around a (typical) wort gravity (WG) of 1.055.  (Tinseth uses the term “wort gravity” and suggests using the average of the (initial) boil gravity and original gravity for wort gravity [Tinseth].)  Because Malowicki measured the production of isomerized alpha acids in water (with a specific gravity of 1.0), we want to think of any increase in gravity as a reduction in the production of isomerized alpha acids, when compared with Malowicki’s work.  Since Noonan’s formula describes higher gravity as always yielding less utilization, his original-gravity correction factor is more suitable for our purposes; it also provides a compromise between the correction factors proposed by Tinseth, Rager, and Garetz [Hall, p. 61].

Other Losses During the Boil:  Isomerized alpha acids are lost during the boil in ways that are not dependent on the wort gravity.  Malowicki says that “trub, and specifically the formation of trub, leads to greatly increased losses of bitter acids” [Malowicki, p. 8; emphasis mine].  He cites work by H.O. Askew in which the use of pre-formed trub produced losses of only 5% to 9%, but the formation of trub created losses of 35% [Malowicki, p. 7-8].  Malowicki also cites Laufer and Brenner who found a 38% loss of bitter acids to trub and a 35% loss to spent hops.  Spetsig reports that about 25-30% of the bitter substances are found in the spent hops and 25-40% are found in the trub [Spetsig 1968, p. 346].  Hall cites Hough et al., who cite Maule (1966), saying that “about 7% of the iso-alpha acids are removed with the breaks” [Hall, p. 57; Hough et al., p. 489].  Garetz says that “8-10% of the iso-alpha acids are adsorbed (meaning they cling to the surface of) the hot and cold breaks.  This number appears to be fairly constant, even given wide variations in the amount of break, composition of the wort, and the method and length of cooling” [Garetz book, p. 126].  In short, the estimated loss of isomerized alpha acids during the boil ranges from 7% to 73%, yielding a correction factor from 0.27 to 0.93, which is a bit too large of a range to be of practical value.

Losses During Fermentation: Isomerized alpha acids are also lost during fermentation [e.g. Hieronymus, p. 190]. Lewis and Young say that “during fermentation, iso-alpha-acids associate with the surface of the yeast cells present… Iso-alpha-acids, being surfactants, react with inert surfaces of all sorts and for example separate on gas bubbles to be deposited on the fermenter walls” [Lewis and Young, p. 267].  Hall describes the same process, saying that “during the fermentation process, iso-alpha acids are scrubbed by the rising CO2 and collect in the foam of the kraeusen.  This sticky foam can be blown off, skimmed off or stuck on the sides of the fermenter … Iso-alpha acids also are bound up by the yeast cells and removed when the yeast flocculates out” [Hall, p. 57].  Daniels says that the amount of loss is dependent on the amount of yeast pitched and the “extent of yeast growth during fermentation” [Daniels, p. 78].  Garetz says that there are two factors, “the total growth of the yeast crop and the amount of time the yeast stays in suspension”, and that there is a 5% variation depending on the flocculation characteristics of the yeast [Garetz book, p. 126].  He also says that if the alpha acids are mixed back into the beer at the right time, utilization is increased by 18% [Garetz book, p. 126], implying typical losses of 18%.  Fix (citing Garetz) estimates loss to yeast sediment at 10% to 20% [Fix, p. 49]. Malowicki (citing Laws et al.) reports losses during fermentation from 5% to 17% [Malowicki, p. 8] and also (citing Laufer and Brenner) losses to yeast of 10% [Malowicki, p. 7].  Spetsig reports losses of 10% to 15% [Spetsig, 1968].  Hieronymus reports losses during fermentation and packaging of 20% [Hieronymus, p. 191].  Tom Nielsen (from Sierra Nevada Brewing Co.) measured the IBUs of wort and finished beer made from 10 types of hops (9 aroma hops and 1 bittering hop) and found a fairly consistent fermentation loss of about 18% (standard deviation approximately 1.6%) [Nielsen, p. 65].  To summarize, there is IAA loss during fermentation ranging from 5% to 20%, yielding a correction factor between 0.80 and 0.95.  A factor of around 0.85 is probably the best compromise between all reported values, and so the model being developed here uses 0.85.  The flocculation factor suggested by Garetz is 0.95 for high-flocculation yeast and 1.05 for low-flocculation yeast [Garetz book, pp. 140-141].

Losses Due to Filtration and Aging:  According to Daniels, “any filtration will remove some bitterness … The addition of clarifying agents such as gelatin or PVPP may have a similar effect.” [Daniels, p. 79].  Garetz says that filtering will reduce utilization by 1.25% to 2.5%, for a filtration loss factor of about 0.98 [Garetz book, p. 141]:

LFfiltering(filtering) = 0.98 for filtered beer or 1.0 for unfiltered beer [25]

where LFfiltering(filtering) is the loss factor due to filtering, if any.

Hall says that “there are oxidation reactions that can reduce the bitterness of beer over extended storage periods” [Hall, p. 58].  According to Kaltner and Mitter, “over a storage time of 12 months, a degradation of bitter substances in various beers in a range of 10% to 15% could be analyzed” [Kaltner and Mitter, p. 37].  According to Peacock, citing results from Forster et al. (2004), beer loses 18% of  isomerized alpha acids and 14% of measured IBUs after 8 months at room temperature [Oliver, pp. 132-133, Peacock p. 164].  I am unaware of an existing model of how IBUs decrease with age for home-brewed beer stored in bottles at room temperature (which may have greater oxidation, less filtering, and other differences with commercially-bottled beer).  I therefore measured the decrease in IBUs for two home-brewed beers after 1, 2, 3, 6, 7, and 13 weeks from the start of fermentation, and fit the measured IBU decrease over time with an exponential-decay function.   If we assume that isomerized alpha acids and non-IAA components are affected by age at the same rate (which is probably an incorrect assumption [Peacock, p. 163], but not unreasonable as a first approximation), we can model the loss factor for isomerized alpha acids using the same age formula determined for IBUs:

LFage(ageweeks) = 0.32 × e0.08 ageweeks + 0.68 [26]

where LFage(ageweeks) is the loss factor due to age of the beer (in weeks).

Summary of IAA Adjustments: We can now express the concentration of IAA in the beer as a function of the concentration of IAA in the wort, multiplied by the various isomerization rate adjustment factors and IAA loss factors discussed above:

RFIAA(T, pH, hopsForm, V) = RFtemp(T) × RFpH(pH) × RFform(hopsForm) × RFsize(V) [27]
LFIAA(OG, flocculation, filtering, ageweeks) = LFOGN(OG) × LFboil × LFferment(flocculation) × LFfiltering(filtering) × LFage(ageweeks) [28]
[IAA]beer = [IAA]wort × RFIAA(T, pH, hopsForm, V) × LFIAA(OG, flocculation, filtering, ageweeks) [29]

where RFIAA is the isomerization rate factor adjustment of isomerized alpha acids, LFIAA is the loss factor for isomerized alpha acids, and [IAA]beer is the concentration of isomerized alpha acids in the finished beer.  The rate factor RFIAA is expressed as a combination of other factors, where RFtemp is a rate factor for temperature (with temperature T still in degrees Kelvin), if desired; RFpH is a rate factor for wort pH (currently a constant 1.0); RFform is the rate factor for the form of the hops (where hopsForm is “pellet”, “loose whole cones”, or “bagged whole cones”); and RFsize is the rate factor for kettle size (specified in this case with volume V).  The loss factor LFIAA is expressed as a combination of other factors, where LFOGN is Noonan’s loss factor as a function of original gravity; LFboil is the loss factor during the boil; LFferment is the loss factor due to fermentation (with flocculation being “high”, “medium”, or “low”); LFfiltering is the loss factor due to filtration (with parameter filtering being “unfiltered” or “filtered”); and LFage is the loss factor due to age (which varies with the age of the beer, ageweeks).  The concentration of IAA in the wort, [IAA]wort, can be computed using Code [2].

The only problem remaining for modeling [IAA]beer is that while we have a good idea of some factors (RFtemp, LFOGN) and a rough approximation of others (RFform, LFferment, LFfiltering, and LFage), we have very little basis for determining the remainder (LFboil,  [AA]limit, and k1_scale).  But we can come back to that problem later.

3.4 A Revised IBU Formula for nonIAA Components
At this point, we have as complete a description as we’re going to get of the concentration of isomerized alpha acids in beer.  The other factor in the IBU formula (Equation [12]) is the concentration of “other bittering substances,” which we call nonIAA.

Alpha acids (before isomerization) are neither soluble [e.g. Lewis and Young, p. 259] nor bitter [Shellhammer, p. 169], but as they age and become oxidized, the resulting oxidized alpha acids (oAA) are soluble in wort and bitter [Algazzali, pp. 14-15, p. 19, p.45; Maye et al, p. 23; Hough et al., pp. 435-436; Hough et al., p. 439; Lewis and Young, p. 265].  Oxidized alpha and beta acids are also produced during the boil [Parkin, p. 11, Algazzali, p. 17; Dierckens and Verzele, p. 454; Oliver p. 471].  Oxidized beta acids (oBA) are also soluble [Algazzali, p. 16] and may be produced and contribute to bitterness in the same way as oxidized alpha acids [Malowicki, p. 2; Peacock, p. 157; Fix, p. 36; Lewis and Young, p. 265; Hall, p. 55; Lewis and Young p. 265; Oliver, p. 132; Oliver, p. 470; Parker, p. 11; Algazzali, p. 17; Hough et al., p. 489].  The formulation of the Hop Storage Index (HSI) implies that oxidized alpha (and beta) acids have optical density at 275 nm [Algazzali, p. 19].  Finally, polyphenols may be a contributing factor to the nonIAA components [e.g. Krogerus]; as Shellhammer states, “the contribution of polyphenols to beer bitterness can not be overlooked” [Shellhammer, p. 177].

I haven’t been able to find definitive (e.g. more than one source) claims on the bitterness or A275nm properties of other substances that might be considered nonIAA.  That leaves us with oxidized alpha acids, oxidized beta acids, and polyphenols as the only nonIAA components that influence the measurement of IBUs.  We can then re-write Equation [12] to be more specific, replacing the generic nonIAA1, nonIAA2, and nonIAA3 with oxidized alpha acids (oAA), oxidized beta acids (oBA) and polyphenols (PP):

IBU = 5/7 × ([IAA]beer + (([oAA]beer × scaleoAA) + ([oBA]beer × scaleoBA) + ([PP]beer × scalePP))) [30]

where [oAA]beer is the concentration of oxidized alpha acids in the beer (in ppm), scaleoAA is the non-IAA scaling factor specific to oxidized alpha acids,  [oBA]beer is the concentration of oxidized beta acids in the beer (in ppm), scaleoBA is the non-IAA scaling factor specific to oxidized beta acids, [PP]beer is the concentration of polyphenols in the beer (in ppm), and scalePP is the non-IAA scaling factor specific to polyphenols.  (Note that we can compute [IAA]beer using Code [2] and Equation [29].)

3.4.1 Oxidized Alpha Acids
As hops age, the alpha and beta acids become oxidized.  The “most important group of oxidized alpha acids formed during hop aging is the humulinones” [Algazzali, p. 13].  The rate at which alpha acids oxidize during storage is determined by the form of the hops (e.g. cones or pellets), hop variety, age, temperature, and amount of exposure to oxygen [Garetz article].  Garetz has a model that predicts the amount of alpha acids remaining in hop cones, given these factors [Garetz article].  (As long as they are properly stored, pellets undergo oxidation at a much slower rate [Hieronymus, p. 230], and so Garetz’s model should only be used for whole hop cones.)  A decrease in the amount of alpha acids is mirrored by a corresponding increase in the amount of oxidized alpha acids.  The alpha acids also undergo some amount of oxidation while still on the bine [Hieronymus, p. 233] and further during the warm and highly oxygenated conditions of hop drying [e.g. Hieronymus, p. 126], and so the level of oxidized alpha acids when we get our newly-dried hops soon after harvest can be greater than zero [Maye, p. 23].  Finally, oxidized alpha acids are created during the boil [Algazzali, p. 17; Dierckens and Verzele, p. 454].

We can model the level of oxidized alpha acids (oAA) in the wort as the sum of three contributions: (1) the oAA present in the freshly-dried hops as a result of oxidation on the bine and during drying, (2) the oAA that accumulate as the hops age and deteriorate, and (3) oAA that is produced during the boil:

oAA = oAAfreshoAAstorageoAAboil(t) [31]

where oAA is the level of oxidized alpha acids (as percent of weight of the hops), oAAfresh is the level of oxidized alpha acids in freshly-dried hops, oAAstorage is the level of oxidized alpha acids produced during storage, and oAAboil(t) is the level of oxidized alpha acids produced during the boil as a function of boil time t; all components are expressed as percent of weight of the hops.

Based on data from Maye et al. [Maye, p. 24], I fit the level of oAA for fresh hops (with a Hop Storage Index (HSI) of 0.25 [Hough et al., p. 434]) to the model of alpha-acid decay proposed by Garetz [Garetz article], and determined that oAAfresh can be modeled reasonably well for the available data with a storage factor of 1 (loose hops), a temperature factor of 1 (20°C or 68°F), and a duration of 0.5 days.  I then fit the data in the Maye paper for higher HSI values to the loss predicted from the Garetz formula multiplied by a scaling factor of 0.022.  (I will go into much more detail on this in a future blog post.)  This leaves oAAboil(t) as the only unknown parameter that must be searched for, expressed as the amount of alpha acids that undergo oxidation relative to the amount of available alpha acids in the boil.  Given a lack of information about oAAboil(t), I’ll assume that it increases linearly from when the hops are added until some time toAAmax, after which it remains at a constant value, oAAboilMax.  These two parameters allow approximation of a wide variety of plausible oxidized alpha acid concentrations produced during the boil.  We can then re-write oAA using different functions to replace oAAfresh, oAAstorage, and oAAboil(t):

oAAboil(t) = oAAboilMax × t / toAAmax if t < toAAmax; otherwise oAAboilMax [32]
oAA = (1 – 1/ek×1×1×0.5) + (oAAagescale × (1 – AAdecayfactor)) + (AA × oAAboil(t)) [33]

where oAAboil(t) is the relative amount of alpha acids that undergo oxidation during the boil, oAA is the same level of oxidized alpha acids in Equation [31], k is the variety-specific hop decay factor from the Garetz model, oAAagescale is the age-related scaling factor of 0.022, AAdecayfactor is the alpha acid decay factor from Equation [18], and AA is the level of alpha acids at the start of the boil (Equation [18]).  This equation is specific to hop cones; some modification would be required for hop pellets, presumably a larger value of oAAfresh but a value close to zero for oAAstorage.  Since oxidized alpha acids are soluble, all of the oxidized alpha acids in the hop cones are in the wort shortly after being added to the kettle, and the oxidized-alpha-acid level peaks at toAAmax.

That leaves us with two other oAA factors that we still need to account for: losses and a scaling factor.  I have not yet been able to find any description of the losses of oxidized alpha acids during the boil and fermentation, so this is a completely unknown factor. It seems reasonable to assume that oxidized alpha acids are lost to trub, yeast, and in other ways, just as isomerized alpha acids are lost in the process of turning wort into beer.  Therefore, because the same types of losses probably occur for oxidized alpha acids as for isomerized alpha acids, we can model the oxidized alpha acid losses as the losses that affect isomerized alpha acids multiplied by some (unknown) scaling factor.  The scaling factor is a high-level correction factor for differences between losses found in isomerized alpha acids and oxidized alpha acids.  In other words,

[oAA]wort = oAA × W × 1000 / V [34]
[oAA]beer = [oAA]wort × LFIAA(OG, flocculation, filtering, ageweeks) × scaleoAAloss [35]

where [oAA]wort is the concentration of oxidized alpha acids in the wort, [oAA]beer is the concentration of oxidized alpha acids in the finished beer, W is (still) the weight of the hops in grams, V is (still) the post-boil volume of the wort in liters, LFIAA is the same IAA loss factor from Equation [28] and scaleoAAloss is the (unknown) loss scaling factor.

We also need a scaling factor in Equation [30] that scales the factor for absorption of light at 275 nm of oxidized alpha acids (unknown) to the factor for absorption of light at 275 nm of isomerized alpha acids (69.68).  Fortunately, Maye et al. provide this data; based on their Figure 7 [Maye, p. 25], the scaling factor is 0.0142/0.0130, or 1.093:

scaleoAA = 1.093 [36]

Despite the large number of parameters for modeling oAA, we end up needing to obtain estimates of only three: oAAboilMaxtoAAmax, and scaleoAAloss.

3.4.2 Oxidized Beta Acids
As with alpha acids, the beta acids oxidize as the hops age.  The most bitter and most prevalent components of the oxidized beta acids are called hulupones [Algazzali, p. 15-16].  The oxidized beta acids are thought to contribute more to beer bitterness than the oxidized alpha acids; as Peacock notes, the “nonIAA bitterness is mostly from oxidation products of the alpha and especially the beta acids of the hops formed during hop storage”. [Peacock, p. 157; emphasis mine].

We can model oxidized beta acids in a way similar to oxidized alpha acids: there are oxidized beta acids occurring in fresh hops, created during storage, and produced during the boil [Algazzali, p. 17; Stevens and Wright p. 496; Hough et al., p. 490]:

oBA = oBAfreshoBAstorageoBAboil(t) [37]

where oBA is the level of oxidized beta acids in the hop cone, oBAfresh is the level of oxidized beta acids in freshly-dried hops, oBAstorage is the level of oxidized beta acids produced during storage, and oBAboil(t) is the level of oxidized beta acids produced during the boil; all components are expressed as percent of weight of the hops.

Stevens and Wright say that oxidized beta acids are present at not more than 0.5% of the weight of the cone [Stevens and Wright, p. 500], Spetsig and Steninger note up to 3% [Spetsig and Steninger, p. 413], and Mussche found oxidized beta acids up to 1% of the weight [Mussche, p. 13].  Peacock implies that the beta acids undergo oxidation losses at approximately the same rate as the alpha acids [Peacock, p. 162].  Given the wide range of reported values of (oBAfreshoBAstorage), I’ll assume that oxidized beta acids are produced at the same levels as oxidized alpha acids both in fresh hops and during aging, but that this sum should be in the ballpark of 0.5% to 3%.  Stevens and Wright provide an estimate of the oxidized-beta-acid boil factor, noting that “after heating colupulone with boiling wort for 2 hr., as much as 10% of the beta acid had been converted into cohulupone.” [Stevens and Wright, p. 500]. I’ll assume that the time to convert beta acids to oxidized beta acids during the boil is the same as for alpha acids, i.e. toBAmax = toAAmax, and Stevens and Wright give the ballpark estimate for oBAboilMax, namely oBAboilMax = 0.10. All of this gives formulas similar to Equations [32] and [33]:

oBAboil(t) = oBAboilMax × t / toBAmax if t < toBAmax; otherwise oBAboilMax [38]
oBA = (1 – 1/ek×1×1×0.5) + (oBAagescale × (1 – AAdecayfactor)) + ((AA / ABratio) × oBAboil(t)) [39]

where oBAboil(t) is the relative amount of beta acids that undergo oxidation during the boil; oBAboilMax is approximately 0.10; oBA is the same level of oxidized beta acids in Equation [37], k is the variety-specific hop decay factor from the Garetz model, oBAagescale is the age-related scaling factor of 0.022, AAdecayfactor is the alpha acid decay factor from Equation [18], AA is the level of alpha acids at the start of the boil (Equation [18]), and ABratio is the ratio of alpha acids to beta acids (see, for example, Tables 2.1 through 2.3 in Principles of Brewing Science [Fix, pp. 60-62]).  This equation is also specific to hop cones; some modification would be required for hop pellets.  As with oxidized alpha acids, since oxidized beta acids are soluble, all of the oxidized beta acids that are present in the hop cones are in the wort shortly after being added to the kettle, and the oxidized-beta-acid level peaks at toBAmax.

That (again) leaves us with two other oxidized beta acid factors that we still need to model: losses and a scaling factor.  It seems reasonable to assume that oxidized beta acids are lost to trub, yeast, and in other ways, just as isomerized alpha acids and (presumably) oxidized alpha acids are lost.  With that assumption, we can model the oxidized beta acid losses as the losses that affect isomerized alpha acids multiplied by some (unknown) scaling factor.  In other words,

[oBA]wort = oBA × W × 1000 / V [40]
[oBA]beer = [oBA]wort × LFIAA(OG, flocculation, filtering, age) × scaleoBAloss [41]

where [oBA]wort is the concentration of oxidized beta acids in the wort, [oBA]beer is the concentration of oxidized beta acids in the finished beer, W is (still) the weight of the hops in grams, V is (still) the post-boil volume of the wort in liters, LFIAA is the same IAA loss factor from Equation [28] and scaleoBAloss is the (unknown) loss scaling factor.

We also need a scaling factor in Equation [30] that scales the factor for absorption of light at 275 nm of oxidized beta acids (unknown) to the factor for absorption of light at 275 nm of isomerized alpha acids (69.68).  Lewis and Young state that “during storage of hops alpha acids decline but presumably new bitter compounds are formed, largely from beta acids.  … if the alpha-acid to beta-acid ratio is about unity as is commonly the case, sensory bitterness remains more or less constant with storage.” [Lewis and Young, p. 261].  Since sensory bitterness and IBUs are correlated [Lewis and Young, p. 266], and since oxidized beta acids are believed to be the second-largest contributor to IBUs (after isomerized alpha acids), this statement implies that the oxidized beta acids have a relationship between light absorption and concentration that is similar to that of the isomerized alpha acids (69.68).  So, the scaling factor for oxidized beta acids (scaleoBA) should be approximately 1, with emphasis on the “approximately”.  According to Hough et al., “hulupones exhibit 80-90% of the absorption of the iso-alpha-acids at [275nm in acid solution]” [Hough et al., p. 491].  In order to convert this absorption to be the equivalent for IAA, a scaling factor of about 1/0.85 or 1.176 is implied:

scaleoBA = 1.176 [42]

Due to the large number of assumptions made and estimates obtained from the literature, we only need to obtain an estimate for three oBA parameters: oBAagescale, oBAboilMax, and scaleoBAloss.  We can also constrain (oBAfresh + oBAstorage) to be between 0.005 and 0.03, and oBAboilMax to be 0.10 or somewhat less.

3.4.3 Polyphenols
The other nonIAA component we need to consider consists of polyphenols, which are “an extraordinarily diverse group of compounds;” the majority of those in brewing are flavonoids [McLaughlin, p. 1].  Polyphenols can come from both barley and hops [Hough et al., p. 471], so we should separate the PP component into PPhops and PPbarley, where PPhops is the amount of polyphenols contributed by the hops and PPbarley is the amount of polyphenols contributed by the barley.

According to Shellhammer, IBUs are in the range of 1 to 3 for unhopped beer [Shellhammer, p. 177].  I brewed a beer with no hops (OG 1.056) and sent it out for IBU analysis three weeks after the start of fermentation; the result was 0 measured IBUs.  For the model being developed, I’ll assume a constant value of 0.5 IBU from barley polyphenols and ignore the potential decrease in IBUs over time.  Setting the other components in Equation 29 to zero, the scaled concentration of barley polyphenols then becomes 0.5 × 7/5 = 0.7:

[PPbarley]beer × scalePPbarley = 0.7 [43]

where [PPbarley]beer is the concentration of barley polyphenols in the finished beer and scalePPbarley is the scaling factor for light absorption at 275 nm.  We don’t need to determine the separate values of these parameters; knowing that their product is 0.7 is sufficient.  We can then update our estimate of IBUs in beer to separate the contributions from hops and barley polyphenols:

IBU = 5/7 × ([IAA]beer + (([oAA]beer × scaleoAA) + ([oBA]beer × scaleoBA) + ([PPhops]beer × scalePPhops) + ([PPbarley]beer × scalePPbarley))) [44]

Hop polyphenol levels are often reported in the range from 2% to 6% of the weight of the hops [Shellhammer, p. 169; Hough et al., p. 422; Algazzali, p. 5], although McLaughlin reports a higher range, from 4% to 14% [McLaughlin, p. 3].  After having been added to the wort, polyphenols are removed “extensively by precipitation with proteins during wort boiling”; 80% of hop flavanols are removed in the trub when boiling hopped wort [McLaughlin, p. 7].  As Noonan phrases it, “the rolling motion of the boil causes the malt proteins to collide with and adhere to the sticky hop polyphenols” [Noonan, p. 158].  (It may be that the polyphenols are not so much removed as largely insoluble in wort.  The largest polyphenol group in hops (prenylflavonoids) are not soluble in water; all other hop polyphenol components are “soluble in water, preferably in hot water” [Forster, p. 124].  The prenylflavonoids make up about 75% to 85% of all hop polyphenols [Forster, p. 124], so only about 20% of the hop polyphenols are soluble, corresponding to 80% removal.)  Then, polyphenols are removed during fermentation, and “it seems possible that this could occur in much the same way as it does with the iso-alpha-acids” [McLaughlin, p. 7].

From this, we can construct a rough model of the concentration of hop polyphenols in wort and in beer, with an initial level of polyphenols at about 4% of the weight of the hops, a loss factor (or solubility factor) for polyphenols in the wort during the boil (LFPP) estimated at 0.20, and the same loss factors for fermentation and packaging that we have for isomerized alpha acids, LFferment and LFpackage:

[PPhops]wortPPrating × W × 1000 / V [45]
LFPP = 0.20 [46]
[PPhops]beer = [PPhops]wort × LFPP × LFferment(flocculation) × LFpackage(filtering, ageweeks) [47]

where [PPhops]wort is the concentration of hop polyphenols in the wort, PPrating is the percent of the hop weight that consists of polyphenols (similar to the AA rating for alpha acids, on the scale from 0 to 1; a value of 0.04 is a reasonable estimate), LFPP is the loss factor for polyphenols precipitated out of the wort (estimated at 0.20), [PPhops]beer is the concentration of hop polyphenols in the finished beer, and LFferment and LFpackage are the same loss factors for isomerized alpha acids.

Finally, we need a scaling factor to use the concentration of hop polyphenols in Equation [44].  According to Ellen Parkin, “an increase of 100 mg/L of polyphenols was predicted to increase the BU value by 2.2” [Parkin, p. 28], so that 1 ppm of hop polyphenols should increase the IBU by 0.022 (Equation [48]). We can consider Equation [44] in terms of hop polyphenols only, with an IAA component of zero, an oAA component of zero, an oBA component of zero, a non-zero hop polyphenol (PPhops) component, and a PPbarley component of zero (Equation [49]).  Since Equations [48] and [49] both measure IBUs from the contribution of only hop polyphenols, we can determine the value of the scaling factor for hop polyphenols (Equation [50]):

IBU = [PPhops]beer × 0.022 [48]
IBU = 5/7 × (0 + 0 + 0 + ([PPhops]beer × scalePPhops) + 0) [49]
scalePPhops = 7/5 × 0.022 = 0.0308 [50]

here [PPhops]beer is the concentration of hop polyphenols in the finished beer (in ppm) and scalePPhops is the scaling factor for hop polyphenols relative to the scaling factor for IAA.

3.4.4 Solubility of nonIAA Components
The nonIAA components (specifically, oxidized alpha and beta acids, and soluble hop and barley polyphenols) are soluble in water [e.g. Lewis and Young, p. 265; Forster, p. 124].  They do not require isomerization, which (for alpha acid isomerization) takes a significant amount of time.  Therefore, they probably contribute quickly to the measured IBU value.  This is of particular significance for hops that are added late in the boil (or at flameout, or after flameout), since they will have a significant amount of their nonIAA components quickly dissolved and contributing to IBUs, whereas the IAA level will be low due to insufficient time for isomerization.  As a result, the ratio of IAA to all bittering substances can be much lower for hops added close to flameout, even for very fresh hops.  In short, the 1960s finding that the concentration of IAA is 5/7 of the total concentration of all bittering substances reflects not only the age and storage conditions of 1960s hops, but also the typical time(s) at which hops were added to the boil in the 1960s.  Freshly-dried hops added at flameout (with 10 minutes of cooling after flameout) may yield 15 IBUs, but only 50% of that from isomerized alpha acids.

4. Available Data, Parameter Estimation, and Results
4.1 Overview
The quantitative description we now have of IBUs is still incomplete, because we don’t have useful estimates for a number of the factors (specifically: LFboil, [AA]limit, k1_scale, oAAboilMaxtoAAmax, scaleoAAloss, oBAagescale, oBAboilMax, and scaleoBAloss).  We do, however, have Tinseth’s model for predicting IBUs under normal circumstances [Tinseth], results from a study by Val Peacock that looks at IBUs and IAA concentration as a function of hop storage conditions [Peacock, p. 162], and results from ten experiments where I’ve measured IBUs with varying hop steeping times, amounts, and temperatures (to be published later on this blog).  We can make assumptions about the conditions of these studies as needed (i.e. boil gravity, post-boil volume, beta acid level, etc.) and use the data and assumptions, along with common techniques for searching a parameter space, to obtain a rough estimate of the nine unknown values.

4.2 Sources of IBU Data
4.2.1 Tinseth Utilization
The Tinseth model is widely used for predicting IBUs.  Tinseth had “access to some handy tools and knowledgeable friends at the USDA hop labs and the Flavor Perception labs at Oregon State University,” [Tinseth] and he has “had quite a few worts and beers analyzed” [Tinseth].  Therefore, whatever model we develop should come up with estimates close to those predicted by the Tinseth model given similar conditions.  Tinseth provides a detailed description of his model and parameters at realbeer.com.  He based his model on a review of the literature and on data from the pilot brewery at Oregon State University and small breweries; he then verified the model by brewing small batches and testing the results [Hieronymus, p. 185].  In the experiments he conducted in order to validate his model, he used hops from vacuum-sealed oxygen barrier bags stored in a freezer, resulting in very low hop degradation [Tinseth emails].  Also, he took small samples at intervals throughout the boil and immediately cooled them, yielding almost no post-boil utilization [Tinseth emails].

4.2.2 Peacock Hop-Storage Conditions
In an article describing IBUs, Peacock provides results of a study that looked at how the storage conditions of hops affected IBU levels [Peacock, p. 162].  He lists four storage conditions (ranging from -20°F (-29°C) to 70°F (21°C)), the relative percent of alpha and beta acids lost (based on the Hop Storage Index), the IAA levels in the finished beer, and the IBUs of the finished beer. He also provides the alpha/beta ratio of the hops used, but not the amount of hops, wort volume, or original gravity.

4.2.3 Personal Experiments
I conducted a series of ten experiments (one in two parts) that look at IBUs as a function of hop steeping time, amount of hops, and wort temperature.  Experiment 1 was a set of “standard” beers with boil times ranging from 10 to 60 minutes and immediate post-flameout cooling, in order to sync up with the Tinseth formula.  Experiments 2a and 2b were a set of beers with hops added only at flameout and held at a constant temperature for 10 or 20 minutes (from 145°F (63°C) to 212°F (100°C)), in order to evaluate the degree of utilization at sub-boiling temperatures.  Experiment 3 was a set of beers with hops added at varying times during the boil (from 0 to 60 min) and a 15-minute post-flameout natural cooling (a.k.a. a hop stand) before forced cooling.  Experiment 4 looked at utilization as a function of kettle material (stainless steel vs. aluminum) and loose vs. bagged hops.  Experiment 5 looked at the decrease in IBUs over time and utilization as a function of the amount of hops, and Experiment 6 varied some factors from Experiment 5 (amount of hops, boil time, and steep temperature) in order to estimate IAA concentrations from IBU values.  Experiments 7 through 10 looked at IBUs as a function of both alpha-acid concentration and boil time.  I will write about Experiments 7 through 10 in more detail in the future.

One of the biggest difficulties in these Experiments 1, 2, and 3 was obtaining accurate alpha-acid levels of the hops at harvest.  As a result of that difficulty, for these three experiments I allowed the IBU model parameter search (Section 4.3) to evaluate ±1 percentage point around the best estimate of alpha-acid levels at harvest.  In all of these personal experiments,  I also provided some flexibility in the alpha-beta ratios (based, when possible, on estimates from analysis of the hops at around the time of brewing) and the value of AAdecayfactor (based on estimates of how well-preserved the hops might be).  Experiments 7 through 10 used hops from the same bag, and so I constrained the alpha-beta ratio and AAdecayfactor to be the same for all four experiments.

4.3 Parameter Estimation and Results
Using 9 IBU values based on Tinseth’s utilization function (from 10 minutes through 90 minutes at 10-minute intervals) (with typical or assumed values for AAharvest, OG, W, and V, and the values of AAdecayfactor and ABratio fit to the data), the 4 IBU values and 4 IAA values from Peacock (assuming values for original gravity and volume, and fitting the boil time, post boil time, and weight of the hops to the data), and 71 measured IBU values from my ten experiments, there are 88 data points with which to estimate the nine unknown parameter values, as well as a number of source-specific parameter values (e.g. weight of the hops in Peacock’s study).  This really isn’t enough data for a reliable estimate of all parameters, but it’s better than nothing.  It helps that these sources of data cover a number of scenarios of interest, including boil time, storage conditions of the hops, weight of hops used, and hop steeping temperature.

Tables 1 through 11 (below) provide the known values, assumptions, estimated values, and IBU (or IAA) results for each set of data.  In addition, flocculation was set to “normal” and filtering was set to “none”.  All other parameters not being estimated were given the best-guess values noted in the previous sections.  For oxidized alpha and beta acids produced during the boil, I previously found that below-boiling temperatures may produce fewer oxidized alpha and beta acids.  I assumed a linear decrease with temperature, from full oxidized-acid production (scale factor 1.0) at boiling to zero production (scale factor 0.0) at room temperature.  For Tinseth and Peacock, I assumed loose whole hops, so that RFform(hopsForm=loose cones)=1.16; for my experiments, I used RFform(hopsForm=loose cones)=1.16 or RFform(hopsForm=bagged cones)=1.05, depending on the form of the hops.  The alpha-acid decay factor in Tables 1 and 2, AAdecayfactor, is the result of the Garetz formula ek×TF×SF×D; I constrained the search range for this factor based on best guesses of the variables k, TF, SF, and D in each condition.

I used an iterative brute-force search over the parameter space to minimize the squared error, starting with the approximate range of each parameter and a coarse search interval.   After each iteration, I used the best estimates of each parameter to specify a smaller range, along with a smaller search interval.  The search process was stopped when best estimates were obtained with a typical search interval of 0.01.  A nested recursion was used to constrain the five unknown model parameters in an iteration to be the same for all data sources, while the unknown parameters from each experiment were searched for individually.  (I will provide java and C-code procedures of the complete IBU model, after I have a chance to publish the remaining experiments and clean up the code.)

The result of this parameter search is not an ideal solution!  We have a very large number of assumptions, a fairly large number of unknown parameters, and a relatively small amount of data.  As a result, the estimates of the parameter values will almost certainly be wrong at some level.  My hope, however, is that a slight overestimate of one factor will be balanced by a small underestimate of another factor, and on average the model will provide a cohesive, general description of the factors that contribute to IBUs.  The model and parameter settings provide a “most likely” set of values given the (limited) data.  Because of the lack of held-out test data, the resulting description of IBUs is descriptive, not predictive.  In other words, I make no guarantee of how well this model will predict your IBU values, even if you know all of the input parameter values (hops weight, volume, alpha acid level at harvest, alpha-beta ratio, storage conditions, steep time, etc.).  This model may, however, help with understanding the various factors and relative contributions of these factors to the IBU measure.

The results of the search for the nine parameters are: LFboil = 0.58, [AA]limit = 230 ppm, k1_scale = 3.9, oAAboilMax = 0.35, toAAmax = 12 minutes, scaleoAAloss = 0.04, oBAagescale = .01, oBAboilMax = 0.08, and scaleoBAloss = 0.76.  The estimated value of LFboil is fairly close to that of Malowicki’s report that the formation of trub causes losses of 35% (translating to a scaling factor of 0.65) [Malowicki, p. 7-8].  The alpha-acid solubility limit of 230 ppm is lower than Spetsig’s estimate of 300 ppm at boiling [Spetsig, p. 1423], but he notes that the value of 300 is an approximation based on extrapolation from two data points at 25°C and 40°C [Spetsig, p. 1424].  The small value of scaleoAAloss, compared with the larger value of scaleoBAloss, results in a smaller contribution of oxidized alpha acids compared with oxidized beta acids, which is also in agreement with the literature (e.g. [Peacock, p. 157]).  The value of oBAagescale, when used to compute oBAstorage, yields (oBAfresh + oBAstorage) in the ballpark of 1% for typical values of oBAfresh, which is in line with reported values from 0.5% to 3% [Stevens and Wright, p. 500; Spetsig and Steninger, p. 413; Mussche, p. 13].  The value of 0.08 (8%) for oBAboilMax is also in line with the reported value of 10% [Stevens and Wright, p. 500].

Tables 1 and 2 provides the known, assumed, and estimated values of parameters that could vary between the sources of data.  Parameters that could vary were constrained to a reasonable search range based on available data. Note that several values in the Tinseth column do not need to be the same as what Tinseth used in his experiments; as long as the same values of these parameters are used in the comparison with the current model, any values can be used.  For the Peacock study, the alpha acid rating at harvest was determined based on the data he published.  I assumed a two-barrel (62 G or 234 liter) volume for Peacock’s experiments; if this assumption is incorrect, then the estimated weight of the hops can be scaled proportionally to give the same results.  I also assumed a fairly slow post-flameout temperature decay for Peacock’s experiments (reaching 200°F (93.5°C) after 30 minutes), under the assumption that a large volume of wort cools slowly; if the actual temperature decay was different, the weight of hops, boil time, and/or post-boil time may need to be adjusted.

Tinseth Peacock Exp. #1 Exp. #2a Exp. #2b Exp. #3
AA at harvest
8.65% (?) 3.9% 8.0% 7.9% 8.4% 6.0%
α/β ratio 1.00 1.35 0.75 1.45 1.0 1.6
AA decay factor
0.95 0.07 to 0.83 0.82 0.87 0.99 0.95
boil time
10 to 90 min 90 min
10 to 60 min 0 min 0 min 0 to 60 min
post-boil time
0 min 50 min 0 min 10 to 20 min 10 min 15 min
post-boil temp.
N/A slow decay N/A 185°F to 212°F 145°F to 212°F fast decay
hops weight
13.0 oz 13.0 oz 0.60 oz 1.60 oz 1.60 oz 0.80 oz
wort volume
62 G (?) 62 G (?) 1.37 to 1.50 G 1.10 to 1.24 G 1.05 to 1.20 G 0.88 to 1.15 G
boil gravity
1.055 (?) 1.035 (?) 1.059 to 1.064 1.064 to 1.066 1.063 to 1.065 1.065 to 1.075

Table 1. Known values, assumed values, and best estimates of parameters that were allowed to vary between the sources of data, for the first six sources of data. If a value has no markings, it is a known value.  If a value has a question mark after it (?), it is an assumed value.  If a value is in bold face and red, it is the best estimate as determined by the parameter search.

Exp. #4 Exp. #5 Exp. #6 Exp. #7 Exp. #8 Exp. #9 Exp. #10
AA at harvest 8.1% 8.1% 8.1% 13.3% 13.3% 13.3% 13.3%
α/β ratio 1.0 1.10 1.10 3.31 3.31 3.31 3.31
AA decay factor 0.95 0.99 0.96 0.55 0.55 0.55 0.55
boil time 20 min 12 min 0 to 26.9 min 0 to 100 min 0 to 103 min 0 to 100 min 0 to 100 min
post-boil time 0 min 0 min 0 to 19 min N/A (0) N/A (0) N/A (0) N/A (0)
post-boil temp. N/A N/A 145°F N/A N/A N/A N/A
hops weight 0.75 oz 0.37 to 2.22 oz 0.37 to 2.22 oz 1.15 oz 2.923 oz 4.25 oz 6.25 oz
wort volume 1.52 to 1.61 G 1.61 to 1.65 G 1.59 to 1.63 G 8.14 to 7.78 G 8.23 to 7.73 G 8.25 to 7.65 G 8.06 to 7.51 G
boil gravity 1.056 to 1.059 1.054 to 1.056 1.055 to 1.056 1.048 to 1.051 1.049 to 1.052 1.048 to 1.052 1.049 to 1.052

Table 2. Known values, assumed values, and best estimates of parameters that were allowed to vary between the sources of data, for the remaining seven sources of data. If a value has no markings, it is a known value.  If a value has a question mark after it (?), it is an assumed value.  If a value is in bold face and red, it is the best estimate as determined by the parameter search.

Tables 3 through 11 show results from the Tinseth, Peacock, and personal experiments.  Table 3 shows the results of IBU estimation based on the Tinseth formula and based on the estimates obtained from the current model:

time 10 min
20 min 30 min 40 min 50 min 60 min 70 min 80 min 90 min
formula 10.9 18.1 23.0 26.3 28.5 29.9 30.9 31.6 32.0
estimate 11.6 17.2 21.2 24.7 27.5 29.9 31.9 33.4 34.7
diff. .07 -1.0 -1.8 -1.6 -1.0 0.0 0.9 1.8 2.6

Table 3. IBU estimates from the Tinseth formula and the current model, as a function of hop steep time, and the difference (error) between the two.

Table 4 shows the IAA and IBU measured values reported by Peacock, and the results of IAA and IBU estimation from the current model:

condition -20°F 25°F 40°F 70°F
measured IAA 19.8 ppm 18.1 ppm 14.4 ppm 2.9 ppm
measured IBU 13.5 12.0 13.5 11.0
estimated IAA 17.6 ppm 15.5 ppm 11.5 ppm 1.5 ppm
estimated IBU 17.0 16.1 14.2 9.7
IAA difference
2.2 ppm 2.6 ppm 2.9 ppm 1.4 ppm
IBU difference
3.6 4.1 0.7 -1.3

Table 4. IAA and IBU measured values and estimates from the current model, as a function of hop storage conditions.  The difference (error) between measured and estimated values is also shown.

Table 5 shows the measured and estimated IBU values from my experiment #1 (mIBU experiment #1), meant to sync up with the Tinseth formula.  The estimate of the alpha-acid rating at harvest (8.0%) is equal to the value written on the package I bought.  The estimate of the alpha/beta ratio (0.75) is not too far off from an estimate obtained by analysis of the hops’ alpha and beta values (0.862).  The degradation factor of 0.82 is reasonably close to the degradation factor estimated from the Hop Storage Index (0.72).

steep time 10 min
20 min
40 min
60 min
measured IBU
22.0 27.1 34.3 35.7
estimated IBU
18.2 24.9 32.1 38.9
IBU difference
-3.8 -2.2 -2.2 3.3

Table 5. Measured IBU values and estimated IBU values from personal experiment #1, as a function of hop steep time.  The difference (error) is also shown.

Table 6 shows the measured and estimated IBU values from my experiment #2 (mIBU experiment #2), which looked at utilization as a function of steep temperature.  In most cases, the steep time was 10 minutes, but in one case the steep time was 20 minutes.

temp/
time
212°F/
10m
200°F/
10m
190°F/
10m
185°F/
10m
192°F/
20m
212°F/
10m
175°F/
10m
165°F/
10m
155°F/
10m
145°F/
10m
meas. 33.3 28.9 30.8 25.5 35.9 40.6 23.6 24.5 23.1 21.8
est. 35.9 30.6 27.9 26.4 33.3 39.4 26.6 24.5 22.4 20.7
diff. 2.6 1.7 -2.9 0.9 -2.6 -1.2 3.0 0.0 -0.7 -1.1

Table 6. Measured IBU values and estimated IBU values from personal experiment #2, as a function of hop steeping temperature and time.  The difference (error) is also shown.

Table 7 shows the measured and estimated IBU values from my experiment #3 (mIBU experiment #3) which combined various hop boil times with a 15-minute hop stand.  The wort was allowed to cool naturally during this 15 minutes, after which it was force-cooled.

time 0 min
7.5 min 15 min 30 min 60 min
measured
16.1 21.2 26.1 35.4 46.4
estimated 14.6 20.6 26.1 35.7 48.9
difference -1.5 -0.6 0.0 0.3 2.5

Table 7. Measured IBU values and estimated IBU values from personal experiment #3, as a function of hop boil time.  The difference (error) is also shown.

Table 8 shows the measured and estimated IBU values from my experiment #4 (utilization experiment #1), which looked at utilization as a function of kettle material and form of the hops.

kettle material,
hop form
stainless steel,
loose
aluminum,
loose
aluminum,
bagged
measured
34 37 36
estimated
33.6 34.0 32.4
difference
-0.4 -3.0 -3.6

Table 8. Measured IBU values and estimated IBU values from personal experiment #4, as a function of kettle material (stainless steel or aluminum) and hop form (loose cones or bagged cones).

Table 9 shows the measured and estimated IBU values from my experiment #5 (utilization experiment #2), which looked at utilization as a function of weight of the hops.

weight
0.37 oz
0.74 oz
1.11 oz
1.48 oz
1.85 oz
2.22 oz
measured
12 23 29 34 41 47
estimated 12.4 23.3 29.1 34.7 40.8 46.6
difference 0.4 0.3 0.1 0.7 -0.2 -0.4

Table 9. Measured IBU values and estimated IBU values from personal experiment #5, as a function of hop weight.  The difference (error) is also shown.

Table 10 shows the measured and estimated IBU values from my experiment #6 (utilization experiment #3), which looked at variety of conditions: Condition H had hop weight of 0.37 oz and boil time of 26.9 min; Condition I had hop weight of 1.11 oz and boil time of 26.9 min; Condition J had hop weight of 1.11 oz and boil time of 12 min; Condition K had hop weight of 2.22 oz and boil time of 19.0 min; and Condition L had hop weight of 2.22 oz, with no boiling but a 19-minute hop stand held at 145°F.  Conditions H through K were immediately cooled upon flameout.

Condition
H
I
J
K
L
measured
18 48 32 58 27
estimated 20.6 47.3 32.4 59.2 27.5
difference 2.6 -0.7 0.4 1.2 0.5

Table 10. Measured IBU values and estimated IBU values from personal experiment #6, as a function of hop boil time.  The difference (error) is also shown.

Table 11 shows the measured and estimated IBU values from my experiments #7 through #10, which looked at IBUs as a function of both boil time and alpha-acid concentration.


Exp. #7
Exp. #8
Exp. #9
Exp. #10
10 min
meas. = 8.0
est. = 7.4
diff. = -0.60
meas. = 19.5
est. = 17.8
diff. = -1.70
meas. = 20.5
est. = 24.1
diff. = 3.62
meas. = 34.0
est. = 30.9
diff. = -3.10
20 min
meas. = 11.0
est. = 10.3
diff. = -0.67
meas. = 27.5
est. = 25.2
diff. = -2.27
meas. = 30.0
est. = 34.4
diff. = 4.38
meas. = 43.0
est. = 41.8
diff. = -1.21
30 min
meas. = 14.5
est. = 12.7
diff. = -1.82
meas. = 32.5
est. = 31.2
diff. = -1.31
meas. = 39.0
est. = 42.6
diff. = 3.60
meas. = 53.0
est. = 51.3
diff. = -1.72
40 min
meas. = 16.5
est. = 14.7
diff. = -1.82
meas. = 38.0
est. = 36.3
diff. = -1.74
meas. = 48.5
est. = 49.6
diff. = 1.08
meas. = 64.0
est. = 59.9
diff. =-4.10
50 min
meas. = 19.5
est. = 16.4
diff. = -3.13
meas. = 42.5
est. = 40.6
diff. = -1.89
meas. = 51.5
est. = 55.5
diff. = 3.99
meas. = 68.0
est. = 67.2
diff. = -0.81
60 min
meas. = 22.5
est. = 22.8
diff. = 0.29
meas. = 48.0
est. = 44.3
diff. = -3.69
meas. = 59.0
est. = 60.5
diff. = 1.50
meas. = 72.5
est. = 73.3
diff. = -2.22
70 min
meas. = 28.0
est. = 27.7
diff. = -0.28
meas. = 47.5
est. = 47.3
diff. = -0.14
meas. = 62.0
est. = 64.6
diff. = 2.58
meas. = 81.0
est. = 78.4
diff. = -2.63
80 min
meas. = 32.5
est. = 31.3
diff. = -1.21
meas. = 50.0
est. = 49.9
diff. = -0.11
meas. = 66.5
est. = 68.0
diff. = 1.47
meas. = 83.0
est. = 82.6
diff. = -0.43
90 min
meas. = 33.5
est. = 34.3
diff. = 0.80
meas. = 54.0
est. = 52.0
diff. = -1.95
meas. = 69.5
est. = 70.8
diff. = 1.25
meas. = 79.0
est. = 85.9
diff. = 6.93
100 min
(103 for Exp #8)
meas. = 37.0
est. = 36.8
diff. = -0.16
meas. = 52.5
est. = 54.1
diff. = 1.60
meas. = 72.5
est. = 72.9
diff. = 0.46
meas. = 74.0
est. = 88.7
diff. =14.67

Table 11. Measured IBU values and estimated IBU values from personal experiments #7 through #10, as a function of hop boil time.  The difference (error) is also shown.  The values from Experiment #10 at times 90 and 100 minutes were not included in the model fitting, because the model is not capable of predicting such a decrease in IBUs.  At this point, I conjecture that the decrease in IBUs is due to changing pH at very high alpha-acid concentrations and long boil times.

5. Discussion of Results
The average difference between observed (or Tinseth model) IBU and IAA values and current model estimates is -0.04, with a standard deviation of 2.0 and a maximum difference of 4.4.  The fact that all 10 IBU values from Experiment #9 are overestimates and all 8 IBU values used in Experiment #10 are underestimates indicates something sub-optimal in the model or in the parameter estimation.  From the data I’ve seen, observed IBU values can deviate quite a bit from expected values (for reasons that are still unclear to me), and so the overall results from the model do not seem excessively bad.  With a human detection threshold of 5 IBU [Daniels, p. 76], none of the errors in the model (with a maximum difference of 4.4 IBU) would be detectable by a human palate. While few, if any, of the model parameters have been estimated with great precision, the overall fit suggests that errors in one parameter estimate are, for the most part, balancing out errors in another estimate.

To the extent that parameter estimation has been reasonable, we can use this model to look at how various factors affect IBUs.  If we assume some typical brew parameters (OG 1.055, volume 5.25 G or 20 liters, a typical AA rating of 8.65%, an alpha/beta ratio of 1.4, exceptionally well-preserved hops with AAdecayfactor of 1.0, post-flameout natural cooling for 10 minutes, and taking IBU measurements one week after the start of fermentation), we can vary the amount and timing of hops additions in the model to look at the impact on IBU and IAA.  For example, 2 oz added at flameout will create 15.6 IBUs with a concentration of 10.4 ppm of IAA (47% of the IBU total), 1.4 ppm of oAA, 7.4 ppm of oBA, and 18.9 ppm of hop polyphenols.  The same 2 oz added at 60 minutes will create 55.9 IBUs with a concentration of 65.2 ppm of IAA (83% of the IBU total), 1.6 ppm of oAA, 8.6 ppm of oBA, and 18.9 ppm of hop polyphenols.  If we triple the amount of hops, from 2 oz to 6 oz, the IBUs only increase from 55.9 to 90.1 (88.2 ppm of IAA, representing 70% of the total; 4.9 ppm of oAA, 25.6 ppm of oBA, and 56.8 ppm of hop polyphenols).  If we add those 6 oz at flameout, we’ll get 31.0 IBUs, with only 10.4 ppm of IAA (24% of the IBU total).  If we have somewhat degraded hops (say, stored at room temperature in airtight packaging for six months) yielding an AAdecayfactor of 0.82, the 2 oz of hops added at 60 minutes will yield 48.9 IBUs, with 53.9 ppm of IAA representing 79% of the IBU total.  Adding these degraded hops at flameout will produce 15.7 IBUs, but with only 8.8 ppm of IAA representing 40% of the IBU total.  The lack of a strong impact on IBUs when using somewhat degraded hops is in line with reported experience [Peacock, p. 162].

Another interesting thing we can do is estimate the contribution of nonIAA components to the Tinseth formula.  While the Tinseth formula uses only the weight and alpha-acid rating of the hops to compute IBUs [Tinseth], the utilization function was fit to observed data [Pyle], which includes nonIAA components.  We can use the current detailed model to separate out the actual IAA contribution to utilization from the (implicit) nonIAA contribution.  For example, at 10 minutes before flameout, the detailed model predicts 14.75 IBUs in a (post-boil volume) alpha-acid concentration of 168.74 ppm using the Tinseth source of data.  (The Tinseth formula predicts 14.20 IBUs using the same data.)  If the IBU value was equivalent to the concentration of isomerized alpha acids, as assumed by the Tinseth equation, then at the final boil volume there would be utilization of 14.75 ppm / 168.74 ppm = 0.0874 (or 8.74% utilization).  The detailed model tells us, however, that at 10 minutes the relative contribution of IAA to the IBU is only 0.50, and that the IAA concentration is 10.30 ppm, for a utilization of 0.0610.  The nonIAA components contribute an “IAA equivalent” 10.35 ppm (obtained by the sum of their estimated concentrations multiplied by their scaling factors), for a total of 20.65 ppm of IAA-equivalents in 168.74 ppm of alpha acids, or an IAA-equivalent utilization of 12.24%.  If we multiply 0.1224 by 5/7, we get the 0.0874 that we estimated by assuming that IBUs are equivalent to the concentration of IAA.  (As a quick example, 6.94 ppm of oBA multiplied by the scaling factor of 1.176 yields 8.16 ppm of IAA equivalents obtained from oBA; the remaining 2.19 ppm comes from oAA and polyphenols.)  In the Tinseth formula, therefore, at 10 minutes about 4.22% of the 8.41% utilization is coming from nonIAA components.  At 5 minutes, 2.58% of the 4.68% utilization is from nonIAA components, and at 12 minutes and above, about 5% is from nonIAA components.

In general, one can think of the nonIAA components as contributing up to 5% of the utilization in the Tinseth formula.  (In other words, if the Tinseth utilization is 0.22 (22%), then 0.05 (5%) can be thought of as coming from nonIAA components, and the remaining 0.17 (17%) from IAA.)  This corresponds fairly well with the Rager IBU formula [Pyle], which has a non-zero and roughly constant utilization of 5% (0.05) from 0 to 5 minutes, presumably accounting for nonIAA components at short boil times.

6. Summary
This post has described the various factors that contribute to the IBU, and quantified each factor as much as possible. Estimates of parameter values that could not be determined from the literature were obtained by fitting a model to the available data.

Despite the length of this post, many things have been left undiscussed.  The current model is restricted to full boil of the wort (i.e. not performing the boil at higher gravity and then diluting).  The topic of dry hopping and its impact on bitterness is left entirely to Ellen Parkin [Parkin], Maye et al. [Maye], and others.  The model is probably much less effective when it comes to the IBUs of darker beers and stouts, since dark malts may affect bitterness and the IBU value (although I’ve seen surprisingly lower-than-expected IBU values in my stouts).  The perception of bitterness is left out entirely (especially at high IBU values), as is the large topic of different bitterness qualities.  I’ve also put off a number of topics (e.g. alpha acid concentration limit at boiling, rate of alpha-acid oxidation based on Maye et al.’s paper [Maye]) for future blog posts.

What’s the take-away message of this post?  If you’re adding hops late in the boil (or at flameout), you will probably not get a lot of bitterness from alpha acid isomerization.  You can, however, get a significant number of IBUs (and bitterness) from this hop addition, with most of the IBU value coming from nonIAA components.  Likewise, if you’re using a large amount of hops, the IBU value may be smaller than you’re expecting (due to what appears to be the solubility limit of alpha acids in boiling wort), but much of that IBU value may come from nonIAA components.  To the extent that the model development and parameter estimation has been correct, most of the contribution to nonIAA components is from oxidized beta acids, and a significant amount of the oxidized beta acids are produced during the boil (or during post-boil steeping).  Hopefully this post and model will help in understanding the relative contributions of isomerized alpha acids and nonIAA components to the IBU.

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