Tag Archives: hops

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 260 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 0.71 (nearly identical to the HSI-based factor of 72%), yielding an AA rating on brew day of 5.7%, which is very close to the AA rating estimated from the Tinseth equation (5.8%).  The estimated alpha/beta ratio was 0.85, very close to the value from AL (0.86).  The RMS error from this model was 2.22 IBUs (about half the error of the Tinseth model), with a maximum difference of 2.9 IBUs.  According to this model, isomerized alpha acids contributed 64%, 58%, 44%, and 30% 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 13.8 IBUs at flameout (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
37.9 31.4 25.2 20.4

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, (a) fitting the AA rating to the IBU data provided a not-terrible fit to the Tinseth model (with an RMS error of 4.32 IBUs) and (b) this estimated AA rating was close to the AA rating estimated by a different, more detailed, model of 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 roughly constant.  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 approximately 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 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 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 estimated alpha-acid solubility limit of 260 ppm by simply limiting the alpha-acid concentration in the Tinseth equation to this value.  (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 applied in a separate blog post: 1.0 – (0.015 × ageweeks), 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 1.3 (somewhat higher than the value of 1.21 reported by KAR), and a decay factor of 1.0 provided the best fit to the data.  With these values, there is an RMS error of 2.44 IBUs and a maximum difference of 4.3 IBUs (for the 60-minute condition).  According to this model, isomerized alpha acids contributed 76%, 69%, 60%, 49%, and 28% 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 high alpha-acid concentration.

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
50.7 37.8 27.9 22.0 14.9

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

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).

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, except for the 60-minute condition with a difference of 4.3 IBUs.  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.

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 storage 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 top 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.

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).  First, he 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 uncharacterized 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).

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 Huang’s 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, before racking.

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 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 about a dozen lines of programming code.  First, we need to 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.  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) [20a]
TC(tf) = -0.74667 tf + 99.244    (for temperature in Celsius) [20b]
TK(tf) = -0.74667 tf + 372.394  (for temperature in Kelvin) [20c]

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 indicates that this function is not significantly dependent on ambient temperature or relative humidity, so this function only needs to be constructed once per 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 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:

totalTime = boilTime + postBoilTime;
integrationTime = 0.001;
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(10.0,11.0)*exp(-11858.0/temp);
    k2 = 4.1*pow(10.0,12.0)*exp(-12994.0/temp);
    dIAA = AA0 * (k1/(k2-k1)) * ((k2*exp(-1.0*k2*t))-(k1*exp(-1.0*k1*t)));
    IAA = IAA + (dIAA * integrationTime);
    time = time + integrationTime;
}

where totalTime is the length of the boil in minutes (boilTime) plus any time after the boil when isomerization might be happening (postBoilTime).  The integration time of 0.001 (called integrationTime) is sufficient for accuracy to at least two places past the decimal point.  Here, IAA is the total concentration of IAA, or [IAA], after time time (in minutes).  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 dIAA is the derivative of [IAA], or change in [IAA] per minute.  The variable AA0 is the initial concentration of alpha acids, in ppm (see Equations [17] and [23]).  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, wort gravity, 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 during the boil, fermentation, filtration, and aging.  We’ll look at each of these briefly in this section.

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) [21]

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.)  An increased wort pH will increase utilization [Lewis and Young, p. 266, Kappler p.334].  The dependence on pH, however, shouldn’t impact the typical homebrewer, who 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].

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]. It is not clear to me if the higher gravity slows the conversion of alpha acids to isomerized alpha acids, or if the higher gravity causes more isomerized alpha acids to bind with trub and settle out of solution.  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:

RFOGN(OG) = (-4.944 × OG) + 6.166    if OG > 1.045, else 1.0 [22]

where RFOGN(OG) is Noonan’s gravity rate-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, RFOGN(OG) is defined as 1.  Glenn Tinseth models the  gravity correction factor as RFWGT(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].

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 would then have a combined correction factor of 1.05 (1.27 × 0.91 × 0.91) [Gartez book, p. 141].  For the model being developed, I’ll assume that bagged hops are always loosely bagged, for a “bagging” correction factor of 0.91.

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 (to be described in a future blog post), 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 simply limiting the alpha acids available for conversion to about 260 ppm, requiring a revision of Equation [17]:

[AA]0 = AA × W × 1000 / V, with maximum [AA]0 = 260 [23]

(The value of 260 was obtained, in part, by fitting the complete quantitative model described in this blog post to available data (see Section 4), so this value is a result of the model development.)  This limit is greater than the solubility of alpha acids at room temperature (around 90 ppm [Malowicki, Appendix A, pp. 51-54]), but it common that solubility increases with temperature [Wikipedia].  Using an alpha-acid concentration limiting factor is also in accord with the conclusion reached by Maule (quoted above [Maule, p. 290]).  Using this approach, utilization increases linearly until the solubility limit is reached (260 ppm), after which utilization is not affected by an increased presence of alpha acids.  This approach is overly simplistic, but seems to work reasonably well on the available data.  One unfortunate complication is that separate hop additions can not be treated independently.

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 During the Boil:  Isomerized alpha acids are lost during the boil.  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].  Malowicki says that “trub, and specifically the formation of trub, leads to greatly increased losses of bitter acids” [Malowicki, p. 8].  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.  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].  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 During 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].  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, 6, 7, and 13 weeks from the start of fermentation, and fit the measured IBU decrease over time to a linear function.  (I will provide more detail about this function in a future blog post.)  A linear fit is probably not optimal, but within the range of two months it provides a reasonable fit to the data available to me.  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 formula determined for IBUs, and include a filtering factor:

LFpackage(filtering, ageweeks) = 0.98 × (1.0 – 0.015 × ageweeks) for filtered beer, or
(1.0 – 0.015 × ageweeks) for unfiltered beer
[24]

where LFpackage(filtering, ageweeks) is the loss factor due to packaging, which encompasses both filtering and age of the beer (in weeks).  The loss factor of 0.98 is applied only to filtered beer, and the decrease in isomerized alpha acids over time is modeled with a factor of -0.015 multiplied by the age of the beer in weeks after the start of fermentation, ageweeks.

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 (based on a maximum alpha acid concentration of 260 ppm), multiplied by the various isomerization rate adjustment factors and IAA loss factors discussed above:

RFIAA(T, OG, hopsForm, V) = RFtemp(T) × RFOGN(OG) × RFform(hopsForm) × RFsize(V) [26]
LFIAA(flocculation, filtering, ageweeks) = LFboil × LFferment(flocculation) × LFpackage(filtering, ageweeks) [27]
[IAA]beer = [IAA]wort × RFIAA(T, OG, hopsForm, V) × LFIAA(flocculation, filtering, ageweeks) [28]

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; RFOGN is Noonan’s rate factor as a function of original gravity; 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 LFboil is the loss factor during the boil; LFferment is the loss factor due to fermentation (with flocculation being “high”, “medium”, or “low”); and LFpackage is the loss factor due to filtration (with parameter filtering being “unfiltered” or “filtered”) and age (which varies with the age of the beer, ageweeks).  Note that 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.

The only problem remaining for modeling [IAA]beer is that while we have a good idea of some factors (RFtemp, RFOGN) and a rough approximation of others (RFform, LFferment, LFpackage), we have very little basis for determining the remainder (LFboil and maximum [AA]0).  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))) [29]

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 [1], Equation [23], and Equation [28].)

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].

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 = oAAfreshoAAstorage + oAAboil [30]

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 is the level of oxidized alpha acids produced during the boil; 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 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:

oAA = (1 – 1/ek×1×1×0.5) + (oAAagescale × (1 – AAdecayfactor)) + (AA × oAAboil) [31]

where oAA is the same level of oxidized alpha acids in Equation [30], 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], AA is the level of alpha acids at the start of the boil (Equation [18]), and oAAboil is the relative amount of alpha acids that undergo oxidation during the boil.  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, I believe that we don’t need to model any dependence on how long the hops are in the kettle; we can assume that all of the oxidized alpha acids are in the wort shortly after being added to the kettle.  (There may be some time dependence for oAAboil, but given a complete lack of data in that regard, I’ll assume for now that the time dependence is minimal.)  That leaves us with two 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 [32]
[oAA]beer = [oAA]wort × LFIAA(flocculation, filtering, ageweeks) × scaleoAAloss [33]

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 [27] and scaleoAAloss is the (unknown) loss scaling factor.

We also need a scaling factor in Equation [29] 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 [34]

Despite the large number of parameters for modeling oAA, we end up needing to obtain estimates of only two: oAAboil 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 = oBAfreshoBAstorage + oBAboil [35]

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 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].  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]. Given a lack of data about oBAfresh, I’ll assume that oxidized beta acids are produced at the same levels as oxidized alpha acids both in fresh hops and during aging.  This gives a formula similar to Equation [31]:

oBA = (1 – 1/ek×1×1×0.5) + (oBAagescale × (1 – AAdecayfactor)) + ((AA / ABratio) × oBAboil) [36]

where oBA is the same level of oxidized beta acids in Equation [35], 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]), 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]), and oBAboil is the relative amount of beta acids that undergo oxidation during the boil, assumed to be 0.10.  This equation is also specific to hop cones; some modification would be required for hop pellets.

As with oxidized alpha acids, we can assume that all of the oxidized beta acids are in the wort shortly after being added to the kettle. That leaves us with two 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 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 [37]
[oBA]beer = [oBA]wort × LFIAA(kettleMaterial, flocculation, filtering, age) × scaleoBAloss [38]

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 [27] and scaleoBAloss is the (unknown) loss scaling factor.

We also need a scaling factor in Equation [29] 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 [39]

Due to the large number of assumptions made and estimates obtained from the literature, we only need to obtain an estimate for one oBA parameter: scaleoBAloss.

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 [40]

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))) [41]

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 really removed, but that they are 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 [42]
LFPP = 0.20 [43]
[PPhops]beer = [PPhops]wort × LFPP × LFferment(flocculation) × LFpackage(filtering, ageweeks) [44]

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 [41].  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 [45]). We can consider Equation [41] 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 [46]).  Since Equations [45] and [46] both measure IBUs from the contribution of only hop polyphenols, we can determine the value of the scaling factor for hop polyphenols (Equation [47]):

IBU = [PPhops]beer × 0.022 [45]
IBU = 5/7 × (0 + 0 + 0 + ([PPhops]beer × scalePPhops) + 0) [46]
scalePPhops = 7/5 × 0.022 = 0.0308 [47]

where [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 very 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 all (or nearly all) 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 very late in the boil, 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 20 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, maximum [AA]0, oAAboil, scaleoAAloss, 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 six 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 five 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 used, wort volume, or original gravity.

4.2.3 Personal Experiments
I conducted a series of six 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 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.

I will write about Experiments 4, 5, and 6 in more detail in the future, but for now I’ll mention that one of the biggest difficulties was obtaining accurate alpha-acid levels of the hops for the first three experiments.  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, and I also provided some flexibility in the alpha-beta ratios (based on estimates from analysis of the hops around the time of brewing) and value of AAdecayfactor.

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 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 the 33 measured IBU values from my six experiments, there are 50 data points with which to estimate the five 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 9 (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 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 Table 1, 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 to be the same for all data sources, while the unknown parameters from each experiment were searched for individually.  (I will provide the C-code procedure of the complete IBU model, after I have a chance to 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 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 five parameters are: LFboil = 0.60, maximum [AA]0 = 265 ppm, oAAboil = 0.07, scaleoAAloss = 0.04, 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 small values of oAAboil and scaleoAAloss, compared with the larger value of scaleoBAloss, result 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]).

Table 1 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 many 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 one-barrel (31 G or 117 liter) volume for Peacock’s experiments; if this assumption is incorrect, then the estimated weight of the hops can be scaled proportionally to give identical results.  I also assumed a slow post-flameout temperature decay of -0.2°C per minute for Peacock’s experiments, 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 Exp. #4 Exp. #5 Exp. #6
AA at harvest
8.65% (?) 3.9% 8.0% 7.4% 8.9% 6.0% 8.1% 8.1% 8.1%
α/β ratio 1.10 1.35 0.85 1.5 1.4 1.3 1.0 1.05 1.05
AA decay factor
0.95 0.07 to 0.83 0.71 0.92 0.94 1.0 1.0 0.975 0.95
boil time
10 to 90 min 90 min
10 to 60 min 0 min 0 min 0 to 60 min 20 min 12 min 0 to 26.9 min
post-boil time
0 min 50 min 0 min 10 to 20 min 10 min 15 min 0 min 0 min 0 to 19 min
post-boil temp.
N/A slow decay N/A 185°F to 212°F 145°F to 212°F fast decay N/A N/A 145°F
hops weight
1.5 oz (?) 5.5 oz 0.60 oz 1.60 oz 1.60 oz 0.80 oz 0.75 oz 0.37 to 2.22 oz 0.37 to 2.22 oz
wort volume
5.25 G (?) 31 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 1.52 to 1.61 G 1.61 to 1.65 G 1.59 to 1.63 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 1.056 to 1.059 1.054 to 1.056 1.055 to 1.056

Table 1. Known values, assumed values, and best estimates of parameters that were allowed to vary between the 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.  The estimated decay factor of 0.51 in Experiment 1 is low, but actually fairly likely as that source of hops was probably stored for months at room temperature.

Tables 2 through 9 show results from the Tinseth, Peacock, and personal experiments.  Table 2 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 14.8 24.7 31.4 35.8 38.8 40.8 42.2 43.1 43.7
estimate 15.8 22.6 28.4 33.3 37.3 40.7 43.5 45.8 47.6
diff. 1.0 -2.1 -3.0 -2.6 -1.5 -0.1 1.3 2.7 3.9

Table 2. 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 3 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 17.0 ppm 15.0 ppm 11.1 ppm 1.4 ppm
estimated IBU 16.2 15.6 14.4 11.5
IAA difference
-2.8 ppm -3.1 ppm -3.3 ppm -1.5 ppm
IBU difference
2.7 3.6 0.9 0.5

Table 3. 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 4 shows the measured and estimated IBU values from my 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 (0.85) is close to an estimate obtained by analysis of the hops’ alpha and beta values (0.862).  The degradation factor of 0.71 is 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
20.4 25.2 31.4 37.9
IBU difference
-1.6 -1.9 -2.9 2.2

Table 4. 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 5 shows the measured and estimated IBU values from my 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. 37.3 30.9 28.2 26.6 31.6 39.2 26.3 24.2 22.3 21.0
diff. 4.0 2.0 -2.6 1.1 -4.3 -1.4 2.7 -0.3 -0.8 -0.8

Table 5. 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 6 shows the measured and estimated IBU values from my 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 14.9 22.0 27.9 37.8 50.7
difference -1.2 0.8 1.8 2.4 4.3

Table 6. 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 7 shows the measured and estimated IBU values from my experiment #4, 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
33.8 34.0 32.3
difference
-0.2 -3.0 -3.7

Table 7. 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 8 shows the measured and estimated IBU values from my experiment #5, 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 24.3 29.7 34.8 40.7 46.1
difference 0.4 1.3 0.7 0.8 -0.3 -0.9

Table 8. Measured IBU values and estimated IBU values from personal experiment #5, as a function of hop weight.  The difference (error) is also shown.

Table 9 shows the measured and estimated IBU values from my experiment #6, which looked at variety of conditions: Condition A had hop weight of 0.37 oz and boil time of 26.9 min; Condition B had hop weight of 1.11 oz and boil time of 26.9 min; Condition C had hop weight of 1.11 oz and boil time of 12 min; Condition D had hop weight of 2.22 oz and boil time of 19.0 min; and Condition E had hop weight of 2.22 oz, with no boiling but a 19-minute hop stand held at 145°F.  Conditions A through D were immediately cooled upon flameout.

Condition
A
B
C
D
E
measured
18 48 32 58 27
estimated 20.8 46.9 32.8 58.5 26.0
difference 2.8 -1.1 0.8 0.5 -0.9

Table 9. Measured IBU values and estimated IBU values from personal experiment #6, as a function of hop boil time.  The difference (error) is also shown.

5. Discussion of Results
The average difference between observed (or Tinseth model) IBU and IAA values and current model estimates is -0.01, with a standard deviation of 2.2 and a maximum difference of 4.3.  The consistent difference of about -3 ppm for the IAA values but an overestimate of the IBU values in the Peacock study is one example of the sub-optimal result of 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 4.3 IBU) would be detectable by a human. 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 19.3 IBUs with a concentration of 11.4 ppm of IAA (52% of the IBU total), 0.5 ppm of oAA, 11.7 ppm of oBA, and 19.1 ppm of hop polyphenols.  The same 2 oz added at 60 minutes will create 61.4 IBUs with a concentration of 70.4 ppm of IAA (82% of the IBU total) and the same concentrations of nonIAA components.  If we triple the amount of hops, from 2 oz to 6 oz, the IBUs only increase from 61.4 to 86.5 (75.6 ppm of IAA, representing 62% of the total; 1.5 ppm of oAA, 35.2 ppm of oBA, and 57.3 ppm of hop polyphenols).  If we add those 6 oz at flameout, we’ll get 41.3 IBUs, with only 12.2 ppm of IAA (21% 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 35.3 IBUs, with 35.8 ppm of IAA representing 72% of the IBU total.  Adding these degraded hops at flameout will produce 13.8 IBUs, but with only 5.8 ppm of IAA representing 30% of the IBU total.

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 15.83 IBUs in a (post-boil volume) alpha-acid concentration of 175.83 ppm using the Tinseth source of data.  (The Tinseth formula predicts 14.80 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 15.83 ppm / 175.83 ppm = 0.0900 (or 9.0% utilization).  The detailed model tells us, however, that at 10 minutes the relative contribution of IAA to the IBU is only 0.511.  Therefore, of the utilization of 0.0900, 0.0460 is alpha-acid utilization (using the standard definition of utilization), and 0.0440 is the effective utilization coming from nonIAA components.  (By “effective”, I mean that the nonIAA components, expressed in ppm, are converted by their scaling factor to be relative to IAA concentrations; e.g. a hop polyphenol contribution of  14 ppm in the finished beer is multiplied by its scaling factor of 0.0308 to yield an effective utilization from hop polyphenols of 0.4312 ppm / 175.83 ppm = 0.0025 or 0.25%.  All of the nonIAA components sum up to yield a total effective utilization from nonIAA components.)  Because the detailed model assumes that nonIAA components contribute to the IBU value in a very short amount of time (unlike the lengthy isomerization process), the effective utilization of 0.0440 for nonIAA components is constant for all boil times.   When the boil time is 60 minutes, the effective utilization from nonIAA components is still 0.0440, but the alpha-acid utilization is 0.1877 (for a total utilization of 0.2317), and so the isomerized alpha acids represent 81% of the IBU value at 60 minutes.  In general, one can think of the utilization part of the Tinseth formula as being a constant 0.0440 from nonIAA components, and the remainder (when the formula yields a value greater than 0.0440) from isomerized alpha acids.  The Tinseth formula predicts utilization of 0.0440 at around the 5-minute mark.  All of this corresponds very 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.

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 a single hop addition, with 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 useless 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 (alpha acid concentration at boiling, decrease in IBUs over time for home-brewed beer, rate of alpha acid oxidation based on Maye et al.’s paper [Maye], and details of my experiments) 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 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 most of that IBU value may come from nonIAA components.  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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  • R. Mussche, “Quantitative Determination of Bitter Substances in Hops by Thin Layer Chromatography”, in Journal of the Institute of Brewing, vol. 81, January-February 1975.
  • T. P. Neilsen, “Character-Impact Hop Aroma Compounds in Ale,” in Hop Flavor and Aroma: Proceedings of the 1st International Brewers Symposium, ed. Thomas H. Shellhammer, Master Brewers Association of the Americas, 2009.
  • G. J. Noonan, New Brewing Lager Beer. Brewers Publications, 1996.
  • G. Oliver, The Oxford Companion to Beer, Oxford University Press, 2011.
  • J. J. Palmer, How to Brew: Everything You Need to Know to Brew Beer Right the First Time. 3rd edition, Brewers Publications, 2006.
  • J. Palmer and C. Kaminski, Water: A Comprehensive Guide for Brewers. Brewers Publications, 2013.
  • E. J. Parkin, The Influence of Polyphenols and Humulinones on Bitterness in Dry-Hopped Beer, Master of Science thesis (advisor: T. H. Shellhammer), Oregon State University, 2014.
  • N. Pyle, “Norm Pyle’s Hops FAQ”. http://realbeer.com/hops/FAQ.html
  • V. Peacock, “The International Bitterness Unit, its Creation and What it Measures,” in Hop Flavor and Aroma: Proceedings of the 1st International Brewers Symposium, ed. Thomas H. Shellhammer, Master Brewers Association of the Americas, 2009.
  • F. G. Priest and G. G. Stewart (eds), Handbook of Brewing. 2nd edition, CRC Press Taylor & Francis Group, 2006.
  • T. H. Shellhammer, “Hop Components and Their Impact on the Bitterness Quality of Beer,” in Hop Flavor and Aroma: Proceedings of the 1st International Brewers Symposium, ed. Thomas H. Shellhammer, Master Brewers Association of the Americas, 2009.
  • B. Smith, “Scaling Beer Recipes for Commercial Use with BeerSmith”, in BeerSmith Home Brewing Blog, June 11, 2014.  http://beersmith.com/blog/2014/06/11/scaling-beer-recipes-for-commercial-use-with-beersmith/
  • J. Spencer, “Small Scale Brewing”, in BYO, Jul/Aug 2007.  https://byo.com/mead/item/1378-small-scale-brewing
  • L. O. Spetsig and M. Steninger, “Hulupones, A New Group of Hop Bitter Substances”, in Journal of the Institute of Brewing, vol. 66, 1960.
  • G. Tinseth, “Glenn’s Hop Utilization Numbers”.  http://realbeer.com/hops/research.html
  • Tinseth emails: personal e-mail communications with Glenn Tinseth on March 16, 2016 and July 5, 2016.  (Many thanks to Prof. Tinseth for his fast and helpful responses to my out-of-the-blue queries.)
  • Wikipedia, “Solubility”. accessed Jan.21, 2017.  http://en.wikipedia.org/wiki/Solubility#Temperature.

An Analysis of Sub-Boiling Hop Utilization

Abstract
In a previous post, “A Modified IBU Calculation (Especially for Late Hopping and Whirlpool Hops)“, one of the components of the modified Tinseth IBU formula is an estimation of relative α-acid utilization at below-boiling temperatures.  The current experiment investigates this relative utilization as a function of temperature.  One result of this experiment is that the measured IBU at 145°F (63°C) is about half the measured IBU value at boiling.  However, IBU values are not the same as iso-α-acid concentrations (especially at low temperatures and short steep times), due to the presence of oxidized alpha acids, oxidized beta acids, and polyphenols.  Therefore, IBU values cannot be used to directly estimate relative α-acid utilization.  Instead, the data from this experiment are applied to a detailed model of IBUs developed in another post to estimate iso-α-acid concentration and, from that, relative α-acid utilization is estimated.  It is shown that this estimate of relative α-acid utilization is consistent with a formula proposed by Huang, Tippmann, and Becker (2013), although because of some dependencies, this estimate is not an independent verification of the formula.  While Huang’s formula is also time-dependent, a reasonable time-independent representation of relative utilization as a function of temperature can be expressed as Urel(T) = 2.39×1011 × e-9773/T (where T is temperature in degrees Kelvin).  Note that this utilization is relative to the amount of utilization at boiling.

Introduction
Hop utilization is defined as isomerized α-acids (iso-α-acids, or IAA) in finished beer divided by total α acids added.  It would be nice to have a model of this utilization as a function of (sub-boiling) temperature, in order to better predict the increase in IBUs that happens after flameout.

I’ve seen reports that utilization decreases as a function of temperature, from maximum utilization at boiling, down to zero utilization at around 180°F (82°C).  (I’ve seen two numbers: 175°F (79°C) according to BYO and a discussion at theelectricbrewery, and 185°F (85 °C) according to a homebrewersassociation discussion and a probrewer discussion).  However, just knowing a maximum (full utilization) and a minimum (zero utilization) doesn’t mean that a straight line is the best fit to the actual utilization.  In addition, I haven’t seen any justification for this lower limit; just because I read it on the Internet doesn’t necessarily mean it’s true.

Next, let’s look at isomerized α-acids, which are the biggest contributor to IBU values and the numerator of the utilization definition.  Malowicki, Huang et al., Kappler et al., and others (e.g. Jaskula) have done a lot of work looking at α-acid isomerization at temperatures other than boiling.  Malowicki provides formulas for the temperature dependence of the loss of α acids (converted into iso-α-acids) and the loss of iso-α-acids (converted into other “uncharacterized degradation products” due to the continued presence of heat).  For the loss of α acids, this first-order reaction has a rate constant k1 = 7.9×1011 e-11858/T (T in degrees Kelvin), e.g. [iso-α-acids] = [iso-α-acidsinitial]ek1, where angle brackets ([]) indicate concentration.  For the loss of iso-α-acids, this first-order reaction has a rate constant k2 = 4.1×1012 e-12994/T (T in degrees Kelvin).

One can take Malowicki’s function for the loss of α acids as a function of temperature and assume a corresponding decrease in utilization.  For example, k1 = 0.01249 at 212°F (100°C) and k1 = 0.00622 at 198°F (92°C), and so the rate of the reaction is halved (reaction time is doubled) at 198°F (92°C).  If one assumes that the concentration of α acids is directly (and inversely) tied to alpha-acid utilization, one can conclude that utilization is also 50% at 198°F (92°C), relative to utilization at 212°F (100°C).

We can improve upon this assumption by including the loss of iso-α-acids during the boil, referring to work by Huang et al..  Huang provides an equation for the concentration of iso-α-acids as a function of time (t) and temperature by combining the two rate constants from Malowicki into a single formula: [iso-α-acids] = [α-acidsinitial](k1/(k2k1))(e-k1t-e-k2t). (The temperature dependence is implicit in the values of k1 and k2.)  We can then plot the concentration of iso-α-acids (relative to the initial concentration of α-acids, not taking into account volume changes produced during the boil) as a function of time for various temperatures (see Figure 1, below).  It can be seen that at 30 minutes, the relative iso-α-acid concentration is 0.2976 at 212°F (100°C) and 0.1696 at 198°F (92°C).  The value 0.1696 is 14% larger than would have been predicted by our first assumption (half the value at boiling, or 0.1488).  Also, according to this formula, there is still noticeable utilization happening at 175°F (79°C), with 5% to 10% utilization between 30 and 60 minutes.

isoAlphaAcidConcentraion

Figure 1: iso-α-acid concentration, relative to initial α-acid concentration, as a function of time and temperature, according to a formula by Huang et al.

We can use this formula to plot relative utilization as a function of temperature for different steep times (Figure 2).  In this case, regardless of the steep time, the relative utilization at boiling is defined to be 1.0, and utilization at other temperatures is relative to 212°F (100°C).

HuangUtilAsFunctionOfTemp

Figure 2: Relative utilization as a function of temperature (boiling = 1.0) and various steep times, according to equation by Huang et al. (2013).

These values of relative utilization are dependent on both time and temperature, although the temperature component has a much larger impact than the time component.  We can approximate this as a function of only temperature, by choosing a single steep time to represent the general case, e.g. 40 minutes.  We can then fit the relative utilization data to an equation.  In this case, a root-mean-squared fitting error of 0.013 can be obtained with the Arrhenius function Urel(T) = 2.39×1011 e-9773/T (where T is temperature in degrees Kelvin).  In this case, at 373.15 Kelvin (or 212°F or 100°C), Urel(T) is close to 1.00; at 194°F (90°C), the utilization is half that of boiling.

The experiment that follows measured IBU values as a function of (sub-boiling) temperature, with hops steeped for 10 minutes, to compare measured IBU values with utilization prediction by this equation.  IBU values are not, however, a substitute for isomerized α acid levels (except for the boil times, hop concentrations, and hop storage conditions of the 1960’s), and so the measured IBU values need to then be converted into estimated isomerized α-acid levels.  This conversion is done using a detailed model of IBUs developed in a separate blog post.  This model uses, in part, the formula from Huang et al. to estimate utilization at sub-boiling temperatures.  Therefore, the IBU values from the model are dependent upon the assumption that this formula is correct.  Because of the dependence of the model on the formula, the results of this experiment don’t provide independent verification of the formula.  However, the results do show that the model can be used to find good estimates of measured IBU values, and therefore this formula can provide a reasonable estimate of temperature-dependent utilization.

Methods
Conditions
Each condition in this experiment consisted of a small batch (1.3 G (4.92 liters) pre-boil volume) of beer brewed with a single 10-minute addition of hops, as described below.  The hops were added (and maintained) at a different target temperature for each condition within a set.

Because of constraints on my time and energy, I divided this experiment into two sets (brewed in September and January).  Within a set, each condition sampled from the same batch of wort and hops.  Since the wort and hops varied between sets, one condition in each set was the reference point, with a target temperature of 212°F (100°C) and a relative utilization (compared with other temperatures) of 1.0.  Other target temperatures ranged from 145°F (63°C) to 200°F (93°C), as listed in the Table 1 (below).

Finished beer from each condition was sent to Analysis Laboratory for analysis of IBUs and original gravity.  (Scott Bruslind from Analysis Laboratory has been very responsive and encouraging with these experiments, providing a full set of measurements (including gravity, pH, and attenuation, in addition to IBUs.))  The IBU level of each condition was divided by the IBU level of the reference condition (target temperature of 212°F (100°C)) in order to obtain a relative IBU level.  Since all other conditions were held as constant as possible (including boil volume, specific gravity, pH, hop steeping time, α acids, oxidized β acids, polyphenols, and fermentation conditions), any difference in IBU levels is due to decreased utilization at the target temperature, an error in measurement (as explained below), or some combination of both. By fitting a smooth function to the data, we’d like to be able to average out errors and estimate utilization in finished beer as a function of temperature.  The problem is that the IBU is not just a measurement of isomerized α acids; it includes other bitter substances that don’t increase at the same rate as isomerized α acids during the boil.  We’ll come back to this problem later in this post.

Sources of Error
This experiment relies on just nine IBU values, with only one value at each sub-boiling temperature, due to limited time and effort.  If one had the luxury (and energy) to repeat this experiment 10 times, one would get a variety of different relative IBU values at a given target temperature, hopefully all clustered together fairly closely.  These differences can be considered errors with respect to the “true” relative IBU value at each temperature.  What causes these errors?   First (and maybe less significantly), there may be errors in the sample analysis.  Second (and maybe more significantly), the small batch size (1.3 G (4.92 liters) pre-boil) makes it very difficult to maintain a consistent target temperature, evaporation rate, and concentration of alpha acids and other bitter substances.  Measured IBU values that do not conform to a simple pattern are very likely off due to such errors, and these errors are unavoidable with my current methodology.  The methods used here are probably sufficient, however, to find a “reasonable” fit to the data by minimizing the error.

Recipe
There were two sets, Set 1 (Conditions A, B, C, and D) and Set 2 (Conditions F, G, H, I, and J).  (No, I’m not very good at fanciful names for these things.  Yes, there was a Condition E, but it was not entirely relevant to this analysis and is omitted here.  The data for Condition E is included in a separate blog post.) Each condition maintained (close to) a target temperature for steeping, listed below in Table 1.

The wort for each set was prepared with 9¼ lbs (4.2 kg) Briess DME dissolved in 7 G (26.5 liters) of water, yielding about 7⅔ G (29 liters) of pre-boil wort.  This wort was heated, boiled for 30 minutes uncovered, and then cooled with a wort chiller.  The cooled wort was stored with a lid on, in order to minimize chances of infection.  For Set 1, the measured specific gravity prior to boiling each condition was 1.060; for Set 2, the specific gravity was 1.061.  For each condition, ~1.3 G (4.9 liters) was taken from the larger pool of wort, heated to boiling, and then cooled to the target temperature.  Once the target temperature was reached, 1.60 oz (45.36 g) of Cascade hops were added, within a large mesh bag.  (The hops were collected in advance from a larger mixture of 8 oz to 9 oz (227 g to 255 g) per set.)  The kettle was covered, and the target temperature was maintained as closely as possible for 10 minutes.  (Temperature readings were taken at one-minute intervals with a long thermometer probe stuck through a very small hole in the lid.)  After 10 minutes, the hops were removed and the wort was cooled as quickly as possible.  This wort was left to settle for 5 minutes, after which 3½ quarts (3.31 liters) were decanted into a 1-gallon (~4 liter) container.  This container was sealed until all conditions within the set were ready.  Once ready, 1½ packets of Safeale US-05 yeast were added to ~0.9 cups of water.  Each condition was aerated for 90 seconds by vigorous shaking, and 1½ oz (42.5 g) from the pool of yeast slurry was added.  Airlocks were applied.  Time passed and beer bubbled.  After 3 weeks, each condition was bottled with a small amount of simple syrup to target about 2.1 volumes CO2.  After 3 more weeks, samples were taken from the bottles (leaving behind the yeast sediment), degassed, and sent for analysis at Analysis Laboratory.   The original gravity and IBU values in Table 1 come from this analysis; the original gravity is converted from degrees Plato.

target temp. average temp. original gravity post-boil volume measured IBUs relative IBUs
Condition A
212°F
100°C
212°F
100°C
1.0658 1.18 G
4.47 l
33.3 1.0
Condition B
200°F
93.3°C
198.8°F
92.7°C
1.0645 1.20 G
4.54 l
28.9 0.868
Condition C
190°F
87.8°C
191.1°F
88.4°C
1.0645 1.23 G
4.66 l
30.8 0.925
Condition D
185°F
85.0°C
185.4°F
85.2°C
1.0641 1.24 G
4.69 l
25.5 0.766
Condition F
212°F
100°C
212°F
100°C
1.0645 1.23 G
4.66 l
40.6 1.0
Condition G
175°F
79.4°C
176.4°F
80.2°C
1.0628 1.26 G
4.77 l
23.6 0.581
Condition H
165°F
73.9°C
166.3°F
74.6°C
1.0628 1.26 G
4.77 l
24.5 0.603
Condition I
155°F
68.3°C
155.6°F
68.7°C
1.0624 1.27 G
4.81 l
23.1 0.569
Condition J
145°F
62.8°C
145.6°F
63.1°C
1.0624 1.27 G
4.81 l
21.8 0.537

Table 1. Target temperature, measured average temperature, original gravity, measured IBU values, and (measured) relative IBU values for each of the nine conditions.

Raw Results
Table 1 shows the target temperature, measured values, and relative IBU values for each condition in the experiment.  (The post-boil volume was computed from the ratio of pre-boil gravity points to post-boil gravity points, multiplied by the initial volume of 1.3 G (4.9 liters)).  The measured IBU values were converted to relative IBU values by dividing the measured IBU of that condition by the IBU value at boiling in that set (Condition A or F).  A plot of these relative IBU values as a function of average steep temperature is shown below in Figure 3.

mibu_exp2_relativeibu

Figure 3. Relative IBU values as a function of temperature (in °C).

Other than the results at 191°F (88.4°C) and 176°F (80.2°C), the data fit quite well to an exponential function.  I assume that the relatively large differences for these two extreme values are due to a relatively higher or lower concentration of α acids (and other components) in the wort, compared with the reference condition, as explained above in the section Sources of Error.  Fitting an exponential function to the eight available data points of relative utilization, we get U(T) = 0.11245 e0.01031T (where T is temperature in °F) or U(T) = 0.15642 e0.01856T (where T is temperature in °C).  The root-mean-squared error of either function is 0.059.

Data Analysis
When comparing the theoretical relative utilization (with about 50% utilization at 194°F (90°C), expressed by the formula above for Urel(T) and plotted in Figure 2) with the relative IBU values from Table 1, it quickly becomes clear that the relative IBU values are quite a bit larger (with a 50% value at around 140°F (60°C)) than the theoretical values.  This puzzled me for quite a while, but it can be explained by the components of the IBU measurement that are not isomerized α acids.  (See “The International Bitterness Unit, its Creation and What it Measures” by Val Peacock, in Hop Flavor and Aroma: Proceedings of the 1st International Brewers Symposium; BYO has an online article by John Palmer that discusses several of the same points as the Peacock article.)

The IBU measures contributions from both isomerized α acids (IAA) and other “interfering substances” (non-IAA components, including oxidized α and β acids and polyphenols, all of which contribute to bitterness).  Normally, the contribution of non-IAA components is much lower than the contribution of IAA.  (In the 1960’s, about 70% of the IBU value was from IAA and 30% was from non-IAA components.  With improvements in the storage conditions of hops over the past decades, the IAA proportion with a 60-minute or greater boil time is now generally higher.)  In this experiment, however, the short boil time (10 minutes), high boil gravity (about 1.064), and relatively large hop additions (1.6 oz in 1.3 G, or 45 g in 4.9 liters) caused the non-IAA contribution to the IBU to be much greater than the IAA contribution, even for the condition at boiling.  As the temperature decreased with each experimental condition, the contribution of IAA to the IBU also decreased, but the non-IAA contribution remained more constant.  Therefore, the IBU values from this experiment cannot be used to directly estimate relative α-acid utilization.

Estimating Alpha-Acid Utilization with a Model of IBUs
In another blog post, I present a model of IBUs that accounts for both α-acid isomerization and the effects of oxidized α acids, oxidized β acids, and polyphenols on IBU values.  This model uses the equation from Huang et al. to estimate the temperature-dependent isomerization of α acids.  It also takes into account the age of the hops, the fact that oxidized α and β  acids are produced during the boil (Algazzali, p. 17), and various losses that impact IBUs.  The model can estimate the IBU values from this experiment with a maximum difference of 4.3 IBUs.  The IBU values in Table 1 vary by as much as 3.5 IBUs from the expected smooth line, and so the error from the model is more or less in line with the observed measurement error.  This model can also be used to also estimate the concentrations of isomerized α acids and nonIAA components in the finished beer.  This gives us three ways to use the measured IBU values (and other data from the experiment) to estimate relative α-acid utilization, all of which produce similar results: (1) determine utilization directly, by dividing the estimated iso-α-acids in the finished beer by total α acids added; (2) multiply the estimated IBU value by the estimated percent of the IBU that comes from isomerized α acids; or (3) multiply the measured IBU value by the estimated percent of the IBU that comes from isomerized α acids.  In all three cases, the result at each temperature is divided by the result at boiling to determine a relative utilization.

In the search for model parameter values, I allowed allowed some flexibility in the AA rating of the hops, the ratio of α to β acids, and the degradation factor due to the age and storage conditions of the hops.  The reason for this flexibility was that I couldn’t determine reliable values for these parameters.  The AA rating on the packages of hops was 8.4% for Set 1 and 7.9% for Set 2.  I set aside some of each set of hops for testing at KAR Laboratories, which came back with 5.75% AA for Set 1 and 6.25% AA for Set 2.  At first I thought that the decrease was caused by degradation of the hops over time, and that the hops in Set 1 that I bought in September were not fresh but just over a year old (and poorly stored, as well).  This would make interpretation of Set 2 values difficult, though: if Set 1 had 31% degradation over 12 to 13 months, Set 2 (purchased in late December) would have 21% degradation over either 3 months or 15 months, which would either be too much (over 3 months) or too little (over 15 months) relative to Set 1.  After sending other samples in for analysis over a longer time period, it seems that there is a wide variation in laboratory-measured AA values; I’ve even seen older hops with a higher AA rating than fresh hops from the same bine (grown in my back yard).  It seems that either the analysis of α-acid percent by weight is not reliable, or that this value is accurate but can vary greatly even between different 30-gram samples taken from the same bine.  (Hough, Briggs, Stevens, and Young say that “sampling of hops is extremely difficult due to their heterogeneous nature” (p. 432).)  Because I can’t determine the AA rating reliably, the ratio of α to β acids is also uncertain.  Finally, without analysis of the Hop Storage Index (HSI), the value for the hop degradation fact0r is also unknown.  As a result, I allowed the search for model parameters to vary the AA rating within one percentage point of the AA rating on the package, the α/β ratio to vary between 1.1 and 1.5 (a range of expected values for Cascade hops), and the degradation factor to vary between 0.50 and 1.0.  Results of fitting the model to the data yielded an AA rating of 7.4%, an α/β ratio of 1.5, and a degradation factor of 0.92 for Set 1, and an AA rating of 8.9%, an α/β ratio of 1.4, and a degradation factor of 0.94 for Set 2.

Table 2 provides, for each condition, (a) the concentration of pre-boil α acids in the volume of wort  at the end of the boil (α-acid concentration, in parts per million (ppm)); (b) IBU values estimated from the model; (c) estimated iso-α-acid concentration (in ppm) in the finished beer; (d) estimated ratio of iso-α-acids contributing to the IBU value (range 0 to 1); and (e) relative utilization determined by multiplying the measured IBU value by the estimated ratio of iso-α-acids contributing to the IBU value.

alpha acid concentration(ppm)
model IBUs
estimated IAA (ppm)
ratio of IAA contributing to IBU
relative utilization
Condition A
691.3 37.3 13.34 0.256 1.0
Condition B
679.8 30.9 7.03 0.163 0.552
Condition C
663.2 28.2 5.01 0.127 0.459
Condition D
657.9 26.6 3.87 0.104 0.311
Condition F
815.0 39.2 13.44 0.245 1.0
Condition G
795.6 26.3 2.51 0.068 0.161
Condition H
795.6 24.2 1.48 0.044 0.108
Condition I
789.3 22.3 0.88 0.028 0.065
Condition J
789.3 21.0 0.58 0.020 0.044

Table 2. Values related to relative utilization that have been determined by fitting the IBU model to available data.

Figure 4 shows the relative utilization determined by two of the three methods discussed above; it can be seen that they all yield similar results, and that these results are close to the values predicted by Huang’s equation.

mibu_exp2_relativeiaa_touchup

Figure 4.  Relative utilization as a function of temperature, estimated by two methods described in the text (method 1 in green; method 3 in blue), and relative utilization predicted from the Huang formula (red).

Discussion: What I’d Do Differently Next Time
I used such a large amount of hops in order to get higher IBU values and thereby (marginally) increase the accuracy of the relative values.  However, I’ve since found that “a high hopping rate reduces extraction efficiency” (Lewis and Young, p. 267), and I now think that the concentration of α acids I used (660 ppm to 815 ppm) was much greater than the α-acid solubility limit at high temperatures (~265 ppm), greatly reducing the amount of isomerized α acid produced but increasing the concentration of nonIAA components.  This experiment used a steep time of 10 minutes, which at boiling would yield a utilization factor of only 0.074 according to the Tinseth model (which doesn’t take into account a high hopping rate).  The greatly reduced degree of α-acid utilization in this experiment, compared with typical beers, resulted in a much lower ratio of IAA to non-IAA components in the resulting IBU values.  If I were to re-do this experiment, I would increase the boil time instead of the hop concentration in order to increase utilization, and target an α-acid concentration of about 200 ppm.  Even better, I would use α-acid extract instead of hops, if I could get it, in order to avoid the non-IAA components entirely… failing that, I’d use the highest α-acid hop I could get.

Conclusion
One obvious result from this experiment is that IBU values are not a direct replacement for isomerized α-acid values, especially at short steep times, high hopping rates, and sub-boiling temperatures.  This is because IBU values reflect not only isomerized α-acid values, but also contributions from oxidized α and β acids and polyphenols.  The function of relative utilization estimated in this blog post is for α-acid utilization, and does not include the contributions of these other components to the IBU.

The results of this experiment don’t provide an independent verification of relative utilization based on Huang’s equation.  However, the results do show that this equation can be used as part of a larger model to provide good estimates of measured IBU values, and that the iso-α-acid levels and relative utilization estimated from measured IBU values conform well to expectations.  By converting Huang’s equation from absolute to relative values and removing the time dependency (using a single representative time point), relative utilization can be modeled with the function Urel(T) = 2.39×1011 e-9773/T (where T is temperature in degrees Kelvin).

Hops Harvest

Summary

year variety harvest weight dry weight harvest date analysis date moisture HSI alpha acids beta acids cohum. colup.
2014 Cascade 63 oz
(1.8 kg)
14.5 oz
(411 g)
Sep. 1, 2014 Dec. 11, 2014 8.8% N/A 7.37% 6.96% 33.2% 51.7%
2014 Cascade Sep. 1, 2014 Oct. 16, 2015 9.1% N/A 6.13% 5.08% 31.9% 50.1%
2015 Cascade 56.1 oz
(1.6 kg)
13.9 oz
(394 g)
Sep. 13, 2015 Oct. 16, 2015 6.20% N/A 6.64% 5.38% 33.7% 52.7%
2016 Cascade 61.75 oz
(1.8 kg)
14.2 oz
(403 g)
Sep. 4, 2016 Sep. 2016 Lab1: 9.03% Lab2: 0.217 Lab1: 8.33%;

Lab2: 9.0%

Lab1: 7.20%

Lab2: 7.5%

Lab1: 32.6% Lab1: 52.0%

Table 1. Summary of hops weight and analysis results.  HSI is the Hop Storage Index.  Moisture, alpha acids, and beta acids are in percent by weight.  Cohumulone is in percent of total α-acids.  Colupulone is in percent of total ß-acids.

2014
This was my third year of growing one Cascade plant, and my first year of growing four Willamettes and four additional Cascades on trellises.  I harvested the older Cascade during the first weekend in September, when the cones were papery and some were starting to turn brown on the edges.  (Some sites suggest harvesting periodically, but I’m not convinced that it’s worth the extra hassle.)  That plant yielded 63 oz (3.94 lbs 1.8 kg) of fresh hops.  I dried them in the basement, which is cool but convenient for me, using several fans to circulate the air.  I put them in several mesh bags or on a mesh sweater hanger.  It took them 9 days to fully dry out; I decided they were done when one day’s weight was the same as the previous day’s weight.  The dried hops weighed 14.5 oz (411 g).  According to Greg Noonan in New Brewing Lager Beer (p. 79), “[hops] contain 70 to 80 percent moisture at harvesting, [and] are dried to 8 to 10 percent moisture”.  In early December, I sent 1¼ oz (35 g) of the older Cascade hops and $35 off to KAR laboratories to get them tested.  The results came back showing 8.76% moisture and an alpha acid level of 7.37%.  With 8.76% moisture remaining, if I do the math correctly that would mean that the hops were 79% moisture at harvesting, which is in line with Noonan’s values, although a bit high.

Here’s a graph of the decrease in weight, per day:

Weight of hops over time

They probably would have dried much faster if they were in a warmer environment (the basement is a fairly steady 70°F/21°C in the summer) or if I had been able to spread them out better.  I’ve heard that a door screen works very well for spreading them out.  According to Noonan in New Brewing Lager Beer (p. 79), commercial hops are dried at ~140°F (60°C) for 8 to 12 hours, and then “cured in cooling bins” for 5 to 10 days, so it seems that a 9-day drying time at room temperature is a reasonable approach that yields similar results (at least in terms of moisture content).

The second weekend in September, I harvested the first-year plants.  The four Willamette plants yielded a total of 4.55 oz (129 g) of fresh hops this first year (about 1 oz or 32 g per plant).  They were dry within 6 days using the same method as with the Cascade, presumably due to the cones being less densely packed.  The dried weight was 1.10 oz (31 g), indicating that the fresh weight was 78% moisture (assuming they were also ~9% moisture when dried).

The four new Cascade plants provide a cautionary tale.  I harvested 11.05 oz (313 g) of fresh hops from the four plants, but there were a number of ants crawling around the bines and cones. I let the plastic bucket of harvested hops sit out overnight, hoping that the ants would go away.  In the morning, the rim of the bucket and all of the hops were covered with many hundreds, possibly thousands, of tiny, pale-green aphids. I had no idea that they could detect and converge on hops that quickly.  The lesson here: spread diatomaceous earth at the base of the plants so that ants won’t be crawling on the hops.  If I hadn’t had the ants in the first place, and then left the bucket out to get rid of the ants, the aphids wouldn’t have had a chance to find my hoppy treats.  (The Cascade plant that is now in its third year had, in its first year, a number of aphids on the underside of the leaves, and it looked like ants were farming the aphids.  I spread diatomaceous earth at the base of the plant and released many, many ladybugs, and by harvest time the ants and aphids were gone.  The second and third years, diatomaceous earth applied at the beginning of the season prevented ants and aphids alike.)

The dried hops were put in vacuum-sealed bags and stored in a chest freezer.

In October 2015, I sent off another sample of the 2014 batch of Cascade to KAR Laboratories, and it came back with 6.13% alpha acids.  The hops were kept in what I believe are close to optimal conditions.  So, to the extent that the measured AA values are correct, even under great storage conditions, the AA level after one year was only 83% of the initial value.

2015
The 2014-2015 winter and the summer of 2015 were warm and dry here in Portland, Oregon.  I think this is why all plants had a lower yield this year.  The Willamettes, in particular, yielded only 64 cones, not even enough worth drying.  In a Willamette Week article that interviewed hop farmer Patrick Leavy, Leavy says that “when you have a warm winter, certain varieties like Willamette don’t get enough ‘chilling hours,’ which regulates their growth hormone.” Oh, well… hopefully they put down lots of roots for next year.

I used a food dehydrator to dry the hops for 12 hours at 95°F (35°C) this year; that worked really well for me.  (Although I’m tempted to try Kai Troester’s hop oast in the future.)  I harvested 46.9 oz (1.3 kg) of fresh hops from the older Cascade plant and 9.18 oz (0.26 kg) from the younger Cascade plants.  This yielded a total of 13.9 oz (394 g) of dry hops.  In October, I sent 1¼ oz (35 g) of hops (from the older plant) and $35 off to KAR laboratories to get them tested.  They came back with 6.20% moisture remaining and 6.64% alpha acids.  This translates into 72% moisture at harvesting, again in line with Noonan’s estimate of between 70% and 80%.

The dried hops were put in vacuum-sealed bags and stored in a chest freezer.

2016
The 2015-2016 winter was fairly mild again here in Portland. The one older Cascade plant yielded 49.75 ounces (1.4 kg) of fresh hops, pretty much in line with previous years.  The hops on the trellises were kind of sad, though.  The four Cascade plants from the trellises yielded only 12.0 ounces (340 g) of fresh hops, even though they’re in their third year.  The four Willamette plans yielded only 10 cones.  It may be some sort of disease, but I’m also starting to think that because I planted them very close to the basement, and inside the basement it never gets below 60°F (16°C), the ground close to the house might be too warm for hops to put down roots in the winter.  For the record, the older Cascade gets morning shade and moderate afternoon sun, the trellis plants get excellent sun in the afternoon and early evening, and all plants get 20 minutes of water each day during the summer through a soaker hose.  I didn’t have time this year to build a hop oast, as planned, so I used the food dehydrator again, drying them for 12 hours at 95°F (35°C).  With 9.03% moisture, 12 hours seems to be a good amount of drying time at this temperature.

This year, I sent samples to two laboratories that estimate alpha acid levels.  I mixed the dried hops thoroughly before separating into two packages for each analysis, so that the AA numbers should be very close.  The results were close, but different enough to have a large impact on IBU calculations: 8.33% from Laboratory 1 and 9.0% from Laboratory 2.