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.
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  • 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).

Techniques for Maximizing Hop Flavor and Aroma

Introduction
This blog post provides a summary of the techniques I’ve found for maximizing hop flavor and aroma, based on my experimental results and general experience.  Others have said many of these things before, and in some cases with better writing skills.  I am not claiming that anything here is a new discovery; I’m just reporting what I’ve found from my personal brewing experience.  (As usual with this blog, the contents of this post may change over time as I revise and update what I’ve learned.)

1. Add Hops Late in the Boil.  Like Really, Really Late
Hops should be added late in the boil.  How late?  That depends on when you start cooling your wort and how quickly it cools.  According to Papazian (The Home Brewer’s Companion, p. 68), flavor is maximized at 10 minutes before flameout.  I’ve found increased hop flavor by adding hops at flameout and then letting the wort cool naturally (with the lid on) for 10 minutes.  There is a lot of room for experimentation here.

Also, according to Greg Noonan, “the bitterness derived from long boiling is coarser than that from a more moderate period” (Noonan, New Brewing Lager Beer, p. 154), which also suggests that more hops for a shorter time in the boil is advantageous.

It seems that boiling temperatures will decrease hop flavor if applied long enough (e.g. greater than 15 minutes), but it may also be that that (near) boiling temperatures are needed in order to bring out hop flavor.

2. Don’t Use Hop Stands for Long Periods of Time
Keeping hops in the wort at various sub-boiling temperatures for 60 minutes adds maybe some extra body, but little to no additional hop flavorA 45-minute hop stand at 170°F seems to add no hop flavor.  On the other hand, a 10-minute hop stand (with hops added at flameout) can produce very nice hop flavor.  The take-away message seems to be that long hop stands (45 minutes or more) don’t add hop flavor; shorter hop stands (around 10 minutes) do, although temperature may also be a factor.  Even at sub-boiling temperatures, steeping for too long removes (or fails to produce) that wonderful hop flavor. Are the sub-boiling temperatures of a hop stand beneficial to flavor, or do the flavor benefits come mostly from the amount of time of the steep?  I don’t know.

3. Cover the Kettle After Late-Hop Additions
The effect is very small, but covering the kettle after late-hop additions (i.e. no greater than 10 minutes before flameout) may provide some increase in hop flavor.  For hop-forward ales, there is minimal risk of high DMS levels caused covering the kettle for a few minutes.

4. Dry Hop for Aroma
I haven’t yet done any controlled experiments on hop aroma, but at this point I’ve found that small to moderate hop aroma requires a large amount of late-hop or flameout additions.  But a big aroma can be obtained through dry hopping with an ounce or two (about 25 to 60 grams).   (Noonan says it better: “the full, fresh aroma of hops is only captured by ‘dry-hopping’.”  (Noonan, New Brewing Lager Beer, p. 78.)

5. Dry Hop using a Weighted Mesh Bag
When I dry hop with whole cones, I sometimes put the hops in a very large mesh bag along with some weights.  The bag makes for easy removal of the hops, and the weights force all of the hops to be submerged.  The large mesh bag allows the hops to move around in the wort.  If you have a narrow-necked carboy, you should probably rack the beer to the keg or bottling bucket before removing the hops.  Removing the hops by squeezing them through a narrow neck, with the beer still in the carboy, may add too much grassy flavor to the beer.  More to come when I have time for a formal experiment.

6. Dry Hop for Shorter Periods of Time
As Palmer notes, dry hopping may yield “a dry aftertaste, like old tea” (Palmer, How To Brew, p. 44).  Strong comments that “it can produce a lingering grassy, vegetal note that some may not like” (Strong, Brewing Better Beer, p. 72). He recommends “limiting the contact time of dry hops in beer to 3 to 7 days” (ibid, p. 72).  More to come on this topic, but a week of dry hopping is not too long, in my experience.

7. Use a Lot of Hops Very Late in the Boil
When using too much hops there is the risk of tannins that can cause a grassy taste (Palmer, How to Brew, p. 44) and of course excessive bitterness.  But big, hoppy beers seem to require (strangely enough) lots of hops.  In a 5-gallon (19-liter) batch, six ounces (170 g) of hops very late in the boil (and/or at flameout) and an additional two ounces (57 g) for dry hopping is good, but maybe even on the low side for a really big beer.

Some beers that I’ve brewed with lots of hops do tend to have what I think is referred to as a “grassy” flavor, as noted by Palmer.  It’s not necessarily bad, but it’s not the character of the hops that I’m looking for.  At this point, I think that this flavor was caused by using improperly-stored hops.  In a six-gallon (23-liter) boil, I’ve added one ounce (28 g) at 15 minutes before flameout, 11 ounces (312 g) at flameout (with 10 minutes of post-flameout natural cooling), and an additional 2 ounces (57 g) during dry-hopping, and not noted any grassy flavor (but lots of wonderful citrusy flavor).  All of these additions were made with properly-stored hops, which may have been a critical factor.  (In fact, that beer turned out quite well and measured only 41 IBUs.)

By adding lots of hops late in the boil, IBUs are difficult to predict using standard models.  On the one hand, additions close to flameout will produce more IBUs than predicted by standard formulas because of continued heat after flameout; on the other hand, once you get above 2 ounces (57 g) of AA 10% hops in a 6-gallon (23-liter) boil, IBUs increase less than standard models will predict.  The mIBU model addresses flameout additions but not high levels of alpha-acid concentration.  More to come on this topic when I have more time.  In the meantime, I’ve learned to not fear large amounts of well-preserved flameout hops with a 10-minute rest period before forced cooling.

Other Techniques
I’ve heard about, but not yet had time to try, the following techniques:
Hop Tea: Add a homemade “hop extract” or “hop tea” to the secondary.  Noonan gives fairly detailed instructions (Noonan, New Brewing Lager Beer, p. 160) and recommends the practice.
Hop Bursting: Divide the flavor hops into several additions, with each addition made at a slightly different time.  For example, instead of adding 2 oz of hops at 10 minutes before flameout, try ½ oz each at 15 min, 10 min, 5 min, and 0 min.

Dry Hopping in a Weighted Mesh Bag

Dry hopping, or adding hops in the secondary fermenter, is the best way I’ve found to get good hop aroma.  When I first started dry hopping, I would push the hop cones through the narrow neck of my 5-gallon glass carboy.  To my dismay, the hops floated on top and most of them didn’t seem to ever touch the beer.  Stirring the beer helped a bit, but the contact between hops and beer was still limited.  And cleaning the hops out of the carboy was a mess.  A lot of people use hop pellets for dry-hopping, but since I grow my own Cascade, I’m interested in using the whole cones.

I now use a large mesh bag for dry hopping, with weights in the bag to pull all of the hops into the beer.  It seems like a good idea that I haven’t seen written about elsewhere, so I’m writing a blog post about the technique.

[Edit, Oct 2015:  I’m having second thoughts about removing the hops before racking, especially with a narrow-necked carboy… the aroma is great, but it might add too much grassy flavor to the beer.  To be safe, I’d rack to the keg/bottling container before removing the bag of hops.  More to come when I have time for a formal experiment.]

[Edit, March 2016: I just read on Norm Pyle’s Hops FAQ: “some brewers use a sanitized hop bag and marbles to sink the hops for maximum contact.”  So, apparently it’s true that “what has been done will be done again; there is nothing new under the sun.”]

1. Large Mesh Bag
I use a large nylon mesh straining bag to hold the hops.  I prefer a fairly large bag, so that the hops are free to roam about within the confines of the mesh enclosure.  Figure 1 shows the mesh bag I use, approximately 15″ × 8″ (38 cm × 20 cm).  I thoroughly wash it and soak it in Star-San before each use.

Large Mesh Bag

Figure 1. Large mesh straining bag for holding weights and hops.

2. Weights
Before adding the hops to the sanitized mesh bag, I add two weights.  The purpose of the weights is to drag the bag down into the beer and drown the hops in beer.  (I clean and sanitize the weights before adding them to the bag.)  The weights must satisfy several criteria: (1) they need to fit through the (small) opening of the carboy; (2) they need to be heavy enough to cause the bag and hops to sink; (3) they should ideally not contain iron or lead.  In my case, I took two brass compression caps measuring about ⅞” (2.22 cm) to 1″ (2.54 cm) across and filled them with lead-free solder.  Each filled cap weighs 1.6 oz (45 g), and the two weights together are just about sufficient to weigh down one ounce of hops.  For two ounces of hops, use two bags and four weights total.  These weights are just small enough to fit through my glass carboy when they are inside the mesh bag.  If you have a secondary fermenter with a large opening, the size of the weights probably doesn’t matter.

Brass weights (compression caps) with lead-free solder.

Figure 2. Brass weights (compression caps) filled with lead-free solder.  A ruler (in inches) shown for scale.

dryHop_brassWeights3

Figure 3. Brass weights (compression caps) filled with lead-free solder and a US quarter shown for scale.

3. Fishing Line
I use fishing line for two purposes: (1) to keep the weights in the bag separate from each other and from the hops, and (2) to provide a “lifeline” to remove the bag when the dry-hop time is up.

I found that allowing the hops and weights to mix together caused clumping, and made removal of the bag a bit difficult.  I now add one weight to a bottom corner of the mesh bag, tie it off with fishing line, add the second weight, tie that off with fishing line, and then add the hops.  This makes adding and removing the bag a little easier.  The result of adding weights and tying them off is shown in Figure 4, using a dry (not sanitized) bag and no hops.

009

Figure 4. The mesh bag with two weights tied into a corner.  40-lb green fishing line is used for visual effect, but 8-lb line works well, too.

Once the bag is filled with hops, I pull the drawstrings of the bag closed and knot them together.  I then tie some fishing line to the bag drawstring (near the knot) and let this line come along the edge of the rubber stopper and outside the carboy.  The rubber stopper seals around the fishing line nicely, allowing no air to escape except through the air lock.  When the time comes to remove the bag, I remove the stopper and pull on the fishing line to pull the bag out of the beer.

008

Figure 5. The mesh bag with fishing line tied to one of the drawstrings.

4. Adding the Hops to the Beer
The procedure of adding the hops is pretty much what you’d guess it is:  I remove the rubber stopper from the carboy, then add the mesh bag filled with weights and hops (starting with the weights).  I let one end of the fishing line tied to the drawstring remain outside the carboy.  I replace the rubber stopper.

xx

Figure 6. Mesh bag containing weights and hops (mostly) sinks into the beer.

dryHop_bagInCarboy2

Figure 7.  Closeup of mesh bag inside the carboy.  (This is a different batch of beer.)  You can see the fishing line extending from the opening of the carboy.

5. Stirring the Hops
I stir the hops in the beer at 12-hour intervals, by tilting the carboy on an edge and sloshing the beer around a small amount.  The idea is just to mix the hops with the beer a little more.  I have no idea if this is really useful or not.

6. Removing the Hops
I used to remove the hops before racking to the next container, but that might yield excessive grassy notes.  Instead, I’ll recommend that you rack the beer to the next container and then remove the hops.

When it’s time to remove the hops, I wash and sanitize my hands, remove the rubber stopper, and pull on the fishing line.  When the hops come out of the carboy, the small neck squeezes most of the beer out of the hops and back into the carboy.  Some beer will rise up and flow over the opening, but by going slowly this can be minimized and wiped up with a paper towel.  The weights come out last, and since they haven’t mixed with the wet hops, they are easily removed.  The fishing line can be easily cut off and discarded.

 

Late Hop Experiment #1 (a.k.a. Hop-Stand Experiment #3)

Abstract
In my quest for lots of hop flavor, I previously found that a hop stand did not provide the increase in flavor I expected.  The current experiment looks at several aspects of the brewing process that might provide an increase hop flavor: covering the pot during the last minutes of the boil, varying the time of late-hop additions, and hop stands with a somewhat different technique than I used previously.

I found that covering the pot to prevent oils escaping with the steam may provide some improvement, but this result was not definitive.  A late-hop addition at flameout (followed by 10 minutes of natural cooling with the lid on) contributed much more hop flavor than additions at 5 or 10 minutes.  Holding the wort (and hops) at 170°F (77°C) for an additional 45 minutes may have contributed something, but not an increase in hop flavor.  It seems that hop flavor is lost with extended contact time with boiling wort, and not increased with below-boiling temperatures.

I’ve also created a summary blog post that describes the techniques I’ve found to be useful at maximizing hop flavor and aroma.

Background
Flameout Hops Additions and Hop Stands

When I first started brewing, I would immediately cool the wort when the 60-minute boil time was up.  That was fine, until I started reading about hops additions at zero minutes/flameout.  Why add a whole bunch of hops, only to immediately cool down the wort and remove them?  I came across a discussion on BeerSmith about adding hops at flameout and then letting the wort sit for a while.  There’s another interesting discussion at BeerAdvocate about how long to let the wort sit before cooling.  There’s also an excellent article in BYO on hop stands, in which it’s explained that “pro brewers [give] their flameout hops extended contact time with the wort“.  Last but not least, there’s an interesting discussion on ProBrewer about how long professional brewers whirlpool their hops after flameout.  In short, the wort is often not cooled immediately, which creates a hop stand (whether or not hops are added at flameout, due to any hops already in the wort that have not yet reached maximum utilization).  This extended contact gives flameout hops time to contribute something to beer flavor (and bitterness) at below-boiling temperatures.  In my previous hop-stand experiments, I added post-flameout hops only after the target temperature (e.g. 170°F (77°C)) had been reached, and steeped for a relatively long period of time (60 minutes).  Since those experiments didn’t demonstrate an increase in hop flavor, maybe higher temperatures or shorter steep times are critical for hop flavor.  In the current experiment, I let all batches sit for 10 minutes after flameout, with the lid on.  (I chose 10 minutes pretty much by chance, but I now think that 10 minutes is a really good time for steeping hops.)

Balancing Bitterness Across Conditions
The goal of the current experiment was to look at hop flavor, but I wanted to examine hop flavor independently of bitterness.  In other words, I wanted to vary the timing of late-hop additions and keep the wort at high temperatures after flameout, but hold the bitterness level of all conditions relatively constant.   If one uses a standard formula for computing IBUs (e.g. Tinseth’s formula), hops additions at 0 minutes contribute no bitterness to the beer.  This is true if one immediately force-cools the wort at flameout, but since I allowed the wort in this experiment to sit for 10 minutes after flameout at high temperatures, there was bitterness that was not accounted for by this formula.  In order to keep the conditions in this experiment at roughly the same bitterness level, I developed a modified version of Tinseth’s IBU formula that predicts bitterness contributions after flameout.  I used this formula to vary the timing and amount of hops added to each condition, in an attempt to equalize bitterness levels. There was a bug in my code at the time I used it for this experiment, and I didn’t have the finished beer tested for IBUs, so despite my good intentions I have no idea how well bitterness was kept constant.

Introduction
This experiment looked at three techniques that may contribute to hop flavor: (1) covering the pot during the last minutes of the boil, (2) varying the time of late-hops additions, and (3) a 45-minute hop stand held at 170°F (77°C), with hops added at flameout instead of when the target temperature is reached.  In all cases, the wort was left to stand for 10 minutes after flameout, which may be a critical detail.

(1) Covering the Pot
It’s well known that volatile oils from the hops escape with the steam during the boil (e.g. Daniels, Designing Great Beers, p. 101; Fix and Fix, An Analysis of Brewing Techniques, p. 33; Lewis and Young, Brewing, 2nd ed., p. 271; Papazian, The Homebrewer’s Companion, p. 63). However, an uncovered boil is essential to drive off the precursors of DMS (e.g. Palmer, How to Brew, p. 82; Fix and Fix, An Analysis of Brewing Techniques, p. 50).  Under the assumption that the risks of DMS outweigh any benefits, I usually leave my pot uncovered during the entire brewing process, in accordance with Papazian’s instructions to “never cover a boiling wort with a lid”. (Papazian, p. 138).  Most ales, however, “have DMS levels well below threshold” (Fix and Fix, p. 50).  Because SMM and DMS are reduced more at ale fermentation temperatures than at lager fermentation temperatures, “any hint of DMS in ales is likely from technical brewing errors, most notably contamination” (Fix, p. 75).  This then brings up the question:  will covering the pot during the last additions of hops yield more (good) hop flavor in the (hop-forward) beer than (bad) DMS?  There’s only one way to find out:  brew one condition with the pot uncovered during the entire boil, then brew a nearly identical batch with the pot covered after the last addition of hops.

(2) Varying the Time of Late Hops Additions
Late hop additions are also well known to provide more hop flavor than early additions.  I’ve seen many general statements to the effect of “Thirty minutes is a traditional cut-off point for flavor hops” (Daniels, p. 101) or “Flavor hops additions are considered to be in the last 10 to 20 minutes of the boil” (Strong, p. 65).  Papazian provides an informative graph, showing an increase in flavor starting at 0 minutes, peaking at 10 minutes, and decreasing to zero at 45 minutes (Papazian, The Homebrewer’s Companion, p. 68).  This graph is a “general guide,” though, and I wanted to examine the effect of hops additions in the final 10 minutes, and include a 10-minute stand after flameout.  Therefore, the current experiment looks at the effect on flavor when adding hops at 10 minutes, 5 minutes, and 0 minutes before flameout.  In all cases, I let the wort cool for 10 minutes after flameout.  This post-flameout wait provided at least a brief hop stand for all batches, but it means that my results will be different from someone who does late hopping and then cools their wort at flameout.

(3) Hop Stand
In my previous attempts at a hop stand, I found that the hops added during the stand contributed very little hop flavor, and that the resulting fuller-bodied beer was most likely the result of non-enzymatic browning of the wort.  Not what I was looking for.  But I added the hops only after the wort had reached the target temperature.  Some (or most?  nearly all?) people conduct a hop stand by adding the hops at flameout, bringing the temperature down (either naturally or by forced cooling), and then (possibly) holding the wort at a target temperature.  In the current experiment, there is an additional condition in which I added the hops at flameout, let the wort cool naturally (while covered) for 10 minutes, force-cooled the wort to the hop-stand target temperature, and then held that temperature for 45 minutes. This allows a direct comparison of how effective a hop stand is for longer time periods at lower temperatures.

Methods
This experiment used five conditions:
(A) The baseline: a beer with a late-hop addition at 10 minutes and no covering of the pot.  This was a pretty generic beer.  The “bittering” hop addition of 0.25 oz in 1.3 G of wort (7 g in 4.9 liters) was made at around the ~20 minute mark (instead of the normal 60 minute mark), under the assumption that at 20 minutes and more, the contribution to hop flavor is minimal.
(B) A late-hop addition at 10 minutes, with the pot covered during the final 10 minutes.  The bittering hop addition of 0.25 oz (7 g) was also around the 20-minute mark.
(C) A late-hop addition at 5 minutes, with the pot covered during the final 5 minutes.  The bittering hop addition was slightly more hops (0.30 oz or 8.5 g) at around the 30-minute mark, to attempt to keep the bitterness level about the same as in other conditions.
(D) A late-hop addition at flameout (0 minutes).  The bittering hop addition was even more hops (0.35 oz or 10 g) at the 45-minute mark, to try to keep the bitterness level about the same as in other conditions.
(E) A late-hop addition at flameout.  The hops additions (amount and timing) were the same as in Condition D.  This condition was different from Condition D in that it was followed by holding the wort at 170°F (77°C) for 45 minutes after the 10-minute natural cooling period.

For all conditions, the wort was left to cool for 10 minutes after flameout with the pot covered. The target OG of all conditions was 1.060.  More details are provided below in Table 1.

Comparisons
Condition A can be compared with B, to determine if covering the pot during the last hop addition (at 10 minutes, in this case) improves hop flavor.  Conditions B, C, and D can be compared with each other to determine which late-hop time (10 minutes, 5 minutes, 0 minutes) yields the most hop flavor (given the subsequent 10-minute hop stand).  Conditions D and E can be compared to determine if a 45-minute hop stand at 170°F (77°C) contributes to increased hop flavor.

I originally intended to compare the bitterness levels across all conditions, as a test of a modification to Tinseth’s IBU formula.  However, due to a bug in my initial calculations, the bitterness level will probably be somewhat different across the batches.  I report on the perceived bitterness levels in the Results: Comparisons section, below.

Recipes
As usual in these experiments, a very simple recipe of Briess liquid malt extract, Cascade hops (8.9% AA), Citra hops (13.9% AA), and Safeale US-05 yeast was used.  Rather than brewing the best beer possible, the idea was to keep things as simple and as replicable as possible.  The target volume of the wort at the end of each boil was 1.3 G (4.9 liters).  The goal was to end up with more than 1 G (3.8 liters) per condition, and to ferment only 3½ quarts (3.3 liters), as it’s better to throw wort away (including wort used in SG readings and settled trub) than to not have enough.  The 3½ quarts (3.3 liters) leaves (just) sufficient head room for fermentation.

condition
A
condition
B
condition
C
condition
D
condition
E
Extract:
2½ lbs (1.13 kg) Briess light LME 2½ lbs (1.13 kg) Briess light LME 2½ lbs (1.13 kg) Briess light LME 2½ lbs (1.13 kg) Briess light LME 2½ lbs (1.13 kg) Briess light LME
Initial Water: 1.80 G
(6.8 liters)
1.68 G
(6.3 liters)
1.80 G
(6.8 liters)
1.95 G
(7.4 liters)
2.0 G
(7.6 liters)
Boil Time: 30 min 30 min 35 min 45 min 45 min
Bittering Hops Addition: 0.25 oz (7 g) Cascade (8.9% AA) at 19 min 0.25 oz (7 g) Cascade (8.9% AA) at 21.3 min 0.30 oz (8.5 g) Cascade (8.9% AA) at 30.5 min 0.35 oz (10 g) Cascade (8.9% AA) at 45 min 0.35 oz (10 g) Cascade (8.9% AA) at 45 min
Aroma/
Flavor Hops Addition:
0.4 oz (11 g) Cascade (8.9% AA) and
0.4 oz (11 g) Citra (13.9% AA)
at 9.3 min,
not covered
0.4 oz (11 g) Cascade (8.9% AA) and
0.4 oz (11 g) Citra (13.9% AA)
at 9.3 min,
covered
0.4 oz (11 g) Cascade (8.9% AA) and
0.4 oz (11 g) Citra (13.9% AA)
at 5.0 min,
covered
0.4 oz (11 g) Cascade (8.9% AA) and
0.4 oz (11 g) Citra (13.9% AA)
at 0 min,
covered
0.4 oz (11 g) Cascade (8.9% AA) and
0.4 oz (11 g) Citra (13.9% AA)
at 0 min,
covered
Hop Stand:
no no no no 45 minutes at 170°F (77°C)
Final Target Volume:
1.3 G
(4.9 liters )
1.3 G
(4.9 liters )
1.3 G
(4.9 liters )
1.3 G
(4.9 liters )
1.3 G
(4.9 liters )
Yeast:
~3.4 g Safeale US-05 in 1.6 oz water added to 3½ quarts (3.3 liters) ~3.4 g Safeale US-05 in 1.6 oz water added to 3½ quarts (3.3 liters) ~3.4 g Safeale US-05 in 1.6 oz water added to 3½ quarts (3.3 liters) ~3.4 g Safeale US-05 in 1.6 oz water added to 3½ quarts (3.3 liters) ~3.4 g Safeale US-05 in 1.6 oz water added to 3½ quarts (3.3 liters)
Priming Sugar:
0.5 oz (14 g)
corn sugar
0.5 oz (14 g)
corn sugar
0.5 oz (14 g)
corn sugar
0.5 oz (14 g)
corn sugar
0.5 oz (14 g)
corn sugar
Target OG:
1.060 1.061 1.061 1.060 1.061

Table 1. Recipes and predicted values for the five conditions.

These recipes assumed an evaporation rate of 0.90 G/hr (3.4 liter/hr) during the uncovered boil, 0.35 G/hr (1.3 liter/hr) at temperatures less than boiling (uncovered), and 0.10 G/hr (0.38 liter/hr) for a covered boil or stand.  (The value for the covered boil was a guess, and assumed some small amount of loss due to various factors.)  The amount of water, the weight of bittering hops, and the timing of all hops additions were varied to attempt to achieve about the same OG, the same post-boil volume, and the same bitterness levels.

At 10 minutes after flameout, each condition was cooled to 75°F (24°C) using a wort chiller and let sit for an additional 10 minutes.  After transferring 3½ quarts (3.3 liters) into a sterile 1 G (4 liter) container (a.k.a. milk jug), the jug was shaken vigorously for 90 seconds, the yeast was pitched, and an airlock was applied.  Fermentation and conditioning proceeded for 3 weeks at around 64°F (18°C), followed by bottling and bottle conditioning for an additional 3 weeks (also around 64°F (18°C)).  Priming used 0.50 oz (14 g) of glucose per condition to yield 2.11 volumes CO2. The yield was 8 12-oz bottles per condition.

I don’t think that the level of precision indicated in these recipes is required in order to obtain perceptually identical beers; a point or two of OG difference or a variation of 5 IBUs (Daniels, p. 76) probably won’t be perceptible.  I tried my best to obtain the target numbers indicated, however, and hoped that any measurement errors would, on average, cancel each other out.

Results
Results: (In)Ability to Follow the Recipes (a.k.a Mistakes)
If I had been able to follow the recipes above to the letter and not had any bugs in my software, then this sub-section wouldn’t be necessary.  But nothing new ever goes completely according to plan, and so there were some unintended deviations from the recipes.  This part discusses what went differently and if I think there may be an impact on results.

(1) Evaporation Rates: Apparently, the 0.90 G/hr (3.4 liter/hr) evaporation rate that I’ve measured in the past (when making 5-gallon batches) was larger than my observed evaporation rate in this experiment.  This may have been because I used a smaller pot (which had a smaller opening), or because I’ve been so worried about too much evaporation that I applied less heat overall.  Likewise, the below-boiling evaporation rate seems to have been slightly overestimated.  Finally, the evaporation rate when the pot was covered was probably much closer to zero.  I realized something was off when Condition A was finished with the boil.  My solution for conditions B, C, D, and E was to wait an additional 5 to 10 minutes during the boil before adding any hops.  Even so, my measured OG values were 1.059 to 1.060 instead of 1.060 to 1.061.  I don’t believe that I can detect the difference of a few points of OG, and the over-estimation of evaporation rate was roughly the same for all conditions, so I don’t think that this will affect results.

(2) Condition A: I mistakenly used 1.85 G (7.0 liters) of water instead of 1.80 G (6.8 liters).  In addition, because the assumed evaporation rates were incorrect, I ended up with 1.75 G (6.6 liters) of wort after the boil instead of 1.60 G (6.0 liters).  My solution was to use a hop-less stand after the boil (at 170°F (77°C)) for 30 minutes in order to evaporate the extra 0.15 G (0.57 liters).  This meant that Condition A probably had a little bit more body than Conditions B, C, and D due to non-enzymatic browning, but body is not one of the factors I’m intending to evaluate in this experiment.

(3) Condition E: By the time I got to Condition E, apparently I was starting to really increase the heat in order to increase evaporation.  I ended up with an OG of 1.061.  Since the other conditions ended up with OGs around 1.059, I added ¼ cup (60 ml) of water to the final 3½ quarts (3.3 liters), which resulted in an OG of 1.060.

(4) Post-Flameout Temperature Decrease: Before brew day, I did a quick experiment in my kitchen to measure how quickly temperatures decrease after flameout.  This test showed that for 1.6 G (6.0 liters) in an uncovered pot, the temperature after 10 minutes was 182°F (83°C), and for 1.6 G (6.0 liters) in a covered pot, the temperature after 10 minutes was 201.5°F (94°C).  Since I planned to keep the lid closed after flameout, I used a line based on the second measurement to predict post-flameout bitterness.  What I forgot to take into account was the minute or so immediately after flameout, when I stirred the wort one last time and took a sample for SG reading.  In this brief time, the temperature quickly dropped while the pot was uncovered.  Also, the temperature in my kitchen (68°F (20°C)) was much greater than in my garage where I brew (around 60°F (15.5°C)).  As a result, I ended up with temperatures between 190°F (88°C) and 195°F (90.5°C) at 10 minutes after flameout.  Because of lower observed temperatures, I achieved less hop utilization during the 10 minutes after flameout than I had predicted.

(5) Bug in the Calculations: While this batch was fermenting, I worked on a blog post to explain a modification to the prediction of IBU values that takes into account post-flameout bitterness.  In the course of this writeup, I found a bug in my code.  As a result of this bug, I was computing less post-utilization flameout than I should have been for earlier hops additions, and so the (hopefully) more correct bitterness levels (mIBU values) decrease with the later hops additions instead of being constant.

After all those mistakes, here is a table summarizing the observed original gravity and final gravity for each batch:

condition
A
condition
B
condition
C
condition
D
condition
E
Original
Gravity
 1.059  1.060  1.059  1.059  1.060
Final
Gravity
 1.013  1.014  1.013  1.013  1.013

Table 2. Measurements of Each Condition

Results: Comparisons
The following table summarizes the results of the comparisons.  The top right half of the table (in blue) is for the “hops flavor” comparison, and the bottom left half of the table (in green) is for a “relative bitterness” comparison. The letter in each box indicates which of the two conditions was preferred; a question mark indicates that no difference could be reliably detected.  Multiple values indicate multiple comparisons of the two conditions, which I did to detect possible random variation.

Condition A
Condition B Condition C Condition D Condition E
Condition A
   ?,B,?  –
 –  –
Condition B  ?,?,A    ?,C,C  D,D  –
Condition C  –  ?,?,?    D,D,D  –
Condition D  –  ?,?  ?,D,C     ?,?,?
Condition E  –
 –  –  ?,?,?
 

A/B Comparison Notes. First tasting: condition A had slightly more body, as expected by the non-enzymatic browning caused by the extra time for evaporation. Condition B had very slightly more hops/citrus flavor, but not enough to be a reliable difference.  It seemed that covering the pot during the last 10 minutes had a negligible effect on flavor.  Second tasting:  Condition B had a very slightly crisper, more citrus flavor than A, as one would expect from Cascade and Citra hops.  The beers were very, very similar, but there was a reliable, detectable difference.  Bitterness levels were the same.  Third tasting:  This time, A seemed slightly more bitter; B more “mellow.”  (In hindsight, it’s likely that I was hallucinating the difference in bitterness; I’m also not sure what would make B more “mellow”.)  I could detect only a very slight difference in hops flavor, with B having slightly more but not enough for me to consider it a reliable difference. In short: B was preferred for hops flavor all three times, but only once did I think it noticeable enough to be considered a “reliable” difference.

B/C Comparison Notes. First tasting: these beers had nearly identical taste.  At first I decided that C was ever so slightly more bitter than B; a half glass later, I decided that B was just slightly more bitter than C.  So I marked it as “no detectable difference” in terms of bitterness.  At first, I could detect no difference in hops flavor.  By the end of the first tasting, I thought that C had slightly more hops flavor than B, but not much.  Second tasting: this time, I could reliably detect a small amount of more hops flavor in C, even from the first sips.  Bitterness levels were about the same, although C seemed maybe just a little more bitter than BThird tasting: C had distinctly more hops flavor than B, although not dramatically more.  The difference was small but noticeable.  I thought B might have been a little bit more bitter than C (the opposite of my second tasting result), but not enough to make it a reliable difference.  In short: bitterness levels were about the same, and C had consistently more hop flavor than B.

C/D Comparison Notes. First tasting: OK, this was the first clear and compelling difference!  D definitely had more hops flavor.  This was a real plus.  On the other hand, it also had more of a tannin flavor.  I had a hard time deciding which was more bitter.  D might have been a tad sweeter, but it also seemed like it might have had more of a tannin or “astringent” bitterness, in contrast with the “clean” bitterness of C.  So in the end the bitterness level seemed about the same.  One unanswered question is whether the astringent bitterness was caused by the longer boil time of the “bittering” hops or the later addition of the “flavor” hops.  Second tasting: D had much more hops flavor, by a wide margin.  C had a definite citrus-hop character, but D brought it out much more.  I thought that D was more bitter, in contrast with the predicted bitterness levels.  Third tasting: Again, and without question, D had more hops flavor.  C seemed to be slightly more bitter, but the bitterness was a “cleaner” bitterness rather than an “astringent” or “grassy” bitterness.   Since these tastings, I’ve found a relevant comment by Greg Noonan: “the bitterness derived from long boiling is coarser than that from a more moderate period” (Noonan, New Brewing Lager Beer, p. 154).  Condition D had a larger amount of bittering hops in the boil for a longer time, and so the difference in bitterness quality probably came from the bittering hop addition rather than the late hop addition.  In short: D clearly had more hop flavor than C; bitterness levels were difficult to judge but about the same.

D/E Comparison Notes. First tasting:  There was almost no difference between these beers.  There was a very slight and subtle difference, but I couldn’t figure out if it E was slightly more astringent, or had more body, or what.  In short, there was no difference between D and E that I could label with any category.  Second tasting:  Same results as the first.  I thought there might be some difference between the two conditions, but I couldn’t quite place what it was.  More bitter?  Fuller?  More sweet?  Crisper?  I really didn’t know.  They were not identical, but not reliably different in either hops flavor or bitterness.  Third tasting:  This time I was able to pin a label on the difference: E was slightly smoother than D.  Once I had decided on that label, I could distinguish them.  Since “smooth” is neither hop flavor nor bitterness, I marked this comparison as “?” in both categories.  The “smoothness” description fits in well with the flavor effects of a hop stand that I observed in Hop Stand Experiment #1In short: bitterness and hop flavor levels were about the same for D and E; E was slightly “smoother”.

B/D Comparison Notes. After the main comparisons (A/B, B/C, C/D, D/E), I had enough bottles left to compare B and D twice, so I did.  First tasting: As expected, D had more hops flavor than B, but I couldn’t detect a difference in bitterness… if anything, D seemed slightly more bitter.  Second tasting: Once again, D had more hops flavor than B.  At first I thought that B was more bitter, then I decided that I really couldn’t tell.  In short: D had more hops flavor than B.

Summary
Covering the lid during the final 10 minutes of the boil (immediately after the last hops addition) had a small impact.  There might be some benefit to covering the pot, resulting in a barely detectable increase in hops flavor.  Certainly there was no downside, and no extra effort.

A hops addition at flameout, with a 10-minute stand, contributed much more hops flavor than otherwise identical additions at 5 and 10 minutes.  A hop addition at 5 minutes contributed more hops flavor than a 10-minute addition, but much less than the flameout addition.  This may be compatible with Papazian’s graph showing a peak in hops flavor at 10 minutes, since his graph may assume cooling at flameout, whereas my batches were kept hot for 10 minutes after flameout.

Holding the hops in the wort at 170°F (77°C) for 45 minutes yielded no reliably-quantifiable effect on hops flavor or bitterness, except for the possibility that the wort held at 170°F (77°C) was slightly smoother.  Unless you’re really trying to squeeze every last possible iota of goodness from your process, when the wort cools to ~180°F (82°C), you might as well force-cool to pitching temperature and get on with the day.

Conclusion and Future Work
Within the constraints of this experimental setup, the best way to maximize hop flavor is to add hops at flameout, cover the pot, and let the wort cool naturally for at least 10 minutes.  Longer hops additions are not as effective as flameout additions.  Covering the pot provides a very small increase in flavor.  Holding the wort at 170°F (77°C) may provide some benefit, but is probably not worth the effort.

Future work: I’d like to see if there is some temperature between boiling and ~180°F (82°C) that maximizes hop flavor, and for how much time the hops should be steeped at that temperature.

A Modified IBU Calculation (Especially for Late Hopping and Whirlpool Hops)

Abstract
The predicted IBU contribution when adding hops at flameout is usually zero.  This is in conflict with widespread experience, which shows that adding hops at flameout does add significant bitterness.  This blog post proposes a modification to the Tinseth IBU formula to account for hops added late in the boil and/or at flameout.  This new metric is referred to as “maximum IBU” (or “mIBU”), for reasons explained below.

Problem
Introduction
If one takes a standard formula for predicting IBUs, such as Tinseth’s, hops additions at flameout contribute nothing to the final IBU measurement.  Tinseth had “access to some handy tools and knowledgeable friends at the USDA hop labs and the Flavor Perception labs at Oregon State University,” and he has “had quite a few worts and beers analyzed.” His formula is very widely used, presumably because it does as good or better a job at predicting bitterness levels compared with other available formulas.  So it’s worth taking his work and formula very seriously.

However, a quick glance at various internet sources indicates that this is one area where formulas and observations do not always line up.  In an excellent BYO article from Mar/Apr 2013, it is reported that large systems (15 bbl = 465 G = 1760 liters) get 16% utilization from hops added at flameout, while smaller systems (11 G or 42 liters) get around 10% utilization.  In a discussion at probrewer, one user says “Matt Brynildson, a former hop chemist, claims to get 22% whirlpool utilization at Firestone Walker (50-bbl system) and to have gotten 15% whirlpool utilization on a 10-bbl brewpub system”.  In another discussion at probrewer, utilization rates between 13% and ~30% are reported for whirlpool additions.  The Beersmith software, based on a discussion from 2013, treats hops at flameout differently from hops added before flameout.  As a result, hops added at 1 minute before flameout will contribute less predicted bitterness than hops added at flameout, which is counter-intuitive.  Brad Smith of Beersmith says in this discussion that “at least this is a step in the right direction”, compared with having flameout hops contribute no bitterness at all.  In short, we have reported post-flameout utilization (and consequent bitterness) ranging from 10% to 30%, and a lot of numbers in between.  There is apparently some utilization happening after flameout; it’s just not quite clear how much. The question then becomes how to model post-flameout utilization in the prediction of IBUs.

Solution
Approach
Hop utilization decreases as a function of temperature, with more utilization at boiling and less utilization at lower temperatures.  (I’ve seen claims that utilization drops to zero at around 180°F (82°C), and claims that the decrease in utilization follows an Arrhenius equation.  In a separate blog post, I look at relative utilization as a function of temperature.)  It’s also very clear that when the gas (or electricity) of the boil is turned off, the temperature of the wort doesn’t instantaneously drop to room temperature.  Therefore, there is additional utilization happening after flameout, and the degree of this additional utilization depends on when the hops were added and how quickly the wort cools.  It seems possible to take Tinseth’s formula for IBUs, combine it with a measure of how utilization is affected by temperature, and calculate the additional IBUs that occur after flameout… and that’s exactly what this section does.

Tinseth’s IBU Formula
I’ll use Glenn Tinseth’s formula for predicting IBUs in this post, because this formula is “considered very accurate” (BeerSmith: Calculating Hop Bitterness: How much Hops to Use?), but also simply because I haven’t yet worked as much with the formulas from Rager or Garetz.  Tinseth’s formula is as follows:

IBU = U(BG,t1) × D(AA,W,V) [1]
U(BG,t1) = b(BG) × f(t1) [2]
D(AA,W,V) = AA × W × 1000 / V [3]
b(BG) = 1.65 × 0.000125(BG − 1) [4]
f(t1) = (1 − e(-0.04t1)) / 4.15 [5]

where U(BG,t1) is the hop utilization (which is a function of both boil gravity and time), D(AA,W,V) is the density of alpha acids in the wort (in mg/l, a function of the alpha-acid rating, weight of hops, and volume of wort), b(BG) is a “bigness factor” that is a function of the boil gravity (BG), f(t1) is a “time factor” that predicts the isomerization of alpha acids as a function of time (in minutes) (t1), AA is the alpha-acid rating (in percent) of the hops added, W is the weight of the hops (in grams), and V is the volume of the wort (in liters).  (Generally I’ll try to use both metric and Imperial units in this post… according to WordPress, about half of the people reading this are from the U.S. and half are not.  In this paragraph, I’ll use only metric for simplicity.)  I use t1 to denote time, instead of just t, because another time variable, t2, will be used later.  Tinseth suggests “fiddling with 4.15 if necessary to match your system; only play with the other three [constants (1.65, 0.000125, and -0.04)] if you like to muck around.”  One complication is that time is usually measured from the point at which hops are added until flameout.  Because of this, any hops that are in the wort after flameout are predicted to contribute nothing to the final IBU value, even though they are in near-boiling wort for some amount of time.

I’ll refer to U(BG,t1) simply as “utilization”; this is the utilization that we believe also decreases as a function of temperature, which isn’t indicated in the current formula.  The utilization is a unitless number less than or equal to 1; usually it’s below 0.3.  Utilization normally represents the relative amount of alpha acids added to the wort that (a) get converted into isomerized alpha acids and (b) end up in the finished beer.  The Tinseth formula (and other formulas) have a direct relationship between utilization, isomerized alpha acids, and IBUs.  This is an oversimplification, since other hop components (oxidized alpha and beta acids, and polyphenols) are not part of the utilization equation, but they still contribute to bitterness and the measured IBU value.  The utilization in these IBU formulas therefore implicitly includes the effects of these other components on the IBU value.  This oversimplification usually works out fairly well, though, for typical hop rates and boiling times.

Plot of Utilization
We can plot the utilization in Equation [2] as a function of time, starting at time 0 and increasing with the length of time that the hops are in the kettle:

Figure 2. Utilization as a function of time, with time increasing from left to right. The utilization increases as the boil gravity (BG) decreases.

Figure 1. Utilization as a function of time, with time increasing from left to right. The utilization increases as the boil gravity (BG) decreases.

This function shows the cumulative effect of utilization after the hops have spent a certain amount of time in the kettle, from t1=0 (hops addition) until some final time.  If we add hops at 40 minutes before flameout to a wort with gravity 1.060, the hops spend 40 minutes in the boiling wort, and cumulative utilization during those 40 minutes is 0.185.  (Here we’re assuming that the utilization drops immediately to zero at flameout.)  We can find the value of 0.185 by simply looking at the utilization value at the 40-minute mark for a boil gravity of 1.060.  Time 0 is when hops are added to the pot, and the selected time (e.g. 40 minutes) is when we remove the hops from the kettle (perhaps at flameout) and/or bring utilization down to zero.

Instantaneous Utilization
Next, we can look at the instantaneous effect of hop utilization at any point in time, by taking the derivative of the utilization function:

U(BG,t1) = 1.65 x 0.000125(BG−1) × (1 − e(-0.04t1)) / 4.15 [6]
dU(BG,t1)/dt1 = -1.65 x 0.000125(BG−1) × -0.04e(-0.04t1)/4.15 [7]

where the formula for U(BG,t1) in Equation [6] is the same (but simplified) utilization function in Equation [2], and dU(BG,t1)/dt1 in Equation [7] is the derivative of this utilization with respect to time.  The instantaneous utilization function in Equation [7] can be plotted like this:

Figure 2: Instantaneous utilization as a function of time.

Figure 2: Instantaneous utilization as a function of time.

This graph tells us how much hop utilization is happening, for example, at exactly the 40-minute mark (in this case, 0.001873 utilization/minute for BG 1.060).  (Note that the rate of utilization is highest when the hops are first added, and this rate decreases as the hops stay in the boil longer.  Tinseth explains that the shape of the utilization curve is because “alpha acid isomerization is a first order, or more likely, a pseudo first order chemical reaction.“)  If we integrate all of these instantaneous utilization values, from time 0 (when we add the hops) up to the time when we remove the hops, we end up with the original utilization value.  For example, if we have a hops addition at 40 minutes for a boil gravity of 1.060, we can compute the area under the instantaneous-utilization curve from 0 to 40 (shown below in green), and that area will equal the total utilization at 40 minutes from the original utilization formula (0.185).

Figure 4: Area under instant utilization curve, up to 40 min

Figure 3: Area under instantaneous utilization curve, up to 40 min

We can also integrate only over a specific time range, to determine how much utilization is happening during a particular period of time; for example, we can look at the utilization between 40 minutes and 50 minutes for a boil gravity of 1.060, which is the area below shown in blue:

Figure 5: Area under instant utilization curve, between 40 and 50 min.

Figure 4: Area under instantaneous utilization curve, between 40 and 50 min.

Additional Utilization
Let’s pretend for a minute that when flameout happens, we forget to turn off the heat for exactly 10 minutes and the boil keeps going.  When we realize our mistake, we immediately and quickly cool the wort (which has an average boil gravity of 1.060).  In this case, utilization keeps happening after our intended flameout; what was a hops addition at 40 minutes becomes really a hops addition at 50 minutes, because the hops are in the boiling wort for an additional 10 minutes:

Figure 6: Area under instant utilization curve, from 0 to 50 min.

Figure 5: Area under instantaneous utilization curve, from 0 to 50 min.

We can look at the utilization of our intended time (40 minutes) and our extra time (10 minutes, between 40 minutes and 50 minutes) separately, by finding the areas under the two separate curves.  In this case, the utilization up to 40 minutes is 0.185 (just as we’d expect), and (by integration of the instantaneous utilization over the range from 40 to 50 minutes) the additional utilization during the final 10 minutes is 0.015.  The total utilization is therefore 0.200.  What really happens (in most cases, when we’re paying attention) is that we do remember to turn off the heat at flameout, but the wort remains hot for some period of time.  We’re not getting maximum utilization after flameout, but we are getting some while the wort is still hot.

Decrease in Utilization as a Function of Temperature and Time.
After flameout, utilization is happening (unless we immediately and very quickly cool the wort), but not at the full rate.  It would be nice to have some function of utilization that shows how it decreases after flameout, as the temperature slowly decreases. We can create such a function from two functions: (a) the temperature of the wort as a function of time after flameout, and (b) the degree of hop utilization as a function of temperature.

If we measure the temperature of the wort after flameout at one-minute intervals, we can map these data points to a function.  Figure 6 shows such a series of measurements and the resulting function, where a straight line provides a good fit to the observed data.

Figure 7: Temperature as a function of time.

Figure 6: Temperature as a function of time after flameout, without forced cooling. This graph shows the temperature of 6 G of wort as it cools in an open kettle.  A line is also fitted to the data points, and this line provides a pretty good fit to the data.

In this case, the line is described by T(t2) = -1.344t2 + 210.64, where T is the estimated temperature in Fahrenheit, -1.344 is the rate of change (degrees per minute), t2 is time after flameout (in minutes), and 210.64 is the approximate temperature at flameout (when t2=0, in °F). (I live at 255 feet (78 meters) above sea level, so I should see water boil at pretty close to 212°F (100°C).  I have always observed, however, a quick drop in temperature as soon as the heat is turned off, so I think a predicted value of 210.64°F (99.24°C) isn’t terribly far off.)  If you’re working in Celsius, the equivalent function is T(t2) = -0.74667t2 + 99.244.

Next, we need a function that describes the relative amount of utilization as a function of temperature, with utilization at boiling defined to have a value of 1.0.  Based on the analysis from this blog post, we can use this function: Urel(T) = 2.39×1011 e-9773/T (where T is temperature in degrees Kelvin).

We can then plot the relative utilization as the temperature decreases after flameout, with 6 G of wort cooling in an open kettle:

Figure 8: Degree of utilization as a function of time.

Figure 7: Degree of utilization as a function of time.

The relative utilization is relative to the amount that would happen at boiling: if the instantaneous utilization at boiling is 0.001873 and the relative utilization at 206°F (96.7°C) is 0.80, then the utilization at this temperature is 0.001873 × 0.80 = 0.001498.

The next formulas summarize the mappings between temperature, time, and degree of utilization, working in Fahrenheit:

Urel(T) = 2.39×1011 e-9773/T degree of utilization as a function of temperature (T is in degrees Kelvin) [8]
T(t2) = -1.344 t2 + 210.64 temperature (in °F) as a function of time, if t2 ≥ 0, for 6G wort in an open kettle. [9]
Urel(t2) = 2.39×1011 e-9773/((5/9)(-1.344 t2 + 210.64)+459.67) degree of utilization as a function of time, if t2 ≥ 0.  Note conversion from °K to °F in the exponent. [10]
Urel(t2) = 1 degree of utilization as a function of time, if t2 < 0 [11]

Equation [8] is the degree of utilization as a function of temperature; Equation [9] is the temperature of the wort as a function of time after flameout.  Equation [10] combines Equations [8] and [9] for time after flameout greater than or equal to 0.  Equation [11] specifies that before flameout (during the boil), the relative degree of utilization is maximum (i.e. the value predicted by the Tinseth formula).

Combining Functions
Now we have a complete picture of instantaneous utilization as a function of time.  Before flameout, we use the instantaneous utilization of Equation [7], where t1 is the amount of time that the hops are in the kettle.  After flameout, we can combine our measure of instantaneous utilization with how much utilization is happening at each instant, by taking the pointwise product of the two formulas.  Since we measure time with two different starting points in the different equations  (when hops are added, t1, and when flameout happens, at which point t2 = 0), we need to convert from one to the other.  If we define tf to be the time, relative to t1, at which flameout happens, then t2t1 − tf.  We can then use t1 and tf in the same formula to measure these two time points on the same scale.  We can write this complete picture with the following formulas:

deg(t1) ⋅ dU(BG,t1)/dt1 = -1.65 x 0.000125(BG−1) × -0.04e(-0.04t1)/4.15 if t1 < tf [12]
deg(t1) ⋅ dU(BG,t1)/dt1Urel(t1 − tf) × (-1.65 x 0.000125(BG−1) × -0.04e(-0.04t1)/4.15) if t1 ≥ tf but
before forced cooling
[13]
deg(t1) ⋅ dU(BG,t1)/dt1 = 0 after forced cooling [14]

Going back to our example, let’s say we turn off the heat at 40 minutes (as intended) and let the kettle cool naturally for 10 minutes.  The graph below shows the instantaneous utilization happening at boiling between 0 and 40 minutes (in green), and the instantaneous utilization that happens between 40 and 50 minutes (in blue) that takes into account the temperature decrease after flameout.  The green area is the same as before, but the blue area is reduced relative to Figure 5.

Figure 9: Area under instant utilization, with cooling between 40 and 50 minutes.

Figure 8: Area under instant utilization, with cooling between 40 and 50 minutes.

If we integrate this plot (or function) from flameout until the utilization becomes zero (usually due to forced cooling), we can determine the total utilization that occurs after flameout.  This value can then be added to the utilization that occurs before flameout (our standard formula), and then we can compute the total IBUs that result from the combined utilization, pre- and post-flameout.

Adjustment for Non-IAA Components
Now we get to a tricky part.  The Tinseth equation has only the hops AA rating, hops weight, boil time, wort volume, and specific gravity as inputs.  From these values, it’s estimating the IBU value.  While the concentration of isomerized alpha acids (IAA) is the primary contributor to the IBU value, other factors also influence the IBU (especially at short boil times): oxidized alpha and beta acids, and polyphenols.  These factors can be referred to collectively as non-IAA components, and they’re not explicitly addressed in the Tinseth model.  These non-IAA components do not require isomerization, and so they affect the IBU value quickly and, it seems, their concentrations remain somewhat constant during the boil.  In another blog post, I estimate that the non-IAA components account for a constant 0.044 of the Tinseth utilization function, which is reached at around the 5-minute mark.  In other words, I think that the Tinseth equation underestimates IBU values below 5 minutes, because it doesn’t account for non-IAA components; at 5 minutes and above, the Tinseth utilization function is modeling the combination of both IAA and non-IAA components.  (The Rager equation has a roughly constant utilization of 0.05 between 0 and 5 minutes, which matches pretty well my estimate of 0.044 during this time period.)

Having the Tinseth equation model a combination of IAA and non-IAA components usually doesn’t matter.  In this case, however, we are modeling the degree of alpha-acid utilization as a function of temperature, not the degree of utilization of alpha acids plus non-IAA components.  The primary problem is that the non-IAA components are much less dependent on temperature than the rate of isomerization.  As a further complication, the non-IAA components have a very different time-dependent contribution to IBUs: shortly after time 0 they contribute about 100%, and after that they contribute very little.  It’s difficult to combine this function with a time-independent but temperature-dependent function of relative alpha-acid utilization.

An expedient solution is to set the relative utilization to 1.0 between 0 and 5 minutes.  By doing this, we are implicitly saying that during the first five minutes of the hop addition, all of the IBU contribution is coming from non-IAA components and that they contribute the same amount regardless of temperature.  After five minutes of steeping, the contribution of isomerized alpha acids dominates, and we switch to the temperature-dependent relative utilization of alpha acids.  This is an over-simplification of what’s really happening, but it requires no modifications to the original Tinseth equation, and it corresponds better with our understanding of the various contributions to the IBU value.  If hops are added more than 5 minutes before flameout, this modification has no impact on estimated IBU values, regardless of any post-boil steeping; only close-to-flameout hop additions are impacted.

Final Algorithm
The final formula for measuring IBUs after flameout might get complicated.  We could integrate Equation [13] over the range t2 values, but I expect that the degree-of-utilization formula will change with different setups (e.g. covered vs. uncovered kettle, boil size, etc.) and it will probably be a little bit different for every brewer.  I’m going to be lazy, since we don’t really need the infinite precision that a mathematical formula gives us. Instead, here is an algorithm that uses the same example of the decrease in temperature as a function of time and the degree-of-utilization formula.  It should be easy to replace these formulas with different ones as needed, and the general algorithm will be the same… no (explicit) integration required.  We’ll just use computer magic to compute the area under a curve with an arbitrary degree of precision.  (Note that this expectation of different degree-of-utilization functions in different brewing systems may explain why we get a number of different opinions on how much utilization there is after flameout.  Larger volumes that cool more slowly will get more utilization.)

Here is the algorithm for computing total IBUs, including the contribution of post-flameout utilization, in C-like pseudocode, with precision to at least two decimal places:

// the following code assumes we know the boil gravity (BG), boil volume
// (volume_gallons, in gallons), the weight of hops added (hopsWeight_oz,
// in ounces), the AA value of the hops (AA, in percent, with values from 
// 0 to 100), the time of the hops in the boil (boilTime_min, in minutes), 
// and when (relative to flameout) there is forced cooling (coolTime_min, in
// minutes.)  The forced cooling is assumed here to be instantaneous.
// There are hard-coded functions that express (a) decrease in temperature 
// as a function of time after flameout and (b) relative utilization as a 
// function of temperature.
// Note that temperature is measured in Fahrenheit, not Celsius.
volume_liters = volume_gallons * 3.78541;
hopsWeight_grams = hopsWeight_oz * 28.3495;
AA_maxOne = AA / 100.0;
boilUtilization = computeBoilUtilization(boilTime_min, BG);
postBoilUtilization = computePostBoilUtilization(boilTime_min, BG, coolTime_min);
totalUtilization = boilUtilization + postBoilUtilization;
IBU = (totalUtilization * AA_maxOne * hopsWeight_grams * 1000.0) /
       volumeLiters;
print("total IBUs, including post-flameout, is %f\n", IBU);

procedure computeBoilUtilization(boilTime_min, BG) {
  bignessFactor = 1.65 * pow(0.000125, (BG-1.0));
  boilTimeFactor = (1.0 - exp(-0.04 * boilTime_min)) / 4.15;
  decimalAArating = bignessFactor * boilTimeFactor;
  return(decimalAArating);
}

// The formula for temp_degF can (and should) be updated to reflect 
// a brewer's observed temperature decrease after flameout.  
procedure computePostBoilUtilization(boilTime_min, BG, coolTime_min) {
  integrationTime = 0.001;
  decimalAArating = 0.0;
  for (t = boilTime_min; t < boilTime_min + coolTime_min; t = t + integrationTime) {
    dU = -1.65 * pow(0.000125, (BG-1.0)) * -0.04 * exp(-0.04*t) / 4.15;
    temp_degF = (-1.344 * (t - boilTime_min)) + 210.64;
    temp_degK = (temp_degF + 459.67) * (5.0/9.0);
    degreeOfUtilization = 2.39*pow(10.0,11.0)*exp(-9773.0/temp_degK);
    if (t < 5.0) degreeOfUtilization = 1.0;  // account for nonIAA components
    combinedValue = dU * degreeOfUtilization;
    decimalAArating += combinedValue * integrationTime;
    }
  }
  return(decimalAArating);
}

For those of you who would like to implement this but aren’t programmers, let me know and I’ll run the numbers and send you the results.  Please let me know by sending an e-mail to the name associated with this blog (no spaces) at yahoo.  There will be some back-and-forth to the communication, but just start by saying that you’re interested in a somewhat different implementation of the formula, and we’ll work it out from there.

Examples
Because the instantaneous utilization is greatest when the hops are first added, the effect of bitterness contributions after flameout is more pronounced for late hop additions than for early additions.  If we have 2 oz (56.7 g) of 10% AA hops added at 6 minutes before flameout in 5.25 G (19.87 liters) of wort with specific gravity 1.060, the kettle then cools naturally for an additional 10 minutes as shown by the graph in Figure 6, and we then force-cool the wort quickly, we get 0.0495 utilization predicted for the 6-minute boil and an additional 0.0521 utilization predicted after flameout.  The combined predicted utilization of 0.1016 is about double that of the prediction that doesn’t take into account post-flameout utilization, resulting in a predicted IBU of 29 instead of 14.

For the same case, but with hops added at 60 minutes instead of 6 minutes before flameout, we get 60 IBU predicted by the standard Tinseth formula and 62 IBUs predicted by the combination of boil utilization and post-boil utilization.

As another example, we can add hops at flameout and perform forced cooling at 30 minutes after flameout.  In this case, we get 11% utilization for a wort with boil gravity 1.060, about equal to the 10% flameout utilization of smaller batches reported by others.  We can get 31 IBUs from 2 oz (56.7 g) of 10% AA hops added at flameout in a boil gravity of 1.060 and volume of 5.25 gallons (19.87 liters), with forced cooling after 30 minutes.  If we keep the kettle covered after flameout, the temperature will decrease more slowly, and we will get even more utilization and IBUs from the flameout addition.

What’s in a Name
One problem with this formula for predicting IBUs is that the predicted values will almost always be larger than what we’re used to… sometimes, quite a bit larger.  I like to aim for an IBU:OG ratio (e.g. 1.0) when planning a recipe, and I’ve developed some intuition for what this ratio means in practice using the Tinseth formula.  When the predicted IBU values become larger by some amount, I’ve lost my point of reference.

I’ve decided to name this combined measure of boil and post-boil IBUs “maximum IBU” or “mIBU”, because it represents a (roughly) maximum bound on IBU values in finished beer.  IBUs will drop during fermentation and over time, but mIBU values should be approximately the highest observed IBU values for a batch of beer.  From a discussion at Brewsmith, by user brewfun: “fermentation, filtration and packaging [can] reduce IBU persistence by 85%”… “it’s a lot easier to lose IBUs than to gain them.”  Daniels says that “fermentation factors can affect the amount of iso-alpha-acids that remain in finished beer in a number of ways”, mostly linked with yeast, aging, and clarification (Daniels, Designing Great Beers, pp. 78-79).

Using a different name allows a separation between the IBUs predicted by the Tinseth formula that I’ve grown accustomed to, and these new, larger values.  I will use both terms in my brewing, and hopefully over time I’ll get a sense of the perceived bitterness of mIBU values and an intuitive feeling for the relationship between Tinseth IBUs and mIBUs.  The separation of terms will hopefully avoid confusion.

Decrease in Wort Temperature as a Function of Time
This post has used only a single example of how wort temperature decreases with time.  This set of data was obtained from a 10 G (37.8 liter) kettle with 6 G (22.7 liters) of water in it, uncovered, in a room at around 68°F (20°C).  I’ve since found that measurements of wort under similar conditions yields a close fit to the same linear function.

I’ve also looked at how 6 G (22.7 liters) of wort cools when the lid is left on.  I measured temperatures of wort at one-minute intervals for 15 minutes, with a long temperature probe through a small hole in the kettle lid.  At one minute after flameout the temperature was 208°F (97.78°C); at 15 minutes after flameout, the temperature was 198°F (98.22°C).  This temperature decrease can be modeled pretty well by the formula T(t2) = -0.7268 t2 + 208.88.  I’ve found several times that the temperature decreases really, really quickly from boiling to around 209°F or 210°F (around 98.5°C) in the initial minute, but the decrease in temperature after that is fairly linear for volumes about 5 gallons (20 liters) or more.

I expect other situations to yield somewhat different functions.  In particular, smaller volumes cool much more quickly, and larger volumes cool much more slowly.  Over time it may become clear that there is typical behavior for different batch sizes and degrees of covering, but there is still a large number of possible combinations (batch size, size of kettle opening, ambient temperature).  At any rate, please be aware that the numbers shown in this post depend quite a bit on this function, and you may need a different function to get good predicted values.

Data/Supporting Evidence
I’ve conducted a series of three experiments to evaluate the ideas in this blog post.  The first experiment simply confirmed that I can get measured IBU values that correlate reasonably well with the Tinseth formula when cooling the wort immediately after flameout.  (I think this experiment would have turned out better if I’d used very fresh hops.)  The second experiment looked at the relative degree of utilization as a function of temperature, and provided the reasoning behind Equation 8.  The third experiment compared measured IBU values with values predicted by the Tinseth equation and the mIBU approach.  This experiment confirmed that the mIBU approach yields, for at least this one set of data, better IBU estimates at short (or zero) boil times.  Another result from this experiment was that I needed to account for the age of the beer and the alpha-acid concentration at the start of the boil, in order for the recommended Tinseth scaling factor of 4.15 to provide a good fit to the data.  (If you have multiple hop additions at different times, and the cumulative alpha-acid concentration becomes quite large with subsequent additions, I’m not aware of any IBU model that can account for the decrease in utilization as a function of AA concentration and time… but that’s a separate topic.)

To get measured IBU values, I sent samples to 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 measurements of hops.

Conclusion
This post has presented an algorithm for modifying Tinseth’s IBU formula to predict post-flameout IBU contributions.  In addition to the standard parameters needed to predict IBUs (boil gravity, volume, etc.), this algorithm needs as input (a) a function (or plot) of how the temperature of the wort in the kettle decreases after flameout, and (b) a time at which forced cooling begins.  Forced cooling is assumed to be instantaneous; if you have a lot of wort, additional information (e.g. a wort transfer rate through a counterflow chiller) and modifications may be needed.