Tag Archives: isomerized alpha acids

Why Do Hop Pellets Produce More IBUs Than Hop Cones?

Abstract
Hop pellets are usually described as having higher utilization than hop cones.  A separate blog post looks at the amount of increase in IBUs caused by using pellets instead of cones. It finds that the amount of increase is constant over a range of hop steep times, instead of increasing with steep time.  This means that the increase in IBUs is not caused by an increase in the rate of alpha-acid isomerization or availability of alpha acids, which would result in longer steep times having a greater increase in IBUs. The first experiment in this post looks at whether this constant increase is more likely to be caused by a greater concentration of isomerized alpha acids (IAA) produced soon after a hop addition, or by other bittering compounds (nonIAA, also called “auxiliary bittering compounds”).  This experiment analyzes the rate at which IAA and nonIAA are removed from beer over time, and a comparison is made with the rate at which the increase in IBUs from pellets decreases over time.  The results indicate that pellets yield increased IBUs from an increase in auxiliary bittering compounds, not from increased IAA.  In other words, the concentration of isomerized alpha acids in finished beer is the same for beer made with cones or pellets, but the concentration of nonIAA is greater in beer made from pellets.  Data from a second experiment indicate that while the concentration of polyphenols is greater with the use of pellets, this greater polyphenol concentration cannot explain the observed increase in IBUs.  In this experiment, the increase in IBUs from pellets does not increase linearly with the amount of hops added, which is consistent with the IBU increase being caused by oxidized alpha acids.  (The same alpha-acid solubility limit that explains relatively lower IAA at higher alpha-acid concentrations can explain the relatively lower production of oxidized alpha acids at higher concentrations.)  The most likely explanation for the increase in IBUs when using pellets is that the pelletization process gives the alpha acids greater surface area, and that these exposed alpha acids oxidize quickly when brought into contact with hot wort, creating an increase in the concentration of oxidized alpha acids during the boil.

1. Background: Utilization, Reported Differences, and IBU Models
1.1 Utilization
Hop utilization, U, is the amount of isomerized alpha acids (IAA) 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) [1]

Utilization refers only to the relative amount of isomerized alpha acids, not to IBUs.  While IAA and IBUs can be considered roughly equivalent as a quick rule of thumb, IBUs measure a number of bitter components in addition to IAA.  These other bitter components are called nonIAA or “auxiliary bittering compounds”.  With short boil times, high hopping rates, low steeping temperatures, improperly-stored hops, and other factors, one can see significant differences between measured IBUs and the concentration of IAA.

1.2 Reported Differences Between Cones and Pellets
Hop pellets are almost always described as having greater utilization than hop cones [e.g. Daniels p. 78].  According to Michael Lewis and Tom Young, “the alpha acids dissolve most easily from extracts, less easily from pellets …, and least with whole hops” [Lewis and Young, p. 266].  It is said that 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].  Greg Noonan says that “with pelletized hops, ruptured and better-exposed lupulin glands give greater utilization” [Noonan, p. 154].

1.3 Modeling IBUs from Pellets with Scaling Factors
A previous blog post describes a model of IBUs based on equations from Val Peacock [Peacock, p. 157] and Mark Malowicki [Malowicik, p. 27].   This model can be used to estimate the scaling factors for isomerized alpha acids (IAA) and auxiliary bittering compounds (nonIAA) in beer made from either cones or pellets.  Another blog post used those scaling factors to show that the increase in IBUs is modeled well by an increase in the concentration of nonIAA, or by some process that adds IBUs at the beginning of the boil but not during the rest of the boil.

2. Introduction
Although the other blog post on pellet-based IBUs found that the increase in IBUs resulting from the use of pellets was modeled well by an increase in nonIAA concentrations, it is still possible that this increase is actually caused by the rapid production of isomerized alpha acids close to the start of a hop addition, instead of the usual time-dependent alpha-acid isomerization.  The model referred to in Section 1.3 groups all compounds that are produced near the beginning of a hop addition as nonIAA compounds, under the assumptions that isomerization is a fairly slow process and that nonIAA compounds are produced quickly.  If IAA are also somehow produced quickly after adding hops, this model would not be able to distinguish these IAA from nonIAA.

Perhaps the process of manufacturing pellets (which includes heat [Srečec, pp. 141-143], a primary factor in isomerization [Verzele and De Keukeleire, pp. 102-109]) transforms alpha acids into an intermediate compound which then quickly results in IAA when the pellets are added to boiling wort.  Such a process would mean that pellets show increased IBUs because of greater utilization, even if this increase in utilization happens much more quickly than the typical isomerization process.  (The existence of such an intermediate compound is postulated simply to explain how the increase in IBUs seen with the use of pellets might be caused by isomerized alpha acids, since the rate of isomerization or availability of alpha acids is not affected when using pellets.)

The rest of this blog post addresses the question of whether the increase in IBUs observed with the use of pellets is more likely to be the result of (a) IAA that are produced soon after a hop addition (i.e. greater utilization), (b) oxidized alpha acids produced when the hops are added to the boiling wort [Algazzali, p. 17; Dierckens and Verzele, p. 454], or (c) hop polyphenols.  (It is highly unlikely that this increase is related to oxidized beta acids because of the negligible impact that oxidized beta acids have on the IBU when using well-preserved hops.)

3. Experiment #1: Experimental Overview and Methods
3.1 Overview of Experiment #1
The IBU level and the concentration of IAA in beer decrease over time, especially at room temperature [Peacock, p. 164].  This decrease may be caused by IAA and possibly nonIAA transforming over time into different products or binding with other compounds and precipitating out of solution.  In either way, IAA and possibly nonIAA compounds are removed from beer over time.  The current analysis assumes that the rate at which IAA and nonIAA compounds are removed from beer is different.  By transforming multiple IBU measurements taken from a single beer at multiple points throughout the boil (both fresh beer and aged beer) into estimates of IAA and nonIAA factors in a model of IBUs, we can evaluate how these factors (and therefore IAA and nonIAA concentrations) change with the age of the beer.  If the increase in IBUs produced by the use of pellets decreases over time at the same rate as IAA loss, we can conclude that this increase in IBUs is probably produced by IAA.  Conversely, if the decrease matches the rate of nonIAA loss, we can conclude that nonIAA compounds are most likely responsible for the increase in IBUs with pellets.  (If the different-rate-of-decay assumption is wrong, then the decrease in IAA will be the same as the decrease in nonIAA, and no conclusions will be possible.)

A picture may help to illustrate the overall concept.  Figure 1 shows hypothetical cases of (a) IBUs produced using hop cones (solid dark blue line), (b) IBUs produced using hop pellets (solid dark green line), (c) the same cone-produced beer after 10 weeks of aging (dotted light-blue line), and (d) the same pellet-produced beer after 10 weeks (dotted light-green line).  This set of hypothetical data is based on two assumptions: (1) the change in IBUs over 10 weeks is due entirely to the loss of IAA; nonIAA compounds do not decrease in beer over time, and (2) the increase in IBUs caused by the use of pellets comes entirely from nonIAA compounds.  These assumptions produce a particular pattern in the IBU levels in Figure 1: (a) the solid green line and solid blue line are different by a constant factor (due to nonIAA compounds), (b) the dotted blue line starts at the same value as the solid blue line at 0 minutes, and then gradually decreases relative to the solid blue line (because only IAA levels decrease with age), and (c) the dotted green line and dotted blue line are different by the same constant factor (because the increase in IBUs with pellets comes only from nonIAA, which does not decrease over time).  Neither of these assumptions may be true, but we can analyze real IBU data using the model mentioned in Section 1.3 to estimate scaling factors.  The scaling factors, which could be used to produce graphs like Figure 1, will tell us how much loss occurs in both IAA and nonIAA over 10 weeks. We can then compare the change in pellet-related IBUs over the 10 weeks to the IAA and nonIAA scaling factors.  Comparing the rates at which losses occur will help us determine if the increase in IBUs from the use of pellets is more likely caused by IAA or nonIAA.

degradation

Figure 1. Hypothetical IBU levels from fresh and aged beer made with cones and pellets.  The data in this figure are made up in order to illustrate the patterns one might see as IBUs change over time in both types of beer.

I previously brewed two batches of beer that were nearly identical in all respects except for the use of cones in one case and pellets in the other, as part of a previous blog post (Hop Cones vs. Pellets: IBU Differences, Experiment #5).  For each batch, I took samples of wort at 10-minute intervals during a 60-minute boil.  Each sample was fermented into beer and 4 oz of each was sent to Oregon BrewLab for IBU analysis about 10 days after the start of fermentation.  I kept whatever wasn’t sent to Oregon BrewLab at room temperature for aging.  Those additional 12 samples were sent to Oregon BrewLab for IBU analysis at 10 weeks after the start of fermentation.

3.2 Methods for Experiment #1
All data for this experiment consisted of two batches of beer brewed on the same day, one batch using hop cones and the other using hop pellets.  I used 7.0 lbs (3.18 kg) of Briess Pilsen DME in 8.0 G (30.28 l) of water, yielding about 8.5 G (32 l) of wort with a specific gravity of about 1.037.  I did not adjust the water profile or pH, which resulted in a pre-boil wort pH of 5.80.

In this experiment, I used Comet cones from Hops Direct (stored in my freezer soon after harvest for about 4 months) and Comet pellets from YCH Hops (lot P92-ZLUCOM5216, about 2½ years old at the time of the experiment).   The previous blog post concluded that the age of the hop pellets did not have any impact on the pellet-based increase in IBUs.

I added hops (i.e. started the steep time at 0) after the wort had been boiling for 5 minutes, to avoid the foam associated with the start of the boil.  The hops were boiled for a total of 60 minutes with the cover on the kettle (except for taking samples) to minimize evaporation and the resulting changes in specific gravity.  I used 1.939 oz (54.96 g) of hop cones (alpha-acid rating 9.70%) and 2.147 oz (60.86 g) of hop pellets (alpha-acid rating 8.76%) to target an initial alpha-acid concentration of 170 ppm in both batches.

Samples were taken every 10 minutes from the start of steeping.  Each sample was taken from the boil in a measuring cup and then transferred to an aluminum cup using a wire mesh sieve to remove larger hop particles.  For the cones condition, 32-oz (0.95-liter) samples were taken; for the pellets condition, 16-oz (0.44-liter) samples were taken.  The aluminum cup was placed in an ice bath and the contents were stirred to cool quickly.  Once cooled to 75°F (24°C), the sample was transferred to a sanitized, sealed, and labeled quart (liter) container.  I aerated each sample by vigorous shaking for 60 seconds, then added .008 or 0.017 oz (0.24 or 0.48 g) of Safale US-05 yeast (depending on the volume of the sample) to target 750,000 viable cells per ml and degree Plato [Fix and Fix, p. 68].  After all samples were taken, the containers were cracked open to vent, and they fermented for eight days.  I swirled the samples every day to remove most of the krausen deposits on the sides of the containers.  After fermentation, I sent 4 oz (0.12 l) of each sample to Oregon BrewLab for IBU measurement.  The remainder of each sample then proceeded to age for 10 weeks at room temperature.  After 10 weeks, another 4 oz (0.12 l) was sent to Oregon BrewLab for IBU measurement.

4. Experiment #1: Results
The estimated room-temperature volume at the start of steeping was 8.34 G (31.57 liters).  The specific gravity after 10 minutes of steeping was about 1.0384.  The specific gravity after a 60-minute steep time was 1.0396.  The small change in specific gravity during the boil (due to keeping the lid on the kettle) means that there is little difference between using the measured IBU values for analysis or normalizing these IBUs by the volume when the sample was taken.  For simplicity and clarity, the measured IBU values are used below.

Figure 2 shows the measured IBU values from this experiment.  The average  difference in IBUs between cones and pellets is shown for both the fresh and aged beer.

AGE2-measuredIBUs-weeks1and10

Figure 2. Measured IBU data from beer made with cones or pellets, at 1 and 10 weeks after the start of fermentation.  The average IBU difference between cones and pellets at week 1 is 11 IBUs, and the average difference at week 10 is 8.5 IBUs.

5. Experiment #1: Analysis
5.1 Average Differences and Visual Analysis
The increase in IBUs caused by the use of pellets decreases from an average of 11.02 IBUs at week 1 to 8.50 IBUs at week 10.  This implies that whatever is causing this increase in IBUs, it does decay as the beer ages.  This pellet-based increase in IBUs decayed by a factor of 0.77 over the 10-week period (0.77 = 8.50/11.02).

It appears that the slope of the line changes between weeks 1 and 10 for both cones and pellets, with less of a difference at 10 minutes and more of a difference at 60 minutes, but the effect is subtle.  This change in slope is caused by the loss of IAA; a 10% loss of IAA will have less of an absolute effect on 20 IBUs than it will on 40 IBUs.  Because the data do not extend back to a steep time of 0, it is difficult to see if the vertical-axis offset of the lines changes with the age of the beer, which would correspond with a decrease in nonIAA concentrations.

In short, whatever is causing the increase in IBUs does decrease with age, and both IAA and nonIAA might decrease with age.  To get a more conclusive answer, we need to distill the data in this graph into a smaller set of numbers for easier comparison.

5.2 Model and Scaling Factors
We can use the technique described in Estimating Isomerized Alpha Acids and nonIAA from Multiple IBU Measurements to split the IBU value into estimates of (a) the concentration of IAA and (b) the concentration of other bitter substances measured with the IBU that are called nonIAA.  Since nonIAA are predominately oxidized alpha acids (oAA), we can use existing models of the other factors (polyphenols and oxidized beta acids) and focus on estimating the concentration of oAA.  (The assumption of oAA as the primary source of nonIAA differences between cones and pellets is examined in Experiment #2.  Even if this assumption is incorrect, the model uses a direct translation between the concentration of oAA and total nonIAA, and so the results of this experiment will still be valid for nonIAA although off by a constant scaling factor.)

We can use multiple IBU values from the same batch of beer, along with an equation that describes the isomerization of alpha acids as a function of time and temperature [Malowicki, p. 27], an equation that describes the IBU as a combination of IAA and nonIAA in the finished beer [Peacock, p. 161], and models of polyphenols and oxidized beta acids, to estimate two scaling factors: scalingIAA and scalingoAA.  The scalingIAA parameter is the scaling factor that accounts for losses of IAA during the boil, fermentation, and aging; scalingoAA is the scaling factor from the initial concentration of alpha acids in the wort to the concentration of oxidized alpha acids in the beer.  With scalingIAA and scalingoAA, as well as the volume of wort, weight of the hops, initial alpha-acid concentration, steep time, original gravity, and models of polyphenols and oxidized beta acids, we can map from IBU value to IAA and oAA concentrations, and vice versa.  The IBU values resulting from this analysis are listed in Table 1.

10
20
30
40
50
60
cones, week 1
(meas., estimate)
16.4,
15.7
21.2,
21.8
26.6,
27.0
31.3,
31.4
35.2,
35.3
39.0,
38.5
cones, week 10
(meas., estimate)
13.5,
12.8
17.5,
18.1
22.1,
22.6
26.6,
26.6
30.0,
30.0
33.2,
32.8
pellets, week 1
(meas., estimate)
26.0,
26.4
33.6,
32.6
37.8,
37.9
41.6,
42.5
46.9,
46.5
49.9,
49.8
pellets, week 10
(meas., estimate)
20.7,
21.4
27.9,
26.6
31.2,
31.2
33.6,
35.1
40.0,
38.4
40.5,
41.3

Table 1. Measured and estimated IBUs for each sample in each condition. Samples are identified by the duration of hop boiling, in minutes (column headings). The type of hops (cones or pellets) and the age of the beer are identified by row headings. Each cell in the table shows measured IBUs followed by estimated IBUs. Estimates are from the model described in Section 5.2.

5.3 Analysis of Cones Data
The analysis of IBU data of the beer made with hop cones and aged one week resulted in scalingIAA = 0.472 and scalingoAA = 0.067.  These results indicate that somewhat less than half of the isomerized alpha acids from this batch made it into the finished beer, and about 7% of the alpha acids added to the wort ended up as oxidized alpha acids in the beer.  The analysis of beer made with hop cones and aged 10 weeks resulted in scalingIAA = 0.414 and scalingoAA = 0.049.

From these results, we can estimate that IAA levels decayed by a factor of 0.877 over the 10 weeks (0.877 = 0.414/0.472), and nonIAA levels decayed by a factor of 0.731 (0.049/0.067).  The decrease over time attributed to pellet-specific factors (0.77 from Section 5.1) is closer to 0.73 than it is to 0.88, and so this suggests that the pellet-based increase in IBUs is more likely to be caused by oxidized alpha acids.

5.4 Analysis of Pellet Data
We can perform a similar analysis on the set of pellet data.  However, we don’t want to include the effect of the increase in IBUs caused by pellets in our analysis results, so when we estimate values for scalingIAA and scalingoAA, we add 11.02 IBUs to the model of week-1 data and 8.50 IBUs to the model of week-10 data.  (Or, equivalently, we can subtract 11.02 from the measured values from week 1 and 8.50 from the measured values at week 10.)  When this is done, the beer made with hop pellets and aged one week results in scalingIAA = 0.484 and scalingoAA = 0.059.  The beer made with hop pellets and aged 10 weeks results in scalingIAA = 0.411 and scalingoAA = 0.047.

From these results of pellet-based IBUs, we can estimate that IAA levels decayed by a factor of 0.849 over the 10 weeks (0.849 = 0.411/0.484) and nonIAA levels decayed by a factor of 0.797 (0.047/0.059).  While this difference between IAA and nonIAA degradation is smaller than that estimated for cones, the decrease over time attributed to pellets (0.77) is even slightly less than the estimated nonIAA decay factor for pellets (.797).  This indicates again that the pellet-based increase in IBUs is more likely to be caused by nonIAA compounds than by IAA.

The IAA scaling factor (scalingIAA), oxidized alpha acid scaling factor (scalingoAA), and root-mean-square (RMS) error resulting from this analysis are listed in Table 2.

IAA scaling factor
oAA scaling factor
RMS error
cones, week 1
0.472 0.067 0.429
cones, week 10
0.414 0.049 0.461
pellets, week 1
0.484 0.059 0.613
pellets, week 10
0.411 0.047 1.106

Table 2. Estimated IAA and oAA scaling factors, and the associated RMS error, for each condition.

5.5 Averaged Analysis
The results in this study rely on parameter estimation that is subject to errors in the model, in the “known” values used in this model (i.e. the concentration of alpha acids at the start of steeping), and in the measured IBU values.  The pellet-based decay factor (0.77) is somewhat higher than the estimated nonIAA factor for cones (0.73), and the pellet-based decay factor is somewhat lower than the estimated nonIAA factor for pellets (0.80).

Assuming that these differences in results for cones and pellets are due to errors in the “known” or measured values, we can average the IAA and nonIAA decay factors (or the scaling factors) to arrive at a more robust combined estimate.  This averaging yields an IAA decay factor of 0.86 and a nonIAA decay factor of 0.76.  From these averaged values, we can conclude that the increase in IBUs caused by pellets (with a decay factor of 0.77) is most likely due entirely to nonIAA.

6. Experiment #2: Experimental Overview and Methods
6.1 Overview of Experiment #2
Having concluded in Experiment #1 that the increase in IBUs is more likely to come from nonIAA than from IAA, Experiment #2 looked at which of the components that are collectively referred to as nonIAA might be responsible for the increase. While Experiment #1 modeled the nonIAA increase assuming oxidized alpha acids are the unknown scaling factor, it is possible that this assumption is not correct, and that (for example) oxidized alpha acids are constant while the concentration of polyphenols is actually responsible for the increase in IBUs.

Malt polyphenols can obviously not be responsible for a change in IBUs caused by the type of hops used, and oxidized beta acids have a negligible impact on IBUs when using well-preserved hops.  This leaves oxidized alpha acids and hop polyphenols as the possible contributors.  It is possible that, even though hop polyphenols normally contribute only a small amount to the IBU [e.g. Shellhammer, p. 177; Almaguer, p. 300], the pelletization process produces such an increase in soluble hop polyphenols that this increase can explain the IBU differences between cones and pellets.

In order to test this theory, Oregon BrewLab measured the polyphenol concentrations in beer made with varying concentrations of hop cones and varying concentrations of hop pellets.  While isomerized alpha acids do not increase linearly with an increase in alpha acids, polyphenols should not have a solubility limit at even fairly high hopping rates.  We can then plot the change in polyphenol levels as a function of concentration to determine (a) the concentration of malt polyphenols, (b) the rate of increase of hop polyphenols when using cones, (c) the rate of increase of hop polyphenols when using pellets, and (d) whether any differences in the polyphenol concentrations between cones and pellets might explain the observed increase in IBUs from pellets.

6.2 Methods for Experiment #2
The data for this experiment consisted of five batches of beer brewed on the same day, two batches using hop cones and the other three batches using hop pellets.  Batch A used 0.76 oz (21.68 g) of cone hops with AA rating 8.32%.  Batch B used 2.04 oz (57.81 g) of the same cone hops.  Batch C used 0.67 oz (18.89 g) of pellet hops with AA rating 9.55%.  Batch D used 1.78 oz (50.37 g) of the same pellet hops.  Finally, Batch E used 2.66 oz (75.55 g) of the same pellet hops.  These weights, when used with the expected volume of wort when hops were added and with the estimated alpha-acid ratings, were designed to result in initial alpha-acid concentrations of 150 ppm, 400 ppm, 150 ppm, 400 ppm, and 600 ppm for Batches A through E, respectively.  Therefore Batches A and C can be directly compared, and Batches B and D can be directly compared.

For each batch, I used 2.88 lbs (1.31 kg) of Briess Pilsen DME in 3.32 G (12.57 l) of water, yielding about 3.47 G (13.14 l) of wort with a specific gravity of about 1.036.  I did not adjust the water profile or pH, which resulted in a pre-boil wort pH of 5.77.

The hops used in this experiment were from the same source as in Experiment #1.  This experiment was conducted nine months after the first; during that time, the hops were stored at about −9°F (−23°C) in vacuum-sealed bags.  I added hops (i.e. started the steep time at 0) after the wort had been boiling for 5 minutes, to avoid the foam associated with the start of the boil.  Samples were taken every 10 minutes from the start of the hop addition, for a total steep time of 40 minutes (4 samples).  Each 15-oz (0.44-liter) sample was taken from the boil in a measuring cup and then transferred to an aluminum cup using a wire mesh sieve to remove larger hop particles.  The aluminum cup was placed in an ice bath and the contents were stirred to cool quickly.  Once cooled to 75°F (24°C), the sample was transferred to a sanitized, sealed, and labeled quart (liter) container.  I aerated each sample by vigorous shaking for 60 seconds, then added .009 oz (0.25 g) of Safale US-05 yeast.  After all samples were taken, the containers were cracked open to vent, and they fermented for nine days. After fermentation, I sent 4 oz (0.12 l) of each sample to Oregon BrewLab for IBU measurement.  The sample taken at 10 minutes of steep time was also analyzed by Oregon BrewLab for polyphenol concentration.

7. Experiment #2: Results
The measured polyphenol levels were, for Batches A through E respectively: 112 mg/L, 125 mg/L, 112 mg/L, 130 mg/L, and 141 mg/L.  Figure 3 shows these polyphenol concentrations plotted as a function of the estimated concentration of total hop matter in the wort at the time the sample was taken (10 minutes of steeping).  The cone polyphenol concentrations are shown with green points and connecting dashed lines, and the pellet concentrations are shown with red points and connecting dashed lines.  The cones data and pellets data were each fit to a linear function (referred to as “model” in Figure 3), which are plotted in lighter green and red with solid lines.

Figure 4 shows the measured IBU values from this experiment, with cones in green and pellets in red.  The average difference in IBUs between Batches A and C is 8.3 IBUs, and the average difference between Batches B and D is 10.1 IBUs.

Figure 3. Concentration of polyphenols as a function of the concentration of total hop matter. Data for cones are plotted in green; data for pellets are plotted in red. The raw data are shown with triangles and dashed lines. The best linear fit to the data is shown using solid lines.

Figure 4. Measured IBU values for the five batches of beer in Experiment #2. Values for hop cones are shown in green, and pellets are shown in red. The average difference between Batches A and C is 8.3 IBUs, and the average difference between B and D is 10.1 IBUs.

8. Experiment #2: Analysis
In Experiment #2, the results from cones indicate a malt polyphenol concentration of 104.20 mg/L (when the hop polyphenol concentration is zero), and the results from pellets indicate a malt polyphenol concentration of 102.84 mg/L.  On average, the results indicate that in this experiment the malt contributed 103.5 mg/L of polyphenols.  The model of polyphenols developed in The Contribution of Malt Polyphenols to the IBU predicts 97.66 mg/L from the specific gravity and boil time, which is within 6% of the measured values.  The model of IBUs developed in that blog post predicts 0.81 IBUs from the malt polyphenols, based on the specific gravity and wort pH.

We can use the slope of the lines in Figure 3 to estimate what percent of the weight of the hops comes from polyphenols.  First, we assume that 20% of polyphenols dissolve in wort [Forster, p. 124] and that there is a fermentation loss factor of 0.70 (estimated in The Contribution of Malt Polyphenols to the IBU and assuming the same loss factor for malt and hop polyphenols).  From those assumptions and the slope of the lines in Figure 3, the hop cone polyphenols are 3.1% of the weight of the hops, and the pellet polyphenols are 4.5% of the weight of the hops.  Both of these values are within published estimates that hop polyphenol levels are in the range from 2% to 6% of the weight of the hops [Shellhammer, p. 169; Hough et al., p. 422; Algazzali, p. 5; Verzele and De Keukeleire, p. 9].  In general, the hop pellets here demonstrate a 43% increase in polyphenol concentrations, compared with hop cones (0.00628 / 0.00439 = 1.43 or 43% increase).

We can then use the slope of the lines in Figure 3 to estimate the IBUs contributed by the hop polyphenols.  Ellen Parkin reports that “an increase of 100 mg/L of polyphenols was predicted to increase the [IBU] value by 2.2” [Parkin, p. 28], and so the increase in hop polyphenols in Figure 2 can be mapped to an increase in IBU levels using a conversion factor of 0.022 from concentration (in mg/L) to IBUs.  Using this conversion results in estimates of 0.17, 0.21, 0.46, 0.57, and 0.85 IBUs for Batches A through E, respectively.

Figure 4 shows the measured IBU values from this experiment.  The average difference between Batches A and C is 8.2 IBUs, and the average difference between Batches B and D is 10.1 IBUs.  The first point of interest is that the observed increase in IBUs from using pellets is at least an order of magnitude greater than the expected increase in IBUs caused by hop polyphenols.  This effectively rules out hop polyphenols as being the primary cause of the increase in IBUs observed with pellets.  The second point of interest is that even though the concentration of hops increased by a factor of 2.67 between Batches A and B and between Batches C and D, the IBUs associated with the use of pellets increased only from 8.2 to 10.1 (on average) with the increase in hop concentration, or a factor of 1.22.  This implies that the increase in IBUs associated with pellets is subject to a solubility limit somewhere between 150 ppm and 400 ppm.  Such a solubility limit is already expected with alpha acids, but is not expected with other auxiliary bittering compounds.  This implied solubility limit is consistent with the hypothesis that the increase in IBUs with pellets is caused by the production of oxidized alpha acids when hops are added to the kettle; this oAA production would be restricted by the same solubility limit that limits the isomerization of alpha acids.

9. Summary and Conclusion
The results of the first experiment indicate that the increase in IBUs associated with the use of pellets is caused by an increased concentration of auxiliary bittering compounds, not by increased availability of alpha acids that quickly become isomerized alpha acids.

Of the possible auxiliary bittering compounds (oxidized alpha acids, oxidized beta acids, hop polyphenols, and malt polyphenols), oxidized alpha acids (especially those produced during the boil [Algazzali, p. 17]) are the only likely candidate, with the increase in IBUs from pellets as a function of hopping rate consistent with an alpha-acid solubility limit.  Oxidized beta acids produced during the boil are highly unlikely because of their very low presence in finished beer when using well-preserved hops.  Hop polyphenols are estimated to contribute about an order of magnitude less to the IBU than observed differences, and the contribution of malt polyphenols is obviously unrelated.

Based on the results of these experiments, oxidized alpha acids appear to be the source of the increase in IBUs when using pellets.  Why would the use of pellets increase the concentration of oxidized alpha acids?  Maye et al. found that oxidized alpha acids make up less than 0.5% by weight of hop pellets before being added to wort [Maye, p. 24], which is not enough to explain the observed increase in IBUs.  However, the “creation of [oxidized alpha acids] occurs when hops are added to boiling wort” [Algazzali, p. 17].  The pelletization process ruptures the luplin glands [Hall, p. 58], and therefore the alpha acids of pellet hops have a much greater surface area (compared with cones).  It seems plausible that the oxidation of alpha acids that happens during the boil is limited by both the initial available surface area of the alpha acids and their solubility; in other words, only those alpha acids that are initially exposed to (and dissolve in) the boiling wort are quickly oxidized.  Therefore, the greater surface area of alpha acids in hop pellets allows more production of oxidized alpha acids during the boil, thereby increasing the IBU value.

10. Acknowledgements
I would, as usual, like to thank Dana Garves at Oregon BrewLab for the IBU and polyphenol analyses for these experiments.  The conclusions reached by these experiments would not be possible without the level of accuracy that Oregon BrewLab provides.

References

  • V. A. Algazzali, The Bitterness Intensity of Oxidized Hop Acids: Humulinones and Hulupones, Master of Science thesis (advisor: T. H. Shellhammer), Oregon State University, 2014.
  • C. Almaguer, C. Schönberger, M. Gastl, E. K. Arendt, and T. Becker, “Humulus lupulus – a story that begs to be told: A review,” in Journal of the Institute of Brewing, vol. 120, pp. 289-314, 2014.
  • R. Daniels, Designing Great Beers: The Ultimate Guide to Brewing Classic Beer Styles.  Brewers Publications, 2000.
  • J. Dierckens and M. Verzele, “Oxidation Products of Humulone and Their Stereoisomerism,” in Journal of the Institute of Brewing, vol. 75, pp. 453-456, 1969.
  • G. J. Fix and L. A. Fix, An Analysis of Brewing Techniques.  Brewers Publications, 1997.
  • A. Forster, “Influence of Hop Polyphenols on Beer Flavor,” in Hop Flavor and Aroma: Proceedings of the 1st International Brewers Symposium, ed. Thomas H. Shellhammer, Master Brewers Association of the Americas, 2009.
  • M. L. Hall, “What’s Your IBU,” in Zymurgy.  Special Edition, 1997.
  • J. S. Hough, D. E. Briggs, R. Stevens, and T. W. Young, Malting and Brewing Science. Volume 2: Hopped Wort and Beer. Springer-Science+Business Media, B. V., 2nd edition, 1982.
  • M. J. Lewis and T. W. Young, Brewing. Springer Science+Business Media, 2nd edition, 2001.
  • M. G. Malowicki, Hop Bitter Acid Isomerization and Degradation Kinetics in a Model Wort-Boiling System, Master of Science thesis (advisor: T. H. Shellhammer), Oregon State University, 2005.
  • J. P. Maye, R. Smith, and J. Leker, “Humulinone Formation in Hops and Hop Pellets and Its Implications for Dry Hopped Beers”, in MBAA Technical Quarterly, vol. 51, no. 1, pp. 23-27, 2016.
  • G. J. Noonan, New Brewing Lager Beer. Brewers Publications, 1996.
  • 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.
  • 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.
  • 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.
  • S. Srečec, T. Rezić, B. Šantek, and V. Marić, “Hop Pellets Type 90: Influence of Manufacture and Storage on Losses of α-Acids,” in Acta Alimentaria. Vol. 38, no. 1, pp. 141–147, 2009
  • M. Verzele and D. De Keukeleire, Chemistry and Analysis of Hop and Beer Bitter Acids.  Developments in Food Science 27.  Elsevier, 1991.

 

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Hop Cones vs. Pellets: IBU Differences

Abstract
Hop pellets are described as having greater utilization than hop cones.  The predicted amount of increase, however, varies quite a bit between different reports.  This blog post compares the IBUs from cones and pellets in a series of five experiments.  While the IBUs from pellets were found to be consistently higher than IBUs from cones, it seems that this increase in IBUs is not caused by an increase in the rate of isomerization (as is typically claimed), but by a greater concentration of bitter substances produced soon after a hop addition.  A separate blog post finds that these bitter substances are most likely not isomerized alpha acids, but probably oxidized alpha acids that are produced when hops are added to the boiling wort.  Furthermore, the amount of increase in IBUs seems to be dependent on the hop variety.   When controlling for the initial concentration of alpha acids, some hop varieties show very little increase (average 1.5 IBUs from 170 ppm of alpha acids), while others have a very large increase (average 10.8 IBUs from 170 ppm of alpha acids).  Because of this variety-dependent increase, predicting IBUs from hop pellets is even more challenging than predicting IBUs from hop cones. For hop cones, it is estimated that about 6% of the alpha acids added to the wort are quickly oxidized and survive into the finished beer.  (One ppm of oxidized alpha acids contributes about 0.7 IBUs.) Considering only the three hop varieties studied here, the increase in oxidized alpha acids from the use of pellet hops varies from a factor of 1.2 to a factor of 3.2. A rough (variety-independent) approximation for predicting IBUs from pellets is that the concentration of oxidized alpha acids produced during the boil doubles in beer made with pellets, from 6% to 12%.

1. Introduction: Reported Differences and IBU Models
1.1 Utilization
Hop utilization, U, is the ratio of the amount of isomerized alpha acids (IAA) 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) [1]

Utilization refers only to the relative amount of isomerized alpha acids, not to IBUs.  While IBUs can be considered roughly equivalent to the concentration of IAA as a quick rule of thumb, IBUs measure a number of bitter compounds in addition to IAA.  With short boil times, high hopping rates, low steeping temperatures, improperly-stored hops, and other factors, one can see significant differences between IBUs and the concentration of IAA.

1.2 Reported Differences Between Cones and Pellets
Hop pellets are almost always described as having greater utilization than hop cones [e.g. Daniels p. 78].  According to Michael Lewis and Tom 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].  Greg Noonan says that “with pelletized hops, ruptured and better-exposed lupulin glands give greater utilization” [Noonan, p. 154].

Expressing pellets as more efficient than whole hops, Noonan provides a pellet correction factor (in table form) that varies from 1.0 to 1.5, based on boil time and gravity [Noonan, p. 215].  Mark Garetz recommends a pellet correction factor of 1.10 for boil times up to 30 minutes, otherwise a correction factor of 1.0 [Garetz, p. 131, 141].  Hieronymus says that hop pellets are 10% to 15% more efficient than cones [Hieronymus, p. 188].  According to Michael Hall, Randy Mosher specifies a correction factor of 1.33 [Hall, p. 62].  This is a wide range of relative increase, from 0% to 50% according to Noonan, and from 0% to 33% according to other sources.

The purpose of this blog post is to get a better understanding of how large an IBU increase there is when using pellets and how this increase can be modeled.  A separate blog post looks at whether this increase is more likely to be the result of a greater concentration of isomerized alpha acids or an increase in other bittering compounds; it finds that the IBU increase is most likely caused by an increased concentration of oxidized alpha acids.

1.3 A Model of the Isomerization of Alpha Acids
Mark Malowicki [p. 27] provides a formula for the concentration of isomerized alpha acids (IAA) as a function of steep time (t, in minutes), temperature, and initial alpha-acid concentration ([AA]0, in ppm):

[IAA]wort = [AA]0 × (k1/(k2k1)) (ek1t-ek2t) [3]

where [IAA]wort is the concentration of isomerized alpha acids in the wort at time t, and e is the constant 2.71828.  The parameters k1 and k2 are two temperature-dependent rate constants.  At boiling, k1 = 0.0125 and k2 = 0.0031.

1.4 Modeling IBUs from Pellets with a Scaling Factor
Figure 1 shows theoretical IBU values based on several scenarios described in this section.  These IBU values are based on Val Peacock’s model of IBUs [Peacock, p. 157], in which

IBU = 5/7 × ([IAA] + [nonIAA]) [2]

where [IAA] is the concentration of isomerized alpha acids in the finished beer and [nonIAA] is the concentration of other bittering substances in the beer.  (This model is described in more detail in the blog post Estimating Isomerized Alpha Acids and nonIAA from Multiple IBU Measurements.)  In Figure 1, the black line shows theoretical IBU values from hop cones using Peacock’s model.  The concentration of isomerized alpha acids (IAA) increases from 8.5 ppm at 10 minutes into the boil to 35.0 ppm at 60 minutes (using the IAA model from Section 1.3 and a loss factor of 0.5), and the dotted gold line (constant at 5 IBUs) shows the contribution of nonIAA in this model.

The effect of pellets is usually expressed in the literature as a scaling factor [Hall, p. 62], for example a factor of 1.20 that is applied to the IBU value predicted for hop cones.  In this case, if an IBU model developed for hop cones predicts 30 IBUs, a pellet correction factor of 1.20 would yield 36 IBUs (36 = 30 × 1.20).  In Figure 1, the blue line shows theoretical IBU values predicted using a scaling factor of 1.20.  Because this scaling factor depends on the IBU value, smaller “cone” IBU values result in a smaller increase, and larger “cone” IBUs result in a larger increase.  For example, in Figure 1 the increase in IBUs is 2.2 IBUs at 10 minutes and 6.0 IBUs at 60 minutes.

Another way to model an increase in IBUs is with a scaling factor that depends on the concentration of isomerized alpha acids.  Because IBUs are correlated with [IAA], the net effect is similar.  In Figure 1, the dashed green line shows theoretical IBU values for pellets using a scaling factor of 1.25 applied to the concentration of IAA.

A third way to model an increase in IBUs is with a scaling factor that doesn’t depend on IBUs or [IAA], but on the concentration of nonIAA (which is also proportional to the total concentration of hops in the boil).  In Figure 1, the red line shows theoretical IBU values predicted by scaling the nonIAA concentration by a factor of 2.0.  In this case, every IBU value is simply increased by 5, because the concentration of nonIAA doesn’t vary with boil time.

Figure1

Figure 1.  Hypothetical IBU values based on (a) Peacock’s model (black line), (b) IBU scaling with a factor of 1.20 (blue line), (c) [IAA] scaling with a factor of 1.25 (dashed green line), (d) [nonIAA] scaling with a factor of 2.0 (red line).  The IBUs predicted when [IAA] is zero are shown with a dotted gold line.

2. Experimental Overview and Methods
2.1 Overview
Five experiments were conducted to look at the relative difference in IBUs between hop cones and pellets.  Within each experiment, two batches of beer were designed to be identical in all respects, except for the use of hop cones in one case (referred to as cones) and hop pellets in the other (referred to as pellets).  The five experiments looked at (a) three varieties of hops, (b) the impact of krausen, and (c) the age of the pellets.

In all experiments, the alpha-acid rating of the cones and pellets was comparable, and adjusted when necessary to yield the same concentration of approximately 170 ppm of alpha acids at the start of the hop addition.  For each batch, I took samples of wort at 10-minute intervals and quickly cooled them in an ice bath.  Each sample was fermented into beer and sent to Oregon BrewLab for IBU analysis.

This set of experiments yielded 37 pairs of IBU values, with the values within a pair being directly comparable in terms of hop variety, boil gravity, initial alpha-acid concentration, boil time, and fermentation conditions.

The first experiment used Citra hops, the second and third used Willamette hops, and the fourth and fifth experiments used Comet hops.  The Comet pellets were very fresh in the fourth experiment and about 2½ years old in the fifth experiment.

2.2 Procedures Common to All Experiments
Each experiment consisted of two batches brewed on the same day.  I used as large a batch size as I dared in my 10 G (38 l) kettle, in order to minimize the effect of measurement errors and evaporation rate.  I used 7.0 lbs (3.18 kg) of Briess Pilsen DME in 8.0 G (30.28 l) of water, yielding about 8.5 G (32 l) of wort with a specific gravity of about 1.037.  I did not adjust the water profile or pH; the local water here in Portland, Oregon has relatively low alkalinity and hardness.  This resulted in a pre-boil wort pH of about 5.70 to 5.80.

I added hops (i.e. started the steep time at 0) after the wort had been boiling for 5 minutes, to avoid the foam associated with the start of the boil.  The hops were boiled for a steep time of 60 to 90 minutes with the cover on the kettle (except for taking samples) to minimize evaporation and the resulting changes in specific gravity.  I did not use a mesh bag with the cones, because I think that it is more standard practice to have the hops freely floating in the wort.  I targeted an initial alpha-acid concentration of 170 ppm in order to not exceed the solubility limit of approximately 200 ppm at boiling, using an estimated volume of about 8.28 G (31.36 l) when adding the hops and the experiment-specific alpha-acid (AA) ratings.  For Experiment #1, an AA rating of about 14.1% for both cones and pellets translated into a hop addition of 1.333 oz (37.81 g).  For Experiments #2 and #3, an AA rating of about 5.05% for both cones and pellets translated into an addition of 3.724 oz (105.57 g).  For Experiment #4, an AA rating of about 10.0% for both cones and pellets translated into an addition of 1.880 oz (53.31 g).  For Experiment #5, an AA rating of 9.70% for cones and 8.76% for pellets translated into additions of 1.939 oz (54.96 g) and 2.147 oz (60.86 g), respectively.

Samples were taken every 10 minutes from the start of steeping.  Each sample (about 15 oz (0.43 l)) was taken from the boil in a measuring cup and then transferred to an aluminum cup using a wire mesh sieve to remove larger hop particles.  The aluminum cup was placed in an ice bath and the contents were stirred to cool quickly.  Samples were cooled below 140°F (60°C) within about 45 seconds.  Once cooled to 75°F (24°C), the sample was transferred to a sanitized, sealed, and labelled quart (liter) container.  I aerated each sample by vigorous shaking for 60 seconds, then added about .01 oz (0.28 g) of Safale US-05 yeast to target 750,000 viable cells per ml and degree Plato [Fix and Fix, p. 68].  (The process of taking a sample, cooling it, transferring it to a sanitized container, aerating, and pitching yeast took between 5 and 10 minutes.)  (For the “cones” condition in Experiment #5, 32-oz (0.95-liter) samples were taken and transferred into 1.6 quart (1.5 liter) sanitized containers for fermentation with 0.017 oz (0.48 g) of Safale US-05 yeast.)  After all samples were taken, the containers were cracked open to vent, and they fermented for nine to ten days.  For every experiment except Experiment #2, I swirled the samples every day to remove most of the krausen deposits on the sides of the containers (mixing the krausen back into the beer).  For Experiment #2, I let krausen deposits accumulate on the sides of the containers.  After fermentation, I sent 4 oz (0.12 l) of each sample to Oregon BrewLab for IBU measurement.

2.3 Hops in Experiment #1
The hop cones in Experiment #1 were Citra from Hops Direct.  The pellets were Citra from Yakima Valley Hops.  Both were from the same harvest year (2017), and were about 3 months old at the time of the experiment.  I purchased both the cones and the pellets soon after they became available and stored them in my freezer until the experiment.  I sent samples to both Alpha Analytics and Brew Laboratory for analysis within 3 weeks of the experiment.  Alpha Analytics used the spectrophotometric method ASBC 6A; Brew Laboratory used high-performance liquid chromatography (HPLC).  The package ratings and analysis results are listed in Table 1.  It can be seen that the analysis results are very consistent with the package ratings, except for the pellets result from Alpha Analytics.  Verzele and De Keukeleire note that “there are easily differences up to 15-20% in alpha acids content between and within bales of a single hop delivery” [Verzele and De Keukeleire, p. 331], and so even this “outlier” (13.3%) is well within the expected variation.  Because of the small number of samples, it is more appropriate to take the median than the mean for a representative value of the alpha acids.  Therefore, the alpha-acid rating on brew day was about 14.2% for cones and 14.0% for pellets.

Cones:
Package Rating
Cones:
Alpha Analytics
Cones:
Brew Laboratory
Pellets:
Package Rating
Pellets:
Alpha Analytics
Pellets:
Brew Laboratory
alpha acids 14.3% 14.2% 14.1% 14.0% 13.3% 14.0%
beta acids N/A 3.6% 3.4% N/A 3.9% 3.8%
HSI N/A 0.265 N/A N/A 0.293 N/A

Table 1. Results of hops analysis for Experiment #1, including alpha acids, beta acids, and (where available) the Hop Storage Index (HSI).

2.4 Hops in Experiments #2 and #3
In Experiments #2 and #3, I used Willamette hops from Yakima Chief Hops.  The cones  were from lot PR2-ZKUWIL5041 and the pellets were from lot P92-ZKUWIL5170, both about 2½ years old at the time of the experiment.  Analysis was performed by Brew Laboratory within two weeks of the experiment.  The package ratings and analysis results are listed in Table 2.  The alpha-acid rating on brew day was about 5.0% for both cones and pellets.

The reason for conducting Experiment #3 was that the results from Experiment #2 were so surprising to me (see Section 3) that I wanted to replicate the results.  In addition, in Experiment #2 I did not remove krausen deposits by swirling, and in Experiment #3 I made sure that fermentation conditions were the same as in Experiments #1, #4, and #5.

Cones:
Package Rating
Cones:
Brew Laboratory
Pellets:
Package Rating
Pellets:
Brew Laboratory
alpha acids 5.0% 5.0% 4.8% 5.1%
beta acids 3.8% 3.1% 4.0% 3.2%
HSI 0.252 N/A 0.298 N/A

Table 2. Results of hops analysis for Experiment #2, including alpha acids, beta acids, and (where available) the Hop Storage Index (HSI).

2.5 Hops in Experiments #4 and #5
In Experiment #4, I used Comet hops from Hops Direct.  (The customer service representative at Hops Direct was very helpful, and they were able to fulfill my request for both hop cones and pellets at close to 10% AA from the most recent (2018) harvest.) These hops were stored in vacuum-sealed packaging in my freezer.  Analysis was performed by Advanced Analytical Research (AAR Lab) within one week of the experiment.  I used an AA rating of 10.0% for both cones and pellets as the best estimates at the time of the experiment.

Cones:
Package Rating
Cones:
AAR Lab
Pellets:
Package Rating
Pellets:
AAR Lab
alpha acids 9.9% 10.8% 10.0% 9.84%
beta acids N/A 3.92% N/A 3.69%
HSI N/A 0.25 N/A 0.33

Table 3. Results of hops analysis for Experiment #4, including alpha acids, beta acids, and (where available) the Hop Storage Index (HSI).

In Experiment #5, conducted four months later, I used the same hop cones, but Comet pellets from YCH Hops (lot P92-ZLUCOM5216) that were 2½ years old at the time of the experiment.  Analysis was performed by AAR Lab with three weeks of the experiment.  Because of the age of the hops, I used the analysis results from AAR Lab (9.70% for cones and 8.76% for pellets) as the best estimates for the AA ratings at the time of the experiment.

Cones:
Package Rating
Cones:
AAR Lab
Pellets:
Package Rating
Pellets:
AAR Lab
alpha acids 9.9% 9.70% 9.5% 8.76%
beta acids N/A 3.17% 4.3% 3.22%
HSI N/A 0.35 0.326 0.42

Table 4. Results of hops analysis for Experiment #5, including alpha acids, beta acids, and (where available) the Hop Storage Index (HSI).

3. Results
The estimated room-temperature volume at the start of steeping was 8.28 G (31.36 liters) for all conditions and all experiments.  The average specific gravity after 10 minutes of steeping was 1.0384 (minimum 1.0378, maximum 1.0392).  The specific gravity after a 90-minute steep time was about 1.0404.  The small change in specific gravity during the boil (due to keeping the lid on the kettle) means that there is little difference between using the measured IBU values for analysis or normalizing these IBUs by the volume when the sample was taken.  For simplicity and clarity, the measured IBU values are used below.

Figures 2, 3, and 4 show the measured IBU values from Experiments 1 through 5.  The average difference in IBUs between cones and pellets is provided in each figure.

conesVsPellets-measuredIBUs-Exp1

Figure 2. Measured IBU values for Citra cones and pellets. The average difference is 5.2 IBUs.

conesVsPellets-measuredIBUs-Exp3-week1

Figure 3.  Measured IBU values for Willamette cones and pellets, in two separate experiments.   The average difference in Experiment #2 is 1.7 IBUs, and the average difference in Experiment #3 is 1.3 IBUs.

conesVsPellets-measuredIBUs-Exp4

Figure 4.  Measured IBU values for Comet cones and pellets, in two separate experiments.  In Experiment #4, both cones and pellets were recently harvested.  In Experiment #5, the pellets were 2.5 years old at the time of the experiment.  The average difference in Experiment #4 is 10.6 IBUs, and the average difference in Experiment #5 is 11.0 IBUs.

In Experiment #2, krausen was allowed to build up on the sides of the fermentation vessels, which  explains the overall lower IBU values when compared with Experiment #3.  (Another blog post looks at the impact of krausen on IBUs; it finds that krausen that adheres to the sides of the fermentation vessel can cause a significant decrease in IBUs.)

The increase in IBUs in Experiment #5 (compared with Experiment #4) may have been caused by the greater weight of hops used in this experiment.  A greater weight of hops in the same volume was used to target the same initial alpha-acid concentration of 170 ppm.  This may have resulted in greater IBU values because (a) the estimated decrease in alpha-acid content over time was greater than the actual decrease, and so the greater weight of hops over-compensated for the decrease in AA levels, (b) variation in AA levels in the hops, (c) the greater weight of hops increased the concentration of nonIAA compounds, thereby increasing IBU levels, or (d) some combination of all of these reasons.

4. Analysis
4.1 Visual Analysis of the Figures
It is easily seen in Figures 2, 3, and 4 that the increase in IBUs from the use of pellets is closer to the pattern associated with nonIAA scaling in Figure 1 than to the pattern of IAA or IBU scaling.  This constant offset is difficult to explain as a relative increase in IBUs or IAA (as illustrated in Figure 1), but very easy to explain as a relative increase in nonIAA concentration.  This may explain why Noonan used different utilization factors for cones and pellets at different steep times [Noonan, p. 215], resulting in a roughly constant increase for pellets regardless of steep time.

It is also clear that the increase in IBUs changes with the use of different hop varieties.  There is an average increase of 5.2 IBUs, 1.5 IBUs, and 10.8 IBUs for Citra, Willamette, and Comet pellets, respectively.  Within a variety, the increase in IBUs from cones to pellets is quite similar.  This topic is discussed more in Section 4.3.  Across varieties, the cone IBU values are much more similar than the pellet IBU values.  For example, at a 10-minute steep time, the cone IBU values are 14.0, 13.9, 14.8, 14.2, and 16.4 (standard deviation 0.9 IBUs) for Experiments #1 through #5, respectively, while the pellet IBU values are 18.7, 16.6, 16.3, 24.0, and 26.0 (standard deviation 4.0 IBUs).

4.2 Modeling Analysis
We can use the technique described in Estimating Isomerized Alpha Acids and nonIAA from Multiple IBU Measurements to split IBU values into estimates of (a) the concentration of IAA and (b) the concentration of other bitter substances measured with the IBU that are called nonIAA.  In brief, we can use multiple IBU values from the same batch of beer, along with (a) the equation in Section 1.3 that describes the isomerization of alpha acids as a function of time and temperature [Malowicki, p. 27] and (b) the equation in Section 1.4 that describes the IBU as a combination of IAA and nonIAA in the finished beer [Peacock, p. 161], in order to estimate two scaling factors: scalingIAA and scalingnonIAAhops.  The scalingIAA parameter is the scaling factor that accounts for losses of IAA during the boil, fermentation, and aging; scalingnonIAAhops is the scaling factor from concentration of total hop particles in the wort to the concentration of hop-related nonIAA in the beer (excluding malt-related nonIAA).  With scalingIAA and scalingnonIAAhops, as well as the weight of the hops, initial alpha-acid concentration, steep time, and original gravity, we can map from IBU value to IAA and nonIAA concentrations, and vice versa.

A separate blog post investigates the reason for the increase in IBUs associated with hop pellets, and concludes that this increase in IBUs is most likely caused by an increase in the concentration of oxidized alpha acids produced when the hops are added to the kettle.  Using models that predict IBUs due to malt polyphenols, hop polyphenols, and oxidized beta acids, we can change the scalingnonIAAhops parameter from a single parameter estimating the combined effect of all hop-related auxiliary bittering compounds to a parameter estimating the effect of only oxidized alpha acids, scalingoAA.

By searching over a large number of values of scalingIAA and scalingoAA to minimize the error on the cones batch of IBU values in Experiment #1, we get scalingIAA = 0.417 and scalingoAA = 0.057.  These results indicate that somewhat less than half of the isomerized alpha acids from this batch made it into the finished beer, and about 6% of the alpha acids were oxidized and survived into the finished beer.  These scaling factors yield a root-mean-square (RMS) error of 0.77 IBUs on the nine IBU values, with a maximum difference of -1.39 IBUs at 90 minutes.  We can do the same search for scalingIAA and scalingoAA using the set of nine values of pellets IBU data from Experiment #1.  In this case, we get scalingIAA = 0.406 and scalingoAA = 0.109, with an RMS error of 0.74 IBUs and a maximum difference of -1.42 IBUs at 50 minutes.  These results indicate that nearly the same percentage of IAA were produced and made it into the finished beer in both the cones and the pellets batches (i.e. hop utilization was the same in both cases), but that the concentration of oxidized alpha acids nearly doubled in the pellets batch.

Table 5 lists the IAA scaling factor (scalingIAA) for both cones and pellets in the five experiments.  It can be seen that the IAA scaling factor is very similar between cones and pellets for all five experiments, slightly higher in some cases and slightly lower in other cases.  (The average cone-to-pellet IAA ratio is 1.05.)  These small differences are probably due to measurement error, and it seems most likely that the IAA scaling factor is basically the same for both cones and pellets.  (The IAA scaling factor in Experiment #2 is expected to be lower than all of the others because the krausen was not mixed back into the beer in this experiment, resulting in greater loss of both IAA and oAA.)

Exp. #1
Exp. #2 Exp. #3 Exp. #4 Exp. #5
IAA scaling factor: cones
0.417 0.339 0.416 0.461 0.472
IAA scaling factor: pellets
0.406 0.307 0.359 0.465 0.476

Table 5. IAA scaling factors for cones and pellets in each experiment.  These values were estimated using the model described in Section 4.2.

We can then set the IAA scaling factor within each experiment to be the average of the IAA scaling factors for cones and pellets, and re-estimate the oAA scaling factors.  Table 6 shows the new estimates for IAA scaling factors (scalingIAA) and oAA scaling factors (scalingoAA).  Table 7 shows the measured IBU values and estimated IBU values using the model and scaling factors from Table 6.  The RMS errors are as follows: Exp #1 cones: 0.78, pellets 0.75; Exp #2 cones: 0.44, pellets 0.39; Exp #3 cones: 1.16, pellets: 1.17; Exp #4 cones: 0.91, pellets: 0.58; Exp #5 cones: 0.43, pellets: 0.62. The RMS error over all experiments is 0.80 IBUs.  Note that the average oAA scaling factor for cones estimated here (6.3%) is close to the value estimated in Section 8.2 of The Relative Contribution of Oxidized Alpha- and Beta-Acids to the IBU (5.9%).

Exp. #1
Exp. #2 Exp. #3 Exp. #4 Exp. #5
IAA scaling factor
0.4115 0.323 0.3875 0.463 0.474
oAA scaling factor: cones
0.059 0.072 0.071 0.046 0.066
oAA scaling factor: pellets
0.107 0.088 0.084 0.148 0.150

Table 6. Averaged IAA scaling factor and oAA scaling factors for cones and pellets in each experiment.

10 min
20 min
30 min
40 min
50 min
60 min
70 min
80 min
90 min
Exp 1: cones
(meas., est.)
14.0,
13.8
18.8,
19.1
24.2,
23.7
26.6,
27.6
31.6,
31.0
32.8,
33.8
36.3,
36.3
37.9,
38.3
41.6,
40.0
Exp 1: pellets
(meas., est.)
18.7,
19.1
24.2,
24.3
29.7,
28.8
32.1,
32.8
37.5,
36.2
39.5,
39.0
41.5,
41.5
42.4,
43.5
44.7,
45.3
Exp 2: cones
(meas., est.)
13.9,
14.1
17.5,
18.2
22.3,
21.8
25.0,
24.9
27.3,
27.6
30.3,
29.8
Exp 2: pellets
(meas., est.)
16.6,
15.8
19.9,
19.9
23.2,
23.5
26.4,
26.5
29.4,
29.2
31.0,
31.4
Exp 3: cones
(meas., est.)
14.8,
14.9
19.1,
19.9
23.7,
24.1
26.7,
27.8
30.4,
30.9
35.1,
33.6
34.8,
35.9
40.1,
37.8
Exp 3: pellets
(meas., est.)
16.3,
16.3
21.5,
21.2
27.6,
25.4
28.3,
29.0
33.2,
32.2
34.7,
34.8
35.1,
37.1
38.5,
39.0
Exp 4: cones
(meas., est.)
14.2,
13.5
19.7,
19.6
25.1,
24.8
28.6,
29.3
31.7,
33.2
35.9,
36.5
39.6,
39.3
43.3,
41.7
Exp 4: pellets
(meas., est.)
24.0,
24.5
31.4,
30.4
34.6,
35.6
40.2,
40.0
44.2,
43.8
46.7,
47.0
49.4,
49.8
52.4,
52.1
Exp 5: cones
(meas., est.)
16.4,
15.7
21.2,
21.7
26.6,
26.9
31.3,
31.4
35.2,
35.3
39.0,
38.6
Exp 5:
pellets
(meas. est.)
26.0,
26.6
33.6,
32.7
37.8,
38.0
41.6,
42.5
46.9,
46.5
49.9,
49.8

Table 7. Measured and estimated IBUs for each sample in each experiment. Samples are identified by the duration of hop steeping, in minutes (column headings). Experiments and condition (cones or pellets) are identified by row headings. Each cell in the table shows measured IBUs followed by estimated IBUs. Estimates are from the model described in Section 4.2.

The change in oAA factor between cones and pellets for each experiment is listed in Table 8, expressed as a ratio of pellets/cones.  It can be seen that these factors vary from an 18% to 222% increase for pellets compared with cones, and that this increase is approximately the same within a hop variety but different between varieties. For example, for Willamette the ratios 1.22 and 1.18 are very similar, and for Comet the ratios 3.22 and 2.27 are more similar to each other than they are to the ratios of other varieties.  The average ratios for each variety are 1.81, 1.20, and 2.74 for Citra, Willamette, and Comet, respectively.  Over all three varieties, the pellet-to-cone ratio is 1.9, representing an approximate doubling in the concentration of oxidized alpha acids in the finished beer.

Exp. #1
Exp. #2 Exp. #3 Exp. #4 Exp. #5
oAA pellet-to-cone ratio
1.81 1.22 1.18 3.22 2.27

Table 8. oAA pellet-to-cone ratios estimated for the five experiments. This ratio expresses the relative increase in oxidized alpha acids that contribute to the observed increase in IBUs with the use of hop pellets.

5. Predicting an Increase in IBUs
5.1 Variety-Specific Factors
The results from Sections 3 and 4 indicate that the increase in IBUs and oAA concentration that results from using hop pellets is dependent on the hop variety. We can check if any quantitative descriptions of these varieties might allow us to predict the amount of increase in oxidized-alpha acid concentration.

Table 9 lists a variety of quantitative descriptions of the three varieties used here. The alpha-acid and beta-acid levels are taken from the averages of cones and pellets in Section 2, and the other descriptions are taken from The Hops List [Healey]. Each cell shows the typical composition (in percent or ml/100g) and the approximate concentration used in these experiments. If a descriptor is associated with the oAA pellet-to-cone ratio, we would expect a correlation between the concentration of this descriptor and the oAA pellet-to-cone ratio for this variety. In other words, we are looking for concentration values that increase in order from Willamette to Citra to Comet. None of these descriptors show such a trend, meaning that we can not currently predict the oAA pellet-to-cone ratio from knowledge of the hop variety or characteristics.

alpha acid (%)
beta acid (%)
cohumulone total oil
storability
Citra
14.1%
170 ppm
3.67%
44 ppm
27.5%
47 ppm
2.25 ml/100g
3% v/v
75%
Comet
10.0%
170 ppm
3.5%
60 ppm
41%
70 ppm
1.98 ml/100g
3% v/v
49%
Willamette
5.0%
170 ppm
3.5%
120 ppm
32.5%
55 ppm
1.25 ml/100g
4% v/v
62%

Table 9. Quantitative descriptions of the three hop varieties used in these experiments. The descriptions are provided as both typical composition (in percent or ml/100g) and approximate concentration in these experiments (in ppm or %v/v). The exception is “storability,” which is the percent of alpha acids remaining after storage for six months at room temperature.

Another possibility is that there is a transformation (other than oxidation) which happens while pellets age in their nitrogen-flushed packaging, and this hypothetical transformation causes less of a pellet-based IBU increase with older hops. The purpose of Experiment #5 was, in fact, to test this hypothesis. In Experiment #1, the Citra pellets were about 3 months old at the time of the experiment, and the pellet-based IBU increase was moderate. In Experiments #2 and #3, the Willamette pellets were several years old at the time of the experiment, and the pellet-based IBU increase was minor. In Experiment #4, the Comet pellets were extremely fresh and the increase was quite large. Therefore, Experiment #5 used Comet pellets that were deliberately several years old at the time of the experiment. If the results of Experiment #5 showed an increase in IBUs similar to that of the Willamette hops, then this hypothesis of older pellet hops having less increase would have been supported. However, the results showed just as large an increase in IBUs as in Experiment #4, indicating that the age of the (properly stored and nitrogen flushed) pellets has no impact on the increase in IBUs.

This leaves us with measuring the variety-specific ratio for each variety of hops. With hundreds of available hop varieties (e.g. [Healey]), this is a nearly impossible task. The more practical but less accurate approach is to treat all hop varieties as having the same increase as the average of the three varieties studied here, i.e. an oAA pellet-to-cone ratio of about 1.9.  (It is also possible that there is no variety-specific increase, but that the differences in the ratios are due to differences in the pellet-production process at each manufacturer.  Checking this hypothesis would require further study of both variety-specific and manufacturer-specific pellets.)

5.2 Modeling an Increase in IBUs
As seen in Table 6, when using hop cones, about 6% of the alpha acids added to the wort are oxidized and survive into the finished beer.  For pellet hops, about 12% of the alpha acids added to the wort are oxidized and survive into the beer.  There is a factor of 0.9155 for scaling the light absorption at 275 nm from oxidized alpha acids to isomerized alpha acids, as seen in Figure 7 of Maye et al. [Maye, p. 25], and a scaling factor of 51.2/69.68 to convert the light absorption of isomerized alpha acids to IBUs [Peacock, p. 161].  Therefore, 1 ppm of oxidized alpha acids will produce 0.67 IBUs.  Let’s consider an example to see how to model the use of pellet hops.  If we have a beer made with 1.50 oz (42.52 g) of 10% AA hops boiled for 60 minutes in 5.50 gallons (20.82 liters) of wort (and ignoring evaporation), when we add the hops we have 204 ppm of alpha acids added to the wort (204 ppm = 0.10 × 42.52 g × 1000 / 20.82 l).  From the 0.150 oz (4.252 g) of alpha acids added to the wort, with hop cones we get 0.009 oz (0.2551 g) of oxidized alpha acids, or 12.25 ppm, in the finished beer (12.25 ppm = 4.252 g × 0.06 × 1000 / 20.82 l), increasing the IBU by 8.24 (0.673 × 12.25 ppm).  With hop pellets, we get 0.018 oz (0.5102 g) of oxidized alpha acids, or 24.51 ppm, increasing the IBU by 16.48.  These oxidized-alpha-acid IBUs are in addition to the IBUs from isomerized alpha acids (e.g. 30 IBUs) and the IBUs from malt and hop polyphenols (e.g. 2 IBUs), resulting in 40 IBUs for cones and 48 IBUs for pellets.  In this example, then, pellets demonstrate a 20% increase in IBUs compared with cones.

6. Summary and Conclusion
The IBU data from these five experiments showed an unexpected but consistent pattern: the increase in IBUs from pellets is constant over a range of steep times, instead of increasing with steep time.  It therefore seems that the increase in IBUs when using pellets is not caused by an increase in the rate of isomerization or availability of alpha acids, and should not be modeled with a multiplication factor applied to [IAA] or IBUs. Instead, this increase in IBUs can be modeled by an increase in the concentration of oxidized alpha acids produced during the boil, as discussed in a separate blog post.  The amount of increase appears to be dependent on the hop variety and is not easily predicted from characteristics within each variety.  Therefore, the most practical way to model this increase in IBUs is to treat the isomerization of alpha acids in the same way as hop cones, but to double the concentration of oxidized alpha acids ending up in the finished beer.

7. Acknowledgement
I greatly appreciate the high-quality IBU analysis provided by Dana Garves at Oregon BrewLab. This accuracy can be seen in the smooth and consistent shape of the IBU plots in Figures 2, 3, and 4.  Without such consistent accuracy, it would not be possible to draw meaningful conclusions from the data.

References

  • R. Daniels, Designing Great Beers: The Ultimate Guide to Brewing Classic Beer Styles.  Brewers Publications, 2000.
  • G. J. Fix and L. A. Fix, An Analysis of Brewing Techniques. Brewers Publications, 1997.
  • M. Garetz, Using Hops: The Complete Guide to Hops for the Craft Brewer. HopTech, 1st edition, 1994.
  • M. L. Hall, “What’s Your IBU,” in Zymurgy.  Special Edition, 1997.
  • J. Healey, The Hops List: 265 Beer Hop Varieties From Around the World. Healey, 1st edition, 2016.
  • S. Hieronymus, For the Love of Hops: The Practical Guide to Aroma, Bitterness, and the Culture of Hops.  Brewers Publications, 2012.
  • M. J. Lewis and T. W. Young, Brewing. Springer Science+Business Media, 2nd edition, 2001.
  • M. G. Malowicki, Hop Bitter Acid Isomerization and Degradation Kinetics in a Model Wort-Boiling System, Master of Science thesis (advisor: T. H. Shellhammer), Oregon State University, 2005.
  • J. P. Maye, R. Smith, and J. Leker, “Humulinone Formation in Hops and Hop Pellets and Its Implications for Dry Hopped Beers,” in Master Brewers Association of the Americas Technical Quarterly, vol. 53, no. 1, pp. 23-27, 2016.
  • G. J. Noonan, New Brewing Lager Beer. Brewers Publications, 1996.
  • 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.
  • M. Verzele and D. De Keukeleire, Chemistry and Analysis of Hop and Beer Bitter Acids.  Developments in Food Science 27.  Elsevier, 1991.

Alpha-Acid Solubility and pH

Abstract
In a previous blog post, Hopping-Rate Correction Based on Alpha Acid Solubility, I describe a model of hopping-rate correction (for predicting IBUs) based on estimated alpha-acid solubility at boiling.  In this model, solubility is not a limiting factor up to 240 ppm, after which solubility increases much more slowly than the concentration of alpha acids.  Any alpha acids that are above this solubility limit are quickly degraded and do not contribute to the isomerized alpha acids in the finished beer.  This model demonstrated a good fit to the available IBU data at a variety of hopping rates and boil times, but was only evaluated at a relatively high wort pH of about 5.75.  According to Spetsig, alpha-acid solubility is greatly influenced by pH.  In this post, I estimate alpha-acid solubility at boiling and a wort pH of 5.2, and show that the alpha-acid solubility limit at boiling does not seem to vary greatly with pH.

1. Introduction
The questions that this blog post attempts to answer are, “Does the solubility limit of alpha acids at boiling change significantly with pH, and if so, how?”  This section presents some background on pH in brewing and alpha-acid solubility.

1.1 Mash and Wort pH
Wort made from two-row malt and low-alkalinity water has a pH of about 5.8 [deLange; Palmer and Kaminski, p. 58], depending in part on the specific gravityBriess Pilsen Light DME, which I usually use for these experiments as an easy-to-use and consistent source of wort, shows the same pH characteristics as two-row malt mashed with low-alkalinity water.  The typical brewer should aim for a mash pH in the ballpark of 5.2 to 5.4 [Palmer and Kaminski, p. 60; Noonan, p 144; Fix, p 49; Troester citing Kunze (2007) and Narziss (2005)], although a pH as high as 5.8 is still acceptable [Troester].  The pH of room-temperature wort can therefore vary from about 5.2 to about 5.8.

1.2 pH and Wort Temperature
Palmer and Kaminski note that wort temperature affects pH in two ways: (a) the response of the pH probe changes as a function of temperature, and (b) the chemical activity of a solution (e.g. wort) also changes with temperature.  A pH meter with ATC will adjust for the first case, but not the second.  They say that “the pH of the wort at mash temperature (~65°C, 150°F) is known to be about 0.3 lower than the same wort when it is cooled to room temperature (~20°C, 68°F).  That is why brewers always refer to pH measurements at room temperature” [Palmer and Kaminski, p. 86].

In a previous blog post, I looked at how the pH of wort varies as a function of both temperature and room-temperature pH (i.e. a baseline pH).  I found that while the pH does decrease with temperature, the amount of decrease is less at lower room-temperature pH values.  If we extrapolate to boiling, a room-temperature pH of 5.75 will have a boiling pH of 5.33, and a room-temperature pH of 5.20 will have a boiling pH of 4.94.  This difference of 0.39 pH units, while less than the original difference of 0.55, suggests that there may still be pH-related differences in wort at boiling, although less than would be expected from the room-temperature pH difference.

1.3 pH and Utilization
It is generally thought that a lower wort pH will decrease utilization [e.g. Lewis and Young, p. 266; Askew 1964, p. 510].  (Utilization is the ratio of isomerized alpha acids (IAA) in the finished beer to the total alpha acids added.)  However, Mark Malowicki looked at IAA produced and degraded during boiling at pH values of 4.8, 5.2, 5.6, and 6.0, and found that “the level of iso-alpha concentrations … was nearly identical for all pH levels” in a buffer solution [Malowicki, p. 37], and that the use of maltose, glucose, or calcium in the solution had no impact on isomerization [Malowicki, p. 39].  He speculated that “the losses to trub would better explain the differences in utilization that are attributed to pH…, since rate of isomerization does not appear to be affected” [Malowicki, p. 41, emphasis mine].  Kappler et al. looked at the recovery rate of IAA, which is inversely related to the losses of IAA that occur during the boil.  They found that while there was a large change in IAA recovery rates between pH 4.0 (58% recovery) and pH 8.0 (95% recovery), the difference in recovery rates (and hence losses) between pH 5.0 and pH 6.0 were much smaller (80% and 86% recovery, respectively) [Kappler et al., p. 334].

In another blog post, I looked at utilization and IBUs as a function of pH in the range of 5.30 to 5.73, and found that the observed reduction in IBUs with lower pH could be modeled primarily by a loss of “auxiliary bitter compounds” (“ABC” or “nonIAA”), which are the bitter components other than IAA contributing to the IBU.

1.4 pH and Alpha-Acid Solubility
There is little previous work looking at pH and alpha-acid solubility.  In 1955, Lars-Olov Spetsig published estimates of alpha-acid solubility as a function of pH and temperature [Spetsig].  He used two temperatures in his measurements, 77°F (25°C) and 104°F (40°C), and from those temperatures he extrapolated to conditions at boiling.  He found that at room temperature and pH 5.2 the alpha-acid solubility is about 70 parts per million (ppm), and that this increases to about 200 ppm at pH 5.75 [Spetsig, p. 1423].   At boiling and pH 5.2, he estimates the solubility at 350 ppm, which increases to about 1000 ppm at pH 5.75 [Spetsig, p. 1423].

D. R. Maule noted that “when humulone was used at rates greater than 200 [ppm] the amount appearing as [alpha acids] and [iso-alpha-acids] on break increased at the expense of the amounts remaining in the wort” [Maule, p. 289].  He concluded that what is not actually adsorbed to the break “represents the difference between the amount of resin present and its solubility in wort under the conditions employed”  [Maule, p. 289].  This suggests an alpha-acid solubility limit close to 200 ppm at boiling with wort pH 5.7 [Maule, p. 287].

Malowicki studied alpha-acid solubility at room temperature and found, in general agreement with Spetsig, a limit of 90 ppm at pH 5.2 [Malowicki, pp. 54].  He also noted that the solubility “curve did not completely plateau, there was a distinct knee in the curve and break from linearity” [Malowicki, p. 53].   In a previous blog post, I noted the same trend with solubility estimates at boiling and pH 5.75, and estimated the solubility limit under these conditions as starting at approximately 240 ppm.

2. Approach
2.1 General Approach
I previously modeled alpha-acid solubility at boiling and a (room-temperature) wort pH of about 5.75 as gradually increasing with initial alpha-acid concentration according to the formulas

[AA]limit = [AA]limitMax × (1 − exp(slope × [AA]0))
slope = log(1 − ([AA]limitMin / [AA]limitMax)) / [AA]limitMin
[AA]limitMin = 240
[AA]limitMax = 490

where [AA]limit is the solubility limit (in ppm), [AA]0 is the concentration of alpha acids when hops are added to the wort (in ppm), slope is a parameter describing the slope of the function above the minimum solubility limit, [AA]limitMin is the smallest concentration at which some of the alpha acids do not dissolve (in ppm), and [AA]limitMax is the maximum concentration of dissolved alpha acids (also in ppm).  (Note: if [AA]0 is less than or equal to [AA]limitMin, then all alpha acids dissolve in the wort.)  In order to extend this model to be dependent on pH, the experiment described here estimates solubility as a function of initial alpha-acid concentration at pH 5.2, using the same general formula but different values for [AA]limitMin and [AA]limitMax.  This general structure of the formula can be used regardless of the pH value, and the two parameters can be made dependent on pH by interpolating between the parameter values at pH 5.2 and 5.75.  The result can then be a set of formulas for alpha-acid solubility that is dependent on both pH and initial alpha-acid concentration.

2.2 Estimating Solubility at pH 5.2
The approach used in this experiment was to create five conditions (i.e. five batches of beer) with different hop concentrations, all at a target pH of 5.2, with one condition having an alpha-acid concentration well below the expected solubility limit.  The wort from each condition was sampled after both 10 and 20 minutes of steep time.  All ten samples were fermented into beer.  The resulting IBU values were then fit to an equation that maps from initial alpha-acid concentration to IAA levels and another set of equations that map between IAA levels and IBU values.   This second of equations has two free parameters, as described below in Section 4.1.  Fitting was done by varying the two parameters of the solubility model ([AA]limitMin and [AA]limitMax) and the two free parameters of the IBU model to minimize the root-mean-square error between modeled and measured IBU values.   The general parameter-estimation technique for the IBU model is described in Estimating Isomerized Alpha Acids and nonIAA from Multiple IBU Measurements.

3. Experimental Methods and Data
I brewed five batches of beer for this experiment.  These five conditions were designed to be identical in all respects except for the concentration of alpha acids.

I created one large pool of wort from which all five conditions were brewed.  Targeting a boil gravity of 1.040, I used 7.0 lbs (3.175 kg) of Briess Pilsen DME in 8.0 G (30.28 l) of water, yielding about 8.52 G (32.25 l) of wort with a specific gravity of 1.0385.  I boiled this wort (uncovered) for about 5 minutes, and then cooled it with a wort chiller to less than 75°F (24°C), yielding 8.28 G (31.34 l) with specific gravity 1.040.  The pH of this wort was 5.75.  I then added phosphoric acid (a total of 6.75 tsp (33.27 ml)) in order to reach a target pH of 5.20 at 70°F (21°C).  Each condition started with 1.50 G (5.68 l) from this larger pool of wort.  Except for taking samples after 10 and 20 minutes of boiling, the wort was boiled with the cover on the kettle in order to minimize evaporation losses.

I used hops from a 1 lb (0.45 kg) bag of Citra HBC394 from Hops Direct that were purchased soon after harvest and stored in a vacuum-sealed bag in my freezer.  This bag had an alpha-acid rating on the package of 14.3%.  I had samples from this bag sent to two laboratories two months before the current experiment, which was conducted in February 2018.   Within three weeks following the current experiment, I had another two samples analyzed, with one laboratory the same and another one different.  Analysis results are shown in Table 1.  Because of the small number of samples, and because of the difficulty of a reliable alpha-acid estimate under the best of circumstances [Hough et al., p. 432; Verzele and De Keukeleire, p. 331], it is more appropriate to take the median than the mean for a representative value of the alpha acid rating.  It can be seen that the median alpha-acid rating is 14.2%, and the median beta-acid rating is 3.35%.  (The mean AA value of 14.3% is quite close to the median, fortunately.  I went a bit overboard with testing this time, which was probably an over-reaction to difficulties I encountered in the blog post Hopping Rate Correction Based on Alpha-Acid Solubility.)

Package Rating Alpha Analytics, within 8 weeks
Brew Laboratory #1, within 8 weeks
AAR Lab, within 3 weeks Brew Laboratory #2, within 3 weeks
alpha acids
14.3% 14.2% 14.1% 13.5% 15.5%
beta acids
N/A 3.6% 3.4% 3.3% 3.7%
HSI N/A 0.265 N/A 0.29 N/A

Table 1. Results of hops analysis, including alpha acids, beta acids, and (where available) the Hop Storage Index (HSI).

During the boil, I contained the hops in a large nylon coarse-mesh bag in order to not include large hop particles in my samples.  Previous experiments (from Brülosophy: 25 IBUs (bagged) vs. 27 IBUs (loose), and Four Experiments on Alpha-Acid Utilization and IBUs: 36 IBUs (bagged) vs. 37 IBUs (loose) and 34 IBUs (bagged) vs. 34 IBUs (loose)) have not shown a significant impact of a mesh bag on measured IBU values.  In order to maximize contact of the hops with the wort, I added brass weights (a total of 3.2 oz (90.7 g)) to the mesh bag so that the hops would be quickly submerged and hydrated.

For all conditions, I took samples of wort after 10 and 20 minutes of steep time.  Each sample (about 14 oz (0.41 l)) was taken from the boil and immediately transferred to an aluminum cup.  The sample was then placed in an ice bath and stirred to cool quickly.  Once cooled to 75°F (24°C), the sample was transferred to a sanitized, sealed, and labeled quart (liter) container.  I aerated each sample by vigorous shaking for 60 seconds, then added .01 oz (0.28 g) of Safale US-05 yeast (age 10 months) to target 750,000 viable cells per ml and degree Plato [Fix and Fix, p. 68].  After all samples were taken, the containers were cracked open to vent and they fermented.  After ten days of fermentation, I sent 4 oz (0.12 l) of each sample to Oregon BrewLab for IBU and original-gravity measurement.  The final gravity of all samples was about 1.0055 (minimum 1.0050; maximum 1.0060).

Table 2 provides data for each condition, including initial wort volume, weight of hops added, estimated initial alpha-acid concentration, and post-boil pH.  Tables 3 and 4 show the original gravity, post-boil volume, and measured IBUs from each condition at 10 minutes and 20 minutes, respectively.  Original gravity values were measured by Oregon BrewLab in degrees Plato, and I converted those values to specific gravity using an equation from Spencer Thomas.  The post-boil volume was estimated from the pre-boil volume, pre-boil gravity, and original gravity.  IBU values were measured by Oregon BrewLab.  Figure 1 shows the measured IBU values for the five conditions at 10 and 20 minutes of hop boiling time.

Condition A Condition B Condition C Condition D Condition E
initial wort volume
1.50 G / 5.678 l 1.50 G / 5.678 l 1.50 G / 5.678 l 1.50 G / 5.678 l 1.50 G / 5.678 l
weight of hops added
0.142 oz / 4.027 g 0.284 oz / 8.054 g 0.426 oz / 12.081 g 0.710 oz / 20.135 g 1.136 oz / 32.216 g
estimated initial alpha-acid concentration
100 ppm 200 ppm 300 ppm 500 ppm 800 ppm
post-boil pH
5.19 @ 67.5°F (19.7°C) 5.17 @ 57.3°F (14.1°C) 5.20 @ 70.5°F (21.4°C) 5.18 @ 61.6°F (16.4°C) 5.19 @ 57.3°F (14.1°C)

Table 2. Measured values for all five conditions.  Measurements include initial wort volume, weight of hops added, initial alpha-acid concentration, and post-boil pH (and temperature at which pH was measured).

Condition: A
B C D E
OG
1.0388 1.0380 1.0380 1.0380 1.0396
volume 1.50 G /
5.678 l
1.50 G /
5.678 l
1.50 G /
5.678 l
1.50 G /
5.678 l
1.50 G /
5.678 l
IBUs 5.6 11.7 13.6 19.3 25.6

Table 3. Values for each condition after 10 minutes steep time.  OG is the original gravity determined by Oregon BrewLab.  Volumes are based on the initial volume, pre-boil specific gravity, and OG.  IBU values were measured by Oregon BrewLab.

Condition: A
B C D E
OG
1.0388 1.0396 1.0396 1.0388 1.0396
volume 1.50 G /
5.678 l
1.50 G /
5.669 l
1.50 G /
5.669 l
1.50 G /
5.674 l
1.50 G /
5.678 l
IBUs 8.4 16.3 19.0 27.4 36.3

Table 4. Values for each condition after 20 minutes steep time.  OG is the original gravity determined by Oregon BrewLab.  Volumes are based on the initial volume, pre-boil specific gravity, and OG.  IBU values were measured by Oregon BrewLab.

solExp-Fig1-measuredIBUs

Figure 1.  Measured IBU values for the five conditions (different initial concentrations of alpha acids) at 10 and 20 minutes of hop boiling time.  The pre-boil pH in all conditions is 5.2.

4. Parameter Estimation Methodology
4.1 Estimating Parameters to Map between IBU and IAA
In order to estimate the alpha-acid solubility limit as a function of initial alpha-acid concentration, we need some way to translate between measured IBU values and estimated IAA concentrations.  Peacock [p. 157] provides just such a formulation in a general form:

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

where IBU is the measured IBU value, [IAA]beer is the concentration of isomerized alpha acids in the finished beer, and [nonIAA]beer is the concentration of other bittering substances that aren’t isomerized alpha acids (also in the finished beer).

The non-IAA compounds (also called “auxiliary bittering compounds” or “ABC”) include oxidized alpha acids (abbreviated as “oAA”; produced during hop storage and during the boil), oxidized beta acids (produced during hop storage), hop polyphenols, and malt polyphenols.  The oxidized alpha acids produced during the boil should be limited to the same extent that isomerization is limited.  To accomplish this, I used estimates of the concentrations of each of these substances based on malt and hop concentrations, and limited the amount of alpha acids available for oxidization using the solubility-limit model.  I left the percent of alpha acids that oxidize during the boil and remain after fermentation as a free parameter called scalingoAA.

We also need some way to translate between estimated IAA concentrations and the concentration of alpha-acids added to the wort.  Malowicki provides a way to estimate IAA in the wort from alpha acids:

k1(T) = 7.9×1011 e-11858/T
k2(T) = 4.1×1012 e-12994/T
[IAA]wort = [AA]0 × (k1(T)/(k2(T) − k1(T))) × (ek1(T)− ek2(T)t)

where k1(T) and k2(T) are empirically-derived rate constants, T is the temperature in Kelvin (i.e. 373.15K for boiling), e is the exponential function, and [AA]0 is the initial concentration of alpha acids in the wort.

That leaves us with needing to find a factor called scalingIAA, which maps between [IAA]wort and [IAA]beer, and the factor scalingoAA, which maps from the concentration of alpha acids in the wort to [oAA]beer.  A method for estimating these two factors is described in a blog post Estimating Isomerized Alpha Acids and nonIAA from Multiple IBU Measurements.

4.2 Estimating Solubility and the Impact of pH
The initial concentration of alpha acids, [AA]0, was reduced to the concentration of dissolved alpha acids using the two parameters of the solubility limit model and the concentration of alpha acids added to the wort.  The concentration of dissolved alpha acids was then used to estimate [IAA]beer and [nonIAA]beer, which was then used to estimate an IBU value.  The four parameters, scalingIAA, scalingoAA, [AA]limitMin, and [AA]limitMax were varied to minimize the error between estimated and measured IBU values.

If the function generated by [AA]limitMin and [AA]limitMax for wort pH 5.2 is not dramatically different from the function estimated at pH 5.75, then we can conclude that pH does not have a clear impact on alpha-acid solubility.  If the functions are quite different, then we can use linear interpolation of these two values between the two pH extremes to estimate the solubility limit at any pH.

5. Results
5.1 Caveat

Before looking at the results in detail, I will note that there were only two measured IBU values per condition in this experiment.  This is the minimum number for IBU parameter estimation, and the cost of a small number of data points per condition is greater uncertainty in the results.  If I were to re-do this experiment with the benefit of hindsight, I would take (at least) four samples per condition, for a total of 20 samples to fit to four parameters.  With more time and energy, I would repeat this experiment at a variety of wort pH levels, with a variety of AA ratings, and at a greater number of [AA]0 values.  Because of the small number of data points currently available, we can expect significant variability in the results and therefore only detect a relatively large effect of pH on alpha-acid solubility.

5.2 Results #1
The search for the four parameters yielded the following values: scalingIAA = 0.21, scalingoAA = 0.040, [AA]limitMin = 180, and [AA]limitMax = 660.  The root-mean-square (RMS) error was 0.67 IBUs.  These values for the solubility limit model result in solubility at pH 5.2 that is greater than the solubility  at pH 5.75.  We expect, from Spetsig’s analysis, that solubility should decrease as the pH decreases, and so these results contradict our expectations.  The solubility-limit model resulting from these parameters is shown with a solid blue line in Figure 2.  The solubility-limit model for pH 5.75 is shown with a dashed gray line.

5.3 Results #2
Because the initial results were unexpected, it is possible that the model has too many parameters (4) for the given number of data points used in analysis (10).  I therefore assumed that the alpha-acid concentration of Condition A is below the solubility limit, and used the technique in Section 4.1 to estimate scalingIAA and scalingoAA for this condition without a solubility-limit model.  This analysis yielded scalingIAA = 0.33 and scalingoAA = 0.015.  Searching for the solubility-limit parameters yielded similar results as in the first analysis, [AA]limitMin = 200 and [AA]limitMin = 680, but with RMS error 1.61.

5.4 Results #3
In an attempt to find a solubility limit that decreases as the pH decreases, I used the value of scalingIAA from Results #2 (0.33) but allowed scalingoAA to be a free parameter.  This parameter search yielded scalingoAA = 0.030, [AA]limitMin = 100, and [AA]limitMax = 520, with RMS error 1.25.  The solubility-limit model resulting from these parameters is shown with a solid green line in Figure 2.

Given the expectation that solubility should decrease as pH decreases, it can be seen that there is not an exceptionally large difference between the estimates at pH 5.2 (green line) and the estimate at 5.75 (dashed-gray line), and so there is no clear effect of pH on alpha-acid solubility.

estimated alpha-acid solubility at pH 5.2

Figure 2. Estimated alpha-acid solubility at pH 5.2 from Results #1 and Results #2 (blue line), at pH 5.2 from Results #3 (green line), and at pH 5.75 (dashed gray line).

6. Conclusion
The results from this experiment do not show a clear effect of wort pH on alpha-acid solubility at boiling.  This was surprising to me, because Spetsig estimated large differences in alpha-acid solubility at boiling as a function of pH [Spetsig, p. 1423].

Spetsig obtained his estimates by extrapolating from measurements at 77°F (25°C) and 104°F (40°C), so it’s possible that the extrapolation of alpha-acid solubility with temperature missed some non-linear changes of solubility with temperature.   As I noted in Section 1.2, the rate of change in pH as a function of temperature is different depending on the room-temperature pH.  Extrapolating these results to boiling results in the difference between room-temperature pH values of 5.20 and 5.75 being 0.39 pH units instead of 0.55 pH units.  Again, it is possible that this extrapolation of pH with temperature is missing some non-linearity as the temperature approaches boiling, and that the pH of boiling wort is similar regardless of whether the room-temperature wort is 5.75 or 5.20.

In the end, I don’t have a good explanation for why the data don’t show an effect of pH.  It is also quite probable that there were too few data points to robustly estimate the necessary model parameters.  Until there is more data on the topic, though, the most plausible explanation seems to be that wort pH does not significantly affect alpha-acid solubility at boiling.

I would like to thank Dana Garves at Oregon BrewLab for her analysis of the samples.  The consistency of the measured IBU values in Figure 1 demonstrates the high quality of her analysis.

References

  • H. O. Askew, “Changes in Concentration of α and β Acids and of Iso-Compounds on Heating Extracts of Hops in Aqueous Solutions and Unhopped Wort,” in Journal of the Institute of Brewing, vol. 71, pp. 10-20, 1965.
  • A. J. deLange, “Understanding pH and Its Application in Small-Scale Brewing − Part 1: Fundamentals and Relevance to Brewhouse Procedures,” in More Beer! Articles, Jul. 18, 2013.  URL: https://www.morebeer.com/articles/understanding_ph_in_brewing.  Accessed most recently Sep. 12, 2018.
  • G. Fix, Principles of Brewing Science. Brewers Publications, 2nd edition, 1999.
  • G. J. Fix and L. A. Fix, An Analysis of Brewing Techniques. Brewers Publications, 1997.
  • J. S. Hough, D. E. Briggs, R. Stevens, and T. W. Young, Malting and Brewing Science.  Volume 2: Hopped Wort and Beer.  Springer-Science+Business Media, B. V., 2nd edition, 1982.
  • S. Kappler, M. Krahl, C. Geissinger, T. Becker, M. Krottenthaler, “Degradation of Iso-alpha-Acids During Wort Boiling,” in Journal of the Institute of Brewing, vol. 116, no. 4, pp. 332-338, 2010.
  • M. J. Lewis and T. W. Young, Brewing. Springer Science+Business Media, 2nd edition, 2001.
  • M. G. Malowicki, Hop Bitter Acid Isomerization and Degradation Kinetics in a Model Wort-Boiling System, Master of Science thesis (advisor: T. H. Shellhammer), Oregon State University, 2005.
  • D. R. Maule, “The Fate of Humulone During Wort Boiling and Cooling”, in Journal of the Institute of Brewing, vol. 72, pp. 285-290, 1966.
  • G. J. Noonan, New Brewing Lager Beer.  Brewers Publications, 1996.
  • J. Palmer and C. Kaminski, Water: A Comprehensive Guide for Brewers. Brewers Publications, 2013.
  • 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.
  • L. O. Spetsig, “Electrolytic Constants and Solubilities of Humulinic Acid, Humulone, and Lupulone,” in Acta Chemica Scandinavica, vol. 9, pp. 1421-1424, 1955.
  • K. Troester, How pH Affects Brewing.  URL:  http://braukaiser.com/wiki/index.php/How_pH_affects_brewing.  Accessed most recently on Sep. 12, 2018.
  • M. Verzele and D. De Keukeleir, Chemistry and Analysis of Hop and Beer Bitter Acids, vol. 27, 1st edition, Elsevier,  ISBN 0-444-88165-4, eBook ISBN 9781483290867, 1991.

Estimating Isomerized Alpha Acids and nonIAA from Multiple IBU Measurements

Abstract
The IBU measures more than just the concentration of isomerized alpha acids (IAA); it includes concentrations of other bittering substances, called nonIAA.  (These substances may also be referred to as “Auxiliary Bittering Compounds”, or ABC).  In trying to get a better understanding of the factors that contribute to the IBU, we would like to estimate the IAA and nonIAA concentrations separately.  It is possible to estimate IAA and nonIAA concentrations in beer from multiple IBU measurements, as long as the conditions in each batch are (nearly) identical except for the concentrations of hops and/or steep times.  This post explains this estimation technique, which I use in a number of blog posts, including Four Experiments on Alpha-Acid Utilization and IBUs, Hopping-Rate Correction Based on Alpha-Acid Solubility, The Effect of pH on Utilization and IBUs, Alpha-Acid Solubility and pH, and The Relative Contribution of Oxidized Alpha- and Beta-Acids to the IBU.

1. Background
1.1 What’s in an IBU?
The International Bitterness Unit (IBU) estimates the concentration of bitter substances in beer, including isomerized alpha acids (IAA) and the other bitter substances that are referred to collectively as “nonIAA” [Peacock, p. 161].  The nonIAA substances include oxidized alpha acids, oxidized beta acids, hop polyphenols, and malt polyphenols.  Measuring the IAA concentration is best done using HPLC, but such tests are of limited availability and great expense.  There is no standardized measurement technique for the concentration of all nonIAA substances.  Therefore, it would be useful to be able to estimate IAA and nonIAA concentrations from the IBU measurement.

1.2 A General Description of IBUs
The IBU is defined in terms of the amount of infrared light absorbed when passing through a sample of acidified beer, but Val Peacock provides a second definition of the IBU in terms of the concentrations of various substances [Peacock, p. 157]:

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

where IBU is the measured IBU value, [IAA]beer is the concentration of isomerized alpha acids in the finished beer (the brackets [] denoting concentration, and the subscript “beer” denoting the type of solution), and [nonIAA]beer is the concentration of other bittering substances that aren’t isomerized alpha acids (also in the finished beer).  Concentrations are expressed in parts per million, or ppm.  The derivation of this formula is outlined in A Summary of Factors Affecting IBUs.

1.3 A Refinement of the General Description of IBUs
We can expand upon Peacock’s general formula to (a) replace 5/7 with the more precise ratio 51.2/69.68 [Peacock, p. 161], (b) express [IAA]beer as the concentration of IAA produced during the boil multiplied by a scaling factor that accounts for the losses of IAA during the boil, fermentation, and aging, (c) express [nonIAA]beer as the concentration of hops in the wort multiplied by a scaling factor that maps from this concentration to the concentration of hop-related nonIAA in the finished beer, and (d) account for the contribution of malt polyphenols to the IBU as a factor separate from the hop-related nonIAA, creating [nonIAAhops] and [nonIAAmaltPP], both of which are concentrations in the finished beer.  We want to separate out the malt factor because the malt polyphenol concentration will be independent of the amount of hops added, but we will assume that there is a linear relationship between amount of hops added and the hop-related nonIAA concentration.

In particular, we can define [IAA]wort as the concentration of IAA in the wort during the boil and define a scaling factor, scalingIAA, that relates [IAA]wort to [IAA]beer. We can also define another scaling factor, scalingnonIAAhops, that relates the concentration of hops in the wort (in ppm) to the concentration of hop-related nonIAA in the finished beer.  (Both of these factors may vary from batch to batch, depending on the various losses and the concentrations of (oxidized) alpha- and beta-acids in the hops.  However, as long as we focus on batches of beer in which these conditions should be the same, then the scaling factors should also be the same for these batches.)  The contribution of malt polyphenols to the IBU, expressed as IAA-equivalent units, can be modeled as

[nonIAAmaltPP]beer = ((OG − 1.0) × 19.0) × ((2.477 × (5.75 − pH)) + 1.0) × (69.68/51.2)

where OG is the original gravity and pH is the post-boil pH, as described in the blog post The Contribution of Malt Polyphenols to the IBU.   The scaling by 69.68/51.2 converts from IBU units to IAA-equivalent parts per million.  These adjustments give us the following formulas as a modification to Peacock’s original formula:

[IAA]beer = [IAA]wort × scalingIAA
[nonIAAhops]beer = [hops]wort × scalingnonIAAhops
IBU = 51.2/69.68 × ([IAA]beer + [nonIAAhops]beer + [nonIAAmaltPP]beer)

where [IAA]wort is the total concentration of IAA produced during the boil, and scalingIAA is the scaling factor that accounts for losses of IAA during the boil, fermentation, and aging.  Also, [nonIAAhops]beer is the hop-related contribution to nonIAA in the finished beer, [hops]wort is the concentration of hops in the wort (in ppm), and scalingnonIAAhops is the scaling factor from concentration of total hop particles to the concentration of hop-related nonIAA in the beer.  The Peacock formula is mostly unchanged, except for the revised ratio, the specification of [nonIAAhops]beer that is specific to hops, and the addition of [nonIAAmaltPP]beer as a separate nonIAA component.

1.4 Conversion of Alpha Acids to Isomerized Alpha Acids
Mark Malowicki wrote his thesis on the conversion of alpha acids into isomerized alpha acids [Malowicki].  In this work, he developed a formula to predict the concentration of IAA from temperature, time, and the initial concentration of alpha acids:

k1(T) = 7.9×1011 e-11858/T
k2(T) = 4.1×1012 e-12994/T
[IAA]wort = [AA]0 × (k1(T)/(k2(T) − k1(T))) × (ek1(T)− ek2(T)t)

where k1(T) is a rate constant for the conversion of alpha acids into isomerized alpha acids, k2(T) is a rate constant for the conversion of isomerized alpha acids into degradation products, T is the temperature (in degrees Kelvin), [AA]0 is the initial concentration of alpha acids (in ppm), and t is time (in minutes) [Malowicki, pp. 25-27].

Malowicki found that this conversion was not influenced by pH or the presence of maltose, glucose, or calcium [Malowicki, p. 39, p. 44].  It’s generally believed that wort gravity [Kappler, p. 335; Lewis and Young, p. 266], pH [Kappler, p. 334; Lewis and Young, p. 266], and calcium [McMurrough, p. 104] can have a significant effect on utilization (the ratio of IAA in finished beer to alpha-acids added to the wort).  Malowicki’s results imply that any of these effects on utilization are due to losses of IAA or nonIAA, not to a different rate of isomerization.  Therefore, we can use his equations to compute [IAA]wort in the modified Peacock equation from the initial alpha-acid concentration, time, and temperature; any other effects on utilization will be reflected in scalingIAA and/or scalingnonIAAhops.

One thing that will affect the production of IAA, other than time and temperature, is the solubility limit of the alpha acids.  I believe that alpha acids in hot wort and above their solubility limit are quickly degraded and do not isomerize, although this has not yet been independently confirmed.  I’ve estimated this solubility limit as starting at approximately 240 ppm.

1.5 Production of nonIAA
The nonIAA contribution to the IBU consists of oxidized alpha acids, oxidized beta acids, hop polyphenols, and malt polyphenols.  The oxidized alpha and beta acids are created during aging of the hops (in the presence of oxygen) [e.g. Algazzali, pp. 12-13, 15; Garetz] and oxidized alpha acids  are produced during the boil [Spetsig 1968, p. 350; Stevens and Wright, p. 500].  The oxidized alpha and beta acids are very soluble in beer (with the oxidized alpha acids more soluble than IAA) [Maye et al., p. 23; Algazzali, p.16], and at least some of the polyphenols are soluble, especially in hot water [Forster, p. 124].  Hough et al. indicate that that the oxidized beta acids are fairly stable in the boil [Hough et al., p. 488-489].  The rate of oxidation of alpha acids during the boil is less clear, but “[oxidized alpha acid] formation on wort boiling can be one of the first things to happen in the complex chemistry of humulone isomerization” [Dierckens and Verzele, p. 454].  All of this suggests that the nonIAA levels may be fairly stable throughout the boil; in other words, there may be no significant time dependency for the production of nonIAA beyond the first five minutes of the boil.  The high level of solubility of the oxidized alpha and beta acids implies that (after oxidation) we don’t need to worry about a solubility limit as we add more and more hops.

2. Methods
We have five parameters in the formula for IBUs but only two unknown values: scalingIAA and scalingnonIAAhops.  The other three parameters, [IAA]wort, [hops]wort, and [nonIAAmaltPP]beer, can be computed or estimated from the known values of steep temperature, steep time, weight of the hops, alpha-acid rating, volume, original gravity, and post-boil pH.  (Other factors (such as beer age or krausen deposits) may affect losses of IAA and nonIAA, and will therefore be reflected in these two scaling factors.)

In theory, we can obtain scalingIAA and scalingnonIAAhops from only two measured IBU values using the technique described below.  In practice, we want more values, because errors or any source of unexpected variability in the measured IBUs will become errors in our IAA and nonIAA estimates.  The more measured IBU values we have, the better we can average out the error in the IAA and nonIAA estimates.  As one example of unexpected variability, consider the case in which we brew two batches of beer that should be identical in all respects, including taking hops from the same bag.  As Verzele and De Keukeleire note, “there are easily differences up to 15-20% in alpha acids content between and within bales of a single hop delivery” [Verzele and De Keukeleire, p. 331].  Small samples of hops from the same source can therefore have relatively large differences in alpha-acid content.  In this example, maybe one beer has hops with 14.5% alpha acids and the other beer has 16.0% alpha acids.  This is a 10% relative difference, well within the 20% possible variation.  In this case, we can get 30.0 IBUs from the first batch and 32.4 IBUs from the second.  While a difference of 2.4 IBUs is not large in terms of IBU prediction, if we’re unable to account for this variability (because we assume that samples of hops from the same source all have the same alpha-acid rating), then our IAA and nonIAA estimates will be impacted.  The best approach, if possible, is to take multiple samples from the same batch at different boil times, cooling and fermenting them with the same procedure.  This approach guarantees the same weight of hops, alpha-acid rating, and initial gravity; one can then account for the differences in volume and gravity that occur over time.  (Using different hops is generally not recommended, because scalingnonIAAhops may vary in this case due to storage conditions.)

If we have M measured IBU values (from the same batch of beer at different boil times, or from different batches of beer with different additions of the same hops), we have M equations (one IBU equation at a specific steep time or hop concentration per measured value) and two unknowns.  Methods for solving such a problem usually aim to minimize the mean-squared error, where the individual error is defined as the difference between the measured and predicted values:

errork = measuredIBUkpredictedIBUk
errorMSE = Σ(errork2) / M

where k is one of the M measured conditions, and we want to minimize errorMSE.  The root-mean-square (RMS) error is often reported, which is the square root of errorMSE:

errorRMS = (errorMSE)

(The RMS error is conceptually similar to the expected difference between the measured and predicted values (i.e. the average absolute difference), but it penalizes outliers more.)

Because we have only two parameters to solve for, the easiest way to find the minimum mean-squared error is to do a brute-force search over all possible values of scalingIAA and scalingnonIAAhops with a resolution of 0.001.  If we search over all values from 0.0 to 1.0 with an increment of 0.001, we have one million combinations (1000 for scalingIAA and 1000 for scalingnonIAAhops).  This is quite reasonable given the current speed of computers; it takes less than 10 seconds per condition k using a slow scripting language, and less than a second with compiled code.  (In practice, because scalingnonIAAhops tends to be very small, I search from 0.0 to 0.1 in increments of 0.0001, which takes the same amount of time.)

Once we have an estimate for scalingnonIAAhops, we can map from measured IBU to estimated nonIAA and IAA concentrations in beer with the formulas:

[nonIAA]beer = ([hops]wort × scalingnonIAAhops) + [nonIAAmaltPP]beer
[IAA]beer =  ((69.68/51.2) × IBU) − ([nonIAA]beer + [nonIAAmaltPP]beer)

where [nonIAA]beer is the total concentration of nonIAA in the finished beer, including contributions from both hops and malt.  (We don’t need scalingIAA unless we want to estimate the IAA concentration in the wort.)

3. Implementation of the Parameter Estimation Process
The most complete way to show the details of the estimation process is to provide pseudo-code that illustrates each step of parameter estimation.  Even if you’re not familiar with programming, this code may still provide some insight into the step-by-step processes.  The programming here isn’t tricky; the most complicated things are a “for” loop, an expression of the form “x = x + y“, and the assumption that there is an array of values for each condition (measured IBU, weight of hops, volume of wort, alpha-acid rating, OG, and steep time), with an index into the array denoted by square brackets, e.g. OG[1] is the first original gravity value in the list of M conditions.  The symbols “//” denote a comment that has no effect on processing.

// units: weight is in grams, volume is in liters, temperature is in Kelvin
//     (373.15=boiling), and the AA rating is in the range from 0.0 to 1.0
//
// key programming concepts: 
//     '{' and '}' denote a block of code
//     ';' marks the end of an instruction
//    'for' loop : a loop with initial, final, increment, and code blocks 
//     x = x + y : the value currently in x is added to y; x is updated
//     values of an array are indexed with square brackets, from 1...M
//     pow(x,y) computes x to the power of y
//     exp(x) computes the e to the power of x

temperature = 373.15;
k1 = 7.9 * pow(10,11) * exp(-11858/temperature);
k2 = 4.1 * pow(10,12) * exp(-12994/temperature);
minimumError = 1000000.0; 
for {IAAscale = 0.0} {IAAscale <= 1.0} {IAAscale = IAAscale + 0.001} {
  for {nonIAAscale = 0.0} {nonIAAscale <= 0.1} {nonIAAscale = nonIAAscale + 0.0001} {
    err_sum = 0.0;
    for {k = 1} {k <= M} {k = k+1} {
      AA0         = AArating[k] * hopsWeight[k] * 1000.0 / volume[k];
      IAA_wort    = AA0 * (k1/(k2-k1)) * (exp(-1.0*k1*time[k]) - exp(-1.0*k2*time[k]));
      IAA_beer    = IAA_wort * IAAscale;
      hopsConcent = hopsWeight[k] * 1000.0 / volume[k]; 
      nonIAA_beer = hopsConcent * nonIAAscale;
      maltPP_beer   = (((OG[k] - 1.0) * 1000.0) * 0.025) * (69.68/51.2); 
      IBU_predicted = (51.2/69.68) * (IAA_beer + nonIAA_beer + maltPP_beer);
      diff = IBU_measured[k] - IBU_predicted;
      err  = diff * diff;
      err_sum = err_sum + err;
    }
    if {err_sum < minimumError} {
      minimumError = err_sum;
      bestIAAscale = IAAscale;
      bestNonIAAscale = nonIAAscale;
    }
  }
}
err_RMS = sqrt(minimumError / M);
print "best IAA scale factor: ", bestIAAscale;
print "best nonIAA hops scale factor: ", bestNonIAAscale;
print "with RMS error: ", err_RMS;

From the arrays of M values, this code will compute the values of scalingIAA and scalingnonIAAhops that minimize the mean-squared error between the measured IBU values and predicted IBU values, and print out the resulting values and error.

4. Example of Mapping IBU to IAA (and Vice Versa)
As one example of how this mapping works, let’s say we brew several batches of beer with different amounts of hops and steep times, using 14.1% AA hops.  From these batches (including measured IBU values) and the above code, we estimate scalingIAA as 0.310 and scalingnonIAAhops as 0.00379.  In one batch, we added 8.054 grams of hops to 5.678 liters of wort, from which we can compute the initial alpha-acid concentration and total hop concentration:

[AA]0 = AArating × weight × 1000 / volume
[AA]0 = 0.141 × 8.054 × 1000 / 5.678 = 200.00 ppm

[hops]wort = weight × 1000 / volume
[hops]wort = 8.054 × 1000 / 5.678 = 1418.457 ppm

(I’m using only metric units here to simplify the presentation.  If you’re using British Imperial units (ounces, gallons, etc.), it’s convenient to convert those to metric first.)  The original gravity after a 20-minute steep time is 1.038 and pH 5.23, and we can use that to estimate the IBUs derived from malt polyphenols:

[nonIAAmaltPP]beer = ((OG − 1.0) × 19.0) × ((2.477 × (5.75 − pH)) + 1.0) × (69.68/51.2)
[nonIAAmaltPP]beer = 0.722 × 2.288 × 1.361 = 2.248 ppm

Let’s say that this batch has a measured IBU value of 15.0 after a 20-minute steep time.  From this information, we can compute the nonIAA concentration,

[nonIAA]beer = ([hops]wort × scalingnonIAAhops) + [nonIAAmaltPP]beer
[nonIAA]beer = (1418.457 × 0.00379) + 2.248) = 7.624 ppm

and the IAA concentration:

[IAA]beer =  ((69.68/51.2) × IBU) − (([hops]wort × scalingnonIAAhops) + [nonIAAmaltPP]beer)
[IAA]beer = ((69.68/51.2) × 15.0) − ((1418.457 × 0.00379) + 2.248) = 12.790 ppm

The hop-derived nonIAA concentration is 5.376 ppm, and the nonIAA concentration from malt is 2.248 ppm; their sum is [nonIAA]beer, or 7.624 ppm.  We can check the accuracy of our math by computing the IBU from the IAA and nonIAA concentrations.  By definition, the IBUs will be equal to the sum of the IAA and nonIAA concentrations, multiplied by about 5/7:

IBU = 51.2/69.68 × ([IAA]beer + [nonIAA]beer)
IBU = 51.2/69.68 × (12.790 + 7.624) = 15.0

In this case, the isomerized alpha acids make up 62.7% of the IBU (because 12.790 / (12.790 + 7.624) = 0.6265).  Even though the hops have been well preserved, the steep time of only 20 minutes reduces the IAA contribution to less than that of a 1960’s beer with more oxidized hops and a 90-minute boil [Peacock, p. 161].

If we don’t know the measured IBU value but we do know the scaling factors, we can estimate the IBU using scalingIAA.  In this case, instead of computing [IAA]wort from IBU, we can estimate it using Malowicki’s equations (with T = 373.15 degrees Kelvin (boiling) and t = 20 minutes) to determine [IAA]wort, and then apply the scaling factor to determine [IAA]beer:

[IAA]wort = [AA]0 × (k1(T)/(k2(T) − k1(T))) × (ek1(T)− ek2(T)t)
[IAA]wort = 200.0 × (0.0125/(0.0031 − 0.0125)) × (e–0.0125×20.0 − e–0.0031×20.0)
[IAA]wort = 200.0 × −1.328 × (e–0.250− e–0.062)
[IAA]wort = 200.0 × −1.328 × (0.779 − 0.940) = 42.842

[IAA]beer = [IAA]wort × scalingIAA
[IAA]beer = 42.842 × 0.310 = 13.281

IBU = 51.2/69.68 × ([IAA]beer + [nonIAA]beer)
IBU = 51.2/69.68 × (13.281 + 7.624) = 15.36

In this particular example, the estimated IBU value (15.36) is very close to the measured value (15.0).

5. Estimating Each Component of nonIAA
We have considered the hop-derived fraction of [nonIAA]beer as a single source of bitterness.  In fact, the hop-derived auxiliary bittering compounds are composed of oxidized alpha acids, oxidized beta acids, and hop polyphenols.  If we use available models of the concentration of hop polyphenols and oxidized beta acids when using well-preserved hops, we can search for the fraction of alpha acids that oxidize during the boil and survive fermentation, instead of the more general scalingnonIAAhops.   In this case, we have:

[nonIAA]beer = [PPhops]beer × scalePPhops + [oAA]beer × scaleoAA + [oBA]beer × scaleoBA + [nonIAAmaltPP]beer

where, as described in the blog post The Relative Contribution of Oxidized Alpha- and Beta-Acids to the IBU,

[PPhops]beer = 0.04 × 0.20 × 0.70 × W × 1000 / V
scalePPhops = 7/5 × 0.022 = 0.0308
[oAA]beer = lossFactoroAA × AA × W × 1000 / V
scaleoAA = 0.0130/0.0142 = 0.9155
[oBA]beer = 0.00195 × BA × W × 1000 / V
scaleoBA = 0.85

where W is the weight of the hops (in grams), V is the post-boil volume of wort (in liters), AA is the alpha-acid rating of the hops (from 0 to 1), and BA is the beta-acid rating of the hops (from 0 to 1).  The parameter lossFactoroAA is the only unknown parameter, and represents the fraction of alpha acids that oxidize during the boil and survive into the finished beer.  Generally, I’ve found that lossFactoroAA is between around 0.02 and 0.06, or 2% to 6% of the available alpha acids.  In the SMPH model of IBUs, the percent of alpha acids that oxidize during the boil is estimated at 11%, and the loss factor of oAA during the boil is estimated at 0.51, for an overall estimate of lossFactoroAA of 0.056.  This estimate doesn’t include other factors such as losses caused by pH, krausen, etc.

6. Conclusion
This blog post presents a technique to estimate IAA and nonIAA concentrations from multiple IBU values.  The IBU values should come from batches of beer that differ only in the concentration or steep time of the hops; all other parameters should be the same or as close as possible (e.g. original gravity, other wort characteristics such as pH, alpha acids, hop storage conditions, and fermentation conditions).  The more IBU values that are available, the better that any differences will average out, and the better the estimates will be.  This post doesn’t provide any experimental data, but the use of this technique can be found in a number of blog posts on this site.

References

  • V. A. Algazzali, The Bitterness Intensity of Oxidized Hop Acids: Humulinones and Hulupones, Master of Science thesis (advisor: T. H. Shellhammer), Oregon State University, 2014.
  • C. Almaguer, C. Schönberger, M. Gastl, E. K. Arendt, and T. Becker, “Humulus lupulus – a story that begs to be told: A review,” in Journal of the Institute of Brewing, vol. 120, pp. 289-314, 2014.
  • H. O. Askew, “Changes in Concentration of α and β Acids and of Iso-Compounds on Heating Extracts of Hops in Aqueous Solutions and Unhopped Wort,” in Journal of the Institute of Brewing, vol. 71, pp. 10-20, 1965.
  • J. Dierckens and M. Verzele, “Oxidation Products of Humulone and Their Stereoisomerism,” in Journal of the Institute of Brewing, vol. 75, pp. 453-456, 1969.
  • A. Forster, “Influence of Hop Polyphenols on Beer Flavor,” in Hop Flavor and Aroma: Proceedings of the 1st International Brewers Symposium, ed. Thomas H. Shellhammer, Master Brewers Association of the Americas, 2009.
  • M. Garetz, “Hop Storage: How to Get – and Keep – Your Hops’ Optimum Value” in Brewing Techniques, January/February 1994, hosted on morebeer.com.
  • J. S. Hough, D. E. Briggs, R. Stevens, and T. W. Young, Malting and Brewing Science.  Volume 2: Hopped Wort and Beer.  Springer-Science+Business Media, B. V., 2nd edition, 1982.
  • S. Kappler, M. Krahl, C. Geissinger, T. Becker, M. Krottenthaler, “Degradation of Iso-alpha-Acids During Wort Boiling,” in Journal of the Institute of Brewing, vol. 116, no. 4, pp. 332-338, 2010.
  • M. J. Lewis and T. W. Young, Brewing. Springer Science+Business Media, 2nd edition, 2001.
  • M. G. Malowicki, Hop Bitter Acid Isomerization and Degradation Kinetics in a Model Wort-Boiling System, Master of Science thesis (advisor: T. H. Shellhammer), Oregon State University, 2005.
  • J. P. Maye, R. Smith, and J. Leker, “Humulinone Formation in Hops and Hop Pellets and Its Implications for Dry Hopped Beers”, in MBAA Technical Quarterly, vol. 51, no. 1, pp. 23-27, 2016.
  • I. McMurrough, K. Cleary, and F. Murray, “Applications of High-Performance Liquid Chromatography in the Control of Beer Bitterness,” in Journal of the American Society of Brewing Chemists, vol. 44, no. 2, pp. 101 – 108, 1986.
  • 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.
  • L. O. Spetsig, “The Bitter Substances of Spent Hops, Trub, and Yeast Cover: A Chromatographic Study,” in Journal of the Institute of Brewing, vol. 74, pp. 346-351, 1968.
  • R. Stevens and D. Wright, “Evaluation of Hops [Part] X. Hulupones and the Significance of β Acids in Brewing,” in Journal of the Institute of Brewing, vol. 67, 1961.
  • M. Verzele and D. De Keukeleire, Chemistry and Analysis of Hop and Beer Bitter Acids, vol. 27, 1st edition, Elsevier,  ISBN 0-444-88165-4, eBook ISBN 9781483290867, 1991.

 

Hopping-Rate Correction Based on Alpha-Acid Solubility

Abstract
It is well known that doubling the amount of hops in the boil can yield less than double the IBUs in the finished beer. When modeling IBUs, therefore, a hopping-rate correction factor is needed, with utilization decreasing as the concentration of hops increases. This post describes a model of alpha-acid solubility at boiling, based on the work of Mark Malowicki and experimental data. This solubility model can be used for hopping-rate correction in IBU prediction. This model is a refinement of the model described in Four Experiments on Alpha-Acid Utilization and IBUs. In this revised model, the alpha-acid solubility at boiling and typical wort pH is not a fixed value, but begins at 240 ppm and increases gradually to 490 ppm. At concentrations below 240 ppm, all of the alpha acids dissolve into the wort. At concentrations greater than 240 ppm, the solubility limit can be modeled with the equation 490 × (1 − e−0.00280×[AA]0) where [AA]0 is the concentration of alpha acids when added to the wort. Any alpha acids that are above this limit are quickly degraded and do not contribute to the isomerized alpha acids (or oxidized alpha acids) in the finished beer. This model demonstrates a good fit to available IBU data at a variety of hopping rates and boil times.

1. Introduction

1.1 Hopping Rates and Utilization
The relative amount of hops 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]. (Utilization is the ratio of isomerized alpha acids present in the finished beer divided by the total alpha acids added.)

Garetz provides a quantitative model of the relationship between amount of hops and utilization. He proposes a hopping-rate correction factor (also described by Hall and Daniels) that depends on volume and “desired IBU” to determine the weight of hops needed [Garetz (b), p. 137; Hall, p. 63; Daniels, p. 86]. If we focus on full boils (instead of boiling a higher-gravity wort and then adding water), we can write the Garetz correction factor as

HF(IBU) = (IBU/260) + 1

where HF is the hopping-rate correction factor that depends on the (desired) IBU value, IBU. If the IBU value is to be estimated from the weight of hops, Hall provides a method to compute this correction factor in two steps rather than through the iterative process suggested by Garetz [Hall, p. 63]. I’ve previously found that this correction factor can overestimate predicted IBU values at high hopping rates.

1.2 Hopping Rates, Utilization, and Alpha-Acid Solubility
Previous work has indicated a relationship between the solubility of alpha acids and the reduced utilization found at high hopping rates. While alpha acids have a solubility limit between 70 and 90 ppm at room temperature and pH 5.2 [Malowicki, p. 54; Spetsig, p. 1423], Spetsig has estimated the solubility at boiling and pH 5.2 to be about 300 ppm [Spetsig, p. 1423]. (Malowicki notes a solubility limit of 200 ppm at boiling and pH 5.0 based on Spetsig’s graph [Malowicki, p. 34], but this seems to be a typo, with actual values of 250 ppm at pH 5.0 and 300 ppm at pH 5.2.) This value of 300 ppm is based on extrapolation from conditions at 77°F (25°C) and 104°F (40°C), and should be considered an approximation [Spetsig, p. 1424]. Maule has noted that “when humulone was used at rates greater than 200 [ppm] the amount appearing as humulone and iso-humulone on break increased at the expense of the amounts remaining in the wort” [Maule, p. 289]. (Humulone is the most prevalent of the alpha acids, and the “chemistry of the other iso-alpha acids is practically identical to that of the isohumulones” [Verzele and De Keukeleire, p. 88], so the terms “humulone” and “iso-humulone” are often considered generally equivalent to the terms “alpha acid” and “isomerized alpha acid,” respectively (to the chagrin of Verzele and De Keukeleire [p. 89]).) Maule then notes that what is not actually adsorbed to the break “represents the difference between the amount of resin present and its solubility in wort under the conditions employed” [Maule, p. 289]. This suggests an alpha-acid solubility limit closer to 200 ppm at boiling. (The solubility limit of the isomerized alpha acids is much higher, at 900 ppm in wort [Rudin, p. 18], and is not considered further in this post.) Hough et al. remark that “as may be anticipated from the solubility of humulone, hops are utilized more efficiently at low rates than at high ones. Indeed, it was concluded that the solubility of humulone was the limiting factor in its utilization” [Hough et al., p. 489]. These statements present a link between alpha-acid solubility and reduced utilization at high hopping rates, with an alpha-acid solubility limit between roughly 200 and 300 ppm.

In a previous blog post, Four Experiments on Alpha-Acid Utilization and IBUs, I suggested a model for hopping-rate correction based on alpha-acid solubility. In this model, the concentration of isomerized alpha acids (IAA) that ends up in the wort is limited by the solubility of alpha acids at boiling, but the bitter substances other than IAA that contribute to measured IBU values (nonIAA) increase linearly with the amount of hops added. These other bitter substances include oxidized alpha acids, hop polyphenols, malt polyphenols, and oxidized beta acids. I estimated the alpha-acid solubility limit in boiling wort at 270 ppm, which is in between the estimates from the literature of 200 to 300 ppm. In this model, the alpha acids only go into solution at concentrations of 270 ppm or less (at boiling and typical wort pH). Above this limit, the alpha acids are quickly degraded (or permanently removed from solution) and do not yield IAA in the finished beer. This model yielded a good fit to the available data, but it is an over-simplification in that oxidized alpha acids produced during the boil should be limited to the same extent that isomerization is limited. (Oxidized alpha acids produced during the boil may be the second-largest contributor to IBUs, after isomerized alpha acids.) The current blog post describes possible revisions to this model and more experiments to evaluate this model at different boil times.

1.3 Isomerization of Undissolved Alpha Acids
Alpha acids generally isomerize according to first-order reactions when dissolved in boiling wort, as described below [Malowicki, p. 24]. But what happens when the concentration of alpha acids is greater than the alpha-acid solubility limit? The dissolved alpha acids presumably undergo isomerization in the usual way, but the fate of those that are not (yet) dissolved is unclear. If the previous assumption that these alpha acids are quickly degraded into some other form is not correct, then the alpha acids may still undergo isomerization, presumably still as a first-order reaction but with an unknown rate constant.

Heat is the only requirement for alpha-acid isomerization; in other words, the presence of wort is not required (although it may be a catalyst). Isomerization can happen not only in the presence of boiling wort or alkaine media [Verzele and De Keukeleire, pp. 102-106], but also by exposure to light (photo-isomerization) [Verzele and De Keukeleire, pp. 106-109], or at high heat with the solid metal salts of humulone [Verzele and De Keukeleire, pp. 109-111]. In thermal isomerization, “humulone resists heating up to 100°C … Above 180°C the thermal transformations and degradations of humulone occur very rapidly” [Verzele and De Keukeleire, p. 109], which implies that the alpha acids might be fairly stable at boiling in the absence of water or oxygen. On the other hand, “heat may also cause other reactions of the sensitive hop bitter acids and high temperatures must therefore be avoided” in the preparation of hop extracts [Verzele and De Keukeleire, p. 13], indicating that high temperatures might degrade the alpha acids through reactions not involving isomerization. It’s also known that isomerization can be catalyzed by “calcium or magnesium ions, either in methanol solution or the solid state” [Hough et al., p. 493; Kappler et al., p. 332]. What are the chemical properties of the resin containing the undissolved alpha acids, and how do these properties affect isomerization? Does a lack of oxygen severely reduce the rate of isomerization? (Oxidation is “one of the first things to happen in the complex chemistry of humulone isomerization” [Dierckens and Verzele, p. 454]; a lack of oxygen might therefore slow down the rate of isomerization.) Does the presence of some catalyst in the resin increase the rate of isomerization? Do other reactions in the presence of heat cause degradation of the alpha acids? At what rate do the undissolved alpha acids dissolve into the wort as the dissolved alpha acids are converted into isomerized alpha acids?

Because the answers to these questions are not currently clear, we can construct a model of isomerization that tests various possibilities, and see which settings of the model produce the best fit to the available data. The resulting model and settings will not have been proven to be correct, but this will provide the most likely explanation (in a statistical sense) given the available data.

1.4 A Model of Isomerization for Dissolved Alpha Acids
I’ll briefly introduce the model of alpha-acid isomerization developed by Mark Malowicki, which forms the foundation of the current hopping-rate correction model. This model of isomerization describes a general process for the conversion of alpha acids (AA) into (bitter) isomerized alpha acids (IAA), and the conversion of IAA into “uncharacterized degradation products” that aren’t bitter (including humulinic acid, isobutyraldehyde, and iso-hexenoic acid) [Malowicki, p. 13, pp. 26-27; Hough et al., p. 480]. In this model, the concentration of IAA at time t can be determined from the initial concentration of alpha acids and two first-order reactions with temperature-dependent rate constants:

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

where [IAA] is the concentration of isomerized alpha acids in the wort at time t and temperature T, in parts per million (ppm), and [AA]0 is the initial concentration of alpha acids in the wort (also in ppm). (e is the mathematical constant 2.71828…) The variable k1(T) is the rate constant for the conversion of alpha acids into isomerized alpha acids and T is the temperature in degrees Kelvin,

k1(T) = 7.9×1011 e-11858/T

and k2(T) is the rate constant for the conversion of isomerized alpha acids into other products,

k2(T) = 4.1×1012 e-12994/T

At boiling (T=373.15°K), k1(T) is 0.0125 and k2(T) is 0.0031. Malowicki also provides a different form of the same equation [Malowicki, p. 27], in which the IAA concentration at time t is not computed from the initial level of alpha acids. Instead, the change in concentration of alpha acids at any time point is computed from the current concentration of alpha acids, and the change in IAA is computed from the current concentrations of alpha acids and IAA:

d([AA])/dt = –k1(T)×[AA]
d([IAA])/dt = k1(T)×[AA] – k2(T)×[IAA]

where d([AA])/dt is the change in alpha-acid concentration as a function of time (e.g. expressed in ppm per minute), and d([IAA])/dt is the change in IAA concentration as a function of time. If we start at time 0 with an alpha-acid concentration [AA]0 and an IAA concentration of zero, these changes in concentration can be integrated over a range of time values to arrive at a total concentration of AA and IAA at time t. We can approximate the integration on a computer using a very small time increment, tδ. While the numerical result is the same in these two forms of equations, one advantage of the second form is that multiple hop additions can be dealt with very easily when applying an alpha-acid solubility limit. Another advantage is that we can express the concentrations of undissolved alpha acids and IAA, and their transformation over time, in the same way.

2. Approach
2.1 Developing a General Model
The proposed model starts with Malowicki’s formulas for the conversion of dissolved alpha acids. We can then predict IAA concentration in the wort from either (a) parallel formulas for the conversion of undissolved alpha acids into undissolved IAA, the production of undissolved degradation products, and the conversion of undissolved into dissolved components, and/or (b) a solubility limit that increases gradually as a function of initial alpha-acid concentration.

For the conversion of undissolved alpha acids into undissolved IAA, if we have alpha acids above the solubility limit at time t, then at the next time instant (t+tδ) some of the alpha acids in solution will have been converted into IAA. This lowers the dissolved alpha-acid concentration at t+tδ, allowing some of the alpha acids not yet in solution to dissolve into the wort at this time, potentially bringing the dissolved alpha-acid level back up to the solubility limit (as long as there are still undissolved alpha acids). This model then has multiple simultaneous processes: the conversion of dissolved alpha acids into (dissolved) IAA, the conversion of undissolved alpha acids into (undissolved) IAA, and the dissolving of the alpha acids and IAA into the wort. For each of these processes, there is an associated rate constant or two. The model is complex, but it can (a) replicate the more simple model in which alpha acids above a constant solubility limit are quickly degraded and (b) test a wide variety of other possible conversions.

The rate of dissolution of alpha acids into the wort can be modeled using the Noyes–Whitney equation, although at the solubility limit this rate can not be faster than the rate of conversion of dissolved alpha acids into IAA. Rather than model each of the five parameters this equation, we can recognize that all components except the surface area are constant at the solubility limit, and (assuming a spherical shape for the undissolved alpha acids) the surface area is proportional to the mass of alpha acids raised to the power of 2/3. Therefore, the rate of dissolution (in mg/second) is some (unknown) factor multiplied by the weight of undissolved alpha acids raised to the power of 2/3.

The solubility data provided by Malowicki at room temperature [Malowicki, p. 53] and estimated in a previous post (Four Experiments on Alpha-Acid Utilization and IBUs) show a gradual rise in solubility as the initial alpha-acid concentration increases above some minimum threshold (see Figure 1). We can therefore also model the solubility limit as a function of initial alpha-acid concentration, instead of using a constant value (e.g. 270 ppm at boiling). In this case, the solubility limit is not reached until some minimum concentration of alpha acids is exceeded, e.g. 200 ppm. As the initial concentration gets larger, the solubility limit also gets somewhat larger, so that an initial concentration of 800 ppm might have solubility of 400 ppm. This approach describes the shape of the observed data better than than a single value, although I don’t have a good explanation for why the solubility would change like this. Because Malowicki observed a similar shape at room temperature, it is unlikely that this shape is a byproduct of the types of isomerization and dissolution discussed in the previous paragraph.

solExp-Fig1-combined

Figure 1. Estimated solubility of alpha acids at room temperature (from Malowicki, p. 53; image reproduced under the fair use doctrine) and at boiling (from Four Experiments on Alpha Acid Utilization and IBUs).

For quantifying a gradually-increasing solubility limit, we can specify a minimum limit at concentration [AA]limitMin and a maximum limit at [AA]limitMax. At concentrations below [AA]limitMin, all of the alpha acids dissolve in the boiling wort. Above this concentration, the solubility increases with concentration according to the formula

[AA]limit = [AA]limitMax × (1 − exp(slope × [AA]0))

where slope is a parameter that expresses how quickly solubility changes with increasing initial concentration, and exp() is the natural exponential function. The slope parameter is defined so that the result of this function at [AA]limitMin equals [AA]limitMin:

slope = log(1 − ([AA]limitMin / [AA]limitMax)) / [AA]limitMin

where log() is a function to take the natural logarithm. This approach allows us to describe the solubility limit with two parameters: [AA]limitMin and [AA]limitMax. For example, we might set the minimum solubility to 200 ppm and the maximum solubility to 500 ppm. If [AA]0 is less than 200 ppm, then all alpha acids dissolve and no adjustment is needed. If [AA]0 is 200 ppm, then the solubility limit, [AA]limit, is 200 ppm. If [AA]0 is 400 ppm, then the solubility limit is 320 ppm, and if [AA]0 is 1000 ppm, then the limit is 461 ppm.  The limit will never be greater than 500 ppm in this example.

2.3 Specific Models
Using the general model of isomerization, solubility, and dissolution described above, we can create ten specific models that reflect different assumptions. Model A has no solubility limit for alpha acids; we would expect this model to have the highest error when evaluated on measured data with high concentrations of [AA]0. Model B has a single (constant) solubility limit for alpha acids, and alpha acids that are not immediately dissolved are quickly turned into degradation products. Model C has a “soft” limit for alpha acids, with solubility a function of the initial alpha-acid concentration and described by two parameters, lower and upper solubility limits. Model D has no isomerization or other transformation of undissolved alpha acids (as hinted at by Verzele and De Keukeleire) with two parameters: a solubility limit and a rate of dissolution. Model E has no isomerization of undissolved alpha acids with three parameters: a lower solubility limit, an upper limit, and a rate of dissolution. Model F has no isomerization of undissolved alpha acids and a rate of dissolution that would be greater than the rate of conversion from alpha acids to IAA. This model has a single parameter, the solubility limit. Model G allows isomerization of the undissolved alpha acids and has a very high potential rate of dissolution. Because of the fast dissolution, the rate constant for transforming undissolved IAA into degradation products doesn’t much matter. This model has two parameters, the solubility limit and the rate of isomerization of undissolved alpha acids. Model H has isomerization of undissolved alpha acids using three parameters: the solubility limit, the rate of isomerization of undissolved alpha acids, and the rate of dissolution. In this model, the undissolved IAA are assumed to be stable. Model I also has isomerization of undissolved alpha acids using three parameters: the solubility limit, the rate of isomerization of undissolved alpha acids, and the rate of dissolution. In this model, the undissolved IAA are assumed to degrade quickly. Finally, Model J has the most flexibility with four parameters: a solubility limit, a rate of conversion of undissolved alpha acids to IAA, a rate of conversion of undissolved IAA to degradation products, and a rate of dissolution.

2.4 Testing the Models
The approach used in this set of experiments was to create four conditions (i.e. four batches of beer) with different concentrations of hops, with one condition having an alpha-acid concentration well below the expected solubility limit. The wort from each condition was sampled at 10-minute intervals (from 10 to 100 minutes) during the boil, and each sample was fermented into beer. The resulting plots of IBUs as a function of boil time and hop concentration were then fit to each model of isomerization.

The best values of the model parameters can be estimated by finding those values that minimize the error between the model and measured IBU values. IBU levels are not the same as IAA levels, but we can use a technique that estimates the loss factors for IAA and auxiliary bittering compounds (particularly oxidized alpha acids), and use these loss factors with the model’s production of IAA and auxiliary bittering compounds to estimate IBU values. The models that have a better description of the actual conversion of undissolved alpha acids to dissolved IAA should have a better fit to the measured IBU values and lower error.

3. Experimental Methods and Data
I brewed four batches of beer for this experiment. The four batches were designed to be identical in all respects, except for the initial concentration of alpha acids. (The first two batches were brewed on the same day, and the last two batches were brewed three weeks later.) The first batch had two hop additions, with 1.15 oz (32.6 g) of hops in 8.45 G (32 liters) added at steep time 0 (100 minutes before flameout) and another 1.15 oz (32.6 g) added at steep time 45 (55 minutes before flameout). For all batches, I took samples of wort at 10-minute intervals during the 100-minute boil, and quickly cooled them in an ice bath. In order to minimize any effects caused by removing samples of wort, I used as large a batch size as I dared in my 10 G (40 l) kettle. Targeting a boil gravity of 1.050, I used 8.5 lbs (3.85 kg) of Briess Pilsen DME in 7.867 G (29.78 l) of water, yielding about 8.6 G (32.55 l) of wort with a specific gravity of 1.047. I added hops after the wort had been boiling for 10 minutes, to avoid the foam associated with the start of the boil. The wort was boiled with the cover mostly on the kettle, except for taking samples and occasionally stirring the wort.

I used a 1 lb (0.45 kg) bag of Citra hops from YCH Hops (now Yakima Chief Hops) for this experiment, with hops from the same bag used in all four batches. (The hops were vacuum-sealed and stored in a freezer during the three weeks between brewing sessions.) This bag (lot number PR2-AAUCIT5065) was from the most recent harvest and had an alpha-acid rating of 13.3% and beta-acid rating of 3.9%. YCH claimed at the time that their hops are stored in nitrogen-flushed packaging to minimize oxidation over time. In a previous experiment using YCH Hops, I confirmed that the alpha-acid rating on my brew day (at about 10 months after harvest) was very similar to the package rating, and I expected the same in this experiment. Unfortunately, the measured IBU values from the first two batches of this experiment were dramatically lower than I was expecting. The most plausible explanation for this is that significant oxidation had occurred over the 7 or 8 months from harvest to brewing, greatly reducing the level of alpha acids (and also increasing the levels of oxidized alpha and beta acids). It’s also unfortunate that, because I then needed all of the rest of this bag of hops for the remaining two experiments, I didn’t have enough left over for testing the Hop Storage Index to confirm this hypothesis. Assuming that oxidation was the culprit, my best guesses are that the nitrogen flushing process didn’t work with this bag, the bag didn’t entirely seal, and/or the bag received a small puncture soon after packaging. The bag then presumably sat at room temperature and exposed to oxygen for 8 months. (Getting the measured IBU values from these experiments and realizing the implication was extremely disheartening, to put it mildly.)

During the boil, I contained the hops in a large nylon coarse-mesh bag in order to not include large hop particles in my samples. Previous experiments (from Brülosophy: 25 IBUs (bagged) vs. 27 IBUs (loose), and Four Experiments on Alpha-Acid Utilization and IBUs: 36 IBUs (bagged) vs. 37 IBUs (loose) and 34 IBUs (bagged) vs. 34 IBUs (loose)) have not shown a significant impact of a mesh bag on measured IBU values. In order to maximize contact of the hops with the wort, I added brass weights (a total of 3.2 oz (90.7 g)) to the mesh bag so that the hops would be quickly submerged and hydrated.

Each sample (about 14 oz (0.41 l)) was taken from the boil in an aluminum cup, which was placed in an ice bath and stirred to cool quickly. Once cooled to 75°F (24°C), the sample was transferred to a sanitized, sealed, and labeled quart (liter) container. I aerated each sample by vigorous shaking for 60 seconds, then added .01 oz (0.28 g) of Safale US-05 yeast (age 9 months) to target 750,000 viable cells per ml and degree Plato [Fix and Fix, p. 68]. (The process of taking a sample, cooling it, transferring it to a sanitized container, aerating, and pitching yeast took between 5 and 10 minutes.) After all samples were taken, the containers were cracked open to vent, and they fermented for a week. After one week of fermentation, I sent 4 oz (0.12 l) of each sample to Oregon BrewLab for IBU measurement. The final gravity of all samples was about 1.008 (minimum 1.0075; maximum 1.0085).

Table 1 shows the measured data for each batch, including initial wort volume, weight of hops added, volume and gravity at steep time 0, post-boil gravity (after 100 minutes), and pre- and post-boil pH. Tables 2 through 5 show the measured data for each sample, including estimated volume, estimated specific gravity, and measured IBUs. The volume at steep time 0 was estimated from the initial wort volume and the change in specific gravity from the initial wort to a sample taken at steep time 0. The specific gravity at each sample time was estimated by linear interpolation between the gravity at time 0 and the post-boil gravity. The volume at each sample time was estimated by interpolation using the time of the sample, volume at time 0, and the ratio of gravity at time 0 to gravity of the sample. Figure 2 shows the measured IBU values from the four batches at each 10-minute interval. Figure 2 also shows, for the first batch, the fit of the model described in Section 4.1 (blue line) using the estimated hops degradation factor of 0.75 and estimated harvest AA rating of 12.0%.

In general, from looking at Figure 2, the measured IBU values seem to fit well with the general concept of a first-order conversion from alpha acids to IAA and another first-order conversion from IAA to degradation products. However, the values at 90 and 100 minutes from the fourth batch (highest concentration of hops) show an unexpected decrease in IBUs. It’s unclear why IBUs would decrease for this batch at these time points but not the other three batches, and I don’t think that any parameter settings from Malowicki’s model or the proposed model would explain this decrease (except for temperatures well in excess of boiling). I will assume for now that this decrease was a consequence of both the very long boil time and very high hopping rate. Because these two data points can not be easily modeled, and because they represent an extreme scenario not likely to be encountered by most brewers, I leave them out of further analysis and modeling.

Batch 1 Batch 2 Batch 3 Batch 4
initial wort volume
8.45 G / 31.99 l 8.45 G / 31.99 l 8.57 G / 32.44 l 8.43 G / 31.91 l
weight of hops added
1.15 oz / 32.60 g at start (t = 0); another 1.15 oz / 32.60 g at t=55 2.923 oz / 82.86 g 4.25 oz / 120.49 g 6.25 oz / 177.18 g
initial specific gravity
1.0467 1.0475 1.0465 1.0468
initial pH of wort
5.80 5.78 5.76 5.76
specific gravity at steep time 0
1.0482 1.0490 1.0475 1.0482
wort volume at steep time 0
8.187 G / 30.99 l 8.187 G / 30.99 l 8.390 G / 31.76 l 8.185 G / 30.98 l
post-boil specific gravity 1.0507 1.0519 1.0521 1.0525
post-boil pH
5.55 5.47 5.43 5.42

Table 1. Measured values for the four batches. Measurements include initial wort volume, weight of hops added (two additions for the first batch), initial specific gravity (before heating the wort), initial pH of wort, specific gravity at steep time 0 (when the hops were added), wort volume at time 0, post-boil specific gravity (after steeping for 100 minutes), and post-boil pH.

time: 10 min
20 min 30 min 40 min 50 min 60 min 70 min 80 min 90 min 100 min
vol.
8.145G
30.83l
8.103G
30.67l
8.062G
30.52l
8.021G
30.36l
7.980G
30.21l
7.940G
30.06l
7.900G
29.90l
7.861G
29.76l
7.822G
29.61l
7.783G
29.46l
SG 1.0485 1.0487 1.0490 1.0492 1.0495 1.0497 1.0499 1.0502 1.0504 1.0507
IBUs 8.0 11.0 14.5 16.5 19.5 22.5 28.0 32.5 33.5 37.0

Table 2. Values for each sample from Batch 1. Volumes are given in gallons (G) and liters (l). SG is the specific gravity estimated at the time the sample was taken.

time: 10 min
20 min 30 min 40 min 50 min 60 min 70 min 80 min 90 min 103 min
vol.
8.143G
30.82l
8.096G
30.65l
8.048G
30.46l
8.002G
30.29l
7.956G
30.12l
7.910G
29.94l
7.865G
29.77l
7.821G
29.61l
7.777G
29.44l
7.734G
29.28l
SG 1.0493 1.0496 1.0499 1.0502 1.0505 1.0507 1.0510 1.0513 1.0516 1.0519
IBUs 19.5 27.5 32.5 38.0 42.5 48.0 47.5 50.0 54.0 52.5

Table 3. Values for each sample from Batch 2. Volumes are given in gallons (G) and liters (l). SG is the specific gravity estimated at the time the sample was taken.

time: 10 min
20 min 30 min 40 min 50 min 60 min 70 min 80 min 90 min 100 min
vol.
8.309G
31.45l
8.230G
31.15l
8.153G
30.86l
8.077G
30.57l
8.002G
30.29l
7.929G
30.01l
7.857G
29.74l
7.786G
29.47l
7.717G
29.21l
7.649G
28.95l
SG 1.0480 1.0484 1.0489 1.0493 1.0498 1.0503 1.0507 1.0512 1.0516 1.0521
IBUs 20.5 30.0 39.0 48.5 51.5 59.0 62.0 66.5 69.5 72.5

Table 4. Values for each sample from Batch 3. Volumes are given in gallons (G) and liters (l). SG is the specific gravity estimated at the time the sample was taken.

time: 10 min
20 min 30 min 40 min 50 min 60 min 70 min 80 min 90 min 100 min
vol.
8.113G
30.71l
8.042G
30.44l
7.972G
30.18l
7.903G
29.92l
7.836G
29.66l
7.769G
29.41l
7.704G
29.16l
7.640G
28.92l
7.577G
28.68l
7.515G
28.45l
SG 1.0486 1.0491 1.0495 1.0499 1.0503 1.0508 1.0512 1.0516 1.0521 1.0525
IBUs 34.0 43.0 53.0 64.0 68.0 75.5 81.0 83.0 79.0 74.0

Table 5. Values for each sample from Batch 4. Volumes are given in gallons (G) and liters (l). SG is the specific gravity estimated at the time the sample was taken.

solExp-measuredIBUs

Figure 2. Measured IBU values from the four batches, and the best estimate from the model for Batch 1 (light blue line). Batch 1 had two hop additions, at 0 and 55 minutes, and no line is plotted between the two measured samples where a discontinuity is expected.

4. Parameter Estimation and Results
4.1 Estimating the Alpha-Acid Rating of the Hops
The measured IBU values from this experiment were far too low to be consistent with the alpha-acid (AA) rating of 13.3% on the package of hops. In order to get a better estimate of the AA rating on brew day, I estimated the hop degradation factor at 0.75, using the Garetz model [Garetz (a)] and assuming 8 months of storage at room temperature, a loss factor of 25% for Citra hops, and with the hops sealed in barrier packaging but not free from oxygen (storage factor 0.75). Then I used the technique described in the blog post Estimating Isomerized Alpha Acids and nonIAA from Multiple IBU Measurements to estimate the levels of IAA and auxiliary bittering compounds in Batch 1 (with all alpha acids presumably dissolved), and adjusted the harvest AA rating to reflect a typical IAA loss factor of about 0.5. The resulting harvest AA rating was 12.0%, lower than the package rating of 13.3% by 10% but well within the possible 20% variation reported by Verzele and De Keukeleire [Verzele and De Keukeleire, p. 331].

While the estimated values of the hop degradation factor and harvest AA rating are plausible, an inability to have real confidence in these values means that the solubility-limit results in this blog post can not be presented with certainty. The results presented here are therefore my best estimate, but further experiments will be needed to gain confidence in (or revise) the results.

The measured IBU values, modeled IBU values (with degradation factor 0.75 and AA rating 12.0%), and differences are listed in Table 6.  The modeled IBU values are shown in Figure 2 with a light-blue line.  The root-mean-square (RMS) error over the 10 data points was 1.27.

time: 10 min
20 min 30 min 40 min 50 min 60 min 70 min 80 min 90 min 100 min
meas. IBU
8.0 11.0 14.5 16.5 19.5 22.5 28.0 32.5 33.5 37.0
model IBU 7.99 10.76 13.15 15.19 16.93 24.05 28.36 32.03 35.14 37.76
diff. -0.01 -0.24 -1.35 -1.31 -2.57 1.55 0.36 -0.47 1.64 0.76

Table 6. Measured IBU values, modeled IBU values, and the difference between these two values, all from Batch 1.

4.2 Estimating the Model Parameters
We have ten different models for the production of IAA from alpha acids, described in Section 2.3. We have a way to convert between IAA values generated by this model and IBU values, described in the blog post Estimating Isomerized Alpha Acids and nonIAA from Multiple IBU Measurements. This conversion is modified by the various ways discussed in Section 2 for expressing solubility, conversion of undissolved alpha acids into IAA, and dissolution of undissolved alpha acids. (One additional modification is that the same solubility limit applied to alpha acids for affecting isomerization is applied to alpha acids that oxidize during the boil.) For each model, we can take a guess at parameter values for the model, compute the IAA and nonIAA concentrations using these values, map from IAA and nonIAA concentrations to an IBU value, and take the difference between estimated and measured IBU values. We do this many, many times with different guesses for the parameter values and all measured IBU values from Batches 2, 3, and 4 to see which values minimize the root-mean-square (RMS) error. In order to base the parameter estimates on a greater variety of initial alpha-acid concentrations, I also used 10 data points from Four Experiments on Alpha Acid Utilization and IBUs, experiments #3 and #4. Once we’ve determined the parameter values that yield the lowest error for a model, we can compare the ten models in terms of how well they fit the data.

For Model A (no solubility limit), the RMS error is 10.4. For Model B (a constant solubility limit and fast degradation of undissolved alpha acids), the RMS error is 3.8 with a limit of 380 ppm. For Model C (lower and upper solubility limits), the RMS error is 1.6 with a lower limit of 240 ppm and an upper limit of 490 ppm. For Model D (no isomerization of undissolved alpha acids), the RMS error is 3.2 with solubility 300 ppm and a rate constant for dissolution (at boiling) of 0.225. For Model E (no isomerization of undissolved alpha acids, with a “soft” solubility limit) the RMS error is 1.6 with a lower limit of 240 ppm, an upper limit of 480 ppm, and a dissolution rate constant of 0.015. For Model F (no isomerization of undissolved alpha acids and fast dissolution) the RMS error is 4.1 and the solubility limit is 280 ppm. For Model G (with isomerization of undissolved alpha acids and fast dissolution), the RMS error is 2.8 with solubility 200 ppm and a rate constant of 0.006 for conversion of alpha acids to IAA. For Model H (with isomerization of undissolved alpha acids, stable undissolved IAA, and a slower dissolution rate), the RMS error is 2.1 with solubility 240 ppm, a rate constant of 0.375 for conversion of alpha acids to IAA, and a dissolution rate constant of 0.112. For Model I (with isomerization of undissolved alpha acids, fast degradation of undissolved IAA, and a slower dissolution rate), the RMS error is 2.8 with solubility 300 ppm, a rate constant of 0.044 for conversion of alpha acids to IAA, and a dissolution rate constant of 0.510. For Model J (with parameters for isomerization and degradation of undissolved alpha acids and IAA), the RMS error is 1.9 with solubility 240 ppm, a rate constant of 0.076 for conversion of alpha acids to IAA, a rate constant of 0.028 for conversion of IAA to degradation products, and a dissolution rate constant of 0.187.

From these results, the best model is Model C, with error 1.6 and two parameters: a lower solubility limit of 240 ppm and an upper solubility limit of 490 ppm. (Model C is preferred over Model E because of its greater simplicity.) While Model C has been selected as the best model, it should be re-stated that this conclusion is tentative because of the difficulty in getting a reliable estimate of the alpha-acid rating and degradation factor of the hops that were used.

Results from this model for Batches 2, 3, and 4 are plotted in Figure 3 with solid green lines. The measured IBU values for Batches 2, 3, and 4 are plotted in this figure with solid blue lines, and Batch 1 is plotted with a dashed gray line. The solubility limit function for Model C is plotted in Figure 4.

solExp-modelIBUs

Figure 3. Estimated IBU values using solubility model C, compared with measured IBU values. The measured IBU values for Batches 2, 3, and 4 are plotted in blue. The model (estimated) values for these batches are plotted with solid green lines. The measured IBU values for Batch 1 are plotted in gray for reference.

solExp-solubilityModel

Figure 4. The “soft” solubility function estimated from this set of data, with a lower limit of 240 ppm and an upper limit of 490 ppm.

5. Conclusion
This blog post has discussed a general model for the isomerization of alpha acids at concentrations greater than the solubility limit. The model can be configured to evaluate different hypotheses, e.g. (a) the alpha acids above the solubility limit are quickly degraded, as I assumed in an earlier model, (b) the alpha acids not yet dissolved in wort are fairly stable, implied as a possibility by a statement by Verzele and De Keukeleire [Verzele and De Keukeleire, p. 109], (c) there are no expectations for the rate of conversion of alpha acids and IAA or the dissolving of alpha acids into wort, other than that they are first-order reactions, or (d) the solubility limit is a function of the initial concentration of alpha acids. The model with a gradually-increasing solubility limit as a function of [AA]0 had the best fit to the entire set of data. Because of difficulties in estimating [AA]0 in the data and the indirect methods used to estimate [IAA], the conclusion is not that this model is correct, but only that it is the most likely explanation for this set of data.

To obtain a more reliable estimate of the solubility of alpha acids at boiling, the best approach would be to use pure alpha acids and measure IAA levels directly, as in Mark Malowicki’s thesis [Malowicki]. A second-best approach would be to re-do the analysis described in this blog post with hops that have a more reliable estimate of alpha-acid content and storage conditions.

In the “soft limit” solubility model, the estimated alpha-acid solubility limit starts at about 240 ppm and increases gradually, with an upper limit at 490 ppm. One complication is that, according to Spetsig’s extrapolations, the solubility limit of alpha acids at the pH of the wort used in these experiments (about 5.75) should be about 1000 ppm. However, another experiment that looks at alpha-acid solubility at boiling and a lower pH indicates that the influence of pH at boiling is not as extreme as indicated by Spetsig’s graph.

Acknowledgements
I am extremely grateful to (in alphabetical order) Nev Ash at Online Brewing Supplies, Dana Garves at Oregon Brewlab, and Hannah McMullen for their contributions to this post, including: feedback on the writing, help with concepts, measuring IBU values, and/or tracking down related published work. Any errors are, of course, the sole responsibility of the author.

References

  • R. Daniels, Designing Great Beers: The Ultimate Guide to Brewing Classic Beer Styles. Brewers Publications, 2000.
  • J. Dierckens and M. Verzele, “Oxidation Products of Humulone and Their Stereo-Isomerism,” in Journal of the Institute of Brewing, vol. 75, pp. 453-456, 1969.
  • G. Fix, Principles of Brewing Science. Brewers Publications, 2nd edition, 1999.
  • G. J. Fix and L. A. Fix, An Analysis of Brewing Techniques. Brewers Publications, 1997.
  • M. Garetz (a), “Hop Storage: How to Get – and Keep – Your Hops’ Optimum Value” in Brewing Techniques, January/February 1994, hosted on morebeer.com.
  • M. Garetz (b), Using Hops: The Complete Guide to Hops for the Craft Brewer. HopTech, 1st edition, 1994.
  • M. L. Hall, “What’s Your IBU,” in Zymurgy. Special Edition, 1997.
  • J. S. Hough, D. E. Briggs, R. Stevens, and T. W. Young, Malting and Brewing Science. Volume 2: Hopped Wort and Beer. Springer-Science+Business Media, B. V., 2nd edition, 1982.
  • M. J. Lewis and T. W. Young, Brewing. Springer Science+Business Media, 2nd edition, 2001.
  • M. G. Malowicki, Hop Bitter Acid Isomerization and Degradation Kinetics in a Model Wort-Boiling System, Master of Science thesis (advisor: T. H. Shellhammer), Oregon State University, 2005.
  • D. R. Maule, “The Fate of Humulone During Wort Boiling and Cooling”, in Journal of the Institute of Brewing, vol. 72, pp. 285-290, 1966.
  • 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.
  • A. D. Rudin, “Solubility of Iso-Compounds in Water and Their State in Solution”, in Journal of the Institute of Brewing, vol. 66, pp. 18-22, 1960.
  • L. O. Spetsig, “Electrolytic Constants and Solubilities of Humulinic Acid, Humulone, and Lupulone,” in Acta Chemica Scandinavica, vol. 9, pp. 1421-1424, 1955.
  • M. Verzele and D. De Keukeleire, Chemistry and Analysis of Hop and Beer Bitter Acids, vol. 27, 1st edition, Elsevier, ISBN 0-444-88165-4, eBook ISBN 9781483290867, 1991.

 

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)
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 Hop Pellets
xxxxxxxxxx3.4.2 Oxidized Alpha Acids
xxxxxxxxxx3.4.3 Oxidized Beta Acids
xxxxxxxxxx3.4.4 Polyphenols
xxxxxxxxxx3.4.5 Solubility of nonIAA Components
xxxxx3.5 The Impact of Dry Hopping on IBUs
4. Available Data, Parameter Estimation, and Results
xxxxx4.1 Overview
xxxxx4.2 Sources of IBU Data
xxxxxxxxxx4.2.1 Peacock Hop-Storage Conditions
xxxxxxxxxx4.2.2 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.  If you want to get a better understanding of what components contribute to an IBU value, how the amount of hops used, wort pH, or krausen 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.  I’ve implemented all of the formulas described here in a model of IBU prediction called SMPH; this model is available at my GitHub page.

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.  You can also 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 many details are not as important as the bigger picture; the details are there to provide support.  Feel free to focus on what interests you.

The IBU measurement itself is not always highly regarded. I’ve heard some people claim that the IBU is basically meaningless. However, the correlation between measured IBUs and perceived bitterness for 91 commercial beers with a trained sensory panel with 19 members has been measured at 0.86 [Hahn, p. 50]. A correlation of 0.86 is considered a “strong” correlation [Taylor, p. 37], and so the IBU is actually meaningful and strongly correlated with perceived bitterness [also, Priest and Stewart, p. 266]. While the perception of bitterness is not linear at higher IBU values, a simple mapping can translate IBU values into a linear perceptual scale, and vice versa [Hahn, pp. 48-50]. Bitterness may have different qualities not captured by the IBU measurement [Peacock, p. 163], and the correlation between IBU levels and bitterness may not hold up under every circumstance (e.g. with extremely high rates of dry-hopping [Maye et al., p. 25]). On the other hand, it is a universally-known and (sometimes grudgingly) accepted quantitative measurement, and there is no other measurement of beer that has a better correlation with perceived bitterness. 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, articles, and internet sources.  If you look at the bibliography, you’ll see many publications produced under the guidance of Thomas Shellhammer.  I’ve tried to cite appropriately, and I’ve put the full bibliography at the bottom of this post.  I’ve omitted a lot of interesting details from these sources, in order to maintain a more focused narrative.

2. Definitions of IBUs
2.1 IBU Definition from the American Society of Brewing Chemists (ASBC)
Because of the complexity of hops and IBUs, it’s probably a good idea to start at a high level of description, which is deceptively simple but not very informative: An IBU is a measurement of the amount of absorption of light at 275 nm (abbreviated as A275nm) in a liquid, multiplied by 50.  The liquid in this case is not just any liquid, but beer that has been combined with twice as much iso-octane (TMP), diluted in octyl alcohol and hydrochloric acid [American Society of Brewing Chemists], and emulsified; 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 [the 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 1950s and 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 are 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 the 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).  We’ll come to Figures 1(b), 1(c), and 1(d) shortly.

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 are covered under the serendipitously-named Beer’s Law or Beer-Lambert Law.  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 light absorption 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 or 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 measure or predict) to the concentrations of different substances (which we can approximate).  This section looks at these three items in more detail.

Before getting too far into this section, this might be a good place to define some terms related to alpha acids and beta acids.  Alpha acids are part of the soft resins in the hop lupulin gland, and the alpha acids contain humulone, cohumulone, and adhumulone  [Oliver, p. 34].   Older work may refer to all alpha acids as humulones [Oliver, p. 462].  Humulone is the most prevalent of the alpha acids, and the “chemistry of the other iso-alpha acids is practically identical to that of the isohumulones” [Verzele and De Keukeleire, p. 88], so the terms “humulone” and “iso-humulone” can usually be considered generally equivalent to the terms “alpha acid” and “isomerized alpha acid,” respectively (to the chagrin of Verzele and De Keukeleire [p. 89]).  The oxidized alpha acids contain humulinone as their most important component [Algazzali, p. 13].  The soft resins also contain beta acids, which are also called lupulones [Oliver, p. 462].  The beta acids are composed of colupulone, adlupulone, lupulone, and prelupulone [Oliver, p. 260].  The oxidized beta acids contain mostly hulupones [Algazzali, p. 15].  The oxidized form of colupulone is called cohulupone [Stevens and Wright, p. 496].

3.1 Concentration of Isomerized Alpha Acids (IAA)
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 (with pH 5.2 and an alpha-acid concentration of 80 ppm).  (These degradation products include humulinic acid, isobutyraldehyde, and iso-hexenoic acid [Hough et al., p. 480].)  Malowicki describes the conversion of alpha acids into isomerized alpha acids as a first-order reaction following an Arrhenius equation with a temperature-dependent rate constant k1:

k1(T) = 7.9×1011 e-11858/T [13]

where k1(T) is the rate constant for the conversion of alpha acids into isomerized alpha acids and T is the temperature in degrees Kelvin.  A first-order reaction is of the form [X] = [X]0ekt (where [X] is the concentration of substance X at time t, [X]0 is the initial concentration of X (at time 0), k is a rate constant, 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 (when the hops are added to the kettle), and k1(T) is the rate constant from Equation [13].  We can assume that the reduction in alpha acids is mirrored by a corresponding increase in isomerized alpha acids (see [Malowicki p. 6]).  Second, Malowicki describes the subsequent conversion of isomerized alpha acids into degradation products, also as a first-order reaction with a temperature-dependent rate constant:

k2(T) = 4.1×1012 e-12994/T [15]

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

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

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

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

isoAlphaAcidConcentraion

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

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

Equation [16] relies on the initial concentration of alpha acids when hops are added to 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 initial 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 (when hops are added to the boil), we don’t have a factor that adjusts for volume changes between that time and the end of the boil.  If we did have such a factor, it would account for the difference between these two volumes, 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 some other volume and then accounted for evaporation.  In short: V should be post-boil, room-temperature wort volume (ideally not including the volume of hops in the wort).

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:

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

where AAharvest is the alpha-acid rating of the hops after harvest and drying (in the range 0 to 1), 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 a temperature factor that describes how degradation is affected by temperature, SF is a 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 soon after harvest, and they’ve been stored in the refrigerator or freezer and in nitrogen-flushed packaging, 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.

One complicating factor in getting a reliable estimate of AA% is that the alpha acids are not evenly distributed throughout the hop cone, and some cones contain more alpha acids than others.  As Verzele and De Keukeleire state, “there are easily differences up to 15 − 20% in alpha acids content between and within bales of a single hop delivery. … Since inhomogeneity is to be expected, a reasonable estimate for the average content of the hops lot can only be obtained by sampling at various positions” [Verzele and DeKeukeleire, p. 331].  Therefore, if you only use an ounce or two of well-preserved hops in your boil, the actual concentration of alpha acids in the wort can be up to 20% different from the expected concentration based on the alpha-acid rating on the package.

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

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

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

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

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

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

A model of how temperature decreases after flameout is presented in a blog post “Predicting Wort Temperature After Flameout“.  This model is described by Equations [22a], [22b], and [22c].  In these equations, the parameter T is temperature (in degrees Kelvin), t is time after flameout (in minutes), b is the rate constant that describes how quickly the temperature decreases, effectiveArea is the “effective” area through which steam ventilates, surfaceArea is the surface area of wort exposed to air (in square centimeters), openingArea is the area of the opening in the kettle (in square centimeters), and volume is the wort volume (in litres).  This rate of natural cooling is primarily influenced by (a) the release of steam, which is in turn influenced by the wort volume, surface area of wort exposed to air, and size of the opening in the kettle through which steam can escape, and (b) radiation of heat from the kettle.  Other factors, such as ambient temperature and kettle material, are of much lesser significance and can be ignored.

T = 53.70 × exp(-b × t) + 319.55 [22a]
b = (0.0002925 × effectiveArea / volume) + .00538 [22b]
effectiveArea = (surfaceArea × openingArea)0.5 [22c]

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

integrationTime = 0.001;
effectiveArea = sqrt(surfaceArea * openingArea);
b = (0.0002925 * effectiveArea / volume) + .00538;
AA = AA0;
IAA = 0.0;
time = 0.0;
while (time <= totalTime) {
    if (time <= boilTime) 
        temp = 373.15; 
    else 
        temp = 53.70 × exp(-b * (time-boilTime)) + 319.55;
    k1 = 7.9 * pow(10,11) * exp(-11858.0/temp);
    k2 = 4.1 * pow(10,12) * exp(-12994.0/temp);
    dAA = -1.0 * k1 * AA;
    AA = AA + (dAA * integrationTime);
    dIAA = (k1 * AA) - (k2 * IAA);
    IAA = IAA + (dIAA * integrationTime);
    time = time + integrationTime;
}

where the integration time of 0.001 (called integrationTime) is sufficient for accuracy to at least two places past the decimal point; a larger value like 0.01 will speed things up and still have very good accuracy.  The variable effectiveArea is the square root of the surface area of the wort and the area of the kettle opening.  The variable b is the rate constant for temperature decay.  The variable AA0 is the initial concentration of alpha acids, in ppm (see Equation [17]).  Here, AA is the total concentration of AA, or [AA], after time time (in minutes).  Likewise, IAA is the total concentration of IAA, or [IAA], after time time.  The value of totalTime is the length of the boil in minutes (boilTime) plus any time after the boil when isomerization might be happening.  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 ‘{‘ and ‘}’ symbols define the beginning and end of what is evaluated during each time point.)  The temp variable is temperature at the current time, in Kelvin.  The k1 and k2 variables are the rate constants from Equations [13] and [15].  The variable dAA is the derivative of [AA], or change in [AA] per minute, as defined in Equation [20].  Likewise, the variable dIAA is the derivative of [IAA], or change in [IAA] per minute, as defined in Equation [21].  The pow() function raises the first argument to the power of the second argument; the exp() function computes the constant e (2.718…) to the power 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.  One difference is a loss of available alpha acids when the amount of alpha acids added to the wort exceeds its solubility limit, as described below (“losses as a function of hopping rate”).  Other differences can usually be described as losses of IAA that are produced in the boiling wort but never make it into the pint glass: losses that are correlated with high wort gravity, other losses during the boil, and losses due to wort turbidity, wort pH, fermentation, flocculation, krausen deposits, filtration, finings, and aging.  We’ll look at each of these briefly in this section.  In general, if we have a loss of x%, the loss factor will be (1 – (x%/100)); for example, a loss of 10% becomes a loss factor of 0.90.  If condition A is x% more efficient than condition B, then the loss factor for condition B is (100/(100+x%)).  For example, if A is 10% more efficient than B, then B has a loss factor of 0.91 compared with A.

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

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

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

Isomerization as a Function of Temperature: 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 k1 and k2 in Equations [13] and [15].  Post-flameout temperature dependencies are discussed above in Section 3.2.  (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.)

Isomerization Differences Based on Form of the Hops: The impact of hop pellets on IBUs, compared with hop cones, is discussed more in Section 3.4.1.  Hop cones can be added to the boil either loose or in a coarse mesh bag.  Garetz 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 [Gartez book, p. 141].  I looked at the effect of a mesh bag vs. loose hops on measured IBUs, and found no significant difference.  Marshall Schott at Brülosophy also looked at bagged vs. loose hops, and found 25 IBUs for the bagged hops and 27 IBUs for the loose hops [Schott].  While this difference is not significant, the ratio of 0.926 (25/27) is close to Garetz’s correction factor of 0.91.  In short, it’s not clear if hop cones in a mesh bag really do have lower utilization.  For the model being developed, I’ll assume that bagged hops have the same utilization as loose hops.

Isomerization as a Function of Kettle Size and/or Geometry: The kettle size and/or kettle geometry may also impact utilization [Daniels, p. 78; Fix, p. 47].  As Hieronymus says, “larger kettles are more efficient, and the difference between a five-gallon homebrew system and even a 10-barrel (310-gallon) commercial brewery is startling” [Hieronymus, p. 188].  There are other claims, however, that recipes should scale linearly with kettle size, indicating no impact on utilization [e.g. Spencer].  If there is an impact, the reason for the change in utilization is not clear to me, especially since Malowicki used only tiny volumes of wort (12 ml) [Malowicki, p. 19] and obtained high utilization rates at boiling (see Figure 2).  The only quantitative description I’ve seen of this impact on utilization is in an article on BeerSmith, which says that “Hop utilization is much higher at craft brewing scales, because large boils simply extract more bitterness. … The Hop Utilization Factor … can easily be 125%, 150% or possibly more for a multi-barrel brewing system” [Smith].  I think that the observed increase in utilization with kettle size is actually 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, and because in numerous experiments I’ve conducted with different volumes of wort (see Section 4.2.3) that show no impact of volume or boil-kettle size on IBUs, I believe that the size of the wort volume or boil kettle has no impact on utilization.

Losses as a Function of Hopping Rate: 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 a quantitative model of the relationship between amount of hops and utilization.  He proposes a hopping-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].  I used this equation when I was initially developing this blog post.  However, after some difficulty fitting the IBU model developed here with available data, and after further experimentation, I concluded that Garetz’s correction factor underestimates the effect of alpha acid concentration on utilization.

A better fit to the data available to me can be obtained by using a “soft” limit on alpha-acid solubility.  In this model, the solubility of alpha acids rises gradually above 240 ppm with increasing [AA]0, according to the equations

slope = log(1 − ([AA]limitMin / [AA]limitMax)) / [AA]limitMin [24]
[AA]limit = [AA]limitMax × (1 − exp(slope × [AA]0)) [25]

where [AA]limitMin equals 240 and [AA]limitMax equals 490.  Any alpha acids above [AA]limit are quickly degraded and don’t contribute to the production of isomerized alpha acids (or oxidized alpha acids).  This “soft” limit is generally consistent with (a) the observations of Maule indicating undissolved alpha acids at concentrations as low as about 200 ppm [Maule, p. 289] and (b) a solubility limit of 300 ppm at boiling and pH 5.2 determined by Spetsig [Spetsig, 1955, p. 1423] for higher concentrations.  (Another preliminary study indicates surprisingly little impact of pH on alpha-acid solubility at boiling.)  Using this approach, utilization increases linearly with alpha-acid concentration until the lowest solubility limit is reached (240 ppm); at higher concentrations, solubility and utilization increase more slowly.  For the model of IBUs developed in this blog post, equations [24] and [25] describe the reduction in utilization as a function of the hopping rate.  This reduction in the available alpha acids can be expressed, if needed, as a loss factor that is a function of [AA]0, LFhoppingRate([AA]0).

Losses Due to Wort Gravity: Utilization decreases with increasing wort gravity, at least at higher gravities [e.g. Lewis and Young, p. 266; Hieronymus, p. 188; Hall, p. 62; Daniels, p. 78; Palmer, p. 55; Malowicki, p. 44; Garetz book, p. 130; Hough et al., p. 489; Kappler, p. 334].  As Lewis and Young state, “iso alpha acids react with proteins of wort whence they are partially removed as trub or hot break” [Lewis and Young, p. 266].  As the gravity increases, the concentration of wort proteins increases, implying a greater loss of isomerized alpha acids with increasing gravity.  Kappler found greater losses of isomerized alpha acids at higher gravities when adding (already) isomerized alpha acids to the boil [Kappler, p. 335], indicating that higher gravity causes more isomerized alpha acids to bind with trub and settle out of solution (as opposed to slowing the rate of conversion from alpha acids to isomerized alpha acids).  Malowicki did not find a significant change in utilization at specific gravities of 1.000 and 1.040 [Malowicki, p. 39], and Garetz indicates that the lower limit for this effect is a specific gravity of 1.050 [Garetz book, p. 130].  Greg Noonan [Noonan, p. 215] provides a table of utilization as a function of boil time, original gravity, and form of the hops.  (His table simply lists “wort density” and “specific gravity”, but he defines wort density as original gravity [Noonan, p. 204]). Glenn Tinseth models the  gravity correction factor as LFOG(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].)

I looked at the impact of specific gravity on IBUs, and while I found the same general trend described by others, the trend was only visible for boil times greater than 40 minutes; at less than 40 minutes boil time, no impact of gravity was seen.  The following function describes a gravity-correction factor as a function of original gravity and boil time:

LFOG(OG, t) = 1 − 2×exp(−1 / (S(t) × (OG − 1))) [26]

where LFOG(OG, t) is a gravity loss-correction factor, OG is the original gravity, t is the time for which the hops are in contact with boiling wort, and S(t) takes on different values depending on the hop steep time.  For steep times 30 minutes and less, S(t) = 1; for steep times 40 minutes and greater, S(t) = 4.9, and for steep times between 30 and 40 minutes, S(t) = 0.39 × (t − 30) + 1.  At steep times 40 minutes and greater, this function provides a compromise between the correction factors proposed by Tinseth, Rager, Mosher, and Noonan [Hall, p. 61].

Other Losses During the Boil:  Isomerized alpha acids are lost during the boil in ways that are not dependent on the wort gravity.  Malowicki says that “trub, and specifically the formation of trub, leads to greatly increased losses of bitter acids” [Malowicki, p. 8; emphasis mine].  He cites work by Askew in which the use of pre-formed trub produced losses of only 5% to 9%, but the formation of trub created losses of 35% [Malowicki, p. 7-8].  Malowicki also cites Laufer and Brenner who found a 38% loss of bitter acids to trub and a 35% loss to spent hops.  Spetsig reports that about 25-30% of the bitter substances are found in the spent hops and 25-40% are found in the trub [Spetsig, 1968, p. 346].  Hall cites Hough et al., who cite Maule (1966), saying that “about 7% of the iso-alpha acids are removed with the breaks” [Hall, p. 57; Hough et al., p. 489].  Garetz says that “8-10% of the iso-alpha acids are adsorbed (meaning they cling to the surface of) the hot and cold breaks.” [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.  While we don’t have a good value for this loss factor yet, we can refer to it as LFboil.

Losses Due to pH: It is generally accepted that a lower wort pH will reduce utilization [e.g. Lewis and Young, p. 266].   Malowicki looked at isomerized alpha acids produced and degraded during boiling at pH values of 4.8, 5.2, 5.6, and 6.0, and found that “the dataset did not show a significant effect of pH on rate of iso-alpha-acids produced” [Malowicki, p. 38].   He speculated that “the losses to trub would better explain the differences in utilization that are attributed to pH…, since rate of isomerization does not appear to be affected[Malowicki, p. 41 (emphasis mine)].  Kappler found a significant loss of isomerized alpha acids during the boil corresponding with a decrease in pH (at pH levels of 4.0, 5.0, 6.0, and 8.0); Huang found similar results [Kappler, p. 334; Huang, p. 50].  However, the difference in recovered IAA between pH 5.0 and 6.0 found by Kappler was only a relative 7%, and this pH range is beyond that found in typical brewing.  In another blog post, “The Effect of pH on Utilization and IBUs“, I measured IBUs as a function of wort pH and concluded that the decrease in IBUs with pH can be modeled as primarily a loss of nonIAA components and secondarily as a loss of IAA.  The loss factor for IAA can be modeled as LFpH_IAA(pH) = 0.071 × pH + 0.592, where LFpH_IAA is the scaling factor to model IAA losses as a function of pH and the variable pH is the post-boil wort pH.  This scaling factors has a value of 1.0 at pH 5.75, and smaller values as the pH decreases.  This loss factor for IAA was based on the data published by Kappler et al. [Kappler, p. 334].

Losses Associated with the Use of Salts:  Kappler et al. (2010) looked at the impact of calcium chloride, magnesium chloride, and calcium sulfate on the recovery of isomerized alpha acids [Kappler et al., p. 335].  They added each salt at various concentrations to pure water with 100 ppm of isomerized alpha acids, and evaluated the recovery rate of IAA after a 60-minute boil.  For calcium chloride and magnesium chloride, they found a very large reduction in the recovery rate of IAA with salt concentrations of 200 and 500 ppm.  They found a smaller effect for calcium sulfate.  This implies that there may be significant losses of isomerized alpha acids with the use of such salts.  I did a similar experiment, but looked at the change in IBU values in finished beer associated with different concentrations of calcium chloride.  I found very little effect of calcium chloride concentration on IBU values (and, by implication, on IAA losses).  Because Kappler’s results were obtained using water instead of wort, it seems for now that there is probably little impact associated with the use of these salts when they are boiled with hops in wort.

Losses Associated with Wort Clarity:  To my surprise, I have measured very significant differences in IBUs when beers differ only in the clarity of the wort.  In this experiment, wort clarity was changed by using different lautering techniques.  (I originally suspected a relationship between the concentration of protein in wort and loss of IAA, but the different conditions in this experiment all had the same protein levels.)  While nonIAA concentrations do not appear to be affected by wort clarity, IAA concentrations can be 30% higher than average in very clear wort and 30% lower in very cloudy wort.  We can then define a loss factor for the loss (or relative gain) of IAA associated with wort clarity, LFwortClarity(clarity), with clarity a somewhat poorly-defined parameter using qualitative descriptions.  For example, LFwortClarity(very clear)=1.30, LFwortClarity(clear)=1.20, LFwortClarity(somewhat clear)=1.10, LFwortClarity(average)=1.00, LFwortClarity(somewhat cloudy)=0.90, LFwortClarity(cloudy)=0.80, and LFwortClarity(very cloudy)=0.70.  The losses (or relative gains) in IAA appear to occur during fermentation.

Losses During Fermentation: Isomerized alpha acids are also lost during fermentation [e.g. Hieronymus, p. 190]. Lewis and Young say that “during fermentation, iso-alpha-acids associate with the surface of the yeast cells present… Iso-alpha-acids, being surfactants, react with inert surfaces of all sorts and for example separate on gas bubbles to be deposited on the fermenter walls” [Lewis and Young, p. 267].  Hall describes the same process, saying that “during the fermentation process, iso-alpha acids are scrubbed by the rising CO2 and collect in the foam of the kraeusen.  This sticky foam can be blown off, skimmed off or stuck on the sides of the fermenter … Iso-alpha acids also are bound up by the yeast cells and removed when the yeast flocculates out” [Hall, p. 57].  Daniels says that the amount of loss is dependent on the amount of yeast pitched and the “extent of yeast growth during fermentation” [Daniels, p. 78].  Garetz says that there are two factors, “the total growth of the yeast crop and the amount of time the yeast stays in suspension”, and that there is a 5% variation depending on the flocculation characteristics of the yeast [Garetz book, p. 126].  He also says that if the alpha acids are mixed back into the beer at the right time, utilization is increased by 18% [Garetz book, p. 126], implying typical losses of 18%.  Fix (citing Garetz) estimates loss to yeast sediment at 10% to 20% [Fix, p. 49]. Malowicki (citing Laws et al.) reports losses during fermentation from 5% to 17% [Malowicki, p. 8] and also (citing Laufer and Brenner) losses to yeast of 10% [Malowicki, p. 7].  Spetsig reports losses of 10% to 15% [Spetsig, 1968, p.346].  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 (LFferment=0.85).  The flocculation factor suggested by Garetz is 0.95 for high-flocculant yeast and 1.05 for low-flocculant yeast [Garetz book, pp. 140-141], so we have LFflocculation(high)=0.95, LFflocculation(medium)=1.00, and LFflocculation(low)=1.05.

As noted by Hall, the concentration of IAA in beer is reduced not only by binding with the yeast that flocculates out, but also by binding with the foam of the krausen.  It is often recommended to remove the krausen during fermentation for a “smooth bitterness.”  Some brewers accomplish this through the use of a blow-off tube and a small headspace in the fermentation vessel. Many brewers do nothing about krausen, allowing most of it to fall back into the beer.  I conducted an experiment which showed that losses of krausen to deposits on the walls of the fermentation vessel can have a small (5% to 10%) impact on IBUs, and that the loss of krausen through a blow-off tube can result in more than a 25% reduction in IBUs.  (The aforementioned “smooth” bitterness might simply be less bitterness.)  The results indicate that both IAA and nonIAA are lost with the removal of krausen, but that the loss of nonIAA is about three times greater than the loss of IAA.  While it is difficult to quantify how much krausen is deposited or lost, mixing the krausen back into the fermenting beer can yield a 13% increase in IAA, and the use of a blow-off tube can yield a 27% loss of IAA. This requires another loss-factor parameter for the loss of IAA due to krausen, LFkrausen_IAA(krausenLoss), with krausenLoss another somewhat poorly-defined parameter with qualitative descriptions.  For example, LFkrausen_IAA(no loss)=1.126, LFkrausen_IAA(minor deposits)=1.05, LFkrausen_IAA(medium deposits)=1.00, LFkrausen_IAA(heavy deposits)=0.95, LFkrausen_IAA(very heavy deposits)=0.90, LFkrausen_IAA(blow-off, slow fermentation)=0.938, LFkrausen_IAA(blow-off, normal fermentation)=0.833, and LFkrausen_IAA(blow-off-vigorous fermentation)=0.729.

The earlier estimate of a typical fermentation loss factor of 0.85 corresponds well with a 12.68% increase in IAA when mixing the krausen back into the fermenter and a 5% increase when using low-flocculant yeast.  Combining these three factors results in an overall loss factor of 1.0 (0.85 × 1.126 × 1.05 = 1.00), suggesting that all of the typical IAA fermentation losses in the factor of 0.85 are accounted for by only krausen and flocculation.

Losses Due to Filtration, Finings, 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].   George and Laurie Fix provide a table of the reduction in IBUs with filter size [Fix and Fix, p. 129].  The IBU loss factors are 0.942, 0.953, and 0.985 at 0.5, 1, and 3 microns, respectively.  If we assume (due to a lack of data) that filtering affects IAA and nonIAA components equally, we can estimate the loss factor for filtering:

LFfiltering(MR) = (0.017 × MR) + 0.934 for MR=0 to 3.83, otherwise 1.0 [27]

where LFfiltering(MR) is the loss factor due to filtering, if any, and MR is the micron rating of the filter.  At micron ratings greater than 3.83, there is no loss of IBUs or IAA due to filtering.  If the beer is unfiltered, then any large value of MR can be used.

Hall says that “there are oxidation reactions that can reduce the bitterness of beer over extended storage periods” [Hall, p. 58].  According to Kaltner and Mitter, “over a storage time of 12 months, a degradation of bitter substances in various beers in a range of 10% to 15% could be analyzed” [Kaltner and Mitter, p. 37].  According to Peacock, citing results from Forster et al. (2004), beer loses 18% of  isomerized alpha acids and 14% of measured IBUs after 8 months at room temperature [Oliver, pp. 132-133, Peacock p. 164].  I am unaware of an existing model of how IBUs decrease with age for home-brewed beer stored in bottles at room temperature (which may have greater oxidation, less filtering, and other differences with commercially-bottled beer).  I therefore measured the decrease in IBUs for two home-brewed beers after 1, 2, 3, 6, 7, 13, and 52 weeks from the start of fermentation, and fit the measured hop-derived IBU decrease over time with an exponential-decay function.   If we assume that isomerized alpha acids and hop-based non-IAA components are affected by age at the same rate (which may be an incorrect assumption [Peacock, p. 163], but not unreasonable as a first approximation), we can model the loss factor for isomerized alpha acids using the same age formula determined for hop-derived IBUs:

LFage(ageweeks) = 0.35 × e-0.073×ageweeks + 0.65    with maximum ageweeks of 16 [28]

where LFage(ageweeks) is the loss factor due to age of the beer (in weeks, stored at room temperature).  After about 16 weeks, the losses seem to stabilize, and so even if the age is greater than 16 weeks, a maximum value of 16 is used.

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

RFIAA = 1.0 [29]
LFIAA([AA]0, OG, t, pH, clarity, flocculation, krausen, MR, ageweeks) = LFhoppingRate([AA]0) × LFboil × LFOG(OG, t) × LFpH_IAA(pH) × LFwortClarity(clarity) × LFferment × LFflocculation(flocculation) × LFkrausen_IAA(krausen) × LFfiltering(MR) × LFage(ageweeks) [30]
[IAA]beer = [IAA]wort × RFIAA × LFIAA(OG, t, pH, clarity, flocculation, krausenLoss, MR, ageweeks) [31]

In these equations, RFIAA describes how the rate of isomerization is affected.  Because (a) the effect of temperature is included in the computation of [IAA]wort (Equations [13] and [15]), (b) the form of the hops is estimated to have no impact on the rate of isomerization (see above and Section 3.4.1), and (c) the kettle size (or geometry) also has no observed impact, the isomerization rate factor is always 1.  The parameter LFIAA is the loss factor for isomerized alpha acids, including any losses from the boil, the hopping rate, gravity, pH, wort clarity, fermentation, flocculation, krausen, filtering, finings, and age, as discussed above.  The concentration of IAA in the wort, [IAA]wort, can be computed using Code [1].  The final concentration of IAA in beer, [IAA]beer, is simply the product of the concentration of IAA in the wort and the various loss factors.

The only problem remaining for modeling [IAA]beer is that while we have a reasonable approximation of most loss factors, we have very little basis for determining LFboil.  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 “auxiliary bittering compounds” (ABC), which we call nonIAA.

Alpha acids (which did not isomerize during the boil) “do not survive to any significant extent into beer” [e.g. Lewis and Young, p. 259] and are not bitter [Shellhammer, p. 169], but as they age and become oxidized, the resulting oxidized alpha acids (oAA) are both 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 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 during storage with exposure to oxygen [Peacock, p. 157; Fix and Fix, p. 36; Lewis and Young, p. 265; Hall, p. 55; 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 are a contributing factor to the nonIAA compounds [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))) [32]

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] and Equation [31].)

3.4.1 Hop Pellets
Before we get to details of oxidized alpha acids, oxidized beta acids, and polyphenols, the topic of hop pellets and extracts should be discussed. It is often said that hop pellets yield higher utilization than hop cones [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 said to be 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].

Expressing pellets as more efficient than whole hops, Noonan provides a pellet correction factor (in table form) that varies from 1.0 to 1.5, based on boil time and gravity [Noonan, p. 215].  Garetz recommends a pellet correction factor of 1.10 for boil times up to 30 minutes, otherwise a correction factor of 1.0 [Garetz book, p. 131, 141].  Hieronymus says that hop pellets are 10% to 15% more efficient than cones [Hieronymus, p. 188].  According to Michael Hall, Mosher specifies a correction factor of 1.33 [Hall, p. 62].  This leaves a wide range of possible correction factors for the use of pellets compared with whole hops (from 1.0 to 1.5), with a median factor of 1.15.

In two posts looking at the impact of hop pellets on IBUs and the reason for the increase in IBUs when using pellets, I concluded that while pellets do yield more IBUs than hop cones, the reason is not because of greater utilization.  Instead, the increase appears to be caused by greater production of oxidized alpha acids during the boil.  It seems plausible that the greater surface area of lupulin glands in pellet hops does not increase the production of IAA or rate of isomerization, but it does increase the production of oAA.  (The surface area being a limiting factor in general may explain why all of the alpha acids do not oxidize immediately when put in boiling wort.)  The amount of oAA produced during the boil is estimated to be approximately twice as much with pellets as it is with cones, but the amount of increase appears to be dependent on the variety of hops.  With Willamette hops the increase is only about 20%, with Citra the increase is about 80%, and with Comet the increase may be as large as 220%.  The wide range of pellet-correction factors reported in the literature may be due to the incorrect assumption that the increase in IBUs from pellets changes with boil time, the influence of different varieties of hops, or some combination of both reasons.

Hough et al. say that alpha-acid extracts are much less efficient than whole hops: “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].   Maule says that “the extent of this effect may however be exaggerated by solution of oxidation products of alpha and beta acids from hops, since these would be measured as iso acids” [Maule, p. 288] using the Rigby method [Peacock, p. 158-159].   Companies that produce hop extract, such as BSG, claim increased utilization with extracts.  Verzele and De Keukeleire note that “the alpha acids utilization yield may be comparable for hops and for hop extracts” [Verzele and De Keukeleire, p. 102].  Given the conflicting claims about the utilization of hop extracts, I will assume that hop extract has the same alpha-acid utilization as whole and pellet hops.

3.4.2 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 hops, given these factors [Garetz article].  (As long as they are properly stored, pellets undergo oxidation at a much slower rate [Hieronymus, p. 230].  A professional inert-gas flush and vacuum packaging results in very little oxidation for whole cones or pellets.)  A decrease in the amount of alpha acids is mirrored by a corresponding increase in the amount of oxidized alpha acids.  The alpha acids also undergo some amount of oxidation while still on the bine [Hieronymus, p. 233] and further during the warm and highly oxygenated conditions of hop drying [e.g. Hieronymus, p. 126], and so the level of oxidized alpha acids when we get our newly-dried hops soon after harvest can be greater than zero [Maye, p. 23].  Finally, oxidized alpha acids are created during the boil [Algazzali, p. 17; Dierckens and Verzele, p. 454].

If it’s correct that oAA production during the boil is limited by the surface area of the lupulin glands (Section 3.4.1), this suggests that the total oxidation of alpha acids is not simply a sum of the oxidation of freshly-dried, stored, and boiled hops.  Instead, with oxidized alpha acids produced during the boil (oAAboil) limited by surface area, there is the implication that oAAboil represents oxidized alpha acids on the lupulin-gland surface that have not yet been oxidized during drying and storage. If storage has oxidized alpha acids below the surface of the lupulin glands, then the boil will not produce any new oxidized alpha acids. So, if X% of the alpha acids oxidize during drying and storage, and Y% of the alpha acids oxidize during the boil, then the total amount of oxidized alpha acids will be the greater of X% or Y%, not their sum.

We can then model the level of oxidized alpha acids (oAA) in the wort as a combination 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 are produced during the boil:

oAA = max(oAAfreshoAAstorage , oAAboil) [33]

where oAA is the total level of oxidized alpha acids in the wort, 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 the weight of the hops.  (The result of the max() function is the maximum of the two values.)

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 3.5 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 0.22.  This leaves oAAboil as the only parameter without any estimated value.  How quickly are oxidized alpha acids produced during the boil?  Dierckens and Verzele say that “humulinone formation on wort boiling can be one of the first things to happen in the complex chemistry of humulone isomerization oxidation” [Dierckens and Verzele, p. 454], so I’ll assume that oAAboil reaches its maximum quickly and does not need to be a function of boil time.  (This assumption is consistent with the data presented in the post The Production of Oxidized Alpha Acids at Hop-Stand Temperatures.)  We can then re-write oAA using different functions to replace oAAfresh, oAAstorage, and oAAboil:

oAAagescale = 0.22 [34]
oAA = max(AAharvest × ((1 – 1/ek×1×1×3.5) + (oAAagescale × (1 – AAdecayfactor))), AAharvest × AAdecayfactor × F(form) × LFhoppingRate([AA]0) × oAAboilFactor) [35]

where oAAagescale is the age-related scaling factor for the oxidized alpha acids in stored hops, oAA is the same level of oxidized alpha acids in Equation [33], AAharvest is the percent of alpha acids in the hops at harvest, k is the variety-specific hop decay factor from the Garetz model, AAdecayfactor is the alpha-acid decay factor from Equation [18], F(form) is the adjustment factor depending on form of the hops (1.0 for cones and approximately 2.0 for pellets, as described in Section 3.4.1), LFhoppingRate([AA]0) is the loss factor at high hopping rates (because alpha acids above the solubility limit are quickly degraded and do not produce oxidized alpha acids) as a function of initial alpha-acid concentration [AA]0, and oAAboilFactor is the relative amount of oAA produced during the boil.  Since oxidized alpha acids are very soluble [e.g. Maye, p. 23], all of the oxidized alpha acids already in the hops and produced during the boil are in the wort shortly after being added to the kettle.

That leaves us with two other oAA factors that we still need to account for: losses and a scaling factor.  I have not yet been able to find any description of the losses of oxidized alpha acids during the boil and fermentation, so this is an 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, and so the loss factors for boiling, gravity, fermentation, flocculation, filtering, and age are approximately the same for oAA and IAA.  I have estimated loss factors specific to nonIAA (different from IAA loss factors) for the effects of pH and krausen.  For pH, I’ve estimated the loss factor as LFpH_nonIAA(pH) = 0.8948 × pH − 4.145, where pH is the post-boil wort pH.  For krausen, the loss factor for nonIAA components, LFkrausen_nonIAA(krausenLoss), is estimated as 3.0 × LFkrausen_IAA(krausenLoss).  I’ve also estimated that oAA accounts for the majority of nonIAA (80% in a “typical” beer), and so these two nonIAA loss factors are mostly oAA loss factors.  We can therefore approximate the concentration of oAA in beer as the concentration in wort multiplied by the product of these various loss factors:

[oAA]wort = oAA × W × 1000 / V [36]
[oAA]beer = [oAA]wort × (LFboil × LFOG(OG, t) × LFpH_nonIAA(pH) × LFferment × LFflocculation(flocculation) × LFkrausen_nonIAA(krausenLoss) × LFfiltering(MR) × LFage(ageweeks)) [37]

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, LFboil is the loss factor during the boil, LFOG is the loss factor for specific gravity, LFpH_nonIAA is the nonIAA-specific loss factor for wort pH, LFferment is the loss factor for fermentation, LFflocculation is the loss factor associated with flocculation characteristics, LFkrausen_nonIAA is the nonIAA-specific loss factor for krausen, LFfiltering is the loss factor for filtering, and LFage is the loss factor for age of the beer at room temperature.

We also need a scaling factor in Equation [32] 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 0.0130/0.0142, or 0.9155:

scaleoAA = 0.9155 [38]

Despite the large number of parameters for modeling oAA, we end up having no estimate of only one: oAAboilFactor.  We may also want to improve upon the initial estimate of oAAagescale.

3.4.3 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 produced during storage potentially contribute as much or 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].  While oxidized beta acids are further oxidized into non-bitter hulupinic acid [Almaguer, p. 295], oxidized beta acids can still contribute significantly to the IBUs of finished beer, and so this second oxidation process must occur over a period of weeks or months. Two experiments that looked at the impact of oxidized beta acids on the IBU found practically no oxidized beta acids produced during the boil, but that oxidized beta acids produced during storage have roughly the same contribution as oxidized alpha acids produced during the boil.  We can model the impact of oxidized beta acids in a way similar to oxidized alpha acids: there are oxidized beta acids occurring in fresh hops, created during storage, and (potentially) produced during the boil [Algazzali, p. 17; Stevens and Wright p. 496; Hough et al., p. 490]:

oBA = max(oBAfreshoBAstorage , oBAboil) [39]

where oBA is the level of oxidized beta acids, 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.  This model assumes that oxidization of beta acids during the boil is limited by the surface area of the lupulin glands, as with alpha acids.

Stevens and Wright say that oxidized beta acids are present at not more than 0.5% of the weight of the cone [Stevens and Wright, p. 500], Spetsig and Steninger note up to 3% [Spetsig and Steninger, p. 413], and Mussche found oxidized beta acids up to 1% of the weight [Mussche, p. 13].  Peacock implies that the beta acids undergo oxidation losses at approximately the same rate as the alpha acids [Peacock, p. 162].  Given the wide range of reported values of (oBAfreshoBAstorage), I’ll assume that oxidized beta acids are produced at the same rate as oxidized alpha acids both in fresh hops and during aging; this sum should be in the ballpark of 0.5% to 3% of the weight of the cone.  Oxidized beta acids may be produced during the boil, but only to a very limited extent, if at all.  All of this gives formulas similar to Equations [34] and [35]:

oBAboilFactor = 0.071 [40]
oBA = oBAboilFactor × BAharvest × ((1 – 1/ek×1×1×3.5) + (oBAagescale × (1 – BAdecayfactor))) [41]

where oBAboilFactor is the amount of oxidized beta acids that survive the boil, which has been estimated at 0.071; oBA is the same level of oxidized beta acids in Equation [39], BAharvest is the percent of beta acids in the hops at harvest (which can be estimated from the alpha-acid rating (AA) divided by 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])), k is the variety-specific hop decay factor from the Garetz model, oBAagescale is the age-related scaling factor (assumed to be the same as oAAagescale), and BAdecayfactor is the beta-acid decay factor for the freshness of the hops (assumed to be the same as AAdecayfactor).

That (again) leaves us with two other oxidized beta-acid factors that we still need to model: losses and a scaling factor.  It seems reasonable to assume that oxidized beta acids are lost to trub, yeast, and in other ways, just as isomerized alpha acids and (presumably) oxidized alpha acids are lost.  We can then model the oxidized beta-acid losses similar to the way losses of oxidized alpha acids are modeled.  In other words,

[oBA]wort = oBA × W × 1000 / V [42]
[oBA]beer = [oBA]wort × (LFoBA_boil × LFOG(OG, t) × LFpH_nonIAA(pH) × LFferment × LFflocculation(flocculation) × LFkrausen_nonIAA(krausenLoss) × LFfiltering(MR) × LFage(ageweeks)) [43]

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, LFoBA_boil is the loss factor for oxidized beta acids during the boil, LFOG is the loss factor for specific gravity, LFpH_nonIAA is the nonIAA-specific loss factor for wort pH, LFferment is the loss factor for fermentation, LFflocculation is the loss factor associated with flocculation characteristics, LFkrausen_nonIAA is the nonIAA-specific loss factor for krausen, LFfiltering is the loss factor for filtering, and LFage is the loss factor for age of the beer at room temperature.

We also need a scaling factor in Equation [32] 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).  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 0.85 is implied:

scaleoBA = 0.85 [44]

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

According to Shellhammer, IBUs are in the range of 1 to 3 for unhopped beer [Shellhammer, p. 177].  In one experiment, I brewed three beers with no hops (OG 1.050, 1.070, and 1.090) and sent them out for IBU analysis at 1, 3, and 5 weeks after the start of fermentation.  From these nine values, I constructed a formula for predicting IBUs from malt polyphenols.  Because the measured values didn’t change much over the 4 weeks, this formula (IBU = (OG − 1.0) × 25) is dependent on original gravity but not time.  In another experiment, I created six samples that varied in original gravity, wort pH, fermentation, and boil time.  I measured the polyphenol concentration and IBU values of each.  I found that malt-derived IBUs (but not polyphenol concentrations) increase as the wort pH decreases, that the process of fermentation decreases both polyphenol levels and IBUs by about 30%, and that the polyphenol concentrations (but not IBUs) increase during the boil.  There is therefore a non-linear relationship between malt polyphenols and malt-derived IBUs, and this relationship depends on both boil time and pH.  If we are more interested in IBUs than actual polyphenol concentrations, we can map from original gravity to malt-derived IBUs as a function of pH.  We can convert this from a prediction of IBUs to a prediction of (scaled) malt polyphenol concentrations if we multiply by 69.68/51.2:

[PPmalt]beer × scalePPmalt = (69.68 / 51.2) × ((OG − 1.0) × 19.0 × ((2.477 × (pH1pH2)) + 1.0) [45]

where [PPmalt]beer is the estimated concentration of malt polyphenols in the finished beer, scalePPmalt is the scaling factor for light absorption at 275 nm, OG is the original gravity of the beer, pH1 is the pH of the wort before any pH adjustment, and pH2 is the post-boil pH.  This estimated concentration of malt polyphenols will not be accurate, because it does not depend on the boil time (and it does depend on pH), but for modeling IBUs this inaccuracy doesn’t matter.  Furthermore, we don’t need to determine the separate values of [PPmalt]beer and scalePPmalt; knowing their product is sufficient.

We can then update our estimate of IBUs in beer to separate out the contributions from hops and malt polyphenols:

IBU = 5/7 × ([IAA]beer + (([oAA]beer × scaleoAA) + ([oBA]beer × scaleoBA) + ([PPhops]beer × scalePPhops) + ([PPmalt]beer × scalePPmalt))) [46]

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; Verzele and De Keukeleire, p. 9], although McLaughlin reports a higher range, from 4% to 14% [McLaughlin, p. 3].  After having been added to the wort, polyphenols are removed “extensively by precipitation with proteins during wort boiling”; 80% of hop flavanols are removed in the trub when boiling hopped wort [McLaughlin, p. 7].  As Noonan phrases it, “the rolling motion of the boil causes the malt proteins to collide with and adhere to the sticky hop polyphenols” [Noonan, p. 158].  (It may be that the polyphenols are not so much removed as largely insoluble in wort.  The largest polyphenol group in hops (prenylflavonoids) are not soluble in water; all other hop polyphenol components are “soluble in water, preferably in hot water” [Forster, p. 124].  The prenylflavonoids make up about 75% to 85% of all hop polyphenols [Forster, p. 124], so only about 20% of the hop polyphenols are soluble, corresponding to 80% removal.)  Then, polyphenols are removed during fermentation, and “it seems possible that this could occur in much the same way as it does with the iso-alpha-acids” [McLaughlin, p. 7].

From this, we can construct a rough model of the concentration of hop polyphenols in wort and in beer, with an initial level of polyphenols at about 4% of the weight of the hops, a precipitation-loss factor (or solubility factor) for polyphenols in the wort during the boil (LFPP) estimated at 0.20, and loss factors for fermentation, flocculation, krausen, and filtering:

[PPhops]wortPPrating × W × 1000 / V [47]
LFPP_precipitate = 0.20 [48]
[PPhops]beer = [PPhops]wort × LFPP_precipitate × LFPP_ferment × LFflocculation(flocculation) × LFkrausen_nonIAA(krausenLoss) × LFfiltering(MR) [49]

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_precipitate 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 LFhop_ferment, LFflocculation, LFkrausen_nonIAA, and LFfiltering are the loss factors for polyphenols associated with fermentation, flocculation, krausen, and filtering, respectively.  Assuming that malt and hop polyphenols share some characteristics, we set the loss factor for hop polyphenols during fermentation (LFPP_ferment) to be the same as malt polyphenols, which has been estimated at 0.70, and assume that polyphenols aren’t lost as the beer ages.  Other loss factors (flocculation, krausen, and filtering) are assumed to be the same as for other nonIAA.

Finally, we need a scaling factor to use with the concentration of hop polyphenols in Equation [46].  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 [50]). We can consider Equation [46] 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 PPmalt component of zero (Equation [51]).  Since Equations [50] and [51] both measure IBUs from the contribution of only hop polyphenols, we can determine the value of the scaling factor for hop polyphenols (Equation [52]):

IBU = [PPhops]beer × 0.022 [50]
IBU = 5/7 × (0 + 0 + 0 + ([PPhops]beer × scalePPhops) + 0) [51]
scalePPhops = 7/5 × 0.022 = 0.03 [52]

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

3.4.5 Solubility of nonIAA Components
The nonIAA components (specifically, oxidized alpha and beta acids, and soluble hop and malt polyphenols) are soluble in water [e.g. Lewis and Young, p. 265; Forster, p. 124].  They do not require isomerization, which (for alpha acids) takes a significant amount of time.  Therefore, they appear to contribute quickly to the measured IBU value.  This is of particular significance for hops that are added late in the boil (or at flameout, or after flameout), since they will have a significant amount of their nonIAA components quickly dissolved and contributing to IBUs, whereas the IAA level will be low due to insufficient time for isomerization.  As a result, the ratio of IAA to all bittering substances can be much lower for hops added close to flameout, even for very fresh hops.  In short, the 1960s finding that the concentration of IAA is 5/7 of the total concentration of all bittering substances reflects not only the age and storage conditions of 1960s hops, but also the typical time(s) at which hops were added to the boil in the 1960s.  Depending on the initial concentration of alpha acids, well-preserved hops added at flameout (with 10 minutes of cooling after flameout) may produce 20 IBUs, but less than half of that value may come from isomerized alpha acids.

3.5. The Impact of Dry Hopping on IBUs
Dry-hopping adds another layer of complexity to the IBU picture. On the one hand, no isomerized alpha acids are produced during dry hopping, and so there is no IAA contribution to the IBU from hops added on the cold side. On the other hand, alpha acids, oxidized alpha- and beta acids, and polyphenols can contribute to an increase in IBUs, while the use of dry hopping has been shown to reduce the concentration of IAA created from kettle hop additions. We will look at each factor in turn.

Alpha acids do not normally survive the boil and fermentation process to end up in finished beer [e.g. Lewis and Young, p. 259]. However, alpha acids added when dry hopping can be present in the final beer [e.g. Maye, p. 26; Lafontaine, p. 56].  Scott Lafontaine measured the concentration of alpha acids in beer as a function of dry hopping rate, and found that about 1% of available alpha acids end up in beer [Lafontaine, p. 56]. A reasonable fit to his data is:

[AA]beer = 14.7 × (1.0 − e(-0.00102 × [AA]added)) [55]

where [AA]added is the concentration of alpha acids added to the beer in the dry hop addition and [AA]beer is the concentration of alpha acids ending up in the finished beer. The solubility limit of alpha acids at the pH of beer (around 4.25 to 4.5) is about 14 ppm [Spetsig (1955) p. 1423; Shellhammer, p. 170]. Alpha acids absorb light at 275 nm and therefore contribute to the IBU measurement, but are not bitter [Shellhammer, p. 169]. This creates a discrepancy between the IBU measurement and perception of bitterness. This discrepancy can be up to 10 IBUs at the solubility limit, which corresponds to a hopping rate of about 2800 g/hL of hops with an AA rating of 10%, or 22 oz in a 5-gallon batch. At less extreme but still very high dosing rates (e.g. 1000 g/hL of 10% AA hops, or 8 oz in 5 gallons) the non-bitter alpha acids will add about 6.3 IBUs.

Oxidized alpha acids increase in hops during storage, with a slower increase when using better preservation techniques (lower temperature, vacuum packaging, and nitrogen flushing) [Garetz article]. These oAA can dissolve into the beer when dry hopping. While the rate of oxidation is variety-specific [Maye, p. 24], fresh hops with a low hop storage index (HSI) have oAA concentrations of about 2% of the alpha-acid concentration [Lafontaine p. 56; Maye p. 24 Fig. 4 and 5]. Over time and with a rate dependent on storage conditions, the HSI and the concentration of oxidized alpha acids increase. For example, with an increase in HSI from 0.27 (corresponding to a “freshness factor” of 0.96) to 0.34 (or a freshness factor of 0.85), oAA concentrations increased from about 2% to over 3% of alpha acids in Galena hops [Maye, p. 24]. While 2% of the alpha acids is a small amount, high rates of dry hopping can yield a significant increase in the IBU. For example, 4 oz (113.4 g) of hops with AA rating 10% added to 5 G (18.93 L) of beer corresponds to 600 ppm of alpha acids. If 2% of those are oxidized, that’s 12 ppm of oAA. With an oAA scaling factor of 0.9155 and multiplying by 51.2/69.68 to convert from IAA-equivalent concentration to IBUs, there will be an increase of 8 IBUs from this addition of oxidized alpha acids. If the oxidized alpha acids are 3% of the alpha acids, then there will be an increase of 12 IBUs from oAA. However, there is a saturation of oAA [Maye, p. 25; Lafontaine p. 56; Hauser, p. 113], and so when the concentration of hops is greater than about 2200 ppm the concentration of oAA may be reduced. A rough model for this saturation, based on data from Maye [p. 25] and Lafontaine [pp. 55-56] is:

saturationFactoroAA = 1.1181 × e(-0.0000506 × [hops]) [56]

where [hops] is the concentration of dry hops added to the beer (in ppm) and saturationFactoroAA is the factor for reducing the concentration of oAA when [hops] is greater than 2200 ppm.

Oxidized beta acids behave similarly to oxidized alpha acids, being produced during storage and dissolving into beer. I have found that about 7% of the available oxidized beta acids end up in beer when added to boiling wort. Assuming that temperature is not a significant factor, we can infer that the same percentage of available oxidized beta acids are added to beer when dry hopping. Very fresh hops may have almost no oxidized beta acids, but with a beta-acid level of 5% at harvest, a freshness factor of 0.75, and a scaling factor of 0.85, 4 oz (113.4 g) of hops added to 5 G (18.93 L) of beer may yield an increase of 4 IBUs.

Polyphenols from dry hopping may also contribute to the IBU. Assuming that polyphenols dissolve into beer in the same way that they dissolve into boiling wort (although perhaps over a longer time period due to the lower temperature), we can use the equations in Section 3.4.3 to model the polyphenol contribution when dry hopping. One modification to this model is that polyphenols may reach saturation when added at very high dosing rates [Maye, p. 25; Hauser, p. 113]. We can use Hauser’s data to construct a quadratic model of the saturation factor:

saturationFactorPP = -7.47×10-10 × [hops]2 + 0.00000542 × [hops] + 0.99733 [57]

where saturationFactorPP is the factor for reducing the concentration of polyphenols. If saturationFactorPP is greater than or equal to 1, then there is no reduction in polyphenol concentration.

Finally, it has been noted that dry hopping may lower the concentration of isomerized alpha acids in beer [Parkin, p. 34; Maye pp. 25-26; Hauser p. 113], thereby reducing the IBU.  The reports are generally consistent that only higher-IBU beers show a reduction in IAA [Maye, p. 25; Lafontaine, p. 55], but the published data are too inconsistent to formulate a reasonable model for IAA reduction.

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.  We do, however, have results from a study by Val Peacock that looks at IBUs and IAA concentration as a function of hop storage conditions [Peacock, p. 162], results from 46 experiments where I’ve measured IBUs with varying hop steeping times, amounts, and temperatures, and measured IBUs from several late-hopped IPAs.  We can use this set of data and the model assumptions described previously, along with common techniques for searching a parameter space, to obtain an estimate of the unknown parameter values.

4.2 Sources of IBU Data
4.2.1 Peacock Hop-Storage Conditions
In an article describing IBUs, Peacock provides results of a study that looked at how the storage conditions of hops affected IBU levels [Peacock, p. 162].  He lists four storage conditions (ranging from -20°F (-29°C) to 70°F (21°C)), the relative percent of alpha and beta acids lost (based on the Hop Storage Index), the IAA levels in the finished beer, and the IBUs of the finished beer. He also provides the alpha/beta ratio of the hops used, but not the amount of hops, wort volume, or original gravity.  I assume (without evidence) that he used a 2-barrel pilot system with a 5% evaporation rate and a counterflow wort chiller that takes 20 minutes to completely transfer all wort.  From these assumptions, I searched for a weight of hops that fits his observed IBU and IAA values.

4.2.2 Personal Experiments
I conducted a series of 46 experiments that evaluated IBUs as a function of hop steeping time, hopping rate, wort temperature, wort pH, krausen, hop form (cones or pellets), original gravity, and other factors.  These experiments yielded over 330 measured IBU values.  In addition, I measured IBUs from three IPAs that used a fairly large amount of late-addition hops with a 10-minute hop stand.  The amount of raw data is too large to post here, but a JSON-format text file with all parameter values and measured IBUs can be found at https://jphosom.github.io/alchemyoverlord/js/trainingData.js.  Most of the experiments are described in detail on this blog.

One of the biggest difficulties in these experiments was obtaining accurate alpha-acid levels of the hops at harvest and good estimates of how storage impacted alpha-acid levels.  Part of this difficulty was caused by the use of improperly-stored hops for several of the earlier experiments, but (as mentioned in Section 3.1) Verzele and De Keukeleire note that “there are easily differences up to 15 − 20% in alpha acids content between and within bales of a single hop delivery” [Verzele and DeKeukeleire, p. 331], which makes precise measurement of the alpha acids in a small amount of hops a nearly futile task.  As a result, for most experiments I allowed the IBU model parameter search (Section 4.3) to evaluate ±10% percent around the best estimate of alpha-acid levels at harvest.  I also allowed some flexibility in the hop degradation factor (AAdecayfactor in Equation [18]).

It is worth noting that the volume of wort used in these experiments varied from about 1 to 8 gallons (4 to 31 liters) (e.g. An Analysis of Sub-Boiling Hop Utilization and Hop Cones vs. Pellets: IBU Differences).  Despite a nearly order-of-magnitude variation in volume between the experiments, the model does not show a significant difference in error as a function of volume.  This supports the claim made in Section 3.3 that the size of the volume or boil kettle has no impact on utilization.

4.3 Parameter Estimation and Results
Using 4 IAA values from Peacock (assuming values for original gravity, volume, boil time, and post-boil time, and fitting the weight of the hops to the data), 327 measured IBU values from my experiments, and 3 IBU values from the IPAs, there are 347 data points with which to estimate the unknown parameter values, as well as a number of experiment-specific parameter values (e.g. weight of the hops in Peacock’s study or alpha-acid rating in most other experiments).

The three parameters with the most uncertain estimates are LFboil, oAAboilFactor, and oAAagescale.  In addition to searching for suitable values for these parameters, I searched for values for [AA]limitMin and [AA]limitMax in order to use all available data to improve the estimates of these two parameters.

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 search interval of 0.01 for the first three factors and 10.0 for the solubility-limit factors.  A nested recursion was used to constrain the five unknown model parameters in an iteration to be the same for all sets of data, while the unknown parameters from each experiment (i.e. specific alpha-acid rating and hop degradation factor) were searched for individually.

The result of this parameter search is not guaranteed to be the correct solution.  Instead, the model and parameter settings provide a “most likely” set of values.  The correctness of the values depends on the quality of the model and on the amount and quality of the data.  We have a very large number of assumptions and a fairly large number of estimated parameters.  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 results from the parameter search are: LFboil = 0.51, oAAboilFactor = 0.11, oAAagescale = 0.33, [AA]limitMin = 200 ppm, and [AA]limitMax = 580 ppm.  The root-mean-square (RMS) error over all 347 values is 1.57 IBUs, with a maximum difference of 7.14 IBUs for one condition in an experiment looking at the use of pellets with a high hopping rate.  This experiment had an initial alpha-acid concentration of about 675 ppm and measured IBU values of 67, 72, 81, and 102, and so the modeled IBU values are still within 10% of the measured values.  The second-largest difference between modeled and measured IBUs was 4.94 in an experiment with a fairly high initial concentration of alpha acids (~485 ppm).

5. Discussion of Results
With a human detection threshold of 5 IBU [Daniels, p. 76], only one of the 347 values is different enough from the measured value to be detectable by a human palate.  Note that these results are on the data used to estimate the model parameters, not on unseen test data.  Results on test data will necessarily be worse.

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, post-boil volume 5.25 G or 19.87 liters, a typical AA rating of 8.8%, exceptionally well-preserved hop cones 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 (56.7 g) added at flameout will create 14.65 IBUs with a concentration of 8.35 ppm of IAA (41% of the IBU total) that creates 5.99 IBUs.  The oAA contribute 7.27 IBUs, the oBA contribute 0.0 IBUs, the hop polyphenols contribute 0.35 IBUs, and the malt polyphenols contribute 1.04 IBUs.  The same 2 oz added at 60 minutes will create 44.6 IBUs with a concentration of 50.24 ppm of IAA (81% of the IBU total).  If we triple the amount of hops, from 2 oz (56.7 g) to 6 oz (170.1 g), the IBUs only increase from 44.6 to 90.90 (103.2 ppm of IAA, representing 81% of the total).  If we add those 6 oz (170.1 g) at flameout, we’ll get 27.8 IBUs, with only 16.2 ppm of IAA (42% 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, in the detailed model, if we have 150 ppm of alpha acids (from 2.0 oz (56.7 g) of 5.26% AA hops in 5.25 G (19.9 L) of wort) and boil them for 0 minutes with instant cooling (creating no IAA), we get 5.98 IBUs (with 4.58 IBUs from oAA, 1.04 IBUs from malt, 0.35 IBUs from hop polyphenols, and 0.0 IBUs from oBA).  This is 8.33 ppm of (scaled) nonIAA (scaled to be equivalent to IAA ppm).  This 8.33 ppm of nonIAA in 150 ppm of alpha acids yields the equivalent of 5.6% utilization (8.33/150.0=0.056).  So, nonIAA in this example contribute to about 6% of the apparent utilization (and this value does not vary with boil time).  In general, then, one can think of the nonIAA components as contributing about 5% or 6% of the utilization in the Tinseth formula.  (In other words, if the Tinseth utilization is 0.22 (22%), then 0.06 (6%) of that can be thought of as coming from nonIAA components, and the remaining 0.16 (16%) from IAA.)  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, presumably implicitly accounting for nonIAA components at short boil times.

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

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

The topics discussed here have been implemented in an IBU calculator called SMPH.

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