This blog post is excessively long. In order to make it somewhat more manageable, here are links to the various sections:
1. Introduction
2. Definitions of IBUs
xxxxx2.1 IBU Definition from the American Society of Brewing Chemists (ASBC)
xxxxx2.2 IBU Definition from Val Peacock
3. A General Description of Factors Affecting IBUs
xxxxx3.1 Concentration of Isomerized Alpha Acids (IAA) Under Ideal Conditions
xxxxx3.2 Accounting for PostBoil Utilization
xxxxx3.3 Adjustments to the Concentration of Isomerized Alpha Acids
xxxxx3.4 A Revised IBU Formula for nonIAA Components
xxxxxxxxxx3.4.1 Oxidized Alpha Acids
xxxxxxxxxx3.4.2 Oxidized Beta Acids
xxxxxxxxxx3.4.3 Polyphenols
xxxxxxxxxx3.4.4 Solubility of nonIAA Components
4. Available Data, Parameter Estimation, and Results
xxxxx4.1 Overview
xxxxx4.2 Sources of IBU Data
xxxxxxxxxx4.2.1 Tinseth Utilization
xxxxxxxxxx4.2.2 Peacock HopStorage Conditions
xxxxxxxxxx4.2.3 Personal Experiments
xxxxx4.3 Parameter Estimation and Results
5. Discussion of Results
6. Summary
References
1. Introduction
This blog post presents a summary of some of the factors that affect the measurement of International Bitterness Units (IBUs) (or simply “Bitterness Units” (BU) if you’re already international). In order to provide as cohesive a summary as possible, I provide both qualitative and quantitative descriptions of these factors. The purpose of the quantitative model is descriptive, not predictive. In other words, the information here may be helpful in understanding how certain factors affect IBU values, but it may not be sufficient to predict the IBU level of your beer much better than existing predictive formulas (e.g. the Tinseth formula). With so many interrelated factors and guesses of appropriate values for many factors, there is a very good chance that IBU values predicted from this quantitative description will not be the same as measured IBU values. If, however, you simply want to get a better understanding of what components contribute to an IBU value, how the storage conditions and amount of hops used may impact IBUs, or how late hopping may decrease the relative proportion of isomerized alpha acids, then this might be the blog post for you.
The more I learn about hops, the more complex the topic becomes, with a seemingly neverending level of detail. If you’re familiar with Alice In Wonderland, then this blog post goes only one level down the IBU rabbit hole, and it looks briefly through a number of open doors at that level without going through them. In other words, there’s a lot of research, chemistry, opinions, known unknowns, unknown unknowns, and contradictions that I’m not going to touch on. If you’re not familiar with Alice In Wonderland, then think of this post as an impressionist painting: if you stand back far enough, you should be able to see a complete picture. If you look too closely and focus too much on the details, however, things that make sense in their relationship to other things may become, when isolated from the larger context, meaningless splotches.
The IBU measurement itself is not always highly regarded. While it is often reported to be correlated with the bitterness of beer (e.g. [Priest and Stewart, p. 266]), the perception of bitterness is not linear (especially at high bitterness levels [Hieronymus, p. 184]), bitterness may have different qualities not captured by the IBU measurement [Peacock, p. 163], and the correlation between IBU levels and bitterness doesn’t hold up under every circumstance (e.g. with dryhopping [Maye et al., p. 25]). On the other hand, it is a universallyknown and (sometimes grudgingly) accepted quantitative measurement. This post doesn’t touch on the pros and cons of the IBU, but, accepting it at face value, tries to break it down into various components and relationships.
This post provides a summary of a large number of sources, including Val Peacock’s article “The International Bitterness Unit, its Creation and What it Measures” in Hop Flavor and Aroma (ed. Shellhammer); Mark G. Malowicki’s Masters thesis, Hop Bitter Acid Isomerization and Degradation Kinetics in a Model WortBoiling System; Michael L. Hall’s article “What’s Your IBU” in Zymurgy (1997); Michael J. Lewis and Tom W. Young’s chapter “Hop Chemistry and Wort Boiling” in Brewing; Mark Garetz’ article “Hop Storage: How to Get – and Keep – Your Hops’ Optimum Value” in Brewing Techniques, and his book Using Hops; Stan Hieronymus’ book For the Love of Hops; J. S. Hough et al.’s Malting and Brewing Science (volume 2); and many other theses, print, and internet sources. If you look at the bibliography, you’ll see many publications produced under the guidance of Thomas Shellhammer. I’ve tried to cite appropriately, and I’ve put the full bibliography at the bottom of this post. I’ve omitted a lot of interesting details from these sources, in order to maintain a more focused narrative.
2. Definitions of IBUs
2.1 IBU Definition from the American Society of Brewing Chemists (ASBC)
Because of the complexity of hops and IBUs, it’s probably a good idea to start at a top level of description, which is deceptively simple but not very informative: An IBU is a measurement of the amount of absorption of light at 275 nm (abbreviated as A_{275nm}) 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 isooctane (TMP) and also diluted in octyl alcohol and hydrochloric acid [American Society of Brewing Chemists], i.e. “acidified beer.” In mathematical form, we can say:
IBU = A_{275nm}(beer) × 50 
[1] 
where IBU is the resulting IBU value, “beer” indicates the substance being analyzed (after proper acidification), and A_{275nm}(beer) is the amount of light absorbed at 275 nm from a sample of acidified beer [Peacock, p. 158].
This measurement has been found to correlate well with the perception of bitterness in beer. As Lewis and Young state, “the value for [IBU] is a good representation of the sensory bitterness of beer.” [Lewis and Young, p. 266]. Why does this correlation exist? There are three intertwining factors: (1) the absorption of light at a particular (ultraviolet) frequency (275 nm) through a sample, (2) the concentration of certain substances in this acidified beer that absorb light at this frequency, and (3) the perception of bitterness that is associated with these substances. This blog post pretty much ignores the first and third factors, assuming that it is predominately those substances that absorb more light at this frequency that have a bitter taste in beer. What this post focuses on, then, is the second factor: the concentration of substances in acidified beer that absorb light at 275 nm. In the development of the IBU measurement, there was a deliberate decision to include not only the bitter isomerized alpha acids (abbreviated here as IAA) that are produced during the boiling of hops in wort, but also other “bittering substances” that contribute to the perception of bitterness [Peacock, p. 159], and which happen (by happy circumstance) to absorb light at 275 nm (as isomerized alpha acids do).
The amount of absorption of light at 275 nm by a sample of acidified beer, multiplied by 50 (or, more precisely, 51.2), was found to provide a good approximation to the concentration of isomerized alpha acids in typical beer of the 1960s (when the IBU measurement was developed) [Peacock, p. 161]. So, we can say:
[IAA]_{beer1960s} = A_{275nm}(beer_{1960s}) × 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 “beer_{1960s}” on the righthand side of the equation indicates that we’re measuring the absorption of a certain type of beer. (Note that beer contains a number of types of substances that absorb light at 275 nm; IAA is the usually predominant, but not only, substance [Peacock, p. 159].) The IBU value can approximately equal the concentration of IAA (i.e. Equations [1] and [2] can be approximately equal), but generally only for hops and boiling times typical of the 1960s, because of the relative concentrations of other bittering substances.
If one has a solution that contains only isomerized alpha acids and no other substances that absorb light at 275 nm, the concentration of IAA can be estimated with the following equation [Peacock, p. 161]:
[IAA]_{IAAsolution} = A_{275nm}(IAAsolution) × 69.68 
[3] 
where [IAA]_{IAAsolution} is the concentration of isomerized alpha acids in this solution, and “IAAsolution” on the righthand side of the equation indicates that the solution being analyzed contains only isomerized alpha acids as the relevant (lightabsorbing) 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).
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 highlevel 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]_{beer1960s} ≈ IBU = 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 “nonisomerized 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 (A_{275nm}(X) = 0), then the concentration of X is zero ([X] = 0). (2) There is a linear relationship between the absorption of light at 275 nm and the concentration of relevant bittering substances in beer, at least within the range of interest. (3) The linear relationship between absorption and concentration is the same for all relevant substances in beer, namely 69.68. The first two assumptions were made by the 1967 Analysis Committee of the European Brewery Convention that developed the unit that became the IBU [Peacock, p. 160161], so they seem reasonable. The third assumption is not necessarily true, but we can modify it for those cases where it isn’t true, so let’s assume it’s true for now.
Let’s start by looking at two beers that have the same amount of absorption of light at 275 nm (i.e. the same level of bitterness): one beer is a (cryogenically preserved) 1960s beer with this bitterness level, and the other beer is something you just brewed:
A_{275nm}(beer_{1960s}) = A_{275nm}(beer) 
[5] 
where beer_{1960s} is our 1960s beer, and beer is the one just brewed.
We can then multiply the numerator and denominator of the lefthand side by 51.2, and multiply the numerator and denominator of the righthand side by 69.89, and the relationship still holds:
(A_{275nm}(beer_{1960s}) × 51.2) / 51.2 = (A_{275nm}(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):
(A_{275nm}(beer_{1960s}) × 51.2) / 51.2 = (A_{275nm}(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 lefthand side and the third assumption for the righthand 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 righthand side equals Peacock’s definition of an IBU, and the lefthand side indicates that this is approximately equal to the concentration of IAA in the 1960s beer:
[IAA]_{beer1960s} ≈ IBU = 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 poorlystored 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} × scale_{nonIAA}:
[IAA]_{beer1960s} ≈ IBU = 5/7 × ([IAA]_{beer} + ([nonIAA]_{beer} × scale_{nonIAA})) 
[11] 
where scale_{nonIAA} is the scaling factor needed to convert the absorptiontoconcentration relationship of nonIAA (696.8 in our example) to the absorptiontoconcentration relationship of IAA (69.68). In our example, scale_{nonIAA} 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]_{beer1960s} ≈ IBU = 5/7 × ([IAA]_{beer} + (([nonIAA1]_{beer} × scale_{nonIAA1}) + ([nonIAA2]_{beer} × scale_{nonIAA2}) + ([nonIAA3]_{beer} × scale_{nonIAA3}))) 
[12] 
where scale_{nonIAA1} is the scaling factor for the first nonIAA substance, scale_{nonIAA2} is the scaling factor for the second nonIAA substance, and scale_{nonIAA3} is the scaling factor for the third nonIAA substance.
The IBU value was designed to be approximately equal to the concentration of isomerized alpha acids (in ppm), given the boiling time, alpha acid levels, and storage conditions of 1960s beer and hops [Peacock, p. 161]. At that time, hops seem to have been stored for long periods of time at cellar or room temperature without special packaging [Peacock, p. 160 and 162]. As Peacock explains, for a typical beer made from typical hops with a typical age and duration of hop boiling, the relative concentration of IAA to all bittering substances (IAA + nonIAA) was about 5/7, or about 71%. In more recent times, it is much more likely that hops are stored at freezing temperatures with less oxygen for less time, which makes the relative concentration of IAA (with a typical 1960s hop boiling time) much higher. So, an IAA concentration of 14 ppm from a 60minute boil might yield an IBU value closer to 12. On the other hand, it is also common now to add a lot more hops much closer to flameout, which increases the relative concentration of nonIAA components in the beer (compared with longer boiling times), as discussed below.
3. A General Description of Factors Affecting IBUs
The preceding descriptions of IBUs actually helped us. Now we know that there are only three things we need to worry about when modeling IBUs: the concentration of isomerized alpha acids (IAA), the concentrations of other bittering substances (nonIAA), and the scaling factors for the nonIAA substances. Thanks to Peacock’s formulation, we’ve moved from the absorption of light at 275 nm (which is very difficult for a homebrewer to predict) to the concentrations of different substances (which we can approximate). This section looks at these three items in more detail.
3.1 Concentration of Isomerized Alpha Acids (IAA) Under Ideal Conditions
A lot of research has been conducted on modeling isomerized alpha acids. We can use this work to estimate the IAA concentration that we need to model IBUs. Mark Malowicki [Malowicki] provides a model for both the conversion of alpha acids into isomerized alpha acids and the subsequent conversion of isomerized alpha acids into other “uncharacterized degradation products”, as functions of time and temperature, under fairly ideal laboratory conditions (with pH 5.2 and an alphaacid concentration of 80 ppm). First, he describes the conversion of alpha acids into isomerized alpha acids as a firstorder reaction following an Arrhenius equation with a temperaturedependent rate constant k_{1}:
k_{1}(T) = 7.9×10^{11} e^{11858/T} 
[13] 
where k_{1}(T) is the rate constant for the conversion of alpha acids into isomerized alpha acids and T is the temperature in degrees Kelvin. A firstorder reaction is of the form [X] = [X]_{0}e^{–kt} (where [X] is the concentration of substance X at time t, [X]_{0} is the initial concentration of X (at time 0), and e is the constant 2.71828…), so we can describe the reduction of alpha acids (due to their conversion into isomerized alpha acids) as:
[AA]_{wort} = [AA]_{0} e^{–k1(T)t} 
[14] 
where [AA]_{wort} is the resulting concentration of alpha acids in the wort at time t (in minutes), [AA]_{0} is the initial concentration of alpha acids (at the start of the boil), and k_{1}(T) is the rate constant from Equation [13]. We can assume that the reduction in alpha acids is mirrored by a corresponding increase in isomerized alpha acids (see [Malowicki p. 6]). Second, Malowicki describes the subsequent conversion of isomerized alpha acids into uncharacterized degradation products, also as a firstorder reaction with a temperaturedependent rate constant:
k_{2}(T) = 4.1×10^{12} e^{12994/T} 
[15] 
where k_{2}(T) is the rate constant for the conversion of isomerized alpha acids into other products (and T is still in degrees Kelvin).
Yarong Huang et al. [Huang 2013] show how to combine these equations into a single model of the cumulative concentration of isomerized alpha acids as a function of time and temperature:
[IAA]_{wort} = [AA]_{0} (k_{1}(T)/(k_{2}(T)k_{1}(T))) (e^{–k1(T)t}e^{–k2(T)t}) 
[16] 
where [IAA]_{wort} is the concentration of isomerized alpha acids in the wort at time t and temperature T. We can plot Huang’s equation in Figure 2, with time on the horizontal axis, relative concentration of isomerized alpha acids (compared with the initial concentration of alpha acids) on the vertical axis, and a few different steeping temperatures represented with different colors:
Figure 2. Theoretical relative concentration of isomerized alpha acids in water, as a function of time and temperature.
This plot at 100°C (212°F) looks reassuringly similar to the utilization of alpha acids in the Tinseth equation for predicting IBUs [Tinseth]; the scale is different, and the shape is somewhat different, but the general trend at boiling is similar.
Equation [16] relies on the initial concentration of alpha acids at the beginning of the boil, which we can determine from the volume of wort (in liters), the weight of hops added (in grams), and either (a) the measured percentage of alpha acids at the time of the boil or (b) the measured percentage of alpha acids at the time of harvest and the degradation of alpha acids over time. These values will give us the concentration of alpha acids in wort (in ppm):
[AA]_{0} = AA × W × 1000 / V 
[17] 
where AA is the alpha acid rating of our hops, scaled to the range 0 to 1 (i.e. AA is the proportion of the hop (cone, pellet, or extract) that is alpha acids, from 0 to 1; e.g. an alpha acid rating of 7% becomes 0.07), W is the weight of the hops in grams, the factor of 1000 converts from grams to milligrams, and V is the volume of the wort in liters. These units combine to give us milligrams of alpha acids per kilogram of wort (since 1 liter of water equals 1 kg; we’ll ignore the extra weight of the extract), or approximately parts per million.
Is V the volume at the beginning, middle, or end of the boil? While [AA]_{0} indicates the initial level of alpha acids (at the beginning of the boil), we don’t have a factor that adjusts for volume changes between the beginning and end of the boil. If we did have such a factor, it would describe the difference between the preboil volume and the postboil volume, since the final concentration of isomerized alpha acids is determined by the postboil 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 postboil volume, and the numbers will come out the same as if we started with preboil volume and then accounted for evaporation. In short: V should be postboil wort volume, before racking.
If we don’t know the alpha acid rating of the hops when we brew our beer, we can use the initial (harvest) estimate with a model of how alpha acids degrade over time, developed by Mark Garetz [Garetz article] to estimate the alpha acid rating for hop cones:
AA = AA_{harvest} × AA_{decayfactor} = AA_{harvest} × 1/e^{k×TF×SF×D} 
[18] 
where AA_{harvest} is the alpha acid rating of the hops after harvest and drying, AA_{decayfactor} is a multiplication factor for how much the AA level has decayed over time (1.0 for fresh hops), k is a value that depends on the percent of alpha acids lost after 6 months at room temperature (which in turn depends on the variety of hops), TF is the temperature factor that describes how degradation is affected by temperature, SF is the storage factor that describes how degradation is affected by storage conditions, and D is the age of the hops, in days. The full definition of all terms is provided in Garetz’s article [Garetz article]. For hop pellets, the rate of deterioration is much slower. Hieronymus says that while whole hops can lose up to 100% of their alpha acids when stored at 68°F (20°C) for one year, pellets lose only 10% to 20% under the same conditions [Hieronymus, p. 230]. If you use pellets that were made immediately after harvest, and they’ve been stored in the refrigerator or freezer, it’s probably safe to assume that losses are somewhere between 5% and negligible, yielding a correction factor between 0.95 and 1.0. If you don’t know how long the hops in your pellets were in wholecone form, or what the storage conditions were, predicting the losses becomes quite difficult.
3.2 Accounting for PostBoil 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 postflameout increase by multiplying the change in IAA concentration at time t by a temperaturedependent 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 postflameout temperature changes. In the current framework, we have a function (Equation [16]) that is already dependent on temperature, so we can take the derivative with respect to time, compute the instantaneous change in concentration at time t and temperature T, and then integrate over time t to arrive back at total concentration of IAA. While the temperature is boiling, we will arrive at the same answer as if we didn’t take the derivative and then integrate. As the kettle cools after flameout, we change the rate constants to reflect the lower rate of isomerization. This can be implemented in about a dozen lines of programming code. First, we need to take the derivative of Equation [16] in order to compute the change in IAA concentration at time t:
d([IAA]_{wort})/dt = [AA]_{0} (k_{1}/(k_{2}–k_{1})) (k_{2}e^{k2t}–k_{1}e^{k1t}) 
[19] 
where d([IAA]_{wort})/dt is the rate at which the IAA concentration changes, in ppm per minute. A model of how temperature decreases after flameout can be obtained by bringing the desired volume of water to a boil, turning off the heat, measuring the temperature at oneminute intervals, and then fitting a line or polynomial to the data. I’ve found that the temperature decrease of a 6gallon (23liter) volume (no lid on the kettle) can be modeled fairly well with a straight line, at least for the first 20 minutes or so:
T_{F}(t_{f}) = 1.344 t_{f} + 210.64 (for temperature in Fahrenheit) 
[20a] 
T_{C}(t_{f}) = 0.74667 t_{f} + 99.244 (for temperature in Celsius) 
[20b] 
T_{K}(t_{f}) = 0.74667 t_{f} + 372.394 (for temperature in Kelvin) 
[20c] 
where T_{F} is the estimated temperature in Fahrenheit, 1.344 is the rate of change (°F per minute), t_{f} is time after flameout (in minutes), and 210.64 is the approximate temperature at flameout (when t_{f} = 0, in °F). Likewise, T_{C} is the estimated temperature in Celsius, 0.7466 is the range of change (°C per minute), and 99.244 is the approximate temperature at flameout (in °C); T_{K} is temperature in Kelvin modeled with 0.74667 degrees Kelvin per minute and a flameout temperature of 372.394 Kelvin. (Note that this formula will only yield reasonable results for a typical homebrewing system with a 6gallon (23liter) volume and an uncovered kettle, and even these “reasonable” results will be affected by factors such as kettle material and size. To maximize accuracy, one should measure the temperature decay of their own system and determine a formula based on systemspecific data. Fortunately, the data I’ve collected indicates that this function is not significantly dependent on ambient temperature or relative humidity, so this function only needs to be constructed once per system.)
We can model total concentration of IAA by integrating the change in [IAA] at each instant, where this amount of change is dependent on the temperature of the wort. Rather than expressing this as a formula, I think a short amount of pseudocode will be easier to understand (referred to as Code [1]), even if you’re not a programmer:
totalTime = boilTime + postBoilTime;
integrationTime = 0.001;
IAA = 0.0;
time = 0.0;
while (time <= totalTime) {
if (time <= boilTime)
temp = 373.15;
else
temp = (0.74667 * (time  boilTime)) + 372.394;
k1 = 7.9*pow(10.0,11.0)*exp(11858.0/temp);
k2 = 4.1*pow(10.0,12.0)*exp(12994.0/temp);
dIAA = AA0 * (k1/(k2k1)) * ((k2*exp(1.0*k2*t))(k1*exp(1.0*k1*t)));
IAA = IAA + (dIAA * integrationTime);
time = time + integrationTime;
}
where totalTime
is the length of the boil in minutes (boilTime
) plus any time after the boil when isomerization might be happening (postBoilTime
). The integration time of 0.001 (called integrationTime
) is sufficient for accuracy to at least two places past the decimal point. Here, IAA
is the total concentration of IAA, or [IAA], after time time
(in minutes). A loop is set up to evaluate (and integrate) all time points from 0.0 to totalTime
in increments of 0.001 minutes, with time
representing the current time instant. The temp
variable is temperature at the current time, in Kelvin. The k1
and k2
variables are the rate constants from Equations [13] and [15]. The variable dIAA
is the derivative of [IAA], or change in [IAA] per minute. The variable AA0
is the initial concentration of alpha acids, in ppm (see Equations [17] and [23]). The pow()
function raises the first argument to the power of the second argument; the exp()
function computes the exponent of its argument. After finishing the loop, IAA
will equal the total concentration of isomerized alpha acids, accounting for both time and (postflameout) temperature.
3.3 Adjustments to the Concentration of Isomerized Alpha Acids
Now we know how to measure the concentration of IAA in wort during the boil under ideal conditions. We can use this as the basis for a quantitative model of IBUs. What we need next is a way to describe the differences between ideal laboratory conditions and (home) brewery conditions. Many factors affect the rate or amount of conversion from alpha acids to isomerized alpha acids: temperature (e.g. boiling at high altitudes), pH of the wort, wort gravity, form of the hops (e.g. extract, pellet or cones; loose or bagged), and alphaacid concentration in the wort. Other factors can be described as losses of IAA that are produced in the boiling wort but never make it into the pint glass: losses during the boil, fermentation, filtration, and aging. We’ll look at each of these briefly in this section.
Before getting into too much detail, this is a good place to define a highlevel term, “utilization.” Hop utilization, U, is the ratio of the amount of isomerized alpha acids in finished beer, divided by the amount of alpha acids added to the kettle, and then multiplied by 100 to convert to percent [e.g. Lewis and Young, p. 266]:
U = 100 × (isomerized alpha acids in beer) / (alpha acids added to kettle) 
[21] 
Temperature and pH: According to Malowicki’s equations (above), a decrease in temperature (e.g. below 100°C) will decrease utilization. If you live at a high enough altitude, your wort will boil at less than 100°C, in which case you might want to adjust k_{2} and k_{2} in Equations [13] and [15], or include a temperaturedependent rate factor, RF_{temp}(T). Postflameout temperature dependencies are discussed above. (Lewis and Young, Palmer, Hieronymus, and others note that the intensity of the boil affects utilization [Lewis and Young, p. 266; Palmer p. 55; Hieronymus, p. 188], which is presumably related to wort temperature.) An increased wort pH will increase utilization [Lewis and Young, p. 266, Kappler p.334]. The dependence on pH, however, shouldn’t impact the typical homebrewer, who should aim for a mash pH in the ballpark of 5.2 to 5.4 [Palmer and Kaminski, p. 60; Noonan, p 144; Fix, p 49].
Wort Gravity: Utilization decreases with increasing wort gravity, at least at higher gravities [e.g. Lewis and Young, p. 266; Hieronymus, p. 188; Hall, p. 62; Daniels, p. 78; Palmer, p. 55; Malowicki, p. 44, Garetz book, p. 130, Hough et al., p. 489]. It is not clear to me if the higher gravity slows the conversion of alpha acids to isomerized alpha acids, or if the higher gravity causes more isomerized alpha acids to bind with trub and settle out of solution. Malowicki did not find a significant change in utilization at specific gravities of 1.000 and 1.040 [Malowicki, p. 39], and Garetz indicates that the lower limit for this effect is a specific gravity of 1.050 [Garetz book, p. 130]. Greg Noonan [Noonan, p. 215] provides a table of utilization as a function of boil time, original gravity, and form of the hops. (His table simply lists “wort density” and “specific gravity”, but he defines wort density as original gravity [Noonan, p. 204].) The original gravity in his table seems to be an independent scaling factor of the other two parameters, with scaling factors of about 1.0, 0.921, 0.865, 0.842, and 0.774 at averaged gravities of 1.040, 1.058, 1.070, 1.080, and 1.090, respectively. A line can be fit through these points to determine an originalgravity correction factor as a function of original gravity:
RF_{OGN}(OG) = (4.944 × OG) + 6.166 if OG > 1.045, else 1.0 
[22] 
where RF_{OGN}(OG) is Noonan’s gravity ratecorrection factor (expressed as an equation instead of the original table form) and OG is the original gravity. If OG is less than or equal to 1.045, RF_{OGN}(OG) is defined as 1. Glenn Tinseth models the gravity correction factor as RF_{WGT}(WG) = 1.65 × 0.000125^{(WG − 1)}, with a scaling factor of 1.0 at around a (typical) wort gravity (WG) of 1.055. (Tinseth uses the term “wort gravity” and suggests using the average of the (initial) boil gravity and original gravity for wort gravity [Tinseth].) Because Malowicki measured the production of isomerized alpha acids in water (with a specific gravity of 1.0), we want to think of any increase in gravity as a reduction in the production of isomerized alpha acids, when compared with Malowicki’s work. Since Noonan’s formula describes higher gravity as always yielding less utilization, his originalgravity correction factor is more suitable for our purposes; it also provides a compromise between the correction factors proposed by Tinseth, Rager, and Garetz [Hall, p. 61].
Form of the Hops: It is often said that whole hops do not provide as much utilization as hop pellets [e.g. Daniels p. 78]. According to Lewis and Young, “the alpha acids dissolve most easily from extracts, less easily from pellets …, and least with whole hops” [Lewis and Young, p. 266]. The higher rate at which alpha acids from pellets dissolve, compared with whole cones, is because “the pelletization process ruptures the lupulin glands and spreads the resins over the hop particles, giving a larger surface area for isomerization” [Hall, p. 58]. Noonan says that “with pelletized hops, ruptured and betterexposed lupulin glands give greater utilization” [Noonan, p. 154]. Garetz indicates that pellets have better utilization up to a boil time of 30 minutes (after which utilization is the same), because after 30 minutes all of the alpha acids have been dissolved, regardless of whether they come from cones or pellets [Garetz book, p. 131].
Hough et al. say that alphaacid extracts are actually much less efficient than whole or pelletized hops: “the solubility of humulone was the limiting factor in its utilization. … In trials using pure humulone, only 5060% of the resin added was isomerized during [the] 1.5 h boil. In contrast, 6575% of the alpha acids present in hops are isomerized in the same period, which supports the view that the isomerization of humulone is catalyzed by the presence of hop cones, break, or even an inert surface such as Celite.” [Hough et al., p. 489, citing Maule, p. 288]. Since Malowicki used alphaacid extract in his experiment (with no added surfaces to serve as a catalyst), the correction factor for the form of the hops in our quantitative description is 1.0 for extracts and about 1.27 (70%/55%) for nonextract forms. (Note that there is 57% isomerization of alpha acids at 90 minutes according to Equation [16], which is very much in line with the statement by Hough et al.)
Expressing whole hops as less efficient than pellets, Noonan provides a wholehop correction factor (in table form) that varies from 0.66 to 1.0, based on boil time and gravity [Noonan, p. 215]. Garetz recommends a correction factor of 0.90 for boil times up to 30 minutes, otherwise a correction factor of 1.0 [Garetz book, p. 141]. Hieronymus says that hop pellets are 10% to 15% more efficient than cones [Hieronymus, p. 188], translating into a correction factor between 0.87 and 0.91 when using whole cones. According to Michael Hall, Mosher specifies a correction factor of 0.75 [Hall, p. 62]. This leaves a wide range of possible correction factors for the use of whole hops compared with pellets (from 0.66 to 1.0), with a median factor of 0.91. For the model of IBUs being built, I’ll assume a factor of 0.91. This wholehop vs. pellet correction factor is in addition to (i.e. multiplied by) the correction factor for nonextracts, 1.27. Therefore, pellets have a correction factor of about 1.27 and whole hops have a correction factor of about 1.16.
Garetz also says that hops kept in a mesh bag during the boil have lower utilization than loose hops, with a correction factor of 0.91 for looselystuffed hops and 0.83 for a full bag. Whole hops in a looselypacked mesh bag would then have a combined correction factor of 1.05 (1.27 × 0.91 × 0.91) [Gartez book, p. 141]. For the model being developed, I’ll assume that bagged hops are always loosely bagged, for a “bagging” correction factor of 0.91.
AlphaAcid Concentration: Along with the form of the hops, the relative amount of hops (and therefore also the relative amount of alpha acids) in the wort affects utilization. As Lewis and Young say, “a high hopping rate reduces extraction efficiency” [Lewis and Young, p. 267]. Daniels phrases this as “simply adding more and more hops does not produce a linear increase in the amount of bitterness produced” [Daniels, p. 85]. Fix also notes that the utilization rate is affected by hop concentration [Fix, p. 47]. Hough et al. say that “hops are utilized more efficiently at low rates” [Hough et al., p. 489]. Maule determined that reduced utilization at higher hop rates can only be accounted for by the “difficulty with which [isomerized alpha acid] enter[s] solution when wort [is] boiled with large amounts of [alpha acid]” [Maule, p. 290], and that “only a small portion of the resin present on the hot break … can be claimed to be truly adsorbed” [Maule, p. 289].
Garetz provides the only source I’m aware of with a quantitative model of the relationship between amount of hops and utilization. He proposes a hoprate correction factor (also described by Hall and Daniels) that depends on volume and “desired IBU” to determine the weight of hops needed [Garetz book, p. 137; Hall, p. 63; Daniels, p. 86]. When I was initially developing this blog post, I used a modified form of his equation to estimate a correction factor based on volume, weight of hops, and alpha acid rating of the hops, since we don’t know the desired IBU when trying to predict an IBU value. However, after some difficulty fitting the IBU model developed in this post to available data, and after further experimentation (to be described in a future blog post), I concluded that Garetz’s correction factor underestimates the effect of alpha acid concentration on utilization. A better fit to the data available to me can be obtained by simply limiting the alpha acids available for conversion to about 260 ppm, requiring a revision of Equation [17]:
[AA]_{0} = AA × W × 1000 / V, with maximum [AA]_{0} = 260 
[23] 
(The value of 260 was obtained, in part, by fitting the complete quantitative model described in this blog post to available data (see Section 4), so this value is a result of the model development.) This limit is greater than the solubility of alpha acids at room temperature (around 90 ppm [Malowicki, Appendix A, pp. 5154]), but it common that solubility increases with temperature [Wikipedia]. Using an alphaacid concentration limiting factor is also in accord with the conclusion reached by Maule (quoted above [Maule, p. 290]). Using this approach, utilization increases linearly until the solubility limit is reached (260 ppm), after which utilization is not affected by an increased presence of alpha acids. This approach is overly simplistic, but seems to work reasonably well on the available data. One unfortunate complication is that separate hop additions can not be treated independently.
Kettle Size and/or Geometry: The kettle size and/or kettle geometry may also impact utilization [Daniels, p. 78; Fix, p. 47]. As Hieronymus says, “larger kettles are more efficient, and the difference between a fivegallon homebrew system and even a 10barrel (310gallon) 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 multibarrel brewing system” [Smith]. It may be that the observed increase in utilization with kettle size is a reflection of longer times between flameout and cooled wort, which is already accounted for in the current model with postboil utilization. In short, kettle size (or wort volume) may (or may not) have an impact on utilization, with a scaling factor ranging from 1.0 (no impact) to 1.5 (large impact). Because of the difficulty of reconciling Malowicki’s use of tiny volumes and resulting high utilization, I assume that kettle size has no impact on utilization. Therefore, the rate factor for kettle size, RF_{size}(V), is assumed to be 1.
Losses During the Boil: Isomerized alpha acids are lost during the boil. As Lewis and Young state, “iso alpha acids react with proteins of wort whence they are partially removed as trub or hot break” [Lewis and Young, p. 266]. Malowicki says that “trub, and specifically the formation of trub, leads to greatly increased losses of bitter acids” [Malowicki, p. 8]. He cites work by H.O. Askew in which the use of preformed trub produced losses of only 5% to 9%, but the formation of trub created losses of 35% [Malowicki, p. 78]. Malowicki also cites Laufer and Brenner who found a 38% loss of bitter acids to trub and a 35% loss to spent hops. Hall cites Hough et al., who cite Maule (1966), saying that “about 7% of the isoalpha acids are removed with the breaks” [Hall, p. 57; Hough et al., p. 489]. Garetz says that “810% of the isoalpha acids are adsorbed (meaning they cling to the surface of) the hot and cold breaks. This number appears to be fairly constant, even given wide variations in the amount of break, composition of the wort, and the method and length of cooling” [Garetz book, p. 126]. In short, the estimated loss of isomerized alpha acids during the boil ranges from 7% to 73%, yielding a correction factor from 0.27 to 0.93, which is a bit too large of a range to be of practical value.
Losses During Fermentation: Isomerized alpha acids are also lost during fermentation [e.g. Hieronymus, p. 190]. Lewis and Young say that “during fermentation, isoalphaacids associate with the surface of the yeast cells present… Isoalphaacids, 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, isoalpha acids are scrubbed by the rising CO_{2} 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 … Isoalpha acids also are bound up by the yeast cells and removed when the yeast flocculates out” [Hall, p. 57]. Daniels says that the amount of loss is dependent on the amount of yeast pitched and the “extent of yeast growth during fermentation” [Daniels, p. 78]. Garetz says that there are two factors, “the total growth of the yeast crop and the amount of time the yeast stays in suspension”, and that there is a 5% variation depending on the flocculation characteristics of the yeast [Garetz book, p. 126]. He also says that if the alpha acids are mixed back into the beer at the right time, utilization is increased by 18% [Garetz book, p. 126], implying typical losses of 18%. Fix (citing Garetz) estimates loss to yeast sediment at 10% to 20% [Fix, p. 49]. Malowicki (citing Laws et al.) reports losses during fermentation from 5% to 17% [Malowicki, p. 8] and also (citing Laufer and Brenner) losses to yeast of 10% [Malowicki, p. 7]. Hieronymus reports losses during fermentation and packaging of 20% [Hieronymus, p. 191]. Tom Nielsen (from Sierra Nevada Brewing Co.) measured the IBUs of wort and finished beer made from 10 types of hops (9 aroma hops and 1 bittering hop) and found a fairly consistent fermentation loss of about 18% (standard deviation approximately 1.6%) [Nielsen, p. 65]. To summarize, there is IAA loss during fermentation ranging from 5% to 20%, yielding a correction factor between 0.80 and 0.95. A factor of around 0.85 is probably the best compromise between all reported values, and so the model being developed here uses 0.85. The flocculation factor suggested by Garetz is 0.95 for highflocculation yeast and 1.05 for lowflocculation yeast [Garetz book, pp. 140141].
Losses During Filtration and Aging: According to Daniels, “any filtration will remove some bitterness … The addition of clarifying agents such as gelatin or PVPP may have a similar effect.” [Daniels, p. 79]. Garetz says that filtering will reduce utilization by 1.25% to 2.5%, for a filtration loss factor of about 0.98 [Garetz book, p. 141]. Hall says that “there are oxidation reactions that can reduce the bitterness of beer over extended storage periods” [Hall, p. 58]. According to Kaltner and Mitter, “over a storage time of 12 months, a degradation of bitter substances in various beers in a range of 10% to 15% could be analyzed” [Kaltner and Mitter, p. 37]. According to Peacock, citing results from Forster et al. (2004), beer loses 18% of isomerized alpha acids and 14% of measured IBUs after 8 months at room temperature [Oliver, pp. 132133, Peacock p. 164]. I am unaware of an existing model of how IBUs decrease with age for homebrewed beer stored in bottles at room temperature (which may have greater oxidation, less filtering, and other differences with commerciallybottled beer). I therefore measured the decrease in IBUs for two homebrewed beers after 1, 2, 6, 7, and 13 weeks from the start of fermentation, and fit the measured IBU decrease over time to a linear function. (I will provide more detail about this function in a future blog post.) A linear fit is probably not optimal, but within the range of two months it provides a reasonable fit to the data available to me. If we assume that isomerized alpha acids and nonIAA components are affected by age at the same rate (which is probably an incorrect assumption [Peacock, p. 163], but not unreasonable as a first approximation), we can model the loss factor for isomerized alpha acids using the same formula determined for IBUs, and include a filtering factor:
LF_{package}(filtering, age_{weeks}) = 0.98 × (1.0 – 0.015 × age_{weeks}) for filtered beer, or
(1.0 – 0.015 × age_{weeks}) for unfiltered beer 
[24] 
where LF_{package}(filtering, age_{weeks}) is the loss factor due to packaging, which encompasses both filtering and age of the beer (in weeks). The loss factor of 0.98 is applied only to filtered beer, and the decrease in isomerized alpha acids over time is modeled with a factor of 0.015 multiplied by the age of the beer in weeks after the start of fermentation, age_{weeks}.
Summary of IAA Adjustments: We can now express the concentration of IAA in the beer as a function of the concentration of IAA in the wort (based on a maximum alpha acid concentration of 260 ppm), multiplied by the various isomerization rate adjustment factors and IAA loss factors discussed above:
RF_{IAA}(T, OG, hopsForm, V) = RF_{temp}(T) × RF_{OGN}(OG) × RF_{form}(hopsForm) × RF_{size}(V) 
[26] 
LF_{IAA}(flocculation, filtering, age_{weeks}) = LF_{boil} × LF_{ferment}(flocculation) × LF_{package}(filtering, age_{weeks}) 
[27] 
[IAA]_{beer} = [IAA]_{wort} × RF_{IAA}(T, OG, hopsForm, V) × LF_{IAA}(flocculation, filtering, age_{weeks}) 
[28] 
where RF_{IAA} is the isomerization rate factor adjustment of isomerized alpha acids, LF_{IAA} is the loss factor for isomerized alpha acids, and [IAA]_{beer} is the concentration of isomerized alpha acids in the finished beer. The rate factor RF_{IAA} is expressed as a combination of other factors, where RF_{temp} is a rate factor for temperature (with temperature T still in degrees Kelvin), if desired; RF_{OGN} is Noonan’s rate factor as a function of original gravity; RF_{form} is the rate factor for the form of the hops (where hopsForm is “pellet”, “loose whole cones”, or “bagged whole cones”); and RF_{size} is the rate factor for kettle size (specified in this case with volume V). The loss factor LF_{IAA} is expressed as a combination of other factors, where LF_{boil} is the loss factor during the boil; LF_{ferment} is the loss factor due to fermentation (with flocculation being “high”, “medium”, or “low”); and LF_{package} is the loss factor due to filtration (with parameter filtering being “unfiltered” or “filtered”) and age (which varies with the age of the beer, age_{weeks}). Note that in general if we have a loss of x%, the loss factor will be (1 – x%/100); for example, a loss of 10% will become a loss factor of 0.90.
The only problem remaining for modeling [IAA]_{beer} is that while we have a good idea of some factors (RF_{temp}, RF_{OGN}) and a rough approximation of others (RF_{form}, LF_{ferment}, LF_{package}), we have very little basis for determining the remainder (LF_{boil} and maximum [AA]_{0}). But we can come back to that problem later.
3.4 A Revised IBU Formula for nonIAA Components
At this point, we have as complete a description as we’re going to get of the concentration of isomerized alpha acids in beer. The other factor in the IBU formula (Equation [12]) is the concentration of “other bittering substances,” which we call nonIAA.
Alpha acids (before isomerization) are neither soluble [e.g. Lewis and Young, p. 259] nor bitter [Shellhammer, p. 169], but as they age and become oxidized, the resulting oxidized alpha acids (oAA) are soluble in wort and bitter [Algazzali, pp. 1415, p. 19, p.45; Maye et al, p. 23; Hough et al., pp. 435436; Hough et al., p. 439; Lewis and Young, p. 265]. Oxidized alpha and beta acids are also produced during the boil [Parkin, p. 11, Algazzali, p. 17; Dierckens and Verzele, p. 454; Oliver p. 471]. Oxidized beta acids (oBA) are also soluble [Algazzali, p. 16] and may be produced and contribute to bitterness in the same way as oxidized alpha acids [Malowicki, p. 2; Peacock, p. 157; Fix, p. 36; Lewis and Young, p. 265; Hall, p. 55; Lewis and Young p. 265; Oliver, p. 132; Oliver, p. 470; Parker, p. 11; Algazzali, p. 17; Hough et al., p. 489]. The formulation of the Hop Storage Index (HSI) implies that oxidized alpha (and beta) acids have optical density at 275 nm [Algazzali, p. 19]. Finally, polyphenols may be a contributing factor to the nonIAA components [e.g. Krogerus]; as Shellhammer states, “the contribution of polyphenols to beer bitterness can not be overlooked” [Shellhammer, p. 177].
I haven’t been able to find definitive (e.g. more than one source) claims on the bitterness or A_{275nm} 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 rewrite 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} × scale_{oAA}) + ([oBA]_{beer} × scale_{oBA}) + ([PP]_{beer} × scale_{PP}))) 
[29] 
where [oAA]_{beer} is the concentration of oxidized alpha acids in the beer (in ppm), scale_{oAA} is the nonIAA scaling factor specific to oxidized alpha acids, [oBA]_{beer} is the concentration of oxidized beta acids in the beer (in ppm), scale_{oBA} is the nonIAA scaling factor specific to oxidized beta acids, [PP]_{beer} is the concentration of polyphenols in the beer (in ppm), and scale_{PP} is the nonIAA scaling factor specific to polyphenols. (Note that we can compute [IAA]_{beer} using Code [1], Equation [23], and Equation [28].)
3.4.1 Oxidized Alpha Acids
As hops age, the alpha and beta acids become oxidized. The “most important group of oxidized alpha acids formed during hop aging is the humulinones” [Algazzali, p. 13]. The rate at which alpha acids oxidize during storage is determined by the form of the hops (e.g. cones or pellets), hop variety, age, temperature, and amount of exposure to oxygen [Garetz article]. Garetz has a model that predicts the amount of alpha acids remaining in hop cones, given these factors [Garetz article]. (As long as they are properly stored, pellets undergo oxidation at a much slower rate [Hieronymus, p. 230], and so Garetz’s model should only be used for whole hop cones.) A decrease in the amount of alpha acids is mirrored by a corresponding increase in the amount of oxidized alpha acids. The alpha acids also undergo some amount of oxidation while still on the bine [Hieronymus, p. 233] and further during the warm and highly oxygenated conditions of hop drying [e.g. Hieronymus, p. 126], and so the level of oxidized alpha acids when we get our newlydried hops soon after harvest can be greater than zero [Maye, p. 23]. Finally, oxidized alpha acids are created during the boil [Algazzali, p. 17].
We can model the level of oxidized alpha acids (oAA) in the wort as the sum of three contributions: (1) the oAA present in the freshlydried hops as a result of oxidation on the bine and during drying, (2) the oAA that accumulate as the hops age and deteriorate, and (3) oAA that is produced during the boil:
oAA = oAA_{fresh} + oAA_{storage} + oAA_{boil} 
[30] 
where oAA is the level of oxidized alpha acids (as percent of weight of the hops), oAA_{fresh} is the level of oxidized alpha acids in freshlydried hops, oAA_{storage} is the level of oxidized alpha acids produced during storage, and oAA_{boil} is the level of oxidized alpha acids produced during the boil; all components are expressed as percent of weight of the hops.
Based on data from Maye et al [Maye, p. 24], I fit the level of oAA for fresh hops (with a Hop Storage Index (HSI) of 0.25 [Hough et al., p. 434]) to the model of alphaacid decay proposed by Garetz [Garetz article], and determined that oAA_{fresh} can be modeled reasonably well for the available data with a storage factor of 1 (loose hops), a temperature factor of 1 (20°C or 68°F), and a duration of 0.5 days. I then fit the data in the Maye paper for higher HSI values to the loss predicted from the Garetz formula multiplied by a scaling factor of 0.022. (I will go into much more detail on this in a future blog post.) This leaves oAA_{boil} as the only unknown parameter that must be searched for, expressed as the amount of alpha acids that undergo oxidation relative to the amount of available alpha acids in the boil:
oAA = (1 – 1/e^{k×1×1×0.5}) + (oAAage_{scale} × (1 – AA_{decayfactor})) + (AA × oAA_{boil}) 
[31] 
where oAA is the same level of oxidized alpha acids in Equation [30], k is the varietyspecific hop decay factor from the Garetz model, oAAage_{scale} is the agerelated scaling factor of 0.022, AA_{decayfactor} is the alpha acid decay factor from Equation [18], AA is the level of alpha acids at the start of the boil (Equation [18]), and oAA_{boil} is the relative amount of alpha acids that undergo oxidation during the boil. This equation is specific to hop cones; some modification would be required for hop pellets, presumably a larger value of oAA_{fresh} but a value close to zero for oAA_{storage}.
Since oxidized alpha acids are soluble, I believe that we don’t need to model any dependence on how long the hops are in the kettle; we can assume that all of the oxidized alpha acids are in the wort shortly after being added to the kettle. (There may be some time dependence for oAA_{boil}, but given a complete lack of data in that regard, I’ll assume for now that the time dependence is minimal.) That leaves us with two oAA factors that we still need to account for: losses and a scaling factor.
I have not yet been able to find any description of the losses of oxidized alpha acids during the boil and fermentation, so this is a completely unknown factor. It seems reasonable to assume that oxidized alpha acids are lost to trub, yeast, and in other ways, just as isomerized alpha acids are lost in the process of turning wort into beer. Therefore, because the same types of losses probably occur for oxidized alpha acids as for isomerized alpha acids, we can model the oxidized alpha acid losses as the losses that affect isomerized alpha acids multiplied by some (unknown) scaling factor. The scaling factor is a highlevel correction factor for differences between losses found in isomerized alpha acids and oxidized alpha acids. In other words,
[oAA]_{wort} = oAA × W × 1000 / V 
[32] 
[oAA]_{beer} = [oAA]_{wort} × LF_{IAA}(flocculation, filtering, age_{weeks}) × scale_{oAAloss} 
[33] 
where [oAA]_{wort} is the concentration of oxidized alpha acids in the wort, [oAA]_{beer} is the concentration of oxidized alpha acids in the finished beer, W is (still) the weight of the hops in grams, V is (still) the postboil volume of the wort in liters, LF_{IAA} is the same IAA loss factor from Equation [27] and scale_{oAAloss} is the (unknown) loss scaling factor.
We also need a scaling factor in Equation [29] that scales the factor for absorption of light at 275 nm of oxidized alpha acids (unknown) to the factor for absorption of light at 275 nm of isomerized alpha acids (69.68). Fortunately, Maye et al provide this data; based on their Figure 7 [Maye, p. 25], the scaling factor is 0.0142/0.0130, or 1.093:
Despite the large number of parameters for modeling oAA, we end up needing to obtain estimates of only two: oAA_{boil} and scale_{oAAloss}.
3.4.2 Oxidized Beta Acids
As with alpha acids, the beta acids oxidize as the hops age. The most bitter and most prevalent components of the oxidized beta acids are called hulupones [Algazzali, p. 1516]. The oxidized beta acids are thought to contribute more to beer bitterness than the oxidized alpha acids; as Peacock notes, the “nonIAA bitterness is mostly from oxidation products of the alpha and especially the beta acids of the hops formed during hop storage”. [Peacock, p. 157, emphasis mine].
We can model oxidized beta acids in a way similar to oxidized alpha acids; there are oxidized beta acids occurring in fresh hops, created during storage, and produced during the boil [Algazzali, p. 17; Stevens and Wright p. 496; Hough et al., p. 490]:
oBA = oBA_{fresh} + oBA_{storage} + oBA_{boil} 
[35] 
where oBA is the level of oxidized beta acids in the hop cone, oBA_{fresh} is the level of oxidized beta acids in freshlydried hops, oBA_{storage} is the level of oxidized beta acids produced during storage, and oBA_{boil} is the level of oxidized beta acids produced during the boil; all components are expressed as percent of weight of the hops.
Stevens and Wright say that oxidized beta acids are present at not more than 0.5% of the weight of the cone [Stevens and Wright, p. 500], Spetsig and Steninger note up to 3% [Spetsig and Steninger, p. 413], and Mussche found oxidized beta acids up to 1% of the weight [Mussche, p. 13]. Peacock implies that the beta acids undergo oxidation losses at approximately the same rate as the alpha acids [Peacock, p. 162]. Stevens and Wright provide an estimate of the oxidized beta acid boil factor, noting that “after heating colupulone with boiling wort for 2 hr., as much as 10% of the beta acid had been converted into cohulupone.” [Stevens and Wright, p. 500]. Given a lack of data about oBA_{fresh}, I’ll assume that oxidized beta acids are produced at the same levels as oxidized alpha acids both in fresh hops and during aging. This gives a formula similar to Equation [31]:
oBA = (1 – 1/e^{k×1×1×0.5}) + (oBAage_{scale} × (1 – AA_{decayfactor})) + ((AA / AB_{ratio}) × oBA_{boil}) 
[36] 
where oBA is the same level of oxidized beta acids in Equation [35], k is the varietyspecific hop decay factor from the Garetz model, oBAage_{scale} is the agerelated scaling factor of 0.022, AA_{decayfactor} is the alpha acid decay factor from Equation [18], AA is the level of alpha acids at the start of the boil (Equation [18]), AB_{ratio} is the ratio of alpha acids to beta acids (see, for example, Tables 2.1 through 2.3 in Principles of Brewing Science [Fix, pp. 6062]), and oBA_{boil} is the relative amount of beta acids that undergo oxidation during the boil, assumed to be 0.10. This equation is also specific to hop cones; some modification would be required for hop pellets.
As with oxidized alpha acids, we can assume that all of the oxidized beta acids are in the wort shortly after being added to the kettle. That leaves us with two oxidized beta acid factors that we still need to model: losses and a scaling factor.
It seems reasonable to assume that oxidized beta acids are lost to trub, yeast, and in other ways, just as isomerized alpha acids and oxidized alpha acids are lost. With that assumption, we can model the oxidized beta acid losses as the losses that affect isomerized alpha acids multiplied by some (unknown) scaling factor. In other words,
[oBA]_{wort} = oBA × W × 1000 / V 
[37] 
[oBA]_{beer} = [oBA]_{wort} × LF_{IAA}(kettleMaterial, flocculation, filtering, age) × scale_{oBAloss} 
[38] 
where [oBA]_{wort} is the concentration of oxidized beta acids in the wort, [oBA]_{beer} is the concentration of oxidized beta acids in the finished beer, W is (still) the weight of the hops in grams, V is (still) the postboil volume of the wort in liters, LF_{IAA} is the same IAA loss factor from Equation [27] and scale_{oBAloss} is the (unknown) loss scaling factor.
We also need a scaling factor in Equation [29] that scales the factor for absorption of light at 275 nm of oxidized beta acids (unknown) to the factor for absorption of light at 275 nm of isomerized alpha acids (69.68). Lewis and Young state that “during storage of hops alpha acids decline but presumably new bitter compounds are formed, largely from beta acids. … if the alphaacid to betaacid ratio is about unity as is commonly the case, sensory bitterness remains more or less constant with storage.” [Lewis and Young, p. 261]. Since sensory bitterness and IBUs are correlated [Lewis and Young, p. 266], and since oxidized beta acids are believed to be the secondlargest contributor to IBUs (after isomerized alpha acids), this statement implies that the oxidized beta acids have a relationship between light absorption and concentration that is similar to that of the isomerized alpha acids (69.68). So, the scaling factor for oxidized beta acids (scale_{oBA}) should be approximately 1, with emphasis on the “approximately”. According to Hough et al., “hulupones exhibit 8090% of the absorption of the isoalphaacids at [275nm in acid solution]” [Hough et al., p. 491]. In order to convert this absorption to be the equivalent for IAA, a scaling factor of about 1/0.85 or 1.176 is implied:
Due to the large number of assumptions made and estimates obtained from the literature, we only need to obtain an estimate for one oBA parameter: scale_{oBAloss}.
3.4.3 Polyphenols
The other nonIAA component we need to consider consists of polyphenols, which are “an extraordinarily diverse group of compounds;” the majority of those in brewing are flavonoids [McLaughlin, p. 1]. Polyphenols can come from both barley and hops [Hough et al., p. 471], so we should separate the PP component into PP_{hops} and PP_{barley}, where PP_{hops} is the amount of polyphenols contributed by the hops and PP_{barley} is the amount of polyphenols contributed by the barley.
According to Shellhammer, IBUs are in the range of 1 to 3 for unhopped beer [Shellhammer, p. 177]. I brewed a beer with no hops (OG 1.056) and sent it out for IBU analysis three weeks after the start of fermentation; the result was 0 measured IBUs. For the model being developed, I’ll assume a constant value of 0.5 IBU from barley polyphenols and ignore the potential decrease in IBUs over time. Setting the other components in Equation 29 to zero, the scaled concentration of barley polyphenols then becomes 0.5 × 7/5 = 0.7:
[PP_{barley}]_{beer} × scale_{PPbarley} = 0.7 
[40] 
where [PP_{barley}]_{beer} is the concentration of barley polyphenols in the finished beer and scale_{PPbarley} is the scaling factor for light absorption at 275 nm. We don’t need to determine the separate values of these parameters; knowing that their product is 0.7 is sufficient. We can then update our estimate of IBUs in beer to separate the contributions from hops and barley polyphenols:
IBU = 5/7 × ([IAA]_{beer} + (([oAA]_{beer} × scale_{oAA}) + ([oBA]_{beer} × scale_{oBA}) + ([PP_{hops}]_{beer} × scale_{PPhops}) + ([PP_{barley}]_{beer} × scale_{PPbarley}))) 
[41] 
Hop polyphenol levels are often reported in the range from 2% to 6% of the weight of the hops [Shellhammer, p. 169; Hough et al., p. 422; Algazzali, p. 5], although McLaughlin reports a higher range, from 4% to 14% [McLaughlin, p. 3]. After having been added to the wort, polyphenols are removed “extensively by precipitation with proteins during wort boiling”; 80% of hop flavanols are removed in the trub when boiling hopped wort [McLaughlin, p. 7]. As Noonan phrases it, “the rolling motion of the boil causes the malt proteins to collide with and adhere to the sticky hop polyphenols” [Noonan, p. 158]. (It may be that the polyphenols are not really removed, but that they are largely insoluble in wort. The largest polyphenol group in hops (prenylflavonoids) are not soluble in water; all other hop polyphenol components are “soluble in water, preferably in hot water” [Forster, p. 124]. The prenylflavonoids make up about 75% to 85% of all hop polyphenols [Forster, p. 124], so only about 20% of the hop polyphenols are soluble, corresponding to 80% removal.) Then, polyphenols are removed during fermentation, and “it seems possible that this could occur in much the same way as it does with the isoalphaacids” [McLaughlin, p. 7].
From this, we can construct a rough model of the concentration of hop polyphenols in wort and in beer, with an initial level of polyphenols at about 4% of the weight of the hops, a loss factor (or solubility factor) for polyphenols in the wort during the boil (LF_{PP}) estimated at 0.20, and the same loss factors for fermentation and packaging that we have for isomerized alpha acids, LF_{ferment} and LF_{package}:
[PP_{hops}]_{wort} = PPrating × W × 1000 / V 
[42] 
LF_{PP} = 0.20 
[43] 
[PP_{hops}]_{beer} = [PP_{hops}]_{wort} × LF_{PP} × LF_{ferment}(flocculation) × LF_{package}(filtering, age_{weeks}) 
[44] 
where [PP_{hops}]_{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), LF_{PP} is the loss factor for polyphenols precipitated out of the wort (estimated at 0.20), [PP_{hops}]_{beer} is the concentration of hop polyphenols in the finished beer, and LF_{ferment} and LF_{package} are the same loss factors for isomerized alpha acids.
Finally, we need a scaling factor to use the concentration of hop polyphenols in Equation [41]. According to Ellen Parkin, “an increase of 100 mg/L of polyphenols was predicted to increase the BU value by 2.2” [Parkin, p. 28], so that 1 ppm of hop polyphenols should increase the IBU by 0.022 (Equation [45]). We can consider Equation [41] in terms of hop polyphenols only, with an IAA component of zero, an oAA component of zero, an oBA component of zero, a nonzero hop polyphenol (PP_{hops}) component, and a PP_{barley} component of zero (Equation [46]). Since Equations [45] and [46] both measure IBUs from the contribution of only hop polyphenols, we can determine the value of the scaling factor for hop polyphenols (Equation [47]):
IBU = [PP_{hops}]_{beer} × 0.022 
[45] 
IBU = 5/7 × (0 + 0 + 0 + ([PP_{hops}]_{beer} × scale_{PPhops}) + 0) 
[46] 
scale_{PPhops} = 7/5 × 0.022 = 0.0308 
[47] 
where [PP_{hops}]_{beer} is the concentration of hop polyphenols in the finished beer (in ppm) and scale_{PPhops} is the scaling factor for hop polyphenols relative to the scaling factor for IAA.
3.4.4 Solubility of nonIAA Components
The nonIAA components (specifically, oxidized alpha and beta acids, and soluble hop and barley polyphenols) are soluble in water [e.g. Lewis and Young, p. 265; Forster, p. 124]. They do not require isomerization, which (for alpha acid isomerization) takes a significant amount of time. Therefore, they probably contribute very quickly to the measured IBU value. This is of particular significance for hops that are added late in the boil (or at flameout, or after flameout), since they will have all (or nearly all) of their nonIAA components quickly dissolved and contributing to IBUs, whereas the IAA level will be low due to insufficient time for isomerization. As a result, the ratio of IAA to all bittering substances can be much lower for hops added very late in the boil, even for very fresh hops. In short, the 1960s finding that the concentration of IAA is 5/7 of the total concentration of all bittering substances reflects not only the age and storage conditions of 1960s hops, but also the typical time(s) at which hops were added to the boil in the 1960s. Freshlydried hops added at flameout (with 10 minutes of cooling after flameout) may yield 20 IBUs, but only 50% of that from isomerized alpha acids.
4. Available Data, Parameter Estimation, and Results
4.1 Overview
The quantitative description we now have of IBUs is still incomplete, because we don’t have useful estimates for a number of the factors (specifically: LF_{boil}, maximum [AA]_{0}, oAA_{boil}, scale_{oAAloss}, and scale_{oBAloss}). We do, however, have Tinseth’s model for predicting IBUs under normal circumstances [Tinseth], results from a study by Val Peacock that looks at IBUs and IAA concentration as a function of hop storage conditions [Peacock, p. 162], and results from six experiments where I’ve measured IBUs with varying hop steeping times, amounts, and temperatures (to be published later on this blog). We can make assumptions about the conditions of these studies as needed (i.e. boil gravity, postboil volume, beta acid level, etc.) and use the data and assumptions, along with common techniques for searching a parameter space, to obtain a rough estimate of the five unknown values.
4.2 Sources of IBU Data
4.2.1 Tinseth Utilization
The Tinseth model is widely used for predicting IBUs. Tinseth had “access to some handy tools and knowledgeable friends at the USDA hop labs and the Flavor Perception labs at Oregon State University,” [Tinseth] and he has “had quite a few worts and beers analyzed” [Tinseth]. Therefore, whatever model we develop should come up with estimates close to those predicted by the Tinseth model given similar conditions. Tinseth provides a detailed description of his model and parameters at realbeer.com. He based his model on a review of the literature and on data from the pilot brewery at Oregon State University and small breweries; he then verified the model by brewing small batches and testing the results [Hieronymus, p. 185]. In the experiments he conducted in order to validate his model, he used hops from vacuumsealed oxygen barrier bags stored in a freezer, resulting in very low hop degradation [Tinseth emails]. Also, he took small samples at intervals throughout the boil and immediately cooled them, yielding almost no postboil utilization [Tinseth emails].
4.2.2 Peacock HopStorage Conditions
In an article describing IBUs, Peacock provides results of a study that looked at how the storage conditions of hops affected IBU levels [Peacock, p. 162]. He lists four storage conditions (ranging from 20°F (29°C) to 70°F (21°C)), the relative percent of alpha and beta acids lost (based on the Hop Storage Index), the IAA levels in the finished beer, and the IBUs of the finished beer. He also provides the alpha/beta ratio of the hops used, but not the amount of hops used, wort volume, or original gravity.
4.2.3 Personal Experiments
I conducted a series of six experiments (one in two parts) that look at IBUs as a function of hop steeping time, amount of hops, and wort temperature. Experiment 1 was a set of “standard” beers with boil times ranging from 10 to 60 minutes and immediate postflameout cooling, in order to sync up with the Tinseth formula. Experiments 2a and 2b were a set of beers with hops added only at flameout and held at a constant temperature for 10 or 20 minutes (from 145°F (63°C) to 212°F (100°C)), in order to evaluate the degree of utilization at subboiling temperatures. Experiment 3 was a set of beers with hops added at varying times during the boil (from 0 to 60 min) and a 15minute postflameout natural cooling (a.k.a. a hop stand) before forced cooling. Experiment 4 looked at utilization as a function of kettle material (stainless steel vs. aluminum) and loose vs. bagged hops. Experiment 5 looked at utilization as a function of the amount of hops, and Experiment 6 varied some factors from Experiment 5 (amount of hops, boil time, and steep temperature) in order to estimate IAA concentrations from IBU values.
I will write about Experiments 4, 5, and 6 in more detail in the future, but for now I’ll mention that one of the biggest difficulties was obtaining accurate alphaacid levels of the hops for the first three experiments. As a result of that difficulty, for these three experiments I allowed the IBU model parameter search (Section 4.3) to evaluate ±1 percentage point around the best estimate of alphaacid levels at harvest, and I also provided some flexibility in the alphabeta ratios (based on estimates from analysis of the hops around the time of brewing) and value of AA_{decayfactor}.
4.3 Parameter Estimation and Results
Using 9 IBU values based on Tinseth’s utilization function (from 10 minutes through 90 minutes at 10minute intervals) (with typical values for AA_{harvest}, OG, W, and V, and the values of AA_{decayfactor} and AB_{ratio} fit to the data), the 4 IBU values and 4 IAA values from Peacock (assuming values for original gravity and volume, and fitting the boil time, post boil time, and weight of the hops to the data), and the 33 measured IBU values from my six experiments, there are 50 data points with which to estimate the five unknown parameter values, as well as a number of sourcespecific parameter values (e.g. weight of the hops in Peacock’s study). This really isn’t enough data for a reliable estimate of all parameters, but it’s better than nothing. It helps that these sources of data cover a number of scenarios of interest, including boil time, storage conditions of the hops, weight of hops used, and hop steeping temperature.
Tables 1 through 9 (below) provide the known values, assumptions, estimated values, and IBU (or IAA) results for each set of data. In addition, flocculation was set to “normal” and filtering was set to “none”. All other parameters not being estimated were given the bestguess values noted in the previous sections. For oxidized alpha and beta acids produced during the boil, I assumed a linear decrease with temperature, from full oxidizedacid production (scale factor 1.0) at boiling to zero production (scale factor 0.0) at room temperature. For Tinseth and Peacock, I assumed loose whole hops, so that RF_{form}(hopsForm=loose cones)=1.16; for my experiments, I used RF_{form}(hopsForm=loose cones)=1.16 or RF_{form}(hopsForm=bagged cones)=1.05, depending on the form of the hops. The alphaacid decay factor in Table 1, AA_{decayfactor}, is the result of the Garetz formula e^{k×TF×SF×D}; I constrained the search range for this factor based on best guesses of the variables k, TF, SF, and D in each condition.
I used an iterative bruteforce search over the parameter space to minimize the squared error, starting with the approximate range of each parameter and a coarse search interval. After each iteration, I used the best estimates of each parameter to specify a smaller range, along with a smaller search interval. The search process was stopped when best estimates were obtained with a typical search interval of 0.01. A nested recursion was used to constrain the five unknown model parameters to be the same for all data sources, while the unknown parameters from each experiment were searched for individually. (I will provide the Ccode procedure of the complete IBU model, after I have a chance to clean up the code.)
The result of this parameter search is not an ideal solution! We have a very large number of assumptions, a fairly large number of unknown parameters, and a relatively small amount of data. As a result, the estimates of the parameter values will almost certainly be wrong at some level. My hope, however, is that a slight overestimate of one factor will be balanced by a small underestimate of another factor, and on average the model will provide a cohesive, general description of the factors that contribute to IBUs. The model and parameter settings provide a “mostlikely” set of values given the (limited) data. Because of the lack of data, the resulting description of IBUs is descriptive, not predictive. In other words, I make no guarantee of how well this model will predict your IBU values, even if you know all of the input parameter values (hops weight, volume, alpha acid level at harvest, alphabeta ratio, storage conditions, steep time, etc.). This model may, however, help with understanding the various factors and relative contributions of these factors to the IBU measure.
The results of the search for the five parameters are: LF_{boil} = 0.60, maximum [AA]_{0} = 265 ppm, oAA_{boil} = 0.07, scale_{oAAloss} = 0.04, and scale_{oBAloss} = 0.76. The estimated value of LF_{boil} is fairly close to that of Malowicki’s report that the formation of trub causes losses of 35% (translating to a scaling factor of 0.65) [Malowicki, p. 78]. The small values of oAA_{boil} and scale_{oAAloss}, compared with the larger value of scale_{oBAloss}, result in a smaller contribution of oxidized alpha acids compared with oxidized beta acids, which is also in agreement with the literature (e.g. [Peacock, p. 157]).
Table 1 provides the known, assumed, and estimated values of parameters that could vary between the sources of data. Parameters that could vary were constrained to a reasonable search range based on available data. Note that many values in the Tinseth column do not need to be the same as what Tinseth used in his experiments; as long as the same values of these parameters are used in the comparison with the current model, any values can be used. For the Peacock study, the alpha acid rating at harvest was determined based on the data he published. I assumed a onebarrel (31 G or 117 liter) volume for Peacock’s experiments; if this assumption is incorrect, then the estimated weight of the hops can be scaled proportionally to give identical results. I also assumed a slow postflameout temperature decay of 0.2°C per minute for Peacock’s experiments, under the assumption that a large volume of wort cools slowly; if the actual temperature decay was different, the weight of hops, boil time, and/or postboil time may need to be adjusted.

Tinseth 
Peacock 
Exp. #1 
Exp. #2a 
Exp. #2b 
Exp. #3 
Exp. #4 
Exp. #5 
Exp. #6 
AA at harvest

8.65% (?) 
3.9% 
8.0% 
7.4% 
8.9% 
6.0% 
8.1% 
8.1% 
8.1% 
α/β ratio 
1.10 
1.35 
0.85 
1.5 
1.4 
1.3 
1.0 
1.05 
1.05 
AA decay factor

0.95 
0.07 to 0.83 
0.71 
0.92 
0.94 
1.0 
1.0 
0.975 
0.95 
boil time

10 to 90 min 
90 min

10 to 60 min 
0 min 
0 min 
0 to 60 min 
20 min 
12 min 
0 to 26.9 min 
postboil time

0 min 
50 min 
0 min 
10 to 20 min 
10 min 
15 min 
0 min 
0 min 
0 to 19 min 
postboil temp.

N/A 
slow decay 
N/A 
185°F to 212°F 
145°F to 212°F 
fast decay 
N/A 
N/A 
145°F 
hops weight

1.5 oz (?) 
5.5 oz 
0.60 oz 
1.60 oz 
1.60 oz 
0.80 oz 
0.75 oz 
0.37 to 2.22 oz 
0.37 to 2.22 oz 
wort volume

5.25 G (?) 
31 G (?) 
1.37 to 1.50 G 
1.10 to 1.24 G 
1.05 to 1.20 G 
0.88 to 1.15 G 
1.52 to 1.61 G 
1.61 to 1.65 G 
1.59 to 1.63 G 
boil gravity

1.055 (?) 
1.035 (?) 
1.059 to 1.064 
1.064 to 1.066 
1.063 to 1.065 
1.065 to 1.075 
1.056 to 1.059 
1.054 to 1.056 
1.055 to 1.056 
Table 1. Known values, assumed values, and best estimates of parameters that were allowed to vary between the sources of data. If a value has no markings, it is a known value. If a value has a question mark after it (?), it is an assumed value. If a value is in bold face and red, it is the best estimate as determined by the parameter search. The estimated decay factor of 0.51 in Experiment 1 is low, but actually fairly likely as that source of hops was probably stored for months at room temperature.
Tables 2 through 9 show results from the Tinseth, Peacock, and personal experiments. Table 2 shows the results of IBU estimation based on the Tinseth formula and based on the estimates obtained from the current model:
time 
10 min

20 min 
30 min 
40 min 
50 min 
60 min 
70 min 
80 min 
90 min 
formula 
14.8 
24.7 
31.4 
35.8 
38.8 
40.8 
42.2 
43.1 
43.7 
estimate 
15.8 
22.6 
28.4 
33.3 
37.3 
40.7 
43.5 
45.8 
47.6 
diff. 
1.0 
2.1 
3.0 
2.6 
1.5 
0.1 
1.3 
2.7 
3.9 
Table 2. IBU estimates from the Tinseth formula and the current model, as a function of hop steep time, and the difference (error) between the two.
Table 3 shows the IAA and IBU measured values reported by Peacock, and the results of IAA and IBU estimation from the current model:
condition 
20°F 
25°F 
40°F 
70°F 
measured IAA 
19.8 ppm 
18.1 ppm 
14.4 ppm 
2.9 ppm 
measured IBU 
13.5 
12.0 
13.5 
11.0 
estimated IAA 
17.0 ppm 
15.0 ppm 
11.1 ppm 
1.4 ppm 
estimated IBU 
16.2 
15.6 
14.4 
11.5 
IAA difference

2.8 ppm 
3.1 ppm 
3.3 ppm 
1.5 ppm 
IBU difference

2.7 
3.6 
0.9 
0.5 
Table 3. IAA and IBU measured values and estimates from the current model, as a function of hop storage conditions. The difference (error) between measured and estimated values is also shown.
Table 4 shows the measured and estimated IBU values from my experiment #1, meant to sync up with the Tinseth formula. The estimate of the alphaacid rating at harvest (8.0%) is equal to the value written on the package I bought. The estimate of the alpha/beta ratio (0.85) is close to an estimate obtained by analysis of the hops’ alpha and beta values (0.862). The degradation factor of 0.71 is close to the degradation factor estimated from the Hop Storage Index (0.72).
steep time 
10 min

20 min

40 min

60 min

measured IBU

22.0 
27.1 
34.3 
35.7 
estimated IBU

20.4 
25.2 
31.4 
37.9 
IBU difference

1.6 
1.9 
2.9 
2.2 
Table 4. Measured IBU values and estimated IBU values from personal experiment #1, as a function of hop steep time. The difference (error) is also shown.
Table 5 shows the measured and estimated IBU values from my experiment #2, which looked at utilization as a function of steep temperature. In most cases, the steep time was 10 minutes, but in one case the steep time was 20 minutes.
temp/
time

212°F/
10m

200°F/
10m

190°F/
10m 
185°F/
10m 
192°F/
20m 
212°F/
10m 
175°F/
10m 
165°F/
10m 
155°F/
10m 
145°F/
10m 
meas. 
33.3 
28.9 
30.8 
25.5 
35.9 
40.6 
23.6 
24.5 
23.1 
21.8 
est. 
37.3 
30.9 
28.2 
26.6 
31.6 
39.2 
26.3 
24.2 
22.3 
21.0 
diff. 
4.0 
2.0 
2.6 
1.1 
4.3 
1.4 
2.7 
0.3 
0.8 
0.8 
Table 5. Measured IBU values and estimated IBU values from personal experiment #2, as a function of hop steeping temperature and time. The difference (error) is also shown.
Table 6 shows the measured and estimated IBU values from my experiment #3, which combined various hop boil times with a 15minute hop stand. The wort was allowed to cool naturally during this 15 minutes, after which it was forcecooled.
time 
0 min

7.5 min 
15 min 
30 min 
60 min 
measured

16.1 
21.2 
26.1 
35.4 
46.4 
estimated 
14.9 
22.0 
27.9 
37.8 
50.7 
difference 
1.2 
0.8 
1.8 
2.4 
4.3 
Table 6. Measured IBU values and estimated IBU values from personal experiment #3, as a function of hop boil time. The difference (error) is also shown.
Table 7 shows the measured and estimated IBU values from my experiment #4, which looked at utilization as a function of kettle material and form of the hops.
kettle material,
hop form

stainless steel,
loose

aluminum,
loose

aluminum,
bagged

measured

34 
37 
36 
estimated

33.8 
34.0 
32.3 
difference

0.2 
3.0 
3.7 
Table 7. Measured IBU values and estimated IBU values from personal experiment #4, as a function of kettle material (stainless steel or aluminum) and hop form (loose cones or bagged cones).
Table 8 shows the measured and estimated IBU values from my experiment #5, which looked at utilization as a function of weight of the hops.
weight

0.37 oz

0.74 oz

1.11 oz

1.48 oz

1.85 oz

2.22 oz

measured

12 
23 
29 
34 
41 
47 
estimated 
12.4 
24.3 
29.7 
34.8 
40.7 
46.1 
difference 
0.4 
1.3 
0.7 
0.8 
0.3 
0.9 
Table 8. Measured IBU values and estimated IBU values from personal experiment #5, as a function of hop weight. The difference (error) is also shown.
Table 9 shows the measured and estimated IBU values from my experiment #6, which looked at variety of conditions: Condition A had hop weight of 0.37 oz and boil time of 26.9 min; Condition B had hop weight of 1.11 oz and boil time of 26.9 min; Condition C had hop weight of 1.11 oz and boil time of 12 min; Condition D had hop weight of 2.22 oz and boil time of 19.0 min; and Condition E had hop weight of 2.22 oz, with no boiling but a 19minute hop stand held at 145°F. Conditions A through D were immediately cooled upon flameout.
Condition

A

B

C

D

E

measured

18 
48 
32 
58 
27 
estimated 
20.8 
46.9 
32.8 
58.5 
26.0 
difference 
2.8 
1.1 
0.8 
0.5 
0.9 
Table 9. Measured IBU values and estimated IBU values from personal experiment #6, as a function of hop boil time. The difference (error) is also shown.
5. Discussion of Results
The average difference between observed (or Tinseth model) IBU and IAA values and current model estimates is 0.01, with a standard deviation of 2.2 and a maximum difference of 4.3. The consistent difference of about 3 ppm for the IAA values but an overestimate of the IBU values in the Peacock study is one example of the suboptimal result of the parameter estimation. From the data I’ve seen, observed IBU values can deviate quite a bit from expected values (for reasons that are still unclear to me), and so the overall results from the model do not seem excessively bad. With a human detection threshold of 5 IBU [Daniels, p. 76], none of the errors in the model (with a maximum difference of 4.3 IBU) would be detectable by a human. While few, if any, of the model parameters have been estimated with great precision, the overall fit suggests that errors in one parameter estimate are, for the most part, balancing out errors in another estimate.
To the extent that parameter estimation has been reasonable, we can use this model to look at how various factors affect IBUs. If we assume some typical brew parameters (OG 1.055, volume 5.25 G or 20 liters, a typical AA rating of 8.65%, an alpha/beta ratio of 1.4, exceptionally wellpreserved hops with AA_{decayfactor} of 1.0, postflameout natural cooling for 10 minutes, and taking IBU measurements one week after the start of fermentation), we can vary the amount and timing of hops additions in the model to look at the impact on IBU and IAA. For example, 2 oz added at flameout will create 19.3 IBUs with a concentration of 11.4 ppm of IAA (52% of the IBU total), 0.5 ppm of oAA, 11.7 ppm of oBA, and 19.1 ppm of hop polyphenols. The same 2 oz added at 60 minutes will create 61.4 IBUs with a concentration of 70.4 ppm of IAA (82% of the IBU total) and the same concentrations of nonIAA components. If we triple the amount of hops, from 2 oz to 6 oz, the IBUs only increase from 61.4 to 86.5 (75.6 ppm of IAA, representing 62% of the total; 1.5 ppm of oAA, 35.2 ppm of oBA, and 57.3 ppm of hop polyphenols). If we add those 6 oz at flameout, we’ll get 41.3 IBUs, with only 12.2 ppm of IAA (21% of the IBU total). If we have somewhat degraded hops (say, stored at room temperature in airtight packaging for six months) yielding an AA_{decayfactor} of 0.82, the 2 oz of hops added at 60 minutes will yield 35.3 IBUs, with 35.8 ppm of IAA representing 72% of the IBU total. Adding these degraded hops at flameout will produce 13.8 IBUs, but with only 5.8 ppm of IAA representing 30% of the IBU total.
Another interesting thing we can do is estimate the contribution of nonIAA components to the Tinseth formula. While the Tinseth formula uses only the weight and alphaacid rating of the hops to compute IBUs [Tinseth], the utilization function was fit to observed data [Pyle], which includes nonIAA components. We can use the current detailed model to separate out the actual IAA contribution to utilization from the (implicit) nonIAA contribution. For example, at 10 minutes before flameout, the detailed model predicts 15.83 IBUs in a (postboil volume) alphaacid concentration of 175.83 ppm using the Tinseth source of data. (The Tinseth formula predicts 14.80 IBUs using the same data.) If the IBU value was equivalent to the concentration of isomerized alpha acids, as assumed by the Tinseth equation, then at the final boil volume there would be utilization of 15.83 ppm / 175.83 ppm = 0.0900 (or 9.0% utilization). The detailed model tells us, however, that at 10 minutes the relative contribution of IAA to the IBU is only 0.511. Therefore, of the utilization of 0.0900, 0.0460 is alphaacid utilization (using the standard definition of utilization), and 0.0440 is the effective utilization coming from nonIAA components. (By “effective”, I mean that the nonIAA components, expressed in ppm, are converted by their scaling factor to be relative to IAA concentrations; e.g. a hop polyphenol contribution of 14 ppm in the finished beer is multiplied by its scaling factor of 0.0308 to yield an effective utilization from hop polyphenols of 0.4312 ppm / 175.83 ppm = 0.0025 or 0.25%. All of the nonIAA components sum up to yield a total effective utilization from nonIAA components.) Because the detailed model assumes that nonIAA components contribute to the IBU value in a very short amount of time (unlike the lengthy isomerization process), the effective utilization of 0.0440 for nonIAA components is constant for all boil times. When the boil time is 60 minutes, the effective utilization from nonIAA components is still 0.0440, but the alphaacid utilization is 0.1877 (for a total utilization of 0.2317), and so the isomerized alpha acids represent 81% of the IBU value at 60 minutes. In general, one can think of the utilization part of the Tinseth formula as being a constant 0.0440 from nonIAA components, and the remainder (when the formula yields a value greater than 0.0440) from isomerized alpha acids. The Tinseth formula predicts utilization of 0.0440 at around the 5minute mark. All of this corresponds very well with the Rager IBU formula [Pyle], which has a nonzero and roughly constant utilization of 5% (0.05) from 0 to 5 minutes.
6. Summary
This post has described the various factors that contribute to the IBU, and quantified each factor as much as possible. Estimates of parameter values that could not be determined from the literature were obtained by fitting a model to the available data.
Despite the length of this post, many things have been left undiscussed. The current model is restricted to a single hop addition, with full boil of the wort (i.e. not performing the boil at higher gravity and then diluting). The topic of dry hopping and its impact on bitterness is left entirely to Ellen Parkin [Parkin], Maye et al. [Maye], and others. The model is probably useless when it comes to the IBUs of darker beers and stouts, since dark malts may affect bitterness and the IBU value (although I’ve seen surprisingly lowerthanexpected IBU values in my stouts). The perception of bitterness is left out entirely (especially at high IBU values), as is the large topic of different bitterness qualities. I’ve also put off a number of topics (alpha acid concentration at boiling, decrease in IBUs over time for homebrewed beer, rate of alpha acid oxidation based on Maye et al.’s paper [Maye], and details of my experiments) for future blog posts.
What’s the takeaway message of this post? If you’re adding hops late in the boil (or at flameout), you will probably not get a lot of bitterness from alpha acid isomerization. You can, however, get a significant number of IBUs from this hop addition, with most of the IBU value coming from nonIAA components. Likewise, if you’re using a large amount of hops, the IBU value may be smaller than you’re expecting (due to what appears to be the solubility limit of alpha acids in boiling wort), but most of that IBU value may come from nonIAA components. Hopefully this post and model will help in understanding the relative contributions of isomerized alpha acids and nonIAA components to the IBU.
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