Tag Archives: mIBU

Predicting Wort Temperature After Flameout

Abstract
In a previous post, I described a method for estimating IBUs that are produced in hot wort after flameout.  This method relies on both (a) relative utilization as a function of temperature, described elsewhere, and (b) a function that describes the decrease in wort temperature after flameout (but before “forced cooling” with a wort chiller).  In this blog post, I describe temperature data collected under a variety of conditions and the resulting formula for predicting the temperature of wort as it naturally cools after flameout.  The data suggest that this rate of natural cooling is primarily influenced by (a) the release of steam, which is in turn influenced by the wort volume, surface area of wort exposed to air, and size of the opening in the kettle through which steam can escape, and (b) radiation of heat from the kettle.  Other factors, such as ambient temperature, are of much lesser significance.  The resulting formula, for homebrew-scale batch sizes, is T = 53.70 × exp(-b × t) + 319.55, where b = (0.0002925 × effectiveArea / volume) + .00538 and effectiveArea = (surfaceArea × openingArea)0.5.  The parameter T is temperature (in degrees Kelvin), t is time after flameout (in minutes), b is the rate constant that describes how quickly the temperature decreases, effectiveArea is the “effective” area through which steam ventilates, surfaceArea is the surface area of wort exposed to air (in square centimeters), openingArea is the area of the opening in the kettle (in square centimeters), and volume is the wort volume (in litres).

1. Motivation
The motivation for the work described here was to predict the temperature decrease of wort after flameout, in order to facilitate computation of the mIBU method of predicting IBUs for homebrew-scale batch sizes.

If one thinks about the various factors that might influence this temperature decrease, many things may come to mind:

  1. The wort volume, with larger volumes potentially cooling more slowly,
  2. The size or surface area of the kettle (which may be much larger than the wort volume), with larger kettles potentially radiating more heat than smaller kettles,
  3. The size of the opening in the kettle (with potentially slower cooling for a smaller opening that traps more heat),
  4. The ambient or room temperature (with wort potentially cooling faster if the room temperature is 10°C (50°F) as opposed to 30°C (86°F)),
  5. The relative humidity (with wort potentially cooling faster in drier conditions),
  6. The specific gravity of the wort (with higher specific gravities potentially cooling differently from water),
  7. The removal of the kettle from the heat source (with potentially slower cooling if the kettle remains on a hot burner),
  8. The kettle material (with materials such as aluminum potentially cooling faster than materials such as stainless steel), and
  9. Whether the kettle is insulated or not (with potentially slower cooling for an insulated kettle).

In order to investigate these possibilities, I tested these factors with either wort or (for simplicity) water, plotted the results, and determined which factors have the greatest impact on the rate of temperature change.  With this information, I then constructed a formula for predicting wort temperature after flameout as a function of time.  This function can be used directly in the mIBU method.

2. Data
I measured the decrease in temperature after boiling for 33 conditions in order to test the various factors listed above; these conditions are listed in Table 1 at the very bottom of this post.  (I did not control for ambient temperature and relative humidity separately; generally, a lower ambient temperature was correlated with a higher relative humidity.)  I measured the temperature of wort or water after flameout in 22 conditions with the kettle uncovered, and additional 11 conditions with the kettle partially or fully covered.   I used wort in 5 cases and water in 28 cases.  I used a Thermapen Mk4 for measuring temperature in all cases except condition AG, in which I used a TelTru analog thermometer with a 30 cm (12″) probe.  I took measurements at 1-minute intervals for the first 15 or 20 minutes after flameout. (Measurements were taken for only 15 minutes for three conditions: T, U, and V.)  For the conditions using water, I measured volume to the nearest 30 ml (1 ounce) using a “Legacy Pro” 4000-ml (128-oz) graduated pitcher (which looks identical to the US Plastics Corp. Accu-Pour™ PP Measuring Pitcher),  recorded the temperature of each addition, and normalized from this volume and temperature to the volume at boiling using Equation 3 in “ITS-90 Density of Water Formulation for Volumetric Standards Calibration” (Jones and Harris, Journal of Research of the National Institute of Standards and Technology, vol 97, no. 3, pp. 335-340 (1992)).  For the conditions using wort, I estimated the volume at close to boiling using a measuring stick or the difference between pre- and post-boil specific gravity.  Twelve of the more interesting conditions are plotted in Figure 1, with time on the horizontal axis and temperature (in degrees Celsius) on the vertical axis.

I will mostly use metric units throughout this blog post in order to simplify the presentation, with apologies to readers in the United States.  The final formula uses degrees Kelvin.

tempDecayExp-Fig1-rawData

Figure 1. Temperature (in degrees Celsius) as a function of time (in minutes) for twelve of the 33 conditions.  The legend for each condition specifies the volume of liquid (water or wort, in litres), the amount by which the kettle was covered (in percent; 0% = uncovered and 100% = completely covered), the size of the kettle (in litres), and any other details about the condition, such as ambient temperature or insulation.  Only one of the conditions in this plot used wort (specific gravity 1.052); the other cases here used water.

3. Parameter Estimation
3.1 Exponential Decay

It can be seen that all of the data in Figure 1 can have a good fit to a function with exponential decay.  Those conditions not plotted in Figure 1 also show a similar goodness of fit to an exponential decay function.  (For many of the cases, a straight line also seems to be a good fit, but the exponential decay function can model nearly-straight lines as well as curved lines of the type seen here.)  An exponential decay function is of the general form a × exp(-b×t) + c, where t is (in this case) time, a, b, and c are parameters that describe the shape of the function, and exp(x) indicates the constant “e” to the power of x, or 2.71828x.  In this case, the parameter b is called the rate constant, and it describes how quickly the function (or temperature) decreases.  (I like using fooplot to visualize different functions and parameter values; one can enter something like “54*exp(-0.03x)+46” on this page to see a representative exponential decay function, setting the graph lower limits to 0, the x-axis upper limit to 50, and the y-axis upper limit to 100.)  The liquid was at boiling in all cases at time 0, with an average measured temperature over all conditions of 100.1°C.  (The expected boiling point of water at my elevation (76 meters above sea level) is 99.7°C.  However, the boiling point of wort is higher than that of water, so the average boiling point over all conditions (with 5 of the 33 cases using wort) was higher than 99.7°C.  The difference of less than 0.4°C is within the  specified accuracy of my Thermapen, which is ±0.4°C.)    If t = 0, then exp(-b×t) is 1 for any value of b, and so a + c must equal 100.1.

In order to simplify the parameter estimation, I searched over conditions A through V (those conditions in which the kettle is uncovered) minimizing the root-mean-square (RMS) error to find the best value of b in each case and the values of a and c that were the best over all conditions.  (In other words, a and c were optimized to have the same value over all conditions, whereas b was optimized per condition.)  If the total RMS error over all conditions is small with constant values of a and c, then the different shapes of each curve can be described well with a single parameter, b.

The search for a and c yielded a=53.70°C and c=46.40°C with an overall RMS error of 0.31°C.   The maximum RMS error was 0.64°C for condition O.  The small RMS error over all conditions indicates that we can, in fact, describe the different rates of temperature decay of these conditions with a single parameter, the rate constant b.  The question then becomes whether we can predict b from the various factors in each condition, and if so, if certain factors are more important than others in predicting b.  (It’s also worth noting that the optimal value of c in this case is not room temperature.  Presumably, if time were measured in hours instead of minutes, the values of a and c would have turned out differently, with c at around room temperature.)

3.2 Predicting the Rate Constant for Uncovered Kettles
Figure 2 plots the values of b that minimize the RMS error in each uncovered-kettle condition with a=53.70°C and c=46.40°C.  The horizontal axis is the volume of wort or water, and the vertical axis is the value of b.  A few clear patterns emerge: the data obtained from a single kettle are grouped in a curved line with negative slope (for the two cases where there are multiple data points per kettle), and these curves (representing different kettles) are separated from each other by possibly constant scaling factors.  The curved line with negative slope for the 38-litre kettle looks like a function of the form 1/x, where x in this case is volume.  This suggests that b can be approximated as a function of scaling/volume, where scaling is some (still unknown) property of the kettle and volume is the wort volume (in litres).

tempDecayExp-Fig2-rateConstVsVolume

Figure 2. Temperature-decay rate constants for all conditions with uncovered kettles, plotted as a function of wort volume.  Each group (e.g. black squares or red diamonds) is for a different kettle (and kettle diameter).

After considering various possibilities for the factor called scaling, the area of the kettle opening (πr2), which equals the surface area of wort exposed to the air, shows a good fit to this set of data.  The black “×” marks in Figure 3 plot the values of area/volume on the horizontal axis for the uncovered kettle (where area is the area of the kettle opening (or πr2, where r is the radius of the kettle) in square centimeters, and volume is the volume of liquid, in litres) and values of the rate constant b on the vertical axis.  The approximately straight line of black × marks in Figure 3 is interesting.  It implies that the rate of temperature decay, represented by the parameter b, can be predicted quite well from only the area of the kettle opening and the volume of liquid.  The value of b when the area is zero implies a rate of cooling caused by heat radiated from the kettle (with an entirely closed system), and the slope of the line implies faster cooling as more steam escapes the kettle with greater wort surface area.  In other words, if b is modeled as a straight line of the form b = slope × (area / volume) + offset, where slope is the slope of the line and offset is the value when area = 0, then offset represents the temperature decay due to heat radiated from the kettle, and slope represents the temperature decay caused by the loss of heat in the steam.  In this case, a good fit can be seen for the line b = 0.0002925 × (area / volume) + 0.00538.

tempDecayExp-Fig3-rateConstVsArea

Figure 3. Temperature-decay rate constant as a function of (kettle opening area) divided by volume, for uncovered kettles. In this case, the kettle opening area equals the area of wort exposed to air.

3.3 The Rate Constant for (Partially) Covered Kettles
I then plotted the values of b for those cases in which the kettle is partially or completely covered, as shown in Figure 4 (with much lower limits on the X and Y axes of this graph).  The cases in which the kettle is completely covered cluster somewhat around the predicted value of b when the area is zero.  Larger kettles and volumes have smaller values of b, implying less radiated heat loss from larger kettles and/or volumes.  Conditions Z and AG are nearly identical except for the size of the kettle; Z used 15.4 litres of water in a covered 18.9 litre kettle, and AG used 15.6 litres of water in a covered 37.9 litre kettle.  The temperature after 20 minutes was very close in both conditions, and the estimated value of b is nearly the same in both cases (0.00371 vs 0.00373).  Therefore, it seems that the size of the kettle has very little impact on the rate of temperature decay through radiated heat, but the volume of liquid does have an impact on radiated heat.

Again looking at Figure 4, the conditions in which the kettle is only partially covered deviate from the predicted line, regardless of whether area (the horizontal axis) is (a) the exposed wort surface area (blue circles) or (b) the kettle opening area (green squares).  The predicted line lies somewhere between these two extremes.  This suggests that if the kettle is partially covered, the amount of steam produced is (still) roughly proportional to the surface area of the wort exposed to air (i.e. the area of the fully-open kettle), but that this steam is not able to escape quite as quickly, leaving more heat trapped in the kettle.  (For an uncovered kettle, the area of the opening in the kettle and the surface area of wort exposed to air are the same.)

One possibility is that the rate of heat loss is proportional to the geometric average of the wort surface area and the opening area.  We can call this the “effective area,” i.e. effectiveArea = (surfaceArea × openingArea)0.5, where surfaceArea is the wort surface area, openingArea is the area of the kettle opening, and (x)0.5 indicates the square root of x.  In this case, when the area of the opening is zero (for a covered kettle), the effective area is also zero.  When the area of the opening equals the surface area of the wort, the effective area is the same as the surface area of the wort.  When we plot effectiveArea / volume on the horizontal axis and b on the vertical axis in Figure 5, we observe that the data from the partially-covered conditions are much closer to the straight line, allowing us to predict temperature decay fairly well with a small number of parameters.

tempDecayExp-Fig4-rateConstCoveredKettle

Figure 4. Rate constants for uncovered kettles (black “×” marks), fully-covered kettles with area = 0 (red triangles), partially-covered kettles with area = area of kettle opening (green squares), and (the same) partially-covered kettles with area = wort surface area (blue circles). The line with the best fit to uncovered-kettle data is also plotted.

tempDecayExp-Fig5-rateConstCoveredKettleEffectiveArea

Figure 5. Rate constants for uncovered kettles (black “×” marks), partially covered kettles (dark red diamonds), and fully covered kettles (light red triangles), plotted as a function of “effective area”. Effective area is the geometric mean of the exposed wort surface area and kettle opening area.

4. Model Accuracy
4.1 Looking at Factors Potentially Influencing Temperature
In the model we have developed, we can predict temperature after flameout using three parameters: wort volume, kettle diameter (to compute exposed wort surface area), and kettle opening diameter (to compute the area of the opening).  Other factors, such as ambient temperature and specific gravity, have a fairly small deviation from the predicted line, indicating that these factors have only a minor impact on the decrease in temperature.  For example, in Figures 3, 4, and 5 there are two rate constants that have the same area/volume value of 46.9 cm2.  (This is most easily seen in the two “×” marks at area/volume=46.9 on the right-hand side of Figure 4.)  The one just below the predicted line, with a value of 0.0190, was from Condition C with water from an uncovered 19-litre kettle and an ambient temperature of 12°C (53°F).   This case has a predicted temperature of 86.9°C after 15 minutes, which is very close to the measured temperature of 86.8°C.  The one even lower than the predicted line, with a value of 0.0176, was obtained under the same conditions except with an ambient temperature of 33°C (91°F), Condition J.  This case has the same predicted temperature of 86.9°C, but a measured temperature of 87.8°C after 15 minutes.  From this, we can conclude that ambient temperature does have an effect on the rate of temperature decrease, with warmer ambient temperatures yielding a slower decrease in temperature.  However, this effect is minor, with a large difference in ambient temperatures (21°C (38°F)) yielding a small difference of 1.0°C (1.8°F) after 15 minutes.  (I also learned that brewing in very hot climates would not be very pleasant for me.  I respect anyone with the dedication to brew when the temperature is above 30°C (86°F).)

Over all conditions, the average absolute difference in temperature at 15 minutes between measured and modeled temperatures is 0.8°C.  The largest difference at 15 minutes, 1.9°C, is for Condition AE, which has a large volume in an entirely closed kettle.

4.2 Factors Potentially Influencing Temperature
In general, we can look at the difference in measured temperatures at 15 minutes between two conditions when only one factor is different between the conditions.  A factor with a larger difference can be considered more important in influencing temperature decay than a factor with a smaller difference.  (The value of 15 minutes is somewhat arbitrary but I think not unreasonable.  It is the largest time point for which I have measured data in all conditions.)  One issue with this metric is that smaller volumes will generally have greater temperature differences over time than larger volumes.  In addition, the diameter of the kettle and area of the kettle opening will have an impact on the magnitude of the measured temperatures.  In order to normalize for these factors, one could look at the difference divided by the temperature of one of the conditions, but this relative error is less intuitive.  I’m not aware of an intuitive error metric that addresses the dependence on volume and kettle characteristics, so I’ll simply report the measurement difference as well as the volume.  Unless otherwise indicated, the kettle diameter and area of the kettle opening are the same within each comparison.

As discussed in the previous section, a high ambient temperature can have a measured temperature difference after 15 minutes of 1.0°C at 15.6 litres.  Removing the kettle from the hot metal burner yields a measured difference of -0.8°C at 25 litres.  A stainless steel kettle (instead of an aluminum kettle) yields a measured difference of 1.0°C at 7.8 litres.  The enamel kettle yields a measured difference of -3.3°C at 11.7 litres, but the two kettles have different exposed surface areas (75.2 cm2 for enamel, 60.6 cm2 for aluminum), and so this difference may appear larger than it is, even after accounting for volume.  (The predicted difference in temperature for the enamel kettle is -1.4°C).  An insulated kettle yields a measured difference of 1.0°C at 15.6 litres.  (The insulation in this case was a combination of closed-cell foam insulation and mylar wrap, around and over the aluminum kettle.)   As noted earlier for covered kettles, larger volumes have slower temperature decay than smaller volumes, with a measured difference of 0.7°C for 31.2 litres compared with 15.6 litres.

To compare the decrease in temperature of wort with water, we can compare (a) the temperature of the 24.6-litre wort case (R) with the 27.3-litre water case (M), with a difference in measured temperatures of -0.3°C; (b) the temperature of the same 24.6-litre wort case (R) with the 23.4-litre water case (D), with a difference of 1.1°C; (c) the temperature of the 29.1-litre wort case (T) with the 31.2-litre water case (E), with a difference of 0.2°C; and (d) the temperature of the 28.9-litre wort case (U) with the 31.2-litre water case (E), with a difference of 0.8°C.  In short, Condition R has no real difference with the temperature of water, while conditions T and U have a small positive difference that is contrary to the expected small negative difference based on different volumes.  The difference between measured and predicted temperatures for conditions R, T, U, and V are 1.5°C, 0.6°C, 1.3°C, and 0.0°C, respectively.  Overall there does not seem to be a large difference between the temperature decrease of wort and of water, although the model may predict slightly lower temperatures than are observed.

4.3 Incorporating Additional Factors into the Model
Because the factors described above seem to have at least some impact on the rate of temperature decrease, should we be modeling them in the temperature-decrease formula?  The answer to that question depends on our purpose (predicting IBUs) and our tolerance for error.  If we have a scenario with fairly typical home-brewing conditions, we can look at how a temperature difference of 3°C after 15 minutes impacts IBUs predicted with the mIBU method.  A difference of 3°C at 15 minutes is somewhat arbitrary, and is 1.5 times larger than the largest observed difference in these 33 conditions, but might be observed with a combination of factors different from those factors used to develop the formula.  Given a post-boil volume of 19.9 litres (5.25 gallons) in an uncovered kettle with diameter 36.8 cm (14.5 inches), a single addition of 28.35 g (1.0 oz) of 10% AA hops at flameout, and a 15-minute hop stand, the predicted temperature after 15 minutes with the formula developed in this blog post is 85.55°C (186.0°F), and we predict 9.92 IBUs using the mIBU method.  If we change the rate constant from 0.02106 to 0.01614 so that the temperature after 15 minutes is 88.55°C (191.4°F), or 3°C warmer, we then predict 10.87 IBUs, or a difference of 0.95 IBUs. If the temperature decreases by 3°C using a rate constant of 0.02638, we predict 9.06 IBUs, or a difference of -0.86 IBUs.  If, instead of a 15-minute whirlpool, we use the same rate constants with a 45-minute whirlpool, we predict 12.48 IBUs when the temperature is 85.55°C after 15 minutes, 14.31 IBUs when the temperature is 88.55°C after 15 minutes, and 10.97 IBUs when the temperature is 83.55°C after 15 minutes, or IBU differences of 1.83 and -1.51 in a 45-minute whirlpool.

Can we tolerate a difference of about 1 to 2 IBUs if our temperature decay model is off by 3°C after 15 minutes?   The short answer to that question is “yes,” for two reasons.  First, it has been reported that people can’t detect a difference less than 5 IBUs (e.g. J. Palmer, How to Brew, p. 56).  So a prediction error of even 2 IBUs is well below our ability to detect with our taste buds.  Second, there are a wide variety of other factors that make IBU prediction so inexact that getting anywhere close to a measured IBU value is cause for celebration.  For example, things that are not accounted for in the Tinseth or mIBU formulas are: (a) the inherent variability (up to 15 to 20%) in alpha acid levels within a single bale of hops (M. Verzele, and D. De Keukeleire, Chemistry and Analysis of Hop and Beer Bitter Acids, p. 331), (b) the hopping rate, which can have a significant impact on IBUs, (c) wort pH, which can affect IBU losses, (d) age of the beer, (e) the effect of pellets instead of hop cones, and (f) the age and storage conditions of the hops.  Any of these factors alone can yield a difference greater than 2 IBUs, and in combination the net effect is a high degree of uncertainty in predicted IBU values.

In summary, factors such as ambient temperature, kettle size, insulation, kettle material, etc. do have an impact on the rate of temperature decay.  However, for our purposes, it does not seem necessary to extend the formula to specifically account for these factors.

5. Summary and Conclusion
The final formula for predicting wort temperature as a function of time after flameout, for homebrew-scale batch sizes, is

T = 53.7 × exp(-b × t) + 319.55
b = (0.0002925 × effectiveArea / volume) + 0.00538
effectiveArea = (surfaceArea × openingArea)0.5

where T is temperature (in degrees Kelvin), t is time after flameout (in minutes), b is the rate constant that describes how quickly the temperature decays, effectiveArea is the “effective” area through which steam ventilates, surfaceArea is the surface area of wort exposed to air (in square centimeters), openingArea is the area of the opening in the kettle (in square centimeters), and volume is the wort volume (in litres).  The area values can be easily determined from the diameters of the kettle and the kettle opening.

It is not clear how well this formula will scale up to commercial-size batches.  If anyone who has such a system is willing to provide me with the necessary parameter values and temperature measurements, I’ll be happy to evaluate the formulas and adjust as necessary.  To contact me for this or any other reason, send an e-mail to the name associated with this blog (no spaces or other punctuation) at yahοο.

Appendix: Specifics of Each Condition
This section lists some details about each condition in table form.  The volume is either of water or wort; if specific gravity is not specified, water was used.  For partially-covered kettles, I constructed cardboard and aluminum-foil “lids” that had openings of 25%, 50%, or 75% of the area of the open kettle.  The kettle size is noted using approximate capacity, in litres.  Unless otherwise noted, the ambient temperature was approximately 13°C (55°F), and the kettle material was aluminum.

Condition volume (litres)
kettle size (litres)
wort surface area (cm2)
percent of kettle covered other notes
measured temp. at 15 minutes (°C)
predicted temp. at 15 minutes (°C)
A
7.8 18.9 710.33 0% 78.4 79.6
B 15.6 37.9 1083.80 0% 81.8 82.9
C 15.6 18.9 710.33 0% 86.8 86.9
D 23.4 37.9 1083.80 0% 87.6 86.8
E 31.2 37.9 1083.80 0% 88.9 88.9
F 7.8 11.4 457.30 0% stainless steel kettle 84.2 84.7
G 7.8 11.4 500.39 0% 83.2 83.7
H 7.8 37.9 1083.80 0% 73.0 73.3
I 15.6 18.9 710.33 0% ambient temp. 27°C 87.8 86.9
J 15.6 18.9 710.33 0% ambient temp. 33°C 87.8 86.9
K 3.9 37.9 1083.80 0% 61.7 61.0
L 11.7 37.9 1083.80 0% 78.6 79.4
M 27.3 37.9 1083.80 0% 89.0 88.0
N 11.7 18.9 710.33 0% 83.9 84.4
O 11.7 18.9 881.21 0% enamel kettle 80.6 82.0
P 19.5 37.9 1083.80 0% 85.2 85.2
Q 15.6 37.9 1083.80 0% insulated kettle 82.8 82.9
R 24.6 37.9 1083.80 0%  wort (SG=1.052) 88.7 87.2
S 25.4 37.9 1083.80 0%  wort (SG=1.052), kettle removed from heat source 87.8 87.5
T 29.1 37.9 1083.80 0%  wort (SG=1.042), loose cones 89.1 88.5
U 28.9 37.9 1083.80 0%  wort (SG=1.042), pellets 89.7 88.4
V 4.6 18.9 710.33 0% wort (SG=1.065), mIBU Exp.#3 71.6 71.6
W 15.4 18.9 710.33 25% 87.3 88.0
X 15.4 18.9 710.33 50% 88.2 89.3
Y 15.5 18.9 710.33 75% 89.7 91.2
Z 15.4 18.9 710.33 100% 97.2 95.9
AA 23.4 37.9 1083.80 50% 87.8 89.3
AB 23.4 37.9 1083.80 75% 90.9 91.1
AC 31.2 37.9 1083.80 50% 90.8 90.9
AD 31.2 37.9 1083.80 75% 92.4 92.3
AE 31.2 37.9 1083.80 100% 97.8 95.9
AF 7.8 11.4 500.39 100% 94.8 95.9
AG 15.6 37.9 1083.80 100% 97.2 95.9

Table 1. Details about each condition in this blog post.

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An On-Line Calculator for the mIBU Technique

1. Overview
In a previous post, I described a method for predicting IBUs, called mIBU, that modifies the Tinseth formula in order to account for utilization from late hopping and whirlpool hops.  After many distractions, I’ve finally implemented this method in an online calculator:

     https://jphosom.github.io/alchemyoverlord/

so you don’t need to program it yourself in order to try it out.  This blog post describes this mIBU calculator in more detail.

2. Information Needed for mIBU
The mIBU method uses the Tinseth formula as a baseline, and modifies it to account for utilization after flameout.  According to the Tinseth formula, the utilization after flameout is zero.  In this modified method, utilization continues after flameout but at a slower rate because of the decrease in wort temperature.  In order to model this post-flameout utilization, we need to know (a) how utilization is affected by temperature, (b) how the wort temperature decreases over time after flameout (before you “force” cooling with an immersion chiller, wort chiller, or ice bath), and (c) how the wort temperature changes during forced cooling.

Sub-Boiling Utilization: A previous blog post looked at sub-boiling utilization and found that Mark Malowicki provides just the information needed for how utilization is affected by temperature.

Wort Temperature Decrease: I’ve collected a fair amount of data on the natural decrease in wort temperature, and found that this decrease can be modeled accurately enough for our purposes using the wort volume, the surface area of wort exposed to air (for how much steam is produced), and the area of the kettle opening (for how quickly the steam can escape).  Other factors, such as ambient temperature or relative humidity, have only a very small impact on predicted IBU values.  Another blog post describes a formula that can be used to predict temperature from the wort volume, wort surface area, and kettle opening area.  The online mIBU calculator has input fields for the kettle diameter and the kettle opening diameter; it uses these to compute the areas and then the function of how temperature decreases over time.  For most homebrewers with an uncovered kettle, the diameter of the kettle is equal to the diameter of the kettle opening.   The decrease over time can be modeled with either a linear function or an exponential decay function.  The linear function is easier to understand, but may not be as accurate as the exponential decay function, especially for small volumes.  If you find that your temperature decrease is not predicted well by the default values in either the linear or exponential decay functions, you can change the function parameters to better model your system.

Forced Cooling: After the natural decrease in temperature of a hop stand or whirlpool, there’s a faster decrease in temperature once you start the forced cooling with an immersion chiller, counterflow chiller, or ice bath.  This function doesn’t need to be as accurate, because in most cases the wort is cooled quickly and during this time the overall contribution to the IBU is small.  I’ve collected a small amount of data on forced cooling, and developed default values based on this smaller dataset.  For the immersion chiller, the exponential decay function has a minimum temperature of 106°F / 41°C.  For the icebath method, the exponential decay function has a minimum temperature of 68°F / 20°C.  The counterflow chiller operates differently, quickly cooling the wort well below isomerization temperatures as it leaves the kettle, and so only the rate at which the wort leaves the kettle needs to be specified.

Determining Your Own Temperature-Decay Parameter Values: If you’d like to determine your own parameter values for modeling the decrease in temperature, I recommend (a) measuring the temperature of wort (or, lacking that, the same volume of water) in your kettle from flameout (time = 0) for 20 minutes (time = 20) at one- or two-minute intervals, and then (b) entering this data into an online curve-fitting calculator to obtain an equation.  I recommend the Colby College Nonlinear Least Squares Curve Fitting page.  If you use this page, just enter the times and temperatures like this:

0 212.1
1 210.9
2 209.2
3 207.9
4 206.0
5 204.4
 ...
18 186.2
19 184.3
20 184.0

with one pair of values per line and spaces in between the values.  Then select either “a exp(-bx) + c” or “ax + b” (down at the bottom of the list) as the function, and click “Fit & Plot”.  It works best if you have initial guesses for the parameters.  If you’re working in Fahrenheit, then good guesses for the exponential function are a=95, b=0.02, c=120, and good guesses for the linear function are a=-1.3, b=212. If you’re working in Celsius, then good guesses for the exponential function are a=50, b=0.02, c=50, and good guesses for the linear function are a=-1, b=100.  The results are somewhat buried in a window of text, for example:

a= 137.8 +- 65.3
b= 0.01173 +- 0.0063
c= 74.6 +- 65

and in this case, you can ignore the values after +-.   The accompanying plot is always interesting, and it should decrease smoothly over time.

3. Default Parameter Values
The online calculator is set up with default values that target a “typical” homebrewer.  If you enter a new value and then want to go back to the default, type d (for “default”) in that field.

The default values for the temperature decay functions are in red, so that it’s clear that they are defaults.  If you specify a value, the value in the field will turn black.  I’ve found that in most cases on a homebrewing scale, these defaults yield reasonable estimates.

4. Bells and Whistles: Partial Boils, Specific Gravity, Constant Hop-Stand Temperature, and Global Scaling
In order to make this calculator as useful as possible to as many brewers as possible, I’ve included a few bells and whistles.

For one, there are input values for (a) the amount of wort and trub left in the kettle after racking and (b) the amount of topoff water added if you’re doing a partial boil.  If you’re doing a full boil (not adding water after the boil), then neither of these fields are necessary.  The IBU value, as a measure of the concentration of bittering substances, depends on volume; if you have two beers, A and B, and B has the same amount of isomerized alpha acids (and other bittering compounds) but double the volume of A, then B will have half of the IBUs of A.  One important question is what volume to use in the IBU calculation: pre-boil (larger volume), post-boil (smaller volume), or something else?  During the boil, the volume will decrease and the concentration will increase.  The IBUs depend on the concentration of bittering compounds at the end of the boil, and so the post-boil volume should be used in IBU calculations.  If you top off this hopped wort with water, the IBUs will decrease as you add more water.

Glenn Tinseth has a “bigness factor” in his equation that modifies the IBU value based on the wort gravity.  He says to “use an average gravity value for the entire boil“.  Rather than ask for an initial and final gravity, this calculator asks you to specify the (post-boil) original gravity and the evaporation rate.  From these two values it computes the average gravity over the entire boil.  If you don’t care that much about such details, you can set the evaporation rate to zero and the calculator will use the original gravity.

There’s a checkbox that allows you to hold a hop stand at a constant temperature.  If this box is not checked (and the text in this field is gray), then the wort cools naturally after flameout during the duration of the whirlpool and/or hop stand.  After the whirlpool/hop stand, the wort is quickly cooled.  If this box is checked (and the text in this field is black), then the wort is quickly cooled to the target temperature using the specified method of forced cooling.  The hop stand is then held at this temperature for the specified hop-stand time.  When this time is up, the wort is again quickly cooled with forced cooling.  This option only works when using an immersion chiller or an icebath for forced cooling, since a counterflow chiller doesn’t cool the entire body of wort simultaneously.

Finally, Prof. Tinseth recommends “fiddling with 4.15 if necessary to match your system“, because IBUs can depend a lot on a brewer’s setup and brewing techniques.  Rather than making this value a variable, I introduce a “global scaling factor” that has the same effect.  The default scaling factor is 1.0, which yields the same results as the standard Tinseth formula or mIBU technique.  If you find that you’re consistently getting more or fewer measured IBUs than predicted, you can adjust this scaling factor.  (I highly recommend getting measurements of the IBUs in your beer; it’s quick, inexpensive, and the only way to really know the IBU value.  There are a number of good laboratories available for testing; I’ve been very happy with Oregon BrewLab.  I send in samples of all of my beers for testing.)

5. Optional Alpha-Acid Solubility Limit
It has been noted that doubling the amount of hops in the boil can yield less than double the IBUs in the finished beer.  With hop-forward beers, I’ve found that the Tinseth formula can greatly overestimate IBU values because it treats each hop addition independently.  At this point, my best (but still incomplete) understanding is that IBUs are not linear with hop concentration because of a limit on the solubility of alpha acids.  I describe this in much greater detail in another blog post.  If you want to try this model of the solubility limit, select “yes” for the field “Apply alpha-acid solubility-limit correction”; otherwise, select “no“.

6. Summary
I hope you find this calculator useful!  If something doesn’t work the way you expect it to, or if you have any questions, feel free to let me know.  If you’re interested in the details of the programming, the source code is available as a link on the github website.

mIBU Experiments #1 and #3

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

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

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

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

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

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

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

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

The average alpha acid concentration of about 185 ppm at the start of the boil for all conditions is just below the lower limit for a linear increase in IBU values with alpha-acid concentration.  Therefore, the Tinseth equation should still yield good results at this concentration.

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

In a separate blog post, I present a more detailed model of IBUs; the values obtained from that model for this experiment are also given in Table 1.  This more detailed model takes into account factors such as original gravity, hopping rate, age and storage conditions of the hops, alpha/beta ratio, age of the beer, and form of the hops.  Using this model, the estimated AA rating at harvest was 8.0% (the same as the value on the package) and the estimated degradation factor was 0.75 (very close to the HSI-based factor of 72%), yielding an AA rating on brew day of 6.0%.  An AA rating of 6.0% is in between the AA rating estimated from the Tinseth equation (5.79%) and the value from AL (6.25%).  The estimated alpha/beta ratio was 0.75, somewhat lower than the value from AL (0.86) but very close to the KAR value (0.761).  The RMS error from this model was 2.31 IBUs (about half the error of the Tinseth model), with a maximum difference of -2.61 IBUs.  According to this model, isomerized alpha acids contributed 65%, 60%, 46%, and 32% to the IBU values of conditions A through D, respectively.  The low percentage for even the 60-minute boil is due to the age, poor storage conditions, and low alpha/beta ratio of the hops.  I used the average boil gravity and average volume over the other four conditions to estimate 13.25 IBUs at a boil time of 0 minutes (0% from isomerized alpha acids); this value is higher than it would typically be, because of the poor storage conditions of the hops.

condition
A
condition
B
condition
C
condition
D
pre-boil SG (from hydrometer)
1.042 1.0425 1.042 1.042
pre-boil volume
2.11 G / 7.99 l 2.13 G / 8.06 l 2.15 G / 8.14 l 2.15 G / 8.14 l
time of hops addition
60 min 40 min 20 min 10 min
post-boil SG (from hydrometer)
1.059 1.058 1.061 1.063
post-boil SG (measured by AL)
1.05986 1.05891 1.06337 1.06417
post-boil volume 1.49 G / 5.64 l 1.54 G / 5.83 l 1.44 G / 5.45 l 1.42 G / 5.38 l
FG (measured by AL)
1.01134 1.00863 1.00928 1.00950
measured IBUs (from AL)
35.7 34.3 27.1 22.0
IBUs from Tinseth
40.0 34.0 24.7 14.9
IBUs from detailed model
38.18 31.69 25.01 20.00

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

 

mIBU-exp1

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

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

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

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

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

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

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

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

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

In this experiment, the alpha-acid concentration of about 290 ppm was (unfortunately) well above the estimated minimum threshold of 210 ppm.  (The alpha-acid concentration can be computed as AA × W × 1000 / V, where AA is the alpha-acid rating of the hops (on a scale from 0 to 1), W is the weight of the hops (in grams), and V is the volume of the wort (in liters).  Therefore, the Tinseth equation will predict IBU values higher than measured IBU values, unless this concentration is taken into account.

I kept a minute-by-minute record of the decrease in temperature after flameout for each condition.  Since the volume of each condition was similar, the temperature decay was also similar for each condition.  I used a single temperature-decay function, fit to the temperatures from all five conditions, to model post-flameout temperature decay in this experiment:  temp = 0.1065t2 – 5.1294t + 211.682, with temperature temp measured in Fahrenheit and time t measured in minutes.  (While larger volumes seem to fit well with a straight line, these small volumes had a temperature decay that fit much better with a quadratic function.  In the future, I’d probably use an exponential decay function.)

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

Another option for fitting the data is to explicitly account for the hopping rate and age of the beer, and use the recommended scaling factor of 4.15 in both the Tinseth and mIBU models.  We can approximate the alpha-acid solubility limit by simply limiting the alpha-acid concentration in the Tinseth equation to 260 ppm.  (Computationally, we can adjust the weight of the hops to an “effective” weight that limits the alpha-acid concentration to no more than 260 ppm at the beginning of the boil.)  We can estimate the impact of age on IBUs using an adjustment factor developed in a separate blog post: 0.33 × e0.076 ageweeks + 0.67, where ageweeks is the age of the beer in weeks.  With these modifications to the Tinseth formula and the recommended scaling factor of 4.15, there is an RMS error of 8.24 IBUs and a maximum difference of 16.1 IBUs (at the 0-minute condition).

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

For the more detailed model, the best fit was obtained by adjusting the AA rating, alpha/beta ratio, and decay factor to fit the data.  An AA rating of 6.0% (somewhat lower than the value of 6.64% reported by KAR), an alpha/beta ratio of 1.6 (higher than the value of 1.21 reported by KAR), and a decay factor of 0.94 provided the best fit to the data.  With these values, there is an RMS error of 1.27 IBUs and a maximum difference of 2.32 IBUs (for the 60-minute condition).  According to this model, isomerized alpha acids contributed 75%, 67%, 57%, 46%, and 25% to the IBU values of conditions A through E, respectively. Given the good storage conditions of the hops, the fairly low iso-alpha percentage for even the 60-minute boil is, in this case, due to the alpha-acid concentration above the solubility limit.

condition
A
condition
B
condition
C
condition
D
condition
E
pre-boil SG (from hydrometer)
1.062 1.062 1.062 1.062 1.062
pre-boil volume
1.30 G / 4.92 l 1.30 G / 4.92 l 1.30 G / 4.92 l 1.30 G / 4.92 l 1.30 G / 4.92 l
time of hops additions
60 min 30 min 15 min 7.5 min 0 min
post-boil SG (from hydrometer)
1.075 1.069 1.067 1.069 1.065
post-boil SG (measured by AL)
1.0760 1.0720 1.0685 1.0689 1.0658
post-boil volume 1.075 G / 4.07 l 1.12 G / 4.42 l 1.18 G / 4.47 l 1.17 G / 4.43 l 1.22 / 4.62 l
FG (measured by AL)
1.01190 1.01114 1.01008 1.01016 1.00944
measured IBUs (from AL)
46.4 35.4 26.1 21.2 16.1
IBUs from Tinseth, scale 6.15
49.2 36.6 22.6 13.0 0.0
IBUs from Tinseth, scale 4.15
44.6 35.0 22.8 13.0 0.0
IBUs from mIBU model, scale 6.60
46.8 37.1 26.3 19.3 12.7
IBUs from mIBU model, scale 4.15
45.5 38.1 28.5 20.7 14.2
IBUs from detailed model
48.73 36.37 26.82 20.50 15.05

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

mIBU-exp3

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

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

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

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