Impress your Friends - Analyzing the Tinseth Formula
Posted Jul 29th 2014 | By:
International Bitterness Units (IBUs) are one of the fundamental concerns of almost any brewer. If you have the resources, a laboratory will be happy to take your money and some of your beer and tell you just how many iso-alpha acids (the main contributor of perceived bitterness) are dissolved into your beer. For those of us who do not want to part with our money or beer, we need some kind of equation so we can make a very good estimate. The Tinseth formula is one of those equations.
You might be asking yourself, "I have brewing software. Why do I care?" That is a valid question. If you are the type of brewer who experiences success and enjoyment from plugging in numbers, this is probably not the article for you. However, you might be the type of brewer who appreciates a deeper understanding. I have gained a better understanding of beer construction from putting this article together and reading it will benefit you. A study of this article will provide:
- An understanding of the basics and behaviors of the Tinseth Formula
- A way to calculate the theoretical maximum IBU contribution for any hop addition
- Information to make informed decisions concerning boil time as it pertains to IBUs
- The Tinseth formula "in reverse" to find a boil time required to achieve desired IBUs.
- An understanding of how the Tinseth formula can be viewed as a function of two variables
- An understanding of the role wort gravity plays in the rate of change of IBUs
- Methods to correct IBUs for late additions of extract and partial boils
I am writing this article for all home brewers, but my goal is to reach those of you less comfortable with math. It is my hope that your ability to read a graph will provide you with more understanding than you had before. I also provide general formulas for you to use.
Introduction to the Tinseth formula (or, How to Impress Your Brew Friends):
For some of you I am sure the Tinseth formula is intimidating:
As ghastly as it might seem, the formula is pretty simple. Yes... there are a lot of pesky decimal places and you might not know what Euler's number (e) is, but the actual math is surprisingly simple.
Side Note: Euler's number (e):
Before I get too far here, I want to make sure you know what the number e is. Consider it like the number pi. Most of us know that pi (represented as p) is approximately equal to 3.14... I say approximately, of course, because the number is irrational (which is a fancy way of saying the decimal goes on forever and never repeats). The number e is Euler's number and is approximately equal to 2.72, because, like pi, it is irrational. Pi is found when you divide the diameter of a circle by its circumference, whereas e is found by using a compound interest formula and seeing what happens as the number of times it is compounded approaches infinity. The number e happens to be really good for describing natural phenomena, which is why you see it all over the place. For our purposes you only need to know where to find the "e" on your calculator or that you can use 2.72 to approximate it.
So what does this function look like?
Sometimes it is better to jump right in. Below I model what happens as time (t) is allowed to change. For this model I am using the following values:
- The post boil specific gravity will be equal to 1.050
- The post boil volume will be equal to 6 gallons
- One ounce of 7.8% alpha acid Palisade hop pellets
What we notice here is that the longer the hop is boiled, the more IBUs that hop contributes. This stands to reason when we consider the common conception of hops and length of boil: the longer the hop boil, the more bitterness. However, maybe the shape of the graph surprised you. This is a general curve shape you will become very familiar with.
Also notice that the formula will equal zero when t equals zero. This due to a special characteristic of this equation: the subtraction portion. The number e (or any number for that matter) that is raised to the power of zero is always equal to one. So when t is equal to zero the subtraction part of our equation looks like "1 1" which is 0. Multiplying anything by 0 provides us with a big fat nothing. This keeps in line with common conceptions concerning IBUs and flameout hops: there simply is no contribution.
It also is easy to see that, eventually, longer boil times do not have a significant impact. We begin to see these diminishing returns at around 60 minutes. If I were to produce two beers with identical grain bills and a 1.050 specific gravity, one with a single addition of Palisade at 60, the other at 90 minutes, I very much doubt even the most experienced taster could tell the difference between 22.5 and 24 IBUs, respectively. This begs the question:
Is there a theoretical limit to IBU contributions as defined by the Tinseth Formula?
IBU Potential: theoretical and impossible to achieve.
I am going to use the term IBU potential to refer to the theoretical limit to IBU contributions. There will always be one upper threshold that the Tinseth curve gets closer and closer to but never touches (in math we say that the graph is asymptotic to that value). This is a tricky number to find, since we can never plug in a value for t that will actually tell us what that value is. Remember that this value is our IBU potential and will be important later on.
With that in mind, let us journey into the absurd to see what we can discover. Imagine that I started a boil and placed an ounce of 7.8% alpha acid Palisade hop pellets in it. Also, let us say that the boil never loses any volume to water vaporization and I have all the energy required to sustain the boil. What would happen if I boiled my wort for an infinite amount of time?
Well the short and obvious answer is I would never get to enjoy my beer. In math, however, this is a question worth asking. We can actually look at a function and examine its pattern as the variable reaches out into infinity. We can, in effect, boil a hop for an infinite amount of time. Let Ipot refer to the IBU potential of a hop addition.
If you do not understand what just happened there, it's okay. Basically all we were concerned with was the number e to a negative power. As the time variable t moves to infinity we see e to a negative power move to being e to a BIGGER negative power. The values we get back are moving closer and closer to zero. Using our finite number system we can never see it actually reach zero. By taking the limit (lim) of that function, we can see that the IBU quantity in our infinite boil actually converges on a number.
This number is the IBU potential of a hop addition. We can never achieve this number, but it is an important number to know. We can never achieve numbers greater than the IBU potential, or even achieve that number. After plugging in the values from my example into the boxed formula above, the IBU potential for my Palisade hops is roughly 27.4 IBUs. That is the most IBUs I could ever possibly get with that hop addition in that recipe.
Boil Time and IBU Utilization:
To find the percentage of the hop bitterness that has been utilized at any given time, we divide the equation that changes with time by the IBU potential, multiplied by 100. Since the two equations share so many terms you can see that most of them cancel out. Basically, the only terms that are going to carry over into our percentage equation are the ones I have highlighted in red. Let U% be the percentage of total IBUs utilized for any given time t.
This shows us some practical information about what we get out of our hop additions. In the past I have increased my boil time on a few occasions to extract more bitterness from my bittering hops. I am probably not the only person to do this, which is why it is important to take a look at the following graph.
You can squeeze out more IBUs from longer boil times. At 120 minutes, we see that we've converted almost as many alpha acids as possible from the hops. At the same time we notice that at 20 minutes, 55.1% will be converted which is better than half of the potential IBUs.
Please note that these values are a percent of the IBU potential, which are different than actual quantities of IBUs. For example, my recipe that uses one ounce of 7.8% alpha acid Palisade hops has a maximum theoretical IBU limit of 27.4. I can utilize 90.9% of 27.4 using a 60 minute hop boil time or 99.2% of 27.4 using a 120 minute hop boil time. In other words, I can get 8.3% more IBUs from my hops if I boil them for 120 minutes as opposed to 60. However, as previously noted, this percentage does not tell us much until we use it to find the difference in IBUs. If we take 8.3% of 27.4, we find that the IBU difference is 2.3 IBUs. This hardly warrants the extra time and energy if achieving greater IBUs is my sole aim.
These values could change depending on the IBU potential of your hop addition, but as a general rule, increasing boil time for the sake of greater IBUs is not efficient. If you consider an additional 10 IBUs worth doubling the boil time from 60 to 120 minutes, your IBU potential would have to be 120.5 IBUs, which is impractical for most applications.
Also notice that as far as sheer IBUs are concerned, adding half of a bag of hops at 120 minutes would achieve fewer IBUs than adding a full bag at 20 minutes. Of course, this greatly affects the flavor and (possibly) aroma characteristics of the beer. This technique of hopping can provide all the required IBUs within the time window normally associated with flavor and aroma additions (flameout to 30 minutes).
Thankfully there are other reasons you may want to increase boil times that will usually trump the need for more IBUs. I believe it is more interesting that a hop addition that has been boiled for a mere 20 minutes has already reached greater than half of its IBU potential.
Solving the Tinseth Formula for t:
Hops come in one ounce packages. If you are like me, you do not always wish to mitigate IBUs by using fractions of a package. Instead, sometimes I use reduced hop addition boil times to provide the remaining bitterness I need. This means we need to change the formula from variable t, dependent I to variable I, dependent t.
It looks ugly, and it certainly is not pretty. Most of the algebra it takes to pull this off you might already know. The big problem is the fact that t is in the exponent: this is where we need "ln". Basically, "ln" is a function that 'undoes' the e to the power of -.04t. It is called the natural logarithm (or natural log)... but let us not get weighed down by any of this.
What we need to know is that you cannot take the natural log of a number that is zero or negative. This is critical because our new equation is not continuous for all values of the new input I. By this I mean that certain values of I will make the value (inside the natural log) zero or negative, which as I mentioned is impossible. This makes sense from our analysis that, given a specific hop, quantity and specific gravity, only a certain amount of IBUs are possible. You might remember this from when we took the limit as t went to infinity. A little algebra can give us a range of acceptable IBU inputs, since we know what we are taking the natural log of must be greater than zero.
That inequality might look familiar to you since the left hand side of the inequality is the equation we achieved for IBU potential. All values of I must be less than that, because it was only by using the technique of "taking the limit" that we were able to look at "infinity time" where that maximum value was achieved. Also, notice that there is only an upper limit imposed here. We know that negative IBUs are patently ridiculous, so therefore the minimum is zero. Now we can mathematically describe the possible I inputs with the following inequality:
Considering the Tinseth Formula as a function of multiple variables:
I mentioned before that for practical purposes we could consider the gravity a constant. This is true because generally speaking when we are examining our hop additions we already have some idea about what our gravity is going to be. This is sound reasoning but sometimes it is worth considering general trends when working on a beer recipe. In this section I will look at what happens as two values are considered variables and can change independently.
In a previous example, we used one ounce of palisade hops in 6 gallons of 1.050 post boil wort. We noticed the exponential trend and observed some traits of this type of function. You might even realize at this point that nothing will change that general shape. This is what we will see when we vary both gravity as well as time in the Tinseth formula. Below is a graph of a range of specific gravities (1.000 to 1.100) and a range of times (0 to 90). Time increases from left to right and the gravity increases from back to front.
To help you see the trends better you can examine the edges and corners of the graph. The left edge of the graph shows that all values for IBU contributions are zero (the blue region). This is when t equals zero. The back left, next to where it says "0", is when t is zero and the specific gravity is 1.000, the front left corner is when t is (still) equal to zero and the specific gravity is 1.100. Time increases as you travel from left to right. The color code helps you see brackets of IBUs: 0-10 IBUs blue, 10-20 IBUs red, 20-30 IBUs green, 30-40 IBUs purple.
If you are having trouble reading the graph, I have provided a table of values to follow. It is important to note that I do not include all of the calculated numbers on the tables that are used to make this graph (or any other graph in this article). In the interest of exactness, I use more data points to make the graph than you would be interested in seeing. I also excluded the points at time zero because the there is no IBU contribution, no matter what the gravity-which is not very interesting.
Look to the far back edge of the surface, then look to the leading edge closest to you. That is the general shape of our function of t, which you can see in the first graph of this article. I will emphasize this by allowing you to look "head on" at the graph in part 2.
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