Level: Alchemist (and maybe then some)
Reading Time: 20 minutes
In the roasting seminar I just attended a week back, I was amazed at how much math and science can be used in determining/developing what are generally considered "artistic" qualities (like in roasting/flavor development). Out of curiosity, what basic math/science skills should I brush up on that will make the biggest impact on my ability to produce the highest grade chocolate and make it a replicable process? Do I need high grade chemistry and calculus? Or is there some other basics that will get me close but still give me an edge over people who are mostly winging it?
I’m finding this decidedly more difficult to answer than I expected. To a large degree it is hard to know what you (me) are using when you already know it and what someone else needs to know. If I recall correctly, and I don’t really care if I get it wrong as the sentiment holds, I heard that both Einstein and Feynman said that if you can’t explain something to someone in simply terms (my memory says Feyman said a 5 year old) then you don’t know it well enough. I find it hard to argue with that.
With that in mind, technically speaking, if I am doing my job right then you don’t need much chemistry or higher maths since I know both and need to get it down to basic levels. You are going to need basic math but we are talking nothing more than addition, subtraction, multiplication, division and exponents, and those are minimal. You should be able to grasp the concept that if the roast goes from 205 to 215 F in one minute then the speed the roast is 10 F/min, i.e. 215-205 is 10 F and 10 divided by 1 minute is 10 F/min. Likewise 205 to 210 in 30 seconds is also 10 F/min as 210-205 is 5 and when divided by 0.5 minutes (30 seconds) you still get 10 F/min.
There are also some chemistry concepts that are helpful regarding rates of reactions, energies of activation and the one way nature of certain chemical reactions that are helpful. Unfortunately most of those are not really taught in general chemistry to any meaningful degree. I think it works just as well to say that many chemical reactions speed up as temperature goes up (this is a rate of reaction concept) and that some chemical reactions won’t happen at all without sufficient energy/temperature input (energy of activation concept). But knowing how to run the actual rate of reaction calculations are nothing you need. The deeper ‘why’ might or might not be helpful for you depending upon how you learn and how your brain is wired. Me – I thrive off those kind of details. Many people are good knowing that you should not drop the temperature of the beans during a roast as it can interrupt certain chemical reactions and once stopped, can’t be started again and we care because those reactions are developing flavors. It is sort of like if you stop baking a cake half way through. The cake falls and won’t rise again if you try and finish baking it some time later. Yes, it will get cooked through but simply won’t be the same.
With that all said, some higher maths can be helpful and I’m going to take this opportunity to springboard into a concept that I’m officially calling Degree of Roast (DoR). I think I can get the concept across to you without the higher maths but for you to really use it to its full potential it will take you understanding some calculus notation, what first and second order equations are (spoiler, a 2nd order equation just has an exponent of 2 in it - you know 3 raised to the 2nd power is also 3 squared and just means 3 times 3) and how to take the integral of those equations (once you solve for them) and how those results give you a way to compare different roasts and evaluate which ones have a higher or lower DoR.
The first thing we should get into is that light and dark roasting does not really have any meaning in cocoa. I understand where it comes from, namely coffee, but it just isn’t applicable since there is effectively no color change as a cocoa roast progresses (fun fact, beans often get lighter as they roast which in the range for chocolate).
It would be great to relate the temperature of the outside of the bean (the end of roast or EOR) to the DoR but unfortunately it isn’t quite that simple either. When you roast, heat migrates from the outside to the inside. The longer the roast, the more time there is for energy to penetrate, and all things being equal (namely the EOR temp) a longer roast will have a high Degree of Roast.
That just means that if you take two roasts to 260 F, generally speaking the one with the longer roast time will be more roasted. This implications of this leads to many other conclusions that are helpful in planning a roast. For instance if you like the DoR of a particular bean (because it properly removed the raw astringency and moisture) but you want to keep it at a lower EOR temperature to preserve the fresh fruit flavors then you need to extend the roast time to get the same DoR.
But how much? For years I have taught in my roasting seminar a qualitative way to estimate it, simply saying to extend it to the farther end of the 3-5 minute window in the finishing phase. This works ok when you have a lot of experience and you can just intuit it by feel as it were. It is sort of like watching the trajectory of a thrown baseball and knowing instinctively how much you need to raise or lower your hand to catch the ball. In truth there is little instinctive about it – it just looks that way from the outside where the observer doesn’t know how much time you have put in to catching balls. But if someone watches throw after throw and works out the trajectory then it is just applied math. The same is true with roasting. This we where we have to pull out some maths and specifically some calculus to get away from what appears instinctual and brings it into the realm of quantitative.
Let’s have a look at what a traditional asymptotic drum roasting profile looks like and what information it holds.
I’ve often said a good roast profile goes asymptotic to some limit but that is not really the case. What you are really doing is roasting along one side of a partial parabola.
The nice part of that is that you can use something like Excel to determine the equation of that portion of the curve, which is a 2nd order equation (there will be an x squared in it).
For those with Excel, you will record the minute by minute progress of your roast, create an XY scatter plot, and then right click on the resulting curve and “add trend line” making sure to pick the 2nd order equation and that you show the equation and R squared value.
Now, the whole curve is clearly not a 2nd order equation and only a portion of it correlates to the DoR. When we drum roast we don’t really care about how long the Drying phase is as no chemical reactions are really happening. You are only preparing the roast to hit a certain Development and Finishing phase during the Drying phase.. For that reason we are only concerned about the portion of the curve between 212 F and the EOR.
Similarly we are only concerned about the portion above 212 F as anything below that is not contributing in any meaningful way to the roast. That leaves us with a portion of curve that looks like this (the darker overlayed part):
To get that area under the curve, bounded by the conditions we have established requires us to integrate the equation of the curve and subtract from it the area below 212 F. So let’s run through what that means and looks like from a calculation standpoint.
The first thing is to plot out your roast profile. Then remove the portion you don’t need so it looks like the partial curve above. Then use Excel to give you the equation of the curve by right clicking on the curve and ‘add tread line’ making sure you make it a 2 order polynomial and that the equation is shown on the line.
If you have done everything right you will get an equation for the line on the graph that is in the form of y = ax^2 + bx + c
At this point we have to integrate the equation. I realize a lot of you don’t know what that means and really, it does not matter. The only thing you have to take away is that it is the way you manipulate the equation so that it spits out the area under the curve (our goal). That transformation looks like this. In short, you add 1 to the exponent of each portion and multiply it by the inverse of that new exponent.
Since we are concerned with only what is happening in the Development phase and the End of Roast, we have to define the integral only in that region based on the x values which are just the times at 212 and the EOR. All that means is you use the equation twice to get a pair of values and subtract the later (EOR) from the former (t @ 212). That looks like this:
The above equation will give you the area of the shaded part below.
The final piece is to get rid of the portion of the area that is below the 212 F level. This area here:
To do that we just have to determine the area and subtract it. I could dive further into how the determine the equation of a flat line, how to get the integral and then apply it but I’ll have a little mercy on you and just tell you that you use this equation:
(Time(EOR) – Time(Drying time) ) x 212
If you look back at the graph you will see it is just a rectangle that is being removed and the area of a rectangle is just height times width. Basically it is just the combined time of the Development and Finishing phase (the width) times the Drying temperature (the height).
The final combined equation looks like this:
Let’s go through an example so you can see what is happening to get the area shown here that is your Degree of Roast.
This is for Fiji Rakiraki. From my classic roasting notation 13:25/15:50/20:00 @ 252 F you can input those numbers into an excel spreadsheet and apply the 2nd order trend line to get this equation and you can pull out all the variable values you need (these values are not from the graphs above so don’t get confused trying to find them).
y = -0.2545x^2 + 14.32x + 68.036
a = -0.2545
b = 14.32
c = 68.036
x@212 & T(Drying) = 13:25 = 13.4167 minutes
x@EOR & T(EOR) = 20.0 minutes
Now, try not to let your eyes glaze over. We are almost done. It is just a matter of plugging everything in and then basic maths and the Degree of Roast value falls out.
DofR = (1/3 * 20.0^3 * -0.2545 + 1/2 * 20.0^2*14.32 + 20.0 * 68.036) - (1/3 * 13.4167 ^3 * -0.2545 + 1/2 * 13.4167^2*14.32 + 13.4167 * 68.036) – ((20.0 – 13.1467)*212)
A little number crunching and you end up with
DofR = 145.8 F-minutes
That is all well and good but that is just a number in isolation. To be meaningful it is helpful to compare it to another roast, so let’s do that.
Catongo @ 10:50/13:30/17:30 @ 255 F When you plug in all the numbers (spoiler – I’m going to give you a spreadsheet you can use for a lot of it a bit later) you get:
DofR = 175.8 F-minutes
What you can immediately see is that the Catongo has a higher degree of roast.
Except it doesn’t………….
……..and now I’ll introduce the final wrinkle – at the end of the day, size does matter. That unit of F-minutes is a unit of energy. A nice way to think about it is like a volume and the bean is a container. If the container is smaller, a given amount of energy will fill it up more than a larger container and how full the container is is really what we care about. What I’m trying to say here is that if you have a two different sized cocoa beans and give them roughly the same amount of energy (they have the same profile), then the smaller one is going to be more roasted. Put a muffin and a loaf of bread in the oven together and it is pretty obvious the muffin will always be more baked. To make use of that concept we need to know the size of the bean and apply it to the DofR number.
Cocoa bean size is classically defined as the weight of 100 beans with the average being 100 grams or 1 gram/bean. Dividing the DofR by this value normalizes the result and lets you compare different sized beans.
Fiji is a smaller bean at about 0.8 g/bean and Catongo is about 1 g/bean. That then looks like this:
Fiji DofR = 145.8 / 0.8 = 182 Fmin/g
Catongo DofR = 175.8 / 1.0 = 175.8 Fmin/g
So what you can see, because the Fiji bean is smaller it is actually more roasted even though the EOR temperature was lower.
Interesting huh?
So how is this all helpful? Say you roast that Fiji and then taste it and determine you like the dried fruit flavor but think it is a little to ‘roasty’. My answer would normally be keep the EOR temperature the same but get there faster so there is not as much time for heat penetration. Sure, you can just randomly push the roast faster and hope you get what you want but the WHOLE point of the above was so you didn’t have to guess and do roast after roast but instead apply some maths and KNOW you are at a lower DoR. What does that look like? Well, you DO just pick a shorter time but instead of doing a roast, you just plug it into the spreadsheet and tweak the numbers until you get a lower DoR.
The original roast’s finishing phase took 4:10. If we kind of arbitrarily reduce that to 3:10 then the spreadsheet spits out a new quadratic curve of y = -0.1526x^2 + 11.77x + 83.490 and for the Profile 13:25/15:50/19:00 @ 252 F the DoR ends up as 152.0 Fmins/g, down from 182 Fmin/g.
Will that be enough? I don’t know as in theory it was your taste preference that said it was too roasty. It will certainly be less roasty but only you will be able to tell if it is enough. It is STILL going to take some iterations but the more you do, the data you are going to accumulate will quickly tell you the DoR that correlates to the roast characteristics you like or don’t like.
Before I leave you to digest all this, I want to point out that you can of course adjust the Development phase too if you want to. Why would you? Maybe you pushed it really hard and it gave a bitter note to the chocolate so you want to slow it down. But as you saw above, slowing it down (giving it more time) will increase the Degree of Roast so you may need to further shorten the Finishing phase so the whole roast is not over roasted.
Finally, so far I am finding a DoR between 135-185 seems to be good range for cocoa. To either side you are probably get astringency (for being too light and not getting rid of the raw astringency) or bitter (from too high of a DoR).
And yes, that was a lot of maths and more than many of you want or even need but should you love this kind of thing and really want to take control of your roasts, this is certainly another half dozen arrows in your proverbial quiver of knowledge, data and roast manipulation that is not just random tweaking. It is math and science!