I recently wrote about the detailed explanations and calculations behind the equations we use to estimate average extraction yield from coffee concentration (often called *TDS*) measured with refractometers. If you have not seen this discussion, I highly recommend reading it before you start reading this blog post, as it introduces a lot of the concepts I will discuss here.

As Scott Rao and Dan Eils pointed out a while ago now, we have almost certainly not been calculating average extraction yield in a very accurate way. They describe in this blog post how (1) retained liquid in a V60 brew does not really have a zero concentration, as the standard percolation equation assumes, and (2) the retained liquid should really be divided in two categories. The first category, which they termed *interstitial liquid*, is the water between coffee particles, which concentration *at the exact moment where the brew ends* we want to count in our average extraction yield calculation.

In my last post, I suggested measuring the concentration of the last few drops to estimate the concentration of this interstitial liquid. I think this is more accurate than sampling the grounds after a brew, because there is a risk that the interstitial liquid concentration keeps going up after the brew ended, in a way that has no effect at all on the taste profile of the beverage. Remember that the taste profile correlates with average extraction yield because *how aggressive the extraction was* will dictate the relative abundances of different chemical compounds in the beverage. Therefore we want to calculate by how much the coffee particles were extracted, exactly when the brew ends, regardless of where the concentrated liquid ends up.

Scott and Dan termed the second category of water retained in the coffee bed *absorbed water*; it consists of water that penetrated the coffee cells inside a coffee particle, but never made it out carrying dissolved coffee solids with it. Hence, this liquid should ** not** be counted in our average extraction yield equations, because by definition it has not extracted any coffee compounds.

The direct effect of this *absorbed water* will be to slightly decrease the average extraction yields calculated for immersion brews, or the immersion term (the one that goes as *W*/*D*, i.e., brew water over dose) in the general equation. If we knew the weight of water that remains trapped in coffee particles (let’s call it *W*_{abs} for *absorbed water*), then implementing it in the general equation would be relatively straightforward.

To do this, we would need to link the concentration of retained water (which we call *C*_{last} because we measure it through the concentration in the last few drops) with the mass of *interstitial water* (let’s call it *W*_{int}) and the mass of coffee liquids dissolved in that retained water (*M*_{ret}), instead of that of all retained water, like this:

and then reversing this equation using some algebra would result in:

The fact that we are now counting only part of the retained water in our equation would also change the relation between beverage mass (*B*) and the mass of brew water (*W*). Remember that this relation also included the mass of coffee solids dissolved in the beverage (*M*_{bev}). That equation now becomes:

and now we can use this relation to express *M*_{ret} as a function of more readily measurable quantities. Skipping some of the detailed algebra, we can then express our general equation for the extraction yield (*E*) as:

Remember that *D* is the mass of the coffee dose, and *C*_{bev} is the beverage concentration (sometimes called the beverage *TDS*). In this equation, we also introduced a term *f*_{abs}, which I’ll call the *absorbed liquid ratio*. It is defined in a way similar to the retained liquid ratio, but counts only the part of the liquid that is absorbed by coffee particles and does not count interstitial liquid in the spent coffee bed:

We already know that *f*_{abs} must be smaller than 2 for V60 and most other percolation brews, because the *liquid retained ratio* is approximately 2 and includes both absorbed and interstitial liquid retained in the spent coffee bed.

Now what we need is a bit of experimentation before we can really use the equations above. We should either come up with an easy way for anyone to directly measure *f*_{abs}, or otherwise hope that it does not strongly depend on roast, brew method and particle size distribution. I suspect that using an aeropress or siphon might generate a scenario where *W*_{int} is close to zero because of the suction. If this is the case, then we would be in a pretty ironic situation where the percolation equation would become more accurate for such methods, while possibly not being accurate at all for V60 brews.

If you’d like to view the detailed algebraic calculations leading to the generalized average extraction yield equation above, you can find it in PDF format here.

Mitch has also just updated his universal extraction calculator to include this new *f*_{abs} term.

I’d like to thank Scott Rao and Mitch Hale for useful discussion.

Hi Jonathan,

Extraction yield, as we measure it for percolation, is a very old and universal measurement, known in other fields as ‘actual yield’ (as opposed to ‘theoretical yield’, or ‘percentage yield’). It is not exclusive to coffee.

The partially extracted liquid in the bed doesn’t affect the taste of the cup, because it is not in the cup, it’s irrelevant. It could be close to 0TDS, it could be average brew strength. It just is more likely to be lower that average brew strength in pulsed brews.

This doesn’t make measurements any more, or less accurate.

Best regards, Mark.

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Hey Mark, thanks for your comment. I disagree with the statement that extracted coffee solids in the spent coffee bed are irrelevant to the taste of the cup. I initially thought the exact same thing when I read Scott’s blog post about extraction yield, but now I am sure it is very important to consider it if we want extraction yield to inform us about the taste profile of the cup.

If you go back to my post on “The Dynamics of Coffee Extraction”, you will see that the profile of chemical compounds (and therefore taste profile) depends on how long the extraction went on (and therefore on average extraction yield), because different chemical compounds extract at different rates. While it is true that extracted compounds in the spent bed have no direct effect on the taste of the beverage, *they make our use of beverage TDS an inaccurate measure of the total extraction that the coffee particles suffered, if you ignore the coffee bed*. This is the key point: if we want to measure something that correlates with the profile of extracted chemicals, we want to measure the true total extracted mass of coffee solids at the very moment where the brew ended, regardless of where these chemicals ended up.

I hope this makes things a bit clearer. It’s very hard to convey the importance of this point, because it is super counter-intuitive.

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Hi Jonathan,

If you want to measure the relationship to extracted chemicals (this phrase is very vague – are you saying it is specific chemicals, if so, what are they & how have you identifies them?), you should maybe start with establishing how much there is to extract to start with (AOAC have an OMA for this purpose), then see how much of that you actually extract. As yet there are no correlations to flavour in the cup, relative to dissolved but not extracted soluble material. You’re not the first to ponder this.

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Hi Mark,

I realize I’m not the first to ponder this, but I also didn’t come up with any of these concepts. Scott and Dan were posting about these retained liquid concerns back in 2017, and I’m sure others were too.

You do not need to know any of the detailed chemical compositions in order to come up with this strategy; just that they extract at different speeds. Once you assume this, it becomes clear that the chemical make-up of a brew (not just its concentration) changes w/r/t brew time.

With that in mind, all we’re trying to do is come up with a number that correlates with that relative chemical make up better than brew time or concentration alone do (these numbers depend on brew method and recipe). If you can estimate the total mass of coffee extracts at the end of the brew, rather than just the mass that ended up in the beverage, then you can potentially compare different brew methods on the same scale.

Hope this clarifies things a bit.

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Hi! Because my English is not good, I did not understand what it is and how do I calculate the fabs? Can you explain it to me more simply? You do a great job. Thanks and sorry for my English.

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Hi Pacamara, fabs is just the mass of “absorbed” water (water that is trapped inside coffee particles that do not participate in extraction) divided by the mass of coffee that you used. I suspect for V60 brews it might be close to fabs ~ 1, but experimentation is needed.

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Thanks you for this post. Could you show the equation in action with exact numbers.

Thanks for your time.

Alfredo

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Dear Alfredo, Yes sure. Let’s say that you make a 1:17 ratio V60 with a 22g dose of coffee (so 374g water) and you measure that the last drops have a TDS of 0.8%, and your beverage has a TDS of 1.42%. We’ll assume you weighed your beverage and its mass was 350 grams. We will assume that f_abs = 1 (i.e., that 22g of coffee will completely absorb 22g of water which does not contribute to extraction; this seems approximately right from some tests Mitch and I did). Remember all % must be expressed as decimal, so 1.42% TDS is C_bev = 1.42/100 = 0.0142. Then, the equation becomes E = (C_bev – C_last)/(1-C_last) * B/D + C_last / (1-C_last) * (W/D – f_abs) = (0.0142-0.008)/(1-.008)*350/22 + 0.008/(1-.008)*(374/22-1) = 0.2285 = 22.85/100 = 22.85%. You can check that you get the same answer with Mitch’s calculator http://awasteof.coffee/tools/universal-extraction-calculator/

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Is there any easy or fast method to calculate accurately the fabs?

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No, not yet. Mitch and I are planning to do some experiments about that. We think it is close to ~ 1. It may depend on roast, grind size, etc. One of the best ways to measure it might be to use a vacuum pump.

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Just because I like to complicate things, what if you squeeze the spent ground by applying, say, 2 bars worth of pressure to get the interstitial fluid out

confounding bits – how does pressure affect water absorption in organic matter like coffee ground

Are we creating more i.f. by squeezing it hence underestimating the Fabs term?

off the top of my head

I thought about dehydrating agents that act quickly but it will be a major pain to separate that agent from coffee grounds/fines – potentially even less accurate

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This is pretty much what happens with Aeropress or a Buchner funnel and a small pump. From the couple tests I have done, it seems to reduce the retained water to a ~1:1 ratio. My interpretation is that typical ~2:1 retained ratios are made up of about 1:1 interstitial water removable by pressure, and 1:1 absorbed water not easily removable by pressure. I don’t know how additional pressure changes extraction dynamics exactly, but it certainly helps dissolving more stuff as happens in espresso.

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