jpender Senior Member Joined: 11 Jul 2011 Posts: 401 Location: California Expertise: I like coffee
Grinder: Kyocera CM-50 Vac Pot: S/S Moka Pot Drip: Aeropress
Posted Mon May 21, 2012, 1:35pm Subject: Re: Coffee Extraction Discussion, Questions for the membership:
Why a Rayleigh distribution?
Maybe extraction rate is proportional to soluble coffee remaining, for a given particle size. Since any grind is going to be a distribution of sizes you might need a sum of decay functions.
Posted Mon May 21, 2012, 1:53pm Subject: Re: Coffee Extraction Discussion, Questions for the membership:
jpender Said:
Why a Rayleigh distribution?
Maybe extraction rate is proportional to soluble coffee remaining, for a given particle size. Since any grind is going to be a distribution of sizes you might need a sum of decay functions.
Occam's Razor. Start with minimum amount of factors/variables, expand only when necessary. It had the approximate shape from memory and is the basis for a Chi^2 distribution, and is easily differentiable.
------------------------------------------ ----------------------------------------- Le café doit être noir comme le diable, chaud comme l'enfer, pur comme un ange, et doux comme l'amour.
"There is no right answer with coffee. There is only the elixir in your cup at the moment you partake."
"...I often say that when you can measure what you are speaking about, and express it in numbers, you know something about it; but when you cannot measure it, when you cannot express it in numbers, your knowledge is of a meagre and unsatisfactory kind;..." - Lord Kelvin RECIPES thread => http://www.coffeegeek.com/forums/coffee/machines/585708
jpender Senior Member Joined: 11 Jul 2011 Posts: 401 Location: California Expertise: I like coffee
Grinder: Kyocera CM-50 Vac Pot: S/S Moka Pot Drip: Aeropress
Posted Mon May 21, 2012, 3:34pm Subject: Re: Coffee Extraction Discussion, Questions for the membership:
Netphilosopher Said:
Occam's Razor. Start with minimum amount of factors/variables, expand only when necessary. It had the approximate shape from memory and is the basis for a Chi^2 distribution, and is easily differentiable.
How about A*ln(t) + B ? It makes no physical sense and fails as t->0 and as t->infinity. But it works well enough if all you want is something simple that fits the data you have so far.
May 22 edit: I manually fit three curves to your data: logarithmic; two term exponential decay; Rayleigh distribution.
jpender Senior Member Joined: 11 Jul 2011 Posts: 401 Location: California Expertise: I like coffee
Grinder: Kyocera CM-50 Vac Pot: S/S Moka Pot Drip: Aeropress
Posted Tue May 22, 2012, 12:09pm Subject: Re: Coffee Extraction Discussion, Questions for the membership:
I suggested the exponential decay function because I thought it was a plausible candidate for a simple physical model of extraction.
For a bimodal distribution two terms might be sufficient. A1/A2 would be the ratio of the mass densities of the two peak sizes. A1+A2 would be the maximum extraction or strength as steep time goes to infinity. T1 and T2 would be extraction time constants and would relate to particle size.
strength = A1*(1-exp(-t/T1)) + A2*(1-exp(-t/T2))
One reasonable possibility is that extraction rate is proportional to total coffee particle surface area. For a given mass of coffee the total surface area would be proportional to 1/diameter of the particles, for a given size. This implies that T1/T2 would be proportional to the ratio of the diameters of the two particle sizes.
To fit your data I used: A1 = 0.85%, A2 = 0.63% (A1/A2 = 1.3; max strength = A1+A2 = 1.48%) T1 = 6.5s, T2 = 65s (T1/T2 = 1/10)
This corresponds to a bimodal distribution with about 30% more mass in the smaller particles and with the larger particles having a diameter 10 times greater than the smaller ones.
If you had sieves to select by particle size or knew the particle size distributions for your grinder at various settings you could test this model.
jpender Senior Member Joined: 11 Jul 2011 Posts: 401 Location: California Expertise: I like coffee
Grinder: Kyocera CM-50 Vac Pot: S/S Moka Pot Drip: Aeropress
Posted Wed May 23, 2012, 12:47pm Subject: Re: Coffee Extraction Discussion, Questions for the membership:
Another way to look at the data is to assume an exponential function and then take the log of both sides. This results in a linear equation. In the attached graph I've done just this. The black line is the linear fit to the data by Excel, with the line equation displayed (slope=0.0154, y-intercept=0.8478).
The inverse slope of the line equals 65 seconds, which is the time constant T2 that I had chosen manually to fit the two-term exponential. The y-intercept represents the contribution from the smaller particles. It can be used to back calculate A1=max_strength*(1-exp(-y_intercept)). It works out to 0.85%, which is what I'd arrived at through trial and error previously.
By the time you start taking measurements the contribution from the smaller particles is essentially complete. From your perspective it's as if the brew water contained that strength of coffee at time zero. Because the temperature is also changing this is likely exacerbated. So you're really only measuring the time dependency of the larger particles.
In the graph the red dashed line represents the larger particles and the blue dashed line the smaller ones.
Extraction rate is almost certainly a function of temperature which isn't being controlled actively. As the Aeropress cools how does this affect the curve? Will it look different in a press pot or on a really cold day? And of course all of this assumes the same brew ratio, grind, coffee and roast.
GlennV Senior Member Joined: 27 Oct 2011 Posts: 28 Location: UK Expertise: I love coffee
Posted Wed May 23, 2012, 3:08pm Subject: Re: Coffee Extraction Discussion, Questions for the membership:
The simplest model of extraction is polynomial rather than exponential. The rate of change of radius of a dissolving sphere is constant at constant temperature if the concentration is far from saturation (Hixon-Crowell). This gives r = r0 - k t, t<r0/k and so the extracted mass is proportional to
1 - (1- t/t0)^3 for t<t0
where t0=r0/k is the time extraction ends. I suspect that doing this with two different radii (and the same k) would fit the data pretty well in spite of the fact that the assumptions are nowhere near satisfied (particularly constant temperature). I also suspect that it would suggest an unreasonable amount of material was present in the smaller particles. An alternative interpretation would be to consider faster and slower dissolving stuff within each particle (ie two different k) - but interestingly this doesn't actually change the mathematical form of the model at all. I think Occam would be smiling.
jpender Senior Member Joined: 11 Jul 2011 Posts: 401 Location: California Expertise: I like coffee
Grinder: Kyocera CM-50 Vac Pot: S/S Moka Pot Drip: Aeropress
Posted Thu May 24, 2012, 11:45am Subject: Re: Coffee Extraction Discussion, Questions for the membership:
It isn't simpler, it's just a different picture.
Hixson-Crowell imagines particles dissolving from the outside in, like a hard candy in a child's mouth. The inner soluble material is untouched by the solvent until the layers outside of it dissolve away.
Exponential decay pretends that all of the soluble material is available to the solvent, which would imply that particle size is irrelevent. Tacking on the additional requirement that surface area determines extraction rate is a kludge.
I'm not certain what you are proposing with faster/slower materials. Can you elaborate? Wouldn't that destroy the picture that Hixson-Crowell draws? If these materials are uniformly distributed then one would determine the dissolution rate of the of other. Alternatively one might imagine the faster material is somehow available to the solvent in advance, such as solvent penetration prior to dissolution of the other material, or maybe a coated shell of faster material outside the slower. This wouldn't make Occam happy at all. So what did you mean?
There are undoubtedly a number of ways to fit the data. Is there a way to distinguish between them? Ultimately a model should be predictive, e.g. determining extraction rate from grind profile. Otherwise parsimony would dictate choosing the simplest curve that fits the data.
Surely someone has already undertaken this research? I found references to one paper on the subject but could not locate the paper itself, only a graph adapted from it (attached below). They appear to be using the exponential model, but it's possible that their data could be just as easily fit with a two-term Hixson-Crowell.
(Adapted from "The Infusion of Coffee Solubles into Water: Effect of Particle Size and Temperature", A.J. Smith and D.L. Thomas, 2005)
Posted Tue May 29, 2012, 5:39am Subject: Re: Coffee Extraction Discussion, Questions for the membership:
jpender Said:
It isn't simpler, it's just a different picture.
Hixson-Crowell imagines particles dissolving from the outside in, like a hard candy in a child's mouth. The inner soluble material is untouched by the solvent until the layers outside of it dissolve away.
Exponential decay pretends that all of the soluble material is available to the solvent, which would imply that particle size is irrelevent. Tacking on the additional requirement that surface area determines extraction rate is a kludge.
I'm not certain what you are proposing with faster/slower materials. Can you elaborate? Wouldn't that destroy the picture that Hixson-Crowell draws? If these materials are uniformly distributed then one would determine the dissolution rate of the of other. Alternatively one might imagine the faster material is somehow available to the solvent in advance, such as solvent penetration prior to dissolution of the other material, or maybe a coated shell of faster material outside the slower. This wouldn't make Occam happy at all. So what did you mean?
There are undoubtedly a number of ways to fit the data. Is there a way to distinguish between them? Ultimately a model should be predictive, e.g. determining extraction rate from grind profile. Otherwise parsimony would dictate choosing the simplest curve that fits the data.
Surely someone has already undertaken this research? I found references to one paper on the subject but could not locate the paper itself, only a graph adapted from it (attached below). They appear to be using the exponential model, but it's possible that their data could be just as easily fit with a two-term Hixson-Crowell.
(Adapted from "The Infusion of Coffee Solubles into Water: Effect of Particle Size and Temperature", A.J. Smith and D.L. Thomas, 2005)
Yah, finding out that for whatever reason, that paper is basically no longer available. I've tried several places that reference that paper, and nobody seems to have a copy of it anywhere. I was interested in what method they used to do their analysis.
Interesting stuff. Longer steep times don't seem to increase the concentration much, if any (the handful of 8:00 brews I did are running between 1.46 and 1.50%). I'm gathering several more brews at 5+minutes same settings before I start getting into temperature, but one interesting thing I did do is brewed a cup to 6 minutes - then dumped the whole shebang into a mason jar and into the fridge it went for 24 hours.
That brew only came up to around 1.52%, and the result was wonderfully chocolately and rich - I was absolutely floored. I can't wait to try this with a Tanzania or Yemen.
the bottom line is that for "normal" (i.e. less than 8 minutes) brewing, at least with immersion, the brew seems to reach an equilibrium. While it's maybe not exactly "saturation", it acts sufficiently like it that I think the analogy is appropriate.
I've also done a few "probe" brews at other grind levels (still same procedure for temperature, essentially "max"). If I infer anything from the data, there's two things that seem to happen with varying the grind:
1) grind size changes the time it takes to reach the "equilibrium" strength, and 2) grind size (all other factors held constant) seems to govern the "equilibrium" strength. (Same essential relationship I found back here: ) Click Here (www.coffeegeek.com) (also, even though I dropped the coffee amount to 15g which changes the thermal portion of this experiment, I'm still getting same strengths)
1 minute strength for a fine grind is pretty high, for example. The resulting equilibrium strength is higher also (and can be consistently duplicated).
At press pot grind, the time it takes to get within a point or two of "equilibrium" is still somewhere around 2:30 to 3:00.
If I had the time, I'd do this time profile for several other grind settings, and then replicate this with varying temperature - but to reduce the runs I need to figure out if the extraction curve can be estimated with only two or three data points. I'm a bit leery of "sampling" the brew during brewing as technically I'm changing the brew ratio. Ideally, I'd love to have a family of extraction curves that overlay and can be tested and predicted based on strike temperature, grind level, and brew ratio.
Of course, to standardize my grind, I'll have to get a set of sieves... LOL (oh, gee, MORE hobby equipment)
------------------------------------------ ----------------------------------------- Le café doit être noir comme le diable, chaud comme l'enfer, pur comme un ange, et doux comme l'amour.
"There is no right answer with coffee. There is only the elixir in your cup at the moment you partake."
"...I often say that when you can measure what you are speaking about, and express it in numbers, you know something about it; but when you cannot measure it, when you cannot express it in numbers, your knowledge is of a meagre and unsatisfactory kind;..." - Lord Kelvin RECIPES thread => http://www.coffeegeek.com/forums/coffee/machines/585708
jpender Senior Member Joined: 11 Jul 2011 Posts: 401 Location: California Expertise: I like coffee
Grinder: Kyocera CM-50 Vac Pot: S/S Moka Pot Drip: Aeropress
Posted Tue May 29, 2012, 12:16pm Subject: Re: Coffee Extraction Discussion, Questions for the membership:
I read a paper by Andrew Stapley (2002) on tea and coffee infusion.
He uses the same basic exponential form, that ln(c_equ/(c_equ-c)) versus time is linear, with a y-intercept. But whereas I was just imagining that a decay function "made sense" he derives this result from Fick's laws of diffusion.
The model he uses is that the solute is distributed uniformly throughout each coffee particle but the rate of dissolution is much faster than the rate of diffusion. This means that the concentration of solute near the surface of each particle initially decreases quickly resulting in a concentration gradient within the particle. The initial surge of dissolution that sets up this gradient is what produces the y-intercept in the logarithmic graph. The slope of the line indicates the ultimate equilibrium concentration. It's worth mentioning that this slope/intercept picture depends on the experimenter being unable to measure (or intentionlly omitting) the early, fast change in concentration. If you could measure the concentration from time zero you'd get a curve rather than a line.
If instead of a single particle size there are two, with one much smaller than the other, the smaller particles would have the same type of effect on initial concentration change as the initial dissolution surge. The assumption that the early effects are due entirely to particle size is likely incorrect, as GlennV noted. Your grind profile is probably not as bad as I was suggesting!
Netphilosopher Said:
1) grind size changes the time it takes to reach the "equilibrium" strength
Stapely quantifies this for spherical particles of uniform radius R. He found that the time constant is proportional to R^2. I had assumed the rate was proportional to total surface area, and thus the time constant proportional to R. Since I don't know the constants of proportionality, using R^2 instead fits your data just as well for the simple reason that it only affects what happens before you were able to make any measurements. An illustration of one of the pitfalls of fitting limited data to a model.
Netphilosopher Said:
2) grind size (all other factors held constant) seems to govern the "equilibrium" strength.
Stapely doesn't address this directly. What does it mean? If there were a concentration gradient in the particle that survived at apparent equilibrium a grind of larger sized particles would retain more solute. But going from a strength of 1.7% (fine grind) to 1.3% (coarse) in a 7.5% brew ratio implies a fair amount of solute trapped by some mechanism. Perhaps in larger particles part of the interior is simply unavailable due to some physical hindrance. Would the same effect be observed in a wash brew method?
Interesting stuff... will it make for better brewing? I don't know, maybe. But I just tossed an unmeasured amount of coffee into the non-inverted AP this morning with water "sort of off the boil" and it tasted great.
Posted Wed May 30, 2012, 7:57am Subject: Re: Coffee Extraction Discussion, Questions for the membership:
jpender Said:
I read a paper by Andrew Stapley (2002) on tea and coffee infusion.
He uses the same basic exponential form, that ln(c_equ/(c_equ-c)) versus time is linear, with a y-intercept. But whereas I was just imagining that a decay function "made sense" he derives this result from Fick's laws of diffusion.
The model he uses is that the solute is distributed uniformly throughout each coffee particle but the rate of dissolution is much faster than the rate of diffusion. This means that the concentration of solute near the surface of each particle initially decreases quickly resulting in a concentration gradient within the particle. The initial surge of dissolution that sets up this gradient is what produces the y-intercept in the logarithmic graph. The slope of the line indicates the ultimate equilibrium concentration. It's worth mentioning that this slope/intercept picture depends on the experimenter being unable to measure (or intentionlly omitting) the early, fast change in concentration. If you could measure the concentration from time zero you'd get a curve rather than a line.
If instead of a single particle size there are two, with one much smaller than the other, the smaller particles would have the same type of effect on initial concentration change as the initial dissolution surge. The assumption that the early effects are due entirely to particle size is likely incorrect, as GlennV noted. Your grind profile is probably not as bad as I was suggesting!
Stapely quantifies this for spherical particles of uniform radius R. He found that the time constant is proportional to R^2. I had assumed the rate was proportional to total surface area, and thus the time constant proportional to R. Since I don't know the constants of proportionality, using R^2 instead fits your data just as well for the simple reason that it only affects what happens before you were able to make any measurements. An illustration of one of the pitfalls of fitting limited data to a model.
Stapely doesn't address this directly. What does it mean? If there were a concentration gradient in the particle that survived at apparent equilibrium a grind of larger sized particles would retain more solute. But going from a strength of 1.7% (fine grind) to 1.3% (coarse) in a 7.5% brew ratio implies a fair amount of solute trapped by some mechanism. Perhaps in larger particles part of the interior is simply unavailable due to some physical hindrance. Would the same effect be observed in a wash brew method?
Interesting stuff... will it make for better brewing? I don't know, maybe. But I just tossed an unmeasured amount of coffee into the non-inverted AP this morning with water "sort of off the boil" and it tasted great.
I think for me, yes it has helped. For the first time I've made a full extraction cold brew coffee - without the insight gained here I would not have been able to do so. In the end, a cup of coffee is a cup of coffee, except now I'm understanding why.
Also, there's a physical limit to practical coffee production and brew ratio. Gravity percolation seems to settle in to around 1.7 absorption. AeroPress seems to be somewhere (depending on a few factors) around 1.05-1.4 for physical absorption.
The maximum possible brew ratio at zero coffee production is ~57% for percolation, and about 95% for a pressed setup (simply given coffee production = Brew Water - (A * coffee), substitute in BR=Coffee/BW, zero coffee production = 1-(A*BR), solve for BR and you get 1/A).
Initially, this seems to make sense. I tried AP of 15g coffee with ~20ish g of hot water - and after a couple minutes managed to squeeze about 3g of intense stuff out of it. Literally, just a few drops. I had enough to taste it, not really enough to settle out and check strength. A filter would have absorbed most of it. It was definitely as strong or stronger than espresso. Of academic value only.
And the formula for making something approaching espresso strength using a one-stage AeroPress would be (using an assumption of 1.1 absorption and at least 18% extraction):
13g coffee 44.3g Brew Water
Expected strength 5% or greater IF extraction of 18% or greater is met.
I'll probably try this tonight or tomorrow morning.
This would be an espresso emulation - it does NOT meet the definition of espresso (producing 30g of espresso from 8 grams using 90-95°C water using 8-9 Bar of pressure in 20-30 seconds), but indications are that this should be pretty good stuff.
What I think I might do next is staged brew, like I did with cold brewed.
Preheat aeropress Get my 8g of coffee ready, add 12g (of 45g on hand) of hot water, steep 3-4 and press. Add 10g, steep and press add 10g, steep and press add remaining, steep and press.
The result should be entertaining, if not cold by the time I'm done. LOL
------------------------------------------ ----------------------------------------- Le café doit être noir comme le diable, chaud comme l'enfer, pur comme un ange, et doux comme l'amour.
"There is no right answer with coffee. There is only the elixir in your cup at the moment you partake."
"...I often say that when you can measure what you are speaking about, and express it in numbers, you know something about it; but when you cannot measure it, when you cannot express it in numbers, your knowledge is of a meagre and unsatisfactory kind;..." - Lord Kelvin RECIPES thread => http://www.coffeegeek.com/forums/coffee/machines/585708
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