A Model of Fines Production – Quantitative Café

In my previous post, we looked at the effect of grinding on coffee density. When we plotted void ratio vs. burr spacing for tamped espresso pucks, we noted that void ratio drops at fine grind settings, and I speculated that this might be due to fines production. In this post I’d like to develop a simple model of grinding which can be used to estimate fines production, and maybe give us some insight into the grinding process.

Two processes

We’re going to look at grinding as two processes that happen in parallel:

A cutting process gradually cuts the coffee bean into smaller and smaller particles, until they are small enough to be ejected from the burr chamber.
As the pieces of the coffee bean rub against each other in the burr chamber, an abrading process gradually turns material from the surface of each particle into “fines”.

These two processes are illustrated in the diagram below.

In each moment, some fraction of the particles is cut into smaller pieces, and at the same time, some of the surface of each particle is converted into fines.

Cutting process

Let’s start by looking at the cutting process. We start with a single particle, and in each moment, some fraction of the particles is cut into smaller pieces. This gives an exponential increase in the number of particles.

$N = e^{kt}$

For simplicity, we assume that the total volume of the particles, $V_0$ , stays the same, so that the volume of each particle is:

$V = \frac{V_0}{N} = V_0 e^{-kt}$

If we assume that the particles are spherical, then the diameter of the nominal particles will be:

$d = d_0 \exp{\left( -\frac{kt}{3} \right)}$

If we solve for $t$ , we get:

$t = \frac{3}{k} \ln{\left( \frac{d_0}{d} \right)}$

This tells us how long it takes a particle to go from an initial diameter $d_0$ to a final diameter $d$ .

Abrading process

Now let’s take a look at the abrasion process. In each moment, the amount of abrasion will be proportional to the total surface area of the particles. The area of each particle is:

$A = A_0 \exp{\left( -\frac{2 kt}{3} \right)}$

So the total surface area is:

$\begin{aligned}A' &= N A \\&= e^{kt} A_0 \exp{\left( -\frac{2 kt}{3} \right)} \\&= A_0 \exp{\left( \frac{kt}{3} \right)}\end{aligned}$

Then the rate of fines production is given by:

$\frac{\partial}{\partial t} V_f = s A' = s A_0 \exp{\left( \frac{kt}{3} \right)}$

We can integrate over time $t$ to get the total volume of fines:

$\begin{aligned}V_f &= \int_0^t{s A_0 \exp{\left( \frac{ku}{3} \right)} du} \\&= s A_0 \frac{3}{k} \left( \exp{\left( \frac{kt}{3} \right)} - 1 \right)\end{aligned}$

If we substitute the expression above for $t$ in terms of $d$ , we get:

$\begin{aligned}V_f &= s A_0 \frac{3}{k} \left( \exp{\left( \frac{k}{3} \frac{3}{k} \ln{\left( \frac{d_0}{d} \right)} \right)} - 1 \right) \\&= s A_0 \frac{3}{k} \left( \frac{d_0}{d} - 1 \right) \\&= A_0 d_0 \cdot 3 \frac{s}{k} \left( \frac{1}{d} - \frac{1}{d_0} \right) \\&= V_0 \cdot 18 \frac{s}{k} \left( \frac{1}{d} - \frac{1}{d_0} \right) \\\end{aligned}$

Then the volume fraction of fines for nominal particle diameter $d$ is given by:

$\phi_f = \frac{V_f}{V_0} = 18 \frac{s}{k} \left( \frac{1}{d} - \frac{1}{d_0} \right)$

Modelling tamped density

We can use this model to fit the tamped density data from my previous post. We’ll focus on measurements made with the logging tamper, since this provides a controlled 15 kgf tamping force. To calculate void ratio, we suppose that the nominal particles have volume $V_0 - V_f$ , and that the nominal particles alone would compress to a void ratio of $e^*$ . Then the void volume not including fines is given by:

$V_{v,0} = e^* (V_0 - V_f)$

Assuming fines occupy the void volume between nominal particles, the void volume not occupied by fines is given by:

$V_v = V_{v,0} - V_f = e^* (V_0 - V_f) - V_f$

Then the void ratio is:

$\begin{aligned}e &= \frac{V_v}{V_0} \\&= e^* - e^* \frac{V_f}{V_0} - \frac{V_f}{V_0} \\&= e^* - (e^* + 1) \phi_f\end{aligned}$

Finally, we suppose that $e^*$ depends on whole bean density as follows:

$e^* = \epsilon_0 + \epsilon_b \rho_b$

The following plot shows the fit for each coffee ground using the Timemore Sculptor 078S:

The code used to calculate this fit can be found on GitHub.

In the plots above, we see values of $s/k$ on the order of 0.2–0.3 µm.

We also see $d_0 \approx 6.3 \times 10^3\text{ mm}$ , which could indicate that the burrs in this grinder produce fines at about the same rate (relative to total surface area) throughout the cutting process.

The values of $e^*$ in the plot above suggest that the nominal particles of lighter roasted coffees pack more tightly than those of darker roasted coffees. A couple of possible explanations come to mind:

It could be that particles of lighter roasted coffees fit together more efficiently. This could happen, for example, if darker roasted coffee particles had a more irregular shape.
It could be that particles of lighter roasted coffee compress more easily than particles of darker roasted coffee. However, this would contradict experimental data on hardness vs. roast level, which suggests that coffee gets softer at darker roast levels.

Modelling particle size distribution

We can also use this model to fit data provided by Lance Hedrick. Lance measured the particle size distribution for a single coffee using three different grinders and burr spacings.

Following the suggestion of Jonathan Gagne, I fit each PSD using three separate log-normal distributions: one for fines, at around 40 µm; one for the nominal particle size; and one between these two. The result is summarized in the following plot:

For each sample, we calculate:

Nominal particle size: The centre of the nominal peak
Volume of fines: The total area under the fines peak and “mystery” peak
Total volume: The total area under all three peaks

This gives us the nominal particle size and volume fraction of fines for each sample. The following plot shows the calculated values along with the fit for each grinder:

The following table gives fit parameters for each grinder:

	d₀	*s/k*
C40	1.49 mm	3.01 µm
Kinu M47	45.6 mm	3.11 µm
Omega	0.840 mm	4.35 µm

There is a fair amount of spread in the data plotted above, so I’m hesitant to ascribe too much meaning to these fit parameters, which are quite similar. However, I think it might be helpful to talk about what these parameters could suggest in general.

Broadly, the parameter $s/k$ describes the relative efficiency of the abrading ( $s$ ) and cutting ( $k$ ) processes. A higher value of $s/k$ indicates faster production of fines relative to the cutting speed of the burr. This could result from a burr geometry which either cuts slowly or encourages interaction between cut particles. The value of $s/k$ likely depends on both the grinder and the coffee.

The parameter $d_0$ , on the other hand, describes the size of the particles when abrasion first becomes a significant factor. I suspect this is related to the geometry of the burr chamber—a geometry which moves cut particles quickly through the burr chamber might result in fewer fines being produced and a lower $d_0$ ; whereas a geometry which restricts the flow of cut particles might result in abrasion earlier in the cutting process, and a higher value of $d_0$ . I suspect that the value of $d_0$ depends mainly on the geometry of the burr chamber.

Both parameters may also be affected by feed rate. If beans are fed one at a time into the grinder, it is likely that abrasion will begin later in the grinding process—fewer particles in the burr chamber means particles will have fewer interactions, and abrasion may not become a significant factor until later in the grinding process, when there are more particles, and less volume for them to move through.

The parameter $s/k$ is about 10 times higher for Lance’s experiment than the values we derived from void ratio measurements. This could suggest that the Sculptor 078S produces fewer fines than these conical grinders, relative to its cutting rate.

It would be interesting to compare fit parameters estimated from particle size distributions to the same parameters estimated from the void ratio of tamped pucks, for the same grinder and coffee. We would expect to get the same value using either method, so any discrepancy would indicate some issue with the methods developed here.

Two processes

Cutting process

Abrading process

Modelling tamped density

Modelling particle size distribution

Leave a comment Cancel reply