Bean Physics

Comparison of Grinding Cross-sections and Extraction Yields Between Conical and Flat Burr Grinders

Published: 2026-06-15 | Read Time: 4 min

Diffusion Saturation Curves of Soluble Components Over Time in French Press Immersion Brewing

Unlike filtration-based methods, the French Press is a representative immersion brewer where ground coffee beans are completely submerged in hot water. The principle of solute extraction in immersion brewing relies solely on Fick's Diffusion Law and dissolution equilibrium, rather than hydrodynamic flow. This implies that convection effects caused by fluid movement are minimized, and diffusion based on component concentration gradients is the dominant phenomenon. In particular, the French Press involves a prolonged contact time between coffee grounds and water, where solid-liquid mass transfer is the core process.

1. Noyes-Whitney Dissolution Kinetics and Early Immersion Dynamics

Immediately after the coffee grounds are submerged in water, the difference in component concentration (Concentration Gradient) between the surface of the coffee particles and the bulk liquid is at its maximum, leading to an explosive release of components during the first minute. This phenomenon can be quantitatively explained by the Noyes-Whitney Dissolution Kinetics. This model describes the rate at which a solute dissolves from a solid surface into a solvent, defining the rate at which soluble components exit the porous internal lattice and surface of the coffee grounds as follows:

$\frac{dC}{dt} = \frac{D A}{h} (C_s - C)$

$D$: Diffusion Coefficient, the key factor determining the movement speed of the solute.
$A$: Effective Surface Area, the surface area of coffee particles in contact with water. Finer grind sizes increase this value, thereby increasing the elution rate.
$h$: Boundary Layer Thickness, the thickness of the region near the coffee particle surface where the solute concentration gradient occurs.
$C_s$: Saturation Concentration, the maximum concentration at which the solvent can dissolve the solute at a given temperature and pressure.
$C$: The actual concentration of the solute in the solution at time $t$.

In the early stages of extraction, the value of $C$ is very small, so the concentration gradient $(C_s - C)$ is at its maximum, resulting in an extremely fast extraction rate for low-molecular-weight polar substances (e.g., citric acid, ester aroma compounds, etc.). However, as time passes, the concentration $C$ of the liquid increases, slowing the driving force of elution, and the Extraction Yield (EY) follows an asymptotic curve toward saturation.

EY(t) = EY_{max}(1 - e^{-kt})

Here, $EY_{max}$ is the maximum extraction yield, and $k$ is the rate constant. This equation accurately describes the pattern where extraction yield gradually reaches a saturation state and component transfer exponentially slows down.

2. Fick's Second Law and Internal Mass Transfer in Coffee Grounds

The mechanism of mass transfer from the solid matrix inside the coffee cell to the porous flow paths can be precisely analyzed using Fick's Second Law. This law explains concentration changes over time and space.

$ \frac{\partial C}{\partial t} = D \cdot \frac{\partial^2 C}{\partial x^2} $

This equation states that the rate of concentration change $(\frac{\partial C}{\partial t})$ is equal to the second derivative of the spatial concentration change $(\frac{\partial^2 C}{\partial x^2})$ multiplied by the diffusion coefficient $D$. This means that mass transfer is active where the concentration gradient is steep. In particular, the diffusion coefficient $D$ has an exponential relationship with temperature. Generally, as temperature increases, the kinetic energy of molecules increases, leading to faster diffusion.

This principle clearly explains, through the difference in diffusion coefficients, why the movement of large organic compounds (e.g., astringent tannins, certain phenolic compounds, bitter chlorogenic acid lactones, etc.) is delayed in the cold-extraction method known as Cold Brew. At low temperatures, the value of $D$ is significantly smaller, requiring much longer extraction times, or preventing certain high-molecular-weight components from eluting at all, thus forming a smoother flavor profile. Conversely, in high-temperature extraction, $D$ is higher, allowing a diverse range of components to elute rapidly within a short time.

3. Reaching Saturation Equilibrium and Over-Extraction Control

From the 4-minute mark of extraction, the solution enters a saturation state where the concentration $C$ in the liquid and the internal concentration $C_s$ of the coffee grounds approach equilibrium, effectively stopping component transfer. This indicates that the solvent can no longer effectively dissolve the solute. In filtration methods, as fresh water is continuously supplied, improper timing can lead to endless over-extraction of bitter compounds. However, the French Press has its own inherent saturation point, so even if the immersion time is slightly delayed beyond 4 minutes, the rate at which additional bitter and acrid flavors are eluted is extremely limited. This is an essential advantage of immersion extraction, providing resilience against over-extraction and making it easier to maintain a consistent taste. Thanks to the piston structure that presses a coarse metal mesh with higher permeability than cloth or paper, oil components and fine lipid particles pass through without loss, creating a very heavy, raw mouthfeel and sweetness nuances.

4. Bypass Dilution and Extraction Optimization

Adding bypass water after coffee extraction is an effective means to independently control Extraction Yield (EY) and Total Dissolved Solids (TDS). The concentration of the brewed liquid $C_{brew}$ can be extracted for a sufficiently long time to achieve a high yield, and the final consumption concentration can then be adjusted through dilution. It satisfies the following concentration dilution equation:

$ C_{final} = C_{brew} \times (1 - \beta) $

$C_{final}$: Concentration of the final beverage after bypass.
$C_{brew}$: Concentration of the extracted concentrate.
$ \beta = \frac{V_{bypass}}{V_{final}} $ : Bypass flow ratio, the volume of added water $(V_{bypass})$ divided by the final beverage volume $(V_{final})$. $(V_{final} = V_{brew} + V_{bypass})$

This technique is particularly important regarding the management of bitter compounds (e.g., chlorogenic acid lactones). Bitter compounds have relatively slow diffusion coefficients, and because the human bitterness receptor threshold concentration $(C_{bitter\_thresh})$ is low, even small amounts can cause unpleasant tastes. Therefore, to achieve high yield, one can terminate the extraction at time $t_{stop}$ and then perform an optimization process of adding the appropriate amount of bypass water to regulate the concentration of bitter substances below the threshold. For example, if the TDS is 2.0% under certain extraction conditions and the bitterness is felt strongly, lowering the TDS to 1.3% through bypass can effectively dilute the bitter components while achieving an optimal balance.

Indicator	Brew (Before Bypass)	After Bypass $(\beta=0.35)$
TDS (%)	2.0%	1.3% $(2.0 \times (1-0.35))$
Bitterness Intensity	Strong	Smooth (Below threshold)
Extraction Yield (EY)	22%	22% (No change)

As shown, bypass dilution allows for the independent control of TDS and specific components (e.g., bitter compounds) while maintaining the extraction yield itself, enabling the realization of more refined and consistent coffee flavors.