The French press is a popular device for brewing coffee, comprising a cylindrical beaker—or “jug”—fitted with a lid and plunger with a fine wire mesh filter. The plunger is used to drive the solid coffee particles to the bottom of the jug, separating these grounds from hot liquid above. When using the French press in this way, a growing permeable pack of ground coffee is pushed through hot water by applying force to the plunger. We use a combination of kitchen-based and laboratory experiments to determine the force required to push on the plunger as a function of the speed of the plunger and the mass of coffee used. We calculate that for the recommended preparation method, the maximum force is $32$ N to complete the pressing action in 50 s. We propose that home coffee preparation provides a fun, low-cost, and relatable learning opportunity for students and for those who are interested in coffee science.

## I. INTRODUCTION

There are many ways to prepare a cup of coffee. The processes that control extraction and flavor profiles in coffee can be divided into chemical and thermodynamic (reaction) processes and hydrodynamic (flow) processes.^{1–6} In the simplest terms, the chemical and thermodynamic processes control reaction rates and the physico-chemical extraction dynamics local to the particles; while the hydrodynamic processes determine the percolative advection of water through the coffee and overall “contact time.”^{4,5} Coffee preparation is a nuanced science precisely because extraction and flavor are due to these processes being coupled.^{4} The pursuit of good coffee can therefore be a complex endeavor, depending on a great many parameters.^{3,7}

Developments in the technology of coffee have focused on the preparation of espresso coffee, including brew methods using espresso machines^{7} and stove-top moka pots.^{1,2} There is less research into what may be perceived as simple coffee preparation methods, including pour-over, drip coffee,^{4,5} and the French press (or cafetière-à-piston) method. In this paper, we focus our attention on the latter system, which is one of the most popular methods of domestic coffee preparation and among the least environmentally impactful.^{8}

The most common French press brew method is one in which the plunger is used to force the suspended coffee down out of the brewed coffee. This process is a percolation problem involving pressing a filter plate submerged in hot water onto a growing pack of coffee grounds and forcing this assembly through the fluid. (We note that in the reference frame of the plunger, this is equivalent to saying that the fluid is pushed through the assembly.) The main objective of this work is to determine the force required to operate the plunger on a French press and to identify the dominant factors that affect its magnitude. From a pedagogic point of view, the analytical and relatively simple nature of our result makes this household problem an excellent way of engaging people in everyday use of mathematics and physics to estimate quantities of genuine general interest. While we do not attempt to answer the subjective question “how do you make the perfect coffee?,” our hope is that our article will help coffee drinkers discuss the physics of coffee preparation in a nuanced and quantitative way.

## II. MATERIAL CHARACTERIZATION AND METHODS

We use a 1-liter glass French press with a standard vertical plunger (Fig. 1(a)). To prepare coffee using this French press, we used the plastic scoop provided with the device to measure out an aliquot of dry coffee. The heaped scoop holds approximately 6.75 g of loosely packed coffee, and eight scoops are recommended, resulting in $54\u2009g$ total recommended coffee mass in the French press. We varied the mass of coffee to achieve different results, ensuring we encompass the recommended dose. We measured the mass of the coffee used before placing it in the bottom of the French press, and we then poured just-boiled water on top to the fill line at 1-liter volume. Once filled, we placed the plunger into the pot, and lowered the filter plate until it was just above the liquid fill level. Before depressing the plunger, we waited 4 or 5 min—the recommended brew time interval.

Using the preparation procedure described above, we performed two types of experiments: (1) at home experiments and (2) laboratory validation experiments. The aim of this two-step approach is to first explore an easy-to-replicate home science experiment, and then validate the experiments in the lab to provide a robust further test of the simple model presented here. The experimental designs are shown in Fig. 1.

### A. Coffee grind radius and texture

The coffee we selected is a commercial coarse-grind variety, designed for use in a French-press. This type of coffee tends to have grains that are larger and more polydisperse in size than espresso-grade ground coffee.^{4} Figure 2(a) shows the particle size distribution measured using a Beckman Coulter™ LS 230 laser refraction particle size analyzer with a measuring range $0.374$– $2000\u2009\mu m$. Although the mean particle radius can be computed to be $104\xb13\u2009\mu m$, the distribution is very polydisperse. It is actually bimodal with two characteristic grain radii of $50\u2009\mu m$ and $300\u2009\mu m$. The particle size was confirmed using an optical microscope in reflected-light mode to examine the coffee grain radius. We sprinkled a small amount of representative dry coffee 1-grain thick on a microscope slide and used a Leica® DM4 B optical microscope with a calibrated on-screen measurement tool to identify the length of 112 coffee grains selected randomly on the slide. The mean particle radius was $102\xb16\u2009\mu m$, consistent with the results from the particle size analyzer.

To assess the extent to which grain swell is a factor in our experiment, we measured the grains' radius before and after they were immersed in hot water. The surface texture of 91 coffee grains was observed, using a Keyence VK-X 1000 laser scanning microscope with a spatial resolution of $\u223c0.1\u2009\mu m$. We found that the grains contained intra-grain porous features, consistent with the previous work that showed that intra-grain porosity can be a relevant factor in coffee preparation.^{4} In Fig. 2, we show images of individual dry coffee grains, captured using the laser scanning microscope. As can be seen, the grains have an aspect ratio of approximate unity. Post-experimental coffee grain radii were within 3% of the pre-experiment values, implying that swelling was negligible. Moreover, using helium pycnometry, we determined the coffee grain density to be $ \rho c=480\u2009kg\u2009 m \u2212 3$, which is less than the density of hot water $ \rho f$, and therefore coffee grains are initially buoyant.

### B. At home experiments

In our home experiments, during a 5-min brew time, a floating coffee pack with discernible top and bottom interfaces developed at the top of the liquid under the plunger. We measured the thickness of this coffee pack using a ruler with millimeter accuracy. For our home experiments, the plunger system had a flat-topped handle (Fig. 1(b)), meaning that we could place objects of known mass onto the plunger system. We performed repeat tests with 0.5 and 1 kg masses on the plunger. Once the masses were placed on its flat-topped handle, the plunger moved downward. Throughout the plunger's motion, we measured the vertical displacement with time, using a ruler and a stopwatch. Uncertainties on the time are dominated by user error associated with logging the displacement and time simultaneously, and are estimated to be maximum 1 s. The displacement and time measurements were used to compute the plunger velocity.

### C. Laboratory validation

In the laboratory, we used a mechanical testing apparatus to directly measure the force required to operate the plunger. For these mechanical tests, after the boiling water was poured onto the coffee, we immediately placed the French press into a mechanical Geocomp™ LoadTracII uniaxial press such that the top of the plunger was in contact with the top piston (Fig. 1(c)). Surrounding the French press was a transparent plastic container used as a safety precaution in case the French press leaked or ruptured during the experiment. For these experiments, we were particularly interested in the force required to maintain a constant plunger velocity. Hence, after a 4-min brew-time, the bottom platen was raised at a constant velocity, which pushed the plunger downward. The force and the vertical displacement of the bottom platen were measured continuously using a load cell and a linear variable differential transducer (LVDT), respectively. These parameters were monitored in real-time using the LoadTracII's data-acquisition system with an acquisition rate of 10 Hz. The evolution of the thickness of the coffee pack (layer) during plunging was monitored using optical video recording via a smartphone. We performed additional experiments using the French press with no coffee, no water, and neither coffee nor water, to check for the force associated with overcoming the frictional resistance between the snug-fitting plunger plate and the glass, as well as with the percolation of water through the filtration plate in the plunger. We measured the temperature of the water throughout, using a digital infrared laser thermometer. We found that the water temperature dropped from $97\u2009\xb0C$ to around $75\u2009\xb0C$ over the 4-min brewing time and the experimental time that followed. In this temperature range, the physical properties of water, such as viscosity or density, scarcely change,^{9} and, in the following, we neglect those variations.

The porosity $\varphi $ of the coffee grain pack was determined using the bulk density $\rho $ of the sample (determined using the total coffee mass $m$ and dimensions of the coffee pack), and the solid density $ \rho c$ of the coffee, measured by the pycnometer: $\varphi =1\u2212\rho / \rho c$.

## III. RESULTS AND ANALYSIS

In this section, we detail the results of our at-home and laboratory experiments. We stress that the at-home experiments are most suitable for use in classrooms or by interested coffee enthusiasts. In contrast, the laboratory experiments represent a validation of the at-home experiments for the purposes of this article.

### A. The coffee pack

During the 4 or 5 min brew time, coffee rises to the top of the water, and rests beneath the plunger. Most of this “coffee pack” is located within a thickness $L$ from the plunger's base plate. In the at-home experiments, we measure this thickness prior to operating the plunger and wait an extra minute for the coffee pack to form fully (5 min brew time). In the laboratory experiments, we extract from the videos the thickness of the pack as the plunger moved down the liquid. When the hot water is poured into the jug, the coffee pack thickness progressively increases before it stabilizes and is then constant to within $\xb13%$ (Fig. 3(a)). In both experiment types, we find that the equilibrium coffee pack thickness is linearly related to the dry mass of coffee $m$ in the jug. Indeed, as can be seen from Fig. 3(b), the coffee pack volume $V=\pi B 2L$ (where $B=4.75\u2009cm$ is the jug's internal radius) scales linearly with $m$. Taking the bulk density of the coffee pack to be $\rho =m/V$, we find $\rho \u2248230\u2009kg\u2009 m \u2212 3$ across all French press experiments. In turn, we can use this to calculate the average porosity of the coffee pack as $\varphi \u22480.53$ (see Sec. II C).

### B. At-home experiments

In our at-home experiments, we find that the displacement of the plunger $d$ varies linearly with time $t$, such that the rate $ u=\Delta d/\Delta t$ is approximately constant, and depends both on the mass applied and on the mass of ground coffee (Fig. 4). Phenomenologically, there is clearly a trade-off between increasing the mass applied to the plunger, which has the effect of increasing the slope of $d(t)$ (increasing $ u$), and increasing the mass of coffee used, which has the effect of decreasing the slope of $d(t)$ (decreasing $ u$). To analyze these results quantitatively, we fit a linear regression to each dataset, to determine the flow velocity for each coffee mass and applied mass. As can be seen from Fig. 4, at-home experiments had flow velocities of the order of $ u\u22480.05\u2009cm\u2009 s \u2212 1$.

### C. Using a laboratory press

Here, we analyze the results of the experiments in which a LoadTracII uniaxial press was used to measure the force $F$ need to move the plunger at constant velocity $ u$. In Fig. 5, we show the raw output force with displacement for two coffee masses: the recommended $m=0.054$ kg, and an extreme value of $m=0.1$ kg. We see that the force quickly equilibrates to a steady state value. In Fig. 5, we indicate the steady state force with a horizontal dashed line, and the standard deviation corresponding to that steady state with grey areas. It is clear from this result that the force required to operate the plunger depends on the mass of coffee used in the coffee preparation. For the recommended eight scoops of coffee ( $m=0.054$ kg), the steady state force is $F\u224812.5\u2009N$ to operate the plunger at a velocity of $ u\u22480.1\u2009mm\u2009 s \u2212 1$.

## IV. A COFFEE PERCOLATION SCALING

In both the at-home and laboratory experiments, we observe that a constant force results in a constant speed of the plunger. Here, we seek to explain that relationship. The force required to operate the plunger can be broken down into components: $F= F h+ F m+ F B+ F f\u2212g m p$, where $ F h$ is the hydrodynamic force required to squeeze the hot water through the pack of coffee, $ F m$ is the force required to squeeze the hot water through the plunger's mesh, $ F B$ is the buoyancy force due to the density difference between the coffee particles and the water, $ F f$ is the frictional force between the plunger and the glass sides of the French press, $ m p$ is the mass of the plunger itself, and $g$ is the acceleration due to gravity. By testing the French press with hot water in it but without coffee, one can find that the operation of the plunger requires substantially less force than when coffee is present, such that we suggest the combination of $ F m$, $ F f$, and $g m p$ are negligible compared with the other components, and for this reason we neglect these contributions. This results in $F= F h+ F B$.

^{10}

Equation (3) shows that the total buoyancy does not depend on the grain radius of the coffee, and instead depends on the total amount of coffee in the pack. A rough calculation shows that the buoyancy force is negligible: if we assume that $\varphi \u22480.5$, $V\u22483\xd7 10 \u2212 4\u2009 m 3$ (justified in Sec. III), and $ \rho f\u22481000\u2009kg\u2009 m \u2212 3$, we find that for the recommended mass of coffee, $ F B=O( 10 \u2212 1)\u2009N$ (where we use $O$ to denote “of order”), which is negligible compared with the pressing force. Therefore, we can conclude that $ F h$ is the dominant contribution to the force felt when operating the French press.

^{11}

## V. THE PERMEABILITY OF PACKED COFFEE

In Sec. IV, we left $k$ as an unknown parameter, which precludes direct use of Eq. (6) for forward-calculations of $F$. Here, we determine $k$ both directly and indirectly. First, we can use the results from the at-home (Fig. 4) and laboratory experiments (Fig. 5) to compute the permeability of the coffee pack during plunging. For the at-home experiments, since $F=g m a$, we can rearrange Eq. (6) to determine $k= um \mu f/(g\rho m a)$. Each French press experiment corresponds to a given value for $ u$ (Fig. 4), which can, in turn, be used for an approximate determination of the permeability of the coffee pack, assuming that the plunging resulted in steady-state flow through the coffee pack. For the laboratory experiments, we can take the equilibrium force (Fig. 5) for each experiment and the set velocity of the press apparatus. In Fig. 6, we plot the equilibrium force, cast as a pressure gradient $\u2207P=\Delta P/L=F/(AL)$ for all experiments. The fit in Fig. 6 is Darcy's law (Eq. 4), where the only fit parameter is $k$. Using a least squares regression method,^{12} we determine that $k= 1.34 \xb1 0.10\xd7 10 \u2212 12\u2009 m 2$. The linear relation between $ u$ and the pressure gradient confirms that the flow is laminar throughout the experiment. Moreover, it is also a clear indication that the relevant velocity is the average velocity $ u$ of the water, and not the local velocities, which are both larger and highly variable in such a porous medium.

^{13}that has been shown to provide a reasonable description of the permeability of ground coffee

^{14}

^{,}

^{14}The Sauter radius can be thought of a characteristic lengthscale for rough particles. While we did not measure the Sauter radius for our coffee, we note that our mean radius $R=104\u2009\mu m$, the shape of the particle size distribution studied here (Fig. 2(a)), and the particle sphericity (see particle shape in Figs. 2(b) and 2(c)) are all within measurement error of a coffee studied by Corrochano

*et al.*

^{14}Therefore, we take their Sauter radius, which they measure to be $ R x\u224880\u2009\mu m$. Previous work has suggested that $C=180$, although we highlight that this is usually an empirical fit-parameter.

^{14}Comparing this prediction with our experimental data shows reasonable agreement (Fig. 7).

## VI. TAKE-HOME MESSAGES FOR HOUSEHOLD USE

We broke down the problem of operating the plunger on a French press into nested physical problems: (1) the buoyancy force of coffee against the plunger and (2) the laminar contributions to the force required to push water through the coffee and plunger system. We show that the buoyancy force contribution is relatively minor. Therefore, to arrive at an order-of-magnitude force estimate, we can use Eq. (6). Using the inputs of $m=0.054$ kg, $\rho =230$ kg m^{−3} (average coffee pack density; Fig. 3), $ \mu f=8.9\xd7 10 \u2212 4$ Pa s (viscosity of water at brew temperature), and the apparent permeability $k\u22481\xd7 10 \u2212 11$ m^{2} (Fig. 7), we can estimate $F$ for any plunging velocity $ u$. If we assume that the brewer would want to complete the plunging action gently in 60 s, and that the distance the plunger must travel is around 10 cm, we can compute $ u=17\u2009mm\u2009 s \u2212 1$. This results in a steady-state force of around $32\u2009N$ (or approximately 3.2 kg equivalent mass applied by the human hand). For a smaller French press (smaller $B$), this force would be lower simply because $m$ would be smaller. Localisation of flow through channels formed in the coffee pack is a common feature of large forces and may reduce the force required to operate the French press.

We propose that the physics of the French press is an accessible problem that can lead to an understanding of percolative flow,^{15,16} the wider suite of problems associated with the physics of coffee,^{17} and could inspire students to apply physics to quantitative studies of the world around them.

## ACKNOWLEDGMENTS

The authors are grateful to Ed Llewellin and Jason Coumans for insightful coffee discussions and to Pauline Harlé for her assistance with aspects of the experimental campaign. Thanks go to anonymous reviewers.

## REFERENCES

*Espresso Coffee: The Science of Quality*

*Bubbles, Drops, and Particles*

*Les Fontaines Publiques de La Ville de Dijon (The Public Fountains of the City of Dijon)*

*The Wonders of Physics*