The modern coffee market aims to provide products which are both consistent and have desirable flavour characteristics. Espresso, one of the most widely consumed coffee beverage formats, is also the most susceptible to variation in quality. Yet, the origin of this inconsistency has traditionally, and incorrectly, been attributed to human variations. This study's mathematical model, paired with experiment, has elucidated that the grinder and water pressure play pivotal roles in achieving beverage reproducibility. We suggest novel brewing protocols that not only reduce beverage variation but also decrease the mass of coffee used per espresso by up to 25%. If widely implemented, this protocol will have significant economic impact and create a more sustainable coffee-consuming future.

Espresso is a beverage brewed using hot, high-pressure water forced through a bed of roasted coffee. Despite being one of the most widely consumed coffee formats, it is also the most susceptible to variation. We report a novel model, complimented by experiment, that is able to isolate the contributions of several brewing variables, thereby disentangling some of the sources of variation in espresso extraction. Under the key assumption of homogeneous flow through the coffee bed, a monotonic decrease in extraction yield with increasingly coarse grind settings is predicted. However, experimental measurements show a peak in the extraction yield versus grind setting relationship, with lower extraction yields at both very coarse and fine settings. This result strongly suggests that inhomogeneous flow is operative at fine grind settings, resulting in poor reproducibility and wasted raw material. With instruction from our model, we outline a procedure to eliminate these shortcomings.

The model enables us to understand the origin of irreproducibility in espresso (namely non-uniform flow), and it also informs us in proposing a novel strategy for minimizing drink variation as well as dry coffee waste. We identify a critical minimum grind size that allows for homogeneous extraction. Below this setting, a counterintuitive reduction in EY and increase in variability is observed. In a departure from the Specialty Coffee Association recommendations, the model suggests that we should ignore brew time and navigate the EY landscape using only mass of coffee and mass of water as independent variables. We demonstrate that we are able to systematically reduce coffee waste and dramatically reducing shot variation, while also saving the cafe both time and money in their production of espresso-based beverages. Our approach is then implemented into a real cafe setting where the economic benefits were monitored. Using these data, paired with those previously reported,we estimate that a 25% reduction in coffee mass per coffee beverage will result in approximately ca. $3.1 million per day.

This paper reports the development of a multi-scale mathematical model for extraction from a granular bed. Here, multi-scale is used to emphasize the fact that the descriptions of the physics spans different length scales (i.e., the size of the coffee grain, which is much smaller than the size of the espresso bed).We apply the model to espresso-style coffee extraction but note that it is readily generalizable to any liquid/granular biphasic system. The model offers scope to independently alter familiar variables such as grind setting, water pressure, flow rate, coffee dose, extraction kinetics, and so forth; these culminate in a prediction of EY. The model's ability to individually change each brewing parameter is crucial to developing enhanced understanding of brewing because altering parameters truly independently is difficult in an experimental setting.

In principle, it is preferable to make objective statements about the flavor of foodstuffs from knowledge of their molecular components. This poses problems for coffee because there are ∼2,000 different compounds extracted from the grounds during brewing.In practice, we are limited to more easily measurable descriptors. The coffee industry uses extraction yield (EY), a ratio of solvated coffee mass to the mass of dry coffee used to produce the beverage, to assess extraction. EY is calculated by first measuring the refractive index, a property that depends on temperature. While a refractive index measurement cannot be used to characterize the beverage composition (i.e., it cannot be used to make qualitative statements about chemical composition; the refractive response is highly molecule specific),it has been shown to accurately correlate with extracted mass.This turn may be related to flavor for a narrow range of brew parameters; we discuss this further in subsequent sections. Accordingly, the Specialty Coffee Association advises that coffee most frequently tastes best when the proportion of extracted dry mass is in the range 17%–23%. Coffee beverages with EYs exceeding 23% typically taste bitter, while those below 17% are often sour. Furthermore, concentration (often referred to as beverage strength) plays another key role in coffee beverage production. Here, we consider this a secondary problem and chose to monitor EY because it is still a descriptor of flavor but also has significant economic implications (i.e., it tells us something about how efficiently we are using our coffee mass). In contrast, one could argue that beverage concentration is related to the consumer’s preference.

There are other variables that have an impact on the beverage quality prior to the ground coffee being exposed to water. The grind setting determines the particle size distribution of the coffee grounds (and therefore the surface area).Once compacted into a granular bed, the particle size distribution plays a role in controlling the permeability of the bed and consequently the flow rate. A decreased flow rate can be achieved in a number of ways: by decreasing the water pressure, grinding finer, packing the bed more tightly, using more coffee, or some combination of these. A further source of variability is that roasted coffee ages through off-gassing, losing roast-generated volatiles thereby altering the resultant beverage density and flavor.

Freshness indices of roasted coffee: monitoring the loss of freshness for single serve capsules and roasted whole beans in different packaging.

Of all of the coffee formats, espresso is by far the most complicated and susceptible to fluctuations in beverage quality. As historically defined by the Specialty Coffee Association, an espresso is a 25–35 mL (ca. 20–30 g) beverage prepared from 7–9 g of ground coffee made with water heated to 92°C–95°C, forced through the granular bed under 9–10 bar of static water pressure and a total flow time of 20–30 s. These metrics have been grandfathered into the industry and are significantly detached from the recipes used in most cafes today. Coffee shops routinely favor higher dry coffee mass (15–22 g), resulting in larger volume beverages (30–60 g beverage mass), produced on machines that dynamically control both water pressure and temperature. The variables of tamp force, flow rate or time, dry mass of coffee, and beverage volume are all determined by the machine's operator.

The past century has seen an increase in the prevalence of coffee consumption, as consumers have gained an appreciation for its complex and exciting flavors, and obvious psychological effects.As a result, the coffee industry contributes significantly to the economic stability of numerous producing and consuming countries. For example, in 2015, the American coffee industry provided over 1.5 million jobs, accounting for $225.2 billion (1.6% gross domestic product), and resulting in ca. $30 billion in tax revenue.However, coffee-producing countries now face new challenges owing to changing climateand shifts in consumer preferences. These challenges highlight the need to maximize the quality and reproducibility of the beverage while minimizing the mass of coffee used to produce it.

Some like it hot: the influence and implications of climate change on coffee berry borer (Hypothenemus hampei) and coffee production in East Africa.

Development of a Rational Model for Espresso Extraction

z ∈ ( 0 , L ) and R ∈ ( 0 , R 0 ) ( Ω s and contain a concentration of soluble coffee, c s . The cylinder also contains inter-granular pore space, Ω l , which is occupied by liquid during extraction, which itself contains a concentration of coffee solubles c l . We use the term coffee solubles to denote the sum of the concentrations of all compounds in coffee; this is in line with the EY measurement. We note that the model could readily be generalized to explicitly track any number of chemicals. However, the utility of doing so is questionable because one would need to also provide or fit kinetic parameters for each individual compound, rendering the model susceptible to overfitting. We would also require knowledge of initial concentrations, and these are difficult to measure for many species. The liquid flow between the inlet ( z = 0 ) and outlet ( z = L ) is driven by an overpressure (the pressure excess relative to atmospheric pressure, P tot ), applied by a pump. The model equations take the form of a system of partial differential equations that describe (1) the transport of coffee solubles from the interior of the grounds to their surface, (2) the exchange or dissolution of the solubles from the grounds into the liquid, and (3) the migration of the solubles in the liquid by a combination of diffusion and advection. Figure 1 A Schematic of the Espresso Basket Geometry Show full caption The coffee grounds are shown in gray ( Ω s ), and the pore space, which is filled with water during extraction, is shown in blue ( Ω l ). The macroscopic spatial coordinate measuring depth through the bed, z, the microscopic spatial coordinate measuring radial position within the spherical coffee particles, r, as well as the basket radius, R 0 , are also indicated. Espresso is brewed in a cylindrical container denoted byand Figure 1 ). The solid coffee grounds occupy part of the cylinder,and contain a concentration of soluble coffee,. The cylinder also contains inter-granular pore space,, which is occupied by liquid during extraction, which itself contains a concentration of coffee solubles. We use the term coffee solubles to denote the sum of the concentrations of all compounds in coffee; this is in line with the EY measurement. We note that the model could readily be generalized to explicitly track any number of chemicals. However, the utility of doing so is questionable because one would need to also provide or fit kinetic parameters for each individual compound, rendering the model susceptible to overfitting. We would also require knowledge of initial concentrations, and these are difficult to measure for many species. The liquid flow between the inletand outletis driven by an overpressure (the pressure excess relative to atmospheric pressure,), applied by a pump. The model equations take the form of a system of partial differential equations that describe (1) the transport of coffee solubles from the interior of the grounds to their surface, (2) the exchange or dissolution of the solubles from the grounds into the liquid, and (3) the migration of the solubles in the liquid by a combination of diffusion and advection.

∂ c l ∂ t = ∇ ⋅ ( D l ∇ c l − u c l ) in Ω l , (Equation 1)

where t, D l , and u are time, the diffusivity of solubles within the liquid, and the velocity of the liquid, respectively. The liquid flow is solved for via the Navier-Stokes equations ∂ u ∂ t + ( u ⋅ ∇ ) u = − 1 ρ ∇ P + μ ρ ∇ 2 u , (Equation 2)

∇ ⋅ u = 0 in Ω l , (Equation 3)

where μ, ρ, and P are liquid viscosity, density, and overpressure, respectively. The solubles in the liquid phase are transported by a combination of diffusion and convection due to the flow of the liquid through the bed. The concentration of solvated coffee is therefore governed by an advection-diffusion equation:where t,, andare time, the diffusivity of solubles within the liquid, and the velocity of the liquid, respectively. The liquid flow is solved for via the Navier-Stokes equationswhere μ, ρ, and P are liquid viscosity, density, and overpressure, respectively.

22 Spiro M.

Selwood R.M. The kinetics and mechanism of caffeine infusion from coffee: the effect of particle size. , 23 Spiro M.

Page C.M. The kinetics and mechanism of caffeine infusion from coffee: hydrodynamic aspects. , 24 Spiro M.

Hunter J.E. The kinetics and mechanism of caffeine infusion from coffee: the effect of roasting. , 25 Spiro M.

Toumi R.

Kandiah M. The kinetics and mechanism of caffeine infusion from coffee: the hindrance factor in intra-bean diffusion. , 26 Spiro M. Modelling the aqueous extraction of soluble substances from ground roast coffee. , 27 Spiro M.

Chong Y.Y. The kinetics and mechanism of caffeine infusion from coffee: the temperature variation of the hindrance factor. , 28 Corrochano B.R.

Melrose J.R.

Bentley A.C.

Fryer P.J.

Bakalis S. A new methodology to estimate the steady-state permeability of roast and ground coffee in packed beds. ∂ c s ∂ t = ∇ ⋅ ( D s ∇ c s ) in Ω s , (Equation 4)

where D s is the diffusivity of solubles within the grains. Here, we treat coffee grounds as spherical dense particles, but we note that the coffee grains themselves may be irregularly shaped and feature intragranular macropores, as previously observed in scanning electron micrographs. 29 Coffee beans. Work by Spiro and colleagues demonstrated that the transport of coffee solubles through the interior of the grounds can be described by a diffusion process.Hence,whereis the diffusivity of solubles within the grains. Here, we treat coffee grounds as spherical dense particles, but we note that the coffee grains themselves may be irregularly shaped and feature intragranular macropores, as previously observed in scanning electron micrographs.As discussed in the next section, most particles in ground coffee are smaller the macropore diameter observed in the micrographs. Nitrogen physisorption was used to assess the microporosity of the coffee grounds; the data suggest that ground coffee does not feature microporosity (see Supplemental Information ). Thus, we expect our description to hold for most espresso grind settings.

z = 0 , include a specified fluid overpressure, the requirement that the water enters the basket with a purely normal velocity, and that the normal flux of dissolved species should be zero: P | z = 0 = P tot , (Equation 5)

u ⋅ t ˆ | z = 0 = 0 , (Equation 6)

( − D l ∇ c l + u c l ) ⋅ n ˆ | z = 0 = 0 , (Equation 7)

where t ˆ and n ˆ are the unit vectors tangent and normal to the surface z = 0 , respectively. At the exit we apply conditions of zero overpressure, zero tangential velocity, and a condition that there is zero diffusive flux of coffee. In summary, P | z = L = 0 , (Equation 8)

u ⋅ t ˆ | z = L = 0 , (Equation 9)

( − D l ∇ c l ) ⋅ n ˆ | z = L = 0 . (Equation 10)

Boundary conditions at the inlet,, include a specified fluid overpressure, the requirement that the water enters the basket with a purely normal velocity, and that the normal flux of dissolved species should be zero:whereandare the unit vectors tangent and normal to the surface, respectively. At the exit we apply conditions of zero overpressure, zero tangential velocity, and a condition that there is zero diffusive flux of coffee. In summary,

R = R 0 , no flux conditions are applied to the liquid coffee concentration, because the liquid cannot exit in these directions: ( − D l ∇ c l + u c l ) ⋅ n ˆ | R = R 0 = 0 , (Equation 11)

u | R = R 0 = 0 . (Equation 12)

At the vertical edges of the cylinder,, no flux conditions are applied to the liquid coffee concentration, because the liquid cannot exit in these directions:

Γ int , there is a flux of solubles per unit area, which we denote by G. Appropriate boundary conditions are ( − D s ∇ c s ) ⋅ n ˆ = G , (Equation 13)

( − D l ∇ c l + u c l ) ⋅ n ˆ = G , (Equation 14)

u = 0 on Γ int , (Equation 15)

where the former two capture mass transfer and the latter imposes that the liquid should be stationary on the grain/pore space interface. On the boundaries between the grains and inter-granular pore space,, there is a flux of solubles per unit area, which we denote by G. Appropriate boundary conditions arewhere the former two capture mass transfer and the latter imposes that the liquid should be stationary on the grain/pore space interface.

c sat ) or (2) when the liquid outside the grain is at the same concentration as the grain (i.e., in equilibrium) or (3) when the grain is depleted of solubles (the experimental upper limit of extraction is approximately 30% by mass). We therefore postulate a rate that satisfies all of the above conditions, namely G = k c s ( c s − c l ) ( c sat − c l ) on Γ int , (Equation 16)

where k is a rate constant. We note that the quantity c sat likely depends on the local temperature. One could readily incorporate a thermal model into the description, but here we assume that the espresso basket is isothermal. This is justified on the basis that the heat capacity of water is relatively high and that espresso basket temperatures are actively controlled in most machines. Determining the form of the reaction rate, G, is non-trivial, and it is something that is not readily measured experimentally. However, it can be reasonably assumed that the rate of transfer of solubles between the phases should depend on the local concentrations of solubles near the interface. Furthermore, the rate of extraction is zero when (1) the liquid immediately outside the grain is saturated (i.e., at a concentration) or (2) when the liquid outside the grain is at the same concentration as the grain (i.e., in equilibrium) or (3) when the grain is depleted of solubles (the experimental upper limit of extraction is approximately 30% by mass). We therefore postulate a rate that satisfies all of the above conditions, namelywhere k is a rate constant. We note that the quantitylikely depends on the local temperature. One could readily incorporate a thermal model into the description, but here we assume that the espresso basket is isothermal. This is justified on the basis that the heat capacity of water is relatively high and that espresso basket temperatures are actively controlled in most machines.

30 Moroney K.M. Heat and mass transfer in dispersed two-phase flows. t = 0 , when extraction begins, the bed is filled with liquid water that is free from solubles. We therefore have c l | t = 0 = 0 , c s | t = 0 = c s 0 (Equation 17)

and note that the errors engendered in making this approximation can be expected to be small because the intrusion stage represents only a small portion of the overall extraction time. In c s 0 is the concentration of solubles in the grains initially. Concurrent with the wetting stage is the potential for the grains in the bed to be rearranged by the invading fluid. 28 Corrochano B.R.

Melrose J.R.

Bentley A.C.

Fryer P.J.

Bakalis S. A new methodology to estimate the steady-state permeability of roast and ground coffee in packed beds. Coffee particulates remain dry until they are connected to the extraction apparatus, at which point water is rapidly introduced to the bed, serving to wet the entire puck and stabilize the particle temperature. Modeling this initial wetting (i.e., pre-infusion) stage poses another series of interesting problems; the model presented here is only valid once liquid infiltration has taken place, and we refer the interested reader to a discussion on pre-infusion.We avoid explicitly modeling this stage by assuming that at, when extraction begins, the bed is filled with liquid water that is free from solubles. We therefore haveand note that the errors engendered in making this approximation can be expected to be small because the intrusion stage represents only a small portion of the overall extraction time. In Equation 17 is the concentration of solubles in the grains initially. Concurrent with the wetting stage is the potential for the grains in the bed to be rearranged by the invading fluid.Rearrangement that may occur during the initial wetting stage will be accounted for later after the equations have been homogenized. One of the results of this procedure is that the geometry is encapsulated in the macroscopic quantity of permeability, and by making this material property inhomogeneous, the model can mimic a non-uniform distributions of grounds.

Particle-Size Distribution of Ground Coffee The model requires knowledge of the distribution of coffee particle sizes produced by the grinder. The population, surface area, and volume fraction of the particles are used to estimate the permeability of the bed, and this is crucial in determining the liquid flow. Moreover, the particle size controls the extraction dynamics, because it determines the typical distance (and in turn the typical time) over which solubles must be transported within the grains before they reach the interface where they can be dissolved into the liquid. 31 Fowler A.C.

Scheu B. A theoretical explanation of grain size distributions in explosive rock fragmentation. 13 Uman E.

Colonna-Dashwood M.

Colonna-Dashwood L.

Perger M.

Klatt C.

Leighton S.

Miller B.

Butler K.T.

Melot B.C.

Speirs R.W.

Hendon C.H. The effect of bean origin and temperature on grinding roasted coffee. S , is reduced, the relative proportion of fines increases, but their size remains constant ( Figure 2 Particle Size Distributions Collected Using the Method Described in the Experimental Procedures Show full caption (A) Surface area and number of coffee particulates produced with a grind setting G S = 2.5. Here, 99% of the particles are <100 μm in diameter and account for 80% of the surface area. (B) The volume percent particle size distribution at G S = 2.5, 2.0, 1.5, and 1.0. Grinding finer reduces the average boulder size and increases the number of fines. Intruders are boulders that are larger than the aperture of the burr set and hence further fractured until they can exit the burrs. Particle size distributions were measured using our described experimental procedure; these data are shown in Figure 2 A. We observe that, similar to the formation of two families of particle sizes found in exploding volcanic rock,there are two groups of particle sizes in ground coffee. Namely, boulders (which we define as larger than 100 μm) and fines (smaller than 100 μm in diameter). This bimodal distribution is caused by large particles fracturing until they are sufficiently small to exit through the grinder burr aperture.The size of the boulders are determined by the burr separation, whereas the fines (much smaller than the burr aperture) are thought to be produced at the fracture interface. One piece of evidence supporting this idea is that as the grind setting, G, is reduced, the relative proportion of fines increases, but their size remains constant ( Figure 2 B).

Multi-scale Homogenization and One-Dimensional Reduction 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 32 Foster J.M.

Chapman S.J.

Richardson G.

Protas B. A mathematical model for mechanically-induced deterioration of the binder in lithium-ion electrodes. 33 Richardson G.

Denuault G.

Please C.P. Multiscale modelling and analysis of lithium-ion battery charge and discharge. Direct solution of Equations 1 16 , and 17 on a realistic packed bed geometry comprising many millions of individual grains is intractable, even using modern high-performance computing. Therefore, rather than tackling the problem directly, we make use of the vast disparity in the length scales between that of a coffee grain (∼10 μm, referred to as the microscopic scale) and that of the whole bed (∼1 cm, referred to as the macroscopic scale) to systematically reduce the system using the asymptotic technique of multiple scales homogenization. Such techniques have been applied to problems with a similar structure in electrochemistry,and rather than present this very involved calculation in full here, we provide a summary in the Supplemental Information and refer the interested reader to Richardson and co-workerswhere the details of an analogous calculation are presented. 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, The macroscopic system of equations, valid on the larger macroscopic length scale of the entire bed, systematically follow from the microscopic Equations 1 16 , and 17 . The application of the multiple scales technique significantly reduces the model complexity, but the requisite information about the microscale variations is retained. For example, because the dissolution rates depend on the concentration of solubles on the microscopic particle surfaces, the multi-scale system contains a series of microscale transport problems that must be solved inside representative grains. It is crucial that this microscopic information is preserved in the multi-scale model, because it will allow us to study the effects of different grind settings on the overall macroscopic behavior of the extraction. a 1 (fines) and a 2 (boulders). Further, we denote the Brunauer-Emmett-Teller (BET) surface area of the different families of particles by b et 1 and b et 2 , respectively. The BET surface area characterizes the amount of interfacial surface area between two intermingled phases per unit volume of the mixture, and therefore has units of 1 / m . We also introduce c s 1 and c s 2 to denote the concentrations of solubles in the two particle families. The resulting macroscopic equation for the concentration of solubles in the liquid is ( 1 − ϕ s ) ∂ c l ∗ ∂ t = ∂ ∂ z ( D eff ∂ c l ∗ ∂ z − q c l ∗ ) + b et , 1 G 1 + b et , 2 G 2 . (Equation 18)

Motivated by the bimodal distribution of particle sizes in the model, it may be assumed that the bed is composed of two families of spherical particles with radii(fines) and(boulders). Further, we denote the Brunauer-Emmett-Teller (BET) surface area of the different families of particles byand, respectively. The BET surface area characterizes the amount of interfacial surface area between two intermingled phases per unit volume of the mixture, and therefore has units of. We also introduceandto denote the concentrations of solubles in the two particle families. The resulting macroscopic equation for the concentration of solubles in the liquid is c l ∗ is the concentration of solubles in the liquid as predicted by the multi-scale modeling approach; whereas c l appearing in the equations in the − D eff ∂ c l ∗ ∂ z + q c l ∗ | z = 0 = 0 , (Equation 19)

− D eff ∂ c l ∗ ∂ z | z = L = 0 . (Equation 20)

Here, the quantityis the concentration of solubles in the liquid as predicted by the multi-scale modeling approach; whereasappearing in the equations in the Supplemental Information is the concentration of solubles in the liquid as predicted by the original microscopic model. The upscaled and reduced versions of Equations 7 and 10 are These assert that there should be no flux of solubles across the inlet and no diffusive contribution to the flux at the outlet. In the next section, we show that parameter estimates indicate that diffusive fluxes are negligible compared with those due to advection in typical espresso brewing conditions. Hence, it is the flow of the liquid through the pores that is primarily responsible for moving solubles through the bed once they have been dissolved. Hence, even though the physical relevance of the latter condition in Equation 19 is not completely clear, it has negligible impact on the model solution. ∂ c s i ∂ t = 1 r 2 ∂ ∂ r ( r 2 D s ∂ c s i ∂ r ) , for i = 1,2 , (Equation 21)

and the symmetry and dissolution rate boundary conditions, which arise from − D s ∂ c s i ∂ r | r = 0 = 0 − D s ∂ c s i ∂ r | r = a i = G i } for i = 1,2 . (Equation 22)

The microscopic equations to be solved areand the symmetry and dissolution rate boundary conditions, which arise from Equation 13 , and which act to couple the micro- and macroscale transport problems, are G i . This has precisely the same form as G i = k c s i ( c s i − c l ∗ ) ( c sat − c l ∗ ) . The problem is closed by supplying the initial conditions Equation 17 and the reaction rates,. This has precisely the same form as Equation 16 but with additional subscripts to differentiate the boulders from the fines, i.e., d M cup d t = π R 0 2 q c l | z = L . (Equation 23)

A formula for EY in terms of the model variables can be derived by first noting that it follows from Equations 18 and 20 that an expression for the rate at which soluble mass enters the cup is given by t = 0 and dividing by the dry mass of coffee initially placed in the basket, M in , we obtain Extraction yield ( EY ) = π R 0 2 q ∫ 0 t shot c l | z = L d t M in , (Equation 24)

where EY is described as the fraction of solvated mass compared with the total mass of available coffee. Here, t shot is the flow time. On integrating this equation along with the initial condition that there is no solvated mass atand dividing by the dry mass of coffee initially placed in the basket,, we obtainwhere EY is described as the fraction of solvated mass compared with the total mass of available coffee. Here,is the flow time. Equation 24 is used in the following sections as a means to compare model predictions of EY with experimental measurements.

Tuning the Model to Espresso Extraction Data R 0 ), the viscosity of heated water (μ), the saturation concentration of heated water ( c sat ), and the concentration of solubles initially in the grounds ( c s 0 ) are readily available in the literature (see the 30 Moroney K.M. Heat and mass transfer in dispersed two-phase flows. ϕ s = 0.8272 and this, along with the density of grounds and the bed radius, allows us to derive a value for the bed depth via the relationship π R 0 2 L = M in ρ grounds ϕ s . (Equation 25)

Initially, simulations of espresso extraction were run using a cafe-relevant recipe of 20 g of dry grounds used to produce a 40 g beverage under 6 bar of static water pressure. Values for the radius of an espresso basket (), the viscosity of heated water (μ), the saturation concentration of heated water (), and the concentration of solubles initially in the grounds () are readily available in the literature (see the Supplemental Information for a summary of values and their sources). Moroney and colleaguesestimate that the volume fraction of grounds in a packed bed isand this, along with the density of grounds and the bed radius, allows us to derive a value for the bed depth via the relationship M in . Values for both the radii and BET surface area for the two differently sizes families of particles in the grounds can be derived from the data shown in q = M out π R 0 2 ρ out t shot , (Equation 26)

where M out is the mass of the beverage (40 g), and we make the assumption that the density of the drink, ρ out , is the same as that of water, but we note that this is an area that could be improved in future model developments. We emphasize the difference between M out and M cup ; the former is the total mass of the beverage, whereas M cup (used in Figure 3 Extraction Yield as a Function of Grind Size, with Varying Coffee Dose and Water Pressure Show full caption (A) The effect of changing the coffee dose M in with constant water pressure shows that reducing the initial coffee mass but keeping the beverage volume constant results in higher extractions. (B) The effect of changing the pump overpressure, P, with a constant brew ratio shows an increase in extraction yield with decrease in water pressure. Figure 4 Espresso Extraction Yield as a Function of Grind Setting Show full caption (A) P W = 6 bar, t F = 98 N shot times are inversely proportional to G S . (B) Extracted mass percent can be described by two regimes. Regime 1: a standard flow system where an expected increase in extraction percent is observed with reducing G S . Regime 2: partially clogged flow is operative when there are too many fines (ca. G S = 1.7), forming aggregates and/or inhomogeneous bed density, effectively reducing the surface area of the granular bed. While it is likely that bed depth varies slightly across the range of grind settings (as the volume fraction changes), we assume that the bed depth is constant for a given dry mass of coffee,. Values for both the radii and BET surface area for the two differently sizes families of particles in the grounds can be derived from the data shown in Figure 3 provided that both families are distributed homogeneously throughout the bed. The Darcy flux, q, determines the flow rate of the liquid through the bed and varies with the grind setting. They are estimated using the shot times presented in Figure 4 from the equationwhereis the mass of the beverage (40 g), and we make the assumption that the density of the drink,, is the same as that of water, but we note that this is an area that could be improved in future model developments. We emphasize the difference betweenand; the former is the total mass of the beverage, whereas(used in Equation 23 ) is the total mass of solubles in the beverage. The parameter values discussed above are tabulated in the tables presented in the Supplemental Information D eff , D s , and k, remain to be specified. The effective macroscopic diffusivity in the liquid is often related to the diffusivity D l via D eff = B D l where B is the permeability factor. This accounts for the reduction in the diffusive fluxes due to the obstacles provided by the intermingled phase, in this case the coffee grains. This permeability factor can either be computed via a homogenization calculation 34 Foster J.M.

Gully A.

Liu H.

Krachkovskiy S.

Wu Y.

Schougaard S.B.

Jiang M.

Goward G.

Botton G.A.

Protas B. Homogenization study of the effects of cycling on the electronic conductivity of commercial lithium-ion battery cathodes. B = ε l 3 / 2 . 35 von Bruggeman D.A.G. Berechnung verschiedener physikalischer konstanten von heterogenen substanzen. i. dielektrizitätskonstanten und leitfähigkeiten der mischkörper aus isotropen substanzen. D l nor D eff have been experimentally characterized. However, if we adopt a value for the diffusivity of a typical compound in water and then compare the expected size of the flux of solubles due to diffusion versus convection, it seems clear that a safe conclusion is that the former is significantly smaller than the latter: D eff c sat L ≪ q c sat . (Equation 27)

Three parameters, namely, and k, remain to be specified. The effective macroscopic diffusivity in the liquid is often related to the diffusivityviawhereis the permeability factor. This accounts for the reduction in the diffusive fluxes due to the obstacles provided by the intermingled phase, in this case the coffee grains. This permeability factor can either be computed via a homogenization calculationor can be estimated using the Bruggemann approximation, which asserts thatUnfortunately, neithernorhave been experimentally characterized. However, if we adopt a value for the diffusivity of a typical compound in water and then compare the expected size of the flux of solubles due to diffusion versus convection, it seems clear that a safe conclusion is that the former is significantly smaller than the latter: 30 Moroney K.M. Heat and mass transfer in dispersed two-phase flows. D s and k, are fitted to the experiment. The results of this fitting are shown in D s = 6.25 × 10 − 10 m 2 / s , k = 6 × 10 − 7 m 7 kg − 2 s − 1 . (Equation 28)

We note that this same conclusion was also reached previously.Henceforth, we assign a small value to the macroscopic diffusivity of solubles in the liquid so that diffusive transport is negligible compared with convection due to liquid flow. The final two parameters,and k, are fitted to the experiment. The results of this fitting are shown in Figure 4 and lead to values of