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Modelling of coﬀee extraction during brewing using multiscale

methods: An experimentally validated model

K.M. Moroneya,∗, W.T. Leea, S.B.G. O’Briena, F. Suijverb, J. Marrab

aMACSI, Department of Mathematics and Statistics, University of Limerick, Ireland

bPhilips Research, Eindhoven, The Netherlands.

Abstract

Accurate and repeatable extraction of solubles from roasted and ground coﬀee with hot

water is vital to produce consistently high quality coﬀee in a variety of brewing techniques.

Despite this, there is an absence in the literature of an experimentally validated model

of the physics of coﬀee extraction. In this work, coﬀee extraction from a coﬀee bed

is modelled using a double porosity model, including the dissolution and transport of

coﬀee. Coﬀee extraction experiments by hot water at 90◦C were conducted in two

situations: in a well stirred dilute suspension of coﬀee grains, and in a packed coﬀee

bed. Motivated by experiment, extraction of coﬀee from the coﬀee grains is modelled

via two mechanisms: an initial rapid extraction from damaged cells on the grain surface,

followed by a slower extraction from intact cells in the grain kernel. Using volume

averaging techniques, a macroscopic model of coﬀee extraction is developed. This model

is parametrised by experimentally measured coﬀee bed properties. It is shown that this

model can quantitatively reproduce the experimentally measured extraction proﬁles. The

reported model can be easily adapted to describe extraction of coﬀee in some standard

coﬀee brewing methods and may be useful to inform the design of future drip ﬁlter

machines.

Keywords: Coﬀee brewing process, Coﬀee extraction experiments, Double porosity

model, Static porous medium, Coﬀee extraction kinetics, Drip ﬁlter coﬀee

1. Introduction

Coﬀee, derived from the seeds (beans) of the coﬀee plant, is among the most popular

beverages consumed worldwide. Typically, after the beans are roasted and ground, some

of their soluble content is extracted by hot water. The resulting solution of hot water

and coﬀee solubles is called coﬀee. Coﬀee extraction is carried out on a large number

of scales varying from large-scale industrial extraction to produce instant coﬀee, right

down to one-cup brewing appliances for domestic use. For the purposes of brewing there

∗Corresponding author

Email addresses: kevin.moroney@ul.ie (K.M. Moroney), william.lee@ul.ie (W.T. Lee),

stephen.obrien@ul.ie (S.B.G. O’Brien), freek.suijver@philips.com (F. Suijver),

johan.marra@philips.com (J. Marra)

Preprint submitted to Journal of Chemical Engineering Science December 11, 2017

are numerous methods of producing a coﬀee beverage, which can be broken into three

main categories: decoction methods, infusion methods and pressure methods. Many of

these brewing techniques are described in refs [1, 2]. The intimate contact of water with

roasted coﬀee solids is the cardinal requirement for producing a coﬀee beverage [1]. In

line with this, all the coﬀee brewing methods mentioned rely on solid-liquid extraction

or leaching, which involves the transfer of solutes from a solid to a ﬂuid. Despite its

widespread consumption, long history and well developed techniques, consistently brew-

ing high quality coﬀee remains a diﬃcult task. This diﬃculty arises from the dependency

of coﬀee quality on a large number of process variables. Some of these include brew ratio

(dry coﬀee mass to water volume used), grind size and distribution, brewing time, water

temperature, agitation, water quality and uniformity of extraction [1, 3]. According to

Clarke [4], “The extraction of roast and ground coﬀee is, in fact, a highly complex oper-

ation, the scientiﬁc fundamentals of which are very diﬃcult to unravel”. This is reﬂected

in the absence of a satisfactory, experimentally validated mathematical system of equa-

tions to model the extraction process accurately. Such a description would have obvious

beneﬁts in quickly and easily investigating the inﬂuence of various parameters on coﬀee

extraction and informing the design of the next generation of coﬀee brewing equipment.

Of course the notion of high quality coﬀee is a rather inexact ideal and to some extent

a matter of taste perception. Relating taste to the physical parameters of extracted

solubles is in itself a non-trivial matter, separate to the issue of consistency. Despite

this, certain correlations have been identiﬁed between coﬀee ﬂavour and extraction yield.

The coﬀee brewing control chart for example, gives target ranges for brew strength and

extraction yield based on preferences observed in organised taste tests [3]. Brew strength

is the ratio of mass of dissolved coﬀee in the beverage to volume. Extraction yield is the

percentage of dry coﬀee grind mass that has extracted as solubles into the water.

The chemistry of coﬀee brewing has received a great deal of attention in recent times

but the physics of the brewing has received relatively little attention. Very often, as

in other food engineering applications, the importance of the microstructure in mass-

transfer is ignored in extraction models, and solids are treated as “black boxes” [5].

Some work has been done modelling the physics of certain brewing systems. Large scale

industrial extractors for production of instant coﬀee have been the subject of detailed in-

vestigations. Early work focused on modelling coﬀee extraction in large packed columns

called diﬀusion batteries with a focus on improving the design of these solid-liquid ex-

tractors [6, 7]. Some of these developments are summarised in [4]. There has also been

some physical modelling of domestic brewing systems. Experimental investigations have

been carried out into the operation and eﬃciency of the Moka pot [8, 9]. Fasano et al.

have developed general multiscale models for the extraction of coﬀee primarily focused

on the espresso coﬀee machine [10, 11, 12, 13]. Voilley et al. [14] conducted a number of

extraction experiments on a well mixed system of coﬀee grounds and water and investi-

gated the inﬂuence of brewing time, granule size, brew ratio and water temperature on

brew strength. The diﬀusion equation in a sphere was also found to be useful to describe

the variations in the brew strength of the coﬀee during the experiments. There has been

very little investigation into the physics of the drip ﬁlter brewing system. There are a

number of aspects in the drip ﬁlter brewing system, where a greater understanding of the

physical process may lead to improved design and increased quality of coﬀee produced.

Some of these aspects were investigated by a group of applied mathematicians working

on a problem posed by Philips Research during the ESGI 87 study group with industry

2

in the University of Limerick. The topics investigated are included in the study group

report [15]. Despite these developments, there is an absence in the literature of a ﬁrst

principles model of coﬀee extraction which is validated by experiment. This paper aims

to address this deﬁcit.

The aim of this study is to formulate a comprehensive experimentally validated model

of the physics of coﬀee extraction. The model should include the dissolution and transport

of coﬀee within the coﬀee bed. It should also take into account the doubly porous nature

of the coﬀee bed, which consists of pores between the coﬀee grains (intergranular), but

also smaller pores within the coﬀee grains (intragranular). In this paper, ﬂow through a

static, saturated coﬀee bed, under the inﬂuence of a pressure gradient is modelled using

a double porosity model. The parameters in this macroscopic model are related to the

microscopic properties of the coﬀee bed by an averaging procedure using Representative

Elementary Volumes. This allows the model to be parametrised from experimentally

measured microscopic quantities. Utilising multiscale modelling of extraction from coﬀee

grains we show that we can quantitatively model extraction from ground coﬀee in two

situations: in a well stirred dilute suspension of coﬀee grains and in a packed coﬀee bed.

In our model, extraction is divided into two regimes. In the ﬁrst, a rapid extraction

occurs from the surface of the coﬀee grains which yields the highest concentrations at

any stage of the brewing process. In the second there is a slow extraction at lower

concentrations from the interior of the grains. The model can be easily generalised to

describe standard coﬀee brewing methods such as french press and drip ﬁlter coﬀee. It

can also be extended to include unsaturated ﬂow in the coﬀee bed.

2. Coﬀee extraction experiments

A large number of experiments were carried out to investigate ﬂow and extraction of

coﬀee from coﬀee beds of various geometries. Two of these experiments will be outlined

here and used to motivate the development of a mathematical model to replicate their

results. The experiments were performed with a number of diﬀerent coﬀee grinds. We

focus on two of these coﬀee grinds. The ﬁrst is a relatively ﬁne grind called Jacobs

Kr¨onung (JK) standard drip ﬁlter coﬀee grind. The second is a coarse grind obtained

with a Cimbali burr grinder from Illy coﬀee beans. The grind used is obtained by setting

the grinder to a coarse setting #20 and will be referred to as Cimbali #20. The grind

size distributions for these grinds are shown in ﬁgure 1. It is apparent from the graph

that both distributions are bimodal, having two peaks. A ﬁrst peak occurs at a particle

size of 25–30 m while the second peak occurs at a larger particle size and gradually

shifts from left to right on the graph with the grind coarseness. The ﬁrst peak accounts

for single cell fragments: the cell size in coﬀee particles is 25–50 m. The second peak

accounts for particles comprising intact coﬀee cells. The grind size distribution is vitally

important in coﬀee extraction in that it aﬀects both the ﬂuid ﬂow through the grind and

the grind’s extraction kinetics.

2.1. Maximum extractable solubles mass from coﬀee grind

Extraction from a coﬀee grain occurs following contact with water. However not all

of the coﬀee grain mass is soluble. Experiments conducted show that extractable mass

of coﬀee grains in water at 90 ◦C can range from 28% for very ﬁne grinds to 32% for

3

0.1

1

10

100

1000

104

0

2

4

6

8

10

Particle size HΜmL

Volume fraction H%L

Figure 1: Coﬀee grind size distributions for JK standard drip ﬁlter grind (–) and Cimbali #20 grind (- -).

Distributions are expressed in terms of volume fraction percentages of particles of a given diameter.

very coarse grinds. These results were obtained using ﬁne and coarse grinds from Douwe

Egberts (DE) coarse drip ﬁlter coﬀee. The extraction was carried out in glass beakers

by continuously stirring the grind through the water with a magnetic stirrer for at least

5 hours to ensure maximum extraction. Increasing the extraction time from 5 hours to

10 hours did not change the extracted amount. Reduced extraction from larger coﬀee

particles probably reﬂects the phenomenon that some solubles cannot be removed from

closed cells inside the larger particle kernels.

2.2. Coﬀee extraction kinetics during batch-wise brewing in a ﬁxed water volume

The extraction kinetics of the two coﬀee grinds considered here were investigated by

mixing 60 grams of coﬀee with a hot water volume, Vwater = 0.5 litres, and measuring

the concentration cbrew of extracted species as a function of time. The temperature of

the liquid during extraction is 80–90 ◦C. This was done by performing batch-wise ex-

traction in a French-press type cylinder and using a piston to separate the brew from the

grounds, after a certain extraction time, by pressing the mixture through a sheet of coﬀee

ﬁlter paper on the cylinder bottom. The solubles concentration cbrew was subsequently

determined by measuring the ◦Brix with a pocket refractometer (PAL-3, Atago, Japan).

It was found for drip ﬁlter coﬀee that 1 ◦Brix corresponds with cbrew = 8.25 g/litre.

The latter calibration factor was obtained by evaporating all the water from the coﬀee

brew and weighing the remaining non-volatile material. The resulting extraction proﬁles

are shown in ﬁgure 2. Extraction proﬁles for some other coﬀee grinds are included in

appendix G.

2.3. Coﬀee extraction proﬁles from a cylindrical brewing chamber

A number of coﬀee extraction experiments were conducted with both a cylindrical

brewing chamber and a conical Melitta ﬁlter. We will focus on the cylindrical brewing

4

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à

à

à

à

à

à

à

0

100

200

300

400

500

600

0

10

20

30

40

timeHsL

cbrewHkgm3L

Figure 2: Coﬀee solubles concentration proﬁles for JK standard drip ﬁlter grind ( s

) and Cimbali #20

grind () during batch extraction experiments. In these experiments 60 g of coﬀee with approximately

4% moisture was mixed with 0.5 l of hot water in a French press type cylinder.

chamber here. In this setup, coﬀee is placed in a cylindrical ﬂow-through cell and 1 litre

of water at 90 ◦C is forced through the coﬀee bed using a rotary vane pump. The system

can operate in a constant ﬂow mode with the pressure diﬀerential across the bed adjusting

itself to the ﬂow. Alternatively the pressure diﬀerential across the coﬀee bed can be ﬁxed

with the ﬂow adjusting itself to the pressure. The coﬀee beverage exiting from the ﬂow

through cell is collected in a coﬀee pot. The solubles concentration of the exiting coﬀee

(cexit), and the coﬀee beverage in the pot (cbrew) is measured throughout the extraction.

The extraction proﬁles of the brewed and exiting coﬀee solubles concentration, given in

milligrams coﬀee solubles per gram of coﬀee beverage here, for both JK standard drip

ﬁlter grind and Cimbali #20 grind are shown in ﬁgure 3. In the case of the JK standard

drip ﬁlter grind, the experiments are repeated for a diﬀerent coﬀee bed mass and for

a diﬀerent absolute pressure of the water. Details of these experiments are included in

appendix G.

We make a number of observations based on the experimental data presented, which

will be used to motivate the development of a mathematical model of the extraction

process. Firstly, it is noted that the experiments tell us nothing about the ﬁlling stage,

where the water initially inﬁltrates the dry coﬀee bed. To this end it will be assumed

that when the ﬁrst coﬀee brew exits, the coﬀee bed, including the intragranular pores,

is saturated with water. It is noted that in the case of the batch extraction experiment,

in ﬁgure 2, the concentration of the coﬀee brew increases rapidly from zero to over half

its maximum value at the beginning of extraction, before a much slower increase over

a longer timescale towards its maximum value. This phenomena can also be seen in

the cylindrical brewing chamber data in ﬁgure 3, where the concentration of the initial

exiting brew is very high and this is maintained for a short time, before there is a large

drop in the exiting concentration to a lower level and this gradually declines over a longer

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Mbrew HgramsL

cbrew HmggramL

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Mbrew HgramsL

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Figure 3: The coﬀee solubles concentration, measured in mg/gram, is plotted against mass of coﬀee

beverage Mbrew (grams) for JK drip ﬁlter grind with a ﬂow rate of 250 ml/min and pressure diﬀerential

of 2.3 bar in (a) the coﬀee pot and (b) the beverage at ﬁlter exit, and for Cimbali #20 grind with a ﬂow

rate of 250 ml/min and pressure diﬀerential of 0.65 bar in (c) the coﬀee pot and (d) the beverage at ﬁlter

exit. The mass of coﬀee used in both cases is 60 g including approximately 4% moisture. The brewing

cylinder diameter is 59 mm. The coﬀee bed heights are 4.05 cm for JK drip ﬁlter grind and 5.26 cm for

Cimbali #20 grind.

period. This is consistent with the ﬁndings in [14] where it was noted that extraction

yield reached 90% of its ﬁnal value within one minute. Large-sized roasted particles in

the coﬀee grind feature a kernel comprising undamaged closed cells and a particle skin

formed by damaged open cells. The grind distribution as seen in ﬁgure 1 also contains

a signiﬁcant proportion of ﬁnes or damaged cells. Thus the reason for the fast initial

extraction may be due to reduced mass transfer resistances in the damaged particle skin

and in the ﬁnes. The slow extraction may then be due to mass diﬀusion from intact cells

in particle kernel. Thus the rate limiting step is diﬀusion from the particle kernel.

3. Mathematical modelling

3.1. Basic modelling assumptions

It is assumed here that the coﬀee brewing process can be broken into three stages.

Initially in the ﬁlling stage, hot water is poured on the dry coﬀee grounds and begins to

6

ﬁll the ﬁlter, but doesn’t leave. Next in the steady state stage the bed is saturated, water

is still entering the bed, but also leaving at the same rate. In the last stage, the draining

stage, no more water enters the bed but it still drains out. In the absence of experimental

data to cover the other stages, only the steady state stage is considered here. However,

the model developed can be easily generalised to include the unsaturated ﬂow, during

the ﬁlling and draining stages. In the steady state stage the coﬀee bed is considered

as a static, saturated porous medium with the ﬂow driven by a pressure gradient. This

pressure gradient may be mechanically applied as in an espresso machine or hydrostatic

as in a drip ﬁlter machine. The bed is composed of a solid matrix of coﬀee grains which

are themselves porous. As the ﬁlling stage is not modelled here, initial conditions for

the steady state stage will have to be estimated or inferred from experimental data.

Furthermore any swelling of the coﬀee grains due to the addition of water will not be

modelled. It is assumed this swelling would occur during the ﬁlling stage and would

manifest itself in the steady state stage possibly as a slight shift to the right of the grind

size distribution. While coﬀee is composed of over 1800 diﬀerent chemical compounds

[1], in this model we will just consider a single entity and model coﬀee concentration or

brew strength in line with the experimental data shown. The model can of course be

generalised to model the concentration of any number of coﬀee constituents. Modelling

a multicomponent system such as coﬀee with a single component is a simpliﬁcation and

requires some justiﬁcation. Firstly, as mentioned in the introduction, relating taste to

the concentration of the diﬀerent coﬀee components in a beverage is a non-trivial matter

and currently no ideal recipe exists. Apart from professional tasters, the most widely

used measure of coﬀee quality is the coﬀee brewing control chart. This chart gives target

ranges for brew strength and extraction yield. This chart is used by both the Speciality

Coﬀee Association of Europe (SCAE) and the Speciality Coﬀee Association of America

(SCAA). Given that the most widely used measure of coﬀee quality considers coﬀee as

a single component, it seems logical to do so as well. Secondly, experiments carried

out suggest that many important coﬀee components have similar extraction kinetics (see

appendix F).

3.2. Coﬀee bed structure

The structure of the coﬀee bed is central to the extraction process. It is immediately

obvious that the bed consists of two phases. Following [16] the highly permeable phase

consisting of the pores between the coﬀee grains is called the h-phase. Similarly the low

permeability phase consisting of the coﬀee grains is called the l-phase. At a microscopic

level there are two phases within the coﬀee grains. The pore or void space within the

grains is called the v-phase, while the solid coﬀee cellular matrix is referred to as the

s-phase. The coﬀee bed has three fundamental length scales. The smallest of these is

the size of the pores within the grains or the size of a coﬀee cell which may be 25–50 m.

The average size of a coﬀee grain in the grind size distribution by volume, excluding

single cell fragments, will be approximately an order of magnitude bigger. Finally the

size (depth) of the coﬀee bed will typically be a few centimetres. In order to model

the transport of coﬀee and water in the bed conservation equations can be formed at

each of these scales. On the microscale (cell size scale), conservation equations can

be formed in the h-phase, v-phase and s-phase. At the intermediate (grain size) scale

conservation equations can be formed in the h-phase and the l-phase. We refer to this as

the mesoscale. At this scale an individual coﬀee grain is represented by two overlapping

7

continua representing the void and solid phases within the grain. At the macroscale

(coﬀee bed scale), the coﬀee bed is represented by three overlapping continua representing

the h-phase, v-phase and s-phase. To reconcile the three representations we can use the

methods of homogenisation or volume averaging. Volume averaging will be adopted here.

A schematic of the volume averaging process is shown in ﬁgure 4. This is very useful

since it relates the averaged macroscopic quantities to the physical parameters at the

microscale. Some macroscopic parameters may be measured by experiment, while others

can be found from their averaged representation in terms of measurable microscopic

quantities. The inﬂuence of microscale properties on the macroscale system parameters

can be easily identiﬁed.

Figure 4: Macroscale equations are matched to microscale equations using volume averaging. At a

macroscopic level the system is represented by three overlapping continua for the intergranular pores

(h-phase), intragranular pores (v-phase) and solid coﬀee (s-phase).

3.3. Coﬀee bed description

The coﬀee bed is represented by a porous medium domain ΩTwith volume VT. The

domain can be split into an intergranular pores domain Ωh, with volume Vhand a coﬀee

grain domain Ωl, with volume Vl. Ωlis further split into an intragranular pore domain

Ωv, with volume Vvand a solid coﬀee domain Ωs, with volume Vs. Clearly the equalities

Vh+Vl=VTand Vv+Vs=Vlhold. The following volume fractions are now deﬁned

8

φh=Vh

VT

, φl=Vl

VT

, φv=Vv

Vl

, φs=Vs

Vl

,(1)

which leads to

φh+φl= 1, φv+φs= 1.(2)

The concentrations (mass per unit volume) of coﬀee in the respective phases are ch,cv

and cs.vhand vvdenote the ﬂuid velocity in the h-phase and v-phase respectively.

The velocity of the solid will be denoted by vs. Further notation will be introduced as

required. The formulation of the equations presented here will follow [16] and [17]. Since

conservation equations will be formed at three diﬀerent scales the variables at each scale

will be denoted as macroscale (˜

·), mesoscale (·∗) and microscale (·) to avoid ambiguity.

3.4. Microscale point balance equations

The point balance equations for coﬀee and liquid within each of the phases on the

microscale are

3.4.1. h-phase

∂ch

∂t =−∇·(chvh+jh),(3)

∂ρh

∂t =−∇·(ρhvh).(4)

3.4.2. v-phase

∂cv

∂t =−∇·(cvvv+jv),(5)

∂ρv

∂t =−∇·(ρvvv).(6)

3.4.3. s-phase

∂cs

∂t = 0.(7)

The microscopic balance equations include the terms jhand jvwhich represent molecular

diﬀusion of coﬀee solubles in the respective phases. Molecular diﬀusion in the solid phase

is assumed negligible.

3.5. Mesoscale point balance equations

The mesoscale balance equations are only required in the grains (l-phase) since only

two scales are needed in the h-phase. Thus the point balance equations for coﬀee and

liquid in the v-phase and the s-phase on the mesoscale are

9

3.5.1. v-phase

∂

∂t (φvc∗

v) = −∇·(φv(c∗

vv∗

v+j∗

v)) −f∗

v→s.(8)

∂

∂t (φvρ∗

v) = −∇·(φv(ρ∗

vv∗

v+i∗

v)).(9)

3.5.2. s-phase

∂

∂t (φsc∗

s) = −∇·(φsj∗

s)−f∗

s→v.(10)

The terms f∗

v→sand f∗

s→vare source/sink terms representing transfer of coﬀee solubles

across the vs-interface and vice-versa. The term i∗

vaccounts for any mechanical dispersion

in the ﬂuid velocity.

3.6. Macroscale point balance equations

For each phase there are macroscopic point balance equations for mass of the coﬀee

and mass of liquid (ﬂow equations). The solid is assumed stationary. The macroscopic

equations take the form:

3.6.1. h-phase

∂

∂t (φh˜ch) = −∇·(φh(˜ch˜vh+˜

jh)) −˜

fh→l,(11)

∂

∂t (φh˜ρh) = −∇·(φh(˜ρh˜vh+˜

ih)) −˜

fw

h→l.(12)

3.6.2. v-phase

∂

∂t (φlφv˜cv) = −∇·(φlφv(˜cv˜vv+˜

jv)) −˜

fv→h−˜

fv→s,(13)

∂

∂t (φlφv˜ρv) = −∇·(φlφv(˜ρv˜vv+˜

iv)) −˜

fw

l→h.(14)

3.6.3. s-phase

∂

∂t (φlφs˜cs) = −∇·(φlφs˜

js)−˜

fs→h−˜

fs→v.(15)

˜

fα→βis transfer of coﬀee solubles from the α-phase to the β-phase across the αβ inter-

face. Similarly ˜

fw

α→βis transfer of liquid from the α-phase to the β-phase across the αβ

interface.

10

3.7. Upscaling from microscale to macroscale

As mentioned the conservation equations at each of the scales can be related by

representing the properties of the medium at a larger scale by averaging the properties

at the smaller scale. This is useful to ﬁnd the forms of the mass transfer terms in the

macroscopic equation. An outline of the general upscaling procedure based on [16, 17]

is included in appendix A. The details of upscaling in this case are included in appendix

B.

4. Developing macroscale equations

The macroscopic balance equations in section 3.6 are in a quite general form. Some

assumptions have already been made but in order to simplify things we make some further

assumptions. We also need to introduce terms to model the transport of ﬂuid and coﬀee

within the bed. Firstly we assume that the density of the liquid is constant and does not

change with coﬀee concentration. This is consistent with [1] where it is noted that the

material extracted from coﬀee has little inﬂuence on liquid density. Thus ˜ρh= ˜ρv=ρ.

It is also assumed that no transport occurs within the l-phase on the macroscale. This

simply means that liquid or coﬀee does not transport directly from grain to grain within

the bed. Any mechanical dispersion in the ﬂow in the h-phase is not considered. Due to

continuity of ﬂux at the interphase boundaries we have ˜

fα→β=−˜

fβ→α. Thus the ﬁve

macroscopic equations reduce to the following

∂

∂t (φh˜ch) = −∇·(φh(˜ch˜vh+˜

jh)) + ˜

fv→h+˜

fs→h,(16)

ρ∂φh

∂t =−ρ∇·(φh˜vh) + ˜

fw

l→h,(17)

∂

∂t (φlφv˜cv) = −˜

fv→h+˜

fs→v,(18)

ρ∂

∂t (φlφv) = −˜

fw

l→h,(19)

∂

∂t (φlφs˜cs) = −˜

fs→h−˜

fs→v.(20)

It now remains to introduce expressions to model the ﬂuid velocity ˜vh, the total macro-

scopic ﬂux ˜

jh, the ﬂuid mass transfer term ˜

fw

l→hand the coﬀee mass transfer terms ˜

fv→h,

˜

fs→vand ˜

fs→hin terms of the system variables. The main transfers occurring in the

coﬀee bed are shown in ﬁgure 5.

4.1. Fluid velocity

Darcy’s Law allows us to relate an averaged velocity or discharge in the pores to the

pressure gradient. The relations in the h-phase are given by

˜uh=φh˜vh,˜uh=−˜

kh

µ(∇˜ph+ρg),˜

kh=˜

kh(φh),(21)

where ˜phis the macroscopic pressure gradient in the h-phase, ˜

khis the permeability and

µis the viscosity of water.

11

Water

Reservoir

Intergranular

Pores

Coffee

Pot

Intragranular

Pores

Coffee Solids

Water in

Coffee

Brew

Out

Water to

Grains

Coffee Dissolving

in grains

Coffee

Dissolving

From

Surface

Coffee Diffusion

From Bulk

Figure 5: Mass transfers occurring in the coﬀee bed.

4.2. Total macroscopic ﬂux

The total macroscopic ﬂux, ˜

jhis made up of the macroscopic average of molecular

diﬀusion ja

hand the dispersive ﬂux jb

h:

˜

jh=hjhih+h˚ch˚vhih=ja

h+jb

h.(22)

For an isotropic porous medium, ja

his often modelled by

ja

h=−D

τ∇˜ch,(23)

where τis the tortuosity deﬁned by

τ=Le

L=actual path length

macroscopic path length.(24)

The tortuosity must be estimated as a function of the porosity. Various functional

relationships are proposed in the literature. Some of these are discussed in [18]. The

expression used here is τ=φ−1

3

hwhich is adopted from [19]. Thus we have

ja

h=−D

τ∇˜ch=−φ

1

3

hD∇˜ch.(25)

4.3. Dispersive ﬂux

Dispersion occurs due to variations in the microscopic velocity of the phase with

respect to the averaged velocity, and molecular diﬀusion [16]. Thus molecular diﬀusion

contributes to the dispersive ﬂux in addition to the diﬀusive ﬂux at the macroscopic level.

In general the dispersive ﬂux is given by

jb

h=h˚ch˚vhih=−˜

Db·∇˜ch,(26)

12

where ˜

Dbis a rank 2 tensor called the dispersion tensor. ˜

Dbis both positive deﬁnite and

symmetric. For an isotropic porous medium the following expression is often used:

Dij =aTδij + (aL−aT)vivj

v2v. (27)

The coeﬃcients aLand aThere are the longitudinal and transverse dispersivities of the

porous medium. vi=hviihis the average velocity in the i-th direction and v=|v|where

vis the average velocity vector in this instance. δij is the Kronecker delta. Further detail

on the diﬀusive and dispersive ﬂuxes is included in appendix C.

4.4. Coﬀee mass transfer terms

Experimental results in section 2 suggest that there are two fundamental extraction

mechanisms from the coﬀee grains. A rapid extraction from ﬁnes (single cells fragments)

and damaged cells on the surface of larger particles and a slower extraction from the

kernels of larger particles. Various diﬀerent models have been applied to represent such

situations, particularly in the area of supercritical ﬂuids. The main models are reviewed

in [20]. These models include linear driving force, bi-linear driving force, shrinking core,

broken plus intact cells in series, broken plus intact cells in parallel and a combined broken

plus intact cells with shrinking core model. These models are widely used in modelling

extraction with supercritical ﬂuids and have been considered in a number of papers, both

theoretical and experimental, including [21, 22, 23, 24]. In [21] it is noted that the broken

plus intact cell model typically features an initial constant extraction period dominated

by extraction from the broken cells, then a falling extraction rate period as broken cells on

the surface are depleted and ﬁnally a diﬀusion controlled period dominated by extraction

from the intact cells. These general features are evident in the extraction experiments

in section 2. The models used here are quite similar to the broken plus intact cells in

parallel. The extraction term ˜

fv→his transfer from the intragranular pores in the grains

to the intergranular pores and is similar to extraction from intact cells. The term ˜

fs→h

is direct extraction from the solid grain matrix surface into the intergranular pores and

is similar to extraction from the broken cells. The term ˜

fs→vmodels dissolution of coﬀee

solubles from the cell walls into the intragranular pores. The transfer ˜

fv→hacross the

interphase boundary is assumed to occur by diﬀusion according to Fick’s ﬁrst law. Thus

˜

fv→h=αvh(˜cv−˜ch).(28)

where αvh is the mass transfer coeﬃcient. The form of αvh can be found from the volume

averaging procedure as

αvh = (1 −φh)φ

4

3

vDv

S∗

hl

∆l

,(29)

where Dvis the diﬀusion coeﬃcient of coﬀee in water, S∗

hl is the speciﬁc surface area of the

l-phase and ∆lis some length scale characterising the distance over which diﬀusion occurs.

The transfers ˜

fs→vand ˜

fv→hare solid-ﬂuid transfer rather than ﬂuid-ﬂuid transfer and

so the modelling is slightly diﬀerent. Under consideration here is the case where the solid

matrix itself is dissolving. It is assumed that there is a thin layer of liquid next to the

solid which is always saturated with solute. This concentration is denoted by csat.csat is

the concentration in the liquid phase that would be in equilibrium with the concentration

13

inside the solid ˜cs. The force of extraction from this thin layer to the bulk of the ﬂuid is

assumed to be proportional to the diﬀerence in concentration between the thin layer and

the bulk of the ﬂuid. Thus, again using volume averaging to determine mass transfer

coeﬃcient, we have

˜

fs→h= (1 −φh)(1 −φv)Dh

S∗

hl

∆s

(csat −˜ch),(30)

where ∆sis some length scale characterising the distance over which diﬀusion occurs.

Similarly

˜

fs→v= (1 −φh)(1 −φv)Dv

S∗

sv

∆s

(csat −˜cv),(31)

where S∗

sv is the speciﬁc surface area of the s-phase. The transfer terms are considered

in more detail in appendix D.

4.5. Liquid transfer term

It is also possible to have a transfer of liquid from the intergranular pores to the

intragranular pores (or vice versa). This could occur for example due to a pressure

imbalance in the phases due to the dissolution of the solid matrix within the grains.

Using Darcy’s Law and volume averaging we ﬁnd that

˜

fw

l→h=−(1 −φh)S∗

lhρkv(φv)

µ

(˜ph−˜pv)

∆l

,(32)

where kv(φv) is the coﬀee grain permeability and ˜phand ˜pvare the water pressures in

the h-phase and the v-phase. Since there is a diﬀerence in concentration in the h and

v-phases the transfer of ﬂuid in either direction would be expected to also result in the

transfer of solute from one phase to another. This can be included as an extra coﬀee

mass transfer term as

fw∗

l→h=(−(1 −φh)S∗

lhρkv(φv)

µ

(˜ph−˜pv)

∆l˜ch,if ˜ph≥˜pv

(1 −φh)S∗

lhρkv(φv)

µ

(˜ph−˜pv)

∆l˜cv,if ˜ph<˜pv

(33)

4.6. Coﬀee bed properties

In order to proceed the quantities such as permeability, surface area, speciﬁc surface

area and distances over which diﬀusion occurs must be expressed in terms of measurable

properties of the coﬀee bed. We can estimate these quantities from the particle size

distribution. In order to simplify things we assume that the intergranular porosity φhis

constant. We note that this means that the only change in porosity occurs within the

l-phase. Thus it seems reasonable to assume that the pressure in the intergranular pores

is always greater than or equal to the pressure in the intragranular pores, i.e. ˜ph≥˜pv.

Thus (33) simpliﬁes to

˜

fw∗

h→l= (1 −φh)S∗

hlρkv(φv)

µ

(˜ph−˜pv)

∆l

˜ch.(34)

14

4.6.1. Speciﬁc surface area of l-phase

Approximating a coﬀee grain of diameter xby a sphere, it has a surface to volume

ratio of 6

x. This can be improved upon if the roundness or sphericity of the coﬀee grain

is known. However in the absence of this we make the spherical approximation. We can

use a number of equivalent spherical diameters to represent the entire size distribution.

One such diameter is the diameter of the spherical particle that has the same speciﬁc

surface area of that distribution. This is called the Sauter mean diameter and is deﬁned

by

ksv =6

Sv

,(35)

where Svis the surface to volume ratio of the distribution which can be found from

the data. The assumption φhis constant means that the Sauter mean diameter doesn’t

change. Thus where required the speciﬁc surface area of the l-phase is given by

S∗

hl =6

ksv

.(36)

In fact we will use two separate Sauter mean diameters. In relation to ﬂow and extraction

from the surface of the grains we need to use the speciﬁc surface area of the entire

distribution. We denote the corresponding Sauter mean diameter ksv1. However when

dealing with extraction from the grain kernel, we should ignore the particles which are

just broken cell fragments and don’t have a kernel of intact cells. In this case we introduce

a second Sauter mean diameter ksv2which is representative of the speciﬁc surface area of

particles above a certain diameter, chosen here to be 50 m. This Sauter mean diameter

is used for extraction from the grain kernel.

4.6.2. Speciﬁc surface area of s-phase

The speciﬁc surface area of the s-phase is more diﬃcult to estimate. Assuming that

the coﬀee grain is made up of solid spherical cells with the same diameter mwe could

approximate the speciﬁc surface area by

S∗

sv =6

m.(37)

However we don’t have any information about the speciﬁc surface area of the s-phase

from the grind size distribution and in practice this will form part of a lumped mass

transfer coeﬃcient which will have to be ﬁtted to the experimental data.

4.6.3. Permeability kh

The permeability can be estimated using the Kozeny-Carman equation for spheres

[25]:

kh=φ3

h

κ(1 −φh)2S∗

hl

2.(38)

Here again S∗

hl is the speciﬁc surface area, while κis a factor which accounts for the

shape and tortuosity. Utilising the derived form for S∗

hl gives

kh=k2

sv1φ3

h

36κ(1 −φh)2=k2

sv1φ3

h

36κ(1 −φh)2.(39)

15

The shape factor κis usually taken to be in the range 2–6. Experiments performed

measuring the pressure drop in an airﬂow through compacted coﬀee beds estimate it at

κ= 3.1.

4.6.4. Permeability kv

Similarly the permeability of the grain is estimated using the Kozeny-Carman equa-

tions for the spherical cells so that

kv=m2φ3

v

180(1 −φv)2.(40)

In the absence of experimental data we have chosen κ= 5 which is often adopted [26].

4.6.5. Average diﬀusion distances

Expressions are required for ∆land ∆s. The distance over which diﬀusion from the

grains to the large pores occurs, is assumed to be equal to the mean radius of the grains

weighted by volume which we denote ∆l=ll. The distance over which diﬀusion occurs

from the surface of the solid to the h-phase is assumed to be equal to the mean radius

of the coﬀee cells. Thus ∆s=ls.

4.6.6. Coﬀee extraction limits

The current forms of the coﬀee mass transfer terms are incomplete as they do not

account for the fact that for a given grind at a given temperature there is a maximum

amount of coﬀee that can be extracted. Looking at our equations, in order to track the

amount of coﬀee extracted we can either allow the grain porosity φvto change or allow

the solid coﬀee concentration ˜csto change, or both. The decision is made here to allow

φvto change as coﬀee is extracted and consider the solid concentration ˜csﬁxed. Thus

φvis allowed to increase as coﬀee is dissolved until it reaches a point when φsconsists of

insoluble material. This maximum value may depend on the grind size and temperature

of extracting water and should be found from the experimentally determined maximum

extractable solubles mass. It is also unrealistic for the mass transfer coeﬃcients from

the solid matrix surface to remain approximately constant even when there is very little

soluble coﬀee left in the solid. This issue is sometimes solved through use of a partition

coeﬃcient. Here we make the simple assumption that the extraction term be proportional

to the amount of coﬀee on the surface. Let φcbe the volume fraction of coﬀee in the

grains. We divide this into coﬀee in the surface of the grains and ﬁnes φs,s, and coﬀee in

the grain kernels φs,b, so that φc=φs,s +φs,b . The initial volume fractions of coﬀee in dry

grains are φc0,φs,s0and φs,b0. Assuming that the soluble coﬀee is uniformly distributed

within the grains the initial volume fraction everywhere is φc0. Thus at a given time the

volume fractions of soluble coﬀee on the surface and in the grain kernels are given by

φc0

φs,s0

φs,s(x, t) = φc0ψs(x, t),(41)

φc0

φs,b0

φs,b(x, t) = φc0ψv(x, t),(42)

16

where xis the position within the coﬀee bed and ψsand ψvare the fractions of the

original amount of coﬀee left on the grain surfaces and in the grain kernels respectively.

We can now substitute φs=φs,i +φs,s +φs,b.φs,i represents the volume fraction of

insoluble solid in the grains. Using the expressions we have developed so far this leads

to two further partial diﬀerential equations for ψsand ψv:

∂ψs

∂t =−12Dhφc0

ksv1mcsat −˜ch

˜csrsψs,(43)

∂ψv

∂t =−12Dvφc0

m2csat −˜cv

˜csrvψv,(44)

where rs=1

φs,s0and rv=1

φs,b0.

4.7. Macroscale equations

Models have now been introduced for the various processes occurring in the coﬀee

bed. The description of the process has been extended to seven coupled partial diﬀerential

equations. These equations are presented in full in (45)-(51). Diﬀerent presentations of

the equations are possible and one equation can be reduced to an algebraic one. Boundary

conditions will depend on the geometry of the problem. Initial conditions will have to

be determined or inferred from experiment following the ﬁlling stage.

φh

∂˜ch

∂t =k2

sv1φ3

h

36κµ(1 −φh)2∇·(˜ch(∇˜ph+ρg))

+φ

4

3

hDh∇2˜ch+φh˜

Db·∇2˜ch

−(1 −φh)φ

4

3

vDv

6

ksv2ll

(˜ch−˜cv)

+ (1 −φh)12Dhφc0

ksv1m(csat −˜ch)ψs

−6(1 −φh)m2

180µksv2ll

φ3

v

(1 −φv)2(˜ph−˜pv)˜ch,(45)

0 = −k2

sv1φ3

h

36κµ(1 −φh)2∇·(∇˜ph+ρg)

+6(1 −φh)m2

180µksv2ll

φ3

v

(1 −φv)2(˜ph−˜pv),(46)

∂

∂t (φv˜cv) = +φ

4

3

vDv

6

ksv2ll

(˜ch−˜cv)

+12φc0Dv

m2(csat −˜cv)ψv

+6(1 −φh)m2

180µksv2ll

φ3

v

(1 −φv)2(˜ph−˜pv)˜ch,(47)

17

∂φv

∂t =6m2

180µksv2ll

φv3

(1 −φv)2(˜ph−˜pv),(48)

∂φv

∂t =−1

rs

∂ψs

∂t −1

rv

∂ψv

∂t ,(49)

∂ψs

∂t =−12Dhφc0

ksv1mcsat −˜ch

˜csrsψs,(50)

∂ψv

∂t =−12Dvφc0

m2csat −˜cv

˜csrvψv,(51)

While these equations give estimates for the diﬀerent coeﬃcients in terms of process

parameters, these parameters may not always be easy to determine accurately. Other

processes which were not considered may also aﬀect the transport of coﬀee and water.

Thus when comparing to experiment it is necessary to introduce ﬁtting parameters par-

ticularly to the terms controlling extraction from the grain surfaces and diﬀusion of coﬀee

from the grain kernel.

5. Numerical simulations

Numerical simulations of the experiments in sections 2.2 and 2.3 are conducted using

the model equations. The equations can be reduced in both of these cases.

5.1. Batch-wise brewing

In the french press type brewing apparatus there is no pressure induced ﬂow and we

assume the solution in the h-phase is well mixed since only the average concentration

in this phase is measured anyway. Thus all spatial derivatives in the equations drop

out leaving a system of ordinary diﬀerential equations. In order to solve the system we

require some initial conditions. Here we need to make some assumptions since we do

not model the initial inﬁltration of the water into the grains when the coﬀee and water

are mixed. Firstly, we assume that the grains are saturated with water initially. We

further assume that the initial concentration in the h-phase is zero, so that none of the

coﬀee has dissolved from the surface of the grains. It is also necessary to give an initial

condition for the concentration in the v-phase. It is assumed initially that all the soluble

coﬀee in the grains has dissolved into the v-phase. This is of course unlikely to be the

case. However, since coﬀee diﬀusion from the v-phase to the h-phase seems to be the

rate limiting process, occurring much slower than dissolution of coﬀee, this assumption

is unlikely to have much impact on the simulation results. We also assume that, only a

change in porosity in the intact cells in the grain kernel will result in a change in pressure

in the v-phase, since any pressure imbalance in the damaged cells on the grain surface

would be almost instantaneously corrected. Thus, as all coﬀee in kernel is assumed

dissolved initially, no pressure diﬀerence between phases will occur. Note the value of

intragranular porosity φv= 0.56 is adopted here for dry coﬀee in air. Allowing for these

assumptions the following system of equations are solved numerically in MATLAB R

.

18

φh

d˜ch

dt =−α∗(1 −φh)φ

4

3

vDv

6

ksv2ll

(˜ch−˜cv)

+β∗(1 −φh)12Dhφc0

ksv1m(csat −˜ch)ψs,(52)

d

dt (φv˜cv)=+α∗φ

4

3

vDv

6

ksv2ll

(˜ch−˜cv),(53)

dφv

dt =−1

rs

∂ψs

∂t ,(54)

dψs

dt =−β∗12Dhφc0

ksv1mcsat −˜ch

˜csrsψs,(55)

with initial conditions

˜ch(0) = 0,˜cv(0) = cv0, φv(0) = φv0, ψs(0) = 1.(56)

The parameters α∗and β∗are used as ﬁtting parameters to ﬁt the experimental results.

The mass transfer coeﬃcients depend on the volume fraction of the interstitual water

in the coﬀee bed in question, so the ﬁtting parameters are just intended to correct for

errors in the other parameters in the mass transfer coeﬃcient. Thus we look for values

of α∗and β∗which ﬁt both experiments equally well. The initial volume fractions of

coﬀee φs,s0and φs,b0can, to some extent, be estimated from the grind size distribution

but here we will use them to allow for any diﬀerences between the batch extraction and

coﬀee ﬂow-through-cell experiments. These diﬀerences may arise, for example, due to

the fact that we do not model the initial water inﬁltration into the coﬀee grains. The

parameters used in the simulations are listed in table 1. The other parameters are all

measured experimentally, estimated from experiments or sourced in the literature. The

value for the coﬀee solubility csat is estimated from the highest observed concentration

across the four experiments. The diﬀusion coeﬃcient is for caﬀeine in water at 80 ◦C [27].

This would be slightly higher at 90 ◦C. However this is only an estimate of the eﬀective

diﬀusion coeﬃcient of coﬀee in water. The ﬁtting parameters are used to correct any

errors in these parameters. The comparison between the numerical solution and the

experimental results is shown in ﬁgure 6.

5.2. Cylindrical brewing chamber

The cylindrical brewing chamber geometry also allows us to make some simpliﬁcations

to the general model. Assuming that the coﬀee bed properties are homogeneous in any

cross section, the equations can be reduced to one spatial dimension, parallel to the

ﬂow direction. Thus the bed depth is labelled by the z-coordinate. The height of the

coﬀee bed is Lwith the bottom (ﬁlter exit) at z= 0 and the top (ﬁlter entrance) at

z=L. For the experiments here it is shown in appendix E that advection dominates over

diﬀusion and mechanical dispersion, so these two processes are neglected. This means

we just require one concentration boundary condition for ˜ch. We use the condition

that the water entering at the top z=Lhas zero concentration. The pressure boundary

conditions are ˜ph= ∆pat z=Land ˜ph= 0 at z= 0. ∆pis the pressure diﬀerence across

19

Parameter JK Drip ﬁlter Cimbali #20

φv00.6444 0.6120

φh0.8272 0.8272

ksv127.35 m 38.77 m

ksv2322.49 m 569.45 m

ll282 m 463 m

Dh=Dv2.2×10−9m2s−12.2×10−9m2s−1

ρ965.3 kg m−3965.3 kg m−3

µ0.315 ×10−3Pa s 0.315 ×10−3Pa s

m30 m 30 m

csat 212.4 kg m−3212.4 kg m−3

˜cs1400 kg m−31400 kg m−3

κ3.1 3.1

φc00.143435 0.122

φs,s00.059 0.07

φs,b00.084435 0.052

α∗0.1833 0.0881

β∗0.0447 0.0086

rs16.94 14.28

cv0183.43 kg m−3118.95 kg m−3

Table 1: Parameters for simulation of the batch extraction experiments.

0 100 200 300 400 500 600

0

5

10

15

20

25

30

35

40

t (s)

Ch (kg/m3)

(a)

0 100 200 300 400 500 600

0

5

10

15

20

25

30

35

t (s)

Ch (kg/m3)

(b)

Figure 6: Comparison between numerical solution (–) with parameters from table 1 and experiment (*)

for the batch extraction experiments for (a) JK drip ﬁlter grind and (b) Cimbali #20 grind.

the bed. In these experiments there is a large pressure diﬀerence across the bed which

can lead to bed compaction and a reduction in the porosity. Thus the porosity cannot be

measured a priori. It assumed that φhadjusts to the pressure while φvstays constant.

We then choose φhby matching the volume ﬂow from Darcy’s Law and the Kozeny-

Carman equation to the experimental volume ﬂow. Once again the initial conditions

need to be inferred. It is assumed that once brewed coﬀee starts to ﬂow from the bed

20

that the bed is fully saturated with liquid. Again it is assumed that all the coﬀee in the

grain kernels has dissolved into the intragranular pores (v-phase) during ﬁlling and this is

uniformly distributed in the bed. It is also necessary to estimate the initial concentration

˜chas a function of z. Based on the initial exiting concentrations we assume for the ﬁne

grind (JK) that initially ˜chis at the coﬀee solubility throughout the h-phase. For the

coarser grain, (Cimbali # 20), we assume a linear concentration proﬁle in the h-phase,

rising from zero at the top to the initial exiting concentration at the bottom of the

bed. Thus it is assumed that some extraction from the ﬁnes and broken surface cells

occurs during ﬁlling. To this end the amount of surface coﬀee is uniformly reduced by

the corresponding amount of coﬀee initially present in the h-phase. In reality we would

expect more extraction would have occurred at the top of the bed during ﬁlling than the

bottom, but, in the absence of experimental guidance, we make the simplest assumption.

As with the batch experiments the assumptions mean that ˜ph= ˜pv. Based on these

assumptions the reduced set of equations to model extraction in the cylindrical brewing

chamber is given by

φh

∂˜ch

∂t =k2

sv1φ3

h

36κµ(1 −φh)2

∂

∂z ˜ch∂˜ph

∂z +ρg

−α∗(1 −φh)φ

4

3

vDv

6

ksv2ll

(˜ch−˜cv)

+β∗(1 −φh)12Dhφc0

ksv1m(csat −˜ch)ψs,(57)

∂2˜ph

∂z2= 0,(58)

∂

∂t (φv˜cv) = +α∗φ

4

3

vDv

6

ksv2ll

(˜ch−˜cv),(59)

∂φv

∂t =−1

rs

∂ψs

∂t ,(60)

∂ψs

∂t =−β∗12Dhφc0

ksv1mcsat −˜ch

˜csrsψs,(61)

for

t > 0,0<z < L, (62)

with initial conditions

˜ch(z , 0) = ch0(z),˜cv(z, 0) = cv0,(63)

φv(z, 0) = φv0, ψs(z, 0) = ψs0,(64)

and boundary conditions

˜ch(L, t) = 0,˜ph(0, t)=0,˜ph(L, t) = ∆p. (65)

21

The initial concentration proﬁle in the JK drip ﬁlter grind is given by ch0(z) = csat.

In the Cimbali #20 grind the initial concentration is given by ch0(z) = cmax

L(L−z).

Some of the parameters are the same as those in the batch extraction case. Any new

parameters or parameters that have changed are included in table 2. The numerical

solution is found using ﬁnite diﬀerences in the spatial direction and the method of lines.

It should be noted that any initial discontinuities in the initial and boundary conditions

will be smoothed out by numerical diﬀusion. If accuracy is required in the initial stages

a large number of steps should be used in the spatial partition. The comparison between

the numerical solution and the experimental results is shown in ﬁgure 7.

Parameter JK Drip ﬁlter Cimbali #20

φv00.6231 0.6218

φh0.2 0.25

cmax −82.63 kg m−3

cs1400 kg m−31400 kg m−3

φs,s00.11 0.07

φs,b00.033435 0.052

α∗0.1833 0.0881

β∗0.0447 0.0086

rs9.09 14.28

cv078.88 kg m−3118.95 kg m−3

∆p230 000 Pa 65 000 Pa

L0.0405 m 0.0526 m

Table 2: Parameters for simulation of the cylindrical brewing chamber extraction experiments.

0 0.2 0.4 0.6 0.8 1

0

50

100

150

200

250

Mbrew (kg)

Cexit (kg/m3)

(a)

0 0.2 0.4 0.6 0.8 1

0

10

20

30

40

50

60

70

80

90

100

Mbrew (kg)

Cexit (kg/m3)

(b)

Figure 7: Comparison between numerical solution (–) with parameters from table 2 and experiment (*)

for the cylindrical brewing chamber extraction experiments for (a) JK drip ﬁlter grind and (b) Cimbali

#20 grind.

22

6. Conclusion

The coﬀee extraction process can be described very eﬀectively using mathematical

models. In this paper a general model is introduced to describe coﬀee extraction by hot

water from a bed of coﬀee grains. The coﬀee bed is modelled as a saturated porous

medium using a double porosity model. The bed consists of two kinds of pores: pores

between the grains (intergranular) and pores within the grains (intragranular). Flow

of liquid within the coﬀee bed is modelled using Darcy’s Law and the Kozeny-Carman

equation. Motivated by experiment, extraction of coﬀee from the coﬀee grains is modelled

using two mechanisms. Coﬀee on the damaged grain surface and in coﬀee cell fragments

or ﬁnes extracts quickly into the intergranular pores due to a relatively low mass transfer

resistance. Coﬀee in intact cells in the grain kernels, ﬁrst extracts into the intragranular

pores and then slowly diﬀuses through the grain into the intergranular pores.

The model is parametrised using experimental data. Numerical simulations are per-

formed and ﬁtted to data. It is shown that the developed models can quantitatively

describe extraction from ground coﬀee in two situations: in a well stirred dilute suspen-

sion of coﬀee grains, and in a packed coﬀee bed. The extraction curves ﬁt the data for

both a ﬁne and a coarse grind as the parameters in the model vary with the surface

area and the mean grain radius of the grind size distribution. Provided the grind size

distribution is known, and the physics of extraction is the same, the model should work

well for an even wider range of grind sizes.

The model described can be easily generalised to describe standard coﬀee brewing

techniques. It can also be extended to include unsaturated ﬂow during water inﬁltration

into the coﬀee bed and drainage of water from the coﬀee bed. The model is also adaptable

to diﬀerent bed geometries. The model is presented for isothermal conditions but may

also be extended in future work to include heat transfers within the coﬀee bed.

Acknowledgements

The authors acknowledge the support of MACSI, the Mathematics Applications Con-

sortium for Science and Industry (www.macsi.ul.ie), funded by the Science Foundation

Ireland Investigator Award 12/IA/1683.

Appendices

A. General upscaling procedure

The following procedure is mainly adapted from [17] and [16]. In order to compare

microscopic and macroscopic equations a procedure is needed for upscaling or averaging

equations from a smaller scale to a larger scale. A general upscaling procedure from a

microscopic scale to a macroscopic scale is outlined here.

Let the general form of a point balance equation for the concentration of a γ-species

per unit volume of the α-phase be

∂cα

∂t =−∇·(cαvα+jα) + Gα,(66)

23

where cαis the concentration of the γ-species, vαis the ﬂuid velocity in the α-phase,

jαis the diﬀusive ﬂux of the γ-species in the α-phase, and Gαis a source or sink of the

γ-species in the α-phase.

Let βrepresent all other phases in the porous medium. The αsubscript will be

dropped where convenient. Equation 66 is the microscopic balance. The corresponding

macroscopic balance equation is

∂

∂t (φαc∗) = −∇·(φαc∗v∗+φαj∗) + φαG∗,(67)

where φαis the volume fraction of the α-phase and the ·∗variables are macroscopic

variables.

To relate (66) and (67) the averaged macroscale properties are deﬁned by averaging

(or integrating) (66) over an appropriate REV or Representative Elementary Volume.

The length scale of an REV is much greater than the pore scale but much less than the

full scale of the system. So if lis the length scale of an REV, dis the pore length scale

and Lis the full problem length scale then

dlL. (68)

The following notation is used

•The volume of an REV is δV .

•The portion of the volume of an REV occupied by the α-phase is δV α.

•The union of the interfacial regions within the REV between the α-phase and a

β-phase is denoted Sαβ .

•The unit vector normal to this surface oriented outward from the α-phase is nα.

To average the microscopic balance equations two theorems that transform the average

of a derivative to the derivative of an average are needed. The time averaging theorem

is given by

ZδV

∂F

∂t dV =∂

∂t ZδV

F dV −X

β6=αZSαβ

nα·wbF|αdS, (69)

where Fis some scalar ﬁeld property of the microscale and F|αjust indicates that the

microscale property Fin the α-phase is being integrated over the αβ-interface. wbis

the velocity of Sαβ. The summation Pβ6=αjust denotes a summation over all phases

except the α-phase.

The divergence averaging theorem is given by

ZδV

∇·BdV =∇·ZδV

BdV +X

β6=αZSαβ

nα·B|αdS, (70)

where Bis some vector ﬁeld property of the microscale and B|αjust indicates that Bis

being integrated over the αβ-interface. The summation Pβ6=αjust denotes a summation

over all phases except the α-phase.

Next some phase averages are deﬁned

24

The Intrinsic Phase Average of a quantity F over the α-phase is deﬁned by

hFiα=Fα=1

δV αZδV α

F dV. (71)

The Phase Average of a quantity F over the α-phase is deﬁned by

hFi=F=1

δV ZδV α

F dV =φαFα.(72)

The Mass Weighted Average of a quantity F over the α-phase is deﬁned by

hF0iα=F0α=1

hρiαδV αZδV α

ρF dV =hρF iα

hρiα.(73)

It will also be necessary to be able to write the average of a product of the form hcviα

in terms of the individual averages hciαand hviα. To do this consider the following

•Let ˚v =v− hviαbe the deviation of the true velocity from the mean velocity.

•Let ˚c=c− hciαbe the deviation of the true concentration from the mean concen-

tration.

Then

h˚viα=hv− hviαiα=hviα− hviα= 0,(74)

h˚ciα=hc− hciαiα=hciα− hciα= 0.(75)

Thus

hcviα=h(hciα+˚c)(hviα+˚v)iα

=hciαhviα+h˚c˚viα+h˚ciαhviα+hciαh˚viα

=hciαhviα+h˚c˚viα.(76)

The term h˚c˚viαis called the dispersive ﬂux.

Now the averaging process can be performed. Firstly (66) integrated over δV αyields

ZδV α

∂c

∂t dV =−ZδV α

∇·(cv+j)dV +ZδV α

GdV. (77)

Now applying theorems (69) and (70) yields

∂

∂t ZδV α

cdV =−∇·(ZδV α

(cv+j)dV )

−X

β6=αZSαβ

nα·(c(v−wb) + j)|αdS

+ZδV α

GdV. (78)

25

This can be written in terms of averaged quantities by dividing across by δV =δV α

φαto

give

∂

∂t (φαhciα) = −∇·φα(hcviα+hjiα)

−1

δV X

β6=αZSαβ

nα·(c(v−wb) + j)|αdS

+φαhGiα.(79)

Utilising the formula for the average of a product it can be seen that

∂

∂t (φαhciα) = −∇·φα(hciαhviα+h˚c˚viα+hjiα)

−1

δV X

β6=αZSαβ

nα·(c(v−wb) + j)|αdS

+φαhGiα.(80)

Comparing this to the macroscopic point balance equation (67) gives the following rela-

tions between the macroscopic and microscopic quantities

c∗=hciα,(81)

v∗=hviα,(82)

j∗=h˚c˚viα+hjiα,(83)

φαG∗=φαhGiα

+1

δV X

β6=αZSαβ

nα·(c(v−wb) + j)|αdS. (84)

Thus it can be seen that the microscale convection and diﬀusion processes at the interfaces

are source terms for the macroscopic equation. It can also be seen that the macroscopic

diﬀusive ﬂux is the sum of the averaged microscopic diﬀusive ﬂux and the dispersive ﬂux.

B. Upscaling from microscale to macroscale equations

The upscaling process basically involves choosing an REV (Representative Elemen-

tary Volume) around every point on the larger scale and representing the properties of

the medium by the averaged properties of the smaller scale over the REV. For these

purposes suitable averages need to be deﬁned including a phase average and an intrinsic

phase average. Also needed are a time averaging theorem and a divergence averaging

theorem. An outline of the upscaling procedure used here is given in appendix A. Before

continuing some notes on the REVs being used are necessary. Two diﬀerent REVs will

be used. One will have a scale between that of a coﬀee cell and a coﬀee grain. The

second will have a scale between that of a coﬀee grain and the coﬀee bed. For the REVs

the following notation is used.

26

V1:Volume of the smaller REV.

V1s:Volume of solid in the smaller REV.

V1v:Volume of void space in smaller REV.

V0:Volume of larger REV.

V0l:Volume of grains (l-phase) in larger REV.

V0h:Volume of void space (h-phase) in larger REV.

Also due to the properties of an REV

φh=V0h

V0

=Vh

VT

, φl=V0l

V0

=Vl

VT

,(85)

φv=V1v

V1

=Vv

Vl

, φs=V1s

V1

=Vs

Vl

.(86)

B.1. Equations for upscaling

For convenience the equations at each of the coﬀee bed length scales from the paper

are reproduced here.

B.1.1. Microscale Point Balance Equations

The point balance equations for coﬀee and liquid within each of the phases on the

microscale are

h-phase.

∂ch

∂t =−∇·(chvh+jh),(87)

∂ρh

∂t =−∇·(ρhvh).(88)

v-phase.

∂cv

∂t =−∇·(cvvv+jv),(89)

∂ρv

∂t =−∇·(ρvvv).(90)

s-phase.

∂cs

∂t = 0.(91)

The microscopic balance equations include the terms jhand jvwhich represent molecular

diﬀusion of coﬀee solubles in the respective phases. Molecular diﬀusion in the solid phase

is assumed negligible.

27

B.1.2. Mesoscale point balance equations

The mesoscale balance equations are only required in the grains (l-phase) since only

two scales are needed in the h-phase. Thus the point balance equations for coﬀee and

liquid in the v-phase and the s-phase on the mesoscale are

v-phase.

∂

∂t (φvc∗

v) = −∇·(φv(c∗

vv∗

v+j∗

v)) −f∗

v→s.(92)

∂

∂t (φvρ∗

v) = −∇·(φv(ρ∗

vv∗

v+i∗

v)).(93)

s-phase.

∂

∂t (φsc∗

s) = −∇·(φsj∗

s)−f∗

s→v.(94)

The terms f∗

v→sand f∗

s→vare source/sink terms representing transfer of coﬀee solubles

across the vs-interface and vice-versa. The term i∗

vaccounts for any mechanical dispersion

in the ﬂuid velocity.

B.1.3. Macroscale point balance equations

For each phase there are macroscopic point balance equations for mass of the coﬀee

and mass of liquid (ﬂow equations). The solid is assumed stationary. The macroscopic

equations take the form:

h-phase.

∂

∂t (φh˜ch) = −∇·(φh(˜ch˜vh+˜

jh)) −˜

fh→l,(95)

∂

∂t (φh˜ρh) = −∇·(φh(˜ρh˜vh+˜

ih)) −˜

fw

h→l.(96)

v-phase.

∂

∂t (φlφv˜cv) = −∇·(φlφv(˜cv˜vv+˜

jv)) −˜

fv→h−˜

fv→s,(97)

∂

∂t (φlφv˜ρv) = −∇·(φlφv(˜ρv˜vv+˜

iv)) −˜

fw

l→h.(98)

s-phase.

∂

∂t (φlφs˜cs) = −∇·(φlφs˜

js)−˜

fs→h−˜

fs→v.(99)

˜

fα→βis transfer of coﬀee solubles from α-phase to β-phase across αβ interface. Similarly

˜

fw

α→βis transfer of liquid from α-phase to β-phase across αβ interface.

28

B.2. Upscaling in l-phase: microscale to mesoscale

B.2.1. Conservation of coﬀee in v-phase

The averaged form of the microscopic point balance equation (89) over the smaller

REV is

∂

∂t (φvhcviv) = −∇·φv(hcvivhvviv+h˚cv˚vviv+hjviv)

−1

V1ZSvs

nv·(cv(vv−wvs) + jv)dS. (100)

Now the surface Svs is a material surface so vv−wvs =0. Comparing this averaged

form with the mesoscale form in (92) it can be seen that

c∗

v=hcviv,(101)

v∗

v=hvviv,(102)

j∗

v=h˚cv˚vviv+hjviv,(103)

f∗

v→s=1

V1ZSvs

nv·jvdS. (104)

B.2.2. Conservation of liquid in v-phase

The averaged form of the microscopic point balance equation (90) over the smaller

REV is

∂

∂t (φvhρviv) = −∇·φv(hρvivhvviv+h˚ρv˚vviv)

−1

V1ZSvs

nv·(ρv(vv−wvs))dS. (105)

Now the surface Svs is a material surface so vv−wvs =0. Comparing this averaged

form with the mesoscale form in (93) it can be seen that

ρ∗

v=hρviv,(106)

v∗

v=hvviv,(107)

i∗

v=h˚ρv˚vviv.(108)

Here in fact ρwill be taken to be constant so i∗

v=0in this case.

B.2.3. Conservation of coﬀee solid in s-phase

The averaged form of the microscopic point balance equation (91) over the smaller

REV is

∂

∂t (φshcsis) = −1

V1ZSsv

ns·(−cswsv)dS. (109)

29

Comparing this averaged form with the mesoscale form in (94) it can be seen that

c∗

s=hcsis,(110)

v∗

s=0,(111)

f∗

s→v=1

V1ZSsv

ns·(−cswsv)dS. (112)

B.3. Upscaling to macroscale

B.3.1. Conservation of coﬀee in h-phase

The averaged form of the microscopic point balance equation (87) over the larger

REV is

∂

∂t (φhhchih) = −∇·φh(hchihhvhih+h˚ch˚vhih+hjhih)

−1

V0ZShl

nh·(ch(vh−whl) + jh)dS. (113)

Comparing this averaged form with the macroscale form in (95) it can be seen that

˜ch=hchih,(114)

˜vh=hvhih,(115)

˜

jh=h˚ch˚vhih+hjhih,(116)

˜

fh→l=1

V0ZShl

nh·(ch(vh−whl) + jh)dS. (117)

Here Shl is the eﬀective surface between the h-phase and the l-phase and includes bound-

aries between the h-phase and the v-phase and the h-phase and the s-phase.

B.3.2. Conservation of liquid in h-phase

The averaged form of the microscopic point balance equation (88) over the larger

REV is

∂

∂t (φhhρhih) = −∇·φh(hρhihhvhih+h˚ρh˚vhih)

−1

V0ZShl

nh·(ρh(vh−whl))dS. (118)

Comparing this averaged form with the macroscale form in (96) it can be seen that

˜ρh=hρhih,(119)

˜vh=hvhih,(120)

˜

ih=h˚ρh˚vhih,(121)

˜

fw

h→l=1

V0ZShl

nh·(ρh(vh−whl))dS. (122)

30

B.3.3. Conservation of coﬀee in v-phase

The averaged form of the microscopic point balance equation (92) over the larger

REV is

∂

∂t (φlφvhc∗

vil) = −∇·φlφv(hc∗

vilhv∗

vil+˚

c∗

v˚

v∗

vl+hj∗

vil)

−φv

V0ZSlh

nl·(c∗

v(v∗

v−wlh) + j∗

v)dS

−φlhf∗

v→sil.(123)

Comparing this averaged form with the macroscale form in (97) it can be seen that

˜ch=hc∗

vil=hhcvivil,(124)

˜vv=hv∗

vil=hhvvivil,(125)

˜

jv=hj∗

vil+˚

c∗

v˚

v∗

vl

=hhjvivil+hh˚cv˚vvivil+˚

c∗

v˚

v∗

vl,(126)

˜

fv→h=φv

V0ZSlh

nl·(hcviv(hvviv−wlh) + hjviv)dS, (127)

˜

f∗

v→s=φlhf∗

v→sil=1

V0ZV0l1

V1ZSvs

nv·jvdSdV0l.(128)

It will be later assumed that ˜vv=0and ˜

jv=0but of course it will still be possible to

have v∗

v6=0and j∗

v6=0.

B.3.4. Conservation of liquid in v-phase

The averaged form of the microscopic point balance equation (93) over the larger

REV is

∂

∂t (φlφvhρ∗

vil) = −∇·φlφv(hρ∗

vilhv∗

vil+˚

ρ∗

v˚

v∗

vl+hi∗

vil)

−φv

V0ZSlh

nl·(ρ∗

v(v∗

v−wlh) + i∗

v)dS. (129)

Comparing this averaged form with the macroscale form in (98) it can be seen that

˜ρh=hρ∗

vil=hhρvivil,(130)

˜vv=hv∗

vil=hhvvivil,(131)

˜

iv=hi∗

vil+˚

ρ∗

v˚

v∗

vl

=hhivivil+hh˚ρv˚vvivil+˚

ρ∗

v˚

v∗

vl,(132)

˜

fw

l→h=φv

V0ZSlh

nl·(hρviv(hvviv−wlh) + hiviv)dS. (133)

31

B.3.5. Conservation of coﬀee in s-phase

The averaged form of the microscopic point balance equation (94) over the larger

REV is

∂

∂t (φlφshc∗

sil) = −∇·φlφs(hj∗

sil)

−φs

V0ZSlh

nl·(c∗

s(−wlh) + j∗

s)dS

−φlhf∗

s→vil.(134)

Comparing this averaged form with the macroscale form in (99) it can be seen that

˜cs=hc∗

sil=hhcsisil,(135)

˜vs=0,(136)

˜

js=hj∗

sil=hhjsisil=0,(137)

˜

fs→h=φs

V0ZSlh

nl·(hcsis(−wlh))dS, (138)

˜

fs→v=φlhf∗

s→vil

=1

V0ZV0l1

V1ZSsv

ns·(−cswlh)dSdV0l.(139)

C. Macroscopic diﬀusion and dispersion ﬂuxes

The total macroscopic ﬂux, ˜

jhis made up of the macroscopic average of molecular

diﬀusion and the dispersive ﬂux:

˜

jh=hjhih+h˚ch˚vhih.(140)

The microscopic diﬀusive ﬂux can be represented by Fick’s Law:

jh=−D∇ch,(141)

where Dis the diﬀusion coeﬃcient of the species in water. The macroscopic equivalent

is obtained by averaging this expression and will generally depend on the structure of

the porous medium. This average is represented by

ja

h=−D˜

Ta(φ)·∇˜ch=−˜

Da(φ)·∇˜ch.(142)

where the ˜

·here on ˜

Tameans this is a tensor of rank two which represents the tortuosity

of the porous medium. For an isotropic porous medium this can be adjusted to

ja

h=−D

τ∇˜ch,(143)

32

where τis the tortuosity deﬁned by

τ=Le

L=actual path length

macroscopic path length.(144)

The tortuosity must be estimated in terms of the porosity. Various estimated are used

in the literature. Some of these include

τ=φ−1

3, τ =φ−1

2, τ =1

1−α(1 −φ).(145)

In the ﬁnal expression α=rσ

Vis a shape factor with rbeing the object radius, σbeing

the object cross sectional area and V being the object volume. Thus for spheres for

example α= 0.75. In this case the ﬁrst expression from [19] will be adopted. The other

expressions and tortuosity in general is discussed in [18]. Thus macroscopic diﬀusion is

approximated by

ja

h=−D

τ∇˜ch=−φ

1

3

hD∇˜ch.(146)

Dispersion occurs due to variations in the microscopic velocity of the phase with

respect to the averaged velocity, and molecular diﬀusion [16]. Molecular diﬀusion con-

tributes to the dispersive ﬂux in addition to the diﬀusive ﬂux at the macroscopic level.

In general the dispersive ﬂux is given by

jb

h=h˚ch˚vhih=−˜

Db·∇˜ch,(147)

where ˜

Dbis a rank 2 tensor called the dispersion tensor. ˜

Dbis both positive deﬁnite and

symmetric. One commonly used expression is

Dij =aijkl

vkvl

v,(148)

where aijkl is a fourth order tensor and vi=hviihis the average velocity in the i-th

direction and v=|v|where vis the average velocity vector in this instance. For an

isotropic porous medium this expression reduces to

Dij =aTδij + (aL−aT)vivj

v2v. (149)

The coeﬃcients aLand aThere are the longitudinal and transverse dispersivities of the

porous medium. For a phase that completely ﬁlls a pore space, aLis a length that should

be of the same order of the pore size. δij is the Kronecker delta. Also it is required that

aL≥0aT≥0.(150)

Laboratory experiments have found that aTis 8 −24 times smaller than aL[16].

33

D. Method of estimating mass transfer terms

It is necessary to estimate the mass transfer terms fα→βwhich govern the transfer of

solute from the solid phase to the liquid phase both within the grains and from the surface

of the grains. It is also necessary to estimate the mass transfer of solute from liquid within

the grains to liquid in the pores between the grains. This subject is dealt with from a food

processing and engineering standpoint in [5]. More general and technical developments

are found in [16] and [28]. The transfer term fα→βmay be due to a number of processes.

Some typical examples are adsorption (from the liquid phase to the solid), evaporation

or volatilization (i.e. a liquid-gas transfer), dissolution (i.e. solid-liquid transfer), and

liquid-liquid transfer. It is possible that a number of these transfers occur simultaneously

so that the transfer term comprises a number of diﬀerent transfer processes. It is assumed

here that there is no source or sink on the interphase boundary, i.e., there is no jump in

the normal ﬂux of the considered species across the boundary so that fα→β=−fβ→α. In

this instance it is assumed that a chemical species can reach the microscopic interphase

boundary by two modes of transport, namely advection and diﬀusion. Hence, as we

have already seen from the averaging procedure in appendices A and B the strength of

a source of a considered species in the α-phase is given by

fα→β=1

U0ZSαβ

nα·(cα(vα−wαβ ) + jα)dS, (151)

where U0is the volume of the REV, Sαβ is the boundary between the α-phase and all

other phases, nαis the unit outward normal vector on this surface and wαβ is the velocity

of points on the interphase boundary. If Sαβ is a material surface as will generally be

the case here than vα−wαβ =0and

fα→β=1

U0ZSαβ

nα·jαdS. (152)

In this case the chemical species crosses the interphase boundary by diﬀusion only. In the

coﬀee bed model developed here non-equilibrium ﬂuid-ﬂuid and mass-ﬂuid transfers will

be considered. The ﬁrst mechanism considered that drives transfer (in an eﬀort to bring

the system closer to equilibrium) is the diﬀerence in concentrations (or more rigorously

the diﬀerence in chemical potentials) at the interface, visualised as a thin ﬁlm. Therefore

the rate of transfer, fα→β, of the mass of the considered species from an α-phase to

an adjacent β-phase across an interface Sαβ is often assumed to be proportional to the

diﬀerence in concentration between the phases. Thus

fα→β=ααβ (cβ−cα).(153)

To estimate the mass transfer coeﬃcient ααβ the following form is used

jα·nα=−Dα(cβ−cα)

∆α

.(154)

Here Dαis the coeﬃcient of diﬀusion of the considered species in the α-phase and ∆αis

the length characterising the mean size of the phase or the length over which diﬀusion

occurs. For example one possible deﬁnition of ∆αis ∆α=U0α

Sαβ , the volume to surface

ratio of the α-phase within the REV. Then

34

fα→β=1

U0ZSαβ

nα·jαdS

=−1

U0ZSαβ

Dα(cβ−cα)

∆α

dS

=−Sαβ

U0Dα(cβ−cα)

∆α

=−S∗

αβ

∆α

φαDα(cβ−cα),(155)

where S∗

αβ =Sαβ

U0αis the speciﬁc surface area of the α-phase or the surface area per unit

volume. Thus it can be seen that the mass transfer coeﬃcient is given by

ααβ =−S∗

αβ

∆α

φαDα.(156)

The above derivation is for a ﬂuid-ﬂuid transfer. For a solid-ﬂuid transfer a slightly

diﬀerent approach is used.

Under consideration here is the case where the solid matrix itself is dissolving (i.e. the

soluble part of the coﬀee grains). It is again assumed that the porous medium is saturated

and a single constituent is considered. It is then assumed that there is a thin layer of

liquid next to the solid which is always saturated with the solute under consideration.

This concentration is denoted by csat.csat is the concentration in the liquid phase that

would be in equilibrium with the concentration inside the solid cs. This assumes that

the dissolution process occurs faster than the transfer from this thin layer to the bulk of

the ﬂuid. It is now assumed that the force of extraction from this thin layer to the bulk

of the ﬂuid is given by

fs→f=αsf (csat −cf).(157)

Proceeding as above it can be shown that the mass transfer coeﬃcient from a solid phase

s to a ﬂuid phase f is given by

αsf =−S∗

sf

∆s

φsDf.(158)

The transfers looked at so far have been transfer of constituent or solute due to

diﬀusion. It is also possible to have a transfer of liquid from the large pores between

the grains to the small pores within the grains (or vice versa). This will occur due to a

pressure imbalance between the phases due to the dissolution of the solid matrix within

the grains. As was seen from the averaging procedure this term has the form

˜

fw

l→h=φv

V0ZSlh

nl·(hρiv(hvviv−wlh) + hiviv)dS. (159)

Now it is assumed that hiviv=0and Darcy’s Law gives

hvviv−wlh =−kv(φv)

φvµ∇p(160)

35

On Slh this means that

hvviv−wlh =−kv(φv)

φvµ

(ph−pv)

∆l

.(161)

Thus

˜

fw

l→h=φv

V0ZSlh

nl·−ρkv(φv)

φvµ

(˜ph−˜pv)

∆ldS

=−Slh

V0

ρkv(φv)

µ

(˜ph−˜pv)

∆l

=−φlS∗

lhρkv(φv)

µ

(˜ph−˜pv)

∆l

.(162)

Again here Slh is the speciﬁc surface area or surface area per unit volume of the l-phase.

Since there is a diﬀerence in concentration in the h and v-phases the transfer of ﬂuid in

either direction will also result in the transfer of solute from one phase to another. This

transfer will be of the following form

fw∗

l→h=(−φlS∗

lhρkv(φv)

µ

(˜ph−˜pv)

∆l˜ch,if ˜ph≥˜pv

φlS∗

lhρkv(φv)

µ

(˜ph−˜pv)

∆l˜cv,if ˜ph<˜pv

(163)

D.1. Form of individual mass transfers

It is now possible to describe the source and sink terms in the macroscopic equations

arising from mass transfers in terms of the microscopic quantities of the system. Now

the total mass transfer from the l-phase to the h-phase can be written as

˜

fl→h=˜

fv→h+˜

fs→h+fw∗

l→h.(164)

˜

fv→hrepresents transfer of solute between the pores within the grains and the pores

between the grains due to diﬀusion. Now considering (127) and (153) it follows that

˜

fv→h=αvh(˜ch−˜cv)

=φlφvD∗

v

S∗

hl

∆l

(˜ch−˜cv)

=φlφvDv

S∗

hl

∆lτ(˜ch−˜cv)

=φlφ

4

3

vDv

S∗

hl

∆l

(˜ch−˜cv),(165)

where τ=φ−1

3

vhas been used. Next transfer of solute from the solid on the surface of

the grains, ˜

fs→his considered. Recalling (138) and (157) and that diﬀusion of solute

into the bulk of the ﬂuid is the rate limiting step it follows that

36

˜

fs→h=αsh(csat −˜ch)

=φlφsDh

S∗

hl

∆s

(csat −˜ch)

=φl(1 −φv)Dh

S∗

hl

∆s

(csat −˜ch).(166)

As outlined above the third transfer term, ˜

fw∗

l→h, arises due to solute being carried in

the ﬂuid that transfers between phases due to pressure diﬀerences between pores this is

given as above by

˜

fw∗

l→h=(−φlS∗

lhρkv(φv)

µ

(˜ph−˜pv)

∆l˜ch,if ˜ph≥˜pv

φlS∗

lhρkv(φv)

µ

(˜ph−˜pv)

∆l˜cv,if ˜ph<˜pv

(167)

The macroscopic transfer term from the s-phase to the v-phase can be arrived at

using (139) and (157) to get

˜

fs→v=αsv(csat −˜cv)

=φlφsDv

S∗

sv

∆s

(csat −˜ch).(168)

E. Dominance of advection over mechanical dispersion and diﬀusion

Considering the dimensional equations from the paper we can consider the relative

importance of the coﬀee transport processes in the intergranular pores by comparing

their magnitudes. This should give us an idea of the dominant transport mechanism

in the bed, although of course there may be narrow regions where other balances hold.

Firstly we compare advection and dispersion. The ratio of the magnitudes of the terms

is

|˜ch˜vh|

˜

Db

h∇˜ch

.(169)

Recall that

(˜

Db

h)ij = aTδij + (aL−aT)vivj

|vh|2!|vh|=aijkl

vivj

|vh|.(170)

We now use some characteristic scales. Note aL∼ll≤500 m. Thus take aijkl ∼aL,

˜ch∼C,|˜vh| ∼ vc,vi∼vc, and z∼L. Thus (169) becomes

Cvc

llvcC

L

=L

ll

∼102.(171)

Thus advection is approximately one hundred times larger than dispersion and so domi-

nates. More generally advection dominates over dispersion when Lllunless there are

very large concentration gradients in the bed.

37

The ratio of advection to diﬀusion is given by

|˜ch˜vh|

φ

4

3

hDh∇˜ch

.(172)

Adopting similar approximations as in the dispersion case we ﬁnd that

Cvc

φ

4

3

hDhC

L

=Lvc

φ

4

3

hDh

.(173)

Now from the experiments typical approximate values for these quantities are L∼0.05 m,

vc∼0.007 m s−1,φh∼0.2 and Dh∼2.2×10−9m2s−1. Thus

Lvc

φ

4

3

hDh

∼106.(174)

These estimates show that unless there are extremely large concentration gradients some-

where within the bed that advection dominates over diﬀusion.

F. Extraction Kinetics of Coﬀee Components

Figure 8: Extraction experiments suggest that a large number of compounds found in coﬀee extract with

similar kinetics [15].

G. Supplementary Coﬀee Extraction Experiments

The experimental results used in this paper are drawn from a much larger collection

of coﬀee extraction experiments. To complement and support these results and satisfy

the interested reader some other relevant experiments are included in this section.

38

G.1. Coﬀee Grinds Used

In the experiments presented here we make use of ﬁve diﬀerent coﬀee grinds ranging

from a ﬁne drip ﬁlter grind to a very coarse grind. The grind size distributions of these

grinds are shown in ﬁgure 9. The ﬁrst grind is a relatively ﬁne grind, called Jacobs

Kr¨onung (JK) standard drip ﬁlter coﬀee grind. The next grinds used are the Douwe

Egberts (DE) standard drip ﬁlter grind and the Douwe Egberts coarse drip ﬁlter grind.

Finally two further grinds were obtained by grinding Illy coﬀee beans using a Cimbali

burr grinder. One very coarse grind was obtained using the #20 setting on the grinder.

A second extremely coarse grind was obtained using the # 30 setting on the grinder.

The grind size distribution of the Cimbali # 30 grind was too coarse to be analysed by

the optical particle size analyzer used (Mastersizer 2000; Malvern Instruments Ltd, UK)

and so is not included in ﬁgure 9.

JK standard drip filter grind

Illy, Cimbali #20 grind

DE standard drip filter grind

DE coarse drip filter grind

0.1

1

10

100

1000

104

0

2

4

6

8

10

Particle size HΜmL

Volume fraction H%L

Figure 9: Grind size distributions of the coﬀee grinds used in experiments.

G.2. Coﬀee extraction kinetics during batch-wise brewing in a ﬁxed water volume

The extraction kinetics of the ﬁve coﬀee grinds were investigated by mixing 60 grams

of coﬀee grounds with a hot water volume, Vwater = 0.5 litres, and measuring the con-

centration cbrew of extracted species as a function of time. The temperature of the liquid

during extraction is 80–90 ◦C. The experimental procedure is identical to that outlined

in the paper but the results for the other grinds are shown here in ﬁgure 10. This

experiment clearly illustrates the key inﬂuence that the grind size distribution has on

extraction.

G.3. Coﬀee extraction proﬁles from a cylindrical brewing chamber under diﬀerent condi-

tions

In this paper we have presented coﬀee extraction proﬁles from a cylindrical brewing

chamber for one ﬁne grind (JK standard drip ﬁlter grind) and one coarse grind (Cimbali

39

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ì

ì

ì

ì

ì

ò

ò

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ò

ò

ò

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æ

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à

Illy, Cimbali #20 grind

ì

Illy, Cimbali #30 grind

ò

DE standard drip filter grind

ô

DE coarse drip filter grind

0

100

200

300

400

500

600

0

10

20

30

40

time HsL

cbrew Hkgm3L

Figure 10: Coﬀee solubles concentration proﬁles for diﬀerent coﬀees and grind size distributions during

batch extraction experiments. In these experiments 60 g of coﬀee with approximately 4% moisture was

mixed with 0.5 l of hot water in a French press type cylinder.

#20 grind). Here we present some ancillary experiments for JK standard drip ﬁlter

grind for a diﬀerent coﬀee bed mass and for a diﬀerent value of absolute pressure in

the coﬀee bed. The experimental apparatus is the same as that outlined in the paper.

To compare results for diﬀerent masses (and hence diﬀerent bed lengths) the extraction

is performed for coﬀee bed masses of 12.5 g and 60 g. These masses correspond to bed

depths of 1.12 cm and 4.05 cm respectively. The ﬂow rate to the coﬀee bed is 250 ml/min

in both cases. The pressure diﬀerence across the bed is measured in both cases. The

solubles concentrations are measured in the coﬀee pot and at the ﬁlter exit. The results

are shown in ﬁgure 11.

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JK, 60g, 2.3bar, 250mlmin

à

JK, 12.5g, 0.5bar, 250mlmin

0

200

400

600

800

1000

0

50

100

150

200

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cbrew HmggramL

(a)

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JK, 60g, 2.3bar, 250mlmin

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JK, 12.5g, 0.5bar, 250mlmin

0

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1000

0

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150

200

Mbrew HgramsL

cexit HmggramL

(b)

Figure 11: The coﬀee solubles concentration, measured in mg/gram, is plotted against mass of coﬀee

beverage Mbrew (grams) for JK drip ﬁlter grind with a ﬂow rate of 250 ml/min in (a) the coﬀee pot and

(b) the beverage at ﬁlter exit for diﬀerent coﬀee bed masses.

40

The inﬂuence of absolute pressure on extraction was also investigated by repeating

experiments in the coﬀee brewing cylinder at diﬀerent values of absolute pressure but

maintaining the same coﬀee bed mass and ﬂow rate. In the case of JK standard drip ﬁlter

grind the absolute pressure in the brewing cylinder is increased from 2.3 bar to 9 bar. The

resulting solubles concentrations proﬁles are plotted in ﬁgure 12. The results are seen

to be virtually identical which indicates that the extraction kinetics are substantially

independent of the absolute pressure (at least for the range of values considered) and

that the results are reproducible to within a few percent. It also suggests that a possibly

faster particle penetration with water at higher pressures does not substantially aﬀect

the observed extraction results.

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JK, 60g, 2.3bar, 250mlmin