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Modelling of coffee extraction during brewing using multiscale methods: An experimentally validated model

December 2015
Chemical Engineering Science 137:216-234

DOI:10.1016/j.ces.2015.06.003

Authors:

Kevin Moroney

University of Limerick

William Lee

University of Limerick

Stephen B.G. O'Brien

University of Limerick

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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 parameterised 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.

…

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).

…

Parameters for simulation of the batch extraction experiments.

…

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Modelling of coffee extraction during brewing using multiscale

methods: An experimentally validated model

K.M. Moroney

, W.T. Lee

, S.B.G. O'Brien

, F. Suijver

, J. Marra

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

Philips Research, Eindhoven, The Netherlands

HIGHLIGHTS

We conduct experiments investigating

extraction of solubles from coffee beds.

We develop a model to simulate

extraction of coffee with hot water

(90 1C).

We model the coffee bed as a satu-

rated, static, doubly porous medium.

Fast and slow release of coffee from

surfaces and kernels of grains are

modelled.

Parameterisedmodelisfoundtoﬁtthe

experimentally measured extraction

curves.

GRAPHICAL ABSTRACT

article info

Article history:

Received 23 January 2015

Received in revised form

21 May 2015

Accepted 1 June 2015

Available online 12 June 2015

Keywords:

Coffee brewing process

Coffee extraction experiments

Double porosity model

Static porous medium

Coffee extraction kinetics

Drip ﬁlter coffee

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 1C 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 parameterised 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.

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, rightdown to one-cup brewing

appliances for domestic use. For the purposes of brewing there 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

Contents lists available at ScienceDirect

journal homepage: www.elsevier.com/locate/ces

Chemical Engineering Science

http://dx.doi.org/10.1016/j.ces.2015.06.003

Corresponding author.

E-mail 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).

Chemical Engineering Science 137 (2015) 216–234

are described in refs. Petracco (2008) and Pictet et al. (1987).The

intimate contact of water with roasted coffee solids is the cardinal

requirement for producing a coffee beverage (Petracco, 2008). 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 brewing

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 (Petracco, 2008; Rao, 2010). According to Clarke et al.

(1987),“The extraction of roast and ground coffee is, in fact, a highly

complex operation, the scientiﬁc fundamentals of which are very

difﬁcult to unravel”.Thisisreﬂected in the absence of a satisfactory,

experimentally validated mathematical system of equations 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 (Rao, 2010). 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”(Aguilera and Stanley, 1999). Some work

has been done modelling the physics of certain brewing systems.

Large scale industrial extractors for the production of instant

coffee have been the subject of detailed investigations. 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 extractors (Sivetz and Foote, 1963; Spaninks,

1979 ). Some of these developments are summarised in Clarke et al.

(1987). 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 (Navarini et

al., 2009; Gianino, 2007). Fasano et al. have developed general

multiscale models for the extraction of coffee primarily focused on

the espresso coffee machine (Fasano and Talamucci, 2000; Fasano

and Farina, 2010; Fasano et al., 2000; Fasano, 2000). Voilley and

Simatos (1979) conducted a number of extraction experiments on

a well mixed system of coffee grounds and water and investigated

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 in the

University of Limerick. The topics investigated are included in the

study group report (Booth et al., 2012). Despite these develop-

ments, 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 experi-

mentally 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 Ele-

mentary Volumes. This allows the model to be parameterised from

experimentally measured microscopic quantities. Utilising multi-

scale 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 geome-

tries. 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 Krönung (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 Fig. 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 μm 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 μm. 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 1C can range from 28% for very ﬁne grinds

to 32% for 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 con-

tinuously stirring the grind through the water with a magnetic

stirrer for at least 5 h to ensure maximum extraction. Increasing

the extraction time from 5 h to 10 h did not change the extracted

K.M. Moroney et al. / Chemical Engineering Science 137 (2015) 216–234 217

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 g of coffee with a hot water

volume, V

water

¼0:5 l, and measuring the concentration c

brew

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

liquid during extraction is 80–90 1C. This was done by performing

batch-wise extraction 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

brew

was subsequently determined by measuring the 1Brix with a

pocket refractometer (PAL-3, Atago, Japan). It was found for drip

ﬁlter coffee that 1 1Brix corresponds with c

brew

¼8:25 g/l. 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 Fig. 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 experimentswere conducted with

both a cylindrical brewing chamber and a conical Melitta ﬁlter. We

will focus on the cylindrical brewing chamber here. In this setup,

coffee is placed in a cylindrical ﬂow-through cell and 1 l of water at

90 1C 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 ðc

exit

Þ, and the coffee beverage in

the pot ðc

brew

Þis measured throughout the extraction. The extrac-

tion proﬁles of the brewed and exiting coffee solubles concentra-

tion, 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 Fig. 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 experi-

ments 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 Fig. 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 cylind-

rical brewing chamber data in Fig. 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 period.

This is consistent with the ﬁndings in Voilley and Simatos (1979)

where it was noted that extraction yield reached 90% of its ﬁnal

value within 1 min. 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 Fig. 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 ﬁll the ﬁlter, but

does not 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 that this swelling would occur during

0.1 1 10 100 1000 104

Particle size m

Volume fraction

Fig. 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.

0 100 200 300 400 500 600

time s

cbrew kg m3

Fig. 2. Coffee solubles concentration proﬁles for JK standard drip ﬁlter grind ðÞ and

Cimbali #20 grind ð□Þduring 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.

K.M. Moroney et al. / Chemical Engineering Science 137 (2015) 216–234218

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 com-

pounds (Petracco, 2008), 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 impor-

tant coffee components have similar extraction kinetics (see Fig. 8,

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 Bear and Cheng (2010) 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 μm. 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

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 aver-

aging will be adopted here. A schematic of the volume averaging

process is shown in Fig. 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.

3.3. Coffee bed description

The coffee bed is represented by a porous medium domain

with volume V

. The domain can be split into an intergranular

pores domain

, with volume V

and a coffee grain domain

with volume V

is further split into an intragranular pore

domain

, with volume V

and a solid coffee domain

, with

volume V

. Clearly the equalities V

þV

¼V

and V

þV

¼V

hold.

The following volume fractions are now deﬁned as

¼V

;ϕ

¼V

;ϕ

¼V

;ϕ

¼V

;ð1Þ

0 200 400 600 800 1000

100

150

200

Mbrew grams

cbrew mg gram

0200 400 600 800 1000

100

150

200

Mbrew grams

cexit mg gram

0 200 400 600 800 1000

100

Mbrew grams

cbrew mg gram

0 200 400 600 800 1000

100

Mbrew grams

cexit mg gram

Fig. 3. The coffee solubles concentration, measured in mg/gram, is plotted against mass of coffee beverage M

brew

(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.

K.M. Moroney et al. / Chemical Engineering Science 137 (2015) 216–234 219

which leads to

þϕ

¼1;ϕ

þϕ

¼1:ð2Þ

The concentrations (mass per unit volume) of coffee in the

respective phases are c

and c

and v

denote the ﬂuid

velocity in the h-phase and v-phase respectively. The velocity of

the solid will be denoted by v

. Further notation will be introduced

as required. The formulation of the equations presented here will

follow (Bear and Cheng, 2010; Gray and Hassanizadeh, 1998). Since

conservation equations will be formed at three different scales the

variables at each scale will be denoted as macroscale ð~Þ, mesoscale

ð

Þand microscale ðÞ 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

∂c

∂t¼∇ðc

þj

Þ;ð3Þ

∂ρ

∂t¼∇ð

Þ:ð4Þ

3.4.2. v-phase

∂c

∂t¼∇ðc

þj

Þ;ð5Þ

∂ρ

∂t¼∇ð

Þ:ð6Þ

3.4.3. s-phase

∂c

∂t¼0:ð7Þ

The microscopic balance equations include the terms j

and j

which 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

3.5.1. v-phase

∂

∂tðϕ

Þ¼∇ðϕ

ðc

þj

ÞÞf

v-s

:ð8Þ

∂

∂tðϕ

Þ¼∇ðϕ

ðρ

þi

ÞÞ:ð9Þ

3.5.2. s-phase

∂

∂tðϕ

Þ¼∇ðϕ

Þf

s-v

:ð10Þ

The terms f

v-s

and f

s-v

are source/sink terms representing

transfer of coffee solubles across the vs-interface and vice-versa.

The term i

accounts 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

∂

∂tðϕ

Þ¼∇ðϕ

ð~

þ~

ÞÞ  ~

h-l

;ð11Þ

∂

∂tðϕ

Þ¼∇ðϕ

ð~

þ~

ÞÞ  ~

h-l

:ð12Þ

3.6.2. v-phase

∂

∂tðϕ

Þ¼∇ðϕ

ð~

þ~

ÞÞ  ~

v-h

~

v-s

;ð13Þ

∂

∂tðϕ

Þ¼∇ðϕ

ð~

þ~

ÞÞ  ~

l-h

:ð14Þ

3.6.3. s-phase

∂

∂tðϕ

Þ¼∇ðϕ

Þ~

s-h

~

s-v

:ð15Þ

is transfer of coffee solubles from the

-phase to the

-phase

across the αβ interface. Similarly ~

is transfer of liquid from the

-phase to the

-phase across the αβ interface.

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 proce-

dure based on (Bear and Cheng, 2010; Gray and Hassanizadeh,

1998) 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.

Fig. 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).

K.M. Moroney et al. / Chemical Engineering Science 137 (2015) 216–234220

This is consistent with Petracco (2008) where it is noted that the

material extracted from coffee has little inﬂuence on liquid

density. Thus ~

¼~

¼ρ: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 ~

¼

. Thus the ﬁve macroscopic

equations reduce to the following:

∂

∂tðϕ

Þ¼∇ðϕ

ð~

þ~

ÞÞ þ ~

v-h

þ~

s-h

;ð16Þ

ρ∂ϕ

∂t¼

ρ∇ðϕ

Þþ~

l-h

;ð17Þ

∂

∂tðϕ

Þ¼~

v-h

þ~

s-v

;ð18Þ

ρ∂

∂tðϕ

Þ¼~

l-h

;ð19Þ

∂

∂tðϕ

Þ¼~

s-h

~

s-v

:ð20Þ

It now remains to introduce expressions to model the ﬂuid

velocity ~

, the total macroscopic ﬂux ~

, the ﬂuid mass transfer

term ~

l-h

and the coffee mass transfer terms ~

v-h

s-v

and ~

s-h

in terms of the system variables. The main transfers occurring in

the coffee bed are shown in Fig. 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

¼ϕ

¼

μð∇~

þρgÞ;~

¼~

ðϕ

Þ;ð21Þ

where ~

is the macroscopic pressure gradient in the h-phase, ~

the permeability and

is the viscosity of water.

4.2. Total macroscopic ﬂux

The total macroscopic ﬂux, ~

is made up of the macroscopic

average of molecular diffusion j

and the dispersive ﬂux j

¼j



þc

○

¼j

þj

:ð22Þ

For an isotropic porous medium, j

is often modelled by

¼

τ∇~

;ð23Þ

where

is the tortuosity deﬁned by

τ¼L

L¼actual path length

macroscopic path length:ð24Þ

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 Pisani (2011). The expression used

here is τ¼ϕ

1=3

which is adopted from Millington (1959). Thus

we have

¼

τ∇~

¼ϕ

1=3

D∇~

:ð25Þ

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 (Bear and Cheng, 2010). Thus 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

¼c

○

¼~

∇~

;ð26Þ

where ~

is a rank 2 tensor called the dispersion tensor. ~

is both

positive deﬁnite and symmetric. For an isotropic porous medium

the following expression is often used:

¼a

þða

a

Þv



v:ð27Þ

The coefﬁcients a

and a

here are the longitudinal and transverse

dispersivities of the porous medium. v

¼v

is the average

velocity in the i-th direction and v¼jvjwhere vis the average

velocity vector in this instance.

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 Oliveira et al.

(2011). 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

Huang (2012),Sovov (2005),Goto et al. (1996),Machmudah et al.

(2012).In(Huang, 2012) 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 ~

v-h

is transferred from the intragranular pores

in the grains to the intergranular pores and is similar to extraction

from intact cells. The term ~

s-h

is direct extraction from the solid

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

Fig. 5. Mass transfers occurring in the coffee bed.

K.M. Moroney et al. / Chemical Engineering Science 137 (2015) 216–234 221

grain matrix surface into the intergranular pores and is similar to

extraction from the broken cells. The term ~

s-v

models dissolution

of coffee solubles from the cell walls into the intragranular pores.

The transfer ~

v-h

across the interphase boundary is assumed to

occur by diffusion according to Fick's ﬁrst law. Thus

v-h

¼α

ð~

~

Þ:ð28Þ

where α

is the mass transfer coefﬁcient. The form of α

can be

found from the volume averaging procedure as

¼ð1ϕ

Þϕ

4=3

;ð29Þ

where D

is the diffusion coefﬁcient of coffee in water, S

is the

speciﬁc surface area of the l-phase and

is some length scale

characterising the distance over which diffusion occurs. The

transfers ~

s-v

and ~

v-h

are 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 c

sat

is the concentration in the liquid phase that

would be in equilibrium with the concentration inside the solid ~

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

s-h

¼ð1ϕ

Þð1ϕ

ÞD

ðc

sat

~

Þ;ð30Þ

where

is some length scale characterising the distance over

which diffusion occurs. Similarly

s-v

¼ð1ϕ

Þð1ϕ

ÞD

ðc

sat

~

Þ;ð31Þ

where S

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

l-h

¼ð1ϕ

ÞS

ρk

ðϕ

ð~

~

;ð32Þ

where k

ðϕ

Þis the coffee grain permeability and ~

and ~

are the

water pressures in the h-phase and the v-phase. Since there is a

difference in concentration in the hand 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

l-h

ð1ϕ

ÞS

ð~

~

if ~

ð1ϕ

ÞS

ð~

~

if ~

ð33Þ

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

is 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. ~

. Thus (33) simpliﬁes to

h-l

¼ð1ϕ

ÞS

ρk

ðϕ

ð~

~

:ð34Þ

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

¼6

;ð35Þ

where S

is the surface to volume ratio of the distribution which

can be found from the data. The assumption

is constant means

that the Sauter mean diameter does not change. Thus where

required the speciﬁc surface area of the l-phase is given by

¼6

:ð36Þ

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 k

sv1

. However when dealing

withextractionfromthegrainkernel,weshouldignoretheparticles

which are just broken cell fragments and do not have a kernel of

intact cells. In this case we introduce a second Sauter mean diameter

sv2

which is representative of the speciﬁc surface area of particles

above a certain diameter, chosen here to be 50 μm. 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

¼6

m:ð37Þ

However we do not 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 k

The permeability can be estimated using the Kozeny–Carman

equation for spheres (Holdich, 2002):

¼ϕ

κð1ϕ

2:ð38Þ

Here again S

is the speciﬁc surface area, while

is a factor which

accounts for the shape and tortuosity. Utilising the derived form

for S

gives

¼k

sv1

36κð1ϕ

¼k

sv1

36κð1ϕ

:ð39Þ

The shape factor

is usually taken to be in the range 2–6.

K.M. Moroney et al. / Chemical Engineering Science 137 (2015) 216–234222

Experiments performed measuring the pressure drop in an airﬂow

through compacted coffee beds estimate it at κ¼3:1.

4.6.4. Permeability k

Similarly the permeability of the grain is estimated using the

Kozeny–Carman equations for the spherical cells so that

¼m

180ð1ϕ

:ð40Þ

In the absence of experimental data we have chosen κ¼5 which is

often adopted (Aubertin and Chapuis, 2003).

4.6.5. Average diffusion distances

Expressions are required for

and

. 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 Δ

¼l

. 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 Δ

¼l

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

to change or allow the solid coffee concentration ~

change, or both. The decision is made here to allow

to change

as coffee is extracted and consider the solid concentration ~

ﬁxed.

Thus

is allowed to increase as coffee is dissolved until it reaches

a point when

consists 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

be the volume fraction of coffee in the grains.

We divide this into coffee in the surface of the grains and ﬁnes ϕ

s;s

and coffee in the grain kernels ϕ

s;b

, so that ϕ

¼ϕ

s;s

þϕ

s;b

. The

initial volume fractions of coffee in dry grains are

,ϕ

s;s0

and

s;b0

. Assuming that the soluble coffee is uniformly distributed

within the grains the initial volume fraction everywhere is

Thus at a given time the volume fractions of soluble coffee on the

surface and in the grain kernels are given by

s;s0

s;s

ðx;tÞ¼ϕ

ðx;tÞ;ð41Þ

s;b0

s;b

ðx;tÞ¼ϕ

ðx;tÞ;ð42Þ

where xis the position within the coffee bed and

and

are the

fractions of the original amount of coffee left on the grain surfaces

and in the grain kernels respectively. We can now substitute

¼ϕ

s;i

þϕ

s;s

þϕ

s;b

.ϕ

s;i

represents the volume fraction of insolu-

ble solid in the grains. Using the expressions we have developed so

far this leads to two further partial differential equations for

and

∂ψ

∂t¼

12D

sv1

sat

~



;ð43Þ

∂ψ

∂t¼

12D

sat

~



;ð44Þ

where r

¼1=ϕ

s;s0

and r

¼1=ϕ

s;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 the experiment following the ﬁlling

stage.

∂~

∂t¼k

sv1

36κμð1ϕ

∇ð

ð∇~

þρgÞÞ

þϕ

∇

þϕ

∇

ð1ϕ

Þϕ

4=3

sv2

ð~

~

þð1ϕ

Þ12D

sv1

mðc

sat

~

Þψ

6ð1ϕ

Þm

180μk

sv2

ð1ϕ

ð~

~

Þ~

;ð45Þ

0¼ k

sv1

36κμð1ϕ

∇ð∇~

þρgÞ

þ6ð1ϕ

Þm

180μk

sv2

ð1ϕ

ð~

~

Þ;ð46Þ

∂

∂tðϕ

Þ¼ϕ

4=3

sv2

ð~

~

þ12ϕ

ðc

sat

~

Þψ

þ6ð1ϕ

Þm

180μk

sv2

ð1ϕ

ð~

~

Þ~

;ð47Þ

∂ϕ

∂t¼6m

180μk

sv2

ð1ϕ

ð~

~

Þ;ð48Þ

∂ϕ

∂t¼1

∂ψ

∂t1

∂ψ

∂t;ð49Þ

∂ψ

∂t¼

12D

sv1

sat

~



;ð50Þ

∂ψ

∂t¼

12D

sat

~



;ð51Þ

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 particularly 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.

K.M. Moroney et al. / Chemical Engineering Science 137 (2015) 216–234 223

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 concentra-

tion 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 instanta-

neously corrected. Thus, as all coffee in kernel is assumed

dissolved initially, no pressure difference between phases will

occur. Note the value of intragranular porosity ϕ

¼0:56 is

adopted here for dry coffee in air. Allowing for these assumptions

the following system of equations are solved numerically in

MATLAB

dt ¼

ð1ϕ

Þϕ

sv2

ð~

~

þβ

ð1ϕ

Þ12D

sv1

mðc

sat

~

Þψ

;ð52Þ

dtðϕ

Þ¼α

4=3

sv2

ð~

~

Þ;ð53Þ

dϕ

dt ¼1

∂ψ

∂t;ð54Þ

dψ

dt ¼

12D

sv1

sat

~



;ð55Þ

with initial conditions

ð0Þ¼0;~

ð0Þ¼c

;ϕ

ð0Þ¼ϕ

;ψ

ð0Þ¼1:ð56Þ

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 interstitial 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 ϕ

s;s0

and ϕ

s;b0

can, 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 c

sat

is estimated from the highest observed concentra-

tion across the four experiments. The diffusion coefﬁcient is for

caffeine in water at 80 1C(Jaganyi and Madlala, 2000). This would

be slightly higher at 90 1C. However this is only an estimate of the

effective diffusion coefﬁcient of coffee in water. The ﬁtting para-

meters are used to correct any errors in these parameters. The

comparison between the numerical solution and the experimental

results is shown in Fig. 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 z¼0 and the top (ﬁlter entrance) at z¼L. 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 ~

. We use the condition

that the water entering at the top z¼Lhas zero concentration. The

pressure boundary conditions are ~

¼Δpat z¼Land ~

¼0at

z¼0. Δpis the pressure difference across 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

adjusts to the pressure while

stays constant. We then choose

by 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 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 ~

as a function of z.

Based on the initial exiting concentrations we assume for the ﬁne

grind (JK) that initially ~

is 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

Table 1

Parameters for simulation of the batch extraction experiments.

Parameter JK drip ﬁlter Cimbali #20

0.6444 0.6120

0.8272 0.8272

sv1

27:35 μm38:77 μm

sv2

322:49 μm 569:45 μm

282 μm 463 μm

¼D

2:210

9

1

2:210

9

1

ρ965:3kgm

3

965:3kgm

3

μ0:315 10

3

Pas 0:315 10

3

Pas

m30 μm30μm

sat

212:4kgm

3

212:4kgm

3

1400 kg m

3

1400 kg m

3

κ3.1 3.1

0.143435 0.122

s;s0

0.059 0.07

s;b0

0.084435 0.052

0.1833 0.0881

0.0447 0.0086

16.94 14.28

183:43 kg m

3

118:95 kg m

3

K.M. Moroney et al. / Chemical Engineering Science 137 (2015) 216–234224

make the simplest assumption. As with the batch experiments the

assumptions mean that ~

¼~

. Based on these assumptions the

reduced set of equations to model extraction in the cylindrical

brewing chamber is given by

∂~

∂t¼k

sv1

36κμð1ϕ

∂

∂z

∂~

∂zþρg



α

ð1ϕ

Þϕ

4=3

sv2

ð~

~

þβ

ð1ϕ

Þ12D

sv1

mðc

sat

~

Þψ

;ð57Þ

∂

∂z

¼0;ð58Þ

∂

∂tðϕ

Þ¼α

4=3

sv2

ð~

~

Þ;ð59Þ

∂ϕ

∂t¼1

∂ψ

∂t;ð60Þ

∂ψ

∂t¼

12D

sv1

sat

~



;ð61Þ

for

t40;0ozoL;ð62Þ

with initial conditions

ðz;0Þ¼c

ðzÞ;~

ðz;0Þ¼c

;ð63Þ

ðz;0Þ¼ϕ

;ψ

ðz;0Þ¼ψ

;ð64Þ

and boundary conditions

ðL;tÞ¼0;~

ð0;tÞ¼0;~

ðL;tÞ¼Δp:ð65Þ

The initial concentration proﬁle in the JK drip ﬁlter grind is

given by c

ðzÞ¼c

sat

. In the Cimbali #20 grind the initial concen-

tration is given by c

ðzÞ¼ðc

max

=LÞðLzÞ. 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 condi-

tions 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 numer-

ical solution and the experimental results is shown in Fig. 7.

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 parameterised using experimental data. Numer-

ical simulations are performed 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 suspension

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

0 100 200 300 400 500 600

t (s)

Ch (kg/m3)

0100 200 300 400 500 600

t (s)

Ch (kg/m3)

Fig. 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.

Table 2

Parameters for simulation of the cylindrical brewing chamber extraction

experiments.

Parameter JK drip ﬁlter Cimbali #20

0.6231 0.6218

0.2 0.25

max

–82:63 kg m

3

1400 kg m

3

1400 kg m

3

s;s0

0.11 0.07

s;b0

0.033435 0.052

0.1833 0.0881

0.0447 0.0086

9.09 14.28

78:88 kg m

3

78:88 kg m

3

Δp230000 Pa 65000 Pa

L0.0405 m 0.0526 m

K.M. Moroney et al. / Chemical Engineering Science 137 (2015) 216–234 225

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 Mathe-

matics Applications Consortium for Science and Industry (www.

macsi.ul.ie ) funded by the Science Foundation Ireland Investigator

Award 12/IA/1683.

Appendix A. General upscaling procedure

The following procedure is mainly adapted from Gray and

Hassanizadeh (1998) and Bear and Cheng (2010). 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

-species per unit volume of the

-phase be

∂c

∂t¼∇ðc

þj

ÞþG

;ð66Þ

where c

is the concentration of the

-species, v

is the ﬂuid

velocity in the

-phase, j

is the diffusive ﬂux of the

-species in

the

-phase, and G

is a source or sink of the

-species in the

phase.

Let

represent all other phases in the porous medium. The

subscript will be dropped where convenient. Eq. (66) is the

microscopic balance. The corresponding macroscopic balance

equation is

∂

∂tðϕ

Þ¼∇ðϕ

þϕ

Þþϕ

;ð67Þ

where ϕ

is the volume fraction of the

-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

d⪡l⪡L:ð68Þ

The following notation is used



The volume of an REV is δV.



The portion of the volume of an REV occupied by the

-phase is

δV



The union of the interfacial regions within the REV between the

-phase and a

-phase is denoted S

αβ



The unit vector normal to this surface oriented outward from

the

-phase is n

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

∂F

∂tdV ¼∂

∂tZ

FdVX

αβ

w

dS;ð69Þ

where Fis some scalar ﬁeld property of the microscale and Fj

just

indicates that the microscale property Fin the

-phase is being

integrated over the αβ-interface. w

is the velocity of S

αβ

. The

summation P

just denotes a summation over all phases

except the

-phase.

The divergence averaging theorem is given by

∇BdV ¼∇Z

BdV þX

αβ

Bj

dS;ð70Þ

where Bis some vector ﬁeld property of the microscale and Bj

just indicates that Bis being integrated over the αβ-interface. The

summation P

just denotes a summation over all phases

except the

-phase.

Next some phase averages are deﬁned

The Intrinsic Phase Average of a quantity F over the

-phase is

deﬁned by

Fhi

¼F

¼1

δV

FdV:ð71Þ

The Phase Average of a quantity F over the

-phase is deﬁned by

¼F¼1

δVZ

FdV¼ϕ

:ð72Þ

The Mass Weighted Average of a quantity F over the

-phase is

deﬁned by



¼F

¼1



δV

ρFdV¼ρF



:ð73Þ

0 0.2 0.4 0.6 0.8 1

100

150

200

250

Mbrew (kg)

Cexit (kg/m3)

00.2 0.4 0.6 0.8 1

100

Mbrew (kg)

Cexit (kg/m3)

Fig. 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.

K.M. Moroney et al. / Chemical Engineering Science 137 (2015) 216–234226

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

product of the form cv

in terms of the individual averages c

and v

. To do this consider the following:



Let v

○

¼vv

be the deviation of the true velocity from the

mean velocity.



Let c

○

¼cc

be the deviation of the true concentration from

the mean concentration.

Then

○

¼vv



¼v

v

¼0;ð74Þ

○

¼cc



¼c

c

¼0:ð75Þ

Thus

cvhi

¼ðchi

þc

○

Þð vhi

þv

○

¼c

þc

○

þc

○

þc

○

¼c

þc

○

:ð76Þ

The term c

○

is called the dispersive ﬂux.

Now the averaging process can be performed. Firstly (66)

integrated over δV

yields

∂c

∂tdV ¼

∇ðcvþjÞdV þZ

GdV:ð77Þ

Now applying theorems (69) and (70) yields

∂

∂tZ

cdV¼∇Z

ðcvþjÞdV



X

αβ

ðcðvw

ÞþjÞj

þZ

GdV:ð78Þ

This can be written in terms of averaged quantities by dividing

across by δV¼

to give

∂

∂tðϕ

Þ¼∇ϕ

ðcv

þj



1

δVX

αβ

ðcðvw

ÞþjÞj

þϕ

Ghi

:ð79Þ

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

that

∂

∂tðϕ

Þ¼∇ϕ

ðc

þc

○

þj



1

δVX

αβ

ðcðvw

ÞþjÞj

þϕ

:ð80Þ

Comparing this to the macroscopic point balance Eq. (67) gives the

following relations between the macroscopic and microscopic

quantities

¼c

;ð81Þ

¼vhi

;ð82Þ

¼c

○

þj



;ð83Þ

¼ϕ

þ1

δVX

αβ

ðcðvw

ÞþjÞj

dS:ð84Þ

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.

Appendix B. Upscaling from microscale to macroscale

equations

The upscaling process basically involves choosing an REV

(Representative Elementary 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 out-

line 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.

: Volume of the smaller REV.

: Volume of solid in the smaller REV.

: Volume of void space in smaller REV.

: Volume of larger REV.

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

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

Also due to the properties of an REV

¼V

;ϕ

¼V

;ð85Þ

¼V

;ϕ

¼V

:ð86Þ

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

B.1.1.1. h-phase.

∂c

∂t¼∇ðc

þj

Þ;ð87Þ

∂ρ

∂t¼∇ð

Þ:ð88Þ

B.1.1.2. v-phase.

∂c

∂t¼∇ðc

þj

Þ;ð89Þ

∂ρ

∂t¼∇ð

Þ:ð90Þ

B.1.1.3. s-phase.

∂c

∂t¼0:ð91Þ

K.M. Moroney et al. / Chemical Engineering Science 137 (2015) 216–234 227

The microscopic balance equations include the terms j

and j

which

represent molecular diffusion of coffee solubles in the respective

phases. Molecular diffusion in the solid phase is assumed negligible.

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

B.1.2.1. v-phase.

∂

∂tðϕ

Þ¼∇ðϕ

ðc

þj

ÞÞf

v-s

:ð92Þ

∂

∂tðϕ

Þ¼∇ðϕ

ðρ

þi

ÞÞ:ð93Þ

B.1.2.2. s-phase.

∂

∂tðϕ

Þ¼∇ðϕ

Þf

s-v

:ð94Þ

The terms f

v-s

and f

s-v

are source/sink terms representing

transfer of coffee solubles across the vs-interface and vice-versa.

The term i

accounts 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

B.1.3.1. h-phase.

∂

∂tðϕ

Þ¼∇ðϕ

ð~

þ~

ÞÞ  ~

h-l

;ð95Þ

∂

∂tðϕ

Þ¼∇ðϕ

ð~

þ~

ÞÞ  ~

h-l

:ð96Þ

B.1.3.2. v-phase.

∂

∂tðϕ

Þ¼∇ðϕ

ð~

þ~

ÞÞ  ~

v-h

~

v-s

;ð97Þ

∂

∂tðϕ

Þ¼∇ðϕ

ð~

þ~

ÞÞ  ~

l-h

:ð98Þ

B.1.3.3. s-phase.

∂

∂tðϕ

Þ¼∇ðϕ

Þ~

s-h

~

s-v

:ð99Þ

is transfer of coffee solubles from

-phase to

-phase

across αβ interface. Similarly ~

is transfer of liquid from

phase to

-phase across αβ interface.

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 Eq. (89)

over the smaller REV is

∂

∂tðϕ

Þ¼∇ϕ

ðc

þc

○

þj



1

c

ðv

w

Þþj



dS:ð100Þ

Now the surface S

is a material surface so v

w

¼0.

Comparing this averaged form with the mesoscale form in (92) it

can be seen that

¼c

;ð101Þ

¼v

;ð102Þ

¼c

○

þj



;ð103Þ

v-s

¼1

j

dS:ð104Þ

B.2.2. Conservation of liquid in v-phase

The averaged form of the microscopic point balance Eq. (90)

over the smaller REV is

∂

∂tðϕ



Þ¼∇ϕ



þρ

○



1

ðρ

ðv

w

ÞÞ dS:ð105Þ

Now the surface S

is a material surface so v

w

¼0. Compar-

ing this averaged form with the mesoscale form in (93) it can be

seen that

¼ρ



;ð106Þ

¼v

;ð107Þ

¼ρ

○

:ð108Þ

Here in fact

will be taken to be constant so i

¼0in this case.

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

The averaged form of the microscopic point balance Eq. (91)

over the smaller REV is

∂

∂tðϕ

Þ¼ 1

ðc

ÞdS:ð109Þ

Comparing this averaged form with the mesoscale form in (94) it

can be seen that

¼c

;ð110Þ

¼0;ð111Þ

s-v

¼1

ðc

ÞdS:ð112Þ

B.3. Upscaling to macroscale

B.3.1. Conservation of coffee in h-phase

The averaged form of the microscopic point balance Eq. (87)

over the larger REV is

∂

∂tðϕ



Þ¼∇ϕ



þc

○

þj





1

ðc

ðv

w

Þþj

ÞdS:ð113Þ

Comparing this averaged form with the macroscale form in (95) it

can be seen that

¼c



;ð114Þ

¼v



;ð115Þ

¼c

○

þj



;ð116Þ

K.M. Moroney et al. / Chemical Engineering Science 137 (2015) 216–234228

h-l

¼1

ðc

ðv

w

Þþj

ÞdS:ð117Þ

Here S

is the effective surface between the h-phase and

the l-phase and includes boundaries 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 Eq. (88)

over the larger REV is

∂

∂tðϕ



Þ¼∇ϕ



þρ

○



1

ðρ

ðv

w

ÞÞ dS:ð118Þ

Comparing this averaged form with the macroscale form in (96) it

can be seen that

¼ρ



;ð119Þ

¼v



;ð120Þ

¼ρ

○

;ð121Þ

h-l

¼1

ðρ

ðv

w

ÞÞ dS:ð122Þ

B.3.3. Conservation of coffee in v-phase

The averaged form of the microscopic point balance Eq. (92)

over the larger REV is

∂

∂tðϕ



Þ¼∇ϕ



þc

○



þj



ϕ

ðc

ðv

w

Þþj

ÞdS

ϕ

v-s



:ð123Þ

Comparing this averaged form with the macroscale form in (97) it

can be seen that

¼c



¼c



;ð124Þ

¼v



¼v



;ð125Þ

¼j



þc

○



¼j



þc

○



þc

○



;ð126Þ

v-h

¼ϕ

ð c

ðv

w

Þþ j



ÞdS;ð127Þ

v-s

¼ϕ

v-s



¼1

j



:ð128Þ

It will be later assumed that ~

¼0and ~

¼0but of course it will

still be possible to have v

a0and j

a0.

B.3.4. Conservation of liquid in v-phase

The averaged form of the microscopic point balance Eq. (93)

over the larger REV is

∂

∂tðϕ



Þ¼∇ϕ



þρ

○



þi



ϕ

ðρ

ðv

w

Þþi

ÞdS:ð129Þ

Comparing this averaged form with the macroscale form in (98) it

can be seen that

¼ρ



¼ρ



;ð130Þ

¼v



¼v



;ð131Þ

¼i



þρ

○



¼i



þρ

○



þρ

○



;ð132Þ

l-h

¼ϕ

ð ρ



ðv

w

Þþ i



ÞdS:ð133Þ

B.3.5. Conservation of coffee in s-phase

The averaged form of the microscopic point balance Eq. (94)

over the larger REV is

∂

∂tðϕ



Þ¼∇ϕ

ðj



ϕ

ðc

ðw

Þþj

ÞdS

ϕ

s-v



:ð134Þ

Comparing this averaged form with the macroscale form in (99) it

can be seen that

¼c



¼c



;ð135Þ

¼0;ð136Þ

¼j



¼j



¼0;ð137Þ

s-h

¼ϕ

ð c

ðw

ÞÞ dS;ð138Þ

s-v

¼ϕ

s-v



¼1

ðc

ÞdS



:ð139Þ

Appendix C. Macroscopic diffusion and dispersion ﬂuxes

The total macroscopic ﬂux, ~

is made up of the macroscopic

average of molecular diffusion and the dispersive ﬂux:

¼j



þc

○

:ð140Þ

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

¼D∇c

;ð141Þ

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

¼D~

ðϕÞ∇~

¼~

ðϕÞ∇~

:ð142Þ

where the ~here on ~

means this is a tensor of rank two which

represents the tortuosity of the porous medium. For an isotropic

K.M. Moroney et al. / Chemical Engineering Science 137 (2015) 216–234 229

porous medium this can be adjusted to

¼

τ∇~

;ð143Þ

where

is the tortuosity deﬁned by

τ¼L

L¼actual path length

macroscopic path length:ð144Þ

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

estimated are used in the literature. Some of these include

τ¼ϕ

1=3

;τ¼ϕ

1=2

;τ¼1

1αð1ϕÞ:ð145Þ

In the ﬁnal expression α¼ðrσÞ=Vis a shape factor with rbeing the

object radius,

being the object cross sectional area and V being

the object volume. Thus for spheres for example α¼0:75. In this

case the ﬁrst expression from Millington (1959) will be adopted.

The other expressions and tortuosity in general are discussed in

Pisani (2011). Thus macroscopic diffusion is approximated by

¼

τ∇~

¼ϕ

1=3

D∇~

:ð146Þ

Dispersion occurs due to variations in the microscopic velocity

of the phase with respect to the averaged velocity, and molecular

diffusion (Bear and Cheng, 2010). 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

¼c

○

¼~

∇~

;ð147Þ

where ~

is a rank 2 tensor called the dispersion tensor. ~

is both

positive deﬁnite and symmetric. One commonly used expression is

¼a

ijkl

v;ð148Þ

where a

ijkl

is a fourth order tensor and v

¼v

is the average

velocity in the i-th direction and v¼jvjwhere vis the average

velocity vector in this instance. For an isotropic porous medium

this expression reduces to

¼a

þða

a

Þv



v:ð149Þ

The coefﬁcients a

and a

here are the longitudinal and transverse

dispersivities of the porous medium. For a phase that completely

ﬁlls a pore space, a

is a length that should be of the same order of

the pore size.

is the Kronecker delta. Also it is required that

Z0a

Z0:ð150Þ

Laboratory experiments have found that a

is 8–24 times smaller

than a

(Bear and Cheng, 2010).

Appendix D. Method of estimating mass transfer terms

It is necessary to estimate the mass transfer terms f

which

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 Aguilera and Stanley (1999). More general and

technical developments are found in Bear and Cheng (2010) and

Cussler (1997). The transfer term f

may 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 f

¼f

. 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

Appendix A and Appendix B the strength of a source of a

considered species in the

-phase is given by

¼1

αβ

ðc

ðv

w

αβ

Þþj

ÞdS;ð151Þ

where U

is the volume of the REV, S

αβ

is the boundary between

the

-phase and all other phases, n

is the unit outward normal

vector on this surface and w

αβ

is the velocity of points on the

interphase boundary. If S

αβ

is a material surface as will generally

be the case here than v

w

αβ

¼0and

¼1

αβ

j

dS:ð152Þ

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 consid-

ered. 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, f

, of the mass of the considered species

from an

-phase to an adjacent

-phase across an interface S

αβ

often assumed to be proportional to the difference in concentra-

tion between the phases. Thus

¼α

αβ

ðc

c

Þ:ð153Þ

To estimate the mass transfer coefﬁcient α

αβ

the following form is

used:

n

¼

ðc

c

:ð154Þ

Here D

is the coefﬁcient of diffusion of the considered species in

the

-phase and Δ

is the length characterising the mean size of

the phase or the length over which diffusion occurs. For example

one possible deﬁnition of Δ

is Δ

¼U

αβ

, the volume to

surface ratio of the

-phase within the REV. Then

¼1

αβ

j

¼1

αβ

ðc

c

¼

αβ

ðc

c



¼

αβ

ðc

c

Þ;ð155Þ

where S

αβ

¼S

αβ

is the speciﬁc surface area of the

-phase or

the surface area per unit volume. Thus it can be seen that the mass

transfer coefﬁcient is given by

αβ

¼

αβ

:ð156Þ

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 c

sat

K.M. Moroney et al. / Chemical Engineering Science 137 (2015) 216–234230

sat

is the concentration in the liquid phase that would be in

equilibrium with the concentration inside the solid c

. 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

s-f

¼α

ðc

sat

c

Þ:ð157Þ

Proceeding as above it can be shown that the mass transfer

coefﬁcient from a solid phase sto a ﬂuid phase fis given by

¼

:ð158Þ

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 proce-

dure this term has the form

l-h

¼ϕ

ρ



ðv

w

Þþ i





dS:ð159Þ

Now it is assumed that i



¼0and Darcy's Law gives

w

¼

ðϕ

μ∇pð160Þ

On S

this means that

w

¼

ðϕ

ðp

p

:ð161Þ

Thus

l-h

¼ϕ

ρk

ðϕ

ð~

~



¼

ρk

ðϕ

ð~

~

¼ϕ

ρk

ðϕ

ð~

~

:ð162Þ

Again here S

is the speciﬁc surface area or surface area per unit

volume of the l-phase. Since there is a difference in concentration

in the hand 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:

l-h

ϕ

ð~

~

if ~

ð~

~

if ~

ð163Þ

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

l-h

¼~

v-h

þ~

s-h

þf

l-h

:ð164Þ

v-h

represents 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

v-h

¼α

ð~

~

¼ϕ

ð~

~

¼ϕ

τð~

~

¼ϕ

ð~

~

Þ;ð165Þ

where τ¼ϕ

1=3

has been used. Next transfer of solute from the

solid on the surface of the grains, ~

s-h

is considered. Recalling

(138) and (157) and that diffusion of solute into the bulk of the

ﬂuid is the rate limiting step it follows that

s-h

¼α

ðc

sat

~

¼ϕ

ðc

sat

~

¼ϕ

ð1ϕ

ÞD

ðc

sat

~

Þ:ð166Þ

As outlined above the third transfer term, ~

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

l-h

ϕ

ð~

~

if ~

ð~

~

if ~

ð167Þ

The macroscopic transfer term from the s-phase to the v-phase

can be arrived at using (139) and (157) to get

s-v

¼α

ðc

sat

~

¼ϕ

ðc

sat

~

Þ:ð168Þ

Appendix 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



∇~



:ð169Þ

Recall that

ð~

¼a

þða

a

Þv



¼a

ijkl



:ð170Þ

We now use some characteristic scales. Note a

l

r500 μm.

Thus take a

ijkl

a

C,~

v

v

, and zL. Thus (169)

becomes

¼L

10

:ð171Þ

Thus advection is approximately one hundred times larger than

dispersion and so dominates. More generally advection dominates

over dispersion when L⪢l

unless there are very large concentra-

tion gradients in the bed.

The ratio of advection to diffusion is given by



4=3

∇~



:ð172Þ

Adopting similar approximations as in the dispersion case we ﬁnd

K.M. Moroney et al. / Chemical Engineering Science 137 (2015) 216–234 231

that

4=3

¼Lv

4=3

:ð173Þ

Now from the experiments typical approximate values for these

quantities are L0:05 m, v

0:007 ms

1

,ϕ

0:2 and

2:2e9m

1

. Thus

4=3

10

:ð174Þ

These estimates show that unless there are extremely large

concentration gradients somewhere within the bed that advection

dominates over diffusion.

Appendix F. Extraction kinetics of coffee components

Appendix G. Supplementary coffee extraction experiments

The experimental results used in this paper are drawn from a

much larger collection of coffee extraction experiments. To com-

plement and support these results and satisfy the interested

reader some other relevant experiments are included in this

section.

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 Fig. 9. The ﬁrst grind is a relatively ﬁne grind, called

Jacobs Krönung (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

Fig. 9.

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

water volume

The extraction kinetics of the ﬁve coffee grinds were investi-

gated by mixing 60 g of coffee grounds with a hot water volume,

water

¼0:5 l, and measuring the concentration c

brew

of extracted

species as a function of time. The temperature of the liquid during

extraction is 80–90 1C. The experimental procedure is identical to

that outlined in the paper but the results for the other grinds are

shown here in Fig. 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 conditions

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 #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 respec-

tively. 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 Fig. 11.

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 concentra-

tions proﬁles are plotted in Fig. 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

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

Particle size m

Volume fraction

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

JK standard drip filter grind

Illy, Cimbali 20 grind

Illy, Cimbali 30 grind

DE standard drip filter grind

DE coarse drip filter grind

0 100 200 300 400 500 600

time s

cbrew kg m3

Fig. 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.

Fig. 8. Extraction experiments suggest that a large number of compounds found in

coffee extract with similar kinetics (Booth et al., 2012).

K.M. Moroney et al. / Chemical Engineering Science 137 (2015) 216–234232

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.

References

Aguilera, J., Stanley, D., 1999. Microstructural Principles of Food Processing and

Engineering, A Chapman & Hall Food Science Book. Springer, USA, URL 〈http://

books.google.ie/books?id=nIeJiL_dLeQC〉.

Aubertin, M., Chapuis, R., 2003. École polytechnique de Montréal, Predicting the

Coefﬁcient of Permeability of Soils Using the Kozeny-Carman Equation, Rapport

Technique. 〈http://books.google.ie/books?id=yk8nNAEACAAJ〉.

Bear, J., Cheng, A.H.-D., 2010. Modeling groundwater ﬂow and contaminant

transport, theory and applications of transport in porous media. Springer,

Heidelberg.

Booth, C., Cummins, C., Dalwadi, M., Dellar, P., Devereux, M., Dewynne, J., Donohue,

J., Duncan, A., Fitzmaurice, F., Gordon, A., Hennessy, M., Hinch, J., Hickey, C.,

Hjorth, P., Kyrke-Smith, T., Leahy, D., Lee, W., Lynch, E., Mercier, O., Miklavcic, S.,

Russell, S., Schwartz, L., Shozi, B., Swierczynski, P., Timoney, C., Tomczyk, J.,

Warneford, E, 2012. In: Technical Report from MACSI's 2012 Problem-Solving

Workshop with Industry. 〈http://www.macsi.ul.ie/esgi87/ReportWeb.pdf〉.

Clarke, R., 1987. Extraction. In: Clarke, R., Macrae, R. (Eds.), Coffee, Springer, The

Netherlands, pp. 109–145. http://dx.doi.org/10.1007/978-94-009-3417-7_5.

Cussler, E., 1997. Diffusion: Mass Transfer in Fluid Systems, Cambridge Series in

Chemical Engineering. Cambridge University Press, Cambridge UK 〈http://

books.google.ie/books?id=TGRmfTrsPTQC〉.

Fasano, A., Farina, A., 2010. Modelling complex ﬂows in porous media by means of

upscaling procedures. Rend. Istit. Mat. Univ. Trieste 42, 65–102.

Fasano, A., 2000. Filtration problems in various industrial processes. In: Fasano, A.

(Ed.), Filtration in 906 Porous Media and Industrial Application, vol. 1734 of

Lecture Notes in Mathematics, Springer 907 Berlin, Heidelberg pp. 79–126.

http://dx.doi.org/10.1007/BFb0103976.

Fasano, A., Talamucci, F., 2000. A comprehensive mathematical model for a multi-

species ﬂow through ground coffee. SIAM J. Math. Anal. 31 (2), 251–273. http:

//dx.doi.org/10.1137/S0036141098336698.

Fasano, A., Talamucci, F., Petracco, M., 2000. The espresso coffee problem. In:

Fasano, A. (Ed.), Complex Flows in Industrial Processes, Modeling and Simula-

tion in Science, Engineering and Technology. Birkhuser, Boston,

pp. 241–280. http://dx.doi.org/10.1007/978-1-4612-1348-2_8.

Gianino, C., 2007. Experimental analysis of the Italian coffee pot Moka. Am. J. Phys.

75 (1), 43–47. http://dx.doi.org/10.1119/1.2358157.

Goto, M., Roy, B.C., Hirose, T., 1996. Shrinking-core leaching model for supercritical-

ﬂuid extraction. J. Supercrit. Fluids 9 (2), 128–133. http://dx.doi.org/10.1016/

S0896-8446(96)90009-1, URL 〈http://www.sciencedirect.com/science/article/

pii/S0896844696900091〉.

Gray, W., Hassanizadeh, S., 1998. Macroscale continuum mechanics for multiphase

porous-media ﬂow including phases, interfaces, common lines and common

points. Adv. Water Resour. 21 (4), 261–281.

Holdich, R., 2002. Fundamentals of Particle Technology, Midland Information

Technology and Publishing, Leistershire, UK. URL 〈http://books.google.ie/

books?id=BiTKAAAACAAJ〉.

Huang, Z., Shi, X. han, Jiang, W. juan, 2012. Theoretical models for supercritical ﬂuid

extraction. J. Chromatogr. A 1250 (0), 2–26. http://dx.doi.org/10.1016/j.

chroma.2012.04.032, supercritical Fluid Extraction and Chromatography URL

〈http://www.sciencedirect.com/science/article/pii/S0021967312005894〉.

Jaganyi, D., Madlala, S.P., 2000. Kinetics of coffee infusion: a comparative study on

the extraction kinetics of mineral ions and caffeine from several types of

medium roasted coffees. J. Sci. Food Agric. 80 (1), 85–90.

Machmudah, S., Martin, A., Sasaki, M., Goto, M., 2012. Mathematical modeling for

simultaneous extraction and fractionation process of coffee beans with super-

critical CO2 and water. J. Supercrit. Fluids 66 (0), 111–119 . http://dx.doi.org/

10.1016/j.supﬂu.2011.11.011, special Edition on the Occasion of Gerd Brunner's

70th Birthday. URL 〈http://www.sciencedirect.com/science/article/pii/

S0896844611004815〉.

Millington, R.J., 1959. Gas diffusion in porous media. Science 130 (3367), 100–102.

http://dx.doi.org/10.1126/science.130.3367.100-a, URL 〈http://www.sciencemag.

org/content/130/3367/100.2.abstract〉arXiv:130/3367/100.2.full.pdf.

Navarini, L., Nobile, E., Pinto, F., Scheri, A., Suggi-Liverani, F., 2009. Experimental

investigation of steam pressure coffee extraction in a stove-top coffee maker.

Appl. Therm. Eng. 29 (56), 998–1004. http://dx.doi.org/10.1016/j.appltherma-

leng.2008.05.014, URL 〈http://www.sciencedirect.com/science/article/pii/

S1359431108002299〉.

Oliveira, E.L., Silvestre, A.J., Silva, C.M., 2011. Review of kinetic models for super-

critical ﬂuid extraction. Chem. Eng. Res. Des. 89 (7), 1104–1117. http://dx.doi.

org/10.1016/j.cherd.2010.10.025, URL 〈http://www.sciencedirect.com/science/

article/pii/S0263876210003278〉.

Petracco, M., 2008. Technology IV Beverage Preparation: Brewing Trends for the

New Millennium. Blackwell Science Ltd., Oxford UK, pp. 140–164.

Pictet, G., 1987. Home and catering brewing of coffee. In: Clarke, R.,

Macrae, R. (Eds.), Coffee. Springer, The Netherlands, pp. 221–256

http://dx.doi.org/10.10 07/978-94-009-3417-7_8.

Pisani, L., 2011. Simple expression for the tortuosity of porous media. Transp.

Porous Media 88 (2), 193–203. http://dx.doi.org/10.1007/s11242-011-9734-9.

Rao, S., 2010. Everything But Expresso, Independent Publisher Scott Rao, Available

from www.scottrao.com.

JK, 60g, 2.3bar, 250ml min

JK, 12.5g, 0.5bar, 250ml min

0 200 400 600 800 1000

100

150

200

Mbrew grams

cbrew mg gram

JK, 60g, 2.3bar, 250ml min

JK, 12.5g, 0.5bar, 250ml min

0200 400 600 800 1000

100

150

200

Mbrew grams

cexit mg gram

Fig. 11. The coffee solubles concentration, measured in mg/gram, is plotted against mass of coffee beverage M

brew

(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.

JK, 60g, 2.3bar, 250ml min

JK, 60g, 9.0bar, 250ml min

0 200 400 600 800 1000

100

150

200

Mbrew grams

cbrew mg gram

JK, 60g, 2.3bar, 250ml min

JK, 60g, 9.0bar, 250ml min

0200 400 600 800 1000

100

150

200

Mbrew grams

cexit mg gram

Fig. 12. The coffee solubles concentration, measured in mg/gram, is plotted against mass of coffee beverage M

brew

(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 values of absolute pressure in the brewing cylinder.

K.M. Moroney et al. / Chemical Engineering Science 137 (2015) 216–234 233

Sivetz, M., Foote, H., 1963. Coffee processing technology, no. v. 2 in Coffee

Processing Technology, Avi Pub. Co. URL 〈http://books.google.ie/books?

id=nbBTAAAAMAAJ〉.

Sovov, H., 2005. Mathematical model for supercritical ﬂuid extraction of natural

products and extraction curve evaluation. J. Supercrit. Fluids 33 (1), 35–52.

http://dx.doi.org/10.1016/j.supﬂu.2004.03.005, URL 〈http://www.sciencedirect.

com/science/article/pii/S0896844604000932.

Spaninks, J.A.M., 1979. Design procedures for solid–liquid extractors and the effect

of hydrodynamic instabilities on extractor performance (Ph.D. thesis), Agricul-

tural University of Washington.

Voilley, A., Simatos, D., 1979. Modeling the solubilization process during coffee

brewing. J. Food Process Eng. 3 (4), 185–198.

K.M. Moroney et al. / Chemical Engineering Science 137 (2015) 216–234234

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Jan 2002

Richard Holdich

Mathematical model for supercritical fluid extraction of natural products and extraction curve evaluation

Article

Full-text available

Jan 2005
J SUPERCRIT FLUID

Helena Sovova

New model for supercritical fluid extraction (SFE) of natural products is presented. Like other models based on the concept of broken and intact cells, it is particularly suited to fit experimental data as it almost independently simulates two extraction periods, the first one governed by phase equilibrium and the second one governed by internal diffusion in particles. Its new feature is a detailed description of the first extraction period where different types of phase equilibrium and solvent flow patterns are taken into account. A simplified approximate form of the model is used to analyse its properties and to estimate its parameters. The number of model parameters is, in dependence on the complexity of the extraction process, 1–3 for phase equilibrium, 2–3 for mass transfer and 1 for flow pattern. The model is versatile, but, as a consequence, more data are necessary than a single extraction curve to determine its parameters in the first period. The evaluation of model parameters from extraction curves is shown on data sets from literature.

Experimental analysis of the Italian coffee pot ``moka''

Article

Full-text available

Jan 2007

Concetto Gianino

I describe an experiment involving the moka Italian coffee pot. The pot is an ingenious device for making coffee that uses the liquid-vapor equation of state of the water and Darcy's law of linear filtration. The filtration coefficient of coffee is measured and a steam engine model is used to estimate the efficiency of the coffee pot.

Extraction

Chapter

Jan 1987

R. J. Clarke

After roasting and grinding, extraction is the key operation in the large- scale manufacture of instant coffee, in which both soluble solids and volatile aroma/flavour compounds are extracted. The direct brewing or infusion by hot water of roast coffee in the home to provide an immediate beverage for drinking is separately discussed in Chapter 8. Large-scale extraction has a number of distinctive features, consequent upon the problems of scale, productivity and economic factors. It is necessary furthermore to provide extracts that can subsequently be dried satisfactorily (as described in Chapter 6) to a convenient instant product, which is also stable and of good flavour on making up to a drinking concentration of solubles with hot water.

Home and Catering Brewing of Coffee

Chapter

Jan 1987

G. Pictet

The proportion of food products and beverages offered to the consumer that are prepared on an industrial scale is increasing. However, in certain cases a significant part of the preparation involving various operations is still carried out domestically, or on a small scale. The precise conditions in these operations, as well as the equipment used, can be very important, and a certain skill or know-how may also be valuable.

Filtration problems in various industrial processes

Article

Antonio Fasano

Mathematical modeling for simultaneous extraction and fractionation process of coffee beans with supercritical CO2 and water

Article

Jun 2012

Mathematical modeling of extraction and fractionation of caffeine from coffee beans using SCCO2 and water was developed. Simulation of the extraction and fractionation process was conducted using a model based on mass transfer balances to estimate recovery of caffeine as non-polar compound both in CO2 and water phases. The model was developed regarding extraction and fractionation systems including batch and counter-current configurations in our previous work. Even though polar compound of chlorogenic acid from coffee beans was extracted in water phase, the simulation was focused on the extraction and fractionation of caffeine because chlorogenic acid was not fractionated but only extracted by water. Effective diffusivity of caffeine in water phase was used as fitting parameter to compare the simulation and experimental results. The value of effective diffusivity used for simulation in batch system was 3 × 10−12, 7 × 10−12 and 1 × 10−11 m2/s; and the value used in counter-current system was 1 × 10−12 m2/s. The simulation results were compared with the previous experimental results [3] at various temperatures, pressures, CO2 flow rates, non-polar recovery section heights and ratios of coffee and water mass. The simulation could describe experimental results almost for all extraction conditions. The model just had lack agreement with the experimental result for the ratio of coffee and water mass in water phase.

Shrinking-core leaching model for supercritical-fluid extraction

Article

Jun 1996
J SUPERCRIT FLUID

Extraction or leaching of a solute from a solid material is a process involving mass transfer in the solid matrix. When the solute content in the solid material is sufficiently large as compared to the solubility in fluid phase, the process is similar to that of irreversible desorption. The shrinking-core model was applied to the modeling of the extraction process. The model including axial dispersion in the extraction column was solved numerically. Quasi-steady-state solution without axial dispersion was derived, and the accuracy was discussed in comparison with the numerical solutions. The model calculations gave a good agreement with the experimental extraction curve reported in literature.

A Comprehensive Mathematical Model for a Multispecies Flow Through Ground Coffee

Article

Jan 2000

A model of filtration in a multispecies porous medium with mechanical and chemical interaction between the flow and the porous matrix is presented. The species removed by the medium are transported either as solids or as solutes. The accumulation of the migrating particles in proximity of the outflow surface gives rise to the growth of a compact layer with high hydraulic resistance. The corresponding mathematical problem is a free boundary problem for parabolic and hyperbolic partial differential equations. Existence and uniqueness globally in time are proved.

Modelling of coffee extraction during brewing using multiscale methods: An experimentally validated model

Abstract and Figures

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