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Modelling of coffee extraction during brewing using multiscale
methods: An experimentally validated model
K.M. Moroney
a,
n
, W.T. Lee
a
, S.B.G. O'Brien
a
, F. Suijver
b
, J. Marra
b
a
MACSI, Department of Mathematics and Statistics, University of Limerick, Ireland
b
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.
&2015 Elsevier Ltd. All rights reserved.
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
0009-2509/&2015 Elsevier Ltd. All rights reserved.
n
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
of
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
c
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
0
2
4
6
8
10
Particle size m
Volume fraction
µ
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
0
10
20
30
40
time s
cbrew kg m3
Fig. 2. Coffee solubles concentration profiles for JK standard drip filter grind ðÞ 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
Ω
T
with volume V
T
. The domain can be split into an intergranular
pores domain
Ω
h
, with volume V
h
and a coffee grain domain
Ω
l
,
with volume V
l
.
Ω
l
is further split into an intragranular pore
domain
Ω
v
, with volume V
v
and a solid coffee domain
Ω
s
, with
volume V
s
. Clearly the equalities V
h
þV
l
¼V
T
and V
v
þV
s
¼V
l
hold.
The following volume fractions are now defined as
ϕ
h
¼V
h
V
T
;ϕ
l
¼V
l
V
T
;ϕ
v
¼V
v
V
l
;ϕ
s
¼V
s
V
l
;ð1Þ
0 200 400 600 800 1000
0
50
100
150
200
Mbrew grams
cbrew mg gram
0200 400 600 800 1000
0
50
100
150
200
Mbrew grams
cexit mg gram
0 200 400 600 800 1000
0
20
40
60
80
100
Mbrew grams
cbrew mg gram
0 200 400 600 800 1000
0
20
40
60
80
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
ϕ
h
þϕ
l
¼1;ϕ
v
þϕ
s
¼1:ð2Þ
The concentrations (mass per unit volume) of coffee in the
respective phases are c
h
,c
v
and c
s
.v
h
and v
v
denote the fluid
velocity in the h-phase and v-phase respectively. The velocity of
the solid will be denoted by v
s
. 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
n
Þand microscale ðÞ 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
h
∂t¼∇ðc
h
v
h
þj
h
Þ;ð3Þ
∂ρ
h
∂t¼∇ð
ρ
h
v
h
Þ:ð4Þ
3.4.2. v-phase
∂c
v
∂t¼∇ðc
v
v
v
þj
v
Þ;ð5Þ
∂ρ
v
∂t¼∇ð
ρ
v
v
v
Þ:ð6Þ
3.4.3. s-phase
∂c
s
∂t¼0:ð7Þ
The microscopic balance equations include the terms j
h
and j
v
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
v
c
n
v
Þ¼∇ðϕ
v
ðc
n
v
v
n
v
þj
n
v
ÞÞf
n
v-s
:ð8Þ
∂
∂tðϕ
v
ρ
n
v
Þ¼∇ðϕ
v
ðρ
n
v
v
n
v
þi
n
v
ÞÞ:ð9Þ
3.5.2. s-phase
∂
∂tðϕ
s
c
n
s
Þ¼∇ðϕ
s
j
n
s
Þf
n
s-v
:ð10Þ
The terms f
n
v-s
and f
n
s-v
are source/sink terms representing
transfer of coffee solubles across the vs-interface and vice-versa.
The term i
n
v
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
h
~
c
h
Þ¼∇ðϕ
h
ð~
c
h
~
v
h
þ~
j
h
ÞÞ ~
f
h-l
;ð11Þ
∂
∂tðϕ
h
~
ρ
h
Þ¼∇ðϕ
h
ð~
ρ
h
~
v
h
þ~
i
h
ÞÞ ~
f
w
h-l
:ð12Þ
3.6.2. v-phase
∂
∂tðϕ
l
ϕ
v
~
c
v
Þ¼∇ðϕ
l
ϕ
v
ð~
c
v
~
v
v
þ~
j
v
ÞÞ ~
f
v-h
~
f
v-s
;ð13Þ
∂
∂tðϕ
l
ϕ
v
~
ρ
v
Þ¼∇ðϕ
l
ϕ
v
ð~
ρ
v
~
v
v
þ~
i
v
ÞÞ ~
f
w
l-h
:ð14Þ
3.6.3. s-phase
∂
∂tðϕ
l
ϕ
s
~
c
s
Þ¼∇ðϕ
l
ϕ
s
~
j
s
Þ~
f
s-h
~
f
s-v
:ð15Þ
~
f
α
-
β
is transfer of coffee solubles from the
α
-phase to the
β
-phase
across the αβ interface. Similarly ~
f
w
α
-
β
is transfer of liquid from the
α
-phase to the
β
-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 ~
ρ
h
¼~
ρ
v
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 ~
f
α
-
β
¼
~
f
β
-
α
. Thus the five macroscopic
equations reduce to the following:
∂
∂tðϕ
h
~
c
h
Þ¼∇ðϕ
h
ð~
c
h
~
v
h
þ~
j
h
ÞÞ þ ~
f
v-h
þ~
f
s-h
;ð16Þ
ρ∂ϕ
h
∂t¼
ρ∇ðϕ
h
~
v
h
Þþ~
f
w
l-h
;ð17Þ
∂
∂tðϕ
l
ϕ
v
~
c
v
Þ¼~
f
v-h
þ~
f
s-v
;ð18Þ
ρ∂
∂tðϕ
l
ϕ
v
Þ¼~
f
w
l-h
;ð19Þ
∂
∂tðϕ
l
ϕ
s
~
c
s
Þ¼~
f
s-h
~
f
s-v
:ð20Þ
It now remains to introduce expressions to model the fluid
velocity ~
v
h
, the total macroscopic flux ~
j
h
, the fluid mass transfer
term ~
f
w
l-h
and the coffee mass transfer terms ~
f
v-h
,~
f
s-v
and ~
f
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
~
u
h
¼ϕ
h
~
v
h
;~
u
h
¼
~
k
h
μð∇~
p
h
þρgÞ;~
k
h
¼~
k
h
ðϕ
h
Þ;ð21Þ
where ~
p
h
is the macroscopic pressure gradient in the h-phase, ~
k
h
is
the permeability and
μ
is the viscosity of water.
4.2. Total macroscopic flux
The total macroscopic flux, ~
j
h
is made up of the macroscopic
average of molecular diffusion j
a
h
and the dispersive flux j
b
h
:
~
j
h
¼j
h
h
þc
h
○
v
h
○
DE
h
¼j
a
h
þj
b
h
:ð22Þ
For an isotropic porous medium, j
a
h
is often modelled by
j
a
h
¼
D
τ∇~
c
h
;ð23Þ
where
τ
is the tortuosity defined by
τ¼L
e
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
h
which is adopted from Millington (1959). Thus
we have
j
a
h
¼
D
τ∇~
c
h
¼ϕ
1=3
h
D∇~
c
h
: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
j
b
h
¼c
h
○
v
h
○
DE
h
¼~
D
b
∇~
c
h
;ð26Þ
where ~
D
b
is a rank 2 tensor called the dispersion tensor. ~
D
b
is both
positive definite and symmetric. For an isotropic porous medium
the following expression is often used:
D
ij
¼a
T
δ
ij
þða
L
a
T
Þv
i
v
j
v
2
v:ð27Þ
The coefficients a
L
and a
T
here are the longitudinal and transverse
dispersivities of the porous medium. v
i
¼v
i
hi
h
is the average
velocity in the i-th direction and v¼jvjwhere vis the average
velocity vector in this instance.
δ
ij
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 ~
f
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 ~
f
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 ~
f
s-v
models dissolution
of coffee solubles from the cell walls into the intragranular pores.
The transfer ~
f
v-h
across the interphase boundary is assumed to
occur by diffusion according to Fick's first law. Thus
~
f
v-h
¼α
vh
ð~
c
v
~
c
h
Þ:ð28Þ
where α
vh
is the mass transfer coefficient. The form of α
vh
can be
found from the volume averaging procedure as
α
vh
¼ð1ϕ
h
Þϕ
4=3
v
D
v
S
n
hl
Δ
l
;ð29Þ
where D
v
is the diffusion coefficient of coffee in water, S
n
hl
is the
specific surface area of the l-phase and
Δ
l
is some length scale
characterising the distance over which diffusion occurs. The
transfers ~
f
s-v
and ~
f
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
.c
sat
is the concentration in the liquid phase that
would be in equilibrium with the concentration inside the solid ~
c
s
.
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
~
f
s-h
¼ð1ϕ
h
Þð1ϕ
v
ÞD
h
S
n
hl
Δ
s
ðc
sat
~
c
h
Þ;ð30Þ
where
Δ
s
is some length scale characterising the distance over
which diffusion occurs. Similarly
~
f
s-v
¼ð1ϕ
h
Þð1ϕ
v
ÞD
v
S
n
sv
Δ
s
ðc
sat
~
c
v
Þ;ð31Þ
where S
n
sv
is the specific surface area of the s-phase. The transfer
terms are considered in more detail in Appendix D.
4.5. Liquid transfer term
It is also possible to have a transfer of liquid from the
intergranular pores to the intragranular pores (or vice versa). This
could occur for example due to a pressure imbalance in the phases
due to the dissolution of the solid matrix within the grains. Using
Darcy's Law and volume averaging we find that
~
f
w
l-h
¼ð1ϕ
h
ÞS
n
lh
ρk
v
ðϕ
v
Þ
μ
ð~
p
h
~
p
v
Þ
Δ
l
;ð32Þ
where k
v
ðϕ
v
Þis the coffee grain permeability and ~
p
h
and ~
p
v
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
f
wn
l-h
¼
ð1ϕ
h
ÞS
n
lh
ρ
k
v
ð
ϕ
v
Þ
μ
ð~
p
h
~
p
v
Þ
Δ
l
~
c
h
if ~
p
h
Z~
p
v
ð1ϕ
h
ÞS
n
lh
ρ
k
v
ð
ϕ
v
Þ
μ
ð~
p
h
~
p
v
Þ
Δ
l
~
c
v
if ~
p
h
o~
p
v
8
<
:
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
ϕ
h
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. ~
p
h
Z~
p
v
. Thus (33) simplifies to
~
f
wn
h-l
¼ð1ϕ
h
ÞS
n
hl
ρk
v
ðϕ
v
Þ
μ
ð~
p
h
~
p
v
Þ
Δ
l
~
c
h
: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
by
k
sv
¼6
S
v
;ð35Þ
where S
v
is the surface to volume ratio of the distribution which
can be found from the data. The assumption
ϕ
h
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
S
n
hl
¼6
k
sv
: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
k
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
S
n
sv
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
h
The permeability can be estimated using the Kozeny–Carman
equation for spheres (Holdich, 2002):
k
h
¼ϕ
3
h
κð1ϕ
h
Þ
2
S
n
hl
2:ð38Þ
Here again S
n
hl
is the specific surface area, while
κ
is a factor which
accounts for the shape and tortuosity. Utilising the derived form
for S
n
hl
gives
k
h
¼k
2
sv1
ϕ
3
h
36κð1ϕ
h
Þ
2
¼k
2
sv1
ϕ
3
h
36κð1ϕ
h
Þ
2
:ð39Þ
The shape factor
κ
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
v
Similarly the permeability of the grain is estimated using the
Kozeny–Carman equations for the spherical cells so that
k
v
¼m
2
ϕ
3
v
180ð1ϕ
v
Þ
2
: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
Δ
l
and
Δ
s
. The distance over which
diffusion from the grains to the large pores occurs, is assumed to
be equal to the mean radius of the grains weighted by volume
which we denote Δ
l
¼l
l
. 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
s
¼l
s
.
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
ϕ
v
to change or allow the solid coffee concentration ~
c
s
to
change, or both. The decision is made here to allow
ϕ
v
to change
as coffee is extracted and consider the solid concentration ~
c
s
fixed.
Thus
ϕ
v
is allowed to increase as coffee is dissolved until it reaches
a point when
ϕ
s
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
ϕ
c
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
c
¼ϕ
s;s
þϕ
s;b
. The
initial volume fractions of coffee in dry grains are
ϕ
c0
,ϕ
s;s0
and
ϕ
s;b0
. Assuming that the soluble coffee is uniformly distributed
within the grains the initial volume fraction everywhere is
ϕ
c0
.
Thus at a given time the volume fractions of soluble coffee on the
surface and in the grain kernels are given by
ϕ
c0
ϕ
s;s0
ϕ
s;s
ðx;tÞ¼ϕ
c0
ψ
s
ðx;tÞ;ð41Þ
ϕ
c0
ϕ
s;b0
ϕ
s;b
ðx;tÞ¼ϕ
c0
ψ
v
ðx;tÞ;ð42Þ
where xis the position within the coffee bed and
ψ
s
and
ψ
v
are the
fractions of the original amount of coffee left on the grain surfaces
and in the grain kernels respectively. We can now substitute
ϕ
s
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
ψ
s
and
ψ
v
:
∂ψ
s
∂t¼
12D
h
ϕ
c0
k
sv1
m
c
sat
~
c
h
~
c
s
r
s
ψ
s
;ð43Þ
∂ψ
v
∂t¼
12D
v
ϕ
c0
m
2
c
sat
~
c
v
~
c
s
r
v
ψ
v
;ð44Þ
where r
s
¼1=ϕ
s;s0
and r
v
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.
ϕ
h
∂~
c
h
∂t¼k
2
sv1
ϕ
3
h
36κμð1ϕ
h
Þ
2
∇ð
~
c
h
ð∇~
p
h
þρgÞÞ
þϕ
4
3
h
D
h
∇
2
~
c
h
þϕ
h
~
D
b
∇
2
~
c
h
ð1ϕ
h
Þϕ
4=3
v
D
v
6
k
sv2
l
l
ð~
c
h
~
c
v
Þ
þð1ϕ
h
Þ12D
h
ϕ
c0
k
sv1
mðc
sat
~
c
h
Þψ
s
6ð1ϕ
h
Þm
2
180μk
sv2
l
l
ϕ
3
v
ð1ϕ
v
Þ
2
ð~
p
h
~
p
v
Þ~
c
h
;ð45Þ
0¼ k
2
sv1
ϕ
3
h
36κμð1ϕ
h
Þ
2
∇ð∇~
p
h
þρgÞ
þ6ð1ϕ
h
Þm
2
180μk
sv2
l
l
ϕ
3
v
ð1ϕ
v
Þ
2
ð~
p
h
~
p
v
Þ;ð46Þ
∂
∂tðϕ
v
~
c
v
Þ¼ϕ
4=3
v
D
v
6
k
sv2
l
l
ð~
c
h
~
c
v
Þ
þ12ϕ
c0
D
v
m
2
ðc
sat
~
c
v
Þψ
v
þ6ð1ϕ
h
Þm
2
180μk
sv2
l
l
ϕ
3
v
ð1ϕ
v
Þ
2
ð~
p
h
~
p
v
Þ~
c
h
;ð47Þ
∂ϕ
v
∂t¼6m
2
180μk
sv2
l
l
ϕ
3
v
ð1ϕ
v
Þ
2
ð~
p
h
~
p
v
Þ;ð48Þ
∂ϕ
v
∂t¼1
r
s
∂ψ
s
∂t1
r
v
∂ψ
v
∂t;ð49Þ
∂ψ
s
∂t¼
12D
h
ϕ
c0
k
sv1
m
c
sat
~
c
h
~
c
s
r
s
ψ
s
;ð50Þ
∂ψ
v
∂t¼
12D
v
ϕ
c0
m
2
c
sat
~
c
v
~
c
s
r
v
ψ
v
;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
v
¼0:56 is
adopted here for dry coffee in air. Allowing for these assumptions
the following system of equations are solved numerically in
MATLAB
s
.
ϕ
h
d~
c
h
dt ¼
α
n
ð1ϕ
h
Þϕ
4
3
v
D
v
6
k
sv2
l
l
ð~
c
h
~
c
v
Þ
þβ
n
ð1ϕ
h
Þ12D
h
ϕ
c0
k
sv1
mðc
sat
~
c
h
Þψ
s
;ð52Þ
d
dtðϕ
v
~
c
v
Þ¼α
n
ϕ
4=3
v
D
v
6
k
sv2
l
l
ð~
c
h
~
c
v
Þ;ð53Þ
dϕ
v
dt ¼1
r
s
∂ψ
s
∂t;ð54Þ
dψ
s
dt ¼
β
n
12D
h
ϕ
c0
k
sv1
m
c
sat
~
c
h
~
c
s
r
s
ψ
s
;ð55Þ
with initial conditions
~
c
h
ð0Þ¼0;~
c
v
ð0Þ¼c
v0
;ϕ
v
ð0Þ¼ϕ
v0
;ψ
s
ð0Þ¼1:ð56Þ
The parameters α
n
and β
n
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
n
and β
n
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 ~
c
h
. We use the condition
that the water entering at the top z¼Lhas zero concentration. The
pressure boundary conditions are ~
p
h
¼Δpat z¼Land ~
p
h
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
ϕ
h
adjusts to the pressure while
ϕ
v
stays constant. We then choose
ϕ
h
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 ~
c
h
as a function of z.
Based on the initial exiting concentrations we assume for the fine
grind (JK) that initially ~
c
h
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
ϕ
v0
0.6444 0.6120
ϕ
h
0.8272 0.8272
k
sv1
27:35 μm38:77 μm
k
sv2
322:49 μm 569:45 μm
l
l
282 μm 463 μm
D
h
¼D
v
2:210
9
m
2
s
1
2:210
9
m
2
s
1
ρ965:3kgm
3
965:3kgm
3
μ0:315 10
3
Pas 0:315 10
3
Pas
m30 μm30μm
c
sat
212:4kgm
3
212:4kgm
3
~
c
s
1400 kg m
3
1400 kg m
3
κ3.1 3.1
ϕ
c0
0.143435 0.122
ϕ
s;s0
0.059 0.07
ϕ
s;b0
0.084435 0.052
α
n
0.1833 0.0881
β
n
0.0447 0.0086
r
s
16.94 14.28
c
v0
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 ~
p
h
¼~
p
v
. Based on these assumptions the
reduced set of equations to model extraction in the cylindrical
brewing chamber is given by
ϕ
h
∂~
c
h
∂t¼k
2
sv1
ϕ
3
h
36κμð1ϕ
h
Þ
2
∂
∂z
~
c
h
∂~
p
h
∂zþρg
α
n
ð1ϕ
h
Þϕ
4=3
v
D
v
6
k
sv2
l
l
ð~
c
h
~
c
v
Þ
þβ
n
ð1ϕ
h
Þ12D
h
ϕ
c0
k
sv1
mðc
sat
~
c
h
Þψ
s
;ð57Þ
∂
2
~
p
h
∂z
2
¼0;ð58Þ
∂
∂tðϕ
v
~
c
v
Þ¼α
n
ϕ
4=3
v
D
v
6
k
sv2
l
l
ð~
c
h
~
c
v
Þ;ð59Þ
∂ϕ
v
∂t¼1
r
s
∂ψ
s
∂t;ð60Þ
∂ψ
s
∂t¼
β
n
12D
h
ϕ
c0
k
sv1
m
c
sat
~
c
h
~
c
s
r
s
ψ
s
;ð61Þ
for
t40;0ozoL;ð62Þ
with initial conditions
~
c
h
ðz;0Þ¼c
h0
ðzÞ;~
c
v
ðz;0Þ¼c
v0
;ð63Þ
ϕ
v
ðz;0Þ¼ϕ
v0
;ψ
s
ðz;0Þ¼ψ
s0
;ð64Þ
and boundary conditions
~
c
h
ðL;tÞ¼0;~
p
h
ð0;tÞ¼0;~
p
h
ðL;tÞ¼Δp:ð65Þ
The initial concentration profile in the JK drip filter grind is
given by c
h0
ðzÞ¼c
sat
. In the Cimbali #20 grind the initial concen-
tration is given by c
h0
ðzÞ¼ðc
max
=LÞðLzÞ. 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
0
5
10
15
20
25
30
35
40
t (s)
Ch (kg/m3)
0100 200 300 400 500 600
0
5
10
15
20
25
30
35
t (s)
Ch (kg/m3)
Fig. 6. Comparison between numerical solution (–) with parameters from Table 1 and experiment (
n
) 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
ϕ
v0
0.6231 0.6218
ϕ
h
0.2 0.25
c
max
–82:63 kg m
3
c
s
1400 kg m
3
1400 kg m
3
ϕ
s;s0
0.11 0.07
ϕ
s;b0
0.033435 0.052
α
n
0.1833 0.0881
β
n
0.0447 0.0086
r
s
9.09 14.28
c
v0
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
γ
-species per unit volume of the
α
-phase be
∂c
α
∂t¼∇ðc
α
v
α
þj
α
ÞþG
α
;ð66Þ
where c
α
is the concentration of the
γ
-species, v
α
is the fluid
velocity in the
α
-phase, j
α
is the diffusive flux of the
γ
-species in
the
α
-phase, and G
α
is a source or sink of the
γ
-species in the
α
-
phase.
Let
β
represent all other phases in the porous medium. The
α
subscript will be dropped where convenient. Eq. (66) is the
microscopic balance. The corresponding macroscopic balance
equation is
∂
∂tðϕ
α
c
n
Þ¼∇ðϕ
α
c
n
v
n
þϕ
α
j
n
Þþϕ
α
G
n
;ð67Þ
where ϕ
α
is the volume fraction of the
α
-phase and the
n
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
α
-phase is
δV
α
.
The union of the interfacial regions within the REV between the
α
-phase and a
β
-phase is denoted S
αβ
.
The unit vector normal to this surface oriented outward from
the
α
-phase is n
α
.
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
Z
δ
V
∂F
∂tdV ¼∂
∂tZ
δ
V
FdVX
β
a
α
Z
S
αβ
n
α
w
b
Fj
α
dS;ð69Þ
where Fis some scalar field property of the microscale and Fj
α
just
indicates that the microscale property Fin the
α
-phase is being
integrated over the αβ-interface. w
b
is the velocity of S
αβ
. The
summation P
β
a
α
just denotes a summation over all phases
except the
α
-phase.
The divergence averaging theorem is given by
Z
δ
V
∇BdV ¼∇Z
δ
V
BdV þX
β
a
α
Z
S
αβ
n
α
Bj
α
dS;ð70Þ
where Bis some vector field property of the microscale and Bj
α
just indicates that Bis being integrated over the αβ-interface. The
summation P
β
a
α
just denotes a summation over all phases
except the
α
-phase.
Next some phase averages are defined
The Intrinsic Phase Average of a quantity F over the
α
-phase is
defined by
Fhi
α
¼F
α
¼1
δV
α
Z
δ
V
α
FdV:ð71Þ
The Phase Average of a quantity F over the
α
-phase is defined by
F
hi
¼F¼1
δVZ
δ
V
α
FdV¼ϕ
α
F
α
:ð72Þ
The Mass Weighted Average of a quantity F over the
α
-phase is
defined by
F
0
α
¼F
0
α
¼1
ρ
α
δV
α
Z
δ
V
α
ρFdV¼ρF
α
ρ
α
:ð73Þ
0 0.2 0.4 0.6 0.8 1
0
50
100
150
200
250
Mbrew (kg)
Cexit (kg/m3)
00.2 0.4 0.6 0.8 1
0
10
20
30
40
50
60
70
80
90
100
Mbrew (kg)
Cexit (kg/m3)
Fig. 7. Comparison between numerical solution (–) with parameters from Table 2 and experiment (
n
) 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
hi
α
in terms of the individual averages c
hi
α
and v
hi
α
. To do this consider the following:
Let v
○
¼vv
hi
α
be the deviation of the true velocity from the
mean velocity.
Let c
○
¼cc
hi
α
be the deviation of the true concentration from
the mean concentration.
Then
v
○
DE
α
¼vv
hi
α
α
¼v
hi
α
v
hi
α
¼0;ð74Þ
c
○
DE
α
¼cc
hi
α
α
¼c
hi
α
c
hi
α
¼0:ð75Þ
Thus
cvhi
α
¼ðchi
α
þc
○
Þð vhi
α
þv
○
Þ
DE
α
¼c
hi
α
v
hi
α
þc
○
v
○
DE
α
þc
○
DE
α
v
hi
α
þc
hi
α
v
○
DE
α
¼c
hi
α
v
hi
α
þc
○
v
○
DE
α
:ð76Þ
The term c
○
v
○
DE
α
is called the dispersive flux.
Now the averaging process can be performed. Firstly (66)
integrated over δV
α
yields
Z
δ
V
α
∂c
∂tdV ¼
Z
δ
V
α
∇ðcvþjÞdV þZ
δ
V
α
GdV:ð77Þ
Now applying theorems (69) and (70) yields
∂
∂tZ
δ
V
α
cdV¼∇Z
δ
V
α
ðcvþjÞdV
X
β
a
α
Z
S
αβ
n
α
ðcðvw
b
ÞþjÞj
α
dS
þZ
δ
V
α
GdV:ð78Þ
This can be written in terms of averaged quantities by dividing
across by δV¼
δ
V
α
ϕ
α
to give
∂
∂tðϕ
α
c
hi
α
Þ¼∇ϕ
α
ðcv
hi
α
þj
α
Þ
1
δVX
β
a
α
Z
S
αβ
n
α
ðcðvw
b
ÞþjÞj
α
dS
þϕ
α
Ghi
α
:ð79Þ
Utilising the formula for the average of a product it can be seen
that
∂
∂tðϕ
α
c
hi
α
Þ¼∇ϕ
α
ðc
hi
α
v
hi
α
þc
○
v
○
DE
α
þj
α
Þ
1
δVX
β
a
α
Z
S
αβ
n
α
ðcðvw
b
ÞþjÞj
α
dS
þϕ
α
G
hi
α
:ð80Þ
Comparing this to the macroscopic point balance Eq. (67) gives the
following relations between the macroscopic and microscopic
quantities
c
n
¼c
hi
α
;ð81Þ
v
n
¼vhi
α
;ð82Þ
j
n
¼c
○
v
○
DE
α
þj
α
;ð83Þ
ϕ
α
G
n
¼ϕ
α
G
hi
α
þ1
δVX
β
a
α
Z
S
αβ
n
α
ðcðvw
b
ÞþjÞj
α
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.
V
1
: Volume of the smaller REV.
V
1s
: Volume of solid in the smaller REV.
V
1v
: Volume of void space in smaller REV.
V
0
: Volume of larger REV.
V
0l
: Volume of grains (l-phase) in larger REV.
V
0h
: Volume of void space (h-phase) in larger REV.
Also due to the properties of an REV
ϕ
h
¼V
0h
V
0
¼V
h
V
T
;ϕ
l
¼V
0l
V
0
¼V
l
V
T
;ð85Þ
ϕ
v
¼V
1v
V
1
¼V
v
V
l
;ϕ
s
¼V
1s
V
1
¼V
s
V
l
: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
h
∂t¼∇ðc
h
v
h
þj
h
Þ;ð87Þ
∂ρ
h
∂t¼∇ð
ρ
h
v
h
Þ:ð88Þ
B.1.1.2. v-phase.
∂c
v
∂t¼∇ðc
v
v
v
þj
v
Þ;ð89Þ
∂ρ
v
∂t¼∇ð
ρ
v
v
v
Þ:ð90Þ
B.1.1.3. s-phase.
∂c
s
∂t¼0:ð91Þ
K.M. Moroney et al. / Chemical Engineering Science 137 (2015) 216–234 227
The microscopic balance equations include the terms j
h
and j
v
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
v
c
n
v
Þ¼∇ðϕ
v
ðc
n
v
v
n
v
þj
n
v
ÞÞf
n
v-s
:ð92Þ
∂
∂tðϕ
v
ρ
n
v
Þ¼∇ðϕ
v
ðρ
n
v
v
n
v
þi
n
v
ÞÞ:ð93Þ
B.1.2.2. s-phase.
∂
∂tðϕ
s
c
n
s
Þ¼∇ðϕ
s
j
n
s
Þf
n
s-v
:ð94Þ
The terms f
n
v-s
and f
n
s-v
are source/sink terms representing
transfer of coffee solubles across the vs-interface and vice-versa.
The term i
n
v
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
h
~
c
h
Þ¼∇ðϕ
h
ð~
c
h
~
v
h
þ~
j
h
ÞÞ ~
f
h-l
;ð95Þ
∂
∂tðϕ
h
~
ρ
h
Þ¼∇ðϕ
h
ð~
ρ
h
~
v
h
þ~
i
h
ÞÞ ~
f
w
h-l
:ð96Þ
B.1.3.2. v-phase.
∂
∂tðϕ
l
ϕ
v
~
c
v
Þ¼∇ðϕ
l
ϕ
v
ð~
c
v
~
v
v
þ~
j
v
ÞÞ ~
f
v-h
~
f
v-s
;ð97Þ
∂
∂tðϕ
l
ϕ
v
~
ρ
v
Þ¼∇ðϕ
l
ϕ
v
ð~
ρ
v
~
v
v
þ~
i
v
ÞÞ ~
f
w
l-h
:ð98Þ
B.1.3.3. s-phase.
∂
∂tðϕ
l
ϕ
s
~
c
s
Þ¼∇ðϕ
l
ϕ
s
~
j
s
Þ~
f
s-h
~
f
s-v
:ð99Þ
~
f
α
-
β
is transfer of coffee solubles from
α
-phase to
β
-phase
across αβ interface. Similarly ~
f
w
α
-
β
is transfer of liquid from
α
-
phase to
β
-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
v
c
v
hi
v
Þ¼∇ϕ
v
ðc
v
hi
v
v
v
hi
v
þc
v
○
v
v
○
DE
v
þj
v
v
Þ
1
V
1
Z
S
vs
n
v
c
v
ðv
v
w
vs
Þþj
v
dS:ð100Þ
Now the surface S
vs
is a material surface so v
v
w
vs
¼0.
Comparing this averaged form with the mesoscale form in (92) it
can be seen that
c
n
v
¼c
v
hi
v
;ð101Þ
v
n
v
¼v
v
hi
v
;ð102Þ
j
n
v
¼c
v
○
v
v
○
DE
v
þj
v
v
;ð103Þ
f
n
v-s
¼1
V
1
Z
S
vs
n
v
j
v
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
v
ρ
v
v
Þ¼∇ϕ
v
ρ
v
v
v
v
hi
v
þρ
v
○
v
v
○
DE
v
1
V
1
Z
S
vs
n
v
ðρ
v
ðv
v
w
vs
ÞÞ dS:ð105Þ
Now the surface S
vs
is a material surface so v
v
w
vs
¼0. Compar-
ing this averaged form with the mesoscale form in (93) it can be
seen that
ρ
n
v
¼ρ
v
v
;ð106Þ
v
n
v
¼v
v
hi
v
;ð107Þ
i
n
v
¼ρ
v
○
v
v
○
DE
v
:ð108Þ
Here in fact
ρ
will be taken to be constant so i
n
v
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
s
c
s
hi
s
Þ¼ 1
V
1
Z
S
sv
n
s
ðc
s
w
sv
ÞdS:ð109Þ
Comparing this averaged form with the mesoscale form in (94) it
can be seen that
c
n
s
¼c
s
hi
s
;ð110Þ
v
n
s
¼0;ð111Þ
f
n
s-v
¼1
V
1
Z
S
sv
n
s
ðc
s
w
sv
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
h
c
h
h
Þ¼∇ϕ
h
c
h
h
v
h
h
þc
h
○
v
h
○
DE
h
þj
h
h
1
V
0
Z
S
hl
n
h
ðc
h
ðv
h
w
hl
Þþj
h
ÞdS:ð113Þ
Comparing this averaged form with the macroscale form in (95) it
can be seen that
~
c
h
¼c
h
h
;ð114Þ
~
v
h
¼v
h
h
;ð115Þ
~
j
h
¼c
h
○
v
h
○
DE
h
þj
h
h
;ð116Þ
K.M. Moroney et al. / Chemical Engineering Science 137 (2015) 216–234228
~
f
h-l
¼1
V
0
Z
S
hl
n
h
ðc
h
ðv
h
w
hl
Þþj
h
ÞdS:ð117Þ
Here S
hl
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
h
ρ
h
h
Þ¼∇ϕ
h
ρ
h
h
v
h
h
þρ
h
○
v
h
○
DE
h
1
V
0
Z
S
hl
n
h
ðρ
h
ðv
h
w
hl
ÞÞ dS:ð118Þ
Comparing this averaged form with the macroscale form in (96) it
can be seen that
~
ρ
h
¼ρ
h
h
;ð119Þ
~
v
h
¼v
h
h
;ð120Þ
~
i
h
¼ρ
h
○
v
h
○
DE
h
;ð121Þ
~
f
w
h-l
¼1
V
0
Z
S
hl
n
h
ðρ
h
ðv
h
w
hl
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
l
ϕ
v
c
n
v
l
Þ¼∇ϕ
l
ϕ
v
c
n
v
l
v
n
v
l
þc
n
v
○
v
n
v
○
l
þj
n
v
l
!
ϕ
v
V
0
Z
S
lh
n
l
ðc
n
v
ðv
n
v
w
lh
Þþj
n
v
ÞdS
ϕ
l
f
n
v-s
l
:ð123Þ
Comparing this averaged form with the macroscale form in (97) it
can be seen that
~
c
h
¼c
n
v
l
¼c
v
hi
v
l
;ð124Þ
~
v
v
¼v
n
v
l
¼v
v
hi
v
l
;ð125Þ
~
j
v
¼j
n
v
l
þc
n
v
○
v
n
v
○
l
¼j
v
v
DE
l
þc
v
○
v
v
○
DE
v
l
þc
n
v
○
v
n
v
○
l
;ð126Þ
~
f
v-h
¼ϕ
v
V
0
Z
S
lh
n
l
ð c
v
hi
v
ðv
v
hi
v
w
lh
Þþ j
v
v
ÞdS;ð127Þ
~
f
n
v-s
¼ϕ
l
f
n
v-s
l
¼1
V
0
Z
V
0l
1
V
1
Z
S
vs
n
v
j
v
dS
dV
0l
:ð128Þ
It will be later assumed that ~
v
v
¼0and ~
j
v
¼0but of course it will
still be possible to have v
n
v
a0and j
n
v
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
l
ϕ
v
ρ
n
v
l
Þ¼∇ϕ
l
ϕ
v
ρ
n
v
l
v
n
v
l
þρ
n
v
○
v
n
v
○
l
þi
n
v
l
!
ϕ
v
V
0
Z
S
lh
n
l
ðρ
n
v
ðv
n
v
w
lh
Þþi
n
v
ÞdS:ð129Þ
Comparing this averaged form with the macroscale form in (98) it
can be seen that
~
ρ
h
¼ρ
n
v
l
¼ρ
v
v
DE
l
;ð130Þ
~
v
v
¼v
n
v
l
¼v
v
hi
v
l
;ð131Þ
~
i
v
¼i
n
v
l
þρ
n
v
○
v
n
v
○
l
¼i
v
v
DE
l
þρ
v
○
v
v
○
DE
v
l
þρ
n
v
○
v
n
v
○
l
;ð132Þ
~
f
w
l-h
¼ϕ
v
V
0
Z
S
lh
n
l
ð ρ
v
v
ðv
v
hi
v
w
lh
Þþ i
v
v
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
l
ϕ
s
c
n
s
l
Þ¼∇ϕ
l
ϕ
s
ðj
n
s
l
Þ
ϕ
s
V
0
Z
S
lh
n
l
ðc
n
s
ðw
lh
Þþj
n
s
ÞdS
ϕ
l
f
n
s-v
l
:ð134Þ
Comparing this averaged form with the macroscale form in (99) it
can be seen that
~
c
s
¼c
n
s
l
¼c
s
hi
s
l
;ð135Þ
~
v
s
¼0;ð136Þ
~
j
s
¼j
n
s
l
¼j
s
s
DE
l
¼0;ð137Þ
~
f
s-h
¼ϕ
s
V
0
Z
S
lh
n
l
ð c
s
hi
s
ðw
lh
ÞÞ dS;ð138Þ
~
f
s-v
¼ϕ
l
f
n
s-v
l
¼1
V
0
Z
V
0l
1
V
1
Z
S
sv
n
s
ðc
s
w
lh
ÞdS
dV
0l
:ð139Þ
Appendix C. Macroscopic diffusion and dispersion fluxes
The total macroscopic flux, ~
j
h
is made up of the macroscopic
average of molecular diffusion and the dispersive flux:
~
j
h
¼j
h
h
þc
h
○
v
h
○
DE
h
:ð140Þ
The microscopic diffusive flux can be represented by Fick's Law:
j
h
¼D∇c
h
;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
j
a
h
¼D~
T
a
ðϕÞ∇~
c
h
¼~
D
a
ðϕÞ∇~
c
h
:ð142Þ
where the ~here on ~
T
a
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
j
a
h
¼
D
τ∇~
c
h
;ð143Þ
where
τ
is the tortuosity defined by
τ¼L
e
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
1αð1ϕÞ:ð145Þ
In the final expression α¼ðrσÞ=Vis a shape factor with rbeing the
object radius,
σ
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
j
a
h
¼
D
τ∇~
c
h
¼ϕ
1=3
h
D∇~
c
h
: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
j
b
h
¼c
h
○
v
h
○
DE
h
¼~
D
b
∇~
c
h
;ð147Þ
where ~
D
b
is a rank 2 tensor called the dispersion tensor. ~
D
b
is both
positive definite and symmetric. One commonly used expression is
D
ij
¼a
ijkl
v
k
v
l
v;ð148Þ
where a
ijkl
is a fourth order tensor and v
i
¼v
i
hi
h
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
D
ij
¼a
T
δ
ij
þða
L
a
T
Þv
i
v
j
v
2
v:ð149Þ
The coefficients a
L
and a
T
here are the longitudinal and transverse
dispersivities of the porous medium. For a phase that completely
fills a pore space, a
L
is a length that should be of the same order of
the pore size.
δ
ij
is the Kronecker delta. Also it is required that
a
L
Z0a
T
Z0:ð150Þ
Laboratory experiments have found that a
T
is 8–24 times smaller
than a
L
(Bear and Cheng, 2010).
Appendix D. Method of estimating mass transfer terms
It is necessary to estimate the mass transfer terms f
α
-
β
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
α
-
β
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
α
-
β
¼f
β
-
α
. In this instance it is assumed that a chemical
species can reach the microscopic interphase boundary by two
modes of transport, namely advection and diffusion. Hence, as we
have already seen from the averaging procedure in Appendices
Appendix A and Appendix B the strength of a source of a
considered species in the
α
-phase is given by
f
α
-
β
¼1
U
0
Z
S
αβ
n
α
ðc
α
ðv
α
w
αβ
Þþj
α
ÞdS;ð151Þ
where U
0
is the volume of the REV, S
αβ
is the boundary between
the
α
-phase and all other phases, n
α
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
α
w
αβ
¼0and
f
α
-
β
¼1
U
0
Z
S
αβ
n
α
j
α
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
α
-
β
, of the mass of the considered species
from an
α
-phase to an adjacent
β
-phase across an interface S
αβ
is
often assumed to be proportional to the difference in concentra-
tion between the phases. Thus
f
α
-
β
¼α
αβ
ðc
β
c
α
Þ:ð153Þ
To estimate the mass transfer coefficient α
αβ
the following form is
used:
j
α
n
α
¼
D
α
ðc
β
c
α
Þ
Δ
α
:ð154Þ
Here D
α
is the coefficient of diffusion of the considered species in
the
α
-phase and Δ
α
is the length characterising the mean size of
the phase or the length over which diffusion occurs. For example
one possible definition of Δ
α
is Δ
α
¼U
0
α
=S
αβ
, the volume to
surface ratio of the
α
-phase within the REV. Then
f
α
-
β
¼1
U
0
Z
S
αβ
n
α
j
α
dS
¼1
U
0
Z
S
αβ
D
α
ðc
β
c
α
Þ
Δ
α
dS
¼
S
αβ
U
0
D
α
ðc
β
c
α
Þ
Δ
α
¼
S
n
αβ
Δ
α
ϕ
α
D
α
ðc
β
c
α
Þ;ð155Þ
where S
n
αβ
¼S
αβ
=U
0
α
is the specific surface area of the
α
-phase or
the surface area per unit volume. Thus it can be seen that the mass
transfer coefficient is given by
α
αβ
¼
S
n
αβ
Δ
α
ϕ
α
D
α
: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
c
sat
is the concentration in the liquid phase that would be in
equilibrium with the concentration inside the solid c
s
. 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
f
s-f
¼α
sf
ðc
sat
c
f
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
α
sf
¼
S
n
sf
Δ
s
ϕ
s
D
f
: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
~
f
w
l-h
¼ϕ
v
V
0
Z
S
lh
n
l
ρ
v
ðv
v
hi
v
w
lh
Þþ i
v
v
dS:ð159Þ
Now it is assumed that i
v
v
¼0and Darcy's Law gives
v
v
hi
v
w
lh
¼
k
v
ðϕ
v
Þ
ϕ
v
μ∇pð160Þ
On S
lh
this means that
v
v
hi
v
w
lh
¼
k
v
ðϕ
v
Þ
ϕ
v
μ
ðp
h
p
v
Þ
Δ
l
:ð161Þ
Thus
~
f
w
l-h
¼ϕ
v
V
0
Z
S
lh
n
l
ρk
v
ðϕ
v
Þ
ϕ
v
μ
ð~
p
h
~
p
v
Þ
Δ
l
dS
¼
S
lh
V
0
ρk
v
ðϕ
v
Þ
μ
ð~
p
h
~
p
v
Þ
Δ
l
¼ϕ
l
S
n
lh
ρk
v
ðϕ
v
Þ
μ
ð~
p
h
~
p
v
Þ
Δ
l
:ð162Þ
Again here S
lh
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:
f
wn
l-h
¼
ϕ
l
S
n
lh
ρ
k
v
ð
ϕ
v
Þ
μ
ð~
p
h
~
p
v
Þ
Δ
l
~
c
h
if ~
p
h
Z~
p
v
ϕ
l
S
n
lh
ρ
k
v
ð
ϕ
v
Þ
μ
ð~
p
h
~
p
v
Þ
Δ
l
~
c
v
if ~
p
h
o~
p
v
8
<
:
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
~
f
l-h
¼~
f
v-h
þ~
f
s-h
þf
wn
l-h
:ð164Þ
~
f
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
~
f
v-h
¼α
vh
ð~
c
h
~
c
v
Þ
¼ϕ
l
ϕ
v
D
n
v
S
n
hl
Δ
l
ð~
c
h
~
c
v
Þ
¼ϕ
l
ϕ
v
D
v
S
n
hl
Δ
l
τð~
c
h
~
c
v
Þ
¼ϕ
l
ϕ
4
3
v
D
v
S
n
hl
Δ
l
ð~
c
h
~
c
v
Þ;ð165Þ
where τ¼ϕ
1=3
v
has been used. Next transfer of solute from the
solid on the surface of the grains, ~
f
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
~
f
s-h
¼α
sh
ðc
sat
~
c
h
Þ
¼ϕ
l
ϕ
s
D
h
S
n
hl
Δ
s
ðc
sat
~
c
h
Þ
¼ϕ
l
ð1ϕ
v
ÞD
h
S
n
hl
Δ
s
ðc
sat
~
c
h
Þ:ð166Þ
As outlined above the third transfer term, ~
f
wn
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
~
f
wn
l-h
¼
ϕ
l
S
n
lh
ρ
k
v
ð
ϕ
v
Þ
μ
ð~
p
h
~
p
v
Þ
Δ
l
~
c
h
if ~
p
h
Z~
p
v
ϕ
l
S
n
lh
ρ
k
v
ð
ϕ
v
Þ
μ
ð~
p
h
~
p
v
Þ
Δ
l
~
c
v
if ~
p
h
o~
p
v
8
<
:
ð167Þ
The macroscopic transfer term from the s-phase to the v-phase
can be arrived at using (139) and (157) to get
~
f
s-v
¼α
sv
ðc
sat
~
c
v
Þ
¼ϕ
l
ϕ
s
D
v
S
n
sv
Δ
s
ðc
sat
~
c
h
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
~
c
h
~
v
h
~
D
b
h
∇~
c
h
:ð169Þ
Recall that
ð~
D
b
h
Þ
ij
¼a
T
δ
ij
þða
L
a
T
Þv
i
v
j
v
h
2
!
v
h
¼a
ijkl
v
i
v
j
v
h
:ð170Þ
We now use some characteristic scales. Note a
L
l
l
r500 μm.
Thus take a
ijkl
a
L
,~
c
h
C,~
v
h
v
c
,v
i
v
c
, and zL. Thus (169)
becomes
Cv
c
l
l
v
c
C
L
¼L
l
l
10
2
:ð171Þ
Thus advection is approximately one hundred times larger than
dispersion and so dominates. More generally advection dominates
over dispersion when L⪢l
l
unless there are very large concentra-
tion gradients in the bed.
The ratio of advection to diffusion is given by
~
c
h
~
v
h
ϕ
4=3
h
D
h
∇~
c
h
: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
Cv
c
ϕ
4=3
h
D
h
C
L
¼Lv
c
ϕ
4=3
h
D
h
:ð173Þ
Now from the experiments typical approximate values for these
quantities are L0:05 m, v
c
0:007 ms
1
,ϕ
h
0:2 and
D
h
2:2e9m
2
s
1
. Thus
Lv
c
ϕ
4=3
h
D
h
10
6
: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,
V
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
0
2
4
6
8
10
Particle size m
Volume fraction
µ
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
0
10
20
30
40
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.
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JK, 60g, 2.3bar, 250ml min
JK, 12.5g, 0.5bar, 250ml min
0 200 400 600 800 1000
0
50
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
0
50
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
0
50
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
0
50
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.
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