ArticlePDF Available

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

Authors:

Abstract and Figures

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.
Content may be subject to copyright.
Modelling of coffee extraction during brewing using multiscale
methods: An experimentally validated model
K.M. Moroneya,, W.T. Leea, S.B.G. O’Briena, F. Suijverb, J. Marrab
aMACSI, Department of Mathematics and Statistics, University of Limerick, Ireland
bPhilips Research, Eindhoven, The Netherlands.
Abstract
Accurate and repeatable extraction of solubles from roasted and ground coffee with hot
water is vital to produce consistently high quality coffee in a variety of brewing techniques.
Despite this, there is an absence in the literature of an experimentally validated model
of the physics of coffee extraction. In this work, coffee extraction from a coffee bed
is modelled using a double porosity model, including the dissolution and transport of
coffee. Coffee extraction experiments by hot water at 90C were conducted in two
situations: in a well stirred dilute suspension of coffee grains, and in a packed coffee
bed. Motivated by experiment, extraction of coffee from the coffee grains is modelled
via two mechanisms: an initial rapid extraction from damaged cells on the grain surface,
followed by a slower extraction from intact cells in the grain kernel. Using volume
averaging techniques, a macroscopic model of coffee extraction is developed. This model
is parametrised by experimentally measured coffee bed properties. It is shown that this
model can quantitatively reproduce the experimentally measured extraction profiles. The
reported model can be easily adapted to describe extraction of coffee in some standard
coffee brewing methods and may be useful to inform the design of future drip filter
machines.
Keywords: Coffee brewing process, Coffee extraction experiments, Double porosity
model, Static porous medium, Coffee extraction kinetics, Drip filter coffee
1. Introduction
Coffee, derived from the seeds (beans) of the coffee plant, is among the most popular
beverages consumed worldwide. Typically, after the beans are roasted and ground, some
of their soluble content is extracted by hot water. The resulting solution of hot water
and coffee solubles is called coffee. Coffee extraction is carried out on a large number
of scales varying from large-scale industrial extraction to produce instant coffee, right
down to one-cup brewing appliances for domestic use. For the purposes of brewing there
Corresponding author
Email addresses: kevin.moroney@ul.ie (K.M. Moroney), william.lee@ul.ie (W.T. Lee),
stephen.obrien@ul.ie (S.B.G. O’Brien), freek.suijver@philips.com (F. Suijver),
johan.marra@philips.com (J. Marra)
Preprint submitted to Journal of Chemical Engineering Science December 11, 2017
are numerous methods of producing a coffee beverage, which can be broken into three
main categories: decoction methods, infusion methods and pressure methods. Many of
these brewing techniques are described in refs [1, 2]. The intimate contact of water with
roasted coffee solids is the cardinal requirement for producing a coffee beverage [1]. In
line with this, all the coffee brewing methods mentioned rely on solid-liquid extraction
or leaching, which involves the transfer of solutes from a solid to a fluid. Despite its
widespread consumption, long history and well developed techniques, consistently brew-
ing high quality coffee remains a difficult task. This difficulty arises from the dependency
of coffee quality on a large number of process variables. Some of these include brew ratio
(dry coffee mass to water volume used), grind size and distribution, brewing time, water
temperature, agitation, water quality and uniformity of extraction [1, 3]. According to
Clarke [4], “The extraction of roast and ground coffee is, in fact, a highly complex oper-
ation, the scientific fundamentals of which are very difficult to unravel”. This is reflected
in the absence of a satisfactory, experimentally validated mathematical system of equa-
tions to model the extraction process accurately. Such a description would have obvious
benefits in quickly and easily investigating the influence of various parameters on coffee
extraction and informing the design of the next generation of coffee brewing equipment.
Of course the notion of high quality coffee is a rather inexact ideal and to some extent
a matter of taste perception. Relating taste to the physical parameters of extracted
solubles is in itself a non-trivial matter, separate to the issue of consistency. Despite
this, certain correlations have been identified between coffee flavour and extraction yield.
The coffee brewing control chart for example, gives target ranges for brew strength and
extraction yield based on preferences observed in organised taste tests [3]. Brew strength
is the ratio of mass of dissolved coffee in the beverage to volume. Extraction yield is the
percentage of dry coffee grind mass that has extracted as solubles into the water.
The chemistry of coffee brewing has received a great deal of attention in recent times
but the physics of the brewing has received relatively little attention. Very often, as
in other food engineering applications, the importance of the microstructure in mass-
transfer is ignored in extraction models, and solids are treated as “black boxes” [5].
Some work has been done modelling the physics of certain brewing systems. Large scale
industrial extractors for production of instant coffee have been the subject of detailed in-
vestigations. Early work focused on modelling coffee extraction in large packed columns
called diffusion batteries with a focus on improving the design of these solid-liquid ex-
tractors [6, 7]. Some of these developments are summarised in [4]. There has also been
some physical modelling of domestic brewing systems. Experimental investigations have
been carried out into the operation and efficiency of the Moka pot [8, 9]. Fasano et al.
have developed general multiscale models for the extraction of coffee primarily focused
on the espresso coffee machine [10, 11, 12, 13]. Voilley et al. [14] conducted a number of
extraction experiments on a well mixed system of coffee grounds and water and investi-
gated the influence of brewing time, granule size, brew ratio and water temperature on
brew strength. The diffusion equation in a sphere was also found to be useful to describe
the variations in the brew strength of the coffee during the experiments. There has been
very little investigation into the physics of the drip filter brewing system. There are a
number of aspects in the drip filter brewing system, where a greater understanding of the
physical process may lead to improved design and increased quality of coffee produced.
Some of these aspects were investigated by a group of applied mathematicians working
on a problem posed by Philips Research during the ESGI 87 study group with industry
2
in the University of Limerick. The topics investigated are included in the study group
report [15]. Despite these developments, there is an absence in the literature of a first
principles model of coffee extraction which is validated by experiment. This paper aims
to address this deficit.
The aim of this study is to formulate a comprehensive experimentally validated model
of the physics of coffee extraction. The model should include the dissolution and transport
of coffee within the coffee bed. It should also take into account the doubly porous nature
of the coffee bed, which consists of pores between the coffee grains (intergranular), but
also smaller pores within the coffee grains (intragranular). In this paper, flow through a
static, saturated coffee bed, under the influence of a pressure gradient is modelled using
a double porosity model. The parameters in this macroscopic model are related to the
microscopic properties of the coffee bed by an averaging procedure using Representative
Elementary Volumes. This allows the model to be parametrised from experimentally
measured microscopic quantities. Utilising multiscale modelling of extraction from coffee
grains we show that we can quantitatively model extraction from ground coffee in two
situations: in a well stirred dilute suspension of coffee grains and in a packed coffee bed.
In our model, extraction is divided into two regimes. In the first, a rapid extraction
occurs from the surface of the coffee grains which yields the highest concentrations at
any stage of the brewing process. In the second there is a slow extraction at lower
concentrations from the interior of the grains. The model can be easily generalised to
describe standard coffee brewing methods such as french press and drip filter coffee. It
can also be extended to include unsaturated flow in the coffee bed.
2. Coffee extraction experiments
A large number of experiments were carried out to investigate flow and extraction of
coffee from coffee beds of various geometries. Two of these experiments will be outlined
here and used to motivate the development of a mathematical model to replicate their
results. The experiments were performed with a number of different coffee grinds. We
focus on two of these coffee grinds. The first is a relatively fine grind called Jacobs
Kr¨onung (JK) standard drip filter coffee grind. The second is a coarse grind obtained
with a Cimbali burr grinder from Illy coffee beans. The grind used is obtained by setting
the grinder to a coarse setting #20 and will be referred to as Cimbali #20. The grind
size distributions for these grinds are shown in figure 1. It is apparent from the graph
that both distributions are bimodal, having two peaks. A first peak occurs at a particle
size of 25–30 m while the second peak occurs at a larger particle size and gradually
shifts from left to right on the graph with the grind coarseness. The first peak accounts
for single cell fragments: the cell size in coffee particles is 25–50 m. The second peak
accounts for particles comprising intact coffee cells. The grind size distribution is vitally
important in coffee extraction in that it affects both the fluid flow through the grind and
the grind’s extraction kinetics.
2.1. Maximum extractable solubles mass from coffee grind
Extraction from a coffee grain occurs following contact with water. However not all
of the coffee grain mass is soluble. Experiments conducted show that extractable mass
of coffee grains in water at 90 C can range from 28% for very fine grinds to 32% for
3
0.1
1
10
100
1000
104
0
2
4
6
8
10
Particle size HΜmL
Figure 1: Coffee grind size distributions for JK standard drip filter grind (–) and Cimbali #20 grind (- -).
Distributions are expressed in terms of volume fraction percentages of particles of a given diameter.
very coarse grinds. These results were obtained using fine and coarse grinds from Douwe
Egberts (DE) coarse drip filter coffee. The extraction was carried out in glass beakers
by continuously stirring the grind through the water with a magnetic stirrer for at least
5 hours to ensure maximum extraction. Increasing the extraction time from 5 hours to
10 hours did not change the extracted amount. Reduced extraction from larger coffee
particles probably reflects the phenomenon that some solubles cannot be removed from
closed cells inside the larger particle kernels.
2.2. Coffee extraction kinetics during batch-wise brewing in a fixed water volume
The extraction kinetics of the two coffee grinds considered here were investigated by
mixing 60 grams of coffee with a hot water volume, Vwater = 0.5 litres, and measuring
the concentration cbrew of extracted species as a function of time. The temperature of
the liquid during extraction is 80–90 C. This was done by performing batch-wise ex-
traction in a French-press type cylinder and using a piston to separate the brew from the
grounds, after a certain extraction time, by pressing the mixture through a sheet of coffee
filter paper on the cylinder bottom. The solubles concentration cbrew was subsequently
determined by measuring the Brix with a pocket refractometer (PAL-3, Atago, Japan).
It was found for drip filter coffee that 1 Brix corresponds with cbrew = 8.25 g/litre.
The latter calibration factor was obtained by evaporating all the water from the coffee
brew and weighing the remaining non-volatile material. The resulting extraction profiles
are shown in figure 2. Extraction profiles for some other coffee grinds are included in
appendix G.
2.3. Coffee extraction profiles from a cylindrical brewing chamber
A number of coffee extraction experiments were conducted with both a cylindrical
brewing chamber and a conical Melitta filter. We will focus on the cylindrical brewing
4
æ
æ
æ
æ
æ
æ
à
à
à
à
à
à
à
0
100
200
300
400
500
600
0
10
20
30
40
timeHsL
cbrewHkgm3L
Figure 2: Coffee solubles concentration profiles for JK standard drip filter grind ( s
) and Cimbali #20
grind () during batch extraction experiments. In these experiments 60 g of coffee with approximately
4% moisture was mixed with 0.5 l of hot water in a French press type cylinder.
chamber here. In this setup, coffee is placed in a cylindrical flow-through cell and 1 litre
of water at 90 C is forced through the coffee bed using a rotary vane pump. The system
can operate in a constant flow mode with the pressure differential across the bed adjusting
itself to the flow. Alternatively the pressure differential across the coffee bed can be fixed
with the flow adjusting itself to the pressure. The coffee beverage exiting from the flow
through cell is collected in a coffee pot. The solubles concentration of the exiting coffee
(cexit), and the coffee beverage in the pot (cbrew) is measured throughout the extraction.
The extraction profiles of the brewed and exiting coffee solubles concentration, given in
milligrams coffee solubles per gram of coffee beverage here, for both JK standard drip
filter grind and Cimbali #20 grind are shown in figure 3. In the case of the JK standard
drip filter grind, the experiments are repeated for a different coffee bed mass and for
a different absolute pressure of the water. Details of these experiments are included in
appendix G.
We make a number of observations based on the experimental data presented, which
will be used to motivate the development of a mathematical model of the extraction
process. Firstly, it is noted that the experiments tell us nothing about the filling stage,
where the water initially infiltrates the dry coffee bed. To this end it will be assumed
that when the first coffee brew exits, the coffee bed, including the intragranular pores,
is saturated with water. It is noted that in the case of the batch extraction experiment,
in figure 2, the concentration of the coffee brew increases rapidly from zero to over half
its maximum value at the beginning of extraction, before a much slower increase over
a longer timescale towards its maximum value. This phenomena can also be seen in
the cylindrical brewing chamber data in figure 3, where the concentration of the initial
exiting brew is very high and this is maintained for a short time, before there is a large
drop in the exiting concentration to a lower level and this gradually declines over a longer
5
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
0
200
400
600
800
1000
0
50
100
150
200
Mbrew HgramsL
cbrew HmggramL
(a)
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
0
200
400
600
800
1000
0
50
100
150
200
Mbrew HgramsL
cexit HmggramL
(b)
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
0
200
400
600
800
1000
0
20
40
60
80
100
Mbrew HgramsL
cbrew HmggramL
(c)
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
0
200
400
600
800
1000
0
20
40
60
80
100
Mbrew HgramsL
cexit HmggramL
(d)
Figure 3: The coffee solubles concentration, measured in mg/gram, is plotted against mass of coffee
beverage Mbrew (grams) for JK drip filter grind with a flow rate of 250 ml/min and pressure differential
of 2.3 bar in (a) the coffee pot and (b) the beverage at filter exit, and for Cimbali #20 grind with a flow
rate of 250 ml/min and pressure differential of 0.65 bar in (c) the coffee pot and (d) the beverage at filter
exit. The mass of coffee used in both cases is 60 g including approximately 4% moisture. The brewing
cylinder diameter is 59 mm. The coffee bed heights are 4.05 cm for JK drip filter grind and 5.26 cm for
Cimbali #20 grind.
period. This is consistent with the findings in [14] where it was noted that extraction
yield reached 90% of its final value within one minute. Large-sized roasted particles in
the coffee grind feature a kernel comprising undamaged closed cells and a particle skin
formed by damaged open cells. The grind distribution as seen in figure 1 also contains
a significant proportion of fines or damaged cells. Thus the reason for the fast initial
extraction may be due to reduced mass transfer resistances in the damaged particle skin
and in the fines. The slow extraction may then be due to mass diffusion from intact cells
in particle kernel. Thus the rate limiting step is diffusion from the particle kernel.
3. Mathematical modelling
3.1. Basic modelling assumptions
It is assumed here that the coffee brewing process can be broken into three stages.
Initially in the filling stage, hot water is poured on the dry coffee grounds and begins to
6
fill the filter, but doesn’t leave. Next in the steady state stage the bed is saturated, water
is still entering the bed, but also leaving at the same rate. In the last stage, the draining
stage, no more water enters the bed but it still drains out. In the absence of experimental
data to cover the other stages, only the steady state stage is considered here. However,
the model developed can be easily generalised to include the unsaturated flow, during
the filling and draining stages. In the steady state stage the coffee bed is considered
as a static, saturated porous medium with the flow driven by a pressure gradient. This
pressure gradient may be mechanically applied as in an espresso machine or hydrostatic
as in a drip filter machine. The bed is composed of a solid matrix of coffee grains which
are themselves porous. As the filling stage is not modelled here, initial conditions for
the steady state stage will have to be estimated or inferred from experimental data.
Furthermore any swelling of the coffee grains due to the addition of water will not be
modelled. It is assumed this swelling would occur during the filling stage and would
manifest itself in the steady state stage possibly as a slight shift to the right of the grind
size distribution. While coffee is composed of over 1800 different chemical compounds
[1], in this model we will just consider a single entity and model coffee concentration or
brew strength in line with the experimental data shown. The model can of course be
generalised to model the concentration of any number of coffee constituents. Modelling
a multicomponent system such as coffee with a single component is a simplification and
requires some justification. Firstly, as mentioned in the introduction, relating taste to
the concentration of the different coffee components in a beverage is a non-trivial matter
and currently no ideal recipe exists. Apart from professional tasters, the most widely
used measure of coffee quality is the coffee brewing control chart. This chart gives target
ranges for brew strength and extraction yield. This chart is used by both the Speciality
Coffee Association of Europe (SCAE) and the Speciality Coffee Association of America
(SCAA). Given that the most widely used measure of coffee quality considers coffee as
a single component, it seems logical to do so as well. Secondly, experiments carried
out suggest that many important coffee components have similar extraction kinetics (see
appendix F).
3.2. Coffee bed structure
The structure of the coffee bed is central to the extraction process. It is immediately
obvious that the bed consists of two phases. Following [16] the highly permeable phase
consisting of the pores between the coffee grains is called the h-phase. Similarly the low
permeability phase consisting of the coffee grains is called the l-phase. At a microscopic
level there are two phases within the coffee grains. The pore or void space within the
grains is called the v-phase, while the solid coffee cellular matrix is referred to as the
s-phase. The coffee bed has three fundamental length scales. The smallest of these is
the size of the pores within the grains or the size of a coffee cell which may be 25–50 m.
The average size of a coffee grain in the grind size distribution by volume, excluding
single cell fragments, will be approximately an order of magnitude bigger. Finally the
size (depth) of the coffee bed will typically be a few centimetres. In order to model
the transport of coffee and water in the bed conservation equations can be formed at
each of these scales. On the microscale (cell size scale), conservation equations can
be formed in the h-phase, v-phase and s-phase. At the intermediate (grain size) scale
conservation equations can be formed in the h-phase and the l-phase. We refer to this as
the mesoscale. At this scale an individual coffee grain is represented by two overlapping
7
continua representing the void and solid phases within the grain. At the macroscale
(coffee bed scale), the coffee bed is represented by three overlapping continua representing
the h-phase, v-phase and s-phase. To reconcile the three representations we can use the
methods of homogenisation or volume averaging. Volume averaging will be adopted here.
A schematic of the volume averaging process is shown in figure 4. This is very useful
since it relates the averaged macroscopic quantities to the physical parameters at the
microscale. Some macroscopic parameters may be measured by experiment, while others
can be found from their averaged representation in terms of measurable microscopic
quantities. The influence of microscale properties on the macroscale system parameters
can be easily identified.
Figure 4: Macroscale equations are matched to microscale equations using volume averaging. At a
macroscopic level the system is represented by three overlapping continua for the intergranular pores
(h-phase), intragranular pores (v-phase) and solid coffee (s-phase).
3.3. Coffee bed description
The coffee bed is represented by a porous medium domain ΩTwith volume VT. The
domain can be split into an intergranular pores domain Ωh, with volume Vhand a coffee
grain domain Ωl, with volume Vl. Ωlis further split into an intragranular pore domain
v, with volume Vvand a solid coffee domain Ωs, with volume Vs. Clearly the equalities
Vh+Vl=VTand Vv+Vs=Vlhold. The following volume fractions are now defined
8
φh=Vh
VT
, φl=Vl
VT
, φv=Vv
Vl
, φs=Vs
Vl
,(1)
which leads to
φh+φl= 1, φv+φs= 1.(2)
The concentrations (mass per unit volume) of coffee in the respective phases are ch,cv
and cs.vhand vvdenote the fluid velocity in the h-phase and v-phase respectively.
The velocity of the solid will be denoted by vs. Further notation will be introduced as
required. The formulation of the equations presented here will follow [16] and [17]. Since
conservation equations will be formed at three different scales the variables at each scale
will be denoted as macroscale (˜
·), mesoscale (·) and microscale (·) to avoid ambiguity.
3.4. Microscale point balance equations
The point balance equations for coffee and liquid within each of the phases on the
microscale are
3.4.1. h-phase
∂ch
∂t =·(chvh+jh),(3)
∂ρh
∂t =·(ρhvh).(4)
3.4.2. v-phase
∂cv
∂t =·(cvvv+jv),(5)
∂ρv
∂t =·(ρvvv).(6)
3.4.3. s-phase
∂cs
∂t = 0.(7)
The microscopic balance equations include the terms jhand jvwhich represent molecular
diffusion of coffee solubles in the respective phases. Molecular diffusion in the solid phase
is assumed negligible.
3.5. Mesoscale point balance equations
The mesoscale balance equations are only required in the grains (l-phase) since only
two scales are needed in the h-phase. Thus the point balance equations for coffee and
liquid in the v-phase and the s-phase on the mesoscale are
9
3.5.1. v-phase
∂t (φvc
v) = ·(φv(c
vv
v+j
v)) f
vs.(8)
∂t (φvρ
v) = ·(φv(ρ
vv
v+i
v)).(9)
3.5.2. s-phase
∂t (φsc
s) = ·(φsj
s)f
sv.(10)
The terms f
vsand f
svare source/sink terms representing transfer of coffee solubles
across the vs-interface and vice-versa. The term i
vaccounts for any mechanical dispersion
in the fluid velocity.
3.6. Macroscale point balance equations
For each phase there are macroscopic point balance equations for mass of the coffee
and mass of liquid (flow equations). The solid is assumed stationary. The macroscopic
equations take the form:
3.6.1. h-phase
∂t (φh˜ch) = ·(φhch˜vh+˜
jh)) ˜
fhl,(11)
∂t (φh˜ρh) = ·(φhρh˜vh+˜
ih)) ˜
fw
hl.(12)
3.6.2. v-phase
∂t (φlφv˜cv) = ·(φlφvcv˜vv+˜
jv)) ˜
fvh˜
fvs,(13)
∂t (φlφv˜ρv) = ·(φlφvρv˜vv+˜
iv)) ˜
fw
lh.(14)
3.6.3. s-phase
∂t (φlφs˜cs) = ·(φlφs˜
js)˜
fsh˜
fsv.(15)
˜
fαβis transfer of coffee solubles from the α-phase to the β-phase across the αβ inter-
face. Similarly ˜
fw
αβis transfer of liquid from the α-phase to the β-phase across the αβ
interface.
10
3.7. Upscaling from microscale to macroscale
As mentioned the conservation equations at each of the scales can be related by
representing the properties of the medium at a larger scale by averaging the properties
at the smaller scale. This is useful to find the forms of the mass transfer terms in the
macroscopic equation. An outline of the general upscaling procedure based on [16, 17]
is included in appendix A. The details of upscaling in this case are included in appendix
B.
4. Developing macroscale equations
The macroscopic balance equations in section 3.6 are in a quite general form. Some
assumptions have already been made but in order to simplify things we make some further
assumptions. We also need to introduce terms to model the transport of fluid and coffee
within the bed. Firstly we assume that the density of the liquid is constant and does not
change with coffee concentration. This is consistent with [1] where it is noted that the
material extracted from coffee has little influence on liquid density. Thus ˜ρh= ˜ρv=ρ.
It is also assumed that no transport occurs within the l-phase on the macroscale. This
simply means that liquid or coffee does not transport directly from grain to grain within
the bed. Any mechanical dispersion in the flow in the h-phase is not considered. Due to
continuity of flux at the interphase boundaries we have ˜
fαβ=˜
fβα. Thus the five
macroscopic equations reduce to the following
∂t (φh˜ch) = ·(φhch˜vh+˜
jh)) + ˜
fvh+˜
fsh,(16)
ρ∂φh
∂t =ρ·(φh˜vh) + ˜
fw
lh,(17)
∂t (φlφv˜cv) = ˜
fvh+˜
fsv,(18)
ρ
∂t (φlφv) = ˜
fw
lh,(19)
∂t (φlφs˜cs) = ˜
fsh˜
fsv.(20)
It now remains to introduce expressions to model the fluid velocity ˜vh, the total macro-
scopic flux ˜
jh, the fluid mass transfer term ˜
fw
lhand the coffee mass transfer terms ˜
fvh,
˜
fsvand ˜
fshin terms of the system variables. The main transfers occurring in the
coffee bed are shown in figure 5.
4.1. Fluid velocity
Darcy’s Law allows us to relate an averaged velocity or discharge in the pores to the
pressure gradient. The relations in the h-phase are given by
˜uh=φh˜vh,˜uh=˜
kh
µ(˜ph+ρg),˜
kh=˜
kh(φh),(21)
where ˜phis the macroscopic pressure gradient in the h-phase, ˜
khis the permeability and
µis the viscosity of water.
11
Water
Reservoir
Intergranular
Pores
Coffee
Pot
Intragranular
Pores
Coffee Solids
Water in
Coffee
Brew
Out
Water to
Grains
Coffee Dissolving
in grains
Coffee
Dissolving
From
Surface
Coffee Diffusion
From Bulk
Figure 5: Mass transfers occurring in the coffee bed.
4.2. Total macroscopic flux
The total macroscopic flux, ˜
jhis made up of the macroscopic average of molecular
diffusion ja
hand the dispersive flux jb
h:
˜
jh=hjhih+h˚ch˚vhih=ja
h+jb
h.(22)
For an isotropic porous medium, ja
his often modelled by
ja
h=D
τ˜ch,(23)
where τis the tortuosity defined by
τ=Le
L=actual path length
macroscopic path length.(24)
The tortuosity must be estimated as a function of the porosity. Various functional
relationships are proposed in the literature. Some of these are discussed in [18]. The
expression used here is τ=φ1
3
hwhich is adopted from [19]. Thus we have
ja
h=D
τ˜ch=φ
1
3
hD˜ch.(25)
4.3. Dispersive flux
Dispersion occurs due to variations in the microscopic velocity of the phase with
respect to the averaged velocity, and molecular diffusion [16]. Thus molecular diffusion
contributes to the dispersive flux in addition to the diffusive flux at the macroscopic level.
In general the dispersive flux is given by
jb
h=h˚ch˚vhih=˜
Db·˜ch,(26)
12
where ˜
Dbis a rank 2 tensor called the dispersion tensor. ˜
Dbis both positive definite and
symmetric. For an isotropic porous medium the following expression is often used:
Dij =aTδij + (aLaT)vivj
v2v. (27)
The coefficients aLand aThere are the longitudinal and transverse dispersivities of the
porous medium. vi=hviihis the average velocity in the i-th direction and v=|v|where
vis the average velocity vector in this instance. δij is the Kronecker delta. Further detail
on the diffusive and dispersive fluxes is included in appendix C.
4.4. Coffee mass transfer terms
Experimental results in section 2 suggest that there are two fundamental extraction
mechanisms from the coffee grains. A rapid extraction from fines (single cells fragments)
and damaged cells on the surface of larger particles and a slower extraction from the
kernels of larger particles. Various different models have been applied to represent such
situations, particularly in the area of supercritical fluids. The main models are reviewed
in [20]. These models include linear driving force, bi-linear driving force, shrinking core,
broken plus intact cells in series, broken plus intact cells in parallel and a combined broken
plus intact cells with shrinking core model. These models are widely used in modelling
extraction with supercritical fluids and have been considered in a number of papers, both
theoretical and experimental, including [21, 22, 23, 24]. In [21] it is noted that the broken
plus intact cell model typically features an initial constant extraction period dominated
by extraction from the broken cells, then a falling extraction rate period as broken cells on
the surface are depleted and finally a diffusion controlled period dominated by extraction
from the intact cells. These general features are evident in the extraction experiments
in section 2. The models used here are quite similar to the broken plus intact cells in
parallel. The extraction term ˜
fvhis transfer from the intragranular pores in the grains
to the intergranular pores and is similar to extraction from intact cells. The term ˜
fsh
is direct extraction from the solid grain matrix surface into the intergranular pores and
is similar to extraction from the broken cells. The term ˜
fsvmodels dissolution of coffee
solubles from the cell walls into the intragranular pores. The transfer ˜
fvhacross the
interphase boundary is assumed to occur by diffusion according to Fick’s first law. Thus
˜
fvh=αvh(˜cv˜ch).(28)
where αvh is the mass transfer coefficient. The form of αvh can be found from the volume
averaging procedure as
αvh = (1 φh)φ
4
3
vDv
S
hl
l
,(29)
where Dvis the diffusion coefficient of coffee in water, S
hl is the specific surface area of the
l-phase and ∆lis some length scale characterising the distance over which diffusion occurs.
The transfers ˜
fsvand ˜
fvhare solid-fluid transfer rather than fluid-fluid transfer and
so the modelling is slightly different. Under consideration here is the case where the solid
matrix itself is dissolving. It is assumed that there is a thin layer of liquid next to the
solid which is always saturated with solute. This concentration is denoted by csat.csat is
the concentration in the liquid phase that would be in equilibrium with the concentration
13
inside the solid ˜cs. The force of extraction from this thin layer to the bulk of the fluid is
assumed to be proportional to the difference in concentration between the thin layer and
the bulk of the fluid. Thus, again using volume averaging to determine mass transfer
coefficient, we have
˜
fsh= (1 φh)(1 φv)Dh
S
hl
s
(csat ˜ch),(30)
where ∆sis some length scale characterising the distance over which diffusion occurs.
Similarly
˜
fsv= (1 φh)(1 φv)Dv
S
sv
s
(csat ˜cv),(31)
where S
sv is the specific surface area of the s-phase. The transfer terms are considered
in more detail in appendix D.
4.5. Liquid transfer term
It is also possible to have a transfer of liquid from the intergranular pores to the
intragranular pores (or vice versa). This could occur for example due to a pressure
imbalance in the phases due to the dissolution of the solid matrix within the grains.
Using Darcy’s Law and volume averaging we find that
˜
fw
lh=(1 φh)S
lhρkv(φv)
µ
(˜ph˜pv)
l
,(32)
where kv(φv) is the coffee grain permeability and ˜phand ˜pvare the water pressures in
the h-phase and the v-phase. Since there is a difference in concentration in the h and
v-phases the transfer of fluid in either direction would be expected to also result in the
transfer of solute from one phase to another. This can be included as an extra coffee
mass transfer term as
fw
lh=((1 φh)S
lhρkv(φv)
µ
ph˜pv)
l˜ch,if ˜ph˜pv
(1 φh)S
lhρkv(φv)
µ
ph˜pv)
l˜cv,if ˜ph<˜pv
(33)
4.6. Coffee bed properties
In order to proceed the quantities such as permeability, surface area, specific surface
area and distances over which diffusion occurs must be expressed in terms of measurable
properties of the coffee bed. We can estimate these quantities from the particle size
distribution. In order to simplify things we assume that the intergranular porosity φhis
constant. We note that this means that the only change in porosity occurs within the
l-phase. Thus it seems reasonable to assume that the pressure in the intergranular pores
is always greater than or equal to the pressure in the intragranular pores, i.e. ˜ph˜pv.
Thus (33) simplifies to
˜
fw
hl= (1 φh)S
hlρkv(φv)
µ
(˜ph˜pv)
l
˜ch.(34)
14
4.6.1. Specific surface area of l-phase
Approximating a coffee grain of diameter xby a sphere, it has a surface to volume
ratio of 6
x. This can be improved upon if the roundness or sphericity of the coffee grain
is known. However in the absence of this we make the spherical approximation. We can
use a number of equivalent spherical diameters to represent the entire size distribution.
One such diameter is the diameter of the spherical particle that has the same specific
surface area of that distribution. This is called the Sauter mean diameter and is defined
by
ksv =6
Sv
,(35)
where Svis the surface to volume ratio of the distribution which can be found from
the data. The assumption φhis constant means that the Sauter mean diameter doesn’t
change. Thus where required the specific surface area of the l-phase is given by
S
hl =6
ksv
.(36)
In fact we will use two separate Sauter mean diameters. In relation to flow and extraction
from the surface of the grains we need to use the specific surface area of the entire
distribution. We denote the corresponding Sauter mean diameter ksv1. However when
dealing with extraction from the grain kernel, we should ignore the particles which are
just broken cell fragments and don’t have a kernel of intact cells. In this case we introduce
a second Sauter mean diameter ksv2which is representative of the specific surface area of
particles above a certain diameter, chosen here to be 50 m. This Sauter mean diameter
is used for extraction from the grain kernel.
4.6.2. Specific surface area of s-phase
The specific surface area of the s-phase is more difficult to estimate. Assuming that
the coffee grain is made up of solid spherical cells with the same diameter mwe could
approximate the specific surface area by
S
sv =6
m.(37)
However we don’t have any information about the specific surface area of the s-phase
from the grind size distribution and in practice this will form part of a lumped mass
transfer coefficient which will have to be fitted to the experimental data.
4.6.3. Permeability kh
The permeability can be estimated using the Kozeny-Carman equation for spheres
[25]:
kh=φ3
h
κ(1 φh)2S
hl
2.(38)
Here again S
hl is the specific surface area, while κis a factor which accounts for the
shape and tortuosity. Utilising the derived form for S
hl gives
kh=k2
sv1φ3
h
36κ(1 φh)2=k2
sv1φ3
h
36κ(1 φh)2.(39)
15
The shape factor κis usually taken to be in the range 2–6. Experiments performed
measuring the pressure drop in an airflow through compacted coffee beds estimate it at
κ= 3.1.
4.6.4. Permeability kv
Similarly the permeability of the grain is estimated using the Kozeny-Carman equa-
tions for the spherical cells so that
kv=m2φ3
v
180(1 φv)2.(40)
In the absence of experimental data we have chosen κ= 5 which is often adopted [26].
4.6.5. Average diffusion distances
Expressions are required for ∆land ∆s. The distance over which diffusion from the
grains to the large pores occurs, is assumed to be equal to the mean radius of the grains
weighted by volume which we denote ∆l=ll. The distance over which diffusion occurs
from the surface of the solid to the h-phase is assumed to be equal to the mean radius
of the coffee cells. Thus ∆s=ls.
4.6.6. Coffee extraction limits
The current forms of the coffee mass transfer terms are incomplete as they do not
account for the fact that for a given grind at a given temperature there is a maximum
amount of coffee that can be extracted. Looking at our equations, in order to track the
amount of coffee extracted we can either allow the grain porosity φvto change or allow
the solid coffee concentration ˜csto change, or both. The decision is made here to allow
φvto change as coffee is extracted and consider the solid concentration ˜csfixed. Thus
φvis allowed to increase as coffee is dissolved until it reaches a point when φsconsists of
insoluble material. This maximum value may depend on the grind size and temperature
of extracting water and should be found from the experimentally determined maximum
extractable solubles mass. It is also unrealistic for the mass transfer coefficients from
the solid matrix surface to remain approximately constant even when there is very little
soluble coffee left in the solid. This issue is sometimes solved through use of a partition
coefficient. Here we make the simple assumption that the extraction term be proportional
to the amount of coffee on the surface. Let φcbe the volume fraction of coffee in the
grains. We divide this into coffee in the surface of the grains and fines φs,s, and coffee in
the grain kernels φs,b, so that φc=φs,s +φs,b . The initial volume fractions of coffee in dry
grains are φc0,φs,s0and φs,b0. Assuming that the soluble coffee is uniformly distributed
within the grains the initial volume fraction everywhere is φc0. Thus at a given time the
volume fractions of soluble coffee on the surface and in the grain kernels are given by
φc0
φs,s0
φs,s(x, t) = φc0ψs(x, t),(41)
φc0
φs,b0
φs,b(x, t) = φc0ψv(x, t),(42)
16
where xis the position within the coffee bed and ψsand ψvare the fractions of the
original amount of coffee left on the grain surfaces and in the grain kernels respectively.
We can now substitute φs=φs,i +φs,s +φs,b.φs,i represents the volume fraction of
insoluble solid in the grains. Using the expressions we have developed so far this leads
to two further partial differential equations for ψsand ψv:
∂ψs
∂t =12Dhφc0
ksv1mcsat ˜ch
˜csrsψs,(43)
∂ψv
∂t =12Dvφc0
m2csat ˜cv
˜csrvψv,(44)
where rs=1
φs,s0and rv=1
φs,b0.
4.7. Macroscale equations
Models have now been introduced for the various processes occurring in the coffee
bed. The description of the process has been extended to seven coupled partial differential
equations. These equations are presented in full in (45)-(51). Different presentations of
the equations are possible and one equation can be reduced to an algebraic one. Boundary
conditions will depend on the geometry of the problem. Initial conditions will have to
be determined or inferred from experiment following the filling stage.
φh
˜ch
∂t =k2
sv1φ3
h
36κµ(1 φh)2·ch(˜ph+ρg))
+φ
4
3
hDh2˜ch+φh˜
Db·2˜ch
(1 φh)φ
4
3
vDv
6
ksv2ll
ch˜cv)
+ (1 φh)12Dhφc0
ksv1m(csat ˜ch)ψs
6(1 φh)m2
180µksv2ll
φ3
v
(1 φv)2(˜ph˜pvch,(45)
0 = k2
sv1φ3
h
36κµ(1 φh)2·(˜ph+ρg)
+6(1 φh)m2
180µksv2ll
φ3
v
(1 φv)2(˜ph˜pv),(46)
∂t (φv˜cv) = +φ
4
3
vDv
6
ksv2ll
ch˜cv)
+12φc0Dv
m2(csat ˜cv)ψv
+6(1 φh)m2
180µksv2ll
φ3
v
(1 φv)2(˜ph˜pvch,(47)
17
∂φv
∂t =6m2
180µksv2ll
φv3
(1 φv)2(˜ph˜pv),(48)
∂φv
∂t =1
rs
∂ψs
∂t 1
rv
∂ψv
∂t ,(49)
∂ψs
∂t =12Dhφc0
ksv1mcsat ˜ch
˜csrsψs,(50)
∂ψv
∂t =12Dvφc0
m2csat ˜cv
˜csrvψv,(51)
While these equations give estimates for the different coefficients in terms of process
parameters, these parameters may not always be easy to determine accurately. Other
processes which were not considered may also affect the transport of coffee and water.
Thus when comparing to experiment it is necessary to introduce fitting parameters par-
ticularly to the terms controlling extraction from the grain surfaces and diffusion of coffee
from the grain kernel.
5. Numerical simulations
Numerical simulations of the experiments in sections 2.2 and 2.3 are conducted using
the model equations. The equations can be reduced in both of these cases.
5.1. Batch-wise brewing
In the french press type brewing apparatus there is no pressure induced flow and we
assume the solution in the h-phase is well mixed since only the average concentration
in this phase is measured anyway. Thus all spatial derivatives in the equations drop
out leaving a system of ordinary differential equations. In order to solve the system we
require some initial conditions. Here we need to make some assumptions since we do
not model the initial infiltration of the water into the grains when the coffee and water
are mixed. Firstly, we assume that the grains are saturated with water initially. We
further assume that the initial concentration in the h-phase is zero, so that none of the
coffee has dissolved from the surface of the grains. It is also necessary to give an initial
condition for the concentration in the v-phase. It is assumed initially that all the soluble
coffee in the grains has dissolved into the v-phase. This is of course unlikely to be the
case. However, since coffee diffusion from the v-phase to the h-phase seems to be the
rate limiting process, occurring much slower than dissolution of coffee, this assumption
is unlikely to have much impact on the simulation results. We also assume that, only a
change in porosity in the intact cells in the grain kernel will result in a change in pressure
in the v-phase, since any pressure imbalance in the damaged cells on the grain surface
would be almost instantaneously corrected. Thus, as all coffee in kernel is assumed
dissolved initially, no pressure difference between phases will occur. Note the value of
intragranular porosity φv= 0.56 is adopted here for dry coffee in air. Allowing for these
assumptions the following system of equations are solved numerically in MATLAB R
.
18
φh
d˜ch
dt =α(1 φh)φ
4
3
vDv
6
ksv2ll
ch˜cv)
+β(1 φh)12Dhφc0
ksv1m(csat ˜ch)ψs,(52)
d
dt (φv˜cv)=+αφ
4
3
vDv
6
ksv2ll
ch˜cv),(53)
v
dt =1
rs
∂ψs
∂t ,(54)
s
dt =β12Dhφc0
ksv1mcsat ˜ch
˜csrsψs,(55)
with initial conditions
˜ch(0) = 0,˜cv(0) = cv0, φv(0) = φv0, ψs(0) = 1.(56)
The parameters αand βare used as fitting parameters to fit the experimental results.
The mass transfer coefficients depend on the volume fraction of the interstitual water
in the coffee bed in question, so the fitting parameters are just intended to correct for
errors in the other parameters in the mass transfer coefficient. Thus we look for values
of αand βwhich fit both experiments equally well. The initial volume fractions of
coffee φs,s0and φs,b0can, to some extent, be estimated from the grind size distribution
but here we will use them to allow for any differences between the batch extraction and
coffee flow-through-cell experiments. These differences may arise, for example, due to
the fact that we do not model the initial water infiltration into the coffee grains. The
parameters used in the simulations are listed in table 1. The other parameters are all
measured experimentally, estimated from experiments or sourced in the literature. The
value for the coffee solubility csat is estimated from the highest observed concentration
across the four experiments. The diffusion coefficient is for caffeine in water at 80 C [27].
This would be slightly higher at 90 C. However this is only an estimate of the effective
diffusion coefficient of coffee in water. The fitting parameters are used to correct any
errors in these parameters. The comparison between the numerical solution and the
experimental results is shown in figure 6.
5.2. Cylindrical brewing chamber
The cylindrical brewing chamber geometry also allows us to make some simplifications
to the general model. Assuming that the coffee bed properties are homogeneous in any
cross section, the equations can be reduced to one spatial dimension, parallel to the
flow direction. Thus the bed depth is labelled by the z-coordinate. The height of the
coffee bed is Lwith the bottom (filter exit) at z= 0 and the top (filter entrance) at
z=L. For the experiments here it is shown in appendix E that advection dominates over
diffusion and mechanical dispersion, so these two processes are neglected. This means
we just require one concentration boundary condition for ˜ch. We use the condition
that the water entering at the top z=Lhas zero concentration. The pressure boundary
conditions are ˜ph= ∆pat z=Land ˜ph= 0 at z= 0. pis the pressure difference across
19
Parameter JK Drip filter Cimbali #20
φv00.6444 0.6120
φh0.8272 0.8272
ksv127.35 m 38.77 m
ksv2322.49 m 569.45 m
ll282 m 463 m
Dh=Dv2.2×109m2s12.2×109m2s1
ρ965.3 kg m3965.3 kg m3
µ0.315 ×103Pa s 0.315 ×103Pa s
m30 m 30 m
csat 212.4 kg m3212.4 kg m3
˜cs1400 kg m31400 kg m3
κ3.1 3.1
φc00.143435 0.122
φs,s00.059 0.07
φs,b00.084435 0.052
α0.1833 0.0881
β0.0447 0.0086
rs16.94 14.28
cv0183.43 kg m3118.95 kg m3
Table 1: Parameters for simulation of the batch extraction experiments.
0 100 200 300 400 500 600
0
5
10
15
20
25
30
35
40
t (s)
Ch (kg/m3)
(a)
0 100 200 300 400 500 600
0
5
10
15
20
25
30
35
t (s)
Ch (kg/m3)
(b)
Figure 6: Comparison between numerical solution (–) with parameters from table 1 and experiment (*)
for the batch extraction experiments for (a) JK drip filter grind and (b) Cimbali #20 grind.
the bed. In these experiments there is a large pressure difference across the bed which
can lead to bed compaction and a reduction in the porosity. Thus the porosity cannot be
measured a priori. It assumed that φhadjusts to the pressure while φvstays constant.
We then choose φhby matching the volume flow from Darcy’s Law and the Kozeny-
Carman equation to the experimental volume flow. Once again the initial conditions
need to be inferred. It is assumed that once brewed coffee starts to flow from the bed
20
that the bed is fully saturated with liquid. Again it is assumed that all the coffee in the
grain kernels has dissolved into the intragranular pores (v-phase) during filling and this is
uniformly distributed in the bed. It is also necessary to estimate the initial concentration
˜chas a function of z. Based on the initial exiting concentrations we assume for the fine
grind (JK) that initially ˜chis at the coffee solubility throughout the h-phase. For the
coarser grain, (Cimbali # 20), we assume a linear concentration profile in the h-phase,
rising from zero at the top to the initial exiting concentration at the bottom of the
bed. Thus it is assumed that some extraction from the fines and broken surface cells
occurs during filling. To this end the amount of surface coffee is uniformly reduced by
the corresponding amount of coffee initially present in the h-phase. In reality we would
expect more extraction would have occurred at the top of the bed during filling than the
bottom, but, in the absence of experimental guidance, we make the simplest assumption.
As with the batch experiments the assumptions mean that ˜ph= ˜pv. Based on these
assumptions the reduced set of equations to model extraction in the cylindrical brewing
chamber is given by
φh
˜ch
∂t =k2
sv1φ3
h
36κµ(1 φh)2
∂z ˜ch˜ph
∂z +ρg
α(1 φh)φ
4
3
vDv
6
ksv2ll
ch˜cv)
+β(1 φh)12Dhφc0
ksv1m(csat ˜ch)ψs,(57)
2˜ph
∂z2= 0,(58)
∂t (φv˜cv) = +αφ
4
3
vDv
6
ksv2ll
ch˜cv),(59)
∂φv
∂t =1
rs
∂ψs
∂t ,(60)
∂ψs
∂t =β12Dhφc0
ksv1mcsat ˜ch
˜csrsψs,(61)
for
t > 0,0<z < L, (62)
with initial conditions
˜ch(z , 0) = ch0(z),˜cv(z, 0) = cv0,(63)
φv(z, 0) = φv0, ψs(z, 0) = ψs0,(64)
and boundary conditions
˜ch(L, t) = 0,˜ph(0, t)=0,˜ph(L, t) = ∆p. (65)
21
The initial concentration profile in the JK drip filter grind is given by ch0(z) = csat.
In the Cimbali #20 grind the initial concentration is given by ch0(z) = cmax
L(Lz).
Some of the parameters are the same as those in the batch extraction case. Any new
parameters or parameters that have changed are included in table 2. The numerical
solution is found using finite differences in the spatial direction and the method of lines.
It should be noted that any initial discontinuities in the initial and boundary conditions
will be smoothed out by numerical diffusion. If accuracy is required in the initial stages
a large number of steps should be used in the spatial partition. The comparison between
the numerical solution and the experimental results is shown in figure 7.
Parameter JK Drip filter Cimbali #20
φv00.6231 0.6218
φh0.2 0.25
cmax 82.63 kg m3
cs1400 kg m31400 kg m3
φs,s00.11 0.07
φs,b00.033435 0.052
α0.1833 0.0881
β0.0447 0.0086
rs9.09 14.28
cv078.88 kg m3118.95 kg m3
p230 000 Pa 65 000 Pa
L0.0405 m 0.0526 m
Table 2: Parameters for simulation of the cylindrical brewing chamber extraction experiments.
0 0.2 0.4 0.6 0.8 1
0
50
100
150
200
250
Mbrew (kg)
Cexit (kg/m3)
(a)
0 0.2 0.4 0.6 0.8 1
0
10
20
30
40
50
60
70
80
90
100
Mbrew (kg)
Cexit (kg/m3)
(b)
Figure 7: Comparison between numerical solution (–) with parameters from table 2 and experiment (*)
for the cylindrical brewing chamber extraction experiments for (a) JK drip filter grind and (b) Cimbali
#20 grind.
22
6. Conclusion
The coffee extraction process can be described very effectively using mathematical
models. In this paper a general model is introduced to describe coffee extraction by hot
water from a bed of coffee grains. The coffee bed is modelled as a saturated porous
medium using a double porosity model. The bed consists of two kinds of pores: pores
between the grains (intergranular) and pores within the grains (intragranular). Flow
of liquid within the coffee bed is modelled using Darcy’s Law and the Kozeny-Carman
equation. Motivated by experiment, extraction of coffee from the coffee grains is modelled
using two mechanisms. Coffee on the damaged grain surface and in coffee cell fragments
or fines extracts quickly into the intergranular pores due to a relatively low mass transfer
resistance. Coffee in intact cells in the grain kernels, first extracts into the intragranular
pores and then slowly diffuses through the grain into the intergranular pores.
The model is parametrised using experimental data. Numerical simulations are per-
formed and fitted to data. It is shown that the developed models can quantitatively
describe extraction from ground coffee in two situations: in a well stirred dilute suspen-
sion of coffee grains, and in a packed coffee bed. The extraction curves fit the data for
both a fine and a coarse grind as the parameters in the model vary with the surface
area and the mean grain radius of the grind size distribution. Provided the grind size
distribution is known, and the physics of extraction is the same, the model should work
well for an even wider range of grind sizes.
The model described can be easily generalised to describe standard coffee brewing
techniques. It can also be extended to include unsaturated flow during water infiltration
into the coffee bed and drainage of water from the coffee bed. The model is also adaptable
to different bed geometries. The model is presented for isothermal conditions but may
also be extended in future work to include heat transfers within the coffee bed.
Acknowledgements
The authors acknowledge the support of MACSI, the Mathematics Applications Con-
sortium for Science and Industry (www.macsi.ul.ie), funded by the Science Foundation
Ireland Investigator Award 12/IA/1683.
Appendices
A. General upscaling procedure
The following procedure is mainly adapted from [17] and [16]. In order to compare
microscopic and macroscopic equations a procedure is needed for upscaling or averaging
equations from a smaller scale to a larger scale. A general upscaling procedure from a
microscopic scale to a macroscopic scale is outlined here.
Let the general form of a point balance equation for the concentration of a γ-species
per unit volume of the α-phase be
∂cα
∂t =·(cαvα+jα) + Gα,(66)
23
where cαis the concentration of the γ-species, vαis the fluid velocity in the α-phase,
jαis the diffusive flux of the γ-species in the α-phase, and Gαis a source or sink of the
γ-species in the α-phase.
Let βrepresent all other phases in the porous medium. The αsubscript will be
dropped where convenient. Equation 66 is the microscopic balance. The corresponding
macroscopic balance equation is
∂t (φαc) = ·(φαcv+φαj) + φαG,(67)
where φαis the volume fraction of the α-phase and the ·variables are macroscopic
variables.
To relate (66) and (67) the averaged macroscale properties are defined by averaging
(or integrating) (66) over an appropriate REV or Representative Elementary Volume.
The length scale of an REV is much greater than the pore scale but much less than the
full scale of the system. So if lis the length scale of an REV, dis the pore length scale
and Lis the full problem length scale then
dlL. (68)
The following notation is used
The volume of an REV is δV .
The portion of the volume of an REV occupied by the α-phase is δV α.
The union of the interfacial regions within the REV between the α-phase and a
β-phase is denoted Sαβ .
The unit vector normal to this surface oriented outward from the α-phase is nα.
To average the microscopic balance equations two theorems that transform the average
of a derivative to the derivative of an average are needed. The time averaging theorem
is given by
ZδV
∂F
∂t dV =
∂t ZδV
F dV X
β6=αZSαβ
nα·wbF|αdS, (69)
where Fis some scalar field property of the microscale and F|αjust indicates that the
microscale property Fin the α-phase is being integrated over the αβ-interface. wbis
the velocity of Sαβ. The summation Pβ6=αjust denotes a summation over all phases
except the α-phase.
The divergence averaging theorem is given by
ZδV
·BdV =·ZδV
BdV +X
β6=αZSαβ
nα·B|αdS, (70)
where Bis some vector field property of the microscale and B|αjust indicates that Bis
being integrated over the αβ-interface. The summation Pβ6=αjust denotes a summation
over all phases except the α-phase.
Next some phase averages are defined
24
The Intrinsic Phase Average of a quantity F over the α-phase is defined by
hFiα=Fα=1
δV αZδV α
F dV. (71)
The Phase Average of a quantity F over the α-phase is defined by
hFi=F=1
δV ZδV α
F dV =φαFα.(72)
The Mass Weighted Average of a quantity F over the α-phase is defined by
hF0iα=F0α=1
hρiαδV αZδV α
ρF dV =hρF iα
hρiα.(73)
It will also be necessary to be able to write the average of a product of the form hcviα
in terms of the individual averages hciαand hviα. To do this consider the following
Let ˚v =v− hviαbe the deviation of the true velocity from the mean velocity.
Let ˚c=c− hciαbe the deviation of the true concentration from the mean concen-
tration.
Then
h˚viα=hv− hviαiα=hviα− hviα= 0,(74)
h˚ciα=hc− hciαiα=hciα− hciα= 0.(75)
Thus
hcviα=h(hciα+˚c)(hviα+˚v)iα
=hciαhviα+h˚c˚viα+h˚ciαhviα+hciαh˚viα
=hciαhviα+h˚c˚viα.(76)
The term h˚c˚viαis called the dispersive flux.
Now the averaging process can be performed. Firstly (66) integrated over δV αyields
ZδV α
∂c
∂t dV =ZδV α
·(cv+j)dV +ZδV α
GdV. (77)
Now applying theorems (69) and (70) yields
∂t ZδV α
cdV =·(ZδV α
(cv+j)dV )
X
β6=αZSαβ
nα·(c(vwb) + j)|αdS
+ZδV α
GdV. (78)
25
This can be written in terms of averaged quantities by dividing across by δV =δV α
φαto
give
∂t (φαhciα) = ·φα(hcviα+hjiα)
1
δV X
β6=αZSαβ
nα·(c(vwb) + j)|αdS
+φαhGiα.(79)
Utilising the formula for the average of a product it can be seen that
∂t (φαhciα) = ·φα(hciαhviα+h˚c˚viα+hjiα)
1
δV X
β6=αZSαβ
nα·(c(vwb) + j)|αdS
+φαhGiα.(80)
Comparing this to the macroscopic point balance equation (67) gives the following rela-
tions between the macroscopic and microscopic quantities
c=hciα,(81)
v=hviα,(82)
j=h˚c˚viα+hjiα,(83)
φαG=φαhGiα
+1
δV X
β6=αZSαβ
nα·(c(vwb) + j)|αdS. (84)
Thus it can be seen that the microscale convection and diffusion processes at the interfaces
are source terms for the macroscopic equation. It can also be seen that the macroscopic
diffusive flux is the sum of the averaged microscopic diffusive flux and the dispersive flux.
B. Upscaling from microscale to macroscale equations
The upscaling process basically involves choosing an REV (Representative Elemen-
tary Volume) around every point on the larger scale and representing the properties of
the medium by the averaged properties of the smaller scale over the REV. For these
purposes suitable averages need to be defined including a phase average and an intrinsic
phase average. Also needed are a time averaging theorem and a divergence averaging
theorem. An outline of the upscaling procedure used here is given in appendix A. Before
continuing some notes on the REVs being used are necessary. Two different REVs will
be used. One will have a scale between that of a coffee cell and a coffee grain. The
second will have a scale between that of a coffee grain and the coffee bed. For the REVs
the following notation is used.
26
V1:Volume of the smaller REV.
V1s:Volume of solid in the smaller REV.
V1v:Volume of void space in smaller REV.
V0:Volume of larger REV.
V0l:Volume of grains (l-phase) in larger REV.
V0h:Volume of void space (h-phase) in larger REV.
Also due to the properties of an REV
φh=V0h
V0
=Vh
VT
, φl=V0l
V0
=Vl
VT
,(85)
φv=V1v
V1
=Vv
Vl
, φs=V1s
V1
=Vs
Vl
.(86)
B.1. Equations for upscaling
For convenience the equations at each of the coffee bed length scales from the paper
are reproduced here.
B.1.1. Microscale Point Balance Equations
The point balance equations for coffee and liquid within each of the phases on the
microscale are
h-phase.
∂ch
∂t =·(chvh+jh),(87)
∂ρh
∂t =·(ρhvh).(88)
v-phase.
∂cv
∂t =·(cvvv+jv),(89)
∂ρv
∂t =·(ρvvv).(90)
s-phase.
∂cs
∂t = 0.(91)
The microscopic balance equations include the terms jhand jvwhich represent molecular
diffusion of coffee solubles in the respective phases. Molecular diffusion in the solid phase
is assumed negligible.
27
B.1.2. Mesoscale point balance equations
The mesoscale balance equations are only required in the grains (l-phase) since only
two scales are needed in the h-phase. Thus the point balance equations for coffee and
liquid in the v-phase and the s-phase on the mesoscale are
v-phase.
∂t (φvc
v) = ·(φv(c
vv
v+j
v)) f
vs.(92)
∂t (φvρ
v) = ·(φv(ρ
vv
v+i
v)).(93)
s-phase.
∂t (φsc
s) = ·(φsj
s)f
sv.(94)
The terms f
vsand f
svare source/sink terms representing transfer of coffee solubles
across the vs-interface and vice-versa. The term i
vaccounts for any mechanical dispersion
in the fluid velocity.
B.1.3. Macroscale point balance equations
For each phase there are macroscopic point balance equations for mass of the coffee
and mass of liquid (flow equations). The solid is assumed stationary. The macroscopic
equations take the form:
h-phase.
∂t (φh˜ch) = ·(φhch˜vh+˜
jh)) ˜
fhl,(95)
∂t (φh˜ρh) = ·(φhρh˜vh+˜
ih)) ˜
fw
hl.(96)
v-phase.
∂t (φlφv˜cv) = ·(φlφvcv˜vv+˜
jv)) ˜
fvh˜
fvs,(97)
∂t (φlφv˜ρv) = ·(φlφvρv˜vv+˜
iv)) ˜
fw
lh.(98)
s-phase.
∂t (φlφs˜cs) = ·(φlφs˜
js)˜
fsh˜
fsv.(99)
˜
fαβis transfer of coffee solubles from α-phase to β-phase across αβ interface. Similarly
˜
fw
αβis transfer of liquid from α-phase to β-phase across αβ interface.
28
B.2. Upscaling in l-phase: microscale to mesoscale
B.2.1. Conservation of coffee in v-phase
The averaged form of the microscopic point balance equation (89) over the smaller
REV is
∂t (φvhcviv) = ·φv(hcvivhvviv+h˚cv˚vviv+hjviv)
1
V1ZSvs
nv·(cv(vvwvs) + jv)dS. (100)
Now the surface Svs is a material surface so vvwvs =0. Comparing this averaged
form with the mesoscale form in (92) it can be seen that
c
v=hcviv,(101)
v
v=hvviv,(102)
j
v=h˚cv˚vviv+hjviv,(103)
f
vs=1
V1ZSvs
nv·jvdS. (104)
B.2.2. Conservation of liquid in v-phase
The averaged form of the microscopic point balance equation (90) over the smaller
REV is
∂t (φvhρviv) = ·φv(hρvivhvviv+h˚ρv˚vviv)
1
V1ZSvs
nv·(ρv(vvwvs))dS. (105)
Now the surface Svs is a material surface so vvwvs =0. Comparing this averaged
form with the mesoscale form in (93) it can be seen that
ρ
v=hρviv,(106)
v
v=hvviv,(107)
i
v=h˚ρv˚vviv.(108)
Here in fact ρwill be taken to be constant so i
v=0in this case.
B.2.3. Conservation of coffee solid in s-phase
The averaged form of the microscopic point balance equation (91) over the smaller
REV is
∂t (φshcsis) = 1
V1ZSsv
ns·(cswsv)dS. (109)
29
Comparing this averaged form with the mesoscale form in (94) it can be seen that
c
s=hcsis,(110)
v
s=0,(111)
f
sv=1
V1ZSsv
ns·(cswsv)dS. (112)
B.3. Upscaling to macroscale
B.3.1. Conservation of coffee in h-phase
The averaged form of the microscopic point balance equation (87) over the larger
REV is
∂t (φhhchih) = ·φh(hchihhvhih+h˚ch˚vhih+hjhih)
1
V0ZShl
nh·(ch(vhwhl) + jh)dS. (113)
Comparing this averaged form with the macroscale form in (95) it can be seen that
˜ch=hchih,(114)
˜vh=hvhih,(115)
˜
jh=h˚ch˚vhih+hjhih,(116)
˜
fhl=1
V0ZShl
nh·(ch(vhwhl) + jh)dS. (117)
Here Shl is the effective surface between the h-phase and the l-phase and includes bound-
aries between the h-phase and the v-phase and the h-phase and the s-phase.
B.3.2. Conservation of liquid in h-phase
The averaged form of the microscopic point balance equation (88) over the larger
REV is
∂t (φhhρhih) = ·φh(hρhihhvhih+h˚ρh˚vhih)
1
V0ZShl
nh·(ρh(vhwhl))dS. (118)
Comparing this averaged form with the macroscale form in (96) it can be seen that
˜ρh=hρhih,(119)
˜vh=hvhih,(120)
˜
ih=h˚ρh˚vhih,(121)
˜
fw
hl=1
V0ZShl
nh·(ρh(vhwhl))dS. (122)
30
B.3.3. Conservation of coffee in v-phase
The averaged form of the microscopic point balance equation (92) over the larger
REV is
∂t (φlφvhc
vil) = ·φlφv(hc
vilhv
vil+˚
c
v˚
v
vl+hj
vil)
φv
V0ZSlh
nl·(c
v(v
vwlh) + j
v)dS
φlhf
vsil.(123)
Comparing this averaged form with the macroscale form in (97) it can be seen that
˜ch=hc
vil=hhcvivil,(124)
˜vv=hv
vil=hhvvivil,(125)
˜
jv=hj
vil+˚
c
v˚
v
vl
=hhjvivil+hh˚cv˚vvivil+˚
c
v˚
v
vl,(126)
˜
fvh=φv
V0ZSlh
nl·(hcviv(hvvivwlh) + hjviv)dS, (127)
˜
f
vs=φlhf
vsil=1
V0ZV0l1
V1ZSvs
nv·jvdSdV0l.(128)
It will be later assumed that ˜vv=0and ˜
jv=0but of course it will still be possible to
have v
v6=0and j
v6=0.
B.3.4. Conservation of liquid in v-phase
The averaged form of the microscopic point balance equation (93) over the larger
REV is
∂t (φlφvhρ
vil) = ·φlφv(hρ
vilhv
vil+˚
ρ
v˚
v
vl+hi
vil)
φv
V0ZSlh
nl·(ρ
v(v
vwlh) + i
v)dS. (129)
Comparing this averaged form with the macroscale form in (98) it can be seen that
˜ρh=hρ
vil=hhρvivil,(130)
˜vv=hv
vil=hhvvivil,(131)
˜
iv=hi
vil+˚
ρ
v˚
v
vl
=hhivivil+hh˚ρv˚vvivil+˚
ρ
v˚
v
vl,(132)
˜
fw
lh=φv
V0ZSlh
nl·(hρviv(hvvivwlh) + hiviv)dS. (133)
31
B.3.5. Conservation of coffee in s-phase
The averaged form of the microscopic point balance equation (94) over the larger
REV is
∂t (φlφshc
sil) = ·φlφs(hj
sil)
φs
V0ZSlh
nl·(c
s(wlh) + j
s)dS
φlhf
svil.(134)
Comparing this averaged form with the macroscale form in (99) it can be seen that
˜cs=hc
sil=hhcsisil,(135)
˜vs=0,(136)
˜
js=hj
sil=hhjsisil=0,(137)
˜
fsh=φs
V0ZSlh
nl·(hcsis(wlh))dS, (138)
˜
fsv=φlhf
svil
=1
V0ZV0l1
V1ZSsv
ns·(cswlh)dSdV0l.(139)
C. Macroscopic diffusion and dispersion fluxes
The total macroscopic flux, ˜
jhis made up of the macroscopic average of molecular
diffusion and the dispersive flux:
˜
jh=hjhih+h˚ch˚vhih.(140)
The microscopic diffusive flux can be represented by Fick’s Law:
jh=Dch,(141)
where Dis the diffusion coefficient of the species in water. The macroscopic equivalent
is obtained by averaging this expression and will generally depend on the structure of
the porous medium. This average is represented by
ja
h=D˜
Ta(φ)·˜ch=˜
Da(φ)·˜ch.(142)
where the ˜
·here on ˜
Tameans this is a tensor of rank two which represents the tortuosity
of the porous medium. For an isotropic porous medium this can be adjusted to
ja
h=D
τ˜ch,(143)
32
where τis the tortuosity defined by
τ=Le
L=actual path length
macroscopic path length.(144)
The tortuosity must be estimated in terms of the porosity. Various estimated are used
in the literature. Some of these include
τ=φ1
3, τ =φ1
2, τ =1
1α(1 φ).(145)
In the final expression α=
Vis a shape factor with rbeing the object radius, σbeing
the object cross sectional area and V being the object volume. Thus for spheres for
example α= 0.75. In this case the first expression from [19] will be adopted. The other
expressions and tortuosity in general is discussed in [18]. Thus macroscopic diffusion is
approximated by
ja
h=D
τ˜ch=φ
1
3
hD˜ch.(146)
Dispersion occurs due to variations in the microscopic velocity of the phase with
respect to the averaged velocity, and molecular diffusion [16]. Molecular diffusion con-
tributes to the dispersive flux in addition to the diffusive flux at the macroscopic level.
In general the dispersive flux is given by
jb
h=h˚ch˚vhih=˜
Db·˜ch,(147)
where ˜
Dbis a rank 2 tensor called the dispersion tensor. ˜
Dbis both positive definite and
symmetric. One commonly used expression is
Dij =aijkl
vkvl
v,(148)
where aijkl is a fourth order tensor and vi=hviihis the average velocity in the i-th
direction and v=|v|where vis the average velocity vector in this instance. For an
isotropic porous medium this expression reduces to
Dij =aTδij + (aLaT)vivj
v2v. (149)
The coefficients aLand aThere are the longitudinal and transverse dispersivities of the
porous medium. For a phase that completely fills a pore space, aLis a length that should
be of the same order of the pore size. δij is the Kronecker delta. Also it is required that
aL0aT0.(150)
Laboratory experiments have found that aTis 8 24 times smaller than aL[16].
33
D. Method of estimating mass transfer terms
It is necessary to estimate the mass transfer terms fαβwhich govern the transfer of
solute from the solid phase to the liquid phase both within the grains and from the surface
of the grains. It is also necessary to estimate the mass transfer of solute from liquid within
the grains to liquid in the pores between the grains. This subject is dealt with from a food
processing and engineering standpoint in [5]. More general and technical developments
are found in [16] and [28]. The transfer term fαβmay 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 fαβ=fβα. In
this instance it is assumed that a chemical species can reach the microscopic interphase
boundary by two modes of transport, namely advection and diffusion. Hence, as we
have already seen from the averaging procedure in appendices A and B the strength of
a source of a considered species in the α-phase is given by
fαβ=1
U0ZSαβ
nα·(cα(vαwαβ ) + jα)dS, (151)
where U0is the volume of the REV, Sαβ is the boundary between the α-phase and all
other phases, nαis the unit outward normal vector on this surface and wαβ is the velocity
of points on the interphase boundary. If Sαβ is a material surface as will generally be
the case here than vαwαβ =0and
fαβ=1
U0ZSαβ
nα·jαdS. (152)
In this case the chemical species crosses the interphase boundary by diffusion only. In the
coffee bed model developed here non-equilibrium fluid-fluid and mass-fluid transfers will
be considered. The first mechanism considered that drives transfer (in an effort to bring
the system closer to equilibrium) is the difference in concentrations (or more rigorously
the difference in chemical potentials) at the interface, visualised as a thin film. Therefore
the rate of transfer, fαβ, of the mass of the considered species from an α-phase to
an adjacent β-phase across an interface Sαβ is often assumed to be proportional to the
difference in concentration between the phases. Thus
fαβ=ααβ (cβcα).(153)
To estimate the mass transfer coefficient ααβ the following form is used
jα·nα=Dα(cβcα)
α
.(154)
Here Dαis the coefficient of diffusion of the considered species in the α-phase and ∆αis
the length characterising the mean size of the phase or the length over which diffusion
occurs. For example one possible definition of ∆αis ∆α=U0α
Sαβ , the volume to surface
ratio of the α-phase within the REV. Then
34
fαβ=1
U0ZSαβ
nα·jαdS
=1
U0ZSαβ
Dα(cβcα)
α
dS
=Sαβ
U0Dα(cβcα)
α
=S
αβ
α
φαDα(cβcα),(155)
where S
αβ =Sαβ
U0αis the specific surface area of the α-phase or the surface area per unit
volume. Thus it can be seen that the mass transfer coefficient is given by
ααβ =S
αβ
α
φαDα.(156)
The above derivation is for a fluid-fluid transfer. For a solid-fluid transfer a slightly
different approach is used.
Under consideration here is the case where the solid matrix itself is dissolving (i.e. the
soluble part of the coffee grains). It is again assumed that the porous medium is saturated
and a single constituent is considered. It is then assumed that there is a thin layer of
liquid next to the solid which is always saturated with the solute under consideration.
This concentration is denoted by csat.csat is the concentration in the liquid phase that
would be in equilibrium with the concentration inside the solid cs. This assumes that
the dissolution process occurs faster than the transfer from this thin layer to the bulk of
the fluid. It is now assumed that the force of extraction from this thin layer to the bulk
of the fluid is given by
fsf=αsf (csat cf).(157)
Proceeding as above it can be shown that the mass transfer coefficient from a solid phase
s to a fluid phase f is given by
αsf =S
sf
s
φsDf.(158)
The transfers looked at so far have been transfer of constituent or solute due to
diffusion. It is also possible to have a transfer of liquid from the large pores between
the grains to the small pores within the grains (or vice versa). This will occur due to a
pressure imbalance between the phases due to the dissolution of the solid matrix within
the grains. As was seen from the averaging procedure this term has the form
˜
fw
lh=φv
V0ZSlh
nl·(hρiv(hvvivwlh) + hiviv)dS. (159)
Now it is assumed that hiviv=0and Darcy’s Law gives
hvvivwlh =kv(φv)
φvµp(160)
35
On Slh this means that
hvvivwlh =kv(φv)
φvµ
(phpv)
l
.(161)
Thus
˜
fw
lh=φv
V0ZSlh
nl·ρkv(φv)
φvµ
(˜ph˜pv)
ldS
=Slh
V0
ρkv(φv)
µ
(˜ph˜pv)
l
=φlS
lhρkv(φv)
µ
(˜ph˜pv)
l
.(162)
Again here Slh is the specific surface area or surface area per unit volume of the l-phase.
Since there is a difference in concentration in the h and v-phases the transfer of fluid in
either direction will also result in the transfer of solute from one phase to another. This
transfer will be of the following form
fw
lh=(φlS
lhρkv(φv)
µ
ph˜pv)
l˜ch,if ˜ph˜pv
φlS
lhρkv(φv)
µ
ph˜pv)
l˜cv,if ˜ph<˜pv
(163)
D.1. Form of individual mass transfers
It is now possible to describe the source and sink terms in the macroscopic equations
arising from mass transfers in terms of the microscopic quantities of the system. Now
the total mass transfer from the l-phase to the h-phase can be written as
˜
flh=˜
fvh+˜
fsh+fw
lh.(164)
˜
fvhrepresents 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
˜
fvh=αvh(˜ch˜cv)
=φlφvD
v
S
hl
l
ch˜cv)
=φlφvDv
S
hl
lτch˜cv)
=φlφ
4
3
vDv
S
hl
l
ch˜cv),(165)
where τ=φ1
3
vhas been used. Next transfer of solute from the solid on the surface of
the grains, ˜
fshis considered. Recalling (138) and (157) and that diffusion of solute
into the bulk of the fluid is the rate limiting step it follows that
36
˜
fsh=αsh(csat ˜ch)
=φlφsDh
S
hl
s
(csat ˜ch)
=φl(1 φv)Dh
S
hl
s
(csat ˜ch).(166)
As outlined above the third transfer term, ˜
fw
lh, arises due to solute being carried in
the fluid that transfers between phases due to pressure differences between pores this is
given as above by
˜
fw
lh=(φlS
lhρkv(φv)
µ
ph˜pv)
l˜ch,if ˜ph˜pv
φlS
lhρkv(φv)
µ
ph˜pv)
l˜cv,if ˜ph<˜pv
(167)
The macroscopic transfer term from the s-phase to the v-phase can be arrived at
using (139) and (157) to get
˜
fsv=αsv(csat ˜cv)
=φlφsDv
S
sv
s
(csat ˜ch).(168)
E. Dominance of advection over mechanical dispersion and diffusion
Considering the dimensional equations from the paper we can consider the relative
importance of the coffee transport processes in the intergranular pores by comparing
their magnitudes. This should give us an idea of the dominant transport mechanism
in the bed, although of course there may be narrow regions where other balances hold.
Firstly we compare advection and dispersion. The ratio of the magnitudes of the terms
is
|˜ch˜vh|
˜
Db
h˜ch
.(169)
Recall that
(˜
Db
h)ij = aTδij + (aLaT)vivj
|vh|2!|vh|=aijkl
vivj
|vh|.(170)
We now use some characteristic scales. Note aLll500 m. Thus take aijkl aL,
˜chC,|˜vh| ∼ vc,vivc, and zL. Thus (169) becomes
Cvc
llvcC
L
=L
ll
102.(171)
Thus advection is approximately one hundred times larger than dispersion and so domi-
nates. More generally advection dominates over dispersion when Lllunless there are
very large concentration gradients in the bed.
37
The ratio of advection to diffusion is given by
|˜ch˜vh|
φ
4
3
hDh˜ch
.(172)
Adopting similar approximations as in the dispersion case we find that
Cvc
φ
4
3
hDhC
L
=Lvc
φ
4
3
hDh
.(173)
Now from the experiments typical approximate values for these quantities are L0.05 m,
vc0.007 m s1,φh0.2 and Dh2.2×109m2s1. Thus
Lvc
φ
4
3
hDh
106.(174)
These estimates show that unless there are extremely large concentration gradients some-
where within the bed that advection dominates over diffusion.
F. Extraction Kinetics of Coffee Components
Figure 8: Extraction experiments suggest that a large number of compounds found in coffee extract with
similar kinetics [15].
G. Supplementary Coffee Extraction Experiments
The experimental results used in this paper are drawn from a much larger collection
of coffee extraction experiments. To complement and support these results and satisfy
the interested reader some other relevant experiments are included in this section.
38
G.1. Coffee Grinds Used
In the experiments presented here we make use of five different coffee grinds ranging
from a fine drip filter grind to a very coarse grind. The grind size distributions of these
grinds are shown in figure 9. The first grind is a relatively fine grind, called Jacobs
Kr¨onung (JK) standard drip filter coffee grind. The next grinds used are the Douwe
Egberts (DE) standard drip filter grind and the Douwe Egberts coarse drip filter grind.
Finally two further grinds were obtained by grinding Illy coffee beans using a Cimbali
burr grinder. One very coarse grind was obtained using the #20 setting on the grinder.
A second extremely coarse grind was obtained using the # 30 setting on the grinder.
The grind size distribution of the Cimbali # 30 grind was too coarse to be analysed by
the optical particle size analyzer used (Mastersizer 2000; Malvern Instruments Ltd, UK)
and so is not included in figure 9.
JK standard drip filter grind
Illy, Cimbali #20 grind
DE standard drip filter grind
DE coarse drip filter grind
0.1
1
10
100
1000
104
0
2
4
6
8
10
Particle size HΜmL
Volume fraction H%L
Figure 9: Grind size distributions of the coffee grinds used in experiments.
G.2. Coffee extraction kinetics during batch-wise brewing in a fixed water volume
The extraction kinetics of the five coffee grinds were investigated by mixing 60 grams
of coffee grounds with a hot water volume, Vwater = 0.5 litres, and measuring the con-
centration cbrew of extracted species as a function of time. The temperature of the liquid
during extraction is 80–90 C. The experimental procedure is identical to that outlined
in the paper but the results for the other grinds are shown here in figure 10. This
experiment clearly illustrates the key influence that the grind size distribution has on
extraction.
G.3. Coffee extraction profiles from a cylindrical brewing chamber under different condi-
tions
In this paper we have presented coffee extraction profiles from a cylindrical brewing
chamber for one fine grind (JK standard drip filter grind) and one coarse grind (Cimbali
39
æ
æ
æ
æ
æ
æ
à
à
à
à
à
à
à
ì
ì
ì
ì
ì
ò
ò
ò
ò
ò
ò
ô
ô
ô
ô
ô
ô
æ
JK standard drip filter grind
à
Illy, Cimbali #20 grind
ì
Illy, Cimbali #30 grind
ò
DE standard drip filter grind
ô
DE coarse drip filter grind
0
100
200
300
400
500
600
0
10
20
30
40
time HsL
cbrew Hkgm3L
Figure 10: Coffee solubles concentration profiles for different coffees and grind size distributions during
batch extraction experiments. In these experiments 60 g of coffee with approximately 4% moisture was
mixed with 0.5 l of hot water in a French press type cylinder.
#20 grind). Here we present some ancillary experiments for JK standard drip filter
grind for a different coffee bed mass and for a different value of absolute pressure in
the coffee bed. The experimental apparatus is the same as that outlined in the paper.
To compare results for different masses (and hence different bed lengths) the extraction
is performed for coffee bed masses of 12.5 g and 60 g. These masses correspond to bed
depths of 1.12 cm and 4.05 cm respectively. The flow rate to the coffee bed is 250 ml/min
in both cases. The pressure difference across the bed is measured in both cases. The
solubles concentrations are measured in the coffee pot and at the filter exit. The results
are shown in figure 11.
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
à
à
à
à
à
à
à
à
à
à
à
à
à
à
æ
JK, 60g, 2.3bar, 250mlmin
à
JK, 12.5g, 0.5bar, 250mlmin
0
200
400
600
800
1000
0
50
100
150
200
Mbrew HgramsL
cbrew HmggramL
(a)
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
à
à
à
à
à
à
à
à
à
à
à
à
à
à
æ
JK, 60g, 2.3bar, 250mlmin
à
JK, 12.5g, 0.5bar, 250mlmin
0
200
400
600
800
1000
0
50
100
150
200
Mbrew HgramsL
cexit HmggramL
(b)
Figure 11: The coffee solubles concentration, measured in mg/gram, is plotted against mass of coffee
beverage Mbrew (grams) for JK drip filter grind with a flow rate of 250 ml/min in (a) the coffee pot and
(b) the beverage at filter exit for different coffee bed masses.
40
The influence of absolute pressure on extraction was also investigated by repeating
experiments in the coffee brewing cylinder at different values of absolute pressure but
maintaining the same coffee bed mass and flow rate. In the case of JK standard drip filter
grind the absolute pressure in the brewing cylinder is increased from 2.3 bar to 9 bar. The
resulting solubles concentrations profiles are plotted in figure 12. The results are seen
to be virtually identical which indicates that the extraction kinetics are substantially
independent of the absolute pressure (at least for the range of values considered) and
that the results are reproducible to within a few percent. It also suggests that a possibly
faster particle penetration with water at higher pressures does not substantially affect
the observed extraction results.
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
à
à
à
à
à
à
à
à
à
à
à
à
à
à
à
à
à
à
à
à
à
à
à
æ
JK, 60g, 2.3bar, 250mlmin