Toward a New Brewing Control Chart for the 21st Century - Journal of

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Food and Beverage Chemistry/Biochemistry

Towards a new Brewing control chart for the 21st century John MELROSE, Borja ROMAN-CORROCHANO, Marcela MONTOYA-GUERRA, and Serafim BAKALIS J. Agric. Food Chem., Just Accepted Manuscript • DOI: 10.1021/acs.jafc.7b04848 • Publication Date (Web): 14 Apr 2018 Downloaded from http://pubs.acs.org on April 15, 2018

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Journal of Agricultural and Food Chemistry

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Towards a new Brewing control chart for the 21st century.

2 3

John MELROSEa*, Borja ROMAN-CORROCHANOa, Marcela MONTOYA-GUERRAa,

4

Serafim BAKALISb.

5

a- Jacobs Douwe Egberts R&D GB Ltd. Ruscote Avenue, Banbury Oxon OX16 2QU, UK

6 7

b - Centre for Formulation Engineering, Department of Chemical Engineering, University of

8

Birmingham, Edgbaston Birmingham, B15 2TT, UK.

9

*Corresponding author: [email protected]

10 11

ABSTRACT

12 13

This paper describes new results from a base model of brewing from a bed of

14

packed coffee grains. The model solves for the diffusion of soluble species out of a

15

distribution of particles into the flow through the bed pore space. It requires a limited

16

set of input parameters. It gives a simple picture of the basics physics of coffee

17

brewing and sets out a set of reduced variables for this process. The importance of

18

bed extraction efficiency is elucidated. A coffee brewing control chart has been

19

widely used to describe the region of ideal coffee brewing for some 50 years. A new

20

chart is needed, however, one that connects actual brewing conditions (weight, flow

21

rate, brew time, grind>.) to the yield and strength of brews. The paper shows a new

22

approach to brewing control charts, including brew time and bed extraction efficiency

23

as control parameters. Using the base model, an example chart will be given for a

24

particular grind ratio of coarse to fine particles, and an ‘espresso regime’ will be

1 ACS Paragon Plus Environment


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picked out. From such a chart yield, volume and strength of a brew can be read off

26

at will.

27 28

KEYWORDS

29

coffee, brewing, diffusion, modelling

30

31

INTRODUCTION

32

The brewing of coffee from a packed bed of particles is complex: it involves many

33

molecular and colloidal species released from a distribution of particle sizes. A

34

challenge is to provide brewers a guide to the effect of changing conditions (flow

35

rate, coffee grind, temperature etc) on brew performance. Modelling the process

36

offers a route to encapsulating experimental observations, but also to establish the

37

basic physics of the process. In general, it is best to start modelling at a simple

38

enough level to provide a base model with few input parameters, one which

39

demonstrates the basic physics with a fair level of accuracy and has a clear route to

40

development and improvement. This paper reports new results from what the

41

authors argue to be such a base model. The model was briefly reported in ASIC

42

conference posters 2-4 and a recent thesis5.

43

The main aim of the paper is to elucidate the basic physics of coffee brewing

44

through use of a simple base model of diffusive release from particles into a bed

45

pore space. It introduces reduced variables, and the concept of bed extraction

46

efficiency. Use of reduced variables, whilst a standard engineering practice, is

47

crucial to allow ease of comparison of different experiments and a comprehensive

48

set is not hitherto evident in the coffee literature. The paper presents a format for a


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new control chart which explicitly includes brew time; this is done for an espresso

50

brewing grind size. This example is consistent with, and predictive of, current

51

optimal espresso practice. Different regimes within the chart are physically

52

described.

53

time response.

54

The paper points out many open questions in particular on the early

Critique of other modelling strategies will be made. By explicitly solving the

55

diffusion equation within particles, the model captures the early time high fluxes of

56

the diffusion equation. It can, in principle, respect both the polydisperse distribution

57

of particle area and volume in the problem, thus allowing a direct interpretation of

58

fitted parameters, in particular diffusion constants.

59

the base model required from experiment are: the diffusion constants of species

60

inside particles, the particle size distribution (the volume averaged dimension and

61

the fraction of volume in fines), the bulk density of the bed, the coffee particle density

62

and the maximal yield - the last parameter is not commonly discussed in the

63

literature. The porosity of the dry bed can be estimated from bulk densities and

64

particle densities; however, it likely varies during brewing. Using particle-based

65

models such as that here, diffusion constants and composition weights can be fit to

66

experiment. Alternatively, predictive modelling of beds can be attempted by using

67

values of diffusion constants fitted to dilute slurry experiments or values estimated

68

from measured bulk values and grain structure. There are a number of effects

69

whose impact can be explored relative to the base model: variable flow rates,

70

partition coefficients and solubility, sphericities, dispersion coefficients, mass transfer

71

layers and boundary conditions, initial wetting of the bed, varying bed porosity and

72

surface composition of particles. In this paper and the base model these are

73

ignored. Directions for improvement of the model will be noted throughout the paper.

The key input parameters for



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The SCAA (Speciality Coffee Association of America) brewing control chart

75

from the 1960’s is still used today as a guide to coffee brewing, it was produced

76

based on research by Lockhart1. The chart overlays various flavour regions on a

77

space of brew strength and brew yield. Sensory data suggests a region of optimal

78

coffee flavour: on the low yield side of this, the brew is said to be under-extracted,

79

generally described as sour or green and lacking in sweetness; on the high yield

80

side, it is over extracted with a bitter, smoky/burnt flavour and sometimes said to give

81

a dry mouthfeel. On the high strength side, these flavours are strong and on the low

82

strength side weak (although in practice desired strength will vary with culture). Note

83

the major flavour variation from under through balanced to over extraction is

84

indicated to just depend on the yield axis, the strength axis delineates weak to strong

85

flavours. The claimed dominant dependence of flavour on yield is not a given

86

feature, if correct it tells us there is something broadly generic about the relative time

87

scales for release of different flavour component families. As a map solely of the

88

outcome of the brewing process, the control chart has some shortfalls. It would be

89

preferable to have a set of charts that given a grind distribution, flow rate and brew

90

time, allows the brewer to look-up the resulting strength and yield. One that could

91

be applied, furthermore, not to just the total mass extracted but also for individual

92

extracting species. The ideal flavour zone found by sensory studies can then be

93

over-laid and related to actual parameters under the brewer’s control. This paper is

94

a step in that direction.

95

In earlier literature, measurement of the release kinetics of total mass and of

96

individual molecular species from coffee grains into dilute solution was reported by a

97

number of authors. 6-14 Spiro and co-workers 7-12 reported extensive work on the

98

release of caffeine and fit their data to models for diffusion out of spherical particles.


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One particular question debated was if the response of the system at early times

100

was consistent or not with a diffusion model.13 Understanding of the early time

101

response is still however in-complete as will be discussed below. Lee et al.14

102

studied the release kinetics from flow through a coffee basket and reported the

103

release kinetics of wide range of molecular species, using half-lives they separated

104

the species into slow and fast extractors.

105

time of molecules from coffee particles and beds has been published. 15-25 These

106

involve the collection of aliquots and subsequent GCMS analysis, or dynamic

107

PTRMS measurement of volatiles in the vapour phase by Sánchez-López et al. 24,25

108

The reader is referred to Kuhn et al.25 for a summary and review of the recent

109

literature. Some papers fit the release profiles to equations. Mateus et al.17 fit both

110

the Weibull function and a single sphere diffusion model, they conclude that for

111

several species, the former equation indicates deviations from simple diffusive

112

release. Mestdagh et al.18 use hyperbolic equations to fit profiles, correlating polarity

113

with the initial release rates. Physical models have been developed based on first

114

order rate equations by Kuhn et al.25 and Moreney et al.26

More recent date on release profiles over

115

116

117

METHODS MODEL. The type of brewing modelled in this paper is a flow through a packed

118

bed of roast and ground coffee particles of dry weight W and involves transfer of

119

releasing species from the particles into the pore space of the bed and convection by

120

the flow through the bed and out into the beverage. An example would be brewing

121

of an espresso. If the volumetric flow rate Q(t) and mass concentration of an



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extracted species in the bed pore space at the exit of the bed C(t) are known at time

123

t, then the total mass extracted into the brew from the grains is given by

124

125

126

() = ( ∗ ) ( ∗ ) ∗

(1)

and the mass of the brew is given by

() = () ( ∗ ) ∗

(2)

127

where is the density of the brew. The strength S(t) and yield Y(t) of the brew

128

at time t are normally expressed as the following percentage values

129

S(t) =

∗ ( )

130

Y(t) =

∗ ( )

(3)

( )

.

(4)

131

Equation 3 and 4 are the common definition of brew strength and yield of the total

132

mass of all extracting species together. In the control chart1 , the regime of ideal

133

flavour lies between 18% to 22% total mass yield. For the strength and yield of a

134

particular species, the numerator in Eq. 3 is defined just with respect to the mass of

135

that species.

136

The full problem involves convective transport through the pore space of the

137

bed, a distribution of particles sizes, a wide variety of diffusing species and the

138

internal structure of coffee grains. These are discussed in turn as the base model

139

with its approximations are introduced.

140

The bed is divided into a set of particles and a bed pore space external to the

141

volume of the particles.

The flow is assumed not to penetrate the bulk of the

142

particles; however, the particles are assumed to wet rapidly by capillary action

143

(observations of gas bubbles on 500 μm grains wetting under a microscope suggest 6 ACS Paragon Plus Environment

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a time scale of a few secs, although complete water ingress into the grain matrix

145

may take longer). The initial wetting of the bed is not modelled, simulations start with

146

a wetted bed and grains at t = 0 and the release of species is assumed to start at

147

that time. In the simulations reported below, the flow is assumed a constant. The

148

transport of extracting species from within particles to their surface is assumed to be

149

due to diffusion driven by concentration gradients, activity, osmotic pressure and

150

solubility effects are ignored. The flow through the bed pore space removes species,

151

eventually into the external brew. The concentration in the pore space sets a

152

changing boundary condition on the diffusion within the particles. In the context of

153

gel chromatography, models for mass transport between particles and beds have

154

been applied27 ; Melrose et al.2-5 have adapted these for coffee brewing.

155

The particle size distribution (PSD) of coffee post grinding measured dry is

156

bimodal with a fine distribution and a coarse distribution. Some recent examples are

157

shown in Wang et al.22

158

these are fit well by a sum of a log-normal distribution for the fine particles, and a

159

log-normal distribution for the coarse particles.

160

approaching mm improved fits are found if a sum of log-normal for the fines and a

161

Weibull distribution for the coarse particles is used. Observation shows that the

162

peak of the fine distribution remains relatively constant in the range of dimensions

163

30-60 μm, but as the coarse size is reduced the volume of fines increases - see for

164

example the data in Wang et al.22

165

three parameters:

166

fraction of fines. The particles are assumed spherical. The coarse particle radius is

167

set at one half the volume averaged dimension, d4,3. This two-particle approximation

168

is the simplest way of both mimicking the longer length scale particles in the PSD but

For coarse size in the range below 0.5 mm, it is found that

For grinds with coarse sizes

The base model approximates the PSD to just

representative fine and coarse particle sizes and the volume



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also the shorter length scale (and fast release) of the fines, together they mimic the

170

dimensions of the surface area of the total grind and the volume averaged

171

dimension.

172

time release kinetics of the system.

173

particles and the use of many particles to represent the PSD is work in progress.

174

Coffee grains are in practice not spheres, sphericities are measured in the range 0.8-

175

0.75

176

unity for the results below. Swelling of grains is assumed absent.

177

5

The volume averaged dimension of the distribution captures the longer It is straightforward to generalise to many

; shape can in part be accounted for with this correction, but this was set to

The results in Figure 2 below are for the model solved for a single diffusing

178

species. Assuming that the species do not interact with each other, the single

179

species solution is not a limitation because multiple species can be handled by

180

rescaling time and superimposing the single species solution. In principle, multiple

181

particles cannot be handled this way as they are coupled together by the common

182

concentration in the bed pore space through the boundary condition - although in

183

dilute enough slurry conditions or fast enough flow through beds (see below) they

184

can be considered decoupled. One of the uses of modelling is to make estimates of

185

when these dilute or fast flow conditions hold.

186 187 188

The concentration of extracting species averaged over the coffee particle volume (that portion of mass extractable) is given by

= ( /100) #

(5)

189

where is the maximal possible yield and # is the density of a coffee particle.

190

In reduced units, the yield is defined by

191

$(̃) = (̃)⁄ .

(6)


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The maximal yield needs be measured, for example by bringing a dilute slurry

193

extraction to equilibrium over a long time.5 For the total mass it was found5 to vary

194

with grind size: at extraction temperatures circa 80 oC total yields are 32-30% for a

195

very fine grind (coarse peak < 200 μm) to values for 25-24% for a larger grind with

196

coarse peak circa 1 mm.

197

extractable may be due to several reasons: diffusing species inside the wet grain

198

may have a probability of being irreversibly trapped or absorbed in the coffee matrix

199

or of being transformed by chemical reaction with the matrix. The maximal yields do

200

vary with temperature; the kinetics of brewing also changes with temperature, but

201

beyond that expected by just the temperature scaling of diffusion constants.

202

The variation with particle size of the total mass

Roast and ground coffee particles have a heterogeneous internal structure28

203

consisting of a nano-porous matrix of structural carbohydrates and proteins

204

surrounding 30-60 μm macro-pores (the pockets in the matrix that held the biological

205

cells before roasting); the surface has exposed sections of macro-pores. It is likely

206

that the extractable material is similarly distributed heterogeneously - the roasted

207

and reacted contents of the original biological cells and oil bodies are most likely on

208

the surface of the macro-pores28 and probably are the majority component. Some

209

species may also release from the matrix, in particular oligomers of the structural

210

carbohydrates, but possibly also small molecules. The fines are fragments of the

211

nano-porous matrix on length scales comparable to that of the macro-pores. This

212

heterogeneity is on the whole ignored in the current model. The model particles,

213

both fines and coarse, are approximated as homogeneous although the impact of

214

heterogeneity on diffusion constants is included (see Eq. 8). The soluble

215

components are assumed to dissolve rapidly as the grain wets. Moroney et al.26 do

216

start with a formalism which separates the matrix and macro-pores, however they 9 ACS Paragon Plus Environment


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also eventually homogenise their model and also assume rapid dissolution of internal

218

species on wetting as is done here.

219

Partitioning may occur between the particles and external bulk. This is may be

220

accounted for by using partition coefficients K: the ratio of total mass concentration

221

averaged over the volume of the particles relative to that of the external bulk at

222

equilibrium in a slurry experiment (see Eq. 11 below). Interpretation of K for coarse

223

coffee particles is complex due to their heterogeneous structure; it is likely that within

224

the particle there is a species dependent partitioning between the particle matrix, oil

225

phase and the macro-pores - given their 30-60 μm scale it seems reasonable to

226

assume that at equilibrium the concentrations in the macro-pores are that of the bulk.

227

K can be found by a series of dilution experiments, Spiro12. Corrochano5 reports K =

228

0.6 on a very fine distribution with 70% of the volume in fines (< 100 μm ) which can

229

be assumed to contain few macro-pores and thus to be an estimate of the

230

partitioning between the matrix and bulk. At typical brewing flow rates, modelling

231

suggests that release kinetics of the total mass is weakly dependent on K. These

232

effects are ignored in the base model. Although, for individual molecular species

233

over typical brewing temperature ranges there may be a strong temperature effect

234

on partition coefficients.

235

Units of time, , length, r0 and concentration c0 are chosen and the model

236

constructed in dimensionless units ̃ = ⁄ , (̃ = ( ⁄( and )̃ = )⁄) ; ( is as

237

defined in Eq. 5. The choice of units of time and length will be discussed later.

238

Within the representative particles, the diffusion equation is solved for the time

239

varying radial grain concentration profile *+ , for the pth particle size fraction


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,-*. , *

240

2

*

*

1+ (, 2-. + 4 ,-.) , = −0 , ̃

(7)

̃ ,̃

241

1+ , is an average over the porous and heterogeneous where the diffusion constant, 0

242

structure of the coffee particle.

243

grain microstructure parameters and known diffusion constants Db. in bulk (water)31 :

1+ can be estimated from measured Values for 0

8

0+ = 56 (7) 9: 0 ,

244

(8a)

245

1+ = 5 ;2 : 0+ , 0

246

(8b)

;

247

where ε and < are averaged particle porosity and tortuosity and 7 is the ratio of pore

248

size to species size - together they set an overall hinderance factor rescaling the

249

bulk diffusion constant to a smaller particle diffusion constant. Equation 8b gives

250

the diffusion constant in the reduced units. The hinderance factor could vary with

251

particle size, in the base model it will be assumed constant for all particles. Spiro and

252

co-workers7-12 reported on the hinderance factor for caffeine. Corrochano5 made

253

estimates based on measurements of particle porosity. The following boundary

254

conditions are applied to Eq. 7:

255 256

= 0; 1+ 0

,-*. ,̃

*+ = *> ; ?0 ≤ )̃ ≤ A$+ B = 0 ; ()̃ = 0, ∀̃ )

(10)

257

where Rp denotes the radius of the pth particle size. Concentration boundary

258

conditions are applied:

259

*+ (A$+ , ̃ ) =

-* ( *) D

(9)

; (̃ > 0) .

(11)



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Equation 7 is hence subject to time varying boundary conditions, and in the

261

polydisperse, packed beds the different particle sizes are coupled together via the

262

common * (̃). As discussed above, K = 1 in the base model.

263

The particle transport equations, Eqs. 7-11, are imbedded in a coarse-grained

264

model of convection through the bed; this is given below in a finite difference form.

265

The bed is assumed axially symmetric and divided into N layers orthogonal to the

266

axis, such that each layer has equal volume and dimensions larger than that of the

267

individual particles. A set of particles are simulated in each layer with the PSD

268

assumed uniform across the bed. The concentration profiles with in each particle in

269

each layer evolve in time according to the boundary condition set by that layer’s pore

270

space concentration and Eqs. 7-11.

271

the ith layer, Cbi , evolves in time according to

272

273

L.

*F (̃ + ∆ ̃ ) = *F (̃) + H3J ∑+

M̃ .N ( *)

̃.

The concentration in the bed pore space of

+ J O PQ 5 *FR (̃)- *F (̃ ):T ∆ ̃ ,

(12)

where (R8U )

274

J=

275

PQ = (

8U

VW X( )

(13)

,

) ,

(14)

276 277

with YZ the bed porosity, W the weight of coffee in the bed, Q the volumetric fluid

278

flow rate through the bed, # the density of a coffee particle, [+F the flux out of the

279

surface of the pth particle class in layer i, \+ the fraction of grind volume in the pth

280

particle class and ∆ ] the computational time step.

281

(for each layer) in Eq. 12 is restricted to just two particles classes sizes: one fine,

In this paper, the of sum fluxes


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one coarse. The pore space concentration in each layer *F (̃) is assumed zero at t

283

= 0. The pore concentration in the final layer i = N is used in Eq. 1 to predict the

284

mass extracted into the brew. The parameter PQ will be termed here the bed

285

extraction efficiency and is discussed in section 2.3. (Eq. 12 is a discretised version

286

of bed transport equations27 with dispersion terms omitted. Dispersion effects can

287

be included in the formulation, the flow rates are typically in the regime where the

288

introduced coefficients will be order N d/L, where L is the bed axial length, and d is a

289

microscopic length of order the particle size. For the problems here, the

290

concentration gradients in the bed are such that these terms are found negligible.)

291 292 293

It remains to make a choice of length and time units. The unit of time is chosen as the diffusive relaxation time for fine particles: ) = A^F_

= ^F_ =

2 `aNb

c.

(15)

294

where A^F_ is the radius of the fine particles. Given that for coffee grinds A^F_ is a

295

relatively fixed length scale, Eq. 15 is a useful definition even if the grind of interest

296

has a low level of fines. The value A^F_ = 20 μm is used in the base model.

297

In general, analytic and trivial solutions are not available for polydisperse and

298

moving boundary condition problems and numerical techniques need be used. The

299

equations were implemented in the scripting language Scilab30 with be-spoke code

300

written by one of the authors. It was found that choosing N=10 layers gave

301

satisfactory convergence of the results vs computational expense. Using a

302

discretised version of Eq. 7 the code updates the concentration profiles for one fine

303

and one coarse particle in each layer according to the updated bed pore space

304

boundary condition. The particles are typically meshed at a scale circa 0.5-1 μm in

305

real units (50 mesh points for the fine particles). The time steps are determined by



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Eq. 12 and are chosen so that the volume flowed is a small fraction ( < 0.1 ) of the

307

layer volume. The time steps so set are such that an explicit scheme was used to

308

solve Eq. 7, and to iterate profiles inside particles. A typical run out to time of 40 in

309

reduced units (an individual curve of Figure 2 below) takes circa 1hr on a Dec-13

310

Pentium 7 laptop with 16GB of RAM. DISCUSSION ON MODELLING. In the models of Kuhn et al.25 and Moreney

311 312

et al.26 transport is modelled by fitting rate constants in first order expressions

313

between particle surfaces and the bed pore space, in contrast the diffusion equation

314

for concentration profiles inside the particles is solved in the model here. First order

315

expressions are a limitation when the overall mass transfer rate in coffee brewing is

316

controlled by diffusion inside the grains, in particular they do not represent the high

317

diffusion fluxes out of particles at early times13, relevant on brewing time scales.

318

Fitted rate constants also limit interpretation in terms of underlying parameters such

319

as diffusion constants and partition coefficients and thus moving from the conditions

320

of the fit to different brewing and grind conditions. The fitted coefficients are not

321

independently measurable, whereas fitted diffusion constants can in principle be

322

independently measured or estimated. As in the base model here and earlier work2-

323

5

324

problem.

325

, Moreney et al.26 do represent separately the coarse and fine length scales in the

The base model does not include brewing pressure as a directly relevant

326

variable. Given the hydrodynamic resistance of the bed, the pressure gradient sets

327

the flow rate but it is assumed in the base model not to directly influence release

328

other than through flow rate. Experiments26 showed that pressure applied to a slurry

329

brewing did not influence release. The grain volume, however, is circa 50% gas

330

(either air or CO2 depending on the level of CO2 degassing before use) and may 14 ACS Paragon Plus Environment

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contain an adsorbed load of CO2. This gas is released as the bed is wetted and does

332

effect coffee crema and organoleptic properties of brew, this process may directly be

333

influenced by the pressure gradient. Multi-phase effects of gas in the beds is

334

beyond the current model and gas release is ignored.

335

In the simulations of this paper, the flow rate is kept constant. The model can,

336

however, be directly coupled to a ‘brewer model’ in which pump behaviour, pressure

337

and flow rate are coupled by the time varying flow resistance of the system3.

338

also feasible to extend the model to include the heterogeneity of the particles: one

339

approach would be to approximate the internal structure of grains to be a series of

340

layers with appropriate thicknesses of matrix and macro-pores with different effective

341

D’s, and distribution of species, in each layer, this would preserve the radial

342

symmetry of Eq. 7.

343

It is

POROSITY AND BED EXTRACTION EFFICIENCY. Given measurements of

344

the density of individual particles5 (# = 550 − 600 kg/m3 ), for packed beds in

345

coffee brewing with dry bulk densities in the range 400-500 kg/m the porosity can

346

crudely be estimated to lie in the range 0.3 - 0.1. Porosity, however, varies during

347

wetting and brewing due to several effects: consolidation of the bed under pressure,

348

an increase in fines level due to those encrusted on the surface of the coarse

349

particles being released into the bed and possible swelling of the grains dependent

350

on water quality. The strategy used in the modelling is to keep it as a separate

351

parameter and to study the sensitivity of the model to it.

352

3

A bed relaxation time is defined as the time by which the volume of fluid flowed

353

through the bed equals the volume of the bed itself. Physically, the optimal

354

definition would be the time to match the pore volume of the bed, but this would

355

complicate the role that bed porosity plays in Eq. 12. Practically, it is just required 15 ACS Paragon Plus Environment


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356

that a definition of the bed relaxation time scales with the bed pore volume; porosity

357

is kept as a separate parameter. The choice in Eq. 14 is that the bed relaxation time

358

is defined relative to the volume of coffee grains:

359

360

361

Z =

/VW

(16)

X

and with the choice Eq. 15 for time units

PQ =

aNb U

.

(17)

362

The bed extraction efficiency, PQ , as defined above can be interpreted as follows. If

363

^F_ > Z species coming out into the pore space of the bed are removed at a

364

faster rate than they come out of the grain, the concentration in the pore space,

365

( , is very low and the rate of release from the grain is maximised (PQ is large,

366

efficient bed extraction). Alternatively, if Z > ^F_ relative to the release rate from

367

the grains, the removal from the bed is slow, concentration builds up in the bed pore

368

space and the release rate from the grain is slowed according to Eq. 11 (PQ is small,

369

inefficient bed extraction). Moreney et al.26 use a similar ratio of bed advection time

370

to grain diffusion time as one of their parameters in an asymptotic analysis of their

371

model.33

372

Extraction efficiency of the bed is a key concept. As beverages are generally

373

produced for fixed target volumes, a variation in brew time is coupled to efficiency:

374

longer brew times occur with lower extraction efficiency and vice-versa, hence yield

375

does not vary as significantly with brew time as might be expected (see below). In a

376

bed extracted at a given flow rate to a target volume, the faster a diffusing species

377

(larger D) the lower (in reduced units) is its bed extraction efficiency, however the 16 ACS Paragon Plus Environment

Page 17 of 38


378

real brew time will map onto a larger reduced time and hence a higher yield for the

379

faster species occurs overall.

380

EXPERIMENTAL RIG. A series of brewing experiments using a range of grinds

381

was done by Corrochano et al.4,5 in both dilute slurry conditions and in a custom-built

382

brewing rig32. The rig drives flow by applying a constant hydrodynamic pressure to a

383

cell and bed of dimensions comparable to that of typical brewing chambers - see

384

Corrochano et al.32 for a more complete description and diagram of the rig.

385

particular experiment from this series is reported below. A blend of Medium-dark

386

roasted arabica beans from Central America (30%) and Brazil (70%) (moisture

387

content 3.5%, medium roast colour) were ground in a Dalla Corte disc grinder

388

(Baranzate, Italy). PSD were measured by using light scattering on a Helos

389

Sympatec.

390

dry. The bed was cylindrical of diameter 3.7 cm; 9.5 g of coffee was packed to a bulk

391

density 480 kg/m (dry at start), resulting in a bed height of 1.8 cm. The rig operates

392

at a fixed applied over pressure; 3.75 bar and 1.75 bar were applied resulting in

393

average flow rates of 3.7 ml/s (HF) and 1.3 ml/s (LF) corresponding to extraction

394

efficiencies of gQ = 0.9, 0.3 respectively. Aliquots were taken over time and brew

395

concentrations measured by refractometer methods (RFM340, Bellingham Stanley

396

Ltd, UK), which had been previously calibrated against the standard drying oven

397

method (103°C during 16-24 hours until constant weight) to measure concentration

398

of soluble solids5; experiments were done in triplicate. Slurry experiments in dilute

399

conditions were used to find the maximal yield of the grind of = 28% and the

400

coffee particle density was estimated # = 580 kg/m3 , these values were used in Eq.

401

5 to set

.

One

The PSD had d4,3 = 325 μm and 20% of the volume in fines measured

3

402 17 ACS Paragon Plus Environment


403

Page 18 of 38

RESULTS AND DISCUSSION COMPARISON WITH AN EXPERIMENT. Figure 1 shows comparison of the rig

404 405

experiment and model predictions. The experimental results plotted are the total

406

mass concentration (all species) in the brew; 3.75 and 1.75 bar were applied

407

resulting in average flow rates of 3.7 (HF) and 1.3 ml/s (LF) corresponding to

408

extraction efficiencies of PQ = 0.9 , 0.3 respectively. Concentrations were found using

409

the average flow rate to define brew volume. Data is averaged over three

410

experiments; errors of the mean are less than the symbol size. Models were run

411

with a single diffusing species and a weighted sum of two diffusing species.

412

Convection through the bed was modelled in this case, by dividing it into 10 layers.

413

The coarse particle radius is set at half d4,3 and a fine particles radius was set at 20

414

Jk.

C kg/m3

100.0 80.0 60.0 40.0 20.0 0.0 0

20

40

60

80

100

120

Time s

415 416

Figure 1. Expt. Data: Squares (LF) and diamonds(HF). Model predictions for a single

417

species of Dp =1.0X10

-10

m2/s representing the total mass are shown by the solid lines. 18 ACS Paragon Plus Environment

Page 19 of 38


418

Also shown by dashed lines is a model fit with two species one with Dp =2.6X 10-10 , and the

419

other with Dp =1.3X10-11 m2/s , with 60-65% of the mass in the larger D species

420

(representing small molecules).

421 422

In early experiments6 , the fitting of diffusion models to slurry/bulk brewing data gave

423

an effective particle diffusion constant for the total soluble solids in the range 0.8-

424

1.1X10 . Figure 1 shows that for a packed bed brewing, with adoption of the value

425

of Dp = 1.0X10

426

for the coarse particle and fines as discussed, the model makes a fair prediction for

427

the extraction of the total mass over early times 20-30 s; but it over predicts the yield

428

and concentrations at longer times.

429

Figure 1. Using two diffusion constants with 60-65% of the mass associated with

430

the larger diffusion constant, fits the data well out to longer times at both flow rates.

431

Two parameters were varied, a base value of D and one a factor 1/20 smaller,

432

motivated by estimates of the bulk values of carbohydrate oligomers (see below).

433

The mass distribution between the two species was then varied to minimise error.

434

The full problem involves a spectrum of relaxation times, due to multiple particle

435

sizes and multiple releasing species. In principle, the model just fits relaxation times,

436

such as that of Eq. 15. The degree to which the diffusion constants used are

437

comparable to that of the actual diffusion of species relies on the success of the

438

choice of length scales to mimic the real system PSD (in the base model here, using

439

just one coarse and one fine particle size). The fitting is also sensitive to the values

440

used for maximal yield, in this case that measured in dilute conditions, the level of

441

fines (in this case that given by the PSD measured on dry grains) and the

442

assumption of homogeneous flow across the bed.

-10

-10

m2/s from ref. 6 when combined with the PSD approximated to d4,3

The model was also used to fit the data of



443

Page 20 of 38

It is now discussed how consistent the values of the two diffusion constants and

444

fitted compositions used in Figure 1 are with separate observations and estimates.

445

The spectrum of releasing species is known to range from minerals, to small

446

molecules (e.g. caffeine), to oligomers of structural carbohydrates (mainly

447

galactomannans) to small colloids up to 100-200 nm in size (e.g. coffee large

448

melonoidins). An average degree of polymerisation of 20 is reported for the

449

galactomannan oligomers.34 This size range corresponds to a range of bulk

450

diffusion constants from l(10Rm ) m2 /s for minerals and small molecules thru to

451

l(10R ) m2 /s for the carbohydrate oligomers and l(10R4 ) m2 /s for the colloids.

452

It is, however, the hindered particle diffusion constants (c.f. Eq. 8a) that determine

453

the release rates rather than bulk diffusion; these can be measured or estimated

454

independently of fitting slurry and brew data. In a series of papers, Spiro and co-

455

workers7-12 measured the infusion of caffeine in slurry conditions. At 80 oC and for

456

particles in the 850-1200 Jk they report hindrance factors of 11 and a particle

457

diffusion constant of 1.7 n10R m2 /s for caffeine at 84 oC 10; Jaganyi and Madlala15

458

report hindrance factors for caffeine in the range 11-14. Values estimated from

459

structural data of coffee particles5 were in the range 6-10 and increased with grind

460

size. Rescaling the expected range of bulk diffusion constants by hindrance factors

461

of this order, suggests small molecules will have particle diffusion constants

462

l(1.0n10R ) m2 /s and the carbohydrates oligomers l(1.0n10R ) m2 /s . These

463

crude range estimates are used to set the values for the two-species fit of Figure 1.

464

There are few studies of the relative composition of species but the limited data

465

available29, suggests 60-70% of the mass extracted into a brew are small molecules

466

and the main mineral potassium. This is consistent with the fitted mass fraction of

467

the faster diffusion species. However, it is likely that the mass fraction resulting in


Page 21 of 38


468

the smaller diffusion constant of the fit is also in part due to small molecules which

469

are hindered by other interactions as discussed below.

470

Given the heterogeneous nature of the real coffee particles, a given species will

471

likely experience varying hindrance factors (c.f. Eq. 8a) as they diffuse through the

472

structure.

473

coarse particles could have diffusion constants closer to bulk values. It would be

474

straightforward to include these effects in a model. However, this outer layer of

475

exposed macro-pores is only 30-60 μm thick and extraction from this layer is over in

476

a few seconds, a time scale comparable to that of the release from the fines. Whilst

477

this is a likely feature, it need be recognised that the spectrum of species rather than

478

grain structure may be the dominant cause of multiple relaxations times evident in

479

the data for total mass release over longer timescales. In support of this, and in

480

contrast to multiple relaxations in the total mass, studies of a single molecule

481

(caffeine) from short to long times (150 min), reported a good fit to a single diffusion

482

constant, hindered from that of bulk values but constant over time.10

Species released from exposed macro-pores at the grain surface of

483

Hindrance may also arise due to interactions other than grain structure. As is

484

evident in recent published data17-21 some species deviate from a simple diffusion

485

release model and these may constitute part of the mass fit by the smaller diffusion

486

constant in Figure 1. Speculation on mechanisms for this includes: slow dissolution

487

due to low water solubility, partitioning with oil phases, adsorption and binding

488

interactions within the grain matrix and other species. A classification of a wide

489

range of molecular release profiles has been made23,24, with low polarity, weakly

490

soluble compounds releasing more slowly18. Species might also interact with each

491

other inside the grains, e.g. phenolic molecules such as chlorogenic acids are known

492

to bind to colloid melanoidin particles.29 Jaganyi and Madlala15 observe that the 21 ACS Paragon Plus Environment


Page 22 of 38

493

mineral ion manganese shows diffusion constants order l(1.0n10R ) m2 /s

494

(comparable to the estimates above for oligomers). They argue this is due to co-

495

valent interactions due it being a transition metal. Guaiacols have been shown to

496

have relatively slow release for their molecular weight and have been argued to be

497

contributing to flavour in over-extraction21. As noted earlier, that maximal yield

498

decreases with grind size may also result from such interactions. Conversely, with

499

size exclusion of large species (light scattering shows the presence of colloidal

500

material in the range 100- 200 nm), it is possible that large and slow diffusing

501

species can only be released from the outer rim of coarse grains and fines, from

502

exposed cell-pockets – via this route such large species may appear to be fast

503

releasing. These many effects may offer routes to new technology and are a rich

504

area for further experimentation and modelling with suitable source/sink terms added

505

to the modelling.

506

A key feature seen in Figure 1 is that the concentration has a peak at early

507

times; this is evident in the model here and in that of Moreney et al.26 In the model

508

here the origin of this effect is simply a very large flux out of the coarse and fine

509

particles at early times. An initial high flux (the so called ‘wash off’) was well known

510

and a contentious issue in the earlier coffee literature. By mistakenly comparing

511

early time behaviour with the long-time (first order) behaviour of diffusion models it

512

was even claimed as evidence against a diffusion model at early times. This issue

513

was resolved by the work of Stapley13 who showed that the early time deviation from

514

first order kinetics was consistent with the early time behaviour also in the diffusion

515

model. Because in coffee systems the early time behaviour may have additional

516

complications, the issue is perhaps not closed.26 As noted by Moreney et al.26 the

517

surface includes both the fines and coarse particle surfaces. In addition to species 22 ACS Paragon Plus Environment

Page 23 of 38


518

at the surface having closer to bulk diffusion constants (as noted above), it is also

519

possible that the surface of grains holds an excess of extractable species over the

520

particle interior: e.g. the contents of cell pockets lost during grinding. Furthermore,

521

the initial flux out of a bed includes the complication of how the front wetting the bed

522

initially wets the grains and convects released species. Densified coffee has fines

523

incorporated into the surface macro-pores of coarse particles. Some of these

524

features can be directly incorporated in models but require detailed calibration

525

against short time data not yet available. Moreney et al.26 explicitly include a first

526

order rate term to model early time release from fines and surfaces, in the model

527

here2-4 fast release at early times is included via the fraction of fine particles.

528 529 530 531

CHARTS OF YIELD AND STRENGTH VS BREW TIME. Figure 2 shows a

532

chart of the model predictions in reduced units for a particular grind in which the

533

coarse particles are 8 times larger than the fine particle and with fines at 20% of the

534

grind volume. For coffee with fines of dimension (diameter) circa 40 μm this is a

535

coarse particle dimension d4,3 circa 320 μm - typical of an espresso grind and the

536

example in Figure 1.

537

bulk density circa 480 kg/m3. The yield is plotted against time (solid lines) for a

538

range of bed extraction efficiencies; overlaid are examples of contours of constant

539

strength (dashed lines) and constant ratio of brew volume to the volume of coffee

540

grind in the bed (dashed-dot lines).

The bed porosity was set at YZ = 0.2, appropriate for a dry



Page 24 of 38

541 542

Figure 2. The solid lines are extraction efficiencies, from the bottom to top : gQ = 0.001,

543

0.002, 0.004, 0.006, 0.008, 0.01, 0.015, 0.02, 0.03, 0.04, 0.05, 0.06, 0.07, 0.08, 0.1, 0.12,

544

0.14, 0.16, 0.18, 0.2, 0.25, 0.3, 0.35, 0.4, 0.5, 0.6, 0.8, 1.0, 2.0, 4.0. The dashed lines are

545

contours of constant strength, at times beyond the peak strength values and the dashed-dot

546

lines are contours of constant ratio of brew volume to grind volume.

547

figure tfine is the fine relaxation time defined in Eq. 15. The crosses show where the peaks

548

of the strength vs time plot occur (see text).

549

In the notation of the

Note that as PQ is increased, the yield vs time curves bunch to a limiting case.

550

This limit is when the flow through the bed is so fast (or equivalently the species

551

diffusion is so slow) that the concentration in the pore space approaches zero, the

552

bed extraction efficiency is high and extraction is just limited by diffusion through the

553

grains into the bed. Note in the high PQ limit with the bed concentration low,


Page 25 of 38


554

different particle sizes are weakly coupled and a polydisperse solution can be built

555

up by time rescaled and weighted sums of the mono-disperse solution. Conversely,

556

in the very low PQ limit, the bed pore space and particles are close to equilibrated on

557

the timescale of the bed relaxation time. The model then is truly trivial, diffusion can

558

be ignored and there is simply a first order decline in the concentration averaged

559

over the whole bed with decay constant given by (1 − YZ ) PQ . In between these

560

extremes (where for most species most brewing is in practice) the solution is more

561

complex.

562

influence. Along each PQ line the brew strength first increases to a maximum and

563

then declines (c.f. Figure 1); the crosses indicate the location of the peaks.

As this regime is approached, polydispersity has increasingly a weak

564

Note the interplay between flow rate and brew time acting along a contour of

565

fixed brew volume relative to grind volume (dashed-dot lines in Figure 1): a longer

566

brew time to a fixed target brew volume, implies a lower flow rate and lower

567

extraction efficiency and vice-versa; it is seen that after a reduced time of 10, the

568

yield along the contour shows a weak dependence on brew time. Overall, however,

569

plotting yield against brew volume does not collapse the data set further.

570

Given the brew weight, W, diffusion constant, Dp, maximal yield Ymax of the

571

species of interest, and the density of the particles, # ; a point ?PQ, ̃, $B of reduced

572

parameters can be mapped to real variables using definitions in Eqs. 6,15-17. and

573

the relationships:

574

= $

,

(18)

575

= ]^F_

,

(19)

576

o = PQ̃

VW

= ( )

(20)

,



577

=

p

q

=

pVW #Q *

Page 26 of 38

(21)

,

578 579

where o is the volume of the brew and is the concentration kg/m3 of

580

released coffee mass in the brew.

581

in the range 550-600 kg/m3 and maximal yields were given earlier. Conversely,

582

inversion of Eqs. 18-21 can be used to map absolute values onto the reduced

583

variables. The dashed-dot lines in Figure 2 are lines of constant PQ̃.

584 585

Typical values for the coffee particle density lie

Commonly, coffee concentration is measured and reported as a strength S as defined in Eq. 3. The relationship between to S is given by

586 587

S=

-

V

≅ 100 ∗

-

Vst (Ru-)

,

(22)

588 589

where is the density of water at the appropriate temperature. The experimental

590

problem is to infer a value of C from a measured S, the simulations have the inverse

591

problem. Up to practical concentrations S < 20%, measurements of vs C or S

592

show a weak linear variation, e.g. using the data for espresso brews reported by

593

Navarini et al.35 one finds v = 0.0002 m3/kg. Using v = 0 is a good approximation

594

and this was used to compute the strength contours in Figures 2 and 3 below.

595

THE ESPRESSO REGIME. In principle, Figure 2 is a control chart of the

596

model grind run for a single species. However, use of 0+ = 1.0n10R m2/s as a

597

proxy for total mass (see Figure 1 and Volliey and Simatos6 ) combined with a

598

measurement of maximal yield , the chart can be read as an predictive estimate of

599

the profile in time of the total mass yield - at least over the time scales of a typical

600

espresso brew. Adopting 0+ = 1.0n10R m2/s , ^F_ = 4s and using Ymax = 28%, 26 ACS Paragon Plus Environment

Page 27 of 38


601

for a system of d4,3 circa 320 μm,

602

chart, plotted in real units.

Figure 3 shows an “espresso region” of the

603 604 605

Figure 3. The ‘Espresso region’, reading from to left to right, the extraction

606

efficiencies (solid lines) are PQ = 4, 2, 1, 0.8, 0.6, 0.5, 0.4, 0.35, 0.3, 0.25, 0.2, 0.18,

607

0.16, 0.14, 0.12; the strength contours (dashed lines) are 3%, 5%, 7% 9%. Two

608

brew volume to grind volume contours (dashed-dot lines) are shown for 40 ml

609

brewed from a weight of 5.5 gm (L) and 9 gm (R). The region shown lies in the box

610

between coordinates (2, 0.6) and (10, 0.8) on the reduced units chart of Figure 2.

611 612 613

A typical café espresso system with a 9 g basket with 40 ml extracted in circa 25 s has PQ = 0.4. An on-demand coffee system, such as the NespressoTM system



Page 28 of 38

614

with capsules containing 5.5 g of coffee brewing to 40 ml in circa 20 s has PQ = 0.8.

615

These brewing systems are predicted to have comparable yields, circa 19% , within

616

the ideal flavour box (18-22%) of the original brew chart. The NespressoTM system

617

has a lower strength circa 3% vs 5%. The strengths (3-5%) are somewhat higher

618

than those in the 1960’s chart for US drip filter brews, but, are appropriate for a

619

modern espresso brew. Figure 3 is a reasonable prediction of actual espresso brew

620

performance and is able to differentiate between know brewing methods.

621

Regions typical of other brewing systems could be extracted from Figure 2 ,

622

e.g. a typical drip filter with 50 g, at 150 ml/min flow would have PQ = 0.14 and brew

623

times in reduced units circa 30; however, the actual flow regime drip filter beds can

624

be quite in-homogeneous, challenging the assumption of homogeneous flow in the

625

model. A drip filter grind size would typically be coarser that that used in Figure 2.

626

In practice, a series of charts are needed for different grind sizes. The prediction of

627

yields for longer times would also need to use a multiple diffusion constant model,

628

c.f. Figure 1.

629

Alternatively, if diffusion constants are known, the model and charts can be

630

used for making predictions and contrasting between individual releasing species. In

631

this case, the flow rate would be considered fixed, and using Eqs. 15 and 17

632

different species map onto different PQ , the larger the diffusion constant the smaller

633

PQ. The model predicts how strength in the brew (or species concentration measured

634

in aliquots over time) will decline post their peak value and this can be used to model

635

experimental data. The form of this decline varies across the chart with flow rate,

636

hence modelling is a vital aid to interpreting experiments.

637

species are independently diffusing, then for a contour of fixed brew volume to grind

638

volume ratio and a fixed flow rate, working out a PQ value for each species and

Assuming that multiple


Page 29 of 38


639

where it would intersect the contour will give its individual yield. The relative

640

composition of different species can thus be predicted. In the model, brews with the

641

same total mass yield achieved by different routes can have different compositions

642

and the degree of such variation can be predicted.

643 644

OUTLOOK

645

The format of charts such as that of Figure 2 and 3 is argued to be superior to the

646

current version1 because given weight, brew time and flow conditions, the modelling

647

and chart makes predictions of yield and strength. The base model used here was

648

found to be consistent with actual practice. Given the many approximations and

649

effects neglected in the base model, this is perhaps surprising. There are many other

650

complications in real systems as outlined throughout this paper: partitioning between

651

grind matrix, oil phase, grind macro-pores and bulk; interactions between species

652

and grind matrix; and early time effects such as wetting time scales, fast release

653

from fines and grind surface composition. On-going work is assessing how some of

654

these impact brewing relative to the prediction of the base model and improving the

655

modelling of the PSD and the spectrum of diffusion constants.

656

building models of individual species characteristic of different components of coffee

657

flavour, flavour significant compositional variations can be quantified. Algorithmic

658

improvements could be made to avoid the expensive solution of the diffusion

659

equations inside particles, alternatively interactions and the particle heterogeneity of

660

matrix and macro-pores could be modelled directly. The model can be combined

661

with a brewer and pump models3 to include realistic changing flow conditions and

662

more accurate predictions for real systems, although the modelling and

It is hoped that by



Page 30 of 38

663

measurement32 of dynamic bed flow resistance is required for this. Multi-phase gas-

664

liquid flow effects through beds which lead to coffee crema are a challenge.

665 666

AUTHOR INFORMATION

667

Corresponding Author

668

Email [email protected]

669

ORCHID

670

John Melrose 0000-0003-4234-9000

671

Funding

672

The authors would like to acknowledge Mondelèz International for sponsorship and

673

the EPSRC (UK) for financial support via the Formulation Engineering Doctorate

674

program.

675

Acknowledgments

676

The authors acknowledge and are grateful too several referees who made significant

677

suggestions for improving and editing the original manuscript. Several questions by

678

the referees have lead to sections of the paper which address and clarify key

679

technical and literature details.

680

acknowledged for the oven-refractometer calibration.

Mr Etienne Arman (Jacobs Douwe Egberts) is

681 682 683

684

REFERENCES 30 ACS Paragon Plus Environment

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685

(1)

Lockhart, E.E. The soluble solids in beverage coffee as an index to cup

686

quality. Tea and Coffee Trade J., 1957, 113, (1) 12; Coffee and Tea Industries,

687

1957, 80,

688

Technology, Westport, Comm.: AVI Pub. Co.

689

US8495950.

690

(2)

691

Bakalis, S. Optimising Coffee Brewing using a Multiscale approach. In ASIC(ed.)

692

proceedings of the 24rd conference International conference on Coffee Science San

693

Jose, Costa-Rica 2012.

694

(3)

695

from packed beds in On-Demand coffee systems. In ASIC(ed.) proceedings of the

696

25th International conference on Coffee Science, Armenia, Colombia 2014.

697

(4)

698

particle microstructure: Numerical modelling and experimental validation in slurry

699

experiments. In ASIC(ed.) proceedings of the 25th International conference on Coffee

700

Science, Armenia, Columbia 2014.

701

(5)

702

Doctorate in Engineering Thesis, University of Birmingham 2017.

703

(6)

704

coffee brewing. J. Food Process Eng. 1979, 3, 185–198.

705

(7)

706

coffee: the temperature variation of the hindrance factor.

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74, 416-420.

16 . See discussion in Sivetz, M.; Foote H.E. Coffee Processing 1963;

see further examples in

Melrose, J. R.; Corrochano, B.; Norton M.; Silanes-Kenny, J.; Fryer, Peter.;

Melrose, J. R.; Corrochano, B.; Bakalis, S. The principles of coffee extraction

Corrochano, B.; Melrose, J.R.; Bakalis, S. Kinetics of coffee extraction and

Corrochano, B. Advancing the engineering understanding of coffee extraction.

Voilley, A.; Simatos, D. (1979). Modeling the solubilization process during

Spiro, M.; Chong, Y. The kinetics and mechanism of caffeine infusion from J. Sci. Food Agric. 1977,



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Spiro, M.; Siddique, S. Kinetics and equilibria of tea infusion: Kinetics of

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extraction of theaflavins, thearubigins and caffeine from Koonsong broken pekoe. J.

710

Sci. Food Agric. 1981, 32, 1135–1139.

711

(9)

712

coffee: Hydrodynamic aspects. J. Sci. Food Agric. 1984, 35, 925–930.

713

(10)

714

caffeine infusion from coffee: The effect of particle size. J. Sci. Food Agric. 1984, 35,

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915–924.

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717

and roast coffee. J. Sci. Food Agric. 1993, 61, 371-372.

718

(12)

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M. Spiro (Eds.), Chem. Biol. Prop. Tea Infusions (pp. 38–62). Frankfurt/Main:

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Umwelt und Medizin 1997.

721

(13)

722

Agric. 2002, 82, 1661-1671.

723

(14)

724

Coffee as a Function of Brewing Time. J. Food Sci. 1992, 57, 1417–1419.

725

(15)

726

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

727

roast coffees. J. Sci. Food. Agric. 2000, 80, 85-90.

728

(16)

729

Liverani, F. Hyper espresso coffee extraction: adding physics to chemistry. In

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ASIC(ed.) proceedings of the 22nd conference Campinas, Brazil 2008.

Spiro, M.; Page, C. M. The Kinetics and mechanism of caffeine infusion from

Spiro, M.;

and Selwood, R. M. (1984). The kinetics and mechanism of

Spiro, M. Modelling the aqueous extraction of soluble substances from ground

Spiro, M. Factors affecting the rate of infusion of black tea. In R. Schubert &

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