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Mathematical modeling of tea bag infusion kinetics
Journal Pre-proof Mathematical modeling of tea bag infusion kinetics Pallavee P. Dhekne, Ashwin W. Patwardhan PII:
S0260-8774(19)30490-X
DOI:
https://doi.org/10.1016/j.jfoodeng.2019.109847
Reference:
JFOE 109847
To appear in:
Journal of Food Engineering
Received Date: 27 February 2019 Revised Date:
16 November 2019
Accepted Date: 27 November 2019
Please cite this article as: Dhekne, P.P., Patwardhan, A.W., Mathematical modeling of tea bag infusion kinetics, Journal of Food Engineering (2019), doi: https://doi.org/10.1016/j.jfoodeng.2019.109847. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Published by Elsevier Ltd.
1
Mathematical Modeling of Tea Bag Infusion Kinetics
2
Pallavee P. Dhekne, Ashwin W. Patwardhan*
3
Department of Chemical Engineering, Institute of Chemical Technology, Matunga,
4
Mumbai-400019, India
5 6 7 8 9 10 11 12 13 14 15 16 17
Corresponding Author: Tel: 91-22-33612018; Fax: 91-22-33611020
18
Email address: [email protected]
19
Abstract
20
The current work explains the development of a mathematical model for the prediction of tea bag
21
infusion kinetics of solid-liquid extraction of polyphenols from tea. The developed model has
22
been used to obtain tea bag infusion profile using equilibrium parameters i.e. partition constant
23
( K ) , the initial content of tea solute ( C si ) and tea bed permeability (κ ) . The Predicted infusion
24
profiles and experimental data are in good agreement having mean relative error < 10 %. For all
25
the conditions (loading in the tea bag, particle size, temperature, and dipping frequency), the
26
Gallic acid equivalence (GAE) per unit equilibrium concentration ( Cveq ) eluted inside tea bag
27
(C¶ ) was found to be higher as compared to that in the vessel ( C¶ ) .
28
increase in particle size for given loading exhibit slower infusion kinetics. However, an increase
29
in dipping frequency (Reynolds number) and temperature leads to an increase in mass transfer
30
coefficient leading to faster infusion kinetics.
31
Keywords: tea bag, infusion kinetics, mathematical model, partition constant, Biot number, time
32
scale analysis
v
b
2
The increased loading and
33
1 Introduction
34
The solid-liquid extraction (leaching) process is widely used in industries like chemical, food,
35
pharmaceutical, biotechnology, etc. for separation of key components from solid matrix to
36
solvent phase. The extraction of phenolic or bioactive compounds from the natural resources has
37
been a subject of several research studies. Among the several extraction processes, the solid-
38
liquid extraction of polyphenols from tea is one of the complicated processes to model. The
39
technique of preparing tea is different in different parts of the world. Most of the researchers
40
have performed the kinetics and equilibrium study of loose tea infusion. Spiro and co-workers
41
lucubrated the partitioning of tea solute and role of tea leaf sizes on the tea infusion kinetics
42
(Price and Spiro, 1985a, 1985b). In further studies, the infusion kinetics of tea solute through
43
the tea bag was also investigated (Spiro and Jaganyi, 2000). Kinetic and infusion studies carried
44
out previously in literature were based on kinetic expression (Spiro and Jago, 1982). The Spiro’s
45
steady-state lumped model reveals the first-order kinetics of the tea infusion process and
46
expressed as follows:
47
(
)
ln C * / ( C * − C ) = kobs t + p
(1)
48
where C * and C are the concentration of tea constituents in the bulk infusion at equilibrium and
49
at a time ‘t’ respectively. In Eq. (1) kobs is the overall rate of infusion and ‘p’ is an empirical
50
constant. Further study by Stapley, explains the significance of intercept (p) in Spiro's kinetic
51
model (Stapley, 2002). The model developed for caffeine extraction from coffee beans reported
52
the caffeine diffusivity as 3.21×10-10 m2/s at 90 °C (Espinoza-Pérez et al.,(2007). The effect of
53
tea bag material on the rate of extraction of caffeine from black tea has been studied (Jaganyi and
54
Mdletshe, 2000). The first-order rate constant for tea bag was found to be 29 % lower than that
3
55
of loose tea. The infusion rate was unaffected by tea bag shape but improved with an increase in
56
tea bag size (Jaganyi and Ndlovu, 2001). The rates of caffeine transport through tea bag paper
57
have been measured using a modified rotating diffusion cell (Spiro and Jaganyi, 2000). The
58
study showed that the transport of tea solute through the tea bag membrane has negligible
59
resistance within the temperature range of 25-80 °C. It was concluded that any motion (stirring
60
the brew around tea bag, dipping of a tea bag) which decreases the thickness of the Nernst layer
61
around the tea bag improves the rate of tea infusion.
62
A computational fluid dynamics (CFD) model was developed and simulated for the tea
63
bag infusion process with different brewing conditions i.e. static and dynamic (Lian and Astill,
64
2002). The hydrodynamics of tea bag infusion was also reported. It was observed that the
65
dynamic condition (stirring) enhances the rate of infusion of the tea solute in water (Jaganyi and
66
Mdletshe, 2000; Lian and Astill, 2002). The swelling kinetics of individual tea particles as well
67
as the bed of tea granules was investigated (Joshi et al., 2016). It was concluded that the
68
swelling of tea particles is a fast process and occurs simultaneously with the infusion process.
69
The effect of particle size and brewing temperature on the rate of tea infusion process has been
70
carried out (Farakte et al., 2016). The rate of infusion was improved with the decrease in particle
71
size and increase in temperature. Moreover, the values of equilibrium constant (partition
72
constant, K) and initial content (Csi) for two types of crush, tear, curl (CTC) teas for different
73
sizes were estimated.
74
A mathematical model has been developed to predict a tea infusion which accounts for
75
swelling kinetics of tea granules (Farakte et al., 2017).
76
constituents inside the tea granule at 60 ºC was reported to be 3.33×10-10 m2/s. The effects of tea
77
particle size, tea bag dipping rate, loading of tea granules and tea bag shapes on infusion rate
4
The estimated diffusivity of tea
78
have been investigated (Yadav et al., 2017). It was observed that the infusion rate decreased
79
with an increase in tea granules loading in the tea bag. The percent fill of tea granules and tea
80
bed height inside the tea bag greatly affect the tea granule swelling and eventually, the infusion
81
rate. Recently, the effect of temperature, particle size and source of tea on the infusion of
82
individual tea components have been studied (Yadav et al., 2018). This study showed that the
83
partition constant (K) depends not only on temperature but also on the tea types (source of tea
84
granules).
85
As discussed above, the previous study deals with the kinetics and equilibrium of tea
86
infusion. Few researchers have focused on modeling of infusion kinetics of loose tea. There is
87
scarce data available on the modeling of tea bag infusion. Due to the rise in demand for tea bags
88
for tea preparation, the tea infusion parameters have to be optimized.
89
parameters include a loading of tea granules in a tea bag, particle size, brewing temperature, and
90
tea bag dipping frequency. Hence, there is a need to develop a model, which can predict the tea
91
bag infusion kinetics, and help to optimize brewing parameters and the tea bag design.
92
2 Materials and Methods
93
2.1 Materials
94
Commonly used crush, tear, curl (CTC) black tea and cellulose acetate paper double-chambered
95
tea bags (6.5 cm × 4 cm) were procured from the local market. For tea bag infusion experiments
96
and analysis, de-ionized (DI) water (Millipore Inc, USA) was used. All the experiments were
97
performed in a infusion vessel of volume 150 ml (Farakte et al., 2016).
5
These tea brewing
98
2.2 Methods
99
The infusion kinetics study was performed using a tea bag dipping set-up (Fig. 1A) (Yadav et al.,
100
2017). The present dipping set-up imitates dipping of the tea bag in a cup of hot water. The
101
vessel containing 100 ml of de-ionized water was heated up to the desired temperature (60, 70,
102
80 and 90, ± 2 ºC) using a constant temperature water bath. When the desired temperature was
103
achieved, the tea bag which consists of 2 g of tea granules was dipped in a vessel with a dipping
104
frequency of 5 dips per minute (dpm).
105
The experiment was conducted for 15 min. 1 ml sample was withdrawn from the vessel
106
at 0.5, 1, 2, 3, 4, 5, 10 and 15 min. Volume of filtrate in the vessel was measured at the end of
107
the experiment. The samples were diluted 100 times using DI water and analyzed using UV-Vis
108
spectrophotometer (Cary 50 ) (Farakte et al., 2016). The UV spectrum of 100X diluted tea
109
samples shows the λmax to be 272 nm. Therefore, absorbance at 272 nm was monitored as a
110
measure of the extent of infusion. Most of the tea polyphenols are Gallic acid (GA) derivatives
111
(Taylor et al., 2010). Thus absorbance of tea infusion samples was expressed as GA equivalence
112
(GAE) by means of calibration curves with standard Gallic acid (Farakte et al., 2016, Spigno and
113
Faveri, 2009). The GAE values reported in this work are the equivalent amount of GA, which
114
would give the same absorbance. The GAE % (percent weight of GAE extracted per unit weight
115
of tea granules) were corrected for volume loss due to evaporation and sampling.
116
The experimental conditions for the tea bag infusion kinetics are shown in Table 1. The
117
effect of dipping frequency on infusion kinetics was studied for 2, 5, 8, 10, 15, 20, 30 and 50
118
dpm at 60 ºC for 15 min. In order to explore the effect of temperature on infusion kinetics,
119
experiments were performed at 60, 70, 80 and 90 ºC for constant dipping frequency of 5 dpm.
120
To study the effect of particle sizes on infusion kinetics, the tea particles were crushed and 6
121
separated (2.12 mm, 1.14 mm and 0.36 mm) by sieving. Moreover, the loading effect of tea
122
granules on the infusion kinetics was assessed with the loading from 1-3 g for 5 dpm at 60 ºC.
123
Here, the loading corresponds to a constant solid-to-liquid ratio (g/ml) of the tea granules to the
124
water.
125
The equilibrium infusion experiments for different sizes of tea granules have been
126
reported in the previous study (Farakte et al., 2016). The value of partition constants (K=0.14)
127
and initial content of tea constituent (Csi = 88 kg/m3) for 2.21 mm particle size is estimated by
128
procedure mentioned elsewhere (Farakte et al., 2016) and used in the present work.
129
3 Model Development
130
Tea infusion through tea bag is a more complex phenomenon than loose tea. In the brewing
131
process, a tea bag is immersed in the hot water. The hot water flows in and out of the tea bag
132
paper, leaching out soluble contents. A schematic of this process is shown in Fig. 2B. The
133
resistance to the flow of water in and out of tea bag is a bed of granules. This process of infusion
134
through the tea bag consists of various steps: (i) flow of water through tea bag paper from bulk
135
water to bed of tea granules; (ii) flow of water through pores of tea bed; (iii) absorption of water
136
by tea granules; (iii) dissolution of tea constituents from solid phase to liquid phase within tea
137
granules; (iv) diffusion of tea constituents from water present inside tea granules to outer surface
138
of granules; (v) transport of dissolved soluble solid from outer surface of granule to pores of tea
139
bed; (vi) convective transfer of soluble solids in the fluid.
140
The present model is based on the physics involved in the infusion process. The mass
141
transfer from tea granule to the surrounding liquid within the tea bag is an important step. The
142
mass transfer inside the tea bag depends upon the liquid ingress into and out of the tea bag. Once 7
143
the infused liquid containing the dissolved components comes out of the tea bag into the vessel,
144
it mixes with the rest of the liquid in the vessel. The model developed thus takes into account the
145
several phenomena actually taking place and it is not just a lumped parameter model.
146
The following assumptions were made in the model formulation: (a) Tea granules are
147
symmetrical. (b) The mass transfer Biot number (Bi) is defined as the ratio of internal to external
148
mass transfer resistances (the detailed derivation is provided in supplementary section) and can
149
be calculated by using following Eq. (Cacace and Mazza, 2003; Karacabey and Mazza, 2008)
Bi =
150
k N KL 2Deff
(2)
151
Where k N is real external mass transfer coefficient and Deff is the effective diffusivity of the tea
152
constitutents. If Biot number far less than unity (Bi << 1), then internal diffusive mass transfer
153
resistance can be negligible as compared to external convective mass transfer resistance at the
154
solid surface. It indicates that solute concentration gradients may not exist within solid particles.
155
(c) The tea solution within a tea bag is well mixed and homogeneous. (d) Tea granules in the
156
bag are considered as a packed bed.
157
With these assumptions, the mass balance for tea solute inside the tea bag can be written as
158
follows,
159
Vb
(
)
dCb = k L S C ∗ − Cb + QC v − QCb dt
160
Eq. (3) describes the rate of change in the concentration of tea solute inside the tea bag (Cb). It
161
can be expressed as follows,
8
(3)
(
)
dCb 1 = k L a C ∗ − Cb + ( Cv − Cb ) dt τb
162
(4)
163
Where k L a (s-1) denotes volumetric mass transfer coefficient ( a = S Vb ), Cv is the concentration
164
of tea constituents in the bulk infusion of volume (Vv ) and
165
tea bag (Vb ) to the volumetric flow rate through the tea bag ( Q ). The release of solute from tea
166
granules to the fluid in the voids of the packed bed is described by the general Eq. of interfacial
167
mass transfer. Hence, the rate of change in solid-phase concentration ( C s ) can be expressed as, −Vs
168
τb is the ratio of water volume inside
(
dC s = k L S C * − Cb dt
)
(5)
169
Substitution of C ∗ = K C s ( K : partition constant, the ratio of the concentration of tea solute in
170
infusion ( C * ) to that in the tea granules at equilibrium) into Eq. (4) and (5) yields,
dCb 1 = k L a ( KCs − Cb ) + ( Cv − Cb ) dt τb
171
−Vs
172
dC s = k L S ( KC s − Cb ) dt
(6)
(7)
173
Once the tea solutes are leached out, their transport occurs by the convective transfer through the
174
tea bag. The corresponding mass transfer Eq. for the tea solutes in the well-mixed liquid phase is
175
written as dC v = QC b − QC v dt
176
Vv
177
dCv 1 = ( Cb − Cv ) dt τ v
9
(8)
(9)
τv
is the space-time based on the volume of solution (τ v = Vv Q ) . Eq. (6) and (9)
178
Where
179
describes the concentration profile with respect to time inside the tea bag and bulk infusion
180
respectively.
181
Overall mass balance gives,
VsCs = VsCsi − VvCv − VbCb
182
(10)
183
The solid phase concentration ( Cs ) can be expressed in terms of the initial content of tea
184
granules ( Csi ) by rearranging Eq. (10) as follows, V C + V C Cs = Csi − b b v v Vs
185
186
(11)
Substituting Cs value from Eq. (11) into Eq. (6) gives, 1 dCb K = k L a KCsi − (Vb Cb + Vv Cv ) − Cb + ( Cv − Cb ) dt Vs τb
187
(12)
KV 1 dCb k aKVb 1 = k L aKCsi − k L a + L + Cb − k L a v − Cv dt Vs τb Vs τ b
188
(13)
189
The non-dimensional form of Eq. (13) can be obtained as follows. Dividing the Eq. (13) by
190
equilibrium concentration ( Cveq ) and volumetric mass transfer coefficient ( k L a ) yields,
191
(
d Cb Cveq d ( tk L a )
) = KC C
si eq v
KVv KVb 1 v − 1 + + Cb Ceq − Vs k L aτ b Vs
(
)
1 − k L aτ b
v Cv Ceq
(
)
(14)
192
¶ = C C eq and non-dimensional time θ = tk a and Introducing non-dimensional concentration, C L b b v
193
rearrangement gives, 10
¶ dC KC KVb 1 b = eqsi − 1 + + dθ Cv Vs k L aτ b
194
¶ KVv C b − Vs
1 − k L aτ b
¶ C v
(15)
195
Similarly, the non-dimensional form of Eq. (9) can be obtained as follows. Dividing the eq. (9)
196
by Cveq and kL a yields,
(
d Cv Cveq
197
d ( tkL a )
)=
1 Cb − Cv k L aτ v Cveq
(16)
198
Rearranging above Eq. in terms of dimensionless concentration C¶v = C v C veq and dimensionless
199
time θ = tk L a
(
¶ dC 1 v ¶ −C ¶ = C b v dθ k L aτ v
200
201
204
205
206
(17)
Differentiating Eq. (17) with respect to non-dimensional time (θ), ¶ d 2C 1 v = 2 dθ k L aτ v
202
203
)
¶ dC ¶ dC b v − dθ dθ
(18)
¶ dθ from Eq. (15) into Eq. (18), it becomes, Substituting for dC b
¶ d 2C 1 v = 2 dθ k L aτ v
KCsi KVb 1 + eq − 1 + Vs k L aτ b Cv
¶ KVv 1 − Cb − k L aτ b Vs
¶ 1 Cv − k L aτ v
¶ dC v dθ
(19)
¶ value from Eq. (17) into Eq. (19) and rearrangement yields following Eq., Substituting C b ¶ ¶ d2C 1 KCsi KVb KVv ¶ KVb 1 1 dC v v = − 1 + + C − 1 + + + v dθ 2 k L aτ v Cveq Vs Vs V k a τ k a τ d θ s L b L v
11
(20)
207
Now, the equilibrium concentration of tea constituents ( Cveq ) can be expressed in terms of the
208
partition constant and initial content of tea granules, Vs C veq = KC si KVv + Vs
209
210
211
(21)
Rearrangement of Eq. (20) yields, ¶ ¶ KVb d2C 1 1 dC 1 v v = − 1 + + + − 2 dθ Vs k L aτ b k L aτ v dθ k L aτ v
KVb KVv ¶ 1 + 1 + Cv + Vs Vs k L aτ v
KVV 1 + VS
(22)
212
The dimensionless variables appearing in Eq. (22) disclose the important characteristics of the 213
physical process. The Damkohler number, Da as the ratio of transport rate to convection rate and 214
can be represented as
kL a (1 τ )
215
Da =
216
The simplified form of non-dimensional model Eq. can be represented as follows:
217
¶ ¶ KVb d 2C 1 1 dC 1 KVb KVv ¶ 1 KVV v v = − + + + − + 1 1 + Cv + 1 + 2 dθ Vs Dab Dav dθ Dav Vs Vs Dav VS
(23)
(24)
218
Eq. (24) can be solved for the concentration of tea constituents in the vessel, with the initial
219
condition as;
220
¶ ¶ = 0 and dCv = 0 at θ =0, C v dθ
(25)
221
The solution of Eq. (24) is obtained using MATLAB (R2015a). Mean relative error (MRE) is
222
used to assess the accuracy of prediction with the experimental data. 12
MRE =
223
((
1 ¶ ¶ abs C ∑ v exp t − C v pred n
)
¶ C v exp t
)
(26)
224
¶ and C ¶ where n is the number of data points, C v expt v pred are the experimental and predicted
225
concentrations respectively. The Runge Kutta method of order 4 (MATLAB, R2015a) was used
226
to solve the set of ODEs. The solution to problems defined by the initial conditions (Eq. 25) was
227
obtained using ODE 45 solver with a known time step.
228
The model Eq. (24) consists of different parameters such as kL , S , Vb , Vv , K , Csi , Vs and
229
Q. The total surface area ( S ) was calculated from the known weight of tea granules (w). The
230
specific surface area ‘a’ is based on the volume of water inside tea bag and can be calculated as;
231
a = S Vb . The volume of water inside the tea bag (Vb ) and in the vessel (Vv ) was measured
232
during the experiment. The value of partition constants ( K = 0.14 ) and initial content of tea
233
constituent Csi = 88 kg/m
234
Farakte et al., (2016) and used in the present work. The Gallic acid diffusion coefficient in water
235
at various temperature were calculated using Wilke–Chang correlation (Treybal, 1980).
236
(
3
) for 2.21 mm particle size is estimated by procedure mentioned by
DGA = 1.173 × 10 −18
T (φ M B )
µ Bν A0.6
0.5
(27)
237
Where, φ is the association factor for solvent (2.26 for water as solvent), vA is solute molar
238
volume, m3/kmol (Gallic acid, C7 H 6O5 ).
239
The hydration of tea and water inside the tea bed (porosity) has the contribution in the
240
swelling of tea bed. The absorbed water and the water inside the pores tend to extricate when the
241
bag is removed from the vessel. Hence, the accurate quantitative measurement of tea bed
242
swelling inside the bag is difficult due to the lateral and longitudinal expansion of tea bag.
13
243
Therefore, in order to quantify the combined effect of swelling and porosity of the bed, tea bed
244
permeability is used as a fitted parameter. The flow-through tea bag ( Q ) can be calculated from
245
the value of tea bed permeability
246
Geankoplis, 2003) ;
(κ , m/s )
by using Darcy's law as follows (Christie J.
dh dl
247
Q = −κ A
248
where dh = v 2 2 g ( dynamic head, m) due to the dipping frequency of tea bag in the vessel, dl
249
is the height of swelled tea bed (m) and v (m/s) can be related to the amplitude (A) and the
250
dipping frequency (f) as: v = Af .
251
Interfacial mass transfer coefficient for the release of the solutes from tea granules can be
252
estimated from the correlation of Thoenes and Kramers (Fogler, 2004).
Sh ' = 2 + 0.5 ( Re ' )
253
254
255
0.5
(28)
Sc 0.33
(29)
Eq. (29) can be written as follows;
kLd p DGA
ε 1− ε
Ud p ρ 1 = 2 + 0.5 γ µ (1 − ε )γ
0.5
µ ρ DGA
0.33
(30)
256
where γ is shape factor (external surface area divided by πdp2 ), ε is a void fraction (porosity) of
257
the packed tea bed and U (m/s) is the superficial liquid velocity through the bed. The Nerst layer
258
resistance (1 kN ) was estimated from the overall mass transfer resistance and the tea leaf
259
diffusive resistance as follows:
14
260
1 1 KL = − k N kL 2Deff
261
The mass transfer Biot number (Bi) was calculated by using Eq. (2) where the effective
262
diffusivity
263
measured thickness of the tea leaf ( L ) is 1.34× 10–4 m .
(31)
( D ) of solid tea constituents is 3.33 × 10–10 m2/s (Farakte et al., 2017) and the eff
15
264
4 Results and Discussion
265
The diffusion of tea components within the tea granule is compared to the convective diffusion
266
from tea to the solution inside the tea bag. During tea bag infusion process, swelling of tea
267
granules, as well as tea bed, takes place. The swelled tea bed leads to an increase in the
268
compactness of the bed in the bag (Yadav et al., 2017). The higher resistance for the external
269
mass transfer is offered by the highly compact tea bed during the tea infusion. This can be
270
observed from the estimated Biot number for intraparticle diffusive resistance to external
271
convective mass transfer resistance. For example: dipping frequency = 2 dpm case;
272
K = 0.14; kL = 4.11 × 10-6 (m/s)
273
(estimated value in the current work), Deff = 3.33 ×10−10 m2 s (Farakte et al., 2017) and
274
L = 1.34 ×10−4 m (measured value in the current work). Therefore, real external mass transfer
275
coefficient is calculated by using Eq. (31).
276
−4 1 1 0.14 ×1.34 ×10 = − kN 4.11×10−6 2 × 3.33 ×10−10
277
k N = 4.648 × 10 −6 m/s
278
The biot number for mass transfer is calculated from Eq. (2) as follows:
Bi =
279
k N K δ 4.648 ×10−6 × 0.14 ×1.34 ×10−4 = = 0.133 2 Deff 2 × 3.33 ×10−10
280
Similarly, the Biot number (Bi) for all cases has been estimated and found to be near unity
281
(Bi~1).
282
mass transfer as well.
This indicates the combined effect of intra-particle diffusion and Nerst layer external
16
283
4.1 Effect of dipping frequency
284
The comparison of predicted (solid line) and experimental (symbol) data for different dipping
285
frequencies is shown in Fig. 2A. This indicates the developed model fits the infusion kinetics
286
data very well with MRE < 10 %. The concentration profile of tea constituents inside the tea bag
287
¶ (dashed line) and in the vessel C ¶ (dotted line) is shown in Fig. 2B. The concentration C b v
288
profile in the vessel shows sigmoid nature (tilted S-shape) at lower dipping frequency which is in
289
good agreement with the literature (Yadav et al., 2017). It implies that the slower infusion
290
kinetics during the initial stages of low dipping frequency. Moreover, C¶b is higher than C¶v for
291
all dipping frequencies interpreting the significance of the external convective mass transfer
292
µ with respect to time (θ ) is shown in the inset of Fig. 2B. It is resistance. The variation of ∆C
293
evident that the increase in dipping frequency leads to a decrease in the change in concentration
294
( ∆Cµ) . The concentration in the vessel (C¶ ) can be attainable as C¶ within 5 min of the infusion
295
period for a higher dipping rate. This is due to the faster infusion kinetics caused by a higher
296
extent and faster swelling kinetics of tea granules (Yadav et al., 2017). Fig. 2C depicts the effect
297
of dipping frequency on the tea bed permeability ( κ , m/s) and flow-through tea bag ( Q , m3/s).
298
From Fig. 2C, it can be seen that the increase in dipping frequency leads to an increase in
299
convective transfer through the tea bag. However, bed permeability decreases with an increase
300
in dipping frequency from 2-50 dpm in the same order. Due to the increase in a flow-through tea
301
bag, swelling of tea bed increases (Joshi et al., 2016); which implies the reduction in bed
302
permeability value. Fig. 2D shows the variation of the volumetric mass transfer coefficient
303
( kL a ) with dipping frequency. It is clear that as dipping rate is increased by a factor of 10 (5
( )
v
b
17
304
dpm to 50 dpm), kLa was improved by 2.2 times due to the increase in Reynolds number (flow
305
provided by higher dipping).
306
4.2 Effect of temperature
307
The shape parameter and physical properties with different temperatures are listed in Table 2.
308
The predictions for tea bag infusion kinetics at different temperatures (60, 70, 80 and 90 ºC) are
309
compared with an experimental data reported previously (Yadav et al., 2017). The comparison
310
between predicted and experimental results is shown in Fig. 3A. The model predictions for
311
infusion at given temperatures fit the experimental data adequately up to 10 min (MRE < 10 %).
312
Fig. 3B depicts the infusion profile of GAE (mg/ml) eluted per unit equilibrium concentration
313
(C )
314
¶ is higher as compared C ¶ . During the initial infusion From Fig. 3B, it is observed that C b v
315
period, a higher infusion rate is observed at higher temperatures. The inset plot in Fig. 3B shows
316
the difference in concentrations between C¶b & C¶v . From Fig. 3C, it is observed that with a rise
317
in temperature, the bed permeability and flow through tea bag decreases. This decrease in flow
318
was caused due to the resistance provided by the swelling of tea bed with a temperature. This
319
observation is supported by Joshi et al. stating that an increase in temperature from 60 to 90 ºC
320
leads to an increase in the rate and extent of swelling (Joshi et al., 2016). From the present
321
model (Fig. 3D), it was found that the estimated values of the mass transfer coefficient ( kL) at 90
322
°C is 1.67 fold than that observed at 60 °C.
eq v
( ) and in the vessel (C¶ ) with respect to dimensionless time (θ ) .
¶ inside the tea bag C b
v
18
323
4.3 Effect of particle size
324
Tea bag infusion kinetics for different particle sizes (2.21 mm, 1.14 mm and 0.36 mm) was
325
reported previously (Yadav et al., 2017).
326
partition constant
327
(Farakte et al., 2016) and used in the present work. It was found that the initial contents of tea
328
constituent ( C si ) and partition constant ( K ) changes with particle size. The possible reasons for
329
the variation in C si and K could be non-uniform plucking and particle size distribution during
330
CTC (crush-tear-curl) process from different parts of the plucked leaf.
The initial contents of tea constituent
( Csi ) and
( K ) for different sizes were measured in the previous tea infusion study
331
The effect of particle size on the tea bag infusion profile is shown in Fig. 4. It is
332
observed from Fig. 4A that the model prediction for infusion using 2.21 mm, 1.14 mm and 0.36
333
mm granules fit the experimental data very well (MRE<10%). According to Yadav et al.,
334
(2017), smaller particle size has improved infusion kinetics. The observation can be explained
335
by the fact that smaller particle size offers a higher interfacial area for mass transfer. However,
336
from Fig. 4A the dimensionless concentration for different particle sizes (2.21, 1.14, 0.36 mm) is
337
not following the trend as like in Fig. 2A and Fig. 3A. The possible reason for this discrepancy
338
is the actual interfacial area, which significantly decreases than that of considered for
339
dimensionless model development based on particle diameter (Dp). This significant decrease in
340
interfacial area is due to the densely packed tea bed of swelled particles. The observed trend of
341
¶ vs. θ in Fig. 4A seems to be reversed with particle size due to the significant effect of C v
342
Damkohler number (Da) given by Eq. (24).
19
343
Fig. 4B shows the concentration of tea constituents inside the tea bag
(C¶ ) (dashed line) b
344
¶ (dotted line). It can be observed from Fig. 4B, for 0.36 mm particle C¶ and in the vessel C v b
345
shows higher as well as a faster infusion to as that of 2.21 mm. The inset Fig. 4B depicts the
346
concentration difference between C¶b and C¶v as a function of time (θ ) . The profile indicates the
347
difference in concentration (∆C) is higher for 0.36 mm particle size as that of 2.21 mm. This is
348
because of the higher extent of infusion inside the tea bag for smaller particle size and resistance
349
to the flow through the bed due to the compactness. The effect of particle size on the flow-
350
through tea bag ( Q ) and tea bed permeability (κ ) is shown in Fig. 4C. The increase in particle
351
size leads to an increase in κ and Q due to the lower bed resistance (decrease in swelling).
352
However, kLa is higher for lower particle size due to the higher interfacial area for mass transfer
353
(shown in Fig. 4D).
354
4.4 Effect of loading
355
The tea bag infusion kinetics with different loading (0.25, 0.5, 0.75, 1, 2 and 3g) was reported by
356
Yadav et al. (Yadav et al., 2017). Fig. 5 (A, B, C, D) shows all the inferences on the effect of
357
loading. Fig. 5A depicts the model fitting of the infusion profile as GAE eluted per unit Cveq for
358
tea bag loading with 1-3 g tea. Model fits the infusion kinetics data very well with MRE < 10%.
359
¶ Fig. 5B depicts the comparison of the concentration of tea constituents inside the tea bag C b
360
¶ with respect to dimensionless time (θ ) . It is evident that, for a given loading, and the vessel C v
361
¶ shows a higher infusion as compared to C ¶ . This interprets the significance of the external C b v
362
convective mass transfer resistance. The difference in concentrations is shown in the inset of
363
Fig. 5B. It is clearly observed that concentration difference is decreased for lower loading of tea
( )
20
Fig. 5C shows the effect of loading on the flow through the tea bag
(Q )
364
granules.
and
365
permeability of tea bed (κ ) . The value of Q (m3/s) obtained for 1 g loading is 1.4 times higher
366
than that of 3 g loading. This indicates that the effect of clogging (resistance to flow) in the tea
367
bed is less dominant with lower loading and difference in concentrations ∆C is lesser. Hence,
368
lower loading resulted in higher infusion kinetics. Moreover, the higher infusion kinetics for low
369
loading is confirmed by an increase in the volumetric mass transfer coefficient ( kL a ) as shown in
370
Fig. 5D.
371
4.5 Statistical Analysis
( )
372
The experimental results reported previously show the mean value of at least three
373
replicates (Yadav et al., 2017). The mean relative error (MRE) (Eq.(26)) estimated by the
374
¶ comparison of experimental C v exp t
375
accuracy of the proposed mathematical model for the prediction of the infusion kinetics. The
376
statistical significance has also been evaluated using the parity plot between the experimental
377
and predicted values of concentrations for all data points, encompassing widely different
378
operating parameters.
379
predictions using the proposed model and the experimental value is ± 10%.
380
4.6 Transport time scale analysis
381
For developing a mathematical model, it is important to know the characteristic time scale of the
382
involved processes. Underlying physics of the tea bag infusion process can be understood from
383
the time scale analysis.
(
) and predicted data (C¶ ) , was calculated to evaluate the v pred
From Fig. 6, it is observed that the mean relative error between
21
384
4.6.1 Mass transfer time scale
385
It is defined as the reciprocal of the volumetric mass transfer coefficient and expressed for loose
386
tea (LT) and tea bag (TB) as follows:
τ MT loose tea =
387
1 ( k L a )LT
and
τ MT tea bag =
1 ( k L a )TB
388
The mass transfer coefficient (MTC) for loose tea is 3.846 × 10-5 (m/s) (Spiro and Jago, 1982).
389
In the current study, MTC for tea bag infusion has been estimated (kL = 4.11 × 10-6 (m/s)). The
390
mass transfer coefficient for loose tea is approximately 9.36 times that of tea bag infusion.
391
Therefore, the mass transfer time scale for loose tea is less as compared to tea bag ( τ MT loose tea <<
392
τ MT tea bag ).
393
4.6.2 Mixing time scale
394
Mixing time scale is defined as the volume of water (V ) in a defined system divided by the
395
volumetric flow rate (Q) through the tea bag.
τv =
396
Vv V and τ b = b Q Q
397
where Vv and Vb are the volume of water in the infusion vessel and inside tea bag respectively.
398
For all conditions, it is observed that the mixing time scale τ v is far greater than τb . Thus we
399
assumed the infusion of tea solute within the tea bag is homogeneous and well mixed.
400
4.6.3 Diffusion time scale
22
401
For the diffusion of tea solute through a tea leaf; the resistance is the ratio of the leaf thickness
402
( L ) to the effective diffusivity ( Deff ) .
τD =
403
404
The diffusive mass transfer time scale is represented as;
L2 Deff
For example: at dipping frequency, =50 dips per minute (dpm)
τ MT =
405
1 L2 = 100 sec ; and τ D = = 54 sec ; Deff ( k N a )TB
406
with L = 1.34 ×10−4 m (measured in the present work) and Deff = 3.33 ×10−10 m2 s (Farakte et al.,
407
2017). For all conditions, it is observed that the diffusive mass transfer time scale (τ D ) is
408
comparable with the convective mass transfer time scale (τ MT ) . Therefore, it can be concluded
409
that the both the resistances i.e. internal diffusive and Nerst layer resistance are significant”. As
410
a result, the intraparticle diffusion and external convective mass transfer has the combined effect
411
on tea bag infusion kinetics.
412
5 Conclusions
413
A mathematical model to predict tea bag infusion kinetics has been developed and explained in
414
the current work. The developed model is physics-based and takes into account the several
415
physical phenomena of the infusion process. The model has been developed incorporating three
416
interacting steps; (i) dissolution of tea constituents from the tea granules to the surrounding fluid;
417
(ii) transfer of the dissolved solids through the porous packed bed of tea granules; (iii)
418
convection of dissolved solids to the bulk water present in the vessel. The solution of the model
419
provides a concentration profile in the bulk infusion with respect to time. The model results are 23
420
in good agreement with the previously published experimental data (MRE <10%). Following
421
conclusions can be made from the current work;
422
(i)
Nerst layer mass transfer resistance (Biot number, Bi ~ 1).
423 424
The tea bag infusion process has the combined effect of intra-particle diffusion and
(ii)
Increase in dipping frequency and the rise in temperature leads to an increase in mass
425
transfer coefficient due to the improved Reynold’s number and GA liquid diffusivity
426
respectively.
427
(iii)
The particle size and loading have an inverse effect on the mass transfer coefficient
428
due to the higher interfacial area for smaller particle and higher swelling for lower
429
loading.
430
The developed model with appropriate modifications to account for geometry and operating
431
parameters can be applied for the extraction of solutes, herbal ingredients and plant metabolites
432
and compounds with medicinal values. With the knowledge of equilibrium parameters (or can
433
be evaluated as discussed in this work) along with the relevant transport equations, the present
434
model can be applied to the extraction of soluble components from the milled grapes, berries,
435
soybean, coffee beans etc.
436
concentrations of individual tea component (catechin, epicatechin gallate, catechin gallate,
437
epigallocatechin gallate, etc.) elution during infusion with time.
438
Acknowledgement
439
Authors would like to thank Pidilite Industries Ltd. for providing financial support under Prof.
440
M.M. Sharma-Pidilite Industries Ltd. Fellowship Scheme.
441
Nomenclature
Future work would be directed towards the prediction of the
24
442
List of symbols
443
a
specific surface area of the tea granule (m2/m3)
444
µ C
dimensionless concentration of tea solute
445
C
concentration of tea solute (kg/m3)
446
C si
initial soluble content of tea granule (kg/m3)
447
Deff
effective diffusivity (m2/s)
448
K
partition constant
449
κ
tea bed permeability (m/s)
450
k obs
overall rate of infusion (1/s)
451
kL
overall mass transfer coefficient (m/s)
452
L
tea leaf thickness (m)
453
M
molecular weight (kg/kmol)
454
p
an empirical parameter in Spiro's kinetic expression
455
Q
flow-through tea bag paper (m3/s)
456
S
total surface of tea granules (m2)
457
t
time (s)
458
T
Temperature (°C)
459
V
volume of water (m3)
460
Vs
volume of solid tea granules (m3)
461
vA
solute molal volume (m3/kmol)
462
w
weight of tea granules (kg)
463
Subscripts
25
464
b
inside the tea bag
465
B
of solvent
466
GA
of gallic acid in water
467
s
in the solid
468
v
in the infusion vessel
469
Superscripts
470
∗
at the solid-liquid interface
471
eq
at equilibrium
472
Dimensionless groups
473
Bi
Biot’s number
474
Da
Damkohler’s number
475
Re
Reynold’s number
476
Sc
Schmidt’s number
477
Sh
Sherwood's number
478
Greek symbols
479
ρ
density of tea granule (kg/m3)
480
µ
solution viscosity (kg/m.s)
481
θ
dimensionless time
482
ε
void fraction (porosity) in a tea bed
483
φ
association constant
484
γ
shape factor
485
τ
space-time based on the volume of water (sec)
486
λmax
maximum wavelength (m) 26
487
27
488
References
489
Cacace, J.E., Mazza, G., 2003. Mass transfer process during extraction of phenolic compounds
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from milled berries. J. Food Eng. 59, 379–389. https://doi.org/10.1016/S0260-
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Alvarado, M.A., 2007. Mathematical modeling of caffeine kinetic during solid-liquid
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Farakte, R.A., Yadav, G., Joshi, B., Patwadhan, A.W., Singh, G., 2016. Role of Particle Size in Tea Infusion Process. Int. J. Food Eng. 12, 1–16. https://doi.org/10.1515/ijfe-2015-0213 Farakte, R.A., Yadav, G.U., Joshi, B.S., Patwardhan, A.W., Singh, G., 2017. Modeling of Tea
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Infusion Kinetics Incorporating Swelling Kinetics. Int. J. Food Eng. 13.
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Fogler, H., 2004. Elements of Chemical Reaction engineering, Third. ed. Prentice Hall of India Private Limited, New Delhi.
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tea in hot water. J. Food Sci. Technol. 53, 315–325. https://doi.org/10.1007/s13197-015-
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Karacabey, E., Mazza, G., 2008. Optimization of Solid - Liquid Extraction of Resveratrol and
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Other Phenolic Compounds from Milled Grape Canes ( Vitis vinifera ) 6318–6325.
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https://doi.org/10.1021/jf800687b
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Lian, G., Astill, C., 2002. Computer simulation of the hydrodynamics of teabag infusion. Food
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Bioprod. Process. Trans. Inst. Chem. Eng. Part C 80, 155–162.
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Price, W.E., Spiro, M., 1985a. Kinetics and equilibria of tea infusion: Theaflavin and caffeine
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concentrations and partition constants in several whole teas and sieved fractions. J. Sci.
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Food Agric. 36, 1303–1308. https://doi.org/10.1002/jsfa.2740361215
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Price, W.E., Spiro, M., 1985b. Kinetics and equilibria of tea infusion: Rates of extraction of
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theaflavin, caffeine and theobromine from several whole teas and sieved fractions. J. Sci.
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Spiro, M., Jago, D.V., 1982. Kinetics and Equilibria of Tea Infusion. Part 3: Rotating-disc
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Experiments Interpreted by a Steady-state Model. J. Chem. Soc. 78, 295–305.
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Spigno, G., Faveri, D.M. De, 2009. Microwave-assisted extraction of tea phenols : A phenomenological study. J. Food Eng. 93, 210–217.
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Stapley, A.G.F., 2002. Modelling the kinetics of tea and coffee infusion. J. Sci. Food Agric. 82, 1661–1671. https://doi.org/10.1002/jsfa.1250
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Yadav, G.U., Joshi, B.S., Patwardhan, A.W., Singh, G., 2017. Swelling and infusion of tea in tea
545
bags. J. Food Sci. Technol. 54, 2474–2484. https://doi.org/10.1007/s13197-017-2690-9
30
Figure Captions
Fig. 1. Schematic representation of (A) tea bag dipping set-up (B) magnified image of the tea bag in an infusion vessel Fig. 2. Effect of dipping frequency on tea infusion: (A) Variation of dimensionless concentration (GAE); (B) Comparison of dimensionless concentration in tea bag and infusion vessel; (C) Variation of flow (m3/s) and tea bed permeability (m/s); (D) Variation of volumetric MTC (1/s). Fig. 3. Effect of temperature on tea infusion: (A) Variation of dimensionless concentration (GAE); (B) Comparison of dimensionless concentration in tea bag and infusion vessel; (C) Variation of flow (m3/s) and tea bed permeability (m/s); (D) Variation of volumetric MTC (1/s) and Sherwood number (Sh). Fig. 4. Effect of particle size on tea infusion: (A) Variation of dimensionless concentration (GAE); (B) Comparison of dimensionless concentration in tea bag and infusion vessel; (C) Variation of flow (m3/s) and tea bed permeability (m/s); (D) Variation of volumetric MTC (1/s). Fig. 5. Effect of loading on tea infusion: (A) Variation of dimensionless concentration (GAE); (B) Comparison of dimensionless concentration in tea bag and infusion vessel; (C) Variation of flow (m3/s) and tea bed permeability (m/s); (D) Variation of volumetric MTC (1/s). Fig. 6. Parity plot for the experimental v/s predicted concentration of tea solute in infusion vessel
Highlights:
Mathematical model development for the prediction of tea bag infusion kinetics
Combined effect of internal diffusive and Nerst layer mass transfer resistance
Higher tea solute concentration inside a tea bag at all times except at equilibrium
The predicted infusion profiles and experimental data are in good agreement
S0260-8774(19)30490-X
DOI:
https://doi.org/10.1016/j.jfoodeng.2019.109847
Reference:
JFOE 109847
To appear in:
Journal of Food Engineering
Received Date: 27 February 2019 Revised Date:
16 November 2019
Accepted Date: 27 November 2019
Please cite this article as: Dhekne, P.P., Patwardhan, A.W., Mathematical modeling of tea bag infusion kinetics, Journal of Food Engineering (2019), doi: https://doi.org/10.1016/j.jfoodeng.2019.109847. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Published by Elsevier Ltd.
1
Mathematical Modeling of Tea Bag Infusion Kinetics
2
Pallavee P. Dhekne, Ashwin W. Patwardhan*
3
Department of Chemical Engineering, Institute of Chemical Technology, Matunga,
4
Mumbai-400019, India
5 6 7 8 9 10 11 12 13 14 15 16 17
Corresponding Author: Tel: 91-22-33612018; Fax: 91-22-33611020
18
Email address: [email protected]
19
Abstract
20
The current work explains the development of a mathematical model for the prediction of tea bag
21
infusion kinetics of solid-liquid extraction of polyphenols from tea. The developed model has
22
been used to obtain tea bag infusion profile using equilibrium parameters i.e. partition constant
23
( K ) , the initial content of tea solute ( C si ) and tea bed permeability (κ ) . The Predicted infusion
24
profiles and experimental data are in good agreement having mean relative error < 10 %. For all
25
the conditions (loading in the tea bag, particle size, temperature, and dipping frequency), the
26
Gallic acid equivalence (GAE) per unit equilibrium concentration ( Cveq ) eluted inside tea bag
27
(C¶ ) was found to be higher as compared to that in the vessel ( C¶ ) .
28
increase in particle size for given loading exhibit slower infusion kinetics. However, an increase
29
in dipping frequency (Reynolds number) and temperature leads to an increase in mass transfer
30
coefficient leading to faster infusion kinetics.
31
Keywords: tea bag, infusion kinetics, mathematical model, partition constant, Biot number, time
32
scale analysis
v
b
2
The increased loading and
33
1 Introduction
34
The solid-liquid extraction (leaching) process is widely used in industries like chemical, food,
35
pharmaceutical, biotechnology, etc. for separation of key components from solid matrix to
36
solvent phase. The extraction of phenolic or bioactive compounds from the natural resources has
37
been a subject of several research studies. Among the several extraction processes, the solid-
38
liquid extraction of polyphenols from tea is one of the complicated processes to model. The
39
technique of preparing tea is different in different parts of the world. Most of the researchers
40
have performed the kinetics and equilibrium study of loose tea infusion. Spiro and co-workers
41
lucubrated the partitioning of tea solute and role of tea leaf sizes on the tea infusion kinetics
42
(Price and Spiro, 1985a, 1985b). In further studies, the infusion kinetics of tea solute through
43
the tea bag was also investigated (Spiro and Jaganyi, 2000). Kinetic and infusion studies carried
44
out previously in literature were based on kinetic expression (Spiro and Jago, 1982). The Spiro’s
45
steady-state lumped model reveals the first-order kinetics of the tea infusion process and
46
expressed as follows:
47
(
)
ln C * / ( C * − C ) = kobs t + p
(1)
48
where C * and C are the concentration of tea constituents in the bulk infusion at equilibrium and
49
at a time ‘t’ respectively. In Eq. (1) kobs is the overall rate of infusion and ‘p’ is an empirical
50
constant. Further study by Stapley, explains the significance of intercept (p) in Spiro's kinetic
51
model (Stapley, 2002). The model developed for caffeine extraction from coffee beans reported
52
the caffeine diffusivity as 3.21×10-10 m2/s at 90 °C (Espinoza-Pérez et al.,(2007). The effect of
53
tea bag material on the rate of extraction of caffeine from black tea has been studied (Jaganyi and
54
Mdletshe, 2000). The first-order rate constant for tea bag was found to be 29 % lower than that
3
55
of loose tea. The infusion rate was unaffected by tea bag shape but improved with an increase in
56
tea bag size (Jaganyi and Ndlovu, 2001). The rates of caffeine transport through tea bag paper
57
have been measured using a modified rotating diffusion cell (Spiro and Jaganyi, 2000). The
58
study showed that the transport of tea solute through the tea bag membrane has negligible
59
resistance within the temperature range of 25-80 °C. It was concluded that any motion (stirring
60
the brew around tea bag, dipping of a tea bag) which decreases the thickness of the Nernst layer
61
around the tea bag improves the rate of tea infusion.
62
A computational fluid dynamics (CFD) model was developed and simulated for the tea
63
bag infusion process with different brewing conditions i.e. static and dynamic (Lian and Astill,
64
2002). The hydrodynamics of tea bag infusion was also reported. It was observed that the
65
dynamic condition (stirring) enhances the rate of infusion of the tea solute in water (Jaganyi and
66
Mdletshe, 2000; Lian and Astill, 2002). The swelling kinetics of individual tea particles as well
67
as the bed of tea granules was investigated (Joshi et al., 2016). It was concluded that the
68
swelling of tea particles is a fast process and occurs simultaneously with the infusion process.
69
The effect of particle size and brewing temperature on the rate of tea infusion process has been
70
carried out (Farakte et al., 2016). The rate of infusion was improved with the decrease in particle
71
size and increase in temperature. Moreover, the values of equilibrium constant (partition
72
constant, K) and initial content (Csi) for two types of crush, tear, curl (CTC) teas for different
73
sizes were estimated.
74
A mathematical model has been developed to predict a tea infusion which accounts for
75
swelling kinetics of tea granules (Farakte et al., 2017).
76
constituents inside the tea granule at 60 ºC was reported to be 3.33×10-10 m2/s. The effects of tea
77
particle size, tea bag dipping rate, loading of tea granules and tea bag shapes on infusion rate
4
The estimated diffusivity of tea
78
have been investigated (Yadav et al., 2017). It was observed that the infusion rate decreased
79
with an increase in tea granules loading in the tea bag. The percent fill of tea granules and tea
80
bed height inside the tea bag greatly affect the tea granule swelling and eventually, the infusion
81
rate. Recently, the effect of temperature, particle size and source of tea on the infusion of
82
individual tea components have been studied (Yadav et al., 2018). This study showed that the
83
partition constant (K) depends not only on temperature but also on the tea types (source of tea
84
granules).
85
As discussed above, the previous study deals with the kinetics and equilibrium of tea
86
infusion. Few researchers have focused on modeling of infusion kinetics of loose tea. There is
87
scarce data available on the modeling of tea bag infusion. Due to the rise in demand for tea bags
88
for tea preparation, the tea infusion parameters have to be optimized.
89
parameters include a loading of tea granules in a tea bag, particle size, brewing temperature, and
90
tea bag dipping frequency. Hence, there is a need to develop a model, which can predict the tea
91
bag infusion kinetics, and help to optimize brewing parameters and the tea bag design.
92
2 Materials and Methods
93
2.1 Materials
94
Commonly used crush, tear, curl (CTC) black tea and cellulose acetate paper double-chambered
95
tea bags (6.5 cm × 4 cm) were procured from the local market. For tea bag infusion experiments
96
and analysis, de-ionized (DI) water (Millipore Inc, USA) was used. All the experiments were
97
performed in a infusion vessel of volume 150 ml (Farakte et al., 2016).
5
These tea brewing
98
2.2 Methods
99
The infusion kinetics study was performed using a tea bag dipping set-up (Fig. 1A) (Yadav et al.,
100
2017). The present dipping set-up imitates dipping of the tea bag in a cup of hot water. The
101
vessel containing 100 ml of de-ionized water was heated up to the desired temperature (60, 70,
102
80 and 90, ± 2 ºC) using a constant temperature water bath. When the desired temperature was
103
achieved, the tea bag which consists of 2 g of tea granules was dipped in a vessel with a dipping
104
frequency of 5 dips per minute (dpm).
105
The experiment was conducted for 15 min. 1 ml sample was withdrawn from the vessel
106
at 0.5, 1, 2, 3, 4, 5, 10 and 15 min. Volume of filtrate in the vessel was measured at the end of
107
the experiment. The samples were diluted 100 times using DI water and analyzed using UV-Vis
108
spectrophotometer (Cary 50 ) (Farakte et al., 2016). The UV spectrum of 100X diluted tea
109
samples shows the λmax to be 272 nm. Therefore, absorbance at 272 nm was monitored as a
110
measure of the extent of infusion. Most of the tea polyphenols are Gallic acid (GA) derivatives
111
(Taylor et al., 2010). Thus absorbance of tea infusion samples was expressed as GA equivalence
112
(GAE) by means of calibration curves with standard Gallic acid (Farakte et al., 2016, Spigno and
113
Faveri, 2009). The GAE values reported in this work are the equivalent amount of GA, which
114
would give the same absorbance. The GAE % (percent weight of GAE extracted per unit weight
115
of tea granules) were corrected for volume loss due to evaporation and sampling.
116
The experimental conditions for the tea bag infusion kinetics are shown in Table 1. The
117
effect of dipping frequency on infusion kinetics was studied for 2, 5, 8, 10, 15, 20, 30 and 50
118
dpm at 60 ºC for 15 min. In order to explore the effect of temperature on infusion kinetics,
119
experiments were performed at 60, 70, 80 and 90 ºC for constant dipping frequency of 5 dpm.
120
To study the effect of particle sizes on infusion kinetics, the tea particles were crushed and 6
121
separated (2.12 mm, 1.14 mm and 0.36 mm) by sieving. Moreover, the loading effect of tea
122
granules on the infusion kinetics was assessed with the loading from 1-3 g for 5 dpm at 60 ºC.
123
Here, the loading corresponds to a constant solid-to-liquid ratio (g/ml) of the tea granules to the
124
water.
125
The equilibrium infusion experiments for different sizes of tea granules have been
126
reported in the previous study (Farakte et al., 2016). The value of partition constants (K=0.14)
127
and initial content of tea constituent (Csi = 88 kg/m3) for 2.21 mm particle size is estimated by
128
procedure mentioned elsewhere (Farakte et al., 2016) and used in the present work.
129
3 Model Development
130
Tea infusion through tea bag is a more complex phenomenon than loose tea. In the brewing
131
process, a tea bag is immersed in the hot water. The hot water flows in and out of the tea bag
132
paper, leaching out soluble contents. A schematic of this process is shown in Fig. 2B. The
133
resistance to the flow of water in and out of tea bag is a bed of granules. This process of infusion
134
through the tea bag consists of various steps: (i) flow of water through tea bag paper from bulk
135
water to bed of tea granules; (ii) flow of water through pores of tea bed; (iii) absorption of water
136
by tea granules; (iii) dissolution of tea constituents from solid phase to liquid phase within tea
137
granules; (iv) diffusion of tea constituents from water present inside tea granules to outer surface
138
of granules; (v) transport of dissolved soluble solid from outer surface of granule to pores of tea
139
bed; (vi) convective transfer of soluble solids in the fluid.
140
The present model is based on the physics involved in the infusion process. The mass
141
transfer from tea granule to the surrounding liquid within the tea bag is an important step. The
142
mass transfer inside the tea bag depends upon the liquid ingress into and out of the tea bag. Once 7
143
the infused liquid containing the dissolved components comes out of the tea bag into the vessel,
144
it mixes with the rest of the liquid in the vessel. The model developed thus takes into account the
145
several phenomena actually taking place and it is not just a lumped parameter model.
146
The following assumptions were made in the model formulation: (a) Tea granules are
147
symmetrical. (b) The mass transfer Biot number (Bi) is defined as the ratio of internal to external
148
mass transfer resistances (the detailed derivation is provided in supplementary section) and can
149
be calculated by using following Eq. (Cacace and Mazza, 2003; Karacabey and Mazza, 2008)
Bi =
150
k N KL 2Deff
(2)
151
Where k N is real external mass transfer coefficient and Deff is the effective diffusivity of the tea
152
constitutents. If Biot number far less than unity (Bi << 1), then internal diffusive mass transfer
153
resistance can be negligible as compared to external convective mass transfer resistance at the
154
solid surface. It indicates that solute concentration gradients may not exist within solid particles.
155
(c) The tea solution within a tea bag is well mixed and homogeneous. (d) Tea granules in the
156
bag are considered as a packed bed.
157
With these assumptions, the mass balance for tea solute inside the tea bag can be written as
158
follows,
159
Vb
(
)
dCb = k L S C ∗ − Cb + QC v − QCb dt
160
Eq. (3) describes the rate of change in the concentration of tea solute inside the tea bag (Cb). It
161
can be expressed as follows,
8
(3)
(
)
dCb 1 = k L a C ∗ − Cb + ( Cv − Cb ) dt τb
162
(4)
163
Where k L a (s-1) denotes volumetric mass transfer coefficient ( a = S Vb ), Cv is the concentration
164
of tea constituents in the bulk infusion of volume (Vv ) and
165
tea bag (Vb ) to the volumetric flow rate through the tea bag ( Q ). The release of solute from tea
166
granules to the fluid in the voids of the packed bed is described by the general Eq. of interfacial
167
mass transfer. Hence, the rate of change in solid-phase concentration ( C s ) can be expressed as, −Vs
168
τb is the ratio of water volume inside
(
dC s = k L S C * − Cb dt
)
(5)
169
Substitution of C ∗ = K C s ( K : partition constant, the ratio of the concentration of tea solute in
170
infusion ( C * ) to that in the tea granules at equilibrium) into Eq. (4) and (5) yields,
dCb 1 = k L a ( KCs − Cb ) + ( Cv − Cb ) dt τb
171
−Vs
172
dC s = k L S ( KC s − Cb ) dt
(6)
(7)
173
Once the tea solutes are leached out, their transport occurs by the convective transfer through the
174
tea bag. The corresponding mass transfer Eq. for the tea solutes in the well-mixed liquid phase is
175
written as dC v = QC b − QC v dt
176
Vv
177
dCv 1 = ( Cb − Cv ) dt τ v
9
(8)
(9)
τv
is the space-time based on the volume of solution (τ v = Vv Q ) . Eq. (6) and (9)
178
Where
179
describes the concentration profile with respect to time inside the tea bag and bulk infusion
180
respectively.
181
Overall mass balance gives,
VsCs = VsCsi − VvCv − VbCb
182
(10)
183
The solid phase concentration ( Cs ) can be expressed in terms of the initial content of tea
184
granules ( Csi ) by rearranging Eq. (10) as follows, V C + V C Cs = Csi − b b v v Vs
185
186
(11)
Substituting Cs value from Eq. (11) into Eq. (6) gives, 1 dCb K = k L a KCsi − (Vb Cb + Vv Cv ) − Cb + ( Cv − Cb ) dt Vs τb
187
(12)
KV 1 dCb k aKVb 1 = k L aKCsi − k L a + L + Cb − k L a v − Cv dt Vs τb Vs τ b
188
(13)
189
The non-dimensional form of Eq. (13) can be obtained as follows. Dividing the Eq. (13) by
190
equilibrium concentration ( Cveq ) and volumetric mass transfer coefficient ( k L a ) yields,
191
(
d Cb Cveq d ( tk L a )
) = KC C
si eq v
KVv KVb 1 v − 1 + + Cb Ceq − Vs k L aτ b Vs
(
)
1 − k L aτ b
v Cv Ceq
(
)
(14)
192
¶ = C C eq and non-dimensional time θ = tk a and Introducing non-dimensional concentration, C L b b v
193
rearrangement gives, 10
¶ dC KC KVb 1 b = eqsi − 1 + + dθ Cv Vs k L aτ b
194
¶ KVv C b − Vs
1 − k L aτ b
¶ C v
(15)
195
Similarly, the non-dimensional form of Eq. (9) can be obtained as follows. Dividing the eq. (9)
196
by Cveq and kL a yields,
(
d Cv Cveq
197
d ( tkL a )
)=
1 Cb − Cv k L aτ v Cveq
(16)
198
Rearranging above Eq. in terms of dimensionless concentration C¶v = C v C veq and dimensionless
199
time θ = tk L a
(
¶ dC 1 v ¶ −C ¶ = C b v dθ k L aτ v
200
201
204
205
206
(17)
Differentiating Eq. (17) with respect to non-dimensional time (θ), ¶ d 2C 1 v = 2 dθ k L aτ v
202
203
)
¶ dC ¶ dC b v − dθ dθ
(18)
¶ dθ from Eq. (15) into Eq. (18), it becomes, Substituting for dC b
¶ d 2C 1 v = 2 dθ k L aτ v
KCsi KVb 1 + eq − 1 + Vs k L aτ b Cv
¶ KVv 1 − Cb − k L aτ b Vs
¶ 1 Cv − k L aτ v
¶ dC v dθ
(19)
¶ value from Eq. (17) into Eq. (19) and rearrangement yields following Eq., Substituting C b ¶ ¶ d2C 1 KCsi KVb KVv ¶ KVb 1 1 dC v v = − 1 + + C − 1 + + + v dθ 2 k L aτ v Cveq Vs Vs V k a τ k a τ d θ s L b L v
11
(20)
207
Now, the equilibrium concentration of tea constituents ( Cveq ) can be expressed in terms of the
208
partition constant and initial content of tea granules, Vs C veq = KC si KVv + Vs
209
210
211
(21)
Rearrangement of Eq. (20) yields, ¶ ¶ KVb d2C 1 1 dC 1 v v = − 1 + + + − 2 dθ Vs k L aτ b k L aτ v dθ k L aτ v
KVb KVv ¶ 1 + 1 + Cv + Vs Vs k L aτ v
KVV 1 + VS
(22)
212
The dimensionless variables appearing in Eq. (22) disclose the important characteristics of the 213
physical process. The Damkohler number, Da as the ratio of transport rate to convection rate and 214
can be represented as
kL a (1 τ )
215
Da =
216
The simplified form of non-dimensional model Eq. can be represented as follows:
217
¶ ¶ KVb d 2C 1 1 dC 1 KVb KVv ¶ 1 KVV v v = − + + + − + 1 1 + Cv + 1 + 2 dθ Vs Dab Dav dθ Dav Vs Vs Dav VS
(23)
(24)
218
Eq. (24) can be solved for the concentration of tea constituents in the vessel, with the initial
219
condition as;
220
¶ ¶ = 0 and dCv = 0 at θ =0, C v dθ
(25)
221
The solution of Eq. (24) is obtained using MATLAB (R2015a). Mean relative error (MRE) is
222
used to assess the accuracy of prediction with the experimental data. 12
MRE =
223
((
1 ¶ ¶ abs C ∑ v exp t − C v pred n
)
¶ C v exp t
)
(26)
224
¶ and C ¶ where n is the number of data points, C v expt v pred are the experimental and predicted
225
concentrations respectively. The Runge Kutta method of order 4 (MATLAB, R2015a) was used
226
to solve the set of ODEs. The solution to problems defined by the initial conditions (Eq. 25) was
227
obtained using ODE 45 solver with a known time step.
228
The model Eq. (24) consists of different parameters such as kL , S , Vb , Vv , K , Csi , Vs and
229
Q. The total surface area ( S ) was calculated from the known weight of tea granules (w). The
230
specific surface area ‘a’ is based on the volume of water inside tea bag and can be calculated as;
231
a = S Vb . The volume of water inside the tea bag (Vb ) and in the vessel (Vv ) was measured
232
during the experiment. The value of partition constants ( K = 0.14 ) and initial content of tea
233
constituent Csi = 88 kg/m
234
Farakte et al., (2016) and used in the present work. The Gallic acid diffusion coefficient in water
235
at various temperature were calculated using Wilke–Chang correlation (Treybal, 1980).
236
(
3
) for 2.21 mm particle size is estimated by procedure mentioned by
DGA = 1.173 × 10 −18
T (φ M B )
µ Bν A0.6
0.5
(27)
237
Where, φ is the association factor for solvent (2.26 for water as solvent), vA is solute molar
238
volume, m3/kmol (Gallic acid, C7 H 6O5 ).
239
The hydration of tea and water inside the tea bed (porosity) has the contribution in the
240
swelling of tea bed. The absorbed water and the water inside the pores tend to extricate when the
241
bag is removed from the vessel. Hence, the accurate quantitative measurement of tea bed
242
swelling inside the bag is difficult due to the lateral and longitudinal expansion of tea bag.
13
243
Therefore, in order to quantify the combined effect of swelling and porosity of the bed, tea bed
244
permeability is used as a fitted parameter. The flow-through tea bag ( Q ) can be calculated from
245
the value of tea bed permeability
246
Geankoplis, 2003) ;
(κ , m/s )
by using Darcy's law as follows (Christie J.
dh dl
247
Q = −κ A
248
where dh = v 2 2 g ( dynamic head, m) due to the dipping frequency of tea bag in the vessel, dl
249
is the height of swelled tea bed (m) and v (m/s) can be related to the amplitude (A) and the
250
dipping frequency (f) as: v = Af .
251
Interfacial mass transfer coefficient for the release of the solutes from tea granules can be
252
estimated from the correlation of Thoenes and Kramers (Fogler, 2004).
Sh ' = 2 + 0.5 ( Re ' )
253
254
255
0.5
(28)
Sc 0.33
(29)
Eq. (29) can be written as follows;
kLd p DGA
ε 1− ε
Ud p ρ 1 = 2 + 0.5 γ µ (1 − ε )γ
0.5
µ ρ DGA
0.33
(30)
256
where γ is shape factor (external surface area divided by πdp2 ), ε is a void fraction (porosity) of
257
the packed tea bed and U (m/s) is the superficial liquid velocity through the bed. The Nerst layer
258
resistance (1 kN ) was estimated from the overall mass transfer resistance and the tea leaf
259
diffusive resistance as follows:
14
260
1 1 KL = − k N kL 2Deff
261
The mass transfer Biot number (Bi) was calculated by using Eq. (2) where the effective
262
diffusivity
263
measured thickness of the tea leaf ( L ) is 1.34× 10–4 m .
(31)
( D ) of solid tea constituents is 3.33 × 10–10 m2/s (Farakte et al., 2017) and the eff
15
264
4 Results and Discussion
265
The diffusion of tea components within the tea granule is compared to the convective diffusion
266
from tea to the solution inside the tea bag. During tea bag infusion process, swelling of tea
267
granules, as well as tea bed, takes place. The swelled tea bed leads to an increase in the
268
compactness of the bed in the bag (Yadav et al., 2017). The higher resistance for the external
269
mass transfer is offered by the highly compact tea bed during the tea infusion. This can be
270
observed from the estimated Biot number for intraparticle diffusive resistance to external
271
convective mass transfer resistance. For example: dipping frequency = 2 dpm case;
272
K = 0.14; kL = 4.11 × 10-6 (m/s)
273
(estimated value in the current work), Deff = 3.33 ×10−10 m2 s (Farakte et al., 2017) and
274
L = 1.34 ×10−4 m (measured value in the current work). Therefore, real external mass transfer
275
coefficient is calculated by using Eq. (31).
276
−4 1 1 0.14 ×1.34 ×10 = − kN 4.11×10−6 2 × 3.33 ×10−10
277
k N = 4.648 × 10 −6 m/s
278
The biot number for mass transfer is calculated from Eq. (2) as follows:
Bi =
279
k N K δ 4.648 ×10−6 × 0.14 ×1.34 ×10−4 = = 0.133 2 Deff 2 × 3.33 ×10−10
280
Similarly, the Biot number (Bi) for all cases has been estimated and found to be near unity
281
(Bi~1).
282
mass transfer as well.
This indicates the combined effect of intra-particle diffusion and Nerst layer external
16
283
4.1 Effect of dipping frequency
284
The comparison of predicted (solid line) and experimental (symbol) data for different dipping
285
frequencies is shown in Fig. 2A. This indicates the developed model fits the infusion kinetics
286
data very well with MRE < 10 %. The concentration profile of tea constituents inside the tea bag
287
¶ (dashed line) and in the vessel C ¶ (dotted line) is shown in Fig. 2B. The concentration C b v
288
profile in the vessel shows sigmoid nature (tilted S-shape) at lower dipping frequency which is in
289
good agreement with the literature (Yadav et al., 2017). It implies that the slower infusion
290
kinetics during the initial stages of low dipping frequency. Moreover, C¶b is higher than C¶v for
291
all dipping frequencies interpreting the significance of the external convective mass transfer
292
µ with respect to time (θ ) is shown in the inset of Fig. 2B. It is resistance. The variation of ∆C
293
evident that the increase in dipping frequency leads to a decrease in the change in concentration
294
( ∆Cµ) . The concentration in the vessel (C¶ ) can be attainable as C¶ within 5 min of the infusion
295
period for a higher dipping rate. This is due to the faster infusion kinetics caused by a higher
296
extent and faster swelling kinetics of tea granules (Yadav et al., 2017). Fig. 2C depicts the effect
297
of dipping frequency on the tea bed permeability ( κ , m/s) and flow-through tea bag ( Q , m3/s).
298
From Fig. 2C, it can be seen that the increase in dipping frequency leads to an increase in
299
convective transfer through the tea bag. However, bed permeability decreases with an increase
300
in dipping frequency from 2-50 dpm in the same order. Due to the increase in a flow-through tea
301
bag, swelling of tea bed increases (Joshi et al., 2016); which implies the reduction in bed
302
permeability value. Fig. 2D shows the variation of the volumetric mass transfer coefficient
303
( kL a ) with dipping frequency. It is clear that as dipping rate is increased by a factor of 10 (5
( )
v
b
17
304
dpm to 50 dpm), kLa was improved by 2.2 times due to the increase in Reynolds number (flow
305
provided by higher dipping).
306
4.2 Effect of temperature
307
The shape parameter and physical properties with different temperatures are listed in Table 2.
308
The predictions for tea bag infusion kinetics at different temperatures (60, 70, 80 and 90 ºC) are
309
compared with an experimental data reported previously (Yadav et al., 2017). The comparison
310
between predicted and experimental results is shown in Fig. 3A. The model predictions for
311
infusion at given temperatures fit the experimental data adequately up to 10 min (MRE < 10 %).
312
Fig. 3B depicts the infusion profile of GAE (mg/ml) eluted per unit equilibrium concentration
313
(C )
314
¶ is higher as compared C ¶ . During the initial infusion From Fig. 3B, it is observed that C b v
315
period, a higher infusion rate is observed at higher temperatures. The inset plot in Fig. 3B shows
316
the difference in concentrations between C¶b & C¶v . From Fig. 3C, it is observed that with a rise
317
in temperature, the bed permeability and flow through tea bag decreases. This decrease in flow
318
was caused due to the resistance provided by the swelling of tea bed with a temperature. This
319
observation is supported by Joshi et al. stating that an increase in temperature from 60 to 90 ºC
320
leads to an increase in the rate and extent of swelling (Joshi et al., 2016). From the present
321
model (Fig. 3D), it was found that the estimated values of the mass transfer coefficient ( kL) at 90
322
°C is 1.67 fold than that observed at 60 °C.
eq v
( ) and in the vessel (C¶ ) with respect to dimensionless time (θ ) .
¶ inside the tea bag C b
v
18
323
4.3 Effect of particle size
324
Tea bag infusion kinetics for different particle sizes (2.21 mm, 1.14 mm and 0.36 mm) was
325
reported previously (Yadav et al., 2017).
326
partition constant
327
(Farakte et al., 2016) and used in the present work. It was found that the initial contents of tea
328
constituent ( C si ) and partition constant ( K ) changes with particle size. The possible reasons for
329
the variation in C si and K could be non-uniform plucking and particle size distribution during
330
CTC (crush-tear-curl) process from different parts of the plucked leaf.
The initial contents of tea constituent
( Csi ) and
( K ) for different sizes were measured in the previous tea infusion study
331
The effect of particle size on the tea bag infusion profile is shown in Fig. 4. It is
332
observed from Fig. 4A that the model prediction for infusion using 2.21 mm, 1.14 mm and 0.36
333
mm granules fit the experimental data very well (MRE<10%). According to Yadav et al.,
334
(2017), smaller particle size has improved infusion kinetics. The observation can be explained
335
by the fact that smaller particle size offers a higher interfacial area for mass transfer. However,
336
from Fig. 4A the dimensionless concentration for different particle sizes (2.21, 1.14, 0.36 mm) is
337
not following the trend as like in Fig. 2A and Fig. 3A. The possible reason for this discrepancy
338
is the actual interfacial area, which significantly decreases than that of considered for
339
dimensionless model development based on particle diameter (Dp). This significant decrease in
340
interfacial area is due to the densely packed tea bed of swelled particles. The observed trend of
341
¶ vs. θ in Fig. 4A seems to be reversed with particle size due to the significant effect of C v
342
Damkohler number (Da) given by Eq. (24).
19
343
Fig. 4B shows the concentration of tea constituents inside the tea bag
(C¶ ) (dashed line) b
344
¶ (dotted line). It can be observed from Fig. 4B, for 0.36 mm particle C¶ and in the vessel C v b
345
shows higher as well as a faster infusion to as that of 2.21 mm. The inset Fig. 4B depicts the
346
concentration difference between C¶b and C¶v as a function of time (θ ) . The profile indicates the
347
difference in concentration (∆C) is higher for 0.36 mm particle size as that of 2.21 mm. This is
348
because of the higher extent of infusion inside the tea bag for smaller particle size and resistance
349
to the flow through the bed due to the compactness. The effect of particle size on the flow-
350
through tea bag ( Q ) and tea bed permeability (κ ) is shown in Fig. 4C. The increase in particle
351
size leads to an increase in κ and Q due to the lower bed resistance (decrease in swelling).
352
However, kLa is higher for lower particle size due to the higher interfacial area for mass transfer
353
(shown in Fig. 4D).
354
4.4 Effect of loading
355
The tea bag infusion kinetics with different loading (0.25, 0.5, 0.75, 1, 2 and 3g) was reported by
356
Yadav et al. (Yadav et al., 2017). Fig. 5 (A, B, C, D) shows all the inferences on the effect of
357
loading. Fig. 5A depicts the model fitting of the infusion profile as GAE eluted per unit Cveq for
358
tea bag loading with 1-3 g tea. Model fits the infusion kinetics data very well with MRE < 10%.
359
¶ Fig. 5B depicts the comparison of the concentration of tea constituents inside the tea bag C b
360
¶ with respect to dimensionless time (θ ) . It is evident that, for a given loading, and the vessel C v
361
¶ shows a higher infusion as compared to C ¶ . This interprets the significance of the external C b v
362
convective mass transfer resistance. The difference in concentrations is shown in the inset of
363
Fig. 5B. It is clearly observed that concentration difference is decreased for lower loading of tea
( )
20
Fig. 5C shows the effect of loading on the flow through the tea bag
(Q )
364
granules.
and
365
permeability of tea bed (κ ) . The value of Q (m3/s) obtained for 1 g loading is 1.4 times higher
366
than that of 3 g loading. This indicates that the effect of clogging (resistance to flow) in the tea
367
bed is less dominant with lower loading and difference in concentrations ∆C is lesser. Hence,
368
lower loading resulted in higher infusion kinetics. Moreover, the higher infusion kinetics for low
369
loading is confirmed by an increase in the volumetric mass transfer coefficient ( kL a ) as shown in
370
Fig. 5D.
371
4.5 Statistical Analysis
( )
372
The experimental results reported previously show the mean value of at least three
373
replicates (Yadav et al., 2017). The mean relative error (MRE) (Eq.(26)) estimated by the
374
¶ comparison of experimental C v exp t
375
accuracy of the proposed mathematical model for the prediction of the infusion kinetics. The
376
statistical significance has also been evaluated using the parity plot between the experimental
377
and predicted values of concentrations for all data points, encompassing widely different
378
operating parameters.
379
predictions using the proposed model and the experimental value is ± 10%.
380
4.6 Transport time scale analysis
381
For developing a mathematical model, it is important to know the characteristic time scale of the
382
involved processes. Underlying physics of the tea bag infusion process can be understood from
383
the time scale analysis.
(
) and predicted data (C¶ ) , was calculated to evaluate the v pred
From Fig. 6, it is observed that the mean relative error between
21
384
4.6.1 Mass transfer time scale
385
It is defined as the reciprocal of the volumetric mass transfer coefficient and expressed for loose
386
tea (LT) and tea bag (TB) as follows:
τ MT loose tea =
387
1 ( k L a )LT
and
τ MT tea bag =
1 ( k L a )TB
388
The mass transfer coefficient (MTC) for loose tea is 3.846 × 10-5 (m/s) (Spiro and Jago, 1982).
389
In the current study, MTC for tea bag infusion has been estimated (kL = 4.11 × 10-6 (m/s)). The
390
mass transfer coefficient for loose tea is approximately 9.36 times that of tea bag infusion.
391
Therefore, the mass transfer time scale for loose tea is less as compared to tea bag ( τ MT loose tea <<
392
τ MT tea bag ).
393
4.6.2 Mixing time scale
394
Mixing time scale is defined as the volume of water (V ) in a defined system divided by the
395
volumetric flow rate (Q) through the tea bag.
τv =
396
Vv V and τ b = b Q Q
397
where Vv and Vb are the volume of water in the infusion vessel and inside tea bag respectively.
398
For all conditions, it is observed that the mixing time scale τ v is far greater than τb . Thus we
399
assumed the infusion of tea solute within the tea bag is homogeneous and well mixed.
400
4.6.3 Diffusion time scale
22
401
For the diffusion of tea solute through a tea leaf; the resistance is the ratio of the leaf thickness
402
( L ) to the effective diffusivity ( Deff ) .
τD =
403
404
The diffusive mass transfer time scale is represented as;
L2 Deff
For example: at dipping frequency, =50 dips per minute (dpm)
τ MT =
405
1 L2 = 100 sec ; and τ D = = 54 sec ; Deff ( k N a )TB
406
with L = 1.34 ×10−4 m (measured in the present work) and Deff = 3.33 ×10−10 m2 s (Farakte et al.,
407
2017). For all conditions, it is observed that the diffusive mass transfer time scale (τ D ) is
408
comparable with the convective mass transfer time scale (τ MT ) . Therefore, it can be concluded
409
that the both the resistances i.e. internal diffusive and Nerst layer resistance are significant”. As
410
a result, the intraparticle diffusion and external convective mass transfer has the combined effect
411
on tea bag infusion kinetics.
412
5 Conclusions
413
A mathematical model to predict tea bag infusion kinetics has been developed and explained in
414
the current work. The developed model is physics-based and takes into account the several
415
physical phenomena of the infusion process. The model has been developed incorporating three
416
interacting steps; (i) dissolution of tea constituents from the tea granules to the surrounding fluid;
417
(ii) transfer of the dissolved solids through the porous packed bed of tea granules; (iii)
418
convection of dissolved solids to the bulk water present in the vessel. The solution of the model
419
provides a concentration profile in the bulk infusion with respect to time. The model results are 23
420
in good agreement with the previously published experimental data (MRE <10%). Following
421
conclusions can be made from the current work;
422
(i)
Nerst layer mass transfer resistance (Biot number, Bi ~ 1).
423 424
The tea bag infusion process has the combined effect of intra-particle diffusion and
(ii)
Increase in dipping frequency and the rise in temperature leads to an increase in mass
425
transfer coefficient due to the improved Reynold’s number and GA liquid diffusivity
426
respectively.
427
(iii)
The particle size and loading have an inverse effect on the mass transfer coefficient
428
due to the higher interfacial area for smaller particle and higher swelling for lower
429
loading.
430
The developed model with appropriate modifications to account for geometry and operating
431
parameters can be applied for the extraction of solutes, herbal ingredients and plant metabolites
432
and compounds with medicinal values. With the knowledge of equilibrium parameters (or can
433
be evaluated as discussed in this work) along with the relevant transport equations, the present
434
model can be applied to the extraction of soluble components from the milled grapes, berries,
435
soybean, coffee beans etc.
436
concentrations of individual tea component (catechin, epicatechin gallate, catechin gallate,
437
epigallocatechin gallate, etc.) elution during infusion with time.
438
Acknowledgement
439
Authors would like to thank Pidilite Industries Ltd. for providing financial support under Prof.
440
M.M. Sharma-Pidilite Industries Ltd. Fellowship Scheme.
441
Nomenclature
Future work would be directed towards the prediction of the
24
442
List of symbols
443
a
specific surface area of the tea granule (m2/m3)
444
µ C
dimensionless concentration of tea solute
445
C
concentration of tea solute (kg/m3)
446
C si
initial soluble content of tea granule (kg/m3)
447
Deff
effective diffusivity (m2/s)
448
K
partition constant
449
κ
tea bed permeability (m/s)
450
k obs
overall rate of infusion (1/s)
451
kL
overall mass transfer coefficient (m/s)
452
L
tea leaf thickness (m)
453
M
molecular weight (kg/kmol)
454
p
an empirical parameter in Spiro's kinetic expression
455
Q
flow-through tea bag paper (m3/s)
456
S
total surface of tea granules (m2)
457
t
time (s)
458
T
Temperature (°C)
459
V
volume of water (m3)
460
Vs
volume of solid tea granules (m3)
461
vA
solute molal volume (m3/kmol)
462
w
weight of tea granules (kg)
463
Subscripts
25
464
b
inside the tea bag
465
B
of solvent
466
GA
of gallic acid in water
467
s
in the solid
468
v
in the infusion vessel
469
Superscripts
470
∗
at the solid-liquid interface
471
eq
at equilibrium
472
Dimensionless groups
473
Bi
Biot’s number
474
Da
Damkohler’s number
475
Re
Reynold’s number
476
Sc
Schmidt’s number
477
Sh
Sherwood's number
478
Greek symbols
479
ρ
density of tea granule (kg/m3)
480
µ
solution viscosity (kg/m.s)
481
θ
dimensionless time
482
ε
void fraction (porosity) in a tea bed
483
φ
association constant
484
γ
shape factor
485
τ
space-time based on the volume of water (sec)
486
λmax
maximum wavelength (m) 26
487
27
488
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489
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Figure Captions
Fig. 1. Schematic representation of (A) tea bag dipping set-up (B) magnified image of the tea bag in an infusion vessel Fig. 2. Effect of dipping frequency on tea infusion: (A) Variation of dimensionless concentration (GAE); (B) Comparison of dimensionless concentration in tea bag and infusion vessel; (C) Variation of flow (m3/s) and tea bed permeability (m/s); (D) Variation of volumetric MTC (1/s). Fig. 3. Effect of temperature on tea infusion: (A) Variation of dimensionless concentration (GAE); (B) Comparison of dimensionless concentration in tea bag and infusion vessel; (C) Variation of flow (m3/s) and tea bed permeability (m/s); (D) Variation of volumetric MTC (1/s) and Sherwood number (Sh). Fig. 4. Effect of particle size on tea infusion: (A) Variation of dimensionless concentration (GAE); (B) Comparison of dimensionless concentration in tea bag and infusion vessel; (C) Variation of flow (m3/s) and tea bed permeability (m/s); (D) Variation of volumetric MTC (1/s). Fig. 5. Effect of loading on tea infusion: (A) Variation of dimensionless concentration (GAE); (B) Comparison of dimensionless concentration in tea bag and infusion vessel; (C) Variation of flow (m3/s) and tea bed permeability (m/s); (D) Variation of volumetric MTC (1/s). Fig. 6. Parity plot for the experimental v/s predicted concentration of tea solute in infusion vessel
Highlights:
Mathematical model development for the prediction of tea bag infusion kinetics
Combined effect of internal diffusive and Nerst layer mass transfer resistance
Higher tea solute concentration inside a tea bag at all times except at equilibrium
The predicted infusion profiles and experimental data are in good agreement