- Email: [email protected]
Spectral method for simulating 3D heat and mass transfer during drying of apple slices
Accepted Manuscript Spectral method for simulating 3D heat and mass transfer during drying of apple slices Atena Pasban, Hassan Sadrnia, Mohebbat Mohebbi, Seyed Ahmad Shahidi PII:
S0260-8774(17)30211-X
DOI:
10.1016/j.jfoodeng.2017.05.013
Reference:
JFOE 8884
To appear in:
Journal of Food Engineering
Received Date: 19 October 2016 Revised Date:
1 May 2017
Accepted Date: 11 May 2017
Please cite this article as: Pasban, A., Sadrnia, H., Mohebbi, M., Ahmad Shahidi, S., Spectral method for simulating 3D heat and mass transfer during drying of apple slices, Journal of Food Engineering (2017), doi: 10.1016/j.jfoodeng.2017.05.013. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. 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.
ACCEPTED MANUSCRIPT
Spectral method for simulating 3D heat and mass transfer during drying of apple slices 1 2
Atena Pasban1, Hassan Sadrnia2*, Mohebbat Mohebbi1, Seyed Ahmad Shahidi3 1
3 4 5 6 7 8 9 10 11 12
Abstract:
13
In the present study, a numerical method is proposed for simulating the coupled three
14
dimensional heat and mass transfer processes during convective drying of apple slices. Spectral
15
collocation method (pseudospectral method) is applied for discretizing both the space and time
16
variables based on the Jacobi Gauss Lobatto (JGL) interpolation points. Also operational
17
matrices of differentiation are implemented for approximating the derivative of the spatial and
18
temporal variables. The external flow and temperature fields were simulated through the Fluent
19
CFD package. The convective heat transfer coefficient is calculated from the lumped system
20
analysis and convective mass transfer coefficient is computed through the analogy between the
21
thermal and concentration boundary layers. The model is validated against experimental data in a
22
range of air temperatures from 60oC up to 90oC. The results illustrate a remarkable agreement
23
between the numerical predictions and experimental results, which confirm robustness,
24
computationally efficient and high accuracy of the proposed approach for predicting the
25
simultaneous heat and mass transfer in apple slices.
27 28 29
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Department of Food Science and Technology, Ferdowsi University of Mashhad, Mashhad, Iran 2 Department of Biosystems Engineering, Ferdowsi University of Mashhad, Mashhad, Iran 3 Department of Food Science and Technology, Ayatollah Amoli Branch, Islamic Azad University, Amol, Iran. *Corresponding author: Email: [email protected], Tel: +985138805839
Key Words: Numerical simulation; Apple drying; Spectral collocation method; Convective coefficients.
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33
Nomenclature
total area of apple slice (m2) moisture content (kgm-3) temperature (K,oC) specific heat (Jkg-1K-1) effective moisture diffusivity (m2s-1) diffusivity of water vapor in air (m2s-1) vaporization latent heat (Jkg-1) thermal conductivity (Wm-1k-1) heat transfer coefficient (Wm-2 K-1) mass transfer coefficient (ms-1)
, , , ,
34 35 36 37 38
!, , " ,"
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Greek symbols viscosity (kg/(m s)) density of food (kg m-3)
time (s) moisture content in dry basis (kg water kg dry mass-1) moisture content in wet basis (kg water kg product-1) axial coordinates (m) half thickness of product in the , , direction (m) Volume (m3) velocities in X, Y and Z direction (m/s) , " number of the collocation nodes
,
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ℎ ℎ
1. Introduction
thermal diffusivity (m2/s) parameters of the Jacobi polynomials
Dimensionless groups Le Lewis number (dimensionless)
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ℎ
Subscripts Air drying air in Initial Al Aluminum F Final
Fruits and vegetables are considered as more perishable foods because of high moisture content
40
(Simal et al., 1997). Drying process is one of the well-known methods for preservation of fruits
41
and vegetables. This process prevents occurrence of unpleasant changes such as microbial
42
spoilage and enzymatic reaction by removing water from food products. Moreover, drying by
43
lowering the mass and volume of food products, reduces the cost of packaging, storage and
44
transportation (Goyal et al., 2006; Mujumdar, 2006).
45
When a moist object is subjected to drying conditions, heat and mass (moisture) transfer happen
46
simultaneously. Heat is transferred by convection from the heated air to the surface of a moist
47
object (food) and by conduction to the interior of food to increase temperatures and to evaporate
48
moisture from the food surface. Moisture transfer is accomplished by diffusion from inside of the
49
food to the surface, and from the food surface to the air by convection due to the heat transfer
50
process (Hernandez et al., 2000; Mujumdar, 2006), although other mechanisms may be involved.
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Heat and mass transfer phenomena in a system are described by governing equations of Fourier
52
law and Fick’s second law of diffusion. These equations are some simplified descriptions of
53
physical reality of heat and mass transfer represented in mathematical terms (Hussain and
54
Dincer, 2003; Tohidi, 2015).
55
Mathematical modeling is an important tool in the design and control of drying process
56
especially in food engineering. Many undesirable changes may occur in foods during the drying
57
process or in dried food after drying process which are associated to the temperature and
58
moisture content distribution. Therefore, simulation and prediction of the temperature and
59
moisture distributions in foods as a function of drying time can help us to prevent the undesirable
60
changes in foods during drying process or during preservation (Mishkin et al., 1983, Wang and
61
Brennan, 1995)
62
The analytical methods and numerical methods (such as the finite difference methods (FDMs),
63
finite element methods (FEMs) and finite volume methods (FVMs)) are the basic mathematical
64
tools that utilized to simulate the model of the heat and mass transfer (Barati and Esfahani, 2011;
65
Lemus Mondaca et al., 2013; Esfahani et al., 2014; García-Alvarado et al., 2014; Esfahani et al.,
66
2015; Vahidhosseini et al., 2016; Tzempelikos et al., 2015).
67
Numerical methods mostly compute the approximate solutions of the governing equations
68
through the localization of spatial and temporal variables that can be more realistic and flexible
69
for simulating the aforementioned phenomena. In contrast, analytical methods require the infinite
70
power series in computations that make them deficient and unfavorable with respect to the
71
numerical methods (Zogheib and Tohidi, 2016). Moreover, in analytical methods, symbolic
72
differentiation and integration are time-consuming operations. Therefore, in numerical
73
techniques, alternative tools such as operational matrices of differentiation and also Gauss
74
quadrature rules are replaced instead of direct symbolic differentiations and integration,
75
respectively for speeding up the operations (Shen and Tang, 2006).
76
In recent years, considerable number of research works have been devoted to numerical
77
simulation of heat and mass transfer phenomena during convective drying of food, such as
78
numerical analysis of coupled heat and mass transfer during drying process in papaya slices and
79
mango with FVM (Villa Corrales et al., 2010; Lemus Mondaca et al., 2013), numerical analysis
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of the transport phenomena occurring during drying process of carrots and mango fruit with
81
FEM (Aversa et al., 2007; Janjai et al., 2008) and numerical simulation of 2D heat and mass
82
transfer during drying of a rectangular object with FDM (Hussain and Dincer, 2003).
83
Among the numerical methods, the spectral methods are popular and robust tools which have
84
been widely implemented during the recent decades for solving smooth partial differential
85
equations (PDEs) with simple domains.
86
Numerical methods for solving PDEs can be classified into the local (like FDMs and FEMs) and
87
global categories. In local methods, derivative approximation of an assumed function at any
88
given point depends only on the information from its neighboring. Whereas in global methods,
89
derivative approximation of an assumed function at any given points depends not only on the
90
information from its neighboring points but also on the information from the entire of the
91
computational domain, which force them to achieve a high precision using a small number of
92
discretization nodes (Costa, 2004; Sun et al., 2012).
93
Spectral methods are global methods and converge exponentially. Spectral methods can provide
94
high accuracy and low computational time and computer memory which make them favorable
95
for solving smooth PDEs such as heat and mass transfer equations. It should be noted that the
96
spectral method becomes less accurate for problems with complex geometries and non-smooth
97
problems, while the FEMs are particularly well suited for solving this problems (shen et al.,
98
2011).
99
Spectral methods have been extended rapidly in the past three decades and have been widely
100
implemented in meteorology (Jang and Hong, 2016), computational fluid dynamics (Canuto et
101
al., 2007), quantum mechanics (Graham et al., 2009) and magnetohydrodynamics (Shan et al,
102
1991). Also, in recent works, researchers have some studies on the implementation of spectral
103
methods for solving radiative heat transfer problem (Kuo et al., 1999, Li et al., 2009, Sun and
104
Li., 2012, Zhou and Li., 2017).
105
According to the authors’ knowledge, there are no results in the literature regarding the
106
application of spectral methods for solving coupled heat and mass transfer equations in food
107
engineering. This partially motivates us to propose such a method for solving the considered
108
systems of coupled heat and mass transfer equations. Moreover, in most of the research works
109
the spectral methods are applied for discretizing spatial variables together with localizing the
110
temporal variable, with low order FDMs, which yields to unbalanced schemes that have high
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accuracy in spatial variables and low accuracy in time variable (Fakhar Izadi and Dehghan,
112
2014). Therefore, another motivation of the present study is to propose a spatial-temporal
113
collocation method for the aforementioned processes which is a balanced numerical approach.
114
The objective of the present study is to extend Jacobi Gauss Lobatto (JGL) spectral collocation
115
method to simulate coupled heat and mass transfer phenomena in three dimensions during
116
convective air drying of apple slices.
117
Operational matrices of differentiation are implemented for approximating the derivative of both
118
spatial and temporal variables. By using this spectral scheme, the coupled 3D heat and mass
119
transfer together with the initial and boundary conditions will be reduced to the associated
120
system of linear algebraic equations, which can be solved by some robust iterative solvers such
121
as GMRES. Moreover, the experimental data are provided to validate the numerical data for the
122
considered models. They confirm the accuracy of the presented numerical method.
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123
2. Material and Methods
125
2.1. Temperature and moisture measurements:
126
The apple fruits were purchased from local markets in Mashhad, Iran, in June 2015 and stored in
127
the refrigerator. Drying experiments were carried out by a convective air dryer equipped with the
128
control unit to set the temperature of the air. The relative humidity and the air velocity were
129
digitally measured by humidity sensor (Rotronic hygropalm, USA) and air velocimeter (Testo
130
425, Germany) which were placed in the dryer chamber. Experiments were performed for drying
131
air temperature of 60, 70, 80 and 900C. In each experiment, apple fruit was sliced with
132
dimensions of 2cm×2cm×1cm and was placed as a thin layer in a stainless steel basket, which
133
hangs on a digital balance with an accuracy of ±0. 01 g. Digital balance connected to a personal
134
computer that records weight loss every 10-second intervals until to reach a stable weight. The
135
center and surface temperatures were measured with T-type thermocouples (RS component, UK)
136
with 0.3 mm diameter and were recorded using a data logger (Pico technology USB TC-08 data
137
logger, RS component, UK) connected to a PC. Under drying conditions, the measured relative
138
humidity and air drying velocity were 6 ± 2.0% and 0.1 m/s respectively. All the drying
139
conditions were repeated three times.
140
The initial moisture content of the apple slices was obtained using the oven method at 105°C for
141
24 h (AOAC 1990). Average moisture content was measured 83.5% (w.b.).
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The thermophysical properties of the apple slices were determined as a function of the moisture
143
content according the following relationship and presented in Table (1) (Krokida and Maroulis,
144
1999; Rao et al., 2005).
145 146
= 770 + 16.18
= 0.148 + 0.493
147
+,
9,
= (1.26 + 2.97
148
− 295.1 × exp (−
9, ) ×
+, )
1000
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142
(1) (2) (3)
2.2. Estimation of heat and mass transfer coefficients
151
The convective heat transfer coefficient (ℎ) can be determined by using Eq. (5), which is a well-
152
known lumped parameter analysis (Ranz and Marshal, 1952; Srikiatden and Roberts, 2008;
153
Holman, 2009). In this method, the aluminum piece with the similar geometry of the apple slices
154
was placed in a drying chamber under all of the mentioned drying conditions (60 up to 90oC).
155
The analysis assumes that the internal temperature gradients are low or the Biot number value is
156
less than 1, therefore the resistance for heat transfer is at the surface of aluminum piece. During
157
heating process at different time intervals ( ), the internal temperature of the aluminum piece
158
increases by the differential amount (
) and recorded. The energy balance at the surface of the
159
aluminum piece during time intervals (
) can be described as follows:
160
ℎ
(
:;<
−
:= )
=
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(
> ):=
(4)
161
Applying the initial condition ( = 0) =
162
surface is:
163
@DF C@ADE
164
Finally, the convective heat transfer coefficient is calculated by using the slope of Eq. (5),
165
namely, when N" [(
166
was obtained (Fig. 1). The convective heat transfer coefficient (ℎ) can be calculated directly from
167
RS :
168
RS =
the integrated form of the energy balance at the
EP
= exp (⎼ (
H I<
(J)AB (KL )AB M
:=
−
(5)
)t)
AC C
@AB C@ADE
;? ,
:;< )⁄( :=
−
:;< )]
is plotted versus time, a straight line with a slope RS
H I<
(6)
(J)AB (KL )AB M
169
Now, the average convective mass transfer coefficient ℎT was computed from the Eq. (7) and by
170
using Chilton – Colburn analogy (Chilton and Colburn, 1934; Incropera et al., 2012).
6
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H (UVA WX Y/[ ) \ADE
171
ℎT =
172
where
173
respectively. N is Lewis number which demonstrate the ratio between thermal and concentration
174
boundary layer thicknesses. It is defined as:
175
N =
176
where
177
The values of convective coefficients (ℎ and ℎT ) are presented in Table (2).
:;<
and,
are diffusivity of water vapor into the air and thermal conductivity of air,
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]:
(7)
^ADE UVA
:;< is
the thermal diffusivity of the air.
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178
(8)
2.3. Calculation of effective moisture diffusivity
180
In works associated to the drying processes, diffusion is generally considered to be the main
181
mechanism during the moisture transfer to the surface object. Fick’s second law of diffusion was
182
applied to compute the effective moisture diffusivity coefficient of apple slices with rectangle
183
cube domain in three dimension. The analytical solution of this equation in the case of symmetric
184
boundary conditions with the assumption of moisture migration being only by diffusion,
185
negligible shrinkage, constant diffusion coefficients and temperature and moisture distribution in
186
material is homogeneous can be written as follows (Crank, 1975; Simal et al., 1997; Hernandez
187
et al., 2000; Rastogi et al., 2004; Zlatanovic et al, 2013): _
e
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179
e
e (2" + 1)d a d 8` 1 = bcc c exp g− d d d d (2" + 1) (2 + 1) (2 + 1) a TfS \fS
EP
?fS
Xhh
Xhh
i exp g−
(2
+ 1)d a d d
Xhh
i
g−
(2 + 1)d a d d
is the effective moisture diffusivity (m2/s),
Where MR is moisture ratio (dimensionless),
189
and " is a positive integer. For long drying times, this equation can be simplified by taking the
190
first term of the series solution:
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j[
191
_=
192
Where
193
respectively.
k
exp m− l
s
=n
=
s
tn
+
kn Uopp q
s
un
+
=n
s
vn
,
r
and ,
(9) and
are the half thickness of product in the , , direction,
7
Xhh
i.
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194
The moisture effective diffusion coefficient is computed by using the slope of the Eq. (9),
195
namely, when natural logarithm MR versus time is plotted, a straight line with a slope
196
obtained:
197
N"( _) = N" m lr − k
where
=
k n Uopp q
kn Uopp w=n
199
(10)
w=n
.
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j[
is
(11)
3. Mathematical model
201
3.1. Modeling of external flow and temperature field
202
Fig. 2 shows the schematic domain of the problem, with its boundary conditions, for the
203
determination of external flow and temperature fields of the drying air around the apple slices. At
204
the left side, inlet velocity is U∞= (0.1 m/s) and inlet temperatures are T∞= (333K, 343K, 353K
205
and 363K).
206
In simulations, side walls are considered at U∞ and T∞, and outlet pressure is assumed similar to
207
the outlet condition of flow field. The governing PDEs for the forced convection motion of a
208
drying fluid in three-dimensional geometry are the mass, momentum and energy conservation
209
equations.
210
The mass conservation is: (Chandra Mohan and Talukdar, 2010; Esfahani et al., 2014):
211
xy
212
And the momentum equations with constant properties are:
213
(!
xy xz
214
(!
x]
215
(!
x9 xz
xz
+
x9 x|
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x]
x{
=0
EP
+
+
xy x{
+
xy ) x|
+
x{
x]
+
x]
+
x9 x{
x|
+
xn y xz n
+
xn y x{ n
+
xn y ) x| n
xn ]
+
x{ n
xn ]
+
xn ]
=−
x> xz
+ (
)=−
x>
+ (
AC C
xz
x9 ) x|
x{
=−
xz n
x> x|
xn 9 xz n
+ (
+
xn 9 x{n
+
xn 9 ) x| n
(15)
217
!
+
x@
x{
+
x@ x|
xn @
= (
xzn
+
xn @
x{ n
+
xn @ x| n
(13)
(14)
The energy equation with constant properties is: xz
(12)
)
x| n
216
x@
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200
)
(16)
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218 219
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3.2. Modeling of the internal temperature and moisture distribution of the object
222
The unsteady 3D temperature and moisture transfer inside the apple slices were computed by the
223
considered model, which is based on Fourier and Fick’s second law. To simplify the problem,
224
the following hypotheses are considered:
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221
•
A transient 3D heat and mass transfer in the apple slices.
226
•
Moisture transfer inside the apple slices only by diffusion.
227
•
Constant thermophysical properties of apple slices.
228
•
Negligible radiation effects around the food and heat generation inside the apple slice.
229
•
Non shrinkage or deformation of apple slices during drying.
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It have been recognized that the influence of shrinkage cannot be neglected in establishing
231
reliable effective water diffusivity in food (Bialobrzewski & Markowski, 2004; Hernandez et al.,
232
2000; Mulet, 1994). But it is shown if shrinkage was ignored, simplified diffusional models still
233
describe satisfactorily the experimental data (Hernandez et al., 2000; Mulet, 1994; Ruiz-Lopez et
234
al., 2004). So, in the present study, despite the shrinkage was appeared in apple slices, the effect
235
of shrinkage in numerical modelling was ignored.
236
By the considered assumptions, the governing equations can be written as follows (Hussain and
237
Dincer, 2003; Chandra Mohan and Talukdar, 2010; Villa Corrales et al., 2010; Lemus Mondaca
238
et al., 2013):
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230
240
Mass Transfer:
241
242
with the following initial conditions:
243 244
∂M ∂t
=
∂
∂x
m
∂M Xhh ∂x r
+
( , , , = 0) =
∂
∂y
m
∂M ∂ ∂M Xhh ∂y r + ∂z m Xhh ∂z r, (
, , , ) ∈ [0, ] × [0, ] × [0, ] × ~0, h •, (17)
;? ,
(18)
and the following Neumann and Robin boundary conditions: 9
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245
x€(z,{,|,q) xz
|zfS = 0, ∈ [0, ] " ∈ [0, ]
(19)
246
x€(z,{,|,q)
247
x€(z,{,|,q) ||fS x|
248
−
Xhh
∂M(z,{,|,q) |z=X = hm ∂x
(M ( , , , ) −
249
−
Xhh
∂M(z,{,|,q) |{=Y = hm ∂y
(M ( , , , , ) −
:;< (
))|{=Y , ∈ [0, ] " ∈ [0, ]
250
−
Xhh
∂M(z,{,|,q) ||=Z = hm ∂z
(M ( , , , , ) −
:;< (
))||=Z ,
ƒ>
∂T ∂t
=
∂ m ∂x
(20)
= 0, ∈ [0, ] " ∈ [0, ]
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:;< (
(21) ∈ [0, ] " ∈ [0, ]
))|z=X ,
∈ [0, ] " ∈ [0, ]
r + m r + m r, ( , , , ) ∈ [0, ] × [0, ] × [0, ] × ~0, h • ∂x ∂y ∂y ∂z ∂z ∂T
∂
∂T
∂
∂T
with the following initial conditions: ( , , , = 0) =
;? ,
257 258
The boundary conditions for describing the heat transfer are as follows:
259
x@(z,{,|,q)
|zfS = 0, ∈ [0, ] " ∈ [0, ],
(23) (24)
(25)
(26)
(27)
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xz
(22)
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Heat Transfer:
255 256
|{fS = 0, ∈ [0, ] " ∈ [0, ]
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251 252 253 254
x{
260
x@(z,{,|,q)
|{fS = 0,
261
x@(z,{,|,q)
||fS = 0, ∈ [0, ] " ∈ [0, ]
262
ℎ„
:;<
− ( , , , )…|zft =
263
ℎ„
:;<
− ( , , , )…|{fu = −
264
ℎ„
:;<
− ( , , , )…||fv =
265
The evaporation effect that imposed in the boundary conditions of the heat equation (Eqs. (30),
266
(31) and (32)), also affect the coupling between the heat and mass transfer equations. ℎ=] (J/kg) is
267
the latent heat of vaporization and was assumed 2257 KJ/kg (Incropera and DeWitt, 1996).
x@(z,{,|,q) xz
(28) (29)
|zft + ℎ=] ℎT ( ( , , , ) −
x@(z,{,|,q)
AC C
x|
∈ [0, ] " ∈ [0, ]
EP
x{
x{
x@(z,{,|,q) x|
|{fu + ℎ=] ℎT ( ( , , , ) −
||fv + ℎ=] ℎT ( ( , , , ) −
:;< )|zft :;< )|{fu :;< )||fv
∈ [0, ] " ∈ [0, ]
(30)
∈ [0, ] " ∈ [0, ] (31) ∈ [0, ] " ∈ [0, ]
(32)
268 269 270
The moisture content of drying air in dry basis ( :;<
R = 2.1667 × „_†‡100… × ˆ ]
:;<‡
(
:;< )
is computed using the following equation: s
‰ × mJ r × 10C` . ADE :;< + 273.15) 10
(33)
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271 272 273
where R] is the partial saturated water vapor pressure and defined as (ASHRAE, 2009): R] = exp [−5.8 × 10 ‡ `
:;<
`
+ 6.545 ln(
:;<
:;< )].
+ 1.391 − 4.864 × 10Cd ×
:;<
− 4.176 × 10CŠ ×
:;<
d
− 1.445 × 10Cj × (34)
275
4. Spectral collocation method
276
In spectral methods the solution ! is approximated by a finite sum:
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274
277
?Ž !( ) ≈ ! ?Ž ( ) = ∑\fS !\ \ ( )
278
where
279
determined. The choice of trial functions
280
the expansion coefficients !\ ) distinguishes the kind of spectral methods. The three main
281
approaches for determination of expansion coefficients are Tau, Galerkin and collocation
282
(pseudospectral) methods (Shen et al., 2011).
283
Spectral collocation methods (or pseudospectral methods) are one of the well-known classes of
284
spectral methods that are easy to implement and are very suitable for solving multi-dimensional
285
PDEs. Moreover, in recent years, the activity on both theory and application of spectral methods
286
has been concentrated on pseudospectral methods (Fornberg and Sloan, 1994; Costa, 2004).
287
Pseudospectral methods has similar property with FDMs. Pseudospectral methods implemented
288
a certain set of mesh points like FDMs, that are called “collocation points” which have more
289
accuracy with respect to FDMs (Costa, 2004).
290
It should be recalled that in pseudospectral methods, the -th derivative of the vector ! at JGL
293
SC
), and test functions (that are used for determining
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\(
points, which will be introduced in the sequel, can be approximated in the form of !(\) ≈ \ (?Ž •s) !,
where
EP
292
) are the trial functions and !\ are the expansion coefficients, which should be
(?Ž •s) is
the differentiation matrix or the derivative matrix associated to the
JGL points with the size of ("z + 1) × ("z + 1). In other words:
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291
\(
(35)
U(FŽ ŸY)
294
›œœœœœœœ•œœœœœœœž S,S S,s … ) !” ( ) S,?Ž ˜ “ !( S ˜ “ ” S ˜ “ ( ) s,s … s,?Ž — ’ !( s ) — ’ ! s — ≈ ’’ s,S — ⋮ ⋮ — ⋱ ⋮ ’ — ’ ⋮ ⋮ —’ ” ! „ … !„ ‘ ?Ž – ?Ž …– ?Ž ,s … ?Ž, ?Ž – ‘ ‘ ?Ž ,S
(36)
295 296
In this study the solutions of the considered equations are approximated by their Lagrange
297
interpolation polynomials based on the interpolation nodes such as JGL points. 11
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298
d ) ¡^•s,¢•s ( ?Ž Cs
It should be recalled that JGL points are the roots of ( ) = (1 − is the Jacobi polynomial of degree "z − 1 with the parameters
300
for choosing such a set of JGL points as the collocation points (in both of the spatial and
301
temporal variables) is the exponential rate of convergence with respect to the uniform collocation
302
points (Boyd, J.P., 1989).
303
To approximate the solution of the aforementioned heat and mass transfer equations, one can
304
write:
(37)
( , , , )≈
?(
?
?Ž ?§ ¨ ¦ ∑\fS ∑TfS ∑?¥fS , , , ) = ∑;fS
„ £; , £\ , ̂T , ¥̂ … ©; ( ) ©\ ( ) ©T ( ) ©¥ ( ),
SC
307
?(
?Ž ?§ ¨ ¦ ∑\fS ∑TfS ∑?¥fS , , , ) = ∑;fS „ £; , £\ , ̂T , ¥̂ … ©; ( ) ©\ ( ) ©T ( ) ©¥ ( ), ?
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( , , , )≈
+ 1. The basic reason
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299
305
+ 1 and
), where ¡?^•s,¢•s Ž Cs
308
(38)
309
where ©; ( ), ©\ ( ), ©T ( ) and ©¥ ( ) are the Lagrange polynomials based on the JGL points and
310
„ £; , £\ , ̂T , ¥̂ … and „ £; , £\ , ̂T , ¥̂ … are the unknown expansion coefficients by assuming 0 ≤ « ≤
"z , 0 ≤
312
these unknown coefficients.
313
As mentioned earlier, in this study the operational matrices of differentiation are implemented
314
for approximating the derivative of the spatial and temporal variables. Applying the operational
315
matrices of differentiation instead of direct symbolic differentiations, speed up the operations
316
(Shen and Tang, 2006).
317
After collocating the main considered 3D coupled heat and mass transfer equations (i.e.,
319 320 321
≤ "| and 0 ≤ ¬ ≤ "q , respectively. It should be noted that, our aim is to find
EP
Equations (17) and (25)) at the JGL points £; , £\ , ̂T and ¥̂ , the following system of linear
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318
≤ "{ , 0 ≤
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311
algebraic equations will be achieved. Taking into account that " = ("z + 1)„"{ + 1…("| + 1)("q + 1):
322
where ® = ˆ ‰,
323
Moreover,
dd
=m
= ¯ d° , s
® = , d?×d?
z d Xhh ˆ( ? )
+„
(39)
=¯
s
{ d ?…
+(
°
d d?×d?
=¯
| d ?) ‰ −
12
ss
0
q ? r,
0
°
dd d?×d?
,
= 0d?×s .
(40)
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ss
= m ƒ>
q ?
− ˆ(
z d ?)
+„
{ d ?…
+(
| d ? ) ‰ r.
324
and
325
Then, the initial and boundary conditions (Eqs. (18) – (24)) and (Eqs. (26) – (32)) should be
326
imposed on the coefficient matrix A and right hand side vector b and finally the linear algebraic
327
equations will be solved by a robust iterative method, known as the GMRES method.
328
The implementations of the spectral method for solving 3D coupled heat and mass transfer
329
equations (Equations (17)-(32)) can be carried out according to the following routine:
330
Step 1: Select the number of collocation points ("z , "{ , "| and "q ), Set " = ("z + 1)„"{ +
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(41)
1…("| + 1)("q + 1) and insert the coefficients and parameters of the model.
332
Step 2: Construct the JGL collocation points and the spatial and temporal operational matrices
333
(
334
Step 3: Set the coefficient matrix
335
Step 4: Imposing the initial condition on matrix coefficient
336
Step 5: Imposing the boundary conditions on matrix coefficient
{ ?,
| ?
and
q ? ).
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z ?,
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331
and vector of ®.
and right hand side vector . and right hand side vector .
Step 6: Solve the update system ® =
338
Step 7: Substituting the components of U (i.e., M and T) in Equations (37) and (38).
339
It should be noted that, all of the computations were accomplished in an i7 PC Laptop with 4
340
kernels with 12 GB of RAM and Cash of 6.
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341
by GMRES and compute the vector ® numerically.
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337
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5. Result and discussion
343
5.1. External flow analysis
344
The result of external flow analysis are presented in this section. Figs. 3(A and B) show the
345
velocity and temperature contours around the moist object in XY plane and at the middle of Z
346
plane for an inlet velocity of 0.1 m/s and inlet temperatures of 363 K.
347
Since the trends of the temperature contours are the same for different drying temperature, only
348
one case is presented. As the results shown, the temperature and velocity contours are seen to be
349
symmetric. It is seen from Fig. 3, that flow domain is wide enough because no change can be
350
seen near side walls. It can be concluded from the Fig. 3B that the temperature is comparatively
351
lower behind face of the moist object compared to the other faces. Consequently, the convective
352
coefficients is lower on the surface facing outlet of the channel. For this reason, average
353
convective heat transfer coefficients is considered for all faces via the lump analysis method
354
(Hussain and Dincer, 2003; Villa Corrales et al., 2010; Lemus Mondaca et al., 2013).
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355
5.2. Validation of numerical solution of heat and mass transfer equations
357
By using Jacobi spectral collocation method, temperature and moisture distributions for the
358
apple slices at different drying air temperature are investigated numerically. All the associated
359
algorithms are written in MATLAB version 7.12.0.635 (R2011b) and all calculations are carried
360
out in double precision and the corresponding matrices are propounded in sparse format. The
361
final linear algebraic systems, which are the results of the proposed discretization method, are
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solved iteratively by the robust GMRES algorithm with the tolerance of 10Cj and the iterations
363
of this algorithm is set to the rows of the coefficients matrices. It should be noted that, we have
364
considered the special case of JGL collocation method assuming
365
Gauss Lobatto collocation method) in our numerical simulations. However, other special cases
366
such as
367
numerical simulations which have similar spectral accuracy.
368
The values of effective moisture diffusion coefficients for different drying air temperature are
369
given in Table (3).
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362
=
=
= 0 (i.e., the Legendre
= −0.5 (i.e., the Chebyshev Gauss Lobatto collocations) can be considered in
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Figs. 4 shows the comparisons between the numerical simulation and the experimental values for
371
the average moisture content and temperature of apple slices during drying at 60 and 90oC. It can
372
be seen that the numerical results are in good agreement with the experimental data from drying
373
process of apple slices. The average relative error of the simulated values on experimental data
374
are 1.5 % and 3.5 % for moisture content and temperature, respectively, which confirmed the
375
high accuracy of the presented model.
376
One can conclude from the Figs. 4, during drying process, moisture content of apple slices
377
reduces rapidly as the warming up the apple slices. The basic reason for this event is the high
378
evaporation at the surface due to the considerable diffusion rate of the internal water towards the
379
surface. Then, the drying rate decreases due to the high internal mass resistance and the lowering
380
the progress of the internal moisture migrates to the surface. At this stage the internal
381
temperature of the apple slices is close to the air drying temperature. This result confirmed the
382
behavior reported by Villa-Corrales et al. (2010), Lemus Mondaca et al. (2013) and Tzempelikos
383
et al. (2015).
384
It should be noted that the CPU time (or the computational time) for the simulation by assuming
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370
"z = "{ = "| = 5 and "q = 12 was about 41 seconds that confirms high rapidity and low
386
computational cost of the proposed numerical method.
387
Fig. 5 shows the comparison between of the experimental data and the numerical simulations
388 389
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385
associated to different values of collocation points, ("q = 10, 12, 14, 16 and ("z = "{ = "| = 6)). Since the temporal interval of this problem is greater than the spatial intervals, we have
assumed larger values of "q with respect to the "z , "{ and "| . It can be observed that, we reach
391
to similar numerical results by increasing the collocation point which indicate the high accuracy
392
and robustness of the suggested numerical approach. Moreover, we can conclude that, only by
393
using small number of collocation nodes yields to accurate numerical results. The slight
394
deviations observed between the simulated and experimental data are probably due to the errors
395
in true performance of the implemented dryer and the lack of accounting shrinkage effects and
396
food properties variation in the model.
397
The numerical model introduced in this paper made reasonable results in prediction of moisture
398
and temperature distribution inside the apple slices at different times during drying. This
399
numerical model could be implemented for other dimensions (one and two), other coordinate
400
systems, such as cylindrical or spherical coordinates and other drying conditions.
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Fig. 6 shows the validation of the present work with experimental data of Kaya et al. (2007) and
402
Zarein et al. (2013). Kaya et al. (2007) evaluated the convective air drying of thin layer apple
403
slices at drying air temperature 60°C. In this study, the relative humidity and air drying velocity
404
were 40% and 0.2 m/s respectively. Zarein et al. (2013) have investigated the drying of apple
405
slices in cylindrical geometry at 80oC, air velocity 1 m/s and relative humidity 24%. It can be
406
seen that a reasonably good agreement exists between experimental data and numerical values
407
that confirm the accuracy of the proposed method for simulating different drying conditions.
408
Nevertheless, the proposed approach is a global approximation method (derivative at a given
409
point depend on all of the mesh points) which thus has disadvantage of requiring the inversion of
410
large system matrices. To remedy this inconvenient, in this paper we have used some robustness
411
iterative algorithms such as GMRES for solving the associated systems of linear algebraic
412
equations, in which computing the inversion of large system of matrices is avoided (Ballestra
413
and Cecere, 2016).
414
GMRES (generalized minimal residual method) is an iterative method for solving nonsymmetric
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401
415
systems of linear algebraic equations in the form of ® = , that approximates the inverse of
416
by some vectors in a Krylov subspace with minimal residual (Saad, 2003).
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417
5.3. Drying process simulation
419
In Figs 7 and 8, the simulated averaged moisture content and the temperature of the apple slices
420
are plotted. Comparison between moisture content curves at different drying air temperature in
421
Fig 7, shows that the moisture elimination is faster at higher drying air temperatures. This
422
behavior strongly is associated to the temperature dependency of moisture effective diffusivity.
423
This can also be illustrated in Table (3), which by increasing the drying air temperature and
424
higher temperature differences between air drying and apple slices, the heat flux to the interior of
425
the apple slices is higher and consequently the values of diffusivity coefficients increases
426
(Tzempelikos et al., 2015).
427
As can be seen in Fig. 8 at the initial stages of the drying process, the temperature of apple slices
428
does not reach close to the equilibrium temperature. This behavior may be related to the
429
evaporative cooling effect owing to the high moisture flux on the apple surface during the initial
430
stages of drying process. The high moisture transfer on the food surface during this period
431
requires more energy for moisture evaporation and hence, less heat transferred within the apple
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slices initially. Then, by increasing the drying times and decreasing the moisture flux to the apple
433
surface, the temperature of apple slices is gradually increased and approaching close to the
434
equilibrium with the drying air temperature.
435
The comparison between temperature and moisture value curves in Figs 7 and 8 shows, the
436
temperature curves change earlier than moisture content curves during drying process. In other
437
words heat transferred more rapidly than moisture within the apple slices. Therefore, the
438
temperature convergence to drying condition is quicker than moisture content. This rapid
439
convergence of temperature than moisture is due to the reality that the rate of heat transfer is
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432
higher than mass transfer or the introduced Lewis number is higher than one 1, N > 1.
441
(Vahidhosseini et al., 2016).
442
One of the specific advantages of numerical simulations of heat and mass transfer equations is
443
providing the high spatial resolution information of temperature and moisture content
444
distribution during the drying process, whereas in practice it is difficult to measure the spatial
445
distribution of moisture inside the material and requires specialized equipment and resources
446
(Aregawi et al., 2013).
447
Fig. 9 shows the spatial distribution of the temperature and moisture content inside the apple
448
slice, as predicted by the numerical model. As can be seen in Fig. 9, moisture diffusion occurred
449
from the inside of the apple slice with high concentration to its boundaries with low
450
concentration. Also the apple slices warming up from its boundaries to the inside, which these
451
phenomena matches with the Fick’s laws of diffusion and Fourier law, respectively. Moreover, at
452
the beginning of the drying process, the moisture and temperature gradients are high. Then
453
moisture and temperature gradients decrease with the advances of the drying process until the
454
internal temperature and moisture pressure of the apple slice will be reaching to the similar
455
condition of the air drying (Janjai et al., 2008; Villa-Corrales et al., 2010; Lemus Mondaca et al.,
456
2013).
457
Also it can be concluded that the temperature profile has slow gradients along the apple slices.
458
This specific behavior is basically imputed to the low heat transfer Biot number, which range
459
between 0.37 and 0.53. These values are of the same order of magnitude compared to the limit of
460
“thermally-thin-materials” set to the order of 0.1. ((Incropera et al., 2007, Tzempelikos et al.,
461
2015). It should be recalled that, the Biot number is a dimensionless group that relates a measure
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462
of the rate of internal heat conduction or mass diffusion in object with a measure of the rate of
463
external convection and are useful to identify controlling mechanisms (Giner et al., 2010).
464
In addition, the values of the mass transfer Biot number for different drying air temperature were
found to be in the ranges of 1.34 × 10Š to 1.5 × 10Š which is compatible with the existence of
466
strong spatial gradients of moisture content inside the apple slices. The mass Biot number higher
467
than 100, illustrates that external resistance to mass transfer is negligible and thus the effective
468
moisture diffusivity is not influenced by external drying conditions (Srikiatden and Roberts,
469
2006).
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465
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470
6. Conclusion:
472
Jacobi spectral collocation method was proposed for simulating distribution of 3D coupled
473
temperature and moisture content inside the apple slices numerically. The presented numerical
474
method needs less computational time with respect to the analytical methods, by making use of
475
the operational matrices in computations. The results of simulations illustrates that the 3D model
476
of the coupled heat and mass transfer provide a better understanding of the transport processes
477
inside the apple slices during drying process. This numerical model can be implemented in
478
automatic control of convective air dryers and energy optimization of dryer operation and
479
improving dryer design. So it is considered a great potential benefit for evaluating the
480
engineering process.
483
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484
Acknowledgments
485
The author thanks from the editor and reviewers of this paper for their constructive comments
486
and nice suggestions, which helped to improve the paper very much.
487 488
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Figure captions:
Fig. 1. Time–temperature (dimensionless) plot for determination of heat transfer coefficient of aluminum cubes. Fig. 2. Apple slices is subjected to convective air drying.
SC
Fig. 3. (A) Velocity and (B) temperature contour around the moist object at XY plane.
Fig. 4. Comparison of the predicted temperature and moisture data with the experimental results during drying at 60 and 90 °C.
M AN U
Fig. 5. Comparison of the numerical simulations of the moisture and temperature associated to 16 and the experimental results.
= 10, 12, 14 and
Fig. 6. Comparison of the experimental data with simulated moisture content.
Fig. 7. Moisture content numerical values in apple slices at selected temperatures. Fig. 8. Temperature numerical values in apple slices at selected temperatures.
Fig. 9. Temperature (Right figures) and moisture (Left figures) distributions inside the apple slice at XY plane at
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90oC for different drying times (a=1500 s, b=5000 s, c=11500s).
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Table 1 Thermophysical properties of the apple slices )
850.47 857.78 858.71 856.25
(
)
0.559 0.565 0.561 0.561
(
)
3733.59 3773.07 3752.53 3750.57
AC C
EP
TE D
M AN U
SC
60 70 80 90
(
RI PT
Tair(°C)
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Table 2 Convective heat and mass transfer coefficients
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0.009347 0.010545 0.012849 0.014128
M AN U TE D EP AC C
60 70 80 90
( )
( ) 10.372 11.525 13.891 14.982
SC
(° )
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Table 3 Effective moisture diffusivity coefficient ( ) -10
0.8149 0.7783 0.9113 0.8514
EP
TE D
M AN U
SC
7.424×10 9.07×10-10 9.3×10-10 1.023×10-10
AC C
60 70 80 90
(
RI PT
(° )
).
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0/5
-0/5
0
200
400
600
800
1000
1200
1400
-1
RI PT
-1/5 -2 y = -0/0026x - 0/0999 R² = 0/9971
-2/5 -3 -3/5 -4
AC C
EP
TE D
M AN U
Time (s)
SC
Ln[(Tal-Tair)/(Tin-Tair)]
0
AC C
EP
TE D
M AN U
SC
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AC C
EP
TE D
M AN U
SC
RI PT
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AC C
EP
TE D
M AN U
SC
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1.2
M AN U
Numerical data (90oC) Experimental data(90oC)
0.8
0.6
TE D
Moisture Ratio(dimension less)
1
0.4
0
2000
AC C
0
EP
0.2
4000
6000
8000
10000 12000 14000 16000 18000
Drying time(s)
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100
M AN U
90
70
Numerical data (90oC) Experimental data(90oC)
60
TE D
50 40 30
0
2000
AC C
20
EP
Temperature( oC)
80
4000
6000 Drying time(s)
8000
10000
12000
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1
M AN U
Numerical data (60oC)
0.8
Experimental data(60oC)
0.7 0.6 0.5
TE D
0.4 0.3 0.2 0.1 0
0.5
AC C
0
EP
Moisture Ratio(dimension less)
0.9
1
1.5
Drying time(s)
2
2.5
3 #10
4
RI PT
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SC
65
M AN U
60
Numerical data (60oC)
50
Experimental data(60oC)
TE D
45
40
35
25
EP
30
0
0.5
AC C
Temperature( oC)
55
1
1.5 time(s)
2
2.5
3 4
x 10
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1
Numerical data with nt=10
0.9
M AN U
Numerical data with nt=12 Numerical data with nt=14 Numerical data with nt=16
0.7
Experimental data
0.6 0.5
TE D
0.4 0.3 0.2 0.1
0
AC C
0
EP
Moisture ratio (dimensionless)
0.8
0.5
1 1.5 Drying times (s)
2
2.5 4
x 10
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M AN U
65 60
Numerical data with nt=10 Numerical data with nt=12
50
Numerical data with nt=14
45
Numerical data with nt=16
40 35 30 0
0.5
AC C
EP
25
TE D
Temperature ( oC)
55
1
1.5 Drying time (s)
Experimental data
2
2.5
3 4
x 10
SC
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7
Numerical data Experimental data (Zarein et al, 2013)
M AN U
5
4
2
1
1000
AC C
0 0
TE D
3
EP
Moisture content (%db)
6
2000
3000 4000 Drying times (s)
5000
6000
7000
SC
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4.5
Numerical data Experimental data (kaya et al, 2007)
M AN U
4
3 2.5 2
TE D
Moisture content (%db)
3.5
1.5 1
0
0.5
AC C
0
EP
0.5
1
1.5 2 2.5 Drying times (s)
3
3.5
4 4
x 10
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SC
1.4
M AN U
1
0.8
80oC 70oC 60oC
EP
0.4
0.2
0
90oC
TE D
0.6
AC C
Moisture Ratio (dimension less)
1.2
0
0.5
1
1.5 Drying time(s)
2
2.5
3
SC
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90
M AN U
80
60 50 40 30
0.2
AC C
0
EP
20 10
90oC 80oC 70oC
TE D
Temperature(oC)
70
0.4
0.6
0.8 1 1.2 Drying time(s)
60oC
1.4
1.6
1.8
2 4
x 10
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EP
TE D
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•
A 3D model is assumed for predicting temperature and moisture transfer numerically.
•
A robust and high accurate numerical scheme for solving 3D coupled heat and mass transfer process has been proposed. Operational matrices of differentiation are used because of reducing symbolic computations w. r. t. the analytical methods
•
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•
Extension and modification of the presented spectral method can be applied for
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simulation other phenomena in Food Engineering.
S0260-8774(17)30211-X
DOI:
10.1016/j.jfoodeng.2017.05.013
Reference:
JFOE 8884
To appear in:
Journal of Food Engineering
Received Date: 19 October 2016 Revised Date:
1 May 2017
Accepted Date: 11 May 2017
Please cite this article as: Pasban, A., Sadrnia, H., Mohebbi, M., Ahmad Shahidi, S., Spectral method for simulating 3D heat and mass transfer during drying of apple slices, Journal of Food Engineering (2017), doi: 10.1016/j.jfoodeng.2017.05.013. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. 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.
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Spectral method for simulating 3D heat and mass transfer during drying of apple slices 1 2
Atena Pasban1, Hassan Sadrnia2*, Mohebbat Mohebbi1, Seyed Ahmad Shahidi3 1
3 4 5 6 7 8 9 10 11 12
Abstract:
13
In the present study, a numerical method is proposed for simulating the coupled three
14
dimensional heat and mass transfer processes during convective drying of apple slices. Spectral
15
collocation method (pseudospectral method) is applied for discretizing both the space and time
16
variables based on the Jacobi Gauss Lobatto (JGL) interpolation points. Also operational
17
matrices of differentiation are implemented for approximating the derivative of the spatial and
18
temporal variables. The external flow and temperature fields were simulated through the Fluent
19
CFD package. The convective heat transfer coefficient is calculated from the lumped system
20
analysis and convective mass transfer coefficient is computed through the analogy between the
21
thermal and concentration boundary layers. The model is validated against experimental data in a
22
range of air temperatures from 60oC up to 90oC. The results illustrate a remarkable agreement
23
between the numerical predictions and experimental results, which confirm robustness,
24
computationally efficient and high accuracy of the proposed approach for predicting the
25
simultaneous heat and mass transfer in apple slices.
27 28 29
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Department of Food Science and Technology, Ferdowsi University of Mashhad, Mashhad, Iran 2 Department of Biosystems Engineering, Ferdowsi University of Mashhad, Mashhad, Iran 3 Department of Food Science and Technology, Ayatollah Amoli Branch, Islamic Azad University, Amol, Iran. *Corresponding author: Email: [email protected], Tel: +985138805839
Key Words: Numerical simulation; Apple drying; Spectral collocation method; Convective coefficients.
30 31 32
1
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33
Nomenclature
total area of apple slice (m2) moisture content (kgm-3) temperature (K,oC) specific heat (Jkg-1K-1) effective moisture diffusivity (m2s-1) diffusivity of water vapor in air (m2s-1) vaporization latent heat (Jkg-1) thermal conductivity (Wm-1k-1) heat transfer coefficient (Wm-2 K-1) mass transfer coefficient (ms-1)
, , , ,
34 35 36 37 38
!, , " ,"
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Greek symbols viscosity (kg/(m s)) density of food (kg m-3)
time (s) moisture content in dry basis (kg water kg dry mass-1) moisture content in wet basis (kg water kg product-1) axial coordinates (m) half thickness of product in the , , direction (m) Volume (m3) velocities in X, Y and Z direction (m/s) , " number of the collocation nodes
,
M AN U
ℎ ℎ
1. Introduction
thermal diffusivity (m2/s) parameters of the Jacobi polynomials
Dimensionless groups Le Lewis number (dimensionless)
TE D
ℎ
Subscripts Air drying air in Initial Al Aluminum F Final
Fruits and vegetables are considered as more perishable foods because of high moisture content
40
(Simal et al., 1997). Drying process is one of the well-known methods for preservation of fruits
41
and vegetables. This process prevents occurrence of unpleasant changes such as microbial
42
spoilage and enzymatic reaction by removing water from food products. Moreover, drying by
43
lowering the mass and volume of food products, reduces the cost of packaging, storage and
44
transportation (Goyal et al., 2006; Mujumdar, 2006).
45
When a moist object is subjected to drying conditions, heat and mass (moisture) transfer happen
46
simultaneously. Heat is transferred by convection from the heated air to the surface of a moist
47
object (food) and by conduction to the interior of food to increase temperatures and to evaporate
48
moisture from the food surface. Moisture transfer is accomplished by diffusion from inside of the
49
food to the surface, and from the food surface to the air by convection due to the heat transfer
50
process (Hernandez et al., 2000; Mujumdar, 2006), although other mechanisms may be involved.
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Heat and mass transfer phenomena in a system are described by governing equations of Fourier
52
law and Fick’s second law of diffusion. These equations are some simplified descriptions of
53
physical reality of heat and mass transfer represented in mathematical terms (Hussain and
54
Dincer, 2003; Tohidi, 2015).
55
Mathematical modeling is an important tool in the design and control of drying process
56
especially in food engineering. Many undesirable changes may occur in foods during the drying
57
process or in dried food after drying process which are associated to the temperature and
58
moisture content distribution. Therefore, simulation and prediction of the temperature and
59
moisture distributions in foods as a function of drying time can help us to prevent the undesirable
60
changes in foods during drying process or during preservation (Mishkin et al., 1983, Wang and
61
Brennan, 1995)
62
The analytical methods and numerical methods (such as the finite difference methods (FDMs),
63
finite element methods (FEMs) and finite volume methods (FVMs)) are the basic mathematical
64
tools that utilized to simulate the model of the heat and mass transfer (Barati and Esfahani, 2011;
65
Lemus Mondaca et al., 2013; Esfahani et al., 2014; García-Alvarado et al., 2014; Esfahani et al.,
66
2015; Vahidhosseini et al., 2016; Tzempelikos et al., 2015).
67
Numerical methods mostly compute the approximate solutions of the governing equations
68
through the localization of spatial and temporal variables that can be more realistic and flexible
69
for simulating the aforementioned phenomena. In contrast, analytical methods require the infinite
70
power series in computations that make them deficient and unfavorable with respect to the
71
numerical methods (Zogheib and Tohidi, 2016). Moreover, in analytical methods, symbolic
72
differentiation and integration are time-consuming operations. Therefore, in numerical
73
techniques, alternative tools such as operational matrices of differentiation and also Gauss
74
quadrature rules are replaced instead of direct symbolic differentiations and integration,
75
respectively for speeding up the operations (Shen and Tang, 2006).
76
In recent years, considerable number of research works have been devoted to numerical
77
simulation of heat and mass transfer phenomena during convective drying of food, such as
78
numerical analysis of coupled heat and mass transfer during drying process in papaya slices and
79
mango with FVM (Villa Corrales et al., 2010; Lemus Mondaca et al., 2013), numerical analysis
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3
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of the transport phenomena occurring during drying process of carrots and mango fruit with
81
FEM (Aversa et al., 2007; Janjai et al., 2008) and numerical simulation of 2D heat and mass
82
transfer during drying of a rectangular object with FDM (Hussain and Dincer, 2003).
83
Among the numerical methods, the spectral methods are popular and robust tools which have
84
been widely implemented during the recent decades for solving smooth partial differential
85
equations (PDEs) with simple domains.
86
Numerical methods for solving PDEs can be classified into the local (like FDMs and FEMs) and
87
global categories. In local methods, derivative approximation of an assumed function at any
88
given point depends only on the information from its neighboring. Whereas in global methods,
89
derivative approximation of an assumed function at any given points depends not only on the
90
information from its neighboring points but also on the information from the entire of the
91
computational domain, which force them to achieve a high precision using a small number of
92
discretization nodes (Costa, 2004; Sun et al., 2012).
93
Spectral methods are global methods and converge exponentially. Spectral methods can provide
94
high accuracy and low computational time and computer memory which make them favorable
95
for solving smooth PDEs such as heat and mass transfer equations. It should be noted that the
96
spectral method becomes less accurate for problems with complex geometries and non-smooth
97
problems, while the FEMs are particularly well suited for solving this problems (shen et al.,
98
2011).
99
Spectral methods have been extended rapidly in the past three decades and have been widely
100
implemented in meteorology (Jang and Hong, 2016), computational fluid dynamics (Canuto et
101
al., 2007), quantum mechanics (Graham et al., 2009) and magnetohydrodynamics (Shan et al,
102
1991). Also, in recent works, researchers have some studies on the implementation of spectral
103
methods for solving radiative heat transfer problem (Kuo et al., 1999, Li et al., 2009, Sun and
104
Li., 2012, Zhou and Li., 2017).
105
According to the authors’ knowledge, there are no results in the literature regarding the
106
application of spectral methods for solving coupled heat and mass transfer equations in food
107
engineering. This partially motivates us to propose such a method for solving the considered
108
systems of coupled heat and mass transfer equations. Moreover, in most of the research works
109
the spectral methods are applied for discretizing spatial variables together with localizing the
110
temporal variable, with low order FDMs, which yields to unbalanced schemes that have high
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4
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accuracy in spatial variables and low accuracy in time variable (Fakhar Izadi and Dehghan,
112
2014). Therefore, another motivation of the present study is to propose a spatial-temporal
113
collocation method for the aforementioned processes which is a balanced numerical approach.
114
The objective of the present study is to extend Jacobi Gauss Lobatto (JGL) spectral collocation
115
method to simulate coupled heat and mass transfer phenomena in three dimensions during
116
convective air drying of apple slices.
117
Operational matrices of differentiation are implemented for approximating the derivative of both
118
spatial and temporal variables. By using this spectral scheme, the coupled 3D heat and mass
119
transfer together with the initial and boundary conditions will be reduced to the associated
120
system of linear algebraic equations, which can be solved by some robust iterative solvers such
121
as GMRES. Moreover, the experimental data are provided to validate the numerical data for the
122
considered models. They confirm the accuracy of the presented numerical method.
M AN U
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111
123
2. Material and Methods
125
2.1. Temperature and moisture measurements:
126
The apple fruits were purchased from local markets in Mashhad, Iran, in June 2015 and stored in
127
the refrigerator. Drying experiments were carried out by a convective air dryer equipped with the
128
control unit to set the temperature of the air. The relative humidity and the air velocity were
129
digitally measured by humidity sensor (Rotronic hygropalm, USA) and air velocimeter (Testo
130
425, Germany) which were placed in the dryer chamber. Experiments were performed for drying
131
air temperature of 60, 70, 80 and 900C. In each experiment, apple fruit was sliced with
132
dimensions of 2cm×2cm×1cm and was placed as a thin layer in a stainless steel basket, which
133
hangs on a digital balance with an accuracy of ±0. 01 g. Digital balance connected to a personal
134
computer that records weight loss every 10-second intervals until to reach a stable weight. The
135
center and surface temperatures were measured with T-type thermocouples (RS component, UK)
136
with 0.3 mm diameter and were recorded using a data logger (Pico technology USB TC-08 data
137
logger, RS component, UK) connected to a PC. Under drying conditions, the measured relative
138
humidity and air drying velocity were 6 ± 2.0% and 0.1 m/s respectively. All the drying
139
conditions were repeated three times.
140
The initial moisture content of the apple slices was obtained using the oven method at 105°C for
141
24 h (AOAC 1990). Average moisture content was measured 83.5% (w.b.).
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5
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The thermophysical properties of the apple slices were determined as a function of the moisture
143
content according the following relationship and presented in Table (1) (Krokida and Maroulis,
144
1999; Rao et al., 2005).
145 146
= 770 + 16.18
= 0.148 + 0.493
147
+,
9,
= (1.26 + 2.97
148
− 295.1 × exp (−
9, ) ×
+, )
1000
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142
(1) (2) (3)
2.2. Estimation of heat and mass transfer coefficients
151
The convective heat transfer coefficient (ℎ) can be determined by using Eq. (5), which is a well-
152
known lumped parameter analysis (Ranz and Marshal, 1952; Srikiatden and Roberts, 2008;
153
Holman, 2009). In this method, the aluminum piece with the similar geometry of the apple slices
154
was placed in a drying chamber under all of the mentioned drying conditions (60 up to 90oC).
155
The analysis assumes that the internal temperature gradients are low or the Biot number value is
156
less than 1, therefore the resistance for heat transfer is at the surface of aluminum piece. During
157
heating process at different time intervals ( ), the internal temperature of the aluminum piece
158
increases by the differential amount (
) and recorded. The energy balance at the surface of the
159
aluminum piece during time intervals (
) can be described as follows:
160
ℎ
(
:;<
−
:= )
=
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149 150
(
> ):=
(4)
161
Applying the initial condition ( = 0) =
162
surface is:
163
@DF C@ADE
164
Finally, the convective heat transfer coefficient is calculated by using the slope of Eq. (5),
165
namely, when N" [(
166
was obtained (Fig. 1). The convective heat transfer coefficient (ℎ) can be calculated directly from
167
RS :
168
RS =
the integrated form of the energy balance at the
EP
= exp (⎼ (
H I<
(J)AB (KL )AB M
:=
−
(5)
)t)
AC C
@AB C@ADE
;? ,
:;< )⁄( :=
−
:;< )]
is plotted versus time, a straight line with a slope RS
H I<
(6)
(J)AB (KL )AB M
169
Now, the average convective mass transfer coefficient ℎT was computed from the Eq. (7) and by
170
using Chilton – Colburn analogy (Chilton and Colburn, 1934; Incropera et al., 2012).
6
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H (UVA WX Y/[ ) \ADE
171
ℎT =
172
where
173
respectively. N is Lewis number which demonstrate the ratio between thermal and concentration
174
boundary layer thicknesses. It is defined as:
175
N =
176
where
177
The values of convective coefficients (ℎ and ℎT ) are presented in Table (2).
:;<
and,
are diffusivity of water vapor into the air and thermal conductivity of air,
RI PT
]:
(7)
^ADE UVA
:;< is
the thermal diffusivity of the air.
SC
178
(8)
2.3. Calculation of effective moisture diffusivity
180
In works associated to the drying processes, diffusion is generally considered to be the main
181
mechanism during the moisture transfer to the surface object. Fick’s second law of diffusion was
182
applied to compute the effective moisture diffusivity coefficient of apple slices with rectangle
183
cube domain in three dimension. The analytical solution of this equation in the case of symmetric
184
boundary conditions with the assumption of moisture migration being only by diffusion,
185
negligible shrinkage, constant diffusion coefficients and temperature and moisture distribution in
186
material is homogeneous can be written as follows (Crank, 1975; Simal et al., 1997; Hernandez
187
et al., 2000; Rastogi et al., 2004; Zlatanovic et al, 2013): _
e
TE D
M AN U
179
e
e (2" + 1)d a d 8` 1 = bcc c exp g− d d d d (2" + 1) (2 + 1) (2 + 1) a TfS \fS
EP
?fS
Xhh
Xhh
i exp g−
(2
+ 1)d a d d
Xhh
i
g−
(2 + 1)d a d d
is the effective moisture diffusivity (m2/s),
Where MR is moisture ratio (dimensionless),
189
and " is a positive integer. For long drying times, this equation can be simplified by taking the
190
first term of the series solution:
AC C
188
j[
191
_=
192
Where
193
respectively.
k
exp m− l
s
=n
=
s
tn
+
kn Uopp q
s
un
+
=n
s
vn
,
r
and ,
(9) and
are the half thickness of product in the , , direction,
7
Xhh
i.
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194
The moisture effective diffusion coefficient is computed by using the slope of the Eq. (9),
195
namely, when natural logarithm MR versus time is plotted, a straight line with a slope
196
obtained:
197
N"( _) = N" m lr − k
where
=
k n Uopp q
kn Uopp w=n
199
(10)
w=n
.
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198
j[
is
(11)
3. Mathematical model
201
3.1. Modeling of external flow and temperature field
202
Fig. 2 shows the schematic domain of the problem, with its boundary conditions, for the
203
determination of external flow and temperature fields of the drying air around the apple slices. At
204
the left side, inlet velocity is U∞= (0.1 m/s) and inlet temperatures are T∞= (333K, 343K, 353K
205
and 363K).
206
In simulations, side walls are considered at U∞ and T∞, and outlet pressure is assumed similar to
207
the outlet condition of flow field. The governing PDEs for the forced convection motion of a
208
drying fluid in three-dimensional geometry are the mass, momentum and energy conservation
209
equations.
210
The mass conservation is: (Chandra Mohan and Talukdar, 2010; Esfahani et al., 2014):
211
xy
212
And the momentum equations with constant properties are:
213
(!
xy xz
214
(!
x]
215
(!
x9 xz
xz
+
x9 x|
M AN U
TE D
x]
x{
=0
EP
+
+
xy x{
+
xy ) x|
+
x{
x]
+
x]
+
x9 x{
x|
+
xn y xz n
+
xn y x{ n
+
xn y ) x| n
xn ]
+
x{ n
xn ]
+
xn ]
=−
x> xz
+ (
)=−
x>
+ (
AC C
xz
x9 ) x|
x{
=−
xz n
x> x|
xn 9 xz n
+ (
+
xn 9 x{n
+
xn 9 ) x| n
(15)
217
!
+
x@
x{
+
x@ x|
xn @
= (
xzn
+
xn @
x{ n
+
xn @ x| n
(13)
(14)
The energy equation with constant properties is: xz
(12)
)
x| n
216
x@
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200
)
(16)
8
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218 219
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220
3.2. Modeling of the internal temperature and moisture distribution of the object
222
The unsteady 3D temperature and moisture transfer inside the apple slices were computed by the
223
considered model, which is based on Fourier and Fick’s second law. To simplify the problem,
224
the following hypotheses are considered:
SC
221
•
A transient 3D heat and mass transfer in the apple slices.
226
•
Moisture transfer inside the apple slices only by diffusion.
227
•
Constant thermophysical properties of apple slices.
228
•
Negligible radiation effects around the food and heat generation inside the apple slice.
229
•
Non shrinkage or deformation of apple slices during drying.
M AN U
225
It have been recognized that the influence of shrinkage cannot be neglected in establishing
231
reliable effective water diffusivity in food (Bialobrzewski & Markowski, 2004; Hernandez et al.,
232
2000; Mulet, 1994). But it is shown if shrinkage was ignored, simplified diffusional models still
233
describe satisfactorily the experimental data (Hernandez et al., 2000; Mulet, 1994; Ruiz-Lopez et
234
al., 2004). So, in the present study, despite the shrinkage was appeared in apple slices, the effect
235
of shrinkage in numerical modelling was ignored.
236
By the considered assumptions, the governing equations can be written as follows (Hussain and
237
Dincer, 2003; Chandra Mohan and Talukdar, 2010; Villa Corrales et al., 2010; Lemus Mondaca
238
et al., 2013):
AC C
239
EP
TE D
230
240
Mass Transfer:
241
242
with the following initial conditions:
243 244
∂M ∂t
=
∂
∂x
m
∂M Xhh ∂x r
+
( , , , = 0) =
∂
∂y
m
∂M ∂ ∂M Xhh ∂y r + ∂z m Xhh ∂z r, (
, , , ) ∈ [0, ] × [0, ] × [0, ] × ~0, h •, (17)
;? ,
(18)
and the following Neumann and Robin boundary conditions: 9
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245
x€(z,{,|,q) xz
|zfS = 0, ∈ [0, ] " ∈ [0, ]
(19)
246
x€(z,{,|,q)
247
x€(z,{,|,q) ||fS x|
248
−
Xhh
∂M(z,{,|,q) |z=X = hm ∂x
(M ( , , , ) −
249
−
Xhh
∂M(z,{,|,q) |{=Y = hm ∂y
(M ( , , , , ) −
:;< (
))|{=Y , ∈ [0, ] " ∈ [0, ]
250
−
Xhh
∂M(z,{,|,q) ||=Z = hm ∂z
(M ( , , , , ) −
:;< (
))||=Z ,
ƒ>
∂T ∂t
=
∂ m ∂x
(20)
= 0, ∈ [0, ] " ∈ [0, ]
RI PT
:;< (
(21) ∈ [0, ] " ∈ [0, ]
))|z=X ,
∈ [0, ] " ∈ [0, ]
r + m r + m r, ( , , , ) ∈ [0, ] × [0, ] × [0, ] × ~0, h • ∂x ∂y ∂y ∂z ∂z ∂T
∂
∂T
∂
∂T
with the following initial conditions: ( , , , = 0) =
;? ,
257 258
The boundary conditions for describing the heat transfer are as follows:
259
x@(z,{,|,q)
|zfS = 0, ∈ [0, ] " ∈ [0, ],
(23) (24)
(25)
(26)
(27)
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xz
(22)
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Heat Transfer:
255 256
|{fS = 0, ∈ [0, ] " ∈ [0, ]
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251 252 253 254
x{
260
x@(z,{,|,q)
|{fS = 0,
261
x@(z,{,|,q)
||fS = 0, ∈ [0, ] " ∈ [0, ]
262
ℎ„
:;<
− ( , , , )…|zft =
263
ℎ„
:;<
− ( , , , )…|{fu = −
264
ℎ„
:;<
− ( , , , )…||fv =
265
The evaporation effect that imposed in the boundary conditions of the heat equation (Eqs. (30),
266
(31) and (32)), also affect the coupling between the heat and mass transfer equations. ℎ=] (J/kg) is
267
the latent heat of vaporization and was assumed 2257 KJ/kg (Incropera and DeWitt, 1996).
x@(z,{,|,q) xz
(28) (29)
|zft + ℎ=] ℎT ( ( , , , ) −
x@(z,{,|,q)
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x|
∈ [0, ] " ∈ [0, ]
EP
x{
x{
x@(z,{,|,q) x|
|{fu + ℎ=] ℎT ( ( , , , ) −
||fv + ℎ=] ℎT ( ( , , , ) −
:;< )|zft :;< )|{fu :;< )||fv
∈ [0, ] " ∈ [0, ]
(30)
∈ [0, ] " ∈ [0, ] (31) ∈ [0, ] " ∈ [0, ]
(32)
268 269 270
The moisture content of drying air in dry basis ( :;<
R = 2.1667 × „_†‡100… × ˆ ]
:;<‡
(
:;< )
is computed using the following equation: s
‰ × mJ r × 10C` . ADE :;< + 273.15) 10
(33)
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271 272 273
where R] is the partial saturated water vapor pressure and defined as (ASHRAE, 2009): R] = exp [−5.8 × 10 ‡ `
:;<
`
+ 6.545 ln(
:;<
:;< )].
+ 1.391 − 4.864 × 10Cd ×
:;<
− 4.176 × 10CŠ ×
:;<
d
− 1.445 × 10Cj × (34)
275
4. Spectral collocation method
276
In spectral methods the solution ! is approximated by a finite sum:
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274
277
?Ž !( ) ≈ ! ?Ž ( ) = ∑\fS !\ \ ( )
278
where
279
determined. The choice of trial functions
280
the expansion coefficients !\ ) distinguishes the kind of spectral methods. The three main
281
approaches for determination of expansion coefficients are Tau, Galerkin and collocation
282
(pseudospectral) methods (Shen et al., 2011).
283
Spectral collocation methods (or pseudospectral methods) are one of the well-known classes of
284
spectral methods that are easy to implement and are very suitable for solving multi-dimensional
285
PDEs. Moreover, in recent years, the activity on both theory and application of spectral methods
286
has been concentrated on pseudospectral methods (Fornberg and Sloan, 1994; Costa, 2004).
287
Pseudospectral methods has similar property with FDMs. Pseudospectral methods implemented
288
a certain set of mesh points like FDMs, that are called “collocation points” which have more
289
accuracy with respect to FDMs (Costa, 2004).
290
It should be recalled that in pseudospectral methods, the -th derivative of the vector ! at JGL
293
SC
), and test functions (that are used for determining
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\(
points, which will be introduced in the sequel, can be approximated in the form of !(\) ≈ \ (?Ž •s) !,
where
EP
292
) are the trial functions and !\ are the expansion coefficients, which should be
(?Ž •s) is
the differentiation matrix or the derivative matrix associated to the
JGL points with the size of ("z + 1) × ("z + 1). In other words:
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291
\(
(35)
U(FŽ ŸY)
294
›œœœœœœœ•œœœœœœœž S,S S,s … ) !” ( ) S,?Ž ˜ “ !( S ˜ “ ” S ˜ “ ( ) s,s … s,?Ž — ’ !( s ) — ’ ! s — ≈ ’’ s,S — ⋮ ⋮ — ⋱ ⋮ ’ — ’ ⋮ ⋮ —’ ” ! „ … !„ ‘ ?Ž – ?Ž …– ?Ž ,s … ?Ž, ?Ž – ‘ ‘ ?Ž ,S
(36)
295 296
In this study the solutions of the considered equations are approximated by their Lagrange
297
interpolation polynomials based on the interpolation nodes such as JGL points. 11
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298
d ) ¡^•s,¢•s ( ?Ž Cs
It should be recalled that JGL points are the roots of ( ) = (1 − is the Jacobi polynomial of degree "z − 1 with the parameters
300
for choosing such a set of JGL points as the collocation points (in both of the spatial and
301
temporal variables) is the exponential rate of convergence with respect to the uniform collocation
302
points (Boyd, J.P., 1989).
303
To approximate the solution of the aforementioned heat and mass transfer equations, one can
304
write:
(37)
( , , , )≈
?(
?
?Ž ?§ ¨ ¦ ∑\fS ∑TfS ∑?¥fS , , , ) = ∑;fS
„ £; , £\ , ̂T , ¥̂ … ©; ( ) ©\ ( ) ©T ( ) ©¥ ( ),
SC
307
?(
?Ž ?§ ¨ ¦ ∑\fS ∑TfS ∑?¥fS , , , ) = ∑;fS „ £; , £\ , ̂T , ¥̂ … ©; ( ) ©\ ( ) ©T ( ) ©¥ ( ), ?
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306
( , , , )≈
+ 1. The basic reason
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299
305
+ 1 and
), where ¡?^•s,¢•s Ž Cs
308
(38)
309
where ©; ( ), ©\ ( ), ©T ( ) and ©¥ ( ) are the Lagrange polynomials based on the JGL points and
310
„ £; , £\ , ̂T , ¥̂ … and „ £; , £\ , ̂T , ¥̂ … are the unknown expansion coefficients by assuming 0 ≤ « ≤
"z , 0 ≤
312
these unknown coefficients.
313
As mentioned earlier, in this study the operational matrices of differentiation are implemented
314
for approximating the derivative of the spatial and temporal variables. Applying the operational
315
matrices of differentiation instead of direct symbolic differentiations, speed up the operations
316
(Shen and Tang, 2006).
317
After collocating the main considered 3D coupled heat and mass transfer equations (i.e.,
319 320 321
≤ "| and 0 ≤ ¬ ≤ "q , respectively. It should be noted that, our aim is to find
EP
Equations (17) and (25)) at the JGL points £; , £\ , ̂T and ¥̂ , the following system of linear
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318
≤ "{ , 0 ≤
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311
algebraic equations will be achieved. Taking into account that " = ("z + 1)„"{ + 1…("| + 1)("q + 1):
322
where ® = ˆ ‰,
323
Moreover,
dd
=m
= ¯ d° , s
® = , d?×d?
z d Xhh ˆ( ? )
+„
(39)
=¯
s
{ d ?…
+(
°
d d?×d?
=¯
| d ?) ‰ −
12
ss
0
q ? r,
0
°
dd d?×d?
,
= 0d?×s .
(40)
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ss
= m ƒ>
q ?
− ˆ(
z d ?)
+„
{ d ?…
+(
| d ? ) ‰ r.
324
and
325
Then, the initial and boundary conditions (Eqs. (18) – (24)) and (Eqs. (26) – (32)) should be
326
imposed on the coefficient matrix A and right hand side vector b and finally the linear algebraic
327
equations will be solved by a robust iterative method, known as the GMRES method.
328
The implementations of the spectral method for solving 3D coupled heat and mass transfer
329
equations (Equations (17)-(32)) can be carried out according to the following routine:
330
Step 1: Select the number of collocation points ("z , "{ , "| and "q ), Set " = ("z + 1)„"{ +
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(41)
1…("| + 1)("q + 1) and insert the coefficients and parameters of the model.
332
Step 2: Construct the JGL collocation points and the spatial and temporal operational matrices
333
(
334
Step 3: Set the coefficient matrix
335
Step 4: Imposing the initial condition on matrix coefficient
336
Step 5: Imposing the boundary conditions on matrix coefficient
{ ?,
| ?
and
q ? ).
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z ?,
SC
331
and vector of ®.
and right hand side vector . and right hand side vector .
Step 6: Solve the update system ® =
338
Step 7: Substituting the components of U (i.e., M and T) in Equations (37) and (38).
339
It should be noted that, all of the computations were accomplished in an i7 PC Laptop with 4
340
kernels with 12 GB of RAM and Cash of 6.
EP AC C
341
by GMRES and compute the vector ® numerically.
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337
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5. Result and discussion
343
5.1. External flow analysis
344
The result of external flow analysis are presented in this section. Figs. 3(A and B) show the
345
velocity and temperature contours around the moist object in XY plane and at the middle of Z
346
plane for an inlet velocity of 0.1 m/s and inlet temperatures of 363 K.
347
Since the trends of the temperature contours are the same for different drying temperature, only
348
one case is presented. As the results shown, the temperature and velocity contours are seen to be
349
symmetric. It is seen from Fig. 3, that flow domain is wide enough because no change can be
350
seen near side walls. It can be concluded from the Fig. 3B that the temperature is comparatively
351
lower behind face of the moist object compared to the other faces. Consequently, the convective
352
coefficients is lower on the surface facing outlet of the channel. For this reason, average
353
convective heat transfer coefficients is considered for all faces via the lump analysis method
354
(Hussain and Dincer, 2003; Villa Corrales et al., 2010; Lemus Mondaca et al., 2013).
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342
355
5.2. Validation of numerical solution of heat and mass transfer equations
357
By using Jacobi spectral collocation method, temperature and moisture distributions for the
358
apple slices at different drying air temperature are investigated numerically. All the associated
359
algorithms are written in MATLAB version 7.12.0.635 (R2011b) and all calculations are carried
360
out in double precision and the corresponding matrices are propounded in sparse format. The
361
final linear algebraic systems, which are the results of the proposed discretization method, are
EP
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356
solved iteratively by the robust GMRES algorithm with the tolerance of 10Cj and the iterations
363
of this algorithm is set to the rows of the coefficients matrices. It should be noted that, we have
364
considered the special case of JGL collocation method assuming
365
Gauss Lobatto collocation method) in our numerical simulations. However, other special cases
366
such as
367
numerical simulations which have similar spectral accuracy.
368
The values of effective moisture diffusion coefficients for different drying air temperature are
369
given in Table (3).
AC C
362
=
=
= 0 (i.e., the Legendre
= −0.5 (i.e., the Chebyshev Gauss Lobatto collocations) can be considered in
14
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Figs. 4 shows the comparisons between the numerical simulation and the experimental values for
371
the average moisture content and temperature of apple slices during drying at 60 and 90oC. It can
372
be seen that the numerical results are in good agreement with the experimental data from drying
373
process of apple slices. The average relative error of the simulated values on experimental data
374
are 1.5 % and 3.5 % for moisture content and temperature, respectively, which confirmed the
375
high accuracy of the presented model.
376
One can conclude from the Figs. 4, during drying process, moisture content of apple slices
377
reduces rapidly as the warming up the apple slices. The basic reason for this event is the high
378
evaporation at the surface due to the considerable diffusion rate of the internal water towards the
379
surface. Then, the drying rate decreases due to the high internal mass resistance and the lowering
380
the progress of the internal moisture migrates to the surface. At this stage the internal
381
temperature of the apple slices is close to the air drying temperature. This result confirmed the
382
behavior reported by Villa-Corrales et al. (2010), Lemus Mondaca et al. (2013) and Tzempelikos
383
et al. (2015).
384
It should be noted that the CPU time (or the computational time) for the simulation by assuming
M AN U
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370
"z = "{ = "| = 5 and "q = 12 was about 41 seconds that confirms high rapidity and low
386
computational cost of the proposed numerical method.
387
Fig. 5 shows the comparison between of the experimental data and the numerical simulations
388 389
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385
associated to different values of collocation points, ("q = 10, 12, 14, 16 and ("z = "{ = "| = 6)). Since the temporal interval of this problem is greater than the spatial intervals, we have
assumed larger values of "q with respect to the "z , "{ and "| . It can be observed that, we reach
391
to similar numerical results by increasing the collocation point which indicate the high accuracy
392
and robustness of the suggested numerical approach. Moreover, we can conclude that, only by
393
using small number of collocation nodes yields to accurate numerical results. The slight
394
deviations observed between the simulated and experimental data are probably due to the errors
395
in true performance of the implemented dryer and the lack of accounting shrinkage effects and
396
food properties variation in the model.
397
The numerical model introduced in this paper made reasonable results in prediction of moisture
398
and temperature distribution inside the apple slices at different times during drying. This
399
numerical model could be implemented for other dimensions (one and two), other coordinate
400
systems, such as cylindrical or spherical coordinates and other drying conditions.
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Fig. 6 shows the validation of the present work with experimental data of Kaya et al. (2007) and
402
Zarein et al. (2013). Kaya et al. (2007) evaluated the convective air drying of thin layer apple
403
slices at drying air temperature 60°C. In this study, the relative humidity and air drying velocity
404
were 40% and 0.2 m/s respectively. Zarein et al. (2013) have investigated the drying of apple
405
slices in cylindrical geometry at 80oC, air velocity 1 m/s and relative humidity 24%. It can be
406
seen that a reasonably good agreement exists between experimental data and numerical values
407
that confirm the accuracy of the proposed method for simulating different drying conditions.
408
Nevertheless, the proposed approach is a global approximation method (derivative at a given
409
point depend on all of the mesh points) which thus has disadvantage of requiring the inversion of
410
large system matrices. To remedy this inconvenient, in this paper we have used some robustness
411
iterative algorithms such as GMRES for solving the associated systems of linear algebraic
412
equations, in which computing the inversion of large system of matrices is avoided (Ballestra
413
and Cecere, 2016).
414
GMRES (generalized minimal residual method) is an iterative method for solving nonsymmetric
M AN U
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401
415
systems of linear algebraic equations in the form of ® = , that approximates the inverse of
416
by some vectors in a Krylov subspace with minimal residual (Saad, 2003).
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417
5.3. Drying process simulation
419
In Figs 7 and 8, the simulated averaged moisture content and the temperature of the apple slices
420
are plotted. Comparison between moisture content curves at different drying air temperature in
421
Fig 7, shows that the moisture elimination is faster at higher drying air temperatures. This
422
behavior strongly is associated to the temperature dependency of moisture effective diffusivity.
423
This can also be illustrated in Table (3), which by increasing the drying air temperature and
424
higher temperature differences between air drying and apple slices, the heat flux to the interior of
425
the apple slices is higher and consequently the values of diffusivity coefficients increases
426
(Tzempelikos et al., 2015).
427
As can be seen in Fig. 8 at the initial stages of the drying process, the temperature of apple slices
428
does not reach close to the equilibrium temperature. This behavior may be related to the
429
evaporative cooling effect owing to the high moisture flux on the apple surface during the initial
430
stages of drying process. The high moisture transfer on the food surface during this period
431
requires more energy for moisture evaporation and hence, less heat transferred within the apple
AC C
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418
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slices initially. Then, by increasing the drying times and decreasing the moisture flux to the apple
433
surface, the temperature of apple slices is gradually increased and approaching close to the
434
equilibrium with the drying air temperature.
435
The comparison between temperature and moisture value curves in Figs 7 and 8 shows, the
436
temperature curves change earlier than moisture content curves during drying process. In other
437
words heat transferred more rapidly than moisture within the apple slices. Therefore, the
438
temperature convergence to drying condition is quicker than moisture content. This rapid
439
convergence of temperature than moisture is due to the reality that the rate of heat transfer is
RI PT
432
higher than mass transfer or the introduced Lewis number is higher than one 1, N > 1.
441
(Vahidhosseini et al., 2016).
442
One of the specific advantages of numerical simulations of heat and mass transfer equations is
443
providing the high spatial resolution information of temperature and moisture content
444
distribution during the drying process, whereas in practice it is difficult to measure the spatial
445
distribution of moisture inside the material and requires specialized equipment and resources
446
(Aregawi et al., 2013).
447
Fig. 9 shows the spatial distribution of the temperature and moisture content inside the apple
448
slice, as predicted by the numerical model. As can be seen in Fig. 9, moisture diffusion occurred
449
from the inside of the apple slice with high concentration to its boundaries with low
450
concentration. Also the apple slices warming up from its boundaries to the inside, which these
451
phenomena matches with the Fick’s laws of diffusion and Fourier law, respectively. Moreover, at
452
the beginning of the drying process, the moisture and temperature gradients are high. Then
453
moisture and temperature gradients decrease with the advances of the drying process until the
454
internal temperature and moisture pressure of the apple slice will be reaching to the similar
455
condition of the air drying (Janjai et al., 2008; Villa-Corrales et al., 2010; Lemus Mondaca et al.,
456
2013).
457
Also it can be concluded that the temperature profile has slow gradients along the apple slices.
458
This specific behavior is basically imputed to the low heat transfer Biot number, which range
459
between 0.37 and 0.53. These values are of the same order of magnitude compared to the limit of
460
“thermally-thin-materials” set to the order of 0.1. ((Incropera et al., 2007, Tzempelikos et al.,
461
2015). It should be recalled that, the Biot number is a dimensionless group that relates a measure
AC C
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440
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462
of the rate of internal heat conduction or mass diffusion in object with a measure of the rate of
463
external convection and are useful to identify controlling mechanisms (Giner et al., 2010).
464
In addition, the values of the mass transfer Biot number for different drying air temperature were
found to be in the ranges of 1.34 × 10Š to 1.5 × 10Š which is compatible with the existence of
466
strong spatial gradients of moisture content inside the apple slices. The mass Biot number higher
467
than 100, illustrates that external resistance to mass transfer is negligible and thus the effective
468
moisture diffusivity is not influenced by external drying conditions (Srikiatden and Roberts,
469
2006).
RI PT
465
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470
6. Conclusion:
472
Jacobi spectral collocation method was proposed for simulating distribution of 3D coupled
473
temperature and moisture content inside the apple slices numerically. The presented numerical
474
method needs less computational time with respect to the analytical methods, by making use of
475
the operational matrices in computations. The results of simulations illustrates that the 3D model
476
of the coupled heat and mass transfer provide a better understanding of the transport processes
477
inside the apple slices during drying process. This numerical model can be implemented in
478
automatic control of convective air dryers and energy optimization of dryer operation and
479
improving dryer design. So it is considered a great potential benefit for evaluating the
480
engineering process.
483
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484
Acknowledgments
485
The author thanks from the editor and reviewers of this paper for their constructive comments
486
and nice suggestions, which helped to improve the paper very much.
487 488
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Figure captions:
Fig. 1. Time–temperature (dimensionless) plot for determination of heat transfer coefficient of aluminum cubes. Fig. 2. Apple slices is subjected to convective air drying.
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Fig. 3. (A) Velocity and (B) temperature contour around the moist object at XY plane.
Fig. 4. Comparison of the predicted temperature and moisture data with the experimental results during drying at 60 and 90 °C.
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Fig. 5. Comparison of the numerical simulations of the moisture and temperature associated to 16 and the experimental results.
= 10, 12, 14 and
Fig. 6. Comparison of the experimental data with simulated moisture content.
Fig. 7. Moisture content numerical values in apple slices at selected temperatures. Fig. 8. Temperature numerical values in apple slices at selected temperatures.
Fig. 9. Temperature (Right figures) and moisture (Left figures) distributions inside the apple slice at XY plane at
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90oC for different drying times (a=1500 s, b=5000 s, c=11500s).
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Table 1 Thermophysical properties of the apple slices )
850.47 857.78 858.71 856.25
(
)
0.559 0.565 0.561 0.561
(
)
3733.59 3773.07 3752.53 3750.57
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60 70 80 90
(
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Tair(°C)
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Table 2 Convective heat and mass transfer coefficients
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0.009347 0.010545 0.012849 0.014128
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60 70 80 90
( )
( ) 10.372 11.525 13.891 14.982
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(° )
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Table 3 Effective moisture diffusivity coefficient ( ) -10
0.8149 0.7783 0.9113 0.8514
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7.424×10 9.07×10-10 9.3×10-10 1.023×10-10
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60 70 80 90
(
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0/5
-0/5
0
200
400
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800
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1400
-1
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-1/5 -2 y = -0/0026x - 0/0999 R² = 0/9971
-2/5 -3 -3/5 -4
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Time (s)
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Ln[(Tal-Tair)/(Tin-Tair)]
0
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1.2
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0.6
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Drying time(s)
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4000
6000 Drying time(s)
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Experimental data(60oC)
0.7 0.6 0.5
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90oC
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•
A 3D model is assumed for predicting temperature and moisture transfer numerically.
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A robust and high accurate numerical scheme for solving 3D coupled heat and mass transfer process has been proposed. Operational matrices of differentiation are used because of reducing symbolic computations w. r. t. the analytical methods
•
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Extension and modification of the presented spectral method can be applied for
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simulation other phenomena in Food Engineering.