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# Heat and Mass Transfer in Convective Drying Processes

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A dynamic mathematical model, based on physical and transport properties and mass and energy balances, was developed for the simulation of unsteady convective drying of agricultural products (fruits and vegetables) in static bed conditions. The local material- averaged drying rate and the heat flux depend on local air humidity and temperature, as well as local mass and heat transfer coefficients in interaction with the moisture and temperature distribution inside the material. The model utilizes water sorption isotherm equations and the change in solid density due to the shrinkage phenomenon. The aim of this article is to describe the modelling and simulation of the dehydration of grapes in a complex drying system processes, using COMSOL Multiphysics Program.
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Excerpt from the Proceedings of the COMSOL Conference 2008 Hannover
Heat and Mass Transfer in Convective Drying Processes

Camelia Gavrila*,1, Adrian Gabriel Ghiaus2, Ion Gruia3
1,2Technical University of Civil Engineering Bucharest, Faculty of Building Services, Romania
3University of Bucharest, Faculty of Physics, Romania
*Camelia Gavrila: 66 Pache Protopopescu Blvd, Sector 2, 021414 Bucharest, Romania,
cgavrila2003@yahoo.com

Abstract: A dynamic mathematical model,
• liquid transport due to capillary
based on physical and transport properties and
forces,
mass and energy balances, was developed for the
• vapour diffusion due to shrinkage and
simulation of unsteady convective drying of
agricultural products (fruits and vegetables) in
law),
static bed conditions. The local material-
• liquid or vapour transport due to the
averaged drying rate and the heat flux depend on
difference in total pressure caused by external
local air humidity and temperature, as well as
pressure and temperature (Poiseuille’s law),
local mass and heat transfer coefficients in
• evaporation and condensation effects
interaction with the moisture and temperature
caused by differences in temperature,
distribution inside the material. The model
• surface diffusion in liquid layers at the
utilizes water sorption isotherm equations and
solid interface due to surface concentration
the change in solid density due to the shrinkage
• liquid transport due to gravity.
describe the modelling and simulation of the
dehydration of grapes in a complex drying
transported inside a material if a suitable
system processes, using COMSOL Multiphysics
Program.
effect), because of thermodynamic coupling of

heat and mass transport processes.
Keywords: heat transfer, mass transfer
Most foods are classified as capillary
convective drying processes, numerical model,
porous rigid or capillary porous colloids, (Bruin,
COMSOL.
1980). Therefore, it is often proposed that a

combination of capillary flow and vapour
1. Introduction
diffusion mechanisms should be used to describe

internal mass transfer.
Dehydration involves the simultaneous
Water activity, rather than moisture
transfer of heat, mass and momentum in which
content, influences biological reactions. In the
heat penetrates into the product and moisture is
regions of water adsorption on polar sites or
removed by evaporation into an unsaturated gas
when a mono-molecular layer exists, there is
phase. Owing to the complexity of the process,
little enzyme activity. Enzyme activity begins
no generalized theory currently exists to explain
only above the region of mono-molecular
the mechanism of internal moisture movement.
adsorption. When the moisture content of a
Although it is now accepted that in most
substrate is reduced below 10 %, micro-
practical situations of air drying of foods the
organisms are no longer active. It is necessary
principal rate-determining step is internal mass
however to reduce the moisture content to below
transfer, there is no agreement on the mechanism
5 % in order to preserve nutrition and flavour.
of internal moisture movement. In the case of

capillary-porous materials such as fruits and
2. Mathematical modelling of drying
vegetables, interstitial spaces, capillaries and
processes
gas-filled cavities exist within the food matrix

and water transport takes place via several
The simulation of various product drying
possible mechanisms acting in various
systems involves solving a set of heat and mass
combinations. The possible mechanisms
transfer equations which describe:
proposed by many workers include:
a) heat and moisture exchange between
• liquid diffusion caused by
product and air,

b) adsorption and desorption rates of heat
specific heat capacity; ? - thermal conductivity;
and moisture transfer,
D - effective mass diffusivity.

c) equilibrium relations between product
eff
and air,
These considerations lead to complex partial

d) psychometrics properties of moist air.
differential equations for the moisture content

Attempts to describe the manner whereby
and temperature fields inside the product. These
moisture is dislodged and evaporates from many
equations incorporate transport coefficients
common materials involve formidable problems
which must be determined experimentally, and
and analysis, and are usually tractable only for
are strong functions of moisture content.
constant drying conditions. However, process

conditions almost always vary from place to
2.2 Modelling of agricultural product
place in a dryer and, in case of batch drying, they
properties
also change with time. It is thus useful to

describe a body of comparatively simple
Variability in composition and physical
structure, through which the movement of
characteristics is typical for all food products.
moisture can be analyzed or experimentally
For example, the composition of fruits and
modelled in a straightforward way, so that the
vegetables depends on variety, location grown,
drying behaviour can be predicted for conditions
climatic conditions, etc. For most engineering
more representative of those in commercial
heat transfer calculations performed in
equipment. This procedure, although very
commercial food dehydration, accuracy greater
approximate in a quantitative sense, nevertheless
than 2-5 % is seldom needed. This is because
provides a number of important clues about
errors due to varying or inaccurately measured
drying behaviour in general and strategies for
boundary conditions such as air temperature and
process operation.
velocity, would overshadow errors caused by
For air-drying of root vegetables (e.g.
inaccurate thermal properties.
carrots, potatoes, sweet potatoes), the core drying

The best sources of thermal property data are
model, which formulates the relation between the
prediction equations based on chemical
changes of the surface areas and moisture
composition, temperature and physical structure
content, assumes the formation of a dried layer at
(density, porosity, size and configuration of void
the outer side of the sample and the existence of
spaces).
an un-dried core at the centre. This model was

Most thermal property models are empirical
found to be in good agreement with the
rather than theoretical, i.e. they are based on
experimental data.
statistical curve fitting rather than theoretical

derivations involving heat transfer analysis. In
2.1 Simultaneous transport of heat and mass
modelling, water is treated as a single, uniform

component of the food product. It could be
The most rigorous methods of describing the
argued that the thermal properties of water in the
drying process are derived from the concepts of
food depend on how it is configured or “bound”
irreversible thermodynamics in which the
within the product.
various fluxes are taken to be directly

proportional to the appropriate “potential”,
Table 1: The parameters used for the Corinthian
(Ghiaus, 1997). The mass balance inside the
grapes
product can be written as:

(??b ?X)
?
?X
=
?
Item Value
div ? ? D
?
?
b
eff
t
?
?
?z ?? (1)
and the heat-energy balance can be set down as:
Water content
75
?
W, %
T
?
?T
?
?
b ? c ?
= div ? ?
?t
?
?
?z ?? (2)
Thermal conductivity
0.5721
?, W/m K
Where X - grape moisture content; T -the
air temperature; ? - bulk bed density; c -
b

Specific heat
until the end of the process. During the first 5
3600
c, J/kg K
hours the temperature gradient is high and
corresponds to the so-called warm-up period of
Effective diffusion
drying. During the next 25 hours the temperature
3.6·10-9
Deff, m2/s
remains practically constant, and during the last
period it starts to increase again. At the end of
bulk bed density
691
the process, the temperature of the grapes
?b kg/m3
reaches 42.5 °C.

3. Results and Discussion

40
The COMSOL Mutiphysics program is used to
simulate the dehydration of grapes in a complex
Co
35
drying system processes which correspond to the
u
r
e,
numerical solution of these model equations. The
p
e
r
at
m 30
above system of non-linear Partial Differential
Te
surface of grape bed
average
Equations, together with the already described
25
bottom of grape bed
set of initial and boundary conditions, has been
0
5
10
15
20
25
30
35
solved by Finite Elements Method implemented
Time, h

by Comsol Multiphysics 3.4. We build the
Figure 2. Evolution of grape temperature during the
geometry of the model, and then we fixed the
drying process.
boundary settings, the mesh parameters and

compute the final solution (Figure 1)
Figure 3 presents the evolution of grape moisture

content at the surface and bottom of the bed and
as an average value. It can be seen that during
the whole process the moisture content of the
grapes is uniform within the bed thickness. This
is due also to the small thickness of the grape
bed.

3.0
surface of grape bed
average
2.5
bottom of grape bed
/
kg-db 2.0
e
n
t
,

kg
1.5
o
nt
1.0
o
i
s
t
u
r
e
c
M 0.5

0
5
10
15
20
25
30
35
Figure 1. Results from the compute solution in
Time, h

COMSOL Multiphysics.

Figure 3. Evolution of grape moisture content during
Figure 2 shows graphically the evolution of
the drying process.
grape temperature during the drying process, at

the surface and bottom of the bed and as an
One of the most important drying parameters
average. the predicted drying time was calculated
is the drying rate (Figure 4) which represents the
to be 38 hours and 21 minutes during which the
rate of evaporated water from one square meter
grapes are dried from 75 % moisture content -
of drying product. At the beginning of the
wet basis to 15 % moisture content - wet basis.
process the drying rate increases from 0.06 g/s
Differences of temperature between the base
m2 to 0.12 g/s m2 and then has a very small
and surface of the bed appear only during the
decreasing slope. At the end of the process the
first period of drying, approx. the first 5 hours.
drying rate decreases rapidly.
After this the bed temperature remains uniform

4. Conclusions
0.14
0.13

0.12
2
In this paper, we have demonstrated the
0.11
/
s m
, g 0.10
versatility of COMSOL Multyphisics with
a
te
r 0.09
regard to the modelling and simulation of the
r
y
i
ng
0.08
D
dehydration of grapes in a complex drying
0.07
0.06
system processes. The model was applied to the
0
5
10
15
20
25
30
35
full scale experimental data with good results.
Time, h

Figure 4. Drying rate vs. drying time for grape
dehydration.
5. References

As the heat exchangers processes do not involve
1. Ghiaus A.-G., Margaris D.P. and Papanikas
mass (water vapors) transfer, the absolute
D.G., Mathematical modelling of the convective
humidity values of the preheated air are equal to
drying of fruits and vegetables, Journal of Food
those of the ambient air, the values of exhaust air
Science, An international journal of the Institute
are the same as those of the air at the outlet of
of Food Technologists, 62, 1154-1157, (1997).
the drying room and also, the values are identical
2. Ghiaus C.-M. and Ghiaus A.-G., Evaluation of
at the inlet and outlet of the main heat exchanger.
the indoor temperature field using a given air
Drying air parameters are predicted at
velocity distribution, Building and Environment
characteristic points of the system: inlet of the
The International Journal of Building Science
fresh air into the economizers (ambient air),
and its Applications, 34, 671-679, (1999).
outlet of the economizers onto the fresh air path
3. Hernàndez, J.A., Pavòn, G., Garcìa, M.A.,
(preheated fresh air), inlet of the main heat
Analytical Solution of Mass Transfer Equation
exchanger (the mixing between preheated and
Journal of Food Engineering, 45, 1-10, (2000).
recycled air), inlet and outlet of the drying room,
4. Wu, Y., & Irudayaraj, J., Analysis of Heat,
and the outlet of the economizers onto the
Mass and Pressure Transfer in Starch Based
exhaust air path. Evolution of drying air
Food Systems, Journal of Food Engineering, 29,
temperature is given in Figure 5 and the relative
399-414, (1996).
humidity of the drying air in Figure 6.
5. Ikediala, J.N., Correira, L.R., Fenton , G.A., &

Abdallah, N.B., Finite Element Modelling of
Heat Transfer in Meat Patties During Single-
70
inlet drying room
Sided Pan-Frying, Journal of Food Science, 61,
65
outlet drying room
inlet heat exchanger
60
796-802, (1996).
C 55
o
50
6. Morgan, M.T., & Okos, M.R., Effects of
u
r
e,
45
erat
Microwave on the Drying, Checking and
40
e
mp
T
preheated
35
Mechanical Strength of Baked Biscuits Journal
exhaust
30
ambient
of Food Engineering, 50, 63-75, (2001).
25
0
5
10
15
20
25
30
35
7. Wang, N., Brennan, J.G., A Mathematical
Time, h

Model of Simultaneous Heat and Moisture
Figure 5. Drying air temperature at different locations
Transfer During Drying of Potato, Journal of
vs. drying time.
Food Engineering, 24, 47-60, (1995).

8. Kalbasi, M., Mehraban, M.R., The Effect of
exhaust
Surface Water Vapour Flux on Drying of Potato,
70
ambient
outlet drying room
Journal Trans IChemE, 78, Part C, (2000).
60
50

i
d
i
t
y
,
%
u
m
40
inlet heat exchanger
e
h
6. Acknowledgements
preheated
tiv 30
inlet drying room
e
la

R
20
This work was supported by Technical
10
University of Civil Engineering Bucharest under
0
5
10
15
20
25
30
35
Time, h

the CEEX contract No 73/2006, project director:
Figure 5. Relative humidity of drying air at different
locations vs. drying time.

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