A FEM Analysis of Transport Phenomena Occurring During Vegetables Drying

Excerpt from the Proceedings of the COMSOL Users Conference 2006 Milano
A FEM Analysis of Transport Phenomena Occurring During
Vegetables Drying

S. Curcio1
1Department of Chemical Engineering and Materials – University of Calabria
*Corresponding author: Ponte P. Bucci – cubo 45/a – 87030 Rende (CS) – ITALY.
E-mail: stefano.curcio@unical.it


Abstract: The aim of the present work is the
moisture content and an increase of its
formulation of a theoretical model describing
temperature. The design and the operation of
the transport phenomena involved in food
industrial drying ovens are, in most cases,
drying process. The attention has been,
based on the utilization of semi-empirical
specifically, focused on the simultaneous
correlations, necessary to calculate – by means
transfer of momentum, heat and mass
of a set of estimated transport coefficients –
occurring in a convective oven where dry and
heat and moisture fluxes between food and air.
hot air flows about a wet and cold food
Different physical, mathematical and
sample. The proposed model represents a
numerical methods have been proposed to
general and predictive tool capable to describe
describe the drying process. Nevertheless,
the real ovens behavior over a wide range of
many different aspects can be still improved
process and fluid-dynamic conditions. In fact,
with respect to the available literature data.
it receives, as inputs, only the initial
These aspects regard the theoretical
conditions, the geometrical characteristics of
formulation of the transport model, the
either the food or the drying chamber and the
numerical procedures used to solve the
expressions of physical and transport
governing equations, a proper definition of the
properties (of both air and food) formulated as
transport properties of both air and food
functions of the local values of temperature
material and the evaluation of heat and mass
and moisture content. The resulting system of
transfer coefficients performed on the basis of
non-linear unsteady-state partial differential
suitable and accurate experimental procedures.
equations has been solved by means of Comsol
An exhaustive analysis is, however, too
Multiphysics with reference to a symmetrical
onerous in terms to properly analyze the
system consisting of a cylindrical-shaped
complex transport phenomena involved in food
vegetable placed in a cylindrical oven.
drying. Therefore, simplified approaches have

been proposed and are still used by process
Keywords:
Food Drying, Transport
designers.
Phenomena, Finite Elements Method, Comsol
Hernàndez, et al. considered the fruits
Multiphysics
drying process as isothermal assuming drying

temperature equal to air temperature and
1. Introduction
accounting for mass transfer only [1]. This

approach was chosen also by Simal, et al. on
The main objective of food drying process
Aloe Vera [2] and by Ben-Yoseph et al. for
is water removal up to a particular moisture
sugar films [3]. Wu & Irudayaraj
content in order to prevent food from microbial
experimentally verified that drying can be
spoilage and deterioration reactions and to
actually supposed as an isothermal process
increase the product shelf life and its safety.
only if the Biot number is very low. When Biot
Forced convection by hot and dry air that
number is, instead, high, internal transport
flows about a wet and cold food sample is the
resistances are also to be considered [4].
most common industrial technique to perform
Ikediala et al. developed more precise and
food drying. Typical values of air temperature
rigorous models that take into account the
range between 40°C and 80°C, whereas air
simultaneous heat and mass transfer within the
velocity generally ranges from 0.5 to 5 m/sec,
food; their model were, for instance, applied to
reaching, in some cases, the value of 10
simulate the cooking process of chicken patties
m/sec; drying time depends on these and other
[5]. Ahmad et al. modeled the transport
parameters, i.e. air humidity and food
phenomena involved in biscuit drying by
characteristics, and it can last up to nearly 20
microwaves ovens accounting for both the
hours. The difference either in temperature or
transfers [6]. Wang & Brennan developed a
in water content promotes a simultaneous
mono-dimensional model for the simultaneous
transfer of heat and water between food and air
heat and mass transfer within potatoes slices
that determines a progressive decrease of food
[7]. The hypothesis of mono-dimensional

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Excerpt from the Proceedings of the COMSOL Users Conference 2006 Milano
transport was experimentally verified in the
that has to be calculated solving the
same study and used by other authors, too [8],
momentum transport coupled to the continuity
[9], [10]. The influence of some of the most
equation. Actually, water vapor transfer by
important operating variables, i.e. humidity
diffusion within dehydrated material to the
and temperature of the drying air, on the
external food surface should also be
performance of vegetables drying process has
considered. However, this mechanism is
been analyzed by means of a transport model
significant especially for highly porous media,
describing the simultaneous bi-dimensional
whereas for vegetables, typically characterized
heat and moisture transfer in the food [11]. The
by void fraction lower than 0.3, can be
authors solved the resulting system of
neglected [13].
unsteady-state partial differential equations by
In this study paper, air velocity field, water
FEMLAB. Since air and food physical
and heat transfers (for both air and food) were
properties may be expressed as functions of
modeled by transient momentum, mass and
temperature and moisture content, being heat
energy balances, with reference to the drying
and mass fluxes coupled together in the
process of cylindrical vegetables. Evaporation/
boundary conditions, the equations describing
condensation effects at the air-food interface
heat and mass transfer form a set of non linear,
were, instead, considered by a set of proper
partial differential equations that can be solved
boundary conditions expressing the continuity
only by numerical procedures. Both finite
of temperature, heat and mass fluxes, whereas
difference (FD) and finite element (FEM)
a thermodynamic equilibrium relationship was
methods have been widely used for this
needed to relate the water concentration in the
purpose. FEM is particularly suitable for
gas phase and the moisture content on food
investigation on domains characterized by
external surfaces. No empirical correlation
irregular geometries in presence of complex
was, therefore, necessary to estimate heat and
boundary conditions and for heterogeneous
mass transfers at the food-air interface.
materials. Nevertheless, it is more complex
The model is based on the following
and computationally expensive than FD [12].
hypotheses:
In the present work, the attention has been,
• Heat and mass transfer in the food occur,
specifically, focused on the simultaneous
respectively, by conduction and diffusion
transfer of momentum (for air only), heat and
only;
mass (for both air and food) occurring in a
• Heat and mass transfer in the air occur also
cylindrical-shaped convective oven. The
by convection;
proposed model represents a general and
• Air is considered as an ideal gas and heat
predictive tool capable to describe the real
generation due to friction is negligible;
ovens behavior over a wide range of process
• The drying air is supplied continuously to the
and fluid-dynamic conditions, since it is not
oven and flows in the axial direction;
based on any semi-empirical correlations for
• Transport of water vapor within dehydrated
the estimation of heat and mass fluxes at food-
material is negligible;
air interface. The main objective of this work
• The effects of food shrinkage were not
is to develop an accurate transport model that
considered;
can be used to define, for different types of
• Heat and mass transfer resistance are
foods, the “optimal” set of operating
insignificant across the net on which food is
conditions in each particular situation. In this
placed [14], [15].
way, it might be possible to minimize
A symmetric system (Fig. 1) was
expensive pilot test-runs and to have good
considered for the analysis of the transport
indications on the characteristics and the
phenomena involved in food drying process.
quality of dried products.

2. Theoretical background

When a moist and cold food is put in
contact with dry and warm air, two different
transport mechanisms simultaneously occur:
the heat transfer from air to the material and
the transfer of water from food to air. Within
the solid material, heat transfer by conduction
and water transfer by diffusion take place. In

the air, also the convective contributions to

heat and mass transfer have to be considered.
Figure 1. Schematic of the convective oven under
This effect is caused by air circulation velocity
consideration.

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Excerpt from the Proceedings of the COMSOL Users Conference 2006 Milano
It is, actually, a cylindrical domain; the
reported, for sake of brevity, in the following
food sample is supposed to have a diameter of
fig. 2.
1 cm and a length of 6 cm, the drying chamber,
instead, has a diameter of 20 cm and is 0.25 m
long.

3. Governing equations

In the following, the governing equations
originating from the above hypotheses are
presented. The momentum balance coupled to
the continuity equation in the drying air lead to
[16]:
? ? =???? u
? t

? ? u =???? uu? ? p?? ? + ? g
? t

where ? is the density of air, expressed in
terms of the local values of temperature and of
water concentration, p is its pressure, u is the

air velocity vector.
Figure 2. Boundary conditions for the system of
The unsteady-state energy balance
PDEs modeling drying process.
equations for both air and food, based on

Fourier’s law, write – respectively – as [16]:
For some of the above boundary conditions
a deeper and more detailed discussion is
? ? Cˆ T
ˆ
p
needed. At food-air interface, where no
= ? ? ? (? k?T + ? Cˆ T u +
+ ?
p
) Dp
DC
T
p
? t
Dt
Dt
accumulation occurs, the continuity of heat

? ?
fluxes actually considers that the heat
Cˆ T
ˆ
2
p2 2 = ???(? k ?T + ?

transported by convection and conduction from
2
2 )
C
D
T
p2
? t
2 2
Dt
air to food is partially used to raise sample
where T is air temperature, Cˆ is its heat
temperature by conduction and partially to
p
allow water evaporation. The latter effect is
capacity and k its thermal conductivity, T2 is
described by considering water latent heat of
food temperature, ?
ˆ
2 is its density, C
is food
p2
vaporization that is expressed in terms of the
heat capacity, and k
interfacial food temperature and of its moisture
2 its thermal conductivity.
The unsteady-state mass balance equations
content. Moreover, a thermodynamic
referred to the water contained in both air and
equilibrium relationship between water
food and based on the Fick’s law, write –
concentration in the gas phase and water
respectively – as [16]:
concentration on food external surfaces has
been used to calculate the values of c
?
eq.on
c = ?? ? (? D?c+ cu)
boundaries 7, 8, and 9:
? t


? c
p?y
0?a
2 = ? ? ? (? D ?c
s = Ps
ws
2
2 )
? t


0
where c is the water concentration in the
where Ps , ys and aws are the vapor pressure,
air and D is the diffusion coefficient of water
the molar fraction, and the activity of water on
in air, c
food external surfaces, respectively. The above
2 is the water concentration in the food
and D
equation is fundamental to properly describe
2 is the effective diffusion coefficient of
water in the food. It should be observed that
the complex steps involved in the drying
the physical and transport properties for either
process. In fact, water activity aws decreases as
air or food are expressed in terms of the local
food moisture content decreases so that a unity
values of temperature and moisture content.
value of aws corresponds only to the free water
The initial values, necessary to perform the
evaporation. When aws < 1, also bounded water
numerical simulations, regard the conditions
is removed. Water activity is a distinctive
of: both air (i.e. humidity, temperature,
parameter of each type of food that, as a
velocity field and pressure) and food (i.e.
function of its own structure, determines how
moisture content and temperature) before
strong the bonds between food and water are.
drying process takes place. As far as the
The boundary conditions used on boundary 2
boundary conditions are concerned, they are
for both energy and mass balances are valid

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Excerpt from the Proceedings of the COMSOL Users Conference 2006 Milano
under the hypothesis that the penetration of
the drying chamber. It is interesting to observe
either temperature or concentration profiles is,
the differences between the front side of food
actually, confined in two very thin regions (the
sample, where air impinges and its rear side
boundary layers) that develop close to the
where a very wide region characterized by low
food-air interface. The above assumption will
values of velocity forms. This phenomenon
be validated or falsified when the simulation
affects the drying rate that is, generally,
results will be analyzed more in detail.
different for each of the surfaces exposed to air
The above system of non-linear Partial
since it depends on the local values of velocity
Differential Equations, together with the
that has a strong influence on both heat and
already described set of initial and boundary
mass transfer at food-air interface.
conditions, has been solved by Finite Elements
Method implemented by Comsol Multiphysics
3.2. The integration domain has been
discretized into 13193 finite elements (see
Fig.3); the resulting number of degrees of
freedom was equal to about 100000. A drying
process duration of 5 hours was, averagely,
simulated in 4 hours on a AMD Athlon(tm),
1,24 GHz Personal Computer running under
LINUX.

Figure 4. Velocity field [m/s] in the air flowing
about the cylindrical carrot sample (u0 = 1 m/s).

In the following Figs. 5-6 temperature and
water concentration profiles in the air, but in a
zone close to the food sample, are shown. The
simulation results confirm that the penetration
of both temperature and concentration profile

is, actually, confined in two thin regions that
Figure 3. Discretization by triangular finite
elements and detail of the mesh.
develop at the air-food interface.



4. Results and Discussion

In the following, some of the most
interesting results obtained by the proposed
model are presented. All the simulations have
been performed assuming that the food sample
is a carrot whose initial temperature and
moisture content (on a wet basis) are equal to
300 K and 87%, respectively. The physical and
transport properties characteristic for the carrot
were expressed in terms of the local values of
temperature and concentration according to the
relationships proposed by Ruìz-Lopez et al.
[17]. The initial conditions for the air,
supposed stagnant, were a temperature of 343
K, and a relative humidity equal to 9.1%. The
proposed model can describe, at any time, all

the profiles in both domains. The following
Figure 5. Temperature [K] profiles in the air (t=
Fig. 4 shows air velocity field that develops in
1200 s)

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Excerpt from the Proceedings of the COMSOL Users Conference 2006 Milano


Figure 6. Water concentration profiles [mol/m3]
Figure 8. Water concentration profiles
in the air (t= 1200 s)
[mol/m3] in the carrot at different drying times


The greatest part of the drying chamber is,
Moreover, it is interesting to notice the
unaffected by the presence of a cold and moist
mobile fronts of both temperature and
solid since it is characterized by constant
concentration in the solid that, as time goes by,
values of either temperature or humidity.
advance either in axial or in radial direction
Moreover, at the rear of the sample, where air
determining a progressive increase of food
circulation is rather limited, temperature tends
temperature and a continuous decrease of its
to increase more slowly than, for instance, at
humidity.
the front side.

The following two figures (Figs. 7 and 8)
5. Conclusions
show the time evolution of both temperature

and concentration profiles in the food sample.
In the present paper a general transport
It can be observed that the carrot front side,
model, describing the drying process
where drying air impingement occurs,
performed in a convective oven, has been
becomes hotter and dryer sooner than the other
formulated and solved by Comsol
exposed surfaces and even much faster than
Multiphysics 3.2. The influence of the most
the inner regions of food sample.
important operating variables on system

performance has been evaluated without
resorting to any empirical correlation for the
estimation of heat and mass transfer at food-air
interface. The developed model could support
process designers either to evaluate product
quality and its dependence on operating
variables or to estimate process economics.
The model will be improved in the future
by accounting also for the volume variation
(shrinkage effects) that takes place during food
drying process. The work is in progress and a
new model formulation, based on the coupling
of the ALE method to the above-described
transport equations, is currently under
investigation.



6. References


Figure 7. Temperature [K] profiles in the carrot
1. Hernàndez, J.A., Pavòn, G., Garcìa, M.A.,
at different drying times
Analytical Solution of Mass Transfer Equation

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Excerpt from the Proceedings of the COMSOL Users Conference 2006 Milano
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The author wants to thank Ing. Maria
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