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Heat and mass transfer during drying process

Article in International Journal of Heat and Technology · January 2008

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 Heat and Technology
Vol. 26, n. 2, 2008

HEAT AND MASS TRANSFER DURING DRYING PROCESS

*Dalel Helel, **Noureddine Boukadida

U.R.E.E, Institut Supérieur des Sciences Appliquées et de Technologie de Sousse, 4003 Ibn Khaldoun, Sousse
Tunisia .
Email: *dalel.helel@enim.rnu.tn **noureddine.Boukadida@issatso.rnu.tn

ABSTRACT

The aim of this work is to study the mechanism of heat and mass transfer during drying process of an unsaturated porous
medium having clay brick characteristics witch is placed in horizontal channel. The porous medium is submitted to a forced
convection laminar flow. We particularly bring out the space time variability effect of heat and mass transfer coefficients on
space-time evolution of temperature, water saturation and gas-pressure in the porous medium, in the solid-fluid interface and
in the channel during the drying process.

1. INTRODUCTION calculated and local coefficients of thermal and mass transfer

are deduced.
The understanding of coupled heat and mass transfer in a Heat and mass transfers are two components holding an
porous media submitted to thermal forced convection has important role on the drying time. There are many works that
been the subject of several theoretical and experimental showed the effect of coefficients of heat and mass transfer
scientific studies and is motivated in more applications and and the thermo-physical characteristics on the drying kinetics
intervenes in varied domains as the energy storage, the [9]. Other works studied the ambient parameters effects
extraction of oil, the thermal drying, etc. During drying (pressure, temperature, vapour concentration) on the process
operation, many problems are encountered as high energy of drying while considering some constant coefficients of
consumption, cracking, colour change, aroma, chemical transfer [10]. Beeby et al. [11] discuss the advantages and
composition change, kinetic of drying and the final shape of disadvantages of using superheated steam over hot air as a
the product. In fact the improvement of technical system drying medium. Many authors reported that in certain cases,
drying is becoming a necessity. So among the main the quality of the product can be enhanced. Yoshida et al.
preoccupations of industrial and researchers is therefore to [12] demonstrated that superheated steam can provide an
optimize drying way and new techniques to obtain an optimal excellent medium for drying food products. In this domain of
drying system permitting to save energy and improve the drying by forced convection, studies are relatively numerous,
quality of the dried product. To attempt this objective, a we can mention the work of Perré et al. [13,14], Fayer et al.
better comprehension of coupled mechanism of heat and [15] and Johansson et al. [16]. Jomaa [17] and Caceres [18]
mass transfer inside and outside porous media in dynamic, in studied the phenomena for a saturated and greatly
thermal and in mass boundaries layers is required. deformable porous media; the objective of their work is to
Among the works concerning the convective drying of control the distortion of the product and his content in water.
unsaturated porous media, we mention the work of Basilico W. Masmoudi et al. [19] interested to the study of the heat
et al. [1,2]. They were interested in phenomena of heat and and mass transfer coefficients at the interface of an
mass transfer inside wood, and to its coupling with the unsaturated porous media exposed to an external air flow
external conditions during a drying by forced convection at during drying process. It is shown that the coefficient differ
high temperature. Among authors who also approached this from the standard values corresponding to the air flow on the
problem, we can cite works of Whitaker [3-5] and of P. Perré flat plate with uniform variables at the interface. Rogers et al.
and Degiovanni [6] who are interested to phenomena of heat [20] examine the process of heat and mass transfer during
and mass transfer inside wood, and to its coupling with the convective drying of saturated granular beds exposed to the
external conditions at the time of a drying by forced ambient air. They proved that the heat and mass transfer
convection at low or high temperature. Amir et al. [7-8] process are not analogous.
studied the drying of a thick plate of a humid porous material Many models for the study of drying have been used for
submitted to laminar forced flow of hot air circulating in the more than 10 years. More recently, pore network models
same way to his surface. The coupling between equations of have been used to study the influence of pore size
boundary layer in air and those describing humidity and heat distributions on saturation and transport parameters as well
transfer in the porous media are considered. The problem is as on drying kinetics [21, 22]. Prat [23] explore the influence
solved numerically by the finite differences scheme. The of pore shape and contact angle on drying rates during the
space-temporal fields of temperature and humidity are isothermal drying of porous materials in relation with the

25
effect of liquid films when viscous effects are important in convective drying of unsaturated porous media. They mainly
the film but not in the liquid saturated pores. It is shown that showed the effect of that limit conditions on mechanism of
the overall drying time is greatly affected by the pore shape convective drying of porous media.
and contact angle when film flows are important, the It should be kept in mind that during drying process heat
quantitative prediction of drying rate becomes very difficult and mass transfer coefficients depend on water saturation of
because it depends on tiny details of the pore space geometry the porous media interface. In order to bring out this effect
and is affected by possible changes in the local humidity Boukadida et al. using the same equations system as
conditions. This contributes to explain why the accurate described in [27] and by assuming that the lower surface of
prediction of drying rate still remains essentially an open the channel is wet, made a preliminary study in which they
question, at least when the effect of films cannot be considered that saturated water vapour pressure at interface is
neglected. function of the wet bulb temperature and of the water
In the porous media convective drying domain, saturation. So they determined evolution of average
Boukadida et al. [24] studied numerically two-dimensional coefficients of heat and mass transfer with water saturation
heat and mass transfer during convective drying of porous for different ambient temperatures. They obtained
media. The established code based on the Whitaker’s theory correlations which are in the following form: ht=Σ4i=1AtiSi and
took into account the effect of gas pressure and permitted to hm=Σ4i=1AmiSi where Ati and Ami are positive constants which
determine the space-time evolution of the medium's state depend on limit conditions at the channel entrance.
variables (temperature, gas pressure, and water saturation). Later, inspired by Whitaker theory and by using
The choice of average heat and mass transfer coefficients BICGSTB (Bi-Conjugate Gradient Stabilised) method [29-
was based on analogy between heat and mass transfer. In 30], Mobarki et al. [31-33] developed a mathematical model
[25], they extended their work by investigating a numerical governing heat and mass transfer during convective and
study to determine effects of the surrounding air's convective-radiative drying of unsaturated porous media.
(temperature, water-vapour concentration, etc.) on drying The main purpose of this model was to resolve many
kinetic and on the space-time evolution of the medium's state numerical problems encountered at the moment of the
variables. appearance of the drying front when the drying operation is
Using by default the analogy between heat and mass conducted at high temperature and high vapour-water
transfer presents insufficiency because it can not really and concentration. By using correlations of local or average heat
precisely describe the real transport phenomena during drying and mass transfer coefficients that were brought out by
process in space and time. Boukadida et al. in case where the heat radiation transfer is
Basing on this, in the same way, Boukadida et al. have taken into account or by assuming that the lower interface is
established a second code permitting to study two wet [27], they presented [31-33] effects of ambient
dimensional coupled heat and mass transfer in laminar forced temperature, ambient vapour-water concentration, interfacial
convection flow inside a horizontal channel [26]. They saturation and effects of variability of fluid thermo-physical
brought out the domain of the validity of analogy between properties on drying process. They depicted that the
heat and mass transfer and they showed that the analogy is interfacial liquid saturation can have a slight effect on the
valid only for low ambient temperature and low water-vapour evolution of medium’s temperature and gas pressure while
concentration. Many other results were obtained as the effect the global drying time is nearly not affected. This slight
of radiation on the different state variables (temperature, effect mainly appears in hygroscopic domain especially for
water-vapour concentration, etc) and on local heat and mass high ambient temperature. They also studied the effect of
transfer coefficients and their variations with ambient state surrounding temperature lower or higher than the inversion
variables. These local coefficients are calculated by temperature and the effect of the variables fluid thermo-
considering that lower surface of the channel is assumed to physical properties on the space time evolution of medium’s
be at the wet bulb temperature which depends mainly on the temperature and gas pressure for different ambient
ambient temperature, ambient gas pressure and water-vapour temperature, gas pressure and vapour-water concentration.
concentration. The lower surface was not considered as a Principal results showed that the effect of fluid thermo-
surface of a porous media in which the drying mechanism is physical properties dependence on temperature and on the
considered. Then the considered saturated water vapour gas pressure affects drying process and modifies in certain
pressure at interface is that of a plane surface. cases the temperature and gas pressure profiles.
In their other paper [27], they studied the effect of the However an insufficiency is noted in the fact that the
radiation heat transfer on inversion temperature in the case of used correlations are independent of time. In reality local
water evaporation into two-dimensional laminar flow inside coefficients of heat and mass transfer principally depend on
the channel. Their results showed that gas pressure and space time evolution of interfacial temperature which is
velocity affect inversion temperature and the inclusion of linked to the thermal, dynamic and mass evolution variables
radiation effect decreases the inversion temperature. in the boundary layers and on initial thermal and liquid
volume fraction in the porous media for every step in the
Believing that the use of average heat and mass transfer time. They also depend on physical properties of the porous
coefficient can not allow a precise idea of the real mechanism media and on thermo-physical properties of the drier fluid.
of space time drying process, then by fitting space-evolutions The present study which extends the previous works
of local coefficients of heat and mass transfer that were consists in studying numerically the effect of thermal, mass
determined in reference [26], Boukadida et al. obtained and dynamic boundaries layers on mechanism of heat and
correlations in the form: htx=atxb and hmx=amxb where x is
t m
mass transfer during drying process of flat plat having a clay
the axial coordinate and at, bt, am and bm are positive brick characteristics (table 1) via the space-time variability
constants. These correlations were injected as thermal and effect of heat and mass transfer coefficients. For each time
mass limit conditions [28] in the first code which permits to step the different state variables inside and outside the porous
simulate mechanism of heat and mass transfer during medium are determined and local coefficients of heat and

26
mass transfer are deduced. We principally describe the space • Heat equation
time evolution of the temperature, the water saturation and  ∂T ∂T  ∂  ∂T  ∂T ∂C v
the pressure fields in particular in the porous medium. The ρg c pg  u +v  =  λ g  + ρg D v c pv − c pa ( ) (3)
space time evolution of local coefficients of heat and mass  ∂x ∂y  ∂y  ∂y  ∂y ∂y
transfer and of the interfacial temperature and water where c pg = (1 − C v )c pa + C v c pv
saturation will also be presented and analysed. • Species equation
∂C v ∂C ∂  ∂C v 
2. BASIC FORMULATIONS ρg u + ρg v v =  ρg D v  (4)
∂x ∂y ∂y  ∂y 
2.1 Position of the problem
2.2.2 In the porous media
The system considered in this work is composed of
Mass conservation equations
unsaturated porous plate composed of an inert and rigid solid
 Liquid phase
phase, a liquid phase (pure water) and a gas phase which
contains both air and water vapour. The two vertical faces of Assuming that liquid density is constant, the mass
the porous plate as well as the low face are assimilated to conservation equation of the liquid phase is:
adiabatic and impervious faces. The high face of the plate ∂εl mɺ
+ ∇ Vl = − v (5)
represents the permeable interface of the horizontal plane ∂t ρl
channel when is exposed of a hot air out-flow with fixed
characteristics (velocity, temperature and humidity). The where m ɺ v is the mass rate of evaporation and ε l is the
geometrical configuration used for the slab is shown in volume fraction of the liquid phase.
figure1.  Gas phase
y
To
BB
For this phase, the average density ρ g is not constant, the
Air flow C vo BB Channel E mass conservation equation is given by:
P go
∂ ρg
BB

Uo L
+ ∇ ρg Vg  = m
BB BB
g
ɺv (6)
∂t  
Porous Medium ℓ  Vapour phase

x
∂ ρv
∂t
(
+ ∇ ρv
g
)
Vv = m
ɺv (7)

Figure 1: Geometrical configuration of the system g g g  ρ 

with ρ v Vv = ρv Vg − ρg D veff ∇  v 
In order to obtain a closed set of governing macroscopic  ρg 
 
equations the following assumptions are made:
where D veff is the coefficient of the effective diffusion of the
- the Soret and Dufour effects are neglected,
BB BB

vapour in the porous media. This coefficient takes into

- the compression work and viscous dissipation effects are
account the resistance to the diffusion due to the tortuosity
neglected,
and the effects of constriction.
- the solid, liquid, gas phases are in local thermodynamic
equilibrium,
The average velocities of the liquid phase Vl and the
- the transfer by radiation is negligible,
- the thermo-physical properties are variable, gas phase Vg are obtained using Darcy's law, which is
- the boundary-layer approximations are valid, generalized by using the concept of relative permeability,
- the porous medium is homogenous and isotropic, defined as the ratio between the effective permeability and
- the air water vapour mixture is a perfect gas, the intrinsic permeability:
- the dispersion and tortuosity terms are interpreted as
diffusion terms,  For the liquid phase
- the fluid flow is considered as laminar.
KK l  
− Pc  + ρl g 
g l
Vl = − ∇ Pg (8)
2.2 Governing equations µ l    
 For the gas phase, without taking the gravitational
Considering these hypotheses, the basic equations of the flow effect, we have:
become:
KK g g
2.2.1 In the channel Vg = − ∇ Pg (9)
µg
• Mass conservation equation g l l g
Pc = Pg − Pl where Pl (resp. Pg
( ) ∂(ρ v)
∂ ρg u
) is the intrinsic
+ =0
g
(1) average pressure of the liquid phase (resp. gas phase), K is
∂x ∂y the intrinsic permeability , Kl (resp. Kg) is the relative
• Momentum equation permeability of the liquid (resp. gaz), Pc is the capillary
pressure [34] considered as a characteristic of the porous
∂u ∂u ∂P ∂  ∂u 
ρg u + ρg v =− g +  µg  (2) media and is function only of the temperature and moisture
∂x ∂y ∂x ∂y  ∂y  constant.

27

Energy conservation equation (ρ l
l
Vl + ρ v
g
Vv) =h (ρ y mx v
g
− ρv 0 ) (13)
∂
∂t
( )
 l g  
ρc p T + ∇  ρ l c pl Vl + ∑ ρ k c pk Vk  T  = λ eff
∂T
+ ∆H vap (ρ V ) =h
l
(T − T ) (14)
 k =a,v   ∂y
l l y tx o

∇(λ eff ∇ T ) − ∆H vap m

ɺv (10) Where, htx and h mx are respectively the local coefficients of
BB

∆H vap is the latent heat of vaporization at temperature T(K): heat and mass transfer, determined by resolution of boundary
( )
layer equations.
∆H vap = ∆H ovap − c pv − c pl T where ∆H ovap is the latent The pressure of gas on surfaces of exchange is equal to the air
heat of vaporization at temperature T = 0 K. g
pressure: Pg = Pg 0
ρc p is the constant pressure heat capacity of the porous
- The equilibrium vapour pressure is a function of the
media given by : ρc p = ρs c ps + ρl cpl + ρ v c pv + ρa c pa temperature and the liquid saturation:
 2σM v 
where ρs c ps , ρl c pl , ρv c pv and ρa c pa are respectively Pv = Pvs exp −  (15)
the mass heat capacity at constant pressure of the solid,  rRρT 
liquid, vapour and air. Where Pvs is the saturating vapour pressure at interface, given
by:
Pvs = 10 5 exp(65.832 − 8.2 ln(Tint ) + 5.71710 −3 Tint −
2.3 Initial and boundary conditions 7235.46
Tint
2.3.1 Initial conditions r is the ray of curvature that depends on the saturation and
σ is the superficial tension.
Initially (t=0), the temperature, the gas pressure and the liquid
saturation are uniform inside the porous medium. 3. NUMERICAL RESOLUTION
2.3.2 Boundary conditions
For the fluid in the channel the system of equation is
• For the fluid in the channel solved numerically by the method of a finite differences
- At the entry of the channel, the temperature, the pressure scheme. The numerical resolution with this method consists
and the water-vapour concentration of the free ambient flow in transforming the system of equations in an algebraic
are constant and uniform. equation system. The resolution takes place plan by plan in
- Assuming that the gas-water interface is semi-permeable, the sense of the out flow. The mesh [26] considered in the
the velocity gas phase at the interface is written as: channel is regular and rectangular one with 180 nodes in the
D v ∂C v x-direction and 180 nodes in the y-direction. The unknown
v g int = − (11) variables (temperature, vapour-water concentration, velocity
1 − C v ∂y
and gas pressure) at column k+1 are calculated using the
- The longitudinal component of the velocity at the interface know variables of column k. For the porous medium the
is equal zero: equations system is numerically solved by a finite volume
For 0<x<L and y = ℓ : u=0 method based on the control domain notion as described by
- At the upper surface of the channel, the velocity of the gas Patankar [35]. In our case, we considered a regular mesh
mixture must respect the condition of adhesion: with 180 nodes in the x-direction and 24 nodes in the y-
For 0< x <L and y = E + ℓ : u=0; v=0 direction. For more details concerning the numeric model,
- The local interfacial evaporating mass flux which is the method of resolution and the thermo-physical properties
evaluated by the following equation: of the fluid, the reader can refer to [25,35].
 ρ D ∂C v 
ɺ vx = −  g v
For 0 ≤ Cv(x,0) < 1: m  (12) 4. RESULTS AND DISCUSSIONS
1 − C v ∂y  y = 0

4.1 Evolution of variables (T, Cv, ρg, u and v) along the

- The upper surface temperature T0 is assumed to be constant:
central line section of the channel
B

T (x,E+ℓ) = T0.
• For the porous medium Figure 2 shows evolution of temperature, water-vapour
- On the adiabatic and impervious sides, the mass and heat concentration, gas density, longitudinal and vertical
velocities in the transverse direction of the channel. These
fluxes are equal zero:
evolutions are drawn in a representative case characterized
For x= 0, x=L and 0 ≤ y ≤ ℓ : by To= 70°C, Cvo=10-3, U0=0.2m.s-1, Pgo=1atm, Sini = 40%
∂ T and Tini= 15°C.
λ eff = 0; Vv = 0; Vg = 0; Vl =0 At a given time (figure 2-a), the temperature that is
∂x x x x
minimal at the interface fluid-porous medium, increases
For y=0 and 0 ≤ x ≤ L : according to the height of the channel and tends to reach the
∂ T ambient temperature (70°C). This increase of the interfacial
λ eff = 0; Vv = 0; Vg = 0; Vl =0 temperature tends to decrease the boundary layer inside the
∂y y y y
channel; it depends on parameters as the drying phase of
- On the permeable side (y=ℓ), heat and mass fluxes can be porous medium, the initial saturation, the initial temperature
written as follows: of the porous medium and the regime of flow, etc. The
evolution of Tint in x=L/2 at different times (45°C for t=84h
and 61°C for t=168h) explains that the interface in this point

28
 is dry. The vapour concentration (Figure 2-b) is maximum at Tint(°C) Sint(%)

the interface and decreasing according to the height of the t=168h t=168h
t=84h t=84h
channel until the ambient vapour-water concentration (10-3). 80
t=10mn t=10mn
40
The interface concentration decreases according to the time
60
and this decrease is bound strongly to the interfacial
saturation of water. The decrease of gas density according to
20
the time (Figure 2-c) is owed to the increase of the interfacial 40

temperature. The profiles of longitudinal and vertical velocity

are independent of the time (Figure 2-d and 2-e). 20

0
X(m) X(m)
T(°C) Cv 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

0.012 (a) (b)

70
0.01
60 t=168h t=168h Figure 3: Evolution of Tint (a), and Sint (b) with X at
t=84h t=84h
50
t=10mn
0.008
t=10mn
different times.
40
0.006

30
0.004
4.3 Space time evolution of local heat and mass transfer
20

0.002
coefficients
10

0 0
0.0 0.2 0.4 0.6 0.8 1.0 Y/E 0.0 0.2 0.4 0.6 0.8 1.0 Y/E Figures 4-a and 4-b illustrate the effect of drying time on
(a) (b) the longitudinal evolution of local heat (htx) and mass (hmx)
ρg(Kg.m-3) transfer coefficients. These values are very important in the
1.3 vicinity of the left surface exchange. Coefficients decrease
t=168h
and tend to be constant far from the side entrance. Their
t=84h
1.2 t=10mn
variation according the time is bound strongly to the out flow
regime, the ambient conditions, and the drying phase (first
1.1
phase characterized by the increase or decrease of the
medium temperature, isenthalpic phase, falling rate period).
1.0
-2 -1 -1
htx (w.m .K ) hmx (m.s )
0.0 0.2 0.4 0.6 0.8 1.0 Y/E 24
(c) 1.0E-2
20
U(m/s) V(m/s) t=168h t=168h
8.0E-3
t=168h t=168h 16 t=10mn t=10mn
t=84h t=84h
0.3 t=10mn t=10mn 6.0E-3
12
1.0E-3
4.0E-3
8
0.2 5.0E-4
4 2.0E-3

0.0E+0
0.1
0.0E+0
X(m)
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 X(m)
-5.0E-4
(a) (b)
0.0
Y/E
-1.0E-3
Y/E
Figure 4: Evolution of local heat and mass transfer
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0
(d) (e) coefficients (htx(a), hmx(b)) with X at different times.

Figure 2 : Evolution of different state variables

(T(a),Cv(b), ρg(c), U(d), V(e)) with Y/E in the transverse 4.4 Space time evolution of the different variables in the
direction of the channel ( X=L/2) at different times. porous medium

Figures 5, 6 and 7, present the space time evolution of the

4.2 Space time evolution of the interface temperature and different state variables in the following cases characterized
water saturation by To= 70°C, Cvo=10-3, U0=0.2m.s-1, Pgo=1atm, Sini = 40%
and Tini= 15°C:
Figure 3 illustrates the effect of drying time on evolution
• In the first case (case 1), coefficients of heat and mass
of interface temperature (Tint) (Figure 3-a) and water
transfer between the permeable interface and the drier fluid
saturation (Sint) (Figure 3-b) along the x-direction. At the
are variable (determined by solving the boundaries layers
beginning of drying (t=10mn), Tint and Sint start to increase
equations).
and to decrease respectively in the vicinity of the upper left
• In the second case (case 2), coefficients of heat and mass
corner witch is placed at the channel entrance. The average
transfer are considered as constant (ht=2.5W.m-2.K-1 and
temperature and water saturation of the interface are about
hm=18.10-4m.s-1).
17°C and 40% respectively. Later an evaporation front takes
The different curves show that, whatever the case, three
place earlier in the vicinity of the upper left corner (t= 48 h
drying phase that were analysed in [36-38] are distinguished.
and t=168 h) dividing the slab on dry zone and wet zone.
In case 1, a sensitive effect on mechanism of drying process is
This front moves step by step towards the end of the slab. At
demonstrated as described. The magnitude of the drying
the end of drying, Tint and Sint evolve respectively towards
kinetic is principally controlled by the magnitude of heat and
constant values (To=70°C and Sirr ).
mass transfer coefficients and the moving nature of the

29
evaporation front sensitively depends of the space time -The interfacial temperature and vapour-water concentration
evolution of transfer coefficients. depend on the phase of drying.
In case 2 where coefficients of heat and mass transfer are -The variability of heat and mass transfer coefficients with
assumed to be constant, whatever the drying phase, at a given time and with x-direction sensitively affects space time
time, for a given y-value, the space time evolution of state temperature, pressure and water saturation profiles of the
variables are uniform along x-direction. The evaporation porous medium during the drying process.
front has a straight line form.
4.4.1 Space time evolution of the temperature inside the
porous medium
In figures 5-a and 5-b, we have presented the space time t=5mn
evolution of medium temperature for different times during
drying kinetic. In case 1, figure 5-a obviously shows that the
temperature of porous medium increases sensitively with time
according to x-and y-direction. This increase is more
important at the left top corner, this is due to the high values
of local coefficients of heat and mass transfer in this zone, it t=70h
is that part of porous medium which enters first in
hygroscopic domain. In case 2, all nodes of a same line in x-
direction and at given period have same value of variables
state (temperature, gas pressure and water saturation). In case
1, the evolution nature of porous medium state variables
sensitively depends on the space-time variability of htx and
hmx. t=168h

4.4.2 Space time evolution of water saturation inside the

porous medium
Figures 6-a and 6-b show the evolution of water
saturation inside the porous medium at different times. In
case 1, the water saturation decreases according time and x-
direction. This is mainly explained by the evaporation of
water at the level of the interface fluid-drying porous
medium. At the beginning of drying process (t=5mn), the
water saturation starts to decrease at the left corner of the Figure 5-a: Space time evolution of temperature in
exchange surface. For t=70h, the moving evaporation front the porous medium. (Case 1)
divides the porous medium in two zones: the first zone is wet
where the saturation approaches its initial value (40%), the
second zone is dry where the saturation decreases sensitively.
The evaporation front takes place earlier and the drying
t=5mn
process is faster than for the right surface exchange. At the
end drying (t=168h), the water saturation of the porous
medium evolves towards Sirr. In case 2, the space time
evolution of water saturation is uniform along x-direction.

4.4.3 Space time evolution of pressure in the porous

medium
t=70h
Figures 7-a and 7-b show the space time evolution of the
gas pressure inside the medium. At the beginning of process
drying (t=5mn) the pressure sustains a depression in the both
cases. When the porous medium enters in the hygroscopic
domain the pressure increases and reaches a maximum value
at the level of the drying front.
t=168h

5. CONCLUSION

This paper presents a numerical study of two

dimensionnel heat and mass transfer during the convective
drying of clay brick, exposed to a laminar flow having
various thermophysical caracteristics. The simulation
principally shows that:
-The different space time evolutions of state variables inside Figure 5-b: Space time evolution of temperature in
the porous medium and in the channel are sensitively affected the porous medium. (Case 2)
when the local behaviour of boundaries layers are taken into
account

30
 t=5mn
t=5mn

t=70h

t=70h

t=168h

t=168h

Figure 6-a: Space time evolution of water saturation Figure 6-b: Space time evolution of water saturation
in the porous medium. (Case 1) in the porous medium. (Case 2)

t=5mn

t=70h

t=168h

Figure 7-a: Space time evolution of gas pressure in Figure 7-b: Space time evolution of gas pressure in
the porous medium. (Case 1) the porous medium. (Case 2)

31
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33. A. Mobarki, Etude des transferts de masse et de chaleur ∆Hvap latent heat of vaporization, J.Kg-1
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1974. µ dynamic viscosity , Kg.m-1.s-1
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325-329, 1971. l liquid
o ambient
7. NOMENCLATURE resp. respectively
s solid
Cv vapour-water concentration v water vapour
cp specific heat of fluid at constant pressure, J.kg-1.K-1 vs saturated vapour
Dv vapor diffusion coefficient into air, m2.s-1 x local
E height of the channel
g gravitational constant, m.s-2 Symbols
hmx local mass transfer coefficient , m.s-1
hm average mass transfer coefficient , m.s-1 average value
htx local heat transfer coefficient, W.m-2.K-1
ht average heat transfer coefficient, W.m-2.K-1
K intrinsic permeability, m2
L Channel length, m
M molecular weight, Kg
P pressure, Pa Porosity ε=0.26
Pc capillary pressure, Pa Density ρs=2600 Kg.m-3
R universal gas constant, J.mole-1.K-1
r curve ray Specific heat Cps=879 J.Kg-1.K-1
S liquid saturation
Thermal conductivity λs=1.44 W.m-1.K-1
T temperature, K
t time, s Intrinsic permeability K=2.510-4m2
u longitudinal velocity, m.s-1
v transverse velocity, m.s-1
m
ɺv masse flow rate, Kg.m-2.s-1 Table 1: Physical characteristics of porous medium.
X longitudinal direction, m
y transverse direction, m

33

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