COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE
EPMESC X, Aug. 21-23, 2006, Sanya, Hainan, China
©2006 Tsinghua University Press & Springer
Development of a Mathematical Model for Heat and Mass Transfer
Inside a Granular Medium
V. J. Petry 1, A. L. De Bortoli 1*, O. Khatchatourian 2
1
2
Department of Pure and Applied Mathematics,UFRGS, PO Box 15080, Porto Alegre, RS - Brazil
Departament of Physics, Statistics and Mathematics, UNIJUÍ, PO Box 560, Ijuí, RS - Brazil
Email: vpetry@mat.ufrgs.br, dbortoli@mat.ufrgs.br, olegkha@admijui.unijui.tche.br
Abstract The aim of the present work is to analise the heat and mass transfer which occurs in a granular medium
during the grains drying process. In the literature we have found several models that try to describe this process. Most
of these models are empiric or semi-empiric and valid, therefore, for a quite limited range of the involved parameters.
Therefore, we present a mathematical model based on the Navier-Stokes equations for the grains in the form of small
spheres inside a chamber. We investigate the mean non dimensional parameters, which mean the Reynolds, the
Schmidt, the Prandtl and the Eckert numbers. Numerical tests, using the Gauss-Seidel central finite difference scheme,
are carried out for non dimensional parameters in the range Re = 200-2000, Sc = 0.7-2, Pr = 0.7-2, Ec = 10−5-5×10−5.
These results contribute to obtain a better understanding of the parameters influence during the drying process of a
granular medium.
Key words: heat and mass transfer, granular medium, fixed bed, mathematical model
INTRODUCTION
During the drying process it is important to control the air and the grains temperature, the heat and mass changes
between the grains and the air, as well as the humidity levels inside the dryer chamber. Such needs, associated to the
high cost for the prototypes construction, have increased the importance of the mathematical models development.
In the literature we can find several models that try to describe these processes. A model for stationary deep bed was
proposed by Boyce [7], wherein a deep bed is modelled as a thin layer of grains using a semi-empirical model.
Srivastava and John [7] reported a deep bed model under unsteady conditions for the simulation of air humidity, grain
temperature and air temperature along the bed. Mhimid, et al. [4] considered two mathematical models for heat and
mass transfer in cylindrical coordinates; they considered local and no local temperature equilibrium.
Recently, Petry et al. [5] presented a mathematical model (a local temperature equilibrium model) to calculate the
temperature distribution in a small nuclear reactor fuel chamber in order to evaluate the passive cooling characteristics
of the Fixed Bed Nuclear Reactor (FBNR), projet which was under the supervision of the International Atomic Energy
Agency (IAEA) [6].
In this work we present a mathematical model, based on the Navier-Stokes equations, which describes the hot air flow
going through the granular middle. We consider the grains in the form of small spheres inside a chamber. During the
drying process the heat transfer in the air occurs by the convection and the conduction processes, while the water mass
transfer (as vapor) in the air, occurs by the convection and the diffusion processes. The heat and mass transfer between
the air and the grains is considered in the source terms of the energy and mass conservation equations for the grains and
the air. Those terms are modeled obeying the Fourier’s and Fick’s laws, respectively.
The governing equations are approximated using first order time and second order space approximations, based on the
finite difference approach. Numerical tests are realized to verify the influence of non dimensional parameters in the
range Re = 200-2000, Sc = 0.7-2, Pr = 0.7-2, Ec = 10−5-5×10−5. We also present a comparison between the calculated
and the experimental data for soy grains [3].
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GOVERNING EQUATIONS AND SOLUTION PROCEDURE
The fixed bed chamber consists of a straight prism of rectangular base inside of which solid spheres of 8mm diameter
are deposited, as shown in Fig. 1; hot air passes among the spheres.
The mathematical model considers a set of partial differential equations that describe (in the grain drying case) the air
and the temperature distribution, the air humidity and the spheres moisture inside the chamber. The following
hypotheses or simplifications are adopted:
(1) the medium porosity is constant;
(2) there is no thermal equilibrium between the spheres and the air inside the chamber;
(3) each sphere has uniform temperature;
(4) the heat transfer from the air to the solid elements occurs by conduction, obeying the Fourier’s law;
(5) the water mass transfer from the spheres to the air occurs by diffusion, obeying the Fick’s law;
(6) the heat transfer in the air occurs by the conduction and the convection processes;
(7) the mass transfer in the air is due to the diffusion and the convection processes;
(8) the chamber walls are isolated.
To describe the air flow we considered the continuity, the momentum and the Poisson equations.
Figure 1: Sketch of the chamber
1. Governing equations The set of governing equations includes the mass conservation, for air and solid, and the
energy, for air and spheres inside the chamber.
1) Mass conservation for air: The mass equation comes from the balance inside a fine layer [1]. It results:
(1)
where
is the ratio of air humidity, rl and rg are the density of liquid (water) and gas (air),
respectively, D is the mass diffusion coefficient and m& is the mass generation rate per unit of time and space.
From mass conservation, the water mass that goes to the air is equal to that which leaves the spheres [3]. Therefore, it
results for the mass generation rate:
(2)
where a is the sphere area to volume ratio, Ds is the mass diffusion coefficient between the solid and the gas, φ the bed
porosity, Vc the control volume and ΔC = ( ρs X − ρg Y ) the mass concentration variation, with X = ml ms the ratio of
spheres humidity.
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Finally, the air humidity balance equation assumes the form:
(3)
2) Mass conservation for solid: From Eq. (2) it follows that:
(4)
where the subscripts s and g represent the solid and gas, respectively.
3) Energy equation for the air inside the chamber: The energy balance for the control volume can be written as:
(5)
where Tg is the gas (air) temperature and a the air thermal diffusivity.
The ratio of energy generation by unit of time and air volume inside de control volume, q& , is given by:
(6)
Considering that the amount of heat that leaves the air is all transferred to the spheres by thermal conduction, it follows
from the Fourier’s law:
(7)
where Ks is the thermal conductivity between the air and the solid, As the unit sphere surface area, Ts the solid
temperature and (1 − φ )Vc Vs the number of spheres inside the control volume. Thus we have:
(8)
and the air energy equation results:
(9)
4) Energy equation for the solid spheres inside the chamber: Since the heat enters the sphere by conduction and
vaporize the water that leaves the sphere in the steam form and warm the sphere mass, we have [3]:
(10)
Notice that ml = ms X and ms = ρsVs , resulting:
(11)
and considering that the spheres don’t move, we have finally:
(12)
The values of mass diffusion coefficients (D and Ds) and the thermal conductivity coefficient Ks were set as constants
when writing the equations; however, they are recalculated when performing the iterations, considering K s ∝ Ts0.7 and
D ∝ Tg1.5 [2]. To obtain the mass diffusion coefficient between solid and air we employed the expression:
(13)
⎯ 652 ⎯
where A and n are defined by experimental data and Xe is the equilibrium moisture.
To describe the air flow among the grains we considered the:
5) Continuity equation:
∂u ∂v ∂w
+ +
=0
∂x ∂y ∂z
(14)
6) Momentum equations:
∂u
1 ∂P
→
−
−
= −→
u .∇u−
+ ν∇2 u
∂t
ρg ∂x
(15)
∂v
1 ∂P
→
−
−
= −→
u .∇v−
+ ν∇2 v
∂t
ρg ∂y
(16)
∂w
1 ∂P
→
−
−
= −→
u .∇w−
+ ν∇2 w
∂t
ρg ∂z
(17)
2. Dimensionless Governing Equations The set of governing Eqs. (3,4,9,12), and (14-17) can be written, respectively,
in the following way:
∗
∗
∗
∂Y ∗
D∗
∗ ∂Y
∗ ∂Y
∗ ∂Y
=
−u
−
v
−
w
+
∂t ∗
∂x∗
∂y∗
∂z∗ ReSc
µ
∂2Y ∗ ∂2Y ∗ ∂2Y ∗
+ ∗2 + ∗2
∂x∗2
∂y
∂z
¶
+
¢
a∗ (1 − φ) ∗ ¡ ∗
Ds ρs X0 X ∗ − ρ∗gY0Y ∗
∗
φρgY0 ReSc
(18)
¡
¢
∂X ∗
a∗
D∗s ρ∗s X0 X ∗ − ρ∗gY0Y ∗
=
−
∗
∗
∂t
ρs X0 ReSc
∂Tg∗
∂Tg∗
∂Tg∗
∂Tg∗
α∗
∗
∗
∗
=
−u
−
v
−
w
+
∂t ∗
∂x∗
∂y∗
∂z∗ RePr
(19)
Ã
∂2 Tg∗ ∂2 Tg∗ ∂2 Tg∗
+ ∗2 + ∗2
∂x∗2
∂y
∂z
!
+
¢
a∗ (1 − φ) ∗ ¡ ∗
Ks Ts − Tg∗
∗
∗
φρgCpg RePr
(20)
¡ ∗
¢
¡
¢
1
a∗ Lv∗
a∗
Ec
∂Ts∗
∗
∗
∗
∗
D
ρ
X
X
−
ρ
Y
Y
−
Ks∗ Ts∗ − Tg∗
=
−
s
s 0
g 0
∗
∗
∗
∗
∗
∗
∗
∗
∗
∂t
ReSc ρs (X0 X Cpw +Cps )
RePr ρs (X0 X Cpw +Cps )
(21)
∂u∗ ∂v∗ ∂w∗
−
→
− →
∇ . u∗ = ∗ + ∗ + ∗ = 0
∂x
∂y
∂z
(22)
∗
∗
∗
∂u∗
∂P∗
1
∗ ∂u
∗ ∂u
∗ ∂u
=
−u
−
v
−
w
−
+
∗
∗
∗
∗
∗
∂t
∂x
∂y
∂z
∂x
Re
∗
∗
∗
∂P∗
1
∂v∗
∗ ∂v
∗ ∂v
∗ ∂v
=
−u
−
v
−
w
−
+
∗
∗
∗
∗
∗
∂t
∂x
∂y
∂z
∂y
Re
µ
µ
∂2 u∗ ∂2 u∗ ∂2 u∗
+
+
∂x∗2 ∂y∗2 ∂z∗2
∂2 v∗ ∂2 v∗ ∂2 v∗
+
+
∂x∗2 ∂y∗2 ∂z∗2
∗
∗
∗
∂w∗
∂P∗
1
∗ ∂w
∗ ∂w
∗ ∂w
=
−u
−
v
−
w
−
+
∗
∗
∗
∗
∗
∂t
∂x
∂y
∂z
∂z
Re
µ
¶
(23)
¶
∂2 w ∗ ∂2 w ∗ ∂2 w ∗
+ ∗2 + ∗2
∂x∗2
∂y
∂z
(24)
¶
(25)
To evaluate the pressure distribution, we employed the Poisson equation:
∇2 P∗ = −
∂ ³→
∂ ³→
∂ ³→
∂ ³→
−∗ ´
−∗ ´
−∗ →
−∗ →
−∗ →
− →
− →
− ∗´
− ∗´
− ∗ ´ 1 2 ³→
∇
∇
.
u
∇
.
u
−
u
.
∇
u
−
u
.
∇
v
−
u
.
∇
w
+
∂t ∗
∂x∗
∂y∗
∂z∗
Re
u
v
w
∗
∗
U0 , v = U0 , w = U0 ,
Ds
D
α
∗
∗
∗
D0 , Ds = D0 , α = α0 , C pw
Notice that we used the following nondimensional variables: u∗ =
ρ
ρs
tU0
g
X
Y
∗
∗
∗
∗
∗
∗
amb
T ∗ = TTair−T
−Tamb , t = Lc , X = X0 , Y = Y0 , ρg = ρ0 , ρs = ρ0 , D =
Lv∗
=
Lv
,
U02
a∗
= aLc and
P∗
=
P−Patm
ρ0 ,
(26)
x∗ =
=
x
Lc ,
y
∗
Lc , z
C ps
∗
C p0 , Ks
y∗ =
C pw
∗
C p0 , C ps
=
Reynolds number, Sc =
=
z
Lc ,
Ks
K0 ,
where Lc is the characteristic length, Tamb the air ambient temperature, Tair the air
temperature in the chamber entrance, X0 the initial spheres moisture and Y0 the initial air humidity. Re =
ν
D0
=
the Schmidt number, Pr =
ν
α0
the Prandtl number and Ec =
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U02
C p0 (Tair −Tamb )
U0 Lc
ν
is the
the Eckert number.
3. Boundary and initial conditions The appropriate implementation of boundary conditions is always important
when solving a set of differential equations. Here some assumptions are adopted as following indicated.
When starting the simulations, the air and the spheres temperature and humidity, the velocity and the pressure are
considered to be:
For the x-direction boundary conditions we have considered:
On the chamber walls, that is, for
Neumann boundary conditions. The pressure
obtained by extrapolation.
and
at
the variables
and the velocity components
and
and
satisfy the
at
are
4. Finite difference approximation The finite difference Gauss-Seidel approach is employed to obtain the numerical
results. First order time and a second order space approximations were used, as follows:
(27)
(28)
(29)
NUMERICAL RESULTS
Numerical results are aimed to do comparisons with experimental data for the soy grains drying process [3]. The
physical properties were obtained in the literature [2].
Fig. 2 shows the grains temperature
and moisture
distributions inside the chamber along the
time in the one dimensional case.
To evaluate the influence of dimensional parameters, numerical simulations were realized for different values of them
and
. It is clearly seen in Fig. 3 that the
in two positions inside the drying chamber:
Reynolds number in the range 200-2000 change the temperature and the moisture considerably.
-2; it results in small spheres
The Schmidt number influence seems to be not significant in the range
temperature and humidity variations, as shows in the Fig. 4. The Prandtl number influence in the range
-2 is
similar to that of Schmidt value.
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Figure 2: Soy grains temperature (left) and moisture (right) distribution along
the time for
and
Figure 3: Influence of Reynolds number on the grains temperature (left) and moisture (right)
distribution along the time at two positions inside the chamber
Figure 4: Influence of Schmidt number on the grains temperature (left) and moisture (right)
distribution along the time at two positions inside the chamber
Figure 5: Influence of Eckert number on the grains temperature (left) and moisture (right)
distribution along the time at two positions inside the chamber
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To evaluate the Eckert number contribution in the heat and mass transfer processes we analyzed the grains temperature
and moisture for Ec = 10−5 and Ec = 5 × 10−5 in the two positions already mentioned. The contributions are indicated
in Fig. 5. We believe that the big temperature and moisture changes happen due to the relationship of the Eckert
number with the temperature difference between the drying air and the closed ambient.
CONCLUSIONS
This work developed a three dimensional mathematical model that describes the heat and the mass transfer processes
in a granular medium. A numerical scheme was presented and simulations were realized to verify the influence of non
dimensional parameters involved in the heat and mass transfer processes.
Obtained results indicate a good agreement between the numerical and the experimental data. Such results contribute
to obtain a better understanding of the drying process which occurs inside a chamber. The moisture distribution
variation near the chamber walls justifies the three dimensional simulation. Besides, this mathematical model can also
be applied to small load nuclear reactors where the spheres are fuel elements. In this case we should consider the
conduction and the radiation effects to the chamber walls.
Acknowledgements
The work reported in this paper has been supported by CNPq (Conselho Nacional de Desenvolviomento Cientfico e
Tecnologico - Brasil). Computations were performed on the computer Cray - T94 of CESUP-UFRGS. The support and
assistance of the staff is gratefully acknowledged.
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of drying of soy grains in camera of fixed bed. in Proceedings of LACAFLUM - V Latin American and
Caribbean Congress on Fluid Mechanics, Caracas, 2001, pp. 1-6.
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with convective and conductive boundary conditions. Int J. of Heat and Mass Transfer, 2000; 43: 2779-2791.
5. Petry VJ, De Bortoli AL, Sefidvash F. Passive cooling of a fixed bed nuclear reactor. in Proceedings of 4th
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