Number 1
Volume 18 January 2012
Journal of Engineering
DOUBLE DIFFUSIVE FREE CONVECTION IN A PACKED BED
SQUARE ENCLOSURE BY USING LOCAL THERMAL
NON-EQUILIBRIUM (LTNE) MODEL
Ahmed N. Mehdy
Assistant Lecturer
Department of Mechanical
Engineering
Engineering college
University of Kufa, Najaf, Iraq
ABSTRACT
In the present study, free convection heat and mass transfer of fluid in a square packed bed enclosure is
numerically investigated. For the considered geometrical shape, the left vertical wall of enclosure was
assumed to be kept at high temperature and concentration while the opposite wall was kept at low
temperature and concentration with insulating both the top and bottom walls of enclosure. The Brinkman–
Forchheimer extended Darcy model was used to solve the momentum equations, while the energy equations
for fluid and solid phases were solved by using the local thermal non-equilibrium (LTNE) model.
Computations are performed for a range of the Darcy number from 10-5 to 10-1, the porosity from 0.5 to 0.9,
and buoyancy ratio from -15 to 15. The results showed that both the buoyancy ratio and the packed bed
characteristics have significant effect on each one of the flow field, heat transfer and mass transfer.
‫الخالصة‬
‫ للنمر ج الهندسر‬.‫حربر تر رراسرتع دردريا‬
‫ انتقال الحرارة والكتلة بالحمل الحرر خرالل طبقرا ححور ة ر تج ير‬,‫خالل الدراسة الحالية‬
‫ححفر ع دنرد ررحرة ةررارة وتركير حررتفعين بينمرا الجردار المقابرل قرد فةفرع دنرد ررحرة‬
‫ر ةرل‬
‫ ت ادتبار الجدار العم ري األيسرر للتج ير‬,‫المفترض‬
‫ رشماير الموتق حن نمر ج رارسر‬-‫ ت استخدام نم ج برنكمان‬. ‫ةرارة وتركي واطئين ح د ل كل حن الجدارين العل ي والسفل‬
‫ لقرد‬.(LTNE) ‫ بينما ت ةل حعارال الطاقة لكل حن الط رين المائ والصلب باستخدام نم ج ددم االت ان الحراري الم قع‬, ‫حعارال ال خ‬
‫ أعهر النتائج بأن كال حن‬,15 ‫ إلى‬-15 ‫ ونسبة الطف حن‬,0.9 ‫ إلى‬0.5 ‫ المساحية حن‬,10-1 ‫ إلى‬10-5 ‫الحسابا لمدى رق رارس حن‬
‫أنج‬
.‫ انتقال الحرارة وانتقال الكتلة‬,‫نسبة الطف وخصائص الطبقا المحو ة تمتلكان تأثيرا حهما دلى كل حن ةقل الجريان‬
Keywords: Double Diffusive, Free Convection, Packed Bed, Local thermal non-equilibrium.
121
DOUBLE DIFFUSIVE FREE CONVECTION IN A PACKED
BED SQUARE ENCLOSURE BY USING LOCAL THERMAL
NON-EQUILIBRIUM (LTNE) MODEL
Ahmed N. Mehdy
buoyancy forces is necessary in order to control
INTRODUCTION
Double-diffusive convection or thermosolutal
these processes. (Chaudhary and Jain, 2007)
convection is generally referred to a fluid flow
studied the MHD flow past an infinite vertical
generated by buoyancy effects due to both
oscillating plate through porous medium with the
temperature and solutal concentration gradients.
presence of free convection and mass transfer
This type of flow is encountered in natural and
analytically
technological applications. Such applications
technique.
include the growth of crystals, solar energy
double-diffusive convection-radiation interaction
systems, welding processes, thermal insulations.
for the unsteady MHD flow over a semi-infinite
by
using
Laplace-transform
(Mohamed, 2009) analyzed the
The term double diffusive-convection is now
vertical moving porous plate embedded in a
widely accepted for all processes which involve
porous medium in the presence of thermal &
simultaneous thermal and solutal concentration
solutal diffusion and heat generation. A numerical
gradients and provides an explanation for a
study of the unsteady free convection and mass
number of natural phenomena. Because of the
transfer flow of an electrically conducting fluid
coupling between the fluid velocity field and the
past an infinite vertical porous plate in the
diffusive (thermal and solutal concentration)
presence of a transverse magnetic field was
fields,
more
presented by (Shariful et. al. , 2005). (Bukhari,
complex than the convection flow which is
2003) applied a linear stability analysis, using the
associated with a single diffusive scalar, and many
spectral Chebyshev polynomial method, to obtain
different behaviours may be expected. Such
numerically the solution of a multi-layer system
double-diffusive processes occur in many fields,
consisting of the finger convection onset in a fluid
including chemical engineering (drying, cleaning
layer overlying a porous layer. (Saha and
operations,
condensation,
Hossain, 2004) studied numerically the laminar
sublimation, deposition of thin films, energy
doubly diffusive free convection flows along an
storage in solar ponds, roll-over in storage tanks
isothermal vertical finite plate immersed in a
containing liquefied natural gas, solution mining
stable thermally stratified fluid by using an
of salt caverns for crude oil storage, casting of
implicit finite difference method and local non-
metal alloys and photosynthesis), oceanography
similarity method. (Hajri et. al. , 2007) presented
(melting and cooling near ice surfaces, sea water
a numerical simulation for the steady double-
intrusion into freshwater lakes and the formation
diffusive natural convection in a triangular cavity
of
during
by using equal finite elements method. A
crystallisation of igneous intrusions in earth's
numerical study was presented from (Mamou et.
crust), geophysics (dispersion of dissolvent
al. , 2001) for the unsteady double-diffusive
materials or particulate matter in flows), etc. A
convection in a two-dimensional horizontal
clear understanding of the nature of the interaction
confined enclosure by using the finite element
between thermal and mass or solutal concentration
technique. (Masuda et. al. , 2002) presented a
double-diffusive
layered
convection is
evaporations,
or
columnar
structures
numerical simulation by using finite differences
122
Number 1
Volume 18 January 2012
Journal of Engineering
method for the unsteady two-dimensional double-
MATHEMATICAL MODEL
diffusive
Geometrical Shape of Studied Problem
convection
in
porous
rectangular
enclosure. After that, (Masuda et. al. , 2008)s
The schematic view of the studied problem is
studied the peculiar oscillating convection which
shown in Fig.1. The square enclosure has a side
is
double-
length (a) and it is filled with a saturated packed
diffusive convection in porous medium is
bed. The left vertical wall of the square enclosure
analyzed numerically. (Khanafer and Vafai,
is kept at high temperature and concentration
2002) presented a numerical study of mixed-
while the opposite wall is kept at low temperature
convection heat and mass transport in a lid-driven
and concentration, finally, the horizontal walls are
square enclosure filled with a non-Darcian fluid-
insulated.
observed
when
two-dimensional
saturated porous medium by using the finite
volumes technique. (Rahli and Bouhadef, 2004)
Governing Equations
studied numerically the double-diffusive natural
In the model development, the following
convection in a partially porous square enclosure
assumptions are adopted; the working fluid has a
with the presence of inclination effect by using the
Prandtl number Pr=0.71 and assumed to be
control volume method and the power law
incompressible and Newtonian fluid, no phase
scheme. (Wang et. al., 2007) presented a
change occurs and the process is in a steady state,
numerical investigation of natural convection of
the thermo physical properties of the fluid are
fluid (without mass transfer) in an inclined square
assumed to be constant except the density
enclosure
variation in the buoyancy force, which is
filled
with porous
medium and
submitted to a strong magnetic field by using
approximated
according
to
the
Boussinesq
(LTNE) model. However, the buoyancy force due
approximation. This variation, due to both
to the double-diffusive effect has received more
temperature and solutal concentration gradients
attention in the literature. But all the previous
can be described as follow (Khanafer and Vafai,
studies which deal with the double-diffusive
2002);
  0 1  T T  TL    C c  cL 
convection in porous mediums assume that the
(1)
porous medium in thermo dynamical equilibrium
Where βT and βC are the coefficients of the
with the fluid which flow inside it, except (Wang
thermal and solutal expansions, which are defined
et. al., 2007) which studied the pure convection
as follow (Khanafer and Vafai, 2002);
without mass transfer in the presence of heat
T  
transfer between the fluid and the porous media.
The present study gives more attention to the
1   
&
 
 0  T  p, c
1   
C    
 0  c  p,T
interaction between the effect of the double-
(2)
diffusive convection and the effect of heat transfer
The Brinkman–Forchheimer extended Darcy
between the porous medium and the fluid which
model is used to solve the momentum equations
flows inside it.
while the energy equations for fluid and solid
phases are solved with the local thermal non-
123
DOUBLE DIFFUSIVE FREE CONVECTION IN A PACKED
BED SQUARE ENCLOSURE BY USING LOCAL THERMAL
NON-EQUILIBRIUM (LTNE) MODEL
Ahmed N. Mehdy
equilibrium (LTNE) model. Thus, the governing
F
equations for the present study will take the
following forms as in (Amiri and Vafai, 1998)
a sf 
and (Khanafer and Vafai, 2002);
(9.a )
150  3
6 (1   )
dp
(9.b)
0.6

 u d  
p 
 f


hsf  k f 2  1.1 Pr 1 3 


 f
 


 

- Continuity equation
u v
 0
x y
1.75
(3)
(9.c)
- Momentum equations
 f  u u 
u  v  
 2  x y 
p  f
 
x 
where the sphere particle diameter dp can be
computed as follow (Amiri and Vafai, 1998);
(4)
F f u u
  2u  2u   f
 2  2   u 
K
 x y  K
d p  (1   )
150 K
(9.d )
3
while the effective thermal conductivity kfeff and
kseff in fluid and solid phase energy equations and
the mean thermal diffusivity αm can be computed
as follow (Wang et. al. , 2007);
(5)
- Fluid phase energy equation
T f
 T f
 f cp f  u
v
y
 x
  2T f
 2T f
k feff 

 x 2
y 2


 


  a sf hsf Ts  T f


k seff  (1   ) k s
(10.b)
Now

k feff  k seff
we
 f cp f
introduce
the
(10.c )
following
non-
dimensional quantities and parameters as in
(Khanafer and Vafai, 2002) and (Wang et. al. ,
2007);
- Solid phase energy equation

  a sf hsf T f  Ts   0

(7)
- Solutal concentration equation
  2c  2c 
c
c
u
v
 D 2  2 
x
y
y 
 x
(10.a )
m 
(6)
  2T
 2Ts
k seff  2s 
y 2
 x
k feff   k f
(8)
The geometric function F, specific surface area
of the packed bed asf and the fluid-to-solid heat
transfer coefficient in a packed bed hsf are
determined as suggested by (Amiri and Vafai,
1998);
124
Number 1
x
y
, Y
a
a
au
av
U
, V 
X 
m
P
f 
s 
m
p a2
 f m
2
(T f  TL )
(TH  TL )
(Ts  TL )
,
(TH  TL )
(c  c L )
C
(c H  c L )
 f cp f
,
Volume 18 January 2012
1  V
V 
U
V

2 
Y 
  X
P Pr   2V  2V




Y
  X 2 Y 2



,















FUV
Pr
V
Da
Da
1   U X
 f 

Y 
1
f

   f  2 f


 X 2
Y 2

2

Pr 

k feff  k seff

3
g a  T (TH  TL ) 

, GrT 
 2f


g a 3  C (c H  c L )
GrC 
, 
2

f


f

K
Da  2 , Sc 

D
a

 C (c H  c L ) GrC
N

, 

 T (TH  TL ) GrT

k feff
a sf hsf a 2 

, 

k seff
k feff



   ( s   f )


  2 s  2 s
0  

2
Y 2
 X
U

    ( f   s )

(16)
C
C Pr   2 C  2 C 


V


X
Y Sc  X 2 Y 2 
(17)
By using the (stream function-vorticity)
formulation we will reduce the dependent
variables to only five variables by differentiating
eq.(13) with respect to (Y) and differentiating
eq.(14) with respect to (X), after that we subtract
the first of the two resulted equations from the
second to eliminate the pressure terms from the
momentum equations, thus, eqs.(12, 13 & 14) will
be transformed to the following equations;
(11.b)
 2  2

 
X 2 Y 2
 
U V 
X
Y
2
    2 
 Pr 2  2 
 X Y 
5, 6, 7 & 8), we get the dimensionless forms of
governing equations as follow;
(12)

 

V
(14)
(15)
By substituting eqs.(11.a & 11.b) in eqs.(3, 4,
U V

0
X Y

 

Pr 2 GrT  f  N C 
(11.a )
f

m
1  U
U 
U
V

2 
Y 
  X
P Pr   2U  2U




X
  X 2 Y 2
Journal of Engineering
 2   f
Pr GrT 
 X

2
 
 Pr F U
 
 Da Da

(13)
FUU
Pr
U
Da
Da
125
(17)
(18)






 U 

F  U
U
 
V

Y 
Da  X




N
C 

X 
DOUBLE DIFFUSIVE FREE CONVECTION IN A PACKED
BED SQUARE ENCLOSURE BY USING LOCAL THERMAL
NON-EQUILIBRIUM (LTNE) MODEL
Ahmed N. Mehdy
where ψ and ω are the (stream function &
NUMERICAL SOLUTION
The governing equations for ψ, ω, θf, θs & C
vorticity) respectively, and they are defined as
can be written in a common form for the
follow;
(convection-diffusion)


&V  
Y
X
V
U


X
Y
U 
(19.a )
(19.b)
variables in the flow field, calculations will be
made for local and mean Nusselt and Sherwood
C
X
 ui   

X i



  X
i



 S

( 22)
diffusion coeffecient Γ and the source term S in
et. al. , 2007) and (Khanafer and Vafai, 2002);
Shl 

X i
the dependent variables under consideration, the
numbers at the hot wall can be found as in (Wang



X 0 
follow
where the general scalar Φ stands for any one of
numbers, where the local Nusselt and Sherwood

 1 s
X
X 0
as
(Versteeg and Malalasekera, 1995);
After getting the final values of all dependent
  f
Nul   
 X

problem
the cartesian form are listed below for each
governing equation;
(20.a)
- Stream function equation
  ,   1, S  
(20.b)
(23.a)
X 0
- Vorticity equation
   ,    Pr , S 
Boundary Conditions
 2
  f
C 

N
Pr GrT 
X 
 X

 2 
 Pr F U 



Da 
 Da


The hydrodynamic boundary conditions for the
present problem at all enclosure walls will obey to
the non-slip condition, while the thermal and
solutal boundary conditions are (the left side wall
was kept at high temperature and solutal

 U 
F  U


V X  U Y 
Da 



concentration, the right side wall was kept at low
temperature and solutal concentration and finally
each one of the top and bottom walls were kept
insulated), thus the boundary conditions will be as
- Fluid phase energy equation
follow;
  f ,  
U  0, V  0,  f  s  C  1


at X  0 ;


U  0, V  0,  f  s  C  0


at X  1;

 f  s C
U  0, V  0,


0 

Y
Y Y

at Y  0 & 1

(23.b)
 ( s   f )
1
,S 
1
1 
1  1
(23.c)
- Solid phase energy equation
(21)
   s ,   1, S    ( f   s )
(23.d )
- Solutal concentration equation
 C ,  
126
Pr
, S 0
Sc
( 23.e)
Number 1
Volume 18 January 2012
The numerical solution of the governing
Journal of Engineering
- Solid phase energy equation
Su     f 


S p    

equations will be made according to the finite
volume method to transform the governing
equations from partial differential form to discrete
( 26.d )
- Solutal concentration equation
algebraic form, this method is based on principle
elements, each one of them is called (control
Su  0 

Sp  0 
volume), after that a discretization process
A computational program was written in
(Versteeg and Malalasekera, 1995) was carried
Fortran-90 language to compute the values of the
out by integrating eq.(22) (of the general
required variables, The discretized algebraic
conservation) over a control volume element,
equations are solved by the tri-diagonal matrix
where this equation will be as follow;
algorithm (TDMA). The used mesh size is
of dividing the flow field to a number of volume
a P  P  a E  E  aW W  a N  N  aS  S  S u
( 26.e)
(60×60). Relaxation factors of about (0.7–0.9) are
(24)
used for all dependent variables, Convergence
was measured in terms of the maximum change in
where;
aP  aE  aW  aN  aS  S P
each variable during an iteration where the
(25)
maximum change allowed for convergence check
was 10-6.
the source coefficients Su and Sp represent the
source terms of the discrete equation and their
RESULTS AND DISCUSSION
values for each governing equation are listed as
All solution were carried out for solution of
follow;
(Pr=0.71 & Sc=0.25) at GrT= 105, where the
- For the stream function equation
Su 
Sp

numerical code which is used in the present

 0
( 26.a )
investigation has been carried out for number of
simulations for a wide range of controlling
- For the vorticity equation
 2
  f
Pr GrT 
 X

2
Su   

 F  U

V
 Da  X


FU
2  Pr
S p   

Da
 Da

C  
 
N
X  

 U  
U

Y  



















parameters such as buoyancy ratio, Darcy number,
and porosity of the packed bed. Figs.(2 - 9) show
the effect of buoyancy ratio for different values of
(-15≤N≤15) on the stream function, fluid phase
(26.b)
temperature and solutal concentration contours
respectively at Da=10-1 and ε=0.9. Figs.[(2.a),
(2.b) & (2.c)] represent these contours for the case
of single diffusing effect at N=0, where there is
only thermal diffusing without solutal diffusing. It
is clear from the stream function contour at N=0
- Fluid phase energy equation
Su
Sp
and as a result of the thermal buoyancy effect, the
 s



1  1 




1  1 

fluid at the left-hand side hot wall will be lighter
( 26.c )
than in other locations while the fluid at the righthand side cold wall will be heavier than in other
locations, so, the fluid particles move upward
127
Ahmed N. Mehdy
DOUBLE DIFFUSIVE FREE CONVECTION IN A PACKED
BED SQUARE ENCLOSURE BY USING LOCAL THERMAL
NON-EQUILIBRIUM (LTNE) MODEL
along the hot wall while they move downward
fluid density will increase with increase of the
along the cold wall, and thus, the flow will take
solutal concentration, and as a result, the flow
the direction of clockwise. The fluid phase
begins to reverse its direction, So, the flow case at
temperature contour at N=0 indicates that the
N=-1 as it is shown in Figs.[(6.a), (6.b) & (6.c)]
temperature levels will decrease gradually from
represents a conversion point in the flow direction
the hot wall towards the cold wall, where the hot
from clockwise to counterclockwise as it is shown
fluid rises up along the left-hand side hot wall and
from the stream function distribution at N=-1,
descends along the right-hand side cold wall
where the clockwise main central vortex which
because of the thermal buoyancy effect. Finally
formed at each of the previous cases of positive
the solutal concentration contour at N=0 indicates
buoyancy ratio will divide into two main vortices
that the solutal concentration will be maximum at
in two opposite directions, also the distribution of
the hot regions and decreases gradually with the
both the fluid phase temperature and the solutal
decrease of the fluid phase temperature. After
concentration at N=-1 will begin to change their
that, and with increasing the positive buoyancy
directions and their gradient will be at the
ratio N>0 as shown in Figs.(3, 4 & 5) which
minimum levels at this case. With increasing the
represent the cases at N=5, 10 & 15, and as it is
value of negative buoyancy ratio as it is shown in
clear from these figures, the solutal buoyancy
Figs.[(7.a), (7.b) & (7.c)] which represent the
force which increases with the buoyancy ratio will
flow case at N=-5, the negative solutal buoyancy
cooperate with the thermal buoyancy and they'll
effect will be greater than the thermal buoyancy
drive the flow in the same direction to form a
effect, and as a result of the total negative
cooperator flow, where the stream function levels
buoyancy effect, the fluid at the hot wall will be
will increase with increasing in positive buoyancy
heavier than it in other locations while the fluid at
ratio because of the increase of the total buoyancy
the cold wall will be lighter than it in other
force due to both the thermal and solutal
locations, so, the fluid particles move downwards
diffusing, while as it is shown from the contours
along the hot wall while they move upwards along
of fluid phase temperature, its distribution will
the cold wall, and thus, the flow will take the
keep on the same previous behavior, but the
direction of counterclockwise. The fluid phase
gradients will be stronger than them at N=0 and
temperature contour at N=-5 indicates that the
increase directly with increase of the buoyancy
temperature levels will decrease gradually from
ratio, also a similar behavior to the fluid
the hot wall towards the cold wall, where the hot
temperature distribution will appear in the
fluid descends downwards along the hot wall and
contours of the solutal concentration, but their
rises up along the cold wall because of the large
gradients will be relatively less than the gradients
negative effect of solutal buoyancy. Finally the
of the fluid phase contours because of the absence
solutal concentration contour at N=-5 indicates
of internal transference sources as it is clear from
that the solutal concentration will be maximum at
eq.(23.e). Figs.(6, 7, 8, 9) the cases of negative
the hot regions and decreases gradually with the
buoyancy ratio N<0, where the negative value of
decrease of the fluid phase temperature. Figs.(8 &
buoyancy ratio means that the value of solutal
9) represent the flow case at N=-10 & -15
expansion coefficient βC is negative too, and as it
respectively, and as it is clear that the levels of
is clear from the density definition in eq.(1), the
counterclockwise stream function will increase
128
Number 1
Volume 18 January 2012
Journal of Engineering
with the increase of negative value of buoyancy
the value of local Nusselt number will be
ratio, while the fluid phase temperature and
maximum at the top of the hot wall and it
solutal concentration keep the same previous
descends gradually with dropping down to the
behaviours but their gradients increase with the
wall bottom because the temperature gradient will
increase of negative value of buoyancy ratio.
be very strong at the top and it decreases
Figs.(10.a & 10.b) indicate the variation of
gradually to the bottom. Fig.(12) represents the
velocity components with buoyancy ratio at the
variation of
intermediate vertical and horizontal locations
buoyancy ratio (which is analog to Nusselt
respectively for Da=10 and ε=0.9, where as it is
number in heat transfer, where it gives an
shown in Fig.(10.a) the horizontal velocity
indication to the rate of mass transfer through the
component at the positive values of buoyancy
solution which happens by solutal concentration
ratio will direct to the right in top half of the
difference), where it is clear that the distribution
enclosure while it will be in opposite direction in
of local Sherwood number will be similar to the
the bottom half of the enclosure because of the
distribution of local Nusselt number and for the
positive total buoyancy effect, while we'll note the
same mentionable reasons previously. Fig.(13)
conversion in flow direction nearly at N=-1 where
explains the variation of the mean Nusselt number
the horizontal velocity begins to reverse its
with buoyancy ratio for different values of Darcy
direction as it happens when the value of negative
number at ε=0.9, generally it is clear that the
buoyancy ratio becomes N<-1, also as it is shown
minimum rate of heat transfer is at N=-1 because
in Fig.(10.b), the vertical velocity component at
of the minimum temperature gradients at that
the positive values of buoyancy ratio will direct
case, after that, heat transfer levels begin to
upwards in the left half of the enclosure while it
increase with the increase of each one of positive
will be in opposite direction in the right half of the
or negative buoyancy ratios, also it is clear that
enclosure because of the positive total buoyancy
the values of Nusselt number will be lower than
effect, where it begins to reverse its direction at
them at the same values of positive buoyancy ratio
the negative buoyancy ratios of N<-1. Fig.(11)
because the thermal and solutal buoyancy effects
represents the variation of local Nusselt number
at negative buoyancy ratios will be in opposite
with buoyancy ratio at Da=10 and ε=0.9, where
direction while they in same direction at positive
it is clear that the value of local Nusselt number at
buoyancy ratios to form (assisting flow), and
N≥0 will be maximum at the bottom of the left-
finally it was noted that the mean Nusselt number
hand vertical wall and it descends gradually with
generally increases with Darcy number increase
rising to the wall top because the temperature
because of the decreasing in overall bed resistance
gradient will be very strong at the bottom and it
to the flow inside it as a result to the increasing in
decreases gradually to the top, also it is clear the
the volume of passable paths of fluid through the
increasing in the local Nusselt number levels with
packed bed. Fig.(14) explains the variation of the
the increase of positive buoyancy ratio, while the
mean Nusselt number with buoyancy ratio for
minimum levels of it will be at N=-1 because the
different values of porosity at Da=10-1, where it
temperature gradients were minimum at that
was noted that when porosity equals to a value in
buoyancy ratio value, also at N<-1 it is clear that
the range of (ε=0.5~0.8), the values of mean
-1
-1
129
local Sherwood
number
with
DOUBLE DIFFUSIVE FREE CONVECTION IN A PACKED
BED SQUARE ENCLOSURE BY USING LOCAL THERMAL
NON-EQUILIBRIUM (LTNE) MODEL
Ahmed N. Mehdy
Nusselt number will increase gradually with the
similarity in Nusselt number behavior with the
increase of porosity in that range, but when the
mentionable study, but there is a small difference
porosity increases above that range, the values of
between these values because of the using of
mean Nusselt number will decrease, and for
(LTNE) model in the present study which causes
explaining this phenomenon clearly we must
more heat transfer due to both the fluid and solid
return to Eq.(20.a), where it is noted that the
phases.
whole
value
of
local
Nusselt
number
is
multiplicand by the porosity while the second
CONCLUSIONS
term in this equation (solid phase term) was
This
paper
has
presented
a
numerical
multiplicand by the inverse of dimensionless
investigation of double-diffusive flow in a packed
thermal conductivity Λ-1 which decreases with the
bed square enclosure by using local thermal non-
increase of porosity 1  (1   ) k s  k f , so,
equilibrium (LTNE) model, and from the obtained


results, the following conclusions are drawn;
when the porosity values equal to or less than
1- At the positive buoyancy ratios N≥0, the flow
(0.8), the value of Nusselt number will increase,
takes the clock wise direction, while at N=-1
while it decreases after that value of porosity
because the value of dimensionless thermal
the flow begins to reverse its direction as it
conductivity will be very small and it causes
happens at N<-1.
decreasing in Nusselt number more than the
2- Generally, levels of flow, heat transfer and
increasing in it because of porosity. Fig.(15)
mass transfer increase with the increase of
shows the variation of the mean Sherwood
both the positive or negative buoyancy ratio
number with buoyancy ratio for different values
as a result of increasing in the total buoyancy
of Darcy number at ε=0.9, where Sherwood
effects.
number will comport a similar behavior to Nusselt
3- The values of Nusselt number increase with
number variation with buoyancy ratio and Darcy
increase of Darcy Number.
number and for the same mentionable reason
4- The values of Nusselt number increase with
previously, while Fig.(16) explains the variation
increase of porosity until the porosity reaches
of the mean Sherwood number with buoyancy
a certain value of about (ε ≈ 0.8) where the
ratio for different values of porosity at Da=10-1,
value of mean Nusselt number will decrease
where it is clear that the values of Sherwood
after that value of porosity.
number will increase with increase of packed bed
5-
porosity because the flow levels will increase as a
The values of Sherwood number increase
with increase of both Darcy Number and the
result of decreasing in the occupied volume by the
porosity.
bed through enclosure, where the overall packed
REFERENCES
bed resistance to the flow will decrease. To
exhibit the reliability of the presented results, the
A. Amiri and K. Vafai, Transient analysis
variation of mean Nusselt number with the
of incompressible flow through a packed bed,
buoyancy ratio at GrT≈1.4×105 was compared
International Journal of Heat and Mass Transfer,
with results of (Rahli and Bouhadef, 2004) as it
41, 4259-4279, 1998.
is shown in Fig.(17), where it is clear the
130
Number 1
A.
K.
Bukhari,
convection
in
a
Volume 18 January 2012
Double
horizontal
diffusive
porous
Journal of Engineering
International Journal of Heat and Mass Transfer
layer
50, 3684-3689, 2007.
superposed by a fluid layer, Umm Al-Qura Univ.
R.
A.
Mohamed,
Double-diffusive
J. Sci. Med. Eng. Vol. 15, No. 2, pp. 95-113,
convection-radiation
interaction
on
unsteady
2003.
MHD flow over a vertical moving porous plate
I. Hajri*, A. Omri and S. Ben Nasrallah,
with heat generation and soret effects, Applied
A numerical model for the simulation of double-
Mathematical Sciences, Vol. 3, no. 13, 629 - 651,
diffusive natural convection in a triangular cavity
2009.
using equal order and control volume based on the
R. C. Chaudhary and Arpita Jain,
finite elements method, Desalination 206, 579-
Combined heat and mass transfer effects on MHD
588, 2007.
free convection flow past an oscillating plate
K. Khanafer and K. Vafai, Double-
embedded in porous medium, Rom. Journ. Phys.,
diffusive mixed convection in a lid-driven
Vol. 52, Nos. 5–7, P. 505–524, Bucharest, 2007.
enclosure filled with a fluid saturated porous
S.C.
Saha,
M.A.
Hossain,
Natural
medium, Numerical Heat Transfer, Part A, 42:
convection flow with combined buoyancy effects
465-486, 2002.
due to thermal and mass diffusion in a thermally
M. Mamou, P. Vasseur and M. Hasnaoui,
stratified media, Nonlinear Analysis: Modelling
On numerical stability analysis of double-
and Control, Vol. 9, No. 1, 89–102, 2004.
diffusive convection in confined enclosures, J.
Versteeg H. K. and Malalasekera W., An
Fluid Mech., vol. 433, pp. 209-250, 2001.
introduction to computational fluid dynamics the
Md. Shariful Alam, M. M. Rahman and Md.
finite volumes method, Longman Group Ltd,
Abdul Maleque, Local similarity solutions for
1995.
unsteady MHD free convection and mass transfer
Y. Masuda, M. Yonaya, T. Ikeshoji, S.
flow past an impulsively started vertical porous
Kimura, F. Alavyoon, T. Tsukada, M. Hozawa,
plate with Dufour and Soret effects, Thammasat
Oscillatory double-diffusive convection in a
Int. J. Sc. Tech., Vol. 10, No. 3, July-Sept., 2005.
porous enclosure due to opposing heat and mass
O. Rahli and K. Bouhadef *, Double-
fluxes on the vertical walls, International Journal
diffusive natural convection in a partially porous
of Heat and Mass Transfer 45, 1365-1369, 2002.
square enclosure; effect of the inclination,
Yoshio Masuda, Michio Yoneya, Akira
Laboratoire LTPMP, Faculté de Génie Mécanique
Suzuki, Shigeo Kimura and Farid Alavyoon,
et Génie des Procédés, USTHB, Alger – Algérie,
Numerical analysis of double-diffusive convection
2004.
in a porous enclosure due to opposing heat and
Q. W. Wang*, M. Zeng, Z. P. Huang, G.
mass fluxes on the vertical walls -Why does
Wang, H. Ozoe, Numerical investigation of
peculiar oscillation occur?-, Research Center for
natural convection in an inclined enclosure filled
Compact chemical Process, National Institute of
with porous medium under magnetic field,
Advanced Industrial Science and Technology, 4-
131
DOUBLE DIFFUSIVE FREE CONVECTION IN A PACKED
BED SQUARE ENCLOSURE BY USING LOCAL THERMAL
NON-EQUILIBRIUM (LTNE) MODEL
Ahmed N. Mehdy
2-1 Nigatake, Miyagino-ku, Sendai 983-8551,
Japan, 2008.
Nomenclature
Greek symbols
a
asf
C
c
cp
D
Da
dp
F
GrC
GrT
g
hsf
k
K
N
Nu
p
P
Pr
Sc
Sh
T
u, v
U, V
x, y
X, Y
α
μ
ν
ρ
βT
βC
θ
ψ
ω
Φ
Γ
ε
Λ
ξ
side length of the enclosure (m)
specific surface area of the packed bed (m-1)
dimensionless solutal concentration
solutal concentration (kmol m-3)
specific heat at constant pressure (J kg-1 K-1)
concentration diffusion coefficient (m2 s-1)
Darcy number
sphere particle diameter (m)
geometric function
solutal Grashof number
Grashof number
gravitational acceleration (m s-2)
solid-fluid heat transfer coefficient(Wm-2K-1)
thermal conductivity (W m-1 K-1)
permeability (m2)
buoyancy ratio
Nusselt number
pressure (Pa)
dimensionless pressure
Prandtl number
Schmidt number
Sherwood number
temperature (K°)
velocity components (m s-1)
dimensionless velocity components
x-, y-coordinates (m)
dimensionless coordinates
thermal diffusivity (m2 s-1)
dynamic viscosity (kg m-1 s-1)
kinematic viscosity (m2 s-1)
density (kg m-3)
thermal expansion coefficient (K-1)
solutal expansion coefficient (kmol-1 m3)
dimensionless temperature
dimensionless stream function
dimensionless vorticity
general scalar dependent variable
diffusion coeffecient
porosity
dimensionless thermal conductivity
dimensionless solid-to-fluid heat transfer
coefficient
Subscripts
f
feff
H
i
L
l
m
s
seff
fluid
effective properties for fluid
high
axis indication
low
local value
mean value
solid
effective properties for solid
Packed Bed
TH
cH
g
TL
cL
a
y
x
Fig.(1) schematic diagram of the physical system
132
Number 1
Volume 18 January 2012
1
1
1
0.9
0.9
0.9
0.7
0.5
0.4
0.3
0.2
0.6
0.5
0.4
0.3
0.2
0.1
0.1
0
0
0
0.25
0.5
0.75
X
a) Streamlines
1
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0.25
0.5
0.75
0
1
1
1
1
0.9
0.9
0.9
0.7
0.6
0.7
0.6
0.5
Y
0.5
0.4
0.3
0.2
0.4
0.3
0.2
0.1
0.1
0
0
0.25
0.5
0.75
X
a) Streamlines
0.9375
0.875
0.8125
0.75
0.6875
0.625
0.5625
0.5
0.4375
0.375
0.3125
0.25
0.1875
0.125
0.0625
0.8
1
0.9375
0.875
0.8125
0.75
0.6875
0.625
0.5625
0.5
0.4375
0.375
0.3125
0.25
0.1875
0.125
0.0625
0.4
0.3
0.2
0.1
0
0.25
0.5
0.75
0
1
0
0.25
0.5
0.75
X temperature
X
b) Fluid phase
c) Solutal concentration
Fig. (3) Properties distribution at; N=5, Da=1×10-1 & ε =0.9
1
1
0.9
0.8
-0.001
-0.03
-0.1
-0.18
-0.25
-0.35
-0.44
-0.5
-0.55
-0.59
-0.62
-0.65
-0.67
0.7
0.6
0.4
0.3
0.2
0.9375
0.875
0.8125
0.75
0.6875
0.625
0.5625
0.5
0.4375
0.375
0.3125
0.25
0.1875
0.125
0.0625
0.8
0.7
0.6
Y
0.5
0.5
0.4
0.3
0.2
0.1
0.1
0
0
0.75
1
0.5
0.9
0.5
0.75
0.6
1
0.25
0.5
0.7
0.9
0
0.25
0.8
Y
-0.001
-0.03
-0.1
-0.17
-0.24
-0.3
-0.35
-0.4
-0.44
-0.47
-0.49
-0.51
-0.52
0
0
X
X temperature
b) Fluid phase
c) Solutal concentration
-1
Fig. (2) Properties distribution at; N=0, Da=1×10 & ε =0.9
0.8
Y
0.7
0.9375
0.875
0.8125
0.75
0.6875
0.625
0.5625
0.5
0.4375
0.375
0.3125
0.25
0.1875
0.125
0.0625
0.8
1
1
0.9375
0.875
0.8125
0.75
0.6875
0.625
0.5625
0.5
0.4375
0.375
0.3125
0.25
0.1875
0.125
0.0625
0.8
0.7
0.6
Y
Y
0.6
0.9375
0.875
0.8125
0.75
0.6875
0.625
0.5625
0.5
0.4375
0.375
0.3125
0.25
0.1875
0.125
0.0625
0.8
Y
-0.001
-0.009
-0.02
-0.035
-0.05
-0.065
-0.08
-0.095
-0.11
-0.12
-0.13
-0.137
-0.142
Y
0.8
Y
Journal of Engineering
0.5
0.4
0.3
0.2
0.1
0
X
a) Streamlines
0.25
b) Fluid
0.5
phaseX
0.75
0
1
0
0.25
0.5
0.75
1
X
c) Solutal concentration
temperature
Fig. (4) Properties distribution at; N=10, Da=1×10-1 & ε =0.9
1
1
1
0.9
0.9
0.9
Y
0.6
0.5
0.4
0.3
0.2
0.1
0
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0.25
0.5
0.75
X
a) Streamlines
1
0
0.9375
0.875
0.8125
0.75
0.6875
0.625
0.5625
0.5
0.4375
0.375
0.3125
0.25
0.1875
0.125
0.0625
0.8
0.7
0.6
Y
0.7
0.9375
0.875
0.8125
0.75
0.6875
0.625
0.5625
0.5
0.4375
0.375
0.3125
0.25
0.1875
0.125
0.0625
0.8
Y
-0.001
-0.03
-0.1
-0.18
-0.25
-0.35
-0.44
-0.52
-0.58
-0.63
-0.67
-0.7
-0.73
-0.75
-0.77
0.8
0.5
0.4
0.3
0.2
0.1
0
0.25
0.5
0.75
1
0
0
0.25
0.5
0.75
X
X
b) Fluid phase temperature
c) Solutal concentration
Fig. (5) Properties distribution at; N=15, Da=1×10 -1 & ε =0.9
133
1
DOUBLE DIFFUSIVE FREE CONVECTION IN A PACKED
BED SQUARE ENCLOSURE BY USING LOCAL THERMAL
NON-EQUILIBRIUM (LTNE) MODEL
Ahmed N. Mehdy
1
1
1
0.9
0.9
0.9
0.08
0.07
0.06
0.05
0.04
0.03
0.02
0.01
0.0025
0.001
-0.001
-0.005
-0.01
-0.013
-0.016
-0.02
-0.023
0.7
0.5
0.4
0.3
0.2
0.1
0
0
0.25
0.5
0.75
X
0.5
0.4
0.3
0.2
0.1
0
0.6
0.5
0.4
0.3
0.2
0.1
0
0.25
0.5
0.75
0
1
1
1
1
0.9
0.9
0.7
0.6
Y
0.5
0.4
0.3
0.2
0.5
0.4
0.3
0.2
0.1
0.1
0
0
0.25
0.5
0.75
X
a) Streamlines
0.9375
0.875
0.8125
0.75
0.6875
0.625
0.5625
0.5
0.4375
0.375
0.3125
0.25
0.1875
0.125
0.0625
0.8
1
0.2
0.1
0
0.25
0.5
0.75
0
1
0.7
0.6
0.7
0.6
Y
0.5
0.4
0.3
0.2
0.5
0.4
0.3
0.2
0.1
0.1
0
0
0.75
1
0.25
0.5
0.75
1
0.9375
0.875
0.8125
0.75
0.6875
0.625
0.5625
0.5
0.4375
0.375
0.3125
0.25
0.1875
0.125
0.0625
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0.25
0.5
0.75
0
1
0
0.25
0.5
0.75
c) Solutal concentration
b) Fluid phaseX temperature
X
-1
Fig. (8) Properties distribution at; N=-10, Da=1×10 & ε =0.9
0.7
0.6
0.5
0.4
0.3
0.2
1
0.9
0.9
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.1
0
0
0.75
1
0.9375
0.875
0.8125
0.75
0.6875
0.625
0.5625
0.5
0.4375
0.375
0.3125
0.25
0.1875
0.125
0.0625
0.8
Y
0.77
0.755
0.73
0.7
0.67
0.62
0.57
0.5
0.42
0.33
0.25
0.17
0.1
0.05
0.01
0.8
1
1
0.9375
0.875
0.8125
0.75
0.6875
0.625
0.5625
0.5
0.4375
0.375
0.3125
0.25
0.1875
0.125
0.0625
0.8
0.7
0.6
Y
1
0.9
0.5
0.9375
0.875
0.8125
0.75
0.6875
0.625
0.5625
0.5
0.4375
0.375
0.3125
0.25
0.1875
0.125
0.0625
0.8
Y
0.67
0.655
0.63
0.6
0.56
0.51
0.46
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0.01
0.8
X
a) Streamlines
0
X
c) Solutal concentration
b) Fluid phaseX temperature
-1
Fig. (7) Properties distribution at; N=-5, Da=1×10 & ε =0.9
1
0.25
0.9375
0.875
0.8125
0.75
0.6875
0.625
0.5625
0.5
0.4375
0.375
0.3125
0.25
0.1875
0.125
0.0625
0.3
0.9
0
1
0.4
1
0.5
0.75
0.5
0.9
X
a) Streamlines
0.5
0.6
1
0.25
0.25
0.7
0.9
0
0
0.8
Y
0.515
0.505
0.49
0.47
0.44
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0.01
0.6
Y
0.7
0.9
0
0.9375
0.875
0.8125
0.75
0.6875
0.625
0.5625
0.5
0.4375
0.375
0.3125
0.25
0.1875
0.125
0.0625
0.8
X
c) Solutal concentration
b) Fluid phaseXtemperature
a) Streamlines
Fig. (6) Properties distribution at; N=-1, Da=1×10-1 & ε =0.9
0.7
Y
0.6
1
0.8
Y
0.7
Y
Y
0.6
0.9375
0.875
0.8125
0.75
0.6875
0.625
0.5625
0.5
0.4375
0.375
0.3125
0.25
0.1875
0.125
0.0625
0.8
Y
0.8
0.5
0.4
0.3
0.2
0.1
0
0.25
0.5
0.75
b) Fluid phaseX temperature
1
0
0
Fig. (9) Properties distribution at; N=-15, Da=1×10-1 & ε =0.9
134
0.25
0.5
0.75
c) Solutal concentration
X
1
Number 1
Volume 18 January 2012
1
Journal of Engineering
10
N=-15
N=-10
N=-5
N=-1
N=0
N=5
N=10
N=15
0.9
7.5
0.8
5
N=-15
N=-10
N=-5
N=-1
N=0
N=5
N=10
N=15
Y
0.6
0.5
0.4
0.3
2.5
V
0.7
0
-2.5
-5
0.2
-7.5
0.1
-10
0
-3 -2.5 -2 -1.5 -1 -0.5
0
0.5
1
1.5
2
2.5
0
3
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
a) U- velocity component at the
b) V- velocity component at the
X
U
intermediate vertical location
intermediate horizontal location
Fig. (10) Variation of velocity components with buoyancy ratio
at the intermediate vertical and horizontal locations for Da=1×10-1 & ε =0.9
1
1
0.9
0.9
0.8
0.8
0.7
0.5
0.4
0.4
0.3
0.2
0.2
0.1
0.1
0
5
10
15
20
25
30
35
40
45
50
Fig. (11) Variation of local Nusselt number
Nu l
with buoyancy ratio at ε=0.9
& Da=1×10-1
55
0
0
2
4
6
8
10
12
14
16
18
20
22
24
26
Fig. (12) Variation of local Sherwood number
Sh l
with buoyancy ratio at ε=0.9
& Da=1×10-1
22
22
22
Da=1.E-5
Da=1.E-5
Da=1.E-4
Da=1.E-4
Da=1.E-3
Da=1.E-3
Da=1.E-2
Da=1.E-2
Da=1.E-1
Da=1.E-1
20
20
18
18
20
18
16
16
16
14
14
14
Nu m
Numm
Nu
0.5
0.3
0
N=-15
N=-10
N=-5
N=-1
N=0
N=5
N=10
N=15
0.6
Y
0.6
Y
0.7
N=-15
N=-10
N=-5
N=-1
N=0
N=5
N=10
N=15
1
12
12
12
10
10
10
8
8
8
6
6
6
4
4
4
2
2
Porosity=0.5
Porosity=0.6
Porosity=0.7
Porosity=0.8
Porosity=0.9
2
-15
-15
-10
-5
0
5
10
Fig.
-10 (13) Variation
-5
0 of mean
5 Nusselt
10
N
N number at ε=0.9
number with Darcy
15
15
135
-15
-10
-5
0
5
10
Fig.
(14) Variation
of mean
Nusselt
N
number with porosity at Da=1.E-1
15
DOUBLE DIFFUSIVE FREE CONVECTION IN A PACKED
BED SQUARE ENCLOSURE BY USING LOCAL THERMAL
NON-EQUILIBRIUM (LTNE) MODEL
Ahmed N. Mehdy
10
10
Da=1.E-5
Da=1.E-4
Da=1.E-3
Da=1.E-2
Da=1.E-1
9
8
7
6
6
Sh m
7
5
5
4
4
3
3
2
2
Porosity=0.5
Porosity=0.6
Porosity=0.7
Porosity=0.8
Porosity=0.9
1
1
-15
-10
-5
0
5
10
Fig. (15) Variation of
N mean Sherwood
number with Darcy number at ε=0.9
11
-15
15
-10
-5
0
5
10
Fig. (16) Variation of
N mean Sherwood
number with porosity at Da=1.E-1
Rahli' s Work at; Da=1.E-5
Present Work at; Da=1.E-5
Rahli' s Work at; Da=1.E-4
Present Work at; Da=1.E-4
10
9
8
7
Nu m
Sh m
8
9
6
5
4
3
2
1
-10
-5
0
5
10
5
Fig. (17) Comparison the variation of mean Nusselt
N number with buoyancy ratio at GrT≈1.4×10
with the work of (Rahli and Bouhadef, (2004)
136
15