Copyright © 2011 Praise Worthy Prize S.r.l. - All rights reserved International Review of Mechanical Engineering, Vol. 5, N. 2 Special Issue on Heat Transfer
356 geometric shape of the roof, we have considered a
conventional roof for reference and two other cases case a sloping roof and the other with mono form chapel.
The results are presented as hydrodynamic and thermal fields for different values of Rayleigh numbers. The
Nusselt number is plotted against the Rayleigh number characterizing the heat exchange and in the room. This
also allowed us to quantify the flow of the heat produced.
II. Mathematical Formulation
In this work, we present a numerical simulation in a two dimensional square cavity. Figure 1 shows the
geometries and boundary conditions of heat transfer from the physical configuration chosen. The upper and lower
walls are adiabatic. The side walls are maintained isothermal T = T
f
cold at T = T
c
in the left surface of contact between the heat source and the fluid air with a
constant temperature that is caused by a flow from the outside in providing heat throughout the study area. The
height of the cavity is H, the form factor A, which is the ratio of height to width is 1 A = 1. The sizing of the
heat source is represented by HS. The heat source and the left vertical wall have a height set at 0.5. All speeds
are limited by the conditions of non-slip.
The form 1 The form 2 The form 3 Fig. 1. Schematic of the three studied configurations
The third dimension of the cavity is assumed large enough that the fluid flow can be in two dimensions. The
fluid properties are evaluated at an average temperature and air flow is low because it is assumed laminar, the
fluid is assumed incompressible and obeys the Boussinesq approximation. Under these assumptions, the
continuity in two dimensions, the equations governing the flow and energy is given by:
Continuity: U
V X
Y ∂
∂ +
= ∂
∂ 1
X-momentum:
2
1 U
U U
P U
V U
t X
Y X
ρ
∂ ∂
∂ ∂
+ +
= − + ∇
∂ ∂
∂ ∂
2 Y-momentum:
2
1 V
V V
P T
U V
V g
t X
Y X
X
β ρ
∂ ∂
∂ ∂
∂ +
+ = −
+ ∇ + ∂
∂ ∂
∂ ∂
3 Energy:
2 p
T T
T U
V T
t X
Y C
λ ρ
∂ ∂
∂ +
+ =
∇ ∂
∂ ∂
4 The derived equation of motion 2 over Y and the
equation of motion 3 by contributing to X, then, after subtracting the two equations obtained, we obtain the
equations dimensionless variables in writing Helmotz in terms of vorticity and stream function formulation are as
follows:
2
T U
V Pr
Ra Pr t
X Y
X ω
ω ω
ω ∂
∂ ∂
∂ +
+ = ∇ +
∂ ∂
∂ ∂
5
2 2
2 2
T T
T T
T U
V t
X Y
X Y
∂ ∂
∂ ∂
∂ +
+ =
+ ∂
∂ ∂
∂ ∂
6
2 2
2 2
X Y
ψ ψ
ω
∂ ∂
+ = −
∂ ∂
7 The stream function and vorticity are related to the
velocity components by the following expressions: V
U U
, V and
Y X
X Y
ψ ψ
ω ∂
∂ ∂
∂ =
= − =
− ∂
∂ ∂
∂ 8
The dimensionless parameters in the equations above are defined as follows:
2 2
x y
X ,Y
, ,
L L
t and t
L L
ψ ψ
α ω
ω α
α =
= =
= =
9
III. Boundary Conditions
The boundary conditions, associated to the problem are as follows.
III.1. Initial Conditions
At time t = 0, U = 0, V = 0 and T = 0 throughout the
domain except on the hot wall at t = 0, U = 0, V = 0 and
T = 1. III.2. Boundary Conditions for T 0
All boundaries satisfy the no-slip velocity conditions: Top wall:
U=V=0, T
Y ∂ =
∂ Bottom wall:
U=V=0, T
Y ∂ =
∂
Copyright © 2011 Praise Worthy Prize S.r.l. - All rights reserved International Review of Mechanical Engineering, Vol. 5, N. 2 Special Issue on Heat Transfer
357 Right and left wall:
U=V=0, T= T
f
=0
TABLE I G
EOMETRICAL
C
HARACTERISTICS
, F
OR
T
HE
P
OSITIONS
O
F
T
HE
H
EATING
E
LEMENTS
W
ITH
L =
1, F
OR
T
HE
C
ONFIGURATIONS
S
TUDIED
Hs L=H Tf Tc Configuration 1
Configuration 2 Configuration 3
0.5 0.5
0.5 1
1 1
1 1
1
IV. Numerical Procedure
The non-linear partial differential governing equations, Eqs 5-7, were discritized using a finite
difference method. The numerical code used was written in FORTRAN
.
The first order upwind scheme was used for the convective terms to avoid possible instabilities
frequently encountered in convection problems. The set of algebraic equations are solved sequentially by TDMA
Tri-Diagonal Matrix Algorithm based on the direct
method [17], based on the implicit technique. The numerical code was validated against the benchmark
results of Vahl Davis [14]. We ran the computer code on a machine PC Pentium
4 with 560 MB and processor speed of RAM is 2.0 GHz. We tested the influence of the mesh on the results and
where we have used five meshes. The choice of the M5 mesh 10000 nodes gave a good compromise between
accuracy and computation time.
The results of the grids used for the three configurations are obtained with different values of
Rayleigh numbers [10
4
≤ Ra ≤10
6
], and Pr=0.71
V. Results and Discussion
Results are presented for natural convection in a cavities for 2D simulations for
Ra in the rang 10
4
-10
6
.
We performed numerical study for the form factor A = L
H = 1 and different values of Rayleigh number by taking the air as fluid filling the cavity. The results of our
study are presented as graphs representing the isotherms, streamlines and velocity fields. Changes in average
Nusselt numbers are also represented.
V.1. Validation To comply with our results we verified and validated
our computer code to adapt to our problem, we first made a classic test of the benchmark, in comparing our
results with the work. Excerpts from the literature [12,16]. Using three variables
U
max
, V
max
and average Nusselt number Table II,
For Rayleigh numbers equal to 10
4
and 10
5
[17]. After testing the reliability of our computer code, so it
can be exploited for other configurations of geometric complexity. We have studied the configurations
presented above, see Fig. 1 and Fig. 3.
TABLE II
C
OMPARISON
O
F
V
ALUES
F
ROM
O
UR
C
ODE
O
F
C
OMPUTING
A
ND
V
ALUES
O
F
V
AHL
D
AVIS
Ra U
max
V
max
Nu [Vhal davis 1983]
[Present ] [Vhal davis 1983]
[Present ] 10
4
10
5
16.189 16.180
36.460 38.497
19.197 19.212
62.790 68.575
2.212 2.257
4.450 4.647
We found distributions lines current and isotherms presented below:
Isotherms
a Pour Ra=10
4
b Pour Ra=10
5
c Pour Ra=10
6
Figs. 2. Isotherms for different types of roofing, for different numbers of Ra, Pr = 0.71
Figures 2 show the isotherms by varying the Rayleigh number of 10
4
, 10
5
and 10
6
in succession for a fixed value of Prandtl number equal to 0.71. We found that for
low Rayleigh numbers, the isotherms are parallel and this representation characterizes the heat transfer is
dominated by conduction, as and when the Rayleigh number increases, the isotherms become more wavy and
intensifies the heat transfer and the flow of natural convection that grows and grows and predominates over
conduction. If the Rayleigh number increases Nusselt increases which justifies the intensification of heat
transfer
.
Copyright © 2011 Praise Worthy Prize S.r.l. - All rights reserved International Review of Mechanical Engineering, Vol. 5, N. 2 Special Issue on Heat Transfer
358
Streamlines
a Pour Ra=10
4
b Pour Ra=10
5
c Pour Ra=10
6
Figs. 3. Streamlines obtained for for different types of roofs and different numbers of Ra, Pr = 0.71
Figures 3 show the streamlines corresponding to Rayleigh numbers: 10
4
.10
5
and 10
6
for the value of Prandtl = 0.71 and for the three roof types studied.
When Ra = 10
4
, the flow is almost symmetric, we see that there are cells that are closed on itself, the movement
of the liquid has been a movement block, which justifies the existence of recirculation zones more near the center
of the cavities, the flow lines are substantially parallel to the vertical walls.
So it also represents the current lines corresponding to the Rayleigh number equal to 10
5
, 10
6
, we find that the current lines are tightened at the vertical walls, this is due
to strong temperature gradient at the vertical walls and therefore the flow is parietal, which characterizes the
great heat transfer by convection to the vertical walls. We also observe the formation of two recirculation cells
in the cavity 10
6
. We also note in this latter case, the expansion of two recirculation zones of a cell in the
clockwise and the other in the opposite direction, with increasing Rayleigh number.
Continuing our investigation, 5 and 6 brought the influence of Rayleigh number on the dimensionless
horizontal velocity on a vertical line x = y = 0.5 and 0.5 horizontal center of the cavity. We found a significant
influence, the speed increases with increasing Rayleigh number. However, there is an increase, even with the
change in the shape of the roof. For example when
Ra = 10
5
for example, the maximum horizontal velocity for the flat roof form is the highest. However, there is growth of
the dimensionless vertical velocity for the three roof shape with the increase of
Ra. Fig. 5 profiles of the dimensionless vertical velocity
on a horizontal line at the center of the cavity y = 0.5 for
different Reynolds numbers and for the first configuration studied.
In continuing our investigation, the Fig. 5 brought the influence of Rayleigh number on the dimensionless
vertical velocity on a horizontal line at the center of the cavity. We found a significant influence, the speed
increases when the Rayleigh number increases. However, there is increased even with the maximum
horizontal velocity change of the position of the heater to an orientation of the latter towards the corners right. In
contrast, there was a decrease in speed at the center of the cavity, In the case of the centred position, it
represents an obstacle, where the flow is delayed.
We present in Figure 7, the influence of Rayleigh number and influence of the shape of roof on the average
Nusselt number. We found considerable influence of Rayleigh number.
Velocity and temperature profiles
Fig. 4. Dimensionless horizontal temperature profiles for the three forms of roofing, for Ra = 10
5
, Pr = 0.71, Y = 0.5
Fig. 5. Dimensionless horizontal velocity profiles for the three forms of roofing, for Ra = 10
5
, Pr = 0.71, x = 0.5
Copyright © 2011 Praise Worthy Prize S.r.l. - All rights reserved International Review of Mechanical Engineering, Vol. 5, N. 2 Special Issue on Heat Transfer
359
Fig. 6. Dimensionless horizontal velocity profiles for the three forms of roofing, for Ra=10
6
, Pr=0.71, y=0.5
Nusselt numbers
Fig. 7. Profiles of average Nusselt number for the three forms of roofing, for different values of Ra, Pr = 0.71
The increase of average Nusselt number with increasing Rayleigh number, whatever the shape of the
roof. The Nusselt number for the flat roof is the most important and which then represents the best heat
transfer in the cavity.
VI.
Conclusio
n
The objective of our work was to study the behavior of air in closed cavities with two-dimensional numerical
study of natural convection in a laminar flow by focusing on the shape of roof. Resolution of the equations
governing natural convection written in variables of and
Helmontz is done by the method of finite deference. We developed a computer code written in
FORTRAN, and we have validated the work found in the literature for hydrodynamic and thermal studies in the
cavities where it showed good consistency. For our simulation, we presented the streamlines, the isotherms
and the influence of major parameters such as roof shape on the characteristics of convection. We have shown by
this simulation the following: 1. Convection affects the structure of the isotherms.
2. The temperature of the fluid increases dramatically
with the growth of Rayleigh number. 3. The roof shape to influence the transfer of heat to
warm the entire field of study in an identical manner, it must be positioned in the center.
4. The shape of the roof has an influence on heat transfer.
5. The best transfer was confirmed for the flat roof.
References
[1] Les investissements économiseurs d’énergie, Page 1-3, Bruxelles,
Janvier 2008. [2]
Réglementation thermique des bâtimen-ts confortables et performants, 2005.
[3] A. Bejan, A.D. Kraus Eds., Heat Transfer Handbook, Wiley,
New York, 2003. [4]
Benachour Elhadj, Simulation Numérique de la Convection Naturelle et Mixte Dans Une Cavité Carrée Avec la Présence d’un
élément de Climatisation, Mémoire de Magister, présenté à lU. S. T. Oran et luniversité de Béchar, Algérie, Mars 2010.
[5] D.B. Ingham, I. Pop Eds., Transport Phenomena in Porous
Media, II, Pergam-on, Oxford, 2002. [6]
K. Vafai Ed., Handbook of Porous Media, Marcel Dekker, New York, 2000.
[7] N.B. Cheikh, B.B. Beya, T. Lili, Aspect ratio effect on natural
convection flow in a cavity submitted to a periodical temperature boundary,J. Heat Transfer 129 2007 1060–1068, 2007.
[8] V.A.F. Costa, M.S.A. Oliveira, A.C.M. Sousa, Control of laminar
natural convection in differentially heated square enclosures using solid inserts at the corners, Int. J. Heat Mass Transfer 46 2003
3529–3537,2003. [9]
G. Accary, I. Raspo, A 3D finite volume method for the prediction of a supercritical fluid buoyant flow in a differentially heated
cavity, Int. J.Heat Mass Transfer 35 2006 1316–1331,2006. [10]
J.K. Platten, M. Marcoux, A. Mojtabi, The Rayleigh–Bénard problem in extremely confined geometries with and without the
Soret effect,C. R. Mecanique 335 2007 638–654,2007. [11]
B. Calgagni, F. Marsili, M. Paroncini, Natural convective heat transfer in square enclosures heated from below, Appl. Therm.
Eng. 25 20052522–2531,2005. [12]
F. Stella, E. Bucchigani, Rayleigh–Bénard convection in limited domains: part 1 – oscillatory flow, Numer. Heat Transfer A 36
1999 1–16,1999. [13]
P. Le Quéré, Accurate solutions to the square thermally driven cavity at high Rayleigh number, Comput. Fluids 20 1991 29–
41,1991. [14]
L G. De Vahl Davis, Natural convection of air in a square cavity: a bench mark numerical solution, Int. J. Numer. Methods Fluids 3
1983249–264,1983. [15]
S. Xin, P. Le Quéré, An extended Chebyshev pseudo-spectral benchmark for the differentially heated cavity, Int. J. Numer.
Methods Fluids 402002 981–998,2002. [16]
M. Hortmann, M. Peric, G. Scheuerer, Finite volume multigrid prediction of laminar natural convection: bench-mark solutions,
Int. J. Numer.Methods Fluids 11 1990 189–207, 1990. [17]
S.V.Patanker. Numerical Heat Transfer and Fluid Flow, Hemisphere, New York,1980.
Copyright © 2011 Praise Worthy Prize S.r.l. - All rights reserved International Review of Mechanical Engineering, Vol. 5, N. 2 Special Issue on Heat Transfer
360
Authors’ information
1
Universitaire de Bechar, Faculté des sciences et technologie,
Département de sciences exactes, BP417 route de Kenedza, Bechar, Algeria.
Tel : 00213 49 81 90 24. E-mails:
benachour_elhadj yahoo.fr bdraouiyahoo.com
rahmani_l1yahoo.fr brahimo12002yahoo.fr
2
Département de Génie Mécanique, Faculté de Génie Mécanique,
USTOran. E-mail:
imine _byahoo.fr
Benachour El Hadj, born on November 20, 1973 in Bechar, Algeria. He received his B.S.
degree in mechanical engineering from the Institute of Mechanics at the University of
Bechar 1998, the MS Degree in Physics and heat the building of the Institute of Mechanics at the
USTOran in 2010. He is currently a PhD candidate at the doctoral School. He is currently
working in the convection and thermal comfort in the building.
Professor Doctor Draoui Belkacem received MEng from the University of Sciences and
Technology of Oran Algeria in January 1988 and PhD thermal degree from University of Nice
French in 1994. His scientific interests are Energy applications in Agriculture and
Horticulture. .
Rahmani Lakhdar, born on July 14, 1975 in Bechar, Algeria. He received his B.S. degree in
Mechanic engineering from the Mechanics Institute of Bechar University in 1998, the M.S.
degree in Energetically Physics from the Mechanics Institute of Bechar University in
2004. Currently, he is a Ph.D. candidate in the Bechar University Graduate School. He
presently works in the Agitate systems.
Mebarki Brahim, born on June 05, 1980 in Mecheria, Algeria. He received his B.S. degree
in Climatic engineering from the Mechanics Institute of Bechar University in 2004, the M.S.
degree in Energetically Physics from the Mechanics Institute of Bechar University in
2007. Currently, he is a Ph.D. candidate in the Bechar University Graduate School. He
presently works in the Agitate systems.
Belloufa Larbi, born on March 6
t h
. 1965 in Beni-ounif, Bechar, Algeria. He received his
graduation degree at the university of Oran in 1989. His ma diploma in 2006 at the University
of Oran. Currently, he is a Ph.D. candidate working on civilization, specialty English.
Special Issue on Heat Transfer, February 2011
Manuscript received and revised January 2011, accepted February 2011 Copyright © 2011 Praise Worthy Prize S.r.l. - All rights reserved
361
Computational Study of the Conjugate Heat Transfer and the Wall Conduction Effects in
a Horizontal Pipe with Temperature Dependent Properties
K. Chahboub, T. Boufendi
Abstract – The heat transfer between a uniformly heated pipe within its entire wall thickness and an internal laminar fluid flow is considered. We present the results of the 3D numerical simulation
of the conjugate heat transfer - mixed convection in the fluid and conduction in the wall. The dynamic viscosity, the thermal conductivity and the density of the fluid are temperature dependent.
The convective and radiative thermal losses between the outer surface of the duct and the ambient are considered. The model equations are discretized by the finite volumes method. The sequential
solution of the discretization equations of the computed variables follows the SIMPLER algorithm. The iterative solution of a system of discretization equations is achieved by the sweeping method
using the Thomas and the tri diagonal cyclic algorithms. Some results are validated by comparison with former published work. The influence of the pipe material conductivity on the
heat transfer and the fluid flow is analyzed for four different materials inconel, tantalum, stainless steel and aluminum. It is found that the Inconel pipe, compared to steel and tantalum
provides better heat transfer between the wall and the fluid. Copyright © 2011 Praise Worthy Prize S.r.l. - All rights reserved.
Keywords:
Numerical Simulation, Conjugate Heat Transfer, Mixed Convection, Variable Properties, Horizontal Duct
Nomenclature
D Pipe diameter,
m g
Gravitational acceleration, m G
Volumetric heat generation,
[
=K
s
Re Pr
-1
] Gr
Modified Grashof number,
[=
g βG
D
i 5
K
s -1
ν
-2
] h
Local heat transfer coefficient, W·m
-2
K h
c
Convective heat transfer coefficient, W·m
-2
K h
r
Radiative heat transfer coefficient, W·m
-2
K K
Fluid thermal conductivity coefficient, W·m
-1
K
s
K
Dimensionless solid thermal conductivity,[=K
s
·K
-1
] L
Pipe length, m Nu
Nusselt number, [=hD
i
K
-1
] P Pressure,
N·m
-2
Pr Prandtl number,
[
= ν·α
-1
]
Ra Rayleigh number,
[
=Gr·Pr
]
Re Reynolds number,
[
=V D
i
· ν
--1
]
r Radial coordinate, m
T Temperature,
o
K t Time,
s V Velocity,
m·s
-1
z Axial coordinate, m
Greek Symbols α
Thermal diffusivity β
Thermal expansion coefficient ε
Emissivity of the outer wall σ
Stephan-Boltzmann constant µ
Dynamic viscosity ν
Kinematic viscosity Θ
Angular coordinate ρ
Density τ
Stress Subscripts Superscripts
b Mean
i, o Inner and outer wall respectively
r, θ z
Radial, angular and axial direction respectively
∞ Ambient air away the outer wall
Entrance of the duct
Dimensionless
I. Introduction
