جاسم محمد
Al-Khwarizmi
Engineering
Journal
Al-Khwarizmi Engineering Journal, Vol. 8, No. 3, PP 81 - 94 (2012)
Natural Convection Heat Transfer in an Inclined Open-Ended
Square Cavity with Partially Active Side Wall
Jasim M. Mahdi
Department of Energy Engineering/ College of Engineering/ University of Baghdad
E-mail: jasim_ned@yahoo.com
(Received 10 November 2011; accepted 7 August 2012)
Abstract
This paper reports a numerical study of flow behaviors and natural convection heat transfer characteristics in an
inclined open-ended square cavity filled with air. The cavity is formed by adiabatic top and bottom walls and partially
heated vertical wall facing the opening. Governing equations in vorticity-stream function form are discretized via finite-
difference method and are solved numerically by iterative successive under relaxation (SUR) technique. A computer
program to solve mathematical model has been developed and written as a code for MATLAB software. Results in the
form of streamlines, isotherms, and average Nusselt number, are obtained for a wide range of Rayleigh numbers 103-106
with Prandtl number 0.71 (air) , inclination angles measured from the horizontal direction 0º-60º , dimensionless lengths
of the active part 0.4-1 ,and different locations of the thermally active part at the vertical wall. The Results show that
heat transfer rate is high when the length of the active part is increased or the active part is located at middle of vertical
wall. Further, the heat transfer rate is poor as inclination angle is increased.
Keywords: Natural convection, square cavity, partially active side wall.
1. Introduction
Natural convection in a cavity with one
vertical side open has received increasing
attention in recent years due to its importance in a
number of practical applications such as heat
transfer and fluid flow in solar thermal receiver
systems [1], fire spread from room [2], and
cooling of electronic equipment [3]. Many
theoretical and experimental studies on natural
convection in the 2-D open square cavity have
been carried out [4]–[10]. For the sake of brevity
literature review will be only restricted to studies
of natural convection in open square cavities with
adiabatic walls and isothermal at the wall facing
the aperture. Chan and Tien [4] performed a
numerical study of laminar natural convection in a
two-dimensional square open cavity with a heated
vertical wall and two insulated horizontal walls.
To overcome the difficulties of unknown
conditions at the opening, they made their
calculations in an extended computational domain
beyond the opening. Results obtained for
Rayleigh numbers ranging 103 - 106 were found to
approach those of natural convection over a
vertical isothermal flat plate. Later on, the same
authors [5] investigated open shallow cavity for
Rayleigh numbers up to 106 to test the validity of
approximate boundary conditions at the opening
for a computational domain restricted to the cavity
instead of using an extended domain. They
concluded that for a square open cavity having an
isothermal vertical side facing the opening and
two adiabatic horizontal sides, satisfactory heat
transfer results could be obtained by the restricted
domain. Polat and Bilgen [6] numerically studied
inclined fully open shallow cavities in which the
side facing the opening was heated by constant
heat flux, and two adjoining walls being adiabatic
over Rayleigh number values ranging 103 - 1010.
The opening was in contact with a reservoir at
constant temperature and pressure; the
computational domain was restricted to the cavity.
They observed that heat transfer approaches
asymptotic values at Rayleigh numbers
independent of the aspect ratio and asymptotic
mailto:jasim_ned@yahoo.com
Jasim M. Mahdi Al-Khwarizmi Engineering Journal, Vol. 8, No. 3, PP 81- 94 (2012)
82
values are close to that for a flat plate with
constant heat flux. Chakroun [7] performed an
experimental investigation to study the effect of
wall conditions as well as its tilt angle on heat
transfer for a fully open tilted cavity.
His study contains seven different wall
configurations over an inclination angle measured
from the vertical direction range −90º to +90º. It
was concluded that thetilt angle, wall
configuration, and number of hot walls are all
factors that strongly affect the natural convection
inside the fully open cavity. Nateghi and Armfield
[8] numerically studied natural convection flow in
a two dimensional inclined open cavity for both
transient and steady-state flow over Rayleigh
numbers ranging 105 - 1010 with Prandtl number
0.71 and inclination angles 10º - 90º. Their results
showed that the flow is steady at the low Rayleigh
number for all angles and becomes unsteady at the
high Rayleigh number for all angles and the
critical Rayleigh number decreases as the
inclination angle increases. Hinojosa et al. [9]
presented results for natural convection and
surface thermal radiation in a square tilted open
cavity. Effects of Rayleigh number 103 - 107 and
inclination angles 0º - 180º are investigated for a
fixed Prandtl number (0.7). The results show that
the convective Nusselt number changes
substantially with the inclination angle of the
cavity, while the radiative Nusselt number is
insensitive to the orientation change of the cavity.
Mohamad et al. [10] simulated the natural
convection in an open cavity using Lattice
Boltzmann Method. The paper demonstrated that
open boundary conditions used at the opening of
the cavity are reliable, where the predicted results
are similar to that of conventional CFD method.
Prandtl number is fixed to 0.71 while Rayleigh
number and aspect ratio of the cavity are changed
in the range of 104 to 106 and of 0.5 to 10,
respectively. The results show that the rate of heat
transfer deceases asymptotically as the aspect
ratio increases and may reach conduction limit for
large aspect ratio.
Most of the above-cited works deal with the
active vertical wall of the cavity at isothermal or
isoflux condition. In real life such as in fields like
solar energy collection and cooling of electronic
components, the active wall may be subject to
abrupt temperature nonuniformities due to
shading or other natural effects[11], therefore only
a part of the wall is either in isothermal or isoflux
condition. The present study deals with the natural
convection in an open square cavity filled with air
as working fluid with partially thermally active
vertical wall, for different lengths of thermally
active part. The active part is located at the top,
middle and bottom to locate the position where
the heat transfer rate is maximum and minimum.
2. Mathematical Formulation
The geometry, the coordinate system, and the
boundary conditions for the problem under
consideration are shown in Fig. (1). A portion of
the right side wall of the cavity is kept at a
temperature Th and the remaining parts are
insulated. Under the assumptions of a two-
dimensional viscous incompressible fluid with
steady-state conditions, the equations governing
the motion and the temperature distribution inside
the cavity may be written as: [12]
Continuity equation ∂u∂Χ + ∂v∂Y = 0 …(1)
Conservation of momentum in x-direction
ρ u ∂u∂Χ + v ∂u∂Y = − ∂p∂x +μ ∂ u∂x + ∂ u∂y + ρgsinφ
…(2)
Conservation of momentum in y-direction
ρ u ∂v∂Χ + v ∂v∂Y = − ∂p∂y +μ ∂ v∂x + ∂ v∂y − ρgcosφ
…(3)
Energy Equation
u ∂T∂Χ + v ∂T∂Y = α ∂ T∂x + ∂ T∂y …(4)
As in many investigations of natural
convection, all fluid properties, except density in
buoyancy term(ρg), are assumed constants and the
Boussinesq approximation is used to incorporate
the temperature dependence of density in the
momentum equation. In this approximation the
density in buoyancy term is assumed to be a linear
function of temperature, ρ = ρ∞ 1 − β(T − T∞)
[12].
The use of vorticity–stream function
formulation can eliminate pressure gradient terms
from momentum equations and simplify the
solution procedure. With the stream function, the
velocity components u and v and vorticity can be
expressed as:
Jasim M. Mahdi Al-Khwarizmi Engineering Journal, Vol. 8, No. 3, PP 81- 94 (2012)
83
u = − ∂ψ∂y , v = ∂ψ∂x and ω = ∂u∂y − ∂v∂x
Furthermore by introducing the following no
dimensional variables: (X, Y) = (x, y) L ,⁄ (U, V) = (u, v) (α L⁄ )⁄ Ψ = ψ α⁄ , Ω = ω (α L ⁄ ),⁄
θ = (T − T∞) (T − T∞)⁄ with T > T∞ ,
the governing equations in terms of vorticity-
stream function form become:
Stream function ∇ Ψ = −Ω …(5)
Vorticity ∇ Ω = 1Pr U ∂Ω∂Χ + V ∂Ω∂Y −Ra ∂θ∂Χ cosφ − ∂θ∂Y sinφ
…(6)
Energy ∇ θ = U ∂θ∂Χ + V ∂θ∂Y …(7)
where U = − ∂Ψ∂Y V = ∂Ψ∂X …(8)
The dimensionless parameters appearing in
equations (5) through (8) are the Prandtl number Pr = ν α⁄ and the Rayleigh number Ra =gβ(T − T∞)L αν⁄ .
Other parameters which enter into this study are
the dimensionless length of the active part H = (0.4 ≤ H ≤ 1) and the dimensionless
position of the active part center S = (H 2⁄ ≤ S ≤ 1 − H 2⁄ ).
The appropriate boundary conditions in
dimensionless form can be formulated as: at X = 0 , S ≤ Y ≤ S + ,
Ψ = 0 Ω = − ∂ Ψ∂X θ = 1
at X = 0, 0 ≤ Y < − , S + < ≤ 1,
Ψ = 0 Ω = − ∂ Ψ∂X ∂θ∂X = 0
at 0 < X < 1 , Y = 0,1 , and
Ψ = 0 Ω = − ∂ Ψ∂Y ∂θ∂Y = 0
at X = 1 , 0 < Y < 1. Ψ = Ω = 0, θ = 0 if U ≤ 0 (in low ) θ = 0 if U > 0(out low)
Local Nusselt number on thermally active part is
defined by: Nu = − h Lk = LT − T∞ ∂T∂x
Where the local heat transfer coefficient hy is
defined by equating heat transfer by convection to
that by conduction at the active part: h(T − T∞) = −k ∂T∂x
by introducing the dimensionless variables
defined above, local Nusselt number will be:
Nu = θ
The average Nusselt number is obtained by
integrating local Nusselt number over the active
part:
Nu = Nu. dY ( ⁄ ) ( ⁄ )
Fig. 1. Schematic of the Square Cavity, the
Coordinate System and Boundary Conditions.
3. Numerical Technique
A finite difference method (FDM) with central
difference scheme is used to discretize the system
of the partial differential equations (5 - 8). The
new algebraic equations system will be solved
using iterative under relaxation method (URM), to
give approximate values of the dependent
variables at a number of discrete points called
(grid points or nodes) in the computational
domain. These nodes are formed by subdividing
the computational domain in the X and Y
directions with vertical and horizontal uniformly
s
h
Th
L
x
y
L
g
φ
Jasim M. Mahdi Al-Khwarizmi Engineering Journal, Vol. 8, No. 3, PP 81- 94 (2012)
84
spaced grid lines. Based on a grid independence
study a grid of 61 × 61 nodes is adopted typically
in this study. However, a careful check for the
grid independence of the numerical solution has
been made to ensure the accuracy and validity of
the numerical schemes. For this purpose, four grid
systems, 45×45, 61×61, 81×81, and 101×101 are
tested. Results show the average Nusselt number
prediction for 61×61 grid points nearly
correspond to that obtained from other grid points,
the maximum relative error (% є) obtained
through the five grid systems is no more than 1%
as shown in Table (1). The accuracy of the
numerical code was assessed by applying it to a
case studied by Mohamad et al. [10], who
considered a cavity configuration similar to that
being considered in this paper with a fully active
hot vertical wall, and good agreement was
observed, as shown in Table (2). The maximum
difference obtained for the average Nusselt
number at the hot vertical wall was 1.7 %.
Furthermore, in order to lend more confidence in
the present numerical results, the results in form
of isothermals contours were compared with those
of Nateghi and Armfield [8] and Mohamad et al.
[10]; and an excellent agreement was achieved,
Fig.(2).The convergence criteria employed to
terminate the computations and reach the solution
were preassigned as (Ψζ - Ψζ-1)/Ψζ ≤ 10-6,
(θζ - θζ-1)/θζ ≤ 10-6, and (Ωζ - Ωζ-1)/ Ωζ ≤ 10-6, the
indices ζ and ζ −1 represent the current and
previous iteration, respectively.
Table 1,
Grid Independence Study Results for Average
Nusselt Number.
% є 101×101 81×81 61×61 45×45 Size
1.0 1.189 1.188 1.180 1.173 Ra = 103
0.5 3.318 3.327 3.319 3.308 Ra = 104
0.3 7.298 7.301 7.301 7.284 Ra = 105
0.7 14.27 14.23 14.22 14.32 Ra = 106
Table 2,
Comparison of the Obtained Results for Average
Nusselt Number with Literature.
Difference (%) Nu [this study] Nu [10] Ra
1.7 3.319 3.377 104
0.2 7.301 7.323 105
1.1 14.22 14.38 106
4. Results and Discussion
In this section, the numerical results in form of
the streamlines, isotherms, and average Nusselt
numbers for various values of the parameters
governing the heat transfer and fluid flow will be
presented and discussed. The parameters are
location of thermally active part, the
dimensionless length of thermally active part, H =
0.4 - 1.0, Rayleigh number, Ra = 103 - 106, and
inclination angle, φ =0º - 60º. Prandtl number, Pr
= 0.71 for air was kept constant. In Figures (3-6),
the darkened area of side wall indicates the
position of thermally active part while the symbol
( ) represents the center of the circulation which
defined as the point of the extremum of the stream
function Ψmax.
The scenario of development natural
convective currents inside open cavity begins
when the air in contact with the active part of side
wall heated by conduction, becomes lighter, and
ascends in adjoin to the side wall to face the
adiabatic upper wall and then through which it
moves horizontally to leave the cavity from the
upper part of the aperture. In order to retrieve
fluid left outside the cavity, fresh air entrained
from the lower part of the aperture toward heated
portion of side wall to be heated and resulting in
clockwise recirculation inside cavity, as shown in
left sides of Figs. (3-6).
4.1. Effect of Active Part Length
The effects of active part length H on flow and
temperature fields are shown in figure (3) for
Rayleigh number, Ra = 105, inclination angle, φ
=0° and position of active part is at middle. The
values of maximum stream functions Ψmax show
that higher values are obtained with increasing the
active part length from H=0.4 to H=1. In fact,
when the active part is wider major portion of the
cavity is occupied by circulating cells (see
streamlines in Fig. (3)) and flow strength
increases making the cavity more significantly
affected by the buoyancy-driven flow. The
corresponding isotherms indicate that as H
increases the isotherms are denser near the active
part that means thermal boundary layer becomes
thinner and temperature gradients becomes
steeper due to strong convection. The amount of
heat transferred from the active part across the
cavity obviously increases as the length H of the
thermally active part increases due to higher heat
transfer surface. Correspondingly, average Nusselt
number which would describe the thermal
Jasim M. Mahdi Al-Khwarizmi Engineering Journal, Vol. 8, No. 3, PP 81- 94 (2012)
85
behavior of the cavity should increase with
increasing H as shown in Fig. (7).
4.2. Effect of Active Part Location
Figure (4) shows the effect of active part
position on the streamlines and isotherms patterns
at Ra = 105, φ =0°, and dimensionless length of
active part, H=0.4. In fact, due to impingement of
circulated fluid to the middle of the vertical wall
and impingement of hot fluid to the top adiabatic
wall, fluid at right top and bottom corner becomes
motionless and also, the half bottom of the cavity
becomes cooler than the upper one. Thus strength
of circulation becomes larger, as clearly denoted
by the higher value of Ψmax, and major portion of
the cavity is occupied by circulating cells as the
active part occupies a position lower in the cavity
(see streamlines in Fig. (4)). In contrast, when the
active part is located at the bottom or the top of
vertical wall the rising fluid by buoyancy force
cannot wash the entire surface of the active part
and thermal boundary layer around the active part
gets thicker (see isotherms in Fig. (4)), thus the
amount of heat exchanged is smaller than that
exchanged when the active part is located at
middle of vertical wall. Subsequently, the average
Nusselt number which describes the overall heat
transfer process through the cavity is higher when
the active part located at middle as shown in Fig.
(8).
4.3. Effect of Inclination
The effect of inclination angle φ in the range
of 0 to 60◦ which covering practical cases in solar
collectors [6] is examined for Ra=105 and H=0.6.
Fig. (5) shows the effect of inclination angle on
streamlines and isotherms patterns. Streamlines
show that as the inclination is increased flow of
the hot fluid from the aperture is choked and a
smaller part of the upper region of the cavity is
occupied by discharged fluid. As a result, the
center of the circulation is impelled upward
indicating significant increase in fluid velocities.
The isotherms show a thicker thermal boundary
layer is formed on the active part of side wall, and
also a thinner hot intrusion on the upper insulated
wall formed as inclination angle increases.
Increasing the inclination angle in the range 0º ≤
φ ≤ 60º leads to a decrease in the average
Nusselt number for the same dimensionless
length of active part as shown in Fig. (9). This
because as φ increases, the thermal boundary
layer on active part get thicker and fluid thermal
stratification in the upper region of the cavity get
poorer (see isotherms in Fig. (5)), since flow of
the hot fluid from the aperture is choked as the
inclination is increased and that leads to slower
replacement of the hot air by fresh air and that
makes convection less rigorous.
4.4. Effect of Rayleigh Number
Figure (6) shows the effect of increasing
Rayleigh number in the range (103-106) on
streamlines and isotherms patterns at H=0.8 and φ
=0°. Usually, the Rayleigh number, is defined by
(Ra = gβL3ΔT/να), and in case of using
Boussinesq approximation with air as a working
fluid, the ν, α and g will be constants. Thus,
only β and ΔT will be the factors that govern
the change in value of Rayleigh number. In case
of Ra =103 , the circulation inside the cavity is so
weak that the viscous forces are dominant over the
buoyancy force and that leads to rather weak
convection, also the effect of the open side does
not seem deep in this case. Consequently, the
corresponding isotherms exhibit rather slight
difference as compared to those of pure heat
conduction which are parallel to the surface of the
thermally active part. On the other hand, when
Rayleigh number increases, the buoyancy force
accelerates the circulation of fluid flow and the
effect of the open side penetrates further into the
flow. As a result, the corresponding isotherms are
greatly deformed and crowded near the active part
leaving the cavity core empty. Therefore, as
Rayleigh number increases, the flow becomes
fully convective dominated, the fresh fluid is
entrained right to the left vertical wall where
high temperature gradients are created, and the
discharging fluid from the upper part of the cavity
occupies smaller section of the opening. It is
interesting to note in case of Ra = 106, the
formation of vortex in the upper portion of the
cavity due to the high heat transfer rate, also the
effect of the open side is quite significant in this
case.
Jasim M. Mahdi Al-Khwarizmi Engineering Journal, Vol. 8, No. 3, PP 81- 94 (2012)
86
(a) Pr = 0.71 Ra = 105 φ = 0º H=1
(b) Pr = 0.71 Ra = 105 φ = 10º H=1
Fig. 2. Comparison of Isotherms with those of (a) Mohamad et al.[10] and (b) Nateghi and Armfield [8].
Jasim M. Mahdi Al-Khwarizmi Engineering Journal, Vol. 8, No. 3, PP 81- 94 (2012)
87
Streamlines
Isotherms
a) H = 0.4
b) H = 0.6
c) H = 0.8
d) H = 1
Fig. 3. Streamlines and Isothermal Contours for φ=0º, location of Active Part at Middle and Ra =105, with
Different Lengths of Thermally Active Part.
1. 6
1.6
1. 6
1.6
3.2
3.2
3.2
3.2
4.8
4.8
4.8
4.8
6.4
6. 4
6.4
6. 4
8
8
8
9.6
9.6
9.6
11.2
11.2 11.2
12.8
12. 8
14.4
0.
1
0.1 0.1
0.
2
0. 2 0.2
0.
3
0. 3 0.3
0.
4
0.4
0.4
0.
5
0.5
0.
6
0.6
0.
7
0.
8
0.
91
1.7
1.7
1.7
1.7
3.4
3.4
3.4
3.4
5.1
5.1
5.1
5.1
6.8
6.8
6.8
6. 8
8. 5
8.5
8.5
10. 2
10.2
10.2
11.9
11.9
11. 9
13.6
13.6
15.3
0.
1
0.1 0.1
0.
2
0. 2 0.2
0.
3
0. 3 0.3
0.
4
0.4 0.4
0.
5
0.5
0.5
0.
6
0. 6
0.
7
0 .
7
0.
8
0.
91
1.8
1.8
1.8
1. 8
3.6
3.6
3.6
3. 6
5.4
5.4
5.4
5.4
7.2
7.2
7.2
7.2
9
9
9
9
10.8
10.8
10.8
12.6
12.6
12.6
14.4
14.4
16. 2
0. 10. 1
0.
1
0.2
0.2
0.
2
0.
3
0.3
0.3
0.
4
0. 4
0.4
0.
5
0.5 0. 5
0.
6
0.6
0 .6
0.
7
0.7
0.
8
0.
8
0.
9
0.9
1
1
1.8
1.8
1.8
1.8
3. 6
3.6
3.6
3.6
5. 4
5.4
5.4
5.4
7.2
7.2
7.2
7.2
9
9
9
9
10.8
10.8
10. 8
12.6
12.6
12.6
14.4
14
.4 14. 4
16.2
0.10.1
0.1
0. 2
0.2
0.
2
0.3
0.
3
0.
3
0. 40.
4
0.
4
0.50.5
0.
5
0.5
0.
6
0.6
0.6
0.
7
0. 7
0.
8
0.8
0.
9
0.
9
1
1
Ψmax = 15.4
Ψmax = 16.5
Ψmax = 17.5
Ψmax = 18.0
Jasim M. Mahdi Al-Khwarizmi Engineering Journal, Vol. 8, No. 3, PP 81- 94 (2012)
88
Streamlines
Isotherms
a) Bottom location
b) Middle location
c) Top location
Fig. 4. Streamlines and Isothermal Contours for φ=0º, H=0.4 and Ra =105 with Different Locations of Thermally
Active Part.
1 .1
1.1
1. 1 1 .1
2. 2
2. 2
2 .2 2 .2
3. 3
3.3
3 .3 3 .3
4.4
4. 4 4 .4
5.5
5.5
5. 5
6.6
6 .6
7. 7
7. 7
8. 8
9.9
0 .1 0 .1
0.2
0. 2
0.
3
0. 3
0 .3
0.
4
0 .4
0. 4
0. 5
0.5
0.
6
0.
6
0.70 .80.
91
1. 6
1.6
1. 6
1 .6
3.2
3 .2
3 .2
3 .2
4 .8
4. 8
4. 8
4 .8
6. 4
6. 4
6. 4
6. 4
8
8
8
9.6
9.6
9. 6
11.2
11.2 11. 2
12. 8
12. 8
14. 4
0.
1
0 .1 0. 1
0.
2
0. 2 0. 2
0.
3
0.3 0. 3
0.
4
0. 4
0. 4
0.
5
0. 5
0.
6
0.6
0.
7
0.
8
0.
91
1 .8
1.8
1. 8
1. 8
3.6
3.6
3 .6
3. 6
5.4
5. 4
5. 4
5. 4
7 .2
7. 2
7. 2
7. 2
9
9
9
9
10 .8
10.
8
10 .8
12. 6
12.
6
12 . 6
14 .4
14
. 4
14 .4
1 6. 2
16 .2
0 .1
0.1
0.1
0. 2
0.
2
0.
2
0.
3
0. 3
0.3
0.4
0.4
0.
5
0.
6
0.
7
0.
8
0.
91
Ψmax = 17.8
Ψmax = 15.4
Ψmax = 10.5
Jasim M. Mahdi Al-Khwarizmi Engineering Journal, Vol. 8, No. 3, PP 81- 94 (2012)
89
Fig. 5. Streamlines and Isotherms for H=0.6, Location of Active Part at Middle and Ra=105 with Different
Values of Inclination Angles.
Streamlines
Isotherms
a) φ=0º
b) φ=30º
c) φ=60º
1.7
1.7
1. 7
1.7
3.4
3.4
3.4
3.4
5.1
5.1
5.1
5. 1
6.8
6.8
6.8
6.8
8. 5
8.5
8.5
10.2
10.2
10.2
11.9
11.9
11.9
13.6
13.6
15.3
0.
1
0.1 0. 1
0.
2
0. 2 0.2
0.
3
0. 3 0.3
0.
4
0.4 0.4
0.
5
0.5
0.5
0.
6
0.6
0.
7
0.
7
0.
8
0.
91
2.7
2.7
2.7 2.7
5.4
5.4
5.4 5.4
8. 1
8.1
8.1 8.1
10.8
10.8
10. 8
13.5
13.
5
13.5
16.2
16.2
16.2
18.9
18.9
21.6
21. 6
24.3
0.
1
0. 1
0. 1
0.
2
0.2 0.2
0.
3
0.3 0.3
0.
4
0.4
0.4
0.
5
0. 5
0.
6
0.6
0.
7
0.7
0.
8
0.
91
2.7
2.7
2. 7 2. 7
5.4
5.4
5.4 5. 4
8.1
8.1
8.1
10.8
10.
8
10.8
13.5
13
.5
13.5
16.2
16. 2
18.9
18.9
21. 6
21. 6
24.3
0.
1
0.1 0.1
0.
2
0. 2 0. 2
0.
3
0.3
0.3
0.
4
0.4
0.4
0.
5
0.5
0.
6
0.6
0.
7
0.7
0.
8
0.
91
Ψmax = 16.5
Ψmax = 26.0
Ψmax = 27.1
Jasim M. Mahdi Al-Khwarizmi Engineering Journal, Vol. 8, No. 3, PP 81- 94 (2012)
90
0.16
0.16
0. 16
0.16
0.32
0.32
0.32
0.32
0.48
0.
48
0.48
0.64
0.
64
0.64
0.8
0.
8
0.8
0.96
0.
96
0.96
1.12
1.
12
1.12
1.28
1.28
1.44
1.44
1.6
0.
1
0.2
0.2
0.3
0.
3
0.4
0.
4
0.5
0.5
0.
6
0.
6
0.
7
0.
7
0.
8
0.
8
0.
9
0.9
1
1
Streamlines
Isotherms
a) Ra = 103
b) Ra = 104
c) Ra = 105
d) Ra = 106
Fig. 6. Streamlines and Isotherms for φ=0º, Location of Active Part at Middle and H =0.8, with Different Values
of Rayleigh Number.
0.7
0.7
0.7 0.7
1.4
1. 4
1.4 1.4
2.1
2.1
2.1 2.1
2.8
2.
8
2.8
3.5
3.
5
3.5
4.2
4.2
4.2
4.9
4.
9
4.9
5.6
5.
6
5.6
6.3
6.3
0.1
0.1
0.2
0.2
0.2
0.3
0.3
0
.3
0.4
0.4
0.
4
0.5
0. 5
0.
5
0.
6
0.6
0.
6
0.
7
0.7
0.
8
0.
8
0.
9
0.9
1
1
1.8
1
.8
1.8
1.8
3.6
3.6
3.6
3.6
5.4
5.4
5.4
5.4
7.2
7.2
7.2
7.2
9
9
9
9
10.8
10.8
10.8
12. 6
12.6
12.6
14.4
14.4
16.2
0.10.1
0.
1
0.2
0.2
0.
2
0.
3
0. 3 0.3
0.
4
0.4
0.4
0.
5
0.5 0.5
0.
6
0.6
0.6
0.
7
0.7
0.
8
0.
8
0.
9
0.9
1
1
3.6
3.6
3.6
3.6
7.2
7.2
7.2
7.2
10.8
10.8
10.8
10.8
14.4
14.4
14.4
14.4
18
18
18
18
21.6
21.6
21.6
21.6
25.2
25.2
25.2
28.8
28.8
32.4
0.
1
0.1 0.1
0.
2
0.2 0.2
0.
3
0.
3
0.3
0.3
0.
4
0.
4 0.4
0.4
0.
5
0.
5
0.5
0.5
0
.6
0.
6
0. 6
0.
7
0. 7
0
.8
0.
8
0
.9
0.
9
1
1
Ψmax = 35.2
Ψmax = 17.5
Ψmax = 6.9
Ψmax = 1.7
Jasim M. Mahdi Al-Khwarizmi Engineering Journal, Vol. 8, No. 3, PP 81- 94 (2012)
91
Fig. 7. Average Nusselt Number as a Function of Rayleigh Number for Different Dimensionless Lengths of
Active Part at φ=0º and Location of Active Part at Middle.
Fig. 8. Average Nusselt Number as a Function of Rayleigh Number for Different Locations of Thermally Active
Part at φ=0º and H=0.4.
Fig. 9. Average Nusselt Number as a Function of Rayleigh Number for Different Inclination Angles at H=0.4
and Location of Active Part at Middle.
10
3
10
4
10
5
10
6
0
2
4
6
8
10
12
14
16
18
20
Ra
N
u
H = 0.4
H = 0.6
H = 0.8
H = 1.0
10
3
10
4
10
5
10
6
0
1
2
3
4
5
6
7
8
9
10
Ra
N
u
Bottom position
Middle position
Top position
10
3
10
4
10
5
10
6
0
2
4
6
8
10
12
Ra
N
u
Φ = 0°
Φ = 30°
Φ = 45°
Φ = 60°
Jasim M. Mahdi Al-Khwarizmi Engineering Journal, Vol. 8, No. 3, PP 81- 94 (2012)
92
5. Conclusions
A steady, two-dimensional natural convection
heat transfer in a square open cavity with partially
active side wall has been studied numerically for
different angles of inclination, different Rayleigh
numbers and different lengths and locations of
thermally active part. The finite-difference
method was employed for the solution of the
present problem. Graphical results in form of
streamlines and isotherms for various parameters
governing heat transfer and fluid flow were
presented and discussed. The results of the present
study lead to the following conclusions:
• Location and length of the active part, Rayleigh
number and inclination angle are important
parameters on flow and temperature fields and
heat transfer in square cavity.
• Heat transfer increases with the increasing of
active part length or Rayleigh number. Values of
stream function (flow strength) also increase with
the increasing of Rayleigh number or active part
length.
• Higher heat transfer rate was obtained when the
active part is at middle location in vertical wall
compared to top and bottom locations.
• Increase of inclination angle increases flow
strength and decreases average Nusselt number.
Nomenclature
L length of the square cavity
g gravitational acceleration
h
H
length of the active part
dimensionless length of the active part
s
S
position of the active part center
dimensionless position of the active part
hy local heat transfer coefficient
Nu local Nusselt number Nu average Nusselt number
p pressure
T dimensional temperature
Pr Prandtl number (ν/α)
Ra Rayleigh number (gβL3ΔT/να)
x,y
X,Y
dimensional Cartesian coordinates
dimensionless coordinates
u
U
v
dimensional velocity in the x-direction
dimensionless velocity in the X-direction
dimensional velocity in the y-direction
V dimensionless velocity in the Y-direction
Greek symbols
α thermal diffusivity
β
μ
thermal expansion coefficient
dynamics viscosity
ν
ω
kinematic viscosity
dimensional vorticity
Ω dimensionless vorticity
ψ dimensional stream-function
Ψ dimensionless stream-function
θ dimensionless temperature
ρ density
Δ laplacian in Cartesian coordinates
Subscripts
∞ ambient
in inflow
out outflow
Superscripts
ζ current iteration number
ζ-1 previous iteration number
6. References
[1] A. M. Clausing, Convective losses from
cavity solar receivers: comparisons between
analytical prediction and experimental
results, J. Solar energy 105 (1983) 29-33.
[2] Abdulkarim H. Abib, Yogesh Jaluria,
penetrative convection in a partially open
enclosure, HTD 198 ASME (1992) 74–83.
[3] S. S. Cha and K. J. Choi. An interferometric
investigation of open cavity natural
convection heat transfer. Exp. J. Heat transfer
2 (1989) 27–40.
[4] Y.L. Chan, C.L. Tien, A numerical study of
two-dimensional natural convection in square
open cavities, Numer. Heat Transfer 8 (1985)
65–80.
[5] Y.L. Chan, C.L. Tien, A numerical study of
two-dimensional laminar natural convection
in shallow open cavities, Int. J. Heat Mass
Transfer 28 (1985) 603–612.
[6] O. Polat, E. Bilgen, Laminar natural
convection in inclined open shallow cavities,
Int. J. Thermal Sciences 41 (2002) 360–368.
[7] Chakroun, W., Effect of Boundary Wall
Conditions on Heat Transfer for Fully
Opened Tilted Cavity, ASME J. Heat
Transfer, 126 (2004) 915–923.
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[8] M. Nateghi, S. W. Armfield, Natural
convection flow of air in an inclined open
cavity, ANZIAM Journal 45 (2004) 870–890.
[9] J.F. Hinojosa, R.E. Cabanillas, G. Alvarez,
C.E. Estrada, Nusselt number for the natural
convection and surface thermal radiation in a
square tilted open cavity , Int.
Communications in Heat and Mass Transfer
32 (2005) 1184–1192.
[10] A.A. Mohamad a, M. El-Ganaoui b, R.
Bennacer c, Lattice Boltzmann simulation
of natural convection in an open ended
cavity, Int. J. Thermal Sciences 48 (2009)
1870–1875.
[11] N. Nithyadevi, P. Kandaswamy, J. Lee,
Natural convection in a rectangular cavity
with partially active side walls, Int. J. Heat
Mass Transfer 50 (2007) 4688–4697.
[12] P.J. Roach, Computational Fluid Dynamics,
Hermosa, Albuquerque, New Mexico,
1985.
)2012( 81 - 94 ، صفحة3، العدد8 مجلة الخوارزمي الھندسیة المجلد د مھدي جاسم محم
94
إنتقال الحرارة بالحمل الحر في تجویف مربع مائل ذو نھایة مفتوحة و بوجود جزء نشط حراریًا
جاسم محمد مھدي
جامعة بغداد/ كلیة الھندسة /قسم ھندسة الطاقة
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الخالصة
ُمحاط بجدارین علوي و سفلي مفتوح من الجانب مائل مربع الشكل وفي حیز الحمل الطبیعي نظریة النتقال الحرارة ب یصف ھذا البحث دراسة
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و تم تحویلھا إلى صورة البعدیة مناسبة و من ثم ُفكت دالة الجریان -المعادالت الحاكمة لجریان المائع و انتقال الحرارة داخل الحیز ُمثلت بصیغة الدوامیة
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طول و موضع الجزء عن خط األفق و لقیم مختلفة لدرجة ٦٠-٠بین تراوحتو زوایا میالن ٦١٠- ٣١٠من أعداد رایلي تراوحت بینالمعدل و لمدى واسع
تشیر النتائج التي تم الحصول علیھا إلى إن معدل انتقال الحرارة یكون أعلى عند یكون الجزء النشط في منتصف الجدار العمودي مقارنة . النشط حراریًا
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