Engineering e-Transaction (ISSN 1823-6379)
Vol. 7, No.2, December 2012, pp 54-61
Online at http://ejum.fsktm.um.edu.my
Received 9 July, 2012; Accepted 19 Sep, 2012
MATHEMATICAL MODELING FOR HEAT TRANSFER ENHANCEMENT IN A VENTILATED
ENCLOSURE WITH A CIRCULAR HEAT-GENERATING BLOCK
M.M. Billaha, M.J.H. Munshib, A.R. Khanc, A. Rashidc and M.M. Rahmanb
a
Department of Arts and Sciences
Ahsanullah University of Engineering and Technology (AUST), Dhaka-1208, Bangladesh
b
Department of Mathematics
Hamdard University Bangladesh, Gozaria, Munshiganj, Bangladesh
c
Department of Mathematics
Jahangirnagar University, Savar, Dhaka-1342, Bangladesh
d
Department of Mathematics, Bangladesh University of Engineering & Technology (BUET),
Dhaka-1000, Bangladesh
Email: mmb.edu@gmail.com
convective heat transfer from periodic open cavities in
a channel with oscillatory through flow. Khanafer et al.
(2002) performed a numerical investigation on mixed
convection heat transfer in open-ended enclosures for
three different flow angles. They found that thermal
insulation of cavity can be achieved through the use of
high horizontal velocity flow. A numerical analysis of
laminar mixed convection in a channel with an open
cavity and a heated wall bounded by a horizontally
insulated plate was presented by Manca et al. (2003),
where the authors considered three heating modes:
assisting flow, opposing flow and heating from below.
Later, a similar problem for the case of assisting forced
flow configuration was tested experimentally by
Manca et al. (2006). Leong et al. (2005) performed a
numerical study on the mixed convection from an open
cavity in a horizontal channel. Authors found that the
heat transfer rate was reduced, and the flow became
unstable in the mixed convection regime.
Aminossadati and Ghasemib (2009) performed a
numerical study on the mixed convection in a
horizontal channel with a discrete heat source in an
open cavity. They considered three different heating
modes and found noticeable differences among the
indicated three heating modes. Wong and Saeid (2009)
numerically investigated the opposing mixed
convection arises from jet impingement cooling of a
heated bottom surface of an open cavity in a horizontal
channel filled with porous medium.
ABSTRACT
In this article, mixed convection flow and heat transfer
characteristics inside a ventilated enclosure have been
investigated. A heat-generating circular block
positioned at the centre of the enclosure. Discretization
of governing equations is achieved using a finite
element technique based on Galerkin weighted
residuals. The computation is carried out for a wide
range of pertinent parameters such as Aspect ratio and
Richardson number. Numerical results are reported for
the effect of above mentioned parameters on the
contours of streamline and isotherm. In addition, the
heat transfer rate in terms of the average Nusselt
number and average temperature of the fluid in the
cavity as well as drag force are presented for the
mentioned parametric values. The obtained results
show that the flow and thermal field are strongly
influenced by the aforesaid parameters.
Keywords: Finite Element Method; circular block;
ventilated cavity and mixed convection.
1.0 INTRODUCTION
Mixed convection is that type of heat transfer in which
there is a noteworthy interaction between free and
forced convection. Mixed convective heat transfer in
open/ventilated cavities has long been considered and
has received increases attention owing to its
application of practical interest, such as nuclear
reactors, solar receiver, thermal storage and open
cavity packaging of semiconductors. Pavlovic and
Penot (1991) made an experimental investigation of
the mixed convection heat transfer in an open
isothermal cubic cavity. The authors found that the
convective heat loss for the central solar receiver.
Fusegi (1997) carried out a numerical study on
A literature review on the subject shows that a sizeable
number of authors had considered mixed convection in
enclosures with heated body. Hsu and How (1999)
studied mixed convection flow in an enclosure with a
finite-size heat source on the sidewall and a heat
conducting body. They observed that both the heat
54
transfer coefficient and the dimensionless temperature
in
the
body
center
strongly
depend
on
the
configurations of the system. Omri and Nasrallah
(1999) performed the mixed convection effect in an
air-cooled cavity with differentially heated vertical
isothermal sidewalls having inlet and exit ports by a
control volume finite element method. They
investigated two different placement configurations of
the inlet and exit ports on the sidewalls. Best
configuration was selected analyzing the cooling
effectiveness of the cavity, which suggested that
injecting air through the cold wall was more effective
in the heat removal and placing inlet near the bottom
and exit near the top produce effective cooling. A
numerical model of laminar mixed convection in an
open cavity with a heated wall bounded by a
horizontally insulated plate was offered by Manca et
al. (2003), where three heating modes were considered:
assisting flow, opposing flow and heating from below.
The effect of the ratio of channel height to the cavity
height was found to play a significant role on
streamline and isotherm patterns for different heating
configurations. Shokouhmand and Sayehvand (2004)
carried out a study of flow and heat transfer in a square
driven cavity numerically. House et al. (1990)
considered natural convection in a vertical square
cavity with heat conducting body, placed on center in
order to understand the effect of the heat conducting
body on the heat transfer process in the cavity. They
concluded that the heat transfer across the enclosure
enhanced by a body with the thermal conductivity ratio
less than unity. Gau and Sharif (2004) conducted
mixed convection in rectangular cavities at various
aspect ratios with moving isothermal side walls and
constant flux heat source on the bottom wall. Bhoite et
al. (2005) presented a numerical model for the problem
of mixed convection flow and heat transfer in a
shallow enclosure with a series of block-like heat
generating component for a range of Reynolds and
Grashof numbers and block-to-fluid thermal
conductivity ratios. Rahman et al. (2009) have done an
analysis on mixed convection in a rectangular
ventilated cavity with a heat conducting circular
cylinder. The authors highlighted the influence of the
mixed convection parameter and the cavity aspect ratio
on the flow structure and temperature distribution. Gau
et al. (2000) performed experiments on mixed
convection in a horizontal rectangular channel with
side heating. Billah et al. (2011) performed a
numerical investigation on heat transfer and flow
characteristics for MHD mixed convection in a lid
driven cavity with heat generating obstacle. The
authors showed that both fluid flow and thermal fields
is strongly affected by magnetic field.
The literature review presented above indicates that not
much attention has been paid to the problem of mixed
convection in a ventilated cavity having a circular heat
generating block. The present work focuses on
conducting a comprehensive study on the effect of
various flow and thermal configurations on mixed
convection for a wide range of pertinent controlling
parameters in a ventilated cavity. These parameters
include Aspect ratio AR, and Richardson number Ri.
2.0 MODEL SPECIFICATION
The geometry of the problem herein investigated is
depicted in Figure 1. A Cartesian co-ordinate system is
used with origin at the lower left corner of the
computational domain. The system consists of a
rectangular cavity with height H and length L, within
which a heat generating solid circular block with
diameter d. The circular block has a thermal
conductivity of k and generates uniform heat q per unit
volume. The side walls of the cavity are considered to
be adiabatic. It is assumed that the incoming flow is at a
uniform velocity, ui and at the ambient temperature, T i.
The inlet opening is located on the bottom of the left
vertical wall, whereas the outlet opening is at the top of
the opposite side wall and the size of the inlet port is w
= 0.1L which is the same as the exit port.
y
w
g
outle
t
d
lx
inlet w
ly
ui,
Ti
x
L of the studied
Fig. 1 Schematic
configuration
3.0 MATHEMATICAL FORMULATION
A two-dimensional, steady, laminar, incompressible,
mixed convection flow is considered inside the
enclosure. The fluid properties are assumed to be
constant except for the density which is considered to
very linearly with temperature according to the
Boussinesq approximation. The working fluid is
assumed to be air (Pr = 0.071). Taking into account the
55
above mentioned assumptions, the governing equations
may be written in the non-dimensional form as follows:
=
U V

0
X Y
(1)
U
U
P
1   2U  2U

U
V



X
Y
X Re  X 2 Y 2

conductivity. The average Nusselt number at the
L
force D  
0
(3)
0
U
d and the average temperature of the
N
3.1 Computational Procedure
The momentum and energy balance equations are the
combination of a mixed elliptic-parabolic system of
partial differential equations that have been solved by
using the Galerkin weighted residual finite element
technique by Rahman et al. (2009). The continuity
equation has been used as a constraint due to mass
conservation. The basic unknowns for the above
differential equations are the velocity components (U,

g  T L
, Pr 
and Ri 
are
V0 2
α

Reynolds number, Prandtl number and Richardson
V0 L
generating
parameter
in
( T  Tb  Tc and   k  C p are

 N d , Drag
cavity volume.
(5)
Q
L
(4)

Q 0


number, respectively and
1
L
fluid is defined as av   dV / V , where V is the
For solid block the energy equation is
Where, Re 
is the dimensionless ratio of the thermal
(2)


1   2  2 


V


X
Y Re Pr  X 2 Y 2 


K   2 s  2 s


RePr  X 2 Y 2

k s / k 
heated surface is calculated as Nu  




V
V
P 1   2V  2V 

  Ri
U
V



X
Y
Y Re  X 2 Y 2 


U
where N is the non-dimensional distances either along
X or Y direction acting normal to the surface and K
q L2
k s T
is the heat
V), the temperature  and the pressure P. The six node
triangular element is used in this work for the
development of the finite element equations. All six
nodes are associated with velocities as well as
temperature; only the corner nodes are associated with
pressure. This means that a lower order polynomial is
chosen for pressure and, which is satisfied through
continuity equation. The velocity component and the
temperature distributions and linear interpolation for the
pressure distribution according to their highest
derivative orders in the differential Equations (2) - (5)
the
solid
block
the
temperature
difference and thermal diffusivity respectively).
The above equations are non-dimensionalized by
employing the following dimensionless quantities
x
y
u
v
p
d
,Y  ,U  ,V  , P 
,D  ,
L
L
ui
ui
L
 ui 2
ly
T  Ti  ,   Ts  Ti 
l
Lx  x , L y  , 
L
L
Th  Ti  s Th  Ti 
X
as U  X , Y   N  U  , V  X , Y   N  V ,   X , Y   N    ,
Where X and Y are the coordinates varying along
horizontal and vertical directions respectively, U and V
are the velocity components in the X and Y directions
respectively, θ is the dimensionless temperature and P
is the dimensionless pressure.
s  X , Y   N s  , P  X , Y   H  P , where β = 1, 2, …
… , 6; λ = 1, 2, 3.
Substituting the element velocity component
distributions, the temperature distribution and the
pressure distribution from Equations (2) - (5) the finite
element equations can be written in the form
The boundary conditions for the present problem are
specified as follows:
At the inlet: U = 1, V = 0,  = 0
U

 0,V  0,
0
X
X
At the solid block boundaries: U  0,V  0,  b

at the cavity walls: U  V 
0
N
  s 
  
U U K
V U  M x P 
 x  
 y  

At the outlet:
K
1 

 S xx  S yy U   Q u
Re  

(6)
U V K
V V  M y P 
 x  
 y  

K
 K
At the fluid-solid interface:  

 N  fluid
 N  solid
1 

 S xx  S yy V  Ri K    Q v
Re  

56
(7)
1


U  K
V  
S
S

 x  
 y   RePr   xx  yy  
K
Table 1 Comparison of average Nusselt numbers for
Re = 1000, Gr = 105, C(x = 0.5, y = 0.5) the present
data with those of Oztop et al. (2009)
(8)
Q 


 S xx  S yy


   Q 

 s
Oztop et al. (2009)
(9)
Where the coefficients in element matrices are in the
form of the integrals over the element area and along
the element edges S0 and Sw as
A N N  , xdA
K y   N N  , y dA

A
K
 N N N dA
 x  A    , x
K
 N N N dA
 y  A    , y
K   N N  dA
A
S xx   N , x N  , x dA

A
S yy   N , y N  , y dA

A
M x   H H  , x dA

A
M y   H H  , y dA

A
Q u   N S x dS0

S0
K
 x

(11)
(12)
(13)
(14)
(15)
(16)
(17)
(18)
Hot wall
Cold wall
Hot wall
7.21
9.13
7.03
9.15
7.14
10.37
7.45
10.59
6.78
11.10
7.87
11.65
(19)
S0 N S ydS0
(20)
Q  

Sw N q1wdSw
(21)
Sw N q2wdSw
Cold wall
right side while the rest of the boundaries of the cavity
were insulated. Moreover, the inner body is isothermal.
Firstly, the comparison of the average Nusselt number
(at the hot wall) between the outcome of the present
code and the results found in the literature Oztop et al.
(2009) for various diameter of the cylinder as shown in
Table 1. The comparisons reveal an excellent
agreement with the reported studies. Finally, the
streamlines and isotherms for four different values
diameter are presented in the Figures 3 and 4
respectively. It can be seen from the figures that the
present results and those reported in Oztop et al.
(2009). This validation boosts the confidence in the
numerical outcome of the present work.
(10)
Q v 

Q s 

Present work
(22)
The set of non-linear algebraic Equations (6) - (9) are
solved using reduced integration technique and
Newton-Raphson method [More details in Reddy
(1993), Zeinkiewicz et al. (1971), Roy and Basak
(2005)]. The convergence of solutions is assumed
when the relative error for each variable between
consecutive iterations is recorded below the
Fig. 2 Grid independency study for different
grid elements, while Re = 100, Ri = 1.0, D =
0.2, K = 5.0, and Q = 1.0.
convergence criterion ε such that  n 1   n  104 , n is
4.0 RESULTS AND DISCUSSION
number of iteration and  is a function of U, V, θ and
θs.
A numerical study on mixed convection inside a vented
cavity having a heat-generating circular block is
governed by different controlling parameters. These
parameters are heat generation Q, solid fluid thermal
conductivity ratio K, Reynolds number Re, Prandtl
number Pr Aspect Ratio AR and Richardson number Ri.
Analysis of the results is made for two parameters AR
and Ri, which affect flow fields and temperature
distribution inside the cavity. Cavity aspect ratio varies
3.2 Code Validation
The present numerical code was validated against the
problem of mixed convection in a lid-driven enclosure
having a circular body studied by Oztop et al. (2009).
The cavity was heated at the left wall and cooled at the
57
at the three different regimes of flow, viz., pure forced
convection, mixed convection and dominating natural
convection with Ri = 0, 1 and 5 respectively. The
variations of the average Nusselt number at the heated
surface, average fluid temperature in the cavity are
plotted for the different values of the aforesaid
parameters.
a
from 0.5 to 2 while other parameters Q = 1, K= 5, Re=
100 and Pr = 7.1 are kept fixed. The results are
presented in terms of streamline and isotherm patterns
c
b
Figure 5 shows the effect of cavity aspect ratio on the
streamlines (on the left) and isotherms (on the right) at
Ri = 0. Four different values of cavity aspect ratios,
AR = 0.5, 1, 1.5, and 2, have been considered for
assessing the effect. At Ri = 0 and AR = 2, it is
observed that a uni-cellular clockwise vortex is
developed at the inlet port which we call primary
vortices generated due to the heat generating block.
The size of the primary cell becomes small with the
decreasing AR. It also be seen that the shape of the
eddy changes while AR changes.
The corresponding effect of cavity aspect ratio for
thermal field in the cavity is shown in the right column
of Figure 5. Thermal boundary layer thickness
decreases and the isothermal lines become closer at the
adjacent area of heat source at Ri = 0 for all values AR.
Besides thermal plume formed on the heat source
changes its shape from inclined to vertical with
decreasing AR.
AR = 2
AR = 1.5
AR = 1
AR = 0.5
Fig. 3 Comparison of the streamlines between the
present work (right) and that of Oztop et al. (2009)
(left) at different diameter (a) D = 0.3L, (b) D = 0.4L
and (c) D = 0.5L.
Fig. 4 Comparison of the isotherms between the
present work (right) and that of Oztop et al.
(2009) (left) at different diameter (a) D = 0.3L,
(b) D = 0.4L and (c) D = 0.5L.
Streamlines
Isotherms
Fig. 5 Streamlines and Isotherms for different
values of cavity aspect ratio at Ri = 0.
of cavity aspect ratio at Ri = 0.
58
It is noticed that flow field for the lower value of AR (=
0.5) is almost identical with all values of Ri. The
corresponding effect of cavity aspect ratio for thermal
field in the cavity is presented in the right column of
Figure 7. It is noticed that the thermal plume formed on
the heat source does not change its shape with changing
AR.
AR = 1
AR = 0.5
Figure 6 illustrates the effect of cavity aspect ratio on
the streamlines (on the left) and isotherms (on the
right) at Ri = 1. It is observed that the primary vortices
remain unchanged for the lower values of AR (= 0.5)
while it compares with those at Ri = 0. The flow field
remains almost identical for higher values of AR (= 2,
and 1.5) except increasing the size of the primary cell.
When AR = 1, a two-cellular secondary vortices
developed near the left vertical wall due to the mixed
convection effect. The corresponding effect of cavity
aspect ratio for thermal field in the cavity is depicted in
the right column of Figure 6. Making a comparison of
the isothermal lines for different values of AR at Ri = 0
and Ri = 1, a slight change is found. The open isotherm
lines increases at the higher values of AR (= 2.0, 1.5).
AR = 2
AR = 1.5
Figure 7 presents the influence of cavity aspect ratio on
the streamlines (on the left) and isotherms (on the
right) at Ri = 10. It is found that the size of the primary
vortex increases rapidly. Further, at Ri = 10 and the
higher values of AR (= 1.0, 1.5 and 2.0), it is seen that
the primary vortex spreads and covers most of the part
of the cavity, indicating a sign of supremacy of natural
convection in the cavity.
AR = 0.5
Streamlines
Isotherms
Fig. 7 Streamlines and Isotherms for different values
of cavity aspect ratio at Ri = 10.
AR = 2
AR = 1.5
AR = 1
The effect of cavity aspect ratio on average Nusselt
number (Nu), Drag force (Δ) and average fluid
temperature (θav) in the cavity at three different values
of Ri, is revealed in Figure 8. From the Figure 8, it is
clearly observed that the distribution of average
Nusselt number (Nu) goes up sharply in the forced
convection dominated region (Ri ≤ 1) and beyond this
region it goes down gradually in the cavity for
increasing values of Ri for AR=1.5 and 2. On the other
hand, average Nusselt number increases monotonically
for the lowest value of AR (= 0.5). Beside this, average
Nusselt number decreases for the value of AR (= 1).
Drag force (Δ) decreases slightly in the forced
convection dominated region (Ri ≤ 1) and beyond this
region it goes up gradually in the cavity for increasing
values of Ri for higher values of AR. In contrast, the
average Nusselt number increases steadily for the
lowest value of AR (= 0.5). On the other hand, for
higher values of AR, the average fluid temperature
(θav) increases
Streamlines
Isotherms
Fig. 6 Streamlines and Isotherms for different values
of cavity aspect ratio at Ri = 1.
59
NOMENCLATURE
(i)
(ii)
d
dimensional diameter of the cylinder (m)
D
dimensionless diameter of the cylinder
g
gravitational acceleration (ms-2)
Gr
Grashof number
h
convective heat transfer coefficient (Wm–2K–1)
k
thermal conductivity of fluid (Wm-1K-1)
ks
thermal conductivity of the solid cylinder (Wm1 -1
K )
K
Solid fluid thermal conductivity ratio
L
length of the cavity (m)
lx
dimensional distance between y-axis and the
cylinder center (m)
ly
dimensional distance between x-axis and the
cylinder center (m)
Lx
dimensionless distance between y-axis and the
cylinder center
Ly
dimensionless distance between x-axis and the
cylinder center
Nu
Nusselt number
N
non-dimensional distances either X or Y
(iii)
direction acting normal to the surface
Fig. 8 Effect of Aspect Ratio AR on (i)
average Nusselt number, (ii) Drag force and
(ii) average fluid temperature in the cavity.
p
dimensional pressure (Nm-2)
P
dimensionless pressure
Pr
Prandtl number
Ra
Rayleigh number
dramatically in the forced convection dominated region
(Ri ≤ 1) and beyond this region it changes gradually in
the cavity for increasing values of Ri. It is also seen that
the average fluid temperature decreases smoothly at the
lowest value of AR (= 0.5).
Re
Reynolds number
Ri
Richardson number
T
dimensional temperature (K)
u, v
dimensional velocity components (ms-1)
5.0 CONCLUSION
V
cavity volume (m3)
A computational study is performed to investigate the
mixed convection flow in a ventilated enclosure with a
heat-generating circular block. The nature of flow and
thermal fields as well as characteristics of heat transfer
process particularly its augmentation due to the
introduction of heat generating circular block has been
evaluated in this study. On the basis of the analysis the
following conclusions have been drawn.
w
height of the opening (m)
U, V dimensionless velocity components
x, y Cartesian coordinates (m)
X, Y
dimensionless Cartesian coordinates
Greek symbols

The influence of cavity aspect ratio on fluid flow
and temperature field is found to be pronounced. The
heat transfer rate for higher cavity aspect ratio is
higher than for lower aspect ratio. Moreover the
maximum average fluid temperature is found for
higher cavity aspect ratio.
α
thermal diffusivity (m2s-1)
β
thermal expansion coefficient (K-1)

kinematic viscosity (m2s-1)
Θ
non dimensional temperature
ρ
density of the fluid (kgm-3)
Subscripts
av
average
i
s
60
inlet state
solid
Abbreviation
CBC convective boundary conditions
enclosures. International Journal of Heat and
Transfer, 45: 5171–90.
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Leong J.C., Brown N.M. and Lai F.C. 2005. Mixed
convection from an open cavity in a horizontal channel.
International Communication Heat and Mass Transfer,
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Aminossadati S.M. and Ghasemib B. 2009. A numerical
study of mixed convection in a horizontal channel with a
discrete heat source in an open cavity, European Journal
Mechanics-B/Fluids, 28(4): 590–8.
Mass
Manca O., Nardini S., Khanafer K. and Vafai K. 2003.
Effect of heated wall position on mixed convection in a
channel with an open cavity, Journal of Numerical Heat
Transfer, 43 (3): 259-82.
Bhoite M.T., Narasimham G..S.V.L. and Murthy M.V.K.
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