effect of discrete heat sources on natural conv in a square cavity by_jaikrishna
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Proceedings of the 37th
National & 4th
International Conference on Fluid Mechanics and Fluid Power
FMFP2010
December 16-18, 2010, IIT Madras, Chennai, India
1
FMFP2010________
EFFECT OF DISCRETE HEAT SOURCES ON NATURAL
CONVECTION IN A SQUARE CAVITY
Jaikrishna. C. R
Student, Department of Mechanical EngineeringPES Institute of Technology,
Bangalore-560 085, Karnataka, INDIA
Rathan Ram. B
Student, Department of Mechanical EngineeringPES Institute of Technology,
Bangalore-560 085, Karnataka, INDIA
[email protected] [email protected]
Aswatha
Department of Mechanical EngineeringBangalore Institute of Technology,
Bangalore-560 004, Karnataka, [email protected]
K. N. Seetharamu
Department of Mechanical Engineering,PES Institute of Technology,
Bangalore - 560 085, Karnataka, [email protected]
ABSTRACTIn the present study a finite volume computational procedure is used to investigate
natural convection in a square cavity. The enclosure used for flow and heat transfer
analysis has been bounded by adiabatic top and bottom walls, constant temperatureright cold wall and discretely heated left wall. Also the computations are carried out
for with 25 % opening of cold wall at the top. The Rayleigh Number (Ra) varying
from 103-107 and Pr = 0.7. When the Rayleigh number is increased, rate of heat
transfer also increases and the maximum temperature at the heater surface decreases.
The effect of convection is more dominant with partial opening. Best heat transfer
was obtained when the heater element is placed at the center with minimum heating
length and opening one.
Key words: Natural convection; square cavity; discrete heat source; electronic cooling
components
1. INTRODUCTION
Natural convection in cavities has gained importance in many electronic applications.
Natural convection cooling is desirable because it doesn’t require energy source for
cooling and hence more reliable. Air is taken as the cooling medium for cooling
electronic components due to its simplicity and low cost. Microprocessors are treated
as heat sources on flat surfaces.
There are numerous studies in the literature regarding natural convection heat transfer
in cavities. Aydin and Yang (2000) numerically investigated natural convection in
enclosures with localized heating from below and symmetrically cooled from side
walls. Nguyen and Prudhomme (2001) have studied convection flows in a rectangular
cavity subjected to uniform heat flux. Salat et al. (2004) studied experimentally andnumerically the turbulent natural convection in a large air-filled cavity.
446
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NOMENCLATURE
g acceleration due to gravity (m/s2)
k thermal conductivity (W m-1 K -1)
L side length of the square cavity (m)
Nu local Nusselt number p dimensional pressure (Pa)
Pr prandtl number
q″ heat flux (W m-2)
Ra Rayleigh number
T temperature (K)
T c temperature of vertical wall (K)
u x- component of velocity
v y- component of velocity
Greek symbols
α thermal conductivity (m2
s-1
) β volume expansion coefficient (K -1)
θ Non - dimensional temperature
γ kinematic viscosity (m2s-1)
ρ density (kg m-3)
ψ stream function
Subscripts
Bilgen and Oztop (2005) investigated numerically natural convection heat transfer in
partially open inclined square cavities. They made an effort to study the steady state
heat transfer by laminar natural convection in a two dimensional partially open
cavities. Natural convection in cavities with constant flux heating at the bottom wall
and isothermal cooling side walls have been studied by Sharif et al. (2005). The effect
of aspect ratio, inclination angles and heat source length on the convection and heat
transfer process in the cavities are analyzed. Kasayapanand (2007) studied the effect
of electric field on natural convection in the partially opened square cavities by finite
volume technique. Nithyadevi et al. (2007) studied the effect of aspect ratio on thenatural convection of a fluid contained in a rectangular cavity with partially active
Fig.1 .Schematic d iagram of the physical system
B
C
D
A
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side walls. The active part of the left side wall was selected at a higher temperature
than that of the right side wall. The top and bottom of the cavity and inactive part of
the side walls were thermally insulated. Yasin Varol et al. (2007) have made efforts to
understand the variable protruding heater length on Natural convection in triangular
enclosures. Recently, Ayla DOGAN et al. (2009) studied numerically the heat transfer
by natural convection from partially open cavities with one wall heated. In this study,the steady state investigation deals with natural convection heat transfer inside the
cavity under uniform heat fluxes, different opening ratios, tilt angles and cavity aspect
ratios for top and center opening positions. They found that the average heat transfer
coefficient increases and the average wall temperature decreases, with the increase in
opening ratio and decrease in the tilt angle. Best heat transfer was obtained with the
aspect ratio 1, for opening ratio of 0.75 and tilt angle of 10° in the clockwise
direction.
The objective of the present paper is to investigate the effect of discrete heat sources
on natural convection in a square cavity with and without opening on the cold wall.
2. MATHEMATICAL FORMULATIONA square cavity illustrated in Fig.1 is chosen for simulating natural convective
flow and heat transfer characteristics. The square cavity of length (L) has left wall
with discrete heat sources of varying lengths and positions. The gravitational force is
acting downwards. A buoyant flow develops because of thermally induced density
gradient.
The governing equations for natural convection flow are conservation of mass,
momentum and energy equations, written as:
Continuity: 0=¶
¶+
¶
¶
y
v
x
u (1)
X-momentum: ÷÷ ø ö
ççè æ
¶¶+
¶¶+
¶¶-=
¶¶+
¶¶
2
2
2
21
y
u
x
uv x
p
y
uv x
uu r
(2)
Y-momentum: ( )cT T g y
v
x
vv
y
p
y
vv
x
vu -+÷
÷
ø
ö
çç
è
æ
¶
¶+
¶
¶+
¶
¶-=
¶
¶+
¶
¶b
r 2
2
2
21 (3)
Energy: =¶
¶+
¶
¶
y
T v
x
T u ÷
÷
ø
ö
çç
è
æ
¶
¶+
¶
¶2
2
2
2
y
T
x
T a (4)
No-slip boundary conditions are specified at all walls.
Left side wall: case 1, 0= x ,4
L y = to
4
3 L =
¶
¶-
x
T k q ″
case 2, 0= x , 0= y to4
L and
4
3 L y = to L =
¶
¶-
x
T k q ″ (5)
case 3, 0= x , 0= y to4
L and
2
L y = to
4
3 L =
¶
¶-
x
T k q ″
case 4, 0= x ,8
3 L y = to
8
5 L =
¶
¶-
x
T k q ″
Remaining lengths of the left side wall are adiabatic
Top and bottom wall: ( ) 0, =¶
¶ L x
y
T and ( ) 00, =
¶
¶ x
y
T
Right side wall: ( ) ( ) cT y LT yT == ,,0
For opening at the right wall: L x = ,
43 L y = to L , 0=P and 0=
¶¶=
¶¶
x
v
x
u
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Where x and y are the dimensional co-ordinates along horizontal and vertical
directions respectively; u and v are dimensional velocity components in x and y -
directions respectively; T is the temperature; p is the dimensional pressure; Here,
the fluid is assumed to be Newtonian and the properties are constant. Only the
Boussinesq approximation is invoked for the buoyancy term.
The changes of variables are as follows:
,k
LqT
"
=D ,a
n =Pr
and
2
3 Pr
n
b TLg Ra
D= (6)
In the present investigation, the selected geometry has been modeled. The modeled
geometry is discretized using Gambit 2.4. The meshed model is saved as data file and
mesh file separately. The saved Gambit files are read in ANSYS FLUENT 6.3 to give
specified boundary conditions, selection of fluid and fluid properties. The several
cases mentioned earlier are solved for Ra ranging from 103 to 107.
3. Stream function and Nusselt number
3.1 Stream function
The motion of buoyant driven fluid inside the cavity is represented by using the
stream function Ψ obtained from velocity components u and v . The relationship
between stream function, Ψ and velocity components for two dimensional flows are
given by Batchelor (1993):
yu
¶
¶= y
and
xv
¶
¶-= y
(7)
which yields to a single equation:
x
v
y
u
y x ¶
¶-
¶
¶=
¶
¶+
¶
¶
2
2
2
2 y y
(8)
The local heat transfer coefficient is defined as h y = q" / [Ts-Tc] at a given point on the
heat source surface where Ts is the local temperature on the surface. Accordingly the
local Nusselt number is obtained as Nu = (h y W)/k. The trapezoidal rule is used for
numerical integration to obtain the average Nusselt number.
4. Numerical procedure
The set of governing equations are integrated over the control volumes, which
produces a set of algebraic equations. The PISO algorithm developed by Issa (1985) is
used to solve the coupled system of governing equations. The set of algebraic
equations are solved sequentially by ADI method. A second-order upwind
differencing scheme is used for the formulation of the convection contribution to the
coefficients in the finite-volume equations. Central differencing is used to discretize
the diffusion terms. The computation is terminated when the entire residuals one
below 10−5.
5. Verification of the present methodology
The verification is made with reference to the results of Sharif and Taquiur (2005).
The cavities used to study are bounded by uniform temperature vertical side walls and
adiabatic top wall. The bottom wall is subjected to a uniform heat flux spread over
from 20 % to 80 % of the length from centre and the remaining length is considered
adiabatic. The Grashof number (Gr) is varied from 10
3
to 10
6
. In order to obtain gridindependent solution, a grid refinement study is performed for a square cavity
T
T T c
D
-=q
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(AR = 1) with heating length = 0.6. Fig. 2(a) shows the convergence of the average
Nu at the heated surface with grid refinement for Gr = 105 of Sharif and Taquiur
(2005). Different grid sizes of 31 x 31, 41 x 41, 51 x 51 and 61 x 61 with uniform
mesh as well as biasing have been studied. The grid 41 x 41 biasing ratio (BR) of 2
(The ratio of maximum cell to the minimum cell is 2, thus making cells finer near the
wall) gave results identical to that of 61 x 61 uniform mesh. In view of this, 41 x 41
grid with biasing ratio 2 is used in all further computations. Fig. 2(b) shows variation
of the Average Nusselt number with Sharif and Taquiur (2005). The percentage of
error was within 2.4 %. This is found to be a good agreement with Sharif and Taquiur
(2005).
6. RESULTS AND DISCUSSION
6.1 Cavity without openingThe flow and heat transfer characteristics in a square cavity have been studied for
four different cases as illustrated in Fig. 1 without opening. Computations are carried
out for Rayleigh number ranging from 103 to 107. The results are presented in the
form of stream lines, isotherms, local Nu and average Nu for all four different cases.
Fig. 3 Contour plots for case 1 without opening of cold wall with Ra = 105.
Fig.2 Conve rgence o f ave rage Nusse l t number w i th (a ) Gr id re f inemen t and
(b ) Sharif and Taquiur (2005).
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Fig. 4 Contour plots for case 2 without opening of cold wall with Ra =105.
Fig.5 Contour plots for case 3 without opening of cold wall with Ra = 105.
Fig. 6 Contour plots for case 4 without opening of cold wall with Ra = 105.
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Figs. 3 to 6 shows the streamlines and isotherms plots for all four different cases. For
case 3 yields higher values of magnitudes of the stream functions as compared to first
two cases and last one. The cells extend towards the bottom right corner of the cavity
for all cases. From the temperature profiles it is observed that the contours are spread
out the entire cavity except case 4 (25% of heating length). The temperature contours
are concentrated at heating length as expected for all the cases. For case 4, the majortemperature contours are settled at the top half of the cav ity, except θ = 0.2 and 0.1.
Figs. 4 & 5 show the plots for the discrete heat sources (Case 2 &3). It is observed
that, the temperature profiles are closer to reach open at the top right portion
indicating higher heat transfer at that location
The variation of local Nusselt number, along the heater element for four different
cases and Rayleigh number 103, 105 & 107 is shown in Fig. 7. The heat transfer rate is
higher at the center of the cavity and reduces towards the heater surface. The local
Nusselt numbers are lower for split heat sources compared to continuous heating.
Case 1 and case 4 yields the similar Nu because of heaters at the centre. In discrete
heating (case 2 and 3) Heater A is giving the maximum Nu as compared to the otherheaters.
Fig.7 Variation of local Nusselt number along the heating length
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Fig. 12 and Table 1 combinedly show
the average Nu for all the cases. It is
observed that the average Nu increase
monotonically with increase of Ra. The
average Nu is maximum for heater (C)
Ra up to 105
, but it is more for Ra = 106
and 107 for heater (B).
Table.1 Average Nusselt Number V/s. Rayleigh Number
6. 2 Cavity with openingIn order to increase the heat transfer rate, the 25 % of the length of the cold wall
is opened at the top. Figs. 9-12 Show the streamlines and temperature plots for the
same.
Case 2 Case 3
Heater - (A) Heater - (B) Heater - (C) Heater - (D)Ra
Closed Open%
increaseClosed Open
%
increaseClosed Open
%
increaseClosed Open
%
increase
103 1.63 1.65 1.21 1.80 1.85 2.70 2.00 2.02 0.99 1.87 1.89 1.06
104 2.16 2.25 4.00 2.81 2.90 3.10 2.83 2.86 1.05 2.84 2.90 2.07
105 3.44 3.69 6.78 4.77 5.09 6.29 4.63 4.79 3.34 4.79 4.99 4.01
106 5.87 6.82 13.93 8.54 9.13 6.46 7.88 8.85 10.96 8.53 9.05 5.75
10
7
10.18 14.06 27.60 15.36 17.07 10.02 13.22 16.92 21.87 15.27 17.03 10.33
Fig.9 Contour plots for case 1 with 25% of opening at top cold wall with Ra = 105.
Fig. 8 Average Nusselt number Vs
Rayleigh Number for cases 1 and 4.
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Fig. 10 Contour plots for case 2 with 25% of opening at top cold wall with Ra = 105.
Fig. 12 Contour plots for case 4 with 25% of opening at top cold wall with Ra = 105.
Fig. 11 Contour plots for case 3 with 25% of opening at top cold wall with Ra = 105.
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It can be clearly seen from the plots drawn the streamlines and the temperature plots flow
out of the cavity at the opening. Fig. 8 and Table 1 combinedly show the variation of
average Nusselt number along the heated surface with Rayleigh Number for constant
heating. The opening at the right cold wall has noticeable effect only after Ra > 5×10 4.
For lower Rayleigh numbers the effect of opening is negligible. For higher Rayleigh
number (107) the opening has a significant effect that can be visualized by the increase in
the local Nusselt number. Table.1. shows the Average Nusselt number for discrete heat
sources (Case 2 & 4) with Rayleigh number. The heater (B) and (D) in case 2 and case 3
respectively have the same values due to its position. Heater (C) has a higher heat
transfer rate than (A). This is because the heat transfer rate is higher at the center and
gradually reduces towards the top wall.
7. Conclusions
The flow and heat transfer characteristics for discrete heating configurations
illustrated in the Fig.1 have been investigated. The following conclusions are drawn from
the present study.
(a) The conduction is dominated, for Ra ≤ 104 for all the cases of with and without
opening in the cold right vertical wall.(b) For higher Rayleigh numbers where convection is dominated the opening has a
significant effect on the average Nusselt number.
(c) With the four different boundary conditions for the vertical wall the maximum heat
transfer occurs for the case of heat source concentrated at the centre.
8. REFERENCES
Aydin, O., Yang, J., 2000. Natural convection in enclosures with localized heating
from below and symmetrical cooling from sides, Int. J. Numer. Methods Heat Fluid
Flow, 10, 518-529.
Ayla DOGAN., BAYSAL, S., Senol BASKAYA., 2009. Numerical analysis of
natural convection heat transfer from partially open cavities heated at one wall, J. of
Thermal Science and Technology 29, 1, 79-90.
Batchelor, G.K., 1993. An introduction to fluid dynamics, Cambridge University
Press, Cambridge, UK. Bilgen, E., Oztop, H., 2005. Natural convection heat transfer in partially open
inclined square cavities, Int. J. of Heat and Mass Transfer, 48, 1470-1479.
Issa, R.I., 1985. Solution of the implicitly discretised fluid flow equations by operator-
splitting, J. Comput. Phys. 62, 40-65.
Kasayapanand, N., 2007. Numerical modeling of natural convection in partially open
cavities under electric field, Int. Comm. Heat Mass Transfer 34, 630-643.
Muhammad, A.R. Sharif., Taquiur Rahman Mohammad., 2005. Natural convection in
cavities with constant flux heating at the bottom wall and isothermal cooling from thesidewalls, Int. J. of Thermal Sciences, 44, 865–878.
Nguyen, T.H., Prudhomme, M., 2001. Bifurcation of convection flows in a
rectangular cavity subjected to uniform heat fluxes, Int. Comm. Heat Mass Transfer, 28,
23-30.
Nithyadevi, N., Kandaswamy, P., Lee J., 2007. Natural convection in a rectangular
cavity with partially active side walls, Int. J. Heat Mass Transfer 50, 4688-4697.
Salat J., Xin S., Joubert P., Sergent A., Penot F., Le Quere P., 2004. Experimental and
numerical investigation of turbulent natural convection in a large air-filled cavity, Int. J.
Heat Fluid Flow 25, 824-832.
Yasin Varol., Hakan, Oztop, F., Tuncay Yilmaz., 2007. Natural convection in
triangular enclosures with protruding isothermal heater, Int. J. of Heat and MassTransfer, 50, 2451-2462.