education ppt for students of mathematics byPurusothaman.ppt

Dr. A. PURUSOTHAMAN
DST-SERB National Post-Doctoral Fellow
Department of Mathematics
National Institute of Technology
Tiruchirappalli- 620 015
Tamilnadu, INDIA
Email: abipurus@gmail.com; apurusoth@nitt.edu
1
Fluid Flow and Heat Transfer Characteristics of Natural
Convection in a Cubical Enclosure with Thermally Active
Source

2
INTRODUCTION
:
Electronic equipment has made its way into
practically every aspect of modern life, from
toys and appliances to high-power computers.
Electronic components depend on the
passage of electric current to perform their
duties, and they become potential sites for
excessive heating, since the current flow
through a resistance is accompanied by heat
generation.
The failure rate of electronic equipment
increases exponentially with temperature.
Both the performance of reliability and life
expectancy of electronic equipment are
inversely related to the component
temperature of the equipment.

3
Long life and reliable performance of a component may be achieved
by effectively controlling the device operating temperature within the
limits.
Therefore, thermal control has become more and more important in
the design and operation of electronic equipment.
In literature, there are several cooling techniques commonly used in
electronic equipment such as conduction cooling, natural convection
and radiation cooling, forced-air cooling, liquid cooling, and immersion
cooling, etc...
Thermal control in electronic components using dielectric liquids has
received increased attention due to inherently high heat removal
capabilities of liquids compared to air.
So, natural convective heat transfer and fluid flow in liquids filled
enclosures with various shapes and wall conditions have been examined
extensively by many researchers for instance, Heindel et.al.(1995), Tou
et.al.(1999) and Tso et.al (2004).

4
 The review of the above literatures indicates that the fluids like
water, oil and dielectric liquids have high thermal conductivity
which leads to maximum heat transfer performance.
 Studies on buoyancy driven convective heat transfer in an enclosure
using dielectric fluids have drawn attraction of many researchers in
recent years.
 Hence, taking this opportunity, the aim of the present study is to
evaluate the natural convection heat transfer and fluid flow
performance in a liquid filled cubical enclosure with an isothermally
active source.
 This work may give some additional knowledge in designing sealed
electronic packages encountered in the microelectronics industry.

L
L
g
c
b
h
L
h
c
b
x
z
y
x
z
y
Physical Configuration
(a) (b)
Fig. 1. Physical configuration.
(a) Vertically located heated plate (b) Horizontally located heated plate.
5

Mathematical Formulation
The governing dimensional form of continuity,
momentum and energy equations can be written as
0
u v w
x y z
  
  
  
(1)
2 2 2
2 2 2
u u u u p u u u
u v w
t x y z x x y z

 
       
      
 
       
 
2 2 2
2 2 2
v v v v p v v v
u v w
t x y z y x y z

 
       
      
 
       
 
(2)
(3)
2 2 2
2 2 2
( )
c
w w w w p w w w
u v w
t x y z z x y z
g T T


 
       
      
 
       
 
  (4)
(5)
2 2 2
2 2 2
T T T T T T T
u v w
t x y z x y z

 
      
     
 
      
  6

The dimensional form of the initial and boundary conditions at
cavity walls and plate are
0 : 0; ; 0 , , ,
0 : 0; ; 0 & L,
0; 0; 0 & ,
0; 0;
c
c
t u v w T T x y z L
t u v w T T x
T
u v w y L
y
T
u v w
z
 
      
 
     

    


   

0 & ,
0; ; on the heater
h
z L
u v w T T

   
The Physical quantities are nondimensionlized by introducing the
following parameters:
2
2 2
3
, , , , ,
, , , ,
( )
, Pr
C
H C
H C
x y z t pL
X Y Z P
L L L L
uL vL wL T T
U V W
T T
g L T T
Ra




  
 
 
    

   


 
7

Mathematical Formulation
The governing nondimensional form of continuity,
momentum and energy equations can be written as
0
U V W
X Y Z
  
  
  
(6)
2 2 2
2 2 2
Pr
U U U U P U U U
U V W
X Y Z X X Y Z

 
       
      
 
       
 
2 2 2
2 2 2
Pr
V V V V P V V V
U V W
X Y Z Y X Y Z

 
       
      
 
       
 
(7)
(8)
2 2 2
2 2 2
Pr
Pr
W W W W P W W W
U V W
X Y Z Z X Y Z
Ra


 
       
      
 
       
 
 (9)
(10)
2 2 2
2 2 2
U V W
X Y Z X Y Z
      

      
     
      
8

The dimensionless form of the initial and boundary conditions at
cavity walls and plate are
0 : 0; 0; 0 , , 1
0 : 0; 0; 0 &1
0; 0; 0 &1
0; 0; 0 &1
U V W X Y Z
U V W X
U V W Y
Y
U V W Z
z
 
 


      
     

    


    

0; 1
U V W 
   
On the plate
9

The nondimensional heat transfer rates at the cold walls are
calculated by the Nusselt numbers. The local Nusselt numbers along
the cold walls are
0&1
0&1
X
X
Nu
X






The average Nusselt numbers are calculated by integrating the local
Nusselt numbers along the cold walls
 
1 1
0&1
0 0 0&1
X
X
Nu Nu dYdZ



The mean Nusselt number is calculated by taking the arithmetic
mean of the average Nusselt numbers along the cold walls.
Nu
Heat Transfer Rate
10
(11)
(12)

 A Finite Volume Method is used to solve the nonlinear governing equations
(conservation of mass, momentum and energy equations for an unsteady, laminar
flow) on a staggered grid system.
 The Power Law Scheme is used to solve the convection and diffusion terms since
it gives a better approximation to the exact solution.
 The SIMPLE (Semi-Implicit Method for Pressure-Linked Equations) algorithm of
Patankar (1980) is used to deal with coupled pressure and velocity fields, which
implicitly takes care of the divergence free nature of incompressible fluid flow.
 Then the set of discretized equations are solved by a line-by-line procedure of the
Tri-Diagonal Matrix Algorithm (TDMA).
To ensure convergence of the numerical algorithm the following criteria was
applied to all dependent variables over the solution domain
5
,
,
,
,
,
,
1
,
,
,
,
10





k
j
i
m
k
j
i
k
j
i
m
k
j
i
m
k
j
i



A ForTran code was developed to implement the above procedure.
U, V, W and θ - Dependent variables
(i, j, k) - space coordinates
‘m’ - current iteration
Numerical Procedure:

12
Code Validation with Tric et al. (2000)
Ra

Tou et al. (1999)
104
105
106
107
101
Present study
Tou et al.
Ra
Row-averaged Nusselt
number against Ra
13

14
Grid Independence
Test
The mean Nusselt number is calculated
using five different computational grids, viz.,
41×41×41, 61×61×61, 81×81×81,
101×101×101 and 121×121×121.
A refinement of the grid from 101×101×101
to 121×121×121 does not have a significant
effect on the results in terms of mean Nusselt
number
The difference in Nusselt numbers between
the two grids 101×101×101 and 21×121×121
is found to be less than 0.25%.
Considering both the accuracy and the computational time, the
present calculations are all performed with a 101×101×101 uniformly
spaced grid system.

Results and Discussion
 A numerical study of three dimensional natural convection in a
cubical cavity induced by a thermally active plate, built in vertically
or horizontally is made for different values of the plate aspect ratio
Ac.
 The active plate dimensions b/L and h/L are fixed to be 0.1 and 0.5
for the vertical plate and 0.5 and 0.1 for the horizontal plate. For this
study, the computations are performed for Ra = 105
, 106
and 107
keeping the plate aspect ratio (Ac =c/L) between 0.1 and 1.0.
 Simulations are carried out for various fluids such as mercury, air
and dielectric liquid FC-77 corresponding to Pr = 0.025,0.71 and 25
respectively.
15

(a) (b)
Fig. 2. ITHVP; Isotherms for Ac
= 0.5 at (a) Y = 0.0, 0.5 &1.0 (b) X = 0.2 & 0.5
with fixed Ra =107
and Pr = 0.71.
16

Fig. 3. ITHHP; Isotherms for Ac
= 0.5 at (a) Y = 0.0, 0.5 &1.0 (b) X = 0.2 & 0.5
with fixed Ra =107
and Pr = 0.71.
17

(a) (b)
Fig. 4. ITHVP; Velocity vectors for Ac
= 0.5 at (a) Y = 0.05, 0.5 &0.95
(b) X = 0.03, 0.25 & 0.5with fixed Ra =107
and Pr = 0.71. 18

(a) (b)
Fig. 5. ITHHP; Velocity vectors for Ac
= 0.5 at (a) Y = 0.05, 0.5 &0.95
(b) X = 0.03, 0.25 & 0.5with fixed Ra =107
and Pr = 0.71.
19

20
Fig. 6. Stream trace for ITHHP with fixed Pr = 0.71 and Ac
= 0.5 (a) Ra =107
;
(b) Ra =106
, Stream trace for ITHHP with fixed Ra =107
and Ac
= 0.5 at;(c)
Pr=0.025 (d) Pr=25.

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
0
2
4
6
8
10
12
14
__
Nu
Ac
__
ITHVP
ITHHP
Pr=0.71
Pr=0.025
Pr=25
Fig. 7. Mean Nusselt number against
Ac with fixed Ra =107
.
0
2
4
6
8
10
12
14
__
Nu
106
Ra
__
105
107
ITHVP
ITHHP
Pr=0.71
Pr=25
Pr=0.025
Fig. 8. Mean Nusselt number against Ra .
21

Conclusion
Based on the findings in the study, the following conclusions are
observed.
 As the aspect ratio Ac of the heated plate is increased, the mean Nusselt
number increases. Further, heat transfer becomes more enhanced for the
vertical plate compare to the horizontal plate.
 With increase of Rayleigh number, the heat transfer rate increases for
both vertical and horizontal plates.
 It is also found that the mean Nusselt number attains its maximum
value for high Pr and minimum for low Pr irrespective of the values of
Ac and Ra.
22

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27

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