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Entropy Generation Analysis of Natural Convection 
in Square Enclosures with Two Isoflux Heat Sources 

 

Saeed Zaidabadi Nejad  
Department of Mechanical Engineering, Kerman Branch, 

Islamic Azad University, Kerman, Iran  
 

 

Mohammad Mehdi Keshtkar 
Department of Mechanical Engineering, Kerman Branch, 

Islamic Azad University, Kerman, Iran 

Abstract-This study investigates entropy generation resulting 
from natural convective heat transfer in square enclosures with 
local heating of the bottom and symmetrical cooling of the 
sidewalls. This analysis tends to optimize heat transfer of two 
pieces of semiconductor in a square electronic package. In this 
simulation, heaters are modeled as isoflux heat sources and 
sidewalls of the enclosure are isothermal heat sinks. The top wall 
and the non-heated portions of the bottom wall are adiabatic. 
Flow and temperature fields are obtained by numerical 
simulation of conservation equations of mass, momentum and 
energy in laminar, steady and two dimensional flows. With 
constant heat energy into the cavity, effect of Rayleigh number, 
heater length, heater strength ratios and heater position is 
evaluated on flow and temperature fields and local entropy 
generation. The results show that a minimum entropy generation 
rate is obtained under the same condition in which a minimum 
peak heater temperature is obtained. 

Keywords- natural convection;enclosure; finite volume method; 
Rayleigh number; entropy 

Nomenclature 
Pr Prandtl number (=ν/α) (dimensionless) 
Ec Eckert number (=α

2 k/h3 Cp q1) 
Ra Rayleigh number (=gβq1 h

4/kαν) 
g Acceleration due to gravity (m/s2) 
h Height of cavity (m) 
k Thermal conductivity (W/mK) 

lh,1 Width of left heater (m) 
lh,2 Width of right heater (m) 
Lh,1 Dimensionless width of left heater (lh,1  /h) 
Ns Dimensionless rate of entropy generation (=Sgen 

h2/k) 

P* Effective pressure (= ( )aP gy P  
P Dimensionless effective pressure (=p* h2/α2) 
q1 Flux over left heater (W/m2) 
q2 Flux over right heater (W/m2) 
qr Heat flux ratio (=q2/q1) 
TC Cold wall temperature (K) 
T  Ambient temperature (K) 

*T  Dimensionless ambient temperature (=TC k/q1 h) 
U Dimensionless horizontal velocity (=uh/α) 
V Dimensionless vertical velocity (=vh/α) 
X Dimensionless horizontal coordinate (=x/h) 
Y Dimensionless vertical coordinate (=y/h) 
α Thermal diffusivity (m2/s) 
β Volumetric expansion coefficient (K-1) 

εr Heater length ratio (lh,2/lh,1) 
μ Dynamic viscosity (Pa.s) 
ν Kinematic viscosity (m2/s) 
θ Dimensionless temperature (=(T-TC)k/q1 h) 
  Ambient density (kg/m

3) 

I. INTRODUCTION  
Entropy Generation Minimization (EGM) in a closed 

enclosure filled with fluid is the interaction of heat transfer, 
fluid mechanics, and engineering thermodynamics. This 
approach is considered as a part of exergy analysis, since 
exergy destruction is proportional to entropy generation. 
Various precise works on natural convection in enclosures have 
been published. In [1-2] authors numerically solved free 
convection problem in a square enclosure with insulated 
horizontal walls and vertical walls under two different 
temperatures. In [3] authors studied laminar free convection in 
a rectangular enclosure with discrete heat sources. They 
showed that constant temperature heat sources are generally 
more significant than constant flux. Passive systems such as fin 
or baffle are simple methods used to control fluid flow and heat 
transfer  due to natural convection in enclosures. In [4], authors 
studied two-dimensional laminar natural convection heat 
transfer in an inclined fin located cavity. They observed that 
the inclination angle of the fin is an important parameter for 
controlling heat and fluid flow. In natural convection process, 
entropy generation should be minimized in thermal processing 
to achieve an optimal processing with minimum irreversibility. 
Optimal conditions can be evaluated by EGM. In [5–6], authors 
comprehensively discussed the analysis of entropy generation 
for various problems. In [7], authors studied entropy generation  
due to heat transfer and fluid low irreversibility in natural 
convection into diverse cavities. In [8] natural convection in 
closed enclosures with heating in corners was evaluated. It was 
documented that heat transfer depends on the length of the 
heating body and that the effect of Prandtl number on average 
Nusselt number is more significant in Prandtl numbers lower 
than 1. In [9], authors studied entropy generation due to natural 
convection in a square enclosure heated by a protruding heat 
source and filled with nanofluid. It was shown that heat transfer 
performance could be maximized and entropy generation could 
be minimized by positioning the heat source on the lower wall 
of the enclosure. In [10], authors investigated free convection 
in a rectangular system with hot sidewall in nine different 
positions of heat source. In [11], authors numerically analyzed 



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free convection in a 3D square enclosure with a cubic body 
producing heat at the center of the enclosure. They found out 
that fluid flow inside the enclosure results from two 
temperature differences; one, temperature difference between 
hot and cold wall of the enclosure and the other temperature 
difference caused by heat generating body. In [12], entropy 
generation due to natural convection in square enclosures with 
two discrete heat sources was generated.  

Recently, different studies have been conducted on mixed 
convection in nanofluid-containing enclosures with moving 
wall and different geometry and boundary conditions [13-15]. 
In [16], authors addressed heat transfer in the space between 
cold outdoor square cylinder and hot indoor elliptical cylinder 
in the presence of a uniform magnetic field and reported the 
results relative to particle volume fraction, Rayleigh number 
and Hartman. In [17], authors simulated natural magne-to-
hydrodynamic convection in a square enclosure full of alumina 
water nanofluid using the Lattice-Boltzmann method. This 
simulation is different from similar cases in a place-dependent 
sine temperature distribution in vertical walls of the enclosure. 
Their results showed that heat transfer is directly proportional 
to Rayleigh number and inversely proportional to Hartmann. In 
[18], authors studied free convection in a steep chamber with 
two adjacent walls at two different temperatures. They found 
that the effect of the enclosure angle is negligible on lines of 
current and temperature in small Rayleigh numbers, while free 
convection considerably increases in large Rayleigh numbers. 
In [19], authors numerically investigated a mixed convection 
fluid flow, heat transfer and entropy generation inside a 
triangular enclosure filled with CuO-water nanofluid with 
variable properties. 

This study tends to minimize operating heater temperatures 
and rate of entropy generation in the system. This study 
optimizes the relationship between locations of heaters, heater 
length ratios as a function of heater strength ratios. 

II. METHODOLOGY 
In this simulation, the electronic components and side walls 

were modeled as constant flux heat sources and isothermal 
thermal heat sinks, respectively. The flow and temperature 
fields were obtained by numerical simulation of mass 
conservation, momentum, and energy equations. 

Simplifying assumptions are: 

• Incompressible and Newtonian fluid. 

• 2D, steady state and laminar flow field. 

• Constant thermo physical properties. 

• Neglect the radiation effects.  

• For elimination of density as a variable, the correlation 
between variation in density and temperature is given 
by Boussinesq approximation.  

The numerical simulation was performed in MATLAB. The 
governing equations were discretized by finite volume method, 
and were solved using the SIMPLER algorithm. In finite 
volume method (FVM) integral forms of governing equations 

derived directly from conservation laws are used for 
discretization in the considered space. FVM contains simple 
discretization and applicability to complex networks. The most 
fundamental problem of FVM is the definition of variables in 
sides of control volume, particularly second order derivatives. 
In this method, interpolation methods are often used to obtain 
flux. It is proved that if Domain mesh resolution is less than 
100×100, inverse matrix method performs better than other 
methods in obtaining linear algebraic equations. The current 
study uses this method. 

A uniform 50×50 staggered grid was used to store 
velocities and scalar variables. The power law scheme was also 
used to define the relationships between convection-diffusion 
terms. At the beginning, the required parameters and matrices 
such as physical parameters, velocity and pseudo-velocity 
matrix are defined. This section defines convergence conditions 
for conservation equations of mass, momentum and energy. 
Then, pseudo-velocity field is obtained by simpler algorithm. 

Step 1: calculate pseudo velocities: 

However, calculations are different for internal nodes and 
boundary conditions. Calculations are done separately. 

The next step of the SIMPLER algorithm is to calculate 
pressure field. 

Step 2: solve pressure equation: 

Since Navier-Stokes equation and consequently discretized 
equation only contains pressure gradient, unknown pressure 
values may be considered as difference of two very large 
numbers (for example, 100100-1000000), which may cause 
problems in numerical solution. To avoid this problem, 
pressure value of a certain node is considered zero in the 
solution. 

In the third step of simpler algorithm, the obtained pressure 
field is considered as speculative pressure field; then, infra-
release factor is applied on pressure field. 

Step 3: p = p* 

In the fourth step, velocity field is calculated by solving 
discretized momentum equation: 

Step 4: solve discretized momentum equations: 

Next, pressure correction field is calculated to correct 
velocity field. Then, temperature field is obtained by corrected 
velocity values. Finally, convergence conditions are checked. 

Entropy field is obtained by calculating velocity and 
temperature fields. FDM and second-order CDS are used for 
discretization. To discretize term of entropy generation due to 
heat convection, temperature derivatives of CDS are 
substituted. To discretize term of entropy generation due to 
fluid friction, velocity derivatives of FDS are substituted. 

Finally, the resulted the linear algebraic set of equations 
was solved using inversion matrix method. Process simulation 
and problem solving is shown in Figure 1. 



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In this optimization, the parameters prevailing in the flow 
field, temperature, and entropy were examined as the 
following: 

• The effect of changing dimensionless parameter of 
Rayleigh number (Ra) on flow field, temperature, and 
entropy was investigated in constant length and 
strength  ratios  of  the  heaters (εr=qr=1). 

•  The effect of changing strength ratio of the heaters (qr) 
on flow field, temperature, and entropy was 
investigated in constant length ratio of the heaters 
(εr=1). 

• The effect of changing length ratio of the heaters (εr) 
on flow field, temperature, and entropy was 
investigated in constant strength ration of the heaters 
(qr=1). 

• The effect of changing the dimensionless parameter of 
distance between centers of the heaters (Xc) on flow 
field, temperature, and entropy was investigated in 
constant length and strength ratios of the heaters 
(εr=qr=1).  

 

 

Fig. 1.  The flow chart of the method steps in this study  

III. GEOMETRY AND BOUNDARY CONDITIONS  
The geometry of the problem and details of boundary 

conditions, illustrated in Figure 2, involves a square chamber 
with 15 cm side length having two heat sources on the bottom 
wall. The length of left-hand and right-hand heaters are Lh,1 and 
Lh,2 , respectively. The distance between centers of the heaters 
is indicated by Xc. The fluid inside the chamber is air with 
Pr=0.7. All chamber walls are fixed, having no movement. The 
temperature of left- and right-hand walls is equal to the 

constant cold wall temperature (Tc). The environment 
temperature is assumed to be 300 K (T∞=300K). The upper 
wall and not-heating part of the lower wall are insulated. The 
following boundary conditions were used to solve the 
governing equations: 

• The boundary condition of no slippage and 
impermeability for all walls (U=V=0). 

• The thermal boundary condition of θ=0 for cold side 
walls. 

• The thermal boundary condition of  0
Y





 for the 

upper wall and non-heating part of the lower wall. 

• The thermal boundary condition of 1
Y


 


 for the 

position of mounting the left-hand heater. 

• The thermal boundary condition of rq
Y


 


 for the 

position of mounting the right-hand heater. 
 

 

Fig. 2.  Problem geometry and boundary conditions  

IV. DIMENSIONLESS GOVERNING EQUATIONS 
Characteristic quantities which are constant in the field of 

flow and temperature are used to make dimensionless constant 
and independent variables; these quantities 
include , ,, , , ,sg L T T P V    . 

Dimensionless independent and constant variable are 
defined as follows: 

   
 

* * *
2

                           
S

P P T TV
V P T

V T TV
 

 

 
  

 
 

* * *                                              
Vg x

g t t x
g L L




    

The subscript ∞ refers to the characteristic distance of the 
object. The dimensionless form of operators is defined as 
follows: 

*
* * *

1 1 1 1
 

L L L Lx y z

  
     

  
 

Molding: modeling of cooling process of two semiconductors in the form of 
natural convection in enclosure. Modeling of electronic components in the 
form of  isoflux heat sources. Modeling  sidewalls in the form of isothermal 
heat sinks

Derivation of governing equations: derived Differential form of 
conservation equations of mass, momentum and energy in state of steady, 
two-dimensional and laminar flow 

Discretization of governing equations: using finite volume method, 
approximation of power law, SIMPLER algorithm and staggered grid 

Solving set of algebraic equations obtained: Using matrix inversion 
methods 

Control convergence and grid independence 

Analysis the results   



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22 *    
2 * *

1
                     

( )

VD D D

D t LL L t D t
D

V





    
 

Dimensionless form of continuity equation: 

*
*

. 0
D

V
D t


    

Dimensionless form of momentum equation: 

2
*

* * * * *
* 2

1DV Gr
T g P V

ReDt Re
      

Dimensionless form of energy equation: 

2
*

* * *1  
D T E c

T
D t R eP r R e

     

Dimensionless Reynolds number: 

V L V L
Re


 
    

Dimensionless Grashof number: 

  3
2

Sg T T L
G r




  

  3
.

Sg T T L
Ra Gr Pr




   

Rayleigh number is defined as the ratio of buoyancy force 
to heat spreader in the fluid. Rayleigh number determines 
laminar or turbulent flow in the free convection. 

V. GOVERNING EQUATIONS 
The governing equations of this this problem include mass 

conservation, momentum, energy conservation, and the rate of 
generated entropy. The stable and dimensionless form of the 
governing equation is: 

0
U V

X Y

 
 

 
 (1) 

2 2

2 2
[ ] 0

U U P U U U V
U V

X Y X X YX Y

      
      

     
 

 

 
(2) 

2 2

2 2
P r[ ] P r

V V P V V
U V R a

X Y Y X Y


    
     

    
 

 

 
(3) 

2 2

2 2
U V

X Y X Y

      
  

   
 

 

 
(4) 

2 2 2 2 2
2

Pr1
[( ) ( ) ] [2{( ) ( ) } ( )

( )
c

s
E u v u v

N
x y x y x yT T

 
  

     
     

      
 

 
(5) 

 Where, X=x/H and Y=y/H are dimensionless horizontal and 
vertical coordinate components respectively. U=uh/α and 
V=vh/α are the dimensionless velocity components in the X 
and Y directions. P is the dimensionless effective pressure, θ 
is the dimensionless temperature, θ=(T−TC)k/q1h , where Tc 
is the cold wall temperature. In (5), The first phrase 
demonstrates entropy generation due to conduction heat 

transfer (Ns,conduction) and the second phrase demonstrates the 
entropy generation due to viscous dissipation (Ns,viscous). 

VI.           RESULTS 
In order to validate the computational code, the results were 
compared with the results of reference [12] and it was observed 
that, in the same conditions, there is good agreement between 
the results. Minimal difference between the results is due to 
differences in applied methods and approximations solution. 
The result of this comparison is shown in Figure 3. The 
summary of the states investigated in this study is shown in 
Table I. Convergence velocity and temperature fields 
controlled with relationship Root square. The remaining 
amount is defined as follow: 

max max
2 8

2 ,
1 1

( ) 10
I I J J

I J
I J

L b
= =

-

= =

= £å å  

 

  
Fig. 3.  Comparing the results of preseny study with reference [12] 

In numerical simulations, one of the important points the 
independence of the results is the number of grid points. To 
evaluate the grid independence, the nondimensional maximum 
temperatures in a particular case and for two different grids 
obtained, and compared with results of the 100×100 grid. 
Result of this investigation are shown in Table II.  Once the 
results were obtained and compared for several grids, it was 
concluded that the 50×50 grid occupied minimal memory or 
lower runtime provided reasonable results compared to other 
grids. 

A. The effect of Rayleigh number on flow field, 
temperature distributuon and entropy generation 

Figure 4 shows the streamlines and isotherms contours for 
different Rayleigh number. In this case, heaters have equal 
length and strength ratios (εr=qr=1). Also the distances 
between heaters and sidewalls are equal. 

The trend of streamlines reveals that in low Rayleigh 
numbers (Ra=103 and Ra=104), the viscous force is partly 
dominant over buoyancy force, and buoyant flow is weak. As 
well as, and the upper half of enclosure hasn't been practically 
affected by the heat sources. The cores of created vortices 
move upward by increasing Rayleigh number due to forming a 
stronger rotating flow and increasing buoyancy force. When 
Ra=106, the isothermal curves are thinner that other states, 
which is due to increased energy transmission. In all states 



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related to Figure 3, heaters have equal length and strength 
ratios and the boundary condition of side walls is also 
symmetrical, therefore, there is similar heat loss from cold side 
walls and expected that heater temperatures are equal.  
dominant over buoyancy force, and buoyant flow is weak. In 
this case, the isothermal curves are rather flat, and the upper 
half hasn't been practically affected by the heat sources. The 
cores of created vortices move upward by increasing Rayleigh 
number due to forming a stronger rotating flow and increasing 

buoyancy force.  Furthermore, by increasing Rayleigh number, 
the isothermal curves divert from those of pure conductance, 
and fluid flow results in transmission of hot air to near the 
upper wall. When Ra=106, the isothermal curves are thinner 
that other states, which is due to increased energy transmission. 
In all states related to Figure 4, heaters have equal length and 
power ratios and the boundary condition of side walls is also 
symmetrical, therefore, there is similar heat loss from cold side 
walls and the temperature of heaters are expected to be equal.  

TABLE I.  THE SUMMARY OF THE STATES INVESTIGATED  

εr qr Lh  XC  Xl Ra Stat
e  

Figure 
No.  

1  
1  
1  
1  

 
0.3 
0.6  
1.5  
2  

 
  
1 
1  
1  
1  

 
 

0.3≤ εr≤ 1.7 
0.3≤ εr≤ 1.7 
0.3≤ εr≤ 1.7 

1 

1  
1  
1  
1  

  
1 
1  
1  
1  

 
  
1 
1  
1  
1  

  
0.5 
1 
2  

  
1 

0.1  
0.1 
0.1 
0.1 

  
0.1 
0.1 
0.1 
0.1 

 
  

0.1 
0.1 
0.1 
0.1 

  
0.1  
0.1  
0.1 
  
0.1 
  

0.5  
0.5  
0.5  
0.5 
  
0.5  
0.5  
0.5  
0.5  

 
 

0.1  
0.3  
0.6  
0.9  

  
0.5  
0.5  
0.5  
  
  

0.25  
0.25  
0.25  
0.25  

  
0.25 
0.25  
0.25  
0.25  
 
  

0.45 
0.35  
0.2  

0.025  
  

0.25 
0.25  
0.25  
  

0.5-( XC/2)  

103  

104  

105  

106  
  

1.5×106 

1.25×106  

8×105  

6.66×105  

 
  
106 

106  

106  

106  
  

2×106/(1+ εr× qr)  
2×106/(1+ εr× qr)  
2×106/(1+ εr× qr)  

  

106  
  

A  
b 
c  
d  

  
a  
b 
c  
d  

 
  
a  
b 
c  
d 

  
a  
b 
b 
  
a  

2 and 3 
2 and 3 
2 and 3  
2 and 3  

  
 5 and 7 

5 and 7 
5 and 7 
5 and 7 

 
 

8 and 10 
8 and 10  
8 and 10 
8 and 10 

   
6 
6 
6 

 
9 and 11  

TABLE II.  THE RESULTS OF GRID INDEPENDENCE 

%Change (absolute) 

in
 max

.Ra q max.Ra q Grid size
  

7.73×104 50×50 3 
0 

0.2 
7.5×104 

7.4×104 
100×100 
150×150 

 

Figure 5 shows the trend of changing entropy generation 
contours resulted from fluid friction irreversibility (FFI) and 
conductive heat transfer irreversibility (HTI) by Rayleigh 
number. The contours are plotted using normalized values. 
These values are obtained by dividing the entropy resulted 
from conductive heat transfer (Ns,conduction) or fluid friction 
(Ns,viscous) to the maximum generated entropy (Ns,max). These 
contours show that the maximum entropy is generated near 
heaters and is due to HTI. The conductive heat transfer process 
was resulted by two temperature differences. These two 
temperature differences, caused by creation of the internal and 
external irreversibility due to heat transfer in the system, 
involve the temperature difference (conduction heat transfer) 
created between heaters and the fluid inside the enclosure, as 
well as the temperature difference created between the hot fluid  

 
and cold enclosure walls. The plot of entropy generated by heat 
transfer shows that the value of HTI is negligible near the 
upper wall, which is due to the little temperature gradient in 
these areas. The rate of entropy generation due to heat transfer 
increases by Rayleigh number. The plots of entropy generated 
by FFI show that the friction irreversibility is higher near the 
walls because of forming a boundary layer and velocity 
gradient. The fluid flow and thickness of the hydrodynamic 
boundary layer formed on the walls increases by Rayleigh 
number, and the FFI value increases as a result. In all states 
investigated in Figure 5, the magnitude of entropy generated by 
heat transfer is higher than the entropy generated by fluid 
friction. Therefore, the Bejan number, which represents the 
relative importance of heat transfer irreversibility over total 
irreversibility, can be considered equal to 1 for all states 
investigated in this study. This issue is consistent with the 
results of [12], whose authors found out that the magnitude of 
entropy generated by fluid friction is several orders of 
magnitude less than the entropy generated by heat transfer. 
Figure 6 shows the contours of changing total entropy 
generation by Rayleigh number. As it can be seen, the total 
entropy generation in the enclosure increases with Rayleigh 
number, which is the result of simultaneous rise in the value of 
entropy generated by fluid friction and conduction heat transfer 
with increasing Rayleigh number. 



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State (b): Ra=104 State (a): Ra=103  

     
State (d): Ra=106 State (c): Ra=105 

     

Fig. 4.  Plots of changing flow curved (left column) and isothermal curves (right column) versus Rayleigh number 

State (b): Ra=104 State (a): Ra=103 

      
State (d): Ra=106 State (c): Ra=105 

     

Fig. 5. The trend of changing entropy resulted from fluid friction (left column) and conductive heat transfer (right column) versus Rayleigh number 

State (d): Ra=106 State (c): Ra=105 State (b): Ra=104 State (a): Ra=103 

      

Fig .6. The trend of changing total entropy versus Rayleigh number 



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B. The effect of length ratio on  flow field, temperature 
distributuon and entropy generation  

Figure 7 shows the effect of inequality of heaters' length a 
flow field and entropy. The power ratio of the heaters are equal 
in all states related to this figure (qr=1), and only their length 
ratio changes. 

The current study differs from [12] in: 

• Evaluating effect of heater length ratio on flow, 
temperature and entropy fields 

• Evaluating entropy generation due to fluid friction 
(FFI) in all scenarios 

• Solving linear algebraic equations obtained 

In practice, the studied problem is similar to [12]; thus, 
structure of this study is also similar to [12]. 

As a result, the length of vortices formed on heaters 
changes an asymmetric heat loss occurs from the sold side 
walls. When εr=0.3, the length of right-hand heater is 0.3 of 
that in the left-hand heater, so the rate of input thermal energy 
to the right-hand heater is 0.3 of that to the left-hand heater. On 
the other hand, the convective flow creates stronger vortices on 
the shorter heater, therefore, more heat would be lost from the 
surface and walls of the right-hand heater. So, for any power 
ratio of the heaters, the temperature of the left-hand heater 
decreases with εr

, while the temperature of the right-hand heater 
increases. Figure 8 confirms the above statements. Comparing 
different states of Figure 8 shows that the temperature of the 
heaters will be equal and at minimum if their length and power 
ratios are equal. In other words, the optimum length and power 
ratios, which prevents creation of hot spot and early failure in 
the system, is equal to 1 (εr= qr=1). Figure 9 shows the trend of 
changing the total entropy generation and the entropy generated 
by fluid friction versus length ratio of the heaters. As it was 
mentioned earlier, changing heaters' length results in changed 

length of the vortices formed on them and an asymmetrical 
heat loss from the cold side walls. So, the area of the surface in 
which the temperature gradient and irreversibility is important 
increases with heaters' length ratio. The contours shown in 
Figure 9, besides confirming the statements above, reveal that 
for the states related to εr<1, increasing the heaters' length ratio 
results in approaching to the optimum state of εr=1 and 
decreasing entropy generation in the chamber. Furthermore, for 
the states related to εr>1, increasing the heaters' length ratio 
results in diverting from the optimal state and increasing 
entropy generation. 

C. The effect of distance between heaters on  flow field, 
temperature distributuon and entropy generation 

Figure 10 shows the effect of positioning of the heaters on 
the flow and temperature fields. In all states related to this 
figure, the length and strength ratios of the heaters are equal 
(εr=qr=1), and their positioning related to the sidewalls are 
equal. The Rayleigh number is also constant and equal to 
Ra=106. So, for a given value of XC, the temperature 
differences between each heater and the fluid inside the 
enclosure are equal, and the vortices formed on the heaters will 
be the same. This can be inferred from the patterns of flow 
curves. The isotherms show that by decreasing XC, the resulted 
interaction between the heaters reduces the rate of heat transfer 
from them, therefore, the surface temperature of the heaters 
increases and the hot fluid can move upward near to the upper 
wall. As Xc increases, the greater distance between the heaters 
results in increasing the rate of heat transfer from them, and the 
surface temperature of the heaters decreases as a results. On the 
other hand, if the distance between the heaters exceeds a 
certain value, the buoyancy force and fluid flow created in the 
enclosure weak, the heat transfer from surface of the heaters 
decreases and their heat loss increases, and heaters temperature 
decreases again. The contrast of these two effects is shown in 
Figure 11. 

State (b): εr=0.6 
 

State (a): εr=0.3
 

      
 State (d): εr=2 State (c): εr=1.5

 

     

Fig .7. Plots of changing flow curves (left column) and isothermal curves (right column) versus heaters' length ratio 



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   State (c): qr=2 State (b): qr=1 State (a): qr=0.5 

   

Fig .8. Plots of heaters' temperature change versus their length ratio 

State (b): εr=0.6 State (a): εr=0.3 

      
State (d): εr=2 State (c): εr=1.5 

      

Fig .9. The trend of changing the total entropy (right column) and the entropy generated by fluid friction (left column) versus the heaters' length ratio 

State (b): Xc=0.3 State (a): Xc=0.1 

     
State (d): Xc=0.9 State (c): Xc=0.6 

     
Fig .10. Plots of flow curves (left column) and isothermal curves (right column) versus distance between the heaters 

Figure 12 shows the contours of the total entropy 
generation versus distance between the heaters. The contours in 
this figure show that by increasing distance between the 

heaters, first, the entropy generation decreases, but then 
increases with approaching the to the side walls. So, the 
changes in the total entropy generation with the distance 



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between the heaters is mainly a function of the entropy 
generated by heat transfer. This can also be inferred from 
Figure 13. Figure 11 shows the trend of changing the heater 
temperature and the rate of entropy generation versus distance 
between the heaters. The rate of total entropy generation is 
derived by integrating the rate of local entropy generation rate. 
This figure shows that changing the rate of entropy generation 
versus distance between the heaters is almost similar to the 
change in the peak temperature of the heaters versus their 
distance. This figure also indicates that the spacing between the 
heaters is the most important parameter in terms of optimal 
thermal management. The most important factor in controlling 
the irreversibility is the positioning of the heaters. Therefore, in 
the perspective of thermal management of the electronic 
components, mounting heaters near the chamber's central line 
is unfavorable since the peak temperature and generated 
entropy are high in this area. On the other hand, on order to 
minimize the peak temperature and entropy generation, heaters 
have to be mounted near cold side walls, or the distance 

between the heaters has to be half of the chamber width. In 
terms of efficiency, when the length and power ratios of the 
heaters are equal, presence of a number of heat sources results 
in equal power loss compared to presence of one heat source in 
the center. The studies carried out in this context confirm this 
issue. 

 
Fig. 11. Plot of changing surface temperature of the heaters versus their 

distance 

 
State (d): Xc=0.9 State (c): Xc=0.6 State (b): Xc=0.3 State (a): Xc=0.1 

    
Fig .12. The trend of changing the total entropy generation versus distance between the heaters 

 
Fig .13. Plot of changing heater temperature and the total entropy generation 

versus distance between the heaters 

VII. CONCLUSIONS 
This study investigates numerically the entropy generated 

by natural convection in a square electronic package with two 
pieces of semiconductor. The flow field, temperature, and 
entropy maps were obtained by numerical simulation of the 
governing equation using finite volume method and SIMPLER 
algorithm in MATLAB. Results show that dominant heat 
transfer mechanism is conduction in low Rayleigh numbers, 
while heat transfer mechanism changes to convection by 
increasing Rayleigh number and effect of conduction 
disappears. Conduction heat transfer and fluid movement 
increased with Rayleigh number; as a result, fluid friction 
irreversibility and heat transfer irreversibility increased. In this 

study, entropy generation mainly resulted from heat transfer 
irreversibility, which was partially associated with fluid 
friction. For all scenarios investigated, the arrangement 
producing minimum peak temperature also produced minimum 
irreversibility. Among the examined scenarios, minimum rate 
of entropy generation was seen in a scenario in which heaters 
had equal length and strength ratios, and the heaters were 
mounted close to cold sidewalls.  Future works can: 

1. Study this subject in three dimensions and compare the 
results with current results 

2. Add a fan to the system and re-optimize the problem in 
order to achieve optimal location of the fan and minimize 
power consumption 

3. Apply a gradient to the enclosure and re-optimize the 
problem in order to achieve optimal gradient (the gradient 
for which entropy generation is minimized in the 
enclosure) 

4. Create an obstacle (baffles) in the enclosure and evaluate 
the role of sizes and location of obstacles in flow, 
temperature and entropy fields 

5. Evaluate different thermal boundary conditions for walls 
in order to achieve uniform temperature distribution and 
to minimize entropy generation in the enclosure 

6. Evaluate the role of round corners in the enclosure and 
compare the results with current results 

 



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