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In this study, heat transfer, flow structure and produced entropy due to natural convection in a threedimensional cavity heated via heat sinks are investigated numerically. One wall of the cavity is heated via pinfins and opposite wall is maintend at lower temperature. Remaining bouandaries are considered as adiabatic. Finite volume method is used to solve governing equations. Three geometrical cases are tested according to number and location of the fins. Other governing parameters are Rayleigh number and fin length. Number and length of the fins were found to be the most effective parameters on both heat transfer and entropy generation.
Entropy Production, 3D Natural Convection, Heat Sinks, Flow Structure
Heated pins are used for different application areas in engineering such as passive cooling or heating of rooms, buildings, radiators, heat exchangers, boilers and some solar applications. In these systems, the predominant heat transfer mechanism is natural convection. Calculation of generated entropy is very important to make good design and energy efficient systems [16].
An experimental comparison study has been done by Kim et al. [7] to compare efficiencies of platefin and pinfin heat sinks. Results indicate better performances of platefin. Yalcın et al. [8] studied the threedimensional heat transfer by testing the clearance gap between fin tips and shroud. The heat transfer was found to increase proportionally to the clearance parameter. Varol et al. [9] worked on isothermal longitudinal heater located in triangular enclosure. They observed that taller cavity and central position of the fin give higher heat exchange. Appadurai and Velmurugan [10] used fins to improve the performances of a solar still. They conducted both theoretical and experimental work to compare conventional still and different types of finned stills. The heat transfer was increased by attaching fins. Fins were also used for heat exchangers as given by Ryu and Lee [11]. Joo and Kim [12] studied the heat transfer of a vertical plate equipped by platefin or pinfin heat sinks. They developed a correlation which they validated experimentally.
Research studies on 3D natural convection are very few. Threedimensional natural convection in a platetype fin attached surface was analyzed by Baskaya et al. [13] using a commercial code. They developed a correlation between geometrical parameters and Rayleigh number to estimate the rate of heat transfer. They found that the increase of fin height enhances the heat transfer. Da Silva and Gosselin [14] studied the 3D natural convection in cubic cavity equipped by a conductive fin. They showed that the geometrical parameters have an important effect on heat transfer and flow structure. Bocu and Altac [15] conducted a threedimensional study on free convection heat transfer and fluid flow with pinfin arrays. They showed heat transfer varies proportionally to Rayleigh number. Recently Kolsi et al [1622] published some paper on natural convection in 3D cavities with inside different shapes active and nonactive obstacles.
In this work the natural convection in a cubic cavity heated by circular fins is exanimated numerically to study heat transfer, flow structure and entropy generation with a focus on the fins number and length.
Figure 1 shows a threedimensional isometric configuration (on the left) and a section of this configuration (in the middle). Three different configurations are chosen and named as Cases
I, II and III as shown at the right side of Figure 1. As shown on the figure, heated pins are mounted on the left hot wall and the rightside wall is maintained at cold temperature. All other walls are considered as adiabatic and gravity acts in ydirection.
Figure 1. Physical Model; (a): 3D configuration; (b): sectional plan; (c): cases
The formulation $(\vec{\psi }\vec{\omega })$ is used for the numerical model. This Formalism is defined by the two following relations:
$\vec{\omega }'=\vec{\nabla }\times \vec{V}'$ and $\vec{V}'=\vec{\nabla }\times \vec{\psi }'$.
The setting for the above relations exists with more details in the work of Kolsi et al. [16]. The dimensionless governing equations are as follow:
$\vec{\omega }={{\nabla }^{2}}\vec{\psi }$ (1)
$\frac{\partial \vec{\omega }}{\partial t}+(\vec{V}.\nabla )\vec{\omega }(\vec{\omega }.\nabla )\vec{V}=\Delta \vec{\omega }+Ra.\Pr .\left[ \frac{\partial T}{\partial z};0;\frac{\partial T}{\partial x} \right]$ (2)
$\frac{\partial T}{\partial t}+\text{ }\vec{V}.\nabla T=\Delta T$ (3)
With $\Pr ={\nu }/{\alpha }\;$ and $Ra=\frac{g.\beta .\Delta T.{{l}^{3}}}{\nu .\alpha }$, the considered boundary conditions are:
temperature
T=1 at x=0 and in the fins, T=0 at
$\frac{\partial T}{\partial n}=0$ on all other walls.
velocity
${{V}_{x,y,z}}=0$ on all walls
The generated entropy is written in the following form:
$S{{'}_{gen}}=\frac{1}{T{{'}^{2}}}.\vec{q}.\vec{}T'+\frac{\mu }{T'}.\varphi '$ (4)
with $\vec{q}=k.gra\vec{d}T$
The dissipation function Φ’ is written in incompressible flow as:
$\begin{align} & \varphi '=2\left[ {{\left( \frac{\partial V{{'}_{x}}}{\partial x'} \right)}^{2}}+{{\left( \frac{\partial V{{'}_{y}}}{\partial y'} \right)}^{2}}+{{\left( \frac{\partial V{{'}_{z}}}{\partial z'} \right)}^{2}} \right] \\ & +{{\left( \frac{\partial V{{'}_{y}}}{\partial x'}+\frac{\partial V{{'}_{x}}}{\partial y'} \right)}^{2}}+{{\left( \frac{\partial V{{'}_{z}}}{\partial y'}+\frac{\partial V{{'}_{y}}}{\partial z'} \right)}^{2}}+{{\left( \frac{\partial V{{'}_{x}}}{\partial z'}+\frac{\partial V{{'}_{z}}}{\partial x'} \right)}^{2}} \\\end{align}$ (5)
The produced entropy is written as [23]:
$S_{gen}^{'}=\frac{k}{T_{0}^{'2}}\left[ {{\left( \frac{\partial T_{0}^{'}}{\partial x_{0}^{'}} \right)}^{2}}+{{\left( \frac{\partial {{T}^{'}}}{\partial {{y}^{'}}} \right)}^{2}}+{{\left( \frac{\partial {{T}^{'}}}{\partial {{Z}^{'}}} \right)}^{2}} \right]$
$+2\frac{\mu }{{{T}_{0}}}\left\{ \left[ {{\left( \frac{\partial V{{'}_{x}}}{\partial x'} \right)}^{2}}+{{\left( \frac{\partial V{{'}_{y}}}{\partial y'} \right)}^{2}}+{{\left( \frac{\partial V{{'}_{z}}}{\partial z'} \right)}^{2}} \right]+{{\left( \frac{\partial V{{'}_{y}}}{\partial x'}+\frac{\partial V{{'}_{x}}}{\partial y'} \right)}^{2}}+{{\left( \frac{\partial V{{'}_{z}}}{\partial y'}+\frac{\partial V{{'}_{y}}}{\partial z'} \right)}^{2}}+{{\left( \frac{\partial V{{'}_{x}}}{\partial z'}+\frac{\partial V{{'}_{z}}}{\partial x'} \right)}^{2}} \right\}$ (6)
Using the dimensionless parameters, the local dimensionless entropy generation can be written as:
${{N}_{s}}=S{{'}_{gen}}\frac{1}{k}{{\left( \frac{l{{T}_{0}}}{\Delta T} \right)}^{2}}$ (7)
from where
${{N}_{s}}=\left[ {{\left( \frac{\partial T}{\partial x} \right)}^{2}}+{{\left( \frac{\partial T}{\partial y} \right)}^{2}}+{{\left( \frac{\partial T}{\partial z} \right)}^{2}} \right]+\phi .\left\{ 2\left[ {{\left( \frac{\partial {{V}_{x}}}{\partial x} \right)}^{2}}+{{\left( \frac{\partial {{V}_{y}}}{\partial y} \right)}^{2}}+{{\left( \frac{\partial {{V}_{z}}}{\partial z} \right)}^{2}} \right]+\left[ {{\left( \frac{\partial {{V}_{y}}}{\partial x}+\frac{\partial {{V}_{x}}}{\partial y} \right)}^{2}}+{{\left( \frac{\partial {{V}_{z}}}{\partial y}+\frac{\partial {{V}_{y}}}{\partial z} \right)}^{2}}+{{\left( \frac{\partial {{V}_{x}}}{\partial z}+\frac{\partial {{V}_{z}}}{\partial x} \right)}^{2}} \right] \right\}$ (8)
$\phi =\frac{\mu {{\alpha }^{2}}{{T}_{m}}}{{{l}^{2}}k\Delta {{T}^{2}}}$ represents the irreversibility coefficient.
The total produced entropy is:
${{S}_{tot}}=\int\limits_{v}{{{N}_{s}}}dv=\int\limits_{v}{\left( {{N}_{sth}}+{{N}_{sfr}} \right)}dv={{S}_{th}}+{{S}_{fr}}$ (9)
With S_{th} and S_{fr} are respectively the thermal and viscous entropy generations.
The local and average Nusselt at the cold wall are given by:
$Nu={{\left. \frac{\partial T}{\partial x} \right}_{x=1}}$ and $N{{u}_{av}}=\int\limits_{0}^{1}{\int\limits_{0}^{1}{Nudydz}}$ (10)
Governing equations [(1)(3)] and (8) are discretized using the finite volume method. Convective terms are treated using a centraldifference scheme and the temporal derivatives are discretized using the fully implicit procedure. The blocked of region method is used to impose fixed temperature and zero velocity in the fins. The grids are uniform in all directions with additional nodes on boundaries. The resolution of the nonlinear algebraic equations is assured by the successive relaxation iteration scheme. After a grid dependency test a spatial mesh of (71$\times $71$\times $71) was retained and the time step is fixed at (10^{4}). The convergence test is based on the following criterion each step of time:
$\sum\limits_{i}^{1,2,3}{\frac{\max \left \psi _{i}^{n}\psi _{i}^{n1} \right}{\max \left \psi _{i}^{n} \right}}+\max \left T_{i}^{n}T_{i}^{n1} \right\le {{10}^{4}}$ (11)
Results were validated by comparing with studies of Wakashima and Saitho [24]) and (Fusegi et al. [25]) for air filled cubic cavity. As seen from table 1, obtained results are acceptable when compared with literature.
Table 1. Validation of results
Ra 
Authors 
${{\psi }_{z}}$ (center) 
${{\omega }_{z}}$ (center) 
${{V}_{x\max }}$ (y) 
${{V}_{y\max }}$ (x) 
$N{{u}_{av}}$ 
10^{4} 
Present work 
0.05528 
1.1063 
0.199 (0.826) 
0.221 (0.112) 
2.062 

Wakashima and saitho [24] 
0.05492 
1.1018 
0.198 (0.825) 
0.222 (0.117) 
2.062 

Fusegi et al. [25] 
 
 
0.201 (0.817) 
0.225 (0.117) 
2.1 
10^{5} 
Present work 
0.034 
0.262 
0.143 (0.847) 
0.245 (0.064) 
4.378 

Wakashima and saitoh [24] 
0.03403 
0.2573 
0.147 (0.85) 
0.246 (0.068) 
4.,366 

Fusegi et al. [25] 
 
 
0.147 (0.,855) 
0.247 (0.065) 
4.361 
10^{6} 
Present work 
0.01972 
0.1284 
0.0832 (0.847) 
0.254 (0.032) 
8.618 

Wakashima and saitho [24] 
0.01976 
0.1366 
0.0811 (0.86) 
0.2583(0.032) 
8.6097 

Fusegi et al. [25] 
 
 
0.0841 (0.856) 
0.259 (0.033) 
8.77 
A numerical study on 3D natural convection and entropy production in a heated pin incorporated enclosure is presented. Results are illustrated via isosurfaces of temperature, local and average Nusselt number, local and total entropy generation, and particle trajectories for three different cases varying length of the fins and Rayleigh number.
Fig. 2 presents the particle trajectories for case 1 and b = 0.25 for different Ra. The flow structure behaves like a differentially heated cavity due to great number of heater in case 1. Almost circular shaped flow trajectory is observed for the lowest value of Rayleigh number. Then, its dimension is increases with Ra and the flow becomes more complex with an intensification of the 3D character.
Fig. 3 illustrates the isosurfaces of temperature for case 1 and b = 0.25 to analyze effect of Ra on temperature distribution. As given in the figure, temperature distribution is almost parallel to the heated pinned wall for the lowest value of Rayleigh number and they twisted with the increasing of Ra. They are almost parallel to the ceiling and bottom horizontal walls at the middle of the cavity. Based on the mechanism of the natural convection, the heated air around the pin rises vertically. Thus, “S” shaped temperature distribution is observed. Near the heated wall, mountainlike distribution is observed due to presence of the heated pins. After that part it resembles to the differentially heated cavity. The isosurfaces of temperature are characterized by a central horizontal stratification for low Ra and a central vertical stratification for high Ra.
Figure 2. Some particles trajectories for case 1 and b = 0.25 ; (a) Ra = 10^{3}; (b) Ra = 10^{4}; (c) Ra = 10^{5}
Figure 3. Isosurfaces of temperature for case 1 and b = 0.25 ; (a) Ra = 10^{3}; (b) Ra = 10^{4}; (c) Ra = 10^{5}
Isocontours of Nu_{loc} on cold wall for case 1 and b = 0.25 are shown in Fig. 4, for different Rayleigh numbers. Due to domination of conductive heat transfer mode, Nu_{loc} is presented almost horizontal at the central region of the wall. Ushaped distribution is observed at the top of the wall due to the existence of an intensive vertical air movement. Values of Nu_{loc} decrease from top to bottom of the cold wall as an expected result.
Figure 4. Local Nusselt number at cold wall, for case 1 and b = 0.25; (a) Ra = 10^{3}; (b) Ra = 10^{4}; (c) Ra=10^{5}
Fig. 5 presents the variation of Nu_{av} with Ra for case 1. As seen from the figure, Nu_{av} is increased almost linearly with increasing of Ra. But values of Nu_{av} are decreased with decreasing of b due to decreasing of incoming energy into system.
Total, friction and thermal local entropy generation are presented in Fig. 6 at z = 0.2 plan. Produced entropy due to heat transfer becomes higher near the edges of fins but the contours are denser near the bottom of the cavity due to rising of the air flow from bottom to top. This effect is clear for higher Rayleigh numbers as seen from the figures. Also, entropy is generated near the top right wall due to the clustering of temperature near that part as noticed from isosurfaces of temperature. Entropy generation due to friction is presented in the second row of the Fig. 6 and it is noticed that it is concentrated near of the walls due to the manifestation of the viscous effect.
Figure 5. Mean Nusselt number for case 1
Figure 6. Local entropy generation in z = 0.2 plan
Figure 7. Projection of the velocity vector in z = 0.2 plan for Ra = 10^{5}.
Entropy generation contours on walls becomes thinner with the increasing of Ra due to decreasing of boundary layer. Also, total entropy generation is presented on the bottom row and it presents similar distribution with other figures. As seen from the figures, edges of the pins are very effective on entropy generation.
Variations of thermal, viscous and total entropy generations are presented in Fig. 7 at different Rayleigh number for case 1. Entropy generation increases almost linearly with Ra due to incoming energy into the system.
For a best understanding of the flow structures vector velocity projections in z = 0.2 plan for Ra = 10^{5} are presented in Fig. 7 for different lengths and numbers of fins. It is noticed that numbers and location of the vortex becomes same fit the same values of fin length. However, flow strength is a function of fin length.
Variations of Nu_{av} as a function of length of fin for different cases are given in Fig. 8. As an expected result, heat transfer is decreased with decreasing of fin number, thus, heat transfer becomes lowest for case 3 and results for case 1 and 2 are almost the same. As a similar manner, total entropy generation is decreased with number of fin and maximum total entropy generation is observed for case 1 as given in Fig. 9. Total entropy generation value is increased with fin length due to increasing of heat transfer surface.
Figure 8. Average Nusselt number as function of b for Ra=10^{5}
Figure 9. Total entropy generation as a function of b for Ra=10^{5}
Effects of heated fin number, fin length and Rayleigh number on heat transfer, fluid flow and entropy generation are studied for threedimensional domain by using a numerical technique. It is noticed that the main effective parameter on heat transfer, fluid flow and entropy generation are length and location of the fins. Edge of the fins plays the dominant role on entropy distribution due to flow friction at that part. Increasing of heat transfer and total entropy generation is almost linear with Rayleigh number. The minimum of produced entropy is observed for the highest number of fins, namely, case 1. Increasing of fin length enhances both heat transfer and entropy generation.
b 
Fin length 
g 
Acceleration due to gravity (m.s^{2}) 
k 
Thermal conductivity (W.m^{−}^{1}.K^{−}^{1}) 
l 
Enclosure Width 
lz 
Enclosure depth 
Nu 
Nusselt number 
N_{s} 
Local dimensionless entropy 
P 
Pressure (N.m^{2}) 
Pr 
Prandtl number 
$\overset{\scriptscriptstyle\rightharpoonup}{q}$ 
Heat flux vector 
Ra 
Rayleigh number 
S 
Generated entropy 
t 
Dimensionless time 
T 
Dimensionless Temperature 
V 
Dimensionless velocity vector 
Greek symbols


α 
Thermal diffusivity (m^{2}·s^{1}) 
β 
Thermal expansion coefficient (K^{1}) 
ΔT 
Temperature difference (K) 
μ 
Dynamic viscosity, (kgm^{1} s^{1}) 
ν 
Kinematic viscosity (m^{2}·s^{1}) 
ρ 
Density (kg·m^{3}) 
Φ’ 
Dissipation function 
φ 
Irreversibility coefficient 
$\psi $ 
vector potential 
$\omega $ 
vorticity 
Subscripts 

c 
cold 
f 
fluid 
fr 
friction 
gen 
generated 
h 
hot 
m 
average 
n 
normal 
th 
thermal 
tot 
total 
x, y, z 
Cartesian coordinates 
0 
Reference 
Superscript 

' 
dimensionnal variable 
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