Natural convection and entropy production in a cubic cavity heated via pin-fins heat sinks

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 [1-6].

An experimental comparison study has been done by Kim et al. [7] to compare efficiencies of plate-fin and pin-fin heat sinks. Results indicate better performances of plate-fin.   Yalcın et al. [8] studied the three-dimensional 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 plate-fin or pin-fin heat sinks. They developed a correlation which they validated experimentally.

Research studies on 3D natural convection are very few. Three-dimensional natural convection in a plate-type 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 three-dimensional study on free convection heat transfer and fluid flow with pin-fin arrays. They showed heat transfer varies proportionally to Rayleigh number. Recently Kolsi et al [16-22] published some paper on natural convection in 3D cavities with inside different shapes active and non-active 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.

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:

${{S}_{tot}}=\int\limits_{v}{{{N}_{s}}}dv=\int\limits_{v}{\left( {{N}_{s-th}}+{{N}_{s-fr}} \right)}dv={{S}_{th}}+{{S}_{fr}}$          (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 central-difference 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 non-linear 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:

$\sum\limits_{i}^{1,2,3}{\frac{\max \left| \psi _{i}^{n}-\psi _{i}^{n-1} \right|}{\max \left| \psi _{i}^{n} \right|}}+\max \left| T_{i}^{n}-T_{i}^{n-1} \right|\le {{10}^{-4}}$         (11)

A numerical study on 3D natural convection and entropy production in a heated pin incorporated enclosure is presented. Results are illustrated via iso-surfaces 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 iso-surfaces 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, mountain-like distribution is observed due to presence of the heated pins. After that part it resembles to the differentially heated cavity. The iso-surfaces of temperature are characterized by a central horizontal stratification for low Ra and a central vertical stratification for high Ra. 

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Figure 2. Some particles trajectories for case 1 and b = 0.25 ; (a) Ra = 10³; (b) Ra = 10⁴; (c) Ra = 10⁵

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Figure 3. Iso-surfaces of temperature for case 1 and b = 0.25 ; (a) Ra = 10³; (b) Ra = 10⁴; (c) Ra = 10⁵

Iso-contours 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. U-shaped 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.

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Figure 4. Local Nusselt number at cold wall, for case 1 and b = 0.25; (a) Ra = 10³; (b) Ra = 10⁴; (c) Ra=10⁵

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 iso-surfaces 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.

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Figure 5. Mean Nusselt number for case 1

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Figure 6. Local entropy generation in z = 0.2 plan

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Figure 7. Projection of the velocity vector in z = 0.2 plan for Ra = 10⁵.

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 = 10⁵ 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.

8.png

Figure 8. Average Nusselt number as function of b for Ra=10⁵

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Figure 9. Total entropy generation as a function of b for Ra=10⁵