Sage Journals: Discover world-class research

Abstract

This study investigates the natural convection induced by a temperature difference between cold outer polygonal enclosure and hot inner circular cylinder. The governing equations are solved numerically using built-in finite element method of COMSOL. The governing parameters considered are the number of polygonal sides, aspect ratio, radiation parameter, and Rayleigh number. We found that the number of contra-rotative cells depended on polygonal shapes. The convection heat transfer becomes constant at $L / D > 0.77$ and the polygonal shapes are no longer sensitive to the Nusselt number profile.

Keywords

Finite element method natural convection polygonal enclosure hot cylinder

Introduction

Fluid flow and natural convection heat transfer from a heated body inside enclosures have long been studied and have received more attention due to its direct relevance to many engineering and technological applications such as flood protection for buried pipes, solidification processes, heat exchange, electronic packaging, and chemical reactors. Natural convection in enclosures containing a heated circular cylinder is studied rigorously in the literature. Different cylinder configurations are studied theoretically or experimentally in the literature.

Yang and Tao¹ investigated the natural convection heat transfer in a cylindrical envelope with an internal concentric slotted hollow cylinder. Yoo et al.² have found multicellular natural convection of a low Prandtl number fluid between horizontal concentric cylinders. Guj and Stella³ numerically considered the natural convection in horizontal eccentric annuli. SW Baek and CY Han⁴ investigated the effects on natural convection phenomena by radiation in concentric and eccentric horizontal cylindrical annuli. An efficient approach to simulate natural convection in arbitrarily eccentric annuli by vorticity-stream function formation has been proposed by Shu et al.⁵ Waheed⁶ discussed the problem of natural convection between two concentric horizontal cylindrical annuli. Abu-Nada et al.⁷ studied the heat transfer enhancement in horizontal concentric annuli using nanofluids. The nanofluids in a lid-driven cavity with a rotating circular cylinder were investigated by Chatterjee et al.⁸

Ghaddar⁹ studied a heated horizontal cylinder placed inside a rectangular enclosure. He found that the maximum air velocity was at a distance of about nine cylinder diameters along the vertical centerline above the heated cylinder. Moukalled and Acharya¹⁰ have analyzed the effect of the radius of the inner circular cylinder and the aspect ratio of the fluid flow and heat transfer rate. Cesini et al.¹¹ performed a numerical and experimental analysis of the cylinder and reported that in general, the numerical and experimental results were in good accordance. Shu et al.¹² used the differential quadrature method and showed that the global circulation, flow separation, and the top space between the square outer enclosure and the cylinder have an important influence on the flow and thermal fields. Shu and Zhu¹³ analyzed the effect of the radius of the inner circular cylinder and the aspect ratio of the fluid flow and heat transfer rate. Kim et al.¹⁴ and Lee et al.¹⁵ showed that the cylinder position could affect the heat transfer quantities. The changes in heat transfer quantities at Rayleigh number of 10⁷ have been presented by Yoon et al.¹⁶ and Yu et al.¹⁷ The different thermal boundary conditions are presented by Roychowdhury et al.¹⁸ Angeli et al.¹⁹ developed a correlation for the average Nusselt number as a function of Rayleigh number and the diameter. The heat transfer enhancement by increasing the cylinder size and/or surface emissivity has been achieved by Mezrhab et al.²⁰ Hussain and Hussein²¹ studied a uniform heat flux which enters through the cylinder instead of being kept at a constant high temperature. They obtained two cellular flow fields and the total average of Nusselt number behaves nonlinearly as a function of locations. The existence of local peaks of the Nusselt number has been investigated by Lee et al.¹⁵ Chatterjee and Halder²² numerically considered two rotating cylinders in a square enclosure filled with an electrically conducting fluid. Recently, Gupta et al.²³ investigated a ventilated cavity in the presence of the heat conducting cylinder.

Xu et al.²⁴ studied the cylinder inside a triangular enclosure. The cross-section geometry was found to have insignificant effects on the overall heat transfer. However, the above studies of heat transfer characteristics were developed for specific enclosures, such as cylindrical, rectangular, and triangular. Hence, this work aims to investigate various polygonal enclosures with inner hot cylinder. The enclosure can be triangular, rectangular, pentagonal, hexagonal, heptagonal, octagonal, nonagonal, or cylindrical. These arbitrarily geometries are expected to be of importance for some engineering applications, for example, thermal management of electronic devices. This work will present the fluid dynamics and thermal performance in the space between hot inner cylinder and cold outer polygonal. The radiative heat flux is taken into consideration.

Mathematical formulation

A schematic diagram of a cold regular polygonal enclosure having an inner hot circular cylinder is shown in Figure 1. The cylinder is located in the center of the enclosure with radius $r_{i}$ , while the radius of the outer enclosure is $r_{o}$ . The polygonal is filled with air. Under the influence of the vertical gravitational field, the inner cylinder and outer polygonal at different levels of temperature lead to a natural convection problem. The fluid is assumed Newtonian while viscous dissipation effects are negligible, but radiative heat flux is taken into consideration. The governing equations for fluid flow and heat transfer are described by the Navier–Stokes and energy equations, respectively. The governing equations in the steady form are transformed into dimensionless form under the following non-dimensional variables:

\begin{matrix} X = \frac{x}{L}, Y = \frac{y}{L}, U = \frac{uL}{α}, \\ V = \frac{vL}{α}, Θ = \frac{T - T_{c}}{T_{h} - T_{c}} \\ \Pr = \frac{ν}{α}, Ra = \frac{g β (T_{h} - T_{c}) L^{3}}{ν α}, \\ P = \frac{p L^{2}}{ρ α}, R_{d} = \frac{4 σ T_{c}^{3}}{β_{R} k}, \end{matrix}

(1)

where Pr is Prandtl number and Ra Rayleigh number. The dimensionless form of the governing equations is expressed as follows:

\frac{\partial U}{\partial X} + \frac{\partial V}{\partial Y} = 0

(2)

U \frac{\partial U}{\partial X} + V \frac{\partial U}{\partial Y} = - \frac{\partial P}{\partial X} + \Pr (\frac{\partial^{2} U}{\partial X^{2}} + \frac{\partial^{2} U}{\partial Y^{2}})

(3)

U \frac{\partial V}{\partial X} + V \frac{\partial V}{\partial Y} = - \frac{\partial P}{\partial X} + \Pr (\frac{\partial^{2} V}{\partial X^{2}} + \frac{\partial^{2} V}{\partial Y^{2}}) + Ra \Pr Θ

(4)

U \frac{\partial Θ}{\partial X} + V \frac{\partial Θ}{\partial Y} = (1 + \frac{4 R_{d}}{3}) \frac{\partial^{2} Θ}{\partial X^{2}} + \frac{\partial^{2} Θ}{\partial Y^{2}}

(5)

where $L = r'_{o} - r_{i}$ and $D = 2 r_{i}$ with $r'_{o}$ as the relative radius of the outer polygonal enclosures and $r_{i}$ as the radius of the inner cylinder. The number of polygon sides is denoted by N, where for $N = 3$ , the shape is triangular; for $N = 4$ , the shape is rectangular; for $N = 5$ , the shape is pentagonal; for $N = 6$ , the shape is hexagonal; for $N = 7$ , the shape is heptagonal; for $N = 8$ , the shape is octagonal; for $N = 9$ , the shape is nonagonal; and for $N \to \infty$ , the shape becomes circular. $U = V = 0$ on the walls of the enclosure and cylinder surface. Rosseland approximation invokes for the radiation, $q_{r} = (4 σ / 3 β_{R}) (\partial T^{4} / \partial x)$ , where $T^{4}$ can be expanded in Taylor series about $T_{c}$ and neglecting higher order terms, $T^{4} \approx 4 {TT}_{c}^{3} - 3 T_{c}^{4}$ . The boundary conditions for the non-dimensional temperatures are

\begin{matrix} On outer walls & Θ = 0 \\ On inner cylinder & Θ = 1 \end{matrix}

(6)

Figure 1.

(a) Schematic representation of the model for pentagonal enclosure and (b) mesh distribution.

Heat transfer rate is very important in thermal engineering applications. Therefore, the rate of heat transfer is computed at inner wall expressed in terms of the local surface Nusselt number $(Nu)$ as

Nu = (1 + \frac{4 R_{d}}{3}) \frac{\partial Θ}{\partial ξ}

(7)

where $ξ$ is the angular location. The surface-averaged Nusselt number $\bar{Nu}$ on the hot circular wall is evaluated as

\bar{Nu} = \frac{1}{2 π} \int_{0}^{360 \circ} Nu (ξ) d ξ

(8)

Computational methodology

The governing equations along with the boundary conditions are solved numerically by the computational fluid dynamics (CFD) software package COMSOL Multiphysics. COMSOL Multiphysics (formerly FEMLAB) is a finite element analysis, solver, and simulation software package for various physics and engineering applications. We consider the following application modes in COMSOL Multiphysics. The incompressible, laminar flow (spf) for equations (2)–(4) and the heat transfer in fluids (ht) for equation (5). P2-P1 Lagrange elements and the Galerkin least-squares method are used to assure stability. In this study, mesh generation on square enclosure having an inner circular cylinder is made using triangles. The triangular mesh distribution is shown in Figure 1(b), and it calibrates for fluid dynamic condition. Several grid sensitivity tests were conducted to determine the sufficiency of the mesh scheme and to ensure that the results are grid independent as shown in Table 1. We use the finer mesh sizes for all the computations done in this article by considering both accuracy and time computation. To validate the computation code, the previously published problems on natural convection in triangular, rectangular, and circular enclosures were solved (see Figure 2). The comparison was made between the resulting figures and the figures provided by Xu et al.,²⁴ Kim et al.,¹⁴ and Abu-Nada et al.⁷ These results provide confidence to measure the accuracy of the current numerical method. In addition, the $\bar{Nu}$ presents current results compared well with that obtained by the literature results for various combination of Rayleigh number and N as shown in Table 2.

Table 1.

Grid sensitivity check at $N = 5$ , $L / D = 0.63$ , $Ra = 10^{5}$ , and $R_{d} = 0.5$ .

Predefined mesh size	Mesh elements	$\bar{Nu}$	CPU time (s)
Extremely coarse	197	9.8889	3
Extra coarse	350	10.1150	2
Coarser	522	10.3072	2
Coarse	855	10.4863	2
Normal	1249	10.5555	2
Fine	1838	10.6079	3
Finer	6106	10.7231	5
Extra fine	17203	10.7694	11
Extremely fine	20019	10.7715	12

Figure 2.

Comparison of computed isotherms of the present work for triangular enclosure (left-top) with Xu et al.²⁴ (right-top) results, rectangular enclosure (left-middle) with Kim et al.¹⁴ (right-middle), and circular enclosure (left-bottom) with Abu-Nada et al.⁷ at $Ra = 10^{5}$ and $R_{d} = 0$ .

Table 2.

Comparison of the $\bar{Nu}$ between the present and the literature results for various combination of Rayleigh number and N.

Parameters	Present	Xu et al.²⁴	Moukalled and Acharya¹⁰ and Kim et al.¹⁴	Shu et al.⁵ and Abu-Nada et al.⁷
$N = 3, Ra = 10^{6}$	20.491	20.911	–	–
$N = 4, Ra = 10^{4}$	3.404	–	3.414, 3.331	–
$N = 4, Ra = 10^{5}$	5.105	–	5.1385, 5.08	–
$N \to \infty, Ra = 0.53 \times 10^{4}$	2.412	–	–	2.470, 2.422
$N \to \infty, Ra = 10^{4}$	3.901	–	–	3.94

Results and discussion

The analysis in the undergoing numerical investigation is performed in the following domain of the associated dimensionless groups: the $L / D$ ratio, $0.25 \leq L / D \leq 1.02$ ; the radiation parameter, $0 \leq R_{d} \leq 1$ ; and the Rayleigh number, $10^{2} \leq Ra \leq 10^{6}$ . Number of polygonal sides are from three up to $\infty$ . It is worth mentioning that the different polygonal shapes have fixed volume at each $L / D$ ratio. Circumscribed circles with dashed lines to all of the non-circular cases were integrated in the contour plots.

Figure 3 shows the effects of various values of radiation parameters and N at $L / D = 0.63$ and $Ra = 10^{5}$ . Figure 3(a) illustrates the effect of changing the flow motion with the boundary condition set for triangular enclosure, $N = 3$ . In general, changing the flow motion and the temperature of outer and inner boundaries lead the streamlines to generate two clockwise and two counterclockwise rotating cells located between the inner cylinder surface and the walls. The counterclockwise cells are located at the left region and the clockwise cells are located at the right region. As the radiation parameter increases (from left to right), the strength of the flow circulation reduces and the circulation cells above the cylinder become larger. The strength of the flow circulation increases with increasing value of N to 4 (square enclosure) as presented in Figure 3(b). The streamlines pattern inside pentagonal enclosure $(N = 5)$ is shown in Figure 3(c). Only single clockwise and single counterclockwise rotating cells are obtained for the considered radiation parameter. The strength of the flow circulation decreases by increasing the radiation intensity. Streamlines of cylindrical enclosure is depicted in Figure 3(d). We observed a single clockwise and a single counterclockwise rotating cell. The eye vortex shrinks by increasing the radiation intensity. This is due to the radiation that tends to render the non-uniform thermal development inside the enclosure.

Figure 3.

Streamlines for different radiation parameter, R_d = 0 (left), R_d = 0.5 (middle), R_d = 1 (right) and values of N at L/D = 0.63, Ra = 10⁵ and (a) N = 3, (b) N = 4, (c) N = 5, (d) $N \to \infty$ .

The effects of radiation parameter and N on isotherms for $L / D = 0.63$ and $Ra = 10^{5}$ are illustrated in Figure 4. Figure 4(a) presents the temperature distribution inside a triangular enclosure. The isotherm pattern is highly distorted by increasing the radiation intensity (from left to right). The average Nusselt number increases as the radiation parameters increases. This is due to the higher energy transmitted by adding the level of radiation exposure. Increasing N to 4 (square enclosure) leads to the reduction in the average Nusselt number. Double upwelling plume are observed at the top cylinder. This is due to the convective flow that is strong enough to distribute along the free space and causes the separation of boundary layer on the bottom wall. Figure 4(c) illustrates the average Nusselt number which decreases as the number of polygon side increases $(N = 5)$ . Increasing the radiation parameter enhances the heat transfer which resulted in an increase in average Nusselt number. It is observed that the pentagonal enclosure produced minimum value of average Nusselt number.

Figure 4.

Isotherms for different radiation parameter, R_d = 0 (left), R_d = 0.5 (middle), R_d = 1 (right) and values of N at L/D = 0.63, Ra = 10⁵ and (a) N = 3, (b) N = 4, (c) N = 5, (d) $N \to \infty$ .

Figure 5 presents surface-averaged Nusselt number $\bar{Nu}$ versus $L / D$ for various value of N at $Ra = 10^{5}$ and $R_{d} = 0$ . The Nusselt number decreases as the cylinder diameter and enclosure length ratio increase. Generally, lower N yields higher average Nusselt number for various ratios. The convection heat transfer becomes constant at $L / D > 0.77$ and the enclosure types are no longer sensitive to the Nusselt number profile. This is due to the higher ratio which creates greater suppression of inner heat circulation.

Figure 5.

Surface-averaged Nusselt number $\bar{Nu}$ versus $L / D$ for various values of N at $Ra = 10^{5}$ and $R_{d} = 0$ .

Figure 6 summarizes the variations in surface-averaged Nusselt number versus the aspect ratio for various value of Rayleigh number at (a) $N = 3$ , (b) $N = 5$ , and (c) $N \to \infty$ for $R_{d} = 0$ . Generally, the convection heat transfer is decreased by the rise of cylinder diameter D or reducing the enclosure size L at $N = 3$ , $N = 5$ , and $N \to \infty$ . A significant decrease in the average Nusselt number occurs at relative low Rayleigh number $(Ra = 10^{3}, 10^{4})$ . Here, at both $Ra = 10^{3}$ and $Ra = 10^{4}$ , the Nusselt number for different aspect ratios is nearly identical for polygon types. This happens due to the lower Rayleigh numbers, and the enhancement of convective flow intensity resulted from the augmented flow area is compensated for the attenuation due to additional viscous friction. Later, a higher Rayleigh number has stronger convection at various cylinder sizes and enclosure lengths. This condition occurs by vanishing the additional viscous friction when the convection becomes dominant.

Figure 6.

Surface-averaged Nusselt number $\bar{Nu}$ versus $L / D$ ratio for various values of Rayleigh number at (a) $N = 3$ , (b) $N = 5$ , and (c) $N \to \infty$ for $R_{d} = 0$ .

Figure 7 depicts the effect of various radiation on the average Nusselt number with $L / D$ at (a) $N = 3$ , (b) $N = 5$ , and (c) $N \to \infty$ for $Ra = 10^{5}$ . As discussed previously, by increasing the geometry ratio, the reduction in heat transfer rate will appear. The convection heat transfer decreases significantly at the highest radiation $(R_{d} = 1)$ for $N = 3$ , $N = 5$ , and $N \to \infty$ . Lower radiation intensity yielded lower surface-averaged Nusselt number at various cylinder sizes and enclosure lengths. A significant Nusselt number enhancement by increasing the radiation magnitude is obtained at a lower ratio for $N = 3$ , $N = 5$ , and $N \to \infty$ .

Figure 7.

Surface-averaged Nusselt number $\bar{Nu}$ versus $L / D$ ratio for various values of radiation parameter at (a) $N = 3$ , (b) $N = 5$ , and (c) $N \to \infty$ for $Ra = 10^{5}$ .

The effects of various Rayleigh number on surface-averaged Nusselt number $\bar{Nu}$ versus N at $L / D = 0.63$ and $R_{d} = 0$ are illustrated in Figure 8. Generally, the convection heat transfer remains fixed by adjusting the number of polygon sides at the Rayleigh number considered. A cautious investigation shows that triangular enclosure has slightly better heat transfer than other types of enclosure. However, the heat transfer rate of square enclosure $N = 4$ is better than other types of enclosure at the highest Rayleigh number. At relative low Rayleigh number, $Ra = 10^{3}$ and $Ra = 10^{4}$ , the Nusselt number for different cross-section geometry is nearly identical. This fact is due to the enlargement of flow area that is neutralized by the viscous friction when the number of polygon sides is calibrated. Later, higher Rayleigh number has better heat transfer for triangular, square, pentagonal, hexagonal, heptagonal, octagonal, nonagonal, and cylindrical enclosures.

Figure 8.

Surface-averaged Nusselt number $\bar{Nu}$ versus N for various values of Rayleigh number at $L / D = 0.63$ and $R_{d} = 0$ .

Figure 9 presents the $\bar{Nu}$ versus N for various value of radiation parameter at $L / D = 0.63$ and $Ra = 10^{5}$ . The values of $\bar{Nu}$ clearly show that the convection heat transfer decreases with increasing the number of polygonal sides. Stronger radiation intensity enhances the convection. This is due to the radiation that generates non-uniform temperature distribution across the enclosure, and the non-uniform temperature structure helps inner heat circulation that leads to the increase in heat transfer performance.

Figure 9.

Surface-averaged Nusselt number $\bar{Nu}$ versus N for various values of radiation parameter at $L / D = 0.63$ and $Ra = 10^{5}$ .

Conclusion

This numerical study investigated the natural convection induced by a temperature difference between the cold outer polygonal enclosure and hot inner cylinder. The dimensionless forms of the governing equations are modeled and solved using the built-in finite element method of COMSOL Multiphysics software. Detailed computational results for flow and temperature fields and the heat transfer are presented in graphical forms. The main conclusions of the present analysis are as follows:

The number of contra-rotative cells depends on polygonal shapes. The flow fields are symmetric for each polygonal shape.

The Nusselt number decreases by increasing the cylinder diameter and enclosure length ratio for each polygonal shape, convection, and radiation parameter. The convection heat transfer becomes constant at $L / D > 0.77$ and the polygonal shapes are no longer sensitive to the Nusselt number profile.

Generally, the Nusselt number decreases by adding the number of polygonal sides.

The results presented here provide an alternative approach to optimize the convective flows and heat transfer rate by a proper choice of the cross-section geometry.

Footnotes

Appendix 1

Academic Editor: Bo Yu

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported partially by the Malaysian Ministry of Education through the Fundamental Research grant scheme no. FRGS/1/2014/SG04/UKM/01/1.

References

Yang

Tao

WQ.

Numerical study of natural convection heat transfer in a cylindrical envelope with internal concentric slotted hollow cylinder. Numer Heat Tr A: Appl 1992; 22: 289–305.

Yoo

Choi

Kim

MU.

Multicellular natural convection of a low Prandtl number fluid between horizontal concentric cylinders. Numer Heat Tr A: Appl 1994; 25: 103–115.

Guj

Stella

Natural convection in horizontal eccentric annuli: numerical study. Numer Heat Tr A: Appl 1995; 27: 89–105.

Baek

Han

CY.

Natural convection phenomena affected by radiation in concentric and eccentric horizontal cylindrical annuli. Numer Heat Tr A: Appl 1999; 36: 473–488.

Shu

Yeo

Yao

An efficient approach to simulate natural convection in arbitrarily eccentric annuli by vorticity-stream function formation. Numer Heat Tr A: Appl 2000; 38: 739–756.

Waheed

MA.

An approach to the simulation of natural-convective heat transfer between two concentric horizontal cylindrical annuli. Numer Heat Tr A: Appl 2002; 53: 323–340.

Abu-Nada

Masoud

Hijazi

Natural convection heat transfer enhancement in horizontal concentric annuli using nanofluids. Int Commun Heat Mass 2008; 35: 657–665.

Chatterjee

Gupta

Mondal

Mixed convective transport in a lid-driven cavity containing a nanofluid and a rotating circular cylinder at the center. Int Commun Heat Mass 2014; 56: 71–78.

Ghaddar

NK.

Natural convection heat transfer between a uniformly heated cylindrical element and its rectangular enclosure. Int J Heat Mass Tran 1992; 35: 2327–2334.

10.

Moukalled

Acharya

Natural convection in the annulus between concentric horizontal circular and square cylinders. J Thermophys Heat Tr 1996; 10: 524–531.

11.

Cesini

Paroncini

Cortella

. Natural convection from a horizontal cylinder in a rectangular cavity. Int J Heat Mass Tran 1999; 42: 1801–1811.

12.

Shu

Xue

Zhu

YD.

Numerical study of natural convection in an eccentric annulus between a square outer cylinder and a circular inner cylinder using DQ method. Int J Heat Mass Tran 2001; 44: 3321–3333.

13.

Shu

Zhu

YD.

Efficient computation of natural convection in a concentric annulus between an outer square cylinder and an inner circular cylinder. Int J Numer Meth Fl 2002; 38: 429–445.

14.

Kim

Lee

. A numerical study of natural convection in a square enclosure with a circular cylinder at different vertical locations. Int J Heat Mass Tran 2008; 51: 1888–1906.

15.

Lee

Yoon

HS.

Natural convection in a square enclosure with a circular cylinder at different horizontal and diagonal locations. Int J Heat Mass Tran 2010; 53: 5905–5919.

16.

Yoon

Kim

. Effect of the position of a circular cylinder in a square enclosure on natural convection at Rayleigh number of 10⁷. Phys Fluids 2009; 21: 047101–047111.

17.

Yoon

MY.

Numerical study of natural convection in a square enclosure with an inner circular cylinder for Rayleigh number of 10⁷. Trans Korea Soc Mech Eng B 2010; 340: 739–747.

18.

Roychowdhury

Das

Sundararajan

Numerical simulation of natural convective heat transfer and fluid flow around a heated cylinder inside an enclosure. Heat Mass Transfer 2002; 38: 565–576.

19.

Angeli

Levoni

Barozzi

GS.

Numerical predictions for stable buoyant regimes within a square cavity containing a heated horizontal cylinder. Int J Heat Mass Tran 2008; 51: 553–565.

20.

Mezrhab

Moussaoui

Naji

Lattice Boltzmann simulation of surface radiation and natural convection in a square cavity with an inner cylinder. J Phys D Appl Phys 2008; 410: 115502.

21.

Hussain

Hussein

AK.

Numerical investigation of natural convection phenomena in a uniformly heated circular cylinder immersed in square enclosure filled with air at different vertical locations. Int Commun Heat Mass 2010; 37: 1115–1126.

22.

Chatterjee

Halder

MHD mixed convective transport in square enclosure with two rotating circular cylinders. Numer Heat Tr A: Appl 2014; 650: 802–824.

23.

Gupta

Chatterjee

Mondal

Investigation of mixed convection in a ventilated cavity in the presence of a heat conducting circular cylinder. Numer Heat Tr A: Appl 2015; 670: 52–74.

24.

. A numerical study of laminar natural convective heat transfer around a horizontal cylinder inside a concentric air-filled triangular enclosure. Int J Heat Mass Tran 2010; 53: 345–355.

Natural convection in polygonal enclosures with inner circular cylinder

Abstract

Keywords

Introduction

Mathematical formulation

Computational methodology

Results and discussion

Conclusion

Footnotes

Appendix 1

Declaration of conflicting interests

Funding

References