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Research Papers: Thermal Systems

# Optimization of the Heat Transfer Rate of Energy Systems of Conductive Bodies Confined to the Center of a Cavity

[+] Author and Article Information
Fatma Habbachi, Afif Elcafsi

Faculté des Sciences de Tunis,
Université Tunis El Manar,
LETTM,
Campus Universitaire El-Manar,
El Manar 2092, Tunisia

Fakhreddine S. Oueslati

Ecole Nationale d'Ingénieurs de Carthage,
Université de Carthage,
45 rue des entrepreneurs Charguia 2,
Tunis-Carthage 2035, Tunisia;
Tianjin Key Lab of Refrigeration Technology,
Tianjin University of Commerce,
Tianjin 300134, China
e-mail: fakhreddine.oueslati@fst.rnu.tn

Rachid Bennacer

LMT/ENS-Paris-Saclay/CNRS/Université
Paris Saclay,
61 Avenue du Président Wilson,
Cachan 94235, France;
Tianjin Key Lab of Refrigeration Technology,
Tianjin University of Commerce,
Tianjin 300134, China
e-mail: rachid.bennacer@ens-cachan.fr

1Corresponding authors.

Contributed by the Heat Transfer Division of ASME for publication in the JOURNAL OF HEAT TRANSFER. Manuscript received May 8, 2017; final manuscript received November 13, 2017; published online June 29, 2018. Assoc. Editor: Zhixiong Guo.

J. Heat Transfer 140(8), 082802 (Jun 29, 2018) (12 pages) Paper No: HT-17-1256; doi: 10.1115/1.4038828 History: Received May 08, 2017; Revised November 13, 2017

## Abstract

This paper is a numerical study conducted to investigate the conjugate flow and heat transfer occurring in three-dimensional (3D) natural convection. A cubical enclosure partially filled with porous block (central cubic) and considered in local thermal equilibrium with the fluid. The physical case considered concerns the existence of a horizontal temperature difference across the enclosure, between the left and the right wall, with the other external surfaces being adiabatic. Under these conditions, flow inside the enclosure is generated by the density (temperature) difference across the enclosure and the interaction between the solid porous blocks and the fluid. The Nusselt number on the hot and cold walls is presented to illustrate the overall characteristics of heat transfer consequence of the constrained flow inside the enclosure. The study focuses on the fluid flow and heat transfer evolution versus the dimensionless thickness of the inserted porous layer (0% ≤ η ≤ 100%) and the relative thermal conductivity of the solid matrix to that of the fluid $(10−3≤λ̃≤103)$. The obtained complex flow structure and the corresponding heat transfer (velocity, temperature profiles) are discussed in a steady-state situation. The numerical results are illustrated in terms of isotherms, velocity, streamlines fields, and averaged Nusselt number. Thus, the results of this work can help developing new tools and to optimize the overall heat transfer rate, which is important in many electronic energy components and other energy recovering systems.

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## References

Beghein, C. , Haghighat, F. , and Allard, F. , 1992, “ Numerical Study of Double-Diffusive Natural Convection in a Square Cavity,” Int. J. Heat Mass Transfer, 35(4), pp. 833–846.
Ostrach, S. , 1980, “ Natural Convection With Combined Driving Forces,” PhysicoChem. Hydrodyn., 1(4), pp. 233–247.
Viskanta, R. , Bergman, T. L. , and Incropera, F. P. , 1985, “ Double Diffusive Natural Convection,” Natural Convection: Fundamentals and Applications, S. Kakac , W. Aung , and R. Viskanta , eds., Hemisphere, Washington, DC, pp. 1075–1099.
Nield, D. A. , and Bejan, A. , 2013, Convection in Porous Media, 4th ed., Springer-Verlag, New York.
Bennacer, R. , 1993, “Convection naturelle thermosolutale: Simulation numérique des transferts et des structures d'écoulement,” Doctoral thesis, Université Pierre et Marie Curie, Paris, France.
Bejan, A. , and Anderson, R. , 1981, “ Heat Transfer Across a Vertical Impermeable Partition Imbedded in Porous Medium,” Int. J. Heat Mass Transfer, 24(7), pp. 1237–1245.
Animasaun, I. L. , 2016, “ Double Diffusive Unsteady Convective Micropolar Flow Past a Vertical Porous Plate Moving Through Binary Mixture Using Modified Boussinesq Approximation,” Ain Shams Eng. J., 7(2), pp. 755–765.
Oueslati, F. , Bennacer, R. , Sammouda, H. , and Belghith, A. , 2008, “ Thermosolutal Convection During Melting in a Porous Medium Saturated With Aqueous Solution,” Numer. Heat Transfer, Part A, 54(3), pp. 315–330.
Bennacer, R. , Tobbal, A. , Beji, H. , and Vasseur, P. , 2001, “ Double Diffusive Convection in a Vertical Enclosure Filled With Anisotropic Porous Media,” Int. J. Therm. Sci., 40(1), pp. 30–41.
Gobin, D. , Goyeau, B. , and Songbe, J. , 1998, “ Double Diffusive Natural Convection in Composite Fluid-Porous Layer,” ASME J. Heat Transfer, 120(1), pp. 234–242.
House, J. , Beckemann, C. , and Theodore, S. , 1990, “ Effect of a Centered Conducting Body on Natural Convection Heat Transfer in an Enclosure,” Numer. Heat Transfer Part A, 18, pp. 213–225.
Hadidi, N. , Ould-Amer, Y. , and Bennacer, R. , 2013, “ Bi-Layered and Inclined Porous Collector: Optimum Heat and Mass Transfer,” Energy, 51, pp. 422–430.
Ha, M. Y. , and Jung, M. J. , 2000, “ A Numerical Study on Three-Dimensional Conjugate Heat Transfer of Natural Convection and Conduction in a Differentially Heated Cubic Enclosure With a Heat-Generating Cubic Conducting Body,” Int. J. Heat Mass Transfer, 43(23), pp. 4229–4248.
Tong, W. , and Sharatchandra, M. C. , 1990, “ Heat Transfer Enhancement Using Porous Inserts,” Heat Transfer Flow Porous Media HTD, Vol. 156, American Society of Mechanical Engineers, New York, pp. 41–46.
Baytas, A. C. , Ingham, D. B. , and Pop, I. , 2009, “ Double Diffusive Natural Convection in an Enclosure Filled With a Step Type Porous Layer: Non-Darcy Flow,” Int. J. Therm. Sci., 48(4), pp. 665–673.
Kramer, J. , Ravnik, J. , Jecl, R. , and Škerget, L. , 2011, “ Simulation of 3D Flow in Porous Media by Boundary Element Method,” Eng. Anal. Boundary Elem., 35(12), pp. 1256–1264.
Varol, Y. , 2011, “ Natural Convection in Porous Triangular Enclosure With a Centered Conducting Body,” Int. Commun. Heat Mass Transfer, 38(3), pp. 368–376.
Jun, Y. , Yuan, C. W. , Xiao, J. Z. , and Yu, P. , 2015, “ Effect of Rayleigh Numbers on Natural Convection and Heat Transfer With Thermal Radiation in a Cavity Partially Filled With Porous Medium,” Procedia Eng., 121, pp. 1171–1178.
Beckerman, C. , Viskanta, S. , and Ramadhyani, S. , 2007, “ A Numerical Study of Non-Darcian Natural Convection in a Vertical Enclosure Filled With a Porous Medium,” Numer. Heat Transfer, 10(6), pp. 557–570.
Hadidi, N. , and Bennacer, R. , 2015, “ Two-Dimensional Thermosolutal Natural Convective Heat and Mass Transfer in a Bi-Layered and Inclined Porous Enclosure,” Energy, 93(Pt. 2), pp. 2582–2592.
Jaballah, S. , Bennacer, R. , and Sammouda, H. , 2006, “ Simulation of Mixed Convection in a Channel Partially Filled With Porous Media,” Prog. Comput. Fluid Dyn., 6(6), pp. 335–341.
Hadidi, N. , and Bennacer, R. , 2016, “ Three-Dimensional Double Diffusive Natural Convection Across a Cubical Enclosure Partially Filled by Vertical Porous Layer,” Int. J. Therm. Sci., 101, pp. 143–157.
Lima, T. , and Ganzarolli, M. M. , 2016, “ A Heat Line Approach on the Analysis of the Heat Transfer Enhancement in a Square Enclosure With an Internal Conducting Solid Body,” Int. J. Therm. Sci., 105, pp. 45–56.
Souayeh, B. , Ben-Cheikh, N. , and Ben-Beya, B. , 2017, “ Effect of Thermal Conductivity Ratio on Flow Features and Convective Heat Transfer,” Part. Sci. Technol., 35(5), pp. 565–574.
Makinde, O. D. , and Animasaun, I. L. , 2016, “ Bioconvection in MHD Nanofluid Flow With Nonlinear Thermal Radiation and Quartic Autocatalysis Chemical Reaction past an Upper Surface of a Paraboloid of Revolution,” Int. J. Therm. Sci., 109, pp. 159–171.
Makinde, O. D. , and Animasaun, I. L. , 2016, “ Thermophoresis and Brownian Motion Effects on MHD Bioconvection of Nanofluid With Nonlinear Thermal Radiation and Quartic Chemical Reaction Past an Upper Horizontal Surface of a Paraboloid of Revolution,” J. Mol. Liq., 221, pp. 733–743.
Boussinesq, J. , 1903, Théorie Analytique De La Chaleur, Gauthier-Villars, Paris, France.
Shang, D.-Y. , and Wang, B.-X. , 1990, “ Effect of Variable Thermophysical Properties on Laminar Free Convection of Gas,” Int. J. Heat Mass Transfer, 33(7), pp. 387–1395.
Patankar, S. V. , 1980, Numerical Heat Transfer and Fluid Flow, Hemisphere, New York.
Hayase, T. , Humphrey, J. A. C. , and Greif, R. , 1992, “ A Consistently Formulated QUICK Scheme for Fast and Stable Convergence Using Finite Volume Iterative Calculation Procedures,” J. Comput. Phys., 98(1), pp. 108–118.
Fusegi, T. , Hyun, J. M. , Kuwahara, K. , and Farouk, B. , 1991, “ A Numerical Study of Three-Dimensional Natural Convection in a Differentially Heated Cubical Enclosure,” Int. J. Heat Mass Transfer, 34(6), pp. 1543–1557.
Bocu, Z. , and Altac, Z. , 2011, “ Laminar Natural Convection Heat Transfer and Air Flow in Three-Dimensional Rectangular Enclosures With Pin Arrays Attached to Hot Wall,” Appl. Therm. Eng., 31(16), pp. 3189–3195.

## Figures

Fig. 1

Physical domain considered and coordinates system (a), streamtrace and thermal fields (b)

Fig. 2

Isotherms, velocity field, and streamlines on vertical (x–z) plane for different y positions, Ra = 105, Da = 10−6, η = 20%, and λ̃=1

Fig. 3

Isotherms, velocity field, and streamlines on horizontal (y–z) plane for different x positions, Ra = 105, Da = 10−6, η = 20%, and λ̃=1

Fig. 4

Isotherms, velocity vector field, and streamlines on vertical (x–y) plane for different z positions, Ra = 105, Da = 10−6, η = 20%, and λ̃=1

Fig. 5

Isotherms, velocity field, and streamlines on vertical (x–z) midplane for different η, Ra = 105, Da = 10−6, and λ̃=1

Fig. 6

Temperature profiles in the vertical (x–z) (y = 0.5) midplane for different thicknesses of the porous media block (η), Ra = 105, Da = 10−6, and λ̃=1

Fig. 7

Velocity component profiles, vertical (a) and horizontal (b) on the vertical midplane, (y = 0.5) for different (η), Ra = 105, Da = 10−6, and λ̃=1

Fig. 8

Evolution of the Nusselt number versus the thickness of the porous block (η), for Ra = 105, Da = 10−6, and λ̃=1

Fig. 9

Isotherms, velocity field, and streamlines in the vertical (x–z) midplane for different thermal (λ̃), Ra = 105, Da = 10−6, and η = 80%

Fig. 10

Isotherms, velocity field, and streamlines in the vertical (x–z) midplane for different y positions for Ra = 105, Da = 10−6, η = 80%, and λ̃=10

Fig. 11

Temperature profiles on the vertical (x–z) midplane (y = 0.5) for Ra = 105, Da = 10−6, η = 80% and different thermal conductivity ratios (λ̃)

Fig. 12

Velocity component profiles, vertical (a) and horizontal (b) on the vertical (x–z) midplane (y = 0.5) for Ra = 105, Da = 10−6, and η = 80%

Fig. 13

Evolution of the Nusselt number (Nu) versus thermal conductivity ratio (λ̃) for different thicknesses of the porous block (η), Ra = 105 and Da = 10−6

Fig. 14

Evolution of the Nusselt (Nu) number versus the porous block thickness (η), for and different Darcy numbers (Da), (λ̃=0.1 and 10), and Ra = 106

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