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Heat Transfer and Entropy Generation in a Porous Square Enclosure in Presence of an Adiabatic Block

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Abstract

The present work investigates the thermal aspects of a differentially heated porous square enclosure in the presence of an adiabatic block of different block sizes utilizing Darcy–Rayleigh number in the range of 1–10,000 with Darcy number \(10^{-2}\)–\(10^{-6}\). Heatlines and Nusselt number, streamlines, and entropy generation are used for the analysis of heat transfer, flow circulation, and irreversibility production in the enclosure. The study reveals that the presence of an adiabatic block affects the heat transfer process severely, and three different zones of heat transfer are identified. These are namely the zone of heat transfer augmentation, the zone of heat transfer augmentation along with entropy generation reduction, and the zone of both heat transfer and entropy generation reduction. It is also found that the presence of an adiabatic block can enhance heat transfer up to a certain critical block size; thereafter, further increasing in block size reduces the heat transfer rate. An optimal block size where the heat transfer enhancement is maximum is observed to be smaller than the critical block size. The study demonstrates the analyses of heat transfer and entropy generation for a better thermal design of a system. This study is also extended for higher Prandtl number fluids.

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Abbreviations

Be :

Bejan number

\(C_P \) :

Specific heat (J/kg K)

Da :

Darcy number

Ec :

Eckert number

EG:

Ethylene glycol

EO:

Unused engine oil

\(F_C\) :

Forchheimer coefficient

g :

Acceleration due to gravity (\(\hbox {m\,s}^{-2 }\))

k :

Thermal conductivity (\(\hbox {W\,m}^{-1}\,\hbox {K}^{-1 }\))

K :

Permeability (\(\hbox {m}^{2 }\))

L :

Length scale (m)

NS :

Dimensionless entropy generation

Nu :

Nusselt number

p :

Pressure, Pa

P :

Dimensionless pressure

\(q_\mathrm{in}\) :

Dimensionless heat input

\(q_\mathrm{out}\) :

Dimensionless heat going out

Pr :

Prandtl number

\(R_k\) :

Thermal diffusivity ratio

Ra :

Fluid Rayleigh number

\({Ra}_{{m}}\) :

Darcy–Rayleigh number

T :

Temperature (K)

\(T_C\) :

Cold wall temperature (K)

\(T_H\) :

Hot wall temperature (K)

\(T^{*}\) :

Dimensionless reference temperature

U, V :

Dimensionless velocities

x, y :

Cartesian coordinates (m)

u, v :

Velocity (\(\hbox {m\,s}^{-1}\))

X, Y :

Dimensionless Cartesian coordinates

\(\alpha \) :

Thermal diffusivity (\(\hbox {m}^{2}\,\hbox {s}^{-1}\))

\(\beta \) :

Thermal expansion coefficient (\(\hbox {K}^{-1}\))

\(|\psi |\) :

Dimensionless absolute streamfunction

\(\theta \) :

Dimensionless temperature

\(\phi \) :

Dimensionless block area

\(\Delta T\) :

\(T_H -T_C\)

\(\nu \) :

Kinematic viscosity (\(\hbox {m}^{2}\,\hbox {s}^{-1}\))

\(\rho \) :

Fluid density (\(\hbox {kg/m}^{3}\))

\(\varPi \) :

Dimensionless heatfunction

\(\eta \) :

Heat transfer parameter

\(\zeta \) :

Entropy generation parameter

\(\varepsilon \) :

Porosity

avg:

Average value

cond :

Conduction

c :

Critical

f :

Fluid

loc :

Local

max :

Maximum value

o :

Optimal

s :

Porous media

tot:

Total

References

  • Basak, T., Roy, S., Paul, T., Pop, I.: Natural convection in a square cavity filled with a porous medium: effects of various thermal boundary conditions. Int. J. Heat Mass Transf. 49, 1430–1441 (2006)

    Article  Google Scholar 

  • Basak, T., Roy, S.: Role of Bejan’s heatlines in heat flow visualization and optimal thermal mixing for differentially heated square enclosures. Int. J. Heat Mass Transf. 51, 3486–3503 (2008)

    Article  Google Scholar 

  • Basak, T., Kaluri, R.S., Balakrishnan, A.R.: Entropy generation during natural convection in a porous cavity:effect of thermal boundary conditions. Numer. Heat Transf. A 62, 336–364 (2012)

    Article  Google Scholar 

  • Baytaş, A.C.: Entropy generation for natural convection in an inclined porous cavity. Int. J. Heat Mass Transf. 43, 2089–2099 (2000)

    Article  Google Scholar 

  • Bejan, A.: Natural convection heat transfer in a porous layer with internal flow obstructions. Int. J. Heat Mass Transf. 26(6), 815–822 (1983)

    Article  Google Scholar 

  • Bejan, A.: Convection Heat Transfer, 3rd edn. Wiley, Canada (2004)

    Google Scholar 

  • Bhave, P., Narshimhan, A., Rees, D.A.S.: Natural convection heat transfer enhancement using adiabatic Block: optimal block size and prandtl number effect. Int. J. Heat Mass Transf. 49, 3807–3818 (2006)

    Article  Google Scholar 

  • Bürger, M., Buck, M., Pohlner, G., Rahman, S., Kulenovic, R., Fichot, F., Ma, W.M., Miettinen, J., Lindholm, I., Atkhen, K.: Coolability of particulate beds in severe accidents: status and remaining uncertainties. Prog. Nucl. Energy 52, 61–75 (2010)

    Article  Google Scholar 

  • Celik, I.B., Ghia, U., Roache, P.J., Freitas, C.J., Coleman, H., Raad, P.E.: Procedure for estimation and reporting of uncertainty due to discretization in CFD applications. J. Fluids Eng. Trans. ASME 130, 078001 (2008)

  • Chandra, P., Satyamurty, V.V.: Non-Darcian and anisotropic effects on free convection in a porous enclosure. Transp. Porous Media 90, 301–320 (2011)

    Article  Google Scholar 

  • Chen, X.B., Yu, P., Sui, Y., Winoto, S.H., Low, H.T.: Natural convection in a cavity filled with porous layers on the top and bottom walls. Transp. Porous Media 78, 259–276 (2009)

    Article  Google Scholar 

  • House, J.M., Beckermann, C., Smith, T.F.: Effect of centered conducting body on natural convection heat transfer in an enclosure. Numer. Heat Transf. A 18, 213–225 (1990)

  • Ingham, D.B., Pop, I.: Transport phenomenon in porous media III. Elsevier Ltd., Amsterdam (2005)

  • Jena, S.K., Mahapatra, S.K., Sarkar, A.: Thermosolutal convection in a rectangular concentric annulus: a comprehensive study. Transp. Porous Media 98, 103–124 (2013)

    Article  Google Scholar 

  • Kaluri, R.S., Basak, T.: Entropy generation due to natural convection in discretely heated porous square cavities. Energy 36, 5065–5080 (2011)

    Article  Google Scholar 

  • Kimura, S., Bejan, A.: The heatline visualization of convective heat transfer. J. Heat Transf. Trans. ASME 105(4), 916–919 (1983)

    Article  Google Scholar 

  • Lam, P.A.K., Prakash, K.A.: A numerical study on natural convection and entropy generation in a porous enclosure with heat sources. Int. J. Heat Transf. 69, 390–407 (2014)

    Article  Google Scholar 

  • Lauriat, G., Prasad, V.: Non-Darcian effects on natural convection in a vertical porous enclosure. Int. J. Heat Mass Transf. 32(11), 2135–2148 (1989)

    Article  Google Scholar 

  • Mahapatra, P.S., De, S., Ghosh, K., Manna, N.K., Mukhopadhyay, A.: Heat transfer enhancement and entropy generation in a square enclosure in the presence of adiabatic and isothermal blocks. Numer. Heat Transf. A 64, 576–597 (2013)

    Article  Google Scholar 

  • Mahmud, S., Fraser, R.A.: Flow, thermal and entropy generation characteristics inside a porous channel with viscous dissipation. Int. J. Therm. Sci. 44, 21–32 (2005)

    Article  Google Scholar 

  • Mahmoodi, M., Sebdani, S.M.: Natural convection in a square cavity containing a nanofluid and an adiabatic square block at the center. Superlattices Microstruct. 52, 261–275 (2012)

    Article  Google Scholar 

  • Mchirgui, A., Hidouri, N., Magherbi, M., Brahim, A.B.: Entropy generation for natural convection in a Darcy-Brinkman porous cavity. World Acad. Sci. Eng. Technol. 76, 360–364 (2013)

    Google Scholar 

  • Merrikh, A.A., Mohamad, A.A.: Blockage effects in natural convection in differentially heated enclosures. Enhanc. Heat Transf. 8, 55–72 (2001)

    Article  Google Scholar 

  • Mukhopadhyay, A.: Analysis of entropy generation due to natural convection in square enclosures with multiple discrete heat sources. Int. Commun. Heat Mass Transf. 37, 867–872 (2010)

    Article  Google Scholar 

  • Nield, D.A., Bejan, A.: Convection in Porous Media, 3rd edn. Springer, New York (2006)

    Google Scholar 

  • Patankar, S.V.: Numerical Heat Transfer and Fluid Flow. Hemisphere, New York (1980)

    Google Scholar 

  • Poulikakos, D.: Buoyancy-driven convection in a horizontal fluid layer extending over a porous substrate. Phys. Fluids 29 (1986)

  • Prasad, V., Kulacki, F.A.: Natural convection in a vertical porous annulus. Int. J. Heat Mass Transf. 27, 207–219 (1984)

    Article  Google Scholar 

  • Roache, P.J.: Perspective: a method for uniform reporting of grid refinement studies. J. Fluids Eng. Trans. ASME 116, 405–413 (1994)

  • Roy, C.: Grid convergence error analysis for mixed-order numerical schemes. AIAA J. 41 (2003)

  • Sathiyamoorthy, M., Basak, T., Roy, S., Pop, I.: Steady natural convection flow in a square cavity filled with a porous medium for linearly heated side wall(s). Int. J. Heat Mass Transf. 50, 1892–1901 (2007)

    Article  Google Scholar 

  • Sathiyamoorthy, M., Narasimman, S.: Control of flow and heat transfer in a porous enclosure due to an adiabatic thin fin on the hot wall. Transp. Porous Media 89, 421–440 (2011)

    Article  Google Scholar 

  • Sheikhzadeh, G.A., Nikfar, M., Fattahi, A.: Numerical study of natural convection and entropy generation of Cu–water nanofluid around an obstacle in a cavity. J. Mech. Sci. Technol. 26, 3347–3356 (2012)

    Article  Google Scholar 

  • Singh, A.K., Roy, S., Basak, T.: Visualization of heat transport during natural convection in a tilted square cavity: effect of isothermal and nonisothermal heating. Numer. Heat Transf. A 61, 417–441 (2012)

    Article  Google Scholar 

  • Sivasankaran, S., Do, Y., Sankar, M.: Effect of discrete heating on natural convection in a rectangular porous enclosure. Transp. Porous Media 86, 261–281 (2011)

    Article  Google Scholar 

  • Tasnim, S.H., Mahmud, S., Dutta, A.: Energy streamlines analyses on natural convection within porous square enclosure with internal obstructions. ASME J. Therm. Sci. Eng. Appl. 5 (2013)

  • Tasnim, S.H., Shohel, M., Mamun, M.A.H.: Entropy generation in a porous channel with hydromagnetic effect. Exergy Int. J. 2, 300–308 (2002)

    Article  Google Scholar 

  • Umavathi, J.C.: Analysis of flow and heat transfer in a vertical rectangular duct using a Non-Darcy model. Transp. Porous Media 96, 527–545 (2013)

    Article  Google Scholar 

  • Vafai, K.: Handbook of Prous Media. Taylor and Francis Group, New York (2005)

    Google Scholar 

  • Varol, Y., Oztop, H.F.: Control of buoyancy-induced temperature and flow fields with an embedded adiabatic thin plate in porous triangular cavities. Appl. Therm. Eng. 2009, 558–566 (2009)

    Article  Google Scholar 

  • Varol, Y., Oztop, H.F., Pop, I.: Entropy analysis due to conjugate-buoyant flow in a right-angle trapezoidal enclosure filled with a porous medium bounded by a solid vertical wall. Int. J. Therm. Sci. 48, 1161–1175 (2009)

    Article  Google Scholar 

  • Varol, Y.: Natural convection in porous triangular enclosure with a centered conducting body. Int. Commun. Heat Mass Transf. 38, 368–376 (2011)

    Article  Google Scholar 

  • Zahmatkesh, I.: On the importance of thermal boundary conditions in heat transfer and entropy generation for natural convection inside a porous enclosure. Int. J. Therm. Sci. 47, 339–346 (2008)

    Article  Google Scholar 

Download references

Acknowledgments

Authors would like to thank anonymous reviewers for their critical comments and valuable suggestions, which have improved the quality of this manuscript.

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Author notes

  1. Pallab Sinha Mahapatra

    Present address: Department of Mechanical and Industrial Engineering, University of Illinois at Chicago, Chicago, IL, USA

Authors and Affiliations

  1. School of Nuclear Studies and Application, Jadavpur University, Kolkata, India

    Priyankan Datta

  2. Department of Applied Mechanics, Indian Institute of Technology Madras, Chennai, 600036, India

    Pallab Sinha Mahapatra

  3. Department of Mechanical Engineering, Jadavpur University, Kolkata, India

    Koushik Ghosh, Nirmal K. Manna & Swarnendu Sen

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  1. Priyankan Datta
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Correspondence to Pallab Sinha Mahapatra.

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Datta, P., Mahapatra, P.S., Ghosh, K. et al. Heat Transfer and Entropy Generation in a Porous Square Enclosure in Presence of an Adiabatic Block. Transp Porous Med 111, 305–329 (2016). https://doi.org/10.1007/s11242-015-0595-5

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