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Conjugate Heat Transfer in an Annulus with Porous Medium Fixed Between Solids

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Abstract

Conjugate heat transfer is an important area of research which has been in demand due to its applications related to various scientific and engineering fields. The current research is focused to study the heat transfer in a porous medium sandwiched between two solid walls of an annular vertical cylinder. The prime focus of the current study was to evaluate the effect of solid wall thickness, the influence of variable wall thickness at inner and the outer radius, the conductivity ratio and the solid wall conductivity ratio on the heat transfer characteristics of the porous medium. The surface at inner and outer radii of the annulus is maintained isothermally at \(T_{h}\) and \(T_{\infty }\) such that \(T_{h}>T_{\infty }\). The governing partial differential equations for the conjugate heat transfer in porous medium and that of solid walls are converted into a set of algebraic equations with the help of finite element method and then solved simultaneously to predict the temperature variation in the solid wall as well as the porous region of the annular domain.

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Abbreviations

H :

Height of cylinder (m)

L :

\(r_{o} - r_{i}\)(m)

Ar :

\(\hbox {Aspect ratio} = H/L\)

\(c_{p}\) :

Specific heat \((\hbox {J/kg}\,{^\circ }\hbox {C})\)

DL :

Thickness ratio of inner solid \((r_{sp1}-r_{i})/(r_{o}-r_{i})\)

DR :

Thickness ratio of outer solid \((r_{o}-r_{sp2})/(r_{o}-r_{i})\)

g :

Gravitational acceleration \((\hbox {m/s}^{2})\)

k :

Thermal conductivity \((\hbox {W/m}\,^{\circ }\hbox {C})\)

K :

Permeability of the porous medium \((\hbox {m}^{2})\)

Kr :

\(\hbox {Thermal conductivity ratio} = k_{s1}/k_{p}\)

\(\textit{Kr}_{o}\) :

\(\hbox {Thermal conductivity ratio} = k_{s2}/k_{p}\)

Krs :

\(\hbox {Thermal conductivity ratio of two solids}\, k_{s1}/k_{s2}\)

\(\overline{Nu} \) :

Average Nusselt number

r, z :

Cylindrical coordinates (m)

\(\overline{r}, \overline{z}\) :

Non-dimensional coordinates

Ra :

Rayleigh number

Rr :

\(\hbox {Radius ratio} = ({r}_{o}-{r}_{i})/ {r}_{i}\)

\(T, \overline{T}\) :

Dimensional \((^{\circ }\hbox {C})\) and non-dimensional temperature, respectively

u, w :

Velocity in r and z directions, respectively (m/s)

\(\upalpha \) :

Thermal diffusivity \((\hbox {m}^{2}/\hbox {s})\)

\(\beta \) :

Coefficient of thermal expansion \((1/^{\circ }\hbox {C})\)

\(\rho \) :

Density \((\hbox {kg/m}^{3})\)

v :

Coefficient of kinematic viscosity

\(\phi \) :

Porosity

\(\varPsi \),:

Stream function

\(\bar{{\psi }}\) :

Non-dimensional stream function

h :

Hot

\(\infty \) :

Conditions at outer radius

i :

Inner

o :

Outer

p :

Porous

s :

Solid

sp :

Solid–Porous interface

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Acknowledgments

The authors would like to thank University of Malaya for providing fund under the UMRG Grant No. RP006A-13AET.

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Authors and Affiliations

  1. Department of Mechanical Engineering, University of Malaya, Kuala Lumpur, 50603, Malaysia

    Irfan Anjum Badruddin, N. Nik-Ghazali, Sarfaraz Kamangar & T. M. Yunus Khan

  2. Faculty of Engineering and Technology, Multimedia University, Bukit Beruang, Malacca, 75450, Malaysia

    Salman Ahmed N. J.

  3. Public Authority for Applied Education and Training, Industrial Training Institute, Kuwait, 13092, Kuwait

    Abdullah A. A. A. Al-Rashed

  4. Department of Civil Engineering, University of Malaya, Kuala Lumpur, 50603, Malaysia

    Mohammed Jameel

  5. Department of Mechanical Engineering, Faculty of Engineering, Islamic University, Madinah Munawwarra, Kingdom of Saudi Arabia

    H. M. T. Khaleed

  6. Department of Mechanical Engineering, BVB College of Engineering and Technology, Hubli, 580031, India

    T. M. Yunus Khan

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  1. Irfan Anjum Badruddin
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  2. Salman Ahmed N. J.
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Correspondence to Irfan Anjum Badruddin.

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Badruddin, I.A., Ahmed N. J., S., Al-Rashed, A.A.A.A. et al. Conjugate Heat Transfer in an Annulus with Porous Medium Fixed Between Solids. Transp Porous Med 109, 589–608 (2015). https://doi.org/10.1007/s11242-015-0537-2

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