Analyzing heat transfer variations at specific locations in a cave filled with porous media, emphasizing non-equilibrium conditions

ABSTRACT

This study focuses on the analysis of conjugate natural convection heat transfer in a cavity filled with a permeable medium. It specifically considers the influence of confined thermal non-equilibrium and the presence of circular solid walls with partially active arcs. The study also explores the bifurcation of heat transfer at the interface of the permeable medium and solid walls. A finite element approach is employed to solve the governing equations for heat transfer in porous spaces, including porous matrices and solid walls, represented as partial differential equations. These equations are transformed into nondimensional forms based on boundary conditions and solved using the finite element method (FEM). The study examines local and overall heat transfer, varying nondimensional parameters, and the influence of active wall positions on flow patterns. Interestingly, the results demonstrate that the position of the active walls has a significant impact on the isotherms and streamlines contours in the permeable region, as well as the isotherms in the concrete walls. Specifically, an angle of ${120}^{\circ }$ between the warm component and the radiator, both positioned vertically and facing each other, results in the maximum overall temperature transmission. Conversely, the lowest total heat transfer rate occurs when the angle between the hot element and cold radiator is 180°, indicating complete alignment. Moreover, the case (e) exhibits the lowest temperature transmission ratio for both the liquid and the concrete stages of the permeable medium. In the case of modifying of $\left(R_k\right)$ from 10 to 0.1, decrease in the temperature transmission by creating isolation regions that perform as thermal walls beside the dynamic fences. This research sheds light on the complex heat transfer behavior in a cavity with permeable media and highlights the crucial role of active wall positioning in shaping flow patterns and temperature distribution.

KEYWORDS:

• Local thermal non-equilibrium (LTNE)
• FEM
• conjugate heat transfer
• porous circular chamber
• natural convection

Nomenclature

Latin symbols	=
$N{u}_y$	=	local Nusselt number in the porous and in the free fluid
$q{ }^{\mathrm{\prime }\mathrm{\prime }}$	=	the surface heat flux $\left(\mathrm{W}{\mathrm{m}}^{-2}\right)$
$Q_{w,y}$	=	local non-dimensional heat transfer at the cavity wall
$a_h$	=	arc length of the thermally active element (hot) (m)
$a_c$	=	non-dimensional arc length of thermally active element(cold)
$Da$	=	Darcy number
$g$	=	gravity constant $Accessisdenied$
$H$	=	porous matrix-pore fluid convection parameter
$h_{fs}$	=	the coefficient of convective heat transfer between the solid matrix and the fluid in porous pores $AccessisdeniedAccessisdenied$
$k$	=	thermal conductivity$\hspace{0.17em}\left(\mathrm{W}{\mathrm{m}}^{-1}{\mathrm{K}}^{-1}\right)$
$K$	=	permeability of the porous medium ($m^2$)
$K_r$	=	porous-fluid thermal conductivity ratio parameter
$L$	=	the length of the cavity wall (m)
$M$	=	mesh size parameter
$Nu$	=	average Nusselt number
$p$	=	the fluid pressure (Pa)
$Pr$	=	Prandtl number
$P$	=	the dimensionless fluid pressure
$Q_w$	=	average of total non-dimensional heat transfer at the cavity wall
$Ra$	=	Rayleigh number
$R_k$	=	the ratio of wall thermal conductivity to fluid thermal conductivity
$u,v$	=	velocity components $\left(\mathrm{m}{\mathrm{s}}^{-1}\right)$
$U,V$	=	non-dimensional velocity
LTNE	=	Local thermal non-equilibrium
FEM	=	finite element method
$r$	=	Cylindrical ray (m)
$R$	=	non dimensional cylindrical ray
$T$	=	temperature variable (K)
Greek symbols	=
$\alpha$	=	effective thermal diffusivity of porous medium and fluid $\left({\mathrm{m}}^2\hspace{0.17em}{\mathrm{s}}^{-1}\right)$
$\beta$	=	volumetric thermal expansion coefficient of the fluid ($K^{-1}$)
$\epsilon$	=	porosity
${\theta }_w$	=	non-dimensional temperature
$\mu$	=	the fluid dynamic viscosity (kg $m^{-1}$ $s^{-1}$)
$v$	=	the fluid kinematic viscosity ($m{ }^2$ $s^{-1}$)
$\rho$	=	the fluid density $(\mathrm{k}\mathrm{g}{m}^{-3}$)
$\rho c$	=	the effective heat capacity of the porous medium and the fluid (J $K^{-1}$ $m^{-3}$)
$\phi$	=	angle of inclination of side walls represented in Table 1 (degree)
$\mathrm{\Psi }$	=	non-dimensional stream-function
Subscripts	=
$0$	=	ambient
$c$	=	cold wall
$eff$	=	effective property
$f$	=	fluid property
$h$	=	hot wall
$max$	=	maximum
$S$	=	solid
$\mathrm{t}$	=	total
$w$	=	wall

Acknowledgements

The financial support from DGRSDT projects (B00L02UN480120230001), ALGERIA

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Notes on contributors

Abdelkader Aissani

Abdelkader Aissani is a PhD Student in Science and Technology University of Oran, His studies focused in computational fluid dynamics and heat transfer.

Redouane Fares

Dr. Redouane Fares has completed his PhD from physics Institute of Science and Technology University of Oran, Algeria . He is a teacher in university of Relizane and member of LIGDD research laboratory. He has published several papers in reputed journals and has been serving as an editorial board member of journals and international and local conferences. Research Interest : Computational fluid dynamics, porous media transport and propagation, pollution, nanofluids, heat and mass transfer.

Rachid Hidki

Rachid Hidki is a Ph.D. student at the Physics Department, Faculty of Sciences Semlalia, Cadi Ayyad University, Marrakech, Morocco. He obtained his Master's degree in 2019 in Fluid Mechanics and Energetics at Cadi Ayyad University. His main research interests are computational fluid dynamics and heat transfer with heating components.

Mohammed Adnane

Mohammed Adnane is a teacher in university of sciences and technology of Oran and member of LMESM research laboratory. Research Interest: Thin solid films, nanomaterials, coating, Solar cells, CZTS.