# Control of Flow and Heat Transfer in a Porous Enclosure due to an Adiabatic Thin fin on the Hot Wall

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## Abstract

Natural convection flow in a differentially heated square enclosure filled with porous matrix with a solid adiabatic thin fin attached at the hot left wall is studied numerically. The Brinkman–Forchheimer-extended Darcy model is used to solve the momentum equations, in the porous medium. The numerical investigation is done through streamlines, isotherms, and heat transfer rates. A parametric study is carried out using the following parameters: Darcy number (*Da*) from 10^{−4} to 10^{−2}, dimensionless thin fin lengths (*L* _{p}) 0.3, 0.5, and 0.7, dimensionless positions (*S* _{p}) 0.25, 0.5, and 0.75 with Prandtl numbers (*Pr*) 0.7 and 100 for *Ra* = 10^{6}. For *Da* = 10^{−3} and *Pr* = 0.7, it is observed that there is a counter clock-wise secondary flow formation around the tip of the fin for *S* _{p} = 0.5 for all lengths of *L* _{p}. Moreover when *Da* = 10^{−2} the secondary circulation behavior has been observed for *S* _{p} = 0.25 and 0.75 and there is another circulation between the top wall and the fin that is separated from the primary circulation. However, these secondary circulations features are not observed for *Pr* = 100. It is also found that the average Nusselt number decreases as the length of the fin increases for all locations. However, the rate of decrease of average Nusselt number becomes slower as the location of fin moves from the bottom wall to the top wall. The overall heat transfer rate can be controlled with a suitable selection of the fin location and length.

## Keywords

Natural convection Non-Darcy flow Thin fin Square cavity## List of Symbols

*g*Acceleration due to gravity, m s

^{−2}*Da*Darcy number

*F*Forchheimer number

*K*Permeability of the porous medium

*l*_{p}Length of the fin, m

*s*_{p}Position of the fin at the bottom wall, m

*L*Length of the square cavity, m

*L*_{p}Dimensionless length of the fin, =

*l*_{p}/*L**S*_{p}Dimensionless position of fin at the left wall, =

*s*_{p}/*L**n*The normal direction on a enclosure wall

*N*Total number of nodes

*Nu*Local Nusselt number

*Nu*_{s}Local Nusselt number at the side wall

- \({\overline {Nu}}\)
Average Nusselt number

- \({\overline {Nu_{\rm s}}}\)
Average Nusselt number at the side wall

*p*Pressure, Pa

*P*Dimensionless fluid pressure

*Pr*Prandtl number

*Ra*Rayleigh number

*T*Fluid temperature, K

*T*_{c}Temperature of cold right wall, K

*T*_{h}Temperature of hot left wall, K

*u**x*component of velocity*U**x*component of dimensionless velocity*v**y*component of velocity*V**y*component of dimensionless velocity*X*Dimensionless distance along

*x*coordinate*Y*Dimensionless distance along

*y*coordinate

## Greek Symbols

*α*Thermal diffusivity, m

^{2}s^{−1}*β*Volume expansion coefficient, K

^{−1}*γ*Penalty parameter

*θ*Dimensionless temperature

*ψ*Stream function

- \({\phi}\)
Porosity

## Subscripts

- l
Left wall

- r
Right wall

- s
Side wall

- p
Partition or fin

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