Numerical and Galerkin’s methods for thermal performance analysis of circular porous fins with various profiles when the surface temperature is higher/lower than the air temperature

ABSTRACT

The main objective of this research is to analyze the optimization and the thermal performance of circular porous fins with three different profiles: rectangular, convex, and triangular under dry surface conditions. In this research, when the surface temperature of fin is higher or lower than the air temperature was studied. Also, modeling is assumed one-dimensional and the temperature changes only in the direction of the radius of the fin. Moreover, the thermal conductivity and heat transfer coefficient are a function of porosity and temperature, respectively. The governing equations are solved using the finite difference method and the Galerkin’s method. In this study, the effect of different parameters including porous parameter (S_H) and porosity, fin efficiency, and fin effectiveness on temperature distribution were investigated. In both cases, the results showed that the fin temperature distribution should be decreased by increasing the porous parameter (S_H) and porosity. In addition, in both cases, the results demonstrated that the efficiency and effectiveness for the rectangular profile are higher than other profiles. In this study, the total average absolute deviation percentage was calculated and its magnitude was 1 to 2.35%.

KEYWORDS:

• Circular porous fin
• porous parameter (S_H)
• thermal performance
• Galerkin’s method
• optimization

Nomenclature

$c_p$	=	Section area of fin, m2
Cp	=	Specific heat, J kg−1K1
$Da$	=	Darcy number, -
$f$	=	Objective function, -
$g$	=	Gravity, ms−2
$h_a$	=	Constant in Eq. (8), Wm−2K1
$h$	=	Heat transfer convective coefficient, Wm−2K1
$k$	=	Thermal conductivity, Wm−1K1
$k_r$	=	Fluid thermal conductivity to solid thermal conductivity ratio, -
$K$	=	Permeability, m2
$m_1$	=	Dry fin constants defined in Eq. (13), m−1
$m_2$	=	Constant defined in Eq. (13), -
$\dot{m}$	=	Mass flow rate, kgs−1
$n$	=	Thickness profile index, -
$q$	=	Actual heat transfer rate of the porous fin, W
$Q$	=	Dimensionless actual heat transfer rate,-
$R$	=	Dimensionless radius,-
$Ra$	=	Rayleigh number,-
$r$	=	Radial direction,-
$S_{{ }^H}$	=	Porous parameter,-
$t_b$	=	Thickness at the fin base, m
$t\left(r\right)$	=	Thickness of the fin, m
$T$	=	Temperature, °C
$U$	=	Dimensionless volume, -
$V$	=	Volume, m3

Greek symbols

$\beta$	=	Coefficient of volumetric thermal expansion, K−1
$\epsilon$	=	Fin effectiveness, -
$\eta$	=	Fin efficiency,-
$\xi$	=	Effective thermal conductivity to solid thermal conductivity ratio, see Eq. (23), -
$\psi$	=	Base thickness to length ratio, -
$\theta$	=	Dimensionless temperature, -
$\upsilon$	=	Velocity of fluid passing through the fin,ms−1
$\gamma$	=	Kinematic viscosity, ms−2
$\rho$	=	Density, kgm−3
$\phi$	=	Porosity,-

Subscripts

$a$	=	Ambient condition
$b$	=	Base condition
$e$	=	End condition
$f$	=	Fluid properties
$eff$	=	Porous properties
$s$	=	Solid properties