An experimental study of natural convection in vertical annulus with helical fin

ABSTRACT

Experimental results of natural convection in an open-ended concentric and eccentric annulus with helical fin on internal tube are presented. The boundary condition investigated is constant temperature for the inner tube while the outer tube is insulated. The effect of pitches and fin diameter under several different constant temperatures applied on the inner tube at concentric and different eccentric conditions are investigated. Helical fins with 2 and 4 mm diameters with 15, 30, and 45 mm pitches have been used in this experimental study. Constant temperature for the inner tube is achieved by passing saturated steam at different pressure through it.

KEYWORDS:

• natural convection
• vertical eccentric annulus
• helical fins
• constant temperature and insulated boundary condition

Nomenclature

A	=	Input duct cross section $(m^2)$, the empirical constant
$C_p$	=	Specific heat capacity at constant pressure $\left(\frac{j}{Kg.K}\right)$
$D_h$	=	Hydraulic diameter$D_h=\frac{4A}{Per}$$(m)$
$D_i$	=	Outside diameter of the inner tube $(mm)$
$D_o$	=	inside diameter of the outer tube$(mm)$
d	=	Fin diameter$(mm)$,
$E$	=	Dimensionless eccentricity $E=\frac{e}{R_o-R_i}$
e	=	Distance between the center of the tubes$(mm)$
$f^{\prime }$	=	Correction factor
$G{r}_D$	=	Mean Grashof number based on hydraulic diameter $G{r}_D=\frac{g\beta \mathrm{\Delta }T{D}_h^3}{{\upsilon }^2}$
$G{r}_D^{\ast }$	=	Modified Grashof number$G{r}_D^{\ast }=G{r}_D\frac{D_h}{L}$
g	=	Gravity constant $(m/s^2)$
$h_m$	=	Average heat transfer coefficient $(w/m^2.{\hspace{0.17em}}^{\circ }C)$
$h_x$	=	Local heat transfer coefficient $(w/m^2.{\hspace{0.17em}}^{\circ }C)$
L	=	Duct’s length $\left(m\right)$
$l_f$	=	Fin’s length $\left(m\right)$
K	=	Thermal conductivity of air $(w/m.{\hspace{0.17em}}^{\circ }C)$
$\dot{m}$	=	Mass flow rate ($\frac{Kg}{s})$
$N{u}_D$	=	Mean Nusselt number based on hydraulic diameter $N{u}_D=\frac{h_m{D}_h}{K}$
n	=	Empirical constant
N	=	The number of thermocouples
Per	=	Wetted perimeter of the cross-section $\left(m\right)$
Pr	=	Prandtl number $Pr=\frac{\upsilon }{\alpha }$
P	=	Pitch of the fin $(mm)$
$R_i$	=	Outer radius of the inner tube$(mm)$
$R_o$	=	Inner radius of the outer tube$(mm)$
$q^{\prime \prime }$	=	Heat flux of the inner tube $\left(\frac{w}{m^2}\right)$
$T_f$	=	mean film temperature $T_f=\frac{{\bar{T}}_s+T_a}{2}$$({\hspace{0.17em}}^{\circ }C)$
$\overline{T_o}$	=	Outlet average fluid temperature$({\hspace{0.17em}}^{\circ }C)$
${\bar{T}}_s$	=	Average wall temperature$({\hspace{0.17em}}^{\circ }C)$
$T_a$	=	Ambient temperature $({\hspace{0.17em}}^{\circ }C)$
$T_{si}$	=	Local wall temperature$({\hspace{0.17em}}^{\circ }C)$
$T_i$	=	Inner tube temperature$({\hspace{0.17em}}^{\circ }C)$
$T_s^{\ast }$	=	Dimensionless local Temperature $T_s^{\ast }=\frac{T_{si}-T_a}{T_a}$
$T$	=	Temperature profile at the channel outlet$({\hspace{0.17em}}^{\circ }C)$
$\bar{T}$	=	average temperature at the channel outlet$({\hspace{0.17em}}^{\circ }C)$
V	=	Average air velocity in the channel $\left(\frac{m}{s}\right)$
$y$	=	Local length (Measured from the bottom of the tube) $\left(m\right)$
$y/L$	=	Dimensionless Length

Greek Symbols

$\alpha$	=	thermal diffusivity $\alpha =\frac{K}{\rho .C_p}$$(m^2/s)$
$\beta$	=	volumetric coefficient of thermal expansion$(1/K)$
$\rho$	=	fluid density$(Kg/m^3)$
$\mu$	=	Dynamic fluid viscosity $(Kg/m.s)$
$\upsilon$	=	Kinematic fluid viscosity $\upsilon =\frac{\mu }{\rho }$$(m^2/s)$