# (219f) Heatline Analysis of Natural Convection In Rhombic Enclosures with Isothermally Hot Side or Bottom Wall

Authors:
Indian Institute of Technology Madras

Heatline method is used to analyze natural convection via differentially
heated enclosure convection [Case 1: hot and cold side walls with
horizontal adiabatic walls] and Rayleigh-Benard convection [Case 2: hot
and cold horizontal walls with vertical adiabatic walls] for rhombic
enclosures with various inclination angles, $\varphi$. An accurate
prediction of the flow structure and heat distribution in such
configurations are of great important due to its significant engineering
applications such as ventilation of rooms, cooling of electronics devices
or air flow in buildings. Simulations are performed for the range of
Rayleigh number, $Ra = 10^{3}-10^{5}$ for various inclination angles
($\varphi=30^\circ$, $45^\circ$, $60^\circ$, $75^\circ$ and $90^\circ$)
using Galerkin finite element method.  Interesting features of heat flow
patterns are visualized by heatlines for various $\varphi$s in both cases.
At $Ra = 10^3$, heatlines and isotherms are less distorted and flow
circulation is very weak at $\varphi=30^\circ$ in both cases. Increase in
$\varphi$ ($\varphi=90^\circ$) shows more distorted heatlines with closed
loop heatline cells due to increase in flow strength compared to
$\varphi=30^\circ$ at $Ra = 10^3$ in case 1 whereas heatlines and
isotherms are found to be parallel and orthogonal to adiabatic walls,
respectively indicating pure conduction dominant heat transfer with
stagnant fluid condition for $\varphi=90^\circ$ in case 2 at $Ra = 10^3$.
At $Ra = 10^5$, strength of fluid and heat flow increases for all
$\varphi$s due to enhanced convection effect and $\varphi=90^\circ$ shows
maximum magnitude of streamfunction ($\psi_{max}$) and heatfunction
($\Pi_{max}$)  values in both cases. It is found that, both $\psi_{max}$
and $\Pi_{max}$ values are comparatively higher in case 2.  Both cases are
compared based on local ($Nu$) and average Nusselt numbers
($\overline{Nu}$) and those are adequately explained based on heatlines.
It is found that $\overline{Nu}$ is independent of $\varphi$ at $Ra = 10^3$ in both cases. Also, $\overline{Nu}$ increases with $Ra$ and shows
its maximum at $Ra = 10^5$ for all $\varphi$s in both cases. It is also
shown that, $\varphi=30^\circ$ shows low heat transfer rate in case 1
compared to case 2 whereas $\varphi=90^\circ$ shows high heat transfer
rate in case 1 compared to case 2. Heat transfer rate is almost similar
for $\varphi=45^\circ$ in both cases. Overall, average heat transfer rate
is maximum for case 1 at $\varphi \ge 45^\circ$, eventhough $\psi_{max}$
and $\Pi_{max}$ values are high in case 2.
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