# (181c) Analysis of Heat Flow Visualization During Conjugate Natural Convection In Square Cavity Via Heatline Method

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Natural convection in a square cavity surrounded by vertical conducting
walls of definite wall thickness ($t_i$) has been studied numerically
using heatline approach.  The phenomenon of conjugate natural convection
is used in many physical situations where heat is transported through
walls, such as building insulations, flat-plate solar collectors, heat
exchangers, etc. The cavity is heated from right wall, cooled from left
wall and the horizontal walls are maintained adiabatic.  Three different
cases are considered based on the location of wall thickness:  Finite wall
thickness on (a) cold side of the cavity ($t_{1}$) [case 1] (b) hot side
of the cavity ($t_{2}$) [case 2] and (c) both hot and cold sides of the
cavity ($t_{1}=t_{2}$) [case 3].  Finite element simulations are carried
out over a range of Rayleigh numbers ($Ra=10^{3}-10^{5}$) for various wall
thicknesses ($t=0.2$ and $t=0.8$ where $t = t_1 + t_2$)  and conductivity
ratios ($K=0.1$ and $10$).  Numerical results are presented in terms of
streamlines ($\psi$), isotherms ($\theta$) and heatlines ($\Pi$). At lower
$Ra$ ($Ra=10^{3}$), the strength of fluid and heat flow is weak and heat
transport is mainly due to conduction in both solid and fluid phases for
low $K$ ($K=0.1$) and $t$ ($t=0.2$) in all three cases. On the other hand,
the strength of fluid and heat flow increase with $K$ ($K=10$) for all
three cases even at $Ra=10^{3}$ and those are independent of $t$ at high
$K$ ($K=10$).  At higher $Ra$ ($Ra=10^{5}$), the strength of fluid and
heat flow increases due to convection. Case 1 shows maximum magnitude of
heatfunction at $K=0.1$, although streamfunction magnitudes are almost
same for all three cases at $Ra=10^{5}$ irrespective of $t$. Increase in
$K$ ($K=10$) shows almost identical streamfunction and heatfunction
magnitude at $t=0.2$ and $Ra=10^{5}$. Increase in wall thickness shows
almost pure conduction dominant heat transport in the fluid phase
irrespective of $Ra$ for all three cases at low $K$. Various qualitative
and quantitative features on variations of local and average Nusselt
numbers for all three cases are adequately explained based on heatlines.
Results show that average Nusselt number is maximum for low $t_i$ and high
$K$ and that is independent of location of wall thickness at $Ra=10^{5}$.
Also, low $Ra$ and high conducting walls show maximum average Nusselt
number than high $Ra$ and low conducting walls. Overall, it is shown that
heatlines give suitable guidelines on location of wall thickness in
conjugate natural convection based on conductivity ratio ($K$).