Consider the steady$-$state heat conduction problem in a square plate with no internal energy

generation illustrated below. The plate is subjected to a convection boundary condition at x$=$L

with H$=($h$)/($k$)$ and the non$-$zero ambient temperature, T$\_(\backslash$infty $),$ given as constants. A zero

temperature is imposed at x$=0,$ a zero temperature boundary condition is imposed at y$=0,$

and an adiabatic boundary condition is imposed at y$=$L$.$

T

Draw an appropriate control volume and derive the boundary condition at x$=$L$.$

Find the solution for the temperature T$($x$,$y$).$ Show that the solution is dimensionally

consistent.

$3.$Find an expression for the heat flux at the right wall, that is find: q$^(\text{'}\text{'})($x$=$L$,$y$).$ Show that

the solution is dimensionally consistent.

Bonus: $[10$ points$]$ Should the problem be now

posed as shown in the sketch, outline only $($no need

to solve$)$ how you would solve this problem. Use T$|\_($t$)=$

appropriate sketches and formulate the solution, i$.$e$.$

GE's, BC$\text{'}$s etc. needed to solve the problem.