The temperature distribution across a plane wall $0\mathrm{.}3$ m thick at a certain instant of time is

T$($x$)=$ a $+$ bx $+$ cx$2,$ where T is in degrees C and x is in metres, a $=200$C$,$ b $=-200$C$/$m$,$

and c $=30$C$/$m$2.$ The wall has a thermal conductivity of $1$ W$/$m$$K$.$

$($a$)$ Without doing any analysis, deduce whether the heat transfer through the wall is

steady or transient. Discuss and explain your answer. State all assumptions.

$($b$)$ Determine the heat fluxes into and out of the wall and the rate of energy per unit area

stored by the wall.

$($c$)$ If the cold surface is exposed to a fluid at $100$C$,$ what is the convection coefficient?

What type of convection would you deduce from your answer and in what type of

fluid?

$($d$)$ Write down the appropriate heat conduction equation and deduce whether the

temperature at every point in the wall is $($i$)$ unchanged with time, $($ii$)$ decreasing with

time, or $($iii$)$ increasing with time.

$[$Ans$.($b$)200$ W$/$m$2,182$ W$/$m$2,18$ W$/$m$2($c$)4\mathrm{.}3$ W$/$m$2$K$]$