A thin electrical heater dissipating $3000\frac{W}{m^2}$ is sandwiched between two $20-mm-$thick plates whose exposed surfaces experience

convection with a fluid for which $T_{}=20C$ and $h=400\frac{W}{m^2}*K.$ The thermophysical properties of the plate material are $=$

$2500k\frac{g}{m^3},c=700\frac{J}{k}g*K,$ and $k=5\frac{W}{m}*K.$

$($a$)$ Calculate values of the temperatures at the surfaces, $x=+-L,$ and the midpoint, $x=0.$

$($b$)$ Consider conditions for which there is a loss of coolant and existence of a nearly adiabatic condition on the $x=+-L$ surface.

Calculate the temperatures at $x=0,+-L.$

$($c$)$ With the system operating as described in part $($b$),$ the surface $x=-L$ also experiences a sudden loss of coolant. This dangerous

situation goes undetected for $15$ min $,$ at which time the power to the heater is deactivated. Assuming no heat losses from the surfaces

of the plates, what is the eventual $(t),$ uniform, steady$-$state temperature distribution in the plates? Hint: Apply the conservation

of energy requirement on a time$-$interval basis, $E_{st}=E_{in}-E_{\text{o}\text{u}\text{t}}+E_g,$ for the initial and final conditions corresponding to part $($b$)$

and part $($c$)$ respectively. $($a$)$:

Temperature at $x=+L,$ in $C$ :

$T_1(+L)=$

Temperature at $x=-L,$ in $C$ :

$T_1(-L)=$

Temperature at $x=0,$ in $C$ :

$T_1(0)=,C$

$($b$)$:

Temperature at $x=+L,$ in $C$ :

$T_2(+L)=$

Temperature at $x=-L,$ in $C$ :

$T_2(-L)=$

Temperature at $x=0,$ in $C$ :

$T_2(0)=,C$

$($c$)$:

Temperature at $x=+L,$ in $C$ :

$T_3(+L)=$

$C$

Temperature at $x=-L,$ in $C$ :

$T_3(-L)=C$

Temperature at $x=0,$ in $C$ :

$T_3(0)=i,$:$C$