Chapter 2, Problem 2.53P

A thin electrical heater dissipating $4000{\text{W/m}}^{\text{2}}$ is sandwiched between two 25-mm-thick plates whose exposed surfaces experience convection with a fluid for which $T_{\infty }=20°\text{C}$ and $h=400{\text{W/m}}^2\cdot \text{K}\text{.}$ The thermophysical properties of the plate material are $\rho =2500{\text{kg/m}}^3,c=700\text{J/kg}\cdot \text{K,}$ and $k=5\text{W/m}\cdot \text{K}\text{.}$

1. On $T-x$ coordinates, sketch the steady-state temperature distribution for $-L\le x\le L.$ Calculate values of the temperatures at the surfaces, $x=\pm L,$ and the midpoint, $x=0.$ Label this distribution as Case 1, and explain its salient features.
2. Consider conditions for which there is a loss of coolant and existence of a nearly adiabatic condition on the $x=+L$ surface. On the $T-x$ coordinates used for part (a), sketch the corresponding steady-state temperature distribution and indicate the temperatures at $x=0,\pm L.$ Label the distribution as Case 2, and explain its key features.
3. With the system operating as described in part (b), the surface $x=-L$ also experiences a sudden loss of coolant. 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 $\left(t\to \infty \right),$ uniform, steady-state temperature distribution in the plates? Show this distribution as Case 3 on your sketch, and explain its key features. Hint: Apply the conservation of energy requirement on a time-interval basis, Eq. 1.12b, for the initial and final conditions corresponding to Case 2 and Case 3, respectively.
4. (d) On $T-t$ coordinates, sketch the temperature history at the plate locations $x=0,\pm L$ during the transient period between the distributions for Cases 2 and 3. Where and when will the temperature in the system achieve a maximum value?