Chapter 2, Problem 2.56P

A large plate of thickness $2L$ is at a uniform temperature of $T_i=200°\text{C,}$ when it is suddenly quenched by dipping it in a liquid bath of temperature $T_{\infty }=20°\text{C}\text{.}$ Heat transfer to the liquid is characterized by the convection coefficient h.

1. If $x=0$ corresponds to the midplane of the wall, on $T-x$ coordinates, sketch the temperature distributions for the following conditions: initial condition $\left(t\le 0\right),$ steady-state condition $\left(t\to \infty \right),$ and two intermediate times.
2. On $q_x^{"}-t$ coordinates, sketch the variation with time of the heat flux at $x=L.$ If $h=100{\text{W/m}}^{\text{2}}\cdot \text{K,}$ what is the heat flux at $x=L$ and $t=0?$ If the wall has a thermal conductivity of $k=50\text{W/m}\cdot \text{K}$ what is the corresponding temperature gradient at $x=L?$ Consider a plate of thickness $2L=20\text{mm}$ with a density of $\rho =2770{\text{kg/m}}^3$ and a specific heat $c_p=875\text{J/kg}\cdot \text{K}\text{.}$ By performing an energy balance on the plate, determine the amount of energy per unit surface area of the plate $\left({\text{J/m}}^2\right)$ that is transferred to the bath over the time required to reach steady-state conditions.
3. From other considerations, it is known that, during the quenching process, the heat flux at $x=+L$ and $x=-L$ decays exponentially with time according to the relation, $q_x^{"}=A\exp \left(-Bt\right),$ where t is in seconds, $A=1.80\times {10}^4{\text{W/m}}^2,$ and $B=4.126\times {10}^{-3}{\text{s}}^{-1}.$ Use this information to determine the energy per unit surface area of the plate that is transferred to the fluid during the quenching process.