Halliday, Resnick & Walker — Temperature, Heat, and the First Law of Thermodynamics: Problem 62

Leidenfrost effect. A water drop will last about 1 s on a hot skillet with a temperature between $100^{\circ}\text{C}$ and about $200^{\circ}\text{C}$. However, if the skillet is much hotter, the drop can last several minutes, an effect named after an early investigator. The longer lifetime is due to the support of a thin layer of air and water vapor that separates the drop from the metal (by distance $L$ in Fig. 18.25). Let $L = 0.100\text{ mm}$, and assume that the drop is flat with height $h = 1.50\text{ mm}$ and bottom face area $A = 4.00 \times 10^{-6}\text{ m}^2$. Also assume that the skillet has a constant temperature $T_s = 300^{\circ}\text{C}$ and the drop has a temperature of $100^{\circ}\text{C}$. Water has density $\rho = 1000\text{ kg/m}^3$, and the supporting layer has thermal conductivity $k = 0.026\text{ W/(m}\cdot\text{K)}$. (a) At what rate is energy conducted from the skillet to the drop through the drop's bottom surface? (b) If conduction is the primary way energy moves from the skillet to the drop, how long will the drop last?