Chapter 5, Problem 5.37P

Thin film coatings characterized by high resistance to abrasion and fracture may be formed by using microscale composite particles in a plasma spraying process. A spherical particle typically consists of a ceramic core, such as tungsten carbide (WC), and a metallic shell, such as cobalt (Co). The ceramic provides the thin film coating with its desired hardness at elevated temperatures, while the metal serves to coalesce the particles on the coated surface and to inhibit crack formation. In the plasma spraying process, the particles are injected into a plasma gas jet that heats them to a temperature above the melting point of the metallic casing and melts the casing before the panicles impact the surface. Consider spherical particles comprised of a WC core of diameter $D_i=16\mu \text{m,}$ which is encased in a Co shell of outer diameter $D_o=20\mu \text{m}\text{.}$ If the particles flow in a plasma gas at $T_{\infty }=10,000\text{K}$ and the coefficient associated with convection from the gas to the particles is $h=20,000{\text{W/m}}^{\text{2}}\cdot \text{K,}$ how long does it take to heat the particles from an initial temperature of $T_i=300\text{K}$ to the melting point of cobalt, $T_{\text{mp}}=1770\text{K?}$ The density and specific heat of WC (the core of the particle) are ${\rho }_c=16,000{\text{kg/m}}^{\text{3}}$ and $c_c=300\text{J/kg}\cdot \text{K,}$ while the corresponding values for Co (the outer shell) are ${\rho }_s=8900{\text{kg/m}}^{\text{3}}$ and $c_s=750\text{J/kg}\cdot \text{K}\text{.}$ Once having reached the melting point, how much additional time is required to completely melt the cobalt if its latent heat of fusion is $h_{sf}=2.59\times {10}^5\text{J/kg?}$ You may use the lumped capacitance method of analysis and neglect radiation exchange between the particle and its surroundings.