Chapter 7, Problem 7.82P

Consider the plasma spray coating process of Problem 5.25. In addition to the prescribed conditions, the argon plasma jet is known to have a mean velocity of $V=400\text{}\text{m/s,}$ While the initial velocity of the injected alumina particles may be approximated as zero. The nozzle exit and the substrate are separated by a distance of $L=100\text{mm,}$ and pertinent properties of the argon plasma may be approximated as $k=0.671\text{W/m}\cdot \text{K,}{c}_p=1480\text{J/kg}\cdot \text{K,}\mu =2.70\times {10}^{-4}\text{kg/s}\cdot \text{m},$ and $v=5.6\times {10}^{-3}{\text{m}}^{\text{2}}\text{/s}\text{.}$

1. Assuming the motion of particles entrained by the plasma jet to be governed by Stokes' law, derive expressions for the particle velocity, $V_p\left(t\right),$ and its distance of travel from the nozzle exit, $x_p\left(t\right),$ as a function of time, t, where $t=0$ corresponds to particle injection. Evaluate the time-in-flight required for a particle to traverse the separation distance, $x_p=L,$ and the velocity $V_p$ at this time.
2. Assuming an average relative velocity of $\overline{\left(V-V_p\right)}=315\text{m/s}$ during the time-of-flight, estimate the convection coefficient associated with heat transfer from the plasma to the particle. Using this coefficient and assuming an initial particle temperature of $T_i=300\text{K,}$ estimate the time-in-flight required to heat a particle to its melting point, $T_{\text{mp}},$ and, once at $T_{\text{mp}},$ for the particle to experience complete melting. Is the prescribed value of L sufficient to ensure complete particle melting before surface impact?