Chapter 4, Problem 4.21P

Laser beams are used to thermally process materials in a wide range of applications. Often, the beam is scanned along the surface of the material in a desired pattern. Consider the laser heating process of Problem 4.18, except now the laser beam scans the material at a scanning velocity of U. A dimensionless maximum surface temperature can be well correlated by an expression of the form [Nissin, Y. 1., A. Lietoila, R. G. Gold, and J. F. Gibbons, J. App/. Phys.. 51, 274, 1980]
$\frac{T_{1,max,U=0}-T_2}{T_{1,\max ,U\ne 0}-T_2}=1+0.301Pe-0.0108P{e}^2$

for the range $0<Pe<10,$ where Pe is a dimensionless velocity known as the Peclet number. For this problem, $Pe=U{r}_b/\sqrt{2}\alpha$ where $\alpha$ is the thermal diffusivity of the material. The maximum material temperature does not occur directly below the laser beam, but at a lag distance behind the center of the moving beam. The dimension-less lag distance can be correlated to Pe by [Sheng, I. C., and Y. Chen, J. Thermal Stresses, 14, 129, 1991]

$\frac{\delta U}{\alpha }=0.944P{e}^{1.55}$

1. For the laser beam size and shape and material of Problem 4.18, determine the laser power required to $T_{1,\max }=200°\text{C}$ for $U=2\text{m/s}\text{.}$ The density achieve and specific heat of the material are $\rho =2000{\text{kg/m}}^{\text{3}}$ and $c=800\text{J/kg}\cdot \text{K},$ respectively.
2. Determine the lag distance $\delta$ associated with $U=2\text{m/s}\text{.}$
3. Plot the required laser power to achieve $T_{\max ,1}=200°\text{C}$ for $0\le U\le 2\text{m/s}\text{.}$