Chapter 4, Problem 4.18P

The temperature distribution in laser-irradiated materials is determined by the power, size, and shape of the laser beam, along with the properties of the material being irradiated. The beam shape is typically Gaussian, and the local beam irradiation flux (often referred to as the laser fluence) is

$q"\left(x,y\right)=q"\left(x=y=0\right)\exp {\left(-x/r_b\right)}^2\exp {\left(-y/r_b\right)}^2$

The x- and y-coordinates determine the location of interest on the surface of the irradiated material. Consider the case where the center of the beam is located at $x=y=r=0.$ The beam is characterized by a radius $r_b,$ defined as the radial location where the local fluence is $q"\left(r_b\right)=q"\left(r=0\right)/e\approx 0.368q"\left(r=0\right).$
A shape factor for Gaussian heating is S = 27112%, where $S=2{\pi }^{1/2}{r}_b,$ is defined in terms of $T_{1,\max }-T_2$ [Nissin, Y. 1., A. Lietoila, R. G. Gold, and J. F. Gibbons, J. Appl. Phys., 51, 274, 19801. Calculate the maximum steady-state surface temperature associated with irradiation of a material of thermal conductivity $k=27\text{W/m}\cdot \text{K}$ and absorptivity $\alpha =0.45$ by a Gaussian beam with $r_b=0.1\text{mm}$ mm and power $P=1\text{W}\text{.}$ Compare your result with the maximum temperature that would occur if the irradiation was from a circular beam of the same diameter and power, but characterized by a uniform fluence (a flat beam). Also calculate the average temperature of the irradiated surface for the uniform fluence case. The temperature far from the irradiated spot is $T_2=25°\text{C}\text{.}$