Solve the below problem:
Q: A laser-materials-processing apparatus encloses a sample in the form of a disk of diameter D = 25 mm and thickness w = 1 mm. The sample has a diffuse surface for which the spectral distribution of the emissivity is prescribed. To reduce oxidation, an inert gas stream of temperature T8 = 500 K and convection coefficient h = 50 W/m2 · K flows over the sample upper and lower surfaces. The apparatus enclosure is large, with isothermal walls at Tenc = 300 K. To maintain the sample at a suitable operating temperature of Ts = 2000 K, a collimated laser beam with an operating wavelength of = 0.5µm irradiates its upper surface.
(a) Determine the total emissivity E of the sample.
(b) Determine the total absorptivity a of the sample for irradiation from the enclosure walls.
(c) Perform an energy balance on the sample and determine the laser irradiation, Glaser, required to maintain the sample at Ts = 2000 K.
(d) Consider a cool-down process, when the laser and the inert gas flow are deactivated. Sketch the total emissivity as a function of the sample temperature, Ts (t), during the process. Identify key features, including the emissivity for the final condition (t=8).
(e) Estimate the time to cool a sample from its operating condition at Ts(0) = 2000 K to a safe-To-touch temperature of Ts(t) = 40°C. Use the lumped capacitance method and include the effect of convection to the inert gas with h = 50 W/m2 · K and T8 = Tenc = 300 K. The thermo physical properties of the sample material are p = 3900 kg/m3, cp = 760 J/kg · K, and k = 45 W/m · K.
