12.76 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,
ε λ
​
(λ)
, is prescribed. To reduce oxidation, an inert gas stream of temperature
T [infinity]
​
=500 K
and convection coefficient
h=50 W/m 2
⋅K
flows over the sample upper and lower surfaces. The apparatus enclosure is large, with isothermal walls at
T enc ​
=300 K
. To maintain the sample at a suitable operating temperature of
T 5
​
=2000 K
, a collimated laser beam with an operating wavelength of
λ=0.5μm
irradiates its upper surface. (a) Determine the total emissivity
ε
of the sample. (b) Determine the total absorptivity
α
of the sample for irradiation from the enclosure walls. (c) Perform an energy balance on the sample and determine the laser irradiation,
G laser ​
, required to maintain the sample at
T s
​
=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,
T s
​
(t)
, during the process. Identify key features, including the emissivity for the final condition
(t→[infinity])
. (e) Estimate the time to cool a sample from its C operating condition at
T s
​
(0)=2000 K
to a safe-to-touch temperature of
T s
​
(t)=40 ∘
C
. Use the lumped capacitance method and include the effect of convection to the inert gas with
h=50 W/m 2
⋅K
and
T [infinity]
​
=T enc ​
=300 K
. The thermophysical properties of the sample material are
rho=3900 kg/m 3
,c p
​
=760 J/kg⋅K
, and
k=45 W/m⋅K