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PROBLEM 4.18
KNOWN:
Power, size and shape of laser beam. Material properties.
FIND:
Maximum surface temperature for a Gaussian beam, maximum temperature for a flat
beam, and average temperature for a flat beam.
SCHEMATIC:
P = 1W
r
b
= 0.1 mm
r
b
Flat
Gaussian
T
max
T
max
T
2
= 25°C
k = 27 W/m•K
α=
0.45
′′
q
′′
q
ASSUMPTIONS:
(1) Steadystate conditions, (2) Constant properties, (3) Semiinfinite medium,
(4) Negligible heat loss from the top surface.
ANALYSIS:
The shape factor is defined in Eq. 4.20 and is q = Sk
Δ
T
12
(1)
From the problem statement and Section 4.3, the shape factors for the three cases are:
Beam Shape
Shape Factor
T
1,avg
or T
1,max
Gaussian
Flat
Flat
b
2r
π
b
r
π
2
b
3r
/
8
π
T
1,max
T
1,max
T
1,avg
For the Gaussian beam,
3
6
1
S =
2
π
× 0.1 × 10 m = 354 × 10 m
For the flat beam (max. temperature),
3
6
2
S =
× 0.1 × 10 m = 314 × 10 m
π
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This note was uploaded on 12/07/2010 for the course MAE Heat Trans taught by Professor Lee,j.s. during the Spring '10 term at Seoul National.
 Spring '10
 LEE,J.S.
 Heat Transfer

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