PROBLEM 1.41
KNOWN:
Hot platetype wafer thermal processing tool based upon heat transfer modes by
conduction through gas within the gap and by radiation exchange across gap.
FIND:
(a) Radiative and conduction heat fluxes across gap for specified hot plate and wafer
temperatures and gap separation; initial time rate of change in wafer temperature for each mode, and
(b) heat fluxes and initial temperaturetime change for gap separations of 0.2, 0.5 and 1.0 mm for hot
plate temperatures 300 < T
h
< 1300
°
C.
Comment on the relative importance of the modes and the
influence of the gap distance.
Under what conditions could a wafer be heated to 900
°
C in less than 10
seconds?
SCHEMATIC:
ASSUMPTIONS:
(1) Steadystate conditions for flux calculations, (2) Diameter of hot plate and
wafer much larger than gap spacing, approximating plane, infinite planes, (3) Onedimensional
conduction through gas, (4) Hot plate and wafer are blackbodies, (5) Negligible heat losses from wafer
backside, and (6)
Wafer temperature is uniform at the onset of heating.
PROPERTIES:
Wafer:
ρ
= 2700 kg/m
3
,
c = 875 J/kg
⋅
K; Gas in gap: k = 0.0436 W/m
⋅
K.
ANALYSIS:
(a) The radiative heat flux between the hot plate and wafer for T
h
= 600
°
C and T
w
=
20
°
C follows from the rate equation,
(
29
(
29
(
29
(
29
4
4
4
4
8
2
4
4
2
rad
h
w
q
T
T
5.67
10
W / m
K
600
273
20
273
K
32.5kW / m
σ

′′
=

×
⋅
+

+
=
=
<
The conduction heat flux through the gas in the gap with L = 0.2 mm follows from Fourier’s law,
(
29
2
h
w
cond
600
20 K
T
T
q
k
0.0436W / m K
126 kW / m
L
0.0002 m


′′
=
=
⋅
=
<
The initial time rate of change of the wafer can be determined from an energy balance on the wafer at
the instant of time the heating process begins,
w
in
out
st
st
i
dT
E
E
E
E
cd
dt
ρ
′′
′′
′′
′′

=
=
±
±
±
±
where
out
E
0
′′
=
and
in
rad
cond
E
q
or q
.
′′
′′
′′
=
Substituting numerical values, find
3
2
w
rad
3
i,rad
dT
q
32.5
10
W / m
17.6 K /s
dt
cd
2700kg / m
875 J / kg K
0.00078 m
ρ
′′
×
=
=
=
×
⋅
×
<
w
cond
i,cond
dT
q
68.4 K /s
dt
cd
ρ
′′
=
=
<
Continued …..
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PROBLEM 1.41 (Cont.)
(b) Using the foregoing equations, the heat fluxes and initial rate of temperature change for each mode
can be calculated for selected gap separations L and range of hot plate temperatures T
h
with T
w
=
20
°
C.
In the lefthand graph, the conduction heat flux increases linearly with T
h
and inversely with L as
expected.
The radiative heat flux is independent of L and highly nonlinear with T
h
, but does not
approach that for the highest conduction heat rate until T
h
approaches 1200
°
C.
The general trends for the initial temperaturetime change, (dT
w
/dt)
i
, follow those for the heat fluxes.
To reach 900
°
C in 10 s requires an average temperaturetime change rate of 90 K/s.
Recognizing that
(dT
w
/dt) will decrease with increasing T
w
, this rate could be met only with a very high T
h
and the
smallest L.
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 Fall '07
 SAVASYAVUZKURT

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