3.044 MATERIALS PROCESSING
LECTURE 8
Radiation:
−k
4
∂T
4
= −εσ Tsurf
− Tsource
∂x
L 3
M = εσ Tsurf
k
∂T
= α ∇2 T
∂t
4
4
− Tsource
)
q = εσ(Tsurf
⇒ Very few analytical solutions, some charts
Date: March 5th, 2012.
1
2
LECTURE 8
CVD: Chemical Vapor Deposition
At steady state the thermocouple outputs a temperature reading TTC
TC is heated by gas (convection)
out
in
4
4
−h(T − Tf )A = εσ(Tsurf
− Tsource
)A
εσ 4
4
− Tsource
)
TTC = Tf − (Tsurf
h
TTC = Tf
Tf ≈ 1000◦ C, Twall ≈ 500◦ C,
W
ε = 0.1, h = 100 2
mK
◦
TT C ≈ 830 C, ΔT ≈ 200◦ C
Conclusions:
1. Objects that “see” cold surroundings are colder than you think
2. If an object “sees” a hot source it can be unexpectedly hot
3. “In vacuum” indicates no convection → radiation must be important
Topics Covered So Far:
Heat
Beat
Mix
heat transfer solid mechanics
diffusion
fluid mechanics phase transition
3.044 MATERIALS PROCESSING
⇒
3
Next step is to discuss heat transfer combined
with diffusion and phase transformations
Solidification: Heat transfer plus phase transition, single component solidification
What are the B.C at the solid/liquid interface?
1. T = Tm
2. heat balance:
q
q
out
in
∂T ∂T −ks
−k
=
l
∂x s
∂s l
3. heat of fusion
Look closely:
∂T qin = −kl
∂x x=s,l
4
LECTURE 8
∂T qout = −ks
∂x x=s,s
kJ
Fusion: − Hf
kg
In time Δt the interface moved Δs ⇒ Volume transformed = AΔs
∂T
∂T
Δs
A + ks
A − Hf ρ
A=0
−kl
∂x
∂x
Δt
in
where
out
Heat of Fusion
Δs
∂s
=
= interface velocity
Δt
∂t
− kl ∂T −ks ∂T
∂x s
∂x l
=
Hf ρ
k
∂T ∂T =
(within factor of two)
−
Hf ρ ∂x s
∂x l
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3.044 Materials Processing
Spring 2013
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