3.044 MATERIALS PROCESSING
LECTURE 1
What is Materials Processing?
- A way to make materials useful: desired chemistry, shape, microstructure
- A way to give materials the desired properties
What Processes are included in copper production?
grinding → colloid / suspension → refining / reducing → casting → electrolysis
→ melting → casting → rolling (hot) → drawing
What thermodynamic variables do we have to work with?
T
heat
heat transfer
P (or σ)
C (composition)
beat (move matter)
mix
solid mechanics
chemical reaction
fluid mechanics phase transformation
diffusion
Topics Covered in this Class
Part I: Heat transfer
Part II: Fluid flow
Part III: Combine all 3
Heat Conduction: Heat flows down the temperature gradient
Date: February 8th, 2012.
1
2
LECTURE 1
Fourier’s Law: q̄ = −kVT ,
:
J
W
q̄ → heat flux mJ2 s = m
s
:
J
−k → thermal conductivity m:WK J
VT → temperature gradient K
m
Compare:
Fourier’s Law
q̄ = −kVT
Fick’s 1st Law
J¯ = −DVc
Why does heat behave this way?
Energy minimization ⇒ Entropy maximization
Heat Conduction Equation:
First think in 1-D
Heat balance in a small element:
= heat accumulation
hheat
ut inn − heat
h utoutn h(+ heat generation)
ut
n
ut
n h
∂H
A · qin
A · qout
V ·
chemical reaction, resistance
A qin − A qout = V
∂H
∂T
∂T
3.044 MATERIALS PROCESSING
3
∂H
∂T
∂H
= Δx
∂T
∂H
= Δx
∂T
k ∂T
∂T
=
( |x+Δx −
|x )
Δx ∂x
∂x
k
∂T
=
Δ( )
Δx ∂x
∂( ∂T )
= k ∂x
∂x
∂2T
=k 2
∂x
qin − qout = Δx
q|x − q|x+Δx
−k
∂T
∂T
|x −
|x+Δx
∂x
∂x
∂H
∂T
∂H
∂T
How does H relate to T?
where cp is heat capacity
and ρ is density
ΔH = ΔT cp ρ
∂H
∂t
∂ 2T
k 2
∂x
k ∂2T
ρ cp ∂x2
∂ 2T
α 2
∂x
∂T
∂t
∂T
∂t
∂T
= ρ cp
∂t
∂T
=
∂t
∂T
=
∂t
∂2T
=α 2
∂x
= ρ cp
α is the Thermal Diffusivity
k
α=
ρ
: W J
mK
:kg K J
k
α=
ρ cp
J
cp
2
m
s
m3
kg
4
LECTURE 1
The values of k, cp and ρ for any material can be looked up in tables and do not
need to be experimentally determined. Therefore α is a materials property.
Compare:
Heat Conduction Equation
∂2T
∂T
= hutn
α
∂x2
∂t
↓
h 2i
thermal diffusivity ms
Fick’s 2nd Law
∂2c
∂c
= hutn
D
∂x2
∂t
↓ h i
2
diffusivity ms
Topic for Future Discussion:
In 3-D ...
ρ cp
∂T
= V · kVT
∂t
∂T
= αV2 T
∂t
** assuming k is constant with respect to the derivative
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3.044 Materials Processing
Spring 2013
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