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Unformatted text preview: , simplifying, separating variables, and integrating both sides, − rs
ri r ln k (T − T )
ri
dr = c m c
r
ρc ∆H f t dt (4) r2
r2
r2 r 1
r
− ln r ri +
= s ln s + ( ri 2 − rs2 )
s
2
4r
2 ri 4 (5) 0 The left side of (4) becomes, r2
r ln ri − r ln r ]dr = ln ri
[
rs
2
ri ri
rs Combining (4) and (5) and integrating the right side gives (1). ri s Exercise 17.33
Subject: Zone melting with a single or partial pass.
Given: A crystal layer undergoing zone melting to remove impurities.
Assumptions: Melt zone of width, l, is perfectly mixed with impurity concentration, w.
Diffusion of the impurity does not occur in the solid phase. Initial impurity concentration is
uniform at wo. Impurity concentration in the melt zone is in equilibrium with that in the solid
phase upstream of the melt zone.
Find: Derive an expression for the average impurity concentration over a particular length of
crystal layer, z2 – z1, after one pass or partial pass of zone melting. Also calculate for Example
17.14, wavg for z1 = 0 and z2/l = 9.
Analysis: The given expression to be derived is:
wavg = wo where, K= l (1 − K ) K ( z2 − z1 ) exp − z2 K
zK
− exp − 1
l
l +1 (1) impurity concentration in the solid phase
impurity concentration in the melt phase Using the definition of wavg with (1773),
z2 wavg = z1 ws dz z2 − z1 z2 = z1 wo 1 − (1 − K ) exp − Kz
l dz
(2) z2 − z1 Integrating (2) and applying the limits, wavg = wo l (1 − K )
l (1 − K )
z2 − z1
zK
zK
+
exp − 2
−
exp − 1
z2 − z1 K ( z2 − z1 )
l
K ( z2 − z1 )
l which simplifies to (1).
From Example 17.14, K = 0.36 and wo = 0.01. From above, (z2 – z1)/l = 9
Substitution into (1) gives, wavg = 0.01 (1 − 0.36 )
0.36 ( 9 ) exp [ −9(0.36] − exp ( 0 ) + 1 = 0.0081 , Exercise 17.34
Subject: Zone melting with a single or partial pass.
Given: A crystal layer undergoing the zone melting of Example 17.14, where the last 20% of
the crystal layer is removed following the first pass to z/l = 9.
Assumptions: Melt zone of width, l, is perfectly mixed with impurity concentration, w.
Diffusion of the impurity does not occur in the solid phase. Initial impurity concentration is
uniform at wo. Impurity concentration in the melt zone is in equilibrium with that in the solid
phase upstream of the melt zone.
Find: The average impurity concentration in the remaining crystal layer using the expression
derived in Exercise 17.33.
Analysis: The expression in Exercise 17.33 is:
wavg = wo l (1 − K ) K ( z2 − z1 ) exp − z2 K
zK
− exp − 1
l
l +1 where, from Example 17.14, K = 0.36 , wo = 0.01 , z1 = 0 , l/L = 0.1, and given z2 = 0.80 .
Therefore, and, l
l/L
0.1
0.1
=
=
=
z2 − z1 z2 / L − z1 / L 0.8 − 0 0.8 z2 z2 / L 0.8
=
=
=8
l
l / L 0.1 Substitution into (1) gives, wavg = 0.01 (1 − 0.36 ) (0.1)
0.36 ( 0.8 ) exp [ −8(0.36] − exp ( 0 ) + 1 = 0.0079 (1) Exercise 17.35
Subject: Zone melting with a single pass.
Given: A bar of 98 wt% Al with 2 wt% Fe impurity subjected to one pass of zone refining.
K = 0.29 for the impurity. The resulting bar is cut off at z2 = 0.75 z and z/l =10.
Assumptions: Melt zone of width, l, is perfectly mixed with impurity concentration, w.
Diffusion of...
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 Spring '11
 Levicky
 The Land

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