errata for solidification (second edition,...
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Errata for SOLIDIFICATION(Second Edition, 2016)
J. A. Dantzig and M. Rappaz
September 6, 2017
NomenclatureThere are several minor typographical errors in this table. Please download thecorrected version from http://www.solidification.org, located under “THEBOOK” tab.
Chapter 1: Overview• Page 12, second paragraph after Key Concept 1.4, line 2: “latter two processes”
should be “latter process”
• Page 21, three lines after Eq. (1.5): the definition of r should have a square root.i.e., r = (ξ2 + y2 + z2)1/2
Chapter 2: Thermodynamics• No corrections at this time.
Chapter 3: Phase Diagrams• No corrections at this time.
Chapter 4: Balance Equations• Page 116, just after Eq. (4.7): “indecial” should be “indicial”
1
Chapter 5: Analytical Solutions• Page 169, Eq. (5.26): the second term in the opening curly braces should have a
“+” sign, rather than a “-” sign. The corrected equation reads as follows:φ exp(φ2)
+cps(T∞ − Tf )
Lf√π
exp ([1− αs/α`]φ2)
erfc(φ√αs/α`
) √k`ρ`cp`ksρscps
×{erf (φ) +
√ksρscpskmρmcpm
}=cps(Tf − T0)
Lf√π
=Ste√π
• Pages 170-171, Example 5.1: The error in the previous item continues in thisexample. The equation under the table should read:{
φ exp(φ2)
+ 0.041exp (−0.968φ2)
erfc (1.403φ)
}· {erf (φ) + 1.310} = f(φ) = 1.079
The correct value of φ is 0.636, and Tms = 453◦C. The corrected form of the fourequations at the top of Page 171:
x∗(t) = 1.06× 10−2√t
Tm = 453 + 428 erf
(96.7
x√t
)Ts = 453 + 327.5 erf
(60.02
x√t
)T` = 700− 190.5 erfc
(84.17
x√t
)Finally, the graphs shown at the end of the example change slightly as well:
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7φ
-1.5
-1
-0.5
0
0.5
f(φ
) - S
te / π
1/2
-0.06 -0.04 -0.02 0 0.02 0.04 0.06Distance from mold-solid interface [m]
0
100
200
300
400
500
600
700
Tem
pera
ture
[o C]
Graphitemold
Solid Liquid
2
Chapter 6: Numerical Methods• No corrections at this time.
Part II: Microstructure• No corrections at this time.
Chapter 7: Nucleation• No corrections at this time.
Chapter 8: Dendritic Growth• Page 356, Figure 8.22 and the text line just above it: “1.5 K” should be “1.0 K”
(a) Tip radius, Rtip
C0 [Wt% Cu]
∆T = 0.5 K∆T = 1.0 KRtip = 20.34 C0
0.25/∆T 1.25
∆T = 0.5 K∆T = 1.0 Kv* = 49.14 ∆T 2.5/C0
1.5
Rtip
[µm
]
v* [µ
m/s
]
0
100
80
60
40
20
0
104
103
102
101
100
10–11 2 3 4 0 1 2 3 4
C0 [Wt% Cu]
(b) Tip velocity, v*
Fig. 8.22 Values of Rtip and v∗, computed with Eqs. (8.91) and (8.94), for Al-Cu alloysat two undercoolings. The dashed lines represent the approximate forms consideringonly the solute, given in Eqs. (99) and (100), valid at low undercooling for compositionsthat are not too small.
• Page 361, Key Concept 8.13: Equation (8.105) should read as follows:
λ1 =
(72π2Γs`D`(∆T
′0)
2
k0∆T0
)1/4
(v∗)−1/4G−1/2
3
and Equation (8.106) should read:
λ1 =
(72π2Γs`D`∆T0
k0
)1/4
(v∗)−1/4G−1/2
Chapter 9: Eutectics and Peritectics• Page 391, final paragraph, line 5 should read as follows: “The figure also shows
Al 〈110〉 and Zn 〈1120〉 pole figure . . . ”
• Page 392, Figure 9.7: The Al pole figures should be labeled 〈110〉. The correctedfigure is:
<1120>Zn
(0001)Zn
(0001)Zn
(111)Al
(111)Al
<1120>Zn
<110>Al20 mm
<110>Al
Fig. 9.7 Two grains of the lamellar Al-Zn regular eutectic with the corresponding polefigures for the fcc Al and hcp Zn phases, transverse cross section. (After Rheme et al.[37].)
• Page 440, Exercise 9.6: There is a minus sign missing from the first term on theright hand side of the second equation. It should read as follows:
v∗β =dx∗βdt
=1(
C∗`β − C∗β`) [−dC∗β`
dt
(λ22− x∗β
)−Dβ
C∗`β − C∗αβx∗β − x∗α
]
Chapter 10: Microsegregation• Page 451, Fig. 10.3: The legend labels do not correspond correctly to the curves
in both graphs. The corrected figure is:
4
(a) Solid fraction vs. temperature
Lever ruleGulliver-Scheil
Lever ruleGulliver-Scheil
Non-equilibrium eutectic
dT/dt = –5 K min–1
dT/dt = –10 K min–1
Solid fraction gs
Tem
pera
ture
[°C
]
gs(b) Microsegregation profiles
640
620
600
580
560
540
520
10
8
6
4
2
0
33.2
0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1
Cs [w
t% C
u]Fig. 10.3 (a) Computed fraction solid vs. temperature curves for an Al-4.5 wt% Cualloy, using the lever rule and the Gulliver-Scheil equation. The experimental datawere obtained by DTA experiments. [8] (b) The corresponding microsegregation pat-terns for the two solidification models.
• Page 460, Eq. (10.43): There is a minus sign missing from the first term on theright hand side. The equation should read as follows:
(C∗`β − C∗`
) dx∗βdt
= −dC∗`
dt
(λ22− x∗β(t)
)−Dβ
∂Cβ∂x
∣∣∣∣x∗β
Chapter 11: Macro-micro Modeling• No corrections at this time.
Chapter 12: Porosity• No corrections at this time.
Chapter 13: Mechanical Behavior and Hot Tearing• No corrections at this time.
5
Chapter 14: Macrosegregation• Page 670, Figure 14.17: Replace the printed image with the following one, which
has higher resolution.
Fig. 14.17 Snapshots of dendritic structure and composition field obtained from insitu X-ray radiography of an In-75wt%Ga alloy solidified against gravity at −0.01 K/sin a gradient G = 1.1 K/mm (i.e. vT = 9.1 µm/s). [34]
• There are typos in Exercises 14.3, 14.4, 14.5 and 14.8. The corrected problemstatements for these three exercises are as follows:
Exercise 14.3. Mean weight composition in Flemings’ criterion .Using the Gulliver-Scheil microsegregation model and the steady staterelationship between liquid fraction and liquid concentration (Eq. 14.32), showthat the mean composition 〈C〉M during solidification is given by Eq. (14.33):
〈C〉M = C0gk0` (ρ` − ρs) + ρsρs(1− g`) + ρ`g`
(1)
Start from Eq. (14.21), replace v`x by −βvT and integrate the equation know-ing that vT∂/∂x = −∂/∂t under steady state conditions. The mean volumetricconcentration 〈ρC〉 has to be integrated first using the Gulliver-Scheil relation.
Exercise 14.4. Flemings’ criterion with lever rule .
6
Assuming lever rule for microsegregation, show that Flemings’ criterion formacrosegregation (Eqs. (14.19) and (14.22)) becomes:
dg`g`
= − ρ`ρs(1− k0)
(1 + k0
ρsgsρ`g`− v` · ∇T
vT · ∇T
)dC`C`
Exercise 14.5. Flemings’ criterion with lever rule: steady state .Under 1D steady state conditions, for which v` · ∇T = −βvT · ∇T , show thatFlemings’ criterion derived in the previous exercise recovers the lever rule:
C` =C0
g`(1− k0) + k0(2)
Start with Eq. (14.21), then show that the mean weight composition 〈C〉M isgiven by
〈C〉MC0
=ρ`g` + k0ρs(1− g`)
(ρs(1− g`) + ρ`g`)(g`(1− k0) + k0)(3)
or:〈ρC〉C0
=ρ`g` + k0ρs(1− g`)g`(1− k0) + k0
(4)
Compare and discuss this result, illustrated in the figure below with that ob-tained using the Gulliver-Scheil model shown in Fig. 14.8.
0 0.2 0.4 0.6 0.8 1Liquid fraction, g
0.95
0.96
0.97
0.98
0.99
1.00
〈CM〉/C
0
k0 = 0.8
k0 = 0.5
k0 = 0.3
Mean weight composition 〈C〉M normalized with the nominal composition C0 duringsteady state directional solidification as a function of the fraction of liquid g` for thelever rule approximation and three values of the partition coefficient.
Exercise 14.8. Grain movement- and deformation-inducedmacrosegregation .Using the definition of the derivative Ds〈·〉/Dt = ∂〈·〉/∂t + vs · ∇〈·〉defined in Sect. 14.6, derive Eq. (14.44) from the average mass balance (Eq.
7
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