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.

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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

(14.22)) and the average solute balance (Eq. (14.21)). Under the assumptionρ` = cst, derive Eq. (14.45). Then combine this result with Eq. (14.21) to finallyobtain the various contributions to macrosegregation given in Eq. (14.46).

Index• No corrections at this time.

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