Errata for SOLIDIFICATION (Second Edition, 2016)

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1 Errata for SOLIDIFICATION (Second Edition, 2016) J. A. Dantzig and M. Rappaz September 6, 2017 Nomenclature There are several minor typographical errors in this table. Please download the corrected version from located under THE BOOK 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 = (ξ 2 + y 2 + z 2 ) 1/2 Chapter 2: Thermodynamics Chapter 3: Phase Diagrams Chapter 4: Balance Equations Page 116, just after Eq. (4.7): indecial should be indicial 1

2 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 ( φ 2) + c ps(t T f ) exp ([1 α s /α l ] φ 2 ) L f π erfc (φ k ) l ρ l c pl α s /α k s ρ s c ps l { } k s ρ s c ps erf (φ) + = c ps(t f T 0 ) = Ste k m ρ m c pm L f π π Pages , Example 5.1: The error in the previous item continues in this example. The equation under the table should read: { φ exp ( φ 2) exp } ( 0.968φ2 ) {erf (φ) } = f(φ) = erfc (1.403φ) The correct value of φ is 0.636, and T ms = 453 C. The corrected form of the four equations at the top of Page 171: x (t) = t T m = erf (96.7 t x ) T s = erf (60.02 t x ) T l = erfc (84.17 t x ) Finally, the graphs shown at the end of the example change slightly as well: f(φ) - Ste / π 1/ Temperature [ o C] Graphite mold Solid Liquid φ Distance from mold-solid interface [m] 2

3 Chapter 6: Numerical Methods Part II: Microstructure Chapter 7: Nucleation Chapter 8: Dendritic Growth Page 356, Figure 8.22 and the text line just above it: 1.5 K should be 1.0 K T = 0.5 K T = 1.0 K R tip = C / T T = 0.5 K T = 1.0 K v * = T 2.5 /C R tip [µm] v * [µm/s] C 0 [Wt% Cu] C 0 [Wt% Cu] (a) Tip radius, R tip (b) Tip velocity, v * Fig Values of R tip and v, computed with Eqs. (8.91) and (8.94), for Al-Cu alloys at two undercoolings. The dashed lines represent the approximate forms considering only the solute, given in Eqs. (99) and (100), valid at low undercooling for compositions that are not too small. Page 361, Key Concept 8.13: Equation (8.105) should read as follows: ( ) 72π 2 Γ sl D l ( T λ 1 = 0) 2 1/4 (v ) 1/4 G 1/2 k 0 T 0 3

4 and Equation (8.106) should read: ( ) 72π 2 1/4 Γ sl D l T 0 λ 1 = (v ) 1/4 G 1/2 Chapter 9: Eutectics and Peritectics k 0 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 corrected figure is: <1120> Zn <1120> Zn (0001) Zn (0001) Zn (111) Al (111) Al <110> Al 20 mm <110> Al Fig. 9.7 Two grains of the lamellar Al-Zn regular eutectic with the corresponding pole figures for the fcc Al and hcp Zn phases, transverse cross section. (After Rhême et al. [37].) Page 440, Exercise 9.6: There is a minus sign missing from the first term on the right hand side of the second equation. It should read as follows: [ ( ) vβ = dx β 1 = ( ) dc β l λ2 dt Cβ l dt 2 C x β l β D ] C α β β x C β β l 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

5 Lever rule Gulliver-Scheil 33.2 Temperature [ C] Lever rule Gulliver-Scheil dt/dt = 5 K min 1 dt/dt = 10 K min 1 C s [wt% Cu] 6 4 Non-equilibrium eutectic Solid fraction g s g s (a) Solid fraction vs. temperature (b) Microsegregation profiles Fig (a) Computed fraction solid vs. temperature curves for an Al-4.5 wt% Cu alloy, using the lever rule and the Gulliver-Scheil equation. The experimental data were obtained by DTA experiments. [8] (b) The corresponding microsegregation patterns for the two solidification models. Page 460, Eq. (10.43): There is a minus sign missing from the first term on the right hand side. The equation should read as follows: ( ) dx ( ) C l β Cl β = dc l λ2 dt dt 2 C β x β(t) D β x Chapter 11: Macro-micro Modeling x β Chapter 12: Porosity Chapter 13: Mechanical Behavior and Hot Tearing 5

6 Chapter 14: Macrosegregation Page 670, Figure 14.17: Replace the printed image with the following one, which has higher resolution. Fig Snapshots of dendritic structure and composition field obtained from in situ X-ray radiography of an In-75wt%Ga alloy solidified against gravity at 0.01 K/s in a gradient G = 1.1 K/mm (i.e. v T = 9.1 µm/s). [34] There are typos in Exercises 14.3, 14.4, 14.5 and The corrected problem statements for these three exercises are as follows: Exercise Mean weight composition in Flemings criterion. Using the Gulliver-Scheil microsegregation model and the steady state relationship between liquid fraction and liquid concentration (Eq ), show that the mean composition C M during solidification is given by Eq. (14.33): g k 0 l C M = C (ρ l ρ s ) + ρ s 0 (1) ρ s (1 g l ) + ρ l g l Start from Eq. (14.21), replace v lx by βv T and integrate the equation knowing that v T / x = / t under steady state conditions. The mean volumetric concentration ρc has to be integrated first using the Gulliver-Scheil relation. Exercise Flemings criterion with lever rule. 6

7 Assuming lever rule for microsegregation, show that Flemings criterion for macrosegregation (Eqs. (14.19) and (14.22)) becomes: ( dg l ρ l ρ s g s = 1 + k 0 v ) l T dcl g l ρ s (1 k 0 ) ρ l g l v T T C l Exercise Flemings criterion with lever rule: steady state. Under 1D steady state conditions, for which v l T = βv T T, show that Flemings criterion derived in the previous exercise recovers the lever rule: C l = C 0 g l (1 k 0 ) + k 0 (2) Start with Eq. (14.21), then show that the mean weight composition C M is given by C M ρ l g l + k 0 ρ s (1 g l ) = C 0 (ρ s (1 g l ) + ρ l g l )(g l (1 k 0 ) + k 0 ) (3) or: ρc = ρ lg l + k 0 ρ s (1 g l ) C 0 g l (1 k 0 ) + k 0 (4) Compare and discuss this result, illustrated in the figure below with that obtained using the Gulliver-Scheil model shown in Fig C M /C k 0 = 0.8 k 0 = 0.5 k 0 = Liquid fraction, g Mean weight composition C M normalized with the nominal composition C 0 during steady state directional solidification as a function of the fraction of liquid g l for the lever rule approximation and three values of the partition coefficient. Exercise Grain movement- and deformation-induced macrosegregation. Using the definition of the derivative D s /Dt = / t + v s defined in Sect. 14.6, derive Eq. (14.44) from the average mass balance (Eq. 7

8 (14.22)) and the average solute balance (Eq. (14.21)). Under the assumption ρ l = cst, derive Eq. (14.45). Then combine this result with Eq. (14.21) to finally obtain the various contributions to macrosegregation given in Eq. (14.46). Index 8