- Email: [email protected]
Inward solidification of cylinders: Reversal in the growth rate and microstructure evolution
Applied Thermal Engineering 61 (2013) 577e582
Contents lists available at ScienceDirect
Applied Thermal Engineering journal homepage: www.elsevier.com/locate/apthermeng
Inward solidification of cylinders: Reversal in the growth rate and microstructure evolution Felipe Bertelli, Noé Cheung, Amauri Garcia* Department of Materials Engineering, University of Campinas, UNICAMP, PO Box 6122, 13083-970 Campinas, SP, Brazil
h i g h l i g h t s The inward solidification of cylinders is investigated numerically and experimentally. The growth rate experiences a reversal during inward solidification of a cylinder. The primary dendritic arm spacing accompanies the trend in the growth rate.
a r t i c l e i n f o
a b s t r a c t
Article history: Received 1 July 2013 Accepted 26 August 2013 Available online 4 September 2013
The inward solidification of cylinders is investigated both numerically and experimentally. Experiments were performed with PbeSb hypoeutectic alloys having quite different freezing ranges. The present study aims to contribute to a better understanding about the correlation between a solidification thermal parameter (the growth rate) and the scale of the dendritic structure along the inward growth of cylindrical castings. It is shown, both by experimental results and numerical heat transfer simulations that the growth rate decreases from the casting surface, stabilizes along a range of positions in the casting and accelerates rapidly from 0.8 of the radius of the cylinder towards the center of the casting. It is also shown that the primary dendritic arm spacing accompanies the trend in the growth rate. Ó 2013 Elsevier Ltd. All rights reserved.
Keywords: Heat transfer Solidification Cylindrical geometry Phase change Alloys
1. Introduction The accurate description of solidification evolution and thermal fields of castings is of both theoretical and technological interest, and requires the accurate solution of the time-dependent heat transfer problem. Numerical models have been widely applied for the analysis of solidification in a number of processes, such as latent heat storage, static and continuous casting, welding, laser processing, etc [1e6]. It is a well-established fact that the structural integrity of alloys castings is closely related to their timeetemperature history during solidification. The solidification kinetics influences the morphological stability of the solid/liquid interface [7] and imposes the microstructure pattern, as well as the scale of cellular, dendritic and interphase spacings [8]. Several studies have been reported in the literature highlighting the role of these microstructural spacings on the final mechanical, corrosion and wear resistances [9e13].
* Corresponding author. Tel.: þ55 19 3521 3320; fax: þ55 19 3289 3722. E-mail address: [email protected] (A. Garcia). 1359-4311/$ e see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.applthermaleng.2013.08.034
The structural and mechanical characteristics of PbeSb alloys as well as their precipitation hardening effect make them a very convenient material for lead-acid battery grids. Finer dendritic arrays are associated with a more homogeneous distribution of the lamellar eutectic mixture (which is located in the interdendritic regions) throughout the alloy microstructure. The eutectic contains the nobler Sb-rich phase, and it has been shown that the Sb-rich lamellae envelope the Pb-rich phase, providing a protective effect against corrosion [12]. Non-conventional geometries such as cylindrical are of particular interest, since this is the shape of a lot of products involving phase change applications e.g.: cool storage systems [14], continuous casting [15] and conventional casting [16]. In the inward solidification of a cylinder the concave mold wall induces a divergent heat flow, which will be able to carry away heat more rapidly than in the solidification of a plate. Furthermore, with the progressive reduction in liquid toward the center of the cylinder, the growth rate which slows initially with distance from the casting surface will tend to accelerate rapidly in the direction of the center of the casting. That is, in the inward solidification of a cylinder the growth rate will experience a reversal trend, which may consequently
578
F. Bertelli et al. / Applied Thermal Engineering 61 (2013) 577e582
affect the microstructure. Despite these evidences, the literature is meager on studies concerning such particular behavior of the kinetics of solidification. The only study found in the literature is limited to the solidification of pure metals and the proposed analytical heat flow model indicated that the growth rate reverses at about 40% of the radius of the cylinder [17]. The aim of this study is to perform both numerical and experimental analyses of the inward solidification of cylinders using two hypoeutectic PbeSb alloys in order to cover particularities such as different transient
metal/mold thermal heat transfer coefficients and solidification ranges. Furthermore, the goal is to examine the influence of a reversal in the growth rate on the scale of the dendritic microstructure. 2. Experimental procedure A water-cooled cylindrical casting assembly made of low carbon steel (SAE 1020) was used in the experiments, as depicted in Fig. 1.
Fig. 1. Experimental water cooled setup used for the inward solidification of cylindrical castings, with positions of thermocouples inside the casting (dimensions in mm).
F. Bertelli et al. / Applied Thermal Engineering 61 (2013) 577e582
579
Table 1 Chemical analyses of metals used to prepare the alloys (wt. %). Metal
Fe
Al
Pb
Pb Sb
e 0.075
0.59 e
Balance 0.215
Sb
Si
Cu
Ni
Ca
Balance
0.0507 0.009
e 0.034
e 0.034
0.1212 e
The experimental device was designed in such a way to promote radial heat transfer through a water-cooled jacket around the casting resulting in inward solidification. The chilled surfaces in contact with the molten alloy were polished with a 1200 grit SiC paper and no coatings had been applied. The alloys were melted in situ and the lateral electric heaters had their power controlled in order to achieve a desired melt temperature, when the heating system is turned off and the water starts flowing to cool the mold surface. Experiments were performed with PbeSb hypoeutectic alloys having quite different freezing ranges (Pb 0.5 wt% Sb and Pb 1.5 wt% Sb alloys). The chemical compositions of metals used to prepare the alloys used in the experimental study are shown in Table 1. The resulting macrostructure and microstructures (at different positions along the cylinder radius) were examined on cross sections of the casting, as shown in Fig. 2. Image processing systems were used to measure the primary dendritic arm spacing (l1) on transverse sections of the castings using the triangle method [18], as shown in Fig. 3. At least 20 measurements were performed for each selected radial position. With a view to determining the evolution of the growth rate during solidification, continuous temperature measurements along the radius of the cylindrical casting were monitored via the output of fine type J thermocouples (0.2 mm diameter wire), and temperature data were collected at a frequency of 100 Hz. 3. Results and discussion
Fig. 3. Representation of measurement of the primary dendritic arm spacing by the triangle method on the cross section of the alloy microstructure. The distances from the center of adjacent primary dendrite trunks are represented by d.
since it is not significant compared with that of the r direction, resulting in a simplified equation:
1 v vT vT $ $ K$r$ ¼ r$c$ r vr vr vt
c’ ¼
cL
vfS vT
(1)
(2) 0
A numerical approach has been used to analyze the inward solidification of binary alloys. Considering that the flow of heat is mostly radial for inward solidification, the general equation of heat conduction [19] can be approximated by the one-dimensional form. The temperature inside the cylindrical ingot depends only on the radius, r, and time t, i.e. T ¼ T(r,t) and the heat extraction along the cylinder longitudinal axis and angular direction can be neglected,
where: K is the thermal conductivity [W m1 K1], c is known as the effective specific heat [J kg1 K1], and accounts for both temperature change and latent heat liberation associated with the phase transformation in the temperature range (TL TS), with TL being the liquidus temperature and TS, the solidus temperature of the alloy. A finite difference form of this simplified equation is obtained for the time-dependent temperature distribution at discrete grid points:
Fig. 2. Schematic representation of samples extracted for metallography.
580
F. Bertelli et al. / Applied Thermal Engineering 61 (2013) 577e582
6
Inward Solidification
Tip growth rate (mm/ m s)
5
Numerica c l Pb-0.5w 5wt.% Sb Numerica c l Pb-1.5w 5wt.% Sb
4
3
2
1
0 0
Fig. 4. Thermal resistances in a cylindrical water-cooled metal/mold system during inward solidification.
Tinþ1 ¼
Dt
r K ri c0i ri Dr2 eq i1 i1 þ Tin for is0
n n Tin þ Keq iþ1 riþ1 Tiþ1 Tin Ti1
(3) where the subscript (i) represents the location of the element in the finite difference mesh, and (n þ 1) represents the time in which the nodal temperature is being calculated. Keq is the equivalent thermal conductivity, which for an adjacent element i þ 1, is given by:
Keq iþ1 ¼
2Kiþ1 Ki Kiþ1 þ Ki
(4)
When the limit of (r) tends to zero, there is an indetermination in the first term of the left side of Eq. (1). L’Hopital rule has been used to solve this indetermination, which states that the differentiation of both the numerator and denominator does not change the resulting limit. This differentiation often simplifies the quotient and/or converts it to a determinate form, allowing the limit to be determined more easily. Applying L’Hopital rule and the finite difference method into Eq. (1), we obtain for i ¼ 0:
4Keq1 Dt T1n T0n þ T0n r0 c00 Dr2
15
20
25
30
35
Fig. 6. Tip growth rate as a function of radial position from the casting surface.
Considering the metal/coolant transient heat transfer, the following equation can be obtained by applying a thermal balance shown by Eq. (6):
KA
vT vT hAðTI TW Þ ¼ rcV vr r¼R vt
(6)
where R is the inner radius of the water-cooled jacket; subscripts I and W refer to the casting surface and the water, respectively; A is the area perpendicular to the heat flux [m2], V is the volume [m3]; h is the overall metal/coolant heat transfer coefficient [W m2 K1], which includes the metal/mold interfacial heat transfer coefficient, the mold thermal conductance, and the mold/cooling-fluid heat transfer coefficient [20e22]. In water cooled molds, the overall equivalent heat flux is affected by a series of thermal resistances, as shown in Fig. 4.The overall thermal resistance 1/h can be expressed by:
1 1 V 1 ¼ þ þ h hW At KM hM=M
(7)
78 72
8000
66
7000
60
6000
54
λ 1 (μ m)
9000
5000
Pb - 0.5 wt% Sb Pb - 1.5 wt% Sb
4000
10
Position from the casting surface (mm)
(5)
2
h (W/m K)
T0nþ1 ¼
5
48 42
-0,35
h = 6500 t
Experimental
36
3000
2
λ1 = - 0.19p + 6.42p +7.97 2
30
2000
R = 0.93
-0,30
h = 5500 t
24
1000 0
10
20
30
40
50
Time (s) Fig. 5. Evolution of the overall metal/coolant heat transfer coefficient with time for the PbeSb alloys cylindrical castings.
0
5
10
15
20
25
30
35
p: Position from the casting surface (mm) Fig. 7. Primary dendrite arm spacing as a function of position from the casting surfacedp [mm]: Pbe0.5 wt% Sb alloy. R2 is the coefficient of determination.
F. Bertelli et al. / Applied Thermal Engineering 61 (2013) 577e582
581
the aforementioned range of positions, and decreases after the reversal in the growth rate for both alloys examined, i.e. a tendency of increasing fineness of dendritic spacings towards the center of the cylinder can be expected accompanying the trend in the growth rate.
80 75 70 65 60
λ (μ m)
55
4. Conclusions
50 45 40 35 Experimental
30
λ = - 0.17p + 6.22.p - 3.1
25
R = 0.94
20 8
12
16
20
24
28
32
p: Position from the casting surface (mm) Fig. 8. Primary dendrite arm spacing as a function of position from the casting surfacedp [mm]: Pbe1.5 wt% Sb alloy. R2 is the coefficient of determination.
where V is the mold volume element (m3) and At is the heat transfer surface area, KM is the mold thermal conductivity (W m1 K1), hW is the mold/cooling fluid heat transfer coefficient (W m2 K1), hM/ 2 1 K ). M is the metal/mold heat transfer coefficient (W m The transient overall metal/coolant heat transfer coefficient, h, was determined by a technique based on the Inverse Heat Conduction Problem (IHCP), which has been detailed in a previous publication [23]. They are: h ¼ 6500t0.35 and h ¼ 5500t0.30 for the Pbe0.5 wt% Sb and Pbe1.5 wt% Sb alloys, respectively (t: time in s). It can be seen in Fig. 5 that the h profiles for Pbe0.5 wt% Sb and the Pbe1.5 wt% Sb alloy castings are quite similar, i.e. only a slight lower h profile is obtained for the Pbe1.5 wt% Sb alloy, which can be due to the influence of the higher solute concentration. The results of thermal analysis in metal have also been used to determine the tip growth rate [24] as a function of position from the casting surface with a view to permitting the reversal in the growth rate to be analyzed. The thermocouples readings have been used to create a plot of position (p) from the casting surface as a function of time (t) corresponding to the liquidus front passing by each thermocouple. The derivative of these curves with respect to time yielded the experimental tip growth rate. These results are shown in Fig. 6 compared with the corresponding numerical simulations, which assumed the corresponding thermophysical properties of each alloy reported elsewhere [23]. Applying an analytical model for the inward solidification of cylinders of pure metals (plane front solidification), Santos and Garcia [17] determined that the growth rate reverses at a specific point lying at about 0.4 of the radius of the cylinder, which is dependent on both Stefan (thermal properties of metal) and Biot (h and the thermal conductivity of the metal) numbers. The results of Fig. 6 show clearly that a characteristic reversal point cannot be observed for any alloy examined in the present study, but rather the growth rate tends to attain a stable value (an essentially constant growth rate) for a range of positions, i.e. 0.4e0.8 and 0.5e0.8 of the radius of the cylinder for the Pb 0.5Sb and Pb 1.5Sb alloys, respectively, and from this range onwards the growth rate accelerates rapidly. Figs. 7 and 8 show experimental values of primary dendritic arm spacings (l1) for the Pb 0.5Sb and Pb 1.5Sb alloys, respectively, fitted to trend functions. Comparing these results with the behavior of the tip growth rate in Fig. 6, one can conclude that this spacing increases initially as the growth rate decreases, becomes quite stable along
The results of growth rate have shown clearly that a characteristic reversal point, as reported previously for pure metals, could not be observed for the alloys examined, but rather the growth rate tends to attain an essentially stable value for a range of radial positions in the cylindrical casting, which depends on the alloy composition, and from this range onwards the growth rate accelerates rapidly towards the center of the casting. The experimental results indicated that the scale of the dendritic microstructure, despite exhibiting an inverse trend, accompanies the tendency of the profile of the growth rate. Acknowledgements The authors acknowledge the financial support provided by FAPESP (São Paulo Research Foundation e grant 2012/16328-2) and CNPq (The Brazilian Research Council). References [1] J. Yang, C.Y. Zhao, Solidification analysis of a single particle with encapsulated phase change materials, Appl. Therm. Eng. 51 (2013) 338e346. [2] N. Cheung, A. Garcia, The use of a heuristic search technique for the optimization of quality of steel billets produced by continuous casting, Eng. Appl. Artif. Intell. 14 (2001) 229e238. [3] F. Bertelli, E.S. Meza, P.R. Goulart, N. Cheung, R. Riva, A. Garcia, Laser remelting of Ale1.5 wt% Fe alloy surfaces: numerical and experimental analyses, Opt. Laser Eng. 49 (2011) 490e497. [4] G.S. Reddy, W.J. Mascarenhas, J.N. Reddy, Numerical simulation of solidification molten aluminum alloys in cylindrical molds, Metall. Trans. B 24 (1993) 677e684. [5] C.A. Santos, J.A. Spim, M.C.F. Ierardi, A. Garcia, The use of artificial intelligence technique for the optimisation of process parameters used in the continuous casting of steel, Appl. Math. Model 26 (2002) 1077e1092. [6] I. Nowak, J. Smolka, A.J. Nowak, An effective 3-D inverse procedure to retrieve cooling conditions in an aluminium alloy continuous casting problem, Appl. Therm. Eng. 30 (2010) 1140e1151. [7] M.W. Chen, X.F. Wang, F. Wang, G.B. Lin, Z.Z. Wang, The effect of interfacial kinetics on the morphological stability of a spherical particle, J. Cryst. Growth 362 (2013) 20e23. [8] Y.H. Zheng, Z.D. Wang, S.M. Zhang, Microstructure of diphase dendrite in Ale 35%La alloy during solidification, J. Cryst. Growth 362 (2013) 33e37. [9] M. Qian, J.N. DuPont, Microsegregation-related pitting corrosion characteristics of AL-6XN superaustenitic stainless steel laser welds, Corros. Sci. 52 (2010) 3548e3553. [10] W.R. Osório, D.M. Rosa, A. Garcia, Electrochemical behaviour of a PbeSb alloy in 0.5 M NaCl and 0.5 M H2SO4 solutions, Mater. Des. 34 (2012) 660e665. [11] W.R. Osório, L.R. Garcia, P.R. Goulart, A. Garcia, Effects of eutectic modification and T4 heat treatment on mechanical properties and corrosion resistance of an Ale9 wt%Si casting alloy, Mater. Chem. Phys. 106 (2007) 343e349. [12] W.R. Osório, D.M. Rosa, A. Garcia, The roles of cellular and dendritic microstructural morphologies on the corrosion resistance of PbeSb alloys for lead acid battery grids, J. Power Sources 175 (2008) 595e603. [13] L. Wang, B.P. Zhang, T. Shinohara, Corrosion behavior of AZ91 magnesium alloy in dilute NaCl solutions, Mater. Des. 31 (2010) 857e863. [14] L. Bilir, Z. Ilken, Total solidification time of a liquid phase change material enclosed in cylindrical/spherical containers, Appl. Therm. Eng. 25 (2005) 1488e1502. [15] M. Okayasu, S. Yoshie, Mechanical properties of AleSi13eNi1.4eMg1.4eCu1 alloys produced by the Ohno continuous casting process, Mat. Sci. Eng. AStruct 527 (2010) 3120e3126. [16] R. Rajaraman, R. Velraj, Comparison of interfacial heat transfer coefficient estimated by two different techniques during solidification of cylindrical aluminum alloy casting, Heat Mass Transfer 44 (2008) 1025e1034. [17] R.G. Santos, A. Garcia, Analytical technique for the determination of solidification rates during the inward freezing of cylinders, J. Mater. Sci. 18 (1983) 3578e3590. [18] M. Gündüz, E. Çadirli, Directional solidification of aluminiumecopper alloys, Mat. Sci. Eng. A-Struct 327 (2002) 167e185.
582
F. Bertelli et al. / Applied Thermal Engineering 61 (2013) 577e582
Nomenclature
c: specific heat [J/kg K] c: effective specific heat [J/kg K] fs: solid fraction [%] h: transient overall metal/coolant heat transfer coefficient [W/m2 K] hW: mold/cooling fluid heat transfer coefficient [W/m2 K] hM/M: metal/mold heat transfer coefficient [W/m2 K] i: element position according to ‘r’ axis [dimensionless] K: thermal conductivity [W/m K] Keq: equivalent thermal conductivity [W/m K] KM: mold thermal conductivity [W/m K] L: latent heat of fusion [J/kg]; n: number of time increment [dimensionless] r: radial coordinate [m] R: inner radius of the water-cooled jacket [m] t: time [s] T: temperature [K] TL: alloy liquidus temperature [K] TS: alloy solidus temperature [K] TW: water temperature [K] V: finite difference volume element [m3]
A: area perpendicular to the heat flux [m2] At: heat transfer surface area [m2]
Greek symbols Dt: time interval [s] r: density [kg/m3]
[19] P. Incropera, D.P. Dewitt, Fundamentals of Heat and Mass Transfer, John Wiley & Sons, New York, 1990. [20] N. Cheung, N.S. Santos, J.M.V. Quaresma, G.S. Dulikravich, A. Garcia, Interfacial heat transfer coefficients and solidification of an aluminum alloy in a rotary continuous caster, Int. J. Heat Mass Transfer 52 (2009) 451e459. [21] D. O’Mahoney, D.J. Browne, Use of experiment and an inverse method to study interface heat transfer during solidification in the investment casting process, Exp. Therm. Fluid Sci. 22 (2000) 111e122. [22] I.L. Ferreira, J.E. Spinelli, J.C. Pires, A. Garcia, The effect of melt temperature profile on the transient metal/mold heat transfer coefficient during solidification, Mat. Sci. Eng. A-Struct 408 (2005) 317e325. [23] F. Bertelli, C. Brito, E.S. Meza, N. Cheung, A. Garcia, Inward and outward solidification of cylindrical castings: the role of the metal/mold heat transfer coefficient, Mater. Chem. Phys. 136 (2012) 545e554. [24] D.M. Rosa, J.E. Spinelli, I.L. Ferreira, A. Garcia, Cellular/dendritic transition and microstructure evolution during transient directional solidification of PbeSb alloys, Metall. Mater. Trans. A 39 (2008) 2161e2174.
Contents lists available at ScienceDirect
Applied Thermal Engineering journal homepage: www.elsevier.com/locate/apthermeng
Inward solidification of cylinders: Reversal in the growth rate and microstructure evolution Felipe Bertelli, Noé Cheung, Amauri Garcia* Department of Materials Engineering, University of Campinas, UNICAMP, PO Box 6122, 13083-970 Campinas, SP, Brazil
h i g h l i g h t s The inward solidification of cylinders is investigated numerically and experimentally. The growth rate experiences a reversal during inward solidification of a cylinder. The primary dendritic arm spacing accompanies the trend in the growth rate.
a r t i c l e i n f o
a b s t r a c t
Article history: Received 1 July 2013 Accepted 26 August 2013 Available online 4 September 2013
The inward solidification of cylinders is investigated both numerically and experimentally. Experiments were performed with PbeSb hypoeutectic alloys having quite different freezing ranges. The present study aims to contribute to a better understanding about the correlation between a solidification thermal parameter (the growth rate) and the scale of the dendritic structure along the inward growth of cylindrical castings. It is shown, both by experimental results and numerical heat transfer simulations that the growth rate decreases from the casting surface, stabilizes along a range of positions in the casting and accelerates rapidly from 0.8 of the radius of the cylinder towards the center of the casting. It is also shown that the primary dendritic arm spacing accompanies the trend in the growth rate. Ó 2013 Elsevier Ltd. All rights reserved.
Keywords: Heat transfer Solidification Cylindrical geometry Phase change Alloys
1. Introduction The accurate description of solidification evolution and thermal fields of castings is of both theoretical and technological interest, and requires the accurate solution of the time-dependent heat transfer problem. Numerical models have been widely applied for the analysis of solidification in a number of processes, such as latent heat storage, static and continuous casting, welding, laser processing, etc [1e6]. It is a well-established fact that the structural integrity of alloys castings is closely related to their timeetemperature history during solidification. The solidification kinetics influences the morphological stability of the solid/liquid interface [7] and imposes the microstructure pattern, as well as the scale of cellular, dendritic and interphase spacings [8]. Several studies have been reported in the literature highlighting the role of these microstructural spacings on the final mechanical, corrosion and wear resistances [9e13].
* Corresponding author. Tel.: þ55 19 3521 3320; fax: þ55 19 3289 3722. E-mail address: [email protected] (A. Garcia). 1359-4311/$ e see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.applthermaleng.2013.08.034
The structural and mechanical characteristics of PbeSb alloys as well as their precipitation hardening effect make them a very convenient material for lead-acid battery grids. Finer dendritic arrays are associated with a more homogeneous distribution of the lamellar eutectic mixture (which is located in the interdendritic regions) throughout the alloy microstructure. The eutectic contains the nobler Sb-rich phase, and it has been shown that the Sb-rich lamellae envelope the Pb-rich phase, providing a protective effect against corrosion [12]. Non-conventional geometries such as cylindrical are of particular interest, since this is the shape of a lot of products involving phase change applications e.g.: cool storage systems [14], continuous casting [15] and conventional casting [16]. In the inward solidification of a cylinder the concave mold wall induces a divergent heat flow, which will be able to carry away heat more rapidly than in the solidification of a plate. Furthermore, with the progressive reduction in liquid toward the center of the cylinder, the growth rate which slows initially with distance from the casting surface will tend to accelerate rapidly in the direction of the center of the casting. That is, in the inward solidification of a cylinder the growth rate will experience a reversal trend, which may consequently
578
F. Bertelli et al. / Applied Thermal Engineering 61 (2013) 577e582
affect the microstructure. Despite these evidences, the literature is meager on studies concerning such particular behavior of the kinetics of solidification. The only study found in the literature is limited to the solidification of pure metals and the proposed analytical heat flow model indicated that the growth rate reverses at about 40% of the radius of the cylinder [17]. The aim of this study is to perform both numerical and experimental analyses of the inward solidification of cylinders using two hypoeutectic PbeSb alloys in order to cover particularities such as different transient
metal/mold thermal heat transfer coefficients and solidification ranges. Furthermore, the goal is to examine the influence of a reversal in the growth rate on the scale of the dendritic microstructure. 2. Experimental procedure A water-cooled cylindrical casting assembly made of low carbon steel (SAE 1020) was used in the experiments, as depicted in Fig. 1.
Fig. 1. Experimental water cooled setup used for the inward solidification of cylindrical castings, with positions of thermocouples inside the casting (dimensions in mm).
F. Bertelli et al. / Applied Thermal Engineering 61 (2013) 577e582
579
Table 1 Chemical analyses of metals used to prepare the alloys (wt. %). Metal
Fe
Al
Pb
Pb Sb
e 0.075
0.59 e
Balance 0.215
Sb
Si
Cu
Ni
Ca
Balance
0.0507 0.009
e 0.034
e 0.034
0.1212 e
The experimental device was designed in such a way to promote radial heat transfer through a water-cooled jacket around the casting resulting in inward solidification. The chilled surfaces in contact with the molten alloy were polished with a 1200 grit SiC paper and no coatings had been applied. The alloys were melted in situ and the lateral electric heaters had their power controlled in order to achieve a desired melt temperature, when the heating system is turned off and the water starts flowing to cool the mold surface. Experiments were performed with PbeSb hypoeutectic alloys having quite different freezing ranges (Pb 0.5 wt% Sb and Pb 1.5 wt% Sb alloys). The chemical compositions of metals used to prepare the alloys used in the experimental study are shown in Table 1. The resulting macrostructure and microstructures (at different positions along the cylinder radius) were examined on cross sections of the casting, as shown in Fig. 2. Image processing systems were used to measure the primary dendritic arm spacing (l1) on transverse sections of the castings using the triangle method [18], as shown in Fig. 3. At least 20 measurements were performed for each selected radial position. With a view to determining the evolution of the growth rate during solidification, continuous temperature measurements along the radius of the cylindrical casting were monitored via the output of fine type J thermocouples (0.2 mm diameter wire), and temperature data were collected at a frequency of 100 Hz. 3. Results and discussion
Fig. 3. Representation of measurement of the primary dendritic arm spacing by the triangle method on the cross section of the alloy microstructure. The distances from the center of adjacent primary dendrite trunks are represented by d.
since it is not significant compared with that of the r direction, resulting in a simplified equation:
1 v vT vT $ $ K$r$ ¼ r$c$ r vr vr vt
c’ ¼
cL
vfS vT
(1)
(2) 0
A numerical approach has been used to analyze the inward solidification of binary alloys. Considering that the flow of heat is mostly radial for inward solidification, the general equation of heat conduction [19] can be approximated by the one-dimensional form. The temperature inside the cylindrical ingot depends only on the radius, r, and time t, i.e. T ¼ T(r,t) and the heat extraction along the cylinder longitudinal axis and angular direction can be neglected,
where: K is the thermal conductivity [W m1 K1], c is known as the effective specific heat [J kg1 K1], and accounts for both temperature change and latent heat liberation associated with the phase transformation in the temperature range (TL TS), with TL being the liquidus temperature and TS, the solidus temperature of the alloy. A finite difference form of this simplified equation is obtained for the time-dependent temperature distribution at discrete grid points:
Fig. 2. Schematic representation of samples extracted for metallography.
580
F. Bertelli et al. / Applied Thermal Engineering 61 (2013) 577e582
6
Inward Solidification
Tip growth rate (mm/ m s)
5
Numerica c l Pb-0.5w 5wt.% Sb Numerica c l Pb-1.5w 5wt.% Sb
4
3
2
1
0 0
Fig. 4. Thermal resistances in a cylindrical water-cooled metal/mold system during inward solidification.
Tinþ1 ¼
Dt
r K ri c0i ri Dr2 eq i1 i1 þ Tin for is0
n n Tin þ Keq iþ1 riþ1 Tiþ1 Tin Ti1
(3) where the subscript (i) represents the location of the element in the finite difference mesh, and (n þ 1) represents the time in which the nodal temperature is being calculated. Keq is the equivalent thermal conductivity, which for an adjacent element i þ 1, is given by:
Keq iþ1 ¼
2Kiþ1 Ki Kiþ1 þ Ki
(4)
When the limit of (r) tends to zero, there is an indetermination in the first term of the left side of Eq. (1). L’Hopital rule has been used to solve this indetermination, which states that the differentiation of both the numerator and denominator does not change the resulting limit. This differentiation often simplifies the quotient and/or converts it to a determinate form, allowing the limit to be determined more easily. Applying L’Hopital rule and the finite difference method into Eq. (1), we obtain for i ¼ 0:
4Keq1 Dt T1n T0n þ T0n r0 c00 Dr2
15
20
25
30
35
Fig. 6. Tip growth rate as a function of radial position from the casting surface.
Considering the metal/coolant transient heat transfer, the following equation can be obtained by applying a thermal balance shown by Eq. (6):
KA
vT vT hAðTI TW Þ ¼ rcV vr r¼R vt
(6)
where R is the inner radius of the water-cooled jacket; subscripts I and W refer to the casting surface and the water, respectively; A is the area perpendicular to the heat flux [m2], V is the volume [m3]; h is the overall metal/coolant heat transfer coefficient [W m2 K1], which includes the metal/mold interfacial heat transfer coefficient, the mold thermal conductance, and the mold/cooling-fluid heat transfer coefficient [20e22]. In water cooled molds, the overall equivalent heat flux is affected by a series of thermal resistances, as shown in Fig. 4.The overall thermal resistance 1/h can be expressed by:
1 1 V 1 ¼ þ þ h hW At KM hM=M
(7)
78 72
8000
66
7000
60
6000
54
λ 1 (μ m)
9000
5000
Pb - 0.5 wt% Sb Pb - 1.5 wt% Sb
4000
10
Position from the casting surface (mm)
(5)
2
h (W/m K)
T0nþ1 ¼
5
48 42
-0,35
h = 6500 t
Experimental
36
3000
2
λ1 = - 0.19p + 6.42p +7.97 2
30
2000
R = 0.93
-0,30
h = 5500 t
24
1000 0
10
20
30
40
50
Time (s) Fig. 5. Evolution of the overall metal/coolant heat transfer coefficient with time for the PbeSb alloys cylindrical castings.
0
5
10
15
20
25
30
35
p: Position from the casting surface (mm) Fig. 7. Primary dendrite arm spacing as a function of position from the casting surfacedp [mm]: Pbe0.5 wt% Sb alloy. R2 is the coefficient of determination.
F. Bertelli et al. / Applied Thermal Engineering 61 (2013) 577e582
581
the aforementioned range of positions, and decreases after the reversal in the growth rate for both alloys examined, i.e. a tendency of increasing fineness of dendritic spacings towards the center of the cylinder can be expected accompanying the trend in the growth rate.
80 75 70 65 60
λ (μ m)
55
4. Conclusions
50 45 40 35 Experimental
30
λ = - 0.17p + 6.22.p - 3.1
25
R = 0.94
20 8
12
16
20
24
28
32
p: Position from the casting surface (mm) Fig. 8. Primary dendrite arm spacing as a function of position from the casting surfacedp [mm]: Pbe1.5 wt% Sb alloy. R2 is the coefficient of determination.
where V is the mold volume element (m3) and At is the heat transfer surface area, KM is the mold thermal conductivity (W m1 K1), hW is the mold/cooling fluid heat transfer coefficient (W m2 K1), hM/ 2 1 K ). M is the metal/mold heat transfer coefficient (W m The transient overall metal/coolant heat transfer coefficient, h, was determined by a technique based on the Inverse Heat Conduction Problem (IHCP), which has been detailed in a previous publication [23]. They are: h ¼ 6500t0.35 and h ¼ 5500t0.30 for the Pbe0.5 wt% Sb and Pbe1.5 wt% Sb alloys, respectively (t: time in s). It can be seen in Fig. 5 that the h profiles for Pbe0.5 wt% Sb and the Pbe1.5 wt% Sb alloy castings are quite similar, i.e. only a slight lower h profile is obtained for the Pbe1.5 wt% Sb alloy, which can be due to the influence of the higher solute concentration. The results of thermal analysis in metal have also been used to determine the tip growth rate [24] as a function of position from the casting surface with a view to permitting the reversal in the growth rate to be analyzed. The thermocouples readings have been used to create a plot of position (p) from the casting surface as a function of time (t) corresponding to the liquidus front passing by each thermocouple. The derivative of these curves with respect to time yielded the experimental tip growth rate. These results are shown in Fig. 6 compared with the corresponding numerical simulations, which assumed the corresponding thermophysical properties of each alloy reported elsewhere [23]. Applying an analytical model for the inward solidification of cylinders of pure metals (plane front solidification), Santos and Garcia [17] determined that the growth rate reverses at a specific point lying at about 0.4 of the radius of the cylinder, which is dependent on both Stefan (thermal properties of metal) and Biot (h and the thermal conductivity of the metal) numbers. The results of Fig. 6 show clearly that a characteristic reversal point cannot be observed for any alloy examined in the present study, but rather the growth rate tends to attain a stable value (an essentially constant growth rate) for a range of positions, i.e. 0.4e0.8 and 0.5e0.8 of the radius of the cylinder for the Pb 0.5Sb and Pb 1.5Sb alloys, respectively, and from this range onwards the growth rate accelerates rapidly. Figs. 7 and 8 show experimental values of primary dendritic arm spacings (l1) for the Pb 0.5Sb and Pb 1.5Sb alloys, respectively, fitted to trend functions. Comparing these results with the behavior of the tip growth rate in Fig. 6, one can conclude that this spacing increases initially as the growth rate decreases, becomes quite stable along
The results of growth rate have shown clearly that a characteristic reversal point, as reported previously for pure metals, could not be observed for the alloys examined, but rather the growth rate tends to attain an essentially stable value for a range of radial positions in the cylindrical casting, which depends on the alloy composition, and from this range onwards the growth rate accelerates rapidly towards the center of the casting. The experimental results indicated that the scale of the dendritic microstructure, despite exhibiting an inverse trend, accompanies the tendency of the profile of the growth rate. Acknowledgements The authors acknowledge the financial support provided by FAPESP (São Paulo Research Foundation e grant 2012/16328-2) and CNPq (The Brazilian Research Council). References [1] J. Yang, C.Y. Zhao, Solidification analysis of a single particle with encapsulated phase change materials, Appl. Therm. Eng. 51 (2013) 338e346. [2] N. Cheung, A. Garcia, The use of a heuristic search technique for the optimization of quality of steel billets produced by continuous casting, Eng. Appl. Artif. Intell. 14 (2001) 229e238. [3] F. Bertelli, E.S. Meza, P.R. Goulart, N. Cheung, R. Riva, A. Garcia, Laser remelting of Ale1.5 wt% Fe alloy surfaces: numerical and experimental analyses, Opt. Laser Eng. 49 (2011) 490e497. [4] G.S. Reddy, W.J. Mascarenhas, J.N. Reddy, Numerical simulation of solidification molten aluminum alloys in cylindrical molds, Metall. Trans. B 24 (1993) 677e684. [5] C.A. Santos, J.A. Spim, M.C.F. Ierardi, A. Garcia, The use of artificial intelligence technique for the optimisation of process parameters used in the continuous casting of steel, Appl. Math. Model 26 (2002) 1077e1092. [6] I. Nowak, J. Smolka, A.J. Nowak, An effective 3-D inverse procedure to retrieve cooling conditions in an aluminium alloy continuous casting problem, Appl. Therm. Eng. 30 (2010) 1140e1151. [7] M.W. Chen, X.F. Wang, F. Wang, G.B. Lin, Z.Z. Wang, The effect of interfacial kinetics on the morphological stability of a spherical particle, J. Cryst. Growth 362 (2013) 20e23. [8] Y.H. Zheng, Z.D. Wang, S.M. Zhang, Microstructure of diphase dendrite in Ale 35%La alloy during solidification, J. Cryst. Growth 362 (2013) 33e37. [9] M. Qian, J.N. DuPont, Microsegregation-related pitting corrosion characteristics of AL-6XN superaustenitic stainless steel laser welds, Corros. Sci. 52 (2010) 3548e3553. [10] W.R. Osório, D.M. Rosa, A. Garcia, Electrochemical behaviour of a PbeSb alloy in 0.5 M NaCl and 0.5 M H2SO4 solutions, Mater. Des. 34 (2012) 660e665. [11] W.R. Osório, L.R. Garcia, P.R. Goulart, A. Garcia, Effects of eutectic modification and T4 heat treatment on mechanical properties and corrosion resistance of an Ale9 wt%Si casting alloy, Mater. Chem. Phys. 106 (2007) 343e349. [12] W.R. Osório, D.M. Rosa, A. Garcia, The roles of cellular and dendritic microstructural morphologies on the corrosion resistance of PbeSb alloys for lead acid battery grids, J. Power Sources 175 (2008) 595e603. [13] L. Wang, B.P. Zhang, T. Shinohara, Corrosion behavior of AZ91 magnesium alloy in dilute NaCl solutions, Mater. Des. 31 (2010) 857e863. [14] L. Bilir, Z. Ilken, Total solidification time of a liquid phase change material enclosed in cylindrical/spherical containers, Appl. Therm. Eng. 25 (2005) 1488e1502. [15] M. Okayasu, S. Yoshie, Mechanical properties of AleSi13eNi1.4eMg1.4eCu1 alloys produced by the Ohno continuous casting process, Mat. Sci. Eng. AStruct 527 (2010) 3120e3126. [16] R. Rajaraman, R. Velraj, Comparison of interfacial heat transfer coefficient estimated by two different techniques during solidification of cylindrical aluminum alloy casting, Heat Mass Transfer 44 (2008) 1025e1034. [17] R.G. Santos, A. Garcia, Analytical technique for the determination of solidification rates during the inward freezing of cylinders, J. Mater. Sci. 18 (1983) 3578e3590. [18] M. Gündüz, E. Çadirli, Directional solidification of aluminiumecopper alloys, Mat. Sci. Eng. A-Struct 327 (2002) 167e185.
582
F. Bertelli et al. / Applied Thermal Engineering 61 (2013) 577e582
Nomenclature
c: specific heat [J/kg K] c: effective specific heat [J/kg K] fs: solid fraction [%] h: transient overall metal/coolant heat transfer coefficient [W/m2 K] hW: mold/cooling fluid heat transfer coefficient [W/m2 K] hM/M: metal/mold heat transfer coefficient [W/m2 K] i: element position according to ‘r’ axis [dimensionless] K: thermal conductivity [W/m K] Keq: equivalent thermal conductivity [W/m K] KM: mold thermal conductivity [W/m K] L: latent heat of fusion [J/kg]; n: number of time increment [dimensionless] r: radial coordinate [m] R: inner radius of the water-cooled jacket [m] t: time [s] T: temperature [K] TL: alloy liquidus temperature [K] TS: alloy solidus temperature [K] TW: water temperature [K] V: finite difference volume element [m3]
A: area perpendicular to the heat flux [m2] At: heat transfer surface area [m2]
Greek symbols Dt: time interval [s] r: density [kg/m3]
[19] P. Incropera, D.P. Dewitt, Fundamentals of Heat and Mass Transfer, John Wiley & Sons, New York, 1990. [20] N. Cheung, N.S. Santos, J.M.V. Quaresma, G.S. Dulikravich, A. Garcia, Interfacial heat transfer coefficients and solidification of an aluminum alloy in a rotary continuous caster, Int. J. Heat Mass Transfer 52 (2009) 451e459. [21] D. O’Mahoney, D.J. Browne, Use of experiment and an inverse method to study interface heat transfer during solidification in the investment casting process, Exp. Therm. Fluid Sci. 22 (2000) 111e122. [22] I.L. Ferreira, J.E. Spinelli, J.C. Pires, A. Garcia, The effect of melt temperature profile on the transient metal/mold heat transfer coefficient during solidification, Mat. Sci. Eng. A-Struct 408 (2005) 317e325. [23] F. Bertelli, C. Brito, E.S. Meza, N. Cheung, A. Garcia, Inward and outward solidification of cylindrical castings: the role of the metal/mold heat transfer coefficient, Mater. Chem. Phys. 136 (2012) 545e554. [24] D.M. Rosa, J.E. Spinelli, I.L. Ferreira, A. Garcia, Cellular/dendritic transition and microstructure evolution during transient directional solidification of PbeSb alloys, Metall. Mater. Trans. A 39 (2008) 2161e2174.