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mold heat transfer coefficient during solidification
Materials Science and Engineering A 408 (2005) 317–325
The effect of melt temperature profile on the transient metal/mold heat transfer coefficient during solidification Ivaldo L. Ferreira a , Jos´e E. Spinelli a , Jos´e C. Pires b , Amauri Garcia a,∗ a
Department of Materials Engineering, State University of Campinas, UNICAMP, P.O. Box 6122, 13083-970 Campinas, SP, Brazil b Federal Center of Technological Education from Minas Gerais, 36700-000 Leopoldina, MG, Brazil Received in revised form 14 August 2005; accepted 14 August 2005
Abstract Modeling of casting solidification can provide a method for improving casting yields. An accurate casting solidification model might be used to predict microstructure and to control the process based on thermal and operational parameters, and for this, it is necessary the previous knowledge of the transient metal/mold heat transfer coefficient, hi . Most investigations concerning the overall heat transfer coefficient between metal and mold have applied numerical methods for the solution of the inverse heat conduction problem (IHCP). In general, such studies consider a constant initial melt temperature in order to reckon the time-dependent hi . In the present work, solidification experiments have been carried with alloys of two metallic systems, and experimentally obtained temperatures were used by a numerical technique in order to determine transient metal/mold heat transfer coefficients, hi . It is shown that hi profiles can be affected significantly by the initial melt temperature distribution. © 2005 Elsevier B.V. All rights reserved. Keywords: Metal/mold heat transfer coefficient; Solidification; Numerical modeling
1. Introduction The computer simulation of freezing patterns in castings has done much to broaden our understanding of casting and mold system design. The structural integrity of shaped castings is closely related to the time–temperature history during solidification, and the use of casting simulation could do much to increase this knowledge in the foundry industry. However, some uncertainties must be eradicated, particularly, heat transfer at the metal/mold interface. The accurate knowledge of interfacial heat transfer coefficients is necessary for accurate modeling of castings. The ability of heat to flow across the casting and through the interface from the casting to the mold directly affects the evolution of solidification and plays a notable role in determining the freezing conditions within the casting, mainly in foundry systems of high thermal diffusivity such as chill castings. Gravity or pressure die castings, continuous casting and squeeze castings are some of the processes where product soundness is more directly affected by heat transfer at the metal/mold interface.
∗
Corresponding author. Tel.: +55 19 37883320; fax: +55 19 32893722. E-mail address: [email protected] (A. Garcia).
0921-5093/$ – see front matter © 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.msea.2005.08.145
Several studies have attempted to quantify the transient interfacial metal/mold heat transfer coefficient, hi , highlighting the different factors affecting heat flow across such interface during solidification [1–6]. These factors include the thermophysical properties of the contacting materials, the casting geometry, the orientation of the casting–mold interface with respect to gravity (contact pressure), mold temperature, pouring temperature, the roughness of mold contacting surface, mold coatings, etc. [3,4,6–9]. Most of the methods of calculation of time-dependent hi existing in the literature are based on numerical techniques generally known as methods of solving the inverse heat conduction problem, IHCP [10–12]. The IHCP method is based on a complete mathematical description of the physics of the process, supplemented with experimentally obtained temperature measurements in metal and/or mold. The inverse problem is solved by adjusting parameters in the mathematical description to minimize the difference between the model-computed values and the experimental measurements. Generally metal/mold heat transfer coefficients are evaluated based on temperature readings during solidification in one or two positions within the casting, considering a constant initial melt temperature [1–4,13–15]. The purpose of this study is to investigate the influence of melt temperature distribution, recorded by
I.L. Ferreira et al. / Materials Science and Engineering A 408 (2005) 317–325
318
Table 1 Thermophysical properties and melt superheats used in the experimental analysis [4,20] Properties
Fig. 1. Schematic ingot initial melt temperature distribution (t = 0).
a bank of thermocouples distributed inside the casting, on the accuracy of time-varying interfacial heat transfer coefficients, hi . Casting experiments have been carried out with Al–Cu and Sn–Pb alloys and the experimental temperatures were compared with simulations furnished by a numerical model for the determination of transient hi profiles. An automatic search selected the best theoretical/experimental fit for each experiment. 2. Numerical modeling
Al–10 wt%Cu
Al–4.5 wt%Cu
Thermal conductivity [W m−1 K−1 ] 65.6 kS kL 32.8
185.7 85.6
193 89
Specific heat [J kg−1 K−1 ] 217 cS cL 253
1086 1028
1092 1059
Density [kg m−3 ] ρS ρL
7475 7181
2870 2633
2654 2488
Thermal diffusivity [m2 /s] 4.04 αS ×10−5 1.81 αL ×10−5
5.96 3.16
6.66 3.38
59214
377830
381773
232
660
660
183
548
548
224.9
633.5
648
255
653.5
664
1.38
3.2
3.2
0.0656
0.1028
0.1100
Latent heat of fusion, H [J kg−1 ] Fusion temperature, TF [◦ C] Solidus temperature (Eutectic), TSol [◦ C] Liquidus temperature, TLiq [◦ C] Initial melt temperature, Tpour [◦ C] Liquidus slope, mL [◦ C/wt%] Partition coefficient, k0
Sn–5 wt%Pb
2.1. Governing equations The numerical model used to simulate inverse segregation profiles is based on that previously proposed by Voller [16]. Modifications to this numerical approach have been incorporated to allow the use of different thermophysical properties for the liquid and solid phases, as well as the mushy zone (it can deal with temperature and concentration-dependent thermophysical
properties), to treat variable metal/mold interface heat transfer coefficient and to account for a space-dependent initial melt temperature profile. A time variable metal/mold interface heat transfer coefficient introduces a non-linearity condition at the z = 0 boundary. In addition, a variable space grid is used to assure the accuracy of simulation results without considerably raising
Fig. 2. Schematic representation of the experimental setup: (1) rotameter; (2) heat-extracting bottom; (3) thermocouples; (4) computer and data acquisition software; (5) data logger; (6) casting; (7) mold; (8) temperature controller; (9) electric heaters; (10) insulating ceramic shielding.
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Fig. 3. (a) Simulated and measured temperature responses for an Al–10 wt%Cu alloy casting at 5, 10, 15, 30 and 50 mm from metal/mold interface adopting a mean melt temperature. (b) Simulated and measured temperature responses for an Al–10 wt%Cu alloy casting at the same positions adopting a melt temperature profile. (c) Evolution of metal/mold interface heat transfer coefficient (hi ) as a function of time for an Al–10 wt%Cu alloy casting (polished mold).
the number of spatial nodes. Considering the previous exposed, the vertical unidirectional solidification of binary hypoeutectic alloys is our target problem. At time t < 0, the alloys are in the molten state at the nominal concentration C0 and with an initial temperature distribution T0 (z) = −az2 + bz + c, contained in the insulated mold defined by 0 < z < Zb according to Fig. 1. Solidification begins by cooling the molten metal at the chill (z = 0) until the temperature drops bellow the eutectic temperature TE . At times t > 0, three transient regions are formed: solid, solid + liquid (mushy zone) and liquid. To develop a numerical solution for the equations of the coupled thermal and solutal fields, the following assumptions were adopted: (i) The domain is one-dimensional, defined by 0 < z < Zb , where Zb is a point far removed from the chill.
(ii) The solid phase is stationary, i.e., once the solid has formed it has zero velocity. (iii) Due to the relatively rapid nature of heat and mass diffusion in the liquid, within a representative elemental averaging volume, the liquid concentration CL , the temperature T, the liquid density ρL and the liquid velocity uL are constant [17]. (iv) The partition coefficient k0 and liquidus slope mL are obtained from the ThermoCalc software.1 (v) Equilibrium conditions exist at the solid/liquid interface, i.e., the temperature and concentrations fulfill the 1 The ThermoCalc software [19] can be used to generate equilibrium diagrams and through ThermoCalc interface for Fortran or C++, it is possible to recall those data generated by the software in order to provide more accurate input values for model simulations.
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320
equations: T = TF − mL CL
(1)
and CS∗ = k0 CL
(2)
where sub-indices S and L refer to solid and liquid phases, respectively, TF is the fusion temperature of the pure solvent in [K] and CS∗ is the solid concentration at the interface; (v) The specific heats, cS and cL , thermal conductivities kS and kL , and the densities ρS and ρL , are constants within each phase, but discontinuous at the solid–liquid boundary. The latent heat of fusion is taken as the difference between phases enthalpies H = HL − HS . (vi) The metal/mold thermal resistance varies with time, and is incorporated in a global heat transfer coefficient defined as hi [18].
heat conduction problem (IHCP) [4]. This method makes a complete mathematical description of the physics of the process and is supported by temperature measurements at known locations inside the heat conducting body. The temperature files containing the experimentally monitored temperatures are used in a finite difference heat flow model to determine hi , as described in a previous article [4]. The process at each time step included the following: a suitable initial value of h is assumed and with this value, the temperature of each reference location in casting at the end of each time interval t is simulated by the numerical model. The correction in h at each interaction step is made by a value hi , and new temperatures are estimated [Test (hi + hi )] or [Test (hi − hi )]. With these values, sensitivity coefficients (φ) are calculated for each interaction given by: φ=
Test (hi + hi ) − Test (hi ) hi
(11)
Using the above assumptions, the mixture equations for binary solidification read:
The procedure determines the value of hi , which minimizes an objective function defined by:
• Energy
F (hi ) =
∂ρcT ∂g + ∇ · (ρL cL uT ) = ∇ · (k∇T ) − ρS H ∂t ∂T • Species
(3)
∂ρC + ∇ · (ρL uCL ) = 0 (4) ∂t • Mass ∂ρ + ∇ · (ρL u) = 0 (5) ∂t where g is the liquid volume fraction and u is the volume averaged fluid velocity defined as: u = guL
n
(Test − Texp )2 ,
(12)
i=1
where Test and Texp are the estimated and the experimentally measured temperatures at various thermocouples locations and times, and n is the iteration stage. The applied method is a simulation assisted one and has been used in recent publications for determining h for a number of solidification situations [3,7–9]. 3. Experimental procedure Solidification experiments were performed with Sn–5 wt%Pb and Al-4.5 and 10 wt%Cu alloys, so that the behavior of metal-
(6)
• Mixture density 1−g ρS dα + gρL ρ=
(7)
0
• Mixture solute density ρC =
1−g
ρS CS dα + gρL CL
(8)
0
where ρC is the volumetric specific heat, taken as volume fraction weighted averages. The boundary conditions at the domain are prescribed as: ∂T ∂CL u = 0, k = hi (T0 − T |z=0 ) and = 0, at z = 0 ∂z ∂z (9) T → Tp and C → C0 ,
at z = Zb ,
(10)
where Tp is the constant initial melt temperature or a melt temperature profile as a function of z. The method used to determine the transient metal/coolant heat transfer coefficient, h, is based on the solution of the inverse
Fig. 4. Temperature distribution status for t = 37.2 s obtained trough ANSYS solidification module considering: (a) hi = 9000t−0.039 and (b) hi = 10800t−0.075 .
I.L. Ferreira et al. / Materials Science and Engineering A 408 (2005) 317–325
lic systems with significantly distinguished thermal diffusivities can be compared. The thermophysical properties and melt superheats employed are summarized in Table 1. Fig. 2a shows the casting assembly used in the experiments. It can be seen that heat is directionally extracted only through a water-cooled bottom made of stainless steel. A data logger and data reading software running in a standard computer were responsible by the thermal data acquisition, collected at a rate of approximately, two measurements per second. Five thermocouples, sheathed in 1.6 mm o.d. stainless steel protection tubes, were used in each experiment. Thus, temperatures in the casting were monitored during solidification for five different positions with regard to the heat extracting surface. While type J thermocouples were placed within the cylindrical ingot of Sn–5 wt%Pb alloy, type K thermocouples were positioned into the Al–Cu ingots. All thermocouples were calibrated considering the melting points of aluminum (for Al–4.5 and10 wt%Cu alloys) and tin (for Sn–5 wt%Pb alloy), exhibiting fluctuations of about 1.0 and 0.4 ◦ C, respectively. The experimental profiles plotted are the averages of three thermocouples readings
321
at each location in casting. Results from repeated experiments have shown differences not greater than 4 ◦ C. A stainless steel split mold was used having an internal diameter of 50 mm, height 110 mm and a 3 mm wall thickness (Fig. 2b). The lateral inner mold surface was covered with a layer of insulating alumina to minimize radial heat losses. The bottom part of the mold was closed with a thin (3 mm) stainless steel sheet. The heat-extracting surface was polished with grinding paper, and the surface roughness was determined with a digital system, where the arithmetic mean of the roughness amplitude profile (Ra in m) was adopted to characterize the surface microgeometry. In all cases, the chill surface roughness was kept constant, with a mean value of about 0.10 m. In some experiments, the heat extracting steel sheet was coated with an alumina based mold wash in order to reduce the metal/mold heat transfer efficiency. The mold wash was applied to the internal surface of the steel sheet with a spray gun, with the coating film thickness standardized at about 100 m. Each alloy was melted in situ and the lateral electric heaters had their power controlled in order to permit a desired superheat
Fig. 5. (a) Simulated and measured temperature responses for an Al–4.5 wt%Cu alloy casting at 5, 10, 15, 30 and 50 mm from metal/mold interface adopting a mean melt temperature. (b) Simulated and measured temperature responses for an Al–4.5 wt%Pb alloy casting at the same positions employing a melt temperature profile. (c) Evolution of metal/mold interface heat transfer coefficient (hi ) as a function of time for the Al–4.5 wt%Pb alloy casting (polished mold).
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to be achieved. To start solidification, the electric heaters were disconnected and at the same time the controlled water flow was initiated. 4. Results and discussion Temperature files containing the experimentally monitored temperatures were coupled to the numerical solidification program for determining the transient metal/mold heat transfer coefficient hi . Thermophysical properties of each alloy and solidification parameters are used as input data for simulations. Fig. 3 shows the temperature data collected in the metal during the course of upward solidification of an Al–10 wt% Cu alloy casting in the vertical water-cooled apparatus, with the bottom heat extracting surface being polished. The experimental thermal responses corresponding to five different positions inside the casting were compared with the predictions furnished by the numerical solidification model. The best theoretical–experimental fit has provided appropriate transient hi profile for two different approaches: (i) a mean initial melt
temperature has been adopted (Fig. 3a) and (ii) a quadratic equation, based on experimental thermal readings, representing the initial melt temperature as a function of position in casting has been used (Fig. 3b). A comparison between hi profiles determined in each case is shown in Fig. 3c. It can be seen that a significant difference exists between the two curves, with the assumption of a constant melt temperature overestimating the metal/mold heat transfer coefficient. The two curves tend to approach each other with increasing time. In order to evaluate the real significance of hi overestimation, additional simulations were conducted using the software ANSYS 8.0 thermal and casting module. Such tool allows the analysis of temperature evolution in the casting trough the observation of temperature layers, which are based on a standard gray pattern. A complex geometry of an Al–10 wt%Cu alloy casting was simulated to evaluate both hi profiles previously determined. Fig. 4a and b show the temperature evolution at the casting cross section for t = 37.2 s considering hi = 9000t−0.039 and 10800t−0.075 , respectively. It can be noticed that solidification was not complete when the lower profile was adopted
Fig. 6. (a) Simulated and measured temperature responses for an Al–4.5 wt%Cu alloy casting at 5, 10, 15, 30 and 50 mm from metal/mold interface adopting a mean melt temperature. (b) Simulated and measured temperature responses for an Al–4.5 wt%Pb alloy casting at the same positions employing a melt temperature profile. (c) Evolution of metal/mold interface heat transfer coefficient (hi ) as a function of time for the Al–4.5 wt%Pb alloy casting (coated mold).
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as shown in Fig. 4a. On the other hand, the overestimated heat transfer coefficient profile provided a complete solidification, since black spots are absent of the casting cross section. There are many practical casting cases where cooling is inhibited by adding thermal barriers at the metal/mold interface, e.g., ceramic mold coatings, gas layers, etc. In the present experimental configuration hi is an overall heat transfer coefficient between the casting surface and the coolant fluid, which can be represented by: 1 1 eC eS 1 = + + + hi hm kC kS hw
(13)
where hm is the heat transfer coefficient between the casting surface and the mold coating, eC and eS the thicknesses of mold coating and bottom steel sheet, respectively, kC and kS the coating and steel thermal conductivities, respectively, and hw is the mold-coolant heat transfer coefficient. The two first components on the right hand side of Eq. (13) are generally the largest. For the previously examined experimental case, i.e., polished mold,
323
eC = 0 and only three components have to be considered in Eq. (13). It has also been reported that hi transient profiles increase with increasing mushy zone length [4]. For long freezing ranges, the interdendritic liquid can feed better the solidification contraction causing a continued presence of liquid at the interface, leading to higher values of hi . To examine the influence of both the increased thermal barrier represented by mold coatings and long solidification freezing ranges on the sensitivity to initial melt superheat variations, experiments with an Al–4.5 wt%Cu alloy on coated and polished molds have also been carried out. Fig. 5 shows the thermal responses compared with the simulations for Al–4.5 wt%Cu castings solidified against a polished metal/mold interface: (a) quadratic melt temperature and (b) constant melt temperature. A comparison between hi profiles determined in each case is shown in Fig. 5c. It can be seen that the resulting behavior is similar to that observed for the Al–10 wt%Cu, with the assumption of a constant melt temperature overestimating the metal/mold heat transfer coefficient. The
Fig. 7. (a) Simulated and measured temperature responses for a Sn–5 wt%Pb alloy casting at 5, 10, 15, 30 and 50 mm from metal/mold interface adopting a mean melt temperature. (b) Simulated and measured temperature responses for a Sn–5 wt%Pb alloy casting at the same positions employing a melt temperature profile. (c) Evolution of metal/mold interface heat transfer coefficient (hi ) as a function of time for the Sn–5 wt%Pb alloy casting (polished mold).
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two curves also tend to approach each other with increasing time. The resulting hi experimental profile is indeed higher than that observed for the Al–10 wt%Cu, confirming that the hi profile is also dependent on mushy zone length for Al–Cu alloys. Fig. 6 presents the series of results for the Al–4.5 wt%Cu casting solidified with the heat extracting steel surface being coated with an alumina based ceramic layer. It can be seen on Fig. 6c that the differences which are observed between hi values obtained considering either a quadratic or a constant melt profile are much less than those observed for the polished mold case (Fig. 5c) along the same time range, i.e., up to about 100 s. This seems to indicate that the sensitivity to initial melt superheat variations also depend on how large the metal/mold heat transfer coefficient is relative to the thermal conductivity of the materials, i.e., on Biot number. For both Al–Cu alloys, experimentally examined hi was shown to depend more readily on initial melt profile conditions under situations which are conducive to relatively high Biot numbers. In order to check the behavior of hi in a metallic system with lower thermal diffusivity, a Sn–5 wt% Pb alloy has been
submitted to a similar experimental analysis. Fig. 7 shows the thermal responses corresponding to the mold bottom surface being polished, compared with the simulations for each case. The resulting hi profiles are compared in Fig. 7c. It can be seen that the tendency is different, with the initial hi initial values being very similar up to about 25 s. After this point, the difference increases significantly with increasing time and again the assumption of a constant initial melt temperature overestimates the metal/mold heat transfer coefficient. Fig. 8 presents the series of results for the Sn–5 wt%Pb casting solidified with the heat extracting steel surface being coated. As can be seen in Fig. 8c, no significant differences in hi can be observed considering either a quadratic or a constant melt profile. As similarly verified for the Al–4.5 wt%Cu casting the sensitivity of hi on the initial melt profile is associated with a relatively high Biot number, being negligible when Biot is significantly reduced by the application of a mold coating. It appears that the observed different behavior of hi profiles for the two metallic systems experimentally examined are associated with their thermal properties, mainly with the significant
Fig. 8. (a) Simulated and measured temperature responses for a Sn–5 wt%Pb alloy casting at 5, 10, 15, 30 and 50 mm from metal/mold interface adopting a mean melt temperature. (b) Simulated and measured temperature responses for a Sn–5 wt%Pb alloy casting at the same positions employing a melt temperature profile. (c) Evolution of metal/mold interface heat transfer coefficient (hi ) as a function of time for the Sn–5 wt%Pb alloy casting (coated mold).
I.L. Ferreira et al. / Materials Science and Engineering A 408 (2005) 317–325
differences in thermal diffusivities, with the wetting of the mold surface by the melt (uncoated mold) and with Biot number for the casting. Accurate knowledge of interfacial heat transfer coefficients is necessary for modeling both casting microstructure and dimensions. However, the present results have shown that care should be exercised when modeling castings using hi profiles determined by an IHCP method based on experimental results of one or two thermocouples inside the casting, i.e., based on a mean value of the initial melt temperature instead of the actual thermal profile. 5. Conclusion Solidification experiments carried out with alloys of two metallic systems have shown that transient metal/mold heat transfer coefficients can be affected significantly by the initial melt temperature profile. The use of IHCP methods based merely on one or two thermocouples inside the casting, i.e., on a mean value for the initial melt temperature, may not be enough for an accurate characterization of the transient hi profile. The sensitivity to initial melt superheat variations was shown to depend on the alloy thermal properties, on the wetting of the mold surface by the melt (uncoated mold) and on how large the metal/mold heat transfer coefficient is relative to the thermal conductivity of the materials, i.e., on Biot number. Acknowledgments The authors acknowledge financial support provided by The Scientific Research Foundation of the State of S˜ao Paulo, Brazil (FAPESP) and The Brazilian Research Council (CNPq).
325
References [1] K. Ho, R.D. Pehlke, Metall. Trans. B. 16 (1998) 585–594. [2] M. Krishnan, D.G.R. Sarma, Int. Comm. Heat Mass Transfer 23 (1996) 203–214. [3] C.A. Santos, C.A. Siqueira, A. Garcia, J.M.V. Quaresma, J.A. Spim, Inv. Probl. Sci. Eng. 12 (2004) 279–296. [4] C.A. Santos, J.M.V. Quaresma, A. Garcia, J. Alloys Compd. 319 (2001) 174–186. [5] M.A. Martorano, J.D.T. Capocchi, Int. J. Heat Mass Transfer 43 (2000) 2541–2552. [6] W.D. Griffiths, Metall. Mater. Trans. B 31 (2000) 285–295. [7] J.E. Spinelli, I.L. Ferreira, A. Garcia, J. Alloys Compd. 384 (2004) 217–226. [8] J.E. Spinelli, D.M. Rosa, I.L. Ferreira, A. Garcia, Mater. Sci. Eng. A 383 (2004) 271–282. [9] E.N. Souza, N. Cheung, C.A. Santos, A. Garcia, Mater. Sci. Eng. A 397 (2005) 239–248. [10] J.V. Beck, Int. J. Heat Mass Transfer 13 (1970) 703–716. [11] T. Loulou, E.A. Artyukhin, J.P. Bardon, Int. J. Heat Mass Transfer 42 (1999) 2119–2127. [12] T.S. Piwonka, K.A. Woodbury, J.M. Wiest, Mater. Des. 21 (2000) 365–372. [13] T.G. Kim, Z.H. Lee, Int. J. Heat Mass Transfer 40 (1997) 3513– 3525. [14] M.A. Taha, N.A. El-Mahallawy, A.W.M. Assar, R.M. Hammouda, J. Mater. Sci. 27 (1992) 3467–3473. [15] C.A. Muojekwu, I.V. Samarasekera, J.K. Brimacombe, Metall. Mater. Trans. B 26 (1995) 361–369. [16] V.R. Voller, Can. Metall. Q. 37 (1998) 169–177. [17] J. Ni, C. Beckermann, Metall. Trans. A 27 (1991) 349–361. [18] I.L. Ferreira, C.A. Santos, V.R. Voller, A. Garcia, Metall. Mater. Trans. B 35 (2004) 285–297. [19] B. Sundman, Q. Chen, STT Foundation, Stockholm, Sweden, 1995. [20] O.L. Rocha, C.A. Siqueira, A. Garcia, Mater. Sci. Eng. A 347 (2003) 59–69.
The effect of melt temperature profile on the transient metal/mold heat transfer coefficient during solidification Ivaldo L. Ferreira a , Jos´e E. Spinelli a , Jos´e C. Pires b , Amauri Garcia a,∗ a
Department of Materials Engineering, State University of Campinas, UNICAMP, P.O. Box 6122, 13083-970 Campinas, SP, Brazil b Federal Center of Technological Education from Minas Gerais, 36700-000 Leopoldina, MG, Brazil Received in revised form 14 August 2005; accepted 14 August 2005
Abstract Modeling of casting solidification can provide a method for improving casting yields. An accurate casting solidification model might be used to predict microstructure and to control the process based on thermal and operational parameters, and for this, it is necessary the previous knowledge of the transient metal/mold heat transfer coefficient, hi . Most investigations concerning the overall heat transfer coefficient between metal and mold have applied numerical methods for the solution of the inverse heat conduction problem (IHCP). In general, such studies consider a constant initial melt temperature in order to reckon the time-dependent hi . In the present work, solidification experiments have been carried with alloys of two metallic systems, and experimentally obtained temperatures were used by a numerical technique in order to determine transient metal/mold heat transfer coefficients, hi . It is shown that hi profiles can be affected significantly by the initial melt temperature distribution. © 2005 Elsevier B.V. All rights reserved. Keywords: Metal/mold heat transfer coefficient; Solidification; Numerical modeling
1. Introduction The computer simulation of freezing patterns in castings has done much to broaden our understanding of casting and mold system design. The structural integrity of shaped castings is closely related to the time–temperature history during solidification, and the use of casting simulation could do much to increase this knowledge in the foundry industry. However, some uncertainties must be eradicated, particularly, heat transfer at the metal/mold interface. The accurate knowledge of interfacial heat transfer coefficients is necessary for accurate modeling of castings. The ability of heat to flow across the casting and through the interface from the casting to the mold directly affects the evolution of solidification and plays a notable role in determining the freezing conditions within the casting, mainly in foundry systems of high thermal diffusivity such as chill castings. Gravity or pressure die castings, continuous casting and squeeze castings are some of the processes where product soundness is more directly affected by heat transfer at the metal/mold interface.
∗
Corresponding author. Tel.: +55 19 37883320; fax: +55 19 32893722. E-mail address: [email protected] (A. Garcia).
0921-5093/$ – see front matter © 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.msea.2005.08.145
Several studies have attempted to quantify the transient interfacial metal/mold heat transfer coefficient, hi , highlighting the different factors affecting heat flow across such interface during solidification [1–6]. These factors include the thermophysical properties of the contacting materials, the casting geometry, the orientation of the casting–mold interface with respect to gravity (contact pressure), mold temperature, pouring temperature, the roughness of mold contacting surface, mold coatings, etc. [3,4,6–9]. Most of the methods of calculation of time-dependent hi existing in the literature are based on numerical techniques generally known as methods of solving the inverse heat conduction problem, IHCP [10–12]. The IHCP method is based on a complete mathematical description of the physics of the process, supplemented with experimentally obtained temperature measurements in metal and/or mold. The inverse problem is solved by adjusting parameters in the mathematical description to minimize the difference between the model-computed values and the experimental measurements. Generally metal/mold heat transfer coefficients are evaluated based on temperature readings during solidification in one or two positions within the casting, considering a constant initial melt temperature [1–4,13–15]. The purpose of this study is to investigate the influence of melt temperature distribution, recorded by
I.L. Ferreira et al. / Materials Science and Engineering A 408 (2005) 317–325
318
Table 1 Thermophysical properties and melt superheats used in the experimental analysis [4,20] Properties
Fig. 1. Schematic ingot initial melt temperature distribution (t = 0).
a bank of thermocouples distributed inside the casting, on the accuracy of time-varying interfacial heat transfer coefficients, hi . Casting experiments have been carried out with Al–Cu and Sn–Pb alloys and the experimental temperatures were compared with simulations furnished by a numerical model for the determination of transient hi profiles. An automatic search selected the best theoretical/experimental fit for each experiment. 2. Numerical modeling
Al–10 wt%Cu
Al–4.5 wt%Cu
Thermal conductivity [W m−1 K−1 ] 65.6 kS kL 32.8
185.7 85.6
193 89
Specific heat [J kg−1 K−1 ] 217 cS cL 253
1086 1028
1092 1059
Density [kg m−3 ] ρS ρL
7475 7181
2870 2633
2654 2488
Thermal diffusivity [m2 /s] 4.04 αS ×10−5 1.81 αL ×10−5
5.96 3.16
6.66 3.38
59214
377830
381773
232
660
660
183
548
548
224.9
633.5
648
255
653.5
664
1.38
3.2
3.2
0.0656
0.1028
0.1100
Latent heat of fusion, H [J kg−1 ] Fusion temperature, TF [◦ C] Solidus temperature (Eutectic), TSol [◦ C] Liquidus temperature, TLiq [◦ C] Initial melt temperature, Tpour [◦ C] Liquidus slope, mL [◦ C/wt%] Partition coefficient, k0
Sn–5 wt%Pb
2.1. Governing equations The numerical model used to simulate inverse segregation profiles is based on that previously proposed by Voller [16]. Modifications to this numerical approach have been incorporated to allow the use of different thermophysical properties for the liquid and solid phases, as well as the mushy zone (it can deal with temperature and concentration-dependent thermophysical
properties), to treat variable metal/mold interface heat transfer coefficient and to account for a space-dependent initial melt temperature profile. A time variable metal/mold interface heat transfer coefficient introduces a non-linearity condition at the z = 0 boundary. In addition, a variable space grid is used to assure the accuracy of simulation results without considerably raising
Fig. 2. Schematic representation of the experimental setup: (1) rotameter; (2) heat-extracting bottom; (3) thermocouples; (4) computer and data acquisition software; (5) data logger; (6) casting; (7) mold; (8) temperature controller; (9) electric heaters; (10) insulating ceramic shielding.
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Fig. 3. (a) Simulated and measured temperature responses for an Al–10 wt%Cu alloy casting at 5, 10, 15, 30 and 50 mm from metal/mold interface adopting a mean melt temperature. (b) Simulated and measured temperature responses for an Al–10 wt%Cu alloy casting at the same positions adopting a melt temperature profile. (c) Evolution of metal/mold interface heat transfer coefficient (hi ) as a function of time for an Al–10 wt%Cu alloy casting (polished mold).
the number of spatial nodes. Considering the previous exposed, the vertical unidirectional solidification of binary hypoeutectic alloys is our target problem. At time t < 0, the alloys are in the molten state at the nominal concentration C0 and with an initial temperature distribution T0 (z) = −az2 + bz + c, contained in the insulated mold defined by 0 < z < Zb according to Fig. 1. Solidification begins by cooling the molten metal at the chill (z = 0) until the temperature drops bellow the eutectic temperature TE . At times t > 0, three transient regions are formed: solid, solid + liquid (mushy zone) and liquid. To develop a numerical solution for the equations of the coupled thermal and solutal fields, the following assumptions were adopted: (i) The domain is one-dimensional, defined by 0 < z < Zb , where Zb is a point far removed from the chill.
(ii) The solid phase is stationary, i.e., once the solid has formed it has zero velocity. (iii) Due to the relatively rapid nature of heat and mass diffusion in the liquid, within a representative elemental averaging volume, the liquid concentration CL , the temperature T, the liquid density ρL and the liquid velocity uL are constant [17]. (iv) The partition coefficient k0 and liquidus slope mL are obtained from the ThermoCalc software.1 (v) Equilibrium conditions exist at the solid/liquid interface, i.e., the temperature and concentrations fulfill the 1 The ThermoCalc software [19] can be used to generate equilibrium diagrams and through ThermoCalc interface for Fortran or C++, it is possible to recall those data generated by the software in order to provide more accurate input values for model simulations.
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equations: T = TF − mL CL
(1)
and CS∗ = k0 CL
(2)
where sub-indices S and L refer to solid and liquid phases, respectively, TF is the fusion temperature of the pure solvent in [K] and CS∗ is the solid concentration at the interface; (v) The specific heats, cS and cL , thermal conductivities kS and kL , and the densities ρS and ρL , are constants within each phase, but discontinuous at the solid–liquid boundary. The latent heat of fusion is taken as the difference between phases enthalpies H = HL − HS . (vi) The metal/mold thermal resistance varies with time, and is incorporated in a global heat transfer coefficient defined as hi [18].
heat conduction problem (IHCP) [4]. This method makes a complete mathematical description of the physics of the process and is supported by temperature measurements at known locations inside the heat conducting body. The temperature files containing the experimentally monitored temperatures are used in a finite difference heat flow model to determine hi , as described in a previous article [4]. The process at each time step included the following: a suitable initial value of h is assumed and with this value, the temperature of each reference location in casting at the end of each time interval t is simulated by the numerical model. The correction in h at each interaction step is made by a value hi , and new temperatures are estimated [Test (hi + hi )] or [Test (hi − hi )]. With these values, sensitivity coefficients (φ) are calculated for each interaction given by: φ=
Test (hi + hi ) − Test (hi ) hi
(11)
Using the above assumptions, the mixture equations for binary solidification read:
The procedure determines the value of hi , which minimizes an objective function defined by:
• Energy
F (hi ) =
∂ρcT ∂g + ∇ · (ρL cL uT ) = ∇ · (k∇T ) − ρS H ∂t ∂T • Species
(3)
∂ρC + ∇ · (ρL uCL ) = 0 (4) ∂t • Mass ∂ρ + ∇ · (ρL u) = 0 (5) ∂t where g is the liquid volume fraction and u is the volume averaged fluid velocity defined as: u = guL
n
(Test − Texp )2 ,
(12)
i=1
where Test and Texp are the estimated and the experimentally measured temperatures at various thermocouples locations and times, and n is the iteration stage. The applied method is a simulation assisted one and has been used in recent publications for determining h for a number of solidification situations [3,7–9]. 3. Experimental procedure Solidification experiments were performed with Sn–5 wt%Pb and Al-4.5 and 10 wt%Cu alloys, so that the behavior of metal-
(6)
• Mixture density 1−g ρS dα + gρL ρ=
(7)
0
• Mixture solute density ρC =
1−g
ρS CS dα + gρL CL
(8)
0
where ρC is the volumetric specific heat, taken as volume fraction weighted averages. The boundary conditions at the domain are prescribed as: ∂T ∂CL u = 0, k = hi (T0 − T |z=0 ) and = 0, at z = 0 ∂z ∂z (9) T → Tp and C → C0 ,
at z = Zb ,
(10)
where Tp is the constant initial melt temperature or a melt temperature profile as a function of z. The method used to determine the transient metal/coolant heat transfer coefficient, h, is based on the solution of the inverse
Fig. 4. Temperature distribution status for t = 37.2 s obtained trough ANSYS solidification module considering: (a) hi = 9000t−0.039 and (b) hi = 10800t−0.075 .
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lic systems with significantly distinguished thermal diffusivities can be compared. The thermophysical properties and melt superheats employed are summarized in Table 1. Fig. 2a shows the casting assembly used in the experiments. It can be seen that heat is directionally extracted only through a water-cooled bottom made of stainless steel. A data logger and data reading software running in a standard computer were responsible by the thermal data acquisition, collected at a rate of approximately, two measurements per second. Five thermocouples, sheathed in 1.6 mm o.d. stainless steel protection tubes, were used in each experiment. Thus, temperatures in the casting were monitored during solidification for five different positions with regard to the heat extracting surface. While type J thermocouples were placed within the cylindrical ingot of Sn–5 wt%Pb alloy, type K thermocouples were positioned into the Al–Cu ingots. All thermocouples were calibrated considering the melting points of aluminum (for Al–4.5 and10 wt%Cu alloys) and tin (for Sn–5 wt%Pb alloy), exhibiting fluctuations of about 1.0 and 0.4 ◦ C, respectively. The experimental profiles plotted are the averages of three thermocouples readings
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at each location in casting. Results from repeated experiments have shown differences not greater than 4 ◦ C. A stainless steel split mold was used having an internal diameter of 50 mm, height 110 mm and a 3 mm wall thickness (Fig. 2b). The lateral inner mold surface was covered with a layer of insulating alumina to minimize radial heat losses. The bottom part of the mold was closed with a thin (3 mm) stainless steel sheet. The heat-extracting surface was polished with grinding paper, and the surface roughness was determined with a digital system, where the arithmetic mean of the roughness amplitude profile (Ra in m) was adopted to characterize the surface microgeometry. In all cases, the chill surface roughness was kept constant, with a mean value of about 0.10 m. In some experiments, the heat extracting steel sheet was coated with an alumina based mold wash in order to reduce the metal/mold heat transfer efficiency. The mold wash was applied to the internal surface of the steel sheet with a spray gun, with the coating film thickness standardized at about 100 m. Each alloy was melted in situ and the lateral electric heaters had their power controlled in order to permit a desired superheat
Fig. 5. (a) Simulated and measured temperature responses for an Al–4.5 wt%Cu alloy casting at 5, 10, 15, 30 and 50 mm from metal/mold interface adopting a mean melt temperature. (b) Simulated and measured temperature responses for an Al–4.5 wt%Pb alloy casting at the same positions employing a melt temperature profile. (c) Evolution of metal/mold interface heat transfer coefficient (hi ) as a function of time for the Al–4.5 wt%Pb alloy casting (polished mold).
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to be achieved. To start solidification, the electric heaters were disconnected and at the same time the controlled water flow was initiated. 4. Results and discussion Temperature files containing the experimentally monitored temperatures were coupled to the numerical solidification program for determining the transient metal/mold heat transfer coefficient hi . Thermophysical properties of each alloy and solidification parameters are used as input data for simulations. Fig. 3 shows the temperature data collected in the metal during the course of upward solidification of an Al–10 wt% Cu alloy casting in the vertical water-cooled apparatus, with the bottom heat extracting surface being polished. The experimental thermal responses corresponding to five different positions inside the casting were compared with the predictions furnished by the numerical solidification model. The best theoretical–experimental fit has provided appropriate transient hi profile for two different approaches: (i) a mean initial melt
temperature has been adopted (Fig. 3a) and (ii) a quadratic equation, based on experimental thermal readings, representing the initial melt temperature as a function of position in casting has been used (Fig. 3b). A comparison between hi profiles determined in each case is shown in Fig. 3c. It can be seen that a significant difference exists between the two curves, with the assumption of a constant melt temperature overestimating the metal/mold heat transfer coefficient. The two curves tend to approach each other with increasing time. In order to evaluate the real significance of hi overestimation, additional simulations were conducted using the software ANSYS 8.0 thermal and casting module. Such tool allows the analysis of temperature evolution in the casting trough the observation of temperature layers, which are based on a standard gray pattern. A complex geometry of an Al–10 wt%Cu alloy casting was simulated to evaluate both hi profiles previously determined. Fig. 4a and b show the temperature evolution at the casting cross section for t = 37.2 s considering hi = 9000t−0.039 and 10800t−0.075 , respectively. It can be noticed that solidification was not complete when the lower profile was adopted
Fig. 6. (a) Simulated and measured temperature responses for an Al–4.5 wt%Cu alloy casting at 5, 10, 15, 30 and 50 mm from metal/mold interface adopting a mean melt temperature. (b) Simulated and measured temperature responses for an Al–4.5 wt%Pb alloy casting at the same positions employing a melt temperature profile. (c) Evolution of metal/mold interface heat transfer coefficient (hi ) as a function of time for the Al–4.5 wt%Pb alloy casting (coated mold).
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as shown in Fig. 4a. On the other hand, the overestimated heat transfer coefficient profile provided a complete solidification, since black spots are absent of the casting cross section. There are many practical casting cases where cooling is inhibited by adding thermal barriers at the metal/mold interface, e.g., ceramic mold coatings, gas layers, etc. In the present experimental configuration hi is an overall heat transfer coefficient between the casting surface and the coolant fluid, which can be represented by: 1 1 eC eS 1 = + + + hi hm kC kS hw
(13)
where hm is the heat transfer coefficient between the casting surface and the mold coating, eC and eS the thicknesses of mold coating and bottom steel sheet, respectively, kC and kS the coating and steel thermal conductivities, respectively, and hw is the mold-coolant heat transfer coefficient. The two first components on the right hand side of Eq. (13) are generally the largest. For the previously examined experimental case, i.e., polished mold,
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eC = 0 and only three components have to be considered in Eq. (13). It has also been reported that hi transient profiles increase with increasing mushy zone length [4]. For long freezing ranges, the interdendritic liquid can feed better the solidification contraction causing a continued presence of liquid at the interface, leading to higher values of hi . To examine the influence of both the increased thermal barrier represented by mold coatings and long solidification freezing ranges on the sensitivity to initial melt superheat variations, experiments with an Al–4.5 wt%Cu alloy on coated and polished molds have also been carried out. Fig. 5 shows the thermal responses compared with the simulations for Al–4.5 wt%Cu castings solidified against a polished metal/mold interface: (a) quadratic melt temperature and (b) constant melt temperature. A comparison between hi profiles determined in each case is shown in Fig. 5c. It can be seen that the resulting behavior is similar to that observed for the Al–10 wt%Cu, with the assumption of a constant melt temperature overestimating the metal/mold heat transfer coefficient. The
Fig. 7. (a) Simulated and measured temperature responses for a Sn–5 wt%Pb alloy casting at 5, 10, 15, 30 and 50 mm from metal/mold interface adopting a mean melt temperature. (b) Simulated and measured temperature responses for a Sn–5 wt%Pb alloy casting at the same positions employing a melt temperature profile. (c) Evolution of metal/mold interface heat transfer coefficient (hi ) as a function of time for the Sn–5 wt%Pb alloy casting (polished mold).
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two curves also tend to approach each other with increasing time. The resulting hi experimental profile is indeed higher than that observed for the Al–10 wt%Cu, confirming that the hi profile is also dependent on mushy zone length for Al–Cu alloys. Fig. 6 presents the series of results for the Al–4.5 wt%Cu casting solidified with the heat extracting steel surface being coated with an alumina based ceramic layer. It can be seen on Fig. 6c that the differences which are observed between hi values obtained considering either a quadratic or a constant melt profile are much less than those observed for the polished mold case (Fig. 5c) along the same time range, i.e., up to about 100 s. This seems to indicate that the sensitivity to initial melt superheat variations also depend on how large the metal/mold heat transfer coefficient is relative to the thermal conductivity of the materials, i.e., on Biot number. For both Al–Cu alloys, experimentally examined hi was shown to depend more readily on initial melt profile conditions under situations which are conducive to relatively high Biot numbers. In order to check the behavior of hi in a metallic system with lower thermal diffusivity, a Sn–5 wt% Pb alloy has been
submitted to a similar experimental analysis. Fig. 7 shows the thermal responses corresponding to the mold bottom surface being polished, compared with the simulations for each case. The resulting hi profiles are compared in Fig. 7c. It can be seen that the tendency is different, with the initial hi initial values being very similar up to about 25 s. After this point, the difference increases significantly with increasing time and again the assumption of a constant initial melt temperature overestimates the metal/mold heat transfer coefficient. Fig. 8 presents the series of results for the Sn–5 wt%Pb casting solidified with the heat extracting steel surface being coated. As can be seen in Fig. 8c, no significant differences in hi can be observed considering either a quadratic or a constant melt profile. As similarly verified for the Al–4.5 wt%Cu casting the sensitivity of hi on the initial melt profile is associated with a relatively high Biot number, being negligible when Biot is significantly reduced by the application of a mold coating. It appears that the observed different behavior of hi profiles for the two metallic systems experimentally examined are associated with their thermal properties, mainly with the significant
Fig. 8. (a) Simulated and measured temperature responses for a Sn–5 wt%Pb alloy casting at 5, 10, 15, 30 and 50 mm from metal/mold interface adopting a mean melt temperature. (b) Simulated and measured temperature responses for a Sn–5 wt%Pb alloy casting at the same positions employing a melt temperature profile. (c) Evolution of metal/mold interface heat transfer coefficient (hi ) as a function of time for the Sn–5 wt%Pb alloy casting (coated mold).
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differences in thermal diffusivities, with the wetting of the mold surface by the melt (uncoated mold) and with Biot number for the casting. Accurate knowledge of interfacial heat transfer coefficients is necessary for modeling both casting microstructure and dimensions. However, the present results have shown that care should be exercised when modeling castings using hi profiles determined by an IHCP method based on experimental results of one or two thermocouples inside the casting, i.e., based on a mean value of the initial melt temperature instead of the actual thermal profile. 5. Conclusion Solidification experiments carried out with alloys of two metallic systems have shown that transient metal/mold heat transfer coefficients can be affected significantly by the initial melt temperature profile. The use of IHCP methods based merely on one or two thermocouples inside the casting, i.e., on a mean value for the initial melt temperature, may not be enough for an accurate characterization of the transient hi profile. The sensitivity to initial melt superheat variations was shown to depend on the alloy thermal properties, on the wetting of the mold surface by the melt (uncoated mold) and on how large the metal/mold heat transfer coefficient is relative to the thermal conductivity of the materials, i.e., on Biot number. Acknowledgments The authors acknowledge financial support provided by The Scientific Research Foundation of the State of S˜ao Paulo, Brazil (FAPESP) and The Brazilian Research Council (CNPq).
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