determination of transient interfacial heat transfer coefficients in chill mold castings

LJournal of Alloys and Compounds 319 (2001) 174–186www.elsevier.com/ locate / jallcom

Determination of transient interfacial heat transfer coefficients in chillmold castings

*C.A. Santos, J.M.V. Quaresma, A. GarciaDepartment of Materials Engineering, State University of Campinas -UNICAMP, PO Box 6122, 13083-970 Campinas, SP, Brazil

Received 29 February 2000; accepted 22 December 2000

Abstract

The present work focuses on the determination of transient mold–environment and metal–mold heat transfer coefficients duringsolidification. The method uses the expedient of comparing theoretical and experimental thermal profiles and can be applied both to puremetals and metallic alloys. A solidification model based on the finite difference technique has been used to provide the theoretical results.The experiments were carried out by positioning the thermocouples in both metal and mold. The comparison between experimental andtheoretical results is made by an automatic search of the best fitting among theoretical and experimental cooling curves simultaneously inmetal and in mold. This has permitted the evaluation of the variation of heat transfer coefficients along the solidification process inunsteady state unidirectional heat flow of Al–Cu and Sn–Pb alloys, as well as the analysis of the effects of the material and the thicknessof the mold and melt superheat. 2001 Elsevier Science B.V. All rights reserved.

Keywords: Interfacial heat transfer coefficients; Mathematical modeling; Chill mold casting; Al–Cu; Sn–Pb

1. Introduction where product quality is more directly affected by theinterfacial heat transfer conditions. Once information in

The structural integrity of shaped castings is closely this area is accurate, foundrymen can effectively optimizerelated to their temperature–time evolution during solidifi- the design of their chilling systems to produce soundcation. A number of analytical and numerical models were castings.developed in the last 2 decades to treat heat transfer during When metal and mold surfaces are brought into contactsolidification, and the consequent simulation of freezing an imperfect junction is formed. While uniform tempera-patterns in castings has provided many improvements in tures gradients can exist in both metal and mold, thefoundry processes. The use of casting solidification simula- junction between the two surfaces creates a temperaturetion could do much to increase knowledge of the process, drop, which is dependent upon the thermophysical prop-however, some uncertainties must be eliminated before erties of the contacting materials, the casting and moldsuch simulations can be widely accepted as realistic geometry, the roughness of mold contacting surface, thedescriptions of the process. The heat transfer at the metal– presence of gaseous and non-gaseous interstitial media, themold interface is one of these uncertainties, and reliable melt superheat, contact pressure and initial temperature ofexperimental values of heat transfer coefficients are re- the mold.quired for various metal–mold combinations and super- Fig. 1a shows a schematic representation of the twoheats, as existing data is sparse. The way the heat flows contacting surfaces. Because the two surfaces in contactacross the metal and mold surfaces directly affects the are not perfectly flat, when the interfacial contact pressureevolution of solidification, and plays a notable role in is reasonably high, most of the energy passes through adetermining the freezing conditions within the metal, limited number of actual contact spots [1,2]. The heat flowmainly in foundry systems of high thermal diffusivity like across the casting–mold interface can be characterized bychill castings. Gravity or pressure die casting, continuous a macroscopic average metal–mold interfacial heat transfercasting, and squeeze casting are some of the processes coefficient (h ), given byi

q*Corresponding author. ]]]]h 5 (1)i A(T 2 T )E-mail address: [email protected] (A. Garcia). IC IM

0925-8388/01/$ – see front matter 2001 Elsevier Science B.V. All rights reserved.PI I : S0925-8388( 01 )00904-5

C.A. Santos et al. / Journal of Alloys and Compounds 319 (2001) 174 –186 175

conductivity of the interfacial gas with declining tempera-ture.

By using measured temperatures in both casting andmold, together with numerical [4–10] or analytical [11,12]solutions of the solidification problem, many researchworkers have attempted to quantify metal–mold interfacialheat transfer in terms either of a heat transfer coefficient orheat flux. In most cases, the numerical techniques general-ly known as the method of solving the inverse heatconduction problem were used to quantify the time depen-dent heat transfer coefficient at the interface. Growth dataobtained from the dendritic microstructure together with anumerical solution have also been used to determinemetal–mold transient heat transfer coefficients [13].

In the present study, the heat flow between the castingand the chill is characterized by the interfacial heat transfercoefficient, h . The variation of h during solidification ofi i

Al–Cu and Sn–Pb alloys, as well as pure aluminum andpure tin, against a vertical mold wall is investigatedexperimentally. The effects of mold material (low carbonsteel and copper) and its thickness and melt superheat arealso investigated. Experimental temperatures in the moldand the metal during solidification are compared withsimulations furnished by a numerical model, and anautomatic search selects the best theoretical–experimentalfitting from a range of values of h . In any case examined,i

expressions are derived representing variation of metal–Fig. 1. Heat flow across mold–casting interface. mold and mold–environment heat transfer coefficients with

time.

where q (W) is the average heat flux across the interfaceand T and T are, respectively casting and mold surface 2. Determination of interfacial heat transferIC IM

temperatures (K). coefficientsThe quantification of heat flux in terms of a heat transfer

coefficient, as indicated in Fig. 1b for the idealized 2.1. Metal–mold heat transfer coefficient — h i

temperature profile, requires that the heat capacity is zeroso that the thermal diffusivity is infinite, and consequently The heat flow across the casting–mold interface can beheat fluxes entering and leaving the interface are equal. characterized by Eq. (1) and h can be determined pro-i

The heat transfer coefficient shows a high value in the vided that all the other terms of the equation, namely q,initial stage of solidification, the result of the good surface T and T , are known. However, these temperatures areIC IM

conformity between the liquid core and the solidified shell. difficult to measure because the accurate location ofAs solidification progresses the mold expands due to the thermocouples of finite mass at the interface is not an easyabsorption of heat and the solid metal shrinks during task, and they can distort the temperature field at thecooling and as a result a gap develops because pressure interface. To overcome this experimental impediment, thebecomes insufficient to maintain a conforming contact at methods of calculation of h existing in the literature arei

the interface. Once the air gaps forms, the heat transfer based on a knowledge of other conditions, such asacross the interface decreases rapidly and a relatively temperature histories at interior points of the casting orconstant value of h is attained. The mode of heat transfer mold, together with mathematical models of heat flowi

across the metal–mold interface has been suggested to be during solidification. Among these methods, those baseddue to both conduction through isolated metal–mold on the solution of the inverse heat conduction problemcontacts and through gases present in the gap and radiation have been widely used in the quantification of the transientbetween the surfaces. During the subsequent stage of interfacial heat transfer. Since solidification of a castingsolidification a slight drop in the interfacial heat transfer involves both a change of phase and temperature variablecoefficient with time can be observed. It is postulated that thermal properties, the inverse heat conduction becomesthis is caused by the growth of oxide films on chill and nonlinear. The nonlinear estimation technique was used bymold surfaces [3], and by a reduction in the thermal Beck for the numerical solution of this class of problem

176 C.A. Santos et al. / Journal of Alloys and Compounds 319 (2001) 174 –186

[14]. It has advantage over the other numerical procedures, T (h 1 Dh ) 2 T (h )est i i est i]]]]]]]in that Beck studied the problem from the standpoint of f 5 (3)

Dh ieffective treatment of experimental data, taking into ac-count inaccuracies concerning the locations of thermocou- The sequence of the solution involves the calculation ofples, statistical errors in temperature measurement and the sensitivity coefficients for measured temperatures. Theuncertainty in material properties. assumed value of h is corrected using the relationi

In the present work, a similar procedure determines theh (new) 5 h (old)6Dh (4)value of h which minimizes an objective function defined i i ii

by the equationThe above-indicated procedure is repeated for a newn

2 value of h , and is continued untilF(h) 5O(T 2 T ) (2) iest expi51

Dh i], 0.01 (5)where T and T are, respectively, the estimated and theest exp h iexperimentally measured temperatures at various ther-

mocouples locations and times, and n is the iteration stage. The calculation of h as a function of time is continuediA suitable initial value of h is assumed and with this until the end of the desired period. The flow chart shown ini

value, the temperature of each reference location in casting Fig. 2 gives an overview of the solution procedure.and mold at the end of each time interval Dt is simulatedby using an explicit finite difference technique. The 2.2. Heat flow modelcorrection in h at each iteration step is made by a valuei

Dh , and new temperatures are estimated [T (h 1 Dh ) ori est i i With adequate insulation of the chill and castingT (h 2 Dh )]. With these values, sensitivity coefficientsest i i chamber, heat flow through the casting can be reasonably(f) are calculated for each iteration, given by approximated as a one-dimensional heat transfer problem,

which can be analyzed by

2≠T ≠ T] ] ~rc 5 k 1 q (6)2≠t ≠x

3where r, c, k are, respectively, density (kg /m ), specificheat (J /kg K) and thermal conductivity (W/m K), T istemperature, t is time (s) and x is distance along the x axis

~(m). The term q on the right hand side of Eq. (6) is a heatsource term which is incorporated to account for the latentheat of solidification, and is given by

≠fS]~q 5 rL (7)≠t

where L is the latent heat of fusion (J /kg) and f is theS

solid fraction. When treating the chill heat flow, the~governing equation is similar to Eq. (6) expect that the q

term is not included.Eq. (7) can be related to temperature as follows

≠f ≠f ≠TS S] ] ]5 ? (8)≠t ≠T ≠t

Substitution of Eqs. (7) and (8) into Eq. (6) gives

2≠f ≠T ≠ TSS ]D] ]r c 2 L ? 5 k (9)S D2≠T ≠t ≠x

≠fS]The term (L ? ) in Eq. (9) can be considered as a pseudo≠T

specific heat and an apparent specific heat (c9) can bedefined, and this equation can be written as

2≠T ≠ T

Fig. 2. Flow chart for the determination of metal–mold heat transfer ] ]rc9 5 k (10)S D2≠tcoefficients. ≠x


The introduction of finite difference terms to Eq. (10) 2.3. Mold–environment heat transfer coefficient — ha

givesThe heat transfer coefficients at the mold–air interfacen11 n n n nT 2 T T 2 2T 1 Ti i i21 i i11 can be calculated as a function of measured mold wallS D]]] ]]]]]]rc9 5 k (11)S D2Dt Dx temperatures (T ) and free-stream air temperature (T )EM 0

[17]. The thermal resistance at this interface is given bywhere the subscripts indicate the node address on the

1spatial network and the superscripts represent time. Mul-]]]]R 5 (22)M /Atiplying Eq. (11) by (Dx Dy Dz) yields (h 1 h )AR C T

n11 n n n n where h and h are, respectively, the radiation andT 2 T T 1 T 2 2T R Ci i i11 i21 iS D S D]]] ]]]]]]A Dxrc9 5 A k convection heat transfer coefficients, calculated as followsT TDt Dx2 2(12) h 5 se(T 1 T ) ? (T 1 T ) (23)R EM 0 EM 0

2 28where A 5 Dy Dz (m ), and y and z are the distances, where s is the Stefan–Boltzmann constant (5.672310T2 4respectively, along y axis and z axis. By applying an W/m K ) and e is the mold emissivity.

analogy between electrical and thermal circuits, the energy The convection heat transfer coefficient is given byaccumulated in a volume element i, is given by

k N´gas u]]h 5 (24)9 9C 5 A Dx r c 5Vr c (13) CTi T i i i i i x

3where V is the finite volume element (m ), and C is the where h are represented in terms of Nusselt number (N ),Ti C uthermal capacitance (J /kg). and for free convection can be calculated as a function of

The thermal resistance at the heat flux line x can be Grashof (G ) and Prandtl (P ) numbers, as follows [15]R Rcalculated for each element, and given by

nN 5 C (G P ) (25)u R RDx i11]]]R 5 (14) where C and n are constants, and x is a characteristici11 2k Ai11 T

length of the solid surface (m), in our particular case theDx chill vertical length. G and P , are given, respectively, byi21 R R]]]R 5 (15)i21 2k Ai21 T 3ggx (T 2 T )EM 0 2]]]]]G 5 ? r (26)R 2 sDx i h

]]R 5 (16)i 2k Ai T h]P 5F ? cG (27)R kIntroducing Eqs. (13–16) into Eq. (12) yields

2where g is the gravitational acceleration (m/s ), g is then11 n n n n nT 2 T T 2 T T 2 Ti i i11 i i21 i volume coefficient of expansion [for ideal gases g 5 1/TS D]]] ]]] ]]] 0C 5 1 (17)S D S DTi 21Dt R 1 R R 1 Ri11 i i21 i (K )], h is fluid viscosity, r is fluid density, k is fluidthermal conductivity and c is the fluid specific heat.or

Dt Dt Dtn11 n n n] ]] ]T 5 ? T 1 1 2 ? T 1 ? TS D F G S Di i11 i i21t t t 3. ExperimentalQi QDi Di

(18)Experiments were performed with Sn, Pb, Al and Sn–Pb

where and Al–Cu alloys, including short and long freezing rangealloys, as well as eutectic compositions. The casting and

t 5 C (R 1 R ) (19)Qi Ti i11 i chill materials selected for experimentation, and the em-ployed thermophysical properties are summarized in Table

t 5 C (R 1 R ) (20)Di Ti i21 i 1.The casting assembly used in solidification experimentst tQi Di

]]]t 5 (21) is shown in Fig. 3. The main design criteria were to ensureQDi t 1 tQi Dia dominant unidirectional heat flow during solidification.

Eq. (18) represents the solution of the explicit form of This objective was achieved by adequate insulation of thethe finite difference method, and will be stable for Dt # chill and casting chamber.t . A three dimensional version of this solution has been Copper and a low carbon steel chills were used, with theQDi

recently applied for cases of complex shaped bodies heat-extracting surfaces being polished. In order to investi-[15,16]. gate the influence of chill thickness on heat flow, four


Table 1aCasting and chill materials used for experimentation and the corresponding thermophysical properties [18–22]

Al Al– Al– Al– Steel Pb Sn– Sn– Sn– Sn– Sn Copper4.5%Cu 15%Cu 33%Cu SAE 39%Pb 20%Pb 10%Pb 5%Pb

1010

k (W/m K) 222 193 179 155 46 34.7 54.7 59 63 64 67 372S

k (W/m K) 92 85 80 71 29.7 31.7 32 33 33 33L

c (J /kg K) 1123 1092 1080 1070 527 129.8 186.2 200 209 221 221 419S

c (J /kg K) 1086 1059 999 895 138.2 212.9 231 243 259 259L3

r (kg /m ) 2550 2650 2910 3410 7860 11 340 8840 8250 7840 7720 7300 8960S3

r (kg /m ) 2380 2480 2760 3240 10 678 8400 7860 7480 7380 7000L2 25

a (m /s) 7.75(310 ) 6.67 5.67 4.25 2.37 3.35 3.58 3.84 3.91 4.15S2 25

a (m /s) 3.36(310 ) 3.24 2.90 2.45 2.04 1.79 1.76 1.81 1.82 1.82L

L (J /kg) 385 000 381 900 274 270 350 000 26 205 47 560 52 580 56 140 57 120 60 710T (8C) 660 660 660 660 327 232 232 232 232 232F

T (8C) 548 548 548 183 183 183 183E

T (8C) 645 618 202 216 220L

e 0.8 0.023K 0.17 0.17 0.0656 0.0656 0.06560

aa, Thermal diffusivity; T , eutetic temperature, T , liquidus temperature; K , partition coefficient.E L o

different thicknesses of chills were used (X57, 17, 28, 39 casting and located 20 mm from the interface, as indicatedand 60 mm). in Fig. 3. All of the thermocouples were connected by

Two chromel–alumel thermocouples were introduced in coaxial cables to a data logger interfaced with a computer,the chill; one near the chill–casting interface and the other and the temperature data were acquired automatically. Theat the outer surface, and a third one was placed in the temperature files were used in a finite-difference heat flow

program to estimate the transient heat transfer coefficients.A schematic representation of the experimental setupconnected to the data acquisition and analysis system isshown in Fig. 4.

Each alloy was melted in an electric resistance-typefurnace until the molten metal reached a predeterminedtemperature. It was then stirred until the temperature wasbrought to a specified value and poured into the castingchamber. The aluminum alloys were degassed with hexa-chloroethane tablets before pouring.

The effect of liquid metal superheat on heat transfercoefficient was also investigated, by using a Sn 10%Pballoy, a 60-mm thick carbon steel chill and differentdegrees of superheat: 20, 40, 70 and 1008C above liquidustemperature. The thermocouples were calibrated at themelting points of aluminum (for Al–Cu alloys) and tin (forSn–Pb alloys), exhibiting fluctuations of about 1.0 and0.48C, respectively. The experimental profiles plotted arethe averages of three thermocouple readings at eachlocation in chill and casting. Results from repeated experi-ments have shown differences not greater than 48C, as canbe seen in Fig. 5 for the case of pure Sn.

4. Results and discussion

4.1. Mold–environment heat transfer coefficients — ha

Figs. 6 and 7 show typical examples of temperature datacollected in chill during the course of solidification experi-

Fig. 3. Experimental set-up (mm). ments with a Sn 10 wt.% Pb alloy. In both cases of carbon


Fig. 4. Schematic representation of the experimental setup connected to the data acquisition and analysis system.

steel and copper chills, a similar trend can be observed. using the above mentioned measured surface temperatures,For the smaller chill thicknesses, the external surface Eqs. (22–27) and the thermophysical properties of Table 1.temperature rises rapidly from the beginning of solidifica- For the 6-mm thick chills (carbon steel and copper) ation until a peak value, and declines thereafter. On the progressive decrease in h were observed for t . 800 s, buta

other hand, a progressive rise is observed when the thicker the differences in h were so small during the course ofa

chills are used. It is obvious that the mold–environment solidification that it was assumed constant and equal to theheat transfer coefficients are expected to follow the same peak value for t . 800 s.trend. The effect of thickness of mold was experimentally

The resulting transient values of h were calculated by investigated, and the typical results are shown in Fig. 8. Noa

significant differences in h can be observed, when thea

steel mold is replaced by a massive copper chill. Similarresults were obtained during experimental investigation ofAl–Cu alloys, so that for purposes of numerical simula-tions performed in next section for the determination ofmetal–mold heat transfer coefficients, the same experimen-tal equation has been adopted for each metallic system

0.15Sn–Pb h 5 5.7 t (28)a

0.27Al–Cu h 5 5.1 t (29)a

2where t (s) and h (W/m K).a

4.2. Metal–mold heat transfer coefficients — h i

4.2.1. Effect of alloy compositionSolidification simulation of each test casting was per-

formed by adopting two different approaches for theliberation of the latent heat of fusion. For eutectic alloysand pure metals, the latent heat (L) was transformed intothe equivalent number of degrees by considering a tem-perature accumulation factor (l) related to L by thespecific heat (l 5 L /c). For short or long freezing rangealloys, the latent heat evolution was taken into account byusing Scheil’s equation until the remaining liquid reachedFig. 5. Repeated experiments for pure Sn.


Fig. 6. Typical experimental temperature responses in steel molds at two locations: external wall temperature and at 3 mm from metal–chill interface: (A)6 mm mold thick and (B) 60 mm mold thick.

the eutectic composition. Temperature was experimentally thermal responses are compared to those numericallymeasured in two locations: in the chill at 3 mm from the simulated by using the transient h profile which providesi

metal–mold interface and in the casting at 20 mm from the best curve fitting.this interface. In Figs. 9 and 10 typical experimental Figs. 11 and 12 show the metal–mold heat transfer

Fig. 7. Typical experimental temperature responses in cooper molds at two locations: external wall temperature and at 3 mm from metal–chill interface:(A) 6 mm mold thick and (B) 60 mm mold thick.


Fig. 8. Calculated mold–environment heat transfer coefficients — h : (A) steel molds with different thicknesses, (B) copper molds with differenta

thicknesses.

coefficients profiles as a function of time, respectively for zone length. For longer mushy zones, the interdendriticthe cases of Sn–Pb and Al–Cu alloys solidifying against a liquid can feed better the solidification contraction causing60 mm thick carbon steel chill. The observed differences in a continued presence of liquid at the interface, leading tothe h profiles between the pure metals and the other alloys higher values of h . This can be taken as a general trend,i i

examined, can be explained by the total shrinkage accom- but care should be exercised when applying this conclusionpanying solidification, the extent of the solidification range to the beginning of solidification.and the wetting of the mold by the melt. For both metallic As can be seen in Fig. 12, the Al 15 wt.% Cu alloysystems, the h profile increases with increasing mushy exhibits initial h values higher than those corresponding toi i


Fig. 9. Typical experimental temperature responses at two locations in casting and chill: in casting at 20 mm from the metal–mold interface and in chill at3 mm from this interface: (A) Sn 10 wt.% Pb, (B) pure Sn both solidified in a 60 mm thick steel chill and a superheat DT 5 0.1 T (10% of liquidus orL

melting temperatures).

the Al 4.5 wt.% Cu alloy, which has a longer mushy zone. heat transfer coefficient. Anyway, a more complex ex-At the initial stage of solidification the wetting of the mold perimental set-up and a numerical technique dealing withby the melt seems to be the dominant factor controlling convection heat transfer would be necessary for an accur-

Fig. 10. Typical experimental temperature responses at two locations in casting and chill: in casting at 20 mm from the metal–mold interface and in chill at3 mm from this interface: (A) Al 4.5 wt.% Cu, (B) pure Al both solidified in a 60 mm thick steel chill and a superheat DT 5 0.1 T (A) and DT 5 0.2 T (B).L L


Fig. 11. Evolution the metal–mold interfacial heat transfer coefficients as a function of alloy composition: Sn–Pb system, 60 mm thick steel chill and asuperheat DT 5 0.1 T .L

Fig. 12. Evolution the metal–mold interfacial heat transfer coefficients as a function of alloy composition: Al–Cu system, 60 mm thick steel chill and asuperheat DT 5 0.1 T (10% of liquidus or melting temperatures).L


Fig. 13. Evolution of the metal–mold heat transfer coefficients as a function of chill thickness: Sn 10 wt.% Pb alloy, steel chill and a superheat DT 5 0.1T .L

ate characterization of initial heat transfer coefficients, as against a carbon steel and copper chills at a superheat offluid flow in the cast alloys, with its associated heat DT 5 0.1 T . As can be seen by comparing Figs. 12 andL

transfer, would be at its strongest. 13, the heat transfer coefficient profiles increase withincreasing thermal diffusivity of the chill material. These

4.2.2. Effects of chill material and chill thickness results are in agreement with other studies in the literatureThe effects of chill material and chill thickness on heat [4,10]. The h profiles increase with decreasing chilli

transfer coefficient are shown in Figs. 13 and 14, for the thickness. The chill temperature rises more rapidly fromcases of a Sn 10 wt.% Pb alloy solidifying, respectively the beginning of solidification with decreasing chill thick-

Fig. 14. Evolution of the metal–mold heat transfer coefficients as a function of chill thickness: Sn 10 wt.% Pb alloy, cooper chill and a superheat DT 5 0.1T .L


Fig. 15. Evolution of the metal–mold heat transfer coefficients as a function of superheat: Sn 10 wt.% Pb alloy, 60 mm thick steel chill.

ness. As a consequence, mold expansion favors the thermal 4.2.3. Effect of superheatcontact between metal and chill surface and as the The heat transfer coefficient increases with increasingsolidified shell is not so thick as for thicker chills, this values of superheat, as can be seen in Figs. 15 and 16,translates to lower contraction away from the chill. Both respectively, for a Sn 10 wt.% Pb alloy and pure Alfactors will contribute to an increase in h values. solidifying against a 60 mm thick carbon steel chill. Thei

Fig. 16. Evolution of the metal–mold heat transfer coefficients as a function of superheat: Al, 60 mm thick steel chill.


fluidity of molten alloys increase with increasing super- • The h profiles increased with increasing thermal dif-i

heat, favoring the wetting of the chill by the melt [23,24]. fusivity of the chill material and with decreasingSome results reported in the literature indicate that the thickness of the chill.surface of solidified shell becomes smoother as the super-heat increases for the same chill microgeometry, thusincreasing the interfacial contact [3]. The influence of Acknowledgementssuperheat is not so significant for the Sn 10 wt.% Pb alloy.In fact some differences can be observed only for values The authors would like to acknowledge financial supporthigher than 408C (Fig. 15). This is not the case for Al, provided by FAPESP (The Scientific Research Foundationwhere fluidity plays a more significant role. The initial ˜of the State of Sao Paulo, Brazil) and CNPq (The Brazilian2value of h rises from about 1500 to 6000 W/m K if thei Research Council).superheat is increased from 10% of the melting point (T )f

to 20% of T (Fig. 16).f

References5. Conclusions

[1] L.S. Fletcher, J. Heat Transfer 110 (1988) 1059.[2] K. Ho, R.D. Pehlke, Afs Trans 92 (1984) 587.Experiments were conducted to analyze the evolution of[3] K. Ho, R.D. Pehlke, Metall. Trans. 16B (1985) 585.

mold–environment (h ) and metal–mold (h ) heat transfera i [4] C.A. Muojekwu, I.V. Samarasekera, J.K. Brimacombe, Metall.coefficients during solidification of Sn–Pb and Al–Cu Trans. 26B (1995) 361.alloys in vertical steel and copper chills. The following [5] M. Krishnan, D.G.R. Sharma, Int. Comm. Heat Mass Transfer 23

(1996) 203.conclusions can be drawn[6] A.V. Reddy, C. Beckermann, Exp. Heat Transfer 6 (1993) 111.[7] M.A. Taha, N.A. El-Mahallawy, A.W.M. Assar, R.M. Hammouda, J.• The transient interfacial heat transfer coefficients (h i Mater. Sci. 27 (1992) 3467.

and h ) have been successfully characterized by usinga [8] T.S. Prasanna, K. Narayan Prabhu, Metall. Trans. 22B (1991) 717.an approach based on measured temperatures along [9] J.F. Evans, D.H. Kirkwood, J. Beech, in: Modeling of Casting,

Welding and Advanced Solidification Processes, Vol. V, The Miner-casting and chill, and analytical calculations (h ) andaals, Metals and Materials Society, 1991, p. 531.numerical simulations provided by a heat flow model

[10] C.H. Huang, M.N. Ozisik, B. Sawaf, Int. J. Heat Mass Transfer 35(h )i . (1992) 1779.• The mold–environment heat transfer coefficient have [11] A. Garcia, T.W. Clyne, M. Prates, Metall. Trans. 10B (1979) 773.

been expressed as a power function of time, given by [12] A. Garcia, T.W. Clyne, in: Proceedings International ConferenceSolidification Technology in the Foundry and Casthouse, Thethe general formMetals Society, 1980, p. 33.

0.15h 5 C (t) [13] R. Caram, A. Garcia, in: Imeche Conf. Trans, Vol. 2, Mechanicala m

Engineers Publications, London, 1995, p. 555.2where h (W/m K), t (s) and C is a constant whicha m [14] J.V. Beck, Int. J. Heat Mass Transfer 13 (1970) 703.depends on thickness of chill. No significant differences [15] J.A. Spim Jr., A. Garcia, Mater. Sci. Eng. A 277 (2000) 198.

[16] J.A. Spim Jr., A. Garcia, Numerical Heat Transfer B: Fundamentalswere found when the steel chill was replaced by a38 (2000) 75.copper chill, considering the physical configuration and

[17] D.R. Poirier, E.J. Poirier, Heat Transfer Fundamentals For Metalsprocessing parameters adopted in the study.Casting, The Minerals, Metals and Materials Society, 1994.

• The metal–mold heat transfer coefficients have also [18] L.F. Mondolfo, Mater. Sci. Technol. 5 (1976) 118.been expressed as a power function of time, given by [19] R.D. Pehlke et al., Summary of Thermal Properties For Casting

Alloys and Mold Materials, University of Michigan, 1982.the general form[20] Y.S. Toloukian et al., in: Thermophysical Properties of Matter, Vol.2nh 5 C (t) 1, IFI /Plenum, New York, 1970.i i

[21] A. Bejan, Heat Transfer, Wiley, New York, 1993.2where h (W/m K), t (s) and C and n are constantsi i [22] D. Bouchard, J.S. Kirkaldy, Metall. Mat. Trans. 28B (1997) 651.which depend on alloy composition, chill material and [23] M. Prates, H. Biloni, Metall. Trans. 3A (1972) 1501.superheat. [24] M.C. Flemings, Solidification Processing, McGraw Hill, 1974.

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