Materials and Design 30 (2009) 3592–3601

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Materials and Design

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Melt characteristics and solidification growth direction with respect to gravityaffecting the interfacial heat transfer coefficient of chill castings

Noé Cheung a, Ivaldo L. Ferreira b, Moisés M. Pariona c, José M.V. Quaresma d, Amauri Garcia a,*

a Department of Materials Engineering, University of Campinas, UNICAMP, P.O. Box 6122, 13083-970 Campinas, SP, Brazilb Department of Mechanical Engineering, Fluminense Federal University, UFF, Av. dos Trabalhadores 420, 27255-125 Volta Redonda, RJ, Brazilc Department of Mathematics and Statistics, State University of Ponta Grossa, UEPG, 84030-900 Ponta Grossa, PR, Brazild Federal University of Pará, UFPA, Augusto Correa 1, Guamá, 66075-110 Belém, PA, Brazil

a r t i c l e i n f o

Article history:Received 15 January 2009Accepted 26 February 2009Available online 5 March 2009

Keywords:Non-ferrous metals and alloys (A)Casting (C)Thermal analysis (G)

0261-3069/$ - see front matter � 2009 Elsevier Ltd. Adoi:10.1016/j.matdes.2009.02.025

* Corresponding author. Tel.: +55 19 35213320; faxE-mail address: amaurig@fem.unicamp.br (A. Garc

a b s t r a c t

For purposes of an accurate mathematical modeling, it is essential to establish trustworthy boundaryconditions. The heat transfer that occurs at the casting/mold interface is one of these important condi-tions, which is a fundamental task during unsteady solidification in permanent mold casting processes.This paper presents an overview of the inverse analysis technique (IHCP) applied to the determinationof interfacial heat transfer coefficients, hi, for a number of alloy solidification situations. A search algo-rithm is used to find the transient metal/mold interface coefficient during solidification which is reportedeither as a function of the casting surface temperature or time. Factors affecting hi such as the direction ofgravity in relation to the growth interface, the initial melt temperature profile, the wettability of theliquid layer in contact with the mold inner surface, were individually analyzed and experimental lawsfor hi have been established.

� 2009 Elsevier Ltd. All rights reserved.

1. Introduction

Inverse problems are encountered in various branches of sci-ence and engineering. Mechanical, materials, aerospace, chemicaland metallurgical engineers, astrophysicists, geophysicists, statisti-cians and specialists of many other disciplines are all interested ininverse problems, each with different application in mind. In thefield of heat and mass transfer, the use of inverse analysis for theestimation of surface conditions such as temperature and heat flux,thermal gradient, or the determination of thermal properties suchas thermal conductivity, heat capacity, enthalpy, latent heat anddensities of solid and liquid by utilizing transient temperaturemeasurements taken within the medium has a wide range of prac-tical applications. The determination of transient metal/mold heattransfer coefficients as a function of position and time during solid-ification of multicomponent alloys is an example of difficultnumerical treatment. In such situations, the inverse method ofanalysis, using transient temperature measurements taken withinthe medium can be applied for the estimation of such quantities.However, difficulties associated with the implementation of in-verse analysis should be also recognized. The main difficulty arisesfrom the fact that inverse solutions are very sensitive to changes inthe input data resulting from measurements and modeling errors,hence may not be unique. Mathematically, the inverse problem be-

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: +55 19 32893722.ia).

longs to the class of problems called the ill-posed problems, that is,their solution does not satisfy the general requirement of exis-tence, uniqueness and stability under small changes to the inputdata. In order to overcome such difficulties, a variety of techniquesfor solving inverse heat transfer problems have been proposed [1].

The way heat flows through the casting/mold interface affectsthe evolution of solidification, and is of notable importance in char-acterizing the ingot cooling conditions, mainly for the majority ofhigh heat diffusivity casting systems such as chill castings [2].When the metal comes into contact with the mold, at the metal/mold interface, the solid bodies are only in contact at isolatedpoints and the actual area of contact is only a small fraction ofthe nominal area, as shown in Fig. 1.

Part of heat flow follows the path of the actual contact, but thereminder must pass through the gaseous and nongaseous intersti-tial media between the surface peaks. The interstices are limited insize, so that convection can be neglected. If temperature differ-ences are not high, radiation does not play a significant role andmost of the energy passes by conduction across the areas of actualphysical contact. The heat flow across a casting/massive moldinterface, can be characterized by a macroscopic averaged metal/mold interfacial heat transfer coefficient (hi) given by,

hi ¼q

AðTIC � TIMÞð1Þ

where q (W) is the global heat flux of the interface; A (m2) is thearea and TIC and TIM are the surface casting temperature and the

Fig. 3. Thermal resistances in a chill mold.

Casting

MoldTp

Tmold

TIC

TIMT

q

Fig. 1. Heat flux at the metal/mold interface.

N. Cheung et al. / Materials and Design 30 (2009) 3592–3601 3593

temperature of the mold inner surface (K), respectively. In water-cooled molds, the global equivalent heat flux is affected by a seriesof thermal resistances, as shown in Fig. 2,

The global thermal resistance 1/hi can be expressed by:

1hi¼ 1

hWþ e

kMþ 1

hM=Mð2Þ

where hi is the global heat transfer coefficient between the castingsurface and the cooling fluid (Wm�2 K�1), e is the mold thickness(m), kM is the mold thermal conductivity (Wm�1 K�1), and finally,hW is the mold/cooling fluid heat transfer coefficient (Wm�2 K�1).The averaged heat flux casting/cooling water is given by [3]:

q ¼ hiðTIC � T0Þ ð3Þ

where T0 is the water temperature (K).The thermal resistance at the mold/air interface, RM/A , can be

calculated as a function of the measured mold wall temperatures(TEM) and the free-stream air temperature (T0), as shown inFig. 3, and is given by:

RM=A ¼1

ðhR þ hCÞATð4Þ

where AT is the chill cross-section area (m2) and hR and hC are theradiation and convection heat transfer coefficients, respectively,given by:

hR ¼ r � eðTEM þ T0ÞðT2EM þ T2

0Þ ð5Þ

where r is the Stefan–Boltzmann constant (5.672 � 10�8

Wm�2 K�4) and e is the mold emissivity.The convection heat transfer coefficient is given by [4]:

hC ¼kNuv ð6Þ

where k is the fluid thermal conductivity (W m�1 K�1) and hC is rep-resented in terms of the Nusselt number (Nu). For free convectionNu can be calculated as a function of Grashof (Gr) and Prandtl (Pr)numbers, as follows:

Nu ¼ CðGr � PrÞn ð7Þ

where C and n are constants, and v is a characteristic length of thesolid surface (m), which in the particular case of Fig. 3 is the chillvertical length . Gr and Pr are given respectively by:

Liquid

Solid

Water e

To

TIC

R3 = 1/ hM/M

R2= e / k

R1 = 1 / h w

Fig. 2. Thermal resistances in a water-cooled metal/mold system.

Gr ¼ g � c � v3ðTEM � T0Þg2 q2

s ð8Þ

Pr ¼ gk� c

h ið9Þ

where g is the gravitational acceleration (m s�2), c is the volumecoefficient of expansion (for ideal gases c = 1/ T0 (K�1)), g is the fluidviscosity, q is the fluid density and c is the fluid specific heat [5].

For successful modeling of casting processes, reliable heattransfer boundary conditions are required, in particular the me-tal/mold heat transfer coefficient. The accurate knowledge of thiscoefficient is necessary for accurate modeling of casting dimen-sions and casting microstructure [6,7]. Many investigations con-cerning the heat transfer coefficient between metal and mold incasting systems have been carried out, and pointed out the impor-tance of the development of appropriate tools to predict the heattransfer coefficient, hi. Most of the methods of calculation of hi

existing in the literature are based on temperature histories atpoints of the casting or mold together with mathematical modelsof heat transfer during solidification. Among these methods, thosebased on the solution of the inverse heat conduction problem(IHCP) have been widely used in the quantification of the transientinterfacial casting/mold heat transfer [8–14]. In general, hi is notconstant but varies during solidification and depends upon a num-ber of factors. These factors include the thermophysical propertiesof the contacting materials, the casting geometry, mold tempera-ture, pouring temperature, the roughness of mold contacting sur-face, mold coatings, etc [15].

The purpose of the present study was to investigate the influ-ence of three important factors on the interfacial heat transfercoefficient: the initial melt temperature profile, the wettability ofthe liquid layer in contact with the mold inner surface, and thedirection of gravity in relation to the growth interface. Tempera-ture readings, recorded by a bank of thermocouples distributed in-side the casting, were used as input data for an inverse heatconduction method in order to determine the time-varying interfa-cial heat transfer coefficient, hi. Casting experiments were carriedout with Al–Cu, Al–Si, Al–Sn, Sn–Pb, and Pb–Sb alloys, which wereunidirectionally solidified in a massive chill mold and in a water-cooled mold under different parametric solidification conditions.Simulations were performed using a two-dimensional version ofa numerical heat transfer solidification model.

2. Numerical modeling

2.1. Governing equations

The numerical model used to simulate the thermal field duringalloy solidification is based on that previously proposed by Voller[16]. Modifications to this numerical approach have been incorpo-

Fig. 4. Schematic casting initial melt temperature distribution (t = 0).

3594 N. Cheung et al. / Materials and Design 30 (2009) 3592–3601

rated to allow the use of different thermophysical properties forthe liquid and solid phases, as well as the mushy zone (it can dealwith temperature and concentration dependent thermophysicalproperties), to treat variable metal/mold interface heat transfercoefficient and to account for a space dependent initial melt tem-perature profile. A time variable metal/mold interface heat transfercoefficient introduces a non-linearity condition at the z = 0 bound-ary. In addition, a variable space grid is used to assure the accuracyof simulation results without considerably raising the number ofspatial nodes. Considering the previous exposed, the solidificationof binary alloys is our target problem. At time t < 0, the alloy is inthe molten state at the nominal concentration C0 and with an ini-tial temperature distribution T0(z) = �a � z2 + b � z + c, contained inthe insulated mold defined by 0 < z < Zb according to Fig. 4. Solidi-fication begins by cooling the molten metal at the chill (z = 0) untilthe temperature drops bellow the eutectic temperature TE. At timest > 0, three transient regions are formed: solid, solid + liquid(mushy zone) and liquid.

To develop a numerical solution for the equations of the cou-pled thermal and solutal fields, the following assumptions wereadopted:

(i) The domain is one-dimensional, defined by 0 < z < Zb, whereZb is a point far removed from the chill.

(ii) The solid phase is stationary, i.e., once the solid has formed ithas zero velocity.

(iii) Due to the relatively rapid nature of heat and mass diffusionin the liquid, within a representative elemental averagingvolume, the liquid concentration CL, the temperature T, theliquid density qL and the liquid velocity uL are constants[17].

(iv) The partition coefficient k0 and liquidus slope mL, areobtained from the ThermoCalc software1.

(v) Equilibrium conditions exist at the solid/liquid interface, i.e.,the temperature and concentrations fulfill the equations:

1 Thethroughgeneratsimulat

T ¼ TF �mLCL ð10Þ

and

C�S ¼ k0CL ð11Þ

ThermoCalc software [19] can be used to generate equilibrium diagrams andThermoCalc interface for Fortran or C++ it is possible to recall those data

ed by the software in order to provide more accurate input values for modelions.

where sub-indices S and L refer to solid and liquid phases,respectively, TF is the fusion temperature of the pure solventin (K) and C�S is the solid concentration at the interface;

(vi) The specific heats, CS and CL, thermal conductivities kS andkL, and the densities qS and qL, are constants within eachphase, but discontinuous at the solid–liquid boundary. Thelatent heat of fusion is taken as the difference betweenphases enthalpies DH = HL–HS.

(vii) The metal/mold thermal resistance varies with time, and isincorporated in a global heat transfer coefficient defined ashi [18].

Using the above assumptions, the mixture equations for binaryalloys solidification read:

� Energy

@qcT@tþr � ðqLcLuTÞ ¼ r � ðkrTÞ � qSDH

@g@T

ð12Þ

� Species

@qC@tþr � ðqLuCLÞ ¼ 0 ð13Þ

� Mass

@q@tþr � ðqLuÞ ¼ 0 ð14Þ

where g is the liquid volume fraction and u is the volume averagedfluid velocity defined as:

u ¼ guL ð15Þ

� Mixture density

q ¼Z 1�g

0qSdaþ gqL ð16Þ

� Mixture solute density

qC ¼Z 1�g

0qSCSdaþ gqLCL ð17Þ

where qC is the volumetric specific heat, taken as volume fractionweighted averages.

The boundary conditions at the domain are prescribed as:

u ¼ 0; k@T@z¼ hi T0 � Tjz¼0ð Þ and

@CL

@z¼ 0 at z ¼ 0 ð18Þ

T ! Tp and C ! C0 at z ¼ Zb; ð19Þ

where Tp is the either a constant initial melt temperature or an ini-tial melt temperature profile as a function of z.

The inverse problem consists on estimating the boundary heattransfer coefficient at the metal/mold interface from experimentaltemperatures in the casting. The inverse problem can be stated asfollows:

– given M measured temperatures Tj (j = 1,2,3, . . . ,N);– estimating the heat transfer coefficient given by its compo-

nents hi (i = 1,2,3, . . . ,N);

In order to solve the problem, the estimated temperature Testi

(i = 1,2,3, . . . ,N) computed from the solution of the direct problemusing the estimated values of the heat transfer coefficient compo-nents hi (i = 1,2,3, . . . ,N), should match the measured temperaturesTexp

i (i = 1,2,3, . . . ,N), as close as possible, as shown by theschematic representation of Fig. 5. This matching can be done by

Fig. 6. Flow chart for the determination of metal/mold heat transfer coefficients.

N. Cheung et al. / Materials and Design 30 (2009) 3592–3601 3595

minimizing the standard least squares norm with respect to eachof the unknown heat transfer coefficient components.

This method makes a complete mathematical description of thephysics of the process and is supported by temperature measure-ments at known locations inside the heat conducting body. Thetemperature files containing the experimentally monitored tem-peratures are used in a finite difference heat flow model to deter-mine hi, as described in a previous article [5]. The process at eachtime step included the following: a suitable initial value of hi isassumed and with this value, the temperature of each referencelocation in casting at the end of each time interval Dt is simulatedby the numerical model. The correction in hi at each interactionstep is made by a value Dhi, and new temperatures are estimated[Test(hi + Dhi)] or [Test(hi � Dhi)]. With these values, sensitivity coef-ficients ð/Þ are calculated for each interaction, given by:

/ ¼ Testðhi þ DhiÞ � TestðhiÞDhi

ð20Þ

The procedure determines the value of hi, which minimizes anobjective function defined by:

FðhiÞ ¼Xn

i¼1

ðTest � TexpÞ2 ð21Þ

where Test and Texp are the estimated and the experimentally mea-sured temperatures at various thermocouples locations and times,and n is the iteration stage. The applied method is a simulation as-sisted one and has been used in recent publications for determininghi for a number of solidification situations [2,20–24].

The flow chart shown in Fig. 6 gives an overview of the solutionprocedure.

3. Experimental procedure

Three different solidification apparatus have been used in theexperimental analysis and the assemblage details of these systemsare shown in Fig. 7.

In order to promote vertical upward solidification, an apparatusdesigned in such a way that the heat was extracted by a water-cooled bottom provoking upward directional solidification wasused (Fig. 7a). A stainless steel cylindrical mold was employed,having an internal diameter (i.d.) of 50 mm, height of 110 mmand wall thickness of 5 mm. The inner vertical surface was coveredwith a layer of insulating alumina to minimize radial heat losses,and a top cover made of insulating material was employed to re-duce heat losses from the metal/air surface. The bottom part ofthe mold was closed with a thin (3 mm) carbon steel sheet.

The use of a water-cooled stainless steel chamber at the top ofthe casting has permitted experiments for downward directionalgrowth to be carried out (Fig. 7b). A stainless steel split moldwas used having an i.d. of 57 mm, height of 150 mm and wallthickness of 10 mm. As mentioned before, alumina was applied

Fig. 5. Diagram showing domain for in

at the mold inner surface in order to prevent radial heat losses.The upper part of the split mold was closed by the cooling chamber(3 mm thick wall).

In the upward and downward systems, the alloys were meltedin situ and the electric heaters had their power controlled in orderto permit a desired melt superheat to be achieved. To begin solid-ification, the electric heaters were disconnected and at the sametime, the water flow was initiated. Temperatures in the castingwere monitored during solidification via the output of a bank oftypes J and K thermocouples accurately positioned with respectto the heat extracting surface. In order to minimize temperaturefield distortions, the thermocouples were installed parallel to theisotherms in the casting [7]. Further, the thermocouple tips wereplaced as near as possible to the transversal geometric center ofthe casting. The thermocouples were also calibrated at the melting

verse heat conduction problems.

Fig. 7. Experimental setups: (a) upward, (b) downward and (c) horizontal systems.

3596 N. Cheung et al. / Materials and Design 30 (2009) 3592–3601

temperatures of aluminum and tin exhibiting fluctuations of about1.0 �C and 0.4 �C, respectively. Thermocouples readings (at inter-vals of 0.5 s) were collected by a data acquisition system andstored in a computer.

Although the correct thermocouple positions with regard to theheat extracting surface were verified before the experiments, adeviation of about ±1 mm from the nominal positions was ob-served for some of them as a result of interaction of sensors withmelt movement and casting shrinkage.

A third casting assembly was used for horizontal solidificationexperiments (Fig. 7c). In order to promote unidirectional heat flowduring solidification, a low carbon steel chill with a wall thicknessof 60 mm was used, with the heat extracting surface being pol-ished. Each alloy was melted in an electric resistance-type furnaceuntil the melt reached a predetermined temperature. It was thenstirred, degassed and poured into the casting chamber as soon asthe desired melt superheat was achieved. Temperatures in the chilland in the casting were monitored during solidification via the out-put of a bank of thermocouples accurately located with respect tothe metal/mold interface. Unidirectional heat flow was achieved byadequate insulation of the casting chamber.

4. Results and discussion

4.1. Influence of melt temperature profile

Temperature files containing the experimentally monitoredtemperatures were coupled to the numerical solidification pro-

gram for determining the transient metal/mold heat transfercoefficient hi. Thermophysical properties of each alloy and solidi-fication parameters are used as input data for simulations. Fig. 8shows the temperature data collected in the metal during thecourse of upward solidification of an Al 10 wt%Cu alloy castingin the vertical water-cooled apparatus, with the bottom heatextracting surface being polished. The experimental thermal re-sponses corresponding to five different positions inside the cast-ing were compared with the predictions furnished by thenumerical solidification model. The best theoretical-experimentalfit has provided appropriate transient hi profile for two differentapproaches: (i) an average initial melt temperature has beenadopted (Fig. 8a), and (ii) a quadratic equation, based on experi-mental thermal readings, representing the initial melt tempera-ture as a function of position in casting has been used (Fig. 8b).A comparison between hi profiles determined in each case isshown in Fig. 8c. It can be seen that a significant difference existsbetween the two curves, with the assumption of a constant melttemperature overestimating the metal/mold heat transfer coeffi-cient. The two curves tend to approach each other with increasingtime.

In order to evaluate the real significance of hi overestimationadditional simulations were conducted considering two-dimen-sional solidification. A regular geometry of an Al 10 wt%Cu alloysquare casting (100 � 100 mm2) was simulated by a 2D versionof the numerical approach described in Section 2, in order to eval-uate the influence of each hi profile previously determined, whichwas imposed at the four faces of the square ingot. Fig. 9a and b

0 20 40 60 80 1000

100

200

300

400

500

600

700

5 mm 10 mm 15 mm 30 mm 50 mm Numerical simulation

Al-10wt%Cu - Polished moldTp = 653.5 °C (mean)

hi = 10800 . t -0.075 [W/m2K]

Tem

pera

ture

[ºC

]

Time [s]

(a)

0 20 40 60 80 1000

100

200

300

400

500

600

700

5 mm 10 mm 15 mm 30 mm 50 mm Numerical Simulation

Al-10wt%Cu - Polished moldTp(z) = -4267.14 z2 + 734.04 z + 910.83 [K]

hi = 9000. t -0.039 [W/m2K]

Tem

pera

ture

[ºC

]

Time [s]

(b)

0 20 40 60 80 1007500

8000

8500

9000

9500

10000

10500

11000

11500 hi = 9000.t-0.039 [W/m2K] - quadratic melt temperature profile hi = 10800.t-0.075 [W/m2K] - constant melt temperature

Time [s]

h i [W

/m2 K

]

(c)

Fig. 8. (a) Simulated and measured temperature responses for an Al 10 wt%Cu alloy casting at 5, 10, 15, 30 and 50 mm from the metal/mold interface adopting an averagemelt temperature; (b) Simulated and measured temperature responses for an Al 10 wt%Cu alloy casting at the same positions adopting a melt temperature profile; and (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).

Fig. 9. Isotherms (�C) distribution for t = 13.75 s obtained considering (a) hi = 9000 � t�0.039 and (b) hi = 10,800 � t�0.075.

N. Cheung et al. / Materials and Design 30 (2009) 3592–3601 3597

show some isotherms at the casting cross-section for t = 13.75 sconsidering hi = 9000 � t�0.039 and hi = 10,800 � t�0.075, respectively.

It can be noticed that the liquid core is larger when the more accu-rate melt profile was adopted as can be seen by comparing Fig. 9a

Fig. 10. (a) Comparison of the resultant experimental hi profiles as a function of time for the Pb–Sb alloys experimentally examined and (b) fluidity behavior of Pb–Sb alloys.

Fig. 11. Isotherms (�C) distribution for t = 48 s considering (a) Pb 2.5 wt%Sb; hi = 4500 � t�0.11 and (b) Pb 3.0 wt%Sb hi = 3700 � t�0.11.

3598 N. Cheung et al. / Materials and Design 30 (2009) 3592–3601

N. Cheung et al. / Materials and Design 30 (2009) 3592–3601 3599

and Fig. 9b, i.e., the adoption of a simplified constant melt profilewill provide a quicker solidification evolution.

4.2. Effect of melt fluidity

Fig. 10a shows the time dependence of the metal/coolant inter-face heat transfer coefficient (hi) during the course of differentexperiments of upward directional solidification of Pb–Sb alloys,including the profile obtained for the eutectic composition. In or-der to permit more accurate values of hi to be determined, a qua-dratic function has been used to characterize the initial meltprofile, as discussed in the preceding section. The thermophysicalproperties, the solidification range and the melt fluidity are someof the factors affecting hi. The surface roughness of the steel sheetwhich separates the metal from the cooling fluid has beenparameterized.

Although a single exponent 0.11 has been found for the powerlaws characterizing the variation of hi with time, different multipli-ers have been obtained. Such multipliers seem to be mainly linkedto the wettability of the liquid layer in contact with the mold innersurface, i.e., connected with the molten alloy fluidity. Both liquidmetal and mold characteristics are involved in determining fluidity[25,26]. Fig. 10b shows the fluidity superimposed to the Pb–Sbphase diagram. The fluidity of Pb–Sb alloys decreases from pure

0 100 200 300 4002000

4000

6000

8000

10000

12000

14000

16000

18000

Time [s]

h i [W

.m-2.K

-1]

hi = 10,500.t-0.1 - Al-20wt% Sn alloy

hi = 6,000.t-0.1 - Al-30wt% Sn alloy

hi = 12,500.t-0.1 - Al-40wt% Sn alloy

Fig. 12. Evolution of metal/coolant interface heat transfer coefficient (hi) as afunction of time (t) during vertical upward solidification.

0 50 100 150 200 250 3001000

2000

3000

4000

5000

6000

7000

8000

9000

10000Sn-5wt%Pb

Time [s]

h i [W/m

2 K]

hi = 18000.t-0.47 [W/m2K] - horizontal solidification hi = 6000.t-0.12 [W/m2K] - upward solidification hi = 1650.t-0.001 [W/m2K] - downward solidification

Fig. 13. Evolution of metal/mold interface heat transfer coefficient (hi) as a functionof time for a Sn 5 wt%Pb alloy solidified vertically upwards, downwards andhorizontally.

lead up to a range of compositions between 3.5 wt%Sb and8.0 wt%Sb increasing again with increasing Sb content toward theeutectic composition. The two extremes of the composition rangeexperimentally examined, i.e., the Pb 2.2 wt%Sb alloy and the eu-tectic composition are associated with the highest hi profiles asshown in Fig. 10. By observing Fig. 10 a correlation between themultiplier (A) of the experimentally determined hi = f(t) equationsand the fluidity’s values can be established.

In Fig. 11, different locations of the isotherms, at t = 48 s, can berealized for the simulation of the two-dimensional solidification oftwo Pb–Sb alloys (Pb 2.5 wt%Sb and Pb 3.0 wt%Sb). Although thecomposition between the two alloys is quite close, the melt fluidityis significantly different which means that specific hi profiles have

Fig. 14. Isotherms (�C) distribution during solidification (for t = 80 s) of a Sn5 wt%Pb alloy casting: hi = 1650 � t�0.001 over the upper surface; hi = 6000 � t�0.12

over the bottom surface; hi = 18,000 � t�0.47 over the lateral surfaces (a) consideringheat transfer in the liquid metal only by conduction (b) considering also fluid flow.

3600 N. Cheung et al. / Materials and Design 30 (2009) 3592–3601

to be considered, i.e., the adoption of a same hi profile for both al-loys can induce important differences. Indeed, the interfacial heattransfer coefficient does influence solidification behavior as it isevident from the simulated isotherms in Fig. 11. Whilst for thePb 2.5 wt%Sb alloy (Fig 11a) the solidification is almost completethe Pb 3.0 wt%Sb alloy casting is not ready to be unmolded.

Fig. 12 shows the time dependence of the overall metal/coolantheat transfer coefficient (hg) during the course of different experi-ments of upward directional solidification of Al–Sn alloys in un-coated cooled molds. Although a same exponent 0.1 has beenfound for the power laws characterizing the hg variation with time,very different multipliers have been obtained. Such multipliers aremainly linked to the wettability of the liquid layer in contact withthe mold inner surface, i.e., connected with fluidity. Both liquidmetal and mold characteristics are involved in determining fluid-ity. The lowest hg profile refers to the Al 30 wt%Sn alloy, whilethe other two alloys present higher hg profiles. It has been demon-strated that when fluidity is superimposed to binary constitutiondiagrams, the best fluidity is attained for pure components, eutec-tics or phases that freeze congruently [26]. It seems that for Al–Snalloys the fluidity decreases from pure aluminum up to a composi-tion about 30 wt%Sn increasing again with increasing Sn contenttoward the eutectic composition. This is reflected by the multipli-ers of the experimentally determined hg = f(t) equations, shown inFig. 12.

4.3. Effect of growth direction with respect to gravity

The influence of the direction of growth on hi during solidifica-tion has been experimentally examined for opposite conditionswith respect to the gravity vector (upward and downward solidifi-cation) and by using alloys of quite different thermal responsesduring solidification (Sn–Pb and Al–Si). For the Sn–Pb alloy thehorizontal configuration has also been examined.

The best theoretical-experimental cooling curves fit has pro-vided an appropriate transient hi profile during solidification of aSn 5 wt%Pb alloy. Fig. 13 shows such profiles during the courseof different experiments involving downward, upward and hori-zontal directional solidification. The heat transfer coefficient isclearly dependent on the orientation of solidification with respectto gravity. In the upward vertical solidification the effect of gravitycauses the casting to rest on the mold surface, but during down-ward solidification, this action causes the solidified portion of thecasting to retreat from the mold surface. It is well known thatthe reduction in the contact pressure between casting and mold

0 20 40 60 80 100 120 140 160 180 200700

1400

2100

2800

3500

4200

4900

hi = 2400 (t)-0.001 - Al-5wt%Si

hi = 2100 (t)-0.001 - Al-7wt%Si

hi = 1100 (t)-0.001 - Al-9wt%Si

Met

al/C

oola

nt H

eat T

rans

fer C

oefic

ient

h i (W

/m2 K)

Time (s)

(a)

Fig. 15. Evolution of metal/coolant interface heat transfer coefficient (hi) as a function osolidification.

surfaces leads to a consequent reduction in the interfacial heattransfer efficiency.

The heat transfer coefficients for both upward and horizontalsolidification are high at the initial stages of solidification, as a re-sult of the good surface conformity between the liquid core and thesolidified shell. The mold expands while solidification progressesdue to the absorption of heat and the solid metal shrinks duringcooling. As a consequence, a gap develops because pressure be-comes insufficient to guarantee a conforming contact betweenthe surfaces. Once the air gaps forms, the heat transfer across theinterface decreases rapidly and a relatively constant value of hi isattained.

In the upward vertical solidification the casting weight will con-tribute to a good metal/mold thermal contact when the lateral con-traction is effective, i.e., when the ingot is able to detach from thelateral walls. This will happen only after a determined solid shell isformed. In contrast, at the early stages of solidification in the hor-izontal apparatus the good thermal contact is assured by the liquidmetal pressure exerted over the solid shell. When the solid shell isable to contract, the air gap is formed and the thermal contactdecreases.

It is a common practice to assume the same interfacial heattransfer coefficient over the whole casting surface when usingsolidification simulation softwares. In order to highlight the impor-tance of using real values of hi according to the gravity vector influ-ence, three different hi profiles were simultaneously applied on thesimulation of solidification of a Sn 5 wt%Pb square casting(100 � 100 mm2). The hi profile determined from the downwardsolidification was applied over the casting upper surface, the onefrom the upward solidification over the casting bottom surface,and the one from the horizontal solidification over the lateral sur-faces. Fig. 14 shows the isotherms shapes, for t = 80 s. It can be seenthat their shapes are not anymore that of concentric circles asshown previously in Figs. 9 and 11. Higher cooling rates at the lat-eral faces changed the isotherm format from circle to ellipseshaped. Similarly, one can conclude that for complex geometries,which are widely used in industrial applications, there is a needfor a realistic description of these coefficients which are used as in-put parameters in softwares for simulation and control of indus-trial casting processes.

In the simulations of Fig. 14a, only heat transfer by conductionin the melt was assumed. In contrast, if the liquid flow duringsolidification is significant but is not taken into account in the sim-ulations, the accuracy of the calculated isotherms will be reduced.Fig. 14b shows results of simulations with the same conditions

0 20 40 60 80 100 120 140 160 180 200700

1400

2100

2800

3500

4200

4900 hi = 4500 (t)-0.09 - Al-5wt%Si

hi = 3900 (t)-0.09 - Al-7wt%Si

hi = 3300 (t)-0.09 - Al-9wt%Si

Time (s)

(b)

f time (t) for Al–Si alloys during vertical (a) downward and (b) upward directional

N. Cheung et al. / Materials and Design 30 (2009) 3592–3601 3601

considered previously for the solidification of the Sn 5 wt%Pb alloy,for t = 80 s, including now the effect of liquid flow during solidifi-cation. It can be seen that the flow inside the mushy zone gives riseto instabilities in the solidification evolution which are responsiblefor changes on the isotherms shape and location. As a consequence,effects on the segregation distribution along the casting are alsoexpected.

The results obtained for three different hypoeutectic Al–Si al-loys for solidification carried out both vertically upwards anddownwards are shown in Fig. 15. Fig. 15a (downward solidifica-tion) shows constant values of hi along solidification. As the castingmoves away from the chamber surface very rapidly due to the cast-ing weight during downward solidification, the sprouting of inter-facial gap is faster than for upward solidification, which causeslower and constant hi values.

5. Conclusions

The following major conclusions can be derived from the pres-ent study:

� When a non-uniform initial melt temperature profile is used asinput data of the IHCP technique in order to derive the corre-sponding interfacial heat transfer coefficient, a more realisticsimulation of the solidification evolution can be achieved.

� The wettability of the liquid layer in contact with the mold innersurface, which is associated to the alloy’s fluidity, was shown tobe important in the characterization of the interfacial heat flow.In this context care should be exercised in the determination ofhi even for small variation of alloy solute content.

� Experimental evidence has shown that hi is strongly dependenton the direction of solidification with respect to the gravity vec-tor. Accurate simulation of freezing patterns in castings willdepend on the experimental determination of hi for importantgrowth directions. The fluid flow when significant was alsoshown to affect the isotherms shape during solidification, andhas also to be included with accurate hi values for a realisticdescription of solidification.

Acknowledgements

The authors acknowledge financial support provided by FAPESP(The Scientific Research Foundation of the State of São Paulo, Bra-zil), CNPq (The Brazilian Research Council) and FAEPEX –UNICAMP.

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