Materials Chemistry and Physics 111 (2008) 444–454
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Materials Chemistry and Physics
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Influences of solute content, melt superheat and growth directionon the transient metal/mold interfacial heat transfer coefficientduring solidification of Sn–Pb alloys
Ivaldo L. Ferreiraa, Jose E. Spinelli a, Britta Nestlerb, Amauri Garciaa,∗
a Department of Materials Engineering, University of Campinas–UNICAMP, PO Box 6122, 13083–970 Campinas, SP, Brazilestras
y comn affeould
on ineratioworkld be
ward% Pb,ectioericalad co
b Institute of Computational Engineering, Karlsruhe University of Applied Science, Moltk
a r t i c l e i n f o
Article history:Received 10 December 2007Received in revised form 5 March 2008Accepted 23 April 2008
Keywords:AlloysSolidificationComputer modeling and simulation
a b s t r a c t
Several factors such as allomold coating layer, etc., caing solidification model shthis previous knowledgebased on thermal and opproperties. In the presentway that thermal data cougravity vector: vertical uphypoeutectic alloys (5 wt.solute content, growth dirtures were used by a numhi rises with decreasing letemperature distribution.
1. Introduction
According to Lewis and Ransing [1], the interfacial heat trans-fer has always been an important area both in terms of numericalmodeling as well as practical operational conditions. Many foundrydesign decisions such as insulation, chill materials, die coatingthickness, etc. directly influence the interfacial heat transfer coeffi-cient, hi. Sahin et al. [2] emphasize that the metal/mold interfacialheat transfer has an important influence on the solidification rate ofmetal castings. The structural integrity of shaped castings is closelyrelated to the time–temperature history during solidification, andthe use of casting simulation, which can be strongly dependent onaccurate values of hi, could do much to increase this knowledge inthe foundry industry [3].
In general, hi is not constant but varies during solidification anddepends upon a number of factors [4]. These factors include thethermophysical properties of the contacting materials, the castinggeometry, mold temperature, pouring temperature, the roughnessof mold contacting surface, mold coatings, etc. [5–12]. It has been
position, melt superheat, mold material, roughness of inner mold surface,ct the transient metal/mold heat transfer coefficient, hi. An accurate cast-be able to unequivocally consider these effects on hi determination. After
terfacial heat transfer, such models might be used to control the processnal parameters and to predict microstructure which affects casting final, three different directional solidification systems were designed in such a
monitored no matter what configuration was tested with respect to theand downward or horizontal. Experiments were carried-out with Sn–Pb10 wt.% Pb, 15 wt.% Pb and 30 wt.% Pb) for investigating the influence of
n and melt superheat on hi values. The experimentally obtained tempera-technique in order to determine time-varying hi values. It was found thatntent of the alloy, and that hi profiles can be affected by the initial melt
found that the heat transfer coefficient is higher when solidifica-tion is vertically upward than when solidification is either vertically
downward or horizontal [13]. Browne and O’Mahoney [4] havestated that, for a given mold, the alloy composition is the mainvariable to be considered in interface heat transfer analysis. Thisdetermines factors such as surface tension of the liquid and alloyfreezing range, both of which can have a significant effect on theinterfacial heat transfer.
Permanent mold castings needs closer control over heat trans-fer and solidification phenomena due to higher cooling rates andbetter surface finish as compared to sand castings. Such control canlead to improvements on mechanical properties, surface corrosionresistance and soundness of the castings [14,15]. Gravity or pressuredie castings, continuous casting and squeeze castings are some ofthe processes where product soundness is more directly affectedby heat transfer at the metal/mold interface.
The methods of calculation of time-dependent hi existing inthe literature are mainly based on numerical techniques gener-ally known as methods of solving the inverse heat conductionproblem, IHCP [16–18]. The IHCP method is based on a completemathematical description of the physics of the process, supple-mented with temperature measurements in metal and/or mold.The inverse problem is solved by adjusting parameters in the
Fig. 1. Schematic two-dimensional solidification problem—g0 is the gravity vector.
mathematical description to minimize the difference between themodel-computed values and the experimental measurements.
The present work investigates the influence of some factors suchas the solidification growth direction with respect to the gravityvector, alloy solute content, melt superheat and initial melt temper-ature distribution on the interfacial heat transfer coefficient profilesduring solidification of Sn–Pb hypoeutectic alloys. Several castingexperiments have been carried-out with Sn–Pb alloys. Such exper-iments were carefully designed to permit a satisfactory control ofthe aforementioned factors. The experimental temperatures werecompared with simulations furnished by a numerical model for thedetermination of transient hi profiles.
2. Mathematical modeling
The numerical model approach used to simulate the macroseg-regation during solidification is based on the model previouslyproposed by Voller [19,20]. For times t < 0, the alloy is at the moltenstate, at the nominal concentration C0, and contained in the insu-lated two-dimensional mold defined by 0 < x < Xb and 0 < y < Yb,according to Fig. 1. Some modifications have been incorporated intothe original numerical approach, such as different thermo-physicalproperties for the liquid and solid phases, transient metal/coolantheat transfer coefficient, and a variable space grid to assure theaccuracy of results without raising the number of nodes, when com-paring with experimental data, since a time variable metal/moldinterface heat transfer coefficient will introduce a non-linearitycondition at the boundary condition.
Solidification begins by cooling the metal at the chill (0 < x < Xb,y = 0) until the temperature drops below the eutectic tempera-ture, TE. At time t > 0, three transient regions are formed: solid,solid + liquid (mushy zone) and liquid. During this process, soluteis rejected into the mushy zone.
In developing the numerical solution for equations of the ther-mal and solute coupled fields, the following assumptions weretaken:
and Physics 111 (2008) 444–454 445
I. The region is two-dimensional, defined by 0 < x < Xb and0 < y < Yb, where Yb is a point far removed from the segregationregion.
II. The segregation region remains free of porosity.III. The solid phase is stationary, i.e., once formed has zero velocity.IV. Due to the rapid nature of heat and liquid mass diffusion, in
a representative elemental averaging volume, the liquid con-centration CL, the temperature T, the liquid density �L and theliquid velocity uL are assumed to be constants [20].
V. The partition coefficient k0 and the liquidus slope mL, mayeither assumed to be constants or variables and read from theThermoCalc data bases.
VI. Equilibrium conditions exist at the solid/liquid interface, i.e. atthis interface we have:
T = Tf − mLCPbL or T = f (CPb
L ) (1)
and
C∗S = k0CL (2)
where T is the temperature, C is the concentration, Tf is thefusion temperature of the pure solvent, C∗
S is the interface solidconcentration.
VII. The specific heats, cS and cL, thermal conductivities, kS and kL,and the densities �S and �L, are constants within each phase,but discontinuous between the solid and liquid phases. Thelatent heat of fusion, �H is taken as the difference betweenphase enthalpies.
VIII. The metal/mold thermal resistance varies with time, and isincorporated in a global heat transfer coefficient defined as hi.
Considering the assumptions previously presented, and car-rying out the following variables transformation in the mass,momentum, energy, species for a two-dimensional set of equations,the mixture equations for multicomponent solidification becomethe following:
�∗ = �
�L(3)
c∗P = cP
cPL(4)
k∗ = k
kL(5)
C∗ = C − C0
CL(6)
x∗ = x
L(7)
y∗ = y
L(8)
V∗ = V
u∞(9)
t∗ = u∞L
t (10)
p∗ = p − p∞�∞u2∞
(11)
D∗ = D
DL(12)
�T = TL − mLC0 − T∞ (13)
and T∗ = T − TS
�T(14)
istry and Physics 111 (2008) 444–454
Table 1Nomenclature
Symbol
Bi Biot number—dimensionless, ratio of the internal thermalresistance of a solid to the boundary layer thermalresistance
Da Darcy number—dimensionless, equal to the fluid velocitytimes the flow path divided by the permeability of themedium
Fr Froude number—dimensionless number comparinginertial and gravitational forces. It quantifies the resistance
446 I.L. Ferreira et al. / Materials Chem
• Mass
∂�∗
∂t∗ + ∂(�∗u∗)∂x∗ + ∂(�∗v∗)
∂y∗ = 0 (15)
• Momentum x
∂(�∗u∗)∂t∗ + ∂(�∗u∗2)
∂x∗ + ∂(�∗u∗v∗)∂y∗ + ∂P∗
∂x∗
= 1Re
[∂
∂x∗
(�∗ ∂u∗
∂x∗
)+ ∂
∂y∗
(�∗ ∂u∗
∂y∗
)]− 1
DaReu∗ + �gX (16)
• Momentum y
∂(�v)∂t
+ ∂(�uv)∂x
+ ∂(�v2)∂y
+ ∂P
∂y
= ∂
∂x
(�L
�
�L
∂v
∂x
)+ ∂
∂y
(�L
�
�L
∂v
∂y
)− �Lv
KY+ �gY (17)
gx and gy, depending on the number of species, can be written fora binary alloy as
�gX,Y = 1Fr2
�g∥∥�g∥∥
[�0(1 − ˇ∗0
T T∗)]
(18)
• Energy
∂(�∗c∗T∗) + ∂
(�∗c∗u∗T∗) + ∂(�∗c∗v∗T∗)
∂t∗ P ∂x∗ P ∂y∗ P
= 1RePr
[∂
∂x∗
(k∗ ∂T∗
∂x∗
)+ ∂
∂y∗
(k∗ ∂T∗
∂y∗
)]− �∗
SSte
∂gS
∂t∗ (19)
• Species
∂
∂t∗ (�∗C∗) + ∂
∂x∗ (�∗C∗u∗) + ∂
∂y∗ (�∗C∗v∗)
= 1ReSc
∂
∂x∗
(�∗D∗ ∂C∗
∂x∗
)+ 1
ReSc
∂
∂y∗
(�∗D∗ ∂C∗
∂y∗
)(20)
All used terms are defined in Table 1.Table 2 summarizes the values of variables inserted in the gen-
eral equations, which were determined for each examined alloy.In the case of multicomponent alloys there must be one species
equation for each solute of the correspondent alloy system.• Mixture density
� =∫ 1−gL
0
�S d˛ + gL�L (21)
Table 2Parameters found for Sn–5 wt.% Pb, 10 wt.% Pb, 15 wt.% Pb and 30 wt.% Pb alloys
Dimensionless and dimensional parameters Symbol Sn–5 wt.% P
TL + mLC0 − T∞ �T 206.3min(XL,YL) L 0.04XL/L X∗
L 1.0YL/L Y∗
L 2.5�Lu∞L/�L Re 251961.4L|g|/u∞ 1/Fr* 0.39224�LCL/kL Pr* 8.793e−03ˇT�T ˇ∗
T 6.905e−03CL�T/�H Ste* 0.883�L/�L/DL Sc* 144u∞t/L t* 7500hLeq/kseq Bi –u∞Lt/K = L2/K Da –YEUT/kS + 2 · (YL − YEUT)/(kS + kL) Leq/keq –
of an object moving through a fluid and compares objectsof different sizes
Pr Prandtl number—ratio of the momentum and thermaldiffusivities
Re Reynolds number—ratio of the inertia and viscous forcesSc Schmidt number—ratio of the momentum and species
diffusivitySte Stefan number—ratio of sensible heat to latent heat�* Dimensionless densityc∗
PDimensionless specific heat
k* Dimensionless thermal conductivityC* Dimensionless concentrationx* Dimensionless x lengthy* Dimensionless y lengthV* Dimensionless velocityt* Dimensionless timep* Dimensionless pressureD* Dimensionless mass diffusivity�T Reference temperature rangeT* Dimensionless temperatureL Dimensional length, i.e., min(XL, YL) lengths
where gL and gS are the liquid and solid volume fractions, fLand fS are the liquid and solid mass fractions, u and v are thevolume averaged fluid velocities, DL and DS are the liquid andsolid mass diffusion coefficients, kL and kS are the liquid andsolid thermal conductivities, cPL and cPS are the liquid and solidheat capacities, and finally, �L and �S are the liquid and soliddensities.
The variables KX and KY represent the permeability in X and Ydirections [21,22]. Sing and Basu present various models to treatthe permeability of the mushy zone [23]. In this paper, an isotropicmodel has been adopted to treat the permeability, described by the
Fig. 2. Flow chart for the determination of metal/mold heat transfer coefficients.
Table 4Chemical composition (wt. Pct) of metals
Metal
Fe Si Mn Zn
Sn 0.041 0.024 a a
Pb 0.051 0.323 a a
a Less than 50 ppm.
Fig. 3. Sn–Pb phase diagram furnished by the software ThermoCalc AB,1 version N.
aforementioned authors as case II.
KX,Y = K0
[g3
L
(1 − gL)2
](30)
Instead of utilizing an average dendrite arm spacing, d, in thepermeability coefficient, K0 = d2/180, an approach recently reported
1 Thermo-Calc software is an exclusive copyright property of the STT Foundation(Foundation of Computational Thermodynamics, Stockholm, Sweden).
istry
448 I.L. Ferreira et al. / Materials Chem
in the literature [24], has been adopted, i.e.:
K0 = �21
180(31)
where �1 is the dendritic primary arm spacing determined byexperimental or theoretical growth models.
A micro-scale model is invoked to extract nodal values of liq-uid concentration CL from each solute density field (�C)Pb. The keyvariable in this calculation is the nodal liquid fraction calculated inthe previous step. A detailed discussion was previously presentedby Voller [20], in which the application of the back diffusion modelproposed by Wang and Beckermann [25] is suggested.
The local liquid concentration of each species is given by,
Pb
[CL ]P
= [�C]PbP − [�C]Pb,old
P + [�LgoldP + ˇPb�S(1 − gold
P )kPb0 ][CPb
L ]oldP
�Lgn+1P + ˇPb�S(1 − gn+1
P )kPb0 + (1 − ˇPb)�SkPb
0 (goldP − gn+1
P )(32)
The segregation parameter ˇ can vary as 0 ≤ ˇ ≤ 1. ˇ = 1 meansthat the lever rule has been adopted and ˇ = 0, provides Scheil’sequation.
The inverse problem consists of estimating the boundary heattransfer coefficient at the metal mold interface from metal temper-atures. This method makes a complete mathematical description ofthe physics of the process and is supported by temperature mea-surements at known locations inside the heat conducting body.The temperature files containing the experimentally monitoredtemperatures are used in a finite difference heat flow model todetermine hi, as described in a previous article [8]. The processat each time step included the following: a suitable initial valueof h is assumed and with this value, the temperature of each ref-erence location in casting at the end of each time interval �t issimulated by the numerical model. The correction in hi at each
Fig. 4. Experimental setups: (a) upward, (b)
and Physics 111 (2008) 444–454
interaction step is made by a value �hi, and new temperaturesare estimated [Test(hi + �hi)] or [Test(hi − �hi)]. With these values,sensitivity coefficients (�) are calculated for each interaction, givenby:
� = Test(hi + �hi) − Test(hi)�hi
(33)
The procedure determines the value of hi, which minimizes anobjective function defined by:
F(hi) =n∑
i=1
(Test − Texp)2 (34)
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 simulationassisted one and has been used in recent publications for determin-ing hi for a number of solidification situations [7]. The flow chartshown in Fig. 2 gives an overview of the solution procedure.
3. Experimental procedure
The composition of the Sn–5 wt.% Pb, Sn–10 wt.% Pb, Sn–15 wt.% Pb andSn–30 wt.% Pb alloys were obtained by thermal analysis, which means that, firstly,proportional amounts of tin and lead were measured by an analytical balance. Sec-ondly, the materials were put in a silicon carbide crucible, heated, melted andmixed. Finally, the cooling curves were obtained by imposing low cooling rates andthe typical experimental transformation temperatures (liquidus and eutectic) werecompared to those of the equilibrium Sn–Pb phase diagram. In a second step thecompositions were checked by X-ray fluorescence spectrometry. If necessary, fur-ther corrections of one of the elements were done until the desired composition wasattained. The corresponding Sn–Pb phase diagram is shown in Fig. 3.
The thermophysical properties of these alloys and the chemical composition ofthe pure elements are summarized in Tables 3 and 4, respectively.
The experimental data concerning Sn–5 wt.% Pb, Sn–10 wt.% Pb, Sn–15 wt.% Pband Sn–30 wt.% Pb alloys were obtained through three different solidification appa-ratus. The assemblage details of these systems are shown in Fig. 4. Furthermore, allmetallic components used in the water-cooled solidification systems are detailed inFig. 5.
downward and (c) horizontal systems.
I.L. Ferreira et al. / Materials Chemistry and Physics 111 (2008) 444–454 449
Fig. 5. Metallic components used in the water-cooled solidi
Fig. 6. Simulated and measured temperature responses for a Sn–5 wt.% Pb alloycasting at 5, 10, 15, 30 and 50 mm from the metal/mold interface.
In order to promote vertical upward solidification, an apparatus designed in sucha way that the heat was extracted by a water-cooled bottom provoking upward direc-tional solidification was used. A stainless steel cylindrical mold was used, having aninternal diameter (i.d.) of 50 mm, height of 110 mm and wall thickness of 5 mm.The inner vertical surface was covered with a layer of insulating alumina to mini-mize radial heat losses, and a top cover made of insulating material was employed
Fig. 7. Evolution of metal/mold interface heat-transfer coefficient (hi) as a functionof time for a Sn–5 wt.% Pb alloy solidified vertically upward and downward, andhorizontally.
fication experiments: (a) upward and (b) downward.
Fig. 8. Evolution of metal–mold interface heat transfer coefficients as a function oftime for Sn–Pb alloys during vertical upward solidification in water-cooled mold.
to reduce heat losses from the metal/air surface. The bottom part of the mold wasclosed with a thin (3 mm) carbon steel sheet.
The use of a water-cooled stainless steel chamber at the top of the casting haspermitted experiments for downward directional growth to be carried out. A stain-less steel split mold was used having an i.d. of 57 mm, height of 150 mm and wallthickness of 10 mm. As mentioned before, alumina was applied at the mold innersurface in order to prevent radial heat losses. The upper part of the split mold wasclosed by the cooling chamber (3 mm wall thickness).
Fig. 9. Typical experimental temperature responses at two locations in casting andchill: in casting at 20 mm from the metal/mold interface and in chill at 3 mm fromthis interface for a Sn–10 wt.% Pb alloy solidified with an initial melt temperature of10% above the liquidus temperature.
istry and Physics 111 (2008) 444–454
Dow
nwar
dH
oriz
onta
l
Sen
sors
(mm
)M
old
con
dit
ion
Mel
tsu
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tSe
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rs(m
m)
Mol
dco
nd
i-ti
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Mel
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Cas
tin
g:20
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30,5
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lish
edca
rbon
stee
l20
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Ch
ill:
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asti
ng:
20
––
5,10
,15,
30,5
0
––
5,10
,15,
30,5
0
450 I.L. Ferreira et al. / Materials Chem
In the upward and downward systems, the alloys were melted in situ and theelectric heaters had their power controlled in order to permit a desired melt super-heat to be achieved. To begin solidification, the electric heaters were disconnectedand at the same time, the water flow was initiated. Temperatures in the casting weremonitored during solidification via the output of a bank of type J thermocouples,sheathed in 1.6 mm outside diameter stainless steel protection tubes, and accuratelypositioned with respect to the heat extracting surface. In order to minimize temper-ature field distortions, the thermocouples were installed parallel to the isothermsin the casting [26]. Further, the thermocouple tips were placed as near as possi-ble to the transversal geometric center of the casting. The thermocouples were alsocalibrated at the melting temperature of tin exhibiting fluctuations of about 0.4 ◦C.Thermocouples readings (at intervals of 0.5 s) were collected by a data acquisitionsystem and stored in a computer.
Although the correct thermocouple positions with regard to the heat extractingsurface were verified before the experiments, a deviation of about ±1 mm from thenominal positions was observed for some of them as a result of interaction of sensorswith melt movement and casting shrinkage.
A third casting assembly was used for horizontal solidification experiments(Fig. 4c). In order to promote unidirectional heat flow during solidification, a lowcarbon steel chill with a wall thickness of 60 mm was used, with the heat extract-ing surface being polished. Each alloy was melted in an electric resistance-typefurnace until the melt reached a predetermined temperature. It was then stirred,degassed and poured into the casting chamber as soon as the desired melt superheatwas achieved. Temperatures in the chill and in the casting were monitored duringsolidification via the output of a bank of type J thermocouples accurately locatedwith respect to the metal/mold interface. Unidirectional heat flow was achieved byadequate insulation of the casting chamber.
Table 5 summarizes the main conditions used during the present experimentalinvestigation of Sn–Pb alloys.
4. Results and discussions
A typical example of the experimental cooling curves for the fivethermocouples inserted into the casting during upward unidirec-tional solidification of a Sn–5 wt.% Pb alloy can be seen in Fig. 6.The results of the experimental thermal analysis in castings werecompared with simulations furnished by a finite difference heatflow program. An automatic search technique selected the besttheoretical–experimental fit from a range of transient heat trans-fer coefficients profiles, hi = f(t). The technique has been detailedin previous articles [8,11,27]. For generating the simulated profilesshown in Fig. 6, a mean initial melt temperature has been adopted.
4.1. Effect of growth direction on hi
As shown in Fig. 6 the best theoretical–experimental fit has pro-vided an appropriate transient hi profile. Fig. 7 shows such profilesduring the course of different experiments involving downward,upward and horizontal directional solidification of hypoeutectic
Sn–Pb alloys against polished steel molds.
The heat transfer coefficient is clearly dependent on the orien-tation of solidification with respect to gravity. In upward verticalsolidification the effect of gravity causes the casting to rest onthe mold surface, but during downward solidification, this actioncauses the solidified portion of the casting to retreat from the moldsurface. It is well known that the reduction in the contact pressurebetween casting and mold surfaces leads to a consequent reductionin heat transfer efficiency.
The heat transfer coefficients for both upward and horizon-tal solidification are high in the initial stages of solidification, asa result of the good surface conformity between the liquid coreand the solidified shell. The mold expands while solidification pro-gresses due to the absorption of heat and the solid metal shrinksduring cooling. As a consequence, a gap develops because pressurebecomes 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 Ta
ble
5Su
mm
ary
ofth
ep
rese
nt
exp
erim
enta
linv
esti
gati
on
All
oys
Up
war
d
Mol
dco
nd
itio
nM
elt
sup
er-h
eat
Sn–5
wt.
%Pb
U-5
Poli
shed
carb
onst
eel
22◦ C
Sn–1
0w
t.%
PbU
-10
Poli
shed
carb
onst
eel
20/1
00
◦ C
Sn–1
5w
t.%
PbU
-15
Poli
shed
carb
onst
eel
21◦ C
Sn–3
0w
t.%
PbU
-30
Poli
shed
carb
onst
eel
20◦ C
istry and Physics 111 (2008) 444–454 451
Pb alloy. The experimental data are compared to those numeri-cally simulated by using the transient hi profile which providesthe best curve fitting. Temperature evolution was experimen-tally measured in two locations: in the chill at 3 mm from themetal/mold interface and in the casting at 20 mm from this inter-face.
I.L. Ferreira et al. / Materials Chem
contraction is effective, i.e., when the ingot is able to detach fromthe lateral walls. This will happen only after a determined solidshell is formed. In contrast, at the early stages of solidification inthe horizontal apparatus the good thermal contact is assured bythe liquid metal pressure exerted over the solid shell. When thesolid shell is able to contract, the air gap is formed and the thermalcontact decreases.
4.2. Effect of solute content on hi
Fig. 8 shows the time dependence of hi during the course of dif-ferent experiments of upward directional solidification of Sn–Pballoys. It can be clearly observed that hi decreases when the solutecontent is increased. Care should be taken when trying to explaina general tendency such as the influence of solute content on hi.Considering parameterized the inner mold surface condition (pol-ished in the present study), a number of other factors influencesuch tendency, such as the alloy fluidity, the solidification rangeand the thermophysical properties all of them varying with solutecontent. Only the integration of such factors will permit to establishthe effect of alloy solute content on hi. Furthermore, the direction ofgrowth will also be influent mainly for wide freezing range alloys.The permeability of the dendritic channels associated with a denserliquid resulting from solute segregation during solidification canhelp the wettability at the metal/mold interface and can even favorinverse segregation.
4.3. Effect of melt superheat on hi
Fig. 9 shows typical experimental thermal responses obtainedalong the horizontal directional solidification of a Sn–10 wt.%
Fig. 10. Time dependence of the metal/mold heat transfer coefficient during verticalupward and horizontal directional solidification of a Sn–10 wt.% Pb alloy.
Fig. 11. (a) Simulated and measured temperature responses for a Sn–5 wt.% Pballoy casting at 5, 10, 15, 30 and 50 mm from the metal/mold interface adoptinga mean melt temperature; (b) simulated and measured temperature responses fora Sn–5 wt.% Pb alloy casting at the same positions employing a initial melt temper-ature profile; (c) evolution of metal/mold interface heat-transfer coefficient (hi) asa function of time for the Sn–5 wt.% Pb alloy casting (polished mold).
istry and Physics 111 (2008) 444–454
Fig. 12. hi values versus wall temperature for upward directionally solidified Sn–Pballoys.
Fig. 13. Wall temperature as function of time for upward directionally solidifiedSn–Pb alloys.
452 I.L. Ferreira et al. / Materials Chem
Fig. 10 shows typical results of time-varying hi obtained dur-ing the course of solidification of a Sn–10 wt.% Pb alloy, in bothupward vertical and horizontal directional solidification experi-ments. It can be seen, that for horizontal solidification experimentshigher hi profiles are obtained as the melt superheat is increased.A reverse situation can be observed during vertical upward solid-ification with hi decreasing as the melt superheat is increased. Inboth cases the superheat delays the solidification evolution. In thehorizontal solidification a higher hi profile is associated with highermelt superheats, since the metal’s contraction from the mold wallwill also be delayed. In the upward solidification the casting weightwill contribute to a better metal/mold thermal contact under con-ditions of effective lateral contraction. This will happen sooner forsolidification without superheat, and as a consequence a higher hiprofile will be associated with lower melt superheats.
4.4. Effect of initial melt temperature profile on hi
Fig. 11 shows the temperature data collected in the metal duringthe course of upward solidification of a Sn–5 wt.% Pb alloy casting inthe vertical water-cooled apparatus, with the bottom heat extract-ing surface being polished. The experimental thermal responsescorresponding to five different positions inside the casting werecompared with the predictions furnished by the numerical solidi-fication model. The best theoretical–experimental fit has providedappropriate transient hi profiles for two different approaches: (i)a mean initial melt temperature (Fig. 11a), and (ii) a quadraticequation, based on experimental thermal readings, representingthe initial melt temperature as a function of position along thecasting length (Fig. 11b). A comparison between hi profiles deter-mined in each case is shown in Fig. 11c. It can be seen that thetendency is different, with the initial hi values being very similar upto about 25 s. After this point, the difference increases significantlywith increasing time and again the assumption of a constant ini-tial melt temperature overestimates the metal/mold heat transfercoefficient.
To make hi data more useful for the mathematical modeling ofthe solidification phenomena in castings, it would be convenient toconvert the hi–time relationships into hi–casting surface temper-ature relationships. Such hi–temperature curves are shown in thefollowing section.
4.5. Influence of hi on the solidification kinetics
There are many practical casting cases where cooling is inhibitedby adding thermal barriers at the metal/mold interface, e.g., ceramicmold coatings, gas layers, etc. In the present experimental study,for the water-cooled solidification apparatus, hi is an overall heattransfer coefficient between the casting surface and the coolantfluid, which can be represented by:
1hi
= 1hm
+ eC
kC+ eS
kS+ 1
hw(35)
where hm is the heat transfer coefficient between the casting sur-face and the mold coating, eC and eS are the thicknesses of moldcoating and steel sheet, respectively, kC and kS are the coatingand steel thermal conductivities, respectively, and hw is the mold-coolant heat transfer coefficient. The two first components on theright hand side of Eq. (35) are generally the largest.
Figs. 12–15 show hi coefficient profiles as a function of the cast-ing wall temperature, wall temperature as a function of time, Biotnumber as a function of solid layer and thermal resistances versussolid layer, respectively, along the unidirectional upward solidifi-cation of Sn–5 wt.% Pb, 15 wt.% Pb and 30 wt.% Pb alloys. The Biotnumber gives an indication of how large is the alloy solid shell ther-
Fig. 14. Biot number as a function of solid layer for upward directionally solidifiedSn–Pb alloys.
I.L. Ferreira et al. / Materials Chemistry and Physics 111 (2008) 444–454 453
r (a) S
Fig. 15. Thermal resistances as a function of the solid layer fo
mal resistance in relation to the interfacial thermal resistance. Byanalyzing Fig. 14, it can be seen that for a same Biot number of 0.5the respective solid layers are 18, 33 and 43 mm as the solute con-tent is increased from 5 wt.% Pb alloy to 30 wt.% Pb alloy. Fig. 15shows the relative significance of heat conduction with respectto the Newtonian heat transfer along the solidification of three
Sn–Pb alloys. It can be seen that the point where the thermalresistances are equal (Bi = 1.0) is displaced to the right side withincreasing alloy solute content. Such kind of plot permits to estab-lish a range of solidifying layers for which the interface resistance isdominant.
5. Conclusions
The following major conclusions can be derived from the presentstudy:
1. The transient metal/coolant heat transfer coefficient, hi, has beensatisfactorily determined as a result of an approach based onmeasured temperatures in metal and/or mold and numericalsimulations furnished by a solidification model.
2. The transient metal/coolant heat transfer coefficient, hi, can beexpressed as a power function of time and rise with decreasinglead content of the alloy.
3. The metal/mold heat transfer coefficient was shown to dependon the melt temperature variations along the casting length,which means that a mean value for the initial melt tempera-
ture may not be enough for an accurate characterization of thetransient hi profile.
4. There is a clear indication that hi is strongly dependent on thedirection of solidification, and that must be considered for anaccurate simulation of freezing patterns in castings.
Acknowledgements
The authors acknowledge financial support provided by FAPESP(The Scientific Research Foundation of the State of Sao Paulo, Brazil),CNPq (The Brazilian Research Council) and FAEPEX –UNICAMP andIAF (Institut fur Angewandte Forschung), HS-Karlsruhe/Germany.
References
[1] R.W. Lewis, R.S. Ransing, Finite Elem. Anal. Des. 34 (2000) 193.[2] H.M. Sahin, K. Kocatepe, R. Kayikci, N. Akar, Energ. Convers. Manage. 47 (2006)
