Heat Transfer Coefficient at Cast-Mold InterfaceDuring Centrifugal Casting: Calculation of Air Gap

JAN BOHACEK, ABDELLAH KHARICHA, ANDREAS LUDWIG, MENGHUAI WU,and EBRAHIM KARIMI-SIBAKI

During centrifugal casting, the thermal resistance at the cast-mold interface represents a mainblockage mechanism for heat transfer. In addition to the refractory coating, an air gap begins toform due to the shrinkage of the casting and the mold expansion, under the continuous influenceof strong centrifugal forces. Here, the heat transfer coefficient at the cast-mold interface h hasbeen determined from calculations of the air gap thickness da based on a plane stress modeltaking into account thermoelastic stresses, centrifugal forces, plastic deformations, and atemperature-dependent Young’s modulus. The numerical approach proposed here is rathernovel and tries to offer an alternative to the empirical formulas usually used in numericalsimulations for a description of a time-dependent heat transfer coefficient h. Several numericaltests were performed for different coating thicknesses dC, rotation rates X, and temperatures ofsolidus Tsol. Results demonstrated that the scenario at the interface is unique for each set ofparameters, hindering the possibility of employing empirical formulas without a precedingexperiment being performed. Initial values of h are simply equivalent to the ratio of the coatingthermal conductivity and its thickness (~ 1000 Wm�2 K�1). Later, when the air gap is formed, hdrops exponentially to values at least one order of magnitude smaller (~ 100 Wm�2 K�1).

https://doi.org/10.1007/s11663-018-1220-0� The Author(s) 2018. This article is an open access publication

I. INTRODUCTION

HORIZONTAL centrifugal casting is an importantindustrial process used especially for the production ofhigh-quality seamless tubes and outer shells of workrolls. In this process, the effect of centrifuging is twofold.First, it is the fictitious centrifugal force making theproduction of axisymmetric hollow castings even possi-ble by pushing the molten metal against the inner wall ofthe cylindrical mold. Second, the interaction betweeninertial forces and the vector of the gravitationalacceleration induces the so-called pumping effect,responsible for thorough mixing,[1] the growth of fineequiaxed grains, and superior mechanical properties ofthe cast.[2,3]

As with many other industrial processes, horizontalcentrifugal casting has been studied with increased

attention, with the help of various numerical techniques,in order to gain a better understanding of the processand underlying physical phenomena. While some of thenumerical studies concentrate more on simulating flowdynamics, such as the mold filling, waves propagatingover the free surface, and complex buoyant flowpatterns inside the molten metal,[4–11] others focus moreon heat transfer and solidification, often assumingcoupling with simple segregation models.[12–14] Thelatter is naturally more frequent within the centrifugalcasting community. Solidification is usually modeled bymeans of applying the enthalpy method with appropri-ate rules for a liquid fraction evolution in the mushyzone. In order to construct useful and realistic heattransfer models, precise and accurate material propertiesand boundary conditions are necessary. Heat transfercoefficients are usually imposed at boundaries, generallybeing determined from empirical formulas for theNusselt number. Materials properties are generallytemperature dependent and must be specified for allzones, i.e., the casting, the mold, and the coating. Thethickness of the coating, a kind of a refractory material,such as ZrO2, is usually small (~ 1 mm); therefore, innumerical models, it is often simplified by an assump-tion of the thin-wall (zero-capacity) model. The coatingis applied on the inner surface of the mold in order toinsulate the mold from high temperatures and also tocontrol to a certain extent the solidification rate. The

JAN BOHACEK, ABDELLAH KHARICHA, and ANDREASLUDWIG are with the Chair of Simulation and ModelingMetallurgical Processes, Metallurgy Department, MontanuniversitaetLeoben, Franz-Josef-Str. 18/III, 8700 Leoben, Austria. Contact e-mail:jan.bohacek@unileoben.ac.at MENGHUAI WU and EBRAHIMKARIMI-SIBAKI are with the Christian Doppler Laboratory for‘‘Advanced Process Simulation of Solidification and Melting,’’,Franz-Josef-Str. 18/III, 8700 Leoben, Austria.

Manuscript submitted October 26, 2016.Article published online March 12, 2018.

METALLURGICAL AND MATERIALS TRANSACTIONS B VOLUME 49B, JUNE 2018—1421

general consensus is that it tends to stick firmly to themold surface. A time-dependent scenario at the contactbetween the casting and the coating attached to themold surface is perhaps one of the weakest points of allcurrently available heat transfer models.

Only during the first seconds of the casting, the moltenmetal is in perfect contact with the coating, as pointed outin Reference 15. Immediately after that, a so-calledmicroscopic air gap is formed, whose properties such asthickness and temporal growth are strongly influenced bya surface roughness of the mold and the coatingeventually. Earlier studies[16] show that the importanceof the surface roughness has been for many yearsunderestimated. Furthermore, in the literature (e.g.,Reference 17), there are interesting numerical worksavailable, taking into account the surface roughness andtrying to evaluate the effective thickness of the micro-scopic air gap by using simple geometrical operations. Astime proceeds, the first layer of the solid has enoughstrength to withstand the metallostatic pressure, wherebythe air gap thickness gradually grows and the microscop-ically thin contact is permanently lost. Such an air gap isoften referred to as the macroscopic air gap. Please notethat while the cast-mold contact has been extensivelystudied in static castings and a large body of experimentalevidence of microscopic and macroscopic air gap behav-ior has been presented, it is not yet clear whether at leastqualitatively the same observations would apply to thecentrifugal casting. Unlike static casting, extreme cen-trifugal forces are exerted on the liquid metal being cast,which strive to delay the subsequent air gap formation.We deduce that the high centrifugal pressure may be ableto significantly reduce the microscopic air gap. Further-more, we also believe—and this has been proven in thisarticle—that once the macroscopic air gap is formed, thecentrifugal force has a negligible impact on its growth.

In order to cope with time-dependent thermal resis-tance induced by the formation of an air gap, variousapproaches have been adopted in earlier numericalstudies of centrifugal casting. Naturally, a simplestapproach would be an assumption of a perfect contactformed throughout the entire casting, which was used,e.g., by Xu et al.,[18] Gao and Wang,[19] and Cooket al.[20] Bohacek et al.[21] pointed out the importance ofan air gap in their findings and conclusions, yet in thenumerical model, it was not taken into account.Humphreys et al.[22] calculated the heat transfer at theinterface by employing a ‘‘virtual wall’’ technique withcumulative thermal resistances; however, they did notprovide parameters to calculate them. Chang et al.[23]

used arbitrary values of heat transfer coefficients at theinterface, constant during the casting and increasing forhigher rotation rates. Other researchers, such as Kanget al.[24] and Kang and Rohatgi,[25] have also usedtime-independent heat transfer coefficients. Ebisu[26] andKamlesh[27] assumed an exponential decay of the radia-tive heat flux through the interface as follows:

q ¼ q0e�bs tð Þ; ½1�

where q0, b, and s(t) are the initial heat flux throughthe interface, a damping coefficient, and the current

solidified thickness, respectively. A similar approachwas adopted by Lajoye and Suery[28] and was laterwidely used by other authors such as Raju andMehrotra,[29] Drenchev et al.,[30] Panda et al.[31]

Instead of the heat flux, a time-dependent heat transfercoefficient h was considered at the interface anddefined by the following formula:

h ¼ h0hfh0

� �sðtÞ=d; ½2�

where h0, hf, and d are the initial and the final heattransfer coefficient and the casting thickness, respec-tively. Naturally, Eqs. [1] and [2] are not equivalent.However, it is worth noting that when the heat flux qwas replaced with the heat transfer coefficient h,Eqs. [1] and [2] would become identical provided thatthe damping coefficient was defined as

b ¼ � 1

dlog

hfh0

� �: ½3�

Recently, Nastac[32] applied a different approachbased on calculating an equivalent convective heattransfer coefficient h to simulate the effect of the coatingand the air gap, which can be written as

h ¼ hakCkC þ dCha

; ½4�

where kC, dC, and ha represent thermal conductivityand thickness of the coating and the heat transfer coef-ficient between the casting and the coating, which isdefined as follows:

ha ¼ h0 þ hf � h0ð Þ 1� min 1;t0t

� �h ic� �; ½5�

where t0, t, and c stand for the time to initiation ofsolidification, the current time, and a constant expo-nent. Table I summarizes the values of h adopted bythe aforementioned authors. Obviously, all of theaforementioned approaches contain at least oneunknown parameter, which needs to be adjusted, e.g.,by means of an experiment. Although especially thechoice of the function given by Eq. [2] appears to be areasonable solution, a careful fine-tuning of hf isrequired in order to reflect, or at least approximate,real-life conditions. According to Vacca et al.,[33] whoperformed a valuable experimental study of the heattransfer coefficient at the interface involving theinverse task, values of the heat transfer coefficientadopted in centrifugal casting simulations are unreli-able and usually arbitrary. A similar approach com-bining a simulation and experiment was employed bySusac et al.[34] and Sahin et al.[35] The inverse task is,however, in general, computationally very intensive.Moreover, time-dependent experimental data are nec-essary at least at one point, located close to the cast-mold interface. While the inverse task cannot be prac-tically applied in a typical centrifugal casting simula-tion, it is an excellent tool for validating othernumerical models or determining constants in

1422—VOLUME 49B, JUNE 2018 METALLURGICAL AND MATERIALS TRANSACTIONS B

empirical models. In Reference 36, the inverse task,the inverse heat conduction problem (IHCP), wassolved by a popular nonlinear estimation technique,originally developed by Beck.[37] The casting material,A356 Al alloy, was cast into a carbon steel mold. TheIHCP proved that the heat transfer to the mold can besignificantly improved by applying the pressure loadduring solidification in terms of restoring a contactbetween the mold and the casting.

In Reference 38, the research concerns the simulationof trip continuous casting. An engineering approach wasemployed to approximate the thermal resistance at thestrip-mold interface from the heat transferred to thecooling water. The water flow rate and temperature wererecorded at different positions along the length of thestrip for this purpose. At selected points in the liquidpool, the calculated cooling curves agreed strongly withthose obtained from Inconel (American Special Metals,Corp., Miami, FL) sheathed thermocouples.

In addition, a direct measurement of the macroscopicair gap can be performed by using linear variabledifferential transformers (LVDTs). However, this tech-nique during the centrifugal casting is limited to staticcastings due to high rotations of the mold duringcentrifugal casting. For example, in Reference 39, theheat transfer coefficient at the cast-mold interface of astatic casting was determined from the inverse task. Theair gap thickness was measured with the help of theLVDTs. Finally, a correlation was found, defining theheat transfer coefficient as a function of the air gapthickness. The effect of the surface roughness of themold was analyzed. As expected, during formation ofthe microscopic air gap, findings showed that the smallerdegree of roughness provides stronger contact with themold and that the heat transfer coefficient is, therefore,higher. Consequently, the smaller the surface roughnessof the mold, the earlier the macroscopic air gap occurs.On the other hand, the ultimate heat transfer coefficient,when solidification is nearly complete, is insignificantlyinfluenced by the surface roughness.

As a numerical alternative of estimating the air gapthickness, one could suggest ignoring stresses built up inthe casting and the mold and using the thermal expansioncoefficient to calculate the shrinkage simply by assumingdisplacements, independent of direction, i.e., uniform rateof deformation of control volume. This technique wasapplied, e.g., by Taha et al.,[40] for a static casting.Accuracy and reliability are, however, doubtful due tothe missing thermal stresses, which may significantly altertotal displacements. In addition, unlike static castings,extreme forces in the centrifugal casting process act onthe casting, leading to yielding of the material especiallyat early stages. In addition, for this reason, the afore-mentioned approach should be avoided. A strategy thatincorporates a more complete and holistic model ofphysics was outlined by Kron,[41] who suggested takinginto account vacancies formed due to the thermalexpansion of the mold and the material being cast, aswell as elastic stresses acting as a consequence ofthermally induced strains. Kron et al.[42] developed thethermomechanical model, based on the plane stressmodel, assuming elastic materials. They showed that themodel predicts accurately the casting scenario only forgrain-refined alloys such as Al-4.5 pct Mg. When thesolid grains are surrounded by liquid, the materialbecomes more ductile and consequently the microscopicair gap is suppressed. On the other hand, in thenon-grain-refined case, the elastic thermomechanicalmodel does not perform particularly effectively. However,one may wish to interpret these findings, the importantmessage of Reference 42 could be formulated as follows:The peak value of the heat transfer coefficient is larger forthe grain-refined alloy due to ductile suppression of themicroscopic air gap, but since the formation of themacroscopic air gap starts earlier in this case, totalsolidification times are almost identical. Lagerstedt, acolleague of Kron, pointed out in the future workchapter of his doctoral thesis[43] that including plasticityin the stress model probably should be the next step indeveloping an accurate shrinkage model.

Table I. Values of h Adopted by Different Authors

Authors Type of h Value (W m�2 K�1)

Xu et al.[18] — perfect contactGao and Wang[19] — perfect contactCook et al.[20] — perfect contactBohacek et al.[21] — perfect contactHumphreys et al.[22] — not availableChang et al.[23] constant 1000 to 2600Kang et al.,[24] Kang and Rohatgi[25] constant 1000Ebisu[26] variable not availableKamlesh[27] variable not availableLajoye and Suery[28] variable 400 to 80,000 (1/10)*Raju and Mehrotra[29] variable 100 to 10,000 (1/10)Drenchev et al.[30] variable 420 to 84,000 (1/10)Panda et al.[31] variable 50 to 5000 (1/10)Nastac[32] variable 90 to 6000 (3/100)Vacca et al.[33] variable (exp.) 50 to 870 (6/100)

*hf/h0 ratio.

METALLURGICAL AND MATERIALS TRANSACTIONS B VOLUME 49B, JUNE 2018—1423

In Reference 44, Schwerdtfeger et al. underlined theimportance of the displacement reference. Classically, instress theory, the displacement is the distance of aspecified atom from the position it had assumed whenthe entire solid body was stress free. Such a situation,however, does not occur during solidification; therefore,they recommended defining the displacement as thedistance of a specified atom from the position where itwas at the moment of solidification. Consequently, oneshould work with stress rates and strain rates rather thanwith stresses and strains. In their study, plastic deforma-tions were also considered and added to the elastic onesby assuming an empirical strain-hardening equation.

Nowadays, most of the commercial software availableon the market, including MAGMASOFT (MAGMA inAachen, North Rhine-Westphalia, Germany), PRO-CAST (ESI Group, Paris, France), and THERCAST(TRANSVALOR S.A., Mougins, France), offers mod-ules for thermomechanical calculations, and often theuser can choose from several elastic-plastic models. InReference 45, in conclusion, Kron et al. stated that anaccurate modeling of the air gap formation can only berealized through fully coupled thermomechanical models.They highlighted that the prediction of the air gap, donewith the commercial codes, is not satisfactory, suggestingthat the solidification shrinkage in the air gap vicinityshould be relaxed by the liquid and, therefore, contributemore to a cavity formed in the top of the casting or to theporosity. In conclusion, the entire strain model needs tobe defined more precisely. Difficulties associated with airgap modeling were summarized in Reference 46 as thefollowing. High-temperature elastic constants are gener-ally hard to obtain. Defining material properties of themushy zone remains a challenging topic. Currently, atransition model is available described by the Percolationtheory. Next, obtaining proper values of rheologicalparameters in the power law equation is difficult. Finally,other difficulties or discrepancies, common to all numer-ical models, are related to oversimplifying assumptionsand numerical errors.

In the article by Nayak and Sundarraj,[47] it wasshown that while it is accurate enough to assume aconstant value of the interface heat transfer coefficientduring the entire casting into the sand mold, it is not thecase with the metal mold. Furthermore, the rate of gapformation significantly affects the solidification process.

The cast-mold interface, namely, the coating and an airgap, represents a significant blockage for the heat transferand solidification. The thermal resistance of the coating isoften negligible compared to that of the air gap.Therefore, a thermal resistance of such an interface hasto be carefully determined in order to allow reliable andtrustworthy numerical simulations. Existing empiricalformulas describing the heat transfer scenario at theinterface should only be applied when validated againstexperimental data. The formation of an air gap dependson many factors such as material and mechanicalproperties of the casting and the mold, the coatingproperties, and the process parameters (initial tempera-tures, the pouring temperature, the casting geometry, andthe rotation rate). Obviously, setting up a generalized andunique formula for the heat transfer coefficient at the

interface a priori would be very hard, if not impossible. Inthe present article, we target developing a simple,computationally cheap, and robust algorithm for calcu-lating the air gap at the cast-mold interface during thecentrifugal casting, which could be used as an alternativeto often doubtful empirical formulas. A schematic of theconfiguration at the interface is shown in Figure 1.

II. NUMERICAL MODEL

During the centrifugal casting of cylindrical parts, it isreasonable to assume that fields of variables and otherproperties are uniform in the tangential direction. Infact, also, axial variations will be often small and,therefore, could be neglected, too. This finding directlysuggests using a plane stress model. Although differentvariations of plane stress models have been used indiverse industrial applications, such as autofrettage ofgun barrels, strain-hardened pressure vessels, and mul-tilayer seamless pipes, in the past,[48–51] they have rarelybeen employed in air gap thickness calculations. Espe-cially, concerning the centrifugal casting, to the best ofour knowledge, not a single match was found in theliterature survey.Radial and tangential stresses, rr and rt, are coupled

through the equilibrium equation:

drrdr

þ rr � rtr

þ qX2r ¼ 0; ½6�

where q, X, and r are the density, the rotation rate,and the radial coordinate, respectively. When onlyelastic deformations are considered, stresses are cou-pled with strains via the Hooke’s law with the ther-moelastic term. Such a relationship takes the followingform:

et ¼ 1E rt � mrr½ � þ aT;

er ¼ 1E rr � mrt½ � þ aT;

½7�

where m, E, a, and T are Poisson’s ratio, Young’s mod-ulus, the thermal expansion coefficient, and the tem-perature, respectively. Strains and total radialdisplacements are related through the following laws:

et ¼u

r; er ¼

du

dr: ½8�

It would, however, be incorrect to only considerelastic strains. Since the casting in a semisolid state caneasily yield under strong centrifugal forces, plasticdeformations also must be taken into account. Then,total strains can be conveniently expressed as a sum ofelastic and plastic strains as follows:

et ¼ 1E rt � mrr½ � þ aTþ ept ;

er ¼ 1E rr � mrt½ � þ aTþ epr ;

½9�

where ept and epr represent plastic strains in correspond-ing directions. In addition, it is also physically mean-ingful to consider a temperature-dependent Young’smodulus. Substituting total strains in Eq. [9] with dis-placements from Eq. [8], and by combining the

1424—VOLUME 49B, JUNE 2018 METALLURGICAL AND MATERIALS TRANSACTIONS B

resulting equations with Eq. [6], we can arrive at thefollowing ordinary differential equation for the totaldisplacement u in the casting:

d2u

dr2þ 1

r

du

dr1þ r

1

E

dE

dr

� �� u

r21� rm

1

E

dE

dr

� �

¼ a 1þ mð Þ dT

drþ T

1

E

dE

dr

� �� qrX2 1� m2

� 1E

þF ept ; epr

� withF ¼ 1

E

dE

drepr þ mept�

þ deprdr

þ mdeptdr

þ 1� mr

epr �1� mr

ept :

½10�In the mold, Eq. [10] is considerably simplified

because one assumes a constant Young’s modulus andpure elastic deformations. The differential equation forthe total displacement u in the mold becomes

d2u

dr2þ 1

r

du

dr� u

r2¼ a 1þ mð Þ dT

dr� qrX2 1� m2

� 1E: ½11�

Note that in Eqs. [10] and [11], subscripts […]S and[…]M denoting the mold and the casting are omitted forthe sake of brevity. Equations [10] and [11] can be solvedprovided that plastic strains epr and ept are known. Inorder to determine them, a universal stress-strain curve,usually assumed to be equivalent to the stress-straincurve obtained from the uniaxial loading test, must beknown in advance. The universal stress-strain curverelates two scalar quantities: the effective stress r andthe effective plastic strain ep. Several models of theeffective stress r exist. Here, the von Mises stress,

calculated by assuming a principal stress loading, wasused in the following form:

r ¼ 1ffiffiffi2

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffirr � rtð Þ2þr2r þ r2t

q: ½12�

The von Mises stress r is coupled with the increment

of the effective plastic strain dep by the Prandtl–Reuss(Levy–Mises) flow rule as follows:

depr ¼ dep

rrr � 0:5rtð Þ ½13�

and

dept ¼ dep

rrt � 0:5rrð Þ: ½14�

Knowing or assuming dep, the plastic strains depr anddept can be easily calculated and then used in Eq. [10] toextrapolate the total displacement u. In Eqs. [13] and

[14], the increment of the effective plastic strain dep isused and not the effective plastic strain ep, which meansthat the plastic strain history (or the loading path) isvery important. Correspondingly, a progressive loadmust be also applied in the simulation. Here, suchloading is automatically realized by a time-dependenttemperature field and a gradual progress ofsolidification.In the present study, a temperature-dependent, per-

fectly elastic-plastic material is used, as shown inFigure 2. At a given temperature, the material deformselastically until a certain threshold of the effective stress,known as the yield strength, is reached, at which pointthe material starts yielding with no further increase ofthe effective stress. Instead of the elastic-perfectly plasticmaterial, any kind of other material could be used suchas a strain-hardening or a strain-softening material.In the following text, we summarize all the facts,

assumptions, and solution strategy necessary to run asuccessful numerical simulation and obtain a reasonableair gap.

Fig. 1—Schematic of the configuration at the cast-mold interface. Inthe case of perfect contact, the air gap disappears.

0 0.005 0.01 0.0150

500

1000

1500

2000

Strain [mm/mm]

Stre

ss [M

Pa]

473K (200°C)

673K (400°C)

873K (600°C)

1373K (1100°C)1073K (800°C)

1473K (1200°C)

Fig. 2—Uniaxial stress-strain curves of elastic-perfectly plasticmaterial used in the present study.

METALLURGICAL AND MATERIALS TRANSACTIONS B VOLUME 49B, JUNE 2018—1425

Facts:

(1) The mold constantly expands during the entirecasting process.

(2) The casting also expands but only during the earlystage of casting. Later, the strength of the solidifiedpart of the casting is sufficient to withstand cen-trifugal forces; therefore, the casting contracts.Consequently, the air gap forms.

Assumptions:

(1) The mold undergoes purely elastic deformations. (Inreality, it may not be true, especially in the vicinityof the cast-mold interface. At the initial stage ofcasting, extreme stresses may occur, causing a typeof damage known as ‘‘fire cracks,’’ to the inner partof the mold.)

(2) Mechanical properties of the mold material areconstant. Thermophysical properties may vary withthe temperature.

(3) The casting may deform both elastically and plasti-cally. A temperature-dependent, perfectly elas-tic-plastic material is assumed (Figure 2).

(4) Mechanical properties (Young’s modulus E andyield strength Y) of the casting material are tem-perature dependent. Thermophysical properties mayalso vary with the temperature.

(5) Only the radiative and the conductive heat transfermechanisms are expected within the air gap. Theconvective mechanism is neglected due to the smallsize of the air gap.

In addition to Eqs. [10] and [11], we also need to solvethe heat conduction for the temperature T. In thecylindrical coordinate system, it takes this form in themold:

qMcpM@T

@t¼ 1

r

@

@rkM

@T

@r

� �; ½15�

where qM, cpM, and kM are the density, specific heat,and thermal conductivity of the mold material, respec-tively. Similarly, in the casting, it can be written as

qScpS@T

@t¼ 1

r

@

@rkS

@T

@r

� �þ qSLf

@gs@t

; ½16�

where qS, cpS, and kS are density, specific heat, andthermal conductivity of the casting material, respec-tively. The last term is a latent heat source term due tothe phase change, in which Lf and gs represent thelatent heat and the solid fraction, respectively. In thepresent study, a simple linear relationship is consideredbetween the solid fraction gs and the temperature T.Other relationships, however, could also be considered(e.g., the lever rule or the Gulliver–Scheil equation).The heat conduction equations, Eqs. [15] and [16], arecoupled via the heat flux at the cast-mold interface. Athin-wall model, also known as a zero-capacity model,was used to numerically simplify the situation at theinterface by considering only a thermal resistance,

exerted by the coating and possibly the air gap. Then,the heat flux at the interface reads as

q ¼ kifcTS � TM

drM þ drS; ½17�

kifc ¼kSkMkCka drM þ drSð Þ

kCka drMkS þ drSkMð Þ þ kSkM dCka þ dakCð Þð Þ ;

½18�

where kifc, kC, and ka denote the effective thermal con-ductivity, the thermal conductivity of the coating, andthe air gap, respectively. Other quantities are explainedin Figure 1. The air gap thermal conductivity ka is, infact, the sum of the thermal conductivity of air ka,physand a thermal conductivity, which is equivalent to theradiative heat transfer through the air gap, given by

ka ¼ ka;phys þ rda T0S þ T��

T02S þ T�2�

; ½19�

where r is the Stefan–Boltzman constant(5.67 9 10�8 W m�2 K�4). The temperatures T’S andT* must be given in Kelvin. Note that without the airgap (da = 0), Eq. [18] is still valid and represents theeffective thermal conductivity only in the presence ofthe coating. The reader should be reminded that in thisstudy, the black body radiation model has been takenfor its simplicity and convenience. However, when tar-geting more accurate results, considering gray bodies isbetter justified and Eq. [19] would then become

ka ¼ ka;phys þ rda T0S þ T��

T02S þ T�2�

1=eS þ 1=eC � 1ð Þ�1;

½20�

where eS and eC are emissivity coefficients of both sur-faces enclosing the air gap, which belong to the castingand the coating, respectively. Using Eq. [20]. insteadof Eq. [19] will naturally reduce the radiative heattransfer. In reality, the air between the casting and theshell may, as a participating gas, further reduce theradiative heat transfer. A description of the corre-sponding mathematical model can be found, e.g., inReference 52.Concerning thermal boundary conditions, both the

free surface of the casting and the outer surface of themold are considered to be adiabatic:

dTdr

rið Þ ¼ 0;

dTdr

roð Þ ¼ 0:½21�

In simulations focused on a comparison with exper-imental data, thermal boundary conditions, however,should be specified more precisely, e.g., with the help ofexisting empirical formulas for the Nusselt number[53,54]

in rotating geometries.In addition to Eq. [21], boundary conditions have to

be specified also for Eqs. [10] and [11]. Here, we have todistinguish between two cases: a perfect contact or an air

1426—VOLUME 49B, JUNE 2018 METALLURGICAL AND MATERIALS TRANSACTIONS B

gap. In the case of the contact, appropriate boundaryconditions take the following form:

rrS rið Þ ¼ 0uS Rð Þ ¼ uM Rð ÞrrM roð Þ ¼ 0:

½22�

Otherwise (the air gap),

rrS rið Þ ¼ 0rrS Rð Þ ¼ 0rrM Rð Þ ¼ 0rrM roð Þ ¼ 0:

½23�

Differential equations for total displacements,Eqs. [10] and [11], and temperature, Eqs. [15] and [16],were all solved using the finite difference method withsecond-order accurate central difference schemes forderivatives (including points at the boundaries). In theradial direction, the casting and the mold were dividedinto NS and NM uniformly spaced grid points with thedimensions drS and drM of 1 mm (Figure 1). The samegrids with uniform spacing were used for both quantitiesu and T. For Eqs. [15] and [16], the implicit backwardEuler method was used for time-stepping. In addition,an iterative approach was necessary for heat conductionequations due to temperature-dependent thermophysi-cal properties, the nonlinear heat flux q at the cast-moldinterface (Eq. [17]), and especially the stiff, latent heatsource term. Treatment of the latent heat source termwas realized by using a semi-implicit method proposedby Voller and Swaminathan.[55] A detailed description ofthe discretization can be found in Reference 56.Although systems of equations are unconditionallystable, the time-step size should be small enough sothat the loading rate still allows finding correct andphysically meaningful increments of plastic strains.Here, the time-step of 0.1 seconds was found to bereasonable.

Solution strategy:

(1) Initialize fields of temperature, solid fraction, plasticstrains, stresses, and air gap thickness.

(2) For time tn+1, solve heat conduction equations,Eqs. [15] and [16], coupled by the heat flux at theinterface (Eq. [17]), with the air gap thickness dafrom the previous time tn and obtain new tempera-ture T and solid fraction gs fields. In our numericaltests, usually between two and five iterations werenecessary to drop scaled residuals below 10 9 10�8

for the latent heat source term and the nonlinearheat flux q, respectively. The residuals were calcu-lated according to the following formula:

res ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiNP

i Ti � Toldi

� 2qP

i Toldi

½24�

where N is the total number of cells (NS+NM). Ti

and Toldi are the current temperature and the

temperature from the previous iteration both taken atthe grid point with the index i(3) Use Figures 3 and 4 to find new values of yield

strength Y and Young’s modulus E of the castingmaterial.

(4) Assume an air gap. For time tn+1, solve Eq. [11] fortotal displacements u of mold with the boundaryconditions for the radial stress rrM in Eq. [23].Similarly, for time tn+1, solve Eq. [10] for totaldisplacements u of the casting with the boundaryconditions for the radial stress rrS in Eq. [23] andthe plastic strains epr and ept from the previous time tn.

(5) If the total displacement of the casting at the inter-face is greater than that of the mold, the casting andthe mold are in perfect contact, i.e., no air gap isformed. Otherwise, the cast-mold contact is lost andthe air gap is formed. If the earlier is true (perfectcontact), recalculate Eqs. [10] and [11] with theboundary conditions given in Eq. [22]. Otherwise,evaluate the air gap thickness as

da ¼ uM Rð Þ � uS Rð Þ ½25�(6) In the casting, evaluate radial and tangential stresses

by using the following explicit formulas, which canbe obtained by straightforward manipulations ofEqs. [8] and [9]:

rrS ¼ E1�m2

dudr

� epr

� �þ m u

r � ept� h i

� aE 11�v T;

rtS ¼ E1�m2

ur � ept�

þ m dudr

� epr

� �h i� aE 1

1�v T:½26�

(7) Using Eq. [26] in Eq. [12], calculate von Misesstresses r in the casting and compare them with yieldstresses Y obtained from the stress-strain curve(Figure 3). Then, identify only the points P thatyield P ¼ r � Yð Þ.

(8) For the points P, new increments dep of the effectiveplastic strain ep must be calculated so that

r� Y ¼ 0 ½27�

This is realized through an optimization loop, in which

Eq. [27] is the objective function and dep is constrainedto values greater than zero. Then, one iterationsequence could have the following form: Estimate

increments dep; using Eqs. [13] and [14], calculateincrements deprand dept and update the plastic strains

eprandept epr ¼ epr þ depr ; e

pt ¼ ept þ dept

� ; solve Eq. [10] for

new displacements u in the casting and get new stresses(Eq. [26]); and recalculate von Mises stresses r andrepeat until the convergence of Eq. [27] is attained. Inthe present article, a nonlinear least-squares optimiza-tion algorithm, known as the trust-region-reflectivealgorithm,[57,58] was applied. In order to reduce disper-

sive errors of dep appearing due to complex loadingand, consequently, yielding on a discrete grid, the Sav-itzky–Golay filter[59] was applied on a temporally over-

laid dep signal.(9) Proceed to the next time-step.

METALLURGICAL AND MATERIALS TRANSACTIONS B VOLUME 49B, JUNE 2018—1427

III. RESULTS AND DISCUSSION

As the main objective of the present study is topropose and test a novel approach of calculating an airgap rather than numerically analyzing a particularprocess, material and mechanical properties used inthe simulations only roughly correspond to those of realmaterials (Table II). The geometry, initial conditions,and other casting parameters are given in Table III.

First, numerical tests were performed for differentcoating thicknesses (0.5, 1, 2, and 4 mm) and all otherparameters were fixed. Total displacements u of themold and the casting at the interface are shown as afunction of time t in Figure 5. As one would expect,initially, the mold and the casting are in contact. At acertain moment, they detach and their displacementsfollow different paths. This is clearly shown by bifur-cating curves in Figure 5. A difference between thecasting and the mold displacement corresponds to the

air gap thickness da. Obviously, the air gap appearsearlier in the case of a thin coating than that of a thickone. Consequently, the heat transfer coefficient h at the

0 200 400 600 800 1000 12000

500

1000

1500

2000

Temperature [°C]

Yie

ld s

treng

th [M

Pa]

Tsol

Fig. 3—Temperature-dependent yield strength of the castingmaterial.[60] The dashed line represents data reconstructed using theextrapolation until the temperature of solidus Tsol.

0 200 400 600 800 1000 12000

50

100

150

200

250

Temperature [°C]

You

ng's

mod

ulus

[MP

a]

Tsol

Fig. 4—Temperature-dependent Young’s modulus of the castingmaterial.[60] The dashed line represents data reconstructed by meansof extrapolation until the solidus temperature Tsol.

Table II. Properties of Materials Used in the Simulations

Material Property Value Unit

Casting[…]S a 5 9 10�6 K�1

q 7860 kg m�3

cp 500 J kg�1 K�1

m 0.5 —E Fig. 4 Pags linear between Tliq and Tsol —k 22 W m�1 K�1

Lf 280 kJ kg�1

Tliq 1593 (1320) K (�C)Tsol 1438 (1165) K (�C)Y Fig. 3 Pa

Mold […]M a 5 9 10�6 K�1

m 0.5 —q 7850 kg m�3

cp 490 J kg�1 K�1

E 200 9 109 Pak 58.6 W m�1 K�1

Coating kC 2 W m�1 K�1

Air ka,phys 0.02 W m�1 K�1

Table III. Geometry, Initial Conditions, and Other CastingParameters

Property Value Unit

X 71 rad s�1

d 65 mmdC 2 mmL 165 mmR 400 mmTfill 1623 (1350) K (�C)Tmold 433 (160) K (�C)

0 50 100 150 200 250 3000

0.05

0.1

0.15

0.2

0.25

0.3dC=4.0 mm2.01.00.5

Time [s]

Tota

l dis

plac

emen

t [m

m]

Fig. 5—Total displacements u at the cast-mold interface for differentcoating thicknesses of 0.5, 1, 2, and 4 mm. Bifurcation correspondsto the first appearance of the air gap.

1428—VOLUME 49B, JUNE 2018 METALLURGICAL AND MATERIALS TRANSACTIONS B

interface will drop faster in the case of a thin coatingthan that of a thick one, which can be seen in Figure 6.The heat transfer coefficient h was simply calculated as

h ¼ q

TS � TM¼ kifc

drM þ drS: ½28�

For each configuration (dC = 0.5, 1, 2, and 4 mm),the time evolution of the heat transfer coefficient h is,therefore, unique. Ultimately, it seems that after300 seconds, the heat transfer coefficients are almostidentical, close to 200 W m�2 K�1.

Similar tests with similar outputs were performed fordifferent values of the rotation rate X (50, 71, and 90 rads�1). Naturally, the higher the centrifugal force, thebetter the contact between the casting and the mold.Concerning the elastic deformations of the mold,displacements u are larger at higher rotation rates X(Figure 7). Higher centrifugal forces are also responsiblefor stronger and longer yielding of the partly solidifiedcasting, which delays a formation of the air gap.Consequently, at a given instance, the heat transfercoefficient h is higher in the case of a higher rotation rateX (Figure 8).

A similar study was carried out for different values ofsolidus temperature Tsol such that Tliq � Tsol = 10 K,50 K, 100 K, and 200 K (10 �C, 50 �C, 100 �C, and 200�C). Although it is a somewhat intuitive, one couldconfidently state that the smaller the difference is, theearlier the air gap occurs. Again, we provide totaldisplacements and heat transfer coefficients at theinterface in Figures 9 and 10, respectively. Since coatingparameters are fixed this time, initial heat transfercoefficients are all identical, equal to 1000 W m�2 K�1.Later, they significantly deviate. While the curvesreferring to mold displacements at the interface have asimilar trend, indicating a continuous thermoelasticexpansion of the mold, those corresponding to castingdisplacements exhibit more complex scenarios due to thecombination of plastic and elastic deformations. In

Figure 9, e.g., the dash-dot line representing Tliq � Tsol

= 200 K (200 �C) indicates the yielding of the castingmaterial within the entire time span simulated. On thecontrary, the dash line Tliq � Tsol = 50 K (50 �C)displays only slight yielding in the beginning, immedi-ately followed by thermoelastic contraction.Although the quantities calculated at the interface,

such as the air gap thickness da and the heat transfercoefficient h, are of primary interest here, the numericalmodel also provides other quantities such as stresses andelastic/plastic strains. In Figure 11, a typical example oftemperature and strains appears at 50 seconds. The grayzone on the right represents the mold. The rest on theleft belongs to the casting. A temperature drop can beseen at the interface due to a large thermal resistance. Asthe time proceeds, total strains grow quite uniformlythroughout the entire thickness of the mold. The sameapplies also to the casting but only at the early stage.Later, when the casting is partly solidified and the yield

0 50 100 150 200 250 3000

500

1000

1500

2000

2500

3000

3500

4000

Time [s]

Hea

t tra

nsfe

r coe

ffici

ent [

Wm

-2K

-1]

dC=4.0 mm2.01.00.5

Fig. 6—Heat transfer coefficients h at the cast-mold interface fordifferent coating thicknesses of 0.5, 1, 2, and 4 mm. Before the airgap is formed, h is constant defined as kCdC.

0 50 100 150 200 250 3000

0.05

0.1

0.15

0.2

0.25

0.3

Time [s]

Tota

l dis

plac

emen

t [m

m]

50 rad s-1

7190

Fig. 7—Total displacements u at the cast-mold interface for differentvalues of rotation rate X such that X = 50, 71, 90 rad s�1.

0 50 100 150 200 250 3000

200

400

600

800

1000

Time [s]

Hea

t tra

nsfe

r coe

ffici

ent [

Wm

−2K

−1] 50 rad s-1

7190

Fig. 8—Heat transfer coefficients h at the cast-mold interface fordifferent values of rotation rate X such that X = 50, 71, and 90 rads�1.

METALLURGICAL AND MATERIALS TRANSACTIONS B VOLUME 49B, JUNE 2018—1429

strength Y increased, total strains start dropping and thecasting contracts consequently.

In addition to strains (Figure 11), stresses are shownin Figure 12. A typical distribution of stresses can beseen in the mold. While the radial stresses are exclusivelycompressive, the tangential stresses are compressiveclose to the inner surface of the mold and become tensileas they approach the outer surface of the mold. Thegreatest stresses the mold must withstand are naturallylocated at the inner surface due to a sudden temperatureloading. In this particular case (Figure 12), they do, in

fact, reach the yield strength of the material. Afterseveral casting cycles, a thermal loading would mostlikely lead to the formation of fire cracks.[61] Concerningstresses in the casting, at the early stage of solidification,they are within the temperature-dependent envelope ofthe yield strength. The (semi-)solid part of the castingclose to the mold can already hold some stresses,whereas the liquid part remains stress free. At laterstages of the casting, the stresses in the casting are wellbelow the yield strength and only the elastic loading ispresent. When the casting procedure is finished and thetemperature field becomes uniform, residual stressesremain in the casting. An analysis of stresses and strainsis, however, beyond the scope of the present article.

0 50 100 150 200 250 3000

0.05

0.1

0.15

0.2

0.25

Time [s]

Tota

l dis

plac

emen

t [m

m]

Tliq -Tsol=10K (10°C)50 (50)100 (100)200 (200)

Fig. 9—Total displacements u at the cast-mold interface for differentvalues of solidus temperature Tsol such that Tliq � Tsol = 10 K, 50K, 100 K, and 200 K (10 �C, 50 �C, 100 �C, and 200 �C).

0 50 100 150 200 250 3000

200

400

600

800

1000

Time [s]

Hea

t tra

nsfe

r coe

ffici

ent [

Wm

−2K

]

−1

Tliq -Tsol=10K (10°C)50 (50)100 (100)200 (200)

Fig. 10—Heat transfer coefficients h at the cast-mold interface fordifferent values of solidus temperature Tsol such that Tliq � Tsol =10 K, 50 K, 100 K, and 200 K (10 �C, 50 �C, 100 �C, and 200 �C).

Fig. 11—Distribution of strains and temperature in the radialdirection for the case with the coating thickness dC of 2.0 mm at 50s. The zones in white and gray stand for the casting and the mold,respectively.

Fig. 12—Distribution of radial and tangential stresses for the casewith the coating thickness dC of 2.0 mm at 50 s.

1430—VOLUME 49B, JUNE 2018 METALLURGICAL AND MATERIALS TRANSACTIONS B

IV. CONCLUSIONS

In centrifugal casting simulations, exponential func-tions are generally used to describe the heat transfercoefficient at the cast-mold interface, varying due to theair gap formation. Such functions contain empiricalconstants, which must be carefully specified. Unfortu-nately, this is not an easy task. An experiment alone isnot sufficient to determine such constants, and compu-tationally expensive inverse methods should beemployed, which is, however, rarely the case. A litera-ture survey performed here reveals an expansive scatterof data used in current and previous research. In thepresent study, we offer an alternative of calculating anair gap thickness and the corresponding heat transfercoefficient at the interface. The heat transfer model iscoupled with a plane stress model, taking into accountthermoelastic stresses, centrifugal forces, plastic defor-mations, and a temperature-dependent Young’s modu-lus. Several numerical tests were performed for differentcoating thicknesses dC, rotation rates X, and solidustemperatures Tsol. Results were analyzed in the sense ofcomparing heat transfer coefficients at the interface andair gap thicknesses as a function of time. The numericalmodel developed here helps demonstrate that thescenario at the interface is unique for each set ofparameters. Therefore, deploying any of the exponentialfunctions that explicitly describe the thermal resistanceat the cast-mold interface will always give rise to thequestion about the actual value of empirical constantsused in that particular function. Although the materialproperties taken for this study do not strictly correspondto any particular material, they are obviously not farfrom material properties of common steels and coatings,and the results obtained here appear to be entirelyreasonable and meaningful. In the near future, we planto verify the current numerical approach against theresults obtained from the inverse task run with theexperimental data. Finally, possible room for improve-ment of the presented model remains. For example,some kind of implicit coupling between the heat transfermodel and the plane stress model would be beneficialand might even be necessary in order to maintainnumerical stability (or suppress unphysical oscillationsof calculated displacements), especially at higher coolingrates, e.g., when an air gap is being just formed. Inaddition, the optimization loop involved in the loadingstep, i.e., the process of calculating the plastic strainscould also be further improved.

ACKNOWLEDGMENTS

Open access funding provided by MontanuniversityLeoben. Financial support from the Austrian FederalGovernment (in particular, from the Bundesministeriumfuer Verkehr, Innovation und Technologie and the Bun-desministerium fuer Wirtschaft, Familie und Jugend)and the Styrian Provincial Government, representedby Oesterreichische Forschungsfoerderungsgesellschaft

mbH and by Steirische Wirtschaftsfoerderungsge-sellschaft mbH, within the research activities of the K2Competence Centre on ‘‘Integrated Research in Materi-als, Processing and Product Engineering,’’ operated bythe Materials Center Leoben Forschung GmbH in theframework of the Austrian COMET Competence Cen-tre Programme, is gratefully acknowledged. This workis also financially supported by the EisenwerkSulzau-Werfen R. & E. Weinberger AG.

OPEN ACCESS

This article is distributed under the terms of theCreative Commons Attribution 4.0 International Li-cense (http://creativecommons.org/licenses/by/4.0/),which permits unrestricted use, distribution, and re-production in any medium, provided you give appro-priate credit to the original author(s) and the source,provide a link to the Creative Commons license, andindicate if changes were made.

NOMENCLATURE

[…]C Subscript referring to the coating[…]M Subscript referring to the mold[…]S Subscript referring to the castingcp Specific heat (J kg�1 K�1)d Casting thickness (mm)da Air gap thickness (mm)dC Coating thickness (mm)E Young’s modulus (Pa)gs Solid fractionh Heat transfer coefficient at the interface

(W m�2 K�1)h0 Initial heat transfer coefficient at the interface

(W m�2 K�1)hf Final heat transfer coefficient at the interface

(W m�2 K�1)ha Heat transfer coefficient between the casting

and the coating (W m�2 K�1)k Thermal conductivity (W m�1 K�1)ka Effective thermal conductivity of the air gap

(W m�1 K�1)ka,phys Thermal conductivity of air (W m�1 K�1)kC Thermal conductivity of the coating

(W m�1 K�1)kifc Effective thermal conductivity of the control

volume built of the mold, coating, air gap, andcasting (W m�1 K�1)

L Thickness of the mold (m)Lf Latent heat of solidification (J kg�1)N Number of grid pointsP Set of yielding pointsq Heat flux through the interface (W m�2)q0 Initial heat flux through the interface (only

coating present) (W m�2)r Radial distance (mm)ri Inner radius of the casting (mm)ro Outer radius of the mold (mm)

METALLURGICAL AND MATERIALS TRANSACTIONS B VOLUME 49B, JUNE 2018—1431

R Inner radius of the mold–radius of theinterface (mm)

s Solidified thickness of the casting (mm)t Time (s)t0 Time of solidification initiation (s)T Temperature (K (�C))Tfill Initial temperature of the liquid metal-filling

temperature (K (�C))Tliq Liquidus temperature of the casting material

(K (�C))Tmold Initial temperature of the mold (K (�C))Tsol Solidus temperature of the casting material (K

(�C))T* Temperature between the coating and the air

gap (Fig. 1) (K (�C))T’M Temperature between the mold and the

coating (K (�C))T’S Temperature between the casting and the air

gap (K (�C))u Radial displacement (mm)Y Yield strength (Pa)

GREEK SYMBOLS

a Thermal expansion coefficient (K�1)b Damping coefficientc Constant exponenter Total strains in radial direction (mm mm�1)et Total strains in tangential direction (mm mm�1)epr Plastic strains in radial direction (mm mm�1)ept Plastic strains in tangential direction (mm mm�1)ep Effective plastic strain (mm mm�1)e Radiative emissivitym Poisson’s ratioq Density (kg m�3)rr Radial stress (Pa)rt Tangential stress (Pa)r von Mises (effective stress) (Pa)r Stefan–Boltzmann constant (W m�2 K�4)X Rate of rotation (rad s�1)

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