*

Department of Mining&Materials Engineering,McGill University,M.H.Wong Building,3610 University Street,Montreal,QC H3A 0C5,Canada

On numerical modeling of low-head direct chill ingot caster for magnesium alloy AZ31

Mainul Hasan*,1,Latifa Begum

Department of Mining&Materials Engineering,McGill University,M.H.Wong Building,3610 University Street,Montreal,QC H3A 0C5,Canada

A comprehensive 3D turbulent CFD study has been carried out to simulate a Low-Head(LH)vertical Direct Chill(DC)rolling ingot caster for the common magnesium alloy AZ31.The model used in this study takes into account the coupled laminar/turbulent melt f l ow and solidif i cation aspects of the process and is based on the control-volume f i nite-difference approach.Following the aluminum/magnesium DC casting industrial practices,the LH mold is taken as 30 mm with a hot top of 60 mm.The previously verif i ed in-house code has been modif i ed to model the present casting process.Important quantitative results are obtained for four casting speeds,for three inlet melt pouring temperatures(superheats)and for three metal-mold contact heat transfer coeff i cients for the steady state operational phase of the caster.The variable cooling water temperatures reported by the industry are considered for the primary and secondary cooling zones during the simulations.Specif i cally,the temperature and velocity f i elds,sump depth and sump prof i les,mushy region thickness,solid shell thickness at the exit of the mold and axial temperature prof i les at the center and at three strategic locations at the surface of the slab are presented and discussed.

Low-head DC caster;Magnesium alloy AZ31;3D CFD modeling;Turbulent melt f l ow;Solidif i cation;Mushy region thickness;Sump prof i le

1.Introduction

Magnesium and many of its alloys are the lightest among all engineering metallic materials in use today.When compared with aluminum,magnesium is about 36 pct lighter, while it is about 78 pct lighter compared to steel[1,2].Magnesium and its alloys have better desirable properties,such as specif i c strength,stiffness,castability,damping capacity, machinability,shielding capacity,recyclability,etc compared to the most widely used materials like steel and aluminum alloys.Demand for magnesium alloys is increasing each year particularly in automotive,electronics and aerospace industries where weight-saving engineering solutionsare continuously being searched for reducing the cost for energy consumption.Among the many casting routes,today about 20 pct of magnesium alloys is cast via the direct chill casting (DCC)process[3].This process allows casting a large volume of magnesium alloys in an energy eff i cient manner.One of the major drawbacks in using this technology for magnesium alloys is that it costs more compared to steel and aluminum alloys[4].It is to be stressed here that DCC process for magnesium varies signif i cantly from plant to plant and unlike DCC of aluminum,the magnesium casting plants in this regard keep their own successful casting practices quite guarded.

The direct chill cast magnesium alloys suffer from a number of unwanted defects,such as hot tears,cold cracks, center cracks,surface folds,etc.Another technical problem is that modeled magnesium alloys have a low volumetric heatcapacity and as a result they freeze quickly making the start-up quite diff i cult.Since magnesium is highly reactive with oxygen hence during the casting operations care is to be taken so that such contact does not take place and in this regard during the DCC operation the top part of the melt delivery system is kept under pressure using a mixture of CO2(90%by volume) and SF6(10%by volume)gases[5].Also,one needs to be very careful while casting these alloys using the DCC process, since they may cause a hydrogen explosion if the alloys come accidentally in contact with the primary or secondary cooling water of the process.

The vertical DCC process,which is used for aluminum, magnesium and a few other non-ferrous metals like zinc, copper,etc,is quite complex due to the interactions of various parameters,likealloy composition,thethermo-physical properties of the alloy,casting speed,pouring temperature of the melt,mold length and cross-section,cooling water temperature and its f l ow rate in the mold and post mold region,etc [6].Though this process has seen many improvements over the past several decades but due to the complex interactions of the above mentioned factors,the successful operation of this process still depends upon the prior experience of the operators gained in earlier years in the plant.

Using a suitable melt delivery system,in a conventional vertical DCC process superheated molten magnesium alloy is supplied from the top to a water-cooled open-ended mold. Before the start of casting,the open bottom end of the mold is kept plugged by a bottom block(also called a starter block). When the supplied molten metal level in the bottom-blocked mold reaches a certain level and the solidif i ed layers form against the top of the bottom block and the mold faces,the partially solidif i ed ingot sitting on the top of the starter block is slowly lowered down towards the casting pit below by the f i tted hydraulic ram on the bottom block.During the lowering of the block-ingot assembly by the ram its speed is increased gradually until a constant casting speed is reached.As the partially solidif i ed ingot containing liquid metal within it exits the bottom of the mold it is further cooled by directly spraying mold water in the form of jets or streams from the bottom of the mold onto the outside surface of the solid shell.The water stream from the impingement region then forms a continuously falling f i lm down the slab.The indirect cooling of the metal that takes place in the mold is called primary cooling where only 10-15%of the heat content of the ingot is removed by the cooling water circulating through the various passages within the mold body[3].Through the direct cooling of the partially solidif i ed ingot by the water from the mold in the post mold region(also called secondary cooling region), the rest(85%-90%)of the total heat content of the ingot is removed[3].The process is initially unsteady.However, depending on the casting speed,the thermo-physical properties of the magnesium alloy and the cooling conditions in the mold and post mold regions after about 0.5 m-1.0 m length of casting the process reaches steady state[3].In a normal slab casting operation after casting of about 8-10 m in length,the melt delivery to the mold is stopped and the casting operation is suspended.The cast is then taken out from the water containing casting pit for further downstream processing.The cast magnesium slabs are usually rolled to produce plates, sheets,foils and strips for various end applications while the billets are extruded and forged to form bars,rods,pipes,tubes, etc.for various practical uses.

Due to the customers'demands for the higher quality products from the castings,the major manufacturers of the aluminum and magnesium castings have adopted a number of technological changes in their DCC practices with a view to improving the cast quality.Because of the interactions of several interlinked parameters in this process it is diff i cult to prevent the formation of defects by controlling one parameter at a time.The use of a low-head mold with an active mold length between 25 and 30 mm has been seen to suppress the formation of an air gap in the mold.And as a result,the reheating of the shell is minimized which signif i cantly reduces exudation,surface macro-segregation and surface cracks,etc. In this regard,the casting company Wagstaff,Inc.(Spokane Valley,WA 99206,USA),a major supplier of DC casting machines,developed a low-head composite(LHC)ingot casting technology in 80s which the company since has been marketing to the various aluminum and magnesium casting industries under their trade name ‘LHC?’[7].In comparison with the conventional DCC mold(which is usually between 70 and 80 mm in length)the LHC mold has been shown to have the following advantages;it produce casts with a better surface quality,it offers safer operations,it minimizes scalp rates,it allows higher casting speeds and results in a longer mold life. The latter improvements have led to the better cast quality for a lower cost.In this regard,the interested readers can consult the recently published book on DCC process[2]for the historical developments,the modern versions of this process as well as for the various experimental and modeling studies related to casting of aluminum and magnesium alloys.It is interesting to note that unlike aluminum alloys,there is not enough modeling studies available in the literature concerning the present problem.This can be verif i ed by looking at the cited references of Hu et al.[8]who very recently,modeled the 3-D temperature f i eld for a DC castslab of 800 mm × 300 mm cross-section of AZ31 alloy for casting speeds ranging from 20 to 70 mm/min all for a f i xed pouring temperature of 600°C.They have reported the sold shell thickness,and the temperature contours and temperature gradients at the wide symmetry plane for various mentioned casting speeds.Although the paper stated that a thermo-f l uid analysis was carried out but no supporting results have been reported,nor have they provided any explanation on how the melt f l ow analysis was carried out.

To the best knowledge of the authors,no one has ever modeled the 3-D coupled laminar/turbulent melt and solidif ication heat transfer which take place in a low-head vertical DCC slab caster for any magnesium alloy and certainly not for the alloy AZ31 modeled in this study.Since casts of AZ31 slabs of various sizes are produced by the DCC industries,the objective of the present investigation is to carry out a comprehensive 3-D numerical simulation study of the L-H DCC process for the alloy AZ31 in order to gain somefundamental understanding of the solidif i cation behavior of this alloy and also to provide some useful guidelines and offer some concrete suggestions to the DCC operators to improve their casting campaigns with regard to the productivity and the quality of the casts from the said alloy.

2.Mathematical model

The system modeled is shown schematically in Fig.1 which represents a low-head rectangular-shaped industrialsized caster with an open-top melt feeding system.Since the modeled geometry has a two-fold symmetry,to save the computational time and cost only the top left-hand quarter of the slab is investigated.A f i xed domain having dimensions of 2500 mm × 865 mm × 330 mm inx,y,andzdirections, respectively with its origin located at the central top part of the ingot with the axial coordinate in the direction of gravity is modeled.The melt of the alloy AZ31 is considered to have been supplied at an uniform temperature and velocity along the entire top cross-section of the mold.The solid-shell which forms in the mold and in the sub-mold regions is assumed to move in the axial direction at the selected but constant casting speed.

2.1.Test material

Fig.1.Schematic of the low-head DC caster and the computational domain for an open-top melt feeding scheme.

As stated earlier,the common magnesium alloy AZ31 was selected to predict the solidif i cation behavior of this alloy in a vertical direct chill slab casting process.The chemical composition limits of this alloy are listed in Table 1[9]and its thermo-physical properties are summarized in Table 2[9].In this magnesium alloy,aluminum and zinc are the two major alloying elements along with other minor elements like manganese,silicon,copper,etc.This alloy possesses good roomtemperature strength and ductility,has good corrosion resistance property and can be welded.The alloy AZ31 f i nds applications in many useful products such as aircraft fuselages, cell phones and laptop cases,intricate components of automobiles,vibration tests equipment,inspection gauges,speaker cones,etc.[9].

2.2.Assumptions and simplif i cations

In order to simplify the modeling effort of the present complex industrial slab DCC problem,the following realistic assumptions were invoked:

[1]A f i xed-coordinatesystem (Eulerian approach)was employed in the simulation.

[2]Local thermodynamic equilibrium during solidif i cation was assumed to prevail.Heat f l ow was taken to be very fast and every point reached equilibrium with its neighboring points instantaneously.

[3]A mushy-f l uid solidif i cation model was assumed and pore formation was ignored.

[4]Flow in the mushy region was modeled similar to a f l ow through a porous medium.

[5]Molten aluminum was assumed to behave as an incompressible Newtonian f l uid and turbulence effects were approximated using the popular two equationk- ε model of turbulence.

[6]Any heat released due to solid-solid transformation was not taken into consideration.The formation of intermetallic compounds during solidif i cations was ignored.

[7]Evolution of latent heat in the solidif i cation domain was not inf l uenced by the microscale species transport,that is, the liquidus and solidus temperatures were f i xed.

[8]Because of the lack of experimental results,the variation of liquid fraction in the mushy zone was assumed to be a linear function of temperature.

[9]The thermo-physical properties of magnesium alloy were invariant with respect to the temperature and there was no viscous dissipation effect.The thermal buoyancy effect wasincorporated in the momentum equationsby employing the Boussinesq approximation.

[10]Only the steady-state phase of the caster's operation was modeled without taking into account the transient start-up condition.

2.3.Numerical formulation

2.3.1.Governing equations for turbulent melt velocity and enthalpy

The 3-D general mean transport equations for an incompressible laminar/turbulent melt velocity and enthalpy in arectangular domain can be expressed in a general form of a dimensional partial differential equation.The conservation equation in the general form at steady state in Cartesian tensor notation can be written as follows:

Table 1Chemical composition limits of AZ31.

Table 2Thermo-physical properties of AZ31.

where,ρ is the melt density anduiis the velocity component in thexi-direction.For continuity equation Φ is equal to 1.For momentum equations it represents the velocity componentsui, while for the energy equation it represents temperatureT,and for the turbulent kinetic energy and its rate of dissipation it representskand ε,respectively. ΓΦis the effective diffusion coeff i cient of Φ andSΦis the source term of Φ.The gravity direction is taken in the positivex-direction.As the gravity term in thex-momentum equation may have an effect on the melt f l ow,the density in the buoyancy force terms is assumed to be dependent on the temperature f i eld based on the Boussinesq approximation.To calculate the turbulence quantities in the liquid pool and mushy region the low-Reynolds number version of the populark-ε eddy viscosity concept proposed by Launder and Sharma[10]were used.The details derivation of the above general equation and turbulence model can be found in Begum[11].

The enthalpy-porosity technique was adopted to account for liquid-mushy and mushy-solid phase changes in the domain, in which a single set of transport equations was appropriately applied for the liquid,mushy and solid regions.In order to model the f l ow in the mushy zone where 0 ＜fl＜ 1,the Darcy's law of porous media based on the Carman-Kozeny equation was adopted[12,13].Equation(1)and the associated boundary conditions were suitably non-dimensionalized to suppress the numerical instabilities which often occur during the solution process of the dimensional discretized equations due to the large change in the values of the common variable Φ.

The following dimensionless variables were employed to non-dimensionalize the governing coupled partial differential equations and the associated boundary conditions in order to ascertain the non-dimensional parameters which govern this process:

whereDis the hydraulic diameter of the caster,uinis the inlet velocity and ΔHfis the latent heat of solidif i cation.All the appropriate variables of the transport equations in the nondimensional form are available in Table 3 in Ref.[14].The non-dimensional form of the boundary conditions is also given in Eqs.(10)-(16)in Ref.[14].In order to keep the article short,the non-dimensional transport equations and boundary conditions are not reported here.The interested readers can consult the recently published PhD dissertation of Begum [11],which can be easily obtained free of charge by Googling the title,for further details.

The values of the effective heat transfer coeff i cient along the length of the DC slab caster were varied and were taken from the very well cited article by Vreeman and Incropera [15],which are given below:

Adiabatic section:

Mold-metal contact region:

Secondary cooling zone:

Table 3Values of the parameter used in open-top melt delivery arrangement.

wherex1=90 mm,x2=100,x3=140 mm and γmax=20.0 W/m2K, γflim=10.0 W/m2K.In the present study,three different cooling water temperatures,namely 10, 20 and 30°C,respectively for the mold,impingement and streaming regions have been considered to ref l ect the reality of the thermal conditions of the cooling water used by the L-H DCC industry[16].A complete description of the solution methods employed in this work is provided in details in Ref. [11].

Before carrying out extensive parametric studies the grid independency tests were f i rst performed using three sets of successively ref i ned meshes(coarse,moderate,and f i ne).The relative difference in the local surface heat f l ux between the coarse and f i ne grids was within 4.0%,indicating that the coarse grid was suff i cient for predicting the results of engineering accuracy.It is to be stressed here that the local surface heat f l ux is the most sensitive quantity with regard to grid in any heat transfer problem and this is particularly true for turbulent convective heat transfer problems.A comparison (see Ref.[11])of the surface temperature prof i les showed less than 1%variation between the f i ne and course grid systems. Therefore,the coarse grid （60 × 42 × 24）was used for all production runs reported in this work.The details of the predicted results along with the comparisons can be found in Ref.[11].

A code validation test was also carried out by simulating the solidif i cation of a rolling ingot for AA-3104 alloy,using the identical experimental casting conditions and set-up for the vertical DCC process performed by Jones et al.[17].The present in-house developed code showed an acceptable agreement with the experimental data.These results were presented in Ref.[11]and are not given here to avoid duplication.

3.Results and discussion

A detailed study of all casting parameters namely,casting speed,inlet melt superheat,HTC,cooling water f l ow rate, mold dimensions,etc.will certainly generate too many results for an arbitrarily selected 3-D geometry.Since the main purpose of this work is to show the inf l uence on the solidif i cation behavior of the most important casting parameters hence an industrial-sized caster with a low-head DC caster was selected for modeling studies by varying three process parameters, namely,casting speed,inlet melt superheat,and HTC at the metal-mold contact region.A set of twelve separate cases for the AZ31 alloy which is listed in Table 3 were selected to convey the most important casting information.

3.1.Temperature and velocity f i elds for cases(1-4)of Table 3

Fig.2.3-D surface plots of temperature f i eld for four cases(1-4).

The surface temperature distributions in f l ooded format are presented in Fig.2(i-iv)for the range of casting speed from 60 to 180 mm/min representing cases(1-4)of Table 3.These numerically predicted temperature data clearly show the progression of the solidif i cation process for the above mentioned cases.The solidif i cation range of magnesium alloy AZ31 is 55°C which is an intermediate freezing range alloy.The present results indicate that the mushy region,which isbounded by the liquidus(630°C)and solidus(575°C)isotherms,is expanded as the casting speed is increased due to the stronger thermal convective effects.Furthermore,a classical parabolic-shaped solidif i cation front is seen to have been developed particularly if one looks from the central axis at each casting speed.Since near the wall the melt loses superheat and as a result the effect of thermal convection reduces compared to thermal conduction effect as the melt moves from the wall region to the central region.The parabolic shape of the solidif i cation prof i le is the manifestation of the above fact. As the casting speed increases the solidif i cation front as well as the other isotherms are becoming steeper and are moving downward,which has already been found by several other authors[18,19].

Fig.3.Longitudinal 2-D view of temperature contours and velocity vectors for case 1.

Fig.2(i-iv)shows only the temperature f i elds for the two symmetric planes(x-yandx-zplanes)and for the top free surface.In order to assess the melt f l ow and heat transfer along the longitudinal planes of the domain,the 2-D projections of the temperature and velocity f i elds at the wide symmetry plane (z=0)and parallel to the wide symmetry plane at two inward locations,viz.,z=62.5 mm,andz=312.5 mm,for two cases(1 and 4)are selected.The right hand panels ofFigs.3(a)and 4(a)are plots for the velocity vectors at wide symmetry plane(z=0)for two casting speeds of 60 and 180 mm/min.As mentioned earlier,in this study it has been assumed that the melt is delivered uniformly at a specif i ed casting speed across the entire top cross-section.Due to this simple melt feeding system,from these f i gures it can be clearly seen that all of the melt f l ows downward.Along the narrow face a part of the vertical melt that are adjacent to the wall is diverted by the developed solidif i ed shell,and then they move along the solidif i cation front.A comparison between these two f i gures shows that the magnitude of the resultant velocity vector increases with the increase of the casting speed.The above observation is due to the conservation of mass into the domain.

The right hand panels of Figs.3(b)and 4(b)are plots for the velocity f i elds on the longitudinal planes at an inward distance ofz=62.5 mm for the same casting speeds.As depicted in those f i gures,the developing solidif i ed shell is still obstructing the corner stream.The right hand panels of Figs.3(c)and 4(c) are showing the velocity f i elds atz=312.5 mm.Notice that the plane atz=312.5 mm is nearer to the rolling face.The difference among the three planes is that forz=312.5 mm the f l ow is moving downward uniformly without any obstruction.This is because on the third plane the melt is almost solidif i ed and the magnitude of the velocity vectors there is slightly higher than the corresponding casting speed,which signif i es that the melt is accelerated in this plane with respect to the imposed casting speed on the solidif i ed shell.

Fig.4.Longitudinal 2-D view of temperature contours and velocity vectors for case 4.

The heat transfer rate increases as one proceeds from the heated ingot center toward the cooled wall.The numerically predicted temperature contours at the three longitudinal planes give a better visualization of this trend.The location of a f i xed isotherm is seen to have been lifted upward progressively at two inward planes atz=62.5 mm andz=312.5 mm with respect to the wide symmetry plane atz=0 mm.The left hand panels of Figs.3(a)and 4(a),Figs.3(b)and 4(b),and Figs.3(c) and 4(c)also show the impact of the change in casting speed on the temperature f i elds.The results in these f i gures depict that all the isotherms have moved downward and the width of the mushy region is increasing with increasing casting speeds.

Among the various possible options to represent the solidif i cation heat transfer phenomena,the 2-D transverse crosssectional planes(y-zplane)are chosen to get a vivid picture of the solidif i cation process happening within the ingot.The liquidus and solidus isotherms are plotted in Fig.5(a)-(d)for four cases(1-4).A total of seven transverse cross-sections starting from the top up to an axial distance of 700 mm have been selected.The numerical results show that a nearly uniform thick solid shell on the narrow and wide faces is formed and as one moves downward the extent of the solid shell increases progressively.Since more heat is being extracted from the ingot through the mold and by the chilled water jets both of the latter factor are contributing in increasing the thickness of the solid-shell along the cast direction.It appears from the progression of the solid-shell thickness that an almost uniform rate of heat extraction is taking place from both the sides.At the corner region heat isbeing extracted from both the rolling and narrow faces,this has resulted in a higher rate of heat extraction at that location compared to the other places.As a result,an almost roundshaped solid layer and mushy zone have formed there.The location of the solidus isotherm with the decrease in the casting speeds has shifted more toward the ingot center over each cross-section compared to the higher casting speed, which can be clearly seen in the above f i gures.The primary reason in the reduction of heat transfer with increasing casting speed is that for a higher casting speed the amount of incoming melt as well as the energy content of the melt are both higher.

Fig.5.Contours of solidus and liquidus temperatures at various transverse cross-sectional planes for four cases(1-4).

3.2.Quantitative analysis forΔT=32°C

3.2.1.Sump depth and mushy layer thickness

The sump depth and mushy thickness are indirect indicators of the quality of the cast.The latter two quantities were obtained by post processing the predicted temperatures from the CFD model.In this regard,the numerically predicted sump depth and the mushy layer thickness at the ingot center for four cases(1-4)are provided in Table 4.The present authors are surprised by the fact that in none of the vast literature on DCC process has reported the quantitative values of the sump depth and mushy thickness for any alloy.A closer look at this table shows that a deeper sump and a thicker mushy layer have developed at a higher casting speed.To get a comparative picture of the effect of the casting speed on sump depth and mushy thickness the lowest casting speed of 60 mm/min was taken as the base case for relative comparisons.The sump depth is found to be approximately 487.803 mm from the top of the mold for a withdrawal speed of 60 mm/min.The relative difference in sump depth for higher casting speeds of 100,140, and 180 mm/min,is respectively about 79.72%,197.47%, 308.75%higher.For the casting speed of 60 mm/min,the mushy thickness is found to be 318.091 mm,whereas for casting speeds of 100 mm/min,140 mm/min,and 180 mm/ min,the mushy thickness increases by 80.52%,215.91%,and 333.614%,respectively compared to the casting speed of 60 mm/min.The reason for the proportionate increase of the mushy layer thickness is diff i cult to explain,since a complex interaction between the f l ow and thermal f i eld prevails in that region.

3.2.2.Predicted shell thickness

As the casting speed inf l uences the growth rate of the solid shell,the predicted shell thickness from the narrow side at the wide symmetry plane and from the wide side at the narrowsymmetry plane at the exit of the mold are plotted in the form of a bar chart in Fig.6 in order to get a quantitative picture of the differences among the results for four cases(1-4).The numerical value of the predicted shell thickness(in mm)is provided inside each bar of the chart of Fig.6.For both locations,with the increase of the casting speed the shell thickness decreases which is consistent with the physics of the problem.The rate of decrease of the shell thickness is smaller for the higher casting speeds(＞100 mm/min)compared to the lower casting speeds(＜100 mm/min).The shell thickness results further show that the rate of growth of the shell at the center of the narrow side is greater compared to the center of the rolling face for all four simulated casting speeds.The reason behind this is the proximity of the middle of the narrow side from the slab corner(region of high heat extraction) compared to the middle of the wide slab side.

Table 4Sump depth and mushy thickness in mm at the ingot center for cases(1-4).

3.2.3.Predicted strand surface temperature

Fig.6.Shell thickness at the middle of the slab faces at mold exit vs.casting speed for cases(1-4).

Fig.7.Variations of surface temperature along the axial direction of the strand at four locations of the caster for:(a)us=60 mm/min(case-1)(b) us=180 mm/min(case-4).

Fig.7(a)and(b)illustrates the temperature distributions along the caster for two cases(1 and 4)which correspond to the casting speed of 60 and 180 mm/min,respectively.It is to be realized that the surface temperatures were not known initially.In order to obtain the surface temperature at each grid location,a heat balance on the control volume corresponding to the grid point was done after the converged solution was obtained.A total of four strategic locations are selected in this study,namely:(a)center;(b)mid-distance of the wide face; (c)mid-distance of the narrow face;and(d)corner point of the caster to portray the surface temperatures.These locations are thought to be the most probable sensitive zone for the developments of the compressive and tensile stresses.The four locations are given on the top of Fig.7(a)and(b).Except for the temperature at the ingot central region,the temperatures are seen to follow the same pattern at three other locations. The temperature drops sharply from the top of the mold up to the initial part of the cooling water streaming zone.i.e.in the region within 60 mm-122 from the top surface.Beyond that region the surface temperatures are seen to rebound due to the imposition of a lower heat transfer coeff i cient and higher cooling water temperature to ref l ect the industrial operating conditions(refer to Section 2.3.1).It is to be noted that in the DCC process the cooling water for the impingement region and for the free streaming region comes from the bottom of the mold and as a results the water temperatures in the two regions increase due to the gain of heat from the ingot.The surface temperatures increased by about 7°C within the length of about 20 mm(142 mm from the top)in the streaming zone and then they decreased gradually up to the end of the simulated casting length.The rebound of the surface temperatures could promote cold crack along with other surface cracks.The corner temperatures in both the f i gures show relatively lower values with respect to the other locations due to the simultaneous extractions of heat through the wide and narrow sides of the slab.The temperatures at the ingot center in the cast direction are the highest compared to the temperatures at three other locations as expected.A comparison of the two f i gures clearly show that for a higher casting speed the surface remains overheated for the same axial location compared to the lower casting speed.This trend is visible in all the four ingot locations.The reason behind this is the fact that the predictions were carried out for the lower and higher casting speeds by invoking the same convective heat transfer boundary conditions without considering the enhanced cooling requirements for a higher casting speed.

3.3.Effect of primary coolant heat transfer coeff i cient

There are serious disagreements among the researchers working in this f i eld regarding the values of the effective heat transfer coeff i cients in the mold region.From the literature one gets conf l icting values of the HTCs in the metal-mold contact region.Even the well refereed literature on this issue has reported a wide range of values for the HTCs.In the metal-mold contact region,a value of HTC ranging from 1000 to 5000 W/ (m2-K)has been reported in the well-cited literature[2,3].In order to ascertain the impact of the heat transfer coeff i cient (HTC)at the mold-metal contact region,for an inlet superheat of 32°C,three HTCs values of 750,1500 and 3000 W/m2K were selected for the two casting speeds namely,100 mm/min and 180 mm/min.Fig.8 reports the values of the shell thickness against the HTCs in the form of a bar chart for the above mentioned two casting speeds.

The shell thickness is estimated from the temperature prof i les at the mold exit at an axial distance of 90 mm from the top surface on the wide symmetry plane from the narrow face. It can be seen that for an increase of HTC the magnitude of the shell thickness increases.Similar trends are found for both casting speeds.For a f i xed HTC,the shell thickness decreases with casting speed as noticed earlier.For a lower casting speed of 100 mm/min,an increase of the HTC from 750 to 1500 W/ (m2-K)has caused an enhancement of 6.67%in solid-shell thickness and for the increase of HTC from 1500 to 3000 W/(m2-K)the increment is almost the same and is approximately 6.79%.For a higher casting speed of 180 mm/ min,the corresponding increase in the shell thickness is about 37.45%,and 28.64%,respectively.Hence,it is clear that the effect of lower HTCs(from 750 to 1500 W/m2-K)on shell thickness is more pronounced at a higher casting speed.

Fig.8.Shell thickness at the middle of the narrow face at mold exit vs. effective HTC(W/m2-K)for six cases(2,4,and 5-8).

Fig.9.Shell thickness at the middle of the narrow face at mold exit vs.melt superheat(°C)for six cases(2,4,and 9-12).

3.4.The effect of inlet melt superheat(ΔT)on shell thickness

The inlet melt superheat is usually f i xed by the casting operator in advance of the casting campaign.If a too low superheat is used there could a possibility of the melt-freeze in the launder-trough assembly during delivery of the melt into the mold which in turn could lead to the suspension of the casting process.On the contrary,if a very high inlet melt superheat is imposed,there could be the possibility of the formation of inter-metallic compounds which would adversely affect the quality of the cast.To study the effect of inlet melt superheat,a total of three superheats,namely 16,32 and 64°C were selected for each of the two casting speeds of 100 and 180 mm/min.The results of the shell thickness at the widesymmetry plane from the narrow face at the exit of the mold are presented in Fig.9 in the form of a bar chart for easy understanding.It can be seen from the above f i gure that for an increase of the superheat,the magnitude of the shell thickness decreases as technically expected.The latter observation is true for both casting speeds.For a lower casting speed of 100 mm/min an increase of the superheat from 16 to 32°C has caused in the reduction of shell thickness of only about 1.45%, but for the increase of superheat from 32 to 64°C the decrease of shell thickness is relatively greater and is about 5.41%.On the contrary,for a higher casting speed of 180 mm/min,the corresponding decrease in shell thickness is about 3.68%and 26.35%,respectively.The predicted results thus indicate that at a higher casting speed the change in superheat from 32 to 64°C causes a greater impact on the solidif i cation process.

4.Conclusions

After investigating the effects of the casting speed,inlet melt superheat and effective heat transfer coeff i cient at the metal-mold contact region through the 3-D turbulent CFD modeling study of a low-head,vertical DC slab casting process for the magnesium alloy AZ31,the following conclusions were drawn:

[1]For the range of casting speeds from 60 to 180 mm/min, for the superheat range of 16°C-64°C and for the metalmold effective heat transfer coeff i cient ranging between 750 W/m2K to 3000 W/m2K the sump depth and the mushy thickness at the center of the ingot do remain conf i ned within the simulated axial length of 2500-mm of the caster.

[2]A shallower sump and a lower depth of the melt pool at the center of the caster are obtained for the lower casting speeds compared to the higher casting speeds.For the standard cases of 1-4 of Table 3,the sump depth(SD in mm)and the melt pool depth(MP in mm)can be represented respectively by the following linear equations with regard to the casting speed (usin mm/min): SD = -325.41 + 12.74us(R2= 0.993)and MP=-264.07+9.02us(R2=0.99)

[3]The vertical extent of the mushy region(which is bounded by the liquidus and solidus temperatures of the alloy)at the ingot center increases with the increase of the casting speed.For the standard cases of 1-4 of Table 3,the mushy thickness(MT in mm)can be represented by the following linear equation with regard to the casting speed(usin mm/min):MT=-264.07+9.02us(R2=0.99)

[4]The solid shell thickness at the exit of the mold decreases with the increase in the casting speed which also resulted in the increase of the surface temperatures of the strand.

[5]The shell thickness at the mold exit decreases with the increase of the pouring temperature(melt superheat)and the reduction in shell thickness is more signif i cant for the higher range of the superheat,particularly at the higher casting speed.

[6]For a constant cooling water temperature in the mold, with a higher imposed HTC at the metal-mold contact region the shell thickness at the mold exit is greater.The enhancement of the shell thickness is more pronounced for the lower HTCs at higher casting speeds compared to the lower casting speeds.

[7]The higher casting speeds have resulted in the stronger thermal convective f l ows in the melt pool and in the mushy region compared to the lower casting speeds.The implication of this is that a greater number of broken dendrites can be expected for the higher casting speeds for this predominantly forced convective melt f l ow system and thus the macro-segregation patterns within the cast will be different for higher casting speeds compared to the lower casting speeds.

Acknowledgments

This work is partially supported from the National Sciences and Engineering Research Council(NSERC)of Canada Discovery Grant RGPIN48158 awarded to M.Hasan of McGill University,Montreal,for which authors are grateful.

Nomenclature

A Darcy coeff i cientap,anb,bcoeff i cients in the discretized governing equationsc1，c2，cμempirical constants for low Reynolds number modelcp specif i c heatC morphology constantD hydraulic diameter of the caster(the entire crosssection of the ingot)Dk extra dissipation term ink-equationEε extra generation term in ε-equationf1,f2,fμempirical constants used in low-Re version ofk-? modelsfl liquid fractionfs solid fractionG production term in turbulent kinetic energy equation Gr Grashof number Grm modif i ed Grashof number(Gr/Ste)h sensible heath∞ ambient enthalpyH total heat(sensible and latent)k turbulent kinetic energyP hydrodynamic pressure Pe Peclet number Pr laminar Prandtl number Re Reynolds number Ret turbulent Reynolds number based on the turbulent quantitiesS source termSΦ source term associated with ΦSk* non-dimensional source term associated with buoyancy term ink-equationSε* non-dimensional source term associated with buoyancy term in ε-equation Ste Stefan numberT temperatureT′ f l uctuation of temperatureTin inlet temperatureTl liquidus temperatureTs solidus temperatureTsurf slab surface temperatureui velocity component in thei-thdirection;corresponding touui time-average velocity component in thei-th directionu′i l uctuation of velocity in thei-th directionuin inlet velocityus casting speedU,V,Wnon-dimensional form of theu,vandwvelocitiesUs non-dimensional form ofusx axial directiony horizontal direction parallel to the wide facez horizontal direction parallel to the narrow faceX,Y,Znon-dimensional form ofx,y,zi,j,k coordinate direction f

Greek symbolsΔH nodal latent heat ΔHf latent heat of solidif i cation Γφ diffusion coeff i cient associated with Φ ρ alloy density μl laminar viscosity μt turbulent viscosity or eddy diffusivity μe effective viscosity equal to μl+ μtΦ generalized dependent variable Γeff effective diffusivity ? rate of energy dissipation ?in inlet rate of energy dissipation γ effective convective heat transfer coeff i cient σk, σ? turbulence model constants σt turbulent Prandtl numberSuperscripts* non-dimensional variables - time-averaged variables′fl uctuation of variables.

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Received 18 September 2014;accepted 20 November 2014 Available online 29 December 2014

*Corresponding author.Tel.:+1 514 398 2524,+1 514 924 8542(mobile); fax:+1 514 398 4492.

E-mail addresses:Mainul.hasan@mcgill.ca(M.Hasan),Latifa.begum@ mail.mcgill.ca(L.Begum).

Peer review under responsibility of National Engineering Research Center for Magnesium Alloys of China,Chongqing University.

1ASME Member(#1964840).

http://dx.doi.org/10.1016/j.jma.2014.11.008.

2213-9567/Copyright 2014,National Engineering Research Center for Magnesium Alloys of China,Chongqing University.Production and hosting by Elsevier B.V.All rights reserved.

Copyright 2014,National Engineering Research Center for Magnesium Alloys of China,Chongqing University.Production and hosting by Elsevier B.V.All rights reserved.