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Applied Thermal Engineering

journal homepage: www.elsevier.com/locate/apthermeng

Research Paper

Influence of process parameters on solidification length of twin-beltcontinuous casting

Pavan Kumar Penumakalaa, Ashok Kumar Nallathambib,⁎, Eckehard Spechtb, Ulrich Urlauc,Doug Hamiltonc, Charlie Dykesd

a Birla Institute of Technology and Science, Hyderabad Campus, IndiabOtto von Guericke University Magdeburg, GermanycUrlau Innomanagement, SwitzerlanddHazelett Strip-Casting Corporation, Colchester, VT, USA

H I G H L I G H T S

• Belt heat flux is estimated using thermal resistance network method.

• Dam block heat flux is estimated using semi-infinite body assumption.

• Total solidifcation length is proportional to the casting speed.

• Under the same casting speed and heat flux, decrease of cross section size decreases the solidification length.

A R T I C L E I N F O

Keywords:Continuous castingtwin-belt castingHazelett castersolidificationcopper

A B S T R A C T

A temperature based Finite Element Model is developed to estimate the evolution of temperature, shell growthprofiles and solidification length of a solidifying ingot in a twin-belt caster. The model is based on the 2-D slicewhich travels through the caster with time. A transient energy equation is solved with the incorporation ofthermal boundary conditions which depends on the position of the slice. The latent heat released during thesolidification is incorporated through the effective specific heat method. The convective heat flux due to thebelts, and dam blocks inside the caster and from spray cooling outside the caster are applied as boundaryconditions. The belt heat flux is estimated by incorporating thermal resistances of the cooling water, the belt, thecoating, the interface between belt and ingot, the solidifying shell and the melt. At the interface between belt andingot, the gap dependent heat transfer is considered which is based on the combined mode of radiation andconduction. The amount of gap is calculated iteratively by considering the thermal contraction of the solidifyingingot. The dam block heat flux is estimated by approximating it as a semi-infinite body. The model is applied tothe state of the art industrial twin-belt continuous copper casting. The simulated temperature and shell thicknessresults are compared with the data measured at the industrial caster. It is found that the travelling slice modelpredicts comparable results with industrial data for high speed copper casting. The model is extremely useful forindustrial engineers to know the design aspects of the twin-belt caster such as final solidification length, belt,dam block and ingot dimensions and casting speed. The effect of the process parameters on temperature evo-lution and shell growth profiles are analysed in detail. The procees length is proportional to casting speed. Whenthe casting speed and heat flux are constant, decrease of cross section size decreases the solidification length

1. Introduction

Integration of casting and rolling processes in a metal processingindustry has a huge potential for energy savings and improved pro-duction [1]. Metal casting industries across the world are using near netshape strip casting technology, which integrates casting and rolling in

series [2,3]. Twin-roll strip casting [4,5] and single belt strip casting[6,7] offers many potential advantages over the conventional con-tinuous casting in mold. However, strip cast products has few qualitydefects such as uneven strip thickness, strip flatness and surface cracks[8].

The twin-belt caster, popularly known as the Hazelett caster,

https://doi.org/10.1016/j.applthermaleng.2018.01.121Received 26 June 2017; Received in revised form 5 January 2018; Accepted 30 January 2018

⁎ Corresponding author at: Otto von Guericke University, Universityplatz 2, 39106 Magdeburg, Germany.E-mail address: ashok.nallathambi@ovgu.de (A.K. Nallathambi).

Applied Thermal Engineering 134 (2018) 275–286

Available online 01 February 20181359-4311/ © 2018 Published by Elsevier Ltd.

T

received major attention for its excellent operating history in the non-ferrous industry due to high production rates with better surface quality[9,10]. In the twin belt caster, the relative motion between the movingbelt and the solidifying shell is almost reduced to a minimum level.Thus, the reduction in friction significantly improves the surface qualitywhen compared to conventional continuous casting molds. As depictedin Fig. 1, Hazelett caster consists of two top and bottom continuouslymoving water-cooled thin metal belts. From the sides, metal blockswhich are linked together and moving at the same belt speed supportsthe molten metal. All four surfaces of the mold are independent fromeach other. The distance between the belts and dam blocks can be ad-justed to cast different sizes. The dam blocks can be adjusted in betweenthe belt for providing the desired taper to accommodate the lateralshrinkage of the ingot. Molten metal is introduced from the tundish intothe cavity formed by the belts and dam blocks. The caster can be set uphorizontally or slightly inclined position [11]. Below the caster exit,secondary cooling starts to promote the solidification rate. Rollers lo-cated at various positions withdraw the solidified product in the castingdirection and gradually bend the solidifying ingot into the horizontalposition. The complete solidification may occur in or beyond the sec-ondary cooling zone depending on the cast metal properties and castingrecipes such as the casting speed and the cooling water flow rate. Thedistance from the meniscus to the point of complete solidification iscalled as solidification length or metallurgical length. This length stronglydepends on the cast metal properties especially thermal conductivity.The twin-belt process has been mostly used for casting thin slab sectionsthat are suitable for direct rolling. Due to the possibilities of achievinghigh casting speeds through rotating belts, twin-belt casters are used inthe industries, where both casting and rolling processes are integrated.The cast product cross section is greatly minimized to a wire due to highpressure imparted by rolling mills. If the ingot is not completely soli-dified, the liquid metal spills in the rolling mills and should not be sentto the rolling. Therefore it is essential that the ingot must be completelysolidified before entry into roll mills. In any continuous caster design,the total solidification length greatly effects the position of roll mills.The position of complete solidification is the major criteria for locatingthe required cooling and rolling related equipments. For integratingboth casting and rolling, design engineers should investigate the soli-dification length under different casting conditions prior to the start ofcasting process. Hence, the effect of process parameters on total soli-dification length is investigated in the present paper.

Numerical models have been applied to different strip casting pro-cesses to investigate and quantify the effect of process parameters onshell growth profiles. Guthrie et al. investigated the solid growth

profiles in twin-roll casting and in horizontal single belt casting andproposed that belt casting is more suitable than twin-roll casting forhigh tonnages production [12]. Many experimental studies analysed thesolidification of metal on a single steel belt to estimate the belt heattransfer coefficient. During the experimental set up, the belt is cooledby the water nozzles from the bottom and liquid metal is poured on thetop of the belt. Schwerdtfeger et al. [13] estimated the overall heattransfer coefficient from the belt as 1000W/m2 K, where as Spitzeret al. [14] estimated as 1200W/m2 K. Farouk et al. [11] used a sta-tionary mold in their experiments and estimated the global heattransfer coefficient between the cast strip and the cooling water as1131W/m2 K.

Even though many experimental studies were presented, they esti-mated the overall heat transfer coefficient from the belt and they arevalid only for the particular set of experimental set up. Those ob-servations are not generalized and not supplemented with appropriatemodel. Many researchers used thermal resistance network method toanalyse the heat transfer at metal/mold interface in static casting by[15,16] and in continuous casting by [17,18,11,19]. In the resistancenetwork, the resistance at the interface is difficult to estimate becauseof the unknown gap size. The previous works did not consider the gapsize explicitly, but used the experimental measurements of belt andwater temperatures to estimate the interface heat transfer coefficient. Inthis work, the gap size is explicitly determined at the interface by in-corporating the thermal contraction of the solidifying ingot. The in-terface heat transfer coefficient is modified based on the gap-dependentinterface heat transfer coefficient relation proposed by [20]. More de-tails of this method are explained in Section 2.

The heat transfer through the dam block was not studied. There isno cooling applied to dam blocks inside the caster. During it’s contactwith the liquid metal inside the caster, the dam block receives heat fromthe liquid metal. The heated dam blocks are cooled separately in awater bath placed below the caster as shown in Fig. 1. In this work, theamount of heat flux at the liquid metal-dam block interface is calculatedby assuming the dam block as a semi-infinite body for heat conduction.More details are given in Section 3.

The heat flux calculated from the belts and dam blocks is used as aninput to the numerical model to calculate the solidifying ingot tem-perature and solid growth profiles. The numerical model solves themultidimensional heat conduction equation using a temperature basedfinite element formulation presented by Celentano et al. [21], which isextended to low Stefan number problems by Nallathambi et al. [22].More details about the application of this finite element model to twin-belt casting is explained in Section 4. The state of the art twin-belt

Nomenclature

Tin initial temperature (°C)Ts solidus temperature (°C)Tl liquidus temperature (°C)Twater cooling water temperature (°C)Tint interface temperature between belt and shell (°C)Tbin temperature of the belt side contact with liquid metal (°C)Tbout temperature of the belt side contact with water (°C)Tm temperature of the dam blockTw contact temperature between dam block and shell (°C)Tm0 dam block initial temperature (°C)

∞T ambient temperature (°C)Tc coating temperature (°C)L latent heat (kJ/kg)x y, coordinates (mm)t time (s)λs thermal conductivity of shell (W/mK)

λb thermal conductivity of belt (W/mK)λcoat thermal conductivity of coating (W/mK)λm thermal conductivity of dam block (W/mK)λa thermal conductivity of air (W/mK)δs shell thickness (mm)δb belt thickness (mm)δm dam block thickness (mm)δcoat coating thickness (mm)hwater cooling water HTC (W/m2 K)hc contact HTC (W/m2 K)hint interface HTC (W/m2 K)ρs density of shell (kg/m3)ρm density of dam block (kg/m3)cs specific heat of the solidifying shell (J/kg K)cm specific heat of dam block (J/kg K)Vc casting speed (m/min)gn normal gap (mm)erf error function

P.K. Penumakala et al. Applied Thermal Engineering 134 (2018) 275–286

276

copper casting process (also abbreviated as CONTIROD1) is simulatedand the obtained temperature and solid growth profiles are discussed inSection 5. Finally, the effect of process parameters on twin-belt casterdesign are discussed.

2. Belt heat transfer

As shown in Fig. 2, the thermal resistance network diagram consistsof thermal resistances of the cooling water, the belt, the coating, theinterface, the solidifying shell and the melt. At a certain position insidethe caster, one dimensional steady state heat conduction situation arisesand the heat rejected by the liquid metal is equal to the heat transferredthrough solidifying shell, at the interface, in the belt and to the coolingwater. The overall heat flux (q) from the liquid metal to the coolingwater can be expressed as

= −q T TR

in water

total (1)

where Rtotal is the total thermal resistance, which can be calculated as

= + + + + +R R R R R R Rtotal melt shell int coat belt water (2)

The thermal resistances from the belt (Rbelt), coating (Rcoat) and shell(Rshell) are calculated using their thermal conductivity and thickness:

=Rbeltδλ

bb, =Rcoat

δλ

coatcoat

and =Rshellδλ

ss. The resistance from the cooling

water can be calculated as =Rwater h1

water. The heat transfer coefficient

from the cooling water can vary between 10 kW/(m2 K) to 50 kW/(m2 K). This is due to nucleate boiling of water. The nucleate boilingheat transfer was studied by many researchers, which is reviewed in[23]. The interface resistance ( =R h1/int int) between the belt and theingot is unknown because of the micro level gap formation due toshrinkage of the shell. During the initial stages of solidification, perfectcontact exists between the solidifying shell and the mold, and the heattransfer between surfaces occurs via pure conduction. After sometime,an air gap develops due to the shrinkage of the solid shell. Now, theinterface heat transfer is assumed to occur mainly due to the combinedheat conduction and radiation in the gap. The interfacial energy bal-ance can be given as

= = −q q h T T( )shell int int int c (3)

where hint is the gap dependent interfacial heat transfer coefficientdefined as

= +h

gλ h

1 1int

n

a c (4)

where gn is the normal gap and hc is the convection-radiation coefficient[20]. The air gap is calculated by considering the linear thermal con-traction of purely solidified material as

= −g α T T δ( ) .n int s s (5)

where α is the linear expansion coefficient. The thermal conductivity ofair λ( )a is taken as 0.05W/(m2.K). Now, to calculate the profile of in-terfacial heat transfer coefficient along the belt, its value at the instanceof liquid metal touching the belt (hc) must be known. Many experimentshave been carried out to estimate the profile of interface heat transfercoefficient in chill mold castings [24,25]. Following [25], an interfacialheat transfer coefficient of 3000W/m2 K is used when the liquid metaltouches the steel belt for the current caster under study. The interfacialheat transfer coefficient profile down the belt is calculated using Eqs.(5) and (4), which is shown in Fig. 3a. The heat flux along the belt sideas a function of distance from the meniscus is calculated by carrying outa transient analysis along casting direction. At each time step, the

Fig. 1. Schematic diagram of the integrated twin-belt continuous casting and rolling.

Fig. 2. Thermal resistance network during the solidification of the metal against the belt.1 CONTIROD® is a registered trademark of Aurubis Belgium

P.K. Penumakala et al. Applied Thermal Engineering 134 (2018) 275–286

277

amount of heat flow along all the thermal resistances is equal. At eachtime step, the shell thickness required to calculate Eqs. (5) and (2) isunknown. Therefore, Eqs. (1)–(5) are solved implicitly by using theprevious steady value of shell thickness. This implicit nature of theproblem creates the numerical difficulty. The estimated heat flux fromthe belt as a function of the distance from the meniscus is shown inFig. 3. The heat flux varies from 2.65MW/m2 to 1.2MW/m2 along thecaster length, which is close to the heat flux measurements at CONTI-ROD caster. In the industrial plant, belt surface temperature measure-ments and ingot surface temperature measurement at caster exit will beused to calculate belt heat flux values. However, the experimentmethod to measure the temperature and computing the heat flux wouldlead the present work beyond the scope. The area under the curve inFig. 3 gives the total heat flux from the belt side. Using this value, theoverall heat transfer coefficient from the belt is estimated as 2200W/m2K, which is close to the experimental values reported in the litera-ture. Now, the belt temperatures on both liquid metal touching side andcooling water touching side can also be determined using this analysis.The thermal gradient through belt thickness is very important to cal-culate the thermal distortion of the belt. The belt temperatures can becalculated using this simple following relations,

= + −T T RR

T T( )bout waterwater

totalin water

(6)

= + −T T RR

T T( )bin boutbelt

totalin water

(7)

where Tbin is the temperature of the belt side which is in contact withthe solid shell and Tbout is the temperature of the belt side which is incontact with the cooling water.

3. Dam block heat transfer

The unique feature of the twin-belt caster is that all four moldsurfaces (belts and dam blocks) are independent. This allows the moldsurfaces to be tapered to remain in contact with the product as itshrinks. The sides of the mold are tapered using spring-loaded guideswhich keep the blocks in contact with the bar. The taper provided bythe dam block arrangement accommodates the gap formed due to so-lidification shrinkage and thermal contraction of the shell, which leadsto a perfect contact assumption between the dam block and shell. Dueto this, the temperature of the dam block side, which is in contact withthe hot metal maintains at the common contact temperature (Tw). Theother side temperature of the dam block is almost constant (Tmo) and noteffected by the heat conducted from inside. Therefore, the dam blockcan be assumed as a semi-infinite body to analyse heat conductionthrough it. Jaluria et al. reviewed that 1-D semi infinite models areadequate for solidification near the boundaries [26]. The temperature

profile inside the dam block can be approximated using the errorfunction as,

−−

=T TT T

erf η( ).m w

mo wm (8)

The parameter ηm is defined as

=η xa t2m

m (9)

where am is the thermal diffusivity of the dam block. Similarly, the endtemperature of the solidifying shell which is in contact with dam blockis at constant contact temperature (Tw). On the other side of the soli-difying shell, the temperature is always constant (Tin) if we approximatea sharp solid–liquid interface. The temperature profiles inside the so-lidifying shell can also be approximated by using the error function as[27],

−−

=T TT T

erf ηerf η

( )( )

.s w

in w

s

δ (10)

The parameter ηs is defined as

=η xa t2s

s (11)

where as being the thermal diffusivity of the shell. The heat flux fromthe dam block can be derived by analysing the heat transfer betweentwo semi-infinite bodies in contact.

The wall temperatureTw at the interface between the block and shellcan be found by equating the heat flux from both sides at the interfaceas

=+

+T

λ ρ c T λ ρ c erf η T

λ ρ c λ ρ c erf η

( )

( ).w

s s s in m m m δ m

s s s m m m δ

0

(12)

The theoretical solution is given elsewhere [27,28]. Finally, the heatflux at the interface between dam and shell surface can be calculated as

= − ∂∂

=q λ Tx

x( 0)mm

(13)

The temperature gradient can be found from Eq. (8) as

∂∂

= = − − =Tx

x T Tπa t

e( 0)m m w

m

η0 m x( 0)2

(14)

Finally, substituting the values for a η,m m in Eq. (14) and expressingthe time t( ) as a function of the casting speed V( )c and the distancebelow the meniscus z( ), the heat flux can be expressed as

= −qλ ρ c

πT T V

z( ) .m m m

w mc

0(15)

a) Shell thickness and Interafce heat transfer coefficient profiles along the belt

b) Heat flux from the belt side

0 0.5 1 1.5 2 2.5 3 3.51.2

1.4

1.6

1.8

2

2.2

2.4

2.6

Hea

t fl

ux

[MW

/m2 ]

Distance from the meniscus [m]

Calculated Heat flux

Measurements at CONTIROD caster

0 0.5 1 1.5 2 2.5 3 3.51000

1500

2000

2500

3000

Distance from the mensicus [m]

Inte

rfac

e H

TC

[W

/m2 K

]

0

5

10

15

20

Sh

ell t

hic

knes

s [m

m]

Shell thickness

Interface HTC

Fig. 3. Shell thickness, interface HTC and belt heat flux profiles.

P.K. Penumakala et al. Applied Thermal Engineering 134 (2018) 275–286

278

For the casting conditions and dam block material properties shown inTable 1, the temperature profiles inside the dam block and the solidi-fying shell are calculated using Eqs. (8) and (10). The non-linear tem-perature profiles of dam block and solid shell are shown in Fig. 4 forvarious times. The contact temperature between the dam block and theshell, which is calculated using Eq. (12) is approximately 800 °C. Thedam block heat flux profile, which is calculated using Eq. (15) is shownin Fig. 4. The heat flux is theoretically infinite at the meniscus because

=z 0. After 1m from the meniscus, the heat flux profile matches wellwith the belt heat flux profile. Using the area under the heat flux curve,the overall heat transfer coefficient from the dam block is estimated as2200W/m2 K. This value may slightly differ from experimental pre-dictions because, semi infinite model is an idealized situation for thedam block. However, the trend of the heat flux curve is similar to theexperimental measurement.

4. Mathematical model for solidifying ingot

Heat is transferred into the belts and dam blocks along the width x( )and thickness y( ) directions of the ingot. In general, the temperaturegradients in the axial/casting direction z( ) cannot be neglected due tothe high thermal conductivity of copper [29]. However, heat advectionoccurs in the casting direction due to the bulk motion of the descendingstrand [30]. In the present high speed twin-belt caster, the advectionphenomenon dominates the conduction phenomenon in the axial

direction because of the bulk motion of casting strand at higher castingspeed. Therefore, the conduction is axial direction can be neglected.The relative importance of these heat flow mechanisms are governed bythe Peclet number (Pe), which is the ratio of advective to conductiveheat flow. The Peclet number is expressed as

=Peρ c V R

λs s c

s (16)

where R is the characteristic length of the casting (minimum lateraldimension). As shown in Fig. 5, for low Pe values (Pe < 100), con-siderable temperature gradients exists along the casting direction andconduction component in this direction cannot be neglected. For highPe values (Pe > 100), there exists no temperature gradient along thecasting direction and the conduction component can be neglected. Forthe current copper casting, for a section size of R=60mm and acasting speed of 10.6m/min, the Pe is 138. For such a high Pecletnumber, energy diffusion in the axial direction is negligible. Therefore,a 2-D slice perpendicular to the casting direction is considered as thecomputational domain to simulate the temperature profiles. The slicetravels in the casting direction with time and boundary conditions areapplied on it based on its current position.

In a Lagrangian frame, the transient energy balance equation thatgoverns the temperature distribution in the solidifying material domainΩ with boundary Γ and in the time interval of analysis ∈t ϒ can bewritten as [21,22],

⎜ ⎟∇ ∇ = ⎛⎝

+∂∂

⎞⎠

∂∂

×∼λ T ρ c LfT

Tt

·( ) inΩ ϒ.sl

(17)

The boundary condition is

→ → = − ×q n q· in Γ ϒ.conv q (18)

The initial conditions are

⎯→⎯=

⎯→⎯=T X t T X( , )| ( ) in Ωt in0 (19)

⎯→⎯==f X t( , )| 1 in Ωl t 0 (20)

Assuming the copper solidifying at a constant temperature (pure metal),the liquid fraction fl can be expressed as

= ⎧⎨⎩

><f T

T TT T( )

1,0, .l

l

l (21)

Using the standard Finite Element procedure, the weak form of Eq. (25)can be written as

Table 1Material and casting parameters.

Parameter Value

Cast cross section 120mm×70mmCasting speed 10.6 m/minInitial temperature 1120 °CLiquidus temperature 1082 °CSolidus temperature 1081 °CLatent heat 205 kJ/kgDensity of casting 8735 kg/m3

Thermal conductivity of casting 300W/(m K)Specific heat capacity of casting 450 J/(kg K)Linear expansion coefficient 16.6× 10−6 1/KCaster length 3.5 mBelt thickness 1.2 mmBelt thermal conductivity 50W/(m K)Dam block thickness 50mmDam block thermal conductivity 120W/(m K)Dam block specific heat 380 J/(kg K)

Thermal conductivity of coating 3W/(m K)

Fig. 4. Schematic drawing of the temperature distribution in the damblock, solid, and liquid during the solidification.

P.K. Penumakala et al. Applied Thermal Engineering 134 (2018) 275–286

279

yUpper belt

Dam x

y

1120Pe=138

Lower belt

block1110

o C]

Pe=1 Pe=10

y 1090

1100

mpe

ratu

re [

Pe=0.1

1080

Tem

x

2D compuatational domain0 1 2 3 4 5 6

1070

Distance from the meniscus z [m]

Fig. 5. Modeling domain and boundary conditions and temperature gradient along the z direction.

Fig. 6. Flow chart of model algorithm.

1500

1400

o C]y

Insulated

t= 5 sec

1200

1300

mpe

ratu

re [o

Mesh size ~ 0.1 mm

Tm = 1495 oCTw=1000oC

t= 25 sec

t=10sec

1100

TemxInsulated

Analytical solutionPresent method

t=10sec

0 5 10 15 201000Distance from slab surface [mm]

Fig. 7. Comparison of the temperature profiles predicted by the numerical method with analytical solution.

P.K. Penumakala et al. Applied Thermal Engineering 134 (2018) 275–286

280

∫ ∫ ∫ ∫∫

∇ ∇ + +∂∂

+

= ⋯

∼ ∼ ∼ ∼

∼∞

T λ T d ρc T T d ρLft

T d h T T d

h T T d d

[ ·( )] Ω ̇ Ω Ω Γ

Γ Ω

sl

q

q

Ω Ω Ω Γ

Γ

q

q (22)

After substituting the finite element discretization, the element matricesand vectors are

∫ ∫

∫∫∫

= +

=

=

= ∞

λ d h d

ρc d

ρLf d

hT d

K B B

C

Ω NN Γ

NN Ω

L N Ω

F N Γ

e T e Tqe

es

T e

el

e

eqe

Ω Γ

Ω

Ω

Γ

eqe

e

e

qe (23)

From the element quantities, the global matrices and vectors can beobtained based on the element-nodal mappings. The final globalequation to be solved is given as

+ + =K CΘ Θ̇ L̇ F (24)

The rate term can be integrated using an Euler-backward method.Assuming Tt as the nodal temperature vector at previous time and +Tt tΔ

as the nodal temperature vector at current time and tΔ as the time in-terval, the time derivative of temperature and latent heat vector can bewritten as

= − = −++

++

t tΘ̇ Θ Θ

Δand L̇ L L

Δt t

t t tt t

t t tΔ

ΔΔ

Δ

(25)

Using Eq. (25), the residual vector at the current time step can bewritten from Eq. (24) as

= + − − − ++ + +t L L tC C KR(Θ ) FΔ Θ ( ) ( Δ )Θt t t t t t t tΔ Δ Δ (26)

The above implicit and highly non-linear equation is solved using aniterative method. Assuming i is the previous iteration and +i 1 is thecurrent iteration, the temperature at +i( 1)th iteration can be written as

= +++ + δΘ Θ Θ.i

t tit t

1Δ Δ (27)

The current iterative-increment temperature δT can be determined as

= − + − +δ JΘ [ (Θ )] R(Θ )it t

it tΔ 1 Δ (28)

where J is the Jacobian matrix, which can be calculated as

(Courtesy: HAZELETT)

2 4

2.6

2

2.2

2.4

[MW

/m2 ]

1.4

1.6

1.8

S d

Hea

t flu

x

Eqaul dendrite growth pattern from 0 1 2 3 4 5 61

1.2

Distance from Meniscus [m]

Caster exit

Secondarycooling

q g pbelt side and dam block sideDistance from Meniscus [m]

Fig. 8. Heat flux applied in both caster and secondary cooling regions.

1100

1200

Core (3) End of solidification

1000

1100

C] Mid surface (2)

1

3

800

900

ratu

re [o

Corner (1)

700

Tem

pe

()

500

600

Caster

0 1 2 3 4 5 6400

Distance from the meniscus [m]

exit

3

2

Fig. 9. Temperature profiles in the copper casting process.

50

60End of solidification

60

5

]

40

35

Widthess

[mm

]

30

Width

Thicknessell t

hick

ne

10

20Thickness

She

0 1 2 3 4 5 60

10

Di t f th i [ ]

Caster exit

Distance from the meniscus [m]

Fig. 10. Shell growth in the copper casting process.

Table 2Comparison of simulated and industrial values.

Parameter Measurements atCONTIROD caster

Simulated

Shell thickness along width at casterexit

30mm 27mm

Shell thickness along thickness atcaster exit

16mm 17mm

Mid surface temperature at casterexit

976 °C 980 °C

Mid surface temperature at exit ofsecondary cooling

890 °C 900 °C

Mid surface temperature at the end ofsolidification

870 °C 840 °C

Metallurgical length 5.5m 5.7 m

P.K. Penumakala et al. Applied Thermal Engineering 134 (2018) 275–286

281

= ∂∂

= − − − ∂∂

++ +

tJ K C(Θ ) RΘ

Δ LΘ

.it t

i

t t

i

t tΔ

Δ Δ

(29)

The derivative of the latent heat vector with respect to the temperatureis denoted as the phase-change matrix Cpc as in [21]. The elemental formof the phase-change matrix is given as in [21]

∫= ∂∂

=∂∂

ρLf

dC LΘ

NΘ

N Ωpce l T e

Ωe (30)

The derivative of the phase fraction introduces non linearity. Furtherdetails are given in the work of Nallathambi et al. [23]. The derivativeof the phase fraction with respect to temperature at every iteration iscalculated as in [21]

∂∂

=−

−∂∂

=+ +

+

+f f f fΘ Θ Θ

andΘ

0.l

i

t tlt t

lt

it t t

lt tΔ Δ

Δ1

Δi

(31)

Finally, the iterative incremental temperature vector can be calculatedfrom the Jacobian Eq. (29) by forcing the residual vector Eq. (28) toreach zero as

= − + − +δ JΘ [ (Θ )] R(Θ )it t

it tΔ 1 Δ (32)

750800850

900

900

9509501000

20

30

600

650700750800

800

850

85090020

30

800

850

00

950

10001050

105010

2000

650

700

750

050

900950

910

20

900

10000

800

850

9501000

0

0

800

850

950

0

1050

1050

Thic

knes

s [m

m]

-10

65070

0750

900

95010001050

-10

750800850

900

0

950950

1000

1000

1050

-30

-20

00650700750800850

850

900

-30

-20

8008900

Width [mm]0 10 20 30 40 50 60

6006507800

Width [mm]0 10 20 30 40 50 60

Caster exit End of solidificationTh

ickn

ess

[mm

]Fig. 11. Temperature contours at the caster exit and at the end of solidification.

280

300

260

280

[o C]

Melt side

220

240 Water side

200

220

Water side

180

Bel

t Tem

pera

ture

Melt side

0 0.5 1 1.5 2 2.5 3 3.5

160

Distance from the meniscus [m]

Fig. 12. Calculated belt temperatures.

12001200

Core(3)Core(3)End of

solidification End ofsolidification

10001000

1100

o C]

Mid surface(2)

()()

Mid surface(2)

600

800

800

900

Tem

pera

ture

[

Corner(1)Corner(1)

12

312

3

0 1 2 3 4 5400

Distance from the meniscus [m]0 1 2 3 4

600

700

Distance from the meniscus [m]

120 50 mm 94 70 mm

o C]

Tem

pera

ture

[

Fig. 13. Temperature profiles for different slab cross sections.

P.K. Penumakala et al. Applied Thermal Engineering 134 (2018) 275–286

282

The convergence criterion at the current iteration is evaluated using thecurrent temperature vector, the residual vector, and the conductancematrix as stated in [21]

< =+

+−TOL

K KR R

( Θ) ( Θ)( 10 ).

T

Ti

t t

1

Δ3

(33)

The flowchart of the algorithm is given in Fig. 6. Pure metal soli-dification problems faces convergence problems because of the suddenjump of liquid fraction fl from one to zero upon solidification. Thebetter convergence is achieved by employing line-search algorithm,which is well explained in [22].

5. Numerical results

A MATLAB algorithm has been developed to solve the temperature

based finite element equations. The algorithm is validated with ananalytical temperature solution presented by Weiner et al.[31] for so-lidification in a semi-infinite slab, which is maintained at a constantsurface temperature. The validation details are given in Fig. 7 and alsoin [32]. The solidification of liquid steel with an initial temperature of1495 °C in a semi-infinite domain is considered. The surface =x( 0) ismaintained at a constant temperature of 1000 °C and all other surfacesare insulated. The solidification progresses along x direction and thetemperature profiles through the thickness of the solidifying shell atdifferent times are shown in Fig. 7. The developed algorithm predictsthe temperature profiles close to the analytical solution.

Now, using the above algorithm the real time state of the art coppercasting process at the CONTIROD caster is simulated. The casting andmaterial parameters related to the process are given in Table 1. Toaccelerate the convergence of numerical model, a small mushy zone of

60

25

End ofsolidification

Width

Thickness

End ofsolidification

Width

Thickness

47

35

Fig. 14. Shell thickness profiles for different slab cross sections.

25 2.5

Vc

15

20

2

W/m

2 ]

Vc -6.89 m/min

Vc -

101.5

Hea

t flu

x [M

Vc -10.6 m/minVc -10.6 m/min

0 0 5 1 15 2 25 3 3 50

5 She

ll th

ickn

ess

[mm

]

0 0 5 1 1 5 2 2 5 3 3 51Vc -6.89 m/min

. . . .Distance from the meniscus [m]

. . . .Distance from the meniscus [m]

14.31 m/min

Vc -14.31 m/min

Fig. 15. Shell growth and heat flux profiles along the belt for different casting speeds.

35 3000

25

30

]

2500

[W/m

2 K]

Vc -14.31 m/min

10

15

20

Gap

[m

2000

rfaci

al H

TC

Vc -10.6 m/min

1 20

5

0 1 2 31000

1500

Inte

Vc -14.31 m/minVc -6.89 m/min

0 3Distance from the meniscus [m] Distance from the meniscus [m]

Vc - 6.89 m/minVc - 10.6 m/min

Fig. 16. Gap development and change of interfacial coefficient for different casting speeds.

P.K. Penumakala et al. Applied Thermal Engineering 134 (2018) 275–286

283

1 °C is assumed leading to a liquidus temperature of 1082 °C and a so-lidus temperature of 1081 °C. The computational domain and theboundary conditions are depicted in Fig. 5. A chamfer of 3mm is givenat the corners of the computational domain. In a 2-D slice, fourfoldsymmetry is considered. At the symmetry line Γ1, thermal insulation isassumed. When the slice is within the caster, on the domain Γ2, the heatflux calculated from the belt cooling is applied along the width (x) andthe heat flux calculated from the dam blocks is applied along thethickness (y). Practically, the same amount of heat flux is aimed on beltsand dam blocks to facilitate the equal growth rate of solid dendritesfrom both directions. The heat removal per unit area of belts and damblocks must be equal in this case. Therefore, a similar heat flux is usedon belts and dam blocks in the simulations. The heat flux in the entireprocess is shown as a function of the distance from the meniscus inFig. 8. Following Wang et al., the liquid thermal conductivity is en-hanced in liquid region to account for the fluid flow effects [33].

The mesh size is 0.5 mm at the corner regions and is about 6.5 mmin the interior. The time step size dt is 0.04 s. The total computationaltime up-to complete solidification is about 35 s. It took around 13minto finish the simulation on a Intel PENTIUM 2.0 GHz work stationrunning on a 4 GB random-access memory. The evolution of tempera-ture profiles at three different material points are plotted in Fig. 9, as afunction of the distance from the meniscus. Point 1 is taken at the baseof the chamfer, point 2 is at the mid surface and point 3 is at the core.The core temperature remains almost constant till the end of solidifi-cation. The corner temperature drops to nearly 800 °C at the caster exit

and then decreases further to 500 °C at the end of solidification. Simi-larly, the mid surface temperature drops to 980 °C at the caster exit andgradually decreases to nearly 800 °C at the end of solidification.

The shell growth profiles are presented in Fig. 10. At the caster exit,the shell thickness is 17mm along the thickness side, and 27mm alongthe width side. This represents that almost half of the liquid metal issolidified at the caster exit. Industrially observed shell thickness alongthe thickness is 16mm and along the width is 30mm at the caster exit.The gradient of the growth profile in the secondary cooling is high. Thecomplete solidification occurs at the solidification length of 5.7 m.Table 2 compares the calculated temperature and shell thickness valueswith the measurements taken at the CONTIROD caster. The final soli-dification point almost matches with the theoretical solidificationlength model presented in [34]. The temperature contours in the soli-difying slice at the caster exit and at the end of solidification areschematically shown in Fig. 11. The belt temperature profiles, whichare calculated from analytical expressions Eqs. (6) and (7), are shown inFig. 12. The temperature of the belt side, which is in contact with themolten copper varies from 290 °C to 170 °C. The temperature of the beltside, which is in contact with the water varies from 240 °C to 150 °C.The belts are made of low carbon steel, held under tension on thecasting machine to ensure flatness and accuracy. However, thermalbending forces resulting from the temperature gradient across the beltcauses bending and distortion of the belt. The bent shaped belt squeezesthe newly solidified material along its surfaces causing the surface de-fects which may turn out to surface cracks on the solidified product.

6. Effect of ingot size

The effect of slab cross section on the temperature and shell growthprofiles is studied by keeping the same casting speed and same heat fluxas that of standard conditions. Two different cases are studied bychanging width in one case and by changing thickness in one case. Twodifferent cross sections studied here are 120mm×50mm and94mm×70mm. Fig. 13 shows the ingot temperature profiles for dif-ferent cross sections. For 120mm×50mm section, at the caster exit,the corner temperature is 770 °C and mid surface temperature is 960 °C.For the standard cross section these temperatures are 800 °C and 980 °C(Fig. 9). It can be seen that, when the thickness of the ingot is decreasedby keeping the width constant, the ingot temperatures decrease at thecaster exit. For 94mm×70mm section, at the caster exit, the cornertemperature is 816 °C and mid surface temperature is 990 °C. Fig. 14shows the effect of slab size on shell growth profiles. The solidificationcompletes at 3.8 m for 120mm×50mm size, whereas complete soli-dification occurs at 4.6 m for 94mm×70mm size. In both cases, thesolidification length is decreased from the standard condition (Fig. 10).

2 6

2.8

3

2.4

.

2 ]

Vc -14.31 m/min

Vc -10.6 m/min

1.8

2

2.2

1.4

1.6 Hea

t flu

x [M

W/m

Vc -6.89 m/min

0 0.5 1 1.5 2 2.5 3 3.51

1.2

Distance from the meniscus [m]

Fig. 17. Dam block side heat flux for different casting speeds.

1200 1200

Core(3)Core(3)End of

solidification End ofsolidification

1000

1100

re [o C

]

900

1000

1100

re [o C

] Mid surface(2)

()

Mid surface(2)

800

900

Tem

pera

tu

700

800

Tem

pera

tu

Corner(1)Corner(1)

12

312

3

0 0.5 1 1.5 2 2.5 3 3.5600

700

Distance from the meniscus [m]0 1 2 3 4 5 6 7500

600

Distance from the meniscus [m]

Vc -6.89 m/min

Fig. 18. Temperature profiles for different casting speeds.

P.K. Penumakala et al. Applied Thermal Engineering 134 (2018) 275–286

284

This is expected because of the reduction of total cross section area fromthe standard condition.

7. Effect of casting speed

The casting speed affects the surface and corner temperatures, shellgrowth and the total metallurgical length of the plant [32]. Numericalstudies are carried for the casting speed of 6.89m/min and 14.31m/min. The three cases correspond to production rates of nearly 30 t/h,45 t/h and 60 t/h respectively.

Fig. 15 shows the shell growth profile along the belt for differentcasting speeds. When the casting speed is increased, the shell is thinnerthan the standard speed case at the caster exit, which may lead to thebreak outs at the mold exit. When the casting speed is decreased, theshell is thicker than the standard speed case at the caster exit, which issafe. The shell thickness at the mold exit is 17mm for the standard caseof 10.6m/min, 23mm for 6.89m/min and 11mm for 14.31m/min. Itcan be inferred that the shell thickness change is almost proportionatewith change in casting speed. Similarly, the heat flux is increased withincrease in the casting speed. This is due to the decrease in thermalresistance of a thinner shell. Similarly, heat flux decreases with increasein casting speed due to the increase in thermal resistance of thickershell. The total area under the heat flux curve increases with increase incasting speed. This represents the lower heat extraction when thecasting speed is increased. Increase in casting speed decreases the re-sidence time of the ingot inside the caster and increases the surfacetemperature and decrease shell thickness at the caster exit.

Fig. 16 shows the gap profiles and interfacial heat transfer coeffi-cient profiles along the belt for different casting speeds. The gap de-veloped is zero in the early stages due to perfect contact between theliquid metal and belt and it increases at the end of caster. The gap sizedecreases with increase in casting speed due to the decrease in shellsize. Similarly, gap size increases with decrease in casting speed due tothe increase in shell size. The gap developed at the caster exit is nearly16 μm for the standard speed of 10.6 m/min, 23μm for 6.89m/min and9μm for 14.31m/min. Therefore, the gap size change is proportional tothe casting speed. Interfacial heat transfer coefficient increases withcasting speed due to decrease in gap size and decreases with castingspeed due to increase in gap size. This can be verified from Eq. (4). Theinterfacial heat transfer coefficient at the caster exit is nearly1500 W/m K2 for the standard speed of 10.6 m/min, 1200 W/m K2 for6.89m/min and 1800 W/m K2 for 14.31m/min. The interfacial heattransfer coefficient change is proportional to the casting speed.

Fig. 17 shows the dam block side heat flux for different castingspeeds. Increase in casting speed increases the heat flux and decrease incasting speed decreases the heat flux. The heat flux from dam block is

proportional to the square root of casting speed (Eq. (15)). Fig. 18shows the temperature profiles at three different locations for low andhigh casting speeds. The low casting speed causes more heat extractionand the complete solidification occurs at a distance of 3m from themeniscus. Higher casting speeds delay the complete solidification dueto a lower heat extraction. The complete solidification occurs almost at7m for the high casting speed. Fig. 19 shows the shell growth profilesalong the width and thickness for low and high casting speeds. Com-plete solidification occurs within the caster for the low casting speed,where as only 57% of the area is solidified at the caster exit for the highcasting speed. For the higher casting speed, the shell thickness along thewidth and thickness are 16mm and 14mm respectively at the casterexit and the final solidification point extends to 7m. Decrease in castingspeed by 35% decreases the solidification length by almost 40% andincrease of casting speed by 35% increases the solidification length bythe same amount.

8. Summary

A method for estimating the belt and dam block heat flux profiles oftwin belt continuous casting is presented. A semi infinite based damblock heat flux and series resistance circuit based belt heat flux arepredicted. Using this heat flux model along with temperature basedfinite element subroutine, a state of the art casting of copper ingot withsize 120mm×70mm is analysed and the temperature and solidgrowth profiles are discussed. The travelling slice model predictedcomparable results with industrial data for high speed copper casting.This heat flux approximation is highly essential for simulating thecasting process and observed that,

1. The heat flux from the belt increases with increase in casting speed.2. The shell thickness and gap developed along the belt decreases with

increase in casting speed.3. The heat flux from dam block is proportional to the square root of

the casting speed.4. The solidification length is directly proportional to mass flow rate.

When the area remains the same, the increase in casting speed in-creases the solidification length.

Acknowledgements

The financial support provided by the German Science Foundation(DFG) through graduate school GRK 1554 is sincerely acknowledged.

6060End of End of

40

50

40

50

60

35

Width

Width

60

35

20

30

20

30

Thickness

10

0

10

She

l lth

ickn

ess

[mm

]

0 2 4 6 80

Distance from the meniscus [m]0 1 2 3 4

Distance from the meniscus [m]

Thickness

She

l lth

ickn

ess

[mm

]

solidification solidification

Fig. 19. Shell thickness for different casting speeds.

P.K. Penumakala et al. Applied Thermal Engineering 134 (2018) 275–286

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Appendix A. Supplementary material

Supplementary data associated with this article can be found, in theonline version, at http://dx.doi.org/10.1016/j.applthermaleng.2018.01.121.

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