Proceedings of the ASME 2016 Dynamic Systems and Control ConferenceDSCC2016

October 12-14, 2016, Minneapolis, Minnesota, USA

DSCC2016-9698

DYNAMIC MODELING OF TWIN-ROLL STEEL STRIP CASTING

Florian Browne, George Chiu, Neera Jain∗School of Mechanical Engineering

Purdue UniversityWest Lafayette, Indiana 47907

Email: fbrowne@purdue.edu, gchiu@purdue.edu, neerajain@purdue.edu

ABSTRACTWe consider the problem of dynamic coupling between the

rapid thermal solidification and mechanical compression of steelin twin-roll steel strip casting. In traditional steel casting, moltensteel is first solidified into thick slabs and then compressed via aseries of rollers to create thin sheets of steel. In twin-roll cast-ing, these two processes are combined, thereby making controlof the overall system significantly more challenging. Therefore,a simple and accurate model that characterizes these coupleddynamics is needed for model-based control of the system. Wemodel the solidification process with explicit consideration forthe mushy (semi-solid) region of steel by using a lumped parame-ter moving boundary approach. The moving boundaries are alsoused to estimate the size and composition of the region of steelthat must be compressed to maintain a uniform strip thickness. Anovelty of the proposed model is the use of a stiffening spring tocharacterize the stiffness of the resultant strip as a function of therelative amount of mushy and solid steel inside the compressionregion. In turn this model is used to determine the force requiredto carry out the compression. Simulation results demonstrate keyfeatures of the overall model.

INTRODUCTIONMotivation and Problem Definition: Near-net-shape manu-

facturing processes are becoming a major contributor in the re-duction of both environmental and economic costs in the indus-trial sector [1]. For the steel industry, twin-roll strip casting is oneof the most prominent near-net-shape manufacturing processes.

∗Address all correspondence to this author.

It requires just one-tenth of the facility space, and it reduces theenergy consumption by a factor of nine, as compared to tradi-tional steel casting [2]. In the latter, molten steel is first solidi-fied into thick slabs and then compressed via a series of rollersto create thin sheets of steel. In contrast, in twin-roll casting,molten steel is poured directly onto the surface of two castingrolls which simultaneously cool and compress the steel into astrip with a thickness of 1−3 millimeters. Combining these twosteps into a single continuous casting process introduces cou-pling between the rapid thermal solidification dynamics and themechanical stiffness of the resulting steel strip. To compensatefor this coupling from a controls perspective, we require a simpleand accurate model that characterizes the system dynamics.

Gaps in Literature: Many researchers have modeled the so-lidification process in twin-roll casting [3–7] but few have con-sidered the coupling between the thermal and mechanical dy-namics [5, 6]. In order to design a controller that achieves thedesired performance objective of uniform strip thickness, we re-quire a simple model that captures the relevant input-output dy-namics of the process. Santos et al. [3] and Liu et al. [4] cre-ated high-resolution simulations of the solidification process, butthese are too complex to be used for control design. For example,Santos et. al derived a model with over 400 states. Furthermore,these models were intended to only capture the solidification pro-cess, and they do not examine the coupling between the solidifi-cation dynamics and the mechanical stiffness of the steel strip.

Other researchers have derived control-oriented models ofthe entire process [5–7]. However, their models assume anabrupt phase transition from liquid to solid steel when, in real-ity, this transition involves the storage of latent heat in a two-

1 Copyright c© 2016 by ASME

NOMENCLATURE

Symbol Description Symbol Description

c Specific heat Ω Rotational speed

fm Mushy fraction Subscript Description

F Force 1 Liquidus

h Heat transfercoefficient

2 Solidus

k Thermalconductivity

Comp Compression

L Latent heat g Gap

R Radius k Kiss

T Temperature ` Liquid

x Horizontaldistance

Lev Steel pool level

y Height m Mushy

Z Transversedirection

O Outer boundary

ε Strain R Steel at the roll

λ Stiffness Roll Roll surface

ρ Density s Solid

θ Angle

phase region known as mushy steel. The amount of mushy steelin the pool plays an essential role in determining the total forcerequired to compress the steel into a uniform strip [8] and mustbe modeled to enable model-based control of the process.

Contribution: In this paper we describe a control-orienteddynamic model of the twin-roll steel casting process that ac-counts for multi-phase steel. A lumped parameter movingboundary approach is used to simplify the dynamics while stillcharacterizing the composition of the steel in the nip region. Asecond model is derived to determine the force required to regu-late the strip thickness.

Outline: The remainder of this article is organized as fol-lows. We present a brief description of twin-roll strip casting inthe Background section followed by an overview of the model-ing approach for both the solidification and compression models.In the section titled Simulation Results, we demonstrate the fea-tures of each model. The paper ends with with a discussion ofour conclusions and future work.

FIGURE 1. A TWIN-ROLL STRIP CASTER WITH DASHEDLINES REPRESENTING THE BOUNDARIES BETWEEN THE LIQ-UID, MUSHY, AND SOLID STEEL PHASES.

FIGURE 2. THE MOVING BOUNDARIES, R1 AND R2, COR-RESPONDING TO THE LIQUIDUS, T1, AND SOLIDUS, T2,ISOTHERMS RESPECTIVELY.

BACKGROUNDFigure 1 shows a schematic of the twin-roll casting pro-

cess after it reaches steady state operation. During the castingprocess, molten steel is poured onto the surface of two castingrolls, thereby forming a liquid steel pool that covers the roll upto an angle θLev. Energy is then transferred from the liquid steel,through the roll, and into a series of cooling channels. As this oc-curs, the liquid steel cools and begins to undergo a phase transi-tion. Initially, it transitions to a semisolid, two-phase state knownas mushy steel. As more heat is extracted from the pool, themushy steel solidifies completely.

The phase transition occurs over the range of temperaturesbetween the liquidus and solidus temperatures. All steel with atemperature above the liquidus temperature, T1, is considered tobe completely liquid. Similarly, all steel below the solidus tem-perature, T2, is considered to be completely solid. This means

2 Copyright c© 2016 by ASME

that the mushy region of the steel pool is defined as the volumeof steel whose bulk temperature lies between the liquidus andsolidus isotherms. With this distinction, the steel pool can be di-vided into three regions – liquid, mushy, and solid – with borderscoinciding with the liquidus and solidus isotherms as shown inFig. 2.

As each of the two rolls rotate, mushy and solid shells beginto form on their surfaces. These shells adhere to the roll sur-face and rotate with the roll while continuing to grow. When themicro-structures of the mushy and solid shells (from each roll)intersect at the center-line, they create a width of steel that maybe wider than the desired strip thickness. When this occurs, thesteel must be compressed to achieve the desired strip thickness.This compression, in turn, is achieved by applying a force, F , toboth rolls in line with the nip, defined as the location at which therolls are the closest to each other, as shown in Fig. 1. The amountof force required for the compression is influenced by the loca-tion of the intersection of the two mushy shells, called the kisspoint. The distance from the kiss point to the nip is dependent onboth the gap distance, xg, and the rolls’ rotational speed, Ω [5,9].We then define the region of compression, also known as the nipregion, as the volume between the nip and the kiss height, yk.Within this region, the force required to compress the steel is de-pendent on the composition of steel which, in turn, is dependenton the growth rate of the solid steel shell and Ω.

From a control systems perspective, we are interested inmodeling how the control inputs - xg and Ω - affect our abilityto manufacture a strip of steel of a desired thickness. To achievethis, we divide the process into two separate models. One modeldescribes how the steel pool geometry and the rolls’ rotationalspeed affect the composition of steel throughout the solidifica-tion process. That composition is then an input into a secondmodel which calculates the force required to maintain a speci-fied gap distance between the rolls.

SOLIDIFICATION MODELThe solidification process is modeled using a lumped pa-

rameter moving boundary approach. The moving boundary ap-proach is commonly used in models of multi-phase flows to trackthe boundary between different phases through the use of ther-modynamic analysis [10, 11]. The twin-roll casting process iswell suited to this approach because the steel pool can be dividedinto three distinct regions – liquid, mushy, and solid – with well-defined boundaries.

The governing equations for the solidification process areconsidered to be the same for each roll. As a result of this as-sumption, the steel pool is divided in half, and we model the pro-cess acting on one roll with the center-line of the pool treated asthe outer boundary of each half pool. This geometry permits thehalf pool to be divided into smaller, discretized control volumes,which we will refer to throughout the manuscript as slices, with

FIGURE 3. A SCHEMATIC OF A HALF STEEL POOL MODELDIVIDED INTO CONTROL VOLUME SLICES WITH ANGULARTHICKNESS δθ .

an angular thickness δθ as shown in Fig. 3. Within each slice,we assume that the mushy and solid shells do not slip along theroll’s surface as the roll rotates. However, the liquid steel withineach slice does not rotate with the roll; instead, the amount ofliquid steel in each slice fluctuates to fill the entire volume.

Governing EquationsWithin each slice we define a lumped parameter set that is

used to calculate its dynamics through a combination of energybalances and boundary continuity equations.

Energy Balance. The energy balance for each region ofsteel is represented by the equation

∂ρcT∂ t

=5(k5T )+S , (1)

where ρ is the density, k is the thermal conductivity, T is thetemperature, and c is the specific heat corresponding to the steelphase. The variable S represents the heat source associatedwith phase transition and is only necessary in the mushy region.Within that region the heat source characterizes the release of la-tent heat needed to fully solidify the steel. Consistent with [3]and [12], the latent heat is characterized using the concept ofpseudo-specific heat. However, here we take a simplified ap-proach and assume that the amount of latent heat released duringsolidification is linearly related to the temperature of the mushysteel. The resulting pseudo-specific heat is given by

cm = c+αL

T1−T2, (2)

where T1 is the liquidus temperature, T2 is the solidus tempera-ture, L is the latent heat of fusion, and α is a tunable parameter

3 Copyright c© 2016 by ASME

FIGURE 4. THE THERMAL IMPEDANCE MODEL FOR THESTEEL POOL.

that approximates the quality within the mushy region. For ex-ample, as α increases, the capacitance of the mushy region in-creases, signifying that there is a higher ratio of solid to liquidsteel within the mushy region.

Within each slice, the total rate of change of the energy ineach steel phase is obtained by integrating Eqn. (1) over the vol-ume of each phase as shown in Eqn. (3). This equation is simpli-fied by substituting dV = rdrdθdz and applying the assumptionthat, for a given slice, the dynamics are uniform in both the an-gular (θ ) and transverse (z) directions of the roll. The result isEqn. (4).

∫∫∫V

∂ρcT∂ t

dV =∫∫∫

V5(k5T )dV (3)

Zδθ

∫∂ρcT

∂ trdr =

∫∫∫V5(k5T )dV . (4)

The left hand side of Eqn. (4) is simplified by applying Leib-niz’s rule and the product rule. The left hand side for the mushyregion reduces to Eqn. (5). The liquid and solid regions are sim-plified similarly.

Zδθ

(ρmcm

2(R2

1−R22)

dTm

dt+ρmcm(Tm−T1)R1

dR1

dt

)+Zδθρmcm(T2−Tm)R2

dR2

dt

(5)

The right hand side of Eqn. (4) is simplified by assumingthat the heat transfer through the steel is dominated by the steel-roll interface and, thus, occurs only in the radial direction. Diver-gence theorem is then applied and the resulting simplification, inits most general form, is

(1r

∂

∂ r

(kZδθr

∂T∂ r

))i+

(1r

∂

∂ r

(kZδθr

∂T∂ r

))o

, (6)

where k is the thermal conductivity of each phase of steel and thesubscripts i and o denote the inner and outer radial boundaries ofeach phase.

Equation (6) is further simplified by assuming the processreaches quasi-steady state and by applying a thermal impedance

model similar to the one shown in Fig. 4. The resulting simplifi-cation is

kpZδθ(Ti−Tp)

ln(

RpRi

) +kpZδθ(To−Tp)

ln(

RoRp

) , (7)

where the subscript p∈ `,m,s denotes parameters that are spe-cific to the phase of steel being modeled.

Furthermore, the water passing through the roll is assumedto be an infinite heat sink, and the roll is considered to have nothermal capacitance. This results in a constant roll temperaturethat serves as an inner boundary for the steel pool model. Thisboundary is modeled by adding a contact resistance between TRand TRoll , as shown in Fig. 4.

Continuity Equation. The amount of energy crossingthe liquidus isotherm should be the same for both the liquid andmushy regions. Likewise, the amount of energy crossing thesolidus isotherm should be the same for both mushy and solidregions. We apply the following two continuity equations to en-sure that these constraints are enforced.

0 = k`T`−T1

ln(

R`R1

) − kmT1−Tm

ln(

R1Rm

) (8)

0 = ksT2−Ts

ln(

R1Rs

) − kmTm−T2

ln(

RmR2

) (9)

In both Eqns. (8) and (9), R` is the radius correspondingto the center of the liquid phase and is calculated by R` =√

0.5(R21 +R2

O). Similarly, Rm is the center of the mushy phaseand Rs is the center of the solid phase, determined by Rm =√

0.5(R21 +R2

2) and Rs =√

0.5(R22 +R2

R), respectively. Equa-tions (8) and (9) are true for all time. Thus, the time derivative ofboth equations is also equal to zero.

State Matrix Equation. The three energy balances andtwo time derivatives of the continuity equations combine to yielda system of state equations that describe the interactions betweenthe state variables, R1, R2, Ts, Tm, and T`. The five state equationscan then be written in the matrix form shown in Eqn. (10) withthe non-zero elements listed in Table 1. This matrix equationapplies for each slice, and the level of discretization can varybased on the needs of the engineer. As such, the total number ofstates needed to capture the solidification dynamics is five timesthe number of slices used.

4 Copyright c© 2016 by ASME

TABLE 1. THE NONZERO VALUES OF THE STATE MATRIXEQUATION (10)

Element Value

M12 ρsc(Ts−T2)R2

M13ρsc2 (R2

2−R2R)

M21 ρmcm(Tm−T1)R1

M22 ρmcm(T2−Tm)R2

M24ρmcm

2 (R21−R2

2)

M31 ρ`c(T1−T`)R1

M35ρ`c2 (R2

O−R21)

M41

− k`(T`−T1)

(0.5R`− R`

R21

)R1

ln(

R`R1

)2R`

+km(T1−Tm)

(1

Rm−0.5R2

1R3

m

)Rm

ln(

R1Rm

)2R1

M42

.5km(T1−Tm)R2

ln(

R1Rm

)2R2

m

M44km

ln(

R1Rm

)M45

k`ln(

R`R1

)M51

0.5km(Tm−T2)R1

ln(

RmR2

)2R2

m

M52

− km(Tm−T2)

(0.5Rm−

RmR2

2

)R2

ln(

RmR2

)2Rm

+ks(T2−Ts)

(1

Rs−0.5R2

2R3

s

)Rs

ln( RsRs )

2R2

M53

ks

ln(

R2Rs

)M54

km

ln(

RmR2

)A1

ks(T2−Ts)

ln(

R2Rs

) − Ts−TRollln(Rs/RR)

ks+ 1

hRR

A2km(T1−Tm)

ln(

R1Rm

) − km(Tm−T2)

ln(

RmR2

)A3

k`(TO−T`ln(

ROR`

) − k`(T`−T1)

ln(

R`R1

)

0 M12 M13 0 0

M21 M22 0 M24 0M31 0 0 0 M35M41 M42 0 M44 M45M51 M52 M53 M54 0

R1R2TsTmT`

=

A1A2A300

(10)

FIGURE 5. A SCHEMATIC OF THE MODEL OF THE COMPRES-SION REGION.

COMPRESSION MODELAfter the steel solidifies into mushy and solid shells on the

surface of the rolls, the compression model is used to character-ize the force required to combine the two shells together withinthe nip region. The force interaction that we model is the forcerequired to compress the shells in order to achieve a desired stripthickness. As the mushy and solid shells grow and rotate throughthe liquid steel pool, there is a point where the shells on both rollshave grown to the extent that they meet at the center-line of thepool and begin to weld together. When this occurs, as shown inFig. 5, the combined shells create a strip that is 2x meters thickerthan the gap distance, xg. The steel must then be compressed tomaintain the gap distance.

The force required to compress the steel is assumed to behighly dependent on the composition of the steel within the nipregion. Therefore, we define a new parameter, the mushy frac-tion, fm, such that

fm =

0, if R2 ≥ RORO−R2RO−RR

, if RR < R2 < RO

1, otherwise .

(11)

Given the composition of the steel in the nip region, an esti-mate for the compressive force required by the process is deter-mined using a mass-spring model of the steel strip. In this modelwe assume the stiffness of mushy steel is substantially lower thanthe stiffness of solid steel which means that the applied force willcompress the mushy steel before compressing the solid steel. Astiffening spring model [13] is used to characterize this behavior.In the spring model, the force is a function of the stiffness of thesteel, λ , and the compressive strain of the steel, εx. The compres-sive force, FComp, is calculated by integrating dFComp = λεxdA

5 Copyright c© 2016 by ASME

FIGURE 6. STIFFNESS CHARACTERISTICS OF THE STIFFEN-ING SPRING AS IT RELATES TO THE STEEL COMPOSITION INTHE COMPRESSION REGION.

over the differential cross-sectional area of the compression re-gion, dA. Substituting dA = ZRR cosθdθ and integrating bothsides results in

FComp = Z∫

θk

0λ (θ)ε(θ)RRcos(θ)dθ , (12)

where the strain, εx is calculated as εx =2δx

2x+xgwith δx and x

defined as shown in Fig. 5.The stiffness of the steel is dependent on the size of the

mushy region, xm, compared to the compressive displacement,δx. If δx is smaller than xm, the compression is considered toonly occur in the mushy steel as shown in Fig. 6. Once thatthreshold is exceeded, the stiffness increases because solid steelis being compressed in addition to mushy steel. The size of themushy region is calculated using Eqn. (13) which relates themushy fraction to the distance between the center-line and R2.The expression for the stiffness of the steel, λ , is then given byEqn. (14).

xm = (1− fm) ·(

x+xg

2

)(13)

λ =

λm, if δx≤ xm

λs, otherwise(14)

SIMULATION RESULTSIn this section we illustrate how the lumped parameter mov-

ing boundary approach captures the formation of both the mushyand solid steel shells with many fewer dynamic states than previ-ous modeling efforts. Additionally, the results demonstrate how

TABLE 2. SIMULATION PARAMETERS

Parameter Value Units

RR 250 mm

Inlet Steel Temperature 1600 C

TRoll 150 C

xg 2 mm

α 0.3

ΘLev 48 deg

Ω 42.7 RPM

TABLE 3. STEEL PROPERTIES

Property Value Units Property Value Units

c 750 J/kg C T1 1524 C

L 272000 J/kg T2 1502 C

k` 4000 W/m K ρ` 7200 kg/m3

km 4000 W/m K ρm 7530 kg/m3

ks 77 W/m K ρs 7860 kg/m3

using the stiffening spring model characterizes the force requiredto compress the steel in the nip region as a function of the solid-ification dynamics.

For this case study we divide the steel pool into 6 sliceswhich results in just 30 dynamic states. The model parametersused in the simulation are listed in Table 2, and the steel proper-ties are listed in Table 3.

Solidification ResultsIn this case study, the liquid steel pool begins to cool and

transition into mushy steel almost immediately after it comesinto contact with the roll as shown in Fig. 7. The solid shellthen gradually grows as more energy is extracted from the mushysteel. As a given slice of steel rotates toward the nip, the growthrate of both shells begins to slow. The slowing is explained byFig. 8 which shows that the heat flux into the roll decreases asthe slice rotates through the pool. The lower heat flux yieldsslower solidification and, thus, slower shell growth. The primarycause for the decreased heat flux is that the temperature of thesteel contacting the roll, TR, decreases over time. When the rollfirst rotates into the steel pool, liquid steel contacts the roll andTR is equal to T`. Then, as the mushy and solid shells form, TRbegins tracking the temperature of whichever phase of steel iscontacting the roll.

6 Copyright c© 2016 by ASME

Time [sec]0 0.05 0.1 0.15 0.2

Isot

herm

Dis

tanc

e F

rom

Rol

l [m

m]

0

0.5

1

1.5

2

R1

R2

FIGURE 7. THE DISTANCE FROM THE LIQUIDUS (R1) ANDSOLIDUS (R2) ISOTHERMS TO THE SURFACE OF THE ROLLFOR ONE SLICE AS IT ROTATES THROUGH THE STEEL POOL.

Time [sec]0 0.05 0.1 0.15 0.2

Nor

mal

ized

Hea

t Flu

xIn

to T

he R

oll

0.5

0.6

0.7

0.8

0.9

1

1.1

FIGURE 8. THE NORMALIZED HEAT FLUX EXTRACTEDFROM ONE SLICE AS IT ROTATES THROUGH THE STEEL POOL.

The mushy and solid steel temperatures, Tm and Ts, are plot-ted as a function of time in Fig. 9. As expected, the mushysteel temperature varies little whereas the solid steel temperaturedecreases significantly as the roll rotates through the pool (ap-proximately 0.19 seconds).

When we examine how all six slices evolve during the dif-ferent stages of their rotation through the pool, we see the profileshown in Fig. 10. This profile shows that both the solid andmushy shells continue to grow as each slice gets closer to thenip, i.e. closer to θ = 0. The mushy shell stops growing whenit intersects the center-line, but the solid shell continues to grow.This behavior is similar to that described by Santos et al. [3].

This model also captures how changing α influences thegrowth of the mushy and solid shells. As α increases, the capac-itance of the mushy region correspondingly increases. This af-fects the location of the liquidus and solidus isotherms as shownin Fig. 11. When α increases from 0.3 to 0.5 there is less so-lidification due to the increased capacitance, cm. The higher cm

Time [sec]0 0.05 0.1 0.15 0.2

Tem

pera

ture

[C]

1200

1250

1300

1350

1400

1450

1500

1550 Liquidus Temperature

Solidus Temperature

Tm

Ts

FIGURE 9. THE MUSHY AND SOLID STEEL TEMPERATURESIN ONE SLICE AS IT ROTATES THROUGH THE STEEL POOL.

Angle Relative to Nip [deg]0 8 16 24 32 40 48

She

ll T

hick

ness

[mm

]

0

0.5

1

1.5

2

MushySolid

FIGURE 10. THE MUSHY AND SOLID SHELL PROFILES FORALL 8 SLICES AT THEIR RESPECTIVE LOCATIONS WITHIN THESTEEL POOL.

indicates that more energy must be extracted from the mushy re-gion in order for the solid shell to grow. This capacitance varieswith the quality of the mushy steel (i.e. the ratio of solid to liq-uid steel), but accurately modeling the quality would require asignificantly more detailed model. Instead, α gives us a simpletuning parameter that we can use to approximate the steel qualityand improve the accuracy of the model.

Compression ResultsThe nip region is formed when the mushy shell intersects the

center-line of the pool as shown in Fig. 5. The mushy fraction,fm, for the nip region is then calculated using the location ofthe solidus isotherm, R2, and Eqn. 11. With that information,the compressive force needed to maintain the gap distance, xg, isdetermined by solving Eqn. 12.

The resulting force calculation is dependent on both the sizeof the nip region as well as fm. As Fig. 12 shows, either an in-crease in the size of the nip region, calculated by θk, or a reduc-

7 Copyright c© 2016 by ASME

Time [sec]0 0.05 0.1 0.15 0.2

Isot

herm

Dis

tanc

e F

rom

Rol

l [m

m]

-0.5

0

0.5

1

1.5

2

R1 , = 0.5

R2 , = 0.5

R1 , = 0.3

R2 , = 0.3

FIGURE 11. THE EFFECT OF VARYING α FROM 0.3 TO 0.5 ONTHE LOCATION OF THE ISOTHERMS IN ONE SLICE AS IT RO-TATES THROUGH THE STEEL POOL.

FIGURE 12. THE COMPRESSIVE FORCE IS A FUNCTION OFθnip AND fm.

tion in fm can result in an increase in the required force. Figure12 also shows that for a very large fm, the compression only oc-curs within the mushy steel. This results in a much lower forcerequirement than a nip region of the same size with a lower fm(e.g. when the size of the solid shell is similar to the size of themushy shell).

As the mushy and solid shell sizes change over time, bothfm and θk can fluctuate, causing the force requirement to alsofluctuate. The proposed model gives us a method to calculate thechange in the required force as a function of time.

CONCLUSIONThis paper describes a control-oriented model of the twin-

roll strip casting process. The proposed model effectively char-acterizes the three regions of the solidification process using onlya fraction of the dynamic states needed in previous solidificationmodels. The division of the pool into liquid, mushy, and solid

phases also assists in determining the size of the nip region andthe amount of force required to compress the steel. By doingso, this modeling approach identifies the key coupling betweenthe thermal solidification dynamics and the mechanical dynam-ics. In future work we will augment this model with the thermaland mechanical dynamics of the rolls into an overall model ofthe process that can be leveraged for multivariable control de-sign. We will also investigate the effects that steel viscosity hason the compression model.

REFERENCES[1] Birat, J.-P., Steffen, R., and Wilmotte, S., 1995. State of the art

and developments in near-net-shape casting of flat steel products:final ECSC report. No. 16671 in EUR Technical Steel Research -Steelmaking. European Commision, Luxembourg.

[2] Matsushita, T., Nakayama, K., Fukase, H., and Osada, S., 2009.“Development and Commercialization of Twin Roll Strip Caster”.IHI Engineering Review, 42(1), pp. 1–9.

[3] Santos, C. A., Spim, J. A., and Garcia, A., 2000. “Modeling ofSolidification in Twin-Roll Strip Casting”. Journal of MaterialsProcessing Technology, 102(1), pp. 33–39.

[4] Liu, L., Liao, B., Guo, J., Liu, L., Hu, H., Zhang, Y., and Yang,Q., 2014. “3d Numerical Simulation on Thermal Flow CouplingField of Stainless Steel During Twin-Roll Casting”. Journal ofMaterials Engineering and Performance, 23(1), Jan., pp. 39–48.

[5] Edwards, J. B., and Cavazos, A., 2005. “Interaction Analysis of theTwin-Roller Strip-Casting Process and the Implications for Pro-cess Control”. Journal of Materials Engineering and Performance,14(3), pp. 395–407.

[6] Zhang, W., Ju, D., Zhao, H., Hu, X., Yao, Y., and Zhang, Y., 2015.“A Decoupling Control Model on Perturbation Method for Twin-Roll Casting Magnesium Alloy Sheet”. Journal of Materials Sci-ence & Technology, 31(5), May, pp. 517–522.

[7] Hong, K.-S., Kim, J.-G., and Tomizuka, M., 2001. “Control ofStrip Casting Process: Decentralization and Optimal Roll ForceControl”. Control Engineering Practice, 9(9), pp. 933–945.

[8] Glowacki, M., 2012. Inverse Analysis Applied to Mushy Steel Rhe-ological Properties Testing Using Hybrid Numerical-AnalyticalModel. INTECH Open Access Publisher.

[9] He, Y., and Essadiqi, E., 2010. “Finite Element Modeling of theTwin-Roll Strip Casting Process: Application to an AZ31 Magne-sium Alloy”. In TMS, pp. 545–552.

[10] Grald, E. W., and MacArthur, J., 1992. “A moving-boundary for-mulation for modeling time-dependent two-phase flows”. Interna-tional Journal of Heat and Fluid Flow, 13(3), Sept., pp. 266–272.

[11] Wedekind, G. L., and Stoecker, W. F., 1968. “Theoretical modelfor predicting the transient response of the mixture-vapor transi-tion point in horizontal evaporating flow”. Journal of Heat Trans-fer, 90(1), pp. 165–174.

[12] Voller, V. R., and Swaminathan, C. R., 1991. “ERAL Source-Based Method for Solidification Phase Change”. Numerical HeatTransfer, Part B: Fundamentals, 19(2), Jan., pp. 175–189.

[13] Clark, K. E., and Hill, S., 1999. “The Effects of a StiffeningSpring”. The College Mathematics Journal, 30(5), Nov., p. 379.

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