1. INTRODUCTION

Aluminum alloys with silicon as a major alloying elementconsist of a class of alloys which provides the most significantpart of all shaped castings manufactured. This is mainly dueto the outstanding effect of silicon in the improvement ofcasting characteristics, combined with other properties suchas mechanical properties and corrosion resistance.

Continous casting is distinguished from othersolidification process by its steady state nature in which themolten metal solidifies against the mold walls while it issimultaneously withdrawn from the bottom of the mold at arate which mentains the solid-liquid interface at a constantposition with time. The process works best when all of itsaspects operate in this steady state manner. Over 90% ofcommercail aluminum alloys are cast by semi-continous andcontinous casting, typically as 0.05-0.5 m daimeter roundsections.

The continuous casting of metals is a very importantprocess in metallurigical industry. Amin and Greif [1] modelthis process using the mass, momentum and energyconservation equations using the average heat capacitymethod. Chen et al. [2] model the heat transfer in thecontinuous casting process, taking an energy conservationequation with a convective term where the velocity isprescribed. Chen and Jiang [3] consider a convective-diffusion equation and they use an implicit method in thediffusion term and explicit method in the convective termtogether with a finite element method in space. A steady-statemodel involving the mass, momentum and energy equations isgiven in [4] using a heat flow method to state the energyequation. They discretize these equations taking a finiteelement method in space with a new finite element calledthermal contact element, an implicit method for the resultingequation. Bermdez and Otero [5] discretized the governingequations using a characteristics method in time and a finiteelement method in space, and they proposed a numericalalgorithm to solve the obtained nonlinear discretized problem.

Aspects of alloy solidification have been extensivelyinvestigated and reported in the literature [6-12].

During casting heat transfer occurs from the hot liquidmetal to the water-cooled mold and the temperature decreasesfrom that of the cast to the surrounding temperature. Theprocess involves three successive stages: the cooling of theliquid metal, the solidification and finally the cooling of thesolid metal. Due to the water cooling (primary and secondary)solidification rate provided by continuous casting is higherthan in other casting methods therefore continuous castingshave more uniform and finer grain structure and enhancedmechanical properties. The way the heat flows across themetal-mold interface strongly affects the evaluation ofsolidification and plays a remarkable role in the structuralintegrity of castings. Product quality is more directly affectedby the interfacial heat transfer conditions.

The structural integrity of shaped castings is closelyrelated to the time temperature history during solidification.Experimental research into such a process is expensive tocarry out and the difficulties of measurement are so great thatthe results are sometimes difficult to interpret. The use ofcasting process simulation could do much to increase thisknowledge in the foundry industry. A mathematical model,therefore, often has the advantage.

In the present paper, we are interested in the analysis ofheat transfer in the metal during solidification. This studyconsists of the description of a mathematical model governingthe different processes mentioned above for hypoeutecticaluminum-silicon alloys. The modeling leads to a system ofnonlinear differential equations representing the conservationof mass, momentum and energy, boundary and initialconditions which depend on the shape of the part (roundpillet), the mold, and the cooling system. A numericaldetermination of the temperature distribution and thesolidification front versus time and the influence of thecooling water flow rate, casting speed, and billet diameter forcylinderical billets is reported.

NUMERICAL STUDY ON THE EFFECT OF SOLIDIFICATION PARAMETERSDURING THE CONTINUOUS CASTING OF AL-SI ALLOYS

Y. Rihan*, B. Abd El-Bary

*Atomic Energy Authority, Hot Lab. Centre, Anshas, B.P. 13759, Egypt. Menoufia University, Fac. Of Engineering, Production Eng. &Mechanical Design Dept., Egypt.

ABSTRACT

A numerical model is developed, by using finite volume method to solve the two-dimensional unsteady flow and energyequations for simulating the flow and heat transfer with solidification in round billet continuous casting. The stronglycoupled set of partial differential equations representing the conservation of mass, momentum and energy are employed. Inaddition of this, a modified form of turbulence model k- is used also during the solution procedure. The thermal profilespredicted by the mathematical model agree with those predicted by another industrial model. The temperature distributionsand solid shell thickness profile were studied in the continuous casting at different conditions. The results from the analysesare applicable to the design of the continuous casting process.

2. MATHEMATICAL MODEL

The numerical model is applied to simulate thesolidification of binary alloys during the continuous castingprocess. Initially, the alloys were assumed to be molten,quiescent and uniformly mixed, with temperatures exceedingthe liquidus temperatures. The top and side walls wereassumed to be insulated while energy was extracted from thebottom at a rate governed by the overall/coolant heat transfercoefficient. The mathematical formulation of thissolidification problem is given by the one-dimensional heatconduction equation [13]:

qxT

xKxt

Tc

(1)

where K is the thermal conductivity (W/m K), c the specificheat (J/kg K), the density (kg/m3), qthe rate of energygeneration (W/m3), T the temperature (K), t the time (s), and xthe rectangular coordinate (m).

The release of latent heat between the liquidus and solidustemperatures is expressed by q:

tf

Lq s (2)

where L is the latent heat (J/kg), and fs the local solid fraction(%).

The fraction of solid in the mushy zone can be estimatedby the Scheil equation, which assumes perfect mixing in theliquid and no solid diffusion. With liquidus and solidus lineshaving constant slopes, fs is often expressed as:

)1(1

0

1

k

lf

fs TT

TTf (3)

where Tf is the melting temperature (K), Tl the liquidustemperature (K), and k0 the partition coefficient.

Eq. (3) is incorporated into the latent heat term (Eq. (2)) bydifferentiating the Scheil equation with respect totemperature. Hence, applying the chain rule of differentiation,we have:

tT

TT

TT

TTktf k

k

lf

f

lf

s

1

2

0

0

0

11 (4)

Substituting Eq. (2) into Eq. (1) gives:

xTxK

xtTc (5)

wherecan be considered as a pseudo-specific heat given by:

Tf

Lcc sM (6)

SsLsM cfcfc 1 (7)where the subscripts S, L and M refer to solid, liquid andmushy, respectively.

The other properties such as thermal conductivity anddensity in the mushy zone are described similarly as thespecific heat in Eq. (7):

SsLsM KfKfK 1 (8) SsLsM ff 1 (9)

A finite difference form of Eq. (5) is obtained for time-dependent temperature distribution at discrete grid points:

xTT

Kx

TTK

xtTT

cn

in

ii

ni

ni

i

ni

ni

i1

2/11

2/1

1 1 (10)

where n and n+1 refer to temperatures before and after theincremental time interval t, respectively, i the elementposition according to x and y axes.Ki+1/2 and Ki-1/2 are given by:

21

2/1ii

i

KKk

(11)

21

2/1

iiiKK

k (12)

The flow is assumed to be fully developed and symmetric.The radial component of velocity is set to zero at thesymmetry axis and at the casting surface next to the mold. Thepouring temperature 700 oC was an initial temperature of theprocess. It was found that the grid size 50100 is sufficient toresolve the details of the flow and temperature distributions.The thermo-physical properties and casting conditions for thecalculations are given in table 1.

Table 1: thermo-physical properties of the casting materialsused in this study [6].

Properties Symbol/units

Al-3wt.%

Si

Al-5wt.%

Si

Al-7wt.%

Si

Al-9wt.%

SiThermal

conductivityKS

(W/m.K)(solid)

KL(W/m.K)(liquid)

121

91

104

90

90

90

81

89

Specific heat cS (J/kg.K)(solid)

cL (J/kg.K)(liquid)

1084

963

1082

963

1080

963

1078

963

Density S (kg/m3)(solid)L (kg/m3)

(liquid)

2695

2385

2690

2389

2680

2394

2670

2399

Latent heatof fusion

L (J/kg ) 389187 393083 397440 405548

Solidustemperature

Ts (oC) 577 577 577 577

Liquidustemperature

Tl (oC) 644 632 610 604

3. RESULTS AND DISCUSSION

Validation of the mathematical models accuracy can berealized by two methods. The first is the direct comparisonwith the experimental data obtained from the continuouscasting machine and the second is the comparison with resultsof other numerical models. The experimental results forSpinelli et al. [14] were employed to compare the thermalprofile obtained in our study. The model was employed withthe same parameters of casting speed, mould shape, rate ofcooling water, heat transfer coefficient, and cast length. Thethermal profile predicted by the suggested mathematicalmodel agrees with the experimental data obtained by Spinelliet al. as shown in Figure 1.

Figures (2-5) present the temperature variation as afunction of time, for the solidification of the studiedaluminum alloys at constant casting speed, uc = 18 cm/minand constant water flow rate, Q = 0.7 m3/hr for billetdiameter, d = 5 cm and cast length, L = 100 cm.

100

200

300

400

500

600

700

0 40 80 120 160 200 240

Time, sec.

Tem

per

atu

re,o

C

r= 27 mm r= 41 mm

r= 62 mm r= 90 mm

Figure 1: Comparison between the presented model withexperimental results of Spinelli et al. [14].

The process involves three successive stages: the coolingof the liquid metal, the solidification and finally the cooling ofthe solid metal. In general, temperature decreases withincreasing time. Heat transfer occurs from the hot liquid metalto the water-cooled mold and temperature decreases from thatof the cast to the surrounding temperature. The solidificationstarts from the walls of mold towards the center. From theresults, one can say that the complete solidification at thesurface occurs after a time of about 30 sec for Al-3 wt% Siand Al-5 wt% Si while it occurs after a time of about 23 secfor Al-7 wt% Si and Al-9 wt% Si and this perhaps can beattributed to the effect of silicon. Also, we notice that thecomplete solidification at the center occurs after a time ofabout 160 sec for Al-3 wt% Si and more than 200 sec for Al-5wt% Si, Al-7 wt% Si and Al-9 wt% Si. This means that assilicon content increases the solidification time increases.When we advance towards the mold center, we observe thesolidification rate decreases.

Figure 6 shows the relationship between solidificationtime versus casting speed for the studied Al-Si alloys fordifferent billet diameters at constant water flow rate and castlength. The figure shows that the solidification time decreaseswith increasing casting speed for all studied Al-Si alloys. Thesolidification time is affected more by the increase of castingspeed up to 40 cm/sec for all studied Al- Si alloys. The morebillet diameter increases the more solidification timeincreases.

0

100

200

300

400

500

600

700

0 40 80 120 160 200

Time, sec.

Tem

per

atu

re,

o C

surface0.625 cm from surface1.250 cm from surface1.875 cm from surfacecenterSolidification line

Figure 2: Cooling curve inside the casting cross-section forAl-3wt % Si, (uc = 18 cm/min., Q = 0.7 m3/hr, d = 5 cm, L =

100 cm).

0

100

200

300

400

500

600

700

0 40 80 120 160 200

Time, sec.

Tem

pera

ture

,oC

surface0.625 cm from surface1.250 cm from surface1.875 cm from surfacecenterSolidification line

Figure 3: Cooling curve inside the casting cross-section forAl-5wt % Si, (uc = 18 cm/min., Q = 0.7 m3/hr, d = 5 cm, L =

100 cm).

0

100

200

300

400

500

600

700

0 40 80 120 160 200Time, sec.

Tem

pera

ture

,oC

surface0.625 cm from surface1.250 cm from surface1.875 cm from surfacecenterSolidification line

Figure 4: Cooling curve inside the casting cross-section forAl-7wt % Si, (uc = 18 cm/min., Q = 0.7 m3/hr, d = 5 cm, L =

100 cm).

0

100

200

300

400

500

600

700

0 40 80 120 160 200Time, sec.

Tem

per

atur

e,oC

surface0.625 cm from surface1.250 cm from surface1.875 cm from surfacecenterSolidification line

Figure 5: Cooling curve inside the casting cross-section forAl-9wt % Si, (uc = 18 cm/min., Q = 0.7 m3/hr, d = 5 cm, L =

100 cm).Figure 7 shows the relationship between solidification

time versus casting speed for the studied Al-Si alloys fordifferent water cooling flow rates at constant billet diameterand cast length. It is clear that solidification time decreaseswith increasing water flow rate for all studied Al-Si alloys.Also, the solidification time is not very affected by theincrease of water flow rate for the studied Al-Si alloys.

50

150

250

350

450

0 20 40 60 80

Casting speed, cm/min

So

lidif

icat

ion

tim

e,s

Al-Si3

Al-Si5

Al-Si7

Al-Si9

d=5 cm

Figure 6: Solidification time vs casting speed , (Q = 0.7 m3/hr,L = 100 cm).

50

150

250

350

0 20 40 60 80

Casting speed, cm/min

Solid

ifica

tion

time,

s Al-Si3Al-Si5Al-Si7Al-Si9Q=0.7 m3/hrQ=4 m3/hr

Figure 7: Solidification time vs casting speed , (d = 5 cm, L =100 cm).

Figure 8 shows the temperature distribution versus time atthe billet surface and the billet center for different values ofbillet diameter with the same value of casting speed, waterflow rate and cast length for Al-Si alloys. The heat removalrate from the billet surface decreases with increasing the billetdiameter due to the bulk of melt and its heat content. At thebillet center the temperature will remain constant for a periodof time and this can be attributed to the latent heat and thedistance from the mold walls.

0

100

200

300

400

500

600

700

0 40 80 120 160 200Time, sec.

Tem

per

atu

re,o

C

d = 5 cmd = 7.5 cmd = 10 cmd = 15 cmd = 20 cmCenterSurface

Figure 8: Surface and center temperatures vs time for Al-3wt% Si at different diameters, (uc = 36 cm/min., Q = 0.7 m3/hr,

L = 100 cm).

The distribution of shell thickness versus length frommeniscus at different conditions of casting speed with thesame cooling condition is shown in Figure 9. The figure

shows that the solidified shell thickness increases withincreasing length from meniscus for all casting speeds. Thefigure also shows that the shell thickness decreases withincreasing the speed value. Figure 10 shows the distributionof shell thickness versus length from meniscus at differentbillet diameters with the same cooling condition. The figureshows that increasing the billet diameter the solidified shellthickness decreases but the distance from meniscus at whichthe solid shell start to form increases.

0

5

10

15

20

25

0 20 40 60

Length from Menisus (cm)

Sh

ell

Th

ickn

ess

(mm

) u=18 cm/minu=36 cm/minu=54 cm/minu=72 cm/min

Figure 9: The shell thickness vs length from meniscus for Al-7wt % Si at different conditions of casting speed, (Q= 4

m3/hr, d = 5 cm).

0

10

20

30

40

50

0 20 40 60 80 100Length from Meniscus (cm)

Sh

ell

Th

ick

nes

s(m

m)

d=5 cmd=10 cmd=15 cmd=20 cm

u=18 cm/min

Figure 10: Shell thickness vs length from meniscus for Al-7wt% Si at different conditions of billet diameter,

(Q= 4 m3/hr).

4. CONCLUSIONS

An unsteady heat transfer model for continuous roundbillet casting was developed to calculate the temperaturedistributions and solid shell thickness profile and to study thesolidification process in the continuous casting. The thermalprofiles predicted by the mathematical model agree withexperimental data for another authors. Results showed thatsuch mathematical analyses of the process can help to controland optimize the process and to investigate the consequencesof parameter changes without the safety and cost limitationsof in-plant experiments. The predicted results showed that thesolidification time increases with increasing silicon contentand decreases with increasing casting speed. The more billetdiameter increases the more solidification time increases. Thecooling water flow rate has less effect on the shell thicknesscompared with the effect of casting speed. The numericalresults show that the method of this work is efficient for

analyzing the flow and heat transfer with solidificationproblem of round billet continuous casting.

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