7_mathematical modeling and optimization strategies

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Engineering Applications of Artificial Intelligence 16 (2003) 511527

Mathematical modeling and optimization strategies

(genetic algorithm and knowledge base) applied to the

continuous casting of steel

C.A. Santosa, J.A. Spimb, A. Garciaa,*aDepartment of Materials Engineering, Faculty of Mechanical Engineering, State University of Campinas (UNICAMP) P.O. Box 6122,

13083-970, Campinas, SP, BrazilbFederal University of Rio Grande do Sul (UFRGS), Center of Technology, P.O. Box 15021, 91501-970, Porto Alegre, RS, Brazil

Abstract

The control of quality in continuous casting products cannot be achieved without a knowledge base which incorporates

parameters and variables of influence such as: equipment characteristics, steel, each component of the system and operational

conditions. This work presents the development of a computational algorithm (software) applied to maximize the quality of steel

billets produced by continuous casting. A mathematical model of solidification works integrated with a genetic search algorithm and

a knowledge base of operational parameters. The optimization strategy selects a set of cooling conditions (mold and secondary

cooling) and metallurgical criteria in order to attain highest product quality, which is related to a homogeneous thermal behavior

during solidification. The results of simulations performed using the mathematical model are validated against both experimental

and literature results and a good agreement is observed. Using the numerical model linked to a search method and the knowledge

base, results can be produced for determining optimum settings of casting conditions, which are conducive to the best strand surface

temperature profile and metallurgical length.

r 2003 Elsevier Ltd. All rights reserved.

Keywords: Continuous casting of steel; Mathematical modeling; Optimization methods; Genetic algorithm

1. Introduction

The continuous casting process is responsible for most

of the steel production in the world, and has largely

replaced conventional ingot casting/rolling for the

production of semi-finished steel shape products.

Fig. 1 shows a schematic representation of a continuous

caster and the different cooling zones along the machine.

The casters have been implemented with modern

equipments for billets, slabs or blooms, multiple casting

and process control.

The quality control of continuous casting is funda-

mental for reducing production costs, processing time,

and to assure reproducibility of the casting operation

and increase of production. This cannot be achieved

without a greater knowledge about the process, in-

corporating both operational parameters, such as

components of the machine, steel composition, casting

temperature, and casting metallurgical constraints, such

as thickness of solidified shell at mold exit and strand

surface temperature profile along the different cooling

zones. The use of optimization strategies, such as genetic

algorithm, heuristic search, knowledge base, working

connected to mathematical models of solidification, can

be seen as a useful tool in the search of operational

parameters that maximize or minimize any aspect of the

dynamic process. The idea of using simulation to

optimize a continuous caster is not just a theoretical

concept and its practicality has already been demon-

strated (Larreq and Birat, 1982; Lally et al., 1991a; Lally

et al., 1991b; Kumar et al., 1993; Samarasekera et al.,

1994; Spim et al., 1997; Filipic and Saler, 1998;

Brimacombe, 1999; Cheung and Garcia, 2001). An

expert system for billet-casting problems has been

developed to guide caster operators in analyzing

quality-related problems and to provide them with a

ready source of fundamental knowledge related to caster

ARTICLE IN PRESS

*Corresponding author. Tel.: +55-19-3788; fax: +55-19-3289-3722.

E-mail address: [email protected] (A. Garcia).

0952-1976/$- see front matterr 2003 Elsevier Ltd. All rights reserved.

doi:10.1016/S0952-1976(03)00072-1


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operation (Kumar et al., 1993). J.K. Brimacombe, I.V.

Samarasekera, S. Kumar and J.A Meech projected thisexpert system. Filipic and Saler proposed and imple-

mented a computational approach to the continuous

casting of steel, which consists of a numerical simulator

of the casting process and a genetic algorithm for real

parameter optimization (Filipic and Saler, 1998).

Cheung, Santos, Spim and Garcia have used a heuristic

search technique for the optimization of the quality of

carbon steel billets (Cheung and Garcia, 2001; Santos

et al., 2002).

The present paper describes a software, which was

developed and is based on the interaction between a

finite difference heat transfer solidification model and a

genetic algorithm and a knowledge base. The heat

transfer model is validated against experimental results

concerning both static casting of AlCu and SnPb

alloys and continuous casting of a low carbon steel slab

and a high carbon steel billet. The software has been

used to explore the space parameter settings in order to

find optimized cooling conditions, which result in best

strand surface temperature profile and minimum me-

tallurgical length.

2. Mathematical model

The mathematical formulation of heat transfer to

predict the temperature distribution and the solid shell

profile during solidification is based on the General

Equation of Heat Conduction in Unsteady State, which

is given for three-dimensional heat flux by

rcqT

qt rk rT q

3

; 1

where r is the material density kg=m3; c is specific heatJ=kg K; k is thermal conductivity W=m K; qT=qt isthe cooling rate K=s; T is temperature (K), t is the time

(s) and q3 represents the term associated to internal heat

generation due to the phase change. It was assumed that

the thermal conductivity and density vary only with

temperature. Then, Eq. (1) can be rewritten in two-

dimensional form as

rc

qT

qt k

q2T

qx2

q2T

qy2

q

3

: 2

Approximating Eq. (2) by finite-difference terms, we

have

rcTn1i; j T

ni; j

Dt k

Tni1; j 2Tni; j T

ni1; j

Dx2

Tni; j1 2T

ni; j T

ni; j1

Dy2

q

3

; 3

where n 1 is the index associated to the future time, n

is the index corresponding the actual time, Dt is the

increment of the time, x;y are the directions, i;j are the

positions and the stability criteria are given by

DtoDx2

2aor

Dy2

2a;

where

a k

rcm2=s:

The objective is to determine the future temperature

of the element i; jTn1i; j as a function of the actualknown temperatures of the elements around the element

i; jTni1; j; Tni1; j; T

ni; j1; T

ni; j1:

2.1. Phase change

In this study, a fixed grid methodology is used with a

heat source term due to the metal phase transformation

(liquid to solid), which is given by an explicit solid

fraction-temperature relationship as

q3

rLqfs

qt; 4

where fs is the solid fraction during phase change along

the solidification range (liquidus and solidus tempera-

tures: TL and TS, respectively) and L is the latent heat

of fusion J=kg. The solid fraction depends on anumber of parameters. However it is quite reasonableto assume fs varying only with temperature in the mushy

zone, and then Eq. (4) can be written as

q3

rLqfs

qT

qT

qt: 5

Substituting q3 into Eq. (3), the specific heat can be

written as c0 c Lqfs=qT; where the termLqfs=qT is called pseudo-specific heat. At the rangeof temperatures where solidification occurs for metallic

alloys, the physical properties will be evaluated taking

into account the amount of liquid and solid that coexists

ARTICLE IN PRESS

Ingot

Sprays

Ladle

Tundish

Mold +Pinch Roll

Flame cut-off

Nozzle

Rolls

Primary

Cooling

Secondary

Cooling

Radiation

Cooling

Unbending Point

Fig. 1. Schematic representation of the continuous casting equipment.

C.A. Santos et al. / Engineering Applications of Artificial Intelligence 16 (2003) 511527512


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in equilibrium at each temperature:

k kS kL :fs kL; 6

c0 cS cL :fs cL L dfs 7

r rS rL :fs rL; 8

where sub-indices S and L; respectively, indicate solidand liquid states. If fs 0; the element is still liquid andonly thermophysical properties of the liquid are

considered, and if fs 1; the element is completelysolid. For carbon steels, the fs is appropriately described

by the lever rule, and Scheils equation applies for

AlCu and SnPb alloys (these alloys will be used in

model validation).

fs 1

1 ko

TL T

Tf T

Lever Rule; 9

fs 1 T

f T

Tf TL

1=ko1Scheils Equation; 10

where Tf is the solvent melting temperature (K) and kois the partition coefficient.

2.2. Analogy between thermal systems and electrical

circuits

In the continuous casting processes, heat is trans-

ferred from the liquid steel to the cooling system (mold,

sprays and free radiation) through various media

namely, the solidified shell, strand/mold interface, mold

wall, cooling water, sprays/strand interface and air/strand interface. The heat transfer through each of the

media can be characterized in terms of a thermal

resistance, analogous to an electrical resistance. To

simplify the development of the mathematical model, is

the analogy between thermal systems and electrical

circuits applied. Multiplying the modified equation (3)

by Dx; Dy; Dz on both sides, considering Dx Dy Dz;At Dy Dz or Dx Dz; and replacing c by c

0; yields

AtDx r c0

Tn1i; j Tni; j

Dt

AtkTni1; j 2T

ni; j T

ni1; j

Dx

Tni; j1 2T

ni; j T

ni; j1

Dy :

11

By analogy, the thermal capacitance CTi; j represents

the energy accumulated in a volume element i; jfrom thegrid, and is given by (Spim and Garcia, 2000),

CTi; j Dxi; jDy Dz ri; j c0i; j; 12

where Dx Dy Dz is the volume of the element i;j:Also by analogy, the thermal flux between central

points has a thermal resistance at the heat flux line (RT)

from point i 1;j or i 1;j to point i;j or i;j 1 or

i;j 1 to point i;j given by (Fig. 2)

RTi Dx

kAtor RTj

Dy

kAt; 13

where Dx and Dy correspond to the distance between

central points of nodes. Each thermal resistance betweenthe central points is given by the sum of the partial

thermal resistance from the center to the boundary and

the boundary to the center, given by

in x: RTi1; ji; j RTi1; j RTi; j;

RTi1; ji; j RTi1; j RTi; j; 14

in y: RTi; j1i; j RTi; j1 RTi; j;

RTi; j1i; j RTi; j1 RTi; j: 15

These terms are given by the sum of thermal

resistances according to the following equations:

RTi1; j Dxi1; j

2ki1; jAt; 16

RTi1; j Dxi1; j

2ki1; jAt; 17

RTi; j1 Dyi; j1

2ki; j1At; 18

RTi; j1

Dyi; j1

2ki; j1At; 19

RTi; j Dxi; j

2ki; jAtor

Dyi; j

2ki; jAt: 20

Then, expanding Eq. (11) and substituting CTi; j; yields

CTi; jTn1i; j T

ni; j

Dt

Tni1; j Tni; j

RTii; ji; j

Tni1; j Tni; j

RTi1; ji; j

Tni; j1 T

ni; j

RTi; j1i; j

Tni; j1 Tni; j

RTi; j1i; j21

ARTICLE IN PRESS

i, j-1

i, j+1

i,j

i+1,ji-1,j

x

y

nodal pointthermal resistance

Fig. 2. Nodal point.

C.A. Santos et al. / Engineering Applications of Artificial Intelligence 16 (2003) 511527 513


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or

Tn1i; j DtTni1; j

ti1; ji; j

Tni1; j

ti1; j;i; j

Tni; j1

ti; j1;i; j

Tni; j1

ti; j1;i; j

1 Dt

ti; j;i; j

Tni; j; 22

where

ti1; j;i; j cTi; jRTi1; j RTi; j; 23

ti1; j;i; j cTi; jRTi1; j RTi; j; 24

ti; j1;i; j cTi; jRTi; j1 RTi; j; 25

ti; j1;i; j cTi; jRTi; j1 RTi; j; 26

1

ti; j;i; j

1

ti1; j;i; j

1

ti1; j;i; j

1

ti; j1;i; j

1ti; j1;i; j

: 27

Eq. (27) is generic and can be applied to any

geometry, by varying only area and volume to be

considered, as well as the thermophysical properties as a

function of the temperature or state of the analyzed

element in the grid. The stability criterion is

Dtpti; j;i; j:

2.3. Boundary conditions

The application of the solidification model to

continuous billet/slab casting operation (Fig. 3) was

based on the following key assumptions:

(1) Two-dimensional heat transfer phenomenon was

considered, with heat flux being admitted to be

negligible along the vertical direction z:

qT

qz

0:

(2) A control volume element, with Dz 1 mm; wasplaced in a transverse section and was analyzed

from the meniscus to the cut-off region. The

distance below the meniscus z is given by

Z VcastingDt z m; t s;

VcastingFcasting speed m=s:

(3) The billet/slab symmetry permits that only one-

quarter of the cross-section modeled for a full

thermal evolution characterization (grid: 100 100

points).

(4) The meniscus surface was assumed to be flat

z 0:(5) Effect of mold oscillation, mold curvature, segrega-

tion, and melt level fluctuation in the mold were

ignored.

(6) The mold is considered uniform and with an

initial temperature equal to the water-cooling

temperature.

(7) The surface temperature of molten metal is con-

sidered equal to the pouring temperature.

(8) The turbulence in liquid metal is analyzed

by a mathematical expedient, where the thermal

conductivity in the liquid is multiplied by a

numerical factor: kef kLA; where A variesbetween 3 and 7 (Toledo et al., 1993; Louhenkilpi,

1994).

(9) The transient mold/strand and sprays/strand

heat transfer coefficients (hm=s and hs=s respectively)

used in this work, are those proposed in the

literature (Samarasekera and Brimacombe, 1988;

Brimacombe et al., 1984; Lait et al., 1974; Hills,

1969; Brimacombe et al., 1980; Mizikar, 1970;

Nozaki et al., 1978; Bolle and Moureau, 1979),

and they are related to the interface thermal

resistances along the different regions on the

machine, given by

RTm=s 1

hm=sAt; 28

RTs=s 1

hs=sAt: 29

3. Optimization strategies

In this work, an algorithm is developed which

incorporates optimization strategies to determine best

ARTICLE IN PRESS

y

xz

mold

sprays

radiation

hm/s

hs/s

hr

Vcasting

z

grid

Fig. 3. Boundary conditions.



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operational parameters to the continuous caster. The

algorithm employs search techniques for finding

these operational parameters, including the type

and characteristics of mold, mold taper, mold and

sprays cooling systems, etc., which are incorporated

in a knowledge base. The search includes finding the

casting objective of maximum production rate as afunction of casting metallurgical constraints. These

constraints represents the product quality and process

feasibility through limits on strand shell thickness

SM; metallurgical length LM; minimum surfacetemperature Tmin surf; casting speed Vcasting ; strandsurface reheating between spray zones DTmin surf

and temperature at the unbending point TLM: Thealgorithm modifies the operational process para-

meters, such as mold and spray cooling efficiencies

and casting speed, with a view to attain the best

conditions for the quality of the cast product at a

maximum production rate without violating the metal-

lurgical constraints.

The functional structure of the algorithm is

basically composed of three operating blocks: the

first consisting of the numerical heat transfer model,

which generates results of simulations as a function

of the input parameters related to operational con

ditions and equipment limitations; the second

block incorporates the knowledge base about the

continuous casting process, and the third block

consists of the decision rules (strategy), which are the

managers of the algorithm. It determines the modifica-

tions on the boundary conditions of the continuous

casting process and is responsible for the insertionof new input variables into the numerical model. This

block has a strong interaction with the results

furnished by the numerical model. The algorithm

works by iteration, and every result given by the

model corresponds to an analysis performed by

the decision rules block, thus indicating any need to

modify the process boundary conditions. The algorithm

includes a database of material properties for various

steels.

3.1. Knowledge base

The knowledge base required to transform molten

steel into quality billets at a high production rate is, of

course, quite large. The present knowledge base was

based on a wide search in the literature on continuous

casting operations and on information obtained

in a continuous casting plant. It was structured in

order to facility the examination of all important

operational parameters. The outline of quality

problems that includes the possible defects, their

origin and suggested preventive techniques has been

prepared as a function of rules and data collected in

the literature and in the industrial practice, and linked

to a solidification mathematical model (solid shell

thickness evolution and surface temperature distribution

along the billet).

The starting input parameters about machine, opera-

tional conditions and casting are first compared with the

knowledge base, and a report with suggestions is

provided. After that, the operating conditions aresubmitted to the decision strategy and inserted into the

numerical model, which generates a simulation repre-

senting the solidification in the continuous casting

equipment. For developing the decision strategy it was

necessary to acquire a knowledge base concerning the

continuous casting of steel, containing two groups of

information: (a) equipment information and (b) process

information.

(a) The equipment information represents the input

variables of the heat transfer model and optimization

program, and generally relates to the physical char-

acteristics of the equipment and the quality of the cast

steel. This information represents characteristics of

operation, such as geometry of caster, casting rate,

composition of steel, casting temperature, type of mold,

mold length, mold taper, metal level, number and length

of sprays zones, water flow rates in the mold and at the

different sprays zones, unbending point and water

temperature.

(b) The process information represents the transient

variables in the process, which can be classified as:

boundary variables: which can be modified within

an operating range to meet specifications of the

desired output, and can eventually be associated with

economic features and a defect-free product; forinstance, casting speed and primary and secondary

cooling efficiencies, and control variables, which are

associated with the results of the continuous casting

process and for instance, solid shell thickness,

surface temperature profiles and metallurgical length

(Fig. 4).

The knowledge base is a set of representation of facts

about the process (rule-based system). Each individual

representation is called a sentence. The sentences are

expressed in a language called a knowledge representa-

tion language. The objective of the knowledge repre-

sentation is to express knowledge in a computer-

tractable form, such that it can be used to help agents

perform well. The logic consists of the Boolean

connectives and quantifiers terms, and the structural

knowledge implements rules and facts. Rulers are

statements and procedures, such as condition state-

ments and search strategies, and facts are classes of

objects and values. Among these objects, various

relations hold. Some of these relations are functions

(relations in which there is only a value for a given

input) with exclusive values and others are restrictions.

The main rules used in knowledge base system are

shown in Table 1.

ARTICLE IN PRESS

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3.2. Genetic algorithm

The decision block contains a set of critical andlimiting operational conditions imposed by metallurgi-

cal constraints, which is systematically compared to the

simulations determining, when necessary, modifications

of the input variables. Such modifications are performed

by observing the functional limits of each variable. The

present study was conducted to attempt maximum

casting speed which depends on the settings of operating

parameters, such as changes in the primary (mold) and

secondary cooling (sprays), reflected in heat transfer

coefficients. These settings are defined as those which

make it possible to run the caster at its maximum

productivity, minimum cost and to cast defect-freeproducts.

3.2.1. Metallurgical criteria

1. Shell thickness at the mold exit SM: The shell

thickness at the mold exit must be greater than some

minimum value Smin; which is considered to be about10% of the casting thickness. This constraint avoids

breakout occurrences caused by extraction stresses and

liquid ferrostatic pressure, and can be written as

Position Lmold exit ) SM > Smin 0:1ecasting:

2. Metallurgical length LM

: The solidification of the

ingot has to be complete before the point where a high

deformation is given (unbending point) in order to avoid

internal and transverse cracking and centerline segrega-

tion. This constraint is

Position LM ) TcenteroTsolidus;

that means that the center of the strand Tcenter must be

at a temperature lower than the solidus temperature

Tsolidus at the unbending point.

3. Temperature at unbending point TLM: The strand

surface Tsurface must be at a temperature outside the

low ductility trough observed in steels and at a

temperature either greater than the high-temperature

limit of the ductility trough (soft cooling) or lower than

the lower limit in order to avoid transverse surfacecracking (hard cooling). The bottom of the ductility

trough for steels is usually located between 700C and

750C; depending on steel composition, mainly in lowcarbon steels, which is the temperature where the g2a

(austeniteferrite) transformation starts, so the surface

temperature must be less than

Position LM ) TsurfaceoTg2a:

The upper limit of the low ductility trough corresponds

to the transition between the transgranular fracture and

intergranular fracture Ttransition : Depending on the

composition of steel, this upper temperature limit canvary between 900C to 1100C:

Position LM ) Tsurface > Ttransition :

Limiting the strand surface above the upper limit of the

low ductility temperature, transversal cracks are also

reduced. Longitudinal cracks at the unbending point are

more usual in steels with carbon contents of about 0.08

0.14%, the maximum value being observed to be about

0.12%C. In this work it was considered that the strand

surface temperature is kept above the upper limit of the

low ductility range, called Tmin surface:

4. Reheating between zones Tmax surface Tmin surface):The reheating effect occurs when the strand passes from

a spray cooling zone with a high cooling efficiency to

one with a lower cooling rate, and must be limited as a

function of steel grade and casting operating para-

meters. This reheating leads to the development of

tensile stresses at the solidification front, which can

induce cracking. The maximum permissible reheating

range along the machine has been chosen to be equal to

100C in order to avoid midway surface cracking

(Brimacombe et al., 1984).

Position Lsprays ) Tmax surface Tmin surfacep100C:

ARTICLE IN PRESS

Mold Sprays UnbendingPoint

Radiation

Tpouring

Tmax

Tmin

Surface Temperature T

Distance from meniscus

Shell Thickness

Smin

Point ofcomplete

solidification

mould zone 1 zone 2 zone 3 radiation zone

Fig. 4. Metallurgical and equipment constraints applied to the continuous casting process.



7/17

Table 1

Definition of the main rules used in the knowledge base system

Classes Objects Relations Consequences

Steel composition

Division:

Low carbon: o0:25% C o0:10% Heat transfer decreases asthe %C increases

Surface cracks, b

0.100.14% Billet surface is rougher

(deeper oscillation mark)

Surface cracks, b

0.12% Lower heat transfer rate(thin solidified shell)

Longitudinal, mibreakout (mold)

0.17% (peritectic) dg phase change (B1490C) External, interna

cracks

0.170.25% Reduced ductility at elevated

temperature

Transversal, long

(solidification fro

Medium carbon: 0:250:50% C 0.250.38% Favor equiaxed grain zone Difficult crack pr

0.42% Higher heat flux Large solidified s

High carbon: > 0:50% C 0.400.77% Large columnar zone(lowest heat transfer)

Breakout (small

0.77% (eutectoid) Long freezing range 100C Breakout (small

cracks

> 0:77% Susceptibility of crack formationat elevated temperatures External, internabreakout, laps, b

Phase transformations

o 0.09% C: L, L+d; d; d g; g; g a; a; a+P dg phase change

(B14001485C)

External, interna

(expansion)

0.090.17% C L, L+d; d g; g; g a; a; aP dg phase change (1485C) External, interna(expansion)

0.170.53% C L, L+d; L+g; g; g a; aP ga phase change

B910727C)

External, interna

(contraction)

0.530.77% C L, L+d; L+g; g; g a; aP

0.77% C L, L+g; g; P> 0:77% C L, L+g; g; g Fe3C; P+Fe3C

Element alloys

Hydrogen (H) o2 ppm Minimize bubbles of gases Pinholes, blowho

(surface/subsurfa

Oxygen (O) o10 ppm

Nitrogen (N) o20 ppm Minimize bubbles of gases Pinholes, blowho

(surface/subsurfa

Mn:S ratio > 2530 Avoid crack formation

in interdendritic liquid

Cracks in grain b

(surfaces are smo


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Phosphorus (P) o0:017% Decreases columnar zone Difficult crack fopropagation

Sulphur (S) o0:015% Formation of FeS

Tf 1200C

Superficial, corne

midway cracks

Copper (Cu) o0:2% Low melting impurities

in grain boundaries

Cracks in grain b

(surfaces are smo

Cu:Sn ratio o4 Minimizes craze crack

formation in surface

Craze crack

Ni:Cu ratio > 1 Form a miscible alloywith a higher Tf

Minimize craze c

Aluminium (Al) o0:02% Formation of AlN900C

Surface transvers

boundaries crack

Niobium and vanadium (Nb, V) o1% Formation of nitrites,

carbides

Surface transvers

boundaries crack

Manganese (Mn) o1% Formation of oxides Transversal crack

Chromium (Cr) > 3% Formation of oxides Internal cracks

Titanium (Ti) o0:004% Minimizes AlN formation Minimize interna

Transformation temperatures TL 1537 88%C 25%S 5%Cu 8%Si

5%Mn 2%Mo 4%Ni 1:5%Cr

18%Ti 2%V 30%P

TS 1535 200%C 12:3%Si 6:8%Mn

124:5%P 183:9%S 4:3%Ni 1:4%Cr

4:1%Al

Cast structure

Columnar grain zone Small section Favor columnar zone Facilitate crack p

Equiaxed grain zone 0.130.20%C and 0.0080.02%P Favor equiaxed zone Difficult crack pr

0.170.38%C Favor equiaxed zone

(medium %C)

Difficult crack p

Large section Favor equiaxed zone Difficult crack pr

Superheat level o30C Favor equiaxed zone

(low, high %C)

Difficult crack pr

Electro-magnetic stirring Favor equiaxed zone Difficult crack pr

Mechanical properties

High temperature zone: TSTx: 020:185%C Tx 40C Cracks due P an

0.45%C Tx 65C

S> 0:025% Tx 80C

Intermediate temperature zone: A3 to 1200C Mn:S rate and carbides

and nitrites

Cracks in grain b

Low temperature zone: 700900C AlN, carbides and nitrites Cracks in grain b

Table 1 (continued)



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Mold

Liquid metal level

Meniscus depth: o100 mm (small depth) Overflow, distortion Rhomboidity, lap

transverse depres

=100 mm Recommended

> 100 mm (very depth) Small solidified shell Breakout (mold)

Fluctuations o75 mm Recommended Surface defects, b

Composition of the mold

Alloy P: 200300 ppm; Ag: 1000 ppm Smaller distortion Minimize rhomb

Smoothing temperature > 500C Smaller distortion Minimize rhomb

su > 400 N=mm2 Smaller distortion Minimize rhomb

HB surface 100500 Smaller distortion Minimize rhomb

Thermal conductivity > 70% to pure Cu Higher heat transfer rate Large solidified s

Mold wall thickness

Section Sections o200 200 mm Recommended Minimize rhomb

Thickness B12:7 mm Recommended Minimize rhomb

Distortion o0:05 mm Minimal Minimize rhomb

0.050:20 mm Unsatisfactory Rhomboidity, co

> 0:20 mm Severe Rhomboidity, ex

internal cracks

Mold constraint system

System Conditions of the corners Boiling at the channel gap Rhomboidity, lo

corner cracks

Slots 2 or 4 (recommended 4) Smaller distortion Rhomboidity

Water channel gap 4:8 mm (recommended) Similar heat flux at 4 faces Rhomboidity

Mold tube alignment

Tolerances o0:5 mm (in all faces) Non-symmetrical cast structure BreakoutCorners o4 mm (same radius of the mold/tube) Different heat flux between

corner/face

Transverse, long

corner cracks, br

Taper mold

Straight No taper Alloy composition

(low %C)

Breakout, rhomb

Single or multiple taper Discrete taper Recommended (high %C) Breakout, rhomb

Parabolic Continuous taper More recommended


10/17

Cooling water quality

Temperature range DT o8C Recommended Rhomboidity, br

cracks (boiling)

Presence of scale deposits Yes or no Heat transfer efficiency

Color and properties Red Iron oxide and corrosion Rhomboidity, br

cracks (boiling)

Black Magnetite Fe3

O4

or carbonaceous:

oil, grease

Light Excessive hardness

Thickness o20 mm Recommended

Impurities o5 ppm Recommended

Cooling water velocity

and pressure

Velocity (v) > 12 m=s Recommended Rhomboidity (bo

Pressure (P) Inlet and outlet Outlet > 135 kPa; inlet> 400 kPa

Rhomboidity (bo

Mold oscillation

Frequency f o4 Hz or 240 cpm Oscillation marks Transversal, long

surface cracksStroke length S 916 mm Oscillation marks Transversal, long

surface cracks

Mold lead (ML) > 4 mmNegative strip time tn 0.120:15 s Oscillation marks Transversal, long

surface cracks, la

Mold lubrication

Composition of the flux Elements SiO2 ; CaO; MgO; Al2O3;TiO2 ; Fe2O3; MnO2;

Na2O; K2O; B2O3 ; Li2O;

F; C; CO2

Inclusion absorption rate (Bi) Bi 1:53%Cao 1:51%MgO 1:94%Na2O

1:48%SiO2 0:10%Al2O3

3:55%Li2O 1:53%CaF21:48%SiO2 0:10%Al2O3

Large Bi Minimize breako

Lubrication index (LI) LI distance from meniscus to position where T Tf

distance from meniscus to bottommoldClose to 1 Minimize breako

Depth of the molten flux (Yp) Yp S sinpf

2

500tnVcasting

f d 612 Minimize breako

Table 1 (continued)



11/17

3.2.2. Equipment constraints

* Water flow: The water flow rate in a given region

(or spray) has a lower and an upper limit depending

on the hydraulic system, which is given in heat

transfer coefficients hs=s (Brimacombe et al., 1980;

Mizikar, 1970; Nozaki et al., 1978; Bolle and

Moureau, 1979):

Position Lsprays ) hmax spraysphs=sphmin sprays:

* Casting speed (Vcasting ): The casting speed is bounded

with a minimum and maximum value, given by

Vmin castingpVcastingpVmax casting:

Objective and constraint functions used in the

optimization framework were formulated to represent

the productivity of the machine, quality of the cast

strand and casting speed. Machine productivity is

characterized by limitation of casting speed, metallurgi-

cal length and spray cooling, and the metallurgicalconstraints are solid shell thickness, metallurgical

length, surface temperatures and reheating between

sprays zones. The objective is to keep a cost function

(J), defined as a sum of individual values of each

constraint (i), close to zero. The process starts with

nominal values of operating parameters, and as a

function of results simulated by the heat transfer

mathematical model (temperature field in the strand),

the cooling conditions are modified in such a way that

the final billet/slab metallurgical quality is assured. Each

violation of any constraint corresponds to one numer-

ical increase in this individual objective function.

When the cost function reaches zero, the castingspeed can be increased by a value DVcasting ; andthe search begins again. The cooling criteria are

formulated in such a way that the lower values of

thermal gradients between cooling zones correspond to

the best situation, withPn

i1Ji 0: For each criterion aweight (w) was used denoting the relative importance of

the criterion. The solid shell thickness at mold exit and

the point of complete solidification have maximum

weight (10), and surface temperature and thermal

gradients at the sprays zones have minimum weight

(1). Eq. (30) presents the formulation for the objective

function J:

J Xni1

wiJi Ji min

Ji max Ji min: 30

The genetic algorithm applied for the continuous

casting optimization consists of:

Step 1: generate an initial population of results

simulated by using the input parameters (nominal);

Step 2: compute cost function;

Step 3: store parameters setting;

Step 4: modify cooling conditions in each region

where the constraint was violated;

ARTICLE IN PRESS

Sprays

Waterflux

Re

lationstoheat-transfercoefficients

Asymmetricalspraycooling

Rhomboidity

(Bom

maraju,

1991)

Minimumsurfacetemperature

>

1100

C

Intermediatelowductilityzone

Midwaycracks

(Brimacombeand

Sorimachi,1977;

Brim

acombeetal.,

1980,

Brim

acombe,1999)

Surfacereheating

o100C

Reheatingofthebilletsurface

(strain/stress)

Midwaycrackcloseto

thesolidificationfront

Radiationzone


>

1100

C


Midwaycracks

Maximumreheating

o100C

Reheatingofthebilletsurface

(strain/stress)

Midwaycrackclosetothe

solidificationfront

Unbendingpoint


>

1100

C


External,

internalcracks

(Brimacombeand

Sorimachi,1977)

Pointofcompletesolidification

TcenteroTs

Endofsolidification

Centralcracks

Centertemperature

o1350

C

Hightemperaturezoneoflow

ductility

Pinch-rollcracks

Note:ddeltaferrite,gaustenite,aalp

haferrite,Pperlite,

Fe3Ccementite,L

liquid,

Ssolid,TLliquidustemperature,TSsolidustemperature,Tffusiontemperature,ffrequency,

Sstrokelength,tnnegativestriptime,

dliquidmetalfluctuations.



12/17

Step 5: apply a genetic operator to determine new

parameter of process;

Step 6: generate new results;

Step 7: if cost function decreases, then SJmin is the

result;

Step 8: ifSJ 0; increase Vcasting and go to step 1;

Step 9: repeat steps 17 until SJ 0:The genetic algorithm was developed using a binary

encoding, in the most common form, the Simple

Genetic Algorithm (SGA). The mathematical model

of the solidification process computes the tempe-

rature field in the strand and the solidified shell

thickness and assesses the metallurgical criteria. Each

set of results of the simulation was used to form an

individual, and a set of individual represents a popula-

tion, where each member has a potential solution

encoded in it. The simulations are performed varying

the values of the sprays water flow, or spray heat

transfer coefficients, and when possible, the casting

speed. In the water flow rate, a step of 0:03 l=sbetween the upper and lower limits was used, and for

the casting rate, a step of 0:001 m=s was used. Fig. 5shows the relation between the knowledge base, the

genetic algorithm and the mathematical solidification

model.

4. Experimental procedure

To validate the proposed solidification mathematical

model and the use of the different fS formulations, the

results of the calculations are compared with experi-

mental data obtained in an experimental setup mon-

itored by thermocouples located both in the mold and in

the metal. The casting assembly used in static solidifica-

tion experiments is shown in Fig. 6. The main design

criterion was to ensure a dominant unidirectional heat

flow during solidification.

ARTICLE IN PRESS

Input of operational

parameters

Heat transfer

mathematical model

Determination of process variables

to be implemented into the

equipment based on metallurgical

constraints

Possible changes

for improvements

Report with thermal

profile, solidified shell

and possible defects

End

Comparison with

knowledge base

Rules conducive to better

operational conditions

(Search/Priority)

wt % carbon:

cracks

oscillation marks

Mold cooling:cracks

rhomboidity

Meniscus:

cracks

inclusions

Lubrification:

breakouts

laps

Sprays cooling:

segregation

cracks

Priority Scale

1 variation of water flow rate

2 variation of casting speed

Tundish:

temperature

area, width,

capacity, steel

composition

Mold:composition, metal

level, support,

thickness, taper,

cooling

Oscillation:frequency, stroke

and negative strip

time

Lubrification:oil or powder flux

Sprays:water temperature

and flow rate

Fig. 5. Block diagram showing the relation between the heat flow model and knowledge base/genetic algorithm.



13/17

Both copper and steel molds were used, with the heat-

extracting surface being polished. Experiments were

performed with Sn210 wt% Pb and Al24:5 wt% Cualloys, with the liquid metal being poured at a

temperature of 10% above the liquidus temperature.

The thermophysical properties of these alloys and chillare summarized in previous articles (Santos et al., 2001;

Quaresma et al., 2000). These alloys were chosen due to

two main reasons: all their thermophysical properties,

which are fundamental for calculations are available in

the literature, and they are quite easy to manipulate in

the laboratory. Temperatures in the chill and in the

casting were monitored during solidification via the

output of a bank of thermocouples (1:6 mm diameter)accurately located with respect to the metal/mold

interface, as indicated in Fig. 6. These sets of exper-

iments were planned for a preliminary validation of the

mathematical model with experimental data of solidifi-

cation.

To demonstrate the applicability of the solidification

mathematical model to the continuous casting process,

simulations will be compared with experimental data

from the literature.

5. Results and discussion

The temperature files containing the experimentally

monitored temperatures during solidification in static

molds were compared to the proposed mathematical

model with the transient metal/mold heat transfer

coefficient, hm=s; described in previous articles (Santoset al., 2001; Quaresma et al., 2000). Fig. 7 shows typical

examples of temperature data collected in metal and

chill during the course of solidification of Sn210wt% Pb

alloy (Fig. 7A) and Al2

4:5 wt% Cu alloy (Fig. 7B).These experimental thermal responses were compared to

those numerically simulated using the fs formulation

given by Scheils equation. In any case a good agreement

can be observed.

To validate the application of the mathematical model

for the continuous casting process, a set of simulations

was performed and the results of both billet and slab

surface temperatures were compared with literature

data. The input parameters used in these simulations

are presented in Table 2. For the billet case, experi-

mental results of a high carbon steel (SAE 1080) were

analyzed, and the simulations were based on metal/

mold heat transfer coefficients proposed by Toledo

et al. (Toledo et al., 1993) and the metal/sprays heat

transfer coefficients proposed by Brimacombe et al.

(Brimacombe et al., 1980). For the slab, the literature

data for a low carbon steel (SAE 1012), as well as the

used formulations for heat transfer coefficients in

mold and in the spray zones, were proposed by

Samarasekera and Brimacombe (1988) and Lait et al.

(1974), respectively.

Fig. 8 shows the comparison between experimental

and simulated surface temperature profiles for the steel

billet. The search space used in this analysis is shown in

ARTICLE IN PRESS

Top view Side view

60

24

24

44

Insulating Material

CastingChamber

Pouring

Channel

SteelChill

Thermocouples

63123 24

100

24

10

203

3

A A

Chill Heat Flux

24

20

Graphicaldisplay

Insulating wallsMold

Chamber

Thermocouples

Insulating

cover IN OUT RS 232

FDM program

T

t

Automatic

SearchData acquisition

Fig. 6. Casting arrangement and position of thermocouples in the mold wall and in the metal (mm).

http://-/?-http://-/?-


14/17

Table 3. The number of candidate parameter settings is

about 36 103: It was considered a population of 50parameter settings, and to alter the parameter vector,

the uniform mutation was applied as the genetic

operator. Although every parameter shown in Table 3

is taken as a variable, the search for each of them was

restricted within a range of values provided on the basis

of the current industrial practice.

It can be seen that a similar simulated profile is

attained for the two considered conditions (nominal and

optimized) up to the end of spray zone 2. From this

spray zone up to the radiation zone, the optimized

surface temperature along the equipment is more

homogenous and with lower thermal gradients between

adjacent spray zones. The water flow rates shown in

Table 4 indicate that the optimized profile is accom-

panied by a decrease of flow rate in spray zones 1 and 2

(18:3% in zone 1 and 30% in zone 2), and an increase inzone 3 (21:4%). In this case, it was not possible toincrease the casting speed because two metallurgical

constraints were violated (solid shell thickness at mold

exit and point of complete solidification).

ARTICLE IN PRESS

0 50 100 150 200 250 300 350 400 450 500

0

30

60

90

120

150

180

210

240

270

Thermocouple (metal) 20 mm interface

Thermocouple (mold) 3 mm interface

Simulated

Temperatur

e[C]

Temperatur

e[C]

Time [s]

0 25 50 75 100 125 150 175 200 225 250

0

100

200

300

400

500

600

700

Thermocouple (metal) 20 mm interface

Thermocouple (mold) 3 mm interface

Simulated

Time [s](A) (B)

Fig. 7. Typical experimental thermal responses of thermocouples at two locations in casting and chill, compared with numerical simulations:

(A) Sn10 wt% Pb and copper mold; (B) Al4.5 wt% Cu and steel mold.

Table 2

Input parameters for billet and slab continuous casting conditions (Louhenkilpi, 1994; El-Bealy et al., 1995)

Units Billet Slab

Dimensions mm 160 160 1680 220

Mold length mm 600 700

Water flow rate l/s 20.08 20.08

Water temperature C 25 25

Metal 1080 Steel 1012 Steel

Specific heat J=kg K cS 678 cL 758 cS 700 cL 700Density kg=m3 rS 7850 rL 7300 rS 7400 rL 7400Thermal conductivity W=m K kS 30:13 kS 34:50 kS 28 kS 28

Latent heat of fusion J=kg 260,000 260,000Solidus temperature C 1360 1471

Liquidus temperature C 1458 1541

Sprays 1 (length/flow rate) (m) (l/s) 2.800 1.47 0.485 3.83

Sprays 2 1.800 1.15 0.900 3.58

Sprays 3 2.700 0.55 1.285 2.66

Sprays 4 1.580 3.33

Sprays 5 1.280 2.10

Sprays 6 1.540 1.66

Sprays 7 2.380 4.66

Sprays 8 4.500 1.96

Casting rate m/s 0.0245 0.0183

Pouring temperature C 1485 1600

Metallurgical length m 10 14

http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-


15/17

The simulated surface temperature profiles for the

slab are compared with experimental results in Fig. 9. It

can be observed that the optimized surface temperature

along the equipment is more homogenous and with

lower thermal gradients from a spray zone to the next,

mainly in zones 6 and 8. As shown in Table 4, the

optimized profile produces a decrease of water flow rate

in zones 1, 2, 4 and 7 (32% in zone 1; 10% in zone 2; 18%

in zone 4, and 14% in zone 7), and an increase in zones5, 6 and 8 (8% in zone 5; 33% in zone 6, and 32% in zone

8), and zone 3 has the same value. In both cases (billet

and slab), the strand is completely solidified just before

the unbending point. For the slab case, it was not

possible to increase the casting speed due to the

violation of a metallurgical constraint (solid shell

thickness at mold exit). This feature of GA acts as a

natural safeguard against any solution which is likely to

appear in other techniques where variables need not be

specified (Chakraborti et al., 2001).

The results of optimized water flow rates for the

different spray zones are presented in Table 4 for both

cases.

6. Conclusions

A mathematical model of solidification working

integrated with a genetic search algorithm and a

knowledge base of operational parameters has permitted

ARTICLE IN PRESS

0 1 2 3 4 5 6 7 8 9 10

Distance from meniscus [m]

500

600

700

800

900

1000

1100

1200

1300

1400

1500

StrandSurfaceTemperature[C]

160x160 mm Billet

600 mm Mold

1080 Steel

Industrial - Louhenkilpi, 1994

Simulated - nominal

Simulated - optimized

Mold

1Sprays

2Sprays

3Sprays

Tmax

Tmin

Fig. 8. Comparison between results of experimental and simulated

(nominal and optimized) strand surface temperature during contin-

uous casting of SAE 1080 steel billet.

Table 3

Parameter space for optimization of a 1080 steel billet

Parameter Minimum Nominal Maximum Discretization

step

Number of

possible values

Casting speed (m/s) 0.0195 0.0245 0.0295 0.001 11

Water flow 1 (l/s) 1.20 1.47 1.74 0.030 19

Water flow 2 (l/s) 0.80 1.15 1.32 0.035 16

Water flow 3 (l/s) 0.43 0.55 0.73 0.030 11

Table 4

Water flow rates for nominal and optimized operational conditions

Billet Slab

Spray

zones

Nominal

water flow

Optimized

water flow

Nominal

water flow

Optimized

water flow

rates (l/s) rates (l/s) rates (l/s) rates (l/s)

1 1.47 1.20 3.83 2.58

2 1.15 0.80 3.58 3.20

3 0.55 0.70 2.66 2.66

4 3.33 2.72

5 2.10 2.30

6 1.66 2.50

7 4.66 4.00

8 1.96 2.92

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Distance from meniscus [m]

0

100

200

300

400

500

600

700

800

900

1000

1100

1200

1300

1400

1500

1600

StrandSurfaceTemperature[C]

1680x220 mm Slab

700 mm Mold

1010 Steel

Mold

1Sprays

2Sprays

3Sprays

4Sprays

5Sprays

6Sprays

7Sprays

8Sprays

Industrial - El-Bealy,1995

Simulated - nominal

Simulated - optimized

1012

Fig. 9. Comparison between results of experimental and simulated

(nominal and optimized) strand surface temperature during contin-

uous casting of an SAE 1012 steel slab.

http://-/?-http://-/?-http://-/?-http://-/?-


16/17

the optimization of cooling conditions (mold and

secondary cooling) for the continuous casting of

steel billets and slabs. The search method supported

by the knowledge base based on metallurgical criteria

permits a more homogenous strand surface temperature

profile to be attained and with lower thermal gradients

between adjacent sprays zones. The reasonable to goodagreement observed between experimental data and

simulations for both billet and slab analyzed in the

present study, permits to conclude that the formulations

used to calculate mold and spray heat transfer

coefficients are able to provide an appropriate descrip-

tion of heat transfer efficiencies along the different

cooling regions, as well as to determine the maximum

casting speed to attain highest product quality. How-

ever, more accurate simulations can be achieved if

particular heat transfer formulations are developed for

each continuous caster by using approaches like

comparison between theoretical-experimental thermal

profiles or data obtained from ingot microstructure.

Acknowledgements

The authors acknowledge the financial support

provided by FAPESP (The Scientific Research Founda-

tion of the State of S*ao Paulo, Brazil) and CNPq

(The Brazilian Research Council). Marco Ol!vio Sotelo

is also acknowledged for helping with the computer

programming.

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