Computational algorithms to simulate the steel continuous casting

International Journal of Minerals, Metallurgy and Materials Volume 17, Number 5, October 2010, Page 596 DOI: 10.1007/s12613-010-0362-0

Computational algorithms to simulate the steel continuous casting A. Ramírez-López1,2), G. Soto-Cortés2), M. Palomar-Pardavé2), M.A. Romero-Romo2), and R. Aguilar-López2) 1) Institute Polytechnic National, Mexico City C.P. 07300, México 2) Department of Materials, Metropolitan Autonomous University, Mexico City C.P. 02200, México (Received: 25 October 2009; revised: 29 November 2009; accepted: 4 December 2009)

Abstract: Computational simulation is a very powerful tool to analyze industrial processes to reduce operating risks and improve profits from equipment. The present work describes the development of some computational algorithms based on the numerical method to create a simulator for the continuous casting process, which is the most popular method to produce steel products for metallurgical industries. The kinematics of industrial processing was computationally reproduced using subroutines logically programmed. The cast steel by each strand was calculated using an iterative method nested in the main loop. The process was repeated at each time step (Δt) to calculate the casting time, simultaneously, the steel billets produced were counted and stored. The subroutines were used for creating a computational representation of a continuous casting plant (CCP) and displaying the simulation of the steel displacement through the CCP. These algorithms have been developed to create a simulator using the programming language C++. Algorithms for computer animation of the continuous casting process were created using a graphical user interface (GUI). Finally, the simulator functionality was shown and validated by comparing with the industrial information of the steel production of three casters. Keywords: continuous casting; simulation; numerical method; computational algorithm

Nomenclature:

lx, ly, lz—Billet dimensions, m;

n—Strand number;

mn—Mass of cast steel, t;

Anozzle—Area of nozzle, m2;

nbn—Number of produced billets on each strand;

An—Area of billet (frontal area), m2;

rc—Radius of the curve zone, m;

lbn—Longitude of the steel out the CCP (used for comparison), m;

tc—Closing time of a strand, s;

dCZ—Length of the curved zone, m; DD—Dimension wanted, m;

vnozzle—Speed of the jet steel (from the tundish to the mold), m/s;

DP,max—Maximum dimension (pixels), pixels;

vn—Casting speed of the strand, m/s;

DR,max—Maximum actual dimension, m;

Vc—Volume of cast steel by all the strands at each step simulation, m3;

LB—Billet length, m;

to—Opening time of a strand, s;

Vcn—Volume of cast steel in each strand at any time, m3;

LF—Length of the free zone (from the end of the curved zone to the flame cutter), m;

Vl—Volume of the ladle, m3;

LSZ—Length of each spray zone, m;

Vn—Volume of cast steel by each strand, m3;

Corresponding author: A. Ramírez-López

E-mail: [email protected]

© University of Science and Technology Beijing and Springer-Verlag Berlin Heidelberg 2010

A. Ramírez-López et al., Computational algorithms to simulate the steel continuous casting

597

VT—Total volume of steel, m3;

work.

VTotal—Total volume originally in the ladle or total volume to be cast, m3;

The first step is to represent the operating of the CCP computationally. This representation is called the kinematics of processing which responds to the questions what happen and how. This work is focused on the development of some computational algorithms to create a simulator for a CCP. Subroutines for reading information about the geometrical configuration and operating conditions of the CCP are explained in detail. The algorithms are also capable to reproduce the steel displacement at each moment during simulation. Furthermore, the subroutines developed are employed as a platform to develop more sophisticated simulators for the calculation of steel thermal behavior in the future works [1-25].

Z—Displacement of the steel in the casting direction, m;

ρsteel—Steel density, kg/m3; Δt—Time step, s.

1. Introduction Computational simulation of industrial processes gives important benefits to steel companies because the production methods used can be analyzed to reduce cost and time. Some authors have been working on the automation of some industrial processes and the development of industrial products and processes [1-5] using computers. Zulch and Grieger [5] have been working on the planning of production simulation systems to create digital factories which provide a powerful tool to support the planning process. Everything from a single workplace to a complete workshop or even a factory can be represented computationally and analyzed to find potential opportunities and dangerous weaknesses. Steel manufacture is a complicated process with many factors involved, and many authors have developed models for describing some of them, such as heat removal, solidification, and grain structure formation [6-18]. They initially used general equations to obtain approaches. Nevertheless, a better description and representation of the process is necessary. First, researchers began to calculate constants for steels with different compositions by evaluating parameters from direct observation to interpolating data and obtaining equations which represent the phenomena during processing such as fluid flow and heat removal [6-10, 19-22]. Although these models were basic and could not be used for different cases, many of those authors continued to develop their original work including more variables and sophisticated routines for calculation [6-10, 19-22]. Recently, many authors [6-10, 12-18, 23] have begun to use computational tools (software packages) to create simulations for the continuous casting process, while some others have been working on developing computational algorithms to treat the chemical and physical phenomena involved [19-22, 24]. Their efforts have been frequently focused on the steel thermal behavior and solidification during processing [4, 6-10, 12-24]. Unfortunately, the analysis is complex due to the physical factors involved, geometrical configuration, and the difference of casting conditions used in each continuous casting plant (CCP). This is the motivation of the present

Independent files including the graphical subroutines for displaying the information, menus for selecting options, and calculation routines, etc., have been compiled separately to avoid unnecessary code. The simulator is compiled using routines and subroutines based on logical algorithms which are the sequences of ordered procedures to be programmed. Whereas the algorithms are illustrated using flow charts which include all the mathematical and logical methods and rules for calculation.

2. Reading data for simulation The defining data process is also called process for reading data (PRD). The process is used to input independent variables that control the simulation. This information is required to generate the graphical sketch of the CCP and the animation of the processing [25]. Fig. 1 illustrates the primary PRD. At the beginning of this process, some data about the geometry of the CCP and the operating conditions are pre-loaded from a previously saved file to be established as the initial information for defining. Then these data are displayed on the screen. Here the user can quickly clean or modify the information required for a particular case. The PRD is shown in a detailed form in the flow chart of Fig. 2. It is developed to modify each independent variable involved in the CCP. The information required for defining the cases includes the following. First, it is possible to select the section to be cast (billets, blooms, round, or slabs) and the dimensions (lx, ly, and lz, or Rp, where Rp is defined as the radius of the rounded cast section). Thus, the nodes should discretize the steel section. The geometrical configuration of the CCP includes two options. The first corresponds to geometrical dimensions

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Fig. 1. General flowchart of the reading data process.

such as the radius of the curved zone (rc), the length of the free zone (LF), the volume in the ladle which is the volume of steel to be cast (Vl), and the number of the working strands (SN) in the CCP, etc. The second option corresponds to the mold geometry information such as the meniscus level, the location angle (θ), and the curvature of the mold sections. This mold receives the liquid steel from the tundish and solidification begins here. This is the reason why the information about each removal coefficient must be read or loaded in this routine. The next step is to define the operating conditions. Particular data for each strand of the CCP is defined here. The information includes the opening and closing times. If any strand is closed before the casting operation finishes, it must be declared before the calculation of the casting time. Finally, the corresponding casting speeds are also declared here.

Fig. 2. Detailed flowchart for the process of reading data.

The geometrical configuration of the secondary cooling system (SCS), such as the spray segments and their corresponding operating conditions including the water flow rates, must also be defined. The option to define the steel composition is the next. Although this information is not used in the present work for the simulation of the CCP, it must be read because it will be used in algorithms of the future work. A two-dimensional array is used to define the variables of each strand (SV=[n][d]), where n is the strand number and d is the data of the independent variables defined by the user. Some of these variables must be declared as the integer data type and they cannot be less or equal to zero such as the

A. Ramírez-López et al., Computational algorithms to simulate the steel continuous casting

working strands, the segments in SCS, the sprays in every segment, the nodes to be used, etc. Others must be declared as the floating data type such as billet dimensions and operating conditions. The particular information for each strand is read using nested loops commanded by the corresponding defined integer data type. Some of these data can be equal to zero but not less than zero. Adequate algorithms with appropriate warnings and logical messages have been included using the programming sentences “if” and “goto” to avoid errors caused by users mismatched during input data. Finally, these data are also used to command the nested loops for the calculation of dependent variables needed during the simulation.

3. Algorithms to simulate the kinematics in CCP A typical configuration of a CCP is illustrated in Fig. 3. During the process, liquid steel is cast from the ladle to the tundish, and then arrives the molds where receive and form the steel section (called billet). After this, the steel is quenched in the secondary cooling system using water jets under the sprayed areas. Finally, the obtained billet is driven through a free zone and cut using a cutter flame. The algorithm for describing the process is based on the solution of an iterative method for calculating the cast steel and the remaining steel at each moment. Then this information is used to illustrate the steel displacement at each

599

moment during the simulation. The model assumes that the steel density is constant, and the volume of cast steel in each strand at any time (Vcn) is a function of the casting speed which is equal to the volume of cast steel flowing from the tundish nozzle towards the mold. The casting speed for each strand (vn) is calculated using Eq. (1). Finally, the volume of cast steel by each strand (Vcn) is calculated as

vn =

v nozzzle ⋅ Anozzle An

Vcn = v n ⋅ An = v nozzle ⋅ Anozzle

(1) (2)

The total volume of steel for each strand (VTn) is the sum of the volumes of cast steel for each strand during the casting time. It can be calculated using Eq. (3), where ton and tcn are the opening and closing times of each strand, respectively.

V Tn =

tc n

∫

An v n dt

(3)

to n

The total volume of cast steel (VT) is the result of the sum of all the partial cast steel volumes by each strand as shown in Eq. (4). If the casting time is obtained, Eq. (4) can be easily solved using a repetitive algorithm included in a pair of nested loops. SN t c n

VT = ∑∑ An v n Δt n =1 t o n

Fig. 3. Sketch of the continuous casting plant.

(4)

600

Nevertheless, this equation can also be solved partially if the partial volume of cast steel for each strand at each moment of the simulation is known. This process is placed in a loop which goes from n=1 to SN, using the sentence “for”, where n corresponds to the strands. Then the procedure is nested in the main loop to be solved step by step. The main loop for calculation and simulation goes from an initial time (t=0) to an ending time (t=tmax). It is a function of the time step (Δt) defined for the calculation, whereby a short time step is recommended to obtain more accurate results. Fig. 4 shows the flow charts for the calculation of casting time and the animation of the process, respectively. Here the shaded zone represents the main subroutine of the kinematics model, which calculates the instant cast steel, the steel in the ladle, and the corresponding position of the steel on each strand.

Int. J. Miner. Metall. Mater., Vol.17, No.5, Oct 2010

Total volumes of cast and non-cast steel are obtained using Eqs. (5) and (6), respectively, for each time step as a function of the casting speed of each strand. t and t−1 refer to the values of the variables in the final and the latest updated step times, respectively. SN

VTt = VTt −1 + ∑ An v n Δt

(5)

n =1

SN

Vlt = Vlt −1 − ∑ An v n Δt

(6)

n =1

This process is repeated until Vl is equal to zero as shown in Fig. 4(a), if the time of the simulation is within the range of the opening and closing times of the strands. When the calculation process is outside of the shaded zone and there is a volume of steel remaining in the ladle, a warning routine

Fig. 4. Flowcharts of the main subroutine for calculating the casting time (a) and the subroutine nested for the animation of processing (b).

A. Ramírez-López et al., Computational algorithms to simulate the steel continuous casting

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indicates that it is necessary to modify the original data for casting the total volume of steel in the ladle. Here, the sentence “if” is employed to make a break in the main loop because the non-cast steel represents economical loses for steel industries. The steel in the CCP is displaced through the casting direction and driven a distance on each strand (Zn). This distance can be obtained using Eq. (7), and the mass of cast steel (mn) can be calculated by Eq. (8).

Z n = v n ⋅ Δt

(7)

m n = V n ⋅ ρ steel

(8)

The algorithm assumes that the steel displacement starts inside the mold at the meniscus level (Z=0) and the ending point is Z=dz, where dz is the distance displaced through the CCP, which is the result of the sum of the curved zone and the free zone until the billet is cut in the cutter flame. The subroutine for the calculation of the cast steel and the displacement in the CCP is unique and can be nested in other loops to eliminate unnecessary code. This subroutine is nested in a loop to be solved for each strand and can be nested again in an open loop (loop without a defined ending) to calculate the casting time. The same subroutine can also be nested in a closed loop in case of knowing the beginning and ending times to display the animation of the process as shown in Fig. 4(b). Geometrical information is calculated to display the sketch of the CCP. The length of the curved zone (dCZ) and the length of each segment (LSZ) can be obtained using Eqs. (9) and (10), respectively. The displacing distance (dTotal) is calculated using Eq. (11).

rc ⋅ π 2 θ ⋅ rc LSZ = 180 r ⋅π d Total = c + LF + LB 2

d CZ =

(9) (10) (11)

Fig. 5 shows the flow chart used to create the tools for displaying the results on the screen. These are divided in three options and the user can select one of them. The sketches of the CCP and the animation of the process are displayed on the screen by executing an algorithm to determine the location of the graphical information using a graphical user interface (GUI). The algorithm translates the coordinates of the real world to positions on the computer screen using a pair of coordinates (x, y) to place the objects.

Fig. 5. Flowchart of the subroutine for displaying the results.

These coordinates are the result of the sum of two members (x1+x2, y1+y2). The origin coordinates of the sketch on the screen are x1 and y1. The origin is the point where the object is placed on the screen. While x2 and y2 are relative positions calculated as a function of the iterative method of the kinematics model. For the position of the steel on each strand, the origin of the trace is the corresponding position of each strand; nevertheless, the relative position is updated at each time step. The members of the coordinates are fitted to the screen using the same scale, solving the second and the third terms of Eq. (12) for each axis, respectively. The effect of representing 3D objects on a computer screen is obtained, drawing the CCP geometry and the steel positions with an isometric location on the screen. D R, max D D Z = = D P, max x1 x2

(12)

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4. Validation of the algorithm for kinematics

the analyzed cases, assuming that each strand is opened at the same time at the beginning of the simulation (t=0). Other assumptions for the simulations are as the following.

The model was tested by comparing with the actual casting operation data for three different CCPs for validation. The general data are shown in Table 1, while Table 2 shows the average data of the casting speeds for each case. The casting speeds shown in Table 2 are used to simulate

Each strand remains open (working) until the volume of steel in the ladle is exhausted.

Table 1. Casting data for the cases analyzed Case

Company

Billet section

Billet size / (mm×mm)

Billet length / m

Strand number

rc / m

Average weight of ladle / kg

1

DeAcero

Squared

160×160

12

4

9.0

98500

2

Sidertul

Squared

130×130

12

3

5.0

55000

3

Sicartsa

Squared

127×127

13

6

5.5

112000

Table 2. Average casting speed of each strand m/min Case

Strand

1

2

3

1

2.44

2.60

2.55

2

2.33

2.45

2.50

3

2.20

2.35

2.45

4

2.14

―

2.40

5

―

―

2.35

6

―

―

2.30

The casting speed is constant for each strand during the simulation. The number of produced billets by each strand is initialized (equal to 0) at the beginning of the simulation and restored when the simulation is over. Nevertheless, these values are updated if the user simulates more than one single casting operation without interruption. The comparison between actual and simulated data is shown in Table 3. The data obtained using the algorithm for kinematics and the actual casting operations are very similar. According to these results, it is assumed that the algorithm can be used to obtain accurate simulations about the continuous casting process. Nevertheless, there are some differences between the actual and simulated values.

The number of produced billets is different from the total obtained billets for one single casting operation because produced billets are the result of the sum of the billets produced by each strand in the corresponding conditions. In contrast, the number of total billets is the quantity of producible billets with the volume of steel in the ladle without scraps. During an actual casting operation, these numbers are equal, because the steel in ladles is constantly cast, the casting times are established by the workers usually without interruption, and ladles are programmed to be cast in an ordered sequence for steel volumes with the same composition. The only note that the workers marked in the history reports is the time when the empty and full ladles are changed for every new casting operation. Fig. 6 shows the animations for the cases analyzed according to their corresponding time step. The figures are snapped screens of the simulation displayed. Here, the time that commands the kinematics algorithm according to the time step (Δt) is illustrated using an analogue clock in the right upper corner of the screen. Below the analogue clock, the volume fraction of steel remained in the ladle is illustrated using a bar. The strand number is shown on the left and the strand status is illustrated as a traffic light signal, and green and red colors are used to indicate if the strand is

Table 3. Comparison between actual and simulated data Case 1 2 3

Billet weight / kg

Number of produced billets

Total billets

Cast time / min

Actual

2365.0

42

42.00

57.00

Simulated

2365.4

40

41.64

55.52

Actual

1560.0

35

35.00

59.00

Simulated

1561.3

34

35.22

58.04

Actual

2364.0

69

68.00

68.00

Simulated

2365.7

68

68.44

66.21

A. Ramírez-López et al., Computational algorithms to simulate the steel continuous casting

603

Fig. 6. Simulation of the CCP for the cases analyzed using kinematics models: (a) case 1; (b) case 2; (c) case 3.

opened (casting) or closed (not casting). This procedure is easily programmed using the sentence “if” included in the displaying routine inside the main loop. Besides, the number of produced billets and the corresponding casting speed are shown by each strand. The animation of the steel driven through the CCP is displayed in the lower part of the screen. The strands are numbered from left to right for an easy identification. The billets produced by each strand are counted when these are cut in the cutter flame. In Table 4, the produced billets by each strand are compared for the simulation and actual process, showing very similar results. Fig. 7 shows the working graphics of each strand for each case, through which the user can verify the strand condition and the moment when a billet is cut. Here the time when a volume of steel is in the CCP is shown in blue, while the time when a volume of steel is ahead of the cutter flame is illustrated with a gray bar. The delay time between these two bars is the billet running

time. Finally, the moment when a billet is cut is printed with a red line. The difference in time between the cuts can be converted in length units in the case of knowing the casting speeds and represents the billet lengths. The difference in the casting speed of each strand can be easily identified by the difference in the times when the billets are cut. The number of produced billets is obtained during the simulation by comparison of the lengths of the steel remained (lbn) in each strand. Here the sentence “if” is used to add a unit to the strand counter. Then, this information is stored in a new initialized two-dimensional array for easy management. The complete process is described in the flow chart of Fig. 8. Here the counter of each strand is initialized to zero. If the distance (Zn) is greater than the length of the CCP (curved zone+free zone), the distance (lbn) is measured and compared with the billet length (LB); if the condition lbn>LB is true, the number of produced billets (nbn) is increased, and the remained steel is updated, returning to zero again to start a new counting.

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Table 4. Comparison between produced billets by each strand Case 1 2 3

Strand 1

2

3

4

5

6

Actual

11

11

10

10

―

―

Simulated

11

10

10

9

―

―

Actual

12

12

11

―

―

―

Simulated

12

11

11

―

―

―

Actual

12

12

12

11

11

11

Simulated

12

12

11

11

11

11

Fig. 8. Flowchart for counting produced billets.

Fig. 7. Working graphics for analyzed cases: (a) case 1; (b) case 2; (c) case 3.

The algorithm developed stores the most relevant information to display the cast steel at each moment of the processing as shown in Fig. 9. Here the shaded area is the remained steel in the ladle. The end of the shaded area means that steel in the ladle is exhausted (casting time). The slope of the line is always constant for the analyzed cases, because the casting speed is always constant and the strands remain opened until the end of the casting operation. Nevertheless, this slope can change if the opening and/or closing times of the strands are different or if the casting speed of any strand is not constant.

5. Analysis of processing The simulator can be used to analyze production using general casting data. Therefore, it is possible to display graphics showing casting costs to know the effect of the factors involved on the casting production.

Fig. 9. Casting steel graphic for case 1.

The casting time and the volume of cast steel are two critical parameters in casting production. These can be obtained as a function of the casting speeds and the strands. The casting time is shorter if the number of working strands is increased. Moreover, the casting time is also reduced if the casting speed of the strand is increased as shown in Fig. 10. Fig. 11 shows the increment in the cast steel volume as a function of the strands used and the casting speeds. Moreover, the complete procedures shown in the flowcharts of Fig. 4 can also be nested in a more complex loop to solve more than one single casting operation. The new main loop is solved under the number of casting operations (co) from 1 to nco, where nco is the number of casting operations. The total casting time is the result of the sum of all the partial casting times of each operation. This process can be

A. Ramírez-López et al., Computational algorithms to simulate the steel continuous casting

605

putationally, changing to a new three-dimensional array for SV=[n][d][co] which allows an easy identification of the data. The volumes in ladle and casting speeds for a sequence of four casting operations are shown in Table 5. Here strand 1 is closed during the third casting operation because of a failure in the SCS and only works 15 min during the fourth casting operation. The rest of the strands remain open without interruption all the time. The casting times of the last two casting operations increase, the number of producible and produced billets is redistributed towards the other strands as indicated in Tables 6 and 7. A delay on the third and fourth casting operations is shown in Table 6. The rows of producible and produced billets correspond to each casting operation, whereas the last columns are the total billets by each strand.

Fig. 10. Reduction in casting time as a function of casting speed and strands for a 112-t ladle.

In Table 7, the number of produced billets can be more than the producible billets due to the algorithms developed, assuming a counting process without interruption for more than one single casting operation. In other words, if there is steel remained in some strands and a new casting operation is initialized, the remained steel will be added to the part of the billets in the new casting operation. These fractions are not considered as scrap fractions. Nevertheless, the total billets produced are always smaller than the producible ones when all the programmed casting operations are simulated due to the steel remainders in strands. Additionally, the information of the cast steel during each casting operation and the scrap fraction is shown in Tables 8 and 9. Table 8 shows some values for a typical casting report. The values are the weight of the cast steel (wsteel) obtained using Eq. (13).

Fig. 11. Volumes of cast steel as a function of casting speed.

represented mathematically using a new sub-index in all the equations presented. A new dimension is added to the original array to store the information of each strand com-

wsteel = An v n ρ steel

(13)

Table 5. Information for continuous multiple casting operations Casting operation

Volume in ladle / kg

1 2

Strand 1

2

3

4

to / min

tc / min

vn / (m·min−1)

to / min

tc / min

vn / (m·min−1)

to / min

tc / min

vn / (m·min−1)

to / min

tc / min

vn / (m·min−1)

97500

0

58.36

2.00

0

58.36

2.15

0

58.36

2.20

0

58.36

2.35

99250

0

56.49

2.25

0

56.49

2.05

0

56.49

2.45

0

56.49

2.40

3

98700

0

0.00

0.00

0

72.40

2.50

0

72.40

2.40

0

72.40

2.20

4

98450

0

15.00

2.10

0

69.74

2.25

0

69.74

2.35

0

69.74

2.30

Table 6. Casting time calculated Casting operation

1

2

3

4

Casting time / min

58.3692

56.4947

72.4031

69.7478

606

Int. J. Miner. Metall. Mater., Vol.17, No.5, Oct 2010 Table 7. Producible billets during each operation by each strand Casting operation

Strand

Total billets by each strand / pieces

1

2

3

4

Producible

Produced

1

9.728208

10.59277

0.00000

2.26500

22.58598

22

2

10.45782

9.651188

15.08399

13.07771

48.27072

48

3

10.70103

11.53435

14.48063

13.65894

50.37495

50

4

11.43064

11.29895

13.27391

13.36833

49.37184

49

Producible billets

42.31771

43.07725

42.83854

42.36999

170.6035

―

Produced billets

40

44

42

43

―

169

this strand is stopped during the third casting operation. In the first two casting operations, the total produced billets are twenty, resulting from the integer sum of the produced billets without interruption (9.728208+10.59277=20.32098) as shown in Table 7; and the rest is a scrap fraction (0.32098). This value is multiplied by the defined billet length (12 m) to know the length of the scrap (12×0.32098=3.85176 m). Finally, the weight of the scrap is obtained using Eq. (13). An intelligent routine is identified by comparing partial scraps between casting operations, using the sentences “if”. The same procedure is used for all the values in the table. This information can be only for statistical analysis, but it also can help workers to open or close a strand to reduce scrap fractions.

Table 8. Weight of cast steel during casting operations kg/min Strand 1 2 3 4 Σ

Casting operation 1

2

3

4

384.0 412.8 422.4 451.2 1670.4

432.0 393.6 470.4 460.8 1756.8

0 480.0 460.8 422.4 1363.2

403.2 432.0 451.2 441.6 1728.0

Table 9 shows the scrap fractions after each casting operation. Here strands 2, 3, and 4 are cast without interruption until the fourth operation; in consequence, there are no partial scraps. Strand 1 is the only one with an interruption, and

Table 9. Scrap fraction of the steel on each strand Strand

Scrap fraction

Scrap length / m

Scrap weight / kg

Partial

Final

Partial

Final

Partial

final

1

0.32098

0.26500

3.85176

3.18000

739.53790

610.5600

2

0

0.27072

0

3.24864

0

623.7389

3

0

0.37495

0

4.49940

0

863.8848

4

0

0.37184

0

4.46208

0

856.7194

Σ

0.32098

1.28251

3.85176

15.39012

739.53790

2954.9030

6. Conclusions

and display the additional chemical and physical phenomena involved during processing.

(1) The results obtained using the algorithms developed are very similar to those taken during actual casting operations whereby the kinematics model can be used to simulate actual casting operations and is capable of describing in detail the steel processing during continuous casting.

(3) The development of advanced software simulators for industrial processes has a direct application and proved advantages. These are powerful tools for the analysis and improvement of industrial processing

(2) The routines and subroutines developed with the algorithms can be used as a platform for developing a more complex simulator in combination with GUIs to animate

Acknowledgments The authors wish to thank the National Council of Science and Technology (CONACyT), Autonomous Metro-

A. Ramírez-López et al., Computational algorithms to simulate the steel continuous casting

politana University (UAM-AZC), Institute Technologic and Autonomous de México (ITAM), and Institute Polytechnic National (IPN).

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