Calibration of the real-time heat transfer - WIT Press · Calibration of the real-time heat...

Calibration of the real-time heat transfer

model for the simulation of the continuous

casting process

E. Laitinen", R. Nieminen", S. Louhenkilpi*

^ University of Jyvdskyld, Department of

Mathematics, Seminaarinkatu 15, SF-40100

Jyvaskyla, Finland

^Helsinki University of Technology, Institution of

Process

Finland

ABSTRACT

In the present paper we consider the problems associated with the cali-bration of the numerical simulation model used in the real-time processcontrol. The aim of the control system is to adjust the spray coolingwater flow amounts in the continuous steel casting process so, that thethermal requirements of the steel will be met.

The numerical simulation model is based on the parabolic partial dif-ferential equation for which the correct estimation of the boundary con-ditions is of crucial importance. In practice the values of the boundaryconditions depend e.g. on the temperature of the boundary and on thespray cooling water pressure and amount. The boundary conditions areusually expressed as heat fluxes or as heat transfer coefficients. A pro-cedure to determine the boundary conditions for the secondary coolingzones from the temperature measurements is presented in this paper.

For the validation of the numerical simulation model, a lot of strandsurface temperature measurements have been carried out in a castingmachine and the measured temperatures were compared with the cal-culated. The results are also presented in this paper.

1. INTRODUCTION

In the continuous casting process the strand thermal state e.g. thick-ness of the solidified shell, depth of the liquid pool and distribution ofsurface temperatures have a considerable influence on cracks and on

Transactions on Modelling and Simulation vol 4, © 1993 WIT Press, www.witpress.com, ISSN 1743-355X

468 Computational Methods and Experimental Measurements

other defects which can be formed in the cast material. Hence, the con-trol of these characteristics are of central importance, especially, whenusing direct hot charging, optimization of casting speed with respectto good productivity or when using soft reduction or electromagneticstirring near the liquid pool length.

To study the thermal state of a continuously cast strand many nu-merical simulation and optimization models of the continuous castingprocess have been developed. They offer an important tool to test thethermal state and to design new cooling strategies. However, most ofthe models can be used only for simulation and optimization of steadystate casting conditions e.g. Lait [1], Rogberg [2 ], Laitinen [3,4 ]andLarrecq [5 ].

Although the steady state models offer important information about theoperational limits of the continuous casting machine, they are not validunder unsteady state operation of the casting machine. However, theunsteady state, which is consequence of changes in casting parameterse.g. casting speed, superheat, mold heat removal, spray water flow ratesetc. plays a very important role in the continuous casting cycle.

In modern casters, casting often continues uninterrupted for many hoursor even days, and sometimes large changes in casting speed are requiredbecause of a late-arriving ladle, ladle or tundish-pouring problems, tem-porary malfunctions of some part of the casting machine, or varioussimilar reasons. The casting machine must be able to respond to thesedisturbances without interruption of casting and without serious degra-dation of product quality.

To avoid product quality deterioration it is necessary to have an intel-ligent system, which adjusts the secondary water flow rates accordingto variations in casting parameters so, that the selected thermal stateof the slab is retained.

Recently, the most sophisticated control systems for slab casting ma-chine are based on the numerical simulation of the slab temperaturefield e.g. Barozzi [6 ], Takawa [7 ], Okuno [8 jand Laitinen [9 ]. Hence,the most fundamental component of the control system is a real-timeheat transfer model. The simulation model must be accurate and fastenough for the control purposes. In this paper a real-time heat trans-fer model, which fulfil the requirements for on-line use is presented.The model is verified with industrial measurements and the first spraycooling control application is under industrial tests in a Finnish steelwork.

The outline of this paper is following: The real-time heat transfer modelis presented in Chapter 2. Our situation requires solutions of a diffusion- convection heat equation, where the location of its phase transition


Computational Methods and Experimental Measurements 469

(free boundary) is not known a-priori. Moreover the boundary condi-tions are nonlinear, which makes the solution algorithm quite compli-cated.

In Chapter 3 we discuss about the parameters of the model. The iden-tification of the correct boundary conditions in the mould and in thesecondary cooling region are in essential importance. The thermophys-ical properties of the steel, as thermal conductivity, specific heat, phasetransformation temperatures etc., depend on the steel chemical compo-sition and can be taken from the data published in the literature.

In Chapter 4 some numerical experiments are presented. For the val-idation of the presented real-time model, a lot of strand surface tem-perature measurements have been carried out in a industrial castingmachine and some simulated results compared with measured are pre-sented.

2. REAL-TIME MODEL OF THE CONTINUOUS CASTINGPROCESS

Solidification process in continuous casting machine involve complexmass and heat transfer phenomena.

In general case the modelling of solidification process requires the com-putational solution of coupled equations of heat, mass and momentumwhich are collectively referred to as the Navier-Stokes equations (cf.Patankar [10 ]). Although this modelling approach gives very accu-rate knowledge of the casting process, it is not suitable for real-timesimulation models of the continuous casting process.

In real-time models many assumptions and special procedures are usu-ally employed to simplify the calculations, while assuring the accuracyof prediction of the model in on-line use.

The conventional modelling approach used in e.g. Okuno [8 ], Laiti-nen [9 ]and Morita [11 ]is to divide the strand into tracking planesand approximate in real-time the temperature through the thicknessof each plane by a one dimensional heat transfer model with movingphase boundaries. The most restrictive assumption of this techniqueis, that the heat convection and conduction to the withdrawal directionis neglected. This makes the real-time solution algorithms (computerimplementation) complicated.

In this paper the modelling approach is more straightforward. Weconsider the problem in a two-dimensional longitudinal cross sectionthrough the middle of the slab illustrated in Figure 1. In this mod-elling approach the thermal state of the cross section can be calculateddynamically by solving the diffusion-convection equation:



d^ T 4- div(pcuT) = div(kVT) + (T), (1)ot

where v — (vx,Vy,Vz) is the velocity of the material flow. Moreover, kis the thermal conductivity, c is the specific heat, p is the density and Tis the temperature. The source term which plays regard to latent heatbeing emitted during the phase change is denoted by q(T}.

In considering convection effects, in addition to the problem of dealingwith the latent heat evolution, the nature of material flow in the vicinityof the phase change needs to be accounted for. The latent heat and ve-locity conditions can be accounted by supplying appropriate definitionsto the source term, i.e.

g(T) = pL + d\v(pvL(I - /.(T))), (2)

where L is the latent heat and fs(T) is the local solid fraction whichdescribes how the fraction of solid phase varies with temperature inmushy region. In practice the form of the /%, versus T curve dependsprimarily on local solute redistribution and in metallurgical literaturethere are a number of possible relationships for /«(T) e.g. Goldsmith[12].

A popular method to model the latent heat evolution is to apply the en-thalpy formulation to the Equation (1). By defining a smooth enthalpyfunction H(T), which takes into account both the latent heat and thespecific heat contribution, the Equation (1) reduces to the form:

= div(fc(T)VT) - v • Vff (T) (3)

wheref

= /Jo

T(x,t)

The solidification can be described as follows: initially the slab is at aconstant temperature and it moves at a velocity v to the withdrawaldirection. The manner in which the solidification proceeds will dependon the boundary conditions and the standard values of the thermaland physical data used in the calculations. Hence, in order to solveEquation (3) the boundary and initial conditions must be specified aswell as other assumptions considering the material and physical data



MOULD

Fig. 1. Geometry of the slab and the calculation domain fl.

used. The following assumptions (i)-(iv) are assumed in consideringthe material and physical data and the boundary conditions:

(i)

The convective heat transfer depends strongly on material flowvelocity. The flow velocity v = (vx,Vy is composed of the castingspeed velocity v" — (0,i>*) and of the velocity of the liquid flow,

V* — (v[,Vy), inside the liquid pool, which approximation requiresthe solution of Navier-Stokes equations. However, the liquid flowhas only a minor effect on the thickness and temperature of thesolid shell because the superheat of the steel is small. In this modelthe liquid flow was taken into account by using an effective thermalconductivity for the liquid and mushy steel. This effective thermalconductivity ^e// is calculated by the equation:

(5)k. 5k,

where kg is the normal value of thermal conductivity in solidustemperature.

The material properties have been measured and published for alarge number of steel grades. In order to use the temperature de-pendent tabulated values presented in literature the specific heatc = c(T), the thermal conductivity k — k(T) of steel the solid frac-tion fs = fs(T) are defined as a temperature dependent functions.



(iii)

The total heat transfer, Qtot, trough the mold wall is calculatedby the equation:

Qtot = PV-CU-W-AT (6)

where p^ and c^ are the density and the specific heat of the moldcooling water, W the flow rate of the mold cooling water and AT isthe temperature difference between incoming and outgoing coolingwater.

However, the air gap formation between the solid shell and themould wall has a very strong influence on the distribution of thetotal heat transfer Qtot- Many investigators have predict experi-mentally the distributed heat transfer profiles, Q(y), in the mold.See Rogberg [1 ]for example.

By fitting the total amount of heat, Qtot, to experimentally ob-tained heat transfer profile the heat flux through the mould wallcan be expressed as:

(7)n n ^

where Am is the area of the mold faces and Q(y,t) is obtained bydistributing Qtot over the mold.

Below the mould, along the secondary cooling region and alongthe air cooling region, the strand is cooled down by conduction tothe support rolls, by convection to the spray cooling water and byradiation to the environment.

The standard technique for calculating the heat flux across thestrand boundary is to define the heat transfer coefficient betweenthe strand surface and the cooling material. The heat flux is thencharacterized as:

r\k(T} — T =

convection, conduction /o\

radiation

where /i(y,£) is the heat transfer coefficient, which takes into ac-count the convective and conductive heat transfer. Moreover,



is the spray water temperature, Text is the temperature of the en-vironment and the parameters a and e are the Stefan-Bolzmanconstant and emissivity, respectively.

The emissivity value of 0.8-0.9 is normally used for the oxidizedstrand surface. In cases, where insulating covers are used to pro-tect the strand against heat loss, the value of emissivity must beadjusted respectively.

Before this boundary condition can be applied in industrial com-putations the relationship between the heat transfer coefficient, /i,and the cooling parameters (spray water flow rates, strand surfacetemperature etc.) must be determined for each cooling zone. A lotof empirical formulas for describing this relationship have been de-rived and presented in literature e.g. Nozaki [13 ]and Brimacombe[14 ]. However, these formulas are only valid for the particular typeof continuous casting machine and steel grade, for which they arederived. They cannot be transferred directly to other casting ma-chines without any changes of their parameters.

3. NUMERICAL APPROXIMATION OF THE CONTINUOUSCASTING MODEL

We consider the following partial differential equation in domain O.

' » JL *. \ -1- I «-/ »...-•. y -•- y \ )%

with boundary conditions

T(x, •; t) = TQ on the top of the strand

{Q(y; t)/Am on the mold;h(%/;f)(T-:rez() + (10)cre(jT* — T f) on the sec. cooling region;VyH(T) on the bottom of the strand

and initial condition

The numerical approximation of the equations (9)-(ll) is performedby finite elements (piecewise linear) in space and by finite differences intime. Let T* = (T\*,..., T£) be the nodal value vector of the temperaturefield T(x,y;t) at time t = t*. Suppose that the simulation time has auniform partition to the time intervals. We thus get a nonlinear systemfor finding T*~*~*



*)

where [M] — {( pi j)} is the mass matrix, [A] — {(V(/?;, Vc j)} isthe stiffness matrix, [C] = {(v • Vy?j,<£>j)} is the convection matrix.Moreover the matrix [B]is defined on the boundary and the vector F isthe load vector.

The notation (u,v) = J^ u(x)v(x)dx defines a Z/^ -inner product and¥>j(x) is the Courant's basis function associated with the node Xj

° ^j

The normal piecewise linear finite element approximation becomes un-stable when the mesh size d is such, that the Peclet number

Pe = > 2. (13)

Condition (13) means that in order to have physically sound numericalresults from the FE- approximation (12) the size of finite elements mustbe small if velocity \v\ is large. If it is feasible to use a sufficiently smallmesh size in a given problem, we can obtain a solution whose qualityis fairly good.

In practice, a very refined finite element model which provides good nu-merical solutions is hard to compute. A standard technique to overcomethe difficulties in numerical stability is to use upwind method, whichhas been extensively studied in computational fluid dynamics. See e.g.Kikuchi [15 ]and Brooks [16 ]. It is independent of the dimension of thedomain 17 and only one parameter #o is involved.

The essential point of the upwind technique is the additional term ofartificial conductivity. The form of this term depends on the givenvelocity field, conductivity, and the mesh size of the FE-model. The factof physics is that the temperature is convected by the flow of material.Thus, those portions of the flow region where the magnitude of velocityvector is large must be convection dominated along the direction of theflow. This means that the artificial conductivity may be added alongthe "stream" line of the flow defined by the unit vector n = r&%z +such that



We shall define the additional artificial conductivity k™ by

where #o G [0,1] ,|u| is the speed of the flow, and d = max x - x|,where x, x G flc- Therefore, when there are portions where the flowvelocity is much larger than the conductivity we may modify the heatequation (12) to read

[M}H(T) + (C + A"]H(T) + (A]K(T)

+ [B}T = F

where[A-], ; = ( V -,V ) (17)

More details on the upwind method can be found in [19] and elsewhere.

In order to solve Equation (16) an iterative method is used. TheNewton-Raphson method is used on such boundary nodes where thenonlinear boundary condition ( Eq.(8) ) appears. The modified S.O.R.-method which takes into account the phase change temperatures is usedon the other nodes. The algorithms are considered in e.g. Laitinen [3

]and Elliot [17].

4. ESTIMATION OF THE SECONDARY COOLING PARAMETERS

For the correct simulation of the heat transfer in continuous casting,the estimation of the secondary cooling parameters (heat transfer coef-ficients) is of crucial importance. In the case of the normal spray watercooling , the heat transfer coefficient depends mainly on the spray waterflow rate and on the surface temperature of the strand.

Before the developed heat transfer model can be applied in industrialcomputations the relationship between the heat transfer coefficient andthe water spray cooling parameters must be determined for each cool-ing zone. This can be done off-line using strand surface temperaturemeasurements. Usually the measurements are done under steady stateconditions. In Figure 2 is presented one steady state thermoelementmeasurement, which is used in this paper when the cooling parametersare estimated.

The first step in the estimation procedure is to calculate the strandtemperature field, which correspond the measured surface temperatureand other steady state conditions. For that we must modify our real-time model (9)-(ll) by neglecting the transient term in the Equation(9) and by replacing the heat flux boundary condition (Eq. (8)) on



5.0 10.0 15.0 20.0DusLance beLow the moLd

25.0 30.

Fig. 2. The thermoelement measurement of the slab surface tempera-ture (dotted line) and the calculated surface temperature (sol-id line).

the spray cooling region by the measured temperature boundary con-dition. Hence, we have the following steady state model for solving thetemperature field of the slab.

(18)(T) - u • V#(T) = 0

with boundary conditions

, •; t) = TO on the top of the strand

,y;£) = Tmeas on the secondary cooling region

dn \ VyH(T) on the bottom of the strand

The second step in the estimation procedure is to calculate the heattransfer coefficients, which corresponds the calculated (Eq. (18)-(19))temperature field of the slab. For that we must calculate the followingoutward normal derivative along the spray cooling region.

-on

(20)



20.0 140.0 160.0 180.

Fig. 3. The calculated heat transfer coefficients and the correspondingwater flow values. The caster has 13 secondary cooling zones.

The heat transfer coefficients can then simply be defined from the heatflux boundary condition, Equation (8). The calculated heat transfercoefficients versus the measured spray water flow amounts are presented

in Figure 3.

The last step in our estimation procedure is to fit an empirical functionformula to the estimated values. A lot of empirical formulas can befound in literature. We have used in this work the following formula:

where W is the spray water flow amount and /i*(W) the correspondingheat transfer coefficient. The parameters a and f3 can be estimated foreach cooling zone by solving the following minimization problem:

min > (a,/?

(22)

where n is number of thermoelement measurements.

For solving the parameters a and /3 for each cooling zone from theEquation (22) more (n at least 2) than one thermoelement measure-ments must be done. In Figure 2 is shown the calculated boundary



temperature when the heat transfer coefficients at each zone are esti-mated from the measurements.

5. TESTING OF THE MODEL

The model was applied to a stainless steel slab casting machine at aFinnish steel work. The caster is of type curved with straight mouldand it has five secondary cooling zones with normal water spray cooling.

(v = 0.92 m/min)

Casting speed

400

0 10 20 30 40 50 GO 70 80 90

Time, min

Fig. 4. Comparison between calculated and measured surface temper-ature with change in casting speed.

To test the accuracy of the model, surface temperatures measured withpyrometers were compared with the calculated values obtained by themodel. Firstly, however, the relationships between the heat transfercoefficients and the cooling parameters for each cooling zone were de-termined.

Some results between the surface temperature calculated by the modeland measured by the pyrometers are shown in the Figures 4-5. It isassumed that the fluctuations in the measured surface temperature are



1200

1100 Calculated 4-40 % waterOo

1000

Q.

0)o(0

(A

900

800

700

Measured

lasting speed = 0.96 m/min, except at the en

10 20 30

Time, min

40 50 60

Fig. 5. Comparison between calculated and measured surface temper-ature with change in the spray water flow rates.

caused due to the scale on the surface of the slab and the top temper-atures are near the actual surface temperatures.

In Fig. 4 the comparison point is in zone 3 on the bottom side ofthe strand. The steel grade was AISI 304L. The casting speed waschanged during simulation. A good correspondence between calculatedand measured temperatures was obtained.

In Figure 5 the comparison point is also in zone 3, but now on the topside of the strand. The steel grade was AISI 321. The casting speedwas kept constant during this trial except at the end of the casting.The water flow rates were decreased by 40 %. Quite good agreementbetween the calculated and measured results is again obtained.

In these calculations the thermal conductivity, the specific heat and thelatent heat of solidification were taken from the published data.

6. DISCUSSION AND SUMMARY

A real-time heat transfer model for continuous slab casting is presented.The model calculates the strand temperature field along the machineas a function of the actual casting variables, strand geometry and steelgrade. The model has been tested by carrying out industrial trials and



a quite good agreement between the calculated and measured results isobtained.

The model is implemented on microcomputer parallelly by using pipelinetechnique, which is considered e.g. in Crichlow [18 ]. The simulationprogram runs not on microcomputer's own processor, but parallellyon two transputer processors on a motherboard plugged into the mi-crocomputer. The user-interface of the program runs simultaneouslyon micro's own processor under Windows environment. In a realis-tic real-time application the computing time for solving the slab (1540FE-nodes) temperature field is about 2-4 seconds.

Summing up it can be said, the model gives reliable results if the bound-ary conditions are correctly determined. It also fulfil the requirementsfor on-line use. As to the future work the main aim is to develop theon-line applications of the model to be used for control of the secondarycooling, on-line quality prediction or optimization of the casting speed.

REFERENCES

1. Lait, J.E. et. al, Mathematical modelling of heat flow in the con-tinuous casting of steel, Ironmaking and steelmaking 2 (1974), p.90.

2. B. Rogberg, "High temperature properties of steel and their influ-ence on the formation of defects on continuous casting," Disserta-tion, The Royal Institute of Technology, Department of Casing ofMetals, 1983.

3. E. Laitinen, P. Neittaanmaki, On Numerical Simulation of theContinuous Casting Process, Journal of Engineering Mathematics22 (1988), 335-354.

4. E. Laitinen, P. Neittaanmaki, On Numerical Solution of the Prob-lem Connected with the Control of the Secondary Cooling in theContinuous Casting Process, Control-Theory and Advanced Tech-nology 4 (1988), 285-305.

5. Larrecq, M., Birat, J.P., Saquez, C. and Henry, J., Optimization ofcasting and cooling conditions on steel continuous casters; Imple-mentation of optimal strategies on slap and bloom casters, IRSID,ACI 83 RE 1004, Juliet 1983 (1983).

6. S. Barozzi et. al, Computer control and optimization of secondarycooling during continuous casting, Iron and Steel Engineer 63 (1986),21-26.

7. T. Takawa et.al, Mathematical Model and Control System of Cool-ing Process, The Sumitomo Search, 1987, No. 34, May, 79-87.



8. Okuno, K. et al, Dynamic spray cooling control system, Iron andSteel Engineer (1987).

9. E. Laitinen, S. Louhenkilpi, T. Mannikko, P. Neittaanmaki, Auto-matic Secondary Cooling Control for the Continuous Casting Pro-cess of Steel, in "Proceedings of the 4th Annual Conference of theEuropean Consortium for Mathematics in Industry (ECMI 89),"Hj. Wacker and W. Zulehner (eds.), B.G.Teubner Stuttgart andKluwer Academic Publishers, 1991, pp. 109-121.

10. Patankar S. V., "Numerical Heat Transfer and Fluid Flow," Wash-ington: Hemisphere Publishing Corporation, 1980.

11. T. Morita et. al, CWZroZ MefW o/ gecoW&n/ CWm# MWer/orBloom Continuous Casting, Technical Bulletin 1109 (1986).

12. F. Richter, "Die wichtigsten physikalischen Eigenschaften von 52Eisenwerkstoffen," Verlag Stahleisen M.B.H., Dysseldorf, 1973.

13. T. Ashworth, D.R. Smith, Thermal Conductivity, No. 18 (1985),

175-185.

14. A. Goldsmith, T.E. Waterman, H.J. Hirschhorn, "Handbook ofThermophysical Properties of Solid Material," Vol II: ALLOYS,The Macmillan Company, New York, 1961.

15. Rogberg, B., High temperature properties of steel and their influ-ence on the formation of defects in continuous casting, Disserta-tion, The Royal Institute of Technology, Department of Casting ofMetals (1982).

16. Nozaki, T. et al, A secondary cooling pattern for preventing surfacecracks of continuous casting slabs, TISIJ 18 (1978) 6. (1978).

17. Brimacombe, J.K., Samarasekera, I.V. and Lait, J.E., ContinuousCWm#, Vol 2, Book Grafters, Ins., Chelsea, MI (1984).

18. Kikuchi N., "Finite Element Methods in Mechanics," CambridgeUniversity Press, 1986.

19. Brooks, A. and Hughes, T. J. R., Streamline upwind/Petrow-Galerkinformulation for convection dominated flows with particular em-phasis on the incompressible Navier-Stokes equations, ComputerMethods in Applied Mechanics and Engineering 32 (1982), 199-259.

20. Elliot, C.M. and Ockendon, J.R., "Weak and variational methodsfor moving boundary problems," Research Notes in Mathematics,59, Pitman, 1982.

21. Joel M. Crichlow, "An introduction to distributed and parallelcomputing," Doctoral Thesis, Report 43, Prentice Hall Interna-tional (UK) Ltd, 1988.


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