Parameter Estimation by Inverse Solution Methodology Using Genetic Algorithms for Real Time Temperature Prediction Model of Ladle Furnace

Abstract

In the process of developing mechanistic dynamic models which faithfully represent characteristics of a process, accurate estimation of parameters is a very crucial step. Inverse solution methodology combined with evolutionary optimization algorithms has been proved to be a very potential technique for offline parameter estimation. Advanced industrial automation systems capable of generating and storing enormous volumes of sensory data have indeed fostered the usage of this approach. In the present work, inverse methodology combined with Genetic Algorithms has been successfully employed for estimating parameter of a dynamic model aimed to predict liquid steel temperature in Ladle Furnace. The parameter evaluated in this study was heat transfer coefficient of ladle refractory walls. The optimal value evaluated was obtained as 10.62 W/m².K.

1. Introduction

Steelmaking community is facing multiple challenges those vary from energy and raw material costs for sustainable business to producing high value-added grades to cater ever increasing demands of customers. Technological advancements facilitate process engineers to meet the challenges to a great extent. Continuously evolving industrial automation systems equipped with sophisticated instrumentation, PLCs, data acquisitions systems and real time process models have demonstrated enormous potential in achieving high quality consistently while optimizing resource utilization. As a result, now-a-day such automation systems became an integral part of the steel manufacturing.

Real time process models are the kernel of systems engineering. Mechanistic dynamic models derived from the first principles are one of class implemented extensively across manufacturing units. However, development of such models which faithfully mimic intricacies of metallurgical processes involves many complex tasks. One of them is correct estimation of model parameters. Many a time, it is observed that accurate determination of model parameters from theory and/or from controlled laboratory experiments is difficult. Sometimes, using data reported in the literature or from experiments may lead to trial-and-error numerical experiments for calibrating the model. Designing experiments in process plant is difficult owing to production constraints and harsh working environment. In this back drop, exploiting the readily available large volumes of plant data for parameter estimation using inverse solution methodology^1,2) (ISM) would be a pragmatic approach. Prof. O. M. Alifanov has explained ISM aptly as: “Solution of an inverse problem entails determining unknown causes based on observation of their effects. This is in contrast to the corresponding direct problem, whose solution involves finding effects based on a complete description of their causes.” In this approach modelers treat the model parameters as unknowns. Since effect of these parameters is reflected in measured data, they can be back-calculated using an optimization framework, wherein an objective function is formulated with an aim to minimize the sum of squares of differences between model predicted values and the actual industrial measurements. The solution finally yields optimum values of the model parameters.

Many researchers have successfully employed this technique. L. Elliott et al.¹⁾ have reviewed various optimization techniques for evaluating reaction rate coefficients for a given reaction mechanism. They have discussed both traditional gradient based methods and the evolutionary algorithms and finally demonstrated the superiority of Genetic Algorithms (GA) over the other methods for combustion of hydrogen, methane and kerosene. Section 2 elaborates the reason for choosing GA over the other traditional methods. A. S. Wade et al.³⁾ illustrated the effectiveness of GA in determining the optimal reaction rate parameters of reaction mechanism for the thermal degradation of aviation fuel and surface fouling. Many other research workers^4,5,6,7) have reported application of GA for estimation of model parameters in steelmaking activities. All these efforts successfully established the potential of ISM combined with GA (ISM-GA).

In the present work, ISM-GA has been applied to evaluate heat transfer coefficient of ladle refractory walls. This parameter is very critical for online temperature prediction model at Ladle Furnace, which predicts liquid steel temperature cyclically. The reasons for choosing ISM-GA is thoroughly explained in sections 3.3 and 4.3 respectively.

2. Framework of ISM-GA

While applying ISM, the first step is building the required process model. From the repository of past data, required number of data sets should be identified. Along with the values of process input variables, the selected data sets should contain measured values of all the variables that the model aims to calculate. Thereafter, the data sets should be classified into two sets, testing and verification, namely.

Subsequent to the model development, optimization framework should be designed. Depending upon the number of parameters to be estimated, it would shape into either a single or a multi-objective problem. In this work, authors used Genetic Algorithms for estimating optimal values of model parameter.

Genetic Algorithms have become popular through the pioneering efforts of J. H. Holland⁸⁾ in early 1970 s. GA⁹⁾ are a mathematical construct that mimic the aspects of evolutionary biology such as inheritance, mutation, selection and crossover. Their search methodology for a candidate solution in the problem space is heuristic and evolutionary. Conventional gradient based optimization techniques may fall in the trap of being converged to local optima. GA have largely surpassed this owing to its element of randomness in its operations. For this reason, GA have been applied successfully for large complex nonlinear problems. As mentioned in the section -1, GA have been widely used in inverse solution problems. Flow chart shown in Fig. 1, illustrates the steps involved in this framework. The steps are described below:

Fig. 1.

Flow chart of ISM-GA framework.

Step 1: Genetic Algorithm initializes random initial population for all the parameters needs to be optimized. Then it sets other criteria like number of generations, population size, crossover fraction etc.

Step 2: With the initial population, it invokes process model and finds the output. This output is compared with measured variables. The sum of squares of the difference is aimed to be minimized

Step 3: Using all the techniques like selection, mutation and cross over, it produces new generations.

Step 4: If the new solution meets the optimum criterion, then the iterations would stop and solution converges; otherwise the optimization search continues.

In subsequent sections, the application of this framework is elaborately described.

3. Application of ISM-GA to Ladle Furnace Temperature Prediction Model

3.1. Ladle Furnace Temperature Prediction Model

The primary aim of Ladle Furnace (LF) is to reheat liquid steel through electric arcing and achieve temperature homogenization through inert gas purging. In addition to this, alloy additions also take place as to attain desired steel chemistry. Since it holds liquid steel, it acts as a buffer between primary steel making process and continuous casting operations.

Precise control of steel temperature at LF is of paramount significance. Temperature values below the target in continuous casting may cause a freezing effect, i.e., steel gets overcooled and hence withdrawal of solidified steel becomes very difficult. Conversely, a high temperature may lead to break out which is even more catastrophic. Correct duration of arcing is necessary for achieving desired temperature of molten steel. In such scenario, dynamic temperature prediction model aiming to predict molten metal temperature in real time plays significant role in guiding the operator to take the right course of action. In the absence of such online process model, operators rely on the rules of thumb those developed through their operating experience.

3.2. Details of Temperature Prediction Model

To develop a thermal model for LF process from first principles, all factors influencing the temperature variations in LF as discussed by N. K. Nath et al.¹⁰⁾ and Huixin TIAN et al.¹¹⁾ are presented below

a) Heat loss through the refractory wall and top surface

b) Heat loss due to purging

c) Heat gain due to arcing

d) Heat effects of addition

While framing the governing equations for all factors mentioned above, it is assumed that the liquid steel is homogenized with respect to temperature and chemistry.

3.2.1. Heat Loss through the Refractory Wall and Top Surface

Figure 2¹²⁾ is the schematic diagram showing the details of the refractory linings for the side and bottom walls of a typical ladle used in LD shop1, Tata Steel at Jamshedpur. Material properties used in the refractory are shown in Table 1.¹²⁾ When a ladle holds liquid steel, heat from the molten metal is lost to its surroundings through its side and bottom walls as well as from top surface. From side and bottom walls heat transfer is largely through conduction and convection whereas from the meniscus, it is essentially by radiation.

Fig. 2.

Schematic representation of the refractory material layout in the Ladle.¹²⁾

Table 1. Material and thermal properties of refractory materials and steel.¹²⁾

Ladle		Side wall					Bottom
Layers	Units	Tar Dolo (A-B)	42% alumina (B-C)	Low Cement Castables, (LC)-70 (C-D)	Ceramic Paper (D-E)	Steel Shell (E-F)	Mag Carbon (G-H)	80% alumina (H-I)	Low Cement Castables, (LC)-70 (I-J)	Steel Shell (J-H)
Thickness	m	0.18	0.032	0.085	0.01	0.028	0.25	0.075	0.113	0.081
Thermal conductivity	W/m°C	2.8	1.55	2.2	0.038	52	3.4	1.8	2.2	52
Density	kg/m³	2920	2400	2720	100	7800	2950	2800	2720	7800
Specific heat	J/kg°C	1000	1130	1100	700	552	1150	1080	1130	552
Coefficient of thermal expansion	10−6/°C	4.9	5.1	6	4	0.11	5	6	6	0.11
Modulus of Elasticity	GPa	5	80	10	4	200	13	70	10	200

The transient heat transfer equation for convection is given as   

$$m_{steel}{C}_p\frac{d{T}_{steel}}{dt}=h\left(A_r+A_b\right)\left(T_{steel}-T_{\infty }\right)$$(1)

While framing the above equation, it is assumed that the ladle inner wall temperature is same as liquid steel. This is a reasonable assumption because liquid steel holdup period from the time of ladle tapping to reaching LF is sufficient to achieve this condition.

Heat transfer coefficient h mentioned in Eq. (1) depends on ladle condition and refractory lining.

The heat loss due to radiation through top surface is given as   

$$\sigma F\epsilon A_t(T_{steel}{}^4-T_{\infty }{}^4)$$(2)

Usage of Eq. (2) has some constraints. The emissivity ε values vary with slag thickness and also with argon mass flow rate. Determining them in real time is a challenging task. H. Tian et al.¹¹⁾ has presented an alternate equation to find the radiation loss which is given as   

$$f_{surf}{C}_w{Q}_w\mathrm{\Delta }{T}_w$$(3)

Combining Eqs. (1) and (3),   

$$m_{steel}{C}_p\frac{d{T}_{steel}}{dt}=h\left(A_r+A_b\right)\left(T_{steel}-T_{\infty }\right)+f_{surf}{C}_w{Q}_w\mathrm{\Delta }{T}_w$$(4)

Integrating Eq. (4),   

$$\mathrm{\Delta }{T}_{wall}=\int \left(\frac{h\left(A_r+A_b\right)\left(T_{steel}-T_{\infty }\right)+f_{surf}{C}_w{Q}_w\mathrm{\Delta }{T}_w}{m_{steel}{C}_p}\right)dt$$(5)

3.2.2. Heat Loss Owing to Purging

Purging causes drop in liquid steel’s temperature because of extraction of heat by inert gas from the steel bath, which increases with mass flow rate of the inert gas. The heat extracted by argon gas is given as   

$$m_g{C}_{p_g}\left(T_{steel}-T_{gas}\right)$$(6)

Integrating Eq. (6),   

$$\mathrm{\Delta }{T}_{purge}=\int \left(\frac{m_g{C}_{p_g}\left(T_{steel}-T_{gas}\right)}{m_{steel}{C}_p}\right)dt$$(7)

3.2.3. Heat Gain Owing to Arcing

The following equation is used for calculating heat gain by arcing.   

$$\frac{d{T}_{steel}}{dt}=\frac{\sqrt{3} VIcos\phi }{m_{steel}{C}_p}$$(8)

Where, cosφ varies with different TAP settings. A TAP number indicates switch position on the power transformer. In LF at Tata steel, TAP numbers vary from one to eight; in the decreasing order of power transmission.

Integrating Eq. (8),   

$$\mathrm{\Delta }{T}_{arc}=\int \left(\frac{\sqrt{3} VIcos\phi }{m_{steel}{C}_p}\right)dt$$(9)

3.2.3 Heat Effects of Addition

The below equation accounts for the heat effects due to material addition   

$$\mathrm{\Delta }{T}_{add}={\sum }_i{W}_{add}^i{Q}_{add}^i$$(10)

Where i denotes a specific addition. $W_{add}^i$ indicates quantity of the material added in Kg, where as $Q_{add}^i$ indicates the temperature change of liquid steel for kg of i. Table 2 shows the statistical average of thermal effects for different material commonly used in LF.

Table 2. Thermal effects of 100 kg addition of below materials in 155 t of steel.

S. No	Material Addition	Chill Factor (ΔT)
1	Lime	−2.500
2	FeSi	+0.400
3	SiMn	−1.200
4	FLUORSPAR	−1.910
5	COKE	−3.500
6	CaSi	−2.500
7	High Carbon FeMn	−1.500
8	Low Carbon FeMN	−1.500
9	LF SLAG	−1.280
10	Aluminium	+4.700
11	FeV	−0.7
12	CaFe	−1
13	FeTi	−0.4
14	FeCr–H	−1.9

3.2.4. Overall Thermal Model

Temperature prediction model for LF can be developed by combining all the thermal effects described thus far. The final expression which can be obtained by combining Eqs. (5), (7), (9) and (10) can be expressed as:   

$$\mathrm{\Delta }{T}_{overall}=\mathrm{\Delta }{T}_{arc}+\mathrm{\Delta }{T}_{purge}+\mathrm{\Delta }{T}_{wall}+\mathrm{\Delta }{T}_{add}$$(11)

Runge Kutta fourth order method was employed for numerical integration of terms expressed in Eqs. (5), (7) and (9). For the satisfactory performance of this model in real time, correctness of the heat transfer coefficient is crucial. Subsequent section elaborates the application of ISM-GA methodology for estimation of reliable heat transfer coefficient. After its estimation, model can be executed online and would predict molten metal temperature values in real time.

3.3. Parameter (Heat Transfer Coefficient) Estimation by the Application of ISM-GA

Evaluation of heat transfer coefficient is a complex task. There can be three ways of evaluating heat transfer coefficient. Each method has its own merits and demerits. In the first approach, for a given ladle filled with liquid steel, a good number of temperature measurements with defined periodicity should be recorded. From this data, rate of heat transfer can be calculated. Once heat transfer rate is known, using Eq. (4), heat transfer coefficient can be calculated. In the second method, using established and relevant heat transfer correlations, the coefficient can be evaluated. A. Tripathi et al.¹³⁾ have used a correlation which contains Nusselt, Rayleigh and Prandtl numbers whereas A. G. Belkovskii et al.¹⁴⁾ have used another correlation. The heat transfer coefficient evaluated by these two approaches works well in the beginning, but gradually leads to prediction error due to the following reasons. Ladle wall thickness, particularly the side wall, decreases with ladle life. Additionally, sculling, slag attack and metal penetration into the refractory brings a significant variation in both thickness and thermal properties of the lining.

The third method using ISM-GA addresses these issues because it is evaluated from plant data and thus captures the nuances. Application of ISM-GA can be repeated from time to time as to track and accommodate the process drifts. Thus, the model prediction accuracy can always be kept intact. Alternately, it can be applied to individual ladles as to track the variation in the heat transfer coefficient along with the ladle life.

As known, LF is batch process and each batch (also called as ‘heat’) is identified with a heat ID. All the process information related to a given heat such as initial chemistry, arcing duration, TAP numbers, material additions along with temperature measurements (also called as probes) are stored in database against its heat ID. Temperature model, when invoked by ISM-GA frame work, receives the input data of selected heats and evaluates liquid steel temperatures. The objective function, as given by Eq. (12), is equal to square of the differences between model and measured temperature values. GA arrives at the optimal solution by minimizing the objective function.

The objective function can be expressed as:   

$$\left(minimize\right)\mathrm{\hspace{0.17em} }Obj={\sum }_{i=1}^{i=N}{\sum }_{j=2}^{j=M}{\left(T_{ij}^{Probe}-T_{ij}^{Model}\right)}^2$$(12)

The only constraint applicable here is that the heat transfer coefficient should be positive, which is expressed as Eq. (13)   

$$h>0$$(13)

In Eq. (12), M and N refer the number of probes and number of heats respectively. For every heat, model calculates temperature values cyclically (at ten seconds interval) for the complete treatment duration. So, each predicted temperature value has a corresponding time stamp. But measurements are taken at discrete intervals. The objective function as shown in Eq. (12) considers only those predictions whose time stamps match with that of measurements. The first probe values of all heats are not taken into account by this framework because these values are used for self-correction by the model. The reason for this is that the model begins with an assumed initial temperature value and needs the actual measurement for adjustment.

3.4. Results and Discussion

Total 50 identified heats from Ladle Furnace #2 of LD shop1 are inputted to ISM-GA algorithm. The operating conditions of Ladle Furnace #2 are shown in Table 3. As explained in section 2 and 3.3, the optimization framework of ISM-GA would try to evaluate optimal heat transfer coefficient by minimizing the objective function. Under this approach, GA produces new candidate solutions with every generation. For each candidate solution, a value of the objective function is calculated. The average of all objective function values and the minimum (best) value is plotted against each generation, which is illustrated in Fig. 3. The difference between average and minimum values is gradually decreasing with the progression of generations and at the generation 40 they became almost the same. Subsequent to 40th generation, the objective function value began converging and finally the optimal value is attained at 55th generation. The evaluated optimum value of heat transfer coefficient was 10.62 W/m².K. It is found that this value is very close to heat transfer values obtained by previous researchers, A. Tripathi et al.¹³⁾ and A. G. Belkovskii et al.¹⁴⁾ The comparison is shown in the subsequent paragraphs.

Table 3. Ladle furnace dimensions and operating conditions.

Parameter	Values
Ladle Dimensions (m)
Top Diameter	3.31
Bottom Diameter	2.53
Height	4.41
Bottom thickness	0.519
Side wall thickness	0.67
Steel weight (ton)	155
Initial Temperature of molten metal (K)	1510–1540
Target Temperature (grade specific) (K)	1580–1610
Argon volumetric flow rate (NM3/hr)	5–15
Slag Weight (ton)	1.5
Carry over slag	0.7
	Carry over slag composition
	Fe (2.2%), CaO (45.91%)
	SiO2(22.34%), MgO (6.92%)
	Al2O3(16.93%), MnO (3.18%)
Added in LF	0.8
Lid cooling water flow rate (m3/hr)	290
Treatment duration (minutes)	~55

Fig. 3.

Evolution of average and best objective function values with generations. (Online version in color.)

Heat transfer from the outer shell of the ladle wall happens through natural convection and radiation, and can be expressed as:   

$$q=h_{natural}\left(T_{outerwall}-T_{\infty }\right)+\sigma ϵ\left(T_{outerwall}{}^4-T_{\infty }{}^4\right)$$(14)

In Eq. (14), ε is the emissivity of steel shell and its value is 0.8.^12,14)

The radiation component mentioned in Eq. (14) can be rearranged as:   

$$\sigma ϵ\left(T_{outerwall}{}^4-T_{\infty }{}^4\right)=h_{radiation}\left(T_{outerwall}-T_{\infty }\right)$$(15)

where,   

$$h_{radiation}=\sigma ϵ\frac{\left(T_{outerwall}{}^4-T_{\infty }{}^4\right)}{\left(T_{outerwall}-T_{\infty }\right)}$$(16)

Substituting Eq. (16) in Eq. (14), the heat loss from the outer wall can be expressed as:   

$$q=\left(h_{natural}+h_{radiation}\right)\left(T_{outerwall}-T_{\infty }\right)$$(17)

From Eq. (17), the overall heat transfer coefficient is:   

$$h=h_{natural}+h_{radiation}$$(18)

The correlation used for obtaining natural convection component by A. Tripathi et al.¹³⁾ is expressed as:   

$$\overline{N{u}_H}=\frac{h_{natural}H}{k}={\left\{0.825+\frac{0.387R{a}^{\frac{1}{6}}}{{\left[1+{\left(\frac{0.492}{Pr}\right)}^{\frac{9}{16}}\right]}^{\frac{8}{27}}}\right\}}^2$$(19)

The correlation for natural convection used by A. G. Belkovskii et al.¹⁴⁾ is expressed as:   

$$h_{natural}=0.135\frac{k}{H}{\left(Ra\right)}^{\frac{1}{3}}$$(20)

In Eq. (19), Nu is the Nusselt number, Pr is Prandtl number. Ra is Rayleigh number defined as the product of the Grashof and Prandtl numbers. Prandtl, Grashof and Rayleigh numbers are expressed as:   

$$Pr=\frac{C_p\mu }{k}$$(21)

  

$$Gr=\frac{g\beta \left(T_{outerwall}-T_{\infty }\right)L^3}{{\text{v}}^2}$$(22)

  

$$Ra=\left(Gr\text{Pr}\right)$$(23)

The film temperature at which all fluid (air) characteristics are evaluated is expressed as:   

$$T_f=\frac{T_{outerwall}+T_{\infty }}{2}$$(24)

T_outerwall value can be obtained either by direct measurement using instruments like handheld pyrometer or by employing mathematical modeling. T. P. Fredman et al.¹⁵⁾ have measured this value and also evaluated through modeling and found it close to 180°C. In the present work, authors have used the same value for film temperature evaluation.

All fluid properties^15,16) and the parameters used in the correlations are mentioned in the Table 4. The characteristic length (L) used in Eq. (22) is equal to the bath height in the ladle. After evaluating the heat transfer coefficients of natural convection component using Eqs. (19) and (20) and heat transfer coefficient of radiation component using Eq. (16), one can obtain the overall heat transfer coefficients. The resultant overall heat transfer coefficients are 16.6 and 17.45 W/m².K respectively. These values are comparable to the heat transfer coefficient obtained though ISM-GA, 10.62 W/m².K.

Table 4. Parameters used in the heat transfer correlations.

S. No	Parameter	Value
1	Ladle furnace outer wall temperature (°C)15)	180
1	Air (film) temperature (°C)	105
2	Thermal conductivity of air W/(m.K)	0.032
3	Kinematic viscosity of air (m2/s)	24.03.10−6
4	Thermal expansion coefficient (1/K)	0.00264
5	gravitational acceleration (m/s2)	9.8
6	Prandtl number	0.706
7	Characteristic length (m)	3.41

Once evaluated by ISM-GA, the heat transfer coefficient (10.62 W/m².K) is substituted in the temperature prediction model. With this optimal value, the prediction accuracy of the LF temperature algorithm has been tested for all the fifty heats. The average value of the number of probes used during heat treatment is close to six. Excluding the first probe of every heat, the rest of the probe values (total 238) are compared with the corresponding temperature values predicted by the thermal model. Figure 4 shows the frequency distribution of predictive errors which are within +15°C. It is shown in the same figure that the hit rate within +5°C is 70.17%. Given the uncertainties in the weights of additions and inflow and outflow temperatures of coolant water and unequal electrode wear, the hit rate achieved is reasonably good. The possible reason for the remaining 29.13% cases lying outside the ±5°C band is the deviation from the standard operating practice of probe measurement, which can potentially introduce error in the measured value. Thermal model, equipped with the optimal heat transfer coefficient, could have predicted either higher or lower temperature value than the actual. This could have resulted into a difference greater than +5°C between prediction and the measurement. But, the number of such deviations is not very large, because the hit rate within +10°C is 94.12%. During the heat treatment, operator gives much importance to the final probe value, which normally gets measured just before the treatment termination. The reason for this is that the last probe indicates the attainment of target temperature. Figure 5 shows the prediction error of only the last probe value. In this case, the hit rate within +5°C is 70.37% and within +10°C is 92.6%. These results are good and have induced confidence in model performance among operators.

Fig. 4.

Distribution of predictive errors evaluated at all measurements.

Fig. 5.

Distribution of predictive errors of final probe with adaptation.

In order to demonstrate a typical temperature profile, the thermal model with the optimum heat transfer coefficient has been executed off-line for a given heat data as input. The resultant temperature values of molten steel along with the time have been plotted in Fig. 6. The flat rectangles on the x axis indicate the arcing duration and the vertical rectangles indicate the details of material additions. Predicted temperatures are shown by the solid line whereas measurements are shown in diamond symbols. Points shown as M-1, M-2 and M-3 specify the drop in temperature due to the thermal effects of material additions. At the point Pr-1, model received the value of the first probe. As said before, model was supposed to adopt this value. In this case, interestingly, the prediction value and first probe were close to each other indicating a perfect initial guess. Subsequently, model received probes at Pr-2 and Pr-3 instances. And towards the end of treatment, model and predicted values are showing a very good match.

Fig. 6.

A typical temperature profile evaluated by model with self-adaptation.

In order to verify the performance of the model, additional 25 heats were tested. Predictive errors with the final probe values are shown in Fig. 7. It can be observed that the hit rate within +5°C is 72% and remained consistent.

Fig. 7.

Distribution of predictive errors of final probe with validation-data-set.

In this exercise, variation of heat transfer coefficient along with ladle life was also studied using ISM-GA. In LD shop1, three Ladle Furnaces, namely 1, 2 and 3 are in operation. A given ladle can go for treatment in any of these furnaces. A ladle is used approximately for 55 heats including all three furnaces. After completion of near about 55 heats from all furnaces, a ladle is sent to repair shop for relining of the worn refractory.

Two ladles, numbered 14 and 19 were selected for this investigation. Data from Ladle Furnace #2 (LF-2) was selected for this study. The reason for this is that all other results mentioned in this work are taken from the same furnace. Out of total 55 heats, around 20 heats are processed in LF-2. Data of each heat was inputted to ISM-GA and the output was the optimal heat transfer coefficient. All these heat transfer coefficients were plotted against the heat numbers. The resultant graphs were shown in Figs. 8 and 9. From the graphs, it can be realized that heat transfer coefficient is marginally reduced with ladle life. In case of ladle 14, the trend line in Fig. 8 shows a marginal decrease in heat transfer coefficient from 11 to 9 W/m².K, whereas the trend line of ladle 19 as shown in Fig. 9 indicates drop from 11.75 to 10 W/m².K. The value decreases with the heat number, which implies that the ladle is getting more insulated. The reasons could be due to skull formation, refractory buckling due to thermal stresses.

Fig. 8.

Variation of heat transfer coefficient with ladle (no 14) life. (Online version in color.)

Fig. 9.

Variation of heat transfer coefficient with ladle (no 19) life. (Online version in color.)

4. Conclusion

In this paper, power of offline parameter evaluation mechanism using inverse solution methodology combined with Genetic Algorithms has been successfully demonstrated. Optimum values of model parameter applied in the secondary steel making area have been calculated.

Heat transfer coefficient, a parameter of real time temperature prediction model of ladle furnace has been evaluated using ISM-GA. This value was comparable with the values obtained by previous researchers.

With this optimum value, when training data set consisting of 50 heats and validation data set consisting of 25 heats are tested, the predictive results are quite satisfactory.

Variation of heat transfer coefficient along with ladle was also investigated. It was found that variation was very small.

From the results, it can be concluded that ISM-GA technique is powerful tool for estimation of parameters of process models used for real time prediction. The future scope of this work is extending the offline-parameter evaluation technique to an online adaptation methodology wherein the parameters can be updated from time-to-time, thus encapsulating the process drifts and nuances automatically.

Nomenclature

A_r, A_b, A_t: Area of radial direction, bottom and top respectively, m²

C_p, C_pg: Specific heat of steel and gas, J/kg.K

C_w: Specific heat of cooling water, J/kg.K

cosφ: A factor that varies according to the TAP number

F: View factor

f_surf: Heat loss coefficient of slag surface and furnace cover.

g: Gravitational acceleration constant, m/s²

h: Heat transfer coefficient, W/m².K

H: Height of liquid steel in the ladle, m

I: Current, A

k: Thermal conductivity, W/m.K

L: Characteristic length, Equal to bath height in the ladle, m

m: Mass of steel, kg

Obj: Objective function

q: Heat flux (loss) through the outer shell of ladle refractory wall, W/m²

Q_add: Heat effect of addition, K/kg

Q_w: Flux of cooling water, m³/s

t: Time, s

T: Temperature, K

ΔT_w: Temperature difference of cooling water, K

V: Voltage, V

$W_{add}^i$ : Quantity of the material addition in Ladle furnace, Kg,

Gr: Grashof number (dimensionless)

Nu: Nusselt number (dimensionless)

Pr: Prandtl number (dimensionless)

Ra: Rayleigh number (dimensionless)

Subscripts

add: Additions

arc: Arcing

f: Film

g: Gas phase

natural: Heat transfer through natural convection

purge: Holding and purging

Overall: Over all thermal effects

Outerwall: Outer surface of ladle refractory wall

radiation: Heat transfer through radiation

steel: Liquid steel

surf: Surface of the steel meniscus

w: Coolant water

wall: Ladle refractory wall

∞: At infinity, i.e. outside condition

Superscripts

g: Gas

Probe: Temperature probe used in Ladle Furnace

Model: Process model

Greek Letters

β: Thermal expansion coefficient, 1/K

σ: Boltzmann constant, W/m² K⁴

μ: Viscosity, Kg/m.s

v: Kinematic viscosity, m²/s

ρ, ρ_g: Density of steel and gas, kg/m³

ε: Emissivity

Abbreviations

LF: Ladle Furnace

Obj: …Objective function that needs to be minimized

Pr: …Temperature probes used in Ladle Furnace

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