# The Inverse Problem-Based Interfacial Heat Transfer between the Billet and Mould and the Crack Sensitiveness Area of Strand.

код для вставкиСкачатьDev. Chem. Eng. Mineral Process. 14(3/4), pp. 473-486, 2006. The Inverse Problem-Based Interfacial Heat Transfer between the Billet and Mould, and the Crack Sensitiveness Area of Strand He-Bi Yin, Man Yao* and Da-Cheng Fang Department of Materials Processing, School of Materials Science and Engineering, Dalian University of Technoloa, Dalian I16024, P R. China -~ ~~ ~~ -~ A coupled thermo-mechanical model has been established including the finite diference method (FDM) simulating the mould heat transfer based on the inverse problem algorithm@om measured temperatures, and thefinite element method (FEM) predicting the strand stress. Calculation results show that the distribution of mould heat flux, which determines the shell thickness, corresponds to that of thermal resistance between the mould and shell. The safety index of billet quality illustrates that the surface cracks are likely to originate near the meniscus and propagate from beneath the surface at the lower part of mould In this stub, the thickness distributions of liquid and solid slag films are also calculated, showing that the heat transfer is determined mainly by the solid slag film in the upper mould. Introduction Numerical simulation is widely used to describe the strand thermo-mechanical behavior and the mould heat transfer in the continuous casting process through building mathematical models [1-4]. However, it still cannot give a precise and comprehensive description of the various phenomena because of the complex phenomena involved and the mutual interactions of the mould process. Often many assumptions are made to reduce the computation time and to avoid complex problems, e.g. intricate interfacial heat transfer from the strand to the mould. In continuous casting process, the behavior of mould heat transfer and the mechanical state of the strand are related. Therefore, many studies have been conducted to develop a coupled mathematical model to calculate the thermal and mechanical behaviors [5-71, and Thomas [ 5 ] developed a coupled 2D thermo-mechanical model to calculate the stress distribution for slab casting. The empirical formula describing heat flux was taken as a boundary condition in many previous studies. However, it varies with various casting conditions [S-lo]. In order to make the results better reflect industrial casting conditions, an inverse problem method (based on measured temperatures from selected thermocouples buried at the center-line) has been developed for slab and billet [ll], around the perimeter is still * Author for correspondence (yaoman@dlut.edu.cn). 4 73 He-Bi En,Man Yao and Da-Cheng Fang not considered. Furthermore, currently there are no examples reported of such research for round billet casting. In order to make the calculation results better reflect actual casting conditions, i.e. the non-uniformity of heat transfer around the perimeter observed in a plant [12], we introduce an inverse problem method based on measured temperatures from the thermocouples located at various transverse and longitudinal sections in the mould. Attention is focused here on the heat flux, solidification state, and mechanical behavior and even the distributions of slag film. The main results to be achieved to reach the fixed targets are: 1. Distributions of mould heat flux and the corresponding thermal resistance. 2. Non-uniform solidified shell thickness. 3. Profiles of liquid and solid slag films thickness. 4. Safety index of cracks based on the stress in the solidified shell. Besides, the experimental method for temperatures used in the inverse problem algorithm is also introduced. Temperature Data Acquisition To obtain the real time distributions of mould temperatures and heat flux, an industrial trial was conducted in a round billet caster machine. The molten steel from the tundish is transferred to the mould through a straight port SEN. To measure the mould temperatures, sheathed thermocouples with 1 mm diameter were buried in the round billet mould (1 78 mm inner diameter, 14 mm wall thickness), and at the same time the mould heat flux was monitored using specially designed heat flux sensor. The thermocouples were on average 7 mm from the cold face. In six transverse sections, whose positions were 95, 155, 245, 365, 515, 650 mm, respectively from the mould top (expressed as L1,L2, ... L6), and six longitudinal sections, whose angles are 60" between any two neighbors (0" stands for inner arc of the mould), there are 36 monitoring points in total. Every point has two thermocouples, having 3.5 mm distance between them. The thermocouples lay out as adopted in the plant trial is presented in Figure 1. To avoid the interference of EMS on the signals, appropriate methods were taken. The compensation lead of thermocouple and the annex cables are shielded, adopting the insolated apparatus such as the amplifier Board, Insolated Digital Output Card and Insolated A/D Card. In order to easily assess and decrease the effect of measurement error, the data used in this paper are time-averaged values under the operation parameters kept almost stable, and the time interval is 10 minutes. It was found in the plant trial that the mould heat flux and temperatures were non-uniform around the perimeter. Based on measured temperatures, the non-uniform heat transfer was calculated by using an inverse problem method that can reflect the real casting conditions. Model for Calculation (0 The FDM/..EM coupled thetmo-mechanicalmodel For calculating the real time thermo-mechanical behavior, the simulation model consists of two parts. Firstly, an inverse problem algorithm is used to calculate the 4 74 Heat Transfer between Billet and Mould, and Crack Sensitiveness Area of Strand mould heat flux with the FDM. Secondly, the coupled thermo-mechanical analysis is based on the FEM having the same geometry and size of grids as the first part to calculate the strand mechanical behavior based on the non-uniform heat flux obtained from the inverse problem calculation as its boundary condition. So, a FDM/FEM coupled thermo-mechanical analysis from experimental temperatures was developed with its schematic diagram shown in Figure 2. Figure 1. Thermocouples layout employed in the mould. Figure 2. Schematic diagram of the coupled thermo-mechanical analysis. 475 He-Bi Yin, Man Yao and Da-Cheng Fang (ii) Establishment of mathematical model The following assumptions have been made according to the characteristics of heat transfer for round billet continuous cast: 1. Mould temperature and heat flux vary around the perimeter, and heat flux is calculated from the differential temperature by two thermocouples (3 .S mm apart) along the radial direction. Heat flow along the drawing direction is negligible. 2. The influence of temperature on the mould physical properties is negligible below 400°C. 3. The mould top and bottom are considered to be adiabatic, and heat absorbed by the mould powder above the meniscus is also negligible. 4. The cooling water extends from the bottom to mould-wall top, and the heat transfer coefficient at the mould/water interface is constant. 5 . The effective thermal resistance is adopted to express the complex heat transfer between the mould and the billet. The governing differential equation for heat conduction in the mould and molten steel in column coordinates is: where a = k / ( p .c P ); k is thermal conductivity; p is density; cp is specific heat; T is temperature; r is radius; $, is radians. To complete the mathematical description of the problem, the following initial and boundary conditions are employed. During casting, the free liquid surface is the upper boundary of the calculation domain. Initial conditions: ...(2) Boundary conditions: Mouldbillet: (1) ...(3) (2) Mouldcooling-water: where T, and T, are the pouring temperature and room-temperature, respectively; r, is inner radius of mould; R~~ is effective thermal resistance between the mould and billet; r, is surface temperature of billet; h, is convection coefficient; T, is the temperature of cooling water; and r,,, and q,, are the temperatures of mold outer and inner walls, respectively. 476 Heat Transfer between Billet and Mould, and Crack Sensitiveness Area of Strand The R~~ between the mould and billet is unknown, so the heat transfer equation is a morbid one. Assuming an initially uniform thermal resistance the temperature field of the mould is calculated at the first step, and subsequently by iteratively modifying the thermal resistance value based on the difference of the calculated and measured temperatures. The calculation is repeated until calculated temperatures are in agreement with measured values. This is a typical inverse heat transfer problem. (iii) Inverse problem algorithm Figure 3 shows the schematic diagram of longitudinal slice and the domain for inverse heat conduction. U Figure 3. Schematic diagram showing domain for inverse heat conduction. The temperature fields vary depending upon local thermal resistances, expressed 5 ) .when the calculated temperatures are in agreement with the by ~ ~ ( ~Only measurements, the given R, (?p, ) meets the requirement. If calculated temperatures ~ 5lower ) are higher than the measured ones, it means the estimated ~ ~ ( are p added; , ) and vice versa, d ~ . ( ~ p , ) comparatively than the actual values, so ~ ~ ( ~ is is subtracted. Given the preceding description, the profile of mould temperature is determined by adjusting the thermal resistance between the billet and mould, ~ ~ ( by an amount, d ~ , ( ~ p according ,), to the following equations: 477 ~ 5 ) ~ He-Bi fin, Man Yao and Da-Cheng Fang where i, j = 1 , 2 , 3 ,...6 , p = 1,2,3, ...n ; ~ ~ ( q r , )is the thermal resistance of the previous iteration; T , ~and q; are the measured and calculated temperatures respectively; 1;; - l;,J is the differential temperature; a,,J is the iteration step. When T; is close to T,,, and 11;; - ~ , ~&,l gthe computation stops. The E is 1°C here. For a given R~(,J,) value, the mould temperature field is calculated. By comparing the calculated temperatures and the measurements, RJ?;) is modified and the temperature field is calculated again. The computation will not stop until the calculated temperatures meet the requirement. The dichotomy method is applied to set at,,. (iv) Calculation of solia7liquid slag jilm A method has been developed to study the thickness of solid and liquid slag films around the mould width based on the temperature fields of mould and strand for slab casting. As the solidifying shell moves down through the mould, the films of solid and liquid slag can be obtained for any transverse section of mould. This method is described in detail in the literature [14]. Similarly, the thickness of solid and liquid film can also be calculated, thus characterizing the non-uniformity due to the non-uniformity of temperature around the perimeter either of mould or of strand as discussed above. The coupled thermo-mechanical analysis The initial conditions applied in the FEM calculation are the same as that in the FDM analysis already discussed, and the elastic-plastic material is considered. Furthermore, the temperature-dependent thermal properties of the steel in both analyses were used. An two-dimensional transient, elastic-plastic stress model was developed to determine the internal stress in the shell arising from the change in temperature gradients (calculated by the heat transfer analysis based on the inverse problem algorithm). (v) Verification of Model The profiles of calculated and experimental heat fluxes along the drawing direction are in agreement with each other, for either 60" or 240" longitudinal mould section as shown in Figure 4(a). Figure 4(b) shows both profiles around the mould perimeter at L2 under various conditions of EMS at the casting speed of 1.79 m/min, and the calculated results are located within the various measurement ranges. The EMS has little influence on the mould heat flux at L2, because at this location it is mainly affected by the steel flow from the nozzle, with the exception of the obvious 4 78 Heat Transfer between Billet and Mould, and Crack Sensitiveness Area of Strand differential heat flux at the outer-arc due to the uneven infiltration of mould flux. Figure 5 indicates the stress profile along radii at 0.11 m below the meniscus, and the profile accords with previously reported results [ 131. The calculated results discussed above show that the inverse problem algorithm and the FDM @EM coupled thermo-mechanical model are valid. 5000 4000 n - 4 3 5 3 E la - A Measurement (2404 30002000- 3 I 1000 -2. 'I 5000 0 - 4500 Measurement (3.9Hz) -A-Calculating (3Hz) :4000 3500 3500 8 2000 z - 2000 c * - 1500 I500 1000 500; - - 1000 ' 60 ' ' I;O ' I i O ' 2iO Position to inner-arc(degree) ' 3AO ' 3:Jo0 (b) Along the mould circumference at L2 Figure 4. Comparison of heatJlux profles between calculatedresults and measurements. 4 79 He-BiEn, Man Yao and Da-Cheng Fang 0,000 0.003 0.006 0.009 Distance beneath surface(rn) 0.012 0.015 Figure 5. Typical temperature and stress distributions through shell thickness. Results and Discussion (0 Proflles of heatjlux and thermal resistance along the mould length Heat flux and the corresponding thermal resistance are plotted against the casting direction for various longitudinal sections in Figure 6. 3 4 2 2 2 3 - -" 16 5 i' Z0 0; -1 ; 3 62 - 4 2 .Y &!32 l 0 6 1 8.0 i 0.1 0.2 0.3 0.4 0.5 0.6 0.7 -1 Distance below meniscus (m) Figure 6. Profiles of heat flux and thermal resistance along the mould length with diflerent angles at the casting speed of 2.0 m/min. 480 Heat Transfer between Billet and Mould, and Crack Sensitiveness Area of Strand The heat flux in the region 0-60 mm below the meniscus is small, and Iess than 1000 kWlm2. This is followed by an increase in heat flux until it reaches the peak value of 3000-4500 kW/m2 in the region of 80-130 mm below the meniscus, where a “high heat flux region” forms in which the heat flux varies over a wide range of 1500-4500 kW/m2 under various process conditions. The mould heat flux changes to become stable along the mould height as the billet moves to the middle/lower parts of mould, and is maintained at approximately 1000 kW/m2. It is interesting to note that in the middle of the mould, about 280-330 mm below the meniscus, the local peak value o f mould heat flux appears. At the mould outlet, the mould heat flux increases gradually for some longitudinal sections. It is suggested that thermal resistance between strand and mould may reflect the distributions of slag film thickness and gap size between the mould and the strand, further determining the heat transfer, i.e. low value of thermal resistance corresponding to high heat flux. (ii) Distributions of slag firm Figures 7(a) and (b) show the distributions of liquid and solid slag films thicknesses calculated in the upper mould before the disappearance of the liquid slag film. In the region from meniscus to 50 mm down the mould length, the liquid slag film is more than 0.056 mm thick, while the solid film is almost uniform around the perimeter and almost constant at 0.26 mm, which can explain the low and uniform heat flux near the meniscus. In the area of 50-100 mm below the meniscus, the solid slag film is significantly thinner, which leads to increasing heat flux along the casting direction. The much thinner liquid slag film cannot provide sufficient lubrication, which may result in many surface defects of the round billet. The subsequent increases of liquid and solid slag films thickness until 200mm below the meniscus correspond to the sharp decline of heat flux as shown in Figure 6. The maximum thickness of 2.1 mm solid film corresponds to the lowest heat flux, while the maximum 0.2 mm thickness of liquid film provides good lubrication. From 200 mm below meniscus to the middle of mould, both liquid and solid slag films decrease, until at approximately 300 mm below the meniscus all the liquid slag solidifies to become a solid slag layer. Figure 7(c) shows that the distribution of the thermal resistance between the strand and mould is similar to that of the solid slag film. Due to the larger thermal resistance of solid slag film compared to the liquid slag film, the former significantly determines the mould heat transfer in the upper mould. Hence predicting the thickness of solid slag can help to evaluate the heat transfer in the upper mould. (iii) Relationship between the heatflux and shell thickness Figure 8 compares the shell thickness with heat flux at four different mould heights below the meniscus. It shows that the mould heat flux and shell thickness vary around the perimeter in the steady casting processes, characterizing the similar distributions around the perimeter at any transverse section. The non-uniformity of the mould heat flux can significantly reflect that of the shell thickness. It is can be concluded that monitoring the heat flux can provide useful information regarding shell thickness and uniformity. 481 He-Bi Yin, Man Yao and Da-Cheng Fang 60 120 180 240 Position to inner-atc (degree) 300 360 _. position to inner-arc(degree) Figure 7. Calculated profiles at the casting speed of 2.3 m/min for: (a) liquid slag film; (b) solid slag film; and (c) thermal resistance. 482 Heat Transfer between Billet and Mould, and Crack Sensitiveness Area of Strand Position to Inner-arc (degree) Figure 8. Comparisons of heat flux with shell thicknessfor difSerent casting times at the casting speed of 2.3 m/min. (iv) Safe0 index distributions of strand A safety index of strand quality, M,, is proposed as: M , = q / u n- 1 ...(7) where oTis the temperature-dependent critical yield strength, and the values for various temperatures are given in [ 151; and C T ~ is the calculated Von-Mises stress of the strand. The profiles of M , ~in the shell surface down the mould length are given in Figure 9(a) for various casting speeds. Figure 9(a) shows that the safety index is low in the vicinity of meniscus, followed by an abrupt increase until it reaches its maximum value at approximately 250 mm below the mould top, and towards the bottom of the mould it stays comparatively high. Hence, the surface cracks are prone to occur near the meniscus. Figure 9(b) shows the variation of safety index averaged around the perimeter along the distance from the surface of the round strand at 483 He-Bi Yin, Man Yao and Da-Cheng Fang various distances below the mould top. At the initial stage of casting, about 133 mm below the mould top, the safety index is low at the surface region. As the solidification proceeds down to 250 mm below the mould top, the safety index at the surface region becomes larger. However, a local minimum safety index appears just below the surface at 647 mm mould height and the mould exit. These results indicate that the surface defects may form at the initial stage of casting, or enlarge from beneath the surface at the lower part. It can also be deduced that the safety indices are comparatively low in the mushy zone at any distance from the mould top, and the internal cracks tend to form at the solidieing front. 7L 1 65 x 4- P.- . a0 3 -3210 1 . 100 1 200 . " ~ . I I I . " ' 300 400 500 600 700 Distance from mould top (m) 800 (a) ................... ............................. I -1 0 I 1 " ' " 1 1 1 1 ' 2 3 4 5 6 7 8 9 1 0 1 1 1 2 Distance beneath surface(m) (b) Figure 9. The calculated distributions of safety index o j (a) the shell surface averaged along mould circumferencefor direrent casting speeds; (b) at various distances from mould top at the casting speed of 2.3 d m i n . 484 Heat Transfer between Billet and Mould, and Crack Sensitiveness Area of Strand Conclusions Based on an inverse problem method, a FDM/FEM coupled thermo-mechanical model was developed in order to calculate the thermal and mechanical behaviors that reflect the industrial continuous casting process of round billet. The conclusions are as follows. 1. The mould heat flux and the shell thickness vary around the perimeter, and the profiles of mould heat flux can reflect that of the shell thickness to some extent. 2. Non-uniform distributions of liquid and solid slag films thicknesses are calculated, and the solid slag film determines the mould heat transfer to a significant extent in the upper mould. 3. By means of the safety index predicting the strand quality, the surface cracks are prone to originate near the meniscus or propagate from beneath the billet surface at the lower part of mould, and the internal cracks may initiate in the mushy zone. Acknowledgements The authors gratefully acknowledge the financial support of NSFC-Steel & Iron Union Research Fund and Science & Technology Research Project of MOE (Ministry of Education) of China. The authors would like to thank Dr. Liu Xiao and Dr. Yu Yan for their dedication and efforts in conducting the plant trials in Baosteel Co. References 1. Mahapatm, R.B., Brimacombe, J.K. and Samarasekera, I.V. 1991. Mold behavior and its influence on quality in the continuous casting of steel slabs: Partl. Industrial trials, mold temperature measurements, and mathematical modeling. Metall. Trans. E,228, 861-874. 2. Samarasekera, I.V. and Brimacombe, J.K. 1982. Thermal and mechanical behaviour of continuous casting billet moulds. 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