DEVELPMENT OF OPTIMIZATION METHOD FOR REHEATING FURNACE OPERATION

Transcription

1 The Pennsylvania State University The Graduate School College of Engineering DEVELPMENT OF OPTIMIZATION METHOD FOR REHEATING FURNACE OPERATION A Thesis in Industrial Engineering by Masahito Kominami 2015 Masahito Kominami Submitted in Partial Fulfillment of the Requirements for the Degree of Master of Science August 2015

2 The thesis of Masahito Kominami was reviewed and approved* by the following: Robert C. Voigt Professor of Industrial and Manufacturing Engineering Thesis Adviser Enrique del Castillo Distinguished Professor of Industrial Engineering and Professor of Statistics Harriet B. Nembhard Professor and Interim Department Head of Industrial and Manufacturing Engineering *Signatures are on file in the Graduate School. ii

3 ABSTRACT The cost of operating reheating furnaces, used for heating mainly billets or blooms in steel rolling mills is quite large. Therefore, reduction of reheating costs is one of the major challenges in rolling mills. The reheating furnaces are usually controlled manually by operators who must respond to changes in downstream rolling conditions. Their reheating furnace control is not consistent and has been observed to depend on operator characteristics, experiences or skills. In many cases, steel billet lots are small, requiring various types of billets/blooms with different specifications to be heated in a furnace at the same time. This means that it is hard to find the optimal heating conditions due to changes in product mix. Additionally, once operational troubles happen at downstream rolling operations, unexpected stoppages are caused. The operators of furnaces are then required to adjust reheating furnace temperatures so that billet/bloom overheating does not occur. It is also difficult to re-establish steady-state reheating condition after the stoppages, because the bulk temperature of the billets/blooms, can be quite different than the observed billet/bloom surface temperature. Therefore, the operators have to rely on their experience when making furnace adjustment during and after stoppages. In this research, a billet simulation model for a walking hearth type reheating furnace was created and an optimization method for economical operation is proposed. The simulation model employs a three dimensional (3-D) difference method and a dynamic programming methodology developed in Matlab. Also, the thermal radiation view factor from bricks inside furnaces to billets/blooms was calculated dynamically. The hearth temperature was approximated using the simulated bottom face temperature of billets. In the optimization method, the extraction temperatures of billets are predicted for current operating conditions. Based on the result, the furnace temperature in each zone of the furnace is controlled. The major feature of this control strategy is having two policies. One is targeting the zone and the time period where billets temperatures can be controlled effectively in changing furnace temperature set points, considering heating and cooling delay and updating the feasible region dynamically. The other is prioritizing the zones for iii

4 increasing furnace temperature. It was first zone 3, then zone 2, then zone1 and finally zone 4, considering the differences in heat transmission efficiency. The final goal of this thesis is to develop an optimization method that can find an optimal solution for furnace temperature control within 10 [min]. This goal was achieved by developing a 2-D billet temperature simulation model, selecting appropriate time increments and mesh size, setting amplifier and lower limiter for temperature increments in optimization, and selective billet tracking for optimization for billet temperature increments. iv

5 TABLE OF CONTENTS LIST OF TABLES... ix LIST OF FIGURES... x ACKNOWLEDGEMENT... xiii Chapter 1. INTRODUCTION Background Reheating during steel rolling Methods to reduce the fuel cost Primary causes of non-optimal furnace operation Prior work on reheating furnace control strategies Objective of this research... 5 Chapter 2. MODELING REHEATING FURNACE Line and reheating furnace performances Mill layout Reheating furnace Billets/blooms movement through the reheating furnace Cycle time of walking hearth furnaces Positions of billets/blooms Heat balance inside the furnace Definition of heat transmission Temperature differences between billets/blooms The unit length of the heating time period is defined as follows (2.7) Estimation of billet/bloom temperature Thermal radiation between billet/bloom and furnace walls Thermal radiation Emissivity View-factor Heat transfer between billets/blooms, the furnace atmosphere and the hearths v

6 Heat transfer Heat transfer from gas to billets/blooms Heat transfer coefficient Thermal conduction Thermal conduction Thermal Properties of Materials Specific heat Emissivity/Absorption rate Thermal conductivity Furnace Modeling Mesh construction Heat balance modeling in each component View factors from furnace walls, hearths and ceiling to a mesh Heat transmission between billets/blooms and the hearths Local temperature of the hearths Interaction between billets/blooms Chapter 3. SIMULATION OF THE MODEL Billet/Bloom initial orders and their parameters Operational conditions Model of thermal property of material Computer specification for simulation Performance of the simulation model Trend of simulated temperature Difference of simulated sectional temperature Heat transmission in billet longitudinal direction Selection of appropriate mesh size Relationship between mesh size and simulated temperature Mesh size and time increments Mesh, time increments and computation time vi

7 3.4. Effect of thermal conductivity on center temperature thermal conductivity effects Impact of thermal conductivity on billet temperature Parameters selection for optimization Estimating extraction temperature of billets/blooms Selection of model and parameters for reheating furnace control Chapter 4. OPTIMIZATION OF FURNACE OPERATION Optimization Problem Objective function Decision variables Constraints Optimization method Outline of the optimization method Determining the initial solution Unit increment of furnace temperature Determination of the schedule matrix and the upper limit of temperature change Effective zone and time period targeting for estimating billet temperature changes Classified searching for efficient temperature changes Updating the feasible region Decrease phase Final treatment for the optimal control solution Initial performance check Shortening computation time Amplifier and lower limiter for furnace temperature changes Selective billet tracking Effects of selective tracking, amplifying and lower limiter vii

8 4.4. Overall Control Performance Fundamental example Effects of initial furnace temperature Effects of inserting billets with higher goal temperatures Initial control action when unexpected stoppage occur Adjustment of furnace temperature Chapter 5. CONCLUSION Conclusion summary Insight for better furnace structure based on simulation results Limitation of this research and further research recommendations APPENDIX A. Dimensions of model furnace APPENDIX B. General calculation of view-factor APPENDIX C. Heat transmission calculation APPENDIX D. View-factor calculation of perpendicular plates APPENDIX E. View-factor calculation from small plate to parallel plate with off-set BIBLIOGRAPHY viii

9 LIST OF TABLES Table 3-1. Coefficient and values used for simulation Table 3-2. Specification of computer and used software for simulation Table 3-4. Simulation properties and operational condition for the simulation Table 3-5. Operational conditions for simulation Table 3-6. Model convergence (C) and divergence (D) for different mesh sizes and modeling time increments Table 4-1. Heating rate and cooling rate of furnace Table 4-2. Computational conditions for optimization Table 4-3. Computational condition for optimization Table 4-4. Different initial furnace temperatures ix

10 LIST OF FIGURES Figure 2-1. Layout of a wire rod mill Figure 2-2. Cyclic motion of a walking hearth reheating furnace Figure 2-3. Structure of a typical reheating furnace Figure 2-4. Furnace temperature trend and billet/bloom furnace temperature experience Figure 2-6. Mesh configuration of a billet/bloom Figure 2-7. Heat transfer into each billet component Figure 2-8. Geometry of furnace wall view factors Figure 2-9. Effective area for thermal radiation view factors Figure Radiation from components with different temperature Figure Temperature difference assumption at each holding time Figure Temperature increase estimation at extraction by radiation from neighboring billets/blooms Figure 3-1. Positions of highlighted portions for analysis Figure 3-2. Simulated temperature trend at the billet front end (z=1) Figure 3-3. Simulated temperature trend at the middle in the billet length Figure 3-4. Simulated temperatures of each portion in the middle section at extraction Figure 3-5. Simulated temperature trend in the middle section Figure 3-6. Difference in simulated billet component temperature at extraction along the length of the billet Figure 3-7. Simulated component temperature difference at extraction in the billet longitudinal direction Figure 3-8. Total transmitted heat until extraction Figure 3-9. Total transmitted heat in the longitudinal direction until extraction Figure Rate of transmitted heat in the z direction to total transmitted heat until extraction Figure Relationship between simulated temperature and unit mesh size Figure Comparison of simulated billet temperatures in different billet positions of a function of simulation mesh size x

11 Figure Simulated temperature difference of each component for different modeling time increments Figure Computation time for various simulation conditions Figure Computation time for various time increments up to 4920 [sec] (=82 [min]) Figure Relationship between temperature and thermal conductivity for various steels 58 Figure Temperature differences for steel with various thermal conductivities Figure Comparison in rolling load between a billet with satisfactory center temperature and a billet with unsatisfactory center temperature Figure 4-1. Comparison of the impact of an increase or a decrease in furnace temperature on billet temperature changes in the various reheating furnace zones Figure 4-2. Constraint example illustration Figure 4-3. Feasible region after consolidating constraints Figure 4-4. Upper limits for descretized variables Figure 4-5. Main optimization steps Figure 4-6. Relationship between an increase of furnace temperature and the resultant increase in billet center temperature Figure 4-7. Influence range of each overheating level Figure 4-8. Converting the updated heat pattern to a discrete expression Figure 4-9. Updated feasible region of furnace temperatures Figure Prolongation of heating and cooling phases Figure Updated discrete lower limits for the variables Figure Obtained heat patterns for each zone before final treatment Figure Obtained optimal heat patterns for each zone after final treatment Figure Improvement of Tex for each billet after optimization Figure dta history of each iteration Figure Relationship between dta and Tex for low furnace temperature Figure Relationship between dta and Tex for high furnace temperature Figure Comparison of computation time based on the number of tracked billets Figure Average Tex and minimum Tex for the various cases Figure Computation time comparison for various amplifiers and lower limiters Figure Average of Tex and ±1σ range for various amplifiers and lower limiters xi

12 Figure Total over-heat for 85 billets for various amplifiers and lower limiters Figure Obtained optimal heat patterns for each zone Figure Difference in heat pattern between lower limiter 0 and 10 [K] (1) Figure Difference in heat pattern between lower limiter 0 and 10 [K] (2) Figure Change of Tex before and after optimization Figure Comparison of Tex between lower limiter 0 and 10 [K] Figure Average Tex and minimum Tex for different lower limiter conditions Figure Heat pattern differences for various initial furnace temperatures (1) Figure Heat pattern differences for various initial furnace temperatures (2) Figure Average Tex and minimum Tex for various initial furnace temperatures Figure Computation time and number of iterations for various initial furnace temperatures Figure Heat pattern of billets with high goal temperatures (1) Figure Heat pattern of billets with high goal temperatures (2) Figure Computation time and number of iterations for a case with high goal temperature billets Figure Average Tex and minimum Tex of a case with high goal temperature billets Figure Change of Tex before and after optimization in a case having high goal temperature billets Figure Change of Tex before and after optimization of tracked billets for a case having high goal temperature billets Figure Change of Tex before and after optimization for a case having high goal temperature billets with shifting the tracked billets Figure Average Tex and minimum Tex for a case having high goal temperature billets with shifting the tracked billets Figure B-1. Thermal radiation from small area da1 to hemisphere Figure D-1. Positional relation of two perpendicular plates Figure E-1. Positional relation of two parallel plates Figure E-2. View factor between parallel plates with off set Figure E-3. View factor between parallel plates without off set xii

13 ACKNOWLEDGEMENT I would like to appreciate my sponsor for all the supports to my study in The Pennsylvania State University. I would like to appreciate Dr. Robert C. Voigt for his continued support throughout my project and Dr. Enrique del Castillo for his greatly helpful suggestions in my project. At last, I would like to thank my wife and my son for their patience and their supports. xiii

14 Chapter 1. INTRODUCTION 1

15 1.1. Background Reheating during steel rolling Steel rolled products, such as plates, rails, wires, bars and so on, are produced from iron charge materials that go through an iron making process, a steel making process and finally a sequence of rolling operations. Among these processes, the fuel cost of reheating furnaces for rolling processes occupies about 10% among the total cost of the steel [1]. Therefore, efficient reheating has been one of the major challenges to reduce the fuel cost. During steelmaking processes, molten steel is initially solidified by continuous casting machines. The solidified intermediate products are usually called slabs or blooms depending on their size. When manufacturing wires and rods, the blooms are sometimes rolled to billets using breakdown mills for quality reasons. These billets are then cooled down before final rolling, because they need to wait until their rolling schedule or prepare for a refining process before rolling. In preparation for a refining process, the temperature of the billets must be cold enough to be inspected by an ultrasonic tester. It usually should be under 373 [K] to avoid boiling the water used for ultrasonic testing. Hence, billets are reheated in a reheating furnace before the start of final rolling Methods to reduce the fuel cost To reduce the fuel cost in reheating furnaces, many strategies have been considered: reinforcing the insulation of furnaces, optimizing air ratios and pressure, improving the efficiency of recuperators, establishing economical heat patterns of products, and optimizing the operation of furnaces [2]. However, optimizing the overall reheating operation is still a difficult production issue, because heating conditions change in various ways in real time. Billets/blooms with different reheating specifications are sometimes heated in a furnace at the same time and the specifications of billets/blooms in the furnace change as new billets/blooms are charged. Also, in practice, unexpected stoppages due to rolling mill downtime occur. These cause billets/blooms reheating variability that impacts final rolling. 2

16 Primary causes of non-optimal furnace operation Once an operational trouble occurs downstream of the reheating furnace, the expected time for fixing the trouble is announced by the operators who are responsible for getting the rolling operations back on line. Based on this expected delay time, the operator of a furnace lowers the furnace temperature to minimize fuel cost and prevent billets/blooms from overheating. The extent of the temperature change from excess time in the reheating furnace is dependent on the operator s experience, personality and preference. If the temperatures of billets/blooms at extraction are not high enough, another operational trouble is caused. Therefore, most operators tend to set the temperature higher than necessary to avoid subsequent rolling issues. After extracting, operators adjust the furnace temperature based on the temperature measured by radiation thermometers equipped in a rolling line. This inevitable conservative action of operators leads to larger reheating energy costs. This is exacerbated by the fact which the operators cannot know the inside temperature of billets/blooms and predict the temperature at extraction precisely. It is difficult to estimate the bulk temperature of all billets/blooms in regular operation, and is even more difficult to estimate during non-steady state conditions, though it can be measured using thermocouples by experiments [3], [4] Prior work on reheating furnace control strategies To overcome this difficulty in knowing the billet/bloom bulk temperature during reheating, simulation models to estimate the bulk temperature of billets or slabs have been suggested so far [3], [5], [6], [7], [8]. However, many of these models usually deal with steady state furnace conditions. In practice, it is necessary to build a dynamic simulation model which can respond to real time furnace condition changes as Watanabe suggests [9]. Modeling real furnace behavior is complex. The thermal properties of steel, such as emissivity, thermal conductivity, heat transfer coefficient and specific heat, are very temperature dependent. For walking beam or hearth type of reheating furnaces, billets/blooms change their position inside the furnaces. These geometric changes affect the thermal condition, especially the thermal radiation view factors. Researchers have also proposed optimization methods for furnace operation. Yoshitani et al. and Steinboeck et al. 3

17 proposed methods in which the furnace temperature is controlled in such a way the products temperature follow their ideal trajectories [10], [11], [12]. However, in practice, product temperature does not need to follow an ideal trajectory and may in fact undergo many acceptable trajectories. This makes the heating pattern more flexible and fuel cost becomes lower as a result. Also, Yang and Lu proposed an optimization model for slabs using dynamic programming [13]. However, it gives only stationary optimal set points of each zone. Therefore, optimization methods which can respond to dynamical condition changes and minimize the fuel cost without using trajectories are to be developed for further energy savings in real furnace operation for billets. 4

18 1.2. Objective of this research In this research, there are two main objectives. The first objective is to develop a simulation model of billet temperature considering the real time changes in thermal conditions, including their thermal properties and thermal radiation view factors. The second goal is to develop a practical furnace control optimization method that responds to real time non-steady state condition changes in the operation of a reheating furnace in rolling mills without using trajectories. By applying these simulation model and control methods to the real operation of reheating furnaces, reheating fuel costs can be minimized and the loss caused by operators differences and conservative actions can also be minimized. 5

19 Chapter 2. MODELING REHEATING FURNACE 6

20 2.1. Line and reheating furnace performances Mill layout Reheating furnace Rolling mills Crop shear Finishing mill Laying head Reforming tub Cooling conveyor Figure 2-1. Layout of a wire rod mill. In wire rod mills, billets are usually reheated up to about 1273 [K]. Those billets are subsequently rolled by multiple rolling mills. Since the front end and the tail end of the billets are unstable in quality, they are cut off by an on-line crop shear. After passing through the final mill, the wire is formed into rings by a laying head. Then, it is fed to a reforming tub through a cooling conveyor and those rings are reformed into a coil. The cooling rate can be controlled at various rates on the conveyor to obtain the required mechanical property. The chosen rolling speed is determined by the rate limiting performance among the rolling machines and operational conditions. Also, the time interval between billets is decided by the rate limiting performance among all the machines in the line and operational condition as well. For example, if the cooling conveyor cannot feed rings quickly, and the next wire comes without enough interval, those wires would collide each other. To avoid such conflicts, a long enough interval between billets must be chosen. If intervals are short and the holding time of billets in the furnace becomes too short, their bulk temperature would not be high enough for rolling. In this case, a stoppage is scheduled to further heat the billets before extracting them from the reheating furnace for avoiding downstream troubles. 7

21 Reheating furnace In rolling mills, two types of reheating furnaces are mostly used -- walking-hearth type and walking-beam type furnaces (more common). Figure 2-2 shows the motion of a walking-hearth type reheating furnace. The walking hearths lift up all of the billets inside the furnace at the same time and move them forward. Then, they are dropped down to the lower limit position. At this point, all the billets are supported by the stationary hearths. The walking hearths then move backward and return to the original position. Walking-beam type furnaces employ the same mechanism for feeding billets, but billets are supported by beams instead of hearths. In this research, a walking-hearth type furnace was considered, because of the geometric complexity. Reheating furnaces usually have multiple zones, preheating zones, a heating zone and a soaking zone. The soaking zone is to homogenize the temperature from the surface to the center of a billet. The temperature set point in the soaking zone is usually lower than that of the heating zone. Furnace zones are segmented by dividing walls. A typical reheating furnace structure is shown in figure 2-3. Because of the dividing walls, the furnace temperature can be controlled independently for each zone. The billet temperature is dependent on the furnace temperature of each zone and the billet holding time in the furnace. The holding time is affected by many factors, including rolling speed, furnace performance in cyclic motion, regular intervals between billets, expected stoppages and unexpected stoppages. Walking hearth Billet/Bloom Stationary hearth 0.Original position 1.Lift up 2.Move forward 3.Lift down 4.Move backward 5.Return to original position Figure 2-2. Cyclic motion of a walking hearth reheating furnace. 8

22 Billet Ceiling Zone 1 Zone 2 Zone 3 Zone 4 Feeding rollers Dividing wall Side wall Hearth Figure 2-3. Structure of a typical reheating furnace Billets/blooms movement through the reheating furnace Billets/blooms are fed through the furnace by the walking hearths. The walking hearths move cyclically with constant stroke. Therefore, once billets/blooms are loaded in their initial position, they are carried through the same portion of the hearths as all of the other billets/blooms. The hearths can be differentiated into two portions, one is where billets/blooms are loaded regularly at each furnace position and the other is where loading positions are empty. In most furnaces, the distance between billets/blooms is constant. It is controlled by pushers or the stroke of the walking hearths. However, in practice, there are cases when billets/blooms are not inserted into the furnace continuously. For example, when the operators of a furnace are expecting stoppages, such as changing the modes of their lines, replacing devices and so on, the operators leave open spaces between certain billets/blooms corresponding to the estimated stoppage timing to minimize furnace holding time variations for billets. 9

23 Cycle time of walking hearth furnaces The furnace cycle time is defined as (2.1). The cycles of billets/blooms inside a furnace are determined by the extracting conditions. t ct = t cw + t cs (2.1) where t cw = { t r + t rv = W + t v rv when a billet/bloom is rolled w t v when there is no billet/bloom to be rolled W: Weight of extracted billet/bloom [tonf] v w : Rolling weight speed [tonf (sec) 1 ] v w = ρv f A f v f : Rolling speed at finishing rolling stand [m (sec) 1 ] A f : Sectional area at finishing rolling stand [m 2 ] ρ: Weight density of extracted billet/bloom [tonf m 3 ] t rv : interval time between cycles when billet/bloom is extracted [sec] t v = t cu + t cf + t cd + t cb + t sv t cu : time for lifting up the hearth [sec] t cf : time for moving forward the hearth [sec] t cd : time for lifing down the hearth [sec] t cb : time for moving backward the hearth [sec] t sv : Additional interval time between cycles when no billet/bloom is extracted [sec] trv and tsv are adjusted by the operators and the specified operational conditions. 10

24 Positions of billets/blooms An example of furnace temperature trends in a typical furnace zone and the furnace temperature that a billet/bloom experiences are shown in figure 2-4. The furnace temperature of each zone always changes and they are mostly different, and are independent unless the temperature gap is quite large. Since the furnace temperature cycle that a billet/bloom experiences depends on the time and the zone where it stays, it is important to track the positions of all the billets/blooms in a furnace when modeling billet/bloom temperature. The initial positions of billet/bloom i, Ii, before charging can be expressed in (2.2). I i = i ( K) G i i = 1,2,, n (2.2) where i: billet/bloom number in order of charge K: Stroke distance [mm] Gi: Initial additional distance from billet/bloom i to i+1 [mm] The stroke distance is determined by the furnace specification and Gi is decided by the operators based on future operations. Their positions after the ncth cycle are obtained using (2.3). P i,nc = I i + n c K (2.3) By finding the number of cycles at time t, the positions of all of the billets/blooms in the reheating furnace are obtained. In order to obtain the number of cycles at time t, expected intervals and stoppages must be known. From a schedule table of stoppages, the times of stoppages tb,nc can be estimated just before the ncth cycle is carried out. Using (2.1), the cumulative time when the ncth cycle is completed can be calculated by (2.4). 11

25 n c t cm,nc = (t ct,k + t b,k ) (2.4) k=1 where tct,nc; Cycle time when the ncth cycle occurs Hence, nc is the completed number of strokes at time t (2.5). t t cm,nc min(t t cm,nc ) (2.5) The position Pi,t of a billet/bloom i after t time periods is estimated by (2.6). P i,t = I i + n c K (2.6) Provided that nc satisfies (2.5). An example of the relationship between time and the number of cycles is shown in table 2-1. If t=20 [min], the number of strokes nc is 0. If t=33 [min], the number of strokes is 2. In a later chapter, another type of holding time will be discussed. Let the time period described in this section be called the computational time period, tcom. 12

26 Zone 1 Furnace temperature Zone 2 Furnace temperature Zone 3 Furnace temperature Zone 4 Furnace temperature Time Billet/bloom experiencing Furnace temperature Time Figure 2-4. Furnace temperature trend and billet/bloom furnace temperature experience. 13

27 Table 2-1. Example of the relationship between the number of cycles and furnace travel distance. Cycle Sequence Number k If extraction occurs 1 o/w 0 tct,k [min] Stoppages tb,k [min] Cumulative Time tcm,k [min] Cumulative Moved Distance Dm=(k-1) SK SK SK SK nc (nc-3) SK nc (nc-2) SK nc (nc-1) SK [m] 14

28 2.2. Heat balance inside the furnace Definition of heat transmission Heat transmission to billets from the furnace has three different types, heat transfer thermal radiation and thermal conduction. Heat transfer is driven by temperature differences. Heat will be transferred only when heat is transmitted from a substance with higher temperature to another substance with a lower temperature. This means that heat transfer occurs between different substances through contact. Thermal radiation is a phenomenon in which heat is transmitted by electromagnetic radiation emitted from the surface of a substance and another substance that absorbs the radiation and converts it to its internal energy. In thermal radiation, heat transmission occurs between different substances without contact. Thermal conduction is the phenomenon by which heat is transmitted within a substance having a temperature gradient. These terminologies are sometimes used in different ways. To avoid confusion, these are used as defined above for further consideration Temperature differences between billets/blooms In figure 2-1, the thermal model used for simulation is illustrated. Billets/blooms are heated by thermal radiation from the ceiling, the hearth and the side-walls. In this model, it was assumed that the combustion gases are non-luminous, so that the thermal radiation from the combustion gases can be ignored. Since each billet inside the furnace is at a different temperature, there is thermal radiation from billets with higher temperature to billets with lower temperature. Billets at downstream locations usually have higher temperature, because of their longer furnace holding time. The other type of heat transmission to billets/blooms is heat transfer from the furnace atmosphere and the hearths. The heat transfer from the hearths is transmitted through the direct contact between the hearths and the bottom face of the billets. The local temperature of a billet is different throughout the length and depth of the billet. Heat is transfered between portions with different temperatures through thermal conduction. Specifically, the center of each billet (except the front end and the tail ends) is heated only by thermal conduction from the surface of the billet. 15

29 T c Ceiling Feeding direction T a q rad,cb Billet/Bloom q tran,ab T w q rad,wb Side wall T bi-1 q rad,bi-1 bi T bi q rad,bi bi+1 T bi+1 T H Hearth q tran,bh <0, if T bi <T H 0, if T bi T H Ta: Temperature of the atmosphere inside the furnace [K] Tbi: Temperature of the billet/bloom i [K] Tc: Temperature of the ceiling [K] Tw: Temperature of the side walls [K] TH: Temperature of the hearths [K] qrad,cb: Transmitted heat by radiation from the ceiling to the billet/bloom [Wm -2 ] qtran,ab: Transmitted heat by heat transfer from the atmosphere to the billet/bloom [Wm -2 ] qrad,bi-1 bi: Transmitted heat by radiation from billet/bloom i to billet/bloom i-1 [Wm -2 ] qrad,bi bi+1: Transmitted heat by radiation from billet i+1 to billet/bloom i [Wm -2 ] qtran,bh: Transmitted heat by heat transfer from the billet/bloom to the hearth [Wm -2 ] Figure 2-5. Heat balance inside the furnace. The unit length of the heating time period is defined as follows (2.7). t p = t L s [sec] (2.7) where 16

30 t L : Furnace holding time of the last billet/bloom in the considered range s: Number of time periods for all zones decided by users Estimation of billet/bloom temperature The specific heat of the billet/bloom must be known to estimate the future temperature of billets/blooms after a certain time period in the furnace. However, the specific heat of steel depends on its temperature [14]. Therefore, the temperature after a certain time period can be estimated by taking the integral of the following equation (2.8). where T estimated Q total = ρv C p (T) dt [W] (2. 8) T current ρ: Mass density [kg/m 3 ] V: Volume [m 3 ] T current : Current temperature [K] T estimated : Estimated temperature after a certain time period[k] C p : Specific heat at constant pressure [J kg 1 K 1 ] = f(t) C v : Specific heat at constant volume If the time period is short, C p can be approximated as a function of T current. In this case, the equation (2.8) can be rewritten as (2.9). Q total = ρv f(t current ) (T estimated T current ) T estimated = T current + Q total (2. 9) ρv f(t current ) 17

31 2.3. Thermal radiation between billet/bloom and furnace walls Thermal radiation Considering the thermal radiation from the furnace walls to the billets/blooms, the transmitted heat is computed using the Stefan-Boltzmann law (2.10) [15]. Q rad = A b σϕ wb (T w 4 T b 4 ) (2. 10) where Q rad: Total heat from the furnace bricks to the billets/blooms by radiation [W] σ: Stefan-Boltzmann constant [Wm 2 K 4 ] A b : Surface area of the billet/bloom [m 2 ] ϕ wb (F wb, F bw, ε w, ε b ): Radiation coefficient F wb : View factor from the furnace bricks to the billets F bw : View factor from the billets to fthe urnace bricks ε w : Emissivity of the furnace bricks ε b : Radiation absorption rate of the billets Emissivity Emissivity indicates how much of thermal energy the surface of a material can emit or absorb. It ranges from 0 to 1.0. If it is 0, it implies that the material is a black body. Also, it is known that polished metal has emissivity values close to 0. The emissivity of oxide steel, appropriate for steel heated in air furnaces, is approximately 0.9 [16]. 18

32 View-factor The view-factor indicates how much radiation can reach geometrically from one surface to another surface. It is defined by (2.11) [15], [17]. See appendix A for the details on the use of view-factors. F 12 = 1 cos φ 1 cos φ 2 A 1 πr 2 da 1 da 2 (2. 11) A 1 A 2 19

33 2.4. Heat transfer between billets/blooms, the furnace atmosphere and the hearths Heat transfer The amount of local heat transfer between substance 1 and substance 2 at two different temperatures can be computed using the following equation (2.12) [15]. q L = h L (T 1 T 2 ) ( 2. 12) where h L : Local heat transfer coefficient [Wm 2 K 1 ] T 1 : Temperature of substance 1 [K] T 2 : Temperature of substance 2 [K] When T1 and T2 do not depend on location, (that is, temperatures are uniform) the total transferred heat through area A is calculated as (2.13). Q = q L A da = (T 1 T 2 ) h L da (2. 13) A When A is constant, Q = h L A(T 1 T 2 ) (2. 14) 20

34 Heat transfer from gas to billets/blooms Since some types of gas emit radiation when they are combusted, billets/blooms are heated by heat transfer and thermal radiation from the combusted gas in the furnace at the same time [15], [18]. However, gas is not distinct in shape. It is difficult to calculate view factors between billets/blooms surface and the furnace gas. Accordingly, the total heat from gas to a billet/bloom was calculated by (2.14), defining μgb as the rate which heat is transmitted to one billet/bloom by thermal radiation. q gb = h gb (T g T b ) + μ gb σ(t g 4 T b 4 ) (2. 14) where h gb : Heat transfer coefficient from gas to billets/blooms T g : Gas temperature T b : Billet/bloom temperature σ: Stefan Boltzman coefficient In this research, it was assumed for simplicity that the combusted furnace gas generates a non-luminous flame, so that it does not simultaneously emit thermal radiation and qgb can be expressed only by its heat transfer term Heat transfer coefficient The total heat transfer coefficient for heat transmission between two substances is mainly affected by four factors, the smoothness of their surfaces, the type of the materials, the extent of pressure on them, and the type of the matter between two substances [19]. Therefore, to obtain accurate heat transfer coefficients for a real furnace, may require experiments corresponding to each heat transfer situation as Fujibayashi et al. showed in steel plate cooling [20]. 21

35 2.5. Thermal conduction Thermal conduction Within a billet/bloom, thermal conduction occurs whenever there is an internal temperature gradient. Conduction follows Fourier s indicated below (2.15) [15]. J = λgradt = λ ΔT ( 2. 15) d where J: Transmitted heat from one portion to another portion within the billet /bloom [Wm 2 ] λ: Thermal conductivity [WK 1 m 1 ] d: Distance between the centers of the portions [m] ΔT: Temperature difference between the portions [K] dv: Volume of the portion [m 3 ] Using equation (2.15), transmitted heat from adjacent portions of a billet by thermal conduction is calculated as (2.16). Q cond = λa ΔT (2. 16) d where A: Area contacting to the adjacent portion [m 2 ] 22

36 2.6. Thermal Properties of Materials To estimate the future temperature of billets/blooms, various coefficients, for instance, specific heat, emissivity and conductivity, of each billet/bloom material must be known. However, these coefficients depend on the temperature of billets/blooms. Hence, the temperature dependence of each coefficient must be estimated Specific heat The specific heat of a steel depends on the alloys present and its temperature [14], [21]. Around the A1 temperature where A1 transformation occurs, the specific heat of steels changes dramatically. Hence, the specific heat of billet/bloom i, Ci, is expressed as a function of the temperature of the steel in (2.17) and in (2.18) separately. C i Tbi,(x,y,z),t T A1 = f i Tbi,(x,y,z),t T A1 (T bi,(x,y,z),t ) (2. 17) C i Tbi,(x,y,z),t <T A1 = f i Tbi,(x,y,z),t <T A1 (T bi,(x,y,z),t ) (2. 18) where T bi,(x,y,z),t : Temperature at (x, y, z) portion of billet/bloom i during time period t 23

37 Emissivity/Absorption rate To know how much heat is transmitted by radiation, the radiation coefficient must be estimated in (2.10). It is a function of view factor and emissivity [17], [18]. Additionally, radiosity must be considered to calculate the radiation coefficient; because emission, absorption or permeation and reflection occur in radiation. For simple geometries, such as plates in parallel, radiation coefficients are easily calculated. However, it is hard to calculate the values in complex systems such as furnaces. In this research, the effect of radiosity was included in the emissivity for simplicity. The simplified radiation coefficient is shown in (2.19). ϕ wb = ε w ε b F wb (2.19) where ε w : Emissivity of bricks including radiosity ε b : Emissivity of billets including radiosity Emissivity and absorption are usually handled together. Substances have their own values. They are affected by the surface conditions, such as smoothness, shape and composition. These properties should be found for applying to simulation models in advance. In this research, it was assumed that the emissivity of bricks and billets/blooms were constant Thermal conductivity Thermal conductivity is also affected by temperature [14], [22]. Therefore, it is expressed as a function of temperature as (2.20). λ i = h i (T bi,(x,y,z),t ) (2. 20) 24

38 2.7. Furnace Modeling Mesh construction To calculate the local temperatures of billets/blooms, they were meshed as shown in figure 2-6. The mesh size was decided by the height, the width and the length of the unit mesh. The billet corners have radius and they are considered when the area and the volume of each mesh are calculated. The bricks of the furnace walls, hearths and ceiling are not meshed by assuming that their temperature is uniform, because of their high thermal insulation performance (l,m,n) n C D E m B I F A H G (1,1,1) l Figure 2-6. Mesh configuration of a billet/bloom. 25

39 Heat balance modeling in each component. The surface of billets/blooms receives heat through thermal radiation and heat transfer, while the inside of billets/blooms is heated by thermal conduction. Figure 2-7 illustrates the model of the billet/bloom heat balance. The component subscripts correspond to those in figure 2-6. Also, the detailed heat transfer calculations are shown in Appendix C. q cond,(1,m,2) q cond,(x,m,2) q cond,(l,m,2) q rad,hb +q rad,cb +q rad,wb +q rad,bi-1 bi +q tran,gb q rad,cb +q rad,wb +q tran,gb q cond,(x-1,m,1) q cond,(l-1,m,1) q cond,(2,m,1) q cond,(x+1,m,1) q rad,hb +q rad,cb +q rad,wb +q rad,bi bi+1 +q tran,gb q cond,(1,m-1,1) q cond,(x,m-1,1) q cond,(l,m-1,1) q rad,hb +q rad,cb q rad,hb +q rad,cb q rad,hb +q rad,cb +q rad,wb +q tran,gb +q rad,wb +q tran,gb +q rad,wb +q tran,gb Component C (1,m,1) Component D (x,m,1) Component E (l,m,1) q cond,(1,y,2) q cond,(x,y,2) q cond,(l,y,2) q cond,(1,y+1,1) q cond,(x,y+1,1) q cond,(l,y+1,1) q rad,hb +q rad,cb +q rad,wb +q rad,bi-1 bi +q tran,gb q cond,(x-1,y,1) q cond,(l-1,y,1) q cond,(2,y,1) q cond,(x+1,y,1) q rad,hb +q rad,cb +q rad,wb +q rad,bi bi+1 +q tran,gb q cond,(1,y-1,1) q cond,(x,y-1,1) q cond,(l,y-1,1) q rad,hb +q rad,cb q rad,hb +q rad,cb q rad,hb +q rad,cb +q rad,wb +q tran,gb +q rad,wb +q tran,gb +q rad,wb +q tran,gb Component B (1,y,1) Component I (x,y,1) Component F (l,y,1) q cond,(1,2,1) q cond,(1,1,2) q cond,(x,2,1) q cond,(x,1,2) q cond,(l,2,1) q cond,(l,1,2) q cond,(x-1,1,1) q cond,(l-1,1,1) q cond,(2,1,1) q cond,(x+1,1,1) q rad,hb +q rad,cb +q rad,wb +q rad,bi-1 bi +q tran,gb q tran,hb q rad,hb +q rad,cb +q rad,wb +q rad,bi bi+1 +q tran,gb q rad,hb +q rad,cb q rad,hb +q rad,cb q rad,hb +q rad,cb +q rad,wb +q tran,gb +q rad,wb +q tran,gb +q rad,wb +q tran,gb Component A (1,1,1) Component H (x,1,1) Component G (l,1,1) Figure 2-7. Heat transfer into each billet component. 26

40 View factors from furnace walls, hearths and ceiling to a mesh Each mesh receives thermal radiation from different regions of the walls, the hearths or the ceiling. The hatched area in figure 2-8 shows the considered furnace regions that emit thermal radiation to small areas on different faces. Based on these configurations, the view factors were calculated for every billet face. (Appendix E and F). At the ends of the billets, the thermal radiation from the hearths is transferred over beyond the dividing walls, because there are open spaces under the dividing walls. In this research, it was assumed for simplicity that the temperature at the dividing walls is uniform and the brick s temperature in a zone where a billet is located is used for the calculation as a representative temperature. In practice, the furnace temperature of each zone is different and the hearth temperature of each zone is expected to be different. The effects by this simplified furnace temperature approximation at the dividing walls are not expected to be significant. When billets are charged continuously, the distance to the adjacent billet is constant. However, if billets are not charged continuously, the distance between billets can vary. In this simulation, the distance to the adjacent billet at both downstream and upstream sides is determined by their initial positions and is always tracked. The regional range of the hearth to be considered can be easily found from the distance. However, the regional ranges of the ceiling, the dividing walls and the side walls must be calculated geometrically. Figure 2-9 indicates the geometric relationship between the ranges and the distance to the adjacent billet. When θ ϕ, the whole range of the ceiling and the dividing walls are effective as the areas which emit thermal radiation to the targeted mesh of a billet. On the other hand, when θ > ϕ, the thermal radiation region depends on the position of the targeted mesh. The effective areas are determined in (2.21) and (2.22). when θ > φ, H we = H w L w tan θ, L ce = H c tan θ (2.21) o/w, H we = H w, L ce = L w (2.22) 27

41 where H we : Effective height of the dividing wall L ce : Effective length of the ceiling H w : Height of the dividing wall L w : Distance from the billet to the dividing wall H c : Height of the ceiling 28

42 Front end and tail end Upper face Upstream side Downstream side Figure 2-8. Geometry of furnace wall view factors. 29

43 ϕ θ ϕ θ ϕ θ ϕ θ Figure 2-9. Effective area for thermal radiation view factors. 30

44 Heat transmission between billets/blooms and the hearths The front ends of the billets/blooms are aligned on the same line as shown in figure 2-3. However, the tail ends may not be aligned on the same line because the length of the billets/blooms varies. At the tail ends, the length of the billets/blooms affects the heat transmission between the billets/blooms and the hearths. If the difference of the heat transmission is considered in the model, the calculation, especially for the temperature of the hearths, becomes more complex. Also, it lengthens the calculation time. Therefore, it was assumed in this research that, for the calculation of the hearths temperature only, the length of the billets/blooms was constant by employing a representative length. Also, the furnace temperature inside the furnace can fluctuate due to various factors such as an imbalance in burner performance, differences in the billets/blooms length, etc.. Thus, the difference of the furnace temperature in axis z should be considered in the model. This can be expressed as a linear model, as follows (2.23). T a,j,z,t = T af,j,t T at,j,t W H (L u z + d F ) + T af,j,t (2. 23) where W H : Width of the furnace [m] T a,j,z,t : Furnace temperature at z in zone j during time period t [K] T af,j,t : Furnace temperature at the front side wall in zone j during time period t [K] T af,j,t : Furnace temperature at the tail side wall in zone j during time period t [K] d F : Set Distance between the front side wall and the front end of billets/blooms [m] L u : Unit length of the mesh in z axis [m] 31

45 Local temperature of the hearth The local temperature of the hearths can vary. For simplicity, the hearth temperature was assumed to be uniform and it was divided into two different temperatures. One is the main zone in which billets/blooms are loaded regularly. The temperature of this area is affected by the heat transfer from billets/blooms when they are loaded, by thermal radiation from the furnace ceiling and the side walls, and by heat transfer from combustion gas when billets/blooms are not loaded. The other temperature is in the reserved zones where billets/blooms are not placed on regularly. It is assumed that the temperature of this area follows the furnace temperature in the same zone and it reaches temperature immediately when the furnace temperature changes, because heat transmission by thermal conduction is small in the bricks which have high thermal insulation performance and the surface temperature of the bricks changes quickly.. For calculating transmitted heat between the main zones and billets/blooms, the average bottom face temperature of the billet/bloom was used. This bottom face temperature varies for every billet/bloom and also changes in real time. Therefore, the temperature of the main zones must be computed every time period. This can influence the temperature whether or not there is a billet/bloom on the main lot in zone j during time t. The transmitted heat is expressed as shown below (2.24) and (2.25). If a billet is placed on, q hb = h hb (T h T b ) (2.24) o/w, q wh = σε w ε w F wh (T 4 w T 4 h ) + h gh (T g T h ) (2.25) where h hb : Heat transfer coefficient from the hearths to the billet h gh : Heat transfer coefficient from the combustion gas to the billet T h : Hearth temperature, T b : billet/bloom temperature, T g : gas temperature, T g : furnace wall temperature 32

46 Interaction between billets/blooms In reheating furnaces multiple billets/blooms are heated at the same time. These billets/blooms can have different temperatures thoroughly and locally. Therefore, temperature interaction by thermal radiation is expected between the billets/blooms. Since they are finely meshed for the calculation of local temperature, the computation time becomes quite large when the interaction of each pair of components is considered. Thus, the effect of billet-to-billet interaction was first computed to investigate how much this interaction affects the temperature change. Figure 2-10 shows the image of thermal radiation from each portion with different temperature of a billet to a portion of another billet. Table 2-2 shows the computational condition to evaluate the interaction. Figure 2-11 shows the assumption for this evaluation in temperature difference between two billets. Figure Radiation from components with different temperature. Table 2-2. Condition for estimating the effect of temperature interaction. Sectional size [m m] Distance between billets/blooms [m] Unit size of components [m m] Specific heat [J/(kg K)] Density [kg/m 3 ] Holing time [min] Stefan Boltzman coefficient [W/(m 2 K4 )]