DEVELPMENT OF OPTIMIZATION METHOD FOR REHEATING …

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

ii

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.

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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

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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.

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TABLE OF CONTENTS

LIST OF TABLES ................................................................................................................... ix

LIST OF FIGURES .................................................................................................................. x

ACKNOWLEDGEMENT ..................................................................................................... xiii

Chapter 1. INTRODUCTION .............................................................................................. 1

1.1. Background ............................................................................................................... 2

1.1.1. Reheating during steel rolling ............................................................................. 2

1.1.2. Methods to reduce the fuel cost .......................................................................... 2

1.1.3. Primary causes of non-optimal furnace operation .............................................. 3

1.1.4. Prior work on reheating furnace control strategies ............................................. 3

1.2. Objective of this research.......................................................................................... 5

Chapter 2. MODELING REHEATING FURNACE ............................................................ 6

2.1. Line and reheating furnace performances ................................................................. 7

2.1.1. Mill layout ........................................................................................................... 7

2.1.2. Reheating furnace ............................................................................................... 8

2.1.3. Billets/blooms movement through the reheating furnace ................................... 9

2.1.4. Cycle time of walking hearth furnaces ............................................................. 10

2.1.5. Positions of billets/blooms ................................................................................ 11

2.2. Heat balance inside the furnace .............................................................................. 15

2.2.1. Definition of heat transmission ......................................................................... 15

2.2.2. Temperature differences between billets/blooms ............................................. 15

The unit length of the heating time period is defined as follows (2.7). ........................... 16

2.2.3. Estimation of billet/bloom temperature ............................................................ 17

2.3. Thermal radiation between billet/bloom and furnace walls .................................... 18

2.3.1. Thermal radiation .............................................................................................. 18

2.3.2. Emissivity ......................................................................................................... 18

2.3.3. View-factor ....................................................................................................... 19

2.4. Heat transfer between billets/blooms, the furnace atmosphere and the hearths ..... 20

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2.4.1. Heat transfer .......................................................................................................... 20

2.4.2. Heat transfer from gas to billets/blooms ............................................................... 21

2.4.3. Heat transfer coefficient .................................................................................... 21

2.5. Thermal conduction ................................................................................................ 22

2.5.1. Thermal conduction .......................................................................................... 22

2.6. Thermal Properties of Materials ............................................................................. 23

2.6.1. Specific heat ...................................................................................................... 23

2.6.2. Emissivity/Absorption rate ............................................................................... 24

2.6.3. Thermal conductivity ........................................................................................ 24

2.7. Furnace Modeling ................................................................................................... 25

2.7.1. Mesh construction ............................................................................................. 25

2.7.2. Heat balance modeling in each component. ..................................................... 26

2.7.3. View factors from furnace walls, hearths and ceiling to a mesh ...................... 27

2.7.4. Heat transmission between billets/blooms and the hearths............................... 31

2.7.5. Local temperature of the hearths....................................................................... 32

2.7.6. Interaction between billets/blooms ................................................................... 33

Chapter 3. SIMULATION OF THE MODEL.................................................................... 36

3.1. Billet/Bloom initial orders and their parameters ..................................................... 37

3.1.1. Operational conditions ...................................................................................... 37

3.1.2. Model of thermal property of material.............................................................. 37

3.1.3. Computer specification for simulation.............................................................. 38

3.2. Performance of the simulation model ..................................................................... 40

3.2.1. Trend of simulated temperature ........................................................................ 40

3.2.2. Difference of simulated sectional temperature ................................................. 45

3.2.3. Heat transmission in billet longitudinal direction ............................................. 48

3.3. Selection of appropriate mesh size ......................................................................... 51

3.3.1. Relationship between mesh size and simulated temperature ............................ 51

3.3.2. Mesh size and time increments ......................................................................... 54

3.3.3. Mesh, time increments and computation time .................................................. 56

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3.4. Effect of thermal conductivity on center temperature ............................................ 58

3.4.1. thermal conductivity effects .............................................................................. 58

3.4.2. Impact of thermal conductivity on billet temperature....................................... 59

3.5. Parameters selection for optimization ..................................................................... 61

3.5.1. Estimating extraction temperature of billets/blooms ........................................ 61

3.5.2. Selection of model and parameters for reheating furnace control .................... 64

Chapter 4. OPTIMIZATION OF FURNACE OPERATION ............................................ 65

4.1. Optimization Problem ............................................................................................. 66

4.1.1. Objective function ............................................................................................. 66

4.1.2. Decision variables ............................................................................................. 70

4.1.3. Constraints ........................................................................................................ 70

4.2. Optimization method .............................................................................................. 74

4.2.1. Outline of the optimization method .................................................................. 74

4.2.2. Determining the initial solution ........................................................................ 76

4.2.3. Unit increment of furnace temperature ............................................................. 77

4.2.4. Determination of the schedule matrix and the upper limit of temperature change

78

4.2.5. Effective zone and time period targeting for estimating billet temperature

changes 80

4.2.6. Classified searching for efficient temperature changes .................................... 81

4.2.7. Updating the feasible region ............................................................................. 86

4.2.8. Decrease phase .................................................................................................. 88

4.2.9. Final treatment for the optimal control solution ............................................... 89

4.2.10. Initial performance check ................................................................................. 90

4.3. Shortening computation time .................................................................................. 93

4.3.1. Amplifier and lower limiter for furnace temperature changes.......................... 93

4.3.2. Selective billet tracking..................................................................................... 96

4.3.3. Effects of selective tracking, amplifying and lower limiter .............................. 97

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4.4. Overall Control Performance ................................................................................ 101

4.4.1. Fundamental example ..................................................................................... 101

4.4.2. Effects of initial furnace temperature ............................................................. 106

4.4.3. Effects of inserting billets with higher goal temperatures .............................. 110

4.4.4. Initial control action when unexpected stoppage occur .................................. 116

4.4.5. Adjustment of furnace temperature ................................................................ 116

Chapter 5. CONCLUSION ............................................................................................... 117

5.1. Conclusion summary ............................................................................................ 118

5.2. Insight for better furnace structure based on simulation results ........................... 120

5.3. Limitation of this research and further research recommendations ...................... 121

APPENDIX A. Dimensions of model furnace. .................................................................... 123

APPENDIX B. General calculation of view-factor. ............................................................. 124

APPENDIX C. Heat transmission calculation. ..................................................................... 126

APPENDIX D. View-factor calculation of perpendicular plates. ........................................ 136

APPENDIX E. View-factor calculation from small plate to parallel plate with off-set. ...... 138

BIBLIOGRAPHY ................................................................................................................. 145

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LIST OF TABLES

Table 3-1. Coefficient and values used for simulation. .......................................................... 38

Table 3-2. Specification of computer and used software for simulation. ............................... 38

Table 3-4. Simulation properties and operational condition for the simulation. .................... 40

Table 3-5. Operational conditions for simulation. .................................................................. 48

Table 3-6. Model convergence (C) and divergence (D) for different mesh sizes and modeling

time increments. ...................................................................................................................... 54

Table 4-1. Heating rate and cooling rate of furnace. .............................................................. 71

Table 4-2. Computational conditions for optimization. .......................................................... 90

Table 4-3. Computational condition for optimization. ......................................................... 101

Table 4-4. Different initial furnace temperatures. ................................................................ 106

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LIST OF FIGURES

Figure 2-1. Layout of a wire rod mill. ...................................................................................... 7

Figure 2-2. Cyclic motion of a walking hearth reheating furnace. ........................................... 8

Figure 2-3. Structure of a typical reheating furnace. ................................................................ 9

Figure 2-4. Furnace temperature trend and billet/bloom furnace temperature experience. .... 13

Figure 2-6. Mesh configuration of a billet/bloom. .................................................................. 25

Figure 2-7. Heat transfer into each billet component. ............................................................ 26

Figure 2-8. Geometry of furnace wall view factors. ............................................................... 29

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

Figure 2-10. Radiation from components with different temperature. ................................... 33

Figure 2-11. Temperature difference assumption at each holding time. ................................ 34

Figure 2-12. Temperature increase estimation at extraction by radiation from neighboring

billets/blooms. ......................................................................................................................... 35

Figure 3-1. Positions of highlighted portions for analysis. ..................................................... 40

Figure 3-2. Simulated temperature trend at the billet front end (z=1). ................................... 41

Figure 3-3. Simulated temperature trend at the middle in the billet length. ........................... 42

Figure 3-4. Simulated temperatures of each portion in the middle section at extraction. ...... 43

Figure 3-5. Simulated temperature trend in the middle section. ............................................. 44

Figure 3-6. Difference in simulated billet component temperature at extraction along the

length of the billet. .................................................................................................................. 46

Figure 3-7. Simulated component temperature difference at extraction in the billet

longitudinal direction. ............................................................................................................. 47

Figure 3-8. Total transmitted heat until extraction ................................................................. 49

Figure 3-9. Total transmitted heat in the longitudinal direction until extraction. ................... 49

Figure 3-10. Rate of transmitted heat in the z direction to total transmitted heat until

extraction................................................................................................................................. 50

Figure 3-11. Relationship between simulated temperature and unit mesh size. ..................... 52

Figure 3-12. Comparison of simulated billet temperatures in different billet positions of a

function of simulation mesh size. ........................................................................................... 53

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Figure 3-13. Simulated temperature difference of each component for different modeling

time increments. ...................................................................................................................... 55

Figure 3-14. Computation time for various simulation conditions ......................................... 56

Figure 3-15. Computation time for various time increments up to 4920 [sec] (=82 [min]). .. 57

Figure 3-16. Relationship between temperature and thermal conductivity for various steels 58

Figure 3-17. Temperature differences for steel with various thermal conductivities. ............ 60

Figure 3-18. Comparison in rolling load between a billet with satisfactory center temperature

and a billet with unsatisfactory center temperature. ............................................................... 63

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. ........................................ 68

Figure 4-2. Constraint example illustration. ........................................................................... 72

Figure 4-3. Feasible region after consolidating constraints. ................................................... 73

Figure 4-4. Upper limits for descretized variables. ................................................................. 73

Figure 4-5. Main optimization steps. ...................................................................................... 75

Figure 4-6. Relationship between an increase of furnace temperature and the resultant

increase in billet center temperature. ...................................................................................... 77

Figure 4-7. Influence range of each overheating level. .......................................................... 83

Figure 4-8. Converting the updated heat pattern to a discrete expression. ............................. 85

Figure 4-9. Updated feasible region of furnace temperatures. ................................................ 86

Figure 4-10. Prolongation of heating and cooling phases....................................................... 87

Figure 4-11. Updated discrete lower limits for the variables.................................................. 87

Figure 4-12. Obtained heat patterns for each zone before final treatment. ............................. 91

Figure 4-13. Obtained optimal heat patterns for each zone after final treatment. .................. 91

Figure 4-14. Improvement of ∆Tex for each billet after optimization. .................................... 92

Figure 4-15. dTa history of each iteration. .............................................................................. 92

Figure 4-16. Relationship between dTa and ∆Tex for low furnace temperature. ..................... 94

Figure 4-17. Relationship between dTa and ∆Tex for high furnace temperature. .................... 95

Figure 4-18. Comparison of computation time based on the number of tracked billets. ........ 97

Figure 4-19. Average ∆Tex and minimum ∆Tex for the various cases. ................................... 98

Figure 4-20. Computation time comparison for various amplifiers and lower limiters. ........ 99

Figure 4-21. Average of ∆Tex and ±1σ range for various amplifiers and lower limiters. ..... 100

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Figure 4-22. Total over-heat for 85 billets for various amplifiers and lower limiters. ......... 100

Figure 4-23. Obtained optimal heat patterns for each zone. ................................................. 101

Figure 4-24. Difference in heat pattern between lower limiter 0 and 10 [K] (1). ................. 102

Figure 4-25. Difference in heat pattern between lower limiter 0 and 10 [K] (2). ................. 103

Figure 4-26. Change of ∆Tex before and after optimization. ................................................ 104

Figure 4-27. Comparison of ∆Tex between lower limiter 0 and 10 [K]. ............................... 105

Figure 4-28. Average ∆Tex and minimum ∆Tex for different lower limiter conditions. ....... 105

Figure 4-29. Heat pattern differences for various initial furnace temperatures (1). ............. 107

Figure 4-30. Heat pattern differences for various initial furnace temperatures (2). ............. 108

Figure 4-31. Average ∆Tex and minimum ∆Tex for various initial furnace temperatures. .... 109

Figure 4-32. Computation time and number of iterations for various initial furnace

temperatures. ......................................................................................................................... 109

Figure 4-33. Heat pattern of billets with high goal temperatures (1).................................... 111

Figure 4-34. Heat pattern of billets with high goal temperatures (2).................................... 112

Figure 4-35. Computation time and number of iterations for a case with high goal

temperature billets. ................................................................................................................ 113

Figure 4-36. Average ∆Tex and minimum ∆Tex of a case with high goal temperature billets.

............................................................................................................................................... 113

Figure 4-37. Change of ∆Tex before and after optimization in a case having high goal

temperature billets. ................................................................................................................ 114

Figure 4-38. Change of ∆Tex before and after optimization of tracked billets for a case having

high goal temperature billets. ................................................................................................ 114

Figure 4-39. Change of ∆Tex before and after optimization for a case having high goal

temperature billets with shifting the tracked billets. ............................................................. 115

Figure 4-40. Average ∆Tex and minimum ∆Tex for a case having high goal temperature billets

with shifting the tracked billets. ............................................................................................ 115

Figure B-1. Thermal radiation from small area dA1 to hemisphere. .................................... 124

Figure D-1. Positional relation of two perpendicular plates. ................................................ 136

Figure E-1. Positional relation of two parallel plates. .......................................................... 138

Figure E-2. View factor between parallel plates with off set. .............................................. 143

Figure E-3. View factor between parallel plates without off set. ......................................... 144

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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.

1

Chapter 1. INTRODUCTION

2

1.1. Background

1.1.1. 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.

1.1.2. 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.

3

1.1.3. 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].

1.1.4. 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.

4

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.

5

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.

6

Chapter 2. MODELING REHEATING FURNACE

7

2.1. Line and reheating furnace performances

2.1.1. Mill layout

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.

Reheating furnace

Rolling mills Crop shear Finishing mill

Laying head

Reforming tub

Cooling conveyor

8

2.1.2. 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.

Figure 2-2. Cyclic motion of a walking hearth reheating furnace.

0.Original position

Billet/Bloom

Stationary

hearth

Walking

hearth

1.Lift up 2.Move forward

3.Lift down 4.Move backward 5.Return to original position

9

Figure 2-3. Structure of a typical reheating furnace.

2.1.3. 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.

Feeding rollers Dividing wall

Zone 1 Zone 2 Zone 3 Zone 4

Side wall Hearth

CeilingBillet

10

2.1.4. 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.

tcT = tcw + tcs ⋯ (2.1)

tcw = {tr + trv =

W

vw+ trv when a billet/bloom is rolled

tv when there is no billet/bloom to be rolled

where

W: Weight of extracted billet/bloom [tonf]

vw: Rolling weight speed [tonf ∙ (sec)−1]

vw = ρvfAf

vf: Rolling speed at finishing rolling stand [m ∙ (sec)−1]

Af: Sectional area at finishing rolling stand [m2]

ρ: Weight density of extracted billet/bloom [tonf ∙ m−3]

trv: interval time between cycles when billet/bloom is extracted [sec]

tv = tcu + tcf + tcd + tcb + tsv

tcu: time for lifting up the hearth [sec]

tcf: time for moving forward the hearth [sec]

tcd: time for lifing down the hearth [sec]

tcb: time for moving backward the hearth [sec]

tsv: 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.

11

2.1.5. 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).

Ii = i × (−K) − Gi 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).

Pi,nc= Ii + nc × 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).

12

tcm,nc= ∑ (tcT,k + tb,k) ⋯ (2.4)

nc

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 ≥ tcm,nc ∩ min(t − tcm,nc

) ⋯ (2.5)

The position Pi,t of a billet/bloom i after t time periods is estimated by (2.6).

Pi,t = Ii + nc × 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.

13

Figure 2-4. Furnace temperature trend and billet/bloom furnace temperature experience.

Bil

let/

blo

om

ex

per

ien

cin

g

Fu

rnac

e te

mp

erat

ure

Time

Zone 1

Fu

rnac

e te

mp

erat

ure

Zone 2

Fu

rnac

e te

mp

erat

ure

Zone 3

Fu

rnac

e te

mp

erat

ure

Zone 4

Fu

rnac

e te

mp

erat

ure

Time

14

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

[m]

1 1 2 20 22 0× SK

2 1 2 0 24 1× SK

3 0 1 10 35 2× SK

⁞ ⁞ ⁞ ⁞ ⁞ ⁞

nc-2 0 1 0 ⁞ (nc-3) × SK

nc-1 1 2 10 ⁞ (nc-2) × SK

nc 1 2 0 ⁞ (nc-1) × SK

15

2.2. Heat balance inside the furnace

2.2.1. 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.

2.2.2. 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.

16

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).

tp =tL

s [sec] ⋯ (2.7)

where

qrad,cb

qtran,ab

qrad,bi-1→bi qrad,bi→bi+1

qtran,bH

Tc

Ta

Tbi Tbi+1Tbi-1

TH

<0, if Tbi<TH

≥0, if Tbi≥TH

Hearth

Ceiling

Billet/Bloom

Feeding direction

qrad,wb

Tw Side wall

17

tL: Furnace holding time of the last billet/bloom in the considered range

s: Number of time periods for all zones decided by users

2.2.3. 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).

Q̇total = ρV ∫ Cp(T)Testimated

Tcurrent

dT [W] ⋯ (2. 8)

where

ρ: Mass density [kg/m3]

V: Volume [m3]

Tcurrent: Current temperature [K]

Testimated: Estimated temperature after a certain time period[K]

Cp: Specific heat at constant pressure [J ∙ kg−1K−1] = f(T)

≅ Cv: Specific heat at constant volume

If the time period is short, Cp can be approximated as a function of Tcurrent. In this case,

the equation (2.8) can be rewritten as (2.9).

Q̇total = ρV ∙ f(Tcurrent) ∙ (Testimated − Tcurrent)

→ Testimated = Tcurrent +Q̇total

ρV ∙ f(Tcurrent) ⋯ (2. 9)

18

2.3. Thermal radiation between billet/bloom and furnace walls

2.3.1. 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].

Q̇rad = Abσϕwb(Tw4 − Tb

4) ⋯ (2. 10)

where

Q̇rad: Total heat from the furnace bricks to the billets/blooms by radiation [W]

σ: Stefan-Boltzmann constant 5.670373 × 10−8 [Wm−2K−4]

Ab: Surface area of the billet/bloom [m2]

ϕwb(Fwb, Fbw, εw, εb): Radiation coefficient

Fwb: View factor from the furnace bricks to the billets

Fbw: View factor from the billets to fthe urnace bricks

εw: Emissivity of the furnace bricks

εb: Radiation absorption rate of the billets

2.3.2. 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].

19

2.3.3. 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.

F12 =1

A1∫ ∫

cos φ1 cos φ2

πr2dA1dA2

A2A1

⋯ (2. 11)

20

2.4. Heat transfer between billets/blooms, the furnace atmosphere

and the hearths

2.4.1. 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].

q̇L = hL(T1 − T2) ⋯ ( 2. 12)

where

hL: Local heat transfer coefficient [Wm−2K−1]

T1: Temperature of substance 1 [K]

T2: 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).

Q̇ = ∫ q̇LA

dA = (T1 − T2) ∫ hLA

dA ⋯ (2. 13)

When A is constant,

Q̇ = hLA(T1 − T2) ⋯ (2. 14)

21

2.4.2. 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.

qgb = hgb(Tg − Tb) + μgbσ(Tg4 − Tb

4) ⋯ (2. 14)

where

hgb: Heat transfer coefficient from gas to billets/blooms

Tg: Gas temperature

Tb: 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.

2.4.3. 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].

22

2.5. Thermal conduction

2.5.1. 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 = λΔT

d⋯ ( 2. 15)

where

J: Transmitted heat from one portion to another portion within the billet

/bloom [Wm−2]

λ: Thermal conductivity [WK−1m−1]

d: Distance between the centers of the portions [m]

ΔT: Temperature difference between the portions [K]

dV: Volume of the portion [m3]

Using equation (2.15), transmitted heat from adjacent portions of a billet by thermal

conduction is calculated as (2.16).

Q̇cond = λAΔT

d⋯ (2. 16)

where

A: Area contacting to the adjacent portion [m−2]

23

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.

2.6.1. 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.

Ci|Tbi,(x,y,z),t≥TA1= fi|Tbi,(x,y,z),t≥TA1

(Tbi,(x,y,z),t) ⋯ (2. 17)

Ci|Tbi,(x,y,z),t<TA1= fi|Tbi,(x,y,z),t<TA1

(Tbi,(x,y,z),t) ⋯ (2. 18)

where

Tbi,(x,y,z),t: Temperature at (x, y, z) portion of billet/bloom i during time period t

24

2.6.2. 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).

ϕwb = ε′wε′bFwb ⋯ (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.

2.6.3. Thermal conductivity

Thermal conductivity is also affected by temperature [14], [22]. Therefore, it is

expressed as a function of temperature as (2.20).

λi = hi(Tbi,(x,y,z),t) ⋯ (2. 20)

25

2.7. Furnace Modeling

2.7.1. 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

Figure 2-6. Mesh configuration of a billet/bloom.

m

ℓ

n

(1,1,1)

(ℓ,m,n)

D

H

B F

C

GA

E

I

26

2.7.2. 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.

Figure 2-7. Heat transfer into each billet component.

Component A (1,1,1)

qcond,(x,1,2)

qcond,(x+1,1,1)

qtran,Hb

qcond,(x-1,1,1)

qcond,(x,2,1)

qcond,(ℓ,1,2)

qcond,(ℓ-1,1,1)

qcond,(ℓ,2,1)

qcond,(1,y,2)

qcond,(2,y,1)

qcond,(1,y+1,1)

qcond,(1,y-1,1)

qcond,(ℓ,y,2)

qcond,(ℓ-1,y,1)

qcond,(ℓ,y+1,1)

qcond,(ℓ,y-1,1)

qcond,(x,y,2)

qcond,(x-1,y,1)

qcond,(x,y+1,1)

qcond,(x,y-1,1)

qcond,(x+1,y,1)

qcond,(ℓ-1,m,1)

qcond,(ℓ,m,2)

qcond,(ℓ,m-1,1)

qcond,(x,m,2)

qcond,(x+1,m,1)

qcond,(x,m-1,1)

qcond,(x-1,m,1)

qcond,(1,m,2)

qcond,(2,m,1)

qcond,(1,m-1,1)

qrad,Hb+qrad,cb+qrad,wb

+qrad,bi-1→bi+qtran,gb

Component H (x,1,1)

Component B (1,y,1) Component I (x,y,1)

Component C (1,m,1) Component D (x,m,1) Component E (ℓ,m,1)

Component F (ℓ,y,1)

Component G (ℓ,1,1)

qcond,(2,1,1)

qcond,(1,1,2)

qcond,(1,2,1)

qrad,Hb+qrad,cb

+qrad,wb+qtran,gb

qrad,cb+qrad,wb

+qtran,gb

qrad,Hb+qrad,cb

+qrad,wb+qrad,bi→bi+1

+qtran,gb

qrad,Hb+qrad,cb

+qrad,wb+qtran,gb

qrad,Hb+qrad,cb

+qrad,wb+qtran,gb

qrad,Hb+qrad,cb

+qrad,wb+qrad,bi-1→bi

+qtran,gb

qrad,Hb+qrad,cb

+qrad,wb+qtran,gb

qrad,Hb+qrad,cb

+qrad,wb+qtran,gb

qrad,Hb+qrad,cb

+qrad,wb+qtran,gb

qrad,Hb+qrad,cb


+qtran,gb

qrad,Hb+qrad,cb

+qrad,wb

+qrad,bi-1→bi

+qtran,gb

qrad,Hb+qrad,cb

+qrad,wb+qtran,gb

qrad,Hb+qrad,cb

+qrad,wb+qtran,gb

qrad,Hb+qrad,cb

+qrad,wb+qtran,gb

qrad,Hb+qrad,cb


+qtran,gb

27

2.7.3. 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, Hwe = Hw − Lw tan θ, Lce =Hc

tan θ⋯ (2.21)

o/w, Hwe = Hw, Lce = Lw ⋯ (2.22)

28

where

Hwe: Effective height of the dividing wall

Lce: Effective length of the ceiling

Hw: Height of the dividing wall

Lw: Distance from the billet to the dividing wall

Hc: Height of the ceiling

29

Figure 2-8. Geometry of furnace wall view factors.

Front end and tail end

Upper face

Upstream side

Downstream side

30

Figure 2-9. Effective area for thermal radiation view factors.

ϕθ

ϕ

θ

ϕθ

ϕθ

31

2.7.4. 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).

Ta,j,z,t =TaF,j,t − TaT,j,t

WH

(Luz + dF) + TaF,j,t ⋯ (2. 23)

where

WH: Width of the furnace [m]

Ta,j,z,t: Furnace temperature at z in zone j during time period t [K]

TaF,j,t: Furnace temperature at the front side wall in zone j during time period t [K]

TaF,j,t: Furnace temperature at the tail side wall in zone j during time period t [K]

dF: Set Distance between the front side wall and the front end of billets/blooms [m]

Lu: Unit length of the mesh in z axis [m]

32

2.7.5. 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, qhb = hhb(Th − Tb) ⋯ (2.24)

o/w, qwh = σεwεwFwh(Tw4 − Th

4) + ℎ𝑔ℎ(Tg − Th) ⋯ (2.25)

where

hhb: Heat transfer coefficient from the hearths to the billet

hgh: Heat transfer coefficient from the combustion gas to the billet

Th: Hearth temperature, T𝑏: billet/bloom temperature,

Tg: gas temperature, Tg: furnace wall temperature

33

2.7.6. 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 2-10. 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/m3]

Holing

time

[min]

Stefan

Boltzman

coefficient

[W/(m2·K4)]

0.165×0.165 0.235 0.055×0.100 452 7.8×103 90 5.6703×10-8

34

Figure 2-11. Temperature difference assumption at each holding time.

200

300

400

500

600

700

800

900

1000

1100

1200

0 3 6 9 12 15 18 21 24 27 30 33 36 39 42 45 48 51 54 57 60 63 66 69 72 75 78 81 84 87 90

Tem

per

ature

[C

°]

Holding time [min]

Billet/Bloom at downstream Billet/bloom at upstream

200

220

240

260

280

300

320

340

360

380

400

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Tem

per

ature

[C

°]

Holding time [min]

35

In figure 2-12, the result of the simulation is shown. As the distance between the

components increases, the extent of temperature interaction becomes smaller. When the

considered range in z is up to 400-500mm, the cumulative temperature increase was about

0.25 °C and almost saturated. Even if the range of ±500 mm in z is considered, the

temperature increase would only be about 0.5 °C. Consequently, the effect of the interaction

by thermal radiation between billets/blooms is negligible in this furnace operation. It was

assumed in this case that the holding time was 90 minutes and the temperature gap with

adjacent billet was 10 [K]. If much larger temperature gap between adjacent billets is

induced, the influence of this interaction might be necessary to be considered.

Figure 2-12. Temperature increase estimation at extraction by radiation from neighboring

billets/blooms.

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.0-0.1 0.1-0.2 0.2-0.3 0.3-0.4 0.4-0.5 0.5-0.6 0.6-0.7 0.7-0.8 0.8-0.9 0.9-1.0

Tem

per

ature

incr

ease

[C

°]

Position of components from the end of billet/bloom [m]

Component y=0.110-0.165 Component y=0.055-0.110 Component y=0-0.055

36

Chapter 3. SIMULATION OF THE MODEL

37

3.1. Billet/Bloom initial orders and their parameters

Using the model that was created in chapter 2, the temperature of billets in the

reheating furnace was simulated. Then, the performance and the reasonability of this model

were evaluated.

3.1.1. Operational conditions

For simplicity, the furnace temperature of each zone in a furnace is independently set

at a constant steady-state temperature over the whole considered duration. Billets are charged

continuously without any additional space between billets. Those billets are the same type of

steel and have the same size. This implies that all of the thermal property and geometry

condition are the same.

3.1.2. Modeling of material thermal properties

For the simulation, a nonlinear regression model shown in (3.1) and (3.2) was

employed to model the specific heat of the steel billets.

C𝑝,𝑖|Tb≤750℃= 𝑎𝑠𝑏Tb,(𝑥,𝑦,𝑧),𝑖

3 + 𝑏𝑠𝑏Tb,(𝑥,𝑦,𝑧),𝑖2 + 𝑐𝑠𝑏Tb,(𝑥,𝑦,𝑧),𝑖 + 𝑑𝑠𝑏 ⋯ (3. 1)

C𝑝,𝑖|Tb≥750℃= 𝑎𝑠𝑎Tb,(𝑥,𝑦,𝑧),𝑖

3 + 𝑏𝑠𝑎Tb,(𝑥,𝑦,𝑧),𝑖2 + 𝑐𝑠𝑎Tb,(𝑥,𝑦,𝑧),𝑖 + 𝑑𝑠𝑎 ⋯ (3. 2)

The thermal conductivity of the billets was approximated using (3.3).

λ𝑖 = 𝑎𝑐Tb,(𝑥,𝑦,𝑧),𝑖3 + 𝑏𝑐Tb,(𝑥,𝑦,𝑧),𝑖

2 + 𝑐𝑐Tb,(𝑥,𝑦,𝑧),𝑖 + 𝑑𝑐 ⋯ (3. 3)

38

The emissivity, the heat transfer coefficient against combusted gas and the heat

conductance against the hearth were assumed constant. The coefficient and the values which

were used for the simulation are summarized in table 3-1. Additionally, the other data for

simulation, such as the furnace structure and the operational condition, are shown in table 3-

3.

Table 3-1. Coefficient and values used for simulation.

Property asb/asa/ac bsb/bsa/bc csb/csa/cc dsb/dsa/dc

Specific heat [J/(kg·K)]]

(Below transformation temperature) 1.1453×10-6 -13.4876×10-4 85.3899×10-2 31.2823×10

Specific heat [J/(kg·K)]]

(Above transformation temperature) -7.0434×10-6 2.9609×10-2 -41.3193 1.9799×10-4

Thermal conductivity [W/(m·K)] 0 -4.4643×10-5 0.022589 54.3889

Emissivity - - - 0.95

Heat transfer coefficient against gas

[W/(m2·K)] - - - 7.4665

Heat conductance against hearth

[W/(m2·K)] - - - 60

3.1.3. Computer specification for simulation

The details of the computer which ran the simulation are shown in table 3-2. These

specifications are for a commercial personal computer with typical performance

characteristics in 2015.

Table 3-2. Specification of the computer and operating system used for simulation.

Category Specification/version

Software Matlab R2012b

CPU Intel(R) Core(TM)2 Quad CPU Q9400 2.66GHz

Memory Installed memory (RAM) 4.00GB (3.87GB usable)

System type 64-bit Operating System

39

Table 3-3. Specification, parameters and conditions for the simulation.

Simbol in programming Value Unit

- 28,000 mm

WF 13,000 mm

Zone1 Lzc(1) 8,000 mm




Zone1 Hc(1) 1,665 mm




Zone 1&2 Hbw(2) & Hfw(1) 1,000 mm

Zone 2&3 Hbw(3) & Hfw(2) 1,000 mm

Zone 3&4 Hbw(4) & Hfw(3) 1,000 mm

Zone 1&2 Wth1 100 mm



df 200 mm

Rb 15 mm

Hb 165 mm

Wb 165 mm

L 10,000 mm

Hu - mm

Wu - mm

Lu - mm

SK 400 mm

- 70

Ih 45 sec

ps 25 sec

v 0 ton/sec

Iv 5 sec

tp - sec

NN - -

Between gas and bricks hbh 6 W/(m2K)

Between bricks and hearth hgh - W/(m2K)

Thermal conductance CH 90 kJ/K

Embr 0.70 -

Fbr 0.001 -

Bricks

Heat transfer coefficient

Operational condition

Stroke distance

Total number of strokes

Minimum interval

Additional pose

Rolling speed

Category

Furnace structure

Furnace length

Furnace width

Zone length

Zone height

Dividing wall

height

Dividing wall

thickness

Front end position

Emissivity of bricks

View factor of hearth spotOther

Length

Billet/bloom size

Corner radius

Sectional height

Sectional width

Regular interval

Computational conditionTime period length

Number of time period

Mesh detail

Unit height

Unit width

Unit length

40

3.2. Performance of the simulation model

3.2.1. Simulated temperature trends

The operational condition and properties for the simulation are shown in Table 3-4.

An example temperature trend for a billet and the atmosphere of each zone is illustrated in

figure 3-2 and 3-3. The number of lines in figure 3-2 corresponds to the portion number in

figure 3-1. Between zone 1 and zone 2 and between zone 2 and zone 3, there are 100 [K]

gaps in the furnace temperature. The billet/bloom temperature also follows this change when

it moves from zone 1 to zone 4.

Table 3-4. Simulation properties and operational conditions for the simulation.

Time increments

[sec]

Unit mesh size

[mm× mm× mm]

Billet/bloom

number

Furnace holding

time [min]

Each zone

temperature

0.5 11×11×50 1

Charged firstly 82

Constant and

uniform

Figure 3-1. Positions of highlighted billet locations for analysis.

#1 #2 #3

#4 #5 #6

#7 #8 #9

x

y

z Front end

Moving direction

Downstream Upstream

41

Figure 3-2. Simulated temperature trend at the billet front end (z=1).

Front end

Down

stream

Up

stream

x

y

z

#7 #9#8

#4 #6#5

#1 #3#2

42

Figure 3-3. Simulated temperature trend at the middle in the billet length.

Front end

Down

stream

Up

stream

x

y

z

#7 #9#8

#4 #6#5

#1 #3#2

43

At the front end, the temperature of the upper corners, #7 and #9, were saturated in

zone 4. However, even at upper corners, the temperature does not reach the furnace

temperature of zone 4. Whereas the front end receives thermal radiation from the side-wall of

the furnace, the middle section does not. As a result, the billet temperature is much lower in

the middle than at the front end.

The temperature of billets was simulated for three different cases in the furnace

temperature of each zone. The result is shown in figure 3-4 and figure 3-5. Case 1 is the case

which zone 1 and zone 3 have relatively high temperature. Case 2 is the average case among

the three cases. Case 3 is the case where the temperature of zone 1 is lower and that of zone 3

is higher. In both of the upper corner at downstream side and the center, the temperature in

case 3 was the lowest until it reached zone 3 due to the lower temperature in zone 1. After

reaching zone 3, the temperature in case 3 exceeds that in case 2, because the furnace

temperature in case 3 is higher than that in case 2.

Consequently, this simulation indicates that this model can respond to furnace

temperature changes well.

Figure 3-4. Simulated temperatures of each portion in the middle section at extraction.

1176

1120

11761182

1134

1185

1229

1189

1235

1175

1118

11751180

1131

1183

1226

1186

1233

1180

1124

11801186

1138

1189

1233

1193

1239

1100

1120

1140

1160

1180

1200

1220

1240

1260

#1 #2 #3 #4 #5 #6 #7 #8 #9

Sim

ula

ted

bil

let

tem

per

atu

re [◦C

]

Case 1 Case 2 Case 3

44

Furnace temperature [K]

Zone 1 Zone 2 Zone3 Zone4

Case 1 1223 1223 1373 1333

Case 2 1223 1273 1323 1333

Case 3 1173 1273 1373 1333

Figure 3-5. Simulated temperature trend in the middle section.

0

100

200

300

400

500

600

700

800

900

1000

1100

1200

1300

1400

1

341

681

102

1

136

1

170

1

204

1

238

1

272

1

306

1

340

1

374

1

408

1

442

1

476

1

510

1

544

1

578

1

612

1

646

1

680

1

714

1

748

1

782

1

816

1

850

1

884

1

918

1

952

1

Sim

ula

ted

bil

let

tem

per

ature

[K

]

Time [×0.5sec]


0

100

200

300

400

500

600

700

800

900

1000

1100

1200

1300

1400

13

41

681

102

11

36

11

70

12

04

12

38

12

72

13

06

13

40

13

74

14

08

14

42

14

76

15

10

15

44

15

78

16

12

16

46

16

80

17

14

17

48

17

82

18

16

18

50

18

84

19

18

19

52

1

Sim

ula

ted

bil

let

tem

per

ature

[K

]

Time [×0.5sec]


At portion #5

At portion #7

Front end

Down

stream

Up

stream

x

y

z

#7 #9#8

#4 #6#5

#1 #3#2

45

3.2.2. Difference of simulated sectional temperature

Figure 3-6 indicates the temperature differences at extraction in different sectional

portions of the billet. In the front end of the billet (z=1), the temperature of the upper corners

was the highest. On the other hand, the temperature of the center and the bottom face shows

the lowest. In the middle section, the upper corners show the highest temperature and the

center and the bottom face shows the lowest as well. The tendency that the temperature of the

corners is higher and the temperature of the center and the bottom face is lower, was the

same for each furnace analysis condition.

As the furnace temperature increases, heat transmission through thermal radiation

becomes significantly large, because it increases nonlinearly at rate of the forth power of the

temperature, while other types of heat transmission increase linearly. The upper face is

heated mainly by thermal radiation from the ceiling and the sidewalls of the furnace.

However, each upper corner is heated by thermal radiation from the ceiling, the sidewalls and

the hearths. From this condition, the upper corners show the highest temperature in both the

front end and the middle section. The center and the bottom face do not receive thermal

radiation and they are heated mainly through thermal conduction, because they are not

exposed to furnace wall radiation.

The temperature change in the longitudinal direction z is shown in figure 3-7. In most

portions, the temperature is the minimum at around 6400 [mm]. Billets are aligned to the

front end and the distance between the sidewall of the furnace and the front end is 200 [mm]

in this model. The width of the furnace is 13,000 [mm]. The minimum temperature position

corresponds to the middle of the furnace width. This seems to indicate that the thermal

radiation is the smallest at the middle of the furnace wide, since it is the farthest from the

both sidewalls of the furnace. At the front end and the tail end, the temperatures are much

higher than other regions. They receive relatively intense thermal radiation from the

sidewalls of the furnace in addition to the ceiling and the hearths. Therefore, these regions

are much higher in temperature than other sections. Compared to the temperature at the front

end, the temperature at the tail end is lower. This is caused by the difference in the distance

from the each sidewall. While the distance to the front end is 200 [mm], the distance to the

46

tail end is 2,800 [mm]. The reheating furnace model expresses these thermal relationships

well.

Furnace temperature [K]

Zone 1 Zone 2 Zone3 Zone4

Case 3 1173 1223 1373 1333

Figure 3-6. Difference in simulated billet component temperature at extraction along the

length of the billet.

1,299

1,264

1,299 1,305

1,278

1,306

1,324

1,312

1,324

1,230

1,240

1,250

1,260

1,270

1,280

1,290

1,300

1,310

1,320

1,330

#1 #2 #3 #4 #5 #6 #7 #8 #9

Front end (z=1)

Sim

ula

ted

tem

per

ature

at

extr

acti

on [

K]

Portion in section

1,180

1,124

1,180 1,186

1,138

1,189

1,233

1,193

1,239

1,060

1,080

1,100

1,120

1,140

1,160

1,180

1,200

1,220

1,240

1,260

#1 #2 #3 #4 #5 #6 #7 #8 #9

Middle (z=50)

Sim

ula

ted

tem

per

ature

at

extr

acti

on [

K]

Portion in section

47

Figure 3-7. Simulated component temperature difference at extraction in the billet longitudinal direction.

48

3.2.3. Heat transmission in billet longitudinal direction

Billets/blooms are usually long products. The possibility of reducing the dimension of

the model was investigated to simplify the model and shorten the computation time, as

Steinboeck et al. proposed a 1-D slab temperature simulation model [21].

The total transmitted heat through thermal conduction is shown in figure 3-8 and the

total transmitted heat through thermal conduction in z is shown in 3-8. The operational

condition is indicated in table 3-5 and the unit mesh size is 55×55×100 [mm× mm× mm].

According to the result in figure 3-9, up to 700 [mm] from the front end and the tale end,

large heat is transmitted in direction z. However, in the region of more than 700 [mm], the

heat became much smaller. In figure 3-10, the rate of the total transmitted heat in z to the

total transmitted heat is shown. Transmitted heat in z occupies large percentages up to 700

[mm]. On the other hand, the heat does not contribute an increase of the temperature in the

region of more than 700 [mm]. If the furnace temperature is higher than that in this

simulation, the region, which the heat transmission in z should be considered, might be

longer than 700 [mm]. If only the middle section is used for the simulation, the transmitted

heat in z is negligible and the model can be reduced to 2-dimension from 3-dimension.

Table 3-5. Operational conditions for simulation.

Billet

Number

Furnace holding

time [min]

Furnace temperature


1 82 1273 1273 1373 1333

49

Figure 3-8. Total transmitted heat until extraction

Figure 3-9. Total transmitted heat in the longitudinal direction until extraction.

0

200

400

600

800

1000

1200

1400

1600

1800

1 5 9

13

17

21

25

29

33

37

41

45

49

53

57

61

65

69

73

77

81

85

89

93

97

To

tal

tran

smit

ted

hea

t [k

J]

#1 #2 #3 #4 #5

#6 #7 #8 #9

0

200

400

600

800

1000

1200

1400

1600

1800

1 5 9

13

17

21

25

29

33

37

41

45

49

53

57

61

65

69

73

77

81

85

89

93

97

To

tal

tran

smit

ted

hea

t in

dir

ecti

on z

[kJ]

#1 #2 #3 #4 #5

#6 #7 #8 #9

50

Figure 3-10. Rate of transmitted heat in the z direction to total transmitted heat until extraction.

0%

5%

10%

15%

20%

25%

30%

35%

1 5 9

13

17

21

25

29

33

37

41

45

49

53

57

61

65

69

73

77

81

85

89

93

97

Rat

e o

f tr

ansm

itte

d h

eat

in z

to

to

tal

tran

smit

ted

hea

t

#1 #2 #3 #4 #5 #6 #7 #8 #9

51

3.3. Selection of appropriate mesh size

3.3.1. Relationship between mesh size and simulated temperature

In this simulation, the sectional size of billets is 165 [mm] × 165 [mm] and the billet

length is 10,000 [mm]. If the unit width of the mesh is 15 [mm], the number of mesh in wide

direction is 165/15=11. It is assumed in this simulation that the temperature of the unit mesh

is uniform. Therefore, the size of the unit mesh may have a large impact on the simulated

temperature.

The temperature of each sectional portion is simulated and compared at various sizes

of the unit mesh. The result is shown in figure 3-11. At the front end of a billet, the bottom

face temperature, #2, and the center temperature, #5, had relatively large variation in 11 ×11

[mm×mm] to 55×55 [mm×mm]. The largest gap was 11 [K] at #2. On the other hand, they

had relatively small gap 2 to 5 [K] in the middle section. This means that even if 55×55

[mm×mm] is chosen as the mesh size, the difference from 11 ×11 [mm×mm] in the center

temperature is about 11 [K] at the front end and about 5 [K] at the middle section. By

leveling or compensating, the center temperature in the middle section can be used for

simulation with little error.

Figure 3-12 shows a comparison of simulated temperature in different length of unit

mesh, 10 [mm] and 100 [mm]. At the front end, differences from 5 to 12 [K] were observed.

However, in the middle section, there were almost no differences between their center

temperatures. As a result, 55×55×100 [mm×mm×mm] can be used as the unit mesh size for

simulation if only the center temperature in the middle section is being considered. If other

billet locations are also to be modeled, the minimum mesh size leads to more precise results,

but the computation time will be dramatically long.

52

Increments

time [sec]

Furnace temperature [K] Furnace holding

time [min] Zone 1 Zone 2 Zone3 Zone4

1 1173 1273 1373 1333 82

Figure 3-11. Relationship between simulated temperature and unit mesh size.

1,120

1,140

1,160

1,180

1,200

1,220

1,240

1,260

1,280

1,300

1,320

1,340

#1 #2 #3 #4 #5 #6 #7 #8 #9

Front end (z=1)

Sim

ula

ted

tem

per

ature

[K

]

Simulated positions

Size 55×55×100 [mm] Size 33×33×100 [mm]

Size 15×15×100 [mm] Size 11×11×100 [mm]

1,120

1,140

1,160

1,180

1,200

1,220

1,240

1,260

1,280

1,300

1,320

1,340

#1 #2 #3 #4 #5 #6 #7 #8 #9

Middle (z=50)

Sim

ula

ted

tem

per

ature

[K

]

Simulated positions

Size 55×55×100 [mm] Size 33×33×100 [mm]

Size 15×15×100 [mm] Size 11×11×100 [mm]

53

Increments

time [sec]



1 1173 1273 1373 1333 82

Figure 3-12. Comparison of simulated billet temperatures in different billet positions of a

function of simulation mesh size.

1,306

1,286

1,302

1,313

1,297

1,310

1,325

1,316

1,323

1,293

1,268

1,288

1,301

1,280

1,297

1,318

1,305

1,315

1,260

1,270

1,280

1,290

1,300

1,310

1,320

1,330

1,340

1,350

1,360

#1 #2 #3 #4 #5 #6 #7 #8 #9

Front end (z=1)

Sim

ula

ted

tem

per

ature

[K

]

Portions in section

Size 55×55×10 [mm] Size 55×55×100 [mm]

1,174

1,138

1,165

1,185

1,152

1,177

1,217

1,187

1,210

1,174

1,138

1,165

1,185

1,152

1,177

1,217

1,187

1,210

1,120

1,130

1,140

1,150

1,160

1,170

1,180

1,190

1,200

1,210

1,220

#1 #2 #3 #4 #5 #6 #7 #8 #9

Middle (z=50)

Sim

ula

ted

tem

per

ature

[K

]

Portions in section

Size 55×55×10 [mm] Size 55×55×100 [mm]

54

3.3.2. Mesh size and time increments

In dynamic programming, time is dealt with as discrete time. That is, until time

moves to the next time period, the current condition is maintained. During each modeled time

increment, the temperature of a unit mesh does not change. If the size of unit mesh becomes

smaller, the volume and the surface area also become smaller. However, the volume and the

surface area do not become smaller at the same rate. Heat by thermal radiation is received

through the surface of the mesh. As a result, the heat by thermal radiation, which the unit

mesh receives per unit time and per volume, becomes larger by choosing smaller mesh sizes.

Therefore, shorter time increments must be chosen when a smaller unit mesh size is selected.

Otherwise, the computation will diverge. In table 3-6, convergence properties were

investigated for four mesh sizes and various modeling time increments. For mesh size 55×

55×100 [mm×mm×mm], the time increments for which computation were completed without

divergence was up to 82 [sec]. In figure 3-13, the simulated temperature in 55×55×100

[mm×mm×mm] mesh sizes and various time increments is shown. At tp=82 [sec], the

temperature was significantly larger than for the others. Even if it is converged, shorter time

increments should be selected for more reasonable modeling estimates.

Table 3-6. Model convergence (C) and divergence (D) for different mesh sizes and modeling

time increments.



1173 1273 1373 1333 82

Mesh size

[mm×mm×mm]

Time increments tp [sec]

0.5 1 2.5 5 30 40 60 82 120

55×55×100 C C C C C C C C D

33×33×100 C C C C D - - - -

15×15×100 C C C C D - - - -

11×11×100 C C C D - - - - -

55

Mesh size

[mm×mm×mm]



55×55×100 1173 1273 1373 1333 82

Figure 3-13. Simulated temperature difference of each component for different modeling

time increments.

1240

1260

1280

1300

1320

1340

#1 #2 #3 #4 #5 #6 #7 #8 #9

Front end (z=1)

Sim

ula

te t

emp

erat

ure

at

extr

acti

on [

K]

tp=0.5 [sec] tp=1 [sec] tp=30 [sec] tp=82 [sec]

1120

1140

1160

1180

1200

1220

1240

#1 #2 #3 #4 #5 #6 #7 #8 #9

Middle (z=50)

Sim

ula

te t

emp

erat

ure

at

extr

acti

on [

K]

tp=0.5 [sec] tp=1 [sec] tp=30 [sec] tp=82 [sec]

56

3.3.3. Mesh, time increments and computation time

In optimization programming, many iterations are usually carried out. However, if the

total computation time is longer than prediction time range in the simulation, the next

computation cannot be done in real time before the next billets/blooms, which are scheduled

in the future out of the prediction range, are charged. In this case, this simulation cannot be

used for real time control of furnaces. Since unit mesh size and time increments influence the

computation time significantly, the extent of this influence was investigated more

thoroughly.

In figure 3-14, the effect of size change in unit mesh on the computation time is

shown for tp=1 and tp=2.5. As the unit mesh size becomes large, the computation time

decreases nonlinearly. At tp=2.5 in unit mesh size 55×55×100 [mm×mm×mm], the

computation time was 42 [sec]. In this case, if there are 20 billets in a furnace and 10

iterations are needed, the total computation time will be 42×20×10=8400 [sec], ≈140 [min].

Assuming the furnace holding time is 82 [min] and the 20 billets are charged continuously

every 1.5[min], the prediction time range becomes 112 [min]. Therefore, the computation

time exceeds the prediction time range. In this case, it is necessary to shorten the

computation time if it is to be used as part of a real-time control strategy.

Figure 3-14. Computation time for various simulation conditions

105 126

373

628

-

100

200

300

400

500

600

700

0 20 40 60

Co

mp

uta

tio

n t

ime

[sec

]

Sectional size of unit mesh [mm]

tp=1 [sec]

42 59

162

277

-

100

200

300

400

500

600

700

0 20 40 60

Co

mp

uta

tio

n t

ime

[sec

]

Sectional size of unit mesh [mm]

tp=2.5 [sec]

57

In figure 3-15, the effect of time increments on the computation time is illustrated. In

33×33×100 [mm×mm×mm], the computation time decreased to 13 [sec] at tp=30 [sec]. In

55×55×100 [mm×mm×mm], the computation time was saturated at tp=30 [sec] and it was

around 11 [sec]. When the computation time is 11 [sec], the total computation time in the

case shown above will be about 11 [sec] × 20 billets × 10 iterations =2200 [sec] and it is

shorter than the prediction time range. In this case, about 30 times iterations are affordable in

this simple estimation. However, for more iterations, it is necessary to shorten the

computation time. It will be discussed in later chapter about optimization whether the

computation should be shortened more or not.

Figure 3-15. Computation time for various time increments up to 4920 [sec] (=82 [min]).

126

59

32 21

13

-

20

40

60

80

100

120

140

0 5 10 15 20 25 30 35

Co

mp

uta

tio

n t

ime

[sec

]

Increments time [sec]

Size=33×33×100

105

42

25 18

11 12 10 12 10

-

20

40

60

80

100

120

140

0 20 40 60 80 100 120 140

Co

mp

uta

tio

n t

ime

[sec

]

Increments time [sec]

Size=55×55×100

58

3.4. Effect of thermal conductivity on center temperature

3.4.1. Thermal conductivity effects

A stainless steel has a much lower thermal conductivity than that of a 0.1% carbon

steel [22]. This implies that the center temperature in billet section at extraction of a stainless

steel is expected to be lower than that of a 0.1% carbon steel under the same conditions.

Hence, it is important to know the effect of thermal conductivity estimates on billet

temperature in order to find required accuracy level for the model of the thermal

conductivity.

In figure 3-16, the relationship between temperature and thermal conductivity is

shown in three examples of steel created artificially. Steel 1 has relatively high value and

steel 2 has middle value up to around 1100 [K]. Steel 3 has lower value, but it exceeds steel 2

at around 1100 [K].

Figure 3-16. Relationship between temperature and thermal conductivity for various steels

0

10

20

30

40

50

60

70

80

90

0 200 400 600 800 1000 1200 1400

Ther

mal

co

nd

uct

ivit

y [

W/(

m·K

)]

Temperature [K]

Steel 1 Steel 2 Steel 3

59

3.4.2. Impact of thermal conductivity on billet temperature

The result of the simulation is shown in figure 3-17. The computational condition is

shown in table 3-7. At the upper corner, #7, the temperature of steel 3 was the lowest and that

of steel 2 was the highest. The gap between those 2 types of steel was approximately 20 [K]

in the middle. Furthermore, at the center, #5, the highest was in steel 3 and the lowest was in

steel 2. The gap was also approximately 20 [K] in the middle. In steel with high thermal

conductivity, heat from the upper corner to the center is transmitted greater distances. As a

result, the temperature of the upper corner in steel 3 was lower than that in steel 2, while the

temperature of the center in steel 3 was higher than that in steel 2.

The difference in thermal conductivity between steel 2 and steel 3 is approximately

27 [W/(m·K)] at all temperature levels. This difference makes about 20 [K] difference in the

temperature at extraction in this simulation (although it can be affected by the furnace

temperature, furnace holding time and other factors).

From these results, a 1 [W/(m·K)] difference in thermal conductivity can make about

a 0.8 [K] difference in the estimated billet temperature at extraction. Considering the

optimization of furnace operation, the accuracy of the model should be within 1 [W/(m·K)].

Table 3-7. Computational condition for thermal conductivity simulations.

Mesh size

[mm×mm×mm]

Time

increments

[sec]



55×55×100 30 1173 1273 1373 1333 82

60

(a) #7 – downstream side upper corner

(b) #5 - center

Figure 3-17. Temperature differences for steel with various thermal conductivities.

1140

1160

1180

1200

1220

1240

1260

1280

1300

1320

1340

1 5 9

13

17

21

25

29

33

37

41

45

49

53

57

61

65

69

73

77

81

85

89

93

97

Sim

ula

ted

tem

per

ature

at

extr

acti

on [

K]

Mesh number in z


1100

1120

1140

1160

1180

1200

1220

1240

1260

1280

1300

1 5 9

13

17

21

25

29

33

37

41

45

49

53

57

61

65

69

73

77

81

85

89

93

97

Sim

ula

ted

tem

per

ature

at

extr

acti

on [

K]

Mesh number in z


61

3.5. Parameters selection for optimization

3.5.1. Estimating extraction temperature of billets/blooms

If billet modeling strategies are focused primarily on the temperature in the middle

section, the dimension of the model can be reduced, since the heat transmission in

longitudinal direction is negligible. For optimization, it is important to balance computation

time and accuracy. For selecting an appropriate model and modeling parameters, the key

temperature which should be focused on in the optimization is discussed below.

The role of reheating furnaces is raising the billets/blooms temperature high enough

for effective rolling. This ‘high enough’ level is determined considering the following

factors.

a) The rolling load of each rolling machine in a line must be under the upper limit of

their specifications which are different for each machine. If the billet bulk

temperature is below the minimum level, rolling machines are damaged or the rollers

are broken. Insufficient bulk temperature leads to other operational trouble, for

instance biting failure in rolling. These troubles cause serious production loss,

especially in steel companies operating for 24 hours/day. The rolling load of each

machine, which is correlated to the billet temperature [23], is sometimes measured

using root mean square (RMS) in their motors which indicates the load on the motors

per unit time.

b) After rolling, when the steel reaches the cooling process in a rolling line, it must have

enough residual temperature to guarantee its quality as a start temperature [24]. If the

temperature is below a threshold temperature, the products are graded down or

rejected and the rolling line is shut down. Then, billets/blooms in a furnace are

reheated until their temperature reaches enough level to re-start the rolling line.

Conversely, overheating the billets/blooms temperature leads to an increase in

reheating fuel costs and steel quality deterioration, such as decarburization, grain coarsening

and so forth. Hence, it is ideal to set the reheating furnace temperature just high enough to

successfully roll.

62

In a furnace, assuming that the furnace temperature is always higher than billets

temperature, once the billet center temperature has reached the required level, it can be said

that the temperature of the other portions of the billet are also sufficient. When the surface

temperature and the center temperature of billets are sufficient, the rolling load at the first

rolling mill and at a mill after several mills is under the upper limits. However, when the

surface temperature is satisfactory and the center temperature is unsatisfactory, the rolling

load at the first rolling mill might be under the upper limit because the surface with

satisfactory temperature of billets is the portion of the billet that is primarily rolled. But, the

rolling load at mills after several mills would exceed their upper limits if the billet center is

cold. An example image describing this situation is shown in figure 3-18. The minimum

center temperature of billets is in the middle of the furnace width. Consequently, the center

temperature in the middle of the furnace width should be focused on for reheating furnace

control optimization. In this case, the 3-D control model can be reduced to 2-D control

model.

63

(a) Center temperature – Satisfactory

(b) Center temperature – Unsatisfactory

Figure 3-18. Comparison in rolling load between a billet with satisfactory center temperature

and a billet with unsatisfactory center temperature.

Billet temperature

Ro

llin

g l

oad

#1 rolling mill

Billet temperature

Ro

llin

g l

oad

#6 rolling mill

Upper limit (rolling mill specification)

Billet temperature

Roll

ing l

oad

#1 rolling mill

Billet temperature

Roll

ing l

oad

#6 rolling mill

64

3.5.2. Selection of model and parameters for reheating furnace control

The computation time of the simulation model is one of the major issues in adapting

its use for the optimization and control of the furnace operation. Although many

computational iterations must be done to obtain an optimal modeling result, the computation

must be done within certain limited time.

In slab simulation, when the dimension of the model was reduced from 2-D to 1-D,

the computation time was shortened dramatically [21]. Also, it is said that 3-D is not

appropriate for billet temperature simulation due to the large computation [25].

Based on the discussion so far, the following condition indicated in table 3-8 was

selected for the optimization, considering the reasonability of the simulated temperature and

computation time constraints.

Table 3-8. Selected model and parameters for furnace control optimization.

Model

Time

increments

[sec]

Size of unit mesh

[mm×mm]

(Number of mesh)

Estimated computation time per

billet during N time period [sec]

2 dimension model 30 55×55

(3×3) ≈10

65

Chapter 4. OPTIMIZATION OF FURNACE OPERATION

66

4.1. Optimization Problem

A simulation model for billets temperature has been introduced. In this chapter, a

furnace control optimization method exploiting the simulation model will be proposed. The

full implementation and use of the detailed reheating furnace billet temperature model,

discussed in the previous chapter, takes longer than one minute computation time to obtain a

complete solution. The goal of this chapter is to develop a simplified method to obtain a

near-optimal solution more quickly that can be used for direct control of the furnace

temperature.

4.1.1. Objective function

Each billet has own goal bulk temperature. When the most economical operation is

considered, the goal temperature becomes the constraint and minimizing the total fuel cost

will be the objective function. In this scenario, the cost function of each furnace zone must be

known. However, there is a difficulty in obtaining this cost function, because the cost in a

zone to increase billets temperature at extraction depends on the initial billets temperature in

the zone, the furnace temperature and the number of billets heated in the zone. Instead of a

direct fuel cost function in each zone, two policies are employed.

The first policy is that the furnace zone and the time period where the total gap

between the goal temperatures and the simulated temperatures of billets is the largest are

targeted for changing set-point furnace temperature.

The second policy is that the priority for increasing furnace temperature is first zone

3, then zone 2, then zone1 and finally zone 4. This prioritization is based on the concept that

heating in zone 3 is likely to be the most efficient. If the furnace temperature in zone 1 is

increased, the temperature of billets in zone 1 increases during certain times. This reduces the

heat transmission after the certain time, since the gap between the billets temperature and the

furnace temperature afterwards in zones 2-4 becomes smaller. This means that earlier action

is not likely to be efficient. Zone 4 is typically referred to as the soaking zone. The main role

of this zone is homogenizing the temperature throughout the billet. In this zone, the furnace

67

temperature is set at lower level than that of zone 3 to avoid surface overheating. The upper

limit of the temperature in this zone is relatively low and this zone is not selected for

increasing the billets temperature. Zone 3 is referred to as the heating zone and the furnace

temperature set point is the highest among the four zones. In this high temperature

environment, thermal radiation has the largest impact on billet temperature. The impact of an

increase of the furnace temperature in zone 3 is expected to be most significant, because the

transmitted heat via thermal radiation is controlled by the fourth power of the furnace

temperature and the billets temperature. As the furnace temperature is increased, the effective

heat transmission to the billet will increase greatly. A comparison of the billet center

temperature increase at the extraction of a billet when the furnace temperature of any zone is

increased 1 [K] from the standard temperature is shown in figure 4-1 (a). Figure 4-1 (b)

shows a comparison of the billet center temperature decrease at extraction of a billet when

the furnace temperature of any zone is decreased 1 [K] from the standard temperature. As a

result, zone 3 had the largest increase and decrease, because the furnace temperature was the

highest. Considered the upper limit temperature constraints for the furnace temperature in

zone 4, the control priority when the furnace temperature is increased is first zone 3,

followed by zone 2, then zone 1 and finally zone 4. However, even if there are billets in zone

3, when the gaps from their goal temperatures are small, choosing zone 3 is not preferable

because effective heat transmission rates in this zone will be small. Overall, the target zone

and time period will be selected by balancing policy 1 and policy 2 in this control method.

68

(a) 1 [K] increase from low temperature (b) 1 [K] decrease from high 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.

Once the priority is decided, minimizing the difference between the estimated billet

temperature and their goal temperature is chosen as the objective function for this problem.

Z = min ∑(Tex,i − Tg,i)

I

i=1

= min ∑ ∆Tex,i ⋯ (4.1)

I

i=1

where

i: Billet number

Tex,i: Sectional center temperature at extraction of billet i

Tg,i: Goal sectional center temperature at the extraciotn of billet i

0.3

0.5 0.50.4

1173

1273

1373

1333

1100

1150

1200

1250

1300

1350

1400

1450

1500

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

Zone1 Zone2 Zone3 Zone4

Atm

osp

her

e te

mp

erat

ure

[K

]

∆T

ex [K

]

0.30.4

0.50.4

1323

1373

1423

1383

1100

1150

1200

1250

1300

1350

1400

1450

1500

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0


Atm

osp

her

e te

mp

erat

ure

[K

]

-∆T

ex [K

]

69

In this simulation model, the maximum number of billets that can fit in the furnace at

the same time is 70. For a full billet load, the whole existing schedule should be considered

to minimize the fuel cost. Billets are continuously charged after the 70th billet is charged. If

the time horizon is infinity, the number of the decision variables becomes infinity, because it

is dependent on the number of the zones and the time periods. Therefore, the time horizon

must be limited. Since the temperatures and the specifications of the billets after the 70th

billet can affect the heat pattern of the 70th billet, the simulation must be extended beyond

just the first 70 billets. If the goal computation time for optimization is 10 [min], the number

of the billets to be increased can be minimized by running the simulation every 10 [min]. In

this research, an extra 15 billets, which is the maximum number of billets zone 3 and zone 4

can hold at the same time, was chosen as the modeling estimation range, considering the goal

computation time. That is, 85 billets are tracked in this optimization.

The same types of billets are usually scheduled for rolling at the same timing for

higher efficiency in downstream rolling operations. Let the group of the same billet type be

called a lot. Since each zone heats multiple billets at the same time, it is not necessary to fully

track all billets for control optimization. For example, by tracking the first billet and the last

billet in a lot and optimizing their heat pattern, the heat pattern of the other billets in the same

lot is also optimized (assuming that the furnace hearths themselves are at their steady state

operating temperature). If the number of the billets in a lot is large, the first billets or the last

billets in a lot are in different furnace zones. To avoid initial and final billet temperature

deviations, every certain number of billets in a lot, billets should be tracked in addition to the

first and the last billets. In this research, it was investigated every how many billets should be

tracked, balancing the computation time and the accuracy loss in the later section.

70

4.1.2. Decision variables

The controllable parameter for optimization is the setting furnace temperature of each

zone during a time period as follows.

Ta|j,𝑡𝑎: Setting furnace temperature of zone j at time period ta

where

j = 1,2 ⋯ : Zone number of the furnace

ta = 1, ⋯ , N: time periods for varialbes from current time

As the length of the time period decreases, the number of variables increases because

the time horizon is constant and the computation time increases. Additionally, there is a delay

in heating and cooling the furnace temperature. Even if the furnace temperature set point is

changed, there is a delay before the furnace temperature reaches the new set point

temperature. The length of this time period must be estimated. In this research, 5 [min] was

chosen as the length of time period ta for a furnace zone temperature set point change to

result in an actual furnace temperature change. In practice, the length of the time period

should be appropriately adjusted based on actual furnace operating performance data.

Although the length of the time period was set at 5 [min], the length of the time

period used for computing the temperature of billets should be shorter to obtain more precise

simulation. In chapter 3, the modeling time increments were evaluated comprehensively. A

modeling time period, of tp=30 [sec], can be used for the estimation of billet temperature.

4.1.3. Constraints

Key constraints in reheating furnace operation are the upper limits of the furnace

temperature in each zone. They are constrained by both furnace specification limits and

upper billet temperature restriction for guaranteeing billets quality. Furnace zone

specification temperature limits are constant but the upper limits required for billets quality

depend on what types of steel being heated in zone j during tp. Since the time period has 5

[min] length, the minimum upper limits among the billets in zone j during ta should be

71

selected as the upper limits in j during ta. Furnace zone lower limits are not restricted, but are

only limited by the goal temperatures of the billets in the furnace.

The heating rate and cooling rate of the furnace temperature are other

constraints. Even if the furnace temperature in zone j during time t is under the upper limit

after an increase, the temperature cannot be increased from time t-1 to t instantaneously.

There is a delay until the temperature reaches the set temperature in heating and in cooling.

Since the temperature during heating or cooling must not exceed the upper limit, these rates

become the constraints. For simplicity, it was assumed in this study that the furnace

temperature rises and declines linearly during set point changes. The rates for this simulation

are shown in table 4-1. Other more complex heat-up and cool-down assumptions could be

made; however, they are not expected to have a significant effect on the modeled billet

temperatures because furnace temperature set point changes during operation are very small.

From these constraints, the feasible region for this problem can be determined. Each

constraint is illustrated in figure 4-2 and the feasible region after consolidating the constraints

is shown in figure 4-3. When the variables in these simulations (the furnace temperature in

zone j during t) are changed, the temperature is assumed to be constant during ta in this

simulation. Thus, the discrete upper limits for variables were generated from the feasible

region. Figure 4-4 illustrates this discrete upper limit approach.

Table 4-1. Heating rate and cooling rate of furnace.

Rate Zone 1 Zone 2 Zone 3 Zone 4

Heating rate [K/sec] 1/6 1/6 1/6 1/6

Cooling rate [K/sec] 1/12 1/12 1/12 1/12

72

(a) Upper temperature limits by quality matter in zone j during time t

(b) Constraint by heating rate

(c) Constraint by cooling rate

Figure 4-2. Constraint example illustration.

Tem

per

atu

re

Time

ta

min(TUL,i,j,t)i

Tem

per

atu

re

Time

Tem

per

atu

re

Time

73

Figure 4-3. Feasible region after consolidating constraints.

Figure 4-4. Upper limits for discretized variables.

Tem

per

atu

re

Time

Tem

per

atu

re

Time

74

4.2. Optimization method

4.2.1. Outline of the optimization method

In practical operation of reheating furnaces for wires and bars, unexpected operational

changes occur frequently. For example, downstream rolling intervals are changed by other

processes in line and short stoppages happen when a small adjustment for machines is

required downstream in other process. In simulation, these unexpected operational changes

are not predictable. Every unexpected change in the billet delivery interval and in the rolling

speed requires a reheating furnace ‘adjustment’ to maintain its optimal operation. Ideally, the

reheating furnace adjustment recalculation is done without waiting time. If it takes 2 hours

for a simulation model to estimate a reheating furnace set point change, but the total furnace

holding time is about 1.5 hour, any corrective action suggestion will be too late. Hence, it is

the goal of this thesis to complete the computation within 10 [min] under practical conditions

with highly economical solution.

Figure 4-5 shows the main steps in this optimization method. Every iteration calls for

the billet temperature simulation program based on the 2-D dynamic programming model

outlined in chapter 2. The key concept of this proposed control method is how to optimize

the simulation model, shortening the computation time with the smallest loss in billet

temperature prediction accuracy. The major characteristics of this model are the following

five functions. Each step will be described individually in later sections.

1. Effective targeting of a furnace zone and a time period for temperature change

adjustments

2. Classified searching for efficient temperature change recommendations

3. Dynamic updating of feasible solution region

4. Amplifier and lower limiter of temperature change adjustments

5. Selective tracking of billets for simulation iterations

75

Figure 4-5. Main optimization steps.

Simulate billets temperature at extractionStep 2

and Obtain ΔTex

Step 3 Obtain the schedule matrix of billets

Step 4 Find the most effective zone and time period on

an increase of billets temperature among the

schedules of billets with ΔTex<0

Increase the temperature in zone j at time tStep 5

Repeat until All

ΔTex becomes

positive or stop at

certain iterations

Fix the variables including the schedule of

billets whose ΔTex<0 initially

and update the feasible region

Step 6

Step 7 Find the most effective zone and time period on

a decrease of billets temperature among the

schedules of billets with ΔTex>0 initially

Step 8 Decrease the temperature in zone j at time t

Repeat until ΔTex

is minimized or

stop at certain

iterations

Step 1 Set initial atmosphere temperature of each zone

76

4.2.2. Determining the initial solution

For the initial solution of this problem, the current furnace temperature, Ta,j,t, is used.

Also, current simulated billet temperatures are input into the established model, when the

simulation is carried out continuously and cyclically. The temperature of billet/bloom i at

extraction, Tex,i, is simulated for the initial state. Based on the target extraction temperatures

for each billet/bloom, Tp,i. The differences between Tex,i and Tp,i are calculated in (4.2).

∆Tex,i = Tex,i − Tp,i ⋯ (4. 2)

If ∆Tex,i is positive, this implies that the billet i will be over-heated. On the other

hand, if ∆Tex,i is negative, this means that it will be under-heated. Over-heating is acceptable

in practice though it is not ideal, whereas under-heating is unacceptable since it causes

operational troubles. Therefore, the furnace variables are updated as negative ∆Tex,i becomes

non negative. An increase of some variables may affect the extraction temperature of billets

with positive ∆Tex,i, because it is possible that billets with positive ∆Tex,i and other billets

with negative ∆Tex,i exist in the same furnace zone at the same time

77

4.2.3. Unit increment of furnace temperature

Targeted zones and time periods are selected as discussed previously. However, it is a

difficult to estimate how much the furnace zone temperature should be increased, because it

is difficult to estimate the effect of an increase of furnace temperature on the final extraction

temperature of billets analytically. Therefore, a simulation model to estimate billets

temperature is employed. To know the extent of the effects of changing the set point furnace

temperature, the simulation has to be run. In figure 4-6, a sample calculation result is shown.

It indicates how much the sectional center billet temperatures increase by increasing the

temperature in zone j by 1 [K], 25 [K], 50 [K]. The billet material properties are the same as

those used previously in chapter 3. According to this result, the change in billet temperature

is not likely to exceed the increment of the furnace temperature change. Therefore, ∆Tex can

be effectively used as the unit increment of furnace temperature.


Furnace temperature [K] 1173 1223 1273 1333

Holding time [min] 23 22.5 18 17.5

Figure 4-6. Relationship between an increase of furnace temperature and the resultant

increase in billet center temperature.

0

10

20

30

40

50

60

0 10 20 30 40 50 60An i

crea

se o

f si

mula

ted

cen

ter

tem

per

ature

at

extr

acti

on o

f a

bil

let

∆T

ex[K

]

An increasae of atmosphere temperature ∆Ta [K]


78

4.2.4. Determination of the schedule matrix and the upper limit of temperature change

Using pi,tcom from chapter 2, the zone where the billet i is staying during the

computational time period tcom can be found. Schedule matrix is defined as Ai,j,t.

Ai,j,t = {1 if billet/bloom i stays in zone j at time period t

0 o/w⋯ (4.3)

The schedules of billets with ∆Tex,i≥0 and billets with ∆Tex,i<0 are separately defined

in (4.4) and (4.5)

B+,i,,j,t = Ai,j,t|∆Tex,i≥0 ⋯ (4. 4)

B−,i,j,t = Ai,j,t|∆Tex,i<0 ⋯ (4. 5)

To find the upper limit of the temperature increment, C+,i,j,t and C-,i,j,t can be defined by (4.6)

and (4.7).

C+,i,j,t = {(∆Tex,ie1,NN) ∘ B+,i,j,t} ⋯ (4. 6)

C−,i,j,t = {(∆Tex,ie1,NN) ∘ B−,i,j,t} ⋯ (4. 7)

where

“ ◦ ” indicates the element-wise product.

e1,NN: defined as a unit matrix with 1 × NN in size and all the elements are 1.

79

The upper limit of the increase of the variable was decided by taking the minus of the

maximum C-,i,j,t among i.

∆TaUL,j,t = −maxIn∋i

C−,i,j,t ⋯ (4.8)

where In indicates which billets have negative ∆Tex,i in zone j during t.

In preparation for finding the effective zone and time period on an increase in billet

temperatures, the number of billets which have scheduled matrices with positive ∆Tex,i is

counted by (4.9).

Npm,j,t = ∑ B+,i,,j,t

I

i=1

⋯ (4.9)

Similarly, the number of billets which have scheduled matrices with negative ∆Tex,i is

counted by (4.10).

Nnm,j,t = ∑ B−,i,,j,t

I

i=1

⋯ (4.10)

80

4.2.5. Effective zone and time period targeting for estimating billet temperature changes

Initially, the most effective zone and time period for an increase in billet temperature

was evaluated from the first policy.

To find the most effective time period for reducing ∆Tex,i, an operational time period

for variables, ta, had to be employed. Initially a unit length of the time period, tu, 5 [min] was

selected, considering practical furnace operation. During this computation, no furnace control

action can be taken. Also, during the first operational period, the furnace temperature cannot

be increased because of the initial temperature adjustment calculation needed to reach the set

temperature at the first operational period. At this point in time, the computation time would

be 10 [min] and the unit length of the operational time period is 5 [min]. The total 15 [min]

initial time period has to be masked for the variables.

The time period of Nnmj,t is converted to operational time period.

Nnm,j,ta= ∑ Nnm,j,t

te

t=ts

⋯ (4.11)

where

ts = tfin + ta × u + 1

te = tfin + (ta + 1) × u

u =tu

tp

ta = 1 ⋯ fix{(NN − tfin − u)/u}

NN: the total number of computational time periods

tfin: the computational time until simulation completes

“fix” means taking the closest integer in direction to zero.

81

The upper limit of variable increment is shown in (4.12).

∆TaUL,j,t𝑎= min(∆TaUL,j,t𝑠

, ⋯ , ∆TaUL,j,t𝑒) ⋯ (4.12)

The score, Scnj,ta, in each zone at each operational time period is defined as (4.13) to

evaluate whether zone j and time period ta are the most effective or not.

Scn,j,ta= Nnm,j,ta

∘ ∆TaUL,j,ta⋯ (4.13)

Selecting zone j and time ta with the highest score directly equals the first policy.

In addition to the first policy, the second policy is reflected in (4.13) by exploiting the

weights of each furnace zone. Since the priority is zone 3, zone 2, zone 1 and zone 4, the

weights, wj, should be decided keeping this order, i.e. zone 3 = 1.5, zone 2 = 1.3, zone 3 =

1.1 and zone 4 =1.0. The scores of (4.13) are multiplied by wj as (4.14).

Scnw,j,ta= Scn,j,ta

∘ wj ⋯ (4.14)

Hence, the zone and the operational time period holding the maximum weighted score

becomes the candidate for an increase in a variable.

4.2.6. Classified searching for efficient temperature changes

Figure 4-7 illustrates an example of an increase in furnace temperature. Once the

furnace temperature in zone j during ta is increased, the furnace temperature before ta and

after ta is also increased because of the delay in heating and cooling. Moreover, the

82

influenced time range is dependent on the amount of the temperature increase. If the increase

is large, the time range becomes wide. Even if ∆TaUL,j,t𝑎 is large, it is not always the most

efficient to increase the temperature up to ∆TaUL,j,t𝑎, because other time periods might be

affected by increasing temperature. This implies that, if there are billets with positive ∆Tex

after and before the targeted period, those billets are heated more even though they have

already achieved their goal temperature. Therefore, not only billets with negative ∆Tex, but

also billets with positive ∆Tex must be evaluated for their extent of the influence by

increasing temperature.

Ideally, the delay effect should be evaluated for all levels of temperature increase.

However, this leads to a large increase in the computation load. To avoid this, five levels, TL,

are considered in this model. The first level is ≤25 [K] (=T1), the second level is ≤50 [K]

(=T2), the third level is ≤75 [K] (=T3), the fourth level is ≤100 [K] (=T4), and the fifth level is

>100 [K] (=T5). Each influence range is illustrated in figure 4-7.

Similarly with Nnmj,t, the number of billets with positive ∆Tex, during ta is counted

using (4.15) to evaluate the extent of over-heating.

Npm,j,ta= ∑ Npm,j,t ⋯ (4.15)

te

t=ts

The score is,

Scp,j,ta= Npm,j,ta

∘ ∆TaUL,j,ta⋯ (4.16)

The weighted score is,

Scpw,j,ta= Scp,j,ta

∘ wj ⋯ (4.17)

The overall score is expressed by (4.18).

Sct,j,ta= Scnw,j,ta

− Scpw,j,ta⋯ (4.18)

83

Figure 4-7. Influence range of each overheating level.

Tem

per

atu

re

Time

Initial temperature

Ea,j,ta-1 Ea,j,ta+1

Ea,j,ta

Tem

per

ature

Time

Tem

per

atu

re

Time

Tem

per

ature

Time

Level1

Level2

Level3

Level4

84

For the next step, the extent of the influence is calculated geometrically. Let the area

in zone j during ta be Ea,j,ta and the affected range of operational time period be from trs to tre.

Sco,j,ta= ∑ (Sct,j,ta

∘ Ea,j,ta)

tre

ta=trs

= ∑ [{(Npm,j,ta− Nnm,j,ta

) ∘ ∆TaUL,j,ta∘ wj} ∘ Ea,j,ta

]

tre

ta=trs

⋯ (4.18)

This is the score when the furnace temperature in zone j during t is increased ∆TaUL,j,ta. By

substituting ∆TaUL,j,ta with TL if all the temperature levels are lower than ∆TaUL,j,ta

, each

score can be calculated. The temperature with the minimum score is selected as an increase

of furnace temperature shown in (4.19). Let the targeted zone be j=jtar, and operational time

period be ta=ttar for further discussion. Updated set furnace temperature can be obtained by

(4.20).

Find j and ta holding the minimum Sco,j,ta

∆Ta = −min (∆TaUL,jtar,ttar, TL|<∆TaUL,jtar,ttar) ⋯ (4.19)

Ta,jtar,ttar = Ta,jtar,ttar + ∆Ta ⋯ (4.20)

Using this new furnace temperature, the temperatures at the extraction of billets are

recalculated, and ∆Tex,i is updated. After increasing the furnace temperature, the temperature

during heating or cooling has a gradient. However, when the time period is chosen for an

increase of temperature for later iterations, the values must be constant within the operational

time period. Thus, the updated heat pattern is expressed in a discrete way as indicated in

figure 4-8 for the next iteration.

85

(a) Base line for increase at the next iteration

(b) Expression in discrete manner

Figure 4-8. Converting the updated heat pattern to a discrete expression.

This process is iterated until the minimum ∆Tex,i ≥0 or any Sct,j,ta >0. When some

billet i has a negative ∆Tex,i but any Sct,j,ta >0, Scnw,j,ta

is used as the score to decide the target

instead of Sct,j,ta. From this point, an increase in furnace temperature always leads to

overheating other billets. Also, it is necessary to run the iteration until the minimum ∆Tex,i ≥0

or any Scn,j,ta =0. If all the furnace temperatures in zone j during ta holding the billets with

∆Tex,i <0, reach their upper limits and the minimum ∆Tex,i <0, this means that there is no

solution under the current furnace condition, and the furnace holding time has to be extended.

After this increase phase finishes, a decrease phase starts as the next step.

Tem

per

ature

Time

Tem

per

atu

re

Time

86

4.2.7. Updating the feasible region

After the increase phase finishes, the feasible region is updated before the decrease

phase. The temperature of some billets reaches their goal temperature by an increase in

furnace temperature. If the furnace temperature during holding them is decreased, the

temperatures of the billets go below their goal temperatures again. However, there is a case

when these billets are confined with the billets that were initially overheated. The

temperature of these billets should be low. The billets that initially reached their goal

temperature and the billets that reached their goal temperature after the increase phase should

be clearly distinguished. Referring to the heating schedule of the billets that reach their goal

temperature after the increase phase, the furnace temperature in zone j during computational

time period t holding the billets is fixed as the new lower limit of the feasible region. Figure

4-6 shows the procedure to update the feasible region after N increase phase.

When a lower limit of furnace temperature is initially set, the feasible region is

updated again. Considering heating and cooling rates, the line indicating their lower limits

during heating or cooling is prolonged until it crosses the lower limit as shown in figure 4-10.

Also, this line is expressed for variables in a discrete manner shown in figure 4-11.

Figure 4-9. Updated feasible region of furnace temperatures.

Tem

per

atu

re

Time

Lower limit

87

Figure 4-10. Prolongation of heating and cooling phases.

Figure 4-11. Updated discrete lower limits for the variables.

Tem

per

ature

Time

Lower limit

Tem

per

atu

re

Time

88

4.2.8. Decrease phase

After the feasible region is updated, the decrease phase starts. The basic control

strategy is the same as the increase phase. The upper limit of a decrease of furnace

temperature is expressed by (4.21).

∆TaUL,j,t = minIp∋i

C+,i,j,t ⋯ (4.21)

Also, the upper limit for the operational time period of the variables is expressed by (4.22).

∆TaUL,j,t𝑎= min(∆TaUL,j,t𝑠

, ⋯ , ∆TaUL,j,t𝑒) ⋯ (4.22)

The total score is calculated by (4.23).

Sct,j,ta= Scpw,j,ta

− Scnw,j,ta⋯ (4.23)

The overall score is calculated by (4.24).

Sco,j,ta= ∑ (Sct,j,ta

∘ Ea,j,ta)

tre

ta=trs

= ∑ [{(Npm,j,ta− Nnm,j,ta

) ∘ ∆TaUL,j,ta∘ wj} ∘ Ea,j,ta

]

tre

ta=trs

⋯ (4.24)

89

Find j and ta holding the maximum Sco,j,ta

dTa = max (∆TaUL,j,ta, TL|<∆TaUL,j,ta

) ⋯ (4.25)

Ta,jtar,ttar = Ta,jtar,ttar + dTa ⋯ (4.26)

When there is long space before the initial charged billet in the simulation, the

temperature of the first billet becomes much higher than other billets at extraction because

there are no obstacles in front of the first billet and it receives the most intense thermal

radiation. In this case, dTa is decided from the temperature of the first billet, due to its high

score. To avoid this, the first billet should be removed from the billet candidates for a

decrease of furnace temperature.

In the increase phase, when all Sct,j,ta becomes positive, Scnw,j,ta

is used instead of

Sct,j,ta. However, in the decrease phase, the temperature at extraction is not allowed to be

lower than the goal temperature. Therefore, once all Sct,j,ta becomes negative, and the

decrease phase finishes.

4.2.9. Final treatment for the optimal control solution

In the range of considered time periods, there are zones and time periods with no

billets (for example, in zone 2 to zone 4 just after the start of charging). During these periods,

the furnace temperature of each zone should be lowered as low as possible. This method

searches the furnace zone and the time periods for which there are no billets and decreases

the temperature considering the heating and the cooling rates.

Eventually, the optimal heat pattern in the furnace is achieved.

90

4.2.10. Initial performance check

By calculating the developed optimization programming, an optimal solution was

found for an example case. The computational condition is shown in table 4-2. Figure 4-12

illustrates the obtained heat pattern of each zone over the entire reheating cycle by this

optimization programming before final treatment. Figure 4-12 illustrates optimal heat pattern

after final treatment.

Figure 4-8 shows a change from the initial ∆Tex to ∆Tex after optimization. All the

initial ∆Tex had negative values. After optimization, all the billets had higher temperature

than their goal temperature. In front of the first billet, there was no billet in the furnace. Thus,

it receives more thermal radiation than other billets, especially from the hearths, so that the

temperature was much higher than that of the second billet.

The average of initial ∆Tex was -23.14 [K] and the average of ∆Tex after optimization

was 2.52 [K]. A large improvement in ∆Tex by this optimization was confirmed.

However, the computation time was 134,532 [sec] ≈ 37.37 [hour]. This is too long for

practical control of reheating operations. The main reason of this long computation time is

that the increment of furnace temperature becomes smaller as the computation is iterated and

∆Tex becomes smaller. Figure 4-15 shows the ∆Tex history for iterations. After 143 iterations,

dTa always became under 1 [K]. This implies that the solutions are almost the same after 143

iterations since the increment is quite small. Even if the computation stops at 143 iterations,

it will take about 1.8 [hour] instead of 37.37 [hour]. Therefore, it is necessary to shorten the

computation time even further for practical operation.

Table 4-2. Computational conditions for optimization.

Goal temperature

of all the billets [K]

Initial furnace temperature [K] Weights

Zone 1 Zone 2 Zone 3 Zone 4 Zone 1 Zone 2 Zone 3 Zone 4

1133 1223 1273 1273 1333 1.2 1.4 1.5 1.0

91

Figure 4-12. Obtained heat patterns for each zone before final treatment.

Figure 4-13. Obtained optimal heat patterns for each zone after final treatment.

1000

1050

1100

1150

1200

1250

1300

1350

1400

1450

1

14

27

40

53

66

79

92

105

118

131

144

157

170

183

196

209

222

235

248

261

274

287

300

313

326

339

352

365

378

Atm

osp

her

e te

mp

erat

ure

[K

]

Time [×30sec]


1000

1050

1100

1150

1200

1250

1300

1350

1400

1450

1

14

27

40

53

66

79

92

105

118

131

144

157

170

183

196

209

222

235

248

261

274

287

300

313

326

339

352

365

378

Atm

osp

her

e te

mp

erat

ure

[K

]

Time [×30sec]


92

Figure 4-14. Improvement of ∆Tex for each billet after optimization.

Figure 4-15. dTa history of each iteration.

-60

-40

-20

0

20

40

60

#1

#4

#7

#10

#13

#16

#19

#22

#25

#28

#31

#34

#37

#40

#43

#46

#49

#52

#55

#58

#61

#64

#67

#70

#73

#76

#79

#82

#85

∆T

ex[K

]

Billet number

∆Tex after optimization Initial ∆Tex

0

5

10

15

20

25

30

35

40

1

91

181

271

361

451

541

631

721

811

901

991

108

1

117

11

26

11

35

1

144

1

153

1

162

1

171

1

180

1

189

1

198

1

207

1

216

12

25

1

234

1

243

1

252

1

261

1

270

1

279

1

288

1

dT

a [K

]

Number of iterations

93

4.3. Shortening computation time

4.3.1. Amplifier and lower limiter for furnace temperature changes

The computation time must be shortened still further for practical use. One of the

solutions is to set lower limits for dTa, since the main cause of the long time computation is

extremely small dTa after some iterations. Figure 4-16 shows the relationship between dTa

and ∆Tex, when the furnace temperature of a zone is increased from relatively low furnace

temperatures. Figure 4-17 illustrates the same relationship when the furnace temperature of a

zone is decreased from relatively higher furnace temperatures. In figure 4-16, if 15 [K] is the

lower limit of dTa, about 4.3 [K] is always expected to be the increment of ∆Tex, when dTa is

less than 10 [K]. If the lower limit is 10 [K] and dTa is less than 10 [K], the expected

increment of ∆Tex is 3 [K]. Also, if the lower limit is 5 [K] and dTa is less than 5 [K], the

expected increment of ∆Tex is 1.5 [K]. In figure 4-17, a similar tendency was also observed.

These expected values can be the maximum over-heat or under-heat for this atmosphere

condition. The level of the lower limit should be decided considering how much over-heat or

under-heat can be accepted. Additionally, when the furnace temperature changes, the

gradient of the lines in figure 4-16 and 4-17 would change. Therefore, it is important to find

well balanced level of the lower limit in the computation time and the over-heat or under-

heat.

In figure 4-16, the gradient of the line in zone 1was the smallest. When dTa=100 [K],

∆Tex=28.5 [K]. The gradient was about 3.5. This indicates that there is an opportunity to

amplify 3.5 to ∆Tex for an increase of furnace temperature. Hence, it is possible to boost dTa

by amplifying for the shorter computation time. The gradient also changes if the furnace

temperature changes, as shown previously. Therefore, this amplifying level also must be

carefully chosen by balancing the computation time and the over-heat or under-heat

thresholds.

94

Figure 4-16. Relationship between dTa and ∆Tex for low furnace temperature.

0

5

10

15

20

25

30

35

40

45

50

0 20 40 60 80 100 120

∆T

ex[K

]

∆Ta [K]


0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

∆T

ex[K

]

∆Ta [K]


Goal temperature




1133 1173 1273 1373 1333 1.2 1.4 1.5 1.0

95

Figure 4-17. Relationship between dTa and ∆Tex for high furnace temperature.

0

5

10

15

20

25

30

35

40

45

50

0 20 40 60 80 100 120

-∆T

ex[K

]

∆Ta [K]


0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

∆T

ex [K

]

∆Ta [K]


Goal temperature




1133 1323 1373 1423 1333 1.2 1.4 1.5 1.0

96

4.3.2. Selective billet tracking

The temperature of all billets in the furnace has been simulated so far. However, if

their reheating specification is the same, it might be possible to remove some billets from the

simulated billets. Selecting tracked billets appropriately can shorten the computation time in

the optimization.

When tracked billets are selected, the hearth temperature is of concern. The

temperatures of the hearths are affected by the temperature of the entering billets. This

implies that all the temperature of billets must be simulated. If tracked billets are selected and

the temperatures of the other billets are not simulated, the hearth temperature cannot be

correctly calculated. However, the computation of all the billets temperature takes hours. To

overcome this difficulty, the bottom face temperature of the billets which are not simulated in

optimization was linearly approximated using the temperature of foregoing billets which are

tracked. From the position of the tracked billets, the bottom face temperature of unfocused

billets is calculated by (4.21).

Tub = Tfb −Tfb − Tofb

Pfb× (Pfb − Pub) ⋯ (4.21)

where

Tub: Average bottom face temperature of untracked billets

Tfb: Average bottom face temperature of foregoing tracked billets

T0fb: Initial average bottom face temperature of foregoing tracked billets

Pfb: Position of foregoing tracked billets

Pub: Position of untracked billets

97

4.3.3. Effects of selective tracking, amplifying and lower limiter

To find appropriate parameters, the effects of selective tracking, amplifying and

lower limiter on the computation time and the accuracy were investigated.

Figure 4-18 shows the comparison of the computation time in 6 different cases to

determine the appropriate number of billets to be tracked. The differences of the cases are

how many billets are tracked and fully simulated for their temperature. When all of the billets

were tracked and the operation was optimized by this method, the computation time was

2460 [sec] with amplifier 3 and a lower limiter of 5 [K]. This computational simplification

exceeds the goal computation time. Considering a case that it takes extra time to complete

computing, case 3 is a candidate for selective billets tracking.

Goal temperature


Furnace

holding

Time [min]

Initial furnace temperature [K] Amplifier Lower

limiter

[K] Zone 1 Zone 2 Zone 3 Zone 4

1133 82 1223 1273 1273 1333 3 5

Case 1 Case 2 Case 3 Case 4 Case 5 Case 6

#1,#2,#85

#1,#2, every 10

billets from #10

and #85

#1,#2, every 7

billets from #8,

and #85

#1,#2, every 5

billets from #6,

and #85

#1,#2, every 3

billets from #4,

and #85

All 85

billets

Figure 4-18. Comparison of computation time based on the number of tracked billets.

33 188

359 545

798

2,460

-

500

1,000

1,500

2,000

2,500

3,000

3

Case 1

11

Case 2

15

Case 3

20

Case 4

30

Case 5

85

Case 6

Co

mp

uta

tio

n t

ime

[sec

]

The number of tracked billets in simulation

Goal computation time: 600 [sec]

98

Figure 4-19 shows the average of ∆Tex and the minimum of ∆Tex in 6 cases. The error

bars indicated ±1σ. Case 6 has relatively high average temperature, but the minimum is

positive while the minimum for the other cases is negative. This implies that all the billets

satisfy the goal temperature in case 6, whereas some billets are over-heated or under-heated

in the other cases. If there is an acceptable range of ∆Tex, such as ±3 [K] or ±5 [K], the other

cases, except case 1, can be used, because the averages are not different from that in case 6.

Now, since case 6 is not realistic due to the computation time, case 3 is the most appropriate

in this condition. In later discussion, case 3 is employed for this reason.

Figure 4-19. Average ∆Tex and minimum ∆Tex for the various cases.

(1.8)

4.1 3.6 3.5

2.0

4.0

-23.2

-1.2 -1.6 -1.4 -1.6

0.3

(25.0)

(20.0)

(15.0)

(10.0)

(5.0)

-

5.0

10.0

Case 1 Case 2 Case 3 Case 4 Case 5 Case 6

Tem

per

ature

[K

]

Average Min

99

Figure 4-20 shows a comparison of the computation time for various lower limits and

amplifiers. If there is no lower limiter, the effective computation time becomes 2106 [sec]

with amplifier 3, even if tracked billets are selected. When 5 [K] or 10 [K] are chosen as the

lower limiter, the computation time was dramatically reduced, and all the computation times

were within 10 [min]. On the other hand, the effect of amplifier on the reduction of

computation time is smaller than that of lower limiter. Figure 4-21 illustrates the average of

∆Tex and ±1σ range in various amplifier and lower limiters. Also, figure 4-22 shows the total

over-heat for various amplifier and lower limiters. The total over-heat was obtained by

multiplying the specific heat of billets to each ∆Tex and taking the summation. As a result,

when lower limiter 10 [K] and amplifier 1.5 to 2.0, the total over-heat was higher, and the

average of ∆Tex was higher as well. When the lower limiter is 5 [K], or when the lower

limiter 10 [K] and the amplifier 1, the total heat, the average of ∆Tex and the computation

time are well balanced. The combination of the lower limiter 10 [K] and the amplifier 2.5 to

3 has a potential for high total over-heat, because higher values were observed in amplifier

1.5 and 2.0. In summary, 10 [K] as the lower limiter, 1 as the amplifier (no amplifier) and

case 3 as selective tracking were selected for control optimization in this thesis.

Goal temperature


Furnace

holding

Time [min]

Initial furnace temperature [K] Tracked

billets Zone 1 Zone 2 Zone 3 Zone 4

1133 82 1223 1273 1273 1333 Case 3

Figure 4-20. Computation time comparison for various amplifiers and lower limiters.

2,106

462 444 393 364 359 364 309 289 257 213

-

500

1,000

1,500

2,000

2,500

3 1 1.5 2 2.5 3 1 1.5 2 2.5 3

Lower

limiter 0

[K]

Lower limiter 5 [K] Lower limiter 10 [K]

Co

mp

uta

tio

n t

ime

[sec

]

Amplifier

100

Figure 4-21. Average of ∆Tex and ±1σ range for various amplifiers and lower limiters.

Figure 4-22. Total over-heat for 85 billets for various amplifiers and lower limiters.

(4.00)

(2.00)

-

2.00

4.00

6.00

8.00

10.00

12.00

14.00

3 1 1.5 2 2.5 3 1 1.5 2 2.5 3

Lower

limiter

0 [K]


Aver

age

of

∆T

ex[K

]

Amplifier

663 625 590 609

501 535

585 718

902

595 602

-

100

200

300

400

500

600

700

800

900

1,000

3 1 1.5 2 2.5 3 1 1.5 2 2.5 3

Lower

limiter

0 [K]


To

tal

over

-hea

t [k

J]

Amplifier

101

4.4. Overall Control Performance

4.4.1. Fundamental example

By calculating the developed optimization programming, an optimal solution was

found. The computational condition is shown in table 4-3. Figure 4-23 illustrates obtained

heat pattern of each zone over the whole reheating time period by this optimization

programming. Figure 4-24 and 4-25 show a comparison in heat pattern between a case with

no amplifier, no lower limiter and all billets tracked, and a case with no amplifier, 10 [K]

lower limiter and case 3 selective billets tracking. The heat patterns are almost similar in all

the zones and the computation time under this condition was 212.8 [sec] ≈ 3.5 [min]. This

computation time is short enough for use in a reheat control strategy.

Figure 4-23. Obtained optimal heat patterns for each zone.

1000

1050

1100

1150

1200

1250

1300

1350

1400

1450

1

15

29

43

57

71

85

99

113

127

141

155

169

183

197

211

225

239

253

267

281

295

309

323

337

351

365

379

Atm

osp

her

e te

mp

erat

ure

[K

]

Time [×30sec]


Table 4-3. Computational condition for optimization.

Goal temperature




1133 1223 1273 1273 1333 1.2 1.4 1.5 1.0

102

(a) Zone 1

(b) Zone 2

Figure 4-24. Difference in heat pattern between lower limiter 0 and 10 [K] (1).

1000

1050

1100

1150

1200

1250

1300

1350

1400

1450

1

14

27

40

53

66

79

92

105

118

131

144

157

170

183

196

209

222

235

248

261

274

287

300

313

326

339

352

365

378

Op

tim

al

atm

osp

her

e te

mp

era

ture

[K

]

Time [×30sec]

Amplifier 1, Lower limter 0 [K] Amplifier 1, Lower limter 10 [K]

1000

1050

1100

1150

1200

1250

1300

1350

1400

1450

1

14

27

40

53

66

79

92

105

118

131

144

157

170

183

196

209

222

235

248

261

274

287

300

313

326

339

352

365

378

Op

tim

al

atm

osp

her

e te

mp

era

ture

[K

]

Time [×30sec]


103

(c) Zone 3

(d) Zone 4

Figure 4-25. Difference in heat pattern between lower limiter 0 and 10 [K] (2).

1000

1050

1100

1150

1200

1250

1300

1350

1400

1450

1

15

29

43

57

71

85

99

113

127

141

155

169

183

197

211

225

239

253

267

281

295

309

323

337

351

365

379

Op

tim

al

atm

osp

her

e te

mp

era

ture

[K

]

Time [×30sec]


1000

1050

1100

1150

1200

1250

1300

1350

1400

1450

1

15

29

43

57

71

85

99

113

127

141

155

169

183

197

211

225

239

253

267

281

295

309

323

337

351

365

379

Op

tim

al

atm

osp

her

e te

mp

era

ture

[K

]

Time [×30sec]


104

Figure 4-26 shows a change from the initial ∆Tex to ∆Tex after optimization in each

billet. Some billets have negative values, but they are small enough to be ignored. Figure 4-

27 shows a comparison of ∆Tex between the two cases. In the case of lower limiter 10 [K],

∆Tex is slightly higher than those of the case of no lower limiter. The average of ∆Tex and the

minimum of ∆Tex are shown in figure 4-28. Based on this result, the temperature difference

was 1.4 [K]. The computation time is shortened by sacrificing this amount of heat under this

operational condition. If this amount of heat is not acceptable, lower limiter must be reduced

or the number of tracked billets should be increased.

Figure 4-26. Change of ∆Tex before and after optimization.

-60

-50

-40

-30

-20

-10

0

10

20

30

40

50

60

#1 #4 #7 #10#13#16#19#22#25#28#31#34#37#40#43#46#49#52#55#58#61#64#67#70#73#76#79#82#85

∆T

ex [

K]

Initial ∆Tex ∆Tex after optimization

105

Figure 4-27. Comparison of ∆Tex between lower limiter 0 and 10 [K].

Figure 4-28. Average ∆Tex and minimum ∆Tex for different lower limiter conditions.

-10

0

10

20

30

40

50

60#

1

#4

#7

#10

#13

#16

#19

#22

#25

#28

#31

#34

#37

#40

#43

#46

#49

#52

#55

#58

#61

#64

#67

#70

#73

#76

#79

#82

#85

∆T

ex [

K]

Billet number


2.5

3.9

-0.7-1.6

(4.0)

(2.0)

-

2.0

4.0

6.0

8.0

10.0

12.0


Tem

per

ature

[K

]

Average of ∆Tex,i Minimum ∆Tex,i

106

4.4.2. Effects of initial furnace temperature

When the initial furnace temperature is set at different level, the optimal heat pattern

might be different. In this section, the performance of this model for various initial furnace

temperatures was evaluated. Table 4-4 indicates four cases with different initial furnace

temperatures.

Figure 4-29 and 4-30 illustrate the optimized heat patterns by this model. Each case

had a different optimal heat pattern. In zone 1 in figure 4-29, the temperature in case 1 and

case 2 decreased from their initial temperature, whereas there was no change in case 3 and

case 4. This implies that the optimal heat pattern is changed slightly if the initial furnace

temperature is different. In figure 4-31, the average of ∆Tex and the minimum of ∆Tex were

compared in the four cases. Case 1 and case 2 showed high average and high minimum

temperatures. On the other hand, case 3 and case 4 showed low average and low minimum

temperatures. In case 3 and case 4, they experience increase phase, that is, the initial ∆Tex are

both negative. Therefore, once ∆Tex enters the acceptable range, the computation stops even

if the value is negative. This algorithm seems to cause the low minimum temperature in case

3 and case 4. Conversely, in case 1 and case 2, ∆Tex starts from positive values. Hence, it is

thought that the average and the minimum were relatively high.

Figure 4-32 shows a comparison of the computation time. Since the optimal heat

pattern was close to the initial furnace temperature in case 3 and case 4, these cases had

shorter computation time.

Overall, because safety in operation is a high priority, the initial furnace temperature

should be set at a slightly high level. If the cost is the priority, then it should be set at a

slightly low level.

Table 4-4. Different initial furnace temperatures.


Case 1 1323 1373 1423 1333

Case 2 1323 1323 1373 1333

Case 3 1223 1273 1323 1333

Case 4 1223 1273 1273 1333

107

(a) Zone 1

(b) Zone 2

Figure 4-29. Heat pattern differences for various initial furnace temperatures (1).

108

(c) Zone 3

(d) Zone 4

Figure 4-30. Heat pattern differences for various initial furnace temperatures (2).

109

Figure 4-31. Average ∆Tex and minimum ∆Tex for various initial furnace temperatures.

Figure 4-32. Computation time and number of iterations for various initial furnace

temperatures.

5.6 6.3

3.7 3.9

-1.02

0.12

-1.68 -1.55

(5.0)

(3.0)

(1.0)

1.0

3.0

5.0

7.0

9.0

11.0

13.0

15.0

Case 1 Case 2 Case 3 Case 4

Tem

per

ature

[K

]

Average of ∆Tex,i Minimumof ∆Tex,i

546 535

333364

62 62

39 38

0

10

20

30

40

50

60

70

80

0

100

200

300

400

500

600

700

800

Case 1 Case 2 Case 3 Case 4

Num

ber

of

iter

atio

ns

Co

mp

uta

tio

n t

ime

[sec

]

Computation time Number of iterations

110

4.4.3. Effects of inserting billets with higher goal temperatures

The case that all the billets have the same specifications has been evaluated so far.

The optimization performance was further investigated in a case which 22 billets with higher

goal temperature, 1213 [K] which is 80 [K] higher, are inserted as the 19th to 40th billets.

Figure 4-33 and 4-34 show the resultant heat pattern. Because of the high goal

temperatures, the temperature in zone 1 and zone 2 showed higher temperature when the

billets entered those zones. Also, figure 4-35 shows the computation time. Since the optimal

heat pattern became further from the initial furnace temperature by inserting billets with a

higher goal temperature, the computation time became longer.

Figure 4-36 indicates the average and the minimum of ∆Tex. Compared to a case that

all the billets have the same goal temperature, the average became higher. Also, the

minimum was -11.3 [K] – a value greatly lower than their goal temperature. Figure 4-36

shows the ∆Tex change of each billet. According to this result, the first 3 billets and the last 3

billets among the billets with high goal temperature had large negative values. In figure 4-37,

the ∆Tex change of tracked billets in the optimization. All ∆Tex were positive. From these

data, the first 3 and the last 3 billets with high goal temperature were not tracked in this

optimization. Therefore, it can be said that they had large negative values in ∆Tex.

Now, to track those billets, the inserting positions were shifted 3 billets later in the

next trial. In this case, the positions of the inserted billets are from 22th to 43th billet. Then,

the obtained ∆Tex of each billet is shown in figure 4-39. All the values became positive. Also,

in figure 4-40, the average and the minimum are indicated. After shifting the billets, the

average was increased and the minimum was positive as mentioned above.

Consequently, if billets with different goal temperature are mixed in a rolling

schedule, the first and the last billets of each type of steel should be tracked to avoid failing

to reach their goal temperature. Also, in a case that billets having high goal temperature are

inserted, the billets with low goal temperature are overheated, because the furnace

temperature starts to be increased before the billets with high goal temperature are fed into

the zone. Also, the initial billets with low goal temperature are overheated, because of the

delay of the furnace temperature. Since energy loss is caused by goal temperature changes

111

during the rolling schedule, consolidating the same type of billets in rolling schedules is

important to reduce this energy loss.

(a) Zone 1

(b) Zone 2

Figure 4-33. Heat pattern of billets with high goal temperatures (1).

1000

1050

1100

1150

1200

1250

1300

1350

1400

14501

13

25

37

49

61

73

85

97

109

121

133

145

157

169

181

193

205

217

229

241

253

265

277

289

301

313

325

337

349

361

373

385

Sim

ula

ted

atm

osp

her

e te

mp

erat

ure

[K

]

Time [×30sec]

Inserted high goal temperature billets Only same type of billets

High goal temperature billets

1000

1050

1100

1150

1200

1250

1300

1350

1400

1450

1

13

25

37

49

61

73

85

97

109

121

133

145

157

169

181

193

205

217

229

241

253

265

277

289

301

313

325

337

349

361

373

385

Sim

ula

ted

atm

osp

her

e te

mp

erat

ure

[K

]

Time [×30sec]



112

(c) Zone 3

(d) Zone 4

Figure 4-34. Heat pattern of billets with high goal temperatures (2).

1000

1050

1100

1150

1200

1250

1300

1350

1400

1450

1

12

23

34

45

56

67

78

89

100

111

122

133

144

155

166

177

188

199

210

221

232

243

254

265

276

287

298

309

320

331

342

353

364

375

386

Sim

ula

ted

atm

osp

her

e te

mp

erat

ure

[K

]

Time [×30sec]



1000

1050

1100

1150

1200

1250

1300

1350

1400

1450

1

12

23

34

45

56

67

78

89

100

111

122

133

144

155

166

177

188

199

210

221

232

243

254

265

276

287

298

309

320

331

342

353

364

375

386

Sim

ula

ted

atm

osp

her

e te

mp

erat

ure

[K

]

Time [×30sec]



113

Figure 4-35. Computation time and number of iterations for a case with high goal

temperature billets.

Figure 4-36. Average ∆Tex and minimum ∆Tex of a case with high goal temperature billets.

478

364

51

38

0

10

20

30

40

50

60

70

80

0

100

200

300

400

500

600

700

800


Num

ber

of

iter

atio

ns

Co

mp

uta

tio

n t

ime

[sec

]

Computation time Number of iterations

14.2

3.9

-11.3

-1.6

-20

-10

0

10

20

30

40


Tem

per

ature

[K

]


114

Figure 4-37. Change of ∆Tex before and after optimization in a case having high goal

temperature billets.

Figure 4-38. Change of ∆Tex before and after optimization of tracked billets for a case having

high goal temperature billets.

-140

-120

-100

-80

-60

-40

-20

0

20

40

60

80

100

#1

#4

#7

#10

#13

#16

#19

#22

#25

#28

#31

#34

#37

#40

#43

#46

#49

#52

#55

#58

#61

#64

#67

#70

#73

#76

#79

#82

#85

∆T

ex,i

[K]

After optimization Before optimization

-140

-120

-100

-80

-60

-40

-20

0

20

40

60

80

100

#1 #2 #8 #15 #22 #29 #36 #43 #50 #57 #64 #70 #71 #78 #85

∆T

ex,i

[K]


115

Figure 4-39. Change of ∆Tex before and after optimization for a case having high goal

temperature billets with shifting the tracked billets.

Figure 4-40. Average ∆Tex and minimum ∆Tex for a case having high goal temperature billets

with shifting the tracked billets.

-140

-120

-100

-80

-60

-40

-20

0

20

40

60

80

100

#1

#4

#7

#10

#13

#16

#19

#22

#25

#28

#31

#34

#37

#40

#43

#46

#49

#52

#55

#58

#61

#64

#67

#70

#73

#76

#79

#82

#85

∆T

ex,i

[K]


14.2

3.9

24

-11.3

-1.6

0.1

-20

-10

0

10

20

30

40

50

60

Inserted high goal temperature

billets

Only same type of billets After shifting

Tem

per

ature

[K

]


116

4.4.4. Initial control action when unexpected stoppage occur

When unexpected stoppages, such as operational troubles, occur, the operator who is

responsible for the stoppage announces the expected time to fix. Unless the time is short, the

furnace temperature of a furnace should be reduced temporally to avoid billet overheating.

Just after the announcement, this simulation starts to find the optimal furnace temperature

based on the operators’ stoppage time estimate. During computing, the heat to the reheating

furnace is stopped. From this, the billet temperature is expected to decline during computing.

Then, the predictively reached temperature after the computation time is determined as the

initial temperature for optimization.

4.4.5. Adjustment of furnace temperature

The actual furnace temperature does not always follow the furnace temperature set

points. Since this causes errors in simulated billet temperatures, adjustment is necessary to

maintain the prediction accuracy of the simulation model every computation. For this

adjustment, the simulation using the real historic data should be run in parallel. The

simulated billet temperature and the hearth temperature should be used at the next

computation.

117

Chapter 5. CONCLUSION

118

5.1. Conclusion summary

In this research, an improved method for reheating furnace operation in rolling mills

has been proposed in order to achieve the most economical operation under various

circumstances. Since furnace temperature is not stable and is not maintained at the same

level, a dynamic programming was developed to estimate billets temperature. However, the

computation time for the initial complex model was the difficulty for real time operation.

This research also has focused on how to shorten the computation time without large loss in

temperature prediction accuracy allowing the model to be used for real time reheating

furnace temperature control.

A simulation model for billet temperature was created using Matlab software and a

commercial personal computer. The major features of this model are;

1. Employing a 3 dimensional difference method and a dynamic programming

2. Tracking the positions of billets in the reheating furnace every time period

3. Estimating hearths temperature and heat transmission from the hearths every time

period

4. Calculating various view factors based on the geometry of billets and furnace

structure

The most important temperature estimate for the accurate control of billet temperature

is the sectional center temperature in the middle of furnace width, which is the lowest

temperature of a billet and is the bottle neck in terms of rolling load during subsequent

rolling.

When the sectional center temperature of billets is considered, the heat transmission

in longitudinal direction of the billet can be considered negligible. Hence, the dimension of

the control model can be reduced from 3-D to 2-D. Also, 55 [mm] × 55 [mm] as the unit

119

mesh size of a billet in the model and 30 [sec] as time increments were selected, balancing

the computation time and the temperature prediction accuracy.

Exploiting this model, an optimization method was developed. This method utilized

two policies.

1. The zone and the timing, where the total gap between goal temperature and simulated

center temperature of billets at extraction is the largest, were selected to increase or

decrease the furnace temperature as a priority.

2. The priority in increasing furnace temperature is first zone 3, then zone 2, then zone1

and finally zone 4, because of the differences in heat transfer efficiency in the various

furnace zones.

The created optimization method has the following characteristics.

1. Effective targeting of a furnace zone and a time period for temperature change

adjustments

2. Classified searching for efficient temperature change recommendations

3. Dynamic updating of feasible solution region

4. Amplifier and lower limiter of temperature change adjustments

5. Selective tracking of billets for simulation iterations

Appropriate selection of parameters and dimension reduction in the billet simulation

model, selective billet tracking, and amplifier and lower limiter considerations in the

optimization could effectively shorten the computation time. Consequently, completing the

computation within 10 [min] was achieved without large prediction accuracy loss under

practical conditions.

120

5.2. Insight for better furnace structure based on simulation results

From the discussion so far, some insights for improved future furnace specification

and operation were obtained.

In high temperature circumstances, the heat transmission by thermal radiation is the

largest on the surface of billets. To improve the heat efficiency, it is possible to use furnace

bricks with higher emissivity for furnace walls. In terms of operation, since the center billet

temperature in the middle of furnace width is the ‘bottle neck’, it is effective to increase the

furnace temperature around the section by increasing input of fuel gas near the center of the

billet. Also, a rise of view factor is effective. For example, the ceiling height should be as

low as possible, considering combusted gas convection.

From optimization point of view, the responsivity of furnace temperature to variables

change is an important factor. Due to heating and cooling delay, untargeted billets are

possibly over-heated or under-heated, if adjacent billets are targeted. If the control

responsivity is raised, the influence will be smaller. To reinforce the responsivity, the

insulation performance of bricks is necessary to be increased for preventing heat transmission

to the outside. Also, the heat capacity should be smaller to increase or decrease the surface

temperature of bricks as quick as possible. It is expected that the selection and the

development of furnace bricks play an important role in furnace performance.

There is a case that billets with high goal temperature and billets with low goal

temperature are scheduled at close timing. This might cause violations of the upper limit of

furnace temperature of billets with low goal temperature or failing to reach goal temperature

for billets with high goal temperature. Thus, this rolling order should be rescheduled. For

economically better operation, not only optimizing the operation, but also scheduling the

rolling order is quite important. By combining the optimization method proposed in this

thesis and the optimization of rolling schedule as Fujii et al. proposed [26], further

improvement in reheating furnace operation is expected.

121

5.3. Limitation of this research and further research

recommendations

In the process of creating a billet temperature model and an optimization method,

many parameters and thermal phenomena were approximated, omitted and assumed to

achieve reasonable optimal solutions by general PC within the goal computation time.

In practice, it is quite hard to measure heat transfer coefficients between atmosphere

and billets, atmosphere and bricks, and bricks and billets, because these heat transmissions

estimates are confined with other type of heat transmission. Therefore, it may be necessary to

adjust the coefficients in the future by the result of measuring billets temperature.

For simplicity, it was assumed that the hearth temperature and other wall temperature

were uniform in the various zones. However, the temperature will not be uniform in practice.

Local billets temperature was calculated every mesh. Similarly, the local temperature of

furnace bricks also should be computed by the difference method for further improvement of

this simulation model. Also, if there are temperature differences in opposite faces of a billet,

the whole shape of a billet may be deformed because of the thermal expansion difference.

Since this deformation influences the state of heat transmission, this deformation also should

be taken into consideration in the model.

The sectional center temperature of billets was used in this thesis because it can be

the representative temperature indicating subsequent rolling load. In practice, the rolling load

is affected by other factors, such as roller gap, roller diameter, sectional size variance of

billets and so forth. To separate these from the effect of the sectional center temperature on

rolling load, these effects on rolling load should be investigated.

Heating and cooling rates in furnace temperature were assumed constant for

simplicity. However, these rates are influenced by other factors, for instance the number of

billets heated in the same zone, their temperature, brick temperature, burners’ performance

and so on. Those rates should be investigated in detail for future model refinement.

Additionally, a constraint of furnace temperature difference between zones was ignored in

122

this thesis. However, if there is huge difference between adjacent zones, the atmosphere with

higher temperature flows into the adjacent zone, because there is an open space under the

dividing wall between them. As a result, the atmosphere with the lower temperature fails to

maintain the optimal temperature. To improve this model, the restriction in temperature

difference between zones should be installed into the constraints of this model. The weights

which were used to decide the priority of zones in this thesis should also be correspondingly

adjusted based on the performance of real furnaces.

In the near future, the validity of these model and improved control method will have

to be verified on the production floor.

.

123

APPENDIX A. Dimensions of model furnace.

WF

Lzc(4

)L

zc(3)

Lzc(2

)L

zc(1)

Hc

Hbw

/Hfw

df

Wth

124

APPENDIX B. General calculation of view-factor.

Generalized equation for finding view factors is obtained by the way shown in [14].

Figure B-1. Thermal radiation from small area dA1 to hemisphere.

𝑖

𝑑𝐴1 cos 𝜑=

𝑑2�̇�

𝑑𝜔

where

i: Radient intensity [Wsr−1]

ω: Solid angle [sr]

dω = sin φ dφdθ

E = ∫ dφ2π

0

∫ i cos φ sin θ dφ

π2

0

where

E: Monochromatic radient intensity [W]

dω

ｒ＝１dA1

dA2

dφ

φ

θdθ

125

When i is uniform for all the emitting directions, the integral above is computed simply.

E = πi

Since E = σεT4,

i =σεT4

π

i1

dA1 cos φ1=

d2Q̇1

dω

dω =dA2 cos φ2

l2

→ d2Q̇1 = i1

cos φ1 cos φ2

l2dA1dA2

→ d2Q̇2 = i2

cos φ1 cos φ2

l2dA1dA2

where

A1: Area which emits thermal radiation [m2]

A2: Area which receives thermal radiation [m2]

d2Q̇ = d2Q̇1 − d2Q̇2 = (i1 − i2)cos φ1 cos φ2

l2dA1dA2

→ d2Q̇ = σ(T14 − T2

4)cos φ1 cos φ2

πl2dA1dA2

→ d2Q̇ = σ(T14 − T2

4)Fd12dA1

Then,

→ Fd12 = ∫cos φ1 cos φ2

πl2dA2

A2

Q̇ = σ(T14 − T2

4)F12A1

F12 =1

A1∫ Fd12dA1 =

1

A1∫ ∫

cos φ1 cos φ2

πl2dA1dA2

A2A1A1

126

APPENDIX C. Heat transmission calculation.

When z=1

Heat balance at component A

Lu ×πr

2× (qrad,Hb,t + qrad,cb,t + qrad,wb,t + qtran,gb,t + qrad,bi+1→bi,t) + Lu × r

× (qcond,(1,2,1),t + qcond,(2,1,1),t)

+πr2

4(qcond,(1,1,2),t + qrad,Heb,t + qrad,cb,t + qrad,wb,t + qtran,gb,t)

= ρi × (Li

n×

πr2

4) × ci × (Ti,(1,1,1),t+1 − Ti,(1,1,1),t)

→ Ti,(1,1,1),t+1 = Ti,(1,1,1),t

+4

πr2Luρici{

πrLu

2(qrad,Hb,t + qrad,cb,t + qrad,wb,t + qtran,gb,t

+ qrad,bi+1→bi,t) + rLu(qcond,(1,2,1),t + qcond,(2,1,1),t)

+πr2

4(qcond,(1,1,2),t + qrad,Heb,t + qrad,cb,t + qrad,wb,t + qtran,gb,t)}

where

Lu =Li

n∶ Unit length of the mesh in z axis

Li =wi

ρi{HbWb − 4 (r2 −

π

4r2)} : Length of billet/bloom i

n: Number of mesh in z axis

ci: Specific heat of billet/bloom i

127

Heat balance at component B

Lu ×Wb − 2r

ℓ − 2× (qtrans,bH + qcond,(x,2,1),t) + Lu × r × (qcond,(x−1,1,1),t + qcond,(x+1,1,1),t)

+Wb − 2r

ℓ − 2× r × (qcond,(x,1,2),t + qrad,bHe,t + qrad,wb,t)

= ρi × (Lu ×Wb − 2r

ℓ − 2× r) × ci × (Ti,(x,1,1),t+1 − Ti,(x,1,1),t)

→ Ti,(x,1,1),t+1 = Ti,(x,1,1),t

+ℓ − 2

rLu(Hb − 2r)ρici{

Lu(Wb − 2r)

ℓ − 2(qtrans,bH + qcond,(x,2,1),t)

+ rLu(qcond,(x−1,1,1),t + qcond,(x+1,1,1),t)

+r(Wb − 2r)

ℓ − 2(qcond,(x,1,2),t + qrad,Heb,t + qrad,cb,t + qrad,wb,t + qtran,gb,t)}

Heat balance at component C

Lu ×πr

2× (qrad,Hb,t + qrad,cb,t + qrad,wb,t + qtran,gb,t − qrad,bi→bi−1,t) + Lu × r

× (qcond,(ℓ,2,1),t + qcond,(ℓ−1,1,1),t)

+πr2

4(qcond,(ℓ,1,2),t + qrad,Heb,t + qrad,cb,t + qrad,wb,t + qtran,gb,t)

= ρi × (Lu ×πr2

4) × ci × (Ti,(ℓ,1,1),t+1 − Ti,(ℓ,1,1),t)

→ Ti,(ℓ,1,1),t+1 = Ti,(ℓ,1,1),t

+4

πr2Luρici{

πrLu


− qrad,bi→bi−1,t) + rLu(qcond,(ℓ,2,1),t + qcond,(ℓ−1,1,1),t)

+πr2

4(qcond,(ℓ,1,2),t + qrad,Heb,t + qrad,cb,t + qrad,wb,t + qtran,gb,t)}

128

Heat balance at component D

Lu ×Hb − 2r

m − 2× (qrad,Hb,t + qrad,cb,t + qrad,wb,t + qtran,gb,t + qrad,bi+1→bi,t + qcond,(2,y,1),t)

+ Lu × r × (qcond,(1,y+1,1),t + qcond,(1,y−1,1),t) +Hb − 2r

m − 2× r

× (qcond,(1,y,2),t + qrad,Heb,t + qrad,cb,t + qrad,wb,t + qtran,gb,t)

= ρi × (Lu ×Hb − 2r

m − 2× r) × ci × (Ti,(1,y,1),t+1 − Ti,(1,y,1),t)

→ Ti,(1,y,1),t+1 = Ti,(1,y,1),t

+m − 2


Lu(Hb − 2r)

m − 2(qrad,Hb,t + qrad,cb,t + qrad,wb,t + qtran,gb,t

+ qrad,bi+1→bi,t + qcond,(2,y,1),t) + rLu(qcond,(1,y+1,1),t + qcond,(1,y−1,1),t)

+r(Hb − 2r)

m − 2(qcond,(1,y,2),t + qrad,Heb,t + qrad,cb,t + qrad,wb,t + qtran,gb,t)}

Heat balance at component E

Lu ×Wb − 2r

ℓ − 2× (qcond,(x,y−1,1) + qcond,(x,y+1,1),t) + Lu ×

Hb − 2r

m − 2

× (qcond,(x−1,y,1),t + qcond,(x+1,y,1),t) +Wb − 2r

ℓ − 2×

Hb − 2r

m − 2

× (qcond,(x,y,2),t + qrad,Heb,t + qrad,cb,t + qrad,wb,t + qtran,gb,t)

= ρi × (Lu ×Wb − 2r

ℓ − 2×

Hb − 2r

m − 2) × ci × (Ti,(x,y,1),t+1 − Ti,(x,y,1),t)

→ Ti,(x,y,1),t+1 = Ti,(x,y,1),t

+(ℓ − 2)(m − 2)

Lu(Wb − 2r)(Hb − 2r)ρici{

Lu(Wb − 2r)

ℓ − 2(qcond,(x,y−1,1)

+ qcond,(x,y+1,1),t) +Lu(Hb − 2r)

m − 2(qcond,(x−1,y,1),t + qcond,(x+1,y,1),t)

+(Wb − 2r)(Hb − 2r)

(ℓ − 2)(m − 2)(qcond,(x,y,2),t + qrad,Heb,t + qrad,cb,t + qrad,wb,t

+ qtran,gb,t)}

129

Heat balance at component F

Lu ×Hb − 2r

m − 2× (qrad,Hb,t + qrad,cb,t + qrad,wb,t + qtran,gb,t − qrad,bi→bi−1,t

+ qcond,(ℓ−1,y,1),t) + Lu × r × (qcond,(ℓ,y+1,1),t + qcond,(ℓ,y−1,1),t) +Hb − 2r

m − 2

× r × (qcond,(ℓ,y,2),t + qrad,Heb,t + qrad,cb,t + qrad,wb,t + qtran,gb,t)

= ρi × (Lu ×Hb − 2r

m − 2× r) × ci × (Ti,(ℓ,y,1),t+1 − Ti,(ℓ,y,1),t)

→ Ti,(ℓ,y,1),t+1 = Ti,(ℓ,y,1),t

+m − 2


Lu(Hb − 2r)


− qrad,bi→bi−1,t + qcond,(ℓ−1,y,1),t) + rLu(qcond,(ℓ,y+1,1),t + qcond,(ℓ,y−1,1),t)

+r(Hb − 2r)

m − 2(qcond,(ℓ,y,2),t + qrad,Heb,t + qrad,cb,t + qrad,wb,t + qtran,gb,t)}

Heat balance at component G

Lu ×πr


× (qcond,(1,m−1,1),t + qcond,(2,m,1),t)

+πr2

4(qcond,(1,m,2),t + qrad,Heb,t + qrad,cb,t + qrad,wb,t + qtran,gb,t)

= ρi × (Lu ×πr2

4) × ci × (Ti,(1,m,1),t+1 − Ti,(1,m,1),t)

→ Ti,(1,m,1),t+1 = Ti,(1,m,1),t

+4

πr2Luρici{

πrLu


+ qrad,bi+1→bi,t) + rLu(qcond,(1,m−1,1),t + qcond,(2,m,1),t)

+πr2

4(qcond,(1,m,2),t + qrad,Heb,t + qrad,cb,t + qrad,wb,t + qtran,gb,t)}

130

Heat balance at component H

Lu ×Wb − 2r

ℓ − 2× (qrad,cb,t + qrad,wb,t + qtran,gb,t + qcond,(x,m−1,1),t) + Lu × r

× (qcond,(x−1,m,1),t + qcond,(x+1,m,1),t) +Wb − 2r

ℓ − 2× r

× (qcond,(x,m,2),t + qrad,Heb,t + qrad,cb,t + qrad,wb,t + qtran,gb,t)

= ρi × (Lu ×Wb − 2r

ℓ − 2× r) × ci × (Ti,(x,m,1),t+1 − Ti,(x,m,1),t)

→ Ti,(x,m,1),t+1 = Ti,(x,m,1),t

+ℓ − 2


Lu(Wb − 2r)

ℓ − 2(qrad,cb,t + qrad,wb,t + qtran,gb,t

+ qcond,(x,m−1,1),t) + rLu(qcond,(x−1,m,1),t + qcond,(x+1,m,1),t)

+r(Wb − 2r)

ℓ − 2(qcond,(x,m,2),t + qrad,Heb,t + qrad,cb,t + qrad,wb,t + qtran,gb,t)}

Heat balance at component I

Lu ×πr


× (qcond,(ℓ,m−1,1),t + qcond,(ℓ−1,m,1),t)

+πr2

4(qcond,(ℓ,m,2),t + qrad,Heb,t + qrad,cb,t + qrad,wb,t + qtran,gb,t)

= ρi × (Lu ×πr2

4) × ci × (Ti,(ℓ,m,1),t+1 − Ti,(ℓ,m,1),t)

→ Ti,(ℓ,m,1),t+1 = Ti,(ℓ,m,1),t

+4

πr2Luρici{

πrLu


− qrad,bi→bi−1,t) + rLu(qcond,(ℓ,m−1,1),t + qcond,(ℓ−1,m,1),t)

+πr2

4(qcond,(ℓ,m,2),t + qrad,Heb,t + qrad,cb,t + qrad,wb,t + qtran,gb,t)}

131

When z=z

Heat balance at component A

Lu ×πr


× (qcond,(1,2,z),t + qcond,(2,1,z),t) +πr2

4(qcond,(1,1,z+1),t + qcond,(1,1,z−1))

= ρi × (Lu ×πr2

4) × ci × (Ti,(1,1,z),t+1 − Ti,(1,1,z),t)

→ Ti,(1,1,z),t+1 = Ti,(1,1,z),t

+4

πr2Luρici{

πrLu


+ qrad,bi+1→bi,t) + rLu(qcond,(1,2,z),t + qcond,(2,1,z),t)

+πr2

4(qcond,(1,1,z+1),t + qcond,(1,1,z−1))}

Heat balance at component B

Lu ×Wb − 2r

ℓ − 2× (qtrans,bH + qcond,(x,2,z),t) + Lu × r × (qcond,(x−1,1,z),t + qcond,(x+1,1,z),t)

+Wb − 2r

ℓ − 2× r × (qcond,(x,1,z+1),t + qcond,(x,1,z−1),t)

= ρi × (Lu ×Wb − 2r

ℓ − 2× r) × ci × (Ti,(x,1,z),t+1 − Ti,(x,1,z),t)

→ Ti,(x,1,z),t+1 = Ti,(x,1,z),t

+ℓ − 2


Lu(Wb − 2r)

ℓ − 2(qcond,bH + qcond,(x,2,z),t)

+ rLu(qcond,(x−1,1,z),t + qcond,(x+1,1,z),t)

+r(Wb − 2r)

ℓ − 2(qcond,(x,1,z+1),t + qcond,(x,1,z−1),t)}

132

Heat balance at component C

Lu ×πr


× (qcond,(ℓ,2,z),t + qcond,(ℓ−1,1,z),t) +πr2

4(qcond,(ℓ,1,z+1),t + qcond,(ℓ,1,z−1),t)

= ρi × (Lu ×πr2

4) × ci × (Ti,(ℓ,1,z),t+1 − Ti,(ℓ,1,z),t)

→ Ti,(ℓ,1,z),t+1 = Ti,(ℓ,1,z),t

+4

πr2Luρici{

πrLu


− qrad,bi→bi−1,t) + rLu(qcond,(ℓ,2,z),t + qcond,(ℓ−1,1,z),t)

+πr2

4(qcond,(ℓ,1,z+1),t + qcond,(ℓ,1,z−1),t)}

Heat balance at component D

Lu ×Hb − 2r

m − 2× (qrad,Hb,t + qrad,cb,t + qrad,wb,t + qtran,gb,t + qrad,bi+1→bi,t + qcond,(2,y,z),t)

+ Lu × r × (qcond,(1,y+1,z),t + qcond,(1,y−1,z),t) +Hb − 2r

m − 2× r

× (qcond,(1,y,z+1),t + qcond,(1,y,z−1))

= ρi × (Lu ×Hb − 2r

m − 2× r) × ci × (Ti,(1,y,z),t+1 − Ti,(1,y,z),t)

→ Ti,(1,y,z),t+1 = Ti,(1,y,z),t

+m − 2


Lu(Hb − 2r)


+ qrad,bi+1→bi,t + qcond,(2,y,z),t) + rLu(qcond,(1,y+1,z),t + qcond,(1,y−1,z),t)

+r(Hb − 2r)

m − 2(qcond,(1,y,z+1),t + qcond,(1,y,z−1))}

133

Heat balance at component E

Lu ×Wb − 2r

ℓ − 2× (qcond,(x,y−1,z) + qcond,(x,y+1,z),t) + Lu ×

Hb − 2r

m − 2

× (qcond,(x−1,y,z),t + qcond,(x+1,y,z),t) +Wb − 2r

ℓ − 2×

Hb − 2r

m − 2

× (qcond,(x,y,z+1),t + qcond,(x,y,z−1),t)

= ρi × (Lu ×Wb − 2r

ℓ − 2×

Hb − 2r

m − 2) × ci × (Ti,(x,y,z),t+1 − Ti,(x,y,z),t)

→ Ti,(x,y,z),t+1 = Ti,(x,y,z),t

+(ℓ − 2)(m − 2)

Lu(Wb − 2r)(Hb − 2r)ρici{

Lu(Wb − 2r)

ℓ − 2(qcond,(x,y−1,z) + qcond,(x,y+1,z),t)

+Lu(Hb − 2r)

m − 2(qcond,(x−1,y,z),t + qcond,(x+1,y,z),t)

+(Wb − 2r)(Hb − 2r)

(ℓ − 2)(m − 2)(qcond,(x,y,z+1),t + qcond,(x,y,z−1),t)}

Heat balance at component F

Lu ×Hb − 2r

m − 2× (qrad,Hb,t + qrad,cb,t + qrad,wb,t + qtran,gb,t − qrad,bi→bi−1,t

+ qcond,(ℓ−1,y,z),t) + Lu × r × (qcond,(ℓ,y+1,z),t + qcond,(ℓ,y−1,z),t) +Hb − 2r

m − 2

× r × (qcond,(ℓ,y,z+1),t + qcond,(ℓ,y,z−1),t)

= ρi × (Lu ×Hb − 2r

m − 2× r) × ci × (Ti,(ℓ,y,z),t+1 − Ti,(ℓ,y,z),t)

→ Ti,(ℓ,y,z),t+1 = Ti,(ℓ,y,z),t

+m − 2


Lu(Hb − 2r)


− qrad,bi→bi−1,t + qcond,(ℓ−1,y,z),t) + rLu(qcond,(ℓ,y+1,z),t + qcond,(ℓ,y−1,z),t)

+r(Hb − 2r)

m − 2(qcond,(ℓ,y,z+1),t + qcond,(ℓ,y,z−1),t)}

134

Heat balance at component G

Lu ×πr


× (qcond,(1,m−1,z),t + qcond,(2,m,z),t) +πr2

4(qcond,(1,m,z+1),t + qcond,(1,m,z−1))

= ρi × (Lu ×πr2

4) × ci × (Ti,(1,m,z),t+1 − Ti,(1,m,z),t)

→ Ti,(1,m,z),t+1 = Ti,(1,m,z),t

+4

πr2Luρici{

πrLu


+ qrad,bi+1→bi,t) + rLu(qcond,(1,m−1,z),t + qcond,(2,m,z),t)

+πr2

4(qcond,(1,m,z+1),t + qcond,(1,m,z−1))}

Heat balance at component H

Lu ×Wb − 2r

ℓ − 2× (qrad,cb,t + qrad,wb,t + qtran,gb,t + qcond,(x,m−1,z),t) + Lu × r

× (qcond,(x−1,m,z),t + qcond,(x+1,m,z),t) +Wb − 2r

ℓ − 2× r

× (qcond,(x,m,z+1),t + qcond,(x,m,z−1))

= ρi × (Lu ×Wb − 2r

ℓ − 2× r) × ci × (Ti,(x,m,z),t+1 − Ti,(x,m,z),t)

→ Ti,(x,m,z),t+1 = Ti,(x,m,z),t

+ℓ − 2


Lu(Wb − 2r)

ℓ − 2(qrad,cb,t + qrad,wb,t + qtran,gb,t

+ qcond,(x,m−1,z),t) + rLu(qcond,(x−1,m,z),t + qcond,(x+1,m,z),t)

+r(Wb − 2r)

ℓ − 2(qcond,(x,m,z+1),t + qcond,(x,m,z−1))}

135

Heat balance at component I

Lu ×πr


× (qcond,(ℓ,m−1,z),t + qcond,(ℓ−1,m,z),t)

+πr2

4(qcond,(ℓ,m,z+1),t + qcond,(ℓ,m,z−1))

= ρi × (Lu ×πr2

4) × ci × (Ti,(ℓ,m,z),t+1 − Ti,(ℓ,m,z),t)

→ Ti,(ℓ,m,z),t+1 = Ti,(ℓ,m,z),t

+4

πr2Luρici{

πrLu


− qrad,bi→bi−1,t) + rLu(qcond,(ℓ,m−1,z),t + qcond,(ℓ−1,m,z),t)

+πr2

4(qcond,(ℓ,m,z+1),t + qcond,(ℓ,m,z−1),t)}

136

APPENDIX D. View-factor calculation of perpendicular plates.

Figure D-1. Positional relation of two perpendicular plates.

(A1 + A2 + A3)F(1+3+5),(2+4+6)

= A1(F12 + F14 + F16) + A3(F32 + F34 + F36) + A5(F52 + F54 + F56)

A1F14 = A3F32

A5F54 = A3F36

A1F16 = A5F52

(A1 + A3)F(1+3),(2+4) = A1(F12 + F14) + A3(F32 + F34)

→ A1F14 = A3F32 =1

2{(A1 + A3)F(1+3),(2+4) − A1F12 − A3F34}

(A3 + A5)F(3+5),(4+6) = A3(F34 + F36) + A5(F54 + F56)

→ A5F54 = A3F36 =1

2{(A5 + A3)F(3+5),(4+6) − A5F56 − A3F34}

A1

A3

A5

A2

A4

A6

137

(A1 + A2 + A3)F(1+3+5),(2+4+6)

= A1(F12 + 2F16) + {(A1 + A3)F(1+3),(2+4) − A1F12 − A3F34} + A3F34

+ {(A5 + A3)F(3+5),(4+6) − A5F56 − A3F34} + A5F56

A1F16 =1

2[(A1 + A2 + A3)F(1+3+5),(2+4+6) − A1F12 − A3F34 − A5F56

− {(A1 + A3)F(1+3),(2+4) − A1F12 − A3F34}

− {(A5 + A3)F(3+5),(4+6) − A5F56 − A3F34}]

A1F16 =1

2{(A1 + A2 + A3)F(1+3+5),(2+4+6) + A3F34 − (A1 + A3)F(1+3),(2+4)

− (A5 + A3)F(3+5),(4+6)}

138

APPENDIX E. View-factor calculation from small plate to

parallel plate with off-set.

Figure E-1. Positional relation of two parallel plates.

From (2.5),

F12 =1

A1∫ ∫

cos φ1 cos φ2

πr2dA1dA2 ⋯ (2.5)

A2A1

Now, since dA2 is small, dA2 is constant. Also, since φ1 =φ2 when two plates are parallel, cos

φ1 =cosφ2.

Then, (2.5) becomes,

φ1

φ2

H

W

dA2

A1

d

139

F12 =dA2

A1∫

cos2 φ1

πr2dA1

A1

Since cosφ1 =d

r and r = √𝑥2+y2 + d2,

F12 =dA2d2

A1∫

1

πr4dA1 =

A1

dA2d2

πA1∫

1

(𝑥2+y2 + d2)2dA1

A1

→ F12 =dA2d2

πA1∫ ∫

1

(𝑥2+y2 + d2)2dxdy

𝑊

0

𝐻

0

Using the following integral formula,

∫1

(x2 + a2)𝑛= In

In =1

2(n − 1)a2{

x

(x2 + a2)n−1+ (2n − 3)In−1}

I1 =1

atan−1

x

a

I2 =1

2a2{

x

x2 + a2+ I1} =

1

2a2{

x

x2 + a2+

1

atan−1

x

a}

Let a2 be y2+d2,

∫1

{𝑥2+y2 + d2}2dx

𝑊

0

= ∫1

(x2+a2)2du

W

0

= [1

2a2{

x

x2 + a2+

1

atan−1

x

a}]

0

W

=1

2a2(

W

W2 + a2+

1

atan−1

W

a)

=1

2(y2 + d2)(

𝑊

𝑊2 + y2 + d2+

1

√y2 + d2tan−1

W

√y2 + d2)

140

Therefore,

F12 =dA2d2

πA1∫ {

1

2(y2 + d2)(

𝑊

𝑊2 + y2 + d2+

1

√y2 + d2tan−1

W

√y2 + d2)}

𝐻

0

dy

=dA2d2

2πA1{∫

W

(y2 + d2)(W2 + y2 + d2)dy + ∫

1

√y2 + d2(y2 + d2)tan−1

W

√y2 + d2

H

0

H

0

dy}

∫W

(y2 + d2)(W2 + y2 + d2)𝑑𝑦

𝐻

0

=1

𝑊∫ (

1

y2 + d2−

1

y2 + W2 + d2) 𝑑𝑦

𝐻

0

=1

W[1

dtan−1

y

d−

1

√W2 + d2tan−1

y

√W2 + d2]

0

H

=1

Wdtan−1

H

d−

1

W√W2 + d2tan−1

H

√W2 + d2

Let y be d tan θ.

dy

dθ=

d

(cos θ)2 , sin θ =

y

√y2 + d2 , cos θ =

d

√y2 + d2

∫1

√y2 + d2(y2 + d2)tan−1

W

√y2 + d2𝑑𝑦

H

0

= ∫1

d3√1 + (tan θ)2(1 + (tan θ)2)tan−1 (

W

d√1 + (tan θ)2)

tan−1Hd

0

d

(cos θ)2dθ

= ∫cos θ

d2tan−1 (

W cos θ

d)

tan−1Hd

0

dθ

f′ = (sin θ

d2)

′

=cos θ

d2

141

g = tan−1 (W cos θ

d)

Since ∫ f′g dy = fg − ∫ fg′dy,

∫cos θ

d2tan−1 (

W cos θ

d)

tan−1Hd

0

dθ

=sin θ

d2tan−1 (

W cos θ

d) − ∫

sin θ

d2{tan−1 (

W cos θ

d)}

′tan−1Hd

0

dθ

Let W cos θ

d be t.

dt

dθ=

−W sin θ

d

{tan−1 (W cos θ

d)}

′

=−

W sin θd

1 + (W cos θ

d)

2

∫cos θ

d2tan−1 (

W cos θ

d)

tan−1Hd

0

dθ

=sin θ

d2tan−1 (

W cos θ

d) − ∫

sin θ

d2

tan−1Hd

0

−W sin θ

d

1 + (W cos θ

d)

2 dθ

= [𝑦

d2√𝑦2 + 𝑑2tan−1 (

W

√𝑦2 + 𝑑2)]

0

𝐻

+𝑊

𝑑∫

(sin 𝜃)2

d2 + (W cos 𝜃)2dθ

tan−1Hd

0

=𝐻

d2√𝐻2 + 𝑑2tan−1 (

W

√𝐻2 + 𝑑2) +

𝑊

𝑑∫

𝑦2

𝑦2 + 𝑑2

𝑑2 +𝑊2𝑑2

𝑦2 + 𝑑2

𝑑

𝑦2 + 𝑑2dy

𝐻

0

=𝐻

d2√𝐻2 + 𝑑2tan−1 (

W

√𝐻2 + 𝑑2) + 𝑊 ∫

𝑦2

𝑑2(𝑦2 + 𝑑2)2 + 𝑊2𝑑2(𝑦2 + 𝑑2)dy

𝐻

0

=𝐻

d2√𝐻2 + 𝑑2tan−1 (

W

√𝐻2 + 𝑑2) +

𝑊

𝑑2∫

𝑦2

(𝑦2 + 𝑑2)(𝑦2 + 𝑑2 + 𝑊2)dy

𝐻

0

142

=𝐻

d2√𝐻2 + 𝑑2tan−1 (

W

√𝐻2 + 𝑑2) +

𝑊

𝑑2∫

𝑦2

𝑊2(

1

𝑦2 + 𝑑2−

1

𝑦2 + 𝑑2 + 𝑊2) dy

𝐻

0

=𝐻

d2√𝐻2 + 𝑑2tan−1 (

W

√𝐻2 + 𝑑2) +

1

𝑊𝑑2∫ {(1 −

𝑑2

𝑦2 + 𝑑2) − (1 −

𝑑2 + 𝑊2

𝑦2 + 𝑑2 + 𝑊2)} dy

𝐻

0

=𝐻

d2√𝐻2 + 𝑑2tan−1 (

W

√𝐻2 + 𝑑2) +

1

𝑊𝑑2∫ {

𝑑2 + 𝑊2

𝑦2 + 𝑑2 + 𝑊2−

𝑑2

𝑦2 + 𝑑2} dy

𝐻

0

=𝐻

d2√𝐻2 + 𝑑2tan−1 (

W

√𝐻2 + 𝑑2)

+1

𝑊𝑑2{(𝑑2 + 𝑊2) ∫

1

𝑦2 + (√𝑑2 + 𝑊2)2 𝑑𝑦 − 𝑑2 ∫

1

𝑦2 + 𝑑2dy

𝐻

0

𝐻

0

}

=𝐻

d2√𝐻2 + 𝑑2tan−1 (

W

√𝐻2 + 𝑑2)

+1

𝑊𝑑2{(𝑑2 + 𝑊2) (

1

√𝑑2 + 𝑊2tan−1

𝐻

√𝑑2 + 𝑊2) − 𝑑2 (

1

𝑑tan−1

𝐻

𝑑)}

=𝐻

d2√𝐻2 + 𝑑2tan−1 (

W

√𝐻2 + 𝑑2) +

√𝑑2 + 𝑊2

𝑊𝑑2tan−1

𝐻

√𝑑2 + 𝑊2−

1

𝑊𝑑tan−1

𝐻

𝑑

Hence,

F12 =dA2d2

2πA1[

1

Wdtan−1

H

d−

1

W√W2 + d2tan−1

H

√W2 + d2

+H

d2√H2 + d2tan−1 (

W

√H2 + d2) +

√d2 + W2

Wd2tan−1

H

√d2 + W2

−1

Wdtan−1

H

d]

=dA2d2

2πA1[

W

d2√W2 + d2tan−1

H

√W2 + d2+

H

d2√H2 + d2tan−1 (

W

√H2 + d2)]

=dA2

2π{

1

𝐻√W2 + d2tan−1

H

√W2 + d2+

1

W√H2 + d2tan−1 (

W

√H2 + d2)}

143

𝐹21 =𝐴1

𝑑𝐴2𝐹12 =

1

2π{

H

√W2 + d2tan−1

H

√W2 + d2+

W

√H2 + d2tan−1 (

W

√H2 + d2)}

When dA2 is off set (x0,y0), F12 becomes

F12 = F32 + F42 + F52 + F62

Figure E-2. View factor between parallel plates with off set.

F12 =dA2

2π{

1

y0√x02 + d2

tan−1y0

√x02 + d2

+1

x0√y02 + d2

tan−1 (x0

√y02 + d2

)}

+dA2

2π{

1

(H − y0)√x02 + d2

tan−1H − y0

√x02 + d2

+1

x0√(H − y0)2 + d2tan−1 (

x0

√(H − y0)2 + d2)}

y0

x0

H

W

dA2

A6

d

A3

A4

A5

144

+dA2

2π{

1

(H − y0)√(W − x0)2 + d2tan−1

H − y0

√(W − x0)2 + d2

+1

(W − x0)√(H − y0)2 + d2tan−1 (

W − x0

√(H − y0)2 + d2)}

=dA2

2π{

1

y0√(W − x0)2 + d2tan−1

y0

√(W − x0)2 + d2

+1

(W − x0)√y02 + d2

tan−1 (W − x0

√y02 + d2

)}

Figure E-3. View factor between parallel plates without off set.

𝐹12 = 𝐹23 − 𝐹24 − 𝐹25 + 𝐹26

𝐹12 =1

2π{

H

√W2 + d2tan−1

H

√W2 + d2+

W

√H2 + d2tan−1 (

W

√H2 + d2)}

H

W

A1

A2

A3

A4 A5

A6

145

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