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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.
iii
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
iv
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.
v
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
vi
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
vii
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
viii
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
ix
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
x
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
xi
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
xii
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
xiii
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
+qrad,wb+qrad,bi→bi+1
+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
+qrad,wb+qrad,bi→bi+1
+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
Zone2 Lzc(2) 8,000 mm
Zone3 Lzc(3) 6,000 mm
Zone4 Lzc(4) 6,000 mm
Zone1 Hc(1) 1,665 mm
Zone2 Hc(2) 1,665 mm
Zone3 Hc(3) 1,665 mm
Zone4 Hc(4) 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
Zone 2&3 Wth2 100 mm
Zone 3&4 Wth3 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]
Case 1 Case 2 Case 3
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]
Case 1 Case 2 Case 3
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
Zone 1 Zone 2 Zone 3 Zone 4
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]
Furnace temperature [K] Furnace holding
time [min] Zone 1 Zone 2 Zone3 Zone4
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.
Furnace temperature [K] Furnace holding
time [min] Zone 1 Zone 2 Zone3 Zone4
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]
Furnace temperature [K] Furnace holding
time [min] Zone 1 Zone 2 Zone3 Zone4
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]
Furnace temperature [K] Furnace holding
time [min] Zone 1 Zone 2 Zone3 Zone4
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
Steel 1 Steel 2 Steel 3
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
Steel 1 Steel 2 Steel 3
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
Zone1 Zone2 Zone3 Zone4
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.
Zone 1 Zone 2 Zone 3 Zone 4
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]
Zone1 Zone2 Zone3 Zone4
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]
Zone 1 Zone 2 Zone 3 Zone 4
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]
Zone 1 Zone 2 Zone 3 Zone 4
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]
Zone1 Zone2 Zone3 Zone4
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]
Zone1 Zone2 Zone3 Zone4
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 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]
Zone 1 Zone 2 Zone 3 Zone 4
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]
Zone 1 Zone 2 Zone 3 Zone 4
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 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
of all the billets [K]
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
of all the billets [K]
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]
Lower limiter 5 [K] Lower limiter 10 [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]
Lower limiter 5 [K] Lower limiter 10 [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]
Zone 1 Zone 2 Zone 3 Zone 4
Table 4-3. Computational condition 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
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]
Amplifier 1, Lower limter 0 [K] Amplifier 1, Lower limter 10 [K]
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]
Amplifier 1, Lower limter 0 [K] Amplifier 1, Lower limter 10 [K]
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]
Amplifier 1, Lower limter 0 [K] Amplifier 1, Lower limter 10 [K]
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
Amplifier 1, Lower limter 0 [K] Amplifier 1, Lower limter 10 [K]
2.5
3.9
-0.7-1.6
(4.0)
(2.0)
-
2.0
4.0
6.0
8.0
10.0
12.0
Amplifier 1, Lower limter 0 [K] Amplifier 1, Lower limter 10 [K]
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.
Zone 1 Zone 2 Zone 3 Zone 4
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]
Inserted high goal temperature billets Only same type of billets
High goal temperature billets
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]
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
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]
Inserted high goal temperature billets Only same type of billets
High goal temperature billets
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
Inserted high goal temperature billets Only same type of billets
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
Inserted high goal temperature billets Only same type of billets
Tem
per
ature
[K
]
Average of ∆Tex,i Minimumof ∆Tex,i
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]
After optimization Before optimization
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]
After optimization Before optimization
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
]
Average of ∆Tex,i Minimumof ∆Tex,i
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
2(qrad,Hb,t + qrad,cb,t + qrad,wb,t + qtran,gb,t
− 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
rLu(Hb − 2r)ρici{
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
rLu(Hb − 2r)ρici{
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) + 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
2× (qrad,Hb,t + qrad,cb,t + qrad,wb,t + qtran,gb,t + qrad,bi+1→bi,t) + Lu × r
× (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
2(qrad,Hb,t + qrad,cb,t + qrad,wb,t + qtran,gb,t
+ 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
rLu(Hb − 2r)ρici{
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
2× (qrad,Hb,t + qrad,cb,t + qrad,wb,t + qtran,gb,t − qrad,bi→bi−1,t) + Lu × r
× (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
2(qrad,Hb,t + qrad,cb,t + qrad,wb,t + qtran,gb,t
− 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
2× (qrad,Hb,t + qrad,cb,t + qrad,wb,t + qtran,gb,t + qrad,bi+1→bi,t) + Lu × r
× (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
2(qrad,Hb,t + qrad,cb,t + qrad,wb,t + qtran,gb,t
+ 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
rLu(Hb − 2r)ρici{
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
2× (qrad,Hb,t + qrad,cb,t + qrad,wb,t + qtran,gb,t − qrad,bi→bi−1,t) + Lu × r
× (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
2(qrad,Hb,t + qrad,cb,t + qrad,wb,t + qtran,gb,t
− 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
rLu(Hb − 2r)ρici{
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) + 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
rLu(Hb − 2r)ρici{
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) + 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
2× (qrad,Hb,t + qrad,cb,t + qrad,wb,t + qtran,gb,t + qrad,bi+1→bi,t) + Lu × r
× (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
2(qrad,Hb,t + qrad,cb,t + qrad,wb,t + qtran,gb,t
+ 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
rLu(Hb − 2r)ρici{
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
2× (qrad,Hb,t + qrad,cb,t + qrad,wb,t + qtran,gb,t − qrad,bi→bi−1,t) + Lu × r
× (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
2(qrad,Hb,t + qrad,cb,t + qrad,wb,t + qtran,gb,t
− 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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